Exergy Int. J. 1(4) (2001) 269–277www.exergyonline.com

Thermodynamic optimization of geometryin engineering flow systems ✩

Adrian BejanDepartment of Mechanical Engineering and Materials Science, Box 90300, Duke University, Durham, NC 27708-0300, USA

(Received 31 December 2000)

Abstract—This review draws attention to an emerging body of work that relies on global thermodynamic optimization in the pursuitof flow system architecture. Exergy analysis establishes the theoretical performance limit. Thermodynamic optimization (or entropygeneration minimization) brings the design as closely as permissible to the theoretical limit. The design is destined to remainimperfect because of constraints (finite sizes, times, and costs). Improvements are registered by spreading the imperfection (e.g., flowresistances) through the system. Resistances compete against each other and must be optimized together. Optimal spreading meansspatial distribution, geometric form, topology, and geography. System architecture springs out of constrained global optimization.The principle is illustrated by simple examples: the optimization of dimensions, spacings, and the distribution (allocation) of heattransfer surface to the two heat exchangers of a power plant. Similar opportunities for deducing flow architecture exist in morecomplex systems for power and refrigeration. Examples show that the complete structure of heat exchangers for environmentalcontrol systems of aircraft can be derived based on this principle. 2001 Éditions scientifiques et médicales Elsevier SAS

1. ARCHITECTURE DEDUCED, NOTASSUMED

The most common method of modeling, analysis andoptimization in thermal engineering begins with assum-ing the system configuration—its geometry, architecture,components and manner in which the components areconnected. Next, the model is a simplified facsimile ofthe assumed system, and the analysis is the mathemat-ical description of the model and its performance. Theoptimization is the simulation of system operation undervarious conditions, and the search for the operating con-ditions (e.g., pressure ratio in a compression stage) mostfavorable for maximum performance (e.g., minimum en-tropy generation, or minimum cost).

The search for optimal design is considerably morechallenging than optimizing the operation of a single (as-sumed) configuration. In principle, one may contemplatea very large number of possible configurations, all theway to large numbers of duct shapes and aspect ratios

✩ This article is based on a keynote paper first presented in the 12thInternational Symposium on Transport Phenomena, 16–20 July 2000,Istanbul, Turkey.

E-mail address: dalford@duke.edu (A. Bejan).

in every detail of a fluid flow network. In practice, onemay assume one or two alternative configurations, op-timize their performance as in the preceding paragraph,and compare in the end the optimized alternatives. Se-lecting the best among two or three is to optimize the‘design’. Important to the subject of this review paper isthat comparing the optimized candidates represents theselection of configuration.

Configuration (geometry, topology) is the chief un-known and major challenge in design. The case-by-casechoices that we are accustomed to making are educatedguesses. Better designers make choices that work, config-urations that have been tested. Experience and longevityare useful when the needed system is envisioned as anassembly of already existing parts. Is this the way to pro-ceed in every application? Has past folklore identifiedall the possible configurations from which today’s time-pressed designer can try only a few?

It would be useful to the designer to have access fromthe beginning to the infinity of configurations that exist.Access means freedom to contemplate them all, with-out constraints based on past experience. It would alsobe useful to have a strategy (route, guide) to which ar-chitectural features lead to more promising configura-tions, and ultimately to better optimized designs. Strat-

2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved 269S1164-0235(01)00028-0/FLA

A. Bejan / Exergy Int. J. 1(4) (2001) 269–277

egy would open the designer’s eyes to the existence ofvaluable shortcuts, or ways to select out of the infinityof configurations a few narrow subclasses that offer mostpromise. This wishful thinking is useful because it rep-resents the reality of design (the limits and frustrationsof those who practice the traditional method; see the firstparagraphs). It is useful in this paper, because it serves asmotive for paying closer attention to the lost opportuni-ties that may come with assuming a single configuration.

My objective in this review is to draw attention toa growing body of work [1] in which configuration isthe unknown to be deduced, not assumed. I do thisin the hope of persuading the reader that freedom inthe selection of configuration makes sense, it is useful,and, in fact, many designers are exercising it. Thepresentation is based on a series of single examplesfrom thermodynamic optimization. The examples are“simple” because the geometric optimization pinpointsoptimal sizes and geometric aspect ratios. More complexproblems of flow topology are for the future, and couldbe attempted based on the approach illustrated in theseexamples.

