We study time-dependent motion of the
solid-liquid interface during melting and solidification
of a material with constant internal heat generation in
cylindrical geometry. We derive a nonlinear,
first-order ordinary differential equation for the
interface that involves Fourier-Bessel series. The model
is valid for all Stefan numbers. One of the primary
applications of this problem is for a nuclear fuel rod
during meltdown. The numerical solutions of the initial
value problem are compared with the solutions of a
previously derived model that was based on the
quasi-steady approximation, which is valid only for Stefan
numbers less than one. The agreement between the two
models is excellent in the low Stefan number regime. For
higher Stefan numbers, where the quasi-steady model is not
accurate, the new model differs from the approximate model
since it incorporates the time-dependent terms for small
times, and as the system approaches steady-state, the
curves converge. At higher Stefan numbers, the system
approaches steady-state faster than for lower Stefan
numbers. During the transient process for both melting and
solidification, the temperature profiles become parabolic.
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