International Journal of Heat and Mass Transfer 98 (2016) 183–192

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International Journal of Heat and Mass Transfer

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

Theoretically and numerically investigation about the novel evaluatingstandard for convective heat transfer enhancement based on theentransy theory

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.02.0810017-9310/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: 09620@buaa.edu.cn (H. Li).

Haidong Yu, Jie Wen, Guoqiang Xu, Haiwang Li ⇑National Key Laboratory of Science and Technology on Aero Engines Aero-thermodynamics, The Collaborative Innovation Center for Advanced Aero-Engine of China,Beihang University, Beijing 100191, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 August 2015Received in revised form 29 December 2015Accepted 24 February 2016Available online 24 March 2016

Keywords:Entransy theoryEnergy dissipationEfficiency factorField synergy principle

This paper theoretically and numerically investigated a novel evaluating standard de for the enhancementof convective heat transfer based on the entransy theory. In the theoretical derivation, the differentialequations about entransy are established using the method of micro-control volume which can givethe obvious physical changing processes. And then, an efficiency factor de is recommended accordingto physical mechanisms of entransy dissipation which is considered the impact of energy dissipation.The principle is verified by numerical simulation. The results indicated that de is more appropriate forevaluating the ability of convective heat transfer enhancement comparing with the traditional methodof FSP (field synergy principle) and EDEP (entransy dissipation extremum principle). The results alsoshowed that the higher the factor of de, the more efficient heat transfer process.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Heat-exchange system is widely used in daily life, industry,energy utilization and so on. In recent years, it was used in aero-nautics and astronautics also.

In order to design and optimize a heat-exchange system, somespecial and important parameters should be focused and calcu-lated. Heat transfer coefficient and flow resistance were alwayscoupled considered in design and optimization because that flowresistance determine the energy consumption of the system andthe heat transfer coefficient reflect the profit of the system. Atthe same time, these two factors are coupled with each other toevaluate the energy utilization efficiency [1].

Lots of researchers paid attentions about how to reasonablyevaluate the energy utilization efficiency of heat exchanger [2].The entropy is one of the most popular factors to be used; it con-sidered that the system is optimum if the system has the mini-mization of entropy generation [3–7]. Chen [8–13] pointed outthat minimum entropy generation principle should be used forthe target of reducing exergy loss. But the entropy generation rep-resents the conversion of heat-to-work. Although the entropy andentransy are all corresponds to microstate number of the system[15], which indicates that both entropy and entransy could

describe the irreversibility of thermal processes, the paradoxwould be derived by utilizing the minimum entropy for some flowand heat transfer optimization problems [39]. Thus, it is not suit-able to evaluate the system without thermodynamic work becausethat the principle reducing the loss of exergy is not always equiv-alent to strengthen the process of the heat transfer system in thesystem without thermodynamic work.

The entransy theory is applied in heat transfer process opti-mization in recent years because that it can represent the heattransport potential capacity of an object [14]. Chen [8–13] alsoindicated that extremum principle of entransy dissipation adoptedwhereas for improving the heat transfer ability.

In aspect of theoretical analysis, Cheng [15–17] proved that theentransy has the property of the extensive quantity in the mona-tomic ideal gas system. At the same time, they investigated thechanging of the entransy in isolated system and the results showedthat the entransy is the uniform function for microscopic statesand its value decreases during the process of the state transition.Following the work of Cheng [15–17], Meng [18–20] deduced thefield coordination equation of the steady laminar flow heat transferby utilizing the variation method.

In aspect of applied analysis, FSP (field synergy principle) andEDEP (entransy dissipation extremum principle) are two popularmethods based on the entransy principle. Chen [21–24] optimizedmany heat transfer problems by using the EDEP. Xu [25] appliedthe minimum thermal resistance based on the entransy dissipation

Nomenclature

t timeh static enthalpyhk kinetic enthalpyu thermodynamic energyeu entransy of thermodynamic energy per unit masseh entransy of enthalpy per unit massek entransy of kinetic enthalpy per unit mass_eq heat entransy fluxm, M masscv specific heat capacity at constant volumecp specific heat at constant pressureT static temperatureTk kinetic temperature

Tvirh ; Tvirc ; Tvirout ; Tvirin ; Tvirvis virtual temperature

DTvirhc ; DTviroi virtual temperature differenceQ ; Qs; Qt heat flowDEc entransy change over time in control volume

