International Journal of Heat and Mass Transfer 53 (2010) 5988–5995

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International Journal of Heat and Mass Transfer

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

Technical Note

Thermal optimization of plate-fin heat sinks with variable fin thickness

Dong-Kwon Kim a,1, Jaehoon Jung b,1, Sung Jin Kim b,*

a Department of Mechanical Engineering, Ajou University, Suwon 443-749, Republic of Koreab Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea

a r t i c l e i n f o

Article history:Received 14 April 2010Received in revised form 10 July 2010Accepted 10 July 2010Available online 20 August 2010

Keywords:Heat sinkThermal optimizationVariable fin thickness

0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.07.052

* Corresponding author. Tel.: +82 42 350 3043; faxE-mail address: sungjinkim@kaist.ac.kr (S.J. Kim).

1 These authors contributed equally to this work.

a b s t r a c t

In the present paper, thermal optimization of a plate-fin heat sink with the fin thickness varying in thedirection normal to the fluid flow was conducted. The model used for this optimization was based onthe volume averaging theory (VAT). It was shown that the thermal resistance of the plate-fin heat sinkcan be reduced by allowing the fin thickness to increase in the direction normal to the fluid flow. Inthe case of a water-cooled heat sink, the thermal resistance decreases by as much as 15%. The amountof the reduction increases as the pumping power increases or as the length of the heat sink decreases.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Recent advances in semiconductor technology have led to a sig-nificant increase in the power densities encountered in microelec-tronic equipment [1]. The possibility of a failure of an electronicdevices increases as the power density increases. Accordingly, ahigher level of performance in cooling technology is essential forthe reliable operation of electronic components [2]. Many ideaspertaining to cooling methods have been proposed. Among thevarious types of developed cooling systems, the plate-fin heat sinkis the most widely used due to the benefits of its simple design andeasy fabrication.

Several research works have concentrated on sizing-optimiza-tion of plate-fin heat sinks. Several optimization methods havebeen proposed based on the fin model [3], the three-dimensionalnumerical model [4], and on the volume averaging theory [5]. Inthese studies, the layout of the heat sink was prescribed as shownin Fig. 1a. The heat sink was optimized by determining the finthickness and channel width which minimize the thermal resis-tance for a given heat sink size. This procedure was based on theassumption that fin thickness and channel width are constantalong the directions parallel and normal to the fluid flow. However,there is no guarantee that a rectangular cross-section fin is themost thermally efficient. It is likely that the thermal resistance ofa plate-fin heat sink may be further reduced by allowing the finthickness to vary in the directions parallel and normal to the fluidflow. Morega and Bejan demonstrated that the thermal resistance

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of an air-cooled heat sink can be reduced by about 15% by increas-ing the fin thickness in the flow direction [6]. However, there hasbeen no study focusing on heat sinks with fin thickness varyingin the direction normal to the fluid flow, to the authors’ knowledge.

Which type of heat sink performs better among the three heatsinks shown in Fig. 1? The present paper is devoted to seeking aproper answer to this question. Thermal optimization of theplate-fin heat sinks was conducted with fin thicknesses varyingin the direction normal to the fluid flow using a model based onthe volume averaging theory (VAT). The results demonstrated thatthe thermal resistance can be reduced by allowing the fin thicknessto increase in the direction normal to the fluid flow, i.e., the heatsink shown in Fig. 1c is the best choice. It will be shown later in thispaper that the reduction of the thermal resistance when using var-iable-thickness-fins increases as either the pumping power in-creases or the length of the heat sink decreases.

2. Mathematical formulation

The problem under consideration in the present paper concernsforced convection through a plate-fin heat sink as depicted inFig. 2a and b. The direction of the fluid flow is parallel to the xdirection. The top surface is insulated and the bottom surface isuniformly heated. Coolant passes through a number of channelsand takes heat away from a heat-dissipating electronic componentattached below. In analyzing the problem, for simplicity, the flow isassumed to be laminar, incompressible, and both hydrodynami-cally and thermally fully-developed. All thermophysical propertiesare assumed to be constant. In addition, the pumping power isassumed to be constant. This condition implies that the powerrequired to drive the fluid through the plate-fin heat sink is given.

