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Modelingheattransfer and liquidflowinmicro-channels

Sabry, M.N. Djebedjian, B.O. Saleh, S.H. Mahgoub, M.M.

Univ. Franqaise d'Egypte, Cairo, Egypt

This paper appears in: Thermal and Mechanical Simulation and Experiments in Microelectronics and

Microsystems, 2004. EuroSimE 2004. Proceedings of the 5th International Conference on

Publication Date: 2004

On page(s): 511 - 518

ISSN:

ISBN: 0-7803-8420-2

INSPEC Accession Number:8001658

Digital Object Identifier: 10.1109/ESIME.2004.1304085

Current Version Published: 2004-10-04

AbstractThe use of micro-channels in order to cool modern high speed electronic circuits is one of the techniques

frequently adopted in current practice. A burst of publications in this area has been observed in the last decade.

However, modeling of fluid flow and heat transfer in micro-channels is still an open problem. In fact, many

deviations have been experimentally observed between well established correlations used for conventional

normally sized channels and the behavior of microchannels. These deviations increase as the channel size

decreases. In this work, observed experimental deviations are first listed, followed by a critical review of different

hypotheses advanced in the literature to explain them. One of these hypotheses is thoroughly developed in order

to build a model that explains both the orders of magnitudes and the trends of observed phenomena.

Index TermsInspec

Controlled Indexing

channel flow circuit simulation cooling flow simulation high-speed integrated circuits

integrated circuit measurement integrated circuit modelling integrated circuit packaging

thermal analysis thermal management (packaging)

Non-controlled Indexing

channel size fluid flow heat transfer modeling high speed electronic circuits cooling liquid

flow modeling micro-channels

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Mod eling heat transfer and liquid flow in micro-channelsSabry,M.N.[l] , Djebedjian,B.0.[2], Saleh, S.H.[2], Mahgoub,M.M.[2]

[11Universite Franqaise d'Egypte, Cairo, Egypt,[2] Mansoura University, Mansoura, [email protected] use of micro-channels in order to cool modemhigh speed electronic circuits is one of the techniquesfrequently adopted in current practice. A burst ofpublications in this area has been observed in the lastdecade. However, modeling of fluid flow and heat transferin micro-channels is still an open problem. In fact, manydeviations have been experimentally observed betweenwell established correlations used for conventionalnormally sized channels and the behavior of micro-channels. These deviations increase as the channel sizedecreases. In this work, observed experimental deviationswill first be listed, followed by a critical review ofdifferent hypotheses advanced in the literature to explainthem. One of these hypotheses will be thoroughlydeveloped in order to build a model that explains both theorders of magnitudes and the trends of observedphenomena.1- IntroductionIn the pioneering work of Tuckerman & Pease[l],micro-channels were proposed as an efficient technique tocool microelectronic circuits. A relatively large number ofsmall channels of hydraulic diameter of the order of 100microns were grooved in the chip, in which purified waterwas used as the cooling fluid. High heat fluxes, of theorder of 700 W/cm2 of the chip projected area werereported. Micro-channels have many other medical andchemical applications and are expected to have a seriousimpact on future technologies [2,3]. Many enhancementswere proposed of micro-channels already highperformance, including the use of special shapes [4], nanoparticles [5], electrostatic fields [6] and two phase flows[7,8]. However it seems that the problem of modeling thebasic single phase straight channel case is still open [9].To clarify this, consider the heat dissipated Q:where h is the heat transfer coefficient, the effectivearea of heat transfer and AT the driving temperaturedifference. Evidently, Q is imposed by electronic activity,and AT is the quantity we wish to minimize, in order toimprove reliability and performance [lo], i.e. maximize hA e ~ . he value of h is calculated from Nusselt number:where 14 is the fluid thermal conductivity and Dh thehydraulic diameter. Nusselt number can be correlated withother parameters, in particular Reynolds number:

Re = Vavg Dh / VIwhere vavgs the average flow velocity and VI is the fluidkinematic viscosity. Let us first assume that correlationsestablished for conventional (relatively large) channelscan be fully applied for micro-channels. In this case, since

