Constructal Optimization of Microchannel Heat Sinks With Noncircular Cross Sections

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Constructal Optimization of Microchannel Heat SinksWith Noncircular Cross SectionsMohammad Reza Salimpour a , Morteza Sharifhasan a & Ebrahim Shirani aa Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, IranAccepted author version posted online: 12 Nov 2012.Published online: 07 Feb 2013.

To cite this article: Mohammad Reza Salimpour , Morteza Sharifhasan & Ebrahim Shirani (2013): Constructal Optimization ofMicrochannel Heat Sinks With Noncircular Cross Sections, Heat Transfer Engineering, 34:10, 863-874

To link to this article: http://dx.doi.org/10.1080/01457632.2012.746552

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Heat Transfer Engineering, 34(10):863–874, 2013Copyright C©© Taylor and Francis Group, LLCISSN: 0145-7632 print / 1521-0537 onlineDOI: 10.1080/01457632.2012.746552

Constructal Optimizationof Microchannel Heat SinksWith Noncircular Cross Sections

MOHAMMAD REZA SALIMPOUR, MORTEZA SHARIFHASAN,and EBRAHIM SHIRANIDepartment of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran

This article reports the numerical geometric optimization of three-dimensional microchannel heat sinks with rectangular,elliptic, and isosceles triangular cross sections. The cross-sectional areas of the mentioned microchannels can changeaccording to the degrees of freedom, that is, the aspect ratio and the solid volume fraction. Actually, the purpose of geometricoptimization is to determine the optimal values of these parameters in such a way that the peak temperature of the wallis minimized. The effects of solid volume fraction and pressure drop upon the aspect ratio, hydraulic diameter, and peaktemperature of the microchannels are investigated. Moreover, these microchannel heat sinks are compared with each otherat their optimal conditions. Considering the constraints and geometric parameters for the optimization of the present study,it is revealed that microchannel heat sinks with rectangular and elliptic cross sections have similar performances, whilemicrochannels with isosceles triangular cross sections show weaker performances. The optimal shapes of all three kinds ofchannels are achieved numerically and compared with the approximate results obtained from scale analysis, for which goodagreements are observed.

INTRODUCTION

The fast development of computing technology has led tothe miniaturization and density integration of electronic de-vices. However, this miniaturization involves more heat gen-eration per unit area. Consequently, the working temperatureof the electronic components may exceed the desired temper-ature level. Since the performance of equipment has a directrelationship with its temperature, it is important to keep it at anacceptable temperature level. Many ideas have been suggestedto improve the cooling technology of electronic componentswith high heat flux generation and compact size. Among theseideas, microchannel heat sinks drew more attention because ofhigh heat transfer coefficients and low coolant requirements.

Microchannel heat sink with a parallel flow was first intro-duced by Tuckerman and Pease [1, 2]. They used direct watercirculation in fabricated microchannels and observed that thesemicrochannels can repulse heat flux up to 790 W/cm2. Weisberg

Address correspondence to Dr. Mohammad Reza Salimpour, Departmentof Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran. E-mail: [email protected]

et al. [3] solved two-dimensional fluid flow and heat transfer withthe assumption of fully developed flow both hydrodynamicallyand thermally. Fedorov and Viskanta [4] developed a three-dimensional model in the microchannel heat sinks using incom-pressible laminar Navier–Stokes equations. They used channelssimilar to those of Kawano et al. [5] experimental work. Ng andPoh [6] used CFD to analyze the fluid flow in a double-layermicrochannel. Wu and Cheng [7] conducted an experimentalstudy on the convective heat transfer in silicon microchannelswith different surface conditions. They showed that the Nusseltnumber and the apparent friction factor depend greatly on thegeometric parameters. Gamrat et al. [8] presented both two- andthree-dimensional numerical analysis of convective heat trans-fer in microchannels. In this work, the thermal entrance effectand the coupling heat transfer modes, conduction and convec-tion, were studied. The results showed that the microchannelswith hydraulic diameters less than 100 μm did not reveal anysignificant scale effect on heat transfer in the microchannel heatsinks.

