OPTIMIZATION AND DESIGN GUIDELINES FOR HIGH FLUX MICRO-CHANNEL HEAT SINKS FOR LIQUID AND GASEOUS SINGLE-PHASE FLOW

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1 OPTIMIZATION AND DESIGN GIDELINES FOR HIGH FLX MICRO-CHANNEL HEAT SINKS FOR LIID AND GASEOS SINGLE-PHASE FLOW Norbert Müller, Luc G. Fréchette Mechanical Engineering Columbia niversity in the City o New York P.O. Box 3 New York, NY, SA, 7 Phone: () 8-96 Fax/Voice: (37) norbertmueller@yahoo.com Web: ABSTRACT A numerical optimization tool is used to optimize orced convection micro-channel heat sinks or minimum pump power at high heat luxes. Results gained with the optimization tool are generalized and optimum conigurations are illustrated on design charts. Physical trends are illustrated analytically using the underlying relations. Investigations are done or air and water in single-phase low with no phase transition; with hydraulic diameters o the micro-channels ranging rom about micron to 8 mm. Optimization is shown to have a tremendous eect. It can reduce pump power by several orders o magnitude, especially or high heat lux devices. sing water and air as coolants, designs or heat luxes o > kw/cm and > W/cm respectively with pump/an power expenses less than % are easily ound with the optimization tool. It appears that large aspect ratios or the low channels are avorable or high heat luxes in micro heat sinks. For each design, there exists an optimum in height and ratio o in thickness to channel width. The design space investigation also suggests that short channels in multiple parallel units are a key eature o compact high heat lux heat sinks. Practical design examples underline the easibility o the results and general design guidelines are derived. KEY WORDS: MEMS, IC cooling, inned surace, maximum heat transer, minimum pump power NOMENCLATRE A area, m c velocity, m/s cf velocity actor, (-) c p speciic heat coeicient, J/kg/K C in constant, /m D h hydraulic diameter, m H in height (channel height), m h heat transer coeicient, W/m /K F unction riction actor, (-) k thermal conductivity, W/m/K L length, m m mass low rate, kg/s N number o channels per in unit, (-) Nu Nusselt number, (-) P an/pump power, W p pressure, Pa q heat lux, W/m heat rate, W Re Reynolds number, (-) S shape actor, (-) T temperature, K number o in units, (-) W channel width, m Greek symbols α in unit angle, dierence ϑ temperature dierence, K δ thickness, m η an/pump eiciency, (-) η in eiciency, (-) η o overall surace eiciency, (-) ν kinematic viscosity, m /s ρ density, kg/m 3 ζ loss coeicient, (-) Subscripts inlet outlet add additional b base corr corrected Cu copper in m logarithmic mean, mean max maximum Opt optimum Si silicon in unit INTRODCTION The rapid increase in power densities o ICs has induced a permanent interest in new reliable and high heat lux cooling technologies [, ]. In parallel, there has been an interest in optimizing conventional heat sinks, in which a single-phase luid is orced over a inned surace [3, ]. These heat sinks are mostly chosen or their low manuacturing costs. So today, extruded heat sinks with ducted ans are the standard or processors in many computer systems. In addition, ongoing

2 research in the area o micro thermo-luid systems raises the need or high lux heat exchangers using a liquid or gaseous working luid, or devices such as micro chemical reactors, rerigeration cycles on-a-chip, microabricated heat engines, and portable uel cells. A comprehensive analysis o studies on heat transer and luid low in micro-channels over the past decade can be ound in Sobhan and Garimella []. For orced convection, the heat lux increases with smaller low channels and - when the channels are rectangular - with higher aspect ratio. But also, the necessary an/pump power or orcing the coolant through the channels increases at smaller scales. In this work, a simple optimization tool is described and used to optimize in geometry and coolant volume low. For a given base and required heat removal, the optimization consists o inding the optimum geometric coniguration o the ins that requires the least pump power to orce the coolant through. nlike Copeland [], Tuckerman and Pease [3], the current analysis assumes no limits or set values or in thickness, channel width and height as well as pump power or low velocity. The objective is to ind the optimum design or each condition in a large design space independent o the chosen manuacturing approach. However, i necessary such limits easily can be implemented. Conigurations