fundamental of exergy analysis.pdf

*Correspondence to: Adrian Bejan, Department of Mechanical Engineering and Materials Science, Duke University,Durham, NC 27708-0300, U.S.A.

Received 16 January 2001Copyright � 2002 John Wiley & Sons, Ltd. Accepted 4 June 2001

INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2002; 26:545}565 (DOI: 10.1002/er.804)

Fundamentals of exergy analysis, entropy generationminimization, and the generation of #ow architecture

Adrian Bejan*

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, U.S.A.

SUMMARY

This paper outlines the fundamentals of the methods of exergy analysis and entropy generation minimiz-ation (or thermodynamic optimization*the minimization of exergy destruction). The paper begins witha review of the concept of irreversibility, entropy generation, or exergy destruction. Examples illustrate theaccounting for exergy #ows and accumulation in closed systems, open systems, heat transfer processes, andpower and refrigeration plants. The proportionality between exergy destruction and entropy generationsends the designer in search of improved thermodynamic performance subject to "nite-size constraints andspeci"ed environmental conditions. Examples are drawn from energy storage systems for sensible heat andlatent heat, solar energy, and the generation of maximum power in a power plant model with "nite heattransfer surface inventory. It is shown that the physical structure (geometric con"guration, topology) of thesystem springs out of the process of global thermodynamic optimization subject to global constraints. Thisprinciple generates structure not only in engineering but also in physics and biology (constructal theory).Copyright � 2002 John Wiley & Sons, Ltd.

KEY WORDS: exergy analysis; entropy generation minimization; EGM; thermodynamic optimization;constructal theory; topology optimization; self-organization in nature; self-optimization innature

1. EXERGY, NOT ENERGY

The energy crisis of the 1970s and the continuing emphasis on e$ciency (conservation of fuelresources) has led to a complete overhaul of the way in which power systems are analyzed andimproved thermodynamically. The new methodology is exergy analysis and its optimizationcomponent is known as thermodynamic optimization, or entropy generation minimization (EGM).This new approach is based on the simultaneous application of the "rst law and the second law inanalysis and design (Evans, 1969; Reistad, 1970; Nerescu and Radcenco, 1970; Brodyanskii, 1973;Haywood, 1980; Ahern, 1980). In the 1990s it has become the premier method of thermodynamic

analysis in engineering education (Moran, 1989; Bejan, 1982, 1988; Feidt, 1987; Sieniutycz andSalamon, 1990; Kotas, 1995; Moran and Shapiro, 1995; Radcenco, 1994; Shiner, 1996; Bejan et al.,1996) and it is now sweeping every aspect of engineering practice (Stecco and Moran, 1990, 1992;Valero and Tsatsaronis, 1992; Bejan and Mamut, 1999; Bejan et al., 1999). It is particularly wellsuited for computer-assisted design and optimization (Sciubba and Melli, 1998; Sciubba,1999a, b). In this paper we review the fundamentals of the method, its current status, and a fewexamples of the ways in which the method can be used for system optimization. Extensive reviewsof the greatly expanding exergy and entropy generation "eld have been published (Bejan,1996a, b, 1997).

To begin with, we must distinguish between exergy and energy in order to avoid any confusionwith the traditional energy-based methods of thermal system analysis and design. Energy #owsinto and out of a system along paths of mass #ow, heat transfer, and work (e.g. shafts, piston rods).Energy is conserved, not destroyed: this is the statement made by the "rst law of thermodynamics.

Exergy is an entirely di!erent concept. It represents quantitatively the &useful' energy, or theability to do or receive work*the work content*of the great variety of streams (mass, heat,work) that #ow through the system. The "rst attribute of the property &exergy' is that it makes itpossible to compare on a common basis di!erent interactions (inputs, outputs, work, heat).

Another bene"t is that by accounting for all the exergy streams of the system it is possible todetermine the extent to which the system destroys exergy. The destroyed exergy is proportional tothe generated entropy. Exergy is always destroyed, partially or totally: this is the statement madeby the second law of thermodynamics. The destroyed exergy, or the generated entropy isresponsible for the less-than-theoretical thermodynamic e$ciency of the system.

By performing exergy accounting in smaller and smaller subsystems, we are able to draw a mapof how the destruction of exergy is distributed over the engineering system of interest. In this waywe are able to pinpoint the components and mechanisms (processes) that destroy exergy the most.This is a real advantage in the search for improving e$ciency (always by "nite means), because ittells us from the start how to allocate engineering e!ort and resources. To the optimal allocation ofresources we return in the example of Section 4.1.

