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From The Law of Conservation of Energy, or Energy = Heat + WorkLets take an engine starting with some hot pressurized air, ready to expand. Energy = HeatInitial. Expand it, get some work. Energy = HeatFinal + Work. How much Work? Work = Energy - HeatFinal. In a vapor, we can say temperature = K*Heat, so Work = Energy - K*TempFinal. But Energy = HeatInitial, so What is Work compared to the beginning total energy? (Wrong, but approximately right) This cannot actually be used to accurately calculate the Work/Heat energy ratio, but oddly that does not matter. Combined with Conservation of Energy, one always knows the total adds to the initial energy. One can always measure the initial heat and final heat (not temperature) and accurately determine the amount of work done.
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By integrating ∫ Volume-1-Bk, the Work Pressure CurveSee the math here. (1-V:final -Bk)/Bk which ought to look familiar-ish. Also, V:final -Bk is relative heat or temperature, so this will simplify further to (1-RelativetemperatureFinal)/Bk or (1-TempFinal/TempInitial)/Bk. Or Work * Bk = 1-TempFinal And Energy total = TempInitial/Bk So Work/Energy = 1 - TempFinal/TempInitial Why Bother with a ratio at all?First thought was that in the era the ratio was developed, there were units for temperature, but not for energy. This may be true. What it is currently still very useful for, is that a given volume change (1 to 2, 1 to 10) always converts the same ratio of energy. So it very useful to characterize an engine. So, a given engine will convert a given ratio of heat to work. But that is NOT its "efficiency" unless you waste the remaining heat. Its not a fundamental limit of the universe any more than any other measurement of progress. "Halfway up the stairs" is a measurement, not a limit. What it is, is a MEASUREMENT. Approximately how Energy is divided between heat and work. An engine can't do better than it did, it can't do worse either. Most important, an engine cannot "lose" or "destroy" energy.
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