Thermal Resistance of a 3-D Flux Channel With Nonuniform Heat Convection in the Sink Plane

Abstract

Heat sinks are designed in different types and shapes depending on temperature distribution along the sink plane, where sometimes intense cooling is only required in the high-temperature regions. Thus, a nonuniform heat transfer coefficient might be present along the sink plane. In this paper, the temperature distribution and thermal resistance of a 3-D flux channel with a nonuniform heat transfer coefficient along the sink plane is modeled and analyzed by solving the problem analytically. A single concentric heat source is considered in the source plane, while the conductance along the sink plane is modeled by a symmetric conductance function. This function is used to define different conductance distributions along the sink plane from the most intense conductance in the central area to a uniform conductance along the sink plane. Furthermore, a convective cooling condition with uniform conductance is assumed along the lateral edges of the channel, which can be turned into adiabatic edges by assuming a very small heat transfer coefficient along the edges. The problem is solved analytically using the method of separation of variables combined with the method of least squares. A dimensionless total thermal resistance is then introduced to study the effect of different aspect ratios between the heat-source dimensions, cross-sectional dimensions, and channel thickness for different Biot numbers and different conductance distributions along the sink plane. For validation of the presented solution, the analytical results were compared with numerical solution results obtained by solving the problem with the finite-element method using the ANSYS commercial software package.