Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-Size Systems and Finite-Time Processes [1 ed.] 0849396514, 9780849396519

Table of contents :
Preface
The author
Acknowledgment
Dedication
Contents
Symbols
1 Thermodynamic concepts and laws
2 Entropy generation and exergy destruction
3 Entropy generation in fluid flow
4 Entropy generation in heat transfer
5 Heat exchangers
6 Insulation systems
7 Storage systems
8 Power generation
9 Solar-thermal power generation
10 Refrigeration
11 Time-dependent operation
Appendix A: Local entropy generation rate
Appendix B: Variational calculus
Author index
Subject index

Citation preview

ENTROPY GENERATION MINIMIZATION The Method of Thermodynamic Optimization of Finite-Size Systems and Finite- Time Processes

ADVANCED TOPICS IN MECHANICAL ENGINEERING SERIES SERIES EDITOR Frank A. Kulacki, University of Minnesota

Books in this series include ENTROPY GENERATION MINIMIZATION Adrian Bejan, Duke University NONLINEAR ANALYSIS OF STRUCTURES M. Sathyamoorthy, Clarkson University FINITE ELEMENT AND COMPOSITE STRUCTURES ANALYSIS USING MATLAB Hyochoong Bang, Naval Postgraduate School MECHANICS OF SOLIDS AND SHELLS Gerald Wempter, Georgia Institute of Technology

ADRIAN BEJAN

J. A. Jones Professor of Mechanical Engineering Duke University

The Method of Thermodynamic Optimization of Finite-Size Systems and Finite-Time Processes

CRCPress Boca Raton New York

Library of Congress Cataloging-in-Publication

Data

Bejan, Adrian, 1948Entropy generation minimization Adrian Bejan. em.- (CRC's mechanical engineering series) p. Includes bibliographical references and indexes. ISBN 0-8493-9651-4 (alk. paper) 1. Thermodynamics-Mathematical models. 2. Heat-TransmissionMathematical models. 3. Entropy. I. Title. II. Series. TJ265.B47 1995 621.402'1-dc20

95-11954 CIP

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 is listed. Reasonable efforts have been made to publish reliable data and information, but the anthor 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. CRC Press LLC's consent 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 for such copying. Direct all inquiries to CRC Press LLC, 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431. © 1996 by CRC Press LLC

No claim to original U.S. Government works International Standard Book Number 0-8493-9651-4 Library of Congress Card Number 95-11954 Printed in the United States of America 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

PREFACE

Entropy Generation Minimization (EGM) is the method of thermodynamic optimization of real systems that owe their thermodynamic imperfection to heat transfer, fluid flow, and mass transfer irreversibilities. It was identified as a self standing field of research in the book by Bejan ( 1982). *The EGM method combines at the most fundamental level the basic principles of thermodynamics, heat and mass transfer, and fluid mechanics. As a field, it is the confluence of thermodynamics, heat transfer, and fluid mechanics (Figure 1); because of its interdisciplinary character, EGM is distinct from each of these classical disciplines. As a method of engineering research and education, EGM was designed to make both thermodynamics and heat transfer more "applicable", i.e., easier to understand, easier to use, and more effective. I was introduced to the power and rigors of the method as a student at MIT, in particular by my graduate theses advisor, Prof. Joseph L. Smith, Jr. Although "good engineering work" has always demanded the simultaneous and correct application of the principles of thermodynamics, heat transfer, and fluid mechanics, I saw a need to bring these principles together systematically into a method defined by its own rules, language, and promise. Entropy Generation Minimization is just that - the minimization of thermodynamic irreversibility in real-world applications by accounting for the finite-size constraints of actual devices and the finite-time constraints of actual processes.

Entropy generation

through

heat and fluid flow

Thermodynamics

Figure 1. The interdisciplinary triangle covered by the EGM method (Bejan, 1982).

* Bejan, A. 1982. Entropy Generation through Heat and Fluid Flow. Wiley, New York.

The arrival of a new field of research is announced by the emergence of a number of simple, fundamental results that hold for entire classes of known and future applications. Entropy Generation Minimization emerged in the 1970s in engineering: its first series of fundamental results formed the backbone of the Bejan (1982) book. These results were obtained by minimizing entropy generation in realistic (heat and fluid flow) models built for several levels of complexity (Figure 2). The minimum entropy generation design was determined for each model, as in the case of the optimal diameter of a duct with flow and heat transfer, or the optimal hot-end temperature of a power plant with bypass heat leak to the ambient. The approach of any other design (Sgen) to the limit of "realistic" thermodynamic ideality represented by the design with minimum entropy generation (Sgen,min) was monitored in terms of the entropy generation number N5 = Sgen/Sgen,min >1, or alternatives of the same ratio. The EGM method is also distinct from exergy analysis. The latter belongs to pure thermodynamics: to calculate exergy, the analyst needs only the first law, the second law, and a convention regarding the values of the intensive properties of the environment. The critically new aspect of the EGM method is the minimization of the calculated entropy generation rate. To minimize the irreversibility of a proposed design the analyst must use the relations between temperature differences and heat transfer rates, and between pressure differences and mass flow rates. He must relate the degree of the thermodynamic nonideality of the design to the physical characteristics of the system, namely, to finite dimensions, shapes, materials, finite speeds, and finite-time intervals of operation. For this he must rely on heat transfer and fluid mechanics principles, in addition to thermodynamics. Only by varying one or more Approach

* E

>VJ

Oi

0

1-

Sgen,min

.'!l cQ) c 0

c.

E

0 (..)

m~

VJ

:;::;J!!

Q)

'I=

VJ

:a.!.I

Power plants

0

a. c.

Solar power and refrigeration plants

Sgen.B' we can say that system A operates more irreversibly than system B. While reviewing the use of Equations (1.16) through (1.19) in the following examples, keep in mind the conditions that are necessary to model an open system as operating in steady flow. The steady operation implies that system properties such as density, specific energy, and specific entropy do not change with time. However, such local properties can vary spatially from one point to another in the control volume. If the system can be described in this manner, then the operation is steady terms in Equations (1.16) through (1.18) vanish. and the

atat

Example 1.2. Consider the steady operation of a steam turbine (Figure 1.5). Across the turbine, the steam pressure drops from a fixed inlet pressure Pin to a fixed outlet pressure Pout· Because this is a steady flow application, the mass conservation statement, Equation (1.16), reduces to

m.m =tit out

=tit

(1.20)

Modeling the control surface drawn around the turbine as adiabatic ( Q = 0), the first law and the second law yield (1.21)

9

(1.22) In writing the ftrst law, Equation (1.21), we neglected the kinetic and gravitational energy changes associated with the inlet and outlet streams.

r------,

I I

I I I

I

L-_._..._ mout

L ______

1

J

Adiabatic surface

Enthalpy

Entropy

Figure 1.5 Thermodynamic analysis of a turbine in steady flow.

10

The right side of Figure 1.5 was drawn for the general case in which Sgen > 0; i.e., the flow of steam through the turbine is irreversible. It is customary to evaluate the performance of a given engineering component such as a steam turbine in the reversible mode. This limiting situation is sketched with a solid line on the h-s diagram of Figure 1.5. Geometric reasoning alone suggests that a reversible turbine delivers the maximum possible shaft power per mass flow rate, provided Pin and Pout are fixed. The first law and the second law in the reversible case yield

w

=

sh,max

s

gen

=

m(h

in

-hout,rev )

m(s out.rev - s )' = o in

(1.23) (1.24)

As in Example 1.1, let us calculate the drop in turbine shaft power caused by irreversibility. Combining Equations (1.21) and (1.23), we find (1.25) Geometric reasoning is again helpful in recognizing that for small departures from reversible operation, the enthalpy deficit hour- hout,rev is proportional to the entropy increase across the turbine, sout- sout,rev (see shaded triangle in Figure 1.5). Because

dh= Tds-vdP

(1.26)

and because from "out,rev" to "rev" dP = 0, we write: hout- hout,rev = Tout (Sout- Sout,rev)

(1.27)

The temperature Tout is an average outlet temperature, defined as Tout =

S

1 out -

Sout,rev

fout out, rev

~utds

(1.28)

The value ofTout falls somewhere between Tout.rev and T001 , because the steam outlet temperature rises as the turbine operates more irreversibly. Having made the observation that led to Equation (1.27), we can now reap the reward by combining this result with Equations (1.25) and (1.24) to express the reduction in shaft power as

w

sh,max

-Wsh =Tout sgen

(1.29)

The entropy generated in the system is directly proportional to the decrease in turbine shaft work. It is well known from classical presentations of turbine thermodynamics that the turbine power output (or the turbine efficiency) decreases with the increasing

11

-

Valve

oro

Container

}

" 'JfJ- ---.L...----, I

Air: 'JQ.

10~-----------~

0 0 0 0 0

Atmosphere:

0

0 0

T0 , P0

Figure 1.6 Evacuated tank being filled with atmospheric air.

degree of turbine irreversibility. The point made in the above example is that subject to Tout= constant [Equation (1.28)], the decrease in shaft power is proportional to the rate of entropy generation.

Example 1.3. The thermodynamic analysis of unsteady flow systems presents further subtleties, reviewed in this example. Figure 1.6 shows a rigid and evacuated container of volume V surrounded by the atmosphere (P 0 , T0 ). At some point, the neck valve opens and atmospheric air gradually fills the container. The wall of the container is thin enough so that eventually the trapped air and the atmosphere reach thermal equilibrium. Let us examine this filling operation and determine the total heat transfer exchanged by container air and atmosphere through the wall of the container. For analysis, we select the container wall as control surface. In the duct communicating with the atmosphere the control port is located upstream of the valve. This choice makes the valve as well as the irreversibility associated with it internal to the system, i.e., part of the control volume. The situation is definitely unsteady, as indicated by the time-dependent air pressure measured in the container. For mass conservation and the first law we write:

. aM

m=-

( 1.30)

ot

. .

au

at

-Q+mh = 0

(1.31)

where M =PVIRTand U =Mcv(T- T0). The enthalpy of atmospheric air,* compatible

* It is tempting to write h = c" (T- T0 ); however, this is not allowed because u = cv (T- T0 ). Note that the definition h = u + Pv must always apply.

12 with having chosen u = cv(T- T0 ), is (1.32)

We can calculate the heat transfer interaction for the filling process by integrating Equation ( 1.31) from the initial time t; to the final equilibrium time ti Q=

i

'f

t

Qdt

I

(1.33)

Because both P 0 and V are positive, the sign of Q is as shown in Figure 1.6. Result Q = P 0 V makes sense physically if we think of the atmosphere as doing work Watm =P 0 Von the batch of air pushed through the valve into the container. This batch eventually stores no energy (Trona! = ~nitia 1 ), so that the work P 0 Vis dissipated via the filling process and is fully ejected back into the atmosphere as heat. The second law of thermodynamics, Equation (1.19), can now be employed to determine the degree of irreversibility of the filling process. In accordance with the sign convention of Figure 1.6 we write

sgen

=

i'f(as-a ';

t

Q . ) dt +--mso T0

(1.34)

We draw the conclusion that the entropy generation increases with the size of the container (V) and with the initial pressure difference (P0). A less obvious conclusion is that the entropy generation is proportional to the work done by the atmosphere to drive the filling process: combining Equation ( 1.34) with the observation that Watm = P 0 Vyields

13 Container (partial vacuum)

Available work (exergy)

Figure 1.7 Apparatus for filling a container while delivering work (exergy).

(1.35) This conclusion is similar to Equations (1.29) and (1.15). The work done by the atmosphere, is lost due to the irreversibility introduced by the valve (Figure 1.6). In order to recover W,1m for our use, we must eliminate Sg•n; in other words, we must replace the valve with a reversible component for steady flow. This alternative is shown schematically in Figure 1.7. The reader is encouraged to analyze the operation and work-producing potential of this new installation.

w.tm.

1.4

THE MOMENTUM THEOREM

It is appropriate to include in this brief review of engineering thermodynamics the relationship between the momentum of a fluid and the forces exerted by the environment on the control surface that surrounds the fluid system. This relationship is embodied in the momentum theorem, which is a modified statement of Newton's second law of motion. As shown in Figure 1.8, the system (control volume) can be subjected to a variety of forces F. Some of these forces act on the control surface, for example, the boundary pressure and the stresses in the cut made by the control surface through a solid object that protrudes into the system. Other forces (termed body forces) act on the control volume as a whole from a distance. For example, in Figure 1.8 the system experiences a gravitational pull in the direction of the gravitational acceleration g. A distinct class of forces arises from the inflow and outflow of fluid through the ports of the surface. The presence of such forces is particularly evident in the handling of garden hoses and water balloons. According to Newton's second law of motion, the resultant of all forces is equal to the time rate of change in the system's momentum. Projecting this statement on one direction of interest, x in Figure 1.8, we write:

~ (MV)

O(

X

CV

="..L..J F .t

+ "(rilV ). ..L..J X

Ul

- "(rilV ). ..L..J X

out

(1.36)

where the subscript "x" indicates the x-component of each vector. The sign appearing in front of each 1hVx tetm in Equation (1.36) depends on the orientation of the ports relative to the x direction (in the present case inlet looks toward negative x and outlet looks toward positive x). As in mechanics, in order to state the momentum theorem correctly one should first construct a precise and unambiguous drawing.

14

;!

Gravitati~nal

acceleratiOn

~Ut

Direction x

Figure 1.8 Forces acting on an open thermodynamic system.

1.5

USEFUL STEPS IN PROBLEM SOLVING

The objective of this chapter so far has been to review the most central concepts and laws introduced by classical thermodynamics in engineering practice. The review is punctuated with three examples in which engineering problems are solved using the concepts and laws of thermodynamics. The conclusions derived from each example have stressed the strong connection that ties the concept of entropy generation (irreversibility) to the notion of lost or destroyed work. This connection is the subject of a more formal development presented in Chapter 2. In addition, Examples 1.1 to 1.3 were designed to demonstrate that the solution to a thermodynamic problem is made easier by a reasoned plan of attack. Problem solving is an issue of great importance in engineering education (Bejan and Paynter, 1976). In my opinion, insufficient attention is being devoted to this aspect of engineering education, especially in the field of engineering thermodynamics and heat transfer. It is therefore useful to highlight a number of steps one could take toward the correct and efficient solution of a thermodynamic problem. 1. Clearly describe the system to be analyzed. The description is always made clearer by a good drawing. This drawing must clearly show the system boundary, that is, the relationship between the system of interest and its neighbors (the environment). 2. Identify all transfer effects (mass, work and heat, entropy) around the system boundary. This step is closely related to step 1. An experienced problem solver will most likely define the system boundary in relation to the presence or absence of one or more interactions (transfer effects). 3. Describe the internal constitution and operation of the system. Questions such as steady vs. unsteady flow, ideal vs. nonideal gas, and so forth, must be addressed. Any geometric constraints in the system also must be identified at this point.

15 4. Apply the three statements (mass conservation, energy conservation, entropy generation) to the problem at hand. These statements and the equations describing the system model (step 3) provide the mathematical apparatus for analyzing the problem and for obtaining a solution. 5. Examine the solution and its meaning with respect to your initial intuitive feel for the problem. This thinking process will strengthen your ability to attack future problems in thermodynamics and in other areas.

1.6

THE TEMPERATURE-ENERGY INTERACTION DIAGRAM AND THE ENTROPY INTERACTION-ENERGY INTERACTION DIAGRAM

The creative use of graphics (step 1 in the preceding list of problem-solving hints) is an impmtant technique that is illustrated and emphasized throughout this book. One example is the temperature-energy interaction diagram (Bejan, 1977), which is presented in detail in Figures 2.3 and 2.4. As the name indicates, this technique consists of plotting the system energy interactions (work, heat) in the twodimensional plane of temperature (T) vs. energy interaction (E). The T-E diagram is particularly useful in visualizing the imperfect thermodynamic performance of power plants and refrigeration plants, i.e., closed systems that operate steadily or in cycles in communication with two temperature reservoirs. The T-E diagrams lose their clarity when applied to open systems in communication with more than two heat reservoirs. In such cases the entropy interactionenergy interaction diagram is more suitable. Three examples of S-E diagrams are illustrated in Bejan (1977). The basic rule in constructing the S-E diagram is to regard each interaction as a carrier of both energy and entropy. Examine again Equations (1.17) to (1.19) and recognize that: of temperature T, bound~ry 1. Each heat transfer interaction Qi that cr.osses a syste~ are considered Both Q and ~ carries with it the entropy interaction Q components positive when they enter the system. The interaction vector Q 1 ~as~ interaction, namely ( Q;, Qj"F; ). the energy interaction and the e~tropy 2. Each work transfer interaction W caiTies zero entropy and is represented by the 0). Recall that the work per unit time W is vector W1 with the components ( considered positive when it is done by the system on the environment. 3. Each mass flow interaction m; through the inlet or outlet port (i) brings in two currents, energy at the rate m;h;, and entropy at the rate m;s;- Both currents are positive when entering the system. In the S-E diagram they are represented by the mass flow interaction vector 1 with the components ( rn;h;, m;s). Note that, for brevity, in writing mihi we have neglected kinetic and potential energy effects [Equation (1.17)]. 4. The time-dependent behavior of the energy and entropy inventories of the system is represented by the vector labeled dldt, which has the components (dE/dt, dS/dt). This vector is zero when the system operates in the steady state or in an integral number of cycles.

/T; .

/T;

\v,

m

To illustrate the generality of the method, Figure 1.9 shows how the interaction vectors add up on the S-E plane for a system with one heat input from a high

16

s

E

Figure 1.9 General outlook of the entropy interaction-energy interaction (s-E) diagram for an open thermodynamic system in the nonsteady state.

temperature source ( Q), heat rejection to the ambient ( Q0 ), one inlet (min), one outlet ( mout), and time-dependent behavior (d/dt). Note that we added vectorially all the interactions except the work interaction. The resultant of these interaction vectors is the balance vector B, which has as components the net power output W and the negative of the entropy generation rate, B(l.f, - Sgen). In any other application the resultant vector will point toward the lower half of the plane because the entropy generation rate cannot be negative. The B vector points to the right in applications that produce power (e.g., power plants), and to the left in applications that consume power (e.g., refrigeration plants). The S-E diagram and the resultant vector show how the system responds when changes are made to improve its thermodynamic performance. The B vector turns and becomes more horizontal as the total rate of entropy generation decreases. When the system produces or consumes zero mechanical power, the resultant vector points vertically downward. All pure heat transfer devices are characterized by such vectors.

REFERENCES Bejan, A. 1977. Graphic techniques for teaching engineering thermodynamics. Mech. Eng. News. May: 26-28. Bejan, A. 1988. Advanced Engineering Thermodynamics. Wiley, New York. Bejan, A. and Paynter, H. M. 1976. Solved Problems in Thermodynamics. Massachusetts Institute of Technology, Cambridge. Moran, M. J. 1989.AvailabilityAnalysis: A Guide to Efficient Energy Use. ASME Press, New York.

PROBLEMS 1.1. Two pools A and Bare filled with equal quantities of water, mA = m8. Initially, pool A is warmer than pool B:TA.o > T8 •0 •

17

a. The two water bodies are now placed in direct thermal contact and eventually reach thermal equilibrium: TA.- = T8 __ = T_. Determine: The final temperature T_. The entropy generated during the heat transfer process, sgen· b. Consider now the alternative process in which a reversible heat engine operates steadily between heat reservoirs TA and T8 • After a sufficient number of cycles, the two pools reach a common temperature T-.rev· Determine: The new final temperature T-,rev· The net work output of the heat engine, W. c. Show that W is proportional to Sgen• and determine the characteristic temperature factor in this proportionality.

Answers a. T_ = ±(TA.o + Ts,o)

b. T-.rev = (TA,OT8,0 )1/ 2

w= me( TA,O + TB,O - 2T__ rev)

w where E = (TA.o - T8 •0 )/T8 ,0• Graphically one can show: T

oo,rev

sgen

-

0

(2.17)

-w

(2.18)

TL

~{

and, for a reversible engine,

Q.H.rev -Q· L.rev S.

gen

max

=0

__ QL,rev _ QH,rev __ 0 TL

~{

(2.19)

Comparing the actual engine to the reversible engine on the basis of equal heat input, QH = QH .rev , we can evaluate the lost available work: (2.20)

27

Note that in order to arrive at result (2.20) we had to eliminate QLrev between Equations (2.18) and (2.19) to obtain wmax ( QH, TL' Tu). Therefore, the low-tempera1j singled out in the gener~ analysis ture reservoir TL assumes the role of res~rvoir based on Figure 2.2. Having evaluated W and Wmax on the basis of equal Q8 means that the heat transfer interaction with the TL reservoir varies with the degree of irreversibility of the engine. In Figure 2.3 the entropy flux QIT is the tangent of the angle subtended by the heat transfer segment Q. For example:

QH =tan a T

H

H

(2.21)

The entropy generation, graphically represented by the difference (tan aL- tan aH), is fixed as soon as T8, Tv QH and W (or QJ are known. Unlike entropy generation, the lost available work varies proportionally with the reference temperature, Equation (2.15). In Figure 2.3 one can see that if the same engine has the option of rejecting heat to a slightly lower temperature, T0 , the lost work decreases at the same rate as the heat rejection QL. A similar discussion may be built around a refrigerating machine using the cold space TL as low-temperature reservoir and the atmosphere TH as high-temperature reservoir (Bejan, 1977, 1988). A comparison of actual vs. reversible operation based onfaed refrigeration load QL is presented graphically in Figure 2.4. This time the The temperalost available work is relative to the heat rejection temperature, ~ost.H. ture-energy interaction (T-E) diagrams of Figure 2.3 and 2.4 were used subsequently by Bucher (1986) and Wallingford (1989) (see also Bejan, 1994). Related graphic constructions are presented later in Figure 8.4 and 10.9. Professor Radcenco just drew my attention to the fact that the T-E diagram also was used independently in the Russian language literature, beginning with Brodianskii (1973). 2.3

ADIABATIC SYSTEMS

When the system of interest is not capable of exchanging heat with its environment, the (destroyed exergy)-(entropy generation) proportionality cannot be expressed as in Equation (2.8). A system of this kind is the steady flow turbine analyzed in Example 1.2. The notions of lost available work and entropy generation imply the comparison of the actual (irreversible) operation of one system with the reversible operation of the same system. From one side to the other in this comparison, things must change inside and along the boundary of the system. One of these changes occurs in the work transfer interaction. As lost available work, this change is the object of the comparison. However, the first law of thermodynamics requires a change in at least one other energy interaction. In an adiabatic system this additional change can be accommodated only by allowing one thermodynamic property at one port to vary (float) with the degree of thermodynamic irreversibility.

28 I

~

QH TH

QL

h

f

I I I I I rev I I

QH,rev

I

o-w 1'

I THI

QH

------

UQH

DoL

QL

- - - h-

I

I

I

/I

I

II I

I

irrev

1/

/I ;I I I I I I 1

1

I

V

I

I

11

I

I

I I

I I

I

w II

jl

I

I

I I

I II

r II

I

I

I I I I

CXH, rev=cxL

I

I/ I I IE wmin

I I I

IIi

"rost,

H

0

Figure 2.4 The T-E diagram. Entropy generation and lost available work (destroyed exergy) in a refrigeration machine (Bejan, 1977).

In Example 1.2 we allowed the turbine outlet temperature to change as the turbine operated less and less irreversibly. One can generalize the analysis of Example 1.2 by considering an adiabatic system with an unspecified number of ports (Figure 2.1, with Q = 0). This generalization is left to the student as an exercise. The end result is .

.

Ht;ost, * = T.Sgen

(2.22)

T. being the average temperature of port labeled (*) where property x is allowed to vary as the irreversibility of the system varies,

(2.23)

In Equation (2.23) y is the remaining thermodynamic property regarded as fixed at port(*). The Gouy-Stodola theorem for adiabatic systems, Equation (2.22), is similar to Equations (2.8) and (2.15). These statements have in common the fact that lost

29

available work W1ostJ is defined relative to the boundary point j where the energy interaction is allowed to vary a'i the system irreversibility changes.

2.4

EXERGY ANALYSIS OF STEADY FLOW PROCESSES

If the open system of Figure 2.1 operates in steady state, the maximum shaft power is simply W.h,max

Im (h- Tas)- Im (h- Tos)

=

(2.24)

out

in

where the specific kinetic and gravitational energies of streams min' mout have been assumed insignificant. Expression (2.24) signals (h - ToS) as an important quantity in the second law analysis of steady flow processes. Following Keenan (1941), we refer to this quantity as the availability function for steady flow:

b = h- T0 s

(2.25)

It is important to bear in mind that b is in fact a thermodynamic property of the system as soon as the environment (T0) is specified. Note the difference between the steady flow availability function, h - T0s, and the Gibbs free energy, h - Ts. For the more general case sketched in Figure 2.5 we write

W= S.

gen

Ic1+flo+ Imh-Imh

=-

L i

in

L.ms+ L.ms

Qi - Qo T L'

(2.26)

out

in

0

(2.27)

out

and, if the system is free of irreversibility,

wmax

Eliminating

fk.rev

=I Qi +f2o.rev +I nth- I in

out

mh

(2.28)

between Equations (2.28) and (2.29) yields

Wmax = W+ ~os~O

(h- :fos)- I,m (h- :fos)+ ~Qi(1-

=~fit ID

U

I

~)

(2.30)

I

Expression (2.30) constitutes a summary of availability flow accounting along the system frontier. The product

30

.

(2

(1-I;J Ta '

(2.30')

represents the "availability of the heat transfer interaction (Q;, 7;) with respect to the reference reservoir T0 ," i.e., the capacity of the heat transfer interaction to produce work. Recalling the heat engine example of Figure 2.3, it is easy to see that Q;O - T0 /T;) represents the maximum work obtained when a reversible engine operates between T; and T0 , with Q; as the heat input. Consequently, the availability of (Q0 , I'a) with respect to T0 is always zero and is not listed in Equation (2.30). Although an equation such as Equation (2.30) may invite the student to assume that an availability "balance" exists in the SyStem, the presence Of the Wlos~O term suggests exactly the opposite. In any real process one finds an availability flow deficit accounted for by the lost available work term. Any real system destroys availability. This feature is illustrated in Figure 2.5, in which the lost availability (W 10 ,~ 0 ) is shown leaving the semi-infinite plane labeled "available". An availability balance exists only in the limiting ca PouJ· This fact is more visible in special limits of pure substance behavior, namely, ideal gas (R, c) (2.47) and incompressible fluid (c, p ):

37

(0'-"'--~--/--~)

,..-Adiabatic

~.-LlP

Figure 2.9 Destruction of exergy in duct flow with friction.

(2.48) The relationship between entropy generation and lost available work is easy to see in the case of fluid friction because, obviously, we must invest useful work to push the fluid through the pipe. The lost available power can be calculated based on exergy flow arguments, Equation (2.30): (2.49) The exergy of the stream decreases along the pipe; the decrease is proportional to the pressure drop per unit length (liP) and the flow rate ( th).

2.6.3

Mixing

Another characteristic irreversible feature is the mixing of two dissimilar batches or streams of fluid. The two streams may differ with regard to thermodynamic state (for example, two water streams at the same pressure but different temperatures) or with regard to chemical composition. As a ftrst example, consider the adiabatic mixing of two streams (thp th2 ) carrying the same substance (Figure 2.10). In steady state the ftrst law and the second law are written as (2.50) (2.51) where we accounted for mass conservation by writing

X=

m1/( m1 + m2 ), m= ~ + m2 •

38

""'p

Figure 2.10 Entropy generation due to isobaric mixing of two gaseous inlet temperatures.

streams with different

If the substance is an ideal gas (cP)' and if the mixing process takes place at constant pressure (P1 = P 2 = P 3), then Equations (2.42) and (2.43) can be written more explicitly as (2.52)

'£3 '£3 -sgen = x In-+ (1- x) In-

titep

I;

I;

(2.53)

Eliminating T3 between these expressions, we obtain

sgen

m·cp

=lnx+-r(l-x) ,..1-x •

(2.54)

Parameter 't is the temperature ratio T2 /T1• Result (2.54) is shown plotted in the lower part of Figure 2.10. It is evident that entropy is being produced through the mixing process as soon as the two inlet temperatures differ, 't 'i; 1. The variation of sgen rhc p with the flow rate ratio x indicates that when the temperature ratio 't is fixed, a critical

I

39 for which the Sgen is a. maximum. This critical value is obtained easily from Equation (2.54) by solving asgen;ax = 0,

X exists

t-1-tlnt (1-t)lnt

(2.55)

X=--:---:---

The entropy generation rate has exactly the same form as in Equation (2.54) if, instead of ideal gas, the mixing involves incompressible fluid. In this case cP is replaced by c, the sole specific heat of the incompressible fluid. One must keep in mind, however, that Equation (2.54) is special, having been derived on the basis of modeling the mixing process as adiabatic and at constant pressure. As a second example, consider the constant pressure mixing of two different (nonreacting) gaseous streams having the same temperature. As in the previous case, the mixing chamber is adiabatic. Here, we recall that the change in specific entropy experienced by each constituent (1,2) from inlet to outlet is given by (s

T

out

.

