Abstract:
A microchannel heat sink used to cool a high power electronic device such as an integrated circuit comprising a plurality of channels in close thermal contact to the integrated circuit and through which a liquid is passed to create either a developing laminar flow or a turbulent flow. The turbulent flow may be either developing or fully developed. The heat sink features a compensation heater surrounding the integrated circuit and heated at the same rate as the integrated circuit to thereby provide a more uniform temperature at the perimeter of the integrated circuit.

Description:
The Government has rights in this invention pursuant to Grant Number AF19628-85-C-0002 awarded by the Air Force. 
    
    
     BACKGROUND OF THE INVENTION 
     This invention relates to an improved method and apparatus for cooling high power microcircuits. 
     Since the invention of the silicon integrated circuit, there has been in excess of a five orders of magnitude increase in circuit integration (the total number of components on a chip). Part of this increase can be traced to growth in chip size, where the characteristic dimensions have gone from about 1 mm in 1965 to about 4 mm to 8 mm in 1985. The remainder of the increase in circuit integration can be primarily traced to the reduction in the intgrated circuit feature size, which is today generally less than 1.0 μm and rapidly approaching the size of 0.1 μm. In fact, silicon MOSFETs have been made with features as small as 0.025 μm, which is on the order of one hundred atoms across. 
     The increase in circuit integration continues to be driven by the need for increased processing speed. Processing speed is dependent upon the signal delay times which are directly proportional to the physical length of interconnections between circuit components. A measure of the processing speed is how many millions of instructions per second (MIPS) can be executed. The number of MIPS and the total power dissipation of a chip are both directly proportional to the packaging density. At the component level, the chip power dissipation has been rapidly increasing due to the tremendous increase in the MIPS/chip. 
     The increase in the chip power dissipation has greatly complicated thermal management of microelectronics. Chips usually must operate at temperatures lower than 120° C. Such a low temperature is dictated by stringent reliability constraints. The failure rate of microelectronic circuitry decreases exponentially with decreasing component temperature. In fact, a commonly used rule of thumb for microelectronic reliability is that a 20° C. decrease in component temperature will typically result in a 50-percent reduction in the failure rate.. Continued pressure to enhance reliability will therefore demand reduction in the component temperatures, as well as more uniform component temperatures. 
     Forced-convection, liquid cooled, microchannel heat sinks have been recently developed to dissipate heat in modern microcircuitry. The first technical publications on this appeared in the early 1980&#39;s as the result of a study done by Tuckerman and Pease at Stanford University, and have caused considerable interest in the heat transfer community. 
     The configuration of a typical microchannel heat sink is shown in FIG. 1. Heat sink 10 is a finned structure which is cooled by forced convection. The coolant flows from inlet plenum 12 through rectangular microchannels 14, and out into the discharge plenum 16. Power is dissipated by microcircuit 20, and the heat is conducted through the substrate to fins 15 (shown in FIG. 2) where it is conducted to the coolant. The coolant can be either a liquid or a gas, but the pumping power requirements for gases are much larger for the same thermal performance. Also shown in FIG. 1 is cover plate 30 and manifold block 32. 
     FIG. 2 is a sectional view of the prior art device of FIG. 1 taken along line A--A, and fins 15 can be seen to create microchannels 14 through which the coolant flows. The width of each microchannel is shown as W c  and the width of each fin is shown as W w . FIG. 3 shows a sectional view of the device of FIG. 1 taken along line B--B. The height of each microchannel is indicated by b, while the distance from circuit 20 and the microchannel is indicated by t. The channel length (or fin length) is indicated in FIG. 1 as L. 
     The microchannels can be fabricated in silicon substrates by precision sawing or orientation-dependent etching, and usually have a larger height b than width W c  This ratio is called the aspect ratio. The channel width W c  is typically about 55 μm, and the fin thickness W w  is typically about 40 to 45 μm. The substrate height t is typically about 100 to 200 μm, and the fin height b about 300 to 400 μm. The channel length is typically 1.4 to 2.0 cm. 
     The hydrodynamic and heat transfer properties of flow through the rectangular microchannels depends on the flow regime, which can be laminar, turbulent, or a transitional stage between the two. The parameter used to define the flow regime is the Reynolds number which is given by ##EQU1## 
     where Re is the Reynolds number, ρ f  the coolant density, μ f  the coolant dynamic viscosity, V c  the average coolant velocity in the duct, and D e  the hydraulic diameter. 
     The transition between laminar and turbulent flow occurs over a range of Reynolds numbers. This range is large if the tube entrance is &#34;bell-mouthed&#34; and much smaller if there is an &#34;abrupt&#34; reduction in the flow cross-sectional area at the entrance. In the latter, the flow tends to &#34;trip&#34; at a &#34;critical&#34; Reynolds number, for which the flow is laminar if Re&lt;Re crit , and the flow is turbulent if Re&gt;Re crit . 
     In order to fully understand the design and operation of microchannel heat sinks, the thermal performance of these devices will be analyzed in detail. The thermal performance of a device is often specified in terms of its thermal resistance. The total thermal resistance R is given by R=ΔT/Q, where ΔT is the peak temperature rise above an ambient reference temperature, and Q is the total heating rate. In analyzing microchannel heat sinks, a modified thermal resistance R tot  is used. It is given by ##EQU2## where A cross  is the surface area over which the heat input Q occurs, and q is the heating rate per unit surface area which is assumed to be uniform over A cross . Using the modified thermal resistance makes it possible to formulate the thermal analysis based on a single channel and an adjacent fin. This is allowed since neighboring channels and fins are symmetrical in thermal response (assuming that the heat source is large enough). Therefore, R tot  will be based on A cross  =L(W w  +W c ). 
     The total thermal resistance can be thought of as being made up of several components. The arrangement of these thermal resistances is similar to an electrical network of resistors in series and/or parallel. The network will be slightly different depending on the aspect ratio. Six main types of thermal resistance will be discussed next. 
     The first thermal resistance is R spread , which is the &#34;constriction resistance&#34; due to thermal spreading from each discrete heat source (e.g., integrated circuit feature or gate) on the surface of the chip. This so-called spreading thermal resistance has been said to be the limiting indicator of transistor speed improvement. This is because the speed of some types of transistors (e.g, bipolar devices) goes up as the supplied power goes up (which is mostly dissipated as heat). Therefore, the transistor speed can be increased only so much because the high heating rate will eventually drive the device temperature too high. 
     The thermal-spreading resistance will be a function of the size and shape of the heat source. For a circular heat source the following equation can be used: 
     
