Summary
The classical two phase Stefan problems for spheres, cylinders and slabs do not, in general, admit exact solutions. For a number of genuine two phase moving boundary problems involving these geometries integral formulation are obtained which generalize known results for idealized one phase Stefan problems and can be utilized to obtain upper and lower bounds on the motion of the moving boundary, even though the analysis is complicated considerably by the presence of a nontrivial second phase. The problems considered are the inward thawing of an initially subcooled solid contained within a finite sphere, cylinder or slab, or within a region bounded by concentric spheres, cylinders or planes, as well as the outward thawing of an initially subcooled solid in the finite region surrounding a sphere, cylinder or plane. The relation of the integral for the boundary motion to the enthalpy is noted, and the bounds obtained for the cases of inward thawing of spheres, cylinders and slabs are compared with exact numerical solutions, generated by the enthalpy method. From these comparisons it becomes apparent that the bounds are adequate when the latent heat is the dominant thermal factor, that is, when more heat is required to produce the change of phase than to warm the substance. However, when this is not the case, that is, when the sensible heat is the dominant thermal factor, further analysis is needed.
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Dewynne, J.N., Hill, J.M. Integral formulations and bounds for two phase Stefan problems initially not at their fusion temperature. Acta Mechanica 58, 201–228 (1986). https://doi.org/10.1007/BF01176600
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DOI: https://doi.org/10.1007/BF01176600