Analytical solutions of heat spreading resistance from a heat source on a finite substrate with isothermal or convective surfaces

Abstract

The objective of this dissertation is to present the analytical solutions to the heat spreading problems that arise due to a flux specified circular heat source on a finite thickness substrate with isothermal or convective surfaces. The solutions to heat spreading resistance of these problems are obtained for the first time by the exact treatment of the mixed boundary conditions present on the substrate at the heat source side. In the case of heat spreading through a substrate with isothermal surfaces the solution method utilizes the two-dimensional axisymmetric equation of thermal conduction allowing for the convective cooling over source region. In the absence of convection over the source, it is shown that the total thermal resistance is composed of spreading resistance of an otherwise isothermal substrate and a correction due to inhomogeneous substrate thermal boundary condition. The application of the method of superposition elucidates the exact definition of source adiabatic temperature that takes care of the correction due to inhomogeneous substrate thermal boundary condition. In the case of heat spreading through a substrate with convective surfaces it is also shown that the expression for the total thermal resistance can be decomposed into a base solution and a correction. Thus the effects of the unequal heat sinks are consolidated in an approximate way to an equivalent or effective heat sink, Stheta1 that contributes the correction of Stheta1 to the base resistance of the homogeneous solution where the upper and lower heat sink temperatures are the same.