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Entropy generation minimization: The new thermodynamics of finitesizedevices and finitetime processesAdrian Bejan Citation: J. Appl. Phys. 79, 1191 (1996); doi: 10.1063/1.362674 View online: http://dx.doi.org/10.1063/1.362674 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v79/i3 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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APPLIED PHYSICS REVIEWS

Entropy generation minimization: The new thermodynamicsof finite-size devices and finite-time processes

Adrian BejanJ. A. Jones Professor of Mechanical Engineering, Department of Mechanical Engineering and MaterialsScience, Duke University, Durham, North Carolina 27708-0300

~Received 12 January 1995; accepted for publication 4 October 1995!

Entropy generation minimization~finite time thermodynamics, or thermodynamic optimization! isthe method that combines into simple models the most basic concepts of heat transfer, fluidmechanics, and thermodynamics. These simple models are used in the optimization of real~irreversible! devices and processes, subject to finite-size and finite-time constraints. The reviewtraces the development and adoption of the method in several sectors of mainstream thermalengineering and science: cryogenics, heat transfer, education, storage systems, solar power plants,nuclear and fossil power plants, and refrigerators. Emphasis is placed on the fundamental andtechnological importance of the optimization method and its results, the pedagogical merits of themethod, and the chronological development of the field. ©1996 American Institute of Physics.@S0021-8979~96!04103-9#

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TABLE OF CONTENTS

I. Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1191II. The method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192III. Cryogenics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193IV. Heat transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197V. Storage systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . .1201VI. Solar power plants. . . . . . . . . . . . . . . . . . . . . . . . . .1203VII. Nuclear and fossil power plants. . . . . . . . . . . . . . . 1206VIII. Refrigeration plants. . . . . . . . . . . . . . . . . . . . . . . .1212IX. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1214

I. OBJECTIVES

Entropy generation minimization~EGM! is the methodof modeling and optimization of real devices that owe thethermodynamic imperfection to heat transfer, mass transand fluid flow irreversibilities. It is also known as ‘‘thermodynamic optimization’’ in engineering, where it was first developed, or more recently as ‘‘finite time thermodynamicsin the physics literature. The method combines from the stthe most basic principles of thermodynamics, heat transand fluid mechanics, and covers the interdisciplinary dompictured in Fig. 1. The most exciting and promising interdiciplinary aspect of the method is that it also combines rsearch interests from engineering and physics.

The objectives of the optimization work may differ fromone application to the next, for example, minimization oentropy generation in heat exchangers,1 maximization ofpower output in power plants,2–8 maximization of an eco-logical benefit,9 and minimization of cost.10 Common inthese applications is the use of models that feature rate pcesses~heat transfer, mass transfer, fluid flow!, the finitesizes of actual devices, and the finite times or finite speedsreal processes. The optimization is then carried out subjec

J. Appl. Phys. 79 (3), 1 February 1996 0021-8979/96/79(3)/11

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physical ~palpable, visible! constraints that are in fact responsible for the irreversible operation of the device. Tcombined heat transfer and thermodynamics model ‘‘visuizes’’ for the analyst the irreversible nature of the devicFrom an educational standpoint, the optimization of suchmodel gives us a feel for the otherwise abstract concepentropy generation, specifically where and how much of itbeing generated, how it flows, and how it impacts thermodnamic performance.

The emergence of a new field of research is markedthe appearance of several fundamental results that holdentire classes of known and future applications. Althouisolated publications had appeared throughout the 1950s1960s, thermodynamic optimization emerged as a sstanding method and field in the 1970s in engineering, wapplications notably in cryogenics, heat transfer engineersolar energy conversion, and education. These first devements were reviewed in Ref. 1, and Refs. 11–13.

The field has experienced tremendous growth during1980s and 1990s. The objective of this article is to reviewfield, and to place its growth in perspective. The explosioninterest that we are witnessing today is due to three ndevelopments: the diversification of the problems tackledengineering after the energy policies of the 1970s, the liftof the Iron Curtain and the absorption of work done by prviously unrecognized pioneers, and the contributions thatpear in the physics literature. The field today is so large aactive that the author is forced to focus this review on juthree aspects, which were stressed by the editors in toriginal invitation:

~1! The fundamental and technological implications of tresults obtained until now.

~2! The pedagogical merits of the method, i.e., how EG

119191/28/$6.00 © 1996 American Institute of Physics

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can make thermodynamicsand heat transfer easier toteach, understand, and apply in actual technical work

~3! The chronological development of the field, based oncombined survey of the engineering and physics literture.

II. THE METHOD

It is instructive to begin the review with a brief look awhy in EGM we need to rely on heat transfer and fluid mchanics, not just thermodynamics. Consider the genesystem–environment configuration shown in Fig. 2. The sytem operates in the unsteady state, and its instantaneouventories of mass, energy, and entropy areM , E, andS. Thesystem experiences the net work transfer rateW, heat trans-fer rates (Q0 ,Q1 ,...,Qn) with n11 temperature reservoirs(T0 ,T1 ,...,Tn), and mass flow rates~min , mout! through anynumber of inlet and outlet ports. For simplicity, only oninlet and one outlet are illustrated in Fig. 2. Noteworthy

FIG. 1. The interdisciplinary field covered by the method of entropy geeration minimization~from Ref. 1!.

1192 J. Appl. Phys., Vol. 79, No. 3, 1 February 1996

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this array of interactions is the heat transfer rate betweensystem and the atmospheric temperature reservoir,Q0, onwhich we focus shortly.

The thermodynamics of the system of Fig. 2 consistsaccounting for the first law and the second law~e.g., Ref.14!,

dE

dt5(

i50

n

Qi2W1(in

mh°2(out

mh°, ~1!

Sgen5dS

dt2(

i50

nQi

Ti2(

inms1(

outms>0, ~2!

whereh° is shorthand for the sum of specific enthalpy, knetic energy, and potential energy of a particular streamthe boundary. In Eq.~2! the total entropy generation rateSgenis simply a definition~notation! for the entire quantity on theleft-hand side of the inequality sign. Soon, we will recognizthat it is advantageous to decreaseSgen, and this can be ac-complished only by changing at least one of the quantiti~properties, interactions! specified along the system boundary.

We selectQ0 as the interaction that is always allowed tfloat asSgenvaries. Historically, this choice was inspired~andjustified! by applications to power plants and refrigeratioplants, because the rejection of heat to the atmosphere wano or little consequence in the overall cost analysis of tdesign. EliminatingQ0 between Eqs.~1! and ~2! we obtain

W52d

dt~E2T0S!1(

i51

n S 12T0Ti

D Qi

1(in

m~h°2T0s!2(out

m~h°2T0s!2T0Sgen. ~3!

The work transfer rate in the limit of reversible operatio~Sgen50! is clearly

n-

FIG. 2. General work transfer, heat transfer, and mass flows between a system and its environment in the unsteady state~from Ref. 14!.

Appl. Phys. Rev.: Adrian Bejan

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Wrev52d

dt~E2T0S!1(

i51

n S 12T0Ti

D Qi

1(in

m~h°2T0s!2(out

m~h°2T0s!. ~4!

In engineering thermodynamics, each of the terms onright-hand side of Eq.~4! is recognized as an exergy of ontype or another~e.g., Ref. 14, Chap. 3!, and the calculationof Wrev is known as exergy analysis. Subtracting Eq.~3!from Eq. ~4! we arrive at the formula

Wrev2W5T0Sgen ~5!

which for most of this century in engineering has beknown as the Gouy–Stodola theorem.1,15,16

Pure thermodynamics~e.g., exergy analysis! ends, andEGM begins, with Eq.~5!. The lost power~Wrev2W! is al-ways positive, regardless of whether the system is a poproducer~e.g., power plant! or a power user~e.g., refrigera-tion plant!. Equation~5! states that the destroyed powerproportional to the total rate of entropy generation. If enneering systems and their components are to operatethat their destruction of work is minimized, then the concetual design of such systems and components must beginthe minimization of entropy generation~Ref. 1, pp. 25,33!.

The critical new aspect of the EGM method—the aspthat makes the use of thermodynamics insufficient, andtinguishes it from exergy analysis—is theminimizationofthe calculated entropy generation rate. To minimize the irversibility of a proposed design the analyst must userelations between temperature differences and heat tranrates, and between pressure differences and mass flow rHe or she must relate the degree of thermodynamic nonality of the design to thephysicalcharacteristics of the system, namely to finite dimensions, shapes, materials, finspeeds, and finite-time intervals of operation. For thisanalyst must rely on heat transfer and fluid mechanics pciples, in addition to thermodynamics. Only by varying oor more of the physical characteristics of the system, cananalyst bring the design closer to the operation characterby minimum entropy generation subject to finite-size afinite-time constraints.

To explain how the following review was structured, itpointed out that the optimization of power and refrigeratisystems has a long and established tradition in engineeThe portion that is based on exergy analysis is voluminoand has been reviewed on several occasions.14,17–23Energyanalysis is distinct from EGM, and is not the object of tpresent review. In fact, to calculateSgen and minimize it, theanalyst does not need to rely on the concept of exergy~Ref.1, pp. 25,33!.

The examples that are presented next illustrate hEGM blends thermodynamics with heat transfer and flumechanics in the optimization of real systems. The minimentropy generation design~Sgen,min! is determined for eachmodel, as in the case of the optimal diameter of a duct wflow and heat transfer, or the optimal hot-end temperaturea power plant with a bypass heat leak to the ambient. Tapproach of any other design~Sgen! to the limit of ‘‘realistic’’

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thermodynamic ideality represented by the design with minmum entropy generation~Sgen,min! is monitored in terms ofthe entropy generation numberNS5Sgen/Sgen,min>1, or alter-natives of the same ratio.1

As a final comment on Fig. 1, note that thermodynamicis a ‘‘foundation’’ that should be visible~i.e., present, andtaught and used! in related disciplines such as heat transfeand thermodynamics. Historically, however, thermodynamicwas formulated after heat transfer, and long after mechan~e.g., Ref. 1!. The interdisciplinary domain that is now beingmapped by the research on EGM or finite-time thermodynamics is finally bridging the gap between thermodynamicand the other thermofluid engineering disciplines. Thiswhy the developments reviewed in this article not only havtechnological relevance but also fundamental and pedagocal value.

III. CRYOGENICS

The field of low temperature refrigeration was the firswhere irreversibility minimization became an establishemethod of optimization and design. As a special applicatioof Eq. ~5!, it is easy to prove that the power required to keea cold space cold is equal to the total rate of entropy genetion times the ambient temperature, with the observation ththe entropy generation rate includes the contribution maby the leakage of heat fromT0 into the cold space. Thestructure of a cryogenic system is in fact dominated by components that leak heat, e.g., mechanical supports, radiatshields, electric cables, and counterflow heat exchangeThe minimization of entropy generation along a heat leapath consists of optimizing the path in harmony24 with therest of the refrigerator of liquefier.

Figure 3 shows a mechanical support of lengthL thatconnects the cold end of the machine~TL! to room tempera-ture ~TH!. The rate of entropy generation inside the supposhown as a vertical column is

Sgen5ETL

TH Q

T2dT, ~6!

FIG. 3. Mechanical support with variable heat leak and intermediate coolineffect ~after Ref. 24!.

1193Appl. Phys. Rev.: Adrian Bejan

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1

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where it is important to note that the heat leakQ is allowedto vary with the local temperatureT. The origin of the inte-grand in Eq. ~6! is the infinitesimal element~shaded inFig. 3!, in which the rate of entropy generationis dSgen5Q/T1dQ/T2(Q1dQ)/(T1dT)5QdT/T2, be-causedT!T. The local heat leak decrementdQ is removedby the rest of the installation, which is modeled as reversibThe heat leak is also related to the local temperature gradand conduction cross-sectionA,

Q5kAdT

dx, ~7!

where the thermal conductivityk(T) decreases toward lowtemperatures. Rearranged and integrated from end to eEq. ~7! places a size constraint on the unknown functioQ(T),

L

A5E

TL

TH k

QdT. ~8!

