Analytical and numerical solutions to the non-diffusive Stefan problem

Abstract

In this work, the Maxwell--Cattaneo--Vernotte (MCV) equation is used to model the one-dimensional hyperbolic Stefan problem in the limit of a small Stefan number (Ste $\ll$ 1). The solutions are approximated with perturbation series expansions using a reformulation in which time is expressed as a function of the solid-liquid interface position. The first proposed solution is derived in a framework that considers diffusive heat transfer at the phase change interface, for analytic tractability. Two rectification strategies are proposed to address the asymptotic divergence present in this formulation: a rescaled inner solution which is then combined with the outer solution to yield a composite solution, and size-dependent thermo-physical system parameters for better capture of hyperbolic effects at the phase change interface. The resulting interface profiles exhibit a characteristic parabolic-like shape, consistent with diffusive Stefan problem findings, with pronounced early-time hyperbolic effects at larger thermal relaxation times. Parametric studies are done over three pertinent variables in the dimensionless system: the Stefan number ($\mathrm{Ste}$), the dimensionless thermal relaxation time ($\widetilde \tau$), and the thermal diffusivity ($\alpha$). The studies suggest that model error scales with the Stefan number in accordance with the theoretical truncation error of the perturbation expansion. Additionally, larger values of $\widetilde \tau$ amplify early-time hyperbolic effects, thereby increasing model error, while larger $\alpha$ extends the relative temporal domain over which these hyperbolic effects remain significant, also corresponding to an increase in model error.