-s.m )i =c Pi ln~-R.ln~ T. m

1

p,. p '

(2.56)

i= 1,2

m

where Pou~i is the partial pressure of constituent i in the mixed stream. The frrst law can easily be applied to show that the outlet and inlet temperatures are equal so that the frrst term in Equation (2.56) vanishes. The rate of entropy generation becomes .

p

p

I

Sgen =-thR I I ln~-m-R p

··-z

2

ln~ p

2

where P =Pin= Pout· Introducing the molal flow rates ~. the entropy generation rate can be written as

(2.57) ~ in place of

ml' m2 , (2.58)

In this expression CZJi is the universal gas constant, R1M1 , and M1 the molal mass of constituent i. We also J!Sed. the fact t.J:Iat the partial pressure ratio ~ou~i IJ! equals the mole fraction 1 = IJljIJl , where IJl is the total molal flow rate ( ~ + ~ in this example). Rearranging Equation (2.58), we can express the entropy generation in dimensionless form:

x

(2.59)

x x

This result is plotted in Figure 2.11, having recognized the fact that 2 + 1 = 1. The entropy generated through mixing is always a positive quantity, as indicated by Equation (2.59), in which 1 < 1. The entropy generation reaches a maximum at the point where the mole fractions are equal.

x

40

Figure 2.11

Entropy generation during isothermal and isobaric mixing of two different gases.

In the case of n different gaseous streams mixed at constant pressure and temperature in an adiabatic chamber one can follow the derivation outlined above to show that the entropy generation rate assumes the more general form (2.60)

subject to

2.6.4

L-7=1 X;= 1 (e.g.,

Further Examples

Reynolds and Perkins, 1977).

of Irreversible

Features

In addition to heat transfer, fluid friction, and mixing, other features can be viewed as definite clues to the irreversible operation of an apparatus. Three additional examples are filling and discharge, compression and expansion, and combustion. In Example 1.3 we calculated the entropy generated in the process of filling an evacuated container with atmospheric air. During the filling process, the pressure drop across the inlet valve is finite. The situation is similar to the duct with friction that is analyzed in this section. Discharge processes occur in much the same manner, their irreversibility being associated with the pressure drop across the outlet. Note that filling and discharge processes are integral features in the operation of reciprocating machinery (piston-cylinder apparatus). The irreversibility associated with the process of compression/expansion is fairly well known from the study of specific engineering components, compressors (pumps) and expanders (turbines). The overall irreversibility is described by the compressor efficiency and the turbine efficiency. The irreversibility of a compressor may be due in part to features such as the filling and discharge mentioned in the

41

preceding paragraph. However, the compressor irreversibility is due also to compressing the working fluid in a manner that departs from the quasi-static limit required for reversible compression. To be more specific, when a gas is compressed rapidly it is set in motion internally. Viscous and inertial effects dictate how fluid currents (eddies) form and decay locally in the compression chamber. The gascylinder-wall interaction is also important. Eventually, the relative kinetic energy of different fluid packets is dissipated locally through friction, the overall result being the generation of entropy. The local production of entropy in fluid flow is examined in Chapters 3 and 4. The irreversibility of combustion processes is an important subject, but it is beyond the objectives of the present treatment. The reader is encouraged to consult Chapter 7 of Bejan (1988), and the combustion chamber optimization problem discussed in Section 8.4 of this book.

REFERENCES Alefeld, G. 1987. Efficiency of compressor heat pumps and refrigerators derived from the second law of thermodynamics. Int. J. Refrig. 70:331-341. Alefeld, G. 1988a. Probleme mit der Exergie. Brennstoff-Waenne-Kraft. 40:72-80. Alefeld, G. 1988b. Die Exergie und der ll. Hauptsatz der Thermodynamik. BrennstojfWaenne-Kraft. 40:458-464. Baehr, H. D. 1973. Thermodynamik. Springer Verlag, Berlin. Bejan, A. 1977. Graphic techniques for teaching engineering thermodynamics. Mech. Eng. News. May: 26-28. Bejan, A. 1982. Entropy Generation through Heat and Fluid Flow. Wiley, New York. Bejan, A. 1988. Advanced Engineering Thermodynamics. Wiley, New York. Bejan, A. 1994. Engineering advances on finite-time thermodynamics. Am. J. Phys. 62:11-12. Bejan, A. and Paynter, H. M. 1976. Solved Problems in Thermodynamics. Mech. Eng. Dept. Massachusetts Institute of Technology, Cambridge. Bosnjacovic, F. 1953. Technische Thermodynamik. Vol. 1, Steinkopf, Dresden. Brodianskii, V. M. 1973. Exergetic Method ofThennodynamic Analysis. Energia, Moscow. Bucher, M. 1986. New diagram for heat flows and work in a Carnot cycle. Am. J. Phys. 54:850-851. Clausius, R. 1887. Die Mechanische Waennetheorie. Braunschweig. Darrieus, G. 1931. L'evolution des centrales thermiques et la notion d'energie utilisable. Sci. Ind. 15:206. Gouy, M. 1889. Sur l'energie utilisable. J. Phys. 8:501. Jin, H. and Ishida, M. 1993. Graphic exergy analysis of complex cycles. Energy. 18:615--625. Keenan, J. H. 1941. Thennodynamics, ch. 18, MIT Press, Cambridge, MA. Kestin, J. 1980. Availability: the concept and associated terminology. Energy. 5:679--692. Mironova, V. A., Tsirlin, A.M., Kazakov, V. A., and Berry, R. S. 1994. Finite-time thermodynamics: exergy and optimization of time-constrained processes. J. Appl. Phys. 76:629-635. Moran, M. J. 1989. Availability Analysis: A Guide to Efficient Energy Use. ASME Press, New York. Rant, Z. 1956. Exergie ein neues Wort flir "technische Arbeitsflihigkeit," Forsch. Ing. Wes. 22:36. Reynolds, W. C. and Perkins, H. C. 1977. Engineering Thennodynamics. McGraw-Hill, New York.

42 Stodola, A. 1910. Steam and Gas Turbines. McGraw-Hill, New York. Tolman, R. C. and Fine, P. C. 1948. On the irreversible production of entropy. Rev. Modem Phys. 20:51-77. Wallingford, J. 1989. Inefficiency and irreversibility in the Bucher diagram. Am. J. Phys. 57:379-381.

PROBLEMS

2.1. A supply of superheated steam (800°C, 2 MPa) is being considered for possible use in a steady flow power apparatus. Taking the dead state at T0 = 25°C, P 0 = 1 atm (0.1013 MPa), calculate the availability (b) and the exergy (e,.) of superheated steam under the above conditions. Report your results in kilojoules per kilogram. 2.2 A heat engine communicating with two heat reservoirs (Tn, TL) can be modeled as a Carnot engine connected in parallel with a conducting strut. In this model the conducting strut accounts for the leakage of heat from Tn to TL around the power-producing components of the engine (Bejan and Paynter, 1976). The bypass heat leak Qc is proportional to the temperature difference across it, Qc = C(Tn- TL), where Cis a measured constant. a. Is the heat engine operating reversibly? Explain. b. For a fixed heat input Qn, determine an expression for the entropy generated in the heat engine, sgen• c. If Tn is a design variable, determine the optimal temperature TH. opt such that the production of entropy is minimized. d. Show that the optimum determined at part (c) corresponds to the case of maximum work output (maximum heat engine efficiency W!Qn).

r------1

I I

Heat engine

~

--------, .----.L..----. I I

Carnot engine

I I I I I - - - - - - ______ L

_j

TL

2.3. A source of geothermal steam is to be utilized to provide heating for a large building (TB > T0). As shown on the left side of the figure, the simplest method is to heat the building directly. At steady state, the steam-building direct heat transfer QD is matched by the building-environment heat leak Q0. The heat leak

43

Q0 is proportional to the building-environment temperature difference, Q0 = C(T8 - T0), where C is the thennal conductance, assumed fixed. A second

heating arrangement is shown on the right side of the figure. Here, the steambuilding heat transfer is used to produce work in a reversible device; the work is then used to power a reversible heat pump installed outside the building. a. Show that the ratio of steam flow rates required to maintain the building at T8 is

b. Compute the ratio m2 /m 1, assuming P;n = 0.6 MPa, Tout= 25°C, Pout= 1 atm (0.1013 MPa), T8 = 22°C, T0 = -5°C. c. Comment on the practical importance of results (a) and (b): What major design change has taken place between the two heating methods considered in this problem?

Saturated vapor

m,

Water (liquid)

~ m

out

Saturated vapor

"7

Water (liquid)

~ rev

building

I

T8

2.4. Consider the refrigerator shown in the figure. The refrigerator operates in cycles between the load (TL) and the ambient (T11 ). The thermal contact between the refrigerator and the two heat reservoirs (TL, T11 ) is imperfect; this means that finite temperature differences, !J.TL and !J.T11 , exist between the cycle (working fluid) and the heat reservoirs. Consequently, the actual temperature extremes faced by the refrigeration cycle are

The refrigeration cycle between T(. and TIf can be modeled as reversible. Assume also that !J.T11 !J.TL. The thetmal contact with T11 or TL can be improved

=

44 by upgrading (adding more surface area to) the heat exchangers that occupy the two temperature gaps (!lTL, !lTH). Suppose you have enough means to decrease either !lTH or !lTL by a small amount OT. Which of the two heat exchangers should you upgrade? Base your answer on thermodynamic analysis.

-o

D

IAmbient Reversible refrigeration cycle

.__ _ _ r._L _ _ .....JI Load

2.5. Express the steady flow exergy ex of an ideal gas in terms of the temperature and pressure ratios TIT0 and PIP 0 , e = _x_

cPT0

fiunctton . (T, ~

Compare this result with the equivalent expression for the nonflow exergy sl(cPT0 ) of the same ideal gas. 2.6. A house is to be heated via a heat pump placed between the house and the environment, as shown in the figure below. The house interior is to be kept at a fixed temperature level T8 • The rate of heat transfer from the house to the ambient (TA) can be modeled as Q = C(T8 - TA), where Cis a known constant. Two designs are being considered. In the first design the heat pump is in communication with the atmosphere; the daily atmospheric temperature variation is described by TA(t) = TA

+ !lT sin(27tt/day)

where TA and !lT are the average temperature and the daily fluctuation amplitude, respectively. In the second design the cold end of the heat pump cycle is buried in the ground, where the temperature is Ta =constant. In both designs the heat pump operates irreversibly, with a known second-law efficiency 'Tln = Wmin(Camot/ Wactual•

45

Calculate and compare the daily heat pump work requirement W corresponding to the two designs. Assuming that the ground temperature Ta is known, under what atmospheric temperature conditions (TA• A.T) is the frrst design preferable? T

0

TG

House

--8 0

1 - - - - - - - -Ground --

Winter time

2.7. A power plant operates irreversibly between the high-temperature reservoir TH and the environment Tv Assume that the power plant operates in the steady state or executes an integral number of cycles. The heat input QH is flxed. Show that to maximize the power output W is the same as minimizing the entropy generation associated with the power plant system delineated in the flgure below.

D QH

TH

Actual (irreversible) power plant

TL DQL

w

3

ENTROPY GENERATION IN FLUID FLOW

In Chapter 2 we became acquainted with the Gouy-Stodola theorem, stating the strong relationship that exists between entropy generation (irreversibility) and lost available work (destroyed exergy). We identified heat transfer across a finite I!.T, flow with friction, and mixing as three major mechanisms responsible for entropy generation in engineering systems. In this chapter we focus on the general topic of fluid flow, with special emphasis on the thermodynamic aspects of the phenomenon. This discussion is designed to bridge the gap between fluid mechanics and thermodynamics, a gap that ordinarily prevails in the mind of the engineering student.

3.1

RELATIONSHIP BETWEEN ENTROPY GENERATION AND VISCOUS DISSIPATION

In the context of steady flow through a pipe with friction we learned that the entropy generation rate is directly proportional to the mechanical power needed to push the flow through the pipe. The mechanical power is proportional to the end-toend pressure drop AP and, in its tum, the pressure drop is proportional to the wall shear stress 'to acting over the wall-fluid interface. Because the pipe (control volume) shown in Figure 2.9 is stationary, a force balance in the longitudinal direction dictates (3.1)

If, as in most engineering applications, the fluid is newtonian, the wall shear stress is

(3.2) where (autay) 0 is the gradient of longitudinal velocity normal to the wall, evaluated at the wall, and Jl is the viscosity. Combining Equations (3.1) and (3.2) we see that the entropy generation rate and the loss of mechanical power are ultimately attributable to the viscous shearing effect present in the fluid. This relationship is brought to light in clearer form by the simple experiment sketched in Figure 3.1. The experimental apparatus contains a viscous fluid such as honey. As a closed thermodynamic system, it is in communication with one heat reservoir (the ambient) at temperature T0 • Suppose that in the steady state

47

48

Rotating disk

Fluid

System

Ambient, T0

Figure 3.1 Destruction of mechanical power and generation of entropy by fluid friction.

Wto the system. The first and second laws

the environment transfers work at the rate of thermodynamics dictate

(3.3)

.

sgen

Q

=->0

Ta

(3.4)

The frrst conclusion is that because sgen > 0, the assumed direction of energy interactions is correct (Figure 3.1). In addition, the entropy generation rate is proportional to the mechanical power introduced into the system through the rotating shaft:

.

sgen

w

=-

Yo

(3.5)

The experiment of Figure 3.1 can be repeated not in a finite-size apparatus, but by considering an infinitesimally small element of size dx dy dz. The entropy generation rate, per unit time and volume, is

S"' sen

W"' = -T-

(3.6)

where W"' is the volumetric rate of mechanical power dissipated in the dx dy dz element by viscous shearing. In the same expression Tis the absolute temperature of

49

the fluid element, assumed in local thermal equilibrium with its immediate surroundings. From the study of fluid mechanics we recall that W"' is equal to the viscosity times the viscous dissipation function (e.g., Bird et al., 1960), hence (3.7) A more formal derivation is used in Chapter 4 to show that in flow fields with heat transfer the local entropy generation formula, Equation (3.7), contains an additional term [see Equation (4.10)]. Special forms of this general expression are compiled in Appendix A.

3.2 LAMINAR FLOW In principle, the velocity distribution in any laminar flow field can be determined analytically or numerically by solving the Navier-Stokes equations subject to an appropriate set of boundary and initial conditions. Once this solution is obtained, the local dissipation function and entropy generation rate can be evaluated in a straightforward manner. The reason for evaluating the local irreversibility rate varies from problem to problem. For example, in lubrication problems involving the flow of highly viscous fluids through narrow relative-motion gaps it is necessary to evaluate because it appears as a "heat source" term in the energy equation. The energy equation must be solved in order to predict the temperature field in the lubricated gap. In this section we evaluate s~:, not with an engineering application in mind, but in order to provide the reader with a basis on which to compare the laminar flow phenomenon with the much more intriguing phenomenon of turbulence. The simplest example of laminar flow is the Couette flow described by

(3.8) The entropy generation rate, Equation (3.7), reduces in this case to

S"' = 1.1(au) 2 = gen

T

ay

Jl(U) T D

2

(3.9)

The velocity and entropy generation profiles are shown in Figure 3.2. As a second example we choose the plane Hagen-Poiseuille flow (e.g., Bejan, 1993, p. 294)

3 u--u 2

avg

[1- (-D/2y-)2]

The entropy generation follows immediately from Equation (3.7):

(3.10)

50 y

D

u

~6~r

7

1

S'" gen

u

Figure 3.2 Velocity and entropy generation profiles in Couette flow (Bejan, 1979).

S"' = )l (du) 2 = 36 )l (u•vg )2 gen T ay T D

(_L) Df2

2

(3.11)

3.3

Figure shows that the entropy generation profile is parabolic and symmetric about the duct centerline. This profile does not change as the Reynolds number u.vp!v varies. On the basis of two simple examples, Figures 3.2 and 3.3, we are able to draw the important conclusion that in laminar flows the generation of entropy takes place en masse, i.e., throughout the flow field. This conclusion is not invalidated by the Hagen-Poiseuille flow example, where S~:, is zero strictly on the centerline (y = 0). In laminar flow then, the entire field participates in the production of entropy. We learn in the next section that this is an important feature that distinguishes laminar flows from nonlaminar (turbulent) flows.

3.3

TURBULENT FLOW

The phenomenon of viscous dissipation is present in turbulent flow as well. Let us focus on the turbulent regime in pipe flow and recognize that entropy is being generated in the pipe as soon as a pressure drop can be detected between the ends of the pipe; note that Equation (2.46) holds, regardless of flow regime. In turbulent flow the end-to-end pressure drop is related to the wall shear stress 'to via Equation (3.1 ); in this case 'to is the time-averaged value of wall shear because the actual wall shear fluctuates due to the buckling of the viscous sublayer (see also the "bursting" phenomenon in Chapter 4 of the book Bejan (1982)). The turbulent entropy generation rate, like everything else in turbulence research prior to the buckling theory, can be determined only on the basis of experiment. For a pipe of diameter D = 2r0 and length L, the overall entropy generation rate is readily calculated as

s·

gen

thM pT

=-

(3.12)

51 y

u

Figure 3.3 Velocity and entropy generation 1979).

profiles in plane Hagen-Poiseuille

flow (Bejan,

where

llP=f4L pV2

D 2

The friction factor f is given by the Kannan-Nikuradse relation (Kays and Perkins, 1973):

(4

))v

2

(3.13)

-0.8 + 0.87ln[Re( 4!) 112 ]

=

which follows empirically from the measurement of the mean axial velocity proflle in the pipe. In the range 3 x 104 < Re < 106, Equation (3.13) is closely approximated by

f

(3.14)

= 0.046 Re-0· 2

The local time-averaged viscous dissipation rate in any turbulent flow (E) may be decomposed into a part associated with the mean velocity gradients (E) plus a part associated with fluctuating velocity gradients (E'), E = v = E + E'. For fully developed turbulent flow in a pipe, the two components of the local dissipation rate are (Hinze, 1975):

v[(oux ) +(aux ) +(~) - -

=

+

ox

(a~

2

or

2

r o

2

+(au, ) +(au,or ) +(~)r o ox

J J :;~ Jl + ( ";;

+(

- 2

2

2

(3.15)

52

30r--,----------------------------------------,

,,

20

~~

.

~\

e'D

2u3

\

\

\

~'

10

,',

Present model, Eq. (3.27). for all values of r~

0 Wall

''
1

Gaussian,

f= exp(-

~2)

Exact,

f =1 - tanh

2

1t112

2

~

(~f

(~r

4/3

0.729 0.693

16/15

The maximum centerline temperature is practically independent of longitudinal position, 11~,max 0.15, while aopt decreases monotonically as .X increases. It is important to note that aopt is of the same order of magnitude as the empirical entrainment coefficient (a. = 0.0367). This observation is particularly valid at longitudinal positions greater than several slit widths, i.e., in the self-similar region of the jet, which is responsible for the measurement of a •.

=

Method 2. With regard to the energy content of the entrained flow, it can also be argued that because the fluid reservoir is at rest, the kinetic energy part (-v~) 2 /2

- - - -- Method 1 --Method 2

0.2

------

0.15 0.1

~Tc,max

' ',CXopt ' '

a.

0.05 0

0

20

Figure 3.11 The optimal entrainment centerline.

40

60

x

80

coefficient for maximum temperature

100

rise on the jet

66

should be zero in Equation (3.54) and (3.55). In this case the centerline temperature distribution that replaces Equation (3.57) is (3.59)

This time, the optimal entrainment coefficient that maximizes the centerline temperature at any given can be determined in closed form,

x

I

a opt = _!_ X'

(3.60)

These results are represented by the solid curves in Figure 3.11, again, by using the constants of the exact jet profile. Little difference exists between the present results and those derived based on Method 1. In fact, the two methods lead to practically the same a.opt and !!.(max values in the downstream region beyond ~everal slit widths, reinforcing the conclusion that the optimum a that maximizes llT, is of the same order of magnitude as the empirical value a.•. This conclusion is backed further by Figure 3.12, which shows the actual centerline temperature of the jet, as a function of x. The single curve in this figure represents the superposition of Equation (3.57) and (3.59), in which a= a., and / 1,2,3 are the constants listed for the exact profile in Table 3.2. In this case the nozzle is located at = 13.6 away from the virtual origin, and the theoretical maximum centerline temperature is !!.~,max = 0.15 [cf. Equation (3.60)]. Figure 3.12 shows that the actual centerline temperature of the jet is nearly the same as !!.~,max for a long section of the jet, beyond some distance downstream from the nozzle. This observation has its explanation in the lower part of Figure 3.11, in which the theoretical coefficient a.opt was found to be equal or comparable to the empirical value a.•. In conclusion, the introduction of an empirical parameter (a.) in the pure fluid mechanics treatment of the jet problem is accompanied by an extrein the first law analysis of the same flow. mum (!!.~.max) / 1,2,3

x

3.6.3

Second

Law of Thermodynamics

With reference to the control volume of thickness dx defined in Figure 3.10, the second law states:

dS

~=-

dx

di~

dx

-~

pusdy-s~-

di~

dx

-~

pudy~O

(3.61)

The two terms that appear on the right side are associated respectively with the LHS and RHS arrows drawn in Figure 3.10. When the two integrals are combined, the local time-averaged entropy change (s- s~) appears as a factor in the integrand. This group can be replaced approximately by s- s~ (T- T~) Cp IT~ because the pressure is uniform and the temperature rise is negligible relative to the absolute temperature.

=

67 0.2

----- L~ ~ ~ -~·=- ---~~-

0.15

8l\

-=--=--=--=-=-..:..:--:..:-.:.:--:.::_j-

0.1 Eqs. (3.57) and (3.59), with a.= a.e

0.05 0

0

Figure 3.12 The effect of

x on

60

40

20

80

the actual centerline temperature

100

distribution (a= a.).

Assuming further that the temperature and velocity profiles are described by the same entropy generation rate (3.61) can be expressed as

/(~).the

ds e = -d _

di

(I

- -) uDAT c

(3.62)

di2c

where (3.63) Integrating Equation (3.62) from the nozzle (where A~ downstream yields

= 0) to any

x situated (3.64)

Note that the physical (dimensional) entropy generation rate Sgen listed in Equation (3.63) represents the rate of entropy generation in the jet section (finite control volume) contained between the plane of the nozzle and a particular constant-x plane situated downstream. Equation (3.64) can be combined with the results obtained previously for uc, .b, and A~ to determine the ways in which a, and !!Ie profile shape/(~) influence the rate of entropy generation. In the case of AT, we have a choice between Equations (3.57) and (3.59); the following expression is based on Equation (3.59):

x,

sgen -_!J_ [1- (~ )1/2] 4a X

(3.65)

[~12

x

x

and shows that Sgen increases monotonically as a and increase. The dependence is illustrated in Figure 3.13, which was drawn using the empirical entrainment coefficient a= a •. The entropy generation rate approaches a ceiling value sgen,Jrul][ as the length of the jet becomes considerably greater than the distance from the virtual origin to the nozzle, x~ l/4a,

68

Figure 3.13 The effect of i and profile shape on the total entropy generation rate (a= a.).

(3.66)

Figure 3.13 shows that the entropy generation rate distribution depends on the shape of the velocity and temperature profile,J(s). Recall that in the integral method employed throughout this section we have the freedom to select any proftle shape that is "reasonable" [i.e., one that satisfies the conditionsf(O) = l,f' (0) =O,f(± oo) = 0]. In the case of the jet of Figure 3.10 we also have the luxury to know the exact proftle supplied by the similarity solution, Equation (3.50). Table 3.1 shows a compilation of reasonable profiles, ranging from the piecewise linear (top-hat, trapezoidal, triangular) to the bell-shaped profiles that are being used routinely in integral jet analyses. The table shows that the group l/Ii12 = sgen,max decreases as the shapef(s) becomes more reasonable, i.e., more like the shape revealed by time-averaged measurements. The smallest value of the group l 3 /lf.12 corresponds to the exact proflle [Equation (3.50)]. The Gaussian profile, which is often used to correlate jet velocity and temperature measurements (Bejan, 1995), yields a Sgen,max value that is only 5% greater than that of the exact profile. On the basis of the evidence assembled in Table 3.2, it can be argued that the proper velocity and temperature profile shape is the one that minimizes the total entropy generation rate of a jet of finite length. Other reasonable profile shapes should be tried and added to Table 3.2 in order to strengthen or refute this argument. Most basic is the conclusion that a correspondence exists between the "empirical constant" of a pure fluid mechanics treatment, and the temperature and entropy generation extrema revealed by thermodynamic analysis. This correspondence deserves further study, for example, by focusing on other turbulent flow configurations. It may be possible to use the thermodynamic extrema to predict (i.e., eliminate) the empirical content of the traditional fluid mechanics treatment.

69 (Zero work and heat transfer I

h

h+dh s+ds

p

p

z

V+dV z+dz

v

Figure 3.14 Stream tube for the derivation of the Bernoulli equation.

3.7

THE BERNOULLI EQUATION

We close this chapter with a classical result that ties together the thermodynamics of a flow in the inviscid limit (Figure 3.9). Consider a stream tube (control volume) in an inviscid (reversible) flow field, as shown in Figure 3.14. In the steady state the first and second laws of thermodynamics dictate, respectively, d (h+

~ 2 + gz) = 0

(3.67)

ds=O

(3.68)

Modeling the fluid as incompressible, we combine

dh

= Tds+ vdP

(3.69)

with Equations (3.67) and (3.68) to write:

d(

~ + ~ 2 + gz) = 0

(3.70)

Equation (3.70) is the Bernoulli equation, which, along a streamline, states: 1 P +- p V2 + 2

pgz = constant

(3.71)

The constant appearing in Equation (3.71) is a parameter of the streamline.

REFERENCES Bejan, A. 1979. A study of entropy generation in fundamental convective heat transfer. J. Heat Transfer. 101:718-725. Bejan, A. 1982. Entropy Generation through Heat and Fluid Flow. Wiley, New York. Bejan, A. 1987. Buckling flows: a new frontier in fluid mechanics. Annu. Rev. Num. Fluid Mech. Heat Transfer. 1:262-304.

70 Bejan, A. 1991a. Predicting the pool fire vortex shedding frequency. J. Heat Transfer. 113:261-263. Bejan, A. 1991b. Thermodynamics of an 'isothermal' flow: the two-dimensional turbulent jet. Int. J. Heat Mass Transfer. 34:407-413. Bejan, A. 1993. Heat Transfer. Wiley, New York. Bejan, A. 1995. Convection Heat Transfer, 2nd ed. Wiley, New York. Bird, R. B., Stewart, W. E., and Lightfoot, E. N. 1960. Transport Phenomena. Wiley, New York. Gibson, C. H. 1991. The direction of the turbulence cascade. In Some Unanswered Questions in Fluid Mechanics. L. M. Trefethen, R. L. Panton, and J. A. C. Humphrey, Eds. ASME paper 91-WA/FE-1. Hinze, J. 0. 1975~ Turbulence. McGraw-Hill, New York. Kays, W. M. and Perkins, H. C. 1973. Forced convection, internal flow in ducts. In Handbook of Heat Transfer. W. M. Rohsenow and J.P. Hartnett, Eds. McGraw-Hill, New York. Laufer, J. 1954. NACA Technical Report 1174. Morton, B., Taylor, G. 1., and Turner, J. S. 1956. Turbulent gravitational acceleration from maintained and instantaneous sources. Proc. R. Soc. London. A234:1-23. Pagni, P. J. 1989. Pool vortex shedding frequencies. In Some Unanswered Questions in Fluid Mechanics. L. M. Trefethen and R. L. Panton, Eds. ASME paper 89-WA/FE-5. Richardson, L. F. 1926. Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. London. All0(756):709-737.

4

ENTROPY GENERATION IN HEAT TRANSFER

In Chapter 2 we saw that heat transfer phenomena are always accompanied by entropy generation, hence by the one-way destruction of available work. Therefore, it makes good engineering sense to focus on the irreversibility of heat transfer processes and to try to understand the function of the entropy generation mechanism. The objective of this study is to devise concrete methods for minimizing entropy generation in engineering equipment for heat transfer. Coincidentally, this goal agrees with the objectives of contemporary thermal design practice. In very broad terms the contemporary heat transfer design objectives fall into two large categories: 1. Heat transfer augmentation problems 2. Thermal insulation problems

Writing for the net heat transfer rate Q between two surfaces (T1, TJ:

Q = hA(1;- Tz) a heat transfer augmentation problem is one in which the thermal conductance hA is to be increased. In such a problem the heat transfer rate is usually prescribed by design, hence increasing the thermal conductance leads to improved thermal contact (lower temperature difference, T1 - T2). At the same time, as shown in Equation (2.41), the entropy generation rate also decreases. In a thermal insulation problem..!. on the other hand, the objective is to minimize the effective thermal conductance hA. In such a problem the two temperatures T1, T2 are fixed; hence, as hA decreases so does the net heat transfer rate (heat leak) between the two surfaces. Therefore, the entropy generation rate, Equation (2.41), also decreases as the thermal conductance decreases. The important observation to be made, with the help of Table 4.1, is that in two seemingly opposite thermal design problems, the first requiring the largest thermal conductance and the second the lowest, the entropy generation rate is universally minimized. In both cases the minimization of entropy generation is the hidden consequence of good engineering thinking applied to two individual problems. In this chapter we consider ftrst the topic of entropy generation minimization in convective heat transfer. This heat transfer mechanism is responsible for entropy generation in an endless list of engineering applications. In light of the irreversibility clues discussed at the end of Chapter 2 the irreversibility of convective heat transfer is seen to be due to two effects: heat transfer across a fmite (nonzero) temperature difference and fluid friction.

