Approximate solutions to the one-phase Stefan problem with non-linear temperature-dependent thermal conductivity

TL;DR

This paper compares approximate analytical methods for solving the one-dimensional one-phase Stefan problem with non-linear, temperature-dependent thermal conductivity, focusing on the heat balance integral method and its variants.

Contribution

It introduces and evaluates approximate solutions using the heat balance integral and refined methods for a non-linear Stefan problem with temperature-dependent conductivity.

Findings

Approximate solutions closely match the exact solution under certain conditions.

Refined balance integral method improves accuracy over standard methods.

Analysis conducted in a dimensionless framework using the Stefan number.

Abstract

In this chapter we consider different approximations for the one-dimensional one-phase Stefan problem corresponding to the fusion process of a semi-infinite material with a temperature boundary condition at the fixed face and non-linear temperature-dependent thermal conductivity. The knowledge of the exact solution of this problem, allows to compare it directly with the approximate solutions obtained by applying the heat balance integral method, an alternative form to it and the refined balance integral method, assuming a quadratic temperature profile in space. In all cases, the analysis is carried out in a dimensionless way by the Stefan number (Ste) parameter.