Thermal Science [1 ed.] 0071772340, 9780071772341

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Erian A. Baskharone, Ph.D., is a Professor Emeritus of Mechanical and Aerospace Engineering at Texas A&M University, and a member of the Rotordynamics/ Turbomachinery Laboratory Faculty. He is a member of the ASME Turbomachinery Executive Committee. After receiving his Ph.D. degree from the University of Cincinnati, Dr. Baskharone became a Senior Engineer with Allied-Signal Corporation (currently Honeywell Aerospace Corporation), responsible for the aerodynamic design of various turbofan and turboprop engines. His research covered a wide spectrum of turbomachinery topics, including unsteady stator/rotor flow interaction, and the fluid-induced vibration problem in the Space Shuttle Main Engine. Dr. Baskharone’s perturbation approach to the problem of turbomachinery fluid-induced vibration was a significant breakthrough. He is the recipient of the General Dynamics Award of Excellence in Engineering Teaching (1991) and the Amoco Foundation Award for Distinguished Teaching (1992).

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Thermal Science Essentials of Thermodynamics, Fluid Mechanics, and Heat Transfer Erian A. Baskharone

New York Chicago San Franc Lisbon London Madrid Mexico Milan New Delhi San Juan Seoul Singapore Sydney T

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The McGraw Hill companies Cataloging-in-Publication Data is on file with the Library of Congress.

McGraw-Hill books are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected]. Thermal Science: Essentials of Thermodynamics, Fluid Mechanics, and Heat Transfer Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in China. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.

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ISBN MHID

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198765432

978-0-07-177234-1 0-07-177234-0

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Preface

xiii

Thermodynamics Definitions

1

3

Properties of Pure Substances Properties of Ideal Gases

11

25

Basic Lawsof Thermodynamics Energy Conversion by Cycles

37 67

Power-Absorbing Cycles: Refrigerators and Heat Pumps

73

Gas Power Cycles 83 Fluid Mechanics: A Control Volume Approach Flow-Governing Equations

107

External and Internal Flow Structures 10 —HW

Rotating Machinery Fluid Mechanics

111

149

Variable-Geometry Turbomachinery Stages Normal Shocks

215

13 — Oblique Shocks

223

12

105

207

14

Prandtl-Meyer Flow

15

Internal Flows: Friction, Pressure Drop, and Heat Transfer

16

Fanno Flow Process for a Viscous Flow Field

17

Rayleigh Flow

Part Ill

Heat Transfer

229 241

253

271 281

18

Heat Conduction

283

19

Heat Convection

303 vii

viii

Brief Contents

20

Heat Exchangers

21

Heat Radiation References Appendix Index

457

361 363

323 343

Preface

xiii —

3-2

:

|

Thermodynamics

Definitions System

3-3

1

“3

oo

State

a

4

7

Gravitational Work Shaft Work

Z

_

a= 10

31 32

32

Other Equations of State

5

Problems

Quasi-Static Process

29

Compressibility Factor: A Measure of Deviation from the Ideal Gas Behavior 32

’

4 —

Equilibrium

34

36

6

System/Surroundings Interaction Modes

Examples of Nonregainable Work Moving Boundary Work The State Postulate

7 |

; 4

Basic Laws of Thermodynamics

7

7

8

=

37

First Law of Thermodynamics

4-1

Second Law of Thermodynamics

39

4-2

Definition of Temperature ae

4-3 4-4

Irreversibility

a9

Efficiencies of Reversible and Irreversible

10

eGnsronees Woe

Phases of a Pure Substance

The Total Energy

is

12

12

Kinetic Energy 13 Potential Energy 13

Internal Energy 13 Enthalpy 14 Equilibrium Diagrams Specific Heats

Problems

: 15 23

24

Efficiency of Work-Producing Heat Engines

Heat Engines

26

25

Second Law in Terms of Reversible Cycles Irreversible and Reversible Processes 49

4-8

Clausius Inequality: A Statement of the

4-9

The T-ds Equations

49

Entropy Change for Ideal Gases

4-13

Entropy Change for a Pure Substance

4-14

The Increase-in-Entropy Principle

4-15

Entropy Change for Compressed Liquids

The T-s Diagram Isentropic Processes

Carnot Cycle 64

62

48

50

4-10 4-11 4-12

Problems

47

48

4-7_

4-16

Properties of Ideal Gases Ideal Gas Relationships

44

Second Law of Thermodynamics

22

The Ideal Rankine Cycle

:

38

Zeroth Law of Thermodynamics:

Properties of Pure

3-1

Polytropic Processes

a

27

29

Details of Moving Boundary Work

3-6

4

Property

Work

3-5

3

27

Modes of System/Surroundings Interaction

“3-4

Boundary 4 iN x

_ Equation of State

51

52 52

53 61

62

Contents

External and Internal Flow Structures 111

Energy Conversion by Cycles 67 Heat-To-Work Conversion

Basic External Flow Structure

68

The Rankine Cycle and T-s Representation Ideal Rankine Cycle Analysis

Potential Flow Fields

69

Rankine Cycle Thermal Efficiency

128

Stream Function

131

Compressibility of a Working Medium:

