High performance pin fin heat sink for electronics cooling

[0002] The present invention is directed to heat sinks in general, and more particularly to heat sinks for use in dissipating waste heat generated by electrical or electronic components and assemblies.

[0003] High power electrical and electronic components continue to have an increasing demand for higher power dissipation within a relatively confined space. In order to provide for such higher power dissipation requirements while remaining suitably compact, several levels of thermal management are usually required at the device, sub-assembly and component level.

[0004] At the component level, various types of heat exchangers and heat sinks have been used that apply natural or forced convection or other cooling methods. A typical heat sink for electrical or electronic components employing pin fins and a foam is depicted in FIG. 1. As shown, the heat sink includes a heat spreader plate 10 to which metal pins 12 are attached. An electronic component is attached to face 14 of spreader plate 10 and a cooling fluid, such as air or water, is passed through foam 16 and across pins 12 to dissipate the heat generated by the electronic component.

[0005] In demanding applications, the method of heat exchange is usually forced convection to the cooling fluid. In such systems, heat exchange can be improved by increasing the pin surface area exposed to the cooling fluid. This is accomplished by increasing the number of the pins per unit volume. However, there are limitations to achievable pin densities based upon manufacturing constraints and cooling fluid flow requirements.

[0006] Reticulated foams are also known in the art for their ability to conduct heat such as the metal foams disclosed in U.S. Pat. Nos. 3,616,841 and 3,946,039 to Walz, and the ceramic foams disclosed in U.S. Pat. No. 4,808,558 to Park et al. Metal foams have been sold under the trade name DUOCEL available from Energy Research and Generation, Inc., Oakland, Calif.

[0007] Until recently, metal and ceramic reticulated foams have not been adapted for use in heat sinks for dissipating waste heat from electronic components. However, these structures, especially when comprised of metal, make excellent heat exchangers because of their conductivity and their extremely high surface area to volume ratio. While earlier porous heat exchangers had up to 100 open cells per square inch, reticulated foam has up to 15,625 open cells per square inch. Reticulated foam is far more porous and has far more surface area per unit volume (1600 square feet/cubic foot) than heat exchangers having other structures. The pressure drop of fluids flowing through reticulated foam is also relatively low so that movement of a cooling fluid through the foam is practical.

[0008] Studies by Bastawros have now shown the efficacy of metallic foams in forced convection heat removal for cooling of electronics. See, Bastawros, A.-F., 1998, Effectiveness of Open-Cell Metallic Foams for High Power Electronic Cooling, ASME Conf. Proc. HTD-361-3/PID-3, 211-217, and Bastawros, A.-F., Evans, A. G. and Stone, H. A., 1998, Evaluation of Cellular Metal Heat Transfer Media, Harvard University report MECH 325, Cambridge, Mass. Bastawros demonstrated that the use of metallic foam improved heat removal rate with a moderate increase in the pressure drop. Bastawros' results were based on thermal and hydraulic measurements (on an open cell aluminum alloy foam having a pore size of 30 pores per inch) used in conjunction with a model based upon a bank of cylinders in cross-flow to understand the effect of various foam morphologies. The model prediction was extrapolated to examine the trade-off between heat removal and pressure drop. The measurements showed that a high performance cellular aluminum heat sink (i.e., aluminum foam) removed 2-3 times the usual heat flux removed by a pin-fin array with only a moderate increase in pressure drop.

[0009] Thus what is desired is a heat sink having pins and a foam surrounding the pins that is optimizable for a given application.

[0010] One aspect of the present invention is a heat sink for electronic devices comprising a spreader plate having a top surface and having a bottom surface wherein a portion thereof is defined for affixing an electronic device to be cooled thereto. A plurality of columnar pins are spaced apart one from the other in a non-uniform manner and affixed to the top surface of the spreader plate substantially perpendicular thereto. A highly porous heat conducting reticulated foam block substantially fills a space defined by said spaced apart columnar pins.

