Shape Optimization of Micro-Channel Heat Sink for Micro-Electronic Cooling

322 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 31, NO. 2, JUNE 2008

Shape Optimization of Micro-Channel HeatSink for Micro-Electronic Cooling

Afzal Husain and Kwang-Yong Kim

Abstract—A numerical investigation of 3-D fluid flow and heattransfer in a rectangular micro-channel has been carried out usingwater as a cooling fluid in a silicon substrate. Navier–Stokes andenergy equations for laminar flow and conjugate heat transfer aresolved using a finite volume solver. Solutions are first carefullyvalidated with available analytical and experimental results; theshape of the micro-channel is then optimized using surrogatemethods. Ratios of the width of the micro-channel to the depthand the width of the fin to the depth are selected as design vari-ables. Design points are selected through a four-level full factorialdesign. A single objective function thermal resistance, formulatedusing pumping power as a constraint, is optimized. Mass flow rateis adjusted by the constant pumping power constraint. Responsesurface approximation, Kriging, and radial basis neural networkmethods are applied to construct surrogates and the optimumpoint is searched by sequential quadratic programming.

Index Terms—Electronic cooling, micro-channel, numerical sim-ulation, optimization, surrogate methods.

NOMENCLATURE

Cross section area of micro-channel.

Surface area of substrate base.

Specific heat.

Hydraulic diameter.

Friction factor.

Convective heat transfer coefficient.

Height of heat sink.

Micro-channel depth.

Thermal conductivity.

Entry length.

Length of heat sink.

Width of heat sink.

Height of heat sink.

Number of micro-channels.

Number of dimensions in design space.

Pressure.

Perimeter.

Pumping power.

Heat flux.

Manuscript received December 21, 2006; revised August 15, 2007. This workwas recommended for publication by Associate Editor S. Bhavnani upon eval-uation of the reviewers comments.

The authors are with the Department of Mechanical Engineering, Inha Uni-versity, Incheon 402-751, Korea (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCAPT.2008.916791

Coolant flow rate.

Re Reynolds number.

Thermal resistance.

Convective resistance.

Adjusted value of R square.

Temperature.

Liquid velocity in micro-channel.

Velocity vector.

Micro-channel width.

Fin width.

Orthogonal coordinate system.

Greek Symbols

Normalized design variables, and ,respectively.Micro-channel aspect ratio.

Density.

Dynamic viscosity.

Kinematic viscosity.

Subscripts

Liquid.

Inlet.

Outlet.

Substrate.

max Maximum value.

avg Average value.

I. INTRODUCTION

RECENT developments in micro electromechanical sys-tems (MEMS) and advanced very large-scale integration

(VLSI) technologies and devices associated with micro minia-turization have led to significant improvement in packingdensities. These developments have helped satisfy growingdemand for higher dissipation of heat flux from electronicdevices. However, it has been observed that operation of mostelectronic devices is strongly influenced by their temperatureand their surrounding thermal environment. Micro-channelheat sink, as an integrated part of silicon based electronicdevice, is a potential solution to this problem. Sophisticatedfabrication processes have yielded economically competitivemicro-channels having a high surface area to volume ratio.

The potential of micro-channel heat sinks as heat transfer de-vices has motivated many researchers to analyze micro-cooling

1521-3331/$25.00 © 2008 IEEE

HUSAIN AND KIM: SHAPE OPTIMIZATION OF MICRO-CHANNEL HEAT SINK 323

phenomena and conduct parametric studies. Tuckerman andPease [1] first realized the potential of this technology andlaid a foundation for silicon based micro-channel heat sinkexperimentation. They experimented on a 56 m wide, 320 mdeep micro-channel fabricated by a chemical etching process.Samalam [2] reported correlations for thermal resistance basedon a theoretical study of experiments of Tuckerman [3]. Fe-dorov and Viskanta [4] carried out a numerical simulationbased on the experiments of Kawano et al. [5]. While numerousinvestigations have been conducted using various techniques[6]–[10], there are significant disparities between the existingexperimental data and the values predicted using classicalmacro techniques [11].

