I am designing a thermionic converter and one of the biggest inefficiencies is the thermal conductivity and limited electrical conductivity of the leads. I'm trying to mitigate the loss of heat besides finding other materials that conduct heat poorer and electricity better than Tungsten. One option open is the shape of the lead. Although I don't know if changing the shape of the lead will make any appreciable difference. I am considering a spherical shaped lead in between two flat electrodes. The point of contact of the sphere should be flattened somewhat to allow a better connection. I don't have the mathematical skill to apply Fourier's law to a semi-sphere. I also need to know its overall resistance and I don't have the skills for this either.
If you think it would be worth calculating please share your answer and formulas. Or if not, please explain your reasoning.
It seems to me that if you had a sphere with 2 opposing contact areas (Ca) it would not conduct heat any faster or slower than a straight cylinder with the same contact area (Ca). The cooling part and the heating part are the same size in area.
How far is my reasoning now?