Figure a shows a regular inned base, where the parallel ins have the same length as the base and the coolant lows rom one end to the other along a single channel ormed between the ins. Instead o extending the ins over the entire length o the base, they can be divided in units, each a raction o the base length. Fresh coolant is introduced at the beginning and heated coolant discharged at the end o each unit (Figure b). As will be shown, the through low velocity and in turn pressure loss and pump power can be reduced or similar or higher heat lux. However, to realize this approach a longer base is necessary to account or the space required or the inlet and discharge o coolant to each unit ((-). L add ) as illustrated in Figure c. This additional spacing can be alleviated, by arranging the in units side-by-side, as shown in Figure d. This coniguration requires a heat spreader i the heat is not generated in long narrow bands, which could be imagined or uture uel cell ribbons and some electronic or electric devices. When the pressure loss o the coolant inlet and outlet between the in units in Figure c is taken into account, it appears that a zigzag arrangement o the in units like in Figure e can reduce the overall pressure loss. Approach In either coniguration, the analysis is centered on determining the heat removal rate and luid power requirements. The approach consists o analytically modeling the heat transer and pressure drop and deining the geometry and low that result in minimal pump/an power or a speciied heat removal requirement and base area. In the ollowing sections, the governing equations or a set number o in units (as in Figures a, b and d) are outlined and algebraically combined to analytically determine the impact o as many design parameters as possible. A numerical optimization is then used or the remaining parameters and the results are presented on design charts. The analysis is then extended to include inlet and exit losses or the zigzag coniguration, where the unit layout introduces additional independent variables, which are numerically optimized, beside the height and aspect ratio o the ins and channels. b) c) d) = =3 e) H a) L b W b δ W L add Fig. : Heat Sink conigurations: a) regular inned base, b) ins divided in in units with resh coolant introduced into each unit, c) a longer base accounts or the space required or coolant inlet/outlet o each unit, d) side-by-side arrangement o in units alleviates additional spacing or inlet/outlet, e) zigzag arrangements o the in units can reduce overall pressure loss compared to c) ANALYSIS Each design point is speciied by the ollowing parameters: the heat rate to be removed, the available heat sink base area A b, base aspect ratio W b /L b and the chosen number o units, which can be combined into a shape actor (Equation 8), the base temperature T b, the thermal conductivity o the heat sink material k, and the chosen cooling luid and its approach temperature T (Table a). The ollowing analysis is reduced to the base surace and the ins themselves. It does not include the thickness o the base (substrate).

3 Hydrodynamic analysis The an/pump power P is calculated or incompressible low, p m P = () η ρ where the quotient o the mass low rate m and density ρ is the volume low rate and η is a overall eiciency o the an/pump. p is the pressure drop across the heat sink, p = L ρ c = Dh ρ c ζ () ρ is the density o the luid, L is the length o a in unit (in length) and ζ=. L /D h is the loss coeicient. The hydraulic diameter D h is deined as : H W D h = (3) H + W with H being the in height and W the channel width (distance between two neighboring ins). Assuming ully developed laminar low, the riction actor is calculated by interpolation o tabulated numerical solutions Re={6.9; 6.; 68.36; 7.9; 78.8; 8.3; 96} or aspect ratios H/W={; ; 3; ; 6; 8; }, = Re(H/W) / Re [6], D c Re = h () ν The kinematic viscosity ν is assumed as the mean o the inlet and outlet values. The through low velocity c is the mean velocity in the low channels, deined as: m c = () ρ H W N where ρ is a mean value or the luid density, N=W /(W+δ ) is the number o channels per in unit and the number o in units. Convection analysis The necessary mass low rate m is calculated rom the heat rate to be removed, m = c p ( ϑ ϑ ) using the speciic heat coeicient c p o the luid and the temperature dierences o the base against the luid ϑ=t b -T at the in unit inlet and outlet, assuming uniorm base temperature T b. The outlet temperature dierence is determined by the logarithmic mean temperature dierence, ϑ m = ϑ ϑ ln ( ϑ ϑ ) that is necessary to remove the desired heat rate (6) (7) ϑ m = A hη rom a certain in geometry, where A is the total heat transer surace o a in unit A = N L ( H + δ W) o + (8) (9) and δ is the in thickness. The heat transer coeicient h is, h = k Nu D h () where k is the thermal conductivity o the luid. Assuming ully developed laminar low, the Nusselt number is calculated by interpolation o tabulated numerical solutions or uniorm wall temperature Nu={.98; 3.8; 3.39; 3.96;.;.6; 7.