Classical two-phase Stefan problems describe freezing (or melting) phenomena initially not at the fusion temperature. Since classical Stefan problems define subcooling and freezing stages only, an extension to characterize cooling stage (i.e., the final stage of freezing) is essential, namely extended Stefan problems. Specifically, these problems with a convective boundary condition in a cylindrical or annular domain can be applied to model ground freezing, heat exchangers, thermal storage systems and many other engineering applications. In recent years, there has been considerable research on numerical methods for such applications. However, high computational power and time are often required in many engineering fields, and thus these numerical schemes are not computationally expensive and, sometime, impractical. It is therefore crucial to develop accurate and reliable analytical solution for relatively fast design and decision makings.

In this paper, an extended two-phase Stefan problem with a convective boundary in an annular region is considered. Particularly, method of multiple scales is applied to separate three different time regimes corresponding to the three freezing stages. At the first stage (i.e., subcooling stage), the moving boundary condition (also known as the Stefan condition) is firstly immobilized by boundary fixing technique and an asymptotic solution is then found by assuming small Stefan numbers. Secondly, a new time scale is introduced at the freezing stage and the initial temperature can be abandoned due to the freezing physics. Moreover, two subscales on time are introduced at the freezing stage to capture behaviors of both phases. An asymptotic solution is formulated in the similar manner until the cooling stage starts. Lastly, the cooling stage uses a solid-based time scale since there is no more liquid. After rearranging the boundary and time conditions, an exact solution can be readily obtained. Combing all aforementioned solutions at different time scales, a final analytical solution is found by multiscale analysis. Non-dimensional temperature profiles and interface motion are extensively studied in terms of thermophysical properties and Stefan numbers.

To verify the analytical solution, a numerical approach called enthalpy method is solved by a control-volume based finite difference scheme. This numerical method takes advantage of a so-called mushy zone which distinguishes the two phases, in order to mitigates the nonlinearity of interface. Comparisons between analytical and numerical solutions are made as regard to non-dimensional temperature profile and interface motion. Furthermore, errors between to solutions are investigated at different thermophysical properties and Stefan numbers. It is found that the presented analytical offers similar results yet with much lesser computational cost.


Analytical Modeling of Outward Solidification With Convective Boundary in Cylindrical Coordinates

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Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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