Development and validation of an asymptotic solution for a two-phase Stefan problem in a droplet subjected to convective boundary condition

Development and validation of an asymptotic solution for a two-phase Stefan problem in a droplet subjected to convective boundary condition

Saad Akhtar, Minghan Xu, Agus P. Sasmito

Abstract

Droplet solidification is governed by classical Stefan problems which have been commonly treated as a single-phase problem by the majority of the studies in the literature. This approach, however, is unable to capture the initial temperature and the start of freezing time correctly. The treatment of two-phase Stefan problem in spherical coordinates is limited. No known exact solution exists, albeit numerical solutions and asymptotics have proven to be useful. We present a singular perturbation solution in the limit of low Stefan number and arbitrary Biot number for the two-phase Stefan problem in a finite spherical domain. An asymptotic solution is developed for a droplet at a non-freezing initial temperature subjected to a convective boundary condition at the surface. The solution is developed for both long-time and short-time scales. The results from asymptotic expansion method are validated with the experimental results in the literature and are further verified by a numerical model of a freezing droplet using enthalpy–porosity method. The sensitivity of the asymptotic solution to the droplet initial temperature, Biot number, and Stefan number has also been studied. The results indicate that the solution from perturbation series and enthalpy–porosity method agrees to within 1%–10% for temperature profile and overall freezing times over a wide range of practical values for initial temperature, Stefan and Biot numbers for the application of spray freezing. Our perturbation series solution is also able to capture the effect of initial temperature on the overall freezing time of the droplet.

Keywords

Perturbation series solution, Asymptotic expansion, Droplet freezing, Two-phase Stefan problem, Enthalpy–porosity method

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