DEbou net entransy variation from boundaryEvis entransy caused by energy dissipationEsour entransy caused by internal heat sourceEdiss entransy consumption rate_m mass flow rateq heat fluxv velocity vector

Greek symbolsU internal heat sourceW energy dissipationq densityk heat conductivity coefficientde efficient factors time

184 H. Yu et al. / International Journal of Heat and Mass Transfer 98 (2016) 183–192

to optimize and design the heat exchanger structure. Feng [26]applied the EDEP to improve the global thermal insulation perfor-mance of thermal insulation constructer. He [27,28] also investi-gated the EDEP, and compared with the FSP. The result provedthat the EDEP and FSP are inherently consistent for the convectiveheat transfer. Guo [29,30] proposed the novel concept of synergyangle to indicate the enhancing convective heat transfer of para-bolic flow, and the results indicated that the convective heat trans-fer can be enhanced by reducing the intersection angle betweenthe velocity and the temperature gradient. Tao [31–34] verifiedthe rationality of synergy angle and extended the theory fromparabolic flow to elliptic flow.

However, the method of EDEP and FSP has some constraint. Liu[35] mentioned that that the optimal solution of convective heattransfer may not be found without taking the energy dissipationinto consideration. Minimum thermal resistance principle maynot always be suitable to optimum the process of convection heattransfer because that minimum thermal resistance principle isgeneralized by heat conduction issues. At the same time, the con-sistent of FSP and EDEP is on the basis of statistic data instead of itsphysical mechanism.

According to the above introduction, there is no suitable factoror method to evaluate the enhancement of heat transfer when theenergy dissipation is considered. This paper hopes to derive a fac-tor to evaluate the enhancement of heat transfer based on theentransy theory. Based on the analogy between heat and masstransfer, the balance equation of entransy is established throughtheoretical derivation, the efficiency factor which takes the energydissipation into consideration is recommended by analyzing thephysical mechanism of integral equations. At last, a numericalexample is provided to verify the applicability of the factor asthe comparison function.

2. Entransy dissipation analysis in mass-control system

Based on the definition of entransy E ¼ 12QvhT , where Qvh is the

thermal energy stored in an object, three concepts about theentransy of thermodynamic energy per unit mass eu, the entransyof enthalpy per unit mass eh and the heat entransy flux _eq are givenaccording to entransy theory [7] as follows

eu ¼Z T

0cvTdT ¼ 1

2cvT

2 ð1Þ

eh ¼Z T

0cpT dT ¼ 1

2cpT

2 ð2Þ

_eq ¼ qT ð3ÞThe mass integral of eu and eh are the entransy stored in closed

and open system respectively, and both of them are related to ther-modynamic state [8]. The area integral of _eq is entransy migrationfrom the boundary which happens during heat transfer process.

2.1. The entransy dissipation analysis for isolated system

An isolated system is shown in Fig. 1. There is no inner heatsource and chemical reaction between any species of them. Threedifferent kinds of gas with the different temperature (T1, T2, T3),mass (m1, m2, m3) and constant-volume specific heat (cv1, cv2,cv3), are kept independently and then are mixed together suddenly.The temperature of system under thermal equilibrium is T . Assumethat the temperature sequence from high to low is T3 > T2 > T1,and all the constant-volume specific heat keep constant under dif-ferent temperature.

For the isolate system, the energy conservation equation of thesystem is

cv1m1T1 þ cv2m2T2 þ cv3m3T3 ¼ cv1m1 þ cv2m2 þ cv3m3ð ÞT ð4ÞBy the previous definition of the entransy of thermodynamic

energy per unit mass eu, the variation of entransy of each gas dur-ing the mixing process is

DEu1 ¼ m1 ebeforeu1 � eafteru1

� �¼ 1

2m1cv1ðT2

1 � T2Þ ð5Þ

Fig. 1. The schematic of the isolated system.

Fig. 3. Two step of heat transfer process model.

H. Yu et al. / International Journal of Heat and Mass Transfer 98 (2016) 183–192 185

DEu2 ¼ m2 ebeforeu2 � eafteru2

� �¼ 1

2m2cv2ðT2

2 � T2Þ ð6Þ

DEu3 ¼ m3 ebeforeu3 � eafteru3

� �¼ 1

2m3cv3ðT2

3 � T2Þ ð7Þ

The entransy changing of the overall system can be calculatedby combing Eqs. (5)–(7) together as follows

DEu ¼ 12m1cv1ðT2

1 � T2Þ þ 12m2cv2ðT2

2 � T2Þ þ 12m3cv3ðT2

3 � T2Þð8aÞ

The equation of Eq. (8a) indicates that the system entransydecrease is always positive because that

12� cv1m1cv2m2ðT1�T2Þ2þ cv1m1cv3m3ðT1�T3Þ2þ cv2m2cv3m3ðT2�T3Þ2

cv1m1þcv2m2þ cv3m3>0

ð8bÞEq. (8b) can be extended into systems with more gas species

from 3 to n, the entransy decrease is generalized as

12�Pn

i–jcv icvjmimjðTi � TjÞ2Pni¼1cvimi

> 0 ð9Þ

It can be seen from Eq. (9) that the entransy variation is alwaysdecreasing when the gas with different temperature was mixedtogether under isolated condition. When the temperatures of gases(Ti) have the same value, Eq. (9) can be deduced into zero. It meansthat there is no entransy dissipation if the temperature of all thegases is equal. So the temperature difference may not only leadto heat flux but also is the essential power to generate the entransydissipation.

2.2. Entransy analysis of mass-control system with boundary heat flow

A mass-control system which exchanges heat with the outsideof the system is shown in Fig. 2. The system which is consideredas the internal part is separated from the atmosphere using the vir-tual wall of the green circle which was shown in Fig. 2. The greencircle is considered as the boundary of the system. The other part isconsidered as outside. The system is filled with ideal gas. Assumethat the temperature of internal ideal gas inside the circle andthe external environment are T1 and T2 respectively, T2 > T1. Totalmass of the inside gas is M, and the specific heat cv keeps constant.Total heat from outside to inside through the boundary is Q whenthe system reaches thermal equilibrium state with the outside.

Based on the analysis of isolated system, it can be inferred thatthe entransy dissipation is inevitable because of the uneven tem-perature. The heat Q is independent to the thermal process andalways maintains constant under the same thermal boundary con-dition. Assume that all the thermal state changing is quasi-staticprocess, and Qs is the heat in each step of changing process. Theequivalent model is established in Fig. 3 to analyze the entransychanging of the inside system. The heat transfer process is

Fig. 2. The heat transfer model of the mass-control system.

separated into two steps below, correspondingly the internal gasis divided into two parts denoted as m1 and m2. The first part ofgas withm1 absorbed the heat Qs firstly. During that time, the tem-perature of the other part is constant. And then, the first part of gasand the second part of gas mix together. The process can be dividedinto two steps:

Step 1: Assume that the inside wall between m1 and m2 is adi-abatic, the gas with m1 is heated to T2 firstly by transferring heatQs, in addition, Qs can heat the temperature of the whole systemup to T . The energy conservation gives

Q s ¼ cvm1ðT2 � T1Þ ¼ cvMðT � T1Þ ð10Þ

where M is the total mass of the system, and M ¼ m1 þm2.According the definition of eu, the entransy increase of m1 is

DEuðm1Þ ¼ m1 eafteru1 � ebeforeu1

� �¼ 1

2cvm1ðT2

2 � T21Þ ð11Þ

Step 2: Assume that the outside wall between m1 and environ-ment is adiabatic. Then, the gases of part one and part two sponta-neous mix together. So the entransy dissipation is calculated as

DEuðm1 m2Þ ¼ cvm1m2ðT2 � T1Þ22M

ð12Þ

Combining the Eq. (11) with Eq. (12), the entransy dissipation ofthe system can be calculated as

DEu ¼ DEuðm1Þ þ DEuðm1 m2Þ ð13ÞEq. (13) can be written as

DEu ¼ 12cvMðT � T1ÞðT2 þ T1Þ � 1

2cvm2ðT � T1ÞðT2 � T1Þ ð13aÞ

According Eq. (10), Eq. (13a) can be rewritten as

DEu ¼ 12QsðT2 þ T1Þ � 1

2m2

MQsðT2 � T1Þ > 0 ð13bÞ

where QsðT2 þ T1Þ=2 is the total entransy transported into the sys-tem from the outside, m2QsðT2 � T1Þ=2M is positive and representsthe entransy consumption rate during heat transfer process.

