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ANALYSIS OF UNIFORM SEMI-POROUS HEAT SINK MEDIAS WITH FREE CONVECTION by Eric R. Savery A Project submitted in partial fulfillment of the requirements for the degree of Masters of Engineering Rensselaer Polytechnic Institute 2011 Approved by ___________________________________________________ Ernesto Guiteerez-Mirravette, Engineering Project Advisor Rensselaer Polytechnic Institute Hartford, Connecticut December, 2011 i Table of Contents: List of Tables .................................................................................................................... iii List of Figures ................................................................................................................... iv Nomenclature .................................................................................................................... v Acknowledgement ............................................................................................................ vi 1. Abstract .................................................................................................................... vii 2. Introduction ............................................................................................................... 1 4. Methodology/Approach ............................................................................................ 3 4.1. Assumption ........................................................................................................ 3 4.2. Theory ................................................................................................................ 4 4.3. Modeling ............................................................................................................ 8 5. Results and Discussion............................................................................................ 11 6. Conclusion ............................................................................................................... 22 7. Appendix 1 ............................................................................................................... 24 8. Appendix 2 ............................................................................................................... 25 9. Appendix 3 ............................................................................................................... 26 10. Reference ............................................................................................................. 27 ii List of Tables Table 1: Temperature vs Density ........................................................................................ 5 Table 2: Porous Heat Sink Characteristics.......................................................................... 8 Table 3: Block Form .......................................................................................................... 9 Table 4: Control Block Heat Sink Results ....................................................................... 11 Table 5: Porous Heat Sink Results................................................................................... 13 iii List of Figures Figure 1: Fin Heat Sink ....................................................................................................... 1 Figure 2: Assumptions ....................................................................................................... 3 Figure 3: Porosity............................................................................................................... 6 Figure 4: Permeability........................................................................................................ 6 Figure 5: No Heat Sink Mesh .......................................................................................... 10 Figure 6: Solid Block Mesh ............................................................................................. 10 Figure 7: Porous Block Mesh .......................................................................................... 10 Figure 8: No Heat Sink T-V............................................................................................. 12 Figure 9: Solid Heat Sink T-V ......................................................................................... 12 Figure 10: Porosity Affect on Heat Transfer ................................................................... 14 Figure 11: Permeability Affect on Heat Transfer ............................................................ 15 Figure 12: Permeability Affect on Velocity..................................................................... 16 Figure 13:Affect Of Increasing Permeability on Total Heat Flux .................................... 17 Figure 14: Affect Of Increasing Permeability on Volume Flow Rate ............................. 