An analytical solution for a phase change nano-capsule cooled by forced convection

Great efforts have been made on the solidification of phase change nano-capsules (PCNCs). However, the analytical solutions received less attention when its initial temperature is not at the fusion temperature. Therefore, this work aims to analytically study the thermal behavior of PCNC with the same solid-liquid phase density during forced convective cooling when its initial temperature is greater than the fusion temperature. Asymptotic solutions are developed for long- and small-time scale. Asymptic solutions in the limit of large Stefan number to the solid phase temperature and the position of the free boundary were derived using the perturbation method with boundary-fixing transformations. The results predicted by the derived solutions are in good agreement with the data reported in the reference. It is shown that the surface temperature decreases negatively and exponentially under the Robin boundary conditions during the pre-cooling process. Increasing the values of the interfacial tension or the large Stefan number has the effect of slowing down the movement of the free boundary. The rate of change of the solidification time with respect to the Stefan number is maintained constant for a stationary free boundary. The analytical solutions can also be used to solve the classical Stefan problem (large Stefan number) by ignoring the surface tension. The analytic estimate provides a more convenient option for engineering applications than the complex numerical solution.