Assuming a fully developed flow, Lee and Garimella [9]proposed a general relationship to predict the Nusselt numberalong the axis of a microchannel, which is useful for designand optimization of microchannel heat sinks. Presenting a

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864 M. R. SALIMPOUR ET AL.

three-dimensional numerical model, Kou et al. [10] studiedthe effect of height and width variations on heat transferparameters of a microchannel heat sink. Naphon and Khonseur[11] investigated the heat transfer characteristics and pressuredrop of microchannel heat sinks experimentally under constantheat flux conditions. The experiments were accomplished forReynolds numbers and heat fluxes ranging from 100 to 200and 1.80 to 5.40 kW/m2, respectively. Chen et al. [12] modeledthe heat transfer and fluid flow in noncircular microchannelheat sinks and showed that the Nusselt number is high at theinlet region but quickly approaches its constant fully developedvalue. They also made a comparison among thermal efficienciesof microchannels with different cross sections. Ng and Tan[13] performed three-dimensional numerical simulations ofdeveloping pressure-driven liquid flow in a microchannelwith an electric double layer (EDL) effect. By combiningthe three-dimensional Poisson–Boltzmann equation and theNavier–Stokes equation, they calculated both the flow entrancelength and Nusselt number in the entry region. Tan and Ng [14]used two EDL models together with Navier-Stokes equationsto compute three-dimensional (3-D) developing microchannelflow. In their work, the Nernst–Planck model (NPM) wasextended for temperature prediction.

Some research studies on microchannel heat sinks and theiroptimization are addressed in the preceding discussion. How-ever, for more study in this regard, one can refer to references[15–21]. For optimization of microchannel heat sinks, differentalgorithms and methods have been used. One of these methodsis the constructal theory.

Bejan [22] and Bejan and Lorente [23] presented a com-prehensive review on the constructal design of heat transfersystems. They focused on the structure and shape generationin freely morphing convective systems. Ghaedamini et al. [24]showed that svelteness has a remarkable effect on the bifurcationangles role in pressure drop and flow uniformity of tree-shapedmicrochannels. Muzychka [25] obtained the optimal passagesize to length ratio for channels with different geometries usingan approximate method. He showed that the optimal duct di-mension is independent of the array structure. This problem wasfurther investigated numerically by Salimpour et al. [26]. Theydetermined the optimal scale of channels with cross-sectionalshapes of square, circular, and isosceles right triangular. Thenthey applied the optimal scales to arrays of passages to achievethe maximum heat transfer rate per unit volume. Moreover, theyselected the optimal geometry among the channels with differ-ent cross-sectional shapes. Bello-Ochende et al. [27] obtainedthe best geometry for a microchannel heat sink numerically byminimizing the peak temperature for a specified pressure dropthrough the elemental volume.

In the present work, invoking the constructal theory and themethod presented by Bello-Ochende et al. [27], the optimal ge-ometries of microchannel heat sinks with rectangular, elliptic,and isosceles triangular cross sections are determined numer-ically such that the global thermal resistances are minimized.In this way, the wall peak temperature is minimized for a fixed

pressure difference through the elemental volume. Then thesemicrochannel heat sinks are compared with each other at theiroptimal conditions. Finally, the results are compared with theapproximate relations available in the literature.

DESCRIPTION OF THE PROBLEM

The problem under consideration in this article is the forcedconvection in microchannel heat sinks with rectangular, elliptic,and isosceles triangular cross sections, as shown in Figure 1.The heat of heat-generating devices like electronic chips arrivesat the bottom of the heat sink and transfers from there by a highlyconductive material like silicon. The heat is then removed by afluid flowing through a number of microchannels. Because of thegeometric symmetry, an elemental volume (unit cell) consistingof a microchannel and its surrounding solid is chosen as thecomputational domain (Figures 1b–d).

Considering the dimensionless global conductance intro-duced in reference [27] as C = q′′L

kf(Tmax−Tin) , we try to maximizeit where q ′′ is the heat flux at the base of the microchannel heatsink, k f is the thermal conductivity of the fluid, and L is thelength of the computational domain of the unit volume.