In engineering thermodynamics today, emphasis is placed on identifying the mechanisms andsystem components that are responsible for thermodynamic losses, the sizes of these losses (exergyanalysis), minimizing the losses subject to the global constraints of the system (entropy generationminimization), and minimizing the total costs associated with building and operating the energysystem (thermoeconomics). To review the fundamentals of thermoeconomics is not the objectiveof this paper. A description of thermoeconomics and its applications, as an evolutionarydevelopment beyond exergy analysis and entropy generation minimization, is available inBejan et al. (1996).

The method of thermodynamic optimization or entropy generation minimization (EGM) isa "eld of activity at the interface between heat transfer, engineering thermodynamics, and #uidmechanics. The position of the "eld is illustrated in Figure 1. The method relies on thesimultaneous application of principles of heat and mass transfer, #uid mechanics, and engineeringthermodynamics, in the pursuit of realistic models of processes, devices, and installations. Byrealistic models we mean models that account for the inherent irreversibility of engineeringsystems and processes.

Thermodynamic optimization may be used by itself (without cost minimization) in thepreliminary stages of design, in order to identify trends and the existence of optimizationopportunities. The optima and structural characteristics identi"ed based on thermodynamic

546 A. BEJAN

Copyright � 2002 John Wiley & Sons, Ltd. Int. J. Energy Res. 2002; 26:545}565

Figure 1. The interdisciplinary "eld covered by the method of thermodynamic optimization, or entropygeneration minimization (Bejan, 1982). The lower diagram is due to Smith (2000).

EXERGY ANALYSIS AND ENTROPY GENERATION MINIMIZATION 547


Figure 2. General de"nition of #ow system (open thermodynamic system) in communicationwith the atmosphere (Bejan, 1988).

optimization can be made more realistic through subsequent re"nements based on global costminimization, by using the method of thermoeconomics. Space limitations do not permit thismethod to be covered in this article, however, examples of its use are mentioned in other articlesin this issue. Modern presentations of thermoeconomics can be found in Benelmir et al. (1991,1992, 1997), Von Spakovsky (1994), Von Spakovsky and Frangopoulos (1993, 1994), Bejan et al.(1996), Benelmir and Feidt (1997), Tsatsaronis (1999), Valero et al. (1999), and Olsommer et al.(1999a, b).

2. ENTROPY GENERATION, OR EXERGY DESTRUCTION

Here is why in thermodynamic optimization we must rely on heat transfer and #uid mechanics,not just thermodynamics. Consider the most general system}environment con"guration, namelya system that operates in the unsteady state, Figure 2. Its instantaneous inventories of mass,energy, and entropy are M, E, and S. The system experiences the net work transfer rate=Q , heattransfer rates (QQ

�, QQ

�,2,QQ

�) with n#1 temperature reservoirs (¹

�, ¹

�,2,¹

�), and mass #ow

rates (mR��, mR

��) through any number of inlet and outlet ports. Noteworthy in this array of

interactions is the heat transfer rate between the system and the environmental (atmospheric)temperature reservoir, QQ

�, on which we focus shortly.

The thermodynamics of the system consists of accounting for the "rst law and the second law(e.g. Moran and Shapiro, 1995; Bejan, 1997),

dE

dt"

��

QQ�!=Q #�

��

mR h!��

mR h (1)

548 A. BEJAN


SQ��

"

dS

dt!

��

QQ�

¹�

!��

mR s#��

mR s*0 (2)

where h is shorthand for the sum of speci"c enthalpy, kinetic energy, and potential energyof a particular stream at the boundary. In Equation (2) the total entropy generation rate SQ

��is simply a de"nition (notation) for the entire quantity on the left-hand side of the inequalitysign. We shall see that it is advantageous to decrease SQ

��; this can be accomplished by

changing at least one of the quantities (properties, interactions) speci"ed along the systemboundary.

We select the environmental heat transfer QQ�

as the interaction that is allowed to #oat asSQ��

varies. Historically, this choice was inspired (and justi"ed) by applications to powerplants and refrigeration plants, because the rejection of heat to the atmosphere was of littleconsequence in the overall cost analysis of the design. Eliminating QQ

�between Equations (1)

and (2) we obtain

=Q "!

d

dt(E!¹

�S)#

��1!

¹�

¹�� QQ

�#�

��

mR (h!¹�s)!�

��

mR (h!¹�s)!¹

�SQ��

(3)

The power output or input in the limit of reversible operation (SQ��

"0) is

=Q��

"!

d

dt(E!¹

�S)#

��1!

