5th Edition

Fundamentals of Engineering Thermodynamics

Michael J. Moran The Ohio State University

Howard N. Shapiro Iowa State University of Science and Technology

John Wiley & Sons, Inc.

Fundamentals of Engineering Thermodynamics

5th Edition

Fundamentals of Engineering Thermodynamics

Michael J. Moran The Ohio State University

Howard N. Shapiro Iowa State University of Science and Technology

John Wiley & Sons, Inc.

Copyright © 2006

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777

Email (for orders and customer service enquiries): [emailprotected] Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to “mailto:[emailprotected]”, or faxed to (+44) 1243 770620. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data Moran, Michael J. Fundamentals of engineering thermodynamics: SI version / Michael J. Moran, Howard N. Shapiro. -- 5th ed. p. cm. Includes bibliographical references and index. ISBN-13 978-0-470-03037-0 (pbk. : alk. paper) ISBN-10 0-470-03037-2 (pbk. : alk. paper) 1. Thermodynamics. I. Shapiro, Howard N. II. Title. TJ265.M66 2006 621.4021--dc22 2006008521 ISBN-13 978-0-470-03037-0 ISBN-10 0-470-03037-2 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Typeset in 10/12 pt Times by Techbooks Printed and bound in Great Britain by Scotprint, East Lothian This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

Preface In this fifth edition we have retained the objectives of the first four editions:

to present a thorough treatment of engineering thermodynamics from the classical viewpoint, to provide a sound basis for subsequent courses in fluid mechanics and heat transfer, and to prepare students to use thermodynamics in engineering practice.

While the fifth edition retains the basic organization and level of the previous editions, we have introduced several enhancements proven to be effective for student learning. Included are new text elements and interior design features that help students understand and apply the subject matter. With this fifth edition, we aim to continue our leadership in effective pedagogy, clear and concise presentations, sound developments of the fundamentals, and state-of-the-art engineering applications.

New in the Fifth Edition

An engaging new feature called “Thermodynamics in the News” is introduced in every chapter. News boxes tie stories of current interest to concepts discussed in the chapter. The news items provide students with a broader context for their learning and form the basis for new Design and Open Ended problems in each chapter. Other class-tested content changes have been introduced: –A new discussion of the state-of-the-art of fuel cell technology (Sec. 13.4). –Streamlined developments of the energy concept and the first law of thermodynamics (Secs. 2.3 and 2.5, respectively). –Streamlined developments of the mass and energy balances for a control volume (Secs. 4.1 and 4.2, respectively). –Enhanced presentation of second law material (Chap. 5) clearly identifies key concepts. –Restructuring of topics in psychrometrics (Chap. 12) and enthalpy of combustion and heating values (Chap. 13) further promotes student understanding. –Functional use of color facilitates data retrieval from the appendix tables. End-of-chapter problems have been substantially refreshed. As in previous editions, a generous collection of problems is provided. The problems are classified under headings to assist instructors in problem selection. Problems range from confidence-building exercises illustrating basic skills to more challenging ones that may involve several components and require higher-order thinking. The end-of-chapter problems are organized to provide students with the opportunity to develop engineering skills in three modes: –Conceptual. See Exercises: Things Engineers Think About. –Skill Building. See Problems: Developing Engineering Skills. –Design. See Design and Open ended Problems: Exploring Engineering Practice. The comfortable interior design from previous editions has been enhanced with an even more learner-centered layout aimed at enhancing student understanding.

Core Text Features This edition continues to provide the core features that have made the text the global leader in engineering thermodynamics education.

Exceptional class-tested pedagogy. Our pedagogy is the model that others emulate. For an overview, see How to Use this Book Effectively inside the front cover of the book. v

vi

Preface

Systematic problem solving methodology. Our methodology has set the standard for thermodynamics texts in the way it encourages students to think systematically and helps them reduce errors. Effective development of the second law of thermodynamics. The text features the entropy balance (Chap. 6) recognized as the most effective way for students to learn how to apply the second law. Also, the presentation of exergy analysis (Chaps. 7 and 13) has become the state-of-the-art model for learning that subject matter. Software to enhance problem solving for deeper learning. We pioneered the use of software as an effective adjunct to learning engineering thermodynamics and solving engineering problems. Sound developments of the application areas. Included in Chaps. 8–14 are comprehensive developments of power and refrigeration cycles, psychrometrics, and combustion applications from which instructors can choose various levels of coverage ranging from short introductions to in-depth studies. Emphasis on engineering design and analysis. Specific text material on the design process is included in Sec. 1.7: Engineering Design and Analysis and Sec. 7.7: Thermoeconomics. Each chapter also provides carefully crafted Design and Open Ended Problems that allow students to develop an appreciation of engineering practice and to enhance a variety of skills such as creativity, formulating problems, making engineering judgments, and communicating their ideas. Flexibility in units. The text allows an SI or mixed SI/English presentation. Careful use of units and systematic application of unit conversion factors is emphasized throughout the text.

Ways to Meet Different Course Needs In recognition of the evolving nature of engineering curricula, and in particular of the diverse ways engineering thermodynamics is presented, the text is structured to meet a variety of course needs. The following table illustrates several possible uses of the text assuming a semester basis (3 credits). Coverage would be adjusted somewhat for courses on a quarter basis depending on credit value. Detailed syllabi for both semester and quarter bases are provided on the Instructor’s Web Site. Courses could be taught in the second or third year to engineering students with appropriate background.

Type of course

Intended audience

Chapter coverage Principles. Chaps. 1–6.

Non-majors

Applications. Selected topics from

Chaps. 8–10 (omit compressible flow in Chap. 9). Surveys

Principles. Chaps. 1–6.

Majors

Applications. Same as above plus selected

topics from Chaps. 12 and 13. First course. Chaps. 1–8.

Two-course sequences

Majors

(Chap. 7 may deferred to second course or omitted.) Second course. Selected topics from

Chaps. 9–14 to meet particular course needs.

Preface

How to Use This Book Effectively

This book has several features that facilitate study and contribute further to understanding: Examples

Numerous annotated solved examples are provided that feature the solution methodology presented in Sec. 1.7.3 and illustrated in Example 1.1. We encourage you to study these examples, including the accompanying comments. Less formal examples are given throughout the text. They open with for example. . . and close with . These examples also should be studied.

Exercises

Each chapter has a set of discussion questions under the heading Exercises: Things Engineers Think About that may be done on an individual or small-group basis. They are intended to allow you to gain a deeper understanding of the text material, think critically, and test yourself. A large number of end-of-chapter problems also are provided under the heading Problems: Developing Engineering Skills. The problems are sequenced to coordinate with the subject matter and are listed in increasing order of difficulty. The problems are also classified under headings to expedite the process of selecting review problems to solve. Answers to selected problems are provided in the appendix (pp. 865–868). Because one purpose of this book is to help you prepare to use thermodynamics in engineering practice, design considerations related to thermodynamics are included. Every chapter has a set of problems under the heading Design and Open Ended Problems: Exploring Engineering Practice that provide brief design experiences to help you develop creativity and engineering judgment. They also provide opportunities to practice communication skills.

Further Study Aids

Each chapter opens with an introduction giving the engineering context and stating the chapter objective. Each chapter concludes with a chapter summary and study guide that provides a point of departure for examination reviews. Key words are listed in the margins and coordinated with the text material at those locations. Key equations are set off by a double horizontal bar, as, for example, Eq. 1.10. Methodology update in the margin identifies where we refine our problem-solving methodology, as on p. 9, or introduce conventions such as rounding the temperature 273.15 K to 273 K, as on p. 20. For quick reference, conversion factors and important constants are provided on the next page. A list of symbols is provided on the inside back cover and facing page.

vii

viii

Preface

Constants

Universal Gas Constant R 8.314 kJ/kmol # K

Standard Atmospheric Pressure 1 atm 1.01325 bar

Standard Acceleration of Gravity g 9.80665 m/s2

Temperature Relations T 1C2 T 1K2 273.15

Acknowledgments We thank the many users of our previous editions, located at more than 200 universities and colleges in the United States and Canada, and over the globe, who contributed to this revision through their comments and constructive criticism. Special thanks are owed to Prof. Ron Nelson, Iowa State University, for developing the EES solutions and for his assistance in updating the end-of-chapter problems and solutions. We also thank Prof. Daisie Boettner, United States Military Academy, West Point, for her contributions to the new discussion of fuel cell technology. Thanks are also due to many individuals in the John Wiley and Sons, Inc., organization who have contributed their talents and energy to this edition. We appreciate their professionalism and commitment. We are extremely gratified by the reception this book has enjoyed, and we have aimed to make it even more effective in this fifth edition. As always, we welcome your comments, criticism, and suggestions. Michael J. Moran [emailprotected] Howard N. Shapiro [emailprotected] John Wiley and Sons Ltd would like to thank Brian J. Woods for his work in adapting the 5th Edition to incorporate SI units.

Contents

CHAPTER

1

CHAPTER

4

Getting Started: Introductory Concepts and Definitions 1

Control Volume Analysis Using Energy 121

1.1 Using Thermodynamics 1 1.2 Defining Systems 1 1.3 Describing Systems and Their Behavior 4 1.4 Measuring Mass, Length, Time, and Force 8 1.5 Two Measurable Properties: Specific Volume and Pressure 10 1.6 Measuring Temperature 14 1.7 Engineering Design and Analysis 18 Chapter Summary and Study Guide 22

4.1 Conservation of Mass for a Control Volume 121 4.2 Conservation of Energy for a Control Volume 128 4.3 Analyzing Control Volumes at Steady State 131 4.4 Transient Analysis 152 Chapter Summary and Study Guide 162

CHAPTER

2

CHAPTER

5

Energy and the First Law of Thermodynamics 29

The Second Law of Thermodynamics 174

2.1 Reviewing Mechanical Concepts of Energy 29 2.2 Broading Our Understanding of Work 33 2.3 Broading Our Understanding of Energy 43 2.4 Energy Transfer By Heat 44 2.5 Energy Accounting: Energy Balance for Closed Systems 48 2.6 Energy Analysis of Cycles 58 Chapter Summary and Study Guide 62

5.1 Introducing the Second Law 174 5.2 Identifying Irreversibilities 180 5.3 Applying the Second Law to Thermodynamic Cycles 184 5.4 Defining the Kelvin Temperature Scale 190 5.5 Maximum Performance Measures for Cycles Operating Between Two Reservoirs 192 5.6 Carnot Cycle 196 Chapter Summary and Study Guide 199

CHAPTER

3

Evaluating Properties

69

3.1 Fixing the State 69 EVALUATING PROPERTIES: GENERAL CONSIDERATIONS 70 3.2 p– v–T Relation 70 3.3 Retrieving Thermodynamic Properties 76 3.4 Generalized Compressibility Chart 94 EVALUATING PROPERTIES USING THE IDEAL GAS MODEL 100 3.5 Ideal Gas Model 100 3.6 Internal Energy, Enthalpy, and Specific Heats of Ideal Gases 103 3.7 Evaluating u and h using Ideal Gas Tables, Software, and Constant Specific Heats 105 3.8 Polytropic Process of an Ideal Gas 112 Chapter Summary and Study Guide 114

CHAPTER

6

Using Entropy

206

6.1 Introducing Entropy 206 6.2 Defining Entropy Change 208 6.3 Retrieving Entropy Data 209 6.4 Entropy Change in Internally Reversible Processes 217 6.5 Entropy Balance for Closed Systems 220 6.6 Entropy Rate Balance for Control Volumes 231 6.7 Isentropic Processes 240 6.8 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps 246 6.9 Heat Transfer and Work in Internally Reversible, Steady-State Flow Processes 254 Chapter Summary and Study Guide 257

ix

x

Contents

CHAPTER

7

Exergy Analysis

272

7.1 Introducing Exergy 272 7.2 Defining Exergy 273 7.3 Closed System Exergy Balance 283 7.4 Flow Exergy 290 7.5 Exergy Rate Balance for Control Volumes 293 7.6 Exergetic (Second Law) Efficiency 303 7.7 Thermoeconomics 309 Chapter Summary and Study Guide 315

CHAPTER

8

Vapor Power Systems

325

8.1 Modeling Vapor Power Systems 325 8.2 Analyzing Vapor Power Systems—Rankline Cycle 327 8.3 Improving Performance—Superheat and Reheat 340 8.4 Improving Performance—Regenerative Vapor Power Cycle 346 8.5 Other Vapor Cycle Aspects 356 8.6 Case Study: Exergy Accounting of a Vapor Power Plant 358 Chapter Summary and Study Guide 365

CHAPTER

9

Gas Power Systems

373

INTERNAL COMBUSTION ENGINES 373 9.1 Introducing Engine Terminology 373 9.2 Air-Standard Otto Cycle 375 9.3 Air-Standard Diesel Cycle 381 9.4 Air-Standard Dual Cycle 385 GAS TURBINE POWER PLANTS 388 9.5 Modeling Gas Turbine Power Plants 388 9.6 Air-Standard Brayton Cycle 389 9.7 Regenerative Gas Turbines 399 9.8 Regenerative Gas Turbines with Reheat and Intercooling 404 9.9 Gas Turbines for Aircraft Propulsion 414 9.10 Combined Gas Turbine—Vapor Power Cycle 419 9.11 Ericsson and Stirling Cycles 424 COMPRESSIBLE FLOW THROUGH NOZZLES AND DIFFUSERS 426

9.12 Compressible Flow Preliminaries 426 9.13 Analyzing One-Dimensional Steady Flow in Nozzles and Diffusers 430 9.14 Flow in Nozzles and Diffusers of Ideal Gases with Constant Specific Heats 436 Chapter Summary and Study Guide 444

CHAPTER

10

Refrigeration and Heat Pump Systems 454 10.1 Vapor Refrigeration Systems 454 10.2 Analyzing Vapor-Compression Refrigeration Systems 457 10.3 Refrigerant Properties 465 10.4 Cascade and Multistage Vapor-Compression Systems 467 10.5 Absorption Refrigeration 469 10.6 Heat Pump Systems 471 10.7 Gas Refrigeration Systems 473 Chapter Summary and Study Guide 479

CHAPTER

11

Thermodynamic Relations

487

11.1 Using Equations of State 487 11.2 Important Mathematical Relations 494 11.3 Developing Property Relations 497 11.4 Evaluating Changes in Entropy, Internal Energy, and Enthalpy 504 11.5 Other Thermodynamic Relations 513 11.6 Constructing Tables of Thermodynamic Properties 520 11.7 Generalized Charts for Enthalpy and Entropy 524 11.8 p–v–T Relations for Gas Mixtures 531 11.9 Analyzing Multicomponent Systems 536 Chapter Summary and Study Guide 548

CHAPTER

12

Ideal Gas Mixtures and Psychrometrics Applications 558 IDEAL GAS MIXTURES: GENERAL CONSIDERATIONS 558 12.1 Describing Mixture Composition 558 12.2 Relating p, V, and T for Ideal Gas Mixtures 562

Contents

12.3 Evaluating U, H, S and Specific Heats 564 12.4 Analyzing Systems Involving Mixtures 566 PSYCHROMETRIC APPLICATIONS 579 12.5 Introducing Psychrometric Principles 579 12.6 Psychrometers: Measuring the Wet-Bulb and Dry-Bulb Temperatures 590 12.7 Psychrometric Charts 592 12.8 Analyzing Air-Conditioning Processes 593 12.9 Cooling Towers 609 Chapter Summary and Study Guide 611

CHAPTER

13

Reacting Mixtures and Combustion 620 COMBUSTION FUNDAMENTALS 620 13.1 Introducing Combustion 620 13.2 Conservation of Energy—Reacting Systems 629 13.3 Determining the Adiabatic Flame Temperature 641 13.4 Fuel Cells 645 13.5 Absolute Entropy and the Third Law of Thermodynamics 648 CHEMICAL EXERGY 655 13.6 Introducing Chemical Exergy 655 13.7 Standard Chemical Exergy 659 13.8 Exergy Summary 664 13.9 Exergetic (Second Law) Efficiencies of Reacting Systems 667 Chapter Summary and Study Guide 669

CHAPTER

xi

14

Chemical and Phase Equilibrium 679 EQUILIBRIUM FUNDAMENTALS 679 14.1 Introducing Equilibrium Criteria 679 CHEMICAL EQUILIBRIUM 684 14.2 Equation of Reaction Equilibrium 684 14.3 Calculating Equilibrium Compositions 686 14.4 Further Examples of the Use of the Equilibrium Constant 695 PHASE EQUILIBRIUM 704 14.5 Equilibrium Between Two Phases of a Pure Substance 705 14.6 Equilibrium of Multicomponent, Multiphase Systems 706 Chapter Summary and Study Guide 711

APPENDIX

Appendix Tables, Figures, and Charts 718 Index to Tables in SI Units 718 Index to Figures and Charts 814 Answers to Selected Problems 822 Index 825

C H A

Getting Started: Introductory Concepts and Definitions

P T E R

1

E N G I N E E R I N G C O N T E X T The word thermodynamics stems from the Greek words therme (heat) and dynamis (force). Although various aspects of what is now known as thermodynamics have been of interest since antiquity, the formal study of thermodynamics began in the early nineteenth century through consideration of the motive power of heat: the capacity of hot bodies to produce work. Today the scope is larger, dealing generally with energy and with relationships among the properties of matter. Thermodynamics is both a branch of physics and an engineering science. The scientist is normally interested in gaining a fundamental understanding of the physical and chemical behavior of fixed quantities of matter at rest and uses the principles of thermodynamics to relate the properties of matter. Engineers are generally interested in studying systems and how they interact with their surroundings. To facilitate this, engineers extend the subject of thermodynamics to the study of systems through which matter flows. The objective of this chapter is to introduce you to some of the fundamental concepts and definitions that are used in our study of engineering thermodynamics. In most instances the introduction is brief, and further elaboration is provided in subsequent chapters.

chapter objective

1.1 Using Thermodynamics

Engineers use principles drawn from thermodynamics and other engineering sciences, such as fluid mechanics and heat and mass transfer, to analyze and design things intended to meet human needs. The wide realm of application of these principles is suggested by Table 1.1, which lists a few of the areas where engineering thermodynamics is important. Engineers seek to achieve improved designs and better performance, as measured by factors such as an increase in the output of some desired product, a reduced input of a scarce resource, a reduction in total costs, or a lesser environmental impact. The principles of engineering thermodynamics play an important part in achieving these goals.

1.2 Defining Systems

An important step in any engineering analysis is to describe precisely what is being studied. In mechanics, if the motion of a body is to be determined, normally the first step is to define a free body and identify all the forces exerted on it by other bodies. Newton’s second 1

2

Chapter 1 Getting Started: Introductory Concepts and Definitions

TABLE 1.1

Selected Areas of Application of Engineering Thermodynamics

Automobile engines Turbines Compressors, pumps Fossil- and nuclear-fueled power stations Propulsion systems for aircraft and rockets Combustion systems Cryogenic systems, gas separation, and liquefaction Heating, ventilating, and air-conditioning systems Vapor compression and absorption refrigeration Heat pumps Cooling of electronic equipment Alternative energy systems Fuel cells Thermoelectric and thermionic devices Magnetohydrodynamic (MHD) converters Solar-activated heating, cooling, and power generation Geothermal systems Ocean thermal, wave, and tidal power generation Wind power Biomedical applications Life-support systems Artificial organs

Solar-cell arrays

Surfaces with thermal control coatings International Space Station

Steam generator

Stack

Combustion gas cleanup

Coal

Air Steam

Turbine Generator

Electric power

Cooling tower

Condenser

Ash Condensate

Cooling water

Electrical power plant

Refrigerator

Automobile engine Trachea Lung

Fuel in Combustor Compressor Air in

Turbine Hot gases out

Heart Turbojet engine

Biomedical applications

1.2 Defining Systems

law of motion is then applied. In thermodynamics the term system is used to identify the subject of the analysis. Once the system is defined and the relevant interactions with other systems are identified, one or more physical laws or relations are applied. The system is whatever we want to study. It may be as simple as a free body or as complex as an entire chemical refinery. We may want to study a quantity of matter contained within a closed, rigid-walled tank, or we may want to consider something such as a pipeline through which natural gas flows. The composition of the matter inside the system may be fixed or may be changing through chemical or nuclear reactions. The shape or volume of the system being analyzed is not necessarily constant, as when a gas in a cylinder is compressed by a piston or a balloon is inflated. Everything external to the system is considered to be part of the system’s surroundings. The system is distinguished from its surroundings by a specified boundary, which may be at rest or in motion. You will see that the interactions between a system and its surroundings, which take place across the boundary, play an important part in engineering thermodynamics. It is essential for the boundary to be delineated carefully before proceeding with any thermodynamic analysis. However, the same physical phenomena often can be analyzed in terms of alternative choices of the system, boundary, and surroundings. The choice of a particular boundary defining a particular system is governed by the convenience it allows in the subsequent analysis.

3

system

surroundings boundary

TYPES OF SYSTEMS

Two basic kinds of systems are distinguished in this book. These are referred to, respectively, as closed systems and control volumes. A closed system refers to a fixed quantity of matter, whereas a control volume is a region of space through which mass may flow. A closed system is defined when a particular quantity of matter is under study. A closed system always contains the same matter. There can be no transfer of mass across its boundary. A special type of closed system that does not interact in any way with its surroundings is called an isolated system. Figure 1.1 shows a gas in a piston–cylinder assembly. When the valves are closed, we can consider the gas to be a closed system. The boundary lies just inside the piston and cylinder walls, as shown by the dashed lines on the figure. The portion of the boundary between the gas and the piston moves with the piston. No mass would cross this or any other part of the boundary. In subsequent sections of this book, thermodynamic analyses are made of devices such as turbines and pumps through which mass flows. These analyses can be conducted in principle by studying a particular quantity of matter, a closed system, as it passes through the device. In most cases it is simpler to think instead in terms of a given region of space through which mass flows. With this approach, a region within a prescribed boundary is studied. The region is called a control volume. Mass may cross the boundary of a control volume. A diagram of an engine is shown in Fig. 1.2a. The dashed line defines a control volume that surrounds the engine. Observe that air, fuel, and exhaust gases cross the boundary. A schematic such as in Fig. 1.2b often suffices for engineering analysis. The term control mass is sometimes used in place of closed system, and the term open system is used interchangeably with control volume. When the terms control mass and control volume are used, the system boundary is often referred to as a control surface. In general, the choice of system boundary is governed by two considerations: (1) what is known about a possible system, particularly at its boundaries, and (2) the objective of the analysis. for example. . . Figure 1.3 shows a sketch of an air compressor connected to a storage tank. The system boundary shown on the figure encloses the compressor, tank, and all of the piping. This boundary might be selected if the electrical power input were

closed system

isolated system

control volume

Gas

Boundary

Figure 1.1 Closed system: A gas in a piston–cylinder assembly.

4

Chapter 1 Getting Started: Introductory Concepts and Definitions Air in

Drive shaft

Air in

Exhaust gas out Fuel in

Fuel in

Drive shaft Exhaust gas out Boundary (control surface)

Boundary (control surface)

(a)

Figure 1.2

(b)

Example of a control volume (open system). An automobile engine.

known, and the objective of the analysis were to determine how long the compressor must operate for the pressure in the tank to rise to a specified value. Since mass crosses the boundary, the system would be a control volume. A control volume enclosing only the compressor might be chosen if the condition of the air entering and exiting the compressor were known, and the objective were to determine the electric power input.

1.3 Describing Systems and Their Behavior

Engineers are interested in studying systems and how they interact with their surroundings. In this section, we introduce several terms and concepts used to describe systems and how they behave. MACROSCOPIC AND MICROSCOPIC VIEWS OF THERMODYNAMICS

Systems can be studied from a macroscopic or a microscopic point of view. The macroscopic approach to thermodynamics is concerned with the gross or overall behavior. This is sometimes called classical thermodynamics. No model of the structure of matter at the molecular, atomic, and subatomic levels is directly used in classical thermodynamics. Although the behavior of systems is affected by molecular structure, classical thermodynamics allows important aspects of system behavior to be evaluated from observations of the overall system. Air

Tank Air compressor

–

+

Figure 1.3

storage tank.

Air compressor and

1.3 Describing Systems and Their Behavior

The microscopic approach to thermodynamics, known as statistical thermodynamics, is concerned directly with the structure of matter. The objective of statistical thermodynamics is to characterize by statistical means the average behavior of the particles making up a system of interest and relate this information to the observed macroscopic behavior of the system. For applications involving lasers, plasmas, high-speed gas flows, chemical kinetics, very low temperatures (cryogenics), and others, the methods of statistical thermodynamics are essential. Moreover, the microscopic approach is instrumental in developing certain data, for example, ideal gas specific heats (Sec. 3.6). For the great majority of engineering applications, classical thermodynamics not only provides a considerably more direct approach for analysis and design but also requires far fewer mathematical complications. For these reasons the macroscopic viewpoint is the one adopted in this book. When it serves to promote understanding, however, concepts are interpreted from the microscopic point of view. Finally, relativity effects are not significant for the systems under consideration in this book.

PROPERTY, STATE, AND PROCESS

To describe a system and predict its behavior requires knowledge of its properties and how those properties are related. A property is a macroscopic characteristic of a system such as mass, volume, energy, pressure, and temperature to which a numerical value can be assigned at a given time without knowledge of the previous behavior (history) of the system. Many other properties are considered during the course of our study of engineering thermodynamics. Thermodynamics also deals with quantities that are not properties, such as mass flow rates and energy transfers by work and heat. Additional examples of quantities that are not properties are provided in subsequent chapters. A way to distinguish nonproperties from properties is given shortly. The word state refers to the condition of a system as described by its properties. Since there are normally relations among the properties of a system, the state often can be specified by providing the values of a subset of the properties. All other properties can be determined in terms of these few. When any of the properties of a system change, the state changes and the system is said to have undergone a process. A process is a transformation from one state to another. However, if a system exhibits the same values of its properties at two different times, it is in the same state at these times. A system is said to be at steady state if none of its properties changes with time. A thermodynamic cycle is a sequence of processes that begins and ends at the same state. At the conclusion of a cycle all properties have the same values they had at the beginning. Consequently, over the cycle the system experiences no net change of state. Cycles that are repeated periodically play prominent roles in many areas of application. For example, steam circulating through an electrical power plant executes a cycle. At a given state each property has a definite value that can be assigned without knowledge of how the system arrived at that state. Therefore, the change in value of a property as the system is altered from one state to another is determined solely by the two end states and is independent of the particular way the change of state occurred. That is, the change is independent of the details of the process. Conversely, if the value of a quantity is independent of the process between two states, then that quantity is the change in a property. This provides a test for determining whether a quantity is a property: A quantity is a property if its change in value between two states is independent of the process. It follows that if the value of a particular quantity depends on the details of the process, and not solely on the end states, that quantity cannot be a property.

property

state

process steady state thermodynamic cycle

5

6

Chapter 1 Getting Started: Introductory Concepts and Definitions

EXTENSIVE AND INTENSIVE PROPERTIES

extensive property

intensive property

Thermodynamic properties can be placed in two general classes: extensive and intensive. A property is called extensive if its value for an overall system is the sum of its values for the parts into which the system is divided. Mass, volume, energy, and several other properties introduced later are extensive. Extensive properties depend on the size or extent of a system. The extensive properties of a system can change with time, and many thermodynamic analyses consist mainly of carefully accounting for changes in extensive properties such as mass and energy as a system interacts with its surroundings. Intensive properties are not additive in the sense previously considered. Their values are independent of the size or extent of a system and may vary from place to place within the system at any moment. Thus, intensive properties may be functions of both position and time, whereas extensive properties vary at most with time. Specific volume (Sec. 1.5), pressure, and temperature are important intensive properties; several other intensive properties are introduced in subsequent chapters. for example. . . to illustrate the difference between extensive and intensive properties, consider an amount of matter that is uniform in temperature, and imagine that it is composed of several parts, as illustrated in Fig. 1.4. The mass of the whole is the sum of the masses of the parts, and the overall volume is the sum of the volumes of the parts. However, the temperature of the whole is not the sum of the temperatures of the parts; it is the same for each part. Mass and volume are extensive, but temperature is intensive. PHASE AND PURE SUBSTANCE

phase

pure substance

The term phase refers to a quantity of matter that is homogeneous throughout in both chemical composition and physical structure. Homogeneity in physical structure means that the matter is all solid, or all liquid, or all vapor (or equivalently all gas). A system can contain one or more phases. For example, a system of liquid water and water vapor (steam) contains two phases. When more than one phase is present, the phases are separated by phase boundaries. Note that gases, say oxygen and nitrogen, can be mixed in any proportion to form a single gas phase. Certain liquids, such as alcohol and water, can be mixed to form a single liquid phase. But liquids such as oil and water, which are not miscible, form two liquid phases. A pure substance is one that is uniform and invariable in chemical composition. A pure substance can exist in more than one phase, but its chemical composition must be the same in each phase. For example, if liquid water and water vapor form a system with two phases, the system can be regarded as a pure substance because each phase has the same composition. A uniform mixture of gases can be regarded as a pure substance provided it remains a gas and does not react chemically. Changes in composition due to chemical reaction are

(a) Figure 1.4

(b)

Figure used to discuss the extensive and intensive property concepts.

1.3 Describing Systems and Their Behavior

considered in Chap. 13. A system consisting of air can be regarded as a pure substance as long as it is a mixture of gases; but if a liquid phase should form on cooling, the liquid would have a different composition from the gas phase, and the system would no longer be considered a pure substance.

EQUILIBRIUM

Classical thermodynamics places primary emphasis on equilibrium states and changes from one equilibrium state to another. Thus, the concept of equilibrium is fundamental. In mechanics, equilibrium means a condition of balance maintained by an equality of opposing forces. In thermodynamics, the concept is more far-reaching, including not only a balance of forces but also a balance of other influences. Each kind of influence refers to a particular aspect of thermodynamic, or complete, equilibrium. Accordingly, several types of equilibrium must exist individually to fulfill the condition of complete equilibrium; among these are mechanical, thermal, phase, and chemical equilibrium. Criteria for these four types of equilibrium are considered in subsequent discussions. For the present, we may think of testing to see if a system is in thermodynamic equilibrium by the following procedure: Isolate the system from its surroundings and watch for changes in its observable properties. If there are no changes, we conclude that the system was in equilibrium at the moment it was isolated. The system can be said to be at an equilibrium state. When a system is isolated, it does not interact with its surroundings; however, its state can change as a consequence of spontaneous events occurring internally as its intensive properties, such as temperature and pressure, tend toward uniform values. When all such changes cease, the system is in equilibrium. Hence, for a system to be in equilibrium it must be a single phase or consist of a number of phases that have no tendency to change their conditions when the overall system is isolated from its surroundings. At equilibrium, temperature is uniform throughout the system. Also, pressure can be regarded as uniform throughout as long as the effect of gravity is not significant; otherwise a pressure variation can exist, as in a vertical column of liquid.

equilibrium

equilibrium state

ACTUAL AND QUASIEQUILIBRIUM PROCESSES

There is no requirement that a system undergoing an actual process be in equilibrium during the process. Some or all of the intervening states may be nonequilibrium states. For many such processes we are limited to knowing the state before the process occurs and the state after the process is completed. However, even if the intervening states of the system are not known, it is often possible to evaluate certain overall effects that occur during the process. Examples are provided in the next chapter in the discussions of work and heat. Typically, nonequilibrium states exhibit spatial variations in intensive properties at a given time. Also, at a specified position intensive properties may vary with time, sometimes chaotically. Spatial and temporal variations in properties such as temperature, pressure, and velocity can be measured in certain cases. It may also be possible to obtain this information by solving appropriate governing equations, expressed in the form of differential equations, either analytically or by computer. Processes are sometimes modeled as an idealized type of process called a quasiequilibrium (or quasistatic) process. A quasiequilibrium process is one in which the departure from thermodynamic equilibrium is at most infinitesimal. All states through which the system passes in a quasiequilibrium process may be considered equilibrium states. Because nonequilibrium effects are inevitably present during actual processes, systems of engineering interest can at best approach, but never realize, a quasiequilibrium process.

quasiequilibrium process

7

8

Chapter 1 Getting Started: Introductory Concepts and Definitions

Our interest in the quasiequilibrium process concept stems mainly from two considerations:

Simple thermodynamic models giving at least qualitative information about the behavior of actual systems of interest often can be developed using the quasiequilibrium process concept. This is akin to the use of idealizations such as the point mass or the frictionless pulley in mechanics for the purpose of simplifying an analysis. The quasiequilibrium process concept is instrumental in deducing relationships that exist among the properties of systems at equilibrium (Chaps. 3, 6, and 11).

1.4 Measuring Mass, Length, Time, and Force

base unit

When engineering calculations are performed, it is necessary to be concerned with the units of the physical quantities involved. A unit is any specified amount of a quantity by comparison with which any other quantity of the same kind is measured. For example, meters, centimeters, kilometers, feet, inches, and miles are all units of length. Seconds, minutes, and hours are alternative time units. Because physical quantities are related by definitions and laws, a relatively small number of physical quantities suffice to conceive of and measure all others. These may be called primary dimensions. The others may be measured in terms of the primary dimensions and are called secondary. For example, if length and time were regarded as primary, velocity and area would be secondary. Two commonly used sets of primary dimensions that suffice for applications in mechanics are (1) mass, length, and time and (2) force, mass, length, and time. Additional primary dimensions are required when additional physical phenomena come under consideration. Temperature is included for thermodynamics, and electric current is introduced for applications involving electricity. Once a set of primary dimensions is adopted, a base unit for each primary dimension is specified. Units for all other quantities are then derived in terms of the base units. Let us illustrate these ideas by considering briefly the SI system of units. 1.4.1 SI Units

SI base units

The system of units called SI, takes mass, length, and time as primary dimensions and regards force as secondary. SI is the abbreviation for Système International d’Unités (International System of Units), which is the legally accepted system in most countries. The conventions of the SI are published and controlled by an international treaty organization. The SI base units for mass, length, and time are listed in Table 1.2 and discussed in the following paragraphs. The SI base unit for temperature is the kelvin, K. The SI base unit of mass is the kilogram, kg. It is equal to the mass of a particular cylinder of platinum–iridium alloy kept by the International Bureau of Weights and Measures near Paris. The mass standard for the United States is maintained by the National Institute of Standards and Technology. The kilogram is the only base unit still defined relative to a fabricated object. The SI base unit of length is the meter (metre), m, defined as the length of the path traveled by light in a vacuum during a specified time interval. The base unit of time is the second, s. The second is defined as the duration of 9,192,631,770 cycles of the radiation associated with a specified transition of the cesium atom. The SI unit of force, called the newton, is a secondary unit, defined in terms of the base units for mass, length, and time. Newton’s second law of motion states that the net force acting on a body is proportional to the product of the mass and the acceleration, written

1.4 Measuring Mass, Length, Time, and Force TABLE 1.2

Units and dimensions for Mass, Length, Time SI

Quantity mass length time

Unit

Dimension

Symbol

kilogram meter second

M L t

kg m s

F ma. The newton is defined so that the proportionality constant in the expression is equal to unity. That is, Newton’s second law is expressed as the equality F ma

(1.1)

The newton, N, is the force required to accelerate a mass of 1 kilogram at the rate of 1 meter per second per second. With Eq. 1.1 1 N 11 kg2 11 m/s2 2 1 kg # m/s2

(1.2)

for example. . . to illustrate the use of the SI units introduced thus far, let us determine the weight in newtons of an object whose mass is 1000 kg, at a place on the earth’s surface where the acceleration due to gravity equals a standard value defined as 9.80665 m/s2. Recalling that the weight of an object refers to the force of gravity, and is calculated using the mass of the object, m, and the local acceleration of gravity, g, with Eq. 1.1 we get

F mg 11000 kg219.80665 m/s2 2 9806.65 kg # m/s2 This force can be expressed in terms of the newton by using Eq. 1.2 as a unit conversion factor. That is F a9806.65

kg # m 1N b` ` 9806.65 N 2 s 1 kg # m/s2

Since weight is calculated in terms of the mass and the local acceleration due to gravity, the weight of an object can vary because of the variation of the acceleration of gravity with location, but its mass remains constant. for example. . . if the object considered previously were on the surface of a planet at a point where the acceleration of gravity is, say, one-tenth of the value used in the above calculation, the mass would remain the same but the weight would be one-tenth of the calculated value. SI units for other physical quantities are also derived in terms of the SI base units. Some of the derived units occur so frequently that they are given special names and symbols, such as the newton. SI units for quantities pertinent to thermodynamics are given in Table 1.3. TABLE 1.3

Quantity Velocity Acceleration Force Pressure Energy Power

Dimensions 1

Lt Lt2 MLt2 ML1 t2 ML2 t2 ML2 t3

Units

Symbol

Name

m/s m/s2 kg m/s2 kg m/s2 (N/m2) kg m2/s2 (N m) kg m2/s3 (J/s)

N Pa J W

newtons pascal joule watt

METHODOLOGY UPDATE

Observe that in the calculation of force in newtons, the unit conversion factor is set off by a pair of vertical lines. This device is used throughout the text to identify unit conversions.

9

10

Chapter 1 Getting Started: Introductory Concepts and Definitions

TABLE 1.4 SI Unit

Prefixes Factor Prefix Symbol 1012 109 106 103 102 102 103 106 109 1012

tera giga mega kilo hecto centi milli micro nano pico

T G M k h c m n p

Since it is frequently necessary to work with extremely large or small values when using the SI unit system, a set of standard prefixes is provided in Table 1.4 to simplify matters. For example, km denotes kilometer, that is, 103 m. 1.4.2 English Engineering Units Although SI units are the worldwide standard, at the present time many segments of the engineering community in the United States regularly use some other units. A large portion of America’s stock of tools and industrial machines and much valuable engineering data utilize units other than SI units. For many years to come, engineers in the United States will have to be conversant with a variety of units.

1.5 Two Measurable Properties: Specific Volume and Pressure

Three intensive properties that are particularly important in engineering thermodynamics are specific volume, pressure, and temperature. In this section specific volume and pressure are considered. Temperature is the subject of Sec. 1.6. 1.5.1 Specific Volume From the macroscopic perspective, the description of matter is simplified by considering it to be distributed continuously throughout a region. The correctness of this idealization, known as the continuum hypothesis, is inferred from the fact that for an extremely large class of phenomena of engineering interest the resulting description of the behavior of matter is in agreement with measured data. When substances can be treated as continua, it is possible to speak of their intensive thermodynamic properties “at a point.” Thus, at any instant the density at a point is defined as m r lim a b VSV ¿ V

(1.3)

where V is the smallest volume for which a definite value of the ratio exists. The volume V contains enough particles for statistical averages to be significant. It is the smallest volume for which the matter can be considered a continuum and is normally small enough that it can be considered a “point.” With density defined by Eq. 1.8, density can be described mathematically as a continuous function of position and time. The density, or local mass per unit volume, is an intensive property that may vary from point to point within a system. Thus, the mass associated with a particular volume V is determined in principle by integration m

r dV

(1.4)

V

specific volume

and not simply as the product of density and volume. The specific volume v is defined as the reciprocal of the density, v 1 r. It is the volume per unit mass. Like density, specific volume is an intensive property and may vary from point to point. SI units for density and specific volume are kg/m3 and m3/kg, respectively. However, they are also often expressed, respectively, as g/cm3 and cm3/g. In certain applications it is convenient to express properties such as a specific volume on a molar basis rather than on a mass basis. The amount of a substance can be given on a

1.5 Two Measurable Properties: Specific Volume and Pressure

molar basis in terms of the kilomole (kmol) or the pound mole (lbmol), as appropriate. In either case we use

n

m M

molar basis

(1.5)

The number of kilomoles of a substance, n, is obtained by dividing the mass, m, in kilograms by the molecular weight, M, in kg/kmol. Appendix Table A-1 provides molecular weights for several substances. To signal that a property is on a molar basis, a bar is used over its symbol. Thus, v signifies the volume per kmol. In this text, the units used for v are m3/kmol. With Eq. 1.10, the relationship between v and v is v Mv

(1.6)

where M is the molecular weight in kg/kmol or lb/lbmol, as appropriate. 1.5.2 Pressure Next, we introduce the concept of pressure from the continuum viewpoint. Let us begin by considering a small area A passing through a point in a fluid at rest. The fluid on one side of the area exerts a compressive force on it that is normal to the area, Fnormal. An equal but oppositely directed force is exerted on the area by the fluid on the other side. For a fluid at rest, no other forces than these act on the area. The pressure p at the specified point is defined as the limit p lim a ASA¿

Fnormal b A

(1.7)

where A is the area at the “point” in the same limiting sense as used in the definition of density. If the area A was given new orientations by rotating it around the given point, and the pressure determined for each new orientation, it would be found that the pressure at the point is the same in all directions as long as the fluid is at rest. This is a consequence of the equilibrium of forces acting on an element of volume surrounding the point. However, the pressure can vary from point to point within a fluid at rest; examples are the variation of atmospheric pressure with elevation and the pressure variation with depth in oceans, lakes, and other bodies of water. Consider next a fluid in motion. In this case the force exerted on an area passing through a point in the fluid may be resolved into three mutually perpendicular components: one normal to the area and two in the plane of the area. When expressed on a unit area basis, the component normal to the area is called the normal stress, and the two components in the plane of the area are termed shear stresses. The magnitudes of the stresses generally vary with the orientation of the area. The state of stress in a fluid in motion is a topic that is normally treated thoroughly in fluid mechanics. The deviation of a normal stress from the pressure, the normal stress that would exist were the fluid at rest, is typically very small. In this book we assume that the normal stress at a point is equal to the pressure at that point. This assumption yields results of acceptable accuracy for the applications considered. PRESSURE UNITS

The SI unit of pressure and stress is the pascal. 1 pascal 1 N/m2

pressure

11

12

Chapter 1 Getting Started: Introductory Concepts and Definitions

However, in this text it is convenient to work with multiples of the pascal: the kPa, the bar, and the MPa. 1 kPa 103 N/m2 1 bar 105 N/m2 1 MPa 106 N/m2 Although atmospheric pressure varies with location on the earth, a standard reference value can be defined and used to express other pressures. 1 standard atmosphere 1atm2 1.01325 105 N/m2 absolute pressure

gage pressure vacuum pressure

Pressure as discussed above is called absolute pressure. Throughout this book the term pressure refers to absolute pressure unless explicitly stated otherwise. Although absolute pressures must be used in thermodynamic relations, pressure-measuring devices often indicate the difference between the absolute pressure in a system and the absolute pressure of the atmosphere existing outside the measuring device. The magnitude of the difference is called a gage pressure or a vacuum pressure. The term gage pressure is applied when the pressure in the system is greater than the local atmospheric pressure, patm. p1gage2 p1absolute2 patm 1absolute2

(1.8)

When the local atmospheric pressure is greater than the pressure in the system, the term vacuum pressure is used. p1vacuum2 patm 1absolute2 p1absolute2

(1.9)

The relationship among the various ways of expressing pressure measurements is shown in Fig. 1.5.

p (gage)

Atmospheric pressure Absolute pressure that is greater than the local atmospheric pressure

p (vacuum)

p (absolute)

patm (absolute)

p (absolute)

Zero pressure Figure 1.5

Absolute pressure that is less than than the local atmospheric pressure Zero pressure

Relationships among the absolute, atmospheric, gage, and vacuum pressures.

1.5 Two Measurable Properties: Specific Volume and Pressure

patm

PRESSURE MEASUREMENT

Gas at pressure p

Two commonly used devices for measuring pressure are the manometer and the Bourdon tube. Manometers measure pressure differences in terms of the length of a column of liquid such as water, mercury, or oil. The manometer shown in Fig. 1.6 has one end open to the atmosphere and the other attached to a closed vessel containing a gas at uniform pressure. The difference between the gas pressure and that of the atmosphere is Tank

p patm rgL

Manometer liquid Figure 1.6 Pressure measurement by a manometer.

Pointer

Pinion gear Support Linkage

Gas at pressure p Figure 1.7

Pressure measurement by a Bourdon tube gage.

Figure 1.8

acquisition.

L

(1.10)

where is the density of the manometer liquid, g the acceleration of gravity, and L the difference in the liquid levels. For short columns of liquid, and g may be taken as constant. Because of this proportionality between pressure difference and manometer fluid length, pressures are often expressed in terms of millimeters of mercury, inches of water, and so on. It is left as an exercise to develop Eq. 1.15 using an elementary force balance. A Bourdon tube gage is shown in Fig. 1.7. The figure shows a curved tube having an elliptical cross section with one end attached to the pressure to be measured and the other end connected to a pointer by a mechanism. When fluid under pressure fills the tube, the elliptical section tends to become circular, and the tube straightens. This motion is transmitted by the mechanism to the pointer. By calibrating the deflection of the pointer for known pressures, a graduated scale can be determined from which any applied pressure can be read in suitable units. Because of its construction, the Bourdon tube measures the pressure relative to the pressure of the surroundings existing at the instrument. Accordingly, the dial reads zero when the inside and outside of the tube are at the same pressure. Pressure can be measured by other means as well. An important class of sensors utilize the piezoelectric effect: A charge is generated within certain solid materials when they are deformed. This mechanical input /electrical output provides the basis for pressure measurement as well as displacement and force measurements. Another important type of sensor employs a diaphragm that deflects when a force is applied, altering an inductance, resistance, or capacitance. Figure 1.8 shows a piezoelectric pressure sensor together with an automatic data acquisition system.

Elliptical metal Bourdon tube

13

Pressure sensor with automatic data

14

Chapter 1 Getting Started: Introductory Concepts and Definitions

1.6 Measuring Temperature

In this section the intensive property temperature is considered along with means for measuring it. Like force, a concept of temperature originates with our sense perceptions. It is rooted in the notion of the “hotness” or “coldness” of a body. We use our sense of touch to distinguish hot bodies from cold bodies and to arrange bodies in their order of “hotness,” deciding that 1 is hotter than 2, 2 hotter than 3, and so on. But however sensitive the human body may be, we are unable to gauge this quality precisely. Accordingly, thermometers and temperature scales have been devised to measure it. 1.6.1 Thermal Equilibrium

thermal (heat) interaction

thermal equilibrium temperature

adiabatic process isothermal process

zeroth law of thermodynamics

A definition of temperature in terms of concepts that are independently defined or accepted as primitive is difficult to give. However, it is possible to arrive at an objective understanding of equality of temperature by using the fact that when the temperature of a body changes, other properties also change. To illustrate this, consider two copper blocks, and suppose that our senses tell us that one is warmer than the other. If the blocks were brought into contact and isolated from their surroundings, they would interact in a way that can be described as a thermal (heat) interaction. During this interaction, it would be observed that the volume of the warmer block decreases somewhat with time, while the volume of the colder block increases with time. Eventually, no further changes in volume would be observed, and the blocks would feel equally warm. Similarly, we would be able to observe that the electrical resistance of the warmer block decreases with time, and that of the colder block increases with time; eventually the electrical resistances would become constant also. When all changes in such observable properties cease, the interaction is at an end. The two blocks are then in thermal equilibrium. Considerations such as these lead us to infer that the blocks have a physical property that determines whether they will be in thermal equilibrium. This property is called temperature, and we may postulate that when the two blocks are in thermal equilibrium, their temperatures are equal. The rate at which the blocks approach thermal equilibrium with one another can be slowed by separating them with a thick layer of polystyrene foam, rock wool, cork, or other insulating material. Although the rate at which equilibrium is approached can be reduced, no actual material can prevent the blocks from interacting until they attain the same temperature. However, by extrapolating from experience, an ideal insulator can be imagined that would preclude them from interacting thermally. An ideal insulator is called an adiabatic wall. When a system undergoes a process while enclosed by an adiabatic wall, it experiences no thermal interaction with its surroundings. Such a process is called an adiabatic process. A process that occurs at constant temperature is an isothermal process. An adiabatic process is not necessarily an isothermal process, nor is an isothermal process necessarily adiabatic. It is a matter of experience that when two bodies are in thermal equilibrium with a third body, they are in thermal equilibrium with one another. This statement, which is sometimes called the zeroth law of thermodynamics, is tacitly assumed in every measurement of temperature. Thus, if we want to know if two bodies are at the same temperature, it is not necessary to bring them into contact and see whether their observable properties change with time, as described previously. It is necessary only to see if they are individually in thermal equilibrium with a third body. The third body is usually a thermometer. 1.6.2 Thermometers

thermometric property

Any body with at least one measurable property that changes as its temperature changes can be used as a thermometer. Such a property is called a thermometric property. The particular

1.6 Measuring Temperature

Thermodynamics in the News... Mercury Thermometers Quickly Vanishing The mercury-in-glass fever thermometers, once found in nearly every medicine cabinet, are a thing of the past. The American Academy of Pediatrics has designated mercury as too toxic to be present in the home. Families are turning to safer alternatives and disposing of mercury thermometers. Proper disposal is an issue, experts say. The present generation of liquid-in-glass fever thermometers for home use contains patented liquid mixtures that are nontoxic, safe alternatives to mercury. Battery-powered digital thermometers also are common today. These devices use the fact that electrical resistance changes predictably with temperature to safely check for a fever.

The safe disposal of millions of obsolete mercury-filled thermometers has emerged in its own right as an environmental issue. For proper disposal, thermometers must be taken to hazardous-waste collection stations rather than simply thrown in the trash where they can be easily broken, releasing mercury. Loose fragments of broken thermometers and anything that contacted its mercury should be transported in closed containers to appropriate disposal sites.

substance that exhibits changes in the thermometric property is known as a thermometric substance. A familiar device for temperature measurement is the liquid-in-glass thermometer pictured in Fig. 1.9a, which consists of a glass capillary tube connected to a bulb filled with a liquid such as alcohol and sealed at the other end. The space above the liquid is occupied by the vapor of the liquid or an inert gas. As temperature increases, the liquid expands in volume and rises in the capillary. The length L of the liquid in the capillary depends on the temperature. Accordingly, the liquid is the thermometric substance and L is the thermometric property. Although this type of thermometer is commonly used for ordinary temperature measurements, it is not well suited for applications where extreme accuracy is required. OTHER TEMPERATURE SENSORS

Sensors known as thermocouples are based on the principle that when two dissimilar metals are joined, an electromotive force (emf ) that is primarily a function of temperature will exist in a circuit. In certain thermocouples, one thermocouple wire is platinum of a specified purity and the other is an alloy of platinum and rhodium. Thermocouples also utilize

L

Liquid (a)

(b)

Figure 1.9 Thermometers. (a) Liquid-in-glass. (b) Infraredsensing ear thermometer.

15

16

Chapter 1 Getting Started: Introductory Concepts and Definitions

Capillary L

Mercury reservoir

Gas bulb

Manometer

Figure 1.10

Constant-volume gas

thermometer.

copper and constantan (an alloy of copper and nickel), iron and constantan, as well as several other pairs of materials. Electrical-resistance sensors are another important class of temperature measurement devices. These sensors are based on the fact that the electrical resistance of various materials changes in a predictable manner with temperature. The materials used for this purpose are normally conductors (such as platinum, nickel, or copper) or semiconductors. Devices using conductors are known as resistance temperature detectors. Semiconductor types are called thermistors. A variety of instruments measure temperature by sensing radiation, such as the ear thermometer shown in Fig. 1.9(b). They are known by terms such as radiation thermometers and optical pyrometers. This type of thermometer differs from those previously considered in that it does not actually come in contact with the body whose temperature is to be determined, an advantage when dealing with moving objects or bodies at extremely high temperatures. All of these temperature sensors can be used together with automatic data acquisition. The constant-volume gas thermometer shown in Fig. 1.10 is so exceptional in terms of precision and accuracy that it has been adopted internationally as the standard instrument for calibrating other thermometers. The thermometric substance is the gas (normally hydrogen or helium), and the thermometric property is the pressure exerted by the gas. As shown in the figure, the gas is contained in a bulb, and the pressure exerted by it is measured by an open-tube mercury manometer. As temperature increases, the gas expands, forcing mercury up in the open tube. The gas is kept at constant volume by raising or lowering the reservoir. The gas thermometer is used as a standard worldwide by bureaus of standards and research laboratories. However, because gas thermometers require elaborate apparatus and are large, slowly responding devices that demand painstaking experimental procedures, smaller, more rapidly responding thermometers are used for most temperature measurements and they are calibrated (directly or indirectly) against gas thermometers. For further discussion of gas thermometry, see box.

M E A S U R I N G T E M P E R AT U R E W I T H T H E G A S THERMOMETER—THE GAS SCALE

It is instructive to consider how numerical values are associated with levels of temperature by the gas thermometer shown in Fig. 1.10. Let p stand for the pressure in a constant-volume gas thermometer in thermal equilibrium with a bath. A value can be assigned to the bath temperature very simply by a linear relation T ap

(1.11)

where is an arbitrary constant. The linear relationship is an arbitrary choice; other selections for the correspondence between pressure and temperature could also be made.

1.6 Measuring Temperature

Measured data for a fixed level of temperature, extrapolated to zero pressure O2

The value of may be determined by inserting the thermometer into another bath maintained at a standard fixed point: the triple point of water (Sec. 3.2) and measuring the pressure, call it ptp, of the confined gas at the triple point temperature, 273.16 K. Substituting values into Eq. 1.16 and solving for a

N2

p p–– tp

273.16 ptp

He H2

The temperature of the original bath, at which the pressure of the confined gas is p, is then T 273.16 a

17

p b ptp

(1.12)

p ptp

ptp Figure 1.11

However, since the values of both pressures, p and ptp, depend in part on the amount of gas in the bulb, the value assigned by Eq. 1.17 to the bath temperature varies with the amount of gas in the thermometer. This difficulty is overcome in precision thermometry by repeating the measurements (in the original bath and the reference bath) several times with less gas in the bulb in each successive attempt. For each trial the ratio pptp is calculated from Eq. 1.17 and plotted versus the corresponding reference pressure ptp of the gas at the triple point temperature. When several such points have been plotted, the resulting curve is extrapolated to the ordinate where ptp 0. This is illustrated in Fig. 1.11 for constant-volume thermometers with a number of different gases. Inspection of Fig. 1.11 shows an important result. At each nonzero value of the reference pressure, the pptp values differ with the gas employed in the thermometer. However, as pressure decreases, the pptp values from thermometers with different gases approach one another, and in the limit as pressure tends to zero, the same value for pptp is obtained for each gas. Based on these general results, the gas temperature scale is defined by the relationship T 273.16 lim

p

T = 273.16 lim p–– tp

(1.13)

where “lim” means that both p and ptp tend to zero. It should be evident that the determination of temperatures by this means requires extraordinarily careful and elaborate experimental procedures. Although the temperature scale of Eq. 1.18 is independent of the properties of any one gas, it still depends on the properties of gases in general. Accordingly, the measurement of low temperatures requires a gas that does not condense at these temperatures, and this imposes a limit on the range of temperatures that can be measured by a gas thermometer. The lowest temperature that can be measured with such an instrument is about 1 K, obtained with helium. At high temperatures gases dissociate, and therefore these temperatures also cannot be determined by a gas thermometer. Other empirical means, utilizing the properties of other substances, must be employed to measure temperature in ranges where the gas thermometer is inadequate. For further discussion see Sec. 5.5.

1.6.3 Kelvin Scale Empirical means of measuring temperature such as considered in Sec. 1.6.2 have inherent limitations. for example. . . the tendency of the liquid in a liquid-in-glass thermometer to freeze at low temperatures imposes a lower limit on the range of temperatures that can be

Readings of constant-volume gas thermometers, when several gases are used.

18

Chapter 1 Getting Started: Introductory Concepts and Definitions

measured. At high temperatures liquids vaporize, and therefore these temperatures also cannot be determined by a liquid-in-glass thermometer. Accordingly, several different thermometers might be required to cover a wide temperature interval. In view of the limitations of empirical means for measuring temperature, it is desirable to have a procedure for assigning temperature values that does not depend on the properties of any particular substance or class of substances. Such a scale is called a thermodynamic temperature scale. The Kelvin scale is an absolute thermodynamic temperature scale that provides a continuous definition of temperature, valid over all ranges of temperature. Empirical measures of temperature, with different thermometers, can be related to the Kelvin scale. To develop the Kelvin scale, it is necessary to use the conservation of energy principle and the second law of thermodynamics; therefore, further discussion is deferred to Sec. 5.5 after these principles have been introduced. However, we note here that the Kelvin scale has a zero of 0 K, and lower temperatures than this are not defined. The Kelvin scale and the gas scale defined by Eq. 1.18 can be shown to be identical in the temperature range in which a gas thermometer can be used. For this reason we may write K after a temperature determined by means of constant-volume gas thermometry. Moreover, until the concept of temperature is reconsidered in more detail in Chap. 5, we assume that all temperatures referred to in the interim are in accord with values given by a constant-volume gas thermometer.

Kelvin scale

1.6.4 Celsius Scale Temperature scales are defined by the numerical value assigned to a standard fixed point. By international agreement the standard fixed point is the easily reproducible triple point of water: the state of equilibrium between steam, ice, and liquid water (Sec. 3.2). As a matter of convenience, the temperature at this standard fixed point is defined as 273.16 kelvins, abbreviated as 273.16 K. This makes the temperature interval from the ice point1 (273.15 K) to the steam point2 equal to 100 K and thus in agreement over the interval with the Celsius scale discussed next, which assigns 100 Celsius degrees to it. The kelvin is the SI base unit for temperature. The Celsius temperature scale (formerly called the centigrade scale) uses the unit degree Celsius (C), which has the same magnitude as the kelvin. Thus, temperature differences are identical on both scales. However, the zero point on the Celsius scale is shifted to 273.15 K, as shown by the following relationship between the Celsius temperature and the Kelvin temperature

triple point

Celsius scale

T1°C2 T1K2 273.15

(1.14)

From this it can be seen that on the Celsius scale the triple point of water is 0.01C and that 0 K corresponds to 273.15C.

1.7 Engineering Design and Analysis

An important engineering function is to design and analyze things intended to meet human needs. Design and analysis, together with systematic means for approaching them, are considered in this section. 1

The state of equilibrium between ice and air-saturated water at a pressure of 1 atm.

2

The state of equilibrium between steam and liquid water at a pressure of 1 atm.

373.15

0.01 0.00 –273.15

Absolute zero

0.00

Kelvin

Ice point

Celsius

Triple point of water

273.16

Steam point

°C

273.15

K

100.0

1.7 Engineering Design and Analysis

Figure 1.12

Comparison of temperature scales.

1.7.1 Design Engineering design is a decision-making process in which principles drawn from engineering and other fields such as economics and statistics are applied, usually iteratively, to devise a system, system component, or process. Fundamental elements of design include the establishment of objectives, synthesis, analysis, construction, testing, and evaluation. Designs typically are subject to a variety of constraints related to economics, safety, environmental impact, and so on. Design projects usually originate from the recognition of a need or an opportunity that is only partially understood. Thus, before seeking solutions it is important to define the design objectives. Early steps in engineering design include pinning down quantitative performance specifications and identifying alternative workable designs that meet the specifications. Among the workable designs are generally one or more that are “best” according to some criteria: lowest cost, highest efficiency, smallest size, lightest weight, etc. Other important factors in the selection of a final design include reliability, manufacturability, maintainability, and marketplace considerations. Accordingly, a compromise must be sought among competing criteria, and there may be alternative design solutions that are very similar.3 1.7.2 Analysis Design requires synthesis: selecting and putting together components to form a coordinated whole. However, as each individual component can vary in size, performance, cost, and so on, it is generally necessary to subject each to considerable study or analysis before a final selection can be made. for example. . . a proposed design for a fire-protection system might entail an overhead piping network together with numerous sprinkler heads. Once an overall configuration has been determined, detailed engineering analysis would be necessary to specify the number and type of the spray heads, the piping material, and the pipe diameters of the various branches of the network. The analysis must also aim to ensure that all components form a smoothly working whole while meeting relevant cost constraints and applicable codes and standards. 3 For further discussion, see A. Bejan, G. Tsatsaronis, and M. J. Moran, Thermal Design and Optimization, John Wiley & Sons, New York, 1996, Chap. 1

design constraints

19

20

Chapter 1 Getting Started: Introductory Concepts and Definitions

Engineers frequently do analysis, whether explicitly as part of a design process or for some other purpose. Analyses involving systems of the kind considered in this book use, directly or indirectly, one or more of three basic laws. These laws, which are independent of the particular substance or substances under consideration, are

engineering model

the conservation of mass principle the conservation of energy principle the second law of thermodynamics

In addition, relationships among the properties of the particular substance or substances considered are usually necessary (Chaps. 3, 6, 11–14). Newton’s second law of motion (Chaps. 1, 2, 9), relations such as Fourier’s conduction model (Chap. 2), and principles of engineering economics (Chap. 7) may also play a part. The first steps in a thermodynamic analysis are definition of the system and identification of the relevant interactions with the surroundings. Attention then turns to the pertinent physical laws and relationships that allow the behavior of the system to be described in terms of an engineering model. The objective in modeling is to obtain a simplified representation of system behavior that is sufficiently faithful for the purpose of the analysis, even if many aspects exhibited by the actual system are ignored. For example, idealizations often used in mechanics to simplify an analysis and arrive at a manageable model include the assumptions of point masses, frictionless pulleys, and rigid beams. Satisfactory modeling takes experience and is a part of the art of engineering. Engineering analysis is most effective when it is done systematically. This is considered next. 1.7.3 Methodology for Solving Thermodynamics Problems A major goal of this textbook is to help you learn how to solve engineering problems that involve thermodynamic principles. To this end numerous solved examples and end-of-chapter problems are provided. It is extremely important for you to study the examples and solve problems, for mastery of the fundamentals comes only through practice. To maximize the results of your efforts, it is necessary to develop a systematic approach. You must think carefully about your solutions and avoid the temptation of starting problems in the middle by selecting some seemingly appropriate equation, substituting in numbers, and quickly “punching up” a result on your calculator. Such a haphazard problem-solving approach can lead to difficulties as problems become more complicated. Accordingly, we strongly recommend that problem solutions be organized using the five steps in the box below, which are employed in the solved examples of this text.

❶

Known: State briefly in your own words what is known. This requires that you read the problem carefully and think about it.

❷

Find: State concisely in your own words what is to be determined.

❸

Schematic and Given Data: Draw a sketch of the system to be considered. Decide whether a closed system or control volume is appropriate for the analysis, and then carefully identify the boundary. Label the diagram with relevant information from the problem statement. Record all property values you are given or anticipate may be required for subsequent calculations. Sketch appropriate property diagrams (see Sec. 3.2), locating key state points and indicating, if possible, the processes executed by the system. The importance of good sketches of the system and property diagrams cannot be overemphasized. They are often instrumental in enabling you to think clearly about the problem.

1.7 Engineering Design and Analysis

❹

Assumptions: To form a record of how you model the problem, list all simplifying assumptions and idealizations made to reduce it to one that is manageable. Sometimes this information also can be noted on the sketches of the previous step.

❺

Analysis: Using your assumptions and idealizations, reduce the appropriate governing equations and relationships to forms that will produce the desired results. It is advisable to work with equations as long as possible before substituting numerical data. When the equations are reduced to final forms, consider them to determine what additional data may be required. Identify the tables, charts, or property equations that provide the required values. Additional property diagram sketches may be helpful at this point to clarify states and processes. When all equations and data are in hand, substitute numerical values into the equations. Carefully check that a consistent and appropriate set of units is being employed. Then perform the needed calculations. Finally, consider whether the magnitudes of the numerical values are reasonable and the algebraic signs associated with the numerical values are correct.

The problem solution format used in this text is intended to guide your thinking, not substitute for it. Accordingly, you are cautioned to avoid the rote application of these five steps, for this alone would provide few benefits. Indeed, as a particular solution evolves you may have to return to an earlier step and revise it in light of a better understanding of the problem. For example, it might be necessary to add or delete an assumption, revise a sketch, determine additional property data, and so on. The solved examples provided in the book are frequently annotated with various comments intended to assist learning, including commenting on what was learned, identifying key aspects of the solution, and discussing how better results might be obtained by relaxing certain assumptions. Such comments are optional in your solutions. In some of the earlier examples and end-of-chapter problems, the solution format may seem unnecessary or unwieldy. However, as the problems become more complicated you will see that it reduces errors, saves time, and provides a deeper understanding of the problem at hand. The example to follow illustrates the use of this solution methodology together with important concepts introduced previously.

EXAMPLE

1.1

Identifying System Interactions

A wind turbine– electric generator is mounted atop a tower. As wind blows steadily across the turbine blades, electricity is generated. The electrical output of the generator is fed to a storage battery. (a) Considering only the wind turbine–electric generator as the system, identify locations on the system boundary where the system interacts with the surroundings. Describe changes occurring within the system with time. (b) Repeat for a system that includes only the storage battery. SOLUTION Known: A wind turbine–electric generator provides electricity to a storage battery. Find: For a system consisting of (a) the wind turbine–electric generator, (b) the storage battery, identify locations where the system interacts with its surroundings, and describe changes occurring within the system with time.

21

22

Chapter 1 Getting Started: Introductory Concepts and Definitions

Schematic and Given Data: Part (a)

Air flow

Turbine–generator

Assumptions: 1. In part (a), the system is the control volume shown by the dashed line on the figure. 2. In part (b), the system is the closed system shown by the dashed line on the figure. Electric current flow

Part (b) Storage battery

3. The wind is steady.

Thermal interaction Figure E1.1

Analysis: (a) In this case, there is air flowing across the boundary of the control volume. Another principal interaction between the system and surroundings is the electric current passing through the wires. From the macroscopic perspective, such an interaction is not considered a mass transfer, however. With a steady wind, the turbine–generator is likely to reach steady-state operation, where the rotational speed of the blades is constant and a steady electric current is generated.

❶

(b) The principal interaction between the system and its surroundings is the electric current passing into the battery through the wires. As noted in part (a), this interaction is not considered a mass transfer. The system is a closed system. As the battery is charged and chemical reactions occur within it, the temperature of the battery surface may become somewhat elevated and a thermal interaction might occur between the battery and its surroundings. This interaction is likely to be of secondary importance.

❶

Using terms familiar from a previous physics course, the system of part (a) involves the conversion of kinetic energy to electricity, whereas the system of part (b) involves energy storage within the battery.

Chapter Summary and Study Guide

In this chapter, we have introduced some of the fundamental concepts and definitions used in the study of thermodynamics. The principles of thermodynamics are applied by engineers to analyze and design a wide variety of devices intended to meet human needs. An important aspect of thermodynamic analysis is to identify systems and to describe system behavior in terms of properties and processes. Three important properties discussed in this chapter are specific volume, pressure, and temperature. In thermodynamics, we consider systems at equilibrium states and systems undergoing changes of state. We study processes during which the intervening states are not equilibrium

states as well as quasiequilibrium processes during which the departure from equilibrium is negligible. In this chapter, we have introduced SI units for mass, length, time, force, and temperature. You will need to be familiar of units as you use this book. Chapter 1 concludes with discussions of how thermodynamics is used in engineering design and how to solve thermodynamics problems systematically. This book has several features that facilitate study and contribute to understanding. For an overview, see How To Use This Book Effectively.

Problems: Developing Engineering Skills

23

The following checklist provides a study guide for this chapter. When your study of the text and the end-of-chapter exercises has been completed you should be able to

work on a molar basis using Eq. 1.5.

write out the meanings of the terms listed in the margin

apply the methodology for problem solving discussed in

throughout the chapter and understand each of the related concepts. The subset of key concepts listed below is particularly important in subsequent chapters.

identify an appropriate system boundary and describe the

interactions between the system and its surroundings. Sec. 1.7.3.

Key Engineering Concepts

surroundings p. 3 boundary p. 3 closed system p. 3 control volume p. 3 property p. 5 state p. 5

process p. 5 thermodynamic cycle p. 5 extensive property p. 6 intensive property p. 6 phase p. 6

pure substance p. 6 equilibrium p. 7 specific volume p. 10 pressure p. 11 temperature p. 14 adiabatic process p. 14

isothermal process p. 14 Kelvin scale p. 18 Rankine scale p. ••

Exercises: Things Engineers Think About 1. For an everyday occurrence, such as cooking, heating or cooling a house, or operating an automobile or a computer, make a sketch of what you observe. Define system boundaries for analyzing some aspect of the events taking place. Identify interactions between the systems and their surroundings. 2. What are possible boundaries for studying each of the following? (a) a bicycle tire inflating. (b) a cup of water being heated in a microwave oven. (c) a household refrigerator in operation. (d) a jet engine in flight. (e) cooling a desktop computer. (f) a residential gas furnace in operation. (g) a rocket launching. 3. Considering a lawnmower driven by a one-cylinder gasoline engine as the system, would this be best analyzed as a closed system or a control volume? What are some of the environmental impacts associated with the system? Repeat for an electrically driven lawnmower. 4. A closed system consists of still air at 1 atm, 20C in a closed vessel. Based on the macroscopic view, the system is in equilibrium, yet the atoms and molecules that make up the air are in continuous motion. Reconcile this apparent contradiction.

5. Air at normal temperature and pressure contained in a closed tank adheres to the continuum hypothesis. Yet when sufficient air has been drawn from the tank, the hypothesis no longer applies to the remaining air. Why? 6. Can the value of an intensive property be uniform with position throughout a system? Be constant with time? Both? 7. A data sheet indicates that the pressure at the inlet to a pump is 10 kPa. What might the negative pressure denote? 8. We commonly ignore the pressure variation with elevation for a gas inside a storage tank. Why? 9. When buildings have large exhaust fans, exterior doors can be difficult to open due to a pressure difference between the inside and outside. Do you think you could open a 3- by 7-ft door if the inside pressure were 1 in. of water (vacuum)? 10. What difficulties might be encountered if water were used as the thermometric substance in the liquid-in-glass thermometer of Fig. 1.9? 11. Look carefully around your home, automobile, or place of employment, and list all the measuring devices you find. For each, try to explain the principle of operation.

Problems: Developing Engineering Skills Exploring System Concepts

1.1 Referring to Figs. 1.1 and 1.2, identify locations on the boundary of each system where there are interactions with the surroundings.

1.2 As illustrated in Fig. P1.2, electric current from a storage battery runs an electric motor. The shaft of the motor is connected to a pulley–mass assembly that raises a mass. Considering the motor as a system, identify locations on the system

24

Chapter 1 Getting Started: Introductory Concepts and Definitions

boundary where the system interacts with its surroundings and describe changes that occur within the system with time. Repeat for an enlarged system that also includes the battery and pulley–mass assembly.

Battery

1.5 As illustrated in Fig. P1.5, water for a fire hose is drawn from a pond by a gasoline engine – driven pump. Considering the engine-driven pump as a system, identify locations on the system boundary where the system interacts with its surroundings and describe events occurring within the system. Repeat for an enlarged system that includes the hose and the nozzle.

Motor

Nozzle

Mass Figure P1.2

1.3 As illustrated in Fig. P1.3, water circulates between a storage tank and a solar collector. Heated water from the tank is used for domestic purposes. Considering the solar collector as a system, identify locations on the system boundary where the system interacts with its surroundings and describe events that occur within the system. Repeat for an enlarged system that includes the storage tank and the interconnecting piping. Hot water supply Intake hose

Pond

Solar collector

Hot water storage tank

Circulating pump

Figure P1.5

Cold water return

1.6 A system consists of liquid water in equilibrium with a gaseous mixture of air and water vapor. How many phases are present? Does the system consist of a pure substance? Explain. Repeat for a system consisting of ice and liquid water in equilibrium with a gaseous mixture of air and water vapor.

+ – Figure P1.3

1.4 As illustrated in Fig. P1.4, steam flows through a valve and turbine in series. The turbine drives an electric generator. Considering the valve and turbine as a system, identify locations on the system boundary where the system interacts with its surroundings and describe events occurring within the system. Repeat for an enlarged system that includes the generator.

1.7 A system consists of liquid oxygen in equilibrium with oxygen vapor. How many phases are present? The system undergoes a process during which some of the liquid is vaporized. Can the system be viewed as being a pure substance during the process? Explain.

Valve

1.8 A system consisting of liquid water undergoes a process. At the end of the process, some of the liquid water has frozen, and the system contains liquid water and ice. Can the system be viewed as being a pure substance during the process? Explain.

Figure P1.4

1.9 A dish of liquid water is placed on a table in a room. After a while, all of the water evaporates. Taking the water and the air in the room to be a closed system, can the system be regarded as a pure substance during the process? After the process is completed? Discuss.

Steam

+ Turbine

Generator

Steam

–

Problems: Developing Engineering Skills

25

Working with Force and Mass

1.10 An object weighs 25 kN at a location where the acceleration of gravity is 9.8 m/s2. Determine its mass, in kg. 1.11

An object whose mass is 10 kg weighs 95 N. Determine

(a) the local acceleration of gravity, in m/s2. (b) the mass, in kg, and the weight, in N, of the object at a location where g 9.81 m/s2. 1.12 Atomic and molecular weights of some common substances are listed in Appendix Table A-1. Using data from the appropriate table, determine the mass, in kg, of 10 kmol of each of the following: air, H2O, Cu, SO2. 1.13 When an object of mass 5 kg is suspended from a spring, the spring is observed to stretch by 8 cm. The deflection of the spring is related linearly to the weight of the suspended mass. What is the proportionality constant, in newton per cm, if g 9.81 m/s2? 1.14 A simple instrument for measuring the acceleration of gravity employs a linear spring from which a mass is suspended. At a location on earth where the acceleration of gravity is 9.81 m/s2, the spring extends 0.739 cm. If the spring extends 0.116 in. when the instrument is on Mars, what is the Martian acceleration of gravity? How much would the spring extend on the moon, where g 1.67 m/s2? 1.15 Estimate the magnitude of the force, in N, exerted by a seat belt on a 25 kg child during a frontal collision that decelerates a car from 8 km/h to rest in 0.1 s. Express the car’s deceleration in multiples of the standard acceleration of gravity, or g’s. 1.16 An object whose mass is 2 kg is subjected to an applied upward force. The only other force acting on the object is the force of gravity. The net acceleration of the object is upward with a magnitude of 5 m/s2. The acceleration of gravity is 9.81 m/s2. Determine the magnitude of the applied upward force, in N. 1.17 A closed system consists of 0.5 kmol of liquid water and occupies a volume of 4 103 m3. Determine the weight of the system, in N, and the average density, in kg/m3, at a location where the acceleration of gravity is g 9.81 m/s2. 1.18 The weight of an object on an orbiting space vehicle is measured to be 42 N based on an artificial gravitational acceleration of 6 m/s2. What is the weight of the object, in N, on earth, where g 9.81 m/s2? 1.19 If the variation of the acceleration of gravity, in m/s2, with elevation z, in m, above sea level is g 9.81 (3.3 106)z, determine the percent change in weight of an airliner landing from a cruising altitude of 10 km on a runway at sea level. 1.20 As shown in Fig. P1.21, a cylinder of compacted scrap metal measuring 2 m in length and 0.5 m in diameter is suspended from a spring scale at a location where the acceleration of gravity is 9.78 m/s2. If the scrap metal density, in kg/m3, varies with position z, in m, according to 7800 360(zL)2, determine the reading of the scale, in N.

L=2m z

D = 0.5 m

Figure P1.20

Using Specific Volume and Pressure

1.21 Fifteen kg of carbon dioxide (CO2) gas is fed to a cylinder having a volume of 20 m3 and initially containing 15 kg of CO2 at a pressure of 10 bar. Later a pinhole develops and the gas slowly leaks from the cylinder. (a) Determine the specific volume, in m3/kg, of the CO2 in the cylinder initially. Repeat for the CO2 in the cylinder after the 15 kg has been added. (b) Plot the amount of CO2 that has leaked from the cylinder, in kg, versus the specific volume of the CO2 remaining in the cylinder. Consider v ranging up to 1.0 m3/kg. 1.22 The following table lists temperatures and specific volumes of water vapor at two pressures: p 1.0 MPa

p 1.5 Mpa

T (C)

3

v (m /kg)

T (C)

v (m3/kg)

200 240 280

0.2060 0.2275 0.2480

200 240 280

0.1325 0.1483 0.1627

Data encountered in solving problems often do not fall exactly on the grid of values provided by property tables, and linear interpolation between adjacent table entries becomes necessary. Using the data provided here, estimate (a) the specific volume at T 240C, p 1.25 MPa, in m3/kg. (b) the temperature at p 1.5 MPa, v 0.1555 m3/kg, in C. (c) the specific volume at T 220C, p 1.4 MPa, in m3/kg. 1.23 A closed system consisting of 5 kg of a gas undergoes a process during which the relationship between pressure and specific volume is pv1.3 constant. The process begins with p1 1 bar, v1 0.2 m3/kg and ends with p2 0.25 bar. Determine the final volume, in m3, and plot the process on a graph of pressure versus specific volume.

26

Chapter 1 Getting Started: Introductory Concepts and Definitions

1.24 A gas initially at p1 1 bar and occupying a volume of 1 liter is compressed within a piston–cylinder assembly to a final pressure p2 4 bar. (a) If the relationship between pressure and volume during the compression is pV constant, determine the volume, in liters, at a pressure of 3 bar. Also plot the overall process on a graph of pressure versus volume. (b) Repeat for a linear pressure–volume relationship between the same end states.

1.29 Figure P1.29 shows a tank within a tank, each containing air. Pressure gage A is located inside tank B and reads 1.4 bar. The U-tube manometer connected to tank B contains mercury. Using data on the diagram, determine the absolute pressures inside tank A and tank B, each in bar. The atmospheric pressure surrounding tank B is 101 kPa. The acceleration of gravity is g 9.81 m/s2.

1.25 A gas contained within a piston–cylinder assembly undergoes a thermodynamic cycle consisting of three processes: Process 1–2: Compression with pV constant from p1 1 bar, V1 1.0 m3 to V2 0.2 m3

patm = 101 kPa

Tank B

Process 2–3: Constant-pressure expansion to V3 1.0 m3 L = 20 cm

Process 3–1: Constant volume Sketch the cycle on a p–V diagram labeled with pressure and volume values at each numbered state. 1.26 As shown in Fig. 1.6, a manometer is attached to a tank of gas in which the pressure is 104.0 kPa. The manometer liquid is mercury, with a density of 13.59 g/cm3. If g 9.81 m/s2 and the atmospheric pressure is 101.33 kPa, calculate (a) the difference in mercury levels in the manometer, in cm. (b) the gage pressure of the gas, in kPa. 1.27 The absolute pressure inside a tank is 0.4 bar, and the surrounding atmospheric pressure is 98 kPa. What reading would a Bourdon gage mounted in the tank wall give, in kPa? Is this a gage or vacuum reading? 1.28 Water flows through a Venturi meter, as shown in Fig. P1.28. The pressure of the water in the pipe supports columns of water that differ in height by 30 cm. Determine the difference in pressure between points a and b, in MPa. Does the pressure increase or decrease in the direction of flow? The atmospheric pressure is 1 bar, the specific volume of water is 103 m3/kg, and the acceleration of gravity is g 9.81 m/s2.

patm = 1 bar g = 9.81 m/s2 L = 30 cm.

Water υ = 10−3 m3/kg

Tank A

Gage A

Mercury ( ρ = 13.59 g/cm3) g = 9.81 m/s2

pgage, A = 1.4 bar Figure P1.29

1.30 A vacuum gage indicates that the pressure of air in a closed chamber is 0.2 bar (vacuum). The pressure of the surrounding atmosphere is equivalent to a 750-mm column of mercury. The density of mercury is 13.59 g/cm3, and the acceleration of gravity is 9.81 m/s2. Determine the absolute pressure within the chamber, in bar. 1.31 Refrigerant 22 vapor enters the compressor of a refrigeration system at an absolute pressure of .1379 MPa.2 A pressure gage at the compressor exit indicates a pressure of 1.93 MPa.2 (gage). The atmospheric pressure is .1007 MPa.2 Determine the change in absolute pressure from inlet to exit, in MPa.2, and the ratio of exit to inlet pressure. 1.32 Air contained within a vertical piston–cylinder assembly is shown in Fig. P1.32. On its top, the 10-kg piston is attached to a spring and exposed to an atmospheric pressure of 1 bar. Initially, the bottom of the piston is at x 0, and the spring exerts a negligible force on the piston. The valve is opened and air enters the cylinder from the supply line, causing the volume of the air within the cylinder to increase by 3.9 104 m3. The force exerted by the spring as the air expands within the cylinder varies linearly with x according to Fspring kx

a

Figure P1.28

b

where k 10,000 N/m. The piston face area is 7.8 103 m2. Ignoring friction between the piston and the cylinder wall, determine the pressure of the air within the cylinder, in bar, when the piston is in its initial position. Repeat when the piston is in its final position. The local acceleration of gravity is 9.81 m/s2.

Problems: Developing Engineering Skills patm

1.37 Derive Eq. 1.10 and use it to determine the gage pressure, in bar, equivalent to a manometer reading of 1 cm of water (density 1000 kg/m3). Repeat for a reading of 1 cm of mercury. Air supply line

x=0 Air

27

The density of mercury is 13.59 times that of water. Exploring Temperature

1.38 Two temperature measurements are taken with a thermometer marked with the Celsius scale. Show that the difference between the two readings would be the same if the temperatures were converted to the Kelvin scale.

Valve

1.39 The relation between resistance R and temperature T for a thermistor closely follows

Figure P1.32

1.33 Determine the total force, in kN, on the bottom of a 100 50 m swimming pool. The depth of the pool varies linearly along its length from 1 m to 4 m. Also, determine the pressure on the floor at the center of the pool, in kPa. The atmospheric pressure is 0.98 bar, the density of the water is 998.2 kg/m3, and the local acceleration of gravity is 9.8 m/s2. 1.34 Figure P1.34 illustrates an inclined manometer making an angle of with the horizontal. What advantage does an inclined manometer have over a U-tube manometer? Explain.

R R0 exp c b a

1 1 bd T T0

where R0 is the resistance, in ohms ( ), measured at temperature T0 (K) and is a material constant with units of K. For a particular thermistor R0 2.2 at T0 310 K. From a calibration test, it is found that R 0.31 at T 422 K. Determine the value of for the thermistor and make a plot of resistance versus temperature. 1.40 Over a limited temperature range, the relation between electrical resistance R and temperature T for a resistance temperature detector is R R0 31 a1T T0 2 4

θ

where R0 is the resistance, in ohms ( ), measured at reference temperature T0 (in C) and is a material constant with units of (C)1. The following data are obtained for a particular resistance thermometer:

Figure P1.34

1.35 The variation of pressure within the biosphere affects not only living things but also systems such as aircraft and undersea exploration vehicles. (a) Plot the variation of atmospheric pressure, in atm, versus elevation z above sea level, in km, ranging from 0 to 10 km. Assume that the specific volume of the atmosphere, in m3/kg, varies with the local pressure p, in kPa, according to v 72.435p. (b) Plot the variation of pressure, in atm, versus depth z below sea level, in km, ranging from 0 to 2 km. Assume that the specific volume of seawater is constant, v 0.956 103 m3/kg. In each case, g 9.81 m/s2 and the pressure at sea level is 1 atm. 1.36 One thousand kg of natural gas at 100 bar and 255 K is stored in a tank. If the pressure, p, specific volume, v, and temperature, T, of the gas are related by the following expression p 3 15.18 103 2T 1v 0.0026682 4 18.91 103 2 v2 where v is in m3/kg, T is in K, and p is in bar, determine the volume of the tank in m3. Also, plot pressure versus specific volume for the isotherms T 250 K, 500 K, and 1000 K.

Test 1 (T0) Test 2

T (C)

R ( )

0 91

(R0) 51.39 51.72

What temperature would correspond to a resistance of 51.47 on this thermometer? 1.41 A new absolute temperature scale is proposed. On this scale the ice point of water is 150S and the steam point is 300S. Determine the temperatures in C that correspond to 100 and 400S, respectively. What is the ratio of the size of the S to the kelvin? 1.42 As shown in Fig. P1.42, a small-diameter water pipe passes through the 6-in.-thick exterior wall of a dwelling. Assuming that temperature varies linearly with position x through the wall from 20C to 6C, would the water in the pipe freeze? T = 6°C

T = 20°C Pipe

3 in. 6 in. x

Figure P1.42

28

Chapter 1 Getting Started: Introductory Concepts and Definitions

Design & Open Ended Problems: Exploring Engineering Practice 1.1D The issue of global warming is receiving considerable attention these days. Write a technical report on the subject of global warming. Explain what is meant by the term global warming and discuss objectively the scientific evidence that is cited as the basis for the argument that global warming is occurring.

1.7D Obtain manufacturers’ data on thermocouple and thermistor temperature sensors for measuring temperatures of hot combustion gases from a furnace. Explain the basic operating principles of each sensor and compare the advantages and disadvantages of each device. Consider sensitivity, accuracy, calibration, and cost.

1.2D Economists and others speak of sustainable development as a means for meeting present human needs without compromising the ability of future generations to meet their own needs. Research the concept of sustainable development, and write a paper objectively discussing some of the principal issues associated with it.

1.8D The International Temperature Scale was first adopted by the International Committee on Weights and Measures in 1927 to provide a global standard for temperature measurement. This scale has been refined and extended in several revisions, most recently in 1990 (International Temperature Scale of 1990, ITS-90). What are some of the reasons for revising the scale? What are some of the principal changes that have been made since 1927?

1.3D Write a report reviewing the principles and objectives of statistical thermodynamics. How does the macroscopic approach to thermodynamics of the present text differ from this? Explain. 1.4D Methane-laden gas generated by the decomposition of landfill trash is more commonly flared than exploited for some useful purpose. Research literature on the possible uses of landfill gas and write a report of your findings. Does the gas represent a significant untapped resource? Discuss. 1.5D You are asked to address a city council hearing concerning the decision to purchase a commercially available 10-kW wind turbine–generator having an expected life of 12 or more years. As an engineer, what considerations will you point out to the council members to help them with their decision? 1.6D Develop a schematic diagram of an automatic data acquisition system for sampling pressure data inside the cylinder of a diesel engine. Determine a suitable type of pressure transducer for this purpose. Investigate appropriate computer software for running the system. Write a report of your findings.

1.9D A facility is under development for testing valves used in nuclear power plants. The pressures and temperatures of flowing gases and liquids must be accurately measured as part of the test procedure. The American National Standards Institute (ANSI) and the American Society of Heating, Refrigerating, and Air Conditioning Engineers (ASHRAE) have adopted standards for pressure and temperature measurement. Obtain copies of the relevant standards, and prepare a memorandum discussing what standards must be met in the design of the facility and what requirements those standards place on the design. 1.10D List several aspects of engineering economics relevant to design. What are the important contributors to cost that should be considered in engineering design? Discuss what is meant by annualized costs. 1.11D Mercury Thermometers Quickly Vanishing (see box Sec. 1.6). Investigate the medical complications of mercury exposure. Write a report including at least three references.

C H A P

Energy and the First Law of Thermodynamics

T E R

2

E N G I N E E R I N G C O N T E X T Energy is a fundamental concept of thermodynamics and one of the most significant aspects of engineering analysis. In this chapter we discuss energy and develop equations for applying the principle of conservation of energy. The current presentation is limited to closed systems. In Chap. 4 the discussion is extended to control volumes. Energy is a familiar notion, and you already know a great deal about it. In the present chapter several important aspects of the energy concept are developed. Some of these you have encountered before. A basic idea is that energy can be stored within systems in various forms. Energy also can be converted from one form to another and transferred between systems. For closed systems, energy can be transferred by work and heat transfer. The total amount of energy is conserved in all conversions and transfers. The objective of this chapter is to organize these ideas about energy into forms suitable for engineering analysis. The presentation begins with a review of energy concepts from mechanics. The thermodynamic concept of energy is then introduced as an extension of the concept of energy in mechanics.

chapter objective

2.1 Reviewing Mechanical Concepts of Energy

Building on the contributions of Galileo and others, Newton formulated a general description of the motions of objects under the influence of applied forces. Newton’s laws of motion, which provide the basis for classical mechanics, led to the concepts of work, kinetic energy, and potential energy, and these led eventually to a broadened concept of energy. The present discussion begins with an application of Newton’s second law of motion. WORK AND KINETIC ENERGY

The curved line in Fig. 2.1 represents the path of a body of mass m (a closed system) moving relative to the x–y coordinate frame shown. The velocity of the center of mass of the body is denoted by V.1 The body is acted on by a resultant force F, which may vary in magnitude from location to location along the path. The resultant force is resolved into a component Fs along the path and a component Fn normal to the path. The effect of the 1

Boldface symbols denote vectors. Vector magnitudes are shown in lightface type.

29

30

Chapter 2 Energy and the First Law of Thermodynamics y

Fs

Path

V ds F s

Body Fn

Figure 2.1 x

Forces acting on a moving

system.

component Fs is to change the magnitude of the velocity, whereas the effect of the component Fn is to change the direction of the velocity. As shown in Fig. 2.1, s is the instantaneous position of the body measured along the path from some fixed point denoted by 0. Since the magnitude of F can vary from location to location along the path, the magnitudes of Fs and Fn are, in general, functions of s. Let us consider the body as it moves from s s1, where the magnitude of its velocity is V1, to s s2, where its velocity is V2. Assume for the present discussion that the only interaction between the body and its surroundings involves the force F. By Newton’s second law of motion, the magnitude of the component Fs is related to the change in the magnitude of V by dV dt

(2.1)

d V ds dV mV ds dt ds

(2.2)

Fs m Using the chain rule, this can be written as Fs m

where V dsdt. Rearranging Eq. 2.2 and integrating from s1 to s2 gives

V2

mV d V

V1

s2

Fs ds

(2.3)

s1

The integral on the left of Eq. 2.3 is evaluated as follows

V2

V1

V2 1 1 mV2 d m 1V22 V21 2 2 2 V1

(2.4)

The quantity 12mV2 is the kinetic energy, KE, of the body. Kinetic energy is a scalar quantity. The change in kinetic energy, KE, of the body is2

kinetic energy

1 ¢KE KE2 KE1 m 1V22 V21 2 2

work

mV d V

(2.5)

The integral on the right of Eq. 2.3 is the work of the force Fs as the body moves from s1 to s2 along the path. Work is also a scalar quantity. With Eq. 2.4, Eq. 2.3 becomes 1 m 1V22 V21 2 2 The symbol always means “final value minus initial value.”

2

s2

s1

F # ds

(2.6)

2.1 Reviewing Mechanical Concepts of Energy

31

Thermodynamics in the News… Hybrids Harvest Energy Ever wonder what happens to the kinetic energy when you step on the brakes of your moving car? Automotive engineers have, and the result is the hybrid electric vehicle combining an electric motor with a small conventional engine. When a hybrid is braked, some of its kinetic energy is harvested and stored in batteries. The electric motor calls on the stored energy to help the car start up again. A specially designed transmission provides the proper split between the engine and the electric motor to minimize fuel use. Because

stored energy assists the engine, these cars get better fuel economy than comparably sized conventional vehicles. To further reduce fuel consumption, hybrids are designed with minimal aerodynamic drag, and many parts are made from sturdy, lightweight materials such as carbon fiber-metal composites. Some models now on the market achieve gas mileage as high as 60–70 miles per gallon, manufacturers say.

where the expression for work has been written in terms of the scalar product (dot product) of the force vector F and the displacement vector ds. Equation 2.6 states that the work of the resultant force on the body equals the change in its kinetic energy. When the body is accelerated by the resultant force, the work done on the body can be considered a transfer of energy to the body, where it is stored as kinetic energy. Kinetic energy can be assigned a value knowing only the mass of the body and the magnitude of its instantaneous velocity relative to a specified coordinate frame, without regard for how this velocity was attained. Hence, kinetic energy is a property of the body. Since kinetic energy is associated with the body as a whole, it is an extensive property. Work has units of force times distance. The units of kinetic energy are the same as for work. In SI, the energy unit is the newton-meter, N # m, called the joule, J. In this book it is convenient to use the kilojoule, kJ. UNITS.

POTENTIAL ENERGY

Equation 2.6 is the principal result of the previous section. Derived from Newton’s second law, the equation gives a relationship between two defined concepts: kinetic energy and work. In this section it is used as a point of departure to extend the concept of energy. To begin, refer to Fig. 2.2, which shows a body of mass m that moves vertically from an elevation z1 to an elevation z2 relative to the surface of the earth. Two forces are shown acting on the system: a downward force due to gravity with magnitude mg and a vertical force with magnitude R representing the resultant of all other forces acting on the system. The work of each force acting on the body shown in Fig. 2.2 can be determined by using the definition previously given. The total work is the algebraic sum of these individual values. In accordance with Eq. 2.6, the total work equals the change in kinetic energy. That is 1 m 1V22 V21 2 2

z2

z1

R dz

z2

mg dz

(2.7)

z1

A minus sign is introduced before the second term on the right because the gravitational force is directed downward and z is taken as positive upward. The first integral on the right of Eq. 2.7 represents the work done by the force R on the body as it moves vertically from z1 to z2. The second integral can be evaluated as follows

z2

z1

mg dz mg 1z2 z1 2

(2.8)

R z2

z

mg

z1

Earth’s surface Figure 2.2 Illustration used to introduce the potential energy concept.

32

Chapter 2 Energy and the First Law of Thermodynamics

where the acceleration of gravity has been assumed to be constant with elevation. By incorporating Eq. 2.8 into Eq. 2.7 and rearranging 1 m 1V22 V21 2 mg 1z2 z1 2 2

z2

R dz

(2.9)

z1

The quantity mgz is the gravitational potential energy, PE. The change in gravitational potential energy, PE, is

gravitational potential energy

¢PE PE2 PE1 mg 1z2 z1 2

(2.10)

The units for potential energy in any system of units are the same as those for kinetic energy and work. Potential energy is associated with the force of gravity and is therefore an attribute of a system consisting of the body and the earth together. However, evaluating the force of gravity as mg enables the gravitational potential energy to be determined for a specified value of g knowing only the mass of the body and its elevation. With this view, potential energy is regarded as an extensive property of the body. Throughout this book it is assumed that elevation differences are small enough that the gravitational force can be considered constant. The concept of gravitational potential energy can be formulated to account for the variation of the gravitational force with elevation, however. To assign a value to the kinetic energy or the potential energy of a system, it is necessary to assume a datum and specify a value for the quantity at the datum. Values of kinetic and potential energy are then determined relative to this arbitrary choice of datum and reference value. However, since only changes in kinetic and potential energy between two states are required, these arbitrary reference specifications cancel. When a system undergoes a process where there are changes in kinetic and potential energy, special care is required to obtain a consistent set of units. for example. . . to illustrate the proper use of units in the calculation of such terms, consider a system having a mass of 1 kg whose velocity increases from 15 m/s to 30 m/s while its elevation decreases by 10 m at a location where g 9.7 m/s2. Then

¢KE

1 m 1V22 V21 2 2

1 m 2 m 2 1N 1 kJ ` 11 kg2 c a30 b a15 b d ` ` ` s s 2 1 kg # m /s2 103 N # m 0.34 kJ ¢PE mg 1z2 z1 2

11 kg2 a9.7

m 1N 1 kJ ` b 110 m2 ` ` ` s2 1 kg # m /s2 103 N # m

0.10 kJ

CONSERVATION OF ENERGY IN MECHANICS

Equation 2.9 states that the total work of all forces acting on the body from the surroundings, with the exception of the gravitational force, equals the sum of the changes in the kinetic and potential energies of the body. When the resultant force causes the elevation to be increased, the body to be accelerated, or both, the work done by the force can be considered

2.2 Broadening Our Understanding of Work

a transfer of energy to the body, where it is stored as gravitational potential energy and/or kinetic energy. The notion that energy is conserved underlies this interpretation. The interpretation of Eq. 2.9 as an expression of the conservation of energy principle can be reinforced by considering the special case of a body on which the only force acting is that due to gravity, for then the right side of the equation vanishes and the equation reduces to 1 m 1V22 V21 2 mg 1z2 z1 2 0 2 or

z

(2.11)

1 1 mV22 mgz2 mV21 mgz1 2 2 Under these conditions, the sum of the kinetic and gravitational potential energies remains constant. Equation 2.11 also illustrates that energy can be converted from one form to another: For an object falling under the influence of gravity only, the potential energy would decrease as the kinetic energy increases by an equal amount. CLOSURE

The presentation thus far has centered on systems for which applied forces affect only their overall velocity and position. However, systems of engineering interest normally interact with their surroundings in more complicated ways, with changes in other properties as well. To analyze such systems, the concepts of kinetic and potential energy alone do not suffice, nor does the rudimentary conservation of energy principle introduced in this section. In thermodynamics the concept of energy is broadened to account for other observed changes, and the principle of conservation of energy is extended to include a wide variety of ways in which systems interact with their surroundings. The basis for such generalizations is experimental evidence. These extensions of the concept of energy are developed in the remainder of the chapter, beginning in the next section with a fuller discussion of work.

2.2 Broadening Our Understanding of Work

The work W done by, or on, a system evaluated in terms of macroscopically observable forces and displacements is

W

s2

F # ds

(2.12)

s1

This relationship is important in thermodynamics, and is used later in the present section to evaluate the work done in the compression or expansion of gas (or liquid), the extension of a solid bar, and the stretching of a liquid film. However, thermodynamics also deals with phenomena not included within the scope of mechanics, so it is necessary to adopt a broader interpretation of work, as follows. A particular interaction is categorized as a work interaction if it satisfies the following criterion, which can be considered the thermodynamic definition of work: Work is done by a system on its surroundings if the sole effect on everything external to the system could have been the raising of a weight. Notice that the raising of a weight is, in effect, a force acting through a distance, so the concept of work in thermodynamics is a natural extension of the

thermodynamic definition of work

mg

33

34

Chapter 2 Energy and the First Law of Thermodynamics

Paddle wheel

System A i

Gas

System B a

b

Battery Figure 2.3

Two examples of

work.

concept of work in mechanics. However, the test of whether a work interaction has taken place is not that the elevation of a weight has actually taken place, or that a force has actually acted through a distance, but that the sole effect could have been an increase in the elevation of a weight. for example. . . consider Fig. 2.3 showing two systems labeled A and B. In system A, a gas is stirred by a paddle wheel: the paddle wheel does work on the gas. In principle, the work could be evaluated in terms of the forces and motions at the boundary between the paddle wheel and the gas. Such an evaluation of work is consistent with Eq. 2.12, where work is the product of force and displacement. By contrast, consider system B, which includes only the battery. At the boundary of system B, forces and motions are not evident. Rather, there is an electric current i driven by an electrical potential difference existing across the terminals a and b. That this type of interaction at the boundary can be classified as work follows from the thermodynamic definition of work given previously: We can imagine the current is supplied to a hypothetical electric motor that lifts a weight in the surroundings.

Work is a means for transferring energy. Accordingly, the term work does not refer to what is being transferred between systems or to what is stored within systems. Energy is transferred and stored when work is done. 2.2.1 Sign Convention and Notation Engineering thermodynamics is frequently concerned with devices such as internal combustion engines and turbines whose purpose is to do work. Hence, in contrast to the approach generally taken in mechanics, it is often convenient to consider such work as positive. That is, sign convention for work

W 7 0: work done by the system W 6 0: work done on the system This sign convention is used throughout the book. In certain instances, however, it is convenient to regard the work done on the system to be positive, as has been done in the discussion of Sec. 2.1. To reduce the possibility of misunderstanding in any such case, the direction of energy transfer is shown by an arrow on a sketch of the system, and work is regarded as positive in the direction of the arrow. To evaluate the integral in Eq. 2.12, it is necessary to know how the force varies with the displacement. This brings out an important idea about work: The value of W depends on the details of the interactions taking place between the system and surroundings during a process

2.2 Broadening Our Understanding of Work

and not just the initial and final states of the system. It follows that work is not a property of the system or the surroundings. In addition, the limits on the integral of Eq. 2.12 mean “from state 1 to state 2” and cannot be interpreted as the values of work at these states. The notion of work at a state has no meaning, so the value of this integral should never be indicated as W2 W1. The differential of work, W, is said to be inexact because, in general, the following integral cannot be evaluated without specifying the details of the process

work is not a property

2

dW W

1

On the other hand, the differential of a property is said to be exact because the change in a property between two particular states depends in no way on the details of the process linking the two states. For example, the change in volume between two states can be determined by integrating the differential dV, without regard for the details of the process, as follows

V2

dV V2 V1

V1

where V1 is the volume at state 1 and V2 is the volume at state 2. The differential of every property is exact. Exact differentials are written, as above, using the symbol d. To stress the difference between exact and inexact differentials, the differential of work is written as W. The symbol is also used to identify other inexact differentials encountered later. 2.2.2 Power Many thermodynamic analyses are concerned with the time rate at which energy # transfer occurs. The rate of energy transfer by work is called power and is denoted by W. When a work interaction involves a macroscopically observable force, the rate of energy transfer by work is equal to the product of the force and the velocity at the point of application of the force # WF#V

(2.13)

# A dot appearing over a symbol, as in W, is used throughout this book to indicate a time rate. In principle, Eq. 2.13 can be integrated from time t1 to time t2 to get the total work done during the time interval W

t2

t1

# W dt

t2

F # V dt

(2.14)

t1

# The same sign convention applies for W as for W. Since power is a time rate of doing work, it can be expressed in terms of any units for energy and time. In SI, the unit for power is J/s, called the watt. In this book the kilowatt, kW, is generally used. for example. . . to illustrate the use of Eq. 2.13, let us evaluate the power required for a bicyclist traveling at 8.94 m/s to overcome the drag force imposed by the surrounding air. This aerodynamic drag force is given by

Fd 12CdArV2

power

35

36

Chapter 2 Energy and the First Law of Thermodynamics

where Cd is a constant called the drag coefficient, A is the frontal area of the bicycle and rider, and is the air density. By Eq. 2.13 the required power is Fd # V or # W 1 12CdArV2 2 V 12CdArV3 Using typical values: Cd 0.88, A 0.362 m2, and 1.2 kg/m3, together with V 8.94 m/s, the power required is # 1 W 10.882 10.362 m2 2 11.2 kg/m3 2 18.94 m/s 2 3 2 136.6 W 2.2.3 Modeling Expansion or Compression Work There are many ways in which work can be done by or on a system. The remainder of this section is devoted to considering several examples, beginning with the important case of the work done when the volume of a quantity of a gas (or liquid) changes by expansion or compression. Let us evaluate the work done by the closed system shown in Fig. 2.4 consisting of a gas (or liquid) contained in a piston–cylinder assembly as the gas expands. During the process the gas pressure exerts a normal force on the piston. Let p denote the pressure acting at the interface between the gas and the piston. The force exerted by the gas on the piston is simply the product pA, where A is the area of the piston face. The work done by the system as the piston is displaced a distance dx is dW pA dx

(2.15)

The product A dx in Eq. 2.15 equals the change in volume of the system, dV. Thus, the work expression can be written as dW p dV

(2.16)

Since dV is positive when volume increases, the work at the moving boundary is positive when the gas expands. For a compression, dV is negative, and so is work found from Eq. 2.16. These signs are in agreement with the previously stated sign convention for work. For a change in volume from V1 to V2, the work is obtained by integrating Eq. 2.16 W

V2

p dV

(2.17)

V1

Although Eq. 2.17 is derived for the case of a gas (or liquid) in a piston–cylinder assembly, it is applicable to systems of any shape provided the pressure is uniform with position over the moving boundary. System boundary Area = A

Average pressure at the piston face = p

F = pA Gas or liquid x

x1

x2

Figure 2.4 Expansion or compression of a gas or liquid.

2.2 Broadening Our Understanding of Work

ACTUAL EXPANSION OR COMPRESSION PROCESSES

p

To perform the integral of Eq. 2.17 requires a relationship between the gas pressure at the moving boundary and the system volume, but this relationship may be difficult, or even impossible, to obtain for actual compressions and expansions. In the cylinder of an automobile engine, for example, combustion and other nonequilibrium effects give rise to nonuniformities throughout the cylinder. Accordingly, if a pressure transducer were mounted on the cylinder head, the recorded output might provide only an approximation for the pressure at the piston face required by Eq. 2.17. Moreover, even when the measured pressure is essentially equal to that at the piston face, scatter might exist in the pressure– volume data, as illustrated in Fig. 2.5. Still, performing the integral of Eq. 2.17 based on a curve fitted to the data could give a plausible estimate of the work. We will see later that in some cases where lack of the required pressure–volume relationship keeps us from evaluating the work from Eq. 2.17, the work can be determined alternatively from an energy balance (Sec. 2.5).

Figure 2.5

37

Measured data Curve fit

V Pressure–

volume data.

QUASIEQUILIBRIUM EXPANSION OR COMPRESSION PROCESSES

An idealized type of process called a quasiequilibrium process is introduced in Sec. 1.3. A quasiequilibrium process is one in which all states through which the system passes may be considered equilibrium states. A particularly important aspect of the quasiequilibrium process concept is that the values of the intensive properties are uniform throughout the system, or every phase present in the system, at each state visited. To consider how a gas (or liquid) might be expanded or compressed in a quasiequilibrium fashion, refer to Fig. 2.6, which shows a system consisting of a gas initially at an equilibrium state. As shown in the figure, the gas pressure is maintained uniform throughout by a number of small masses resting on the freely moving piston. Imagine that one of the masses is removed, allowing the piston to move upward as the gas expands slightly. During such an expansion the state of the gas would depart only slightly from equilibrium. The system would eventually come to a new equilibrium state, where the pressure and all other intensive properties would again be uniform in value. Moreover, were the mass replaced, the gas would be restored to its initial state, while again the departure from equilibrium would be slight. If several of the masses were removed one after another, the gas would pass through a sequence of equilibrium states without ever being far from equilibrium. In the limit as the increments of mass are made vanishingly small, the gas would undergo a quasiequilibrium expansion process. A quasiequilibrium compression can be visualized with similar considerations. Equation 2.17 can be applied to evaluate the work in quasiequilibrium expansion or compression processes. For such idealized processes the pressure p in the equation is the pressure of the entire quantity of gas (or liquid) undergoing the process, and not just the pressure at the moving boundary. The relationship between the pressure and volume may be graphical or analytical. Let us first consider a graphical relationship. A graphical relationship is shown in the pressure–volume diagram ( p–V diagram) of Fig. 2.7. Initially, the piston face is at position x1, and the gas pressure is p1; at the conclusion of a quasiequilibrium expansion process the piston face is at position x2, and the pressure is reduced to p2. At each intervening piston position, the uniform pressure throughout the gas is shown as a point on the diagram. The curve, or path, connecting states 1 and 2 on the diagram represents the equilibrium states through which the system has passed during the process. The work done by the gas on the piston during the expansion is given by p dV, which can be interpreted as the area under the curve of pressure versus volume. Thus, the shaded area on Fig. 2.7 is equal to the work for the process. Had the gas been compressed from 2 to 1 along the same path on the p–V diagram, the magnitude of the work would be

quasiequilibrium process

Incremental masses removed during an expansion of the gas or liquid

Gas or liquid Boundary Figure 2.6

Illustration of a quasiequilibrium expansion or compression.

38

Chapter 2 Energy and the First Law of Thermodynamics 1

p1

Path

Pressure

δ W = p dV

2

p2

Area = 2 ∫1 p dV V1

dV

V2

Volume Gas or liquid

x

p 1

A B Area = work for process A

2

V Figure 2.8

Illustration that work depends on the process.

polytropic process

EXAMPLE

2.1

x1

x2

Figure 2.7 Work of a quasiequilibrium expansion or compression process.

the same, but the sign would be negative, indicating that for the compression the energy transfer was from the piston to the gas. The area interpretation of work in a quasiequilibrium expansion or compression process allows a simple demonstration of the idea that work depends on the process. This can be brought out by referring to Fig. 2.8. Suppose the gas in a piston–cylinder assembly goes from an initial equilibrium state 1 to a final equilibrium state 2 along two different paths, labeled A and B on Fig. 2.8. Since the area beneath each path represents the work for that process, the work depends on the details of the process as defined by the particular curve and not just on the end states. Using the test for a property given in Sec. 1.3, we can conclude again (Sec. 2.2.1) that work is not a property. The value of work depends on the nature of the process between the end states. The relationship between pressure and volume during an expansion or compression process also can be described analytically. An example is provided by the expression pV n constant, where the value of n is a constant for the particular process. A quasiequilibrium process described by such an expression is called a polytropic process. Additional analytical forms for the pressure–volume relationship also may be considered. The example to follow illustrates the application of Eq. 2.17 when the relationship between pressure and volume during an expansion is described analytically as pV n constant.

Evaluating Expansion Work

A gas in a piston–cylinder assembly undergoes an expansion process for which the relationship between pressure and volume is given by pV n constant The initial pressure is 3 bar, the initial volume is 0.1 m3, and the final volume is 0.2 m3. Determine the work for the process, in kJ, if (a) n 1.5, (b) n 1.0, and (c) n 0.

2.2 Broadening Our Understanding of Work

SOLUTION Known: A gas in a piston–cylinder assembly undergoes an expansion for which pV n constant. Find: Evaluate the work if (a) n 1.5, (b) n 1.0, (c) n 0. Schematic and Given Data: The given p–V relationship and the given data for pressure and volume can be used to construct the accompanying pressure–volume diagram of the process.

3.0

1

2c

❶

p (bar)

Gas 2.0 pV n = constant

2b 1.0

p1 = 3.0 bar V1 = 0.1 m3 V2 = 0.2 m3

2a

Area = work for part a

0.1

0.2 3)

V (m

Figure E2.1

Assumptions: 1. The gas is a closed system. 2. The moving boundary is the only work mode.

