Theory 4 - Optimal buffer pressure of the ideal Stirling Cycle

Ideal Stirling cycle

The ideal stirling cycle is composed of two isotherms (at T_h and T_c) and two isochore transformations (at V_min and V_max).

Two important parameters are:

• The temperature ratio: τ = T_c / T_h
• The volume (compression) ratio: r = V_max / V_min

Type of ideal cycles

For ideal Stirlings they are two main classes of cycles:

1) the efficacious cylce

For a given range of buffer pressures, the whole cycle can be performed without any work taken back from the energy buffer (flywheel) to the mechanism (W_- = 0). This case occurs when τ*r < 1

2) the non-efficacious cylce

For any buffer pressure the energy buffer has to give back some work to the mechanism during some part of the cycle (W_- > 0).

This case occurs when τ*r > 1

Optimal buffer pressure

For the efficacious cycles the optimal pressure is any value within the range where W_- = 0

According the the mechanical efficiency E_meca defined previously we get

E_{meca_efficacious} =  E

For the non-efficacious cycle we can define an optimal buffer pressure resulting in the highest overall work production. We can show that this pressure corresponds merely to the mean cycle pressure. Fortunately, due to leaks, a steady stirling will adjust its internal mean cycle pressure to the buffer pressure, i.e. it will adjust itself to the optimal working conditions depending on the buffer pressure.

Using the law of ideal gas we can deduce the parameter W_-/W for any optimally buffered stirling cycle:

Here are some curves showing W_-/W versus compression ratio r for various temperature ratio τ

With these curves of W_-/W and the general mechanical efficiency equation (see previous chapter) we can get the mechanical efficiency of an non-efficacious ideal engine  with optimal buffer pressure:

Following graph shows the result for a constant mechanism effectiveness E = 0.7: