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In this study, the optimum design dimensions and performance analyses of convective-radiative cooling fin subjected to magnetic field are presented using finite element method. The numerical solutions are verified by the exact analytical solution for the linearized models using Laplace transform. The optimum dimensions for the optimum performance of the convection-radiative fin with variable thermal conductivity are investigated and presented graphically. Also, the effects of convective, radiative, and magnetic parameters as well as Biot number on the thermal performance of the cooling fin are analyzed using the numerical solutions. From the results, it is established that the optimum length of the fin and the thermogeometric parameter increases as the nonlinear thermal conductivity term increases. Further analyses also reveal that as the Biot number, convective, radiative, and magnetic parameters, increases, the rate of heat transfer from the fin increases and consequently improves the efficiency of the fin. Additionally, effects of the thermal stability values for the various multiboiling heat transfer modes are established. It is established that, in order to ensure stability and avoid numerical diffusion of the solution by the Galerkin finite element method, the thermogeometric parameter must not exceed some certain values for the different multiboiling heat transfer modes. It is hope that the present study will enhance the understanding of thermal response of solid fin under various factors and fin design considerations.

Optimum Design and Performance Analyses of Convective-Radiative Cooling Fin under the Influence of Magnetic Field Using Finite Element Method