2. OPTIMAL DISTRIBUTION OFIMPERFECTION

The principle that generates geometric structure is thepursuit of a global objective subject to specified globaland local constraints [1]. In the following examplesthe global objective happens to be the maximizationof thermodynamic performance, or the minimization ofthermodynamic irreversibility. A global objective couldalso be the minimization of total cost. Examples of globalconstraints are total volume, total volume, total weight,and total cost. This design method is known as entropygeneration minimization (EGM) [2–4]. The generation ofarchitecture on the basis of the “objective and constraintsprinciple” in engineering is also known as constructaldesign [1]. The thought that the same principle accountsfor the generation of geometric structure in natural flowsystems is constructal theory [1].

Here is a brief review of what thermodynamic opti-mization requires, this, on the background provided bythe disciplines of thermal engineering. Today we speak ofthree relatively new developments in engineering thermo-dynamics: the methods of exergy analysis (EA), entropygeneration minimization (EGM) and thermoeconomics(TE). Work is being done on identifying the mechanismsand system components that are responsible for thermo-dynamic losses (EA), the size of each loss (EA), the min-

imization of losses subject to global constraints (EGM),the development of system architecture based on globalthermodynamic optimization (EGM), and the minimiza-tion of the costs due to building and operating the energysystem (TE).

Exergy analysis is pure thermodynamics. It relies onthe laws of thermodynamics to establish the theoreticallimit of ideal (reversible) operation, and the extent towhich the operation of the given (actual, specified) sys-tem departs from the ideal. The departure is measuredby the calculated quantity called destroyed exergy, or ir-reversibility. This quantity is proportional to the gener-ated entropy. The proportionality is known as the Gouy-Stodola theorem [2–4]. Exergy is the thermodynamicproperty that describes the “useful energy” content, or the“work producing potential” of substances and streams. Inreal systems exergy is always destroyed, partially or to-tally, when components and streams interact.

The minimization of entropy generation requires theuse of more than thermodynamics: fluid mechanics, heatand mass transfer, materials, constraints, and geometryare also needed in order to establish the relationshipsbetween the physical configuration and the destructionof exergy. Reductions in exergy destruction are pursuedthrough changes in configuration. The resistances facedby various streams are minimized and balanced together.

In sum, the imperfection is optimally distributed.In the end, the configuration emerges as the resultof the global optimization of performance subject toglobal constraints. The following examples illustrate thisprinciple.

3. OPTIMAL DIMENSIONS

In the field of heat transfer, the EGM method bringsout the inherent competition between heat-transfer andfluid-flow irreversibilities in the optimization of devicessubjected to overall constraints. Consider the most ele-mentary heat exchanger passage [2] which is representedby a duct of arbitrary cross-section (A) and arbitrary wet-ted perimeter (p). The engineering function of the pas-sage is specified in terms of the heat transfer rate per unitlength (q ′) that is to be transmitted to the stream (m);that is, both q ′ and m are fixed. In the steady state, theheat transfer q ′ crosses the temperature gap �T formedbetween the wall temperature (T + �T ) and the bulktemperature of the stream (T ). The stream flows withfriction in the x direction; hence, the pressure gradient(−dP/dx) > 0.

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A. Bejan / Exergy Int. J. 1(4) (2001) 269–277

Taking as thermodynamic system a passage of lengthdx , the first law and the second law state that

mdh= q ′ dx and S′gen = m

ds

dx− q ′

T +�T� 0 (1)

where S′gen is the entropy generation rate per unit length.

Combining these statements with the assumption that thefluid is single phase (dh = T ds + v dP), the design-important quantity S′

gen becomes

S′gen = q ′�T

T 2(1 +�T/T )+ m

ρT

(−dP

dx

)

∼= q ′�TT 2 + m

ρT

(−dP

dx

)� 0 (2)

The denominator of the first term on the right-hand sidehas been simplified by assuming that the local tempera-ture difference�T is negligible when compared with thelocal absolute temperature T . The heat exchanger pas-sage is a site for both flow with friction and heat trans-fer across a finite �T ; the S′

gen expression thus has twoterms, each accounting for one irreversibility mechanism.We record this observation by rewriting equation (2) as

