The equation,
I believe, is something like T2 = T1 * R ^ xxx
T1 is the starting gas temp, T2 is the temp after expansion or compression,
R is the compression ratio -- the ratio of beginning and ending pressure.
Temperatures are absolute -- degrees K, or if you are a hardcore imperial unit guy, degrees Rankine.
Using a compression ratio of 1/200 (going from 200 bar to 1 bar) and assuming that the exponent is 2/7, you get a possible minimum temp of 66K or about -200C. Using a compression ratio of 1/20 for the drop across a 1st stage of 200bar down to 10 bar, the coldest theoretical outlet temp would be -145 C.
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I'm not 100% sure of what the proper exponent of the equation would be, I get confused a bit going through specific heat capacity, gamma, gamma-1, etc. I think that for a diatomic gas like nitrogen and oxygen, the exponent is (7/5-1)/(7/5), which is 2/7 or about 0.286. (For ideal diatomic gas the adiabatic index is 7/5 =1.4. The measured number for air is about 1.395 to 1.403 depending upon temp&pressure) (For a monoatomic gas the exponent is (5/3-1)/(5/3) = 0.4)
Since I'm not sure that is the correct equation, to see if these numbers make sense I calculated the compression heating effect in a gasoline engine and a typical diesel engine. a gas engine will "knock" if the compression heating lights off the fuel-air mixture, while a diesel engine relies upon compression heating to ignite the injected diesel fuel.
A gasoline engine compression rato of 8 would result in a theoretical max temperature at the end of the compression stroke of 300K * 8^0.286 = 544K = 270C = 519F. This a bit above the gasoline self ignition temp of 246C, but heat losses apparently give a bit of margin. OTOH, a diesel engine compression ratio is in the range of 14 to 24. Taking the midpoint of 19, and assuming a chilly 0C, the equation works out at T2 = 273 * 19^0.286 = 633K = 360C = 681F , which is apparently enough to ignite the injected diesel fuel which has a self ignition temp of 210C.
I'm by no means an expert in this field. Comments and correction from someone who actually understands this are welcomed.
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http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/diesel.html includes a calculator that shows heat of compression in a diesel engine as being calculated with an exponent of (7/5 - 1) = 0.4 for air rather than the exponent I used above of (7/5-1)/(7/5) = 2/7 = 0.286 which is what I interpret from some Wiki articles on heat capacity. Using 0.4 makes the heating and cooling effects of compession and expansion greater than my calculations.
