the thermal resistance of pin fin heat sinks in transverse flow

grid CFD simulations will provide a more accurate prediction of fullyducted heat sink behavior than most practical CFD models of the completeheat sink.Nu , f Ratio10.90.80.70.60.5Figure 4. Grid-sensitivity of friction factor and NusseltnumberA CFD model of the single row of pins with the grid density of 225from Figure 4 was used to develop the building block correlations forthe analytical model. While not entirely grid independent, the forthcomingdiscussion will show that the CFD models strike a reasonable balancebetween providing good agreement with empirical data andeconomy of computational effort.Results obtained from the CFD simulations include the per-row frictionfactor, Nusselt number, and the thermal wake function. Figures 5aand 5b respectively show the per pin friction factor results for squareand non-square arrays using the same format as Zukauskas[12]. Forsquare arrays the per pin friction factor for each value of X Lis computedas defined in the nomenclature. The friction factor ratio χ fornon-square arrays is computed by dividing the friction factor result forthe X T≠ X Lsimulation by the result for the square array with the sameX L. As shown in Figure 5(a), the CFD results for square arrays are ingood agreement with the empirical data compiled by Zukauskas over therange X L= X T= 1.5…2.5 . Results shown in Fig. 5(b) for the nonsquarearrays agree quite well with the empirical data for X T< X Lbutare low for X T> X L. For calculating the friction factor we use equation(4) and the coefficients given in Table 3. This correlation is based onempirical data from Zukauskas [12] except for the last row in the tablewhich is based on pin fin heatsink data from Jonsson and Moshfegh[13].where X =Nuf0 100 200 300 400 500Grid Density1/mf = X – 0.754 ⋅ (( C1Re – 0.8 ) m + C2 m ) 10 < Re < 3000( X T– 1)-------------------( X L– 1)f/χ10TABLE 3. Coefficients for the friction factorcorrelationX L C1 C2 m1.5 48 0.30 2.52.0 23 0.25 2.02.5 12 0.18 1.74.3 9 0.10 1.11Zukauskas (XL=1.5)Zukauskas (XL=2.0)Zukauskas (XL=2.5)0.110 100 1000 10000(a)Reχ(b)10.01.00.10.1 1 10(X T-1)/(X L-1)Re=2000Re=200Re=20Figure 5. Friction factor results from single-row CFDmodelFigure 6 compares the Nusselt numbers at the fourth pin in the flowdirection with empirical correlations over the Reynolds number rangefrom 10 to 3000 and the combinations of X L and X T listed in Table 2.The predictions agree very well with the empirical correlations ofZukauskas over the whole range studied except for the intermediaterange 100 < Re < 1000 where the correlation equation appears to below probably due to a typographical error in [12]. The CFD results arewell represented by slightly modifying and blending the correlationsprovided by Zukauskas for Reynolds number ranges above and belowthis intermediate range using the following equation also shown in Figure6.Nu = (( 0.85Re 0.4 Pr 0.36 ) 10 + ( 0.26Re 0.63 Pr 0.36 ) 10 ) 110 /(4)4

This correlation is used for all the pins in the array based on a pin bypin Nusselt number study at Re ~ 750 which showed that the fourth pindata was representative of all the pins in the array within +−2 percentexcept for the first pin which was only 5 percent higher. The reasonableagreement seen between CFD and empirical results for the friction factorand Nusselt number provides support for the thermal wake functiondetermined from the same CFD simulations.Nu10010XT=2 XL=2XT=2 XL=2.5XT=2.5 XL=2XT=1.5 XL=2XT=2.5 XL=2.5XT=1.5 XL=1.5XT=2 XL=1.5ZukauskasEquation (5)have higher values of . This means that with increasing transversepitch the bulk temperature decreases much faster (as 1/S T ) than the adiabatictemperature rise in the wake of the heated pin. This is just whatwould be expected when fluid convection dominates. Because our CFDresults are not entirely grid independent the reader should note that thetrue magnitude of is somewhat greater than the values reported here.