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Re: [Phys-L] Carnot (?) efficiency of non-Carnot cycles




On Mar 3, 2015, at 2:53 PM, John Denker <jsd@av8n.com> wrote:

On 03/03/2015 01:28 PM, Herbert Schulz wrote:

You can easily use the method you mentioned (breaking up an
non-Carnot engine in differentially small Carnot engines) to prove
that the efficiency of that engine MUST be lower than a Carnot engine
acting between the same T(cold) and T(hot).

I beg to differ.

The efficiency of any *reversible* heat engine operating
between a given Tc and Th *must* be the same as the
efficiency of any other reversible heat engine operating
between the same two temperatures.

One-line proof: Otherwise you could hook the two of
them in tandem and have a perpetual motion machine.

This concept is spelled out in more detail in Feynman
and probably a hundred other places. See figure 44-7,
and maybe 44-8 also.
http://www.feynmanlectures.caltech.edu/I_44.html

Secondly, we know that this efficiency has to be.

Carnot's epochal book used both of these arguments:
He proved that if you know the efficiency of one
reversible heat engine you know them all ... and
then he used a particularly convenient cycle to
calculate what this efficiency has to be.

The Stirling cycle is infamously inconvenient to
calculate, but it can be done, and sure enough the
efficiency is the same as any other reversible heat
engine. The Stirling cycle is conspicuously different
from the Carnot cycle, but the efficiency is the same.

Howdy,

The interesting thing about the engines shown is that all the heat in occurs at T(hot) and all the heat out is at T(cold). So, all those cycles are Carnot cyclaes. That is NOT true of most reversible cycles.

Make a P-V diagram for an arbitrary, reversible cycle. At the point of highest temperature, T(hot), draw a isotherm; similarly at T(cold), the lowest temperature of operation. Then draw adiabatics so they are tangential to the cycle at the right and left. That will give you a Carnot cycle between those same two temperatures. Break up that Carnot cycle into many small (micro)cycles and you will see that each of the sub-cycles cut through the original cycle and you can make a (micro)cycle within the original cycle (i.e., the original cycle can be broken up into many (micro)cycles that approximate tiny Carnot cycles). However the T(hot)_i of the (micro)cycle is less (or at best equal to) than for the enveloping Carnot cycle and the T(cold)_i of the (micro)cycle is greater than (or at best equal to) that of the enveloping Carnot cycle so its efficiency is less than that of the enveloping Carnot cycle operating between the same T(hot) and T(cold).

Good Luck,

Herb Schulz
(herbs at wideopenwest dot com)