18 Figure 15: Porous Heat Sink T-V .................................................................................... 19 Figure 11: Porous Heat Sink P-V .................................................................................... 19 Figure 17: Mesh Affect on Total Flux .............................................................................. 20 Figure 18: Mesh Affect on Volume Flow Rate ............................................................... 21 iv Nomenclature Symbols Cp = K = ρ = ε = q = P = T = K = u = g = Heat Capacity at Constant Pressure Permeability Density Porosity Heat Flux Pressure Temperature Thermal Conductivity Fluid Velocity Field Gravity Subscripts Air = AL = c = eq = 0 = Air Aluminum Heat sink Initial/Cold Temperature Equivalent Initial v Acknowledgement vi 1. Abstract This work investigates what affects the use of metallic porous materials has when used as a heat sink. Specifically a comparison of how differences in the porous material characteristics affect the performance of a heat sink while comparing it to a traditional block heat sink and no heat sink at all. The modeled heat sinks will use the same volume mass of material, material thermal conductivity (ie Material) and have the same dimensional footprint. The boundary chip is held at a constant temperature and by varying the heat sinks porosity and permeability the total heat flux of the chip will be calculated. The results of the each heat sink are then compared and analyzed against each other to determine the affect of porous material as a heat sink. vii 2. Introduction Since the discovery of the microprocessor computing power is becoming more powerful and consequently the chips have been creating more heat. The heat in which more powerful processors create is required to be dissipated else the processor will become less reliable or fail1. For every 10ºc that you are able to reduce a transistors temperature the failure of the electronic component is halved1. Therefore the more heat that is able to be removed from the transistor the greater the reliability is. To transfer the waste heat from the transistor to the environment requires the use of a heat sink. A heat sink is a device which dissipates energy from a component to the ambient environment by use of natural or forced convection. The heat sink’s ability to dissipate the heat energy is a function of the material properties, geometry and the environmental conditions2. Increasing the amount of heat energy which is dissipated by the heat sink will allow for higher processing speeds at a lower temp. Below is an example of a simple fin heat sink that would be used on a typical integrated circuit. Figure 1: Fin Heat Sink This paper looks at how the use of semi-porous media with a significant amount of surface area affects the efficiency of a heat sink to dissipate heat energy from a computer chip. With the use of comsol modeling this paper will compare the efficiency of a block heat sink against a heat sink that uses a semi-porous material. 1 3. Problem Description The design of a heat sink is dependant on the physical space, cooling needs and cost of the component. In a semiconductor application increasing capabilities results in the need for additional cooling, increase cooling requires a larger heat sink and as a result increases the overall size and cost of the final component. For portable electronics the goal is to provide the most capability, in the smallest package for the lowest cost. Using a semi-porous material for a heat-sink may provide increased cooling per cubic inch of space resulting in a smaller and more capable device at a lower cost when compared to a traditional fin heat sink. To determine the most efficient porous material heat sink a comparison of the porous material characteristics is required in addition to comparing the porous material heat sink to the control case. 2 4. Methodology/Approach A systematic approach is used to build and analyze 2 different heat sinks and a flat plate. The heat sinks are be made of the same base material and have the following physics; porous and non-porous blocks. Each heat sink is made from the same base material to ensure that the differences in the results are not due to material thermal properties. The solid block heat sink and no heat sink are control cases while the porous block heat sink are subjected to differing porous material characteristics to see there effects on the heat sinks ability transfer heat at a standard temperature. 