70

The Definition of Sonic Speed

135

Power Apserning Cycles: Refrigerators an

Compressibility of the Flow Field: Definition

Heat Pumps

Introduction of the Critical Mach Number

of the Mach Number

73

Energy Conservation for a Reversed Cycle

Performance Measures

Problem

Air-Standard Assumptions

80

10-1

33

10-2 10-3 10-4

85

Diesel Cycle: Ideal Cycle for

The Ideal Brayton Cycle

89

90

Isentropic Efficiency of a Process

98

Brayton Cycle with Regeneration

101

Problems

103

Fluid Mechanics: A Control

Volume Approach

Flow-Governing Equations 107 Continuity Equation

Energy Equation Problems

110

108

109

105

90

10-5 10-6 10-7 10-8

150

Velocity Diagrams

154

Compressor- and Turbine-Rotor Directions

154

Axial Momentum Equation Radial Momentum Equation

Cross-Flow Area Variation

157

157

158

Variable-Geometry Stators Design-Related Variables

Euler’s Equation

159 162

166

Introduction of the Total Relative Properties

Incidence and Deviation Angles

170

Means of Assessing Turbomachinery

Performance

10-15 10-16

156

Total Pressure Variation Across Multistage

Turbomachines

10-9 10-10 10-11 10-12 10-13 10-14

152

Sign Convention

of Rotation 87

Thermal Efficiency of Brayton Cycle

Real-Life Brayton Cycle

Classification of Turbomachinery

Components

84

Compression-Ignition Engines

147

Rotating Machinery Fluid Mechanics 149

84

Mean Effective Pressure

140

78

81

Gas Power Cycles Otto Cycle

74

Problems

Choice of the Working Medium

136

Isentropic Flow Through Varying-Area Passages 144

74

Vapor-Compression Refrigeration Cycle

Part Il

125

Introduction of the Velocity Potential and

70

Methods of Efficiency Enhancement: Regeneration and Reheat

112

External Flows: Boundary Layer Buildup

68

170

Supersonic Stator Cascades

180

Sign Convention Governing Radial

Turbomachines Problems

203

185

166

Contents

11

Variable-Geometry Turbomachinery Stages

16

207

16-1 16-2— 16-3 16-4 16-5 16-6 16-7 16-8 16-9

Definition of a Variable-Geometry

Turbomachine

11-2

Examples of Variable-Geometry Turbomachines 208 Problems

12 12-1 12-2 12-3 12-4

13-1 13-2

208

Normal Shocks Introduction

215

216

Shock Analysis

216

Normal Shock Tables

219

17

221

Oblique Shocks

17-1 17-2 17-3

223

Oblique Shock Tables and Charts

227

Boundary Condition of Flow Direction Problems

227

14-1 14-2 14-3 14-4 14-5-

15.

Momentum Equation

257

Working Equations for an Ideal Gas Reference State and Fanno Tables

Friction Choking

260

Moody Diagram

262

2a 258

265

Losses in Constant-Area Annular Ducts

270

Rayleigh Flow Introduction

271

272

Analysis for a General Fluid

272

Working Equations for an Ideal Gas

2tt

278

Isentropic Turns from Infinitesimal Shocks

Analysis of Prandtl-Meyer Flow Prandtl-Meyer Tables

235

231 231

18-1 18-2 18-3 18-4 18-5 18-6 18-7

Internal Flows: Friction,

Pressure Drop, and 19 242

Mechanical Energy Conservation 252

242

244

Energy Conservation

284

Simple One-Dimensional Problems

Equivalent Thermal Network

285

286

Network Resistance for Heat-Convection

286

Multilayer Plane Walls

287

Thermal Contact Resistance Transient Heat Conduction Problems

Pressure Drop and Wall Shear Stress Friction Factors

Introduction

233

289 293

301

241

242

Mass Conservation

281

Boundary Conditions

238

239

Assumptions

Heat Transfer

Heat Conduction

230

Pressure and Entropy Changes Versus Deflection Angles for Weak Oblique Shocks

Problems

256

Limiting Point

229

Argument for Isentropic Turning Flows

Heat Transfer 15-1 15-2 15-3 15-4 15-5 15-6

254

228

Prandtl-Meyer Flow

Problems

254

Analysis for a General Fluid

Problems

Part fll

14

Introduction

Problems

220

Shocks in Nozzles

Problems

13

208

Fanno Flow Process for a Viscous Flow Field 253

245

244

19-1 19-2 19-3 19-4 19-5

Heat Convection Introduction

303

304

Convection Heat Transfer Coefficient

Natural Convection

309

Lumped Parameter Analysis

318

Forced Convection Analysis

321

Problems

322

305

xi

xii

Contents

20 20-1 20-2 20-3 20-4 20-5 20-6 20-7 20-8

Heat Exchangers Introduction

21-1 21-2 21-3

324

Log Mean Temperature Difference Overall Heat Transfer Coefficient Fouling Factor

325 329

331

Analysis of Heat Exchangers The Similitude Principle Dynamic Similarity

334

View Factor Relations

348

Radiation Functions

348

Radiation Properties

350

Total Radiation Properties Radiation Shape Factor

350 352

Directional Radiation Properties Problems

352

360

References

361

338

341

Appendix

Heat Radiation

Tables and Charts

Thermal Radiation: Blackbody

Planck’s Law

21-4 21-5 21-6 21-7 21-8 21-9

337

337

Dimensionless Parameters Problems

21

323

Wien’s Displacement Law

344 Index

346 346

457

363

In writing this book, special care was given to the fact that it targets undergraduate students who are not majoring in mechanical engineering. Conciseness, therefore, was a must, as the student is introduced to the major subject of thermal science. In all three subdisciplines of thermodynamics, fluid mechanics, and heat transfer, emphasis is placed on the basics of each subject as clearly as possible. As the worldwide interest in energy generation/conversion grows daily, it becomes essential for all engineering students to gain a keen understanding of the basic principles of thermal science, and this is what is behind this book’s importance. In covering the thermodynamics topic, attention is focused on the three laws of thermodynamics and their applications in real-life problems. Special emphasis is placed on the well-known thermodynamic cycles of Carnot, Rankine, and Brayton, as well as those related to internal combustion engines. In all cases, the working medium (be that