❷

3. The expansion is a polytropic process. Analysis: The required values for the work are obtained by integration of Eq. 2.17 using the given pressure–volume relation. (a) Introducing the relationship p constantV n into Eq. 2.17 and performing the integration W

V2

p dV

V1

V2

V1

constant dV Vn

1constant2 V 1n 1constant2 V 1n 2 1 1n

The constant in this expression can be evaluated at either end state: constant p1V 1n p2V 2n. The work expression then becomes W

1p1V 1n 2 V 1n 1 p2V 2n 2 V 1n 2 1 1n

p2V2 p1V1 1n

(1)

This expression is valid for all values of n except n 1.0. The case n 1.0 is taken up in part (b). To evaluate W, the pressure at state 2 is required. This can be found by using p1V1n p2V2n, which on rearrangement yields p2 p1 a

V1 n 0.1 1.5 b 13 bar2 a b 1.06 bar V2 0.2

Accordingly

❸

Wa

11.06 bar2 10.2 m3 2 13210.12

17.6 kJ

1 1.5

b`

105 N/m2 1 kJ ` ` 3 # ` 1 bar 10 N m

39

40

Chapter 2 Energy and the First Law of Thermodynamics

(b) For n 1.0, the pressure–volume relationship is pV constant or p constantV. The work is W constant

V2

V1

V2 V2 dV 1constant2 ln 1 p1V1 2 ln V V1 V1

(2)

Substituting values W 13 bar2 10.1 m3 2 `

❹

0.2 105 N/m2 1 kJ ` ` 3 # ` ln a b 20.79 kJ 1 bar 0.1 10 N m

(c) For n 0, the pressure–volume relation reduces to p constant, and the integral becomes W p(V2 V1), which is a special case of the expression found in part (a). Substituting values and converting units as above, W 30 kJ.

❶ In each case, the work for the process can be interpreted as the area under the curve representing the process on the accompanying p–V diagram. Note that the relative areas are in agreement with the numerical results.

❷ The assumption of a polytropic process is significant. If the given pressure–volume relationship were obtained as a fit to experimental pressure–volume data, the value of p d V would provide a plausible estimate of the work only when the measured pressure is essentially equal to that exerted at the piston face.

❸ Observe the use of unit conversion factors here and in part (b). ❹ It is not necessary to identify the gas (or liquid) contained within the piston–cylinder assembly. The calculated values for W are determined by the process path and the end states. However, if it is desired to evaluate other properties such as temperature, both the nature and amount of the substance must be provided because appropriate relations among the properties of the particular substance would then be required.

2.2.4 Further Examples of Work To broaden our understanding of the work concept, we now briefly consider several other examples. EXTENSION OF A SOLID BAR. Consider a system consisting of a solid bar under tension, as shown in Fig. 2.9. The bar is fixed at x 0, and a force F is applied at the other end. Let the force be represented as F A, where A is the cross-sectional area of the bar and the normal stress acting at the end of the bar. The work done as the end of the bar moves a distance dx is given by W A dx. The minus sign is required because work is done on the bar when dx is positive. The work for a change in length from x1 to x2 is found by integration

W

x2

sA dx

(2.18)

x1

Equation 2.18 for a solid is the counterpart of Eq. 2.17 for a gas undergoing an expansion or compression. STRETCHING OF A LIQUID FILM. Figure 2.10 shows a system consisting of a liquid film suspended on a wire frame. The two surfaces of the film support the thin liquid layer inside by the effect of surface tension, owing to microscopic forces between molecules near the liquid–air interfaces. These forces give rise to a macroscopically measurable force perpendicular to any line in the surface. The force per unit length across such a line is the surface tension. Denoting the surface tension acting at the movable wire by , the force F indicated on the figure can be expressed as F 2l, where the factor 2 is introduced because two film surfaces act at the wire. If the movable wire is displaced by dx, the work is given by W 2l dx. The minus sign is required because work is done on the system when dx is

2.2 Broadening Our Understanding of Work

41

Rigid wire frame Surface of film

Movable wire

Area = A

F

l F

x

dx x1

x

x2 Figure 2.9

Figure 2.10

Elongation of a solid bar.

Stretching of a liquid film.

positive. Corresponding to a displacement dx is a change in the total area of the surfaces in contact with the wire of dA 2l dx, so the expression for work can be written alternatively as W dA. The work for an increase in surface area from A1 to A2 is found by integrating this expression W

A2

t dA

(2.19)

A1

A rotating shaft is a commonly encountered machine element. Consider a shaft rotating with angular velocity and exerting a torque t on its surroundings. Let the torque be expressed in terms of a tangential force Ft and radius R: t FtR. The velocity at the point of application of the force is V R , where is in radians per unit time. Using these relations with Eq. 2.13, we obtain an expression for the power transmitted from the shaft to the surroundings # W FtV 1tR21Rv2 tv (2.20) POWER TRANSMITTED BY A SHAFT.

˙ shaft W

+ Motor

–

,ω

A related case involving a gas stirred by a paddle wheel is considered in the discussion of Fig. 2.3. ELECTRIC POWER. Shown in Fig. 2.11 is a system consisting of an electrolytic cell. The cell is connected to an external circuit through which an electric current, i, is flowing. The current is driven by the electrical potential difference e existing across the terminals labeled a and b. That this type of interaction can be classed as work is considered in the discussion of Fig. 2.3. The rate of energy transfer by work, or the power, is # (2.21) W ei

Since the current i equals dZdt, the work can be expressed in differential form as dW e dZ

(2.22)

where dZ is the amount of electrical charge that flows into the system. The minus signs are required to be in accord with our previously stated sign convention for work. When the power is evaluated in terms of the watt, and the unit of current is the ampere (an SI base unit), the unit of electric potential is the volt, defined as 1 watt per ampere. Let us next refer briefly to the types of work that can be done on systems residing in electric or magnetic fields, known as the work of polarization and magnetization, respectively. From the microscopic viewpoint, WORK DUE TO POLARIZATION OR MAGNETIZATION.

–

i

+ a

Ᏹ

b

System boundary Electrolytic cell

Figure 2.11 Electrolytic cell used to discuss electric power.

42

Chapter 2 Energy and the First Law of Thermodynamics

electrical dipoles within dielectrics resist turning, so work is done when they are aligned by an electric field. Similarly, magnetic dipoles resist turning, so work is done on certain other materials when their magnetization is changed. Polarization and magnetization give rise to macroscopically detectable changes in the total dipole moment as the particles making up the material are given new alignments. In these cases the work is associated with forces imposed on the overall system by fields in the surroundings. Forces acting on the material in the system interior are called body forces. For such forces the appropriate displacement in evaluating work is the displacement of the matter on which the body force acts. Forces acting at the boundary are called surface forces. Examples of work done by surface forces include the expansion or compression of a gas (or liquid) and the extension of a solid. 2.2.5 Further Examples of Work in Quasiequilibrium Processes Systems other than a gas or liquid in a piston–cylinder assembly can also be envisioned as undergoing processes in a quasiequilibrium fashion. To apply the quasiequilibrium process concept in any such case, it is necessary to conceive of an ideal situation in which the external forces acting on the system can be varied so slightly that the resulting imbalance is infinitesimal. As a consequence, the system undergoes a process without ever departing significantly from thermodynamic equilibrium. The extension of a solid bar and the stretching of a liquid surface can readily be envisioned to occur in a quasiequilibrium manner by direct analogy to the piston–cylinder case. For the bar in Fig. 2.9 the external force can be applied in such a way that it differs only slightly from the opposing force within. The normal stress is then essentially uniform throughout and can be determined as a function of the instantaneous length: (x). Similarly, for the liquid film shown in Fig. 2.10 the external force can be applied to the movable wire in such a way that the force differs only slightly from the opposing force within the film. During such a process, the surface tension is essentially uniform throughout the film and is functionally related to the instantaneous area: (A). In each of these cases, once the required functional relationship is known, the work can be evaluated using Eq. 2.18 or 2.19, respectively, in terms of properties of the system as a whole as it passes through equilibrium states. Other systems can also be imagined as undergoing quasiequilibrium processes. For example, it is possible to envision an electrolytic cell being charged or discharged in a quasiequilibrium manner by adjusting the potential difference across the terminals to be slightly greater, or slightly less, than an ideal potential called the cell electromotive force (emf). The energy transfer by work for passage of a differential quantity of charge to the cell, dZ, is given by the relation dW e dZ

(2.23)

In this equation e denotes the cell emf, an intensive property of the cell, and not just the potential difference across the terminals as in Eq. 2.22. Consider next a dielectric material residing in a uniform electric field. The energy transferred by work from the field when the polarization is increased slightly is dW E # d1V P2 (2.24) where the vector E is the electric field strength within the system, the vector P is the electric dipole moment per unit volume, and V is the volume of the system. A similar equation for energy transfer by work from a uniform magnetic field when the magnetization is increased slightly is dW m0H # d1V M2 (2.25) where the vector H is the magnetic field strength within the system, the vector M is the magnetic dipole moment per unit volume, and 0 is a constant, the permeability of free space.

2.3 Broadening Our Understanding of Energy

The minus signs appearing in the last three equations are in accord with our previously stated sign convention for work: W takes on a negative value when the energy transfer is into the system. GENERALIZED FORCES AND DISPLACEMENTS

The similarity between the expressions for work in the quasiequilibrium processes considered thus far should be noted. In each case, the work expression is written in the form of an intensive property and the differential of an extensive property. This is brought out by the following expression, which allows for one or more of these work modes to be involved in a process dW p dV sd 1A x2 t dA e dZ E # d1V P2 m0H # d1V M2 # # # (2.26) where the last three dots represent other products of an intensive property and the differential of a related extensive property that account for work. Because of the notion of work being a product of force and displacement, the intensive property in these relations is sometimes referred to as a “generalized” force and the extensive property as a “generalized” displacement, even though the quantities making up the work expressions may not bring to mind actual forces and displacements. Owing to the underlying quasiequilibrium restriction, Eq. 2.26 does not represent every type of work of practical interest. An example is provided by a paddle wheel that stirs a gas or liquid taken as the system. Whenever any shearing action takes place, the system necessarily passes through nonequilibrium states. To appreciate more fully the implications of the quasiequilibrium process concept requires consideration of the second law of thermodynamics, so this concept is discussed again in Chap. 5 after the second law has been introduced.

2.3 Broadening Our Understanding of Energy

The objective in this section is to use our deeper understanding of work developed in Sec. 2.2 to broaden our understanding of the energy of a system. In particular, we consider the total energy of a system, which includes kinetic energy, gravitational potential energy, and other forms of energy. The examples to follow illustrate some of these forms of energy. Many other examples could be provided that enlarge on the same idea. When work is done to compress a spring, energy is stored within the spring. When a battery is charged, the energy stored within it is increased. And when a gas (or liquid) initially at an equilibrium state in a closed, insulated vessel is stirred vigorously and allowed to come to a final equilibrium state, the energy of the gas is increased in the process. In each of these examples the change in system energy cannot be attributed to changes in the system’s overall kinetic or gravitational potential energy as given by Eqs. 2.5 and 2.10, respectively. The change in energy can be accounted for in terms of internal energy, as considered next. In engineering thermodynamics the change in the total energy of a system is considered to be made up of three macroscopic contributions. One is the change in kinetic energy, associated with the motion of the system as a whole relative to an external coordinate frame. Another is the change in gravitational potential energy, associated with the position of the system as a whole in the earth’s gravitational field. All other energy changes are lumped together in the internal energy of the system. Like kinetic energy and gravitational potential energy, internal energy is an extensive property of the system, as is the total energy. Internal energy is represented by the symbol U, and the change in internal energy in a process is U2 U1. The specific internal energy is symbolized by u or u, respectively, depending on whether it is expressed on a unit mass or per mole basis.

Paddle wheel

Gas

F

i

Battery

internal energy

43

44

Chapter 2 Energy and the First Law of Thermodynamics

The change in the total energy of a system is

or

E2 E1 1KE2 KE1 2 1PE2 PE1 2 1U2 U1 2 (2.27)

¢E ¢KE ¢PE ¢U

microscopic interpretation of internal energy for a gas

All quantities in Eq. 2.27 are expressed in terms of the energy units previously introduced. The identification of internal energy as a macroscopic form of energy is a significant step in the present development, for it sets the concept of energy in thermodynamics apart from that of mechanics. In Chap. 3 we will learn how to evaluate changes in internal energy for practically important cases involving gases, liquids, and solids by using empirical data. To further our understanding of internal energy, consider a system we will often encounter in subsequent sections of the book, a system consisting of a gas contained in a tank. Let us develop a microscopic interpretation of internal energy by thinking of the energy attributed to the motions and configurations of the individual molecules, atoms, and subatomic particles making up the matter in the system. Gas molecules move about, encountering other molecules or the walls of the container. Part of the internal energy of the gas is the translational kinetic energy of the molecules. Other contributions to the internal energy include the kinetic energy due to rotation of the molecules relative to their centers of mass and the kinetic energy associated with vibrational motions within the molecules. In addition, energy is stored in the chemical bonds between the atoms that make up the molecules. Energy storage on the atomic level includes energy associated with electron orbital states, nuclear spin, and binding forces in the nucleus. In dense gases, liquids, and solids, intermolecular forces play an important role in affecting the internal energy.

Gas

2.4 Energy Transfer by Heat

Hot plate

energy transfer by heat

Thus far, we have considered quantitatively only those interactions between a system and its surroundings that can be classed as work. However, closed systems also can interact with their surroundings in a way that cannot be categorized as work. for example. . . when a gas in a rigid container interacts with a hot plate, the energy of the gas is increased even though no work is done. This type of interaction is called an energy transfer by heat. On the basis of experiment, beginning with the work of Joule in the early part of the nineteenth century, we know that energy transfers by heat are induced only as a result of a temperature difference between the system and its surroundings and occur only in the direction of decreasing temperature. Because the underlying concept is so important in thermodynamics, this section is devoted to a further consideration of energy transfer by heat. 2.4.1 Sign Convention, Notation, and Heat Transfer Rate

The symbol Q denotes an amount of energy transferred across the boundary of a system in a heat interaction with the system’s surroundings. Heat transfer into a system is taken to be positive, and heat transfer from a system is taken as negative. sign convention for heat transfer

Q 7 0: heat transfer to the system Q 6 0: heat transfer from the system This sign convention is used throughout the book. However, as was indicated for work, it is sometimes convenient to show the direction of energy transfer by an arrow on a sketch of

2.4 Energy Transfer by Heat

the system. Then the heat transfer is regarded as positive in the direction of the arrow. In an adiabatic process there is no energy transfer by heat. The sign convention for heat transfer is just the reverse of the one adopted for work, where a positive value for W signifies an energy transfer from the system to the surroundings. These signs for heat and work are a legacy from engineers and scientists who were concerned mainly with steam engines and other devices that develop a work output from an energy input by heat transfer. For such applications, it was convenient to regard both the work developed and the energy input by heat transfer as positive quantities. The value of a heat transfer depends on the details of a process and not just the end states. Thus, like work, heat is not a property, and its differential is written as Q. The amount of energy transfer by heat for a process is given by the integral Q

45

heat is not a property

2

dQ

(2.28)

1

where the limits mean “from state 1 to state 2” and do not refer to the values of heat at those states. As for work, the notion of “heat” at a state has no meaning, and the integral should never be evaluated as Q2 Q1. # The net rate of heat transfer is denoted by Q. In principle, the amount of energy transfer by heat during a period of time can be found by integrating from time t1 to time t2 Q

t2

# Q dt

rate of heat transfer

(2.29)

t1

To perform the integration, it would be necessary to know how the rate of heat transfer varies with time. # In some cases it is convenient to use the heat flux, q, which is the heat transfer rate per # # unit of system surface area. The net rate of heat transfer, Q, is related to the heat flux q by the integral # Q

q# dA

(2.30)

A

where A represents the area on the boundary of the system where heat transfer occurs. # # The units for Q and Q are the same as those introduced previously for W and W, respectively. The units for the heat flux are those of the heat transfer rate per unit area: kW/m2 or Btu /h # ft2. UNITS.

2.4.2 Heat Transfer Modes Methods based on experiment are available for evaluating energy transfer by heat. These methods recognize two basic transfer mechanisms: conduction and thermal radiation. In addition, empirical relationships are available for evaluating energy transfer involving certain combined modes. A brief description of each of these is given next. A detailed consideration is left to a course in engineering heat transfer, where these topics are studied in depth.

. Qx

T1

L

CONDUCTION

Energy transfer by conduction can take place in solids, liquids, and gases. Conduction can be thought of as the transfer of energy from the more energetic particles of a substance to adjacent particles that are less energetic due to interactions between particles. The time rate of energy transfer by conduction is quantified macroscopically by Fourier’s law. As an elementary application, consider Fig. 2.12 showing a plane wall of thickness L at steady state,

T2 Area, A x

Figure 2.12 Illustration of Fourier’s conduction law.

46

Chapter 2 Energy and the First Law of Thermodynamics

where the temperature T(x) varies linearly with position # x. By Fourier’s law, the rate of heat transfer across any plane normal to the x direction, Qx, is proportional to the wall area, A, and the temperature gradient in the x direction, dTdx # dT Qx kA dx

Fourier’s law

(2.31)

where the proportionality constant is a property called the thermal conductivity. The minus sign is a consequence of energy transfer in the direction of decreasing temperature. for example. . . in this case the temperature varies linearly; thus, the temperature gradient is T 2 T1 dT 16 02 dx L and the rate of heat transfer in the x direction is then # T2 T1 Qx kA c d L

(2.32)

Values of thermal conductivity are given in Table A-19 for common materials. Substances with large values of thermal conductivity such as copper are good conductors, and those with small conductivities (cork and polystyrene foam) are good insulators. RADIATION

Stefan–Boltzmann law

Thermal radiation is emitted by matter as a result of changes in the electronic configurations of the atoms or molecules within it. The energy is transported by electromagnetic waves (or photons). Unlike conduction, thermal radiation requires no intervening medium to propagate and can even take place in a vacuum. Solid surfaces, gases, and liquids all emit, absorb,# and transmit thermal radiation to varying degrees. The rate at which energy is emitted, Qe, from a surface of area A is quantified macroscopically by a modified form of the Stefan–Boltzmann law # Qe esAT b4 (2.33) which shows that thermal radiation is associated with the fourth power of the absolute temperature of the surface, Tb. The emissivity, , is a property of the surface that indicates how effectively the surface radiates (0 1.0), and is the Stefan–Boltzmann constant. In general, the net rate of energy transfer by thermal radiation between two surfaces involves relationships among the properties of the surfaces, their orientations with respect to each other, the extent to which the intervening medium scatters, emits, and absorbs thermal radiation, and other factors. CONVECTION

Energy transfer between a solid surface at a temperature Tb and an adjacent moving gas or liquid at another temperature Tf plays a prominent role in the performance of many devices of practical interest. This is commonly referred to as convection. As an illustration, consider Fig. 2.13, where Tb Tf. In this case, energy is transferred in the direction indicated by the arrow due to the combined effects of conduction within the air and the bulk motion of the air. The rate of energy transfer from the surface to the air can be quantified by the following empirical expression: # Qc hA1Tb Tf 2 (2.34)

2.4 Energy Transfer by Heat Cooling air flow Tf < Tb

47

. Qc

Tb A Wire leads Transistor

Figure 2.13 Circuit board

Illustration of Newton’s law of cooling.

known as Newton’s law of cooling. In Eq. 2.34, A is the surface area and the proportionality factor h is called the heat transfer coefficient. In subsequent applications of Eq. 2.34, a minus sign may be introduced on the right side to conform to the sign convention for heat transfer introduced in Sec. 2.4.1. The heat transfer coefficient is not a thermodynamic property. It is an empirical parameter that incorporates into the heat transfer relationship the nature of the flow pattern near the surface, the fluid properties, and the geometry. When fans or pumps cause the fluid to move, the value of the heat transfer coefficient is generally greater than when relatively slow buoyancy-induced motions occur. These two general categories are called forced and free (or natural) convection, respectively. Table 2.1 provides typical values of the convection heat transfer coefficient for forced and free convection. 2.4.3 Closure The first step in a thermodynamic analysis is to define the system. It is only after the system boundary has been specified that possible heat interactions with the surroundings are considered, for these are always evaluated at the system boundary. In ordinary conversation, the term heat is often used when the word energy would be more correct thermodynamically. For example, one might hear, “Please close the door or ‘heat’ will be lost.” In thermodynamics, heat refers only to a particular means whereby energy is transferred. It does not refer to what is being transferred between systems or to what is stored within systems. Energy is transferred and stored, not heat. Sometimes the heat transfer of energy to, or from, a system can be neglected. This might occur for several reasons related to the mechanisms for heat transfer discussed above. One might be that the materials surrounding the system are good insulators, or heat transfer might not be significant because there is a small temperature difference between the system and its surroundings. A third reason is that there might not be enough surface area to allow significant heat transfer to occur. When heat transfer is neglected, it is because one or more of these considerations apply. Typical Values of the Convection Heat Transfer Coefficient TABLE 2.1

Applications Free convection Gases Liquids Forced convection Gases Liquids

h (W/m2 # K) 2–25 50–1000 25–250 50–20,000

Newton’s law of cooling

48

Chapter 2 Energy and the First Law of Thermodynamics

In the discussions to follow the value of Q is provided, or it is an unknown in the analysis. When Q is provided, it can be assumed that the value has been determined by the methods introduced above. When Q is the unknown, its value is usually found by using the energy balance, discussed next.

2.5 Energy Accounting: Energy Balance for Closed Systems

As our previous discussions indicate, the only ways the energy of a closed system can be changed are through transfer of energy by work or by heat. Further, based on the experiments of Joule and others, a fundamental aspect of the energy concept is that energy is conserved; we call this the first law of thermodynamics. These considerations are summarized in words as follows:

first law of thermodynamics

change in the amount net amount of energy net amount of energy of energy contained transferred in across transferred out across D within the system T Dthe system boundary byT D the system boundary T during some time heat transfer during by work during the interval the time interval time interval This word statement is just an accounting balance for energy, an energy balance. It requires that in any process of a closed system the energy of the system increases or decreases by an amount equal to the net amount of energy transferred across its boundary. The phrase net amount used in the word statement of the energy balance must be carefully interpreted, for there may be heat or work transfers of energy at many different places on the boundary of a system. At some locations the energy transfers may be into the system, whereas at others they are out of the system. The two terms on the right side account for the net results of all the energy transfers by heat and work, respectively, taking place during the time interval under consideration. The energy balance can be expressed in symbols as

E2 E1 Q W

(2.35a)

Introducing Eq. 2.27 an alternative form is

energy balance

¢KE ¢PE ¢U Q W

(2.35b)

which shows that an energy transfer across the system boundary results in a change in one or more of the macroscopic energy forms: kinetic energy, gravitational potential energy, and internal energy. All previous references to energy as a conserved quantity are included as special cases of Eqs. 2.35. Note that the algebraic signs before the heat and work terms of Eqs. 2.35 are different. This follows from the sign conventions previously adopted. A minus sign appears before W because energy transfer by work from the system to the surroundings is taken to be positive. A plus sign appears before Q because it is regarded to be positive when the heat transfer of energy is into the system from the surroundings. OTHER FORMS OF THE ENERGY BALANCE

Various special forms of the energy balance can be written. For example, the energy balance in differential form is dE dQ dW

(2.36)

2.5 Energy Accounting: Energy Balance for Closed Systems

where dE is the differential of energy, a property. Since Q and W are not properties, their differentials are written as Q and W, respectively. The instantaneous time rate form of the energy balance is # # dE QW dt

(2.37)

The rate form of the energy balance expressed in words is time rate of change net rate at which net rate at which of the energy energy is being energy is being D contained within T D transferred in T D transferred out T the system at by heat transfer by work at time t at time t time t Since the time rate of change of energy is given by dE d KE d PE dU

dt dt dt dt Equation 2.37 can be expressed alternatively as # # d KE d PE dU

QW dt dt dt

(2.38)

Equations 2.35 through 2.38 provide alternative forms of the energy balance that may be convenient starting points when applying the principle of conservation of energy to closed systems. In Chap. 4 the conservation of energy principle is expressed in forms suitable for the analysis of control volumes. When applying the energy balance in any of its forms, it is important to be careful about signs and units and to distinguish carefully between rates and amounts. In addition, it is important to recognize that the location of the system boundary can be relevant in determining whether a particular energy transfer is regarded as heat or work. for example. . . consider Fig. 2.14, in which three alternative systems are shown that include a quantity of a gas (or liquid) in a rigid, well-insulated container. In Fig. 2.14a, the gas itself is the system. As current flows through the copper plate, there is an energy transfer from the copper plate to the gas. Since this energy transfer occurs as a result of the temperature difference between the plate and the gas, it is classified as a heat transfer. Next, refer to Fig. 2.14b, where the boundary is drawn to include the copper plate. It follows from the thermodynamic definition of work that the energy transfer that occurs as current crosses the boundary of this system must be regarded as work. Finally, in Fig. 2.14c, the boundary is located so that no energy is transferred across it by heat or work. CLOSING COMMENT. Thus far, we have been careful to emphasize that the quantities symbolized by W and Q in the foregoing equations account for transfers of energy and not transfers of work and heat, respectively. The terms work and heat denote different means whereby energy is transferred and not what is transferred. However, to achieve economy of expression in subsequent discussions, W and Q are often referred to simply as work and heat transfer, respectively. This less formal manner of speaking is commonly used in engineering practice.

ILLUSTRATIONS

The examples to follow bring out many important ideas about energy and the energy balance. They should be studied carefully, and similar approaches should be used when solving the end-of-chapter problems.

time rate form of the

energy balance

49

50

Chapter 2 Energy and the First Law of Thermodynamics Copper plate Gas or liquid

Rotating shaft

+

Gas W or liquid

Q –

+ –

W=0 Electric generator System boundary

System boundary

Insulation

Q=0

Mass (a)

(b)

+

Gas or liquid

System boundary

–

Q = 0, W = 0 (c)

Figure 2.14

Alternative choices for system boundaries.

In this text, most applications of the energy balance will not involve significant kinetic or potential energy changes. Thus, to expedite the solutions of many subsequent examples and end-of-chapter problems, we indicate in the problem statement that such changes can be neglected. If this is not made explicit in a problem statement, you should decide on the basis of the problem at hand how best to handle the kinetic and potential energy terms of the energy balance. The next two examples illustrate the use of the energy balance for processes of closed systems. In these examples, internal energy data are provided. In Chap. 3, we learn how to obtain thermodynamic property data using tables, graphs, equations, and computer software. PROCESSES OF CLOSED SYSTEMS.

EXAMPLE

2.2

Cooling a Gas in a Piston–Cylinder

Four kilograms of a certain gas is contained within a piston–cylinder assembly. The gas undergoes a process for which the pressure–volume relationship is pV1.5 constant The initial pressure is 3 bar, the initial volume is 0.1 m3, and the final volume is 0.2 m3. The change in specific internal energy of the gas in the process is u2 u1 4.6 kJ/kg. There are no significant changes in kinetic or potential energy. Determine the net heat transfer for the process, in kJ.

2.5 Energy Accounting: Energy Balance for Closed Systems

SOLUTION Known: A gas within a piston–cylinder assembly undergoes an expansion process for which the pressure–volume relation and the change in specific internal energy are specified. Find: Determine the net heat transfer for the process. Schematic and Given Data:

p

1 u2 – u1 = – 4.6 kJ/kg pV 1.5 = constant

Gas

❶

pV 1.5 = constant

2 Area = work

V

Figure E2.2

Assumptions: 1. The gas is a closed system. 2. The process is described by pV1.5 constant. 3. There is no change in the kinetic or potential energy of the system. Analysis: An energy balance for the closed system takes the form 0

¢KE ¢PE ¢U Q W where the kinetic and potential energy terms drop out by assumption 3. Then, writing U in terms of specific internal energies, the energy balance becomes m1u2 u1 2 Q W where m is the system mass. Solving for Q Q m1u2 u1 2 W The value of the work for this process is determined in the solution to part (a) of Example 2.1: W 17.6 kJ. The change in internal energy is obtained using given data as m1u2 u1 2 4 kg a4.6

kJ b 18.4 kJ kg

Substituting values

❷

Q 18.4 17.6 0.8 kJ

❶

The given relationship between pressure and volume allows the process to be represented by the path shown on the accompanying diagram. The area under the curve represents the work. Since they are not properties, the values of the work and heat transfer depend on the details of the process and cannot be determined from the end states only.

❷

The minus sign for the value of Q means that a net amount of energy has been transferred from the system to its surroundings by heat transfer.

51

52

Chapter 2 Energy and the First Law of Thermodynamics

In the next example, we follow up the discussion of Fig. 2.14 by considering two alternative systems. This example highlights the need to account correctly for the heat and work interactions occurring on the boundary as well as the energy change. EXAMPLE 2.3

Considering Alternative Systems

Air is contained in a vertical piston–cylinder assembly fitted with an electrical resistor. The atmosphere exerts a pressure of 1 bar on the top of the piston, which has a mass of 45 kg and a face area of .09 m2. Electric current passes through the resistor, and the volume of the air slowly increases by .045 m3 while its pressure remains constant. The mass of the air is .27 kg, and its specific internal energy increases by 42 kJ/kg. The air and piston are at rest initially and finally. The piston–cylinder material is a ceramic composite and thus a good insulator. Friction between the piston and cylinder wall can be ignored, and the local acceleration of gravity is g 9.81 m/s2. Determine the heat transfer from the resistor to the air, in kJ, for a system consisting of (a) the air alone, (b) the air and the piston. SOLUTION Known: Data are provided for air contained in a vertical piston–cylinder fitted with an electrical resistor. Find: Considering each of two alternative systems, determine the heat transfer from the resistor to the air. Schematic and Given Data: Piston

Piston

patm = 1 bar

System boundary for part (a)

System boundary for part (b)

m piston = 45 kg A piston = .09 m2 +

+

Air

Air –

– m air = .27 kg V2 – V1 = .045 m3

(a)

∆u air = 42 kJ/kg.

(b) Figure E2.3

Assumptions: 1. Two closed systems are under consideration, as shown in the schematic. 2. The only significant heat transfer is from the resistor to the air, during which the air expands slowly and its pressure remains constant.

❶

3. There is no net change in kinetic energy; the change in potential energy of the air is negligible; and since the piston material is a good insulator, the internal energy of the piston is not affected by the heat transfer. 4. Friction between the piston and cylinder wall is negligible. 5. The acceleration of gravity is constant; g 9.81 m/s2. Analysis: (a) Taking the air as the system, the energy balance, Eq. 2.35, reduces with assumption 3 to 1¢KE ¢PE ¢U2 air Q W 0

Or, solving for Q Q W ¢Uair

2.5 Energy Accounting: Energy Balance for Closed Systems

For this system, work is done by the force of the pressure p acting on the bottom of the piston as the air expands. With Eq. 2.17 and the assumption of constant pressure W

V2

V1

p dV p 1V2 V1 2

To determine the pressure p, we use a force balance on the slowly moving, frictionless piston. The upward force exerted by the air on the bottom of the piston equals the weight of the piston plus the downward force of the atmosphere acting on the top of the piston. In symbols p Apiston mpiston g patmApiston Solving for p and inserting values p p

mpiston g Apiston

patm

145 kg2 19.81 m/s2 2 .09 m2

`

1 bar ` 1 bar 1.049 bar 105 N /m2

Thus, the work is W p 1V2 V1 2

11.049 bar2 1.045 m2 2 `

105 N/m2 1 kJ ` ` 3 # ` 4.72 kJ 1 bar 10 N m

With Uair mair(uair), the heat transfer is Q W mair 1¢uair 2

4.72 kJ 11.07 kJ 15.8 kJ

(b) Consider next a system consisting of the air and the piston. The energy change of the overall system is the sum of the energy changes of the air and the piston. Thus, the energy balance, Eq. 2.35, reads 1 ¢KE ¢PE ¢U2 air 1¢KE ¢PE ¢U 2 piston Q W 0

where the indicated terms drop out by assumption 3. Solving for Q Q W 1 ¢PE2 piston 1¢U2 air For this system, work is done at the top of the piston as it pushes aside the surrounding atmosphere. Applying Eq. 2.17 W

V2

V1

p dV patm 1V2 V1 2

11 bar2 1.045 m2 2 `

105 N/m2 1 kJ ` ` 3 # ` 4.5 kJ 1 bar 10 N m

The elevation change, z, required to evaluate the potential energy change of the piston can be found from the volume change of the air and the area of the piston face as ¢z

V2 V 1 .045 m3 .5 m Apiston .09 m2

Thus, the potential energy change of the piston is 1 ¢PE2 piston mpiston g¢z

145 kg2 19.81 m/s2 2 10.5 m2 .22 kJ

53

54

Chapter 2 Energy and the First Law of Thermodynamics

Finally Q W 1 ¢PE2 piston mair ¢uair

4.5 kJ .22 kJ 11.07 kJ 15.8 kJ

❷

which agrees with the result of part (a).

❶

Using the change in elevation z determined in the analysis, the change in potential energy of the air is about 103 Btu, which is negligible in the present case. The calculation is left as an exercise.

❷

Although the value of Q is the same for each system, observe that the values for W differ. Also, observe that the energy changes differ, depending on whether the air alone or the air and the piston is the system.

A system is at steady state if none of its properties change with time (Sec. 1.3). Many devices operate at steady state or nearly at steady state, meaning that property variations with time are small enough to ignore. The two examples to follow illustrate the application of the energy rate equation to closed systems at steady state.

STEADY-STATE OPERATION.

EXAMPLE

2.4

Gearbox at Steady State

During steady-state operation, a gearbox receives 60 kW through the input shaft and delivers power through the output shaft. For the gearbox as the system, the rate of energy transfer by convection is # Q hA1Tb Tf 2 where h 0.171 kW/m2 K is the heat transfer coefficient, A 1.0 m2 is the outer surface area of the gearbox, Tb 300 K (27C) is the temperature at the outer surface, and Tf 293 K (20C) is the temperature of the surrounding air away from the immediate vicinity of the gearbox. For the gearbox, evaluate the heat transfer rate and the power delivered through the output shaft, each in kW. SOLUTION Known: A gearbox operates at steady state with a known power input. An expression for the heat transfer rate from the outer surface is also known. Find: Determine the heat transfer rate and the power delivered through the output shaft, each in kW. Schematic and Given Data:

Tb = 300 K W˙ 1 = –60 kW Tf = 293 K

1

h = 0.171 kW/m2 · K

Input shaft

2

Assumption: The gearbox is a closed system at steady state.

Gearbox Outer surface A = 1.0

m2

Output shaft Figure E2.4

2.5 Energy Accounting: Energy Balance for Closed Systems

❶

# Analysis: Using the given expression for Q together with known data, the rate of energy transfer by heat is # Q hA1Tb Tf 2 a0.171

kW b 11.0 m2 21300 2932 K m2 # K

1.2 kW # The minus sign for Q signals that energy is carried out of the gearbox by heat transfer. The energy rate balance, Eq. 2.37, reduces at steady state to 0

# # dE QW dt

❷

or

# # WQ

# # # The symbol W represents the net power from the system. The net power is the sum of W1 and the output power W2 # # # W W1 W2 # With this expression for W, the energy rate balance becomes # # # W1 W2 Q # # # Solving for W2, inserting Q 1.2 kW, and W1 60 kW, where the minus sign is required because the input shaft brings energy into the system, we have # # # W2 Q W1

❸ ❹

11.2 kW2 160 kW2

58.8 kW # The positive sign for W2 indicates that energy is transferred from the system through the output shaft, as expected.

❶

In accord with #the sign convention for the heat transfer rate in the energy rate balance (Eq. 2.37), Eq. 2.34 is written with a minus sign: Q is negative when Tb is greater than Tf.

❷

Properties of a system at steady state do not change with time. Energy E is a property, but heat transfer and work are not properties.

❸ ❹

For this system energy transfer by work occurs at two different locations, and the signs associated with their values differ. At steady state, the rate of heat transfer from the gear box accounts for the difference between the input and output power. This can be summarized by the following energy rate “balance sheet” in terms of magnitudes: Input 60 kW (input shaft) Total: 60 kW

EXAMPLE

2.5

Output 58.8 kW (output shaft) 1.2 kW (heat transfer) 60 kW

Silicon Chip at Steady State

A silicon chip measuring 5 mm on a side and 1 mm in thickness is embedded in a ceramic substrate. At steady state, the chip has an electrical power input of 0.225 W. The top surface of the chip is exposed to a coolant whose temperature is 20C. The heat transfer coefficient for convection between the chip and the coolant is 150 W/m2 # K. If heat transfer by conduction between the chip and the substrate is negligible, determine the surface temperature of the chip, in C.

55

56

Chapter 2 Energy and the First Law of Thermodynamics

SOLUTION Known: A silicon chip of known dimensions is exposed on its top surface to a coolant. The electrical power input and convective heat transfer coefficient are known. Find: Determine the surface temperature of the chip at steady state. Schematic and Given Data: Coolant h = 150 W/m2 · K Tf = 20° C 5 mm

Assumptions:

5 mm

1. The chip is a closed system at steady state.

Tb

+ W˙ = –0.225 W

–

2. There is no heat transfer between the chip and the substrate.

1 mm

Ceramic substrate

Figure E2.5

Analysis: The surface temperature of the chip, Tb, can be determined using the energy rate balance, Eq. 2.37, which at steady state reduces as follows 0

❶

# # dE QW dt

❷

With assumption 2, the only heat transfer is by convection to the coolant. In this application, Newton’s law of cooling, Eq. 2.34, takes the form # Q hA1Tb Tf 2 Collecting results # 0 hA1Tb Tf 2 W Solving for Tb # W Tb

Tf hA # In this expression, W 0.225 W, A 25 106 m2, h 150 W/m2 # K, and Tf 293 K, giving Tb

10.225 W2

1150 W/m2 # K2125 106 m2 2

293 K

353 K 180°C2

❶ Properties of a system at steady state do not change with time. Energy E is a property, but heat transfer and work are not properties.

❷ In accord # with the sign convention for heat transfer in the energy rate balance (Eq. 2.37), Eq. 2.34 is written with a minus sign: Q is negative when Tb is greater than Tf.

Many devices undergo periods of transient operation where the state changes with time. This is observed during startup and shutdown periods. The next example illustrates the application of the energy rate balance to an electric motor during startup. The example also involves both electrical work and power transmitted by a shaft. TRANSIENT OPERATION.

2.5 Energy Accounting: Energy Balance for Closed Systems

EXAMPLE

2.6

Transient Operation of a Motor

The rate of heat transfer between a certain electric motor and its surroundings varies with time as # Q 0.23 1 e10.05t2 4 # where t is in seconds and Q is in kW. The shaft of the motor rotates at a constant speed of 100 rad/s (about 955 revo# lutions per minute, or RPM) and applies a constant torque# of t # 18 N m to an external load. The motor draws a constant electric power input equal to 2.0 kW. For the motor, plot Q and W, each in kW, and the change in energy E, in kJ, as functions of time from t 0 to t 120 s. Discuss. SOLUTION Known: A motor operates with constant electric power input, shaft speed, and applied torque. The time-varying rate of heat transfer between the motor and its surroundings is given. # # Find: Plot Q, W, and E versus time, Discuss. Schematic and Given Data:

= 18 N · m ω = 100 rad/s

W˙ elec = –2.0 kW +

W˙ shaft

Assumption: The system shown in the accompanying sketch is a closed system.

Motor –

˙ = – 0.2 [1 – e (–0.05t)] kW Q Figure E2.6a

Analysis: The time rate of change of system energy is # # dE QW dt # # W represents the net power from the # system: the sum of the power associated with the rotating shaft, Wshaft, and the power associated with the electricity flow, Welec # # # W Wshaft Welec # # The rate Welec is known from the problem statement: W# elec 2.0 kW, where the negative sign is required because energy is carried into the system by electrical work. The term Wshaft can be evaluated with Eq. 2.20 as # Wshaft tv 118 N # m2 1100 rad /s2 1800 W 1.8 kW Because energy exits the system along the rotating shaft, this energy transfer rate is positive. In summary # # # W Welec Wshaft 12.0 kW2 1 1.8 kW2 0.2 kW where the minus sign means that the than the power transferred out along the shaft. # electrical power input is greater # With the foregoing result for W and the given expression for Q, the energy rate balance becomes dE 0.231 e10.05t2 4 10.22 0.2e10.05t2 dt Integrating t

¢E

0.2e

10.05t2

dt

t 0.2 e10.05t2 d 4 31 e10.05t2 4 10.052 0

57

❷

# # The accompanying plots, Figs. E2.6b, c, are developed using the given expression for Q and the for W and # expressions # E obtained in the analysis. Because of our sign conventions for heat and work, the values of Q and W are negative. In the first few seconds, the net rate energy is carried in by work greatly exceeds the rate energy is carried out by heat transfer. # Consequently, # the energy stored in the motor increases rapidly as the motor “warms up.” As time elapses, the value of Q approaches W, and the rate of energy storage diminishes. After about 100 s, this transient operating mode is nearly over, and there is little further change in the amount of energy stored, or in any other property. We may say that the motor is then at steady state. 5 –0.05

4

˙ , kW Q˙ , W

❶

Chapter 2 Energy and the First Law of Thermodynamics

∆E, kJ

58

3 2

· Q

–0.15

˙ W

–0.20

1 0

–0.10

10 20 30 40 50 60 70 80 90 100

–0.25

Time, s

10 20 30 40 50 60 70 80 90 100 Time, s

Figure E2.6b, c

❶ ❷

Figures E.2.6b, c can be developed using appropriate software or can be drawn by hand. # At steady state, the value of Q is constant at 0.2 kW. This constant value for the heat transfer rate can be thought of as the portion of the electrical power input that is not obtained as a mechanical power output because of effects within the motor such as electrical resistance and friction.

2.6 Energy Analysis of Cycles

In this section the energy concepts developed thus far are illustrated further by application to systems undergoing thermodynamic cycles. Recall from Sec. 1.3 that when a system at a given initial state goes through a sequence of processes and finally returns to that state, the system has executed a thermodynamic cycle. The study of systems undergoing cycles has played an important role in the development of the subject of engineering thermodynamics. Both the first and second laws of thermodynamics have roots in the study of cycles. In addition, there are many important practical applications involving power generation, vehicle propulsion, and refrigeration for which an understanding of thermodynamic cycles is necessary. In this section, cycles are considered from the perspective of the conservation of energy principle. Cycles are studied in greater detail in subsequent chapters, using both the conservation of energy principle and the second law of thermodynamics. 2.6.1 Cycle Energy Balance The energy balance for any system undergoing a thermodynamic cycle takes the form ¢Ecycle Qcycle Wcycle

(2.39)

where Qcycle and Wcycle represent net amounts of energy transfer by heat and work, respectively, for the cycle. Since the system is returned to its initial state after the cycle,

2.6 Energy Analysis of Cycles

Hot body

59

Hot body Q out

Q in System

System Wcycle = Q in – Q out

Q out Cold body

Q in

Wcycle = Q out – Q in

Cold body

(a)

(b)

Figure 2.15 Schematic diagrams of two important classes of cycles. (a) Power cycles. (b) Refrigeration and heat pump cycles.

there is no net change in its energy. Therefore, the left side of Eq. 2.39 equals zero, and the equation reduces to Wcycle Qcycle

(2.40)

Equation 2.40 is an expression of the conservation of energy principle that must be satisfied by every thermodynamic cycle, regardless of the sequence of processes followed by the system undergoing the cycle or the nature of the substances making up the system. Figure 2.15 provides simplified schematics of two general classes of cycles considered in this book: power cycles and refrigeration and heat pump cycles. In each case pictured, a system undergoes a cycle while communicating thermally with two bodies, one hot and the other cold. These bodies are systems located in the surroundings of the system undergoing the cycle. During each cycle there is also a net amount of energy exchanged with the surroundings by work. Carefully observe that in using the symbols Qin and Qout on Fig. 2.15 we have departed from the previously stated sign convention for heat transfer. In this section it is advantageous to regard Qin and Qout as transfers of energy in the directions indicated by the arrows. The direction of the net work of the cycle, Wcycle, is also indicated by an arrow. Finally, note that the directions of the energy transfers shown in Fig. 2.15b are opposite to those of Fig. 2.15a.

METHODOLOGY UPDATE

When analyzing cycles, we normally take energy transfers as positive in the directions of arrows on a sketch of the system and write the energy balance accordingly.

2.6.2 Power Cycles Systems undergoing cycles of the type shown in Fig. 2.15a deliver a net work transfer of energy to their surroundings during each cycle. Any such cycle is called a power cycle. From Eq. 2.40, the net work output equals the net heat transfer to the cycle, or Wcycle Qin Qout

1power cycle2

(2.41)

where Qin represents the heat transfer of energy into the system from the hot body, and Qout represents heat transfer out of the system to the cold body. From Eq. 2.41 it is clear that Qin

power cycle

60

Chapter 2 Energy and the First Law of Thermodynamics

must be greater than Qout for a power cycle. The energy supplied by heat transfer to a system undergoing a power cycle is normally derived from the combustion of fuel or a moderated nuclear reaction; it can also be obtained from solar radiation. The energy Qout is generally discharged to the surrounding atmosphere or a nearby body of water. The performance of a system undergoing a power cycle can be described in terms of the extent to which the energy added by heat, Qin, is converted to a net work output, Wcycle. The extent of the energy conversion from heat to work is expressed by the following ratio, commonly called the thermal efficiency h

thermal efficiency

Wcycle Qin

1power cycle2

(2.42)

Introducing Eq. 2.41, an alternative form is obtained as h

Qout Qin Qout 1 Qin Qin

1power cycle2

(2.43)

Since energy is conserved, it follows that the thermal efficiency can never be greater than unity (100%). However, experience with actual power cycles shows that the value of thermal efficiency is invariably less than unity. That is, not all the energy added to the system by heat transfer is converted to work; a portion is discharged to the cold body by heat transfer. Using the second law of thermodynamics, we will show in Chap. 5 that the conversion from heat to work cannot be fully accomplished by any power cycle. The thermal efficiency of every power cycle must be less than unity: 1. 2.6.3 Refrigeration and Heat Pump Cycles refrigeration and heat pump cycles

Next, consider the refrigeration and heat pump cycles shown in Fig. 2.15b. For cycles of this type, Qin is the energy transferred by heat into the system undergoing the cycle from the cold body, and Qout is the energy discharged by heat transfer from the system to the hot body. To accomplish these energy transfers requires a net work input, Wcycle. The quantities Qin, Qout, and Wcycle are related by the energy balance, which for refrigeration and heat pump cycles takes the form Wcycle Qout Qin

1refrigeration and heat pump cycles2

(2.44)

Since Wcycle is positive in this equation, it follows that Qout is greater than Qin. Although we have treated them as the same to this point, refrigeration and heat pump cycles actually have different objectives. The objective of a refrigeration cycle is to cool a refrigerated space or to maintain the temperature within a dwelling or other building below that of the surroundings. The objective of a heat pump is to maintain the temperature within a dwelling or other building above that of the surroundings or to provide heating for certain industrial processes that occur at elevated temperatures. Since refrigeration and heat pump cycles have different objectives, their performance parameters, called coefficients of performance, are defined differently. These coefficients of performance are considered next. REFRIGERATION CYCLES

The performance of refrigeration cycles can be described as the ratio of the amount of energy received by the system undergoing the cycle from the cold body, Qin, to the net

2.6 Energy Analysis of Cycles

61

work into the system to accomplish this effect, Wcycle. Thus, the coefficient of performance, , is

b

Qin Wcycle

1refrigeration cycle2

(2.45)

coefficient of performance: refrigeration

Introducing Eq. 2.44, an alternative expression for is obtained as b

Qin Qout Qin

1refrigeration cycle2

(2.46)

For a household refrigerator, Qout is discharged to the space in which the refrigerator is located. Wcycle is usually provided in the form of electricity to run the motor that drives the refrigerator. for example. . . in a refrigerator the inside compartment acts as the cold body and the ambient air surrounding the refrigerator is the hot body. Energy Qin passes to the circulating refrigerant from the food and other contents of the inside compartment. For this heat transfer to occur, the refrigerant temperature is necessarily below that of the refrigerator contents. Energy Qout passes from the refrigerant to the surrounding air. For this heat transfer to occur, the temperature of the circulating refrigerant must necessarily be above that of the surrounding air. To achieve these effects, a work input is required. For a refrigerator, Wcycle is provided in the form of electricity.

HEAT PUMP CYCLES

The performance of heat pumps can be described as the ratio of the amount of energy discharged from the system undergoing the cycle to the hot body, Qout, to the net work into the system to accomplish this effect, Wcycle. Thus, the coefficient of performance, , is

g

Qout Wcycle

1heat pump cycle2

(2.47)

Introducing Eq. 2.44, an alternative expression for this coefficient of performance is obtained as g

Qout Qout Qin

1heat pump cycle2

(2.48)

From this equation it can be seen that the value of is never less than unity. For residential heat pumps, the energy quantity Qin is normally drawn from the surrounding atmosphere, the ground, or a nearby body of water. Wcycle is usually provided by electricity. The coefficients of performance and are defined as ratios of the desired heat transfer effect to the cost in terms of work to accomplish that effect. Based on the definitions, it is desirable thermodynamically that these coefficients have values that are as large as possible. However, as discussed in Chap. 5, coefficients of performance must satisfy restrictions imposed by the second law of thermodynamics.

coefficient of performance: heat pump

62

Chapter 2 Energy and the First Law of Thermodynamics

Chapter Summary and Study Guide

In this chapter, we have considered the concept of energy from an engineering perspective and have introduced energy balances for applying the conservation of energy principle to closed systems. A basic idea is that energy can be stored within systems in three macroscopic forms: internal energy, kinetic energy, and gravitational potential energy. Energy also can be transferred to and from systems. Energy can be transferred to and from closed systems by two means only: work and heat transfer. Work and heat transfer are identified at the system boundary and are not properties. In mechanics, work is energy transfer associated with macroscopic forces and displacements at the system boundary. The thermodynamic definition of work introduced in this chapter extends the notion of work from mechanics to include other types of work. Energy transfer by heat is due to a temperature difference between the system and its surroundings, and occurs in the direction of decreasing temperature. Heat transfer modes include conduction, radiation, and convection. These sign conventions are used for work and heat transfer: # 7 0: work done by the system W, W e 6 0: work done by the system Q, Q e

#

7 0: heat transfer to the system 6 0: heat transfer from the system

Energy is an extensive property of a system. Only changes in the energy of a system have significance. Energy changes are accounted for by the energy balance. The energy balance

for a process of a closed system is Eq. 2.35 and an accompanying time rate form is Eq. 2.37. Equation 2.40 is a special form of the energy balance for a system undergoing a thermodynamic cycle. The following checklist provides a study guide for this chapter. When your study of the text and end-of-chapter exercises has been completed, you should be able to write out the meanings of the terms listed in the margins

throughout the chapter and understand each of the related concepts. The subset of key concepts listed below is particularly important in subsequent chapters. evaluate these energy quantities

–kinetic and potential energy changes using Eqs. 2.5 and 2.10, respectively. –work and power using Eqs. 2.12 and 2.13, respectively. –expansion or compression work using Eq. 2.17 apply closed system energy balances in each of several

alternative forms, appropriately modeling the case at hand, correctly observing sign conventions for work and heat transfer, and carefully applying SI and English units. conduct energy analyses for systems undergoing

thermodynamic cycles using Eq. 2.40, and evaluating, as appropriate, the thermal efficiencies of power cycles and coefficients of performance of refrigeration and heat pump cycles.

Key Engineering Concepts

kinetic energy p. 30 potential energy p. 32 work p. 33 power p. 35

internal energy p. 43 heat transfer p. 44 first law of thermodynamics p. 48

energy balance p. 48 power cycle p. 59

refrigeration cycle p. 60 heat pump cycle p. 60–61

Exercises: Things Engineers Think About 1. What forces act on the bicyle and rider considered in Sec. 2.2.2? Sketch a free body diagram.

5. List examples of heat transfer by conduction, radiation, and convection you might find in a kitchen.

2. Why is it incorrect to say that a system contains heat?

6. When a falling object impacts the earth and comes to rest, what happens to its kinetic and potential energies?

3. An ice skater blows into cupped hands to warm them, yet at lunch blows across a bowl of soup to cool it. How can this be interpreted thermodynamically? 4. Sketch the steady-state temperature distribution for a furnace wall composed of an 8-inch-thick concrete inner layer and a 1/2inch-thick steel outer layer.

7. When you stir a cup of coffee, what happens to the energy transferred to the coffee by work? 8. What energy transfers by work and heat can you identify for a moving automobile?

Problems: Developing Engineering Skills

9. Why are the symbols U, KE, and PE used to denote the energy change during a process, but the work and heat transfer for the process represented, respectively, simply as W and Q? 10. If the change in energy of a closed system is known for a process between two end states, can you determine if the energy change was due to work, to heat transfer, or to some combination of work and heat transfer? 11. Referring to Fig. 2.8, can you tell which process, A or B, has the greater heat transfer?

63

12. What form does the energy balance take for an isolated system? Interpret the expression you obtain. 13. How would you define an appropriate efficiency for the gearbox of Example 2.4? 14. Two power cycles each receive the same energy input Qin and discharge energy Qout to the same lake. If the cycles have different thermal efficiencies, which discharges the greater amount Qout? Does this have any implications for the environment?

Problems: Developing Engineering Skills Applying Energy Concepts from Mechanics

2.1 An automobile has a mass of 1200 kg. What is its kinetic energy, in kJ, relative to the road when traveling at a velocity of 50 km/h? If the vehicle accelerates to 100 km/h, what is the change in kinetic energy, in kJ? 2.2 An object whose mass is 400 kg is located at an elevation of 25 m above the surface of the earth. For g 9.78 m/s2, determine the gravitational potential energy of the object, in kJ, relative to the surface of the earth. 2.3 An object of mass 1000 kg, initially having a velocity of 100 m /s, decelerates to a final velocity of 20 m/s. What is the change in kinetic energy of the object, in kJ? 2.4 An airplane whose mass is 5000 kg is flying with a velocity of 150 m/s at an altitude of 10,000 m, both measured relative to the surface of the earth. The acceleration of gravity can be taken as constant at g 9.78 m/s2. (a) Calculate the kinetic and potential energies of the airplane, both in kJ. (b) If the kinetic energy increased by 10,000 kJ with no change in elevation, what would be the final velocity, in m/s? 2.5 An object whose mass is 0.5 kg has a velocity of 30 m/s. Determine (a) the final velocity, in m/s, if the kinetic energy of the object decreases by 130 J. (b) the change in elevation, in ft, associated with a 130 J change in potential energy. Let g 9.81 m/s2. 2.6 An object whose mass is 2 kg is accelerated from a velocity of 200 m/s to a final velocity of 500 m/s by the action of a resultant force. Determine the work done by the resultant force, in kJ, if there are no other interactions between the object and its surroundings. 2.7 A disk-shaped flywheel, of uniform density , outer radius R, and thickness w, rotates with an angular velocity , in rad/s. (a) Show that the moment of inertia, I vol rr2 dV, can be expressed as I wR 42 and the kinetic energy can be expressed as KE I 22.

(b) For a steel flywheel rotating at 3000 RPM, determine the kinetic energy, in N # m, and the mass, in kg, if R 0.38 m and w 0.025 m. (c) Determine the radius, in m, and the mass, in kg, of an aluminum flywheel having the same width, angular velocity, and kinetic energy as in part (b). 2.8 Two objects having different masses fall freely under the influence of gravity from rest and the same initial elevation. Ignoring the effect of air resistance, show that the magnitudes of the velocities of the objects are equal at the moment just before they strike the earth. 2.9 An object whose mass is 25 kg is projected upward from the surface of the earth with an initial velocity of 60 m/s. The only force acting on the object is the force of gravity. Plot the velocity of the object versus elevation. Determine the elevation of the object, in ft, when its velocity reaches zero. The acceleration of gravity is g 9.8 m/s2. 2.10 A block of mass 10 kg moves along a surface inclined 30 relative to the horizontal. The center of gravity of the block is elevated by 3.0 m and the kinetic energy of the block decreases by 50 J. The block is acted upon by a constant force R parallel to the incline and by the force of gravity. Assume frictionless surfaces and let g 9.81 m/s2. Determine the magnitude and direction of the constant force R, in N. 2.11 Beginning from rest, an object of mass 200 kg slides down a 10-m-long ramp. The ramp is inclined at an angle of 40 from the horizontal. If air resistance and friction between the object and the ramp are negligible, determine the velocity of the object, in m/s, at the bottom of the ramp. Let g 9.81 m /s2. Evaluating Work

2.12 A system with a mass of 5 kg, initially moving horizontally with a velocity of 40 m/s, experiences a constant horizontal deceleration of 2 m/s2 due to the action of a resultant force. As a result, the system comes to rest. Determine the length of time, in s, the force is applied and the amount of energy transfer by work, in kJ.

64

Chapter 2 Energy and the First Law of Thermodynamics

2.13 The drag force, Fd, imposed by the surrounding air on a vehicle moving with velocity V is given by Fd CdA12rV2 where Cd is a constant called the drag coefficient, A is the projected frontal area of the vehicle, and is the air density. Determine the power, in kW, required to overcome aerodynamic drag for a truck moving at 110 km/h, if Cd 0.65, A 10 m2, and 1.1 kg/m3.

varies linearly from an initial value of 900 N to a final value of zero. The atmospheric pressure is 100 kPa, and the area of the piston face is 0.018 m2. Friction between the piston and the cylinder wall can be neglected. For the air, determine the initial and final pressures, in kPa, and the work, in kJ. A = 0.018 m2

patm = 100 kPa

2.14 A major force opposing the motion of a vehicle is the rolling resistance of the tires, Fr, given by Air

Fr f w where f is a constant called the rolling resistance coefficient and w is the vehicle weight. Determine the power, in kW, required to overcome rolling resistance for a truck weighing 322.5 kN that is moving at 110 km /h. Let f 0.0069. 2.15 Measured data for pressure versus volume during the expansion of gases within the cylinder of an internal combustion engine are given in the table below. Using data from the table, complete the following: (a) Determine a value of n such that the data are fit by an equation of the form, pV n constant. (b) Evaluate analytically the work done by the gases, in kJ, using Eq. 2.17 along with the result of part (a). (c) Using graphical or numerical integration of the data, evaluate the work done by the gases, in kJ. (d) Compare the different methods for estimating the work used in parts (b) and (c). Why are they estimates? Data Point

p (bar)

V (cm3)

1 2 3 4 5 6

15 12 9 6 4 2

300 361 459 644 903 1608

2.16 One-fourth kg of a gas contained within a piston–cylinder assembly undergoes a constant-pressure process at 5 bar beginning at v1 0.20 m3/kg. For the gas as the system, the work is 15 kJ. Determine the final volume of the gas, in m3. 2.17 A gas is compressed from V1 0.3 m3, p1 1 bar to V2 0.1 m3, p2 3 bar. Pressure and volume are related linearly during the process. For the gas, find the work, in kJ. 2.18 A gas expands from an initial state where p1 500 kPa and V1 0.1 m3 to a final state where p2 100 kPa. The relationship between pressure and volume during the process is pV constant. Sketch the process on a p–V diagram and determine the work, in kJ. 2.19 Warm air is contained in a piston–cylinder assembly oriented horizontally as shown in Fig. P2.19. The air cools slowly from an initial volume of 0.003 m3 to a final volume of 0.002 m3. During the process, the spring exerts a force that

Spring force varies linearly from 900 N when V1 = 0.003 m3 to zero when V2 = 0.002 m3 Figure P2.19

2.20

Air undergoes two processes in series:

Process 1–2: polytropic compression, with n 1.3, from p1 100 kPa, v1 0.04 m3/kg to v2 0.02 m3/kg Process 2–3: constant-pressure process to v3 v1 Sketch the processes on a pv diagram and determine the work per unit mass of air, in kJ/kg. 2.21 For the cycle of Problem 1.25, determine the work for each process and the net work for the cycle, each in kJ. 2.22 The driveshaft of a building’s air-handling fan is turned at 300 RPM by a belt running on a 0.3-m-diameter pulley. The net force applied by the belt on the pulley is 2000 N. Determine the torque applied by the belt on the pulley, in N # m, and the power transmitted, in kW. 2.23 An electric motor draws a current of 10 amp with a voltage of 110 V. The output shaft develops a torque of 10.2 N # m and a rotational speed of 1000 RPM. For operation at steady state, determine (a) the electric power required by the motor and the power developed by the output shaft, each in kW. (b) the net power input to the motor, in kW. (c) the amount of energy transferred to the motor by electrical work and the amount of energy transferred out of the motor by the shaft, in kW # h during 2 h of operation. 2.24 A 12-V automotive storage battery is charged with a constant current of 2 amp for 24 h. If electricity costs $0.08 per kW # h, determine the cost of recharging the battery. 2.25 For your lifestyle, estimate the monthly cost of operating the following household items: microwave oven, refrigerator, electric space heater, personal computer, hand-held hair drier, a 100-W light bulb. Assume the cost of electricity is $0.08 per kW # h. 2.26 A solid cylindrical bar (see Fig. 2.9) of diameter 5 mm is slowly stretched from an initial length of 10 cm to a final length of 10.1 cm. The normal stress in the bar varies according to C(x x0)x0, where x is the length of the bar, x0 is the initial length, and C is a material constant (Young’s modulus).

Problems: Developing Engineering Skills

For C 2 107 kPa, determine the work done on the bar, in J, assuming the diameter remains constant. 2.27 A wire of cross-sectional area A and initial length x0 is stretched. The normal stress acting in the wire varies linearly with strain, , where e 1x x0 2 x0

and x is the length of the wire. Assuming the cross-sectional area remains constant, derive an expression for the work done on the wire as a function of strain. 2.28 A soap film is suspended on a 5 cm 5 cm wire frame, as shown in Fig. 2.10. The movable wire is displaced 1 cm by an applied force, while the surface tension of the soap film remains constant at 25 105 N/cm. Determine the work done in stretching the film, in J. 2.29 Derive an expression to estimate the work required to inflate a common balloon. List all simplifying assumptions. Evaluating Heat Transfer

2.30 A 0.2-m-thick plane wall is constructed of concrete. At steady state, the energy transfer rate by conduction through a 1-m2 area of the wall is 0.15 kW. If the temperature distribution is linear through the wall, what is the temperature difference across the wall, in K?

2.35 A closed system of mass 5 kg undergoes a process in which there is work of magnitude 9 kJ to the system from the surroundings. The elevation of the system increases by 700 m during the process. The specific internal energy of the system decreases by 6 kJ/kg and there is no change in kinetic energy of the system. The acceleration of gravity is constant at g 9.6 m/s2. Determine the heat transfer, in kJ. 2.36 A closed system of mass 20 kg undergoes a process in which there is a heat transfer of 1000 kJ from the system to the surroundings. The work done on the system is 200 kJ. If the initial specific internal energy of the system is 300 kJ/kg, what is the final specific internal energy, in kJ/kg? Neglect changes in kinetic and potential energy. 2.37 As shown in Fig. P2.37, 5 kg of steam contained within a piston–cylinder assembly undergoes an expansion from state 1, where the specific internal energy is u1 2709.9 kJ/kg, to state 2, where u2 2659.6 kJ/kg. During the process, there is heat transfer to the steam with a magnitude of 80 kJ. Also, a paddle wheel transfers energy to the steam by work in the amount of 18.5 kJ. There is no significant change in the kinetic or potential energy of the steam. Determine the energy transfer by work from the steam to the piston during the process, in kJ.

2.31 A 2-cm-diameter surface at 1000 K emits thermal radiation at a rate of 15 W. What is the emissivity of the surface? Assuming constant emissivity, plot the rate of radiant emission, in W, for surface temperatures ranging from 0 to 2000 K. The Stefan–Boltzmann constant, , is 5.67 108 W/m2 # K4. 2.32 A flat surface having an area of 2 m2 and a temperature of 350 K is cooled convectively by a gas at 300 K. Using data from Table 2.1, determine the largest and smallest heat transfer rates, in kW, that might be encountered for (a) free convection, (b) forced convection. 2.33 A flat surface is covered with insulation with a thermal conductivity of 0.08 W/m # K. The temperature at the interface between the surface and the insulation is 300C. The outside of the insulation is exposed to air at 30C, and the heat transfer coefficient for convection between the insulation and the air is 10 W/m2 # K. Ignoring radiation, determine the minimum thickness of insulation, in m, such that the outside of the insulation is no hotter than 60C at steady state. Using the Energy Balance

2.34 Each line in the following table gives information about a process of a closed system. Every entry has the same energy units. Fill in the blank spaces in the table. Process

Q

W

E1

a b c d e

50

50 40

20

20

20

50

90

20

E2

E

50

60

50

20 0 100

65

5 kg of steam Wpiston = ?

Wpw = –18.5 kJ

Q = +80 kJ

u1 = 2709.9 kJ/kg u2 = 2659.6 kJ/kg

Figure P2.37

2.38 An electric generator coupled to a windmill produces an average electric power output of 15 kW. The power is used to charge a storage battery. Heat transfer from the battery to the surroundings occurs at a constant rate of 1.8 kW. Determine, for 8 h of operation (a) the total amount of energy stored in the battery, in kJ. (b) the value of the stored energy, in $, if electricity is valued at $0.08 per kW # h. 2.39 A closed system undergoes a process during which there is energy transfer from the system by heat at a constant rate of 10 kW, and the power varies with time according to # 8t W e 8

0 6 t 1h t 7 1h

# where t is time, in h, and W is in kW.

66

Chapter 2 Energy and the First Law of Thermodynamics

(a) What is the time rate of change of system energy at t 0.6 h, in kW? (b) Determine the change in system energy after 2 h, in kJ. 2.40

A storage battery develops a power output of # W 1.2 exp1t 602 # where W is power, in kW, and t is time, in s. Ignoring heat transfer (a) plot the power output, in kW, and the change in energy of the battery, in kJ, each as a function of time. (b) What are the limiting values for the power output and the change in energy of the battery as t S ? Discuss.

2.41 A gas expands in a piston–cylinder assembly from p1 8 bar, V1 0.02 m3 to p2 2 bar in a process during which the relation between pressure and volume is pV1.2 constant. The mass of the gas is 0.25 kg. If the specific internal energy of the gas decreases by 55 kJ/kg during the process, determine the heat transfer, in kJ. Kinetic and potential energy effects are negligible.

2.46 A gas contained within a piston–cylinder assembly is shown in Fig. P2.46. Initially, the piston face is at x 0, and the spring exerts no force on the piston. As a result of heat transfer, the gas expands, raising the piston until it hits the stops. At this point the piston face is located at x 0.06 m, and the heat transfer ceases. The force exerted by the spring on the piston as the gas expands varies linearly with x according to Fspring kx where k 9,000 N/m. Friction between the piston and the cylinder wall can be neglected. The acceleration of gravity is g 9.81 m/s2. Additional information is given on Fig. P2.70. patm = 1 bar Apist = 0.0078 m2 m pist = 10 kg

2.42 Two kilograms of air is contained in a rigid well-insulated tank with a volume of 0.6 m3. The tank is fitted with a paddle wheel that transfers energy to the air at a constant rate of 10 W for 1 h. If no changes in kinetic or potential energy occur, determine

x=0 Gas m gas = 0.5 g

(a) the specific volume at the final state, in m3/kg. (b) the energy transfer by work, in kJ. (c) the change in specific internal energy of the air, in kJ/kg. 2.43 A gas is contained in a closed rigid tank. An electric resistor in the tank transfers energy to the gas at a constant rate of 1000 W. Heat transfer# between the gas #and the surroundings occurs at a rate of Q 50t, where Q is in watts, and t is time, in min. (a) Plot the time rate of change of energy of the gas for 0 t 20 min, in watts. (b) Determine the net change in energy of the gas after 20 min, in kJ. (c) If electricity is valued at $0.08 per kW # h, what is the cost of the electrical input to the resistor for 20 min of operation? 2.44 Steam in a piston–cylinder assembly undergoes a polytropic process, with n 2, from an initial state where p1 3.45 MPa, v1 .106 m3/kg, u1 3171 kJ/kg to a final state where u2 2304 kJ/kg. During the process, there is a heat transfer from the steam of magnitude 361.8. The mass of steam is .544 kg. Neglecting changes in kinetic and potential energy, determine the work, in kJ. 2.45 Air is contained in a vertical piston–cylinder assembly by a piston of mass 50 kg and having a face area of 0.01 m2. The mass of the air is 5 g, and initially the air occupies a volume of 5 liters. The atmosphere exerts a pressure of 100 kPa on the top of the piston. The volume of the air slowly decreases to 0.002 m3 as the specific internal energy of the air decreases by 260 kJ/kg. Neglecting friction between the piston and the cylinder wall, determine the heat transfer to the air, in kJ.

Figure P2.46

(a) What is the initial pressure of the gas, in kPa? (b) Determine the work done by the gas on the piston, in J. (c) If the specific internal energies of the gas at the initial and final states are 210 and 335 kJ/kg, respectively, calculate the heat transfer, in J. Analyzing Thermodynamic Cycles

2.47 The following table gives data, in kJ, for a system undergoing a thermodynamic cycle consisting of four processes in series. For the cycle, kinetic and potential energy effects can be neglected. Determine (a) the missing table entries, each in kJ. (b) whether the cycle is a power cycle or a refrigeration cycle. Process 1–2 2–3 3–4 4–1

U

Q

600 1300

600 700

W

0 500

700

Design & Open Ended Problems: Exploring Engineering Practice

2.48 A gas undergoes a thermodynamic cycle consisting of three processes: Process 1–2: compression with pV constant, from p1 1 bar, V1 1.6 m3 to V2 0.2 m3, U2 U1 0 Process 2–3: constant pressure to V3 V1 Process 3–1: constant volume, U1 U3 3549 kJ There are no significant changes in kinetic or potential energy. Determine the heat transfer and work for Process 2–3, in kJ. Is this a power cycle or a refrigeration cycle? 2.49 A gas undergoes a thermodynamic cycle consisting of three processes: Process 1–2: constant volume, V 0.028 m3, U2 U1 26.4 kJ Process 2–3: expansion with pV constant, U3 U2 Process 3–1: constant pressure, p 1.4 bar, W31 10.5 kJ There are no significant changes in kinetic or potential energy. (a) (b) (c) (d)

Sketch the cycle on a p–V diagram. Calculate the net work for the cycle, in kJ. Calculate the heat transfer for process 2–3, in kJ. Calculate the heat transfer for process 3–1, in kJ.

Is this a power cycle or a refrigeration cycle? 2.50 For a power cycle operating as in Fig. 2.15a, the heat transfers are Qin 50 kJ and Qout 35 kJ. Determine the net work, in kJ, and the thermal efficiency. 2.51 The thermal efficiency of a power cycle operating as shown in Fig. 2.15a is 35%, and Qout 40 MJ. Determine the net work developed and the heat transfer Qin, each in MJ.

67

2.53 A power cycle has a thermal efficiency of 35% and generates electricity at a rate of 100 MW. The electricity is valued at $0.08 per kW # h. Based on the cost of fuel, the cost to # supply Qin is $4.50 per GJ. For 8000 hours of operation annually, determine, in $, (a) the value of the electricity generated per year. (b) the annual fuel cost. 2.54 For each of the following, what plays the roles of the hot body and the cold body of the appropriate Fig. 2.15 schematic? (a) Window air conditioner (b) Nuclear submarine power plant (c) Ground-source heat pump 2.55 In what ways do automobile engines operate analogously to the power cycle shown in Fig. 2.15a? How are they different? Discuss. 2.56 A refrigeration cycle operating as shown in Fig. 2.15b has heat transfer Qout 2530 kJ and net work of Wcycle 844 kJ. Determine the coefficient of performance for the cycle. 2.57 A refrigeration cycle operates as shown in Fig. 2.15b with a coefficient of performance 1.5. For the cycle, Qout 500 kJ. Determine Qin and Wcycle, each in kJ. 2.58 A refrigeration cycle operates continuously and removes energy from the refrigerated space at a rate of 3.5 kW. For a coefficient of performance of 2.6, determine the net power required. 2.59 A heat pump cycle whose coefficient of performance is 2.5 delivers energy by heat transfer to a dwelling at a rate of 20 kW.

2.52 A power cycle receives energy by heat transfer from the combustion of fuel at a rate of 300 MW. The thermal efficiency of the cycle is 33.3%.

(a) Determine the net power required to operate the heat pump, in kW. (b) Evaluating electricity at $0.08 per kW # h, determine the cost of electricity in a month when the heat pump operates for 200 hours.

(a) Determine the net rate power is developed, in MW. (b) For 8000 hours of operation annually, determine the net work output, in kW # h per year. (c) Evaluating the net work output at $0.08 per kW # h, determine the value of the net work, in $/year.

2.60 A household refrigerator with a coefficient of performance of 2.4 removes energy from the refrigerated space at a rate of 200 W. Evaluating electricity at $0.08 per kW # h, determine the cost of electricity in a month when the refrigerator operates for 360 hours.

Design & Open Ended Problems: Exploring Engineering Practice 2.1D The effective use of our energy resources is an important societal goal. (a) Summarize in a pie chart the data on the use of fuels in your state in the residential, commercial, industrial, and transportation sectors. What factors may affect the future availability of these fuels? Does your state have a written energy policy? Discuss. (b) Determine the present uses of solar energy, hydropower, and wind energy in your area. Discuss factors that affect

the extent to which these renewable resources are utilized. 2.2D Among several engineers and scientists who contributed to the development of the first law of thermodynamics are: (a) James Joule. (b) James Watt. (c) Benjamin Thompson (Count Rumford). (d) Sir Humphrey Davy. (e) Julius Robert Mayer.

68

Chapter 2 Energy and the First Law of Thermodynamics

Write a biographical sketch of one of them, including a description of his principal contributions to the first law.

such bottles and explain the basic principles that make them effective.

2.3D Specially designed flywheels have been used by electric utilities to store electricity. Automotive applications of flywheel energy storage also have been proposed. Write a report that discusses promising uses of flywheels for energy storage, including consideration of flywheel materials, their properties, and costs.

2.8D A brief discussion of power, refrigeration, and heat pump cycles is presented in this chapter. For one, or more, of the applications listed below, explain the operating principles and discuss the significant energy transfers and environmental impacts:

2.4D Develop a list of the most common home-heating options in your locale. For a 2500-ft2 dwelling, what is the annual fuel cost or electricity cost for each option listed? Also, what is the installed cost of each option? For a 15-year life, which option is the most economical?

(a) coal-fired power plant. (b) nuclear power plant. (c) refrigeration unit supplying chilled water to the cooling system of a large building. (d) heat pump for residential heating and air conditioning. (e) automobile air conditioning unit

2.5D The overall convective heat transfer coefficient is used in the analysis of heat exchangers (Sec. 4.3) to relate the overall heat transfer rate and the log mean temperature difference between the two fluids passing through the heat exchanger. Write a memorandum explaining these concepts. Include data from the engineering literature on characteristic values of the overall convective heat transfer coefficient for the following heat exchanger applications: air-to-air heat recovery, airto-refrigerant evaporators, shell-and-tube steam condensers.

2.9D Fossil-fuel power plants produce most of the electricity generated annually in the United States. The cost of electricity is determined by several factors, including the power plant thermal efficiency, the unit cost of the fuel, in $ per kW # h, and the plant capital cost, in $ per kW of power generated. Prepare a memorandum comparing typical ranges of these three factors for coal-fired steam power plants and natural gas–fired gas turbine power plants. Which type of plant is most prevalent in the United States?

2.6D The outside surfaces of small gasoline engines are often covered with fins that enhance the heat transfer between the hot surface and the surrounding air. Larger engines, like automobile engines, have a liquid coolant flowing through passages in the engine block. The coolant then passes through the radiator (a finned-tube heat exchanger) where the needed cooling is provided by the air flowing through the radiator. Considering appropriate data for heat transfer coefficients, engine size, and other design issues related to engine cooling, explain why some engines use liquid coolants and others do not.

2.10D Lightweight, portable refrigerated chests are available for keeping food cool. These units use a thermoelectric cooling module energized by plugging the unit into an automobile cigarette lighter. Thermoelectric cooling requires no moving parts and requires no refrigerant. Write a report that explains this thermoelectric refrigeration technology. Discuss the applicability of this technology to larger-scale refrigeration systems.

2.7D Common vacuum-type thermos bottles can keep beverages hot or cold for many hours. Describe the construction of

2.11D Hybrids Harvest Energy (see box Sec. 2.1). Critically compare and evaluate the various hybrid electric vehicles on the market today. Write a report including at least three references.

C H A P

Evaluating Properties

T E R

3

E N G I N E E R I N G C O N T E X T To apply the energy balance to a system of interest requires knowledge of the properties of the system and how the properties are related. The objective of this chapter is to introduce property relations relevant to engineering thermodynamics. As part of the presentation, several examples are provided that illustrate the use of the closed system energy balance introduced in Chap. 2 together with the property relations considered in this chapter.

chapter objective

3.1 Fixing the State

The state of a closed system at equilibrium is its condition as described by the values of its thermodynamic properties. From observation of many thermodynamic systems, it is known that not all of these properties are independent of one another, and the state can be uniquely determined by giving the values of the independent properties. Values for all other thermodynamic properties are determined once this independent subset is specified. A general rule known as the state principle has been developed as a guide in determining the number of independent properties required to fix the state of a system. For most applications considered in this book, we are interested in what the state principle says about the intensive states of systems. Of particular interest are systems of commonly encountered pure substances, such as water or a uniform mixture of nonreacting gases. These systems are classed as simple compressible systems. Experience shows that the simple compressible systems model is useful for a wide range of engineering applications. For such systems, the state principle indicates that the number of independent intensive properties is two.

state principle

simple compressible systems

for example. . . in the case of a gas, temperature and another intensive property such as a specific volume might be selected as the two independent properties. The state principle then affirms that pressure, specific internal energy, and all other pertinent intensive properties could be determined as functions of T and v: p p(T, v), u u(T, v), and so on. The functional relations would be developed using experimental data and would depend explicitly on the particular chemical identity of the substances making up the system. The development of such functions is discussed in Chap. 11.

Intensive properties such as velocity and elevation that are assigned values relative to datums outside the system are excluded from present considerations. Also, as suggested by the name, changes in volume can have a significant influence on the energy of simple 69

70

Chapter 3 Evaluating Properties

simple system

compressible systems. The only mode of energy transfer by work that can occur as a simple compressible system undergoes quasiequilibrium processes, is associated with volume change and is given by p dV. To provide a foundation for subsequent developments involving property relations, we conclude this introduction with more detailed considerations of the state principle and simple compressible system concepts. Based on considerable empirical evidence, it has been concluded that there is one independent property for each way a system’s energy can be varied independently. We saw in Chap. 2 that the energy of a closed system can be altered independently by heat or by work. Accordingly, an independent property can be associated with heat transfer as one way of varying the energy, and another independent property can be counted for each relevant way the energy can be changed through work. On the basis of experimental evidence, therefore, the state principle asserts that the number of independent properties is one plus the number of relevant work interactions. When counting the number of relevant work interactions, it suffices to consider only those that would be significant in quasiequilibrium processes of the system. The term simple system is applied when there is only one way the system energy can be significantly altered by work as the system undergoes quasiequilibrium processes. Therefore, counting one independent property for heat transfer and another for the single work mode gives a total of two independent properties needed to fix the state of a simple system. This is the state principle for simple systems. Although no system is ever truly simple, many systems can be modeled as simple systems for the purpose of thermodynamic analysis. The most important of these models for the applications considered in this book is the simple compressible system. Other types of simple systems are simple elastic systems and simple magnetic systems.

EVALUATING PROPERTIES: GENERAL CONSIDERATIONS This part of the chapter is concerned generally with the thermodynamic properties of simple compressible systems consisting of pure substances. A pure substance is one of uniform and invariable chemical composition. Property relations for systems in which composition changes by chemical reaction are considered in Chap. 13. In the second part of this chapter, we consider property evaluation using the ideal gas model.

3.2 p–v–T Relation

p–v–T surface

We begin our study of the properties of pure, simple compressible substances and the relations among these properties with pressure, specific volume, and temperature. From experiment it is known that temperature and specific volume can be regarded as independent and pressure determined as a function of these two: p p(T, v). The graph of such a function is a surface, the p–v–T surface. 3.2.1 p –v–T Surface Figure 3.1 is the p–v–T surface of a substance such as water that expands on freezing. Figure 3.2 is for a substance that contracts on freezing, and most substances exhibit this characteristic. The coordinates of a point on the p–v–T surfaces represent the values that pressure, specific volume, and temperature would assume when the substance is at equilibrium.

3.2 p–v–T Relation

Liquid

Pressure

Critical point

Liq uid Tri va ple po lin r e

Solid

So

Va p

lid

Sp

or

-va

eci

fic vol u

por

Tc

re atu per m Te

me

Solid

(a)

Solid

Critical point Critical Liquid point L

V Vapor S Triple point V Temperature (b)

Pressure

Pressure

S L

Liquidvapor Triple line Solid-vapor

Vapor

T > Tc Tc T < Tc

Specific volume (c)

Figure 3.1 p–v–T surface and projections for a substance that expands on freezing. (a) Three-dimensional view. (b) Phase diagram. (c) p–v diagram.

There are regions on the p–v–T surfaces of Figs. 3.1 and 3.2 labeled solid, liquid, and vapor. In these single-phase regions, the state is fixed by any two of the properties: pressure, specific volume, and temperature, since all of these are independent when there is a single phase present. Located between the single-phase regions are two-phase regions where two phases exist in equilibrium: liquid–vapor, solid–liquid, and solid–vapor. Two phases can coexist during changes in phase such as vaporization, melting, and sublimation. Within the twophase regions pressure and temperature are not independent; one cannot be changed without changing the other. In these regions the state cannot be fixed by temperature and pressure alone; however, the state can be fixed by specific volume and either pressure or temperature. Three phases can exist in equilibrium along the line labeled triple line. A state at which a phase change begins or ends is called a saturation state. The domeshaped region composed of the two-phase liquid–vapor states is called the vapor dome. The lines bordering the vapor dome are called saturated liquid and saturated vapor lines. At the top of the dome, where the saturated liquid and saturated vapor lines meet, is the critical point. The critical temperature Tc of a pure substance is the maximum temperature at which liquid and vapor phases can coexist in equilibrium. The pressure at the critical point is called

two-phase regions

triple line saturation state vapor dome critical point

71

Chapter 3 Evaluating Properties

Pressure

Solid

Solid-Liq uid

72

Liquid Critical point

Constantpressure line

Va p

or

So lid -va po Sp r eci fic vo lum e

ure

Tc

t era mp e T

Solid Solid-liquid

(a)

S

Liquid Critical point

Solid

L V S V

Triple point Temperature (b)

Vapor

Critical point

Pressure

Pressure

L

Liquidvapor Triple line Solid-vapor

Vapor

T > Tc Tc T < Tc

Specific volume (c)

Figure 3.2 p–v–T surface and projections for a substance that contracts on freezing. (a) Three-dimensional view. (b) Phase diagram. (c) p–v diagram.

the critical pressure, pc. The specific volume at this state is the critical specific volume. Values of the critical point properties for a number of substances are given in Tables A-1 located in the Appendix. The three-dimensional p–v–T surface is useful for bringing out the general relationships among the three phases of matter normally under consideration. However, it is often more convenient to work with two-dimensional projections of the surface. These projections are considered next. 3.2.2 Projections of the p–v–T Surface THE PHASE DIAGRAM

phase diagram

If the p–v–T surface is projected onto the pressure–temperature plane, a property diagram known as a phase diagram results. As illustrated by Figs. 3.1b and 3.2b, when the surface is projected in this way, the two-phase regions reduce to lines. A point on any of these lines represents all two-phase mixtures at that particular temperature and pressure.

3.2 p–v–T Relation

The term saturation temperature designates the temperature at which a phase change takes place at a given pressure, and this pressure is called the saturation pressure for the given temperature. It is apparent from the phase diagrams that for each saturation pressure there is a unique saturation temperature, and conversely. The triple line of the three-dimensional p–v–T surface projects onto a point on the phase diagram. This is called the triple point. Recall that the triple point of water is used as a reference in defining temperature scales (Sec. 1.6). By agreement, the temperature assigned to the triple point of water is 273.16 K. The measured pressure at the triple point of water is 0.6113 kPa. The line representing the two-phase solid–liquid region on the phase diagram slopes to the left for substances that expand on freezing and to the right for those that contract. Although a single solid phase region is shown on the phase diagrams of Figs. 3.1 and 3.2, solids can exist in different solid phases. For example, seven different crystalline forms have been identified for water as a solid (ice).

saturation temperature saturation pressure

triple point

p – v DIAGRAM

Projecting the p–v–T surface onto the pressure–specific volume plane results in a p–v diagram, as shown by Figs. 3.1c and 3.2c. The figures are labeled with terms that have already been introduced. When solving problems, a sketch of the p–v diagram is frequently convenient. To facilitate the use of such a sketch, note the appearance of constant-temperature lines (isotherms). By inspection of Figs. 3.1c and 3.2c, it can be seen that for any specified temperature less than the critical temperature, pressure remains constant as the two-phase liquid–vapor region is traversed, but in the single-phase liquid and vapor regions the pressure decreases at fixed temperature as specific volume increases. For temperatures greater than or equal to the critical temperature, pressure decreases continuously at fixed temperature as specific volume increases. There is no passage across the two-phase liquid–vapor region. The critical isotherm passes through a point of inflection at the critical point and the slope is zero there.

p–v diagram

T–v DIAGRAM

Projecting the liquid, two-phase liquid–vapor, and vapor regions of the p–v–T surface onto the temperature–specific volume plane results in a T–v diagram as in Fig. 3.3. Since consistent patterns are revealed in the p–v–T behavior of all pure substances, Fig. 3.3 showing a T–v diagram for water can be regarded as representative.

pc = 22.09 MPa 30 MPa 10 MPa

Temperature

Tc Liquid

Critical point

Vapor 1.014 bar

Liquid-vapor

s

100°C g

f 20°C

l Specific volume

Figure 3.3

Sketch of a temperature– specific volume diagram for water showing the liquid, two-phase liquid–vapor, and vapor regions (not to scale).

T–v diagram

73

74

Chapter 3 Evaluating Properties

As for the p–v diagram, a sketch of the T–v diagram is often convenient for problem solving. To facilitate the use of such a sketch, note the appearance of constant-pressure lines (isobars). For pressures less than the critical pressure, such as the 10 MPa isobar on Fig. 3.3, the pressure remains constant with temperature as the two-phase region is traversed. In the single-phase liquid and vapor regions the temperature increases at fixed pressure as the specific volume increases. For pressures greater than or equal to the critical pressure, such as the one marked 30 MPa on Fig. 3.3, temperature increases continuously at fixed pressure as the specific volume increases. There is no passage across the two-phase liquid–vapor region. The projections of the p–v–T surface used in this book to illustrate processes are not generally drawn to scale. A similar comment applies to other property diagrams introduced later. 3.2.3 Studying Phase Change It is instructive to study the events that occur as a pure substance undergoes a phase change. To begin, consider a closed system consisting of a unit mass (1 kg) of liquid water at 20C contained within a piston–cylinder assembly, as illustrated in Fig. 3.4a. This state is represented by point l on Fig. 3.3. Suppose the water is slowly heated while its pressure is kept constant and uniform throughout at 1.014 bar. LIQUID STATES

subcooled liquid compressed liquid

As the system is heated at constant pressure, the temperature increases considerably while the specific volume increases slightly. Eventually, the system is brought to the state represented by f on Fig. 3.3. This is the saturated liquid state corresponding to the specified pressure. For water at 1.014 bar the saturation temperature is 100C. The liquid states along the line segment l–f of Fig. 3.3 are sometimes referred to as subcooled liquid states because the temperature at these states is less than the saturation temperature at the given pressure. These states are also referred to as compressed liquid states because the pressure at each state is higher than the saturation pressure corresponding to the temperature at the state. The names liquid, subcooled liquid, and compressed liquid are used interchangeably. TWO-PHASE, LIQUID–VAPOR MIXTURE

When the system is at the saturated liquid state (state f of Fig. 3.3), additional heat transfer at fixed pressure results in the formation of vapor without any change in temperature but with a considerable increase in specific volume. As shown in Fig. 3.4b, the system would

Water vapor

Liquid water

Liquid water

(a)

(b)

Water vapor

(c)

Figure 3.4 Illustration of constant-pressure change from liquid to vapor for water.

3.2 p–v–T Relation

now consist of a two-phase liquid–vapor mixture. When a mixture of liquid and vapor exists in equilibrium, the liquid phase is a saturated liquid and the vapor phase is a saturated vapor. If the system is heated further until the last bit of liquid has vaporized, it is brought to point g on Fig. 3.3, the saturated vapor state. The intervening two-phase liquid–vapor mixture states can be distinguished from one another by the quality, an intensive property. For a two-phase liquid–vapor mixture, the ratio of the mass of vapor present to the total mass of the mixture is its quality, x. In symbols, x

mvapor mliquid mvapor

two-phase liquid–vapor mixture

(3.1) quality

The value of the quality ranges from zero to unity: at saturated liquid states, x 0, and at saturated vapor states, x 1.0. Although defined as a ratio, the quality is frequently given as a percentage. Examples illustrating the use of quality are provided in Sec. 3.3. Similar parameters can be defined for two-phase solid–vapor and two-phase solid–liquid mixtures.

VAPOR STATES

Let us return to a consideration of Figs. 3.3 and 3.4. When the system is at the saturated vapor state (state g on Fig. 3.3), further heating at fixed pressure results in increases in both temperature and specific volume. The condition of the system would now be as shown in Fig. 3.4c. The state labeled s on Fig. 3.3 is representative of the states that would be attained by further heating while keeping the pressure constant. A state such as s is often referred to as a superheated vapor state because the system would be at a temperature greater than the saturation temperature corresponding to the given pressure. Consider next the same thought experiment at the other constant pressures labeled on Fig. 3.3, 10 MPa, 22.09 MPa, and 30 MPa. The first of these pressures is less than the critical pressure of water, the second is the critical pressure, and the third is greater than the critical pressure. As before, let the system initially contain a liquid at 20C. First, let us study the system if it were heated slowly at 10 MPa. At this pressure, vapor would form at a higher temperature than in the previous example, because the saturation pressure is higher (refer to Fig. 3.3). In addition, there would be somewhat less of an increase in specific volume from saturated liquid to vapor, as evidenced by the narrowing of the vapor dome. Apart from this, the general behavior would be the same as before. Next, consider the behavior of the system were it heated at the critical pressure, or higher. As seen by following the critical isobar on Fig. 3.3, there would be no change in phase from liquid to vapor. At all states there would be only one phase. Vaporization (and the inverse process of condensation) can occur only when the pressure is less than the critical pressure. Thus, at states where pressure is greater than the critical pressure, the terms liquid and vapor tend to lose their significance. Still, for ease of reference to such states, we use the term liquid when the temperature is less than the critical temperature and vapor when the temperature is greater than the critical temperature.

MELTING AND SUBLIMATION

Although the phase change from liquid to vapor (vaporization) is the one of principal interest in this book, it is also instructive to consider the phase changes from solid to liquid (melting) and from solid to vapor (sublimation). To study these transitions, consider a system consisting of a unit mass of ice at a temperature below the triple point temperature. Let us begin

superheated vapor

75

76

Chapter 3 Evaluating Properties

Critical point

Liquid a´´ b´´

c´´

Melting Pressure

Vaporization a

b

c

Solid Triple point

Sublimation a´

b´

Vapor

c´

Figure 3.5 Temperature

Phase diagram for water (not to

scale).

with the case where the system is at state a of Fig. 3.5, where the pressure is greater than the triple point pressure. Suppose the system is slowly heated while maintaining the pressure constant and uniform throughout. The temperature increases with heating until point b on Fig. 3.5 is attained. At this state the ice is a saturated solid. Additional heat transfer at fixed pressure results in the formation of liquid without any change in temperature. As the system is heated further, the ice continues to melt until eventually the last bit melts, and the system contains only saturated liquid. During the melting process the temperature and pressure remain constant. For most substances, the specific volume increases during melting, but for water the specific volume of the liquid is less than the specific volume of the solid. Further heating at fixed pressure results in an increase in temperature as the system is brought to point c on Fig. 3.5. Next, consider the case where the system is initially at state a of Fig. 3.5, where the pressure is less than the triple point pressure. In this case, if the system is heated at constant pressure it passes through the two-phase solid–vapor region into the vapor region along the line a–b–c shown on Fig. 3.5. The case of vaporization discussed previously is shown on Fig. 3.5 by the line a–b–c.

3.3 Retrieving Thermodynamic Properties

steam tables

Thermodynamic property data can be retrieved in various ways, including tables, graphs, equations, and computer software. The emphasis of the present section is on the use of tables of thermodynamic properties, which are commonly available for pure, simple compressible substances of engineering interest. The use of these tables is an important skill. The ability to locate states on property diagrams is an important related skill. The software available with this text, Interactive Thermodynamics: IT, is also used selectively in examples and endof-chapter problems throughout the book. Skillful use of tables and property diagrams is prerequisite for the effective use of software to retrieve thermodynamic property data. Since tables for different substances are frequently set up in the same general format, the present discussion centers mainly on Tables A-2 through A-6 giving the properties of water; these are commonly referred to as the steam tables. Tables A-7 through A-9 for Refrigerant 22, Tables A-10 through A-12 for Refrigerant 134a, Tables A-13 through A-15 for ammonia, and Tables A-16 through A-18 for propane are used similarly, as are tables for other substances found in the engineering literature.

3.3 Retrieving Thermodynamic Properties

3.3.1 Evaluating Pressure, Specific Volume, and Temperature VAPOR AND LIQUID TABLES

The properties of water vapor are listed in Tables A-4 and of liquid water in Tables A-5. These are often referred to as the superheated vapor tables and compressed liquid tables, respectively. The sketch of the phase diagram shown in Fig. 3.6 brings out the structure of these tables. Since pressure and temperature are independent properties in the single-phase liquid and vapor regions, they can be used to fix the state in these regions. Accordingly, Tables A-4 and A-5 are set up to give values of several properties as functions of pressure and temperature. The first property listed is specific volume. The remaining properties are discussed in subsequent sections. For each pressure listed, the values given in the superheated vapor table (Tables A-4) begin with the saturated vapor state and then proceed to higher temperatures. The data in the compressed liquid table (Tables A-5) end with saturated liquid states. That is, for a given pressure the property values are given as the temperature increases to the saturation temperature. In these tables, the value shown in parentheses after the pressure in the table heading is the corresponding saturation temperature. for example. . . in Tables A-4 and A-5, at a pressure of 10.0 MPa, the saturation temperature is listed as 311.06C. for example. . . to gain more experience with Tables A-4 and A-5 verify the following: Table A-4 gives the specific volume of water vapor at 10.0 MPa and 600C as 0.03837 m3/kg. At 10.0 MPa and 100C, Table A-5 gives the specific volume of liquid water as 1.0385 103 m3/kg.

The states encountered when solving problems often do not fall exactly on the grid of values provided by property tables. Interpolation between adjacent table entries then becomes necessary. Care always must be exercised when interpolating table values. The tables provided in the Appendix are extracted from more extensive tables that are set up so that linear interpolation, illustrated in the following example, can be used with acceptable accuracy. Linear interpolation is assumed to remain valid when using the abridged tables of the text for the solved examples and end-of-chapter problems.

Compressed liquid tables give v, u, h, s versus p, T

Pressure

Liquid

Solid

Critical point

Vapor Superheated vapor tables give v, u, h, s versus p, T

Temperature

Figure 3.6 Sketch of the phase diagram for water used to discuss the structure of the superheated vapor and compressed liquid tables (not to scale).

linear interpolation

77

Chapter 3 Evaluating Properties 3 ——

(240°C, 0.2275 mkg ) v (m3/kg)

78

p = 10 bar T(°C) v (m3/kg) 200 0.2060 215 v=? 240 0.2275

(215°C, v)

3 ——

(200°C, 0.2060 mkg ) 200 Figure 3.7

215 T(°C)

240

Illustration of linear interpolation.

for example. . . let us determine the specific volume of water vapor at a state where p 10 bar and T 215C. Shown in Fig. 3.7 is a sampling of data from Table A-4. At a pressure of 10 bar, the specified temperature of 215C falls between the table values of 200 and 240C, which are shown in bold face. The corresponding specific volume values are also shown in bold face. To determine the specific volume v corresponding to 215C, we may think of the slope of a straight line joining the adjacent table states, as follows

slope

10.2275 0.20602 m3/kg 1v 0.20602 m3/kg 1240 2002°C 1215 2002°C

Solving for v, the result is v 0.2141 m3/kg. SATURATION TABLES

The saturation tables, Tables A-2 and A-3, list property values for the saturated liquid and vapor states. The property values at these states are denoted by the subscripts f and g, respectively. Table A-2 is called the temperature table, because temperatures are listed in the first column in convenient increments. The second column gives the corresponding saturation pressures. The next two columns give, respectively, the specific volume of saturated liquid, vf, and the specific volume of saturated vapor, vg. Table A-3 is called the pressure table, because pressures are listed in the first column in convenient increments. The corresponding saturation temperatures are given in the second column. The next two columns give vf and vg, respectively. The specific volume of a two-phase liquid–vapor mixture can be determined by using the saturation tables and the definition of quality given by Eq. 3.1 as follows. The total volume of the mixture is the sum of the volumes of the liquid and vapor phases V Vliq Vvap Dividing by the total mass of the mixture, m, an average specific volume for the mixture is obtained v

Vliq Vvap V

m m m

Since the liquid phase is a saturated liquid and the vapor phase is a saturated vapor, Vliq mliq vf and Vvap mvap vg, so va

mliq m

b vf a

mvap m

b vg

3.3 Retrieving Thermodynamic Properties

79

Introducing the definition of quality, x mvapm, and noting that mliqm 1 x, the above expression becomes v 11 x2 vf xvg vf x 1vg vf 2

(3.2)

The increase in specific volume on vaporization (vg vf) is also denoted by vfg. for example. . . consider a system consisting of a two-phase liquid–vapor mixture of water at 100C and a quality of 0.9. From Table A-2 at 100C, vf 1.0435 103 m3/kg and vg 1.673 m3/kg. The specific volume of the mixture is

v vf x 1vg vf 2 1.0435 103 10.92 11.673 1.0435 103 2 1.506 m3/kg

To facilitate locating states in the tables, it is often convenient to use values from the saturation tables together with a sketch of a T–v or p–v diagram. For example, if the specific volume v and temperature T are known, refer to the temperature table, Table A-2, and determine the values of vf and vg. A T–v diagram illustrating these data is given in Fig. 3.8. If the given specific volume falls between vf and vg, the system consists of a two-phase liquid–vapor mixture, and the pressure is the saturation pressure corresponding to the given temperature. The quality can be found by solving Eq. 3.2. If the given specific volume is greater than vg, the state is in the superheated vapor region. Then, by interpolating in Table A-4 the pressure and other properties listed can be determined. If the given specific volume is less than vf, Table A-5 would be used to determine the pressure and other properties. for example. . . let us determine the pressure of water at each of three states defined by a temperature of 100C and specific volumes, respectively, of v1 2.434 m3/kg, v2 1.0 m3/kg, and v3 1.0423 103 m3/kg. Using the known temperature, Table A-2 provides the values of vf and vg: vf 1.0435 103 m3/kg, vg 1.673 m3/kg. Since v1 is greater than vg, state 1 is in the vapor region. Table A-4 gives the pressure as 0.70 bar. Next, since v2 falls between vf and vg, the pressure is the saturation pressure corresponding to 100C, which is 1.014 bar. Finally, since v3 is less than vf, state 3 is in the liquid region. Table A-5 gives the pressure as 25 bar. EXAMPLES

The following two examples feature the use of sketches of p–v and T–v diagrams in conjunction with tabular data to fix the end states of processes. In accord with the state principle, two independent intensive properties must be known to fix the state of the systems under consideration.

Temperature

Critical point

Liquid

Saturated liquid v < vf

Saturated vapor

vf < v < vg

f

Vapor v > vg

g

vf

vg Specific volume

Figure 3.8 Sketch of a T–v diagram for water used to discuss locating states in the tables.

T

100°C

3 f 2

g 1

v

80

Chapter 3 Evaluating Properties

EXAMPLE

Heating Water at Constant Volume

3.1

A closed, rigid container of volume 0.5 m3 is placed on a hot plate. Initially, the container holds a two-phase mixture of saturated liquid water and saturated water vapor at p1 1 bar with a quality of 0.5. After heating, the pressure in the container is p2 1.5 bar. Indicate the initial and final states on a T–v diagram, and determine (a) the temperature, in C, at each state. (b) the mass of vapor present at each state, in kg. (c) If heating continues, determine the pressure, in bar, when the container holds only saturated vapor. SOLUTION Known: A two-phase liquid–vapor mixture of water in a closed, rigid container is heated on a hot plate. The initial pressure and quality and the final pressure are known. Find: Indicate the initial and final states on a T–v diagram and determine at each state the temperature and the mass of water vapor present. Also, if heating continues, determine the pressure when the container holds only saturated vapor. Schematic and Given Data: T

p1 x1 p2 x3

= 1 bar = 0.5 = 1.5 bar = 1.0

3 V = 0.5

1.5 bar

m3 2

1 bar

1

+

–

Hot plate

v

Figure E3.1

Assumptions: 1. The water in the container is a closed system. 2. States 1, 2, and 3 are equilibrium states. 3. The volume of the container remains constant. Analysis: Two independent properties are required to fix states 1 and 2. At the initial state, the pressure and quality are known. As these are independent, the state is fixed. State 1 is shown on the T–v diagram in the two-phase region. The specific volume at state 1 is found using the given quality and Eq. 3.2. That is v1 vf1 x 1vgl vfl 2 From Table A-3 at p1 1 bar, vfl 1.0432 10

3

m3/kg and vg1 1.694 m3/kg. Thus

v1 1.0432 103 0.5 11.694 1.0432 103 2 0.8475 m3/kg At state 2, the pressure is known. The other property required to fix the state is the specific volume v2. Volume and mass are each constant, so v2 v1 0.8475 m3/kg. For p2 1.5 bar, Table A-3 gives vf2 1.0582 103 and vg2 1.159 m3/kg. Since

❶ ❷

vf2 6 v2 6 vg2 state 2 must be in the two-phase region as well. State 2 is also shown on the T–v diagram above.

3.3 Retrieving Thermodynamic Properties

(a) Since states 1 and 2 are in the two-phase liquid–vapor region, the temperatures correspond to the saturation temperatures for the given pressures. Table A-3 gives T1 99.63°C

and

T2 111.4°C

(b) To find the mass of water vapor present, we first use the volume and the specific volume to find the total mass, m. That is m

V 0.5 m3 0.59 kg v 0.8475 m3/kg

Then, with Eq. 3.1 and the given value of quality, the mass of vapor at state 1 is mg1 x1m 0.5 10.59 kg2 0.295 kg The mass of vapor at state 2 is found similarly using the quality x2. To determine x2, solve Eq. 3.2 for quality and insert specific volume data from Table A-3 at a pressure of 1.5 bar, along with the known value of v, as follows x2

v vf2 vg2 vf2 0.8475 1.0528 103 0.731 1.159 1.0528 103

Then, with Eq. 3.1 mg2 0.731 10.59 kg2 0.431 kg

❸

(c) If heating continued, state 3 would be on the saturated vapor line, as shown on the T–v diagram above. Thus, the pressure would be the corresponding saturation pressure. Interpolating in Table A-3 at vg 0.8475 m3/kg, we get p3 2.11 bar.

❶ ❷ ❸

The procedure for fixing state 2 is the same as illustrated in the discussion of Fig. 3.8. Since the process occurs at constant specific volume, the states lie along a vertical line. If heating continued at constant volume past state 3, the final state would be in the superheated vapor region, and property data would then be found in Table A-4. As an exercise, verify that for a final pressure of 3 bar, the temperature would be approximately 282C.

EXAMPLE

3.2

Heating Ammonia at Constant Pressure

A vertical piston–cylinder assembly containing 0.05 kg of ammonia, initially a saturated vapor, is placed on a hot plate. Due to the weight of the piston and the surrounding atmospheric pressure, the pressure of the ammonia is 1.5 bars. Heating occurs slowly, and the ammonia expands at constant pressure until the final temperature is 25C. Show the initial and final states on T–v and p–v diagrams, and determine (a) the volume occupied by the ammonia at each state, in m3. (b) the work for the process, in kJ. SOLUTION Known: Ammonia is heated at constant pressure in a vertical piston–cylinder assembly from the saturated vapor state to a known final temperature. Find: Show the initial and final states on T–v and p–v diagrams, and determine the volume at each state and the work for the process.

81

82

Chapter 3 Evaluating Properties

Schematic and Given Data:

T

25°C

2 Ammonia

25.2°C

1

+

–

Hot plate

v p

Assumptions: 1. The ammonia is a closed system. 25°C

2. States 1 and 2 are equilibrium states. 3. The process occurs at constant pressure.

2

1.5 bars 1

v

Figure E3.2

Analysis: The initial state is a saturated vapor condition at 1.5 bars. Since the process occurs at constant pressure, the final state is in the superheated vapor region and is fixed by p2 1.5 bars and T2 25C. The initial and final states are shown on the T–v and p–v diagrams above. (a) The volumes occupied by the ammonia at states 1 and 2 are obtained using the given mass and the respective specific volumes. From Table A-14 at p1 1.5 bars, we get v1 vg1 0.7787 m3/kg. Thus V1 mv1 10.05 kg210.7787 m3/kg2 0.0389 m3

Interpolating in Table A-15 at p2 1.5 bars and T2 25C, we get v2 .9553 m3/kg. Thus V2 mv2 10.05 kg2 1.9553 m3/kg2 0.0478 m3

(b) In this case, the work can be evaluated using Eq. 2.17. Since the pressure is constant W

V2

V1

p dV p1V2 V1 2

Inserting values W 11.5 bars210.0478 0.03892m3 `

❶

W 1.335 kJ

❶

Note the use of conversion factors in this calculation.

