Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat

TL;DR

This paper compares various approximate methods for solving one-phase Stefan-like problems with space-dependent latent heat, analyzing their accuracy against exact solutions under different boundary conditions and parameters.

Contribution

It introduces and evaluates the heat balance integral, modified heat balance integral, and refined integral methods for these problems, including a least-squares error approach.

Findings

The refined integral method shows high accuracy in approximations.

Approximate solutions effectively recover classical Stefan problem results.

Numerical simulations identify the most optimal integral method for different scenarios.

Abstract

The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), a modified heat balance integral method, the refined integral method (RIM) . Taking advantage of the exact analytical solutions we compare and test the accuracy of the approximate solutions. The analysis is carried out using the dimensionless generalized Stefan number (Ste) and Biot number (Bi). It is also studied the case when Bi goes to infinity in the problem with a convective condition, recovering the approximate solutions when a temperature condition is imposed at the fixed face. Some numerical simulations are provided in order to assert which of the approximate integral methods turns out to be optimal. Moreover, we pose an approximate technique based on minimizing the least-squares error, obtaining also approximate solutions for the classical Stefan problem.