Do All Engines Have the Same Efficiency as Carnot Engines?

[annetjelie]

My book states:

"To prove that no real heat engine operating between two reservoirs is more efficient than a carnot engine between the same reservoirs, imagine a more efficient engine to drive a less efficient carnot refrigerator. For the combination of the engine and refrigerator you get a net transfer of energy from the cold to the hot reservoir without work being done on the combination, which is a violation of the Clausius statement."

But what if i combine two *real* engines in the same way? This violates the Clausius statement too.
Does that mean that *any*(real or reversible) two or more engines between the same reservoirs have the same efficiency? I thought that was only true for Carnot engines.

help would be much appreciated, thanks in advance!

 

[Andrew Mason]

My book states:

"To prove that no real heat engine operating between two reservoirs is
more efficient than a carnot engine between the same reservoirs, imagine
a more efficient engine to drive a less efficient carnot refrigerator.
For the combination of the engine and refrigerator you get a net
transfer of energy from the cold to the hot reservoir without work being
done on the combination, which is a violation of the Clausius statement."

But what if i combine two *real* engines in the same way? This violates
the Clausius statement too.
Does that mean that *any*(real or reversible) two or more engines
between the same reservoirs have the same efficiency? I thought that was
only true for Carnot engines.

The quoted statement only leads to this conclusion if the refrigeration cycle is reversible. Here is the explanation:

Since the Carnot refrigerator cycle is reversible, after it delivers heat from the cold to the hot reservoir it can return to its original state by reversing the cycle and using the heat delivered to the hot reservoir to produce the same amount of work that was used in the cooling cycle (ie a Carnot engine). If that energy is stored, (eg. by raising a weight or stretching a spring) the cooling cycle can be repeated followed again by the engine cycle etc.ad infinitum.

But suppose the engine cycle (which takes the heat and produces work, which is stored and used to drive the Carnot refrigeration cycle) is more efficient than the Carnot cycle. This means that it produces MORE work with the heat Qh than the refrigerator cycle needs to deliver that Qh back to the hot reservoir. If the refrigerator then uses all of the engine's work output, it would deliver MORE heat than Qh from the cold reservoir back to the hot reservoir. This means the hot register would get hotter and the cold reservoir colder using no net energy (ie using the same energy over and over). And this violates the second law.

AM