θ13.53.02.52.0θ 1θ 1XT=2.5 XL=2XT=2, XL=2XT=1.5 XL=2Equation (6)1.5110 100 1000 10000ReFigure 6. Nusselt number results from single-row CFDsimulationsThe wake function is modeled in two parts, the first accounts for theadiabatic temperature rise of the first pin downstream of the heated pinand the second part accounts for the decay of the wake function downstreamfrom the first pin. In order to arrive at a suitable correlation welooked for asymptotes for the wake function at low and high Reynoldsnumbers and then blended the two together using the method ofChurchill and Usagi [15]. Following Zukauskas, we first addressedsquare arrays where X T= X Land then looked at deviations for configurationswhere X T≠ X L. The resulting correlations are inspired by theunderlying physics of the data rather than a multi-parameter fit to a largeset of data. The wake function for the first pin is well represented by thefollowing equation which is compared in Figure 8 with CFD results forthree configurations which represent the upper and lower bound valuesof θ 1in our CFD study.θ 11 ϕ 1 1 0.015Reϕ 1 0.71.00 500 1000 1500 2000 2500 3000ReFigure 7. Wake function at first pin downstream of heatedpinThe decay of the thermal wake at the downstream pins is well representedby the following correlation function for square and non-squarearrays.( θ i– 1)( ------------------ =θ 1– 1)e ni–1– ( ) + ( 1 – X)The value of the exponent n may be determined for a given longitudinalpitch by interpolation from the values tabulated in Table 4.TABLE 4. Coefficients for Wake DecayX L n1.5 1.702.0 1.052.5 0.904.3 0.85(7)where=+[ + ( / ) 2 ] – 0.7/2{ –}ϕ = ( X L– 1) ( 1.2X 2 – 0.64X + 0.48)As shown in Figure 7, the correlation equation provides a good fitfor the data over the whole range studied. The figure shows that thevalue of θ 1starts at 1.0 (i.e. adiabatic temperature rise is equal to thebulk temperature rise) at low Re where heat diffusion is dominant,increases sharply up to an Re ~ 200 as convection becomes more andmore dominant and then more slowly at Re > 500 towards a high Reasymptote of 1 + ϕ . A comparison between the data for the three differentgeometries shows that configurations with a larger transverse pitch(5)(6)Heat Sink Model FrameworkWe consider a heat sink composed of an array of circular pin fins asshown in Figure 2. The length L of each pin is constant in the z direction.The airflow is assumed to approach the pin fin array in the x-directionwith a uniform velocity and temperature and to remain twodimensionalwithin the array (no velocity in the z-direction). Thermalwake effects are limited to fins situated along the same flow stream, i.e.within the same y and z segments.5

.U aT inSegmentNumberOdzT fkPin Number 1 2 ... i ... N21T 0q kFigure 8. Heat transfer from an interior fin in the arrayWe examine heat transfer from a fin within the array with the aid ofFigure 8. The fin is divided into several segments along its length startingfrom the base. Heat transfer from each segment of the fin is drivenby the same overall temperature potential; the difference between thebase temperature of the fin and the inlet temperature of the air. Thispotential can be separated into three distinct parts comprising the conductionwithin the fin, convection from the fin surface and upstreamheatup of the air as follows:These separate pieces can be written in terms of the heat transferredfrom each segment of the fin. For segment k on the i th fin in the flowdirection, these terms become:z kT adT 0– T in= ( T 0– T f) + ( T f– T ad) +( T ad– T in)(8)path. The path is constrained only in the sense that it represent a flowstreamtube which must be known apriori.Equations (10) through (12) can be substituted in equation (9) toyield the heat balance equation for each segment of each fin∑1 ≤ l≤kz l⎛-----------⎝λ mA f⎠, 1+ ------------ + --------------------------- q (12)hPdz ρU ac pS Tdz ∑ l, kθ i–l– T 0= T inOne more equation is needed to specify the heat input or temperatureboundary condition at the base of each fin as follows:(13)where is a switch with a value of either 1 for a specified heat inputboundary condition or 0 for a specified base temperature boundary conditionfor the i th pin.The above equations (13) and (14) comprise a set of N ⋅ ( O + 1)equations for a similar number of unknowns (q for O segments on eachof N fins plus T 0 for N fins). This set of equations can be written inmatrix form and solved using standard matrix inversion