Heat balance integral methods applied to the one-phase Stefan problem with a convective boundary condition at the fixed face

TL;DR

This paper evaluates the accuracy of classical and refined heat balance integral methods for a one-phase Stefan problem with convective boundary conditions, using exact solutions and numerical simulations.

Contribution

It develops and compares variations of heat balance integral methods tailored for a Stefan problem with convective boundary conditions, including optimal versions.

Findings

Refined integral methods improve accuracy over classical methods.

Optimal variations depend on Stefan and Biot numbers.

Numerical results quantify approximation errors.

Abstract

In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method, assuming a quadratic temperature profile in space. We develop variations of these methods which turn out to be optimal in some cases. Throughout this paper, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to estimate the errors committed by each approach for the corresponding free boundary and temperature profiles.