Non-local effects and size-dependent properties in Stefan problems with Newton cooling

We model the growth of a one-dimensional solid by considering a modified Fourier law with a size-dependent effective thermal conductivity and a Newton cooling condition at the interface between the solid and the cold environment. In the limit of a large Biot number, this condition becomes the commonly used fixed-temperature condition. It is shown that in practise the size of this non-dimensional number is very small. We study the effect of a small Biot number on the solidification process with numerical and asymptotic solution methods. The study indicates that non-local effects become less important as the Biot number decreases.