International Journal of Heat and Mass...

International Journal of Heat and Mass Transfer xxx (2011) xxx–xxx

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Entransy expression of the second law of thermodynamics and its applicationto optimization in heat transfer process

W. Liu a,⇑, Z.C. Liu a, H. Jia a, A.W. Fan a, A. Nakayama b

a School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, Chinab Department of Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561, Japan

a r t i c l e i n f o

Article history:Received 7 October 2010Received in revised form 17 February 2011Accepted 17 February 2011Available online xxxx

Keywords:EntransyIrreversibilityConvective heat transferOptimization

0017-9310/$ - see front matter � 2011 Published bydoi:10.1016/j.ijheatmasstransfer.2011.02.041

⇑ Corresponding author. Tel.: +86 27 87541998; faxE-mail address: [email protected] (W. Liu).

Please cite this article in press as: W. Liu et al., Efer process, Int. J. Heat Mass Transfer (2011), do

a b s t r a c t

Based on theories of thermodynamics, the energy equation in terms of entransy in heat transfer process isintroduced, which not only describes the change of entransy, but also defines the entransy consumptionrate. According to the regularity of entransy change in heat transfer process and the effect of entransyconsumption rate on the irreversibility of heat transfer process, it can be found that entransy is a statevariable, from which a new expression for the second law of thermodynamics is presented. Then by set-ting entransy consumption rate and power consumption rate as optimization objective and constraintcondition for each other, the Lagrange conditional extremum principle is used to deduce momentumequation, constraint equation and boundary condition for optimizing flow field of convective heat trans-fer, which are applied to simulate convective heat transfer coupling with energy equation in an enclosedcavity. Through the numerical simulation, the optimized flow field under different constraint conditionsis obtained, which shows that the principle of minimum entransy consumption is more suitable than theprinciple of minimum entropy generation for optimizing convective heat transfer process.

� 2011 Published by Elsevier Ltd.

1. Introduction

Past centuries have witnessed the gradual perfection of classicalthermodynamics theories including irreversible thermodynamics.The second law of thermodynamics in particular, which is regardedas a fundamental law of physics, has found wide applications inengineering and scientific fields. However, in the field of heattransfer which is most closely related to thermodynamics, noimportant theoretical breakthrough has been achieved to heattransfer optimization over the past decades. The reason for thismay be that the second law of thermodynamics has not beenclosely integrated with heat transfer theories in developing newconcept and methodology.

In order to evaluate the degree of heat transfer enhancement,Guo afresh surveyed the mechanism for convective heat transferand proposed the field synergy principle [1,2]. This novel conceptattracted much attention of researchers [3–8]. Through theirconsistent effort, a systematic assessment for performance of heattransfer enhancement was gradually established. As we know thatto make heat exchange equipment work efficiently with high heattransfer coefficient and lower flow resistance or lower energyconsumption, the key factor is to optimize heat transfer process.For this purpose, Guo et al. newly proposed a new physical quan-

Elsevier Ltd.

: +86 27 87540724.

ntransy expression of the seconi:10.1016/j.ijheatmasstransfer.

tity ‘‘entransy’’ based on the analogy between heat conductionand electrical conduction as well as thermodynamics theories [9].They deducted this concept theoretically and validated it throughmodeling and numerical computation [10–16]. Although introduc-tion of the concept of entransy has shown the advantages in heattransfer optimization, it is still in its initial stage and needs to beperfected. By resorting to the second law of thermodynamics, thispaper attempts to give new insights into the concept of entransywhich is treated as a basic physical quantity, and to reaffirm theimportance of entransy in developing optimization theories forheat transfer process.

2. The Irreversibility of heat transfer process

The transfer of energy, momentum and mass are three mainforms of transport phenomena in the nature or engineering. Energytakes many forms, such as thermal energy, mechanical energy,electronic energy, optical energy, sound energy, etc. When energyin any form other than thermal energy is transferred, a part of itwill be transformed into thermal energy, and when thermal energyis transferred, a certain amount of it will lose or dissipate to some-where. This is the so-called irreversibility of energy conversion andtransport processes. Therefore, in order to reduce the irreversibilityof heat transfer process, it is necessary to explore its physicalmechanism and optimize the process by quantitatively analyzingthe changes of physical quantities and recording the traces leftby these changes.

d law of thermodynamics and its application to optimization in heat trans-2011.02.041