4.1. Assumption To effectively analyze the affects of using porous material as a heat sink the following diagram should be referenced: Open Boundary Insulated Wall Heat Sink Air Insulated Wall Open Boundary at Tc Figure 2: Assumptions In addition the following assumptions should be used: Bounded area is 50mm by 50mm. No single side of the heat sink can be larger than 10mm. Air inlet temperature is Tc. Tc is 293.15 K. 3 Insulated Wall Wall with Temperatur e of Th Wall has a constant temperature Th. Th is 310 4.2. Theory Heat Transfer: There are three different modes of heat transfer conduction, convection and radiation. In the case of the model without a heat sink the heat transfer of interest is pure convection. In addition to convection the porous and non porous material goes under conduction. The system without a heat sink results in a pure convection process where heat from the wall transfers directly into the fluid without an intermediate step. When a heat sink is used an intermediate step is present where the energy from the wall must pass through the block under conduction and then into the liquid via convection. Equation 1 is the governing equation for conduction and equation 2 is the governing equation for convection for non-porous material heat transfer. Qcond kA T x Equation 1:Conduction Qconv hAs T Equation 2: Convection One might think that adding an addition heat transfer step might decrease the efficiency of removing heat from a chip or heat source but the exact opposite true. The advantage of the intermediate step is that the block will increase the surface area in which convective heat transfer is able to occur. Increasing the surface area in which convection can occur can have a substantial increase in the rate in which heat can be transferred from a heat source. Buoyancy: Natural circulation is a function of the buoyancy of the cooling fluid. In cases where there isn’t a fan or another motive force to circulate coolant a natural force known as buoyancy causes the fluid to circulate. This fluid movement is consequence of the density gradient caused by the differing temperatures between the incoming air and the air which has been heated. As the air is heated the density of the fluid decreases causing an upward force opposite of gravity, this is also known as buoyancy force. The 4 forces cause an upward velocity of the cooling fluid. For modeling this phenomenon the following equation is used to determine the volume force. F g( 0 ) Equation 3 The equation 3 shows that as the density of a point decreases below the initial density a force is created. This density is directly proportional to the temperature of the cooling fluid at a particular point. Table 1 below show how density of air changes as a function of air. Temperature (degrees C) 35 30 25 20 15 10 5 0 -5 -10 -15 -20 -25 Density (kg/m^3) 1.1455 1.1644 1.1839 1.2041 1.225 1.2466 1.269 1.292 1.3163 1.3413 1.3673 1.3943 1.4224 Table 1: Temperature vs Density3 Porous Material Characteristics: There are two characteristics of significant importance when analyzing porous heat sinks. Those characteristics are porosity and permeability. Permeability (kappa) is the measure of the ability of a material to transmit fluid though itself. Porosity (epsilon) is also known as void fraction, this is the measure of voids or spaces in a material. This is defined in terms of a fraction of the total volume of the material. An increase in porosity means the heat sink is more air than the base material. See Figure 3 for an example of the difference of porosity and Figure 4 for an explanation of permeability. 5 Figure 3: Porosity Figure 4 shows the porosity of the block on the left is 4 time the porosity of the block on the right. Permeability of both of the blocks is approximately the same because only one hole passes all the way through the blocks. Assuming that the each hole equates to a tenth of the total area of the block, the block on the left has a porosity of .4 while the block on the right has a porosity of .1. Below is an example and explanation of permeability. Figure 4: Permeability Figure 4 demonstrates how two blocks could have the same porosity of approximately .4 but have two completely different permeability’s. The block on the left has one small hole passing through the entire way, while the block on the right has a large hole passing all the way through. Therefore the ability to pass fluid through the block on the right is much easier than passing fluid through the block on the left. This mean the permeability of the block on the right is greater than the block on the left. Heat Transfer in Porous Media: Heat transfer through a porous material is similar to heat transfer through a solid block. A heat equation similar to that used for heat transfer through a solid block heat sink is used to calculate heat transfer through the porous block. Equation 4 is the heat transfer equation that is used for the porous block heat sink: * Cp eq T * Cp * u * * T * k eq * * T Q t Equation 4 6 4 The difference between the simple heat equation and the heat equation for porous material are the