Q=h&ffAT (1.1)

Nu = h DI,kl (1.2)

(1.3)

Dh is small, Re is so small and the flow can be assumedlaminar. Hence, if channel length L is high compared toDh, flow is fully developed and Nu is a constant. Thetheoretical value NQ does not depend on Re, but onchannel cross sectional shape (circular, trapezoidal . )and heating type (uniform heat flux or uniformtemperature). It follows from (1.2) that h is inverselyproportional to b. ence, surprisingly large values of hcould thus be obtained, even for laminar flow, as I&decreases to 100 pm. This also leads to high pressuredrops, because in the same above conditions (i.e. fullydeveloped laminar flow) the Poiseuille number:(where f is the Darcy's friction coefficient) is constant.The theoretical value Po* depends only on channel shape.Hence, whether we fix Re, fluid velocity v,,, or dischargethe pressure drop will always increase for smallera.

In addition, micro-channels largely increase &E. Infact, channel side areas are large compared to the outsidearea of the block in which they were grooved. Theseadditional side areas act as a fin array that efficientlycools the block. The central question in this work is now:Are the assumptions made above, based on thebehavior of large channels, still valid for micro-channels?

2. ObserveddeviationsAs the size of a phenomenon changes by severalorders of magnitude, the extension of physical modelsused before to the new size becomes questionable, evenafter rescaling. New physical effects may be involved thatwere irrelevant in normal size. Experimentalinvestigations have confirmed the following deviations:

Po=fRe (1.4)

2.a Friction losses in laminar flowFor micro-channels the ratio cf of observed Po to Pothhas reported to be less than 1 [11-13] greater than 1[14,15] or both less or greater than 1 depending on Re[9,16]. All agreed upon the fact that Po increases with Refor laminar flow, in contradiction with classical theory.2.b Transition to turbulenceEarlier transition to turbulence in micro-channels [19-201has been reported compared to conventional channels.2.c Heat transfer in laminar flowFor micro-channels, the ratio ch of observed Nu toNUth has also been reported to increase with Re [17,18,20,211, in contradiction with theory. Values of C, can be lessor greater than 1.2.d Turbulent flowExperiments gave contradicting results hence thissubject will not be treated here.

0 - 78 0 3 -8 4 2 0- 2 /0 4 /$ 2 0 .0 0 0 2 0 0 4 IEE E -511-5th. Int. ConJ:on Thermal and Mechanical Simulation and Experiments in Micro-electronics an d Micro-System s, EuroSimE2004
mailto:[email protected]:[email protected]


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3. Physical effects involveddeviations:The following hypotheses were advanced to explain

3.a Non-Continuum effectsBased on the Knudsen number Kn (ratio of mean freepath to duct dimension) Eringen [22] has theoreticallystudied these effects. It is well established that for Kn


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Figure 1 -Different roughness shapes

L W

I-- PlPinlet ~ TadT au inlet ....-.. &/& inletI43210

-1-2-3

Figure 2 - Flow over circular rough elementsI-P/P inlet __ au/Tau inlet--- - .- .--- .. Qw/ Qwin let---

I n I I

-3-4

I

Figure 3 - Flow over rectangular rough elements

i- IP inlet __ TaulTau inlet ---- N/& inlet --- I6420

-2I

Figure 4 - Flow over triangular rough elementsIt will also be proved in the next section that

!