The results of the aforementioned studies indicate that thecross section parameters and fin width of the micro-channel cangreatly influence the behavior of the fluid flow and resultingheat transfer capabilities of rectangular micro-channels [9],[10], [12]. In addition, experimental investigations [1], [7], [9],[10] reported a pressure drop in the convective heat transfer forwater flowing in rectangular micro-channels of various aspectratios. These results suggest that the aspect ratio is an importantparameter in the determination of flow friction and convectiveheat transfer.

Some analytical studies [13]–[15] have focused on modelingof the heat transfer and optimization of the micro-channel geom-etry. Weisberg et al. [16] presented a design algorithm for theselection of a rectangular micro-channel heat exchanger using a2-D conjugate heat transfer model. Qu and Mudawar [17] com-puted the 3-D fluid flow and heat transfer for a micro-channelwith a rectangular cross section. Li et al. [18] numerically inves-tigated 3-D conjugate heat transfer in a silicon based micro-heatsink. Toh et al. [19] carried out a detailed numerical study ofvariation of local thermal resistance and friction factor along theflow direction in micro-channels by solving 3-D Navier–Stokesequations. They extended the length of the micro-channel to en-sure realistic boundary conditions at the exit. As an attempt tooptimize the shape of the micro-channel, Li and Peterson [20]carried out a parametric study on the geometry of the micro-channel. An exhaustive review of the literature, however, showsthat systematic optimization techniques have not yet been ap-plied to this problem.

With the aid of high performance computers, the last decadehas witnessed rapid development of design optimization tech-niques. Numerical optimization methods [21] are regarded asgeneral design tools and offer a number of advantages, includingautomated design capability, varieties of constraints, and mul-tidisciplinary applications. However, due to large computingtime, coupling with Navier–Stokes analyses has recently provenpractical. Surrogate models are widely used in multidisciplinaryoptimizations. Queipo et al. [22] reviewed various surrogatebased models used in aerospace applications.

The current investigation explores the application of surro-gate based optimization techniques for the shape optimizationof a rectangular micro-channel cross-section to minimizethermal resistance. Response Surface Approximation (RSA),Kriging (KRG), and radial basis neural network (RBNN)methods are used to construct surrogates, and the minimum

Fig. 1. Schematic diagram of micro-channel heat sink and design variables.

thermal resistance of the micro-channel is then searched bysequential quadratic programming.

II. PROBLEM DESCRIPTION AND GEOMETRIC CONSTRAINTS

A schematic of the rectangular micro-channel heatsink optimized in the current study is shown in Fig. 1.The dimensions of the heat sink under consideration are10 mm 10 mm 0.5 mm. The thickness of the base of themicro-channel is 100 m while the depth of the micro-channelis kept constant at 400 m. Simulations are performedfor varying fin width and channel width. A uniform heat flux( 100 W/cm ) is applied at the bottom of the heat sink toelucidate the effect of micro-channel geometry on the thermalresistance and friction factor. The flow is assumed to be lam-inar and fully developed and is maintained by low flow ratesand low Reynolds numbers. Since the focus of the study is tooptimize the microchannel geometry with surrogate analysis,thermodynamic and hydrodynamic properties of the substanceare assumed to be constant with a reference temperature of27 C in all simulations.

One of the major challenges in micro-channel optimization ismanufacturing feasibility. The optimal design should not be im-practical from manufacturing and design points of view. A sil-icon based micro-channel can be fabricated with an aspect ratio

of up to 20:1 using DRIE [23], [24] and 6:1 usingKOH wet etching [1], [5], [24]. On the basis of strength, theconstraint thickness of the base of the micro-channel is kept as100 m, which is well above the minimum required thickness,as suggested by Li and Peterson [20]. Li and Peterson [20] foundthat thermal resistance decreases with an increase in the aspectratio, and therefore is kept constant for all optimization casesin order to assess the effect of the micro-channel width and finwidth on the thermal resistance and friction factor. A constantpumping power is applied to the micro-channel heat sink, andis defined as

(1)

where is the volumetric flow rate across the heat sink andis the pressure drop. indicates the number of channels,is the average velocity, and is the cross-sectional area of themicro-channel.


Fig. 2. Computational domain.

