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[AAMAIK]

Carnot's postulate: one cannot build an engine whose sole effect is to transfer heat from a cold body to a hot body. How granted this postulate can I prove that no engine beats Carnot's engine?
From this postulate, I can conclude that work must be done from the surroundings and that Carnot's hypothetical device can run as both an engine/refrigerator.
I have a Carnot engine which takes QH calories, delivers W1 calories in work, and rejects QL1. Someone claims to have an engine with better efficiency than that of Carnot's engine, this implies that for the same heat input
W2>W1 and QL2<QL1
If I run Carnot's' engine backward, the refrigerator requires W1 calories of work input, to reject QH Calories to the high-temperature reservoir. The two devices are connected such that the work input to Carnot's refrigerator is satisfied. But in doing so I am violating the first law If I define my system to be Carnot's' refrigerator because I am supplying more work than it needs. Also If I define my system to be the two devices altogether excluding the temperature I will violate both the 1st law of thermodynamics and the postulate.
So is my analysis generic and correct?

 

[Andrew Mason]

Hi AAMAIK

If I understand your question correctly, you want to prove that no heat engine operating between two reservoirs at temperatures T_h > T_c respectively, will perform more efficiently than a Carnot engine relying only on the premise that the Carnot's postulate is correct: that you cannot build an engine whose sole effect is to transfer heat from a cold body to a hot body.

IF I follow your answer correctly, you are suggesting doing this:

Assume you have a heat engine that is more efficient than a Carnot. You run it for one (or several) cycle(s) and store the work as, say, gravitational potential energy and then use that energy to perform work on a Carnot refrigerator between the same reservoirs (i.e. Carnot engine cycle run backward).

Since your engine it is more efficient than the Carnot engine, this means that: for the same amount of heat flow from the hot to the cold reservoir, you have produced more work and, therefore, have stored more gravitational potential energy than a Carnot engine would have produced.

Since a Carnot refrigerator would be able to return that same amount of heat flow back to the hot reservoir using only the energy that a Carnot engine would have produced (which is less than the amount that your engine produced), you will be able to transfer more heat flow back to the hot reservoir than flowed out of the hot reservoir in your heat engine's forward cycle(s). So, the result is a device (combination) whose sole effect is to cause a net flow of heat from cold to hot, which violates your premise.

In other words, if your engine was more efficient that a Carnot engine you could have it running a Carnot refrigerator between the same two reservoirs with the net result that more heat would be transferred from the cold to the hot reservoir than your engine transfers from the hot to the cold reservoir.

If that is what you are saying, then your proof would be correct.

AM