An approximate solution for planar solidification with Newton cooling at the surface

Abstract

The classical moving boundary problem for the planar freezing of a semi-infinite saturated liquid with Newton cooling at the wall is well known not to admit an exact solution. Existing perturbation solutions are valid only when the Stefan number is large. Further, since the implementation of the Newton cooling condition involves the step size, numerical solutions are only accurate if extremely small sizes are taken which involves large computing times. Here two new approximate analytic solutions are obtained, the first is an initial or starting solution while the second is valid for subsequent times. In the limit of large Stefan numberα the pseudo-steady state and first order corrected motions arise from both approximations. Further, in the limit of no Newton cooling at the wall the large time solution gives rise to precisely the well known Stefan or Neumann solution. The validity of the approximations is illustrated numerically with reference to previous work and known upper and lower bounds.