DRAFT ANALYSIS OF PUMPING REQUIREMENTS FOR MICROCHANNEL COOLING SYSTEMS. Vishal Singhal, Dong Liu and Suresh V. Garimella

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1 Proeedings of IPACK03 International Eletroni Pakaging Tehnial Conferene and Exhibition July 6 11, 003, Maui, Haaii, USA DRAFT InterPak ANALYSIS OF PUMPING REQUIREMENTS FOR MICROCHANNEL COOLING SYSTEMS Vishal Singhal, Dong Liu and Suresh V. Garimella NSF Cooling Tehnologies Researh Center Shool of Mehanial Engineering, Purdue University 585 Purdue Mall, West Lafayette, Indiana USA sureshg@en.purdue.edu ABSTRACT Large pressure drops, and the assoiated pumping requirements, are often onsidered the most ritial fator hindering idespread ommerial use of mirohannel heat sinks. Analytial methods are used in the present ork to arrive at the pumping requirements for any given mirohannel heat sink. A graphial method to hek the suitability of a pump to a mirohannel heat sink appliation has been devised. The size of the mirohannels is also optimized so that for a speified heat removal rate, the pumping requirements are minimized. A number of ommerially available pumps as ell as several miropumps presented in the literature are ompared based on their flo rate, pressure head and physial size to assess their suitability for a speifi representative ooling appliation. NOMENCLATURE A surfae area of mirohannels p speifi heat C onstant d/dx gradient along length of mirohannel D diameter f frition fator g funtion h height k thermal ondutivity L length m mass flo rate n number of hannels Nu Nusselt number P perimeter q heat removal rate Q volume flo rate Re Reynolds number t thikness T temperature U W mean flo veloity idth Greek Symbols aspet ratio of mirohannels µ dynami visosity ρ density of fluid p pressure drop T temperature differene Subsripts and Supersripts * optimal 1 number 1 number hannel d die f fluid i inlet max maximum o outlet all INTRODUCTION Mirohannel heat sinks have been investigated for over to deades for appliation in eletronis ooling [1,]. Despite their proven potential for handling high heat fluxes, they have not found idespread ommerial use. This is believed to be largely due to the very high pressure drops enountered in the mirohannels. High pressure drops neessitate the use of relatively large pumps ith signifiant poer requirements. In the present ork, the pumping requirements of mirohannel heat sinks are analyzed, and the dimensions of the mirohannels are optimized suh that for a given thermal load, the pumping requirements of the heat sink are minimized. A simple analytial approah is employed in hih the flo rate required for ooling and the pressure drop 1 Copyright 003 by ASME

2 through the hannels are related to the size of the mirohannels. The suitability of various pumps and miropumps (omprehensively revieed in [3]) for use in mirohannel heat sink appliations is evaluated in a graphial format. DERIVATION OF THE PUMPING REQUIREMENTS An analysis frameork is developed in this setion to optimize the dimensions of a mirohannel heat sink suh that the desired heat removal rate is ahieved hile employing the minimum possible pumping poer. The minimum flo rate and pressure head required to ahieve the desired heat removal rate are alulated under speifi thermal onstraints and these are in turn related to the dimensions and the number of mirohannels in a heat sink. A mirohannel heat sink used to ool a high poer density hip is onsidered, ithout regard to the mehanism used for pumping the fluid or the heat exhanger required for ultimately transferring the heat from the fluid to the ambient. The dimensions of the heat sink and the mirohannels are denoted in the shemati in Figure 1. The length and the idth of the heat sink are the same as that of the die being ooled, and hene are fixed at L d and W d. The idth of a mirohannel and its aspet ratio (and hene its height h = ) are variables. The number of hannels n depends on the hannel idth and hene is also a variable. The number of hannels ould also depend on the