Simultaneous Determination of Two Unknown Thermal Coefficients Through a Mushy Zone Model With an Overspecified Convective Boundary Condition

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 SolomonWilson-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 phasechange problems arise. The study of all of them is presented and explicit formulae for * aceretani@austral.edu.ar † dtarzia@austral.edu.ar 1 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.

Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model

“…Thus, it is reasonable to expect that the solution to problem (1) converges to the solution to problem (1 ⋆ ) ∞ when the heat transfer coefficient increases its value. In this section we will analyse this sort of convergence, which was already proved for some other Stefan problems in [8][9][10].…”

Determination of unknown thermal coefficients in a Stefan problem for Storm’s type materials

“…Several papers on the determination of thermal coefficients in free boundary problems are found in the bibliography, see Briozzo et al (1999), Ceretani and Tarzia (2015), Ceretani and Tarzia (2016), Tarzia (1982Tarzia ( , 1983Tarzia ( , 1998. Determination of one and two unknown constant thermal coefficients through an inverse one-phase Stefan problem with a temperature boundary condition and an overspecied heat flux condition at fixed face were considered in Tarzia (1982Tarzia ( , 1983).…”

“…That is a similar Stefan problem to the one considered here, but this manuscript differs from that in the over condition imposed on the fixed face. In Ceretani and Tarzia (2015) and Ceretani and Tarzia (2016) determination of one and two constant unknown thermal coefficients through a mushy zone model with a convective over-specified boundary condition, for the free boundary problem and for the moving boundary problem, respectively, were considered. In Tarzia (2015) explicit expression for one unknown thermal coefficient of the semi-infinite material through the onephase fractional Lamé-Clapeyron Stefan problem with an over-specified boundary condition on the fixed face x = 0 is obtained.…”

Determination of Two Unknown Thermal Coefficients Through an Inverse One-Phase Fractional Stefan Problem

“…Determination of thermal coefficients for phase-change materials through inverse Stefan problems has been widely studied during the last decades [5,6,14,[27][28][29]. Especially, phase-change processes involving solidification or melting have been extensively studied because of their scientific and technological applications [1,2,4,7,8,10,18,26,31].…”