corresponding affect of the volumetric heat capacity and thermal conductivity. In the case of the porous material both volumetric heat capacity and thermal conductivity is a function of the material porosity. As the material becomes more porous the composition of both the thermal conductivity and the volumetric heat capacity becomes more like the cooling fluid. Alternatively as the porosity nears zero the conductivity and volumetric heat capacity becomes more like that of the base material. Below equation 5 are the equation to determine the equivalent thermal conductivity and volumetric heat capacity. keq p k p F k F * Cp eq p pCp p F F Cp F p F 1 Equation 5 It should be noted equation 5 shows how the fraction of porous base material and fraction of cooling fluid must add up to 1. Free and Porous Media Flow: Since in porous material the cooling fluid is able to move through the heat sink, a way to model the flow of liquid through it is required. For a pure porous system flow the Darcy-Brinkman equation is the accepted method of approximating the flow through the media. Since the system modeled has both free and porous flow the interactions between the porous media and the free flow region must be taken into account. For this reason a modified Darcy-Brinkman equation is need. The modified equation mixes the Strokes-Darcy equation as described in Reference 5. The mixed equation for modeling the Stroke-Darcy is equation 6. u 2 T u * pl u u * u t F u Qbr u F p p p 3 p kbr Equation 6 7 It should be noted that equation 6 is affected by both permeability and porosity of the heat sink material. This equation is used to model all cases where a porous heat sinks is used. 4.3. Modeling Using Comsol a heat sink is modeled to be 10 mm by 10 mm in a 50mm by 50 mm system. The heat sink sits on a wall that is heated to 310K. This wall mimics a computer chip in that it provides a heat flux that must be dissipated. The chip is design to run only at 310K so depending on the efficiency of the heat the speed of the chip will be determined. The faster the chip is able to operate the higher the heat flux will be. The heat sink will consist of two forms; a porous block, a solid block and a third condition with no heat sink at all. The porous heat sinks will consist of different characteristics; 10 different porosities and 3 different permeabilities for a total of 30 different porous heat sink options. In addition one solid block heat sink and one system without a heat sink will be analyzed for a total of 32 different cases all of which are provided below in Table 2 (Porous Heat Sink) and Table 3 (Block Heat Sink). Table 2: Porous Heat Sink Characteristics Form Porosity 1a 1b 1c 1d 1e 1f 1g 1h 1i 1j 2a 2b 2c 2d 2e 2f 2g 2h 2i 2j Permeability 0.05 0.1544 0.2589 0.3633 0.4678 0.5722 0.6767 0.7811 0.8856 0.99 0.05 0.1544 0.2589 0.3633 0.4678 0.5722 0.6767 0.7811 0.8856 0.99 8 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j 0.05 0.1544 0.2589 0.3633 0.4678 0.5722 0.6767 0.7811 0.8856 0.99 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 Table 3: Block Form Form Depth (mm) 5-a 0 5-b 10 The characteristics chosen above for the porous heat sink are those that easily show the affect of permeability and porosity to heat transfer. The porous material characteristics analyzed may not be producible in the real world but they provide a useful understanding of the affect of porous materials has when used as a heat sink. Each heat sink will be made of the same base material in addition each system will utilize the same cooling fluid (air). The models will be subjected to the same wall temperature or 310K. The models are analyzed to determine the total heat flux that is able to be dissipated for the prescribed wall temperature. In addition plots of the flow path and temperature gradients will be produced. Using the resulting data it is then determined which heat sink has the potential of transferring the most heat. In addition mesh control can significantly affect the results of the analysis. To ensure that meshing is controlled in the same manner between all of the models the same mesh setting are used. For all three systems the mesh will be a physic-controlled mesh with normal sized element. This set up allows Comsol to optimize the number elements, the mesh size and the mesh geometry. This optimization is based on the geometry of the model and studies which are being completed. Figures 5, 6 and 7 below are the meshes in which Comsol has determined to be the most effective meshes based on the system geometry. 