International Journal of Heat and Mass Transfer 52 (2009) 3510–3517 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmtComparison of thermal performances of plate-ﬁn and pin-ﬁn heat sinks subjectto an impinging ﬂowDong-Kwon Kim, Sung Jin Kim *, Jin-Kwon BaeDepartment of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Koreaa r t i c l e i n f o a b s t r a c tArticle history: In this paper, we compare thermal performances of two types of heat sinks commonly used in the elec-Received 16 September 2008 tronic equipment industry: plate-ﬁn and pin-ﬁn heat sinks. In particular, heat sinks subject to an imping-Received in revised form 26 February 2009 ing ﬂow are considered. For comparison of the heat sinks, experimental investigations are performed forAccepted 26 February 2009 various ﬂow rates and channel widths. From experimental data, we suggest a model based on the volumeAvailable online 15 April 2009 averaging approach for predicting the pressure drop and the thermal resistance. By using the model, ther- mal resistances of the optimized plate-ﬁn and pin-ﬁn heat sinks are compared. Finally, a contour map,Keywords: which depicts the ratio of the thermal resistances of the optimized plate-ﬁn and pin-ﬁn heat sinks as aHeat sinkImpinging ﬂow function of dimensionless pumping power and dimensionless length, is presented. The contour map indi-Thermal resistance cates that optimized pin-ﬁn heat sinks possess lower thermal resistances than optimized plate-ﬁn heat sinks when dimensionless pumping power is small and the dimensionless length of heat sinks is large. On the contrary, the optimized plate-ﬁn heat sinks have smaller thermal resistances when dimensionless pumping power is large and the dimensionless length of heat sinks is small. Ó 2009 Elsevier Ltd. All rights reserved.1. Introduction ychka  also developed a simple semi-empirical model for pre- dicting the heat transfer coefﬁcient of plate-ﬁn heat sinks. Recent advances in semiconductor technology have led to a sig- Meanwhile, many studies have also dealt with the pin-ﬁn heat sinkniﬁcant increase in power densities encountered in microelec- subject to an impinging jet. A numerical simulation of the three-tronic equipment [1,2]. Accordingly, effective cooling technology dimensional ﬂow and temperature ﬁeld was performed by Satheis essential for reliable operation of electronic components [3,4]. et al. . Ledezma et al.  presented a simpliﬁed numerical modelVarious types of cooling systems have been developed as thermal to obtain correlations for the optimal ﬁn spacing of pin-ﬁn heatsolutions. In this regard, a heat sink is the most widely used type sinks. Kondo et al.  suggested a semi-empirical zonal modelof cooling device and has received much attention due to its for a pin-ﬁn heat sink. The ﬂow behavior and heat transfer duecost-effective performance. Two common types of heat sinks to air jet impingement on pin-ﬁn heat sinks was experimentallywhich are widely used in the industry are plate-ﬁn heat sinks studied by Issa and Ortega .and pin-ﬁn heat sinks. The former offers simple design and easy As noted above, the plate-ﬁn and pin-ﬁn heat sinks are bothfabrication, while the latter has an advantage of providing higher commonly used and studied as cooling solutions for electronicheat transfer coefﬁcient at the expense of higher production cost. equipment. Then, an obvious question is which type of heat sinkGiven that the two types of heat sinks have respective strengths, has better performance? To date, several researchers have beena considerable body of research has been concentrated on both interested in this question and sought proper answers. Kondoconﬁgurations. et al. used a zonal approach in optimizing the thermal performance Numerous studies have focused on the plate-ﬁn heat sink sub- of plate-ﬁn and pin-ﬁn heat sinks under the constraint of ﬁxedject to an impinging ﬂow. Biber  numerically studied a number pumping power and heat sink size. They showed that optimizedof different combinations of channel parameters and presented a plate-ﬁn heat sinks provide 40% lower thermal resistances com-correlation for the average Nusselt number for a plate-ﬁn heat sink. pared to optimized pin-ﬁn heat sinks . Li et al.  comparedKondo et al.  used a semi-empirical zonal approach for deter- thermal