105 N/m2 kJ ` ` 3 # ` 1 bar 10 N m

3.3 Retrieving Thermodynamic Properties

3.3.2 Evaluating Specific Internal Energy and Enthalpy In many thermodynamic analyses the sum of the internal energy U and the product of pressure p and volume V appears. Because the sum U pV occurs so frequently in subsequent discussions, it is convenient to give the combination a name, enthalpy, and a distinct symbol, H. By definition H U pV (3.3)

enthalpy

Since U, p, and V are all properties, this combination is also a property. Enthalpy can be expressed on a unit mass basis h u pv

(3.4)

h u pv

(3.5)

and per mole Units for enthalpy are the same as those for internal energy. The property tables introduced in Sec. 3.3.1 giving pressure, specific volume, and temperature also provide values of specific internal energy u, enthalpy h, and entropy s. Use of these tables to evaluate u and h is described in the present section; the consideration of entropy is deferred until it is introduced in Chap. 6. Data for specific internal energy u and enthalpy h are retrieved from the property tables in the same way as for specific volume. For saturation states, the values of uf and ug, as well as hf and hg, are tabulated versus both saturation pressure and saturation temperature. The specific internal energy for a two-phase liquid–vapor mixture is calculated for a given quality in the same way the specific volume is calculated u 11 x2uf xug uf x 1ug uf 2

(3.6)

The increase in specific internal energy on vaporization (ug uf) is often denoted by ufg. Similarly, the specific enthalpy for a two-phase liquid–vapor mixture is given in terms of the quality by h 11 x2hf xhg hf x 1hg hf 2

(3.7)

The increase in enthalpy during vaporization (hg hf) is often tabulated for convenience under the heading hfg. for example. . . to illustrate the use of Eqs. 3.6 and 3.7, we determine the specific enthalpy of Refrigerant 22 when its temperature is 12C and its specific internal energy is 144.58 kJ/kg. Referring to Table A-7, the given internal energy value falls between uf and ug at 12C, so the state is a two-phase liquid–vapor mixture. The quality of the mixture is found by using Eq. 3.6 and data from Table A-7 as follows:

x

u uf 144.58 58.77 0.5 ug uf 230.38 58.77

Then, with the values from Table A-7, Eq. 3.7 gives h 11 x2hf xhg 11 0.52 159.352 0.51253.992 156.67 kJ/kg In the superheated vapor tables, u and h are tabulated along with v as functions of temperature and pressure. for example. . . let us evaluate T, v, and h for water at 0.10 MPa and a

83

84

Chapter 3 Evaluating Properties

Thermodynamics in the News… Natural Refrigerants—Back to the Future Naturally-occurring refrigerants like hydrocarbons, ammonia, and carbon dioxide were introduced in the early 1900s. They were displaced in the 1920s by safer chlorine-based synthetic refrigerants, paving the way for the refrigerators and air conditioners we enjoy today. Over the last decade, these synthetics largely have been replaced by hydroflourocarbons (HFC’s) because of uneasiness over ozone depletion. But studies now indicate that natural refrigerants may be preferable to HFC’s because of lower overall impact on global warming. This has sparked renewed interest in natural refrigerants. Decades of research and development went into the current refrigerants, so returning to natural refrigerants creates challenges, experts say. Engineers are revisiting the concerns of flammability, odor, and safety that naturals present, and are meeting with some success. New energy-efficient refrigerators

using propane are now available in Europe. Manufacturers claim they are safer than gas-burning home appliances. Researchers are studying ways to eliminate leaks so ammonia can find more widespread application. Carbon dioxide is also being looked at again with an eye to minimizing safety issues related to its relatively high pressures in refrigeration applications. Neither HFCs nor natural refrigerants do well on measures of direct global warming impact. However, a new index that takes energy efficiency into account is changing how we view refrigerants. Because of the potential for increased energy efficiency of refrigerators using natural refrigerants, the naturals score well on the new index compared to HFCs.

specific internal energy of 2537.3 kJ/kg. Turning to Table A-3, note that the given value of u is greater than ug at 0.1 MPa (ug 2506.1 kJ/kg). This suggests that the state lies in the superheated vapor region. From Table A-4 it is found that T 120C, v 1.793 m3/kg, and h 2716.6 kJ/kg. Alternatively, h and u are related by the definition of h h u pv kJ N m3 1 kJ

a105 2 b a1.793 b ` 3 # ` kg kg 10 N m m 2537.3 179.3 2716.6 kJ/kg

2537.3

Specific internal energy and enthalpy data for liquid states of water are presented in Tables A-5. The format of these tables is the same as that of the superheated vapor tables considered previously. Accordingly, property values for liquid states are retrieved in the same manner as those of vapor states. For water, Tables A-6 give the equilibrium properties of saturated solid and saturated vapor. The first column lists the temperature, and the second column gives the corresponding saturation pressure. These states are at pressures and temperatures below those at the triple point. The next two columns give the specific volume of saturated solid, vi, and saturated vapor, vg, respectively. The table also provides the specific internal energy, enthalpy, and entropy values for the saturated solid and the saturated vapor at each of the temperatures listed. REFERENCE STATES AND REFERENCE VALUES

reference states reference values

The values of u, h, and s given in the property tables are not obtained by direct measurement but are calculated from other data that can be more readily determined experimentally. The computational procedures require use of the second law of thermodynamics, so consideration of these procedures is deferred to Chap. 11 after the second law has been introduced. However, because u, h, and s are calculated, the matter of reference states and reference values becomes important and is considered briefly in the following paragraphs. When applying the energy balance, it is differences in internal, kinetic, and potential energy between two states that are important, and not the values of these energy quantities at each of the two states. for example. . . consider the case of potential energy. The numerical value of potential energy determined relative to the surface of the earth is different from the value relative to the top of a tall building at the same location. However, the difference in

3.3 Retrieving Thermodynamic Properties

potential energy between any two elevations is precisely the same regardless of the datum selected, because the datum cancels in the calculation. Similarly, values can be assigned to specific internal energy and enthalpy relative to arbitrary reference values at arbitrary reference states. As for the case of potential energy considered above, the use of values of a particular property determined relative to an arbitrary reference is unambiguous as long as the calculations being performed involve only differences in that property, for then the reference value cancels. When chemical reactions take place among the substances under consideration, special attention must be given to the matter of reference states and values, however. A discussion of how property values are assigned when analyzing reactive systems is given in Chap. 13. The tabular values of u and h for water, ammonia, propane, and Refrigerants 22 and 134a provided in the Appendix are relative to the following reference states and values. For water, the reference state is saturated liquid at 0.01C. At this state, the specific internal energy is set to zero. Values of the specific enthalpy are calculated from h u pv, using the tabulated values for p, v, and u. For ammonia, propane, and the refrigerants, the reference state is saturated liquid at 40C. At this reference state the specific enthalpy is set to zero. Values of specific internal energy are calculated from u h pv by using the tabulated values for p, v, and h. Notice in Table A-7 that this leads to a negative value for internal energy at the reference state, which emphasizes that it is not the numerical values assigned to u and h at a given state that are important but their differences between states. The values assigned to particular states change if the reference state or reference values change, but the differences remain the same. 3.3.3 Evaluating Properties Using Computer Software The use of computer software for evaluating thermodynamic properties is becoming prevalent in engineering. Computer software falls into two general categories: those that provide data only at individual states and those that provide property data as part of a more general simulation package. The software available with this text, Interactive Thermodynamics: IT, is a tool that can be used not only for routine problem solving by providing data at individual state points, but also for simulation and analysis (see box).

U S I N G I N T E R A C T I V E T H E R M O DY N A M I C S : I T

The computer software tool Interactive Thermodynamics: IT is available for use with this text. Used properly, IT provides an important adjunct to learning engineering thermodynamics and solving engineering problems. The program is built around an equation solver enhanced with thermodynamic property data and other valuable features. With IT you can obtain a single numerical solution or vary parameters to investigate their effects. You also can obtain graphical output, and the Windows-based format allows you to use any Windows word-processing software or spreadsheet to generate reports. Other features of IT include:

a guided series of help screens and a number of sample solved examples from the text to help you learn how to use the program. drag-and-drop templates for many of the standard problem types, including a list of assumptions that you can customize to the problem at hand. predetermined scenarios for power plants and other important applications. thermodynamic property data for water, refrigerants 22 and 134a, ammonia, air–water vapor mixtures, and a number of ideal gases.

85

86

Chapter 3 Evaluating Properties

the capability to input user-supplied data. the capability to interface with user-supplied routines.

The software is best used as an adjunct to the problem-solving process discussed in Sec. 1.7.3. The equation-solving capability of the program cannot substitute for careful engineering analysis. You still must develop models and analyze them, perform limited hand calculations, and estimate ranges of parameters and property values before you move to the computer to obtain solutions and explore possible variations. Afterward, you also must assess the answers to see that they are reasonable.

IT provides data for substances represented in the Appendix tables. Generally, data are retrieved by simple call statements that are placed in the workspace of the program. for example. . . consider the two-phase, liquid–vapor mixture at state 1 of Example 3.1 for which p 1 bar, v 0.8475 m3/kg. The following illustrates how data for saturation temperature, quality, and specific internal energy are retrieved using IT. The functions for T, v, and u are obtained by selecting Water/Steam from the Properties menu. Choosing SI units from the Units menu, with p in bar, T in C, and amount of substance in kg, the IT program is p = 1 // bar v = 0.8475 // m3/kg T = Tsat_P(“Water/Steam”,p) v = vsat_Px(“Water/Steam”,p,x) u = usat_Px(Water/Steam”,p,x)

Clicking the Solve button, the software returns values of T 99.63C, x 0.5, and u 1462 kJ/kg. These values can be verified using data from Table A-3. Note that text inserted between the symbol // and a line return is treated as a comment. The previous example illustrates an important feature of IT. Although the quality, x, is implicit in the list of arguments in the expression for specific volume, there is no need to solve the expression algebraically for x. Rather, the program can solve for x as long as the number of equations equals the number of unknowns. Other features of Interactive Thermodynamics: IT are illustrated through subsequent examples. The use of computer software for engineering analysis is a powerful approach. Still, there are some rules to observe:

Software complements and extends careful analysis, but does not substitute for it. Computer-generated values should be checked selectively against hand-calculated, or otherwise independently determined values. Computer-generated plots should be studied to see if the curves appear reasonable and exhibit expected trends.

3.3.4 Examples In the following examples, closed systems undergoing processes are analyzed using the energy balance. In each case, sketches of p–v and/or T–v diagrams are used in conjunction with appropriate tables to obtain the required property data. Using property diagrams and table data introduces an additional level of complexity compared to similar problems in Chap. 2.

3.3 Retrieving Thermodynamic Properties

EXAMPLE

3.3

Stirring Water at Constant Volume

A well-insulated rigid tank having a volume of .25 m3 contains saturated water vapor at 100C. The water is rapidly stirred until the pressure is 1.5 bars. Determine the temperature at the final state, in C, and the work during the process, in kJ. SOLUTION Known: By rapid stirring, water vapor in a well-insulated rigid tank is brought from the saturated vapor state at 100C to a pressure of 1.5 bars. Find: Determine the temperature at the final state and the work. Schematic and Given Data:

Water

Boundary p

T 2 1.5 bars

2

❶

T2

1.5 bars T2 1.014 bars 100C

1.014 bars 1

1

100°C v

v

Figure E3.3

Assumptions: 1. The water is a closed system. 2. The initial and final states are at equilibrium. There is no net change in kinetic or potential energy. 3. There is no heat transfer with the surroundings. 4. The tank volume remains constant. Analysis: To determine the final equilibrium state, the values of two independent intensive properties are required. One of these is pressure, p2 1.5 bars, and the other is the specific volume: v2 v1. The initial and final specific volumes are equal because the total mass and total volume are unchanged in the process. The initial and final states are located on the accompanying T–v and p–v diagrams. From Table A-2, v1 vg(100C) 1.673 m3/kg, u1 ug(100C) 2506.5 kJ/kg. By using v2 v1 and interpolating in Table A-4 at p2 1.5 bars. T2 273°C,

u2 2767.8 kJ/kg

Next, with assumptions 2 and 3 an energy balance for the system reduces to 0

¢U ¢KE ¢PE Q W On rearrangement

W 1U2 U1 2 m1u2 u1 2

87

88

Chapter 3 Evaluating Properties

To evaluate W requires the system mass. This can be determined from the volume and specific volume V 0.25 m3 b 0.149 kg a v1 1.673 m3/kg

m

Finally, by inserting values into the expression for W W 1.149 kg212767.8 2506.52 kJ/kg 38.9 kJ where the minus sign signifies that the energy transfer by work is to the system.

❶

Although the initial and final states are equilibrium states, the intervening states are not at equilibrium. To emphasize this, the process has been indicated on the T–v and p–v diagrams by a dashed line. Solid lines on property diagrams are reserved for processes that pass through equilibrium states only (quasiequilibrium processes). The analysis illustrates the importance of carefully sketched property diagrams as an adjunct to problem solving.

EXAMPLE

3.4

Analyzing Two Processes in Series

Water contained in a piston–cylinder assembly undergoes two processes in series from an initial state where the pressure is 10 bar and the temperature is 400C. Process 1–2: The water is cooled as it is compressed at a constant pressure of 10 bar to the saturated vapor state. Process 2–3: The water is cooled at constant volume to 150C. (a) Sketch both processes on T–v and p–v diagrams. (b) For the overall process determine the work, in kJ/kg. (c) For the overall process determine the heat transfer, in kJ/kg. SOLUTION Known: Water contained in a piston–cylinder assembly undergoes two processes: It is cooled and compressed while keeping the pressure constant, and then cooled at constant volume. Find: Sketch both processes on T–v and p–v diagrams. Determine the net work and the net heat transfer for the overall process per unit of mass contained within the piston–cylinder assembly. Schematic and Given Data:

Water 10 bar

Boundary p

1 T

10 bar

2

400°C

179.9°C

1

400°C

2

179.9°C

4.758 bar

4.758 bar

150°C 3

150°C

v

3 v

Figure E3.4

3.3 Retrieving Thermodynamic Properties

Assumptions: 1. The water is a closed system. 2. The piston is the only work mode. 3. There are no changes in kinetic or potential energy. Analysis: (a) The accompanying T–v and p–v diagrams show the two processes. Since the temperature at state 1, T1 400C, is greater than the saturation temperature corresponding to p1 10 bar: 179.9C, state 1 is located in the superheat region. (b) Since the piston is the only work mechanism W

3

p dV

1

2

p dV

1

3

p dV

2

The second integral vanishes because the volume is constant in Process 2–3. Dividing by the mass and noting that the pressure is constant for Process 1–2 W p 1v2 v1 2 m The specific volume at state 1 is found from Table A-4 using p1 10 bar and T1 400C: v1 0.3066 m3/kg. Also, u1 2957.3 kJ/kg. The specific volume at state 2 is the saturated vapor value at 10 bar: v2 0.1944 m3/kg, from Table A-3. Hence m3 105 N/m2 1 kJ W 110 bar210.1944 0.30662 a b ` ` ` 3 # ` m kg 1 bar 10 N m 112.2 kJ/kg The minus sign indicates that work is done on the water vapor by the piston. (c) An energy balance for the overall process reduces to m1u3 u1 2 Q W By rearranging Q W 1u3 u1 2

m m To evaluate the heat transfer requires u3, the specific internal energy at state 3. Since T3 is given and v3 v2, two independent intensive properties are known that together fix state 3. To find u3, first solve for the quality x3

v3 vf3 0.1944 1.0905 103 0.494 vg3 vf3 0.3928 1.0905 103

where vf3 and vg3 are from Table A-2 at 150C. Then

u3 uf3 x3 1ug3 uf3 2 631.68 0.49412559.5 631.982 1583.9 kJ/kg

where uf3 and ug3 are from Table A-2 at 150C. Substituting values into the energy balance Q 1583.9 2957.3 1112.22 1485.6 kJ/kg m The minus sign shows that energy is transferred out by heat transfer.

The next example illustrates the use of Interactive Thermodynamics: IT for solving problems. In this case, the software evaluates the property data, calculates the results, and displays the results graphically.

89

90

Chapter 3 Evaluating Properties

EXAMPLE

3.5

Plotting Thermodynamic Data Using Software

For the system of Example 3.1, plot the heat transfer, in kJ, and the mass of saturated vapor present, in kg, each versus pressure at state 2 ranging from 1 to 2 bar. Discuss the results. SOLUTION Known: A two-phase liquid–vapor mixture of water in a closed, rigid container is heated on a hot plate. The initial pressure and quality are known. The pressure at the final state ranges from 1 to 2 bar. Find: Plot the heat transfer and the mass of saturated vapor present, each versus pressure at the final state. Discuss. Schematic and Given Data: See Figure E3.1. Assumptions: 1. There is no work. 2. Kinetic and potential energy effects are negligible. 3. See Example 3.1 for other assumptions. Analysis:

The heat transfer is obtained from the energy balance. With assumptions 1 and 2, the energy balance reduces to 0

¢U ¢KE ¢PE Q W

or Q m1u2 u1 2 Selecting Water/Steam from the Properties menu and choosing SI Units from the Units menu, the IT program for obtaining the required data and making the plots is

❶

// Given data—State 1 p1 = 1 // bar x1 = 0.5 V = 0.5 // m3 // Evaluate property data—State 1 v1 = vsat_Px(“Water/Steam”,p1,x1) u1 = usat_Px(“Water/Steam”,p1,x1) // Calculate the mass m = V/v1 // Fix state 2 v2 = v1 p2 = 1.5 // bar // Evaluate property data—State 2 v2 = vsat_Px(“Water/Steam”,p2,x2) u2 = usat_Px(“Water/Steam”,p2,x2) // Calculate the mass of saturated vapor present mg2 = x2 * m // Determine the pressure for which the quality is unity v3 = v1 v3 = vsat_Px(“Water/Steam”,p3,1) // Energy balance to determine the heat transfer m * (u2 – u1) = Q – W W=0 Click the Solve button to obtain a solution for p2 1.5 bar. The program returns values of v1 0.8475 m3/kg and m 0.59 kg. Also, at p2 1.5 bar, the program gives mg2 0.4311 kg. These values agree with the values determined in Example 3.1. Now that the computer program has been verified, use the Explore button to vary pressure from 1 to 2 bar in steps of 0.1 bar. Then, use the Graph button to construct the required plots. The results are:

600

0.6

500

0.5

400

0.4 mg, kg

Q, kJ

3.3 Retrieving Thermodynamic Properties

300

0.3

200

0.2

100

0.1

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Pressure, bar

2

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Pressure, bar

2 Figure E3.5

We conclude from the first of these graphs that the heat transfer to the water varies directly with the pressure. The plot of mg shows that the mass of saturated vapor present also increases as the pressure increases. Both of these results are in accord with expectations for the process.

❶

Using the Browse button, the computer solution indicates that the pressure for which the quality becomes unity is 2.096 bar. Thus, for pressures ranging from 1 to 2 bar, all of the states are in the two-phase liquid–vapor region.

3.3.5 Evaluating Specific Heats cv and cp Several properties related to internal energy are important in thermodynamics. One of these is the property enthalpy introduced in Sec. 3.3.2. Two others, known as specific heats, are considered in this section. The specific heats are particularly useful for thermodynamic calculations involving the ideal gas model introduced in Sec. 3.5. The intensive properties cv and cp are defined for pure, simple compressible substances as partial derivatives of the functions u(T, v) and h(T, p), respectively cv

0u b 0T v

(3.8)

cp

0h b 0T p

(3.9)

where the subscripts v and p denote, respectively, the variables held fixed during differentiation. Values for cv and cp can be obtained via statistical mechanics using spectroscopic measurements. They also can be determined macroscopically through exacting property measurements. Since u and h can be expressed either on a unit mass basis or per mole, values of the specific heats can be similarly expressed. SI units are kJ/kg # K or kJ/kmol # K. The property k, called the specific heat ratio, is simply the ratio k

cp cv

(3.10)

The properties cv and cp are referred to as specific heats (or heat capacities) because under certain special conditions they relate the temperature change of a system to the amount of energy added by heat transfer. However, it is generally preferable to think of cv and cp in terms of their definitions, Eqs. 3.8 and 3.9, and not with reference to this limited interpretation involving heat transfer.

specific heats

91

92

Chapter 3 Evaluating Properties

In general, cv is a function of v and T (or p and T ), and cp depends on both p and T (or v and T ). Figure 3.9 shows how cp for water vapor varies as a function of temperature and pressure. The vapor phases of other substances exhibit similar behavior. Note that the figure gives the variation of cp with temperature in the limit as pressure tends to zero. In this limit, cp increases with increasing temperature, which is a characteristic exhibited by other gases as well. We will refer again to such zero-pressure values for cv and cp in Sec. 3.6. Specific heat data are available for common gases, liquids, and solids. Data for gases are introduced in Sec. 3.5 as a part of the discussion of the ideal gas model. Specific heat values for some common liquids and solids are introduced in Sec. 3.3.6 as a part of the discussion of the incompressible substance model. p = constant Saturated liquid

Special methods often can be used to evaluate properties of liquids and solids. These methods provide simple, yet accurate, approximations that do not require exact compilations like the compressed liquid tables for water, Tables A-5. Two such special methods are discussed next: approximations using saturated liquid data and the incompressible substance model.

p = constant

f T = constant

APPROXIMATIONS FOR LIQUIDS USING SATURATED LIQUID DATA

v(T, p) ≈ vf (T )

Approximate values for v, u, and h at liquid states can be obtained using saturated liquid data. To illustrate, refer to the compressed liquid tables. Tables A-5. These tables show

v

9

60

8

70

7

80 or

6

ted vap

90 100

60 70 80 90 100 MPa

Satura

v vf

3.3.6 Evaluating Properties of Liquids and Solids

cp , kJ/kg·K

T

5

50 Pa

40

M

4 30 25 20

15 10

3

5 2

1

2

0 Zero pressure limit

1.5 100

200

300

400

500

600

700

800

T, °C Figure 3.9

cp of water vapor as a function of temperature and pressure.

3.3 Retrieving Thermodynamic Properties

that the specific volume and specific internal energy change very little with pressure at a fixed temperature. Because the values of v and u vary only gradually as pressure changes at fixed temperature, the following approximations are reasonable for many engineering calculations: v1T, p2 vf 1T 2 u1T, p2 uf 1T 2

(3.11) (3.12)

That is, for liquids v and u may be evaluated at the saturated liquid state corresponding to the temperature at the given state. An approximate value of h at liquid states can be obtained by using Eqs. 3.11 and 3.12 in the definition h u pv; thus h1T, p2 uf 1T 2 pvf 1T 2

This can be expressed alternatively as h1T, p2 hf 1T 2 vf 1T 2 3p psat 1T 2 4

(3.13)

where psat denotes the saturation pressure at the given temperature. The derivation is left as an exercise. When the contribution of the underlined term of Eq. 3.13 is small, the specific enthalpy can be approximated by the saturated liquid value, as for v and u. That is h1T, p2 hf 1T 2

(3.14)

Although the approximations given here have been presented with reference to liquid water, they also provide plausible approximations for other substances when the only liquid data available are for saturated liquid states. In this text, compressed liquid data are presented only for water (Tables A-5). Also note that Interactive Thermodynamics: IT does not provide compressed liquid data for any substance, but uses Eqs. 3.11, 3.12, and 3.14 to return liquid values for v, u, and h, respectively. When greater accuracy is required than provided by these approximations, other data sources should be consulted for more complete property compilations for the substance under consideration. INCOMPRESSIBLE SUBSTANCE MODEL

As noted above, there are regions where the specific volume of liquid water varies little and the specific internal energy varies mainly with temperature. The same general behavior is exhibited by the liquid phases of other substances and by solids. The approximations of Eqs. 3.11–3.14 are based on these observations, as is the incompressible substance model under present consideration. To simplify evaluations involving liquids or solids, the specific volume (density) is often assumed to be constant and the specific internal energy assumed to vary only with temperature. A substance idealized in this way is called incompressible. Since the specific internal energy of a substance modeled as incompressible depends only on temperature, the specific heat cv is also a function of temperature alone cv 1T 2

du dT

1incompressible2

This is expressed as an ordinary derivative because u depends only on T.

(3.15)

incompressible substance model

93

94

Chapter 3 Evaluating Properties

Although the specific volume is constant and internal energy depends on temperature only, enthalpy varies with both pressure and temperature according to h 1T, p2 u 1T 2 pv

1incompressible2

(3.16)

For a substance modeled as incompressible, the specific heats cv and cp are equal. This is seen by differentiating Eq. 3.16 with respect to temperature while holding pressure fixed to obtain 0h du b 0T p dT The left side of this expression is cp by definition (Eq. 3.9), so using Eq. 3.15 on the right side gives c p cv

1incompressible2

(3.17)

Thus, for an incompressible substance it is unnecessary to distinguish between cp and cv, and both can be represented by the same symbol, c. Specific heats of some common liquids and solids are given versus temperature in Tables A-19. Over limited temperature intervals the variation of c with temperature can be small. In such instances, the specific heat c can be treated as constant without a serious loss of accuracy. Using Eqs. 3.15 and 3.16, the changes in specific internal energy and specific enthalpy between two states are given, respectively, by u2 u1

T2

T2

T1

c1T 2 dT

1incompressible2

(3.18)

c1T 2 dT v1 p2 p1 2

1incompressible2

(3.19)

h2 h1 u2 u1 v1 p2 p1 2

T1

If the specific heat c is taken as constant, Eqs. 3.18 and 3.19 become, respectively, u2 u1 c1T2 T1 2 h2 h1 c1T2 T1 2 v1p2 p1 2

(incompressible, constant c)

(3.20a) (3.20b)

In Eq. 3.20b, the underlined term is often small relative to the first term on the right side and then may be dropped.

3.4 Generalized Compressibility Chart

The object of the present section is to gain a better understanding of the relationship among pressure, specific volume, and temperature of gases. This is important not only as a basis for analyses involving gases but also for the discussions of the second part of the chapter, where the ideal gas model is introduced. The current presentation is conducted in terms of the compressibility factor and begins with the introduction of the universal gas constant.

3.4 Generalized Compressibility Chart

UNIVERSAL GAS CONSTANT, R

Let a gas be confined in a cylinder by a piston and the entire assembly held at a constant temperature. The piston can be moved to various positions so that a series of equilibrium states at constant temperature can be visited. Suppose the pressure and specific volume are measured at each state and the value of the ratio pv T 1v is volume per mole) determined. These ratios can then be plotted versus pressure at constant temperature. The results for several temperatures are sketched in Fig. 3.10. When the ratios are extrapolated to zero pressure, precisely the same limiting value is obtained for each curve. That is, lim pS0

pv R T

(3.21)

where R denotes the common limit for all temperatures. If this procedure were repeated for other gases, it would be found in every instance that the limit of the ratio pv T as p tends to zero at fixed temperature is the same, namely R. Since the same limiting value is exhibited by all gases, R is called the universal gas constant. Its value as determined experimentally is R 8.314 kJ/kmol # K

pv T

Measured data extrapolated to zero pressure

R T3 T4 p Figure 3.10

Sketch of pv T versus pressure for a gas at several specified values of temperature.

universal gas constant

Having introduced the universal gas constant, we turn next to the compressibility factor.

COMPRESSIBILITY FACTOR, Z

Z

pv RT

(3.23)

As illustrated by subsequent calculations, when values for p, v, R, and T are used in consistent units, Z is unitless. With v Mv (Eq. 1.11), where M is the atomic or molecular weight, the compressibility factor can be expressed alternatively as Z

pv RT

(3.24)

R

R M

(3.25)

where

R is a constant for the particular gas whose molecular weight is M. The unit for R is kJ/kg # K. Equation 3.21 can be expressed in terms of the compressibility factor as lim Z 1 pS0

(3.26)

T1 T2

(3.22)

The dimensionless ratio pv RT is called the compressibility factor and is denoted by Z. That is,

95

compressibility factor

96

Chapter 3 Evaluating Properties 1.5

35 K

100 K

50 K 60 K 200 K

1.0 300 K Z

0.5

100 p (atm)

200

Figure 3.11

Variation of the compressibility factor of hydrogen with pressure at constant temperature.

That is, the compressibility factor Z tends to unity as pressure tends to zero at fixed temperature. This can be illustrated by reference to Fig. 3.11, which shows Z for hydrogen plotted versus pressure at a number of different temperatures. In general, at states of a gas where pressure is small relative to the critical pressure, Z is approximately 1.

GENERALIZED COMPRESSIBILITY DATA, Z CHART

reduced pressure and temperature

Figure 3.11 gives the compressibility factor for hydrogen versus pressure at specified values of temperature. Similar charts have been prepared for other gases. When these charts are studied, they are found to be qualitatively similar. Further study shows that when the coordinates are suitably modified, the curves for several different gases coincide closely when plotted together on the same coordinate axes, and so quantitative similarity also can be achieved. This is referred to as the principle of corresponding states. In one such approach, the compressibility factor Z is plotted versus a dimensionless reduced pressure pR and reduced temperature TR, defined as

pR

generalized compressibility chart

p pc

and

TR

T Tc

(3.27)

where pc and Tc denote the critical pressure and temperature, respectively. This results in a generalized compressibility chart of the form Z f(pR, TR). Figure 3.12 shows experimental data for 10 different gases on a chart of this type. The solid lines denoting reduced isotherms represent the best curves fitted to the data. A generalized chart more suitable for problem solving than Fig. 3.12 is given in the Appendix as Figs. A-1, A-2, and A-3. In Fig. A-1, pR ranges from 0 to 1.0; in Fig. A-2, pR ranges from 0 to 10.0; and in Fig. A-3, pR ranges from 10.0 to 40.0. At any one temperature, the deviation of observed values from those of the generalized chart increases with pressure. However, for the 30 gases used in developing the chart, the deviation is at most

3.4 Generalized Compressibility Chart 1.1 TR = 2.00

1.0 0.9

TR = 1.50 0.8 TR = 1.30

pv Z = ––– RT

0.7 0.6

TR = 1.20 Legend Methane Isopentane Ethylene n-Heptane Ethane Nitrogen Propane Carbon dioxide n-Butane Water Average curve based on data on hydrocarbons

0.5 TR = 1.10 0.4 TR = 1.00

0.3 0.2 0.1

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

Reduced pressure pR Figure 3.12

Generalized compressibility chart for various gases.

on the order of 5% and for most ranges is much less.1 From Figs. A-1 and A-2 it can be seen that the value of Z tends to unity for all temperatures as pressure tends to zero, in accord with Eq. 3.26. Figure A-3 shows that Z also approaches unity for all pressures at very high temperatures. Values of specific volume are included on the generalized chart through the variable v¿R, called the pseudoreduced specific volume, defined by v¿R

v RTc pc

(3.28)

For correlation purposes, the pseudoreduced specific volume has been found to be preferable to the reduced specific volume vR v vc, where vc is the critical specific volume. Using the critical pressure and critical temperature of a substance of interest, the generalized chart can be entered with various pairs of the variables TR, pR, and v¿R: 1TR, pR 2, 1pR, v¿R 2, or 1TR, v¿R 2. Tables A-1 list the critical constants for several substances. The merit of the generalized chart for evaluating p, v, and T for gases is simplicity coupled with accuracy. However, the generalized compressibility chart should not be used as a substitute for p–v–T data for a given substance as provided by a table or computer software. The chart is mainly useful for obtaining reasonable estimates in the absence of more accurate data. The next example provides an illustration of the use of the generalized compressibility chart. 1

To determine Z for hydrogen, helium, and neon above a TR of 5, the reduced temperature and pressure should be calculated using TR T(Tc 8) and pR p( pc 8), where temperatures are in K and pressures are in atm.

pseudoreduced specific volume

97

98

Chapter 3 Evaluating Properties

EXAMPLE

3.6

Using the Generalized Compressibility Chart

A closed, rigid tank filled with water vapor, initially at 20 MPa, 520C, is cooled until its temperature reaches 400C. Using the compressibility chart, determine (a) the specific volume of the water vapor in m3/kg at the initial state. (b) the pressure in MPa at the final state. Compare the results of parts (a) and (b) with the values obtained from the superheated vapor table, Table A-4. SOLUTION Known: Water vapor is cooled at constant volume from 20 MPa, 520C to 400C. Find: Use the compressibility chart and the superheated vapor table to determine the specific volume and final pressure and compare the results. Schematic and Given Data: p1 = 20 MPa T1 = 520°C T2 = 400°C Closed, rigid tank

1.0 1

pv Z = ––– RT

Z1

2

TR = 1.3 TR = 1.2 Water vapor

v´R = 1.2 v´R = 1.1 TR = 1.05

Cooling

pR2

0.5

Block of ice

0.5 pR

1.0 Figure E3.6

Assumptions: 1. The water vapor is a closed system. 2. The initial and final states are at equilibrium. 3. The volume is constant. Analysis:

❶

❷

(a) From Table A-1, Tc 647.3 K and pc 22.09 MPa for water. Thus 793 20 TR1 1.23, pR1 0.91 647.3 22.09 With these values for the reduced temperature and reduced pressure, the value of Z obtained from Fig. A-1 is approximately 0.83. Since Z pvRT, the specific volume at state 1 can be determined as follows: RT1 RT1 v 1 Z1 0.83 p1 Mp1 N#m 8314 kmol # K 793 K ≤ 0.0152 m3/ kg 0.83 ± kg ° 6 N ¢ 18.02 20 10 2 kmol m The molecular weight of water is from Table A-1.

3.4 Generalized Compressibility Chart

Turning to Table A-4, the specific volume at the initial state is 0.01551 m3/kg. This is in good agreement with the compressibility chart value, as expected. (b) Since both mass and volume remain constant, the water vapor cools at constant specific volume, and thus at constant v¿R. Using the value for specific volume determined in part (a), the constant v¿R value is

v¿R

vpc RTc

m3 N b a22.09 106 2 b kg m 1.12 8314 N # m a b 1647.3 K2 18.02 kg # K

a0.0152

At state 2 673 1.04 647.3 Locating the point on the compressibility chart where v¿R 1.12 and TR 1.04, the corresponding value for pR is about 0.69. Accordingly TR2

p2 pc 1 pR2 2 122.09 MPa210.692 15.24 MPa Interpolating in the superheated vapor tables gives p2 15.16 MPa. As before, the compressibility chart value is in good agreement with the table value.

❶

Absolute temperature and absolute pressure must be used in evaluating the compressibility factor Z, the reduced temperature TR, and reduced pressure pR.

❷

Since Z is unitless, values for p, v, R, and T must be used in consistent units.

EQUATIONS OF STATE

Considering the curves of Figs. 3.11 and 3.12, it is reasonable to think that the variation with pressure and temperature of the compressibility factor for gases might be expressible as an equation, at least for certain intervals of p and T. Two expressions can be written that enjoy a theoretical basis. One gives the compressibility factor as an infinite series expansion in pressure: ˆ 1T 2 p3 # # # Z 1 Bˆ 1T 2 p Cˆ 1T 2 p2 D (3.29) ˆ , . . . depend on temperature only. The dots in Eq. 3.29 reprewhere the coefficients Bˆ , Cˆ , D sent higher-order terms. The other is a series form entirely analogous to Eq. 3.29 but expressed in terms of 1 v instead of p Z1

B1T 2 C1T 2 D1T 2

2 3 # # # v v v

(3.30)

Equations 3.29 and 3.30 are known as virial equations of state, and the coefficients ˆ , . . . and B, C, D, . . . are called virial coefficients. The word virial stems from the Bˆ , Cˆ , D Latin word for force. In the present usage it is force interactions among molecules that are intended. The virial expansions can be derived by the methods of statistical mechanics, and physical significance can be attributed to the coefficients: B v accounts for two-molecule interactions, C v2 accounts for three-molecule interactions, etc. In principle, the virial coefficients can be calculated by using expressions from statistical mechanics derived from consideration of the force fields around the molecules of a gas. The virial coefficients also can be determined from experimental p–v–T data. The virial expansions are used in Sec. 11.1 as a point of departure for the further study of analytical representations of the p–v–T relationship of gases known generically as equations of state.

virial equations

99

100

Chapter 3 Evaluating Properties

The virial expansions and the physical significance attributed to the terms making up the expansions can be used to clarify the nature of gas behavior in the limit as pressure tends to zero at fixed temperature. From Eq. 3.29 it is seen that if pressure decreases at fixed temperature, the terms Bˆ p, Cˆ p2, etc. accounting for various molecular interactions tend to decrease, suggesting that the force interactions become weaker under these circumstances. In the limit as pressure approaches zero, these terms vanish, and the equation reduces to Z 1 in accordance with Eq. 3.26. Similarly, since volume increases when the pressure decreases at fixed temperature, the terms B v, C v2, etc. of Eq. 3.30 also vanish in the limit, giving Z 1 when the force interactions between molecules are no longer significant.

EVALUATING PROPERTIES USING THE IDEAL GAS MODEL As discussed in Sec. 3.4, at states where the pressure p is small relative to the critical pressure pc (low pR) and/or the temperature T is large relative to the critical temperature Tc (high TR), the compressibility factor, Z pvRT, is approximately 1. At such states, we can assume with reasonable accuracy that Z 1, or ideal gas equation of state

pv RT

(3.32)

Known as the ideal gas equation of state, Eq. 3.32 underlies the second part of this chapter dealing with the ideal gas model. Alternative forms of the same basic relationship among pressure, specific volume, and temperature are obtained as follows. With v Vm, Eq. 3.32 can be expressed as pV mRT

(3.33)

In addition, since v vM and R R M, where M is the atomic or molecular weight, Eq. 3.32 can be expressed as or, with v V n, as

pv RT

(3.34)

pV nRT

(3.35)

3.5 Ideal Gas Model

For any gas whose equation of state is given exactly by pv RT, the specific internal energy depends on temperature only. This conclusion is demonstrated formally in Sec. 11.4. It is also supported by experimental observations, beginning with the work of Joule, who showed in 1843 that the internal energy of air at low density depends primarily on temperature. Further motivation from the microscopic viewpoint is provided shortly. The specific enthalpy of a gas described by pv RT also depends on temperature only, as can be shown by combining the definition of enthalpy, h u pv, with u u(T ) and the ideal gas equation of state to obtain h u(T ) RT. Taken together, these specifications constitute the ideal gas model, summarized as follows

ideal gas model

pv RT u u1T 2 h h1T 2 u1T 2 RT

(3.32) (3.36) (3.37)

3.5 Ideal Gas Model

The specific internal energy and enthalpy of gases generally depend on two independent properties, not just temperature as presumed by the ideal gas model. Moreover, the ideal gas equation of state does not provide an acceptable approximation at all states. Accordingly, whether the ideal gas model is used depends on the error acceptable in a given calculation. Still, gases often do approach ideal gas behavior, and a particularly simplified description is obtained with the ideal gas model. To verify that a gas can be modeled as an ideal gas, the states of interest can be located on a compressibility chart to determine how well Z 1 is satisfied. As shown in subsequent discussions, other tabular or graphical property data can also be used to determine the suitability of the ideal gas model. A picture of the dependence of the internal energy of gases on temperature at low density can be obtained with reference to the discussion of the virial equations in Sec. 3.4. As p S 0 (v S ), the force interactions between molecules of a gas become weaker, and the virial expansions approach Z 1 in the limit. The study of gases from the microscopic point of view shows that the dependence of the internal energy of a gas on pressure, or specific volume, at a specified temperature arises primarily because of molecular interactions. Accordingly, as the density of a gas decreases at fixed temperature, there comes a point where the effects of intermolecular forces are minimal. The internal energy is then determined principally by the temperature. From the microscopic point of view, the ideal gas model adheres to several idealizations: The gas consists of molecules that are in random motion and obey the laws of mechanics; the total number of molecules is large, but the volume of the molecules is a negligibly small fraction of the volume occupied by the gas; and no appreciable forces act on the molecules except during collisions. The next example illustrates the use of the ideal gas equation of state and reinforces the use of property diagrams to locate principal states during processes. MICROSCOPIC INTERPRETATION.

EXAMPLE

3.7

101

METHODOLOGY UPDATE

To expedite the solutions of many subsequent examples and end-ofchapter problems involving air, oxygen (O2), nitrogen (N2), carbon dioxide (CO2), carbon monoxide (CO), hydrogen (H2), and other common gases, we indicate in the problem statements that the ideal gas model should be used. If not indicated explicity, the suitability of the ideal gas model should be checked using the Z chart or other data.

Air as an Ideal Gas Undergoing a Cycle

One pound of air undergoes a thermodynamic cycle consisting of three processes. Process 1–2: constant specific volume Process 2–3: constant-temperature expansion Process 3–1: constant-pressure compression At state 1, the temperature is 300K, and the pressure is 1 bar. At state 2, the pressure is 2 bars. Employing the ideal gas equation of state, (a) sketch the cycle on p–v coordinates. (b) determine the temperature at state 2, in K; (c) determine the specific volume at state 3, in m3/kg. SOLUTION Known: Air executes a thermodynamic cycle consisting of three processes: Process 1–2, v constant; Process 2–3, T constant; Process 3–1, p constant. Values are given for T1, p1, and p2. Find: Sketch the cycle on p–v coordinates and determine T2 and v3.

102

Chapter 3 Evaluating Properties

Schematic and Given Data: p

2 p2 = 2 bars

Assumptions:

❶

1. The air is a closed system.

v=C

T=C

2. The air behaves as an ideal gas.

1 p1 = 1 bar

3 p=C 600 K 300 K v

Figure E3.7

Analysis: (a) The cycle is shown on p–v coordinates in the accompanying figure. Note that since p RTv and temperature is constant, the variation of p with v for the process from 2 to 3 is nonlinear. (b) Using pv RT, the temperature at state 2 is T2 p2v2 R To obtain the specific volume v2 required by this relationship, note that v2 v1, so v2 RT1 p1 Combining these two results gives T2

❷

p2 2 bars T a b 1300 K2 600 K p1 1 1 bar

(c) Since pv RT, the specific volume at state 3 is Noting that T3 T2, p3 p1, and R RM v3

v3 RT3 p3

RT2 Mp1

±

kJ kmol # K 1 bar 103 N # M 600 K ≤a ba 5 b a b kg 1 bar 1 kJ 10 N/m2 28.97 kmol

8.314

1.72 m3/kg where the molecular weight of air is from Table A-1.

❶

Table A-1 gives pc 37.3 bars, Tc 133 K for air. Therefore pR2 .053, TR2 4.51. Referring to A-1 the value of the compressibility factor at this state is Z 1. The same conclusion results when states 1 and 3 are checked. Accordingly, pv RT adequately describes the p–v–T relation for gas at these states.

❷ Carefully note that the equation of state pv RT requires the use of absolute temperature T and absolute pressure p.

3.6 Internal Energy, Enthalpy, and Specific Heats of Ideal Gases

3.6 Internal Energy, Enthalpy, and Specific Heats of Ideal Gases

For a gas obeying the ideal gas model, specific internal energy depends only on temperature. Hence, the specific heat cv, defined by Eq. 3.8, is also a function of temperature alone. That is, cv 1T 2

1ideal gas2

du dT

(3.38)

This is expressed as an ordinary derivative because u depends only on T. By separating variables in Eq. 3.38 du cv 1T 2 dT

(3.39)

On integration u1T2 2 u1T1 2

T2

T1

cv 1T 2 dT

1ideal gas2

(3.40)

Similarly, for a gas obeying the ideal gas model, the specific enthalpy depends only on temperature, so the specific heat cp, defined by Eq. 3.9, is also a function of temperature alone. That is cp 1T 2

1ideal gas2

dh dT

(3.41)

Separating variables in Eq. 3.41 dh cp 1T 2 dT

(3.42)

On integration h1T2 2 h1T1 2

T2

T1

cp 1T 2 dT

1ideal gas2

(3.43)

An important relationship between the ideal gas specific heats can be developed by differentiating Eq. 3.37 with respect to temperature dh du

R dT dT and introducing Eqs. 3.38 and 3.41 to obtain cp 1T 2 cv 1T 2 R

1ideal gas2

(3.44)

1ideal gas2

(3.45)

On a molar basis, this is written as cp 1T 2 cv 1T 2 R

Although each of the two ideal gas specific heats is a function of temperature, Eqs. 3.44 and 3.45 show that the specific heats differ by just a constant: the gas constant. Knowledge of either specific heat for a particular gas allows the other to be calculated by using only the gas constant. The above equations also show that cp cv and cp 7 cv, respectively.

103

104

Chapter 3 Evaluating Properties

For an ideal gas, the specific heat ratio, k, is also a function of temperature only k

cp 1T 2

1ideal gas2

cv 1T 2

(3.46)

Since cp cv, it follows that k 1. Combining Eqs. 3.44 and 3.46 results in cp 1T 2

kR k1

(3.47a)

(ideal gas)

R cv 1T 2 k1

(3.47b)

Similar expressions can be written for the specific heats on a molar basis, with R being replaced by R. The foregoing expressions require the ideal gas specific heats as functions of temperature. These functions are available for gases of practical interest in various forms, including graphs, tables, and equations. Figure 3.13 illustrates the variation of cp (molar basis) with temperature for a number of common gases. In the range of temperature shown, cp increases with temperature for all gases, except for the monatonic gases Ar, Ne, and He. For these, cp is closely constant at the value predicted by kinetic theory: cp 52R. Tabular specific heat data for selected gases are presented versus temperature in Tables A-20. Specific heats are also available in equation form. Several alternative forms of such equations are found in the engineering literature. An equation that is relatively easy to integrate is the polynomial form USING SPECIFIC HEAT FUNCTIONS.

cp R

a bT gT 2 dT 3 eT 4

(3.48)

Values of the constants , , , , and are listed in Tables A-21 for several gases in the temperature range 300 to 1000 K.

CO2 7

H2O

6

cp 5

O2 CO

R 4

H2

Air

3 Ar, Ne, He 0

1000

2000

3000

Temperature, K Figure 3.13 Variation of cp R with temperature for a number of gases modeled as ideal gases.

3.7 Evaluating ⌬u and ⌬h Using Ideal Gas Tables, Software, and Constant Specific Heats

for example. . . to illustrate the use of Eq. 3.48, let us evaluate the change in specific enthalpy, in kJ/kg, of air modeled as an ideal gas from a state where T1 400 K to a state where T2 900 K. Inserting the expression for cp(T) given by Eq. 3.48 into Eq. 3.43 and integrating with respect to temperature

h2 h1

R M

T2

T1

1a bT gT 2 dT 3 eT 4 2 d T

b g R d e c a1T2 T1 2 1T 22 T 21 2 1T 32 T 31 2 1T 42 T 41 2 1T 52 T 51 2 d M 2 3 4 5 where the molecular weight M has been introduced to obtain the result on a unit mass basis. With values for the constants from Table A-21

h2 h1

8.314 1.337 e 3.6531900 4002 19002 2 14002 2 28.97 21102 3 3.294 1.913 19002 3 14002 3 19002 4 14002 4 6 41102 9 31102 0.2763

19002 5 14002 5 f 531.69 kJ/kg 51102 12

Specific heat functions cv(T) and cp(T) are also available in IT: Interactive Thermodynamics in the PROPERTIES menu. These functions can be integrated using the integral function of the program to calculate u and h, respectively. for example. . . let us repeat the immediately preceding example using IT. For air, the IT code is cp = cp_T (“Air”,T) delh = Integral(cp,T)

Pushing SOLVE and sweeping T from 400 K to 900 K, the change in specific enthalpy is delh 531.7 kJ/kg, which agrees closely with the value obtained by integrating the specific heat function from Table A-21, as illustrated above. The source of ideal gas specific heat data is experiment. Specific heats can be determined macroscopically from painstaking property measurements. In the limit as pressure tends to zero, the properties of a gas tend to merge into those of its ideal gas model, so macroscopically determined specific heats of a gas extrapolated to very low pressures may be called either zero-pressure specific heats or ideal gas specific heats. Although zeropressure specific heats can be obtained by extrapolating macroscopically determined experimental data, this is rarely done nowadays because ideal gas specific heats can readily be calculated with expressions from statistical mechanics by using spectral data, which can be obtained experimentally with precision. The determination of ideal gas specific heats is one of the important areas where the microscopic approach contributes significantly to the application of thermodynamics.

3.7 Evaluating ⌬u and ⌬h Using Ideal Gas Tables, Software, and Constant Specific Heats

Although changes in specific enthalpy and specific internal energy can be obtained by integrating specific heat expressions, as illustrated above, such evaluations are more easily conducted using the means considered in the present section.

105

106

Chapter 3 Evaluating Properties

USING IDEAL GAS TABLES

For a number of common gases, evaluations of specific internal energy and enthalpy changes are facilitated by the use of the ideal gas tables, Tables A-22 and A-23, which give u and h (or u and h) versus temperature. To obtain enthalpy versus temperature, write Eq. 3.43 as h1T 2

T

Tref

cp 1T 2 dT h1Tref 2

where Tref is an arbitrary reference temperature and h(Tref) is an arbitrary value for enthalpy at the reference temperature. Tables A-22 and A-23 are based on the selection h 0 at Tref 0 K. Accordingly, a tabulation of enthalpy versus temperature is developed through the integral2 h1T 2

T

cp 1T 2 dT

(3.49)

Tabulations of internal energy versus temperature are obtained from the tabulated enthalpy values by using u h RT. For air as an ideal gas, h and u are given in Table A-22 with units of kJ/kg. Values of molar specific enthalpy h and internal energy u for several other common gases modeled as ideal gases are given in Tables A-23 with units of kJ/kmol. Quantities other than specific internal energy and enthalpy appearing in these tables are introduced in Chap. 6 and should be ignored at present. Tables A-22 and A-23 are convenient for evaluations involving ideal gases, not only because the variation of the specific heats with temperature is accounted for automatically but also because the tables are easy to use. for example. . . let us use Table A-22 to evaluate the change in specific enthalpy, in kJ/kg, for air from a state where T1 400 K to a state where T2 900 K, and compare the result with the value obtained by integrating cp 1T 2 in the example following Eq. 3.48. At the respective temperatures, the ideal gas table for air, Table A-22, gives kJ kJ h1 400.98 , h2 932.93 kg kg Then, h2 h1 531.95 kJ/kg, which agrees with the value obtained by integration in Sec. 3.6. USING COMPUTER SOFTWARE

Interactive Thermodynamics: IT also provides values of the specific internal energy and enthalpy for a wide range of gases modeled as ideal gases. Let us consider the use of IT, first for air, and then for other gases. AIR. For air, IT uses the same reference state and reference value as in Table A-22, and the values computed by IT agree closely with table data. for example. . . let us reconsider the above example for air and use IT to evaluate the change in specific enthalpy from a state where T1 400 K to a state where T2 900 K. Selecting Air from the Properties menu, the following code would be used by IT to determine h (delh), in kJ/kg

h1 = h2 = T1 = T2 = delh 2

h_T(“Air”,T1) h_T(“Air”,T2) 400 // K 900 // K = h2 – h1

The simple specific heat variation given by Eq. 3.48 is valid only for a limited temperature range, so tabular enthalpy values are calculated from Eq. 3.49 using other expressions that enable the integral to be evaluated accurately over wider ranges of temperature.

3.7 Evaluating ⌬u and ⌬h Using Ideal Gas Tables, Software, and Constant Specific Heats

Choosing K for the temperature unit and kg for the amount under the Units menu, the results returned by IT are h1 400.8, h2 932.5, and h 531.7 kJ/kg, respectively. As expected, these values agree closely with those obtained previously. IT also provides data for each of the gases included in Table A-23. For these gases, the values of specific internal energy u and enthalpy h returned by IT are determined relative to different reference states and reference values than used in Table A-23. Such reference state and reference value choices equip IT for use in combustion applications; see Sec. 13.2.1 for further discussion. Consequently the values of u and h returned by IT for the gases of Table A-23 differ from those obtained directly from the table. Still, the property differences between two states remain the same, for datums cancel when differences are calculated. OTHER GASES.

for example. . . let us use IT to evaluate the change in specific enthalpy, in kJ/kmol, for carbon dioxide (CO2) as an ideal gas from a state where T1 300 K to a state where T2 500 K. Selecting CO2 from the Properties menu, the following code would be used by IT: h1 = h2 = T1 = T2 = delh

h_T(“CO2”,T1) h_T(“CO2”,T2) 300 // K 500 // K = h2 – h1

Choosing K for the temperature unit and moles for the amount under the Units menu, the results returned by IT are h1 3.935 105, h2 3.852 105, and ¢h 8238 kJ/kmol, respectively. The large negative values for h1 and h2 are a consequence of the reference state and reference value used by IT for CO2. Although these values for specific enthalpy at states 1 and 2 differ from the corresponding values read from Table A-23: h1 9,431 and h2 17,678, which give ¢h 8247 kJ/kmol, the difference in specific enthalpy determined with each set of data agree closely. ASSUMING CONSTANT SPECIFIC HEATS

When the specific heats are taken as constants, Eqs. 3.40 and 3.43 reduce, respectively, to u 1T2 2 u 1T1 2 cv 1T2 T1 2 h 1T2 2 h 1T1 2 cp 1T2 T1 2

(3.50) (3.51)

Equations 3.50 and 3.51 are often used for thermodynamic analyses involving ideal gases because they enable simple closed-form equations to be developed for many processes. The constant values of cv and cp in Eqs. 3.50 and 3.51 are, strictly speaking, mean values calculated as follows

cv

T2

T1

cv 1T 2 dT

T2 T1

,

cp

T2

T1

cp 1T 2 dT

T2 T1

However, when the variation of cv or cp over a given temperature interval is slight, little error is normally introduced by taking the specific heat required by Eq. 3.50 or 3.51 as the arithmetic average of the specific heat values at the two end temperatures. Alternatively, the specific heat at the average temperature over the interval can be used. These methods are particularly convenient when tabular specific heat data are available, as in Tables A-20, for then the constant specific heat values often can be determined by inspection. The next example illustrates the use of the ideal gas tables, together with the closed system energy balance.

107

Chapter 3 Evaluating Properties

108

EXAMPLE

3.8

Using the Energy Balance and Ideal Gas Tables

A piston–cylinder assembly contains 0.9 kg of air at a temperature of 300K and a pressure of 1 bar. The air is compressed to a state where the temperature is 470K and the pressure is 6 bars. During the compression, there is a heat transfer from the air to the surroundings equal to 20 kJ. Using the ideal gas model for air, determine the work during the process, in kJ. SOLUTION Known: 0.9 kg of air are compressed between two specified states while there is heat transfer from the air of a known amount. Find: Determine the work, in kJ. Schematic and Given Data: p

2

p2 = 6 bars 0.9 kg of air

❶ T2 = 470 1 p1 = 1 bar

T1 = 300K v

Figure E3.8

Assumptions: 1. The air is a closed system. 2. The initial and final states are equilibrium states. There is no change in kinetic or potential energy.

❷

3. The air is modeled as an ideal gas. Analysis: An energy balance for the closed system is 0

¢KE ¢PE ¢U Q W where the kinetic and potential energy terms vanish by assumption 2. Solving for W W Q ¢U Q m 1u2 u1 2

❸

From the problem statement, Q 20 kJ. Also, from Table A-22 at T1 300 K, u1 214.07 kJ/ kg, and at T2 470K, u2 337.32 kJ/kg. Accordingly W 20 10.92 1337.32 214.072 130.9 kJ

The minus sign indicates that work is done on the system in the process.

❶

Although the initial and final states are assumed to be equilibrium states, the intervening states are not necessarily equilibrium states, so the process has been indicated on the accompanying p–v diagram by a dashed line. This dashed line does not define a “path” for the process.

❷

Table A-1 gives pc 37.7 bars, Tc 133K for air. Therefore, at state 1, pR1 0.03, TR1 2.26, and at state 2, pR2 0.16, TR2 3.51. Referring to Fig. A-1, we conclude that at these states Z 1, as assumed in the solution.

❸

In principle, the work could be evaluated through p dV, but because the variation of pressure at the piston face with volume is not known, the integration cannot be performed without more information.

3.7 Evaluating ⌬u and ⌬h Using Ideal Gas Tables, Software, and Constant Specific Heats

109

The following example illustrates the use of the closed system energy balance, together with the ideal gas model and the assumption of constant specific heats.

EXAMPLE

3.9

Using the Energy Balance and Constant Specific Heats

Two tanks are connected by a valve. One tank contains 2 kg of carbon monoxide gas at 77C and 0.7 bar. The other tank holds 8 kg of the same gas at 27C and 1.2 bar. The valve is opened and the gases are allowed to mix while receiving energy by heat transfer from the surroundings. The final equilibrium temperature is 42C. Using the ideal gas model, determine (a) the final equilibrium pressure, in bar (b) the heat transfer for the process, in kJ. SOLUTION Known: Two tanks containing different amounts of carbon monoxide gas at initially different states are connected by a valve. The valve is opened and the gas allowed to mix while receiving a certain amount of energy by heat transfer. The final equilibrium temperature is known. Find: Determine the final pressure and the heat transfer for the process. Schematic and Given Data: Assumptions: 1. The total amount of carbon monoxide gas is a closed system.

❶

2. The gas is modeled as an ideal gas with constant cv.

Carbon monoxide

3. The gas initially in each tank is in equilibrium. The final state is an equilibrium state.

2 kg, 77°C, 0.7 bar

Carbon monoxide Valve

8 kg, 27°C, 1.2 bar

4. No energy is transferred to, or from, the gas by work. Tank 1

5. There is no change in kinetic or potential energy.

Tank 2

Figure E3.9

Analysis: (a) The final equilibrium pressure pf can be determined from the ideal gas equation of state pf

mRTf V

where m is the sum of the initial amounts of mass present in the two tanks, V is the total volume of the two tanks, and Tf is the final equilibrium temperature. Thus pf

1m1 m2 2RTf V1 V2

Denoting the initial temperature and pressure in tank 1 as T1 and p1, respectively, V1 m1RT1p1. Similarly, if the initial temperature and pressure in tank 2 are T2 and p2, V2 m2RT2p2. Thus, the final pressure is pf

Inserting values pf

a

1m1 m2 2RTf

m1RT1 m2 RT2 b a b p1 p2

a

1m1 m2 2Tf

m1T1 m2 T2 b a b p1 p2

110 kg2 1315 K2 1.05 bar 12 kg2 1350 K2 18 kg21300 K2

0.7 bar 1.2 bar

110

Chapter 3 Evaluating Properties

(b) The heat transfer can be found from an energy balance, which reduces with assumptions 4 and 5 to give ¢U Q W

or Q Uf Ui Ui is the initial internal energy, given by

Ui m1u 1T1 2 m2u 1T2 2

where T1 and T2 are the initial temperatures of the CO in tanks 1 and 2, respectively. The final internal energy is Uf Uf 1m1 m2 2 u 1Tf 2

Introducing these expressions for internal energy, the energy balance becomes Q m1 3u 1Tf 2 u 1T1 2 4 m2 3u 1Tf 2 u 1T2 2 4 Since the specific heat cv is constant (assumption 2) Q m1cv 1Tf T1 2 m2cv 1Tf T2 2 Evaluating cv as the mean of the values listed in Table A-20 at 300 K and 350 K, cv 0.745 kJ/kg # K. Hence Q 12 kg2 a0.745

❷

kJ kJ b 1315 K 350 K2 18 kg2 a0.745 b 1315 K 300 K2 37.25 kJ kg # K kg # K

The plus sign indicates that the heat transfer is into the system.

❶

By referring to a generalized compressibility chart, it can be verified that the ideal gas equation of state is appropriate for CO in this range of temperature and pressure. Since the specific heat cv of CO varies little over the temperature interval from 300 to 350 K (Table A-20), it can be treated as constant with acceptable accuracy.

❷

As an exercise, evaluate Q using specific internal energy values from the ideal gas table for CO, Table A-23. Observe that specific internal energy is given in Table A-23 with units of kJ/kmol.

The next example illustrates the use of software for problem solving with the ideal gas model. The results obtained are compared with those determined assuming the specific heat cv is constant.

EXAMPLE

3.10

Using the Energy Balance and Software

One kmol of carbon dioxide gas (CO2) in a piston–cylinder assembly undergoes a constant-pressure process at 1 bar from T1 300 K to T2. Plot the heat transfer to the gas, in kJ, versus T2 ranging from 300 to 1500 K. Assume the ideal gas model, and determine the specific internal energy change of the gas using (a) u data from IT. (b) a constant cv evaluated at T1 from IT. SOLUTION Known: One kmol of CO2 undergoes a constant-pressure process in a piston–cylinder assembly. The initial temperature, T1, and the pressure are known. Find: Plot the heat transfer versus the final temperature, T2. Use the ideal gas model and evaluate ¢u using (a) u data from IT, (b) constant cv evaluated at T1 from IT.

3.7 Evaluating ⌬u and ⌬h Using Ideal Gas Tables, Software, and Constant Specific Heats

111

Schematic and Given Data: Assumptions: Carbon dioxide

T1 = 300 K n = 1 kmol p = 1 bar

1. The carbon dioxide is a closed system. 2. The process occurs at constant pressure. 3. The carbon dioxide behaves as an ideal gas. 4. Kinetic and potential energy effects are negligible.

Figure E3.10a

Analysis: The heat transfer is found using the closed system energy balance, which reduces to U2 U1 Q W Using Eq. 2.17 at constant pressure (assumption 2) W p 1V2 V1 2 pn 1v2 v1 2

Then, with ¢U n1u2 u1 2, the energy balance becomes

n 1u2 u1 2 Q pn 1v2 v1 2

Solving for Q Q n 3 1u2 u1 2 p 1v2 v1 2 4

❶ With pv RT, this becomes

Q n 3 1u2 u1 2 R1T2 T1 2 4

The object is to plot Q versus T2 for each of the following cases: (a) values for u1 and u2 at T1 and T2, respectively, are provided by IT, (b) Eq. 3.50 is used on a molar basis, namely u2 u1 cv 1T2 T1 2 where the value of cv is evaluated at T1 using IT. The IT program follows, where Rbar denotes R, cvb denotes cv, and ubar1 and ubar2 denote u1 and u2, respectively. // Using the Units menu, select “mole” for the substance amount. // Given Data T1 = 300 // K T2 = 1500 // K n = 1 // kmol Rbar = 8.314 // kJ/kmol # K // (a) Obtain molar specific internal energy data using IT. ubar1 = u_T(“CO2”, T1) ubar2 = u_T(“CO2”, T2) Qa = n*(ubar2 – ubar1) + n*Rbar*(T2 – T1) // (b) Use Eq. 3.50 with cv evaluated at T1. cvb = cv_T(“CO2”, T1) Qb = n*cvb*(T2 – T1) + n*Rbar*(T2 – T1) Use the Solve button to obtain the solution for the sample case of T2 1500 K. For part (a), the program returns Qa 6.16 104 kJ. The solution can be checked using CO2 data from Table A-23, as follows: Qa n 3 1u2 u1 2 R1T2 T1 2 4

11 kmol2 3 158,606 69392 kJ/kmol 18.314 kJ/kmol # K211500 3002 K4 61,644 kJ

Chapter 3 Evaluating Properties

112

Thus, the result obtained using CO2 data from Table A-23 is in close agreement with the computer solution for the sample case. For part (b), IT returns cv 28.95 kJ/kmol # K at T1, giving Qb 4.472 104 kJ when T2 1500 K. This value agrees with the result obtained using the specific heat cv at 300 K from Table A-20, as can be verified. Now that the computer program has been verified, use the Explore button to vary T2 from 300 to 1500 K in steps of 10. Construct the following graph using the Graph button: 70,000 60,000

internal energy data cv at T1

Q, kJ

50,000 40,000 30,000 20,000 10,000 0 300

❷

500

700

900 T2, K

1100

1300

1500 Figure E3.10b

As expected, the heat transfer is seen to increase as the final temperature increases. From the plots, we also see that using constant cv evaluated at T1 for calculating ¢u, and hence Q, can lead to considerable error when compared to using u data. The two solutions compare favorably up to about 500 K, but differ by approximately 27% when heating to a temperature of 1500 K.

❶

Alternatively, this expression for Q can be written as Q n 3 1u2 pv2 2 1u1 pv1 2 4

Introducing h u pv, the expression for Q becomes

Q n 1h2 h1 2

❷

It is left as an exercise to verify that more accurate results in part (b) would be obtained using cv evaluated at Taverage (T1 T2)2.

3.8 Polytropic Process of an Ideal Gas

Recall that a polytropic process of a closed system is described by a pressure–volume relationship of the form pV n constant (3.52) where n is a constant (Sec. 2.2). For a polytropic process between two states p1V n1 p2V n2 or p2 V1 n a b p1 V2

(3.53)

The exponent n may take on any value from to , depending on the particular process. When n 0, the process is an isobaric (constant-pressure) process, and when n , the process is an isometric (constant-volume) process.

3.8 Polytropic Process of an Ideal Gas

113

For a polytropic process

2

p dV

1

p2V2 p1V1 1n

1n 12

(3.54)

for any exponent n except n 1. When n 1,

2

p dV p1V1 ln

1

V2 V1

1n 12

(3.55)

Example 2.1 provides the details of these integrations. Equations 3.52 through 3.55 apply to any gas (or liquid) undergoing a polytropic process. When the additional idealization of ideal gas behavior is appropriate, further relations can be derived. Thus, when the ideal gas equation of state is introduced into Eqs. 3.53, 3.54, and 3.55, the following expressions are obtained, respectively p2 1n12n T2 V1 n1 a b a b p1 T1 V2 2 mR 1T2 T1 2 p dV 1n 1 2 V2 p dV mRT ln V1 1

1ideal gas2

(3.56)

1ideal gas, n 12

(3.57)

1ideal gas, n 12

(3.58)

For an ideal gas, the case n 1 corresponds to an isothermal (constant-temperature) process, as can readily be verified. In addition, when the specific heats are constant, the value of the exponent n corresponding to an adiabatic polytropic process of an ideal gas is the specific heat ratio k (see discussion of Eq. 6.47). Example 3.11 illustrates the use of the closed system energy balance for a system consisting of an ideal gas undergoing a polytropic process. EXAMPLE

3.11

Polytropic Process of Air as an Ideal Gas

Air undergoes a polytropic compression in a piston–cylinder assembly from p1 1 bar, T1 22C to p2 5 bars. Employing the ideal gas model, determine the work and heat transfer per unit mass, in kJ/kg, if n 1.3. SOLUTION Known: Air undergoes a polytropic compression process from a given initial state to a specified final pressure. Find: Determine the work and heat transfer, each in kJ/kg. Schematic and Given Data: p 5 bars

Assumptions:

2

1. The air is a closed system.

❶

Air p1 = 1 bar T1 = 22C p2 = 5 bars

pv1.3 = constant

1 bar

1

2. The air behaves as an ideal gas. 3. The compression is polytropic with n 1.3. 4. There is no change in kinetic or potential energy.

v

Figure E3.11

114

Chapter 3 Evaluating Properties

Analysis: The work can be evaluated in this case from the expression W

2

p dV

1

With Eq. 3.57 R1T2 T1 2 W m 1n The temperature at the final state, T2, is required. This can be evaluated from Eq. 3.56 T2 T1 a

p2 1n12n 5 11.3121.3 b 295 K a b 428°K p1 1

The work is then R1T2 T1 2 W 8.314 kJ 428°K 295°K a ba b 127.2 kJ/kg m 1n 28.97 kg # K 1 1.3 The heat transfer can be evaluated from an energy balance. Thus Q W

1u2 u1 2 127.2 1306.53 210.492 31.16 kJ/kg m m where the specific internal energy values are obtained from Table A-22.

❶

The states visited in the polytropic compression process are shown by the curve on the accompanying p–v diagram. The magnitude of the work per unit of mass is represented by the shaded area below the curve.

Chapter Summary and Study Guide

In this chapter, we have considered property relations for a broad range of substances in tabular, graphical, and equation form. Primary emphasis has been placed on the use of tabular data, but computer retrieval also has been considered. A key aspect of thermodynamic analysis is fixing states. This is guided by the state principle for pure, simple compressible systems, which indicates that the intensive state is fixed by the values of two independent, intensive properties. Another important aspect of thermodynamic analysis is locating principal states of processes on appropriate diagrams: p–v, T–v, and p–T diagrams. The skills of fixing states and using property diagrams are particularly important when solving problems involving the energy balance. The ideal gas model is introduced in the second part of this chapter, using the compressibility factor as a point of departure. This arrangement emphasizes the limitations of the ideal gas model. When it is appropriate to use the ideal gas model, we stress that specific heats generally vary with temperature, and feature the use of the ideal gas tables in problem solving.

The following checklist provides a study guide for this chapter. When your study of the text and end-of-chapter exercises has been completed you should be able to write out the meanings of the terms listed in the margins

throughout the chapter and understand each of the related concepts. The subset of key concepts listed below is particularly important in subsequent chapters. retrieve property data from Tables A-1 through A-23,

using the state principle to fix states and linear interpolation when required. sketch T–v, p–v, and p–T diagrams, and locate principal

states on such diagrams. apply the closed system energy balance with property

data. evaluate the properties of two-phase, liquid–vapor

mixtures using Eqs. 3.1, 3.2, 3.6, and 3.7. estimate the properties of liquids using Eqs. 3.11, 3.12,

and 3.14.

Problems: Developing Engineering Skills apply the incompressible substance model. use the generalized compressibility chart to relate p–v–T

data of gases.

115

is warranted, and appropriately using ideal gas table data or constant specific heat data to determine u and h.

apply the ideal gas model for thermodynamic analysis,

including determining when use of the ideal gas model

Key Engineering Concepts

state principle p. 69 simple compressible system p. 69 p–v–T surface p. 70 phase diagram p. 72

saturation temperature p. 73 saturation pressure p. 73 p–v diagram p. 73 T–v diagram p. 73

two-phase, liquid–vapor mixture p. 75 quality p. 75 superheated vapor p. 75

enthalpy p. 83 specific heats p. 91 ideal gas model p. 100

Exercises: Things Engineers Think About 1. Why does food cook more quickly in a pressure cooker than in water boiling in an open container? 2. If water contracted on freezing, what implications might this have for aquatic life? 3. Why do frozen water pipes tend to burst? 4. Referring to a phase diagram, explain why a film of liquid water forms under the blade of an ice skate. 5. Can water at 40C exist as a vapor? As a liquid? 6. What would be the general appearance of constant-volume lines in the vapor and liquid regions of the phase diagram? 7. Are the pressures listed in the tables in the Appendix absolute pressures or gage pressures? 8. The specific internal energy is arbitrarily set to zero in Table A-2 for saturated liquid water at 0.01C. If the reference value for u at this reference state were specified differently, would there be any significant effect on thermodynamic analyses using u and h? 9. For liquid water at 20C and 1.0 MPa, what percent difference would there be if its specific enthalpy were evaluated using Eq. 3.14 instead of Eq. 3.13? 10. For a system consisting of 1 kg of a two-phase, liquid–vapor mixture in equilibrium at a known temperature T and specific

volume v, can the mass, in kg, of each phase be determined? Repeat for a three-phase, solid–liquid–vapor mixture in equilibrium at T, v. 11. By inspection of Fig. 3.9, what are the values of cp for water at 500C and pressures equal to 40 MPa, 20 MPa, 10 MPa, and 1 MPa? Is the ideal gas model appropriate at any of these states? 12. Devise a simple experiment to determine the specific heat, cp, of liquid water at atmospheric pressure and room temperature. 13. If a block of aluminum and a block of steel having equal volumes each received the same energy input by heat transfer, which block would experience the greater temperature increase? 14. Under what circumstances is the following statement correct? Equal molar amounts of two different gases at the same temperature, placed in containers of equal volume, have the same pressure. 15. Estimate the mass of air contained in a bicycle tire. 16. Specific internal energy and enthalpy data for water vapor are provided in two tables: Tables A-4 and A-23. When would Table A-23 be used?

Problems: Developing Engineering Skills Using p–v–T Data

3.1 Determine the phase or phases in a system consisting of H2O at the following conditions and sketch p–v and T–v diagrams showing the location of each state. (a) p 5 bar, T 151.9C. (b) p 5 bar, T 200C. (c) T 200C, p 2.5 MPa.

(d) T 160C, p 4.8 bar. (e) T 12C, p 1 bar. 3.2 Plot the pressure–temperature relationship for two-phase liquid–vapor mixtures of water from the triple point temperature to the critical point temperature. Use a logarithmic scale for pressure, in bar, and a linear scale for temperature, in C.

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Chapter 3 Evaluating Properties

3.3 For H2O, plot the following on a p–v diagram drawn to scale on log–log coordinates:

the final mass of vapor in the tank, in kg, and the final pressure, in bar.

(a) the saturated liquid and saturated vapor lines from the triple point to the critical point, with pressure in MPa and specific volume in m3/kg. (b) lines of constant temperature at 100 and 300C.

3.15 Two thousand kg of water, initially a saturated liquid at 150C, is heated in a closed, rigid tank to a final state where the pressure is 2.5 MPa. Determine the final temperature, in C, the volume of the tank, in m3, and sketch the process on T–v and p–v diagrams.

3.4 Plot the pressure–temperature relationship for two-phase liquid–vapor mixtures of (a) Refrigerant 134a, (b) ammonia, (c) Refrigerant 22 from a temperature of 40 to 100C, with pressure in kPa and temperature in C. Use a logarithmic scale for pressure and a linear scale for temperature. 3.5 Determine the quality of a two-phase liquid–vapor mixture of (a) H2O at 20C with a specific volume of 20 m3/kg. (b) Propane at 15 bar with a specific volume of 0.02997 m3/kg. (c) Refrigerant 134a at 60C with a specific volume of 0.001 m3/kg. (d) Ammonia at 1 MPa with a specific volume of 0.1 m3/kg. 3.6 For H2O, plot the following on a p–v diagram drawn to scale on log–log coordinates: (a) the saturated liquid and saturated vapor lines from the triple point to the critical point, with pressure in KPa and specific volume in m3/kg 150C (b) lines of constant temperature at 300 and 560C. 3.7 Two kg of a two-phase, liquid–vapor mixture of carbon dioxide (CO2) exists at 40C in a 0.05 m3 tank. Determine the quality of the mixture, if the values of specific volume for saturated liquid and saturated vapor CO2 at 40C are vf 0.896 103 m3/kg and vg 3.824 102 m3/kg, respectively. 3.8 Determine the mass, in kg, of 0.1 m3 of Refrigerant 134a at 4 bar, 100C. 3.9 A closed vessel with a volume of 0.018 m3 contains 1.2 kg of Refrigerant 22 at 10 bar. Determine the temperature, in C. 3.10 Calculate the mass, in kg, of 1 m3 of a two-phase liquid– vapor mixture of Refrigerant 22 at 1 bar with a quality of 75%. 3.11 A two-phase liquid–vapor mixture of a substance has a pressure of 150 bar and occupies a volume of 0.2 m3. The masses of saturated liquid and vapor present are 3.8 kg and 4.2 kg, respectively. Determine the mixture specific volume in m3/kg. 3.12 Ammonia is stored in a tank with a volume of 0.21 m3. Determine the mass, in kg, assuming saturated liquid at 20C. What is the pressure, in kPa? 3.13 A storage tank in a refrigeration system has a volume of 0.006 m3 and contains a two-phase liquid–vapor mixture of Refrigerant 134a at 180 kPa. Plot the total mass of refrigerant, in kg, contained in the tank and the corresponding fractions of the total volume occupied by saturated liquid and saturated vapor, respectively, as functions of quality. 3.14 Water is contained in a closed, rigid, 0.2 m3 tank at an initial pressure of 5 bar and a quality of 50%. Heat transfer occurs until the tank contains only saturated vapor. Determine

3.16 Steam is contained in a closed rigid container with a volume of 1 m3. Initially, the pressure and temperature of the steam are 7 bar and 500C, respectively. The temperature drops as a result of heat transfer to the surroundings. Determine the temperature at which condensation first occurs, in C, and the fraction of the total mass that has condensed when the pressure reaches 0.5 bar. What is the volume, in m3, occupied by saturated liquid at the final state? 3.17 Water vapor is heated in a closed, rigid tank from saturated vapor at 160C to a final temperature of 400C. Determine the initial and final pressures, in bar, and sketch the process on T–v and p–v diagrams. 3.18 Ammonia undergoes an isothermal process from an initial state at T1 80F and v1 10 ft3/lb to saturated vapor. Determine the initial and final pressures, in lbf/in.2, and sketch the process on T–v and p–v diagrams. 3.19 A two-phase liquid–vapor mixture of H2O is initially at a pressure of 30 bar. If on heating at fixed volume, the critical point is attained, determine the quality at the initial state. 3.20 Ammonia undergoes a constant-pressure process at 2.5 bar from T1 30C to saturated vapor. Determine the work for the process, in kJ per kg of refrigerant. 3.21 Water vapor in a piston–cylinder assembly is heated at a constant temperature of 204C from saturated vapor to a pressure of .7 MPa. Determine the work, in kJ per kg of water vapor, by using IT. 3.22 2 kg mass of ammonia, initially at p1 7 bars and T1 180C, undergo a constant-pressure process to a final state where the quality is 85%. Determine the work for the process, kJ. 3.23 Water vapor initially at 10 bar and 400C is contained within a piston–cylinder assembly. The water is cooled at constant volume until its temperature is 150C. The water is then condensed isothermally to saturated liquid. For the water as the system, evaluate the work, in kJ/kg. 3.24 Two kilograms of Refrigerant 22 undergo a process for which the pressure–volume relation is pv1.05 constant. The initial state of the refrigerant is fixed by p1 2 bar, T1 20C, and the final pressure is p2 10 bar. Calculate the work for the process, in kJ. 3.25 Refrigerant 134a in a piston–cylinder assembly undergoes a process for which the pressure–volume relation is pv1.058 constant. At the initial state, p1 200 kPa, T1 10C. The final temperature is T2 50C. Determine the final pressure, in kPa, and the work for the process, in kJ per kg of refrigerant.

Problems: Developing Engineering Skills Using u–h Data

3.26 Using the tables for water, determine the specified property data at the indicated states. Check the results using IT. In each case, locate the state by hand on sketches of the p–v and T–v diagrams. (a) (b) (c) (d) (e) (f) (g) (h)

At p 3 bar, T 240C, find v in m3/kg and u in kJ/kg. At p 3 bar, v 0.5 m3/kg, find T in C and u in kJ/kg. At T 400C, p 10 bar, find v in m3/kg and h in kJ/kg. At T 320C, v 0.03 m3/kg, find p in MPa and u in kJ/kg. At p 28 MPa, T 520C, find v in m3/kg and h in kJ/kg. At T 100C, x 60%, find p in bar and v in m3/kg. At T 10C, v 100 m3/kg, find p in kPa and h in kJ/kg. At p 4 MPa, T 160C, find v in m3/kg and u in kJ/kg.

3.27 Determine the values of the specified properties at each of the following conditions. (a) For Refrigerant 134a at T 60C and v 0.072 m3/kg, determine p in kPa and h in kJ/kg. (b) For ammonia at p 8 bar and v 0.005 m3/kg, determine T in C and u in kJ/kg. (c) For Refrigerant 22 at T 10C and u 200 kJ/kg, determine p in bar and v in m3/kg. 3.28 A quantity of water is at 15 MPa and 100C. Evaluate the specific volume, in m3/kg, and the specific enthalpy, in kJ/kg, using (a) data from Table A-5. (b) saturated liquid data from Table A-2. 3.29 Plot versus pressure the percent changes in specific volume, specific internal energy, and specific enthalpy for water at 20C from the saturated liquid state to the state where the pressure is 300 bar. Based on the resulting plots, discuss the implications regarding approximating compressed liquid properties using saturated liquid properties at 20C, as discussed in Sec. 3.3.6. 3.30 Evaluate the specific volume, in m3/kg, and the specific enthalpy, in kJ/kg, of ammonia at 20C and 1.0 MPa. 3.31 Evaluate the specific volume, in m3/kg, and the specific enthalpy, in kJ/kg, of propane at 800 kPa and 0C.

117

8 bar to 50C. For the refrigerant, determine the work and heat transfer, per unit mass, each in kJ/kg. Changes in kinetic and potential energy are negligible. 3.35 Saturated liquid water contained in a closed, rigid tank is cooled to a final state where the temperature is 50C and the masses of saturated vapor and liquid present are 0.03 and 1999.97 kg, respectively. Determine the heat transfer for the process, in kJ. 3.36 Refrigerant 134a undergoes a process for which the pressure–volume relation is pvn constant. The initial and final states of the refrigerant are fixed by p1 200 kPa, T1 10C and p2 1000 kPa, T2 50C, respectively. Calculate the work and heat transfer for the process, each in kJ per kg of refrigerant. 3.37 A piston–cylinder assembly contains a two-phase liquid–vapor mixture of Refrigerant 22 initially at 24C with a quality of 95%. Expansion occurs to a state where the pressure is 1 bar. During the process the pressure and specific volume are related by pv constant. For the refrigerant, determine the work and heat transfer per unit mass, each in kJ/kg. 3.38 Five kilograms of water, initially a saturated vapor at 100 kPa, are cooled to saturated liquid while the pressure is maintained constant. Determine the work and heat transfer for the process, each in kJ. Show that the heat transfer equals the change in enthalpy of the water in this case. 3.39 One kilogram of saturated solid water at the triple point is heated to saturated liquid while the pressure is maintained constant. Determine the work and the heat transfer for the process, each in kJ. Show that the heat transfer equals the change in enthalpy of the water in this case. 3.40 A two-phase liquid–vapor mixture of H2O with an initial quality of 25% is contained in a piston–cylinder assembly as shown in Fig. P3.40. The mass of the piston is 40 kg, and its diameter is 10 cm. The atmospheric pressure of the surroundings is 1 bar. The initial and final positions of the piston are shown on the diagram. As the water is heated, the pressure inside the cylinder remains constant until the piston hits the stops. Heat transfer to the water continues until its pressure is patm = 100 kPa

Applying the Energy Balance

3.32 A closed, rigid tank contains 2 kg of water initially at 80C and a quality of 0.6. Heat transfer occurs until the tank contains only saturated vapor. Kinetic and potential energy effects are negligible. For the water as the system, determine the amount of energy transfer by heat, in kJ. 3.33 A two-phase liquid–vapor mixture of H2O, initially at 1.0 MPa with a quality of 90%, is contained in a rigid, wellinsulated tank. The mass of H2O is 2 kg. An electric resistance heater in the tank transfers energy to the water at a constant rate of 60 W for 1.95 h. Determine the final temperature of the water in the tank, in C. 3.34 Refrigerant 134a vapor in a piston–cylinder assembly undergoes a constant-pressure process from saturated vapor at

4.5 cm 1 cm

Q Diameter = 10 cm Mass = 40 kg

Initial quality x1 = 25% Figure P3.40

118

Chapter 3 Evaluating Properties

3 bar. Friction between the piston and the cylinder wall is negligible. Determine the total amount of heat transfer, in J. Let g 9.81 m/s2. 3.41 Two kilograms of Refrigerant 134a, initially at 2 bar and occupying a volume of 0.12 m3, undergoes a process at constant pressure until the volume has doubled. Kinetic and potential energy effects are negligible. Determine the work and heat transfer for the process, each in kJ. 3.42 Propane is compressed in a piston–cylinder assembly from saturated vapor at 40C to a final state where p2 6 bar and T2 80C. During the process, the pressure and specific volume are related by pvn constant. Neglecting kinetic and potential energy effects, determine the work and heat transfer per unit mass of propane, each in kJ/kg. 3.43 A system consisting of 2 kg of ammonia undergoes a cycle composed of the following processes: Process 1–2: constant volume from p1 10 bar, x1 0.6 to saturated vapor Process 2–3: constant temperature to p3 p1, Q23 228 kJ Process 3–1: constant pressure Sketch the cycle on p–v and T–v diagrams. Neglecting kinetic and potential energy effects, determine the net work for the cycle and the heat transfer for each process, all in kJ. 3.44 A system consisting of 1 kg of H2O undergoes a power cycle composed of the following processes: Process 1–2: Constant-pressure heating at 10 bar from saturated vapor. Process 2–3: Constant-volume cooling to p3 5 bar, T3 160C. Process 3–4: Isothermal compression with Q34 815.8 kJ. Process 4–1: Constant-volume heating. Sketch the cycle on T–v and p–v diagrams. Neglecting kinetic and potential energy effects, determine the thermal efficiency. 3.45 A well-insulated copper tank of mass 13 kg contains 4 kg of liquid water. Initially, the temperature of the copper is 27C and the temperature of the water is 50C. An electrical resistor of neglible mass transfers 100 kJ of energy to the contents of the tank. The tank and its contents come to equilibrium. What is the final temperature, in C? 3.46 An isolated system consists of a 10-kg copper slab, initially at 30C, and 0.2 kg of saturated water vapor, initially at 130C. Assuming no volume change, determine the final equilibrium temperature of the isolated system, in C. 3.47 A system consists of a liquid, considered incompressible with constant specific heat c, filling a rigid tank whose surface area is A. Energy transfer by work from a paddle wheel to the liquid occurs at a constant rate. Energy transfer # by heat occurs at a rate given by Q hA (T T0), where T is the instantaneous temperature of the liquid, T0 is the temperature of the surroundings, and h is an overall heat

transfer coefficient. At the initial time, t 0, the tank and its contents are at the temperature of the surroundings. Obtain a differential equation for temperature T in terms of time t and relevant parameters. Solve the differential equation to obtain T(t). Using Generalized Compressibility Data

3.48 Determine the compressibility factor for water vapor at 200 bar and 470C, using (a) data from the compressibility chart. (b) data from the steam tables. 3.49 Determine the volume, in m3, occupied by 40 kg of nitrogen (N2) at 17 MPa, 180 K. 3.50 A rigid tank contains 0.5 kg of oxygen (O2) initially at 30 bar and 200 K. The gas is cooled and the pressure drops to 20 bar. Determine the volume of the tank, in m3, and the final temperature, in K. 3.51 Five kg of butane (C4H10) in a piston–cylinder assembly undergo a process from p1 5 MPa, T1 500 K to p2 3 MPa, T2 450 K during which the relationship between pressure and specific volume is pvn constant. Determine the work, in kJ. Working with the Ideal Gas Model

3.52 A tank contains 0.05 m3 of nitrogen (N2) at 21C and 10 MPa. Determine the mass of nitrogen, in kg, using (a) the ideal gas model. (b) data from the compressibility chart. Comment on the applicability of the ideal gas model for nitrogen at this state. 3.53 Show that water vapor can be accurately modeled as an ideal gas at temperatures below about 60C. 3.54 For what ranges of pressure and temperature can air be considered an ideal gas? Explain your reasoning. 3.55 Check the applicability of the ideal gas model for Refrigerant 134a at a temperature of 80C and a pressure of (a) 1.6 MPa. (b) 0.10 MPa. 3.56 Determine the temperature, in K, of oxygen (O2) at 250 bar and a specific volume of 0.003 m3/kg using generalized compressibility data and compare with the value obtained using the ideal gas model. 3.57 Determine the temperature, in K, of 5 kg of air at a pressure of 0.3 MPa and a volume of 2.2 m3. Verify that ideal gas behavior can be assumed for air under these conditions. 3.58 Compare the densities, in kg/m3, of helium and air, each at 300 K, 100 kPa. Assume ideal gas behavior. 3.59 Assuming ideal gas behavior for air, plot to scale the isotherms 300, 500, 1000, and 2000 K on a p–v diagram.

Problems: Developing Engineering Skills

3.60 By integrating cp(T ) obtained from Table A-21, determine the change in specific enthalpy, in kJ/kg, of methane (CH4) from T1 320 K, p1 2 bar to T2 800 K, p2 10 bar. Check your result using IT. 3.61 Show that the specific heat ratio of a monatomic ideal gas is equal to 53. Using the Energy Balance with the Ideal Gas Model

3.62 One kilogram of air, initially at 5 bar, 350 K, and 3 kg of carbon dioxide (CO2), initially at 2 bar, 450 K, are confined to opposite sides of a rigid, well-insulated container, as illustrated in Fig. P3.62. The partition is free to move and allows conduction from one gas to the other without energy storage in the partition itself. The air and carbon dioxide each behave as ideal gases. Determine the final equilibrium temperature, in K, and the final pressure, in bar, assuming constant specific heats.

Air 1 kg 5 bar 350 K

Partition

CO2 3 kg 2 bar 450 K

Insulation

Figure P3.62

3.63 Consider a gas mixture whose apparent molecular weight is 33, initially at 3 bar and 300 K, and occupying a volume of 0.1 m3. The gas undergoes an expansion during which the pressure–volume relation is pV1.3 constant and the energy transfer by heat to the gas is 3.84 kJ. Assume the ideal gas model with cv 0.6 (2.5 104)T, where T is in K and cv has units of kJ/kg # K. Neglecting kinetic and potential energy effects, determine (a) (b) (c) (d)

the final temperature, in K. the final pressure, in bar. the final volume, in m3. the work, in kJ.

3.64 Helium (He) gas initially at 2 bar, 200 K undergoes a polytropic process, with n k, to a final pressure of 14 bar. Determine the work and heat transfer for the process, each in kJ per kg of helium. Assume ideal gas behavior. 3.65 Two kilograms of a gas with molecular weight 28 are contained in a closed, rigid tank fitted with an electric resistor. The resistor draws a constant current of 10 amp at a voltage of 12 V for 10 min. Measurements indicate that when equilibrium is reached, the temperature of the gas has increased by 40.3C. Heat transfer to the surroundings is estimated to occur at a constant rate of 20 W. Assuming ideal gas behavior, determine an average value of the specific heat cp, in kJ/kg # K,

119

of the gas in this temperature interval based on the measured data. 3.66 A gas is confined to one side of a rigid, insulated container divided by a partition. The other side is initially evacuated. The following data are known for the initial state of the gas: p1 5 bar, T1 500 K, and V1 0.2 m3. When the partition is removed, the gas expands to fill the entire container, which has a total volume of 0.5 m3. Assuming ideal gas behavior, determine the final pressure, in bar. 3.67 A rigid tank initially contains 3 kg of air at 500 kPa, 290 K. The tank is connected by a valve to a piston–cylinder assembly oriented vertically and containing 0.05 m3 of air initially at 200 kPa, 290 K. Although the valve is closed, a slow leak allows air to flow into the cylinder until the tank pressure falls to 200 kPa. The weight of the piston and the pressure of the atmosphere maintain a constant pressure of 200 kPa in the cylinder; and owing to heat transfer, the temperature stays constant at 290 K. For the air, determine the total amount of energy transfer by work and by heat, each in kJ. Assume ideal gas behavior. 3.68 A piston–cylinder assembly contains 1 kg of nitrogen gas (N2). The gas expands from an initial state where T1 700 K and p1 5 bar to a final state where p2 2 bar. During the process the pressure and specific volume are related by pv1.3 constant. Assuming ideal gas behavior and neglecting kinetic and potential energy effects, determine the heat transfer during the process, in kJ, using (a) a constant specific heat evaluated at 300 K. (b) a constant specific heat evaluated at 700 K. (c) data from Table A-23. 3.69 Air is compressed adiabatically from p1 1 bar, T1 300 K to p2 15 bar, v2 0.1227 m3/kg. The air is then cooled at constant volume to T3 300 K. Assuming ideal gas behavior, and ignoring kinetic and potential energy effects, calculate the work for the first process and the heat transfer for the second process, each in kJ per kg of air. Solve the problem each of two ways: (a) using data from Table A-22. (b) using a constant specific heat evaluated at 300 K. 3.70 A system consists of 2 kg of carbon dioxide gas initially at state 1, where p1 1 bar, T1 300 K. The system undergoes a power cycle consisting of the following processes: Process 1–2: constant volume to p2, p2 p1 Process 2–3: expansion with pv1.28 constant Process 3–1: constant-pressure compression Assuming the ideal gas model and neglecting kinetic and potential energy effects, (a) sketch the cycle on a p–v diagram. (b) plot the thermal efficiency versus p2p1 ranging from 1.05 to 4.

120

Chapter 3 Evaluating Properties

3.71 A closed system consists of an ideal gas with mass m and constant specific heat ratio k. If kinetic and potential energy changes are negligible, (a) show that for any adiabatic process the work is W

mR1T2 T1 2

3.72 Steam, initially at 5 MPa, 280C undergoes a polytropic process in a piston–cylinder assembly to a final pressure of 20 MPa. Plot the heat transfer, in kJ per kg of steam, for polytropic exponents ranging from 1.0 to 1.6. Also investigate the error in the heat transfer introduced by assuming ideal gas behavior for the steam. Discuss.

1k

(b) show that an adiabatic polytropic process in which work is done only at a moving boundary is described by pVk constant.

Design & Open Ended Problems: Exploring Engineering Practice 3.1D This chapter has focused on simple compressible systems in which magnetic effects are negligible. In a report, describe the thermodynamic characteristics of simple magnetic systems, and discuss practical applications of this type of system. 3.2D The Montreal Protocols aim to eliminate the use of various compounds believed to deplete the earth’s stratospheric ozone. What are some of the main features of these agreements, what compounds are targeted, and what progress has been made to date in implementing the Protocols? 3.3D Frazil ice forming upsteam of a hydroelectric plant can block the flow of water to the turbine. Write a report summarizing the mechanism of frazil ice formation and alternative means for eliminating frazil ice blockage of power plants. For one of the alternatives, estimate the cost of maintaining a 30-MW power plant frazil ice–free. 3.4D Much has been written about the use of hydrogen as a fuel. Investigate the issues surrounding the so-called hydrogen economy and write a report. Consider possible uses of hydrogen and the obstacles to be overcome before hydrogen could be used as a primary fuel source. 3.5D A major reason for the introduction of CFC (chlorofluorocarbon) refrigerants, such as Refrigerant 12, in the 1930s was that they are less toxic than ammonia, which was widely used at the time. But in recent years, CFCs largely have been phased out owing to concerns about depletion of the earth’s stratospheric ozone. As a result, there has been a resurgence of interest in ammonia as a refrigerant, as well as increased interest in natural refrigerants, such as propane. Write a report outlining advantages and disadvantages of ammonia and natural refrigerants. Consider safety issues and include a summary of any special design requirements that these refrigerants impose on refrigeration system components. 3.6D Metallurgists use phase diagrams to study allotropic transformations, which are phase transitions within the solid

region. What features of the phase behavior of solids are important in the fields of metallurgy and materials processing? Discuss. 3.7D Devise an experiment to visualize the sequence of events as a two-phase liquid–vapor mixture is heated at constant volume near its critical point. What will be observed regarding the meniscus separating the two phases when the average specific volume is less than the critical specific volume? Greater than the critical specific volume? What happens to the meniscus in the vicinity of the critical point? Discuss. 3.8D One method of modeling gas behavior from the microscopic viewpoint is known as the kinetic theory of gases. Using kinetic theory, derive the ideal gas equation of state and explain the variation of the ideal gas specific heat cv with temperature. Is the use of kinetic theory limited to ideal gas behavior? Discuss. 3.9D Many new substances have been considered in recent years as potential working fluids for power plants or refrigeration systems and heat pumps. What thermodynamic property data are needed to assess the feasibility of a candidate substance for possible use as a working fluid? Write a paper discussing your findings. 3.10D A system is being designed that would continuously feed steel (AISI 1010) rods of 0.1 m diameter into a gas-fired furnace for heat treating by forced convection from gases at 1200 K. To assist in determining the feed rate, estimate the time, in min, the rods would have to remain in the furnace to achieve a temperature of 800 K from an initial temperature of 300 K. 3.11D Natural Refrigerants–Back to the Future (see box Sec. 3.3). Although used for home appliances in Europe, hydrocarbon refrigerants have not taken hold in the United States thus far owing to concerns about liability if there is an accident. Research hydrocarbon refrigerant safety. Write a report including at least three references.

C H A P

Control Volume Analysis Using Energy

T E R

4

E N G I N E E R I N G C O N T E X T The objective of this chapter is to develop and illustrate the use of the control volume forms of the conservation of mass and conservation of energy principles. Mass and energy balances for control volumes are introduced in Secs. 4.1 and 4.2, respectively. These balances are applied in Sec. 4.3 to control volumes at steady state and in Sec. 4.4 for transient applications. Although devices such as turbines, pumps, and compressors through which mass flows can be analyzed in principle by studying a particular quantity of matter (a closed system) as it passes through the device, it is normally preferable to think of a region of space through which mass flows (a control volume). As in the case of a closed system, energy transfer across the boundary of a control volume can occur by means of work and heat. In addition, another type of energy transfer must be accounted for—the energy accompanying mass as it enters or exits.

chapter objective

4.1 Conservation of Mass for a Control Volume

In this section an expression of the conservation of mass principle for control volumes is developed and illustrated. As a part of the presentation, the one-dimensional flow model is introduced. 4.1.1 Developing the Mass Rate Balance The mass rate balance for control volumes is introduced by reference to Fig. 4.1, which shows a control volume with mass flowing in at i and flowing out at e, respectively. When applied to such a control volume, the conservation of mass principle states £

conservation of mass

time rate of change of time rate of flow time rate of flow mass contained within § £ of mass in across § £ of mass out across § the control volume at time t inlet i at time t exit e at time t

Denoting the mass contained within the control volume at time t by mcv(t), this statement of the conservation of mass principle can be expressed in symbols as dmcv # # mi me dt

(4.1)

121

122

Chapter 4 Control Volume Analysis Using Energy Dashed line defines the control volume boundary Inlet i Exit e

Figure 4.1

One-inlet, one-exit control volume.

# # where dmcvdt is the time rate of change of mass within the control volume, and mi and me are mass flow rates at the inlet and exit, respectively. As for the symbols # the instantaneous # # # W and Q, the dots in the quantities mi and me denote time rates of transfer. In SI, all terms in Eq. 4.1 are expressed in kg/s. For a discussion of the development of Eq. 4.1, see box. In general, there may be several locations on the boundary through which mass enters or exits. This can be accounted for by summing, as follows

mass flow rates

dmcv # # a mi a me dt i e

mass rate balance

(4.2)

Equation 4.2 is the mass rate balance for control volumes with several inlets and exits. It is a form of the conservation of mass principle commonly employed in engineering. Other forms of the mass rate balance are considered in discussions to follow.

DEVELOPING THE CONTROL VOLUME MASS BALANCE Dashed line defines the control volume boundary mi

mcv(t)

Region i Time t

mcv(t + ∆t)

me Region e

Time t + ∆t

For each of the extensive properties mass, energy, and entropy (Chap. 6), the control volume form of the property balance can be obtained by transforming the corresponding closed system form. Let us consider this for mass, recalling that the mass of a closed system is constant. The figures in the margin show a system consisting of a fixed quantity of matter m that occupies different regions at time t and a later time t t. The mass under consideration is shown in color on the figures. At time t, the mass is the sum m mcv(t) mi, where mcv(t) is the mass contained within the control volume, and mi is the mass within the small region labeled i adjacent to the control volume. Let us study the fixed quantity of matter m as time elapses. In a time interval t all the mass in region i crosses the control volume boundary, while some of the mass, call it me, initially contained within the control volume exits to fill the region labeled e adjacent to the control volume. Although the mass in regions i and e as well as in the control volume differ from time t to t t, the total amount of mass is constant. Accordingly mcv 1t2 mi mcv 1t ¢t2 me

(a)

mcv 1t ¢t2 mcv 1t2 mi me

(b)

or on rearrangement

Equation (b) is an accounting balance for mass. It states that the change in mass of the control volume during time interval t equals the amount of mass that enters less the amount of mass that exits.

4.1 Conservation of Mass for a Control Volume

123

Equation (b) can be expressed on a time rate basis. First, divide by t to obtain mcv 1t ¢t2 mcv 1t2 mi me ¢t ¢t ¢t

(c)

Then, in the limit as t goes to zero, Eq. (c) becomes Eq. 4.1, the instantaneous control volume rate equation for mass dmcv # # mi me dt

(4.1)

where dmcvdt denotes the time rate of change of mass within the control volume, and # # mi and me are the inlet and exit mass flow rates, respectively, all at time t.

EVALUATING THE MASS FLOW RATE

# An expression for the mass flow rate m of the matter entering or exiting a control volume can be obtained in terms of local properties by considering a small quantity of matter flowing with velocity V across an incremental area dA in a time interval t, as shown in Fig. 4.2. Since the portion of the control volume boundary through which mass flows is not necessarily at rest, the velocity shown in the figure is understood to be the velocity relative to the area dA. The velocity can be resolved into components normal and tangent to the plane containing dA. In the following development Vn denotes the component of the relative velocity normal to dA in the direction of flow. The volume of the matter crossing dA during the time interval t shown in Fig. 4.2 is an oblique cylinder with a volume equal to the product of the area of its base dA and its altitude Vn t. Multiplying by the density gives the amount of mass that crosses dA in time t amount of mass

£ crossing dA during § r1Vn ¢t2 dA

the time interval ¢t

Dividing both sides of this equation by t and taking the limit as t goes to zero, the instantaneous mass flow rate across incremental area dA is £

instantaneous rate of mass flow § rVn dA across dA

When this is integrated over the area A through which mass passes, an expression for the mass flow rate is obtained # m

rV dA n

(4.3)

A

Equation 4.3 can be applied at the inlets and exits to account for the rates of mass flow into and out of the control volume. 4.1.2 Forms of the Mass Rate Balance The mass rate balance, Eq. 4.2, is a form that is important for control volume analysis. In many cases, however, it is convenient to apply the mass balance in forms suited to particular objectives. Some alternative forms are considered in this section.

Vn ∆t Volume of matter

V ∆t

dA A

Figure 4.2 Illustration used to develop an expression for mass flow rate in terms of local fluid properties.

124

Chapter 4 Control Volume Analysis Using Energy e

Area = A Air

V, T, v

Air compressor Air

i

–

+

Figure 4.3 Figure illustrating the one-dimensional flow model.

ONE-DIMENSIONAL FLOW FORM

one-dimensional flow

When a flowing stream of matter entering or exiting a control volume adheres to the following idealizations, the flow is said to be one-dimensional:

METHODOLOGY UPDATE

In subsequent control volume analyses, we routinely assume that the idealizations of onedimensional flow are appropriate. Accordingly the assumption of onedimensional flow is not listed explicitly in solved examples.

The flow is normal to the boundary at locations where mass enters or exits the control volume. All intensive properties, including velocity and density, are uniform with position (bulk average values) over each inlet or exit area through which matter flows.

for example. . . Figure 4.3 illustrates the meaning of one-dimensional flow. The area through which mass flows is denoted by A. The symbol V denotes a single value that represents the velocity of the flowing air. Similarly T and v are single values that represent the temperature and specific volume, respectively, of the flowing air.

When the flow is one-dimensional, Eq. 4.3 for the mass flow rate becomes # m rAV

(4.4a)

1one-dimensional flow2

(4.4b)

or in terms of specific volume AV # m v

volumetric flow rate

1one-dimensional flow2

When area is in m2, velocity is in m/s, and specific volume is in m3/kg, the mass flow rate found from Eq. 4.4b is in kg/s, as can be verified. The product AV in Eqs. 4.4 is the volumetric flow rate. The volumetric flow rate is expressed in units of m3/s. Substituting Eq. 4.4b into Eq. 4.2 results in an expression for the conservation of mass principle for control volumes limited to the case of one-dimensional flow at the inlet and exits dmcv AiVi AeVe a a vi ve dt i e

1one-dimensional flow2

(4.5)

Note that Eq. 4.5 involves summations over the inlets and exits of the control volume. Each individual term in either of these sums applies to a particular inlet or exit. The area, velocity, and specific volume appearing in a term refer only to the corresponding inlet or exit.

4.1 Conservation of Mass for a Control Volume

125

STEADY-STATE FORM

Many engineering systems can be idealized as being at steady state, meaning that all properties are unchanging in time. For a control volume at steady state, the identity of the matter within the control volume changes continuously, but the total amount present at any instant remains constant, so dmcvdt 0 and Eq. 4.2 reduces to # # a mi a me i

steady state

(4.6)

e

That is, the total incoming and outgoing rates of mass flow are equal. Equality of total incoming and outgoing rates of mass flow does not necessarily mean that a control volume is at steady state. Although the total amount of mass within the control volume at any instant would be constant, other properties such as temperature and pressure might be varying with time. When a control volume is at steady state, every property is independent of time. Note that the steady-state assumption and the one-dimensional flow assumption are independent idealizations. One does not imply the other. INTEGRAL FORM

We consider next the mass rate balance expressed in terms of local properties. The total mass contained within the control volume at an instant t can be related to the local density as follows mcv 1t2

r dV

(4.7)

V

where the integration is over the volume at time t. With Eqs. 4.3 and 4.7, the mass rate balance Eq. 4.2 can be written as d dt

r dV a a rV dAb a a rV dAb n

V

i

A

(4.8)

n

i

e

A

e

where the area integrals are over the areas through which mass enters and exits the control volume, respectively. The product Vn appearing in this equation, known as the mass flux, gives the time rate of mass flow per unit of area. To evaluate the terms of the right side of Eq. 4.8 requires information about the variation of the mass flux over the flow areas. The form of the conservation of mass principle given by Eq. 4.8 is usually considered in detail in fluid mechanics.

mass flux

EXAMPLES

The following example illustrates an application of the rate form of the mass balance to a control volume at steady state. The control volume has two inlets and one exit.

EXAMPLE

4.1

Feedwater Heater at Steady State

A feedwater heater operating at steady state has two inlets and one exit. At inlet 1, water vapor enters at p1 7 bar, T1 200C with a mass flow rate of 40 kg/s. At inlet 2, liquid water at p2 7 bar, T2 40C enters through an area A2 25 cm2. Saturated liquid at 7 bar exits at 3 with a volumetric flow rate of 0.06 m3/s. Determine the mass flow rates at inlet 2 and at the exit, in kg/s, and the velocity at inlet 2, in m/s.

Chapter 4 Control Volume Analysis Using Energy

126

SOLUTION Known: A stream of water vapor mixes with a liquid water stream to produce a saturated liquid stream at the exit. The states at the inlets and exit are specified. Mass flow rate and volumetric flow rate data are given at one inlet and at the exit, respectively. Find: Determine the mass flow rates at inlet 2 and at the exit, and the velocity V2. Schematic and Given Data:

2

1 T1 = 200 °C p1 = 7 bar m1 = 40 kg/s

A2 = 25 cm2 T2 = 40 °C p2 = 7 bar 3

Assumption: The control volume shown on the accompanying figure is at steady state.

Control volume boundary

Saturated liquid p3 = 7 bar (AV)3 = 0.06 m3/s

Figure E4.1

# Analysis: The principal relations to be employed are the mass rate balance (Eq. 4.2) and the expression m AV v (Eq. 4.4b). At steady state the mass rate balance becomes 0

dmcv # # # m1 m2 m3 dt

❶ # Solving for m2

# # # m2 m3 m1

# The mass flow rate m1 is given. The mass flow rate at the exit can be evaluated from the given volumetric flow rate 1AV2 3 # m3 v3

where v3 is the specific volume at the exit. In writing this expression, one-dimensional flow is assumed. From Table A-3, v3 1.108 103 m3/kg. Hence # m3

0.06 m3/s 54.15 kg /s 11.108 103 m3/kg2

The mass flow rate at inlet 2 is then # # # m2 m3 m1 54.15 40 14.15 kg /s # For one-dimensional flow at 2, m2 A2V2 v2, so

# V2 m2v2 A2

State 2 is a compressed liquid. The specific volume at this state can be approximated by v2 vf 1T2 2 (Eq. 3.11). From Table A-2 at 40C, v2 1.0078 103 m3/kg. So V2

❶

114.15 kg/s211.0078 103 m3/kg2 104 cm2 ` ` 5.7 m/s 25 cm2 1 m2

At steady state the mass flow rate at the exit equals the sum of the mass flow rates at the inlets. It is left as an exercise to show that the volumetric flow rate at the exit does not equal the sum of the volumetric flow rates at the inlets.

4.1 Conservation of Mass for a Control Volume

127

Example 4.2 illustrates an unsteady, or transient, application of the mass rate balance. In this case, a barrel is filled with water.

EXAMPLE

4.2

Filling a Barrel with Water

Water flows into the top of an open barrel at a constant mass flow rate of 7 kg/s. Water exits through a pipe near the base # with a mass flow rate proportional to the height of liquid inside: me 1.4 L, where L is the instantaneous liquid height, in m. The area of the base is 0.2 m2, and the density of water is 1000 kg/m3. If the barrel is initially empty, plot the variation of liquid height with time and comment on the result. SOLUTION Known: Water enters and exits an initially empty barrel. The mass flow rate at the inlet is constant. At the exit, the mass flow rate is proportional to the height of the liquid in the barrel. Find: Plot the variation of liquid height with time and comment. Schematic and Given Data:

mi = 30 lb/s

Boundary of control volume

Assumptions: 1. The control volume is defined by the dashed line on the accompanying diagram. 2. The water density is constant.

L (ft)

A = 3 ft2

me = 9L lb/s Figure E4.2a

Analysis: For the one-inlet, one-exit control volume, Eq. 4.2 reduces to dmcv # # mi me dt The mass of water contained within the barrel at time t is given by mcv 1t2 rAL1t2 where is density, A is the area of the base, and L(t) is the instantaneous liquid height. Substituting this into the mass rate balance together with the given mass flow rates d 1rAL2 dt

7 1.4L

Since density and area are constant, this equation can be written as 1.4 7 dL

a bL dt rA rA

Chapter 4 Control Volume Analysis Using Energy

128

which is a first-order, ordinary differential equation with constant coefficients. The solution is L 5 C exp a

❶

1.4t b rA

where C is a constant of integration. The solution can be verified by substitution into the differential equation. To evaluate C, use the initial condition: at t 0, L 0. Thus, C 5.0, and the solution can be written as L 5.031 exp 11.4trA2 4

Substituting r 1000 kg/m and A 0.2 m results in 2

2

L 5 31 exp 10.007t2 4

Height, m

This relation can be plotted by hand or using appropriate software. The result is

20

40

60 80 Time, s

100 120 Figure E4.2b

From the graph, we see that initially the liquid height increases rapidly and then levels out. After about 100 s, the height stays nearly constant with time. At this point, the rate of water flow into the barrel nearly equals the rate of flow out of the barrel. L S 5.

❶

Alternatively, this differential equation can be solved using Interactive Thermodynamics: IT. The differential equation can be expressed as der(L,t) + (1.4 * L)/(rho * A) = 7/(rho * A) rho = 1000 // kg/m3 A = 0.2 // m2 where der(L,t) is dLdt, rho is density , and A is area. Using the Explore button, set the initial condition at L 0, and sweep t from 0 to 200 in steps of 0.5. Then, the plot can be constructed using the Graph button.

4.2 Conservation of Energy for a Control Volume

In this section, the rate form of the energy balance for control volumes is obtained. The energy rate balance plays an important role in subsequent sections of this book. 4.2.1 Developing the Energy Rate Balance for a Control Volume We begin by noting that the control volume form of the energy rate balance can be derived by an approach closely paralleling that considered in the box of Sec. 4.1, where the control volume mass rate balance is obtained by transforming the closed system form. The present

4.2 Conservation of Energy for a Control Volume

Q

Energy transfers can occur by heat and work W

Inlet i

Ve2 ue + ___ + gze 2

mi

me

Control volume Vi2 + gzi ui + ___ 2 zi

Exit e ze Dashed line defines the control volume boundary Figure 4.4

Figure used to develop Eq. 4.9.

development proceeds less formally by arguing that, like mass, energy is an extensive property, so it too can be transferred into or out of a control volume as a result of mass crossing the boundary. Since this is the principal difference between the closed system and control volume forms, the control volume energy rate balance can be obtained by modifying the closed system energy rate balance to account for these energy transfers. Accordingly, the conservation of energy principle applied to a control volume states: time rate of change net rate at which net rate at which net rate of energy of the energy energy is being energy is being transfer into the D contained within T D transferred in T D transferred out T D control volume T the control volume at by heat transfer by work at accompanying time t at time t time t mass flow For the one-inlet one-exit control volume with one-dimensional flow shown in Fig. 4.4 the energy rate balance is # # dEcv V2i V2e # # Q W mi aui

gzi b me aue

gze b (4.9) dt 2 2 # # where Ecv denotes the energy of the control volume at time t. The terms Q and W account, respectively, for the net rate of energy transfer by heat and work across the boundary of the control volume at t. The underlined terms account for the rates of transfer of internal, kinetic, and potential energy of the entering and exiting streams. If there is no mass flow in or out, the respective mass flow rates vanish and the underlined terms of Eq. 4.9 drop out. The equation then reduces to the rate form of the energy balance for closed systems: Eq. 2.37.

