Carnot Principles
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[edit] Definition
- 1 : No heat engine is more efficient than a reversible heat engine operating between the same two temperature.
- 2 : All reversible heat engines are equally efficient when operating between the same two temperatures.
- where the efficiency of a heat engine over a cycle is defined as the net work output divided by the heat input - not please note, the net heat input (that ratio would always be 1) but the quantity of heat drawn into the system from the higher temperature of the two thermal reservoirs between which it operates. So the thermal efficiency of the heat engine shown in the diagram is given by ηth = W / QH , or, equivalently, since W = QH - QC , we may write ηth = 1 - ( QC / QH ).
[edit] Proof of the Carnot Principles
We shall prove the first Carnot Principle by demonstrating in a "thought experiment" that if you had a heat engine more efficient than a reversible heat engine, you could violate the Second Law of Thermodynamics.
So, suppose you have a heat engine more efficient than a reversible heat engine working between the same thermal reservoirs. First, scale one or the other of the two engines up until they both discharge the same quantity of waste heat QC in the same time. Then for the more efficient engine to be more efficient, the quantity of heat QH that it draws in from the high-temperature thermal reservoir must be greater than the heat QL drawn in by the reversible engine. And so it must also be the case that the work output by the more efficient engine, given by QH - QC, is greater than the work output by the reversible heat engine, given by QL - QC.
Finally, we connect the heat input of our reversed heat engine (fridge) to the heat exhaust of the more efficient engine, as shown on the diagram opposite. The First Law of Thermodynamics is satisfied, since the net effect of the system as a whole is the conversion of a quantity of heat QH - QL into an equivalent quantity of work.
But we have violated the Second Law of Thermodynamics, since the system as a whole is a heat engine which produces net work while attached to only one thermal reservoir, which is forbidden explictly by the Kelvin-Planck statement of the Second Law (and implicitly by all the other ways of expressing the Second Law).
So if we had a machine which violated the First Carnot Principle, we could have a machine which violated the Second Law of Thermodynamics. So there can be no such machine: no heat engine operates more efficiently between two thermal reservoirs than a reversible heat engine.
The Second Carnot Principle may be derived immediately from the First Carnot Principle. If you have two reversible heat engines, then by the First Carnot Principle, neither of them can be more efficient than the other, so both are equally efficient.
[edit] Further Discussion
Besides its theoretical use in defining Thermodynamic Temperature Scales, this result illustrates why imaginary devices such as reversible heat engines are of interest to engineers. A reversible heat engine is not merely an idealization, but also an ideal - something to be approached as closely as is possible in practice.
In the same way, if we define the coefficient of performance of a fridge to be the quantity of heat it removes from the low-temperature reservoir divided by the amount of work input, it can be shown that no fridge has a better coefficient of performance than a reversible fridge operating between the same two temperatures, and any two reversible fridges have the same coefficient of performance. The proof is not dissimilar to the proof of the Carnot Principles given above, and is left as an exercise for the reader.