9 Figure 5: No Heat Sink Mesh Figure 6: Solid Block Mesh Figure 7: Porous Block Mesh 10 5. Results and Discussion Control Cases: The first analysis that was completed was the comparison of the use of no heat sink and use of a solid block heat sink. Table 4: Control Block Heat Sink Results below provides the important data from the model analysis. Table 4 provides the corresponding heat flux associated with a non-porous block and the model with no heat sink. Table 4: Control Block Heat Sink Results Trial Depth (mm) No Heat Sink Solid Heat Sink 0 10 Heat Flux (W/m) 1.513 2.9278 Mass Flow Rate (m^2/sec) 0.0012 0.001 As expected the heat flux out of the chip with the solid heat sink is larger than the case without any heat sink. This is due to the fact that there is a greater amount of surface area for heat to transfer to occur between the source and the cooling fluid. Figure 8 shows how the block heat sink has a large area of material at the 310K temperature. Alternatively Figure 7 shows that the wall is the only location where the temperature is 310K. If the block was to have a smaller width dimension the heat flux would be less than that of the 10mm block but more than the system without a heat sink. The associated mass flow rate of the model without the heat sink is greater than that of the model with the block heat sink. The mass flow rate is a function of the buoyancy force and the drag caused by the heat sink. Increasing the heat flux results in a greater buoyancy force therefore increase the flow rate. Alternatively the larger the block is the greater the drag force acting on the fluid moving by it. As can be seen in figure 8 the flow path for the model with the block heat sink requires the fluid to flow around the heat sink while figure 7 shows how with no heat sink flows right across the chip without a disruption in the flow path. 11 Figure 8: No Heat Sink T-V Figure 9: Solid Heat Sink T-V Porous Heat Sink: The second analysis compared porous heat sinks with differing permeabilities and porosities. Table 5: Porous Heat Sink Results below provide the data from the model analysis of porous material heat sinks. Specific data of interest is the total heat flux and volumetric flow rate of the systems. 12 Table 5: Porous Heat Sink Results Form 1a 1b 1c 1d 1e 1f 1g 1h 1i 1j 2a 2b 2c 2d 2e 2f 2g 2h 2i 2j 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j Porosity Permeability 0.05 0.1544 0.2589 0.3633 0.4678 0.5722 0.6767 0.7811 0.8856 0.99 0.05 0.1544 0.2589 0.3633 0.4678 0.5722 0.6767 0.7811 0.8856 0.99 0.05 0.1544 0.2589 0.3633 0.4678 0.5722 0.6767 0.7811 0.8856 0.99 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 1.00E-08 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 5.05E-07 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 Total Heat Flux Volumetric Flow (W/m) Rate (m^2/s) 3.1391 3.1443 3.1456 3.1452 3.1443 3.1429 3.1406 3.1361 3.1233 2.8752 7.6455 11.5721 10.6431 10.04 9.7644 9.6461 9.585 9.5285 9.4072 7.5302 7.9756 14.0391 14.9183 15.1979 15.1892 15.0854 14.9662 14.8254 14.543 10.7887 0.0011 0.0011 0.0011 0.0011 0.0011 0.0011 0.0011 0.0011 0.0011 0.001 0.002 0.0027 0.0025 0.0024 0.0024 0.0024 0.0024 0.0024 0.0023 0.0021 0.002 0.0031 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033 0.0028 For ease of analysis the affect of porosity on the heat sinks performance will be compared independent of permeability. Figure 6 was created from the data in Table 4 for the differing permeability’s. By fluctuating the porosity the affect of porosities between 0 and 1 the corresponding heat flux is plotted. 13 Porosity Affect on Heat Transfer 16 14 Total Heat Flux (W/m) 12 10 kappa = 1e-8 kappa = 5.05e-7 kappa =1e-6 8 6 4 2 0 0 0.2 0.4 0.6 0.8 1 1.2 Porosity (epsilon) Figure 10: Porosity Affect on Heat Transfer From Figure 10 the affect of porosity can be analyzed independent of permeability. At the extremes of porosity (ie 0-.1 and .9-1) a detrimental affect on the performance of the heat sink is seen. By looking at appendix 2 flow models, it is obvious why the extreme cases cause a decrease in the heat sinks ability to transfer heat. In those cases where the porosity is large the flow path of the coolant is similar to that of block heat sink case where fluid does not flow through the heat sink. This is also simlar to the case where the heat sink has a low permeability. As flow through the heat sink diminishes until the heat sink act as a boundary and no longer allows flow though it the advantage of being a porous media is lost. On the other hand looking at the temperature plots in appendix 2 it is apparent that when the porosity is increases above .9 the propagation of heat through the heat sink is not as uniformed when compared with material with less porosity. From the governing equation for heat conduction through a porous media it is apparent that once porosity becomes too large the equivalent conductivity becomes more like the cooling fluid than the base material. This causes heat not to propagate though the heat sink as readily. This 14 diminishment of heat propagation results in the decrease