the fins should be on the liquid side or phase transition(say from liquid to vapor) side?
As a very general rule, where you have a boiling liquid or condensing vapor, heat transfer will be about 10x what you can get with simple forced convection. So I say put the fins on the side NOT having the phase change. Of course, the physical properties of the materials may counteract this; if the liquid has a gigantic heat capacity, very low viscosity, etc, while the changing phase has a huge latent heat, huge viscosity, tiny density change with change of phase, etc, you'd want the fins on the side with the changing phase. Now if the phase change involves a solid phase, you are going to have fouling problems, so you will not want fins there, since it will be hard to clean.
The purpose of fins is usually to provide extra surface for heat transfer. Boiling liquids tend to have a fairly high heat transfer rate due to the action of the boiling process. With this in mind, I would think the fins should be on the liquid side
Convection is air that goes up and then comes back down. Radiation is like a fire, the heat is just being sent up. Conduction is when the heat is being transferred to something else that absorbs the heat. So it's conduction because the heat from the stove is being absorbed by the pan and is heating the water. Hope that helps, and good luck!
