Chapter 5, Problem 5.97P

When a molten metal is cast in a mold that is a poor conductor, the dominant resistance to heat flow is within the mold wall. Consider conditions for which a liquid metal is solidifying in a thick-walled mold of thermal conductivity $k_w$ and thermal diffusivity ${\alpha }_w.$ The density and latent heat of fusion of the metal are designated as $\rho$ and $h_{sf},$ respectively, and in both its molten and solid states, the thermal conductivity of the metal is very much larger than that of the mold.

Just before the start of solidification $\left(S=0\right),$ the mold wall is everywhere at an initial uniform temperature $T_i$ and the molten metal is everywhere at its fusion (melting point) temperature of $T_f.$ Following the start of solidification, there is conduction heat transfer into the mold wall and the thickness of the solidified metal S increases with time t.

1. Sketch the one-dimensional temperature distribution, $T\left(x\right),$ in the mold wall and the metal at $t=0$ and at two subsequent times during the solidification. Clearly indicate any underlying assumptions.
2. Obtain a relation for the variation of the solid layer thickness S with time t, expressing your result in terms of appropriate parameters of the system.