EVALUATING WORK FOR A CONTROL VOLUME

Next, we will place Eq. 4.9 in an alternative form that is more convenient for subsequent ap# plications. This will be accomplished primarily by recasting the work term W, which represents the net rate of energy transfer by work across all portions of the boundary of the control volume. Because work is always done on or by a control volume where matter flows across the # boundary, it is convenient to separate the work term W into two contributions: One contribution is the work associated with the fluid pressure as mass is introduced at inlets and

129

130

Chapter 4 Control Volume Analysis Using Energy

# removed at exits. The other contribution, denoted by Wcv, includes all other work effects, such as those associated with rotating shafts, displacement of the boundary, and electrical effects. Consider the work at an exit e associated with the pressure of the flowing matter. Recall from Eq. 2.13 that the rate of energy transfer by work can be expressed as the product of a force and the velocity at the point of application of the force. Accordingly, the rate at which work is done at the exit by the normal force (normal to the exit area in the direction of flow) due to pressure is the product of the normal force, peAe, and the fluid velocity, Ve. That is time rate of energy transfer

£ by work from the control § 1 peAe 2Ve

(4.10)

volume at exit e

where pe is the pressure, Ae is the area, and Ve is the velocity at exit e, respectively. A similar expression can be written for the rate of energy transfer by work into the control volume at inlet i. # With these considerations, the work term W of the energy rate equation, Eq. 4.9, can be written as # # W Wcv 1 peAe 2 Ve 1 piAi 2Vi (4.11)

flow work

where, in accordance with the sign convention for work, the term at the inlet has a negative sign because energy is transferred into the control volume there. A positive sign precedes the work term at the exit because energy is transferred out of the control volume there. With # AV mv from Eq. 4.4b, the above expression for work can be written as # # # # W Wcv me 1 peve 2 mi 1 pivi 2 (4.12) # # where mi and me are the mass flow rates and vi and ve are the specific volumes evaluated at # # the inlet and exit, respectively. In Eq. 4.12, the terms mi( pivi) and me(peve) account for the work associated with the pressure at # the inlet and exit, respectively. They are commonly referred to as flow work. The term Wcv accounts for all other energy transfers by work across the boundary of the control volume. 4.2.2 Forms of the Control Volume Energy Rate Balance Substituting Eq. 4.12 in Eq. 4.9 and collecting all terms referring to the inlet and the exit into separate expressions, the following form of the control volume energy rate balance results # # dEcv V2i V2e # # Qcv Wcv mi aui pivi

gzi b me aue peve

gze b (4.13) dt 2 2 # The subscript “cv” has been added to Q to emphasize that this is the heat transfer rate over the boundary (control surface) of the control volume. The last two terms of Eq. 4.13 can be rewritten using the specific enthalpy h introduced in Sec. 3.3.2. With h u pv, the energy rate balance becomes # # dEcv V2i V2e # # Qcv Wcv mi ahi

gzi b me ahe

gze b dt 2 2

(4.14)

The appearance of the sum u pv in the control volume energy equation is the principal reason for introducing enthalpy previously. It is brought in solely as a convenience: The algebraic form of the energy rate balance is simplified by the use of enthalpy and, as we have seen, enthalpy is normally tabulated along with other properties.

4.3 Analyzing Control Volumes at Steady State

131

In practice there may be several locations on the boundary through which mass enters or exits. This can be accounted for by introducing summations as in the mass balance. Accordingly, the energy rate balance is # # dEcv V2i V2e # # Qcv Wcv a mi ahi

gzi b a me ahe

gze b dt 2 2 i e

(4.15)

Equation 4.15 is an accounting balance for the energy of the control volume. It states that the rate of energy increase or decrease within the control volume equals the difference between the rates of energy transfer in and out across the boundary. The mechanisms of energy transfer are heat and work, as for closed systems, and the energy that accompanies the mass entering and exiting. OTHER FORMS

As for the case of the mass rate balance, the energy rate balance can be expressed in terms of local properties to obtain forms that are more generally applicable. Thus, the term Ecv(t), representing the total energy associated with the control volume at time t, can be written as a volume integral Ecv 1t2

re dV r au

V

V

V2

gzb dV 2

(4.16)

Similarly, the terms accounting for the energy transfers accompanying mass flow and flow work at inlets and exits can be expressed as shown in the following form of the energy rate balance d dt

V

# # re dV Qcv Wcv a c i

A

ah

V2

gzb rVn dA d 2 i

(4.17)

2

V a c ah

gzb r Vn dA d 2 e A e

Additional forms of the energy rate balance can be obtained by expressing the heat transfer # rate Q as the integral of the heat flux over the boundary of the control volume, and the work cv # Wcv in terms of normal and shear stresses at the moving portions of the boundary. In principle, the change in the energy of a control volume over a time period can be obtained by integration of the energy rate balance with respect to time. Such integrations require information about the time dependences of the work and heat transfer rates, the various mass flow rates, and the states at which mass enters and leaves the control volume. Examples of this type of analysis are presented in Sec. 4.4. In Sec. 4.3 to follow, we consider forms that the mass and energy rate balances take for control volumes at steady state, for these are frequently used in practice.

4.3 Analyzing Control Volumes at Steady State

In this section steady-state forms of the mass and energy rate balances are developed and applied to a variety of cases of engineering interest. The steady-state forms obtained do not apply to the transient startup or shutdown periods of operation of such devices, but only to periods of steady operation. This situation is commonly encountered in engineering.

energy rate balance

METHODOLOGY UPDATE

Equation 4.15 is the most general form of the conservation of energy principle for control volumes used in this book. It serves as the starting point for applying the conservation of energy principle to control volumes in problem solving.

132

Chapter 4 Control Volume Analysis Using Energy

4.3.1 Steady-State Forms of the Mass and Energy Rate Balances For a control volume at steady state, the conditions of the mass within the control volume and at the boundary do not vary with time. The mass flow rates and the rates of energy transfer by heat and work are also constant with time. There can be no accumulation of mass within the control volume, so dmcvdt 0 and the mass rate balance, Eq. 4.2, takes the form # a mi

i

1mass rate in2

# a me

(4.18)

e

1mass rate out2

Furthermore, at steady state dEcvdt 0, so Eq. 4.15 can be written as

# # V2i V2e # # 0 Qcv Wcv a mi ahi

gzi b a me ahe

gze b 2 2 i e

(4.19a)

Alternatively # # V2i V2e # # Qcv a mi ahi

gzi b Wcv a me ahe

gze b 2 2 i e 1energy rate in2 1energy rate out2

(4.19b)

Equation 4.18 asserts that at steady state the total rate at which mass enters the control volume equals the total rate at which mass exits. Similarly, Eqs. 4.19 assert that the total rate at which energy is transferred into the control volume equals the total rate at which energy is transferred out. Many important applications involve one-inlet, one-exit control volumes at steady state. It is instructive to apply the mass and energy rate balances to this special case. The mass rate # # balance reduces simply to m1 m 2. That is, the mass flow rate must be the same at the # exit, 2, as it is at the inlet, 1. The common mass flow rate is designated simply by m. Next, applying the energy rate balance and factoring the mass flow rate gives # # 1V21 V22 2 # 0 Qcv Wcv m c 1h1 h2 2

g1z1 z2 2 d 2

(4.20a)

Or, dividing by the mass flow rate

# # 1V21 V22 2 Wcv Qcv 0 # # 1h1 h2 2

g1z1 z2 2 m m 2

(4.20b)

The enthalpy, kinetic energy, and potential energy terms all appear in Eqs. 4.20 as differences between their values at the inlet and exit. This illustrates that the datums used to assign values to specific enthalpy, velocity, and elevation cancel, provided the same ones are # # # # used at the inlet and exit. In Eq. 4.20b, the ratios Qcv m and Wcv m are rates of energy transfer per unit mass flowing through the control volume. The foregoing steady-state forms of the energy rate balance relate only energy transfer quantities evaluated at the boundary of the control volume. No details concerning properties within the control volume are required by, or can be determined with, these equations. When applying the energy rate balance in any of its forms, it is necessary to use the same units for all terms in the equation. For instance, every term in Eq. 4.20b must have a unit such as kJ/kg or Btu/lb. Appropriate unit conversions are emphasized in examples to follow.

4.3 Analyzing Control Volumes at Steady State

4.3.2 Modeling Control Volumes at Steady State In this section, we provide the basis for subsequent applications by considering the modeling of control volumes at steady state. In particular, several examples are given in Sec. 4.3.3 showing the use of the principles of conservation of mass and energy, together with relationships among properties for the analysis of control volumes at steady state. The examples are drawn from applications of general interest to engineers and are chosen to illustrate points common to all such analyses. Before studying them, it is recommended that you review the methodology for problem solving outlined in Sec. 1.7.3. As problems become more complicated, the use of a systematic problem-solving approach becomes increasingly important. When the mass and energy rate balances are applied to a control volume, simplifications are normally needed to make the analysis manageable. That is, the control volume of interest is modeled by making assumptions. The careful and conscious step of listing assumptions is necessary in every engineering analysis. Therefore, an important part of this section is devoted to considering various assumptions that are commonly made when applying the conservation principles to different types of devices. As you study the examples presented in Sec. 4.3.3, it is important to recognize the role played by careful assumption making in arriving at solutions. In each case considered, steady-state operation is assumed. The flow is regarded as one-dimensional at places where mass enters and exits the control volume. Also, at each of these locations equilibrium property relations are assumed to apply. # In several of the examples to follow, the heat transfer term Qcv is set to zero in the energy rate balance because it is small relative to other energy transfers across the boundary. This may be the result of one or more of the following factors:

The outer surface of the control volume is well insulated. The outer surface area is too small for there to be effective heat transfer. The temperature difference between the control volume and its surroundings is so small that the heat transfer can be ignored. The gas or liquid passes through the control volume so quickly that there is not enough time for significant heat transfer to occur. # The work term Wcv drops out of the energy rate balance when there are no rotating shafts, displacements of the boundary, electrical effects, or other work mechanisms associated with the control volume being considered. The kinetic and potential energies of the matter entering and exiting the control volume are neglected when they are small relative to other energy transfers. In practice, the properties of control volumes considered to be at steady state do vary with time. The steady-state assumption would still apply, however, when properties fluctuate only slightly about their averages, as for pressure in Fig. 4.5a. Steady state also might be assumed in cases where periodic time variations are observed, as in Fig. 4.5b. For example, in p

p

pave

pave

t

t (a) Figure 4.5

(b)

Pressure variations about an average. (a) Fluctuation. (b) Periodic.

133

Chapter 4 Control Volume Analysis Using Energy

UNIT ED

Thermodynamics in the News...

NUM

R

QU

Engineers are developing miniature systems for use where weight, portability, and/or compactness are critically important. Some of these applications involve tiny micro systems with dimensions in the micrometer to millimeter range. Other somewhat larger meso-scale systems can measure up to a few centimeters. Microelectromechanical systems (MEMS) combining electrical and mechanical features are now widely used for sensing and control. Medical applications of MEMS include minute pressure sensors that monitor pressure within the balloon inserted into a blood vessel during angioplasty. Air bags are triggered in an automobile crash by tiny acceleration sensors. MEMS are also found in computer hard drives and printers. Miniature versions of other technologies are being investigated. One study aims at developing an entire gas turbine

ES STEAPTLURIBUOSFA

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Smaller Can Be Better

LA

134

reciprocating engines and compressors, the entering and exiting flows pulsate as valves open and close. Other parameters also might be time varying. However, the steady-state assumption can apply to control volumes enclosing these devices if the following are satisfied for each successive period of operation: (1) There is no net change in the total energy and the total mass within the control volume. (2) The time-averaged mass flow rates, heat transfer rates, work rates, and properties of the substances crossing the control surface all remain constant. 4.3.3 Illustrations In this section, we present brief discussions and examples illustrating the analysis of several devices of interest in engineering, including nozzles and diffusers, turbines, compressors and pumps, heat exchangers, and throttling devices. The discussions highlight some common applications of each device and the important modeling assumptions used in thermodynamic analysis. The section also considers system integration, in which devices are combined to form an overall system serving a particular purpose. NOZZLES AND DIFFUSERS

nozzle diffuser

A nozzle is a flow passage of varying cross-sectional area in which the velocity of a gas or liquid increases in the direction of flow. In a diffuser, the gas or liquid decelerates in the direction of flow. Figure 4.6 shows a nozzle in which the cross-sectional area decreases

1

V 2 > V1 p2 < p1

V 2 < V1 p2 > p1 2

2

1

Figure 4.6 Nozzle

Diffuser

diffuser.

Illustration of a nozzle and a

4.3 Analyzing Control Volumes at Steady State

135

Flow-straightening screens

Acceleration

Nozzle

Deceleration Test section

Diffuser

Figure 4.7

Wind-tunnel test

facility.

in the direction of flow and a diffuser in which the walls of the flow passage diverge. In Fig. 4.7, a nozzle and diffuser are combined in a wind-tunnel test facility. Nozzles and diffusers for high-speed gas flows formed from a converging section followed by diverging section are studied in Sec. 9.13. For nozzles and diffusers, the only work is flow work at locations where mass enters and # exits the control volume, so the term Wcv drops out of the energy rate equation for these devices. The change in potential energy from inlet to exit is negligible under most conditions. At steady state the mass and energy rate balances reduce, respectively, to 0

dmcv # # m1 m2 dt 0

# # 0 dEcv V12 V22 # # Qcv Wcv m1 ah1

gz1 b m2 ah2

gz2 b dt 2 2 where 1 denotes the inlet and 2 the exit. By combining these into a single expression and dropping the potential energy change from inlet to exit # Qcv V21 V22 0 # 1h1 h2 2 a b (4.21) m 2 # # # where m is the mass flow rate. The term Qcv m representing heat transfer with the surroundings per unit of mass flowing through the nozzle or diffuser is often small enough relative to the enthalpy and kinetic energy changes that it can be dropped, as in the next example.

EXAMPLE

4.3

Calculating Exit Area of a Steam Nozzle

Steam enters a converging–diverging nozzle operating at steady state with p1 40 bar, T1 400C, and a velocity of 10 m/s. The steam flows through the nozzle with negligible heat transfer and no significant change in potential energy. At the exit, p2 15 bar, and the velocity is 665 m/s. The mass flow rate is 2 kg/s. Determine the exit area of the nozzle, in m2. SOLUTION Known: Steam flows at steady state through a nozzle with known properties at the inlet and exit, a known mass flow rate, and negligible effects of heat transfer and potential energy. Find: Determine the exit area.

136

Chapter 4 Control Volume Analysis Using Energy

Schematic and Given Data:

T

T1 = 400 °C

1

m = 2 kg/s Insulation p = 40 bar

❶

2

p2 = 15 bar V2 = 665 m/s

p1 = 40 bar T1 = 400 °C V1 = 10 m/s

p = 15 bar 1

Control volume boundary

2 v

Figure E4.3

Assumptions: 1. The control volume shown on the accompanying figure is at steady state. # 2. Heat transfer is negligible and Wcv 0. 3. The change in potential energy from inlet to exit can be neglected. # Analysis: The exit area can be determined from the mass flow rate m and Eq. 4.4b, which can be arranged to read A2

# mv2 V2

To evaluate A2 from this equation requires the specific volume v2 at the exit, and this requires that the exit state be fixed. The state at the exit is fixed by the values of two independent intensive properties. One is the pressure p2, which is known. The other is the specific enthalpy h2, determined from the steady-state energy rate balance # 0 # 0 V21 V22 # # 0 Qcv Wcv m ah1

gz1 b m ah2

gz2 b 2 2 # # where Qcv and Wcv are deleted by assumption 2. The change in specific potential energy drops out in accordance with as# sumption 3 and m cancels, leaving 0 1h1 h2 2 a

V21 V22 b 2

Solving for h2 h2 h1 a

V21 V22 b 2

From Table A-4, h1 3213.6 kJ/kg. The velocities V1 and V2 are given. Inserting values and converting the units of the kinetic energy terms to kJ/kg results in

❷

h2 3213.6 kJ/kg c

1102 2 16652 2

2 3213.6 221.1 2992.5 kJ/kg

da

m2 1N 1 kJ b` ` ` 3 # ` 2 2 # s 1 kg m /s 10 N m

4.3 Analyzing Control Volumes at Steady State

137

Finally, referring to Table A-4 at p2 15 bar with h2 2992.5 kJ/kg, the specific volume at the exit is v2 0.1627 m3/kg. The exit area is then

❸

A2

12 kg/s210.1627 m3/ kg2 665 m/s

4.89 104 m2

❶

Although equilibrium property relations apply at the inlet and exit of the control volume, the intervening states of the steam are not necessarily equilibrium states. Accordingly, the expansion through the nozzle is represented on the T–v diagram as a dashed line.

❷ ❸

Care must be taken in converting the units for specific kinetic energy to kJ/kg. # The area at the nozzle inlet can be found similarly, using A1 mv1 V1.

Stationary blades

Figure 4.8

Rotating blades

Schematic of an axial-

flow turbine.

TURBINES

A turbine is a device in which work is developed as a result of a gas or liquid passing through a set of blades attached to a shaft free to rotate. A schematic of an axial-flow steam or gas turbine is shown in Fig. 4.8. Turbines are widely used in vapor power plants, gas turbine power plants, and aircraft engines (Chaps. 8 and 9). In these applications, superheated steam or a gas enters the turbine and expands to a lower exit pressure as work is developed. A hydraulic turbine installed in a dam is shown in Fig. 4.9. In this application, water falling through the propeller causes the shaft to rotate and work is developed.

Water level

Water flow Water level Propeller

Figure 4.9

Hydraulic turbine installed in a dam.

turbine

138

Chapter 4 Control Volume Analysis Using Energy

For a turbine at steady state the mass and energy rate balances reduce to give Eq. 4.20b. When gases are under consideration, the potential energy change is typically negligible. With a proper selection of the boundary of the control volume enclosing the turbine, the kinetic energy change is usually small enough to be neglected. The only heat transfer between the turbine and surroundings would be unavoidable heat transfer, and as illustrated in the next example this is often small relative to the work and enthalpy terms.

EXAMPLE 4.4

Calculating Heat Transfer from a Steam Turbine

Steam enters a turbine operating at steady state with a mass flow rate of 4600 kg/h. The turbine develops a power output of 1000 kW. At the inlet, the pressure is 60 bar, the temperature is 400C, and the velocity is 10 m/s. At the exit, the pressure is 0.1 bar, the quality is 0.9 (90%), and the velocity is 50 m/s. Calculate the rate of heat transfer between the turbine and surroundings, in kW. SOLUTION Known: A steam turbine operates at steady state. The mass flow rate, power output, and states of the steam at the inlet and exit are known. Find: Calculate the rate of heat transfer. Schematic and Given Data: 1 T

m· 1 = 4600 kg/h p1 = 60 bar T1 = 400°C V1 = 10 m/s 1

T1 = 400°C

p = 60 bar

· Wcv = 1000 kW p = 0.1 bar 2 p2 = 0.1 bar x2 = 0.9 (90%) V2 = 50 m/s

2 v Figure E4.4

Assumptions: 1. The control volume shown on the accompanying figure is at steady state. 2. The change in potential energy from inlet to exit can be neglected. Analysis: To calculate the heat transfer rate, begin with the one-inlet, one-exit form of the energy rate balance for a control volume at steady state # # V21 V22 # #

gz1 b m ah2

gz2 b 0 Qcv Wcv m ah1

2 2 # # where m is the mass flow rate. Solving for Qcv and dropping the potential energy change from inlet to exit # # V22 V21 # Qcv Wcv m c 1h2 h1 2 a bd 2 To compare the magnitudes of the enthalpy and kinetic energy terms, and stress the unit conversions needed, each of these terms is evaluated separately.

4.3 Analyzing Control Volumes at Steady State

139

First, the specific enthalpy difference h2 h1 is found. Using Table A-4, h1 3177.2 kJ/kg. State 2 is a two-phase liquid–vapor mixture, so with data from Table A-3 and the given quality h2 hf 2 x2 1hg2 hf 2 2

191.83 10.92 12392.82 2345.4 kJ/kg

Hence h2 h1 2345.4 3177.2 831.8 kJ/kg Consider next the specific kinetic energy difference. Using the given values for the velocities

❶

a

1502 2 1102 2 m2 V22 V21 1N 1 kJ b c da 2b` ` ` ` 2 2 s 1 kg # m/s2 103 N # m 1.2 kJ/kg

# Calculating Qcv from the above expression # kg kJ 1h 1 kW Qcv 11000 kW2 a4600 b 1831.8 1.22 a b ` ` ` ` h kg 3600 s 1 kJ/s

❷

61.3 kW

❶ ❷

The magnitude of the change in specific kinetic energy from inlet to exit is much smaller than the specific enthalpy change. # The negative value of Qcv means that there is heat transfer from the turbine to its surroundings, as would be expected. The # magnitude of Qcv is small relative to the power developed.

COMPRESSORS AND PUMPS

Compressors are devices in which work is done on a gas passing through them in order to raise the pressure. In pumps, the work input is used to change the state of a liquid passing through. A reciprocating compressor is shown in Fig. 4.10. Figure 4.11 gives schematic diagrams of three different rotating compressors: an axial-flow compressor, a centrifugal compressor, and a Roots type. The mass and energy rate balances reduce for compressors and pumps at steady state, as for the case of turbines considered previously. For compressors, the changes in specific kinetic and potential energies from inlet to exit are often small relative to the work done per unit of mass passing through the device. Heat transfer with the surroundings is frequently a secondary effect in both compressors and pumps. The next two examples illustrate, respectively, the analysis of an air compressor and a power washer. In each case the objective is to determine the power required to operate the device.

Inlet

Outlet

Figure 4.10

Reciprocating compressor.

compressor pump

140

Chapter 4 Control Volume Analysis Using Energy Outlet Rotor Stator

Impeller Impeller

Inlet

Driveshaft (a)

(b) Outlet

Inlet (c) Figure 4.11

EXAMPLE

4.5

Rotating compressors. (a) Axial flow. (b) Centrifugal. (c) Roots type.

Calculating Compressor Power

Air enters a compressor operating at steady state at a pressure of 1 bar, a temperature of 290 K, and a velocity of 6 m/s through an inlet with an area of 0.1 m2. At the exit, the pressure is 7 bar, the temperature is 450 K, and the velocity is 2 m/s. Heat transfer from the compressor to its surroundings occurs at a rate of 180 kJ/min. Employing the ideal gas model, calculate the power input to the compressor, in kW. SOLUTION Known: An air compressor operates at steady state with known inlet and exit states and a known heat transfer rate. Find: Calculate the power required by the compressor. Schematic and Given Data: Assumptions: 1. The control volume shown on the accompanying figure is at steady state. 2. The change in potential energy from inlet to exit can be neglected.

❶

3. The ideal gas model applies for the air.

· Wcv = ? p1 = 1 bar T1 = 290 K 1 V1 = 6 m/s A1 = 0.1m2

Air compressor

2 p2 = 7 bar T2 = 450 K V2 = 2 m/s

· Q cv = –180 kJ/min

Figure E4.5

4.3 Analyzing Control Volumes at Steady State

141

Analysis: To calculate the power input to the compressor, begin with the energy rate balance for the one-inlet, one-exit control volume at steady state: # # V21 V22 # # 0 Qcv Wcv m ah1

gz1 b m ah2

gz2 b 2 2 Solving # # V21 V22 # Wcv Qcv m c 1h1 h2 2 a bd 2 The change in potential energy from inlet to exit drops out by assumption 2. # The mass flow rate m can be evaluated with given data at the inlet and the ideal gas equation of state. 10.1 m2 216 m/s2 1105 N/m2 2 A1V1p1 A1V1 # 0.72 kg/s m v1 1RM2T1 8314 N # m a b 1290 K2 28.97 kg # K

The specific enthalpies h1 and h2 can be found # from Table A-22. At 290 K, h1 290.16 kJ/kg. At 450 K, h2 451.8 kJ/kg. Substituting values into the expression for Wcv # kg kJ 1 min kJ Wcv a180 b` ` 0.72 c 1290.16 451.82 min 60 s s kg

a

162 2 122 2 2

ba

m2 1N 1 kJ `d b` ` ` s2 1 kg # m/s2 103 N # m

❷

kg kJ kJ 3

0.72 1161.64 0.022 s s kg

❸

119.4

❶ ❷ ❸

kJ 1 kW ` ` 119.4 kW s 1 kJ/s

The applicability of the ideal gas model can be checked by reference to the generalized compressibility chart. The contribution of the kinetic energy is negligible in this case. Also, the heat transfer rate is seen to be small relative to the power input. # # In this example Qcv and Wcv have negative values, indicating that the direction of the heat transfer is from the compressor and work is done on the air passing through the compressor. The magnitude of the power input to the compressor is 119.4 kW.

EXAMPLE

4.6

Power Washer

A power washer is being used to clean the siding of a house. Water enters at 20C, 1 atm, with a volumetric flow rate of 0.1 liter/s through a 2.5-cm-diameter hose. A jet of water exits at 23C, 1 atm, with a velocity of 50 m/s at an elevation of 5 m. At steady state, the magnitude of the heat transfer rate from the power unit to the surroundings is 10% of the power input. The water can be considered incompressible, and g 9.81 m/s2. Determine the power input to the motor, in kW. SOLUTION Known: A power washer operates at steady state with known inlet and exit conditions. The heat transfer rate is known as a percentage of the power input. Find: Determine the power input.

142

Chapter 4 Control Volume Analysis Using Energy

Schematic and Given Data:

p2 = 1 atm T2 = 23°C V2 = 50 m/s z2 = 5 m

2

Assumptions: 1. A control volume enclosing the power unit and the delivery hose is at steady state.

5m

2. The water is modeled as incompressible. p1 = 1 atm T1 = 20°C (AV)1 = 0.1 liter/s D1 = 2.5 cm 1

Hose +

–

Figure E4.6

Analysis: To calculate the power input, begin with the one-inlet, one-exit form of the energy balance for a control volume at steady state

❶

# # V21 V22 # 0 Qcv Wcv m c 1h1 h2 2 a b g1z1 z2 2 d 2 # # # Introducing Qcv 10.12Wcv, and solving for Wcv # 1V21 V22 2 # m c 1h1 h2 2

g 1z1 z2 2 d Wcv 0.9 2 # The mass flow rate m can be evaluated using the given volumetric flow rate and v vf (20C) 1.0018 103 m3/kg from Table A-2, as follows # m 1AV2 1 v 10.1 L /s2 11.0018 103 m3/kg2 `

103 m3 ` 1L

0.1 kg/s

❷

Dividing the given volumetric flow rate by the inlet area, the inlet velocity is V1 0.2 m/s. The specific enthalpy term is evaluated using Eq. 3.20b, with p1 p2 1 atm and c 4.18 kJ/kg # K from Table A-19 h1 h2 c 1T1 T2 2 v1p1 p2 2 14.18 kJ/kg # K2 13 K2 12.54 kJ/kg 0

Evaluating the specific kinetic energy term V21

2

V22

m 2 3 10.22 2 1502 2 4 a b s 1N 1 kJ ` ` 1.25 kJ/kg ` ` 2 1 kg # m /s2 103 N # m

4.3 Analyzing Control Volumes at Steady State

143

Finally, the specific potential energy term is g1z1 z2 2 19.81 m /s2 2 10 52m ` Inserting values

❸

1N 1 kJ ` ` 3 # ` 0.05 kJ/kg 2 # 1 kg m /s 10 N m

# 10.1 kg/s2 kJ 1 kW Wcv 3 112.542 11.252 10.052 4 a b ` ` 0.9 kg 1 kJ/s

Thus # Wcv 1.54 kW where the minus sign indicates that power is provided to the washer.

❶ ❷ ❸

# Since power is required to operate the washer, Wcv is negative in# accord with our sign convention. The energy # transfer by heat is# from the control volume to the surroundings, and thus is negative as well. Using the value of Q W cv cv found be# low, Qcv 10.12Wcv 0.154 kW. The power washer develops a high-velocity jet of water at the exit. The inlet velocity is small by comparison. The power input to the washer is accounted for by heat transfer from the washer to the surroundings and the increases in specific enthalpy, kinetic energy, and potential energy of the water as it is pumped through the power washer.

HEAT EXCHANGERS

Devices that transfer energy between fluids at different temperatures by heat transfer modes such as discussed in Sec. 2.4.2 are called heat exchangers. One common type of heat exchanger is a vessel in which hot and cold streams are mixed directly as shown in Fig. 4.12a. An open feedwater heater is an example of this type of device. Another common type of heat

Figure 4.12

(a)

(b)

(c)

(d)

Common heat exchanger types. (a) Direct contact heat exchanger. (b) Tube-within-a-tube counterflow heat exchanger. (c) Tube-withina-tube parallel flow heat exchanger. (d ) Cross-flow heat exchanger.

heat exchanger

144

Chapter 4 Control Volume Analysis Using Energy

exchanger is one in which a gas or liquid is separated from another gas or liquid by a wall through which energy is conducted. These heat exchangers, known as recuperators, take many different forms. Counterflow and parallel tube-within-a-tube configurations are shown in Figs. 4.12b and 4.12c, respectively. Other configurations include cross-flow, as in automobile radiators, and multiple-pass shell-and-tube condensers and evaporators. Figure 4.12d illustrates a cross-flow heat exchanger. The only work interaction at the boundary of a control volume enclosing a heat exchanger # is flow work at the places where matter enters and exits, so the term Wcv of the energy rate balance can be set to zero. Although high rates of energy transfer may be achieved from stream to stream, the heat transfer from the outer surface of the heat exchanger to the surroundings is often small enough to be neglected. In addition, the kinetic and potential energies of the flowing streams can often be ignored at the inlets and exits. The next example illustrates how the mass and energy rate balances are applied to a condenser at steady state. Condensers are commonly found in power plants and refrigeration systems.

EXAMPLE

4.7

Power Plant Condenser

Steam enters the condenser of a vapor power plant at 0.1 bar with a quality of 0.95 and condensate exits at 0.1 bar and 45C. Cooling water enters the condenser in a separate stream as a liquid at 20C and exits as a liquid at 35C with no change in pressure. Heat transfer from the outside of the condenser and changes in the kinetic and potential energies of the flowing streams can be ignored. For steady-state operation, determine (a) the ratio of the mass flow rate of the cooling water to the mass flow rate of the condensing stream. (b) the rate of energy transfer from the condensing steam to the cooling water, in kJ per kg of steam passing through the condenser. SOLUTION Known: Steam is condensed at steady state by interacting with a separate liquid water stream. Find: Determine the ratio of the mass flow rate of the cooling water to the mass flow rate of the steam and the rate of energy transfer from the steam to the cooling water. Schematic and Given Data: Condensate 0.1 bar 2 45°C

Steam 0.1 bar x = 0.95

1

T Cooling water 20°C

3

4

Control volume for part (a)

Condensate

2

1

Cooling water 35°C

0.1 bar 45.8°C

Steam

1

2 4 3

Energy transfer to cooling water

v

Control volume for part (b) Figure E4.7

4.3 Analyzing Control Volumes at Steady State

145

Assumptions: 1. Each of the two control volumes shown on the accompanying sketch is at steady-state. # 2. There is no significant heat transfer between the overall condenser and its surroundings, and Wcv 0. 3. Changes in the kinetic and potential energies of the flowing streams from inlet to exit can be ignored. 4. At states 2, 3, and 4, h hf (T ) (see Eq. 3.14). Analysis: The steam and the cooling water streams do not mix. Thus, the mass rate balances for each of the two streams reduce at steady state to give # # # # m1 m2 and m3 m4 # # (a) The ratio of the mass flow rate of the cooling water to the mass flow rate of the condensing steam, m3 m1, can be found from the steady-state form of the energy rate balance applied to the overall condenser as follows: # # V23 V21 # #

gz1 b m3 ah3

gz3 b 0 Qcv Wcv m1 ah1

2 2 V22 V24 # #

gz2 b m4 ah4

gz4 b m2 ah2

2 2 The underlined terms drop out by assumptions 2 and 3. With these simplifications, together with the above mass flow rate relations, the energy rate balance becomes simply # # 0 m1 1h1 h2 2 m3 1h3 h4 2 Solving, we get # m3 h1 h2 # m1 h4 h3 The specific enthalpy h1 can be determined using the given quality and data from Table A-3. From Table A-3 at 0.1 bar, hf 191.83 kJ/kg and hg 2584.7 kJ/kg, so h1 191.83 0.9512584.7 191.832 2465.1 kJ/kg

❶

Using assumption 4, the specific enthalpy at 2 is given by h2 hf (T2) 188.45 kJ/kg. Similarly, h3 hf (T3) and h4 hf (T4), giving h4 h3 62.7 kJ/kg. Thus # m3 2465.1 188.45 36.3 # m1 62.7 (b) For a control volume enclosing the steam side of the condenser only, the steady-state form of energy rate balance is

❷

# # V21 V22 # # 0 Qcv Wcv m1 ah1

gz1 b m2 ah2

gz2 b 2 2 # # The underlined terms drop out by assumptions 2 and 3. Combining this equation with m1 m2, the following expression for the rate of energy transfer between the condensing steam and the cooling water results: # # Qcv m1 1h2 h1 2 # Dividing by the mass flow rate of the steam, m1, and inserting values # Qcv # h2 h1 188.45 2465.1 2276.7 kJ/kg m1 where the minus sign signifies that energy is transferred from the condensing steam to the cooling water.

❶ ❷

Alternatively, (h4 h3) can be evaluated using the incompressible liquid model via Eq. 3.20b. Depending on where the boundary of the control volume is located, two different formulations of the energy rate balance are obtained. In part (a), both streams are included in the control# volume. Energy transfer between them occurs internally and not across the boundary of the control # volume, so the term Qcv drops out of the energy rate balance. With the control volume of part (b), however, the term Qcv must be included.

146

Chapter 4 Control Volume Analysis Using Energy

Excessive temperatures in electronic components are avoided by providing appropriate cooling, as illustrated in the next example.

EXAMPLE

4.8

Cooling Computer Components

The electronic components of a computer are cooled by air flowing through a fan mounted at the inlet of the electronics enclosure. At steady state, air enters at 20C, 1 atm. For noise control, the velocity of the entering air cannot exceed 1.3 m/s. For temperature control, the temperature of the air at the exit cannot exceed 32C. The electronic components and fan receive, respectively, 80 W and 18 W of electric power. Determine the smallest fan inlet diameter, in cm, for which the limits on the entering air velocity and exit air temperature are met. SOLUTION Known: The electronic components of a computer are cooled by air flowing through a fan mounted at the inlet of the electronics enclosure. Conditions are specified for the air at the inlet and exit. The power required by the electronics and the fan are also specified. Find: Determine for these conditions the smallest fan inlet diameter. Schematic and Given Data: +

Electronic components

–

T2 ≤ 32°C Air out 2

Fan

1

Air in

T1 = 20°C p1 = 1 atm V1 ≤ 1.3 m/s

Figure E4.8

Assumptions: 1. The control volume shown on the accompanying figure is at steady state. # 2. Heat transfer from the outer surface of the electronics enclosure to the surroundings is negligible. Thus, Qcv 0.

❶ ❷

3. Changes in kinetic and potential energies can be ignored. 4. Air is modeled as an ideal gas with cp 1.005 kJ/kg # K. # Analysis: The inlet area A1 can be determined from the mass flow rate m and Eq. 4.4b, which can be rearranged to read # mv1 A1 V1 The mass flow rate can be evaluated, in turn, from the steady-state energy rate balance # # V 21 V 22 # b g1z1 z2 2 d 0 Qcv Wcv m c 1h1 h2 2 a 2 The underlined terms drop out by assumptions 2 and 3, leaving # # 0 Wcv m 1h1 h2 2

4.3 Analyzing Control Volumes at Steady State

147

# # where Wcv accounts for the total electric power provided to the electronic components and the fan: Wcv (80 W)

# (18 W) 98 W. Solving for m, and using assumption 4 with Eq. 3.51 to evaluate (h1 h2) # 1Wcv 2 # m cp 1T2 T1 2 Introducing this into the expression for A1 and using the ideal gas model to evaluate the specific volume v1 # 1Wcv 2 RT1 1 A1 c da b p1 V1 cp 1T2 T1 2

From this expression we see that A1 increases when V1 and/or T2 decrease. Accordingly, since V1 1.3 m/s and T2 305 K (32C), the inlet area must satisfy

A1

1 ≥ 1.3 m/s

98 W kJ a1.005 # b 1305 2932 K kg K

a

8314 N # m b 293 K 1 kJ 1 J/s 28.97 kg # K ` 3 ` ` ≤ ` ¥± 1.01325 105 N/m2 10 J 1 W

0.005 m2 Then, since A1 pD21 4

14210.005 m2 2 102 cm ` 0.08 m ` p 1m B D1 8 cm D1

For the specified conditions, the smallest fan inlet diameter is 8 cm.

❶ ❷

Cooling air typically enters and exits electronic enclosures at low velocities, and thus kinetic energy effects are insignificant. The applicability of the ideal gas model can be checked by reference to the generalized compressibility chart. Since the temperature of the air increases by no more than 12C, the specific heat cp is nearly constant (Table A-20).

THROTTLING DEVICES

A significant reduction in pressure can be achieved simply by introducing a restriction into a line through which a gas or liquid flows. This is commonly done by means of a partially opened valve or a porous plug, as illustrated in Fig. 4.13. For a control volume enclosing such a device, the mass and energy rate balances reduce at steady state to # # 0 m1 m2 # # 0 V21 V22 # # 0 Qcv Wcv m1 ah1

gz1 b m2 ah2

gz2 b 2 2

Inlet Inlet

Partially open valve

Figure 4.13

Exit

Examples of throttling devices.

Porous plug

Exit

148

Chapter 4 Control Volume Analysis Using Energy

There is usually no significant heat transfer with the surroundings and the change in potential energy from inlet to exit is negligible. With these idealizations, the mass and energy rate balances combine to give V21 V22 h2

2 2 Although velocities may be relatively high in the vicinity of the restriction, measurements made upstream and downstream of the reduced flow area show in most cases that the change in the specific kinetic energy of the gas or liquid between these locations can be neglected. With this further simplification, the last equation reduces to h1

h1 h2

throttling process

When the flow through a valve or other restriction is idealized in this way, the process is called a throttling process. An application of the throttling process occurs in vapor-compression refrigeration systems, where a valve is used to reduce the pressure of the refrigerant from the pressure at the exit of the condenser to the lower pressure existing in the evaporator. We consider this further in Chap. 10. The throttling process also plays a role in the Joule–Thomson expansion considered in Chap. 11. Another application of the throttling process involves the throttling calorimeter, which is a device for determining the quality of a two-phase liquid–vapor mixture. The throttling calorimeter is considered in the next example.

throttling calorimeter

EXAMPLE

(4.22)

Measuring Steam Quality

4.9

A supply line carries a two-phase liquid–vapor mixture of steam at 20 bars. A small fraction of the flow in the line is diverted through a throttling calorimeter and exhausted to the atmosphere at 1 bar. The temperature of the exhaust steam is measured as 120C. Determine the quality of the steam in the supply line. SOLUTION Known: Steam is diverted from a supply line through a throttling calorimeter and exhausted to the atmosphere. Find: Determine the quality of the steam in the supply line. Schematic and Given Data:

Steam line, 20 bars

p

Thermometer 1

1

p1 = 20 bars

Calorimeter p2 = 1 bar 2 T2 = 120°C

2

p2 = 1 bar T2 = 120°C

v Figure E4.9

4.3 Analyzing Control Volumes at Steady State

149

Assumptions: 1. The control volume shown on the accompanying figure is at steady state. 2. The diverted steam undergoes a throttling process.

❶

Analysis: For a throttling process, the energy and mass balances reduce to give h2 h1, which agrees with Eq. 4.22. Thus, with state 2 fixed, the specific enthalpy in the supply line is known, and state 1 is fixed by the known values of p1 and h1. As shown on the accompanying p–v diagram, state 1 is in the two-phase liquid–vapor region and state 2 is in the superheated vapor region. Thus h2 h1 hf1 x1 1hg1 hf1 2 Solving for x1 x1

h2 hf1 hg1 hf1

From Table A-3 at 20 bars, hf1 908.79 kJ/kg and hg1 2799.5 kJ/kg. At 1 bar and 120C, h2 2766.6 kJ/kg from Table A-4. Inserting values into the above expression, the quality in the line is x1 0.956 (95.6%).

❶

For throttling calorimeters exhausting to the atmosphere, the quality in the line must be greater than about 94% to ensure that the steam leaving the calorimeter is superheated.

SYSTEM INTEGRATION

Thus far, we have studied several types of components selected from those commonly seen in practice. These components are usually encountered in combination, rather than individually. Engineers often must creatively combine components to achieve some overall objective, subject to constraints such as minimum total cost. This important engineering activity is called system integration. Many readers are already familiar with a particularly successful system integration: the simple power plant shown in Fig. 4.14. This system consists of four components in series, a turbine-generator, condenser, pump, and boiler. We consider such power plants in detail in subsequent sections of the book. The example to follow provides another illustration. Many more are considered in later sections and in end-of-chapter problems.

˙ in Q

Boiler

W˙ p

Pump

Turbine

W˙ t

Condenser

˙ out Q

Figure 4.14

Simple vapor power plant.

150

Chapter 4 Control Volume Analysis Using Energy

Thermodynamics in the News... Sensibly Built Homes Cost No More Healthy, comfortable homes that cut energy and water bills and protect the environment cost no more, builders say. The “I have a Dream House,” a highly energy efficient and environmentally responsible house located close to the Atlanta boyhood home of Dr. Martin Luther King Jr., is a prime example. The house, developed under U.S. Department of Energy auspices, can be heated and cooled for less than a dollar a day, and uses 57% less energy for heating and cooling than a conventional house. Still, construction costs are no more than for a conventional house. Designers used a whole-house integrated system approach whereby components are carefully selected to be complementary in achieving an energy-thrifty, cost-effective outcome.

EXAMPLE

4.10

The walls, roof, and floor of this 1565 square-foot house are factory-built structural insulated panels incorporating foam insulation. This choice allowed designers to reduce the size of the heating and cooling equipment, thereby lowering costs. The house also features energy-efficient windows, tightly sealed ductwork, and a high-efficiency air conditioner that further contribute to energy savings.

Waste Heat Recovery System

An industrial process discharges gaseous combustion products at 478K, 1 bar with a mass flow rate of 69.78 kg/s. As shown in Fig. E 4.10, a proposed system for utilizing the combustion products combines a heat-recovery steam generator with a turbine. At steady state, combustion products exit the steam generator at 400K, 1 bar and a separate stream of water enters at .275 MPa, 38.9C with a mass flow rate of 2.079 kg/s. At the exit of the turbine, the pressure is 0.07 bars and the quality is 93%. Heat transfer from the outer surfaces of the steam generator and turbine can be ignored, as can the changes in kinetic and potential energies of the flowing streams. There is no significant pressure drop for the water flowing through the steam generator. The combustion products can be modeled as air as an ideal gas. (a) Determine the power developed by the turbine, in kJ/s. (b) Determine the turbine inlet temperature, in C. SOLUTION Known: Steady-state operating data are provided for a system consisting of a heat-recovery steam generator and a turbine. Find: Determine the power developed by the turbine and the turbine inlet temperature. Evaluate the annual value of the power developed. Schematic and Given Data:

p1 = 1 bar T1 = 478°K m1 = 69.78 kg/s 1

Assumptions: 1. The control volume shown on the accompanying figure is at steady state. Turbine 4

2 T2 = 400°K p2 = 1 bar

Steam generator 5 3 p3 = .275 MPa T3 = 38.9°C m3 = 2.08 kg/s

Power out

2. Heat transfer is negligible, and changes in kinetic and potential energy can be ignored. 3. There is no pressure drop for water flowing through the steam generator. 4. The combustion products are modeled as air as an ideal gas.

p5 = .07 bars x5 = 93% Figure E4.10

4.3 Analyzing Control Volumes at Steady State

151

Analysis: (a) The power developed by the turbine is determined from a control volume enclosing both the steam generator and the turbine. Since the gas and water streams do not mix, mass rate balances for each of the streams reduce, respectively, to give # # m1 m2,

# # m3 m5

The steady-state form of the energy rate balance is # # V 23 V 21 # # 0 Qcv Wcv m1 ah1

gz1 b m3 ah3

gz3 b 2 2 V 25 V 22 # # m2 ah2

gz2 b m5 ah5

gz5 b 2 2 The underlined terms drop out by assumption 2. With these simplifications, together with the above mass flow rate relations, the energy rate balance becomes # # # Wcv m1 1h1 h2 2 m3 1h3 h5 2 # # where m1 69.78 kg/s, m3 2.08 kg/s The specific enthalpies h1 and h2 can be found from Table A-22: At 478 K, h1 480.35 kJ/kg, and at 400 K, h2 400.98 kJ/kg. At state 3, water is a liquid. Using Eq. 3.14 and saturated liquid data from Table A-2, h3 hf 1T3 2 162.9 kJ /kg. State 5 is a two-phase liquid–vapor mixture. With data from Table A-3 and the given quality h5 161 0.9312571.72 1612 2403 kJ/kg # Substituting values into the expression for Wcv # Wcv 169.78 kg/s2 1480.3 400.982 kJ/kg

12.079 kg/s2 1162.9 24032 kJ/kg

876.8 kJ/s 876.8 kW

❶

(b) To determine T4, it is necessary to fix the state at 4. This requires two independent property values. With assumption 3, one of these properties is pressure, p4 0.275 MPa. The other is the specific enthalpy h4, which can be found from an energy rate balance for a control volume enclosing just the steam generator. Mass rate balances for each of the two streams give # # # # m1 m2 and m3 m4. With assumption 2 and these mass flow rate relations, the steady-state form of the energy rate balance reduces to # # 0 m1 1h1 h2 2 m3 1h3 h4 2 Solving for h4 # m1 h4 h3 # 1h1 h2 2 m3 162.9

❷

kJ 69.78 kJ

a b 1480.3 400.982 kg 2.079 kg

2825 kJ/kg Interpolating in Table A-4 at p4 .275 MPa with h4, T4 180C.

❶ ❷

Alternatively, to determine h4 a control volume enclosing just the turbine can be considered. This is left as an exercise. The decision about implementing this solution to the problem of utilizing the hot combustion products discharged from an industrial process would necessarily rest on the outcome of a detailed economic evaluation, including the cost of purchasing and operating the steam generator, turbine, and auxiliary equipment.

152

Chapter 4 Control Volume Analysis Using Energy

4.4 Transient Analysis

transient

Many devices undergo periods of transient operation in which the state changes with time. Examples include the startup or shutdown of turbines, compressors, and motors. Additional examples are provided by vessels being filled or emptied, as considered in Example 4.2 and in the discussion of Fig. 1.3. Because property values, work and heat transfer rates, and mass flow rates may vary with time during transient operation, the steady-state assumption is not appropriate when analyzing such cases. Special care must be exercised when applying the mass and energy rate balances, as discussed next. MASS BALANCE

First, we place the control volume mass balance in a form that is suitable for transient analysis. We begin by integrating the mass rate balance, Eq. 4.2, from time 0 to a final time t. That is

t

a

dmcv b dt dt

t

# a a mi b dt i

t

a a m# b dt e

e

This takes the form mcv 1t2 mcv 102 a a i

t

# mi dtb a a e

t

m# dtb e

Introducing the following symbols for the underlined terms

mi

t

t

# mi dt

amount of mass entering the control d volume through inlet i, from time 0 to t

# me dt

amount of mass exiting the control d volume through exit e, from time 0 to t

me

the mass balance becomes mcv 1t2 mcv 102 a mi a me i

(4.23)

e

In words, Eq. 4.23 states that the change in the amount of mass contained in the control volume equals the difference between the total incoming and outgoing amounts of mass. ENERGY BALANCE

Next, we integrate the energy rate balance, Eq. 4.15, ignoring the effects of kinetic and potential energy. The result is Ucv 1t2 Ucv 102 Qcv Wcv a a i

t

# mi hi dtb a a e

t

m# h dtb e e

(4.24a)

where Qcv accounts for the net amount of energy transferred by heat into the control volume and Wcv accounts for the net amount of energy transferred by work, except for flow work.

4.4 Transient Analysis

153

The integrals shown underlined in Eq. 4.24a account for the energy carried in at the inlets and out at the exits. For the special case where the states at the inlets and exits are constant with time, the respective specific enthalpies, hi and he, would be constant, and the underlined terms of Eq. 4.24a become t

t

m# dt h m m# h dt h m# dt h m # mihi dt hi

0 t

i

i

i

t

e e

e

e

e

e

Equation 4.24a then takes the following special form Ucv 1t2 Ucv 102 Qcv Wcv a mihi a mehe i

(4.24b)

e

Whether in the general form, Eq. 4.24a, or the special form, Eq. 4.24b, these equations account for the change in the amount of energy contained within the control volume as the difference between the total incoming and outgoing amounts of energy. Another special case is when the intensive properties within the control volume are uniform with position at each instant. Accordingly, the specific volume and the specific internal energy are uniform throughout and can depend only on time, that is v(t) and u(t). Thus mcv 1t2 Vcv 1t2 v1t2 Ucv 1t2 mcv 1t2u1t2

(4.25)

When the control volume is comprised of different phases, the state of each phase would be assumed uniform throughout. The following examples provide illustrations of the transient analysis of control volumes using the conservation of mass and energy principles. In each case considered, we begin with the general forms of the mass and energy balances and reduce them to forms suited for the case at hand, invoking the idealizations discussed in this section when warranted. The first example considers a vessel that is partially emptied as mass exits through a valve.

EXAMPLE

4.11

Withdrawing Steam from a Tank at Constant Pressure

A tank having a volume of 0.85 m3 initially contains water as a two-phase liquid—vapor mixture at 260C and a quality of 0.7. Saturated water vapor at 260C is slowly withdrawn through a pressure-regulating valve at the top of the tank as energy is transferred by heat to maintain the pressure constant in the tank. This continues until the tank is filled with saturated vapor at 260C. Determine the amount of heat transfer, in kJ. Neglect all kinetic and potential energy effects. SOLUTION Known: A tank initially holding a two-phase liquid–vapor mixture is heated while saturated water vapor is slowly removed. This continues at constant pressure until the tank is filled only with saturated vapor. Find: Determine the amount of heat transfer.

Chapter 4 Control Volume Analysis Using Energy

154

Schematic and Given Data: Pressure regulating valve e

T

1

2, e

Saturated watervapor removed, while the tank is heated

260°C

v

e

Initial: two-phase liquid-vapor mixture

Final: saturated vapor

Figure E4.11

Assumptions: 1. The control volume is defined by the dashed line on the accompanying diagram. # 2. For the control volume, Wcv 0 and kinetic and potential energy effects can be neglected. 3. At the exit the state remains constant.

❶

4. The initial and final states of the mass within the vessel are equilibrium states. Analysis: Since there is a single exit and no inlet, the mass rate balance takes the form dmcv # me dt With assumption 2, the energy rate balance reduces to # dUcv # Qcv mehe dt Combining the mass and energy rate balances results in # dUcv dmcv Qcv he dt dt By assumption 3, the specific enthalpy at the exit is constant. Accordingly, integration of the last equation gives ¢Ucv Qcv he ¢mcv Solving for the heat transfer Qcv Qcv ¢Ucv he ¢mcv or

❷

Qcv 1m2u2 m1u1 2 he 1m2 m1 2 where m1 and m2 denote, respectively, the initial and final amounts of mass within the tank. The terms u1 and m1 of the foregoing equation can be evaluated with property values from Table A-2 at 260C and the given value for quality. Thus u1 uf x1 1ug uf 2

1128.4 10.7212599.0 1128.42 2157.8 kJ/kg

4.4 Transient Analysis

155

Also, v1 vf x1 1vg uf 2

1.2755 103 10.7210.04221 1.2755 103 2 29.93 103 m3/kg

Using the specific volume v1, the mass initially contained in the tank is m1

V 0.85 m3 28.4 kg v1 129.93 103 m3/kg2

The final state of the mass in the tank is saturated vapor at 260C, so Table A-2 gives u2 ug 1260°C2 2599.0 kJ/kg,

v2 vg 1260°C2 42.21 103 m3/kg

The mass contained within the tank at the end of the process is m2

V 0.85 m3 20.14 kg v2 42.21 103 m3/kg

Table A-2 also gives he hg 1260°C2 2796.6 kJ/kg. Substituting values into the expression for the heat transfer yields Qcv 120.14212599.02 128.4212157.82 2796.6120.14 28.42 14,162 kJ

❶

In this case, idealizations are made about the state of the vapor exiting and the initial and final states of the mass contained within the tank.

❷

This expression for Qcv could be obtained by applying Eq. 4.24b together with Eqs. 4.23 and 4.25

In the next two examples we consider cases where tanks are filled. In Example 4.12, an initially evacuated tank is filled with steam as power is developed. In Example 4.13, a compressor is used to store air in a tank.

EXAMPLE

4.12

Using Steam for Emergency Power Generation

Steam at a pressure of 15 bar and a temperature of 320C is contained in a large vessel. Connected to the vessel through a valve is a turbine followed by a small initially evacuated tank with a volume of 0.6 m3. When emergency power is required, the valve is opened and the tank fills with steam until the pressure is 15 bar. The temperature in the tank is then 400C. The filling process takes place adiabatically and kinetic and potential energy effects are negligible. Determine the amount of work developed by the turbine, in kJ. SOLUTION Known: Steam contained in a large vessel at a known state flows from the vessel through a turbine into a small tank of known volume until a specified final condition is attained in the tank. Find: Determine the work developed by the turbine.

156

Chapter 4 Control Volume Analysis Using Energy

Schematic and Given Data: Valve Steam at 15 bar, 320°C

Control volume boundary

Turbine

V= 0.6 m3 Initially evacuated tank

Figure E4.12

Assumptions: 1. The control volume is defined by the dashed line on the accompanying diagram. # 2. For the control volume, Qcv 0 and kinetic and potential energy effects are negligible.

❶

3. The state of the steam within the large vessel remains constant. The final state of the steam in the smaller tank is an equilibrium state. 4. The amount of mass stored within the turbine and the interconnecting piping at the end of the filling process is negligible. Analysis: Since the control volume has a single inlet and no exits, the mass rate balance reduces to dmcv # mi dt The energy rate balance reduces with assumption 2 to # dUcv # Wcv mi hi dt Combining the mass and energy rate balances gives # dUcv dmcv Wcv hi dt dt Integrating ¢Ucv Wcv hi ¢mcv

❷

In accordance with assumption 3, the specific enthalpy of the steam entering the control volume is constant at the value corresponding to the state in the large vessel. Solving for Wcv Wcv hi ¢mcv ¢Ucv Ucv and mcv denote, respectively, the changes in internal energy and mass of the control volume. With assumption 4, these terms can be identified with the small tank only. Since the tank is initially evacuated, the terms Ucv and mcv reduce to the internal energy and mass within the tank at the end of the process. That is ¢Ucv 1m2u2 2 1m1u1 2 , 0

¢mcv m2 m1

where 1 and 2 denote the initial and final states within the tank, respectively. Collecting results yields Wcv m2 1hi u2 2

(1)

4.4 Transient Analysis

157

The mass within the tank at the end of the process can be evaluated from the known volume and the specific volume of steam at 15 bar and 400C from Table A-4 m2

V 0.6 m3 2.96 kg v2 10.203 m3/kg2

The specific internal energy of steam at 15 bar and 400C from Table A-4 is 2951.3 kJ/kg. Also, at 15 bar and 320C, h1 3081.9 kJ/kg. Substituting values into Eq. (1)

❸

Wcv 2.96 kg13081.9 2951.32kJ/kg 386.6 kJ

❶

In this case idealizations are made about the state of the steam entering the tank and the final state of the steam in the tank. These idealizations make the transient analysis manageable.

❷

A significant aspect of this example is the energy transfer into the control volume by flow work, incorporated in the pv term of the specific enthalpy at the inlet.

❸

If the turbine were removed and steam allowed to flow adiabatically into the small tank, the final steam temperature in the tank would be 477C. This may be verified by setting Wcv to zero in Eq. (1) to obtain u2 hi, which with p2 15 bar fixes the final state.

EXAMPLE

4.13

Storing Compressed Air in a Tank

An air compressor rapidly fills a .28m3 tank, initially containing air at 21C, 1 bar, with air drawn from the atmosphere at 21C, 1 bar. During filling, the relationship between the pressure and specific volume of the air in the tank is pv1.4 constant. The ideal gas model applies for the air, and kinetic and potential energy effects are negligible. Plot the pressure, in atm, and the temperature, in F, of the air within the tank, each versus the ratio mm1, where m1 is the initial mass in the tank and m is the mass in the tank at time t 7 0. Also, plot the compressor work input, in kJ, versus mm1. Let mm1 vary from 1 to 3. SOLUTION Known: An air compressor rapidly fills a tank having a known volume. The initial state of the air in the tank and the state of the entering air are known. Find: Plot the pressure and temperature of the air within the tank, and plot the air compressor work input, each versus mm1 ranging from 1 to 3. Schematic and Given Data: Air

i

Ti = 21C pi = 1 bar

Tank Air compressor V = .28 m3 T1 = 21C p1 = 1 bar pv1.4 = constant

–+

Figure E4.13a

158

Chapter 4 Control Volume Analysis Using Energy

Assumptions: 1. The control volume is defined by the dashed line on the accompanying diagram. # 2. Because the tank is filled rapidly, Qcv is ignored. 3. Kinetic and potential energy effects are negligible. 4. The state of the air entering the control volume remains constant. 5. The air stored within the air compressor and interconnecting pipes can be ignored.

❶

6. The relationship between pressure and specific volume for the air in the tank is pv1.4 constant. 7. The ideal gas model applies for the air. Analysis: The required plots are developed using Interactive Thermodynamics: IT. The IT program is based on the following analysis. The pressure p in the tank at time t 7 0 is determined from pv1.4 p1v1.4 1 where the corresponding specific volume v is obtained using the known tank volume V and the mass m in the tank at that time. That is, v Vm. The specific volume of the air in the tank initially, v1, is calculated from the ideal gas equation of state and the known initial temperature, T1, and pressure, p1. That is 8314 N # m b 1294°K2 28.97 kg # °K RT1 1 bar v1 ` 5 ` .8437 m3/kg p1 11 bar2 10 N/m2 a

Once the pressure p is known, the corresponding temperature T can be found from the ideal gas equation of state, T pvR. To determine the work, begin with the mass rate balance for the single-inlet control volume dmcv # mi dt Then, with assumptions 2 and 3, the energy rate balance reduces to # dUcv # Wcv mihi dt Combining the mass and energy rate balances and integrating using assumption 4 gives ¢Ucv Wcv hi ¢mcv Denoting the work input to the compressor by Win Wcv and using assumption 5, this becomes Win mu m1u1 1m m1 2hi

(1)

where m1 is the initial amount of air in the tank, determined from m1

V .28 m3 0.332 kg v1 0.8437 m3/kg

As a sample calculation to validate the IT program below, consider the case m 0.664 kg, which corresponds to mm1 2. The specific volume of the air in the tank at that time is v

V 0.28 m3 0.422 m3/kg m 0.664 kg

4.4 Transient Analysis

159

The corresponding pressure of the air is p p1a

0.8437 m3/kg 1.4 v1 1.4 b 11 bar2 a b v 0.422 m3/kg

2.64 bars and the corresponding temperature of the air is T

12.64 bars2 1.422 m3/kg2 105 N/m2 pv ` ` £ ≥ 1 bar R 8314 J a b 28.97 kg ⴢ °K 388°K 1114.9°C2

Evaluating u1, u, and hi at the appropriate temperatures from Table A-22, u1 209.8 kJ/kg, u 277.5 kJ/kg, hi 294.2 kJ/kg. Using Eq. (1), the required work input is Win mu m1u1 1m m1 2hi 10.664 kg2 a277.5

kJ kJ kJ b 10.332 kg2 a209.8 b 10.332 kg2 a294.2 b kg kg kg

16.9 kJ IT Program. Choosing SI units from the Units menu, and selecting Air from the Properties menu, the IT program for solving the problem is // Given data p1 = 1 // bar T1 = 21 // C Ti = 21 // C V = .28 // m3 n = 1.4 // Determine the pressure and temperature for t > 0 v1 = v_TP(“Air”, T1, p1) v = V/m p * v ^n = p1 * v1 ^n v = v_TP(“Air”, T, p) // Specify the mass and mass ratio r v1 = V/m1 r = m/m1 r=2 // Calculate the work using Eq. (1) Win = m * u – m1 * u1 – hi * (m–m1) u1 = u_T(“Air”, T1) u = u_T(“Air”, T) hi = h_T(“Air”, Ti) Using the Solve button, obtain a solution for the sample case r mm1 2 considered above to validate the program. Good agreement is obtained, as can be verified. Once the program is validated, use the Explore button to vary the ratio mm1 from

Chapter 4 Control Volume Analysis Using Energy

160

1 to 3 in steps of 0.01. Then, use the Graph button to construct the required plots. The results are: 5

400 350

4 T, °C

p, bars

300 3 2

250 200 150

1 0

100 1

1.5

2 m/m1

2.5

50

3

1

1.5

2 m/m1

2.5

3

60 50

Win, kJ

40 30 20 10 0

1

1.5

2 m/m1

2.5

3

Figure E4.13b

We conclude from the first two plots that the pressure and temperature each increase as the tank fills. The work required to fill the tank increases as well. These results are as expected.

❶

This pressure-specific volume relationship is in accord with what might be measured. The relationship is also consistent with the uniform state idealization, embodied by Eqs. 4.25.

The final example of transient analysis is an application with a well-stirred tank. Such process equipment is commonly employed in the chemical and food processing industries.

EXAMPLE

4.14

Temperature Variation in a Well-Stirred Tank

A tank containing 45 kg of liquid water initially at 45C has one inlet and one exit with equal mass flow rates. Liquid water enters at 45C and a mass flow rate of 270 kg/h. A cooling coil immersed in the water removes energy at a rate of 7.6 kW. The water is well mixed by a paddle wheel so that the water temperature is uniform throughout. The power input to the water from the paddle wheel is 0.6 kW. The pressures at the inlet and exit are equal and all kinetic and potential energy effects can be ignored. Plot the variation of water temperature with time.

4.4 Transient Analysis

161

SOLUTION Known: Liquid water flows into and out of a well-stirred tank with equal mass flow rates as the water in the tank is cooled by a cooling coil. Find: Plot the variation of water temperature with time. Schematic and Given Data:

318

m1 = 270 kg/h Water temperature, K

Mixing rotor Constant liquid level

Tank Boundary Cooling coil

296

m2 = 270 kg/h

0.5 Time, h

1.0 Figure E4.14

Assumptions: 1. The control volume is defined by the dashed line on the accompanying diagram. 2. For the control volume, the only significant heat transfer is with the cooling coil. Kinetic and potential energy effects can be neglected.

❶

3. The water temperature is uniform with position throughout: T T(t). 4. The water in the tank is incompressible, and there is no change in pressure between inlet and exit. Analysis: The energy rate balance reduces with assumption 2 to # # dUcv # Qcv Wcv m 1h1 h2 2 dt # where m denotes the mass flow rate. The mass contained within the control volume remains constant with time, so the term on the left side of the energy rate balance can be expressed as d1mcvu2 dUcv du mcv dt dt dt Since the water is assumed incompressible, the specific internal energy depends on temperature only. Hence, the chain rule can be used to write du du dT dT c dt dT dt dt where c is the specific heat. Collecting results dUcv dT mcvc dt dt With Eq. 3.20b the enthalpy term of the energy rate balance can be expressed as h1 h2 c1T1 T2 2 v1p1 p2 2 0

162

Chapter 4 Control Volume Analysis Using Energy

where the pressure term is dropped by assumption 4. Since the water is well mixed, the temperature at the exit equals the temperature of the overall quantity of liquid in the tank, so h1 h2 c1T1 T 2 where T represents the uniform water temperature at time t. With the foregoing considerations the energy rate balance becomes mcvc

# # dT # Qcv Wcv mc 1T1 T 2 dt

As can be verified by direct substitution, the solution of this first-order, ordinary differential equation is # # # Qcv Wcv m T C1 exp a tb a b T1 # mcv mc The constant C1 is evaluated using the initial condition: at t 0, T T1. Finally # # # Qcv Wcv m T T1 a tb d b c 1 exp a # mcv mc Substituting given numerical values together with the specific heat c for liquid water from Table A-19 3 7.6 10.62 4 kJ/s 270 tb d c 1 exp a S C 270 kg 45 kJ b a4.2 # b a 3600 s kg K 318 22 31 exp16t2 4

T 318 K

❷

where t is in hours. Using this expression, we can construct the accompanying plot showing the variation of temperature with time.

❶

In this case idealizations are made about the state of the mass contained within the system and the states of the liquid entering and exiting. These idealizations make the transient analysis manageable.

❷

As t S , T S 296 K. That is, the water temperature approaches a constant value after sufficient time has elapsed. From the accompanying plot it can be seen that the temperature reaches its constant limiting value in about 1 h.

Chapter Summary and Study Guide

The conservation of mass and energy principles for control volumes are embodied in the mass and energy rate balances developed in this chapter. Although the primary emphasis is on cases in which one-dimensional flow is assumed, mass and energy balances are also presented in integral forms that provide a link to subsequent fluid mechanics and heat transfer courses. Control volumes at steady state are featured, but discussions of transient cases are also provided. The use of mass and energy balances for control volumes at steady state is illustrated for nozzles and diffusers, turbines, compressors and pumps, heat exchangers, throttling devices, and integrated systems. An essential aspect of all such applications is the careful and explicit listing of appropriate assumptions. Such model-building skills are stressed throughout the chapter. The following checklist provides a study guide for this chapter. When your study of the text and end-of-chapter exercises

has been completed you should be able to write out the meanings of the terms listed in the margins

throughout the chapter and understand each of the related concepts. The subset of key concepts listed below is particularly important in subsequent chapters. list the typical modeling assumptions for nozzles and

diffusers, turbines, compressors and pumps, heat exchangers, and throttling devices. apply Eqs. 4.18–4.20 to control volumes at steady state,

using appropriate assumptions and property data for the case at hand. apply mass and energy balances for the transient analysis

of control volumes, using appropriate assumptions and property data for the case at hand.

Problems: Developing Engineering Skills

163

Key Engineering Concepts

mass flow rate p. 122 mass rate balance p. 122 one-dimensional flow p. 124

volumetric flow rate p. 124 steady state p. 125 flow work p. 130

energy rate balance p. 131 nozzle p. 134 diffuser p. 134 turbine p. 137

compressor p. 139 pump p. 139 heat exchanger p. 143 throttling process p. 148

Exercises: Things Engineers Think About 1. Why does the relative velocity normal to the flow boundary, Vn, appear in Eqs. 4.3 and 4.8?

energy rate balances are important to describe steady-state operation?

2. Why might a computer cooled by a constant-speed fan operate satisfactorily at sea level but overheat at high altitude?

9. When air enters a diffuser and decelerates, does its pressure increase or decrease?

3. Give an example where the inlet and exit mass flow rates for a control volume are equal, yet the control volume is not at steady state. # 4. Does Qcv accounting for energy transfer by heat include heat transfer across inlets and exits? Under what circumstances might heat transfer across an inlet or exit be significant?

10. Even though their outer surfaces would seem hot to the touch, large steam turbines in power plants might not be covered with much insulation. Why not?

5. By introducing enthalpy h to replace each of the (u pv) terms of Eq. 4.13, we get Eq. 4.14. An even simpler algebraic form would result by replacing each of the (u pv V22

gz) terms by a single symbol, yet we have not done so. Why not?

12. A hot liquid stream enters a counterflow heat exchanger at Th,in, and a cold liquid stream enters at Tc,in. Sketch the variation of temperature with location of each stream as it passes through the heat exchanger.

6. Simplify the general forms of the mass and energy rate balances to describe the process of blowing up a balloon. List all of your modeling assumptions.

13. What are some examples of commonly encountered devices that undergo periods of transient operation? For each example, which type of system, closed system or control volume, would be most appropriate?

7. How do the general forms of the mass and energy rate balances simplify to describe the exhaust stroke of a cylinder in an automobile engine? List all of your modeling assumptions. 8. Waterwheels have been used since antiquity to develop mechanical power from flowing water. Sketch an appropriate control volume for a waterwheel. What terms in the mass and

11. Would it be desirable for a coolant circulating inside the engine of an automobile to have a large or a small specific heat cp? Discuss.

14. An insulated rigid tank is initially evacuated. A valve is opened and atmospheric air at 20C, 1 atm enters until the pressure in the tank becomes 1 bar, at which time the valve is closed. Is the final temperature of the air in the tank equal to, greater than, or less than 20C?

Problems: Developing Engineering Skills Applying Conservation of Mass

4.1 The mass flow rate at the inlet of a one-inlet, one-exit con# trol volume varies with time according to mi 10011 e 2t 2, # where mi has units of kg/h and t is in h. At the exit, the mass flow rate is constant at 100 kg/h. The initial mass in the control volume is 50 kg. (a) Plot the inlet and exit mass flow rates, the instantaneous rate of change of mass, and the amount of mass contained in the control volume as functions of time, for t ranging from 0 to 3 h. (b) Estimate the time, in h, when the tank is nearly empty. 4.2 A control volume has one inlet and one exit. The mass flow # # rates in and out are, respectively, mi 1.5 and me # 0.002t 1.511 e 2, where t is in seconds and m is in kg/s. Plot the time rate of change of mass, in kg/s, and the net change

in the amount of mass, in kg, in the control volume versus time, in s, ranging from 0 to 3600 s. 4.3 A 0.5-m3 tank contains ammonia, initially at 40C, 8 bar. A leak develops, and refrigerant flows out of the tank at a constant mass flow rate of 0.04 kg/s. The process occurs slowly enough that heat transfer from the surroundings maintains a constant temperature in the tank. Determine the time, in s, at which half of the mass has leaked out, and the pressure in the tank at that time, in bar. 4.4 A water storage tank initially contains 400 m3 of water. The average daily usage is 40 m3. If water is added to the tank at an average rate of 20[exp(t20)] m3 per day, where t is time in days, for how many days will the tank contain water? 4.5 A pipe carrying an incompressible liquid contains an expansion chamber as illustrated in Fig. P4.5.

164

Chapter 4 Control Volume Analysis Using Energy

(a) Develop an expression for the time rate of change of liquid level in the chamber, dLdt, in terms of the diameters D1, D2, and D, and the velocities V1 and V2. # (b) Compare the relative magnitudes of the mass flow rates mi # and m2 when dLdt 0, dLdt 0, and dLdt 0, respectively.

D Expansion chamber

L

V2 m·

V1 m· 1

2

D1

D2

Figure P4.5

is 350 m/s. The air behaves as an ideal gas. For steady-state operation, determine (a) the mass flow rate, in kg/s. (b) the exit flow area, in cm2. 4.9 Infiltration of outside air into a building through miscellaneous cracks around doors and windows can represent a significant load on the heating equipment. On a day when the outside temperature is –18°C, 0.042 m3/s of air enters through the cracks of a particular office building. In addition, door openings account for about .047 m3/s of outside air infiltration. The internal volume of the building is 566 m3, and the inside temperature is 22°C. There is negligible pressure difference between the inside and the outside of the building. Assuming ideal gas behavior, determine at steady state the volumetric flow rate of air exiting the building through cracks and other openings, and the number of times per hour that the air within the building is changed due to infiltration. 4.10 Refrigerant 134a enters the condenser of a refrigeration system operating at steady state at 9 bar, 50°C, through a 2.5-cm-diameter pipe. At the exit, the pressure is 9 bar, the temperature is 30°C, and the velocity is 2.5 m/s. The mass flow rate of the entering refrigerant is 6 kg/min. Determine (a) the velocity at the inlet, in m/s. (b) the diameter of the exit pipe, in cm.

4.6 Velocity distributions for laminar and turbulent flow in a circular pipe of radius R carrying an incompressible liquid of density are given, respectively, by VV0 31 1rR2 2 4

VV0 31 1rR2 4 17

where r is the radial distance from the pipe centerline and V0 is the centerline velocity. For each velocity distribution (a) plot VV0 versus rR. (b) derive expressions for the mass flow rate and the average velocity of the flow, Vave, in terms of V0, R, and , as required. (c) derive an expression for the specific kinetic energy carried through an area normal to the flow. What is the percent error if the specific kinetic energy is evaluated in terms of the average velocity as (Vave)22? Which velocity distribution adheres most closely to the idealizations of one-dimensional flow? Discuss. 4.7 Vegetable oil for cooking is dispensed from a cylindrical can fitted with a spray nozzle. According to the label, the can is able to deliver 560 sprays, each of duration 0.25 s and each having a mass of 0.25 g. Determine (a) the mass flow rate of each spray, in g/s. (b) the mass remaining in the can after 560 sprays, in g, if the initial mass in the can is 170 g. 4.8 Air enters a one-inlet, one-exit control volume at 8 bar, 600 K, and 40 m/s through a flow area of 20 cm2. At the exit, the pressure is 2 bar, the temperature is 400 K, and the velocity

4.11 Steam at 160 bar, 480°C, enters a turbine operating at steady state with a volumetric flow rate of 800 m3/min. Eighteen percent of the entering mass flow exits at 5 bar, 240°C, with a velocity of 25 m/s. The rest exits at another location with a pressure of 0.06 bar, a quality of 94%, and a velocity of 400 m/s. Determine the diameters of each exit duct, in m. 4.12 Air enters a compressor operating at steady state with a pressure of 1 bar, a temperature of 20°C, and a volumetric flow rate of 0.25 m3/s. The air velocity in the exit pipe is 210 m/s and the exit pressure is 1 MPa. If each unit mass of air passing from inlet to exit undergoes a process described by pv1.34 constant, determine the exit temperature, in °C. 4.13 Air enters a 0.6-m-diameter fan at 16°C, 101 kPa, and is discharged at 18°C, 105 kPa, with a volumetric flow rate of 0.35 m3/s. Assuming ideal gas behavior, determine for steadystate operation (a) the mass flow rate of air, in kg/s. (b) the volumetric flow rate of air at the inlet, in m3/s. (c) the inlet and exit velocities, in m/s. 4.14 Ammonia enters a control volume operating at steady state at p1 14 bar, T1 28°C, with a mass flow rate of 0.5 kg/s. Saturated vapor at 4 bar leaves through one exit, with a volumetric flow rate of 1.036 m3/min, and saturated liquid at 4 bar leaves through a second exit. Determine (a) the minimum diameter of the inlet pipe, in cm, so the ammonia velocity does not exceed 20 m/s. (b) the volumetric flow rate of the second exit stream, in m3/min.

Problems: Developing Engineering Skills

4.15 At steady state, a stream of liquid water at 20°C, 1 bar is mixed with a stream of ethylene glycol (M 62.07) to form a refrigerant mixture that is 50% glycol by mass. The water molar flow rate is 4.2 kmol/min. The density of ethylene glycol is 1.115 times that of water. Determine (a) the molar flow rate, in kmol/min, and volumetric flow rate, in m3/min, of the entering ethylene glycol.

165

(b) the diameters, in cm, of each of the supply pipes if the velocity in each is 2.5 m/s. 4.16 Figure P4.16 shows a cooling tower operating at steady state. Warm water from an air conditioning unit enters at 49°C with a mass flow rate of 0.5 kg/s. Dry air enters the tower at 21°C, 1 atm with a volumetric flow rate of 1.41 m3/s. Because of evaporation within the tower, humid air exits at the

Humid air m· 4 = 1.64 kg/s Cooling tower 4 Fan

Warm water inlet m· 1 = 0.5 kg/s

Spray heads

1

Air conditioning unit

T1 = 49°C Dry air T3 = 21°C 3 p = 1 bar 3 (AV)3 = 1.41 m3/s

2 Liquid

Pump

Return water

T2 = 27°C m· 2 = m· 1 5 Makeup water

top of the tower with a mass flow rate of 1.64 kg/s. Cooled liquid water is collected at the bottom of the tower for return to the air conditioning unit together with makeup water. Determine the mass flow rate of the makeup water, in kg/s. Energy Analysis of Control Volumes at Steady State

4.17 Air enters a control volume operating at steady state at 1.05 bar, 300 K, with a volumetric flow rate of 12 m3/min and exits at 12 bar, 400 K. Heat transfer occurs at a rate of 20 kW from the control volume to the surroundings. Neglecting kinetic and potential energy effects, determine the power, in kW. 4.18 Steam enters a nozzle operating at steady state at 30 bar, 320°C, with a velocity of 100 m/s. The exit pressure and temperature are 10 bar and 200°C, respectively. The mass flow rate is 2 kg/s. Neglecting heat transfer and potential energy, determine (a) the exit velocity, in m/s. (b) the inlet and exit flow areas, in cm2. 4.19 Methane (CH4) gas enters a horizontal, well-insulated nozzle operating at steady state at 80°C and a velocity of 10 m/s. Assuming ideal gas behavior for the methane, plot the temperature of the gas exiting the nozzle, in °C, versus the exit velocity ranging from 500 to 600 m/s. 4.20 Air enters an uninsulated nozzle operating at steady state at 420°K with negligible velocity and exits the nozzle at 290°K

+ – Figure P4.16

with a velocity of 460 m/s. Assuming ideal gas behavior and neglecting potential energy effects, determine the heat transfer per unit mass of air flowing, in kJ/kg. 4.21 Air enters an insulated diffuser operating at steady state with a pressure of 1 bar, a temperature of 300 K, and a velocity of 250 m/s. At the exit, the pressure is 1.13 bar and the velocity is 140 m/s. Potential energy effects can be neglected. Using the ideal gas model, determine (a) the ratio of the exit flow area to the inlet flow area. (b) the exit temperature, in K. 4.22 The inlet ducting to a jet engine forms a diffuser that steadily decelerates the entering air to zero velocity relative to the engine before the air enters the compressor. Consider a jet airplane flying at 1000 km/h where the local atmospheric pressure is 0.6 bar and the air temperature is 8°C. Assuming ideal gas behavior and neglecting heat transfer and potential energy effects, determine the temperature, in °C, of the air entering the compressor. 4.23 Refrigerant 134a enters an insulated diffuser as a saturated vapor at 7 bars with a velocity of 370 m/s. At the exit, the pressure is 16 bars and the velocity is negligible. The diffuser operates at steady state and potential energy effects can be neglected. Determine the exit temperature, in °C.

166

Chapter 4 Control Volume Analysis Using Energy

4.24 Air expands through a turbine from 10 bar, 900 K to 1 bar, 500 K. The inlet velocity is small compared to the exit velocity of 100 m/s. The turbine operates at steady state and develops a power output of 3200 kW. Heat transfer between the turbine and its surroundings and potential energy effects are negligible. Calculate the mass flow rate of air, in kg/s, and the exit area, in m2. 4.25 A well-insulated turbine operating at steady state develops 23 MW of power for a steam flow rate of 40 kg/s. The steam enters at 360°C with a velocity of 35 m /s and exits as saturated vapor at 0.06 bar with a velocity of 120 m /s. Neglecting potential energy effects, determine the inlet pressure, in bar. 4.26 Nitrogen gas enters a turbine operating at steady state with a velocity of 60 m/s, a pressure of 0.345 Mpa, and a temperature of 700 K. At the exit, the velocity is 0.6 m/s, the pressure is 0.14 Mpa, and the temperature is 390 K. Heat transfer from the surface of the turbine to the surroundings occurs at a rate of 36 kJ per kg of nitrogen flowing. Neglecting potential energy effects and using the ideal gas model, determine the power developed by the turbine, in kW. 4.27 Steam enters a well-insulated turbine operating at steady state with negligible velocity at 4 MPa, 320°C. The steam expands to an exit pressure of 0.07 MPa and a velocity of 90 m/s. The diameter of the exit is 0.6 m. Neglecting potential energy effects, plot the power developed by the turbine, in kW, versus the steam quality at the turbine exit ranging from 0.9 to 1.0. 4.28 The intake to a hydraulic turbine installed in a flood control dam is located at an elevation of 10 m above the turbine exit. Water enters at 20°C with negligible velocity and exits from the turbine at 10 m/s. The water passes through the turbine with no significant changes in temperature or pressure between the inlet and exit, and heat transfer is negligible. The acceleration of gravity is constant at g 9.81 m/s2. If the power output at steady state is 500 kW, what is the mass flow rate of water, in kg/s? 4.29 A well-insulated turbine operating at steady state is sketched in Fig. P4.29. Steam enters at 3 MPa, 400°C, with a volumetric flow rate of 85 m3/min. Some steam is extracted from the turbine at a pressure of 0.5 MPa and a temperature of 180°C. The rest expands to a pressure of 6 kPa and a quality of 90%. The total power developed by the turbine is 11,400 kW. Kinetic and potential energy effects can be neglected. Determine

1 Power out p1 = 3MPa T1 = 400°C (AV)1 = 85 m3/min

Turbine

2 p2 = 0.5 MPa T2 = 180°C V2 = 20 m/s

Figure P4.29

3 p3 = 6 kPa x3 = 90%

(a) the mass flow rate of the steam at each of the two exits, in kg/h. (b) the diameter, in m, of the duct through which steam is extracted, if the velocity there is 20 m/s. 4.30 Air is compressed at steady state from 1 bar, 300 K, to 6 bar with a mass flow rate of 4 kg/s. Each unit of mass passing from inlet to exit undergoes a process described by pv1.27 constant. Heat transfer occurs at a rate of 46.95 kJ per kg of air flowing to cooling water circulating in a water jacket enclosing the compressor. If kinetic and potential energy changes of the air from inlet to exit are negligible, calculate the compressor power, in kW. 4.31 A compressor operates at steady state with Refrigerant 22 as the working fluid. The refrigerant enters at 5 bar, 10°C, with a volumetric flow rate of 0.8 m3/min. The diameters of the inlet and exit pipes are 4 and 2 cm, respectively. At the exit, the pressure is 14 bar and the temperature is 90°C. If the magnitude of the heat transfer rate from the compressor to its surroundings is 5% of the compressor power input, determine the power input, in kW. 4.32 Refrigerant 134a enters an air conditioner compressor at 3.2 bar, 10°C, and is compressed at steady state to 10 bar, 70°C. The volumetric flow rate of refrigerant entering is 3.0 m3/min. The power input to the compressor is 55.2 kJ per kg of refrigerant flowing. Neglecting kinetic and potential energy effects, determine the heat transfer rate, in kW. 4.33 A compressor operating at steady state takes in 45 kg/min of methane gas (CH4) at 1 bar, 25°C, 15 m/s, and compresses it with negligible heat transfer to 2 bar, 90 m/s at the exit. The power input to the compressor is 110 kW. Potential energy effects are negligible. Using the ideal gas model, determine the temperature of the gas at the exit, in K. 4.34 Refrigerant 134a is compressed at steady state from 2.4 bar, 0°C, to 12 bar, 50°C. Refrigerant enters the compressor with a volumetric flow rate of 0.38 m3/min, and the power input to the compressor is 2.6 kW. Cooling water circulating through a water jacket enclosing the compressor experiences a temperature rise of 4°C from inlet to exit with a negligible change in pressure. Heat transfer from the outside of the water jacket and all kinetic and potential energy effects can be neglected. Determine the mass flow rate of the cooling water, in kg/s. 4.35 Air enters a water-jacketed air compressor operating at steady state with a volumetric flow rate of 37 m3/min at 136 kPa, 305 K and exits with a pressure of 680 kPa and a temperature of 400 K. The power input to the compressor is 155 kW. Energy transfer by heat from the compressed air to the cooling water circulating in the water jacket results in an increase in the temperature of the cooling water from inlet to exit with no change in pressure. Heat transfer from the outside of the jacket as well as all kinetic and potential energy effects can be neglected.

Problems: Developing Engineering Skills

4.40 Carbon dioxide gas is heated as it flows steadily through a 2.5-cm-diameter pipe. At the inlet, the pressure is 2 bar, the temperature is 300 K, and the velocity is 100 m /s. At the exit, the pressure and velocity are 0.9413 bar and 400 m /s, respectively. The gas can be treated as an ideal gas with constant specific heat cp 0.94 kJ/kg K. Neglecting potential energy effects, determine the rate of heat transfer to the carbon dioxide, in kW. 4.41 A feedwater heater in a vapor power plant operates at steady state with liquid entering at inlet 1 with T1 45°C and p1 3.0 bar. Water vapor at T2 320°C and p2 3.0 bar enters at inlet 2. Saturated liquid water exits with a pressure of p3 3.0 bar. Ignore heat transfer with the surroundings and all kinetic and potential energy effects. If the mass flow rate

4.44 Figure P4.44 shows a solar collector panel with a surface area of 2.97 m2. The panel receives energy from the sun at a rate of 1.5 kW. Thirty-six percent of the incoming energy is lost to the surroundings. The remainder is used to heat liquid water from 40C to 60C. The water passes through the solar collector with a negligible pressure drop. Neglecting kinetic and potential energy effects, determine at steady state the mass flow rate of water, in kg. How many gallons of water at 60C can eight collectors provide in a 30-min time period?

l

A = 2.97 m2

or

pa

ne

1.5 kW

Water in at 40°C

ct

4.39 A steam boiler tube is designed to produce a stream of saturated vapor at 200 kPa from saturated liquid entering at the same pressure. At steady state, the flow rate is 0.25 kg/min. The boiler is constructed from a well-insulated stainless steel pipe through which the steam flows. Electrodes clamped to the pipe at each end cause a 10-V direct current to pass through the pipe material. Determine the required size of the power supply, in kW, and the expected current draw, in amperes.

4.43 Refrigerant 134a flows at steady state through a long horizontal pipe having an inside diameter of 4 cm, entering as saturated vapor at 8°C with a mass flow rate of 17 kg/min. Refrigerant vapor exits at a pressure of 2 bar. If the heat transfer rate to the refrigerant is 3.41 kW, determine the exit temperature, in °C, and the velocities at the inlet and exit, each in m/s.

le

4.38 Ammonia enters a heat exchanger operating at steady state as a superheated vapor at 14 bar, 60°C, where it is cooled and condensed to saturated liquid at 14 bar. The mass flow rate of the refrigerant is 450 kg/h. A separate stream of air enters the heat exchanger at 17°C, 1 bar and exits at 42°C, 1 bar. Ignoring heat transfer from the outside of the heat exchanger and neglecting kinetic and potential energy effects, determine the mass flow rate of the air, in kg/min.

(a) the mass flow rate of refrigerant, in kg/min. (b) the rate of energy transfer, in kJ/min, from the air to the refrigerant.

rc ol

4.37 An oil pump operating at steady state delivers oil at a rate of 5.5 kg/s and a velocity of 6.8 m/s. The oil, which can be modeled as incompressible, has a density of 1600 kg/m3 and experiences a pressure rise from inlet to exit of .28 Mpa. There is no significant elevation difference between inlet and exit, and the inlet kinetic energy is negligible. Heat transfer between the pump and its surroundings is negligible, and there is no significant change in temperature as the oil passes through the pump. If pumps are available in 14-horsepower increments, determine the horsepower rating of the pump needed for this application.

4.42 The cooling coil of an air-conditioning system is a heat exchanger in which air passes over tubes through which Refrigerant 22 flows. Air enters with a volumetric flow rate of 40 m3/min at 27°C, 1.1 bar, and exits at 15°C, 1 bar. Refrigerant enters the tubes at 7 bar with a quality of 16% and exits at 7 bar, 15°C. Ignoring heat transfer from the outside of the heat exchanger and neglecting kinetic and potential energy effects, determine at steady state

la

4.36 A pump steadily delivers water through a hose terminated by a nozzle. The exit of the nozzle has a diameter of 2.5 cm and is located 4 m above the pump inlet pipe, which has a diameter of 5.0 cm. The pressure is equal to 1 bar at both the inlet and the exit, and the temperature is constant at 20°C. The magnitude of the power input required by the pump is 8.6 kW, and the acceleration of gravity is g 9.81 m/s2. Determine the mass flow rate delivered by the pump, in kg/s.

# of the liquid entering at inlet 1 is m1 3.2 105 kg/h, de# termine the mass flow rate at inlet 2, m2, in kg/h.

36% loss

So

(a) Determine the temperature increase of the cooling water, in K, if the cooling water mass flow rate is 82 kg/min. (b) Plot the temperature increase of the cooling water, in K, versus the cooling water mass flow rate ranging from 75 to 90 kg/min.

167

Water out at 60C

Figure P4.44

4.45 As shown in Fig. P4.45, 15 kg/s of steam enters a desuperheater operating at steady state at 30 bar, 320C, where it is mixed with liquid water at 25 bar and temperature T2 to produce saturated vapor at 20 bar. Heat transfer between the device and its surroundings and kinetic and potential energy effects can be neglected.

168

Chapter 4 Control Volume Analysis Using Energy

# (a) If T2 200C, determine the mass flow rate of liquid, m2, in kg/s. 3

Valve 1 p1 = 30 bar T1 = 320°C m· 1 = 15 kg/s

Desuperheater

cal power. The rate of energy transfer by convection from the plate-mounted electronics is estimated to be 0.08 kW. Kinetic and potential energy effects can be ignored. Determine the tube diameter, in cm.

p3 = 20 bar Saturated vapor Valve 2

Convection cooling on top surface

p2 = 25 bar T2 Figure P4.45

# (b) Plot m2, in kg/s, versus T2 ranging from 20 to 220C. 4.46 A feedwater heater operates at steady state with liquid water entering at inlet 1 at 7 bar, 42C, and a mass flow rate of 70 kg/s. A separate stream of water enters at inlet 2 as a two-phase liquid–vapor mixture at 7 bar with a quality of 98%. Saturated liquid at 7 bar exits the feedwater heater at 3. Ignoring heat transfer with the surroundings and neglecting kinetic and potential energy effects, determine the mass flow rate, in kg/s, at inlet 2. 4.47 The electronic components of Example 4.8 are cooled by air flowing through the electronics enclosure. The rate of energy transfer by forced convection from the electronic components to the air is hA(Ts Ta), where hA 5 W/K, Ts denotes the average surface temperature of the components, and Ta denotes the average of the inlet and exit air temperatures. Referring to Example 4.8 as required, determine the largest value of Ts, in C, for which the specified limits are met. 4.47 The electronic components of a computer consume 0.1 kW of electrical power. To prevent overheating, cooling air is supplied by a 25-W fan mounted at the inlet of the electronics enclosure. At steady state, air enters the fan at 20C, 1 bar and exits the electronics enclosure at 35C. There is no significant energy transfer by heat from the outer surface of the enclosure to the surroundings and the effects of kinetic and potential energy can be ignored. Determine the volumetric flow rate of the entering air, in m3/s. 4.49 Ten kg/min of cooling water circulates through a water jacket enclosing a housing filled with electronic components. At steady state, water enters the water jacket at 22C and exits with a negligible change in pressure at a temperature that cannot exceed 26C. There is no significant energy transfer by heat from the outer surface of the water jacket to the surroundings, and kinetic and potential energy effects can be ignored. Determine the maximum electric power the electronic components can receive, in kW, for which the limit on the temperature of the exiting water is met. 4.50 As shown in Fig. P4.50, electronic components mounted on a flat plate are cooled by convection to the surroundings and by liquid water circulating through a U-tube bonded to the plate. At steady state, water enters the tube at 20C and a velocity of 0.4 m /s and exits at 24C with a negligible change in pressure. The electrical components receive 0.5 kW of electri-

2 T2 = 24°C

1

+ –

Electronic components

T1 = 20°C V1 = 0.4 m/s Water Figure P4.50

4.51 Electronic components are mounted on the inner surface of a horizontal cylindrical duct whose inner diameter is 0.2 m, as shown in Fig. P4.51. To prevent overheating of the electronics, the cylinder is cooled by a stream of air flowing through it and by convection from its outer surface. Air enters the duct at 25C, 1 bar and a velocity of 0.3 m /s and exits with negligible changes in kinetic energy and pressure at a temperature that cannot exceed 40C. If the electronic components require 0.20 kW of electric power at steady state, determine the minimum rate of heat transfer by convection from the cylinder’s outer surface, in kW, for which the limit on the temperature of the exiting air is met.

Convection cooling on outer surface

Air T1 = 25°C p1 = 1 bar 1 V1 = 0.3 m/s D1 = 0.2 m

2 +

–

T2 ≤ 40°C p2 = 1 bar

Electronic components mounted on inner surface

Figure P4.51

4.52 Ammonia enters the expansion valve of a refrigeration system at a pressure of 1.4 MPa and a temperature of 32C and exits at 0.08 MPa. If the refrigerant undergoes a throttling process, what is the quality of the refrigerant exiting the expansion valve?

Problems: Developing Engineering Skills

4.56 Refrigerant 134a enters the flash chamber operating at steady state shown in Fig. P4.56 at 10 bar, 36C, with a mass flow rate of 482 kg/h. Saturated liquid and saturated vapor exit as separate streams, each at pressure p. Heat transfer to the surroundings and kinetic and potential energy effects can be ignored.

4.53 Propane vapor enters a valve at 1.6 MPa, 70C, and leaves at 0.5 MPa. If the propane undergoes a throttling process, what is the temperature of the propane leaving the valve, in C? 4.54 A large pipe carries steam as a two-phase liquid–vapor mixture at 1.0 MPa. A small quantity is withdrawn through a throttling calorimeter, where it undergoes a throttling process to an exit pressure of 0.1 MPa. For what range of exit temperatures, in C, can the calorimeter be used to determine the quality of the steam in the pipe? What is the corresponding range of steam quality values?

(a) Determine the mass flow rates of the exiting streams, each in kg/h, if p 4 bar. (b) Plot the mass flow rates of the exiting streams, each in kg/h, versus p ranging from 1 to 9 bar.

4.55 As shown in Fig. P4.55, a steam turbine at steady state is operated at part load by throttling the steam to a lower pres-

3 Saturated vapor, pressure p

Valve 1

Valve 1 p1 = 1.5 MPa T1 = 320C

Power out

2

169

Turbine

p1 = 10 bar T1 = 36°C m· 1 = 482 kg/h

p2 = 1 MPa

Flash chamber

Saturated liquid, pressure p

3 p3 = .08 bar x3 = 90%

2

Figure P4.56

Figure P4.55

sure before it enters the turbine. Before throttling, the pressure and temperature are, respectively, 1.5 MPa and 320C. After throttling, the pressure is 1 MPa. At the turbine exit, the steam is at .08 bar and a quality of 90%. Heat transfer with the surroundings and all kinetic and potential energy effects can be ignored. Determine

4.57 Air as an ideal gas flows through the turbine and heat exchanger arrangement shown in Fig. P4.57. Data for the two flow streams are shown on the figure. Heat transfer to the surroundings can be neglected, as can all kinetic and potential energy effects. Determine T3, in K, and the power output of the second turbine, in kW, at steady state.

(a) the temperature at the turbine inlet, in C. (b) the power developed by the turbine, in kJ/kg of steam flowing.

4.58 A residential heat pump system operating at steady state is shown schematically in Fig. P4.58. Refrigerant 134a circulates through the components of the system, and property data at the numbered locations are given on the figure. The mass

· Wt1 = 10,000 kW

Turbine 1

Air in

1 T1 = 1400 K p1 = 20 bar

T2 = 1100 K p2 = 5 bar

p3 = 4.5 bar T3 = ?

2 6 T6 = 1200 K p6 = 1 bar

· Wt 2 = ?

Turbine 2

3

T4 = 980 K p4 = 1 bar 4 T5 = 1480 K 5 p5 = 1.35 bar m· 5 = 1200 kg/min

Heat exchanger Air in

Figure P4.57

170

Chapter 4 Control Volume Analysis Using Energy

flow rate of the refrigerant is 4.6 kg/min. Kinetic and potential energy effects are negligible. Determine

flow rate of the water is 109 kg/s. Kinetic and potential energy effects are negligible as are all stray heat transfers. Determine

(a) rate of heat transfer between the compressor and the surroundings, in kJ/min. (b) the coefficient of performance.

(a) the thermal efficiency. (b) the mass flow rate of the cooling water passing through the condenser, in kg/s. Transient Analysis

Return air from house at 20°C

Heated air to house at T > 20°C

T3 = 30°C p3 = 8 bar

3

p2 = 8 bar h 2 = 270 kJ/kg

2 Condenser

Expansion valve

Power input to compressor = 2.5 kW

Compressor

Evaporator 4

1

T 4 = –12°C

Air exits at T < 0°C

p1 = 1.8 bar T1 = –10°C

Outside air enters at 0°C

Figure P4.58

4.59 Figure P4.59 shows a simple vapor power plant operating at steady state with water circulating through the components. Relevant data at key locations are given on the figure. The mass

· Q in

p1 = 100 bar T1 = 520°C

Power out Turbine

1

p2 = 0.08 bar 2 x2 = 90%

Steam generator

4.60 A tiny hole develops in the wall of a rigid tank whose volume is 0.75 m3, and air from the surroundings at 1 bar, 25C leaks in. Eventually, the pressure in the tank reaches 1 bar. The process occurs slowly enough that heat transfer between the tank and the surroundings keeps the temperature of the air inside the tank constant at 25C. Determine the amount of heat transfer, in kJ, if initially the tank (a) is evacuated. (b) contains air at 0.7 bar, 25C. 4.61 A rigid tank of volume 0.75 m3 is initially evacuated. A hole develops in the wall, and air from the surroundings at 1 bar, 25C flows in until the pressure in the tank reaches 1 bar. Heat transfer between the contents of the tank and the surroundings is negligible. Determine the final temperature in the tank, in C. 4.62 A rigid, well-insulated tank of volume 0.5 m3 is initially evacuated. At time t 0, air from the surroundings at 1 bar, 21C begins to flow into the tank. An electric resistor transfers energy to the air in the tank at a constant rate of 100 W for 500 s, after which time the pressure in the tank is 1 bar. What is the temperature of the air in the tank, in C, at the final time? 4.63 The rigid tank illustrated in Fig. P4.63 has a volume of 0.06 m3 and initially contains a two-phase liquid–vapor mixture of H2O at a pressure of 15 bar and a quality of 20%. As the tank contents are heated, a pressure-regulating valve keeps the pressure constant in the tank by allowing saturated vapor to escape. Neglecting kinetic and potential energy effects (a) determine the total mass in the tank, in kg, and the amount of heat transfer, in kJ, if heating continues until the final quality is x 0.5. (b) plot the total mass in the tank, in kg, and the amount of heat transfer, in kJ, versus the final quality x ranging from 0.2 to 1.0. Pressure-regulating valve

Cooling water in at 20°C Condenser

4

p4 = 100 bar T4 = 43°C

Cooling water out at 35°C

V = 0.06 m3 p = 15 bar xinitial = 20%

Pump Power in Figure P4.59

3 p3 = 0.08 bar Saturated liquid Figure P4.63

Problems: Developing Engineering Skills

4.64 A well-insulated rigid tank of volume 10 m3 is connected to a large steam line through which steam flows at 15 bar and 280C. The tank is initially evacuated. Steam is allowed to flow into the tank until the pressure inside is p. (a) Determine the amount of mass in the tank, in kg, and the temperature in the tank, in C, when p 15 bar. (b) Plot the quantities of part (a) versus p ranging from 0.1 to 15 bar. 4.65 A tank of volume 1 m3 initially contains steam at 6 MPa and 320C. Steam is withdrawn slowly from the tank until the pressure drops to p. Heat transfer to the tank contents maintains the temperature constant at 320C. Neglecting all kinetic and potential energy effects

4.68 A well-insulated piston–cylinder assembly is connected by a valve to an air supply line at 8 bar, as shown in Fig. P4.68. Initially, the air inside the cylinder is at 1 bar, 300 K, and the piston is located 0.5 m above the bottom of the cylinder. The atmospheric pressure is 1 bar, and the diameter of the piston face is 0.3 m. The valve is opened and air is admitted slowly until the volume of air inside the cylinder has doubled. The weight of the piston and the friction between the piston and the cylinder wall can be ignored. Using the ideal gas model, plot the final temperature, in K, and the final mass, in kg, of the air inside the cylinder for supply temperatures ranging from 300 to 500 K.

(a) determine the heat transfer, in kJ, if p 1.5 MPa. (b) plot the heat transfer, in kJ, versus p ranging from 0.5 to 6 MPa. 4.66 A 1 m3 tank initially contains air at 300 kPa, 300 K. Air slowly escapes from the tank until the pressure drops to 100 kPa. The air that remains in the tank undergoes a process described by pv1.2 constant. For a control volume enclosing the tank, determine the heat transfer, in kJ. Assume ideal gas behavior with constant specific heats. 4.67 A well-insulated tank contains 25 kg of Refrigerant 134a, initially at 300 kPa with a quality of 0.8 (80%). The pressure is maintained by nitrogen gas acting against a flexible bladder, as shown in Fig. P4.67. The valve is opened between the tank and a supply line carrying Refrigerant 134a at 1.0 MPa, 120C. The pressure regulator allows the pressure in the tank to remain at 300 kPa as the bladder expands. The valve between the line and the tank is closed at the instant when all the liquid has vaporized. Determine the amount of refrigerant admitted to the tank, in kg.

Pressure-regulating valve

N2

Nitrogen supply

Flexible bladder

Refrigerant 134a 300 kPa Tank

Line: 1000 kPa, 120 °C

Figure P4.67

171

patm = 1 bar

Diameter = 0.3 m L

Initially: L1 = 0.5 m T1 = 300 K p1 = 1 bar Valve Air supply line: 8 bar

Figure P4.68

4.69 Nitrogen gas is contained in a rigid 1-m tank, initially at 10 bar, 300 K. Heat transfer to the contents of the tank occurs until the temperature has increased to 400 K. During the process, a pressure-relief valve allows nitrogen to escape, maintaining constant pressure in the tank. Neglecting kinetic and potential energy effects, and using the ideal gas model with constant specific heats evaluated at 350 K, determine the mass of nitrogen that escapes, in kg, and the amount of energy transfer by heat, in kJ. 4.70 The air supply to a 56 m3 office has been shut off overnight to conserve utilities, and the room temperature has dropped to 4C. In the morning, a worker resets the thermostat to 21C, and 6 m3/min of air at 50C begins to flow in through a supply duct. The air is well mixed within the room, and an equal mass flow of air at room temperature is withdrawn through a return duct. The air pressure is nearly 1 bar everywhere. Ignoring heat transfer with the surroundings and kinetic and potential energy effects, estimate how long it takes for the room temperature to reach 21C. Plot the room temperature as a function of time.

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Chapter 4 Control Volume Analysis Using Energy

Design & Open Ended Problems: Exploring Engineering Practice 4.1D What practical measures can be taken by manufacturers to use energy resources more efficiently? List several specific opportunities, and discuss their potential impact on profitability and productivity. 4.2D Methods for measuring mass flow rates of gases and liquids flowing in pipes and ducts include: rotameters, turbine flowmeters, orifice-type flowmeters, thermal flowmeters, and Coriolis-type flowmeters. Determine the principles of operation of each of these flow-measuring devices. Consider the suitability of each for measuring liquid or gas flows. Can any be used for two-phase liquid–vapor mixtures? Which measure volumetric flow rate and require separate measurements of pressure and temperature to determine the state of the substance? Summarize your findings in a brief report. 4.3D Wind turbines, or windmills, have been used for generations to develop power from wind. Several alternative wind turbine concepts have been tested, including among others the Mandaras, Darrieus, and propeller types. Write a report in which you describe the operating principles of prominent wind turbine types. Include in your report an assessment of the economic feasibility of each type. 4.4D Prepare a memorandum providing guidelines for selecting fans for cooling electronic components. Consider the advantages and disadvantages of locating the fan at the inlet of the enclosure containing the electronics. Repeat for a fan at the enclosure exit. Consider the relative merits of alternative fan types and of fixed- versus variable-speed fans. Explain how characteristic curves assist in fan selection. 4.5D Pumped-hydraulic storage power plants use relatively inexpensive off-peak baseload electricity to pump water from a lower reservoir to a higher reservoir. During periods of peak demand, electricity is produced by discharging water from the upper to the lower reservoir through a hydraulic turbinegenerator. A single device normally plays the role of the pump during upper-reservoir charging and the turbine-generator during discharging. The ratio of the power developed during discharging to the power required for charging is typically much less than 100%. Write a report describing the features of the pump-turbines used for such applications and their size and cost. Include in your report a discussion of the economic feasibility of pumped-hydraulic storage power plants. 4.6D Figure P4.6D shows a batch-type solar water heater. With the exit closed, cold tap water fills the tank, where it is heated by the sun. The batch of heated water is then allowed to flow to an existing conventional gas or electric water heater. If the batch-type solar water heater is constructed primarily from salvaged and scrap material, estimate the time for a typical family of four to recover the cost of the water heater from reduced water heating by conventional means. 4.7D Low-head dams (3 to 10 m), commonly used for flood control on many rivers, provide an opportunity for electric power generation using hydraulic turbine-generators. Estimates of this hydroelectric potential must take into account the

Reflective surface Glass or clear plastic

Tap water Inlet Warm water to storage

Tank Exit

Insulated walls and hinged covers Figure P4.6D

available head and the river flow, each of which varies considerably throughout the year. Using U.S. Geological Survey data, determine the typical variations in head and flow for a river in your locale. Based on this information, estimate the total annual electric generation of a hydraulic turbine placed on the river. Does the peak generating capacity occur at the same time of year as peak electrical demand in your area? Would you recommend that your local utility take advantage of this opportunity for electric power generation? Discuss. 4.8D Figure P4.8D illustrates an experimental apparatus for steady-state testing of Refrigerant 134a shell-and-tube evaporators having a capacity of 100 kW. As shown by the dashed lines on the figure, two subsystems provide refrigerant and a water-glycol mixture to the evaporator. The water-glycol mixture is chilled in passing through the evaporator tubes, so the water-glycol subsystem must reheat and recirculate the mixture to the evaporator. The refrigerant subsystem must remove the energy added to the refrigerant passing through the evaporator, and deliver saturated liquid refrigerant at 20C. For each subsystem, draw schematics showing layouts of heat exchangers, pumps, interconnecting piping, etc. Also, specify

Refrigerant subsystem

Waterglycol subsystem

Shell-and-tube evaporator Figure P4.8D

the mass flow rates, heat transfer rates, and power requirements for each component within the subsystems, as appropriate.

4.10D Smaller can be Better (see box Sec. 4.3). Investigate the scope of current medical applications of MEMS. Write a report including at least three references.

4.9D The stack from an industrial paint-drying oven discharges 30 m3/min of gaseous combustion products at 240C. Investigate the economic feasibility of installing a heat exchanger in the stack to heat air that would provide for some of the space heating needs of the plant.

4.11D Sensibly Built Homes Cost No More (see box Sec. 4.3). In energy-efficient homes, indoor air quality can be a concern. Research the issue of carbon monoxide and radon buildup in tightly sealed houses. Write a report including at least three references.

C H A P

5 chapter objective

T E R

The Second Law of Thermodynamics

E N G I N E E R I N G C O N T E X T The presentation to this point has considered thermodynamic analysis using the conservation of mass and conservation of energy principles together with property relations. In Chaps. 2 through 4 these fundamentals are applied to increasingly complex situations. The conservation principles do not always suffice, however, and often the second law of thermodynamics is also required for thermodynamic analysis. The objective of this chapter is to introduce the second law of thermodynamics. A number of deductions that may be called corollaries of the second law are also considered, including performance limits for thermodynamic cycles. The current presentation provides the basis for subsequent developments involving the second law in Chaps. 6 and 7.

5.1 Introducing the Second Law

The objectives of the present section are to (1) motivate the need for and the usefulness of the second law, and (2) to introduce statements of the second law that serve as the point of departure for its application. 5.1.1 Motivating the Second Law It is a matter of everyday experience that there is a definite direction for spontaneous processes. This can be brought out by considering the three systems pictured in Fig. 5.1.

174

System a. An object at an elevated temperature Ti placed in contact with atmospheric air at temperature T0 would eventually cool to the temperature of its much larger surroundings, as illustrated in Fig. 5.1a. In conformity with the conservation of energy principle, the decrease in internal energy of the body would appear as an increase in the internal energy of the surroundings. The inverse process would not take place spontaneously, even though energy could be conserved: The internal energy of the surroundings would not decrease spontaneously while the body warmed from T0 to its initial temperature. System b. Air held at a high pressure pi in a closed tank would flow spontaneously to the lower pressure surroundings at p0 if the interconnecting valve were opened, as illustrated in Fig. 5.1b. Eventually fluid motions would cease and all of the air would be

5.1 Introducing the Second Law Atmospheric air at T0

Q

Time

Body at Ti > T0

Time

T0 < T < Ti

T0

(a)

Atmospheric air at p0 Valve

Air at pi > p0

Air at p0

Air p0 < p < pi

(b)

Mass zi

Mass 0 < z < zi

Mass

(c) Figure 5.1

Illustrations of spontaneous processes and the eventual attainment of equilibrium with the surroundings. (a) Spontaneous heat transfer. (b) Spontaneous expansion. (c) Falling mass.

at the same pressure as the surroundings. Drawing on experience, it should be clear that the inverse process would not take place spontaneously, even though energy could be conserved: Air would not flow spontaneously from the surroundings at p0 into the tank, returning the pressure to its initial value. System c. A mass suspended by a cable at elevation zi would fall when released, as illustrated in Fig. 5.1c. When it comes to rest, the potential energy of the mass in its initial condition would appear as an increase in the internal energy of the mass and its surroundings, in accordance with the conservation of energy principle. Eventually, the mass also would come to the temperature of its much larger surroundings. The inverse process would not take place spontaneously, even though energy could be conserved: The mass would not return spontaneously to its initial elevation while its internal energy or that of its surroundings decreased.