resistances of several plate-ﬁn and pin-ﬁn heat sinks withmining the thermal resistance as well as the pressure drop for a various ﬁn heights and Reynolds numbers experimentally usingplate-ﬁn heat sink. In their model, the heat sink was divided into infrared thermography. Contrary to the results presented by Kondosix regions, each of which was solved separately. Duan and Muz- et al., they concluded that the thermal performance of the pin-ﬁn heat sink is superior to that of the plate-ﬁn heat sink. As indicated * Corresponding author. Tel.: +32 350 3043; fax: +32 350 8207. by these contradictory results, the selection between the plate-ﬁn E-mail address: firstname.lastname@example.org (S.J. Kim). and pin-ﬁn heat sinks is still not obvious. Furthermore, previous0017-9310/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2009.02.041
D.-K. Kim et al. / International Journal of Heat and Mass Transfer 52 (2009) 3510–3517 3511 Nomenclature a wetted area per volume Greek symbols c heat capacity of ﬂuid e porosity (wc/(wc + ww)) h heat transfer coefﬁcient based on one-dimensional bulk g similarity variable mean temperature geff ﬁn efﬁciency H channel height l viscosity k thermal conductivity h dimensionless temperature K permeability q density L length of heat sink (=width of heat sink) W stream function L* dimensionless length of heat sink (L/H) p pressure Subscripts PPump pumping power bm bulk mean PÃPump dimensionless pumping power (PPump =l3 qÀ2 HÀ1 ) f f f ﬂuid q heat transfer rate in inlet R total thermal resistance opt optimized T temperature pin pin-ﬁn heat sink u,v,w velocity plate plate-ﬁn heat sink U,V,W characteristic velocity s solid v0 inlet velocity w wall wc channel width wÃc dimensionless channel width (wc/H) Superscripts ww ﬁn thickness f averaged value over the ﬂuid region x,y,z Cartesian coordinate system s averaged value over the solid region hi averaged value hTib,f one-dimensional bulk mean temperature for the ﬂuid phaseworks did not compare the optimized plate-ﬁn and pin-ﬁn heat connected to a manometer (FCO510, Furness). To measure the tem-sinks systematically for various pumping powers and the heat sink peratures at the base of the ﬁns, seven 30 gauge J-type thermocou-sizes. Kondo et al. did not show the effect of the size, while Li et al. ples (Omega) are mounted through 3 mm deep holes of a basedid not optimize the geometry of the heat sink. The objectives of the present paper are to eliminate ambiguityassociated with the selection of a heat sink type and to comparethe thermal performance of the optimized plate-ﬁn and pin-ﬁnheat sinks for various pumping powers and the heat sink sizes.For this comparison, experimental investigations are performedfor various ﬂow rates and channel widths. Then, from experimentaldata, we suggest a model based on the volume averaging methodfor predicting the pressure drop and the thermal resistance. Byusing the model, thermal resistances of the optimized plate-ﬁnand pin-ﬁn heat sinks are compared. Finally, a contour map, whichdepicts the ratio of the thermal resistances of the optimized plate-ﬁn and pin-ﬁn heat sinks as a function of dimensionless pumpingpower and dimensionless length, is presented. This study will helpengineers to choose a heat sink with better performance betweenthe two types of heat sinks subject to an impinging jet.2. Experiments In this section, the experimental portion of the present study isdescribed. Two types of ﬁn designs, plate-ﬁn and inline square pin-ﬁn heat sinks, have been tested as shown in Figs. 1(a), 2(a), and 3.The dimensions of the heat sinks are shown in Table 1. The testedheat sinks are made of aluminum alloy 6061 (k = 170 W/m K) andno additional surface treatment is applied. The general layout and a photo of the experimental apparatusare shown in Figs. 4 and 5, respectively. The gas ﬂows into the windtunnel from a pressure tank through a metering valve and a massﬂow meter. The ﬂow meter is used for measuring the ﬂow rate inthe wind tunnel (D08-8C, Sevenstar electronics). The wind tunnelwalls are made of acrylic (k = 0.86 W/m K). In order to measurethe pressure difference between the inlet and the outlet of the heatsink, a pressure tap is positioned on the wind tunnel wall and is Fig. 1. Schematic diagrams for plate-ﬁn heat sink subject to an impinging ﬂow.
