Thermal Resistance in a Rectangular Flux Channel With Nonuniform Heat Convection in the Sink Plane

In this paper, thermal resistance of a 2D flux channel with nonuniform convection coefficient in the heat sink plane is studied using the method of separation of variables and the least squares technique. For this purpose, a two-dimensional flux channel with discretely specified heat flux is assumed. The heat transfer coefficient at the sink boundary is defined symmetrically using a hyperellipse function which can model a wide variety of different distributions of heat transfer coefficient from uniform cooling to the most intense cooling in the central region. The boundary condition along the edges is defined with convective cooling. As a special case, the heat transfer coefficient along the edges can be made negligible to simulate a flux channel with adiabatic edges. To obtain the temperature profile and the thermal resistance, the Laplace equation is solved by the method of separation of variables considering the applied boundary conditions. The temperature along the flux channel is presented in the form of a series solution. Due to the complexity of the sink plane boundary condition, there is a need to calculate the Fourier coefficients using the least squares method. Finally, the dimensionless thermal resistance for a number of different systems is presented. Results are validated using the data obtained from the finite element method (FEM). It is shown that the thick flux channels with variable heat transfer coefficient can be simplified to a flux channel with the same uniform heat transfer coefficient.