¹�

¹�� QQ

�#�

��

mR (h!¹�s)!�

��

mR (h!¹�s) (4)

In engineering thermodynamics each of the terms on the right-hand side of Equation (4) isrecognized as an exergy of one type or another (see Section 3), and the calculation of =Q

��is known as exergy analysis. Subtracting Equation (3) from Equation (4) we arrive at theGouy-Stodola theorem,

=Q��

!=Q "¹�SQ��

(5)

In Equation (5) =Q��

is "xed because all the heat and mass #ows (other than QQ�) are "xed.

Pure thermodynamics (e.g., exergy analysis) ends, and the method of entropy generationminimization (EGM) begins with Equation (5). The lost power (=Q

��!=Q ) is always positive,

regardless of whether the system is a power producer (e.g. power plant) or a power user (e.g.refrigeration plant). To minimize lost power when=Q

��is "xed is the same as maximizing power

output in a power plant, and minimizing power input in a refrigeration plant. This operation isalso equivalent to minimizing the total rate of entropy generation.

The critically new aspect of the EGM method*the aspect that makes the use of thermo-dynamics insu$cient, and distinguishes EGM from pure exergy analysis*is the minimization ofthe calculated entropy generation rate. Optimization and design (the generation of structure)are the di!erence. To minimize the irreversibility of a proposed con"guration, the analyst mustuse the relations between temperature di!erences and heat transfer rates, and between pressuredi!erences and mass #ow rates. The analyst must express the thermodynamic nonideality ofthe design SQ

��as a function of the topology and physical characteristics of the system, namely,

"nite dimensions, shapes, materials, "nite speeds, and "nite-time intervals of operation. For this



Figure 3. Closed system and process en route to thermal and mechanical equilibrium with the environment.

the analyst must rely on heat transfer and #uid mechanics principles, in addition to thermo-dynamics. Only by varying one or more of the physical characteristics of the system, can theanalyst bring the design closer to the operation characterized by minimum entropy generationsubject to size and time constraints. We illustrate this technique by means of a few very basicmodels in Section 4.

3. EXERGY ANALYSIS

There is a rich nomenclature and mathematical apparatus associated with de"ning and calculat-ing the exergies of various entities. Most of these exergy names are attached to the four types ofterms shown on the right side of Equation (4). This nomenclature must be used with care,especially now as the method is applied for the "rst time to areas where energy-based methods arestill the norm. The key feature is this: exergy is the maximum (theoretical) work that can beextracted (or the minimum work that is required) from the entity (e.g. stream, amount of matter)as the entity passes from a given state to one of equilibrium with the environment. As such, exergyis a measure of the departure of the given state from the environmental state*the larger thedeparture, the greater the potential for doing work.

To illustrate the calculation of exergy, consider the following examples in which the environ-mental state is represented by the atmospheric temperature ¹

�and pressure P

�. If the entity is

a closed system ("xed mass, Figure 3) at an initial state represented by the energy E, entropy S andvolume <, then its exergy � (expressed in joules) relative to the environment is

�"E!E�!¹

�(S!S

�)#P

�(<!<

�) (6)

In this expression � is known as the non-ow exergy of the given mass (Moran, 1989; Bejan, 1988),and the subscript 0 indicates the system properties in the state of thermal and mechanicalequilibrium with the environment. The environmental state is also known as the restricted dead

550 A. BEJAN


Figure 4. Open system, steady state, and one stream en route to thermal and mechanicalequilibrium with the environment.

state*&dead' because once in this state the system cannot deliver any more work relative to theenvironment. It is &restricted' because in this state the system is in thermal and mechanicalequilibrium with the environment, but not in chemical equilibrium. Equation (6) is general in thesense that the internal construction and materials (e.g. single phase vs. multi phase) of the givenmass are not speci"ed. Equation (6) can be generalized further for cases where the chemicalcomposition of the given mass may change en route to chemical equilibrium with the environ-ment (Moran, 1989; Bejan, 1988).

Note that the non-#ow exergy � has its origin in the "rst term identi"ed on the right-hand sideof Equation (3) or Equation (4). The system &accumulates' the quantity (E!¹

�S ) as potential

work, and, in going from left to right in Figure 3, this work potential decreases from (E!¹�S ) to

(E�!¹

�S�). The di!erence, which is E!E

�!¹

�(S!S

�), represents all the work that could be

produced during the process. From this quantity we must subtract the work fraction done by thesystem against the atmosphere, P

�(<

�!<). The resulting expression is Equation (6).