OPTIMIZATION CONSTRAINTS AND PARAMETERS

As mentioned before, Bello-Ochende et al. [27] optimized arectangular microchannel heat sink using the constructal theory.They determined the optimal geometric shape of the microchan-nel in a way such that the peak temperature was minimized. Forthis purpose, five degrees of freedom, namely, L, t1/t2, t2/t3,H/G, and ϕ, were considered, while three of them, namely, L,t1/t2, and t2/t3, were left unchanged.

In this study, the optimal geometry of microchannel heatsinks with rectangular, elliptic, and isosceles triangular crosssections is determined invoking the constructal theory and thealready-mentioned method of reference [27] in such a way thatoverall global thermal conductance is maximized.

The elemental volume constraint for a given computationalcell of microchannel heat sinks for all three cross sections isdefined as [27]:

GHL = V = constant (1)

To calculate the volume of the solid substrate, Eqs. (2a)–(2c)are used for rectangular, elliptic, and isosceles triangular crosssections, respectively.

2HLt1 + (G − 2t1)Lt2 + (G − 2t1)Lt3 = Vs = constant (2a)

2HLt1 + (G − 2t1)Lt2 + (G − 2t1)Lt3 + (1 − π/4)(G − 2t1)

(H − t2 − t3)L = Vs = constant (2b)

heat transfer engineering vol. 34 no. 10 2013

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M. R. SALIMPOUR ET AL. 865

Figure 1 Structure of a microchannel heat sink and the 3-D numerical domain.

2HLt1 + (G − 2t1) Lt2+ (G − 2t1) Lt3 + (H − t2 − t3)

× (G − 2t1) L/2 = Vs = constant (2c)

For a fixed length of microchannel, we have

GH = V/L = A = constant (3)

and

Vf = V − Vs = constant (4)

Referring to reference [27] and Figure 1a, the total numberof channels in a microchannel heat sink with a fixed total width,


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W, is obtained from Eq. (5):

num = W

B + 2t1(5)

ANALYTICAL SOLUTION

The following analysis is an application of the intersectionof asymptotes method [28]. In this method, the relationshipbetween the global thermal conductance, C, and the distancebetween the walls, D, is achieved analytically in the two ex-tremes, D → 0 and D → ∞. Then these asymptotes are used toargue that C can be maximized by selecting appropriate D. Ac-tually, the optimal D resides approximately on the interceptionpoint of the asymptotes.

In this analysis, the following assumptions are adopted: uni-form flow distribution, Prandtl number greater than 0.5, negli-gible axial conduction, and ks � kf. The aforementioned twoextremes are addressed in the following.

Small-Sized Channels

In this extreme, the heat sink consists of channels with smallcross sections, and thus the length of the microchannels ismainly covered by fully developed flow regime, which is mainlyHagen–Poiseuille. As shown in reference [27], in this regimewe have

qsmall = (Tw,L − Tin)ρCpA2c�PDh

μPcLPODh

(6)

where Ac is the cross-sectional area of an elemental channel, Cp

is the specific heat of the fluid, and PODhis Poiseuille number

based on the hydraulic diameter of the channel defined as Dh =4Ac/Pc. Poiseuille number is a constant value (fReDh = 2PODh

)varying with the cross-section shape. Shah and London [29]reported the Poiseuille numbers of about 40 different shapes,from which they observed that for most of the geometric shapes,6 < PODh

< 12. The resulting expression for dimensionless globalthermal conductance is [27]

Csmall = 0.25D2

hBe

LV1/3PO(7)

In Eq. (7), Be is the Bejan number based on the unit volume(�PV2/3/μα). From Eq. (7), it can be concluded that

Csmall ∼ D2h (8)

The first conclusion from Eq. (8) is that the dimensionlessglobal thermal conductance, C, decreases as D → 0. We areinterested in larger C and because of this we turn our attentionto how C varies with D in the opposite limit.