[0011] Another aspect of the present invention is a method of making a heat sink comprising a spreader plate, a plurality of columnar pins and a reticulated foam block for filling the space between the columnar pins, the method comprises the steps of selecting the spreader plate and defining a portion on the bottom surface of the spreader plate for affixing to an electronic device. A plurality of columnar pins are selected and affixed to the top surface of the spreader plate such that the columnar pins are arranged in a non-uniform density on the top surface of the spreader plate. A reticulated foam block is placed to fill the spaces between the columnar pins affixed to the spreader plate and is affixed to the spreader plate and to the plurality of columnar pins.

[0012] Yet another aspect of the present invention is a method of cooling electronic components with a heat sink of the type having a spreader plate with a plurality of columnar pins affixed in a non-uniform density to a top surface of the spreader plate and having a reticulated heat conducting foam filling the space between the plurality of columnar pins. The method comprises the steps of affixing the electronic device to a bottom surface of the spreader plate and arranging a first plurality of columnar pins to the top surface of the spreader plate over the electronic device at a greater density than a second plurality of columnar pins over the remainder of the spreader plate. The heat conducting foam is affixed to the spreader plate to fill the space between adjacent ones of the first and second columnar pins. A cooling fluid is passed over the top surface of the spreader plate and through the heat conducting foam.

[0013] These and other advantages of the invention will be further understood and appreciated by those skilled in the art by reference to the following written specification, claims and appended drawings.

[0014] FIG. 1 is a perspective view of a prior art heat sink;

[0015] FIG. 2 is a perspective view of a heat sink embodying the present invention, wherein the density of pins is greater over the electronic device.

[0016] FIG. 3 is a cross-sectional view of the heat sink shown in FIG. 2 and taken along the line 3—3 of FIG. 2.

[0017] FIG. 4 is a segmented view of a square pin.

[0018] FIG. 5 is a segmented view of a triangular pin.

[0019] FIG. 6 is a segmented view of a rectangular pin.

[0020] FIG. 7 is a segmented view of a round pin.

[0021] FIG. 8 is a graph of an idealized thermal distribution across the spreader plate.

[0022] FIG. 9 is a graph of an actual thermal distribution across the spreader plate.

[0023] FIG. 10 is a graph of the non-uniform heat flux over the spreader plate as a cosine function.

[0024] FIG. 11 is a graph of the non-uniform heat flux over the spreader plate as a step function.

[0025] For purposes of description herein, the terms “upper”, “lower”, “left”, “rear”, “right”, “front”, “vertical”, “horizontal”, and derivatives thereof shall relate to the invention as oriented in FIG. 2. However, it is to be understood that the invention may assume various alternative orientations and step sequences, except where expressly specified to the contrary. It is also to be understood that the specific devices and processes illustrated in the attached drawings, and described in the following specification, are simply exemplary embodiments of the inventive concepts defined in the appended claims. Hence, specific dimensions and other physical characteristics relating to the embodiments disclosed herein are not to be considered as limiting, unless the claims expressly state otherwise.

[0026] It has been discovered that there are constraints on the volume and geometry of reticulated foam beyond which the use of additional foam will have little impact on the overall heat sink's ability to dissipate thermal power at a given flow rate (i.e., the performance). This is because the reticulated foam is not a fully dense material (e.g., aluminum foam has a density of about 10% of solid aluminum). Therefore, a high convective heat transfer surface area is achieved at the expense of reduced thermal conductivity.

[0027] Additionally, in microelectronic cooling applications such as for example for microprocessors, practical considerations relative to package size, air flow rate, pressure drop and noise limits can impose further constraints on possible configurations and dimensions. Nonetheless, using the methods of the present invention, suitable heat sinks can be produced.

[0028] Heat sinks of the present invention achieve very high convective heat transfer surface per unit volume. These heat sinks comprise a spreader plate, a plurality of columnar pins, and a porous reticulated foam block that fill the space between the pins. This basic structure may be expanded to any configuration comprising foam blocks filling the space between the columnar pins that are mounted onto the surface of a spreader plate.

