An Alternative Criterion in Heat Transfer Optimization

7/30/2019 An Alternative Criterion in Heat Transfer Optimization

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Proc. R. Soc. A (2011) 467, 10121028doi:10.1098/rspa.2010.0293

Published online 13 October 2010

An alternative criterion in heattransfer optimization

BY QUN CHEN1,3,*, HONGYE ZHU2, NING PAN3 AND ZENG-YUAN GUO1

1Key Laboratory for Thermal Science and Power Engineering ofMinistry of Education, Department of Engineering Mechanics, and 2Institute

of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084,Peoples Republic of China

3Biological and Agricultural Engineering Department, University of California,Davis, CA 95616, USA

Entropy generation is recognized as a common measurement of the irreversibility indiverse processes, and entropy generation minimization has thus been used as the criterionfor optimizing various heat transfer cases. To examine the validity of such entropy-basedirreversibility measurement and its use as the optimization criterion in heat transfer,both the conserved and non-conservative quantities during a heat transfer process areanalysed. A couple of irreversibility measurements, including the newly defined conceptentransy, in heat transfer process are discussed according to different objectives. Itis demonstrated that although thermal energy is conserved, the accompanied systementransy and entropy in heat transfer process are non-conserved quantities. When the

objective of a heat transfer is for heating or cooling, the irreversibility should be measuredby the entransy dissipation, whereas for heat-work conversion, the irreversibility shouldbe described by the entropy generation. Next, in Fouriers Law derivation using theprinciple of minimum entropy production, the thermal conductivity turns out to beinversely proportional to the square of temperature. Whereas, by using the minimumentransy dissipation principle, Fouriers Law with a constant thermal conductivity asexpected is derived, suggesting that the entransy dissipation is a preferable irreversibilitymeasurement for heat transfer.

Keywords: heat transfer; irreversibility; optimization; entransy dissipation; entropy generation

1. Introduction

Heat transfer occurs in about 80 per cent of all the energy utilization systems,so improving the heat transfer performance significantly promotes the energyconservation in most thermal systems, through either increasing the heat flowrate for a given volume of facility, or reducing the cost of equipment with givenheat load (Webb & Bergies 1983; Bergles 1997, 1988). Besides, enhancing the

*Author for correspondence ([email protected]).

Received 8 June 2010Accepted 14 September 2010 This journal is 2011 The Royal Society1012
mailto:[email protected]:[email protected]


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Criterion in heat transfer optimization 1013

heat transfer effectively raises the operation reliability of electronic devices whereelectricity-generated heat has frequently posed a serious problem (Chen 1996).Thus, heat transfer improvement (or optimization) has become one of the criticalissues for the efficiency of energy utilization, and heat transfer research haspropelled rapid advancement in various heat transfer enhancement technologies,

including using extended surfaces, stirrers and external electric or magnetic fields(Bergles 1988, 1997; Webb 1994; Karcz et al. 2005; Schfer et al. 2005).In the theoretical study of heat transfer, based on the concepts of entropy and

entropy generation, Gyarmati (1970) derived Fouriers Law with the criterionof minimum entropy generation and showed that entropy generation is theirreversibility measurement for any heat transfer process. Bejan (1979, 1996)and Charach & Rubinstein (1989) used this criterion for the optimization ofconvective heat transfer process and heat exchangers for different applications.On the other hand, there are some scholars who questioned if the entropygeneration is the universal irreversibility measurement for heat transfer or

if the minimum entropy generation is the general optimization criterion forall heat transfer processes, regardless of the nature of the applications. Forinstance, Bertola & Cafaro (2008) found that when satisfying the Onsagerreciprocal relation, the principle of minimum entropy production (Prigogine1967) could be tenable only if there is zero generalized flow under anon-zero generalized force, or the thermal conductivity should be inverselyproportional to the square of the absolute temperature during steady-stateheat conduction. By analysing the relationship between the efficiency andthe entropy generation in 18 heat exchangers with different structures,Shah & Skiepko (2004) demonstrated that even when the system entropygeneration reaches the extremum, the efficiency of the heat exchangers canbe at either the maximum or the minimum, or anything in between. Inaddition, the so-called entropy generation paradox (Bejan 1996; Hesselgreaves2000) exists when the entropy generation minimization is used as theoptimization criterion for counter-flow heat exchanger. That is, enlargingthe heat exchange area from zero simultaneously increases the heat transferrate and improves the heat exchanger efficiency, but does not reduce theentropy generation rate monotonouslythe entropy generation rate increasesat first and then decreases. Therefore, it was speculated that the optimizationcriterion of minimum entropy generation is not always consistent with the heattransfer improvement.

