Superheated two-phase Stefan similarity ($a=2\lambda \sqrt{\kappa t}$)

For equal phase thermal properties, a roof undercooling $\Delta T$ and liquid superheat $\Theta$, the planar Stefan problem has interface $a=2\lambda \sqrt{\kappa t}$. The solid error function profile and liquid complementary-error-function profile give $S\lambda =e^{-{\lambda }^2}[1/\mathrm{erf}\lambda -(\Theta /\Delta T)/\mathrm{erfc}\lambda ]/\sqrt{\pi }$. Here $S=L/(c_p\Delta T)$ is the latent-to-sensible heat ratio. Liquid heat supplied to the interface reduces crust growth.

The small-$\lambda$ limit of superheated two-phase Stefan similarity gives $a\simeq A\sqrt{\kappa t}$, where $A=2/[\sqrt{{\vartheta }^2/\pi +2S}+\vartheta /\sqrt{\pi }]$ and $\vartheta =\Theta /\Delta T$. For fixed superheat at large latent-to-sensible heat ratio, $A\sim \sqrt{2/S}$. For much larger superheat, $A\sim \sqrt{\pi }/\vartheta$. The approximation retains the leading effect of a hot interior and its coefficient need not be exact when $S$ is only order one.