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If you mean that the ambient temp air flow is affected, I wouldn't expect that to be much. Say ambient is 20C, how much lower can it get by injecting mist of water that's 30C+ (the bottle is in the engine bay!). I suspect that after the pump the water is even warmer.
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Actually it can be very substantial. The cooling is by evaporation, and typically will exceed 10 - 20 deg C . The initial temp of the water is of little consequence. The cooling due to evaporation (latent heat of evaporation) is very much larger than the latent heat of the liquid water.
As you can see below the evaporation of a gram of water will absorb 542 x the heat energy required to lower that same quantity of liquid water one degree in temperature. That means that even if the water were nearly at boiling temps when injected, evaporative cooling would reduce the air flow temp below ambient temperatures long before all the water changed to vapor.
Specific heat capacity of liquid water - 4.187 kJ/kgK
Latent heat of evaporation - 2,270 kJ/kg
Since the evaporation of the water and the alcohol will essentially stop when the air becomes saturated (ie. 100% humidity) there is a practical limit to the cooling or around 20-30 deg C for alcohol water mixes, and in real systems you seldom get much more than about 15-20 deg C.
For every 5 deg C you drop the inlet air temp you will increase mass flow by about 1% due to density increases.
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I would have thought that this 'efficiency gain' is the effect of water droplets inside the compressor blades, as they try to squeeze the air.
Am I right?
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Only in a very crude sense. Your thinking in a mechanical piston pushes on air sort of way, but what happens inside the compressor impeller occurs at a molecular level.
Think of it this way If you could freeze frame time, and stop what was happening inside the impeller while its spinning at 120,000 rpm. Each impeller passage between adjacent pairs of compressor blades contains a wedge shaped parcel of air. When spinning at 120,000 rpm the air is subject to huge centrifugal forces as it moves away from the hub of the impeller and toward the rim of the compressor. The trapped air would like very much to be slung out of the impeller but like a crowd at a stadium after a match it simply cannot all get out as fast as it would like. As a result it stacks up (compresses) as it gets near the exit. In this process a lot of internal friction occurs. The air near the tips of the compressor might be moveing near the speed of sound at maximum flow, this heating makes the air try to expand. This increases the pressure which fights the outward movement of the air. Eventually a balance is achieved between the centrifugal forces trying to throw the air out of the impeller and the pressure build up due to the compression and the pressure build up due to the heating. The addition of the water mist removes a very large fraction of the pressure gain due to heating. As a result more air can exit the impeller over a given period of time, and more of the pressure gain is real compression rather than waste heat. The net result is a more isothermic compression which is always more effecient than an adiabatic compression.
Larry