thisscenario may successfully model all other deviationsenumerated above. In fact, micro-eddies that will beprovoked by roughness tips will cause a Re numberdependence of both Po and Nu, which explains anotherobserved deviation. Finally, it is evident that micro-eddiesare responsible for the observed early transition toturbulence in micro-channels. Moreover, the order ofmagnitude analysis will also show that these effectscannot be detected for conventional channels.5. Modeling frictionFully developed flow will be assumed since L/9, (Lbeing the channel length in flow direction) is usually veryhigh. Modeling will be done in two steps. First, very lowRe will be assumed to obtain a simple model withoutmicro-eddies. Second, the effect of micro-eddies will beadded. Subscript cl will be used for classical results, i.e.conventional channels, lo for low Re results and mc forthe final model of micro-channels. As an idealization ofthe peaks and valleys caused by wall roughness, it will beassumed that wall will be separated into two distinctportions (Fig. 5) . The first portion is shielded by the gasblanket and occupies a portion 5 of the wall area, where 5is the shielding coefficient that should typically be around0.5. The unshielded portion occupies 1-5. The gasblanket is of thickness 6 , which is equal to the surfaceroughness. This scheme is applicable as long as Re issmall enough to prevent (or at least reduce) the effect ofmicro-eddies on the mean flow. Hence we need first asimple friction model in the shielded portion, thencombine it with friction in the unshielded portion andfinally add effects of micro-eddies.

5.a Friction in the shielded portionThe cross section constitutes a 2D domain that will besubdivided into 2 sub-domains, one occupied by the liquid0 1 , which is surrounded by the domainOgoccupied by thegas (Fig. 5). Fully developed flow implies that pressure isconstant over both RI and Og and has a constant axial

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gradient -B. Hence, Navier-Stokes equation and boundaryconditions in both domains are:

a n

.x

n

Figure 5 - Simplified problem geometryV2v,= -B/p, in Ri (i=g or 1)v, = 0 on dR , i= v, on do, (5.1)

where vi is the axial velocity of gas (i=g) or liquid (i=l), pis the dynamic viscosity and dCli is the liquidgasinterface. Since the gas layer has a small thicknesscompared to section dimensions, it may be approximatedas a 2D channel of infinite width, having a solid boundaryat it s bottom, and a liquid boundary at its top. Across Qthe shear stress should vary linearly with y the coordinatenormal to the channel wall, due to the fully developedassumption. Momentum balance over any layer extendingfrom 0 to y gives:where x is the coordinate along the flow. Momentumbalance over the whole cross sectionR ields:

(L - ~ ( y ) )x1 P y (5.2)

~~p dx = dP Ai.e.T~ =dP/dx Alp = -B Dh/4 (5.3)where p is the section perimeter and A its cross-sectionalarea. From Eqs (5.2,3) we get an expression for z(y):~ ( y ) B (y-Dd4) (5.4)~ ( y ) - pg dv,/dy; ~ ~ ( 0 )0 ( 5 . 5 )VdY)= B(YI)IJ4-Y2/2)/P, (5.6)

v = E B D2/4 pg (5.7)

Integrating Eq. (5.4), and noting that:yields:Hence, the velocity at the interface is constant and

(assuming a small ratio &=6/Dh)s equal to:If the domain figwas filled with liquid, the velocity at

y= F would have been the same as that given by Eq. (5.7),but with pl instead of CL,. Hence, the presence of a gaslayer increases the liquid velocity at the interface, for thesame wall shear stress, by the constant amount:

AV = E B D2/4 (l/p,-l/pl) (5.8)The variable liquid velocity in the whole domainRIatlow Re, V I o can thus be approximated by:

VI o = VI Ci + AVBut, from the definition of Po, and Eq. (5.3):Po = -8 T,., Re/@ vavg2)= 2BD?/(pl vlavg)

(5.9)(5.10)where avg denotes the average lover the cross section.Hence, by taking the average of (5 9 ) and using (5.10):

PO & PoCl (l+AV/v,i) = PO,! / (1+p) (5.1 1)p = PoCl (p1/pg-l)/85.b- Friction for low ReCombining friction on shielded and unshielded

(5.12)where Po5 denotes Po for the shielded portion.

portions gives:POI,=P O ~ I1-6) + 6 PO< =PoCl l-\P/(p+l)) (5.13)5 .c-Effect of micro-eddiesAccording to the mixing length theory [39], turbulent

like eddies can produce an additional shear Ted:Ted = -p h2 Idv/dyl dv/dy (5.14)

where h is the mixing length. It can be expressed as:h = c 6 (5.15)

where c is the mixing coefficienl that is expected to bearound 0.5 as seen from Fig. 5.The average dv/dy at thewall could be obtained from the viscous wall shear stresscorresponding to the low Re case obtained above (5.10):Hence:pldv/dylw=-w, PolopvavpZ/( 8Re)

