One is open at the bottom and collects hot water from the vent. The density of the hot water is less than the sea in general, so, if the hydrostatic pressure at the bottom is equal to the hydrostatic pressure of the sea, then the pressure in the pipe at the surface will be less than atmospheric, and the hot water will boil and spew, naturally, from the open top. What am I missing here? Do you expect the hot water to flash into steam at some depth and arrive at the surface as somewhat cooled, by expansion, steam at low pressure, using only the heat (via heat exchanger) and not the pressure of the natural steam? If you want liquid hot water at the top of the pipe, it will have to be under pressure. That pressure, added to the hydrostatic pressure of the pipe full of water, would reverse the flow, would it not?
A second pipe brings cold water from the bottom to the surface to maximize the temperature difference between the hot water/steam and the condneser temperature. The cold water is more dense than the sea around the pipe. How do you lift the extra weight? You explain it, I'm sure, but I don't see it on your site. You show a pump on the oil platform, but how do you "suck" water up?
The third pipe has to handle the cooled hot water, "waste" minerals, the warmed cooling water, and the condensed steam. Since the density of this warmish waste water will be less than the sea, it will require energy to pump it downward, right? You show a pump in the diagram, but won't it consume a great deal of energy?
Clearly you have put a lot of thought into this, and I'm missing something. Could you, perhaps, tag your diagram with pressures, etc., so we can see how the plumbing works?
