An upper bound for the steady-state temperature for a class of heat conduction problems wherein the thermal conductivity is temperature dependent

This article presents an a priori upper bound estimate for the steady-state temperature distribution in a body with a temperature-dependent thermal conductivity. The discussion is carried out assuming linear boundary conditions (Newton law of cooling) and a piecewise constant thermal conductivity (when regarded as a function of the temperature). These estimates consist of a powerful tool that may circumvent an expensive numerical simulation of a nonlinear heat transfer problem, whenever it suffices to know the highest temperature value. In these cases the methodology proposed in this work is more effective than the usual approximations that assume thermal conductivities and heat sources as constants.