Numerical Study of Heat Transfer in a High Temperature Heat Sink

Transcription

1 International Journal of Scientific & Engineering Research Volume 4, Issue 2, February Numerical Study of Heat Transfer in a High Temperature Heat Sink ABSTRACT Dr. Ahmed F. Khudheyer, 2013, Iraq, ahyaja@yahoo.com, A finite volume based numerical scheme is developed to estimate the temperature distribution within a high temperature rectangular fin heat sink. A coupled conduction, convection, and radiation problem is solved with conduction occurring through the substrate and radiation and convection boundary conditions on the surfaces of the fins. Those boundaries involving both radiation and convection to the ambient are accounted for by linearizing radiation terms and updating the temperature-based radiosities at each iteration. The convection coefficient and heat sink thermal conductivity are assumed to be uniform and independent of temperature. A geometric multi-grid method is implemented to drastically decrease computation time. A number of test cases are analyzed for a heat sink with overall dimensions of 50 mm x 50 mm x 5 mm while varying the fin thickness and number of channels. Using the performance metric of thermal resistance based on differences between base and ambient temperatures, the optimum design composes of 6 channels with 1 mm thick fins corresponding to a thermal resistance of 13.6 K/W. INTRODUCTION Heat sinks are commonly used in many industries to increase the rate of heat removal over which forced or free convection can occur. They can most often be found in various electronic devices such as microprocessors and high performance video cards. In many cases, heat sinks are coated with high emissivity paint to further increase the heat removal rate. Radiation then becomes important and determining the temperature distribution within the heat sink is non-trivial and must account for radiosities of all exposed surfaces and view factors from one fin to the next. The current work provides a numerical approach to solving this problem. the problem is first described in more detail including treatment of all boundary conditions. Discretized equations representing the heat transfer throughout the entire domain are presented in forms most easily implemented into line by line tridiagonal matrix (LBLTDMA) solvers. The exchange of radiation is then discussed with methods for calculating view factors and radiosities of all surfaces exposed to ambient conditions. The overall solution procedure is given as well as a description of the multi-grid scheme implemented to accelerate convergence. This is followed by a code validation comparing the present work with similar results obtain using Fluent. Finally results from a number of test cases are given showing the ability to optimize the geometric dimensions of the heat sink. NOMENCLATURE A Coefficient for q b Ai Cell surface area a Cell coefficient B Coefficient for q b b Source term E b Black body emissive power F ij View factor from surface I to surface j g Gravitational constant H Overall height of heat sink h Convection coefficient k Thermal conductivity of substrate kf Thermal conductivity of fluid Overall length and width of heat sink L 0 Q in q q 0 '' Ra S R th S T x x y y Greek Total heat input into the base of the heat sink Heat flow Heat flux into base of heat sink Raleigh number based on channel width Overall thermal resistance of heat sink Channel width Temperature Cell width Distance in x-direction Cell height Distance in y-direction Thermal diffusivity Coefficient of thermal expansion Emissivity Kinematic viscosity Stefan-Boltzmann constant Subscripts/Superscripts b Boundary conv Convecti on E East neighboring cell e East face i,j Surface indices N North neighboring cell n North face nb Near boundary p Particular cell rad Radiation S s Tb W w South neighboring cell South face Linearized with respect to T b West neighboring cell West face w 1 Arc length of boundary surface