Recently, Guo et al. (2007) introduced the concepts of entransy and entransydissipation to measure, respectively, the heat transfer capacity of a systemand the loss of such capacity during the process. Moreover, Guo et al. (2007)proposed the entransy dissipation extremum as an alternative optimizationcriterion for a heat transfer process not involved in a thermodynamic cycle,and, consequently, developed the extremum principle of entransy dissipation tooptimize the processes in heat conduction (Guo et al. 2007; Chen et al. 2009a),heat convection (Meng et al. 2005; Chen et al. 2007) and thermal radiation(Wu & Liang 2008).

The contribution of this present paper is to further examine the physical

implications of both entropy generation and entransy dissipation, compare theirdifferences and, more importantly, examine their applicability to heat transferoptimization in applications of different nature.

Proc. R. Soc. A (2011)
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1014 Q. Chen et al.

2. Irreversibility of heat transfer

(a) Conserved and non-conserved quantities in a transport process

After intensive study, we found that all transport processes contain two differenttypes of physical quantities owing to the existing irreversibility, i.e. the conservedones and the non-conserved ones, and the loss or dissipation in the non-conserved quantities can then be used as the measurements of the irreversibilityin the transport process. Taking an electric system as an example, althoughboth the electric charge and the total energy are conserved during electricconduction, the electric energy, however, is not conserved and it is partlydissipated into the thermal energy owing to the existence of the electricalresistance. Consequently, the electrical energy dissipation rate is often regarded asthe irreversibility measurement in the electric conduction process. Similarly, fora viscous fluid flow, both the mass and the momentum of the fluid, transportedduring the fluid flow, are conserved, whereas the mechanical energy, including

both the potential and kinetic energies, of the fluid is turned into the thermalenergy owing to the viscous dissipation. As a result, the mechanical energydissipation is a common measure of irreversibility in a fluid flow process. Theabove two examples show that the mass, or the electric quantity, is conservedduring the transport processes, while some form of the energy associated withthem is not. This loss or dissipation of the energy can be used as the measurementof irreversibility in these transport processes. However, an irreversible heattransfer process seems to have its own particularity, for the thermo-energy alwaysremains constant during transfer and it does not appear to be readily clearwhat the non-conserved quantity is in a heat transfer process. Non-equilibrium

thermodynamics (Gyarmati 1970; Kreuzer 1981) seems to offer an answer bysuggesting that the entropy or available energy (exergy) is the non-conservedquantity in a heat transfer process; that is, entropy can be generated or exergycan be dissipated during the process. Also it is interesting that, in general, allother physical energies (mechanical, electric, acoustic, etc.) can turn into thermalenergy as the final form, except we rarely, if ever, use entropy or exergy to measurethe irreversibility in other physical processes besides heat.

(b) Irreversibility measurements in heat transfer

Before discussing the irreversibility in a heat transfer process, it is necessary todistinguish the objectives of heat transfer, for, as shown below, the irreversibilitymay have different implications for different objectives. The objectives in heattransfer can be classified into two categories: one is to use thermal energy asa form of energy to perform work, and the other is directly using thermalenergy for warming up or cooling down the temperature. When performingwork, i.e. in heatwork conversion, the heat transfer process is a link in athermodynamic cycle. Whereas for heating or cooling, the heat transfer processis apparently much simpler. We will demonstrate that it is necessary to usedifferent concepts and quantities to describe the irreversibilities in two such

distinctive processes.Based on the analogy between heat conduction and electric conduction, Guoet al. (2007) introduced a physical quantity, termed entransy, to study a heat



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transfer process not involved in heatwork conversion. The definition of entransyG is

G=1

2UT, (2.1)

where T is the temperature and U the internal energy of the system. The

expression of entransy is analogous to that of electric energy Ee in a capacitor,