3512 D.-K. Kim et al. / International Journal of Heat and Mass Transfer 52 (2009) 3510–3517 Fig. 3. Heat sinks used for experimental study: (a) plate-ﬁn heat sink, (b) pin-ﬁn heat sink. Table 1 Tested heat sinks. Fig. 2. Schematic diagrams for pin-ﬁn heat sink subject to an impinging ﬂow. Type of heat sink wc (mm) ww (mm) H (mm) L (mm) Plate-ﬁn 1 0.5 1 25 40 Plate-ﬁn 2 0.95plate having 5 mm thickness. The relative positions of thermocou- Plate-ﬁn 3 1.44ples for temperature measurement are shown in Fig. 4. The maxi- Plate-ﬁn 4 2 Plate-ﬁn 5 3.33mum heat sink base temperature is measured from readings of Pin-ﬁn 1 0.5 1 25 40seven thermocouples. An Agilent 34970A DAQ is used to transport Pin-ﬁn 2 0.95data measured by the thermocouples. From the analysis of the Pin-ﬁn 3 1.44experimental uncertainties, maximum uncertainties for measure- Pin-ﬁn 4 2 Pin-ﬁn 5 3.33ment of the pressure drop and the temperature are estimated tobe 5% and 2%, respectively. The heat sinks are heated by electricalheaters fabricated using stainless steel with 0.25 mm thicknesssandwiched by KaptonÒ ﬁlms. To reduce heat loss, the bottom of 3. Mathematical modeling using the volume averagingthe heater is insulated with Teﬂon. The heat loss is determined approachfrom a surface temperature measurement of the test section. Tocalculate the net heat ﬂux at the bottom of the heat sink, the esti- In the present study, detailed information, such as velocity andmated heat loss is subtracted from the electric power supplied to temperature distributions, is of secondary importance. Instead wethe heater. The heat supplied from the power supply is calculated are concerned with estimating the macroscopic aspects of a prob-by measuring the voltage drop and the current through the heater. lem such as the pressure drop across a channel and the overallThe maximum heat loss is about 5% of the supplied heat. thermal resistance of a heat sink. Given that our main interest lies A typical test procedure is as follows: Flow rate is set by regu- in the macroscopic quantities, it would be wise to average the gov-lating the metering value. The heater is then powered up to a heat erning equations in the direction normal to the ﬂow. This approachload of 30 W and allowed to stabilize. The temperature distribution can lead to considerable simpliﬁcation, especially when the origi-at the base of the heat sink is measured until the change in the nal problem requires a great deal of time and money for obtainingtemperature is smaller than ±0.1 °C in a 2 min period. We con- a complete solution. Since a heat transfer device is modeled as aducted experiments in the laminar ﬂow regime with ﬂow rates ﬂuid-saturated porous medium in this technique, it is frequentlyof 50, 100, 150, and 200 Standard Liter per Minute (SLM). The called a porous medium approach. This approach has been success-pumping power is calculated by multiplying the measured ﬂow fully applied in thermal design and optimization of heat transferrate and pressure drop. The range of the pumping power (PPump) devices such as internally ﬁnned tubes and microchannel heatfrom 0.00082 to 0.096 W is covered in the present experiment. sinks with a parallel ﬂow [14,15]. In this section, a modeling tech-Each experiment was conducted four times. nique based on the volume averaging approach is proposed.