In each case considered, the initial condition of the system can be restored, but not in a spontaneous process. Some auxiliary devices would be required. By such auxiliary means the object could be reheated to its initial temperature, the air could be returned to the tank

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and restored to its initial pressure, and the mass could be lifted to its initial height. Also in each case, a fuel or electrical input normally would be required for the auxiliary devices to function, so a permanent change in the condition of the surroundings would result. The foregoing discussion indicates that not every process consistent with the principle of energy conservation can occur. Generally, an energy balance alone neither enables the preferred direction to be predicted nor permits the processes that can occur to be distinguished from those that cannot. In elementary cases such as the ones considered, experience can be drawn upon to deduce whether particular spontaneous processes occur and to deduce their directions. For more complex cases, where experience is lacking or uncertain, a guiding principle would be helpful. This is provided by the second law. The foregoing discussion also indicates that when left to themselves, systems tend to undergo spontaneous changes until a condition of equilibrium is achieved, both internally and with their surroundings. In some cases equilibrium is reached quickly, in others it is achieved slowly. For example, some chemical reactions reach equilibrium in fractions of seconds; an ice cube requires a few minutes to melt; and it may take years for an iron bar to rust away. Whether the process is rapid or slow, it must of course satisfy conservation of energy. However, that alone would be insufficient for determining the final equilibrium state. Another general principle is required. This is provided by the second law.

OPPORTUNITIES FOR DEVELOPING WORK

By exploiting the spontaneous processes shown in Fig. 5.1, it is possible, in principle, for work to be developed as equilibrium is attained. for example. . . instead of permitting the body of Fig. 5.1a to cool spontaneously with no other result, energy could be delivered by heat transfer to a system undergoing a power cycle that would develop a net amount of work (Sec. 2.6). Once the object attained equilibrium with the surroundings, the process would cease. Although there is an opportunity for developing work in this case, the opportunity would be wasted if the body were permitted to cool without developing any work. In the case of Fig. 5.1b, instead of permitting the air to expand aimlessly into the lower-pressure surroundings, the stream could be passed through a turbine and work could be developed. Accordingly, in this case there is also a possibility for developing work that would not be exploited in an uncontrolled process. In the case of Fig. 5.1c, instead of permitting the mass to fall in an uncontrolled way, it could be lowered gradually while turning a wheel, lifting another mass, and so on. These considerations can be summarized by noting that when an imbalance exists between two systems, there is an opportunity for developing work that would be irrevocably lost if the systems were allowed to come into equilibrium in an uncontrolled way. Recognizing this possibility for work, we can pose two questions:

What is the theoretical maximum value for the work that could be obtained? What are the factors that would preclude the realization of the maximum value?

That there should be a maximum value is fully in accord with experience, for if it were possible to develop unlimited work, few concerns would be voiced over our dwindling fuel supplies. Also in accord with experience is the idea that even the best devices would be subject to factors such as friction that would preclude the attainment of the theoretical maximum work. The second law of thermodynamics provides the means for determining the theoretical maximum and evaluating quantitatively the factors that preclude attaining the maximum.

5.1 Introducing the Second Law

SECOND LAW SUMMARY

The preceding discussions can be summarized by noting that the second law and deductions from it are useful because they provide means for 1. 2. 3. 4.

predicting the direction of processes. establishing conditions for equilibrium. determining the best theoretical performance of cycles, engines, and other devices. evaluating quantitatively the factors that preclude the attainment of the best theoretical performance level.

Additional uses of the second law include its roles in 5. defining a temperature scale independent of the properties of any thermometric substance. 6. developing means for evaluating properties such as u and h in terms of properties that are more readily obtained experimentally. Scientists and engineers have found many additional applications of the second law and deductions from it. It also has been used in economics, philosophy, and other areas far removed from engineering thermodynamics. The six points listed can be thought of as aspects of the second law of thermodynamics and not as independent and unrelated ideas. Nonetheless, given the variety of these topic areas, it is easy to understand why there is no single statement of the second law that brings out each one clearly. There are several alternative, yet equivalent, formulations of the second law. In the next section, two equivalent statements of the second law are introduced as a point of departure for our study of the second law and its consequences. Although the exact relationship of these particular formulations to each of the second law aspects listed above may not be immediately apparent, all aspects listed can be obtained by deduction from these formulations or their corollaries. It is important to add that in every instance where a consequence of the second law has been tested directly or indirectly by experiment, it has been unfailingly verified. Accordingly, the basis of the second law of thermodynamics, like every other physical law, is experimental evidence. 5.1.2 Statements of the Second Law Among many alternative statements of the second law, two are frequently used in engineering thermodynamics. They are the Clausius and Kelvin–Planck statements. The objective of this section is to introduce these two equivalent second law statements. The Clausius statement has been selected as a point of departure for the study of the second law and its consequences because it is in accord with experience and therefore easy to accept. The Kelvin–Planck statement has the advantage that it provides an effective means for bringing out important second law deductions related to systems undergoing thermodynamic cycles. One of these deductions, the Clausius inequality (Sec. 6.1), leads directly to the property entropy and to formulations of the second law convenient for the analysis of closed systems and control volumes as they undergo processes that are not necessarily cycles. CLAUSIUS STATEMENT OF THE SECOND LAW

The Clausius statement of the second law asserts that: It is impossible for any system to operate in such a way that the sole result would be an energy transfer by heat from a cooler to a hotter body.

Clausius statement

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Q Hot

Yes!

Metal bar Cold

No! Q

The Clausius statement does not rule out the possibility of transferring energy by heat from a cooler body to a hotter body, for this is exactly what refrigerators and heat pumps accomplish. However, as the words “sole result” in the statement suggest, when a heat transfer from a cooler body to a hotter body occurs, there must be some other effect within the system accomplishing the heat transfer, its surroundings, or both. If the system operates in a thermodynamic cycle, its initial state is restored after each cycle, so the only place that must be examined for such other effects is its surroundings. for example. . . cooling of food is accomplished by refrigerators driven by electric motors requiring work from their surroundings to operate. The Clausius statement implies that it is impossible to construct a refrigeration cycle that operates without an input of work.

KELVIN–PLANCK STATEMENT OF THE SECOND LAW

thermal reservoir

Kelvin–Planck statement Thermal reservoir

Qcycle

No!

Wcycle

System undergoing a thermodynamic cycle

Before giving the Kelvin–Planck statement of the second law, the concept of a thermal reservoir is introduced. A thermal reservoir, or simply a reservoir, is a special kind of system that always remains at constant temperature even though energy is added or removed by heat transfer. A reservoir is an idealization of course, but such a system can be approximated in a number of ways—by the earth’s atmosphere, large bodies of water (lakes, oceans), a large block of copper, and a system consisting of two phases (although the ratio of the masses of the two phases changes as the system is heated or cooled at constant pressure, the temperature remains constant as long as both phases coexist). Extensive properties of a thermal reservoir such as internal energy can change in interactions with other systems even though the reservoir temperature remains constant. Having introduced the thermal reservoir concept, we give the Kelvin–Planck statement of the second law: It is impossible for any system to operate in a thermodynamic cycle and deliver a net amount of energy by work to its surroundings while receiving energy by heat transfer from a single thermal reservoir. The Kelvin–Planck statement does not rule out the possibility of a system developing a net amount of work from a heat transfer drawn from a single reservoir. It only denies this possibility if the system undergoes a thermodynamic cycle. The Kelvin–Planck statement can be expressed analytically. To develop this, let us study a system undergoing a cycle while exchanging energy by heat transfer with a single reservoir. The first and second laws each impose constraints:

A constraint is imposed by the first law on the net work and heat transfer between the system and its surroundings. According to the cycle energy balance Wcycle Qcycle

analytical form: Kelvin–Planck statement

In words, the net work done by the system undergoing a cycle equals the net heat transfer to the system. Although the cycle energy balance allows the net work Wcycle to be positive or negative, the second law imposes a constraint on its direction, as considered next. According to the Kelvin–Planck statement, a system undergoing a cycle while communicating thermally with a single reservoir cannot deliver a net amount of work to its surroundings. That is, the net work of the cycle cannot be positive. However, the Kelvin–Planck statement does not rule out the possibility that there is a net work transfer of energy to the system during the cycle or that the net work is zero. Thus, the analytical form of the Kelvin–Planck statement is Wcycle 0

1single reservoir2

(5.1)

5.1 Introducing the Second Law

where the words single reservoir are added to emphasize that the system communicates thermally only with a single reservoir as it executes the cycle. In Sec. 5.3.1, we associate the “less than” and “equal to” signs of Eq. 5.1 with the presence and absence of internal irreversibilities, respectively. The concept of irreversibilities is considered in Sec. 5.2. The equivalence of the Clausius and Kelvin–Planck statements can be demonstrated by showing that the violation of each statement implies the violation of the other (see box).

D E M O N S T R AT I N G T H E E Q U I VA L E N C E O F T H E C L A U S I U S A N D K E LV I N – P L A N C K S TAT E M E N T S

The equivalence of the Clausius and Kelvin–Planck statements is demonstrated by showing that the violation of each statement implies the violation of the other. That a violation of the Clausius statement implies a violation of the Kelvin–Planck statement is readily shown using Fig. 5.2, which pictures a hot reservoir, a cold reservoir, and two systems. The system on the left transfers energy QC from the cold reservoir to the hot reservoir by heat transfer without other effects occurring and thus violates the Clausius statement. The system on the right operates in a cycle while receiving QH (greater than QC) from the hot reservoir, rejecting QC to the cold reservoir, and delivering work Wcycle to the surroundings. The energy flows labeled on Fig. 5.2 are in the directions indicated by the arrows. Consider the combined system shown by a dotted line on Fig. 5.2, which consists of the cold reservoir and the two devices. The combined system can be regarded as executing a cycle because one part undergoes a cycle and the other two parts experience no net change in their conditions. Moreover, the combined system receives energy (QH QC) by heat transfer from a single reservoir, the hot reservoir, and produces an equivalent amount of work. Accordingly, the combined system violates the Kelvin–Planck statement. Thus, a violation of the Clausius statement implies a violation of the Kelvin–Planck statement. The equivalence of the two second-law statements is demonstrated completely when it is also shown that a violation of the Kelvin–Planck statement implies a violation of the Clausius statement. This is left as an exercise.

System undergoing a thermodynamic cycle QC

Hot reservoir

QH

Wcycle = Q H – Q C

QC

Cold reservoir

QC

Dotted line defines combined system

Figure 5.2

Illustration used to demonstrate the equivalence of the Clausius and Kelvin–Planck statements of the second law.

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5.2 Identifying Irreversibilities

One of the important uses of the second law of thermodynamics in engineering is to determine the best theoretical performance of systems. By comparing actual performance with the best theoretical performance, insights often can be gained into the potential for improvement. As might be surmised, the best performance is evaluated in terms of idealized processes. In this section such idealized processes are introduced and distinguished from actual processes involving irreversibilities.

IRREVERSIBLE PROCESSES

irreversible process reversible processes

A process is called irreversible if the system and all parts of its surroundings cannot be exactly restored to their respective initial states after the process has occurred. A process is reversible if both the system and surroundings can be returned to their initial states. Irreversible processes are the subject of the present discussion. Reversible processes are considered again later in the section. A system that has undergone an irreversible process is not necessarily precluded from being restored to its initial state. However, were the system restored to its initial state, it would not be possible also to return the surroundings to the state they were in initially. As illustrated below, the second law can be used to determine whether both the system and surroundings can be returned to their initial states after a process has occurred. That is, the second law can be used to determine whether a given process is reversible or irreversible. It might be apparent from the discussion of the Clausius statement of the second law that any process involving a spontaneous heat transfer from a hotter body to a cooler body is irreversible. Otherwise, it would be possible to return this energy from the cooler body to the hotter body with no other effects within the two bodies or their surroundings. However, this possibility is contrary to our experience and is denied by the Clausius statement. Processes involving other kinds of spontaneous events are irreversible, such as an unrestrained expansion of a gas or liquid considered in Fig. 5.1. Friction, electrical resistance, hysteresis, and inelastic deformation are examples of effects whose presence during a process renders it irreversible. In summary, irreversible processes normally include one or more of the following irreversibilities:

irreversibilities

Heat transfer through a finite temperature difference Unrestrained expansion of a gas or liquid to a lower pressure Spontaneous chemical reaction Spontaneous mixing of matter at different compositions or states Friction—sliding friction as well as friction in the flow of fluids Electric current flow through a resistance Magnetization or polarization with hysteresis Inelastic deformation

Although the foregoing list is not exhaustive, it does suggest that all actual processes are irreversible. That is, every process involves effects such as those listed, whether it is a naturally occurring process or one involving a device of our construction, from the simplest mechanism to the largest industrial plant. The term “irreversibility” is used to identify any of these effects. The list given previously comprises a few of the irreversibilities that are commonly encountered.

5.2 Identifying Irreversibilities

As a system undergoes a process, irreversibilities may be found within the system as well as within its surroundings, although in certain instances they may be located predominately in one place or the other. For many analyses it is convenient to divide the irreversibilities present into two classes. Internal irreversibilities are those that occur within the system. External irreversibilities are those that occur within the surroundings, often the immediate surroundings. As this distinction depends solely on the location of the boundary, there is some arbitrariness in the classification, for by extending the boundary to take in a portion of the surroundings, all irreversibilities become “internal.” Nonetheless, as shown by subsequent developments, this distinction between irreversibilities is often useful. Engineers should be able to recognize irreversibilities, evaluate their influence, and develop practical means for reducing them. However, certain systems, such as brakes, rely on the effect of friction or other irreversibilities in their operation. The need to achieve profitable rates of production, high heat transfer rates, rapid accelerations, and so on invariably dictates the presence of significant irreversibilities. Furthermore, irreversibilities are tolerated to some degree in every type of system because the changes in design and operation required to reduce them would be too costly. Accordingly, although improved thermodynamic performance can accompany the reduction of irreversibilities, steps taken in this direction are constrained by a number of practical factors often related to costs. for example. . . consider two bodies at different temperatures that are able to communicate thermally. With a finite temperature difference between them, a spontaneous heat transfer would take place and, as discussed previously, this would be a source of irreversibility. It might be expected that the importance of this irreversibility would diminish as the temperature difference approaches zero, and this is the case. From the study of heat transfer (Sec. 2.4), we know that the transfer of a finite amount of energy by heat between bodies whose temperatures differ only slightly would require a considerable amount of time, a larger (more costly) heat transfer surface area, or both. To eliminate this source of irreversibility, therefore, would require an infinite amount of time and/or an infinite surface area.

Whenever any irreversibility is present during a process, the process must necessarily be irreversible. However, the irreversibility of the process can be demonstrated using the Kelvin–Planck statement of the second law and the following procedure: (1) Assume there is a way to return the system and surroundings to their respective initial states. (2) Show that as a consequence of this assumption, it would be possible to devise a cycle that produces work while no effect occurs other than a heat transfer from a single reservoir. Since the existence of such a cycle is denied by the Kelvin–Planck statement, the initial assumption must be in error and it follows that the process is irreversible. This approach can be used to demonstrate that processes involving friction (see box), heat transfer through a finite temperature difference, the unrestrained expansion of a gas or liquid to a lower pressure, and other effects from the list given previously are irreversible. However, in most instances the use of the Kelvin–Planck statement to demonstrate the irreversibility of processes is cumbersome. It is normally easier to use the entropy production concept (Sec. 6.5).

D E M O N S T R AT I N G I R R E V E R S I B I L I T Y: F R I C T I O N

Let us use the Kelvin–Planck statement to demonstrate the irreversibility of a process involving friction. Consider a system consisting of a block of mass m and an inclined plane. Initially the block is at rest at the top of the incline. The block then slides down

internal and external irreversibilities

Q Hot, TH Cold, TC Area

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Chapter 5 The Second Law of Thermodynamics

the plane, eventually coming to rest at a lower elevation. There is no significant heat transfer between the system and its surroundings during the process. Applying the closed system energy balance 1Uf Ui 2 mg1zf zi 2 1KEf KEi 2 Q W 0

or Uf Ui mg1zi zf 2 where U denotes the internal energy of the block-plane system and z is the elevation of the block. Thus, friction between the block and plane during the process acts to convert the potential energy decrease of the block to internal energy of the overall system. Since no work or heat interactions occur between the system and its surroundings, the condition of the surroundings remains unchanged during the process. This allows attention to be centered on the system only in demonstrating that the process is irreversible. When the block is at rest after sliding down the plane, its elevation is zf and the internal energy of the block–plane system is Uf. To demonstrate that the process is irreversible using the Kelvin–Planck statement, let us take this condition of the system, shown in Fig. 5.3a, as the initial state of a cycle consisting of three processes. We imagine that a pulley–cable arrangement and a thermal reservoir are available to assist in the demonstration. Process 1: Assume that the inverse process can occur with no change in the surroundings. That is, as shown in Fig. 5.3b, assume that the block returns spontaneously to its initial elevation and the internal energy of the system decreases to its initial value, Ui. (This is the process we want to demonstrate is impossible.) Process 2: As shown in Fig. 5.3c, use the pulley–cable arrangement provided to lower the block from zi to zf, allowing the decrease in potential energy to do work by lifting another mass located in the surroundings. The work done by the system equals the decrease in the potential energy of the block: mg(zi zf). Process 3: The internal energy of the system can be increased from Ui to Uf by bringing it into communication with the reservoir, as shown in Fig. 5.3d. The heat transfer required is Q Uf Ui. Or, with the result of the energy balance on the system given above, Q mg(zi – zf). At the conclusion of this process the block is again at elevation zf and the internal energy of the block–plane system is restored to Uf. The net result of this cycle is to draw energy from a single reservoir by heat transfer and produce an equivalent amount of work. There are no other effects. However, such a cycle is denied by the Kelvin–Planck statement. Since both the heating of the system by the reservoir (Process 3) and the lowering of the mass by the pulley–cable while work is done (Process 2) are possible, it can be concluded that it is Process 1 that is impossible. Since Process 1 is the inverse of the original process where the block slides down the plane, it follows that the original process is irreversible.

REVERSIBLE PROCESSES

A process of a system is reversible if the system and all parts of its surroundings can be exactly restored to their respective initial states after the process has taken place. It should be evident from the discussion of irreversible processes that reversible processes are purely hypothetical. Clearly, no process can be reversible that involves spontaneous heat transfer

5.2 Identifying Irreversibilities

zi

Block

zf (a)

(b)

Reservoir

Q Figure 5.3 (c)

(d)

Figure used to demonstrate the irreversibility of a process involving friction.

through a finite temperature difference, an unrestrained expansion of a gas or liquid, friction, or any of the other irreversibilities listed previously. In a strict sense of the word, a reversible process is one that is perfectly executed. All actual processes are irreversible. Reversible processes do not occur. Even so, certain processes that do occur are approximately reversible. The passage of a gas through a properly designed nozzle or diffuser is an example (Sec. 6.8). Many other devices also can be made to approach reversible operation by taking measures to reduce the significance of irreversibilities, such as lubricating surfaces to reduce friction. A reversible process is the limiting case as irreversibilities, both internal and external, are reduced further and further. Although reversible processes cannot actually occur, they can be imagined. Earlier in this section we considered how heat transfer would approach reversibility as the temperature difference approaches zero. Let us consider two additional examples:

A particularly elementary example is a pendulum oscillating in an evacuated space. The pendulum motion approaches reversibility as friction at the pivot point is reduced. In the limit as friction is eliminated, the states of both the pendulum and its surroundings would be completely restored at the end of each period of motion. By definition, such a process is reversible. A system consisting of a gas adiabatically compressed and expanded in a frictionless piston–cylinder assembly provides another example. With a very small increase in the external pressure, the piston would compress the gas slightly. At each intermediate volume during the compression, the intensive properties T, p, v, etc. would be uniform throughout: The gas would pass through a series of equilibrium states. With a small decrease in the external pressure, the piston would slowly move out as the gas expands. At each intermediate volume of the expansion, the intensive properties of the gas would be at the same uniform values they had at the corresponding step during the compression. When the gas volume returned to its initial value, all properties would be restored to their initial values as well. The work done on the gas during the compression would equal the work done by the gas during the expansion. If the work between the system and its surroundings were delivered to, and received from, a frictionless pulley–mass assembly, or the equivalent, there would also be no net change in the surroundings. This process would be reversible.

Gas

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Chapter 5 The Second Law of Thermodynamics

INTERNALLY REVERSIBLE PROCESSES

internally reversible process

METHODOLOGY UPDATE

Using the internally reversible process concept, we refine the definition of the thermal reservoir introduced in Sec. 5.1.2. In subsequent discussions we assume that no internal irreversibilities are present within a thermal reservoir. That is, every process of a thermal reservoir is internally reversible.

In an irreversible process, irreversibilities are present within the system, its surroundings, or both. A reversible process is one in which there are no internal or external irreversibilities. An internally reversible process is one in which there are no irreversibilities within the system. Irreversibilities may be located within the surroundings, however, as when there is heat transfer between a portion of the boundary that is at one temperature and the surroundings at another. At every intermediate state of an internally reversible process of a closed system, all intensive properties are uniform throughout each phase present. That is, the temperature, pressure, specific volume, and other intensive properties do not vary with position. If there were a spatial variation in temperature, say, there would be a tendency for a spontaneous energy transfer by conduction to occur within the system in the direction of decreasing temperature. For reversibility, however, no spontaneous processes can be present. From these considerations it can be concluded that the internally reversible process consists of a series of equilibrium states: It is a quasiequilibrium process. To avoid having two terms that refer to the same thing, in subsequent discussions we will refer to any such process as an internally reversible process. The use of the internally reversible process concept in thermodynamics is comparable to the idealizations made in mechanics: point masses, frictionless pulleys, rigid beams, and so on. In much the same way as these are used in mechanics to simplify an analysis and arrive at a manageable model, simple thermodynamic models of complex situations can be obtained through the use of internally reversible processes. Initial calculations based on internally reversible processes would be adjusted with efficiencies or correction factors to obtain reasonable estimates of actual performance under various operating conditions. Internally reversible processes are also useful in determining the best thermodynamic performance of systems.

5.3 Applying the Second Law to Thermodynamic Cycles

Several important applications of the second law related to power cycles and refrigeration and heat pump cycles are presented in this section. These applications further our understanding of the implications of the second law and provide the basis for important deductions from the second law introduced in subsequent sections. Familiarity with thermodynamic cycles is required, and we recommend that you review Sec. 2.6, where cycles are considered from an energy, or first law, perspective and the thermal efficiency of power cycles and coefficients of performance for refrigeration and heat pump cycles are introduced. 5.3.1 Interpreting the Kelvin–Planck Statement Wcycle 0 1single reservoir2

Let us reconsider Eq. 5.1, the analytical form of the Kelvin–Planck statement of the second law. Equation 5.1 is employed in subsequent sections to obtain a number of significant deductions. In each of these applications, the following idealizations are assumed: The thermal reservoir and the portion of the surroundings with which work interactions occur are free of irreversibilities. This allows the “less than” sign to be associated with irreversibilities within the system of interest. The “equal to” sign is employed only when no irreversibilities of any kind are present. (See box.)

5.3 Applying the Second Law to Thermodynamic Cycles

Thermal reservoir Heat transfer

System Boundary Mass

Figure 5.4 System undergoing a cycle while exchanging energy by heat transfer with a single thermal reservoir.

A S S O C I AT I N G S I G N S W I T H T H E K E LV I N – P L A N C K S TAT E M E N T

Consider a system that undergoes a cycle while exchanging energy by heat transfer with a single reservoir, as shown in Fig. 5.4. Work is delivered to, or received from, the pulley–mass assembly located in the surroundings. A flywheel, spring, or some other device also can perform the same function. In subsequent applications of Eq. 5.1, the irreversibilities of primary interest are internal irreversibilities. To eliminate extraneous factors in such applications, therefore, assume that these are the only irreversibilities present. Hence, the pulley–mass assembly, flywheel, or other device to which work is delivered, or from which it is received, is idealized as free of irreversibilities. The thermal reservoir is also assumed free of irreversibilities. To demonstrate the correspondence of the “equal to” sign of Eq. 5.1 with the absence of irreversibilities, consider a cycle operating as shown in Fig. 5.4 for which the equality applies. At the conclusion of one cycle,

The system would necessarily be returned to its initial state. Since Wcycle 0, there would be no net change in the elevation of the mass used to store energy in the surroundings. Since Wcycle Qcycle, it follows that Qcycle 0, so there also would be no net change in the condition of the reservoir.

Thus, the system and all elements of its surroundings would be exactly restored to their respective initial conditions. By definition, such a cycle is reversible. Accordingly, there can be no irreversibilities present within the system or its surroundings. It is left as an exercise to show the converse: If the cycle occurs reversibly, the equality applies. Since a cycle is either reversible or irreversible, it follows that the inequality sign implies the presence of irreversibilities, and the inequality applies whenever irreversibilities are present.

5.3.2 Power Cycles Interacting with Two Reservoirs A significant limitation on the performance of systems undergoing power cycles can be brought out using the Kelvin–Planck statement of the second law. Consider Fig. 5.5, which shows a system that executes a cycle while communicating thermally with two thermal reservoirs, a hot reservoir and a cold reservoir, and developing net work Wcycle. The thermal

185

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Chapter 5 The Second Law of Thermodynamics Hot reservoir

QH

Boundary

Wcycle = Q H – Q C

Cold reservoir

Figure 5.5 System undergoing a power cycle while exchanging energy by heat transfer with two reservoirs.

QC

efficiency of the cycle is h

METHODOLOGY UPDATE

The energy transfers labeled on Fig 5.5 are positive in the directions indicated by the arrows.

Wcycle QH

1

QC QH

(5.2)

where QH is the amount of energy received by the system from the hot reservoir by heat transfer and QC is the amount of energy discharged from the system to the cold reservoir by heat transfer. If the value of QC were zero, the system of Fig. 5.5 would withdraw energy QH from the hot reservoir and produce an equal amount of work, while undergoing a cycle. The thermal efficiency of such a cycle would have a value of unity (100%). However, this method of operation would violate the Kelvin–Planck statement and thus is not allowed. It follows that for any system executing a power cycle while operating between two reservoirs, only a portion of the heat transfer QH can be obtained as work, and the remainder, QC, must be discharged by heat transfer to the cold reservoir. That is, the thermal efficiency must be less than 100%. In arriving at this conclusion it was not necessary to (1) identify the nature of the substance contained within the system, (2) specify the exact series of processes making up the cycle, or (3) indicate whether the processes are actual processes or somehow idealized. The conclusion that the thermal efficiency must be less than 100% applies to all power cycles whatever their details of operation. This may be regarded as a corollary of the second law. Other corollaries follow. Since no power cycle can have a thermal efficiency of 100%, it is of interest to investigate the maximum theoretical efficiency. The maximum theoretical efficiency for systems undergoing power cycles while communicating thermally with two thermal reservoirs at different temperatures is evaluated in Sec. 5.5 with reference to the following two corollaries of the second law, called the Carnot corollaries.

CARNOT COROLLARIES.

Carnot corollaries

The thermal efficiency of an irreversible power cycle is always less than the thermal efficiency of a reversible power cycle when each operates between the same two thermal reservoirs. All reversible power cycles operating between the same two thermal reservoirs have the same thermal efficiency.

A cycle is considered reversible when there are no irreversibilities within the system as it undergoes the cycle and heat transfers between the system and reservoirs occur reversibly. The idea underlying the first Carnot corollary is in agreement with expectations stemming from the discussion of the second law thus far. Namely, the presence of irreversibilities during the execution of a cycle is expected to exact a penalty. If two systems operating between the same reservoirs each receive the same amount of energy QH and one executes a reversible cycle while the other executes an irreversible cycle, it is in accord with intuition that the net work developed by the irreversible cycle will be less, and it will therefore have the smaller thermal efficiency.

5.3 Applying the Second Law to Thermodynamic Cycles

The second Carnot corollary refers only to reversible cycles. All processes of a reversible cycle are perfectly executed. Accordingly, if two reversible cycles operating between the same reservoirs each receive the same amount of energy QH but one could produce more work than the other, it could only be as a result of more advantageous selections for the substance making up the system (it is conceivable that, say, air might be better than water vapor) or the series of processes making up the cycle (nonflow processes might be preferable to flow processes). This corollary denies both possibilities and indicates that the cycles must have the same efficiency whatever the choices for the working substance or the series of processes. The two Carnot corollaries can be demonstrated using the Kelvin–Planck statement of the second law (see box).

D E M O N S T R AT I N G T H E C A R N O T C O R O L L A R I E S

The first Carnot corollary can be demonstrated using the arrangement of Fig. 5.6. A reversible power cycle R and an irreversible power cycle I operate between the same two reservoirs and each receives the same amount of energy QH from the hot reservoir. The reversible cycle produces work WR while the irreversible cycle produces work WI. In accord with the conservation of energy principle, each cycle discharges energy to the cold reservoir equal to the difference between QH and the work produced. Let R now operate in the opposite direction as a refrigeration (or heat pump) cycle. Since R is reversible, the magnitudes of the energy transfers WR, QH, and QC remain the same, but the energy transfers are oppositely directed, as shown by the dashed lines on Fig. 5.6. Moreover, with R operating in the opposite direction, the hot reservoir would experience no net change in its condition since it would receive QH from R while passing QH to I. The demonstration of the first Carnot corollary is completed by considering the combined system shown by the dotted line on Fig. 5.6, which consists of the two cycles and the hot reservoir. Since its parts execute cycles or experience no net change, the combined system operates in a cycle. Moreover, the combined system exchanges energy by heat transfer with a single reservoir: the cold reservoir. Accordingly, the combined system must satisfy Eq. 5.1 expressed as Wcycle 6 0

1single reservoir2

where the inequality is used because the combined system is irreversible in its operation since irreversible cycle I is one of its parts. Evaluating Wcycle for the combined system in terms of the work amounts WI and WR, the above inequality becomes WI WR 6 0 which shows that WI must be less than WR. Since each cycle receives the same energy input, QH, it follows that I R and this completes the demonstration. The second Carnot corollary can be demonstrated in a parallel way by considering any two reversible cycles R1 and R2 operating between the same two reservoirs. Then, letting R1 play the role of R and R2 the role of I in the previous development, a combined system consisting of the two cycles and the hot reservoir may be formed that must obey Eq. 5.1. However, in applying Eq. 5.1 to this combined system, the equality is used because the system is reversible in operation. Thus, it can be concluded that WR1 WR2, and therefore, R1 R2. The details are left as an exercise.

187

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Chapter 5 The Second Law of Thermodynamics Dotted line defines combined system Hot reservoir QH

WR

R

QH

I

WI Figure 5.6 Sketch for demonstrating that a reversible cycle R is more efficient than an irreversible cycle I when they operate between the same two reservoirs.

QC = QH – WR Q′C = QH – WI Cold reservoir

5.3.3 Refrigeration and Heat Pump Cycles Interacting with Two Reservoirs The second law of thermodynamics places limits on the performance of refrigeration and heat pump cycles as it does for power cycles. Consider Fig. 5.7, which shows a system undergoing a cycle while communicating thermally with two thermal reservoirs, a hot and a cold reservoir. The energy transfers labeled on the figure are in the directions indicated by the arrows. In accord with the conservation of energy principle, the cycle discharges energy QH by heat transfer to the hot reservoir equal to the sum of the energy QC received by heat transfer from the cold reservoir and the net work input. This cycle might be a refrigeration cycle or a heat pump cycle, depending on whether its function is to remove energy QC from the cold reservoir or deliver energy QH to the hot reservoir. For a refrigeration cycle the coefficient of performance is b

QC QC Wcycle QH QC

(5.3)

The coefficient of performance for a heat pump cycle is g

QH QH Wcycle QH QC

(5.4)

As the net work input to the cycle Wcycle tends to zero, the coefficients of performance given by Eqs. 5.3 and 5.4 approach a value of infinity. If Wcycle were identically zero, the system of Fig. 5.7 would withdraw energy QC from the cold reservoir and deliver energy QC to the hot reservoir, while undergoing a cycle. However, this method of operation would violate the Clausius statement of the second law and thus is not allowed. It follows that these coefficients of performance must invariably be finite in value. This may be regarded as another corollary of the second law. Further corollaries follow.

Hot reservoir

QH = QC + Wcycle

Boundary

Wcycle = Q H – Q C

Cold reservoir

QC

Figure 5.7 System undergoing a refrigeration or heat pump cycle while exchanging energy by heat transfer with two reservoirs.

5.3 Applying the Second Law to Thermodynamic Cycles

Thermodynamics in the News... Urban Heat Islands Worrisome Warm blankets of pollution-laden air surround major cities. Sunlight-absorbing rooftops and expanses of pavement, together with little greenery, conspire with other features of city living to raise urban temperatures as much as 10F above adjacent rural areas. Health-care professionals worry about the impact of these “heat islands,” especially on the elderly. Paradoxically, the hot exhaust from the air conditioners city dwellers use to keep cool also make sweltering neighborhoods even hotter. Recent studies indicate that airconditioner exhaust accounts for as much as 2F of the urban temperature rise. Vehicles and commercial activity also contribute.

City planners are combating heat islands. In Chicago, the once gravelroofed City Hall is now topped by a garden, with 20,000 vines, shrubs, and trees that absorb solar energy without increasing the roof temperature. The roof temperature in August is as much as 75F cooler than on nearby buildings, city sources report.

The maximum theoretical coefficients of performance for systems undergoing refrigeration and heat pump cycles while communicating thermally with two reservoirs at different temperatures are evaluated in Sec. 5.5 with reference to the following corollaries of the second law: COROLLARIES FOR REFRIGERATION AND HEAT PUMP CYCLES.

The coefficient of performance of an irreversible refrigeration cycle is always less than the coefficient of performance of a reversible refrigeration cycle when each operates between the same two thermal reservoirs. All reversible refrigeration cycles operating between the same two thermal reservoirs have the same coefficient of performance.

By replacing the term refrigeration with heat pump, we obtain counterpart corollaries for heat pump cycles. The first of these corollaries agrees with expectations stemming from the discussion of the second law thus far. To explore this, consider Fig. 5.8, which shows a reversible refrigeration cycle R and an irreversible refrigeration cycle I operating between the same two reservoirs. Each cycle removes the same energy QC from the cold reservoir. The net work input required to operate R is WR, while the net work input for I is WI. Each cycle discharges energy by heat transfer to the hot reservoir equal to the sum of QC and the net work input. The directions of the energy transfers are shown by arrows on Fig. 5.8. The presence of irreversibilities during the operation of a refrigeration cycle is expected to exact a penalty. If two refrigerators working

QH = QC + WR Q′H = QC + WI Hot reservoir

WR

R

I

WI Figure 5.8

QC

Cold reservoir

QC

Sketch for demonstrating that a reversible refrigeration cycle R has a greater coefficient of performance than an irreversible cycle I when they operate between the same two reservoirs.

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Chapter 5 The Second Law of Thermodynamics

between the same reservoirs each receive an identical energy transfer from the cold reservoir, QC, and one executes a reversible cycle while the other executes an irreversible cycle, we expect the irreversible cycle to require a greater net work input and thus to have a smaller coefficient of performance. By a simple extension it follows that all reversible refrigeration cycles operating between the same two reservoirs have the same coefficient of performance. Similar arguments apply to the counterpart heat pump cycle statements. These corollaries can be demonstrated formally using the Kelvin–Planck statement of the second law and a procedure similar to that employed for the Carnot corollaries. The details are left as an exercise.

5.4 Defining the Kelvin Temperature Scale

The results of Sec. 5.3 establish theoretical upper limits on the performance of power, refrigeration, and heat pump cycles communicating thermally with two reservoirs. Expressions for the maximum theoretical thermal efficiency of power cycles and the maximum theoretical coefficients of performance of refrigeration and heat pump cycles are developed in Sec. 5.5 using the Kelvin temperature scale defined in the present section. From the second Carnot corollary we know that all reversible power cycles operating between the same two thermal reservoirs have the same thermal efficiency, regardless of the nature of the substance making up the system executing the cycle or the series of processes. Since the efficiency is independent of these factors, its value can be related only to the nature of the reservoirs themselves. Noting that it is the difference in temperature between the two reservoirs that provides the impetus for heat transfer between them, and thereby for the production of work during the cycle, we reason that the efficiency depends only on the temperatures of the two reservoirs. From Eq. 5.2 it also follows that for such reversible power cycles the ratio of the heat transfers QCQH depends only on the reservoir temperatures. That is a

QC b rev c1uC, uH 2 QH cycle

(5.5)

where H and C denote the temperatures of the reservoirs and the function is for the present unspecified. Note that the words “rev cycle” are added to this expression to emphasize that it applies only to systems undergoing reversible cycles while operating between two thermal reservoirs. KELVIN SCALE

Kelvin scale

Equation 5.5 provides a basis for defining a thermodynamic temperature scale: a scale independent of the properties of any substance. There are alternative choices for the function that lead to this end. The Kelvin scale is obtained by making a particularly simple choice, namely, TCTH, where T is the symbol used to denote temperatures on the Kelvin scale. With this, Eq. 5.5 becomes a

QC TC b QH rev TH

(5.6)

cycle

Thus, two temperatures on the Kelvin scale are in the same ratio as the values of the heat transfers absorbed and rejected, respectively, by a system undergoing a reversible cycle while communicating thermally with reservoirs at these temperatures.

5.4 Defining the Kelvin Temperature Scale

If a reversible power cycle were operated in the opposite direction as a refrigeration or heat pump cycle, the magnitudes of the energy transfers QC and QH would remain the same, but the energy transfers would be oppositely directed. Accordingly, Eq. 5.6 applies to each type of cycle considered thus far, provided the system undergoing the cycle operates between two thermal reservoirs and the cycle is reversible. Equation 5.6 gives only a ratio of temperatures. To complete the definition of the Kelvin scale, it is necessary to proceed as in Sec. 1.6 by assigning the value 273.16 K to the temperature at the triple point of water. Then, if a reversible cycle is operated between a reservoir at 273.16 K and another reservoir at temperature T, the two temperatures are related according to T 273.16 a

Q b Qtp rev

(5.7)

cycle

where Qtp and Q are the heat transfers between the cycle and reservoirs at 273.16 K and temperature T, respectively. In the present case, the heat transfer Q plays the role of the thermometric property. However, since the performance of a reversible cycle is independent of the makeup of the system executing the cycle, the definition of temperature given by Eq. 5.7 depends in no way on the properties of any substance or class of substances. In Sec. 1.6 we noted that the Kelvin scale has a zero of 0 K, and lower temperatures than this are not defined. Let us take up these points by considering a reversible power cycle operating between reservoirs at 273.16 K and a lower temperature T. Referring to Eq. 5.7, we know that the energy rejected from the cycle by heat transfer Q would not be negative, so T must be nonnegative. Equation 5.7 also shows that the smaller the value of Q, the lower the value of T, and conversely. Accordingly, as Q approaches zero the temperature T approaches zero. It can be concluded that a temperature of zero on the Kelvin scale is the lowest conceivable temperature. This temperature is called the absolute zero, and the Kelvin scale is called an absolute temperature scale. INTERNATIONAL TEMPERATURE SCALE

When numerical values of the thermodynamic temperature are to be determined, it is not possible to use reversible cycles, for these exist only in our imaginations. However, temperatures evaluated using the constant-volume gas thermometer introduced in Sec. 1.6 are identical to those of the Kelvin scale in the range of temperatures where the gas thermometer can be used. Other empirical approaches can be employed for temperatures above and below the range accessible to gas thermometry. The Kelvin scale provides a continuous definition of temperature valid over all ranges and provides an essential connection between the several empirical measures of temperature. To provide a standard for temperature measurement taking into account both theoretical and practical considerations, the International Temperature Scale (ITS) was adopted in 1927. This scale has been refined and extended in several revisions, most recently in 1990. The International Temperature Scale of 1990 (ITS-90) is defined in such a way that the temperature measured on it conforms with the thermodynamic temperature, the unit of which is the kelvin, to within the limits of accuracy of measurement obtainable in 1990. The ITS-90 is based on the assigned values of temperature of a number of reproducible fixed points (Table 5.1). Interpolation between the fixed-point temperatures is accomplished by formulas that give the relation between readings of standard instruments and values of the ITS. In the range from 0.65 to 5.0 K, ITS-90 is defined by equations giving the temperature as functions of the vapor pressures of particular helium isotopes. The range from 3.0 to 24.5561 K is based on measurements using a helium constant-volume gas thermometer. In the range from 13.8033 to 1234.93 K, ITS-90 is defined by means of certain platinum resistance thermometers. Above 1234.9 K the temperature is defined using Planck’s equation for blackbody radiation and measurements of the intensity of visible-spectrum radiation.

191

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Chapter 5 The Second Law of Thermodynamics

Defining Fixed Points of the International Temperature Scale of 1990

TABLE 5.1

T (K)

Substancea

Stateb

3 to 5 13.8033 17 20.3 24.5561 54.3584 83.8058 234.3156 273.16 302.9146 429.7485 505.078 692.677 933.473 1234.93 1337.33 1357.77

He e-H2 e-H2 e-H2 Ne O2 Ar Hg H2 O Ga In Sn Zn Al Ag Au Cu

Vapor pressure point Triple point Vapor pressure point Vapor pressure point Triple point Triple point Triple point Triple point Triple point Melting point Freezing point Freezing point Freezing point Freezing point Freezing point Freezing point Freezing point

a He denotes 3He or 4He; e-H2 is hydrogen at the equilibrium concentration of the ortho- and para-molecular forms. b Triple point: temperature at which the solid, liquid, and vapor phases are in equilibrium. Melting point, freezing point: temperature, at a pressure of 101.325 kPa, at which the solid and liquid phases are in equilibrium.

Source: H. Preston-Thomas, “The International Temperature Scale of 1990 (ITS-90),” Metrologia 27, 3–10 (1990).

5.5 Maximum Performance Measures for Cycles Operating Between Two Reservoirs

The discussion of Sec. 5.3 continues in this section with the development of expressions for the maximum thermal efficiency of power cycles and the maximum coefficients of performance of refrigeration and heat pump cycles in terms of reservoir temperatures evaluated on the Kelvin scale. These expressions can be used as standards of comparison for actual power, refrigeration, and heat pump cycles. 5.5.1 Power Cycles The use of Eq. 5.6 in Eq. 5.2 results in an expression for the thermal efficiency of a system undergoing a reversible power cycle while operating between thermal reservoirs at temperatures TH and TC. That is

Carnot efficiency

hmax 1

which is known as the Carnot efficiency.

TC TH

(5.8)

5.5 Maximum Performance Measures for Cycles Operating Between Two Reservoirs

1.0

η max

Recalling the two Carnot corollaries, it should be evident that the efficiency given by Eq. 5.8 is the thermal efficiency of all reversible power cycles operating between two reservoirs at temperatures TH and TC, and the maximum efficiency any power cycle can have while operating between the two reservoirs. By inspection, the value of the Carnot efficiency increases as TH increases and/or TC decreases. Equation 5.8 is presented graphically in Fig. 5.9. The temperature TC used in constructing the figure is 298 K in recognition that actual power cycles ultimately discharge energy by heat transfer at about the temperature of the local atmosphere or cooling water drawn from a nearby river or lake. Note that the possibility of increasing the thermal efficiency by reducing TC below that of the environment is not practical, for maintaining TC lower than the ambient temperature would require a refrigerator that would have to be supplied work to operate. Figure 5.9 shows that the thermal efficiency increases with TH. Referring to segment a–b of the curve, where TH and are relatively low, we can see that increases rapidly as TH increases, showing that in this range even a small increase in TH can have a large effect on efficiency. Though these conclusions, drawn as they are from Fig. 5.9, apply strictly only to systems undergoing reversible cycles, they are qualitatively correct for actual power cycles. The thermal efficiencies of actual cycles are observed to increase as the average temperature at which energy is added by heat transfer increases and/or the average temperature at which energy is discharged by heat transfer is reduced. However, maximizing the thermal efficiency of a power cycle may not be the only objective. In practice, other considerations such as cost may be overriding.

A more complete discussion of power cycles is provided in Chaps. 8 and 9. 5.5.2 Refrigeration and Heat Pump Cycles Equation 5.6 is also applicable to reversible refrigeration and heat pump cycles operating between two thermal reservoirs, but for these QC represents the heat added to the cycle from the cold reservoir at temperature TC on the Kelvin scale and QH is the heat discharged to the hot reservoir at temperature TH. Introducing Eq. 5.6 in Eq. 5.3 results in the following expression for the coefficient of performance of any system undergoing a reversible refrigeration cycle while operating between the two reservoirs

(5.9)

Similarly, substituting Eq. 5.6 into Eq. 5.4 gives the following expression for the coefficient of performance of any system undergoing a reversible heat pump cycle while operating between the two reservoirs gmax

TH TH TC

(5.10)

a 298 1000 2000 3000 TH (K)

Figure 5.9 Carnot efficiency versus TH, for TC 298 K.

Conventional power-producing cycles have thermal efficiencies ranging up to about 40%. This value may seem low, but the comparison should be made with an appropriate limiting value and not 100%. for example. . . consider a system executing a power cycle for which the average temperature of heat addition is 745 K and the average temperature at which heat is discharged is 298 K. For a reversible cycle receiving and discharging energy by heat transfer at these temperatures, the thermal efficiency given by Eq. 5.8 is 60%. When compared to this value, an actual thermal efficiency of 40% does not appear to be so low. The cycle would be operating at two-thirds of the theoretical maximum.

TC TH TC

b

0.5 0

COMMENT.

bmax

193

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Chapter 5 The Second Law of Thermodynamics

The development of Eqs. 5.9 and 5.10 is left as an exercise. Note that the temperatures used to evaluate max and max must be absolute temperatures on the Kelvin or Rankine scale. From the discussion of Sec. 5.3.3, it follows that Eqs. 5.9 and 5.10 are the maximum coefficients of performance that any refrigeration and heat pump cycles can have while operating between reservoirs at temperatures TH and TC. As for the case of the Carnot efficiency, these expressions can be used as standards of comparison for actual refrigerators and heat pumps. A more complete discussion of refrigeration and heat pump cycles is provided in Chap. 10. 5.5.3 Power Cycle, Refrigeration and Heat Pump Applications In this section, three examples are provided that illustrate the use of the second law corollaries of Secs. 5.3.2 and 5.3.3 together with Eqs. 5.8, 5.9, and 5.10, as appropriate. The first example uses Eq. 5.8 to evaluate an inventor’s claim. In the next example, we evaluate the coefficient of performance of a refrigerator and compare it with the maximum theoretical value. EXAMPLE

5.1

Evaluating a Power Cycle Performance Claim

An inventor claims to have developed a power cycle capable of delivering a net work output of 410 kJ for an energy input by heat transfer of 1000 kJ. The system undergoing the cycle receives the heat transfer from hot gases at a temperature of 500 K and discharges energy by heat transfer to the atmosphere at 300 K. Evaluate this claim. SOLUTION Known: A system operates in a cycle and produces a net amount of work while receiving and discharging energy by heat transfer at fixed temperatures. Find: Evaluate the claim that the cycle can develop 410 kJ of work for an energy input by heat of 1000 kJ. Schematic and Given Data: Qin = 1000 kJ 500 K Power cycle

Qout

Assumptions: W = 410 kJ

1. The system is shown on the accompanying figure. 2. The hot gases and the atmosphere play the roles of hot and cold reservoirs, respectively.

300 K Figure E5.1

Analysis: Inserting the values supplied by the inventor into Eq. 5.2, the cycle thermal efficiency is h

410 kJ 0.41 141%2 1000 kJ

The maximum thermal efficiency any power cycle can have while operating between reservoirs at TH 500 K and TC 300 K is given by Eq. 5.8.

❶

hmax 1

TC 300 K 1 0.40 140%2 TH 500 K

Since the thermal efficiency of the actual cycle exceeds the maximum theoretical value, the claim cannot be valid.

❶

The temperatures used in evaluating max must be in K or R.

5.5 Maximum Performance Measures for Cycles Operating Between Two Reservoirs

EXAMPLE

5.2

195

Evaluating Refrigerator Performance

By steadily circulating a refrigerant at low temperature through passages in the walls of the freezer compartment, a refrigerator maintains the freezer compartment at 5C when the air surrounding the refrigerator is at 22C. The rate of heat transfer from the freezer compartment to the refrigerant is 8000 kJ/h and the power input required to operate the refrigerator is 3200 kJ/h. Determine the coefficient of performance of the refrigerator and compare with the coefficient of performance of a reversible refrigeration cycle operating between reservoirs at the same two temperatures. SOLUTION Known: A refrigerator maintains a freezer compartment at a specified temperature. The rate of heat transfer from the refrigerated space, the power input to operate the refrigerator, and the ambient temperature are known. Find: Determine the coefficient of performance and compare with that of a reversible refrigerator operating between reservoirs at the same two temperatures. Schematic and Given Data:

Surroundings at 22°C (295 K) · QH · Wcycle = 3200 kJ/h

Assumptions: 1. The system shown on the accompanying figure is at steady state. 2. The freezer compartment and the surrounding air play the roles of cold and hot reservoirs, respectively.

System boundary · Q C = 8000 kJ/h Freezer compartment at –5°C (268 K)

Figure E5.2

Analysis: Inserting the given operating data into Eq. 5.3, the coefficient of performance of the refrigerator is # QC 8000 kJ/h b # 2.5 3200 kJ/h Wcycle Substituting values into Eq. 5.9 gives the coefficient of performance of a reversible refrigeration cycle operating between reservoirs at TC 268 K and TH 295 K

❶

bmax

❶

TC 268 K 9.9 T H TC 295 K 268 K

The difference between the actual and maximum coefficients of performance suggests that there may be some potential for improving the thermodynamic performance. This objective should be approached judiciously, however, for improved performance may require increases in size, complexity, and cost.

In Example 5.3, we determine the minimum theoretical work input and cost for one day of operation of an electric heat pump.

Chapter 5 The Second Law of Thermodynamics

196

EXAMPLE

Evaluating Heat Pump Performance

5.3

A dwelling requires 5 10 5 kJ per day to maintain its temperature at 22C when the outside temperature is 10C. (a) If an electric heat pump is used to supply this energy, determine the minimum theoretical work input for one day of operation, in kJ. SOLUTION Known: A heat pump maintains a dwelling at a specified temperature. The energy supplied to the dwelling, the ambient temperature, and the unit cost of electricity are known. Find: Determine the minimum theoretical work required by the heat pump and the corresponding electricity cost. Schematic and Given Data:

Assumptions: Heat pump Wcycle

QH

Dwelling at 22°C (295°K)

1. The system is shown on the accompanying figure. 2. The dwelling and the outside air play the roles of hot and cold reservoirs, respectively.

QC at 10°C (283 K)

Figure E5.3

Analysis: (a) Using Eq. 5.4, the work for any heat pump cycle can be expressed as Wcycle QH . The coefficient of performance of an actual heat pump is less than, or equal to, the coefficient of performance max of a reversible heat pump cycle when each operates between the same two thermal reservoirs: max. Accordingly, for a given value of QH, and using Eq. 5.10 to evaluate max, we get QH Wcycle gmax a1

TC bQ TH H

Inserting values Wcycle a1

❶

283°K b 15 105 kJ2 2.03 104 kJ/day 295°K

The minimum theoretical work input is 2.03 104 kJ/day.

❶ ❷

Note that the reservoir temperatures TC and TH must be expressed here in K. Because of irreversibilities, an actual heat pump must be supplied more work than the minimum to provide the same heating effect. The actual daily cost could be substantially greater than the minimum theoretical cost.

5.6 Carnot Cycle

Carnot cycle

The Carnot cycle introduced in this section provides a specific example of a reversible power cycle operating between two thermal reservoirs. Two other examples are provided in Chap. 9: the Ericsson and Stirling cycles. Each of these cycles exhibits the Carnot efficiency given by Eq. 5.8. In a Carnot cycle, the system executing the cycle undergoes a series of four internally reversible processes: two adiabatic processes alternated with two isothermal processes.

5.6 Carnot Cycle

Figure 5.10 shows the p–v diagram of a Carnot power cycle in which the system is a gas in a piston–cylinder assembly. Figure 5.11 provides details of how the cycle is executed. The piston and cylinder walls are nonconducting. The heat transfers are in the directions of the arrows. Also note that there are two reservoirs at temperatures TH and TC, respectively, and an insulating stand. Initially, the piston–cylinder assembly is on the insulating stand and the system is at state 1, where the temperature is TC. The four processes of the cycle are Process 1–2: The gas is compressed adiabatically to state 2, where the temperature is TH. Process 2–3: The assembly is placed in contact with the reservoir at TH. The gas expands isothermally while receiving energy QH from the hot reservoir by heat transfer. Process 3–4: The assembly is again placed on the insulating stand and the gas is allowed to continue to expand adiabatically until the temperature drops to TC. Process 4–1: The assembly is placed in contact with the reservoir at TC. The gas is compressed isothermally to its initial state while it discharges energy QC to the cold reservoir by heat transfer. For the heat transfer during Process 2–3 to be reversible, the difference between the gas temperature and the temperature of the hot reservoir must be vanishingly small. Since the reservoir temperature remains constant, this implies that the temperature of the gas also remains constant during Process 2–3. The same can be concluded for Process 4–1. For each of the four internally reversible processes of the Carnot cycle, the work can be represented as an area on Fig. 5.10. The area under the adiabatic process line 1–2 represents the work done per unit of mass to compress the gas in this process. The areas under process lines 2–3 and 3–4 represent the work done per unit of mass by the gas as it expands in these processes. The area under process line 4–1 is the work done per unit of mass to compress the gas in this process. The enclosed area on the p–v diagram, shown shaded, is the net work developed by the cycle per unit of mass. The Carnot cycle is not limited to processes of a closed system taking place in a piston–cylinder assembly. Figure 5.12 shows the schematic and accompanying p–v diagram of a Carnot cycle executed by water steadily circulating through a series of four interconnected components that has features in common with the simple vapor power plant shown in Fig. 4.14. As the water flows through the boiler, a change of phase from liquid to vapor at constant temperature TH occurs as a result of heat transfer from the hot reservoir. Since

Adiabatic compression

Isothermal expansion

Adiabatic expansion

Isothermal compression

Gas

Insulating stand

Process 1–2 Figure 5.11

assembly.

QH Hot reservoir, TH

Process 2–3

Insulating stand Boundary Process 3–4

QC Cold reservoir, TC

Process 4–1

Carnot power cycle executed by a gas in a piston–cylinder

197

p 2

TC

3 1

TH 4 v

Figure 5.10 p–v diagram for a Carnot gas power cycle.

Chapter 5 The Second Law of Thermodynamics

198

Hot reservoir, TH QH 4

1

Boiler

p Pump

Work

Turbine Work

3

Condenser

TC

1

4

2

QC Cold reservoir, TC Figure 5.12

TH

TH 3

2

TC v

Carnot vapor power cycle.

temperature remains constant, pressure also remains constant during the phase change. The steam exiting the boiler expands adiabatically through the turbine and work is developed. In this process the temperature decreases to the temperature of the cold reservoir, TC, and there is an accompanying decrease in pressure. As the steam passes through the condenser, a heat transfer to the cold reservoir occurs and some of the vapor condenses at constant temperature TC. Since temperature remains constant, pressure also remains constant as the water passes through the condenser. The fourth component is a pump, or compressor, that receives a twophase liquid–vapor mixture from the condenser and returns it adiabatically to the state at the boiler entrance. During this process, which requires a work input to increase the pressure, the temperature increases from TC to TH. Carnot cycles also can be devised that are composed of processes in which a capacitor is charged and discharged, a paramagnetic substance is magnetized and demagnetized, and so on. However, regardless of the type of device or the working substance used, the Carnot cycle always has the same four internally reversible processes: two adiabatic processes alternated with two isothermal processes. Moreover, the thermal efficiency is always given by Eq. 5.8 in terms of the temperatures of the two reservoirs evaluated on the Kelvin or Rankine scale. If a Carnot power cycle is operated in the opposite direction, the magnitudes of all energy transfers remain the same but the energy transfers are oppositely directed. Such a cycle may be regarded as a reversible refrigeration or heat pump cycle, for which the coefficients of performance are given by Eqs. 5.9 and 5.10, respectively. A Carnot refrigeration or heat pump cycle executed by a gas in a piston–cylinder assembly is shown in Fig. 5.13. The cycle consists of the following four processes in series:

p TH 4

TC

3 1 2 v

Figure 5.13

p–v diagram for a Carnot gas refrigeration or heat pump cycle.

Process 1–2: The gas expands isothermally at TC while receiving energy QC from the cold reservoir by heat transfer. Process 2–3: The gas is compressed adiabatically until its temperature is TH. Process 3–4: The gas is compressed isothermally at TH while it discharges energy QH to the hot reservoir by heat transfer. Process 4–1: The gas expands adiabatically until its temperature decreases to TC. It will be recalled that a refrigeration or heat pump effect can be accomplished in a cycle only if a net work input is supplied to the system executing the cycle. In the case of the cycle shown in Fig. 5.13, the shaded area represents the net work input per unit of mass.

Exercises: Things Engineers Think About

199

Chapter Summary and Study Guide

In this chapter, we motivate the need for and usefulness of the second law of thermodynamics, and provide the basis for subsequent applications involving the second law in Chaps. 6 and 7. Two equivalent statements of the second law, the Clausius and Kelvin–Planck statements, are introduced together with several corollaries that establish the best theoretical performance for systems undergoing cycles while interacting with thermal reservoirs. The irreversibility concept is introduced and the related notions of irreversible, reversible, and internally reversible processes are discussed. The Kelvin temperature scale is defined and used to obtain expressions for the maximum performance measures of power, refrigeration, and heat pump cycles operating between two thermal reservoirs. Finally, the Carnot cycle is introduced to provide a specific example of a reversible cycle operating between two thermal reservoirs.

The following checklist provides a study guide for this chapter. When your study of the text and end-of-chapter exercises has been completed you should be able to write out the meanings of the terms listed in the margins

throughout the chapter and understand each of the related concepts. The subset of key concepts listed below is particularly important in subsequent chapters. give the Kelvin–Planck statement of the second law,

correctly interpreting the “less than” and “equal to” signs in Eq. 5.1. list several important irreversibilities. apply the corollaries of Secs. 5.3.2 and 5.3.3 together

with Eqs. 5.8, 5.9, and 5.10 to assess the performance of power cycles and refrigeration and heat pump cycles. describe the Carnot cycle.

Key Engineering Concepts

Kelvin–planck statement p. 178 irreversible process p. 180

irreversibilities p. 180 internal and external irreversibilities p. 181

internally reversible process p. 184 Carnot corollaries p. 186

Kelvin scale p. 190 Carnot efficiency p. 192

Exercises: Things Engineers Think About 1. Explain how work might be developed when (a) Ti is less than T0 in Fig. 5.1a, (b) pi is less than p0 in Fig. 5.1b.

Would you expect the power input to the compressor to be greater in an internally reversible compression or an actual compression?

2. A system consists of an ice cube in a cup of tap water. The ice cube melts and eventually equilibrium is attained. How might work be developed as the ice and water come to equilibrium?

9. If a window air conditioner were placed on a table in a room and operated, would the room temperature increase, decrease, or remain the same?

3. Describe a process that would satisfy the conservation of energy principle, but does not actually occur in nature.

10. To increase the thermal efficiency of a reversible power cycle operating between thermal reservoirs at temperatures TH and TC, would it be better to increase TH or decrease TC by equal amounts?

4. Referring to Fig. 2.3, identify internal irreversibilities associated with system A. Repeat for system B. 5. What are some of the principal irreversibilities present during operation of (a) an automobile engine, (b) a household refrigerator, (c) a gas-fired water heater, (d) an electric water heater? 6. For the gearbox of Example 2.4, list the principal irreversibilities and classify them as internal or external. 7. Steam at a given state enters a turbine operating at steady state and expands adiabatically to a specified lower pressure. Would you expect the power output to be greater in an internally reversible expansion or an actual expansion? 8. Air at a given state enters a compressor operating at steady state and is compressed adiabatically to a specified higher pressure.

11. Electric power plants typically reject energy by heat transfer to a body of water or the atmosphere. Would it be advisable to reject heat instead to large blocks of ice maintained by a refrigeration system? 12. Referring to Eqs. 5.9 and 5.10, how might the coefficients of performance of refrigeration cycles and heat pumps be increased? 13. Is it possible for the coefficient of performance of a refrigeration cycle to be less than one? To be greater than one? Answer the same questions for a heat pump cycle.

200

Chapter 5 The Second Law of Thermodynamics

Problems: Developing Engineering Skills Exploring the Second Law

5.1 A heat pump receives energy by heat transfer from the outside air at 0C and discharges energy by heat transfer to a dwelling at 20C. Is this in violation of the Clausius statement of the second law of thermodynamics? Explain. 5.2 Air as an ideal gas expands isothermally at 20C from a volume of 1 m3 to 2 m3. During this process there is heat transfer to the air from the surrounding atmosphere, modeled as a thermal reservoir, and the air does work. Evaluate the work and heat transfer for the process, in kJ/kg. Is this process in violation of the second law of thermodynamics? Explain. 5.3 Complete the demonstration of the equivalence of the Clausius and Kelvin–Planck statements of the second law given in Sec. 5.1 by showing that a violation of the Kelvin–Planck statement implies a violation of the Clausius statement. 5.4 An inventor claims to have developed a device that undergoes a thermodynamic cycle while communicating thermally with two reservoirs. The system receives energy QC from the cold reservoir and discharges energy QH to the hot reservoir while delivering a net amount of work to its surroundings. There are no other energy transfers between the device and its surroundings. Using the second law of thermodynamics, evaluate the inventor’s claim. 5.5 A hot thermal reservoir is separated from a cold thermal reservoir by a cylindrical rod insulated on its lateral surface. Energy transfer by conduction between the two reservoirs takes place through the rod, which remains at steady state. Using the Kelvin–Planck statement of the second law, demonstrate that such a process is irreversible. 5.6 Methane gas within a piston–cylinder assembly is compressed in a quasiequilibrium process. Is this process internally reversible? Is this process reversible? 5.7 Water within a piston–cylinder assembly cools isothermally at 120C from saturated vapor to saturated liquid while interacting thermally with its surroundings at 20C. Is the process internally reversible? Is it reversible? Discuss. 5.8 Complete the discussion of the Kelvin–Planck statement of the second law in Sec. 5.3.1 by showing that if a system undergoes a thermodynamic cycle reversibly while communicating thermally with a single reservoir, the equality in Eq. 5.1 applies. 5.9 A power cycle I and a reversible power cycle R operate between the same two reservoirs, as shown in Fig. 5.6. Cycle I has a thermal efficiency equal to two-thirds of that for cycle R. Using the Kelvin-Planck statement of the second law, prove that cycle I must be irreversible. 5.10 A reversible power cycle R and an irreversible power cycle I operate between the same two reservoirs. (a) If each cycle receives the same amount of energy QH from the hot reservoir, show that cycle I necessarily discharges

more energy QC to the cold reservoir than cycle R. Discuss the implications of this for actual power cycles. (b) If each cycle develops the same net work, show that cycle I necessarily receives more energy QH from the hot reservoir than cycle R. Discuss the implications of this for actual power cycles. 5.11 Provide the details left to the reader in the demonstration of the second Carnot corollary given in Sec. 5.3.2. 5.12 Using the Kelvin–Planck statement of the second law of thermodynamics, demonstrate the following corollaries: (a) The coefficient of performance of an irreversible refrigeration cycle is always less than the coefficient of performance of a reversible refrigeration cycle when both exchange energy by heat transfer with the same two reservoirs. (b) All reversible refrigeration cycles operating between the same two reservoirs have the same coefficient of performance. (c) The coefficient of performance of an irreversible heat pump cycle is always less than the coefficient of performance of a reversible heat pump cycle when both exchange energy by heat transfer with the same two reservoirs. (d) All reversible heat pump cycles operating between the same two reservoirs have the same coefficient of performance. 5.13 Before introducing the temperature scale now known as the Kelvin scale, Kelvin suggested a logarithmic scale in which the function of Eq. 5.5 takes the form c exp uC exp uH where H and C denote, respectively, the temperatures of the hot and cold reservoirs on this scale. (a) Show that the relation between the Kelvin temperature T and the temperature on the logarithmic scale is u ln T C where C is a constant. (b) On the Kelvin scale, temperatures vary from 0 to . Determine the range of temperature values on the logarithmic scale. (c) Obtain an expression for the thermal efficiency of any system undergoing a reversible power cycle while operating between reservoirs at temperatures H and C on the logarithmic scale. 5.14 Demonstrate that the gas temperature scale (Sec. 1.6) is identical to the Kelvin temperature scale. 5.15 To increase the thermal efficiency of a reversible power cycle operating between reservoirs at TH and TC, would you increase TH while keeping TC constant, or decrease TC while keeping TH constant? Are there any natural limits on the increase in thermal efficiency that might be achieved by such means?

Problems: Developing Engineering Skills

5.16 Two reversible power cycles are arranged in series. The first cycle receives energy by heat transfer from a reservoir at temperature TH and rejects energy to a reservoir at an intermediate temperature T. The second cycle receives the energy rejected by the first cycle from the reservoir at temperature T and rejects energy to a reservoir at temperature TC lower than T. Derive an expression for the intermediate temperature T in terms of TH and TC when

201

(c) Letting T H¿ TC T0, plot Q¿HQH versus THT0 for T ¿C T0 0.85, 0.9, and 0.95, and versus T ¿C T0 for TH T0 2, 3, and 4. 5.22 Figure P5.22 shows a system consisting of a power cycle driving a heat pump. At steady state, the power cycle receives # Qs by heat# transfer at Ts from the high-temperature source and # delivers Q1 to a dwelling at Td. The heat pump receives Q0 # from the outdoors at T0, and delivers Q2 to the dwelling.

(a) the net work of the two power cycles is equal. (b) the thermal efficiencies of the two power cycles are equal. 5.17 If the thermal efficiency of a reversible power cycle operating between two reservoirs is denoted by max, develop an expression in terms of max for the coefficient of performance of (a) a reversible refrigeration cycle operating between the same two reservoirs. (b) a reversible heat pump operating between the same two reservoirs. 5.18 The data listed below are claimed for a power cycle operating between reservoirs at 527C and 27C. For each case, determine if any principles of thermodynamics would be violated. (a) QH 700 kJ, Wcycle 400 kJ, QC 300 kJ. (b) QH 640 kJ, Wcycle 400 kJ, QC 240 kJ. (c) QH 640 kJ, Wcycle 400 kJ, QC 200 kJ. 5.19 A refrigeration cycle operating between two reservoirs receives energy QC from a cold reservoir at TC 280 K and rejects energy QH to a hot reservoir at TH 320 K. For each of the following cases determine whether the cycle operates reversibly, irreversibly, or is impossible: (a) (b) (c) (d)

QC 1500 kJ, Wcycle 150 kJ. QC 1400 kJ, QH 1600 kJ. QH 1600 kJ, Wcycle 400 kJ. 5.

5.20 A reversible power cycle receives QH from a hot reservoir at temperature TH and rejects energy by heat transfer to the surroundings at temperature T0. The work developed by the power cycle is used to drive a refrigeration cycle that removes QC from a cold reservoir at temperature TC and discharges energy by heat transfer to the same surroundings at T0. (a) Develop an expression for the ratio QCQH in terms of the temperature ratios THT0 and TCT0. (b) Plot QCQH versus THT0 for TCT0 0.85, 0.9, and 0.95, and versus TCT0 for THT0 2, 3, and 4. 5.21 A reversible power cycle receives energy QH from a reservoir at temperature TH and rejects QC to a reservoir at temperature TC. The work developed by the power cycle is used to drive a reversible heat pump that removes energy Q¿C from a reservoir at temperature T ¿C and rejects energy Q¿H to a reservoir at temperature T H¿ . (a) Develop an expression for the ratio Q¿H QH in terms of the temperatures of the four reservoirs. (b) What must be the relationship of the temperatures TH, TC, T ¿C, and T H¿ for Q¿HQH to exceed a value of unity?

· Q1

Dwelling at Td

· Qs

Air Ts

· Q2

Heat pump

Power cycle Driveshaft

Fuel

· Q0 Outdoors at T0

Figure P5.22

(a) Obtain an expression for the maximum value # # theoretical # of the performance parameter 1Q1 Q2 2 Qs in terms of the temperature ratios TsTd and T0Td. (b) Plot the result of part (a) versus TsTd ranging from 2 to 4 for T0Td 0.85, 0.9, and 0.95. Power Cycle Applications

5.23 A power cycle operates between a reservoir at temperature T and a lower-temperature reservoir at 280 K. At steady state, the cycle develops 40 kW of power while rejecting 1000 kJ/min of energy by heat transfer to the cold reservoir. Determine the minimum theoretical value for T, in K. 5.24 A certain reversible power cycle has the same thermal efficiency for hot and cold reservoirs at 1000 and 500 K, respectively, as for hot and cold reservoirs at temperature T and 1000 K. Determine T, in K. 5.25 A reversible power cycle whose thermal efficiency is 50% operates between a reservoir at 1800 K and a reservoir at a lower temperature T. Determine T, in K. 5.26 An inventor claims to have developed a device that executes a power cycle while operating between reservoirs at 800 and 350 K that has a thermal efficiency of (a) 56%, (b) 40%. Evaluate the claim for each case. 5.27 At steady state, a cycle develops a power output of 10 kW for heat addition at a rate of 10 kJ per cycle of operation from a source at 1500 K. Energy is rejected by heat transfer to cooling water at 300 K. Determine the minimum theoretical number of cycles required per minute. 5.28 At steady state, a power cycle having a thermal efficiency of 38% generates 100 MW of electricity while discharging energy by heat transfer to cooling water at an average temperature

202

Chapter 5 The Second Law of Thermodynamics

of 70F. The average temperature of the steam passing through the boiler is 900F. Determine (a) the rate at which energy is discharged to the cooling water, in Btu /h. (b) the minimum theoretical rate at which energy could be discharged to the cooling water, in Btu/h. Compare with the actual rate and discuss. 5.29 Ocean temperature energy conversion (OTEC) power plants generate power by utilizing the naturally occurring decrease with depth of the temperature of ocean water. Near Florida, the ocean surface temperature is 27C, while at a depth of 700 m the temperature is 7C. (a) Determine the maximum thermal efficiency for any power cycle operating between these temperatures. (b) The thermal efficiency of existing OTEC plants is approximately 2 percent. Compare this with the result of part (a) and comment. 5.30 During January, at a location in Alaska winds at 30C can be observed. Several meters below ground the temperature remains at 13C, however. An inventor claims to have devised a power cycle exploiting this situation that has a thermal efficiency of 10%. Discuss this claim. 5.31 Figure P5.31 shows a system for collecting solar radiation and utilizing it for the production of electricity by a power cycle. The solar collector receives solar radiation at the rate of 0.315 kW per m2 of area and provides energy to a storage unit whose temperature remains constant at 220C. The power cycle receives energy by heat transfer from the storage unit, generates electricity at the rate 0.5 MW, and discharges energy by heat transfer to the surroundings at 20C. For operation at steady state,

Solar radiation Solar collector

5.32 The preliminary design of a space station calls for a power cycle that at steady state receives energy by heat transfer at TH 600 K from a nuclear source and rejects energy to space by thermal radiation according to Eq. 2.33. For the radiative surface, the temperature is TC, the emissivity is 0.6, and the surface receives no radiation from any source. The thermal efficiency of the power cycle is one-half that of a reversible power cycle operating between reservoirs at TH and TC. # (a) For TC 400 K, determine Wcycle A, the net power developed per unit of radiator surface area, in kW/m2, and the thermal efficiency. # (b) Plot Wcycle A and the thermal efficiency versus TC, and de# termine the maximum value of Wcycle A. (c) Determine the range of temperatures TC, in K, for which # Wcycle A is within 2 percent of the maximum value obtained in part (b). The Stefan-Boltzmann constant is 5.67 108 Wm2 K4. Refrigeration and Heat Pump Cycle Applications

5.33 An inventor claims to have developed a refrigeration cycle that requires a net power input of 1.2 kW to remove 25,000 kJ/h of energy by heat transfer from a reservoir at 30C and discharge energy by heat transfer to a reservoir at 20C. There are no other energy transfers with the surroundings. Evaluate this claim. 5.34 Determine if a tray of ice cubes could remain frozen when placed in a food freezer having a coefficient of performance of 9 operating in a room where the temperature is 32C (90F). 5.35 The refrigerator shown in Fig. P5.35 operates at steady state with a coefficient of performance of 4.5 and a power input of 0.8 kW. Energy is rejected from the refrigerator to the surroundings at 20C by heat transfer from metal coils whose average surface temperature is 28C. Determine (a) the rate energy is rejected, in kW. (b) the lowest theoretical temperature inside the refrigerator, in K. (c) the maximum theoretical power, in kW, that could be developed by a power cycle operating between the coils and

Surroundings at 20°C Refrigerator β = 4.5

Area

Storage unit at 220°C

Power cycle

Surroundings, 20°C

+ –

Coils, 28°C · QH

Figure P5.31

(a) determine the minimum theoretical collector area required, in m2. (b) determine the collector area required, in m2, as a function of the thermal efficiency and the collector efficiency, defined as the fraction of the incident energy that is stored. Plot the collector area versus for collector efficiencies equal to 1.0, 0.75, and 0.5.

+ –

0.8 kW

Figure P5.35

Problems: Developing Engineering Skills

the surroundings. Would you recommend making use of this opportunity for developing power? 5.36 Determine the minimum theoretical power, in W, required at steady state by a refrigeration system to maintain a cryogenic sample at 126C in a laboratory at 21C, if energy leaks by heat transfer to the sample from its surroundings at a rate of 900 W. 5.37 For each kW of power input to an ice maker at steady state, determine the maximum rate that ice can be produced, in kg/h, from liquid water at 0C. Assume that 333 kJ/kg of energy must be removed by heat transfer to freeze water at 0C, and that the surroundings are at 20C. 5.38 At steady state, a refrigeration cycle removes 18,000 kJ/h of energy by heat transfer from a space maintained at 40C and discharges energy by heat transfer to surroundings at 20C. If the coefficient of performance of the cycle is 25 percent of that of a reversible refrigeration cycle operating between thermal reservoirs at these two temperatures, determine the power input to the cycle, in kW. 5.39 A refrigeration cycle having a coefficient of performance of 3 maintains a computer laboratory at 18C on a day when the outside temperature is 30C. The thermal load at steady state consists of energy entering through the walls and windows at a rate of 30,000 kJ/h and from the occupants, computers, and lighting at a rate of 6000 kJ/h. Determine the power required by this cycle and compare with the minimum theoretical power required for any refrigeration cycle operating under these conditions, each in kW. 5.40 A heat pump operating at steady state is driven by a 1-kW electric motor and provides heating for a building whose interior is to be kept at 20C. On a day when the outside temperature is 0C and energy is lost through the walls and roof at a rate of 60,000 kJ/h, would the heat pump suffice? 5.41 A heat pump maintains a dwelling at 20C when the outside temperature is 0C. The heat transfer rate through the walls and roof is 3000 kJ/h per degree temperature difference between the inside and outside. Determine the minimum theoretical power required to drive the heat pump, in kW. 5.42 A building for which the heat transfer rate through the walls and roof is 400 W per degree temperature difference between the inside and outside is to be maintained at 20C. For a day when the outside temperature is 4C, determine the power required at steady state, kW, to heat the building using electrical resistance elements and compare with the minimum theoretical power that would be required by a heat pump. Repeat the comparison using typical manufacturer’s data for the heat pump coefficient of performance. 5.43 Plot the coefficient of performance max given by Eq. 5.9 for TH 298 K versus TC ranging between 235 and 298 K. Discuss the practical implications of the decrease in the coefficient of performance with decreasing temperature TC. 5.44 At steady state, a refrigerator whose coefficient of performance is 3 removes energy by heat transfer from a freezer compartment at 0C at the rate of 6000 kJ/h and discharges

203

energy by heat transfer to the surroundings, which are at 20C. Determine the power input to the refrigerator and compare with the power input required by a reversible refrigeration cycle operating between reservoirs at these two temperatures. 5.45 By supplying energy to a dwelling at a rate of 25,000 kJ/h, a heat pump maintains the temperature of the dwelling at 20C when the outside air is at 10C. If electricity costs 8 cents per kW # h, determine the minimum theoretical operating cost for each day of operation. Carnot Cycle Applications

5.46 Two kilograms of water execute a Carnot power cycle. During the isothermal expansion, the water is heated until it is a saturated vapor from an initial state where the pressure is 40 bar and the quality is 15%. The vapor then expands adiabatically to a pressure of 1.5 bar while doing 491.5 kJ/kg of work. (a) Sketch the cycle on p–v coordinates. (b) Evaluate the heat and work for each process, in kJ. (c) Evaluate the thermal efficiency. 5.47 One kilogram of air as an ideal gas executes a Carnot power cycle having a thermal efficiency of 60%. The heat transfer to the air during the isothermal expansion is 40 kJ. At the end of the isothermal expansion, the pressure is 5.6 bar and the volume is 0.3 m3. Determine (a) the maximum and minimum temperatures for the cycle, in K. (b) the pressure and volume at the beginning of the isothermal expansion in bar and m3, respectively. (c) the work and heat transfer for each of the four processes, in kJ. (d) Sketch the cycle on p–v coordinates. 5.48 The pressure–volume diagram of a Carnot power cycle executed by an ideal gas with constant specific heat ratio k is shown in Fig. P5.48. Demonstrate that (a) V4V2 V1V3. (b) T2T3 ( p2p3)(k–1)k. (c) T2T3 (V3V2)k–1. p 1 Isothermal

2 Q41 = 0 4

Q23 = 0

Isothermal

3 v

Figure P5.48

Chapter 5 The Second Law of Thermodynamics

204

5.49 One-tenth kilogram of air as an ideal gas with k 1.4 executes a Carnot refrigeration cycle, as shown in Fig. 5.13. The isothermal expansion occurs at 23C with a heat transfer to the air of 3.4 kJ. The isothermal compression occurs

at 27C to a final volume of 0.01 m3. Using the results of Prob. 5.64 as needed, determine (a) the pressure, in kPa, at each of the four principal states. (b) the work, in kJ, for each of the four processes.Design

Design & Open Ended Problems: Exploring Engineering Practice 5.1D Write a paper outlining the contributions of Carnot, Clausius, Kelvin, and Planck to the development of the second law of thermodynamics. In what ways did the now-discredited caloric theory influence the development of the second law as we know it today? What is the historical basis for the idea of a perpetual motion machine of the second kind that is sometimes used to state the second law? 5.2D The heat transfer rate through the walls and roof of a building is 3570 kJ/h per degree temperature difference between the inside and outside. For outdoor temperatures ranging from 15 to 20C, make a comparison of the daily cost of maintaining the building at 20C by means of an electric heat pump, direct electric resistance heating, and a conventional gas-fired furnace. 5.3D To maintain the passenger compartment of an automobile traveling at 13.4 m/s at 21C when the surrounding air temperature is 32C, the vehicle’s air conditioner removes 5.275 kW by heat transfer. Estimate the amount of engine horsepower required to drive the air conditioner. Referring to typical manufacturer’s data, compare your estimate with the actual horsepower requirement. Discuss the relationship between the initial unit cost of an automobile air-conditioning system and its operating cost. 5.4D Prepare a memorandum discussing alternative means for achieving the required cooling of a 1000 MW power plant located on the river of Problem 5.38. Discuss environmental issues related to each of your alternatives. 5.5D Figure P5.5D shows that the typical thermal efficiency of U.S. power plants increased rapidly from 1925 to 1960, but has increased only gradually since then. Discuss the most important factors contributing to this plateauing of thermal efficiency and the most promising near-term and long-term

Thermal efficiency, %

40

30

20

1925 1935 1945 1955 1965 1975 1985 Year

Figure P5.5D

technologies that might lead to appreciable thermal efficiency gains. 5.6D Abandoned lead mines near Park Hills, Missouri are filled with an estimated 2.5 108 m3 of water at an almost constant temperature of 14C. How might this resource be exploited for heating and cooling of the town’s dwellings and commercial buildings? A newspaper article refers to the water-filled mines as a free source of heating and cooling. Discuss this characterization. 5.7D The Minto Wheel is a power-producing device activated by evaporating and condensing a working substance. The only energy input would be from a waste heat or solar source. Write a paper explaining the operating principles of the device. Indicate whether the Minto wheel operates as a thermodynamic power cycle, and if so give the range of thermal efficiencies that might be achieved. Evaluate propane as a working substance and suggest an alternative. How would the wheel diameter and the volumes of the containers holding the working substance affect performance? Are there any practical applications for such a device? Discuss. 5.8D Figure P5.8D shows a device for pumping water without the use of an electrical or fuel input. The container (1) holds a suitable liquid working substance separated from a quantity of air by a flexible bladder (2). During the daytime, heat transfer from the warm surroundings vaporizes some of the liquid, thereby displacing the bladder and forcing air through the pipe (3) into the top of the chamber (4) below. As the air enters the lower chamber, it pushes against the top of the piston (5). The displacement of the piston pumps water from the chamber through the lift pipe (6) and into the collection tank (7). At night, heat transfer to the cooler surroundings causes the vapor to condense, thereby restoring the bladder to its original position and recharging the lower chamber. Critically evaluate this device for pumping water. Specify a suitable working substance. Does the device operate in a thermodynamic power cycle? If so, estimate the range of thermal efficiencies that might be expected. Propose a means for pumping water with this type of device more than once a day. Write a report of your findings. 5.9D A method for generating electricity using gravitational energy is described in U.S. Patent No. 4,980,572. The method employs massive spinning wheels located underground that serve as the prime mover of an alternator for generating electricity. Each wheel is kept in motion by torque pulses transmitted to it via a suitable mechanism from vehicles passing overhead. What practical difficulties might be encountered in implementing such a method for generating electricity? If the vehicles are trolleys on an existing urban transit system, might this be a cost-effective way to generate electricity? If the

Design & Open Ended Problems: Exploring Engineering Practice

Trapped air

1 2

205

Working substance 7 Valve

6 3

1. Container 2. Flexible bladder 3. Air pipe 4. Chamber with one-way (check) valve 5. Piston 6. Water pipe with one-way valve 7. Water collection tank

5 4 One-way valve

Figure P5.8D

vehicle motion were sustained by the electricity generated, would this be an example of a perpetual motion machine? Discuss. 5.10D A technical article considers hurricanes as an example of a natural Carnot engine (K. A. Emmanuel, “Toward a General Theory of Hurricanes,” American Scientist, 76, 371–379, 1988). A subsequent U.S. Patent (No. 4,885,913) is said to have

been inspired by such an analysis. Does the concept have scientific merit? Engineering merit? Discuss. 5.11D Urban Heat Islands Worrisome (see box Sec. 5.3). Investigate adverse health conditions that might be exacerbated for persons living in urban heat islands. Write a report including at least three references.

C H A P

6 chapter objective

T E R

Using Entropy

E N G I N E E R I N G C O N T E X T Up to this point, our study of the second law has been concerned primarily with what it says about systems undergoing thermodynamic cycles. In this chapter means are introduced for analyzing systems from the second law perspective as they undergo processes that are not necessarily cycles. The property entropy plays a prominent part in these considerations. The objective of the present chapter is to introduce entropy and show its use for thermodynamic analysis. The word energy is so much a part of the language that you were undoubtedly familiar with the term before encountering it in early science courses. This familiarity probably facilitated the study of energy in these courses and in the current course in engineering thermodynamics. In the present chapter you will see that the analysis of systems from a second law perspective is conveniently accomplished in terms of the property entropy. Energy and entropy are both abstract concepts. However, unlike energy, the word entropy is seldom heard in everyday conversation, and you may never have dealt with it quantitatively before. Energy and entropy play important roles in the remaining chapters of this book.

6.1 Introducing Entropy

Corollaries of the second law are developed in Chap. 5 for systems undergoing cycles while communicating thermally with two reservoirs, a hot reservoir and a cold reservoir. In the present section a corollary of the second law known as the Clausius inequality is introduced that is applicable to any cycle without regard for the body, or bodies, from which the cycle receives energy by heat transfer or to which the cycle rejects energy by heat transfer. The Clausius inequality provides the basis for introducing two ideas instrumental for analyses of both closed systems and control volumes from a second law perspective: the property entropy (Sec. 6.2) and the entropy balance (Secs. 6.5 and 6.6). The Clausius inequality states that for any thermodynamic cycle

Clausius inequality

a T b dQ

~

(6.1)

b

where Q represents the heat transfer at a part of the system boundary during a portion of the cycle, and T is the absolute temperature at that part of the boundary. The subscript “b” serves as a reminder that the integrand is evaluated at the boundary of the system executing the cycle. The symbol indicates that the integral is to be performed over all parts of the boundary and 206

6.1 Introducing Entropy

over the entire cycle. The equality and inequality have the same interpretation as in the Kelvin–Planck statement: the equality applies when there are no internal irreversibilities as the system executes the cycle, and the inequality applies when internal irreversibilities are present. The Clausius inequality can be demonstrated using the Kelvin–Planck statement of the second law (see box).

DEVELOPING THE CLAUSIUS INEQUALITY

The Clausius inequality can be demonstrated using the arrangement of Fig. 6.1. A system receives energy Q at a location on its boundary where the absolute temperature is T while the system develops work W. In keeping with our sign convention for heat transfer, the phrase receives energy Q includes the possibility of heat transfer from the system. The energy Q is received from (or absorbed by) a thermal reservoir at Tres. To ensure that no irreversibility is introduced as a result of heat transfer between the reservoir and the system, let it be accomplished through an intermediary system that undergoes a cycle without irreversibilities of any kind. The cycle receives energy Q from the reservoir and supplies Q to the system while producing work W. From the definition of the Kelvin scale (Eq. 5.6), we have the following relationship between the heat transfers and temperatures: dQ¿ dQ a b Tres T b

(a)

As temperature may vary, a multiplicity of such reversible cycles may be required. Consider next the combined system shown by the dotted line on Fig. 6.1. An energy balance for the combined system is dEC dQ¿ dWC where WC is the total work of the combined system, the sum of W and W, and dEC denotes the change in energy of the combined system. Solving the energy balance for WC and using Eq. (a) to eliminate Q from the resulting expression yields dWC Tres a

dQ b dEC T b

Now, let the system undergo a single cycle while the intermediary system undergoes one or more cycles. The total work of the combined system is WC

res a

T ~

dQ b T b

dE ~

0 C

Tres

aTb dQ

(b)

~

b

Since the reservoir temperature is constant, Tres can be brought outside the integral. The term involving the energy of the combined system vanishes because the energy change for any cycle is zero. The combined system operates in a cycle because its parts execute cycles. Since the combined system undergoes a cycle and exchanges energy by heat transfer with a single reservoir, Eq. 5.1 expressing the Kelvin–Planck statement of the second law must be satisfied. Using this, Eq. (b) reduces to give Eq. 6.1, where the equality applies when there are no irreversibilities within the system as it executes the cycle and the inequality applies when internal irreversibilities are present. This interpretation actually refers to the combination of system plus intermediary cycle. However, the intermediary cycle is regarded as free of irreversibilities, so the only possible site of irreversibilities is the system alone.

207

208

Chapter 6 Using Entropy Reservoir at Tres δQ´ Intermediary cycle δW´

Combined system boundary δQ T System

δW Figure 6.1 Illustration used to develop the Clausius inequality.

System boundary

Equation 6.1 can be expressed equivalently as

aTb dQ

scycle

~

(6.2)

b

where cycle can be viewed as representing the “strength” of the inequality. The value of cycle is positive when internal irreversibilities are present, zero when no internal irreversibilities are present, and can never be negative. In summary, the nature of a cycle executed by a system is indicated by the value for cycle as follows: scycle 0 scycle 7 0 scycle 6 0

no irreversibilities present within the system irreversibilities present within the system impossible

Accordingly, cycle is a measure of the effect of the irreversibilities present within the system executing the cycle. This point is developed further in Sec. 6.5, where cycle is identified as the entropy produced (or generated) by internal irreversibilities during the cycle. 2 C B A 1 Figure 6.2

Two internally reversible cycles.

6.2 Defining Entropy Change

A quantity is a property if, and only if, its change in value between two states is independent of the process (Sec. 1.3). This aspect of the property concept is used in the present section together with Eq. 6.2 to introduce entropy. Two cycles executed by a closed system are represented in Fig. 6.2. One cycle consists of an internally reversible process A from state 1 to state 2, followed by internally reversible process C from state 2 to state 1. The other cycle consists of an internally reversible process B from state 1 to state 2, followed by the same process C from state 2 to state 1 as in the first cycle. For the first cycle, Eq. 6.2 takes the form a

2

1

1

dQ b a T A

a

2

(6.3a)

2

0 dQ b scycle T C

dQ b a T B

1

0 dQ b scycle T C

(6.3b)

and for the second cycle

1

2

In writing Eqs. 6.3, the term cycle has been set to zero since the cycles are composed of internally reversible processes.

6.3 Retrieving Entropy Data

When Eq. 6.3b is subtracted from Eq. 6.3a a

2

1

dQ b a T A

2

1

dQ b T B

This shows that the integral of QT is the same for both processes. Since A and B are arbitrary, it follows that the integral of QT has the same value for any internally reversible process between the two states. In other words, the value of the integral depends on the end states only. It can be concluded, therefore, that the integral represents the change in some property of the system. Selecting the symbol S to denote this property, which is called entropy, its change is given by S2 S1 a

2

1

dQ b int T rev

definition of entropy (6.4a)

change

where the subscript “int rev” is added as a reminder that the integration is carried out for any internally reversible process linking the two states. Equation 6.4a is the definition of entropy change. On a differential basis, the defining equation for entropy change takes the form dS a

dQ b int T rev

(6.4b)

Entropy is an extensive property. The SI unit for entropy is J/K. However, in this book it is convenient to work in terms of kJ/K. Units in SI for specific entropy are kJ/kg # K for s and kJ/kmol # K for s. Since entropy is a property, the change in entropy of a system in going from one state to another is the same for all processes, both internally reversible and irreversible, between these two states. Thus, Eq. 6.4a allows the determination of the change in entropy, and once it has been evaluated, this is the magnitude of the entropy change for all processes of the system between the two states. The evaluation of entropy change is discussed further in the next section. It should be clear that entropy is defined and evaluated in terms of a particular integral for which no accompanying physical picture is given. We encountered this previously with the property enthalpy. Enthalpy is introduced without physical motivation in Sec. 3.3.2. Then, in Chap. 4, enthalpy is shown to be useful for thermodynamic analysis. As for the case of enthalpy, to gain an appreciation for entropy you need to understand how it is used and what it is used for.

6.3 Retrieving Entropy Data

In Chap. 3, we introduced means for retrieving property data, including tables, graphs, equations, and the software available with this text. The emphasis there is on evaluating the properties p, v, T, u, and h required for application of the conservation of mass and energy principles. For application of the second law, entropy values are usually required. In this section, means for retrieving entropy data are considered.

units for entropy

209

210

Chapter 6 Using Entropy

6.3.1 General Considerations The defining equation for entropy change, Eq. 6.4a, serves as the basis for evaluating entropy relative to a reference value at a reference state. Both the reference value and the reference state can be selected arbitrarily. The value of the entropy at any state y relative to the value at the reference state x is obtained in principle from S y Sx a

y

x

dQ b int T rev

(6.5)

where Sx is the reference value for entropy at the specified reference state. The use of entropy values determined relative to an arbitrary reference state is satisfactory as long as they are used in calculations involving entropy differences, for then the reference value cancels. This approach suffices for applications where composition remains constant. When chemical reactions occur, it is necessary to work in terms of absolute values of entropy determined using the third law of thermodynamics (Chap. 13).

ENTROPY DATA FOR WATER AND REFRIGERANTS

Tables of thermodynamic data are introduced in Sec. 3.3 for water and several refrigerants (Tables A-2 through A-18). Specific entropy is tabulated in the same way as considered there for the properties v, u, and h, and entropy values are retrieved similarly. VAPOR DATA. In the superheat regions of the tables for water and the refrigerants, specific entropy is tabulated along with v, u, and h versus temperature and pressure. for example. . . consider two states of water. At state 1 the pressure is 3 MPa and the temperature is 500C. At state 2, the pressure is 0.3 MPa and the specific entropy is the same as at state 1, s2 s1. The object is to determine the temperature at state 2. Using T1 and p1, we find the specific entropy at state 1 from Table A-4 as s1 7.2338 kJkg # K. State 2 is fixed by the pressure, p2 0.3 MPa, and the specific entropy, s2 7.2338 kJkg # K. Returing to Table A-4 at 0.3 MPa and interpolating with s2 between 160 and 200C results in T2 183C.

For saturation states, the values of sf and sg are tabulated as a function of either saturation pressure or saturation temperature. The specific entropy of a twophase liquid–vapor mixture is calculated using the quality SATURATION DATA.

s 11 x2sf xsg sf x1sg sf 2

(6.6)

These relations are identical in form to those for v, u, and h (Sec. 3.3). for example. . . let us determine the specific entropy of Refrigerant 134a at a state where the temperature is 0C and the specific internal energy is 138.43 kJ/kg. Referring to Table A-10, we see that the given value for u falls between uf and ug at 0C, so the system is a two-phase liquid–vapor mixture. The quality of the mixture can be determined from the known specific internal energy x

u uf 138.43 49.79 0.5 ug uf 227.06 49.79

Then with values from Table A-10 s 11 x2sf xsg 10.5210.19702 10.5210.91902 0.5580 kJ/kg # K

6.3 Retrieving Entropy Data

211

Compressed liquid data are presented for water in Tables A-5. In these tables s, v, u, and h are tabulated versus temperature and pressure as in the superheat tables, and the tables are used similarly. In the absence of compressed liquid data, the value of the specific entropy can be estimated in the same way as estimates for v and u are obtained for liquid states (Sec. 3.3.6), by using the saturated liquid value at the given temperature

LIQUID DATA.

s1T, p2 sf 1T 2

(6.7)

for example. . . suppose the value of specific entropy is required for water at 25 bar, 200C. The specific entropy is obtained directly from Table A-5 as s 2.3294 kJkg # K. Using the saturated liquid value for specific entropy at 200C from Table A-2, the specific entropy is approximated with Eq. 6.7 as s 2.3309 kJkg # K, which agrees closely with the previous value.

The specific entropy values for water and the refrigerants given in Tables A-2 through A-18 are relative to the following reference states and values. For water, the entropy of saturated liquid at 0.01C is set to zero. For the refrigerants, the entropy of the saturated liquid at 40C is assigned a value of zero. COMPUTER RETRIEVAL. The software available with this text, Interactive Thermodynamics: IT, provides data for the substances considered in this section. Entropy data are retrieved by simple call statements placed in the workspace of the program. for example. . . consider a two-phase liquid–vapor mixture of H2O at p 1 bar, v 0.8475 m3/kg. The following illustrates how specific entropy and quality x are obtained using IT

p = 1 // bar v = 0.8475 // m3/kg v = vsat_Px(“Water/Steam”,p,x) s = ssat_Px(“Water/Steam”,p,x)

The software returns values of x 0.5 and s 4.331 kJ/kg # K, which can be checked using data from Table A-3. Note that quality x is implicit in the list of arguments in the expression for specific volume, and it is not necessary to solve explicitly for x. As another example, consider superheated ammonia vapor at p 1.5 bar, T 8C. Specific entropy is obtained from IT as follows: p = 1.5 // bar T = 8 // C s = s_PT(“Ammonia”,p,T)

The software returns s 5.981 kJ/kg # K, which agrees closely with the value obtained by interpolation in Table A-15.

METHODOLOGY UPDATE

Note that IT does not provide compressed liquid data for any substance. IT returns liquid entropy data using the approximation of Eq. 6.7. Similarly, Eqs. 3.11, 3.12, and 3.14 are used to return liquid values for v, u, and h, respectively.

USING GRAPHICAL ENTROPY DATA

The use of property diagrams as an adjunct to problem solving is emphasized throughout this book. When applying the second law, it is frequently helpful to locate states and plot processes on diagrams having entropy as a coordinate. Two commonly used figures having entropy as one of the coordinates are the temperature–entropy diagram and the enthalpy– entropy diagram. TEMPERATURE–ENTROPY DIAGRAM. The main features of a temperature–entropy diagram are shown in Fig. 6.3. For detailed figures for water in SI, see Fig. A-7. Observe

T–s diagram

Chapter 6 Using Entropy

h = constant

T

v = constant p = const ant p = constan t

212

h

p = constant

T = constant

Critical point

T = constant

Satu rate d va por p= con sta nt p= con sta nt

Sat u ra ted liqu id

p = constant

v = constant

Sat

ura ted

x = 0.9

v

x = 0.2

ap

or

x= x=

0 .9

0.9

6

Critical point s

Figure 6.3

Temperature–entropy diagram.

s Figure 6.4

Enthalpy–entropy diagram.

that lines of constant enthalpy are shown on these figures. Also note that in the superheated vapor region constant specific volume lines have a steeper slope than constantpressure lines. Lines of constant quality are shown in the two-phase liquid–vapor region. On some figures, lines of constant quality are marked as percent moisture lines. The percent moisture is defined as the ratio of the mass of liquid to the total mass. In the superheated vapor region of the T–s diagram, constant specific enthalpy lines become nearly horizontal as pressure is reduced. These states are shown as the shaded area on Fig. 6.3. For states in this region of the diagram, the enthalpy is determined primarily by the temperature: h(T, p) h(T). This is the region of the diagram where the ideal gas model provides a reasonable approximation. For superheated vapor states outside the shaded area, both temperature and pressure are required to evaluate enthalpy, and the ideal gas model is not suitable.

Mollier diagram

ENTHALPY–ENTROPY DIAGRAM. The essential features of an enthalpy–entropy diagram, commonly known as a Mollier diagram, are shown in Fig. 6.4. For detailed figures for water in SI, see Figs. A-8. Note the location of the critical point and the appearance of lines of constant temperature and constant pressure. Lines of constant quality are shown in the twophase liquid–vapor region (some figures give lines of constant percent moisture). The figure is intended for evaluating properties at superheated vapor states and for two-phase liquid– vapor mixtures. Liquid data are seldom shown. In the superheated vapor region, constanttemperature lines become nearly horizontal as pressure is reduced. These states are shown, approximately, as the shaded area on Fig. 6.4. This area corresponds to the shaded area on the temperature–entropy diagram of Fig. 6.3, where the ideal gas model provides a reasonable approximation.

for example. . . to illustrate the use of the Mollier diagram in SI units, consider two states of water. At state 1, T1 240C, p1 0.10 MPa. The specific enthalpy and

6.3 Retrieving Entropy Data

quality are required at state 2, where p2 0.01 MPa and s2 s1. Turning to Fig. A-8, state 1 is located in the superheated vapor region. Dropping a vertical line into the two-phase liquid–vapor region, state 2 is located. The quality and specific enthalpy at state 2 read from the figure agree closely with values obtained using Tables A-3 and A-4: x2 0.98 and h2 2537 kJ/ kg. USING THE T dS EQUATIONS

Although the change in entropy between two states can be determined in principle by using Eq. 6.4a, such evaluations are generally conducted using the T dS equations developed in this section. The T dS equations allow entropy changes to be evaluated from other more readily determined property data. The use of the T dS equations to evaluate entropy changes for ideal gases is illustrated in Sec. 6.3.2 and for incompressible substances in Sec. 6.3.3. The importance of the T dS equations is greater than their role in assigning entropy values, however. In Chap. 11 they are used as a point of departure for deriving many important property relations for pure, simple compressible systems, including means for constructing the property tables giving u, h, and s. The T dS equations are developed by considering a pure, simple compressible system undergoing an internally reversible process. In the absence of overall system motion and the effects of gravity, an energy balance in differential form is 1dQ2 int dU 1dW2 int rev

rev

(6.8)

By definition of simple compressible system (Sec. 3.1), the work is 1dW2 int p dV rev

(6.9a)

On rearrangement of Eq. 6.4b, the heat transfer is 1dQ2 int T dS rev

(6.9b)

Substituting Eqs. 6.9 into Eq. 6.8, the first T dS equation results T dS dU p dV

(6.10)

first T dS equation

The second T dS equation is obtained from Eq. 6.10 using H U pV. Forming the differential dH dU d1 pV2 dU p dV V dp On rearrangement dU p dV dH V dp Substituting this into Eq. 6.10 gives the second T dS equation T dS dH V dp

(6.11)

The T dS equations can be written on a unit mass basis as T ds du p dv

(6.12a)

T ds dh v dp

(6.12b)

second T dS equation

213

214

Chapter 6 Using Entropy

or on a per mole basis as T ds du pdv T d s d h v dp

(6.13a) (6.13b)

Although the T dS equations are derived by considering an internally reversible process, an entropy change obtained by integrating these equations is the change for any process, reversible or irreversible, between two equilibrium states of a system. Because entropy is a property, the change in entropy between two states is independent of the details of the process linking the states. To show the use of the T dS equations, consider a change in phase from saturated liquid to saturated vapor at constant temperature and pressure. Since pressure is constant, Eq. 6.12b reduces to give ds

dh T

Then, because temperature is also constant during the phase change sg sf

hg hf

(6.14)

T

This relationship shows how sg sf is calculated for tabulation in property tables. for example. . . consider Refrigerant 134a at 0C. From Table A-10, hg hf 197.21 kJ/kg, so with Eq. 6.14 sg sf

197.21 kJ/kg kJ 0.7220 # 273.15 K kg K

which is the value calculated using sf and sg from the table.

6.3.2 Entropy Change of an Ideal Gas In this section the T dS equations are used to evaluate the entropy change between two states of an ideal gas. It is convenient to begin with Eqs. 6.12 expressed as p du

dv T T v dh dp ds T T

ds

(6.15) (6.16)

For an ideal gas, du cv(T) dT, dh cp(T) dT, and pv RT. With these relations, Eqs. 6.15 and 6.16 become, respectively ds cv 1T 2

dT dv

R v T

and

ds cp 1T 2

dp dT R p T

(6.17)

Since R is a constant, the last terms of Eqs. 6.17 can be integrated directly. However, because cv and cp are functions of temperature for ideal gases, it is necessary to have information about

6.3 Retrieving Entropy Data

the functional relationships before the integration of the first term in these equations can be performed. Since the two specific heats are related by cp 1T 2 cv 1T 2 R

(3.44)

where R is the gas constant, knowledge of either specific heat function suffices. On integration, Eqs. 6.17 give, respectively s1T2, v2 2 s1T1, v1 2

T2

s1T2, p2 2 s1T1, p1 2

T2

T1

T1

cv 1T 2

v2 dT

R ln v1 T

(6.18)

cp 1T 2

p2 dT R ln p1 T

(6.19)

As for internal energy and enthalpy changes, the evaluation of entropy changes for ideal gases can be reduced to a convenient tabular approach. To introduce this, we begin by selecting a reference state and reference value: The value of the specific entropy is set to zero at the state where the temperature is 0 K and the pressure is 1 atmosphere. Then, using Eq. 6.19, the specific entropy at a state where the temperature is T and the pressure is 1 atm is determined relative to this reference state and reference value as USING IDEAL GAS TABLES.

s°1T 2

T

cp 1T 2 T

dT

(6.20)

The symbol s(T ) denotes the specific entropy at temperature T and a pressure of 1 atm. Because s depends only on temperature, it can be tabulated versus temperature, like h and u. For air as an ideal gas, s with units of kJ/kg # K is given in Tables A-22.Values of s ° for several other common gases are given in Tables A-23 with units of kJ/kmol # K. Since the integral of Eq. 6.19 can be expressed in terms of s

T2

T1

cp

T2 T1 dT dT dT cp cp T T T 0 0 s°1T2 2 s°1T1 2

it follows that Eq. 6.19 can be written as s1T2, p2 2 s1T1, p1 2 s°1T2 2 s°1T1 2 R ln

p2 p1

(6.21a)

p2 p1

(6.21b)

or on a per mole basis as s1T2, p2 2 s1T1, p1 2 s °1T2 2 s °1T1 2 R ln

Using Eqs. 6.21 and the tabulated values for s or s °, as appropriate, entropy changes can be determined that account explicitly for the variation of specific heat with temperature. for example. . . let us evaluate the change in specific entropy, in kJkg # K, of air modeled as an ideal gas from a state where T1 300 K and p1 1 bar to a state where

215

216

Chapter 6 Using Entropy

T2 1000 K and p2 3 bar. Using Eq. 6.21a and data from Table A-22 s2 s1 s°1T2 2 s°1T1 2 R ln

p2 p1 kJ 3 bar 8.314 kJ 12.96770 1.702032 # ln # kg K 28.97 kg K 1 bar # 0.9504 kJ/kg K

If a table giving s (or s °) is not available for a particular gas of interest, the integrals of Eqs. 6.18 and 6.19 can be performed analytically or numerically using specific heat data such as provided in Tables A-20 and A-21. ASSUMING CONSTANT SPECIFIC HEATS. When the specific heats cv and cp are taken as constants, Eqs. 6.18 and 6.19 reduce, respectively, to

s1T2, v2 2 s1T1, v1 2 cv ln

T2 v2

R ln v1 T1 p2 T2 s1T2, p2 2 s1T1, p1 2 cp ln R ln p1 T1

(6.22) (6.23)

These equations, along with Eqs. 3.50 and 3.51 giving u and h, respectively, are applicable when assuming the ideal gas model with constant specific heats. for example. . . let us determine the change in specific entropy, in kJ/kg # K, of air

as an ideal gas undergoing a process from T1 300 K, p1 1 bar to T2 400 K, p2 5 bar. Because of the relatively small temperature range, we assume a constant value of cp evaluated at 350 K. Using Eq. 6.23 and cp 1.008 kJ/kg # K from Table A-20 ¢s cp ln

p2 T2 R ln p1 T1

a1.008

kJ 400 K 8.314 kJ 5 bar b ln a ba b ln a b kg # K 300 K 28.97 kg # K 1 bar 0.1719 kJ/kg # K

COMPUTER RETRIEVAL. For air and other gases modeled as ideal gases, IT directly returns s(T, p) based upon the following form of Eq. 6.19

s1T, p2 s1Tref, pref 2

cp 1T 2

T

Tref

T

dT R ln

p pref

and the following choice of reference state and reference value: Tref 0 K (0R), pref 1 atm, and s(Tref, pref) 0, giving s1T, p2

T

cp 1T 2 T

dT R ln

p pref

Changes in specific entropy evaluated using IT agree with entropy changes evaluated using ideal gas tables. for example. . . consider a process of air as an ideal gas from T1 300 K, p1 1 bar to T2 1000 K, p2 3 bar. The change in specific entropy, denoted

6.4 Entropy Change in Internally Reversible Processes

as dels, is determined in SI units using IT as follows: p1 = 1 // bar T1 = 300 // K p2 = 3 T2 = 1000 s1 = s_TP(“Air”,T1,p1) s2 = s_TP(“Air”,T2,p2) dels = s2 – s1

The software returns values of s1 1.706, s2 2.656, and dels 0.9501, all in units of kJ/kg # K. This value for s agrees with the value obtained using Table A-22 in the example following Eqs. 6.21. Note that IT returns specific entropy directly and does not use the special function s. 6.3.3 Entropy Change of an Incompressible Substance The incompressible substance model introduced in Sec. 3.3.6 assumes that the specific volume (density) is constant and the specific heat depends solely on temperature, cv c(T). Accordingly, the differential change in specific internal energy is du c(T) dT and Eq. 6.15 reduces to 0

c1T 2 dT c1T 2 dT p dv ds

T T T On integration, the change in specific entropy is s 2 s1

c1T 2 dT T

T2

T1

1incompressible2

When the specific heat is assumed constant, this becomes s2 s1 c ln

T2 T1

1incompressible, constant c2

(6.24)

Equation 6.24, along with Eqs. 3.20 giving u and h, respectively, are applicable to liquids and solids modeled as incompressible. Specific heats of some common liquids and solids are given in Table A-19.

6.4 Entropy Change in Internally Reversible Processes

In this section the relationship between entropy change and heat transfer for internally reversible processes is considered. The concepts introduced have important applications in subsequent sections of the book. The present discussion is limited to the case of closed systems. Similar considerations for control volumes are presented in Sec. 6.9. As a closed system undergoes an internally reversible process, its entropy can increase, decrease, or remain constant. This can be brought out using Eq. 6.4b dS a

dQ bint T rev

217

218

Chapter 6 Using Entropy

which indicates that when a closed system undergoing an internally reversible process receives energy by heat transfer, the system experiences an increase in entropy. Conversely, when energy is removed from the system by heat transfer, the entropy of the system decreases. This can be interpreted to mean that an entropy transfer accompanies heat transfer. The direction of the entropy transfer is the same as that of the heat transfer. In an adiabatic internally reversible process, the entropy would remain constant. A constant-entropy process is called an isentropic process. On rearrangement, the above expression gives

isentropic process

Qint = rev

T (δQ) = T dS int

2 1

1dQ2 int T dS rev

T dS 2

Integrating from an initial state 1 to a final state 2

rev

1

Qint rev

S Figure 6.5

Area representation of heat transfer for an internally reversible process of a closed system.

