COMPUTATIONAL MECHANICS and APPLIED ANALYSIS

ADVANCED

THERMODYNAMICS

ENGINEERING

CRC Series in

COMPUTATIONAL MECHANICS

and APPLIED ANALYSIS

Series Editor: J.N. Reddy Texas A&M University

Published Titles APPLIED FUNCTIONAL ANALYSIS

J. Tinsley Oden and Leszek F. Demkowicz THE FINITE ELEMENT METHOD IN HEAT TRANSFER

AND FLUID DYNAMICS, Second Edition J.N. Reddy and D.K. Gartling

MECHANICS OF LAMINATED COMPOSITE PLATES:

THEORY AND ANALYSIS J.N. Reddy

PRACTICAL ANALYSIS OF COMPOSITE LAMINATES

J.N. Reddy and Antonio Miravete SOLVING ORDINARY and PARTIAL BOUNDARY

VALUE PROBLEMS in SCIENCE and ENGINEERING

Karel Rektorys

Library of Congress Cataloging-in-Publication Data

Annamalai, Kalyan. Advanced thermodynamics engineering / Kalyan Annamalai & Ishwar K. Puri. p. cm. — (CRC series in computational mechanics and applied analysis) Includes bibliographical references and index. ISBN 0-8493-2553-6 (alk. paper)

1. Thermodynamics. I. Puri, Ishwar Kanwar, 1959- II. Title. III. Series. TJ265 .A55 2001

621.402 ′1—dc21 2001035624

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher.

The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying.

Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for

identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com

© 2002 by CRC Press LLC No claim to original U.S. Government works

International Standard Book Number 0-8493-2553-6 Library of Congress Card Number 2001035624 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

KA dedicates this text to his mother Kancheepuram Pattammal Sunda- ram, who could not read or write, and his father, Thakkolam K. Sunda- ram, who was schooled through only a few grades, for educating him in all aspects of his life. He thanks his wife Vasanthal for companionship throughout the cliff–hanging journey to this land of opportunity and his children, Shankar, Sundhar and Jothi for providing a vibrant source of “energy” in his career.

IKP thanks his wife Beth for her friendship and support and acknowl- edges his debt to his sons Shivesh, Sunil, and Krishan, for allowing him to take time off from other pressing responsibilities, such as playing catch. His career has been a fortunate journey during which his entire family, including his parents Krishan and Sushila Puri, has played a vital role.

PREFACE

We have written this text for engineers who wish to grasp the engineering physics of thermodynamic concepts and apply the knowledge in their field of interest rather than merely digest the abstract generalized concepts and mathematical relations governing thermodynam- ics. While the fundamental concepts in any discipline are relatively invariant, the problems it faces keep changing. In many instances we have included physical explanations along with the mathematical relations and equations so that the principles can be relatively applied to real world problems.

The instructors have been teaching advanced thermodynamics for more than twelve years using various thermodynamic texts written by others. In writing this text, we acknowl- edge that debt and that to our students who asked questions that clarified each chapter that we wrote. This text uses a “down–to–earth” and, perhaps, unconventional approach in teaching advanced concepts in thermodynamics. It first presents the phenomenological approach to a problem and then delves into the details. Thereby, we have written the text in the form of a self–teaching tool for students and engineers, and with ample example problems. Readers will find the esoteric material to be condensed and, as engineers, we have stressed applications throughout the text. There are more than 110 figures and 150 engineering examples covering thirteen chapters.

Chapter 1 contains an elementary overview of undergraduate thermodynamics, mathematics and a brief look at the corpuscular aspects of thermodynamics. The overview of microscopic thermodynamics illustrates the physical principles governing the macroscopic behavior of substances that are the subject of classical thermodynamics. Fundamental concepts related to matter, phase (solid, liquid, and gas), pressure, saturation pressure, temperature, en- ergy, entropy, component property in a mixture and stability are discussed.

Chapter 2 discusses the first law for closed and open systems and includes problems involving irreversible processes. The second law is illustrated in Chapter 3 rather than pre- senting an axiomatic approach. Entropy is introduced through a Carnot cycle using ideal gas as the medium, and the illustration that follows considers any reversible cycle operating with any medium. Entropy maximization and energy minimization principles are illustrated. Chapter 4 introduces the concept of availability with a simple engineering scheme that is followed by the most general treatment. Availability concepts are illustrated by scaling the performance of various components in a thermodynamic system (such as a power plant or air conditioner) and determining which component degrades faster or outperforms others. Differential forms of energy and mass conservation, and entropy and availability balance equations are presented in Chapters 2 to 4 using the Gauss divergence theorem. The differential formulations allow the reader to determine where the maximum entropy generation or irreversibility occurs within a unit so as to pinpoint the major source of the irreversibility for an entire unit. Entropy genera- tion and availability concepts are becoming more important to energy systems and conserva- tion groups. This is a rapidly expanding field in our energy–conscious society. Therefore, a number of examples are included to illustrate applications to engineering systems. Chapter 5 contains a postulatory approach to thermodynamics. In case the reader is pressed for time, this chapter may be entirely skipped without loss of continuity of the subject.

Chapter 6 presents the state equation for real gases including two and three parameter, and generalized equations of state. The Kessler equation is then introduced and the methodol-

ogy for determining Z (1) and Z is discussed. Chapter 7 starts with Maxwell’s relations fol- lowed by the development of generalized thermodynamic relations. Illustrative examples are

presented for developing tables of thermodynamic properties using the Real Gas equations. Chapter 8 contains the theory of mixtures followed by a discussion of fugacity and activity. Following the methodology for estimating the properties of steam from state equations, a methodology is presented for estimating partial molal properties from mixture state equations. Chapter 9 deals with phase equilibrium of multicomponent mixtures and vaporization and boiling. Applications to engineering problems are included. Chapter 10 discusses the regimes presented for developing tables of thermodynamic properties using the Real Gas equations. Chapter 8 contains the theory of mixtures followed by a discussion of fugacity and activity. Following the methodology for estimating the properties of steam from state equations, a methodology is presented for estimating partial molal properties from mixture state equations. Chapter 9 deals with phase equilibrium of multicomponent mixtures and vaporization and boiling. Applications to engineering problems are included. Chapter 10 discusses the regimes

Chapter 11 deals with reactive mixtures dealing with complete combustion, flame temperatures and entropy generation in reactive systems. In Chapter 12 criteria for the direc- tion of chemical reactions are developed, followed by a discussion of equilibrium calculations using the equilibrium constant for single and multi-phase systems, as well as the Gibbs mini- mization method. Chapter 13 presents an availability analysis of chemically reacting systems. Physical explanations for achieving the work equivalent to chemical availability in thermody- namic systems are included. The summary at the end of each chapter provides a brief review of the chapter for engineers in industry.