S′gen = S′

gen,�T + S′gen,�P (3)

A remarkable feature of the S′gen expression (2) and of

many like it for other simple devices is that a proposeddesign change (for example, making the passage nar-rower) induces changes of opposite signs in the two termsof the expression. An optimal trade-off then exists be-tween the two irreversibility contributions, an optimal de-sign for which the overall measure of exergy destruction(S′

gen) is minimal while the system continues to serve itsspecified function (q ′, m). To illustrate this trade-off, letus use the standard definition of friction factor, Stantonnumber, mass velocity, Reynolds number, and hydraulicdiameter

f = ρDh

2G2

(−dP

dx

)St = q ′/(p�T )

cPG

G= m

ARe = GDh

µDh = 4A

p

(4)

where q ′/(p�T ) is the average heat transfer coefficient.The entropy generation rate formula (2) becomes

S′gen = (q ′)2Dh

4T 2mcP St+ 2m3f

ρ2TDhA2(5)

Since q ′ and m are fixed, we note that the thermodynamicdesign of the heat exchanger passage has two degrees

of freedom, the perimeter p and the cross-sectional areaA, or any other pair of independent parameters, such as(Re,Dh) or (G,Dh).

The trade-off between the two irreversibilities and theminimum value of the overall S′

gen can be illustrated byassuming a special case of passage geometry, namely thestraight tube with circular cross section. In this case, pand A are related through the pipe inner diameter, D,the only degree of freedom left in the design process.WritingDh =D,A= πD2/4, and p = πD, equation (4)becomes

S′gen = (q ′)2

πT 2kNu+ 32m3f

π2ρ2TD5 (6)

where Re = 4m/(πµD) and Nu = hDh/k = St Re Pr.Invoking two reliable correlations for fully developedturbulent pipe flow, such as Nu = 0.023Re0.8Pr0.4 andf = 0.046Re−0.2, yields an expression for S′

gen that

depends only on Re. Solving dS′gen/d(Re) = 0, we find

that the entropy generation rate is minimized when theReynolds number (or pipe diameter) reaches the optimalvalue Reopt = 2.023Pr−0.07B0.36

0 . This compact formulaallows the designer to select the optimal tube size forminimal irreversibility. Parameter B0 is fixed as soonas q ′, m, and the working fluid are specified: B0 =mq ′ρ/[µ5/2(kT )1/2]. The effect of Re on S′

gen can be

expressed relative to S′gen,min = S′

gen(Reopt),

S′gen

S′gen,min

= 0.856

(Re

Reopt

)−0.8

+ 0.144

(Re

Reopt

)4.8

(7)

where the ratio on the left-hand side is known as the en-tropy generation numberNS [2]. The literature contains agrowing number of internal flow configurations that havebeen subjected to the same type of thermodynamic opti-mization: reviews can be found in Refs. [2–4].

External flow configurations are governed by the samecompetition between heat-transfer and fluid-flow irre-versibilities. The overall entropy generation rate associ-ated with external flow is [2]

S′gen = QB(TB − T∞)

T∞TB+ FDU∞

T∞(8)

where QB , TB , T∞, FD , and U∞ are, respectively, theinstantaneous heat transfer rate between the body and thefluid reservoir, the body surface temperature, the fluidreservoir temperature, the drag force, and the relativespeed between body and reservoir. This remarkably sim-ple result proves again that poor thermal contact (the first

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term) and fluid friction (the second term) contribute handin hand to degrading the thermodynamic performance ofthe external convection and configuration. One area inwhich equation (8) has found application is the problemof selecting the size and number (density) of fins for thedesign of extended surfaces [2–4].

Figure 1 shows a collection of optimal-size resultsbodies with external convection. The ordinate shows theoptimal sizes in dimensionless terms,

ReL,opt =U∞Lopt/ν and ReD,opt =U∞Dopt/ν.

Plotted on the abscissa are first, the duty parameter for theplate and cylinder, B = QB/[WU∞(kµT∞Pr1/3)1/2],where W is the dimension perpendicular to the plane offigure 1, and, second, the duty parameter for the sphere,Bs = QB/[ν(kµT∞Pr1/3)1/2].