Ted / Twi0= Polo (c&)*Re/8The total wall shear stress will thus be assumed as the

(5.16)sum of viscous and eddy stresses:

5m c = cwlo + Ted = Twlo (1+Ted/Kwlo)i.e. Po, = Po,,( 1+Pol,(cs)Rel8) (5.17)This equation (where POI,has, been defined by 5.13)constitutes the full friction model of micro-channels. Itexplains why friction has been found to be less thanexpected in investigations performed at low Re, and whyit was higher at high Re. In the next paragraph it will bequantitatively compared with published results.5.d- Experimental verification

Test results will be confronted with the postulatedmodel given by Eq. (5.17). It contains 2 parameters thatdepend on the nature or shape of roughness elements,namely 6 and c. Note that these are NOT adjustablecoefficients that may take any value to fit test results.They do have a physical meaning that implies a narrowrange of expected values around 0.5 for each of them. Thefirst comparison will be made with the results of Pfahler etal. [121 that were obtained at very low Re.

Figure 6 shows the effect of size on friction. Typicalvalues of G=c=0.5 were taken. It is clear that forIsopropanol there is a perfect match. But for Silicon oil,the trend was correct with a small shift. This means that asmall modification of 6 may be needed for this particulartype of fluid/wall combination. Effect of Re on friction iscompared in Fig. 7with experimental results [14,15,20]. Itis astonishing to notice that ai1 available test resultscompared well with (5.17) if we take k=c=0.5, except onecase (Harms et a1 [20]) with 5=0.4 and ~ 0 . 5 . hisconstitutes a strong proof of the validity of the proposed

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scenario, as well as the simple model issued from it.Intensive test series may give a more precise estimate of 6and c depending on wall andor liquid type as well asmachining procedure used. But, in all cases, Eq. (5.17)seems a suitable form for a practical precise correlation,as well as a rough estimate for designers with the typicalvalues of t=c=0.5.

2.0

-0- Pfahler et al( l991) Isopropanol- fahler et al (1991) Silicon oilModel IsopropanolModel Silicon oi l- - -

A...

1.1 I I I 1

0.60.1 1 10 100Dh bm)

Figure 6 - Effect of channel size on friction for low Re

Modelarms et al.. -- .Model

-Mala & Li 50micr. . - . * - e - Model- ala & Li 80micr. .- . k . . - Modeli-- ala& Li 100micr . - - + - - - odel

...*...Papautsky et al

0.0 I I I0 500 1000 1500 2000ReFigure 7-Dependence of friction on Re

Surface roughness was not specified by Papautsky etal. [I41 hence, a value of 1% was assumed which istypical for this fabrication technology. Surface roughnessreported by Mala & Li [151 was not measured, but basedon manufacturer data. Hence, the smallest channel (50pm) had a greater error. They have stated that Po maydecrease with Re in a certain range, which is not normal.The decrease lies however within error bars. They havealso presented a model based on surface roughness. Itused an added viscosity, called roughness viscosity. Itcontained an adjustable coefficient that varied from 0.01

to 1. The coefficient was correlated in terms of testsection geometry and Re against their experiments using 4adjustable coefficients and an expression that was notbased on a physical model or scenario. The advantage ofsuch expressions is that they can easily fi t a given set oftest conditions, but the disadvantage is that one cannotguarantee the satisfaction of other sets. In particular, thiscorrelation cannot predict cases for which POP% < 1.The model presented here was derived from problemphysics according to a given scenario. Hence it containedonly 2 parameters having a physical meaning as well as avery narrow range of variation.6. Modeling transition to turbulence

