Method of entropy generation minimization, or modeling and optimization based on combined heat transfer and thermodynamics

Rev Ge’n Therm (1996) 35, 637-646 0 Elsevier, Paris

Method of entropy generation minimization, or modeling and optimization based on

combined heat transfer and thermodynamics

Adrian Bejan

Dept of Mechanical Engineering, Duke University, Durham, North Carolina 27708-0300, USA

R&urn6 - Wthode de minimisation de la production entropique ou modklisation et optimisation basCes sur la combinaison du transfert thermique et de la thermodynamique. La minimisation de la production d’entropie (ECM, optimisation thermodynamique, ou thermodynamique en temps fini) est une methode de modelisation de procedes reels (irreversibles) et d’equipements. Les modeles integrent des principes de base en thermodynamique et en transfert thermique et I’optimisation est assujettie a des contraintes liees a des dimensions finies et des temps finis. L’etonnant essor de I’ECM est illustre par des exemples extraits d’installations cryogeniques, de transfert thermique, de stockage, de conversion d’inergie solaire, de centrales de production electrique et d’installations frigorifiques. Une attention particuliere est portee sur la valeur de I’ingenierie dans les modeles ECM, la contribution de tCte menee par Chambadal et Novikov en 1957, et les nouvelles directions comme I’ECM des procedes dependent du temps.

Nomenclature

A

2 D EGM

f FD

;

L 7h Ns P 4

: sg,n T u Ux ti

area specific heat at constant pressure thermal conductance diameter method of entropy generation minimization friction factor drag force irreversibility factor, equation (12) thermal conductivity length mass flow rate entropy generation number perimeter of cross-section heat transfer rate ratio of thermal conductances (UA),/(UA)i

heat transfer rate entropy generation rate thermodynamic temperature overall heat transfer coefficient based on A free stream velocity power

Greek symbols

AP pressure drop AT temperature difference 77 first-law efiiciency QI second-lax efficiency f? ratio of temperatures

Subscripts

11: body collector

[I: Carnot, or reversible part high temperature

oi internal

I;:in low temperature minimum

opt

L optimum reversible

0’ per unit length 00 environment

1 I INTRODUCTION

Entropy generation minimization (EGM, thermodynamic optimization, or finite-time thermodynamics) is a method for modeling actual (irreversible) processes and devices. The models incorporate basic principles of thermodynamics and heat transfer, and the optimization is subjected to finite-size and finite-time constraints. The astonishing growth of the EGM field is illustrated with examples drawn from cryogenics, heat transfer, storage, solar energy conversion, power plants, and refrigeration plants. Emphasis is placed on the engineering value of

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EGM models, the pioneering contributions made by Chambadal and Novikov in 1957, and new directions such as the EGM of time- dependent processes.

2 I METHOD

During the last two decades Entropy Generation Minimization (EGM) has become an established method and field in thermal science and engineering. The EGM method relies on the simultaneous application of heat transfer and engineering thermodynamics principles in the pursuit of realistic models for heat transfer processes, devices and installations. By realistic models we mean models that account for the inherent thermodynamic irreversibility of the heat, mass and fluid flow processes. The method is also known as thermodynamic optimization, second-law analysis, and thermodynamic design, or by new names such as finite-time, endoreversible, or exoirreversible thermodynamics.

The development of EGM first became evident in engineering, specifically, in the fields or refrigeration (cryogenics), heat transfer, storage, solar thermal power conversion and thermal science edu- cation. These developments were first recognized in textbook form in 1982 [ll. The field has experienced astonishing growth during the 1980s and 199Os, in both engineering and physics. The objective of this article is to illustrate how large and diverse the EGM field has become. Emphasis is placed on the engineering (practical) value of the method, and some of the more recent applications such as the optimization of time-dependent processes.