Exercise problems are placed at the end. This is followed by several tables containing thermodynamic properties and other useful information. The field of thermodynamics is vast and all subject areas cannot be covered in a sin- gle text. Readers who discover errors, conceptual conflicts, or have any comments, are encour- aged to E–mail these to the authors (respectively, kannamalai@tamu.edu and ikpuri@uic.edu). The assistance of Ms. Charlotte Sims and Mr. Chun Choi in preparing portions of the manu- script is gratefully acknowledged. We wish to acknowledge helpful suggestions and critical comments from several students and faculty. We specially thank the following reviewers: Prof. Blasiak (Royal Inst. of Tech., Sweden), Prof. N. Chandra (Florida State), Prof. S. Gollahalli (Oklahoma), Prof. Hernandez (Guanajuato, Mexico), Prof. X. Li. (Waterloo), Prof. McQuay (BYU), Dr. Muyshondt. (Sandia National Laboratories), Prof. Ochterbech (Clemson), Dr. Pe- terson, (RPI), and Prof. Ramaprabhu (Anna University, Chennai, India).

KA gratefully acknowledges many interesting and stimulating discussions with Prof. Colaluca and the financial support extended by the Mechanical Engineering Department at Texas A&M University. IKP thanks several batches of students in his Advanced Thermody- namics class for proofreading the text and for their feedback and acknowledges the University of Illinois at Chicago as an excellent crucible for scientific inquiry and education.

Kalyan Annamalai, College Station, Texas Ishwar K. Puri, Chicago, Illinois

ABOUT THE AUTHORS

Kalyan Annamalai is Professor of Mechanical Engineering at Texas A&M. He received his B.S. from Anna University, Chennai, and Ph.D. from the Georgia Institute of Technology, Atlanta. After his doctoral degree, he worked as a Research Associate in the Division of Engi- neering Brown University, RI, and at AVCO-Everett Research Laboratory, MA. He has taught several courses at Texas A&M including Advanced Thermodynamics, Combustion Science and Engineering, Conduction at the graduate level and Thermodynamics, Heat Transfer, Com- bustion and Fluid mechanics at the undergraduate level. He is the recipient of the Senior TEES Fellow Award from the College of Engineering for excellence in research, a teaching award from the Mechanical Engineering Department, and a service award from ASME. He is a Fel- low of the American Society of Mechanical Engineers, and a member of the Combustion In- stitute and Texas Renewable Industry Association. He has served on several federal panels. His funded research ranges from basic research on coal combustion, group combustion of oil drops and coal, etc., to applied research on the cofiring of coal, waste materials in a boiler burner and gas fired heat pumps. He has published more that 145 journal and conference arti- cles on the results of this research. He is also active in the Student Transatlantic Student Ex- change Program (STEP).

Ishwar K. Puri is Professor of Mechanical Engineering and Chemical Engineering, and serves as Executive Associate Dean of Engineering at the University of Illinois at Chicago. He re- ceived his Ph.D. from the University of California, San Diego, in 1987. He is a Fellow of the American Society of Mechanical Engineers. He has lectured nationwide at various universities and national laboratories. Professor Puri has served as an AAAS-EPA Environmental Fellow and as a Fellow of the NASA/Stanford University Center for Turbulence Research. He has been funded to pursue both basic and applied research by a variety of federal agencies and by industry. His research has focused on the characterization of steady and unsteady laminar flames and an understanding of flame and fire inhibition. He has advised more than 20 gradu- ate student theses, and published and presented more than 120 research publications. He has served as an advisor and consultant to several federal agencies and industry. Professor Puri is active in international student educational exchange programs. He has initiated the Student Transatlantic Engineering Program (STEP) that enables engineering students to enhance their employability through innovative international exchanges that involve internship and research experiences. He has been honored for both his research and teaching activities and is the re- cipient of the UIC COE’s Faculty Research Award and the UIC Teaching Recognition Pro- gram Award.

NOMENCLATURE *

Symbol Description

SI

English Conversion

SI to English

A Helmholtz free energy

kJ

BTU 0.9478

2 A 2 area m ft 10.764

a acceleration

ms –2

ft s –2 3.281

a –1 specific Helmholtz free energy kJ kg BTU lb

a attractive force constant

a specific Helmholtz free energy kJ kmole –1 BTU lbmole –1 , 0.4299

b body volume constant m 3 kmole –1 ft 3 lbmole –1 16.018

c –1 specific heat kJ kg K BTU/lb R 0.2388 COP

Coefficient of performance

E energy, (U+KE+PE)

kJ

BTU 0.9478

E T Total energy (H+KE+PE)

kJ

BTU 0.9478

e –1 specific energy kJ kg BTU lb

e methalpy = h + ke + pe kJ kg T –1 BTU lb –1 m 0.4299

F force

kN

lb f 224.81

f fugacity kPa(or bar) lb f in –2 0.1450

G Gibbs free energy

kJ

BTU 0.9478

BTU lb –1 m 0.4299 (mass basis)

g specific Gibbs free energy

kJ kg –1

g gravitational acceleration

ms –2

ft s –2 3.281

g c gravitational constant

g Gibbs free energy (mole basis) kJ kmole –1 BTU lbmole –1 0.4299 partial molal Gibb's function,

