# Efficiency of Carnot engine [duplicate]

Why is the efficiency of Carnot engine regarded as the maximum efficiency possible for a heat engine?

• – PM 2Ring Sep 7 '19 at 12:01
• – PM 2Ring Sep 7 '19 at 12:04

Because you can prove that if there were a more efficient engine then it could be used, in conjunction with a Carnot engine, to bring about a net result that breaks the 2nd law of thermodynamics. One could, for example, transfer heat from one body to a hotter body without requiring any net work to be done.

The proof is based on making a heat engine using the impossibly efficient one. It can be found in any basic thermodynamics textbook. I would guess it's in wikipedia too.

There are several statements of the second law that would be violated by a heat engine being more efficient than a Carnot engine. @Andrew Steene mentioned transferring heat from a cold body to a hot body without requiring any net work to be done. That would violate the Clausius' statement of the second law, or:

No refrigerator or heat pump cycle can operate without a net work input

COROLLARY: No refrigerator or heat pump can have a higher COP than a Carnot Cycle refrigerator or heat pump.

The example I give below applies the Kelvin-Plank statement of the second law:

No heat engine can operate in a cycle while transferring heat with a single heat reservoir.

COROLLARY: No heat engine can higher efficiency than a Carnot Cycle operating between the same reservoirs

The example described below illustrates violation of the Kelvin-Plank statement and its corollary.

Figure 1 below shows a Carnot heat engine whose work output runs a Carnot refrigerator or heat pump. Note that the work out of the heat engine equals the work into the refrigerator. Also the heat transfers into and out of the two temperature reservoirs are equal. The overall effect is that there is no net work done and no net heat transferred.

Refer now to Figure 2. We replace the Carnot heat engine with a "super efficient" engine, that is, an engine more efficient that the Carnot heat engine, and use it to operate the Carnot refrigerator. To be more efficient than the Carnot engine, our super efficient engine will produce more work out for a given heat in $$Q_H$$. We will call the total work output $$W + dW$$. Since more of the heat into the engine produces work, the engine therefore rejects less heat to the low temperature reservoir. We will call the heat rejected $$Q_{L}-dQ$$ where, for conservation of energy, $$dQ=dW$$. Part of this work, $$W$$, is used to operate the refrigerator, as in the case of Fig 1. The rest, $$dW$$, is available to be used for some other purpose.

We see that our super efficient heat engine still complies with the first law. But let's look at the overall result of the super efficient engine operating the refrigerator.

1. The heat removed from the high temperature reservoir by the super efficient heat engine equals the heat put back by the Carnot refrigerator. That means there is no net heat transfer to the high temperature reservoir, $$T_H$$.

2. The net heat transfer to the low temperature reservoir, $$T_L$$, is $$-dQ$$, which means net heat is removed from low temperature reservoir.

3. The net work output of the combined devices is $$dW$$.

CONCLUSION: The combined effect of the supper efficient heat engine operating the Carnot refrigerator is to take heat $$dQ$$ from a single reservoir, $$T_L$$ and produce an equivalent amount of work, $$dW$$. This is in violation of the Kelvin-Plank statement of the second law. Therefore, no heat engine has a greater efficiency than the Carnot heat engine.

Hope this helps.