Carnot cycle

T dS 2

(6.25)

1

From Eq. 6.25 it can be concluded that an energy transfer by heat to a closed system during an internally reversible process can be represented as an area on a temperature–entropy diagram. Figure 6.5 illustrates the area interpretation of heat transfer for an arbitrary internally reversible process in which temperature varies. Carefully note that temperature must be in kelvins or degrees Rankine, and the area is the entire area under the curve (shown shaded). Also note that the area interpretation of heat transfer is not valid for irreversible processes, as will be demonstrated later. To provide an example illustrating both the entropy change that accompanies heat transfer and the area interpretation of heat transfer, consider Fig. 6.6a, which shows a Carnot power cycle (Sec. 5.6). The cycle consists of four internally reversible processes in series: two isothermal processes alternated with two adiabatic processes. In Process 2–3, heat transfer to the system occurs while the temperature of the system remains constant at TH. The system entropy increases due to the accompanying entropy transfer. For this process, Eq. 6.25 gives Q23 TH(S3 S2), so area 2–3–a–b–2 on Fig. 6.6a represents the heat transfer during the process. Process 3–4 is an adiabatic and internally reversible process and thus is an isentropic (constant-entropy) process. Process 4–1 is an isothermal process at TC during which heat is transferred from the system. Since entropy transfer accompanies the heat transfer, system entropy decreases. For this process, Eq. 6.25 gives Q41 TC(S1 S4), which is negative in value. Area 4–1–b–a–4 on Fig. 6.6a represents the magnitude of the heat transfer Q41. Process 1–2, which completes the cycle, is adiabatic and internally reversible (isentropic).

TH

2

3

T

TH

4

3

1

2

T TC

1

4

b

a

TC

b

a

S

S

(a)

(b)

Figure 6.6 Carnot cycles on the temperature–entropy diagram. (a) Power cycle. (b) Refrigeration or heat pump cycle.

6.4 Entropy Change in Internally Reversible Processes

219

The net work of any cycle is equal to the net heat transfer, so enclosed area 1–2–3–4–1 represents the net work of the cycle. The thermal efficiency of the cycle may also be expressed in terms of areas: h

Wcycle Q23

area 1–2–3–4–1 area 2–3–a–b–2

The numerator of this expression is (TH TC)(S3 S2) and the denominator is TH(S3 S2), so the thermal efficiency can be given in terms of temperatures only as 1 TCTH. If the cycle were executed as shown in Fig. 6.6b, the result would be a Carnot refrigeration or heat pump cycle. In such a cycle, heat is transferred to the system while its temperature remains at TC, so entropy increases during Process 1–2. In Process 3–4 heat is transferred from the system while the temperature remains constant at TH and entropy decreases. To further illustrate concepts introduced in this section, the next example considers water undergoing an internally reversible process while contained in a piston–cylinder assembly.

EXAMPLE

6.1

Internally Reversible Process of Water

Water, initially a saturated liquid at 100C, is contained in a piston–cylinder assembly. The water undergoes a process to the corresponding saturated vapor state, during which the piston moves freely in the cylinder. If the change of state is brought about by heating the water as it undergoes an internally reversible process at constant pressure and temperature, determine the work and heat transfer per unit of mass, each in kJ/kg. SOLUTION Known: Water contained in a piston–cylinder assembly undergoes an internally reversible process at 100C from saturated liquid to saturated vapor. Find: Determine the work and heat transfer per unit mass. Schematic and Given Data:

p

T

f Water

System boundary

g W –– m

100°C 100°C

g

f Q –– m

v

s

Figure E6.1

Assumptions: 1. The water in the piston–cylinder assembly is a closed system. 2. The process is internally reversible. 3. Temperature and pressure are constant during the process. 4. There is no change in kinetic or potential energy between the two end states.

220

Chapter 6 Using Entropy

Analysis: At constant pressure the work is W m

g

f

p dv p1vg vf 2

With values from Table A-2 W 1 kJ m3 105 N/m2 ` ` 3 # ` 11.014 bar2 11.673 1.0435 103 2 a b ` m kg 1 bar 10 N m 170 kJ/kg Since the process is internally reversible and at constant temperature, Eq. 6.25 gives Q

g

T dS m

f

g

T ds

f

or Q T 1sg sf 2 m With values from Table A-2 Q 1373.15 K217.3549 1.30692 kJ/ kg # K 2257 kJ/kg m

❶

As shown in the accompanying figure, the work and heat transfer can be represented as areas on p–v and T–s diagrams, respectively.

❶

The heat transfer can be evaluated alternatively from an energy balance written on a unit mass basis as ug u f

Q W m m

Introducing Wm p(vg vf) and solving Q 1ug uf 2 p1vg vf 2 m

1ug pvg 2 1uf pvf 2

h g hf

From Table A-2 at 100C, hg hf 2257 kJ/kg, which is the same value for Qm as obtained in the solution.

6.5 Entropy Balance for Closed Systems

2 R I

1 Figure 6.7

Cycle used to develop the entropy balance.

In this section, the Clausius inequality expressed by Eq. 6.2 and the defining equation for entropy change are used to develop the entropy balance for closed systems. The entropy balance is an expression of the second law that is particularly convenient for thermodynamic analysis. The current presentation is limited to closed systems. The entropy balance is extended to control volumes in Sec. 6.6. 6.5.1 Developing the Entropy Balance Shown in Fig. 6.7 is a cycle executed by a closed system. The cycle consists of process I, during which internal irreversibilities are present, followed by internally reversible process R. For this cycle, Eq. 6.2 takes the form

2

1

a

dQ b

T b

1

2

a

dQ b int s T rev

(6.26)

6.5 Entropy Balance for Closed Systems

where the first integral is for process I and the second is for process R. The subscript b in the first integral serves as a reminder that the integrand is evaluated at the system boundary. The subscript is not required in the second integral because temperature is uniform throughout the system at each intermediate state of an internally reversible process. Since no irreversibilities are associated with process R, the term cycle of Eq. 6.2, which accounts for the effect of irreversibilities during the cycle, refers only to process I and is shown in Eq. 6.26 simply as . Applying the definition of entropy change, we can express the second integral of Eq. 6.26 as S1 S2

1

2

a

dQ b int T rev

With this, Eq. 6.26 becomes dQ b 1S1 S2 2 s T b 1 Finally, on rearranging the last equation, the closed system entropy balance results

2

a

S2 S 1 entropy change

dQ b

s T b 1 entropy entropy transfer production

2

a

(6.27)

If the end states are fixed, the entropy change on the left side of Eq. 6.27 can be evaluated independently of the details of the process. However, the two terms on the right side depend explicitly on the nature of the process and cannot be determined solely from knowledge of the end states. The first term on the right side of Eq. 6.27 is associated with heat transfer to or from the system during the process. This term can be interpreted as the entropy transfer accompanying heat transfer. The direction of entropy transfer is the same as the direction of the heat transfer, and the same sign convention applies as for heat transfer: A positive value means that entropy is transferred into the system, and a negative value means that entropy is transferred out. When there is no heat transfer, there is no entropy transfer. The entropy change of a system is not accounted for solely by the entropy transfer, but is due in part to the second term on the right side of Eq. 6.27 denoted by . The term is positive when internal irreversibilities are present during the process and vanishes when no internal irreversibilities are present. This can be described by saying that entropy is produced within the system by the action of irreversibilities. The second law of thermodynamics can be interpreted as requiring that entropy is produced by irreversibilities and conserved only in the limit as irreversibilities are reduced to zero. Since measures the effect of irreversibilities present within the system during a process, its value depends on the nature of the process and not solely on the end states. It is not a property. When applying the entropy balance to a closed system, it is essential to remember the requirements imposed by the second law on entropy production: The second law requires that entropy production be positive, or zero, in value s: e

7 0 irreversibilities present within the system (6.28) 0 no irreversibilities present within the system The value of the entropy production cannot be negative. By contrast, the change in entropy of the system may be positive, negative, or zero: 7 0 S2 S1: • 0 (6.29) 6 0 Like other properties, entropy change can be determined without knowledge of the details of the process.

closed system entropy

balance

entropy transfer accompanying heat transfer

entropy production

221

222

Chapter 6 Using Entropy This portion of the boundary is at temperature Tb

Gas or liquid Reservoir at Tb

Q

Q/Tb Figure 6.8 Illustration of the entropy transfer and entropy production concepts.

for example. . . to illustrate the entropy transfer and entropy production concepts, as well as the accounting nature of entropy balance, consider Fig. 6.8. The figure shows a system consisting of a gas or liquid in a rigid container stirred by a paddle wheel while receiving a heat transfer Q from a reservoir. The temperature at the portion of the boundary where heat transfer occurs is the same as the constant temperature of the reservoir, Tb. By definition, the reservoir is free of irreversibilities; however, the system is not without irreversibilities, for fluid friction is evidently present, and there may be other irreversibilities within the system. Let us now apply the entropy balance to the system and to the reservoir. Since Tb is constant, the integral in Eq. 6.27 is readily evaluated, and the entropy balance for the system reduces to

S2 S1

Q

s Tb

(6.30)

where QTb accounts for entropy transfer into the system accompanying heat transfer Q. The entropy balance for the reservoir takes the form ¢S 4 res

0 Qres

sres Tb

where the entropy production term is set equal to zero because the reservoir is without irreversibilities. Since Qres Q, the last equation becomes ¢S 4 res

Q Tb

The minus sign signals that entropy is carried out of the reservoir accompanying heat transfer. Hence, the entropy of the reservoir decreases by an amount equal to the entropy transferred from it to the system. However, as shown by Eq. 6.30, the entropy change of the system exceeds the amount of entropy transferred to it because of entropy production within the system. If the heat transfer were oppositely directed in the above example, passing instead from the system to the reservoir, the magnitude of the entropy transfer would remain the same, but its direction would be reversed. In such a case, the entropy of the system would decrease if the amount of entropy transferred from the system to the reservoir exceeded the amount of entropy produced within the system due to irreversibilities. Finally, observe that there is no entropy transfer associated with work.

6.5 Entropy Balance for Closed Systems

223

6.5.2 Other Forms of the Entropy Balance The entropy balance can be expressed in various forms convenient for particular analyses. For example, if heat transfer takes place at several locations on the boundary of a system where the temperatures do not vary with position or time, the entropy transfer term can be expressed as a sum, so Eq. 6.27 takes the form S2 S 1 a j

Qj Tj

s

(6.31)

where QjTj is the amount of entropy transferred through the portion of the boundary at temperature Tj. On a time rate basis, the closed system entropy rate balance is # Qj dS # a

s dt j Tj

(6.32)

closed system entropy rate balance

# where dSdt is the time rate of change of entropy of the system. The term Qj Tj represents the time rate of entropy transfer through the portion of the boundary whose instantaneous # temperature is Tj. The term s accounts for the time rate of entropy production due to irreversibilities within the system. It is sometimes convenient to use the entropy balance expressed in differential form dS a

dQ b ds T b

(6.33)

Note that the differentials of the nonproperties Q and are shown, respectively, as Q and . When there are no internal irreversibilities, vanishes and Eq. 6.33 reduces to Eq. 6.4b. 6.5.3 Evaluating Entropy Production and Transfer Regardless of the form taken by the entropy balance, the objective in many applications is to evaluate the entropy production term. However, the value of the entropy production for a given process of a system often does not have much significance by itself. The significance is normally determined through comparison. For example, the entropy production within a given component might be compared to the entropy production values of the other components included in an overall system formed by these components. By comparing entropy production values, the components where appreciable irreversibilities occur can be identified and rank ordered. This allows attention to be focused on the components that contribute most to inefficient operation of the overall system. To evaluate the entropy transfer term of the entropy balance requires information regarding both the heat transfer and the temperature on the boundary where the heat transfer occurs. The entropy transfer term is not always subject to direct evaluation, however, because the required information is either unknown or not defined, such as when the system passes through states sufficiently far from equilibrium. In such applications, it may be convenient, therefore, to enlarge the system to include enough of the immediate surroundings that the temperature on the boundary of the enlarged system corresponds to the temperature of the surroundings away from the immediate vicinity of the system, Tf. The entropy transfer term is then simply QTf. However, as the irreversibilities present would not be just for the system of interest but for the enlarged system, the entropy production term would account for the effects of internal irreversibilities within the original system and external irreversibilities present within that portion of the surroundings included within the enlarged system.

Boundary of enlarged system T > Tf

Temperature variation

Tf

Chapter 6 Using Entropy

224

6.5.4 Illustrations The following examples illustrate the use of the energy and entropy balances for the analysis of closed systems. Property relations and property diagrams also contribute significantly in developing solutions. The first example reconsiders the system and end states of Example 6.1 to demonstrate that entropy is produced when internal irreversibilities are present and that the amount of entropy production is not a property.

EXAMPLE

Irreversible Process of Water

6.2

Water initially a saturated liquid at 100C is contained within a piston–cylinder assembly. The water undergoes a process to the corresponding saturated vapor state, during which the piston moves freely in the cylinder. There is no heat transfer with the surroundings. If the change of state is brought about by the action of a paddle wheel, determine the net work per unit mass, in kJ/kg, and the amount of entropy produced per unit mass, in kJ/kg # K. SOLUTION Known: Water contained in a piston–cylinder assembly undergoes an adiabatic process from saturated liquid to saturated vapor at 100C. During the process, the piston moves freely, and the water is rapidly stirred by a paddle wheel. Find: Determine the net work per unit mass and the entropy produced per unit mass. Schematic and Given Data:

p

T g

f

100°C

❶

Area is not work

g

f

System boundary

Water 100°C

v

Area is not heat s

Figure E6.2

Assumptions: 1. The water in the piston–cylinder assembly is a closed system. 2. There is no heat transfer with the surroundings. 3. The system is at an equilibrium state initially and finally. There is no change in kinetic or potential energy between these two states. Analysis: As the volume of the system increases during the process, there is an energy transfer by work from the system during the expansion, as well as an energy transfer by work to the system via the paddle wheel. The net work can be evaluated from an energy balance, which reduces with assumptions 2 and 3 to 0

¢U ¢KE ¢PE Q W On a unit mass basis, the energy balance reduces to W 1ug uf 2 m

6.5 Entropy Balance for Closed Systems

225

With specific internal energy values from Table A-2 at 100C kJ W 2087.56 m kg The minus sign indicates that the work input by stirring is greater in magnitude than the work done by the water as it expands. The amount of entropy produced is evaluated by applying an entropy balance. Since there is no heat transfer, the term accounting for entropy transfer vanishes 0

¢S

2

1

dQ a b s T b

On a unit mass basis, this becomes on rearrangement s sg sf m With specific entropy values from Table A-2 at 100C

❷

kJ s 6.048 # m kg K

❶

Although each end state is an equilibrium state at the same pressure and temperature, the pressure and temperature are not necessarily uniform throughout the system at intervening states, nor are they necessarily constant in value during the process. Accordingly, there is no well-defined “path” for the process. This is emphasized by the use of dashed lines to represent the process on these p–v and T–s diagrams. The dashed lines indicate only that a process has taken place, and no “area” should be associated with them. In particular, note that the process is adiabatic, so the “area” below the dashed line on the T–s diagram can have no significance as heat transfer. Similarly, the work cannot be associated with an area on the p–v diagram.

❷

The change of state is the same in the present example as in Example 6.1. However, in Example 6.1 the change of state is brought about by heat transfer while the system undergoes an internally reversible process. Accordingly, the value of entropy production for the process of Example 6.1 is zero. Here, fluid friction is present during the process and the entropy production is positive in value. Accordingly, different values of entropy production are obtained for two processes between the same end states. This demonstrates that entropy production is not a property.

As an illustration of second law reasoning, the next example uses the fact that the entropy production term of the entropy balance cannot be negative.

EXAMPLE

6.3

Evaluating Minimum Theoretical Compression Work

Refrigerant 134a is compressed adiabatically in a piston–cylinder assembly from saturated vapor at 0C to a final pressure of 0.7 MPa. Determine the minimum theoretical work input required per unit mass of refrigerant, in kJ/kg. SOLUTION Known: Refrigerant 134a is compressed without heat transfer from a specified initial state to a specified final pressure. Find: Determine the minimum theoretical work input required per unit of mass.

Chapter 6 Using Entropy

226

Schematic and Given Data:

T 2s

Accessible states Internal energy 2 decreases Actual compression

Insulation

Internally reversible compression 1

R-134a

s

Figure E6.3

Assumptions: 1. The Refrigerant 134a is a closed system. 2. There is no heat transfer with the surroundings. 3. The initial and final states are equilibrium states. There is no change in kinetic or potential energy between these states. Analysis: An expression for the work can be obtained from an energy balance. By applying assumptions 2 and 3 0

¢U ¢KE ¢PE Q W When written on a unit mass basis, the work input is then W a b u2 u1 m The specific internal energy u1 can be obtained from Table A-10 as u1 227.06 kJ/kg. Since u1 is known, the value for the work input depends on the specific internal energy u2. The minimum work input corresponds to the smallest allowed value for u2, determined using the second law as follows. Applying an entropy balance 0

¢S

2

1

dQ a b s T b

where the entropy transfer term is set equal to zero because the process is adiabatic. Thus, the allowed final states must satisfy s2 s1

s 0 m

The restriction indicated by the foregoing equation can be interpreted using the accompanying T–s diagram. Since cannot be negative, states with s2 s1 are not accessible adiabatically. When irreversibilities are present during the compression, entropy is produced, so s2 s1. The state labeled 2s on the diagram would be attained in the limit as irreversibilities are reduced to zero. This state corresponds to an isentropic compression. By inspection of Table A-12, we see that when pressure is fixed, the specific internal energy decreases as temperature decreases. Thus, the smallest allowed value for u2 corresponds to state 2s. Interpolating in Table A-12 at 0.7 MPa, with s2s s1 0.9190 kJ/kg K, we find that u2s 244.32 kJ/kg. Finally, the minimum work input is W a b u2s u1 244.32 227.06 17.26 kJ/kg m min

❶ ❶

The effect of irreversibilities exacts a penalty on the work input required: A greater work input is needed for the actual adiabatic compression process than for an internally reversible adiabatic process between the same initial state and the same final pressure.

6.5 Entropy Balance for Closed Systems

227

To pinpoint the relative significance of the internal and external irreversibilities, the next example illustrates the application of the entropy rate balance to a system and to an enlarged system consisting of the system and a portion of its immediate surroundings.

EXAMPLE

6.4

Pinpointing Irreversibilities

# Referring to Example 2.4, evaluate the rate of entropy production s, in kW/K, for (a) the gearbox as the system and (b) an enlarged system consisting of the gearbox and enough of its surroundings that heat transfer occurs at the temperature of the surroundings away from the immediate vicinity of the gearbox, Tf 293 K (20C). SOLUTION Known: A gearbox operates at steady state with known values for the power input through the high-speed shaft, power output through the low-speed shaft, and heat transfer rate. The temperature on the outer surface of the gearbox and the temperature of the surroundings away from the gearbox are also known. # Find: Evaluate the entropy production rate s for each of the two specified systems shown in the schematic. Schematic and Given Data:

At this boundary the temperature is Tf = 293 K

System boundary

Temperature variation Tb

Q = –1.2 kW 60 kW

60 kW Tf

58.8 kW 58.8 kW

Tb = 300 K Gearbox (a)

(b)

Figure E6.4

Assumptions: 1. In part (a), the gearbox is taken as a closed system operating at steady state, as shown on the accompanying sketch labeled with data from Example 2.4. 2. In part (b) the gearbox and a portion of its surroundings are taken as a closed system, as shown on the accompanying sketch labeled with data from Example 2.4. 3. The temperature of the outer surface of the gearbox and the temperature of the surroundings are each uniform. Analysis: (a) To obtain an expression for the entropy production rate, begin with the entropy balance for a closed system on a time rate basis: Eq. 6.32. Since heat transfer takes place only at temperature Tb, the entropy rate balance reduces at steady state to # 0 Q dS #

s dt Tb Solving # Q # s Tb

228

Chapter 6 Using Entropy

# Introducing the known values for the heat transfer rate Q and the surface temperature Tb 11.2 kW2 # 4 103 kW/K s 1300 K2

(b) Since heat transfer takes place at temperature Tf for the enlarged system, the entropy rate balance reduces at steady state to # 0 Q dS #

s dt Tf Solving

# Q # s Tf

# Introducing the known values for the heat transfer rate Q and the temperature Tf

11.2 kW2 # s 4.1 103 kW/K 1293 K2

❶ ❶

The value of the entropy production rate calculated in part (a) gauges the significance of irreversibilities associated with friction and heat transfer within the gearbox. In part (b), an additional source of irreversibility is included in the enlarged system, namely the irreversibility associated with the heat transfer from the outer surface of the gearbox at Tb to the surroundings at Tf. In this case, the irreversibilities within the gearbox are dominant, accounting for 97.6% of the total rate of entropy production.

6.5.5 Increase of Entropy Principle Our study of the second law began in Sec. 5.1 with a discussion of the directionality of processes. In the present development, it is shown that the energy and entropy balances can be used together to determine direction. The present discussion centers on an enlarged system comprising a system and that portion of the surroundings affected by the system as it undergoes a process. Since all energy and mass transfers taking place are included within the boundary of the enlarged system, the enlarged system can be regarded as an isolated system. An energy balance for the isolated system reduces to ¢E4 isol 0

(6.34a)

because no energy transfers take place across its boundary. Thus, the energy of the isolated system remains constant. Since energy is an extensive property, its value for the isolated system is the sum of its values for the system and surroundings, respectively, so Eq. 6.34a can be written as ¢E4 system ¢E4 surr 0

(6.34b)

In either of these forms, the conservation of energy principle places a constraint on the processes that can occur. For a process to take place, it is necessary for the energy of the system plus the surroundings to remain constant. However, not all processes for which this constraint is satisfied can actually occur. Processes also must satisfy the second law, as discussed next. An entropy balance for the isolated system reduces to 0

¢S 4 isol or

dQ a b sisol T b 1 2

¢S 4 isol sisol

(6.35a)

6.5 Entropy Balance for Closed Systems

where isol is the total amount of entropy produced within the system and its surroundings. Since entropy is produced in all actual processes, the only processes that can occur are those for which the entropy of the isolated system increases. This is known as the increase of entropy principle. The increase of entropy principle is sometimes considered an alternative statement of the second law. Since entropy is an extensive property, its value for the isolated system is the sum of its values for the system and surroundings, respectively, so Eq. 6.35a can be written as ¢S 4 system ¢S 4 surr sisol

229

increase of entropy principle

(6.35b)

Notice that this equation does not require the entropy change to be positive for both the system and surroundings but only that the sum of the changes is positive. In either of these forms, the increase of entropy principle dictates the direction in which any process can proceed: the direction that causes the total entropy of the system plus surroundings to increase. We noted previously the tendency of systems left to themselves to undergo processes until a condition of equilibrium is attained (Sec. 5.1). The increase of entropy principle suggests that the entropy of an isolated system increases as the state of equilibrium is approached, with the equilibrium state being attained when the entropy reaches a maximum. This interpretation is considered again in Sec. 14.1, which deals with equilibrium criteria. The example to follow illustrates the increase of entropy principle.

EXAMPLE

6.5

Quenching a Hot Metal Bar

A 0.3 kg metal bar initially at 1200K is removed from an oven and quenched by immersing it in a closed tank containing 9 kg of water initially at 300K. Each substance can be modeled as incompressible. An appropriate constant specific heat value for the water is cw 4.2 kJ/kg K, and an appropriate value for the metal is cm 0.42 kJ/kg K. Heat transfer from the tank contents can be neglected. Determine (a) the final equilibrium temperature of the metal bar and the water, in K, and (b) the amount of entropy produced, in kJ/. SOLUTION Known: A hot metal bar is quenched by immersing it in a tank containing water. Find: Determine the final equilibrium temperature of the metal bar and the water, and the amount of entropy produced. Schematic and Given Data: System boundary

Assumptions: 1. The metal bar and the water within the tank form a closed system, as shown on the accompanying sketch. 2. There is no energy transfer by heat or work: The system is isolated. 3. There is no change in kinetic or potential energy. Metal bar: Tmi = 1200°Κ cm = 0.42 kJ/kg · Κ mm = 0.3 kg

Water: Twi = 300°K cw = 4.2 kJ/kg · Κ mw = 9 kg

4. The water and metal bar are each modeled as incompressible with known specific heats.

Figure E6.5

230

Chapter 6 Using Entropy

Analysis: (a) The final equilibrium temperature can be evaluated from an energy balance 0

¢KE ¢PE ¢U Q W

where the indicated terms vanish by assumptions 2 and 3. Since internal energy is an extensive property, its value for the overall system is the sum of the values for the water and metal, respectively. Thus, the energy balance becomes ¢Uwater ¢Umetal 0 Using Eq. 3.20a to evaluate the internal energy changes of the water and metal in terms of the constant specific heats mwcw 1Tf Twi 2 mmcm 1Tf Tmi 2 0 where Tf is the final equilibrium temperature, and Twi and Tmi are the initial temperatures of the water and metal, respectively. Solving for Tf and inserting values Tf

mw 1cwcm 2Twi mmTmi mw 1cw cm 2 mm

19 kg2 1102 1300°K2 10.3 kg211200°K2

303°K

19 kg2 1102 10.3 kg2

(b) The amount of entropy production can be evaluated from an entropy balance. Since no heat transfer occurs between the system and its surroundings, there is no accompanying entropy transfer, and an entropy balance for the system reduces to 0

¢S

2

1

dQ a b s T b

Entropy is an extensive property, so its value for the system is the sum of its values for the water and the metal, respectively, and the entropy balance becomes ¢Swater ¢Smetal s Evaluating the entropy changes using Eq. 6.24 for incompressible substances, the foregoing equation can be written as s mwcw ln

Tf Tf

mmcm ln Twi Tmi

Inserting values

❶

s 19 kg2 a4.2

❷

a0.3761

kJ 303 kJ 303 b ln

10.3 kg2 a0.42 # b ln kg # °K 300 kg °K 1200

kJ kJ b a0.1734 b 0.2027 kJ/°K °K K

❶

The metal bar experiences a decrease in entropy. The entropy of the water increases. In accord with the increase of entropy principle, the entropy of the isolated system increases.

❷

The value of is sensitive to roundoff in the value of Tf.

STATISTICAL INTERPRETATION OF ENTROPY

In statistical thermodynamics, entropy is associated with the notion of disorder and the second law statement that the entropy of an isolated system undergoing a spontaneous process tends to increase is equivalent to saying that the disorder of the isolated system tends to increase. Let us conclude the present discussion with a brief summary of concepts from the microscopic viewpoint related to these ideas.

6.6 Entropy Rate Balance for Control Volumes

231

Thermodynamics in the News… Roll Over Boltzmann Physicist Constantino Tsallis has proposed a new statistical definition of entropy some say will shake the foundations of science as we know it. The Boltzmann relation, providing the link between the microscopic interpretation of entropy and observations of macroscopic system behavior for over a century, is being called into question. Scientists are weighing in on whether the theory provides new fundamental understanding or simply amplifies our thinking. Supporters of the new definition of entropy say it extends the scope of statistical thermodynamics to new and exciting classes of problems. They claim it fills a gap in theory first noted by Einstein and allows researchers to explain phenomena ranging from the movement of microorganisms to the motions of stars, and even swings in the stock market. Skeptics believe that it only provides an empirical “fudge” factor with

RIP no new fundamental S = k ln w understanding. The new statistical definition reduces to the Boltzmann relation for systems at equilibrium. But for systems at states far from equilibrium the new definition shines, say its supporters. The theory uses the mathematical notion of fractals to account for nonequilibrium behavior and describes systems on the verge of chaos by new statistical descriptions. Time will tell whether the new definition is accepted as the basis of statistical mechanics, or takes its place on a long list of bright ideas that fell by the wayside of science.

Viewed macroscopically, an equilibrium state of a system appears to be unchanging, but on the microscopic level the particles making up the matter are continually in motion. Accordingly, a vast number of possible microscopic states correspond to any given macroscopic equilibrium state. The total number of possible microscopic states available to a system is called the thermodynamic probability, w. Entropy is related to w by the Boltzmann relation S k ln w

(6.36)

where k is Boltzmann’s constant. From this equation we see that any process that increases the number of possible microscopic states of a system increases its entropy, and conversely. Hence, for an isolated system, processes only occur in such a way that the number of microscopic states available to the system increases. The number w is referred to as the disorder of the system. We can say, then, that the only processes an isolated system can undergo are those that increase the disorder of the system.

6.6 Entropy Rate Balance for Control Volumes

Thus far the discussion of the entropy balance concept has been restricted to the case of closed systems. In the present section the entropy balance is extended to control volumes. Like mass and energy, entropy is an extensive property, so it too can be transferred into or out of a control volume by streams of matter. Since this is the principal difference between the closed system and control volume forms, the control volume entropy rate balance can be obtained by modifying Eq. 6.32 to account for these entropy transfers. The result is # Qj dScv # # # a

a misi a mese scv dt j Tj j e rate of entropy change

rates of entropy transfer

rate of entropy production

(6.37)

control volume entropy rate balance

232

Chapter 6 Using Entropy

entropy transfer accompanying mass flow

where dScvdt represents the time rate of change of entropy within the control volume. # # The terms misi and me se account, respectively, for rates of entropy transfer accompany# ing mass flow into and out of the control volume. The term Qj represents the time rate of heat transfer # at the location on the boundary where the instantaneous temperature # is Tj. The ratio Qj Tj accounts for the accompanying rate of entropy transfer. The term scv denotes the time rate of entropy production due to irreversibilities within the control volume. INTEGRAL FORM

As for the cases of the control volume mass and energy rate balances, the entropy rate balance can be expressed in terms of local properties to obtain forms that are more generally applicable. Thus, the term Scv(t), representing the total entropy associated with the control volume at time t, can be written as a volume integral Scv 1t2

rs dV

V

where and s denote, respectively, the local density and specific entropy. The rate of entropy transfer accompanying heat transfer can be expressed more generally as an integral over the surface of the control volume time rate of entropy £ transfer accompanying § heat transfer

# q a b dA A T b

# where q is the heat flux, the time rate of heat transfer per unit of surface area, through the location on the boundary where the instantaneous temperature is T. The subscript “b” is added as a reminder that the integrand is evaluated on the boundary of the control volume. In addition, the terms accounting for entropy transfer accompanying mass flow can be expressed as integrals over the inlet and exit flow areas, resulting in the following form of the entropy rate balance d dt

V

rs dV

# q a b dA a a A T b i

srV dAb a a srV dAb s# n

n

A

i

e

A

cv

(6.38)

e

where Vn denotes the normal component in the direction of flow of the velocity relative to the flow area. In some cases, it is also convenient to express the entropy production rate as a volume integral of the local volumetric rate of entropy production within the control volume. The study of Eq. 6.38 brings out the assumptions underlying Eq. 6.37. Finally, note that for a closed system the sums accounting for entropy transfer at inlets and exits drop out, and Eq. 6.38 reduces to give a more general form of Eq. 6.32.

6.6.1 Analyzing Control Volumes at Steady State Since a great many engineering analyses involve control volumes at steady state, it is instructive to list steady-state forms of the balances developed for mass, energy, and entropy. At steady state, the conservation of mass principle takes the form # # a mi a me i

e

(4.27)

6.6 Entropy Rate Balance for Control Volumes

The energy rate balance at steady state is # # V2e V2i # # 0 Qcv Wcv a mi ahi

gzi b a me ahe

gze b 2 2 i e

(4.28a)

Finally, the steady-state form of the entropy rate balance is obtained by reducing Eq. 6.37 to give # Qj

steady-state entropy

# # # 0a

a misi a mese scv T j j i e

(6.39) rate balance

These equations often must be solved simultaneously, together with appropriate property relations. Mass and energy are conserved quantities, but entropy is not conserved. Equation 4.27 indicates that at steady state the total rate of mass flow into the control volume equals the total rate of mass flow out of the control volume. Similarly, Eq. 4.28a indicates that the total rate of energy transfer into the control volume equals the total rate of energy transfer out of the control volume. However, Eq. 6.39 requires that the rate at which entropy is transferred out must exceed the rate at which entropy enters, the difference being the rate of entropy production within the control volume owing to irreversibilities.

ONE-INLET, ONE-EXIT CONTROL VOLUMES

Since many applications involve one-inlet, one-exit control volumes at steady state, let us also list the form of the entropy rate balance for this important case: # Qj # # 0a

m 1s1 s2 2 scv T j j # Or, on dividing by the mass flow rate m and rearranging # Qj 1 s2 s1 # a a b

m j Tj

# scv # m

(6.40)

The two terms on the right side of Eq. 6.40 denote, respectively, the rate of entropy transfer accompanying heat transfer and the rate of entropy production within the control volume, each per unit of mass flowing through the control volume. From Eq. 6.40 it can be concluded that the entropy of a unit of mass passing from inlet to exit can increase, decrease, or remain the same. Furthermore, because the value of the second term on the right can never be negative, a decrease in the specific entropy from inlet to exit can be realized only when more entropy is transferred out of the control volume accompanying heat transfer than is produced by irreversibilities within the control volume. When the value of this entropy transfer term is positive, the specific entropy at the exit is greater than the specific entropy at the inlet whether internal irreversibilities are present or not. In the special case where there is no entropy transfer accompanying heat transfer, Eq. 6.40 reduces to # scv s2 s1 # (6.41) m

233

234

Chapter 6 Using Entropy

Accordingly, when irreversibilities are present within the control volume, the entropy of a unit of mass increases as it passes from inlet to exit. In the limiting case in which no irreversibilities are present, the unit mass passes through the control volume with no change in its entropy—that is, isentropically. 6.6.2 Illustrations The following examples illustrate the use of the mass, energy, and entropy balances for the analysis of control volumes at steady state. Carefully note that property relations and property diagrams also play important roles in arriving at solutions. In the first example, we evaluate the rate of entropy production within a turbine operating at steady state when there is heat transfer from the turbine.

EXAMPLE

6.6

Entropy Production in a Steam Turbine

Steam enters a turbine with a pressure of 30 bar, a temperature of 400C, and a velocity of 160 m/s. Saturated vapor at 100C exits with a velocity of 100 m /s. At steady state, the turbine develops work equal to 540 kJ per kg of steam flowing through the turbine. Heat transfer between the turbine and its surroundings occurs at an average outer surface temperature of 350 K. Determine the rate at which entropy is produced within the turbine per kg of steam flowing, in kJ/kg # K. Neglect the change in potential energy between inlet and exit. SOLUTION Known: Steam expands through a turbine at steady state for which data are provided. Find: Determine the rate of entropy production per kg of steam flowing. Schematic and Given Data: 30 bar T 1

p1 = 30 bar T1 = 400°C V1 = 160 m/s 1

Wcv ––– = 540 kJ/kg m 100°C

Tb = 350 K

400°C

2

T2 = 100°C Saturated vapor V2 = 100 m/s

2

s Figure E6.6

Assumptions: 1. The control volume shown on the accompanying sketch is at steady state. 2. Heat transfer from the turbine to the surroundings occurs at a specified average outer surface temperature. 3. The change in potential energy between inlet and exit can be neglected. Analysis: To determine the entropy production per unit mass flowing through the turbine, begin with mass and entropy rate balances for the one-inlet, one-exit control volume at steady state: # # 0 m1 m2 # Qj # # # 0 a

m1s1 m2s2 scv j Tj

6.6 Entropy Rate Balance for Control Volumes

235

# Since heat transfer occurs only at Tb 350 K, the first term on the right side of the entropy rate balance reduces to QcvTb. Combining the mass and entropy rate balances # Qcv # # 0

m1s1 s2 2 scv Tb # # # where m is the mass flow rate. Solving for scvm # # # scv Qcvm

1s2 s1 2 # m Tb # # The heat transfer rate, Qcvm, required by this expression is evaluated next. Reduction of the mass and energy rate balances results in # # Qcv Wcv V22 V21 b # # 1h2 h1 2 a m m 2 where the potential energy change from inlet to exit is dropped by assumption 3. From Table A-4 at 30 bar, 400C, h1 3230.9 kJ/kg, and from Table A-2, h2 hg(100C) 2676.1 kJ/kg. Thus # 11002 2 11602 2 m2 Qcv kJ kJ 1N 1 kJ `

12676.1 3230.92 a b c da 2b ` ` ` # 540 m kg kg 2 s 1 kg # m /s2 103 N # m 540 554.8 7.8 22.6 kJ/kg From Table A-2, s2 7.3549 kJ/kg # K, and from Table A-4, s1 6.9212 kJ/kg # K. Inserting values into the expression for entropy production # 122.6 kJ/ kg2 scv kJ

17.3549 6.92122 a # b # m 350 K kg K 0.0646 0.4337 0.4983 kJ/kg # K

❶ ❶

If the boundary were located to include a portion of the immediate surroundings so heat transfer would take place at the temperature of the surroundings, say Tf 293 K, the entropy production for the enlarged control volume would be 0.511 kJ/kg # K. It is left as an exercise to verify this value and to explain why the entropy production for the enlarged control volume would be greater than for a control volume consisting of the turbine only.

In Example 6.7, the mass, energy, and entropy rate balances are used to evaluate a performance claim for a device to produce hot and cold streams of air from a single stream of air at an intermediate temperature.

EXAMPLE

6.7

Evaluating a Performance Claim

An inventor claims to have developed a device requiring no energy transfer by work or heat transfer, yet able to produce hot and cold streams of air from a single stream of air at an intermediate temperature. The inventor provides steady-state test data indicating that when air enters at a temperature of 39C and a pressure of 5.0 bars, separate streams of air exit at temperatures of 18C and 79C, respectively, and each at a pressure of 1 bar. Sixty percent of the mass entering the device exits at the lower temperature. Evaluate the inventor’s claim, employing the ideal gas model for air and ignoring changes in the kinetic and potential energies of the streams from inlet to exit.

236

Chapter 6 Using Entropy

SOLUTION Known: Data are provided for a device that at steady state produces hot and cold streams of air from a single stream of air at an intermediate temperature without energy transfers by work or heat. Find: Evaluate whether the device can operate as claimed. Schematic and Given Data:

1 2

T1 = 21°C p1 = 5.1 bars

Inlet

T2 = 79°C p2 = 1 bar Hot outlet

3 Cold outlet

T3 = 18°C p3 = 1 bar

Figure E6.7

Assumptions: 1. The control volume shown on the accompanying sketch is at steady state. # # 2. For the control volume, Wcv 0 and Qcv 0.

❶

3. Changes in the kinetic and potential energies from inlet to exit can be ignored. 4. The air is modeled as an ideal gas with constant cp 1.0 kJ/kg # K. Analysis: For the device to operate as claimed, the conservation of mass and energy principles must be satisfied. The second law of thermodynamics also must be satisfied; and in particular the rate of entropy production cannot be negative. Accordingly, the mass, energy and entropy rate balances are considered in turn. With assumptions 1–3, the mass and energy rate balances reduce, respectively, to # # # m1 m2 m3 # # # 0 m1h1 m2h2 m3h3 # # # # Since m3 0.6m1, it follows from the mass rate balance that m2 0.4m1. By combining the mass and energy rate balances and evaluating changes in specific enthalpy using constant cp, the energy rate balance is also satisfied. That is

❷

# # # # 0 1m2 m3 2 h1 m2h2 m3h3 # # m2 1h1 h2 2 m3 1h1 h3 2 # # 0.4m1cp 1T1 T2 2 0.6m1cp 1T1 T3 2 0.411052 0.61702 0 Accordingly, with the given data the conservation of mass and energy principles are satisfied. Since no significant heat transfer occurs, the entropy rate balance at steady state reads # 0 Qj

# # # #

m1s1 m2s2 m3s3 scv 0a Tj j

6.6 Entropy Rate Balance for Control Volumes

237

Combining the mass and entropy rate balances # # # # # 0 1m2 m3 2s1 m2s2 m3s3 scv # # # m2 1s1 s2 2 m3 1s1 s3 2 scv # # # 0.4m1 1s1 s2 2 0.6m1 1s1 s3 2 scv

# # Solving for scvm1 and using Eq. 6.23 to evaluate changes in specific entropy # scv T3 p3 p2 T2 # 0.4 c cp ln R ln d 0.6 c cp ln R ln d p1 p1 m1 T1 T1 0.4 c a1.0

❸

kJ 352 8.314 kJ 1 b ln a b ln d # # kg K 294 28.97 kg °K 5.0

0.6 c a1.0

kJ 255 8.314 kJ 1 b ln a b ln d kg # °K 294 28.97 kg # °K 5.0 0.454 kJ/kg # °K

❹ ❺

Thus, the second law of thermodynamics is also satisfied. On the basis of this evaluation, the inventor’s claim does not violate principles of thermodynamics.

❶

Since the specific heat cp of air varies little over the temperature interval from 0 to 79C, cp can be taken as constant. From Table A-20, cp 1.0 kJ /kg # K.

❷ ❸ ❹

Since temperature differences are involved in this calculation, the temperatures can be either in C or K.

❺

Such devices do exist. They are known as vortex tubes and are used in industry for spot cooling.

In this calculation involving temperature ratios, the temperatures must be in K. If the value of the rate of entropy production had been negative or zero, the claim would be rejected. A negative value is impossible by the second law and a zero value would indicate operation without irreversibilities.

In Example 6.8, we evaluate and compare the rates of entropy production for three components of a heat pump system. Heat pumps are considered in detail in Chap. 10.

EXAMPLE

6.8

Entropy Production in Heat Pump Components

Components of a heat pump for supplying heated air to a dwelling are shown in the schematic below. At steady state, Refrigerant 22 enters the compressor at 5C, 3.5 bar and is compressed adiabatically to 75C, 14 bar. From the compressor, the refrigerant passes through the condenser, where it condenses to liquid at 28C, 14 bar. The refrigerant then expands through a throttling valve to 3.5 bar. The states of the refrigerant are shown on the accompanying T–s diagram. Return air from the dwelling enters the condenser at 20C, 1 bar with a volumetric flow rate of 0.42 m3/s and exits at 50C with a negligible change in pressure. Using the ideal gas model for the air and neglecting kinetic and potential energy effects, (a) determine the rates of entropy production, in kW/K, for control volumes enclosing the condenser, compressor, and expansion valve, respectively. (b) Discuss the sources of irreversibility in the components considered in part (a). SOLUTION Known: Refrigerant 22 is compressed adiabatically, condensed by heat transfer to air passing through a heat exchanger, and then expanded through a throttling valve. Steady-state operating data are known. Find: Determine the entropy production rates for control volumes enclosing the condenser, compressor, and expansion valve, respectively, and discuss the sources of irreversibility in these components.

238

Chapter 6 Using Entropy

Schematic and Given Data: Indoor return air T5 = 20°C 5 p5 = 1 bar (AV)5 = 0.42 m3/s 3 p3 = 14 bar T3 = 28°C Expansion valve

Supply air T = 50°C 6 p 6 = 1 bar 6 T Condenser

2

2

p2 = 14 bar T2 = 75°C

75°C

14 bar 3

Compressor

p4 = 3.5 bar 4

1

28°C

T1 = –5°C p1 = 3.5 bar

3.5 bar 4

1

–5°C

s

Outdoor air Evaporator

Figure E6.8

Assumptions: 1. Each component is analyzed as a control volume at steady state. 2. The compressor operates adiabatically, and the expansion across the valve is a throttling process. # # 3. For the control volume enclosing the condenser, Wcv 0 and Qcv 0. 4. Kinetic and potential energy effects can be neglected.

❶

5. The air is modeled as an ideal gas with constant cp 1.005 kJ/kg # K. Analysis: (a) Let us begin by obtaining property data at each of the principal refrigerant states located on the accompanying schematic and T–s diagram. At the inlet to the compressor, the refrigerant is a superheated vapor at 5C, 3.5 bar, so from Table A-9, s1 0.9572 kJ/kg # K. Similarly, at state 2, the refrigerant is a superheated vapor at 75C, 14 bar, so interpolating in Table A-9 gives s2 0.98225 kJ/kg # K and h2 294.17 kJ/kg. State 3 is compressed liquid at 28C, 14 bar. From Table A-7, s3 sf (28C) 0.2936 kJ/kg # K and h3 hf (28C) 79.05 kJ/kg. The expansion through the valve is a throttling process, so h3 h4. Using data from Table A-8, the quality at state 4 is x4

1h4 hf4 2 1hfg 2 4

179.05 33.092 1212.912

0.216

and the specific entropy is

s4 sf4 x4 1sg4 sf4 2 0.1328 0.21610.9431 0.13282 0.3078 kJ/kg # K

Condenser. Consider the control volume enclosing the condenser. With assumptions 1 and 3, the entropy rate balance reduces to # # # 0 mref 1s2 s3 2 mair 1s5 s6 2 scond # # # To evaluate scond requires the two mass flow rates, mair and mref , and the change in specific entropy for the air. These are obtained next. Evaluating the mass flow rate of air using the ideal gas model (assumption 5) 1AV2 5 p5 # 1AV2 5 mair v5 RT5 11 bar2 m3 105 N/m2 1 kJ ` ` ` 3 # ` 0.5 kg/s a0.42 b s 1 bar 10 N m 8.314 kJ a b 1293 K2 28.97 kg # K

6.6 Entropy Rate Balance for Control Volumes

239

The refrigerant mass flow rate is determined using an energy balance for the control volume enclosing the condenser together with assumptions 1, 3, and 4 to obtain # mair 1h6 h5 2 # mref 1h2 h3 2 With assumption 5, h6 h5 cp(T6 T5). Inserting values

❷

# mref

a0.5

kg kJ b a1.005 # b 1323 2932K s kg K 0.07 kg/s 1294 .17 79 .052 kJ/kg

Using Eq. 6.23, the change in specific entropy of the air is s6 s5 cp ln

T6 p6 R ln p5 T5 0

kJ 323 1.0 a1.005 # b ln a b R ln a b 0.098 kJ/kg # K kg K 293 1.0 # Finally, solving the entropy balance for scond and inserting values # # # scond mref 1s3 s2 2 mair 1s6 s5 2 c a0.07

kg kJ 1 kW b 10.2936 0.982252 # 10.52 10.0982 d ` ` s kg K 1 kJ/s

7.95 104

kW K

Compressor. For the control volume enclosing the compressor, the entropy rate balance reduces with assumptions 1 and 3 to # # 0 mref 1s1 s2 2 scomp or # # scomp mref 1s2 s1 2 kg kJ 1 kW a0.07 b 10.98225 0.95722 a # b ` ` s kg K 1 kJ/s 17.5 104 kW/K Valve. Finally, for the control volume enclosing the throttling valve, the entropy rate balance reduces to # # 0 mref 1s3 s4 2 svalve # Solving for svalve and inserting values kg kJ 1 kW # # svalve mref 1s4 s3 2 a0.07 b 10.3078 0.29362 a # b ` ` s kg K 1 kJ/s 9.94 104 kW/K (b) The following table summarizes, in rank order, the calculated entropy production rates:

❸

Component

. cv (kW/K)

compressor valve condenser

17.5 104 9.94 104 7.95 104

240

Chapter 6 Using Entropy

Entropy production in the compressor is due to fluid friction, mechanical friction of the moving parts, and internal heat transfer. For the valve, the irreversibility is primarily due to fluid friction accompanying the expansion across the valve. The principal source of irreversibility in the condenser is the temperature difference between the air and refrigerant streams. In this example, there are no pressure drops for either stream passing through the condenser, but slight pressure drops due to fluid friction would normally contribute to the irreversibility of condensers. The evaporator lightly shown in Fig. E6.8 has not been analyzed.

❶ ❷ ❸

Due to the relatively small temperature change of the air, the specific heat cp can be taken as constant at the average of the inlet and exit air temperatures. # Temperatures in K are used to evaluate mref, but since a temperature difference is involved the same result would be obtained if temperatures in C were used. Temperatures in K must be used when a temperature ratio is involved, as in Eq. 6.23 used to evaluate s6 s5. By focusing attention on reducing irreversibilities at the sites with the highest entropy production rates, thermodynamic improvements may be possible. However, costs and other constraints must be considered, and can be overriding.

6.7 Isentropic Processes

The term isentropic means constant entropy. Isentropic processes are encountered in many subsequent discussions. The object of the present section is to explain how properties are related at any two states of a process in which there is no change in specific entropy. 6.7.1 General Considerations

The properties at states having the same specific entropy can be related using the graphical and tabular property data discussed in Sec. 6.3.1. For example, as illustrated by Fig. 6.9, temperature–entropy and enthalpy–entropy diagrams are particularly convenient for determining properties at states having the same value of specific entropy. All states on a vertical line passing through a given state have the same entropy. If state 1 on Fig. 6.9 is fixed by pressure p1 and temperature T1, states 2 and 3 are readily located once one additional property, such as pressure or temperature, is specified. The values of several other properties at states 2 and 3 can then be read directly from the figures. Tabular data also can be used to relate two states having the same specific entropy. For the case shown in Fig. 6.9, the specific entropy at state 1 could be determined from the superheated vapor table. Then, with s2 s1 and one other property value, such as p2 or T2, T

h

p1

1

1

T1

p1 T1 p2

2 2

p2 T2

T3 p3

3

p3

T2

3

T3 s

Figure 6.9

specific entropy.

T–s and h–s diagrams showing states having the same value of

s

6.7 Isentropic Processes

241

state 2 could be located in the superheated vapor table. The values of the properties v, u, and h at state 2 can then be read from the table. An illustration of this procedure is given in Sec. 6.3.1. Note that state 3 falls in the two-phase liquid–vapor regions of Fig. 6.9. Since s3 s1, the quality at state 3 could be determined using Eq. 6.6. With the quality known, other properties such as v, u, and h could then be evaluated. Computer retrieval of entropy data provides an alternative to tabular data. 6.7.2 Using the Ideal Gas Model Figure 6.10 shows two states of an ideal gas having the same value of specific entropy. Let us consider relations among pressure, specific volume, and temperature at these states, first using the ideal gas tables and then assuming specific heats are constant.

T 2

v2 p2 T2

IDEAL GAS TABLES

For two states having the same specific entropy, Eq. 6.21a reduces to p2 0 s°1T2 2 s°1T1 2 R ln p1

1

(6.42a)

Equation 6.42a involves four property values: p1, T1, p2, and T2. If any three are known, the fourth can be determined. If, for example, the temperature at state 1 and the pressure ratio p2p1 are known, the temperature at state 2 can be determined from s°1T2 2 s°1T1 2 R ln

p2 p1

s°1T2 2 s°1T1 2 d R

Two states of an ideal gas where s2 s1.

(6.42b)

(6.42c)

Equations 6.42 can be used when s (or s°) data are known, as for the gases of Tables A-22 and A-23. AIR. For the special case of air modeled as an ideal gas, Eq. 6.42c provides the basis for an alternative tabular approach for relating the temperatures and pressures at two states having the same specific entropy. To introduce this, rewrite the equation as

exp3s°1T2 2 R4 p2 p1 exp3s°1T1 2 R4 The quantity exp[s(T )R] appearing in this expression is solely a function of temperature, and is given the symbol pr(T ). A tabulation of pr versus temperature for air is provided in Tables A-22.1 In terms of the function pr, the last equation becomes p2 pr2 p1 pr1

1

1s1 s2, air only2

(6.43)

The values of pr determined with this definition are inconveniently large, so they are divided by a scale factor before tabulating to give a convenient range of numbers.

T1

s Figure 6.10

Since T1 is known, s(T1) would be obtained from the appropriate table, the value of s(T2 ) would be calculated, and temperature T2 would then be determined by interpolation. If p1, T1, and T2 are specified and the pressure at state 2 is the unknown, Eq. 6.42a would be solved to obtain p2 p1 exp c

v1 p1

242

Chapter 6 Using Entropy

where pr1 pr(T1) and pr2 pr(T2). The function pr is sometimes called the relative pressure. Observe that pr is not truly a pressure, so the name relative pressure is misleading. Also, be careful not to confuse pr with the reduced pressure of the compressibility diagram. A relation between specific volumes and temperatures for two states of air having the same specific entropy can also be developed. With the ideal gas equation of state, v RTp, the ratio of the specific volumes is p1 v2 RT2 a ba b p2 v1 RT1 Then, since the two states have the same specific entropy, Eq. 6.43 can be introduced to give pr 1T1 2 v2 RT2 c d c d v1 pr 1T2 2 RT1

The ratio RTpr(T ) appearing on the right side of the last equation is solely a function of temperature, and is given the symbol vr(T ). Values of vr for air are tabulated versus temperature in Tables A-22. In terms of the function vr, the last equation becomes

v2 vr2 v1 vr1

1s1 s2, air only2

(6.44)

where vr1 vr(T1) and vr2 vr(T2). The function vr is sometimes called the relative volume. Despite the name given to it, vr(T ) is not truly a volume. Also, be careful not to confuse it with the pseudoreduced specific volume of the compressibility diagram.

METHODOLOGY UPDATE

When applying the software IT to relate two states of an ideal gas having the same value of specific entropy, note that IT returns specific entropy directly and does not employ the special functions s, pr and vr.

ASSUMING CONSTANT SPECIFIC HEATS

Let us consider next how properties are related for isentropic processes of an ideal gas when the specific heats are constants. For any such case, Eqs. 6.22 and 6.23 reduce to the equations p2 T2 R ln p1 T1 T2 v2 0 cv ln R ln v1 T1

0 cp ln

Introducing the ideal gas relations R k1

(3.47)

p2 1k12k T2 a b p1 T1

1s1 s2, constant k2

(6.45)

v1 k1 T2 a b v2 T1

1s1 s2, constant k2

(6.46)

cp

kR , k1

cv

these equations can be solved, respectively, to give

The following relation can be obtained by eliminating the temperature ratio from Eqs. 6.45 and 6.46:

p2 v1 k a b p1 v2

1s1 s2, constant k2

(6.47)

6.7 Isentropic Processes p

T

n = –1

=

co ns tan

t

n=k

T

p =

s=

co

ns

ta ns co

ta n

o =c

n st

an

t

n=1

t n=1

n=k v

Figure 6.11

n=±∞ n = –1 n=0

nt

n=±∞

v

n=0

243

s

Polytropic processes on p–v and T–s diagrams.

From the form of Eq. 6.47, it can be concluded that a polytropic process pvk constant of an ideal gas with constant k is an isentropic process. We noted in Sec. 3.8 that a polytropic process of an ideal gas for which n 1 is an isothermal (constant-temperature) process. For any fluid, n 0 corresponds to an isobaric (constant-pressure) process and n corresponds to an isometric (constant-volume) process. Polytropic processes corresponding to these values of n are shown in Fig. 6.11 on p–v and T–s diagrams. The foregoing means for evaluating data for an isentropic process of air modeled as an ideal gas are considered in the next example.

EXAMPLE

Isentropic Process of Air

6.9

Air undergoes an isentropic process from p1 1 bar, T1 300K to a final state where the temperature is T2 650K. Employing the ideal gas model, determine the final pressure p2, in atm. Solve using (a) pr data from Table A-22 (b) Interactive Thermodynamics: IT, and (c) a constant specific heat ratio k evaluated at the mean temperature, 475K, from Table A-20. SOLUTION Known: Air undergoes an isentropic process from a state where pressure and temperature are known to a state where the temperature is specified. Find: Determine the final pressure using (a) pr data, (b) IT, and (c) a constant value for the specific heat ratio k. Schematic and Given Data:

T p2 = ? 2

T2 = 650 K

Assumptions: 1. A quantity of air as the system undergoes an isentropic process.

1

p1 = 1 bar T1 = 300 Κ

2. The air can be modeled as an ideal gas. 3. In part (c) the specific heat ratio is constant.

s

Figure E6.9

244

Chapter 6 Using Entropy

Analysis: (a) The pressures and temperatures at two states of an ideal gas having the same specific entropy are related by Eq. 6.43 p2 pr2 p1 pr1 Solving p2 p1

pr2 pr1

With pr values from Table A-22 p 11 bar2

21.86 15.77 bars 1.3860

(b) The IT solution follows: T1 = 300 // K p1 = 1 // atm T2 = 650 // K s_TP(“Air“, T1,p1) = s_TP(“Air”,T2,p2)

❶

// Result: p2 = 15.77 bar (c) When the specific heat ratio k is assumed constant, the temperatures and pressures at two states of an ideal gas having the same specific entropy are related by Eq. 6.45. Thus p2 p1 a

T2 k1k12 b T1

From Table A-20 at 202C (475K), k 1.39. Inserting values into the above expression

❷

p2 11 bar2 a

650 1.390.39 b 15.81 bar 300

❶

IT returns a value for p2 even though it is an implicit variable in the argument of the specific entropy function. Also note that IT returns values for specific entropy directly and does not employ special functions such as s, pr and vr.

❷

The close agreement between the answer obtained in part (c) and that of parts (a), (b) can be attributed to the use of an appropriate value for the specific heat ratio k.

Another illustration of an isentropic process of an ideal gas is provided in the next example dealing with air leaking from a tank.

EXAMPLE

6.10

Air Leaking from a Tank

A rigid, well-insulated tank is filled initially with 5 kg of air at a pressure of 5 bar and a temperature of 500 K. A leak develops, and air slowly escapes until the pressure of the air remaining in the tank is 1 bar. Employing the ideal gas model, determine the amount of mass remaining in the tank and its temperature. SOLUTION Known: A leak develops in a rigid, insulated tank initially containing air at a known state. Air slowly escapes until the pressure in the tank is reduced to a specified value. Find: Determine the amount of mass remaining in the tank and its temperature.

6.7 Isentropic Processes

245

Schematic and Given Data:

Assumptions: 1. As shown on the accompanying sketch, the closed system is the mass initially in the tank that remains in the tank.

System boundary Slow leak Mass initially in the tank that remains in the tank

2. There is no significant heat transfer between the system and its surroundings.

Mass initially in the tank that escapes

3. Irreversibilities within the tank can be ignored as the air slowly escapes. 4. The air is modeled as an ideal gas.

Initial condition of tank

Figure E6.10

Analysis: With the ideal gas equation of state, the mass initially in the tank that remains in the tank at the end of the process is m2

p2V

1RM2 T2

where p2 and T2 are the final pressure and temperature, respectively. Similarly, the initial amount of mass within the tank, m1, is m1

p1V

1RM2 T1

where p1 and T1 are the initial pressure and temperature, respectively. Eliminating volume between these two expressions, the mass of the system is m2 a

❶

p2 T1 b a b m1 p1 T2

Except for the final temperature of the air remaining in the tank, T2, all required values are known. The remainder of the solution mainly concerns the evaluation of T2. For the closed system under consideration, there are no significant irreversibilities (assumption 3), and no heat transfer occurs (assumption 2). Accordingly, the entropy balance reduces to 0

¢S

2

1

dQ 0 a b s 0 T b

Since the system mass remains constant, S m2 s, so ¢s 0 That is, the initial and final states of the system have the same value of specific entropy. Using Eq. 6.43 p2 pr2 a b pr1 p1 where p1 5 bar and p2 1 bar. With pr1 8.411 from Table A-22 at 500 K, the previous equation gives pr2 1.6822. Using this to interpolate in Table A-22, T2 317 K. Finally, inserting values into the expression for system mass m2 a

❶

1 bar 500 K ba b 15 kg2 1.58 kg 5 bar 317 K

This problem also could be solved by considering a control volume enclosing the tank. The state of the control volume would change with time as air escapes. The details of the analysis are left as an exercise.

246

Chapter 6 Using Entropy

6.8 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps

Engineers make frequent use of efficiencies and many different efficiency definitions are employed. In the present section isentropic efficiencies for turbines, nozzles, compressors, and pumps are introduced. Isentropic efficiencies involve a comparison between the actual performance of a device and the performance that would be achieved under idealized circumstances for the same inlet state and the same exit pressure. These efficiencies are frequently used in subsequent sections of the book. ISENTROPIC TURBINE EFFICIENCY

To introduce the isentropic turbine efficiency, refer to Fig. 6.12, which shows a turbine expansion on a Mollier diagram. The state of the matter entering the turbine and the exit pressure are fixed. Heat transfer between the turbine and its surroundings is ignored, as are kinetic and potential energy effects. With these assumptions, the mass and energy rate balances reduce, at steady state, to give the work developed per unit of mass flowing through the turbine # Wcv # h1 h2 m Since state 1 is fixed, the specific enthalpy h1 is known. Accordingly, the value of the work depends on the specific enthalpy h2 only, and increases as h2 is reduced. The maximum value for the turbine work corresponds to the smallest allowed value for the specific enthalpy at the turbine exit. This can be determined using the second law. Since there is no heat transfer, the allowed exit states are constrained by Eq. 6.41 # scv # s2 s1 0 m # # Because the entropy production scvm cannot be negative, states with s2 s1 are not accessible in an adiabatic expansion. The only states that actually can be attained adiabatically are those with s2 s1. The state labeled “2s” on Fig. 6.12 would be attained only in the limit of no internal irreversibilities. This corresponds to an isentropic expansion through the turbine. For fixed exit pressure, the specific enthalpy h2 decreases as the specific entropy s2

p1 h T1

1

Actual expansion h1 – h2 Isentropic expansion

h1 – h2s 2 2s

Accessible states p2 s

Figure 6.12 Comparison of actual and isentropic expansions through a turbine.

6.8 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps

decreases. Therefore, the smallest allowed value for h2 corresponds to state 2s, and the maximum value for the turbine work is # Wcv a # b h1 h2s m s In an actual expansion through the turbine h2 h2s, and thus less work than the maximum would be developed. This difference can be gauged by the isentropic turbine efficiency defined by # # Wcv m ht # # 1Wcvm2 s

(6.48)

isentropic turbine efficiency

Both the numerator and denominator of this expression are evaluated for the same inlet state and the same exit pressure. The value of t is typically 0.7 to 0.9 (70–90%). ISENTROPIC NOZZLE EFFICIENCY

A similar approach to that for turbines can be used to introduce the isentropic efficiency of nozzles operating at steady state. The isentropic nozzle efficiency is defined as the ratio of the actual specific kinetic energy of the gas leaving the nozzle, V22 2, to the kinetic energy at the exit that would be achieved in an isentropic expansion between the same inlet state and the same exhaust pressure, 1V22 22 s hnozzle

V22 2 1V22 22 s

(6.49)

Nozzle efficiencies of 95% or more are common, indicating that well-designed nozzles are nearly free of internal irreversibilities. ISENTROPIC COMPRESSOR AND PUMP EFFICIENCIES

The form of the isentropic efficiency for compressors and pumps is taken up next. Refer to Fig. 6.13, which shows a compression process on a Mollier diagram. The state of the matter entering the compressor and the exit pressure are fixed. For negligible heat transfer with

Accessible states p2

h

2 2s Actual compression Isentropic compression

h2 – h1 h2s – h1

p1 1

s

Figure 6.13 Comparison of actual and isentropic compressions.

isentropic nozzle efficiency

247

248

Chapter 6 Using Entropy

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the surroundings and no appreciable kinetic and potential energy effects, the work input per unit of mass flowing through the compressor is # Wcv a # b h2 h1 m Since state 1 is fixed, the specific enthalpy h1 is known. Accordingly, the value of the work input depends on the specific enthalpy at the exit, h2. The above expression shows that the magnitude of the work input decreases as h2 decreases. The minimum work input corresponds to the smallest allowed value for the specific enthalpy at the compressor exit. With similar reasoning as for the turbine, the smallest allowed enthalpy at the exit state that would be achieved in an isentropic compression from the specified inlet state to the specified exit pressure. The minimum work input is given, therefore, by # Wcv a # b h2s h1 m s In an actual compression, h2 h2s, and thus more work than the minimum would be required. This difference can be gauged by the isentropic compressor efficiency defined by isentropic compressor efficiency

isentropic pump efficiency

# # 1Wcv m2 s # hc # 1Wcv m2

(6.50)

Both the numerator and denominator of this expression are evaluated for the same inlet state and the same exit pressure. The value of c is typically 75 to 85% for compressors. An isentropic pump efficiency, p, is defined similarly. The series of four examples to follow illustrate various aspects of isentropic efficiencies of turbines, nozzles, and compressors. Example 6.11 is a direct application of the isentropic turbine efficiency t to a steam turbine. Here, t is known and the objective is to determine the turbine work.

6.8 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps

EXAMPLE

6.11

249

Evaluating Turbine Work Using the Isentropic Efficiency

A steam turbine operates at steady state with inlet conditions of p1 5 bar, T1 320C. Steam leaves the turbine at a pressure of 1 bar. There is no significant heat transfer between the turbine and its surroundings, and kinetic and potential energy changes between inlet and exit are negligible. If the isentropic turbine efficiency is 75%, determine the work developed per unit mass of steam flowing through the turbine, in kJ/kg. SOLUTION Known: Steam expands through a turbine operating at steady state from a specified inlet state to a specified exit pressure. The turbine efficiency is known. Find: Determine the work developed per unit mass of steam flowing through the turbine. Schematic and Given Data: p1 = 5 bar h 1

T1 Actual expansion Isentropic expansion

h1 – h2 h1 – h2s

Assumptions: 1. A control volume enclosing the turbine is at steady state.

2

2. The expansion is adiabatic and changes in kinetic and potential energy between the inlet and exit can be neglected.

2s Accessible states

p2 = 1 bar s

Figure E6.11

Analysis: The work developed can be determined using the isentropic turbine efficiency, Eq. 6.48, which on rearrangement gives # # Wcv Wcv # ht a # b ht 1h1 h2s 2 m m s

❶

From Table A-4, h1 3105.6 kJ/kg and s1 7.5308 kJ/kg # K. The exit state for an isentropic expansion is fixed by p2 1 bar and s2s s1. Interpolating with specific entropy in Table A-4 at 1 bar gives h2s 2743.0 kJ/kg. Substituting values # Wcv # 0.7513105.6 2743.02 271.95 kJ/kg m

❶

The effect of irreversibilities is to exact a penalty on the work output of the turbine. The work is only 75% of what it would be for an isentropic expansion between the given inlet state and the turbine exhaust pressure. This is clearly illustrated in terms of enthalpy differences on the accompanying h–s diagram.

The next example is similar to Example 6.11, but here the working substance is air as an ideal gas. Moreover, in this case the turbine work is known and the objective is to determine the isentropic turbine efficiency.

250

Chapter 6 Using Entropy

EXAMPLE

6.12

Evaluating the Isentropic Turbine Efficiency

A turbine operating at steady state receives air at a pressure of p1 3.0 bar and a temperature of T1 390 K. Air exits the turbine at a pressure of p2 1.0 bar. The work developed is measured as 74 kJ per kg of air flowing through the turbine. The turbine operates adiabatically, and changes in kinetic and potential energy between inlet and exit can be neglected. Using the ideal gas model for air, determine the turbine efficiency. SOLUTION Known: Air expands adiabatically through a turbine at steady state from a specified inlet state to a specified exit pressure. The work developed per kg of air flowing through the turbine is known. Find: Determine the turbine efficiency. Schematic and Given Data:

3.0 bar T

Air turbine p1 = 3.0 bar T1 = 390 K 1

T1 = 390 K Wcv ––– = 74 kJ/kg m

Actual expansion

p2 = 1.0 bar

Isentropic expansion

2

1.0 bar 2 2s

s

Figure E6.12

Assumptions: 1. The control volume shown on the accompanying sketch is at steady state. 2. The expansion is adiabatic and changes in kinetic and potential energy between inlet and exit can be neglected. 3. The air is modeled as an ideal gas. Analysis: The numerator of the isentropic turbine efficiency, Eq. 6.48, is known. The denominator is evaluated as follows. The work developed in an isentropic expansion from the given inlet state to the specified exit pressure is # Wcv a # b h1 h2s m s From Table A-22 at 390 K, h1 390.88 kJ/kg. To determine h2s, use Eq. 6.43 pr 1T2s 2 a

p2 b p 1T 2 p1 r 1

With p1 3.0 bar, p2 1.0 bar, and pr1 3.481 from Table A-22 at 390 K pr 1T2s 2 a

1.0 b 13.4812 1.1603 3.0

6.8 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps

251

Interpolation in Table A-22 gives h2s 285.27 kJ/kg. Thus # Wcv a # b 390.88 285.27 105.6 kJ/kg m s Substituting values into Eq. 6.48 # # 74 kJ/kg Wcv m ht # # 105.6 kJ/kg 0.70 170%2 1Wcvm2 s

In the next example, the objective is to determine the isentropic efficiency of a steam nozzle. EXAMPLE

6.13

Evaluating the Isentropic Nozzle Efficiency

Steam enters a nozzle operating at steady state at p1 1.0 MPa and T1 320C with a velocity of 30 m/s. The pressure and temperature at the exit are p2 0.3 MPa and T2 180C. There is no significant heat transfer between the nozzle and its surroundings, and changes in potential energy between inlet and exit can be neglected. Determine the nozzle efficiency. SOLUTION Known: Steam expands through a nozzle at steady state from a specified inlet state to a specified exit state. The velocity at the inlet is known. Find: Determine the nozzle efficiency. Schematic and Given Data:

h p1 = 1.0 MPa T1 = 320°C V1 = 30 m/s

1.0 MPa 320°C

1 Isentropic expansion

Actual expansion

p2 = 0.3 MPa T2 = 180°C

0.3 MPa 180°C 2

Steam nozzle

2s 2

1

s Figure E6.13

Assumptions: 1. The control volume shown on the accompanying sketch operates adiabatically at steady state. # 2. For the control volume, Wcv 0 and the change in potential energy between inlet and exit can be neglected. Analysis: The nozzle efficiency given by Eq. 6.49 requires the actual specific kinetic energy at the nozzle exit and the specific kinetic energy that would be achieved at the exit in an isentropic expansion from the given inlet state to the given exit pressure. The mass and energy rate balances for the one-inlet, one-exit control volume at steady state reduce to give V22 V21 h1 h2

2 2 This equation applies for both the actual expansion and the isentropic expansion.

252

Chapter 6 Using Entropy

From Table A-4 at T1 320C and p1 1.0 MPa, h1 3093.9 kJ/kg, s1 7.1962 kJ /kg # K. Also, with T2 180C and p2 0.3 MPa, h2 2823.9 kJ/kg. Thus, the actual specific kinetic energy at the exit in kJ/kg is 1302 2 m 2 V22 kJ 1N 1 kJ b (3093.9 2823.9)

a b a ba 2 kg 2 s 1 kg # m/s2 103 N # M 270.45

kJ kg

Interpolating in Table A-4 at 0.3 MPa, with s2s s1, results in h2s 2813.3 kJ/kg. Accordingly, the specific kinetic energy at the exit for an isentropic expansion is a

V22 302 281.05 kJ /kg b 3093.9 2813.3

2 s 21103 2

Substituting values into Eq. 6.49

❶

hnozzle

1V22 22

1V22

22 s

270.45 0.962 196.2%2 281.05

❶ The principal irreversibility in nozzles is friction between the flowing gas or liquid and the nozzle wall. The effect of friction is that a smaller exit kinetic energy, and thus a smaller exit velocity, is realized than would have been obtained in an isentropic expansion to the same pressure.

In Example 6.14, the isentropic efficiency of a refrigerant compressor is evaluated, first using data from property tables and then using IT.

EXAMPLE

Evaluating the Isentropic Compressor Efficiency

6.14

For the compressor of the heat pump system in Example 6.8, determine the power, in kW, and the isentropic efficiency using (a) data from property tables, (b) Interactive Thermodynamics: IT. SOLUTION Known: Refrigerant 22 is compressed adiabatically at steady state from a specified inlet state to a specified exit state. The mass flow rate is known. Find: Determine the compressor power and the isentropic efficiency using (a) property tables, (b) IT. Schematic and Given Data:

2s

T

2 T2 = 75°C

14 bar

Assumptions: 1. A control volume enclosing the compressor is at steady state.

3.5 bar

1 T1 = –5°C s

2. The compression is adiabatic, and changes in kinetic and potential energy between the inlet and the exit can be neglected.

Figure E6.14

6.8 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps

253

Analysis: (a) By assumptions 1 and 2, the mass and energy rate balances reduce to give # # Wcv m 1h1 h2 2 From Table A-9, h1 249.75 kJ/kg and h2 294.17 kJ/kg. Thus # 1 kW Wcv 10.07 kg/s2 1249.75 294.172 kJ/kg ` ` 3.11 kW 1 kJ/s The isentropic compressor efficiency is determined using Eq. 6.50 # # 1h2s h1 2 1Wcvm2 s hc # # 1h h 2 1Wcv m2 2 1 In this expression, the denominator represents the work input per unit mass of refrigerant flowing for the actual compression process, as calculated above. The numerator is the work input for an isentropic compression between the initial state and the same exit pressure. The isentropic exit state is denoted as state 2s on the accompanying T–s diagram. From Table A-9, s1 0.9572 kJ/kg # K. With s2s s1, interpolation in Table A-9 at 14 bar gives h2s 285.58 kJ/kg. Substituting values

❶

hc

1285.58 249.752

1294.17 249.752

0.81181%2

# # (b) The IT program follows. In the program, Wcv is denoted as Wdot, m as mdot, and c as eta_c. // Given Data: T1 = –5 // °C p1 = 3.5 // bar T2 = 75 // °C p2 = 14 // bar mdot = 0.07 // kg/s // Determine the specific enthalpies. h1 = h_PT(“R22”,p1,T1) h2 = h_PT(“R22”,p2,T2) // Calculate the power. Wdot = mdot * (h1 – h2) // Find h2s: s1 = s_PT(“R22”,p1,T1) s2s = s_Ph(“R22”,p2,h2s) s2s = s1

❷

// Determine the isentropic compressor efficiency. eta_c = (h2s – h1) / (h2 – h1) # Use the Solve button to obtain: Wcv 3.111 kW and hc 80.58%, which agree closely with the values obtained above.

❶

The minimum theoretical power for adiabatic compression from state 1 to the exit pressure of 14 bar would be # # 1Wcv 2 s m 1h1 h2s 2 10.0721249.75 285.582 2.51 kW The magnitude of the actual power required is greater than the ideal power due to irreversibilities.

❷

Note that IT solves for the value of h2s even though it is an implicit variable in the argument of the specific entropy function.

254

Chapter 6 Using Entropy

6.9 Heat Transfer and Work in Internally Reversible, Steady-State Flow Processes

This section concerns one-inlet, one-exit control volumes at steady state. The objective is to derive expressions for the heat transfer and the work in the absence of internal irreversibilities. The resulting expressions have several important applications. HEAT TRANSFER

For a control volume at steady state in which the flow is both isothermal and internally reversible, the appropriate form of the entropy rate balance is # Qcv # # 0

m 1s1 s2 2 s cv 0 T # where 1 and 2 denote the inlet and exit, respectively, and m is the mass flow rate. Solving this equation, the heat transfer per unit of mass passing through the control volume is # Qcv # T 1s2 s1 2 m More generally, the temperature would vary as the gas or liquid flows through the control volume. However, we can consider the temperature variation to consist of a series of infinitesimal steps. Then, the heat transfer per unit of mass would be given as # Qcv a # b int m rev

· Qcv ––– m·

( ) T

=

int rev

2 1

2

T ds

(6.51)

1

The subscript “int rev” serves to remind us that the expression applies only to control volumes in which there are no internal irreversibilities. The integral of Eq. 6.51 is performed from inlet to exit. When the states visited by a unit mass as it passes reversibly from inlet to exit are described by a curve on a T–s diagram, the magnitude of the heat transfer per unit of mass flowing can be represented as the area under the curve, as shown in Fig. 6.14.

T ds 2

1

WORK s Figure 6.14

Area representation of heat transfer for an internally reversible flow process.

The work per unit of mass passing through the control volume can be found from an energy rate balance, which reduces at steady state to give # # Qcv Wcv V21 V22 b g1z1 z2 2 # # 1h1 h2 2 a m m 2 This equation is a statement of the conservation of energy principle that applies when irreversibilities are present within the control volume as well as when they are absent. However, if consideration is restricted to the internally reversible case, Eq. 6.51 can be introduced to obtain # 2 Wcv V21 V22 a # bint T ds 1h1 h2 2 a b g1z1 z2 2 (6.52) m rev 2 1

6.9 Heat Transfer and Work in Internally Reversible, Steady-State Flow Processes

255

where the subscript “int rev” has the same significance as before. Since internal irreversibilities are absent, a unit of mass traverses a sequence of equilibrium states as it passes from inlet to exit. Entropy, enthalpy, and pressure changes are therefore related by Eq. 6.12b T ds dh v dp which on integration gives

2

1

T ds 1h2 h1 2

2

v dp

1

Introducing this relation, Eq. 6.52 becomes # Wcv a # bint m rev

2

1

v dp a

V21 V22 b g1z1 z2 2 2

When the states visited by a unit of mass as it passes reversibly from inlet to exit are described by a curve on a p–v diagram as shown in Fig. 6.15, the magnitude of the integral v dp is represented by the shaded area behind the curve. Equation 6.53a may be applied to devices such as turbines, compressors, and pumps. In many of these cases, there is no significant change in kinetic or potential energy from inlet to exit, so # Wcv a # bint m rev

2

1

v dp

1¢ke ¢pe 02

p

(6.53a)

(6.53b)

2

2 1

vdp 1 v

Figure 6.15 Area representation of 12 v dp.

This expression shows that the work is related to the magnitude of the specific volume of the gas or liquid as it flows from inlet to exit. for example. . . consider two devices: a pump through which liquid water passes and a compressor through which water vapor passes. For the same pressure rise, the pump would require a much smaller work input per unit of mass flowing than would the compressor because the liquid specific volume is much smaller than that of vapor. This conclusion is also qualitatively correct for actual pumps and compressors, where irreversibilities are present during operation. If the specific volume remains approximately constant, as in many applications with liquids, Eq. 6.53b becomes # Wcv a # bint v1 p2 p1 2 m rev

1v constant, ¢ke ¢pe 02

(6.53c)

Equation 6.53a # can also be applied to study the performance of control volumes at steady state in which Wcv is zero, as in the case of nozzles and diffusers. For any such case, the equation becomes

2

1

v dp a

V22 V21 b g1z2 z1 2 0 2

which is a form of the Bernoulli equation frequently used in fluid mechanics.

(6.54) Bernoulli equation

256

Chapter 6 Using Entropy

WORK IN POLYTROPIC PROCESSES

When each unit of mass undergoes a polytropic process as it passes through the control volume, the relationship between pressure and specific volume is pvn constant. Introducing this into Eq. 6.53b and performing the integration # 2 2 dp Wcv a # bint v dp 1constant2 1n 1n m rev 1 1 p n 1p v p1v1 2 1polytropic, n 12 (6.55) n1 2 2

for any value of n except n 1. When n 1, pv constant, and the work is # 2 2 dp Wcv a # bint v dp constant m rev 1 1 p

1p1v1 2 ln1p2 p1 2

1polytropic, n 12

(6.56)

Equations 6.55 and 6.56 apply generally to polytropic processes of any gas (or liquid). For the special case of an ideal gas, Eq. 6.55 becomes # Wcv nR a # bint 1T T1 2 1ideal gas, n 12 m rev n1 2

IDEAL GAS CASE.

(6.57a)

For a polytropic process of an ideal gas, Eq. 3.56 applies: p2 1n12n T2 a b p1 T1 Thus, Eq. 6.57a can be expressed alternatively as # Wcv p2 1n12n nRT1 a # bint ca b 1d m rev n 1 p1 For the case of an ideal gas, Eq. 6.56 becomes # Wcv a # bint RT ln1 p2 p1 2 m rev

1ideal gas, n 12

1ideal gas, n 12

(6.57b)

(6.58)

In the next example, we consider air modeled as an ideal gas undergoing a polytropic compression process at steady state.

EXAMPLE

6.15

Polytropic Compression of Air

An air compressor operates at steady state with air entering at p1 1 bar, T1 20C, and exiting at p2 5 bar. Determine the work and heat transfer per unit of mass passing through the device, in kJ/kg, if the air undergoes a polytropic process with n 1.3. Neglect changes in kinetic and potential energy between the inlet and the exit. Use the ideal gas model for air. SOLUTION Known: Air is compressed in a polytropic process from a specified inlet state to a specified exit pressure. Find: Determine the work and heat transfer per unit of mass passing through the device.

6.9 Heat Transfer and Work in Internally Reversible, Steady-State Flow Processes

257

Schematic and Given Data:

p

2

T2 = 425 K 5 bar

Assumptions: 1. A control volume enclosing the compressor is at steady state.

pv1.3 = constant

2. The air undergoes a polytropic process with n 1.3.

❶

3. The air behaves as an ideal gas. 1 1 bar Shaded area = magnitude of (Wcv/m) int

4. Changes in kinetic and potential energy from inlet to exit can be neglected.

rev

v

Figure E6.15

Analysis: The work is obtained using Eq. 6.57a, which requires the temperature at the exit, T2. The temperature T2 can be found using Eq. 3.56 T2 T 1 a

p2 1n12n 5 11.3121.3 b 293 a b 425 K p1 1

Substituting known values into Eq. 6.57a then gives # Wcv nR 1.3 8.314 kJ 1T T1 2 a b 1425 2932 K # m n1 2 1.3 1 28.97 kg # K 164.2 kJ/kg The heat transfer is evaluated by reducing the mass and energy rate balances with the appropriate assumptions to obtain # # Qcv Wcv # # h2 h1 m m Using the temperatures T1 and T2, the required specific enthalpy values are obtained from Table A-22 as h1 293.17 kJ/kg and h2 426.35 kJ/ kg. Thus # Qcv # 164.15 1426.35 293.172 31 kJ/kg m

❶

The states visited in the polytropic compression process are shown by the curve on the accompanying p–v diagram. The magnitude of the work per unit of mass passing through the compressor is represented by the shaded area behind the curve.

Chapter Summary and Study Guide

In this chapter, we have introduced the property entropy and illustrated its use for thermodynamic analysis. Like mass and energy, entropy is an extensive property that can be transferred across system boundaries. Entropy transfer accompanies both heat transfer and mass flow. Unlike mass and energy, entropy is not conserved but is produced within systems whenever internal irreversibilities are present. The use of entropy balances is featured in this chapter. Entropy balances are expressions of the second law that account for the entropy of systems in terms of entropy transfers and

entropy production. For processes of closed systems, the entropy balance is Eq. 6.27, and a corresponding rate form is Eq. 6.32. For control volumes, rate forms include Eq. 6.37 and the companion steady-state expression given by Eq. 6.39. The following checklist provides a study guide for this chapter. When your study of the text and end-of-chapter exercises has been completed you should be able to write out meanings of the terms listed in the margins

throughout the chapter and understand each of the

258

Chapter 6 Using Entropy

–determining s of ideal gases using Eq. 6.21 for variable specific heats together with Tables A-22 and A-23, and using Eqs. 6.22 and 6.23 for constant specific heats. –evaluating isentropic efficiencies for turbines, nozzles, compressors, and pumps from Eqs. 6.48, 6.49, and 6.50, respectively, including for ideal gases the appropriate use of Eqs. 6.43–6.44 for variable specific heats and Eqs. 6.45–6.47 for constant specific heats.

related concepts. The subset of key concepts listed below in is particularly important in subsequent chapters. apply entropy balances in each of several alternative

forms, appropriately modeling the case at hand, correctly observing sign conventions, and carefully applying SI and English units. use entropy data appropriately, to include

–retrieving data from Tables A-2 through A-18, using Eq. 6.6 to evaluate the specific entropy of two-phase liquid–vapor mixtures, sketching T–s and h–s diagrams and locating states on such diagrams, and appropriately using Eqs. 6.7 and 6.24.

apply Eq. 6.25 for closed systems and Eqs. 6.51 and

6.53 for one-inlet, one-exit control volumes at steady state, correctly observing the restriction to internally reversible processes.

Key Engineering Concepts

entropy change p. 209 T–s diagram p. 211 Mollier diagram p. 212 entropy balance p. 221

entropy transfer p. 221, 232 entropy production p. 221

entropy rate balance p. 223, 231, 233

isentropic efficiencies p. 247–248

Exercises: Things Engineers Think About 1. Of mass, energy, and entropy, which are conserved? 2. Both entropy and enthalpy are introduced in this text without accompanying physical pictures. Can you think of other such properties? 3. How might you explain the entropy production concept in terms a child would understand? 4. Referring to Fig. 2.3, if systems A and B operate adiabatically, does the entropy of each increase, decrease, or remain the same? 5. If a closed system would undergo an internally reversible process and an irreversible process between the same end states, how would the changes in entropy for the two processes compare? How would the amounts of entropy produced compare? 6. Is is possible for the entropy of both a closed system and its surroundings to decrease during a process? 7. Describe a process of a closed system for which the entropy of both the system and its surroundings increase. 8. How can entropy be transferred into, or out of, a closed system? A control volume? 9. What happens to the entropy produced in a one-inlet, one-exit control volume at steady state? 10. The two power cycles shown to the same scale in the figure are composed of internally reversible processes. Compare the net work developed by these cycles. Which cycle has the greater thermal efficiency?

T

T

2

1

3

2

3

1 S

S

11. Sketch T–s and p–v diagrams of a gas executing a power cycle consisting of four internally reversible processes in series: constant specific volume, constant pressure, isentropic, isothermal. 12. Sketch the T–s diagram for the Carnot vapor cycle of Fig. 5.12. 13. All states of an adiabatic and internally reversible process of a closed system have the same entropy, but is a process between two states having same entropy necessarily adiabatic and internally reversible? 14. Discuss the operation of a turbine in the limit as isentropic efficiency approaches 100%; in the limit as isentropic efficiency approaches 0%. 15. What can be deduced from energy and entropy balances about a system undergoing a thermodynamic cycle while receiving energy by heat transfer at temperature TC and discharging energy by heat transfer at a higher temperature TH, if these are the only energy transfers the system experiences? 16. Reducing irreversibilities within a system can improve its thermodynamic performance, but steps taken in this direction are usually constrained by other considerations. What are some of these?

259

Problems: Developing Engineering Skills

Problems: Developing Engineering Skills Surroundings

Exploring Fundamentals

T0

6.1 A system executes a power cycle while receiving 1000 kJ by heat transfer at a temperature of 500 K and discharging energy by heat transfer at a temperature of 300 K. There are no other heat transfers. Applying Eq. 6.2, determine cycle if the thermal efficiency is (a) 60%, (b) 40%, (c) 20%. Identify the cases (if any) that are internally reversible or impossible.

Q0 Qs

6.2 A reversible power cycle R and an irreversible power cycle I operate between the same two reservoirs. Each receives QH from the hot reservoir. The reversible cycle develops work WR, while the irreversible cycle develops work WI. The reversible cycle discharges QC to the cold reservoir, while the irreversible cycle discharges QC. (a) Evaluate cycle for cycle I in terms of WI, WR, and temperature TC of the cold reservoir only. (b) Demonstrate that WI WR and QC QC. 6.3 A reversible refrigeration cycle R and an irreversible refrigeration cycle I operate between the same two reservoirs and each removes QC from the cold reservoir. The net work input required by R is WR, while the net work input for I is WI. The reversible cycle discharges QH to the hot reservoir, while the irreversible cycle discharges QH. Show that WI WR and QH QH. 6.4 A reversible power cycle receives energy Q1 and Q2 from hot reservoirs at temperatures T1 and T2, respectively, and discharges energy Q3 to a cold reservoir at temperature T3. (a) Obtain an expression for the thermal efficiency in terms of the ratios T1T3, T2T3, q Q2Q1. (b) Discuss the result of part (a) in each of these limits: lim q S 0, lim q S , lim T1 S . 6.5 Complete the following involving reversible and irreversible cycles: (a) Reversible and irreversible power cycles each discharge energy QC to a cold reservoir at temperature TC and receive energy QH from hot reservoirs at temperatures TH and TH, respectively. There are no other heat transfers. Show that TH TH. (b) Reversible and irreversible refrigeration cycles each discharge energy QH to a hot reservoir at temperature TH and receive energy QC from cold reservoirs at temperatures TC and TC, respectively. There are no other heat transfers. Show that TC TC. (c) Reversible and irreversible heat pump cycles each receive energy QC from a cold reservoir at temperature TC and discharge energy QH to hot reservoirs at temperatures TH and TH, respectively. There are no other heat transfers. Show that TH TH. 6.6 The system shown schematically in Fig. P6.6 undergoes a cycle while receiving energy # at the rate Q0 from the surroundings at temperature T0, Qs from a source at temperature

Qu

Air

Fuel

Ts

Tu

Figure P6.6

# Ts, and delivering energy at the rate Qu at a use temperature Tu. There are no other energy transfers. For Ts Tu T0,# obtain an expression for the maximum theoretical value of Qu in # terms of Qs and the temperatures Ts, Tu, and T0. 6.7

Answer the following true or false. If false, explain why.

(a) The change of entropy of a closed system is the same for every process between two specified states. (b) The entropy of a fixed amount of an ideal gas increases in every isothermal compression. (c) The specific internal energy and enthalpy of an ideal gas are each functions of temperature alone but its specific entropy depends on two independent intensive properties. (d) One of the T ds equations has the form T ds du p dv. (e) The entropy of a fixed amount of an incompressible substance increases in every process in which temperature decreases. 6.8 A closed system consists of an ideal gas with constant specific heat ratio k. (a) The gas undergoes a process in which temperature increases from T1 to T2. Show that the entropy change for the process is greater if the change in state occurs at constant pressure than if it occurs at constant volume. Sketch the processes on p–v and T–s coordinates. (b) Using the results of (a), show on T–s coordinates that a line of constant specific volume passing through a state has a greater slope than a line of constant pressure passing through that state. (c) The gas undergoes a process in which pressure increases from p1 to p2. Show that the ratio of the entropy change for an isothermal process to the entropy change for a constant-volume process is (1 k). Sketch the processes on p–v and T–s coordinates. 6.9

Answer the following true or false. If false, explain why.

(a) A process that violates the second law of thermodynamics violates the first law of thermodynamics.

260

Chapter 6 Using Entropy

(b) When a net amount of work is done on a closed system undergoing an internally reversible process, a net heat transfer of energy from the system also occurs. (c) One corollary of the second law of thermodynamics states that the change in entropy of a closed system must be greater than zero or equal to zero. (d) A closed system can experience an increase in entropy only when irreversibilities are present within the system during the process. (e) Entropy is produced in every internally reversible process of a closed system. (f ) In an adiabatic and internally reversible process of a closed system, the entropy remains constant. (g) The energy of an isolated system must remain constant, but the entropy can only decrease. 6.10 A fixed mass of water m, initially a saturated liquid, is brought to a saturated vapor condition while its pressure and temperature remain constant. (a) Derive expressions for the work and heat transfer in terms of the mass m and properties that can be obtained directly from the steam tables. (b) Demonstrate that this process is internally reversible. 6.11 A quantity of air is shown in Fig. 6.8. Consider a process in which the temperature of the air increases by some combination of stirring and heating. Assuming the ideal gas model for the air, suggest how this might be done with (a) minimum entropy production. (b) maximum entropy production. 6.12 Taken together, a certain closed system and its surroundings make up an isolated system. Answer the following true or false. If false, explain why. (a) No process is allowed in which the entropies of both the system and the surroundings increase. (b) During a process, the entropy of the system might decrease, while the entropy of the surroundings increases, and conversely. (c) No process is allowed in which the entropies of both the system and the surroundings remain unchanged. (d) A process can occur in which the entropies of both the system and the surroundings decrease. 6.13 An isolated system of total mass m is formed by mixing two equal masses of the same liquid initially at the temperatures T1 and T2. Eventually, the system attains an equilibrium state. Each mass is incompressible with constant specific heat c. (a) Show that the amount of entropy produced is s mc ln c

T1 T2

21T1T2 2 12

d

(b) Demonstrate that must be positive. 6.14 A cylindrical rod of length L insulated on its lateral surface is initially in contact at one end with a wall at temperature TH

and at the other end with a wall at a lower temperature TC. The temperature within the rod initially varies linearly with position z according to T 1z2 TH a

TH TC bz L

The rod is then insulated on its ends and eventually comes to a final equilibrium state where the temperature is Tf. Evaluate Tf in terms of TH and TC and show that the amount of entropy produced is s mc a1 ln Tf

TC TH ln TC ln TH b TH TC T H TC

where c is the specific heat of the rod. 6.15 A system undergoing a thermodynamic cycle receives QH at temperature TH and discharges QC at temperature TC. There are no other heat transfers. (a) Show that the net work developed per cycle is given by Wcycle QH a1

T ¿C b T ¿C s T ¿H

where is the amount of entropy produced per cycle owing to irreversibilities within the system. (b) If the heat transfers QH and QC are with hot and cold reservoirs, respectively, what is the relationship of TH to the temperature of the hot reservoir TH and the relationship of TC to the temperature of the cold reservoir TC? (c) Obtain an expression for Wcycle if there are (i) no internal irreversibilities, (ii) no internal or external irreversibilities. 6.16 A system undergoes a thermodynamic power cycle while receiving energy by heat transfer from an incompressible body of mass m and specific heat c initially at temperature TH. The system undergoing the cycle discharges energy by heat transfer to another incompressible body of mass m and specific heat c initially at a lower temperature TC. Work is developed by the cycle until the temperature of each of the two bodies is the same, T. (a) Develop an expression for the minimum theoretical final temperature, T, in terms of m, c, TH, and TC, as required. (b) Develop an expression for the maximum theoretical amount of work that can be developed, Wmax, in terms of m, c, TH, and TC, as required. (c) What is the minimum theoretical work input that would be required by a refrigeration cycle to restore the two bodies from temperature T to their respective initial temperatures, TH and TC? 6.17 At steady state, an insulated mixing chamber receives two liquid streams of the same substance at temperatures T1 and T2 # # and mass flow rates m1 and m2, respectively. A single stream

Problems: Developing Engineering Skills

# exits at T3 and m3. Using the incompressible substance model with constant specific heat c, obtain an expression for # # (a) T3 in terms of T1, T2, and the ratio of mass flow rates m1 m3. (b) the rate of entropy production per unit of mass exiting the # # chamber in terms of c, T1T2, and m1m3. (c) For fixed values of c and T1T2, determine the value of # # m1 m3 for which the rate of entropy production is a maximum. Using Entropy Data

6.18 Using the tables for water, determine the specific entropy at the indicated states, in kJ/kg # K. Check the results using IT. In each case, locate the state by hand on a sketch of the T–s diagram. (a) (b) (c) (d)

p 5.0 MPa, T 400C p 5.0 MPa, T 100C p 5.0 MPa, u 1872.5 kJ/kg p 5.0 MPa, saturated vapor

261

(b) air as an ideal gas, T1 27C, p1 1.5 bar, T2 127C. Find p2 in bar. (c) Refrigerant 134a, T1 20C, p1 5 bar, p2 1 bar. Find v2 in m3/kg. 6.23 One kilogram of oxygen (O2) modeled as an ideal gas undergoes a process from 300 K, 2 bar to 1500 K, 1.5 bar. Determine the change in specific entropy, in kJ/kg # K, using (a) (b) (c) (d)

Equation 6.19 with cp 1T 2 from Table A-21. Equation 6.21b with s° from Table A-23. Equation 6.23 with cp at 900 K from Table A-20. IT.

6.24 Five kilograms of ammonia undergo a process from an initial state where the pressure is 8 bar and the temperature is 100C to a final state of 8 bar, 0C. Determine the entropy change of the ammonia, in kJ/K, assuming the process is (a) irreversible. (b) internally reversible.

6.19 Using the appropriate table, determine the change in specific entropy between the specified states, in kJ/kg # K. Check the results using IT.

6.25 A quantity of liquid water undergoes a process from 80C, 5 MPa to saturated liquid at 40C. Determine the change in specific entropy, in kJkg # K, using

(a) water, p1 10 MPa, T1 400C, p2 10 MPa, T2 100C. (b) Refrigerant 134a, h1 111.44 kJ/kg, T1 40C, saturated vapor at p2 5 bar. (c) air as an ideal gas, T1 7C, p1 2 bar, T2 327C, p2 1 bar. (d) hydrogen (H2) as an ideal gas, T1 727C, p1 1 bar, T2 25C, p2 3 bar.

(a) Tables A-2 and A-5. (b) saturated liquid data only from Table A-2. (c) the incompressible liquid model with a constant specific heat from Table A-19. (d) IT.

6.20 One kilogram of Refrigerant 22 undergoes a process from 0.2 MPa, 20C to a state where the pressure is 0.06 MPa. During the process there is a change in specific entropy, s2 s1 0.55 kJ/kg # K. Determine the temperature at the final state, in C, and the final specific enthalpy, in kJ/kg. 6.21 Employing the ideal gas model, determine the change in specific entropy between the indicated states, in kJ/kg # K. Solve three ways: Use the appropriate ideal gas table, IT, and a constant specific heat value from Table A-20. (a) air, p1 100 kPa, T1 20C, p2 100 kPa, T2 100C. (b) air, p1 1 bar, T1 27C, p2 3 bar, T2 377C. (c) carbon dioxide, p1 150 kPa, T1 30C, p2 300 kPa, T2 300C. (d) carbon monoxide, T1 300 K, v1 1.1 m3/kg, T2 500 K, v2 0.75 m3/kg. (e) nitrogen, p1 2 MPa, T1 800 K, p2 1 MPa, T2 300 K. 6.22 Using the appropriate table, determine the indicated property for a process in which there is no change in specific entropy between state 1 and state 2. Check the results using IT. (a) water, T1 10C, x1 0.75, saturated vapor at state 2. Find p2 in bar.

6.26 One-quarter lbmol of nitrogen gas (N2) undergoes a process from p1 20 lbf/in.2, T1 500R to p2 150 lbf/in.2, T2 800R. For the process W 500 Btu. Employing the ideal gas model, determine (a) the heat transfer, in Btu. (b) the change in entropy, in Btu/R. (c) Show the initial and final states on a T–s diagram. 6.27 Methane gas (CH4) enters a compressor at 298 K, 1 bar and exits at 2 bar and temperature T. Employing the ideal gas model, determine T, in K, if there is no change in specific entropy from inlet to exit. Analyzing Internally Reversible Processes

6.28 A quantity of air amounting to 2.42 102 kg undergoes a thermodynamic cycle consisting of three internally reversible processes in series. Process 1–2: constant-volume heating at V 0.02 m3 from p1 0.1 MPa to p2 0.42 MPa Process 2–3: constant-pressure cooling Process 3–1: isothermal heating to the initial state Employing the ideal gas model with cp 1 kJ/kg # K, evaluate the change in entropy, in kJ/K, for each process. Sketch the cycle on p–v and T–s coordinates. 6.29 One kilogram of water initially at 160C, 1.5 bar undergoes an isothermal, internally reversible compression process

262

Chapter 6 Using Entropy

to the saturated liquid state. Determine the work and heat transfer, each in kJ. Sketch the process on p–v and T–s coordinates. Associate the work and heat transfer with areas on these diagrams.

Process 2–3: adiabatic compression to 550 K, 6.25 bar Process 3–1: constant-pressure compression Employing the ideal gas model,

6.30 A gas initially at 14 bar and 60C expands to a final pressure of 2.8 bar in an isothermal, internally reversible process. Determine the heat transfer and the work, each in kJ per kg of gas, if the gas is (a) Refrigerant 134a, (b) air as an ideal gas. Sketch the processes on p–v and T–s coordinates.

(a) sketch the cycle on p–v and T–s coordinates. (b) determine T1, in K (c) If the cycle is a power cycle, determine its thermal efficiency. If the cycle is a refrigeration cycle, determine its coefficient of performance.

6.31 Reconsider the data of Problem 6.37, but now suppose the gas expands to 2.8 bar isentropically. Determine the work, in kJ per kg of gas, if the gas is (a) Refrigerant 134a, (b) air as an ideal gas. Sketch the processes on p–v and T–s coordinates.

6.37 Figure P6.37 gives the schematic of a vapor power plant in which water steadily circulates through the four components shown. The water flows through the boiler and condenser at constant pressure, and flows through the turbine and pump adiabatically.

6.32 Nitrogen (N2) initially occupying 0.5 m3 at 1.0 bar, 20C undergoes an internally reversible compression during which pV1.30 constant to a final state where the temperature is 200C. Determine assuming the ideal gas model

(a) Sketch the cycle on T–s coordinates. (b) Determine the thermal efficiency and compare with the thermal efficiency of a Carnot cycle operating between the same maximum and minimum temperatures.

(a) the pressure at the final state, in bar. (b) the work and heat transfer, each in kJ. (c) the entropy change, in kJ/K.

Hot reservoir · QH

6.33 Air initially occupying a volume of 1 m3 at 1 bar, 20C undergoes two internally reversible processes in series Saturated 4 liquid at 10 bar

Process 1–2: compression to 5 bar, 110C during which pV n constant Process 2–3: adiabatic expansion to 1 bar (a) (b) (c) (d)

0.2 bar, x = 18%

6.34 One-tenth kilogram of water executes a Carnot power cycle. At the beginning of the isothermal expansion, the water is a saturated liquid at 160C. The isothermal expansion continues until the quality is 98%. The temperature at the conclusion of the adiabatic expansion is 20C. (a) Sketch the cycle on T–s and p–v coordinates. (b) Determine the heat added and net work, each in kJ. (c) Evaluate the thermal efficiency. 6.35 One pound mass of air as an ideal gas undergoes a Carnot power cycle. At the beginning of the isothermal expansion, the temperature is 880K and the pressure is 8 MPa. The isothermal compression occurs at 280K and the heat added per cycle is 42 kW. Assuming the ideal gas model for the air, determine (a) the pressures at the end of the isothermal expansion, the adiabatic expansion, and the isothermal compression, each in MPa. (b) the net work developed per cycle, kW. (c) the thermal efficiency. 6.36 A quantity of air undergoes a thermodynamic cycle consisting of three internally reversible processes in series. Process 1–2: isothermal expansion from 6.25 to 1.0 bar

Pump

Work

Sketch the two processes on p–v and T–s coordinates. Determine n. Determine the temperature at state 3, in C. Determine the net work, in kJ.

1

Boiler

Saturated vapor at 10 bar

Turbine Work

3

Condenser

2

0.2 bar, x = 88%

· QC Cold reservoir Figure P6.37 Applying the Entropy Balance: Closed Systems

6.38 A closed system undergoes a process in which work is done on the system and the heat transfer Q occurs only at temperature Tb. For each case, determine whether the entropy change of the system is positive, negative, zero, or indeterminate. (a) (b) (c) (d) (e) (f)

internally reversible process, Q 0. internally reversible process, Q 0. internally reversible process, Q 0. internal irreversibilities present, Q 0. internal irreversibilities present, Q 0. internal irreversibilities present, Q 0.

6.39 For each of the following systems, specify whether the entropy change during the indicated process is positive, negative, zero, or indeterminate. (a) One kilogram of water vapor undergoing an adiabatic compression process.

Problems: Developing Engineering Skills

(b) Two kg mass of nitrogen heated in an internally reversible process. (c) One kilogram of Refrigerant 134a undergoing an adiabatic process during which it is stirred by a paddle wheel. (d) One kg mass of carbon dioxide cooled isothermally. (e) Two kg mass of oxygen modeled as an ideal gas undergoing a constant-pressure process to a higher temperature. (f ) Two kilograms of argon modeled as an ideal gas undergoing an isothermal process to a lower pressure. 6.40 A piston–cylinder assembly initially contains 0.04 m3 of water at 1.0 MPa, 320C. The water expands adiabatically to a final pressure of 0.1 MPa. Develop a plot of the work done by the water, in kJ, versus the amount of entropy produced, in kJ/K. 6.41 An insulated piston–cylinder assembly contains Refrigerant 134a, initially occupying 0.017 m3 at .6 MPa, 40C. The refrigerant expands to a final state where the pressure is .32 MPa. The work developed by the refrigerant is measured as 5.275 kJ. Can this value be correct? 6.42 One kilogram of air is initially at 1 bar and 450 K. Can a final state at 2 bar and 350 K be attained in an adiabatic process? 6.43 Air as an ideal gas is compressed from a state where the pressure is 0.1 MPa and the temperature is 27C to a state where the pressure is 0.5 MPa and the temperature is 207C. Can this process occur adiabatically? If yes, determine the work per unit mass of air, in kJ/kg, for an adiabatic process between these states. If no, determine the direction of the heat transfer. 6.44 Two kilograms of Refrigerant 134a initially at 1.4 bar, 60C are compressed to saturated vapor at 60C. During this process, the temperature of the refrigerant departs by no more than 0.01C from 60C. Determine the minimum theoretical heat transfer from the refrigerant during the process, in kJ. 6.45 A patent application describes a device that at steady state receives a heat transfer at the rate 1 kW at a temperature of 167C and generates electricity. There are no other energy transfers. Does the claimed performance violate any principles of thermodynamics? Explain. 6.46 One-half kilogram of propane initially at 4 bar, 30C undergoes a process to 14 bar, 100C while being rapidly compressed in a piston–cylinder assembly. Heat transfer with the surroundings at 20C occurs through a thin wall. The net work is measured as 72.5 kJ. Kinetic and potential energy effects can be ignored. Determine whether it is possible for the work measurement to be correct. 6.47 One kilogram mass of air in a piston–cylinder assembly is compressed adiabatically from 4C, 1 bar to 5 bars. (a) If the air is compressed without internal irreversibilities, determine the temperature at the final state, in C, and the work required, in kJ. (b) If the compression requires 20% more work than found in part (a), determine the temperature at the final state, in F, and the amount of entropy produced, in kJ/K. (c) Show the processes of parts (a) and (b) on T–s coordinates. Employ the ideal gas model.

263

6.48 For the silicon chip of Example 2.5, determine the rate of entropy production, in kW/K. What is the cause of entropy production in this case? 6.49 An electric water heater having a 200 liter capacity employs an electric resistor to heat water from 23 to 55C. The outer surface of the resistor remains at an average temperature of 80C. Heat transfer from the outside of the water heater is negligible and the states of the resistor and the tank holding the water do not change significantly. Modeling the water as incompressible, determine the amount of entropy produced, in kJ/K, for (a) the water as the system. (b) the overall water heater including the resistor. Compare the results of parts (a) and (b), and discuss. 6.50 At steady state, a 20-W curling iron has an outer surface temperature of 80C. For the curling iron, the rate of entropy production, in Btu/ h R. 6.51 An electric motor operating at steady state draws a current of 10 amp with a voltage of 220 V. The output shaft rotates at 1000 RPM with a torque of 16 N # m applied to an external load. The rate of heat transfer from the motor to its surroundings is related to the surface temperature Tb and the ambient temperature T0 by hA(Tb T0), where h 100 W/m2 # K, A 0.195 m2, and T0 293 K. Energy transfers are considered positive in the directions indicated by the arrows on Fig. P6.51. (a) Determine the temperature Tb, in K. (b) For the motor as the system, determine the rate of entropy production, in kW/K. (c) If the system boundary is located to take in enough of the nearby surroundings for heat transfer to take place at temperature T0, determine the rate of entropy production, in kW/K, for the enlarged system.

T0 = 293 K

· Welec

+ –

· Q

Tb = ?

· Wshaft

Figure P6.51

6.52 At steady state, work at a rate of 25 kW is done by a paddle wheel on a slurry contained within a closed, rigid tank. Heat transfer from the tank occurs at a temperature of 250C to surroundings that, away from the immediate vicinity of the tank, are at 27C. Determine the rate of entropy production, in kW/K, (a) for the tank and its contents as the system. (b) for an enlarged system including the tank and enough of the nearby surroundings for the heat transfer to occur at 27C.