methods todetermine the heat flow and temperature distribution within the row offins. These equations were implemented in a commercial mathematicscode MathCad [16]. Correlation equations described earlier for the frictionfactor, Nusselt number and thermal wake for circular pin fins wereimplemented in the model. The solution results for q and T 0 can be usedwith equations (10) and (11) to determine the fin temperature and adiabatictemperature for each segment of each fin.S iq⎞ qi , lS iq∑lq i,lq ik1 ≤ l < iq q q+ ( 1–S i)T 0= S i Qi + ( 1–S i)Tb iT 0–T f=∑1 ≤ l ≤kz l⎛-----------⎝λ mA f⎠⎞ qi , l(9)Heat Transfer from Fin Segments to AirFin temperatures in ArrayT f–T ad=q i k------------ ,hPdz(10)1T ad– T in= --------------------------- qρU ac pS Tdz ∑ lk1 ≤ l < i,θ i–l(11)whereq i,kθ i – lz kis the heat transfer from segment k on pin iis the wake function on pin i due to heat transferred from pin lis the distance from the base to the center of segment k of the findz kis the length of segment k of the finAπDfis the cross-sectional area of the fin ⎛=---------2 ⎞⎝ 4 ⎠Note that value of T ad can be set equal to the bulk temperature by specifyingθ i–l= 1 . The adiabatic temperature rise expressed in equation(12) contains information on the path of the fluid flow through the pinfin array and the order in which the various pins are arranged along thatqFigure 9. Sample solution from a one-row pin fin arraymodel (D=1 mm, S T =S L =3 mm, L=25 mm, U a =4m/s, T a =0 C, Tb=100 C)Figure 9 presents a typical result from the model for a row with tenpins each with five segments. This model computes the solution in under2 seconds on a personal computer with an Intel P3 CPU running at 650MHz. As seen in Figure 9, this model provides a detailed accounting ofthe heat transfer rates and temperature distribution for each fin in therow. A check of the temperature distribution on the first pin showedexcellent agreement with 1-D models for conduction in a fin with a uniformconvective heat transfer coefficient from the fin surface to fluid ata uniform temperature. On downstream fins, the fluid temperatureT6

ecomes non-uniform so temperature predictions depart from the 1-Dfin model.Moving a step further we developed the model to represent a heatsink having a conducting base and a two dimensional array of fins. Heattransfer from each row of fins is treated as before but each fin is nowconnected to a heat sink base of finite thickness. Heat conduction withinthe base is solved using a standard finite volume approach [17] withboundary conditions supplied on the lower surface of the heat sink base.A simple version of base conduction was implemented by setting thesize of each finite volume equal to the volume of the base under eachfin, i.e., S T× S L× tb . The base temperature T 0 for a fin is now the basetemperature at the center of the finite volume below that fin. Thermalconductance terms connect the temperature in a control volume to adjacentcontrol volumes in the x and y directions and to the boundary conditionsspecified on the lower surface of the heat sink base. The heatconduction equation for the base control volume below the i th pin in thej th row can be written as follows:( q2C x+ 2C y+ S i, jCz)T + i j 0–λwhere the conductance terms are mS Ttb λ= -----------------, C mS Ltby= ----------------- , andC zAn additional resistance R ⎛ c =⎝, ,q ijk , ,k(14)is added to the term⎛-----------⎞in equation (13) to account for conduction between the position⎝λ mA f⎠at which T 0 is now computed and the base of the actual fin. This termincludes the constriction resistance at the base of the fin. The heat transferfrom the top surface of the heat sink base around the fins is includedin the convective heat transfer expression for the first segment of the fin.The heat transfer coefficient on the base area was assumed to be 0.5times the heat transfer coefficient on the surface of the fins based on datareported by Metzger et. al [18]. The constant of proportionality willmainly affect the thermal resistance predictions for heat sinks withsparse pin fin arrays where heat transfer from the wall area is significant.This wall heat transfer coefficient is very difficult to measure experimentallyand