Phase-Change in Finite Slabs With Time-Dependent Convective Boundary Conditions

The heat balance integral method is used to solve one-dimensional phase-change problem in a finite slab with time-dependent convective boundary condition, [T_∞,1(t)], applied at the left face. The temperature, T_∞,1(t), decreases linearly with time; the other face of the slab is subjected to a constant convective boundary condition with T_∞,2 held fixed at the ambient temperature. Two initial conditions are investigated: temperature of the solid below the melting point (subcooled), and initially at the fusion temperature (T_f). The temperature, T_∞,1(t) at time t = 0 is so chosen such that convective heating takes place and the slab begins to melt (i.e., T_∞,1(0)> T_f> T_∞,2). Thus the solid-liquid interface proceeds forward to the right. As time continues, and T_∞,1(t) decreases with time, the phase-change front slows, stops, and may even reverse direction. Hence this problem features sequential melting and freezing of the slab with partial penetration of the solid-liquid front before reversal of the phase-change process. It should, however, be noted that the study is limited to only one solid-liquid interface at any given time during the phase-change process (either melting or freezing) and that slight subcooling of the melt is allowed. The effect of varying the Biot number at the right face of the slab, for both the initial conditions, is also investigated to determine its impact on the growth/recession of the solid-liquid interface. Temperature profiles in both regions (liquid and solid) are reported in detail. The effect of a slower decay rate of T_∞,1(t) on the phase-change process is also analyzed for the initial condition of the slab being at the fusion temperature.