The Stirling engine

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:

1. Isothermal expansion from to at temperature .
2. Isochoric lowering of temperature from to at volume .
3. Isothermal contraction from to at temperature .
4. 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.