Heat transfer in rapidly solidifying supercooled pure melt during final transient

TL;DR

This paper develops a heat transfer model for the final transient phase of rapid solidification in supercooled pure melts, solving a hyperbolic Stefan problem and connecting it to classical parabolic models.

Contribution

It introduces a hyperbolic heat transfer model for supercooled melt solidification and links it to traditional parabolic models through a limiting process.

Findings

Solution of hyperbolic Stefan problem obtained

Transition from hyperbolic to parabolic heat transfer model demonstrated

Analytical expressions for temperature distribution derived

Abstract

The heat transfer model for a one-dimensional supercooled melt during the final stage of solidification is considered. The Stefan problem for the determination of the temperature distribution is solved under the condition that (i) the interface approaches the specimen surface with a constant velocity $V$; (ii) the latent heat of solidification linearly depends on the interface temperature; (iii) all the physical quantities given at the phase boundary are presented by linear combinations of the exponential functions of the interface position. First we find the solution of the corresponding hyperbolic Stefan problem within the framework of which the heat transfer is described by the telegraph equation. The solution of the initial parabolic Stefan problem is then found as a result of the limiting transition $V/V_H \rightarrow 0$ $(V_H \rightarrow \infty)$, where $ V_H $ is the velocity of the propagation of the heat disturbances, in which the hyperbolic heat model teds to the parabolic one.