Large-Sized Channels

In the opposite extreme, that is, for large channels, the flow isessentially a boundary-layer flow. As shown by Bello-Ochendeet al. [27], in this situation we have

qlarge∼= 0.5192kf(Tw,L − Tin)

(�PAcP2

cLPr

ρν2

)1/3

(9)

In dimensionless form, the global thermal conductance be-comes [27]

Clarge = 1.31V1/9L1/3

D2/3h

Be1/3 (10)

Therefore,

Clarge ∼ D−2/3h (11)

The conclusion drawn from Eq. (11) is that in the large chan-nels, C decreases as D increases.

Intersection of Asymptotes

According to Eqs. (8) and (11), C decreases in both extremes;therefore, it must have a maximum at an intermediate value ofD. This value can be found approximately by intersecting theasymptotes, as shown in Figure 2. The optimal channel shapecan be approximated as the value of Dh corresponding to theintersection point [27]:

Dh,opt = 1.86(L3V)1/6Be−1/4P3/8ODh

(12)

Introducing Eq. (12) to Eq. (10) or Eq. (7), the maximumdimensionless global thermal conductance can be obtained as

Cmax,theoretical ≈ 0.864Be1/2

P1/4O

(13)

Figure 2 The dimensionless global thermal conductance in the limit of largeand small channels (asymptote method).


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The Poiseuille number for rectangular cross section is

PODh = 12

(1 + ε)2[1 − 192ε

π5 tanh(

π2ε

)] (14)

where ε = B/Hc is the channel aspect ratio. Equation (14) rep-resents a single term approximation of the Poiseuille numberwith a maximum error of 0.5%, which occurs at the limit ofε = 1 [25].

The Poiseuille number for elliptic cross section is

PODh = (1 + ε)2

[π

E(ε′)

]2

(15)

where E(ε′) is the complete elliptic integral of the second kindof complementary modulus ε′ = √

1 − ε2, while ∈= B/Hc isthe duct aspect ratio [25].

The Poiseuille number for isosceles triangular cross sectionis

PODh = 6(ε3 + 0.2595ε2 − 0.2046ε + 0.0552)/ε3

(1 ≤ ε ≤ ∞) (16)

Because of the nature and the limits of approximate methods,Eqs. (12) and (13) should be compared with their numericalcounterparts.

NUMERICAL SIMULATION

Physical Model and Computational Domain

Figure 1 shows the physical model and the computational do-main for a microchannel heat sink. Heat is supplied to a highlyconductive silicon substrate with known thermal conductivityfrom a heating area located at the bottom of the heat sink. Theheat is then removed by a fluid flowing through a number of mi-crochannels. Because of the geometric symmetry, an elementalvolume (unit cell) consisting of a microchannel and its surround-ing solid is chosen as the computational domain, as in Figures1b–d.

Governing Equations and Boundary Conditions

The heat transfer mechanism in a microchannel heat sinkmainly consists of conduction in the solid region and forcedconvection in the cooling fluid. Coolant fluid with entrance tem-perature of Tin = 20◦C is driven into the microchannel by a fixedpressure drop of�P = P(z = 0) − P(z = L). The purpose ofthis analysis is to determine the best possible geometric shapeto reach the maximal global thermal conductance or minimalglobal thermal resistance. To model the flow and heat transferin the elemental volume, the following assumptions are takeninto consideration.

1. Heat transfers due to radiation and natural convection arenegligible.

2. As the hydraulic diameter of the analyzed microchannel isgreater than 10 μm, the continuum regime is present forwater; therefore, the Navier–Stokes and Fourier equations arestill held and can be used for transport processes description[30].

3. Steady-state conditions are present for both flow and heattransfer.

4. The properties of fluid are constant.5. Flow is incompressible.6. The number of elemental microchannels is large.

According to these assumptions, the continuity and momen-tum equations for the coolant fluid are written as:

∇ · ( �U ) = 0 (17)

ρ( �U · ∇ �U ) = −∇ P + μ∇2 �U (18)

The hydrodynamic boundary conditions are no-slip conditionat channel walls, that is, u = v = w = 0, where u, v and w arecomponents of the velocity vector �U, and given pressures atinlet, P = Pin, and outlet, P = Pout.

