Heat Conduction with Phase Change#
Mathematical Model:
\[\begin{split}
\begin{align*}
&\text{Heat Conduction:} \quad \rho c \frac{\partial T}{\partial t} - \nabla \cdot (k \nabla T) = Q \\
&\text{Phase Change:} \quad \frac{\partial \phi}{\partial t} = M \nabla^2 T
\end{align*}
\end{split}\]
Weak Formulation:
\[\begin{split}
\begin{align*}
&\text{Heat Conduction:} \quad \int_{\Omega} \rho c \frac{\partial T}{\partial t} \phi \,d\Omega + \int_{\Omega} k \nabla T \cdot \nabla \phi \,d\Omega = \int_{\Omega} Q \phi \,d\Omega \\
&\text{Phase Change:} \quad \int_{\Omega} \frac{\partial \phi}{\partial t} \psi \,d\Omega = \int_{\Omega} M \nabla^2 T \psi \,d\Omega
\end{align*}
\end{split}\]