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Nomenclature

A, B, C0 Lagrange multipliersc specific heat [J/(kg � K)]cp specific heat at constant pressure [J/(kg � K)]h enthalpy [J/kg]J, J0 functionalp pressure [Pa]Q heat flux [W]_Q 000 internal heat source [W/m3]q heat flux vector [W/m2]S entropy [J/K]s specific entropy [J/(kg � K)]Sg,p entropy generation induced by fluid viscosity [J/K]Sg,T entropy generation induced by heat transfer [J/K]T temperature [K]

t time [s]U velocity vector [m/s]V volume [m3]Z entransy [J � K]z specific entransy [J � K/kg]Ze,T entransy consumption [J � K]

Greek symbolsk thermal conductivity [W/(m � K)]q fluid density [kg/m3]l viscosity coefficient [kg/(m � s)]U dissipated heat from fluid viscosity [W/m3]X control volume [m3]

Fig. 1. Sketch of an enclosed cavity with geometry and boundary conditions(L = H = 15 mm).

2 W. Liu et al. / International Journal of Heat and Mass Transfer xxx (2011) xxx–xxx

2.1. Thermodynamic definition of entransy and its deduction

For a non-equilibrium convective heat transfer process, itsenergy equation can be expressed in terms of enthalpy as:

qDhDt¼ �r � qþUþ _Q 000: ð1Þ

Where q is the fluid density, h is the fluid enthalpy, �r � q is theheat flux transferring into and out of fluid elementary volume, Uis the dissipated heat from fluid viscosity, and _Q 000 is the internalheat source.

Eq. (1) can be rewritten in terms of entropy as:

qDsDt¼ �r � q

T

� �þ kðrTÞ2

T2 þUTþ

_Q 000

T: ð2Þ

Where s is the entropy of fluid, �r � qT

� �is the entropy flow transfer-

ring into and out of fluid elementary volume, kðrTÞ2

T2 is the entropygeneration rate induced by heat transfer process, U

T is the entropygeneration rate caused by the heat generation from viscous dissipa-tion of mechanical energy, which can also be defined as analogicalentropy source, and _Q 000

T is the internal heat source in form of entropyexpression. The differential entropy for incompressible fluid can beexpressed as: ds ¼ dh

T ¼ cdTT .

The principle of entropy generation minimization is widely uti-lized to optimize the thermodynamics process based on the secondlaw of thermodynamics [17]. According to finite time thermody-namics theory [18], the entropy generation rate induced by heattransfer is a representation of the irreversibility of the process, soit can be denoted as the dot product of entropy flow and force,i.e. kðrTÞ2

T2 ¼ qT � � rT

T

� �, while the entropy generation rate U

T denotesmechanical energy dissipation induced by fluid viscosity. Accord-ingly, thermodynamic equilibrium of entropy is given as:

DS ¼ S2 � S1 ¼Z 2

1

dQTþ Sg;T þ Sg;p: ð3Þ

Where the integral term represents entropy flow, Sg,T represents en-tropy generation induced by heat transfer, and Sg,p represents entro-py generation induced by fluid viscosity. When boundary heat fluxis zero, entropy flow is zero. Then Eq. (3) reduces to:

DS ¼ Sg;T þ Sg;p: ð4Þ

Multiplying both sides of Eq. (2) by T2 and making transformationyields:

qcTDTDt¼ �r � ðqTÞ � kðrTÞ2 þUT þ _Q 000T: ð5Þ

Please cite this article in press as: W. Liu et al., Entransy expression of the seconfer process, Int. J. Heat Mass Transfer (2011), doi:10.1016/j.ijheatmasstransfer

Referencing to the definition of entropy, the entransy of incom-pressible fluid, termed z, can be defined in the following differen-tial expression:

dz ¼ Tdh ¼ cTdT: ð6Þ

It can be seen that Eq. (6) is a differential definition for entransysimilar to thermodynamics entropy, which shows that the relationbetween differential enthalpy and temperature is expressed asproduct other than quotient. Thus the energy equation of convec-tive heat transfer can be expressed in terms of entransy as:

qDzDt¼ �r � ðqTÞ � kðrTÞ2 þUT þ _Q 000T: ð7Þ

Where z is the entransy of fluid, �r � (qT) is the entransy flowtransferring into and out of fluid elementary volume, k(rT)2 is theentransy consumption rate induced by heat transfer process, UTis defined as the analogical entransy source induced by dissipatedheat from fluid mechanical energy, and _Q 000T is the internal heatsource in form of entransy expression.

