Modified perturbation solutions for Stefan problems with convective boundary conditions at high Stefan numbers

Abstract

The classical Stefan problem is one of the formulations to represent the moving boundary problems, such as solidification and melting processes. The nonlinearity of the differential equation that governs the moving boundary, i.e., the Stefan condition, makes finding the exact solutions a difficult task. Hence, perturbation theory is often applied to generate approximate analytical solutions by assuming a small Stefan number, i.e., $\text{Ste}\le 0.01$, which indicates the ratio of the sensible heat over latent heat in phase change processes. This assumption, however, limits the thermal engineering application of the approximate solution. The present study introduces a modified perturbation solution by adding a correction term after the leading-order solution that extends the validity to a wider range of Stefan numbers (i.e., 0.01 $\le$ Ste $\le$ 1). Specifically, the Stefan problem is first formulated in Cartesian, cylindrical, and spherical coordinates subject to a realist Robin boundary condition. Then, the leading-order perturbation solution is calculated and a correction term is added by using the Monte-Carlo statistical method and a multi-variant regression analysis. Results indicate that the correction term changes linearly with the Stefan number and is not significantly influenced by the Biot number. The proposed modified solution represents a rapid and precise method to predict the nonlinear moving boundary and temperature profiles in phase change processes.