The energy equation for the liquid is

ρCp( �U · ∇T) = kf∇2T (19)

while the heat conduction in the silicon substrate can be ex-pressed as

∇2T = 0 (20)

Assuming uniform heat flux at the bottom of the heat sink,the following thermal boundary condition holds for Eq. (20):

ks∂T

∂y= −q′′ (21)

The other walls of the heat sink and its plane of symmetryare modeled as adiabatic. From the continuity of temperatureand heat flux at the interface of the solid and fluid, we can write

−ks∂T

∂n

∣∣∣∣�

= −kf∂T

∂n

∣∣∣∣�

(22)

Numerical Method and Validation

In the present investigation, a finite-volume method [31] isused to resolve continuity, momentum, and energy equations.The Simple algorithm is used to resolve the velocity–pressurecoupling, while a second-order upwind scheme is invoked todiscretize the equations.

The convergence is obtained when the residuals of mass,momentum, and energy equations are less than 1e-5, 1e-5, and1e-9, respectively. To check the grid independency, the numberof grids is doubled until a deviation of less than 1% is observed inthe maximum velocity and temperature profiles resultant fromtwo grids. To validate our numerical work, the results of thepresent study are compared with the numerical results of Qu


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and Mudawar [32]. In their case, the heating component ofFigure 1 was placed on top of a microchannel heat sink withrectangular cross section. Figure 3 compares inlet and outletthermal resistances of the present work with those of reference[32]. In this figure, parameters Rt,out, Rt,in, and ReDh are definedas

Rt,in = Tw,in − Tin

q′′ (23)

Rt,out = Tw,out − Tin

q(24)

Figure 3 Comparison of the numerical results of the present study with theresults of [32]: (a) inlet thermal resistance and (b) outlet thermal resistance.

ReDh = UDh

ν(25)

where Dh is the hydraulic diameter of the channel, defined as

Dh = 4Ac

Pc= 4BHc

2(B + Hc)(26)

As is evident from Figure 3, the present numerical workhas good agreement with the results of reference [32] with adeviation of less than 1.5%. Note also that Qu and Mudawar[32] had validated their results with the experimental data ofKawano et al. [5].

RESULTS AND DISCUSSION

The solid substrate of the present heat sink is made of sili-con with thermal conductivity of ks = 148 W/m − K. Thermo-physical properties of water used in this study are evaluatedat 20◦C.

The uniform heat flux of 100 W/cm2 arrives at the bottomsurface of the microchannel heat sink. In the Analytical So-lution section, an approximate method was used to determinethe optimal channel shape that minimizes the global thermalresistance. In this section, following a series of numerical op-timizations, the effects of pressure drop, solid material, andexternal aspect ratio (the internal aspect ratio is consideredfixed) on the optimal geometry of microchannel heat sinks withrectangular, elliptic, and isosceles triangular cross sections areinvestigated.

As mentioned before, the microchannel heat sink has five de-grees of freedom, namely, L, t1/t2, t2/t3, H/G, and ϕ. In this study,two degrees of freedom (H/G and ϕ) are allowed to change whilethe other three degrees are left unchanged. Setting t1/t2 = 0.08and t2/t3 = 1, we fix the internal structure of the microchannel.The total volume and the length of the unit microchannel areset to 0.9 mm3 and 10 mm, respectively. For the first stage ofoptimization, the pressure drop through the microchannel is setto 50 kPa.

The variations of Tmax with the external shape, H/G, arerepresented in Figure 4. This figure confirms that for each crosssection there is an optimal value for H/G in which maximumtemperature is minimized.