         R.sub.spread =ΔT.sub.c /q=1/(2 k.sub.w a√π) 
    
     where a is the radius of the heat source, ΔT c  is the constriction effect, and K w  is the thermal conductivity. For a square heat source the following equation is used: ##EQU3## where k w  is the thermal conductivity of the material evaluated at the device temperature, and a is the characteristic length. 
     Currently, thermal spreading accounts for on the order of 10° C. of the total temperature rise of about 100° C. in typical silicon chips. However, as the level of circuit integration continues to increase, the temperature rise due to thermal spreading should decrease making this resistance much less significant. Therefore, the effect of thermal spreading will not be included in the thermal resistance models, discussed below. 
     The second type of thermal resistance is R solid , which is due to the conduction of heat through the solid material between the heating surface (e.g., integrated circuit) and the fin base and channel base plane. The modified thermal resistance is given by: ##EQU4## where k w  is the wall thermal conductivity evaluated at the average substrate temperature, and t is the substrate thickness. Note that if there is a thermal interface then R solid  will be the sum of the contributions between the heating surface and the interface, and between the interface and the fin base and channel base plane. 
     The third type of thermal resistance is R int , which is due to the thermal interface (if any) between the microchannel heat sink and the heat source (e.g., an integrated circuit chip). If there is an interface, then the microchannel heat sink may be considered a &#34;cold plate&#34;. Attachment of the heat source can be done using various types of bonding (solder, epoxy, etc), thermal grease, gas layers, etc. The thermal resistance models discussed below assume that the microchannel heat sink is manufactured directly into the substrate of the heat source, and therefore do not include R int . 
     The fourth type of thermal resistance is R cont , which is due to the &#34;constriction effect&#34; at the base of the fin (if there is a fin). If fins are used, it is intended that they transfer more heat than if the fin base surface area were exposed directly to the fluid. This necessitates that the heat flow be &#34;funneled&#34; into the base of the fin. The fins act as a long-strip heat sink for which the constriction thermal resistance may be obtained from the following equation: ##EQU5## where k w  is the material thermal conductivity which is evaluated at the average substrate temperature. This equation applies for large aspect-ratio channels. For moderate aspect-ratio channels, some heat is convected from the channel base thereby making the constriction thermal resistance somewhat smaller. Compared to the overall thermal resistance, this reduction is small and therefore will be ignored for simplicity and conservatism. The constriction thermal resistance is zero for small aspect-ratio channels because the fins are used for structural purposes and are assumed to transfer no heat. The fin thickness is assumed to be much smaller than the channel width, and therefore there is also no constriction thermal resistance for the heat flow into the channel base. 
     The fifth type of thermal resistance is R conv , which is due to the convection of heat from the channel base and/or the fin. If there is a fin, R conv   also includes the thermal resistance of heat conduction in the fin. Therefore the convective thermal resistance is given by: ##EQU6## 
     where M=(2h/K w  W w ) 0 .5, A bc  is the area of the channel base (LW c ), A bf  is the area of the fin base (LW w ), h is the average heat transfer coefficient, and b is the height of the fin. Using the definition of the fin efficiency, n f  =[tanh(mb)]/mb, and doing some rearranging gives: ##EQU7## which represents the convective thermal resistance between the fin base and channel base plane and the local coolant, for moderate aspect-ratio channels. This equation assumes that the temperatures of the fin base and the channel base are equal. Note that for large aspect-ratio ducts, hw c  ˜0, and for small aspect-ratio ducts 2hbn f  ˜0. 
     The overall thermal resistance is based on the difference between the peak chip surface temperature and the inlet coolant temperature. Thus, an effective thermal resistance can be defined which is to be added to the local thermal resistance at a given distance from the channel entrance. Therefore, the sixth type of thermal resistance is R bulk , which is due to the bulk temperature rise of the coolant from the channel entrance caused by absorption of the heat from the microchannel heat sink. The bulk thermal reistance is given by: ##EQU8## where V is the volumetric flow rate of the coolant per channel, V c  is the velocity of the coolant in the channel, Cp f  is the specific heat, and ρ f  is the coolant density. 
     There is one phenomenon that cannot be put into the form of a thermal resistance since it is independent of the heating rate. This is the temperature rise of the coolant due to viscous heating. The coolant temperature rises due to the conversion of mechanical energy (fluid pressure) into thermal energy (fluid temperature rise). The temperature rise is defined as ΔT pump  and is given by: 
     
         ΔT.sub.pump =ΔP/ρ.sub.f Cp.sub.f J 
    
     where ΔP is the coolant pressure drop (between the inlet plenum and the channel exit), and J is the mechanical equivalent of heat. 
     To summarize, the total modified thermal resistance is give by the sum of six thermal resistance terms as: 
     
         R.sub.tot =R.sub.spread +R.sub.solid +R.sub.int +R.sub.cont +R.sub.conv +R.sub.bulk 
    
