Power piston sizing for Stirling engines

One of the more difficult yet critical issues in designing your first Stirling engine is deciding on the correct power piston displacement. My earlier discussion of power piston sizing did not satisfy a lot of people. So I’m going to make this one very simple to apply. For me to suggest the right size power piston displacement to match the displacer, I only need to know the absolute temperature ratio of your engine’s hot and cold ends and the minimum gas volume of your engine.

Absolute temperature ratio

Specifically, the absolute temperature ratio is the temperature ratio of the hot end divided by the temperature of the cold end using temperature measured on an absolute scale. That means either the Kelvin scale (Kelvin = degrees Celsius + 273) or the Rankine scale (Rankine = degrees F + 460). Your temperature ratio will be larger than 1 and less than 2.5 unless you are getting into seriously high or low temperatures. Keep in mind that if your engine will be air cooled, the cold end will be hotter (possibly much hotter) than the ambient air temperature. Almost all the heat you put into the engine will have to be rejected from the cold end.

For the above temperature ratios you don’t need the actual gas temperature, the temperature of the heat conducting surface that is heating or cooling the gas is required, because that is what I’ll be comparing it to.

Gas volume

You do need to know all the gas volume including the regenerator (if any) and any dead volume in hoses, clearance spaces around the displacer, clearance above the power piston, etc. This volume will be the minimum gas volume. It really is vital that you make an accurate accounting of all the gas volume. This will be all the working gas volume except the displacement volume of the power piston. You can use any measurement units you like because we’ll only be dealing with volume ratio.

Once you have the temperature ratio and the minimum gas volume, then you can use the following graph to make a reasonably good estimate of the volume ratio needed for your engine to operate. The engines marked on this plot are all gamma designs I have built and tested. The results should also be applicable to alpha and beta kinematic Stirling engine configurations.

Here’s a detail plot that amplifies the low end

The plots for the four different engines show a range of temperature ratios for each engine. The lowest temperature ratio is for sustained no-load operation. The highest temperature ratio is for a measured power output for each engine with the exception of the smallest engine. For that engine the high temperature ratio is just an arbitrary “good output”.

I just want to add an aside here. If I designed these engines with better heat transfer, those long horizontal lines covering each engine’s temperature ratio range would be shorter. The lines wouldn’t need to go so far to the right on the plot to develop good power.

How to use the chart

The way I would use the above chart is to first find your desired operating temperature ratio. Use either the temperature ratio for minimum operation or something closer to the power output temperature ratio. