AN ANALYSIS OF POROUS MEDIA HEAT SINKS FOR NATURAL CONVECTION COOLED MICROELECTRONIC SYSTEMS. Eric R. Savery

Transcription

1 AN ANALYSIS OF POROUS MEDIA HEAT SINKS FOR NATURAL CONVECTION COOLED MICROELECTRONIC SYSTEMS. by Eric R. Savery Engineering Project submitted in partial fulfillment of the requirements for the degree of Master of Engineering in Mechanical Engineering Approved by Ernesto Gutierrez-Miravete, Engineering Project Advisor Rensselaer Polytechnic Institute Hartford, Connecticut December, 2011 i

2 Table of Contents: List of Tables... iii List of Figures... iv Nomenclature... v Abstract... vi 1. Introduction Methodology/Approach Assumptions Theory Finite Element Modeling Results and Discussion Conclusion Appendix Appendix Appendix Appendix Reference ii

3 List of Tables Table 1: Porous Heat Sink Characteristics Table 2: Block Form Table 3: Control Block Heat Sink Results Table 4: Porous Heat Sink Results Table 5: Number of Element for Mesh Setting iii

4 List of Figures Figure 1: Fin Heat Sink Figure 2: Schematic Representation of Microelectronic Heat Sink... 2 Figure 3: No Heat Sink... 3 Figure 4: Solid Block Heat Sink... 3 Figure 5: Porous Heat Sink... 4 Figure 6: Schematic Representation of system... 5 Figure 7: Temperature vs. Density... 7 Figure 8: Different Porosity, Same Permeability... 8 Figure 9: Same Porosity, Different Permeability... 9 Figure 10: No Heat Sink Mesh Figure 11: Solid Block Mesh Figure 12: Porous Block Mesh Figure 13: No Heat Sink T-V (Isotherms and Velocity Vectors) Figure 14: Solid Heat Sink T-V (Isotherms and Velocity Vectors) Figure 15: Porous Heat Sink Isotherms and Velocity Field Figure 16: Porous Heat Sink Streamlines and Velocity Magnitude Figure 17: Effect of Porosity Effect on Heat Removal Figure 18: Permeability Effect on Heat Transfer Figure 19: Permeability Effect on Volumetric Flow Rate Figure 20: Effect of Permeability on Total Heat Flux Figure 21: Effect of Permeability on Volume Flow Rate Figure 22: Effect of Mesh on Heat Removal Rate Figure 23: Effect of Mesh on Volumetric Flow Rate iv

5 Nomenclature Symbols Cp = J/kg K = Heat Capacity at Constant Pressure K = m^2 = Permeability ρ = g/m^3 = Density ε = - = Porosity = W/m^2 = Heat Flux P = N/m^2 = Pressure T = K = Temperature k = W/(m*K) = Thermal Conductivity u = m/s = Fluid Velocity g = m/s^2 = Gravity t = sec = Time v = m/s = Velocity Field µ = N*s/m^2 = Fluid Viscosity Subscripts Air = Air Al = Aluminum Heat sink c = Initial/Cold Temperature eq = Equivalent 0 = Initial p = Porous Base Material f = Cooling Fluid v

6 Abstract This work investigates what effects the use of metallic porous materials has on heat transfer when it is used as a heat sink numerically using the finite element model. The study is motivated by the problem of cooling microelectronic components. Specifically a comparison of how differences in the porous material characteristics affect the performance of a heat sink is analyzed. In addition a comparison of the porous media results to a traditional block heat sink and a system with no heat sink at all is completed. The heat sinks used for the porous and non-porous heat sinks have the same volume mass of material, material and dimensional footprint. Where the heat sink attaches to the outer boundary a constant temperature is applied. By varying the heat sinks porosity and permeability the total heat flux out of the chip is calculated. The results of the all case are compared to quantify the overall effect of porous material characteristics has on heat dissipation. vi