D.-K. Kim et al. / International Journal of Heat and Mass Transfer 52 (2009) 3510–3517 3513 The momentum and energy equations are given as follows. ! @u @u @u @p @2u @2u @2u q u þv þw ¼À þl þ þ ð1Þ @x @y @z @x @x2 @y2 @z2 ! @v @v @v @p @2v @2v @2v q u þv þw ¼À þl þ þ ð2Þ @x @y @z @y @x2 @y2 @z2 ! @T @T @T @2T @2T @2T qc u þ v þ w ¼k þ þ ð3Þ @x @y @z @x2 @y2 @z2 In the volume averaging approach, the governing equations for the averaged velocity and temperature are established by the averaging momentum equations and energy equation. Before averaging these equations, it is necessary to deﬁne the quantities to be averaged. Since the present system has a periodic structure in the z direction, the representative elementary volume (REV) for averaging can be visualized as a rectangular parallelepiped aligned parallel to the z-axis, as shown in Fig. 1(b). The averaged quantities over the ﬂuid and solid phases of the REV are then deﬁned, respectively, as follows: Z wc Z wc þww 1 1 h/if ¼ /dz; h/is ¼ /dz ð4Þ wc 0 ww wc For the averaged ﬂuid temperature, the one-dimensional bulk mean temperature is more suitable than the mean temperature to calcu- late the thermal performance of the heat sinks. The one-dimen- sional bulk mean temperature is deﬁned as Fig. 4. Layout of the experimental apparatus. R wc Tudz hTif ;b ¼ R wc 0 ð5Þ 0 udz As pointed out by Slattery , some information is lost when gov- erning equations are averaged: in the present case, the dependence of the velocity and temperature distributions in the averaging direc- tion is lost. For the present conﬁguration, the lost information is recovered with an approximation. For this, it is assumed that the velocity and temperature proﬁles for the ﬂuid in the averaging direction are similar to those of a Poiseuille ﬂow between two inﬁ- nite parallel plates. This assumption is similar to the assumption made for the proﬁle shape function in obtaining integral solutions for a laminar boundary layer ﬂow . The velocity and tempera- ture distributions for the Poiseuille ﬂow in this conﬁguration can be easily determined to be z z u ¼ 6huif Á 1À ð6Þ wc wc 4 3 ! 420 1 z 1 z 1 z T¼ ðhTib;f À hTis Þ À þ þ hTis ð7Þ 17 6 wc 3 wc 6 wc Fig. 5. Photo of the experimental apparatus. The governing equations for the averaged velocity and temperature are established by averaging the momentum and energy equations with Eqs. (6) and (7). x directional momentum equation:3.1. Plate-ﬁn heat sink ! 6 @huif @huif @hpif qf huif þ hv if þ The problem under consideration is an impinging ﬂow through 5 @x @y @x !a plate-ﬁn heat sink as shown in Fig. 1(a) and (b). The bottom elf @ 2 huif @ 2 huifsurface is kept constant at a high temperature. Air impinges on ¼À huif þ l þ ð8Þ K @x2 @y2the heat sink along the y-axis and then ﬂows parallel to x-axis.Air, employed as a coolant, passes through the heat sink, thusremoving the heat generated by the component attached at the y directional momentum equation:bottom of the heat sink substrate. In analyzing this problem, ! 6 @hv if @hv if @hpifthe ﬂow is assumed to be steady and laminar. In addition, all qf huif þ hv if þ 5 @x @y @ythermal properties are evaluated at the ﬁlm temperature of ﬂuid. !In addition, it is assumed that the aspect ratio of the channel is el 2 @ hv if @ hv if 2higher than 1 and the solid conductivity is higher than the ﬂuid ¼À hv if þ l þ ð9Þ K @x2 @y2conductivity.
3514 D.-K. Kim et al. / International Journal of Heat and Mass Transfer 52 (2009) 3510–3517Energy equation for the ﬂuid phase: Partial differential governing equations (Eqs. (11) and (23)–(25)) ! can be transformed into an ordinary differential equation with a b;f b;f @hTi @hTi similarity variable. The similarity variable g, the stream functioneqf cf huif þ hv if @x @y W, and the dimensionless temperatures are deﬁned as ! sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s b;f @hTib;f @hTib;f v 0 qf ¼ haðhTi À hTi Þ þ ekf þ ð10Þ g¼y ð26Þ @x2 @y2 Hlf sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃEnergy equation for the solid phase: v 0 lf W¼ FðgÞx ð27Þ @hTis @hTis H qfð1 À eÞks 2 þ 2 þ haðhTib;f À hTis Þ ¼ 0 ð11Þ @x @y hTib;f À T w hb;f ðgÞ ¼ ð28Þwhere e, K, h, and a are the porosity, permeability, heat transfer T in À T wcoefﬁcient, and wetted area per volume, respectively. hTis À T w hs ðgÞ ¼ ð29Þ wc T in À T we¼ ð12Þ ww þ wc With Eqs. (26)–(29), the governing equations (Eqs. (11) and (23)– 2 (25)) and the boundary conditions (Eqs. (16) and (17)) are given asa¼ ð13Þ ww þ wc