According to variational calculus~e.g., Ref. 14!, the heatleak function that minimizes theSgen integral ~6! subject tothe finite-size constraint~8! is obtained by finding the extre-mum of the aggregate integral*TL

THFdT whose integrandF isa linear combination of the integrands of Eqs.~6! and ~8!,F5Q/T21lk/Q, andl is a Lagrange multiplier. The Eulerequation reduces in this case to]F/]Q50, which yieldsQopt5(lk)1/2T. The Lagrange multiplier is finally deter-mined by substitutingQopt into the finite-size constraint~8!.The results are24

Qopt5SAL ETL

TH k1/2

TdTD k1/2T, ~9!

Sgen,min5A

L S ETL

TH k1/2

TdTD 2. ~10!

Equation~6! was provided by thermodynamics and Eq~8! by heat transfer: together they prescribe the optimal dsign @Eqs.~9! and ~10!#, which is characterized by a certaindistribution of intermediate cooling effect (dQ/dT)opt. Anyother design,Q(T), will generate more entropy and will re-quire more power in order to maintain the cold end of thsupport atTL . Quantitative examples are given in Refs.13, and 24. Together, Eqs.~6! and~8! illustrate the method ofthermodynamic optimization subject to a physical constraiand, to paraphrase some of the more recent physics termiogy, they constitute one of the earliest examples of ‘‘finitsize thermodynamics,’’ this in 1974 in engineering.

The technological applications of the variable heat leoptimization principle are numerous and important. In thcase of a mechanical support, the optimal design is appromated in practice by placing a stream of cold helium gascounterflow ~and in thermal contact! with the conductionpath, Fig. 4. The heat leak varies asdQ/dT5mcp , wheremcp is the capacity flow rate of the stream. The practicvalue of the theory@Eqs. ~9! and ~10!# is that it guides thedesigner to an optimal flow rate for minimum entropy geeration. To illustrate, if the support conductivity is temper



le.ient

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ture independent, then the optimal flow rate ismopt5(Ak/Lcp)ln(TH/TL). In reality the conductivity ofcryogenic structural materials varies strongly with the temperature, and the single-stream intermediate cooling tecnique can approachSgen,minonly approximately. The optimalflow rates to be used in conjunction with common structuramaterials are reported in Ref. 25.

The fabrication of the heat exchanger between the suport and the coolant is less difficult and more economic if thcontinuous contact~Fig. 4! is replaced by a succession ofdiscrete cooling stations~Fig. 5!. The optimal heat leak func-tion required by Eq.~9! must be approximated by a stepwisevarying function. The problem consists of determining noonly the optimal cooling rate for each station, but also thoptimal spatial position of each station. Hilal and Boom26

solved this problem using the method of Lagrange multiplers, and reported optimal structural designs for several supconducting coil applications of very large scales.

The optimization of conduction heat leak paths was considered in more general circumstances by Bisio.27,28An evenmore practical technique for optimizing the support of a vessel filled with cryogen is to place the natural stream of vapo~boil off! in contact with the support at a number of coolingstations~Fig. 6!. The optimization problem is also simplerbecause it reduces to finding only the optimal positions o

FIG. 4. The intermediate cooling effect provided by a cold stream in couterflow with the conduction current through a structural support~fromRef. 25!.

FIG. 5. Structural support cooled at several locations~from Ref. 26!.



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the cooling stations. The optimal design~Sgen,min! corre-sponds to the minimum boil off rate, or the minimum coend heat leak into the vessel. This problem was solvedLagrange multipliers in Ref. 29, which also reports the opmal positions and boil off rates for designs with up to sintermediate cooling stations and structural materials wk/T5constant.

The thermodynamic optimization of a stack of radiatioshields in vacuum, Fig. 7, follows essentially the same roas in Eqs.~6!–~10!. To see this at a glance, assume that

FIG. 6. Structural support with discrete stations for helium boil off cooli~from Ref. 1!.

FIG. 7. Discrete intermediate cooling of a stack of radiation shields~fromRef. 14!.



dbyti-ixith

ntehe

number of shields~N21! is sufficiently large such that thenet radiation heat transfer between two adjacent shieldsapproximately1,14

Qi5sAF~Ti114 2Ti

4!>4sAFTi3 DTi

D i, ~11!

whereDTi5Ti112Ti , D i5( i11)2 i andA, F, ands arethe shield area, effective view factor, and Stefan–Boltzmaconstant. Integrating Eq.~11! across the stack we obtain afinite-size constraint that replaces Eq.~8!,

N

A5E

TL

T0 4sT3F

QdT, ~12!

whereT0 is the same as the room temperatureTH used ear-lier. The result of minimizing the entropy generation ratintegral ~6! is an optimal radiation heat leak variationQopt(T), i.e., an optimal way of cooling each shield. Thepractical techniques of providing this intermediate effec~single stream, discrete stations, boil off! have also been ap-plied to stacks of radiation shields. The optimization efforequires computer assistance when the number of shieldssmall and the approximation made in Eq.~11! does not hold.Martynovskii et al.30 determined the optimal intermediatecooling ~and optimal shield temperatures! for minimum en-tropy generation rate in stacks with one, two, and threshields. Eyssa and Okasha31 optimized stacks of radiationshields where the space between shields is filled with supinsulation. Chato and Khodadadi32 minimized the entropygeneration rate in stacks with shield-to-shield spaces occpied by a structural material. Further aspects of the thermdynamic optimization of cryogenic insulation and suppotechniques are presented in Chenet al.33

Superconducting devices must communicate with throom temperature environment not only mechanically balso electrically. For example, the electric cables~currentleads! that connect the liquid helium temperature windings tthe room temperature network can be modeled as a heat cducting path of lengthL, cross-sectionA, and constant ther-mal conductivityk. In addition, the cable conducts a totaelectric currentI , against an electric resistivityre . The factthat the electric power dissipated via Joule heating in thcable is inversely proportional toA, and that the conductiveheat leak is proportional toA, guarantees the existence of anoptimal cable cross section for which the two sourcesirreversibility add up to a minimum. Furthermore, when thcable geometry and material are specified, it is possibleequip the cable with the optimal intermediate cooling effeso that the refrigerator power needed to keep the cable cis minimized.

The optimization method is the same as in Eqs.~6!–~10!.The optimal intermediate cooling regime for cryogenicables was developed by Agsten34 who minimized the totalpower~refrigerator power1electric power! required to oper-ate the cooled cable. When the single stream cooling tecnique of Fig. 4 is used, the optimal flow rate of cold heliumgas is1,35

moptcpL

kA5 ln

THTL

11

ln~TH /TL!S ILL 0

1/2

kA D 2, ~13!

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where L052.4531028 ~W/A K !2 is the constant in theWiedemann–Franz law,L05kre/T. An additional advantageto using the flow rate~13! is that it guarantees the thermstability of the cable.36 The optimal cable cross section fominimum entropy generation rate is1

Aopt5ILL 0

1/2

k ln~TH /TL!. ~14!

The current cable with Joule heating and longitudinheat leak is just one of the optimization application probleat the interface between cryogenics and electrical engining. Another wide class of applications deals with the coing of heat generating electric components that must opeat a certain temperature. Examples are the supercondutransition,37 superconducting windings for stationary manets and rotating machines,38–40 and electronic packages icomputers.41,42

Another class of engineering components that has boptimized based on the irreversibility minimization priciples ~6!–~10! is the counterflow heat exchangers that conect the coldest regions of refrigerators and liquefiers toroom temperature compressor.43 The counterflow sketched inFig. 8 was intentionally oriented and labeled in the same was the mechanical support of Fig. 3 to stress the anabetween the two devices. The entropy generation rate aciated with the two streams and the space between them~Fig.8! is given by Eq.~6! in which Q is now the longitudinalconvective heat leak

Q5mcpDT ~15!

andDT is the transversal~stream-to-stream! temperature dif-ference. The equivalent of the finite-size constraint~8! isobtained by writing that the enthalpy gained by the coldstream is equal to the local stream-to-stream heat tranrate,

mcpdT5~pdx!UDT, ~16!

FIG. 8. The longitudinal convective heat leak carried by a counterflow hexchanger~from Ref. 43!.


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wherep is the perimeter of the duct that carriesm, andU isthe overall heat transfer coefficient based onp. The totalstream-to-stream heat transfer areapL is fixed. We accountfor this constraint by eliminatingDT between Eqs.~15! and~16!, and integrating fromx50 to x5L,

pL5ETL

TH mcp

QUdT. ~17!

The integral constraint~17! plays the same role as Eq.~8!,and the result of the EGM analysis is again an optimal heleak variation,

Qopt5S mcpUpLlnTHTL

D mcpT, ~18!

Sgen,min5S mcpUpLlnTHTL

D 2UA ~19!

or an optimal regime of intermediate cooling, (dQ/dT)opt.The way this cooling effect is built into the practical

design of the counterflow heat exchanger is by bleedingfraction (me) of the high pressure stream, expanding it inwork producing device~cylinder and piston, or turbine!, andusing this cold stream to cool the counterflow heat exchanger, Fig. 9. The optimal flow rate through the expandis known from Eq.~18! andme,optcp 5 (dQ/dT)opt. Whenthe pressure ratioPH/PL is not large enough for the ex-panded fractionme to become as cold as the cold end of thcounterflow (TL), the engineering solution is to install two orthree expanders along the counterflow. The optimizationthe temperature locations of such a sequence of expanderdescribed in Ref. 14. Another fundamental developmentthe optimal temperature staging of cryogenic refrigerators.44

One interesting characteristic of the counterflow heat echanger with optimal intermediate cooling effect is that thstream-to-stream temperature differenceDTopt decreasesproportionally withT, Fig. 9,

S DT

T Dopt

5mcpUA

lnTHTL

. ~20!

This rule follows from Eqs.~15! and ~18! and is widelyrecognized in the design of cryogenic counterflow heaexchangers.45 Another interesting aspect is that the convective heat leak~18! and entropy generation rate~19! decreasewhen the stream-to-stream thermal conductanceUpL in-creases. In other words, by promoting heat transfer in thtransversal direction, counterflow heat exchangers serveeffective insulationsin the longitudinal direction.

The structure of any low temperature installation contains more than one heat leak path. The intermediate cooliregime for parallel heat leak paths has been optimized46

based on the EGM approach presented in this section. Oconclusion is that parallel heat leak paths must be fitted idividually with optimal continuous distributions of interme-diate cooling. References such as 24, 30, and 45 are furthexamples of earlyfinite size thermodynamicswork in engi-neering.

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FIG. 9. The optimal intermediate cooling of a counterflow heat exchanger~from Ref. 14!.

of

IV. HEAT TRANSFER

The field of heat transfer engineering adopted the teniques developed in cryogenics and applied them to mclasses of devices for promoting heat transfer. The optimtion was carried out at two levels of complexity: complecomponents~e.g., heat exchangers!, and elemental feature~e.g., fins, ducts!. The field is vast, therefore in this sectiowe review only some of the most basic examples.

Consider first the flow of a single-phase streammthrough a heat exchanger tube of internal diameterD. Theheat transfer rate per unit tube length, between the tubeand the stream,q8, is given. The entropy generation rate punit tube length is47,48

Sgen8 5q82

pkT2Nu1

32m3f

p2r2TD5 , ~21!

where the Nusselt number Nu5hD/k is a nondimensionaway of expressing the wall–stream heat transfer coeffich. Similarly, the friction factorf accounts for the frictionapressure drop along the tube,f5(2dP/dx)rD/(2G2),whereG5m/(pD2/4). The propertiesr, T, and k are thebulk density, temperature, and thermal conductivity offluid, andx increases in the downstream direction.

The Nusselt number is a result taken from the fieldheat transfer, e.g., Nu50.023 ReD

0.8Pr0.4 when 2500,ReD,106, ReD5VD/n andV5m/(rpD2/4). The frictionfactor is a result taken from fluid mechanics, namef50.046 ReD

20.2 when 23104,ReD,106. In this way Eq.~21! shows at a glance the meaning of Fig. 1, i.e., how thmodynamics is combined from the start with heat transand fluid mechanics in the calculation and minimizationSgen8 . The first term on the right-hand side is the contributi

9, No. 3, 1 February 1996

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er-ferofon

made by heat transfer,Sgen,DT8 , while the second term is thecontribution due to fluid friction,Sgen,DP8 , in other words

Sgen8 5Sgen,DT8 1Sgen,DP8 . ~22!