III. NUMERICAL ANALYSIS

In the present study, steady, incompressible, laminar flowand conjugate heat transfer are considered with constantfluid (water) properties (density 997 kg/m , spe-cific heat capacity 4.179 J/kg K, dynamic viscosity

8.55 10 kg/ms, and thermal conductivity0.613 W/mK at a reference temperature of 27 C). Continuumequations for conservation of mass, momentum, and energy forthe convective heat transfer in the micro-channel can be writtenin vector form asMass

(2)

Momentum

(3)

Energy

(4)

(5)

where represents the velocity vector, is the fluid temper-ature, and and are the substrate temperature and thermalconductivity, respectively.

A numerical model is formulated to solve the 3-D conjugateheat transfer in micro-channels using commercial code CFX5.7 [25]. The code uses finite volume discretization of gov-erning differential equations and the solution is based on theSIMPLE algorithm [26]. In the numerical solution, the con-vective terms are discretized using a first-order upwind schemefor all equations. Due to symmetry of the problem, half of thesingle micro-channel is selected for computation, as shown inFig. 2. A hexahedral mesh is generated in the specified domainthrough ICEM 5.7 [25] and a 401 61 16 grid is used forthe current simulation cases. Grid independency is checked bytaking different grids: it was found that for a 501 71 21 gridthe change in the highest temperature in the substrate was only0.03% while for a 301 41 11 grid the difference in highesttemperature attained was roughly 2%. A finite volume solver isused under the following boundary conditions.

Water flows into the micro-channel at the inlet of the heatsink and leaves at the outlet; the remainder of the heat sink isoccupied by the silicon substrate. The silicon part of the heatsink at the inlet and outlet of the channel is maintained as anadiabatic boundary. No-slip condition is applied at the interiorwalls of the channel, i.e., 0. Average velocity obtained bythe method described later in this section is used at the inletof the microchannel, and velocity gradients are neglected at theexit. The thermal conditions in the z-direction are

The left and right surfaces in Fig. 2 are assigned as symmetricboundary conditions.

Formulation of the average velocity in terms of pumpingpower is described as follows. Fanning friction is defined as

(6)

where is the hydraulic diameter. The Reynolds number interms of average velocity can be defined as

(7)

where

For a fully developed flow, Knight et al. [13] determined that

(8)

where

The average velocity can be obtained from (6), (7), and (8) asfollows:

(9)

Substituting (1) into (9) we obtain

(10)

where the pumping power should be realistic and can be takenas being in a range of 0.01 to 0.8 considering the capacity ofstate-of-the-art micro pumps [14]. For comparison of the resultswith experimental data, the thermal resistance is calculated as

(11)


where is the area of the substrate subjected to heat flux andis the maximum temperature rise in the heat sink and is

defined as

(12)

where is the substrate temperature near the outlet andis the fluid inlet temperature. Thermal resistance at the inlet isdefined as

and at the outlet, thermal resistance is defined as

where and are the chip surface temperatures at the inletand outlet, respectively.

IV. OPTIMIZATION TECHNIQUES

Fig. 3 represents the various steps involved in the optimiza-tion algorithm. Design points are selected using a four-level fullfactorial design. Two design variables, and , are chosen forthe optimization methodology, where is defined asand is defined as . Hence, 16 design points chosenfrom four-level full factorial design assisted with 4 more de-sign points are used to construct the surrogates. These levelsare equally spaced within the design range for all variables. De-sign space is specified performing some preliminary calcula-tions within the geometric constraints discussed in Section IIto obtain minimum of thermal resistance. The design space isgiven in Table I. Design variables are normalized from 0 to 1to construct surrogate models. Objective functions are calcu-lated by solving Navier–Stokes and heat conduction equationsat specified design points and optimized using surrogate models.In the current study, thermal resistance is the foremost con-sideration for optimization and is the objective function for thesurrogate based optimization i.e., .

Three surrogate models, response surface approximation(RSA), Kriging (KRG), and radial basis neural network(RBNN), are applied to predict the optimal design point. InRSA [27], a least square curve fitting by regression analysisis performed on the data obtained by computational fluid dy-namic (CFD) calculations using Navier–Stokes equations. Thefollowing polynomial function is fitted to obtain the responsesurface function. If the regression coefficients are ’s, thepolynomial function becomes

(13)

where is the number of design variables, and ’s are the de-sign variables. For a second order polynomial model, used inthe current study, the number of regression constants is

.