!À1 6 00 6 e H lf FF þ ð1 À F 0 F 0 Þ þ ð1 À F 0 Þ ¼ 0 ð30Þ @u
z¼0 17wc ð1 À eÞks 00 hs þ haðhb;f À hs Þ ¼ 0 ð32Þ HlfBoundary conditions are given as F ¼ 0; hs ¼ 0 for g ¼ 0 ð33Þ f f f s sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃhui ¼ hv i ¼ 0; hTi ¼ hTi ¼ T w for y ¼ 0 ð16Þ 0 v 0 Hqf F 0 ¼ 1; hb;f ¼ 1; hs ¼ 0 for g ¼ ð34Þ f f @hTis lfhv i ¼ À v 0 ; hTi ¼ T in ; ¼ 0 for y ¼ H ð17Þ @y By solving Eq. (30), the following equations for the velocity distribu-Eqs. (8)–(10) can be further simpliﬁed. For this, it is assumed that tion are obtained.the aspect ratio of the channel is much higher than 1 and the solidconductivity is higher than the ﬂuid conductivity. F¼g ð35Þ @W v 0 @W v0H; L ) wc ; ks ) kf ð18Þ u¼ ¼ x; v ¼À ¼À y ð36Þ @y H @x HEstimating the order of magnitude of each term appearing on the In general, Eqs. (31) and (32) cannot be solved analytically. How-right-hand side of Eqs. (8) and (9), we have ever, when we assume the ﬁn efﬁciency is 1, the following equa- Uf Uf Uf tions for the temperature distribution can be obtained.lf ) lf ; lf ð19Þ w2 c L2 H2 0 1eqhac H Vf Vf Vf f f v0lf ) lf ; lf 2 ð20Þ B g C w2 c L2 H hs ¼ 0; hb;f ¼ @qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃA ð37ÞCombining Eqs. (10) and (11), we have v 0 Hqf =lf ! y eqhac H @hTib;f f @hTib;f @hTis s b;f f f v0eqf cf hui þ hv if ¼ ð1 À eÞks hTi ¼ T w ; hTi ¼ ðT in À T w Þ þ Tw ð38Þ @x @y @x2 H @hTib;f @hTis @hTib;f Finally, the pressure drop between the inlet and the outlet of the þ ekf 2 þ ð1 À eÞks 2 þ ekf ð21Þ heat sink and the thermal resistance are obtained, respectively, @x @y @y2 from Eqs. (36) and (38).Estimating the order of magnitude of each term appearing on the Z L=2 Zright-hand side of Eq. (21), it follows that 1 1 H f Dpplate ¼ hpify¼H dx À hpix¼L=2 dy LÀL=2 H 0ks T kf T ks T kf T ! ! 2 ) 2 ; 2 ) ð22Þ 1 L2 2 elf H 1 L2 1L L H H2 ¼ qv 02 À þ v0 þ ð39Þ 10 H2 5 K 12 H2 3From the relations shown in Eqs. (19), (20), and (22), the simpliﬁed 0 0 !À1 11À1forms of Eqs. (8)–(10) are given as T w À T in @ 2@ haH AA Rplate ¼ ¼ eqf cf v 0 L 1 À 1 þ ð40Þ ! q eqf cf v 0 f f f6 @hui @hui @hpi elf f q huif þ hv if þ ¼À hui ð23Þ5 f @x @y @x K When the ﬁn efﬁciency is not 1, the approximate value of the ther- ! mal resistance is given from Eq. (40) by using the ﬁn model as f f f6 @hv i @hv i @hpi el q huif þ hv if þ ¼ À hv if ð24Þ follows:5 f @x @y @y K 0 0 ! !À1 11À1 @hTi b;f @hTi b;f geff haHeqf cf huif þ hv if ¼ haðhTis À hTib;f Þ ð25Þ Rplate ¼ @eqf cf v 0 L2 @1 À 1 þ AA ð41Þ @x @y eqf cf v 0
D.-K. Kim et al. / International Journal of Heat and Mass Transfer 52 (2009) 3510–3517 3515 geff haHwhere geff is the ﬁn efﬁciency. On the other hand, when eqf cf v 0 ( 1, Eq. (47) becomes qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ha tanh H ks ð1ÀeÞ T w À T in Rpin ¼ ¼ ðgeff haHpR2 ÞÀ1 0 ð51bÞgeff ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð42Þ q ha H ks ð1ÀeÞ Eqs. (51a) and (51b) correspond to low and high pumping-powerWhen geff haH ) 1, Eq. (41) becomes conditions, respectively. eqf cf v 0 The value of the present model lies in its simplicity. The pres-Rplate ¼ ðeqf cf v 0 L2 ÞÀ1 ð43aÞ sure drop and thermal resistance for the plate-ﬁn heat sink are eas- geff haH ily obtained by solving Eqs. (39) and (41), respectively, togetherOn the other hand, when eqf cf v 0 ( 1, Eq. (41) becomes with Eqs. (44) and (45). On the contrary, the model suggested by 2 À1 Kondo et al.  consists of dozens of equations and requires te-Rplate ¼ ðgeff haHL Þ ð43bÞ dious calculation