A medieval-tech desert city relies on a nearby subterranean lake for their water source. In lieu of employing thousands of slaves carrying pots up a long series of tunnels, could the intrepid desert settlers boil the lake water and force it through narrow tubes to the surface, where it is cooled, condensed, and collected?
I can think of at least three ways to generate the heat necessary.
1) A vast system of convex mirrors collects solar radiation (plentiful and cheap in the desert) and directs it down a series of tunnels to the lake's surface.
2) The desert-dwellers manipulate magma vents to boil the water.
3) The people have a contract with the local deity of magma, who allows them access to as much heat as they care to generate.
Would this be economical? Would the amount of heat needed to boil a lake (or sections of it) be ridiculously prohibitive? Is there any possible way to keep the water from cooling as it travels up the narrow tunnels? Would the steam arising form the tunnels make the ground surface on which the city sits unbearably hot? Any reason why these people shouldn't just dig wells to tap the lake?
Any other conceivable difficulties?