kJ kmole ˆg –1 BTU lbmole –1 0.4299

H enthalpy

kJ

BTU 0.9478

h fg enthalpy of vaporization

kJ kg –1

BTU lb –1 m 0.4299

h –1 specific enthalpy (mass basis) kJ kg BTU lb

h o,h * ideal gas enthalpy

kJ kg –1

BTU lb –1 m 0.4299

I irreversibility

kJ

BTU 0.9478

I irreversibility per unit mass

kJ kg –1

BTU lb –1 m 0.4299

I electrical current

amp

J Joules' work equivalent of heat (1 BTU = 778.14 ft lb f ) J k

fluxes for species, heat etc kg s –1 , kW BTU s –1 0.9478 J k

fluxes for species, heat etc kg s –1 , kW lb s –1 0.4536 K

equilibrium constant KE

kinetic energy

ke –1 specific kinetic energy kJ kg BTU lb

ratio of specific heats L

3.281 l

length, height

ft

intermolecular spacing

3.281 l m mean free path

BTU 0.9478 LW

lost work

kJ

ft lb f 737.52 M

lost work

kJ

m lbmole m mass kg lb m

molecular weight, molal mass

kg kmole –1 lb

* Lower case (lc) symbols denote values per unit mass, lc symbols with a bar (e.g., h ) denote values on mole basis, lc symbols with a caret and tilde (respectively, ˆh and ˜ h ) denote values

on partial molal basis based on moles and mass, and symbols with a dot (e.g. ˙ Q ) denote rates.

m 2.2046 Y

mass fraction N

lbmole 2.2046 N Avag

number of moles

kmole

Avogadro number molecules molecules 0.4536 kmole –1 lbmole -1 n

polytropic exponent in PV n = constant P

kPa lb f in –2 0.1450 PE

pressure

kN m –2

BTU 0.9478 pe

potential energy

kJ

specific potential energy Q

BTU 0.9478 q

heat transfer

kJ

BTU lb –1 0.4299 q c charge R

heat transfer per unit mass

kJ kg –1

gas constant kJ kg –1 K –1 BTU lb –1 R –1 0.2388 R

universal gas constant –1 kJ kmole BTU lbmole 0.2388

BTU R –1 0.5266 s

entropy

kJ K –1

specific entropy (mass basis) kJ kg –1 K –1 BTU lb –1 R –1 0.2388 s

specific entropy (mole basis) kJ kmole –1 K –1 BTU lbmole –1 R –1

°F, °R (9/5)T+32 T

T temperature

U internal energy

BTU 0.9478 u

kJ

BTU lb –1 0.4299 u

specific internal energy

kJ kg –1

internal energy (mole basis) kJ kmole –1 BTU lbmole –1 0.4299

V volume m 3 ft 3 35.315

V volume m 3 gallon 264.2

ft s –1 3.281 v

V velocity

ms –1

specific volume (mass basis)

m 3 kg –1

ft 3 lb –1 m 16.018

3 v –1 specific volume (mole basis) m kmole ft lbmole 16.018 W

BTU 0.9478 W

work

kJ

ft lb f 737.5 w

work

kJ

BTU lb –1 0.4299 w

work per unit mass

kJ kg –1

Pitzer factor ω

1b m –1 lb m x

specific humidity

kg kg –1

quality x k

mole fraction of species k Y k

mass fraction ofspecies k z

compressibility factor

Greek symbols

α ˆ k activity of component k,

/f k

β P , β T , compressibility K –1 , atm –1 R –1 , bar –1 0.555, 1.013

bar 1.013 γ k

s atm

activity coefficient, ˆ α k /ˆ α id k ˆφ k / φ k Gruneisen constant λ

thermal conductivity kW m –1 K –1 BTU ft –1 R –1 0.1605 η

First Law efficiency

ω specific humidity ρ

density –3 kg m 1b m ft 0.06243 φ

equivalence ratio, fugacity coefficient φ

relative humidity, Φ

BTU 0.9478 Φ'

absolute availability(closed system)

kJ

BTU lb –1 0.4299 φ

relative availability or exergy

kJ kg –1

fugacity coefficient

J –1

ºR atm 1.824 µ

Joule Thomson Coefficient

K bar

chemical potential kJ kmole –1 BTU lbmole –1 0.4299 ν

stoichiometric coefficient σ

BTU R –1 0.2388 Ψ

entropy generation

kJ K –1

BTU lb –1 0.2388 Ψ'

absolute stream availability

kJ kg –1

relative stream availability or exergy

Subscripts

a air

b boundary

c critical chem

chemical c.m.

control mass c.v.

control volume

e exit

f flow

f saturated liquid (or fluid)

f formation

fg saturated liquid (fluid) to vapor

g saturated vapor (or gas)

H high temperature

I inlet inv

inversion id

ideal gas iso

isolated (system and surroundings) L

low temperature max

maximum possible work output between two given states (for an expansion process)

m mixture min

minimum possible work input between two given states net

net in a cyclic process p

at constant pressure p,o

at constant pressure for ideal gas R

reduced, reservoir rev

reversible r

relative pressure, relative volume s

isentropic work, solid sf

solid to fluid (liquid) sh

shaft work Th

Thermal TM

Thermo-mechanical Thermo-mechanical

at constant volume for ideal gas

v vapor (Chap. 5)

0 or o ambient, ideal gas state

Superscripts

(0) based on two parameters (1)

Pitzer factor correction α

alpha phase β

beta phase id

ideal mixture ig

ideal gas Ρ

liquid

g gas l

liquid res

residual sat

saturated o

pressure of 1 bar or 1 atm - molal property of k, pure component

^ molal property when k is in a mixure

Mathematical Symbols

δ( ) differential of a non-property, e.g., δ Q, W δ , etc.

d () differential of property, e.g., du, dh, dU, etc. ∆

change in value

Acronyms

CE Carnot Engine c.m.