As a second exergy calculation example, consider the steady-#ow system with a stream of mass#ow rate mR , where the given (inlet) state is represented by the speci"c enthalpy h, entropy s, kineticenergy �

�<�, and gravitational potential energy gz, where z is the altitude of the inlet. The speci"c

-ow exergy is expressed in J kg��, and is evaluated relative to the environment (¹�, P

�):

e�"(h#�

�<�#gz)!(h#�

�<�#gz)

�!¹

�(s!s

�) (7)

As shown in Figure 4, the subscript 0 indicates the properties of the stream that reached thermaland mechanical equilibrium with the environment. In other words, e

�is the change in the value of

the group h#��<�#gz!¹

�s, in going from the inlet to the outlet. The exergy #ow rate of this

stream is the product mR e�. More general versions of Equation (7) are available for streams that

can also exchange chemical species with the environment (Moran, 1989; Bejan, 1988).The streams of #ow exergy were identi"ed already in the third and fourth terms of Equations

(3) and (4), where h was shorthand for the sum (h#��<�#gz). Continuing with this shorthand

notation, we note that the group (h!¹�s) represents the speci"c -ow availability of the stream.

The di!erence between the #ow availability at the indicated state (inlet, Figure 4) and the #ow



availability of the same stream at the environmental state (outlet, Figure 4) is the #ow exergy,namely e

�"(h!¹

�s)!(h

�!¹

�s�), which is the same as Equation (7).

A third example is the exergy content of heat transfer. If the heat transfer rate QQ enters thesystem by crossing a boundary of local temperature ¹, then its exergy stream relative to theenvironment (¹

�) is

EQ�"QQ �1!

¹�

¹ � (8)

Alternative notations used for EQ�

are EQ�and EQ

��. The heat-transfer exergy #ow rate EQ

�is zero

when ¹"¹�, i.e. as in the case of the heat rejected to the ambient by power and refrigeration

plants. In Figure 2 an exergy stream is associated with each of the heat inputs QQ�,2,QQ

�, while the

QQ�

stream carries no exergy EQ��

"QQ�(1!¹

�/¹

�)"0. The exergy #ows associated with heat

transfer are accounted for by the second term on the right-hand side of Equations (3) and (4).It is important to stress that in Figure 2 and Equation (8) ¹

�is the temperature of those regions

of the environment that are su$ciently close to the system but not a!ected by the discharge. Thepurpose of this modeling decision is to place inside &the system' all the irreversibilities associatedwith the internal and external e!ect of the physical installation that resides inside the system. Thesystem comprised the installation and the surrounding regions that are a!ected (e.g. heated) bythe installation. In an actual power or refrigeration plant, the rejected heat current QQ leaves theinstallation and enters the neighbouring environmental #uid (air, water) at a temperaturesomewhat higher than ¹

�. Further down the line, the same QQ reaches the true environment at

¹�

(i.e. it crosses the ¹�

boundary, Figure 2). The interaction between energy systems and theirsurrounding #uids forms the subject of the environmental #ows documented in several articles inthis issue.

Exergy accounting and the spatial distribution of exergy destruction are illustrated in Figure 5.The top "gure shows the traditional, energy-#ow analysis of a simple Rankine cycle power plant.The heat input is QQ

�, and the net power output is =Q

�!=Q

"�

��E��

. The fraction ��

is thesecond-law or rational e.ciency of the power plant. It is a relative measure of the combinedimperfections of the power plant. The corresponding exergy wheel for the #ows through andaround a refrigeration plant is illustrated in Figure 6.

The calculated widths of the exergy destruction streams indicate a ranking of the componentsas candidates for thermodynamic optimization. The exergy #ow analysis (Figures 5 and 6,bottom) can be performed inside each component in order to determine the particular features(e.g. combustion, fouling, heat transfer, pressure drop) that dominate the irreversibility of thatcomponent. Finally, the success of the thermodynamic improvements that are implemented canonly be evaluated by repeating the exergy analysis and registering the changes in exergydestruction and second-law e$ciency.

The current literature reviewed in references shows that exergy principles are being applied toa wide variety of thermal/chemical processes. Avoidable destructions of exergy represent thewaste of exergy sources such as oil, natural gas and coal. By devising ways to avoid thedestruction of exergy, better use can be made of fuels. Exergy analysis determines the location,type, and true magnitude of the waste of fuel resources, and plays a central role in developingstrategies for more e!ective fuel use.

552 A. BEJAN


Figure 5. The conversion and partial destruction of exergy in a power plant based on the simple Rankinecycle. Top: the traditional notation and energy-interaction diagram. Bottom: the exergy wheel diagram

(Bejan, 1988), and the de"nition of the second law e$ciency ��.