At a certain intensity level, micro-eddies may bestrong enough to induce and sustain turbulence in themean flow. The corresponding model could be inspiredfrom experimental results of turbulence initiation due to acylindrical obstacle (a wire) having an axis normal to flowdirection and attached to a wall [39]. Turbulence isinitiated when the cylinder diameter (seen here as theroughness height) reaches a value that was experimentallydetermined:whereK is an empirical constant that equals 7 for the caseof a single wire, and U* is the frictional velocity: U*= (IT,,l/p)12.Using this definition as well as (5.10) gives:from which the criterion for transition to turbulence couldbe derived in terms of the critical Re:

6 = K p (pu*) (6.1)

6 = K p (p2 (Po/8)vav,Z/Re)12

Recr= 8 (Us)POI,

(6.2)

(6.3)The value of K for the case of distributed roughnessstill has to be determined experimentally. It is evident thatit should be much smaller than the case of a single wire(i.e. 7) but still probably of the order of 1. Unlike the caseof friction, literature is both unanimous about the earlytransition and lacking data on roughness values. To theauthors knowledge, only one article [20] has providedboth the value of roughness (~=2%)nd the transition Re(Re,,=1510). Substituting in (6.3) gives K=1.92. It iscertainly not possible to recommend a value of K basedon only one experiment, especially for the transition toturbulence. But the reasonable value of K pledges in favorof the proposed scenario. It is hoped that once the form ofa suitable criterion has been derived (Eq. 6.3), newexperiments could use it to obtain reliable correlations.

It is interesting to note that an equivalent criterioncould be obtained using another approach. In fact, if wedefine transition to turbulence as the point at which eddyand viscous shear stresses are of equal importance, thenwe get from (5.16):Recr= 8 / ((cE)~OI,) (6.4)The amazing fact is that if we apply it with therecommended value of c=0.5, that has been validated

from friction data, we get ReCr=1630,which is very closeto the observed value. The value of c=OS in factcorresponds to the value of K=2 in equation (6.3).Although this result was obtained for the only quantifiedtest case available, which does not constitute a formal

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proof, it is still a good indication of the validity of thescenario proposed.7. Modeling heat transferAvailable heat transfer data are much less than thoseof friction, and in addition some of them are unusable.This is due to different definitions of effectivetemperature differences and areas, as well as omittingroughness data. Hence, the heat transfer model that will bedeveloped here is not intended to validate the proposedscenario. It is believed that above arguments weresufficient. But it is rather intended to understand theimplication of the proposed scenario on thermalresistance. At very low Re, the gas blanket acts as aninsulating layer. Shielded and unshielded areas can beconsidered as two parallel thermal resistances. Theunshielded portion has a Nu number that is the same asthat of the classical problem Nbl. The resistance of theshielded portion can be estimated as 2 series resistances.The thin gas blanket could be considered as a layerconveying heat mainly by conduction. The liquid core hasa Nu number that is slightly greater than N u . But thedifference is not big and may be neglected, especiallywhen added in series with the rather insulating gasblanket. Adding these three resistances yields:

Nulo= N&i( 1 ty/( 1+y))y = E N& I (klkg)

(7.1)( 7 4which is analogous to Eq. (5.13). Modeling the effects ofmicro-eddies could also be done using mixing lengthapproach in an analogous way as for friction:

This explains why Nu was less than expected in somecases (low Re), and higher in others. It also explains thedependence of Nu on Re in laminar flow.At this level, we need to analyze the notion of'expected' Nu. In fact, the value of Nucl is not as evidentas it may seem at first sight. It is defined as the value ofNu for a conventional channel of large size, having theSAME geometry AND boundary conditions as theconsidered micro-channel. But boundary conditions formicro-channels are peculiar in 2 respects. First, micro-channels fabricated by etching, either rectangular ortrapezoidal with an angle of 54.7", have one adiabatic side(called here the topside). If we assume that heat suppliedthrough channel walls is at constant heat flux $, as isdone by most investigators, then channel walls at anycross-section should have a constant temperature alongthe contour that linearly increases with the axial distance[40]. The adiabatic topside should evidently be treatedotherwise. As heat is supplied through 3 sides only, Nuclshould be less than that used for channels of homogeneousside walls. For rectangular channels, it is easy to obtainthe corresponding Nbl by solving analytically the energyequation on a cross-section:

8=0 for x=O,W, and y=Ode/dy=O for y=H

Nu,,= Nul,( 1+Pol0(ce)*RePr/8) (7.3)

V 28= 2(2*H+W)/(H+W)v/vaV, (7.4)

1- arms et al-- Model II I

0 1000 2000 3000 4000ReFigure 8- Comparison of model with Harms et al.