all thikness ; hoever, this parameter is onsidered to be a onstant in this ork. Finally, the mean flo veloity U, and hene the total volume flo rate through all the hannels Q, as ell as the pressure drop p, are also variables. W H h L d W d W W Figure 1. Desriptors of the mirohannel heat sink geometry. The to primary thermal onsiderations in the design of a ooling solution for an integrated iruit hip relate to the limits on the maximum temperature and the maximum temperature gradient on the hip. These may be expressed as: 1) The maximum temperature at any point on the hip should be less than T max, i.e., T d < T max. ) The maximum temperature gradient on the hip is limited dt dt dt to, therefore, <. dx max dx d dx max These to thermal limits are used in the folloing to alulate the limiting flo rate and the pressure head for given values of heat removal rate and hannel dimensions. Assuming that all the heat from the hip goes toards inreasing the temperature of the ooling fluid (q d = q f = q), energy onservation requires that q = ρqp Tf (1) Further, assuming thermally fully developed onditions under a onstant heat flux boundary ondition, the temperature profile in the mirohannels in the axial diretion ould be as shon in Figure. T -T f T T,i T f,i Wall Fluid T,o T f,o T -T f T f Figure. Temperature profiles in the mirohannels. Therefore, the heat transfer oeffiient is onstant throughout the hannel, and the Nusselt number based on hydrauli diameter is a funtion only of the hannel aspet ratio [4], i.e., hdh Nu h = = g1 () k The loal heat flux at any axial loation in the hannel an no be ritten as q A= h T T (3) ( f ) in hih A = npl is the internal area of the mirohannels. Sine the differene beteen the all temperature and the mean fluid temperature is also onstant, the limit on the maximum temperature gradient on the hip an be ritten in terms of the maximum temperature gradient in the ooling fluid, as ( dt dx) f < ( dt dx) max. Sine the fluid temperature varies linearly along the length of the hannel, Equation (1) yields the limit on the minimum flo rate as, q Q > (4) ρpld ( dt dx) max Equation (3) an also be used to derive an expression for the maximum temperature on the hip, hih ill our at the mirohannel outlet: q Dh T = do, T + fo, Tmax npl d g1 k < (5) Using the expressions T, = T, + q ρ Q, P = ( h + ), f o f i p = ( + ) and n W ( ) x Dh h h = d + the above equation an be ritten as q q( + ) Tf, i+ + < T max ρq p g1 klw d d ) (6) Sine the inlet temperature of the ooling fluid is limited by the ambient temperature, another limit on the minimum flo rate exists, and is given by rearranging Equation (6): Copyright 003 by ASME

3 q 1 Q > (7) ρ p q( + ) Tmax Tf, i g 1 kldwd ) It may be noted that the minimum flo rate in Equation (7) is a funtion of the mirohannel idth, idth of the all ( ) and the hannel aspet ratio, hereas the minimum flo rate in Equation (4) is independent of these parameters. Equation (7) also provides a limit on the maximum idth of the mirohannels for a given flo rate: g 1 kldw q d 1 T max T < f, i + q ρq (8) p For fully developed flo through a hannel of retangular ross-setion, the pressure drop through the hannels an be related to the mean flo veloity and the size of the hannels using standard hydrodynami relations. The produt of the frition fator f and Reynolds number Re (based on hydrauli diameter) is a funtion only of the aspet ratio of the hannels: f Re = g (9) Using standard definitions of f and Re, Equation (9) beomes pdh µ Ld = g U (10) The mean flo veloity an be related to the volume flo rate and the number of hannels using Q = nhu. Using this relation along ith some algebrai manipulation results in the folloing expression relating the pressure drop and the flo rate in mirohannels: 3 p ( h ) µ Ld = g (11) Q ( ) 8W + h + d Representing the height of the hannels in terms of the aspet ratio, Equation (11) an be ritten as: 4 p µ Ld ) g 3 Q = (1) + 8Wd For a presribed hannel aspet ratio, all the variable parameters are on the left hand side of Equation (1), hile