7 1. Introduction Since the discovery of the microprocessor computing power is becoming more powerful and consequently the chips have been creating more heat. The heat the most powerful processors create must be dissipated; else the processor will become less reliable or fail 1. For every 10ºC reduction in temperature, the failure rate of the electronic component is halved 1. Therefore the more heat removed from the transistor, the greater the reliability. 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 thermal energy is a function of the material properties, geometry and the environmental conditions 2. Increasing the amount of thermal energy dissipated by the heat sink allows for higher processing speeds. Below is an example of a simple fin heat sink used on a typical integrated circuit. Figure 1: Fin Heat Sink 2 An alternative to the standard non-porous heat sinks which are used today is the use of semi-porous material. The use of porous material as a heat sink base material allows for the cooling fluid to flow through the heat sink similar to a fin heat sink with and infinite number of fins. This report describes a study designed to show 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. The COMSOL Multiphysics finite element modeling program is used to compare the efficiency of a heat sink that uses a semi-porous material with that of a block heat sink and also with the absence of a heat sink. 1

8 2. Problem Description The design of a heat sink is dependent on the physical space, cooling needs and cost of the component. In a semiconductor application increasing capabilities results in the need for additional cooling. The increased cooling requirements can result in larger heat sinks, increasing 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. Ideally cooling should occur by natural convection alone and this is the situation considered in this study. Figure 2 is a schematic representation of the system investigated. The chip is not included in the figure but it thermal effect is represented by fixing the temperature at the base of the heat sink. The Figure then shows the heat sink surrounded by free space occupied by air. Open Boundary Insulated Wall Chip Heat Sink Open Space (air) Insulated Wall Insulated Wall Open Boundary at Tc Figure 2: Schematic Representation of Microelectronic Heat Sink During normal operation the high temperature at the base of the heat sink will produce heat flow into the surrounding air. As the temperature increases, natural convection will set in. 2

9 Current methods of transferring the heat from a component to the environment are with a heat sink and without them. Below Figure 3 and Figure 4 show the current methods of remove heat from a system. Qout Qin Qout Qin Qout Tc Tc Qout Figure 3: No Heat Sink Figure 4: Solid Block Heat Sink Figure 3 represents a system in which has no heat sink to transfer the heat from the heat source to the atmosphere. Figure 4 on the other hand has a large block heat sink that transfer the heat from the heat source to the outer walls of the heat sink and then to the atmosphere. In both cases the sum of the energy in (Qin) and the sum of the energy out (Qout) equal each other. If the fluid flowing through the system is not pushed by a pump or a fan then a natural circulation of the fluid will move the fluid away from the heat source. In Figure 3 it is apparent that the amount of heat transferred out of the system is based on the surface area of the heat source and the velocity of the fluid. Alternatively in Figure 4 the amount of heat that is able to be dissipated is based on the outer surface of the heat sink and the fluid velocity. Since the heat sink provides a larger area for heat to transfer to the atmosphere it is accurate to assume that the system with the heat sink will transfer more heat. Therefore a correlation between an increase in heat sink total surface area and an increase in heat transfer can made. A way to increase this surface area is to cut slots or holes into the block. If an infinite number of holes were to be cut the material would become porous. Figure 4 below provides a general diagram of the porous case. 3

10 Qout Qin Qout Qout Tc Figure 5: Porous Heat Sink In the porous case the fluid would flow through the heat sink transferring heat to the cooling fluid as it goes through. Since the interior of the porous material is filled with voids there is an increase in the surface area and therefore an increase in the total heat flux 3 / 4. On the other hand too many holes in the porous system would result in the system becoming similar to the system with no heat sink. In that case the lack of the porous base material will cause a decrease in heat flux. Therefore it is desired to determine the optimum characteristics of the porous material which provide the best heat transfer. 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 i.e. the system with a block heat sink and the system with no heat sink. 4

11 3. Methodology/Approach A systematic approach is used to model and analyze three different heat transfer systems, one system without a heat sink and two systems with heat sinks. The two systems with heat sinks will be made with porous and non-porous material. The heat sinks are made of the same base material and the cooling fluid is the same (air) in all the systems. 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 but due to the porous characteristics. The solid block heat sink and no heat sink are control cases while the various porous block heat sinks are subjected to differing porous material characteristics investigate their effects on the heat sink s ability transfer heat Assumptions Figure 6 shows a schematic representation of the system considered in this study. Open Boundary Insulated Wall Wall with Temperature of T h Heat Sink Air Insulated Wall Insulated Wall Open Boundary at Tc Figure 6: Schematic Representation of system The following assumptions were used: 2D system Steady State Conditions 5

