Casting
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Mould wall temperature
We define the temperature difference as:
T = TW − T
so that
$${{dT} \over {dt}} = {h \over {L{C_V}}}\left( {{T_W} - T} \right)$$
rearranging and integrating gives:
$$\int {{1 \over {\left( {{T_W} - T} \right)}}dT} = \int {{h \over {L{C_V}}}dt} $$
$$ - \ln \left| {{T_W} - T} \right| = {h \over {L{C_V}}}t + c$$
where c is a constant.
Applying the boundary condition that T = TP when t = 0, we get:
$$c = - \ln \left| {{T_W} - {T_P}} \right|$$
so that the time taken for the liquid to cool to a given temperature is given by:
$$t = - {{L{C_V}} \over h}\ln \left| {{{{T_W} - T} \over {{T_W} - {T_P}}}} \right|$$