A Lump-Integral Model Based Freezing and Melting of A bath Material Onto a Cylindrical Additive of Negligible Resistance
Abstract
In a theoretical analysis, a lump-integral model for freezing and melting of the bath material onto a cylindrical additive having its thermal resistance negligible with respect to that of the bath is developed. It is regulated by independent non-dimensional parameters, namely the Stefan number, St the heat capacity ratio, Cr and the modified conduction factor, Cofm. Series solutions associated with short times for time variant growth of the frozen layer and rise in interface temperature between the additive and the frozen layer are obtained. For all times, numerical solutions concerning the frozen layer growth with its melting and increase in the interface temperature are also found. Time for freezing and melting is estimated for different values of Cr, St and Cofm. It is predicted that for lower total time of freezing and melting Cofm<2 or Cr<1 needs to be maintained. When the bath temperature equals the freezing temperature of the bath material, the model is governed by only Cr and St and gives closed-form expressions for the growth of the frozen layer and the interface temperature. For the interface attaining the freezing temperature of the bath material the maximum thickness of the frozen layer becomes
ξmax= (Cr)1/2(Cr+St)1/2. The model is validated once it is reduced to a problem of heating of the additive without freezing of the bath material onto the additive. Its closed-form solution is exactly the same as that reported in the literature.
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