In this presentation, I rely on a point of view acquired during two just completed book projects. One was done in collaboration with Moran and Tsatsaronis [21, where we presented in a gradual and evolutionary manner the current status of three methods: exergy analysis, EGM and thermoeconomics. The other [3] focused strictly on the EGM method and the current status of the EGM literature in both engineering and physics. Because the size limit of this paper does not permit an extensive list of references, the reader is referred to [3] for a complete and up-to-date bibliography.

It is instructive to begin with a brief look at why in EGM we need to rely on heat transfer and fluid mechanics, not just thermodynamics. Consider the most general definition of a thermodynamic system that functions while in thermal contact with the ambient (TO). The Gouy-Stodola theorem [1,41, WmJ - I+ = T&en, states that the destroyed power (tiT,, - ti) is proportional to the total rate of entropy generation. If engineering systems and their components are to operate in such a way that their destruction of work is minimized, then the conceptual design of such systems and

components must begin with the minimization of entropy generation.

The critically new aspect of the EGM method (the aspect that makes the use of thermodynamics in- sufficient and distinguishes EGM from exergy analysis) is the minimization of the calculated entropy generation rate. To minimize the irreversibility of a proposed design, the analyst must use the re- lations between temperature differences and heat transfer rates, and between pressure differences and mass flow rates. He must relate the degree of thermodynamic non-ideality of the design to the physical characteristics of the system, namely to finite dimensions, shapes, materials, finite speeds and finite-time intervals of operation. For this he must rely on heat transfer and fluid mechanics principles, in addition to thermodynamics.

3 I CRYOGENICS

As a special application of the Gouy-Stodola theorem, it is easy to prove that the power required to keep a cold space cold is equal to the total rate of entropy generation times the ambient temperature, with the observation that the entropy generation rate includes the contribution made by the leakage of heat from To into the cold space. The structure of a cryogenic system is in fact dominated by components that leak heat, for example, mechanical supports, radiation shields, electric cables and counterflow heat exchangers. The minimization of entropy generation along a heat leak path consists of optimizing the path in harmony with the rest of the refrigerator of liquefier.

Figure la shows a mechanical support of length L that connects the cold end of the machine (TL) to room temperature (TH). The rate of entropy generation inside the support shown as a vertical column is:

PTH ir

where the heat leak 4 is allowed to vary with the local temperature T. The local heat leak change do is removed by the rest of the installation, which is modeled as reversible. The heat leak is also related to the local temperature gradient and conduction cross-section A:

Q=kAg

where the thermal conductivity k(T) decreases toward low temperatures. Rearranged and integrated from end to end, equation (?) places a size constraint on the unknown function Q(T):

L I TH k -_= A

,dT Tr. Q

(3)

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Modeling based on combined heat transfer and thermodynamics

The heat leak function that minimizes the $,,n integral (1) subject to the finite-size constraint (4) is obtained based on variational calculus:

In summary, equation (1) was provided by thermodynamics and equation (3) by heat transfer: together they prescribe the optimal design (4), which is characterized by a certain distribution of intermediate cooling effect (d&/dT),,t. Any other design, Q(T), will generate more entropy and will require more power in order to maintain the cold end of the support at TL. Together, equations (1) and (3) illustrate the method of thermodynamic optimization subject to a physical constraint, and, to paraphrase some of the current terminology, they constitute one of the earliest examples of finite-size thermodynamics. The technological applications of the variable heat leak optimization principle are numerous and important. In the case of a mechanical support, the optimal design is approximated in practice by placing a stream of cold helium gas in counterflow (and in thermal contact) with the conduction path. The fabrication of the heat exchanger between the support and the coolant is less difficult and more economic if the continuous contact is replaced by a succession of discrete cooling stations. In this case the optimal heat leak function required by equation (4) must be approximated by a stepwise-varying function.

The thermodynamic optimization of a stack of radiation shields in vacuum follows essentially the same route as in equations (l)-(4). For example, Martynovskii et al [51 .determined the optimal intermediate cooling (and optimal shield temperatures) for minimum entropy generation rate in stacks with one, two and three shields. The optimal intermediate cooling regime for cryogenic current cables was developed based on EGM [ll.

Another class of engineering components that have been optimized based on the EGM method are the counterflow heat exchangers that connect the coldest regions of refrigerators and liquefiers to the room temperature compressor. The counterflow sketched in figure lb was intentionally oriented in the same way as the mechanical support of figure la to stress the analogy between the two devices. The entropy generation rate associated with the two streams and the space between them (fig lb) is given by equation (1) in which & is now the longitudinal convective heat leak Q = ticPAT, and AT is the transversal (stream-to- stream) temperature difference. The equivalent of the finite-size constraint (3) is obtained eliminating AT from tic,dT = (pdx)UAT where p is the perimeter of the duct that carries 7iz, and u is the overall heat transfer coefficient based on p. The total stream-to-stream heat transfer area P_L is

fixed. The result of the EGM analysis is again an optimal heat leak variation,

L, TH I I

x+dx, T+dT--

x. T--

0, TI. I I

(a)

T+dT X+d%------

X-----Tj----f T+‘,T

I t . =P

o---___ --- - Ti. -I t

(b)

Fig 1. (a) Mechanical support with variable heat leak; Wkc[of;nterflow heat exchanger with longitudinal heat

This cooling effect can be built into the practical design of the counter-flow heat exchanger by bleed- ing a fraction (tie) of the high pressure stream, expanding it in a work producing device (cylinder & piston, or turbine), and using the cold stream to cool the counterflow heat exchanger. The optimal flow rate through the expander is known from equation (5), ti,,,,tcP = (dQ/dT),,t. When the pressure ratio PHI PL is not large enough for the expanded fraction tie to become as cold as the cold end of the counterflow (TL), the engineering solution is to install two or three expanders along the counterflow. The optimization of the temperature locations of such a sequence of expanders is described in [4]. One interesting characteristic of the counter-flow heat

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exchanger with optimal intermediate cooling effect is that the stream-to-stream temperature difference ATopt decreases proportionally with T, namely (AT/T) = (rjZ+/UA) In (TH/TL). This rule is widely recognized in the design of cryogenic counter-flow heat exchangers.

4 I HEAT TRANSFER

Workers in the mainstream of heat transfer adopted the techniques developed in cryogenics and applied them to many classes of devices for promoting heat transfer. The optimization was carried out at two levels of complexity: complete components (eg, heat exchangers), and elemental features (eg, fins, ducts). For example, the flow of a single-phase stream riz through a heat exchanger tube of internal diameter D. The heat transfer rate per unit tube length, between the tube wall and the stream, q’, is given. The entropy generation rate per unit tube length is:

$,, = & + 327i23f x2p2TD5

The Nusselt number is a result taken from the field of heat transfer, while the friction factor is from fluid mechanics. In this way equation (6) shows at a glance how thermodynamics is combined from the start with heat transfer and fluid mechanics in EGM. The first term on the right side is the contribution made by heat transfer, while the second term is the contribution due to fluid friction:

s’ $?en = &!n,aT + $?n,aP (7)

A characteristic of all heat transfer devices with fluid flow is that ~?&+r competes against $en,AP: for example, in the smooth tube with fixed q’ and riz, the changes in the two terms have opposing signs as D changes. The optimal tube diameter that minimizes (6) can be obtained analytically. The performance of any other design (D, S&,) relative to the optimal design (Dopt, $en,min) is described by the entropy generation number Ns, which for the turbulent regime assumes the form:

s’ -0.8 t?en Ns = -

S&nin 4.6

(8)

Another large and diverse group of heat transfer devices relies on external convection, that is heat transfer between a stream and a body immersed in the stream. The total entropy generation rate associated with heat transfer and drag on an immersed body is:

sgm = C~B(TB - Toa) + F~uce - T T

T, (9) Boc

where Qn, TB, T, , FD and U, are the heat transfer rate, body temperature, free stream temperature, drag force and free stream velocity. The relation for calculating 0~ is provided by the field of heat transfer. Similarly, the fluid flow information required for evaluating FD comes from fluid mechanics. The 5&, expression has the same two-term structure as equation (7). The competition between the two terms points to an optimal body size for minimum entropy generation rate. Such optimal sizes have been reported for the flat plate in parallel flow (laminar and turbulent), the sphere and the cylinder in cross-flow. Considerable EGM work has been done on the optimization of heat transfer augmentation techniques, as shown by the review in 131.