} and aspect ratios H/W={; ; 3; ; 6; 8; } [6]. The overall surace eiciency o one in unit η o in Equation 8 is calculated by η o = ( η ) with the in area o one in unit A A = N L ( H + δ ) and the in eiciency η [7], η = A () A () ( H ) tanh C CH corr corr (3) where H corr =H+δ / is a corrected in height or an active in tip [8]. C = k h δ () The in constant C contains an approximated in perimeter L [9]. Combined analysis For analysis purposes, it is convenient to combine the above equations to the extent possible. Since Equations 7 and 3 cannot be solved algebraically or the desired variables, the above equations are combined into three subsets. Introducing the dimensionless parameters H/W, δ /W and W b /L b Equations -6 can reduce to: with: P. η = ( ϑ ϑ ) ν Geo ρc Geo = ( ) ( ) + H/ W ( + δ ) / W Re H/ W p b b (a) (b) 8 W /L H

4 Figure shows the relationship ϑ =F(ϑ m ) which is implicit in Equation 7. K ϑ K 3 K ϑ = K K K ϑ m K Fig. : Logarithmic mean temperature (Equation 7) Equations 8- can combine to: with: ϑ m = k A Geo b Geo = ( ) ( + H/ W ) [ + ( H/ W + δ / W ) ] η Nu H/ W H( + δ / W) (6a) (6b) Introducing the same dimensionless parameters H/W and δ /W in Equation 3, gives: with: tanh η ( Geo3 k/k ) = Geo3 Geo3 = Nu( H/ W) ( + H/ W) η k/k (7a) H/ W δ / W + (7b) δ / W H/ W.9. Geo3 = k/k 3. Fig. 3: Fin eiciency η (k/k, Geo3) in relevant range (Equation 7) Figure 3 shows, that a linear equation η =F(Geo3) could approximate Equation 7a very well. But in the optimization Equation 7 is used, since with such a linear approximation the combined Equation 6 and 7 is still too complex to be solved analytically or H/W and δ /W. OPTIMIZATION The goal o the optimization is to ind the optimum in coniguration expressed by H, H/W and δ /W or least pump power required. During the optimization, the optimum low parameters such as velocity c and mass low rate m are ound implicitly. The optima or the chosen design parameter are ound by irst analyzing Equations -7 analytically. These three equations are combined in order to contain only the chosen design parameters and the geometric parameters to be optimized or least pump power. Table a lists the design parameters that monotonically aect the pump power and the trend to minimize it, or a constant heat removal rate and base area. Thus, Table a tells the designer how to change the design point to obtain initially a lower an/pump power beore perorming an optimization or in geometry and low parameters. All these parameters speciy a nominal design point and are kept constant or the subsequent numerical optimization o the in geometry. Symbol Optimum ν Min ρ c p k k/k Min Min ϑ A b W b /L b Table a: Optimum values or least pump power (initial values determining the design point) The analytical analysis o Equations -7 gives no clear optimum or the ollowing geometric design parameters: H, H/W, δ /W. The separate evaluation o these equations suggests conlicting trends or minimal pump power, as summarized in Table b. From Equation a it is obvious that ϑ and Geo need to be minimized. ϑ is minimum, i ϑ m is minimum (Figure ); e.g. Geo needs to be maximized (Equation 6a). Geo is maximum i η is maximum, which is the case i Geo3 is minimum as shown in Figure 3. Eq. Opt. H H/W δ /W Geo (a) Min Min Min Geo (6a) Min Min Geo3 (7a) Min - Min Table b: Optimum geometric values or least pump power The optimum values or H, H/W and δ /W can only be determined using a numerical optimization, speciic to a set o values or the design parameters listed in Table a. Numerical Optimization The optimum values o H, H/W, δ /W are ound numerically by Newton method. H, H/W and δ /W in Equations -7 are varied, while all design parameters in Table a are kept constant until the minimum pump power is ound. In this optimization process, Equation 7 needs to be solved or ϑ iteratively and all relevant low parameters are ound implicitly. The in height, H, was ound to be the parameter o greatest importance, and will be discussed in the ollowing paragraph.

