How big should your Stirling engine power piston be?
(Be sure to see my more recent post on power piston sizing. It provides more specific information. — Doug Conner)
A common problem Stirling engine designers face is how much volume should the power piston sweep in comparison with the displacer? This of course only applies to a Beta or Gamma engine; an alpha engine designer faces a related problem of what phase angle to use between the hot and cold pistons. The answer for an ideal Stirling engine is that a larger piston will always give you more power. This isn’t true for a real Stirling engine.
The problem centers around the ideal isothermal compression and expansion cycles. In a real engine you can’t realize perfect isothermal compression. It’s probably safe to say that the higher the cycle rate and the more powerful the engine is, the more difficult it is to get close to isothermal compression.
A ready example of the difficulty in achieving isothermal compression is a typical shop air compressor. Compression cylinders normally have cooling fins that get very hot when the compressor is operating coninuously.The air inside the compression cylinder is even hotter than the cylinder fins. Those cooling fins are used to get as close to isothermal compression as practical because isothermal compression uses the least amount of power.
An ideal Stirling engine uses isothermal compression and expansion cycles. Concentrating for the moment on the compression cycle, the gas has moved through the regenerator and cooling tubes (if used) and into the cold end of the displacer cylinder before the compression cycle begins. The compression cycle (besides increasing pressure) adds heat uniformly to all of the working gas. Remember this is after the gas has gone through cooling in the regenerator and the cooling tubes. During the compression cycle the only components to help cool the gas are the cylinder walls and the piston. Isothermal compression is just not the reality for real engines.
On one hand we have isothermal compression as an ideal best case. On the other hand we have the corresponding ideal worst case of adiabatic compression. In the adiabatic or isentropic compression case, no heat leaves the gas during compression. This would be the case if you had all material contacting the gas be a perfect insulator. The isentropic compression case is computed using the equation:
T2/T1 = (V1/V)^(k-1)
where k = cp/cv (about 1.4 for air at typical temperatures)
T1, T2 are beginning and ending absolute temperatures
V1/V2 is volume ratio (maximum volume to minimum volume)
If you have a volume of air at 0 degrees C (273K) and compress it down to half the volume, the air would double in pressure for isothermal compression and of course stay at 0 degrees C. For isentropic compression the temperature would increase to about 87 degrees C. Because the temperature is higher in the isentropic case, the pressure would be more than double the initial pressure.
Real world compression gives you a result somewhere between the isothermal and isentropic cases. Machinery’s Handbook (27th edition) suggests air compressors typically operate about midway between isothermal and adiabatic compression. A Stirling engine may perform better or worse than that depending on its design and cycle rate.
Also note that everything I’ve covered here that applies to the compression cycle applies equally to the expansion cycle. The only difference is that during expansion while the gas is doing work pushing the power piston, you’d like the gas to remain isothermal for the most power, but the gas will be cooling due to the expansion.
In my simulator, I include an output labeled isentropic compression heating that is listed as a percentage. This value is calculated using the engine volume ratio and assumes completely isentropic compression and expansion. What this means is that if you have a delta T (=Th – Tc) of 100 degrees and you see 25% for the isentropic compression heating, then 25% of the temperature swing (25 degrees) would be caused by compression heating and expansion cooling. The power output value listed by the simulator is based on strictly isothermal compression and expansion. To correct for the isentropic compression heating and expansion cooling losses, I subtract the 25% value (or whatever the simulator shows) from the power output value to get a more accurate value of the power output. This result approximates adiabatic rather than isothermal compression and expansion. Using this corrected value gives a conservative estimate although you might want to use something midway between isothermal and adiabatic if you have a reason to be optimistic.
To show how the results are affected, the plot below shows the power output based on both isothermal and adiabatic compression and expansion as the power piston varies in size and alters the % compression heating on my engine 3F.
You might be tempted to design your engine for isentropic compression heating values of up to 40% or even higher because the plot even for the worst case adiabatic compression has not reached a peak value. In reality that may not give you the most power. Typically as you increase the piston diameter or stroke to increase the compression, two things happen.
First, you’ll pick up more friction with the larger piston or longer stroke. As you approach the nearly level part of the power curve, you may end up with less net power due to friction.
Second, the higher the compression, the more likely you’ll experience power loss due to gas leaks past the piston. Gas leaks past the piston (or leaks anywhere) reduce the pressure ratio and will reduce engine power even at low compression. The losses become greater as the compression increases.
Once you have an operating engine you can determine the lowest temperature ratio the engine will operate at. This will give you more insight into how close you are to the adiabatic curve vs the isothermal. Of course the engine must overcome all sources of friction to run, so you aren’t seeing the real zero-power level, but you’ll get an idea. Another problem is that the effect of any gas leaks is magnified when the engine is operated at a minimum rpm condition. Yet another problem is actually measuring the hot and cold gas temperatures in the displacer. I end up measuring the external temperature of the displacer cylinder which is not an accurate measure of the gas temperature. For the engines of mine that I’ve tested, the minimum operating condition seems to be in the 30% to 50% compression heating region (based on external temperatures).
You can’t just blindly assume some ratio of piston diameter to displacer diameter, even for engines operating at the same temperature ratio. The dead volume can have a huge effect on the volume ratio and thus the appropriate piston diameter. A high temperature engine can easily work with a power piston displacement as large or even larger than the displacer swept volume while a low temperature engine may need a power piston displacement that is a small fraction of the displacer swept volume.