2 International Journal of Scientific & Engineering Research Volume 4, Issue 2, February Ambient PROBLEM DESCRIPTION The heat sink of interest in this paper is shown in Figure 1 where dashed lines reveal the domain considered during analysis. This domain is shown in Figure 2 where the heat flux at the base is specified (q 0 ''). In addition, due to symmetry, both the far left and right boundaries can be considered insulated yielding a total of three pure Neumann boundary conditions. The discrete equations for these cells are given in Eq. (1) where the coefficient for any particular cell (a P ) is simply the sum of the coefficients for all neighboring cells to the east (E), west (W), north (N), and south (S). For the case of a specified non-zero heat flux (as is the case with the heat sink base), the b term simply becomes q'' x as shown in Eq. (1), but in all other cases, this is zero. Figure 1: Illustration of heat sink with overall dimensions of L * L 0 * H and heat flux entering through base. Figure 2: Heat sink subsection used for analysis with temperature of surroundings equal to ambient conditions. Surfaces 1-5 experience heat transfer to ambient through both convection (qconv) and radiation (qrad). The treatment of the remaining boundaries (surfaces 1-5 of Figure 2) however is more difficult because both convection and radiation contribute to the heat leaving those boundaries. The radiation in particular requires careful consideration and due to the enclosure type problem, radiosities must be computed to accurately describe the total heat transfer occurring at these surfaces. Moreover, once the domain is discretized these procedures become increasingly more complex and are described separately in the next section. MULTIMODE BOUNDARY CONDITIONS We begin the analysis of boundaries where both convection and radiation occur by observing the heat leaving a discretized surface as shown in Figure 3 where T P and T b are the cell centroid and boundary temperatures, respectively. The total heat leaving (q b ) through this surface is expressed in Eq. (2) where h is the convection coefficient, is the surface emissivity, and J b* is the radiosity. The notation (*) denotes a value which is computed using prevailing values, or those from a previous iteration. Also, it is important to note that this expression is strictly representative of a cell with a vertical surface exposed to ambient conditions, hence the use of y. For a cell with a horizontal boundary, one needs to simply replace this with x for this equation and all equations hereafter. The detailed treatment of J b* will follow in subsequent sections and is based on view factors and temperatures of all surfaces within the enclosure. For the time being, we will simply accept that this quantity is calculated from prevailing values and treat it as a known parameter. The expression in Eq. (2) can be rearranged grouping terms common to T b and those based on specified parameters as is shown in Eq. (3). The radiation term containing T b4 must now be linearized to obtain discrete equations which ultimately form a linear set of equations. This is done by expressing the non-linear quantity in terms of prevailing and current iteration values of T b as is shown in Eq. (4) where the non-linear term involving T b in Eq. (3) has been redefined as q Tb. It is worth noting that once convergence is met (T b = T b* ), the expression simplifies to the original definition of q Tb. Upon substitution of Eq. (4) into Eq. (3), the overall expression for heat leaving the boundary takes the form shown in Eq. (5), where again the terms involving T b and those based on known parameters are grouped together. The T b coefficient and the constant are redefined as B * and A *, respectively and it is important note that regardless of the initial guess or the accuracy of the current iteration, both B * and A * remain positive quantities. This will prove useful once the final discretized equations are presented.

3 International Journal of Scientific & Engineering Research Volume 4, Issue 2, February RADIATION EXCHANGE As discussed in the previous section, J b must be evaluated to determine the q rad portion of the boundary heat flux. The radiosity from each discretized surface can be evaluated by solving the system of linear equations [1]: Rewriting Eq. (10) in a form more convenient for programming yields: The next step is to balance the expression for q b with the conduction from the cell centroid to the boundary as is shown in Eq. (6), which can then be solved for T b as shown in Eq. (7). This result is then substituted back into Eq. (5) and used in conjunction with the overall energy balance within the cell, namely all the heat leaving through conduction to neighboring cells plus the heat leaving through the boundary must equal zero. For the particular cell configuration shown in Figure 3, the three neighboring cells are to the west, north, and south. The conductive fluxes are balanced as previously described and the result is shown in Eq. (8), which represents the discrete equation used for all boundary cells where both convection and radiation occur. A more convenient form is presented in Eq.(9), which can easily be adapted to cells with horizontal surfaces exposed to ambient conditions as well. Notice that because both B * and A * are positive quantities, a P is greater than the sum the neighboring coefficients (a W, a N, and a S ) thereby satisfying the scarborough criterion in the inequality enabling the use of iterative techniques for solving the system of equations. where i is the emissivity, A i is the surface area, is the Stefan-Boltzmann constant, E bi is the black body emissive power, and T bi is the surface temperature. The form of Eq. (11) is similar to a diffusion equation. However, instead of being linked to just the near boundary cells, J i is dependent on the radiosity from all surfaces j in the domain that surface i undergoes radiation exchange with. In the event that i = 0, the source term, b i, goes to 0 and a i evaluates to just the sum of the a ij coefficients. In the other extreme, if i = 1, a i and b i goes to infinity. Clearly this is not desirable from a computational standpoint. From Eq. (10), it is apparent that J i = E bi when i = 1. By preserving the form of Eq. (11), the coefficients can be evaluated as: Eq. (11) satisfies the Scarborough Criteria unconditionally in the equality and conditionally in the inequality as long as i >0 for at least one surface in the domain. Since i > 0 for practical engineering surfaces,