     R spread  is ignored since it should become small as the level of circuit integration increases, and because it is highly dependent on the transistor technology. R int  is in general not required for integrated circuits and is also ignored. Therefore, the thermal resistance models discussed below will consider the total thermal resistance as being given by: 
     
         R.sub.tot =R.sub.solid +R.sub.cont +R.sub.conv +R.sub.bulk 
    
     Since viscous heating cannot be accounted for using a thermal resistance, the total temperature rise at the channel exit is given by: 
     
         ΔT.sub.tot =R.sub.tot q+ΔT.sub.pump 
    
     where it is implied that R tot  and ΔT pump  are independent of q. To be accurate, the properties must be evaluated at proper average temperatures. 
     To facilitate comparison between heat sink designs, it is necessary to have a common ambient reference temperature (i.e., the inlet coolant temperature). But this is not enough. It is obvious that different heating rates will result in different average properties of the liquid coolant and the heat sink material as well. Therefore, the total thermal resistance of the same heat sink with the same inlet coolant temperature will be different for different heating rates--especially if a comparison is attempted between very small and very large heating rates. Therefore, when comparing various heat sink designs, the heating rate must also be prescribed. 
     Thermal resistance models will now be analyzed for various channel aspect ratios (α=b/W c ). They are large (α&gt;10), moderate (0.1&lt;α&lt;10), and small (α&lt;0.1). 
     Large aspect-ratio channels have large fin heights. The surface area for convective heat transfer of the channel base is small compared to the surface area of the fins. Therefore, the contribution of the channel base to the overall heat transfer is small and can be ignored even though the channel base surface temperature is on average higher than the fin temperature. Ignoring the contribution of the channel base (hw c  ˜0) in R conv , gives the following model for the total thermal resistance: ##EQU9## where R tot ,large is the total modified thermal resistance for large aspect-ratio channels. 
     Moderate aspect-ratio channels have shorter fin heights such that the heat transfer from the channel base is also significant (and therefore not ignored). The total thermal resistance is given by: ##EQU10## where R tot ,moderate is the total modified thermal resistance for moderate aspect-ratio channels. 
     Small aspect-ratio channels have fins which are primarily used for structural purposes. The fins are small in thickness compared to the channel width (w w  &lt;&lt;w c ), and are very short in height (b&lt;&lt;w c ) Therefore, the contribution of the fins to the overall heat transfer is small and can be ignored. Since w w  &lt;&lt;w c , it is assumed that w w  ˜0, which results in R cont  ˜0. Therefore, the total thermal resistance is: ##EQU11## where R tot ,small is the total modified thermal resistance for small aspect-ratio ducts. 
     The analysis performed by D. Tuckerman and R. Pease of Stanford University considered similar thermal resistance formulas and developed an &#34;optimum&#34; design for microchannel heat sinks. Their analysis was the basis for prior art microchannel heat sink designs and every microchannel heat sink built to date has been constructed in accordance with their conclusions. The Tuckerman and Pease analysis is discussed in one or more of the following publications, each of which are incorporated herein by reference: Tuckerman, D. B., 1984, &#34;Heat-Transfer Microstructures for Integrated Circuits&#34;, PhD Thesis, Stanford University, Stanford, California; Tuckerman, D. B. and Pease, R. F. W., 1981, &#34;High Performance Heat Sinking for VLSI&#34;, IEEE Electron Device Lett. EDL-2, pp. 126-129; Tuckerman, D. B. and Pease, R. F. W., 1981, &#34;Ultrahigh Thermal Conductance Microstructures for Cooling Integrated Circuits&#34;, 32nd Electronics Components Conf. Proc., pp. 145-149; Tuckerman, D. B. and Pease, R. F. W., 1982, &#34;Optimized Convective Cooling Using Micromachined Structures&#34;, Electrochemical Society Extended Abstract No. 125,82, pp. 197-198; Tuckerman, D. B. and Pease, R. F. W., 1983, &#34;Microcapillary Thermal Interface Technology for VLSI Packaging&#34;, Symposium on VLSI Technology, Digest of Technical Papers, pp. 60-61. Background material is also available in Philips, R. J., 1988, &#34;Forced Convection Liquid Cooled Microchannel Heat Sinks&#34;, Technical Report 787, Massachusetts Institute of Technology, also incorporated herein by reference. The following is a summary of Tuckerman and Pease&#39;s analysis and conclusions. 
     For large aspect-ratio channels, there is a surface area multiplication factor δ=2b/(w w  +w c ) due to the fins Assuming fully developed laminar flow at the channel exit provides the friction factor as f=24/Re=24μ f  /(ρ f  V c  2w c ). The Nusselt number is constant for a given channel aspect ratio. The coolant pressure drop is a design constraint and is given by ΔP=4f(L/D e )ρ f  Vc 2  /2gc. Upon substitution into the thermal resistance equation, the following thermal resistance model is obtained: ##EQU12## When δ is large, it can be shown that w w  =w c  maximizes n f  (which is less than 1.0 and therefore minimizes R). The surface area enhancement is given by δ=(k w  /k f  Nu) 0 .5 which results in n f  =0.76. With these values for δ and n f  substituted into the above equation, the &#34;optimum&#34; channel width is given by taking dR/dw c  =0 which gives: 
     
         W.sub.c ≈2.29(μ.sub.f L.sup.2 k.sub.f Nu/ρ.sub.f Cp.sub.f ΔP).sup.0.25 
    
     for which the &#34;optimum&#34; modified thermal resistance is given by: ##EQU13## 
     As a result of this analysis, prior art microchannel heat sinks are designed to operate with fully developed laminar flow and have channel widths typically less than 100 μm. 
     SUMMARY OF THE INVENTION 
     The present invention is directed to a method of using a microchannel heat sink that includes the step of passing a liquid through a plurality of channels in the heat sink in such a manner as to create a turbulent flow (either developing or fully developed). A second embodiment of the invention utilizes a developing laminar flow instead of a turbulent flow. 
     In order to create more uniform temperatures about the periphery of the integrated circuit the present invention utilizes a &#34;compensation heater&#34; comprising a heater in close thermal contact to the perimeter of the IC chip. 
     In the preferred embodiments, the microchannel heat sink is formed from indium phosphide with channel widths between 125 μm and 300 μm. The heating rate of the compensation heater is the same as the heating rate of the IC chip. 
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The drawings will first briefly be described. 
    
