We will calculate the efficiency of a different type of cyclic reversible thermodynamic process called the Stirling cycle. We saw this machine in the previous lecture.

The Stirling cycle has the following steps:

- Isothermal expansion from to at temperature .
- Isochoric lowering of temperature from to at volume .
- Isothermal contraction from to at temperature .
- Isochoric increase of temperature from to

We first calculate the amount of work done. Using the result previously
derived for the isothermal process we can immediately write

As opposed to the Carnot cycle where heat only flows to the system in
isothermal processes heat flows to the working gas of a Stirling engine
both during the isochoric heating process and during the isothermal
expansion. For these processes we can write

We can then derive the efficiency of the engine for converting heat into
mechanical work:

where we defined as the efficiency of the corresponding
Carnot cycle and we used the equipartition theorem to
introduce the number of degrees of freedom of the
working substance. Thus we clearly see that the Stirling cycle is less
efficient than the Carnot cycle. The reduction in efficiency
results from the isochoric process where heat goes in but no work
gets done. The efficiency of the Stirling engine is greater
the fewer degrees of freedom per atom in the working substance,
so a machine based on a Mon-atomic gas would work better than one based on
a more complicated molecular gas. We also see that the larger the ratio of
volumes involved the close the Stirling process approximates the Carnot
cycle.

We can also readily calculate the efficiency of a heat pump operating as a Stirling
engine:

We see that the Stirling machine is actually more efficient than the Carnot cycle
()
as a heat pump and that the efficiency is enhanced for gases with more degrees
of freedom per mole and for the smallest ratio of volumes.

Similarly we calculate
the efficiency of a Stirling machine operating as a refrigerator:

Here we obtain a smaller efficiency than the Carnot cycle ()
and a requirement that
the degrees of freedom of the gas, *s*,
and the relative volume change be as large as possible.
Clearly a system which is to cool as well as heat cannot be optimally designed for
both of its chaws.

Mon Dec 8 11:41:49 EST 1997