A two-phase Stefan problem with power-type temperature-dependent thermal conductivity. Existence of a solution by two fixed points and numerical results

Abstract

A one-dimensional two-phase Stefan problem for the melting of a semi-infinite material with a power-type temperature-dependent thermal conductivity is considered. The assumption of tak- ing thermal parameters as functions of temperature find it bases in physical and industries applications, allowing for a more precise and realistic description of phase change processes. By, imposing a Dirich- let condition at the fixed face, a theoretical and approximate study is developed. Through a similarity transformation an equivalent ordinary differential problem is obtained from which an integral problem is deduced. The existence of at least one analytical solution is guaranteed by using the Banach fixed point theorem. Due the unavailability of an analytical solution, a heat balance integral method is ap- plied, assuming a quadratic temperature profile in space, to simulate temperature variations and the location of the interface during the melting process. For constant thermal conductivity, results can be compared with the exact solution available in the literature to check the accuracy of the approximate method.