The problem I am currently working on involves rapidly cooling liquid aluminum in a cylindrical graphite crucible and getting the radial temperature distribution over time, possibly using finite difference methods.
This is a 1D transient problem in only the radial direction using the following equation and I am only concerned about the liquid portion of aluminum (no source term and constant thermal properties).
$$\frac{1}{a}\frac{dT}{dt} = \frac{1}{r}\frac{d}{dr}\left(r\frac{dT}{dr}\right)$$
The main problem I am having is determining the boundary conditions, especially at the interface between the graphite inner wall with the liquid aluminum (as graphite and liquid aluminum have different thermal diffusivities). The boundary condition at the outer graphite wall involves convection and the BC at the centerline of the cylinder should have no heat flux for symmetry but I am not sure how to get a boundary condition at the interface (I am thinking of equating heat flux of graphite to aluminum).
I was wondering if anyone here has any idea on how to get the boundary condition at the interface or have any credible sources that I can look at. Thank you.