[0029] Primary heat transfer to the cooling fluid is by convection from the foam, with the pins and spreader plate being used to conduct heat from the connected heat source (i.e., the electronic component) into the foam. In air-to-air heat exchange (i.e., where air is being used as the cooling fluid), ambient air may be drawn in through the foam's open vertical side walls and exhausted through the foam's top surface, or vice versa. Providing such air can be accomplished using methods well known in the art. These methods include methods such as mounting an electrically powered fan to the top or a side of the heat sink or by attaching an air duct, through which cooling air is flowing, to the top or side of the heat sink.

[0030] Turning to the drawings, FIG. 2 shows a heat sink 20, which is one of the preferred embodiments of the present invention and illustrates its various components. As shown, heat sink 20 comprises a heat spreader plate 22, with outer columnar pins 24 and central columnar pins 26 mounted on a top surface 21 of the spreader plate 22 in a substantially perpendicular manner. The density of the outer pins 24 is lower than the density of the central pins 26. A reticulated foam block 28 fills the space in-between the pins. The pins 24, 26 and foam block 28 form an assembly that is mounted onto one surface of the spreader plate 22, leaving an opposing surface free for contact with an electronic component to be cooled.

[0031] Referring to FIGS. 2-3, columnar pins 24, 26 are mounted so that they are substantially perpendicular to the top surface 21 of the spreader plate 22. Foam block 28 is mounted on spreader plate 22 and surrounds columnar pins 24, 26 to fill the region that defines the horizontal space between adjacent columnar pins. The foam block 28 also preferably fills the height region that defines the vertical space between adjacent columnar pins 24, 26 to the height of the columnar pins 24, 26. While FIGS. 2-3 show that the foam block 28 fills the height region, it is contemplated that in alternative embodiments the foam block 28 may partially fill or overfill the height region.

[0032] Columnar pins 24, 26 are generally columnar in nature, meaning that the height of each pin is substantially greater than its cross-sectional width. Columnar pins 24, 26 can assume a variety of shapes, such as: a square pin 42 having a width ‘d’ as shown in FIG. 4; an equilateral triangular pin 46 having a side dimension ‘d’ as shown in FIG. 5; a rectangular pin 44 having a width ‘d’ as shown in FIG. 6; or, a circular pin 40 having a diameter ‘d’ as shown in FIG. 7. Those practiced in the art will understand that other cross-section geometries can also be used.

[0033] Referring again to FIG. 3, An electronic device 30 such as a computer chip or other electronic device that requires heat dissipation is mounted on bottom surface 23 of spreader plate 22. Device 30 can be directly mounted to surface 23, or alternatively can be received in a recess (not shown) in spreader plate 22 defined by surface 23. A thermal conductive layer 32, commonly referred to as “thermal grease”, is interposed between device 30 and spreader plate 22 to enhance the transfer of heat from device 30 to spreader plate 22.

[0034] FIG. 8 illustrates the temperature distribution on the pin side of a spreader plate caused by an idealized chip generating uniform heat flux, and FIG. 9 shows the temperature distribution on the pin side of the spreader plate caused by a real chip generating non-uniform heat flux.

[0035] Since the pins conduct heat from the spreader plate for dissipation into the cooling fluid, it is evident from the temperature distributions of the graphs in FIGS. 8 and 9 that the pin density should be greater in the center of the spreader plate where the temperature is the highest than closer to the edge where the temperature is significantly lower. The prior art pins with uniform pin density over the spreader plate are thus not properly disposed to dissipate heat in an efficient fashion. Furthermore, the prior art spreader plates do not employ any heat transfer coefficient augmentation means to dissipate heat more efficiently into the flowing fluid stream.

[0036] In heat sink 20, the aforementioned deficiencies of the prior art heat sinks for electronics cooling are removed. This is achieved by the following means:

[0037] 1. High density of pin pins where needed most.

[0038] 2. Pin fins of optimized dimensions.

[0039] 3. Pin fins embedded in a metallic foam block to enhance the heat transfer coefficient for efficient heat dissipation into the flowing fluid stream.