4 International Journal of Scientific & Engineering Research Volume 4, Issue 2, February the Gauss-Seidel method is guaranteed to converge and is the solution method chosen in the present work. A TDMA method is not recommended for solving Eq. (11) since the radiosity of surface i is linked to all of the visible surfaces j in the enclosure, not just the neighbors. Eq. (11) also guarantees boundedness of the solution; J i is bounded by min [J j for all j i, E bi ] and max[j j for all j i, E bi ]. The view factors for 2D surfaces, infinite in one extent, can be conveniently evaluated using the cross-strings method [1]. The view factor between any two surfaces 1 and 2 (see Figure 4) is given by the expression: where w 1 is the arc length along surface 1. The absolute value in Eq. (13) is to emphasize that whether ac and bd are cross-distances or direct-distances (shown as ad and bc in Figure 4), the result only differs in sign. By implementing the absolute value in the program, no elaborate algorithms are required to ensure that the crossdistances are evaluated first. SOLUTION PROCEDURE The calculation of the cell values, T P, requires that the boundary surface radiosities, J b, are known a priori. However, the surface radiosities are dependent on the boundary surface temperatures, T b, which are unknowns. Therefore, the basic methodology behind the solution procedure is to first calculate the J b based on a guess of T b, calculate T P based on the estimated J b, update T b based on the new T P values, and iterate. The detailed solution procedure is given below. Again, the starred (*) terms indicate values based on the previous iterate. here k f is the thermal conductivity of the fluid, S is the channel width, and Ra S* is the Raleigh number based on the channel width given by: where g is the gravitational constant, is the coefficient of thermal expansion with 1/T for air, is the kinematic viscosity and is the thermal diffusivity. 7) Calculate the remaining cell coefficients, a P and b, from Eq. (9). 8) Solve for the cell centroid temperatures, T p, using T b* and J b* (see Eq. (9)) with a Line-by-Line TDMA routine. The Line-by-Line TDMA routine performs a user specified number of sweeps through the domain so that a reasonably accurate temperature field is obtained based on the current values of T b* and J b*. This was found to improve computation efficiency compared to evaluating J b * and T b * after every sweep. 9) Calculate the boundary temperatures, T b, based on the newly calculated values of T P using Eq. (7). 10) Check for converge. If the convergence criterion is met, the solution is complete. Otherwise, go back to step 5) and iterate. It is worthwhile discussing the data structures used in the program; most notably, how to store the view factors, radiosities, and boundary temperatures and link them to their respective boundary cells. Figure 5 shows the labeling scheme used in the program. Each of the labeled surfaces 1 through 6 is divided into discrete surfaces based on the grid used in the domain. Surface 6 is a virtual surface representing the enclosure for surfaces 3 through 5. The node points of surface 6 are determined by projected the nodes from surface 4 onto 6. 1) Generate grid based on user inputs for the number of cells spanning the fin width and fin height and the number of cells spanning the overall domain height and width. 2) Calculate view factors, F ij, using Eq. (13). 3) Calculate the cell coefficients a W, a E, a N, and a S using Eq. (1). 4) Use a guess to initialize the cell centroid temperatures, T P, and boundary temperatures, T b. 5) Solve the system of equations in Eq. (11) to find the surface radiosities, J b, based on T b* starting with an initial guess for J b. Iterate until convergence using Gauss-Seidel iteration. 6) Estimate the convection coefficient, h, using the Elenbass correlation [1] based on T b*. This step was deemed necessary since the convection coefficient changes considerably with the channel dimensions. The Elanbass correlation is given below.