    
     DRAWINGS 
     FIGS. 1-3 are various views of a prior art microchannel heat sink. 
     FIGS. 4 and 5 are two views of a microchannel heat sink in accordance with the present invention. 
     FIG. 6 is a flowchart outlining a method for computing thermal resistance models in accordance with the theory of the present invention. 
     FIG. 7 is a flowchart which shows when subroutines are called in the program used for computing thermal resistance models. 
     FIG. 8 is a graph of thermal resistance and pumping power v. channel width for the preferred embodiment. 
     FIG. 9 is a graph of thermal resistance v. the distance from the upstream heater edge. 
     FIG. 10 is a graph of thermal resistance v. channel width for the preferred embodiment and for the prior art. 
     FIG. 11 is a graph of thermal resistance v. distance from the upstream compensation heater edge with and without the compensation heater energized. 
    
    
     STRUCTURE 
     The microchannel heat sink structure of the present invention is shown in FIG. 4 and comprises heat sink 100 having a plurality of microchannels 114 through which coolant flows as indicated by the arrows. Heat sink 100 is mounted on cover plate 130. Integrated circuit (IC) 120 is surrounded by compensation heater 150 shown in greater detail in FIG. 5. The device includes input and output ports identical to those described above in connection with the prior art device shown in FIG. 1 and are therefore not shown in FIG. 4. 
     FIG. 5 illustrates a more detailed view of compensation heater 150. Compensation heater 150 extends a distance Zs on both sides of IC 120 in the direction of fluid flow as shown, and extends a distance Zt in a direction perpendicular to fluid flow. The dimensions of Zs and Zt will vary according to the specific design parameters. These dimensions apply regardless of the size of IC 120. Power is supplied to compensation heater 120 through four positive contacts 122 and four negative contacts 124, which may be formed of any high conductivity material such as silver or gold. Compensation heater 120 may be formed of any standard resistive material. 
     OPERATION 
     The operation of the microchannel heat sink of this invention was developed as the result of a detailed analysis of the thermal resistance models that departs substantially from the analysis of Tuckerman and Pease upon which the prior art microchannel heat sinks are based. A summary of the analysis upon which the present invention is based is presented below. 
     As noted above, the Tuckerman and Pease analysis began by assuming that the flow was laminar and fully developed. These assumptions produced the &#34;optimum&#34; designs that are the basis for today&#39;s microchannel heat sinks The analysis upon which the present invention is based, however, makes no such assumptions. Through the use of a sophisticated computer program discussed below and attached hereto as appendix A, it was discovered that the &#34;optimum&#34; of Tuckerman and Pease was illusory. As the following detailed discussion makes clear, vastly superior results have been shown to be achievable if the microchannel heat sink is operated in the turbulent regime (fully developed or developing flow) or in the laminar regime with developing flow. 
     The solution of the thermal resistance models will now be analyzed for three design constraints on the coolant flow rate. 
     The first design constraint is fixed pressure drop ΔP, which may be required due to supply pressure limitations of the circulation pump or due to structural (stress) limitations of the heat-sink design. The second design constraint is fixed volumetric flow rate per unit heater surface area V, which may be required since the circulation pump may be of the positive-displacement type. The third design constraint is fixed pumping power per unit heater surface area P=ΔPV, which may be required to limit the power consumption of the cooling system. These three constraints are not totally independent when a practical heat-sink design is being sought. In general, all three constraints must be satisfied in that the pressure drop does not exceed ΔP, that the volumetric flow rate does not exceed V, and that the pumping power does not exceed P. 
     To determine the thermal performance of a heat sink, several analysis parameters must be known in advance. They are: the heat-sink material, the liquid coolant, the inlet temperature of the liquid coolant T f ,in, the heating rate q, the channel width w c , the fin thickness w w  (or the ratio w w  /w c ), the fin height b (or the channel aspect ratio α=b/w c ), the channel length L, the ratio of the channel and plenum cross-sectional areas Ac/Ap, the substrate thickness t, the ratio of the channel and plenum cross-sectional areas Ac/Ap, the substrate thickness t, and one design constraint (ΔP, V, or P). This adds up to ten variables plus the design constraint. If the channels are roughened with repeated-ribs, there are four additional analysis parameters which must also be known in advance. These roughness parameters are: the ratio of the rib height to the channel hydraulic diameter e/D e , the ratio of the distance between repeated-ribs to the rib height p/e, the rib shape angle φ, and the flow attack angle ψ. The computer program used in this study can only treat one design constraint at a time. Therefore, the solution procedures used to obtain the coolant velocity V c  for each design constraint are discussed separately. 
     Since the coolant velocity is not in general known in advance, the Reynolds number also is not known. To avoid problems with satisfying the Reynolds number criterion for laminar vs turbulent flow, the flow is first assumed to be laminar. The entire solution procedure is then completed assuming that the flow is laminar. The flow is then assumed to be turbulent and the entire solution procedure is repeated. If the laminar case Reynolds number turns out to be greater than Re crit , the numerical results are discarded. Similarly, if the turbulent case Reynolds number turns out to be lower than Re crit , the numerical results are discarded. 