The EGM method was also applied to heat exchangers as complete systems. Counterflow heat exchangers were optimized subject to fixed area, fixed volume, and fixed area and volume. Similar EGM work as done in recent years on phase-change heat exchangers, cross-flow heat exchangers, regen- erators, and heat exchanger networks. For example, Tondeur and Kvaalen [61 showed that, for a given duty, the best configuration of a heat and mass process is that in which the entropy generation rate is distributed in the most uniform way possible. Several authors have recommended that commercial computational fluid dynamics packages should have the built-in capability of displaying local entropy generation rate fields in both laminar and turbulent flow (eg, Paoletti et al [71). Fun- damental EGM studies have also focused on convective mass transfer, reacting flows, radiative heat transfer, convection through porous media, and conduction through non-homogeneous and anisotropic media.

5 I STORAGE SYSTEMS

The optimization of time-dependent heating and cooling processes has generated a subfield of its own. Common to all these applications is the search for optimal strategies for executing heating and cooling processes, ie, the search for optimal histories, or optimal evolutions in time (see also 5 9). The earliest work of this type focused on the sensible heating of a storage element (solid or incompressible liquid) of mass M and specific heat C, by using a stream of hot gas (ti, cp, T,) as shown in figure 2. Initially, the storage element is at the ambient temperature TO.

The sensible heating process has two sources of entropy generation, the heat transfer between the stream and the storage element, and the heat transfer between the exhaust and the ambient. It was shown that the total entropy generated is minimum at a certain time (duration) of the heating

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Internal EXterflal irreversibility irrevenibility

/\/

--------- -j

QO t

0-l To

_-_- ___-_-_ SVStWll 7& ambient

Fig 2. The irreversibilities of sensible-heat storage [I].

process, topt. For engineering design purposes, the optimal heating strategy is such that the process must be terminated when the thermal inertia of the hot gas used (ti+) becomes of the same order as the thermal inertia of the storage element (MC). The complete storage and removal cycle was optimized in subsequent EGM studies.

The optimization of heating and cooling processes continued in two additional directions. One is the optimization subject to a finite-time constraint, and addresses the fundamental question of how to cool or heat a mass to a prescribed temperature level during a fixed time interval t,, while using the minimum quantity of coolant or heating agent [l].

The other direction is concerned with phase- change storage elements, eg, latent heating instead of sensible heating. This activity began with Bjurstrom and Carlsson @I who showed that the entropy generated during heating (melting) is minimum when the melting temperature of the storage material has the optimal value T,,,, =

(T,To)l/*. Numerous studies accounted for the details of the actual (time-dependent) melting and solidification processes, including the effect of natural convection, the optimization of two or more phase-change elements in series, the optimization of shell-and-tube heat exchangers with storage in the shell, and the optimization of a latent heat element coupled in series with a power plant.

6 I SOLAR POWER PLANTS

The generation of mechanical or electrical power has been subjected to thermodynamic optimization in many studies that cover a vast territory, The first power generation area to use EGM models regularly was that of solar driven power plants [1,41. An example of an early model is the power plant driven by a solar collector with convective heat leak to the ambient (fig 3). The heat leak was modeled as proportional to the collector-ambient temperature difference, & = (UA),(T, - To). The internal heat exchanger between the collector and the hot end of the power cycle (the user) was modeled similarly, Q = (UA)i(Tc - T,,). It was found that 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 temperature:

(10) 10 1tfi

where R = (UA),/UA)i, Omaz = T,,,,,/To and T,,,,, is the maximum (stagnation) temperature of the collector. There is a corresponding coupling between a collector and a refrigeration cycle. More complete collector models also accounted for the radiation effect on the heat loss to the ambient.

lo reversible power plant

Fig 3. Solar power plant model with collector-ambient heat loss and collector-power cycle heat exchanger [II.