its dependence on the array geometry, aspect ratio (L/D) ofthe pins, Reynolds number etc. is not known sufficiently and is a candidatefor further study, maybe using CFD. The heat transfer from the fintips has been left out of the present model but it can be easily included inthe expression for the last segment of the fin if the fin tips are exposed toair flow.Extensions of the Heat Sink modelAlthough we have focused on a simple transverse flow aligned within-line rows of pin fins to develop the model, the approach presented∑C xT – i – 1, j,0C xT – C T C i + 1, j,0 y i , j – 1 , 0– T y ij , 1 ,=qS i, jQij2λ mS LS= --------------------- T.tbz l,+ ( 1 – S ij)C zTb i,j,qC xS Ltb---------------------- + --------------1 ⎞2λ mS TS K2λ mD⎠+ 0S There is readily extendable to non-orthogonal transverse flows includingimpinging flows. This would be done by partitioning the flow field intostreamtubes (see [7]) which delineate the corresponding thermal wakelinkages between segments, representing new effective pin rows, whichmay even be curved. Equation (12) provides the means to implementany “flow stream connectivity” implied by the flow streamtubes thatdescribe the flow path through the array. If the wake function correlationsare not available, a suitable first step is to set the wake function tounity.Other types of fins such as interrupted plate, elliptical, wing shapedetc. can be modeled using the same heat sink model by simply providingthe applicable correlation equations for Nusselt number and friction factor,e.g. those described by Muzychka and Yovanovich [19] for offsetstrip fin arrays.MODEL RESULTS AND DISCUSSIONThe heat sink model was compared against the experimental data ofJonsson and Moshfegh [13] for a heat sink consisting of a 9 x 9 array of10 mm long 1.5 mm diameter pin fins arranged on a square grid with apitch of 6.5 mm. We assumed a base thickness of 4 mm since this wasnot provided in [13]. Pressure drop and thermal resistance predictionswere made for a fully ducted configuration for approach velocities rangingfrom 1.7 to 9 m/s corresponding to a Reynolds number range fromabout 200 to 1100.CFD simulations were conducted using Icepak [14] for the aboveheat sink geometry using three levels of grid refinement. Even at the finestgrid which used ~450,000 control volumes the CFD results were notgrid independent. Further grid refinement was not conducted becausethe computational array sizes exceeded the physical memory on thedesktop PC (~512 MB). Each CFD simulation took 44 minutes to complete.Thermal Resistance(C/W)32.521.51R-Data [13]R-ModelR-CFD 449,320 gridsDP-Data [13]DP-ModelDP-CFD 449,320 grids0.500.00 3.00 6.00 9.00Approach Velocity Ua (m/s)Figure 10. Model predictions for a pin fin heat sinkcompared to experimental data of [13]200160120Figure 10 shows that the predictions from the model are in excellentagreement with the experimental data while the CFD simulations showconsiderable differences. For the model, the excellent agreement withthe pressure drop data is to be expected since the data itself was used todetermine the coefficients for the friction factor model (shown in Table3). However, the good agreement over the whole velocity range showsthe good correlation that is achieved between the Reynolds number andthe friction factor through equation (4).The excellent agreement between the model predictions and experimentaldata for thermal resistance validates the Nusselt number correla-8040Pressure Drop (Pa)7

tion used and provides confidence in the overall model including thechoice of the proportionality constant for the heat transfer coefficient onthe base surface around the fins. If the proportionality constant were setto 1.0 instead of 0.5, the predicted thermal resistance would decrease by6.5% at 1.7 m/s and by 10% at 9 m/s. The influence of the thermal wakefunction in this case is more significant. If the adiabatic temperature risewere replaced by the bulk temperature rise (by setting θ i – l= 1 ) in thecalculation of heat transfer from the fins, the predicted thermal resistancewould be reduced by 18% at 1.7 m/s and by 12% at 9 m/s airvelocity.CONCLUSIONSA physics based model for pin fin heat sinks in forced convectionhas been developed. Key