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How would you cool a nuclear power plant on the moon, no water?
Good question. Not all reactors by design need water as coolant. Some use molten salt, others are gas cooled.
These alternate cooling materials would likely still need to be imported though, so it doesn't particularly make it any easier.
My main question is the effect of the lower gravity on cooling the reactor (thermal hydraulic effects). All of our current reactors are designed for 1 g use, not 0.1654 g that's on the moon. Heat mixture rates in fluids would be different, which is important when you're calculating effective heat dissipation.
Just for fun, go check out xkcd's new What If video. They go into heat dissipation of a nuclear reactor in space (not the moon, but still incredibly interesting, informative, and entertaining).
I should have shared, but I'm lazy.
https://www.youtube.com/watch?v=EsUBRd1O2dU
Maybe only turn it on at night when the surface gets cold and conduct it into the ground or land it in a permanent shadow? The article said they planned to do it fully automated so I am guessing digging of any form is out.
You don't just turn a nuclear power plant off.
I believe they mean inserting control rods to dampen the reactions to subcritical.
If sun is an issue, it might be simpler to set up a “beach umbrella”
You cool it with liquid thermal transfer and radiators. Here's what a kilopower plant looks like, the big disk is a radiator.