I think we agree. I'm talking about computing the number of moles (mass of air) and you are talking about mass flow, but we are both talking about mass.
In the end we are asking: does the compressor carry more air particles if cooling occurs during compression, than if cooling occurs after compression?
I believe the answer is yes, and here is my lastest thought experiment. It's gets right to the heart of the matter: number of air molecules compressed per rotation.
Imagine this case. The ambient air is -20C, when compressed this air, with each turn of the compressor wheel, contains more air molecules (cold is denser) than if the air were at 20C (less dense few air molecules/unit area).
Likewise, if during compaction, temperature (i.e. kinetic energy) is kept constant, the air being compacted will be more dense, and hence again with each turn of the compressor more air molecules are compacted.
The work that we are interested in is the work of compression, which is all about the pressure ratio given by the centrifugal forces than are acting on the air. Imagine that each time the impeller blade swings by is analogous to a shovel full of air. If each shovel full wieghs twice as much, then you end up with twice as much air shoveled per unit time. Similarly, with a pressure ratio of 2 (isothermic) each swoop of the impeller results in a compression of twice as much gas.
If without water the turbo is compressing air at a PR of 2, and we could completely suppress all heat during compression, we will have gained back all of the adiabatic heat loss and all of the compressor's inefficiency. The percent gain will be directly related to the PR. The larger the PR, the more gain can be had for each compression event--just like the shovels of air, 4X density air is more than 2X air.
As has been said before, the gain could be huge (40%+) if all the heat is absorbed by a small amount of water. To tie this back to yesterday's discussion. We would need a much larger turbo to compress say 40% more air, that was then cooled by WI post injection. It's all in the air density per rotation of the compressor--more dense air (cooler) leads to more air molecules compressed per rotation of the impeller.
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