5 International Journal of Scientific & Engineering Research Volume 4, Issue 2, February The boundary temperatures, T b, and radiosities, J b, are stored into 1xP arrays with P = P i representing the total number of boundary surfaces and P i represents the number of boundary surfaces on the labeled surface i shown in Figure 5a. The structure of the array is shown in Figure 5b. The direction of the arrows on Figure 5a indicates the direction of increasing index for the array along each labeled surface. The view factors, F ij, are stored in a PxP matrix with each index indicating the i th and j th surface. Since the number of cells spanning the fins and overall domain is known, the cell to which each surface belongs can be readily determined. GEOMETRIC MULTIGRID At the boundary cells, T p is linked to T through the a P and b coefficients of Eq. (9). If these coefficients are small compared to the a nb coefficients, T p will be more dependent on the current values of the near boundary temperatures than T. For the heat sink properties and conditions used in the present study, the convergence rate was quite slow. Therefore, it was decided to implement a rudimentary multi-grid method to accelerate convergence. stall on the finer meshes. For the present study, a relative error between iterates of 10-6 was found to allow fast convergence and accurate solutions. The computational acceleration due to the multi-grid method is astonishing. Prior to the multigrid, one case took 2 hrs, 45 min, and 58.3 sec to converge using a 40x40 mesh. The same case can now be completed in 2.02 sec, almost a 5000x decrease in computation time. MODEL VERIFICATION To ensure that our results are meaningful, our numerical code must be validated. This was achieved by benchmarking our code against a commercial CFD package, Fluent. Since the boundary conditions of the current problem (convection and radiation) are difficult to implement directly in Fluent, it was decided to benchmark the radiation and convection parts separately. Table 1: Properties and dimensions of heat sink used in model verification The dimensions and properties of the heat sink used in the model validation are given in Table 1. To benchmark the convection part of the code, the radiation was turned off. The convection coefficient was set to a constant of 15 W/m 2 -K with an ambient temperature of 300 K. The results from the Fluent simulations and our code is shown in Table 2. Table 2: Comparison of results from numerical code and Fluent An example of the meshing using in the geometric multigrid method is shown in Figure 6. The program first iterates on a coarse mesh such as the one shown in Figure 6a using a user supplied initial guess. Once the solution has converged on the coarse mesh, the cell values are interpolated onto a finer mesh. The process repeats until the finest mesh level is achieved. Five grid levels were implemented in the present work: 2x3, 5x5, 10x10, 20x20, and 40x40 meshes. There are some limitations to the multi-grid method as implemented. For instance, if the program stalls on a fine mesh, it will not go back to a coarser mesh. The program simply proceeds from the coarsest to finest mesh. In some instances, it was found that if the convergence criterion was set too small, the program would The radiation part of the code was verified by turning convection off and setting the emissivity to 0.9. The computational domain and boundary conditions used in the Fluent model is shown in Figure 7. A dummy fluid was specified to fill the channel region, which allows the surface-to-surface (s2s) radiation model to be implemented. The fluid motion was specified to be stagnant and the thermal conductivity was set to a negligible value to eliminate the effects of the fluid on the heat transfer. Surfaces 3-6 used the s2s model. Surfaces 1 and 2 were set to radiation boundary