Another research direction is the optimal allocation of a fixed heat transfer area in a solar power plant. Another is the area-constrained optimization of a model with phase-change energy storage at the hot end of the cycle, between the collector and the working fluid. One useful result is that the melting material must be such that its melting point is the geometric average of the collector and ambient temperatures, Tm,opt = (TcTo)‘/*. The growing literature on solar power plant models includes non- isothermal collectors, time-varying solar input, and sensible heat storage in the collector. Extraterres- trial solar power plant models have been optimized subject to fixed total (collector + radiator) area [4,91: the optimal collector/radiator area ratio is 0.54.

The common message of these models is that several extremely basic tradeoffs exist in the thermodynamic optimization of power plants driven by heat transfer from the sun. The models share the feature that heat loss always occurs between the collector and the ambient. The thermodynamic tradeoffs are of two kinds. When the overall size of the installation is constrained, there is an optimal way of allocating the hardware between the various components. When the daily variation of the solar heat input is known, there is an optimal time- dependent strategy of operating the power plant.

7 I OTHER POWER PLANTS

In 1957 papers and several engineering textbooks in French, Russian and English, Chambadal [lo]

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A Bejan

and Novikov [ill showed independently that the hot end temperature of a power plant can be optimized such that the power output is maximum. Chambadal’s analytical argument corresponds to the model drawn in figure 4. The power plant is driven by a stream of hot single-phase fluid of inlet temperature TH and constant specific heat c,. The power plant model has two compartments. The one sandwiched between the surface of temperature THC and the ambient (TL) operates reversibly. The area of the THC surface is assumed sticiently large such that the outlet temperature of the stream is equal to THC . There is only one degree of freedom in the optimization of the power plant: the hot end of the inner compartment, or the exhaust temperature THC. It is not difficult to express W as a function of THC, and to show that the conversion efficiency at maximum power is:

THcyopt = (THTL)"~,

l/2

(11)

ticp

_____________

Fig 4. The sources of entropy generation in Chambadal’s power plant model 131.

It can be shown that the efficiency formula (11) also holds when the heat exchanger area is small and the exhaust temperature is higher than the hot end temperature of the reversible compartment [31. Equation (11) also holds when the unmixed stream of figure 4 is replaced with a single-temperature (mixed) fluid inside that heat exchanger 131.

The maximum-power efficiency (11) can also be derived by minimizing the total entropy generation rate associated with the power plant. One source of entropy generation in figure 4 is the heat exchanger. The other is the dumping of the used stream into the ambient: THC is a degree of freedom only when the exhaust (THC) is free to float, ie, when it is not required (used) by someone else downstream. The external irreversiblity indicated in figure 4 is an essential part of the physics of the optimization process: without it the plant design cannot be optimized. This additional irreversibility is what gives the design room to moue, therefore it can be called the room-to-moue irreversibility [31.

It is worth noting is that if we had overlooked the room-to-move irreversiblity, that is, if we had

written only the entropy generation associated with the visible confines of the power plant, then we would have found that &, has a minimum at a THC value that differs from the maximum-power value (11). These THC values differ not because maximum power and minimum entropy generation rate are two different designs, but because an error has occurred in the evaluation of the total rate of entropy generation. This observation contradicts the claim [121 that, in power plants with the maximum power efficiency (ll), minimum entropy generation and maximum power are two different design conditions.

A maximum power design similar to Cham- badal’s is the optimal combustion chamber temperature that was derived independently in [41. It was shown that when the specific heats of all the products of combustion are assumed sufficiently constant (independent of temperature), the maximum power design corresponds to a hot end (flame) temperature equal to the geometric average of the adiabatic flame temperature and the ambient temperature, as in the first of equation (11).

