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The intercooler's purpose is to reduce the temperature of recently-compressed air.
Regardless of the case, the presence of ANY compressor will result in the air temperature rising. Even without an OUNCE of boost, the heat that comes through the housing (particularly in a turbo) and the heat resulting from violent thrashing, is put into the air... but compressing packs more air in a small space, and all the latent heat IN that air becomes concentrated, thus, the effective TEMPERATURE of the incoming air is higher. Now, we don't have to worry about DENSITY of the charge, because the COMRPESSOR is increasing the density... but the empirical result of Combined Gas Law is that compressed gasses experience temperature rise, because all the latent heat in the air BEFORE compression, is now packed into a much smaller area, so the thermometer will indicate much HIGHER temp.
In a gasoline engine, the intercooler's greatest benefit, is that since the incoming charge is COOLER, it will be less likely for the engine to exhibit PREIGNITION. Even if there's darned near NO fuel in the cylinder, all that heat will cause the charge to ignite WAY too early, which means you're gonna LOSE power (at least, until you burn a hole through a piston, right?)
IN a DIESEL engine, the same applies... but wait... the diesel is a COMPRESSION IGNITION engine, so there's no spark plug 'hot spot' to worry about with preignition, right? OF course, but can a diesel 'preignite'? Not if there's no fuel... if there's enough oil blowby though... it CAN preignite.
So what's this mean with respect to horsepower? As Doc noted, you can cram all the air in you want... it will only burn as much fuel as you feed it, and your power output is entirely dependant upon your fuel burn rate. This means that it doesn't matter if you go from 12 pounds of boost, to 30 pounds, to 130 pounds, if you don't increase the fuel flow, the boost will simply be a waste of time.
And the converse is true- IF you look and see black smoke coming out the pipe, you're not getting good burn, and one of the FIRST things that cause that... is overfueling. When you see black soot, you can throw all the FUEL you want in there, it will NOT give you any more power, because there's simply NO OXYGEN LEFT to burn that fuel... it's goin' up as a dark cloud... money thrown away...
If you cram more air in there... that stuff will burn, the pipe's discharge cleans up, and you get more effective cylinder pressure.
Back to the intercooler...
Internal combustion reciprocating piston engines are HEAT-INTO PRESSURE devices. When you burn a mix of fuel and air in a small space, the heat causes gases to expand, exerting pressure. The more fuel and air you can pack in, the greater the pressure...
But the piston and cylinder work on a basic physics principle:
The cylinder fires at one temperature, and the temp increases, forcing gas pressure up, and the rod and crank turn that into rotating kinetic energy.
It is the DIFFERENCE in pressure that makes the engine RUN. Start at a low pressure, close the valves, and change it to a HIGH pressure.
Obviously, if the intake charge is at 2700F, and the burned fuel charge is 2700F, you're not gonna make much power, right?
That intake example is pretty extreme, but let's say it's oh... 400F, and finishes at 2400F? Well, the total differential pressure is significantly HIGHER, so your output power will be much higher.
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