in the overall heat flux being dissipated. The second characteristic that was of importance for heat transfer in a porous material is the materials permeability. Figure 11 was created from the results in Table 4 for 5. By fluctuating the permeability the corresponding heat flux is able to be compared between each other. Permeability Affect on Heat Transfer 16 14 Total Heat Flux (W/m) 12 10 Porosity 0.6 Porosity 0.795 Porosity 0.99 Wall Block 8 6 4 2 0 0.00E+00 2.00E-07 4.00E-07 6.00E-07 8.00E-07 1.00E-06 1.20E-06 Permeability (1/m^2) Figure 11: Permeability Affect on Heat Transfer From Figure 11 it is determined that increasing permeability will increase the performance of the heat sink. By looking at Appendix 1 plots of flow it is apparent that increasing the permeability causes the flow to be similar to that of the control case with no heat sink at all. The increase heat flux is due to the ability of the fluid to flow through the heat sink, while the heat sink thermal conductivity is unaffected. This differs from decreasing porosity in that the optimum flow pattern is able to be achieved without changing the thermal conductivity of the heat sink. That flow pattern contributes to the volumetric flow rate being so high. 15 As presented in Figure 12 the permeability and porosity also affect the volumetric flow rate as well as the heat transfer. The higher the permeability is will result is a higher coolant flow rate. Permeability Affect on Velocity 0.0035 0.003 Velorcity (m/sec) 0.0025 Porosity 0.6 Porosity 0.99 Porosity 0.795 Block No Heatsink 0.002 0.0015 0.001 0.0005 0 1.00E-08 5.05E-07 1.00E-06 Permeability (1/m^2) Figure 12: Permeability Affect on Velocity The reason for the increase of flow rate due to the increase of permeability can be easily seen in appendix 1 flow rate plots. The greater the permeability is results in less flow being diverted around the heat sinks. The less diverting of flow results in less drag and an increase in the volumetric flow rate. For the same amount of buoyancy force a system with no drag due to a block will have more flow rate than one with a block in its flow path. On the other hand an increase of porosity decreases the volumetric flow rate. This is due to the fact that flow is a function of buoyancy force. Buoyancy force is a function of heat flux since the force is a comparison of incoming density to a point density, and density at a point in the system is dependant on temperature. An increase in heat flux will correspond to an increase in temperature, therefore increase the buoyancy force. Since a 16 low porosity has a negative affect on heat flux it will also have a negative affect on velocity. It should also be noted that there is a diminishing affect on the heat sinks ability to transfer heat once the flow is able to pass through the heat sink with out being disturbed. Figure 13 and Figure 14 below show how once permeability gets above a certain point increasing it further does not has as large of an affect. At the point where neither velocity nor heat flux increase with respect to permeability is where the flow path is similar to that of the control condition with no heat sink. This is due to the fact that once flow of the coolant become unobstructed there is no increase in its velocity. Since velocity doesn’t increase heat transfer will not increase either. Figure 13:Affect Of Increasing Permeability on Total Heat Flux 17 Figure 14: Affect Of Increasing Permeability on Volume Flow Rate Comparison of Porous and Control Conditions: From the analysis above it is evident that the porous heat sinks have a district advantage over the non-porous heat sink when comparing heat dissipation. This was expected due to the ability of the cooling fluid to flow through the heat sink in addition to the increased surface area between the cooling fluid and the heat sink. The characteristic of the flow through the system without a heat sink can be seen in figure 8 and appendix 3 which shows the fluid flowing straight through the system heating up as it passes the chip. Figure 9 and Appendix 2 show the fluid flow of the block heat sink and how the fluid rises from the inlet with cool air continues to flow around the heat sink while the fluids temperature rises until it exist from the top of the system. Alternatively in Figure 15, 16 and Appendix 1 the cooling air flows through as the heat sink. In addition Figure 12: Permeability Affect on Velocity and Tables 4 and 5 show how the fluid volumetric flow rate is increased with porosity. This increase in volumetric flow rate is due to two reasons. First the drag produced by the porous material versus the solid block is less. Since the air can pass through the block instead of around it the volume can pass with less pressure drop. Secondly the increased 18 buoyancy force due to a