Updated: Jul 7
The two Carnot principles are as follows:
The efficiency of an irreversible heat engine is always less than the efficiency of a reversible engine operating between the same two reservoirs(The source and the sink).
The efficiencies of all the reversible heat engines operating between the two reservoirs are the same.
Now, let's find out what these principles really mean. As I have already discussed in the "isentropic process and isentropic efficiency" article, no process in real life is reversible and every spontaneous process in nature is always irreversible. The reversible process is an ideal process and it is used as a reference for comparing the real processes.
Suppose that we have a real practical heat engine whose processes are irreversible and it has frictional and all other kinds of losses and it operates between two reservoir temperatures TH and TL. We imagine another heat engine that is an ideal engine without having any losses and all of its processes are reversible in nature (we wish!) operating between the same two temperatures TH and TL.
Which one do you think is more efficient?
The answer is obvious. A reversible heat engine is always more efficient than any irreversible heat engine. But, there is another fact that a reversible process/cycle does not exist in nature. So, we accept both the facts and use the less efficient heat engines which are practical and do the intended work than just keep imagining the highly efficient but impractical reversible heat engines.
The second principle is straightforward. The reversible heat engines have exactly the same maximum efficiencies. (Of course, these engines do not exist in reality!)
As we know, the efficiency of the reversible heat engine can be directly calculated using the source and the sink temperature.
Hence, if the two reversible heat engines are operating between the same two reservoir temperatures TH and TL, their efficiency is also the same.
But, in the case of the irreversible heat engines efficiency can not be calculated using temperature values only. In irreversible engines, efficiency is the ratio of the work output and heat input. Hence, the more the irreversibilities less will be the work output, and in turn, it results in less efficiency.
What can you do with all this information?
Whenever someone comes up with a heat engine model of an exceptionally high (and doubtful) thermal efficiency just ask them what is the source and sink temperature and calculate the Carnot cycle efficiency between these two temperatures using the above formula. The Carnot cycle efficiency should always be higher than any real practical engine. You can immediately discard any design which violates this principle.