5 The ratio o in thickness to channel width δ /W, exhibited similar trends, but o less importance. The aspect ratio o the channels H/W, came out to be maximized or minimum pump power within the investigated ield. One explanation or this is the trend o Nusselt number and riction actor with aspect ratio. For numerical reason the maximum limit was set to H/W=. In test optimizations with the maximum limit set to H/W=, H/W also met the maximum limit. While channel width W and the ratio o in thickness to channel width δ /W then were approximately the same as with H/W=, in height H increased by actor and an/pump power decreased by same actor. Accordingly, or any other high aspect ratio H/W all results presented here may be adapted by scaling in height and an/pump power with the ratio o chosen upper limit H/W to. Optima o H/W less than the upper limit could be ound when H was constrained, as shown in the reerence results below. Fin height eect Since the in height dominates the optimization, it will be discussed here in more detail. The heat transer and the hydraulics introduce two competing eects that cause an optimum in height, as illustrated in Figure. Combining Equations and, an power is determined by P=m. (ζ/). c or η=. On one hand, the heat transer coeicient h decreases with increasing hydraulic diameter D h (Equation ), leading to increasing low rate (Equation 6-8 and Figure ). This is shown in Figure by the increase in m as H approaches cm. From the perspective o the hydraulics, the loss coeicient ζ (which is part o the pressure drop Equation ) increases with smaller hydraulic diameter. The product o m and ζ gives a curve with a minimum. The eect is ampliied by the curve o the through low velocity c (Equation ) shown in Figure, when calculating the pump power with P=m. (ζ/). c (Figure ). This basic trade-o between hydrodynamics and heat transer is central in determining the optimal channel geometry. Fin eiciency η, overall eiciency η o and total heat transer area o a in unit A appear to stay constant throughout the optimization since H/W is constant. P [W] 8 6 P = = ζ -% H Opt +%. H [cm]. Fig. : Eect o in height H on an power P: air cooled, approach temperature 3 C, base temperature 8. C, base area cm =, =6W, η=.3, H/W=, δ /W=.7 m m [g/s] 8 ζ 3 c [m/s] 3 c = = -% H Opt +% H. W. N H. W. N [mm ]. H [cm]. Fig. : Eect o in height H on through low velocity c: air cooled, approach temperature 3 C, base temperature 8. C, base area cm =, =6W, η=.3, H/W=, δ /W=.7 P an [W] Fig. 6: Eect o nit number on an power P: air cooled, approach temperature 3 C, base temperature 8. C, base area cm =6W, η=.3, H/W=, (δ /W) Si =.7, (δ /W) Cu =.6 nit number eect For a given heat removal and base area, the unit number changes only the through low velocity as c~/ (Equation ), while mass low rate m and loss coeicient ζ stay constant. Thus, has a dramatic impact on the required an/pump power as Figure and already have shown or one and two units and Figure 6 shows or to units. It can reduce an/pump power by several orders avoring conigurations with multiple in units in parallel arrangement. It should be mentioned here that this optimizing eect comes out so clear, because ully developed laminar low is assumed and no entry length eect is considered, which implies that Nu and Re are independent o velocity c and constant or constant aspect ratio H/W. However, the general trend will be similar i entry length eects are considered. Values may change, especially or extremely short units. RESLTS In this section, the results o the above optimization are condensed in two design charts one or air cooled and one or water cooled heat sinks. Some general eects are outlined. Design charts Figures 7 and 8 show the results o the numerical optimization or the minimized an/pump power, in a ive dimensional (, Si Cu