A characteristic of all heat transfer devices with fluidflow is thatSgen,DT8 competes againstSgen,DP8 : for example, inthe smooth tube with fixedq8 and m, the changes inSgen,DT8 and Sgen,DP8 have opposing signs asD changes. Theoptimal tube diameter that minimizes theSgen8 expression~21! is given by48

ReD,opt52.023B00.36Pr20.07, ~23!

where B0 is a heat and fluid flow ‘‘duty’’ parameter thataccounts for the constraints (q8,m):

B05q8m

~kT!1/2m5/2/r. ~24!

The minimum entropy generation rateSgen,min8 is ob-tained by combining Eq.~23! with Eq. ~21!. The perfor-mance of any other design (D,Sgen8 ) relative to the optimaldesign (Dopt,Sgen,min8 ) is described bythe entropy generationnumber NS ,

NS5Sgen8

Sgen,min850.856S ReD

ReD,optD 20.8

10.144S ReDReD,opt

D 4.8,~25!

where ReD /ReD,opt5Dopt/D because the mass flow rate isfixed. This ratio is plotted in Fig. 10, which shows that therate of entropy generation increases sharply on either sidethe optimum. The irreversibility distribution ratio plotted onthe curve is defined asf 5 Sgen,DP8 /Sgen,DT8 . The entropy gen-



c

h

n

t

e

aohtle1a

-

k

the

llyalso

ew

oferi-

eeat

esn aionen

yan

,and

reto.th

s

m

eration rate ratioSgen8 /Sgen,min8 is used to monitor the approachof any design relative to the best design that can be coceived subject to the same constraints. This performanceterion was used extensively in the engineering literatu~e.g., Refs. 1 and 14! and, just last year, was also recognizein the physics literature.49

The minimization of entropy generation in ducts witheat transfer attracted considerable interest from engineworking on heat transfer augmentation techniques. In sutechniques the main objective is to increase the wall–fluheat transfer coefficient relative to the coefficient of the uaugmented~i.e., untouched! surface. A parallel objective,however, is to register this improvement without causingdamaging increase in the pumping power demanded byforced-convection arrangement. These two objectives revthe conflict that accompanies the implementation of any aumentation technique: a design modification that improves tthermal contact~e.g., roughening the heat transfer surface! islikely to also augment the mechanical pumping power rquirement.

The true effect of a proposed augmentation techniquethermodynamic performance can be evaluated by comparthe entropy generation rate of the heat-exchange apparbefore and after the implementation of the augmentatitechnique. This method of optimizing augmentation tecniques was proposed in Ref. 50, where it was applied todesign of ducts with sand-grain roughness and transversaroughness. Spiral tubes, twisted tape inserts, propeller insand tubes with internal spiral ribs were optimized in Ref. 5Nelson52 used this method to evaluate the effect of hetransfer augmentation on optimized full-size heat exchaners. Sekulicet al.53 documented the thermodynamic optimization of several ducts~smooth and enhanced!, and showedthat the minimum entropy generation design differs maredly from the design based on conventional methodPerez-Blanco54 integrated the entropy generation rate alonthe entire surface of the heat exchanger, and then evalua

FIG. 10. Entropy generation numberNS , or relative entropy generation ratein a smooth tube with heat transfer~from Ref. 48!.



n-ri-red

erschid-

aheealg-he

-

oningtusn-heribrts.tg-

-s.gted

the effect of the heat transfer augmentation technique ontotal entropy generation rate. Zimparov and Vulchanov55 ap-plied the EGM method to assess the merits of using spiracorrugated tubes. Performance evaluation studies wereconducted by Chen and Huang56 and Prasad and Shen.57,58

The EGM method was recognized in the most recent reviof heat transfer augmentation techniques.59

The opportunity of minimizing the entropy generation tdetermine the optimal sizes of ducts and other heat transdevices has been noted by practitioners in the field of chemcal process engineering.60 The same topic became a sizablcomponent in the new edition of a classic undergraduate htransfer textbook.61

Another large and diverse group of heat transfer devicrelies on external convection, that is, heat transfer betweestream and a body immersed in the stream. The minimizatof entropy generation for members of this class has beperformed by adapting the internal flow analyses~21!–~25!to external flow. The start is the formula for the total entropgeneration rate associated with heat transfer and drag onimmersed body62

Sgen5QB~TB2T`!

TBT`1FDU`

T`, ~26!

whereQB , TB , T` , FD , andU` are the heat transfer ratebody temperature, free stream temperature, drag force,free stream velocity. The relation for calculatingQB is pro-vided by the field of heat transfer. Similarly, the fluid flowinformation required for evaluatingFD comes from fluid me-chanics. TheSgenexpression has the same two-term structuas Eq.~22!. The competition between the two terms pointsan optimal body size for minimum entropy generation rate

The simplest example is the selection of the swept lengL of a plate immersed in a parallel stream~Fig. 11 inset!. Theoptimization procedure for laminar and turbulent flow wa

FIG. 11. The optimal size of a plate, cylinder, and sphere for minimuentropy generation~from Ref. 64!.



tds.n

-d

is

.

-

yt

di-

r

given in Ref. 1, and for turbulent flow in Ref. 63. The resultfor ReL,opt5U`Lopt/n are shown in Fig. 11 whereB is theduty parameter

B5QB /W

U`~kmT`Pr1/3!1/2

~27!

andW is the plate dimension perpendicular to Fig. 11. Thsame figure shows the corresponding results for the optimdiameter of a cylinder in cross flow,63,64 where ReD,opt5 U`Dopt/n, andB is given by Eq.~27!. The optimal diam-eter of the sphere1,63,64 is referenced to the sphere duty parameter defined by

Bs5QB

n~kmT`Pr1/3!1/2

. ~28!

The fins planted on the surfaces of heat exchanges acbodies with heat transfer in external flow. The size of a fin ogiven shape can be optimized by minimizingSgenof Eq. ~26!while accounting for the internal heat transfer characteristi~longitudinal conduction!65 of the fin. Figure 12 shows oneexample of how to select the optimal length and diametera pin fin ~cylindrical spine!, where ReL,opt5U`Lopt/n,ReD,opt5U`Dopt/n, and B5rn3kT`/QB

2. The figure wasdrawn forM5(k/l)1/2Pr21/65100 wherek andl are the finand fluid thermal conductivities. Optimal sizes for other fishapes are reported in Ref. 62.

The technological benefit of applying the EGM methoto heat transfer devices is that such devices are notorioustheir large number of dimensions, which have to be selectby the designer. The thermodynamic optimization methoshows how to select certain dimensions such that the dev

FIG. 12. Optimal pin fin diameter and length for minimum entropy genertion ~from Ref. 62!.



s

eal

-

t asf

cs

of

n

dforeddice

destroys minimum power while performing its assigned heatand fluid flow duty. The design is improved and, at the sametime, simplified. Even the teaching of the discipline of heattransfer benefits from this approach.

Elemental features such as ducts and fins are what heaexchangers are made of. The EGM method was also applieto heat exchangers as complete, more complex systemConsider for example the counterflow heat exchanger seen iFig. 8, where the inlet conditions (TH ,PH) and (TL ,PL) arefixed. The two streams have the same capacity ratemcp , andthe fluid is the same ideal gas (R,cp). Each stream flowswith pressure drop through a space~passage! of length(LH ,LL) and hydraulic diameter (Dh,H ,Dh,L). The sub-scriptsH andL refer to the high pressure side and, respec-tively, low pressure side of the heat transfer surface. It hasbeen shown that the rate of entropy generation of each passage is minimized if the passage slenderness ratio is selecteoptimally66

S LDhDopt

5t

4G* @~R/cp! fSt#1/2, ~29!

Sgen,minmcp

52tS RcpD1/2

G* S fStD1/2

, ~30!

where t5uTH2TLu/(THTL)1/2, G

*5G/(2rP)1/2, andG is

the mass velocity through the passage flow cross-sectionAf ,namelyG5m/Af . As a first approximation, the ratiof /St isa constant for a given type of heat exchanger surface.66 ThesubscriptsH and L are dropped from Eqs.~29! and ~30!because one such set can be written for either passage. Thfurther illustrates the power of the method: the same optimi-zation rule applies on both sides of the heat transfer surface

In Eq. ~29! we see just one of the geometric optima thathave been developed. The balanced counterflow heat exchanger subject to fixed heat transfer area was optimized inRef. 66, which also reported the optimization subject to fixedheat exchanger volume. The doubly constrained optimizationof the counterflow heat exchanger with fixed area and vol-ume was reported in Ref. 14. In Refs. 47 and 66 we see earlexamples of finite size thermodynamics applications in heatransfer. The optimization subject to fixed total number ofheat transfer units~Ntu! was developed in a noteworthy studyby Sekulic and Herman.67 Another line of research focusedon the thermodynamic optimization of several types of heatexchangers where the fluid friction irreversibility isnegligible.68–71Heat exchangers with phase change~boilers,condensers! were optimized by London and Shah,72 Zubairet al.,73 and Lauet al.74

Extensions to the study of entropy generation in coun-terflow heat exchangers66 have been performed by Sarangiand Chowdhury68 and Huang.75 Grazzini and Gori76 recon-sidered the entropy generation analysis, and distinguishebetween three separate entropy generation number defintions, two of which are new. They investigated the extremaof these numbers, and applied their results to an air-to-aicounterflow heat exchanger. In chemical engineering,Tondeur77 and Tondeur and Kvaalen78 showed that, for a

a-



e

s

n

h

a

x

-

-s.

-x-isd-i-slly

h-

a

given duty, the best configuration of a heat and mass procis that where the entropy generation rate is distributed in tmost uniform way possible.

A study of crossflow heat exchangers was conductedBaclic and Sekulic.79 Their study reveals once again thetradeoff between heat transfer and fluid flow irreversibilitieas well as the remanent~imbalance! irreversibility associatedpurely with the crossflow arrangement. Several basic studof the thermodynamics of forced convection heat transferheat exchangers were undertaken by Soumerai.80,81The rela-tionship between entropy generation minimization and cominimization was illustrated by Wepferet al.82 in the prob-lem of deciding the optimal size of a steam pipe and iinsulation.

The generation of entropy at the local~differential! levelin heat exchangers was documented by El-Sayed,83 Liangand Kuehn,84 Evans and von Spakovsky,85 and Drost andZaworski.86 The calculation and display87 of entropy genera-tion rate distributions through heat and fluid flow fields warecommended by Paolettiet al.,88 White and Drost,89 Chengand Huang,90 Drost and White,91,92Benedetti and Sciubba,93

and Chenget al.94 Taken together, these studies argue thcommercial computational fluid dynamics packages shouhave the capability of displaying entropy generation ramaps for both laminar and turbulent flows.

For example, Figs. 13~a! and 13~b! show the flow andtemperature fields in the entrance region to a vertical chanwith mixed ~forced1natural! convection. The right wall isheated~T1!, and is fitted with horizontal fins. The left wall iscooled to the ambient temperature~T0!, which is also thetemperature of the fluid that enters through the bottom of tchannel. New in this display of numerical information is Fig13~c!, which shows the contour lines of the local~volumet-ric! entropy generation rate. For the latter, Paolettiet al.88

proposed plotting contour lines for constant values of thdimensionless groupSgen,DT- /(Sgen,DT- 1Sgen,DP- ), to show therelative importance of heat transfer irreversibility in the locatotal entropy generation rate~see also Refs. 93 and 95!.