264

Chapter 6 Using Entropy

6.53 A system consists of 2 m3 of hydrogen gas (H2), initially at 35C, 215 kPa, contained in a closed rigid tank. Energy is transferred to the system from a reservoir at 300C until the temperature of the hydrogen is 160C. The temperature at the system boundary where heat transfer occurs is 300C. Modeling the hydrogen as an ideal gas, determine the heat transfer, in kJ, the change in entropy, in kJ/K, and the amount of entropy produced, in kJ/K. For the reservoir, determine the change in entropy, in kJ/K. Why do these two entropy changes differ? 6.54 An isolated system consists of a closed aluminum vessel of mass 0.1 kg containing 1 kg of used engine oil, each initially at 55C, immersed in a 10-kg bath of liquid water, initially at 20C. The system is allowed to come to equilibrium. Determine (a) the final temperature, in degrees centigrade. (b) the entropy changes, each in kJ/K, for the aluminum vessel, the oil, and the water. (c) the amount of entropy produced, in kJ/K. 6.55 An insulated cylinder is initially divided into halves by a frictionless, thermally conducting piston. On one side of the piston is 1 m3 of a gas at 300 K, 2 bar. On the other side is 1 m3 of the same gas at 300 K, 1 bar. The piston is released and equilibrium is attained, with the piston experiencing no change of state. Employing the ideal gas model for the gas, determine (a) the final temperature, in K. (b) the final pressure, in bar. (c) the amount of entropy produced, in kJ/ kg. 6.56 An insulated, rigid tank is divided into two compartments by a frictionless, thermally conducting piston. One compartment initially contains 1 m3 of saturated water vapor at 4 MPa and the other compartment contains 1 m3 of water vapor at 20 MPa, 800C. The piston is released and equilibrium is attained, with the piston experiencing no change of state. For the water as the system, determine (a) the final pressure, in MPa. (b) the final temperature, in C. (c) the amount of entropy produced, in kJ/K. 6.57 A system consisting of air initially at 300 K and 1 bar experiences the two different types of interactions described below. In each case, the system is brought from the initial state to a state where the temperature is 500 K, while volume remains constant. (a) The temperature rise is brought about adiabatically by stirring the air with a paddle wheel. Determine the amount of entropy produced, in kJ/kg # K. (b) The temperature rise is brought about by heat transfer from a reservoir at temperature T. The temperature at the system boundary where heat transfer occurs is also T. Plot the amount of entropy produced, in kJ/kg # K, versus T for T 500 K. Compare with the result of (a) and discuss. 6.58 A cylindrical copper rod of base area A and length L is insulated on its lateral surface. One end of the rod is in contact with a wall at temperature TH. The other end is in contact with a wall at a lower temperature TC. At steady state, the rate at

which energy is conducted into the rod from the hot wall is kA1TH TC 2 # QH L where is the thermal conductivity of the copper rod. (a) For the rod as the system, obtain an expression for the time rate of entropy production in terms of A, L, TH, TC, and k. (b) If TH 327°C, TC 77°C, # 0.4 kWm # K, A 0.1 m2, plot the heat transfer rate QH, in kW, and the time rate of entropy production, in kW/K, each versus L ranging from 0.01 to 1.0 m. Discuss. 6.59 A system undergoes a thermodynamic cycle while receiving energy by heat transfer from a tank of liquid water initially at 90C and rejecting energy by heat transfer at 15C to the surroundings. If the final water temperature is 15C, determine the minimum theoretical volume of water in the tank, m3, for the cycle to produce net work equal to 1.6 105 kJ. 6.60 The temperature of an incompressible substance of mass m and specific heat c is reduced from T0 to T (T0) by a refrigeration cycle. The cycle receives energy by heat transfer at T from the substance and discharges energy by heat transfer at T0 to the surroundings. There are no other heat transfers. Plot (WminmcT0) versus TT0 ranging from 0.8 to 1.0, where Wmin is the minimum theoretical work input required by the cycle. 6.61 The temperature of a 12-oz (0.354-L) can of soft drink is reduced from 20 to 5C by a refrigeration cycle. The cycle receives energy by heat transfer from the soft drink and discharges energy by heat transfer at 20C to the surroundings. There are no other heat transfers. Determine the minimum theoretical work input required by the cycle, in kJ, assuming the soft drink is an incompressible liquid with the properties of liquid water. Ignore the aluminum can. 6.62 As shown in Fig. P6.62, a turbine is located between two tanks. Initially, the smaller tank contains steam at 3.0 MPa, 280C and the larger tank is evacuated. Steam is allowed to flow from the smaller tank, through the turbine, and into the larger tank until equilibrium is attained. If heat transfer with the surroundings is negligible, determine the maximum theoretical work that can be developed, in kJ.

Initially: steam at 3.0 MPa, 280°C

Initially evacuated Turbine

100 m3 Figure P6.62

1000 m3

Problems: Developing Engineering Skills Applying the Entropy Balance: Control Volumes

6.63 A gas flows through a one-inlet, one-exit control# volume operating at steady state. Heat transfer at the rate Qcv takes place only at a location on the boundary where the temperature is Tb. For each of the following cases, determine whether the specific entropy of the gas at the exit is greater than, equal to, or less than the specific entropy of the gas at the inlet: # (a) no internal irreversibilities, Q# cv 0. (b) no internal irreversibilities, Q# cv 0. (c) no internal irreversibilities, # Qcv 0. (d) internal irreversibilities, Q# cv 0. (e) internal irreversibilities, Qcv 0. 6.64 Steam at 10 bar, 600C, 50 m/s enters an insulated turbine operating at steady state and exits at 0.35 bar, 100 m/s. The work developed per kg of steam flowing is claimed to be (a) 1000 kJ/kg, (b) 500 kJ/kg. Can either claim be correct? Explain. 6.65 Air enters an insulated turbine operating at steady state at 6.5 bar, 687C and exits at 1 bar, 327C. Neglecting kinetic and potential energy changes and assuming the ideal gas model, determine (a) the work developed, in kJ per kg of air flowing through the turbine. (b) whether the expansion is internally reversible, irreversible, or impossible. 6.66 Methane gas (CH4) at 280 K, 1 bar enters a compressor operating at steady state and exits at 380 K, 3.5 bar. Ignoring heat transfer with the surroundings and employing the ideal gas model with cp 1T 2 from Table A-21, determine the rate of entropy production within the compressor, in kJ/kg # K. 6.67 Figure P6.67 provides steady-state operating data for a well-insulated device with air entering at one location and exiting at another with a mass flow rate of 10 kg/s. Assuming ideal gas behavior and negligible potential energy effects, determine the direction of flow and the power, in kW.

Power shaft ? ? p = 1 bar T = 600 K V = 1000 m/s

p = 5 bar T = 900 K V = 5 m/s

Figure P6.67

6.68 Hydrogen gas (H2) at 35C and pressure p enters an insulated control volume operating at steady state for which # Wcv 0. Half of the hydrogen exits the device at 2 bar and 90C and the other half exits at 2 bar and 20C. The effects of kinetic and potential energy are negligible. Employing the ideal gas model with constant cp 14.3 kJ/kg # K, determine the minimum possible value for the inlet pressure p, in bar.

265

6.69 An inventor claims to have developed a device requiring no work input or heat transfer, yet able to produce at steady state hot and cold air streams as shown in Fig. P6.69. Employing the ideal gas model for air and ignoring kinetic and potential energy effects, evaluate this claim. · · Qcv = 0, Wcv = 0

Air at 60°C, 2.7 bar

Air at 20°C, 2.74 bar Air at 0°C, 2.7 bar Figure P6.69

6.70 Ammonia enters a valve as a saturated liquid at 7 bar with a mass flow rate of 0.06 kg/min and is steadily throttled to a pressure of 1 bar. Determine the rate of entropy production in kW/K. If the valve were replaced by a power-recovery turbine operating at steady state, determine the maximum theoretical power that could be developed, in kW. In each case, ignore heat transfer with the surroundings and changes in kinetic and potential energy. Would you recommend using such a turbine? 6.71 Air enters an insulated diffuser operating at steady state at 1 bar, 3C, and 260 m/s and exits with a velocity of 130 m/s. Employing the ideal gas model and ignoring potential energy, determine (a) the temperature of the air at the exit, in C. (b) The maximum attainable exit pressure, in bar. 6.72 According to test data, a new type of engine takes in streams of water at 200C, 3 bar and 100C, 3 bar. The mass flow rate of the higher temperature stream is twice that of the other. A single stream exits at 3.0 bar with a mass flow rate of 5400 kg/h. There is no significant heat transfer between the engine and its surroundings, and kinetic and potential energy effects are negligible. For operation at steady state, determine the maximum theoretical rate that power can be developed, in kW. 6.73 At steady state, a device receives a stream of saturated water vapor at 210C and discharges a condensate stream at 20C, 0.1 MPa while delivering energy by heat transfer at 300C. The only other energy transfer involves heat transfer at 20C to the surroundings. Kinetic and potential energy changes are negligible. What is the maximum theoretical amount of energy, in kJ per kg of steam entering, that could be delivered at 300C? 6.74 A patent application describes a device for chilling water. At steady state, the device receives energy by heat transfer at a location on its surface where the temperature is 540F and discharges energy by heat transfer to the surroundings at another location on its surface where the temperature is 100F. A warm liquid water stream enters at 100F, 1 atm and a cool stream exits at temperature T and 1 atm. The device requires no power input to operate, there are no significant effects of kinetic and potential energy, and the water can be modeled as

266

Chapter 6 Using Entropy

incompressible. Plot the minimum theoretical heat addition required, in Btu per lb of cool water exiting the device, versus T ranging from 60 to 100F. 6.75 Figure P6.75 shows a proposed device to develop power using energy supplied to the device by heat transfer from a high-temperature industrial process together with a steam input. The figure provides data for steady-state operation. All surfaces are well insulated except for the one at 527C, through which heat transfer occurs at a rate of 4.21 kW. Ignoring changes in kinetic and potential energy, evaluate the maximum theoretical power that can be developed, in kW. · Qcv = 4.21 kW

Steam at 3 bar 500°C, 1.58 kg/min

· Wcv

527°C

1

2

1 bar

Figure P6.75

6.76 Figure P6.76 shows a gas turbine power plant operating at steady state consisting of a compressor, a heat exchanger, and a turbine. Air enters the compressor with a mass flow rate of 3.9 kg/s at 0.95 bar, 22C and exits the turbine at 0.95 bar, 421C. Heat transfer to the air as it flows through the heat exchanger occurs at an average temperature of 488C. The compressor and turbine operate adiabatically. Using the ideal gas model for the air, and neglecting kinetic and potential energy effects, determine the maximum theoretical value for the net power that can be developed by the power plant, in MW.

T = 488°C Heat exchanger

Compressor

Turbine · Wnet

1 Air at 0.95 bar, 22°C

2 Air at 0.95 bar, 421°C

Figure P6.76

30-ohm resistor, T = 127°C + Air in at 15°C, 1 atm

– Air out at 25°C

Figure P6.77

The air enters the duct at 15C, 1 atm and exits at 25C with a negligible change in pressure. Kinetic and potential energy changes can be ignored. (a) For the resistor as the system, determine the rate of entropy production, in kW/K. (b) For a control volume enclosing the air in the duct and the resistor, determine the volumetric flow rate of the air entering the duct, in m3/s, and rate of entropy production, in kW/K. Why do the entropy production values of (a) and (b) differ? 6.78 For the computer of Example 4.8, determine the rate of entropy production, in W/K, when air exits at 32C. Ignore the change in pressure between the inlet and exit. 6.79 For the computer of Problem 4.70, determine the rate of entropy production, in kW/K, ignoring the change in pressure between the inlet and exit. 6.80 For the water-jacketed electronics housing of Problem 4.71, determine the rate of entropy production, in kW/K, when water exits at 26C. 6.81 For the electronics-laden cylinder of Problem 4.73, determine the rate of entropy production, in W/K, when air exits at 40C with a negligible change in pressure. Assume convective cooling occurs on the outer surface of the cylinder in accord with hA 3.4 W/K, where h is the heat transfer coefficient and A is the surface area. The temperature of the surroundings away from the vicinity of the cylinder is 25C. 6.82 Steam enters a horizontal 15-cm-diameter pipe as a saturated vapor at 5 bar with a velocity of 10 m/s and exits at 4.5 bar with a quality of 95%. Heat transfer from the pipe to the surroundings at 300 K takes place at an average outer surface temperature of 400 K. For operation at steady state, determine (a) the velocity at the exit, in m/s. (b) the rate of heat transfer from the pipe, in kW. (c) the rate of entropy production, in kW/K, for a control volume comprising only the pipe and its contents. (d) the rate of entropy production, in kW/K, for an enlarged control volume that includes the pipe and enough of its immediate surroundings so that heat transfer from the control volume occurs at 300 K. Why do the answers of parts (c) and (d) differ?

6.77 Figure P6.77 shows a 30-ohm electrical resistor located in an insulated duct carrying a stream of air. At steady state, an electric current of 15 amp passes through the resistor, whose temperature remains constant at 127C.

6.83 Steam enters a turbine operating at steady state at a pressure of 3 MPa, a temperature of 400C, and a velocity of 160 m/s. Saturated vapor exits at 100C, with a velocity of 100 m/s. Heat transfer from the turbine to its surroundings

Problems: Developing Engineering Skills

takes place at the rate of 30 kJ per kg of steam at a location where the average surface temperature is 350 K. (a) For a control volume including only the turbine and its contents, determine the work developed, in kJ, and the rate at which entropy is produced, in kJ/K, each per kg of steam flowing. (b) The steam turbine of part (a) is located in a factory where the ambient temperature is 27C. Determine the rate of entropy production, in kJ/K per kg of steam flowing, for an enlarged control volume that includes the turbine and enough of its immediate surroundings so that heat transfer takes place from the control volume at the ambient temperature. Explain why the entropy production value of part (b) differs from that calculated in part (a). 6.84 Carbon dioxide (CO2) enters a nozzle operating at steady state at 28 bar, 267C, and 50 m /s. At the nozzle exit, the conditions are 1.2 bar, 67C, 580 m /s, respectively. (a) For a control volume enclosing the nozzle only, determine the heat transfer, in kJ, and the change in specific entropy, in kJ/K, each per kg of carbon dioxide flowing through the nozzle. What additional information would be required to evaluate the rate of entropy production? (b) Evaluate the rate of entropy production, in kJ/K per kg of carbon dioxide flowing, for an enlarged control volume enclosing the nozzle and a portion of its immediate surroundings so that the heat transfer occurs at the ambient temperature, 25C. 6.85 Air enters a compressor operating at steady state at 1 bar, 22C with a volumetric flow rate of 1 m3/min and is compressed to 4 bar, 177C. The power input is 3.5 kW. Employing the ideal gas model and ignoring kinetic and potential energy effects, obtain the following results: (a) For a control volume enclosing the compressor only, determine the heat transfer rate, in kW, and the change in specific entropy from inlet to exit, in kJ/kg # K. What additional information would be required to evaluate the rate of entropy production? (b) Calculate the rate of entropy production, in kW/K, for an enlarged control volume enclosing the compressor and a portion of its immediate surroundings so that heat transfer occurs at the ambient temperature, 22C. 6.86 Air is compressed in an axial-flow compressor operating at steady state from 27C, 1 bar to a pressure of 2.1 bar. The work input required is 94.6 kJ per kg of air flowing through the compressor. Heat transfer from the compressor occurs at the rate of 14 kJ per kg at a location on the compressor’s surface where the temperature is 40C. Kinetic and potential energy changes can be ignored. Determine (a) the temperature of the air at the exit, in C. (b) the rate at which entropy is produced within the compressor, in kJ/K per kg of air flowing. 6.87 Ammonia enters a counterflow heat exchanger at 20C, with a quality of 35%, and leaves as saturated vapor at 20C. Air at 300 K, 1 atm enters the heat exchanger in a separate

267

stream with a flow rate of 4 kg/s and exits at 285 K, 0.98 atm. The heat exchanger is at steady state, and there is no appreciable heat transfer from its outer surface. Neglecting kinetic and potential energy effects, determine the mass flow rate of the ammonia, in kg/s, and the rate of entropy production within the heat exchanger, in kW/K. 6.88 A counterflow heat exchanger operates at steady state with negligible kinetic and potential energy effects. In one stream, liquid water enters at 15C and exits at 23C with a negligible change in pressure. In the other stream, Refrigerant 22 enters at 12 bar, 90C with a mass flow rate of 150 kg/h and exits at 12 bar, 28C. Heat transfer from the outer surface of the heat exchanger can be ignored. Determine (a) the mass flow rate of the liquid water stream, in kg/h. (b) the rate of entropy production within the heat exchanger, in kW/K. 6.89 Steam at 0.7 MPa, 355C enters an open feedwater heater operating at steady state. A separate stream of liquid water enters at 0.7 MPa, 35C. A single mixed stream exits as saturated liquid at pressure p. Heat transfer with the surroundings and kinetic and potential energy effects can be ignored. (a) If p 0.7 MPa, determine the ratio of the mass flow rates of the incoming streams and the rate at which entropy is produced within the feedwater heater, in kJ/K per kg of liquid exiting. (b) Plot the quantities of part (a), each versus pressure p ranging from 0.6 to 0.7 MPa. 6.90 Air as an ideal gas flows through the compressor and heat exchanger shown in Fig. P6.90. A separate liquid water stream also flows through the heat exchanger. The data given are for operation at steady state. Stray heat transfer to the surroundings can be neglected, as can all kinetic and potential energy changes. Determine (a) the compressor power, in kW, and the mass flow rate of the cooling water, in kg/s. (b) the rates of entropy production, each in kW/K, for the compressor and heat exchanger.

Compressor

Air in

1

p1 = 96 kPa T1 = 27°C (AV)1 = 26.91 m3/min

Figure P6.90

TB = 40°C Water out

(A)

(B)

Heat exchanger 2

p2 = 230 kPa T1 = 127°C

TA = 25°C Water in

3 T3 = 77°C p3 = p2

268

Chapter 6 Using Entropy

6.91 Determine the rates of entropy production, each in kW/K, for the turbines and the heat exchanger of Problem 4.82. Place these in rank order beginning with the component contributing most to inefficient operation of the overall system. 6.92 Determine the rates of entropy production, each in kW/K, for the turbine, condenser, and pump of Problem 4.85. Place these in rank order beginning with the component contributing most to inefficient operation of the overall system. 6.93 For the control volume of Example 4.12, determine the amount of entropy produced during filling, in kJ/K. Repeat for the case where no work is developed by the turbine. 6.94 Steam is contained in a large vessel at 100 lbf/in.2, 450F. Connected to the vessel by a valve is an initially evacuated tank having a volume of 1 ft3. The valve is opened until the tank is filled with steam at pressure p. The filling is adiabatic, kinetic and potential energy effects are negligible, and the state of the large vessel remains constant. (a) If p 100 lbf/in.2, determine the final temperature of the steam within the tank, in F, and the amount of entropy produced within the tank, in Btu/R. (b) Plot the quantities of part (a) versus presssure p ranging from 10 to 100 lbf/in.2 6.95 A well-insulated rigid tank of volume 10 m3 is connected by a valve to a large-diameter supply line carrying air at 227C and 10 bar. The tank is initially evacuated. Air is allowed to flow into the tank until the tank pressure is p. Using the ideal gas model with constant specific heat ratio k, plot tank temperature, in K, the mass of air in the tank, in kg, and the amount of entropy produced, in kJ/K, versus p in bar. Using Isentropic Processes/Efficiencies

6.96 A piston–cylinder assembly initially contains 0.1 m3 of carbon dioxide gas at 0.3 bar and 400 K. The gas is compressed isentropically to a state where the temperature is 560 K. Employing the ideal gas model and neglecting kinetic and potential energy effects, determine the final pressure, in bar, and the work in kJ, using (a) data from Table A-23. (b) IT (c) a constant specific heat ratio from Table A-20 at the mean temperature, 480 K. (d) a constant specific heat ratio from Table A-20 at 300 K. 6.97 Air enters a turbine operating at steady state at 6 bar and 1100 K and expands isentropically to a state where the temperature is 700 K. Employing the ideal gas model and ignoring kinetic and potential energy changes, determine the pressure at the exit, in bar, and the work, in kJ per kg of air flowing, using (a) data from Table A-22. (b) IT. (c) a constant specific heat ratio from Table A-20 at the mean temperature, 900 K. (d) a constant specific heat ratio from Table A-20 at 300 K.

6.98 Methane (CH4) undergoes an isentropic expansion from an initial state where the temperature is 1000 K and the pressure is 5 bar to a final state where the temperature is T and the pressure is p. Using the ideal gas model together with cp (T ) from Table A-21, determine (a) p when T 500 K (b) T when p 1 bar. (c) Check the results of parts (a) and (b) using IT. 6.99 An ideal gas with constant specific heat ratio k enters a nozzle operating at steady state at pressure p1, temperature T1, and velocity V1. The air expands isentropically to a pressure of p2. (a) Develop an expression for the velocity at the exit, V2, in terms of k, R, V1, T1, p1, and p2, only. (b) For V1 0, T1 1000 K, plot V2 versus p2p1 for selected values of k ranging from 1.2 to 1.4. 6.100 An ideal gas undergoes a polytropic process from T1, p1 to a state where the temperature is T2. (a) Derive an expression for the change in specific entropy in terms of n, R, T1, T2, s(T1), and s(T2). (b) Using the result of part (a), develop an expression for n if s 1 s2 . 6.101 A rigid well-insulated tank having a volume of 0.2 m3 is filled initially with Refrigerant 134a vapor at a pressure of 10 bar and a temperature of 40C. A leak develops and refrigerant slowly escapes until the pressure within the tank becomes 1 bar. Determine (a) the final temperature of the refrigerant within the tank, in C. (b) the amount of mass that exits the tank, in kg. 6.102 A rigid tank is filled initially with 5.0 kg of air at a pressure of 0.5 MPa and a temperature of 500 K. The air is allowed to discharge through a turbine into the atmosphere, developing work until the pressure in the tank has fallen to the atmospheric level of 0.1 MPa. Employing the ideal gas model for the air, determine the maximum theoretical amount of work that could be developed, in kJ. Ignore heat transfer with the atmosphere and changes in kinetic and potential energy. 6.103 Air enters a 3600-kW turbine operating at steady state with a mass flow rate of 18 kg/s at 800C, 3 bar and a velocity of 100 m/s. The air expands adiabatically through the turbine and exits at a velocity of 150 m/s. The air then enters a diffuser where it is decelerated isentropically to a velocity of 10 m/s and a pressure of 1 bar. Employing the ideal gas model, determine (a) the pressure and temperature of the air at the turbine exit, in bar and C, respectively. (b) the rate of entropy production in the turbine, in kW/K. (c) Show the processes on a T–s diagram. 6.104 Steam at 5 MPa and 600C enters an insulated turbine operating at steady state and exits as saturated vapor at 50 kPa. Kinetic and potential energy effects are negligible.

Problems: Developing Engineering Skills

Determine (a) the work developed by the turbine, in kJ per kg of steam flowing through the turbine. (b) the isentropic turbine efficiency. 6.105 Nitrogen (N2) at 3.8 atm and 170C enters an insulated turbine operating at steady state and expands to 1 atm. If the isentropic turbine efficiency is 83.2%, determine the temperature at the turbine exit, in C, using the ideal gas model for the nitrogen and ignoring kinetic and potential energy changes. 6.106 Figure P6.106 provides steady-state operating data for a throttling valve in parallel with a steam turbine having an isentropic turbine efficiency of 90%. The streams exiting the valve and the turbine mix in a mixing chamber. Heat transfer with the surroundings and changes in kinetic and potential energy can be neglected. Determine (a) the power developed by the turbine, in horsepower. (b) the mass flow rate through the valve, in lb/s. (c) Locate the four numbered states on an h–s diagram. ηt = 90% Turbine

3

p3 = 1 MPa

Mixing chamber

1 p1 = 4 MPa T1 = 360°C m· 1 = 11 kg/s

Valve

2

4 p4 = 1 MPa T4 = 240°C

p2 = 1 MPa

Figure P6.106

6.107 Water vapor enters an insulated nozzle operating at steady state at 0.7 MPa, 320C, 35 m/s and expands to 0.15 MPa. If the isentropic nozzle efficiency is 94%, determine the velocity at the exit, in m /s. 6.108 Argon enters an insulated nozzle at 2.77 bar, 1300 K, 10 m/s and exits at 1 bar, 645 m/s. For steady-state operation, determine (a) the exit temperature, in K. (b) the isentropic nozzle efficiency. (c) the rate of entropy production, in kJ/K per kg of argon flowing. 6.109 Air enters an insulated compressor operating at steady state at 0.95 bar, 27C with a mass flow rate of 4000 kg/h and exits at 8.7 bar. Kinetic and potential energy effects are negligible. (a) Determine the minimum theoretical power input required, in kW, and the corresponding exit temperature, in C. (b) If the exit temperature is 347C, determine the power input, in kW, and the isentropic compressor efficiency. 6.110 Refrigerant 134a enters a compressor operating at steady state as saturated vapor at 4C and exits at a pressure of

269

14 bar. The isentropic compressor efficiency is 75%. Heat transfer between the compressor and its surroundings can be ignored. Kinetic and potential energy effects are also negligible. Determine (a) the exit temperature, in C. (b) the work input, in kJ per kg of refrigerant flowing. 6.111 Air enters an insulated compressor operating at steady state at 1 bar, 350 K with a mass flow rate of 1 kg/s and exits at 4 bar. The isentropic compressor efficiency is 82%. Determine the power input, in kW, and the rate of entropy production, in kW/K, using the ideal gas model with (a) data from Table A-22. (b) IT. (c) a constant specific heat ratio, k 1.39. 6.112 A compressor operating at steady state takes in atmospheric air at 20C, 1 bar at a rate of 1 kg/s and discharges air at 5 bar. Plot the power required, in kW, and the exit temperature, in C, versus the isentropic compressor efficiency ranging from 70 to 100%. Assume the ideal gas model for the air and neglect heat transfer with the surroundings and changes in kinetic and potential energy. 6.113 In a gas turbine operating at steady state, air enters the compressor with a mass flow rate of 5 kg/s at 0.95 bar and 22C and exits at 5.7 bar. The air then passes through a heat exchanger before entering the turbine at 1100 K, 5.7 bar. Air exits the turbine at 0.95 bar. The compressor and turbine operate adiabatically and kinetic and potential energy effects can be ignored. Determine the net power developed by the plant, in kW, if (a) the compressor and turbine operate without internal irreversibilities. (b) the compressor and turbine isentropic efficiencies are 82 and 85%, respectively. Analyzing Internally Reversible Flow Processes

6.114 Air enters a compressor operating at steady state at 17C, 1 bar and exits at a pressure of 5 bar. Kinetic and potential energy changes can be ignored. If there are no internal irreversibilities, evaluate the work and heat transfer, each in kJ per kg of air flowing, for the following cases: (a) isothermal compression. (b) polytropic compression with n 1.3. (c) adiabatic compression. Sketch the processes on p–v and T–s coordinates and associate areas on the diagrams with the work and heat transfer in each case. Referring to your sketches, compare for these cases the magnitudes of the work, heat transfer, and final temperatures, respectively. 6.115 Air enters a compressor operating at steady state with a volumetric flow rate of 8 m3/min at 23C, 0.12 MPa. The air is compressed isothermally without internal irreversibilities, exiting at 1.5 MPa. Kinetic and potential energy effects can be ignored. Evaluate the work required and the heat transfer, each in kW.

270

Chapter 6 Using Entropy

6.116 Refrigerant 134a enters a compressor operating at steady state at 1.8 bar, 10C with a volumetric flow rate of 2.4 102 m3/s. The refrigerant is compressed to a pressure of 9 bar in an internally reversible process according to pv1.04 constant. Neglecting kinetic and potential energy effects, determine (a) the power required, in kW. (b) the rate of heat transfer, in kW. 6.117 Compare the work required at steady state to compress water vapor isentropically to 3 MPa from the saturated vapor state at 0.1 MPa to the work required to pump liquid water isentropically to 3 MPa from the saturated liquid state at 0.1 MPa, each in kJ per kg of water flowing through the device. Kinetic and potential energy effects can be ignored. 6.118 An electrically-driven pump operating at steady state draws water from a pond at a pressure of 1 bar and a rate of 40 kg/s and delivers the water at a pressure of 4 bar. There is no significant heat transfer with the surroundings, and changes in kinetic and potential energy can be neglected. The isentropic pump efficiency is 80%. Evaluating electricity at 8 cents per kW # h, estimate the hourly cost of running the pump. 6.119 Figure P6.119 shows three devices operating at steady state: a pump, a boiler, and a turbine. The turbine provides the power required to drive the pump and also supplies power to other devices. For adiabatic operation of the pump and turbine, and ignoring kinetic and potential energy effects, determine, in kJ per kg of steam flowing · Qin

3

2 8 bar Boiler

8 bar, saturated vapor Turbine ηt = 90%

· Wnet

Pump ηp = 70% 1 Feedwater 1 bar, 30°C

4 Steam 1 bar

Figure P6.119

(a) the work required by the pump. (b) the net work developed by the turbine. (c) the heat transfer to the boiler. 6.120 As shown in Fig. P6.120, water flows from an elevated reservoir through a hydraulic turbine. The pipe diameter is constant, and operation is at steady state. Estimate the minimum mass flow rate, in kg/s, that would be required for a turbine power output of 1 MW. The local acceleration of gravity is 9.8 m/s2.

1

p1 = 1.3 bar p2 = 1.0 bar

· Wt = 1 MW 100 m

2

5m

Figure P6.120

6.121 A 5-kilowatt pump operating at steady state draws in liquid water at 1 bar, 15C and delivers it at 5 bar at an elevation 6 m above the inlet. There is no significant change in velocity between the inlet and exit, and the local acceleration of gravity is 9.8 m/s2. Would it be possible to pump 7.5 m3 in 10 min or less? Explain. 6.122 A 4-kW pump operating at steady state draws in liquid water at 1 bar, 16C with a mass flow rate of 4.5 kg/s. There are no significant kinetic and potential energy changes from inlet to exit and the local acceleration of gravity is 9.81 m/s2. Would it be possible for the pump to deliver water at a pressure of 10 bar? Explain. 6.123 Carbon monoxide enters a nozzle operating at steady state at 5 bar, 200C, 1 m/s and undergoes a polytropic expansion to 1 bar with n 1.2. Using the ideal gas model and ignoring potential energy effects, determine (a) the exit velocity, in m/s. (b) the rate of heat transfer between the gas and its surroundings, in kJ per kg of gas flowing.

Design & Open Ended Problems: Exploring Engineering Practice 6.1D Of increasing interest today are turbines, pumps, and heat exchangers that weigh less than 1 gram and have volumes of 1 cubic centimeter or less. Although many of the same design considerations apply to such micromachines as to corresponding full-scale devices, others do not. Of particular interest to designers is the impact of irreversibilities on the performance of such tiny devices. Write a report discussing the influence of

irreversibilities related to heat transfer and friction on the design and operation of micromachines. 6.2D The growth of living organisms has been studied and interpreted thermodynamically by I. Prigogine and others, using the entropy and entropy production concepts. Write a paper summarizing the main findings of these investigations.

Design & Open Ended Problems: Exploring Engineering Practice

271

6.3D The theoretical steam rate is the quantity of steam required to produce a unit amount of work in an ideal turbine. The Theoretical Steam Rate Tables published by The American Society of Mechanical Engineers give the theoretical steam rate in lb per kW # h. To determine the actual steam rate, the theoretical steam rate is divided by the isentropic turbine efficiency. Why is the steam rate a significant quantity? Discuss how the steam rate is used in practice.

6.7D Water is to be pumped from a lake to a reservoir located on a bluff 290 ft above. According to the specifications, the piping is Schedule 40 steel pipe having a nominal diameter of 1 inch and the volumetric flow rate is 10 gal/min. The total length of pipe is 580 ft. A centrifugal pump is specified. Estimate the electrical power required by the pump, in kW. Is a centrifugal pump a good choice for this application? What precautions should be taken to avoid cavitation?

6.4D Figure P6.4D illustrates an ocean thermal energy conversion (OTEC) power plant that generates power by exploiting the naturally occurring decrease of the temperature of ocean water with depth. Warm surface water enters the evaporator # with a mass flow rate of mw at temperature Tw 28C and exits at T1 Tw. Cool water brought from a depth of 600 m # enters the condenser with a mass flow rate of mc at temperature Tc 5C and exits at T2 Tc. The pumps for the ocean water flows and other auxiliary equipment typically require 15% of the gross power generated. Estimate the mass flow rates # # mw and mc, in kg/s, for a desired net power output of 125 MW.

6.8D Elementary thermodynamic modeling, including the use of the temperature–entropy diagram for water and a form of the Bernoulli equation has been employed to study certain types of volcanic eruptions. (See L. G. Mastin, “Thermodynamics of Gas and Steam-Blast Eruptions,” Bull. Volcanol., 57, 85–98, 1995.) Write a report critically evaluating the underlying assumptions and application of thermodynamic principles, as reported in the article.

m· w , Tw

T1 Evaporator

Turbine

Pump

Condenser m· c , Tc

T2

Figure P6.4D

6.5D How might the principal sources of irreversibility be reduced for the heat pump components analyzed in Example 6.8? Carefully consider the effects of a change in any one component on the performance of each of the others and on the heat pump as a whole. Consider the economic consequences of proposed changes. Summarize your findings in a memorandum. 6.6D The Bernoulli equation can be generalized to include the effects of fluid friction in piping networks in terms of the concept of head loss. Investigate the head loss formulation as it applies to incompressible flows through common pipes and fittings. Using this information, estimate the head, in ft, a booster pump would need to overcome because of friction in a 2-in. galvanized steel pipe feeding water to the top floor of a 20-story building.

6.9D An inventor claims to have conceived of a second lawchallenging heat engine. (See H. Apsden, “The Electronic Heat Engine,” Proceedings 27th International Energy Conversion Engineering Conference, 4.357–4.363, 1992. Also see U.S. Patent No. 5,101,632.) By artfully using mirrors the heat engine would “efficiently convert abundant environmental heat energy at the ambient temperature to electricity.” Write a paper explaining the principles of operation of the device. Does this invention actually challenge the second law of thermodynamics? Does it have commercial promise? Discuss. 6.10D Noting that contemporary economic theorists often draw on principles from mechanics such as conservation of energy to explain the workings of economies, N. Georgescu-Roegen and like-minded economists have called for the use of principles from thermodynamics in economics. According to this view, entropy and the second law of thermodynamics are relevant for assessing not only the exploitation of natural resources for industrial and agricultural production but also the impact on the natural environment of wastes from such production. Write a paper in which you argue for, or against, the proposition that thermodynamics is relevant to the field of economics. 6.11D Roll Over Boltzman (see box Sec. 6.5). The new statistical definition of entropy is said to give insights about physical systems verging on chaos. Investigate what is meant by chaos in this context. Write a report including at least three references. 6.12D Star Guides Consumer Choices (see box Sec. 6.8). Using the ENERGY STAR® Home Improvement Toolbox, obtain a rank-ordered list of the top three cost-effective improvements that would enhance the overall energy efficiency of your home. Develop a plan to implement the improvements. Write a report including at least three references.

C H A P

7 chapter objective

T E R

Exergy Analysis

E N G I N E E R I N G C O N T E X T The objective of this chapter is to introduce exergy analysis, a method that uses the conservation of mass and conservation of energy principles together with the second law of thermodynamics for the design and analysis of thermal systems. Another term frequently used to identify exergy analysis is availability analysis. The importance of developing thermal systems that make effective use of nonrenewable resources such as oil, natural gas, and coal is apparent. The method of exergy analysis is particularly suited for furthering the goal of more efficient resource use, since it enables the locations, types, and true magnitudes of waste and loss to be determined. This information can be used to design thermal systems, guide efforts to reduce sources of inefficiency in existing systems, and evaluate system economics.

7.1 Introducing Exergy

Energy is conserved in every device or process. It cannot be destroyed. Energy entering a system with fuel, electricity, flowing streams of matter, and so on can be accounted for in the products and by-products. However, the energy conservation idea alone is inadequate for depicting some important aspects of resource utilization. for example. . . Figure 7.1a shows an isolated system consisting initially of a small container of fuel surrounded by air in abundance. Suppose the fuel burns (Fig. 7.1b) so that finally there is a slightly warm mixture of combustion products and air as shown in Fig. 7.1c. Although the total quantity of energy associated with the system would be unchanged, the initial fuel–air combination would have a greater economic value and be intrinsically more useful than the final warm mixture. For instance, the fuel might be used in some device to generate electricity or produce superheated steam, whereas the uses to which the slightly warm combustion products can be put would be far more limited in scope. We might say that the system has a greater potential for use initially than it has finally. Since nothing but a final warm mixture would be achieved in the process, this potential would be largely wasted. More precisely, the initial potential would be largely destroyed because of the irreversible nature of the process.

272

7.2 Defining Exergy Boundary of the isolated system Air at temperature Ti

Fuel

(a)

Air and combustion products at temperature Ti + dT

Fuel

(b)

(c)

Time Energy quantity constant Potential for use decreases Figure 7.1

Illustration used to introduce exergy.

Anticipating the main results of this chapter, we can read exergy as potential for use wherever it appears in the text. The foregoing example illustrates that, unlike energy, exergy is not conserved. Subsequent discussion shows that exergy not only can be destroyed by irreversibilities but also can be transferred to a system or from a system, as in losses accompanying heat transfers to the surroundings. Improved resource utilization can be realized by reducing exergy destruction within a system and/or losses. An objective in exergy analysis is to identify sites where exergy destructions and losses occur and rank order them for significance. This allows attention to be centered on the aspects of system operation that offer the greatest opportunities for improvement.

7.2 Defining Exergy

The basis for the exergy concept is present in the introduction to the second law provided in Chap. 5. A principal conclusion of Sec. 5.1 is that an opportunity exists for doing work whenever two systems at different states are brought into communication. In principle, work can be developed as the systems are allowed to come into equilibrium. When one of the two systems is a suitably idealized system called an exergy reference environment or simply, an environment, and the other is some system of interest, exergy is the maximum theoretical work obtainable as they interact to equilibrium. The definition of exergy will not be complete, however, until we define the reference environment and show how numerical values for exergy can be determined. These tasks are closely related because the numerical value of exergy depends on the state of a system of interest, as well as the condition of the environment. 7.2.1 Exergy Reference Environment Any system, whether a component of a larger system such as a steam turbine in a power plant or the larger system itself (power plant), operates within surroundings of some kind. It is important to distinguish between the environment used for calculating exergy and a system’s surroundings. Strictly speaking, the term surroundings refers to everything not included in the system. However, when considering the exergy concept, we distinguish between the

exergy reference environment exergy

273

274

Chapter 7 Exergy Analysis Environment – Intensive properties of this portion of the surroundings are not affected by any process within the power plant or its immediate surroundings

Stack gases

Immediate surroundings–intensive properties may vary in interactions with power plant

Delineates the immediate plant surroundings from the environment

Boundary of plant Fuel Power

Power

Air Cooling water out

Figure 7.2

Cooling water in

River or other body of water – the portion not interacting with the power plant would be in the environment

Schematic of a power plant and its surroundings.

immediate surroundings, where intensive properties may vary during interactions with the system, and the larger portion of the surroundings at a distance, where the intensive properties are unaffected by any process involving the system and its immediate surroundings. The term environment identifies this larger portion of the surroundings. for example. . . Fig. 7.2 illustrates the distinction between a system consisting of a power plant, its immediate surroundings, and the environment. In this case, the environment includes portions of the surrounding atmosphere and the river at a distance from the power plant. Interactions between the power plant and its immediate surroundings have no influence on the temperature, pressure, or other intensive properties of the environment. MODELING THE ENVIRONMENT

The physical world is complicated, and to include every detail in an analysis is not practical. Accordingly, in describing the environment, simplifications are made and a model results. The validity and utility of an analysis using any model are, of course, restricted by the idealizations made in formulating the model. In this book the environment is regarded to be a simple compressible system that is large in extent and uniform in temperature, T0, and pressure, p0. In keeping with the idea that the environment represents a portion of the physical world, the values for both p0 and T0 used throughout a particular analysis are normally taken as typical environmental conditions, such as 1 atm and 25C. The intensive properties of each phase of the environment are uniform and do not change significantly as a result of any process under consideration. The environment is also regarded as free of irreversibilities. All significant irreversibilities are located within the system and its immediate surroundings. Although its intensive properties do not change, the environment can experience changes in its extensive properties as a result of interactions with other systems. Changes in the extensive properties internal energy Ue, entropy Se, and volume Ve of the environment are

7.2 Defining Exergy

related through the first T dS equation, Eq. 6.10. Since T0 and p0 are constant, Eq. 6.10 takes the form ¢Ue T0 ¢Se p0 ¢Ve

(7.1)

In this chapter kinetic and potential energies are evaluated relative to the environment, all parts of which are considered to be at rest with respect to one another. Accordingly, as indicated by the foregoing equation, a change in the energy of the environment can be a change in its internal energy only. Equation 7.1 is used below to develop an expression for evaluating exergy. In Chap. 13 the environment concept is extended to allow for the possibility of chemical reactions, which are excluded from the present considerations. 7.2.2 Dead State Let us consider next the concept of the dead state, which is also important in completing our understanding of the property exergy. If the state of a fixed quantity of matter, a closed system, departs from that of the environment, an opportunity exists for developing work. However, as the system changes state toward that of the environment, the opportunity diminishes, ceasing to exist when the two are in equilibrium with one another. This state of the system is called the dead state. At the dead state, the fixed quantity of matter under consideration is imagined to be sealed in an envelope impervious to mass flow, at rest relative to the environment, and internally in equilibrium at the temperature T0 and pressure p0 of the environment. At the dead state, both the system and environment possess energy, but the value of exergy is zero because there is no possibility of a spontaneous change within the system or the environment, nor can there be an interaction between them. With the introduction of the concepts of environment and dead state, we are in a position to show how a numerical value can be determined for exergy. This is considered next.

dead state

7.2.3 Evaluating Exergy The exergy of a system, E, at a specified state is given by the expression E 1E U0 2 p0 1V V0 2 T0 1S S0 2

(7.2) exergy of a system

Wc Closed system

System boundary

Boundary of the combined system. Heat and work interactions with the environment

Environment at T0 , p0

Figure 7.3

Combined system of closed system and environment.

275

276

Chapter 7 Exergy Analysis

METHODOLOGY UPDATE

In this book, E and e are used for exergy and specific exergy, respectively, while E and e denote energy and specific energy, respectively. Such notation is in keeping with standard practice. The appropriate concept, exergy or energy, will be clear in context. Still, care is required to avoid mistaking the symbols for these concepts.

where E( U KE PE), V, and S denote, respectively, the energy, volume, and entropy of the system, and U0, V0, and S0 are the values of the same properties if the system were at the dead state. By inspection of Eq. 7.2, the units of exergy are seen to be the same as those of energy. Equation 7.2 can be derived by applying energy and entropy balances to the combined system shown in Fig. 7.3, which consists of a closed system and an environment. (See box).

E VA L U AT I N G T H E E X E R G Y O F A S Y S T E M

Referring to Fig. 7.3, exergy is the maximum theoretical work that could be done by the combined system if the closed system were to come into equilibrium with the environment—that is, if the closed system passed to the dead state. Since the objective is to evaluate the maximum work that could be developed by the combined system, the boundary of the combined system is located so that the only energy transfers across it are work transfers of energy. This ensures that the work developed by the combined system is not affected by heat transfer to or from it. Moreover, although the volumes of the closed system and the environment can vary, the boundary of the combined system is located so that the total volume of the combined system remains constant. This ensures that the work developed by the combined system is fully available for lifting a weight in its surroundings, say, and is not expended in merely displacing the surroundings of the combined system. Let us now apply an energy balance to evaluate the work developed by the combined system. ENERGY BALANCE.

An energy balance for the combined system reduces to 0

¢Ec Qc Wc

(7.3)

where Wc is the work developed by the combined system, and ¢Ec is the energy change of the combined system, equal to the sum of the energy changes of the closed system and the environment. The energy of the closed system initially is denoted by E, which includes the kinetic energy, potential energy, and internal energy of the system. Since the kinetic energy and potential energy are evaluated relative to the environment, the energy of the closed system when at the dead state would be just its internal energy, U0. Accordingly, ¢Ec can be expressed as ¢Ec 1U0 E2 ¢Ue

Using Eq. 7.1 to replace ¢Ue, the expression becomes

¢Ec 1U0 E2 1T0 ¢Se p0 ¢Ve 2

(7.4)

Substituting Eq. 7.4 into Eq. 7.3 and solving for Wc gives

Wc 1E U0 2 1T0 ¢Se p0 ¢Ve 2

As noted previously, the total volume of the combined system is constant. Hence, the change in volume of the environment is equal in magnitude but opposite in sign to the volume change of the closed system: Ve (V0 V ).With this substitution, the above expression for work becomes Wc 1E U0 2 p0 1V V0 2 T0 ¢Se

(7.5)

This equation gives the work developed by the combined system as the closed system passes to the dead state while interacting only with the environment. The maximum theoretical value for the work is determined using the entropy balance as follows.

7.2 Defining Exergy

ENTROPY BALANCE.

277

The entropy balance for the combined system reduces to give ¢Sc sc

where the entropy transfer term is omitted because no heat transfer takes place across the boundary of the combined system, and sc accounts for entropy production due to irreversibilities as the closed system comes into equilibrium with the environment. ¢Sc is the entropy change of the combined system, equal to the sum of the entropy changes for the closed system and environment, respectively, ¢Sc 1S0 S2 ¢Se where S and S0 denote the entropy of the closed system at the given state and the dead state, respectively. Combining the last two equations 1S0 S2 ¢Se sc

(7.6)

Eliminating ¢Se between Eqs. 7.5 and 7.6 results in Wc 1E U0 2 p0 1V V0 2 T0 1S S0 2 T0sc

(7.7)

The value of the underlined term in Eq. 7.7 is determined by the two end states of the closed system—the given state and the dead state—and is independent of the details of the process linking these states. However, the value of the term T0sc depends on the nature of the process as the closed system passes to the dead state. In accordance with the second law, T0sc is positive when irreversibilities are present and vanishes in the limiting case where there are no irreversibilities. The value of T0sc cannot be negative. Hence, the maximum theoretical value for the work of the combined system is obtained by setting T0sc to zero in Eq. 7.7. By definition, the extensive property exergy, E, is this maximum value. Accordingly, Eq. 7.2 is seen to be the appropriate expression for evaluating exergy.

7.2.4 Exergy Aspects In this section, we consider several important aspects of the exergy concept, beginning with the following:

Exergy is a measure of the departure of the state of a system from that of the environment. It is therefore an attribute of the system and environment together. However, once the environment is specified, a value can be assigned to exergy in terms of property values for the system only, so exergy can be regarded as a property of the system. The value of exergy cannot be negative. If a system were at any state other than the dead state, the system would be able to change its condition spontaneously toward the dead state; this tendency would cease when the dead state was reached. No work must be done to effect such a spontaneous change. Accordingly, any change in state of the system to the dead state can be accomplished with at least zero work being developed, and thus the maximum work (exergy) cannot be negative. Exergy is not conserved but is destroyed by irreversibilities. A limiting case is when exergy is completely destroyed, as would occur if a system were permitted to undergo a spontaneous change to the dead state with no provision to obtain work. The potential to develop work that existed originally would be completely wasted in such a spontaneous process.

E Constantexergy line

p T

E = 0 at T0, p0

278

Chapter 7 Exergy Analysis

Exergy has been viewed thus far as the maximum theoretical work obtainable from the combined system of system plus environment as a system passes from a given state to the dead state while interacting with the environment only. Alternatively, exergy can be regarded as the magnitude of the minimum theoretical work input required to bring the system from the dead state to the given state. Using energy and entropy balances as above, we can readily develop Eq. 7.2 from this viewpoint. This is left as an exercise.

Although exergy is an extensive property, it is often convenient to work with it on a unit mass or molar basis. The specific exergy on a unit mass basis, e, is given by e 1e u0 2 p0 1v v0 2 T0 1s s0 2

(7.8)

where e, v, and s are the specific energy, volume, and entropy, respectively, at a given state; u0, v0, and s0 are the same specific properties evaluated at the dead state. With e u V 22 gz, e 3 1u V2 2 gz2 u0 4 p0 1v v0 2 T0 1s s0 2

and the expression for the specific exergy becomes e 1u u0 2 p0 1v v0 2 T0 1s s0 2 V22 gz

specific exergy

(7.9)

By inspection, the units of specific exergy are the same as those of specific energy. Also note that the kinetic and potential energies measured relative to the environment contribute their full values to the exergy magnitude, for in principle each could be completely converted to work were the system brought to rest at zero elevation relative to the environment. Using Eq. 7.2, we can determine the change in exergy between two states of a closed system as the difference exergy change

E2 E1 1E2 E1 2 p0 1V2 V1 2 T0 1S2 S1 2

(7.10)

where the values of p0 and T0 are determined by the state of the environment. When a system is at the dead state, it is in thermal and mechanical equilibrium with the environment, and the value of exergy is zero. We might say more precisely that the thermomechanical contribution to exergy is zero. This modifying term distinguishes the exergy concept of the present chapter from a more general concept introduced in Sec. 13.6, where the contents of a system at the dead state are permitted to enter into chemical reaction with environmental components and in so doing develop additional work. As illustrated by subsequent discussions, the thermomechanical exergy concept suffices for a wide range of thermodynamic evaluations. 7.2.5 Illustrations We conclude this introduction to the exergy concept with examples showing how to calculate exergy and exergy change. To begin, observe that the exergy of a system at a specified state requires properties at that state and at the dead state. for example. . . let us use Eq. 7.9 to determine the specific exergy of saturated water vapor at 120C, having a velocity of 30 m/s and an elevation of 6 m, each relative to an exergy reference environment where T0 298 K (25C), p0 1 atm, and g 9.8 m/s2. For water as saturated vapor at 120C, Table A-2 gives v 0.8919 m3/kg, u 2529.3 kJ/kg, s 7.1296 kJ/kg # K. At the dead state, where T0 298 K (25C) and p0 1 atm, water is a liquid. Thus, with Eqs. 3.11, 3.12, and

7.2 Defining Exergy

6.7 and values from Table A-2, we get v0 1.0029 103 m3/kg, u0 104.88 kJ/kg, s0 0.3674 kJ/kg # K. Substituting values V2 e 1u u0 2 p0 1v v0 2 T0 1s s0 2

gz 2 kJ c 12529.3 104.882 d kg N m3 1 kJ

c a1.01325 105 2 b 10.8919 1.0029 103 2 d` 3 # ` kg 10 N m m kJ d c 1298 K2 17.1296 0.36742 kg # K 130 m/s2 2 m 1N 1 kJ

c

a9.8 2 b 16 m2 d ` ` ` ` 2 s 1 kg # m/s2 103 N # m kJ kJ 12424.42 90.27 2015.14 0.45 0.062 500 kg kg

279

Saturated vapor at 120°C 30 m/s

6m

z

p0 = 1 atm T0 = 298 K g = 9.8 m/s2

The following example illustrates the use of Eq. 7.9 together with ideal gas property data.

EXAMPLE

7.1

Exergy of Exhaust Gas

A cylinder of an internal combustion engine contains 2450 cm3 of gaseous combustion products at a pressure of 7 bar and a temperature of 867C just before the exhaust valve opens. Determine the specific exergy of the gas, in kJ/kg. Ignore the effects of motion and gravity, and model the combustion products as air as an ideal gas. Take T0 300 K (27C) and p0 1.013 bar. SOLUTION Known: Gaseous combustion products at a specified state are contained in the cylinder of an internal combustion engine. Find: Determine the specific exergy. Schematic and Given Data:

2450 cm3 of air at 7 bar, 867°C

Assumptions: 1. The gaseous combustion products are a closed system. 2. The combustion products are modeled as air as an ideal gas. 3. The effects of motion and gravity can be ignored. 4. T0 300 K (27C) and p0 1.013 bar.

Figure E7.1

Analysis: With assumption 3, Eq. 7.9 becomes e u u0 p0 1v v0 2 T0 1s s0 2

280

Chapter 7 Exergy Analysis

The internal energy and entropy terms are evaluated using data from Table A-22, as follows: u u0 880.35 214.07 666.28 kJ/kg s s0 s°1T 2 s°1T0 2

p R ln M p0

3.11883 1.70203 a

7 8.314 b ln a b 28.97 1.013

0.8621 kJ/kg # K

T0 1s s0 2 1300 K2 10.8621 kJ/kg # K2 258.62 kJ/kg

The p0(v v0) term is evaluated using the ideal gas equation of state: v (R M )Tp and v0 (R M )T0p0, so p0 1v v0 2

R p0T a T0 b M p 8.314 11.0132 111402 c 300 d 28.97 7 38.75 kJ/kg

Substituting values into the above expression for the specific exergy e 666.28 138.752 258.62

❶

368.91 kJ/kg

❶

If the gases are discharged directly to the surroundings, the potential for developing work quantified by the exergy value determined in the solution is wasted. However, by venting the gases through a turbine some work could be developed. This principle is utilized by the turbochargers added to some internal combustion engines.

The next example emphasizes the fundamentally different characters of exergy and energy, while illustrating the use of Eqs. 7.9 and 7.10.

EXAMPLE

7.2

Comparing Exergy and Energy

Refrigerant 134a, initially a saturated vapor at 28C, is contained in a rigid, insulated vessel. The vessel is fitted with a paddle wheel connected to a pulley from which a mass is suspended. As the mass descends a certain distance, the refrigerant is stirred until it attains a state where the pressure is 1.4 bar. The only significant changes of state are experienced by the suspended mass and the refrigerant. The mass of refrigerant is 1.11 kg. Determine (a) the initial exergy, final exergy, and change in exergy of the refrigerant, each in kJ. (b) the change in exergy of the suspended mass, in kJ. (c) the change in exergy of an isolated system of the vessel and pulley–mass assembly, in kJ. Discuss the results obtained, and compare with the respective energy changes. Let T0 293 K (20C), p0 1 bar. SOLUTION Known: Refrigerant 134a in a rigid, insulated vessel is stirred by a paddle wheel connected to a pulley–mass assembly. Find: Determine the initial and final exergies and the change in exergy of the refrigerant, the change in exergy of the suspended mass, and the change in exergy of the isolated system, all in kJ. Discuss the results obtained.

7.2 Defining Exergy

281

Schematic and Given Data: Isolated system Q=W=0

Refrigerant 134a m R = 1.11 kg

T

Saturated vapor

1.4 bar 1.0 bar

2 Initially, saturated vapor at –28°C. p2 = 1.4 bar

mass

initially

20°C

0.93 bar

Dead state –28°C z

mass

1

finally

v

T0 = 293 K, p0 = 1 bar

Figure E7.2

Assumptions: 1. As shown in the schematic, three systems are under consideration: the refrigerant, the suspended mass, and an isolated system consisting of the vessel and pulley–mass assembly. For the isolated system Q 0, W 0. 2. The only significant changes of state are experienced by the refrigerant and the suspended mass. For the refrigerant, there is no change in kinetic or potential energy. For the suspended mass, there is no change in kinetic or internal energy. Elevation is the only intensive property of the suspended mass that changes. 3. For the environment, T0 293 K (20C), p0 1 bar. Analysis: (a) The initial and final exergies of the refrigerant can be evaluated using Eq. 7.9. From assumption 2, it follows that for the refrigerant there are no significant effects of motion or gravity, and thus the exergy at the initial state is E1 mR 3 1u1 u0 2 p0 1v1 v0 2 T0 1s1 s0 2 4

The initial and final states of the refrigerant are shown on the accompanying T–v diagram. From Table A-10, u1 ug(28C) 211.29 kJ/kg. v1 vg(28C) 0.2052 m3/kg, s1 sg(28C) 0.9411 kJ/kg # K. From Table A-12 at 1 bar, 20C, u0 246.67 kJ/kg, v0 0.23349 m3/kg, s0 1.0829 kJ/kg # K. Then E1 1.11 kg c 1211.29 246.672

kJ N m3 1 kJ kJ

a105 2 b 10.2052 0.233492 ` d ` 293 K10.9411 1.08292 kg kg 103 N # m kg # K m

1.11 kg 3 135.382 12.832 141.552 4

kJ 3.7 kJ kg

The final state of the refrigerant is fixed by p2 1.4 bar and v2 v1. Interpolation in Table A-12 gives u2 300.16 kJ/kg, s2 1.2369 kJ/kg # K. Then

❶

E2 1.11 kg 3 153.492 12.832 145.122 4

kJ 6.1 kJ kg

For the refrigerant, the change in exergy is

❷

1¢E2 refrigerant E2 E1 6.1 kJ 3.7 kJ 2.4 kJ The exergy of the refrigerant increases as it is stirred.

282

Chapter 7 Exergy Analysis

(b) With assumption 2, Eq. 7.10 reduces to give the exergy change for the suspended mass 1 ¢E2 mass 1 ¢U p0 ¢V T0 ¢S ¢KE ¢PE2 mass 0

1 ¢PE2 mass

❸

Thus, the exergy change for the suspended mass equals its change in potential energy. The change in potential energy of the suspended mass is obtained from an energy balance for the isolated system as follows: The change in energy of the isolated system is the sum of the energy changes of the refrigerant and suspended mass. There is no heat transfer or work, and with assumption 2 we have 1¢KE ¢PE ¢U 2 refrigerant 1 ¢KE ¢PE ¢U 2 mass Q W 0

Solving for (PE)mass and using previously determined values for the specific internal energy of the refrigerant 1 ¢PE2 mass 1¢U2 refrigerant

11.11 kg2 1300.16 211.292 a

kJ b kg

98.6 kJ Collecting results, (E)mass 98.6 kJ. The exergy of the mass decreases because its elevation decreases. (c) The change in exergy of the isolated system is the sum of the exergy changes of the refrigerant and suspended mass. With the results of parts (a) and (b) 1¢E2 isol 1¢E2 refrigerant 1¢E2 mass 12.4 kJ2 198.6 kJ2 96.2 kJ The exergy of the isolated system decreases. To summarize

Refrigerant Suspended mass Isolated system

❹

Energy Change

Exergy Change

98.6 kJ 98.6 kJ

2.4 kJ 98.6 kJ

0.0 kJ

96.2 kJ

For the isolated system there is no net change in energy. The increase in the internal energy of the refrigerant equals the decrease in potential energy of the suspended mass. However, the increase in exergy of the refrigerant is much less than the decrease in exergy of the mass. For the isolated system, exergy decreases because stirring destroys exergy.

❶

Exergy is a measure of the departure of the state of the system from that of the environment. At all states, E 0. This applies when T T0 and p p0, as at state 2, and when T T0 and p p0, as at state 1.

❷

The exergy change of the refrigerant can be determined more simply with Eq. 7.10, which requires dead state property values only for T0 and p0. With the approach used in part (a), values for u0, v0, and s0 are also required.

❸

❹

The change in potential energy of the suspended mass, (PE)mass, cannot be determined from Eq. 2.10 (Sec. 2.1) since the mass and change in elevation are unknown. Moreover, for the suspended mass as the system, (PE)mass cannot be obtained from an energy balance without first evaluating the work. Thus, we resort here to an energy balance for the isolated system, which does not require such information. As the suspended mass descends, energy is transferred by work through the paddle wheel to the refrigerant, and the refrigerant state changes. Since the exergy of the refrigerant increases, we infer that an exergy transfer accompanies the work interaction. The concepts of exergy change, exergy transfer, and exergy destruction are related by the closed system exergy balance introduced in the next section.

7.3 Closed System Exergy Balance

7.3 Closed System Exergy Balance

A system at a given state can attain a new state through work and heat interactions with its surroundings. Since the exergy value associated with the new state would generally differ from the value at the initial state, transfers of exergy across the system boundary can be inferred to accompany heat and work interactions. The change in exergy of a system during a process would not necessarily equal the net exergy transferred, for exergy would be destroyed if irreversibilities were present within the system during the process. The concepts of exergy change, exergy transfer, and exergy destruction are related by the closed system exergy balance introduced in this section. The exergy balance concept is extended to control volumes in Sec. 7.5. These balances are expressions of the second law of thermodynamics and provide the basis for exergy analysis.

7.3.1 Developing the Exergy Balance The exergy balance for a closed system is developed by combining the closed system energy and entropy balances. The forms of the energy and entropy balances used in the development are, respectively

dQ W dQ a b s T 2

E2 E1

1

S2 S 1

2

b

1

where W and Q represent, respectively, work and heat transfers between the system and its surroundings. These interactions do not necessarily involve the environment. In the entropy balance, Tb denotes the temperature on the system boundary where Q is received and the term accounts for entropy produced by internal irreversibilities. As the first step in deriving the exergy balance, multiply the entropy balance by the temperature T0 and subtract the resulting expression from the energy balance to obtain 1E2 E1 2 T0 1S2 S1 2

2

dQ T0

1

2

1

a

dQ b W T0s T b

Collecting the terms involving Q and introducing Eq. 7.10 on the left side, we can rewrite this expression as 1E2 E1 2 p0 1V2 V1 2

2

1

a1

T0 b dQ W T0s Tb

Rearranging, the closed system exergy balance results

E2 E1

2

1

exergy change

a1

T0 b dQ 3W p0 1V2 V1 2 4 T0s Tb exergy exergy transfers destruction

(7.11) closed system

Since Eq. 7.11 is obtained by deduction from the energy and entropy balances, it is not an independent result, but it can be used in place of the entropy balance as an expression of the second law.

exergy balance

283

284

Chapter 7 Exergy Analysis

INTERPRETING THE EXERGY BALANCE

For specified end states and given values of p0 and T0, the exergy change E2 E1 on the left side of Eq. 7.11 can be evaluated from Eq. 7.10. The underlined terms on the right depend explicitly on the nature of the process, however, and cannot be determined by knowing only the end states and the values of p0 and T0. The first underlined term on the right side of Eq. 7.11 is associated with heat transfer to or from the system during the process. It can be interpreted as the exergy transfer accompanying heat. That is exergy transfer

exergy transfer R accompanying heat

B

accompanying heat

2

1

a1

T0 b dQ Tb

(7.12)

The second underlined term on the right side of Eq. 7.11 is associated with work. It can be interpreted as the exergy transfer accompanying work. That is exergy transfer

B

accompanying work

exergy transfer R 3W p0 1V2 V1 2 4 accompanying work

(7.13)

The exergy transfer expressions are discussed further in Sec. 7.3.2. The third underlined term on the right side of Eq. 7.11 accounts for the destruction of exergy due to irreversibilities within the system. It is symbolized by Ed. Ed T0s

exergy destruction

(7.14)

To summarize, Eq. 7.11 states that the change in exergy of a closed system can be accounted for in terms of exergy transfers and the destruction of exergy due to irreversibilities within the system. When applying the exergy balance, it is essential to observe the requirements imposed by the second law on the exergy destruction: In accordance with the second law, the exergy destruction is positive when irreversibilities are present within the system during the process and vanishes in the limiting case where there are no irreversibilities. That is Ed: e

7 0 0

irreversibilities present with the system no irreversibilities present within the system

(7.15)

The value of the exergy destruction cannot be negative. It is not a property. By contrast, exergy is a property, and like other properties, the change in exergy of a system can be positive, negative, or zero 7 0 E2 E1: • 0 6 0

(7.16)

To close our introduction to the exergy balance concept, we note that most thermal systems are supplied with exergy inputs derived directly or indirectly from the consumption of fossil fuels. Accordingly, avoidable destructions and losses of exergy represent the waste of these resources. By devising ways to reduce such inefficiencies, better use can be made of fuels. The exergy balance can be used to determine the locations, types, and magnitudes of energy resource waste, and thus can play an important part in developing strategies for more effective fuel use.

7.3 Closed System Exergy Balance

OTHER FORMS OF THE EXERGY BALANCE

As in the case of the mass, energy, and entropy balances, the exergy balance can be expressed in various forms that may be more suitable for particular analyses. A form of the exergy balance that is sometimes convenient is the closed system exergy rate balance. # # T0 # dE dV a a1 b Qj aW p0 b Ed dt Tj dt j

closed system (7.17) exergy rate balance

# where dEdt is the time rate of change of exergy. The term (1 T# 0 Tj)Qj represents the time rate of exergy transfer accompanying heat transfer at the rate Qj occurring at the location # on the boundary where the instantaneous temperature is Tj. The term W represents the time rate # of energy transfer by work. The accompanying rate of exergy transfer is given # by (W p0 dVdt), where dVdt is the time rate of change of system volume. The term Ed accounts for the time rate of exergy destruction due to irreversibilities within the system and # # is related to the rate of entropy production within the system by the expression Ed T0s. For an isolated system, no heat or work interactions with the surroundings occur, and thus there are no transfers of exergy between the system and its surroundings. Accordingly, the exergy balance reduces to give ¢E4 isol Ed 4 isol

(7.18)

Since the exergy destruction must be positive in any actual process, the only processes of an isolated system that occur are those for which the exergy of the isolated system decreases. For exergy, this conclusion is the counterpart of the increase of entropy principle (Sec. 6.5.5) and, like the increase of entropy principle, can be regarded as an alternative statement of the second law. 7.3.2 Conceptualizing Exergy Transfer Before taking up examples illustrating the use of the closed system exergy balance, we consider why the exergy transfer expressions take the forms they do. This is accomplished through simple thought experiments. for example. . . consider a large metal part initially at the dead state. If the part were hoisted from a factory floor into a heat-treating furnace, the exergy of the metal part would increase because its elevation would be increased. As the metal part was heated in the furnace, the exergy of the part would increase further as its temperature increased because of heat transfer from the hot furnace gases. In a subsequent quenching process, the metal part would experience a decrease in exergy as its temperature decreased due to heat transfer to the quenching medium. In each of these processes, the metal part would not actually interact with the environment used to assign exergy values. However, like the exergy values at the states visited by the metal part, the exergy transfers taking place between the part and its surroundings would be evaluated relative to the environment used to define exergy. The following subsections provide means for conceptualizing the exergy transfers that accompany heat transfer and work, respectively. EXERGY TRANSFER ACCOMPANYING HEAT

Consider a system undergoing a process in which a heat transfer Q takes place across a portion of the system boundary where the temperature Tb is constant at Tb T0. In accordance with Eq. 7.12, the accompanying exergy transfer is given by T0 exergy transfer R a1 bQ accompanying heat Tb

B

(7.19)

285

286

Chapter 7 Exergy Analysis

Gas p

Q Volume constant

T0 Dead state p0 Final state, 2 T2 < T0

Initial state, T1 < T2 < T0

1

V

Q

Figure 7.4 Illustration used to discuss an exergy transfer accompanying heat transfer when T T0.

Tb

R

WR = T 1 – __0 Q Tb

( T0

)

The right side of Eq. 7.19 is recognized from the discussion of Eq. 5.8 as the work, WR, that could be developed by a reversible power cycle R receiving Q at temperature Tb and discharging energy by heat transfer to the environment at T0. Accordingly, without regard for the nature of the surroundings with which the system is actually interacting, we may interpret the magnitude of an exergy transfer accompanying heat transfer as the work that could be developed by supplying the heat transfer to a reversible power cycle operating between Tb and T0. This interpretation also applies for heat transfer below T0, but then we think of the magnitude of an exergy transfer accompanying heat as the work that could be developed by a reversible power cycle receiving a heat transfer from the environment at T0 and discharging Q at temperature Tb T0. Thus far, we have considered only the magnitude of an exergy transfer accompanying heat. It is necessary to account also for the direction. The form of Eq. 7.19 shows that when Tb is greater than T0, the heat transfer and accompanying exergy transfer would be in the same direction: Both quantities would be positive, or negative. However, when Tb is less than T0, the sign of the exergy transfer would be opposite to the sign of the heat transfer, so the heat transfer and accompanying exergy transfer would be oppositely directed. for example. . . refer to Fig. 7.4, which shows a system consisting of a gas heated at constant volume. As indicated by the p–V diagram, the initial and final temperatures of the gas are each less than T0. Since the state of the system is brought closer to the dead state in this process, the exergy of the system must decrease as it is heated. Conversely, were the gas cooled from state 2 to state 1, the exergy of the system would increase because the state of the system would be moved farther from the dead state. In summary, when the temperature at the location where heat transfer occurs is less than the temperature of the environment, the heat transfer and accompanying exergy transfer are oppositely directed. This becomes significant when studying the performance of refrigerators and heat pumps, where heat transfers can occur at temperatures below that of the environment. EXERGY TRANSFER ACCOMPANYING WORK

We conclude the present discussion by taking up a simple example that motivates the form taken by the expression accounting for an exergy transfer accompanying work, Eq. 7.13. for example. . . consider a closed system that does work W while undergoing a process in which the system volume increases: V2 V1. Although the system would not necessarily interact with the environment, the magnitude of the exergy transfer is evaluated

7.3 Closed System Exergy Balance

Initial location of boundary, volume = V1

Final location of boundary, volume = V2

Expanding system does work W Environment at T0 , p0

287

Wc = W – p0(V2 – V1) Environment displaced as system expands. Work done by system on environment = p0(V2 – V1)

Boundary of the combined system of system plus environment

Figure 7.5 Illustration used to discuss the expression for an exergy transfer accompanying work.

as the maximum work that could be obtained were the system and environment interacting. As illustrated by Fig. 7.5, all the work W of the system in the process would not be available for delivery from a combined system of system plus environment because a portion would be spent in pushing aside the environment, whose pressure is p0. Since the system would do work on the surroundings equal to p0(V2 V1), the maximum amount of work that could be derived from the combined system would thus be Wc W p0 1V2 V1 2 which is in accordance with the form of Eq. 7.13. As for heat transfer, work and the accompanying exergy transfer can be in the same direction or oppositely directed. If there were no change in the system volume during the process, the transfer of exergy accompanying work would equal the work W of the system. 7.3.3 Illustrations Further consideration of the exergy balance and the exergy transfer and destruction concepts is provided by the two examples that follow. In the first example, we reconsider Examples 6.1 and 6.2 to illustrate that exergy is a property, whereas exergy destruction and exergy transfer accompanying heat and work are not properties.

EXAMPLE

7.3

Exploring Exergy Change, Transfer, and Destruction

Water initially a saturated liquid at 100C is contained in a piston–cylinder assembly. The water undergoes a process to the corresponding saturated vapor state, during which the piston moves freely in the cylinder. For each of the two processes described below, determine on a unit of mass basis the change in exergy, the exergy transfer accompanying work, the exergy transfer accompanying heat, and the exergy destruction, each in kJ/kg. Let T0 20C, p0 1.014 bar. (a) The change in state is brought about by heating the water as it undergoes an internally reversible process at constant temperature and pressure. (b) The change in state is brought about adiabatically by the stirring action of a paddle wheel.

288

Chapter 7 Exergy Analysis

SOLUTION Known: Saturated liquid at 100C undergoes a process to the corresponding saturated vapor state. Find: Determine the change in exergy, the exergy transfers accompanying work and heat, and the exergy destruction for each of two specified processes. Schematic and Given Data: See Figs. E6.1 and E6.2. Assumptions: 1. For part (a), see the assumptions listed for Example 6.1. For part (b), see the assumptions listed for Example 6.2. 2. T0 20C, p0 1.014 bar. Analysis: (a) The change in specific exergy is obtained using Eq. 7.9 ¢e ug uf p0 1vg vf 2 T0 1sg sf 2 Using data from Table A-2 ¢e 2087.56

kJ N m3 1 kJ kJ

a1.014 105 2 b a1.672 b ` 3 # ` 1293.15 K2 a6.048 b kg kg 10 N m kg # K m

484 kJ kg Using the expression for work obtained in the solution to Example 6.1, Wm pvfg, the transfer of exergy accompanying work is W exergy transfer p0 1vg vf 2 R accompanying work m

B

1 p p0 2vfg 0

Although the work has a nonzero value, there is no accompanying exergy transfer in this case because p p0. Using the heat transfer value calculated in Example 6.1, the transfer of exergy of accompanying heat transfer in the constanttemperature process is B

T0 Q exergy transfer R a1 b accompanying heat T m a1

293.15 K kJ b a2257 b 373.15 K kg

484 kJ/kg The positive value indicates that exergy transfer occurs in the same direction as the heat transfer. Since the process is accomplished without irreversibilities, the exergy destruction is necessarily zero in value. This can be verified by inserting the three exergy quantities evaluated above into an exergy balance and evaluating Edm.

❶

(b) Since the end states are the same as in part (a), the change in exergy is the same. Moreover, because there is no heat transfer, there is no exergy transfer accompanying heat. The exergy transfer accompanying work is W exergy transfer p0 1vg vf 2 R accompanying work m

B

With the net work value determined in Example 6.2 and evaluating the change in specific volume as in part (a) kJ N m3 1 kJ exergy transfer R 2087.56 a1.014 105 2 b a1.672 b ` 3 # ` accompanying work kg kg 10 N m m

B

2257 kJ/kg The minus sign indicates that the net transfer of exergy accompanying work is into the system.

7.3 Closed System Exergy Balance

289

Finally, the exergy destruction is determined from an exergy balance. Solving Eq. 7.11 for the exergy destruction per unit mass Ed W ¢e c p0 1vg vf 2 d 484 122572 1773 kJ/kg m m

❷

The numerical values obtained can be interpreted as follows: 2257 kJ/kg of exergy is transferred into the system accompanying work; of this, 1773 kJ/kg is destroyed by irreversibilities, leaving a net increase of only 484 kJ/kg.

❶

Exergy is a property and thus the exergy change during a process is determined solely by the end states. Exergy destruction and exergy transfer accompanying heat and work are not properties. Their values depend on the nature of the process.

❷

Alternatively, the exergy destruction value of part (b) could be determined using Edm T0(m), where m is obtained from the solution to Example 6.2. This is left as an exercise.

In the next example, we reconsider the gearbox of Examples 2.4 and 6.4 from an exergy perspective to introduce exergy accounting.

EXAMPLE

7.4

Exergy Accounting for a Gearbox

For the gearbox of Examples 2.4 and 6.4(a), develop a full exergy accounting of the power input. Let T0 293 K. SOLUTION Known: A gearbox operates at steady state with known values for the power input, power output, and heat transfer rate. The temperature on the outer surface of the gearbox is also known. Find: Develop a full exergy accounting of the input power. Schematic and Given Data: See Fig. E6.4a. Assumptions: 1. See the solution to Example 6.4(a). 2. T0 293 K. # Analysis: Since the gearbox volume is constant, the rate of exergy transfer accompanying power, namely 1W p0 dV dt2, reduces to the power itself. Accordingly, exergy is transferred into the gearbox via the high-speed shaft at a rate equal to the power input, 60 kW, and exergy is transferred out via the low-speed shaft at a rate equal to the power output, 58.8 kW. Additionally, exergy is transferred out accompanying heat transfer and destroyed by irreversibilities within the gearbox. Let us evaluate the rate of exergy transfer accompanying heat transfer. Since the temperature Tb at the outer surface of the gearbox is uniform with position T0 # time rate of exergy R a1 b Q transfer accompanying heat Tb

B

# With Q 1.2 kW and Tb 300 K from Example 6.4a, and T0 293 K B

293 time rate of exergy R a1 b 11.2 kW2 transfer accompanying heat 300 0.03 kW

where the minus sign denotes exergy transfer from the system.

290

❶

Chapter 7 Exergy Analysis

# # # Next, the rate of exergy destruction is calculated from Ed T0s, where s is the rate of entropy production. From the so# 3 lution to Example 6.4(a), s 4 10 kW/K. Then # # Ed T0s 1293 K214 103 kW/K2 1.17 kW

The analysis is summarized by the following exergy balance sheet in terms of exergy magnitudes on a rate basis: Rate of exergy in: high-speed shaft Disposition of the exergy: • Rate of exergy out low-speed shaft heat transfer • Rate of exergy destruction

❷

60.00 kW (100%)

58.80 kW (98%) 0.03 kW (0.05%) 1.17 kW (1.95%) 60.00 kW (100%)

❶

Alternatively, the rate of exergy destruction can be determined from the steady-state form of the exergy rate balance, Eq. 7.17. This is left as an exercise.

❷

The difference between the input and output power is accounted for primarily by the rate of exergy destruction and only secondarily by the exergy transfer accompanying heat transfer, which is small by comparison. The exergy balance sheet provides a sharper picture of performance than the energy balance sheet of Example 2.4, which ignores the effect of irreversibilities within the system and overstates the significance of the heat transfer.

7.4 Flow Exergy

The objective of the present section is to develop the flow exergy concept. This concept is important for the control volume form of the exergy rate balance introduced in Sec. 7.5. When mass flows across the boundary of a control volume, there is an exergy transfer accompanying mass flow. Additionally, there is an exergy transfer accompanying flow work. The specific flow exergy accounts for both of these, and is given by

specific flow exergy

ef h h0 T0 1s s0 2

V2

gz 2

(7.20)

In Eq. 7.20, h and s represent the specific enthalpy and entropy, respectively, at the inlet or exit under consideration; h0 and s0 represent the respective values of these properties when evaluated at the dead state.

EXERGY TRANSFER ACCOMPANYING FLOW WORK

As a preliminary to deriving Eq. 7.20, it is necessary to account for the exergy transfer accompanying flow work. When one-dimensional flow is assumed, the work at the inlet or exit of a control volume, # # the flow work, is given on a time rate basis by m(pv), where m is the mass flow rate, p is

7.4 Flow Exergy

291

the pressure, and v is the specific volume at the inlet or exit (Sec. 4.2.1). The following expression accounts for the exergy transfer accompanying flow work c

# time rate of exergy transfer d m 1pv p0v2 accompanying flow work

(7.21)

exergy transfer: flow work

For the development of Eq. 7.21, see box.

Control volume boundary

ACCOUNTING FOR EXERGY TRANSFER A C C O M PA N Y I N G F L O W W O R K

Let us develop Eq. 7.21 for the case pictured in Fig. 7.6. The figure shows a closed system that occupies different regions at time t and a later time t t. The fixed quantity of matter under consideration is shown in color. During the time interval t, some of the mass initially within the region labeled control volume exits to fill the small region e adjacent to the control volume, as shown in Fig. 7.6b. We assume that the increase in the volume of the closed system in the time interval t is equal to the volume of region e and, for further simplicity, that the only work is associated with this volume change. With Eq. 7.13, the exergy transfer accompanying work is B

exergy transfer R W p0 ¢V accompanying work

(a) Time t.

(7.22a)

where V is the volume change of the system. The volume change of the system equals the volume of region e. Thus, we may write V meve, where me is the mass within region e and ve is the specific volume, which is regarded as uniform throughout region e. With this expression for V, Eq. 7.22a becomes exergy transfer R W me 1 p0ve 2 accompanying work

B

(7.22b) (b) Time t + ∆t.

Equation 7.22b can be placed on a time rate basis by dividing each term by the time interval t and taking the limit as t approaches zero. That is B

me W time rate of exergy R lim a b lim c 1 p0ve 2 d transfer accompanying work ¢t S 0 ¢t ¢t S 0 ¢t

(7.23)

In the limit as t approaches zero, the boundaries of the closed system and control volume coincide. Accordingly, in this limit the rate of energy transfer by work from the closed system is also the rate of energy transfer by work from the control volume. For the present case, this is just the flow work. Thus, the first term on the right side of Eq. 7.23 becomes lim a

¢tS0

W # b me 1 peve 2 ¢t

(7.24)

# where me is the mass flow rate at the exit of the control volume. In the limit as t approaches zero, the second term on the right side of Eq. 7.23 becomes lim c

¢tS0

me # 1 p0ve 2 d m e 1 p0ve 2 ¢t

Region e me, ve

(7.25)

Figure 7.6 Illustration used to introduce the flow exergy concept.

292

Chapter 7 Exergy Analysis

In this limit, the assumption of uniform specific volume throughout region e corresponds to the assumption of uniform specific volume across the exit (one-dimensional flow). Substituting Eqs. 7.24 and 7.25 into Eq. 7.23 gives B

# # time rate of exergy transfer R me 1 peve 2 me 1 p0ve 2 accompanying flow work # me 1 peve p0ve 2

(7.26)

Extending the reasoning given here, it can be shown that an expression having the same form as Eq. 7.26 accounts for the transfer of exergy accompanying flow work at inlets to control volumes as well. The general result applying at both inlets and exits is given by Eq. 7.21.

DEVELOPING THE FLOW EXERGY CONCEPT

With the introduction of the expression for the exergy transfer accompanying flow work, attention now turns to the flow exergy. When mass flows across the boundary of a control volume, there is an accompanying energy transfer given by B

time rate of energy transfer # R me accompanying mass flow V2 # m au

gzb 2

(7.27)

where e is the specific energy evaluated at the inlet or exit under consideration. Likewise, when mass enters or exits a control volume, there is an accompanying exergy transfer given by time rate of exergy transfer # R me accompanying mass flow # m 3 1e u0 2 p0 1v v0 2 T0 1s s0 2 4

B

(7.28)

where e is the specific exergy at the inlet or exit under consideration. In writing Eqs. 7.27 and 7.28, one-dimensional flow is assumed. In addition to an exergy transfer accompanying mass flow, an exergy transfer accompanying flow work takes place at locations where mass enters or exits a control volume. Transfers of exergy accompanying flow work are accounted for by Eq. 7.21. Since transfers of exergy accompanying mass flow and flow work occur at locations where mass enters or exits a control volume, a single expression giving the sum of these effects is convenient. Thus, with Eqs. 7.21 and 7.28, time rate of exergy transfer # R m 3e 1 pv p0v2 4 accompanying mass flow and flow work # m 3 1e u0 2 p0 1v v0 2 T0 1s s0 2 (7.29)

B

1 pv p0v2 4

The underlined terms in Eq. 7.29 represent, per unit of mass, the exergy transfer accompanying mass flow and flow work, respectively. The sum identified by underlining is the specific flow exergy ef. That is ef 1e u0 2 p0 1v v0 2 T0 1s s0 2 1 pv p0v2

(7.30a)

7.5 Exergy Rate Balance for Control Volumes

293

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The specific flow exergy can be placed in a more convenient form for calculation by introducing e u V22 gz in Eq. 7.30a and simplifying to obtain V2

gz u0 b 1 pv p0v0 2 T0 1s s0 2 2 V2 1u pv2 1u0 p0v0 2 T0 1s s0 2

gz 2

ef au

(7.30b)

Finally, with h u pv and h0 u0 p0v0, Eq. 7.30b gives Eq. 7.20, which is the principal result of this section. Equation 7.20 is used in the next section where the exergy rate balance for control volumes is formulated. A comparison of the current development with that of Sec. 4.2 shows that the flow exergy evolves here in a similar way as does enthalpy in the development of the control volume energy rate balance, and they have similar interpretations: Each quantity is a sum consisting of a term associated with the flowing mass (specific internal energy for enthalpy, specific exergy for flow exergy) and a contribution associated with flow work at the inlet or exit under consideration.

7.5 Exergy Rate Balance for Control Volumes

In this section, the exergy balance is extended to a form applicable to control volumes. The control volume form is generally the most useful for engineering analysis. GENERAL FORM

The exergy rate balance for a control volume can be derived using an approach like that employed in the box of Sec. 4.1, where the control volume form of the mass rate balance is obtained by transforming the closed system form. However, as in the developments of the energy and entropy rate balances for control volumes, the present derivation is conducted less formally by modifying the closed system rate form, Eq. 7.17, to account for the exergy transfers accompanying mass flow and flow work at the inlets and exits.

294

Chapter 7 Exergy Analysis

The result is the control volume exergy rate balance

control volume

exergy rate balance

# # T0 # dEcv dVcv # # a a1 b Qj aWcv p0 b a miefi a meefe Ed dt T dt j j i e rate of rate of rate of exergy exergy exergy change transfer destruction

(7.31)

As for control volume rate balances considered previously, i denotes inlets and e denotes exits. In Eq. 7.31 the term# dEcvdt represents the time rate of change of the exergy of the control volume. The term Qj represents the time rate of heat transfer at the location on the boundary where the instantaneous temperature is Tj. The accompanying exergy transfer rate is given # # by (1 T0 Tj)Qj. The term Wcv represents the time rate of energy transfer rate by work other # than flow work. The accompanying exergy transfer rate is given by (Wcv p0 dVcv dt), where # dVcvdt is the time rate of change of volume. The term miefi accounts for the time rate of ex# ergy transfer accompanying mass flow and flow work at inlet i. Similarly, miefe accounts for the time rate of exergy transfer accompanying mass flow and flow work at exit e. The flow exergies efi and efe appearing in these expressions are evaluated using Eq. 7.20. In writing Eq. 7.31, #one-dimensional flow is assumed at locations where mass enters and exits. Finally, the term Ed accounts for the time rate of exergy destruction due to irreversibilities within the control volume. STEADY-STATE FORMS

Since a great many engineering analyses involve control volumes at steady state, steadystate forms of the exergy rate balance are particularly important. At steady state, dEcvdt dVcvdt 0, so Eq. 7.31 reduces to the steady-state exergy rate balance steady-state exergy rate balance:

0 a a1

j

control volumes

# # T0 # # # b Qj Wcv a miefi a meefe Ed Tj i e

(7.32a)

This equation indicates that the rate at which exergy is transferred into the control volume must exceed the rate at which exergy is transferred out, the difference being the rate at which exergy is destroyed within the control volume due to irreversibilities. Equation 7.32a can be expressed more compactly as # # # # # 0 a Eq j Wcv a Efi a Efe Ed j

i

(7.32b)

e

where # T0 # Eq j a1 b Qj Tj # # Efi mi efi # # Efe mee fe

(7.33) (7.34a) (7.34b)

are # exergy transfer rates. At steady state, the rate of exergy transfer accompanying the power Wcv is the power itself.

7.5 Exergy Rate Balance for Control Volumes

295

If there is a single inlet and a single exit, denoted by 1 and 2, respectively, Eq. 7.32a reduces to # # T0 # # 0 a a1 b Qj Wcv m 1ef1 ef2 2 Ed Tj j

(7.35)

# where m is the mass flow rate. The term (ef1 ef2) is evaluated using Eq. 7.20 as V21 V22 ef1 ef2 1h1 h2 2 T0 1s1 s2 2

g1z1 z2 2 2

METHODOLOGY UPDATE

(7.36)

ILLUSTRATIONS

The following examples illustrate the use of the mass, energy, and exergy rate balances for the analysis of control volumes at steady state. Property data also play an important role in arriving at solutions. The first example involves the expansion of a gas through a valve (a throttling process, Sec. 4.3). From an energy perspective, the expansion of the gas occurs without loss. Yet, as shown in Example 7.5, such a valve is a site of thermodynamic inefficiency quantified by exergy destruction.

EXAMPLE

7.5

When the rate # of exergy destruction Ed is the objective, it can be determined either from an exergy# rate balance or # from Ed T0scv, where # scv is the rate of entropy production evaluated from an entropy rate balance. The second of these procedures normally requires fewer property evaluations and less computation.

Exergy Destruction in a Throttling Valve

Superheated water vapor enters a valve at 3.0 MPa, 320C and exits at a pressure of 0.5 MPa. The expansion is a throttling process. Determine the specific flow exergy at the inlet and exit and the exergy destruction per unit of mass flowing, each in kJ/kg. Let T0 25C, p0 1 atm. SOLUTION Known: Water vapor expands in a throttling process through a valve from a specified inlet state to a specified exit pressure. Find: Determine the specific flow exergy at the inlet and exit of the valve and the exergy destruction per unit of mass flowing. Schematic and Given Data:

Assumptions: 1

2

Steam 3.0 MPa 320°C

0.5 MPa

1. The control volume shown in the accompanying figure is at steady state. # # 2. For the throttling process, Qcv Wcv 0, and kinetic and potential energy effects can be ignored. 3. T0 25C, p0 1 atm.

Figure E7.5

296

Chapter 7 Exergy Analysis

Analysis: The state at the inlet is specified. The state at the exit can be fixed by reducing the steady-state mass and energy rate balances to obtain h2 h 1 Thus, the exit state is fixed by p2 and h2. From Table A-4, h1 3043.4 kJ/s, s1 6.6245 kJ/kg # k. Interpolating at a pressure of 0.5 MPa with h2 h1, the specific entropy at the exit is s2 7.4223 kJ/kg # k. Evaluating h0 and s0 at the saturated liquid state corresponding to T0, Table A-2 gives h0 104.89 kJ/kg, s0 0.3674 kJ/kg # k. Dropping V22 and gz, we obtain the specific flow exergy from Eq. 7.20 as ef h h0 T0 1s s0 2

Substituting values into the expression for ef, the flow exergy at the inlet is

ef1 13043.4 104.892 29816.6245 0.36742 1073.89 kJ/kg

At the exit ef2 13043.4 104.892 29817.4223 0.36742 836.15 kJ/kg With assumptions listed, the steady-state form of the exergy rate balance, Eq. 7.35, reduces to 0

#0 # T0 # # 0 a a1 b Qj Wcv m 1ef1 ef2 2 Ed Tj j

❶

# Dividing by the mass flow rate m and solving, the exergy destruction per unit of mass flowing is # Ed # 1ef1 ef2 2 m Inserting values # Ed # 1073.89 836.15 237.7 kJ/kg m

❷ ❶

Since h1 h2, this expression for the exergy destruction reduces to # Ed # T0 1s2 s1 2 m entropies s1 and s2. The foregoing equaThus, the exergy destruction can be determined knowing only T0 and # the specific # tion can be obtained alternatively beginning with the relationship Ed T0scv and then evaluating the rate of entropy pro# duction scv from an entropy balance.

❷

Energy is conserved in the throttling process, but exergy is destroyed. The source of the exergy destruction is the uncontrolled expansion that occurs.

Although heat exchangers appear from an energy perspective to operate without loss when stray heat transfer is ignored, they are a site of thermodynamic inefficiency quantified by exergy destruction. This is illustrated in the next example.

EXAMPLE

❶

7.6

Exergy Destruction in a Heat Exchanger

Compressed air enters a counterflow heat exchanger operating at steady state at 610 K, 10 bar and exits at 860 K, 9.7 bar. Hot combustion gas enters as a separate stream at 1020 K, 1.1 bar and exits at 1 bar. Each stream has a mass flow rate of 90 kg/s. Heat transfer between the outer surface of the heat exchanger and the surroundings can be ignored. Kinetic and potential energy effects are negligible. Assuming the combustion gas stream has the properties of air, and using the ideal gas model for both streams, determine for the heat exchanger

7.5 Exergy Rate Balance for Control Volumes

297

(a) the exit temperature of the combustion gas, in K. (b) the net change in the flow exergy rate from inlet to exit of each stream, in MW. (c) the rate exergy is destroyed, in MW. Let T0 300 K, p0 1 bar. SOLUTION Known: Steady-state operating data are provided for a counterflow heat exchanger. Find: For the heat exchanger, determine the exit temperature of the combustion gas, the change in the flow exergy rate from inlet to exit of each stream, and the rate exergy is destroyed. Schematic and Given Data:

∆Tave

T3 T2

T4 T1

Assumptions: T3 = 1020 K 3 p3 = 1.1 bar

p4 = 1 bar 4 Combustion gases 2 1 T1 = 610 K p1 = 10 bar

Compressor

Fuel

1. The control volume shown in the accompanying figure is at steady state. # # 2. For the control volume, Qcv 0, Wcv 0, and changes in kinetic and potential energy from inlet to exit are negligible.

Combustor

3. Each stream has the properties of air modeled as an ideal gas.

T2 = 860 K p2 = 9.7 bar

4. T0 300 K, p0 1 bar.

Turbine

Air

Figure E7.6

Analysis: (a) The temperature T4 of the exiting combustion gases can be found by reducing the mass and energy rate balances for the control volume at steady state to obtain # # V23 V24 V21 V22 # # 0 Qcv Wcv m c 1h1 h2 2 a b g1z1 z2 2 d m c 1h3 h4 2 a b g1z3 z4 2 d 2 2 # where m is the common mass flow rate of the two streams. The underlined terms drop out by listed assumptions, leaving # # 0 m 1h1 h2 2 m 1h3 h4 2 # Dividing by m and solving for h4 h4 h 3 h 1 h 2 From Table A-22, h1 617.53 kJ/kg, h2 888.27 kJ/kg, h3 1068.89 kJ/kg. Inserting values h4 1068.89 617.53 888.27 798.15 kJ/kg

❷

Interpolating in Table A-22 gives T4 778 K (505C).

298

Chapter 7 Exergy Analysis

(b) The net change in the flow exergy rate from inlet to exit for the compressed air stream can be evaluated using Eq. 7.36, neglecting the effects of motion and gravity. With Eq. 6.21a and data from Table A-22 # # m 1ef2 ef1 2 m 1h2 h1 2 T0 1s2 s1 2

p2 # m c 1h2 h1 2 T0 as °2 s°1 R ln b d p1 kg kJ 8.314 9.7 kJ c 1888.27 617.532 300 K a2.79783 2.42644 ln b # d 90 s kg 28.97 10 kg K 14,103

kJ 1 MW ` ` 14.1 MW s 103 kJ/s

Thus, as the air passes from 1 to 2, its flow exergy increases. Similarly, the change in the flow exergy rate from inlet to exit for the combustion gas is p4 # # m 1ef4 ef3 2 m c 1h4 h3 2 T0 as °4 s °3 R ln b d p3 90 c 1798.15 1068.892 300 a2.68769 2.99034 16,934

8.314 1 ln b d 28.97 1.1

kJ 1 MW ` ` 16.93 MW s 103 kJ/s

Thus, as the combustion gas passes from 3 to 4, its flow exergy decreases.

❸

(c) The rate of exergy destruction within the control volume can be determined from an exergy rate balance 0

#0 # T0 # # # 0 a a1 b Qj Wcv m 1ef1 ef2 2 m 1ef3 ef4 2 Ed Tj j # Solving for Ed and inserting known values # # # Ed m 1ef1 ef2 2 m 1ef3 ef4 2

114.1 MW2 116.93 MW2 2.83 MW

❹

Comparing results, we see that the exergy increase of the compressed air stream is less than the exergy decrease of the combustion gas stream, even though each has the same energy change. The difference is accounted for by exergy destruction. Energy is conserved but exergy is not.

❶ ❷ ❸

Heat exchangers of this type are known as regenerators (see Sec. 9.7).

❹

Exergy is destroyed by irreversibilities associated with fluid friction and stream-to-stream heat transfer. The pressure drops for the streams are indicators of frictional irreversibility. The temperature difference between the streams is an indicator of heat transfer irreversibility.

The variation in temperature of each stream passing through the heat exchanger is sketched in the schematic. # # # Alternatively, the rate of exergy destruction can be determined using Ed T0scv, where scv is the rate of entropy production evaluated from an entropy rate balance. This is left as an exercise.

The next two examples provide further illustrations of exergy accounting. The first involves the steam turbine with stray heat transfer considered previously in Ex. 6.6.

7.5 Exergy Rate Balance for Control Volumes

EXAMPLE

7.7

299

Exergy Accounting of a Steam Turbine

Steam enters a turbine with a pressure of 30 bar, a temperature of 400C, a velocity of 160 m/s. Steam exits as saturated vapor at 100C with a velocity of 100 m /s. At steady state, the turbine develops work at a rate of 540 kJ per kg of steam flowing through the turbine. Heat transfer between the turbine and its surroundings occurs at an average outer surface temperature of 350 K. Develop a full accounting of the net exergy carried in by the steam, per unit mass of steam flowing. Neglect the change in potential energy between inlet and exit. Let T0 25C, p0 1 atm. SOLUTION Known: Steam expands through a turbine for which steady-state data are provided. Find: Develop a full exergy accounting of the net exergy carried in by the steam, per unit mass of steam flowing. Schematic and Given Data: See Fig. E6.6. Assumptions: 1. The turbine is at steady state. 2. Heat transfer between the turbine and the surroundings occurs at a known temperature. 3. The change in potential energy between inlet and exit can be neglected. 4. T0 25C, p0 1 atm. Analysis: The net exergy carried in per unit mass of steam flowing is obtained using Eq. 7.36 ef1 ef2 1h1 h2 2 T0 1s1 s2 2 a

V21 V22 b 2

where the potential energy term is dropped by assumption 3. From Table A-4, h1 3230.9 kJ/kg, s1 6.9212 kJ/kg # K. From Table A-2, h2 2676.1 kJ /kg, s2 7.3549 kJ/kg # K. Hence, the net rate exergy is carried in is ef1 ef2 c 13230.9 2676.12 29816.9212 7.35492

11602 2 11002 2 2 103

d

691.84 kJ/kg The net exergy carried in can be accounted for in terms of exergy transfers accompanying work and heat from the control volume and destruction within the control volume. At steady state, the exergy transfer accompanying work is the work # exergy # # # itself, or Wcvm 540 kJ/kg. The quantity is evaluated in the solution to Example 6.6 using the steady-state forms of m Q cv # # the mass and energy rate balances: Qcvm 22.6 kJ/kg. The accompanying exergy transfer is # # Eq T0 Qcv a1 b a # # b m Tb m a1

kJ 298 b a22.6 b 350 kg

3.36

❶

kJ kg

where Tb denotes the temperature on the boundary where heat transfer occurs. The exergy destruction can be determined by rearranging the steady-state form of the exergy rate balance, Eq. 7.35, to give # # # T0 Qcv Wcv Ed # a1 b a # b # 1ef1 ef2 2 m Tb m m Substituting values # Ed # 3.36 540 691.84 148.48 kJ/kg m

Chapter 7 Exergy Analysis

300

The analysis is summarized by the following exergy balance sheet in terms of exergy magnitudes on a rate basis: Net rate of exergy in: Disposition of the exergy: • Rate of exergy out work heat transfer • Rate of exergy destruction

691.84 kJ/kg (100%)

540.00 kJ/kg (78.05%) 3.36 kJ/kg (0.49%) 148.48 kJ/kg (21.46%) 691.84 kJ/kg (100%)

Note that the exergy transfer accompanying heat transfer is small relative to the other terms.

❶

# # # The exergy destruction can be determined alternatively using Ed T0scv, where scv is the rate of entropy production from # # an entropy balance. The solution to Example 6.6 provides scvm 0.4983 kJ/kg # K.

The next example illustrates the use of exergy accounting to identify opportunities for improving thermodynamic performance.

EXAMPLE

7.8

Exergy Accounting of a Waste Heat Recovery System

Suppose the system of Example 4.10 is one option under consideration for utilizing the combustion products discharged from an industrial process. (a) Develop a full accounting of the net exergy carried in by the combustion products. (b) Discuss the design implications of the results. SOLUTION Known: Steady-state operating data are provided for a heat-recovery steam generator and a turbine. Find: Develop a full accounting of the net rate exergy is carried in by the combustion products and discuss the implications for design. Schematic and Given Data: m· 1 = 69.78 kg/s p1 = 1 bar 1 T1 = 478°K

Turbine 2 T2 = 400°K p2 = 1 bar

Steam generator

4 T4 = 180°C p4 = 0.275 MPa

· Wcv = 877 kW

Assumptions: 1. See solution to Example 4.10. 2. T0 298K.

5 3 p3 = .275 MPa Water in T3 = 38.9°C m· 3 = 2.08 kg/s

p5 = 0.07 bar x5 = 93% Figure E7.8

7.5 Exergy Rate Balance for Control Volumes

301

Analysis: (a) We begin by determining the net rate exergy is carried into the control volume. Modeling the combustion products as an ideal gas, the net rate is determined using Eq. 7.36 together with Eq. 6.21a as # # m1 3ef1 ef2 4 m1 3h1 h2 T0 1s1 s2 2 4 p1 # m1 c h1 h2 T0 as °1 s °2 R ln b d p2 With data from Table A-22, h1 480.35 kJ/kg, h2 400.97 kJ/kg, s °1 2.173 kJ/kg # °K, s °2 1.992 kJ/kg # °K, and p2 p1, we have kJ kJ # 298°K12.173 1.9922 m1ef1 ef2 69.8 kg/s c 1480.35 400.972 d kJ/s kg kg # °C 1775.78 kJ/s Next, we determine the rate exergy is carried out of the control volume. Exergy is carried out of the control volume by work at a rate of 876.8 kJ/s, as shown on the schematic. Additionally, the net rate exergy is carried out by the water stream is # # m3ef5 ef3 m3h5 h3 T0 1s5 s3 2

From Table A-2E, h3 hf 139°C2 162.82 kg/s, s3 sf 139°C2 0.5598 kJ/kg # °K. Using saturation data at 0.07 bars from Table A-3 with x5 0.93 gives h5 2403.27 kJ/kg and s5 7.739 kJ/kg # °K. Substituting values kg kJ kJ # c 12403.27 162.822 29817.739 0.55982 d m3ef5 ef3 2.08 s kg kg # °K 209.66 kJ/s

Next, the rate exergy is destroyed in the heat-recovery steam generator can be obtained from an exergy rate balance applied to a control volume enclosing the steam generator. That is 0

#0 # T0 # # # 0 a a1 b Qj Wcv m1 1ef1 ef2 2 m3 1ef3 ef4 2 Ed T j j # Evaluating 1ef3 ef4 2 with Eq. 7.36 and solving for Ed # # # Ed m1 1ef1 ef2 2 m3h3 h4 T0 1s3 s4 2 The first term on the right is evaluated above. Then, with h4 2825 kJ/kg, s4 7.2196 kJ/kg # °K at 180°C, .275 MPa from Table A-4, and previously determined values for h3 and s3 # kg kJ kJ kJ Ed 1775.78 2.08 c 1162 28252 2981.559 7.21962 # d s s kg kg °K 366.1 kJ/s Finally, the rate exergy is destroyed in the turbine can be obtained from an exergy rate balance applied to a control volume enclosing the turbine. That is 0

# # T0 # # 0 a a1 b Qj Wcv m4 1ef4 ef5 2 Ed T j j

❶

# Solving for Ed, evaluating 1ef4 ef5 2 with Eq. 7.36, and using previously determined values # # # Ed Wcv m4 h4 h5 T0 1s4 s5 2 kg kJ kJ kJ 876.8 2.08 c 12825 24032 298°K 17.2196 7.7392 # d s s kg kg °C 320.2 kJ /s

302

Chapter 7 Exergy Analysis

The analysis is summarized by the following exergy balance sheet in terms of exergy magnitudes on a rate basis: Net rate of exergy in: Disposition of the exergy: • Rate of exergy out power developed water stream • Rate of exergy destruction heat-recovery steam generator turbine

1772.8 kJ/s (100%)

876.8 kJ/s (49.5%) 209.66 kJ/s (11.8%) 366.12 kJ/s (20.6%) 320.2 kJ/s (18%)

(b) The exergy balance sheet suggests an opportunity for improved thermodynamic performance because about 50% of the net exergy carried in is either destroyed by irreversibilities or carried out by the water stream. Better thermodynamic performance might be achieved by modifying the design. For example, we might reduce the heat transfer irreversibility by specifying a heat-recovery steam generator with a smaller stream-to-stream temperature difference, and/or reduce friction by specifying a turbine with a higher isentropic efficiency. Thermodynamic performance alone would not determine the preferred option, however, for other factors such as cost must be considered, and can be overriding. Further discussion of the use of exergy analysis in design is provided in Sec. 7.7.1.

❶

Alternatively, the rates of # exergy destruction in control volumes enclosing the heat-recovery steam generator and turbine # # can be determined using Ed T0 scv, where scv is the rate of entropy production for the respective control volume evaluated from an entropy rate balance. This is left as an exercise.

In previous discussions we have noted the effect of irreversibilities on thermodynamic performance. Some economic consequences of irreversibilities are considered in the next example.

EXAMPLE

7.9

Cost of Exergy Destruction

For the heat pump of Examples 6.8 and 6.14, determine the exergy destruction rates, each in kW, for the compressor, condenser, and throttling valve. If exergy is valued at $0.08 per kW # h, determine the daily cost of electricity to operate the compressor and the daily cost of exergy destruction in each component. Let T0 273 K (0C), which corresponds to the temperature of the outside air. SOLUTION Known: Refrigerant 22 is compressed adiabatically, condensed by heat transfer to air passing through a heat exchanger, and then expanded through a throttling valve. Data for the refrigerant and air are known. Find: Determine the daily cost to operate the compressor. Also determine the exergy destruction rates and associated daily costs for the compressor, condenser, and throttling valve. Schematic and Given Data: See Examples 6.8 and 6.14. Assumptions: 1. See Examples 6.8 and 6.14. 2. T0 273 K (0C). Analysis: The rates of exergy destruction can be calculated using # # E d T0 s

7.6 Exergetic (Second Law) Efficiency

303

together with data for the entropy production rates from Example 6.8. That is # kW b 0.478 kW 1Ed 2 comp 1273 K2117.5 104 2 a K # 4 1Ed 2 valve 12732 19.94 10 2 0.271 kW # 1Ed 2 cond 12732 17.95 104 2 0.217 kW The costs of exergy destruction are, respectively a

$0.08 24 h Daily cost of exergy destruction due b 10.478 kW2 a b` ` $0.92 to compressor irreversibilities kW # h day

a

Daily cost of exergy destruction due to b 10.271210.082 024 0 $0.52 irreversibilities in the throttling valve

a

Daily cost of exergy destruction due to b 10.271210.082 024 0 $0.42 irreversibilities in the condenser

From the solution to Example 6.14, the magnitude of the compressor power is 3.11 kW. Thus, the daily cost is a

❶

$0.08 24 h Daily cost of electricity b 13.11 kW2 a b` ` $5.97 to operate compressor kW # h day

Note that the total cost of exergy destruction in the three components is about 31% of the cost of electricity to operate the compressor.

❶

Associating exergy destruction with operating costs provides a rational basis for seeking cost-effective design improvements. Although it may be possible to select components that would destroy less exergy, the trade-off between any resulting reduction in operating cost and the potential increase in equipment cost must be carefully considered.

7.6 Exergetic (Second Law) Efficiency

The objective of this section is to show the use of the exergy concept in assessing the effectiveness of energy resource utilization. As part of the presentation, the exergetic efficiency concept is introduced and illustrated. Such efficiencies are also known as second law efficiencies. 7.6.1 Matching End Use to Source

Tasks such as space heating, heating in industrial furnaces, and process steam generation commonly involve the combustion of coal, oil, or natural gas. When the products of combustion are at a temperature significantly greater than required by a given task, the end use is not well matched to the source and the result is inefficient use of the fuel burned. To illustrate this simply, refer to Fig. 7.7, which shows a closed system receiving a heat transfer · Q1 T1

· Qs

Air

· Qu

Ts Fuel

Tu System boundary

Figure 7.7 Schematic used to discuss the efficient use of fuel.

exergetic efficiency

304

Chapter 7 Exergy Analysis

# # at the rate Qs at a source temperature Ts and delivering # Qu at a use temperature Tu. Energy is lost to the surroundings by heat transfer at a rate Ql across a portion of the surface at Tl. All energy transfers shown on the figure are in the directions indicated by the arrows. Assuming that the system of Fig. 7.7 operates at steady state and there is no work, the closed system energy and exergy rate balances reduce, respectively, to 0

# # # #0 dE 1Qs Qu Ql 2 W dt 0

#0 # T0 # T0 # T0 # dE dV c a1 b Qs a1 b Qu a1 b Q1 d c W p0 d Ed dt Ts Tu T1 dt These equations can be rewritten as follows # # # Qs Qu Ql (7.37a) # T0 # T0 # T0 # a1 b Qs a1 b Qu a1 b Ql Ed (7.37b) Ts Tu Tl # # Equation 7.37a indicates that # the energy carried in by heat transfer, Qs, is either used, Qu, or lost to the surroundings, Ql. This can be described by an efficiency in terms of energy rates in the form product/input as # Qu h # (7.38) Qs In principle, the value # of h can be increased by applying insulation to reduce the loss. The limiting value, when Ql 0, is h 1 (100%). Equation # 7.37b shows that the exergy carried into the system accompanying # the heat # transfer Qs is either transferred from the system accompanying the heat transfers Qu and Ql or destroyed by irreversibilities within the system. This can be described by an efficiency in the form product/input as # 11 T0 Tu 2Qu # e (7.39a) 11 T0Ts 2Qs Introducing Eq. 7.38 into Eq. 7.39a results in e ha

1 T0 Tu b 1 T0 Ts

(7.39b)

The parameter e, defined with reference to the exergy concept, may be called an exergetic efficiency. Note that and e each gauge how effectively the input is converted to the product. The parameter does this on an energy basis, whereas e does it on an exergy basis. As discussed next, the value of e is generally less than unity even when 1. Equation 7.39b indicates that a value for as close to unity as practical is important for proper utilization of the exergy transferred from the hot combustion gas to the system. However, this alone would not ensure effective utilization. The temperatures Ts and Tu are also important, with exergy utilization improving as the use temperature Tu approaches the source temperature Ts. For proper utilization of exergy, therefore, it is desirable to have a value for as close to unity as practical and also a good match between the source and use temperatures. To emphasize the central role of temperature in exergetic efficiency considerations, a graph of Eq. 7.39b is provided in Fig. 7.8. The figure gives the exergetic efficiency e versus the use temperature Tu for an assumed source temperature Ts 2200 K. Figure 7.8 shows that e tends to unity (100%) as the use temperature approaches Ts. In most cases, however, the

7.6 Exergetic (Second Law) Efficiency 1.0 ∋ → 1 (100%) as Tu → Ts Heating in industrial furnaces

∋ 0.5

Process steam generation

Space heating

1000 K 1500 K 300 500 K Tu 540 900°R 1800°R 2700°R Figure 7.8 Effect of use temperature Tu on the exergetic efficiency e

(Ts 2200 K, h 100% ).

use temperature is substantially below Ts. Indicated on the graph are efficiencies for three applications: space heating at Tu 320 K, process steam generation at Tu 480 K, and heating in industrial furnaces at Tu 700 K. These efficiency values suggest that fuel is used far more effectively in the higher use-temperature industrial applications than in the lower use-temperature space heating. The especially low exergetic efficiency for space heating reflects the fact that fuel is consumed to produce only slightly warm air, which from an exergy perspective has considerably less utility. The efficiencies given on Fig. 7.8 are actually on the high side, for in constructing the figure we have assumed to be unity (100%). Moreover, as additional destruction and loss of exergy would be associated with combustion, the overall efficiency from fuel input to end use would be much less than indicated by the values shown on the figure. For the system in Fig. 7.7, further the # it is instructive to consider # rate of exergy loss accompanying the heat loss Ql, that is (1 T0 T1)Q1. This expression measures the true thermodynamic value of the heat loss and is graphed in Fig. 7.9. The figure shows that the thermodynamic value of the heat loss depends significantly on the tem# perature at which the heat loss occurs. Stray heat transfer, such as Ql, usually occurs at relatively low temperature, and thus has relatively low thermodynamic value. We might expect that the economic value of such a loss varies similarly with temperature, and this is the case. for example. . . since the source of the exergy loss by heat transfer is the fuel input (see Fig. 7.7), the economic value of the loss can be accounted for in terms of the unit cost of fuel based on exergy, cF (in $/kW # h, for example), as follows COSTING HEAT LOSS.

