Heat Sink Optimizarion With Application to Microchannels

7/29/2019 Heat Sink Optimizarion With Application to Microchannels

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83 2 IEEE TRANSACTIONS ON COMPONENTS, HYBRIDS, AN D MANUFACTURING TECHNOLOGY, VOL. 15, NO. 5, OCTOBER 1992

Heat Sink Optimization with Applicationto MicrochannelsRoy W. Knight , Donald J . Hal l , John S. Go o d l i n g , an d R i ch ard C. Jaeger, Fellow, IEEE

Abstruct- The equations governing the fluid dynamics andcombined conduction/convection heat transfer in a heat sink arepresented in dimensionless form for both laminar and turbulentflow. A scheme presented for solving these equations e nablesthe determination of heat sink dimensions that display the lowestthermal resistance between the hottest portion of the heat sinkand the incoming fluid.Results from the present method are applied to heat sinksreported by previous investigators mckerman and Pease [l],[2], Goldberg [3], and Phillips [4], [ 5 ] ) o study effects of their re-strictions regarding the nature of the flow (laminar or turbulent),the ratio of fin thickness to channel width, or the aspect ratio ofthe fluid channel. Present results indicate that when the pressuredrop through the channels is small, laminar solutions yield lowerthermal resistance than turbulent solutions. Conversely, when thepressure drop is large, the optimal thermal resistance is foundin the turbulent region. With the relaxation of these constraints,configurations and dimensions found using the present procedureproduce significant improvement in thermal resistance over thosepresented by all three previous studies. These improvementsrange from 10 to 35% in values for design thermal resistance.NOMENCLATURE

ABCCPC1DDe,

fDhGhkeLm

Area.Constant defined in (31).Constant defined in (32).Specific heat at constant pressure.Coefficient defined as n+r (n- 1) .Depth of heat sink.Equivalent laminar diameter of the fluid flowchannel.Hydraulic diameter of the fluid flow channel.Friction factor.Parameter defined in (23).Heat transfer coefficient.Thermal conductivity.Channel width.Length of heat sink in the direction of fluid flow.[ ( h P f i n / k f i n A c , f i n ) ] 1'2*

Manuscript received August 3 1, 199 1; revised June 25, 1992. This workwas supported by the National Scienc e Foundation under Grant CBT-8805 607,by the Alabama Microelectronics Science and Technology Center of AubumUniversity, and by the Alabama Research Institute under Grant ARI-90-201.R. W . Knight and J. S. Goodling are with the Mechanical Engineering De-partment and the Alabama Microelectronics Science and Technology Center,Auburn University, Aubum , AL 36849.D. J. Hall was with the Mechanical Engineering Department, AubumUniversity, Auburn, AL. He is now with Compaq Computers, Houston, TX77070.R. C. Jaeger is with the Electrical Engineering Department and the AlabamaMicroelectronics Science and Technology Center, Aubum University, Auburn,AL 36849.IEEE Log Number 9202704.

Total mass flow rate of coolant throughchannels.Number of cooling channels.Pressure difference number, (Ap / L)W 3/ (pu2).Work rate number, tbW/ (p u 3 ) .Nusselt number, h D h / k f i u i d .Nusselt number for fully developed flow.Fin perimeter.Pressure drop through the heat sink channels.Prandtl number, u/a .Heat source power.Reynolds number based on hydraulic diameter.Temperature.Largest temperature difference between coolantand source.Improvement in thermal resistance.Mean fluid velocity.Volumetric flow rate.Pumping power.Width of heat sink.width of channel, I?.Width of fin, re.

Greek SymbolsQ

71rrl Fin efficiency.e Thermal resistance, AT/q.0 Dimensionless thermal resistance,

Thermal diffusivity of fluid or aspect ratio.Percent of infinite fin performance.Coefficient defined by (24).Ratio of fin thickness to channel width.P

( A T / Q ) / h u i d W .U Kinematic viscosity of fluid.P Mass density.Subscriptsbase Fin base.C , fin Cross-sectional of fin.ch Channel.f, Fluid inlet.f ,o Fluid outlet.fin Fin.fluid Fluid.h Hydraulic.S , is , o

C Cross sectional available for flow.

S Surface available for heat transfer.Surface at the fluid inlet face.Surface at the fluid outlet face.

I. INTRODUCTIONHE advent of high density components has requiredT nvestigation of innovative techniques for removing heat

01484411/92$03.00 0 1992 IEEE


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KNIGHT et al.: HEAT SINK OPTIMIZATION 833

from these devices [l]. One method is through the use offorced convection heat spreaders called a heat sinks. This paperis inspired by the pioneering heat sink work offered first byTuckerman and Pease [l] , wherein they reported a methodfor cooling a chip by forcing coolant through closed channelsetched onto the backside of a silicon wafer. They observedthat for laminar flow in a channel, the heat transfer coefficientis inversely proportional to the channel width. A minimum inthe thermal resistance expression was used to size the coolantchannels sub ject to sev eral restrictions. A sample heat sink wasfabricated and tested with a resulting modest chip temperaturerise above ambient for very high flux levels of thermal input.This paper serves to show that, under certain circumstances,conditions imposed in previous studies are not desirable inoptimizing the fin configurations subject to a specified pres-sure drop through the heat sink channels. Since the problempresented here lends itself to generalization through nondimen-sionalization, the technique described herein is applicable toany closed, forced convection heat exchanger.

11. REVIEW F LITERATUREIn the original work by Tuckerman and Pease [ l ] , theoptimization of the heat sink design was done subject toseveral constraints: the flow through the channels was fullydeveloped and laminar in nature; the laminar Nusselt numberand the channel to fin width ratio (W &/ W fin ) were fixed, aswere the pressure drop through the fin array, pumping work,planar dimensions, and fin efficiency. Subject to the designeddimensions, a heat sink was constructed in a silicon wafer anda heat flux of 790 W/cm2 was achieved with a temperaturerise of 71C using water as the coolant.Since that landmark paper, many studies have been pub-lished using a similar optimization technique, but with fewvariations on the constraints. Most notably are the works ofGoldberg [3], Mahalingam [6], Sasaki and Kishimoto [7],Kishimoto and Ohsaki [8 ] Hwang et al. [9] and Phillips [4].

With the exception of the latter two referenced works, all wereiterations on the theme set forth by the Tuckerman and Pease[ l ] and showed that such a design has practical applicationsand can in fact be implemented.Goldberg [3] constructed three air cooled, forced convectionheat sinks and tested each one. Each heat sink had a differentfin thickness, with the Wch/Wfin ratio restricted to unity, andthe flow limited to the laminar regime. The air flow for eachheat sink was adjusted to provide a rate of 30 L/min. Asexpected, the design w ith the largest pressure drop and sm allestchannel width yielded the smallest thermal resistance.Mahalingam [6] constructed a microchannel heat sink ina 5 cm by 5 cm by 0.20 cm silicon substrate. The channelswere about 200 pm wide with a depth of 1700 pm, separatedby a 10 0 bm fin. A 3.8 cm by 3. 8 cm thin film heat sourcewas mounted on the topside and experiments were carried outusing water and air as coolants. The water experiments, at aflux of 1100 W , ielded thermal resistances of O.O3C/W andO.O2OC/W for flow rates of 12cm3/s and 63 cm3/s, respec-tively. Results for the air experiments were also presented, butyielded higher thermal resistances, as expected.

Sasaki and Kishimoto [7] optimized the channel dimensionsof a finned heat sink constructed on a silicon chip for a givenpressure drop. The Wch/Wfin ratio again was restricted tounity, and the optimal channel widths were found to be 40 0and 250 pm for a pressure drop of 200 and 2000 kg/m2,respectively, again subject to the laminar flow constraint. Theexperimental results were claimed to match the analysis well;however, the latter was not presented.Kishimoto and Ohsaki [8] discussed a packaging techniquewherein VLSI chips are mounted on a multilayered alumnasubstrate. Coolant channels (800 pm wide by 400 pm high)were made in the substrate at a staggered pitch of 2.54 mm . Athermal resistance of O.3l0C/W was obtained for a pressuredrop of 19.6 kPa and flowrate of 1 L/min of water.Hwang et al. [9] designed a multichip, water cooled modulesuitable fo r VLSI packaging. An analytically designed 25 chipmodule was modeled by a 9 chip experimental module withthe coolant flow rate scaled to match the 25 chip model. Thelaminar and turbulent flow regimes were considered with thelatter yielding the least thermal resistance. The turbulent caseyielded a maximum temperature rise per chip of 18C at achip power level of 42 W. A pressure drop of 55.2 kPa with aflow rate of 126 cm3/s was experienced through the heat sinkwith channel dimensions of 5870 pm wide by 1000 pm high,separated by a wall 1270 pm thick. This design is significantlydifferent from the fin concept utilized in this paper because thechannels are parallel to the heat source instead of perpendicularto it.Phillips [4 ] reported an innovative design method for watercooled, microchannel heat sinks in which both laminar andturbulent flow regimes were considered for hydrodynami-cally developing and thermally fully developed flow whererestrictions were placed only on the Wch/Wfin ratio and theaspect ratio. All assumptions made by Phillips a re thoroughlydiscussed in his paper. Most results are presented pictoriallywith thermal resistances displayed as a function of channelwidth. For the test case presented, the design which allowedturbulent flow yielded the smallest thermal resistance.Bar-Cohen and Jelinek [lo] optimized arrays of air cooledrectangular fins to maximize heat transfer or minimize finmass. In their scheme, fin material, air flow rate, availablepressure head, and either the fin depth or thickness is specified,and the remaining fin dimension (depth or thickness) andnumber of fins are determined by a computerized searchprocedure. In this manner, optimal designs of two air cooledfin systems are reported.Landram [ l l ] identified optimal configurations of heat sinksby the use of a computational scheme. T he temperature profilesfor both the cooling fluid and the conductive heat sink weresimultaneously determined. Optimal designs were found tobe strong functions of coolant to wall thermal conductivityratio and channel aspect ratio. The flow considered was fullydeveloped and laminar in nature.In a recent publication, Knight, Goodling, and Hall [12]present an optimization method for sizing coolant channelsfor a given pressure drop or pumping power, given pla-nar dimensions, specified fin efficiency or fin length, andcoolant and solid properties. The governing equations were


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834 IEEE TRANSACTIONS ON COMPONENTS, HYBRIDS, AND MANUFACTURING TECHNOLOGY, VOL. 15, NO. 5, OCTOBER 1992

Fig. 1. Schematic of the fin system.

made dimensionless, and characteristic groups (not unlikethe Reynolds number) were formed. An analytical solutionwas presented for the idealized case of infinitesimally thinfins which showed that the optimal turbulent case offerssignificantly better thermal resistance than the optimal laminarflow for some pressure drops. The optimization technique isthen applied to a heat sink with fins of finite thickness withsome simplifying assumptions, most notably of which is thatof fully developed flow. The design technique is applied to thedesigns of Tuckerman and Pease, and Goldberg. In that samepaper [12], it was shown that the optimal solution occurs inthe laminar regime when the pressure drop through the finarray is low and in the turbulent region when that pressuredrop is high.In this paper, the optimization method delineated in [12]is further generalized and includes developing flow and betterheat transfer correlations. Results using the present method arecompared to those of three previous investigations.

111. T H E MODELA schematic of the modeled heat sink being considered isshown in Fig. 1.It consists of a flat rectangular energy sourcewith a series of channels and fins extending from a base plate.The structure has n channels and ( n- ) fins. The tips of the

fins are insulated with a flat plate used solely for containingthe coolant flow.The equations presented in the following show that oncecertain variables are specified, the number of fins and thefin channel width ratio can be found such that the thermalresistance between the hottest point on the heat source and thecoldest point in the coolant is a minimum. The factors to bespecified are:1) the thermal conductivity of the material used for the2) the maximum allowable planar size ( L by W ) f the3) the properties of the coolant used ( p , c p , ,Pr, Icfluid);4) the maximum allowable pressure drop and/or pumping5 ) the desired fraction of infinite fin performance or spec-The first three require no elaboration in that the emphasishere is on the fin and channel geometry. Regarding condition

4), it was shown by [12] that if the design is constrainedby the maximum allowable pressure drop through the fins,

heat sink;heat sink;

work used;ified fin length.

AT

Le ng t hFig. 2 . Temperature variation in a constant flux heat exchanger.

then the device with the minimum thermal resistance couldrequire a pumping power which is comparable in magnitudeto the amount of heat dissipated in the heat source. This highpumping power requirement is due to the high volumetric flowrates found for a minimum thermal resistance. Since sucha solution is undesirable, a upper limit on pumping poweris advisable in the optimization scheme. Condition 5) is aconvenient option in the design procedure.The heat sink can be analyzed as a two-dimensional flowthrough narrow rectangular channels with a constant heat fluxboundary c ondition at the base of the fins. Temperature profilesin the L-direction for the heat sink and coolant are shown inFig. 2, if the heat transfer coefficient is assumed constant.The temperature difference between the solid heat sourceand the bulk fluid is the same at any plane in the L direction(Fig. 2) . Thus the total heat transfer rate equivalent to theelectrical circuit power is written as

q =h&(Ts, , - Tf,%)=A S ( T s p-Tf ,o). (1)This relation holds true even if the heat transfer coefficient isspatially variable, as long as the average value of h is used in(1).Furthermore, this same amount of power is transferred tothe fluid and is expressed by

4=+&,o -TfJ. (2)These two equations, written in the standard notation of heatexchanger terminology [13], are combined to yield thermalresistance defined as the maximum temperature differencebetween the source and coolant AT =(Ts,oT f , * ) ividedby the electrical power of the source.

(3 )The two terms on the right side of (3 ) are referred toas the convective resistance and capacity resistance terms,respectively, in heat exchanger terminology. The latter wasdesignated the caloric resistance by Tuckerman and Pease.

T


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IV. DIMENSIONLESSARAMETERSAs with many important problems, the use of dimensionlessgroups generalizes the problem and allows for scaling to anyphysical dimension. In a previous paper [12], each of theappropriate groups for this paper were presented in algebraicdetail. Each is briefly reviewed here.Equation (3 ) with the usual dimen sions of thermal resistance

(degree per watt) when m ultiplied by kfiuid W becomes anexpression for dimensionless thermal resistance:+-T--.=------ (4)f l u i dw k f l u i dw-T hAS mcP

A dimensionless group, N A ~ ,s related to the maximumallowable pressure drop experienced over the length of theheat sink. This pressure drop is dictated by the coolant pumpor fan capacity and is defined as

( 5 )p 1 ~ 3Nap =-PV 2

For realistic pressure drops through microchips, this numberis typically of order lo 8 toAnother group, Nwork,epresents the amount of pumpingwork used by a specified coolant pump or fan. It is defined as

(6 )If only Nap is specified for a given design, the pumpingwork required to attain minimum thermal resistance mightexceed the cooling required of the heat source, which is undermany circumstances unacceptable. Thus the procedure re-quired that this parameter have an upper bound. By definition,Nap and Nworkare related as follows.

Some other applicable dimensionless groups commonlyused in heat transfer andfluid dynamics studies are the:UmDhReynolds number: ReDh = vhD hNusselt number: NUDh =-fluid

Prandtl number: P r = Nfriction factor:

Additional groups relevant to this paper are the thermalconductivity ratio kfluid/kfin;ratio of fin width to channelwidth r = Wfin /Wch , the aspect ratio cy = D/l , the hea tsink length to width ratio L I W and the heat sink depth towidth ratio DI W .As a matter of convenience, other geometrical parametersare expressed through those described previously. For inter-nally confined fluid flow and heat transfer, the ex pressions for

friction and heat transfer coefficient are written in terms oftwo parameters; the hydraulic diameter, Dh, which for onechannel is expressed as2w

n+r ( n- 1)+( W I D )h =and the fluid flow channel aspect ratio

D - n + r ( n - l )- _c W I D . (9)The frontal cross-sectional area available for coolant flow inthe heat sink A , becomes

n W DA -, n + ( n - 1)while the surface area available for heat transfer,A,, is writtenas:

+2 q D L ( n - 1). (11)n W Ln+ ( n - 1)s =The first term is the area available for heat transfer at the baseand between the channels. T he second term is the effective finarea with the fin efficiency included.

V. FIN EQUATIONSHeat transfer enhancement due to the presence of fins canbe quantified with the assumptions of one-dimensional con-duction along the fin length only, constant material properties,constant heat transfer coefficient, no radiation, and uniform finbase temperature. First, a convenient grouping of terms, m, sdefined.

