The inward solidification of spheres and circular cylinders

An analytical study is presented of the inward freezing of a sphere or a circular cylinder, initially molten and at the fusion temperature, when the outside surface is suddenly cooled. The treatment assumes, among other things, constant thermal properties and that the parameter β, the ratio of the latent heat to the sensible heat of the substance, is large. Basic series solutions are first derived and these are then followed by a two-region analysis which is needed to accommodate the sharp change in thermal profile just before the centre of the sphere or the cylinder solidifies. Although the theory is strictly asymptotic in nature, results compare well with numerical solutions of the full problem for β = 10 and 20 (cylinder) and β = 10 (sphere).