Ee =1

2QeV, (2.2)

where Qe and V are the electric quantity and the electric potential, respectively.As shown in equations (2.1) and (2.2), the entransy G in the heat transfershould be viewed as a special form of energy with the dimension (JK). Further,accompanying the electric charge, the electric energy is transported during electricconduction. Similarly, along with the heat, the entransy is transported duringheat transfer too. For example, in a heat conduction process, the thermal energy

conservation equation can be expressed as

rcvvT

vt= V q + Q, (2.3)

where r is the density, cv the constant-volume specific heat, t the time, q the heatflow density and Q the internal heat source. Multiplying both sides of equation(2.3) by temperature T gives an equation that can be viewed as the conservationequation of the entransy in the heat conduction:

rcvTvT

vt= V (qT) + q VT + QT, (2.4)

that isvg

vt= V (qT) fh + g, (2.5)

where g= G/V = (1/2) uT is the specific entransy, V the volume, u the specificinternal energy, qT the entransy flow density, g the entransy change owing toheat source and fh can be taken as the entransy dissipation rate per unit volume,expressed as

fh = q VT. (2.6)

The left term in either equation (2.4) or (2.5) is the time variation of the entransy

stored per unit volume, consisting of three items shown on the right: the firstrepresents the entransy transferred from one (or part of the) system to another(part), the second term can be considered as the local entransy dissipation duringthe heat conduction and the third is the entransy input from the internal heatsource. It is clear from equation (2.5) that the entransy is dissipated when heatis transferred from high temperature to low temperature. Thus, heat transfer isirreversible from the viewpoint of entransy, and the dissipation of entransy canhence be used as a measurement of the irreversibility in heat transfer.

Figure 1 is a schematic diagram of a case of one-dimension steady-state heatconduction. During the heat transfer, the entropy (or exergy) and entransy

are transported with the heat. Among the system parameters, the thermalenergy is conserved during the entire process, i.e. q1 q2, but neither theentropy nor the entransy is conservedthe entropy is generated and the entransy



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1016 Q. Chen et al.

T1 >T2, q1 =q2,

q2

g2

s2

q1 g1 s1

T2

T1

g1 >g2, s1 g2. Now we have two physical quantitiesfor measuring the irreversibility in heat transfer, and we will demonstratethat, because of the intrinsic complex nature of heat transfer, we need bothin dealing with the aforementioned different objectives in heat transfer. For

heatwork conversion, the entropy generation or the exergy dissipation is abetter irreversibility measurement, whereas for heating or cooling, the entransydissipation is preferable. The physical distinctions between them will be furtherelucidated later in this article.

3. Irreversibility of heat transfer and Fouriers Law

In non-equilibrium thermodynamics, the thermodynamic force and thermo-dynamic flow for heat transfer are chosen to ensure that the scalar product of

them equals the entropy generation rate Sgen=

k|V

T|2

/T

2

. Thus, based on theOnsager theory, the linear phenomenological heat transfer law can be generallyexpressed as

q= LV

1

T

, (3.1)

where L is the phenomenological coefficient for heat transfer process and aconstant unrelated to temperature. The phenomenological law in equation (3.1)is also termed the entropy picture of heat transfer by Gyarmati (1970), which isdifferent from the Fourier picture of heat transfer, i.e. Fouriers Law:

q= kV

T (3.2)where k is the thermal conductivity. Comparison between equations (3.1) and(3.2) gives the relationship between the different phenomenological coefficients inthe two representations

L = kT2 or k=L

T2=

constant

T2(3.3)

That is, if the phenomenological law of heat transfer in non-equilibriumthermodynamics is to be consistent with Fouriers Law, the thermal conductivityk has to be proportional to 1/T2.

According to the entropy picture of heat transfer, Prigogine (1967) proposed in1947, the least energy dissipation principle, i.e. the minimum entropy generationprinciple, which is expressed for systems that satisfy the linear phenomenological



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law and the Onsager reciprocal relations as: during evolution towards astationary state the entropy production decreases and takes its lowest valuecompatible with external constraints when this stationary state is reached (p.85). From this minimum entropy generation principle, Fouriers Law was thenderived. However, in the derivation, the phenomenological coefficient L in the

entropy picture was treated as a constant, requiring the thermal conductivitybe inversely proportional to the square of the absolute temperature duringsteady-state heat conduction. But in actuality, the thermal conductivity of mostmaterials used in engineering is largely a constant, independent of temperatureunder normal conditions. This led to the conclusion by Prigogine (1967) himselfthat Fouriers Law derived using the minimum entropy production principle isnot the most desirable.