The approximate equality results from the assumption that

There are at least two methods of quantifying the degreeto which fins enhance heat transfer. The first is through theuse of fin efficiency, defined as the ratio of actual fin heattransfer to that of a similarly shaped fin with infinite thermalconductivity. For straight rectangular fins of uniform crosssections and insulated tips, the actual fin heat transfer canbe found analytically [13] as

L >> ri.

while the heat transfer of the ideal fin with infinite conductivityis

The ratio of these two expressions forms the fin efficiency.(15)t a n h ( m D )r l = mD .

A second method follows the concept set forth by Tucker-man and Pease wherein performance was defined by the ratioof actual heat transfer of a fin of finite length compared to that


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836 IEEE TRANSACTIONS ON COMPONENTS, HYBRIDS, AND MANUFACTURING TECHNOLOGY, VOL. 15, NO. 5, OCTOBER 1992

of a fin made of the same material and having the same crosssection, but of infinite length. The infinitely long fin transfersheat according to the following relation:A . Laminar Flowanalytically as a simple function of Reo,.For fully developed laminar flow, f can be expressed

The ratio of these is,h=t a n h ( m D ) .

f=-. 71Reo,

The value of 71 is determined from the aspect ratio of a givenchannel. Bejan [14] defined a parameter G as:(17)G = ( [ lo )' +1 (23)( l / D +1)' 'Equation (11) incorporates the symbol q, in efficiency asdefined in (15), as a convenient way of expressing effectivefin heat transfer area. Equation (1 7) is also used in the currentoptimization scheme.Once a value of 77 or p is specified, m D can be found from(15) or (17). Using geometric relations and the definition of

A least squares fit of a straight line in G to available valuesfor 71 yields

Nusselt number in (12) yields the following relation: 71 =4.70 +19.64G. (24)Equation (24) agrees with exact values [15] within 23%.Algebraic combination of (20)-(22) along with the definitionsof NAP and Dh allow for the Reynolds number to be expressedas a function of the NAP and other parameters that are afunction of the geometry only:

["' ~ ~ ~ ' 1mD)2=Nuoh kfluidlkfin) (I8)which can be rewritten as a quadratic equation in D / W

where C1 is defined as n+r ( n - 1) .Consequently (19) canbe readily solved for dimensionless fin length.VI . HEAT TRANSFER

The fins shown in Fig. 1 are cooled by forcing a fluidthrough the channels formed by adjacent fins. Depending onthe value of the Reynolds number, this forced convective flowis either laminar or turbulent in nature, or it will be in transitionfrom the former to the latter. The magnitudes of pressure dropand the heat transfer coefficient depend on which of theseregimes the flow is in.The Reynolds number is defined as:

Th e heat transfer coefficient, h, is obtained from the Nusseltnumber. It is a function only of the channel aspect ratio. Theparameter, G , used in com puting the friction factor is .usedagain. As before, a least squares fit to the available exactvalues gives(26)uD,, =-1.047 +9.326G.

Equation (26) agrees well with analytical results [15].resistances are formulated respectively asLaminar dimensionless capacity and convective thermal

andkf lu idW-hA,N u D , ( L / W ) [ C 1 +(w/D) l [n +2 q ( D / w ) ( n - ) c l ]

where U,,, is the mean velocity, Dh is the hydraulic equivalentdiameter of the enclosed conduit, and v is the kinematicviscosity of the coolant. The generally accepted value of thefrom laminar to turbulent flow begins is Reo, E 2300 [13].For the flow to be assured turbulent, Reo, 2 4000. The

(28)Reynolds n umber fo r internal duct flow at which the transition 2C1-where C1 =n +r ( n- 1).function of dimensionless pressure drop, Nap, as follows:

transitional regime exists between these two values.dimensionless Fanning friction factor:Pressure drop can be obtained through the definition of a Dimensionless work, Nwork, is expressed as an explicit


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837NIGHT et al.: HEAT SINK OF'TIMIZATION

B . Turbulent Flowbe channelled downstream as is the case here, the velocity

As with velocity profiles, the thermal profiles must also de-When flow originates from a reservoir or manifold to velop from the entrance downstream. For thermally develop ingZukauskus [191 Offers the

profiles will grow to their final steady-state form some distancedownstream from the entrance. For this hydrodynamicallydeveloping turbulent flow, Phillips [4] recommends a frictionfactor of the form [ d m (37)3600 1= +o.48(L/Dh)-0.25 +Nu, . exp ( -0 .17LIDh)which is valid for 4000 5 ReDh 5 5 X l o 5 , 0. 7 5where Pr 5 1.0; and L / D h 20.0 6. The fully developed Nusselt

For thermally d eveloping turbulent flow using coolants withPrandtl numbers not covered by the Zukauskus correlation,the Gneilinski equation for thermally developing flow is used.Gneilinski [18] offers the correlation

f =B R eC (30)

(31) numbe r is Nu,..01612( L I D h )0.31930(L/Dh)

B =0.09290 +and

(32)=-0.26800 - -.When L/Dh becomes large, (30) yields values similar to thosefound by other investigators for fully developed turbulent flow[16]. Equation (30) is valid for 2300 5 R e 5 28000 and isused for hydrodynamically developing turbulent flow in thispaper.Equation (30) is for round enclosures. Jones [17] recom-mended that an equivalent diameter ( De s) should be used inlieu of the hydraulic diameter (Dh) for rectangular ducts. Theexpression for their ratio developed by [17] is

This is valid when the channel aspect ratio, l/ D , is less thanone. The equivalent diameter concept used in circular duct

which is valid for 2300 5 ReDh 5 l o 6 ; and 0.6 5 Pr 5 l o 5 .This correlation does not include the effects of ReDh or Pr.For thermally developing turbulent flow (37) is used for gases(low Prandtl Number fluids) and (38) for liquids (high PrandtlNumber fluids).The dimensionless capacity and convective thermal resis-tances for turbulent flow are written, respectively, as

correlations reduces the scatter of rectangular duct, turbulentflow friction factors from 520% to 25% [16].Once a friction factor is determined from the pressure dropand geometry, the mean velocity and the Reynolds num ber canandk f l u i d ~hA,

(40)be found. Whenf iswritten in the form of (30), U , becomes 2C1

1 N uD h( L/ W) [ Cl +( w / D ) ] [ n +2 ' d D / W ) ( n - )C11'[ % ] D r - c ) v c a+c (34) As was done for laminar flow, an expression for the relation-ship between N a p and Nwo rk or turbulent flow is derived:Nwork =N ~ ~ ( L / W ) ( D / W ) ( ~ / C ~ ) { ~ [ ~(W/D)]}-1/2 .