If the time of heat exchange is infinite until steady state, whenthe m2 is close to zero and Qsis to Q , Eq. (13b) can reduced into

DEu ¼ 12QðT2 þ T1Þ ¼ 1

2cvMðT2

2 � T21Þ ð14Þ

Through the analysis mentioned above, it can be inferred thatthe entransy of heat is not merely promotes the entransy of gas,but also engenders the entransy dissipation during heat transferprocess.

Fig. 4. Three-dimensional micro-body model.

186 H. Yu et al. / International Journal of Heat and Mass Transfer 98 (2016) 183–192

3. Deduction of the entransy balance differential equations

Take the micro-control volume as shown in Fig. 4. Some physi-cal quantities such as temperature and heat flux are graded dis-tributed in micro-body. From the analysis in Sections 2.1 and 2.2,the entransy is changing because of unbalanced potentials, andthe entransy dissipation always exists in heat transfer process.Ignore the chemical reaction hear and thermal radiation, thechange of entransy for the system concludes four parts:

DEc ¼ Ebou þ DEvis þ Esour � Ediss ð15Þwhere DEc is the total entransy change in control volume, DEbou isthe net entransy flow through the boundary, Evis is defined as theanalogic entransy source induced by viscous dissipated, Esour is theentransy expression generated by the internal heat source, Ediss isthe entransy dissipation during heat transfer process.

3.1. Derivation process of the heat conduction

The system in this section is considered as no mass transfer, sothe entransy flow through the boundary is just induced by heat

transfer, and Evis ¼ 0. DEcondbou can be expressed on the dimension

of x according _eq

DEcondbouðxÞ ¼ _eqðxÞ � _eqðxþdxÞ

� �dydzds

¼ qxTdydzds� qxþdx T þ @T@x

dx� �

dydzds ð16aÞ

Base on the definition of heat flux, qx ¼ �k @T@x Eq. (16a) can be

written as

DEcondbouðxÞ ¼ �T

@

@x�k

@T@x

� �� k � @T

@x

� �2" #

dxdydzds

¼ � @

@xðqxTÞdxdydzds ð16bÞ

Similarly the dimension of y and z, DEcondbou ¼ DEcond

bouðxÞþDEcond

bouðyÞ þ DEcondbouðzÞ. According that, DEcond

bou can be represented bytensor as

DEcondbou ¼ �r � ðqTÞdxdydzds ð17ÞBased on the analysis of Eq. (13b), the term of Ediss can be

expressed on the x dimension

EdissðxÞ ¼ qxT � qx T þ @T@x

dx� ��

dydzds ¼ k@T@x

� �2

dxdydzds ð18Þ

Similarly with the definition of DEbou, Ediss can be represented bytensor as

Ediss ¼ �q � rTdxdydzds ð19ÞAccording the definition of entransy, changes of DEc over time

in micro-control volume can be described as

DEc ¼ q@ 1

2uT� �@s

dxdydzds ¼ qcvT@T@t

dxdydzds ð20Þ

Assume that internal heat source is U. According the definitionof entransy, the heat source causes the entransy is Esour ¼ UTds.Utilizing the method of tensor expresses, balance equation of theentransy can be calculated by integrating with Eqs. (17), (19) and(20), The relationship of Eq. (15) can be written as

qcvT@T@s

¼ �r � ðqTÞ þ q � rT þUT ð21Þ

Eq. (21) can be reduced to classical conservation equation if U iszero.

3.2. Derivation process of the convection heat transfer

Expect that the heat transfer makes contribution to theentransy changing. At the same time, the enthalpy transport alsohas an impact on the entransy in system. Besides, the heat gener-ated by flow loss enlarges the entransy.