In contrast, according to the expression of entransy dissipation in equation(2.6), the temperature gradient is thought as the generalized force and the heatflux as the generalized flow for heat conduction, i.e.

X =VT (3.4)and

J = q (3.5)

Here, the scalar product of the generalized force and generalized flow equals tothe entransy dissipation rate. Then based on the Onsager theory, the generalizedforce representation of the dissipation function j can be expressed as

j = 12 L(VT)2, (3.6)

where the constant, L is the heat transfer phenomenological coefficient.

Moreover, the linear phenomenological law of the generalized force andgeneralized flow is

q= LVT. (3.7)

Substituting equation (3.7) into equation (3.6) yields

j = 12 q VT. (3.8)

Comparing equation (2.6) with equation (3.8) gives

2j fh = 0. (3.9)

In order to search for the extremum of the dissipation function in equation (3.6)with the constraint from equation (3.9), a Lagrange function, F, is constructed as

F = j + A(2j fh). (3.10)

where A is the Lagrange multiplier. The variation ofF with respect to the generalforce, as shown in equation (3.4), is

dF =

(2A 1)

vj

vVT A

vfh

vVT

dVT. (3.11)

For F to be stationary, the term within the square bracket must vanish, that is,

(2A 1)vj

vVT A

vfh

vVT= 0. (3.12)



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Meanwhile, the partial differential of the entransy dissipation and the dissipationfunction, as shown in equations (2.6) and (3.6), with respect to the temperaturegradient VT are, respectively

vfh

vVT

= q=fh

VT

(3.13)

andvj

vVT= LVT =

2j

VT. (3.14)

Substituting equations (3.13) and (3.14) into equation (3.12) yields

(2A 1)2j Afh = 0. (3.15)

Equating equation (3.9) to equation (3.15), we obtain the Lagrange multiplierA = 1. Then equation (3.11) can be rewritten as

dF =

vj

vVT

vfh

vVT

dVT. (3.16)

This is the requirement for the extreme value of the force representation inthe dissipation function and satisfies both the linear phenomenological law andthe Onsager reciprocal relation. Because j is a positive-definite function andd2F = d2(j fh)J = L > 0, the extreme value determined by equation (3.16) isthe minimum, i.e.

(j fh)

q= min, dq= 0, dVT = 0 (3.17)

Since the temperature gradient VT is arbitrary, the term in the parenthesesshould be equal to 0.

LVT + q= 0. (3.18)

Now when the linear phenomenological coefficient L* is chosen to equal thethermal conductivity k, equation (3.18) is exactly Fouriers Law for heat transfer.In other words, for a given heat flux distribution, of all the possible temperaturedistributions, the actual solution is the one that satisfies the minimum entransydissipation. Similarly, from the general flux representation of the dissipation

function, we can also prove that for a given temperature distribution, comparedwith all the other possible heat flux distributions, the actual solution satisfiesthe least action principle based on the entransy dissipation. Therefore, for anarbitrary boundary condition, heat will always flow along the path with the leastaction of the minimum entransy dissipationa truly natural law as expected.

The aforementioned deduction process is similar to that based on non-equilibrium thermodynamics, and the differences between them lie in that theleast action here refers to the entransy dissipation rate, while the least action usedin non-equilibrium thermodynamics is in terms of the entropy generation rate.Furthermore, Fouriers Law derived here requires a constant thermal conductivity,

whereas in non-equilibrium thermodynamics, the thermal conductivity has to beinversely proportional to the squared temperaturesomething deviating grosslyfrom existing facts.



