Abstract:
A heat sink for electronic devices comprises 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.

Description:
TECHNICAL FIELD 
     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. 
     BACKGROUND OF THE INVENTION 
     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. 
     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. 
     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. 
     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. 
     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. 
     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&#39; 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. 
     Thus what is desired is a heat sink having pins and a foam surrounding the pins that is optimizable for a given application. 
     SUMMARY OF THE INVENTION 
     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. 
     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. 
     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. 
    
    
     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. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a perspective view of a prior art heat sink; 
     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. 
     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 . 
     FIG. 4 is a segmented view of a square pin. 
     FIG. 5 is a segmented view of a triangular pin. 
     FIG. 6 is a segmented view of a rectangular pin. 
     FIG. 7 is a segmented view of a round pin. 
     FIG. 8 is a graph of an idealized thermal distribution across the spreader plate. 
     FIG. 9 is a graph of an actual thermal distribution across the spreader plate. 
     FIG. 10 is a graph of the non-uniform heat flux over the spreader plate as a cosine function. 
     FIG. 11 is a graph of the non-uniform heat flux over the spreader plate as a step function. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     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. 
     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&#39;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. 
     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. 
     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. 
     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&#39;s open vertical side walls and exhausted through the foam&#39;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. 
     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. 
     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. 
     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. 
     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 . 
     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. 
     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. 
     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: 
     1. High density of pin pins where needed most. 
     2. Pin fins of optimized dimensions. 
     3. Pin fins embedded in a metallic foam block to enhance the heat transfer coefficient for efficient heat dissipation into the flowing fluid stream. 
     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 . 
     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. 
     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 h  represents a ratio of the surface area of the heat source A h  with 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 in 2  and 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 in 2 . 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. 
     The optimum height b of a pin fin of uniform cross section along its height is determined using the relation:              b   =     0.919              k   f     h            A   c     p                   (   1   )                                
     where 
     k f  is the thermal conductivity of the pin material, Btu/ft s° F. 
     h is the convective heat transfer coefficient for the foam-filled space bounded by the pins and the spreader plate, Btu/ft 2  s° F. 
     A c  is the cross sectional area of the pin, ft 2    
     p is the perimeter of the pin, ft. 
     The ratio of the cross sectional area A c  to perimeter p for some simple pins are presented in Table 1. 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Ratio of Cross Sectional Area to Perimeter for Different pin Shapes 
               
             
          
           
               
                   
                 Fin cross section 
                 A c /p 
               
               
                   
                   
               
               
                   
                 Circular with diameter d 
                 d/4 
               
               
                   
                 Rectangular with thickness d 
                 d/2 
               
               
                   
                 Square with side d 
                 d/4 
               
               
                   
                 Equilateral triangular with side d 
                 d/43 
               
               
                   
                   
               
             
          
         
       
     
     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. 
     The heat transfer coefficient h entering Equation (1) for the foam-filled space can be determined using the formula:              h   =       1.2704        [       n   0.50         (     1   -   φ     )     0.25       ]            (         ρ   0.50          k   0.63          c   p   0.37         μ   0.13       )          u   m   0.50               (   2   )                                
     where: 
     n is the linear density of the foam material, pores per ft 
     φ is the foam porosity expressed as a fraction 
     ρ is the density of the flowing fluid, lb m /ft 3    
     k is the thermal conductivity of the flowing fluid, Btu/ft s° F. 
     c p  is the isobaric specific heat of the flowing fluid, Btu/lb m ° F. 
     μ is the dynamic viscosity of the flowing fluid, lb m /ft s 
     u m  is the mean velocity of the flowing fluid, ft/s. 
     Referring to FIG. 10, the non-uniform heat flux {dot over (q)}″ over the spreader plate can be represented by a cosine function as:                  q   .     ″     =         q   .     o   ″          [     1   +     αcos        (       π                 x     a     )         ]               (   6   )                                
     where 
     {dot over (q)}″ o  is the mean heat flux over the spreader plate, Btu/ft 2    
     α is a dimensionless constant dependent on the maximum and minimum values {dot over (q)}″ max  and {dot over (q)}″ min  of the heat flux {dot over (q)}″ (vide infra) 
     x is the variable distance measured from the center of the spreader plate along its width, ft 
     a is the spreader plate half-width, ft. 
     The mean heat flux {dot over (q)}″ o  and the dimensionless constant α are expressible in terms of {dot over (q)}″ min  and {dot over (q)}″ max  as                  q   .     o   ″     =       1   2          (         q   .     min   ″     +       q   .     max   ″       )               (   7   )               α   =       1   -         q   .     min   ″     /       q   .     max   ″           1   +         q   .     min   ″     /       q   .     max   ″                   (   8   )                                
     The non-uniform pin density β is related to the non-uniform heat flux {dot over (q)}″ simply as              β   =       β   o          (         q   .     ″         q   .     o   ″       )               (   9   )                                
     where 
     β o  is the mean pin density over the spreader plate in the flow direction, pins/ft. The mean pin density β o  can be interpreted as the linear pin density in the flow direction with uniform heat flux over the spreader plate. 
     Introducing Equation (6) into Equation (9), the non-uniform pin density β can be expressed as              β   =       β   o          [     1   +     αcos        (       π                 x     a     )         ]               (   10   )                                
     which mirrors the non-uniform heat flux {dot over (q)}″ distribution given in Equation (6). 
     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 
     