According the definition of eh, the entransy of enthalpy which is

contained in DEflowbou can be expressed as

DEflowbouðxÞ ¼ _mx ehðxÞ � ehðxþdxÞ

� �ds

¼ qvxhxT � qþ @q@x dx

� �vx þ @vx

@x dx� �

hxþdx T þ @T@x dx

� �2

dydzds

ð22aÞwhere _mx is the mass flow of x dimension. Eq. (22a) can be simpli-fied as

DEflowbouðxÞ ¼ �qcpTvx

@T@x

dxdydzds ð22bÞ

where cp is specific heat at constant pressure.Similarly the dimension of y and z, flow component in

DEflowbou ¼ DEflow

bouðxÞ þ DEflowbouðyÞ þ DEflow

bouðzÞ can be represented by tensor as

DEflowbou ¼ �qcpTv � rTdxdydzds ð23ÞAccording to the relationship between enthalpy and thermody-

namic energy, h ¼ uþ p=q, changes of DEc over time in micro-control volume can be written as

DEc ¼ q@ 1

2hT � 12

pq T

� �@s

dxdydzds

¼ qcpT@T@s þ qT

@ 12v � v� �@s

� �dxdydzds ð24Þ

Considering the compressible property of gas, the entransy vari-ation caused by the change of kinetic energy should be taken intoaccount. The kinetic energy is always expressed as kineticenthalpy, entransy of kinetic energy can be derived based on theEq. (2), formulated as

ek ¼Z T

0

12v � v

� �dT ¼

Z T

0hkdT ¼

Z T

0cpTkdT ¼ cpTTk ð25Þ

where hk is kinetic enthalpy per unit mass, Tk is equivalent temper-ature as called kinetic temperature. It is noticeable that Tk has norelevant to the integral temperature. Expanding will reduce thethermodynamic energy in control volume, thus, the entransy

H. Yu et al. / International Journal of Heat and Mass Transfer 98 (2016) 183–192 187

decrease by expanding under the previous temperature of x dimen-sion is described as

DEcompbouðxÞ ¼ _mx ekðxÞ � ekðxþdxÞ

� �ds ¼ _mxT cpTkðxÞ � cpTkðxþdxÞ

� �ds

¼ T q12v2

� �vx � qþ @q

@xdx

� �12

v2 þ @v2

@xdx

� ��

� vx þ @vx

@xdx

� �dydzds ð26aÞ

Eq. (26a) can be simplified as

DEcompbouðxÞ ¼ �qvxT

@ 12v

2� �@x

dxdydzds ð26bÞ

Similarly the dimension of y and z, the compressible issueDEcomp

bou ¼ DEcompbouðxÞ þ DEcomp

bouðyÞ þ DEcompbouðzÞ can be represented by tensor

as

DEcompbou ¼ �qTv � r 1

2v � v

� �dxdydzds ð27Þ

Assume that the dissipated heat due to viscosity is W, whichcauses the entransy is Evis ¼ WTds. Integrating Eqs. (23), (24) and(27) to modify Eq. (21) by choosing the same micro-control vol-ume, the balance equation of entransy (Eq. (15)) for convectionheat transfer is

qcpT@T@s

þ qT@ 1

2v � v� �@s

¼ �qcpTv � rT �r � ðqTÞ � qTv

� r 12v � v

� �þ q � rT þUT þWT ð28aÞ

For incompressible flow with low Mach number, the itemr 1

2v � v� �is negligible and q ¼ constant. Then, Eq. (28a) can be

written as

qcpT@T@s

¼ �r � 12qcpT2v

� ��r � ðqTÞ þ q � rT þUT þWT

ð28bÞ

4. The efficiency factor for evaluating the enhancement ofconvective heat transfer

Enhancing the convective heat transfer and reducing the flowresistance are two important aspects to improve the performanceof heat transfer equipment; however, they always contradict witheach other, which imply that there exist some synergetic relationsbetween them. In order to give a reasonable optimization goal forenhancing heat transfer, the physical mechanism of entransy bal-ance equation should be considered again.

In the condition of incompressible gas with steady-stateð@T=@s ¼ 0Þ and no internal heat sources ðU ¼ 0Þ, the Eq. (28b)can be reduced into

�q � rT ¼ �r � ðqTÞ � r � 12qcpT2v

� �þWT ð29aÞ

Using Gauss transformation method, the control volume inte-gral of Eq. (29a) can be expressed as

�ð�q � rTÞdv ¼ �tðqTÞds�t12qcpvT2

� �ds

þ�ðWTÞdv ð29bÞwhere m is velocity vector, and s is area vector.

According the definition of Eq. (15), �ð�q � rTÞdv equals to

Ediss, �tðqTÞds�t 12qcpvT

2� �

ds equals to Ebou and �ðWTÞdvequals to Evis. Then Eq. (29b) can be reduced into

Ediss ¼ DEbou þ Evis ð30ÞAssume that Qt is the total heat transfer rate between the inter-

nal fluid and the outside. The physical meaning of each item in Eq.(29b) will be analyzed.

