https://space.stackexchange.com/questions/19906/constant-lun...
What is the heat conductivity though ? If it is low, you might have to sink too many heat transfer channels, not to mention with possibly ending up with a hot "bubble" that would no longer provide a temperature gradient.
We all know how cold it is to stand on a stone floor - unlike wood or plastics, it keeps sucking that heat off.
1. https://en.wikipedia.org/wiki/Internal_structure_of_the_Moon
2. https://en.wikipedia.org/wiki/List_of_thermal_conductivities
UPDATE: I forgot about not-water coolants. Molten salt or liquid metal reactors can work at 500 C, so would radiate anyway, right.
A 10kW reactor - easily; a very limited amount of engineering to design a heat distribution system will be required, but 10kW is also not enough for anything useful.
I suppose the future of Moon energy will be in megawatt-class reactors, which are still possible to cool like that, but provide enough heat+energy to be useful for colonies.
In the 70s and 80s, the USSR launched a whole series of radar satellites that were powered by several generations of nuclear reactors: https://en.m.wikipedia.org/wiki/RORSAT
Liquid metal coolant, as you suggested. Also, direct thermionic conversion to electricity -- no heat engine cycle. No control rods either -- control reflectors outside the core. Very different kind of nuclear engineering than is needed down here on Earth. No shielding!
I think this was the video https://youtu.be/9Xs3mGhQGxM