[0040] FIGS. 2-3 show the perspective and elevation views of heat sink 20. The density of columnar pins 26 in the center of the plate where the heat flux and hence the temperature is greater than the density of columnar pins 24 around the periphery of spreader plate 22 where the temperature is lower as illustrated by FIG. 9. Foam material 28 surrounds columnar pins 24, 26. The primary function of the columnar pins 24, 26 is to conduct heat from spreader plate 22 into the space between pins 24, 26 which is populated with foam block 28. The primary function of foam 28 is to enhance the heat transfer coefficient of the fluid stream that removes heat from the space between columnar pins 24, 26.

[0041] The selection of spreader plate size for a given power dissipation requirement, follow those techniques known in the art. The overall dimensions of the spreader plate are generally fixed based on the amount of heat to be dissipated from the surface of the heat source (such as a computer chip). The spreader plate surface area should be such that, for a prescribed flow rate of the cooling fluid flowing over the spreader plate, the heat from the heat source is able to spread to the edges of the spreader plate. Additional considerations may also be determinative of the spreader plate surface area such as packaging constraints.

[0042] Generally, however, spreader plate surface area is selected by multiplying the surface area of the heat source with the area magnification factor. The area magnification factor F=A_s/A_hrepresents a ratio of the surface area of the heat source A_hwith the surface area of the spreader plate A_s. Typical values of F are in the range of 8 to 12, and are generally used in calculating spreader plate surface area for a given surface area of a heat source. From the standpoint of heat removal efficiency, F should be as low as possible. Highly effective heat transfer surfaces such as highly conductive pins of optimized dimensions and/or the use of heat transfer augmentation means such as reticulated foam provide for relatively low values of F. For example, if the surface area of the heat source is 1.5 in²and the selected area magnification factor is taken as F=8 (for highly efficient transfer), then the surface area of the spreader plate will be 8×1.5=12 in². For a spreader plate of this area, packaging considerations could prescribe a length of the plate in the flow direction to be 4 in. Then the width of the spreader plate will be 3 in.

[0043] The optimum height b of a pin fin of uniform cross section along its height is determined using the relation: $\begin{array}{cc}b=0.919\sqrt{\frac{k_f}{h}\frac{A_c}{p}}& \left(1\right)\end{array}$

[0044] where

[0045] k_fis the thermal conductivity of the pin material, Btu/ft s° F.

[0046] h is the convective heat transfer coefficient for the foam-filled space bounded by the pins and the spreader plate, Btu/ft²s° F.

[0047] A_cis the cross sectional area of the pin, ft²

[0048] p is the perimeter of the pin, ft.

[0049] The ratio of the cross sectional area A_cto perimeter p for some simple pins are presented in Table 1.

[0050] Since for a given perimeter, a circle encloses the maximum area, it is evident that for a given perimeter, the ratio A_c/p is the maximum for circular pin. Thus, in accordance with Equation (1), the optimum height b is the maximum for the circular pin.

[0051] The heat transfer coefficient h entering Equation (1) for the foam-filled space can be determined using the formula: $\begin{array}{cc}h=1.2704\left[\frac{n^{0.50}}{{\left(1-\phi \right)}^{0.25}}\right]\left(\frac{{\rho }^{0.50}{k}^{0.63}{c}_p^{0.37}}{{\mu }^{0.13}}\right)u_m^{0.50}& \left(2\right)\end{array}$

[0052] where:

[0053] n is the linear density of the foam material, pores per ft

[0054] φ is the foam porosity expressed as a fraction

[0055] ρ is the density of the flowing fluid, lb_m/ft³

[0056] k is the thermal conductivity of the flowing fluid, Btu/ft s° F.

[0057] c_pis the isobaric specific heat of the flowing fluid, Btu/lb_m° F.

[0058] μ is the dynamic viscosity of the flowing fluid, lb_m/ft s

[0059] u_mis the mean velocity of the flowing fluid, ft/s.