71

72 Table 4.1

The Two Main Problems

Problem

1. Thermal contact enhancement (heat transfer augmentation) 2. Thermal insulation (heat leak minimization)

4.1

T

~

LJ.T

.[].

Q

T

in Heat Transfer Temperature difference (aT)

Heat transfer rate (Q)

Entropy generation rate ( S gon = Q &T/P)

To be reduced

Fixed

Reduced

Fixed

To be reduced

Reduced

5T

I -------- _t_ ~

T

~

LJ.T

l.~

.[J.Q .(}q

LOCAL RATE OF ENTROPY GENERATION IN CONVECTIVE HEAT TRANSFER

Consider a point (x, y) in a fluid engaged in convective heat transfer (Figure 4.1). The fluid element dx dy surrounding this point is part of a considerably more complex convective heat transfer arrangement. In the following derivation we consider a twodimensional flow, although a similar analysis may be carried out for the more general three-dimensional case. We regard the small element dx dy as an open thermodynamic system subjected to mass fluxes, energy transfer, and entropy transfer interactions that penetrate the fixed control surface formed by the dx dy rectangle of Figure 4.1. The element size is small enough so that the thermodynamic state of the fluid inside the element may be regarded as uniform (independent of position). However, the thermodynamic state of the element may change with time. The fluid is in local thermodynamic equilibrium. Based on the above model, the entropy generation rate per unit volume [W/m3 K] may be estimated writing the second law of thermodynamics for dx dy as an open system:

s::"

oq oq. q +-X dx q +-) dy dx d X OX d y oy d qx d q)' d S., gcn y= oT y+ -..,T X-- y-- X T+-dx T 0 d T T + oy Y ox

+(s+ ~~dx) (vx+ ~: dx) (p+ ~:dx)dy + (s + OS dyJ (v '+ ov dyJ (p + op dyJ dx oy ) oy oy y

(4.1)

73 y

y+dy

x+dx

X

X

Figure 4.1 The local generation of entropy in a flow with convective heat transfer.

In this expression the first four terms account for the entropy transfer associated with heat transfer, the next four terms represent the entropy convected into and out of the system, and the last term represents the time rate of entropy accumulation in the dx dy control volume. Dividing by dx dy, the local rate of entropy generation becomes:

+s[ap+v ap+v ap+p(avx + avy)]

at

X

ax

y

ay

ax

ay

(4.2)

Note that the last term (in square brackets) vanishes identically based on the mass conservation principle (e.g., Bejan, 1995),

Dp +pV·v=O

(4.3)

Dt

where D/Dt is the substantial (material) derivative

v

a

at

a

ax

a

-=-+v - + v Dt X yay

(4.4)

and v is the velocity vector (vx, vy). In vectorial notation the volumetric rate of entropy generation can be expressed as 1 1 Ds S"' =-V·q--q·VT+pgen T T2 Dt

(4.5)

74 We now tum our attention to the first law of thermodynamics, written for one point in the convective medium (e.g., Bejan, 1995):

Du p - = - V · q - P (V · v) + Jl Dt

(4.6)

In this expression the rate of change in internal energy per unit volume is equal to, in order, the net heat transfer rate by conduction, plus the work transfer rate due to compression, plus the work transfer rate per unit volume associated with viscous dissipation. In writing Equation (4.6) we assume that the fluid is Newtonian, Jl being the viscosity and the viscous dissipation function whose units are [s-2]. Writing the canonical relation du = T ds- P d(llp) and using the substantial derivative notation, Equation (4.4), we obtain:

Ds _ £. Du _ _!_ Dp p T Dt

p Dt - T Dt

(4.7)

Combining Equation (4.7) with pDs/Dt given by Equation (4.5) and pDu/Dt given by Equation (4.6), we arrive at: (4.8)

Finally, if the Fourier law of heat conduction for an isotropic medium applies,

q=-kVT

(4.9)

the volumetric rate of entropy generation becomes: (4.10) For the two-dimensional Cartesian system of Figure 4.1, Equation (4.10) states:

S"' = gen

_!_[(aT) + (ar) iJx 2

T2

+i{z[(~

i:)y

2

]

)' +(~; )}(";;+~ J}

(4.11)

Complete forms of s~:n for other systems of coordinates are compiled in Appendix A These forms show that locally the irreversibility is due to two effects, conductive (k) and viscous (Jl). Furthermore, s~;n is positive and finite as soon as temperature or velocity gradients are present in the medium.

75

4.2

FLUID FRICTION VS. HEAT TRANSFER IRREVERSIBILITY

An important observation concerns the order of magnitude of the viscous term relative to the heat transfer term in the makeup of s~:n. This issue requires some care because in a large number of convective heat transfer problems the viscous dissipation term J..L is routinely neglected in the first law, Equation (4.6). If in an actual problem the characteristic dimensions (scales) for velocity gradient and temperature gradient are v*/L * and !J.T* /L*, then the size of the viscous term J..L relative to the conduction term k V · (VI) in the energy equation, Equation (4.6), is

( fluid friction) heat transfer Istlaw

(4.12)

On the other hand, the size of fluid friction irreversibility relative to heat transfer irreversibility, Equation (4.1 0), is ( fluid friction) heat transfer 2nd law

(4.13)

It is clear now that although J..L(V*) 2/(k !J.T*) may be much smaller than unity in many applications, the fluid friction irreversibility term is not necessarily negligible in Equation (4.10). As shown in Equation (4.13), the size of the viscous term depends also on the ratio of the characteristic absolute temperature divided by the characteristic temperature difference, T* I!J.T*. In many heat transfer applications this ratio is considerably greater than unity, as the reader will discover in subsequent examples and problems. In view of this discussion we recognize two new dimensionless parameters in the entropy generation analysis of convective heat transfer problems. These parameters are the dimensionless temperature difference:

!J.T

'C=-

T

(4.14)

and the irreversibility distribution ratio*:

s;:n (fluid friction)

=-!-.=-.:_ _ _ _ _ s~: (heat transfer)

(4.15)

where the entropy generation expression, Equation (4.10), has the basic form:

S"' gen = !;'" gen (heat transfer) + S"' gen (fluid friction)

(4.15')

*For this new dimensionless group I chose the symbol cp a.~ a reminder that the irreversibility distribution ratio increases as the friction irreversibility contribution increases.

76

More. recently, the alternative irreversibility distribution parameter S~:, (heat transfer)/ S~~n = (1 + q>)-1 was named the Bejan number (Be) by Paoletti et al. (1989) and subsequent investigators (e.g., Benedetti and Sciubba, 1993; Petrescu, 1994). Accordingly, Be = 1 is the limit at which the heat transfer irreversibility dominates, Be = 0 is the opposite limit at which the irreversibility is dominated by fluid friction effects, and Be= 1/ 2 is the case (e.g., one point in the flow field) in which the heat transfer and fluid friction entropy generation rates are equal.

Example 4.1. The volumetric entropy generation rate formula (4.10) may be used to derive irreversibility profiles or maps for convective heat transfer arrangements in

which the temperature and velocity are known at each point in the medium. Consider as an example the classic solution for Hagen-Poiseuille flow through a round smooth tube of radius r 0 with uniform heat flux q" [W/m2] at the wall (seeinsetofFigure 4.2). In the fully developed regime the velocity and temperature are given by (e.g., Bejan, 1995): (4.16)

(4.17)

2

NO ...... IN:.,.

R

Rgure 4.2 Entropy generation (Bejan, 1979}.

profiles for Hagen-Poiseuille

flow with uniform wall heat flux

77 where T0 is the absolute temperature at a point of origin located on the tube axis. The volumetric rate of entropy generation in cylindrical coordinates (Appendix A) reduces to:

S"'

gen

=

_.!:._[(aT) ]+ Tll (avx()r ) T ax +(aT) ()r

Combining this form with

2

2

2

Vz

2

(4.18)

and T given by Equation (4.16) and (4.17) yields: (4.19)

where (4.20) is the Peclet number. Also:

R

= r I r0

(4.21)

and (4.22) In the derivation of Equation (4.19) it is assumed that the local temperature difference T- T0 , Equation (4.17), is considerably smaller than the local absolute temperature. On the right side of Equation (4.19), the first term represents the entropy generated by beat transfer in the radial direction, the second term accounts for axial conduction, and the last term is the fluid friction contribution. Parameter cj>, Equation (4.22), is the irreversibility distribution ratio discussed earlier in this section. Figure 4.2 shows a number of radial entropy generation profiles in the limit where the irreversibility due to axial conduction is negligible, 16/Pe2 ~ 1. The centerline region (R = 0) is free of entropy generation because the velocity and temperature gradients are zero in the center of the tube. On the other hand, the wall region (R = 1) acts as a strong producer of entropy. As viscous effects become important (i.e., as 4> increases), the center of gravity of the s~:n profile shifts closer to the wall. Entropy generation profiles or maps may be constructed using Equation (4.10) so long as the velocity and temperature gradients are known at each point in the convective medium. The backbone of this construction was outlined in Example 4.1. The reader will find a wider selection of entropy generation profiles in the original paper (Bejan, 1979) as well as in the more recent literature (Drost and Zaworski, 1988; White and Drost, 1989; Paoletti et al., 1989; Benedetti and Sciubba, 1993). A

78

recent study concluded that the local entropy generation rate should be calculated and incorporated into computational fluid dynamics codes (Drost and White, 1991).

4.3

INTERNAL FLOWS

In a large number of convective heat transfer situations the velocity and temperature fields are not known at each point in the medium. This is the case when the flow regime is turbulent or when the flow geometry is so complicated that an exact description of velocity and temperature is not available in analytical or numerical form. However, most of these heat transfer arrangements are documented in the heat transfer archives on the basis of heat transfer and fluid friction data measured along the solid boundaries surrounding the flow. For example, the heat transfer to fully developed turbulent flow in a pipe is described by Stanton number and friction factor correlations derived from a large number of experiments (e.g., Kays and London, 1964). In this section we derive an important result for the rate of entropy generation in internal flow, i.e., in a duct of arbitrary geometry with heat transfer at the wall. This calculation is aimed primarily at duct flows in which S"' cannot be evaluated at individual points throughout the medium. Instead, we calculate the entropy generation rate per unit length, S~en [W/m K], based on information contained in the usual correlations for average heat transfer and fluid friction (Bejan, 1978). Consider the flow passage of arbitrary cross-section A and wetted perimeter p shown schematically in Figure 4.3. The bulk properties of the stream mare T, P, h, s, p. In general, this heat transfer arrangement is characterized by a finite frictional pressure gradient -dP/dx > 0 and, when heat is transferred to the stream at a rate q' [W/m], by a finite wall-bulk fluid temperature difference !J.T. Focusing on a slice of thickness dx as a system, the rate of entropy generation is given by the second law:

. . q'dx dSgen = m ds- - - T+!J.T

(4.23)

The first law of thermodynamics applied to the same system is mdh = q'dx

Rgure 4.3 Forced convection heat transfer in a duct of arbitrary geometry.

(4.24)

79 In addition, for any pure substance we write the canonical relation as: (4.25) Substituting ds given by Equation (4.23) and dh given by Equation (4.24) into the canonical form of Equation (4.25) yields the entropy generation rate per unit duct length: (4.26) where s~en = dSgenjdx. In applications in which the dimensionless temperature difference 't is negligible as compared to unity we have the simpler form: (4.27) Result (4.27) can now be related to average heat transfer and fluid friction information, which may be obtained experimentally or numerically for most duct geometries. We recall that in heat transfer, the relationship between heat transfer rate q and wall-bulk fluid temperature difference is expressed in the form of Stanton number correlations: (4.28) where G is the mass velocity:

G=m

A

(4.29)

Note that q'l(pAT) appearing in Equation (4.28) is the average heat transfer coefficient, whose symbol h should not be confused with that of specific enthalpy, which is also used in the present treatment. For most duct geometries, the Stanton number depends on the Reynolds number: Re=DG

Jl

(4.30)

and on a number of other parameters describing the fluid and the specific geometry (Kays and London, 1964). The Reynolds number Re is based on the hydraulic diameter, defined as:

D= 4A p

(4.31)

80 The fluid friction characteristics of a certain duct are reported usually in the form of friction factor correlations:*

f = _E!!___(2G 2

dP) dx

(4.32)

The friction factor depends on the Reynolds number plus other geometric parameters of the duct and its surface. In order to illustrate the dependence of S~en on Stanton number and friction factor information, consider the case in which the heat transfer rate per unit length q and the mass flow rate m are specified. Combining definitions (4.28) and (4.32) with formula (4.27) yields (4.33) Under the present assumptions the duct configuration has two degrees of freedom, the wetted perimeter p and the cross-sectional area A or any other couple of independent parameters such as (Re, D) or (G, D). Examining Equation (4.33), it becomes evident that a high Stanton number contributes to the reduction of the heat transfer share of s~en, while a high friction factor has the effect of increasing the entropy generation rate due to viscous effects. Example 4.2. In a round tube of diameter D, the rate of entropy generation per unit

length, Equation (4.33), assumes the form

(4.34) In this expression Rev is the Reynolds number based on D:

4th Re = - -

(4.35)

Nu = St RevPr

(4.36)

v

1t~D

and Nu is the Nusselt number:

Note that unlike the general form, Equation (4.33), which depends on two parameters (A, p), the formula for a round tube, Equation (4.34), depends on only one geometric parameter (D or Rev). As the tube diameter increases, Rev decreases; the interesting effect on s~.n is that while the heat transfer contribution increases, the fluid friction term decreases. This is one example in which a variation in one geometric parameter

* Note that some heat transfer books define "friction factor" as four times the quantity on the right side of Equation (4.32).

81 always has opposite effects on the two terms of s:en. Consequently, it is possible to determine the optimal tube diameter, or Rev, which leads to minimum irreversibility. Ifthe pipe flow is turbulent and fully developed, the Nusselt number and friction factor are given by the well-known correlations (e.g., Bejan, 1995): Nu = 0.023 Re~

8

Pr

(4.37)

0 ·4

f = 0.046 Re -:;· 2

(4.38)

Combining these expressions with Equation (4.34) and setting as:en/'dRev = 0, yields (Bejan, 1979): Re D,opt = 2•023 Pr...o·071 B00358 where B0 is a duty parameter, fixed as soon as q', specified:

(4.39)

m, and the working fluid are (4.40)

In conclusion, the optimal Reynolds number increases with the product mq'. This means that the optimal tube diameter D steadily decreases as the heat exchange duty parameter B0 increases. Finally, we can substitute result (4.39) into Equation (4.34) to calculate the s:en,min. An interesting result o~ this entire minimum rate of entropy gen~ration analysis is the relative size of s:en (Rev) with respect to the minimum s:en (Rev.opJ· Skipping the algebraic details, we find the entropy generation number Ns: N = _sgen "I

s

S'gen,min

=0.856

(

Rev Re D,opt

J...Q.8 +0.144 (

Rev Re D,opt

]4.8

(4.41)

This dimensionless ratio* is plotted in Figure 4.4, showing that the rate of entropy generation increases sharply on either side of the optimum. The graph also shows the irreversibility distribution ratio as it varies along the V-shaped curve. When s:en = s:en.min• the irreversibility distribution ratio is opt= 0.144/0.856 = 0.168.

* The dimensionless group Sgen ISgen.min was introduced in the 1970s, and was used extensively in the book Bejan (1982) and the engineering literature that followed. This is why I am very pleased to note that just recently, in the physics literature, Mironova et al. (1994) recognized that the Sgen I Sgen,min criterion is the most appropriate measure of a real system's approach to thermodynamic ideality. This is an important and timely development for 1994, in view of the parallel development of the physics branch of our field, in which the emphasis has been quite different: namely, the maximization of power in macrosystems (power or refrigeration plants) divided by the modeler into irreversible and reversible compartments. Early examples of such modeling can also be seen here in Problems 2.2 and 2.4, and Figures 5.14 and 6.2, which were also present in Bejan (1982).

82

- s~n N s. s~en,

min

Figure 4.4 Relative entropy generation rate in a smooth tube (Bejan, 1980).

This fundamental issue of thermodynamic irreversibility at the heat exchanger passage level was reconsidered by Kotas and Shakir (1986). They took into account the temperature dependence of transport properties and showed that the operating temperature of the heat exchanger passage has a profound effect on the thermodynamic optimum. For example, the optimal Reynolds number increases as the absolute temperature T decreases. The minimum entropy generation rate corresponding to this optimal design also increases as T decreases. The opportunity to minimize the entropy generation to determine the optimal sizes of ducts and other heat transfer devices was noted by practitioners in the field of chemical process engineering (Kenney, 1989). The same topic became a sizable component in the new edition of a classical heat transfer textbook (Kreith and Bohn, 1986).

4.4

EXTERNAL FLOWS

Another large class of convective heat transfer arrangements includes the heat transfer between a stream and a solid body (surface) suspended in the stream. Classic examples of such arrangements are the fins of various geometries which extend into the heat exchange fluid circulating through a finned-surface heat exchanger. Consider the configuration presented in Figure 4.5. A body of arbitrary shape and surface area A is suspended in a uniform stream with velocity U_ and absolute temperature T_. The local heat transfer rate between body and stream is q'' [W/m2l and the local wall temperature is Tw. Fluid friction manifests itself in the form of the drag force F 0 , which is the sum of all forces distributed around A and projected on the direction of flow, U_. In view of the discussion that concluded Chapter 2 we

83

u_, '{ Figure 4.5 Convective heat transfer in external flow.

expect the irreversibility of this arrangement to be due to heat transfer across the nonzero temperature difference Tw- T_ and also to frictional drag.

4.4.1

Total Rate of Entropy Generation

We can use more than one analytical approach to obtain the rate of entropy generation associated with the external flow of Figure 4.5. Imagine, for example, a stream tube (control volume) surrounding the solid surface A, as shown in Figure4.6. The radius of the tube is considerably larger than the characteristic linear dimension of the solid body, hence the fluid outside and immediately inside the tube surface belongs to the undisturbed flow U_, T_. The stream tube surface can then be regarded as adiabatic and shear free (i.e., no shear work transfer occurs across it). For the purpose of analysis, it is convenient to describe the thermodynamic state across the inlet and outlet tube cross-sections by using the bulk properties P, h, s. The following analysis follows to some extent the steps taken in the entropy generation analysis of internal flow; therefore, we present here only the highlights. The three thermodynamic statements for the stream tube as an open system in steady flow are

m.. =m m

out

(4.42)

=lit

mh in + JJ q"dA - mhout = 0

. . . If

sgen

The canonical form dh

=

msout -msin-

q"dA

~

(4.43) (4.44)

= T ds + (1/p) dP may be written:

hout-hin =T_(sout-sin)+

1

p_ (Pout-Pin)

(4.45)

84

I

I I I

I. lm I I I \

\

out Figure 4.6 Stream tube for calculating the entropy generation external flow.

rate due to heat transfer in

where it was assumed that the temperature and density do not vary appreciably between inlet and outlet. Combining Equations (4.43) through (4.45), we write for the entropy generation rate:

Sgen =

1 ) dA 1 ffq" (T_ -Tw

_!!!_(P p_T_ oot -P) m

(4.46)

A

Recognizing that: (4.47) and, from a force balance on the control volume, (4.48) we obtain (4.49)

In writing this result we have already assumed that the temperature difference TwT_ is much smaller than either absolute temperature, Tw or T_. The heat transfer part of sgen, the first term in Equation (4.49), can be transformed further if we consider the following special extremes. Ifthe body temperature is uniform (independent of position around A), then (Tw - T_) is constant, and Equation (4.49) becomes: (4.50) Expression (4.50) takes into account that:

ff A

q"dA = q = hA(Tw-T_)

(4.51)

85 where h is the average heat transfer coefficient based on A. We conclude that when difference is fixed, the only way to minimize the heat the body-ambient tempe~ature transfer contri.!?ution to sgcn is by insulating the body, i.e., by decreasing the thermal conductance hA. The other extreme that needs to be considered is when the local heat flux q" is uniform around A. In this case the entropy generation rate (4.49) becomes

~A+_]_FU h

A

T

~

D ~

(4.52)

We again used the fact that

Tw -T

(4.53)

~

where hA is the local heat transfer coefficient. Note tha~_ in order to reduce the heat transfer entropy generation we must strive to increase hA; in other words, we must enhance or augment the heat transfer from body to ambient. We return to this observation in Section 4.8, where we analyze a number of specific techniques for convective heat transfer augmentation.

4.4.2

Flat Plate

A classic example of convective heat transfer in external flow is sketched in Figure 4.7. Heat is transferred from a flat plate of length Land breadth W to a uniform flow U_, L parallel to the plate. The heat flux q" [W/m 2] is uniformly distributed over the surface LW. The arrangement is usually modeled as two-dimensional whenever W ~ L and when U~ is parallel to the x direction. Consider the case in which U~' L and the external fluid are specified and when the heat transfer rate per unit breadth is fixed by design:

q' =

i

L

(4.54)

q"dx = q"L

()

If the boundary layer flow developing over the plate is laminar, the heat transfer and fluid friction (momentum transfer) characteristics of the arrangement are condensed in local Nusselt number and skin friction coefficient information (e.g., Bejan, 1995): Nu C

X

f.x

= 0.458 Pr 113 Re 112 = h,x k X

=

10

(1/2)pu:

= 0 664 Re- 112 x .

(4.55) (4.56)

86

-- ----------q"

:;...--

-----0- -o-o-----7--

---------~~~-----0

L --~X

Figure 4.7 Analysis of entropy generation in laminar boundary layer flow over a flat plate.

where Re.. = U_ xI v. In expressions (4.55) and (4.56) h.. and t 0 are the local heat transfer coefficient [W/m2 K] and wall shear stress [N/m2]. The aggregate entropy generation function, Equation (4.52), becomes

. S

gen

q" 2

WlL -+---pU dx u_w 1 2lL C h T 2 -

= -2T -

0

.X

0

-

f,x

dx

(4.57)

and, after substituting Equations (4.54) through (4.56),

U2

(4.58)

kT - Rel/ 2 q'2 L

+ 0.664 - J.l.

This result is in dimensionless form. The left side of Equation (4.58) may be interpreted as an entropy generation number Ns, while the group q' 2 WikT~ plays the role of characteristic entropy generation rate for the convective heat transfer arrangement. It is now clear that the length of the plate is the only design variable capable of inducing changes in the rate of entropy generation. Setting 'dNs /'d(ReJ =0 we find the optimal plate length (Reynolds number based on L) such that S8en reaches a minimum Re

L,opt

= 2.193 Pr-1/3 ----.~q

12

__

U?. J.l. k T_

(4.59)

87 In conclusion, when the external flow is fixed (fluid, U_, T_), the optimal plate length is proportional to the total heat transfer rate squared, q' 2 • This result is important in the local optimization of plate-finned surfaces* in heat exchangers. The minimum irreversibility rate is obtained by combining Equations (4.58) and (4.59), hence, (4.60)

It can be verified easily that (4.61) or (4.62) This result allows an immediate estimate of the relative increase in irreversibility associated with using a plate of length L -:!- Lopt· Note that at the optimum (L = LopJ, the heat transfer irreversibility is equal to the fluid friction contribution; in other words, the irreversibility distribution ratio is equal to unity. Equation (4.62) is represented by the "plate, laminar" curve plotted in Figure 4.8, which is a graphic device of the same type as that pioneered in Figure 4.4. Note also thatNs INs.mm of Equation (4.62) is the same as S8en/Sg•n.min indicated on the ordinate of Figure 4.8. It is fitting to close the discussion of the plate in laminar flow with an illustration of local distribution of entropy generation in the boundary layer. The flow is shown in the horizontal plane of the isometric drawing presented in Figure 4.9. Far away from the solid wall (y =0), the velocity and temperature are uniform, U_ and T_. The wall temperature is constant, T0 • For a fluid with Pr = 1, where the velocity and temperature distributions have the same analytical form, the local rate of entropy generation is (Bejan, 1979)

s~:, (

v ) -_ ( 1 + Ec-Pr) f"--- k 'tU_ 't Rex 2

2

(4.63)

In this expression we used the notation Ec = U_ 21cP (T0 - T_), 't = (T0 - T_)IT0 and Re.. = x U_!v, whilefis Blasius' function of the similarity variable y (U_!vx) 112 • The

* As shown in Section 4.8.4, in order to tl1ermodynamically optimize a fin we must also account for the entropy generated inside the fin. In tl1e present example we considered only the entropy generated around the solid plate of Figure 4.7.

88

Cylinder Plate, turbulent Plate, laminar

0.01

10

0.1

100

Figure 4.8 The relative entropy generation rate of several bodies engaged in external forced convection heat transfer (Fowler and Bejan, 1994).

Local entropy generation rate

f" 2 Rex

0.01

Figure 4.9 Local entropy generation rate in laminar boundary layer flow and heat transfer over a flat plate (Bejan, 1979).

89

a

0 _j

Q)

a:

-

a0

6 Q)

a:

Figure 4.10 The optimal size of a plate, cylinder, and sphere for minimum entropy generation in external forced convection heat transfer (Fowler and Bejan, 1994}.

left hand side of Equation (4.63) plays the role of entropy generation number, which appears plotted as a surface in Figure 4.9. The solid wall is clearly the place where the generation of entropy is concentrated. In addition, S~~n varies as 1/x in the longitudinal direction and blows up at the tip of the flat plate. The analytical procedure that led to the optimal dimension expressed by Equation (4.59) can be applied to other external flows. The flat plate with turbulent boundary layer flow was treated in Bejan (1979}, Bejan (1982, Problem 5.4}, and in Poulikakos and Johnson (1989). The results were reviewed in Fowler and Bejan (1994), from which we reproduce the curve for the optimal plate length, or ReL.opt (Figure 4.10). The dimensionless parameter used as abscissa in Figure 4.10 is defined as (4.64) where q is the total heat transfer rate from both sides of the plate. The result for laminar flow, Equation (4.59), is also shown in Figure 4.10. The relative entropy generation rate and the sharpness of the sgen minimum in turbulent boundary layer flow are shown in Figure 4.8.

4.4.3

Cylinder in Cross-Flow

The entropy generation rate associated with a single cylinder in cross-flow was presented in Bejan (1979). The optimal cylinder diameter for minimum entropy generation was reported by Poulikakos and Johnson (1989). Figure 4.10 shows the optimal diameter, ReD, opt= u~D op/V, as a function of the B group defined in Equation (4.64), where W is the cylinder length. The sharpness of the corresponding entropy generation minimum is illustrated in Figure 4.8.

90

4.4.4

Sphere

The minimization of the entropy generation rate associated with heat transfer from a sphere was treated in Bejan (1982, Problem 5.3). Results for entropy generation and optimal diameter were generated numerically by Fowler and Bejan (1994), and added to Figures 4.8 and 4.10. In the case of the sphere the abscissa parameter is defined differently:

-

B, -

(

q

v k Jl T~Prl/3

)1/2

(4.65)

In summary, Figure 4.10 shows that the optimal body size (ReL,opt or ReD,opJ increases with the respective duty parameter (B orB,). In Figure 4.11 Fowler and Bejan (1994) demonstrated that it is possible to correlate the three curves of Figure 4.10 by nondimensionalizing the body size as a Reynolds number based on the transversal length scale ofthe flow, and by replacing the body with an equivalent one in two-dimensional flow. This means that for the plate Reo,opt is based on the boundary layer thickness at the distance L downstream, oopt := 4.92 Lapt Re~~~ for laminar flow, and oopt := 0.37 Lopt Re~~~~ for turbulent flow. The transversal flow dimension for the cylinder and sphere continues to beD, hence the use of ReD.opt on the ordinate of Figure 4.11. The new abscissa parameter for the sphere. (4.66)

-Cylinder

Figure 4.11 The optimal sizes of Figure 4.10 correlated in terms of a universal duty parameter (B or B.) and a Reynolds number based on the transversal length scale of the flow (Fowler and Bejan, 1994).

91

is of the same type as the plate and cylinder parameter, B [Equation (4.64)]. The B. group was constructed by recognizing the spherical surface rrD2 as equal to the lateral surface of a short cylinder (D = W) with insulated ends. Figure 4.11 shows that the consistent use of the transversal flow scale and the duty parameter of type B brings the curves much closer than in the original plot (Figure 4.10).