As we know that a state variable or parameter can be defined torepresent the state of a thermodynamics system. If the values of allstate variables of a system are known, the state of this system canbe well described. The entransy is such a variable introduced toexpress the nature of a system, in which the heat is transferred fromthe high to the low temperature sites. The entransy consumptionrate is induced by heat transfer temperature difference, which can

d law of thermodynamics and its application to optimization in heat trans-.2011.02.041


0.0001

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(c) (d)

Fig. 2. Vortex structure and optimized fields with minimum entransy consumption (C0 = �2.0 � 10�8). (a) Stream function field; (b) temperature field; (c) constraint variablefield; (d) pressure field.

W. Liu et al. / International Journal of Heat and Mass Transfer xxx (2011) xxx–xxx 3

be denoted as the dot product of entransy flow and force, i.e.kðrTÞ2 ¼ qT � � rT

T

� �. Clearly, k(rT)2 and UT are totally different,

the former reflects the irreversibility of thermal energy transportprocess, while the latter reflects the irreversibility of mechanicalenergy dissipation. Since the value of UT is much less than that ofk(rT)2, it can be neglected in the case of simplification.

Corresponding to Eq. (7), thermodynamic equilibrium of en-transy can be expressed as:

DZ ¼ Z2 � Z1 ¼ �Z 2

1TdQ � Ze;T : ð8Þ

Where the integral term is entransy flow, Ze,T is entransy consump-tion. When boundary heat flow is zero, entransy flow is zero. ThenEq. (8) reduces to:

DZ ¼ �Ze;T : ð9Þ

If the motion is excluded from consideration, the amount of vis-cous dissipation is zero, and heat transfer process can be regardedas pure heat conduction. Then enthalpy equation is:

Please cite this article in press as: W. Liu et al., Entransy expression of the seconfer process, Int. J. Heat Mass Transfer (2011), doi:10.1016/j.ijheatmasstransfer.

q@h@t¼ �r � qþ _Q 000; ð10Þ

entropy equation is:

q@s@t¼ �r � q

T

� �þ kðrTÞ2

T2 þ_Q 000

T; ð11Þ

and entransy equation is:

q@z@t¼ �r � ðqTÞ � kðrTÞ2 þ _Q 000T: ð12Þ

Therefore we can say that, for any heat transfer process,changes in enthalpy take place when heat flow is transported;changes in entropy take place when entropy flow is transported,and the changes is tracked by the entropy generation rate; changesin entransy take place when entransy flow is transported, and thechanges is tracked by entransy consumption rate. Thus, the goal ofreducing transport process irreversibility and enlarging heat trans-fer efficiency actually becomes to optimize entransy consumptionrate k(rT)2.



0.0001186477.28409E-05

1.69364E-05

0.000132235

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2.2. A new statement for the second law of thermodynamics

In non-equilibrium heat transfer process, both entropy andentransy would change as variables. From Eqs. (2) and (7), weknow that entropy in an irreversible process tends to increasewhile entransy changes in the opposite direction. The trace leftin a transport process over time and space takes the form of entro-py generation rate or entransy consumption rate. The higher thedegree of irreversibility is, the larger the entropy generation rateor entransy consumption rate would be. Therefore, entransy, sim-ilar to entropy, is actually a state variable or parameter reflectingthe irreversibility of a transport process, from which a new state-ment for the second law of thermodynamics would be expressedas: entransy never increases when heat is transferred from higherto lower temperature in non-equilibrium or equilibrium state. Thisstatement can be called as the principle of entransy decrease inheat transfer process. For a system in non-equilibrium state,entransy tends to decrease over time until the system reaches atequilibrium state; and for a system in equilibrium state, when heat


is transferred from heat source at high temperature to heat sink atlow temperature, entransy will decrease as well.

