Simultaneous determination of two unknown thermal coefficients through a mushy zone model with an overspecified convective boundary condition

TL;DR

This paper develops explicit formulas for simultaneously determining two unknown thermal coefficients in a phase-change process with a mushy zone, using an overspecified boundary condition in a semi-infinite material.

Contribution

It introduces a comprehensive method to solve fifteen phase-change problems with explicit formulas for unknown thermal coefficients under various conditions.

Findings

Explicit formulas for unknown thermal coefficients are derived.

Necessary and sufficient data conditions for solution existence are provided.

The approach covers all fifteen possible phase-change problem configurations.

Abstract

The simultaneous determination of two unknown thermal coefficients for a semi-infinite material under a phase-change process with a mushy zone according to the Solomon-Wilson-Alexiades model is considered. The material is assumed to be initially liquid at its melting temperature and it is considered that the solidification process begins due to a heat flux imposed at the fixed face. The associated free boundary value problem is overspecified with a convective boundary condition with the aim of the simultaneous determination of the temperature of the solid region, one of the two free boundaries of the mushy zone and two thermal coefficients 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. The another free boundary of the mushy zone, the bulk temperature and the heat flux and heat transfer coefficients at the fixed face are assumed to be known. According to the choice of the unknown thermal coefficients, fifteen phase-change problems arise. The study of all of them is presented and explicit formulae for the unknowns are given, beside necessary and sufficient conditions on data in order to obtain them. Formulae for the unknown thermal coefficients, with their corresponding restrictions on data, are summarized in a table.