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4. Optimization of heat transfer with different ob jectives

(a) Entransy dissipation extremum principle

Heat transfer optimization aims for minimizing the temperature difference at agiven heat transfer rate,

d(DT) = df(x, y, z, t, T, k, q, r, cp, ) = 0 (4.1)

or maximizing the heat transfer rate at a given temperature difference

dQ= dg(x, y, z, t, T, k, q, r, cp, ) = 0. (4.2)

In conventional heat transfer analysis, it is difficult to establish the relationshipbetween the local temperature difference (or local heat transfer rate) andthe other related physical variables over the entire heat transfer area, so thevariational methods in equations (4.1) and (4.2) are not practically usable.

However, in addition, to be the expression of least action in heat transfer, theentransy dissipation in equation (2.6) is also a function of the local heat flux andlocal temperature gradient in the heat transfer area, and thus the variationalmethod will become usable if written in terms of the entransy dissipation(Cheng 2004).

Integrating the conservation equation of the entransy equation (2.4) over theentire heat transfer area gives

U

rcTvT

vtdV =

U

V (qT)dV +

U

q VTdV +

U

QTdV. (4.3)

For a steady-state heat conduction problem, the left term in equation (4.3)vanishes, i.e.

0 =

U

V (qT)dV +

U

q VTdV +

U

QTdV. (4.4)

If there is no internal heat source in the heat conduction domain, equation (4.4)is further reduced into

U

q VTdV =

U

V (qT)dV. (4.5)

By transforming the volume integral to the surface integral on the domainboundary according to Gausss Law, the total entransy dissipation rate in theentire heat conduction domain is deduced as

Fh =

U

q VTdV =G

qGT dS =G+

qinTin dS G

qoutTout dS, (4.6)

where G+ and G represent the boundaries of the heat flow input and output,respectively.

The continuity of the total heat flowing requires a constant total heat flow Qt,

Qt =

G+

qin dS =G

qout dS. (4.7)



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We further define the ratio of the total entransy dissipation and the total heatflow as the heat flux-weighted average temperature difference DT

DT =Fh

Qt=

G+

qin

QtTin dS

G

qout

QtTout dS. (4.8)

For one-dimensional heat conduction, equation (4.8) is reduced into DT =(Tin Tout), exactly the conventional temperature difference between the hot andcold ends. Using the heat flux-weighted average temperature difference definedin equation (4.8) and applying the divergence theorem, a new expression foroptimization of a steady-state heat conduction at a given heat flow rate can beconstructed as

Qtd(DT) = d

U

k|VT|2dV = 0. (4.9)

It shows that when the boundary heat flow rate is given, minimizing the

entransy dissipation leads to the minimum in temperature difference, that is,the optimized heat transfer. Conversely, to maximize the heat flow at a giventemperature difference, equation (4.9) can be rewritten as

DTdQt = d

U

1

k|q|2dV = 0, (4.10)

showing that maximizing the entransy dissipation leads to the maximum inboundary heat flow rate.

Similarly, for a steady-state heat dissipating process with internal heat sourcein equation (4.4), the total entransy dissipation rate in the entire heat conduction

domain is derived asFh =

U

q VTdV =

U

QT dV G

qoutTout dS. (4.11)

The heat generated in the entire domain will be dissipated through theboundaries, i.e.

Qt =

U

QdV =G

qout dS. (4.12)

Again the heat flux-weighted average temperature is defined as the entransydissipation over the heat flow rate

DT =Fh

Qt=

U

Q

QtTdV

G

qout

QtTout dS. (4.13)

And thus the optimization of the process is achieved when

Qtd(DT) = d

U

k|VT|2dV = 0, (4.14)

which means that in a heat dissipating process, minimizing the entransydissipation leads to the minimum-averaged temperature over the entire domain.

Based on the results from equations (4.9), (4.10) and (4.14), it can be concludedthat the extremes in entransy dissipation lead to the optimized heat transferperformance at different boundary conditions.



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T0

uniform heat source

Q

L

high-

conductivity

material

H

W

adiabatic

Figure 2. Two-dimensional heat conduction with a uniformly distributed internal heat source.

(b) Application to a two-dimensional volume-point heat conduction

We will apply our proposed approach to practical cases where heat transfer isused for heating or cooling such as in the so-called volume-point problems (Bejan1997) of heat dissipating for electronic devices as shown in figure 2. A uniforminternal heat source Q distributes in a two-dimensional device with length andwidth of L and H, respectively. Owing to the tiny scale of the electronic device,the joule heat can only be dissipated through the surroundings from the point

boundary area such as the cooling surface in figure 2, with the opening W and thetemperature T0 on one boundary. In order to lower the unit temperature, a certainamount of new material with high thermal conductivity is introduced inside thedevice. As the amount of the high thermal conductivity material (HTCM) isgiven, we need to find an optimal arrangement so as to minimize the averagetemperature in the device.