liquid or gas) will naturally undergo substantial changes in pressure and temperatures as it proceeds from one thermodynamic state to the next. The medium, in the Rankine cycle, will even change its phase from liquid to wet saturated steam to superheated vapor. In all cases, the pure substance (be that water or Freon) is carefully monitored, and its thermodynamic properties defined. In our coverage of the open version of the Brayton cycle, the simple turbojet engine is introduced first. The topics of regeneration and reheat techniques, as they generally apply to the Rankine cycle, are discussed in detail. Next, important derivatives of the the turbojet engine, namely the turbofan and turboprop engines are introduced, with the distinction of the turboprop engine, being categorically a turboshaft engine, emphasized. In the latter category, attention is focused on the fact that the turbine-produced power is utilized not only in providing the thrust force but also in driving a propeller. Note that the cross-oceanic air carriers are exclusively turbofan engines. Turboprop engines, on the other hand, are utilized in short missions (e.g., Phoenix to Los Angeles), for they are

most efficient in this distance range. In covering the fluid mechanics subject, the student is introduced to the principle of control volume, with part of the boundary being flow-permeable. Under focus in the entire section are the flow-governing equations, including those of mass, linear momentum, and energy. Clearly presented in this section of the book is the topic of flow behavior through turbomachinery components, such as compressors and turbines. Owing to the author’s industrial experience, this topic was covered to reveal a wide range of industrial applications. Such important phenomena as subsonic-supersonic flow conversion and flow “choking” are comprehensively covered through meaningful examples. The consequences of friction and boundary layer buildup are introduced under the classical topic of Fanno flow structure, with interesting applications in the turbomachinery area. The other major topic is that of the Rayleigh flow pattern, which addresses heat exchange effects on flow behavior and is presented with applications concerning annular combustors. The examples and graphical illustrations in this case bring the student closer to the physical problem at hand. xiii

xiv

Contents

The third part of the book presents the three modes of heat transfer: conduction, convection, and radiation. Special attention in this case is paid to the various practical applications of each mode. The purpose here is to deepen the student’s comprehension of how basic principles of heat transfer apply to real-life practical problems. On the subject of heat conduction, such real-life problems as contact resistance in a solid with multiple layers are discussed, with the subtopic of electrical analogy highlighted. On the topic of heat convection, the distinction between free (or natural) and forced convection is clearly made, with practical applications. Also covered in detail is the problem of heat exchangers, a real industry-related problem in terms of sizing and analysis. Aside from all factors governing radiation heat transfer, the topic of shape factor, which is basically how a heat-emitting surface “sees” another, is particularly emphasized.

Note to Instructors If you are an instructor using this book as a textbook, please be aware that an Instructor’s Manual with chapter problem solutions and exam problems is available at www.mhprofessional.com/thermalscience. Please visit this site for information about how you can download this material to assist your teaching. Erian A. Baskharone

PART

Thermodynamics Thermodynamics is the mother of the general discipline of thermal science. In addition to the zeroth law of thermodynamics, through which temperature, as a property, is defined, we discuss two major laws: the first law of thermodynamics, which stands for an energy conservation principle, and the second law of thermodynamics, which takes into account the system degradation sources during a real-life process, This leads to the definition of anew property, called entropy, with which the so-called irreversibilities of a process are gauged. The working medium, in this context, is either treated as an ideal gas such as air at

a “sufficiently” low pressure, or a pure substance such as water and refrigerant 12. The latter can exist in one, two, or even three phases: namely, solid (meaning ice), liquid, or

water vapor, depending on the conditions to which the water substance is exposed, or dry superheated vapor. Under specific situations, such as saturated liquid and saturated vapor co-existing, both the temperature and pressure are dependent on one another. In this case, a new property termed the steam quality (or dryness factor) is needed to identify the state of that mixture. At the heart of the whole thermodynamics topic is what is referred to as a system in either the closed kind (e.g., a rigid container or the piston/cylinder device in Figure 1—1) or the open one (e.g., a flowing medium through a diffuser as in Figure 1-2). Of course, open systems embrace a virtually unlimited number of other configurations, such as pipes and sudden-enlargement ducts.

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Definitions Chapter Outline 1-1

System

4

1-2

Boundary

1-3

Property

1-4

State

1-5

Equilibrium

1-6

Quasi-Static Process

1-7

System/ Surroundings Interaction Modes

1-8

Examples of Nonregainable Work

1-9

Moving Boundary Work

4 4

4

5 6

7

7

1-10

The State Posiulate

8

1-11

Zeroth Law of Thermodynamics

10

7

4

Thermodynamics

i

System A thermodynamic system is generically defined as a collection of matter contained within a boundary that is either solid or even fictional for closed and open systems, respectively.

Ryd

Boundary A boundary is what separates the system from the so-called universe (Figure 1-1).