The thermodynamic optimization of complete heat exchanger systems was performed by Tapia and Moran,96 Ra-nasingheet al.,97 and Li et al.98 Heat exchanger networkswere optimized by Hesselmann,99 Chato and Damianides,100

and Sekulic and Milosevic.101 Regenerators were optimizedby Tsujikawa et al.102 for gas-turbine power plants, byHutchinson and Lyke103 for Stirling cycle machines, andMatsumoto and Shiino104 for cryogenic plants. Heat ex-changers with offset strip-fin surfaces were optimized morecently by Schenoneet al.105

The method was also extended in several fundamendirections of heat transferscience. The generation of entropyby pure heat conduction, in the absence of fluid flow, hcome under close scrutiny in more recent papers. Bisio106

focused on one-dimensional heat conduction in systems wtime-dependent boundary conditions. The thermal conductity was a function of temperature and the coordinate of thheat flux direction. He examined homogeneous as wellmultilayered systems, and pointed out relations between etropy generation extrema and the solution to the heat coduction problem~the temperature distribution!. Extensions of



sshe

by

,

iesin

st

ts

s

atldte

el

e.

e

l

-

re

tal

s

ithiv-easn-n-

this work cover conduction in a medium with thermal con-ductivity ~or its derivative! that is a piecewise continuousfunction of temperature,107 and thermal conductivity that isalso a function of the temperature gradient in the heat fludirection.108

An entirely new area of application of entropy generation by conduction is being charted by the work of Kinra andhis associates109–113at Texas A&M University. They used theentropy generation calculations to explain and predict damping in homogeneous and inhomogeneous elastic systemThey named their theoryelastothermodynamic damping.

Kinra’s approach begins with the observation that a material that is stressed reversibly and adiabatically always eperiences local changes in temperature, however small. Ththermoelastic effect can be predicted using the first law anthe second law. The new observation is that since the temperature field and the stress field are coupled, nonuniformties in stress and materials properties induce nonuniformitiein temperature. As a consequence, heat is conducted locafrom regions of relatively high temperature to regions of lowtemperature. The entropy generated by conduction througout the material is responsible for the damping effect.

Kinra, Bishop, and Milligan have used this approach todemonstrate that it is now possible todesigna composite

FIG. 13. Flow, temperature, and volumetric entropy generation rate invertical duct with mixed convection~from Ref. 94: Courtesy of Prof. C.-H.Cheng, Tatung Institute of Technology, Taipei!.



h

s

i

c

e

m

i

n

e

sfere-on

sssr-

.

ofd iny-tn-

eats-

elts

r-shethe

en

material that has certain, desired damping characterisTheirs is an interdisciplinary example of the importancethe concept of entropy generation in the field of materidesign. This is an important development, because the oapplications collected in this review come strictly from tmainstream of thermal engineering~Fig. 1!.

As an analogy to the convective heat transfer irreveibility illustrated in Fig. 10, the competition between matransfer and fluid flow irreversibilities was demonstratedSanet al.114 The irreversibility of combined heat and matransfer was minimized in internal flow by Sanet al.115 andin external flow by Poulikakos and Johnson.63 Carringtonand Sun116,117 reconsidered the internal and external floproblems. Sun and Carrington118 developed a general analysis of the flow of a fluid mixture with heat conduction, madiffusion, fluid friction, and chemical reactions.

The generation of entropy in reacting flows with radtion has been studied by Arpaci and his associates119–124andPuri.125 For example, Arpaci and Selamet122 focused on pre-mixed flames stabilized above a flat flame burner, ashowed that the tangency condition~i.e., the minimumquench distance! is related to an extremum of the entropgeneration rate, which is inversely proportional to the Penumber. Puri125 minimized the entropy generation rate ofdroplet burning in a stream. He showed that the minimentropy generation rate corresponds to a tradeoff betwdrag and mass loss from the droplet, and to maximum enper unit mass flowrate at the combustor exit.

The entropy generation rate in convection throughsaturated porous medium was presented in Ref. 126.general form for the local entropy generation rate duecombined heat, mass, and fluid flow through a porousdium is given in Ref. 14.

V. STORAGE SYSTEMS

The optimization of time-dependent heating and coolprocesses has generated a subfield of its own. The apptions are diverse, for example, sensible heating versus laheating, or the temporary storage of ‘‘heating’’ versus tstorage of ‘‘refrigeration.’’ Common to all these applicatiois the search for optimal strategies for executing heatingcooling processes, i.e., the search foroptimal histories, oroptimal evolutions in time.

The earliest work of this type127 focused on the sensiblheating of a storage element~solid or incompressible liquid!of massM and specific heatC, by using a stream of hot ga(m,cp ,T`) as shown in Fig. 14. Initially, the storage elemeis at the ambient temperatureT0. Gradually, the element temperatureT and the gas outlet temperatureTout rise and ap-proachT` . The temperature historiesT(t) andTout(t) can bewritten analytically as functions of the dimensionless timeuand the number of heat transfer unitsNtu based on thestream-element thermal conductanceUA ~fixed!,

u5tmcpMC

, Ntu5UA

mcp. ~31!

Figure 14 shows that the sensible heating processtwo sources of entropy generation, the heat transfer betw



tics.ofalsthere

rs-ssbys

w-ss

a-

nd

yletaumeenrgy

aThetoe-

nglica-tenthesand

snt-

haseen

the stream and the storage element, and the heat tranbetween the exhaust and the ambient. A third source, nglected here but treated in Ref. 127, is the pressure dropthe gas side of theUA heat exchanger. It was shown that thetotal entropy generated fromt50 to t5t is minimum at acertain time~duration! of the heating process,topt or uopt. Inthe limit t!topt, the generated entropy is due mainly to theinternal source, while in the limitt@topt the dominant sourceis the external thermal mixing~Fig. 14!. Charts for calculat-ing the optimal heating timeuopt as a function ofNtu and(T`2T0)/T0 are available in Refs. 1 and 127. For(T`2T0)!T0 , the optimal heating time is available inclosed form:

uopt51.256

12exp~2Ntu!. ~32!

Since in most heat exchangersNtu is of the order of 1 orgreater, Eq.~32! shows thatuopt is of the order of 1, or thattopt;MC/(mcp). In conclusion, for engineering design pur-poses, the optimal heating strategy is such that the procemust be terminated when the ‘‘thermal inertia’’ of the hot gaused (mcpt) becomes of the same order as the thermal inetia of the storage element (MC).

The thermodynamic optimization of the element of Fig14 during a complete storage and removal cycle~i.e., heatingfollowed by cooling back toT0! was performed by Krane.

128

An improvement to the single-element storage methodFig. 14 is the use of several elements in series, as proposeRef. 1. This proposal was investigated in great detail by Talor et al.129 based on a solid distributed-storage-elemenmodel in which the storage material temperature varied cotinuously along the stream. Tayloret al.showed among otherthings that the longitudinal conduction of heat through thstorage material during the periodic operation of the heexchanger can have a major impact on the overall irreveribility of the installation. The overall entropy generation isagain a strong function of the time interval required by thstorage part of the cycle: the identification of the optimastorage time interval is critical. The series of storage uniwas optimized further by Sekulic and Krane.130

Closely related to the continuous one-dimensional stoage scheme with periodic counterflow circulation is the clasof periodic heat exchangers recognized as regenerators. Tdesign of this type of heat exchanger was approached onbasis of EGM by Sanet al.131 Their model consists of two-dimensional parallel-plate channels sandwiched betwe

FIG. 14. Solid or liquid-bath element for sensible heat storage~fromRef. 127!.



u-rel-

n-

e

nse

ars.din

ofinkbe

eyithn-

aorinhe

istentth

theoren-

ti-

it

nd

slabs of storage material. The longitudinal conduction of hthrough the storage material is neglected. An importantference between this regenerator model and the continustorage system analyzed by Tayloret al.129 is that in the re-generator the stream exhausted during the storage phanot dumped into the atmosphere.

The minimization of entropy generation in periodic-floregenerative heat exchangers was studied also by Hutchiand Lyke,103 Shen and Worek,132 Mathiprakasam andBeeson,133 Das and Sahoo,134 and Sahoo and Das.135 Thethermodynamic arguments of this section were combinwith overall cost minimization arguments into athermoeco-nomicsextension of the sensible heat storage process bydaret al.136 and Kotas and Jassim.137

An overview of the thermodynamic optimization of sesible heat storage methods was presented by Rosenet al.138

Treated were the individual storage and removal processewell as the complete cycle. The emphasis was on the deopment of simple and consistent ways~conventions! toevaluate and compare the performance of competing desThis work was continued by Rosen139 who showed that dif-ferent efficiency measures are suitable for different applitions, and that it is important to agree on a common eciency definition before comparing designs of the same cl

The thermodynamic optimization of heating and cooliprocesses continued in two additional directions. One isoptimization subject to a finite-time constraint. Figure 15lustrates the fundamental question of how to cool a massprescribed temperature level (TL) during a fixed time inter-val tc , while using the minimum quantity of coolant140

m5E0

tcm~ t !dt. ~33!

FIG. 15. Batch cooling and temperature history during a cooldown proc~from Ref. 140!.



eatdif-ous

se is

wnson

ed

Ba-

n-

s asvel-

igns.

ca-ffi-ass.ngtheil-to a

This question is essential in the operation of large scale sperconducting systems, which must be cooled down befothey can function. Note that to minimize the amount of cooing agent~cryogen! is equivalent to minimizing the refrig-erator work needed to produce the cryogen, or the total etropy generated in the cold space.

This problem was solved by variational calculus for thoptimal flow rate historymopt(t) that minimizes them inte-gral ~33! subject to the time constrainttc . The details can befound in Refs. 1 and 140. The end result is

mopt~ t !5F U~T!A

C* cp~T!G1/2

, ~34!

where T(t) is the companion result for the optimal bodytemperature history. The constantC* can be evaluated byusing the time constrainttc . Equation~34! is a compact re-sult with important engineering implications. Bearing inmind that the heat transfer coefficientU varies asT de-creases, we learn that during poor heat transfer conditio~low U! the flow rate should be lower. Furthermore, sincduring cooldown the coolantcp increases, the flow rateshould decrease even more as the end of the process ne

The engineering work on the optimization of heating ancooling subject to time constraints was continued recentlythe physics literature by Andresen and Gordon,141 who mini-mized the generation of entropy instead of the amountheating or cooling agent used. As heat source or heat sthey assumed a heat reservoir of temperature that canvaried at will, instead of the coolant flow ratem(t) of Fig.15. Between the heat reservoir and the thermal inertia thassumed several heat transfer rate laws, e.g., convection wa constant heat transfer coefficient, and radiation with costant~temperature-independent! emissivities. In a companionpaper, Andresen and Gordon142 considered the related prob-lem where the heating of the body of interest is effected bystream, while the heat reservoir temperature may changeremain constant. The result of the EGM procedure is agaan optimal flow rate of the heating agent. In both papers teffect of pressure drop was neglected.

The other direction of the work on storage systemsconcerned with phase-change storage elements, e.g., laheating instead of sensible heating. This activity began withe papers of Bjurstrom and Carlsson143 and Adebiyi andRussell144who applied to a phase-change storage elemententropy generation analysis given in Ref. 127 and Fig. 14 fthe single-phase element. These authors showed that thetropy generated during heating~melting! is minimum whenthe melting temperature of the storage material has the opmal value

Tm,opt5~T`T0!1/2. ~35!

Technologically, this result is extremely valuable becauseguides the designer in the selection ofthe type of phase-change material.

The details of the real~time-dependent! melting and so-lidification processes were accounted for in Refs. 145 a146, where it was shown that Eq.~35! is a good approxima-tion for the optimal phase-change temperature. Limet al.147

showed that Eq.~35! holds for the entire melting and solidi-

ess



eed

theles.ase-an

-e-idce

-to

end-s

nn-

sd-de

alld.ra.o

at

h

s

atm-

at

fication cycle when the fluid is modeled as well mixed insithe storage element. The same authors showed that themodynamic performance of the latent heat storage proccan be improved by using two or more different phaschange materials in series. Two such elements are pictureFig. 16, for which there are two optimal melting tempertures ~i.e., two materials! as shown in Fig. 17. The dimensionless optimal temperatures in Fig. 17 are referenced to~35!, namely t1,opt5Tm,1,opt/(T`T0)

1/2 and t2,opt5Tm,2,opt/(T`T0)

1/2. Lim et al.147 similarly optimized an in-finite number of phase-change materials heated in series,a single material melted by an unmixed stream.