4. ENTROPY GENERATION MINIMIZATION

Most of the newest work and opportunities for advances are in developing strategies for theoptimal allocation (con"guration, topology) of resources. This work is known as exergy destruc-tion minimization, irreversibility (entropy generation) minimization, or thermodynamic optim-ization. It is often subjected to overall constraints such as "nite sizes, "nite times, materialtypes, and shapes. Recent reviews of the literature (Bejan, 1996a, b) show that thermodynamic



Figure 6. The conversion and partial destruction of exergy in a refrigeration plant based on the simple vaporcompression cycle. Top: the traditional notation and energy-interaction diagram. Bottom: the exergy wheel

diagram (Bejan, 1988), and the de"nition of the second law e$ciency ��.

optimization is making fast progress in cryogenics, heat transfer engineering, energy storagesystems, solar power plants, fossil-fuel power plants, and refrigeration plants (Feidt, 1998). Theexamples collected in this section illustrate the opportunities for devising strategies of optimalallocation.

4.1. Optimal allocation of heat transfer area

Heat transfer principles combined with thermodynamics shed light on why energy systems areimperfect, and why they possess geometric structure*why their hardware is arranged in certain

554 A. BEJAN


Figure 7. Model of power plant with two heat transfer surfaces, and the maximization of power output(or the minimization of entropy generation) subject to "xed heat input (Q

�) and "xed total heat

transfer surface (C�#C

�"C, constant).

amounts, and in certain ways in space. A power plant owes its irreversibility to many factors, oneof which is the transfer of heat across "nite temperature di!erences. This e!ect has been isolatedin Figure 7. The power plant is the vertical segment marked between the high temperature ¹

�and

the ambient temperature ¹�. The heat input Q

�("xed) and the rejected heat Q

�must be driven by

temperature di!erences: the temperature gaps ¹�!¹

�and ¹

�!¹

�account for some of the

space occupied by the power plant. Heat transfer surfaces reside in these spaces. The rest of thespace is reserved for the rest of the power plant: for simplicity, this inner space is assumed to beirreversibility free (endoreversible),

S��

"

Q�

¹�

!

Q�

¹�

"0 (9)

The entire irreversibility of this power plant model is concentrated in the spaces occupied by thetwo temperature gaps. The simplest heat transfer model for these spaces is the proportionalitybetween heat current and temperature di!erence,

Q�"C

�(¹

�!¹

�) (10)

Q�"C

�(¹

�!¹

�) (11)

Each thermal conductance (C�, C

�) is proportional to its area for heat transfer, e.g. C

�";

�A

�and C

�";

�A

�, where each heat transfer area (A

�, A

�) is multiplied by its corresponding overall

heat transfer coe$cient (;�,;

�). When;

�and;

�are equal (or of the same order of magnitude),

the total area constraint (A�#A

�"A, constant) is represented adequately by the total conduc-

tance constraint (Bejan, 1988, 1997)

C�#C

�"C (12)



where C is "xed. More general constraints, valid for unequal heat transfer coe$cients, can also beused (Bejan, 1996a, b).

The analytical model is completed by the "rst law, written for the power plant as a closedsystem operating in steady state or in an integral number of cycles, ="Q

�!Q

�. Combining

this with the preceding relations, we obtain the power output as a function of the conductanceallocation fraction x"C

�/C (in heat exchanger terminology this dimensionless parameter is also

known as C*),

=

Q�

"1!

¹�/¹

�

1!(Q�/¹

�C) �

1

x#

1

1!x�(13)

The minimization of entropy generation (S��

) is equivalent to the maximization of power output(=), cf. Equation (5). The= expression (13) can be maximized with respect to x, and the result isx��

"1/2 (Bejan, 1988), or

C��

"C��

(14)

In conclusion, there is an optimal way to allocate the constrained hardware (C) to the two endsof the power plant, that is, if the maximization of power output subject to "xed heat input (Q

�)

and "xed size (C) is the purpose. Equation (14) also holds for refrigerating machines modeled inthe same way (Bejan, 1996a, b).

The maximization of= is shown graphically in Figure 7. Small conductances strangle the #owof heat, and demand large temperature di!erences. The power output is small when thetemperature di!erence across the innermost (reversible) compartment is small. The "rst and thirdframes of Figure 7 show that when the two conductances are highly dissimilar in size, largetemperature gaps are present, and the power output is small. The best irreversible performance issomewhere in the middle, where the conductances are comparable in size.

