A Variational Analysis of Freezing or Melting in a Finite Medium Subject to Radiation and Convection

Abstract

The objective of this work is to develop an approximate analytical solution for the heat transfer in a finite medium with a change in phase. The medium is subjected to radiative and aerodynamic cooling (or heating) at one side and is thermally insulated at the other side. Furthermore, the initial temperature is different from the fusion (or melting) temperature. It is assumed that all the physical properties are constant for each phase but may be different (except density) for different phases. In this analysis, Biot’s variational method is employed. With this technique, the complicated nonlinear problem is reduced to an initial value problem which is then solved by the Runge Kutta method. The calculated temperature histories at both surfaces and the time variant fusion (or melting) line are in terms of dimensionless parameters such as radiation number, Biot’s number, Stefan number, ratio of freezing (or melting) to initial temperatures and the thermal properties ratios in both phases. Some limiting solutions of the present work are found to agree with the earlier analysis.