the right hand side of the equation is a onstant. Hene the pressure drop in the hannels an be plotted against the total volume flo rate as a funtion of the idth of the hannels (sine is a onstant). Different values of the aspet ratio ill result in different urves, all passing through the origin of the axes. Moreover, aording to Equation (1), the minimum pressure drop in a mirohannel heat sink depends on the minimum volume flo rate. Hene to limits on minimum pressure drop an be obtained by ombining Equations (4) and (7) respetively ith Equation (1): qµ ) + p> g 3 (13) 4 8ρW p d ( dt dx) max ) + g 3 4 qµ L d p > (14) 8ρW p d q( + ) Tmax Tf, i g kldwd ) The minimum thikness of the mirohannel alls ( ) and the maximum depth and aspet ratio of the mirohannels are also generally limited by fabriation onstraints and onstraints related to strutural integrity, and ill limit the maximum heat removal rate. A typial plot of the pressure drop versus total volume flo rate for analyzing the pumping requirements of a mirohannel heat sink is shon in Figure 3. The minimum volume flo rate and the minimum pressure drop set by the limit on the maximum temperature gradient on the hip are plotted from Equations (4) and (13) by varying the hannel idth for a given aspet ratio. It may be noted that sine the limit on the minimum volume lo rate from Equation (4) is independent of the aspet ratio and the hannel size, this limit is a straight line parallel to the pressure axis. The end points of the line are determined from the limit on the minimum pressure head (Equation (13)) and hene are dependent on the hannel size. The minimum flo rate and the minimum pressure drop set by the limit on the maximum temperature of the hip are also plotted from Equations (7) and (14). Both these limits are dependent on the aspet ratio and the hannel idth. This trendline is also plotted for a presribed aspet ratio over a range of hannel idths. As the idth of the mirohannels inreases, the minimum flo rate alulated from Equation (7) inreases, hile the minimum pressure head obtained from Equation (14) first dereases, reahes a minimum value and then inreases. P (Pressure Head) * inreasing Operating Region Limit from Eqs. (4) & (13) Limit from Eqs. (7) & (14) Minimum Operating Limit inreasing Q (Florate) Figure 3. Pumping requirements for a mirohannel heat sink ith mirohannels of fixed aspet ratio. The operating region of the mirohannel heat sink has also been identified in Figure 3. If the operating point of the mirohannel heat sink, i.e., the flo rate and pressure drop at hih the mirohannel heat sink operates, lies ithin the operating region, the desired heat transfer rate ill be ahieved and the limits on the maximum hip temperature and maximum temperature gradient ill be met; otherise, one or more of these three onditions ould not be satisfied. If the operating region of a mirohannel heat sink of given dimensions (aspet ratio and hannel idth), ere to be plotted, it ould be a straight line, hose starting point ould be determined by the limits on the minimum flo rate (Equations (4) and (7)) and the orresponding limits on the minimum pressure drop (Equations (13) and (14)). But sine the flo rate and the pressure drop in a mirohannel heat sink are related by Equation (1), the four limits for a mirohannel 3 Copyright 003 by ASME

4 heat sink of given dimensions may be fully represented by the limits on the flo rates, or alternatively, by the limits on the pressure head. Therefore, in effet, the greater of the to flo rate limits, or the to pressure drop limits, ill identify the operating region for a mirohannel heat sink of given dimensions. With referene to Figure 3, for mirohannel heat sinks here the mirohannel idth is less than the optimal value * marked in the figure, the flo rate from the limit on the maximum temperature gradient gives the limiting ondition, hile for ases here the mirohannel idth is greater than *, the pressure drop from the limit on the maximum hip temperature gives the limiting ondition. The resulting area