4.2. Optimal latent-heat storage temperature

Here is an example of how the type of energy-storage material can be chosen based onthermodynamic optimization. A simple way to perform the thermodynamic optimization of thelatent heat storage process was proposed by Lim et al. (1992), Figure 8. The hot stream ofinitial temperature ¹

�comes in contact with a single phase-change material through a "nite

thermal conductance UA, assumed known, where A is the heat transfer area between the meltingmaterial and the stream, and ; is the overall heat transfer coe$cient based on A. Thephase-change material (solid or liquid) is at the melting point ¹

�. The stream is well mixed

at the temperature ¹��

, which is also the temperature of the stream exhausted into theatmosphere (¹

�).

The steady operation of the installation modelled in Figure 8 accounts for the unsteady (cyclic)operation in which every in"nitesimally short storage (melting) stroke is followed by a shortenergy retrieval stroke: mR is stopped, and the recently melted phase-change material is solidi"edto its original state by the cooling e!ect provided by the heat engine positioned between ¹

�and

¹�. In this way, the steady-state equivalent model of Figure 8 represents the complete cycle*that

is, storage followed by retrieval.

556 A. BEJAN


Figure 8. The generation of power using a phase-change material and a hot stream that is ultimatelydischarged into the ambient (Lim et al., 1992).

The steady cooling e!ect of the power plant can be expressed in two ways:

QQ�";A (¹

��!¹

�) (15)

QQ�"mR c

�(¹

�!¹

��) (16)

By eliminating ¹��

between these two equations we obtain

QQ�"mR c

�

N��

1#N��

(¹�

!¹�) (17)

in which N��

is the number of heat transfer units of the heat exchanger surface,

N��

"

;A

mR c�

(18)

Of interest is the minimization of entropy generation, or the calculation of the maximum rate ofexergy (useful work,=Q in Figure 8) that can be extracted from the phase-change material. Forthis, we model as reversible the cycle executed by the working #uid between ¹

�and ¹

�

=Q "QQ� �1!

¹�

¹�� (19)

and, after combining with Equation (17), we obtain

=Q "mR c�

N��

1#N��

(¹�

!¹�) �1!

¹�

¹�� (20)



Figure 9. Model of a solar thermal power plant with collector-ambient heat loss and a heat exchangerbetween the collector and a Carnot energy conversion cycle (Bejan, 1982).

By maximizing=Q with respect to ¹�*that is, with respect to the type of phase-change material,

we obtain the optimal melting and solidi"cation temperature:

¹��

"(¹�

¹�)� � (21)

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

=Q��

"mR c�¹

�

N��

1#N��1!�

¹�

¹��

� �

��

(22)

The same results, Equations (21) and (22), could have been obtained by minimizing the total rateof entropy generation. One way to improve the power output of the single-element installation ofFigure 8 is by placing the exhaust in contact with one or more phase-change elements of lowertemperatures. This method is illustrated in Lim et al. (1992).

4.3. Optimal solar collector temperature

The optimization of solar energy conversion has been studied under the banners of twofundamental problems. One is concerned with establishing the theoretical limits of convertingthermal radiation into work, or calculating the exergy content of radiation. The other problemdeals with the delivery of maximum power from a solar collector of "xed size (Bejan, 1982, DeVos,1992). This problem has also been solved in many subsequent applications (Bejan, 1996a), whichare united by a characteristic and important design feature: the collector operating temperaturecan be optimized.

This optimization opportunity is illustrated in Figure 9. A power plant is driven by a solarcollector with convective heat loss to the ambient. The heat loss is assumed to be proportional tothe collector-ambient temperature di!erence, QQ

�"(;A)

�(¹

�!¹

�). The internal heat exchanger

between the collector and the hot end of the power cycle (the user) is modelled similarly,QQ "(;A)

�(¹

�!¹

�). There is an optimal coupling between the collector and the power cycle

such that the power output is maximum. This design is represented by the optimal collector

558 A. BEJAN


Figure 10. Two sources of irreversibility in the heating (charging) stroke of asensible-heat storage process (Bejan, 1982).

temperature (Bejan, 1982)

¹��¹

�

"

��

#R��

1#R(23)

where R"(;A)�/(;A)

�and �

�� "¹

�� /¹

�is the uppermost (&stagnation') temperature of the

collector. This optimum has its origin in the trade-o! between the Carnot e$ciency of thereversible part of the power plant (1!¹

�/¹

�) and the heat loss to the ambient, QQ

�. The power

output is the product QQ (1!¹�/¹

�). When ¹

�(¹

��the Carnot factor is too small. When

¹�'¹

��the heat input QQ drawn from the collector is too small, because the heat loss to the

ambient QQ�

is large.Corresponding optimal couplings have been identi"ed for solar-driven power plants of many

power-cycle designs, extraterrestrial power plants, and refrigeration systems driven by solarpower (Bejan, 1996a). These optima are obtained based on the maximization of power output, orthe minimization of the total rate of entropy generation.

