I pulled out an Engineering Thermo book by Look and Sauer (PWS Engineering). They treat several cycles. One example problem deals with the Sterling efficiency. I'll use Carl's numbering system with the high temp, high pressure point as #1 and moving clockwise (the book starts at low temp/low pressure). The problem is to derive an expression for the eff. of the Stirling cycle. Also, using an ideal monotomic gas with c_v=(3/2)R
Their result is
RT_h ln (V2/V1)-RT_l ln(V3/V4) / (RT_h ln (V2/V1 + 1.5R(T_h-T_l)
which with Carl's examples of temp and volume changes yields 0.2401.
However!! Following this result they state: "If it were possible to use the heat rejected in the process [2-3, the isochoric decompression] to be added to the cycle during process [4-1, the isochoric compression] in a reversible fashion (this is called regeneration), the resulting efficiency would be ... 1-T_l/T_h. Note that this efficiency is equal to that of a Carnot engine."
So my guess is that Carl's analysis is irreversible at some juncture, namely, the heat rejected to the cold reservoir in the isochoric decompression.
-> -----Original Message-----
-> From: Phys-l [mailto:phys-l-bounces@www.phys-l.org] On Behalf Of John
-> Denker
-> Sent: Tuesday, March 03, 2015 2:54 PM
-> To: Phys-L@Phys-L.org
-> Subject: Re: [Phys-L] Carnot (?) efficiency of non-Carnot cycles
->
-> 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.
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