A quick way to estimate the power of a Stirling Engine is with the Beale formula:
Where:
W is the power output (W)
Bn is the Beale number which is generally in the range 0.11 – 0.15 (ref: http://en.wikipedia.org/wiki/Beale_number)
P is the average engine pressure (Pa) over one cycle
V is the swept volume of the expansion space (m3)
ƒ is the engine speed (Hz)
The figure below shows a comparison between predicted power versus engine speed, using the Beale formula and a numerical model I created in a spreadsheet. It also shows the predicted engine torque versus engine speed, using the same numerical model.
The Beale formula for power is a straight line, whereas the numerical model for power is a curve that peaks at around 25 Hz and then falls sharply. This indicates that engine power is a function of speed, which makes sense since pumping (flow) losses inside the engine increase with increasing speed until a maximum is reached, and then the power falls. This physical loss is modeled in the spreadsheet model I used.
The power curve shown applies to air as the working gas. If a working gas such as hydrogen is used, the maximum power would occur at a higher speed. It is well known that hydrogen as the working gas results in greater engine power than using air.
The engine torque curve (shown in red) gives insight into how the torque changes as the speed increases. It clearly shows that the torque of a Stirling Engine is highest at start-up. The highest engine torque is at lower speeds, which can be useful depending on the application.
The Beale formula best matches the numerical curve up to about 20 Hz. The values used in the Beale formula are:
Bn = 0.2
P = 555000 Pa
V = 0.000055 m3
P and V also correspond to the values in the numerical model. Bn is chosen to best fit the numerical model.
Other points to make regarding the numerical model are:
• the swept volume in the expansion and compression space are equal
• the hot engine temperature is 400 degrees Celsius and the cold engine temperature is 20 degrees Celsius
• for maximum power, the optimum phase difference between the hot and cold volume space is 90 degrees
• the average engine pressure increases only a little bit with temperature increase. For example, the average pressure inside an engine operating at a hot side temperature of 300 degrees Celsius is only a bit lower than the pressure of the same engine operating at 800 degrees Celsius
If the engine temperature changes, then the Beale number Bn should also change to reflect the change in engine power. As a rough estimate:
Bnnew = (Cnew/Cold)xBnold
Where:
Cnew = (1-Tcold/Thot)new
Cold = (1-Tcold/Thot)old
The temperatures are in Kelvin. Note that (Cnew/Cold) is the ratio of Carnot efficiency at the new and old temperatures.
The Beale formula is a good estimate for engine power at lower engine speeds, perhaps up to 15-20 Hz, and doesn’t apply past the speed where the actual engine power peaks.
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