Why can no other engine be more efficient than a Carnot engine?

[member 392791]

Hello,

I am having difficulty understanding the paragraph in my textbook explaining why no other engine can be more efficient than a carnot engine. What does it mean?

To prove the validity of this theorem, imagine two heat engines operating
between the same energy reservoirs. One is a Carnot engine with efficiency eC, and
the other is an engine with efficiency e, where we assume e > eC. Because the cycle
in the Carnot engine is reversible, the engine can operate in reverse as a refrigerator.
The more efficient engine is used to drive the Carnot engine as a Carnot refrigerator.
The output by work of the more efficient engine is matched to the input by
work of the Carnot refrigerator. For the combination of the engine and refrigerator,
no exchange by work with the surroundings occurs. Because we have assumed the
engine is more efficient than the refrigerator, the net result of the combination is
a transfer of energy from the cold to the hot reservoir without work being done on
the combination. According to the Clausius statement of the second law, this process
is impossible. Hence, the assumption that e > eC must be false.

 

[mfb]

It means that no engine can be more efficient than a Carnot engine. Such a machine could be used to reduce entropy (move energy from cold to hot reservoir), as shown in the text.

 

[jtbell]

That paragraph assumes in advance that the Clausius statement of the second law is valid, i.e. it is impossible to transfer energy from a cold reservoir to a hot one without an input of work. It then uses the method of contradiction to prove that no heat engine can have an efficiency greater than that of a Carnot engine. That is, we imagine that such an engine exists, and show that it would allow us to violate the Clausius statement.