Fe)

Property A property is any characteristic of the system. However, such characteristics as color do not constitute thermodynamic properties. The term property includes two distinct families: e Extensive properties, which depend on the contents (meaning mass) of the system. e Intensive properties, which are independent of the mass, mostly the property magnitude per unit mass, such as the specific volume. Obvious intensive properties are temperature and pressure.

4

State This is the condition of the system as defined by all of its properties. The question here arises: How many, in the least, are required to define the state of a particular system? For a single-phase substance, two independent properties suffice. An example of two dependent properties is the volume and specific volume (defined as the volume per unit mass). As for a two-phase substance, an additional property is required and is commonly termed the quality of the mixture (defined as the mass of vapor in the mixture divided by the entire mixture mass in the coexisting vapor and liquid phase).

Figure 1-1 Stopper

A rising piston within a cylinder.

Piston displacement

Final state —»

eee

Initial'state =

|e

ee

ee

eee

ee

ee

ee

eee ee ae eee

Ke

Definitions

Figure 1-2

Closed system

5

Open system

Open and closed systems.

Inlet velocity profile

1-5

Exit velocity profile

Equilibrium Consider a system that is not undergoing any process. At the beginning of a process, such as that of a rising piston inside a cylinder (Figure 1-3), the system possesses a set of properties that completely define the state of the system. At a given state, all of the system’s properties have fixed magnitudes. If the value of any property changes, the state will change to a different one, as well.

Thermodynamics, as a topic, deals with equilibrium states. Equilibrium implies a state of balance. In an equilibrium state, there are no unbalanced potentials (or driving force) within the system. A system in equilibrium experiences no change when it is isolated from the surroundings. There are many kinds of equilibrium, and a given system is not in thermodynamic equilibrium unless the conditions of all relevant types of equilibrium are satisfied. For example, a system is in a state of thermal equilibrium if the temperature has the same magnitude throughout the entire system. That is, the system involves no temperature

Figure 1-3

Stopper

Expansion through a piston/cylinder apparatus.

Cylinder

6

Thermodynamics

differentials (the driving force for heat flow) anywhere. Mechanical equilibrium is attained if the pressure is the same at any point within the system. The pressure may vary within the system elevation as a result of gravitational effects. But the pressure at the bottom layer is balanced by the extra weight it must carry, and therefore there is no imbalance of forces. The variation of pressure as a result of gravity in most thermodynamic systems is relatively small and is usually ignored. If the system involves two phases, it is a phase of equilibrium when the mass of each phase reaches an equilibrium level and stays there. Furthermore, a system is in a state of chemical equilibrium if its chemical composition does not differ with time, that is, no chemical reaction is occurring. A system will not be in thermodynamic equilibrium unless all of these relevant criteria are satisfied.

Quasi-Static Process This is a process where the system progresses from one equilibrium. state to another. An example of such a process is shown in Figure 1-4, where the piston-applied weights are gradually increased over a rather long (theoretically infinite) period of time.

Figure 1-4 Example of a quasi-static

process.

Quasi-static process

Progression of loading the piston

Definitions

Figure 1-5

1

Two examples of nonregainable work.

7

t] U

Electric current Paddle-wheel device

sy

System/Surroundings Interaction Modes These are two, namely heat transfer and work, interactions that can take place. The former will occur due to a finite nonzero temperature differential between the system and its surroundings. As for work, we have two distinct categories. For a closed system, such as a piston-cylinder device, work will be exerted on or by the system due to expansion or compression of the boundary, respectively. Figure 1-5 shows such system/environment exchanges for two closed systems. For open systems, work is simply shaft work that is done due to the existence of a turbomachinery component such as a pump, a compressor, or a turbine.

8

Examples of Nonregainable Work Figure 1-5 shows two types of processes where the work done is unregainable. The first involves electric current controlled from outside the system. The second is one that involves a paddle-wheel apparatus.

Moving Boundary Work This is the most common type of work where the displacement of a piston, say, within a cylinder, will give rise to a so-called displacement type of work. The following expression quantifies this type of work (Figure 1-1):

wia=

2 | pdV 1

where p is the instantaneous magnitude of pressure and dV change due to the piston’s movement.

EXAMPLE 1-1

is the differential volume

An equal amount of work of 50 kJ/kg is supplied to the two systems shown in Figure 1—5. The torque exerted in the first system is 415 N.m., and the piston radius, in the second case, is 14.5 cm. Calculate the rotational speed and piston displacement in these two cases, respectively.

8

Thermodynamics

Solution

(a)

W, =TW

which yields: @ = 120.5 rad/s = 6, 904 r/min

(b) Apiston

=

mr?

=> 0.066

m?

We now proceed to calculate the piston displacement as follows: w=

pAv=

PApiston AA

which yields the piston displacement: W

IN fy P

== Sti)

Apiston

e —— e

1-10

The State Postulate As noted earlier, the state of a given system is defined by all of its properties. But we know from experience that we do not need to specify all properties to define a state. Once a sufficient number of properties is specified, the rest assume specific magnitudes correspondingly. That is, specifying a certain number of properties is sufficient to define a thermodynamic state. The number of such properties is given by the so-called state postulate principle: “The state of a simple compressible, single-phased system is totally defined by two independent intensive properties.” A system is referred to as a simple compressible system in the absence of electric, magnetic, and gravitational energy as well as surface tension. These effects are due to external force fields and are negligible for most engineering applications. Otherwise, an additional property needs to be specified for such effects that are significant. If the gravitational effects are supposed to be included, for example, the elevation (z) needs to be specified in addition to the two properties needed to define a state. The state postulate requires that the two properties be independent to define a unique state. Two properties are independent if one is allowed to vary while the other is held constant. Temperature and specific volume, for instance, are always independent of ene another.