6 k, η,, W b /L b ) design space. Given the desired heat removal rate rom a base area o cm (abscise), and the conductivity ratio k/k (let ordinate) resulting rom the chosen in material, the optimum in height H (dashed parameter curves), the optimum in thickness/channel width ratio δ /W (right ordinate) and the an power needed (solid parameter curves) can be ound. (The optimum channel aspect ratio is maximum H/W=.) The actual power consumption o the an/pump can be calculated knowing the an/pump eiciency η, choosing the number o units and the base aspect ratio W b /L b. The last two are expressed by the shape actor S:. k/k. S =. 3 W L b b = =, W = P. η. S W b (8) Lb,.. 7. H = mm. be reduced by one order o magnitude..7 Si.8 Design Robustness Al Figures and show a % margin or the optimum in..7 Cu height. Within this margin, the change o an power is very. small. So, the design is considered to be robust within this range. A similar robustness can be observed or a change o q in W/cm inlet temperature dierence ϑ in a range o +/-K. Fig. 7: Design chart or optimized heat sinks: air cooled, approach temperature 3 C, base temperature 8. C, base area cm k/k W = P. η. S Si Al Cu 3, 3,, q in W/cm Fig. 8: Design chart or optimized heat sinks: water cooled, approach temperature C, base temperature 8. C, base area cm 3.. H = mm Eects Changing the chosen unit number in S shows how dramatically multi-unit conigurations (Figure b-d) can reduce an/pump power compared to the coniguration in Figure a with =. The optimum in height H decreases remarkably with heat rate and the conductivity ratio k/k. The optimum ratio o in thickness to channel width δ /W depends only on the conductivity ratio luid/in k/k, and can hence be plotted directly on the right ordinate. In the design space or air, the optimum δ /W changes only slightly around. with a maximum o.8 or k/k =.6, which lies between the conductivity ratio or silicon and aluminum. In the design space or water, the optimum δ /W varies over a larger range o.. and the maximum exists or a in conductivity greater than that o cooper. For the same heat rate and in material using water instead o air as a coolant the optimum in height H is approximately 8 times greater, where as optimum δ /W is approximately %.9 less. δ /W The sketches in Figures 7 and 8 are not to scale concerning H and H/W=, although H/W is constant or the sketches in. each igure. These sketches shall give an idea how the heat.3 sink design changes in each design space..3.. The eect o in conductivity k on the power is shown in Figure 6. By using Cu instead o Si the an/pump power can Examples Figure 7 indicates that with air as a coolant in a single-unit heat sink a heat lux o q= W/cm can be achieved with a. an power o W using a silicon heat sink, 6W or δ /W aluminum and 8W or cooper, assuming η=. With a still easily practical two-unit arrangement, an power reduces to W, W and W and with in units to.mw,.7mw and.8mw. Figure 8 indicates that with water as a coolant in a single-unit heat sink a heat lux o q= kw/cm can be achieved with a pump power o mw using silicon, mw using aluminum and mw using cooper, assuming again η=. With a two-unit arrangement, pump power reduces to 6mW,.mW and mw. q=kw/cm can be achieved with single in unit and a pump power o.kw using silicon, kw using aluminum and.kw using cooper. ZIGZAG CONFIGRATION Zigzag analysis extension The pressure drop in the inlet and outlet o the in units (when multiple units are used) was not considered in the above analysis. It appears that a zigzag arrangement o the in units allows a better velocity distribution in the unit inlet/outlet than a parallel arrangement o the units (Figures e and 9). Hence, the pressure loss will be less due to smaller velocity peaks.