empirical data and correlations for the pressuredrop and heat transfer in in-line arrays of circular pin fin arrays werevalidated through CFD simulations. New correlation equations weredeveloped to represent the empirical friction factor data. New knowledgeabout the thermal wake within the pin fin arrays was developedfrom the CFD simulations. Correlations to represent the thermal wakewere developed. An attempt was made to accurately represent within themodel framework the most relevant mechanisms governing heat transportwithin the heatsink structure including heat transfer from the finsand the fluid wetted wall of the base, heat spreading in the base and thermalwake effects due to fluid routing through the fin array. The modelwas implemented using the commercial mathematics software Mathcad.Predictions from the model were in good agreement with experimentaldata, much better than grid-limited CFD results for the full heat sink.The model provides a framework for putting existing literature correlationsand information developed from detailed fine-grid CFD simulationsto work in the design of pin fin heat sinks. At a minimum, themodel can be expected to predict heat sink performance similar to adetailed CFD model with a fraction of the data entry and computationaleffort. Better yet, the incorporation of empirical information allows thepresent model to capture complex physical effects which would requirehuge detailed CFD models to simulate from scratch each time. Last butnot least, the model provides detailed but easily interpreted resultsregarding heat transport and temperatures within the heat sink structure.It is anticipated that the model will become a valuable tool to enableconsistent performance comparisons and selection between varioustypes of pin fin and plate fin heat sinks.REFERENCES[1] Bejan, A., and Sciubba,E., 1992, “The optimal spacing of parallelplates cooled by forced convection,” Int. J. Heat Mass Transfer,Vol. 35, pp. 3259-3264[2] Bejan, A. and Morega, A. M., 1993, “Optimal arrays of pin fins andplate fins in laminar forced convection,” ASME Journal of HeatTransfer, Vol. 115, pp. 75-81.[3] Culham, J.R. and Muzychka, Y. S., 2001, “Optimization of plate finheat sinks using entropy generation minimization,” IEEE Trans.Comp. Packag. Technol., Vol 24, pp. 159-165.[4] Iwasaki, H., Sasaki, T., and Ishizuka, M., 1994, “Cooling performanceof plate fins for multichip modules,” in Proc. Fourth Intersoc.Conf. Thermal Phenomena in Electronic Systems, ITHERM1994, pp. 144-147.[5] Teertstra P.M., Yovanovich, M. M., Culham, J. R. and Lemczyk, T.F., 1999, “Analytical forced convection modeling of plate fin heatsinks,” in Proc. 15th Annu. IEEE Semicon. Thermal Meas. Manag.Symp., San Diego, CA, March 9-11, pp. 34-41.[6] S. Lee, “Optimum design and selection of heat sinks”, in Proc. 11thIEEE SEMI-THERM Symposium, 1995, pp. 48-54.[7] Holahan, M. F., Kang, S. S., and Bar-Cohen, A. 1996, “A flowstreambased analytical model for design of parallel plate heatsinks”,in Proc. 31st National Heat Transfer Conference, ASMEHTD-Vol. 329-7, pp 63-71.[8] Chen,C.L., Peppi, K., Day, S., Liao, C., 1998, “Thermal analysisdesign tool for parallel plate heatsinks”, in Proc. Sixth Intersoc.Conf. on Thermal Phenomena, ITHERM 1998, pp371-377.[9] H. Shaukatullah, W. R. Storr, B. J. Hansen, M.A. Gaynes, 1996,“Design and optimization of pin fin heat sinks for low velocityapplications”, 12th IEEE SEMI-THERM Symposium, pp.151-163.[10] Jonsson H. and Palm, B. 1996, “Experimental comparison of differentheat sink designs for cooling of electronics,” in Proc. 31stNational Heat Transfer Conference, ASME HTD-Vol. 329-7, pp27-34.[11] Dogruoz M. B., Urdaneta, M., Ortega, A. and Westphal, R. V.,2002, “Experiments and modeling of the hydraulic resistance of inlinesquare pin fin heat sinks with top bypass flow,” in Proc. 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W., 1984, “Effects of PinShape and Array Orientation on Heat Transfer and Pressure Loss inPin Fin Arrays,” Journal of Engineering for Gas Turbines andPower, Vol. 106, pp. 252-257.[19] Muzychka, Y. S. and Yovanovich, M. M., 1999, “Modeling the fand j Characteristics of the Offset Strip Fin Array,” ASME HTD-Vol. 364-1, pp 91-100.8