[ 2 p B ] 'From the definition of the Reynolds number, hydraulic diame-ter, and (34), an expression for the turbulent Reynolds numberis written as (41)

There exists a multitude of heat transfer correlations forturbulent flow in enclosures. The Gneilinski equation forhydrodynamically developed turbulent flow [181 is chosen heredue to its excellent corroboration with experimental data [161.(36)( f / 2 ) ( R e ~ ,- 1000)Pr1.0+1 2 . 7 0 Pr2l3- 1)Nu, =

Table I is a recapitulation of the applicable equationsfor designing a heat sink. For any given n r, mD, andmaximum al lowable N ap , Nworkand dimensionless fin length(D IW ), there are five unknowns to be found: Reynoldsnumber, friction factor, Nusselt number, actual fin length, andactua l N a p or Nwork.The unknowns are found from thesimultaneous , terative solution of the five nonlinear eq uationsfor friction factor, Reynolds number, Nusselt number, finlength relation, and pumping work given in the table. Thesolution method is delineated in the following.The solution procedure for laminar flow for a given n, I?,m D, and maximum al lowable N a p , Nwork,and D/W is asfollows.


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838 IEEE TRANSACTIONS ON COMPONENTS, HYBRIDS, AN D MANUFACTURING TECHNOLOGY, VOL. 15, NO. 5,OCTOBER 1992

TABLE IEQUATIONSOR THE OF'TIMlZATlON OF HEATSINK DESIGNHeat Sink Relations

Laminar: f =e:1 =4.70 +1 9 . 6 4 6 : 6 = ( f /D+ l )e D hFriction Factor Turbulent: f =B R e b h : B =0 .09290 +w: =-0 .26800 - L I D h )

Laminar: R ~ D ,=(4/*,1 ).vA,[n + ( n- 1)+( u - /D ) ] -~Reynolds Number

1Turbulent: R e o , ={ ( 4 / B ) > V ~ , l n+r(n- 1)+ W' /D ) ] - ' }=Laminar: Kunh=-1 .047 +9 . 3 2 6 6

Nusselt Number(f/z) .e., -inon ~r1 0+12 7@[Pr2/3-1]

1urbulent: Xu, =

Fin Length Relation

Capacity ThermalResistance

Convective ThermalResistanceI

Assume a D / W and N a p .Maximum allowable val-ues are convenient starting points.Find t/Dh from (9).Find G from (23).Use (24) to find 71.Calculate the Reynolds number from (25). The re-sulting Reynolds number is checked to ensure that itfalls in the range of applicability for the laminar flowcorrelations used here.Use (29) to find Nwork. If Nwork exceeds the maxi-mum allowable value, solve (29) for the N a p valuewhich results in the maximum allowable Nwork. IfN a p is changed here, return to step L5).Calculate the Nusselt number from (26).Solve (19) for D/W. If D / W exceeds the maximumallowable value, use the maximum allowable D /W .

L9) Repeat steps L2)-L8) until D / W and N a p areconverged.L10) Calculate convective resistance from (40), capacity

resistance from (39), and sum them to obtain totalthermal resistance.

Steps L1)-LlO) are repeated for a wide range of n and rvalues, and the geometry resulting in laminar flow which yieldsthe minimum laminar thermal resistance is found.

The solution procedure for turbulent flow for a given n, r,m D , and maximum allowable N a p , Nwork, a nd D / W i s a sfollows.T1) Assume a D/Wand N a p . Maximum allowable val-T2) Find L/Dh from (8).T3) Find B and C from (31) and (32).

ues are convenient starting points.


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Researcher R r kPa (i n t i l , Watt ir, liter/min A@H 2 0 )-oldberg 25 1 1.17 (4.68) 0.583 300.435 1.17 (4.68) 1.747 89.9 32.4%Present Study same U: 0.390 0.73 (2.92) 0.583 48.2 15.4%c ~ ~ \ ! ~ ~ f ~ ~ sresent Study same A pI

c a ~ = l ~ , ~ ~ ~ s

-oldberg 12.5 1 0.29 (1.17) 0.146 300.315 0.29 (1.17) 0.239 49.2 18.4%I Present Study same U 0.390 0.26 (1.06) 0.146 32.8 11.4%c ~ = l ~ , ~ ~ ~ sresent Study same A p-oldberg 1 0.047 (0.19) 0.024 30

0.250 0.047 (0.19) 0.018 22.4 38.6%0.320 0.075 (0.30) 0.024 19.1 46.2%Present Study same ApPresent Study same U.

Calculate the Reynolds number from (35). The re-sulting Reynolds number is checked to ensure thatit falls in the range of applicability for the turbulentflow correlations used here.Find friction factor from (30).Use (41) to wind Nwork. If Nwork exceeds the max-imum allowable value, solve (41) for the N a p valuewhich results in the maximum allowable Nwork.fNap is changed here, return to step T4).Calculate the Nusselt number from (33), (36) and (37)or (38).Solve (19) for D/W. f D/W exceeds the maximumallowable value, use the maximum allowable D / .Repeat steps T2)-T8) until D/W and N a p ar econverged.Calculate convective resistance from (40), capacityresistance from (39), and sum them to obtain totalthermal resistance.