control mass c.s

control surface c.v

control volume ES

Equilibrium state HE Heat engine

IPE,ipe

Intermolecular potential energy

IRHE Irreversible HE KE

Kinetic energy ke

kinetic energy per unit mass LHS

Left hand side KES

Kessler equation of state MER

Mechanical energy reservoir mph

miles per hour NQS/NQE

non-equilibrium PC

piston cylinder assembly PCW

piston cylinder weight assembly

PE Potential energy pe

potential energy per unit mass

PR Peng Robinson RE, re

Rotational energy RHE

Reversible HE

RKS

Redlich Kwong Soave

QS/QE Quasi-equilibrium ss

steady state sf

steady flow TE, te

translational TER

Thermal energy reservoir

TM thermo-mechanical equilibrium TMC

Thermo-mechanical-chemical equilibrium uf

uniform flow us

uniform state VE,ve

Vibrational energy VW

Van der Waals

Laws of Thermodynamics in Lay Terminology

First Law: It is impossible to obtain something from nothing, but one may break even Second Law: One may break even but only at the lowest possible temperature Third Law: One cannot reach the lowest possible temperature

Implication: It is impossible to obtain something from nothing, so one must optimize resources

The following equations, sometimes called the accounting equations, are useful in the engi- neering analysis of thermal systems.

Accumulation rate of an extensive property B: dB/dt = rate of B entering a volume ( ˙B i ) – rate of B leaving a volume ( ˙B e ) + rate of B generated in a volume ( ˙B gen ) – rate of B de- stroyed or consumed in a volume ( ˙B des/cons ).

Mass conservation: dm cv / dt = m ˙ i − m ˙ e .

First law or energy conservation: dE cv / dt = Q ˙ − W ˙ + me ˙ i Ti , − me ˙ e Te , ,

where e T = h + ke + pe, E = U + KE + PE, δw rev, open = –v dP, δw rev, closed = P dv.

Second law or entropy balance equation: dS cv / dt = QT ˙/ b + ms ˙ i i − ms ˙ e e +σ ˙ cv , where σ ˙ cv > 0 for an irreversible process and is equal to zero for a reversible process. Availability balance: dE ( cv − TS o cv / dt = Q ( 1 − T 0 / T R ) + m ˙ i ψ i − m ˙ e ψ e − WT ˙ − o σ ˙ cv , where ψ = (e T –T 0 s) = h + ke + pe – T 0 s, and E = U + KE + PE.

Third law: S → 0 as T → 0.

CONTENTS

Preface Nomenclature

1. Introduction

A. Importance, Significance and Limitations

B. Limitations of Thermodynamics

1. Review

a. System and Boundary

b. Simple System

c. Constraints and Restraints

d. Composite System

e. Phase

f. Homogeneous

g. Pure Substance

h. Amount of Matter and Avogadro Number i.

Mixture j.

Property k.

State l.

Equation of State m.

Standard Temperature and Pressure n.

Partial Pressure o.

Process p.

Vapor–Liquid Phase Equilibrium

C. Mathematical Background

1. Explicit and Implicit Functions and Total Differentiation

2. Exact (Perfect) and Inexact (Imperfect) Differentials

a. Mathematical Criteria for an Exact Differential

3. Conversion from Inexact to Exact Form

4. Relevance to Thermodynamics

a. Work and Heat

b. Integral over a Closed Path (Thermodynamic Cycle)