4.5

CONDUCTION HEAT TRANSFER

The generation of entropy by heat conduction, in the absence of fluid flow, has come under close scrutiny in more recent papers. Bisio (1988) focused on onedimensional heat conduction in systems with time-dependent boundary conditions. The thermal conductivity was a function of temperature and the coordinate of the heat flux direction. He examined homogeneous as well as multilayered systems, and pointed out relations between entropy generation extrema and the solution to the heat conduction problem (the temperature distribution). Extensions of this work cover the fields of conduction in a medium with thermal conductivity (or its derivative) that is a piecewise continuous function of temperature (Bisio, 1990), and thermal conductivity that is also a function of the temperature gradient in the heat flux direction (Bisio, 1992). An entirely new area of application of entropy generation by conduction is being charted by the work of Prof. Vikram Kinra and associates at Texas A&M University. They used the entropy generation calculations to explain and predict damping in homogeneous and inhomogeneous elastic systems (Bishop and Kinra, 1993a, b; Milligan and Kinra, 1993a, b; Kinra and Milligan, 1994). They named their theory

elastothermodynamic damping.

Kinra's approach begins with the observation that a material that is stressed reversibly and adiabatically always experiences local changes in temperature, however small. This thermoelastic effect can be predicted using the first and second laws of thermodynamics. The new observation is that because the temperature field and the stress field are coupled, nonuniformities in stress and material properties induce nonuniformities in temperature. As a consequence, heat is conducted locally from regions of relatively high temperature to regions of low temperature. The entropy generated by conduction throughout the material is responsible for the damping effect, i.e., the destruction of available work (exergy). Kinra, Bishop, and Milligan have used this approach to demonstrate that it is now possible to design a composite material that has certain desired damping characteristics. Theirs is an interdisciplinary example of the importance of the concept of entropy generation in the field of design. This is an important development, because the applications collected in Bejan (1982) (and in this book starting with Chapter 3) came strictly from the mainstream of thermal engineering.

4.6

CONVECTIVE MASS TRANSFER

As an analogy to the convective heat transfer irreversibility presented in Sections 4.1 and 4.2, the competition between mass transfer and fluid flow irreversibilities was demonstrated by San et al. (1987a). The irreversibility of combined heat and mass

92 transfer was minimized in internal flow by San et al. (1987b) and in external flow by Poulikakos and Johnson (1989). Carrington and Sun (1991, 1992) reconsidered the internal and external flow problems by pointing out that the local entropy generation rate formula developed by San et al. (1987a, b) contained a spurious coupling term between heat and mass transfer. Sun and Carrington (1991) developed a general exergy balance equation for the flow of a fluid mixture with heat conduction, mass diffusion, fluid friction, and chemical reactions. The generation of entropy in reacting flows with radiation was studied by Arpaci (1986, 1987, 1991), Arpaci and Selamet (1988, 1992), Selamet and Arpaci (1990), and Pori (1992). For example, Arpaci and Selamet (1988) focused on premixed flames stabilized above a flat flame burner, and showed that the tangency condition (i.e., the minimum quench distance) is related to an extremum of the entropy generation rate, which is inversely proportional to the Peclet number. Pori (1992) minimized the entropy generation rate of a droplet burning in a stream. He showed that the minimum entropy generation rate corresponds to a trade off between drag and mass loss from the droplet, and to maximum energy per unit mass flow rate at the combustor exit. The entropy generation rate in convection through a saturated porous medium was first presented by Bejan (1984, p. 355). The general form for the local entropy generation rate due to combined heat, mass. and fluid flow through a porous medium is given in Bejan (1988, p. 657). 4.7

GENERAL HEAT-EXCHANGER PASSAGE

An important conclusion that follows from Figure 4.4 is that the dependence between irreversibility and design parameters can be nonmonotoni. For this reason, it is difficult to predict in advance (e.g., on the basis of a "rule of thumb") the change induced in the overall irreversibility figure by a certain design modification. A more general example of how rules of thumb can fail us in the realm of thermodynamic design is illustrated in Figure 4.12. Plotted in the horizontal plane is the dimensionless temperature difference 't

= f..T/T

(4.67)

while the intent of the figure is to show that the pure heat transfer rule that a small wall-fluid !:iT might be desirable is not necessarily correct. The temperature difference !:iT is defined in Figure 4.3. In the case of steady forced convection through a most general heat-exchanger passage we begin with Equation (4.26) and write it in dimensionless form as (Bejan, 1978) (4.68)

93

Figure 4.12 The entropy generation number for a general duct with forced convection heat transfer, as a function of the wall-fluid temperature difference ('t) and the additional parameter A, (Bejan, 1978).

where (4.69) The left side of Equation (4.68) shows the entropy generation number defmed in this case as

s~en

(4.70)

N =-s q'/T

It is apparent that the heat transfer and fluid flow contributions toNs, Equation (4.68), are coupled through the temperature difference number t. The minimization of Ns requires the optimal selection oft, and, as shown in Figure 4.12, a well-defined optimal t exists, especially at low values of the A 1 parameter: A_ J ( _lL) I-

In the limit A 1

~

32 St

V2

Re

St

(4.71)

0, i.e .. in the limit of small ATs, the optimum is described by

94

(4.72)

(4.73) When 't < 'topt• the 11T irreversibility is small relative to the fluid friction irreversibility. Conversely, when 't > 'topt• the entropy generation rate is dominated by the effect of inadequate thermal contact. The lack of monotoneity between Ns and 't means that decreasing the wall-fluid !l.Tis a good idea only for those designs that fall on the right side (high-'t side) of the Ns- 't valley illustrated in Figure 4.12. Another example of a rule of thumb that clashes with the thermodynamic objective of entropy generation minimization is summarized in Figure 4.13. In many instances designers seek to maximize the ratio of the heat transfer coefficient divided by the pumping power. A dimensionless group that is proportional to this ratio is

R=

hppT

m(-dPJdx)

(4.74)

The entropy generation number (4.68) can be put now in terms of Rand A1: (4.75)

~

Region dominated by fluid friction dP losses

Figure 4.13 The entropy generation number for a general duct with forced convection heat transfer, as a function of the ratio of the heat transfer coefficient divided by pumping power (R) and the additional parameter A1 (Bejan, 1978).

95

and, as shown in Figure 4.13, the features of the surface Ns(R, A 1) are very similar to the features of the N8 ('t, A1) surface of Figure 4.12. IfA1 is constant, an optimal ratio R exists for which the entropy generation rate is minimized, for example, (4.76) In conclusion, the decision to increase the ratio of heat transfer coefficient divided by the pumping power is not sufficient if the objective is to improve thermodynamic performance. Because N8 depends on more thanjustR, the true effect of a proposed design change can be evaluated only by estimating first the changes induced in both R and A1, and eventually in N8 • A related research direction was explored by Nag and Mukherjee (1987), who minimized the entropy generation rate of a finite-length section of duct along which the bulk temperature of the fluid decreases exponentially. They showed that the trade off between heat transfer and fluid friction irreversibilities results in an optimal value for the inlet temperature difference, i.e., for the difference between the inlet temperature of the fluid and the constant temperature of the duct wall. This result is equivalent to the discovery of an optimal distribution of wall-fluid AT along the duct. The entropy generation minimization in the entrance region of a duct with swirling flow was analyzed similarly by Mukherjee et al. (1987).

4.8

HEAT TRANSFER AUGMENTATION TECHNIQUES

Another example of the competition between different irreversibility mechanisms occurs in connection with the general problem of heat transfer augmentation (Bergles, 1985), in which the main objective is to devise techniques that increase the wall-fluid heat transfer coefficient relative to the coefficient of the unaugmented (i.e., untouched) surface. A parallel objective, however, is to register this improvement without causing a damaging increase in the pumping power demanded by the forced-convection arrangement. These two objectives reveal the conflict that accompanies the application of any augmentation technique: a design modification that improves the thermal contact (e.g., roughening the heat transfer surface) is likely to augment also the mechanical pumping power requirement. The true effect of a proposed augmentation technique on thermodynamic performance can be evaluated by comparing the entropy generation rate of the heatexchange apparatus before and after the implementation of the augmentation technique (Bejan and Pfister, 1980).

4.8.1

Entropy Generation

Analysis

Consider a heat exchanger passage of length dx, heat transfer per unit length if, and mass flow rate tit (Figure 4.3). The passage geometry is described in terms of the hydraulic diameter D and the flow cross-sectional area A. We learned in Equation (4.27) that the entropy generation rate per unit length is

96

S'

gen

= q'AT

T2

+!!!:._(dP) pT dx

(4.77)

where the first term, S:U., represents the irreversibility duet~ heat transfer across the wall-fluid temperature difference, and the second term, S~, is the irreversibility caused by fluid friction. In writing Equation (4.77) we assumed that the wall-fluid AT is considerably smaller than the local (wall or fluid) absolute temperature. In termS Of the irreversibility distributiOn ratiO, = S:V,/S~T, Equation (4.77) becomes (4.78) The impact of an augmentation technique on the irreversibility of a known heatexchanger passage can be evaluated directly by calculating the entropy generation rate in the augmented passage, s~en.a' an~ comparing it to the entropy generation rate in the original (unaugmented) pa T. Combining Equations (5.80) and (5.81), we can write sgen in terms of known quantities plus the unknown a(x): (5.82) It can be verified that the denominator of the integrand is always positive,* i.e., q < Ua THR· As summarized in Appendix B, the function a(x), which minimizes sgen subject to the total area constraint, Equation (5.79), is the same function that minimizes the aggregate integral I=

iL( o

2q

2

Ua~~R-qTHR

+Aa ) dx

(5.83)

subject to no constraints. Parameter A. appearing in the aggregate integrand is a Lagrange multiplier. According to variational calculus (Appendix B), the optimal distribution of heat transfer area is a

= opt

iqi

{A.U) 1/2 THR

+-qU THR

(5.84)

The Lagrange multiplier ').. is calculated finally by substituting aoplx) into the total area constraint, Equation (5.79); the result is (5.85) A simpler expression for aop 1(x) is obtained in the limit iTnR- Ti ; dQ

x+dx, T+dT-x, T - -

1

T

II ~ :I I

Figure 6.4 Analysis of entropy generation continuously over its length L.

Q

., ' ,.... ,.

-

nTL

I

in a one-dimensional

thermal insulation cooled

161

Qopt =(koAlnT")T L T

(6.16)

L

(6.17) The heat current Qopt is proportional to the absolute temperature, which means that the cooling effect is distributed evenly along the temperature scale TL ~ Tn. The relative reduction in entropy generation is expressed by the entropy generation number obtained by dividing Equations (6.17) and (6.2): (6.18) This result has been plotted in Figure 6.5, showing again that the irreversibility reduction is sizable if Tn!TL is much greater than 1. Comparing Figures 6.5 and 6.3, we find that the continuous cooling regime is more effective than the discrete (one point) cooling regime studied initially. For example, if Tn!TL = 10, continuous cooling reduces the irreversibility rate by 35% as compared to 27% attainable with discrete cooling. 6.4

COUNTERFLOW HEAT EXCHANGERS AS ONE-DIMENSIONAL INSULATIONS

The success of the entropy generation minimization method discussed up to this point depends on our ability to integrate the cooling effect (heat extracted from the insulation) into a power-refrigeration cycle that operates close to the reversible limit. The process of integration is feasible in the case of insulation systems for power

Ns

Figure 6.5 The entropy generation reduction due to optimal continuous cooling effect.

162

L ------

----

T+dT x+dx------

TH

-T+dT+f).T

x-----T

-T+f).T

Figure 6.6 Counterflow heat exchanger as a one-dimensional insulation system oriented in the warm-end to cold-end direction.

plants and refrigeration (liquefaction) plants. In such cases the thennodynamic cycle and the insulation can be designed as one unit, the connection between cycle and insulation being expressed in the optimum cooling regime dQ 0 P1kfT [Equation (6.14)]. In this section we learn that counterflow heat exchangers, which very often serve as cycle components, also can serve a thennal insulation function. This observation is very useful in the process of cycle-insulation integration, which is considered further in Chapters 8 to 11. Figure 6.6 shows a balanced counterflow heat exchanger operating between TH and TL as extreme temperatures. Neglecting the fluid friction irreversibility, and writing AT for the stream-to-stream temperature difference, the rate of entropy generation in the dashed segment dT is

ds.gen

.

= tneP

I T+dT . n --T- + tneP In

T+AT T+dT+AT (6.19)

whenllT~T

Comparing this result to Equation (6.10), we draw the conclusion that mc~Tplays the role of heat leak in the end-to-end direction of the heat exchanger: (6.20)

163

The heat exchanger heat leak Q flows from the warm end toward the cold end, as shown in Figure 6.6. Because Q - 11T heat exchangers with superior stream-tostream thermal contact (low 117) have low heat leak in the end-to-end direction. This fact appears more clearly if we consider the relationship between 11T and p (heat transfer area per unit length) (6.21) where U is the overall heat transfer coefficient based on p. Eliminating 11T between Equations (6.20) and (6.21), we find

Q=

(mcpf ar Up

dx

(6.22)

This result demonstrates that the group ( mcP)2f(Up) plays the same role as (kA) in the one-dimensional insulation system considered in Equation (6.12). Likewise, the length of the heat exchanger plays the role of insulation thickness L. The longitudinal heat transport is inversely proportional to the heat transfer area pL and to the number of heat transfer units N1" = UpLI( mcp). By promoting effective heat transfer in the stream-to-stream direction, counterflow heat exchangers serve as effective insulations in the end-to-end direction (Bejan, 1979). The general results obtained in the preceding sections apply also to counterflow heat exchangers. The irreversibility of this type of insulation is improved first by increasing the stream-to-stream contact area pL. When this approach is no longer feasible, intermediate cooling can further improve the thermodynamic performance. In many applications the effective conductance ( mcP) 2/(UpL) is temperature independent; hence, the simplified entropy generation minimization principle, Equation (6.16), holds. This principle recommends an evenly distributed cooling effect so that Q becomes proportional to the local absolute temperature. Because Q -11T [Equation (6.20)], counterflow heat exchangers must be cooled such that

11T T

-=constant

(6.23)

As shown in Section 10.4.5, this conclusion is in excellent agreement with the practical design of helium liquefaction plants (Kropschotet al., 1968). It is interesting also that a result similar to Equation (6.23) was reported in the German literature by Grassmann and Kopp (1957); see also Szargut (1980).

6.5

PARALLEL INSULATIONS

The process of thermodynamically matching a power-refrigeration cycle to its insulation system is the same as the minimization of entropy generation in a system of two parallel insulations. For example, in a helium liquefier we first distinguish an

164 elaborate insulating structure that consists of radiation shields, evacuated space, and low heat leak mechanical supports (see Chapter 10). The second, more subtle, insulation is the main counterflow heat exchanger that connects the room-temperature compressor with the low temperature end of the liquefaction process. As shown in the preceding section, the main heat exchanger is an insulation in the hot-end to cold-end direction. Therefore, it is quite reasonable to model power-refrigeration cycles and their insulations as systems of parallel insulations (e.g. Section 8.2). We examine in this section how the in-eversibility of such insulations may be reduced on the basis of simple design. Perhaps the simplest way to effect the optimal continuous cooling of one insulation is to channel a stream of single-phase (cold) fluid from TL to T8 , i.e., against the heat cun-ent Q that penetrates the insulation. In such an arrangement

dQ . -=me dT

(6.24)

P

Hence, using Equation (6.14) with k =constant, the optimal flow rate for minimum irreversibility is

(rite

)

Pop!

Ak L

T

(6.25)

= - I n _!!_

TL

In a system of two parallel insulations we face two optimization problems: We must determine the optimum coolant flow rate, as in Equation (6.25), and we must determine the optimum sharing of this flow rate between the two insulations. Figures 6.7 and 6.8 show two possible arrangements for providing intermediate cooling to a parallel insulation system. The minimization of entropy generation in these two arrangements was undertaken by Schultz and Bejan (1983), from which we reproduced Figures 6.9 and 6.10. The minimum entropy generation rate shown in Figure 6.9 emerges as a function of end-to-end temperature ratio and thermal conductance ratio, N

. = function ( TH M 1 ) T 'M

S.mm

L

(6.26)

2

where

( ri!CPL)

M - 1,2kA

(6.27) 1.2

The optimal coolant flow rate responsible for the minimum irreversibility, Equation (6.26), is shown in Figure 6.10. It should be noted that in addition to the optimal flow rate, Figure 6.9 is based on having optimally determined the optimal flow fraction {X) for the arrangement of Figure 6.7, and the optimal crossover locations {z1, z2) for the single-stream arrangement of Figure 6.8.

165

D QL1

1-X

X

,;,, cP

D

Tv

o

QL2

Figure 6.7 Parallel insulations cooled continuously by two streams (Schultz and Bejan, 1983).

A,

T

T

0,

1i

D

Figure 6.8 Parallel insulations cooled only partially by a single stream (Schultz and Bejan, 1983}.

The details of the numerical optimization procedure leading to Figures 6.9 and 6.10 are reported in Schultz and Bcjan (1983). Examining the numerical results we conclude that the single-stream arrangement of Figure 6.8 is thermodynamically

166

100

NS,min

10

0.1

1.0

10

Figure 6.9 Minimum entropy generation rates in the systems of Figure 6.7 (solid lines) and Figure 6.8 (dashed lines) (Schultz and Bejan, 1983).

inferior to the continuous-cooling arrangement of Figure 6. 7. The optimal flow rate appears to be relatively insensitive to whether the stream is divided (Figure 6.7) or undivided (Figure 6.8). The chief conclusion is that for minimum irreversibility, parallel insulations, like single insulations, must be cooled continuously (as in Figure 6.7). This contradicts Hilal and Eyssa (1980), who reached the opposite conclusion while thermodynamically optimizing the design of a large-scale cryogenic system.

6.6

INTERMEDIATE COOLING OR HEATING OF INSULATION SYSTEMS FOR POWER AND REFRIGERATION PLANTS

As an introduction to the more specific power and refrigeration applications of Chapters 8 to 11, it is appropriate to conclude with the summarizing view provided by Figure 6.11. The figure shows the concept of intermediate heat transfer between the insulation and the insulated system. The heat transfer can have either sign, i.e., the insulation may be heated or cooled intermediately. Large-scale insulation systems are most indispensable when used in association with power applications at very high temperatures and with refrigeration applications at very low temperatures. The intermediate heat transfer effect is the result of placing the insulation system in thermal communication with the power or refrigeration

167

Figure 6.10 Optimal coolant flow rates for the cooling methods of Figure 6.7 (solid lines) and Figure 6.8 (dashed lines) (Schultz and Bejan, 1983). INTERMEDIATE HEATING

INTERMEDIATE COOLING

I

T•

~

POWER PLANT INSULATION

I ~

_____ ...

-----,

6-•

6-'

'

I ~

I

~ REFRIGERATOR INSULATION

'

:

T,

T•

'

C)=

I

~

I

I

_____ .J'

I

~

cr

------, ' T,

I ~

Figure 6.11 The optimization of the insulation system in harmony with the power or refrigeration plant that is being insulated (Bejan, 1979).

168

plant. Depending on whether the intermediate heat transfer represents cooling or heating, the insulation serves as heat source or heat sink for the power or refrigeration cycle. Therefore, to be truly optimized, the insulation system must become an integral part of the power or refrigeration cycle. The chief conclusion of this chapter is also the main principle for the thermodynamic optimization of advanced power and refrigeration plants. An insulation system that faces an extreme absolute temperature ratio must be optimized in harnwny with the insulated system. The distributed heat transfer is the connection between the insulated power or refrigeration cycle and its insulation system.

REFERENCES Bejan, A. 1979. A general variational principle for thermal insulation system design. Int. J. Heat Mass Transfer. 22:219-228. Bejan, A. 1982. Second-law analysis in heat transfer and thermal design. Adv. Heat Transfer. 15:1-58. Bejan, A. and Smith, J. L., Jr. 1974. Thermodynamic optimization of mechanical supports for cryogenic apparatus. Cryogenics. 14:158-163. Grassmann, P. and Kopp, J. 1957. Zur gunstigsten Wahl der Temperaturdifferenz und der Wiirmeiibergangszahl in Wiirmeaustauschern. 157. Kaeltetechnik. 9:306-308. Hila!, M.A. and Eyssa, Y. M. 1980. Minimization of refrigeration power for large cryogenic systems. Adv. Cryogenic Eng. 25:350--357. Kropschot, R. H., Birmingham, B. W., and Mann, D. B., Eds. 1968. Technology of Liquid Heliwn, Monogr. No. 111, National Bureau of Standards, Washington, D.C. Schultz, W. and Bejan, A. 1983. Exergy conservation in parallel thermal insulation systems. Int. J. Heat Mass Transfer. 26:335-340. Szargut, J. 1980. International progress in second law analysis. Energy. 5:709-718.

PROBLEMS 6.1. Consider an insulated vessel (Dewar) containing M [kg] of saturated liquid helium at atmospheric pressure (TL = 4.2 K). The insulation allows the heat leak Q (W) to reach the liquid pool; as a result, the liquid boils and the boiloff (saturated vapor, 4.2 K) escapes directly into the atmosphere (the boiloff does not cool the insulation). Demonstrate analytically that the instantaneous boil-off rate is in proportion to the rate of entropy generation in the system. (Warning: this is an open system in unsteady state.) Discuss the mechanism for entropy generation, and what the designer can do to reduce the boil-off rate.

169

r----1

------,

I

I

I I I

I I

T8 1

I

I

I

Ql I

I

I

I I

ISVstem

I

I I I

L------------_

I

_J

6.2. Consider the minimization of entropy generation in the one-dimensional insulation (k, A, L, T8 , TL) sketched in Figure 6.2. Assume that the insulation is cooled only at one intermediate point (location x, temperature Tm). Determine the optimal x and Tm for minimum entropy generation. 6.3. A special kind of unidirectional thermal insulation is a radiation shield that separates two surfaces (T8 , TL). Model the net heat transfer rate between each surface and the shield as Q8_s = a. (T~ - 'fs4 ), and Qs-L = a. ('fs4 - r:), where a. is a known constant and Ts is the absolute temperature of the shield. Assuming that the shield temperature can be controlled externally (by contact with a reversible cyclic device), determine the optimal Ts for minimum irreversibility in the space between T8 and TL.

OQS-L

7

STORAGE SYSTEMS

In this chapter we focus on the special class of heat exchangers intended for use in batch heating and batch cooling processes. The batch heating process appears frequently in the design of systems for "thermal energy" storage (Kreith and Kreider, 1978). The batch cooling process, on the other hand, is frequently used in metallurgy, chemical engineering, and, beginning with the past 20 years, the cooldown of largescale superconducting devices. One important characteristic of this class of heat exchangers is their time-dependent operation; time becomes an important design parameter which, as shown below, plays a significant role in determining the thermodynamic irreversibility of the process (Bejan, 1978). The title of this chapter was chosen to call the reader's attention to an engineering technique of great current interest. In what follows, however, we concentrate not on energy storage but on exergy storage. We study in detail the effect played by timehistory on the loss of exergy during the exergy storage process.

7.1 7.1.1

SENSIBLE HEAT STORAGE Time-Dependent

Operation

of a Storage

Unit

Consider the operation of the sensible heat storage system shown schematically in Figure 7.1. The system consists of a large liquid bath of mass M and specific heat C placed in an insulated vessel. Hot gas enters the system through one port, is cooled while flowing through a heat exchanger immersed in the bath, and is eventually discharged into the atmosphere. Gradually, the bath temperature T as well as the gas outlet temperature Tout rises, approaching the hot gas inlet temperature T~. The bath is filled with an incompressible liquid such as water or oil. The stream mcarries an ideal gas, for example, high-temperature steam or air. The bath liquid is thermally well mixed so that at any given time its temperature is uniform, T(t). It is assumed that initially the bath temperature equals the environment temperature T0 • The time dependence of the bath temperature T(t) and gas outlet temperature Tou. 0

"Refrigeration" storage

Exergy storage, 3 > 0

188

tI

Exergy storage

4

2

T!T0

Figure 7.13 Exergy storage via energy (hot) storage or refrigeration (cold) storage.

(7.37)

This expression was plotted in Figure 7.13, which shows that the system stores exergy as long as T #- T0 • The engineering role of the two basic problems considered in this chapter is to show how to place the storage process in time, so that minimum exergy is destroyed while the amount 3 [Equation (7.37)] is being stored. Additional examples of the thermodynamic optimization of time-dependent processes are given in Chapter 11. The minimization of entropy generation during cold storage has received considerable coverage in the recent literature. Sahoo (1989) used the method of Section 7 .1.2 to optimize refrigeration storage in units plagued by heat leak from room temperature. Pedinelli et al. (1993) optimized cold storage systems that operate in cycles (charging and discharging), while the storage material experiences sensible cooling or phase change. They also accounted for the effects of stratification. Dumas et al. (1992) optimized cold storage units for air conditioning and refrigeration, which are based on freezing and melting processes. The next section focuses on the optimization of this important class of time-dependent processes.

7.4 7.4.1

LATENT HEAT STORAGE

Single Storage

Element

We now tum our attention to the storage process in which the heated material melts, i.e., to latent heat storage as opposed to the sensible heat storage discussed in the preceding sections. The simplest way to perform the thermodynamic optimization of the latent heat storage process is shown in Figure 7.14 (Lim et al., 1992). The

189

m.T~

Figure 7.14 The steady production of power using a single phase-change stream (Lim et al., 1992).

material and a mixed

hot stream of initial temperature T~ comes in contact with a single phase-change material through a finite thermal conductance UA, assumed known, where A is the contact area between the melting material and the stream, and U is the overall heat transfer coefficient based on A. The phase-change material (solid or liquid) is at the melting point Tm. The stream is well mixed at the temperature Tout• which is also the temperature of the stream exhausted into the atmosphere (T0 ). The "steady" operation of the installation described in Figure 7.14 accounts for the cyclic operation in which every infinitesimally short storage (melting) stroke is followed by a short energy retrieval (solidification) stroke. During the solidification stroke the flow is stopped, and the recently melted phase-change material is solidified to its original state by tl1e cooling effect provided by the heat engine positioned between Tm and T0 • In this way the steady-state model of Figure 7.14 represents the complete cycle, i.e., storage followed by retrieval. The steady cooling effect of the power plant can be expressed in two ways:

m

n~

By eliminating

Tout

= UA (Tout -T) m

(7.38)

Q.m =me p (T- -Tout )

(7.39)

between these two equations we obtain N tu (T T ) - . Q.m-mcPI+N ~m

(7.40)

tu

in which Ntu is the number of heat transfer units of tlle heat exchanger surface,

N = UA 1"

the p

(7.41)

Of interest here is tlle maximum rate of exergy, or useful work ( W in Figure 7.14), tllat can be extracted from the phase-change material. For tllis, we model as reversible tlle cycle executed by tlle working fluid between Tm and T0 ,

190

. . ( W=Qm

1-T:

T. )

(7.42)

and, after combining with Equation (7 .40), we obtain

W=thc P ~(T 1+ N

~

IU

-T)(l- To) "'

Tm

(7.43)

By maximizing W with respect to Tm, i.e., with respect to the type of phase-change material, we obtain the optimal melting and solidification temperature,

Tm,opt = (Too T.0 ) 112

(7.44)

The maximum power output that corresponds to this optimal choice of phase-change material is

Wmax =thcT~1-(Ta) p ~1+N

[ lu

1/2]2 T

(7.45)

~

The same results, Equations (7 .44) and (7 .45), could have been obtained by minimizing the total rate of entropy generation, as done in the preceding sections for sensible heat storage. Equation (7.44) was frrst reported by Bjurstrom and Carlsson (1985) and Adebiyi and Russell (1987), who analyzed the heating (melting) portion of the process based on a lumped model and the entropy generation minimization approach used in Bejan (1978) for sensible heat storage. The details of the actual melting and solidification process were taken into account by De Lucia and Bejan (1990, 1991). The effect of temperature distribution in the melt layer, or the effect of liquid superheating during melting, was analyzed by De Lucia and Bejan (1991) using the unidirectional time-dependent conduction process shown in Figure 7.15. The degree of liquid superheating is expressed in dimensionless terms by the Stefan number (e.g., Bejan, 1995): Ste =

c

P

(T~ - T) m h,f

(7.46)

where cP is the specific heat of the heating agent, and h, 1 is the latent heat of melting of the storage material. In an earlier study De Lucia and Bejan (1990) showed that in the limit Ste ~ 1 the optimal melting temperature is the same as the constant indicated by Equation (7.44). When Ste is of order 1 or greater, Tm.opt is no longer a constant but depends on Ste, the absolute temperature ratio T_IT0 , the number of heat transfer units during melting N,., and the duration (t) of the melting process, (7.47)

191

U,A Liquid

X

Solid

X

Figure 7.15 Unidirectional time-dependent Lucia and Bejan, 1991).

heating of a layer of phase-change

material (De

where k is the thermal conductivity of the liquid phase, and p is the density of the liquid and solid phases. Figure 7.16 shows that typically, the effects of Ste and 'tare relatively weak, and that Equation (7 .44) is a good approximation, even when liquid superheating is significant. The complete cycle of melting during the finite time't, Equation (7.47), followed by solidification during the next time interval t was optimized by De Lucia and Bejan (1991). The solidification dimensionless parameters are defined by (7.48) where the asterisks indicate the corresponding physical parameters of the installation used during the solidification process, and k, is the thermal conductivity of the solid

Tm,opt

(T-To)V2

0.9

L,...__ _....__ _.....___ __.__ ___.__ ____._ ____,J

0

2

4

6

8

10

12

Figure 7.16 The effects of liquid superheating (Ste) and the duration of the melting process ('t) on the optimal melting temperature (De Lucia and Bejan, 1991).