[0060] Referring to FIG. 10, the non-uniform heat flux {dot over (q)}″ over the spreader plate can be represented by a cosine function as: $\begin{array}{cc}{\stackrel{.}{q}}^{″}={\stackrel{.}{q}}_o^{″}\left[1+\mathrm{\alpha cos}\left(\frac{\pi x}{a}\right)\right]& \left(6\right)\end{array}$

[0061] where

[0062] {dot over (q)}″_ois the mean heat flux over the spreader plate, Btu/ft²

[0063] α is a dimensionless constant dependent on the maximum and minimum values {dot over (q)}″_maxand {dot over (q)}″_minof the heat flux {dot over (q)}″ (vide infra)

[0064] x is the variable distance measured from the center of the spreader plate along its width, ft

[0065] a is the spreader plate half-width, ft.

[0066] The mean heat flux {dot over (q)}″_oand the dimensionless constant α are expressible in terms of {dot over (q)}″_minand {dot over (q)}″_maxas $\begin{array}{cc}{\stackrel{.}{q}}_o^{″}=\frac{1}{2}\left({\stackrel{.}{q}}_{\min }^{″}+{\stackrel{.}{q}}_{\max }^{″}\right)& \left(7\right)\\ \alpha =\frac{1-{\stackrel{.}{q}}_{\min }^{″}/{\stackrel{.}{q}}_{\max }^{″}}{1+{\stackrel{.}{q}}_{\min }^{″}/{\stackrel{.}{q}}_{\max }^{″}}& \left(8\right)\end{array}$

[0067] The non-uniform pin density β is related to the non-uniform heat flux {dot over (q)}″ simply as $\begin{array}{cc}\beta ={\beta }_o\left(\frac{{\stackrel{.}{q}}^{″}}{{\stackrel{.}{q}}_o^{″}}\right)& \left(9\right)\end{array}$

[0068] where

[0069] β_ois the mean pin density over the spreader plate in the flow direction, pins/ft. The mean pin density β_ocan be interpreted as the linear pin density in the flow direction with uniform heat flux over the spreader plate.

[0070] Introducing Equation (6) into Equation (9), the non-uniform pin density β can be expressed as $\begin{array}{cc}\beta ={\beta }_o\left[1+\mathrm{\alpha cos}\left(\frac{\pi x}{a}\right)\right]& \left(10\right)\end{array}$

[0071] which mirrors the non-uniform heat flux {dot over (q)}″ distribution given in Equation (6).

[0072] In some cases, the non-uniform heat flux {dot over (q)}″ over the spreader plate can be represented by the step function as shown in FIG. 11. Analytically, this non-uniform heat flux can be represented as

[0073] {dot over (q)}″={dot over (q)}″_maxfor 0≦x≦λa  (11)

[0074] {dot over (q)}″={dot over (q)}″_minfor λa<x≦a  (12)

[0075] where

[0076] λ is the fraction of the spreader plate width with maximum heat flux.

[0077] The non-uniform pin density β in this case also is given in Equation (9). Introducing, Equations (11) and (12) into Equation (9), we obtain the non-uniform pin density β over the spreader plate as $\begin{array}{cc}\beta ={\beta }_o\left(\frac{{\stackrel{.}{q}}_{\max }^{″}}{{\stackrel{.}{q}}_o^{″}}\right)\mathrm{for}0\le x\le \lambda a& \left(13\right)\\ \beta ={\beta }_o\left(\frac{{\stackrel{.}{q}}_{\min }^{″}}{{\stackrel{.}{q}}_o^{″}}\right)\mathrm{for}\lambda a<x\le a& \left(14\right)\end{array}$

[0078] where the mean heat flux {dot over (q)}″_oover the spreader plate is given in Equation (7).

[0079] Knowing the non-uniform pin density β from the above relations, the number of pins N at any distance x along the spreader plate width can be determined simply from the relation

[0080] N=βl  (15)

[0081] where l is the spreader plate length normal to the spreader plate width 2a.

[0082] Four numerical examples are given below to illustrate how to apply the teachings of the present invention with the use of Equations (1)-(15).

[0083] It is required to determine the optimum height b of the pin fin of circular cross section with diameter d=0.125 inch to dissipate heat from an electronic device employing aluminum pins with thermal conductivity k_f=133 Btu/ft hr° F. and commercially available open cell foam with linear density n=20 pores per inch and porosity φ=0.90. The cooling medium is the ambient air flowing with a mean velocity u_m=10 ft/s. The transport properties of the ambient air are as follows.