Nomenclature

a wetted area per volumec heat capacity of a fluidDh,fr aspect ratio of the frontal area (2HW/(H+W))f friction factorh interstitial heat transfer coefficient based on the one-

dimensional bulk mean temperatureH channel heightk thermal conductivitykse effective thermal conductivity of the solidK permeabilityL length of the heat sink_m mass flow rate

n number of channelsN node numberNu Nusselt number (hDh/kf)p fin pitch (wc+ww)P pressurePpump pumping powerPr Prandtl numberq heat transfer rateq00 heat fluxq00sf heat flux from the fins to the fluidR thermal resistanceRe Reynolds numberT temperatureu velocityW width of the heat sink

wc channel widthww fin thicknessx,y,z Cartesian coordinate system<> averaged value<>b,f one-dimensional bulk mean temperature for the fluid

phase

Greek symbolsa aspect ratio of the channel (H/wc)afr aspect ratio of the frontal area (H/W)e porosity (wc/(ww+wc))l viscosityq density

Subscripts and superscriptsbm bulk meancap capacitiveconv convectiveuni uniform-thickness-fin heat sinkf fluidfr frontal areai node numberopt optimizeds solidtot totalvar variable-thickness-fin heat sink

D.-K. Kim et al. / International Journal of Heat and Mass Transfer 53 (2010) 5988–5995 5989

The channel width and the fin thickness vary in the direction nor-mal to the fluid flow (wc = wc(y), ww = ww(y)), but the fin pitch i.e.the sum of the channel width and the fin thickness is constant.

The momentum and energy equations for the fully developedflow are given as follows:

@p@x¼ l @2u

@y2 þ@2u@z2

!; ð1Þ

qc@ðuTÞ@x

¼ k@2T@y2 þ

@2T@z2

!: ð2Þ

In the model based on the volume averaging theory, the governingequations for the averaged velocity and temperature are establishedby averaging the momentum and energy equations in the z-direc-tion over the analysis domain shown in Fig. 2c. When the solid con-ductivity is higher than the fluid conductivity and the aspect ratio ofthe channel is much higher than 1, the governing equations for theaveraged quantities and the appropriate boundary conditions aregiven as follows [5]:

� dpdx¼

lf

Kehuif ; ð3Þ

eqf cf huif@hTif ;bm

@x¼ haðhTis � hTif ;bmÞ; ð4Þ

@

@ykse

@hTis

@y

� �¼ haðhTis � hTif ;bmÞ; ð5Þ

huif ¼ 0 at y ¼ 0;H; ð6ÞhTis ¼ hTif ;bm ¼ Tw at y ¼ 0; ð7Þ@hTis

@y¼ @hTi

f ;bm

@y¼ 0 at y ¼ H; ð8Þ

where the average fluid velocity, average fluid temperature, andaverage solid temperature are defined as shown below,respectively:

huif ¼ 2wc

Z ðwwþwcÞ=2

ww=2udz; hTif ;bm ¼

R ðwwþwcÞ=2ww=2 TudzR ðwwþwcÞ=2ww=2 udz

;

hTis ¼ 2ww

Z ww=2

0T dz: ð9Þ

In Eq. (9), for the average fluid temperature, the one-dimensionalbulk mean is employed in order to calculate the bulk mean temper-ature of the fluid easily. e, a, kse, K, and h are the porosity, wettedarea per volume, effective thermal conductivity of the solid, perme-ability, and interstitial heat transfer coefficient. These can be repre-sented as

e ¼ wc

wc þww; a ¼ 2

wc þww; kse ¼

ksww

wc þww;

K � ewchuif

2@u@z

����z¼ww=2

!�1

¼ w3c

12ðwc þwwÞ;

h �q00sf

hTis � hTif ;bm¼ 70kf

17wc: ð10Þ

where e, a, kse, K, and h are functions of y, because they are the func-tions of wc and ww.

