OK, I haven’t done this in forever, but I muddled through it some numbers. Spoiler - there is no free lunch. It costs110kJ of energy to prime the system for 1 cycle, but operating it only yields 103kJ of energy.
Can someone check that I’ve decomposed the problem correctly (or at least sufficiently) and I’m thinking in the right units?
Given normal fresh water at STP, using a 10 meter column of water, and a 1 m³ ball with density of 50 kg/m³, we will extract energy from the force of the rising buoyant object, shift so that it will fall in air and pull a chain to recapture that energy. Then we will insert the ball at the bottom and start over.
Buoyant force = density of liquid in kg/m³ * volume of object * gravitational acceleration.
Our 1m³ ball has a buoyant force of 9807N. Traveling over a distance of 10 meters, we have done 98,070J of work. Assume we store it all.
Falling force = F*ma. Our 1m³ ball of polystyrene weighs 50kg. That’s 490.35N of force, over 10 meters again, we end up with 4904J. Assume we store it all.
Our combined stored energy is now at 102974J. We cannot exceed this budget as we are resetting the system. How much energy does that require?
The diameter of our 1m³ ball is 1.24m. Again, the buoyant force from the earlier calculation is 9807N, so to submerge it, multiply those two. 12160J merely to put that ball underwater, disregarding pressure.
To reset the system, we must lift the displaced 1 cubic meter of water back to the top.
F = 1000kg * 9.807 = 9807N
work = 98,070J
Adding the cost of submersion plus cost of refill:
98070J + 12160J = 110230J.
Total energy captured via rising buoyancy and falling object = 102974J.
The stored energy has a deficit of 7256J without even factoring friction. This system does not generate enough energy to reset itself for another cycle.