+ Wangsec.3 ~ Nbdel 0.5mm- - t w a n g s e c 5 ~ Pdbdel 0.3mm

10, k

0 -0 500 R~ 1000 1500

Figure 9- Comparison of model with Wang et al.where 8 is the dimensionless temperature k(T-T,)/(qwDh),v is the axial velocity obtained from solving (5.1)with no slip boundary conditions, and T, is the constantwall temperature. Finally, W an'd H are the width anddepth respectively. Note that q, here is the average takenover non-adiabatic walls only. The analytical solution of[40] using series expansion is standard, and need not bedeveloped here. The departure of'the resulting NLLI romthat of homogeneous walls increases with the ratio W/H.For low aspect ratio (W/H), another problem appears.Channel side walls at x=O and W, start acting as fins withvariable temperature along the contour. Weisberg et al.[41] have studied the 2 dimensional conjugate problem ina channel cross-section and showed that the total thermalresistance may be largely affected at low (W/H). Sabry[42] has also studied the 2D-temperature field on channelside walls along and normal to flow direction to show thatthis effect may not always be negligible. In this work,solving a 3D conjugate problem may not be appropriatesince it will only blurs the main goal, which is explainingmicro-channel deviations. Hence only a very simplecorrection will be adopted:

where N&I or is the corrected Nu rrumber taking fin effectsinto consideration where q is the &n efficiency. Using thissimple correction as well as the same 5 and c used forfriction, comparison with the results of Harms et a1 [20] isgiven in Fig. 8. Correspondence is quite satisfactorydespite of the crude correction done for conjugate heat

N u E I ~ ~ ~ =uEI(~~H+W)/(~H+W) (7 .5)

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transfer effects. Another test series is due to Wang et a1[17]. Unfortunately they did not report roughness values.A value of 2% was assumed here. They have tested 6different test sections. Sections 1 and 2 were disregardedsince the flow was not fully developed.The temperature difference used to calculate h wasunusual: outlet wall temperature-inlet fluid temperature.Fortunately, enough data were given to enable us to re-estimate it by the standard way for constant heat flux case:

wall-fluid temperature, both taken at outlet. Comparisonresults are shown in Fig. 9 for test sections 3 and 5 .Taking into consideration the large scatter in experimentaldata, the model was at least able to predict the correctorder of magnitude as well as the correct trend. Sections 4and 6, were identical to 3 and 5 respectively, but withthinner walls. Differences were significant, thus provingthat conjugate heat transfer effect is important, asindicated elsewhere [311.8. ConclusionsLiquid flow and heat transfer in micro-channels werestudied. Deviations of flow and heat transfer from those ofnormal size channels were reviewed. A critical analysiswas made of previously advanced hypotheses to explaindeviations of liquid flow. A hypothesis relying on surfaceroughness was made, based on which simple models werederived to predict most of the phenomena encountered inmicro-channels. In particular, the reduction of bothfriction and heat transfer compared to predictions ofclassical theory for low Re, as well as the increase of bothPo and Nu numbers with Re in the laminar range were allexplained. At relatively high Re, both Po and Nu may behigher than the classical theory as has also been observed.A criterion was also derived for the transition toturbulence that predicted the observed early transition.The model was successfully confronted with available testresults from different authors giving both the order ofmagnitudes and the trends. It is important to note that themodel contained no adjustable coefficients. It containedthough two parameters having a physical meaning andthus an expected limited range of values. Very goodagreement between model and experiments was obtainedfor parameter values lying in the middle of this range.Future work in this area may give a more precise value ofthese parameters depending on the nature of walls, liquidand machining procedure used.References[l] Tuckerman, D.B. & Pease, R.F.W., Highperformance heat sinking for VLSI, IEEE ElectronDevice Letters, Vol. EDL-2, No 5 (1981) pp, 126-129.[2] Kandlikar, S.G. & Grande, W.J., Evolution ofmicrochannel flow passages thermohydraulicperformance and fabrication technology, ASME Int.Mechanical Engineering Congress & Exposition,November 2002, New Orleans, Louisiana[3] Erickson, D.& Li, D. , Numerical simulations of alow power microchannel thermal cycling reactor,International Journal of Heat and Mass Transfer, 45(2002) pp. 3759-3770