Most of the literature on the thermodynamic optimiza-tion of heat transfer devices is devoted to more com-plex systems that incorporate the elemental features il-lustrated in this section. Examples include heat exchang-ers of several configurations (counterflow, parallel flow,cross-flow) with single-phase or two-phase flow. For thisbody of work the reader is directed to current reviews[3, 4].

Figure 1. The optimal size of a plate, cylinder, sphere forminimal entropy generation rate in external convection [4].

4. OPTIMAL INTERNAL SPACINGS

Even simpler is the sizing of a system in which asingle transport mechanism causes the irreversibility, forexample, heat transfer. When the heat current is imposed,minimizing the entropy generation means minimizing theresistance to heat flow.

In the cooling of electronic packages, both the volumeand the heat generation rate that is distributed uniformlyover that volume are fixed. The heat current is removedby a single-phase stream with natural or forced convec-tion.

The geometric arrangement of heat-generating com-ponents can be optimized such that the hot-spot temper-ature is minimal. In attempting such an arrangement, theplate-to-plate spacing (or number of columns) is free tovary. If the spacing is too large, however, there is notenough heat-transfer area and the hot-spot temperaturebecomes high; when the spacing is too small, the coolantflow rate decreases and the hot-spot temperature is againhigh. Between the two lies an optimal spacing—an op-timal package architecture—that minimizes the thermalresistance between the system and the environment. Thismethod of geometric optimization of internal architectureis known as the intersection of the asymptotes method [1,5].

The first example in this class of geometric optimawas the plate-to-plate spacing (D) in a stack of verticalplates that fill a fixed volume [5]. When the surroundingfluid (T∞) cools this assembly by natural convection, theoptimal spacing is

Dopt

H∼= 2.3Ra−1/4

H (9)

where H is the height of the assembly, and RaH is theRayleigh number based on height and hot-spot excesstemperature, RaH = gβ(Thot − T∞)H 3/(αν). The liter-ature contains optimal-spacing results for other internalcomponents cooled by natural convection, e.g., horizon-tal cylinders in laminar and turbulent flow. This activityis reviewed in Ref. [6].

The forced convection counterpart of equation (9) isequally simple. The volume contains a stack of parallelplates of spacing D and length L parallel to the flowdirection. When the flow is laminar and the pressuredifference maintained across the volume is �P , theoptimal plate-to-plate spacing is [7]

Dopt

L∼= 2.7Π−1/4

L (10)

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A. Bejan / Exergy Int. J. 1(4) (2001) 269–277

whereΠL =�P ·L2/(µα) is a new dimensionless grouprevealed for the first time by the simple analysis thatyielded equation (10). Petrescu [8] pointed out that thenew group ΠL is important because it is the forced-convection analog of the Rayleigh number. This can beseen by comparing equations (9) and (10).

If the volume is immersed in a stream that ap-proaches with the free-stream velocity U∞ , then, byusing �P ∼ ρU2∞/2 equation (10) becomes Dopt/L ∼=3.2Pr−1/4Re1/2

L , where ReL = U∞L/ν. This and otherinternal configurations have been the subject of recent re-views [6, 9]. The same method was used recently to pre-dict the formation of optimally-spaced cracks in solids(e.g., wet mud) that shrink upon cooling or drying [1].

5. OPTIMAL ALLOCATION OF HEATTRANSFER AREA INVENTORY

Much of what has been contributed fundamentally toconvective heat transfer benefits the process of designingheat exchangers. This process requires information on therelationship between heat transfer rates, temperature dif-ferences and types of surfaces. Interesting and very basicconclusions follow if we think beyond fundamental heattransfer, and ask questions inspired by thermodynamicsand potential fields of application.

Consider the fact that heat exchangers are alwayspresent in the design of power plants and refrigerationplants. In the model of figure 2, the operation of an actual(irreversible) power plant can be approximated by thetwo heat exchangers and the inner (reversible) powercycle executed by the working fluid. The simplest heatexchanger models are [2]

QH =UHAH�TH and QL =ULAL�TL (11)

for which the thermal conductances UHAH and ULALare furnished by the field of fundamental heat transfer.A global constraint that accounts for the size and costof the heat transfer area (AH ,AL) and heat transferaugmentation (UH ,UL) is the total thermal conductanceconstraint [3]

UHAH +ULAL =UA (constant) (12)

In this model, the reversible cycling of the working fluidis represented by the second law

QH

TH −�TH= QL

TL +�TL(13)

Figure 2. Model of steady-state irreversible power plant, withfinite hot-end and cold-end heat exchangers [2].