[0084] Density ρ=0.0749 lb_m/ft³

[0085] Thermal conductivity k=0.0000041 Btu/ft s° F.

[0086] Isobaric specific heat c_p=0.2410 Btu/lb_m° F.

[0087] Dynamic viscosity μ=0.0000123 lb_m/ft s.

[0088] As a prelude to the calculation of the optimum pin height b using Equation (1), the appropriate convective heat transfer coefficient h is calculated using Equation (2). Upon introducing into Equation (2), n=20×12=240 pores per ft., φ=0.90, ρ=0.0749 lb_m/ft³, k=0.0000041 Btu/ft s° F., c_p=0.2410 Btu/lb_m° F., μ=0.0000123 lb_m/ft s and u_m=10 ft/s, the heat transfer coefficient h=0.0313 Btu/ft²s° F. is obtained.

[0089] Next, the ratio of the cross sectional area A_cto perimeter p for the circular pin fin is determined. As per Table 1, for circular pin A_c/p=d/4 where d is the diameter of the pin. With d=0.125 inch=0.0104 ft, we obtain A_c/p=0.0026 ft.

[0090] Finally, introducing h=0.0313 Btu/ft²s° F., A_c/p=0.0026 ft. and k_f=133/3600=0.0369 Btu/ft s° F. into Equation (1), the optimum pin height b=0.0509 ft=0.6110 inch is obtained.

[0091] It is required to determine the optimum height b of the pin fin of equilateral triangular cross section to dissipate heat from an electronic device employing aluminum pins with thermal conductivity k_f=133 Btu/ft hr° F. and commercially available open cell foam with linear density n=20 pores per inch and porosity φ=0.90. The side d of the equilateral triangular pin is to be selected in such a fashion that the perimeter p of the pin is the same as the perimeter of the circular pin of Example 1 with diameter 0.125 inch.

[0092] The cooling medium is the ambient air flowing with a mean velocity u_m=10 ft/s. The transport properties of the ambient air are as follows.

[0093] Density ρ=0.0749 lb_m/ft³

[0094] Thermal conductivity k=0.0000041 Btu/ft s° F.

[0095] Isobaric specific heat c_p=0.2410 Btu/lb_m° F.

[0096] Dynamic viscosity μ=0.0000123 lb_m/ft s.

[0097] As a prelude to the calculation of the optimum pin height b using Equation (1), the appropriate convective heat transfer coefficient h is calculated using Equation (2). Upon introducing into Equation (2), n=20×12=240 pores per ft., φ=0.90, ρ=0.0749 lb_m/ft³, k=0.0000041 Btu/ft s° F., c_p=0.2410 Btu/lb_m° F., μ=0.0000123 lb_m/ft s and u_m=10 ft/s, the heat transfer coefficient h=0.0313 Btu/ft²s° F. is obtained.

[0098] In order to determine the ratio of the cross sectional area A_cto perimeter p for the equilateral triangular pin fin, the side d of the pin must first be determined. As per the example statement, the perimeter p of the equilateral triangular pin must match the perimeter of the circular pin of Example 1. Knowing that the diameter of the circular pin of Example 1 is 0.125 inch, it follows that the perimeter of the circular pin is π×0.125=0.3927 inch. Since the perimeter p of the equilateral triangular pin is 3d, it follows that 3d=0.3927 inch whence d=0.1309 inch=0.0109 ft. As per Table 1, for equilateral triangular pin A_c/p=d/4{square root over (3)}. Introducing d=0.0109 ft, the resulting A_c/p=0.0016 ft is obtained.

[0099] Finally, introducing h=0.0313 Btu/ft²s° F., A_c/p=0.0016 ft. and k_f=133/3600=0.0369 Btu/ft s ° F. into Equation (1), the optimum pin height b=0.0396 ft=0.4751 inch is obtained. On comparing this height with the height 0.6110 inch of the circular pin in Example 1, it is noticed that the optimum height of the equilateral triangular pin is less than that of the circular pin. This is to be expected in view of the fact that the optimum height is proportional to {square root over (A_c/p)} as per Equation (1) and for a given perimeter the ratio A_c/p is the maximum for circular pin.