Steps Tl)-T10) are repeated for a wide range of n and rvalues, and the geometry resulting in turbulent flow whichgives the minimum turbulent thermal resistance is found.The geometry yielding the minimum thermal resistance,considerin g both laminar and turbulent flow, can be determinedfrom a comparison of the laminar and turbulent thermalresistances.

VII. RESULTSThe optimization scheme described previously was appliedto three published studies; one by Goldberg [3] where aircooled, copper heat sinks were designed and tested; one byTuckerman and Pease [ l ] using a laminar flow, water cooled,silicon heat sink; and one by Phillips [4] which was also awater cooled, silicon heat sink designed without the laminarflow restriction. In all three of these cases , the original authorsrestricted the channel to fin width ratio (r) o unity. Further

a (the aspect ratio of the coolant channels) was fixed in bothGoldbergs and Phillips studies. The results presented hereshow notable improvement in the performance of the heat sinkby relaxation of the previously imposed restrictions.Goldberg designed, built, and tested three narrow channelforced convection copper heat sinks cooled by room tempera-ture air. The size of the square heat source was 0.635 cm by

0.635 cm (1/4 by 1/4 in) with a fixed fin length of 1.27 cm(1/2 in). The designs by Goldberg mandated a flow rate of 30L/min, thereby establishing the capacity component of thermalresistance. The heat sink designs were not optimized, but ratherthe channel and fin thicknesses were systematically varied atvalues of 5 , 10 and 25 mils (corresponding to 0.127, 0.254,and 0.635 mm and aspect ratios of a = 100, 50, and 20).The most severe restrictions were mandatory laminar flow andr = 1. In Goldbergs design procedure the Nusselt numberwas fixed at a value of eight. When the restrictions of avolumetric flow rate of 30 L/m in, N usselt number of eight,and the geometries specified by Goldberg are examined in (3)or (4), the obtained predicted thermal resistances agree withthose found by Goldberg.In the present analysis, the overall heat sink dimensions ofGoldbergs design were kept, but those regarding fixed r, a,NuDh , flow regime and volumetric flow rate were relaxed. Theresults for each case are presented in Table 11. For each of thethree cases compared, the best design was determined either atthe same pressure drop through the device or the same powerconsumed by the fan as those set or measured by Goldberg. Ineach case, the optimal solution occurred in the laminar regiondue to the allowable pressure drops. Likewise, the choice ofvalue was significantly less than unity. The last column inthe table indicates that significant improvement in the optimaldesign (11-39% in thermal resistance) is achieved when ther = 1 restriction is dropped. These results show that thiscondition is neither necessary nor desirable.

Tuckerman and Pease designed, built, and tested a 1 cmby 1 cm water cooled silicon, microchannel heat sink. Theiroptimization procedure included assumptions of laminar flow,fixed pressure drop through the heat sink, r = 1, a fullydeveloped and fixed Nusselt number ( N u D ~ )=6, p =0.76,and a friction factor for infinite parallel plates (71 96).Their study indicated that the best design occurred withchannel widths of 57 pm and channel depths of 365 pmfor the pressure drop considered. At a power level of 79 0W/cm2 through the heat spreader, a resulting AT of 71OCwas measured. Pumping power was 0.3% of dissipation, or2.27 W (corresponding to 11cm3/s flow rate at 206.8 kPa,or 30 lbf/in2 pressure drop). In a previous study by Knightet al. [12], the optimal laminar flow design under the above


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840 IEEE TRANSACTIONS ON COMPONENTS, HYBRIDS, AND MANUFACTURING TECHNOLOGY,VOL. 15, NO. 5, OCTOBER 1992

TABLE 111COMPARISONF PRESENTESULTS O THOSE F TUCKERMANND PEASE l]

Tuckerman and Pease Present StudyCONSTRAINTSSize, Length (L) by Width (W )pressure drop, Ap% infinite fin performance, 3Fin length, pmCoolantPrandtl number, PrFin Materialk f l u l d l k f i nFin to channel thickness ratio, r'Nusselt NumberType of flow-fl, laminar friction factor

~ ~

1 cm x 1 c m206.8 kPa76 %unrestrictedwater3.71Silicon0.0046416laminar96

samesameunrestricted365samesamesamesameunrestrictedunrestrictedlaminar or turbulentnot applicable

DIMENSIONLESS GROUPS

CALCULATED RESULTSn, number of channelsDepth, D , pmFin to channel width ratio, r'Fin thickness, pmChannel thickness, pmReynolds NumberVolumetric Flow Rate, cm3/sAspect Ratio, cyNusselt NumberLaminar fr iction factor -r i1YA pAVwork

Capacity Thermal Res, -C/WConvective Thermal Res, OC/WTotal Thermal Resistance, " C / WChange in Thermal Resistance, A@

12.82 x 10"3.62 x 1013Laminar883651575173 0116. 46962 .82 x 10''3 .62 x 10130.0220.0640.086

samesameunrestrictedTurbulent223650.21581317845959.20.9785.6not applicable2.82 x 10"1.95 x 10140.0060.0500.05635%

restrictions was obtained using the design method reported inthis paper. The obtained dim ensions and results were a ll within5% of those obtained by Tuckerman and Pease.In the present paper, the pressure drop and overall packagedimensions (planar dimensions of heater and fin length) weremaintained the same while the restrictions on the state of flowdevelopment (fully) and the values of I?, yl, nd NuDh wererelaxed in order to find optimal channel and fin dimensions.An optimal solution for turbulent flow was sought. Table 111is a presentation of results.When turbulent flow is allowed, the thermal resistance isreduced by 35% from that of Tuckerman and Pease. Therelatively wide channels found for the best turbulent solu-tion allow, for fixed pressure drop and same fin height, agreatly increased mass flow, thereby significantly reducing thecapacity resistance term. Likewise, the heat transfer coefficientincreases due to the presence of turbulence, reducing theconvective term by 22%.