EXAMPLE

1-2

The system in Figure 1-6 is composed of a trapped amount of gas inside a pistoncylinder force of sectional calculate

Solution

Sanne nn en nn

device. The piston, which weighs 4.0 kg, is held in place by the stiffness a compressed spring which provides a force of 60 Newtons. The crossarea of the piston is 35 m*. If the local atmospheric pressure is 0.95 bar, the gas pressure.

Considering the piston balance, we have: F spring oP Patmospheric A piston = PgasApiston ls Mpiston§ =

0

which, upon substitution, yields:

Peas = 1.234 bars SSS

sss

Definitions

Figure 1-6 Piston balance under the

Cylinder

stiffness of a compressed

spring.

Compressed spring

EXAMPLE

1-3 _ Referring to the previous example, the following changes are made: e The gas pressure is 1.2 bars. e The piston is allowed to move up against a spring force which has a stiffness factor (k) of 1800 N/m (Figure 1-7).

Now calculate the piston’s displacement. Solution

Again, considering the piston balance, we have:

Mpiston® + Kspring AZ + PatmosphericApiston — PgasApiston = 0 which yields: AZ = 2:7 em

Figure 1-7 Different configuration of the same problem.

Cylinder :

Compressed spring

Piston displacement

9

10

Thermodynamics

la

Zeroth Law of Thermodynamics: Definition of Temperature We all recognize temperature as a measure of “hotness” at a given point within the system. This leaves this intensive property with no numerical magnitude. Furthermore, our sense of this property may be misleading. A metal chair, for instance, will feel much colder than a wooden one even when both are at the same temperature. Fortunately, several properties of materials change with temperature in a repeatable way, and this forms the basis for an accurate temperature measurement. The commonly used mercury-in-glass thermometer, for example, is based on the expansion of mercury with temperature. Temperature is also measured by using several other temperaturedependent properties. It is common experience that a cup of hot coffee left on the table eventually cools off and a cold drink eventually warms up. That is, when a body is brought into contact with another body which is at a different temperature, heat is transferred from the body which is at a higher temperature to that at a lower one, until both bodies attain the same temperature. At this point, the heat transfer terminates, and the two bodies are said to have reached a state of thermal equilibrium. The equality of temperature is the only requirement for thermal equilibrium. The zeroth law of thermodynamics states that if two bodies are in a state of thermal equilibrium with a third body, then they are also in thermal equilibrium with each other. It may seem silly that such an obvious fact is referred to as one of the basic laws of thermodynamics. However, it can’t be concluded from the other laws of thermodynamics,

and it serves as a basis for the validity of temperature measurements. By replacing the third body with a thermometer, the zeroth law of thermodynamics can be restated as: two bodies are in thermal equilibrium if both have the same temperature reading even if they are not in contact with one another.

Properties of Pure Substances Chapter Outline Phases of a Pure Substance The Total Energy Kinetic Energy

Potential Energy Internal Energy.

Enthalpy

12

12 13

13 13

14

Equilibrium Diagrams

Specitic Heats

15

22

The Ideal Rankine Cycle

23

12

Thermodynamics

A pure substance is one that is property-homogeneous. Examples of that include water, nitrogen, and lithium.

A pure substance does not have to be of a single chemical element. Air, for instance, would constitute a pure substance as long as the mixture of gases is homogeneous. A mixture of two or more phases of a pure substance is still a pure substance. For example, a mixture of liquid water and superheated (gaseous) vapor is not homogeneous since part of the container would be occupied by the liquid while the other is occupied by vapor.

lis

Phases of a Pure Substance We all know that under standard magnitudes of temperature and pressure, copper exists in its solid phase, water in the liquid phase, and oxygen in the gaseous phase. However, under different magnitudes, temperature and pressure, a substance could be a mixture of two or even three of its phases. The substance molecules in a solid are arranged in a three-dimensional pattern (lattice) that is repeated throughout. Because of the small distances between molecules in this phase, the attractive forces of molecules on each other are large and keep molecules at fixed positions. The molecules’ spacings in the liquid phase are not that much different from their counterparts in the solid phase. The distance between molecules generally experiences a slight increase as the solid turns into liquid, with water being a rare exception. In the gas phase, the molecules are far apart from one another, and a molecular order is nonexistent. Gas molecules move around in random, continually colliding with one another and the wall of the container. Particularly at low magnitudes of density, the intermolecular forces are very small, and collision is the only mode of interaction

between molecules. Molecules in the gaseous phase are not at a considerably higher energy level than they are in a liquid. Therefore, a gas must release a large amount of its energy before it can condense or freeze.

pie

The Total Energy Work and heat are defined as modes of energy transfer across the system boundary. Heat is defined as the energy transfer across the boundary due to a temperature differential between the system and its surroundings. Work is defined as the energy transfer arising from an effect that is solely equivalent to a force acting through a distance. Neither heat nor work are properties, and they cannot be represented by exact differentials since their values depend on the process path followed during a change of state. Many forms of energy, other than heat and work, are equally important. In a classification of all different forms of energy that play a role in thermodynamic systems, distinguishing between microscopic and macroscopic forms of energy is helpful. Microscopic forms of energy are those related to the energy possessed by the individual molecules and to the interaction between molecules and to the interaction between molecules that comprise the system at hand. Macroscopic forms of energy, on the other hand, are related to the gross characteristics of a substance on a scale that is large compared to the mean free path of molecules. The total energy (£) is a property of a system and is defined as the sum of all macroscopic forms of energy plus the total of the microscopic forms as follows: E=

E microscopic + E macroscopic

Properties of Pure Substances A thermodynamic energy of a system system experience of energy that sum

2-3

13

analysis usually includes a determination of the change in the total during a process or a series of processes. However, only rarely does a significant changes in more than only a few of many different forms up to the total energy of the system.