12 Bounded area is 50mm by 50mm. Chip Size 10mm Heat sink size 10mm x 10mm Air inlet temperature is Tc = K. Chip wall has a constant temperature Th = 310 Laminar incompressible Newtonian flow 3.2. 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 for the porous and non porous material cases conduction heat transfer is also modeled. 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 conduction and then into the fluid via convection. Equation 1 is the governing equation for conduction and convection in nonporous media. T t = ( k T) u T + Φ [1] One might mistakenly think that adding an additional 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 heat sink increases 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. This is the principle behind the widespread use of fins to assist energy dissipation in thermal engineering. Natural Convection: In cases where there is not a fan or another motive force to circulate coolant, a natural force known as buoyancy causes the fluid to move. The 6

13 ability of natural convection is a function of the buoyancy of the cooling fluid. 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 air is heated its density decreases causing an upward force opposite to that of gravity, this is the buoyancy force. The force then causes an upward velocity of the cooling fluid. For modeling this phenomenon the following equation is used to determine the volume force. F = g( ρ ρ ) b 0 [2] Where ρ0 is a reference value of the fluid density. Equation 2 shows that as the density of a point decreases below the initial density an upward force is created. Since density is directly proportional to the temperature of the cooling fluid at a particular point, the force will also vary from point to point. Figure 7 below show how density of air changes as a function of air 5. Figure 7: Temperature vs. Density The force that is caused by the buoyancy force is then introduced into the momentum balance equation. Equation 3 below is the general equation governing the flow of the air. As the buoyancy force F increases the mass flow rate increases and therefore the momentum also increases. 7

14 υ 2 ρ0 + ( υ ) υ = Fb + p+ µ υ [3] t Porous Material Characteristics: There are two characteristics of significant importance when analyzing the porous heat sinks. Those characteristics are the materials porosity and permeability. Permeability (κ) is the measure of the ability of a material to transmit fluid though itself. Porosity (ε) is also known as void fraction, is the measure of voids or spaces inside 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 has more void space than the solid material. Figure 8 and 9 helps explain the nature of and the difference between porosity and permeability. Figure 8: Different Porosity, Same Permeability Figure 8 shows the porosity of the block on the left is 4 times the porosity of the block on the right. However in the figure on the left the three shaded holes are blocked and do not allow through flow of fluid. The permeability of both blocks will be approximately the same because only one hole passes all the way through the blocks. Assuming that 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 but both blocks would have the same permeability. 8

15 Figure 9: Same Porosity, Different Permeability Figure 9 shows how two blocks could have the same porosity but completely different permeabilities. 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 mass flow rate of fluid through the block on the right would be greater than that through the block on the left. This would mean the permeability of the block on the right is greater than that of the block on the left. Heat Transfer in Porous Media: Heat transfer through a porous material shows similarities with heat transfer through a solid block and also with heat flow in a fluid medium. Equation 4 governs heat transfer in the porous block heat sink: 5 T t ( ρ Cp) + ρcpu T = ( k T) + φ eq eq [4] The difference between the simple heat equation and the heat equation for porous material is associated with the corresponding effect of the volumetric heat capacity and thermal conductivity. In the case of the porous material both volumetric heat capacity and thermal conductivity are functions of the material porosity. As the material becomes more porous both the thermal conductivity and the volumetric heat capacity become more like that of the cooling fluid. Alternatively, as the porosity nears zero the conductivity and volumetric heat capacity become more like those of the base material. Equation 5a,b below are used to determine the equivalent thermal conductivity and volumetric heat capacity. 9