A maximum pressure drop of 206.8 kPa (3 0/bf/i n2) iscommon to both cases. Pumping power for the turbulent case isincreased to 12.2 W or 1.6% of heater power, still a negligibleamount.

The results here differ from those presented previously[12]; not in the magnitude of improvement but rather inthe heat sink design configuration. This is due primarilyto the inclusion of developing flow for both the fluid flowand heat transfer equations. The current modeling programnow includes developing turbulent flow, a laminar equivalenthydraulic diameter, and a better correlation for turbulent heattransfer in rectangular channels.Fig. 3 graphically quantifies the influence these limitationshave on the optimal thermal design of heat sinks. Three setsof dimensionless thermal resistance values are plotted as afunction of number of channels, n. Common to all are thematerials, fluids, pressure drop, and overall dimensions ofthe heat sink (1 cm by 1 cm by 365 pm fin length) usedby Tuckerman and Pease [l].The equations employed foranalyses are those presented in Table I.

A dashed line is drawn for flow with r =1.The optimumin the laminar regime occurs at n =93, a value close to thatfound by [l], he difference being due to the laminar Nusseltnumber in this paper being variab le with aspect ratio. The solidline in the laminar regime is drawn with the J? restraint liftedand results in a 17% improvement in thermal performance


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KNIGHT et al . : HEAT SINK OPTIMIZATION

-----

TABLE IVCOMPARISONF PRESENTRESULTSTO THE WORKF PHILLIPS[4 ]

1.00.8 r0.6

0.4

0.2

841

Phillips Present StudyCONSTRAINTSSize, Length ( L )by Width ( W )pressure drop, ApAspect RatioFin Length, micronsCoolantFin MaterialFin to channel thickness ratio, rNusselt Number CorrelationType of flowDIMENSIONLESS GROUPS

1 c m x 1 c m68.9 kPa4.0unspecifiedwaterSilicon1laminar or turbulent(37).

samesameunspecified1889samesameunrestrictedsamesame

L/WMaximum NhpMaximum NworkCALCULATED RESULTSn, umber of channelsDepth, D , pmFin to channel width ratio, rFin thickness, pmChannel thickness, pmVolumetric Flow Rate, cm3/sNAPNwork

Capacity Thermal Res, "C/ WConvective Thermal Res, "C/WTotal Thermal Resistance, "C/WChange in Thermal Resistance, A 0

18.86 x l o 91.45 x 1014

1118891472.1472.11458.86 x l o 91.45 x 10140.00160.0640.066

samesamesame

1918890.6621532314 58.86 x IO 91.45 x 10140.00160.0490.05221 %

0. r=i0 , r=OptimumOptimum0.00100.0009

0.0008 1 / 11.2\ /

0.00070 0.0006

0.00050.0004

0.0003I I I 1 I I0.0002 ' '0.00 25 50 75 100 125 150 175Number of Channels

Fig, 3. Thernial resistance as a function of the number of channels for theheat sink described in Table 111

with = 0.32 being the best value. The solid line in theturbulent regime depicts thermal resistance values for turbulentflow, again with no constraint on r. Here the performance isimproved by 32 % over that for laminar flow and fixed r,withthe optimal value of r= 0.215.

The dotted lines in Fig. 3 shows the value of J? whichyielded the lowest thermal resistance for a given n. The slopeof these lines is seen to change sharply near an n value of45 and again near an n value of 70. Between these two nvalues, the r value which minimizes the equation for thermalresistance yields a Reynolds number between 2300 and 4000.Since the correlations used are not valid in this regime, suchresults are viewed as unacceptable. So,between n of about 45and the end of the laminar flow region, the reported r gives thehighest allowable Reynolds number for laminar flow, 2300.This yields the lowest thermal resistance with the Reynoldsnumber in the range where the correlations used are accurate.In the turbulent flow regime, the given I? yields the lowestacceptable Reynolds number for turbulent flow when n is lessthan 70 for the same reason. Since these breaks in slope arenot near the laminar or turbulent minima, there is no effect onthe resulting predicted overall optimal design.

Phillips devised a scheme for microchannel heat sink designto include the possibility of turbulent flow and accountedfor hydrodynamic flow development in both regimes, butmaintained the restrictions of r = 1 and a specified aspectratio. Phillips used water as the coolant and silicon as the heatsink material with all properties evaluated at 27C. When theoptimal geometry determined by Phillips is examined usingthe equations presented in this paper, excluding the effect ofthermally developing flow, a thermal resistance is predicted


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whose value is within 1% of that given by Phillips. W hen thegeometry recomm ended by Phillips is analyzed, accounting fordeveloping flow, the predicted thermal resistance is loweredby 18% from the thermally fully developed case.Only a comparison with his turbulent optimal solutions ismade here to show the effects of mandating r and a. For ourcase, the fin length is identical to that of Phillips, but the finthickness to channel width ratio (r), nd therefore, the aspectratio is determined for the best turbulent case.The pressure drop, volumetric flow rate, and pumping workare identical to those used by Phillips. The present design hasthe same overall exterior dimensions as that of Phillips butas Table IV shows, improved performance by 21%. The mainreason for this improvement comes from the inclusion of theeffect of thermally developing flow. Additional improvementcomes from the relaxation of the r constraint. The currentdesign has a much smaller I (0.66 compared to 1 for Phillips),more channels (19 versus 1 ), and consequently substantiallymore area available for heat transfer for the same fin length.It is recognized that the optimal design scheme describedhere could lead to a design which could be impractical dueto the channels or fins being too thin to manufacture. Thisoptimization scheme necessarily incorporates a constraint onminimum thickness of either fin or channel.