Kinetic Energy The kinetic energy of a mass m with velocity V is defined as follows: Ex =

Le 2

—mV

The kinetic energy and the velocity of the center of the mass are physical properties and, as such, must be measured with respect to some physical external frame of reference. Often the most convenient frame of reference is one that is stationary relative to the earth. For this reference frame, a quantity of mass with no motion “relative” to the earth has a relative velocity of zero, and its kinetic energy would obviously be zero as well. Thermodynamic analyses are most often concerned with determining the change in kinetic energy from one state to another. Since the kinetic energy is a property, the change in it is independent of the process path that the system may get into. This magnitude of change is solely dependent on the mass and velocity of the system at the end states. The kinetic energy per unit mass, which is an intensive property, is consistently defined as: y2 PE

Obviously the preceding definition is well suited to evaluate the kinetic energy associated with the mass flow across the boundary of an open system.

24

Potential Energy A quantity of mass (m) possesses potential energy in a gravitational field with acceleration of gravity g, by virtue of its elevation z above some arbitrary datum. The potential energy is defined as follows: Ep = mg-Z

From which the intensive property can be expressed as follows: Cp = 8%

Internal Energy The energy associated with a substance on a molecular scale can consist of several forms. Molecules possess kinetic energy due to their individual mass and velocity as they move about a linear path. The molecules also possess vibrational and rotational angles as they rotate and vibrate as a consequence of their random motion, which is another form of energy associated with the intermolecular forces between molecules. The sum of all of these molecular or microscopic energies is called the internal energy of a substance. Internal energy is a thermodynamic property. In other words, the internal energy change during a process depends solely on the end states. The symbol for internal energy

14

Thermodynamics

Figure 2-1 Determination of the

specific heats.

Determination of cy

Determination of c,

Tangent

Tangent

of a quantity of mass is U. This is an extensive property of which the corresponding intensive property is defined as follows:

re

m

and its variation with temperature is shown in Figure 2-1.

Enthalpy In the analysis of open systems (known as control-volume analysis), the combination: h=u+pv

is rather important as it combines the internal energy with that which is required to “push” the flow across any cross-flow plane in the fluid stream, and in other applications as well. This is the same property shown versus the temperature in Figure 2—1. The tangents in this figure define two new intensive properties; namely the specific heats under constant volume and constant pressure, respectively, both of which will be discussed next. Note that the preceding combination is a new property termed enthalpy. As seen, the value of enthalpy at any state depends on other thermodynamic properties. Therefore it is, as well, a valid property. The constant-volume specific heat c,, by reference to Figure 2-1, is defined as follows:

and the constant-pressure specific heat is defined by:

ped fi sae a These are very useful intensive properties. They are independent of any system/ surroundings interaction. Figure 2—1 shows that c, is the slope of the constant-volume

Properties of Pure Substances

15

line on a u versus T plot, whereas c, is the slope of a constant-pressure line on an h versus T plot. The ratio of specific heats is particularly important variable and is only a function of temperature: Mi

aE Cy

y varies, for instance, across a simple turbojet engine from approximately 1.4 across the compressor to approximately 1.33 in the turbine.

Equilibrium Diagrams All known substances can exist in several phases, based on the temperature and pressure.

Referring to Figure 2—2, a mixture of more than one phase of a substance in equilibrium is also common. Because of this complex behavior, a single relationship between the pressure, temperature, and specific volume that is valid for all possible states of a substance cannot be developed. However, the qualitative aspects of these substances can be discussed in order to gain insight into their p-v-T behavior. Consider the following simple experiment devised to monitor the changes in temperature and specific volume of a substance such as water as it is heated at constant pressure. For this purpose, imagine a transparent cylinder initially filled with (liquid) water at 20°C and maintains a pressure of 101.35 kPa (the standard ambient pressure) by means of a piston of constant weight as shown in Figure 2-2. As heat is transferred to the water, the system temperature will begin to increase and, at the same time, water will expand. In other words, the specific volume of the

cylinder’s contents will increase. Since the cylinder is closed, the expansion will also cause the piston to move. If energy is continually added to the water, a plot of the temperature as a function of specific volume could be constructed for this process. A sketch of the results is shown in Figure 2—2. Note that the water temperature continues to increase up to a state (the saturated liquid state) at which the monitor will register the first appearance of a

Figure 2-2 Constant pressure lines on the p-v diagram.

p = constant

16

Thermodynamics

vapor bubble. During subsequent heating, the specific volume continues to increase as well, up to a state where the water temperature reaches the magnitude of 100°C. The fact that water at atmospheric pressure begins to evaporate is common knowledge, with the appearance of the first vapor bubble signaling the beginning of a process during which the pressure will remain constant. As more energy is transferred to the now two-phase working medium, the temperature will remain constant as well. At some state, a point is reached where approximately one-half of the mixture is liquid water while the other half (by mass) is vapor. Eventually, the liquid component is all gone. Further heating will cause the temperature to increase. Upon continuing the heating process, the vapor inside the cylinder then is referred to as superheated vapor.