process. However, the applicability of the sug-Eqs. (43a) and (43b) correspond to low and high pumping-power gested model is limited because the permeability K and the heatconditions, respectively. transfer coefﬁcient h are obtained from experimental data. The Eqs. (39) and (41) are derived without adopting correlations for present model is applicable forthe friction factor and the Nusselt number for a parallel ﬂow. In-stead, it is assumed that the velocity and temperature proﬁles for 5 Â 109 PÃ Pump 1011 ; 1 LÃ 2 ð52Þthe ﬂuid in the averaging direction are similar to those of a Poiseu- for which the experiment is conducted.ille ﬂow between two inﬁnite parallel plates, as shown in Eqs. (6)and (7). This assumption is reasonable but is not exact, because 4. Results and discussionthe velocity and temperature proﬁles for an impinging ﬂow arenot exactly the same as those for the ﬂow between two inﬁnite The pressure drop and the thermal resistance obtained fromparallel plates. Therefore, the permeability K and the heat transfer experiments are compared with those from the proposed model.coefﬁcient h presented in Eqs. (14) and (15) should be modiﬁed in As shown in Fig. 6, the experimental results of the pressure droporder to better predict the pressure drop and the thermal resis-tance. In the present study, the appropriate values for K and h 100are determined from the experimental data for the pressure dropand the thermal resistance with Eqs. (39) and (41), respectively. Fi-nally, the modiﬁed parameters obtained from experimental inves-tigations are given as follows. 10 Pressure drop (Pa) ew2 cK¼ ð44Þ 12ðð0:289 þ 16:0wÃ Þ þ ð0:00263 þ 0:133wÃ ÞRewc Þ c c 70kf Ã 2 À3h¼ ðð17:4wc À 63:4ðwc Þ Þ þ ð7:67 Â 10 ÞwÃ Rewc Þ Ã c ð45Þ 1 17wc Experimental data From the model wc=3.3mm3.2. Pin-ﬁn heat sink wc=2.0mm 0.1 wc=1.44mm The ﬂuid ﬂow in the pin-ﬁn heat sink is axisymmetric in nature wc=0.95mmrather than two-dimensional. In the present study, the pin-ﬁn heat wc=0.50mmsink is modeled as an equivalent porous cylinder which has the 0.001 0.002 0.003same porosity, wetted surface, and base area. The top view for 3 Flow rate (m /s)the equivalent porous cylinder is shown in Fig. 2(b). By solvingEqs. (11) and (23)–(25) in cylindrical coordinates, the pressure (a) Plate-fin heat sinksdrop between the inlet and the outlet of the heat sink and the ther-mal resistance are given, respectively, as 100 ! ! 3 R2 2 elf H 1 R2 1Dppin ¼ qv 2 0 0 À þ v0 0 þ ð46Þ 40 H2 5 K 8 H2 3 0 0 10 !À1 11À1 Pressure drop (Pa) geff haH AARpin ¼ @eqf cf v 0 pR0 1 À 1 þ 2@ ð47Þ eqf cf v 0 1where 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Experimental data From the model ww 4ww wc=3.3mme¼1À ; ; R0 ¼ L2 =p a¼ ð48Þ wc=2.0mm ww þ wc ðww þ wc Þ2 0.1 wc=1.44mm ew2 c wc=0.95mmK¼ ð49Þ wc=0.50mm 12ðð0:232 þ 10:4wcÃ Þ þ ð0:000376 þ 0:0840wÃ ÞRe Þ c wc 70kf 0.001 0.002 0.003h¼ ðð9:96wÃ À 28:3ðwÃ Þ2 Þ þ ð6:41 Â 10À4 ÞRewc Þ c c ð50Þ 3 17wc Flow rate (m /s) geff haHWhen eqf cf v 0 ) 1, Eq. (47) becomes (b) Pin-fin heat sinksRpin ¼ ðeqf cf v 0 pR2 ÞÀ1 0 ð51aÞ Fig. 6. Pressure drop (L = 40 mm, H = 25 mm, ww = 1 mm).
3516 D.-K. Kim et al. / International Journal of Heat and Mass Transfer 52 (2009) 3510–3517are in close agreement with the predictions from the present mod- sults for the thermal resistance obtained from the experiments andel within a maximum error of 31% for the plate-ﬁn heat sinks and the model match well within a maximum error of 20% for the22% for the pin-ﬁn heat sinks, respectively. As shown in Fig. 7, re- plate-ﬁn heat sinks and 13% for the pin-ﬁn heat sinks, respectively. The close agreement between experiment and prediction is be- 10 cause the permeability and heat transfer coefﬁcient are deter- Experimental data From the model mined from the experimental side