The principle of entransy decrease corresponds to the principleof entropy increase. It is well known that entropy is a measure ofthe degree of disorder of a system in the micro-level. The largerthe entropy is, the more chaotic the microscopic motion ofmolecules or particles will be. By analogy with this microscopicexplanation for entropy, entransy can be regarded as a physicalquantity measuring the degree of order of a system in the micro-le-vel. When the entransy of a system is small, it means that thesystem is not well ordered and its capability to transfer thermalenergy is not high. Conversely, an optimized system will be in goodorder and will have higher heat transfer capability. We take a gassystem with high temperature as an example. When the systemtransfers heat to its surroundings, the gas temperature in thesystem drops, and the molecular motion in the gas becomes moreand more disordered, so that the system becomes less and lessordered. When the gas temperature is lowered to the temperatureof surroundings, the degree of order of the system tends to be least,



2.9884E-06

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and its heat transfer capability approximates zero. The entransy ofthe system will be least at this time.

3. Application of the principle of entransy decrease to heattransfer optimization

Optimizing a heat transfer process is to reduce the irreversibil-ity of the process by controlling the entransy consumption and thedegree of order of the system. In order to optimize flow fields indifferent channels, the Lagrange conditional extremum principleis used to construct functional. Then by functional variation andseeking extremum, the momentum equation, constraint equationand boundary conditions for the optimized flow field are obtained.

3.1. Optimization method

The principle of minimum entropy generation proposed foroptimizing the heat-work conversion process contributes to theoptimization of the thermal energy transport process. From Eq.(2), one can find that the entropy generation rate is positively


correlated with temperature gradient and negatively correlatedwith absolute temperature. However, Eq. (7) shows that entransyconsumption rate is only correlated with temperature gradient.Lower entransy consumption rate ensures smaller temperaturegradient, which will even up temperature profile of the fluid,reduce heat resistance, and thereby enhance convective heattransfer. For the problem of heat transfer optimization, we wishthe temperature field of the fluid to be uniform. This can beachieved by minimizing temperature gradient rT or entransyconsumption rate k(rT)2, but not by minimizing entropygeneration rate kðrTÞ2

T2 . Therefore, although both entropy genera-tion rate and entransy consumption rate reflects the processirreversibility, the principle of minimum entransy consumptionis more appropriate for the optimization of the heat transferprocess.

Enhancing convective heat transfer and reducing fluid flowresistance are two aspects of improving the performance of heattransfer equipment, but they always contradict each other.Enhancing convective heat transfer may lead to the increase inflow resistance, and much higher flow resistance may weakenconvective heat transfer, which implies that there exists some kind



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of synergetic relation between them. Therefore, when constructingfunctional according to the Lagrange conditional extremum princi-ple, two aspects need to be considered: (1) to set minimum en-transy consumption rate as optimization objective and powerconsumption rate as constraint condition; (2) to set minimumpower consumption rate as optimization objective and entransyconsumption rate as constraint condition.

3.2. Optimization equation for convective heat transfer

For convective heat transfer process, the fluid momentum equa-tion is:

qU � rU ¼ �rpþ lr2U: ð13Þ

Multiplying both sides of Eq. (13) by velocity vector U yields:

�rp � U ¼ ðqU � rU � lr2UÞ � U: ð14Þ

Where �rp � U represents the work consumed by the fluid. Eq. (14)reflects the conservation of mechanical energy, i.e. the pump powerconsumed by the fluid is the sum of kinetic energy loss and viscous


dissipation power. Thus, if power consumption is small, the kineticenergy loss and viscous dissipation power are both small in thesystem. In the mean time, if the viscous dissipation can be reduced,the irreversibility of the process will be reduced.

For a heat transfer process in any channel, if the optimizationobjective is to achieve the temperature uniformity of fluid, thenwith minimum entransy consumption rate and fixed power con-sumption rate, the Lagrange function can be made by variationalcalculus:

J ¼Z

XkðrTÞ2 þ C0ðqU � rU � lr2UÞ � U þ Ar � Uh

þ Bðkr2T � qcpU � rTÞidV : ð15Þ

Where A, B and C0 are Lagrange multipliers. Multiplier C0, thermalconductivity k, fluid density q, specific heat cp and viscosity coeffi-cient l are taken as constant respectively.