According to the new extremum principle based on entransy dissipation,for this volume-point heat conduction problem, the optimization objective isto minimize the volume-average temperature, the optimization criterion is theminimum entransy dissipation, the optimization variable is the distribution of

the HTCM and the constraint is the fixed amount of the HTCM, i.e.U

k(x, y)dV = constant. (4.15)

By the variational method, a Lagrange function, P, is constructed

=

U

[k|VT|2 + Bk]dV, (4.16)

where the Lagrange multiplier B remains constant because of a constant thermalconductivity. The variation ofP with respect to temperature T gives

G

kVTdT ndS

U

V (kVT)dTdV = 0. (4.17)



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Because the boundaries are either adiabatic or isothermal, the surface integralon the left side of equation (4.17) vanishes, that is,

G

kVTdT ndS = 0. (4.18)

Moreover, owing to a constant entransy output and a minimum entransydissipation rate, the entransy input reaches the minimum when

d

U

QT dV =

U

QdTdV = 0. (4.19)

Substituting equations (4.18) and (4.19) into equation (4.17) in fact gives thethermal energy conservation equation based on Fouriers Law

V (kVT) + Q= 0. (4.20)

This result shows again that the irreversibility of heat transfer can be measured

by the entransy dissipation rate Fh. The variation ofP with respect to thermalconductivity k gives

|VT|2 = B= constant (4.21)

This means that in order to optimize the heat dissipating process, i.e. tominimize the volume-average temperature, the temperature gradient should beuniform. This in turn requires that the thermal conductivity be proportional tothe heat flow in the entire heat conduction domain, i.e. the HTCM be placed atthe area with the largest heat flux.

As an example, the cooling process in a low-temperature environment isanalysed here. For the unit shown in figure 2, L = H = 5cm, Q= 100Wcm2,W = 0.5 cm and T0 = 10 K. The thermal conductivity of the unit is 3 W (mK)1,and that for the HTCM is 300 W (mK)1 occupying 10 per cent of the whole heattransfer area. The implementary steps are as follows:

(A) Local optimization

(1) Divide the entire heat transfer region into several parts; (2) set allthe parts composed by the original substrate material with low thermalconductivity; (3) numerically simulate the temperature field and theentransy dissipation rate in the entire heat transfer region; (4) find thepart with the highest temperature gradient, and fill it with the HTCM; (5)numerically simulate again the temperature field and the total entransydissipation rate in the heat transfer region; (6) compare the total entransydissipation rates with and without the HTCM element just filled in. If thetotal entransy dissipation rate is reduced, go to step 8. Otherwise go tostep 7; (7) remove the HTCM element from the part that is just filled inand fill it in the part where the temperature gradient is the next largest.Then go to step 5; (8) judge whether all the HTCM has been used up. Ifso, go to (B) global optimization. Otherwise, go to step 4.

(B) Global optimization

In the above steps, the HTCM is distributed by iteration. Since localoptimum may not assure a global optimum, the results obtained aboveneed further adjustment. The detailed steps are:



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(a)

(c)

(b)

Figure 3. Different arrangements of HTCM. (a) Simple uniform HTCM arrangement, (b) HTCMarrangement using the extremum principle of entransy dissipation and (c) HTCM arrangementusing the principle of minimum entropy generation.

(9) Choose one of the parts filled with the HTCM one after another;(10) remove the HTCM element from the chosen part and fill it in withthe original substrate material; (11) numerically simulate the temperature

field and the entransy dissipation rate in the entire heat transfer region;(12) find the part with the highest temperature gradient, and fill it withthe HTCM; (13) numerically simulate again the temperature field and thetotal entransy dissipation rate in the heat transfer region; (14) comparethe total entransy dissipation rates with and without using the HTCM.If the total entransy dissipation rate is reduced, go to step 16. Otherwisego to step 15; (15) remove the HTCM element from the part that is justfilled in and fill it in the part where the temperature gradient is the nextlargest. Then go to step 13; (16) judge if all the HTCM filled parts in serieshave been checked and the arrangement of them has not been changed:

if not, go to step 9 and continue the optimization steps, or else endthe optimization.