16 k eq = θ k p p + θ k F F ( ρ * Cp) eq = θ ρ Cp + θ ρ Cp p p p F F F [5] θ + θ p F = 1 Equation 5c shows that the fractions of porous base material and fraction of cooling fluid must add up to 1. Free and Porous Media Flow: Since in a porous material the cooling fluid is able to move through the pores of 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 porous 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 governing Stokes equation for the fluid flow in porous media is provide in equation 6 and described in Reference 6. ρ ε p u ε p µ ε ( u * ) = pl+ u+ ( u) p T 2µ ( ) ( u) t µ * + β u + Q u+ F 3ε p k F br [6] 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 porous heat sinks are used Finite Element Modeling Using COMSOL Multiphysics, 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 at the base to 310K. This wall mimics a computer chip in that it provides a heat flux that must be dissipated. The chip is designed to run at a maximum temperature of 310K so depending on the efficiency of the heat removal mechanism the speed of the chip will be determined. The higher the value of the dissipated heat flux the faster the chip is able to operate. As 10

17 mentioned before the heat sink will consist of two forms; a porous block in one case and a solid non-porous block in the other. A third condition will be modeled 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 1 (Porous Heat Sink) and Table 2 (Block Heat Sink). The characteristics chosen for the porous heat sinks are those that most easily show the effect of permeability and porosity on heat transfer. Some of the porous material characteristics analyzed may not be readily producible in the real world but they provide a useful understanding of the effect that porous materials has when used as a heat sink. Each heat sink is made of the same base material and each system utilizes the same cooling fluid (air). All the models are subjected to the same wall temperature of 310K representing the chip. The models are analyzed to determine the total amount of thermal energy dissipated for the prescribed wall temperature. In addition plots of the flow path and temperature gradients are produced. The resulting data is then used to determine which heat sink has the potential of transferring the most heat. Mesh control can have a significantly effect on 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 meshing is physics-controlled with normal sized elements. This resulted in models with between 300 and 1200 total elements. 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 the physics involved. Figures 10, 11 and 12 below are the meshes that COMSOL has determined to be the most effective based on the system geometry. 11

18 Table 1: Porous Heat Sink Characteristics Case Number Porosity Permeability 1a E-08 1b E-08 1c E-08 1d E-08 1e E-08 1f E-08 1g E-08 1h E-08 1i E-08 1j E-08 2a E-07 2b E-07 2c E-07 2d E-07 2e E-07 2f E-07 2g E-07 2h E-07 2i E-07 2j E-07 3a E-06 3b E-06 3c E-06 3d E-06 3e E-06 3f E-06 3g E-06 3h E-06 3i E-06 3j E-06 Table 2: Block Form Form Depth (mm) 4-a 0 4-b 10 12

19 Figure 10: No Heat Sink Mesh Figure 11: Solid Block Mesh Figure 12: Porous Block Mesh Detailed information about all the models including the material properties used can be found in Appendix 4. 13

20 4. Results and Discussion Control Cases (No Heat Sink and Solid Block Heat Sink): The first analysis that was completed was the comparison of the use of no heat sink and the use of a solid block heat sink. (Table 3) Results below provide the important data from the model analysis. Table 3 provides the corresponding heat removal rates and cooling fluid volumetric flow rates (both per unit width) associated with a non-porous block and the model with no heat sink. The heat removal rate (per unit width) was determined by integrating the heat flow across the heated wall (chip). Similarly the volumetric flow rate (per unit width) was determined by integrating the velocity over the inlet. Table 3: Control Block Heat Sink Results Trial Depth (mm) Heat Removal Rate (W/m) Volumetric Flow Rate (m^2/sec) No Heat Sink Solid Heat Sink As expected, the heat removal rate out of the chip with the solid heat sink is larger than the case without the 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 of heat (chip) and the cooling fluid. Figure 14 shows how the block heat sink has a large area of material at the 310K temperature. Alternatively Figure 13 shows that the boundary wall is the only location where the temperature is 310K. If the block was to have a smaller width dimension the heat removal rate would be less than that of the 11mm block but more than the system without a heat sink. The associated volumetric 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 becomes. As can be seen in Figure 14 the flow path for the model with the block heat sink requires the fluid to flow around the heat sink 14

21 while Figure 13 shows how with no heat sink flow proceeds right across the chip without any disruptions in the flow. Figure 13: No Heat Sink T-V (Isotherms and Velocity Vectors) Figure 14: Solid Heat Sink T-V (Isotherms and Velocity Vectors) 15

22 Porous Heat Sink: Figures 15, 16 show typical results obtained for a porous heat sink. Figure 15 shows isotherms and velocity field while Figure 16 shows pressure contours. Figure 15: Porous Heat Sink Isotherms and Velocity Field 16