The basic question for the designer is how to split theUA inventory between the two heat exchangers such thatthe power output W = QH − QL is maximal. It is notdifficult to show that the energy conversion efficiency forthis model (η=W/QH) can be written as [1, 3]

η= 1 −(TL

TH

)/[1 − QH

THUA

(1

x+ 1

1 − x

)](14)

where x is the inventory allocation ratio, x = UHAH/

(UA). We assume that the heat input QH is fixed, i.e.,we optimize the performance on a per-unit (heat input,or fuel) basis. Equation (14) shows that the optimal xfor maximal η is xopt = 1/2, which means that UHAH =ULAL and

ηmax = 1 −(TL

TH

)/(1 − 4QH

THUA

)(15)

The maximized efficiency (15) is expectedly lower thanthe Carnot efficiency associated with the same tempera-ture extremes, ηC = 1−TL/TH . In fact, when the dimen-sionless group QH/(THUA) is a constant of the orderof 0.1, the ηmax values furnished by equation (15) agreewell with the reported η data of ten power plants reviewed

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A. Bejan / Exergy Int. J. 1(4) (2001) 269–277

in [3]. The constancy of the QH/(THUA) group makessense because in the real world both QH and UA scalewith the overall size of the power plant.

6. STRUCTURE IN HEAT EXCHANGERS

More complex structures can be derived via thermo-dynamic optimization when the system models are morecomplex. As an example, consider a counterflow heat ex-changer with channels formed by parallel plates [10]. Thetwo fluids are ideal gases, and the flow is fully developed,laminar or turbulent. The constraints are the total volumeand the total weight (or the total volume of wall mater-ial). The core parameters and dimensions are defined infigure 3.

The geometry was optimized by minimizing the en-tropy generation rate Sgen, which was calculated in twoways. First, the Sgen formula accounted only for the ir-reversibility associated with the core—the rectangularspace in the upper part of figure 3,

Sgen = (mcp

)a

[lnT2

T1−

(R

cp

)a

lnP2

P1

]

Figure 3. Counterflow heat exchanger structure [10].

+ (mcp

)e

[lnT4

T3−

(R

cp

)e

lnP4

P3

](16)

The thermofluid performance of the core was modeledaccording with the classical NTU-effectiveness formula-tion, and the pressure drop calculations accounted for en-trance, exit and flow acceleration (density changes) [11].In summary, the parallel-plates geometry is characterizedby six dimensionless parameters, the number of channels(n) and the five dimensions(

H , L, W , Da, De) = (H,L,W,Da,De)/V

1/3 (17)

where V is the core volume. Only three parameters arefree to vary, because of the geometric relations

H = n

2

(De + Da

)(18)

H LW = 1 (19)

S = nLW (S = S/V 2/3) (20)

Equation (19) is the volume constraint. Because thethickness of each plate is assumed given and negligible,the mass constraint is the fixed total plate surface S =nLW , or equation (20).

Table I lists the physical parameters that were heldconstant during numerical optimization, in which weminimized the dimensionless entropy generation numberNS = Sgen/(mcp)e. Capacity rates are also denoted byC = mcp. The properties of the two air streams wereestimated at the specified inlet conditions. The propertiesµ, cp and Pr were assumed constant. Figure 4 showsthat there exists an optimal ratio of channel spacingsDe/Da such that NS is minimal. This effect is dueto the entropy generation due to pressure drops (thedifferenceNS −NST =NSP in figure 4) increases in bothextremes, Da/De � 1 and Da/De 1. The entropygeneration rate contributed by imperfect thermal contact(NST , the dashed line) is relatively insensitive to varying

TABLE IPhysical parameters held constant during the numericaloptimization of the heat exchanger of figure 3 [10].

cp,a = 998 J·kg−1·K−1 Re = 287 J·kg−1·K−1

cp,e = 1,010 J·kg−1·K−1 T1 = 244 Kme = 0.1 kg·s−1 T3 = 359 KPra = 0.72 V = 1 m3

Pre = 0.71 µa = 1.6 × 10−5 Pa·sP1 = 0.031 MPa µe = 2.14 × 10−5 Pa·sP3 = 0.27 MPa ρa = 0.442 kg·m−3