Novikov 1111 equally deserves credit for equation (11) (see fig 5). The hot end heat exchanger of finite- thermal conductance UA drives the heat transfer rate QH 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 irreversil$lity by writing (sd - sa) = (1 + i)(Sd,rev - sa), or QL = (1 + ~)QL,~~~, where (1 + i) > 1 and QL is the heat transferred to the ambient (TL). The rest of the power plant operates reversibly. Novikov’s optimal heating temperature and efficiency for maximum power output:

THqopt = (1 + i)1'2(T~T~)"2,

~=l-(l+i)““(g2 (12)

match Chambadal’s equation (11) in the limit where the expansion is executed reversibly (i = 0). It has been shown [31 that in Novikov’s model the maximum-power efficiency can also be derived by minimizing the total entropy generation rate: the latter must tak? into account the variability of the heat iFput QH over a certain design interval 0 < QH < Q (fixed). Again, this development contradicts the report 1121 that maximum W and minimum S,,, are two different designs.

The Chambadal-Novikov efficiency (11) was re- discovered almost two decades later in the physics literature by Curzon and Ahlborn [131. Another way of modeling the irreversible operation of a power plant is shown in figure 6. The loss of heat from the hot end of the power plant was modeled as a thermal resistance (bypass heat leak) in parallel with an irreversiblity free compartment that produces the actual power. The heat leak was modeled as proportional to the temperature difference between the hot end and the ambient, &C = c(TH - TL), where C is the thermal conductance of the leaky insulation of the power plant. The power is maximum

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THC

TL

TH

Tnc

T

TI.

0 b

S

Fig 5. Novikov’s model for a steady-state power plant with heat transfer and expander irreversibilities [3].

II QH r---- --- ------1

Heat 1 engine 1

11

I Carnot engine

TI.

Fig 6. Bejan’s power plant model with bypass heat leak and variable hot-end temperature 111.

when the variable hot end temperature TH reaches the optimal level:

QH

( ‘>

l/2 TH,opt = TL 1 + m (13)

A large segment of the EGM literature is concerned with power plant models with the features of figures 4-6, or combinations of such features (reviews can be found, for example, in [3,4,14,151). One topic of continuing interest is the optimal allocation of a heat exchanger inventory in a model with two heat exchangers. For example, the question addressed in 111 was: given the two heat exchangers (UHAH and ULAL) and an additional unit of heat transfer area (AA), should AA be placed at the hot end, or at the cold end? If the total thermal conductance UA = UHAH + ULAL is fixed, the optimal conductance allocation rule is:

(UH-‘bf)opt = (ULAdopt (14)

If the total area is constrained, A = AH + AL, the optimal area allocation result is:

(15) AT - [1+ (g,“‘]-’

If the heat input OH is treated as fixed in a model with two heat exchangers, the only degree of freedom left in the optimization is the allocation of the finite-heat exchanger inventory. For example [33, if the UA inventory is constrained, then the optimal allocation rule continues to be equation (141, with the corresponding maximum efficiency:

wrna, Tj=.

&H =l-$(l-$&-l (16)

The optimal way to allocate the fixed thermal conductance inventory of a combined cycle power plant with three heat exchangers, UA =

UHAH+UMAM+ULAL~~(UHAH)~~~ =(UMAM)+ = (UL&)opt. Another example is the engine model with bypass heat leak (fig 61, in which the insulation (the heat leak) is optimized in harmony with the power-producing compartment, ie, in ac- cordance with the cryogenics method of figure la: the maximum power efficiency of such a model:

> (17)

is in approximate numerical agreement with the Chambadal-Novikov-Curzon-Ahlborn efficiency (11). This agreement suggests that an actual power plant may be viewed as an obstacle to direct heat transfer from source (TH) to sink (TL), ie, as an insulation designed to produce maximum power when its size is constrained.