     For the fixed pressure drop constraint, V c  can be obtained after rearrangement as: ##EQU14## where K 90  ˜1.2, K c  is the entrance pressure loss coefficient, and K e  is the exit pressure loss coefficient. For laminar flow in smooth channels, the apparent friction factor f app  is obtained as a function of α=b/w c  and L +=L/D   e  Re. For turbulent flow in smooth channels, the apparent friction factor is obtained as a function of L/D e  and Re. For fully developed turbulent flow in repeated-rib-roughened channels, the apparent friction factor is given by f 1  or f 2  which are obtained as a function of D e , the roughness parameters (e/D e , p/e, φ, and ψ), and Re. Therefore, Kc, Ke, and f app  are all functions of Re and other known analysis parameters. Since Re is a function of V c  and other known analysis parameters, it becomes clear that trial-and-error solution using successive guesses of V c  is required. 
     For the fixed volumetric flow rate constraint, the solution for the coolant velocity is easy. The volumetric flow rate per unit heater surface area V must first be converted to a volumetric flow rate per channel. This is done by multiplying V by the heater surface area directly over one channel and an adjacent fin L(w w  +w c )]. The velocity is directly obtained by dividing the result by the cross-sectional flow area of one channel (bw c ). Therefore, the coolant velocity is given by: ##EQU15## where V is the design constraint, and the geometrical parameters (L, w w , w c , and b) are all known. 
     For the fixed pumping power constraint, the coolant velocity can be obtained using: ##EQU16## where the solution procedure is the same as that discussed above with the fixed pressure drop constraint. 
     Now that the coolant velocity is known, the heat-transfer coefficient can be obtained from the Nusselt number Nu=hD e  /k f . For laminar flow in smooth channels, Nu is obtained as a function of α and x*=L/D e  RePr (where Pr is the coolant Prandtl number). For turbulent flow in smooth channels, the heat-transfer coefficient is obtained as a function of Pr and Re. For fully developed turbulent flow in repeated-rib-roughened channels, Nu is obtained as a function of D e , the roughness parameters (e/D e , p/e, φ, and ψ), the coolant parameters (ρf, μf, and Pr),and Re. 
     The total thermal resistance can now be obtained by substituting the values for the coolant velocity and the heat-transfer coefficient into the thermal resistance models. This solution is &#34;accurate&#34; if the correct values of the heat-sink material properties and the coolant properties are known. It was previously mentioned that the heating rate q and the inlet coolant temperature T f ,in both affect the properties of the heat sink and the coolant. To account for the effect of the heating rate, the following procedure is used. First the properties of the coolant and the heat-sink material are evaluated at T f ,in and assumed constant over the entire channel length. Then the coolant velocity, the friction factor, and the heat-transfer coefficient (at the channel exit) are obtained using the solution procedure discussed above. At this point, the individual thermal resistance components are computed and summed to give the total thermal resistance. Several temperatures are then computed using the heating rate, the thermal resistances, and ΔT pump . The relative error between the &#34;new&#34; and &#34;old&#34; temperatures is computed, and the solution is obtained if all the errors are sufficiently small. If one or more of the errors is too large, then average values for the temperatures are computed and used to evaluate the heat sink and coolant properties. 
     The entire solution procedure is then repeated using these new properties and calculations are performed to account for variations of Nu and f app  due to nonuniform coolant properties. This process continues until all the temperature errors are small enough. It is only after this final solution is obtained that the Reynolds number can be checked for being in the flow regime that was assumed in the beginning. If Re is not in the proper flow regime, the solution is discarded because it is not valid. 
     FIG. 6 summarizes the iterative solution procedure. The solution procedure requires considerable numerical computation. Because of this, a new computer program called &#34;MICROHEX&#34; has been written to compute the thermal and fluid performance of forced-convection, liquid-cooled, microchannel heat sinks (appendix A). MICROHEX will handle small, moderate, and large aspect-ratio channels with fully developed and developing flow in the laminar and turbulent regimes. The channel surfaces can be smooth or roughened with repeated-ribs. Variable property effects are included for the chip material and the liquid coolant (compressibility effects have not been included). The effect of viscous dissipation in the liquid coolant is included. Thermal spreading at the periphery of the heat source is also taken into account with simplified one-dimensional models. 
     Program MICROHEX and its 19 subroutines are written in Fortran IV. It has been successfully run on an IBM 3081, a MICROVAX II, and an IBM PC. Table 1 briefly discusses what each subroutine does, and FIG. 7 presents the program flowchart which shows when each subroutine is called. 
     The input for MICROHEX is provided via initializing up to 25 variables in the MICROHEX program data statements. These variables will now be introduced (see Table 2 for a summary of the input variables). 
     The variable NCURVE determines how many generic heat-sink designs will be analyzed. From one to five designs are allowed, and each design will have its thermal and fluid performance plotted using a unique curve type. 
     The variable IX determines if either the channel width w c  or the channel length x is fixed (constant). If the channel length is constant (IX=1), then the thermal and fluid performance will be plotted as a function of the channel width with all results provided at the channel exit (thermal spreading at the heater periphery is not included here since the channel exit is not the location of the peak, total thermal resistance if thermal spreading is considered). If the channel width is constant (IX=2), then the thermal and fluid performance will be plotted as a function of the channel length (thermal spreading along the channel length at the heater periphery will be included). 
     The variable NSINK is used to specify the heat-sink material. MICROHEX contains data for gallium arsenide, germanium, indium phosphide, silicon, aluminum, copper and silver. The user can easily include other materials simply by adding additional PROPS# subroutines. 