By finding functional variation with respect to velocity U andtemperature T respectively, the momentum equation is obtainedas:



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Fig. 6. Vortex structure and optimized fields with minimum power consumption (C0 = �3.0 � 10�5). (a) Stream function field; (b) temperature field; (c) constraint variablefield; (d) pressure field.


qU � rU ¼ �rpþ lr2U þ qcpBC0rT; ð16Þ

where qcpBC0rT denotes the virtual volume force induced to optimize

flow field, and the constraint equation of parameter B is deduced as:

qcpU � rBþ kr2B� 2kr2T ¼ 0: ð17Þ

For Eq. (17), the boundary condition at constant wall tempera-ture is:

B ¼ 0; ð18Þ

and the boundary condition at constant heat flux is:

2krT � krB ¼ 0: ð19Þ

In addition, if the optimization objective is to reduce flow resis-tance of the fluid, we can construct Lagrange function with mini-mum power consumption rate and fixed entransy consumptionrate as:


J0 ¼Z

XðqU � rU � lr2UÞ � U þ C0kðrTÞ2 þ Ar � Uh

þ Bðkr2T � qcpU � rTÞidV : ð20Þ

After finding functional variation with respect to velocity U andtemperature T respectively, we can have the momentum equationas:

qU � rU ¼ �rpþ lr2U þ qcpBrT; ð21Þ

where qcpBrT represents the virtual volume force, similar to theone in Eq. (16), and the constraint equation of parameter B as:

qcpU � rBþ kr2B� 2C0kr2T ¼ 0: ð22Þ

For Eq. (22), the boundary condition at constant wall tempera-ture is:

B ¼ 0; ð23Þ

and the boundary condition at constant heat flux is:

2C0krT � krB ¼ 0: ð24Þ



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The coupling of above momentum equation, constraint equa-tion and energy equation makes it possible to simulate convectiveheat transfer in any channel with different boundary conditions.Based on the simulation results, the optimal flow field structuredepending on different optimization objectives and constraint con-ditions can be obtained to achieve the ultimate goal of enhancingconvective heat transfer and controlling fluid flow resistance,which may guide the design for heat transfer unit and heatexchanger.

4. Calculation and analysis for the optimized flow field in anenclosed cavity

A 2D enclosed cavity model [19] with geometric size of15 � 15 mm2 is shown in Fig. 1. The left and right wall tempera-tures are constant: T1 = 300K, Th = 315K. The upper and lower wallsare in adiabatic condition: qw = 0. Normally speaking, in anenclosed cavity with constant wall temperature or constant heatflux, the driving force for convective heat transfer comes fromthe density difference of the fluid. Buoyancy force, however, is


not included in the model we established, the fluid is driven by avirtual volume force induced by heat transfer optimization. Thus,by solving above controlling equations, we can inspect the effectof virtual force field on the flow organization and optimization tovalidate the model. Noting that the value of coefficient C0 is relatedto the intensity of virtual force field, we find that the magnitude ofC0 value would result in different flow filed structure. In addition,since C0 value also reflects the degree of constraint, it is restrictedto a certain range depending on the geometric size and thermalboundary conditions. The FLUENT 6.3 is used to the simulation.The velocity and pressure are linked using the SIMPLEC algorithm.The convection and diffusion terms are discretized using the QUICKscheme. The user defined function (UDF) in the FLUENT is utilizedfor solving the governing Eqs. (16) and (21) respectively. The con-vergence solutions are obtained, when the residuals of all the cou-pled equations are less than 10�8.

Figs. 2–5 show the calculation results for the case in which min-imum entransy consumption is set as optimization objective andfixed power consumption as constraint condition. The cavity sizesare chosen as non-dimension in the form of center symmetry. Inthe calculation, a different value for C0 is taken. It can be found that



0.00012

0.0003

2E-05 3.78198E-06

0.00012

0.00016

8E-05

3.78198E-06

0.000352629

0.00016

X

Y

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01 303

308

304.798

312

304

306

310 31

2

304

308

311

X

Y

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01

6

-1

66 5

-5

-3-1

1

4

3

0-3

-5-5

-3

13

5

X

Y

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01

-0.004

-0.002

-0.004-0.002

-0.0003044930.000

799483

-0.002

-0.000304493

-0.004

-0.000304493

-0.000304493

-0.00285337

X

Y

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01

(a) (b)

(c) (d)



with the increase in absolute value of C0, the number of vortex willincrease from one to four, and if absolute value of C0 keeps grow-ing, more vortexes would appear as a result of growing virtual vol-ume force. Apparently, the increase in vortex number wouldintensify the fluid disturbance, which would even up temperatureprofile, leading to enhancing heat transfer. Accordingly, the virtualvolume force is acting as a vortex generator in the enclosed cavitywithout considering buoyancy force.