The effect of hydraulic diameter on the maximum temper-ature of different microchannels is portrayed in Figure 5. Thisfigure shows that for each microchannel there is an optimal Dh

such that the overall global thermal conductance is maximized.This matter is in accordance with what was expected from theanalytical solution, Eq. (12). From Figure 5, it is observed thatthe microchannel diameters less than Dh,opt are detrimental forthermal performance, because when Dh < Dh,opt the maximumtemperature increases more rigorously than when Dh > Dh,opt.This is because for the small values of Dh, the effective distancebetween walls squeezes the flow and it becomes overworked,and hence the peak temperature increases [27]. Figure 5 also


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H/G

Tm

ax,o C

4 6 8 10 12 1436

38

40

42

44

46

Isosceles triangularEllipticRectangular

Figure 4 The effect of aspect ratio on the maximum temperature, for mi-crochannel heat sinks with different cross sections (�P = 50 kPa, t1/t2 = 0.08,t2/t3 = 1, and ϕ = 0.8).

shows that this effect is more drastic in isosceles triangular mi-crochannels.

As illustrated in Figure 6, there is an optimal allocationof solid fraction for rectangular and elliptic microchannel

Dh, μm

Tm

ax,o C

40 60 80 100 120 14036

38

40

42

44

46

Isosceles triangularEllipticRectangular

Figure 5 The effect of hydraulic diameter on the maximum temperature, for amicrochannel heat sink with different cross sections (�P = 50 kPa, t1/t2 = 0.08,t2/t3 = 1, and ϕ = 0.8).

Dh, μm

Tm

ax,o C

60 80 100 120 140 160 180 20030

32

34

36

38

40

42

44φ=0.8φ=0.6φ=0.5φ=0.4φ=0.3

(a)

Dh, μm

Tm

ax,o C

60 80 100 120 140 160 180 20030

32

34

36

38

40

42

44φ=0.8φ=0.6φ=0.5φ=0.4φ=0.3

(b)

Dh, μm

Tm

ax,o C

40 60 80 100 12030

32

34

36

38

40

42

44

46φ=0.8φ=0.6

(c)

Figure 6 The effects of hydraulic diameter and solid volume fraction on themaximum temperature: (a) rectangular microchannel, (b) elliptic microchannel,and (c) isosceles triangular microchannel (�P = 50 kPa, t1/t2 = 0.08, andt2/t3 = 1).


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heat sinks that minimizes the maximum temperature, that is,0.4 < ϕopt < 0.5. It is noted that according to the dimen-sions of the microchannels of the present study, the valueof ϕ is more than 0.6 for isosceles triangular cross sections.From Figure 6, it is evident that for all of the microchan-nel heat sinks when ϕ > 0.6, there is a jump in the peaktemperature. The reason is that as ϕ increases, Dh decreases,which causes the fluid to get overworked. Moreover, this is-sue is stronger in triangular microchannels, because the effec-tive distance between the walls decreases more rapidly with Dh

reduction.Figure 7 contains the graphs of the optimal external aspect

ratio, (H/G)opt, variations versus the pressure drop. In rectangu-lar and elliptic microchannels for a fixed solid fraction, (H/G)optincreases with pressure drop in the low values of �P, but when�P ≥ 50 kPa, this increase gets slight, and is almost invariantat ϕ = 0.8. In triangular microchannels, when ϕ = 0.8, the trendof (H/G)opt variations is similar to those of rectangular andelliptic microchannels, but when ϕ = 0.6, (H/G)opt increaseswith pressure drop in the whole pressure drop range. Also, itis clear from this figure that for all three kinds of microchannelheat sinks, the optimal external aspect ratio increases whenthe solid volume fraction decreases. Also, from Figure 7 it isseen that in two cases, under the same pressure drop, differentsolid fractions have same optimal aspect ratios. This is notstrange because although (H/G)opt is the same for two differentvolume fractions, the hydraulic diameters of these cases aredifferent.

For better comparison of thermal performance in thesethree kinds of microchannels, the variations of the mini-mized maximum temperature difference, �Tmin, with pres-sure drop is shown in Figure 8 for ϕ = 0.6 and ϕ = 0.8.As seen from this figure, �Tmin decreases with �P incre-ment and ϕ decrement. Regarding the parameters and con-straints of the present study, the rectangular and elliptic mi-crochannel heat sinks have similar performance, while the tri-angular microchannels have lower global thermal conductance.However, the thermal performance of isosceles triangular mi-crochannel heat sinks improves at ϕ = 0.6 and gets close tothat of rectangular and elliptic microchannels, as shown inFigure 8.