23 Figure 16: Porous Heat Sink Streamlines and Velocity Magnitude The second analysis compared the porous heat sinks with differing permeabilities and porosities. Table 4 below provides the results from the model analysis of porous material heat sinks. Specific data of interest is the total heat removal rate and volumetric flow rate of the systems. For ease of analysis the effect of porosity on the heat sinks performance will be first compared independent of permeability. Figure 17 shows the effect of porosity on the computed heat flux at various permeabilities. The increase in heat flow with increasing permeability is clearly shown. 17

24 Table 4: Porous Heat Sink Results Form Porosity Permeability Heat Removal Rate (W/m) Volumetric Flow Rate (m^2/s) 1a E b E c E d E e E f E g E h E i E j E a E b E c E d E e E f E g E h E i E j E a E b E c E d E e E f E g E h E i E j E

25 Porosity Effect on Heat Transfer Heat Removal Rate (W/m) (W/m) kappa = 1e-8 kappa = 5.05e-7 kappa =1e Porosity (epsilon) Figure 17: Effect of Porosity Effect on Heat Removal At the extremes of porosity (ie 0-.1 and.9-1) a detrimental effect on the performance of the heat sink is seen. Appendix 1 shows why the extreme cases produce a decrease in the sink s ability to remove heat. In those cases where the porosity is small, the flow path of the coolant is similar to that of the solid block heat sink case where fluid does not flow through the heat sink. This is also similar to the case where the heat sink has a low permeability. As flow through the heat sink diminishes the heat sink acts as a boundary which doesn t allow convection to occur inside the wall boundaries. On the other hand looking at the temperature plots in Appendix 1 it is apparent that when the porosity increases above 0.9 the propagation of heat through the heat sink is not as uniform as compared with materials 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 and it begins 19

26 to approach the no heat sink case. This reduction in heat propagation results in a decrease in the overall heat being dissipated. The second characteristic of importance for heat transfer in a porous material is the permeability. Figure 18shows the effect of permeability on the heat flux for various porosity values. Permeability Effect on Heat Transfer Heat Removal Rate W/m Porosity 0.6 Porosity Porosity 0.99 Wall Block E E E E E E E-06 Permeability (1/m^2) Figure 18: Permeability Effect on Heat Transfer The increase heat flux is due to the ability of the fluid to flow through the heat sink and to extract heat from the surfaces of pores. 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. 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. Figure 19 shows the effect of permeability and porosity on the volumetric flow rate. As expected higher permeability results in higher coolant flow rate. 20

27 Permeability Affect on Volumetric Flow Rate Volumetric Flow Raty (m^s/sec) Porosity 0.6 Porosity 0.99 Porosity Block No Heatsink E E E-06 Permeability (1/m^2) Figure 19: Permeability Effect on Volumetric Flow Rate The greater the permeability less flow is diverted around the heat sink. 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 the high permeability in the block will have more flow rate than one with a block with low permeability. The reason for the increase of volumetric flow rate due to the increase in permeability can be clearly seen in Appendix 1 flow rate plots. On the other hand a decrease in porosity decreases the volumetric flow rate. Since a low porosity has a negative effect on heat flux there will be also a negative effect on volumetric flow. This is due to the fact that flow is a function of buoyancy force and buoyancy force is a function of heat flux. Since the heat flux associated with the lower permeabilities is lower the volumetric flow rate is also lower. It should also be noted that there is a diminishing effect on the heat sink s ability to transfer heat. Once the flow is able to pass through the heat sink with out being disturbed 21

28 increasing permeability has little effect. Figure 20 and Figure 21 below shows how once permeability gets above a certain point, further increase does not have much of an effect. At the permeability value where velocity and heat flux stop increasing the flow path is similar to that of the control system with no heat sink. This is due to the fact that once flow of the coolant become unobstructed there is no further increase in its velocity. Since velocity does not increase, heat transfer will not increase either. Heat Removal Rate (W/m) Figure 20: Effect of Permeability on Total Heat Flux Figure 21: Effect of Permeability on Volume Flow Rate 22

