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We consider the one-dimensional solidification of a finite slab of a pure substance which is initially in liquid state. Convective cooling is applied at one end of the slab while the other end is kept adiabatic. There is a precooling stage before solidification begins. A void, which has substantial overall effect on the solidification of the slab, is formed at the cooled end due to shrinkage and grows during solidification. Perturbation analysis for small Stefan numbers ϵ is used to find the interface position and temperature distributions in both solid and liquid phases. The order of magnitude of the Biot number at the cold end, Bi, determines the nature of the solution. Regular and two time-scale perturbation solutions are obtained for Bi∼O(ϵ−1/2) and Bi∼O(1), respectively. The results can be extended to other orders of magnitude, as well as to slowly-varying time-dependent Bi. The results agree well with special cases in the literature, and with numerical solutions calculated using the enthalpy method. |
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