8 I REFRIGERATION PLANTS

The modeling features used for power plants (5 6 and 7) have also been used in the optimization of refrigeration plants. This extensive work is reviewed in 131. The model that was studied the most is the refrigerator composed of a cold end heat exchanger (eg, evaporator), a reversible

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A Bejan

7’ ’ II

Fig 7. (a) Refrigeration plant model with two finite size heat exchangers [l]; (b) model with bypass heat leak

x3- e Reversible refrigeration

cycle

7; Load

T- H

T- 1

irreversibility 141.

compartment that receives power and moves the entropy stream toward higher temperatures, and a room-temperature heat exchanger (eg, condenser), (see fig 7~). In ref [l], the question was where to invest an additional unit of heat transfer area (AA) when the cold end conductance ULAL and room temperature conductance UHAH are given. Goth and Feidt [161 assumed that the total heat transfer area A is constrained and showed that the power input is minimum when A is divided as shown in equation (15). When the optimization is based on the UA constraint, the thermal conductance inventory must be split evenly between the two heat exchangers (cf Eq 14). The same optimization rule applies when the UA inventory is minimized subject to flxed power input. The allocation of a finite UA or A can also be optimized in more complex refrigerator models, such as that of a modern defrosting refrigerator based on the vapor compression cycle and fluids R-12 or R-134a 1171.

Another practical merit of EGM models is that they explain the trends in the data reported on the performance of existing refrigeration plants. To correlate the data is important because correlations are needed for making design projections on the refrigeration needs of future large scale projects. A 3-line analysis of the model of figure 7b predicts that the second-law efficiency (vi) should vary as:

m= [l+z$ql-g)]-l (18)

where Ci is the thermal conductance of the leaky insulation, 0~ = Ci (TH - TL ) . A comparison with the reported performance data shows that equation (18) with C~THIQL M 5 is an adequate curvefit. Equation (18) predicts several of the observed trends. First, mr decreases monotonically as Ci increases, which should be expected because leakier refrigerated enclosures make less efficient refrigeration plants. Second, 91 decreases as TL/TH decreases. Third, the second-law efficiency (or the COP) increases as

0~ increases, which means that larger machines are more efficient.

9 I TIME-DEPENDENT OPERATION

One growth area in EGM is the optimization of the time-dependent operation of a process or installation. In each case the objective is to determine the optimal evolution of the process in time, ie, the optimal history. This subfield began with the problems illustrated in § 5. In this section we take a brief look at four of the most recent examples from our group’s work 131.

Figure 8 shows what is perhaps the simplest model of a modern defrosting refrigerator with frost accumulation on the evaporator surface. The frost layer is detrimental to refrigerator thermodynamic performance. The refrigerator operates in on and off fashion, so that the frost is removed during the off intervals. It has been shown that when the off interval (cleaning time) is specified, the on interval can be optimized in such a way that the refrigerator power requirement (averaged over the on and off cycle) is minimum. More complicated (realistic) defrosting refrigerator models can also be optimized in this way.

The corresponding on and off optimization prob- lem in the power generation area consists of de- termining the time to shut down a power plant and remove the scale from its heat exchangers [ 181. Fouling, or scale formation, decreases the heat exchanger conductances and conversion efficiency of the power cycle. There is an optimal on interval (ie, power generation) for a specified heat exchanger cleaning time interval. This is true for fouling ei- ther in the hot end heat exchanger or in the cold end heat exchanger.

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ambient To

refrigerated TL

irreversiblt : space refrigerata I

_w

evaporator Tmin I

Fig 8. Model of a refrigerator with on and off operation and frost accumulation on the evaporator surface [3, IS].


The formation of frost on cold heat exchangers is detrimental to refrigerator thermodynamic performance, which is why the optimization of the freezing and frost-removal cycle is an important technological issue. Most interesting is that the same phenomenon - the same cycle - can be viewed and optimized as something beneficial, namely, the maximization of ice production in industrial ice making installations. This optimization principle is relevant to the production by solidification of other materials, not just ice. On the basis of a simple EGM model, it was shown [31 that the frequency of the ice formation and removal cycle can be optimized for maximum time-average rate of ice (or solid) production.