The thermal performance of the heat sink was evaluated byintroducing the concept of thermal resistance. Thermal resistanceis defined here as the difference between the base temperatureof the heat sink at the outlet and the fluid bulk mean temperatureat the inlet per unit of heat flow rate. The thermal resistance can bedecomposed into the capacitive resistance, which is responsible forthe temperature rise of the coolant from the inlet to the exit, andthe convective resistance, which is related to the heat transfer fromthe fins to the coolant:

(a) Rectangular-fin heat sink

(b) Triangular-fin heat sink

(c) Reverse-trapezoidal-fin heat sink

Fig. 1. Plate-fin heat sinks.

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Rtot ¼ Rcap þ Rconv ; ð11Þ

Rcap ¼1_mcf

; Rconv ¼Tw � Tbm

q: ð12Þ

In Eq. (12), _m and Tbm are the mass flow rate and the bulk meantemperature of the fluid, respectively. These values can be calcu-lated by integrating the average velocity and temperature, as shownbelow.

_m ¼ qnZ H

0huif wcdy; Tbm ¼

R H0 hTi

f ;bmhuif wcdyR H0 hui

f wcdy: ð13Þ

In the present study, the thermal resistance is calculated from theaveraged velocity and temperature obtained by solving Eqs. (3)–(8) numerically. The fin and the channel are discretized as shownin Fig. 2c. Obviously, the averaged velocity and temperature dependon the discretized channel width wc,i and fin thickness ww,i wherei = 0, 1, . . . , N. Therefore the thermal resistance is a function of wc,i

and ww,i.

3. Results and discussion

In order to validate the solution presented in the previous sec-tion, thermal resistances obtained from the proposed model werecompared with results of a two-dimensional direct numerical sim-ulation, the solutions of which were obtained by solving themomentum equation and energy equation (Eqs. (1) and (2)) usingthe control-volume-base finite difference method. Fig. 3a depictsthe thermal resistances for the uniform-thickness-fin heat sinkand variable-thickness-fin heat sink, whose porosities vary linearlyfrom 0.4 at the base to 0.6 at the top. Fig. 3b depicts the thermalresistances for variable-thickness-fin heat sinks whose porositiesvary as shown in Table 1. The thermal resistances obtained fromthe proposed model are in good agreement with the numerical re-sults with a relative error of less than 10%. However, it is shownthat the thermal resistance obtained from the proposed modeldeviates from numerical results as the channel height decreases.It is because the proposed model is based on the assumption thatthe aspect ratio of the channel is much higher than 1. In addition,the present results were also compared with Nusselt numbers re-sults for rectangular and triangular channels given by Shah [7], asshown in Fig. 4. The present results and the data from Shah matchwith a maximum error of 10%.

To design an optimized heat sink, the fin thickness and channelwidth which minimize the thermal resistance should be deter-mined. In the present study, the optimal values of wc,i and ww,,i

for which the thermal resistance is minimized for a given height,length, width, and pumping power were numerically obtainedwith the gradient descent method. The sequence of numerical sim-ulation can be stated as

(1) Start with the guessed channel width w�c;i and fin pitch p*.(2) Calculate partial derivatives of Rtot with respect to wc,i and p.

@Rtot

@wc;i’

Rtotðwc;i ¼ w�c;i þ cÞ � Rtotðwc;i ¼ w�c;i � cÞ2c

; ð14Þ

� �

@Rtot

@p’ Rtotðp ¼ p þ cÞ � Rtotðp ¼ p � cÞ

2c: ð15Þ

Here, c is a positive small number.

(3) Calculate wc,i and ww,i from the following equations.

wc;i ¼ w�c;i � c@Rtot

@wc;i; p ¼ p� � c

@Rtot

@p: ð16Þ

(4) Treat wc,i and p as newly guessed w�c;i and p*, return to step 2.Repeat step 2–4 until converged values for wc,i and p areobtained.

(5) Calculate ww,i from the following equations.

ww;i ¼ p�wc;i: ð17Þ

Subsequently, second degree polynomial equations for the optimalchannel width wc(y) and fin thickness wc(y) were calculated fromthe wc,i and ww,i values using a least square fitting.

