If the temperature inside a body passes a melting point T_P (subscript P for “phase change”), the isothermal surface T = T_P subdivides the body into two regions, each with a different phase and properties. The thermal fields in both phases are coupled (see Section 161C). Additionally, the position of the interface may change with time. Exact analytical solutions for these problems are known only for some simple special cases (Carslaw and Jaeger, 1959, chapter XI). For practical applications it is often sufficient to know only the time required to solidify or melt a body completely, with the details of the temperature field of minor interest.

Solidifying times may be calculated by a simple approximation if the enthalpy change of the solid shell is small when compared with the latent enthalpy of solidification. Figure 1 shows the problem. It is assumed that the liquid (subscript 1) is initially at the phase-change temperature T_P.

Figure 1 Solidification (or melting) of simple bodies

The energy balance for this system may be written

\[\label{eq1} -\dot{Q}\cong\rho_1 \Delta h_{12} \frac{dV_1(x)}{dx}\frac{dx}{dt}\tag{1}\]