Work done by small expanding bubbles in TrombePump

First, it occurred to me that it is possible to avoid making the hot air subject to direct compression by expanding bubbles. There was no good reason for the gas output of the CCHEX to be connected to the same plenum as the water output. The revised design separates the cold water from the cold gas before they enter the CCHEX, and the hot gas enters the hot water flow at the bubble injectors. Therefore, aside from a very small exposure at the bubble injectors, there is no place where the water can do work on the hot gas after the CCHEX. The downward force from expanding bubbles is transmitted through water to the cold air collecting at the bottom of the trompe.

Unfortunately, some of the force also continues up the trompe, around the inverted U siphon tube, down through the hot to cold water CCHEX, and pushes up on water in the air-free column of hot water which flows out of the bubble pump.

Is this force mostly a balanced force which does no work?

The reason the bubble expands is that the weight of the water above it is less than the force exerted by the gas. This happens because the bubble has moved and is at a shallower depth. But it is also easier for the bubble to push the water down, because there is more water below counterbalancing the water in the downflow column. The hydrostatic pressure pushing upward equals the hydrostatic pressure pushing downwards.

However, this is dynamics, not statics. The mass of the large quantity of water below dictates that an equal force pushing downward will decelerate the large mass below much less than the upward force accelerates the smaller mass of water above. The equal forces act through different distances. More work is done on the water above than on the water below.

(Given equal forces pushing a big mass and a small mass, more work is done on the small mass because the force can act through a larger distance. This is in contrast to gravity, which applies equal acceleration, not equal force, to different masses).

My conclusion is that the small bubble design (which easily achieves isothermal expansion because of the large thermal mass of the water) captures nearly all of the work done by the expanding hot gas and stores it as gravitational potential energy in water moved from the top of the rising bubble-filled column over to the top of the heavier air-free column. From there, gravity does work on the water again, accelerating it. A force appears at the top of the bubbles of cold air, compressing them.

Of the smaller fraction of work done by expanding gas on the hot water below the bubbles, most results in compressing the cold air already at the bottom of the trompe. A smaller amount acts to compress cold air bubbles descending in the trompe. An even smaller amount acts to decelerate hot water flowing downward in the air-free column on the bubble pump output.

And there is a final triumph. It looks to me like even that small amount of work ends up compressing cold gas, because what difference is there between water which went up the bubble pump and over to the heavy column, or water which was pushed backward up the heavy column? So long as the surface of the hot water isn't rising and doing work on atmospheric pressure air, I see none. And actually, the surface is not rising.

I seem to be reaching the point where I can design an experiment to test this theory out.

Update:  probably most of the inefficiency in a bubble pump occurs when raised water falls back down when dynamics is being relied on to do the pumping rather than statics.  If the bubble pump is just supposed to raise the water a tiny amount, it should be quite efficient, even if the input side is not sealed, according to Archimerged's latest thinking…  Need to do some experiments…

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