For example, figure 9 shows the growth of an ice sheet on a plane wall in contact with water at solidification temperature, when S increases as t112. If the freezing time is tl and the specified ice removal time is t2, the time averaged rate of ice production is proportional to ti’2/(tl + t2): this quantity is maximized when tl is selected such that t1,opt = t2.

Another example of time-dependent operation is the case of a power plant driven by heating from a deep hot-dry-rock deposit. The water stream circulated through the rock fissure cools the rock by time-dependent conduction. When the lifetime of power plant operation is fixed 131, there is an optimal water flow rate such that the total work produced is maximum.

IO I CONCLUSIONS

A quiet revolution is taking place in thermodynamics, and it amounts to the closing of the gap between

thermodynamics and heat transfer. The method and field that unite these classical disciplines is entropy generation minimization. Today EGM is an established method in both fundamental and applied heat transfer (eg, 119,201). I am delighted that this, historically an engineering method, has spread even outside engineering to attract contributions from physicists, mathematicians and optimal control scientists.

64 I-

0 ‘1 ‘1 + L1 L

Fig 9. The periodic followed by the ice

production of ice: the freezing time tl, removal time t2 [31.

On the practical side, the EGM method has shown repeatedly that fundamental optima (tradeoffs) exist when the thermodynamic optimization is subjected to finite-size and finite-time constraints. For example, a fixed heat exchanger inventory can be divided optimally among the components of an installation. If these optima exist in EGM models as simple as the ones reviewed in this article, then real opportunities exist for the optimal distribution of hardware in the design of actual installations. The practical contribution of EGM is the same as that of any other fundamental method: it is to show

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the way, to uncover new opportunities for the industrial work that will follow. In the classroom, EGM has changed the way in which we teach and apply Thermodynamics [l-4,21].

Most recently, the method has been used to predict self-organization and self-optimization in Nature [221. It was found that the tree network is the geometric-optimization result for minimal flow resistance between one point and a finite volume 1221.

REFERENCES

[l I Bejan A (1982) Entropy generation through heat and fluid flow. Wiley & Sons, New York, USA

[2] Bejan A, Tsatsaronis C, Moran M (1996) Thermal design and optimization. Wiley & Sons, New York, USA

[3] Bejan A (1995) Entropy generation minimization. CRC Press, Boca Raton, FL, USA

[4] Bejan A (1988) Advanced engineering thermodynamics. Wiley & Sons, New York, USA

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[6] Tondeur D, Kvaalen E (1987) Equipartition of entropy production. An optimality criterion for transfer and separation processes. Ind Eng Chem Res 26, 50-56

[7] Paoletti S, Rispoli F, Sciubba E (1989) Calculation of exergetic losses in compact heat exchanger passages. ASME AES 1 O-2, 21-29

[8] Bjurstrom H, Carlsson B (1985) An exergy analysis of sensible and latent heat storage. Heat Recovery Cyst 5, 233-250

[9] Bejan A (1993) Heat Transfer. Wiley & Sons, New York, USA

A Bejan

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[l 11 Novikov II (1957) The efficiency of atomic power stations. Atomnayu energiya 3(1 1), 409, English translation in: I Nuclear Energy /I 7(1958), 125-l 28

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[17] Radcenco V, Vargas JVC, Bejan A, Lim JS (1995) Two design aspects of defrosting refrigerators. Int J Refrig 18, 76-86

1181 Bejan A, Vargas JVC, Lim JS (1994) When to defrost a refrigerator and when to remove the scale from the heat exchanger of a power plant. Int J Heat Mass Transfer 37, 523-532

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[20] Bejan A (1995) Convection heat transfer. Second ed, Wiley & Sons, New York, USA

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[22] Bejan A (1997) Constructai-theory network of con- structing paths for cooling a heat generating volume. Int J Heat Mass Transfer 40, 799-816

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Documents

Method of entropy generation minimization, or modeling and optimization based on combined heat transfer and thermodynamics