Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face

TL;DR

This paper derives an exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition, generalizing previous models and exploring limits leading to a new free boundary problem.

Contribution

It provides an explicit solution using Kummer functions for a two-phase Stefan problem with position-dependent latent heat and convective boundary conditions, extending prior work.

Findings

Exact solution using Kummer functions under certain conditions

Generalization of previous one-phase free boundary problems

Limit analysis leading to a new free boundary problem

Abstract

Recently it was obtained in [Tarzia, Thermal Sci. 21A (2017) 1-11] for the classical two-phase Lam\'e-Clapeyron-Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding one-phase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in [Zhou-Shi-Zhou, J. Engng. Math. (2017) DOI 10.1007/s10665-017-9921-y].