of the work. From Figs. 6 and wc=3.3mm 7, it is concluded that Eqs. (44), (45), (49), and (50) are reasonable wc=2.0mm choices for the permeability and the heat transfer coefﬁcient. wc=1.44mm Using the models for the plate-ﬁn and pin-ﬁn heat sinks, their Thermal resistance ( C/W) wc=0.95mm thermal resistances are calculated. Then, we numerically optimize wc=0.50mm o the dimensions for the plate-ﬁn and pin-ﬁn heat sinks by using the gradient descent algorithm. The optimized structures for which the thermal resistances are minimized under the ﬁxed pumping power are then obtained, as listed in Table 2. The constraint for the ﬁxed pumping power physically means that the power required to drive 1 the coolant through the heat sinks is ﬁxed. Finally, by comparing the thermal resistances of the optimized plate-ﬁn and pin-ﬁn heat sinks, a contour map for an impinging ﬂow conﬁguration is pre- sented in Fig. 8. Fig. 8 depicts the ratio of the thermal resistances of the optimized plate-ﬁn and pin-ﬁn heat sinks Ropt,plate/Ropt,pin as 0.001 0.002 0.003 a function of dimensionless pumping power and dimensionless 3 length. This ratio was successfully used for comparing thermal per- Flow rate (m /s) formances of plate-ﬁn and pin-ﬁn heat sinks with parallel-ﬂow (a) Plate-fin heat sinks conﬁguration . In Fig. 8, in the region where the ratio is above 1, the thermal resistance of the optimized pin-ﬁn heat sink is smal- 10 ler than that of the optimized plate-ﬁn heat sink. On the other Experimental data From the model hand, the thermal resistance of the optimized plate-ﬁn heat sink wc=3.3mm is smaller than that of the optimized plate-ﬁn heat sink when wc=2.0mm the ratio is below 1. wc=1.44mm The contour map for an impinging ﬂow conﬁguration indicates Thermal resistance ( C/W) wc=0.95mm the following: Optimized pin-ﬁn heat sinks have lower thermal wc=0.50mm resistances than optimized plate-ﬁn heat sinks only when the o dimensionless pumping power is small and the dimensionless length of heat sinks is large. In other words, the pin-ﬁn heat sink shows better performance than the plate-ﬁn heat sink as the pumping power per unit of heat sink volume decreases. It is worth 1 mentioning that Fig. 8, which describes the comparison of the ther- mal performance of the optimized plate-ﬁn and pin-ﬁn heat sinks, is helpful in determining the best designs for heat sinks. For the plate-ﬁn heat sink, impinging air ﬂows out only along the y-axis. On the contrary, air can ﬂow out in every direction for 0.001 0.002 0.003 the pin-ﬁn heat sink. Hence, the ﬂow impedance of the optimized 3 pin-ﬁn heat sink is typically lower than that of the optimized plate- Flow rate (m /s) ﬁn heat sink, and the inlet velocity of the optimized pin-ﬁn heat (b) Pin-fin heat sinks sink is higher than that of the optimized plate-ﬁn heat sink under the ﬁxed pumping-power condition, as shown in Fig. 9. From Eqs. Fig. 7. Thermal resistance (L = 40 mm, H = 25 mm, ww = 1 mm). (43a) and (51a), it can be shown that Ropt,plate/Ropt,pin is proportionalTable 2Comparison of geometries and thermal resistances for optimized heat sinks.Dimensionless Dimensionless Type of Optimum channel Optimum ﬁn Optimum thermalpumping power length heat sink width (mm) thickness (mm) resistance (K/W) P pump Ll3 qÀ2 HÀ1 H wc,opt ww,opt Ropt3 Â 1010 1 Plate-ﬁn 0.763 0.202 1.76 Pin-ﬁn 0.716 0.807 1.936 Â 1010 1 Plate-ﬁn 0.639 0.185 1.48 Pin-ﬁn 0.578 0.677 1.65 109 Â 10 1 Plate-ﬁn 0.576 0.175 1.34 Pin-ﬁn 0.511 0.614 1.50 103 Â 10 2 Plate-ﬁn 1.15 0.233 0.747 Pin-ﬁn 1.09 1.10 0.7346 Â 1010 2 Plate-ﬁn 0.977 0.217 0.617 Pin-ﬁn 0.881 0.917 0.6209 Â 1010 2 Plate-ﬁn 0.884 0.207 0.553 Pin-ﬁn 0.779 0.830 0.562H = 25 mm, qf = 1.1774 kg/m3, lf = 1.8462 Â 10À5 kg/m s, cf = 1.0057 Â 103 J/kg K, kf = 0.02624 W/m K, ks = 171 W/m K.