(b) Boundary conditions

(a) Schematic diagram

(c) Analysis domain

Fig. 2. Heat sink with variable fin thickness.

D.-K. Kim et al. / International Journal of Heat and Mass Transfer 53 (2010) 5988–5995 5991

In Table 2, the channel width, fin thickness, fin pitch, fin num-ber, fin surface area, friction factor, capacitive thermal resistance,convective thermal resistance, and total thermal resistance forthe optimized variable-thickness-fin heat sink are presented andcompared with those of the optimized uniform-thickness-fin heatsink under the same constraints. As indicated in Table 2, the fric-tion factor is lower for the variable-thickness-fin heat sink. As a

result, a smaller fin pitch and a larger number of fins are allowedfor the variable-thickness-fin heat sink without compromisingthe capacitive thermal resistance. The larger number of fins lowersthe convective resistance, as the convective resistance generallydecreases as the fin surface area, which is proportional to the num-ber of fins, increases. Consequently, the variable-thickness-fin heatsink can have lower convective resistance than the uniform-thick-

(a) Thermal resistances for rectangular and reverse-trapezoidal heat sink

(b) Thermal resistances for heat sink with variable fin thickness

Fig. 3. Thermal resistances for plate-fin heat sinks. (L = W = 1 cm, lf = 0.000855 kg/m s, cf = 4179 J/kg K, qf = 997 kg/m3, p = 1 mm, ks = 148 W/m K, kf = 0.613 W/m K,Ppump = 2.56 W).

Table 1Geometrical description of typical variable-thickness-fin heat sinks.

Length (L) �width (W) Fin pitch (mm) Schematic of heat sinks (not to scale) Porosity

Case A 1 cm � 1 cm 0.1 0.16(y/H)2 � 0.42(y/H) + 0.57

Case B 1 cm � 1 cm 0.1 �0.16(y/H)2 + 0.42(y/H) + 0.43

Case C 1 cm � 1 cm 0.1 0.21(y/H)2 � 0.59(y/H) + 0.55

Case D 1 cm � 1 cm 0.1 �0.21(y/H)2 + 0.59(y/H) + 0.45

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Fig. 4. Nusselt numbers for rectangular and triangular channels.

Table 2Comparison of the optimized heat sinks with variable-thickness-fins and uniform-thickness-fins.

Variable-thickness-fins Uniform-thickness-fins

Present model 2D numerical results Present model 2D numerical results

ConstraintsLength (L) �width (W) 4 cm � 4 cmHeight (H) 0.4 cmPumping power 2.56WSolid AluminumFluid Water

ResultsFin pitch (mm) 0.276 0.427Fin number 144 93Channel width (mm) 0.0552�((y/H)2 � 3(y/H) + 2.75) 0.128Fin thickness (mm) 0.0552�(�(y/H)2 + 3(y/H) + 2.25) 0.299Surface area (m2) 0.501 0.300f Re 16.4 23.8Rcap(�C/W) 0.0040 0.0042 0.0034 0.0034Rconv(�C/W) 0.0065 0.0066 0.0098 0.0098Rtot(�C/W) 0.0105 0.0108 0.0132 0.0132Schematic of channel (not to scale)

D.-K. Kim et al. / International Journal of Heat and Mass Transfer 53 (2010) 5988–5995 5993

ness-fin heat sink while their capacitive resistances are similar. Fi-nally, the total resistance of the optimized variable-thickness-finheat sink is lower than that of the optimized uniform-thickness-fin heat sink. In the case presented in Table 2, the total thermalresistance shows a decrease of about 15% after allowing the finthickness to vary in the direction normal to the fluid flow.

The optimized geometries and thermal resistances of the vari-able-thickness-fin and uniform-thickness-fin heat sinks for variouspumping powers and various lengths are listed in Table 3. Through

a comparison of the thermal resistances of the optimized variable-thickness-fin and uniform-thickness-fin heat sinks, a contour mapwas drawn, as shown in Fig 5. Fig. 5 depicts the ratio of the thermalresistances of the variable-thickness-fin and uniform-thickness-finheat sinks (Ropt,var/Ropt,uni) for water-cooled systems. In Fig. 5, in theregion where the ratio is less than 1, the optimized variable-thick-ness-fin heat sink performs better than the optimized uniform-thickness-fin heat sink. Additionally, the opposite is true whenthe ratio is greater than 1. Therefore, the contour map indicates

Table 3Comparison of geometries and thermal resistances for optimized heat sinks.