For example, after dividing the whole heat transfer area into 40 40 parts,figure 3b shows the distribution of the HTCM according to the extremumprinciple of entransy dissipation, where the black area represents the HTCMthesame hereinafter. The HTCM with a tree structure absorbs the heat generated bythe internal source and transports it to the isothermal outlet boundarysimilarin both the shape and function of actual tree roots.

For a fixed amount of HTCM, figure 4a,b compares the temperature

distributions between a uniform distribution of HTCM shown in figure 3a,and the optimized distribution in figure 3b based on the extremum principleof entransy dissipation. The average temperature in the first case is 544.7 K



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1024 Q. Chen et al.

600

650

(a)

(c)

(b)

550

500

55

25 25

35

45 45

5555

65 65

7575

35

55

350450

250150

180 180

170 170

160 160

150 150

150150

130 130

140140

120

11025

120

Figure 4. The temperature fields obtained from different arrangements of HTCM. (a) From theuniform HTCM arrangement in figure 3a, (b) from the optimized arrangement of HTCM infigure 3b and (c) from the HTCM arrangement in figure 3c.

while the temperature in the second-optimized case is 51.6 K, a 90.5 per centreduction! It clearly demonstrates that the optimization criterion of entransy

dissipation extremum is highly effective for such applications. Furthermore, asshown in figure 4b, the temperature gradient field is also less fluctuating in theoptimized case.

(c) The same case optimized using the minimum entropy approach

For comparison, we also treated the same problem using the minimum entropygeneration principle. Again, the constraint is

U

k(x, y)dV = constant, and whatwe seek is also the optimal HTCM arrangement, except in this case that:(i) the optimization criterion is the minimum entropy generation and (ii) the

corresponding energy conservation equation should be added as a constraint,because it is not implied in the principle of minimum entropy generation whenthe thermal conductivity is constant.



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Introducing the corresponding Lagrange function

=

U

k

|VT|2

T2+ Bk + C(V kVT + Q)

dV, (4.22)

where B and C are also the Lagrange multipliers. The constraint of thermalconductivity is the isoperimetric condition, and consequently B is a constant.C is a variable related to space coordinates. The variation ofP with respect totemperature T gives

V (kVC) =2k|VT|2

T3+

2Q

T2. (4.23)

While the variation ofP with respect to thermal conductivity k yields

VC VT V

T

2

T2= B = constant (4.24)

Likewise, equation (4.24) gives the guideline for optimization based on thecriterion of minimum entropy generation. Meanwhile, the most optimization stepsused are the same as in the first case, except that: (i) change finding the partwith extreme temperature gradient to that with the extreme in absolute value ofVA3 VT (VT2/T2); (ii) replace the criterion of the total entransy dissipationrate by that of the total entropy generation rate.

Figure 3c shows the distribution of HTCM based on the principle ofminimum entropy generation. Comparison offigure 3b,c shows that although thedistributions of HTCM are similar between the two results in most areas, theroot-shape structure from the extremum principle of entransy dissipation is notdirectly connected to the heat flow outlet, leaving some parts with the originalmaterial in between them so that the heat cannot be transported smoothlyto the isothermal outlet boundary. Figure 4c gives the optimized temperaturedistribution obtained by the minimum entropy generation. Because the low-thermal conductivity material is adjacent to the heat outlet, the temperaturegradient grows larger and thus lowers the entire heat transfer performance. Theaveraged temperature of the entire area is 150.899.2 K higher than that obtainedby the extremum principle of entransy dissipation. From the definition, it is

easy to find that in order to decrease the entropy generation, we have to bothreduce the temperature gradient and raise the temperature, thus leading to thearrangement of HTCM shown in figure 3c.

