In Larry Niven's Ringworld, the Outsiders lie with one half of their body in the shade and one half of their body in direct sunlight and use the temperature difference to generate energy. Would this be at all viable on a tidally-locked planet, with one end on the cold half and one end on the hot half, and humans living on or near the terminator, or am I better off sticking to a method like nuclear power?
Yes, you can use the temperature difference. The high tech device to harness the power of temperature gradient on different side of the planet is called windmills. Under such conditions it would be even working better than on Earth (more stable wind patterns).
What's better purely depends on local factors. With AD 2019 technology windmills as the dominating source of energy would be doable, but somewhat problematic:
- they are connected to the grid in asynchronous way, thus do not provide grid stability like any big turbine.
- their energy output is highly variable, so you require either huge pumped-storage hydroelectricity or you accept huge waste.
You could do this cheaply and robustly with a solar concentrator and a steam generator.
Solar: finiky silicon sandwiches, rare earth elements; bah. Wind - stuff moving at high speed, getting clogged with dirt and hawks; bah.
Steam is what you want. Plain foil mirrors, no moving parts out in the environment and cheap cheap cheap.
Your shady side is used only to condense steam back to water. But you might be able to do that on the sunny side depending on how hot it is. I am not sure how the depicted concentrator is recondensing its working fluid - I would think they could use the shade under the mirrors but it does not look like there is anything under there. If it is hot enough on your world to boil water without the concentrator then you will either need the shady side of your world to effect the phase change back to liquid, or use a liquid with a higher boiling point.
It sounds like a good idea. Given that the cold side would be at very low temperature, one can get a high termodinamic yield just because of this.
Remember that the ideal yield of a thermodinamic cycle between $T_h$ and $T_l$ is given by $\eta = 1 - $$T_l\over T_h$
The tricky part will be the losses due to the piping from he hot to the cold side, but engineering can deal with that.