Ra = 287 J·kg−1·K−1 ρe = 2.61 kg·m−3

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A. Bejan / Exergy Int. J. 1(4) (2001) 269–277

Figure 4. The optimized ratio of channel spacings, the corre-sponding number of channels, and the effect of varying theaspect ratio of each heat transfer surface (W/L) [10].

the channel spacings. The optimized spacing ratio isalso reported as channel aspect ratios in the lower partof figure 4. The minimized entropy generation rate NSvaries monotonically with the remaining free parameters,H/L and W/L.

NS can be minimized for a second time by varying thecontact area S , i.e., by relaxing the weight constraint (20):see the upper part of figure 5. The lower part of the figureshows a sample of results for configurations in whichboth S and De/Da have been optimized. The number ofheat transfer units is N = UA/(mcp)e, where (mcp)e <(mcp)a . The charts for calculating the correspondingdimensions are reported in [10].

The thermodynamic optimization described in fig-ures 4 and 5 was concerned solely with the irreversibilityof the counterflow heat exchanger. There is another wayof approaching this optimization problem, and it is basedon the view that in some applications the function of theheat exchanger is to extract exergy from the hot streamand to deliver this quantity to the cold stream. In such

Figure 5. The effectiveness, number of channels and numberof heat transfer units that correspond to the optimized S andDe/Da [10].

cases the spent hot stream (state 4 in figure 3) is not use-ful anymore, and is discharged into the ambient.

The mixing triggered by the discharge serves asan additional source of irreversibility. This situation issketched in figure 6, where the total entropy generatedinside the heat exchanger and the neighboring regionaffected by the discharge (the dashed line) is

Sgen = me(s0 − s3)+ ma(s2 − s1)+ Q

T0(21)

The cooling rate experienced by the exhaust is Q =(mcp)e(T4 − T0). A new feature is that the capacityratio (mcp)e/(mcp)a covers a wider range, taking valuessmaller and larger than 1, and consequently the ε-NTUrelations had to be modified when (mcp)e and (mcp)aswitched places in the Cmin definition. The hot stream ispractically at atmospheric pressure, because the pressureof the exhaust is atmospheric, P4 = P0.

The step-by-step minimization of the total NS isillustrated in figure 7. First, NS has a shallow minimumwith respect to De/Da : the top frame of figure 7 is verysimilar to the corresponding step in the minimization

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Figure 6. Heat exchanger and neighboring region affected bydischarging the spent hot stream [10].

Figure 7. The three-way minimization of the total entropygeneration rate of the aggregate system defined in figure 6[10].

of the heat exchanger irreversibility alone (figure 4).Important is that the optimal geometry, (De/Da)opt, canbe expected to be nearly the same as in the first part ofthis section.

Next, the total entropy generation rate minimized withrespect to De/Da exhibits a minimum with respect to thesize of the contact surface, S. New is the third minimumshown in the bottom frame of figure 7, where the twice-

minimized total entropy generation rate NS,mm is plottedversus the capacity rate ratio. The lowest NS,mm valueis finite, and in the case of figure 7 it corresponds tothe case of nearly balanced counterflow, Ce/Ca ∼= 1. Theexistence of an optimum with respect to the capacity rateratio is a characteristic of systems driven by the exergydrawn from a stream of hot gas, where the stream iseventually discharged into the ambient. This particularoptimum was demonstrated in a recent study [12], inwhich the arrangement and geometric aspect ratios of theheat exchanger were not considered.

7. CONCLUSIONS

Putting the first two frames of figure 7 together andcomparing them with figures 4 and 5, we concludethat the optimization of the internal architecture of theheat exchanger is robust with respect to whether theexternal thermal-mixing irreversibility is included inthe calculation. Robustness is valuable because it cansimplify the design. The broader issue is the observationthat geometric “structure” springs out of the process ofthermodynamic optimization subject to constraints. Toillustrate this principle, we used the simplest possiblemodel of counterflow heat exchanger: smooth parallelplates with fully developed flow on both sides.