192 1.1 .----------.------.------.----..,..-------,

Tm,opt

(r~ror

--

----------~~~---2

0

10

20

Figure 7.17 The optimal melting temperature for minimum entropy generation during a com· plete melting & solidification cycle with Ntu = 1, Ntu = 1, 't = i, and k. U = k U. (De Lucia and Bejan, 1991).

phase. The melting and solidification heat transfer problems were formulated as unidirectional time-dependent conduction processes. The liquid superheating during melting and the solid subcooling during solidification were assumed small enough such that their respective Stefan numbers were < 1. De Lucia and Bejan (1991) showed that the entropy generated during the melting and solidification cycle (t + t.) depends on seven dimensionless parameters: 't, Nru, Tm/T0 , T_IT0 , :t. Nru and k,UI(kU.). The melting temperature Tm.opt that minimizes the time-integrated entropy generation depends on the remaining six parameters. Figure 7.17 shows the optimal melting temperature in the special case in which Nru = 1, Nru = 1, 't = t, and k,U = kU•. Once again, the Tm.opt constant furnished by Equation (7 .44) is a fairly good approximation, although, as expected, the entropy generated during the complete melting and solidification cycle is significantly greater than during the melting process alone.

7.4.2

Two Phase-Change

Materials in Series

One way to improve the power output of the single-element installation of Figure 7.14 is by placing the exhaust in contact with a second phase-change element of a lower temperature (Lim et al., 1992). This method is illustrated in Figure 7 .18, in which, individually, each phase-change element has the features of the element described in Figure 7.14. In general, these two elements contain different phase-change materials (Tm.I• Tm. 2), and their heat exchanger surfaces are not identical [(UA)1, (UA)~. The well-mixed gas temperatures above each heat exchanger surface are T1 and T2• The total power output from the arrangement of Figure 7.18 is

193

11 Figure 7.18 Power production based on melting and solidification in two phase-change rials placed in series (lim et al.. 1992).

w< 2> = 0

m,l

[1-~)+T

m,l

0

m,2

mate-

[~-_!Q_) T

m,2

(7.49)

An analysis equivalent to that contained between Equations (7.38) and (7.41) yields (7.50) where the respective numbers of heat transfer units of the two elements are

N tu,l

=

(UA)t

me

p

N

tu,2

=

(UA)2

me

(7.51)

p

For a meaningful comparison of the performance of the two-element system (N,•. 1, N,., 2) relative to the reference single-element system (N,.), it is reasonable to adopt the overall heat exchanger size constraint (7.52) This constraint can be rewritten in terms of a thermal conductance allocation ratio x: (7.53) Beyond this point, we can skip the algebra and report only the resulting expression for the total power output w(2>. We present this in relative terms (dimensionless) by 2 ) by the single-element power maximum derived in Equation (7.45): dividing

w
1 and 't2,opt < 1 indicates that the optimal melting temperature of the upstream element must be higher than the single value recommended by Equation (7.44), and, at the same time, the optimal melting temperature of the downstream element must be lower than (T~T 0 ) 112 • The departure of both 'tt,opt and 't2,opt from the value 1 [i.e., from Equation (7.44)] becomes more accentuated as N,. increases and

195 1.15,----------.------,

= Ntu,2

Ntu, 1

1.10

1.05

0.6

0.7

0.8

0.9

y = (To1T~)112

Figure 7.19 The optimal melting temperature of the phase-change stream element of Figure 7.18 (Lim et al., 1992).

material used in the up-

0.95 't2,opt 0.90

Ntu,1

=

Ntu,2

0.85

0.80 +-----,-----.-----.-----1 0.9 0.8 0.6 0.7 0.5

Figure 7.20 The optimal melting temperature of the phase-change stream element of Figure 7.18 (Lim et al., 1992).

material used in the down-

as y decreases. This means that as the total area A and the inlet temperature T~ increase, the optimization of the two-element system gains in importance relative to simply using Equation (7.44) for the selection of both materials. The total relative power output that corresponds to the optimal temperatures calculated in Figures 7.19 and 7.20 has been plotted in Figure 7.21. The subscript of Wmax,max on the ordinate is a reminder that Wof Equation (7 .55) was maximized twice, with respect to 't 1 and 't2• Figure 7.21 shows that the relative gain in power output increases as N1" andy= (T 0 /T~) 112 increase. For example, when TjT0 = 2 and N1u 1t> 1, the temperature span parameter is y = 0.707, and Wmax,max = 1.282. In other words, the optimal two-element design promises to produce 28.2% more power than the optimal single-element design. The ceiling value of 4/3 reached by Wmax.max at y = 1 is somewhat deceiving, because in the limit y ---+ 1 the power W produced by either the scheme of Figure 7.14 or the scheme of Figure 7.18 approaches zero. In the analysis reported in this section the two heat exchangers were assumed to have the same size (Nru,t = Nru. 2 , or x = 1/ 2), and the relative power function W was

196

wmax, max

1.1

1----o.s--------1 Ntu,1 = Ntu,2

1.0-1-----.----.----.------l 0.5 0.6 0.7 0.8 0.9

Figure 7.21 The maximum relative power output that corresponds to the optimal melting temperatures reported in Figures 7.19 and 7.20 (lim et al., 1992).

maximized only with respect to 't 1 and 't 2• In a subsequent effort Lim et al. (1992) relaxed the assumption that x is fixed, and proceeded to maximize W with respect to x, 't 1, and 't 2 simultaneously. They carried out this work numerically while searching for xopt• 't 1,opt• and 'tz,opt as functions of N'" andy. In the domain represented by 0.5 $; N'" $; 10 and 0.5 $; y $; 0.9 they found that the optimal value of x varies about 0.5 ± 0.0001, i.e., that 't 1,opt• 'tz,opt• and the three times maximized Whave practically the same values as in Figures 7.19 to 7.21. In conclusion, the x = 1/ 2 value assumed at the start of this section is a very good approximation for the optimal value that maximizes W. For minimum entropy generation, then, the total heat exchanger inventory N ru must be divided equally between the two phase-change elements.

7.4.3

Infinite Number of Phase-Change

Materials in Series

In view of the improvement in thermodynamic performance registered in going from the single-element scheme (Figure 7.14) to the two-element scheme (Figure 7 .18) it is reasonable to think of the limit in which we employ not two but an infinite number of elements (materials) in series (Lim et al., 1992). The melting points of these elements vary infinitesimally from one element to the next, so that the longitudinal distribution of melting points is represented by the function Tm(x). This new arrangement is analyzed in Figure 7.23, where x indicates the position of each element along the stream rh. As a preliminary step in the analysis of Figure 7 .23, it is useful to consider the performance of the simpler scheme shown in Figure 7 .22. Here, the melting temperature is unique; in other words, from the inlet to the outlet of the hot stream the melting and solidification processes occur in a layer of the same phase-change material. The essential difference between Figures 7.14 and 7.22 is that in Figure 7.22 the hot stream is not mixed: its temperature varies smoothly along the layer of phase-change material. The power that can be extracted from the scheme of Figure 7.22 can be calculated with Equation (7.42), in which the heating rate is now given by

197

To-------------------Figure 7.22 Single phase-change

material in contact with an unmixed stream (Lim et al., 1992).

(7.60)

Finally, by combining Equations (7.42) and (7.60) we obtain (7.61)

aw

By solving Fig. 7.22 I aTm = 0, we find that the maximum power output occurs at the same optimal melting point as in Equation (7 .44). The power output that corresponds to this optimum is

(7.62) Now we tum our attention to the original purpose of this section, namely, the scheme with unmixed stream and infinite series of elements shown in Figure 7.23. The total power output is the sum of the power produced by each slice of infinitesimal thickness dx, (7.63) In this expression p is the heat transfer area per unit length, while T(x) is the bulk temperature of the hot stream entering the slice (control volume) of thickness dx. Also important is the observation that the dx-thin control volume is conceptually analogous to the system of Figure 7.22. They are analogous in the sense that each of these systems has a unique Tm value for the phase-change material. The only difference between them is the length of the area of heat exchange, namely dx in Figure 7.23 vs. L=Aip in Figure 7.22. While maximizing the expression (7.61), we learned that the optimal melting temperature Tm,opt = (T_T0 ) 112 is not a function of the length

198

T(x)

Tm,in~•n:

.v: •

Tm(X)

;

Tm,out

0 ..... To----------~~-------0

X

X+dX

Figure 7.23 Series of infinitesimal phase-change (Lim et al., 1992).

dW

L

elements in contact with an unmixed stream

of thermal contact. This means that the optimal local melting point for the dx-thin element of Figure 7.23 is simply (7.64) The two temperature distributions T(x) and Tm.op,(x) can be obtained by combining Equation (7.64) with the usual analysis of the beat exchange between T and Tm: (7.65) In particular, by eliminating T(x) between Equations (7.64) and (7.65), and by integrating from x =0 to x =L we obtain the relationship between the temperatures of the two ends of the string of phase-change elements:

Tm,in- To

(7.66)

Tm,out- TO

In the above equation we set Tm.in = (T_T0 ) 112 in accordance with Equation (7.64). The total power delivered by the scheme of Figure 7.23 is the result of integrating from x =0 to x =L. In that integral we replace T(x) with T;.,op/T0, and after some algebra we obtain

wH = 2 .

mcp

max

rmJ.i• ( Tm -

T.

rm.out

=

t) ar

m

0

1- (Tm,m 2. 2mcp [2To

(7.67) 2 Tm.out

) -

Tm.1n. + Tm,out ]

199

The superscript (oo) indicates that this maximum power output is based on an infinite number of elements in series. Finally, by writing Tm.in = (T_T0 ) 1n., and by eliminating Tm.out between Equations (7.66) and (7.67), it is possible to show that

(7.68) What is surprising at this point is that Equations (7.68) and (7.62) are identical; i.e., the maximum power from the scheme of Figure 7.23 is the same as that produced by the scheme of Figure 7.22. How this maximum power output compares to that of the single-element arrangement (Figure 7 .14) is illustrated in Figure 7 .24. Plotted on the ordinate is the ratio (7.69) which depends only on the total number of heat transfer units. Its maximum value, 1.298, occurs atNru = 1.793, indicating that relative to the arrangement of Figure 7.14, the optimal designs in Figures 7.23 and 7.22 can produce up to 30% more power.

7.4.4

More Advanced

Phase-Change

Storage

Models

The thermodynamic optimization of storage installations with melting and solidification has attracted a lot of interest. The progress was reviewed recently by Charach (1994). For example, Adebiyi (1991) considered a bed with particles with several ratios of latent heat to sensible heat storage capability. He modeled the heat conduction as one dimensional in the cylindrical pellet geometry. For the complete storage and removal cycle, he found that the optimal phase-change temperature is equal to the arithmetic average of the heat source and ambient temperatures. Charach and Zemel (1992a) optimized the thermodynamic performance of latent heat storage in the shell of a shell-and-tube heat exchanger. The focus was on the effect of two-dimensional heat transfer, as a step beyond the one-dimensional model 1.4

w max

W!!

1.3

1.2

1.1

1.0

10

0

Ntu

Figure 7.24 The relative performance of the optimized arrangements (Lim et al., 1992).

of Figures 7.23 and 7.14

200 employed by De Lucia and Bejan (1990). Charach and Zemel (1992a) also considered the effect of pressure drop on the stream side of the heat exchanger. Their analysis was extended by Charach and Zemel (1992b) and Charach (1993) to the complete melting and solidification cycle by using a quasi-steady treatment of the phase change process occurring in the shell. These latest studies showed that the optimal phase-change temperature is bounded from above and from below by the arithmetic and, respectively, geometric averages of the source and ambient temperatures. Another interesting direction has been the study of latent heat storage units coupled in series with a power plant, and optimized over the entire storage and removal cycle. Bellecci and Conti (1994a, b) showed that minimization of entropy generation and operation stability are two competing criteria in the optimization of the aggregate installation. The arithmetic average of the extreme temperatures again emerges as the optimal phase-change temperature. Detailed modeling and numerical simulations of the heat transfer behavior of the shell-and-tube phase-change heat exchanger were performed by Bellecci and Conti (1993a, b). Aceves-Saborio et al. (1994) developed a systematic way of modeling phasechange storage systems by applying the lumped model to many independent elements. They also considered the more general case in which the phase-change material melts over a finite temperature range. In an earlier study Aceves-Saborio et al. (1993) optimized a single latent-heat cell by first simulating numerically the phase-change process. This is an important and difficult task because the natural convection currents that occur in the liquid control the shape and movement of the liquid-solid interface (e.g., Section 10.4 in Bejan, 1995). The shape of the melting front in a phase-change element heated from the side is illustrated in Figure 7.25, which should be compared to the unidirectional model shown in Figure 7.15. It was shown by De Lucia and Bejan (1990) that when the melting process is controlled by natural convection the optimal melting temperature is equal to the geometric average shown in Equation (7.44). Adebiyi et al. (1992) constructed a numerical model for simulating and then optimizing the performance of storage systems with multiple phase-change materials. They found that the second law efficiency of systems with multiple materials can exceed by 13 to 26% the efficiency of systems employing a single material. This conclusion agrees in an order of magnitude sense with the results reviewed in Sections 7 .4.2 and 7 .4.3. Fundamental studies of entropy generation minimization in time-dependent unidirectional heat conduction were conducted by Charach and Rubinstein (1989) and Gordon et al. (1990). Sections 7.1 and 7.2 reviewed the early work on the fundamental thermodynamic problem of how to cool or heat a body in the least irreversible manner possible while subject to size and time constraints. The literature surveyed in this chapter showed that this fundamental problem triggered a significant amount of work in the 1980s and 1990s in engineering. It is interesting to see that just recently the same basic question was deemed worthy of study in physics (Andresen and Gordon, 1992a, b). These authors minimized the generation of entropy instead of the amount of heating or cooling agent used. As heat source or heat sink they assumed a heat reservoir of temperature that can be varied at will (instead of the coolant flow rate of Figure 7 .11). Between the heat reservoir and the thermal inertia they assumed several heat transfer rate laws, for example, convection, with a constant heat transfer

m

201

/Liquid

.. ·.

::·.=·

·:....:· ....... ~

ir::

·.=.···•,:··

··...

:·~:;· ~ ;:;'/.

~->·.·.

Solid

:=: .:·.· ...... .

_m_cP_·

T-~- -1+: :.:·

='.::·}t

Figure 7.25 Melting dominated by natural convection in a single storage element heated from the side (De Lucia and Bejan, 1990).

coefficient, and radiation, with constant (temperature independent) emissivities. In a companion paper Andresen and Gordon (1992b) considered a related problem -the heating of the body of interest is effected by a stream, while the heat reservoir temperature may change or remain constant. The result of the entropy generation minimization procedure is again an optimal flow rate of heating agent. In both papers the effect of pressure drop was neglected. The minimization of entropy generation in other classes of time-dependent processes and installations is illustrated further in Chapter 11.

REFERENCES Aceves-Saborio, S., Hernandez-Guerrero, A., and Nakamura, H. 1993. Heat transfer and exergy analysis of the charge process of a heat storage cell. ASME HTD. 266:73-80. Aceves-Saborio, S., Nakamura, H., and Reistad, G. M. 1994. Optimum efficiencies and phasechange temperatures in latent heat storage systems. 1. Energy Resources TechnoL 116:79--86. Adebiyi, G. 1991. A second law study of packed bed energy storage systems utilizing phasechange materials. 1. Solar Energy Eng. 113:146-156. Adebiyi, G. and Russell, L. D. 1987. A second law analysis of phase-change thermal energy storage systems. ASME HTD. 80:9-20. Adebiyi, G. A., Hodge, B. K., Steele, W. G., Jalalzadeh, A., and Nsofor, E.C. 1992. Computer simulation of high temperature thermal energy storage system employing multiple families of phase-change materials. ASMEAES. 27:1-11. Andresen, B. and Gordon, J. M. 1992a. Optimal paths for minimizing entropy generation in a common class of finite-time heating and cooling processes. Int. 1. Heat Fluid Flow. 13:294-299. Andresen, B. and Gordon, J. M. 1992b. Optimal heating and cooling strategies for heat exchanger design. 1. Appl. Phys. 71:76-79. Badar, M.A., Zubair, S.M., and Al-Farayedhi, A. A. 1993. Second-law-based thermoeconomic optimization of a sensible heat thermal energy storage system. Energy. 18:641-649. Bejan, A. 1978. Two thermodynamic optima in the design of sensible heat units for energy storage. 1. Heat Transfer. 100:708-712. Bejan, A. 1988. Advanced Engineering Thermodynamics. Wiley, New York. Bejan, A. 1995. Convection Heat Transfer. 2nd ed. Wiley, New York.

202 Bejan, A. and Schultz, W. 1982. Optimwn flowrate history for cooldown and energy storage processes. Int. J. Heat Mass Transfer. 25:1087-1092. Bellecci, C. and Conti, M. 1993a. Latent heat thermal storage for solar dynamic power generation. Solar Energy. 51:169-173. Bellecci, C. and Conti, M. 1993b. Phase change thermal storage: transient behavior analysis of a solar receiver/storage module using enthalpy method. Int. J. Heat Mass Transfer. 36:2157-2163. Bellecci, C. and Conti, M. 1994a. Thermal storage for solar dynamic power generation: performance indicators in a second law perspective. SOLCOM-1 Int. Conf Comparative AssessmentofSolar Power Technologies. Program and Abstracts, p. 3. A. Roy, Ed. Israel Ministry of Science and Technology, Jerusalem. Bellecci, C. and Conti, M. 1994b. Phase change energy storage: entropy production, irreversibility and second law efficiency. Solar Energy. 53:163-170. Bisio, G. 1993. Analysis of possibilities of thermal energy recovery from furnace molten slags with reducing environmental pollution. Proc. Int. Conf Energy Systems and Ecology (ENSEC '93). pp. 731-738. Cracow, Poland. Bjurstrom, H. and Carlsson, B. 1985. An exergy analysis of sensible and latent heat storage. Heat Recovery Syst. 5:233-250. Charach, Ch. 1993. Second-law efficiency of an energy storage-removal cycle in a phasechange material shell-and-tube heat exchanger. J. Solar Energy Eng. 115:240-243. Charach, Ch. 1994. On the second law efficiency of thermal energy storage. In Proc. Int. Conf Comparative Assessments of Solar Power Technologies. A. Roy and W. Grasse, Eds. Muller Verlag, Karlsruhe. Charach, Ch. and Rubinstein, I. 1989. On entropy generation in phase-change heat conduction. J. Appl. Phys. 66:4053-4061. Charach, Ch. and Zemel, A. 1992a. Thermodynamic analysis of latent heat storage in a shelland-tube heat exchanger. J. Solar Energy Eng. 114:93-99. Charach, Ch. and Zemel, A. 1992b. Irreversible thermodynamics of phase-change heat transfer: basic principles and applications latent heat storage. Open Syst. Inform. Dyn. 1:423458. Das, S. K. and Sahoo, R. K. 1991. Thermodynamic optimization of regenerators. Cryogenics. 31:862-868. De Lucia, M. and Bejan, A. 1990. Thermodynamics of energy storage by melting due to conduction or natural convection. J. Solar Energy Eng. 112:110-116. De Lucia, M. and Bejan, A. 1991. Thermodynamics of phase-change energy storage: the effects of liquid superheating during melting, and irreversibility during solidification. J. Solar Energy Eng. 113:2-10. Dwnas, J. P., Strub, F., Bedecarrats, J. P., Zeraouli, Y., Broto, F., and Lenotre, Ch. 1992. Influence of undercooling on cold storage systems. Proc. Int. Symp. Efficiency, Costs, Optimization and Simulation of Energy Systems (ECOS '92). p. 571-574. Zaragoza, Spain. Geskin, E. S. 1980. Second law analysis of fuel consumption in furnaces. Energy. 5:949-954. Gordon, J. M., Rubinstein, I., and Zarmi, Y. 1990. On optimal heating and cooling strategies for melting and freezing. J. Appl. Phys. 67:81-84. Hutchinson, R. A. and Lyke, S. E. 1987. Microcomputer analysis of regenerative heat exchangers for oscillating flows. Proc. /987 ASME/JSMEThermal Eng. Joint Conf 2:653, ASME, New York. Kotas, T. J. and Jassim, R. K. 1993. Costing of exergy flows in the thermoeconomic optimization of the geometry of rotary regenerators. Proc. Int. Conf Energy Systems and Ecology (ENSEC '93). p. 313-322. Cracow, Poland.

203 Krane, R. J. 1985. A Second Law Analysis of a Thermal Energy Storage System with Joulean Heating of the Storage Element. ASME paper No. 85-WA/HT-19. Krane, R. J. 1987. A second law analysis of the optimum design and operation of thermal energy storage systems. Int. J. Heat Mass Transfer. 30:43-57. Krane, R. J. and Krane, M. J. M. 1991. The optimum design of stratified thermal energy storage systems. Parts 1 and 2. Analysis ofThermal and Energy Systems: Proc. Int. Conf p. 197-218. Athens, Greece. Kreith, F. and Kreider, J. F. 1978. Principles ofSolar Engineering. p. 193, 428. McGraw-Hill, New York. Lim, J. S., Bejan, A., and Kim, J. H. 1992. Thermodynamic optimization of phase-change energy storage using two or more materials. J. Energy Resources Techno[. 114:84-90. Mathiprakasam, B. and Beeson, J. 1983. Second law analysis of thermal energy storage devices. Proc. A!ChE Symp. Ser., National Heat Transfer Conf p. 161-168. Matsumoto, K. and Shiino, M. 1989. Thermal regenerator analysis: analytical solution for effectiveness and entropy production in regenerative process. Cryogenics. 29:888-894. Pedinelli, N., Rosen, M. A., and Hooper, F. C. 1993. Thermodynamic assessment of cold capacity thermal energy storage systems. Proc. Int. Conf Energy Systems and Ecology (ENSEC '93). p. 705-712. Cracow, Poland. Rosen, M.A. 1992. Appropriate thermodynamic performance measures for closed systems for thermal energy storage. J. Solar Energy Eng. 114:100-105. Rosen, M.A., Hooper, F. C., and Barbaris, L. N. 1988. Exergy analysis for the evaluation of closed thermal energy storage systems. J. Solar Energy Eng. 110:255-261. Sahoo, R. K. 1989. Exergy maximization in refrigeration storage units with heat leak. Cryogenics. 29:59-64. Sahoo, R. K. and Das, S. K. 1994. Exergy maximization in cryogenic regenerators. Cryogenics. 34:475-482. San, J. Y., Worek, W. M., and Lavan, Z. 1987. Second-law analysis of a two-dimensional regenerator. Energy. 12:485-496. Sekulic, D. P. and Krane, R. J. 1992. The use of multiple storage elements to improve the second law efficiency of a thermal energy storage system. Parts I and II. Proc. Int. Symp. Efficiency, Costs, Optimization and Simulation ofEnergy Systems (ECOS '92). p. 61-72. Zaragoza, Spain. Shen, C. M. and Worek, W. M. 1993. Second-law optimization of regenerative heat exchangers, including the effect of matrix heat conduction. Energy. 18:355-363. Taylor, M. J. 1986. Second Law Optimization of a Sensible Heat Thermal Energy Storage System with a Distributed Storage Element. M.S. thesis, University of Tennessee, Knoxville. Taylor, M. J., Krane, R. J., and Parsons, J. R. 1990. Second law optimization of a sensible heat thermal energy storage system with a distributed storage element. Parts 1 and 2. In A Future for Energy: Proc. Florence World Energy Symp. S.S. Stecco and M.J. Moran, Eds. p. 885-908. Pergamon Press, Oxford.

8

POWER GENERATION

This chapter takes a systematic look at power plants and how their inherently irreversible operation can be modeled in terms of the simplest and most basic heat transfer components possible. As we have seen repeatedly in the earlier chapters, such models allow us to identify thermodynamic trade offs; i.e., ways of adjusting the physical characteristics of the system such that the system performs its intended function while generating minimum entropy. In this chapter the application is specific (a power plant); therefore, we may conduct the thermodynamic optimization by maximizing the power output of the plant. It is important to recognize from the start that to maximize the power output is simply a matter of preference. It is equivalent to minimizing the entropy generation rate of the power plant, cf. the Gouy-Stodola theorem, Equation (2.8), and, more specifically, Problem 2.7. This equivalence is illustrated further in Section 8.6. The model consists of combining several features of heat transfer irreversibility recognized in the Bejan (1982) book, namely the bypass heat leak (Problem 2.2) and the two finite-size heat exchangers (Figure 5.14). In the simplest version of the model (Bejan, 1988a), Section 8.1, it is assumed as in Problem 2.2 that the heat leak proceeds from the heat source to the heat sink unattenuated. Section 8.2 improves the model by allowing the heat leak (the power plant insulation) to interact with the rest of the power plant (Bejan, 1995). In this way the thermodynamic performance can be optimized further, along the general course traced in the 1970s for thermal insulations (Chapter 6). With these models we continue to pursue the fundamental question considered first in Section 5.6: how to distribute a finite heat exchanger inventory over the temperature range spanned by the power plant.

8.1 8.1.1

MODEL WITH BYPASS HEAT LEAK AND TWO FINITE-SIZE HEAT EXCHANGERS Optimal Heat Exchanger

Temperature

Differences

Consider the model outlined in Figure 8.1 (Bejan, 1988a). The power plant delineated by the solid boundary operates between the high temperature T8 and low temperature TL. The modeling challenge consists of providing the minimum construction detail that allows the outright identification of the irreversible and reversible compartments of the power plant. Two sources of irreversibility are the heat exchangers, or the heat transfer across the finite temperature gaps, 205

206

0H TH QHC

-

THe

0;

c

w

Figure 8.1 Power plant model with bypass heat leak and two finite-size heat exchangers (Bejan, 1988a).

Qnc

=(UA)H(Tn- Tnc)

QLc = (UA)L(TLC- TJ

(8.1) (8.2)

The proportionality coefficients (UA)n and (UA)L are the hot-end and cold-end thermal conductances. In heat exchanger design terms, for example, (UA)n represents the product of the hot-end heat transfer area A8 times the overall heat transfer coefficient based on that area, Un. Because both (UA)8 and (UA)L are commodities in short supply, it makes sense to recognize as a constraint the total thermal conductance inventory UA (Bejan, 1988a, b) (8.3)

or, in terms of the external conductance allocation ratio x, (8.4)

Another source of irreversibility is the heat transfer rate C2; that leaks directly through the machine and, especially, around the power producing compartment labeled (C). In order to distinguish this irreversibility from the external irreversibilities due to (T8 - Tnd and (TLC- TL), we will refer to C2; as the internal heat transfer rate (heat leak) through the plant. The internal heat leak was first identified as a modeling feature of power plant irreversibility by Bejan and Paynter (1976). Many features of an actual power plant fall under the umbrella represented by C!;. for example, the heat transfer lost through the wall of a boiler or combustion chamber, the heat transfer removed by the cooling system of an internal combustion engine, and the streamwise convective heat leak channeled toward room temperature

207

by the counterflow beat exchanger of a regenerative Brayton cycle (Bejan, 1979). The simplest internal heat leak model that is consistent with the linear models of Equations (8.1) and (8.2) is (8.5)

where R; is the internal thermal resistance of the power plant. To summarize the three irreversibility sources that have been identified so far, we note that the model of Figure 8.1 relies analytically on three parameters, namely [(UA)11 , (UA)L,R;] or(UA ,x, R;). The remaining power plant "compartment", which is labeled C, from Carnot, is assumed irreversibility free. The second law for compartment C is (8.6)

The analytical representation of Figure 8.1 is completed by the first-law statements (8.7) .

.

.

Q/1 = Qi + QI/C

(8.8)

(8.9) where W, Q11 , and QL are the interactions of the power plant as a whole, namely the power output, the rate of heat input, and the heat rejection rate. As shown in Section 8.6, a fourth source of entropy ge1,1eration is present on the outside of the power plant. It accounts for the fact that Q11 (or T11c) must be free to vary for the degree of freedom in the optimization process. This fourth source must be added to the first three when the power plant is optimized by minimizing the total rate of entropy generation. In this section we pursue the same objective by maximizing the instantaneous power output. Consider now the conditions under which the instantaneous power output W is maximum. Combining Equations (8.4) to (8.9) leads to · ( l-x) UA ,TL ( TTL LC -1 ) (THe W= TLC -l)

(8.10)

in which the unknown is the ratio TLC/TL. From Equations (8.6) and (8.4) we find that TLC

't

TL

'tc

-=1-x+x-

(8.11)

208 where t

c

THC

(8.12)

=-

TLC

Taken together, Equations (8.10) and (8.11) yield (8.13)

This expression shows that the instantaneous power output per unit of external conductance inventory UA can be maximized in two ways with respect to tc and x. By solving aW/ate= 0 we obtain the optimal temperature ratio across the Carnot compartment (C), -

tC,opt- 't

1f2

or

(

;c ) LC

= (;

opt

)1!

