Integral balance methods applied to a non-classical Stefan problem

TL;DR

This paper investigates two non-classical Stefan problems with heat flux-dependent sources, comparing exact solutions to approximate integral methods under various boundary conditions, and analyzing the effects of key parameters through numerical simulations.

Contribution

It introduces approximate solutions via heat balance integral methods for a non-classical Stefan problem with convective boundary conditions, extending previous work with new boundary scenarios.

Findings

Approximate solutions closely match exact solutions for various parameters.

The heat balance integral methods are effective for non-classical Stefan problems.

Numerical results validate the accuracy of the proposed approximate methods.

Abstract

In this paper we consider two different Stefan problems for a semi-infinite material for the non classical heat equation with a source which depends on the heat flux at the fixed face x = 0. One of them (with constant temperature on x = 0) was studied in [4] where it was found a unique exact solution of similarity type and the other (with a convective boundary condition at the fixed face) is presented in this work. Due to the complexity of the exact solution it is of interest to obtain some kind of approximate solution. For the above reason, the exact solution of each problem is compared with approximate solutions obtained by applying the heat balance integral method and the refined heat balance integral method, assuming a quadratic temperature profile in space. In all cases, 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 verify the accuracy of the approximate methods