Consider the heat conduction equation:

dT/dx = K * d2T/dx2

I'm interested in the semi-infinite solid. I want to be able to compare four different cases. In all cases the temperature throughout is constant at t=0

The first two are.

1a. Specified temperature applied at x=0, for t>0. 2a. Convection to fluid at a known temperature and convection coefficient (both of which are constant, and applied for t>0).The other two cases (1b and 2b) have the same boundary conditions, but are complicated by the presence of a phase change. There is no mass transfer, and the density of the solid and liquid phases is identical, though the conductivity and specific heat may be different. The phase change is assumed to occur at a single temperature.

I have analytical solutions for 1a, 2a and 1b. What I lack is the analytical solution to 2b (phase change, convective boundary condition). Can anyone provide me with a source for this? (Or the solution itself?)

Also, I fully understand the derivations for 1a and 1b in Carslaw & Jaeger (Conduction of Heat in Solids, 1959), but I was unable to follow along with the derivation of 2a. Can anyone recommend a source with a clearer derivation for the single-phase convective BC case?

Thanks,

-Paul