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# multi-mode-ht - NUMERIC SOLVER FOR MULTIMODE HEAT TRANSFER...

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1 NUMERIC SOLVER FOR MULTIMODE HEAT TRANSFER IN A HIGH TEMPERATURE HEAT SINK Ben Jones Purdue University Mark Kimber Purdue University ABSTRACT 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 a 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. For reference, the code in its entirety is included in Appendix A and is based on a finite volume approach. In the remainder of this paper, 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.

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multi-mode-ht - NUMERIC SOLVER FOR MULTIMODE HEAT TRANSFER...

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