The thermodynamic optimization of practical storagestallations with melting and solidification has attracted aof interest in both engineering and physics. For exampAdebiyi148 considered a bed with particles with several ratiof latent heat to sensible heat storage capability. He modthe heat conduction as one dimensional in the cylindripellet geometry. For the complete storage and removal cyhe found that the optimal phase-change temperature is eto the arithmetic average of the heat source and ambtemperatures.

Charach and Zemel149 optimized the thermodynamicperformance of latent heat storage in the shell of a shell-atube heat exchanger. The focus was on the effect of tdimensional heat transfer, that is, as a step beyond thedimensional model employed by De Lucia and Bejan.145

Charach and Zemel149 also considered the effect of pressudrop on the stream side of the heat exchanger. Their anahas been extended by Charach and Zemel150 and Charach151

to the complete melting and solidification cycle, by usingquasisteady treatment of the phase change process occuin the shell. These latest studies showed that the optiphase-change temperature is bounded from above andbelow by the arithmetic and, respectively, geometric avages of the source and ambient temperatures.

Another interesting direction has been the study of latheat storage units coupled in series with a power plant,optimized over the entire storage and removal cycle. Belleand Conti152 showed that minimization of entropy generatioand operation stability are two competing criteria in the otimization of the aggregate installation. The arithmetic avage of the extreme temperatures emerges again as themal phase-change temperature. Detailed modeling

FIG. 16. Latent heat storage by using two phase-change materials in s~from Ref. 147!.


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numerical simulations of the heat transfer behavior of thshell-and-tube phase-change heat exchanger were performby Bellecci and Conti.153,154

Aceves-Saborioet al.155 developed a systematic way ofmodeling phase-change storage systems, by applyinglumped model to many independent phase-change capsuThey also considered the more general case where the phchange material melts over a finite temperature range. Inearlier study, Aceves-Saborioet al.156 optimized a singlelatent-heat cell by first simulating numerically the phasechange process. This is an important and difficult task bcause the natural convection currents that occur in the liqucontrol the shape and movement of the liquid–solid interfa~e.g., Sec. 10.4 in Ref. 157!. It was shown by De Lucia andBejan145 that when the melting process is controlled by natural convection the optimal melting temperature is equalthe geometric average shown in Eq.~35!.

Adebiyi et al.158 constructed a numerical model forsimulating and then optimizing the performance of storagsystems with multiple phase-change materials. They fouthat the second law efficiency of systems with multiple materials can exceed by 13%–26% the efficiency of systememploying a single material. A fundamental study of EGM itime-dependent unidirectional heat conduction was coducted by Charach and Rubinstein.159

VI. SOLAR POWER PLANTS

The generation of mechanical or electrical power habeen subjected to thermodynamic optimization in many stuies that cover a vast territory. Therefore in this section anSec. VII we cover only some of the central ideas and thliterature that followed. We do this chronologically, i.e., inthe order in which the use of the method reached criticmass in the various sectors of the power generation fieThis occurred first in the optimization of solar driven poweplants, as exemplified by the ‘‘solar’’ theme chosen forrecent book160 dedicated to the application of the methodBook chapters on the application of the EGM method tsolar power plants appeared earlier in Refs. 1 and 14.

In a 1957 paper, Mu¨ser161 optimized the power producedby an engine driven by solar heating. Mu¨ser’s model isequivalent to the upper portion drawn betweenTs andTL inFig. 18. The hot end of the heat engine cycle receives hefrom a solar collector (TH) that sees not only the solar disc(Ts) but also the cold universe~T`!. The heat engine cycle ismodeled as reversible, with the cold end in equilibrium witthe ambient. Mu¨ser161 and, independently, Castans,162

Jeter,163 and De Vos and Pauwels164 showed that the poweroutput W is maximum when the collector temperature hathe optimal value given by

4TH,opt5 23TLTH,opt

4 2Ts4TL50. ~36!

This was an important first step in the demonstration than optimal design can be found based on a model that cobines only the thermodynamics of the (TH2TL) compart-ment with the irreversibility of radiation heat transfer in thethree-surface enclosure formed byTH , Ts , and T` . Thesame combination of thermodynamics and radiation he

eries



l

e,elat

p-

e-

e

s

the

tg

e

transfer was the method used independently in 1971 by Mtynovskii et al.30 in the optimization of radiation shields.

Another important observation is that to maximize thpower outputW is equivalent to minimizing the total entropygeneration rate associated with the solar power plant~Ref.14!. Note that asTH is varied during the optimization pro-cess, the net solar heat input to the collector~Q, from Ts toTH! also varies. And ifQ is to float freely, then the actuaheat input available from the sun must be greater than anyQvalue that might be required in the course ofW maximiza-tion. Let Q† be this sufficiently large~and fixed! heat transferrate. A portion ofQ† is intercepted by the collector (Q),while the remainder (Q†2Q) must necessarily ‘‘fall on theground,’’ i.e., be rejected to the ambientTL . Both portionsvary with the lone degree of freedom in the power pladesign~TH , Q, or W!. The total entropy generation rate

Sgen5Q

TH2Q†

Ts1Q†2Q

TL52

W

TL1Q†S 1TL2

1

TsD ~37!

FIG. 17. The optimal melting temperatures of the phase-change materiathe series arrangement of Fig. 16~From Ref. 147!.



ar-

e

nt

shows that when the external irreversibility due to (Q†2Q)is taken into account, the maximization ofW is indeed analo-gous to the minimization ofSgen, in agreement with theGouy-Stodola theorem~5!.

De Vos’s book160 is also a review of the models such asFig. 18 that have been used in physics. In this section wreview some of the engineering contributions to the fieldwhich are not covered in Ref. 160. For example, the modof Fig. 18 refers to an extraterrestrial solar power plant thuses a radiator to reject heat to the universe~T`!. The heattransfer fromTL to T` is by radiation in a two-surface en-closure. In Refs. 14 and 65 this power plant model was otimized subject to the total area constraint

AH1AL5A. ~38!

The design has two degrees of freedom, and the twicmaximized power output is represented by

AH,opt50.35A, AL,opt50.65A, ~39!

FHsS TsTH,opt

D 451.538, ~40!

Wmax50.0414sAFHsTs4, ~41!

wheres andFHs are the Stefan–Boltzmann constant and thcollector-sun view factor. If we substituteTs55762 K andFHs;1024 into this solution we obtainTH,opt > 520 K andTL,opt > 340 K for the recommended extreme temperature~boiler, condenser! of the power cycle executed by the work-ing fluid. Another technological aspect of this result is thathe radiator area should be roughly twice as large as tcollector area.

As a way to think, the extraterrestrial solar power planmodel of Fig. 18 is related to the conversion of solar heatininto wind power~natural convection! on earth. This power is

ls in

FIG. 18. Model of solar power plant with heat transfer irreversibilities at thhot end and the cold end~after Refs. 14 and 65!.



b-dyithle,-gnt

s--lof

snt

rali-n.ursic-g

utp-

i-

ele

-rke

n

destroyed almost totally by friction and heat transfer acrfinite temperature differences. The thermodynamic heatgine that drives any natural convection process was anoted in the field of natural convection heat transfer~Ref.126!. The theoretical limits of the conversion of terrestrisolar heating into wind power were investigated by Gordand Zarmi,165 De Vos and Flater,166 and De Vos.160

Another engineering model is the power plant drivena solar collector with convective heat leak to the ambient167

which is shown in Fig. 19. The heat leak was modeledproportional to the collector–ambient temperature diffeence,Q05(UA)c(Tc2T0). The internal heat exchanger between the collector and the hot end of the power cycle~theuser! was modeled similarly,Q5(UA) i(Tc2Tu). It wasfound that there is an optimal coupling between the collecand the power cycle such that the power output is maximuThis design is represented by the optimal collector tempeture

Tc,optT0

5umax1/2 1Rumax11R

, ~42!

whereR5(UA)c/(UA) i , umax5Tc,max/T0, andTc,max is themaximum~stagnation! temperature of the collector. The coresponding optimal coupling between a collector with heloss and a refrigerator was documented by Sokolov aHershgal.168

Radiation heat transfer contributes significantly to tcollector–ambient heat loss mechanism when the colleoperating temperature rises above the 100–200 °C raThe radiation effect on the heat loss and the optimizationthe solar power plant were analyzed by Howell aBannerot.169 A sample of their results for optimal collectotemperature is presented in Fig. 20 for four classes of coltor designs~A, B, C, D!, which are described in Ref. 169The dimensionless radiative heat loss parametera and con-vective heat loss parameterb are also defined in Ref. 169Howell and Bannerot produced similar graphs for the opmal thermodynamic design of solar driven power cyc~Stirling, Ericsson, Brayton!, heat pumps, and absorption refrigerators.

The solar power plant model with collector–ambieheat loss and heat exchanger between the cold end of

FIG. 19. Solar power plant model with collector-ambient heat loss acollector-power cycle heat exchanger~from Ref. 167!.



ossen-lso

alon

by,asr--

torm.ra-

r-atnd

hectornge.ofndrlec-.

.ti-les-

ntthe

power cycle and the ambient was optimized in Ref. 170 suject to a total heat transfer area constraint. The same stupresents the area-constrained optimization of a model wphase-change energy storage at the hot end of the cycbetween the collector and the working fluid. One useful result is that the melting material must be such that its meltinpoint is the geometric average of the collector and ambietemperatures,

Tm,opt5~TcT0!1/2. ~43!

Solar power plant models with nonisothermal collectorwere optimized in Ref. 167. Models with collectors operating under time-varying conditions dictated by the daily insolation cycle were optimized in Refs. 171 and 172. A modewith time-dependent solar heating and variable amountfluid in the collector was optimized in Ref. 172. In thismodel the collector has the ability to store the solar input asensible heat, and to deliver it to the rest of the power plawhen the solar heating effect is less intense.

The common message of these models is that seveextremely basic tradeoffs exist in the thermodynamic optimzation of power plants driven by heat transfer from the suThe models share the feature that heat loss always occbetween the collector and the ambient. The thermodynamtradeoffs are of two kinds. When the overall size of the installation is constrained, there is an optimal way of allocatinthe hardware~e.g., heat transfer area! between the variouscomponents. When the daily variation of the solar heat inpis known, there is an optimal time-dependent strategy of oerating the power plant.

Thermodynamic tradeoffs have been found in the optmization of the direct~photovoltaic! conversion of solar ra-diation. This work was reviewed in the book by De Vos160

and continues today. For example, Baruch and Parrott173 ex-amined the possibility of constructing a Carnot cycle for thconversion of photovoltaic energy by using electron–hoplasma as the working substance.

The basic thermodynamic limits to the conversion of solar radiation have attracted considerable attention. This wowas reviewed in Ref. 174, where it was also shown that th

nd

FIG. 20. Optimal collector temperatures for maximum power generatio~after Ref. 169!.



,

p

he

b

a

y

n

r

e

tre

at

m-le

e

ssi-n-s

.l

-

n

tant

he

r-ther-

different ideal conversion efficiencies reported by Petela175

Spanner,176 and Jeter163 are complementary, and can be unfied into a single theory. Related aspects were considemore recently by Badescu.177–180 For example Badescu178

showed that the maximum efficiency decreases abruwhen the collector concentration ratio decreases.

Roy and Grasse181 reviewed comparatively the practicamerits of solar power technologies across the entire sptrum, from photovoltaic to solar–thermal applications. Tmechanisms of entropy generation in solar collectors walso investigated by Fujiwara,182 Suzuki,183 and Han andLee.184 The thermodynamic optimization of collectorcoupled with power plants was considered furtherGrazzini,185 Borner et al.,186 De Vos et al.,187 and Yan andChen.188 Models with radiation-dominated collector and rdiator were optimized by De Vos and van der Wel189 andGotkun et al.190

A more applied line of inquiry concerns the thermodnamic optimization of specific designs of processes apower plants driven by solar heating. Recent examplesthe optimization of flat-plate solar air heaters,191,192 phase-change energy storage,193 and solar assisted heat pumps.194

The optimal coupling between solar collectors and Stirliand Ericsson cycles was also documented by Badescu.195

Another type of power plant driven by a renewable eergy source is the power plant with heating from a hot-drock system. The optimal time-dependent strategy of runnsuch a power plant~e.g., the optimal water flow rate! wasdeveloped in Ref. 196.