     The variable NFLUID is used to specify the liquid coolant, MICROHEX contains data for FC-77, Freon 12 (CCl 2  F 2 ), and water. The user can easily include other liquid coolants simply by adding additional PROPF# subroutines. 
     The variable ICASE is used to specify the flow regimes(s) and channel surface type. Three ICASEs are currently used. They are: (a) fully developed laminar flow everywhere (smooth surfaces), (b) fully developed/developing, laminar/turbulent flow (smooth channels). and (c) fully developed turbulent flow everywhere (repeated-rib-roughened surfaces). 
     The variable Q is used to specify the surface heat input q in watts per square centimeter of heater surface area. The variable TFLUIN is used to specify the inlet temperature of the liquid coolant in degrees Kelvin. 
     The variable WCSTAR is used to specify the smallest channel width w c  in microns that will be analyzed. The variable WCEND is used to specify the largest channel width in microns that will be analyzed. The variable WCINCR is used to specify the increment in channel width between WCSTAR and WCEND for intermediate channel widths that will be analyzed. If IX=1, then the thermal performance will be computed for channel widths w c  =WCSTAR, WCSTAR+1WCINCR, WCSTAR+2WCINCR, . . . ,WCEND. If IX=2, then WCSTAR=WCEND, and WCINCR=0 is required since the channel width is constant. 
     The variable WWBYWC is used to specify the ratio of the fin width w w  to the channel width w c . 
     The variable IZ is used to determine whether the channel height (fin height) b is constant, or if the channel aspect ratio α=b/w c  is constant. If IZ=1, then the variable B is used to specify the channel height (fin height) b in microns. If IZ=2, then the variable ASPECT is used to specify the constant channel aspect ratio. 
     The variable L is used to specify the channel length in meters. The variable T is used to specify the thickness in microns of the solid material between the surface of heat input and the channel base and fin base plane. 
     The variable IKLOSS is used to determine if the inlet and exit header pressure losses will be included or not. If IKLOSS=0, then Kc=Ke=K 90  =0, and only the channel friction will be used to determine the overall pressure drop. If IKLOSS=1, then Kc, Ke, and K 90  will be computed and included in the overall pressure drop. 
     The variable ICONS is used to determine which flow constraint will be used to determine the velocity of the liquid in the microchannels. If ICONS=1, then the variable DELP is used to specify the coolant pressure drop ΔP in psi. If ICONS=2, then the variable VOLUME is used to specify the coolant flow rate V in cubic centimeters per second per square centimeter of heater surface area. If ICONS=3, then the variable POWER is used to specify the coolant pumping power P in watts per square centimeter of heater surface area. 
     If the channel surfaces are roughened with repeated-ribs (α&lt;0.1 or α&gt;10), then the following four variables must be initialized. The variable EBYDE is used to specify the ratio of the rib height e to the channel equivalent diameter D e . The variable PBYE is used to specify the ratio of the spacing between repeated ribs p and the rib height e. The variable PHI is used to specify the rib shape angle φ in degrees. The variable ZI is used to specify the flow attack angle ψ in degrees. 
     Typical thermal and fluid performance predictions that can be obtained using MICROHEX will now be described. A &#34;reference&#34; microchannel heat sink design is utilized The reference design is a water-cooled silicon heat sink. The modified &#34;echo&#34; (produced by subroutine PINPUT) of the input variables is provided as Table 3. From this table, it can be seen that all thermal and fluid performance calculations were done at the channel exit and therefore the effect of thermal spreading at the heater perimeter was not included. The calculations were done for fully developed/developing, laminar/turbulent flow in smooth channels. The inlet coolant temperature is T f ,in =300° K. The surface heat flux is assumed negligible, and therefore the properties of the heat-sink material and the liquid coolant are constant all along the channel. The computations begin with a channel width of 5 μm and proceed in 5-μm increments up to w c  =500 μm. The ratio of the fin thickness to the channel width is unity, and the channel aspect ratio is constant at 4.0. The channel length is 1.0 cm, and the substrate thickness t is 100 μm. The pressure drop through the 90° bends, the inlet contraction, and the exit expansion are assumed to be small compared to the friction pressure drop along the 1.0 cm-long channel. The coolant velocity is determined for a 68.9 kPa (10 psi) pressure drop for all channel widths. 
     The thermal and fluid performance predictions are plotted in FIG. 8 for the reference case. The upper graph is a plot of the total thermal resistance per unit area as a function of the channel width. The lower graph is a plot of the pumping power required per unit surface area as a function of the channel width. On both graphs, the laminar flow results begin with open circles (small w c ), and end with open squares (large w c ) (the open circle is not visible on the lower graph). Similarly, the turbulent flow results begin with closed circles (smallest turbulent w c ), and end with closed squares (largest w c ) The transition Reynolds number determines where the laminar flow curve ends, and where the turbulent flow curve beings. The upward pointing triangular symbols indicate the separation between fully developed flow (to the left) and developing flow (to the right). The downward pointing triangular symbols on the laminar flow curves indicate where the thermal entrance length x*=L/D e  RePr becomes&lt;0.005 (to the right). 
     Three thermal resistance curves are shown for the reference case. They are the coolant bulk temperature rise thermal resistance R bulk ), the convective thermal resistance (R conv ), and the total thermal resistance (R tot ). The total thermal resistance also includes the solid substrate thermal resistance (R solid ) and the contraction thermal resistance (R cont ). The solid substrate thermal resistance was not plotted since it is constant for all channel widths (q=0). The contraction thermal resistance was not plotted since it is small and nearly constant. 
