Without a constant source of energy, heat pumps cease to work.
Say you start with two boxes with equal temperatures, the heat pump moves heat energy from one side to the other so one side gets colder and the other warmer. After a short while, the pump stops working because it can no long move heat energy – one side is too hot and the other too cold. If you use the hot side to “power” something like boiling water to make steam and then run a turbine, the warm side just cools down quickly again. So there is no solution here. This is assuming 100% efficiency which is impossible. Energy is not created – it is simply moved around.
This is like trying to run an infinite turbine by letting a ball roll down a hill, then pushing the ball back up the hill again. Clearly you’d have to expend more energy pushing the ball back up the hill than would ever be generated from the ball rolling down.
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