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



On 03/03/2015 02:51 PM, Herbert Schulz wrote:
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's not the accepted definition of what a
Carnot cycle is. A Stirling cycle is dramatically
different from a Carnot cycle, yet it uses a single
heat source at T(hot) and a single heat sink at
T(cold).

A Carnot cycle is a rectangle in (T, S) space.
A Stirling cycle is a rectangle in (T, V) space.

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.

You could draw such things. However that's got
little to do with the question that was asked.

The Carnot efficiency formula involves two temperatures,
namely the temperature of /the/ heat source and /the/
heat sink. If you have N different sources and M
different sinks, you can model it with NxM different
heat engines ... and yes, the efficiency of the most
efficient one is an upper bound on the efficiency
of the whole mess, and obviously so ... but again
that's got precious little to do with the question
that was asked.

The point remains that a Stirling cycle is quite
different from a Carnot cycle. It is not however
an "arbitrary" cycle.
a) Like a Carnot cycle, it is reversible.
b) Also like a Carnot cycle, it has a single
T(hot) and a single T(cold).

All heat engines that meet these two conditions (a)
and (b) have exactly the same efficiency. Otherwise
you could hook them in tandem and make a perpetual
motion machine.

This is the glory of classical thermodynamics. It
is one of the most elegant and most useful ideas
in the history of the world.

===================

It must be emphasized that all this applies to
_heat engines_ and not otherwise. In particular,
consider an electrochemical fuel cell. If it's
done right, the efficiency of such a thing vastly
exceeds 1 - Tc/Th. That's OK, because it's not
a heat engine. Even though it takes in fuel and
puts out work, it is not a heat engine.

As an even more familiar example, consider a plain
old electrochemical battery driving an electric
motor. If it's done right, the efficiency of such
a thing vastly exceeds 1 - Tc/Th.

It is not worth the trouble to try to understand
such things in terms of heat, or in terms of
classical thermodynamics generally. I doubt it
is even possible.

It is however straightforward to understand such
things in terms of modern (post-1898) thermodynamics,
i.e. statistical mechanics. Rather than getting
bogged down trying to define "heat", formalize
everything in terms of energy and entropy instead.
Entropy is defined in terms of probability, as a
sum over states. In a battery, the state that
stores the energy doesn't even have a temperature,
and it isn't in equilibrium with your thermometer.

================

A Carnot cycle is a rectangle in (T, S) space.
A Stirling cycle is a rectangle in (T, V) space.
An Otto cycle is a rectangle in (V, S) space.
A Rankine cycle has no simple picture AFAIK.

Nobody in the real world builds ideal Carnot
engines or Stirling engines. The Stirling-ish
engines of commerce don't even try to keep
the constant-V legs constant; there's no
operational advantage. (There is at least in
theory an advantage to operating at constant
high source temperature and constant low sink
temperature.)