# Cost # rate of heat lossR cF 11 T0Tl 2Ql Ql at temperature Tl

B

· Ql

(7.40)

· · · · ·

T · 1 – __0 Q l Tl

1

2

3

4

5

6

Tl / T0

Figure 7.9 Effect of the temperature ratio TlT0 on the exergy loss associated with heat transfer.

305

306

Chapter 7 Exergy Analysis

Equation 7.40 shows that the cost of such a loss is less at lower temperatures than at higher temperatures. The above example illustrates what we would expect of a rational costing method. It would not be rational to assign the same economic value for a heat transfer occurring near ambient temperature, where the thermodynamic value is negligible, as for an equal heat transfer occurring at a higher temperature, where the thermodynamic value is significant. Indeed, it would be incorrect to assign the same cost to heat loss independent of the temperature at which the loss is occurring. For further discussion of exergy costing, see Sec. 7.7.2. 7.6.2 Exergetic Efficiencies of Common Components Exergetic efficiency expressions can take many different forms. Several examples are given in the current section for thermal system components of practical interest. In every instance, the efficiency is derived by the use of the exergy rate balance. The approach used here serves as a model for the development of exergetic efficiency expressions for other components. Each of the cases considered involves a control volume at steady state, and we assume no heat transfer between the control volume and its surroundings. The current presentation is not exhaustive. Many other exergetic efficiency expressions can be written. For a turbine operating at steady state with no heat transfer with its surroundings, the steady-state form of the exergy rate balance, Eq. 7.35, reduces as follows:

TURBINES.

# # T0 # # 0 a a1 b Qj Wcv m 1ef1 ef2 2 Ed Tj j This equation can be rearranged to read # # Wcv Ed ef1 ef2 # # m m

(7.41)

The term on the left of Eq. 7.41 is the decrease in flow exergy from turbine inlet to# exit. # The equation shows that the exergy decreases because the turbine develops work, Wcv m, # flow # and exergy is destroyed, Ed m. A parameter that gauges how effectively the flow exergy decrease is converted to the desired product is the exergetic turbine efficiency # # Wcv m e ef1 ef2

(7.42)

This particular exergetic efficiency is sometimes refered to as the turbine effectiveness. Carefully note that the exergetic turbine efficiency is defined differently from the isentropic turbine efficiency introduced in Sec. 6.8. for example. . . the exergetic efficiency of the turbine considered in Example 6.11 is 81.2% when T0 298 K. It is left as an exercise to verify this value.

For a compressor or pump operating at steady state with no heat transfer with its surroundings, the exergy rate balance, Eq. 7.35, can be placed in the form # # Wcv Ed a # b ef2 ef1 # m m

COMPRESSORS AND PUMPS.

7.6 Exergetic (Second Law) Efficiency

Hot stream, m· h Cold stream, m· c

2

1

3

4 Figure 7.10

Counterflow heat exchanger.

# # Thus, the exergy input to the device, Wcvm, is accounted for either as an increase in the flow exergy between inlet and exit or as exergy destroyed. The effectiveness of the conversion from work input to flow exergy increase is gauged by the exergetic compressor (or pump) efficiency e

ef2 ef1 # # 1Wcvm2

(7.43)

for example. . . the exergetic efficiency of the compressor considered in Example 6.14 is 84.6% when T0 273 K. It is left as an exercise to verify this value.

The heat exchanger shown in Fig. 7.10 operates at steady state with no heat transfer with its surroundings and both streams at temperatures above T0. The exergy rate balance, Eq. 7.32a, reduces to

HEAT EXCHANGER WITHOUT MIXING.

# # T0 # # # # # 0 a a1 b Qj Wcv 1m hef1 mcef3 2 1m hef2 m cef4 2 Ed Tj j # # where mh is the mass flow rate of the hot stream and mc is the mass flow rate of the cold stream. This can be rearranged to read # # # mh 1ef1 ef2 2 mc 1ef4 ef3 2 Ed (7.44) The term on the left of Eq. 7.44 accounts for the decrease in the exergy of the hot stream. The first term on the right accounts for the increase in exergy of the cold stream. Regarding the hot stream as supplying the exergy increase of the cold stream as well as the exergy destroyed, we can write an exergetic heat exchanger efficiency as # mc 1ef4 ef3 2 e # mh 1ef1 ef2 2

(7.45)

for example. . . the exergetic efficiency of the heat exchanger of Example 7.6 is 83.3%. It is left as an exercise to verify this value. DIRECT CONTACT HEAT EXCHANGER. The direct contact heat exchanger shown in Fig. 7.11 operates at steady state with no heat transfer with its surroundings. The exergy rate balance, Eq. 7.32a, reduces to 0

# # T0 # # # # 0 a a1 b Qj Wcv m1ef1 m2ef2 m3ef3 Ed Tj j

307

308

Chapter 7 Exergy Analysis 1 Hot stream, m· 1

3 Mixed stream, m· 3

2 Cold stream, m· 2

Figure 7.11

Direct contact heat

exchanger.

# # # With m3 m1 m2 from a mass rate balance, this can be written as # # # m1 1ef1 ef3 2 m2 1ef 3 ef 2 2 Ed

(7.46)

The term on the left Eq. 7.46 accounts for the decrease in the exergy of the hot stream between inlet and exit. The first term on the right accounts for the increase in the exergy of the cold stream between inlet and exit. Regarding the hot stream as supplying the exergy increase of the cold stream as well as the exergy destroyed by irreversibilities, we can write an exergetic efficiency for a direct contact heat exchanger as # m2 1ef3 ef2 2 e # m1 1ef1 ef3 2

(7.47)

7.6.3 Using Exergetic Efficiencies

cogeneration

power recovery waste heat recovery

Exergetic efficiencies are useful for distinguishing means for utilizing energy resources that are thermodynamically effective from those that are less so. Exergetic efficiencies also can be used to evaluate the effectiveness of engineering measures taken to improve the performance of a thermal system. This is done by comparing the efficiency values determined before and after modifications have been made to show how much improvement has been achieved. Moreover, exergetic efficiencies can be used to gauge the potential for improvement in the performance of a given thermal system by comparing the efficiency of the system to the efficiency of like systems. A significant difference between these values would suggest that improved performance is possible. It is important to recognize that the limit of 100% exergetic efficiency should not be regarded as a practical objective. This theoretical limit could be attained only if there were no exergy destructions or losses. To achieve such idealized processes might require extremely long times to execute processes and/or complex devices, both of which are at odds with the objective of profitable operation. In practice, decisions are usually made on the basis of total costs. An increase in efficiency to reduce fuel consumption, or otherwise utilize resources better, normally requires additional expenditures for facilities and operations. Accordingly, an improvement might not be implemented if an increase in total cost would result. The tradeoff between fuel savings and additional investment invariably dictates a lower efficiency than might be achieved theoretically and may even result in a lower efficiency than could be achieved using the best available technology. Various methods are used to improve energy resource utilization. All such methods must achieve their objectives cost-effectively. One method is cogeneration, which sequentially produces power and a heat transfer (or process steam) for some desired use. An aim of cogeneration is to develop the power and heat transfer using an integrated system with a total expenditure that is less than would be required to develop them individually. Further discussions of cogeneration are provided in Secs. 7.7.2 and 8.5. Two other methods employed to improve energy resource utilization are power recovery and waste heat recovery. Power recovery can be accomplished by inserting a turbine into a pressurized gas or liquid stream to capture some of the exergy that would otherwise be destroyed in a spontaneous expansion. Waste heat recovery contributes to overall efficiency by using some of the exergy that would otherwise be discarded to the surroundings, as in the exhaust gases of large internal combustion engines. An illustration of waste heat recovery is provided by Example 7.8.

7.7 Thermoeconomics

7.7 Thermoeconomics

Thermal systems typically experience significant work and/or heat interactions with their surroundings, and they can exchange mass with their surroundings in the form of hot and cold streams, including chemically reactive mixtures. Thermal systems appear in almost every industry, and numerous examples are found in our everyday lives. Their design involves the application of principles from thermodynamics, fluid mechanics, and heat transfer, as well as such fields as materials, manufacturing, and mechanical design. The design of thermal systems also requires the explicit consideration of engineering economics, for cost is always a consideration. The term thermoeconomics may be applied to this general area of application, although it is often applied more narrowly to methodologies combining exergy and economics for optimizing the design and operation of thermal systems. 7.7.1 Using Exergy in Design

To illustrate the use of exergy in design, consider Fig. 7.12 showing a thermal system consisting of a power-generating unit and a heat-recovery steam generator. The power-generating unit develops an electric power output and combustion products that enter the heat recovery unit. # Feedwater also enters the heat-recovery steam generator with a mass flow rate of m w, receives exergy by heat transfer from the combustion gases, and exits as steam at some desired condition for use in another process. The combustion products entering the heat-recovery steam generator can be regarded as having economic value. Since the source of the exergy of the combustion products is the fuel input (Fig. 7.12), the economic value can be accounted for in terms of the cost of fuel, as we have done in Sec. 7.6.1 when costing heat loss. From our study of the second law of thermodynamics we know that the average temperature difference, Tave, between two streams passing through a heat exchanger is a measure of irreversibility and that the irreversibility of the heat transfer vanishes as the temperature difference approaches zero. For the heat-recovery steam generator in Fig. 7.12, this source of exergy destruction exacts an economic penalty in terms of fuel cost. Figure 7.13 shows the annual fuel cost attributed to the irreversibility of the heat exchanger as a function of Tave. The fuel cost increases with increasing Tave, because the irreversibility is directly related to the temperature difference. From the study of heat transfer, we know that there is an inverse relation between Tave and the surface area required for a specified heat transfer rate. More heat transfer area means a larger, more costly heat exchanger—that is, a greater capital cost. Figure 7.13 also shows the annualized capital cost of the heat exchanger as a function of Tave. The capital cost decreases as Tave increases. · We

Fuel Power-generating unit

Heat-recovery steam generator

Air

Steam Figure 7.12

Feedwater, m· w

Figure used to illustrate the use of exergy in design.

Combustion gases

thermoeconomics

309

Chapter 7 Exergy Analysis

Annualized cost, dollars per year

310

Total cost = Capital Cost + Fuel Cost

a´

a

a´´

Capital cost

Fuel cost 0 0

Nearly optimal Average temperature difference, ∆Tave

Figure 7.13

Cost curves for a single heat exchanger.

The total cost is the sum of the capital cost and the fuel cost. The total cost curve shown in Fig. 7.13 exhibits a minimum at the point labeled a. Notice, however, that the curve is relatively flat in the neighborhood of the minimum, so there is a range of Tave values that could be considered nearly optimal from the standpoint of minimum total cost. If reducing the fuel cost were deemed more important than minimizing the capital cost, we might choose a design that would operate at point a. Point a would be a more desirable operating point if capital cost were of greater concern. Such trade-offs are common in design situations. The actual design process can differ significantly from the simple case considered here. For one thing, costs cannot be determined as precisely as implied by the curves in Fig. 7.13. Fuel prices may vary widely over time, and equipment costs may be difficult to predict as they often depend on a bidding procedure. Equipment is manufactured in discrete sizes, so the cost also would not vary continuously as shown in the figure. Furthermore, thermal systems usually consist of several components that interact with one another. Optimization of components individually, as considered for the heat exchanger, usually does not guarantee an optimum for the overall system. Finally, the example involves only Tave as a design variable. Often, several design variables must be considered and optimized simultaneously. 7.7.2 Exergy Costing of a Cogeneration System Another important aspect of thermoeconomics is the use of exergy for allocating costs to the products of a thermal system. This involves assigning to each product the total cost to produce it, namely the cost of fuel and other inputs plus the cost of owning and operating the system (e.g., capital cost, operating and maintenance costs). Such costing is a common problem in plants where utilities such as electrical power, chilled water, compressed air, and steam are generated in one department and used in others. The plant operator needs to know the cost of generating each utility to ensure that the other departments are charged properly according to the type and amount of each utility used. Common to all such considerations are fundamentals from engineering economics, including procedures for annualizing costs, appropriate means for allocating costs, and reliable cost data. To explore further the costing of thermal systems, consider the simple cogeneration system operating at steady state shown in Fig. 7.14. The system consists of a boiler and a turbine, with each having no significant heat transfer to its surroundings. The figure is labeled with exergy

7.7 Thermoeconomics · EfP Combustion products

Boiler Fuel · EfF, cF

Turbine-electric generator

High-pressure steam · Zb

Air · Efa

· Zt

· We, ce

1 · Ef1, c1

2 Low-pressure steam

Feedwater · Efw

· Ef2, c2

Figure 7.14 Simple cogeneration system.

transfer rates associated with the flowing streams, where the subscripts F, a, P, and w denote fuel, combustion air, combustion products, and feedwater, respectively. The subscripts 1 and 2 denote high- and low-pressure steam, respectively. Means for evaluating the exergies of the fuel and combustion products are introduced # in Chap. 13. The cogeneration system has two principal products: electricity, denoted by We, and low-pressure steam for use in some process. The objective is to determine the cost at which each product is generated. BOILER ANALYSIS. Let us begin by evaluating the cost of the high-pressure steam produced by the boiler. For this, we consider a control volume enclosing the boiler. Fuel and air enter the boiler separately and combustion products exit. Feedwater enters and high-pressure steam exits. The total cost to produce the exiting streams equals the total cost of the entering streams plus the cost of owning and operating the boiler. This is expressed by the following cost rate balance for the boiler # # # # # # C1 CP CF Ca Cw Zb (7.48) # # where C is the cost rate of the respective stream and Zb accounts for the cost rate associated with owning and operating the boiler (each in $ per hour, for example). In the present dis# cussion, the cost rate Zb is presumed known from a previous economic analysis. # Although the cost rates denoted by C in Eq. 7.48 are evaluated by various means in practice, the present discussion features the use of exergy for this purpose. Since exergy measures the true thermodynamic values of the work, heat, and other interactions between a system and its surroundings as well as the effect of irreversibilities within the system, exergy is a rational basis for assigning costs. With exergy costing, each of the cost rates is evaluated in terms of the associated rate of exergy transfer and a unit cost. Thus, for an entering or exiting stream, we write # # C cEf (7.49) # # where c denotes the cost per unit of exergy (in cents per kW h, for example) and Ef is the associated exergy transfer rate. For simplicity, we assume the feedwater and combustion air enter the boiler with negligible exergy and cost, and the combustion products are discharged directly to the surroundings with negligible cost. Thus Eq. 7.48 reduces as follows

# # 0 # # 0 # 0 # C1 CP CF Ca Cw Zb Then, with Eq. 7.49 we have # # # c1Ef1 cFEf F Zb

(7.50a)

cost rate balance

exergy unit cost

311

312

Chapter 7 Exergy Analysis

Solving for c1, the unit cost of the high-pressure steam is # # Zb Ef F c1 cF a # b # Ef1 Ef1

(7.50b)

This equation shows that the unit cost of the high-pressure steam is determined by two contributions related, respectively, to the cost of the fuel and the cost of owning and operating the boiler. Due to exergy destruction and loss,# less# exergy exits the boiler with the highpressure steam than enters with the fuel. Thus, Ef F Ef1 is invariably greater than one, and the unit cost of the high-pressure steam is invariably greater than the unit cost of the fuel. Next, consider a control volume enclosing the turbine. The total cost to produce the electricity and low-pressure steam equals the cost of the entering high-pressure steam plus the cost of owing and operating the device. This is expressed by the cost rate balance for the turbine # # # # Ce C2 C1 Zt (7.51) # # # where Ce is the cost rate associated with the electricity, C1 and # C2 are the cost rates associated with the entering and exiting steam, respectively, and Zt accounts for the cost rate associated with# owning and operating the turbine. With exergy costing, each of the cost rates # # Ce, C1, and C2 is evaluated in terms of the associated rate of exergy transfer and a unit cost. Equation 7.51 then appears as # # # # ceWe c2Ef2 c1Ef1 Zt (7.52a) TURBINE ANALYSIS.

The unit cost c1 in Eq. 7.52a is given by Eq. 7.50b. In the present discussion, the same unit cost is assigned to the low-pressure steam; that is, c2 c1. This is done on the basis that the purpose of the turbine is to generate electricity, and thus all costs associated with owning and operating the turbine should be charged to the power generated. We can regard this decision as a part of the cost accounting considerations that accompany the thermoeconomic analysis of thermal systems. With c2 c1, Eq. 7.52a becomes # # # # ceWe c1 1Ef1 Ef2 2 Zt (7.52b) The first term on the right side accounts for the cost of the exergy used and the second term accounts for the cost of the system itself. Solving Eq. 7.52b for ce, and introducing the exergetic turbine efficiency e from Eq. 7.42 # Zt c1 ce # (7.52c) e We This equation shows that the unit cost of the electricity is determined by the cost of the highpressure steam and the cost of owning and operating the turbine. Because of exergy destruction within the turbine, the exergetic efficiency is invariably less than one, and therefore the unit cost of electricity is invariably greater than the unit cost of the high-pressure steam. By applying cost rate balances to the boiler and the turbine, we are able to determine the cost of each product of the cogeneration system. The unit cost of the electricity is determined by Eq. 7.52c and the unit cost of the low-pressure steam is determined by the expression c2 c1 together with Eq. 7.50b. The example to follow provides a detailed illustration. The same general approach is applicable for costing the products of a wideranging class of thermal systems.1

SUMMARY.

1

See A. Bejan, G. Tsatsaronis, and M. J. Moran, Thermal Design and Optimization, John Wiley & Sons, New York, 1996.

7.7 Thermoeconomics

EXAMPLE

7.10

313

Exergy Costing of a Cogeneration System

A cogeneration system consists of a natural gas-fueled boiler and a steam turbine that develops power and provides steam for an industrial process. At steady state, fuel enters the boiler with an exergy rate of 100 MW. Steam exits the boiler at 50 bar, 466C with an exergy rate of 35 MW. Steam exits the turbine at 5 bar, 205C and a mass flow rate of 26.15 kg/s. The unit cost of the fuel is 1.44 cents per kW # h of exergy. The costs of owning and operating the boiler and turbine are, respectively, $1080/h and $92/h. The feedwater and combustion air enter with negligible exergy and cost. The combustion products are discharged directly to the surroundings with negligible cost. Heat transfer with the surroundings and kinetic and potential energy effects are negligible. Let T0 298 K. (a) For the turbine, determine the power and the rate exergy exits with the steam, each in MW. (b) Determine the unit costs of the steam exiting the boiler, the steam exiting the turbine, and the power, each in cents per kW # h of exergy. (c) Determine the cost rates of the steam exiting the turbine and the power, each in $/h. SOLUTION Known: Steady-state operating data are known for a cogeneration system that produces both electricity and low-pressure steam for an industrial process. Find: For the turbine, determine the power and the rate exergy exits with the steam. Determine the unit costs of the steam exiting the boiler, the steam exiting the turbine, and the power developed. Also determine the cost rates of the low-pressure steam and power. Schematic and Given Data:

Boiler Gaseous fuel · EfF = 100 MW cents cF = 1.44 _____ kW·h

· Zb = $1080/h

Air

Turbine-electric generator

Combustion products

· Zt = $92/h

1

· We, ce

· Ef1 = 35MW p1 = 50 bar T1 = 466°C

Feedwater

Process steam 2

p2 = 5 bar T2 = 205°C m· 2 = 26.15 kg/s

Figure E7.10

Assumptions: 1. Each control volume shown in the accompanying figure is at steady state. # 2. For each control volume, Qcv 0 and kinetic and potential energy effects are negligible. 3. The feedwater and combustion air enter the boiler with negligible exergy and cost. 4. The combustion products are discharged directly to the surroundings with negligible cost. 5. For the environment, T0 298 K. Analysis: (a) With assumption 2, the mass and energy rate balances for a control volume enclosing the turbine reduce at steady state to give # # We m 1h1 h2 2

314

Chapter 7 Exergy Analysis

From Table A-4, h1 3353.54 kJ/kg and h2 2865.96 kJ/kg. Thus # kg kJ 1 MW We a26.15 b 13353.54 2865.962 a b ` 3 ` s kg 10 kJ/s 12.75 MW Using Eq. 7.36, the difference in the rates exergy enters and exits the turbine with the steam is # # # Ef2 Ef1 m 1ef2 ef1 2 # m h2 h1 T0 1s2 s1 2 # Solving for Ef2 # # # Ef2 Ef1 m h2 h1 T0 1s2 s1 2 # # With known values for Ef1 and m, and data from Table A-4: s1 6.8773 kJ/kg # K and s2 7.0806 kJ/kg # K, the rate exergy exits with the steam is # kg 1 MW kJ kJ 298 K 17.0806 6.87732 # d ` 3 Ef2 35 MW a26.15 b c 12865.96 3353.542 ` s kg kg K 10 kJ/s 20.67 MW (b) For a control volume enclosing the boiler, the cost rate balance reduces with assumptions 3 and 4 to give # # # c1Ef1 cFEfF Zb # EfF is the exergy rate of the entering fuel, cF and c1 are the unit costs of the fuel and exiting steam, respectively, and where # Zb is the cost rate associated with the owning and operating the boiler. Solving for c1 and inserting known values # # Zb EfF c1 cF a # b # Ef1 Ef1 a1.44

1080 $/h 1 MW 100 cents cents 100 MW ba b a b ` 3 ` ` ` kW # h 35 MW 35 MW 1$ 10 kW

14.11 3.092

cents cents 7.2 kW # h kW # h

The cost rate balance for the control volume enclosing the turbine is # # # # ceWe c2Ef2 c1Ef1 Zt

❶

# where ce and c2 are the unit costs of the power and the exiting steam, respectively, and Zt is the cost rate associated with owning and operating the turbine. Assigning the same unit cost to the steam entering and exiting the turbine, c2 c1 7.2 cents/kW # h, and solving for ce ce c1 c

# # # Zt Ef1 Ef2 d # # We We

Inserting known values

❷

ce a7.2

135 20.672 MW cents 92$/h 1 MW 100 cents bc d a b` ` ` ` # kW h 12.75 MW 12.75 MW 103 kW 1$

18.09 0.722

cents cents 8.81 # kW h kW # h

Key Engineering Concepts

315

(c) For the low-pressure steam and power, the cost rates are, respectively # # C2 c2Ef2 a7.2

cents 103 kW 1$ b 120.67 MW2 ` ` ` ` # kW h 1 MW 100 cents

$1488/h # # Ce ceWe

❸

a8.81

cents 103 kW 1$ b 112.75 MW2 ` ` ` ` # kW h 1 MW 100 cents

$1123/h

❶ ❷ ❸

The purpose of the turbine is to generate power, and thus all costs associated with owning and operating the turbine are charged to the power generated. Observe that the unit costs c1 and ce are significantly greater than the unit cost of the fuel. Although the unit cost of the steam is less than the unit cost of the power, the steam cost rate is greater because the associated exergy rate is much greater.

Chapter Summary and Study Guide

In this chapter, we have introduced the property exergy and illustrated its use for thermodynamic analysis. Like mass, energy, and entropy, exergy is an extensive property that can be transferred across system boundaries. Exergy transfer accompanies heat transfer, work and mass flow. Like entropy, exergy is not conserved. Exergy is destroyed within systems whenever internal irreversibilities are present. Entropy production corresponds to exergy destruction. The use of exergy balances is featured in this chapter. Exergy balances are expressions of the second law that account for exergy in terms of exergy transfers and exergy destruction. For processes of closed systems, the exergy balance is Eq. 7.11 and a corresponding rate form is Eq. 7.17. For control volumes, rate forms include Eq. 7.31 and the companion steady-state expressions given by Eqs. 7.32. Control volume analyses account for exergy transfer at inlets and exits in terms of flow exergy. The following checklist provides a study guide for this chapter. When your study of the text and end-of-chapter exercises has been completed you should be able to

write out meanings of the terms listed in the margins

throughout the chapter and understand each of the related concepts. The subset of key concepts listed below is particularly important. evaluate exergy at a given state using Eq. 7.2 and exergy

change between two states using Eq. 7.10, each relative to a specified reference environment. apply exergy balances in each of several alternative

forms, appropriately modeling the case at hand, correctly observing sign conventions, and carefully applying SI and English units. evaluate the specific flow exergy relative to a specified

reference environment using Eq. 7.20. define and evaluate exergetic efficiencies for thermal

system components of practical interest. apply exergy costing to heat loss and simple cogenera-

tion systems.

Key Engineering Concepts

exergy p. 273 exergy reference environment p. 273 dead state p. 275

closed system exergy balance p. 283 exergy transfer p. 284, 291

exergy destruction p. 284 flow exergy p. 290 exergy rate balance p. 285, 294

exergetic efficiency p. 303

316

Chapter 7 Exergy Analysis

Exercises: Things Engineers Think About 1. When you hear the term “energy crisis” used by the news media, do the media really mean exergy crisis?

7. Can an energy transfer by heat and the associated exergy transfer be in opposite directions? Repeat for work.

2. For each case illustrated in Fig. 5.1 (Sec. 5.1), identify the relevant intensive property difference between the system and its surroundings that underlies the potential for work. For cases (a) and (b) discuss whether work could be developed if the particular intensive property value for the system were less than for the surroundings.

8. When evaluating exergy destruction, is it necessary to use an exergy balance?

3. Is it possible for exergy to be negative? For exergy change to be negative? 4. Does an airborne, helium-filled balloon at temperature T0 and pressure p0 have exergy? 5. Does a system consisting of an evacuated space of volume V have exergy? 6. When an automobile brakes to rest, what happens to the exergy associated with its motion?

9. For a stream of matter, how does the definition of flow exergy parallel the definition of enthalpy? 10. Is it possible for the flow exergy to be negative? 11. Does the exergetic efficiency given by Eq. 7.45 apply when both the hot and cold streams are at temperatures below T0? 12. A gasoline-fueled generator is claimed by its inventor to produce electricity at a lower unit cost than the unit cost of the fuel used, where each cost is based on exergy. Comment. 13. A convenience store sells gasoline and bottled drinking water at nearly the same price per gallon. Comment.

Problems: Developing Engineering Skills Evaluating Exergy

7.1 A system consists of 5 kg of water at 10C and 1 bar. Determine the exergy, in kJ, if the system is at rest and zero elevation relative to an exergy reference environment for which T0 20C, p0 1 bar. 7.2 Determine the exergy, in kJ, at 0.7 bar, 90C for 1 kg of (a) water, (b) Refrigerant 134a, (c) air as an ideal gas with cp constant. In each case, the mass is at rest and zero evaluation relative to an exergy reference environment for which T0 20C, p0 1 bar. 7.3 Determine the specific exergy, in kJ/kg, at 0.01C of water as a (a) saturated vapor, (b) saturated liquid, (c) saturated solid. In each case, consider a fixed mass at rest and zero elevation relative to an exergy reference environment for which T0 20C, p0 1 bar. 7.4 A balloon filled with helium at 20C, 1 bar and a volume of 0.5 m3 is moving with a velocity of 15 m /s at an elevation of 0.5 km relative to an exergy reference environment for which T0 20C, p0 1 bar. Using the ideal gas model, determine the specific exergy of the helium, in kJ. 7.5 Determine the specific exergy, in kJ/kg, at 0.6 bar, 10C of (a) ammonia, (b) Refrigerant 22, (c) Refrigerant 134a. Let T0 0C, p0 1 bar and ignore the effects of motion and gravity. 7.6 Consider a two-phase solid–vapor mixture of water at 10C. Each phase present has the same mass. Determine the specific exergy, in kJ/kg, if T0 20C, p0 1 atm, and there are no significant effects of motion or gravity. 7.7 Determine the exergy, in kJ, of the contents of a 2-m3 storage tank, if the tank is filled with

(a) air as an ideal gas at 400C and 0.35 bar. (b) water vapor at 400C and 0.35 bar. Ignore the effects of motion and gravity and let T0 17C, p0 1 atm. 7.8 Air as an ideal gas is stored in a closed vessel of volume V at temperature T0 and pressure p. (a) Ignoring motion and gravity, obtain the following expression for the exergy of the air: E p0V a1

p p p

ln b p0 p0 p0

(b) Using the result of part (a), plot V, in m3, versus pp0 for E 1 kW # h and p0 1 bar. (c) Discuss your plot in the limits as pp0 S , pp0 S 1, and p p0 S 0. 7.9 An ideal gas is stored in a closed vessel at pressure p and temperature T. (a) If T T0, derive an expression for the specific exergy in terms of p, p0, T0, and the gas constant R. (b) If p p0, derive an expression for the specific exergy in terms of T, T0, and the specific heat cp, which can be taken as constant. Ignore the effects of motion and gravity. 7.10 Equal molar amounts of carbon dioxide and helium are maintained at the same temperature and pressure. Which has the greater value for exergy relative to the same reference environment? Assume the ideal gas model with constant cv for each gas. There are no significant effects of motion and gravity.

Problems: Developing Engineering Skills

7.11 Refrigerant 134a vapor initially at 1 bar and 20C fills a rigid vessel. The vapor is cooled until the temperature becomes 32C. There is no work during the process. For the refrigerant, determine the heat transfer per unit mass and the change in specific exergy, each in kJ/kg. Comment. Let T0 20C, p0 0.1 MPa. 7.12 As shown in Fig. P7.12, two kilograms of water undergo a process from an initial state where the water is saturated vapor at 120C, the velocity is 30 m /s, and the elevation is 6 m to a final state where the water is saturated liquid at 10C, the velocity is 25 m /s, and the elevation is 3 m. Determine in kJ, (a) the exergy at the initial state, (b) the exergy at the final state, and (c) the change in exergy. Take T0 25C, p0 1 atm and g 9.8 m /s2.

30 m/s

2

Saturated liquid at 10°C 25 m/s

6m

3m z

Applying the Exergy Balance: Closed Systems

7.15 One kilogram of water initially at 1.5 bar and 200C cools at constant pressure with no internal irreversibilities to a final state where the water is a saturated liquid. For the water as the system, determine the work, the heat transfer, and the amounts of exergy transfer accompanying work and heat transfer, each in kJ. Let T0 20C, p0 1 bar. 7.16 One kilogram of air initially at 1 bar and 25C is heated at constant pressure with no internal irreversibilities to a final temperature of 177C. Employing the ideal gas model, determine the work, the heat transfer, and the amounts of exergy transfer accompanying work and heat transfer, each in kJ. Let T0 298 K, p0 1 bar. 7.17 One kilogram of helium initially at 20C and 1 bar is contained within a rigid, insulated tank. The helium is stirred by a paddle wheel until its pressure is 1.45 bar. Employing the ideal gas model, determine the work and the exergy destruction for the helium, each in kJ. Neglect kinetic and potential energy and let T0 20C, p0 1 bar.

Saturated vapor at 120°C

1

317

p0 = 1 atm, T0 = 25°C, g = 9.8 m/s2

7.18 A rigid, well-insulated tank consists of two compartments, each having the same volume, separated by a valve. Initially, one of the compartments is evacuated and the other contains 0.25 kmol of nitrogen gas at 0.35 MPa and 38C. The valve is opened and the gas expands to fill the total volume, eventually achieving an equilibrium state. Using the ideal gas model for the nitrogen (a) determine the final temperature, in C, and final pressure, in MPa. (b) evaluate the exergy destruction, in kJ. (c) What is the cause of exergy destruction in this case? Let T0 21C, p0 1 bar.

Figure P7.12

7.13 Consider 1 kg of steam initially at 20 bar and 240C as the system. Determine the change in exergy, in kJ, for each of the following processes: (a) The system is heated at constant pressure until its volume doubles. (b) The system expands isothermally until its volume doubles. Let T0 20C, p0 1 bar. 7.14 A flywheel with a moment of inertia of 6.74 kg # m2 rotates as 3000 RPM. As the flywheel is braked to rest, its rotational kinetic energy is converted entirely to internal energy of the brake lining. The brake lining has a mass of 2.27 kg and can be regarded as an incompressible solid with a specific heat c 4.19 kJ/kg # K. There is no significant heat transfer with the surroundings. (a) Determine the final temperature of the brake lining, in C, if its initial temperature is 16C. (b) Determine the maximum possible rotational speed, in RPM, that could be attained by the flywheel using energy stored in the brake lining after the flywheel has been braked to rest. Let T0 16C.

7.19 One kilogram of Refrigerant 134a is compressed adiabatically from the saturated vapor state at 10C to a final state where the pressure is 8 bar and the temperature is 50C. Determine the work and the exergy destruction, each in kJ/kg. Let T0 20C, p0 1 bar. 7.20 Two solid blocks, each having mass m and specific heat c, and initially at temperatures T1 and T2, respectively, are brought into contact, insulated on their outer surfaces, and allowed to come into thermal equilibrium. (a) Derive an expression for the exergy destruction in terms of m, c, T1, T2, and the temperature of the environment, T0. (b) Demonstrate that the exergy destruction cannot be negative. (c) What is the cause of exergy destruction in this case? 7.21 As shown in Fig. P7.21, a 0.3 kg metal bar initially at 1200 K is removed from an oven and quenched by immersing it in a closed tank containing 9 kg of water initially at 300 K. Each substance can be modeled as incompressible. An appropriate constant specific heat for the water is cw 4.2 kJ/kg # K, and an appropriate value for the metal is cm .42 kJ/kg # K. Heat transfer from the tank contents can be neglected. Determine the exergy destruction, in kJ. Let T0 25C.

318

Chapter 7 Exergy Analysis System boundary

For each device, evaluate, in kW, the rates of exergy transfer accompanying heat and work, and the rate of exergy destruction. Can either device operate as claimed? Let T0 27C. 7.25 For the silicon chip of Example 2.5, determine the rate of exergy destruction, in kW. What causes exergy destruction in this case? Let T0 293 K.

Metal bar: Tmi = 1200 K cm = 0.42 kJ/kg·K mm = 0.3 kg

Water: Twi = 300 K cw = 4.2 kJ/kg·K mw = 9 kg

Figure P7.21

7.22 As shown in Fig. P7.22, heat transfer equal to 5 kJ takes place through the inner surface of a wall. Measurements made during steady-state operation reveal temperatures of T1 1500 K and T2 600 K at the inner and outer surfaces, respectively. Determine, in kJ (a) the rates of exergy transfer accompanying heat at the inner and outer surfaces of the wall. (b) the rate of exergy destruction. (c) What is the cause of exergy destruction in this case? Let T0 300 K.

7.26 Two kilograms of a two-phase liquid–vapor mixture of water initially at 300C and x1 0.5 undergo the two different processes described below. In each case, the mixture is brought from the initial state to a saturated vapor state, while the volume remains constant. For each process, determine the change in exergy of the water, the net amounts of exergy transfer by work and heat, and the amount of exergy destruction, each in kJ. Let T0 300 K, p0 1 bar, and ignore the effects of motion and gravity. Comment on the difference between the exergy destruction values. (a) The process is brought about adiabatically by stirring the mixture with a paddle wheel. (b) The process is brought about by heat transfer from a thermal reservoir at 900 K. The temperature of the water at the location where the heat transfer occurs is 900 K. 7.27 For the water heater of Problem 6.48, determine the exergy transfer and exergy destruction, each in kJ, for (a) the water as the system. (b) the overall water heater including the resistor as the system. Compare the results of parts (a) and (b), and discuss. Let T0 20C, p0 1 bar. 7.28 For the electric motor of Problem 6.50, evaluate the rate of exergy destruction and the rate of exergy transfer accompanying heat, each in kW. Express each quantity as a percentage of the electrical power supplied to the motor. Let T0 293 K.

· Q1 = K kJ

T1 = 1500 K

· Q2

T2 = 600 K

Figure P7.22

7.23 For the gearbox of Example 6.4(b), develop a full exergy accounting of the power input. Compare with the results of Example 7.4 and discuss. Let T0 293 K. 7.24 The following steady-state data are claimed for two devices: Device 1. Heat transfer to the device occurs at a place on its surface where the temperature is 52C. The device delivers electricity to its surroundings at the rate of 10 kW. There are no other energy transfers. Device 2. Electricity is supplied to the device at the rate of 10 kW. Heat transfer from the device occurs at a place on its surface where the temperature is 52C. There are no other energy transfers.

7.29 A thermal reservoir at 1200 K is separated from another thermal reservoir at 300 K by a cylindrical rod insulated on its lateral surfaces. At steady state, energy transfer by conduction takes place through the rod. The rod diameter is 2 cm, the length is L, and the thermal conductivity is 0.4 kW/m # K. Plot the following quantities, each in kW, versus L ranging from 0.01 to 1 m: the rate of conduction through the rod, the rates of exergy transfer accompanying heat transfer into and out of the rod, and the rate of exergy destruction. Let T0 300 K. 7.30 A system undergoes a refrigeration cycle while receiving QC by heat transfer at temperature TC and discharging energy QH by heat transfer at a higher temperature TH. There are no other heat transfers. (a) Using an exergy balance, show that the net work input to the cycle cannot be zero. (b) Show that the coefficient of performance of the cycle can be expressed as ba

TC THEd b a1 b TH TC T0 1QH QC 2

where Ed is the exergy destruction and T0 is the temperature of the exergy reference environment.

Problems: Developing Engineering Skills

(c) Using the result of part (b), obtain an expression for the maximum theoretical value for the coefficient of performance. Applying the Exergy Balance: Control Volumes

7.31 The following conditions represent the state at the inlet to a control volume. In each case, evaluate the specific exergy and the specific flow exergy, each in kJ/kg. The velocity is relative to an exergy reference environment for which T0 20C, p0 1 bar. The effect of gravity can be neglected. (a) water vapor at 100 bar, 520C, 100 m/s. (b) Ammonia at 3 bar, 0C, 5 m /s. (c) nitrogen (N2) as an ideal gas at 50 bar, 527C, 200 m/s. 7.32 For an ideal gas with constant specific heat ratio k, show that in the absence of significant effects of motion and gravity the specific flow exergy can be expressed as p 1k12k ef T T 1 ln ln a b p0 cpT0 T0 T0 (a) For k 1.2 develop plots of efcpT0 versus TT0 for pp0 0.25, 0.5, 1, 2, 4. Repeat for k 1.3 and 1.4. (b) The specific flow exergy can take on negative values when pp0 1. What does a negative value mean physically? 7.33 A geothermal source provides a stream of liquid water at temperature T ( T0) and pressure p. Using the incompressible liquid model, develop a plot of efcT0, where ef is the specific flow exergy and c is the specific heat, versus TT0 for pp0 1.0, 1.5, and 2.0. Neglect the effects of motion and gravity. Let T0 60F, p0 1 atm. 7.34 The state of a flowing gas is defined by h, s, V, and z, where velocity and elevation are relative to an exergy reference environment for which the temperature is T0 and the pressure is p0. Determine the maximum theoretical work, per unit mass of gas flowing, that could be developed by any one-inlet, one-exit control volume at steady state that would reduce the stream to the dead state at the exit while allowing heat transfer only at T0. Using your final expression, interpret the specific flow exergy. 7.35 Steam exits a turbine with a mass flow rate of 2 105 kg/h at a pressure of 0.008 MPa, a quality of 94%, and a velocity of 70 m /s. Determine the maximum theoretical power that could be developed, in MW, by any one-inlet, one-exit control volume at steady state that would reduce the steam to the dead state at the exit while allowing heat transfer only at temperature T0. The velocity is relative to an exergy reference environment for which T0 15C, p0 0.1 MPa. Neglect the effect of gravity. 7.36 Water at 25C, 1 bar is drawn from a mountain lake 1 km above a valley and allowed to flow through a hydraulic turbine-generator to a pond on the valley floor. For operation at steady state, determine the minimum theoretical mass flow rate, in kg /s, required to generate electricity at a rate of 1 MW. Let T0 25C, p0 1 bar.

319

7.37 Water vapor enters a valve with a mass flow rate of 2.7 kg/s at a temperature of 280C and a pressure of 30 bar and undergoes a throttling process to 20 bar. (a) Determine the flow exergy rates at the valve inlet and exit and the rate of exergy destruction, each in kW. (b) Evaluating exergy at 8 cents per kW · h, determine the annual cost associated with the exergy destruction, assuming 7500 hours of operation annually. Let T0 25C, p0 1 atm. 7.38 Steam enters a turbine operating at steady state at 6 MPa, 500C with a mass flow rate of 400 kg/s. Saturated vapor exits at 8 kPa. Heat transfer from the turbine to its surroundings takes place at a rate of 8 MW at an average surface temperature of 180C. Kinetic and potential energy effects are negligible. (a) For a control volume enclosing the turbine, determine the power developed and the rate of exergy destruction, each in MW. (b) If the turbine is located in a facility where the ambient temperature is 27C, determine the rate of exergy destruction for an enlarged control volume that includes the turbine and its immediate surroundings so the heat transfer takes place from the control volume at the ambient temperature. Explain why the exergy destruction values of parts (a) and (b) differ. Let T0 300 K, p0 100 kPa. 7.39 Air at 1 bar, 17C, and a mass flow rate of 0.3 kg/s enters an insulated compressor operating at steady state and exits at 3 bar, 147C. Determine, the power required by the compressor and the rate of exergy destruction, each in kW. Express the rate of exergy destruction as a percentage of the power required by the compressor. Kinetic and potential energy effects are negligible. Let T0 17C, p0 1 bar. 7.40 Refrigerant 134a at 10C, 1.4 bar, and a mass flow rate of 280 kg/h enters an insulated compressor operating at steady state and exits at 9 bar. The isentropic compressor efficiency is 82%. Determine (a) the temperature of the refrigerant exiting the compressor, in C. (b) the power input to the compressor, in kW. (c) the rate of exergy destruction expressed as a percentage of the power required by the compressor. Neglect kinetic and potential energy effects and let T0 20C, p0 1 bar. 7.41 Water vapor at 4.0 MPa and 400C enters an insulated turbine operating at steady state and expands to saturated vapor at 0.1 MPa. Kinetic and potential energy effects can be neglected. (a) Determine the work developed and the exergy destruction, each in kJ per kg of water vapor passing through the turbine. (b) Determine the maximum theoretical work per unit of mass flowing, in kJ/kg, that could be developed by any oneinlet, one-exit control volume at steady state that has water vapor entering and exiting at the specified states, while allowing heat transfer only at temperature T0.

320

Chapter 7 Exergy Analysis

Compare the results of parts (a) and (b) and comment. Let T0 27C, p0 0.1 MPa. 7.42 Air enters an insulated turbine operating at steady state at 8 bar, 500 K, and 150 m/s. At the exit the conditions are 1 bar, 320 K, and 10 m/s. There is no significant change in elevation. Determine (a) the work developed and the exergy destruction, each in kJ per kg of air flowing. (b) the maximum theoretical work, in kJ per kg of air flowing, that could be developed by any one-inlet, one-exit control volume at steady state that has air entering and exiting at the specified states, while allowing heat transfer only at temperature T0. Compare the results of parts (a) and (b) and comment. Let T0 300 K, p0 1 bar. 7.43 For the compressor of Problem 6.84, determine the rate of exergy destruction and the rate of exergy transfer accompanying heat, each in kJ per kg of air flow. Express each as a percentage of the work input to the compressor. Let T0 20C, p0 1 atm. 7.44 A compressor fitted with a water jacket and operating at steady state takes in air with a volumetric flow rate of 900 m3/h at 22C, 0.95 bar and discharges air at 317C, 8 bar. Cooling water enters the water jacket at 20C, 100 kPa with a mass flow rate of 1400 kg/h and exits at 30C and essentially the same pressure. There is no significant heat transfer from the outer surface of the water jacket to its surroundings, and kinetic and potential energy effects can be ignored. For the waterjacketed compressor, perform a full exergy accounting of the power input. Let T0 20C, p0 1 atm. 7.45 Steam at 1.4 MPa and 350C with a mass flow rate of 0.125 kg/s enters an insulated turbine operating at steady state and exhausts at 100 kPa. Plot the temperature of the exhaust steam, in C, the power developed by the turbine, in kW, and the rate of exergy destruction within the turbine, in kW, each versus the isentropic turbine efficiency ranging from 0 to 100%. Neglect kinetic and potential energy effects. Let T0 20C, p0 0.1 MPa. 7.46 If the power-recovery device of Problem 6.102 develops a net power of 6 kW, determine, in kW (a) the rate exergy enters accompanying heat transfer. (b) the net rate exergy is carried in by the steam. (c) the rate of exergy destruction within the device. Let T0 293 K, p0 1 bar. 7.47 A counterflow heat exchanger operating at steady state has ammonia entering at 60C, 14 bar with a mass flow rate of 0.5 kg/s and exiting as saturated liquid at 14 bar. Air enters in a separate stream at 300 K, 1 bar and exits at 335 K with a negligible change in pressure. Heat transfer between the heat exchanger and its surroundings is negligible as are changes in kinetic and potential energy. Determine (a) the change in the flow exergy rate of each stream, in kW. (b) the rate of exergy destruction in the heat exchanger, in kW. Let T0 300 K, p0 1 bar.

7.48 Saturated water vapor at 0.008 MPa and a mass flow rate of 2.6 105 kg/h enters the condenser of a 100-MW power plant and exits as a saturated liquid at 0.008 MPa. The cooling water stream enters at 15C and exits at 35C with a negligible change in pressure. At steady state, determine (a) the net rate energy exits the plant with the cooling water stream, in MW. (b) the net rate exergy exits the plant with the cooling water stream, in MW. Compare these values. Is the loss with the cooling water significant? What are some possible uses for the exiting cooling water? Let T0 20C, p0 0.1 MPa. 7.49 Air enters a counterflow heat exchanger operating at steady state at 22C, 0.1 MPa and exits at 7C. Refrigerant 134a enters at 0.2 MPa, a quality of 0.2, and a mass flow rate of 30 kg/h. Refrigerant exits at 0C. There is no significant change in pressure for either stream. (a) For the Refrigerant 134a stream, determine the rate of heat transfer, in kJ/h. (b) For each of the streams, evaluate the change in flow exergy rate, in kJ/h. Compare the values. Let T0 22C, p0 0.1 MPa, and ignore the effects of motion and gravity. 7.50 Determine the rate of exergy destruction, in kW, for (a) the computer of Example 4.8, when air exits at 32C. (b) the computer of Problem 4.70, ignoring the change in pressure between the inlet and exit. (c) the water-jacketed electronics housing of Problem 4.71, when water exits at 24C. Let T0 293 K, p0 1 bar. 7.51 Determine the rate of exergy destruction, in kW, for the electronics-laden cylinder of Problems 4.73 and 6.108. Let T0 293 K, p0 1 bar. 7.52 Helium gas enters an insulated nozzle operating at steady state at 1300 K, 4 bar, and 10 m/s. At the exit, the temperature and pressure of the helium are 900 K and 1.45 bar, respectively. Determine (a) the exit velocity, in m/s. (b) the isentropic nozzle efficiency. (c) the rate of exergy destruction, in kJ per kg of gas flowing through the nozzle. Assume the ideal gas model for helium and ignore the effects of gravity. Let T0 20C, p0 1 atm. 7.53 As shown schematically in Fig. P7.53, an open feedwater heater in a vapor power plant operates at steady state with liquid entering at inlet 1 with T1 40C and p1 7.0 bar. Water vapor at T2 200C and p2 7.0 bar enters at inlet 2. Saturated liquid water exits with a pressure of p3 7.0 bar. Ignoring heat transfer with the surroundings and all kinetic and potential energy effects, determine # # (a) the ratio of mass flow rates, m1 m2. (b) the rate of exergy destruction, in kJ per kg of liquid exiting. Let T0 25C, p0 1 atm.

Problems: Developing Engineering Skills

Liquid water 40°C, 7 bar

1 2

Water vapor 200°C, 7 bar

321

7.60 For the compressor and heat exchanger of Problem 6.88, develop a full exergy accounting, in kW, of the compressor power input. Let T0 300 K, p0 96 kPa. Using Exergetic Efficiencies

3 Saturated liquid 7 bar Figure P7.53

7.54 Reconsider the open feedwater heater of Problem 6.87a. For an exiting mass flow rate of 1 kg/s, determine the cost of the exergy destroyed for 8000 hours of operation annually. Evaluate exergy at 8 cents per kW · h. Let T0 20C, p0 1 atm. 7.55 Steam at 3 MPa and 700C is available at one location in an industrial plant. At another location, steam at 2 MPa, 400C, and a mass flow rate of 1 kg/s is required for use in a certain process. An engineer suggests that steam at this condition can be provided by allowing the higher-pressure steam to expand through a valve to 2 MPa and then flow through a heat exchanger where the steam cools at constant pressure to 400C by heat transfer to the surroundings, which are at 20C. (a) Determine the total rate of exergy destruction, in kW, that would result from the implementation of this suggestion. (b) Evaluating exergy at 8 cents per kW # h, determine the annual cost of the exergy destruction determined in part (a) for 8000 hours of operation annually. Would you endorse this suggestion? Let T0 20C, p0 0.1 MPa.

7.61 Plot the exergetic efficiency given by Eq. 7.39b versus TuT0 for TsT0 8.0 and 0.4, 0.6, 0.8, 1.0. What can be learned from the plot when TuT0 is fixed? When e is fixed? Discuss. 7.62 The temperature of water contained in a closed, wellinsulated tank is increased from 15 to 50C by passing an electric current through a resistor within the tank. Devise and evaluate an exergetic efficiency for this water heater. Assume that the water is incompressible and the states of the resistor and the enclosing tank do not change. Let T0 15C. 7.63 Measurements during steady-state operation indicate that warm air exits a hand-held hair dryer at a temperature of 83C with a velocity of 9.1 m/s through an area of 18.7 cm2. As shown in Fig. P7.63, air enters the dryer at a temperature of 22C and a pressure of 1 bar with a velocity of 3.7 m/s. No significant change in pressure between inlet and exit is observed. Also, no significant heat transfer between the dryer and its surroundings occurs, and potential energy effects can be ignored. Let T0 22C. For the hair dryer (a) evaluate the power # Wcv, in kW, and (b) devise and evaluate an exergetic efficiency.

2 Air T1 = 22°C 1 p1 = 1 bar V1 = 3.7 m/s

T2 = 83°C p2 = 1 bar V2 = 9.1 m/s A2 = 18.7 cm2

7.56 For the compressor and turbine of Problem 6.110, determine the rates of exergy destruction, each in kJ per kg of air flowing. Express each as a percentage of the net work developed by the power plant. Let T0 22C, p0 0.95 bar. 7.57 For the turbines and heat exchanger of Problem 4.57, determine the rates of exergy destruction, each in kW. Place in rank order, beginning with the component contributing most to inefficient operation of the overall system. Let T0 300 K, p0 1 bar. 7.58 For the turbine, condenser, and pump of Problem 4.59, determine the rates of exergy destruction, each in kW. Place in rank order, beginning with the component contributing most to inefficient operation of the overall system. Let T0 293 K, p0 1 bar. 7.59 If the gas turbine power plant of Problem 6.74 develops a net power output of 0.7 MW, determine, in MW, (a) the rate of exergy transfer accompanying heat transfer to the air flowing through the heat exchanger. (b) the net rate exergy is carried out by the air stream. (c) the total rate of exergy destruction within the power plant. Let T0 295 K (22C), p0 0.95 bar.

+ – Figure P7.63

7.64 From an input of electricity, an electric resistance furnace operating at steady state delivers energy by heat transfer to a # process at the rate Qu at a use temperature Tu. There are no other significant energy transfers. (a) Devise an exergetic efficiency for the furnace. (b) Plot the efficiency obtained in part (a) versus the use temperature ranging from 300 to 900 K. Let T0 20C. 7.65 Hydrogen at 25 bar, 450C enters a turbine and expands to 2 bar, 160C with a mass flow rate of 0.2 kg/s. The turbine operates at steady state with negligible heat transfer with its surroundings. Assuming the ideal gas model with k 1.37 and neglecting kinetic and potential energies, determine (a) the isentropic turbine efficiency.

322

Chapter 7 Exergy Analysis

(b) the exergetic turbine efficiency. Let T0 25C, p0 1 atm. 7.66 An ideal gas with constant specific heat ratio k enters a turbine operating at steady state at T1 and p1 and expands adiabatically to T2 and p2. When would the value of the exergetic turbine efficiency exceed the value of the isentropic turbine efficiency? Discuss. Ignore the effects of motion and gravity. 7.67 Air enters an insulated turbine operating at steady state with a pressure of 4 bar, a temperature of 450 K, and a volumetric flow rate of 5 m3/s. At the exit, the pressure is 1 bar. The isentropic turbine efficiency is 84%. Ignoring the effects of motion and gravity, determine (a) the power developed and the exergy destruction rate, each in kW. (b) the exergetic turbine efficiency. Let T0 20C, p0 1 bar. 7.68 Argon enters an insulated turbine operating at steady state at 1000C and 2 MPa and exhausts at 350 kPa. The mass flow rate is 0.5 kg/s. Plot each of the following versus the turbine exit temperature, in C (a) the power developed, in kW. (b) the rate of exergy destruction in the turbine, in kW. (c) the exergetic turbine efficiency. Neglect kinetic and potential energy effects. Let T0 20C, p0 1 bar. 7.69 A compressor operating at steady state takes in 1980 kg/h of air at 1 bar and 25C and compresses it to 10 bar and 200C. The power input to the compressor is 160 kW, and heat transfer occurs from the compressor to the surroundings at an average surface temperature of 60C. (a) Perform a full exergy accounting of the power input to the compressor. (b) Devise and evaluate an exergetic efficiency for the compressor. (c) Evaluating exergy at 8 cents per kW · h, determine the hourly costs of the power input, exergy loss associated with heat transfer, and exergy destruction. Neglect kinetic and potential energy changes. Let T0 25C, p0 1 bar. 7.70 In the boiler of a power plant are tubes through which water flows as it is brought from 0.8 MPa, 150C to 240C at essentially constant pressure. The total mass flow rate of the water is 100 kg/s. Combustion gases passing over the tubes cool from 1067 to 547C at essentially constant pressure. The combustion gases can be modeled as air as an ideal gas. There is no significant heat transfer from the boiler to its surroundings. Assuming steady state and neglecting kinetic and potential energy effects, determine (a) the mass flow rate of the combustion gases, in kg/s. (b) the rate of exergy destruction, in kJ/s. (c) the exergetic efficiency given by Eq. 7.45. Let T0 25C, p0 1 atm.

7.71 Liquid water at 95C, 1 bar enters a direct-contact heat exchanger operating at steady state and mixes with a stream of liquid water entering at 15C, 1 bar. A single liquid stream exits at 1 bar. The entering streams have equal mass flow rates. Neglecting heat transfer with the surroundings and kinetic and potential energy effects, determine for the heat exchanger (a) the rate of exergy destruction, in kJ per kg of liquid exiting. (b) the exergetic efficiency given by Eq. 7.47. Let T0 15C, p0 1 bar. 7.72 Refrigerant 134a enters a counterflow heat exchanger operating at steady state at 20C and a quality of 35% and exits as saturated vapor at 20C. Air enters as a separate stream with a mass flow rate of 4 kg/s and is cooled at a constant pressure of 1 bar from 300 to 260 K. Heat transfer between the heat exchanger and its surroundings can be ignored, as can all changes in kinetic and potential energy. (a) As in Fig. E7.6, sketch the variation with position of the temperature of each stream. Locate T0 on the sketch. (b) Determine the rate of exergy destruction within the heat exchanger, in kW. (c) Devise and evaluate an exergetic efficiency for the heat exchanger. Let T0 300 K, p0 1 bar. 7.73 Determine the exergetic efficiencies of the turbines and heat exchanger of Problem 4.57. Let T0 300 K, p0 1 bar. 7.74 Determine the exergetic efficiencies of the compressor and condenser of the heat pump system of Examples 6.8 and 6.14. Let T0 273 K, p0 1 bar. 7.75 Determine the exergetic efficiencies of the compressor and heat exchanger of Problem 6.88. Let T0 300 K, p0 96 kPa. 7.76 Determine the exergetic efficiencies of the steam generator and turbine of Examples 4.10 and 7.8. Let T0 298C, p0 1 atm. Considering Thermoeconomics

7.77 The total cost rate for a device varies with the pressure drop for flow through the device, ( p1 p2), as follows: # C c1 1 p1 p2 2 13 c2 1 p1 p2 2 where the c’s are constants incorporating economic factors. The first term on the right side of this equation accounts for the capital cost and the second term on the right accounts for the operating cost (pumping power). # (a) Sketch a plot of C versus ( p1 p2). (b) At the point of minimum total cost rate, evaluate the contributions of the capital and operating cost rates to the total cost rate, each in percent. Discuss. 7.78 A system # operating at steady state generates# electricity # at the rate We. The cost rate of the fuel input is CF cFEfF,

Problems: Developing Engineering Skills

where cF is the unit cost of fuel based on exergy. The cost of owning and operating the system is # Z ca

# e b We 1e

# # where e # WeEfF# , and c is a constant incorporating economic factors. CF and Z are the only significant cost rates for the system. (a) Derive an expression for the unit cost of electricity, ce, # based on We in terms of e and the ratios cecF and ccF only. (b) For fixed ccF, derive an expression for the value of e corresponding to the minimum value of cecF. (c) Plot the ratio cecF versus e for ccF 0.25, 1.0, and 4.0. For each specified ccF, evaluate the minimum value of cecF and the corresponding value of e. 7.79 At steady state, a turbine with an exergetic efficiency of 85% develops 18 107 kW · h of work annually (8000 operating hours). The annual cost of owning and operating the turbine is $5.0 105. The steam entering the turbine has a specific flow exergy of 645 Btu/lb, a mass flow rate of 32 104 lb/h, and is valued at $0.0182 per kW · h of exergy. (a) Evaluate the unit cost of the power developed, in $ per kW # h. (b) Evaluate the unit cost based on exergy of the steam entering and exiting the turbine, each in cents per lb of steam flowing through the turbine. 7.80 Figure P7.80 shows a boiler at steady state. Steam having a specific flow exergy of 1300 kJ/kg exits the boiler at a mass flow rate of 5.69 104 kg/h. The cost of owning and operating the boiler is $91/h. The ratio of the exiting steam exergy to the entering fuel exergy is 0.45. The unit cost of the fuel based on exergy is $1.50 per 106 kJ. If the cost rates of the combustion air, feedwater, heat transfer with the surroundings, and exiting combustion products are ignored, develop (a) an expression for the unit cost based on exergy of the steam exiting the boiler. (b) Using the result of part (a), determine the unit cost of the steam, in cents per kg of steam flowing.

Boiler

Combustion products

Fuel cF = $1.50 per 106 kJ

· Zb = $91/h

Air

Feedwater Figure P7.80

Steam

ef = 1300 kJ/kg m· = 5.69 × 104 kg/h

323

7.81 A cogeneration system operating at steady state is shown schematically in Fig. P7.81. The exergy transfer rates of the entering and exiting streams are shown on the figure, in MW. The fuel, produced by reacting coal with steam, has a unit cost of 5.85 cents per kW # h of exergy. The cost of owning and operating the system is $1800/h. The feedwater and combustion air enter with negligible exergy and cost. The combustion products are discharged directly to the surroundings with negligible cost. Heat transfer with the surroundings can be ignored. (a) Determine the rate of exergy destruction within the cogeneration system, in MW. (b) Devise and evaluate an exergetic efficiency for the system. (c) Assuming the power and steam each have the same unit cost based on exergy, evaluate the unit cost, in cents per kW # h. Also evaluate the cost rates of the power and steam, each in $/h.

Cogeneration system

Combustion products · EfP = 5 MW 2 Steam · Ef 2 = 15 MW

1 Fuel · EfF = 80 MW cF = 5.85 cents per kW·h

· Z = $1800/h 3

Combustion air

Power · We = 25 MW

Feedwater

Figure P7.81

7.82 Consider an overall control volume comprising the boiler and steam turbine of the cogeneration system of Example 7.10. Assuming the power and process steam each have the same unit cost based on exergy: ce c2, evaluate the unit cost, in cents per kW # h. Compare with the respective values obtained in Example 7.10 and comment. 7.83 The table below gives alternative specifications for the state of the process steam exiting the turbine of Example 7.10. The cost of owning and operating the turbine, in# $/h, varies # # with the power We, in MW, according to Zt 7.2We. All other data remain unchanged. p2 1bar2

T21C2

40

30

20

9

5

2

1

436

398

349

262

205

128

sat

Plot versus p2, in bar # (a) the power We, in MW. (b) the unit costs of the power and process steam, each in cents per kW # h of exergy. (c) the unit cost of the process steam, in cents per kg of steam flowing.

324

Chapter 7 Exergy Analysis

Design & Open Ended Problems: Exploring Engineering Practice 7.1D A utility charges households the same per kW · h for space heating via steam radiators as it does for electricity. Critically evaluate this costing practice and prepare a memorandum summarizing your principal conclusions. 7.2D For what range of steam mass flow rates, in kg/s, would it be economically feasible to replace the throttling valve of Example 7.5 with a power recovery device? Provide supporting calculations. What type of device might you specify? Discuss. 7.3D A vortex tube is a device having no moving mechanical parts that converts an inlet stream of compressed air at an intermediate temperature into two exiting streams, one cold and one hot. (a) A product catalogue indicates that 20% of the air entering a vortex tube at 21C and 5 bar exits at 37C and 1 bar while the rest exits at 34.2C and 1 bar. An inventor proposes operating a power cycle between the hot and cold streams. Critically evaluate the feasibility of this proposal. (b) Obtain a product catalogue from a vortex tube vendor located with the help of the Thomas Register of American Manufacturers. What are some of the applications of vortex tubes? 7.4D A government agency has solicited proposals for technology in the area of exergy harvesting. The aim is to develop small-scale devices to generate power for rugged-duty applications with power requirements ranging from hundreds of milliwatts to several watts. The power must be developed from only ambient sources, such as thermal and chemical gradients, naturally occurring fuels (tree sap, plants, waste matter, etc.), wind, solar, sound and vibration, and mechanical motion including human motion. The devices must also operate with little or no human intervention. Devise a system that would meet these requirements. Clearly identify its intended application and explain its operating principles. Estimate its size, weight, and expected power output. 7.5D In one common arrangement, the exergy input to a power cycle is obtained by heat transfer from hot products of combustion cooling at approximately constant pressure, while exergy is discharged by heat transfer to water or air at ambient conditions. Devise a theoretical power cycle that at steady state develops the maximum theoretical net work per cycle from the exergy supplied by the cooling products of combustion and

discharges exergy by heat transfer to the natural environment. Discuss practical difficulties that require actual power plant operation to depart from your theoretical cycle. 7.6D Define and evaluate an exergetic efficiency for an electric heat pump system for a 2500 ft2 dwelling in your locale. Use manufacturer’s data for heat pump operation. 7.7D Using the key words exergetic efficiency, second law efficiency, and rational efficiency, develop a bibliography of recent publications discussing the definition and use of such efficiencies for power systems and their components. Write a critical review of one of the publications you locate. Clearly state the principal contribution(s) of the publication. 7.8D The initial plans for a new factory space specify 1000 fluorescent light fixtures, each with two 8-ft conventional tubes sharing a single magnetic ballast. The lights will operate from 7 AM to 10 PM, 5 days per week, 350 days per year. More expensive high-efficiency tubes are available that require more costly electronic ballasts but use considerably less electricity to operate. Considering both initial and operating costs, determine which lighting system is best for this application, and prepare a report of your findings. Use manufacturer’s data and industrial electric rates in your locale to estimate costs. Assume that comparable lighting levels would be achieved by the conventional and high-efficiency lighting. 7.9D A factory has a 120 kW screw compressor that takes in 0.5 m3/s of ambient air and delivers compressed air at 1 MPa for actuating pneumatic tools. The factory manager read in a plant engineering magazine that using compressed air is more expensive than the direct use of electricity for operating such tools and asks for your opinion. Using exergy costing with electric rates in your locale, what do you say? 7.10D Pinch analysis (or pinch technology) is a popular methodology for optimizing the design of heat exchanger networks in complex thermal systems. Pinch analysis uses a primarily graphical approach to implement second-law reasoning. Write a paper discussing the role of pinch analysis within thermoeconomics. 7.11D Wind Power Looming Large (see box Sec. 7.4) Identify sites in your state where wind turbines for utility-scale electrical generation are feasible, considering both engineering and societal issues. Write a report including at least three references.

C H A P

Vapor Power Systems

T E R

8

E N G I N E E R I N G C O N T E X T An important engineering goal is to devise systems that accomplish desired types of energy conversion. The present chapter and the next are concerned with several types of power-generating systems, each of which produces a net power output from a fossil fuel, nuclear, or solar input. In these chapters, we describe some of the practical arrangements employed for power production and illustrate how such power plants can be modeled thermodynamically. The discussion is organized into three main areas of application: vapor power plants, gas turbine power plants, and internal combustion engines. These power systems, together with hydroelectric power plants, produce virtually all of the electrical and mechanical power used worldwide. The objective of the present chapter is to study vapor power plants in which the working fluid is alternately vaporized and condensed. Chapter 9 is concerned with gas turbines and internal combustion engines in which the working fluid remains a gas.

chapter objective

8.1 Modeling Vapor Power Systems

The processes taking place in power-generating systems are sufficiently complicated that idealizations are required to develop thermodynamic models. Such modeling is an important initial step in engineering design. Although the study of simplified models generally leads only to qualitative conclusions about the performance of the corresponding actual devices, models often allow deductions about how changes in major operating parameters affect actual performance. They also provide relatively simple settings in which to discuss the functions and benefits of features intended to improve overall performance. The vast majority of electrical generating plants are variations of vapor power plants in which water is the working fluid. The basic components of a simplified fossil-fuel vapor power plant are shown schematically in Fig. 8.1. To facilitate thermodynamic analysis, the overall plant can be broken down into the four major subsystems identified by the letters A through D on the diagram. The focus of our considerations in this chapter is subsystem A, where the important energy conversion from heat to work occurs. But first, let us briefly consider the other subsystems. The function of subsystem B is to supply the energy required to vaporize the water passing through the boiler. In fossil-fuel plants, this is accomplished by heat transfer to the working fluid passing through tubes and drums in the boiler from the hot gases produced by the combustion of a fossil fuel. In nuclear plants, the origin of the energy is a controlled nuclear reaction taking place in an isolated reactor building. Pressurized water, a liquid metal, 325

326

Chapter 8 Vapor Power Systems

Stack

A

D

Combustion gases to stack Turbine Boiler

Electric generator

C

+ – Cooling tower

Fuel

Condenser

Air

Warm water B Feedwater pump

Figure 8.1

Cooled water Pump Makeup water

Components of a simple vapor power plant.

or a gas such as helium can be used to transfer energy released in the nuclear reaction to the working fluid in specially designed heat exchangers. Solar power plants have receivers for concentrating and collecting solar radiation to vaporize the working fluid. Regardless of the energy source, the vapor produced in the boiler passes through a turbine, where it expands to a lower pressure. The shaft of the turbine is connected to an electric generator (subsystem D). The vapor leaving the turbine passes through the condenser, where it condenses on the outside of tubes carrying cooling water. The cooling water circuit comprises subsystem C. For the plant shown, the cooling water is sent to a cooling tower, where energy taken up in the condenser is rejected to the atmosphere. The cooling water is then recirculated through the condenser. Concern for the environment and safety considerations govern what is allowable in the interactions between subsystems B and C and their surroundings. One of the major difficulties in finding a site for a vapor power plant is access to sufficient quantities of cooling water. For this reason and to minimize thermal pollution effects, most power plants now employ cooling towers. In addition to the question of cooling water, the safe processing and delivery of fuel, the control of pollutant discharges, and the disposal of wastes are issues that must be dealt with in both fossil-fueled and nuclear-fueled plants to ensure safety and operation with an acceptable level of environmental impact. Solar power plants are generally regarded as nonpolluting and safe but as yet are not widely used. Returning now to subsystem A of Fig. 8.1, observe that each unit of mass periodically undergoes a thermodynamic cycle as the working fluid circulates through the series of four interconnected components. Accordingly, several concepts related to thermodynamic power cycles introduced in previous chapters are important for the present discussions. You will recall that the conservation of energy principle requires that the net work developed by a power cycle equals the net heat added. An important deduction from the second law is that the thermal efficiency, which indicates the extent to which the heat added is converted to a net work output, must be less than 100%. Previous discussions also have indicated that

8.2 Analyzing Vapor Power Systems—Rankine Cycle

327

improved thermodynamic performance accompanies the reduction of irreversibilities. The extent to which irreversibilities can be reduced in power-generating systems depends on thermodynamic, economic, and other factors, however.

8.2 Analyzing Vapor Power Systems—Rankine Cycle

All of the fundamentals required for the thermodynamic analysis of power-generating systems already have been introduced. They include the conservation of mass and conservation of energy principles, the second law of thermodynamics, and thermodynamic data. These principles apply to individual plant components such as turbines, pumps, and heat exchangers as well as to the most complicated overall power plants. The object of this section is to introduce the Rankine cycle, which is a thermodynamic cycle that models the subsystem labeled A on Fig. 8.1. The presentation begins by considering the thermodynamic analysis of this subsystem.

Rankine cycle

8.2.1 Evaluating Principal Work and Heat Transfers The principal work and heat transfers of subsystem A are illustrated in Fig. 8.2. In subsequent discussions, these energy transfers are taken to be positive in the directions of the arrows. The unavoidable stray heat transfer that takes place between the plant components and their surroundings is neglected here for simplicity. Kinetic and potential energy changes are also ignored. Each component is regarded as operating at steady state. Using the conservation of mass and conservation of energy principles together with these idealizations, we develop expressions for the energy transfers shown on Fig. 8.2 beginning at state 1 and proceeding through each component in turn. TURBINE. Vapor from the boiler at state 1, having an elevated temperature and pressure, expands through the turbine to produce work and then is discharged to the condenser at state 2 with relatively low pressure. Neglecting heat transfer with the surroundings, the mass and energy rate balances for a control volume around the turbine reduce at steady state to give 0

# 0 # V21 V22 # 0 Qcv Wt m c h1 h2

g 1z1 z2 2d 2 or # Wt # h1 h2 m

(8.1)

˙t W

Turbine 1 2

˙ in Q

˙ out Q

Boiler

Cooling water

Pump

Condenser 4 3

˙p W

Figure 8.2 Principal work and heat transfers of subsystem A.

METHODOLOGY UPDATE

When analyzing vapor power cycles, we take energy transfers as positive in the directions of arrows on system schematics and write the energy balance accordingly.

328

Chapter 8 Vapor Power Systems

# # # where m denotes the mass flow rate of the working fluid, and Wt m is the rate at which work is developed per unit of mass of steam passing through the turbine. As noted above, kinetic and potential energy changes are ignored. CONDENSER. In the condenser there is heat transfer from the vapor to cooling water flowing in a separate stream. The vapor condenses and the temperature of the cooling water increases. At steady state, mass and energy rate balances for a control volume enclosing the condensing side of the heat exchanger give # Qout (8.2) # h2 h 3 m # # where Qout m is the rate at which energy is transferred by heat from the working fluid to the cooling water per unit mass of working fluid passing through the condenser. This energy transfer is positive in the direction of the arrow on Fig. 8.2.

The liquid condensate leaving the condenser at 3 is pumped from the condenser into the higher pressure boiler. Taking a control volume around the pump and assuming no heat transfer with the surroundings, mass and energy rate balances give # Wp (8.3) # h4 h3 m # # where Wp m is the rate of power input per unit of mass passing through the pump. This energy transfer is positive in the direction of the arrow on Fig. 8.2. PUMP.

The working fluid completes a cycle as the liquid leaving the pump at 4, called the boiler feedwater, is heated to saturation and evaporated in the boiler. Taking a control volume enclosing the boiler tubes and drums carrying the feedwater from state 4 to state 1, mass and energy rate balances give # Qin (8.4) # h1 h4 m # # where Qin m is the rate of heat transfer from the energy source into the working fluid per unit mass passing through the boiler. BOILER.

feedwater

thermal efficiency

PERFORMANCE PARAMETERS. The thermal efficiency gauges the extent to which the energy input to the working fluid passing through the boiler is converted to the net work output. Using the quantities and expressions just introduced, the thermal efficiency of the power cycle of Fig. 8.2 is # # # # Wt m Wp m 1h1 h2 2 1h4 h3 2 # h (8.5a) # h1 h4 Qin m

The net work output equals the net heat input. Thus, the thermal efficiency can be expressed alternatively as # # # # # # Qout m Qinm Qoutm # # 1 h # # Qinm Qin m 1

1h2 h3 2 1h1 h4 2

(8.5b)

8.2 Analyzing Vapor Power Systems—Rankine Cycle

329

Thermodynamics in the News… Cleaner Is Better The United States relies heavily upon abundant coal reserves to generate electric power, but these systems require a great deal of clean-up, experts say. Awareness of the health and environmental impacts of coal have led to increasinglystringent regulations on coal-burning power plants. As a result, the search for new clean-coal technologies has intensified. According to industry sources, controlling particulate emissions and safely disposing of millions of tons of coal waste once were the main concerns. Sulfur dioxide removal then became an issue due to concern over acid rain. More recently, nitric oxide (NOX), mercury, and fine particle (less than 3 microns) emissions were recognized as especially harmful. Strides have been made in developing more effective sulfur dioxide scrubbers and particulate capture devices, but stricter environmental standards demand new approaches.

One promising technology is fluidized bed combustion, where a powdered coal-limestone mixture churns in air to enhance the burning. The limestone removes some sulfur during combustion rather than waiting to remove the sulfur after combustion, as in conventional boilers. Nitric oxide formation is also less because of the relatively low temperatures of fluidized bed combustion. Another innovation is the integrated gasification combined-cycle plant, or IGCC, that promises to be cleaner and have higher thermal efficiency than conventional plants. In an IGCC, coal is converted to a cleanerburning combustible gas that is used in a power plant combining gas and steam turbines.

Another parameter used to describe power plant performance is the back work ratio, or bwr, defined as the ratio of the pump work input to the work developed by the turbine. With Eqs. 8.1 and 8.3, the back work ratio for the power cycle of Fig. 8.2 is # # Wp m 1h4 h3 2 bwr # # (8.6) 1h1 h2 2 Wt m

back work ratio

Examples to follow illustrate that the change in specific enthalpy for the expansion of vapor through the turbine is normally many times greater than the increase in enthalpy for the liquid passing through the pump. Hence, the back work ratio is characteristically quite low for vapor power plants. Provided states 1 through 4 are fixed, Eqs. 8.1 through 8.6 can be applied to determine the thermodynamic performance of a simple vapor power plant. Since these equations have been developed from mass and energy rate balances, they apply equally for actual performance when irreversibilities are present and for idealized performance in the absence of such effects. It might be surmised that the irreversibilities of the various power plant components can affect overall performance, and this is the case. Even so, it is instructive to consider an idealized cycle in which irreversibilities are assumed absent, for such a cycle establishes an upper limit on the performance of the Rankine cycle. The ideal cycle also provides a simple setting in which to study various aspects of vapor power plant performance.

8.2.2 Ideal Rankine Cycle If the working fluid passes through the various components of the simple vapor power cycle without irreversibilities, frictional pressure drops would be absent from the boiler and condenser, and the working fluid would flow through these components at constant pressure. Also, in the absence of irreversibilities and heat transfer with the surroundings, the processes through the turbine and pump would be isentropic. A cycle adhering to these idealizations is the ideal Rankine cycle shown in Fig. 8.3.

ideal Rankine cycle

330

Chapter 8 Vapor Power Systems T

1′ a

1

4 3

2

c

b

2′ s

Figure 8.3

Temperature–entropy diagram of the ideal Rankine cycle.

Referring to Fig. 8.3, we see that the working fluid undergoes the following series of internally reversible processes: Process 1–2: Isentropic expansion of the working fluid through the turbine from saturated vapor at state 1 to the condenser pressure. Process 2–3: Heat transfer from the working fluid as it flows at constant pressure through the condenser with saturated liquid at state 3. Process 3–4: Isentropic compression in the pump to state 4 in the compressed liquid region. Process 4–1: Heat transfer to the working fluid as it flows at constant pressure through the boiler to complete the cycle. The ideal Rankine cycle also includes the possibility of superheating the vapor, as in cycle 1–2–3–4–1. The importance of superheating is discussed in Sec. 8.3. Since the ideal Rankine cycle consists of internally reversible processes, areas under the process lines of Fig. 8.3 can be interpreted as heat transfers per unit of mass flowing. Applying Eq. 6.51, area 1–b–c–4–a–1 represents the heat transfer to the working fluid passing through the boiler and area 2–b–c–3–2, is the heat transfer from the working fluid passing through the condenser, each per unit of mass flowing. The enclosed area 1–2–3–4–a–1 can be interpreted as the net heat input or, equivalently, the net work output, each per unit of mass flowing. Because the pump is idealized as operating without irreversibilities, Eq. 6.53b can be invoked as an alternative to Eq. 8.3 for evaluating the pump work. That is # 4 Wp a # b int v dp (8.7a) m rev 3

METHODOLOGY UPDATE

For cycles, we modify the problem-solving methodology: The Analysis begins with a systematic evaluation of required property data at each numbered state. This reinforces what we know about the components, since given information and assumptions are required to fix the states.

where the minus sign has been dropped for consistency with the positive value for pump work in Eq. 8.3. The subscript “int rev” has been retained as a reminder that this expression is restricted to an internally reversible process through the pump. No such designation is required by Eq. 8.3, however, because it expresses the conservation of mass and energy principles and thus is not restricted to processes that are internally reversible. Evaluation of the integral of Eq. 8.7a requires a relationship between the specific volume and pressure for the process. Because the specific volume of the liquid normally varies only slightly as the liquid flows from the inlet to the exit of the pump, a plausible approximation to the value of the integral can be had by taking the specific volume at the pump inlet, v3, as constant for the process. Then # Wp a # b int v3 1 p4 p3 2 (8.7b) m rev The next example illustrates the analysis of an ideal Rankine cycle.

8.2 Analyzing Vapor Power Systems—Rankine Cycle

EXAMPLE

8.1

331

Ideal Rankine Cycle

Steam is the working fluid in an ideal Rankine cycle. Saturated vapor enters the turbine at 8.0 MPa and saturated liquid exits the condenser at a pressure of 0.008 MPa. The net power output of the cycle is 100 MW. Determine for the cycle # (a) the thermal efficiency, (b) the back work ratio, (c) the mass flow rate of the steam, in kg/h, #(d) the rate of heat transfer, Qin, into the working fluid as it passes through the boiler, in MW, (e) the rate of heat transfer, Qout, from the condensing steam as it passes through the condenser, in MW, (f) the mass flow rate of the condenser cooling water, in kg/ h, if cooling water enters the condenser at 15C and exits at 35C. SOLUTION Known: An ideal Rankine cycle operates with steam as the working fluid. The boiler and condenser pressures are specified, and the net power output is given. Find: Determine the thermal efficiency, the back work ratio, the mass flow rate of the steam, in kg/h, the rate of heat transfer to the working fluid as it passes through the boiler, in MW, the rate of heat transfer from the condensing steam as it passes through the condenser, in MW, the mass flow rate of the condenser cooling water, which enters at 15C and exits at 35C. Schematic and Given Data:

˙ in Q p1 = 8.0 MPa

Boiler

˙t W

Turbine 1 Saturated vapor

T 8.0 MPa

2

1

Condenser

˙ out Q Cooling water

Pump

4

0.008 MPa

3

4

˙p W

3 Saturated liquid at 0.008 MPa

2 s

Figure E8.1

Assumptions: 1. Each component of the cycle is analyzed as a control volume at steady state. The control volumes are shown on the accompanying sketch by dashed lines. 2. All processes of the working fluid are internally reversible. 3. The turbine and pump operate adiabatically. 4. Kinetic and potential energy effects are negligible. 5. Saturated vapor enters the turbine. Condensate exits the condenser as saturated liquid.

❶

Analysis: To begin the analysis, we fix each of the principal states located on the accompanying schematic and T–s diagrams. Starting at the inlet to the turbine, the pressure is 8.0 MPa and the steam is a saturated vapor, so from Table A-3, h1 2758.0 kJ/kg and s1 5.7432 kJ/kg # K. State 2 is fixed by p2 0.008 MPa and the fact that the specific entropy is constant for the adiabatic, internally reversible expansion through the turbine. Using saturated liquid and saturated vapor data from Table A-3, we find that the quality at state 2 is x2

s2 sf 5.7432 0.5926 0.6745 sg sf 7.6361

332

Chapter 8 Vapor Power Systems

The enthalpy is then h2 hf x2hfg 173.88 10.674522403.1 1794.8 kJ/kg

State 3 is saturated liquid at 0.008 MPa, so h3 173.88 kJ/kg. State 4 is fixed by the boiler pressure p4 and the specific entropy s4 s3. The specific enthalpy h4 can be found by interpolation in the compressed liquid tables. However, because compressed liquid data are relatively sparse, it is more convenient to solve Eq. 8.3 for h4, using Eq. 8.7b to approximate the pump work. With this approach # # h4 h3 Wp m h3 v3 1 p4 p3 2 By inserting property values from Table A-3 h4 173.88 kJ/kg 11.0084 10 3 m3/kg2 18.0 0.0082 MPa `

1 kJ 106 N/m2 ` ` 3 # ` 1 MPa 10 N m

173.88 8.06 181.94 kJ/kg (a) The net power developed by the cycle is # # # Wcycle Wt Wp Mass and energy rate balances for control volumes around the turbine and pump give, respectively # # Wp Wt and # h1 h 2 # h4 h3 m m # where m is the mass flow rate of the steam. The rate of heat transfer to the working fluid as it passes through the boiler is determined using mass and energy rate balances as # Qin # h1 h4 m The thermal efficiency is then # # Wt Wp 1h1 h2 2 1h4 h3 2 h # h1 h4 Qin 3 12758.0 1794.82 1181.94 173.882 4 kJ/kg 12758.0 181.942 kJ/kg 0.371 137.1%2

(b) The back work ratio is

❷

# Wp 1181.94 173.882 kJ/kg h4 h3 bwr # h h 12758.0 1794.82 kJ/kg Wt 1 2 8.06 8.37 103 10.84%2 963.2 (c) The mass flow rate of the steam can be obtained from the expression for the net power given in part (a). Thus # m

# Wcycle

1h1 h2 2 1h4 h3 2

1100 MW2 103 kW/MW 3600 s/h 1963.2 8.062 kJ/kg

3.77 105 kg/h

8.2 Analyzing Vapor Power Systems—Rankine Cycle

333

# (d) With the expression for Qin from part (a) and previously determined specific enthalpy values # # Qin m 1h1 h4 2 13.77 105 kg/ h2 12758.0 181.942 kJ/kg 3600 s/h 103 kW/MW 269.77 MW (e) Mass and energy rate balances applied to a control volume enclosing the steam side of the condenser give # # Qout m 1h2 h3 2 13.77 105 kg/h211794.8 173.882 kJ/kg 3600 s/h 103 kW/MW

❸

169.75 MW # # Note that the ratio # of Qout to Qin is 0.629 (62.9%). Alternatively, Qout can be determined from an energy rate balance on the overall vapor power plant. At steady state, the net power developed equals the net rate of heat transfer to the plant # # # Wcycle Qin Qout Rearranging this expression and inserting values # # # Qout Qin Wcycle 269.77 MW 100 MW 169.77 MW The slight difference from the above value is due to round-off. (f) Taking a control volume around the condenser, the mass and energy rate balances give at steady state

# where mcw

#0 #0 # # 0 Qcv Wcv mcw 1hcw,in hcw,out 2 m 1h2 h3 2 # is the mass flow rate of the cooling water. Solving for mcw # m 1h2 h3 2 # mcw 1hcw,out hcw,in 2

The numerator in this expression is evaluated in part (e). For the cooling water, h hf (T), so with saturated liquid enthalpy values from Table A-2 at the entering and exiting temperatures of the cooling water 1169.75 MW2 103 kW/MW 3600 s/h # mcw 7.3 106 kg/h 1146.68 62.992 kJ/kg

❶

Note that a slightly revised problem-solving methodology is used in this example problem: We begin with a systematic evaluation of the specific enthalpy at each numbered state.

❷

Note that the back work ratio is relatively low for the Rankine cycle. In the present case, the work required to operate the pump is less than 1% of the turbine output.

❸

In this example, 62.9% of the energy added to the working fluid by heat transfer is subsequently discharged to the cooling water. Although considerable energy is carried away by the cooling water, its exergy is small because the water exits at a temperature only a few degrees greater than that of the surroundings. See Sec. 8.6 for further discussion.

8.2.3 Effects of Boiler and Condenser Pressures on the Rankine Cycle In discussing Fig. 5.9 (Sec. 5.5.1), we observed that the thermal efficiency of power cycles tends to increase as the average temperature at which energy is added by heat transfer increases and/or the average temperature at which energy is rejected decreases. (For elaboration, see box.) Let us apply this idea to study the effects on performance of the ideal Rankine cycle of changes in the boiler and condenser pressures. Although these findings are obtained with reference to the ideal Rankine cycle, they also hold qualitatively for actual vapor power plants.

334

Chapter 8 Vapor Power Systems

C O N S I D E R I N G T H E E F F E C T O F T E M P E R AT U R E O N THERMAL EFFICIENCY

Since the ideal Rankine cycle consists entirely of internally reversible processes, an expression for thermal efficiency can be obtained in terms of average temperatures during the heat interaction processes. Let us begin the development of this expression by recalling that areas under the process lines of Fig. 8.3 can be interpreted as the heat transfer per unit of mass flowing through the respective components. For example, the total area 1–b–c–4–a–1 represents the heat transfer into the working fluid per unit of mass passing through the boiler. In symbols, # 1 Qin a # b int T ds area 1–b–c–4–a–1 m rev 4

The integral can be written in terms of an average temperature of heat addition, Tin, as follows: # Qin a # b int Tin 1s1 s4 2 m rev where the overbar denotes average. Similarly, area 2–b–c–3–2 represents the heat transfer from the condensing steam per unit of mass passing through the condenser # Qout a # b int Tout 1s2 s3 2 area 2–b–c–3–2 m rev Tout 1s1 s4 2 where Tout denotes the temperature on the steam side of the condenser of the ideal Rankine cycle pictured in Fig. 8.3. The thermal efficiency of the ideal Rankine cycle can be expressed in terms of these heat transfers as # # 1Qout m2 int Tout rev hideal 1 # 1 (8.8) # Tin 1Qin m2 int rev

By the study of Eq. 8.8, we conclude that the thermal efficiency of the ideal cycle tends to increase as the average temperature at which energy is added by heat transfer increases and/or the temperature at which energy is rejected decreases. With similar reasoning, these conclusions can be shown to apply to the other ideal cycles considered in this chapter and the next.

Figure 8.4a shows two ideal cycles having the same condenser pressure but different boiler pressures. By inspection, the average temperature of heat addition is seen to be greater for the higher-pressure cycle 1–2–3–4–1 than for cycle 1–2–3–4–1. It follows that increasing the boiler pressure of the ideal Rankine cycle tends to increase the thermal efficiency. Figure 8.4b shows two cycles with the same boiler pressure but two different condenser pressures. One condenser operates at atmospheric pressure and the other at less than atmospheric pressure. The temperature of heat rejection for cycle 1–2–3–4–1 condensing at atmospheric pressure is 100C (212F). The temperature of heat rejection for the lowerpressure cycle 1–2–3–4–1 is correspondingly lower, so this cycle has the greater thermal efficiency. It follows that decreasing the condenser pressure tends to increase the thermal efficiency.

8.2 Analyzing Vapor Power Systems—Rankine Cycle T

335

T Fixed boiler pressure

Increased boiler pressure

1′

1

1

Decreased condenser pressure

4 4′ 4

Fixed condenser pressure

3

100° C (212° F)

2′ 2

patm

2 4″ 3″

3

p < patm

2″

Ambient temperature

s

s

(a)

(b)

Figure 8.4 Effects of varying operating pressures on the ideal Rankine cycle. (a) Effect of boiler pressure. (b) Effect of condenser pressure.

The lowest feasible condenser pressure is the saturation pressure corresponding to the ambient temperature, for this is the lowest possible temperature for heat rejection to the surroundings. The goal of maintaining the lowest practical turbine exhaust (condenser) pressure is a primary reason for including the condenser in a power plant. Liquid water at atmospheric pressure could be drawn into the boiler by a pump, and steam could be discharged directly to the atmosphere at the turbine exit. However, by including a condenser in which the steam side is operated at a pressure below atmospheric, the turbine has a lower-pressure region in which to discharge, resulting in a significant increase in net work and thermal efficiency. The addition of a condenser also allows the working fluid to flow in a closed loop. This arrangement permits continual circulation of the working fluid, so purified water that is less corrosive than tap water can be used economically. COMPARISON WITH CARNOT CYCLE. Referring to Fig. 8.5, the ideal Rankine cycle 1–2–3–4–4–1 has a lower thermal efficiency than the Carnot cycle 1–2–3–4–1 having the same maximum temperature TH and minimum temperature TC because the average temperature between 4 and 4 is less than TH. Despite the greater thermal efficiency of the Carnot

Cooling curve for the products of combustion

TH

p≈

con

sta nt

T

4′

1

4 TC

3

3′

2 Figure 8.5 s

Illustration used to compare the ideal Rankine cycle with the Carnot cycle.

336

Chapter 8 Vapor Power Systems

cycle, it has two shortcomings as a model for the simple vapor power cycle. First, the heat passing to the working fluid of a vapor power plant is usually obtained from hot products of combustion cooling at approximately constant pressure. To exploit fully the energy released on combustion, the hot products should be cooled as much as possible. The first portion of the heating process of the Rankine cycle shown in Fig. 8.5, Process 4–4, is achieved by cooling the combustion products below the maximum temperature TH. With the Carnot cycle, however, the combustion products would be cooled at the most to TH. Thus, a smaller portion of the energy released on combustion would be used. The second shortcoming of the Carnot vapor power cycle involves the pumping process. Note that the state 3 of Fig. 8.5 is a two-phase liquid–vapor mixture. Significant practical problems are encountered in developing pumps that handle two-phase mixtures, as would be required by Carnot cycle 1–2–3– 4–1. It is far easier to condense the vapor completely and handle only liquid in the pump, as is done in the Rankine cycle. Pumping from 3 to 4 and constant-pressure heating without work from 4 to 4 are processes that can be closely achieved in practice. 8.2.4 Principal Irreversibilities and Losses Irreversibilities and losses are associated with each of the four subsystems shown in Fig. 8.1. Some of these effects have a more pronounced influence on performance than others. Let us consider the irreversibilities and losses associated with the Rankine cycle. TURBINE. The principal irreversibility experienced by the working fluid is associated with the expansion through the turbine. Heat transfer from the turbine to the surroundings represents a loss, but since it is usually of secondary importance, this loss is ignored in subsequent discussions. As illustrated by Process 1–2 of Fig. 8.6, an actual adiabatic expansion through the turbine is accompanied by an increase in entropy. The work developed per unit of mass in this process is less than for the corresponding isentropic expansion 1–2s. The isentropic turbine efficiency ht introduced in Sec. 6.8 allows the effect of irreversibilities within the turbine to be accounted for in terms of the actual and isentropic work amounts. Designating the states as in Fig. 8.6, the isentropic turbine efficiency is

# # 1Wt m2 h1 h2 ht # # h1 h2s 1Wt m2 s

(8.9)

where the numerator is the actual work developed per unit of mass passing through the turbine and the denominator is the work for an isentropic expansion from the turbine inlet state

T

1

4 4s 3

2s 2

s

Figure 8.6 Temperature–entropy diagram showing the effects of turbine and pump irreversibilities.

8.2 Analyzing Vapor Power Systems—Rankine Cycle

to the turbine exhaust pressure. Irreversibilities within the turbine significantly reduce the net power output of the plant. The work input to the pump required to overcome frictional effects also reduces the net power output of the plant. In the absence of heat transfer to the surroundings, there would be an increase in entropy across the pump. Process 3– 4 of Fig. 8.6 illustrates the actual pumping process. The work input for this process is greater than for the corresponding isentropic process 3– 4s. The isentropic pump efficiency p introduced in Sec. 6.8 allows the effect of irreversibilities within the pump to be accounted for in terms of the actual and isentropic work amounts. Designating the states as in Fig. 8.6, the isentropic pump efficiency is PUMP.

# # 1Wp m2 s h4s h3 hp # # h4 h3 1Wp m2

(8.10)

In this expression, the pump work for the isentropic process appears in the numerator. The actual pump work, being the larger quantity, is the denominator. Because the pump work is so much less than the turbine work, irreversibilities in the pump have a much smaller impact on the net work of the cycle than do irreversibilities in the turbine. The turbine and pump irreversibilities mentioned above are internal irreversibilities experienced by the working fluid as it flows around the closed loop of the Rankine cycle. The most significant sources of irreversibility for a fossil-fueled vapor power plant, however, are associated with the combustion of the fuel and the subsequent heat transfer from the hot combustion products to the cycle working fluid. These effects occur in the surroundings of the subsystem labeled A on Fig. 8.1 and thus are external irreversibilities for the Rankine cycle. These irreversibilities are considered further in Sec. 8.6 and Chap. 13 using the exergy concept. Another effect that occurs in the surroundings is the energy discharge to the cooling water as the working fluid condenses. Although considerable energy is carried away by the cooling water, its utility is extremely limited. For condensers in which steam condenses near the ambient temperature, the cooling water experiences a temperature rise of only a few degrees over the temperature of the surroundings in passing through the condenser and thus has limited usefulness. Accordingly, the significance of this loss is far less than suggested by the magnitude of the energy transferred to the cooling water. The utility of condenser cooling water is considered further in Sec. 8.6 using the exergy concept. In addition to the foregoing, there are several other sources of nonideality. For example, stray heat transfers from the outside surfaces of the plant components have detrimental effects on performance, since such losses reduce the extent of conversion from heat input to work output. Frictional effects resulting in pressure drops are sources of internal irreversibility as the working fluid flows through the boiler, condenser, and piping connecting the various components. Detailed thermodynamic analyses would account for these effects. For simplicity, however, they are ignored in the subsequent discussions. Thus, Fig. 8.6 shows no pressure drops for flow through the boiler and condenser or between plant components. Another effect on performance is suggested by the placement of state 3 on Fig. 8.6. At this state, the temperature of the working fluid exiting the condenser would be lower than the saturation temperature corresponding to the condenser pressure. This is disadvantageous because a greater heat transfer would be required in the boiler to bring the water to saturation. In the next example, the ideal Rankine cycle of Example 8.1 is modified to include the effects of irreversibilities in the turbine and pump.

OTHER NONIDEALITIES.

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Chapter 8 Vapor Power Systems

338

EXAMPLE

8.2

Rankine Cycle with Irreversibilities

Reconsider the vapor power cycle of Example 8.1, but include in the analysis that the turbine and the pump each have an isentropic efficiency of 85%. Determine for the modified cycle (a) the # thermal efficiency, (b) the mass flow rate of steam, in kg/h, for a net power output of 100 MW, # (c) the rate of heat transfer Qin into the working fluid as it passes through the boiler, in MW, (d) the rate of heat transfer Qout from the condensing steam as it passes through the condenser, in MW, (e) the mass flow rate of the condenser cooling water, in kg/h, if cooling water enters the condenser at 15C and exits as 35C. Discuss the effects on the vapor cycle of irreversibilities within the turbine and pump. SOLUTION Known: A vapor power cycle operates with steam as the working fluid. The turbine and pump both have efficiencies of 85%. Find: Determine the thermal efficiency, the mass flow rate, in kg/h, the rate of heat transfer to the working fluid as it passes through the boiler, in MW, the heat transfer rate from the condensing steam as it passes through the condenser, in MW, and the mass flow rate of the condenser cooling water, in kg/h. Discuss. Schematic and Given Data: Assumptions:

T 8.0 MPa

1. Each component of the cycle is analyzed as a control volume at steady state.

1

2. The working fluid passes through the boiler and condenser at constant pressure. Saturated vapor enters the turbine. The condensate is saturated at the condenser exit.

4 4s 3

3. The turbine and pump each operate adiabatically with an efficiency of 85%.

0.008 MPa

2s 2

4. Kinetic and potential energy effects are negligible. s

Figure E8.2

Analysis: Owing to the presence of irreversibilities during the expansion of the steam through the turbine, there is an increase in specific entropy from turbine inlet to exit, as shown on the accompanying T–s diagram. Similarly, there is an increase in specific entropy from pump inlet to exit. Let us begin the analysis by fixing each of the principal states. State 1 is the same as in Example 8.1, so h1 2758.0 kJ/kg and s1 5.7432 kJ/kg # K. The specific enthalpy at the turbine exit, state 2, can be determined using the turbine efficiency # # Wt m h1 h2 ht # # h1 h2s 1Wt m2 s

where h2s is the specific enthalpy at state 2s on the accompanying T–s diagram. From the solution to Example 8.1, h2s 1794.8 kJ/ kg. Solving for h2 and inserting known values h2 h1 ht 1h1 h2s 2

2758 0.8512758 1794.82 1939.3 kJ/kg

State 3 is the same as in Example 8.1, so h3 173.88 kJ/kg. To determine the specific# enthalpy at the pump exit, state 4, reduce mass and energy rate balances for a control volume # around the pump to obtain Wpm h4 h3. On rearrangement, the specific enthalpy at state 4 is # # h4 h3 Wpm To determine h4 from this expression requires the pump work, which can be evaluated using the pump efficiency hp, as follows. By definition # # 1Wp m2 s hp # # 1Wp m2

8.2 Analyzing Vapor Power Systems—Rankine Cycle

# # # # The term 1Wpm2 s can be evaluated using Eq. 8.7b. Then solving for Wp m results in # Wp v3 1 p4 p3 2 # hp m The numerator of this expression was determined in the solution to Example 8.1. Accordingly, # Wp 8.06 kJ/kg 9.48 kJ/kg # m 0.85 The specific enthalpy at the pump exit is then # # h4 h3 Wp m 173.88 9.48 183.36 kJ/kg (a) The net power developed by the cycle is # # # # Wcycle Wt Wp m 3 1h1 h2 2 1h4 h3 2 4 The rate of heat transfer to the working fluid as it passes through the boiler is # # Qin m 1h1 h4 2 Thus, the thermal efficiency is h

1h1 h2 2 1h4 h3 2 h1 h4

Inserting values h

12758 1939.32 9.48 2758 183.36

0.314 131.4%2

(b) With the net power expression of part (a), the mass flow rate of the steam is # Wcycle

# m

1h1 h2 2 1h4 h3 2

1100 MW2 3600 s/h 103 kW/MW

1818.7 9.482 kJ/kg

4.449 105 kg/h

# (c) With the expression for Qin from part (a) and previously determined specific enthalpy values # # Qin m 1h1 h4 2 14.449 105 kg/h212758 183.362 kJ/kg

3600 s/h 103 kW/MW

318.2 MW

(d) The rate of heat transfer from the condensing steam to the cooling water is # # Qout m 1h2 h3 2

14.449 105 kg/h2 11939.3 173.882 kJ/kg 3600 s/h 103 kW/MW

218.2 MW

(e) The mass flow rate of the cooling water can be determined from # mcw

# m 1h2 h3 2

1hcw,out hcw,in 2

1218.2 MW2 103 kW/MW 3600 s/h 1146.68 62.992 kJ/kg

9.39 106 kg/h

339

340

Chapter 8 Vapor Power Systems

The effect of irreversibilities within the turbine and pump can be gauged by comparing the present values with their counterparts in Example 8.1. In this example, the turbine work per unit of mass is less and the pump work per unit of mass is greater than in Example 8.1. The thermal efficiency in the present case is less than in the ideal case of the previous example. For a fixed net power output (100 MW), the smaller net work output per unit mass in the present case dictates a greater mass flow rate of steam. The magnitude of the heat transfer to the cooling water is greater in this example than in Example 8.1; consequently, a greater mass flow rate of cooling water would be required.

8.3 Improving Performance—Superheat and Reheat

The representations of the vapor power cycle considered thus far do not depict actual vapor power plants faithfully, for various modifications are usually incorporated to improve overall performance. In this section we consider two cycle modifications known as superheat and reheat. Both features are normally incorporated into vapor power plants. Let us begin the discussion by noting that an increase in the boiler pressure or a decrease in the condenser pressure may result in a reduction of the steam quality at the exit of the turbine. This can be seen by comparing states 2 and 2 of Figs. 8.4a and 8.4b to the corresponding state 2 of each diagram. If the quality of the mixture passing through the turbine becomes too low, the impact of liquid droplets in the flowing liquid–vapor mixture can erode the turbine blades, causing a decrease in the turbine efficiency and an increased need for maintenance. Accordingly, common practice is to maintain at least 90% quality (x 0.9) at the turbine exit. The cycle modifications known as superheat and reheat permit advantageous operating pressures in the boiler and condenser and yet offset the problem of low quality of the turbine exhaust. superheat

SUPERHEAT. First, let us consider superheat. As we are not limited to having saturated vapor at the turbine inlet, further energy can be added by heat transfer to the steam, bringing it to a superheated vapor condition at the turbine inlet. This is accomplished in a separate heat exchanger called a superheater. The combination of boiler and superheater is referred to as a steam generator. Figure 8.3 shows an ideal Rankine cycle with superheated vapor at the turbine inlet: cycle 1–2–3–4–1. The cycle with superheat has a higher average temperature of heat addition than the cycle without superheating (cycle 1–2–3–4–1), so the thermal efficiency is higher. Moreover, the quality at turbine exhaust state 2 is greater than at state 2, which would be the turbine exhaust state without superheating. Accordingly, superheating also tends to alleviate the problem of low steam quality at the turbine exhaust. With sufficient superheating, the turbine exhaust state may even fall in the superheated vapor region.

reheat

REHEAT. A further modification normally employed in vapor power plants is reheat. With reheat, a power plant can take advantage of the increased efficiency that results with higher boiler pressures and yet avoid low-quality steam at the turbine exhaust. In the ideal reheat cycle shown in Fig. 8.7, the steam does not expand to the condenser pressure in a single stage. The steam expands through a first-stage turbine (Process 1–2) to some pressure between the steam generator and condenser pressures. The steam is then reheated in the steam generator (Process 2–3). Ideally, there would be no pressure drop as the steam is reheated. After reheating, the steam expands in a second-stage turbine to the condenser pressure (Process 3–4). The principal advantage of reheat is to increase the quality of the steam at the turbine exhaust. This can be seen from the T–s diagram of Fig. 8.7 by comparing state 4 with

8.3 Improving Performance—Superheat and Reheat

341

Reheat section

Low-pressure turbine

3 2

˙ in Q

W˙ t 1

1

Highpressure turbine

4

3

Condenser

Figure 8.7

˙ out Q

Pump

W˙ p

T3

2

Steam generator

6

T1

T

6 5

4′ 4 s

5

Ideal reheat cycle.

state 4, the turbine exhaust state without reheating. When computing the thermal efficiency of a reheat cycle, it is necessary to account for the work output of both turbine stages as well as the total heat addition occurring in the vaporization/superheating and reheating processes. This calculation is illustrated in Example 8.3. The temperature of the steam entering the turbine is restricted by metallurgical limitations imposed by the materials used to fabricate the superheater, reheater, and turbine. High pressure in the steam generator also requires piping that can withstand great stresses at elevated temperatures. Although these factors limit the gains that can be realized through superheating and reheating, improved materials and methods of fabrication have permitted significant increases over the years in the maximum allowed cycle temperatures and steam generator pressures, with corresponding increases in thermal efficiency. This has progressed to the extent that vapor power plants can be designed to operate with steam generator pressures exceeding the critical pressure of water 22.1 MPa, and turbine inlet temperatures exceeding 600C. Figure 8.8 shows an ideal reheat cycle with a supercritical steam generator pressure. Observe that no phase change occurs during the heat addition process from 6 to 1. In the next example, the ideal Rankine cycle of Example 8.1 is modified to include superheat and reheat. SUPERCRITICAL CYCLE.

EXAMPLE

8.3

1

T

3

2

6 5

4 s

Figure 8.8 Supercritical ideal reheat cycle.

Ideal Reheat Cycle

Steam is the working fluid in an ideal Rankine cycle with superheat and reheat. Steam enters the first-stage turbine at 8.0 MPa, 480C, and expands to 0.7 MPa. It is then reheated to 440C before entering the second-stage turbine, where it expands to the condenser pressure of 0.008 MPa. The net power output is 100 MW. # Determine (a) the thermal efficiency of the cycle, (b) the mass flow rate of steam, in kg/h, (c) the rate of heat transfer Qout from the condensing steam as it passes through the condenser, in MW. Discuss the effects of reheat on the vapor power cycle.

342

Chapter 8 Vapor Power Systems

SOLUTION Known: An ideal reheat cycle operates with steam as the working fluid. Operating pressures and temperatures are specified, and the net power output is given. Find: Determine the thermal efficiency, the mass flow rate of the steam, in kg/h, and the heat transfer rate from the condensing steam as it passes through the condenser, in MW. Discuss. Schematic and Given Data:

T1 = 480°C p1 = 8.0 MPa 1

Steam generator

Turbine 1 Turbine 2

p2 = 0.7 MPa 2 3

1

T

T1

4

3

T3 = 440°C

T3

8.0 MPa Condenser

0.7 MPa

2

pcond = 0.008 MPa 6 Pump 5

6

0.008 MPa

5 Saturated liquid

4 s

Figure E8.3

Assumptions: 1. Each component in the cycle is analyzed as a control volume at steady state. The control volumes are shown on the accompanying sketch by dashed lines. 2. All processes of the working fluid are internally reversible. 3. The turbine and pump operate adiabatically. 4. Condensate exits the condenser as saturated liquid. 5. Kinetic and potential energy effects are negligible. Analysis: To begin, we fix each of the principal states. Starting at the inlet to the first turbine stage, the pressure is 8.0 MPa and the temperature is 480C, so the steam is a superheated vapor. From Table A-4, h1 3348.4 kJ/kg and s1 6.6586 kJ/kg # K. State 2 is fixed by p2 0.7 MPa and s2 s1 for the isentropic expansion through the first-stage turbine. Using saturated liquid and saturated vapor data from Table A-3, the quality at state 2 is x2

s2 sf 6.6586 1.9922 0.9895 sg sf 6.708 1.9922

The specific enthalpy is then h2 hf x2hfg

697.22 10.989522066.3 2741.8 kJ/kg

State 3 is superheated vapor with p3 0.7 MPa and T3 440C, so from Table A-4, h3 3353.3 kJ/kg and s3 7.7571 kJ/kg # K.

8.3 Improving Performance—Superheat and Reheat

343

To fix state 4, use p4 0.008 MPa and s4 s3 for the isentropic expansion through the second-stage turbine. With data from Table A-3, the quality at state 4 is x4

s4 sf 7.7571 0.5926 0.9382 sg sf 8.2287 0.5926

The specific enthalpy is h4 173.88 10.938222403.1 2428.5 kJ/kg State 5 is saturated liquid at 0.008 MPa, so h5 173.88 kJ/kg. Finally, the state at the pump exit is the same as in Example 8.1, so h6 181.94 kJ/kg. (a) The net power developed by the cycle is # # # # Wcycle Wt1 Wt2 Wp Mass and energy rate balances for the two turbine stages and the pump reduce to give, respectively # # Turbine 1: Wt1m h1 h2 # # Turbine 2: Wt2m h3 h4 # # Pump: Wpm h6 h5 # where m is the mass flow rate of the steam. The total rate of heat transfer to the working fluid as it passes through the boiler–superheater and reheater is # Qin # 1h1 h6 2 1h3 h2 2 m Using these expressions, the thermal efficiency is h

1h1 h2 2 1h3 h4 2 1h6 h5 2 1h1 h6 2 1h3 h2 2

13348.4 2741.82 13353.3 2428.52 1181.94 173.882 13348.4 181.942 13353.3 2741.82

1523.3 kJ/kg 606.6 924.8 8.06 0.403 140.3%2 3166.5 611.5 3778 kJ/kg

(b) The mass flow rate of the steam can be obtained with the expression for net power given in part (a). # Wcycle # m 1h1 h2 2 1h3 h4 2 1h6 h5 2

1100 MW2 3600 s/h 103 kW/MW 1606.6 924.8 8.062 kJ/kg

2.363 105 kg/h

(c) The rate of heat transfer from the condensing steam to the cooling water is # # Qout m 1h4 h5 2 2.363 105 kg/h 12428.5 173.882 kJ/kg 148 MW 3600 s/h 103 kW/MW To see the effects of reheat, we compare the present values with their counterparts in Example 8.1. With superheat and reheat, the thermal efficiency is increased over that of the cycle of Example 8.1. For a specified net power output (100 MW), a larger thermal efficiency means that a smaller mass flow rate of steam is required. Moreover, with a greater thermal efficiency the rate of heat transfer to the cooling water is also less, resulting in a reduced demand for cooling water. With reheating, the steam quality at the turbine exhaust is substantially increased over the value for the cycle of Example 8.1.

Chapter 8 Vapor Power Systems

344

The following example illustrates the effect of turbine irreversibilities on the ideal reheat cycle of Example 8.3.

EXAMPLE

Reheat Cycle with Turbine Irreversibility

8.4

Reconsider the reheat cycle of Example 8.3, but include in the analysis that each turbine stage has the same isentropic efficiency. (a) If t 85%, determine the thermal efficiency. (b) Plot the thermal efficiency versus turbine stage efficiency ranging from 85 to 100%. SOLUTION Known: A reheat cycle operates with steam as the working fluid. Operating pressures and temperatures are specified. Each turbine stage has the same isentropic efficiency. Find: If t 85%, determine the thermal efficiency. Also, plot the thermal efficiency versus turbine stage efficiency r