5. Homogeneous Functions

a. Relevance of Homogeneous Functions to Thermodynamics

6. Taylor Series

7. LaGrange Multipliers

8. Composite Function

9. Stokes and Gauss Theorems

a. Stokes Theorem

b. Gauss–Ostrogradskii Divergence Theorem

c. The Leibnitz Formula

D. Overview of Microscopic Thermodynamics

1. Matter

2. Intermolecular Forces and Potential Energy

3. Internal Energy, Temperature, Collision Number and Mean Free Path

a. Internal Energy and Temperature

b. Collision Number and Mean Free Path

4. Pressure

a. Relation between Pressure and Temperature

5. Gas, Liquid, and Solid

6. Work

7. Heat

8. Chemical Potential

a. Multicomponent into Multicomponent

b. Single Component into Multicomponent

9. Boiling/Phase Equilibrium

a. Single Component Fluid

b. Multiple Components

10. Entropy

11. Properties in Mixtures – Partial Molal Property

E. Summary

F. Appendix

1. Air Composition

2. Proof of the Euler Equation

3. Brief Overview of Vector Calculus

a. Scalar or Dot Product

b. Vector or Cross Product

c. Gradient of a Scalar

d. Curl of a Vector

2. First Law of Thermodynamics

A. Introduction

1. Zeroth Law

2. First Law for a Closed System

a. Mass Conservation

b. Energy Conservation

c. Systems with Internal Motion

d. Cyclical Work and Poincare Theorem

e. Quasiequilibrium Work

f. Nonquasiequilibrium Work

g. First Law in Enthalpy Form

3. First Law for an Open System

a. Conservation of Mass

b. Conservation of Energy

c. Multiple Inlets and Exits

d. Nonreacting Multicomponent System

4. Illustrations

a. Heating of a Residence in Winter

b. Thermodynamics of the Human Body

c. Charging of Gas into a Cylinder

d. Discharging Gas from Cylinders

e. Systems Involving Boundary Work

f. Charging of a Composite System

B. Integral and Differential Forms of Conservation Equations

1. Mass Conservation

a. Integral Form

b. Differential Form

2. Energy Conservation

a. Integral Form

b. Differential Form

c. Deformable Boundary

C. Summary

D. Appendix

1. Conservation Relations for a Deformable Control Volume

3. Second law and Entropy

A. Introduction

1. Thermal and Mechanical Energy Reservoirs

a. Heat Engine

b. Heat Pump and Refrigeration Cycle

B. Statements of the Second Law

1. Informal Statements

a. Kelvin (1824-1907) – Planck (1858-1947) Statement

b. Clausius (1822-1888) Statement

C. Consequences of the Second Law

1. Reversible and Irreversible Processes

2. Cyclical Integral for a Reversible Heat Engine

3. Clausius Theorem

4. Clausius Inequality

5. External and Internal Reversibility

6. Entropy

a. Mathematical Definition

b. Characteristics of Entropy

7. Relation between ds, δq and T during an Irreversible Process

a. Caratheodary Axiom II

D. Entropy Balance Equation for a Closed System

1. Infinitesimal Form

a. Uniform Temperature within a System

b. Nonuniform Properties within a System

2. Integrated Form

3. Rate Form

4. Cyclical Form

5. Irreversibility and Entropy of an Isolated System

6. Degradation and Quality of Energy

a. Adiabatic Reversible Processes

E. Entropy Evaluation

1. Ideal Gases

a. Constant Specific Heats

b. Variable Specific Heats

2. Incompressible Liquids

3. Solids

4. Entropy during Phase Change

a. T–s Diagram

5. Entropy of a Mixture of Ideal Gases

a. Gibbs–Dalton´s Law

b. Reversible Path Method

F. Local and Global Equilibrium

G. Single–Component Incompressible Fluids

H. Third law

I. Entropy Balance Equation for an Open System

1. General Expression

2. Evaluation of Entropy for a Control Volume

3. Internally Reversible Work for an Open System

4. Irreversible Processes and Efficiencies

5. Entropy Balance in Integral and Differential Form

a. Integral Form

b. Differential Form b. Differential Form

b. Solids J.

Maximum Entropy and Minimum Energy

1. Maxima and Minima Principles

a. Entropy Maximum (For Specified U, V, m)

b. Internal Energy Minimum (for specified S, V, m)

c. Enthalpy Minimum (For Specified S, P, m)

d. Helmholtz Free Energy Minimum (For Specified T, V, m)

e. Gibbs Free Energy Minimum (For Specified T, P, m)

2. Generalized Derivation for a Single Phase

a. Special Cases K.

Summary L.

Appendix

1. Proof for Additive Nature of Entropy

2. Relative Pressures and Volumes

3. LaGrange Multiplier Method for Equilibrium

a. U, V, m System

b. T, P, m System

4. Availability

A. Introduction

B. Optimum Work and Irreversibility in a Closed System

1. Internally Reversible Process

2. Useful or External Work

3. Internally Irreversible Process with no External Irreversibility

a. Irreversibility or Gouy–Stodola Theorem

4. Nonuniform Boundary Temperature in a System

C. Availability Analyses for a Closed System

1. Absolute and Relative Availability under Interactions with Ambient

2. Irreversibility or Lost Work

a. Comments

D. Generalized Availability Analysis

1. Optimum Work

2. Lost Work Rate, Irreversibility Rate, Availability Loss

3. Availability Balance Equation in Terms of Actual Work

a. Irreversibility due to Heat Transfer

4. Applications of the Availability Balance Equation

5. Gibbs Function

6. Closed System (Non–Flow Systems)

a. Multiple Reservoirs

b. Interaction with the Ambient Only

c. Mixtures

7. Helmholtz Function

E. Availability Efficiency

1. Heat Engines

a. Efficiency

b. Availability or Exergetic (Work Potential) Efficiency

2. Heat Pumps and Refrigerators

a. Coefficient of Performance

3. Work Producing and Consumption Devices

a. Open Systems:

4. Graphical Illustration of Lost, Isentropic, and Optimum Work

5. Flow Processes or Heat Exchangers

a. Significance of the Availability or Exergetic Efficiency

b. Relation between η Avail,f and η Avail,0 for Work Producing Devices

F. Chemical Availability

1. Closed System

2. Open System

a. Ideal Gas Mixtures

b. Vapor or Wet Mixture as the Medium in a Turbine

c. Vapor–Gas Mixtures

d. Psychometry and Cooling Towers

G. Integral and Differential Forms

1. Integral Form

2. Differential Form

3. Some Applications

H. Summary

5. Postulatory (Gibbsian) Thermodynamics

A. Introduction

B. Classical Rationale for Postulatory Approach

1. Simple Compressible Substance

C. Legendre Transform

1. Simple Legendre Transform

a. Relevance to Thermodynamics

2. Generalized Legendre Transform

3. Application of Legendre Transform

D. Generalized Relation for All Work Modes

1. Electrical Work

2. Elastic Work

3. Surface Tension Effects

4. Torsional Work

5. Work Involving Gravitational Field

6. General Considerations

E. Thermodynamic Postulates for Simple Systems

1. Postulate I

2. Postulate II

3. Postulate III

4. Postulate IV

F. Entropy Fundamental Equation

G. Energy Fundamental Equation

H. Intensive and Extensive Properties

I. Summary

6. State Relationships for Real Gases and Liquids

A. Introduction

B. Equations of State

C. Real Gases

1. Virial Equation of State

a. Exact Virial Equation

b. Approximate Virial Equation

2. Van der Waals (VW) Equation of State

a. Clausius–I Equation of State

4. Other Two–Parameter Equations of State

5. Compressibility Charts (Principle of Corresponding States)

6. Boyle Temperature and Boyle Curves

a. Boyle Temperature

b. Boyle Curve

c. The Z = 1 Island

7. Deviation Function

8. Three Parameter Equations of State

a. Critical Compressibility Factor (Z c ) Based Equations

b. Pitzer Factor

c. Evaluation of Pitzer factor, ω

9. Other Three Parameter Equations of State

a. One Parameter Approximate Virial Equation

b. Redlich–Kwong–Soave (RKS) Equation

c. Peng–Robinson (PR) Equation

10. Generalized Equation of State

11. Empirical Equations of State

a. Benedict–Webb–Rubin Equation

b. Beatie – Bridgemann (BB) Equation of State

c. Modified BWR Equation

d. Lee–Kesler Equation of State

e. Martin–Hou

12. State Equations for Liquids/Solids

a. Generalized State Equation

b. Murnaghan Equation of State

c. Racket Equation for Saturated Liquids

d. Relation for Densities of Saturated Liquids and Vapors

e. Lyderson Charts (for Liquids)