overed by these to limits thus delineates the operating region. The loer limit of the operating region, labelled the minimum operating limit in Figure 3, represents the flo rate and pressure head ombinations belo hih either the desired heat transfer rate or one of the thermal onstraints ould remain unmet for any mirohannel idth at a given aspet ratio. P (Pressure Head) Q (Florate) Minimum Operating Limit Mirohannel 1 Mirohannel Mirohannel 3 Pump 1 Pump Figure 4. Pumping requirements for mirohannel heat sinks, relative to the minimum operating limit identified in Figure 3 and hypothetial pump urves. The suitability of a pump to a mirohannel heat sink design an be assessed by superimposing the pump urve and required flo rate versus pressure drop harateristis for the heat sink on a plot of the pumping requirements for the heat sink, as illustrated shematially in Figure 4. The pump urve refers to the flo rate versus pressure head harateristis of the pump, and is usually obtained experimentally. The flo rate versus pressure drop harateristis of the mirohannel heat sink are obtained from Equation (1); these are straight lines of positive slope passing through the origin. The slope of the line is a funtion of the mirohannel dimensions. The point of intersetion of these to urves ould be the operating point of the pump and heat sink. Only if the operating point lies in the operating region an the pump meet the desired onditions for the mirohannel heat sink. Hene, for the example in Figure 4, pump 1 in onjuntion ith mirohannel heat sink and 3 or pump ith mirohannel heat sink 3 an dissipate the given heat load hile satisfying the speified thermal onstraints (points marked ith open irles). The other pump-heat sink ombinations ould not ork (points marked ith solid irles). This approah provides a very simple ay of assessing the suitability of any pump and mirohannel heat sink for a desired appliation. The point of intersetion of the to limiting urves also represents the minimum pumping requirements of the mirohannel heat sink, hile satisfying the given thermal onstraints. The mirohannel idth orresponding to this point, *, ould be optimal for a presribed aspet ratio, and the heat sink ith these dimensions ould impose the minimum requirements on the pump used to drive the fluid through the mirohannels. This optimal idth an be alulated by equating the flo rates from Equations (4) and (7): * g1 kldwd dt 1 T max T f, i L + = d q dx (15) max It may be noted that the foregoing analysis as performed under the folloing assumptions: fully developed hydrodynami and thermal boundary layers in the mirohannel; onstant heat flux boundary ondition; all four alls of eah mirohannel partiipating in heat transfer (lid not separately aounted for); onstant all thikness beteen adjaent mirohannels; and negligible deterioration in heat transfer due to fin ineffiienies. All these assumptions an be readily relaxed in the analysis frameork; hoever, losed-form solutions may not be obtainable in some ases, and a numerial approah may be needed for solution. ILLUSTRATIVE EXAMPLE A speifi example of an integrated iruit hip (given size and heat flux) is no used to illustrate the proedure developed above to obtain representative numerial values for optimal mirohannel dimensions and pumping requirements. The pumping requirements alulated are ompared to the pump urves of several small ommerially available pumps. The pumping requirements are also ompared to the apabilities of various miropumps presented in literature. Table I. Inputs used in the example. Parameter Value Coolant Density, ρ kg/m 3 Speifi heat, p 4184 J/kg.K Visosity, µ N.s/m Condutivity, k 0.65 W/m.K Chip Length, L 1 m Width, W 1 m Mirohannels Aspet ratio, 6 Channel idth, µm Wall thikness, 100 µm Thermal Parameters Heat load, q 100 W Maximum hip temperature, T max 80 C Maximum hip temperature gradient, ( dt dx 5 C/m ) max 4 Copyright 003 by ASME