The p-T Diagram Figure 2-3 shows the p-T diagram of a pure substance. This diagram is often called the phase diagram since all three phases are shown separated from one another by three lines. The sublimation line separates the solid and the vapor regions, the vaporization line separates the liquid and vapor regions, and the melting (or fusion) line separates the solid and liquid regions. These three lines meet at the triple point, where all three phases would coexist in equilibrium. The vaporization line ends at the critical point, above which no distinction can be made between the liquid and vapor phases.

The p-v-T Surface As earlier indicated, the state of a simple compressible substance is determined and fixed by any two independent intensive properties. Once the two appropriate properties are determined, all other properties become dependent on them. Remembering that any equation with two independent variables in the form z = z(x, y) represents a surface in space, we can represent the p-v-T behavior of a substance as a surface in space, as shown

Figure 2-3 The p-T diagram for a pure substance.

Substance expands on freezing

i) \ \ \ \ \ \

Critical state

\

\

,

Vaporization

‘

Triple point 6 Sublimation

Properties of Pure Substances

17

Figure 2-4 A p-v-T surface for a

substance that expands on freezing.

Pressure

in Figure 2-4. Here T and v may be viewed as the independent variables (the base) and p as the dependent variable (height). Going back to Figure 2—2, and focusing on the horizontal segments in this figure as well as in Figure 2-4, a new property needs to be introduced, the steam quality (x) or dryness factor, and is defined as follows:

Me ‘

M mixture

_

Meg Mg +My

where m, and m - refer to the masses of saturated liquid and saturated vapor, respectively. Knowing all necessary data during the experiment, one can similarly construct a pressure-specific volume chart. Should the preceding experiment be repeated at a different (say higher) magnitude of pressure, the distance between the two saturation points would begin to shrink. In fact, under a specific pressure and temperature combination, the convergence from liquid to vapor states will shrink down to a point (the critical state). Upon repetition of the same experiment, the locus of the saturation lines will come to form a “dome,” with the L + V states all existing inside the dome as shown in Figure 2-5. Figure 2-5 also shows two constant temperature lines on the p-v diagram, where the upper line is associated with the higher of the two temperatures. Note that, away from the liquid-vapor dome, on the right-hand side, the constant-temperature lines are hyperbolic in shape, as will be discussed later in conjunction with ideal gases.

18

Thermodynamics

Figure 2-5

P

Constant-temperature line on the p-v diagram.

T = constant

EXAMPLE 2-1

Referring to Figure 2-6, saturated water vapor is kept at 40 bars. The vapor undergoes a constant-pressure process until the volume is doubled. The vapor is then returned to the saturated vapor status through a constant-volume process. Calculate the net change in enthalpy (h3 — h;).

Solution

Referring to the water table in the Appendix at 40 bars, we get:

v1 = 0.049 m3/kg h, = 2801 kJ/kg Now we move to state 2: P2 = 40 bars (given)

v2 = 0.098 m?/kg Ins= 6006 hz = 2800.5 kJ/kg Now we focus on state 3:

v3 = 0.098 m3/kg Finally:

hz — h; = 0.5 kJ/kg

LEE sss

Properties of Pure Substances

Figure 2-6

P

Constant-pressure and constant-volume processes.

Constant pressure process

Figure 2-7

P

Wet steam conversion into the superheated vapor region.

Constant pressure process

1

EXAMPLE 2-2

Referring to Figure 2-7, water wet steam is kept under a pressure of 20 bars and a quality of 0.4. The steam then expands until the volume is tripled. Calculate the change in internal energy.

19

20

Thermodynamics

Solution

Let us consider the initial state: Uy = XjUg + (1 — xj)uz = 1584.0 kI/kg v1) = X1V_g + (1 — x1)v¢ = 0.06 m/kg

vy = 3v; = 0.018 me/kg P2 = 20 bars

uz = 3203.5 kJ/kg Finally:

uy — uy = 1619.5 kJ/kg

EXAMPLE 2-3

Referring to Figure 2-8, superheated water vapor is maintained at 25 bars and 700°C. The vapor undergoes a constant-volume process until the pressure is 10 bars. Calculate the final steam quality.

Solution

— Using the superheated water vapor tables:

v, = 0.017 m’/kg hy = 3777.5 kI/kg Now we focus on the final state: P2 = 10 bars

v2 = 0.017 me/kg = x2v, + (1 — x2)vf which yields:

x2 = 0.09 SS 2S

Figure 2-8 Constant-volume conversion of superheated vapor.

SS

SP

SSS

Properties of Pure Substances

Figure 2-9

Pp

Constant-pressure condensation process.

EXAMPLE 2-4 _ Referring to Figure 2-9, superheated water vapor is initially at a pressure of 25 bars and a temperature of 500°C. The vapor undergoes a constant-pressure condensation process until the volume is one-half of its initial magnitude. Calculate the steam final quality.

Solution

v; = 0.14 m/kg Now we focus on the final state:

v2 = 0.07 m3/kg

EXAMPLE 2-5

State the phase or phases of water that may exist at the given states:

(a) T = 215°C, p = 2.0 MPa (b) T = 240°C, x = 0.4 (c) T = 260°C, v = 0.40 m3/kg Solution

(a) At T = 215°C, and using the water tables in the Appendix, we find that: — Psat.vapor = 0.021 bar.