f. Incompressible Approximation

D. Summary

E. Appendix

1. Cubic Equation

a. Case I: γ>0

b. Case II: γ <0

2. Another Explanation for the Attractive Force

3. Critical Temperature and Attraction Force Constant

7. Thermodynamic Properties of Pure Fluids

A. Introduction

B. Ideal Gas Properties

C. James Clark Maxwell (1831–1879) Relations

1. First Maxwell Relation

a. Remarks

2. Second Maxwell Relation

a. Remarks

3. Third Maxwell Relation

a. Remarks

4. Fourth Maxwell Relation

a. Remarks

5. Summary of Relations

2. Internal Energy (du) Relation

a. Remarks

3. Enthalpy (dh) Relation

a. Remarks

4. Relation for (c p –c v )

a. Remarks

E. Evaluation of Thermodynamic Properties

1. Helmholtz Function

2. Entropy

3. Pressure

4. Internal Energy

a. Remarks

5. Enthalpy

a. Remarks

6. Gibbs Free Energy or Chemical Potential

7. Fugacity Coefficient

F. Pitzer Effect

1. Generalized Z Relation

G. Kesler Equation of State (KES) and Kesler Tables

H. Fugacity

1. Fugacity Coefficient

a. RK Equation

b. Generalized State Equation

2. Physical Meaning

a. Phase Equilibrium

b. Subcooled Liquid

c. Supercooled Vapor

I. Experiments to measure (u o – u) J.

Vapor/Liquid Equilibrium Curve

1. Minimization of Potentials

a. Helmholtz Free Energy A at specified T, V and m

b. G at Specified T, P and m

2. Real Gas Equations

a. Graphical Solution

b. Approximate Solution

3. Heat of Vaporization

4. Vapor Pressure and the Clapeyron Equation

a. Remarks

5. Empirical Relations

a. Saturation Pressures

b. Enthalpy of Vaporization

6. Saturation Relations with Surface Tension Effects

a. Remarks

b. Pitzer Factor from Saturation Relations

K. Throttling Processes

1. Joule Thomson Coefficient

a. Evaluation of µ JT

b. Remarks

2. Temperature Change during Throttling

4. Inversion Curves

a. State Equations

b. Enthalpy Charts

c. Empirical Relations

5. Throttling of Saturated or Subcooled Liquids

6. Throttling in Closed Systems

7. Euken Coefficient – Throttling at Constant Volume

a. Physical Interpretation L.

Development of Thermodynamic Tables

1. Procedure for Determining Thermodynamic Properties

2. Entropy M.

Summary

8. Thermodynamic Properties of Mixtures

A. Partial Molal Property

1. Introduction

a. Mole Fraction

b. Mass Fraction

c. Molality

d. Molecular Weight of a Mixture

2. Generalized Relations

a. Remarks

3. Euler and Gibbs–Duhem Equations

a. Characteristics of Partial Molal Properties

b. Physical Interpretation

4. Relationship Between Molal and Pure Properties

a. Binary Mixture

b. Multicomponent Mixture

5. Relations between Partial Molal and Pure Properties

a. Partial Molal Enthalpy and Gibbs function

b. Differentials of Partial Molal Properties

6. Ideal Gas Mixture

a. Volume

b. Pressure

c. Internal Energy

d. Enthalpy

e. Entropy

f. Gibbs Free Energy

7. Ideal Solution

a. Volume

b. Internal Energy and Enthalpy

c. Gibbs Function

d. Entropy

8. Fugacity

a. Fugacity and Activity

b. Approximate Solutions for ˆg k

c. Standard States

d. Evaluation of the Activity of a Component in a Mixture

e. Activity Coefficient e. Activity Coefficient

Excess Property l.

Osmotic Pressure

B. Molal Properties Using the Equations of State

1. Mixing Rules for Equations of State

a. General Rule

b. Kay’s Rule

c. Empirical Mixing Rules 25

d. Peng Robinson Equation of State

e. Martin Hou Equation of State

f. Virial Equation of State for Mixtures

2. Dalton’s Law of Additive Pressures (LAP)

3. Law of Additive Volumes (LAV)

4. Pitzer Factor for a Mixture

5. Partial Molal Properties Using Mixture State Equations

a. Kay’s Rule

b. RK Equation of State

C. Summary

9. Phase Equilibrium for a Mixture

A. Introduction

1. Miscible, Immiscible and Partially Miscible Mixture

2. Phase Equilibrium

a. Two Phase System

b. Multiphase Systems

c. Gibbs Phase Rule

B. Simplified Criteria for Phase Equilibrium

1. General Criteria for any Solution

2. Ideal Solution and Raoult’s Law

a. Vapor as Real Gas Mixture

b. Vapor as Ideal Gas Mixture

C. Pressure And Temperature Diagrams

1. Completely Miscible Mixtures

a. Liquid–Vapor Mixtures

b. Relative Volatility

c. P–T Diagram for a Binary Mixture

d. P–X k(l) –T diagram

e. Azeotropic Behavior

2. Immiscible Mixture

a. Immiscible Liquids and Miscible Gas Phase

b. Miscible Liquids and Immiscible Solid Phase

3. Partially Miscible Liquids

a. Liquid and Gas Mixtures

b. Liquid and Solid Mixtures

D. Dissolved Gases in Liquids

1. Single Component Gas

2. Mixture of Gases

3. Approximate Solution–Henry’s Law

E. Deviations From Raoult’s Law E. Deviations From Raoult’s Law

b. Two Phases

c. Three Phases

d. Theory

2. General Phase Rule for Multicomponent Fluids

3. Raoult’s Law for the Vapor Phase of a Real Gas

10. Stability

A. Introduction

B. Stability Criteria

1. Isolated System

a. Single Component

2. Mathematical Criterion for Stability

a. Perturbation of Volume

b. Perturbation of Energy

c. Perturbation with Energy and Volume

d. Multicomponent Mixture

e. System with Specified Values of S, V, and m

f. Perturbation in Entropy at Specified Volume

g. Perturbation in Entropy and Volume

h. System with Specified Values of S, P, and m

i.

System with Specified Values of T, V, and m

j.