4.4. Optimal sensible-heat storage time interval

The opportunity for minimizing the destruction of exergy during exergy storage becomes evidentif we examine the system shown in Figure 10. The storage system (the left side of the "gure)contains a batch of liquid (m, c). The liquid is held in an insulated vessel. The hot-gas streammR enters the system through one port and is gradually cooled as it #ows through a heat exchangerimmersed in the liquid bath. The spent gas is discharged directly into the atmosphere. As timepasses, the bath temperature ¹ and the gas outlet temperature ¹

��approach the hot-gas inlet

temperature, ¹�

.If we model the hot gas (steam, products of combustion) as an ideal gas with constant speci"c

heat c�, the temperature (history) of the storage system is expressed in closed form by the

equations

¹(t)!¹�

¹�

!¹�

"1!exp (!y�) (24)



¹��

(t)!¹�

¹�

!¹�

"1!y exp (!y�) (25)

where y and the dimensionless time � are de"ned as

y"1!exp (!N��), N

��"

hM�A

�mR c

�

, �"

m� c�

mct (26)}(28)

In these equations, A�

is the total heat-exchanger surface separating the stream from the liquidbath, and hM

�is the mean heat transfer coe$cient (in time and space) based on A

�. Built into the

model is the assumption that the liquid bath is well mixed, i.e., that the liquid temperature (¹ ) isa function of the time (t) only. As expected, both ¹ and ¹

��approach ¹

�asymptotically. They

approach ¹�

faster when N��

is higher.Turning our attention to the irreversibility of the energy-storage process, Figure 10 shows that

the irreversibility is due to two distinct parts of the apparatus. First, there is the "nite-�¹ (t)irreversibility associated with the heat transfer between the hot stream and the cold liquid bath.Second, the stream exhausted into the atmosphere is eventually cooled down to ¹

�, again by heat

transfer across a "nite �¹� (t). Neglected in the present model is the irreversibility due to thepressure drop across the heat exchanger traveled by the stream mR .

The combined e!ect of the competing irreversibilities noted in Figure 10 is a characteristic of allsensible-heat storage systems. Because of it, only a fraction of the exergy content of the hot streamcan be stored in the liquid bath. In order to see this, consider the instantaneous rate of entropygeneration in the overall system delineated in Figure 10,

SQ��

"mR c�ln

¹�

¹�

#

QQ�

¹�

#

d

dt(mc ln ¹) (29)

where QQ�"mR c

�(¹

��!¹

�). Important is the entropy generated during the entire charging-time

interval 0!t, which, using Equations (24)}(29), can be put in dimensionless form as

1

mc ��

�

SQ��

dt"��ln¹

�¹

�

#��#ln (1#��)!��

�(30)

where the "rst-law e$ciency ��is shorthand for the right-hand side of Equation (24), and where

�"(¹�

!¹�)/¹

�.

Multiplied by ¹�, the entropy-generation integral �

�SQ��

dt calculated above represents thedestroyed exergy*the bite taken by irreversibilities out of the total exergy supply brought intothe system by the hot stream

E�"tEQ

�"tmR c

�ln [¹

�!¹

�!¹

�ln (¹

�/¹

�)] (31)

On this basis, we de"ne the entropy-generation number N�as the ratio of the lost exergy divided

by the total exergy invested during the time interval 0!t:

N�(�, �, N

��)"

¹�

E��

�

�

S��

dt"1!

��!ln (1#��

�)

� [�!ln (1#�)](32)

560 A. BEJAN


The entropy-generation number takes values in the range 0}1, the N�"0 limit representing the

elusive case of reversible operation. Note the relation N�"1!�

��, where �

��is the second-law

e$ciency of the installation during the charging process.Charts of the N

�(�, �, N

��) surface show that N

�decreases steadily as the heat-exchanger size

(N��) increases (Bejan, 1982). This e!ect is expected. Less expected is the fact that N

�goes through

a minimum as the dimensionless time � increases. For example, the optimal time for minimumN

�can be calculated analytically in the limit �;1, where Equation (32) becomes

N�"1![1!exp (!y�)]�/� (33)

The solution of the equation N�/�"0 is

��

"1.256 [1!exp (!N��)]�� (34)

In other words, for the common range of N��

values (1}10), the optimal dimensionless chargingtime is consistently a number of order 1. This conclusion continues to hold as � takes valuesgreater than 1.