Figs. 6–9 show the calculation results for the case in which min-imum power consumption is set as optimization objective andfixed entransy consumption as constraint condition. It can befound that, similar to the case in Figs. 2–5, with the increase inabsolute value of C0, the vortex number inside the enclosed cavitywill increase from one to four, and more vortexes would appearwhen C0 keeps growing. By comparing the two cases, we can findthat whether the optimization objective is minimum entransy con-sumption or minimum power consumption, the optimization re-sults do not differ greatly. If the comparison is made closely, itcan be found that temperature distribution in the case of minimum


power consumption appears more even and heat transfer in thiscase is more effective.

Fig. 10(a) shows the change curve of heat transfer capacity inthe enclosed cavity when different optimization objectives areset. If there is no volume force acting inside, the process is pureheat conduction with heat transfer capacity of 0.365W. If the opti-mization objective is minimum entransy consumption, comparedto the process of pure heat conduction, heat transfer capacity in-creases from 97% to 611%. If the optimization objective is mini-mum power consumption, heat transfer capacity increases from111% to 683%. Fig. 10 (b) shows the change curve of Nu numberwith different optimization objective. In case of pure heat conduc-tion, equivalent Nu number is 1. When certain optimization objec-tive is taken, Nu number will be larger than 1, which indicates acertain degree of heat transfer enhancement. After comparingbetween Figs. 10(a) and 10(b), we find that under the same vertexnumber in the enclosed cavity, heat transfer capacity withminimum power consumption as optimization objective is greaterthan that with minimum entransy consumption as optimization



0.000168E-05

4E-05

0.000260.00034

0.00012E-05

0.000240.00032

X

Y

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01303

305 306 307

308 309 310 311

307306304

309311

304

X

Y

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01

-4

2

4

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3

-4

3

-5

-5

-5

5

5

-1

-1

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Y

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01

0.002

0.004

0.006

0.004

0

0.002 00.004

0.002

-0.002

0.006

0.002

X

Y

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01

(a) (b)

(c) (d)


Fig. 10. Heat flux (a) and Nu number (b) under different optimization objective. C0,1: for minimum power consumption rate; C0,2: for minimum entransy consumption rate.


Please cite this article in press as: W. Liu et al., Entransy expression of the second law of thermodynamics and its application to optimization in heat trans-fer process, Int. J. Heat Mass Transfer (2011), doi:10.1016/j.ijheatmasstransfer.2011.02.041



objective. This suggests that to effectively reduce flow resistance isconducive to enhancing heat transfer.

5. Conclusions

Entransy is a state variable that measures the degree of order of asystem in which heat is transferred. Entransy consumption rate re-flects the degree of irreversibility of heat transfer process, which canbe expressed as a dot product of entransy flow and force. The prin-ciple of entransy decrease in heat transfer process states thatentransy would never increase when heat is transferred from higherto lower temperature in the non-equilibrium or equilibrium state.

Since entransy consumption rate is only correlated with tem-perature gradient of the fluid, and temperature uniformity is oneof most important objective pursued by convective heat transferoptimization, the principle of minimum entransy consumption isaccordingly more suitable for optimizing heat transfer processthan the principle of minimum entropy generation. The lowerthe entransy consumption rate is, the better the temperature uni-formity of the flow field will be, and the less the irreversibility ofconvective heat transfer process will be.

As power consumption of the fluid is both related with flowresistance and momentum loss, to achieve minimum power con-sumption is considered as an appropriate optimization objective.The lower the power consumption rate is, the less the flow resis-tance will be, and the smaller the momentum loss will be, whichleads to enhancing heat transfer.

When optimizing a heat transfer process, entransy consumptionrate and power consumption rate may not be reduced simulta-neously. By setting minimum power consumption rate as optimi-zation objective and entransy consumption rate as constraintcondition, better optimization results can be obtained than thecase by setting minimum entransy consumption rate as optimiza-tion objective and power consumption rate as constraint condition,for the parameters selected in the paper, such as Re number, geo-metric sizes and boundary conditions, etc.

Acknowledgements

This work is financially supported by the National NaturalScience Foundation of China (Grant No. 51036003, 50721005)


and the National Key Fundamental R & D Program of China (GrantNo. G2007CB206903).

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