The advantage of triangular microchannels compared to therectangular and elliptic ones is that in the optimal conditions,less coolant fluid is required to reach a predetermined �Tmin.That is, if the mass flow rate of coolant fluid is one of theconstraints (the pressure drop constraint is removed), then anisosceles triangular microchannel is the best choice when ϕ =0.6, as observed from Figure 9.

Figure 10 compares the numerical and approximate analyt-ical solutions of Cmax for different microchannels at ϕ = 0.6.As is seen from the figure, the trend of Cmax variations withBejan number is similar for both solutions but the values aredifferent. This is because the intersection of asymptotes andscale analysis methods are strong tools that allow researchersto evaluate the trends and optimal configurations [28]. More-

ΔP, kPa

(H/G

) opt

0 20 40 60 804

8

12

16

20φ=0.8φ=0.6φ=0.5φ=0.4φ=0.3

(a)

ΔP, kPa

(H/G

) opt

0 20 40 60 804

8

12

16

20φ=0.8φ=0.6φ=0.5φ=0.4φ=0.3

(b)

ΔP, kPa

(H/G

) opt

0 20 40 60 802

6

10

14

18

22

26φ=0.8φ=0.6

(c)

Figure 7 The effect of pressure drop and solid volume fraction on the opti-mized aspect ratio: (a) rectangular microchannel, (b) elliptic microchannel, and(c) isosceles triangular microchannel (t1/t2 = 0.08 and t2/t3 = 1).


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ΔP, kPa

ΔTm

in,o C

0 20 40 60 800

10

20

30

40

50

60

70

Rectangular,φ=0.8Rectangular,φ=0.6Elliptic, φ=0.8Elliptic, φ=0.6Isosceles triangular,φ=0.8Isosceles triangular,φ=0.6

Figure 8 The effect of pressure drop and solid fraction on the minimized peaktemperature difference for a microchannel heat sink with different cross sections(t1/t2 = 0.08 and t2/t3 = 1).

over, the limitations accompanying the theoretical results ofEq. (13) shall be considered. Actually, the analytical results arejust rough estimates, which should be validated with the resultsobtained by numerical schemes. In fact, we can expect the scale

Figure 9 Coolant mass flow rate in the microchannel heat sink with optimalgeometry.

Be

Cm

ax

5E+07 3.25E+08 6E+080

10000

20000Analytical, Isosceles triangularNumerical, Isosceles triangularAnalytical, RectangularNumerical, RectangularAnalytical, EllipticNumerical, Elliptic

Figure 10 The effect of dimensionless pressure drop on the dimensionlessglobal thermal conductance (t1/t2 = 0.08, t2/t3 = 1, and ϕ = 0.6).

analysis to predict the trends of variations of Cmax with Be,correctly.

The comparison between the channels’ optimal shapes,Dh,opt, obtained from the present numerical solution and theapproximate analytical solution, Eq. (12), is presented inFigure 11 for different microchannels at ϕ = 0.6. It is observedfrom this figure that the trends of the two solutions are similarand there is acceptable agreement with maximum 25% deviationbetween the results.

Figure 11 The optimal channel size for maximal global thermal conductance(t1/t2 = 0.08, t2/t3 = 1, and ϕ = 0.6).


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CONCLUSIONS

In this study, we showed theoretically and numerically thatfor microchannel heat sinks with rectangular, elliptic, andisosceles triangular cross sections, there is an optimal hydraulicdiameter at which the global thermal conductance is maxi-mal. The following results were achieved from the presentinvestigation:

1. In microchannel heat sinks with rectangular, elliptic, andisosceles triangular cross sections, (H/G)opt increases withpressure drop.

