How do I calculate the energy balance of a trompe? I was quite fascinated by the concept of an ancient type of air compressor, called a trompe. It entrains air bubbles into a falling stream of water via the Venturi effect, and extracts the air at a lower level where it is pressurized by the hydraulic head. More details on the Wikipedia page.
Now, the output of such a device in comparison to its size, combined with the fact that you need to have sufficient amounts of water conveniently falling down somewhere, means that this device is not competitive with air compressors we use. 
However, I was wondering how it performs on the energy level, and noticed that I'm not quite sure how I should calculate that. If I take the work that the air does when decompressing as my energy output, what would be my energy input? Could I say that it's the work I need to do to overcome the buoyancy of the pressurized air at the lower level?
 A: Actually, the trompe is competitive with other technologies in certain situations. Bruce Leavitt pioneered the practical use of trompes for aeration at remote sites without electricity. Grady Hillhouse put together a great video about trompes and it shows how a simple garden hose can compress air.
Trompes don't need massive amounts of water. Trompes require pressure head. Inspired by these projects, I've been building an even smaller trompe that uses an electrical water pump to compress air. This recirculating, electrical trompe is a closed system whose inputs and outputs can be easily measured. Notably, a recirculating electrical trompe is quieter than an electrical compressor such as an aquarium aeration pump. Indeed, the water pump is quieter than the noise of the water falling down the 1/2" venturi pipe. Another advantage of trompes is that the compressed air does not require special attention to cooling that typical air compressors have to deal with.
Trompes are simple to build, but difficult to calculate given the number of contributing factors. Trompes can't even be calculated using Bernoulli's equation since friction is definitely an issue. Trompes also mix two fluids, one of which is compressible and the other incompressible--the density varies. Even the Venturi flow equation isn't much help because it doesn't really give us how much air is entrained so that we might calculate compressor output capacity.
We can't even calculate output pressure directly except to say, "oh, it's about 80% of input pressure head or less". Additionally, increasing venturi stages improves air entrainment and is a major factor for trompe efficiency.
Empirically, a 12V, 0.7A water pump can effectively generate almost 1 litre of air a minute at zero pressure using a 7-stage venturi and can easily produce 0.4 PSI per foot of 1/2" down pipe as limited by the pump head. More powerful pumps would require larger pipes, since the down pipe friction becomes significant at higher flow rates.
In summary, trompes are easy to build and measure but hard to calculate exactly.

