Relationship between two solidification problems in order to determine unknown thermal coefficients when the heat transfer coefficient is very large

Abstract

The phase-change processes are found in a wide variety of dynamic systems, for example in the study of snow avalanches. When a thermal property of the material is unknown, we can add a boundary condition to formulate an Inverse Stefan Problem, and determine this property. In this paper we study a heat conduction phase-change problem with Robin and Neumann boundary condition at a fixed face. This overspecified condition allows to simultaneously determine two unknown thermal coefficients through a moving boundary problem or a free boundary problem. Formulae for different cases where obtained by Ceretani and Tarzia (2015) [6]. The formulation with these type of boundary conditions is a more realistic one than the heat conduction phase-change problems with Dirichlet and Neumann boundary condition at the fixed face, considered by Tarzia, (1982-1983). Therefore we propose to study the relationship between the problems with Robin-Neumann conditions, and the problems with Dirichlet-Neumann conditions. The main result of this work is the convergence analysis of these problems, when the heat transfer coefficient h of the Robin condition is very large. We present for each case of the free and moving boundary problems, an upper bound for the error of the two unknown parameters, obtaining in every case a bound of order o(1h). Finally we show a numerical example of the convergence, for a phase change material commonly used in heating or cooling processes.