2

(8.14)

L

or the optimal temperature differences across the heat exchangers, ( TH - THC ) opt = TH (1- x) (1-

t-1!

2)

(8.15) (8.16)

We return at the end of the next section to the physical meaning of these temperature differences. The corresponding maximum power output is (8.17) The optimization with respect to the temperature ratio across the reversible compartment, Equation (8.14), has an interesting history which began in 1957.1t was frrst reported in engineering independently by Chambadal in French and by Novikov in Russian and English. Chambadal's work is described at the end of Section 8.4. Novikov (1957, 1958) used a model of a steady-state power plant with finite-size heat exchanger only at the hot end (relative to Figure 8.1, a model without heat leak (R; ~ oo) and with an infinite cold-end heat exchanger). Novikov's analysis was reprinted in books in English by El-Wakil (1962, 1971, 1982), and in books in Russian by Vukalovich and Novikov (1972), Novikov and Voskresenskii (1977), and Novikov (1984). It is discussed further in Section 8.6.2. The principle of operation at maximum power output was also described by Odum and Pinkerton (1955), who gave several examples from the fields of engineering, physics, and biology. The same idea [Equation (8.14)] resurfaced almost 2 decades later in the physics literature in a paper by Curzon and Ahlborn (1975), who used a four-process (two stroke) model as in Sadi Carnot' s original work; however, this time the working fluid (piston and cylinder apparatus) made contact during finite time intervals with the two

209

heat reservoirs across finite temperature differences. The Curzon-Ahlbom paper triggered a series of papers on engine power maximization in the physics literature, summarized as "finite time" thermodynamics in Andresen et al. (1984). To these authors finite time meant real rate (transport) processes such as heat transfer, mass transfer, and fluid flow, the same modeling ingredients as in entropy generation minimization. Noteworthy is the first paper in this series (Andresen et al., 1977a), in which the engine is modeled in the steady state (as in Novikov's work), and where it is accounted for additional work dissipation (e.g., shaft friction, Joule heating) that must be rejected as heat to one of the finite temperature reservoirs. Although Andresen et al. called the rejected heat a "heat leak", it must be noted that the origin of that energy interaction in their model is always the work reservoir, or, in Andresen and co-workers' own terminology, a reservoir with "infinite temperature" (or energy transfer accompanied by zero entropy transfer). Their leak is actually a work leak, or "friction leak", as stressed by Grazzini ( 1991 ), not a heat leak in the sense of Problem 2.2, Chapter 6, and Figure 8.1. Using this literature as background, the contribution of the model of Figure 8.1 is that it shows that Equation (8.14) holds even when the heat leak Qi cannot be neglected (Bejan, 1988a). I am pleased to note that more recently the model put forth in Figure 8.1 was found useful in the physics literature by Gordon and Huleihil (1992), Gordon and Orlov (1993), and Pathria et al. (1993).

8.1.2

Optimal Allocation

of Heat Exchanger

Inventory

Another new aspect that is brought to light by the model of Figure 8.1 is that an optimal way exists to allocate the UA inventory between the hot and cold ends, so that the power output is maximized once more. By solving aWmax/ax= 0 we obtain the optimal thermal conductance allocation fraction (8.18) and the corresponding (twice maximized) instantaneous power output wmaxmax

=~UA

TL('tV2 -lr

(8.19)

In conclusion, in order to operate at maximum power there must be not only a balance between the thermodynamic temperature ratios 'tc and 't [Equation (8.14)], but also a balance between the sizes of the hot-end and cold-end heat exchangers. This conclusion was first reported in Bejan (1988a, b), although the area allocation issue was also analyzed in Bejan (1982). It can be shown that Equation (8.18) also holds when the total thermal conductance inventory UA is minimized subject to fixed power output W (Bejan, 1993). It is interesting to compare this conclusion to the results reached when the total thermal conductance constraint of Equation (8.3) is replaced by the constraint that the physical size (or weight) of the total heat transfer area is fixed (Bejan, 1994a, 1995)

210

(8.20) The area constraint can be restated as (8.21) (8.21) into the W where y is now the area allocation ratio. By substituting Equatio~ expression derived from Equations (8.6) to (8.9), and solving a WI ay = 0 yields the optimal area allocation ratio

(A= constant)

(8.22)

Equation (8.22) is plotted as a solid line in Figure 8.2. This result shows that a larger fraction of the area supply should be allocated to the heat exchanger whose overall heat transfer coefficient is lower. Only when U11 is equal to UL should A be divided equally between the two heat exchangers. The optimal design based on the total thermal conductance constraint, Equation (8.18), can be rewritten as 1 AH,opr = , A l+U 11 JUL

(UA =constant)

(8.23)

which is represented by the dashed line in Figure 8.2. In conclusion, the optimal area allocation ratio derived based on the thermal conductance constraint is qualitatively the same as the ratio recommended when the total area is fixed. Relative to the area constraint, the use of the UA constraint tends to exaggerate the inequality between An.opt and AL.opt· The two constraints are equivalent if U11 = UL' The design based on the area constraint is pursued further in Bejan (1994a, 1995). The rest of the material in Section 8.1 refers to optimizations based on the total thermal conductance constraint. In summary, we maximized the instantaneous power output of the model of Figure 8.1 in two ways, with respect to the temperature range spanned by the working fluid ('tc) and the allocation of the total thermal conductance (x). The practical implications of xopt are clear and immediate - the heat exchanger inventory must be divided in a certain way. The practical implications of 'tc,opt are more subtle. One way to see them is by looking at Equations (8.15) and (8.16), where 'tc,opt has been used to calculate T11c.opt and TLC,opt; one set of T11c,opt and TLC,opt values exists for each set of specified heat exchanger sizes, (UA)11 and (UA)L' The message to the designer is that the working fluid must be selected in such a way that it can be heated while at a certain temperature (e.g., boil at T11 c.opt in a simple ideal Rankine cycle) and be cooled while at another optimal temperature (e.g., condense at TLC.opt) for each given pair of (UA)11 and (UA)L' The designer is considerably less free to experiment with the fluid type (e.g., to abandon water) than to divide the UA inventory. I believe this reality explains at least in part why Novikov' s (1957) 'tc optimization was not noted

211

0.5 A"" constant

UA ; constant ' ' _

Figure 8.2 The optimal way to divide the total heat transfer area between the two heat exchangers of a power plant (Bejan, 1994a, 1995).

more in engineering. Another reason is that large scale power plants are optimized for fixed heat input (Problem 8.9), not variable heat input.

8.1.3

Trade Off between Insulation

Heat Exchanger

Inventory and Thermal

In this section we tum our attention to a third way of maximizing the power output: by balancing the investment in total heat exchanger equipment UA and the investment in power plant insulation R; (Bejan, 1988a). We begin with a look at the second law efficiency that corresponds to the twice-maximized power output (8.19), (8.24)

where Tin is the ratio of the first law efficiency 11 = Tic= 1- TL!Tn,

W/On

to the Camot efficiency

11

(8.25)

Tln=Tic

Equation (8.24) shows that Tin depends on two parameters, the overall temperature ratio 1: =Ta!TL and the dimensionless group R;UA. As expected, the efficiency decreases as R; UA decreases, i.e., as more of the heat input bypasses the powerproducing compartment of the power plant. It is desirable to increase both R; and UA; however, these moves are resisted by economic constraints. An interesting trade off exists between R; and UA when the total resources available for R; and UA are fixed. Let us assume that in the overall cost of the power plant these two features compete against one another in a cost constraint of the type

On

212

Pc UA + Pr R. = constant

(8.26)

1

where Pc and P, are the unit costs (prices) associated with, respectively, adding more thermal conductance and more insulation thermal resistance. Expressed in thermal conductance units, the constraint (8.26) reads p UA+....L.R. p I =C

(8.27)

c

where C is the total thermal conductance that could be built with the resources available for both UA and R;. The splitting of the C inventory is described by the investment allocation ratio z, (8.28)

UA=zC The product R;UA that appears in the efficiency formula (8.24),

R; UA=z (1-z); C2

(8.29)

r

reaches its maximum at z opt = 1/2, i.e., when PcUA = P,R;. The corresponding maximum efficiency is (8.30) where 1t is the dimensionless price ratio, 1t

PfP

=-'--c

c2

(8.31)

What we have achieved in Equation (8.30) is a means to anticipate the ceiling value of Tln as a function of 't and only one additional parameter, 1t. The latter is not expected to vary much in any given technological age; therefore, it can be held constant while drawing the curves shown in Figure 8.3. Although in principle the entire 0 to 1 range is accessible to 'flu, the theory on which Equation (8.30) is based suggests that the efficiencies of actual power plants should fall below the 1t = 0 curve. Note that 1t =0 represents the limit of infinitely plentiful thermal insulation (or zero heat leak), for which Equation (8.30) reduces to

11 = 1 -

't

-1/2

( TL )

= 1 - TH

1/2

(8.32)

Note also that in this limit the first law efficiency 11 is the same as in the optimized models of Novikov (1957), Chambadal (1957), and Curzon and Ahlborn (1975), in which the bypass heat leak was not one of the features of irreversible operation.

213

7'

Figure 8.3 Theoretical and empirical power plant second-law efficiencies (Bejan, 1988a). The numbers indicate the performance data of the plants tabulated in Bejan (1988a, b).

Figure 8.3 shows the efficiencies of ten existing power plants, based on operating parameters tabulated in Bejan (1988a, b). With only one exception, the empirical Tln values fall under the 1t:::: 0 curve, in agreement with the theory. This agreement suggests that the simple model of Figure 8.1 captures the most essential features of an actual (irreversible) power plant. Another interesting observation is that the ten power plants projected in Figure 8.3 correspond to a remarkably narrow band of 1t values. The average 1t value is on the order ofO.Ol, which corresponds to an R; UA of approximately 25. If, in addition, we regard 't - 2.5 as the representative order of magnitude of the 't values of the ten power plants, we find that the heat leak ratio Qj Q8 is roughly 12%. Continuing this numerical example, we can show that the contributions to the entropy generation rate of the optimized power plant model of Figure 8.1 are the bypass heat leak, 40%; the hot-end heat exchanger, 23%; and the cold-end heat exchanger 37%.

8.1.4

Graphic Representation

of the Thermodynamic

Optimum

The double maximization of W described in Sections 8.1.1 and 8.1.2 means that the temperature differences across the heat exchangers [cf. Equations (8.15) and (8.16)] are in the same proportion as the temperatures across the reversible compartment of the power plant, (8.33) Bejan (1988a) showed that this proportionality, or

214

QHITHC flow of entropy

t

QLITLC

N

0

0

a

Q/T

b

c

Figure 8.4 The flow and growth of the entropy stream through a power plant operating at maximum power output (Bejan, 1995}.

(8.34) can be used to determine T8 c and TLC graphically, in a construction that requires only a ruler and a compass. The resulting diagram is reproduced in Figure 8.4a, where we used tan a= TJ/{2 -TJ) and 11 = 1 - (TL/T8 ) 112 , cf. Equation (8.32). New in Figure 8.4 is the third drawing, which shows the flow and growth of the entropy current as it proceeds through the power plant (Bejan, 1995). The graphic construction of Figure 8.4c is detailed in Figure 8.4b. These two drawings correspond to the limit Ri ~ oo, where the irreversibility of the power plant of Figure 8.1 is due almost entirely to the two heat exchangers. The abscissas of these new drawings show the entropy transfer rate Q/T as it progresses toward lower temperatures through the engine. The growth of the entropy current is a general feature of energy conversion systems (Smith, 1993). Figure 8.4b shows that the Q/T stream starts as Q8 /T8 as it enters the power plant at T8 • It is then augmented by an amount dictated by the angle a while it flows across the first heat exchanger. Across the inner compartment, the entropy stream remains unchanged, Q8 /T8 c = Q/TLc· Finally, across the second heat exchanger,

215 the entropy stream is augmented again by the irreversibility of the transfer of QL from Tu: to TL. This second augmentation is dictated also by a. The analysis that stands behind the construction of Figure 8.4b is omitted for the sake of brevity. The analytical version of this figure can be deduced from the drawing, for example, by noting that QH!Tnc = ( QH!Tn) (1 +tan a). The construction of Figure 8.4b begins from the top of the drawing, by selecting the unit horizontal length QH!Tn. The construction progresses downward in the direction indicated by the arrows. The construction ends at point M. Note that point P must be selected on the horizontal line such that the arc NM meets the line RM at M. Figure 8.4c is a symmetric version of the entropy stream constructed graphically in Figure 8.4b. The contribution of Figures 8.4b and cis to show that the power plant is a system through which entropy flows from the hot end to the cold end. The entropy stream increases as it traverses the heat exchangers; however, the larger of the two increases occurs across the cold-end heat exchanger, in agreement with the numerical example at the end of the preceding section. This conclusion, however, may seem unexpected because the cold-end heat exchanger has the smaller temperature difference (Tu: - TL < Tn - Tnc) and the smaller heat transfer rate ( QL < Qn ).

8.2 8.2.1

POWER PLANT VIEWED AS AN INSULATION BETWEEN HEAT SOURCE AND AMBIENT Optimal Distribution

of Heat Leak Through the Insulation

In this section we focus exclusively on the irreversibility due to the bypass heat leak Q;, and assume that the irreversibilities of the two heat exchangers are negligible (Bejan, 1995). The resulting power plant model is shown in Figure 8.5a. The power plant has two compartments, the bypass thermal conductance from Tn to Tv and the rest of the plant (the actual cycle), which is irreversibility free. The total heat

Q.H

a

b

Figure 8.5 (a) Power plant model with bypass heat leak to the ambient and (b) extraction of power from the heat that flows through the insulation (Bejan, 1995).

216

input to the power plant On is divided between the power producing compartment, O.n and the leaky insulation, Qill. . The question, again, is under what conditions is the total power output W maximum? Consider first the heat leak through the insulation, and the physical constraint that the insulation thickness (L), cross-section (A;). and effective thermal conductivity (k;) are fixed. If, as in the model of Figure 8.1, the heat leak (}; is conserved from Tn to TL, then the internal resistance R; is simply L/(k;A;). In general, however, the heat leak may vary across the insulation, (!; = 0;(1), because some of the heat leak may be intercepted (and used) by the reversible compartment. This possibility is illustrated in Figure 8.5b. The total power that the reversible compartment can extract from the insulation is (8.35) At every intermediate temperature level T the heat leak is driven by the local temperature gradient, (8.36) where xis the position in the insulation, X= 0 at T= TL, andx =L; at T= Tn. Equation (8.36) can be integrated across the insulation to obtain the physical or fmite-size constraint (constant,

R;)

(8.37)

where it is assumed that k; is temperature independent. The problem consists of finding the function Q;(T) that maximizes the W; integral (8.35) subject to the integral constraint (8.37). The solution is readily available by variational calculus; therefore, we list only the results for the optimal heat leak function and the maximum power, ·

Qi,opt

WunR = _!_In(TH) T i

L

(T)

T =-In _!!_ Ri TL

[T _ H

(8.38)

T _T L

L

Jn(THJ] T

(8.39)

L

Equation (8.39) represents the maximum power that can be extracted from the insulation, i.e., from the internal resistance R;. At this stage it is instructive to C?mpare the maximum power W;m to the heat leak received by R; from Tn, namely Chn,opt = R;- 1Tn In( TH /TJ. The ratio of these quantities is the efficiency of the insulation R; at maximum power,

217

I-T] /

/

/

/

/

/

/

/

/

/

/

/

/

/

/

/

/

Camot

/

/ 0.2 .__..__L......L-L------"----'----'----'--'-..__L......L-J 0.06 0.1

Figure 8.6 The maximum-power efficiency of a power plant viewed as an insulation between its heat source and the ambient, Equation (8.40) (Bejan, 1995).

(8.40)

which has been plotted as 1 - Tl; vs. TL!Tn in Figure 8.6. This is an important plot because it shows that Tl; is comparable to the maximum-power efficiency derived by Novikov, Chambadal, and Curzon and Ahlborn, namely Equation (8.32). The dashed line is the Carnot efficiency 1 - Tl = TL!Tn. The ten points plotted in Figure 8.6 reproduce the reported efficiencies shown earlier in Figure 8.3. In Figure 8.6 we see that Equation (8.40) also agrees with the empirical data. What we have proposed in this section is a new interpretation of the irreversible operation of actual power plants. The agreement between Equation (8.40) and reported data suggests that an actual power plant may be viewed as an obstacle to the direct heat transfer from the heat source Tn to the heat sink Tv i.e., an "insulation" designed to produce maximum power when its overall size (R;) is constrained. Indeed, a power plant acts as an insulation between its heat source and heat sink in the same way that a refrigeration plant insulates its coldest compartment from the ambient.

8.2.2

Optimal Heat Source Temperature

We can now return to the model of Figure 8.5a, in which the insulation resistance R; is in parallel with a reversible cycle. Let us assume that the resistance R; has been optimized according to Equation (8.38) and Figure 8.5b. This means that t~ Wim listed in Equation. (8.39), thC? power plant delivers the in addit~on power W. = Q,n (1 - TL!Tn), where Qen = QH - C?;H,opt. The total power output(W' = wim + W.) can be nondimensionalized as an overall efficiency ratio, (8.41)

218

Where the OVerall beat input conductance,

QH

iS fixed, and

Ci

iS the nondimensional insulation

(8.42) It can be shown numerically that the total power (8.41) bas a maximum with respect to THITL, i.e., with respect to the beat source temperature TH, because TL is fixed. The power is zero in the limit TH =TL. It increases as TH becomes greater than TL; however, when TH is high enough the power starts to decrease because an increasing fraction of QH flows through R;. The existence of such a power maximum was noted first in Problem 2.2, in which the beat leak was constant (conserved) as it flowed through the insulation. In the present case the beat leak itself generates the maximum power of which it is capable, and the maximum W is the absolute limit that the power cycle with bypass conductance can reach. The numerical results are shown in Figure 8.7, in which C. represents the size of the insulation conductance R;-1relative to the overall size of th~ plant ( QH ). For each physical configuration C; we have~ optimal heat source temperature (TH/TL)ap 1 and a maximum total power output ( W/ The beat source temperature and the maximum power decrease as the insulation" conductance increases.

QH) .

8.3

COMBINED-CYCLE POWER PLANT

Now we focus on a most interesting extension of the double maximization of the power output encountered in the study of Figure 8.1. That power plant model is repeated in Figure 8.8a, where it is assumed that the bypass beat leak irreversibility is negligible (R; ~ oo). The question we address is bow to maximize the total power output of a combined-cycle power plant that operates between the same temperatures

(~L

0.05

0.1

c,

Figure 8.7 The optimal heat source temperature and the maximum power output of the power plant with bypass heat leak shown in Figure 8.5a (Bejan, 1995).

219 TH

TH

UHAH O Q H

UHAHOQH THI

c

THe

r---wl

Tu

c

UMAMo~

r--w

Tml TLc

c

TL2

ULALOQL

h

rw,

ULAL O Q L TL

a

b

Figure 8.8 (a) Power plant model with two heat exchangers and (b) combined-cycle power plant with three heat exchangers (Bejan, 1995). , TL) and uses the same heat exchange inventory as the original plant (Bejan, 1995). The combined-cycle power plant of Figure 8.8b has three heat exchangers,

(T8

(8.43) and two reversible compartments, the high-temperature compartment between Tm and TL1 and the low-temperature compartment between Tm and T1.2· The optimization of the power plant sandwich of Figure 8.8a was also considered by Rubin and Andresen (1982). The new aspect considered in this section is the UA constraint (8.43) and the optimal allocation of the UA inventory between the three heat exchangers. This problem can be solved analytically in two steps. In the first step we assume that the temperature level TL1 is fixed, and then we maximize ~ and ~ . The analysis in this first step is the same as in Section 8.1, therefore, we can omit the details. For the part contained between T8 and TL1 we find that the optimal Tm temperature is (8.44)

220 and that the corresponding maximum power output of the upper cycle is

(8.45) For the part contained between TL1 and TL the optimization rules are also known, namely (Tm/TL2 )opt = (TL/TL)112 , and UMAM = ULAv The corresponding maximum power output of the lower cycle is [cf. Equation (8.19)]

(8.46)

In the second step of the analysis we pursue the maximization of lt; + W2 with respect to two parameters, TLI and the relative size of ULAL. By adding Equations (8.45) and (8.46) we obtain

(8.47) Maximizing this with respect to TLIITu we find (8.48) and (8.49)

lti

with respect to ULAL/UA while noting that the Finally, we maximize lt; + constraint (8.43) now reads UA = UuAu + 2ULAv The result is (8.50) which means that the UA inventory must be divided equally between UuA H• UMAM, and ULAL. The maximum power output is (8.51) The conclusion that UA must be divided equally between all the heat exchangers is not too surprising because when we have only two heat exchangers (Figure 8.8a),

221 the same UA allocation rule applies. The efficiency of the combined-cycle optimized in Equation (8.51), (8.52) is the same as the efficiency of the original power plant of Figure 8.8a. Rubin and Andresen (1982) found the same efficiency formula. Equation (8.52) may seem unexpected because one of the main reasons for the development of combined-cycle power plants is the desire to increase the energy conversion efficiency (e.g., Bejan, 1988b, pp. 450 to 454). In conclusion, when the total heat exchange inventory is fixed and the design is optimized for maximum power, the Chambadal-Novikov-CurzonAhlbom efficiency characterizes both plants. Another interesting result is that the maximum power output of the combinedcycle, Equation (8.51), is considerably smaller than the corresponding value of the single-cycle power plant of Figure 8.8a [cf. Equation (8.19)],

(8.53) The ratio between Equations (8.51) and (8.53) is 4/9. To understand this result, we note that when we switch from Figure 8.8a to 8.8b and hold UA constant, the allocation of one third of UA to the new heat exchanger (UMAM) means that we must reduce UnAn and ULAv This in tum leads to a smaller heat input for the power plant of Figure 8.8b. Another reason is that in Figure 8.8a the innermost compartment is reversible, while in Figure 8.8b that compartment contains the irreversibility due to the middle heat exchanger UMAM.

8.4

OPTIMAL COMBUSTION CHAMBER TEMPERATURE

When a power plant is driven by the heating effect provided by a combustion chamber, the power output is maximum when the products of the combustion are at a certain temperature (Bejan, 1988b, pp. 382 to 386). In this section we examine this important case of thermodynamic optimization by using the combustion chamber model shown in Figure 8.9. The reactants enter the chamber at environmental conditions (T0 , P 0 ), combustion occurs, the products of combustion of "flame" temperature 7Jheat the working fluid of the rest of the power plant, and, finally, the products of temperature TP are discharged into the atmosphere. The energy rates Hr, HP' QH, Qv and Ware expressed per mole of fuel burned. Note also that H, and HP are the cumulative enthalpies of the streams of reactants and products. In the model of Figure 8.9 we focus strictly on the irreversibility associated with the combustion chamber and the dumping of products; therefore, we model the rest of the power plant (the square between 1f and T0) as free of irreversibilities. The simplest model for the temperature distribution inside the combustion chamber is to assume that the products of combustion are well mixed, so that their temperature is

222 Combustion chamber

Products

-w Carnot-cycle power plant

Figure 8.9 Power plant driven by heating from a combustion chamber with isothermal products of combustion (Bejan, 1988b).

uniform and equal to that of the exiting products, TP. According to this model, the absolute temperature of the heat source seen by the power cycle can only be TP; therefore,

TI =TP

(8.54)

The adiabatic flame temperature Tat is the highest value that 1j can have and is defined by the first law statement in the limit where Q8 = 0, (8.55) There exists an optimal combustion chamber temperature TP (or T1) that maximizes the output of the power plant, W. Although it is not necessary for making this point, superior illustration is made possible by the additional assumption that in the range (Tar TP)' the mole fractions and the specific heats of all the ideal gas products of combustion are constant. In this case because H, equals the HP value calculated at the adiabatic flame temperature, Equation (8.55), the heat transfer drawn by the power plant from the combustion chamber is

(8.56)

where i indicates products, vpi are the stoichiometric coefficients of the n products, and ( cp)p; is the specific heat of each product, expressed per mole of product. The products constant CP is shorthand for L7= 1 (vcp)p;· Combining Equations (8.54) to (8.56) with W = Q8 (1 - T0 11j) and Q8 = H,- HP, we find that W reaches a peak as TP varies between Tat and T0 :

223 (8.57) namely, (8.58) at an optimal effective flame temperature of (8.59) At flame temperatures higher than this optimum, the work production decreases because the heat input to the power plant (H, - HP) decreases. When the flame temperature decreases below this optimal level, the work output W decreases because of the decreasing Carnot efficiency (1 - T 0 /~). The optimal temperature discovered in Equation (8.59) is a "high" temperature relative to the temperature range spanned by the power cycle itself. For example, if T0 = 298 K and Taf = 3000 K, Equation (8.59) yields Ii,opt = 946 K. If the combustion reaction is followed by dissociation, then the Tat constant of this theory must be replaced by the corresponding temperature (another constant) for adiabatic and zero-work steady-flow combustion followed by dissociation. An alternate combustion chamber model is shown in Figure 8.10. It stems from the question of what might be done to improve the performance of the single-T1 scheme of Figure 8.9 beyond the optimal conditions identified above. One option is to use the still-hot exhaust (HP of Figure 8.9) as an additional heat source in the power cycle. In the new scheme of Figure 8.10 the temperature of the products varies along the combustion chamber. The temperature decrease from Tat to TP is caused by the cooling effect provided by the reversible power plant positioned underneath. In hardware terms the arrangement can be realized by building a counterflow heat exchanger in which the products of combustion flow to the right, while the working fluid of the hot end of the power cycle flows to the left. Anywhere along this counterflow, the temperature difference between the gaseous products and the working fluid is zero. Combustion chamber

Tar Working fluid Reversible power plant

-w

Figure 8.10 Power plant driven by the continuous counterllow heating effect provided by the stream of products of combustion (Bejan, 1988b).

224 The work extracted by the reversible part of the plant from the stream of products is equal to the drop in the flow exergy of that stream,

w= cA:z:f- Tp)- :foCP In;!

(8.60)

p

in which we have assumed once again that in the range Taf to TP the products are ideal gases with temperature-independent specific heats and mole fractions. The important feature of this new W estimate is that it increases monotonically as TP decreases. The maximum W is reached when TP = T0 ,

(:z:/

:z:/)

Wmax =CpTO T -:- l - I n7:0

(8.61)

0

This quantity can be compared to that of Equation (8.58) to see that the counterflow heat exchanger scheme of Figure 8.10 is indeed superior to the best of the singletemperature products scheme of Figure 8.9. I originally derived the optimal flame temperature based on the model of Figure 8.9 while writing my graduate-level textbook (Bejan, 1988b). Professor Feidt (1994) recently drew my attention to a related optimum published by Chambadal (1957, 1958, 1963) for a power plant driven by a stream of hot ideal gas. Actually, Chambadal's analysis corresponds to the model shown in the drawing accompanying Problem 1.2, which was also present in the Bejan (1982) book. A difference exists, however, because the exit temperature of the stream is higher than T0 • Chambadal did not present a drawing of his model. My own interpretation of the physical apparatus that corresponds to Chambadal' s analysis is described in Problem 8.2, which is based on the assumption that the heat exchanger size is infinite. Chambadal showed that the optimal exhaust temperature of the gas is (THT0 ) 112 , where THis the inlet temperature of the hot stream. Furthermore, the first law efficiency of the optimized power plant is 11 = 1 - (T0 /TH) 112 (the same as in Novikov's model). We return to Chambadal's model in Section 8.6.1.

8.5

OTHER POWER PLANT OPTIMIZATION STUDIES

The thread that links the results described in Sections 8.1 to 8.4 is that fundamental thermodynamic trade offs exist in the design and operation of power plants. One recurring theme is that the power output of a power plant can be maximized by dividing the finite supply of heat transfer equipment among the heat transfer components involved. The engineering implications of this step are that if the hardware inventory can be allocated optimally in simple models such as Figures 8.1, 8.5, and 8.8, then an opportunity exists to distribute the hardware optimally in the design of actual, much more complicated, installations. The contribution of simple models and the method of entropy generation minimization is to show the way, to uncover new opportunities for the more applied work that will follow in industrial research and development.