The item �tðqTÞds, represents the net heat entransy flowtransported with the outside. q This item should range in appropri-ate values. For instance, as for boundary temperature T holds inconstant (first boundary condition), heat transfer is enhancedascribe to higher heat flux q, therefor this item should maintain ahigher value. Similarly, heat transfer is enhanced when the itemkeeps lower level in the condition of constant heat flux (secondboundary condition). As for third boundary condition, the itemchanges in a reasonable region according to the limitation on thetemperature or heat flux.

In order to simple the derivation, the new factor of virtual tem-perature is used in the following derivation

DTvirhc ¼ Tvirh � Tvirc ¼RR

inðqTÞdsQt

�RR

outðqTÞdsQt

ð31Þ

where Tv irh is the virtual temperature of heat source, and Tvirc is the

virtual temperature of cold source. DTvirhc means the equivalent heattransfer temperature difference between heat source and coldsource. The virtual temperature includes the total effect of temper-ature in three dimensions, it is traditional temperature difference inone-dimensional heat conduction. This item is correlation with theboundary conditions. Obviously, this value will remains constantunder the first boundary conditions.

The item t 12qcpvT

2� �

ds, which represents the net enthalpy

entransy flow around boundary. Based on the theory of convectiveheat field synergy [29,30], the non-dimension integral of �v � rT ispositive correlation with Nu, or, put another way, FSP is the differ-ent expression forms of Nu. It can be written as follows:

�v � rTd�y / Nu ð32Þ

The item t 12qcpvT

2� �

ds can be transformed into

� qcpTðv � rTÞ �dxdydz by using the first Green’s theorem.

According to mean value theorem of integrals, there must exist atemperature ~T to set the Eq. (33a) to hold

� qcpTðv � rTÞ �dxdydz ¼ ~T� qcpðv � rTÞ �

dxdydz ð33aÞMaking Eq. (33a) non-dimension form as

cZZ

RePrZ

�v � rTd�y� �

dxdz ð33bÞ

where c is the constant number which does not affect the integralprocess. Combining Eqs. (32) and (33b), it can be inferred that the

itemt 12qcpvT

2� �

ds is positive correlation with Nu. Thus, the higher

this item, the better convection heat transfer enhancement process.Introducing the following parameters

DTviroi ¼ Tvirout � Tvirin ¼RR

out12qcpvT

2� �

ds

Qt�RR

in12qcpvT

2� �

ds

Qtð34Þ

where Tv irin is the virtual inlet temperature of fluid, Tvirout is the virtual

outlet temperature of fluid, DTv iroi represents the entransy variationof fluid transferring per unit heat. Considering heat transferenhancement problem from the perspective of fluid, the issue canbe equivalent to heat or cool the fluid. The higher absolute valuethis item represents the increasing for the degree of heat transferpotential capacity or decreasing per unit heat transferred. The

188 H. Yu et al. / International Journal of Heat and Mass Transfer 98 (2016) 183–192

higher value of DTviroi also means the better heat transfer enhance-ment process under the same flow and heat boundary conditions.

The item Evis ¼ �ðWTÞdv , is the analogical entransy sourceinduced by dissipated heat from fluid mechanical energy. It bringsa positive effect by maintaining it a low proportion in entransy dis-

sipation. Introducing the parameter Tvirvis is

Tvirvis ¼�ðWTÞdv

Qtð35Þ

The increase of heat transfer Qt is always at the expense of

boost of flow losses W. This virtual temperature Tvirvis representsthe ability of weaken heat transfer enhancement which shouldkeep relatively low level.

The item Ediss ¼ �ð�q � rTÞdv describes the loss of heat

transfer ability. As mentioned above, DTvirhc is constant under thefirst boundary condition, heat transfer will be enhanced by higher

value of DTviroi and lower Tvirvis . Thus, with the same Qt value, heattransfer performance will be strengthened for the small value ofEdiss. In other words, the less entransy consumption per unit effec-tive energy transferred, the more optimal heat transfer process willget.

A diagram of heat transfer model is shown in Fig. 5. Based onthe above-mentioned analysis of Eqs. (31), (34) and (35),Eq. (29b) can be converted into

Fig. 6. Schematic describe of the numerical model with/without step (a) geometries(b) grid systems.