Dimensionlesspumping power

Dimensionlesslength

Type of fins Dimensionlesschannel width

Dimensionlessfin thickness

Thermalresistance (�C/W)

Schematic ofchannel(not to scale)Ppump

l3q�2D�1h;fr

LDh;fr

wcDh;fr

wwDh;fr

R

1011 1 Uniform thickness 0.0024 0.0057 0.289

Variable thickness 0.0011�((y/H)2 � 2.8(y/H) + 2.62) 0.0011�(�(y/H)2 + 2.8(y/H) + 2.14) 0.239

1011 10 Uniform thickness 0.0089 0.0057 0.105

Variable thickness 0.0101�((y/H)2 � 2.43(y/H) + 9.43) 0.0101�(�(y/H)2 + 2.43(y/H) + 4.86) 0.103

1012 1 Uniform thickness 0.0016 0.0056 0.222

Variable thickness 0.0012�((y/H)2 � 2.41(y/H) + 1.56) 0.0012�(�(y/H)2 + 2.41(y/H) + 1.38) 0.153

1012 10 Uniform thickness 0.0059 0.0056 0.063

Variable thickness 0.0120�((y/H)2 � 2.55(y/H) + 5.55) 0.0120�(�(y/H)2 + 2.55(y/H) + 3.55) 0.061

1013 1 Uniform thickness 0.001 0.0057 0.182

Variable thickness 0.0014�((y/H)2 � 1.84(y/H) + 0.86) 0.0014�(�(y/H)2 + 1.84(y/H) + 0.86) 0.099

1013 10 Uniform thickness 0.0040 0.0055 0.041

Variable thickness 0.0013�((y/H)2 � 2.63(y/H) + 3.56) 0.0013�(�(y/H)2 + 2.63(y/H) + 2.69) 0.037

(Dh,fr = 0.01 m, lf = 0.000855 kg/m s, cf = 4179 J/kg K, qf = 997 kg/m3, ks = 148 W/m K, kf = 0.613 W/m K).

5994 D.-K. Kim et al. / International Journal of Heat and Mass Transfer 53 (2010) 5988–5995

that optimized variable-thickness-fin heat sinks have lower ther-mal resistances than optimized uniform-thickness-fin heat sinksin practical situations. The thermal resistance is reduced by asmuch as nearly 15% by employing variable-thickness-fins. The dif-ference between the thermal resistances increases as the length

decreases and as the pumping power increases because the con-vective thermal resistance becomes dominant over the capacitivethermal resistance as either the pumping power increases or thelength decrease. Moreover the variable-thickness-fin heat sinkscan reduce the convective thermal resistance effectively without

Fig. 5. Contour plots of Ropt,var/Ropt,uni (ks/kf = 2.41 � 103, Pr = 0.707, afr = H/W, Dh,fr = (2HW/(H + W))).

D.-K. Kim et al. / International Journal of Heat and Mass Transfer 53 (2010) 5988–5995 5995

compromising the capacitive thermal resistance, as mentionedabove. The ratio of the thermal resistances decreases as the aspectratio of the frontal area increases as well.

4. Conclusion

In the present paper, thermal optimization of a plate-fin heatsink was conducted with the fin thickness varying in the directionnormal to the fluid flow. A model based on the volume averagingtheory (VAT) was used for this optimization. The thermal resis-tance was reduced by as much as 15% compared to uniform-thick-ness-fin heat sinks by employing variable-thickness-fins in thecase of water-cooled heat sink. The amount of the reduction in-creases as either the pumping power increases or the length ofthe heat sink decreases. Due to its high thermal performance, thevariable-thickness-fin heat sink is expected to be suitable for as anext generation cooling solution.

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