System with Specified Values of T, P, and m

k. Multicomponent Systems

C. Application to Boiling and Condensation

1. Physical Processes and Stability

a. Physical Explanation

2. Constant Temperature and Volume

3. Specified Values of S, P, and m

4. Specified Values of S (or U), V, and m

D. Entropy Generation during Irreversible Transformation

E. Spinodal Curves

1. Single Component

2. Multicomponent Mixtures

F. Determination of Vapor Bubble and Drop Sizes

G. Universe and Stability

H. Summary

11. Chemically Reacting Systems

A. Introduction

B. Chemical Reactions and Combustion

1. Stoichiometric or Theoretical Reaction

2. Reaction with Excess Air (Lean Combustion)

3. Reaction with Excess Fuel (Rich Combustion)

4. Equivalence Ratio, Stoichiometric Ratio

5. Dry Gas Analysis

C. Thermochemistry

1. Enthalpy of Formation (Chemical Enthalpy)

2. Thermal or Sensible Enthalpy

3. Total Enthalpy

1. First Law

2. Adiabatic Flame Temperature

a. Steady State Steady Flow Processes in Open Systems

b. Closed Systems

E. Combustion Analyses In the case of Nonideal Behavior

1. Pure Component

2. Mixture

F. Second Law Analysis of Chemically Reacting Systems

1. Entropy Generated during an Adiabatic Chemical Reaction

2. Entropy Generated during an Isothermal Chemical Reaction

G. Mass Conservation and Mole Balance Equations

1. Steady State System

H. Summary

12. Reaction Direction and Chemical Equilibrium

A. Introduction

B. Reaction Direction and Chemical Equilibrium

1. Direction of Heat Transfer

2. Direction of Reaction

3. Mathematical Criteria for a Closed System

4. Evaluation of Properties during an Irreversible Chemical Reaction

a. Nonreacting Closed System

b. Reacting Closed System

c. Reacting Open System

5. Criteria in Terms of Chemical Force Potential

6. Generalized Relation for the Chemical Potential

C. Chemical Equilibrium Relations

1. Nonideal Mixtures and Solutions

a. Standard State of an Ideal Gas at 1 Bar

b. Standard State of a Nonideal Gas at 1 Bar

2. Reactions Involving Ideal Mixtures of Liquids and Solids

3. Ideal Mixture of Real Gases

4. Ideal Gases

a. Partial Pressure

b. Mole Fraction

5. Gas, Liquid and Solid Mixtures

6. van’t Hoff Equation

a. Effect of Temperature on K o (T)

b. Effect of Pressure

7. Equilibrium for Multiple Reactions

8. Adiabatic Flame Temperature with Chemical Equilibrium

a. Steady State Steady Flow Process

b. Closed Systems

9. Gibbs Minimization Method

a. General Criteria for Equilibrium

b. Multiple Components

D. Summary

E. Appendix

13. Availability Analysis for Reacting Systems

2. Adiabatic Combustion

3. Maximum Work Using Heat Exchanger and Adiabatic Combustor

4. Isothermal Combustion

5. Fuel Cells

a. Oxidation States and electrons

b. H 2 -O 2 Fuel Cell

D. Fuel Availability

E. Summary

14. Problems

A. Chapter 1 Problems

B. Chapter 2 Problems

C. Chapter 3 Problems

D. Chapter 4 Problems

E. Chapter 5 Problems

F. Chapter 6 Problems

G. Chapter 7 Problems

H. Chapter 8 Problems

I. Chapter 9 Problems J.

Chapter 10 Problems K.

Chapter 11 Problems L.

Chapter 12 Problems M.

Chapter 13 Problems

Appendix A. Tables Appendix B. Charts Appendix C. Formulae Appendix D. References

Chapter 1

1. INTRODUCTION

A. IMPORTANCE, SIGNIFICANCE AND LIMITATIONS

Thermodynamics is an engineering science topic,which deals with the science of “motion” (dynamics) and/or the transformation of “heat” (thermo) and energy into various other energy–containing forms. The flow of energy is of great importance to engineers in- volved in the design of the power generation and process industries. Examples of analyses based on thermodynamics include:

The transfer or motion of energy from hot gases emerging from a burner to cooler water in

a hot–water heater. The transformation of the thermal energy, i.e., heat, contained in the hot gases in an auto- mobile engine into mechanical energy, namely, work, at the wheels of the vehicle. The conversion of the chemical energy contained in fuel into thermal energy in a com- bustor.

Thermodynamics provides an understanding of the nature and degree of energy trans- formations, so that these can be understood and suitably utilized. For instance, thermodynam- ics can provide an understanding for the following situations:

In the presence of imposed restrictions it is possible to determine how the properties of a system vary, e.g., The variation of the temperature T and pressure P inside a closed cooking pot upon heat addition can be determined. The imposed restriction for this process is the fixed volume V of the cooker, and the pertinent system properties are T and P. It is desirable to characterize the variation of P and T with volume V in an automobile en-

gine. During compression of air, if there is no heat loss, it can be shown that PV 1.4 ≈ con- stant (cf. Figure 1 ).

Inversely, for a specified variation of the system properties, design considerations may re- quire that restrictions be imposed upon a system, e.g.,

A gas turbine requires compressed air in the combustion chamber in order to ignite and burn the fuel. Based on a thermodynamic analysis, an optimal scenario requires a com- pressor with negligible heat loss ( Figure 2a ). During the compression of natural gas, a constant

P temperature must be maintained. Therefore, it is necessary to transfer heat, e.g., by using cooling water (cf. Figure 2b ). It is also possible to determine the types of proc-

esses that must be chosen to make the best use of T resources, e.g.,

Q=0

To heat an industrial building during winter, one option might be to burn natural gas while another might involve the use of waste heat from a power plant. In this case a thermodynamic analysis will assist in making the appropriate decision based on rational scientific bases. For minimum work input during a compression process, should a process with no heat loss be util-

e.g : p v k = c o n st., fo r

i d e a l g a s, C p0 ized or should one be used that maintains a con- co n st stant temperature by cooling the compressor? In a i sen tr o p i c p ro ce s s

later chapter we will see that the latter process re- Figure 1: Relation between pres- quires the minimum work input.

sure and volume

The properties of a substance can be determined using the relevant state equations. Ther- modynamic analysis also provides relations among nonmeasurable properties such as en- ergy, in terms of measurable properties like P and T (Chapter 7). Likewise, the stability of

a substance (i.e., the formation of solid, liquid, and vapor phases) can be determined under given conditions (Chapter 10). Information on the direction of a process can also be obtained. For instance, analysis shows that heat can only flow from higher temperatures to lower temperatures, and chemical reactions under certain conditions can proceed only in a particular direction (e.g.,

under certain conditions charcoal can burn in air to form CO and CO 2 , but the reverse process of forming charcoal from CO and CO 2 is not possible at those conditions).