Away from the optimal charging time (i.e. when �P0 or �PR), the entropy-generationnumber N

�approaches unity. In the short-time limit (�;�

��), the entire exergy content

of the hot stream is destroyed by heat transfer to the liquid bath, which was initially atenvironmental temperature ¹

�. In the long-time limit (�<�

��), the external irreversibility takes

over: the used stream exits the heat exchanger as hot as it enters (¹��

"¹�

), and its exergycontent is destroyed entirely by the heat transfer (or mixing) with the ¹

�atmosphere. The

traditional ("rst-law) rule of thumb of increasing the time of communication between heat sourceand storage material is counterproductive from the point of view of avoiding the destruction ofexergy.

5. STRUCTURE SPRINGS OUT OF THERMODYNAMIC OPTIMIZATION

The summarizing conclusion that unites the examples reviewed in Section 4 is that the physicalresult of global optimization of thermodynamic performance is structure (con"guration, topo-logy, geometry, architecture, pattern). The examples covered structure in space (Section 4.1),temperature (Sections 4.2 and 4.3) and time (Section 4.4) This structure-generating principledeserves to be pursued further, in increasingly more complex system con"gurations. The genera-tion of structure in engineering has been named constructal method; the thought that the sameprinciple accounts for the generation of shape and structure in natural #ow systems is constructaltheory (Bejan, 2000).

The principle of organizing structure for the purpose of extracting and using maximum exergyfrom a hot stream is particularly relevant to the integrative conceptual design of energy #owsystems for aircraft. The same principle applies to systems in which all the functions are drivenby the exergy drawn from the limited fuel installed on board: ships, automobiles, military vehicles,environmental-control suits, portable power tools, etc. Additional support for this view isprovided by the record on powered #ight, engineered and natural. Figure 11 shows the cruisingspeeds of insects, birds and airplanes, next to the theoretical speed obtained by minimizingthe power (exergy rate) destroyed during #ight (Bejan, 2000). The speeds of the aircraft



Figure 11. Cruising speeds of insects, birds and airplanes, and the theoretical speed (<&M� �) forminimum rate of exergy destruction (Bejan, 2000).

compiled in this "gure refer to the optimal cruising conditions*the speed for minimal powerconsumption*not the maximal speed of the aircraft.

The performance record of the natural and engineered #ow systems (e.g., Figure 11) suggeststhat the constructal principle is important not only in engineering but also in physics and biologyin general. In this theoretical framework the airplane emerges as a physical extension of man, inthe same way that the body of the #ying animal (e.g., bat, bird) developed its own well adaptedextensions. All such extensions are discrete marks on a continuous time axis that pointstoward the better and the more complex. This theoretical line of inquiry is explored in a new book(Bejan, 2000).

ACKNOWLEDGEMENTS

I want to thank my colleagues Michel Feidt (University Henri PoincareH , Nancy), Enrico Sciubba (Universityof Rome I, La Sapienza) and Richard A. Smith (Air Force Research Laboratory) for their constructivecontributions to the revised form of this article.

NOMENCLATURE

A "area (m�)C "thermal conductance (W K��)c, c

�"speci"c heat (J kg�� K��)

562 A. BEJAN


E "energy (J)EQ�

"exergy transfer rate (W)e

"speci"c #ow exergy (J kg��)E

"#ow exergy (J)EQ

"#ow exergy rate (W)EGM "entropy generation minimizationg "gravitational acceleration (m s��)h "speci"c enthalpy (J kg��)hM "overall heat transfer coe$cient (Wm��K��)m, M "mass (kg)mR "mass #ow rate (kg s��)P "pressure (Pa)Q "heat transfer (J)QQ "heat transfer rate (W)R "ratio of thermal conductances, Equation (23)s "speci"c entropy (J kg��K��)S "entropy (J K��)SQ��

"entropy generation rate (WK��)t "time (s)¹ "temperature (K); "overall heat transfer coe$cient (Wm��K��)< "velocity (m s��)< "volume (m�)=Q "power (W)x "conductance allocation fractiony "dimensionless group, Equation (26)z "elevation (m)

Greek symbols

��

""rst law e$ciency��

"second law e$ciency� "dimensionless time, Equation (28)� "temperature ratio, Equation (23)� "speci"c non#ow exergy (J kg��)� "non-#ow exergy (J)� "dimensionless temperature di!erence, Equation (30)

Subscripts

b "bathc "collectorC "Carnot, reversibleH "highL "lowm "meltingmax "maximum



opt "optimumout "outletp "pumprev "reversiblet "turbineu "user0 "environment

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