Similar geometric features are discovered in the opti-mization of increasingly more complex systems that em-ploy heat exchangers. An example is the crossflow heatexchanger used in the environmental control system foran aircraft, figure 8. The heat exchanger brings together ahot air stream (drawn from the engine) with a cold ram-air stream. The latter passes through a diffuser beforeentering the heat exchanger, and is discharged througha nozzle. In the first part of the study [13] the heat ex-changer entropy generation rate was minimized alone,subject to fixed total volume and volume fraction occu-pied by solid walls. Optimized geometric features suchas the ratio of channel spacings and flow lengths werereported. It was found that the optimized features are rel-atively insensitive to changes in other physical parame-ters of the installation. In the second part of the studythe entropy generation rate also accounted for the irre-versibility due to discharging the ram-air stream into theatmosphere. The optimized geometric features were rel-atively insensitive to this additional effect, emphasizingthe robustness of the thermodynamic optimum.

This optimization opportunity deserves to be pursuedin designs of greater complexity. In this direction, thechallenge that we must face is the number of degrees of

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Figure 8. Crossflow heat exchanger with engine air on the hot side and ram air on the cold side [13].

freedom, which increases as the complexity of the con-templated system increases. Relevant to this is whetherthe thermodynamic optima of simpler systems are robustenough to reappear at least approximately in larger as-semblies. In such cases the results obtained for simplersystems can be used as shortcuts in the optimization oflarger and more complex constructs.

Continuing on the path from isolated heat exchang-ers (figure 3) to assemblies for environmental control sys-tems for aircraft (figure 8), it is possible to contemplatethe thermodynamic optimization of architecture at theaircraft-system level. Indeed, by minimizing the powerrequired for sustaining flight, it is possible to predict theobserved cruising speeds of all the bodies that fly, nat-ural and engineered [1]. The theoretical cruising speed Vis proportional to the body mass M raised to the power1/6. The same principles apply to designs in which all thefunctions are driven by the exergy drawn from the limitedfuel installed on board: ships, automobiles, military ve-hicles, environmental-control suits, portable power tools,weapons systems, etc.

The performance record of the natural and engineereddesigns [1] suggests that the principle invoked in this pa-per (constructal law) is important not only in engineer-ing but also in physics and biology in general. In thistheoretical framework the airplane emerges as a physicalextension of man, in the same way that the body of theflying animal (e.g., bat, bird, insect) developed its ownwell adapted extensions. All such extensions are discretemarks on a continuous time axis that points toward thebetter and the more complex [1].

REFERENCES

[1] Bejan A., Shape and Structure, from Engineering toNature, Cambridge Univ. Press, Cambridge, UK, 2000.

[2] Bejan A., Entropy Generation through Heat andFluid Flow, Wiley, New York, 1982.

[3] Bejan A., Advanced Engineering Thermodynamics,2nd edn., Wiley, New York, 1997.

[4] Bejan A., Entropy Generation Minimization, CRCPress, Boca Raton, FL, 1996.

[5] Bejan A., Convection Heat Transfer, Wiley, NewYork, 1984, Problem 11, p. 157.

[6] Bejan A., Convection Heat Transfer, 2nd edn.,Wiley, New York, 1995.

[7] Bejan A., Sciubba E., The optimal spacing of parallelplates cooled by forced convection, Internat. J. Heat MassTransfer 35 (1992) 3259–3264.

[8] Petrescu S., Comments on the optimal spacing ofparallel plates cooled by forced convection, Internat. J. HeatMass Transfer 37 (1994) 1283.

[9] Kim S.J., Lee S.W. (Eds.), Air Cooling Technology forElectronic Equipment, CRC Press, Boca Raton, FL, 1996.

[10] Ordonez J.C., Bejan A., Entropy generation mini-mization in parallel-plates counterflow heat exchangers, In-ternat. J. Energy Res. 24, 843–864.

[11] Kays W.M., London A.L., Compact Heat Exchangers,3rd edn., McGraw-Hill, New York, 1984.

[12] Bejan A., Errera M.R., Maximum power from a hotstream, Internat. J. Heat Mass Transfer 41 (1998) 2025–2036.

[13] Alebrahim A., Bejan A., Entropy generation mini-mization in a ram-air cross-flow heat exchanger, Internat.J. Appl. Thermodynamics 2 (1999) 145–157.

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