       
           {dot over (q)}″={dot over (q)}″   max  for 0 ≦x≦λa   (11) 
       
     
     
       
           {dot over (q)}″={dot over (q)}″   min  for λ a&lt;x≦a   (12) 
       
     
     where 
     λ is the fraction of the spreader plate width with maximum heat flux. 
     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              β   =           β   o          (         q   .     max   ″         q   .     o   ″       )                     for                 0     ≤   x   ≤     λ                 a               (   13   )               β   =           β   o          (         q   .     min   ″         q   .     o   ″       )                     for                 λ                 a     &lt;   x   ≤   a             (   14   )                                
     where the mean heat flux {dot over (q)}″ o  over the spreader plate is given in Equation (7). 
     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 
     
       
           N=βl   (15) 
       
     
     where l is the spreader plate length normal to the spreader plate width  2   a.    
     Four numerical examples are given below to illustrate how to apply the teachings of the present invention with the use of Equations (1)-(15). 
     EXAMPLE 1 
     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. 
     Density ρ=0.0749 lb m /ft 3    
     Thermal conductivity k=0.0000041 Btu/ft s° F. 
     Isobaric specific heat c p =0.2410 Btu/lb m ° F. 
     Dynamic viscosity μ=0.0000123 lb m /ft s. 
     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 /ft 3 , 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/ft 2 s° F. is obtained. 
     Next, the ratio of the cross sectional area A c  to 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. 
     Finally, introducing h=0.0313 Btu/ft 2 s° 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. 
     EXAMPLE 2 
     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. 
     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. 
     Density ρ=0.0749 lb m /ft 3    
     Thermal conductivity k=0.0000041 Btu/ft s° F. 
     Isobaric specific heat c p =0.2410 Btu/lb m ° F. 
     Dynamic viscosity μ=0.0000123 lb m /ft s. 
     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 /ft 3 , 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/ft 2 s° F. is obtained. 
     In order to determine the ratio of the cross sectional area A c  to 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. 
     Finally, introducing h=0.0313 Btu/ft 2 s° 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. 
     EXAMPLE 3 
     It is required to determine the non-uniform pin fin density over the surface of a square spreader plate with side  2   a =5 in. It is found that with uniform heat flux {dot over (q)}″ o  over the spreader plate the mean pin density β o  in 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)}″ max  at the center of the spreader plate is twice the minimum heat flux {dot over (q)}″ min  at 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. 
     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  2   a  since the plate is a square. Thus l= 2   a =5 in. The calculated values of β and N at representative distances x (vide FIG. 10) are presented in the following table: 
     
       
         
               
               
               
               
               
               
               
             
           
               
                   
                   
               
             
             
               
                   
                 x, in. 
                 0 
                 ±0.625 
                 ±1.250 
                 ±1.875 
                 ±2.500 
               
               
                   
                 β, 
                 13.3 
                 12.4 
                 10.0 
                 7.6 
                 6.6 
               
               
                   
                 pins/in. 
               
               
                   
                 N, pins 
                 67 
                 62 
                 50 
                 38 
                 33 
               
               
                   
                   
               
             
          
         
       
     
     EXAMPLE 4 
     It is required to determine the pin fin density over the surface of a square spreader plate with side  2   a =5 in. It is found that with uniform heat flux {dot over (q)}″ o  over 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)}″ max  at the center of the spreader plate is twice the minimum heat flux {dot over (q)}″ min  at 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)}″ max  is imposed over 25% of the spreader plate width so that the fraction λ=0.25. 
     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)}″ o  and {dot over (q)}″ min /{dot over (q)}″ o  must 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  2   a  since the plate is a square. Thus, l= 2   a =5 in. The calculated values of β and N for the appropriate ranges of x (vide FIG. 11) are presented in the following table: 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                   
               
               
                   
                   
                 β, 
                   
               
               
                   
                 x, in. 
                 pins/in. 
                 N, pins 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 −0.625 ≦×≦ 0.625 
                 13.3 
                 67 
               
               
                   
                   0.625 &lt;×≦ 2.5 
                 6.7 
                 33 
               
               
                   
                   −2.5 ≦×&lt; −0.625 
                 6.7 
                 33 
               
               
                   
                   
               
             
          
         
       
     
     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.