     Some interesting features of the curves should be noted. The magnitude of R bulk  continuously gets smaller with channel width since more coolant can be forced through the larger channels at the same pressure drop. It is quite interesting to note that the turbulent flow R bulk  appears to be an extension of the laminar flow results. 
     The laminar flow R conv  continuously gets larger with increasing channel width. This is an intuitively correct result. For example, for fully developed laminar flow, Nu=hD e  /k f  =const. Since the channel hydraulic diameter increases with channel width, the heat-transfer coefficient must decrease, thereby causing the thermal resistance to increase. For developing laminar flow, the Nusselt number increases as the flow at the channel exit becomes less fully developed, and therefore R conv  increases less rapidly. The turbulent flow R conv  is shown initially to decrease slowly with increasing channel width, and then to increase slowly. This is due to a steady increase in the flow Reynolds number which is required to meet the specified pressure drop. The Nusselt number gets bigger quite rapidly due to the increase in the Reynolds number. In fact, this effect is initially stronger than the reduction in h due to increases in D e . One should also remember that the Nusselt number was determined assuming that the flow was fully developed, when in this case it really is not fully developed at the channel exit. 
     Perhaps the most interesting fact to note here is that the laminar flow R tot  curve does not show any distinct optimum as predicted by the Tuckerman and Pease analysis. Instead, the total thermal resistance tends to more-or-less level off with sufficiently large channel widths. In fact, it is found that the total thermal resistance for turbulent flow is lower! It is significantly lower--on the order of 20 to 30 percent. The turbulent flow thermal and fluid performance predictions should be conservative (too large a friction factor, and too small a Nusselt number). This result is completely contrary to the results achieved by the prior art analysis and is the basis for the superior performance of the present invention. These facts will become more evident as the analysis is completed. 
     Four separate output files are provided as output by MICROHEX The numerical results contained in those output files for the reference case have been included as Appendix B of this report. At the top of each output column is the variable name and the units of the output results. The description of what each variable name represents is included in the comment header section of the RTOTAL subroutine (see Apendix A). 
     The typical values of several analysis parameters for the reference case will now be reviewed. For simplicity, the following channel widths are selected: fully developed laminar flow (w c  =50 μm), developing laminar flow (w c  =100 μm),and developing turbulent flow (w c  =300 μm). The liquid coolant and heat-sink material properties are uniformly constant since the surface heat input is zero (q=0). The properties are evaluated at TFLUIN=300° K. The silicon thermal conductivity is k w  ˜148 W/m°C. The water thermal conductivity is k f  ˜0.613 W/m°C., the specific heat is C pf  ˜4177.6 J/kg°C., the Prandtl number is Pr˜6.033, the density is ρ f  ˜9995.5 kg/m 3 , and the dynamic viscosity is μ f  ˜0.00088 kg/ms. The coolant temperature rise due to viscous dissipation is ΔT pump  ˜0.017° C. This is a very small temperature change, and therefore the properties are still esentially constant. 
     The heat sinks have a channel height b=200, 400, and 1200 μm, respectively, for w c  =50, 100, 25 and 300 μm (α=4.0). The corresponding fin widths are w w  =50, 100, and 300 μm. The channel hydraulic diameters are D e  =4A/P=80, 160, and 480 μm for the same channel widths. The laminar equivalent diameter is D 1  =416.3 μm for w c  =300 μm (turbulent flow). 
     The average coolant velocity in the channels is Vc=1.40, 4.67, and 11.59 m/s, respectively, for w c  =50, 100, and 300 μm. The corresponding Reynolds numbers are Re=127, 845, and 6290. The laminar equivalent Reynolds number is Re*=5450 for w c  =300 μm (turbulent flow). The hydrodynamic entrance length is given by L +  =L/D e  Re=0.99, 0.074, and 0.0033, respectively. The corresponding thermal entrance lengths are x*=0.164, 0.0123, and 0.0005. 
     The laminar flow friction factor can be obtained as f app  =[(f app  Re)/Re]=0.142 and 0.0254, respectively, for w c  =50 and 100 μm. The turbulent flow friction factor is f app  =0.0124. Using the specified pressure drop and these friction factors, the aforementioned coolant velocities are obtained (assuming Kc=Ke=K 90  =0). The coolant volumetric flow per unit heater surface area is V=1.4, 9.35, and 69.53 (cm 3  /s)/cm 3 , respectively, for w c  =50, 100, and 300 μm. The corresponding coolant pumping power per unit heater surface area is P=ΔPV=0.10, 0.64, and 4.79 W/cm 2 . 
     The laminar flow Nusselt number is approximately 5.86 and 7.17, respectively, for w c  =50 and 100 μm. The turbulent flow Nusselt number is approximately 42.8 for w c  =300 μm. The corresponding heat-transfer coefficients are h=Nuk f  /D e  =44,900, 27,470, and 54,600 W/m 2  °C. The corresponding fin efficiencies are 86.4, 84.0 and 50.7 percent. The corresponding values of the fin criterion are 2k w  /hw w  =132, 108, and 18.1. 
     The thermal resistance was computed for a moderate-aspect ratio since α=4.0. The solid material thermal resistance between the heater surface and the fin base and channel base plane is R solid  =0.0068° C./(W/cm 2 ) for all three channel widths. The contraction thermal resistance is R cont  =0.0007, 0.0015, and 0.0045° C./(W/cm 2 ). The convective thermal resistance is R conv  =0.0563, 0.0943, and 0.0724° C./(W/cm 2 ). The coolant bulk temperature rise thermal resistance is R bulk  =0.1718, 0.0257, and 0.0035° C./(W/cm 2 ). The total thermal resistance is R tot  =0.2356, 0.1282, and 0.0871° C./(W/cm 2 ). 
     The total thermal resistance of the three representative channel widths for the reference case is plotted in FIG. 9 as a function of the distance from the upstream heater edge. Two curves are shown for turbulent flow. The dashed curve assumes that the Nusselt number is fully developed everywhere. The solid dot curve multiplies the turbulent Nusselt number by [1+(D e  /L) 2/3  ] to approximate the effect of developing flow on the Nusselt number. A comparison of the two curves shows that the flow at the channel exit is nearly fully developed for w c  =300 μm. A comparison of the curves shows that the effect of R bulk  is very important for laminar flow. This leads to greater nonuniformity in R tot  along the channel. The average R tot  is consequently much smaller for turbulent flow. &#34;Compensation heaters&#34; can be used to enhance the uniformity in R tot , as discussed more fully below. 