7 The hydraulic diameter in the inlet and outlet is D h =H. W E /(W E +H), where W E is the inlet width or one unit. Since the hydraulic diameter and the aspect ratio H/(W E ) in the inlet and outlet region are greater than or the channels between the ins the inlet/outlet velocity c E can be higher by the velocity actor c E =c. cf or a comparable inlet/outlet pressure drop p E <= p. Thus, the necessary entrance width or each in unit is W E =N. W/cF and with W =L b +W E the unit width W >L b can be approximated to obtain a closed procedure by: W add W = 3 i= i= W W E Lb ( δ + W) ( δ + W) ( W/cF) α L W b Fig. 9: Zigzag coniguration 3 c E Fig. : Calculation inlet/outlet low L b (9) For the low calculation in the inlet and outlet region, it is assumed that the coolant lows through the zigzag arrangement in a parallel low as Figure shows. For one unit the low is divided in I straight low tubes o same width, with the aspect ratio H/ W E. The intersection with the in unit divides the low tubes in I length section L b, which gives an array IxI. The entry width or the hydraulic diameter D h,i o each section i in low direction is determined by W E, i =W E,I +(W E, -W E, I ). (I-i)/I, where W E, =W E and or numerical reasons W E, I =W. So Re i (H/W E,i ) is determined as or Equation. Equation can be rewritten to (ζ. c) i =Re. i L. b ν/d h,i, where D h,i =H. W E, i //(W E +H). The accuracy o incremental product (ζ. c) i improves with (ζ. c) i = (Re i +Re i- )/. L. b ν/[(d h,i +D h,i- )/] The sum or the low length L b is calculated by: (ζ. c) sum,i = i I i ( ζ c) i + ( ζc) j ( ζ c) j + ( ζ c) j + () j= j= where the irst two sums are or the inlet and outlet region (gray in Figure ) and the last term accounts or both triangle regions at the in unit. Hence, Equation can be written as: c i = p guess. (ζ. c) sum,i. /ρ () where c i is the velocity in the stream tube i and p guess is a irst guess or the pressure drop in the inlet/outlet region. p guess is calculated rom an assumed ully developed laminar velocity proile, or which c max =c m and c m =c. cf. p guess = c cf Min ( c) sum, j) j= I ζ ρ () The minimum unction selects the smallest (ζ. c) sum, where the maximum velocity c max exists. Assuming the same pressure drop occurs across each stream tube, now p guess is corrected by p E = I p guessccf (3) I c j j= so that mass conservation is ulilled. The overall pressure drop over the entire heat sink is then p sum = p+ p E and p is calculated rom Equation. In the analysis o Equations - is substituted by =W b /(W add +W E ), with W add =L. W /W b. Analogous to the outlined hydraulic analysis, a convection analysis can be added or the inlet/outlet region. But or the optimized zigzag heat sinks, the contribution is typically around % and will be neglected, where as the pressure drop o the inlet/outlet region p E adds around % to the overall pressure drop p sum. Zigzag results Figures a and b show the optimization results o the zigzag coniguration. Figure a shows the parameters used previously, while Figure b shows the parameters unique to the zigzag coniguration. The numerical optimization must now determine and L in addition to H, H/W and δ W /W. In the illustrated optimization, H/W reached again the upper limit which was set to. From Figures a it appears that the zigzag heat sink optimization results in more compact heat sinks with less an/pump power consumption, although results in Figures 7 and 8 are obtained neglecting unit inlet/outlet pressure losses. For example, or a 9W heat removal rate rom a cmxcm base the optimized in height and an power or silicon and cooper heat sinks are H Si =.3(9.)mm, P Si =.3(88)W, H Cu =7.8(6.)mm, P Cu =.3()W, where the values in parentheses are rom Figure 7 or a single unit heat sink with S=. The optimized zigzag heat sinks have typically a small unit angle α=arcsin(l E /W ) in the order o α=. With higher heat lux q the an power, number o units and velocity actor cf increase and the in height H, in unit length L, and the in unit angle α decrease. In the investigated ield, the ratio δ /W decreases negligible with δ /W~/.