VII. NUCLEAR AND FOSSIL POWER PLANTS

In 1957 in nuclear engineering, Chambadal2 andNovikov5 showed independently that the hot-end tempeture of a power plant can be optimized such that the powoutput is maximum. Chambadal’s analytical argument corsponds to the model drawn by the author in Fig. 21. Tpower plant is driven by a stream of hot single-phase fluidinlet temperatureTH and constant specific heatcp . Thepower plant model has two compartments. The one sawiched between the surface of temperatureTHC and the am-bient (TL) operates reversibly. The area of theTHC surface isassumed to be sufficiently large such that the outlet temp

FIG. 21. The sources of entropy generation in Chambadal’s power pmodel ~from Ref. 197!.



i-red

tly

lec-ere

sy

-

-ndare

g

n-y-ing

ra-erre-heof

nd-

ra-

ture of the stream is equal toTHC . There is only one degreeof freedom in the optimization of the power plant: the hoend of the inner compartment, or the exhaust temperatuTHC . It is not difficult to expressW as a function ofTHC andto show thatW is maximized when

THC,opt5~THTL!1/2. ~44!

Chambadal’s corresponding energy conversion efficiencymaximum power is

h512S TLTHD 1/2. ~45!

It can be shown that the same efficiency formula~45! holdswhen the heat exchanger area is finite and the exhaust teperature is higher than hot-end temperature of the reversibcompartment.197 Equation~45! also holds when the unmixedstream of Fig. 21 is replaced with a single-temperatur~mixed! fluid inside that heat exchanger.197

The maximum-power efficiency~45! can also be derivedby minimizing the total entropy generation rate associatedwith the power plant.197 One obvious source of entropy gen-eration in Fig. 21 is the heat exchanger. The other is leobvious: it is the dumping of the used stream into the ambent. This second source was featured prominently by an etire subfield dedicated to the optimization of storage system~Sec. IV, Fig. 14!. The dumping of theTHC-hot exhaust is anessential part of the optimization process:THC is a degree offreedom only when the exhaust (THC) is free to float, i.e.,when it is not required~used! by someone else downstreamThe external irreversibility indicated in Fig. 21 is an essentiapart of thephysicsof the optimization process: without it theplant design cannot be optimized. This additional irreversibility is what gives the designroom to move, therefore it canbe called the ‘‘room-to-move irreversibility.’’

With reference to Fig. 21, the total entropy generatiorate due to the power plant is

~46!

where the stream was treated as an ideal gas at conspressure, QH5mcp(TH2THC) and Qe5mcp(THC2TL).Equation~46! shows thatSgenhas a minimum with respect toTHC , and that the Sgen,min design corresponds to themaximum-power formulas~44! and ~45!.

It is worth noting that if we had overlooked the room-to-move irreversibility, that is, if we had written only theentropy generation associated with the visible confines of tpower plant, then we would have found thatSgenhas a mini-mum at aTHC value that differs from the maximum-powervalue ~44!. TheseTHC values differ not because maximumpower and minimum entropy generation rate are two diffeent designs, but because an oversight has occurred inevaluation of the total rate of entropy generation. This obsevation sheds light on the physics literature claim198 that, in

lant



hvd

e

7

n

’s

.e.

l-lsssinil-

–

in

hot

t.

the same power plant, minimum entropy generation amaximum power are two different design conditions.

A maximum power design similar to Chambadal’s is toptimal combustion chamber temperature that was deriindependently in Ref. 14. Figure 22 shows a two-part moof a power plant with isobaric and isothermal~well mixed!combustion chamber. It was shown that when the specheats of all the products of combustion are assumed tosufficiently constant~independent of temperature!, the maxi-mum power design corresponds to a hot-end~flame! tem-perature (Tf) equal to the geometric average of the adiabaflame temperature~Taf! and the ambient temperature~T0!,

Tf ,opt5~TafT0!1/2. ~47!

The maximization of power output was also describby Odum and Pinkerton,199 who gave several examples fromengineering, physics, and biology, without deriving resusuch as Eqs.~44! and~45!. Equal credit for such results goeto Novikov5 who, like Chambadal, published them in 195Novikov’s analysis was reprinted in English in NorthAmerican engineering textbooks200,201as well as in Russianengineering textbooks.7,202,203His model is shown in Fig. 23.The hot-end heat exchanger of finite thermal conductaUA drives the heat transfer rateQH into the working fluid,which is heated at constant temperature (THC) from state (b)to state (c). The fluid is expanded irreversibly from (c) to(d): Novikov accounted for this irreversibility by writing

FIG. 22. Power plant driven by heating from a combustion chamber wwell mixed products of combustion~from Ref. 14!.



nd

eedel

ificbe

tic

d

ltss.-

ce

(sd2 sa)5 (11 i )(sd,rev2 sa),orQL 5 (11 i )QL,rev, where~11i !>1 andQL is the heat transferred to the ambient (TL).The rest of the power plant operates reversibly. Novikovoptimal heating temperature and efficiency for maximumpower output,

THC,opt5~11 i !1/2~THTL!1/2, ~48!

h512~11 i !1/2S TLTHD 1/2 ~49!

match Chambadal’s Eqs.~44! and~45! in the limit where theexpansion is executed reversibly~i50!.

The efficiency formula~45! was rediscovered in 1975 inthe physics literature by Curzon and Ahlborn.8 Their modeldiffered from Chambadal’s and Novikov’s in two respectsFirst, the power plant operated in unsteady fashion, in timIt executed a four-process~two-stroke! cycle modeled as inSadi Carnot’s original memoir, however, the piston and cyinder apparatus made contact during finite time intervawith the two heat reservoirs. This contact occurred acrofinite temperature differences. The second new featureCurzon and Ahlborn’s model is the heat transfer irreversibity ~finite thermal conductance! placed at the cold end of thecycle. In summary, the maximum-power efficiency~45! maybe called, chronologically, the Chambadal–NovikovCurzon–Ahlborn efficiency, orthe CNCA efficiencyforshort.

An entirely different way of modeling the irreversibleoperation of a power plant was proposed in 1976engineering.204 The loss of heat from the hot end of thepower plant was modeled as a thermal resistance~bypassheat leak! in parallel with an irreversibility free compartmentthat produces the actual power outputW, Fig. 24. The hot-end temperatureTH could vary. The heat leak was modeledas proportional to the temperature difference between theend and the ambient,QC5C(TH2TL), whereC is the ther-mal conductance of the leaky insulation of the power planThe power is maximum when the hot-end temperatureTHreaches the optimal level

ith

FIG. 23. Novikov’s model for a steady-state power plant with heat transfer and expander irreversibilities~from Ref. 197!.



-

b

e

n

d

-

tor

rs

r-h

--

c-

d

TH,opt5TLS 11QH

CTLD 1/2. ~50!

The corresponding efficiency~Wmax/QH! is expectedly lowerthan the Carnot efficiency,

h512TL /TH,opt11TL /TH,opt

5~11r !1/221

~11r !1/211, ~51!

wherer5QH/(CTL) is the dimensionless thermal resistancof the power plant insulation. An optimal hot-end temperture exists because whenTH , TH,opt the Carnot efficiency ofthe right-hand side of the model of Fig. 24 is too low, whilwhenTH . TH,opt too much of the fixed heat inputQH by-passes the power producing component.

An interesting aspect of this power plant model is thwhen we minimize the entropy generated inside the dashbox betweenTH andTL in Fig. 24, we find an optimum thatdiffers from Eqs.~50! and ~51! ~Ref. 1!. This paradox is anindication that the entropy generation calculation is incorect, as noted earlier under Eq.~46!. For if TH is a degree offreedom in the design, then the hot end of the power plamust be free to float between the ambient and another~nec-essary! heat reservoir whose temperatureTf ~fixed! must bealways higher thanTH . The neglected room-to-move irreversibility occurs as QH crosses the temperature ga(Tf2TH). It is easy to show that by minimizing the entropgenerated betweenTf(.TH) andTL in Fig. 24, the optimalhot-end temperature and efficiency match Eqs.~50! and~51!~Ref. 12!.

Curzon and Ahlborn’s work8 triggered a series of paperson power maximization in the physics literature. Reviewsthis specialized line of research were publishedAndresen205 and Andresenet al.206 Noteworthy in this seriesis the first follow up paper,207which appeared in 1977 and inwhich the power plant was modeled in the steady state. Copared with Novikov’s model~Fig. 23!, the model of Ref. 207had three new features:~i! a finite thermal conductance at thcold end,~ii ! a speed-dependent dissipation of mechanicpower caused by shaft friction, and~iii ! a speed-independen

FIG. 24. Power plant model with bypass heat leak~after Ref. 204!.



ea-

e

ated

r-

nt

py

ofy

m-

alt

contribution to power dissipation. Feature~i! makes thesteady state power plant model look like the one drawn iFig. 25, which will be discussed shortly. Feature~ii ! illus-trates the interplay between friction, thermodynamics, anheat transfer in the application of the method~Fig. 1!. Re-garding feature~iii !, to which Andresenet al.207 refer as‘‘heat leak,’’ it must be noted that the origin of that energyinteraction is the work reservoir, or, in the authors’ own terminology, a reservoir with ‘‘infinite temperature’’~i.e., anenergy interaction accompanied by zero entropy transfer!.Feature~iii ! is actually power dissipation~e.g., electric, Jouleheating! or a ‘‘friction leak’’ as stressed by Grazzini,208 not abypass heat leak in the sense of Fig. 24.

The steady-state power plant model with two finite heaexchangers was proposed independently in Ref. 1, not fderiving the CNCA efficiency~45!, but for answering a moredirect technological question: Given the two heat exchange~UHAH andULAL! and an additional unit of expensive heattransfer area (DA), shouldDA be placed at the hot end, or atthe cold end? The finite-area allocation problem was genealized by considering a power plant that makes contact witan infinity of heat reservoirs distributed over a finite tem-perature interval, and by deriving based on variational calculus the optimal distribution of area over the same temperature interval~Ref. 1!.

The optimal allocation of heat exchanger inventory inthe power plant model of Fig. 25 was illustrated in Refs. 10and 14, where it was assumed that the total thermal condutance is fixed,

UHAH1ULAL5UA. ~52!

FIG. 25. Steady-state power plant model with finite hot-end and cold-enthermal conductances~from Ref. 1!.



a

o

-

s

s

t

o

eat

The power output, which was maximized once to arriveEq. ~45!, can be maximized one more time by choosing tproper ratioUHAH/(ULAL) subject to constraint~52!. Theoptimal conductance allocation rule is

~UHAH!opt5~ULAL!opt. ~53!

This optimization result also holds when the conductaninventoryUA is minimized subject to fixed power output.209

If, instead of Eq.~52!, the optimization is performed subjecto the total area constraint

AH1AL5A ~54!

the optimal way to divide the area inventory is210

AH,opt

A5

1

11~UH /UL!1/2. ~55!

Optimization results such as Eqs.~45!, ~53!, and~55! canalso be derived by minimizing the entropy generation ratethe power plant model of Fig. 25. In this alternate approait is essential to note that the optimization requires thatQH

vary, i.e., that the appropriate room-to-move entropy genetion rate197 must be included in the calculation of the totentropy generation rate. On the other hand, if the heat inQH is treated asfixed,197 the only degree of freedom left inthe optimization of the model of Fig. 25 is the allocationthe finite heat exchanger inventory. For example, if theUAinventory is constrained in accordance with Eq.~52!, thenthe optimal allocation rule continues to be Eq.~53!, with thecorresponding maximum efficiency197 @see also the discussion under Eq.~63!#:

h5Wmax

QH

512TL

THS 12

4QH

UATHD 21

. ~56!

Rubin and Andresen211 optimized the temperature staging of two power plants fitted with three finite-size heat echangers, Fig. 26. They showed that the efficiency at mamum total power (W11W2) is given by the same formula ain Eq. ~45!. The power output was maximized further in Re170, where the total thermal conductance inventory was cstrained

UHAH1UMAM1ULAL5UA. ~57!

The optimal way to allocateUA to the three heat exchangerand the resulting maximum power are

~UHAH!opt5~UMAM !opt5~ULAL!opt, ~58!