D.-K. Kim et al. / International Journal of Heat and Mass Transfer 52 (2009) 3510–3517 3517 5. Conclusion In this paper, we compared thermal performances of plate-ﬁn and pin-ﬁn heat sinks subject to an impinging ﬂow. Experimental investigations were performed for various ﬂow rates and channel widths. From experimental data, we suggested a model based on the volume averaging method for predicting the pressure drop and the thermal resistance. Using the proposed model, thermal resistances of the optimized plate-ﬁn and pin-ﬁn heat sinks were compared under ﬁxed pumping-power conditions. It was shown that optimized pin-ﬁn heat sinks possess lower thermal resistances than optimized plate-ﬁn heat sinks when dimensionless pumping power is small and the dimensionless length of heat sinks is large. On the contrary, the optimized plate-ﬁn heat sinks have smaller thermal resistances when dimensionless pumping power is large and the dimensionless length of heat sinks is small. Acknowledgements This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the National Research Lab. Program funded by the Ministry of Science and Technology (No.Fig. 8. Ratios of the thermal resistances of the optimized plate-ﬁn and pin-ﬁn heat M1060000022406J000022410).sinks (ks/kf = 6.4 Â 103, Pr = 0.707). References  S. Oktay, R.J. Hannemann, A. Bar-Cohen, High heat from a small package, Mech. Eng. 108 (3) (1986) 36–42.  A. Bar-Cohen, Thermal management of electric components with dielectric liquids, in: J.R. Lloyd, Y. Kurosaki (Eds.), Proceedings of ASME/JSME Thermal Engineering Joint Conference, vol. 2, 1996, pp. 15–39.  F.P. Incropera, Convection heat transfer in electronic equipment cooling, J. Heat Transfer 110 (1988) 1097–1111.  W. Nakayama, Thermal management of electronic equipment: a review of technology and research topics, Appl. Mech. Rev. 39 (12) (1986) 1847–1868.  C.R. Biber, Pressure drop and heat transfer in an isothermal channel with impinging ﬂow, IEEE Trans. Components Pack. Manuf. Technol. A 20 (1997) 458–462.  Y. Kondo, M. Behnia, W. Nakayama, H. Matsushima, Optimization of ﬁnned heat sinks for impingement cooling of electronic packages, J. Electronic Pack. (1998) 259–266.  Z. Duan, Y.S. Muzychka, Experimental investigation of heat transfer in impingement air cooled plate ﬁn heat sinks, J. Electronic Pack. 128 (2006) 412–418.  S. Sathe, K.M. Kelkar, K.C. Karki, C. Tai, C. Lamb, S.V. Patankar, Numerical prediction of ﬂow and heat transfer in an impingement heat sink, J. Electronic Pack. 119 (1997) 58–63.  G. Ledezma, A.M. Morega, A. Bejan, Optimal spacing between pin ﬁns with impinging ﬂow, J. Heat Transfer 118 (1996) 570–577.  Y. Kondo, H. Matsushima, T. Komatsu, Optimization of pin ﬁn heat sinks for impingement cooling of electronic packages, J. Electronic Pack. 122 (2000)Fig. 9. Ratios of the thermal resistances, inlet velocities, and heat transfer 240–246.coefﬁcients of the optimized plate-ﬁn and pin-ﬁn heat sinks (ks/kf = 6.4 Â 103,  J.S. Issa, A. Ortega, Experimental measurements of the ﬂow and heat transfer ofPr = 0.707, L/H = 2). a square jet impinging on an array of square pin ﬁns, J. Electronic Pack. 128 (2006) 61–70.  Y. Kondo, H. Matsushima, S. Ohashi, Optimization of heat sink geometries for impingement air-cooling of LSI packages, Heat Transfer Asian Res. 28 (1999) 138–151.to v0opt,pin/v0opt,plate when the pumping power per unit of heat sink  H.-Y. Li, S.-M. Chao, G.-L. Tsai, Thermal performance measurement of heatvolume is small. Therefore, the pin-ﬁn heat sink shows better per- sinks with conﬁned jet by infrared thermography, Int. J. Heat Mass Transfer 48formance than the plate-ﬁn heat sink when the pumping power (2005) 5386–5394.  S.J. Kim, J.W. Yoo, S.P. Jang, Thermal optimization of a circular-sectored ﬁnnedper unit of heat sink volume is small. On the other hand, Eqs. tube using a porous medium approach, J. Heat Transfer 124 (2002) 1026–1033.(43b) and (51b), it can be shown that Ropt,plate/Ropt,pin is proportional  D.-K. Kim, S.J. Kim, Averaging approach for microchannel heat sinks subject toto hopt,pin/hopt,plate when the pumping power per unit of heat sink the uniform wall temperature condition, Int. J. Heat Mass Transfer 49 (2006) 695–706.volume is large. As shown in Fig. 9, the optimized plate-ﬁn heat  V.C. Slattery, Advanced Transport Phenomena, Cambridge University Press,sink has a higher heat transfer coefﬁcient than the optimized Cambridge, 1999.pin-ﬁn heat sink for high dimensionless pumping powers. There-  A. Bejan, Convective Heat Transfer, third ed., John Wiley Sons, Inc., New York,fore, the plate-ﬁn heat sink shows better performance than the 2004. pp. 42–49.  S.J. Kim, D.-K. Kim, H.H. Oh, Comparison of ﬂuid ﬂow and thermalpin-ﬁn heat sink when the pumping power per unit of heat sink characteristics of plate-ﬁn and pin-ﬁn heat sinks subject to a parallel ﬂow,volume is large. Heat Transfer Eng. 29 (2008) 169–177.