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The Stirling engine

We will calculate the efficiency of a different type of cyclic reversible thermodynamic process called the Stirling cycle. We also have at hand demonstration of such a Stirling engine which we will test after having gone through the calculation.

The Stirling cycle has the following steps:

  1. Isothermal expansion from tex2html_wrap_inline271 to tex2html_wrap_inline273 at temperature tex2html_wrap_inline253.
  2. Isochoric lowering of temperature from tex2html_wrap_inline253 to tex2html_wrap_inline257 at volume .
  3. Isothermal contraction from tex2html_wrap_inline273 to tex2html_wrap_inline271 at temperature tex2html_wrap_inline257.
  4. Isochoric increase of temperature from tex2html_wrap_inline257 to tex2html_wrap_inline253

We first calculate the amount of work done. Using the result previously derived for the isothermal process we can immediately write
equation109
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
equation117
We can then derive the efficiency of the engine for converting heat into mechanical work:
eqnarray121
where we defined tex2html_wrap_inline291 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 (Note that a previous version of this document had a misprint in this formula which was repeated on the blackboard in class). 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.

After understanding the mathematical description of the Stirling cycle we explain and test an actual Stirling engine.


next up previous
Next: About this document Up: Efficiency of reversible cyclic Previous: The Carnot cycle

Collin Broholm
Mon Dec 8 01:33:45 EST 1997