greater heat flux results in a greater volume force, increasing velocity and therefore increasing the volumetric flow rate. v Figure 15: Porous Heat Sink T-V Figure 16: Porous Heat Sink P-V How does meshing Affect the Results: As described in prior sections it was determined to use a normal mesh setting for analysis of the systems. This helped to ensure that the models were able to achieve a result in a reasonable amount of time while achieving results of significance. To ensure that the 19 results are of significance Figures 17 and 18 are provided below to compare the results of a system with a coarser mesh a normal mesh, a finer mesh and an extremely fine mesh. Mesh Affect on Total Energy Flux 12 Total Energy Flux (W/m) 10 8 6 4 2 0 Coarser Normal Finer Mesh Mesh Affect on Total Energy Flux Figure 17: Mesh Affect on Total Flux 20 Extremely Fine Affect of Mesh On Volume Flow Rate 0.0026 0.0025 Volume Flow Rate 0.0024 0.0023 0.0022 0.0021 0.002 0.0019 Coarser Normal Finer Extremely Fine Mesh Epsilon .9 Kappa 5.05e-7 Figure 18: Mesh Affect on Volume Flow Rate It is determined that the results using a mesh setting of normal is numerically significant. The results of normal mesh setting and the extremely fine setting are within 10% of each other. For the purpose of this analysis that is close enough. In addition the graphs show that the results begin to flatten out right past the normal mesh criteria. Therefore the results determined in this analysis are considered to be of significance. 21 6. Conclusion The comparison between the use of porous and non-porous heat sinks confirmed the hypothesis that there is a substantial benefit of using porous material in lieu of nonporous material for heat sinks. The benefit between porous and non-porous material heat sinks is only relevant to the initial efficiency of the heat sink. Over time fouling and material degradation may have a greater affect on the porous material versus the nonporous heat sink. In addition neither a cost comparison nor material availability research has been completed which may diminish the porous material overall advantage. To compare the advantage of porous material versus non-porous, an analysis of the affects of permeability and porosity characteristics was evaluated. Unlike the simple comparison between porous and non-porous material, the evaluation of the porous material characteristics a more complicated result ensued. The first characteristic analyzed was permeability, which is the ease in which fluid can pass through the heat sink. It was determined that the more permeability the porous media is the more efficient the heat sink becomes. Although the increase of permeability helped only so much, once there was little resistance for the cooling fluid to pass through the heat sink there was little advantage to increase the permeability. At that point increasing and decreasing porosity had a greater affect. The second material characteristic that was analyzed was the affect of porosity to the performance of the porous heat sink. It was determined that if the porosity is too high or too low there performance of the heat sink is adversely affected. Performance of the heat sink diminishes with increase of porosity due to the fact that the equivalent thermal conductivity of the heat sink diminishes to be close to the thermal conductivity of the cooling fluid. On the other hand, decreasing the porosity too much causes surface area for heat transfer to occur to diminish. Once porosity is diminished enough its flow path become similar to that of a solid block. Overall it has been determined a heat sink with high permeability and a porosity not near the extremes of the porosity range would perform best. This analysis looked at the 22 hypothetical cases for material porosity and permeability, therefore the ability to make the material analyzed may not be possible. For future analysis it would be beneficial to look at the possible porous materials that are able to be created and to perform actual testing of the material. 23 7. Appendix 1 Data from Porous Material Analysis 12-4-2011 24 8. Appendix 2 Data from Block Analysis 12-4-2011 25 9. Appendix 3 Data from No Heat Sink Analysis 12-4-2011 26 10. Reference 1 Fundamentals of Thermal-Fluid Sciences; Second Edition, Yunus A. Cengel & Robert H. Turner, © 2005 http://en.wikipedia.org/wiki/Heat_sink 3 Medal Foam and Finned Metal Foam Heat Sinks for Electronic Cooling in Buoyancy-Induced Convection. A. Bhattacharya & R.L Mahajan, © 2006 4 An analytical study of local thermal equilibrium in porous heat sinks using fin theory. Tzer-Ming Jeng, Sheng-Chung Tzeng, Ying-Huei Hung; © January 10, 2006 3 http://en.wikipedia.org/wiki/Density_of_air 4 Comsol 5 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. M. Le Bars and M. GRAE Worster, ©8 August 2005. 2 27