Fig. 3. Optimization procedure.

TABLE IDESIGN VARIABLES AND DESIGN SPACE

The KRG model, also known as Kriging metamodelling [28],is a deterministic technique for optimization. A linear polyno-mial function with a Gauss correlation function is used for themodel construction. The Kriging postulation is a combinationof a global model and departures of the following form:

(14)

where represents the unknown function, is the knownfunction of , and is the realization of a stochastic processwith mean zero and non-zero covariance. A linear function,

, is fitted first, and real points are interpolated on it to ob-tain the mean zero. Here, is the global design space while

is the localized deviations. The covariance matrix can bewritten as

(15)

where is the number of dimensions in the set of design vari-ables is the standard deviation of the population, and is


the measure of degree of correlation among the data along thedirection .

A radial basis neural network (RBNN) [29] uses a linear com-bination of n radially symmetric functions, , for the re-sponse function as

(16)

where is the coefficients of the linear combination, g is the setof radial basis functions (typically Gaussian), and is the set oferrors with equal variance, . Due to its linear nature, RBNNhas simpler mathematics and lower computational cost as com-pared to back-propagation neural network (BPNN) [29]. RBNNis a two layer network that consists of a hidden layer of the radialbasis function and a linear output layer. The design parametersfor this function are spread constant (SC) and a user definederror goal (EG). The SC value is selected such that it shouldnot be so large that each neuron does not respond in the samemanner for the all inputs, and that it should not be so small thatthe network becomes highly sensitive for every input within thedesign space. EG or mean square error goal selection is also im-portant. A very small error goal will produce over-training of thenetwork while a large error goal will affect the accuracy of themodel. The allowable error goal is decided from the allowableerror from the mean input responses. Cross-validation [22] isperformed to check the quality of constructed surrogates. Thesesurrogate methods are implemented using MATLAB [30].

V. RESULTS AND DISCUSSION

The flow is assumed to be steady, incompressible, and fullydeveloped laminar flow. The velocity obtained from (10) is ap-plied at the inlet of the micro-channel. In order to ensure afully developed flow, lower pumping power, 0.05 W, toa 10 mm 10 mm chip has been chosen for the optimizationcases. To validate the assumption of fully developed flow, theentrance length is calculated for the micro-channel flow. Fora typical case of 0.15 and 0.1, the entrance lengthis found to be 6% of the total channel length which is in goodagreement with the relation developed by Langhaar [31] for flowin a circular tube.

0.057 (17)

It is found that except the extreme case the entrance length isless than one fifth of the length of micro-channel heat sink,therefore assumption of fully developed flow is acceptable atlow pumping power [20]. The numerical model is validated ina number of ways to ensure the accuracy of the numerical solu-tions and the design optimization. Fig. 4 presents a comparisonof the numerical model with available analytical results [32] attwo different planes perpendicular to each other. Non-dimen-sional velocity profiles are plotted along the y andz directions, respectively. This comparison reveals good agree-ment between the numerical and analytical results for the ve-locity distribution in different directions. Experimental valida-tion of the numerical results is done with the experiment of

Fig. 4. Comparison of numerically predicted fully developed velocity profileswith analytical results: (a) on z middle plane and (b) on y middle plane.

Kawano et al. [5]. Fig. 5 shows a comparison of the numeri-cally calculated inlet thermal resistance and outlet thermal resis-tance with available experimental results [5] for a wide range ofReynolds number. The present results show appreciable agree-ment with the experimental results, with most of the predictionslying within the experimental uncertainties (indicated by errorbars in the figure). The present model underpredicts the inletthermal resistance only for a low Reynolds number. One of thepossible reasons for this underprediction may be heat loss tothe ambient by the solid substrate, as discussed by Qu and Mu-dawar [17]. Moreover, the low Reynolds number conditions areunreliable because of the larger temperature-induced viscositygradients near the inlet portion [5]. Further validation of the nu-merical model is done with experimental results of Tuckermanand Pease [1] for three different cases of different channel depthand heat flux, as given in Table II. In light of the numerical re-production of these experimental results by Toh et al. [19] andLiu et al. [8], the present model shows good agreement with theexperimental results, even for higher heat flux.