225 The work that has been published on the thermodynamic optimization of power plant models is sizeable. Novikov's model, unlike Chambadal's and Curzon and Ahlborn's, included the effect of irreversible expansion through the turbine of the steam cycle (Problem 8.1). He accounted for this example of "internal" irreversibility through a dimensionless factor. Lu (1980), Grazzini (1991), Ibrahim et al. (1991, 1992), and Wu and Kiang (1992) extended this model by also accounting for the irreversibility of the compression process and for the fact that the temperature of the working fluid changes along the two heat exchangers. Most recently, Petrescu et al. (1994) generalized the model by including the effect of piston speed and type of working fluid. The model of Figure 8.1 was extended by Swanson (1989) to account for the capacity flow rates and effectiveness-Ntu relations of the two heat exchangers. Swanson showed that in this case the optimal thermal conductance xopt allocation ratio depends on the total number of heat transfer units. A similar model was optimized by Ibrahim and Klein (1989) and Lee and Kim (1991, 1992), who extended it further to the optimization of Lorentz cycles, and by Ibrahim et al. ( 1991 ), who also considered the life cycle economic optimization of the power plant. The four-process (two stroke) model of Curzon and Ahlborn (1975) and its steady-state counterpart (introduced by Novikov, 1957, and independently by Andresen et al., 1977a; Lu, 1980; and Figure 5.11 in Bejan, 1982) was pursued along several lines. These were reviewed by Andresen et al. (1984), Wu et al. (1993), and Feidt et al. (1994). In several studies beginning with Gutkowicz-Krusin et al. (1978) the assumption that the heat transfer rates are proportional to the local temperature differences was replaced with more general, nonlinear heat transfer models that account for natural convection, radiation (e.g., Problem 6.3), and temperature-dependent properties (De Vos, 1985; Chen and Yan, 1989; Angulo-Brown and PaezHernandez, 1993). The maximum power efficiency formula (8.32) does not hold when the heat transfer model is not linear. Early studies were also contributed by Rubin (1979), Lucca (1981), Rozonoer and Tsirlin (1983), Mozurkewich and Berry (1983), and Tsirlin (1991). The effect of speed on optimal performance was studied by Spence and Harrison (1985) and Petrescu et al. (1994). There is an important engineering issue to contemplate in connection with the four-process model optimized by Curzon and Ahlborn and their followers in the physics literature. Sadi Camot's 1824 essay, which was interpreted analytically and graphically by Emile Clapeyron 10 years later, told of a gas contained in a cylinder and piston apparatus that underwent a cycle composed of four processes, two quasistatic and isothermal processes interspersed with two quasistatic and adiabatic processes. Curzon and Ahlborn added finite thermal resistances between the cylinder and the respective temperature reservoirs, and in this way described and optimized the time-dependent evolution of the cycle. Although the original cycle described by Camot is a good instrument for classroom teaching, Curzon and Ahlborn's model is a questionable roadmap to improvements in the thermodynamic performance of real heat engines. Recall that the once-maximized power output, Equation (8.17), can be increased further by increasing the thermal conductances associated with the isothermal processes. Can this be accomplished in a real four-process heat engine in which the same cylinder wall is asked to be a perfect insulator during one process and a very

226 good thermal conductor during the next process in the same piston stroke? Engineers faced this question early in the development of practical heat engines. Examples are Watt's 1769 separate condenser, Brayton's 1873 external combustion chamber, and Otto and Langen's 1876 internal combustion engine. The history of the step-by-step improvements in engine thermodynamic performance is illustrated and referenced in Bejan (1988b, pp. 51 and 405). Returning to the review of the current literature, it can be noted that another line of research focused on individual features of the four-process model. Important in this respect is the paper by Band et al. (1981) which dealt with the optimization of the heating process undergone by a fluid in a piston and cylinder apparatus. Richter and Ross (1978) and Fairen and Ross (1981a, b) considered the effect of timedependent operation and inertia. Orlov and Berry (1991) optimized an engine model in which the working fluid is nonisothermal and viscous (with pressure drop) while in contact with the heat reservoirs. The mechanical optimization of the kinematics of engines is an entirely new and interesting direction that is defined by the work of Senft (1987, 1991, 1993). Related to this is the cylindroids rotary engine of Vargas and Florea (1994). The maximization of work output as opposed to power output was pursued by Grazzini and Gori (1987) and Wu et al. (1993). Subtle differences between the maximum power in time-dependent (reciprocating) vs. steady-flow power plant models were clarified by Kiang and Wu (1994). As figure of merit in power plant optimi~tion, Angulo-Brown (1991) proposed to maximize the function TL sgen where sgen is the entropy generation rate of the power plant and TL is the heat sink temperature. Nomenclature innovations included the introduction of the term "endoreversible" (Rubin, 1979) to describe the reversibility of the innermost compartment such as the one shown in Figure 5.11, or alternatively, the term "exoirreversible" for the external irreversibilities that surround the same compartment (Radcenco, 1990). Radcenco also emphasized the equivalence between maximizing power output and minimizing entropy generation rate (see also Radcenco et al., 1993). It should be noted that the concept of internal reversibility (or external irreversibility) is as old as thermodynamics itself because it is the same as the local thermodynamic equilibrium model (Berg, 1992) that serves as the foundation for modern heat transfer and fluid mechanics. The term "finite time" thermodynamics was introduced in 1977 to describe the optimization of thermodynamic processes subjected to time constraints (Andresen et al., 1977b). Alefeld (1989) analyzed an entire steam-cycle modern power station and showed how the entropy generation rates contributed by the components can be summed up to evaluate the overall performance of the power plant. A similar procedure was illustrated using the simple Rankine cycle in Bejan (1988b, pp. 414 to 426). The Rankine cycle power plant was also analyzed and optimized by Wilson and Radwan (1977), Roche (1983), Habib and Zubair (1992), Smith (1992), and Radcenco et al. (1993). Several of these studies emphasized the importance of matching the (variable) temperature of the working fluid to the temperature of the heating agent. A comprehensive treatment of the distribution of sources of entropy generation in a gas-turbine power plant was present by Denton (1993). The effect of turbine blade

w-

227 cooling on entropy generation was investigated by Farina and Donatini (1993). Fundamental studies of power maximization in simple Brayton (Joule) cycles were conducted by Leff (1987), Landsberg and Leff (1989), Bejan (1988a, b), and Wu (1992). Organ (1987a) presented a detailed analysis of the distribution of entropy generation in a Stirling-cycle power plant. This topic and the maximization of power received considerable attention in subsequent papers by Organ (1988), Organ and Jung (1989), Radcenco et al. (1993), Ladas and Ibrahim (1994), and Blanket al. (1994). Reviews of this field were published by Organ (1987b), Reader (1991), and by Organ (1992). A systematic treatment of the Otto, Brayton (Joule), Diesel, and Atkinson cycles was made by Leff (1987) and, in a generalized form, by Landsberg and Leff (1989). The Diesel cycle was optimized for maximum power by Hoffman et al. (1985), and the distribution of entropy generation was studied by Primus and Flynn (1986). Papers on Otto-cycle power plants were written by Mozurkewich and Berry (1982) and Angulo-Brown et al. (1994). Internal combustion power plants were also optimized by Orlov and Berry (1993). Potentially important in practice is the fact that optimized stroke-by-stroke cycles such as the Otto cycle require optimal timedependent piston motions, which, in tum, require optimal kinematics (linkages, shapes) between piston and crankshaft. The entropy generation minimization method was extended to ocean thermal energy conversion, or OTEC power plants, by Johnson (1983) and Wu (1987). The MHD power cycle was treated by Aydin and Yavuz (1993) and Human (1994).

8.6

WHY MAXIMUM POWER MEANS MINIMUM ENTROPY GENERATION RATE

In a letter to the physics community I pointed out that some of the thermodynamics ideas that are presented as new in physics journals appeared earlier in the engineering literature (Bejan, 1994b). This observation applies with peculiar force to the method itself - the maximization of power in heat engine models with heat transfer irreversibilities - because maximum power is equivalent to minimum entropy generation rate [cf. the Gouy-Stodola theorem (2.8), or Problem 2.7]. It has been written, however, that maximum power and minimum entropy generation rate are two distinct optimization criteria for power plants (e.g., Salamon and Nitzan, 1981; Andresen et al., 1984; Andresen, 1990). For this reason I think it is instructive to demonstrate in this section the equivalence between the maximum power and minimum entropy generation rate designs. I do this by using the earliest and simplest models of power plants in which the efficiency at maximum power was found to be 1 - (TL!TH)112 , that is, precisely the class of models that attracted interest in physics.

8.6.1

Chambadal's

Power Plant Model

Consider the power plant shown in Figure 8.11. The heat source is the hot stream The reversible compartment is sandwiched between THe and Tv and is powered by the heating effect provided by a heat exchanger. Chambadal

m of temperature TH.