B. LIMITATIONS OF THERMODYNAMICS

It is not possible to determine the rates of transport processes using thermodynamic analyses alone. For example, thermodynamics demonstrates that heat flows from higher to lower temperatures, but does not provide a relation for the heat transfer rate. The heat conduc- tion rate per unit area can be deduced from a relation familiarly known as Fourier’s law, i.e.,

(1) where ∆T is the driving potential or temperature difference across a slab of finite thickness,

q ˙ ′′ = Driving potential ÷ Resistance = ∆T/R H ,

and R H denotes the thermal resistance. The Fourier law cannot be deduced simply with knowl- edge of thermodynamics. Rate processes are discussed in texts pertaining to heat, mass and momentum transport.

1. Review

a. System and Boundary

A system is a region containing energy and/or matter that is separated from its sur- roundings by arbitrarily imposed walls or boundaries.

A boundary is a closed surface surrounding a system through which energy and mass may enter or leave the system. Permeable and process boundaries allow mass transfer to occur. Mass transfer cannot occur across impermeable boundaries. A diathermal boundary al- lows heat transfer to occur across it as in the case of thin metal walls. Heat transfer cannot occur across the adiabatic boundary. In this case the boundary is impermeable to heat flux, e.g., as in the case of a Dewar flask.

P 1 Q=0 P 1 Q

P 2 >P 1 T 2 >T 1 P 2 >P 1 ,T 2 =T 1

To Combustion

Storage tanks

Chamber

Figure 2: (a) Compression of natural gas for gas turbine appli- cations; (b) Compression of natural gas for residential applica- tions.

System Boundary

Control Volume

Room air (A)

Hot Water (W)

Figure 3. Examples of: (a) Closed system. (b) Open system (filling of a water tank with drainage at the bottom). (c) Composite system.

A moveable/deforming boundary is capable of performing “boundary work”. No boundary work transfer can occur across a rigid boundary. However energy transfer can still occur via shaft work, e.g., through the stirring of fluid in a blender.

A simple system is a homogeneous, isotropic, and chemically inert system with no exter- nal effects, such as electromagnetic forces, gravitational fields, etc. Surroundings include everything outside the system (e.g. dryer may be a system; but the surroundings are air in the house + lawn + the universe) An isolated system is one with rigid walls that has no communication (i.e., no heat, mass, or work transfer) with its surroundings.

A closed system is one in which the system mass cannot cross the boundary, but energy can, e.g., in the form of heat transfer. Figure 3a contains a schematic diagram of a closed system consisting of a closed–off water tank. Water may not enter or exit the system, but heat can . A philosophical look into closed system is given in Figure 4a . An open system is one in which mass can cross the system boundary in addition to energy (e.g., as in Figure 3b where upon opening the valves that previously closed off the water tank, a pump now introduces additional water into the tank, and some water may also flow out of it through the outlet).

A composite system consists of several subsystems that have one or more internal con- straints or restraints. The schematic diagram contained in Figure 3c illustrates such a sys- tem based on a coffee (or hot water) cup placed in a room. The subsystems include water (W) and cold air (A)

b. Simple System

A simple system is one which is macroscopically homogeneous and isotropic and involves a single work mode. The term macroscopically homogeneous implies that properties such as the density ρ are uniform over a large dimensional region several times larger than the

mean free path (l ) during a relatively large time period, e.g., 10 –6 m s (which is large compared to the intermolecular collision time that, under standard conditions, is approximately 10 –15 s, as we will discuss later in this chapter). Since,

ρ = mass ÷ volume, (2) where the volume V » l 3 m , the density is a macroscopic characteristic of any system.

Closed

Open

System

System

C.V.

RIP

Exhaust and

Air and

Excretions

Food

(a)

(b)

Figure 4 : Philosophical perspective of systems: (a) Closed system. (b) Open system. An isotropic system is one in which the properties do not vary with direction, e.g., a cy-

lindrical metal block is homogeneous in terms of density and isotropic, since its thermal conductivity is identical in the radial and axial directions.

A simple compressible system utilizes the work modes of compression and/or expansion, and is devoid of body forces due to gravity, electrical and magnetic fields, inertia, and capillary effects. Therefore, it involves only volumetric changes in the work term.

c. Constraints and Restraints Constraints and restraints are the barriers within a system that prevent some changes from occurring during a specified time period.

A thermal constraint can be illustrated through a closed and insulated coffee mug. The in- sulation serves as a thermal constraint, since it prevents heat transfer. An example of a mechanical constraint is a piston–cylinder assembly containing com- pressed gases that is prevented from moving by a fixed pin. Here, the pin serves as a me- chanical constraint, since it prevents work transfer. Another example is water storage be- hind a dam which acts as a mechanical constraint. A composite system can be formulated by considering the water stores behind a dam and the low–lying plain ground adjacent to the dam.

A permeability or mass constraint can be exemplified by volatile naphthalene balls kept in

a plastic bag. The bag serves as a non–porous impermeable barrier that restrains the mass transfer of naphthalene vapors from the bag. Similarly, if a hot steaming coffee mug is capped with a rigid non–porous metal lid, heat transfer is possible whereas mass transfer of steaming vapor into the ambient is prevented.