Before carrying out the actual optimization, the effect ofthe design variables on the thermal resistance and frictionconstant is assessed. Figs. 6 and 7 show the variations of thethermal resistance and friction constant with changes in thedesign variables. Convective thermal resistance is definedas 1 , where is the convective heat


Fig. 5. Comparison of numerical model predictions with experimental data: (a)inlet thermal resistance and (b) outlet thermal resistance.

TABLE IICOMPARISON OF THERMAL RESISTANCES BETWEEN

COMPUTATIONAL AND EXPERIMENTAL RESULTS

transfer coefficient. With an increase of , the average velocityincreases. Furthermore, the area subjected to convective heattransfer reduces as the number of micro-channels decreasesin a specified heat sink width for a constant under constantpumping power. Therefore, the thermal resistance decreases as

is increased. This trend continues until the velocity dominatesthe convective heat transfer and then the thermal resistanceincreases when the area subjected to convective heat transferis significantly reduced. The increased velocity and reduced

Fig. 6. Variation of (a) thermal resistance and (b) friction constant with designvariable �.

pressure drop result in a decreased friction constant with anincrease of . For lower values of , where the flow resistanceis significant, an increase in further increases the flow re-sistance, resulting in increased thermal resistance. For highervalues of , an increase in reduces pressure drop but increasesthe velocity, which results in lower thermal resistance. Forintermediate values of , a decrease in the thermal resistance isobserved with an increase in up to a certain value, becausean increase in the fin thickness results in higher velocity, whichin turn results in higher convective heat transfer. A reducedarea subjected to convective heat transfer results from a furtherincrease in , and decreases convective heat transfer throughthe micro-channel. A higher velocity reduces the friction factorbut results in a higher effective friction constant. It is naturalthat the friction constant appears to be unaffected by any changein the value of over the entire range of . An increase inpressure drop is countered by an increase in average velocity.

The RSA, KRG, and RBNN surrogates are constructed usingthe training data from a four-level full factorial design. Someadditional CFD data is added to enhance the performance of thesurrogate models. The results of optimization of the thermal re-sistance using the surrogate models are shown in Table III. In theRSA method, analysis of variance (ANOVA) and a regressionanalysis, provided by t-statistics in [7], are implemented to mea-sure the uncertainty in the set of coefficients in the polynomial.


Fig. 7. Variation of (a) thermal resistance and (b) friction constant with designvariable �.

TABLE IIIOPTIMAL POINTS (NORMALIZED) PREDICTED BY DIFFERENT SURROGATES

AND CORRESPONDING CFD CALCULATED VALUES

In the present study, was maintained at 0.967, which can beconsidered reliable in reference to the value of 0.91 1.0suggested by Guinta [33] for accurate prediction of the responsesurface model. The functional relationship between objectivefunction and design variables is established by RSA as

(18)

The KRG model is prepared with the help of toolbox (DACE)[34] in MATLAB. Correlation function parameters are adjustedcarefully to ensure consistent performance of the model. Theacceptable variance of the method was kept at 3.93 10 .

Fig. 8. Sensitivity analysis of objective function F (thermal resistance) near theoptimal point.

The accuracy of the RBNN is checked by varying the errorgoal and the spread constant in order to obtain the minimumPRESS (Prediction Error Sum of Squares) for training data.Prediction errors are calculated by cross-validation of theconstructed model predictions at the design points. In theRBNN model, spread constant and error goals a set at 0.51and 1 10 , respectively, to train the network. Consistentperformance of the network is checked with variation of thespread constant. Objective function values are calculated at thesurrogate predicted optimal points, as shown in Table III. Theoptimum values obtained by a Navier–Stokes analysis at thedesign points predicted by all methods are almost identical.RSA gives the optimum design as 0.671 0.430.In contrast, the optimum design from the KRG and RBNNmodels was 0.492 0.226 and 0.490 0.306,respectively. The optimum design can help the designer findthe design variables the geometry of his/her choice. For amicro-channel heat sink of channel depth 360 m, theadjusted number of channels for the optimum design calculatedfrom the RBF model is 120, which is in line with the findingsof Li and Peterson [20].