Table 1Independent solution of mesh.

Total elements Maximum yplus Nusselt number Deviation of Nu (%)

Smooth tube12,000 5.43 11.3226,600 2.97 11.94 5.48

Ediss

Qt¼ �ð�q � rTÞdv

Qt¼ DTvirhc � DTviroi þ DTvirvis > 0 ð36Þ

Under the first boundary condition, the higher value of DTviroi

and lower Tvirvis is, the less consumptions Ediss=Qt will get. It should

stand out the impact of DTviroi because that unlike the FSP, whichdescribes the heat transfer intensity but not capacity, choosing

the item DTviroi to replace FSP (or Nu) is more appropriate to evalu-ate heat transfer enhancement problems which kernel is potential

changes but not superficial. Besides, the three items, DTvirhc DTviroi and

Tvirvis , are interrelated and affected with each other. For instance, the

item DTvirhc keeps constant under constant wall temperature bound-

ary condition, raising DTviroi will boost Tvirvis , which lead to uncer-tainty changes of Ediss=Qt for their reciprocal relationships. Thus,by comprehensive logical reasoning, the lower Ediss=Qt , which isonly the necessary condition, cannot indicate convective heattransfer enhancement. Therefore, the efficiency factor which is rep-resented the non-dimension ratio of benefits and consumptions isrecommend

45,000 0.93 12.38 3.6877,650 0.37 12.41 0.242

Step tube42,400 2.72 17.2592,780 1.43 18.16 5.28165,600 0.87 18.44 1.54229,430 0.21 18.47 0.163

de ¼ DTviroiEdissQt

ð37Þ

Apparently, the higher this parameter is, the better convectiveheat transfer process will get.

Fig. 5. Abstract heat

5. Numerical validation

In the following presentation a numerical example was pro-vided to validate de, in addition, the comparisons of the differentparameters such as FSP and EDEP [25,26] were also discusseddetails.

In the following discussion, the flow and heat transfer behaviorsof air in the smooth tube and the step tube were investigated. In

transfer model.

H. Yu et al. / International Journal of Heat and Mass Transfer 98 (2016) 183–192 189

the discussion, the flow is fully developed, the flow and heat trans-fer are in steady state and fluid thermo-physical properties areconstants, besides, the wall temperature for both of the models is

0 10000 20000 300000

20

40

60

80

100

120

140

160

180

Nu

Re

Fig. 7. Variation of Nu v

0 10000 20000 30000

64

128

256

512

1024

2048

E diss

/W·K

Re

SmoothStep

(a)

0 10000 20000 30000

0

200

400

600

800

1000

1200

1400

E vis/W

·K

Re

Smooth-Evis

Step-Evis

(b)

Smooth-ΔEbou

Step-ΔEbou

Fig. 8. Variation of (a) Ediss vs. and (b

constant. The dimension of step tube is e=d ¼ 0:05; p=e ¼ 50;p0=p ¼ 0:25 [35,36]. The physical mean of e, d, and p were shownin Fig. 6. The grid system generated by ICEM is presented in

40000 50000 60000

Smooth- Nu Step-Nu

89.45

89.50

89.55

89.60

89.65

89.70

89.75

Smooth-θ Step-θ

θ

s. and h vs. with Re.

40000 50000 60000

40000 50000 60000

0

200

400

600

800

1000

ΔEbo

u/W·K

) Ev is vs. and DEbou vs. with Re.

0 10000 20000 30000 40000 50000 60000

3

4

5

6

7

8

9

10

11

12

δ e

Re

Smooth Step

(a)

0 10000 20000 30000 40000 50000 60000270

275

280

285

290

295

ΔTvi

roi

/K

Re

Smooth-ΔTviroi

Step-ΔTviroi

(b)

0

10

20

30

40

50

60

70

Smooth-Tvirvis

Step-Tvirvis

Tvir

vis/K

Fig. 9. Variation of (a) de vs. and (b) DTv iroi vs. and Tv irv is vs. with Re.

0 10000 20000 30000 40000 50000 600000

5

10

15

20

25

30

Ther

mal

Res

ista

nce

(K/W

)

Re

SmoothStep

Fig. 10. Variation of thermal resistance.