     A comparison between the prior art thermal and fluid performance prediction theory (i.e., the Tuckerman and Pease theory), and that theory which is the basis of this invention, is shown in FIG. 10. The solid curve is for the reference case where the calculations are done for fully developed/developing, laminar/turbulent flow. The dotted curve is for the same reference case except that the flow is assumed to be fully developed and laminar for all channel widths (i.e., the Tuckerman and Pease assumption). When the flow really is laminar and fully developed (to the left of the upward pointing triangles), the two computational methods yield virtually identical results, which indeed they should. The small difference between the two curves in this region is due to the inherent disagreement between using polynomial equations to model fully developed laminar flow (see subroutines SFFD and SNUFD in Appendix A), and using interpolation in the lookup tables used in this study (see subroutines SLAMF and SLAMNU in Appendix A). 
     To the right of the upward pointing triangles, the flow is not fully developed. The R tot  for the case where the flow is always assumed to be laminar and fully developed is higher than that for the reference case. The error increases with increasing channel width. In fact, there is a distinct minimum (or &#34;optimum&#34;) in the thermal resistance, which is an artifact of the assumption that the flow is always laminar and fully developed. This optimum led the industry to believe that the best possible results were obtained using fully developed laminar flow when, in fact, as can be clearly seen from FIG. 10, the total thermal resistance is less for developing laminar flow and substantially less for developing or fully developed turbulent flow. Therefore, a microchannel heat sink operating in accordance with the present invention will achieve results that are superior to anything achieved by the prior art since the prior art thought such results were impossible. 
     FIG. 10 also presents thermal and fluid performance results for a channel aspect ratio of α=10.01 (the solid-dot and dashed curves). The solid-dot curve gives the results where the flow is assumed to be fully developed and laminar for all channel widths. The dashed curve gives the results for fully developed/developing, laminar/turbulent flow. The computational differences between the two models are greater for α=4.0 than for α=10.01. This difference is due to the fact that when α is greater than 10.0 it is assumed to be equivalent to flow between infinite parallel plates. The difference is notably small (less than 10 percent), which is the reason why α=10.0 is used to delineate between moderate-and large-aspect-ratio channels. The thermal resistance in o laminar flow is about 50 percent smaller for α=10.01 than for α=4.0. The turbulent flow thermal performance &#34;crosses over&#34; at w c  =240 μm. This effect is due to the fact that the fin efficiency for α=10.01 is much smaller than that for α=4.0 (even though the α=10.01 Nusselt number is larger). 
     Thermal spreading at the edge of an IC heat source can cause significant variation in temperature across an IC. This nonuniformity in temperature can lead to reliability problems owing to thermal stress caused by differential thermal expansion across the IC. Timing problems may also arise because the path lengths between devices will change due to differential thermal expansion. If the IC temperature is more uniform, these problems can be virtually eliminated. 
     One known method used to make the IC temperature more uniform is to keep the coolant bulk temperature rise as small as possible. The coolant bulk temperature rise can be reduced by increasing the coolant flow rate per unit surfce area V. This can be achieved by increasing the coolant pressure drop for a specified channel width, and by increasing the channel width for a specified coolant pressure drop. The effect of thermal spreading at the heater perimeter can still be a serious problem even if the coolant flow is large. 
     In the preferred embodiment. a &#34;compensation heater&#34; is used to make the IC temperature more uniform. As shown in FIG. 5, if an IC is sourrounded by a heater with the same heating rate q, then the zone of thermal spreading can be displaced outward from the IC. The heater which surrounds the IC heat source is called a &#34;compensation heater&#34; because it compensates for thermal spreading at the IC heat source perimeter. 
     FIG. 11 represents the thermal performance prediction obtained by MICROHEX for a comparison of the IC with and without the compensation heater energized. The dotted and solid-dot curves are for the IC without the compensation heater energized, and the solid and dashed curves are for both the IC and the compensation heater energized. The solid and dotted curves are for the center of the IC in the streamwise direction. The dashed and solid-dot curves are for the edge of the IC in the streamwise direction. Note that the effect of the two-dimensional heat transfer near the corners of the IC and the compensation heater has been crudely modeled by superimposing the one-dimensional models for the two directions. 
     From FIG. 11, it is apparent that the thermal resistance over the region of the IC heat source is much more uniform when the compensation heater is energized. The average total thermal resistance, though, is somewhat larger than when the compensation heater is off. The average thermal resistance is about R tot  =0.0825 and 0.05° C./(W/cm 2 ) when the compensation heater is on and off, respectively. This means that the inlet coolant temperature would need to be only about 3.0° to 3.5° C. lower when the compensation heater is on (q=100 W/cm 2 ) to maintain the same average surface temperature. In this example, the additional power required for the compensation heater is 2.75 times that of the IC heat source (the IC requires 16 W and the &#34;compensation heater&#34; requires 44 W). 
     Additional embodiments of the invention are within the scope of the appended claims. ##SPC1## ##SPC2## ##SPC3##