[0100] It is required to determine the non-uniform pin fin density over the surface of a square spreader plate with side 2a=5 in. It is found that with uniform heat flux {dot over (q)}″_oover the spreader plate the mean pin density β_oin the fluid flow direction is given as β_o=10 pins/inch. It is required to determine the non-uniform pin density β across the spreader plate width if a non-uniform heat flux {dot over (q)}″, represented by the cosine function in Equation (6). This requirement is imposed by the electronics device bonded on the opposite unfinned surface of the spreader plate. The maximum heat flux {dot over (q)}″_maxat the center of the spreader plate is twice the minimum heat flux {dot over (q)}″_minat the edge of the spreader plate, i.e., {dot over (q)}″_max/{dot over (q)}″_min=2 or {dot over (q)}″_min/{dot over (q)}″_max½.

[0101] As a prelude to the determination of the non-uniform pin density β, first the dimensionless constant cc with the use of Equation (8) is determined. By the problem statement, {dot over (q)}″_min/{dot over (q)}″_max=0.5. Introducing this value into Equation (8), a value of α=⅓ is obtained. Introducing this value of α together with the spreader plate half-width a=2.5 inches and the mean pin density β_o=10 pins/inch, the local pin density across the spreader plate width with the use of Equation (10) is determined. Thus, knowing the local pin density β, the number of pins N at any distance x is calculable with the use of Equation (15). The spreader plate length l entering Equation (15) is equal to the spreader plate width 2asince the plate is a square. Thus l=2a=5 in. The calculated values of β and N at representative distances x (vide FIG. 10) are presented in the following table:

[0102] It is required to determine the pin fin density over the surface of a square spreader plate with side 2a=5 in. It is found that with uniform heat flux {dot over (q)}″_oover the spreader plate the mean pin density in the fluid flow direction β_o=10 pins/in. It is required to determine the non-uniform pin density β across the spreader plate width if a non-uniform heat flux {dot over (q)}″, represented by the step function of Equations (11) and (12), is imposed by the electronics device bonded on the opposite unfinned surface of the spreader plate. The maximum heat flux {dot over (q)}″_maxat the center of the spreader plate is twice the minimum heat flux {dot over (q)}″_minat the edge of the spreader plate i.e., {dot over (q)}″_max/{dot over (q)}″_min=2 or {dot over (q)}″_min/{dot over (q)}″_max=½. The maximum heat flux {dot over (q)}″_maxis imposed over 25% of the spreader plate width so that the fraction λ=0.25.

[0103] As a prelude to the determination of the non-uniform pin density β using Equations (13) and (14), the flux ratios {dot over (q)}″_max/{dot over (q)}″_oand {dot over (q)}″_min/{dot over (q)}″_omust first be determined. This is easily done by introducing the flux ratio {dot over (q)}″_min/{dot over (q)}″_max=0.5 into Equation (7). Thus, the values {dot over (q)}″_max/{dot over (q)}″_o={fraction (4/3)} and {dot over (q)}″_min/{dot over (q)}″_o=⅔ are obtained. Introducing these values together with spreader plate half-width a=2.5 in. and the mean pin density in the fluid flow direction β_o=10 pins/inch into Equations (13) and (14), the desired values of the local pin density β in the two regions of the spreader plate is obtained. Therefore, knowing the local pin density β, the number of pins N at any distance x are calculable with the use of Equation (15). The spreader plate length l entering Equation (15) is equal to the spreader plate width 2asince the plate is a square. Thus, l=2a=5 in. The calculated values of β and N for the appropriate ranges of x (vide FIG. 11) are presented in the following table:

[0104] In the foregoing description those skilled in the art will readily appreciate that modifications may be made to the invention without departing from the concepts disclosed herein. Such modifications are to be considered as included in the following claims, unless these claims expressly state otherwise.