Now, since p > Psat.vapor, the water substance exists in the superheated gas domain.

(b) Since the steam quality has a finite magnitude (0.4) between 0 and 1.0, the state

is that of a wet liquid-vapor mixture. (c) Searching the saturated-gas leg of the L + V dome, we find that at T = 260°C, Vsat = 0.0205 m*/kg. Therefore, that state exists in the subcooled liquid subregion (by just comparing the specific volumes). ———————

————————————————— EE

21

22

Thermodynamics

TE

EXAMPLE 2-6

A tank contains 1 kg of liquid water and 0.1 kg of water vapor at 200°C. Determine the following properties: (a) The quality of the water substance (b) The total volume of the container

(c) The pressure of the water substance

Solution

(a) x= —-£— =0.091 Mm, +m

(b)

V=mv = 1.1 [xvg + 1 — x)myz] = 14.291 m?

(c) The water pressure is that associated with T = 200°C within the L + V dome,

which is 0.015 bar.

EXAMPLE 2-7

A

rigid tank with a volume of 2.5 m® contains 5 kg of a saturated liquid-vapor

mixture of a water at 75°C. Now the water substance is slowly heated. Determine the temperature at which the liquid in the tank is completely vaporized.

Solution

V vy, = — = 05 m/kg m

Vy = Xivg + (1 — x1)ve

Using the tables we get x; = 0.119 It follows that: uy = X\Ug + (1 — x) )uz¢ = 571.2 kI/kg

Searching the saturated vapor leg of the L + V dome, we find:

Veapor, = 0.3829 an, ks also

uz = 2545.0 kJ/kg At this state we find:

T, = 140°C

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Specific Heats The state postulate (presented earlier) was used to conclude that the state of a simple compressible substance is determined by the values of two independent intensive properties. As a result, the internal energy of a simple compressible fluid could be considered a function of temperature and specific volume, i.e., =

Chea)

For a “small” process where the end states are sufficiently close to one another, we get:

Properties of Pure Substances

23

The first partial derivative in this expression is a thermodynamic property termed the specific heat at constant volume (c,). Similarly the property enthalpy can be treated as a function of temperature and pressure, with the following similar relationship:

dh =|

oh oh — |] dT+|{—] Oly,

Op)

r+

dv

The first derivative on the right-hand side is referred to by the specific heat under constant pressure (c,). Both specific heats are graphically shown in Figure 2-1 as tangents to u versus T and h versus T curves, respectively.

The Ideal Rankine Cycle Figures 2-10 and 2-11 show the different components and their functions in an ideal Rankine cycle. First the water substance (usually saturated liquid) is compressed up to

Figure 2-10 Components comprising the Rankine cycle.

Electrical

generator

Boiler

Components of Rankine cycle Condenser

Liquid water

Figure 2-11

Ideal (reversible) expansion

Components’ functions.

Power shaft

Electrical generator

Constant pressure heat addition

Constant condensation

Liquid water

Ideal (reversible) compression

24

Thermodynamics

Figure 2-12 Constant-pressure process under the piston’s weight.

Piston displacement

|

re

a high pressure magnitude through the pump component. In the boiler, heat is added at constant pressure. The substance (usually superheated vapor) then expands across the steam turbine to a significantly low pressure. Finally, and across the condenser, the substance (usually wet steam) undergoes a condensation process back to the initial state.

PROBLEMS Water steam is kept under a pressure of 5.0 bars and a quality of 0.9. Shaft work is imparted (Figure 2-12) at constant volume through a paddle-wheel device until the pressure is 10 bars. Calculate the final temperature. Two kilograms of water substance at 200°C and 300 kPa is contained in a weighted piston-cylinder device. As a result of heating under constant pressure, the temperature rises to 400°C. Determine the change in volume (AV), internal energy, and enthalpy.

2-3

Determine the pressure and specific volume of a water substance at 20°C that has a specific internal energy of 1200 kJ/kg.

2-4

A rigid tank contains 50 kg of saturated liquid water at 200°C. Determine the pressure in the tank as well as the specific volume. Water vapor at 5.0 bars and 500°C follows a constant-pressure (Figure 2-10) process, until its temperature increases to 800°C. Determine the enthalpy and internal energy changes that occur

during the process.

i

Dry saturated steam at 200°C is heated in a constant-pressure process. Determine the amount of work per kilogram of the steam performed on the surroundings if the final temperature of the steam

is SOO°C. Find the quality of a liquid-vapor mixture where v ¢ = 0.00101 m3/kg, Ve = 0.00526 m°?/kg, and the total mass is 2.0 kg, which occupies a volume of 0.01 m?.

Properties of Ideal Gases Chapter Outline 3-1

Ideal Gas Relationships

3-2

Equation of State

3-3

Modes of System/Surroundings Interaction

3-4

Work

3-5

Details of Moving Boundary Work

3-6

Polytropic Processes

31

3-7

Gravitational Work

32

3-8

Shatt Work

3-9

Compressibility Factor: A Measure of Deviation from the Ideal Gas Behavior

3-10

26

27

27

29

29

32

Other Equations of State

32

34

25

Thermodynamics (ee) 500

«

Validity of the ideal gas

status to superheated water vapor.