davefitz (and others),
you mentioned that the droplets should be less than 200 microns to avoid erosion of compressor blades. i am interested in looking into this. could you provide some references, papers, etc. on this topic? thanks.
1) the problem is 2D-axisymmetric.
2) do you have sufficient mesh resolution to solve for the boundary layer?
3) is the flow laminar or turbulent?
4) the heat transfer coeff is not a constant, but a function of space for steady state.
5) at the surface, there's only balance of conduction heat...
you can assume 1D heat transfer problem, T(x). take an element with length dx and apply energy balance {q(x) = q(x+dx) + dq(radiation)}. apply Taylor series along with Fourier's law and you should arrive at the equation you want to solve. you might have to solve this numerically since...