Determination of one unknown thermal coefficient through a mushy zone model with a convective overspecified boundary condition

TL;DR

This paper develops explicit formulas to determine an unknown thermal coefficient in a mushy zone model of solidification, using overspecified boundary conditions and solving related free boundary problems.

Contribution

It introduces a method to simultaneously identify a thermal coefficient and free boundaries in a mushy zone model with convective boundary conditions, providing explicit solutions.

Findings

Explicit formulas for unknowns are derived.

Necessary and sufficient data conditions are established.

Relationships between different overspecified boundary conditions are analyzed.

Abstract

A semi-infinite material under a solidification process with the Solomon-Wilson- Alexiades's mushy zone model with a heat flux condition at the fixed boundary is considered. The associated free boundary problem is overspecified through a convective boundary condition with the aim of the simultaneous determination of the temperature, the two free boundaries of the mushy zone and one thermal coefficient among the latent heat by unit mass, the thermal conductivity, the mass density, the specific heat and the two coefficients that characterize the mushy zone. Bulk temperature and coefficients which characterize the heat flux and the heat transfer at the boundary are assumed to be determined experimentally. Explicit formulae for the unknowns are given for the resulting six phase-change problems, beside necessary and sufficient conditions on data in order to obtain them. In addition, relationship between the phase-change process solved in this paper with an analogous process overspecified by a temperature boundary condition is presented, and this second problem is solved by considering a large heat transfer coefficient at the boundary in the problem with the convective boundary condition. Formulae for the unknown thermal coefficients corresponding to both problems are summarized in two tables.