~W11W2!max51

9UATHF12S TLTHD 1/2G2. ~59!

A power plant model that combines the irreversibilimechanisms illustrated in Figs. 24 and 25 is the model wheat leak and two heat exchangers,10 Fig. 27. The poweroutput can be maximized in three ways. First, the optimtemperature ratio across the imagined reversible compment turns out to be the same as in Ref. 8,

S THCTLCDopt

5S THTL D1/2

~60!



athe

ce

t

ofch

ra-lput

f

-x-xi-

f.on-

,

yith

alart-

leading to the CNCA efficiency, Eq.~45!. Second, the finiteUA inventory~52! must be divided equally between the twheat exchangers, cf. Eq.~53!. Third, there is a tradeoff be-tween investing more in the heat exchanger equipment (UA)and in the resistanceRi to the bypass heat leak

FIG. 26. Combined-cycle power plant with three heat exchangers~fromRef. 170!.

FIG. 27. Power plant model with bypass heat leak and two finite-size hexchangers~from Ref. 10!.



u

ect

e

h

e

r

ealin-am-

eisrt-icsof

aythe

on-eut

both.ferbe-fer-

to

at-

odyEq.of

ndass

ew

Qi5(TH2TL)/Ri . It was shown that if the totalUA andRi

compete against one another in the cost constraint

pcUA1prRi5K, ~61!

wherepc andpr are the unit costs of conductance and restance, then the optimal way of allocating the cost is

pc~UA!opt5prRi ,opt. ~62!

The model of Fig. 27 was used later in the physics literatby Gordon and Huleihil,212Gordon and Orlov,213 and Pathriaet al.214

The technological applicability of the modeling featurpresented so far in this section varies. The practical implitions of results such as Eqs.~53! and~62! are clear: the heaexchanger inventory and the total cost must be dividedcertain ways. The practical meaning of the optimal tempeture staging~60! is less clear. Equation~60! means that thereis one set of optimal internal temperatures (THC,opt,TLC,opt)for each set of specified heat exchanger sizes. The mesto the designer is that the working fluid must be selectedsuch a way that it can be heated while at a certain tempture ~e.g., boil atTHC,opt in a simple ideal Rankine cycle!,and be cooled while at another optimal temperature~e.g.,condense atTLC,opt! for each given pair of heat exchangesizes,UHAH andULAL . The designer is considerably lesfree to play around with the fluid type~e.g., to abandon theuse of water! than to divide theUA inventory. This reality,the author believes, explains at least in part wChambadal’s2–4 and Novikov’s5–7,200–203optimal boiler tem-perature~44! was not noted more in engineering. Anothreason is that large scale power plants are designed~opti-mized! per unit of rate of fuel [email protected]., fixedQH ,Eq. ~56!#, not with a freely varying~infinitely abundant! heatinput.

Another way to summarize the modeling featuresviewed in this section is by comparing the reported efficiecies of actual power plants~the dots in Fig. 28! with theCNCA efficiency ~45!. The efficiency data are based oncompilation started in Ref. 8 and continued in Refs. 10 a14. The agreement between the data and Eq.~45! tends tosuggest that the irreversible nature of power plants is ctured by simple models with finite heat exchangers~e.g.,

FIG. 28. The efficiencies of: actual power plants, models with finite hexchangers, and a model with bypass heat leak optimized in harmonythe power-producing compartment~From Ref. 170!.



is-

re

sa-

inra-

sageinra-

rs

y

r

e-n-

and

ap-

Figs. 21–23 and 25–27!. However, in the presentation of thempirical efficiency dataTH was erroneously taken as equto the highest temperature reached by the working fluid,stead of the equivalent temperature of the combustion chber as an exergy source~Ref. 14!. An alternative explanationfor the position of the empirical data is provided by thmodel of Fig. 29, in which the bypass heat leak of Fig. 24optimized in harmony with the power-producing compament, i.e., in accordance with the method used in cryogen~Fig. 3!. When the entropy generated inside the insulationresistanceRi is minimized as in Eqs.~6!–~10!, the powerplant model of Fig. 29~b! delivers maximum power with theefficiency170

h512TLTH

S 11 lnTHTL

D . ~63!

The agreement between Eq.~63! and the efficiencies re-ported in Fig. 28 suggests that an actual power plant malso be viewed as an obstacle to direct heat transfer fromheat sourceTH to the heat sinkTL , i.e., as aninsulationdesigned to produce maximum power when its size is cstrained. A third alternative to explaining the position of threported efficiencies is provided by the fixed-heat-inpmodel: Equation~56! agrees well with all the plottedh dataif the dimensionless groupQH/(UATH) has a value of order0.1. The constancy of this group makes sense becauseQH andUA scale with the overall size of the power plant

Two notes on the history of the discipline of heat transcan be made at this point. The assumed proportionalitytween convective heat transfer rate and temperature difence, e.g.,

QH5UHAH~TH2TH1! ~64!

in Fig. 26, in the physics literature is sometimes referredas ‘‘Newton’s law of cooling.’’ A study of the original writ-ings shows that there is no published basis on which totribute such a formula to Newton~Ref. 65!. Newton statedthat the cooling rate of a body~i.e., the derivativedT/dt! isproportional to the temperature difference between the band the ambient. To use an analytical statement such as~64!, Newton would have needed the concepts of quantity

FIG. 29. ~a! Power plant model with bypass heat leak to the ambient, a~b! the optimization of the heat transfer interaction between the bypconductance and the power-producing compartment~From Ref. 170!.

atith



-

o

e

w

ad

is

s

inninm-ndnd

m-edentin

byso-eat

al

el-9ionon

s

a

-

y

r

e-

f

n

nt

atid

heat, specific heat and convective heat transfer coefficiewhich were conceived roughly a hundred years laterBlack,14 Wilcke,14 and Fourier,65 respectively. Proportionali-ties such as Eq.~64! are the invention of Fourier, who defined in this way the convective~fluid flow! heat transfercoefficient as a concept distinct from that of thermal condutivity. To his contemporaries and the subsequent develment of heat transfer science, Fourier’s heat transfer coecient was revolutionary: it is an important reason why hand not Biot, won the race for the development of a succeful analytical theory of conduction heat transfer.

The second observation concerns the nonlinear altertive to Eq.~64!, which in the physics literature is sometimereferred to as the ‘‘Dulong & Petit law.’’ This terminologybelongs to the mid-1800s, when it was knownempiricallythat the heat transfer relation can be more complicated thin Eq. ~64!. The readers may be interested to know that moern heat transfer is now a mature science, which is capablanticipating the nonlinear heat transfer rates based on higsuccessful theories of forced convection, natural convectiradiation, mixed convection, and conjugate heat transfer. Tpredictive powers of heat transfer science were developover the past 200 years on the back of analytical advanmade, chronologically, in heat conduction, hydrodynamicthermal radiation, aerodynamics, boundary layer theory, aconvection in porous media. The most recent overviewsthe current status of heat transfer science are presenteRefs. 65, 157, and 215.

The work that has been published on the thermodynamoptimization of power plant models is sizable. As we sawFig. 23, Novikov’s model included the effect of irreversiblexpansion through the turbine of the steam cycle. Lu,216

Grazzini,208 Ibrahim et al.,217,218 and Wu and Kiang219 ex-tended this model by also accounting for the irreversibilitythe compression process and for the fact that the temperaof the working fluid changes along the two heat exchangePetrescuet al.220 generalized the model by including the effect of piston speed and type of working fluid.

The model of Fig. 27 was extended by Swanson221 toaccount for the capacity flow rates and effectiveness—Ntu

relations of the heat exchangers. Swanson showed that incase the optimal thermal conductance allocation ratio dpends on the total number of heat transfer units of the theat exchangers. A similar model was optimized by Ibrahand Klein222 and Lee and Kim,223,224who extended it furtherto the optimization of Lorentz cycles, and by Ibrahimet al.,217 who also considered the life-cycle economic optmization of the power plant.

The four-process~two-stroke! model of Curzon andAhlborn8 and its steady-state counterpart~Fig. 25! was pur-sued along several lines. These were reviewed most receby Wu et al.,225 Feidt et al.,226 and Arias-Hernandez andAngulo-Brown,227 to whom the reader is directed. For example, in several studies beginning with Gutkowicz-Kruset al.,228 the assumption that the heat transfer rates are pportional to the local temperature differences was replacwith more general, nonlinear heat transfer models thatcount for natural convection, radiation, and temperaturependent properties.229–231 The maximum power efficiency



nt,by

c-p-ffi-e,ss-

na-s

and-ofhlyon,heedcess,ndofd in

icine

ofturers.-

thise-o

im

i-

ntly

-inro-edc-e-

formula ~45! does not hold when the heat transfer modelnot linear. Early studies were also contributed by Rubin,232

Lucca,233 Rozonoer and Tsirlin,234 and Mozurkewich andBerry.235 The effect of speed on optimal performance wastudied by Spence and Harrison236 and Petrescuet al.220

There is an important technological issue to considerconnection with the two-stroke model inspired by Ref. 8. ISadi Carnot’s 1824 essay we were told of a gas containeda cylinder and piston apparatus that underwent a cycle coposed of two strokes and four processes: two quasistatic aisothermal processes interspaced with two quasistatic aadiabatic processes. Curzon and Ahlborn8 added finite ther-mal resistances between the cylinder and the respective teperature reservoirs, and, in this way, described and optimizthe time-dependent evolution of the cycle. Although thcycle described by Curzon and Ahlborn is a good instrumefor teaching, it is a questionable roadmap to improvementsthe thermodynamic performance ofreal heat engines. Recallthat the maximized power output can be further increasedincreasing the thermal conductances associated with the ithermal processes. Can this be accomplished in a real hengine in which thesame cylinder wallis asked to be aperfect insulator during one process and a very good thermconductor during the next processin the same stroke? Theengine builders have faced this question early in the devopment of practical machines. Examples are Watt’s 176separate condenser, Brayton’s 1873 external combustchamber, and Otto and Langen’s 1876 internal combustiengine~Ref. 14!.

Another line of research focused on individual featureof the four-process model. Bandet al.237 performed the op-timization of the heating process undergone by a fluid inpiston and cylinder apparatus. Richter and Ross238 andFairen and Ross239,240 considered the effect of time-dependent operation and inertia. Orlov and Berry241 opti-mized an engine model where the working fluid is nonisothermal and viscous~with pressure drop! while in contactwith the heat reservoirs. Themechanicaloptimization of thekinematics of engines is an interesting direction defined bthe work of Senft.242–244Related to this is the cylindroidsrotary engine of Vargas and Florea.245

The maximization of work output as opposed to poweoutput was pursued by Grazzini and Gori246 and Wuet al.225

Subtle differences between the maximum power in timdependent~reciprocating! versus steady-flow power plantmodels were clarified by Kiang and Wu.247As an ecologicalfigure of merit in power plant optimization, Angulo-Brown9

proposed to maximize the functionW2TLSgen, whereSgen isthe entropy generation rate of the power plant andTL is theheat sink temperature.