228

I·

internal irreversibility

------------ c~o~~

external

irreversibility

·I

~~~=~------------------------------------

heat exchanger mcP~~--+-----------------r---~----------------~

compartment

-'--r-----Hi----,-------1. - - - - - - - - - - - - - --"'--- - - - - - - J QL

I

Figure 8.11 The calculation of the total rate of entropy generation in Chambadal's power plant model.

(1957) assumed that the heat exchanger is sufficiently large so that the temperature of the source stream matches the hot-end temperature of the reversible compartment (Tnc), before the stream is discharged into the ambient. The optimization of the power plant has only one degree of freedom: the exhaust temperature Tnc· It is a simple matter to express the power output W as a function of Tnc• and to show that W is maximized when (8.62) The W maximization analysis is proposed as an exercise (Problem 8.2). The energy conversion efficiency at maximum power is (8.63) Now let us derive the same result by minimizing the total entropy generation rate associated with the power plant. One source of entropy generation is the heat exchanger. The other is less obvious, the dumping of the used stream into the ambient. This second irreversibility was featured prominently in the sensible heat storage literature (e.g., Figure 7.2). The dumping of the Tnc-hot stream is an integral part of the optimization process: Tnc is a degree of freedom in the maximization of W only when the exhaust (Tnc) is free to float, i.e., wl!en it is not required (used) by anybody else downstream. The external irreversibility indicated in Figure 8.11 is an essential part of the physics of the power plant. Without it the plant design does not have even a single degree of freedom and cannot be optimized for maximum power. The entropy generation rate associated with the total system is

229

(8.64) where the stream is treated as an ideal gas at constant pressure, and (8.65) Combining Equations (8.64) and (8.65) we obtain (8.66)

This expression shows that the total entropy generation rate has a minimum with respect to THe· Solving sg.J()TIIe = 0 leads straight to Equation (8.62) and the efficiency formula (8.63). Maximum power indeed means minimum entropy generation rate. The same conclusion is reached if the infinite-size beat exchanger of Figure 8.11 is replaced by a finite-size heat exchanger in which the stream is either unmixed or mixed (Problem 8.3). If we had made the mistake of overlooking the external irreversibility in the calculation of the entropy generation rate, we would have calculated only

a

· S·8.... =Sgen

internal

. ( ln THe =mcP - + TH --1 Til

Tile

l

(8.67)

a

This expression also has a minimum with respect to THe· Solving sgen taTHe = 0 yields T11 e,opr = T11 and, as a consequence, TJ = 1 - TL/TH. These results differ from Equations (8.62) and (8.63), not because maximum power and minimum entropy generation are two different designs, but because an error has been made in the calculation of sgen in Equation (8.67).

8.6.2

Novikov's and Curzon and Ahlborn's

Power Plant Model

The simplest power plant model of the Novikov and Curzon and Ahlborn type is shown in the box drawn with heavy line .in Figure 8.12. One finite-size heat exchanger is at the hot end of the power plant, QH = UA(TH- THe), where the thermal conductance UA is fixed. The design has only one degree of freedom represented by the high temperature of the reversible compartment (THe) or by the heat transfer input QH. The variability of QH is an absolutely essential feature on which we focus next. In view of the power maximization analyses presented in Sections 8.1 to 8.4 it is not difficult to maximize the power output Wand to show that the same Equations (8.62) and (8.63) describe the maximum power design of the model of Figure 8.12. A somewhat more complicated version of this maximum-power analysis is proposed in Problem 8.1. The question we face here is whether Equations (8.62) and (8.63) can be derived by minimizing the entropy generation rate associated with the optimization of the power plant.

230

beat exchanger T HC 1----------1

reversible

compartment

----

TL~--------~-

L__

internal irreversibility

~--

external irreversibility~

_____j

Figure 8.12 The calculation of the total rate of entropy generation in a simpler version of Novikov's and Curzon and Ahlborn's power plant model.

In the model of Figure 8.12 we first recognize the heat exchanger as a source of entropy generation because Q11 crosses the finite temperature gap (T11 - T11c). Because the input Q11 must be varied at will to optimize the power plant, however, it. must mean that at the designer's disposal at T11 is not Q11 but a higher heat input Q, such that always

Q> Q11

(

Q= constant)

(8.68)

This inequality represents the degree of freedom that means that Q11 (and T 11c) can be varied. This freedom also means that the difference ( Q - Q11 ) must not used the _optin).ifor any other purpose because it might be needed to augment Q11 du~g zation process. When not used, the external (standby) heat transfer Q. = Q - Q11 is rejected to the ambient, as shown in Figure 8.12, or in the optimization of power plants driven by solar collectors (Bejan, 1988b, pp. 503 to 505). In conclusion, a power plant that must have a free-floating heat input must also have an external irreversibility associated with the unused portion of the heat input. This feature is illustrated in Figure 8.12 and helps us calculate the total entropy generation rate sgen

=

sgen

internal

+S gen

external

·(1 1) ·(1 1)

=Q

11

- - - +Q - - TIIC ~I e TL Til

(8.69)

231

After using

Q.

=

Q - Q8 and QII

= UA(T8 - T8c) we obtain

1_

l)

Sgen =UA(T.II -T.1/C ) (T. - __!_)+Q·(__!_ __ T. T. T. HC L L II

(8.70)

a

where the last term is a constant. Finally, by solving Sgen/dTIIc = 0 we obtain Equation (8.62), reconfirmation of the general theorem that maximum power means minimum entropy generation rate. It is tempting, of course, to overlook the external irreversibility and minimize only the visible entropy generation rate due to the heat exchanger,

s.gen = s.gcn

internal.

= UA

(r.T

_lL

\

HC

+ T.TIIC - 2 )

(8.71)

H

This expression certainly has a minimum with respect to T8 c, located at Tnc.opt = T8 , which is different than the maximum power design (8.62). It is different because Equation (8.71) is incorrect. It denie~ an essential part of the physics of the optimization process, namely, the fact that QII must be free to vary. To deny that maximum power is the same as minimum entropy generation rate is to deny that the power plant can be optimized. It is important to note that to account for Q. and Q > Q8 in Figure 8.12 does not mean that the power plant model has been "altered" into one with bypass heat leak, as . in Figure 8.1. The excess heat input Q can be made infinitesimally smaller* tqan QH.opt after the optimiz~tion process has been completed: The main point is that Q. was on standby so th.at QII could be varied en route to QII,opt. Incidentally, the same can be said about Q8 in Figure 8.1. To be able to vary it, we must have access to a larger heat input Q from T8 • In other words, Figure 8.1 contains four sources of entropy generation: the two heat exchangers, the bypass heat leak, and the rejection of the unused heat input Q - QII to the ambient

.

8.6.3

The "Room to Move" Entropy Generation

Rate

The common message of the entropy generation minimization analyses centered on Figures 8.11 and 8.12 is that a degree of freedom in the design requires an additional source of entropy generation on the outside of the visible confines of the power plant. This additional irreversibility is what gives the design room to move. My first encounter with this principle was in 1976 in Problem 2.2, where to vary the heat source temperature T8 meant that the heat input QII was available from a fixed temperature level always higher than TH. Another example from Bcjan and Paynter (1976) is Problem VIII G, which is also exhibited here as Problem 1.3. That example shows that the minimization of the power input to a refrigeration plant is equivalent to the design with minimum entropy generation rare, when the external entropy generation rate associated with the existence

* By constructing a sufficiently large insulation resistance.

232 of a refrigeration load is also taken into account. The load first enters the cold space from room temperature as a heat leak and generates entropy. The fact that the maximization of and the minimization of the total sgen le~d to the same optimal design does not mean that we should rush and replace all W analyses with Sgen analyses. Each of us is free to choose the method (Bejan, 1984). In the case of power plants it is easier to explain the optimization process in terms of W maximization. In the optimization of solar and cryogenic installations and many basic components such as heat exchangers, fins, ducts, and storage elements the method that brings these applications together is the minimization of the entropy generation rate.

w

8.7

MAXIMUM POWER FROM FLUID FLOW

This section considers the fundamental thermodynamic problem of how to extract maximum instantaneous power from a fluid flow driven from a high-pressure reservoir (P 1) to a low-pressure reservoir (P2). The corresponding maximum power problem of heat engines operating with thermal resistances between two temperature reservoirs has been treated in great detail in the literature, and forms the subject of Problem 8.4. The objectives of the following analysis are 1. To extend to the field of fluid power conversion the thermodynamic optimization principles developed for thermal power conversion 2. To show the analogy between the fluid and thermal maximum power designs 3. To illustrate the physical meaning of the heat engine power maximum by using the purely mechanical analog and language of the flow driven between two pressure reservoirs 4. To illustrate the physical meaning of the fluid power maximum by optimizing actual steady-flow shaft-work components such as turbines, compressors, and pumps

8.7.1

Fluid Flow between

Two Pressure

Reservoirs

Consider the piston and cylinder apparatus shown in Figure 8.13. The piston moves with friction under the influence of the pressure difference P 1c- P 2c maintained across its two faces. The instantaneous power delivered by the piston to an external system (not shown) is (8.72)

where A, A1 , and V are the piston frontal area, the lateral (friction) area, and the instantaneous speed. The ratio 1118 accounts for Couette flow in the relative motion gap of thickness 8, where 11 is the viscosity of the lubricant. The piston inertia is assumed negligible. The working fluid experiences a pressure drop (P 1 - P 1c) as it is admitted from the reservoir P 1 to the chamber on the driven side of the piston. Similarly, the fluid ejected from the chamber positioned on the driving side of the piston experiences

233

Figure 8.13 Piston and cylinder apparatus for extracting mechanical power from the flow of a fluid between two pressure reservoirs.

another pressure drop, (P 2c- P 2). By analogy with the simplest beat transfer model used in power maximization studies of thermal energy conversion, Equations (8.1) and (8.2), we assume that the pressure differences are proportional to the respective flow rates, which in tum are proportional to V: ~ -~c

=RIV

(8.73)

~c-~

=R2V

(8.74)

In these expressions R1 and R2 are the two instantaneous fluid resistances. The fluid is being assumed incompressible on both sides of the piston. The linear model (8.73) to (8.74) is appropriate for laminar flow such as in capillary ducts for micromecbanical energy converters. A model for higher Reynolds number flows is presented in Section 8.7.4. The reservoir pressures P 1 and P 2 are fixed, however, P tc and P 2c depend on the piston speed, which is the only degree of freedom in the operation of the mechanical energy conversion device shown in Figure 8.13. The optimal speed for maximum instantaneous power delivery can be obtained by eliminating P 1c and V between Equations (8.73) and (8.74), substituting into Equation (8.72), and solving the equation oW /oP 2c = 0. The result for the optimal downstream pressure is (8.75) where AP1 = A1 Jli(A8) is the pressure drop due solely to piston-cylinder friction. The remaining parts of the maximum-power solution are obtained by substituting Equation (8.75) back into Equations (8.72) to (8.74):

P.

P2 + (1 + 2 R2 fR 1 )~

---=-~___,.---=.:.....__:~

IC,opt -

_+ AP1

2 {1+ R2 / Rl)

___t_

(8.76)

(8.77)

(8.78)

234 One way to interpret the maximum power condition is to calculate the pressure difference across the piston, and compare it to the overall pressure difference: (8.79) We learn that in the limit of negligible piston friction the pressure difference across the piston must be exactly half of the reservoir-to-reservoir pressure difference. Symmetry exists between this result and the corresponding result for a power plant sandwiched between two thermal resistances (Figure 8.14). Pressure differences play the role of absolute temperature ratios. Furthermore, 1/ 2 appears as a factor in Equation (8.79) and as an exponent in the case of a thermal power plant optimized for maximum power [Equation (8.32)]. If the optimization on the right side of Figure 8.14 is called by some finite-time thermodynamics, then the left side of the same figure is an introduction to "finite-time fluidodynamics", which is the mission of this entire section. Another way is to compare the optimal instantaneous speed (Vopt) with the piston speed in the limit of zero power delivery (V0). The latter is obtained by combining W = 0 with Equations (8.72) to (8.74), (8.80) Equations (8.77) and (8.80) show that the piston speed at maximum power is exactly one half of the piston speed at zero power, Vopt = V0 /2, regardless of whether piston friction is negligible.

8.7.2

Efficiency at Maximum Power

The work transfer rate received from the P 1 reservoir is P 1AV. This quantity is analogous to the heat transfer rate Q1 of the corresponding heat engine model, Figure 8.14. In the reversible limit (llP1 =0, R1 =0, R2 =0) the mechanical power drawn from P 1AV and delivered by the moving piston is only "':ev = (P 1 - P 2 )AV because the portion P 2 AVis being absorbed by the P2 reservoir. The power conversion efficiency in the reversible limit is _ w;ev -}- ~ 'Tlrev - P.AV P. 1

(8.81)

I

The symmetry between llrev and the Carnot efficiency of a heat engine (1- T2 /T1) is evident. The conversion efficiency of the device of Figure 8.13 under conditions of maximum power delivery is Tl

_ max

w

max

P.AV I

opt

(8.82)

235

Mechanical power conversion (41'f =0)

Thermomecbanical power conversion

(T'T2Cc) =(!!.) T2

112

opt

TJ,...=l-11

Figure 8.14 The analogy between the maximum power conditions for fluid power conversion vs. thermal power conversion.

or, after using Equations (8.77) and (8.78),

11

= max

_!_(1- p2 2

~

llP,) ~

(8.83)

The maximum power efficiency is exactly one half of the reversible-limit efficiency when piston friction is negligible. This case is compared in Figure 8.14 to the maximum-power efficiency of a power plant, 1 - (T2 /T1) 112• Again, as in the frrst line of the table, the 1/ 2 factor of the formula for fluid power conversion becomes an exponent in the formula for thermal power conversion. In general, the effect of increasing piston friction is to shift the maximum power design toward lower llmax• wmax. and vopt·

8.7.3

Overall Size Constraints

Another issue that relates the two columns of Figure 8.14 concerns the overall size constraint that must be faced by the design of the actual device. On the heat engine side of the figure, this issue has been studied extensively for the purpose of determining the optimal allocation of a finite heat transfer area (or thermal

236

L ---------------+

~--------------

(a)

(b)

Figure 8.15 Examples of overall size constraints. (a) Fixed total length and pressure reservoirs on opposite sides of the piston; (b) fixed total length and thickness and pressure reservoirs on the same side of the piston.

conductance) between the two heat exchangers (Section 8.1). In the case of the general fluid power converter shown in Figure 8.13 the impact of the overall size constraint depends on the shape (layout) of the system. To illustrate this point, consider the long and thin tube of length L shown in Figure 8.15 and assume that APt= 0. The instantaneous position of the piston divides L into two sections, the driving fluid column L 1 and the driven column ~· The corresponding flow resistances of these two sections are (8.84)

where C is a Hagen-Poiseuille flow constant that depends on viscosity and tube diameter. The important observation is that Cis the same on both sides of the piston, which means that

R1 + R2 = CL,

(constant)

(8.85)

In conclusion, Equations (8.77) and (8.78) show that Vopt and Wmax do not depend on the position occupied by the piston along L. In other words, in the device of Figure 8.15a no optimum exists with respect to the way in which Lis divided into L 1 and ~· The absence of such an optimum distinguishes the mechanical device of Figure 8.15a from the heat engine model optimized in Section 8.1.2. In the second example, Figure 8.15b, we continue to assume that APt= 0. This time the two pressure reservoirs are positioned to the left of the piston, while the two fluid columns are oriented in counterflow. Let us assume that the overall size of the device is fixed, L x D, and that the fluid columns flow through narrow parallel-plate channels of spacing D 1 and D 2 • The overall thickness constraint

237 D 1 + D 2 =D.

(constant)

(8.86)

makes the relative thickness x of one channel the only degree of freedom, (8.87) Assuming that the piston is close to the end-tum region, and that the flow is of the Hagen-Poiseuille type in both channels, we have (8.88) (8.89) where V2 is the mean velocity in the D 2 channel. Mass conservation requires VD 1 = V2D 2; after comparing Equations (8.88) and (8.89) to Equations (8.73) and (8.74) we conclude that the flow resistances are and

(8.90)

To further maximize Wmax of Equation (8.78) we must minimize (R 1 + R 2 ), which is equivalent to minimizing the function (D) 2 + D/Di), or [x-2 + x/(1 - x)3]. The optimal relative spacing is xopt = 0.454, which means that the downstream channel should be 20% wider than the upstream channel. The optimal spacings D 1 and D 2 are not equal because the geometry of Figure 8.15b is not symmetric with respect to the two channels. (The piston is inside one of the channels, the D 1 channel.)

8.7.4

Nonlinear Flow Resistance

Relations

The power maximization principles discussed up to now also apply at higher Reynolds numbers, where the linear flow resistance model (8.73) and (8.74) is replaced by nonlinear relations. A general model that relates the pressure drops to the instantaneous flow rate is (8.91) (8.92)

where 1 < n < 2 and (r1, r~ are constant coefficients that depend on duct geometry and fluid properties. As in turbulent flow through a duct with rough walls, through an orifice, or through a porous medium, the n exponent increases as the Reynolds number increases. The n = 1 limit represents the regime analyzed in the preceding sections.

238 Let us assume that at high Reynolds numbers the piston friction term is negligible on the right side of Equation (8.72). Combining Equation (8.72) with Equations (8.91) and (8.92) we find that the operation at maximum power is described by

P.

~(1 + 2r2f'i)- ~[n2+ (n -1)1jjr2] 2 (1 + r2f'i)

(8.93)

~ + P2[n+(n+1)r 1 jr2] 2 (1+1jfr2)

(8.94)

-~~~~~~~--------~~ IC,opt -

P.

2C,opt

=--'---='""--,-------'---,.-:.:~

(8.95)

(8.96)

The maximum power output can be calculated using The efficiency at maximum power is given by

Tlmax

Wmax

= (P 1c- P 2 c)optAVopt·

= wmax =.!_{1[n+(n-1)3..]} 2 r p2

~AVopt

~

(8.97)

2

which shows that Tlmax decreases as n increases. Finally, Equation (8.97) can be restated as a second law efficiency at maximum power,

Tl ll,max = Tlmax 'll 'lrev

=

_l)_L]

.!_[1(n -1) (1 + 2 r, P.-P. 2 I 2

(8.98)

In then= 1limit the second law efficiency is equal to 1/ 2, which can also be seen by dividing Equation (8.83) by Equation (8.81) when AP1 =0. The second law efficiency drops below 1/ 2 as n becomes > 1.

8.7.5

Applications

to Steady-Flow

Shaft-Work Components

The maximization of power extraction from fluid flow has potential applications in the optimization of power plants. This opportunity was pointed out by Radcenco (1994), who analyzed the effect of finite inlet and outlet flow resistances in internal combustion engines, gas turbine power plants, and reciprocating compressors and expanders by assuming linear and nonlinear pressure drop relations and isentropic expansion and compression. These research opportunities are illustrated in Problems 8.6 and 8.7 by extending the simple model of Figure 8.13 to steady-state shaft-work machines such as turbines, compressors, and pumps.

239 One interesting conclusion is that compressors and pumps do not have a minimum power input with respect to the inlet or outlet pressure drop or to the pressure ratio across the inner compartment of the model. The absence of such an optimum is analogous to the absence of an optimal temperature ratio across the inner (reversible) compartment of a refrigerator with warm-end and cold-end thermal resistances (Section 10.3). This analogy is due to the fact that both compressors and refrigerators are power consumers. In contrast to these, turbines and power plants have optimal inner pressure and temperature ratios because they are producers ofpower. The applications of the principle of maximum power from fluid flow deserve to be explored in future optimizations of flow components and complete power plants that contain such components.

REFERENCES Alefeld, G. 1989. The COP of thermal power stations derived from the second law. ASME AES. 10-2:61-70. Andresen, B. 1990. Finite-time thermodynamics. In Finite-Time Thermodynamics and Thermoeconomics. S. Sieniutycz and P. Salamon, Eds. p. 66-94. Taylor & Francis, New York. Andresen, B., Salamon, P., and Berry, R. S. 1977a. Thermodynamics in finite time: extremals for imperfect engines. J. Chem. Phys. 66:1571-1577. Andresen, B., Berry, R. S., Nitzan, A., and Salamon, P. 1977b. Thermodynamics in finite time. I. The step Carnot cycle. Phys. Rev. A. 15:2086-2093. Andresen, B., Salamon, P. and Berry, R. S. 1984. Thermodynamics in finite time. Phys. Today. September: 62-70. Angulo-Brown, F. 1991. An ecological optimization criterion for finite-time heat engines. J. Appl. Phys. 69:7465-7469. Angulo-Brown, F. and Paez-Hernandez, R. 1993. Endoreversible thermal cycle with nonlinear heat transfer law. J. Appl. Phys. 74:2216-2219. Angulo-Brown, F., Fernandez-Betanzos, J. and Diaz-Pico, C. A. 1994. Compression ratio of an optimized air standard Otto-cycle model. Eur. J. Phys. 15:38-42. Aydin, M. and Yavuz, H. 1993. Application of finite-time thermodynamics to MHO power cycles. Energy. 18:907-911. Band, Y. B., Kafri, 0., and Salamon, P. 1981. Optimal heating of the working fluid in a cylinder equipped with a moving piston. J. Appl. Phys. 52:3745-3749. Bejan, A. 1979. A general variational principle for thermal insulation system design. Int. J. Heat Mass Transfer. 22:219-228. Bejan, A. 1982. Entropy Generation through Heat and Fluid Flow. Wiley, New York. Bejan, A. 1984. Convection Heat Transfer. p. 53. Wiley, New York (2nd ed., 1995, p. 55). Bejan, A. 1988a. Theory of heat transfer-irreversible power plants. Int. J. Heat Mass Transfer. 31:1211-1218. Bejan, A. 1988b. Advanced Engineering Thermodynamics. Wiley, New York. Bejan, A., 1993. Power and refrigeration plants for minimum heat exchanger size. J. Energy Res. Technol. 115:148-150. Bejan, A. 1994a. Power generation and refrigeration models with heat transfer irreversibilities. Thermal Sci. Eng. 33(128):68-75. Bejan, A. 1994b. Engineering advances on finite-time thermodynamics. Am J. Phys. 62:11-12. Bejan, A. 1995. Theory of heat transfer-irreversible power plants. IT. The optimal allocation of heat exchange equipment. Int. J. Heat Mass Transfer. 38:433-444.

240 Bejan, A. and Paynter, H. M. 1976. Solved Problems in Thermodynamics. Problem VII D. Massachusetts Institute of Technology, Cambridge. Berg, C. 1992. Conceptual issues in energy efficiency. Proc. lnJ. Symp. Efficiency, Costs, Optimization and Simulation of Energy Systems (ECOS '92). pp. 7-16. Zaragoza, Spain. Blank, D. A., Davis, G. W., and Wu, C. 1994. Power optimization of an endoreversible Stirling cycle with regeneration. Energy. 19:125-133. Chambadal, P. 1957. Les Centrales Nucleaires. pp. 41-58. Armand Colin, Paris. Chambadal, P. 1958. Le choix du cycle thermique dans une usine generatrice nucleaire. Rev. Gen. Electr. 67:332-345. Chambadal, P. 1963. Evolution et Applications du Concept d'Entropie. Section 30. Dunod, Paris. Chen, L. and Yan, Z. 1989. The effect of heat transfer law on performance of a two-heat-source endoreversible cycle. J. Chem. Phys. 90:3740-3743. Curzon, F. L. and Ahlborn, B. 1975. Efficiency of a Camot engine at maximum power output. Am. J. Phys. 43:22-24. DeVos, A. 1985. Efficiency of some heat engines at maximum-power conditions. Am. J. Phys. 53:570-573. Denton, J. D. 1993. Loss mechanisms in turbomachines. J. Turbomach. 115:621-656. El-Wakil, M. M. 1962. Nuclear Power Engineering. pp. 162-165. McGraw-Hill, New York. El-Wakil, M. M. 1971. Nuclear Energy Conversion. pp. 31-35, International Textbook Company, Scranton, PA (2nd ed., 1982). Fairen, V. and Ross, J. 1981a. On the efficiency of thermal engines with power output: consideration of inertial effects. J. Chem. Phys. 75:5485-5489. Fairen, V. and Ross, J. 1981b. On the efficiency of thermal engines with power output: harmonically driven engines. J. Chem. Phys. 75:5490-5496. Farina, F. and Donatini, F. 1993. Second Law Approach to the Analysis of Blade Cooling Effects on Gas Turbine Performance. ASME paper No. 93-JPGC-GT-7. Feidt, M. 1994. Private communication to A. Bejan. Feidt, M., Ramany-Bala, P., and Benelmir, R. 1994. Synthesis, unification and generalization of preceding finite time thermodynamic studies devoted to Carnot cycles. In FLOWERS '94, Proc. Florence World Energy Res. Symp. E. Carnevalle, G. Manfrida, and F. Martelli, Eds. pp. 429-439. SGEditoriali, Padova. Gordon, J. M. and Huleihil, M. 1992. General performance characteristics of real heat engines. J. Appl. Phys. 72:829-837. Gordon, J. M. and Orlov, V. N. 1993. Performance characteristics of endoreversible chemical engines. J. Appl. Phys. 74:5303-5309. Grazzini, G. 1991. Work from irreversible heat engines. Energy. 16:747-755. Grazzini, G. and Gori, F. 1987. Influence of thermal irreversibilities on work producing systems. Rev. Gen. Therm. 312:637-639. Gutkowicz-Krusin, D., Procaccia, I., and Ross, J. 1978. On the efficiency of rate processes. Power and efficiency of heat engines. J. Chem. Phys. 69:3898-3906. Habib, M.A. and Zubair, S.M. 1992. Second-law-based thermodynamic analysis of regenerative-reheat Rankine-cycle power plants. Energy. 17:295-301. Hoffman, K. H., Watowich, S. J., and Berry, R. S. 1985. Optimal paths for thermodynamic systems: the ideal diesel cycle. J. Appl. Phys. 58:2125-2134. Human, M. 1994. Second law analysis of a liquid metal magnetohydrodynamic power system. ASMEAES.33:113-123. Ibrahim, 0. M. and Klein, S. A. 1989. Optimum power of Carnot and Lorentz cycles. ASME HTD. 124:91-96. Ibrahim, 0. M., Klein, S. A., and Mitchell, J. W. 1991. Optimum heat power cycles for specified boundary conditions. J. Eng. Gas Turbines Power. 113:514-521.

241 Ibrahim, 0. M., Klein, S. A., and Mitchell, J. W. 1992. Effects of irreversibility and economics on the performance of a heat engine. J. Solar Energy Eng. 114:267-271. Johnson, D. H. 1983. The exergy of the ocean thermal resource and analysis of second-law efficiencies of idealized ocean thermal energy conversion power cycles. Energy. 8:927946. Kiang, R. L. and Wu, C. 1994. Clarification of finite-time thermodynamic-cycle analysis. Int. J. Power Energy Syst. 14:68-71. Ladas, H. G. and Ibrahim, 0. M. 1994. Finite-time view of the Stirling engine. Energy. 19:837-843. Landsberg, P. T. and Leff, H. S. 1989. Thermodynamic cycles with nearly universal maximum-work efficiencies. J. Phys. A: Math. Gen. 22:4019--4026. Lee, W. Y. and Kim, S. S. 1991. Power optimization of an irreversible heat engine. Energy. 16:1051-1058. Lee, W. Y. and Kim, S. S. 1992. The maximum power from a finite reservoir for a Lorenz cycle. Energy. 17:275-281. Leff, H. S. 1987. Thermal efficiency at maximum work output: new results for old heat engines. Am. J. Phys. 55:602-610. Lu, P.-C. 1980. On optimum disposal of waste heat. Energy. 5:993-998. Lucca, G. 1981. Un corollario al teorema di Carnot. Cond. Aria Rise. Refrig. January:19--42. Mozurkewich, M. and Berry, R. S. 1982. Optimal paths for thermodynamic systems: the ideal Otto cycle. J. Appl. Phys. 53:34--42. Mozurkewich, M. and Berry, R. S. 1983. Optimization of a heat engine based on a dissipative system. J. Appl. Phys. 54:3651-3661. Novikov, I. I. 1958. The efficiency of atomic power stations. J. Nucl. Energy II. 7:125-128; translated from 1957. At. Energ. 3(11):409. Novikov, I. I. 1984. Thennodynamics. Mashinostroenie, Moscow. Novikov, 1.1. and Voskresenskii, K. D. 1977. Thermodynamics and Heat Transfer. Atomizdat, Moscow. Odum, H. T. and Pinkerton, R. C. 1955. Time's speed regulator: the optimum efficiency for maximum power output in physical and biological systems. Am. Sci. 43(1):331-343. Organ, A. J. 1987a. Thermodynamic design of Stirling cycle machines. Proc. Inst. Mech. Eng. 201(C2):107-116. Organ, A. J. 1987b. Thermodynamic analysis of the Stirling cycle machine- a review of the literature. Proc. Inst. Mech. Eng. 201(C6):381--402. Organ, A. J. 1988. Two-dimensional thermodynamic flow analysis of the simple hot-air engine. Proc. lnst. Mech. Eng. 202(C1):31-38. Organ, A. J. and Jung, P. S. 1989. The Stirling cycle as a linear wave phenomenon. Proc. /nst. Meek Eng. 203:301-309. Organ, A. J. 1992. Thermodynamics and Gas Dynamics of the Stirling Cycle Machine. Cambridge University Press, Cambridge, U.K. Orlov, V. N. and Berry, R. S. 1991. Power output from an irreversible heat engine with a nonuniform working fluid. Phys. Rev. A. 42:7230-7235. Orlov, V. N. and Berry, R. S. 1993. Power and efficiency limits for internal combustion engines via methods of finite-time thermodynamics. J. Appl. Phys. 74:4317--4322. Pathria, R. K., Nulton, J. D., and Salamon, P. 1993. Carnot-like processes in finite time. IT. Applications to model cycles. Am. J. Phys. 61:916-924. Petrescu, S., Harman, C., and Bejan, A. 1994. The Carnot cycle with external and internal irreversibilities. In FLOWERS '94, Proc. Florence World Energy Res. Symp. E. Carnevale, G. Manfrida, and F. Martelli, Eds. pp. 441--452. SGEditoriali, Padova. Primus, R. J. and Flynn, P. F. 1986. The assessment of losses in diesel engines using second law analysis. ASMEAES. 2-3:61-68.

242 Radcenco, V. 1990. Definition of exergy, heat and mass fluxes based on the principle of minimum entropy generation. Rev. Chim. 41:40-49 (in Romanian). Radcenco, V. 1994. Generalized Thermodynamics. Editura Tehnica, Bucharest (in English). Radcenco, V., Popescu, G., Apostol, V., and Feidt, M. 1993. Thermodynamique en temps fini appliquee aux machines motrices. Etudes de cas: machine a vapeur et moteur de Stirling. Rev. Gen. Therm. Fr. 382:509-514. Reader, G. T. 1991. Performance criteria for Stirling engines at maximum power output. /SEC, 5th Int. Stirling Engine Conf, pp. 353-359, Dubrovnik, Yugoslavia. Richter, P. H. and Ross, J. 1978. The efficiency of engines operating around a steady state at finite frequencies. J. Chern. Phys. 69:5521-5531. Roche, M. 1983. Recuperation de chaleur. Le cycle thermodynamique ideal. Rev. Gen. Thenn. Fr. 260-261:561-571. Rozonoer, L. I. and Tsirlin, A. M. 1983. Optimal control of thermodynamic processes ill. Avtom. Telemekh. 44:50-64. Rubin, M. H. 1979. Optimal configuration of a class of irreversible heat engines. I. Phys. Rev. A. 19:1272-1276. Rubin, M. H. and Andresen, B. 1982. Optimal staging of endoreversible engines. J. Appl. Phys. 53:1-7. Salamon, P. and Nitzan, A. 1981. Finite time optimization of a Newton's law Camot cycle. J. Chern. Phys. 74:3546-3560. Senft, J. R. 1987. Mechanical efficiency of kinematic heat engines. J. Franklin lnst. 324:273290. Senft, J. R. 1991. Pressurization effects in kinematic heat engines. J. Franklin /nst. 328:255279. Senft, J. R. 1993. General analysis of the mechanical efficiency of reciprocating heat engines. J. Franklin Inst. 330:967-984. Smith, I. K. 1992. Matching and work ratio in elementary thermal power plant theory. Proc. lnst. Mech. Eng. 206:257-262. Smith, J. L., Jr. 1993. Entropy flow and generation in energy conversion systems. ASMEHTD. 266:181-188. Spence, R. D. and Harrison, M. J. 1985. Speed dependence of the efficiency of heat engines. Am. J. Phys. 53:890-893. Swanson, L. W. 1989. Thermodynamic optimization of irreversible power cycles with constant external reservoir temperatures. ASME HTD. 125:31-37. Tsirlin, A. M. 1991. Optimal control of irreversible heat and mass transfer processes. Tekh. Kybernet. 2:171-177. Vargas, J. V. C. and Florea, R. 1994. Analysis of a cylindroids rotary engine. In Heavy Duty Engines: A Look at the Future. ASMEICE. 22:215-229. Vukalovich, M.P. and Novikov, I. I. 1972. Thermodynamics. Mashinostroenie, Moscow. Wilson, S. S. and Radwan, M.S. 1977. Appropriate thermodynamics for heat engine analysis and design. Int. J. Mech. Eng. Ed. 5:68-80. Wu, C. 1987. A performance bound for real OTEC heat engines. Ocean Eng. 14:349-354. Wu, C. 1992. Power optimization in a finite-time closed gas-turbine power plant. Int. J. Energy Environ. Econ. 2:57-62. Wu, C. and Kiang, R. L. 1992. Finite-time thermodynamic analysis of a Camot engine with internal irreversibility. Energy. 17:1173-1178. Wu, C., Kiang, R. L., Lopardo, V. J., and Karpouzian, G. N. 1993. Finite-time thermodynamics and endoreversible heat engines. Int. J. Mech. Eng. Ed. 21:337-346.

243 PROBLEMS 8.1. Novikov's steady-state model of an irreversible power plant is shown below. The heat exchanger of finite thermal conductance UA drives the heat transfer rate Q, into the working fluid, which is heated at constant temperature from (b) to (c). The fluid is expanded irreversibly from (c) to (d). Novikov accounted for this irreversibility by writi~g (sd - Sa) = (1 + i) (Sd.rev - s.), Or Q2 = (1 + i) Q2 rev' where (1 + i) ~ 1, and Q2 is the heat transferred to the ambient T2 • The rest or'the power plant operates reversibly. Determine the optimal hot-end temperature T1c for maximum power. Show that the efficiency at maximum power is

_r:::-1power plant heat exchanger

reversible

compartment

~.!I

.....

.I~

8.2. Chambadal's optimization of the steady-state power plant can be reproduced by considering the model shown below. The power plant is powered by a stream of hot single-phase fluid of inlet temperature T1 and constant specific heat cr The power plant model consists of two compartments. The lower compartment sandwiched between the surface of temperature T2 and the ambient T0 operates reversibly. The area of the heat transfer surface T2 is infinite, and, as a consequence, the outlet temperature of the stream is equal to T2• Determine the optimal hot-end temperature T2 such that the instantaneous power output W is maximized. Show that when the power is maximum the efficiency TJ = W I Q is equal to 1- (T0 /T1) 112• max power plant heat exchanger

mcP

----t---------+---

reversible

compartment

244 8.3. Chambadal's maximum-power efficiency is based on an analysis in which the heat exchanger area is treated as infinite (Problem 8.2). Show that the same efficiency formula holds when the heat exchanger size is finite. The stream is unmixed, the finite thermal conductance between the stream and the T2 surface is specified (UA), and the outlet temperature varies according to UA in the figure below (part a). The second model (b) is the same as the first, except that the stream is well mixed at the temperature Tout inside the heat exchanger. power plant

power plant TI

rilcp

beat exchanger

Tout

-----t---------+--

TI

rilcp

beat exchanpr

-+----1

UA

reversible

compartment

a

reversible

compartment

b

8.4. The power plant model shown in the next figure contains three compartments: the hot-end heat exchanger, an inner compartment that houses the circulating working fluid, and the cold-end heat exchanger. Assume that the heat transfer rates are proportional to the respective temperature differences,

and that the thermal resistance R characterizes both heat exchangers. Assume also that the inner compartment operates reversibly. Derive an expression for the instantaneous power output Was a function of T 1c, T1, T2 , and R. Maximize W with respect to T1c apd show that T1ciT2c = (T1/T2) 112 • Show further that the efficiency 11 = W/Q1 at maximum power is equal to 1 - (T2 /T1) 112 • Finally, calculate the ratio of the entropy generated in the hot heat exchanger divided by the entropy generated in the cold heat exchanger, and show that the cold heat exchanger generates more entropy.

245

R heat exchanger Tic 1 - - - - - - - - - - - - - l

reversible

compartment

T2c 1 - - - - - - - - - - - 1 heat exchanger

8.5. The conclusions reached in Section 8.1 were made possible by the simplicity of the power plant model proposed in Figure 8.1. More realistic models require more complex analytical and numerical work. For illustration, consider the ideal Brayton cycle shown in the figure below. The working fluid (ideal gas with constant cP) is heated at constant pressure PH as it flows from the compressor outlet (1) to the turbine inlet (2). The compressor and the turbine process the fluid reversibly and adiabatically, i.e., isentropically. Between the turbine outlet (3) and the compressor inlet (4), the working fluid is cooled at constant pressure Pv The power plant is sandwiched between two temperature reservoirs, TH and TL. The Brayton cycle is said to be "ideal" because the only irreversibilities accounted for by this model are external to the space occupied by the working fluid itself. These irreversibilities are associated with the heat transfer from (T8 ) to the heater tube (1) ~ (2), and with the heat transfer from the cooler (3) ~ (4) to the low-temperature reservoir (TL). Assume that the total thermal conductance UA of the two heat exchangers is fixed. Show analytically that the entropy generation rate of the power plant is minimized when UA is divided equally between the heat exchangers.

T

I

I

246 8.6. Consider the adiabatic steady flow turbine shown below. An ideal gas of flow rate mexpands from P 1 to P 2, as shown on the attached T-s diagram. The stream experiences the pressure drop !J.P 1 as it is ducted and distributed to the first turbine stage. It then expands through the turbine stage (or sequence of stages) with the isentropic efficiency 11 1 < 1. The final pressure drop is due to the discharge and ducting of the stream to the next component in the power plant (the cooler, or condenser). Assume that the relations between m and pressure drops are nonlinear, !J.P 1 = r1thn and !J.P 2 = r2 mn, where r 1 and r 2 are the inlet and outlet flow resistances. Determine the optimal pressure drops and flow rate for maximum turbine power output. Compare the maximum power with the power (at the same flow rate) in the limit of zero pressure drops (r 1 = r2 = 0). T

T2 T 2,rev

8.7. The steady-flow adiabatic compressor shown below has pressure drops at the inlet and the outlet, !J.P 1 = 1jthn and M 2 = r2 mn. The isentropic efficiency of the compressor stage (or sequence of stages) is Tlc < 1. The fluid is a ideal gas. Derive an expression for the compressor power input Wc as a function of M 1, and show • that W, does not have a minimum with respect to M 1 (or m, or !J.P2).

247 8.8. In Section 8.6.2 we used a power plant model with one heat exchanger to demonstrate that maximizing the power output W is equivalent to minimizing the total entropy generation rate Sg•n. Consider the model shown below, where the hot-end and cold-end thermal conductances are generally not equal. Recognize the availability of a larger heat input Q such that QH can be varied. Derive an expression for the total rate of entropy generation rate, and show that the minimization of sgen is analytically the same as the maximization of

w.

r:

I__

1

Q~~T~-- - - -

~'! ~y~t~~-- - - - -

~~ D~

internal irreversibility

external

irreversibility~

_____]

8.9. In the power plant model shown in the following figure the heat input QH is jaed. The power plant can be modeled as a reversible compartment sandwiched between two heat exchangers, Qn = (UAMTn-Tnc) and QL = (UAMTLc-TL). The total conductance inventory is fixed, (UA)n + (UA)L = UA. Show that to maximize W we have only one degree of freedom, namely the conductance allocation ratio x = (UA)u!UA. Show further that the maximum power and efficiency are described by

~max

Ql/

=

1_TL [1 _4QH )-I ~I

THUA

and provide in this way an alternative explanation as to why the reported efficiencies (Figure 8.6) fall at a certain level below the Camot limit (1- TL!Tn).

248 This explanation is at least ~l'> plausible m; the other theories given in Chapter 8, because real power plants are designed for steady power production at ftxed heat (fuel rate) input.

heat exchanger (UA)H

reversible

compartment

heat exchanger (UA)L

~

9

SOLAR-THERMAL POWER GENERATION

In this chapter we consider the thermodynamic optimization of power plants that receive their heat input from solar radiation. The treatment is limited to solar-thermal applications, that is, power plants in which the radiative input has been converted by a collector into heat transfer to the hot end of the power cycle. The direct (photovoltaic) conversion of solar radiation and the ideal thermodynamic limit (exergy content of solar radiation) have been reviewed on several occasions, and are discussed further in Section 9.8. The solar-thermal power plant models constructed in this chapter continue the sequence begun in Problem 2.2 and Chapter 8, and reinforce the trends unveiled by that sequence. For this reason, as well as brevity, the focus in this chapter is on the optimal allocation of heat transfer equipment between the components of the installation and the time-dependent strategies for operating the power plant during the daily cycle.

9.1 9.1.1

MODELS WITH COLLECTOR HEAT LOSS TO THE AMBIENT Convective

Heat Loss

The most basic thermodynamic aspects of the operation of a power plant driven by a solar collector are illustrated by considering the collector model shown in Figure 9.1 (Bejan et al., 1981). The collector has a cross-sectional areaAc that receives solar radiation at the rate Q* from the sun. The net solar heat transfer Q* is proportional to the collector cross-section Ac, the proportionality factor being q* [W/m2], which varies with geographical position on the earth, the orientation of the collector, meteorological conditions, and the time of day. In the present analysis it is assumed that q* is a constant and that the system is in steady state. This constraint is relaxed in Section 9.7. The incident solar radiation is partly delivered to a user (e.g., power cycle) as heat transfer Q at the collector temperature Tc. The remaining fraction, Q0 , represents the collector-ambient heat loss, ~ =Q· -Q

(9.1)

Equation (9 .1) represents the energy conservation statement for the solar collector as a system in steady state. As an approximation for "low temperature" collectors, we

249

250

n

T.

}il

___SL._ Ac

----

Oo

-

e

a·=q·Ac

--a

''

~O,Tc

~o

reversible power plant

''

''

'''

0

''

'

T]c

b

Figure 9.1 Power plant model with collector heat loss to the ambient (a) and the collector efficiency according to the linear heat loss model (b) (Bejan et al., 1981 ).

assume that the heat loss Q0 is proportional to the collector-ambient temperature difference and to the collector cross-section, or (9.2)

where Uc is the overall heat transfer coefficient based on Ac; note that Uc is a characteristic constant of the collector. Combining Equations (9.1) and (9.2), it is apparent that the maximum collector temperature occurs when Q =0, i.e., when the entire solar heat transfer Q* is lost to the ambient. The maximum collector temperature is given in dimensionless form by (9.3)

where Tc,lilJJJ( is also known as the stagnation temperature of the collector. In dimensionless form the collector temperature 9 = TJT0 will vary between 1 and elilJJJ(, depending on the heat delivery rate Q. The stagnation temperature emax is the parameter that describes the performance of the collector with regard to collectorambient heat loss. According to the above model, the collector efficiency Tlc is a linear function of collector temperature e. (9.4)

The collector efficiency is plotted on the right side of Figure 9.1, showing that the point of maximum efficiency (Tlc = 1) corresponds to a collector in thermal equilibrium with the environment. At this point the heat transfer Q carries zero exergy, or zero potential for mechanical power production.

251

9.1.2

Maximum Power or Minimum Entropy Generation

Rate

The operation of any solar collector is thermodynamically irreversible due to three main features, namely, the sun-.

0.4

0.3 5

0

Figure 10.11

No

10

The optimal allocation of a fixed total thermal conductance model of Figure 10.10 (Radcenco et al., 1995).

in the refrigerator

(10.40) This rule is general, and holds for all values of C0 and Cv It is important to note that only under special conditions is Klein's optimization rule (10.40) the same as the U0 A0 = UA rule recommended by the total thermal conductance constraint. Only when the heat exchangers are sufficiently small that (10.41) do Equations (10.33) and (10.34) approach £ 0 = U0 A0 /C 0 and EL = UA/CL, such that Equation (10.40) becomes identical to Equation (10.38). At the opposite extreme, when the surfaces are sufficiently large that (10.42) the rule (10.40) reduces to C0 = Cv which is certainly not the same as Equation (10.38). Furthermore, because C0 = CL says nothing about the optimal allocation of heat exchange hardware between condenser and evaporator, it is appropriate to question the physical meaning of the C constraint postulated in Equation (10.39). Finally, we tum our attention to how the area constraint

Aa +A= Atot'

(fixed)

(10.43)

influences the optimal design of the refrigerator modeled in Figure 10.10. By minimizing numerically F with respect to x =A0 /A 101 , Xopt emerges as a function of C0 /CL, U0 /U, and

N

= UoAtot tot

Co

(10.44)

294 Figure 10.12 shows that the optimal A0 is generally not equal to A. The optimal way of dividing A101 among the heat exchangers depends on the total surface available (N10 J, and the relative magnitude of the external parameters of the heat exchangers, namely the ratios CLIC0 and UIU0 • The results of the optimization subject to the A101 constraint (Figure 10.12) attain a simple form in the limit in which the two heat exchanger surfaces are bathed by isothermal (well mixed) fluids, e.g., ambient air at T0 and TL. This limit is the same as setting (N0 , NL) --+ 0 in the model of Figure 10.1 0, and the minimization ofF with respect to x = A0 IA101 yields reviewed specifically in Problem 8.1. For example, if the power cycle executed between Tmax and T0 is reversible, the instantaneous power output W is maximized if Tmax has the constant value (11.20) The fact that Tmax.opt is constant and independent of t1 is an important feature that must

be kept in mind as we tum our attention to the optimal selection of the second free

parameter, t1• The instantaneous power output maximized with respect to 0 - ]· . TJ 11 ( 1 - -T.W= Q

Tmax

is (11.21)

Tmax,opt

where the second law efficiency constant TJn accounts for the assumed irreversibility of the power cycle, and where Tmax,opr is a constant depending on T0 , Tn, and Tln· By combining Equations (11.18) to ( 11.21) it can be shown that the time-averaged power output is given by the expression

~=TJu(1-~] t! + (2

Tmax,opt

(Tn- Tmax,opr) hA·F(O,Bi.)

(11.22)

where

. ) _ In( I+ e Bi.) F (e,Bt. - ( ) . 1 + e n,.

(11.23)

and

h b t2 Bi = - • k

(11.24)

332 0.8 Bi. =0.1

0.6

F

0.4

0.2

0

Figure 11.13

0

2

6

3

a

The effect of the time of operation and the rate of fouling on the average power output (Bejan et al., 1994).

Noteworthy in these definitions are the nondimensional time interval of power plant operation, 8, and the nondimensional rate of fouling, Bi•. The effect of 8 and Bi. on the average power output is conveyed by the function F(S, Bi.), which is shown plotted in Figure 11.13. The average power output reaches a maximum at a distinct operating time eopt• Figure 11.14 shows how the group Bi. influences the optimal operating time and the corresponding maximum average power. The Bi. number may be seen as a way of nondimensionalizing the known cleaning time t 2 • The figure shows that as the cleaning time decreases the maximum average power and the time ratio eopt increase. The optimal time interval of power plant operation, t 1,opt• is proportional to the product eopt • Bi •. The dashed line in Figure 11.14 shows that this time interval decreases as the cleaning time decrea is repeated many times. The question we address next is what is the optimal freezing time t1 so that the time-averaged rate of ice production is maximum? To show that an optimal freezing time exists, it is a good idea to begin with the simplest freezing model possible. We assume therefore that the ice layer is sufficiently thin relative to the radius of curvature of the wall, so that the wall may be regarded as plane. Accordingly, the thickness of the ice layer at the end of the freezing interval is (Bejan, 1993), (11.51) where the constant b is shorthand for [2 k (Tm- Tw) /p hS/] 112• It is also assumed that the Stefan number c (Tm - Tw) I hst is smaller than 1. The properties p, c, and k are ice properties. The evolution of the ice layer thickness o(t) and the freezing and removal production cycle are shown qualitatively in Figure 11.23. Our objective is to maximize the amount of ice produced over the entire duration of one cycle (11.52)

343

ice layer

o(t)

Figure 11.23 The periodic production of ice: the freezing time t1 , followed by the ice removal time~-

The lone degree of freedom is the freezing time t 1, or its dimensionless counterpart (11.53) By solving

dS/dt = 0 we find 't P = 1, or 0 1

(11.54) In conclusion, the optimal freezing time interval of the cycle must be as long as the ice removal interval. This conclusion will change somewhat as we include in the model additional features such as the wall curvature (e.g., freezing inside a tube), the fmite heat transfer coefficient between the coolant and the outer surface of the water container, and the dependence between the ice removal time and the final thickness of the ice layer. The essential point made by this simple analysis is that an optimal freezing time exists, and that a