All the surrogate models under study predict almost same ob-jective function value, although their predicted values of the de-sign variables are different. Moreover, the CFD predicted objec-tive function values are also very close. This characteristic of thedesign space shows the relatively less sensitivity of the objectivefunction near the optimum point. Surrogate predictions dependupon the nature and suitability of the problem, therefore eachsurrogate predicts different optimum point. The use of multiplesurrogates helps to understand the insight of the design spaceand suitability of the surrogate for the kind of problem understudy.

A sensitivity analysis of the objective function is performedby varying the design variables around the optimum design.Each design variable is varied from the optimum point in bothdirections while keeping the other variables fixed. The objectivefunction values at these sets of design variables are calculatedusing a surrogate model (typically RSA). The objective function(thermal resistance) increases sharply with a change in whilekeeping fixed. On the other hand, change of has a smallereffect on the objective function for a fixed , as shown in Fig. 8.


Fig. 9. Temperature contours on the middle x–z plane: (a) for [1] case and (b)for optimum case.

It can be seen that the optimal design is highly sensitive toas compared to in the specified range. Therefore, the designvariable can be suitably adjusted for the optimum number ofchannels in order to obtain minimum thermal resistance for thespecified channel depth of the heat sink.

Figs. 9 and 10 present a comparison of the temperature dis-tributions between [1] and optimum shapes on x–z and y–zplanes, respectively. These temperature distributions for the ge-ometry of case 1 in [1] and for the optimum shape predicted byresponse surface approximation are calculated under constantpumping power and heat source. By minimizing the thermal re-sistance, the optimum shape shows lower maximum tempera-ture, which occurs at the bottom of the substrate near the exit ofthe micro-channel, in comparison with the reference shape.

VI. CONCLUSION

A 3-D rectangular micro-channel heat sink has been geomet-rically optimized for minimum thermal resistance using sur-rogate models. Fluid flow and heat transfer analyses are con-ducted by solving 3-D Navier–Stokes and heat conduction equa-tions to find the overall thermal resistance of the heat sink. The

Fig. 10. Temperature contours on the middle y–z plane: (a) for [1] case and (b)for optimum case.

thermal resistance of a micro-channel heat sink fabricated ona silicon wafer is minimized for a constant heat source andconstant pumping power. Three different surrogate models, i.e.,RSA, KRG, and RBNN, are employed for the optimization. De-sign variables related to micro-channel depth and fin width areselected to construct the surrogates, which are used to predictthe minimum of the objective function (thermal resistance ofthe heat sink). The three surrogate models yielded somewhatdifferent optimum geometries, but predicted almost the sameobjective function values. The objective function is found to bemore sensitive to channel width to depth ratio than fin width todepth ratio around the optimal point.

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Afzal Husain received the B.E. and M.Tech. degreesin mechanical engineering with specialization inthermal sciences from Aligarh Muslim University,Aligarh, India, in 2003 and 2005, respectively, andis currently pursuing the Ph.D. degree in thermo-dynamics and fluid mechanics in Inha University,Incheon, Korea.

His research interests are thermal analysis of mi-crosystems (MEMS), electronic cooling, and surro-gate based analysis and optimization.

Kwang-Yong Kim received the Ph.D. degree fromthe Korea Advanced Institute of Science and Tech-nology (KAIST), Daejon, Korea, in 1987.

Presently, he is a Professor in School of Me-chanical Engineering, Inha University, Incheon,Korea. He published more than 120 research pa-pers in professional journals, and presented 76papers at international conferences and 156 pa-pers at domestic conferences. He also published60 technical reports which were supported by avariety of research grants from government and

industries, and has four domestic patents. He is presently the Editor-in-Chiefof the Transactions of Korean Society of Mechanical Engineers (KSME). Hisrecent research works have been concentrated on applications of numericaloptimization techniques using computational fluid dynamics (especially thethree-dimensional Reynolds-averaged Navier–Stokes analysis techniques) todesigns of engineering systems or devices, such as heat transfer augmentationdevices, components of thermal-hydraulics system in various nuclear reactors,turbomachinery blades, micro-mixers, micro heat sinks, etc., where the use ofnumerical optimization techniques was not popular.

Dr. Kim is Chief Vice President of the Korean Fluid Machinery Association(KFMA).

Documents

Shape Optimization of Micro-Channel Heat Sink for Micro-Electronic Cooling