Nomenclature innovations included the introduction othe term ‘‘endoreversible’’~Rubin!232 to describe the revers-ibility of the innermost compartment such as the one showin Fig. 25, or alternatively, the term ‘‘exoirreversible’’ for theexternal irreversibilities that surround the same compartme~Radcenco!.248 As pointed out by Berg,249 the concept ofinternal reversibility~or external irreversibility! is basicallythe same as the local thermodynamic equilibrium model thserves as foundation for all modern heat transfer and flu



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mechanics. The term ‘‘thermodynamics in finite time’’ waintroduced in 1977 specifically for the optimization of themodynamic processes subjected to time constraints.250

Alefeld251 analyzed an entire steam-cycle modern powstation, and showed how the entropy generation rates ctributed by the components can be summed up to evaluthe overall performance of the power plant. A similar procdure was illustrated using the simple Rankine cycle in R14. The Rankine cycle power plant was also analyzedoptimized by Wilson and Radwan,252 Roche,253 Habib andZubair,254 Smith,255 and Radcencoet al.256 Several of thesestudies emphasized the importance of matching the variatemperature of the working fluid to the temperature of theating agent. The graphic presentation and optimizationcomplex power cycles was illustrated by Jin and Ishida.257

A comprehensive treatment of the distribution of sourcof entropy generation in a gas-turbine power plant wpresent by Denton.258 The effect of turbine blade cooling onentropy generation was investigated by Farina aDonatini.259 Fundamental studies of power maximizationsimple Brayton~Joule! cycles were conducted by Leff,260

Landsberg and Leff,261Wu,262 as well as in Refs. 10 and 14Organ263 presented a detailed analysis of the distributi

of entropy generation in a Stirling-cycle power plant. Thtopic and the maximization of power received consideraattention in subsequent papers by Organ,264 Organ andJung,265Radcencoet al.,256 Ladas and Ibrahim,266 and Blanket al.267 Reviews of this field were published by Organ,268

Reader269 and, in a recent book, by Organ.270

A systematic treatment of the Otto, Brayton~Joule!, Die-sel, and Atkinson cycles was made by Leff260 and, in a gen-eralized form, by Landsberg and Leff.261 The Diesel cyclewas optimized for maximum power by Hoffmanet al.,271

and the distribution of entropy generation was studiedPrimus and Flynn.272 Papers on Otto-cycle power planwere written by Mozurkewich and Berry235 and Angulo-Brown et al.273 Internal combustion power plants were alsoptimized by Orlov and Berry:274 potentially important inpractice is the fact that optimized stroke-by-stroke cycsuch as the Otto cycle require optimal time-dependent pismotions, which, in turn, require optimal kinematics~link-ages, shapes! between piston and crankshaft.

Ocean thermal energy conversion~OTEC! power plantswere optimized by Johnson275 and Wu.276 The MHD powercycle was treated by Aydin and Yavuz277 and Human.278 Theon and off operation of power plants that have to be sdown to have their heat exchangers cleaned~defouled! wasoptimized in Ref. 279. An example of the model of timdependent operation is shown in Fig. 30, where foulingassumed to occur on the surfaces of the hot-end heatchanger. The thickness of the scale increases with time,d(t).It was shown that the period-averaged power output is mamized when the ‘‘on’’ time interval has an optimal duratioThe optimal design conditions are reported in dimensionlcharts that also hold for power plants in which fouling occuon the surfaces of the cold-end heat exchanger.



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VIII. REFRIGERATION PLANTS

The modeling features used for power plants~Secs. VIand VII! have also been used in the optimization of refrigeration plants. This body of work parallels in several respecthe work done on power plants, therefore its review in tharticle will be more brief. It is interesting that by ending thiarticle with a second look at refrigeration we are in faccompleting a circle that started in cryogenic engineerin~Sec. III!. This is another indicator of how established thmethod is, and how well rounded the field has become.

The model that was studied the most is the refrigeratcomposed of a cold-end heat exchanger~e.g., evaporator!, areversible compartment that receives power and movesentropy stream toward higher temperatures, and a rootemperature heat exchanger~e.g., condenser!. This model isshown in Fig. 31~b!. It was used first by Andresenet al.207

who focused on the optimal temperature staging of the thr

FIG. 30. Model of power plant with on and off operation and scale~fouling!accumulating on the hot-end heat exchanger surface~from Ref. 197!.

FIG. 31. ~a! Actual steady-state refrigeration plant, and~b! model with twofinite size heat exchangers~from Refs. 1 and 209!.



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compartments: unlike in the power-plant equivalent~Fig.25!, in a refrigerator there is no optimalTHC/TLC ratio forminimum power input.

The model of Fig. 31~b! was proposed independently inRef. 1 for determining the optimal allocation of a finite heaexchanger inventory. This topic was pursued along sevelines in the literature. In Ref. 1, the initial question wawhere to invest an additional unit of heat transfer area (DA)when the cold-end conductanceULAL and room temperatureconductanceUHAH are given. Goth and Feidt280 assumedthat the total heat transfer areaA is constrained according toEq. ~54!, and showed that the power input is minimum wheA is divided as shown in Eq.~55!. An independent study281

was based on the optimization based on theUA constraint~52!, and led to the conclusion that the thermal conductaninventory must be split evenly between the two heat echangers, cf. Eq.~53!. The same optimization rule applieswhen theUA inventory is minimized subject to fixed poweinput.209,282

Of technological importance is that compact design rulsuch as Eqs.~53! and ~54! lead to savings in power input,and that they apply to power plants as well. The modelFig. 31~b! was also analyzed by Yan and Chen283,284 andGrazzini.285 The optimal allocation of eitherA or UA in amodern defrosting refrigerator based on the vapor comprsion cycle with fluids R-12 or R-134a is documented numecally in Ref. 286.

A model that combines the heat exchanger irreversibties of Fig. 31~b! with the heat leak irreversibility of Fig. 3was proposed in Ref. 281~Fig. 32!. The bypass heat leak wasmodeled as proportional to the room-load temperature diffence,Qi5(TH2TL)/Ri . There are two degrees of freedomin the maximization of the refrigeration effectQL , the allo-cation of the heat exchanger inventory between the two enof the machine, and the allocation of cost between the toheat exchanger inventory and the thermal insulation. Speccally, when the heat transfer rates are modeledQHC5UHAH(THC2TH) and QLC5ULAL(TL2TLC), andthe heat exchanger inventory is constrained according to~52!, the optimalUA allocation rule is Eq.~53!. If the ther-mal conductance inventoryUA and the thermal resistance o

FIG. 32. Refrigerator model with bypass heat leak and two finite-size hexchangers~from Ref. 281!.



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the insulationRi compete in a cost formula that is the samas Eq.~61!, then the optimal cost allocation rule is given byEq. ~62!. We see again the power of compact design optimzation results such as Eqs.~53! and ~62!: they apply acrossthe board, to power plants and refrigeration plants. Thmodel of Fig. 32 was used subsequently by Agrawal anMenon287 and Chen.288 The optimal allocation of heat ex-changer inventory subject to a constraint that is qualitativerelated to Eq.~52! was documented by Klein.289

Another practical merit of this modeling method is withrespect to correlating and explaining the trends in the dareported on the performance of existing refrigeration plantTo correlate the data is important because correlationsneeded for making projections on the refrigeration needsfuture large scale projects~e.g., Ref. 40!. This line or workwas pursued in Refs. 14 and 281 for low temperature refrierators, and in Ref. 290 for near ambient refrigerators aheat pumps~chillers!.

The power of the method can be seen by examining t‘‘cloud’’ of performance data of actual refrigerators,291 Fig.33. The data cover refrigerators in the temperature rangeK,TL,90 K and capacity~size! range 0.1 W,QL,106 W.The ordinate shows the reported second-law efficiencihII5COP/~COP!rev where COP5QL/W. A three-line analysisof the model of Fig. 34 predicts thathII should vary as14,281

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, ~65!

whereCi is the thermal conductance of the leaky insulationQL5Ci(TH2TL). A comparison with Fig. 33 shows that Eq.~65! with CiTH/QL;5 is an adequate curve fit for the re-ported performance data. Equation~65! predicts several ofthe observed trends. First,hII decreases monotonically asCi

increases, which should be expected because leakier referated enclosures make less efficient refrigeration planSecond,hII decreases asTL/TH decreases. Third, the secondlaw efficiency~or the COP! increases asQL increases, whichmeans that larger machines are more efficient. Finally, tfact that the empirical data of Fig. 33 are fitted byCiTH/QL;5 means that in modern low temperature refrigerators the heat leak is of the order of five times the acturefrigeration load. Some of these trends were also anticipaby Gordon and Ng290 and Ref. 281 based on more complemodels such as Fig. 32.

Other aspects of improving the thermodynamic perfomance of refrigeration machines are discussedAlefeld.292,293 In contrast to using exergy analysis, Alefelddescribes the effectiveness of components in terms of threspective contributions to the total entropy generation raThe optimal distribution of a finite amount of insulation ovea cold enclosure of nonuniform temperature was developin Ref. 294. The application of this insulation allocation principle to a modern domestic refrigerator with two temperatucompartments~fridge and freezer! is described in Ref. 295.Even closer to modern refrigeration technology is the opmization of time-dependent refrigerators that must be dfrosted periodically. The model of Fig. 35 was sufficient fopinpointing an optimal rhythm of on and off operation whethe growth of the frost layer thicknessd(t) is known. Opti-

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FIG. 33. Compilation of the second-law efficiencies of existing refrigerators and liquifiers~from Ref. 291!.

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mization results that are directly applicable to current desand manufacture are presented in Ref. 286 for refrigerabased on the vapor compression cycle. These results cnot only the optimal control of the defrosting cycle but athe optimal allocation of a finite heat transfer area toevaporator and the condenser.

The formation of frost on cold heat exchangers is demental to refrigerator thermodynamic performance, whicwhy the optimization of the freezing and frost removal cyis an important technological issue. Most interesting is tthe same phenomenon—the same on and off cycle—caviewed and optimized as something beneficial, namely,maximization of ice production in industrial ice making istallations. Models of the same class as Fig. 35 have bused to determine the optimal freezing and removal

FIG. 34. Model of refrigeration plant with heat leak irreversibility~fromRefs. 14 and 281!.

l. Phys., Vol. 79, No. 3, 1 February 1996

13 to 152.3.194.11. This article is copyrighted as indicated in the abstra

igntorsoversothe

tri-islehatn bethe-eenfre-

quency for the manufacture of ice on the outside of horizotal tubes296 and on the inside or outside of vertical surfacof several shapes.297 The analyses also require the usecontact melting theory.298 This optimization principle is rel-evant to the production by intermittent solidification of othmaterials, not just ice.

IX. CONCLUSION

A revolution is taking place in thermodynamics, andamounts to the bridging of the gaps between thermodynaics, heat transfer and fluid mechanics. Today a commmethodology unites these seemingly separate disciplinand, when viewed as self-standing, the method and its vous models are surprisingly simple. This review showed tnew contributions are being added at an accelerated pac

FIG. 35. Model of refrigerator with unsteady operation and frost accumution on the evaporator surface~from Ref. 197!.



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both engineering and physics. The contributors are almosdiverse as the topics: engineers, physicists, chemists, andtimal control scientists. My hope299 is that this review willlead to more effective interactions between the various stors of the field, and, certainly, to a broader and more acrate view of the field.

A bird’s-eye-view of the territory we just covered ishown in Fig. 36. The field has a two-dimensional structuwith specificapplicationsadded steadily on the vertical, anwith the optimizationlevel indicated on the horizontal. Theoptimization can be approached from left, by focusing frothe start on the total system and dividing it into compaments that account for identifiable irreversibilities. In thcase, success depends greatly on the analyst’s intuition~feelfor irreversibility sources!: this stresses the importance oteachingthe entropy generation minimization method asintegral part of thermodynamics. The optimization can abe approached from the right in Fig. 36, by recognizing thsystems are made of actual components, that each comnent may contain a large number of one or more elemefeatures, and that each elemental feature owes its irreversity to processes that occur at the differential level.

The present review emphasized the fundamentaltechnological implications of the optimal results deliveredthe method. At a fundamental and pedagogical level,method EGM is changing thermodynamics itself, i.e., tway in which we think and apply thermodynamic principleFor additional review material, the reader is directed tovolume of papers edited by Sieniutycz and Salamon,300 thebooks of De Vos,160 Feidt,301 and Radcenco,302 as well as tothe books 1, 14, 23, and 197.

On the practical side, we learned repeatedly that a fiinventory of hardware can be divided optimally among twor more components of an installation. If this can be donemodels as simple as Figs. 27 and 32, then there is a

FIG. 36. The diversity and structure of the field of entropy generation mmization ~finite time thermodynamics, or thermodynamic optimizatio!~from Ref. 197!.



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opportunity to distribute the hardware optimally in the design of actual installations. The contribution made by simpmodels and the method of entropy generation minimizatiois to show the way, i.e., to uncover new opportunities for thework that will follow in industrial research and development

ACKNOWLEDGMENTS

I want to thank the Editors, Dr. J. M. Poate, and Dr. S. JRothman, for inviting me to prepare this review article. I alsthank the National Science Foundation for supporting mcurrent research on basic thermodynamics, heat transfer, afluid mechanics.

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