5 The heat load, thermal onstraints, dimensions of the mirohannel heat sink and the oolant properties used in the example are summarized in Table I, and are onsidered to be realisti estimates from pratial appliations. The oolant properties are those of ater at 330 K [4]. The pumping requirements of the mirohannel heat sink onsidered are plotted in Figure 5. The flo rate versus pressure drop harateristis for to hannel idths (100 and 400 µm) are also plotted. In addition, pump urves for several miniature onventional pumps, namely entrifugal, gear and flexible impeller pumps [5], as ell as for a number of miropumps are also plotted so that their suitability to this appliation may be assessed. The miropumps onsidered are a valveless (nozzle-diffuser) miropump using piezoeletri atuation [6], an injetion-type EHD miropump [7], an eletroosmoti miropump [8], a rotary miropump [9] and a piezoeletri miropump [10]. Both the flo rate and pressure head provided by the mini entrifugal pump are greater than the range of values on the axes in Figure 5, and are not visible in the plot. Beause of the ide variation in the pressure heads provided by different pumps and miropumps, a semilogarithmi sale is seleted. All the pump urves are plotted using the maximum flo rate (at zero pressure head) and the maximum pressure head (at zero flo rate) provided by the pump and are assumed to be linear beteen those end-points; this is adequate for the illustration here, but atual pump urves should be used henever available. Sine none of the miropumps presented in literature are able to meet the requirements of the mirohannel heat sink onsidered, espeially the flo rates, several miropump units ould need to be ombined in parallel to ahieve the desired flo rate. The pump urves for the miropumps presented in Figure 5 are for suh parallel sets of pumps. For the present demonstration of the analysis approah, no additional onsiderations are inluded in terms of the feasibility or ineffiienies of onneting these miropumps in parallel. The pumps and miropumps are also ompared in Table II based on their maximum flo rate, maximum pressure drop and size. For miropumps, the size of the individual miropumps is listed along ith the number of miropumps needed. Hene 5 injetion-type EHD pumps or 1000 rotary pumps ould be needed in parallel to provide the flo rate needed for ahieving the desired heat removal rate Miniature Gear Pump Eletroosmoti Miropump Flexible Impeller Pump Miniature Gear Pump Required Mirohannel Charateristis = 100 µm P (Pressure Head, Pa) 1000 Rotary Miropump Valveless Miropump Injetion-type EHD Miropump Piezoeletri Miropump Required Mirohannel = 400 µm 100 Minimum Operating Limit 10 Inreasing Q (Florate, ml/min) Figure 5. Pumping requirements for mirohannel heat sinks ( = 6, 50µm < < 800µm) and assessment of the suitability of various pumps; a large number of eah of the miropumps are used in parallel (as shon in Table II) to ahieve the desired flo rates. 5 Copyright 003 by ASME

6 Clearly, the flo rate and pressure head provided by the onventional pumps is muh higher than that provided by several hundreds of miropumps in parallel. Hoever, the size of eah of the onventional pumps is muh larger than that of the miropump ombinations. The smallest onventional pump, miniature gear pump 1, has a volume of approximately 300 m 3. In omparison, the volume of the multiple injetiontype EHD pumps, rotary miropumps and piezoeletri pumps required ould be 0.17, 6.03 and m 3, respetively, again assuming that no additional volume is neessary for implementing multiple pumps in parallel. Interpreting Figure 5, for the mirohannels of hannel idth 100 µm, only the onventional pumps and the eletroosmoti and the valveless miropump ombinations, among the ones onsidered, ould be suitable for the present design. It may be noted that the points of intersetion of the pump urves of onventional pumps ith the mirohannel harateristis lie outside the graph (to the right). Hoever, for the heat sink ith 400 µm ide mirohannels, all the pumps and miropump ombinations ould provide the desired heat removal rate, hile satisfying the thermal onstraints. Table