6 International Journal of Scientific & Engineering Research Volume 4, Issue 2, February conditions and surface 6 was set to a constant temperature. The results of the Fluent simulations and computational model are shown in table 2. It is apparent that our code is performing as expected. The mesh independence of the solution was also explored using the mesh sizes as stated before: 2x3, 5x5, 10x10, 20x20, and 40x40. Due to the high thermal conductivity of the heat sink, the temperature in the heat sink is close to uniform so the results are relatively insensitive to mesh size. In fact, even the 2x3 mesh yielded temperatures very close to the 40x40 mesh (within 0.05 K). However it was decided to use the 40x40 mesh both due to the computational efficiency of the multi-grid algorithm and because the temperature contour plots are more aesthetically pleasing on the 40x40 mesh than the coarser meshes. RESULTS FROM OPTIMIZATION To further explore the performance of an entire family of heat sinks, the thermal resistance was analyzed varying the fin thickness and number of channels keeping all other parameters fixed at specified values. Figure 8 illustrates the heat sink orientation considered and the various parameters which are kept fixed during the optimization routines. The values used for these parameters are listed in Table 3 along with the specified channel height, which also remains fixed. The thermal and radiative properties of the heat sink are based on aluminum painted black. The range considered for fin thickness and number of channels can also be found in Table 3 and analysis was performed at 21 and 19 discrete levels over each respective range. The channel width was constrained to be greater than 1 mm to prevent unrealistically small or negative values. For the case of 1 mm thick fins and 10 channels, the converged temperature contours are shown in Figure 9. For this case and all cases considered during optimization, the result is very near isothermal (a temperature difference of 0.24 K for the case shown in Figure 9). This is due to the geometric sizes considered as well as the high thermal conductivity of the heat sink. Thermal resistance (R th ) defined by Eq. (16) is used as the performance metric, where T bs is the average base temperature and Q in is the overall heat input to the heat sink. The thermal resistance was computed for each unique case and a surface plot of the results is shown in Figure 10 where a minimum occurs with a fin thickness of 1 mm and 6 channels yielding an overall resistance of 13.6 K/W. For heat sinks with 9 channels or greater, the thermal resistance could not be computed up to fin thicknesses of 5 mm because the channel width was below the constraint of 1 mm. As the number of channels increases for any given fin thickness, a drop in thermal resistance is observed for a small number of channels. A minimum thermal resistance is reached before the resistance increases with increasing number of channels. The thermal resistance of the heat sink is a balance between the convective and

7 International Journal of Scientific & Engineering Research Volume 4, Issue 2, February radiative fluxes and the overall surface area. The natural convection coefficient (given by Eq.(14)) typically decreases with increasing number of channels due to the decreased fluid motion caused by the constrained channel dimensions. The radiative heat flux also diminishes due to the narrowing channel widths. However, the surface area of the heat sink increases with increasing number of channels. For a small number of channels, the increase in surface area more than compensates for the decreasing convective and radiative fluxes. However, after an optimum is reached, the increase in surface area is no longer able to compensate for the decreasing boundary fluxes resulting in an increase in thermal resistance. The optimum number of channels is also dependent on the fin thickness. However, the high thermal conductivity of the heat sink material results in high fin efficiency for the fin thicknesses under consideration. Therefore, the thermal resistance of the heat sink increases for increasing fin thickness (i.e. decreasing channel width) due to the reduced boundary fluxes. CONCLUSION A numerical approach to multimode heat transfer in a high temperature heat sink has been developed and validated against a commercial CFD package. This code developed uses linearization techniques to treat boundaries that experience heat transfer through both convection and radiation. Radiosities are computed using a surface to surface radiation exchange model. A geometric multi-grid method was found to significantly improve computational performance (a 5000x speedup for the cases explored). The model was used to explore the optimal dimensions of a high temperature heat sink that undergoes radiation and natural convection. The optimal design of a 50 mm x 50 mm x 5 mm aluminum heat sink with a heat input of 12.5 W consists of 6 channels with 1 mm fin widths. The code could be easily implemented to study the heat transfer in other geometries where convection and radiation are important. However, the numerical model is currently formulated for rectangular, structured meshes and the formulation may require modification for other meshing geometries. REFERENCES [1] Incropera, F.P., DeWitt, D.P., Fundamentals of Heat and Mass Transfer, 5 th ed., John Wiley & Sons, NY.

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