Thermal spreading resistance from an isothermal source into a finite-thickness body

Abstract

Thermal spreading resistance is encountered during two-dimensional thermal conduction from a hot surface into a body. Thermal spreading and the opposite problem of thermal constriction resistance are of much technological relevance in the design of heat sinks and thermal spreaders for thermal management of microelectronics and other heat-generating devices. Much of the past work in the theoretical analysis of thermal spreading is based on a source with given heat flux. In contrast, the isothermal source problem presents difficulties due to the resulting mixed nature of the boundary condition, due to which, only approximate solutions are available. This work derives the steady-state thermal spreading resistance from an isothermal source into a finite-thickness slab or cylinder. The mixed boundary condition is handled by posing it in the form of a spatially varying convective boundary condition, with a sufficiently large Biot number over the source to represent its isothermal nature. A series solution for the problem is derived, along a sufficient set of linear algebraic equations to determine the series coefficients. Results are shown to agree well with finite-element simulations. Results are compared with previously-reported approximate solutions within the parametric range of validity of the approximate solutions. The impact of key non-dimensional parameters on thermal spreading resistance is quantified. It is shown that thermal spreading resistance increases with decreasing size of the isothermal heat source, as expected. A third-order polynomial correlation with very good accuracy is proposed. This work advances the theoretical understanding of a problem for which only approximate solutions have been reported in the past. Results presented here offer a practical tool for thermal design and optimization of a variety of practical thermal management problems involving spreading or constriction.