3 4 5 12 20 27 35 40 45 50 55 600

5

10

15

20

25

30

ΔNu

Re×10-3

Smooth Step

Fig. 11. The increment of Nu with Re.

190 H. Yu et al. / International Journal of Heat and Mass Transfer 98 (2016) 183–192

Fig. 6(b), the smooth tube with total 45,000 hexahedral meshesand the step tube 165,600. The independent of mesh solutions isgiven as shown in Table 1. The criterion of grid independent solu-tion is mainly considered two aspects, one is that the height of firstlayer should be applicative to low Reynolds number model

(yplus < 1), the other is small difference of Nusselt number(deviation < 1%).

Utilizing k—e model and SIMPLEC algorithms were adopted todeal with the linkage between velocity and pressure. Numerical

H. Yu et al. / International Journal of Heat and Mass Transfer 98 (2016) 183–192 191

solutions are conducted by using the software FLUENT. After theconverged solutions are obtained, the domain averaged parametersabovementioned is determined by a UDF incorporated into FLUENT,and the characteristics of flow resistance and heat transfer for bothof tubes are all verified by correlative empirical formula [36–38].

From the numerical results of temperature and velocity fields,

the variations of Nusselt number Nu ¼ QdkAðTw�Tf Þ

� �and FSP

(h ¼ cos�1P

v �rTPjv jjrTj

� �) with Reynolds number are shown in Fig. 7.

It can be seen that, Nusselt number for the surface of step tube ishigher than that of smooth tube; its synergy angle is lower thansmooth tube.

The entransy of Ediss, Evis and DEbou in the process with Reynoldsnumber are shown in Fig. 8(a) and (b), respectively. It can beobserved from Fig. 8(a) that the entransy dissipation of step tubeis lower than smooth tube when the Re is less than 12,000; the ten-dency is going into reverse when the Re is higher than 12,000. As isshown in Fig. 8(b), DEbou for both of two tube increases linearly, andEvis grows exponentially with Re. According to Eq. (30), it can beinferred that the impact of Evis on Ediss can be neglected when theRe is lower than 12,000, after which is gradually becoming a dom-inant item with high Re.

The efficiency factor de as well as the change of DTviroi and Tvirviswith Reynolds number is shown in Fig. 9(a) and (b). DTvirhc is con-

stant for the given wall temperature case, DTviroi of step tube is

always higher than the smooth tube, also, Tvirvis is getting higherwith an exponentially growing. According to Eq. (37), it can beobserved that de of step tube is higher than smooth tube underthe condition of low Reynolds number when the proportion of

Tvirvis is very small. However the Reynolds number is larger than45,000, de of step tube is lower than smooth, the major influence

factor DTviroi which is almost keeping constant is replaced by Tvirvis .It can be indicated that the effect of the convective heat transferenhancement of step tube becomes worse than smooth.

According to EDEP, the thermal resistance Ediss=Q2t , which prin-

ciple is keeping this parameter to minimum, with Reynolds num-ber is shown in Fig. 10. Obviously, the step tube is always betterthan smooth, which indicated that the thermal resistance cannotreflect the intersection point when Re is very high (45,000).

The increment of Nu in the same interval of Re are presented inFig. 11. It can be observed that the growth rate of Nu for the steptube is faster than smooth before Re surpasses 45,000, the effectof heat transfer enhancement is approximate same around theregion, after that, this enhancement is reversed.

It should be noted that the synergy angle line is always match-ing with Re, and there is no intersection point for these two lines,but de has the intersection point. It inferred that the factor dereflects efficiency of convection heat transfer when the energy dis-sipation is taken into account.

6. Conclusion

The new standards for the enhancement of heat transfer wereintroduced based on the balance difference equation of entransyis introduced by detailed theoretical inference which is consideredthe physical mechanism of heat transfer phenomena. According toanalyzing each item in the integrated entransy differential balanceequation of convective heat transfer, it is recommend that choos-ing the efficiency factor de for evaluating convective heat transferenhancement in which the impact of energy dissipation due tothe shear stress is taken into consideration rather than the fieldsynergy angle (FSP) and thermal resistance (EDEP). The higher thisde is, the better effect of the heat transfer enhancement will be. Thenumerical results indicate that the principle of the maximum de is

verified and its assessment ability is more effective than FSP, inaddition, the entransy caused by viscous dissipation cannot beneglected in high Re region.

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