8 P [W] 3 δ /W Cu H [cm] q in W/cm Fig. a: Optimized zigzag heat sinks: air cooled, approach temperature 3 C, base temperature 8. C, base area cm, η=.3, cf δ /W Si L,Cu L,Si H Si H Cu c Si ccu P Si P Cu Si Cu δ /W q in W/cm Fig. b: Optimized zigzag heat sinks: air cooled, approach temperature 3 C, base temperature 8. C, base area cm, η=.3 c [m/s] 3 7 L [µm] 9 cf Si cf Cu α Cu REFERENCE RESLTS Tuckerman and Pease [3] optimized a heat sink o cmxcm made o silicon or water as a coolant. They ound an optimum with δ =W=7micron and H=36micron. With the reported cm 3 /s and a pressure drop o 6.8kPa(3psi), the pump power would be P=.8W or η=. In tests, the device reached up to =79W. For the reported ϑ =7K, the same geometry and cm 3 /s the analyses ater Equations - gives or k =8W/m/K: P=.3W, =88W, p=3.kpa. Matching the reported pressure drop o 6.8kPa(3psi) the results are: P=3.6W, =7W and V=7.cm 3 /s. The optimization or =79W results in P=.mW, p=kpa, H=6.38mm, H/W=, δ /W=., V=.cm 3 /s. Copeland [] perorms an optimization or an air-cooled heat sink with ixed dimensions W b =9cm, L b =6.cm, H=.cm (approximate size o near-uture processor modules). He allows a maximum pressure drop o Pa and a maximum an power o.w and inds a minimal thermal resistance o.7c/w. sing the inlet temperature dierence o.k (as used in this work or air) this would give a heat rate o about W. The optimization or a single-unit heat sink ater Equations - with η= and k =W/m/K assuming α Si aluminum results or P=.W in =36W, p=3pa, c=3.m/s (velocity in micro-channel), H/W=., δ /W=.. This is equivalent to in pitch.3mm and in thickness o.mm. It has to be mentioned that Copeland limited the in pitch and thickness to.mm and.3mm because those represent practical minima or corrugated and skived in technology. However, Copeland ound optima at in pitch.3.mm and in thickness o.mm. These dimensions give with Equations - and =W: P=.3W, p=pa. For the same base coniguration, the described zigzag optimization results with P=.W and η= in =878W, p=.7pa, c=.9m/s (in micro-channel), H/W=, δ /W=.86, L =.76mm, α=.6. SMMARY & CONCLSIONS The described optimization or least an/pump power shows that or every heat removal, base area and aspect ratio there is an optimum in height and ratio in thickness/channel width. α [ ] The channel aspect ratio always tends to be maximized, when. the in height is not ixed during the optimization. With increased heat removal, the optimum in height decreases, while an/pump power increases. The optimum ratio o in thickness to channel width using water is about. and using air.7. Design charts or both coolants show the trends o optimized parameters such as an/pump power, in height and the ratio in thickness to channel width. Multi in-units in parallel low arrangement can reduce an/pump power by several orders o magnitude. With the optimizing tool, conigurations or heat removal o W/cm with air and kw with water and a pump power expense o less than % could easily be ound. Optimized zigzag arrangements are preerable to parallel arrangement o in units. They are smaller and require less pump power or the same heat removal. REFERENCES [] I. Mudawar: Assessment o high-heat-lux thermal management schemes, IEEE Transactions on Components and Packaging, Vol. /,, pp. -. [] International roadchart o semiconductors, [3] D. B. Tuckerman and R. F. W. Pease, High-Perormance Heat Sinking or VLSI, IEEE Electron Device Letters, Vol. EDL-, No., May 98, pp [] D. Copeland Optimization o parallel plate heatsinks or orced convection, Semiconductor Thermal Measurement and Management Symposium, -3 March. (6th Annual IEEE), pp [] C. B. Sobhan and S. V. Garimella, A comparative analysis o studies on heat transer and luid low in microchannles, Microscale Thermophysical Engineering, Vol.,, pp [6] Y. A. Cengel. Introduction to thermodynamics and heat transer, McGraw-Hill, New York, 997. [7] F. P. Incropera Introduction to heat transer, Wiley, New York, 996. [8] D. R. Harper and Brown W. B. Mathematical equations or heat conduction in the ins o air cooled engines, NACA Report No. 8, 9. [9] P. J. Schneider. Conduction heat transer, Addison- Wesley, Reading MA, 9.