29 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 in regards to heat dissipation capability. 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 typical flow through the system without a heat sink can be seen in Figure 13 (see also Appendix 3) which shows the fluid flowing straight through the system heating up as it passes the chip. Figure 14 (see also Appendix 2) shows 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 fluid temperature rises until it exist from the top of the system. Alternatively Figures 15, 16 (see also Appendix 1) show how the cooling air flows through the porous heat sink. The increase in volumetric flow rate with porosity 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 fluid can flow with less pressure drop. Secondly the increased buoyancy force due to a greater heat flux results in an increase in velocity and therefore an increase in the volumetric flow rate. The results of the modeling are comparable with research which has been done in the field already. In particular other research has shown that increasing permeability while limiting porosity has improved the heat transfer 7. The conclusion is that increasing permeability while keeping porosity between.1 and.9 improves the heat sink efficiency. System Meshing: 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 obtaining accurate results. To ensure that the results are of significance Figures 22 and 23 below compare the results of a model with a coarser mesh, a normal mesh, a finer mesh and an extremely fine mesh. 23

30 Affect on Total Energy Flux Meshing Effect on Total Heat Removal Rate (W/m)) Coarser Normal Finer Extremely Fine Mesh Mesh Affect on Total Energy Flux Figure 22: Effect of Mesh on Heat Removal Rate Affect of Mesh On Volume Flow Rate (m^2/sec) Volume Flow Rate Coarser Normal Finer Extremely Fine Mesh Epsilon.9 Kappa 5.05e-7 Figure 23: Effect of Mesh on Volumetric Flow Rate 24

31 The results of normal mesh setting and the extremely fine setting are within 10% of each other. 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 numerically significant. Table 5 shows that an increase in accuracy of 10% would require 20 times as many elements. It is thus determined that the results obtained using a mesh setting of normal are numerically significant Table 5: Number of Element for Mesh Setting Mesh Setting Number of Elements Coarser 416 Normal 1170 Finer 4776 Extremely Fine

32 5. 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 effect on the porous material versus the nonporous heat sink. In addition neither a cost comparison nor material availability research was completed which may diminish the porous material overall advantage. To compare the advantages of porous material versus non-porous one, an analysis of the effects of permeability and porosity characteristics was performed. Unlike the simple comparison between porous and non-porous material, the evaluation of the porous material characteristics was a bit less intuitive. The first characteristic analyzed was permeability, which is the ease with which fluid can pass through the heat sink. It was determined that the more permeability the porous media has the more efficient a heat sink becomes. Although the increase of permeability helped initially, once there was little resistance for the cooling fluid to pass through the heat sink there was little advantage in increasing the permeability any further. At that point increasing and decreasing porosity had a greater effect. The second material characteristic that was analyzed was the effect of porosity on the performance of the porous heat sink. It was determined that if the porosity was too high or too low the performance of the heat sink was adversely affected. Performance of the heat sink diminished with increase in porosity due to the fact that the equivalent thermal conductivity of the heat sink diminishes and approaches the thermal conductivity of the cooling fluid. On the other hand, decreasing the porosity causes surface area for heat transfer to diminish. Once porosity is diminished enough the flow path of coolant becomes similar to that around a solid block. Overall it has been determined a heat sink with high permeability and porosity not near the extremes of the porosity range would perform best. 26

33 6. Appendix 1 Data from Porous Material Analysis

34 7. Appendix 2 Data from Block Analysis

35 8. Appendix 3 Data from No Heat Sink Analysis

36 9. Appendix 4 Modeling Characteristic and Heat Sink Properties

37 10. Reference 1 Fundamentals of Thermal-Fluid Sciences; Second Edition, Yunus A. Cengel & Robert H. Turner, Wikipedia (December 9, 2011) Heat Sink. Received From 3 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, Medal Foam and Finned Metal Foam Heat Sinks for Electronic Cooling in Buoyancy-Induced Convection. A. Bhattacharya & R.L Mahajan, Porous Heat Transfer; Multiphysics Modeling, Finite Element Analysis, and Engineering Simulation Software, COMSOL inc 6 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. M. Le Bars and M. GRAE Worster, August Sintered porous medium heat sink for cooling of high-power mini-devices. G Hetsroni, M Gurevich, R Rozenblit. 18 August