Perturbation solutions for melting or freezing in annular regions initially not at the fusion temperature

Abstract

This paper deals with the inward solidification of liquid in an annular space which is initially not at the fusion temperature. The outer cylindrical surface is maintained at a subfreezing temperature while the inner cylindrical boundary is assumed to be either insulated or maintained at constant temperature. New perturbation solutions are obtained for the temperature distribution and the interface motion. The perturbation parameter ε = C s(T f — T a) L is the ratio of the sensible heat of the solid-phase to the latent heat of fusion. The non-uniformity of the long-time scale solutions is treated by constructing inner expansions in the short-time scale. The two solutions are matched using asymptotic theory. The solutions for the insulated case do not depart markedly, except on the short time scale, from the corresponding solutions for liquids which are initially at the fusion temperature T f . In contrast, the solutions for the isothermal inner boundary depart substantially from those with an initial temperature equal to T f . This is true even if the sensible heat is small compared to the latent heat of fusion. Similarly, curvature plays a minor role in the interface motion for the insulated case while its effect is dramatic in the isothermal case.