II. Comparison of the apabilities and sizes of the pumps onsidered. Pump Mini Centrifugal Magneti Drive Pump Miniature Gear Pump 1 Miniature Gear Pump Flexible Impeller Pump Valveless Miropump Injetion-type EHD Miropump Eletroosmoti Miropump Rotary Miropump Piezoeletri Miropump Maximum Florate (L/min) Maximum Pressure (kpa) Size (mm 3 ) (150X ) (5X) (400X) (1000X) (50X) Indiates that 150 miropumps onneted in parallel ould be needed to ahieve the desired performane Miropumps other than the eletroosmoti and valveless ones an also be used for the heat sink ith 100 µm ide mirohannels; hoever this ould require attahing miropumps in series to ahieve the high pressure drops in these smaller hannels. For example for the ombined parallel set of rotary miropumps, 50 suh sets ould need to be onneted in series, assuming eah additional miropump enhanes overall pressure head by the same amount ithout any derease in flo rate. Sine 1000 rotary miropumps are needed in parallel, a total of 50,000 ( ) units ould be needed to ahieve the desired flo rate and pressure head. The volume of suh ombination pumps ould then exeed the smallest onventional pump onsidered. The feasibility of implementation of suh a large number of miropumps in the series-parallel ombination is also unertain. The optimal mirohannel idths alulated using Equation (15) to satisfy the imposed thermal onstraints are listed in Table III for different hannel aspet ratios. For the speified heat removal rate and thermal onstraints, the pumping requirements of the mirohannel heat sinks are minimal at these mirohannel idths. Table III. Optimal mirohannel idths for minimum pumping requirements. Aspet ratio, Optimal mirohannel idth, * (µm) CONCLUSIONS A method has been developed to analyze the pumping requirements of mirohannel heat sinks for speifi appliations. A graphial method to assess the suitability of a given pump for a partiular mirohannel heat sink appliation has also been devised. In addition, a relationship has been developed to optimize the size of the mirohannels so that the requirements on the pump are minimized for a given heat removal rate. The methods to analyze the pumping requirements and to assess the suitability of pumps to partiular mirohannel heat sink designs have been explained ith an illustrative example. Aknoledgments The authors aknoledge the finanial support of the members of the Cooling Tehnologies Researh Center ( a National Siene Foundation Industry/University Cooperative Researh Center. REFERENCES [1] Tukerman, D. B., and Pease, R. F. W., 1981, High performane heat sinking for VLSI, IEEE Eletron Devie Letters, Vol. EDL-, pp [] Garimella, S. V., and Sobhan, C. B., 00, Transport in mirohannels A ritial revie, Annual Revie of Heat Transfer, Vol. 13. [3] Singhal, V., Garimella, S. V., and Raman, A., Mirosale Pumping Tehnologies for Mirohannel Cooling Systems, Applied Mehanis Revie (in revie). [4] Inropera, F. P., and Deitt, D. P., 1998, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, Ne York. [5] Cole-Parmer Instrument Company, 001, Cole-Parmer Produt Manual 001/0, Cole-Parmer Instrument Company, Vernon Hills, IL. [6] Olsson, A., Enoksson, P., Stemme, G., and Stemme, E., 1997, Miromahined flat-alled valveless diffuser 6 Copyright 003 by ASME

7 pumps, Journal of Miroeletromehanial Systems, 6, pp [7] Rihter, A., Plettner, A., Hofmann, K. A., and Sandmaier, H., 1991, A miromahined eletrohydrodynami (EHD) pump, Sensors and Atuators A: Physial, 9, pp [8] Zeng, S., Chen, C.-H., Santiago, J. G., Chen, J.-R., Zare, R. N., Tripp, J. A., Sve, F., and Fréhet, J. M. J., 00, Eletroosmoti flo pumps ith polymer frits, Sensors and Atuators B: Chemial, 8, pp [9] Dea, A. S., Deng, K., Ritter, D. C., Bonham, C., Gukel, H., and Massood-Ansari, S., 1997, Development of LIGAfabriated, self-priming, in-line gear pumps, Transduers 97, Chiago, pp [10] Zengerle, R., Stehr, M., Messner, S., and Sandmaier, H., 1996, The VAMP - A ne devie for handling liquids or gases, Sensors and Atuators A: Physial, 57, pp Copyright 003 by ASME