One-dimensional solidification of supercooled melts

Abstract

In this paper a one-phase supercooled Stefan problem, with a nonlinear relation between the phase change temperature and front velocity, is analysed. The model with the standard linear approximation, valid for small supercooling, is first examined asymptotically. The nonlinear case is more difficult to analyse and only two simple asymptotic results are found. Then, we apply an accurate heat balance integral method to make further progress. Finally, we compare the results found against numerical solutions. The results show that for large supercooling the linear model may be highly inaccurate and even qualitatively incorrect. Similarly as the Stefan number β → 1 + the classic Neumann solution which exists down to β = 1 is far from the linear and nonlinear supercooled solutions and can significantly overpredict the solidification rate.

The Stefan problem with variable thermophysical properties and phase change temperature

“…This in turn depends on the interface velocity, s t , which is determined by the Stefan condition (19). The problem formulation is completed by specifying appropriate initial and boundary conditions.…”

“…However, even with a flat interface T I may vary. For example when dealing with the solidification of supercooled materials [1, §2.4F], [42] there exists a nonlinear relation between the degree of supercooling and the front velocity [2,19] ds…”

“…All consistent choices can conserve energy provided the Stefan condition is used in the form (13). A common error in a number of papers is the use of the form (19), which is based on setting the interface temperature T s = T l = T I (t): the subsequent choice T s = T m then leads to loss of energy conservation.…”

“…Unfortunately, as discussed in §2.3 the models used in both [21,43] were derived from that of the textbook [1] which incorrectly represents the kinetic energy. As shown by equation (19) the term (ρ s /2)(1 − ρ s /ρ l ) 2 R 3 t used in the previous studies should be replaced with (ρ s /2)[1 − (ρ s /ρ l ) 2 ]R 3 t . By comparing the coefficients of R 3 t in these terms (with the parameter values of Table 1) we find that kinetic energy in the present model is roughly 18 times stronger in the case of gold, and 70 times stronger in the case of tin, indicating that the role of density variation needs to be re-assessed.…”

A one-phase Stefan problem with size-dependent thermal conductivity

“…curvature-induced melting point depression [12,13,14,15,16]), the speed of the moving boundary (e.g. supercooling [10,11,18]), or the temperature itself [17]. Motivated by recent experimental studies on Silicon nanofilms and nanowires showing that the thermal conductivity decreases as the size of the physical system decreases [19,20], in this work we will consider the effect of size-dependent thermal conductivity on the solidification process of a onedimensional slab.…”