2. The thermal performances of rectangular and elliptic mi-crochannel heat sinks are almost the same, and the optimalsolid volume fraction, ϕopt, of these heat sinks resides in therange of 0.4 ≤ ϕopt ≤ 0.5; we cannot conclude about theisosceles triangular microchannels as a result of geometriclimitations (ϕ ≥ 0.6).

3. Regarding the parameters and constraints of the presentstudy, the triangular microchannels have lower global ther-mal conductance than their rectangular and elliptic counter-parts. However, the thermal performance of isosceles trian-gular microchannel heat sinks improves at ϕ = 0.6 and getsclose to that of rectangular and elliptic microchannels.

4. If the mass flow rate of coolant fluid is one of the constraints(the pressure drop constraint is removed), then an isoscelestriangular microchannel is the best choice when ϕ = 0.6.

5. Numerical optimization results showed that Dh,opt and Cmax

are functions of the applied pressure difference and solidvolume fraction.

6. An agreement with maximum 25% deviation was observedbetween the present numerical and approximate analyticalsolutions of the optimal shapes of the microchannels.

NOMENCLATURE

A channel cross-sectional area, m2

Be Bejan number based on a unit volume, = �PV2/3/μα

B channel width, mC dimensionless global conductanceCp specific heat, J/kg-KD distance between the walls, mDh hydraulic diameter, = 4Ac/Pc

f friction factor, = τ/( 12ρU2)

G width of computational domain, mH height, mHc channel height, mk thermal conductivity, W/m-KL axial length, mm mass flow rate, kg/sn normalnum number of channelsPC perimeter of microchannel, mPO Poiseuille number, = τDh/μU

P pressure, N/m2

Pr Prandtl number, = ν/αq heat transfer rate, Wq′′ heat flux, W/m2

Rt thermal resistance, K-cm2/WReDh Reynolds number based on hydraulic diameterT temperature, KTw,L exit temperature, Kt1 half thickness of vertical solid, mt2 base thickness, mt3 top thickness, m�U velocity vector, m/sU average velocity, m/su velocity along the x-axisv velocity along the y-axisV volume, m3

w velocity along the z-axisW heat sink width, mx,y,z Cartesian coordinates, m

Greek Symbols

α thermal diffusivity, m2/sε aspect ratio, = B/Hc

� differenceμ dynamic viscosity, N-s/m2

ν kinematic viscosity, m2/sρ fluid density, kg/m3

ϕ volume fraction of solid materialτ shear stress, N/m2

� interface between the solid and fluid

Subscripts

c cross-sectionalf fluidin inletmax maximummin minimumopt optimumout outlets solidw wall

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[30] Karniadakis, G., Beskok, A., and Aluru, N., Microflowsand Nanoflows Fundamentals and Simulation, Springer,New York, NY, 2005.

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Mohammad Reza Salimpour is an associate pro-fessor in the Department of Mechanical Engineer-ing, Isfahan University of Technology, Isfahan, Iran.He received his B.Sc., M.Sc., and Ph.D. in mechan-ical engineering from University of Tehran in 2001,2003, and 2007, respectively. His major research ar-eas are constructal design, electronics cooling, two-phase flow heat transfer, and thermodynamic design.

Morteza Sharifhasan is an M.Sc. student of mechan-ical engineering in Isfahan University of Technology,Isfahan, Iran. He received his B.Sc. in mechanicalengineering in 2008 from University of Ferdowsi,Mashhad, Iran. He is currently working on the opti-mization of microchannel heat sinks using constructaltheory.

Ebrahim Shirani is a professor in the Departmentof Mechanical Engineering, Isfahan University ofTechnology, Isfahan, Iran. He received his B.Sc. inmechanical engineering in 1975 from Sharif Univer-sity of Technology, Tehran, Iran. He also receivedhis M.Sc. and Ph.D. in thermofluids from StanfordUniversity in 1977 and 1981, respectively. He haspublished/presented more than 100 papers in highlyranked journals/conferences. His main research areasare CFD, micro- and nanofluid dynamics, turbulence,

and turbomachinery.


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Documents

Constructal Optimization of Microchannel Heat Sinks With Noncircular Cross Sections