VIII. CONCLUSIONSThe governing equations for fluid dynamics and heat trans-fer have been presented in a generalized, dimensionless formalong with applicable geometrical relationships. These can beused to determine the dimensions of any microchanneled heatsink of rectangular coordinates such that the resulting thermalresistance is minimized.Comparisons of present results with those obtained byprevious investigations show that unnecessary and undesirablerestraints were imposed on their design procedures (laminarflow, fixed fin thickness to channel width ratio and/or fixedaspect ratio) and that relaxation of these restraints leads to

significant improvements in designed thermal resistances.Depending on whether the pressure drop or pumping powerwas maintained the same, the Goldberg heat sink designimprovemen t ranged from 11.4 to 46.2% in therm al resistance.The most notable changes occurred in the designed valuesof I?(- 0.3) and the number of fins. The redesign of theTuckerman and Pease laminar heat sink rendered a muchsmaller value for r (0.215 as compared to I), the numberof fins (reduced from 88 to 22), and the nature of the flow(turbulent rather than laminar) with a resulting decrease inthermal resistance of 35%. Refinement in Phillips designeffected a decrease in the r value from unity to 0.66 andmore than a 50% increase in the number of fins resulting in apredicted decrease of 21% in thermal resistance.REFERENCES

[ I ] D. B. Tuckerman and R. F. W. Pease, High-performance heat sinkingfor VLSI, IEEE Electronic Device Lett.. vol. EDL-2. DD. 126-129.

[2 ] D. B. Tuckerman, Heat transfer micro-structures for integrated cir-cuits, SRC Tech. Rep. No. 032, SRC Cooperative Research, ResearchTriangle Park, NC, 1984.[3] N. Goldberg, Narrow channel forced air heat sink IEEE Trans. Comp.Hybrids Manuf. Technol., vol. CHMT-7, pp. 154-159, Mar. 1984.(41 R. J. Phillips, Micro-channel heat sinks, Advances in Thermal Mod-eling of Electronic Components and Systems, Volume 2. A. Bar-Cohenan d A. D . Kraus, eds. New York: ASME, Ch. 3, 1990.[5] A. Bar-Cohen and A. D. Kraus, Advances in Thermal Modeling ofElectronic Components and Systems, Volume2, New York: ASME, 1990.[6] M. Mahalingam, Thermal management in semicond uctor device pack-aging, Proc. IEEE, vol. 73, pp. 1396-1404, Sept. 1985.[7] S. Sasaki and T. Kishimoto, Optimal structure for micro-groovedcooling fin for high-power LSI devices, Electron. Lett., vol. 22, no. 25 ,pp. 1332-1334, 1986.[8] T. Kishimoto and T. Ohsaki, VLSI packaging technique using liquid-cooled channels, 36th Electronics Components Conf. P roc., May 1986,pp. 595-601.[9] L. T. Hwang, I. Turlik, and A. Reisman, A thermal module design foradvanced packaging, J . Electron. Mat., vol. 16, no . 5, pp. 347-355,May 1987.[lo] A. Bar-Cohen and M. Jelinek, Optimum arrays of longitudinal, rect-angular fins in convective heat transfer, Heat Transfer Eng., vol. 6 ,no. 3, pp. 68-78, 1986.[ l l ] C . S . Landram, Computational model for optimizing longitudinal finheat transfer in laminar internal flows, Heat Transfer in Electron.Equipment, vol. 171, pp. 127-134, 1991.[I21 R. W. Knight, J. S . Goodling, and D. J. Hall, Optimal thermal design offorced convection heat sinks-Analytical, J . Electron. Pack., vol. 113,no. 3, pp. 313-321 , Sept. 1991.[13] F. P. Incropera and D . P. DeWitt, Fundamentals of Heat and MassTransfer. New York: Wiley, 1990.[14] A. Bejan, Convection Heat Transfer.1151 W. M . Kays, and M. E. Crawford, Convective Heat and Mass Transfer,New York McGraw-Hill, 1990.116) S. Kakac, R. K. Shah, and W. Aung, Handbook of Single-phaseConvective Heat Transfer,[17] 0. C. Jones, Jr., A n improvement in the calculation of turbulent frictionin rectangular ducts,J. Ffuids Eng., vol. 98, pp. 173-81, June 1976.[18] V. Gneilinski, New equatio ns for heat and mass transfer in turbulentpipe and channel flow, Int. Chem. Eng., vol. 16, pp. 359-368, 1976.[191 A. Zukauskas, High-Performance Single-phase Heat Exchangers, J.Karni, ed.

New York: Wiley, pp. 75-82.

New York: Wiley, 1987.

New York: Hemisphere, 1989

Donald J. Hall was born in Columbus, GA, in 1965.He received the B.S. degree in applied mathematicsand the M.S. degree in mechanical engineering fromAuburn University in 1987 and 1991, respectively.Since 1991, he has been employed by CompaqComputer Corporation of Houston, TX , as a me-chanical engineer.Mr. Hall is a member of the A merican Society ofMechanical Engineers.

Roy W. Knight, for a photograph and biography, please see page 760 ofthis issue.

John S. Goodling, for a photograph and biography, please see page 760 ofthis issue.

Richard C. Jaeger, (S68-M169-SM78-F86), for a photog raph and biog-I IMay 1981. raphy, please see page 821 of this issue.

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Heat Sink Optimizarion With Application to Microchannels