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.

update 6/29/14

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.

Calculating the resistance was fairly easy to do in slices in javascript. And the resistance reduction from a straight cylinder with contact areas (Ca) on either side to a sphere with a limited contact areas (Ca) was considerable.

How far is my reasoning now?

  • $\begingroup$ I think your question is an engineering question, not a physics question. It asks for a way to best arrange a specific instrument. $\endgroup$
    – ACuriousMind
    Commented Jun 29, 2014 at 17:32

1 Answer 1


Turns out the reduced electrical conductivity ruins this idea. I've posted my solution here:


Actually a little more inspection shows that there is a small improvement if (and only if) you use larger diameter sphere leads and optimize the cutoff ratio. So it doesn't look like a complete waste of time...


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