[4] Ueno, K., Kim, H-B. & Kitamura, N., Channel shapeeffects on the solution-flow characteristics and theliquidliquid extraction efficiency in polymermicrochannel chips, Analytical Sciences, 19, (2003),[5] Keblinski, P. Phillpot, S.R., Choi,S.U.S. & Eastman,J.A., Mechanism of heat flow in suspensions of nano-sized particles (nanofluids), Int. J. Heat MassTransfer,45, (2002), pp. 855-863.[6] Souders, D., Khan, I ., Yao, G.F., Incognito, A. &Corrado, M. A numerical model for simulation ofcombined electroosmotic and pressure driven flow inmicrodevices. 7Ih Int Symp on Fluid Control,Measurement and Visualization,Flucom03, Perronte.[7] Pamula, V.K. & Chakrabarty, K. Cooling ofintegrated circuits using droplet-based microfluidics,GLSVLSIO3,April 2003, Washington, DC, USA.[8] Wexler, E., Lout@, R., Ortega, A., Wallinger, D. &Prindiville, T. Novel Boiling-Enhanced Multi-Channel Heat Sinks, The 6Ih ASME-JSME ThermalEngineering Joint Conference, March 2003[9] Papautsky, I, Ameel, T & Frazier, A.B. A review oflaminar single-phase flow in microchannels, ASMEinternational Mechanical Engineering Congress and

Exposition, November 2001[101 Sabry, M.N., Dynamic compact thermal modelsused for electronic design: A review of recentprogress, INTERPACK2003-35 185, July 2003,Hawaii[ l l ] Pfahler, J., Harley, J., Bau, H.H. & Zemel, J.Liquid and gas transport in small channels, ASME-DSC, vol. 19, ed. CHO et al., (1990) pp. 149-157.[121 Pfahler, J., Harley, J., Bau, H.H. & Zemel, J., Gasand liquid flow in small channels, ASME-DSC, vol.32, ed. CHO et al., (1991) pp. 49-60.[13] Jiang, X.N., Zhou, Z.Y., Huang, X.Y. & Liu, C.Y.,Laminar flow through microchannels used formicroscale cooling, Proc. 1s t Electronic PackagingTechnology Conference, 1997, pp. 119-122[14] Papautsky, I. , Brazzle, J., Ameel, T.A. & Frazier,A.B., Microchannel fluid behavior using micropolarfluid theory, Proceedings of the Eleventh AnnualInternational Workshop on Micro Electro MechanicalSystems, 1998, pp. 54 4 -549.[151 Mala, G.M. & Li, D., Flow characteristics of waterin microtubes, Int. J Heat and Fluid Flow,Vol. 20,[16] Choi, S.B., Barron, R.F. & R.O. Warrington, Fluidflow and heat transfer in microtubes, ASME DSC-vol.32, ed. Cho et al., (1991) pp. 123-134.[17] Wang, B.X. & Peng, X.F. ExperimentalInvestigation on Liquid Forced Convection HeatTransfer through Micro-channels, Int. J. Heat MassTransfer,Vol37, Suppl, 1, (1994) pp 73-82.[18] Peng X.F. & Peterson, G.P. Convective HeatTransfer and Flow Friction for Water Flow inMicrochannel Structures, Int. J. Heat Mass Transfer,

pp. 391-394.

(1999) pp 142-148.

V0139, NO. 12, (1996) pp 2599-2608.

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