In addition, according to the principle of minimum entropy generation,the optimization objective of a steady-state heat dissipating process can beexpressed as:

Qd

D

1

T

m

= d

U

k|VT|2

T2dV = 0, (4.25)

where (D(1/T))m = ((1/T) (1/T0))m is the equivalent thermodynamicspotential difference that represents the generalized force in the entropy picturefor heat transfer. Thus, minimizing the entropy generation equals to minimizing



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Table 1. Optimized results obtained by the optimization criteria of minimum entropy generationand entransy dissipation extremum.

optimization results

optimization criterion Fh (WK)1 Sgen (W K1) Tm (K) Tmax (K)D

1

T

m/(1/K)

extremum entransydissipation

5.5 104 100.7 51.6 83.0 2.2 102

minimum entropygeneration

1.58 105 81.7 150.8 194.9 7.1 103

the equivalent thermodynamics potential difference (D(1/T))m, leading to the

highest exergy transfer efficiency. That is, the minimum entropy generationprinciple is equivalent to the minimum exergy dissipation during a heattransfer process.

To facilitate the comparison between the two results from figure 4b,c,table 1 lists the key findings side by side, obtained, respectively, by theoptimization criteria of the minimum entropy generation and the entransydissipation extremum. It indubitably shows in the table that the proposedentransy-based approach is more effective than the entropy-based one in heattransfer optimization, for the former leads to a result with significantly reducedmean temperature than that by the latter (51.6 versus 150.8 K), and muchlower maximum temperature (83.0 versus 194.9 K). Whereas the entropy-based approach is preferred in exergy transfer optimization, as it results in asignificantly lower equivalent thermodynamic potential (7.1 103 K1 versus2.2 102 K1).

Besides heat conduction, we (Chen et al. 2009b) also compared the twocriteria in heat convection optimization. Our results indicate that both principlesare applicable to convective heat transfer optimization, subject, however, todifferent objectives. The minimum entropy generation principle works better insearching for the minimum exergy dissipation during a heatwork conversion,whereas the entransy dissipation extremum principle is more effective forprocesses not involving heatwork conversion, in minimizing the heat transfer

ability dissipation.In addition, based on the concept of the entransy dissipation rate, we (Chen

et al. 2009a) introduced the non-dimensional entransy dissipation rate andemployed it as an objective function to analyse the thermal transfer processin a porous material. Moreover, some of us (Guo et al. 2010) defined theequivalent thermal resistance of a heat exchanger to measure the irreversibilityof heat transfer in the processes of heating or cooling. After establishing therelationship between the heat exchanger effectiveness and the thermal resistance,Guo et al. found that reducing the thermal resistance leads to a monotonicincrease in the heat exchange effectiveness. Guo et al. also demonstrated that the

irreversibility in a heat exchanger is more effectively represented by its thermalresistance, while the so-called entropy generation paradox occurs if using theentropy generation criterion.

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5. Conclusions

In an electric conduction process or in a fluid flow process, although theelectron or the fluid mass is conserved, the electric energy or the mechanicalenergy is dissipated. These two non-conserved quantities are used as the

irreversibility measurements for their respective processes. Similarly, inheat transfer processes, the thermal energy is conserved. When the heattransfer involves heatwork conversion, exergy is the dissipated quantity,whereas in the heating or cooling process, we have demonstrated that thenewly defined entransy is the dissipated quantity.

Correspondingly, the objectives of heat transfer are classified into twodifferent categories: one is direct heating or cooling and the other isfor heatwork conversion. The irreversibility of the former case shouldbe measured by the entransy dissipation rate while the latter should bemeasured by the rate of entropy generation or exergy dissipation.

More generally, Fouriers Law derived by the minimum entransydissipation principle satisfies a constant thermal conductivity, whereas bythe principle of minimum entropy production, the thermal conductivity hasto be inversely proportional to the squared temperaturesomething thatcontradicts the existing facts. Thus, the entransy dissipation is a betteralternative for the measurement of irreversibility in heat transfer.

For the testing case of a volume-point heat conduction problem withuniform internal heat source, if a fixed amount of HTCM is usedto minimize the volume-average temperature, the entransy dissipationextremum has shown to be a more effective optimization criterion thanthe minimum entropy generation.

The present work is supported by the National Natural Science Foundation of China (grant no.51006060) and the Postdoctoral Scientific Fund of China (grant no. 200902080).

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An Alternative Criterion in Heat Transfer Optimization

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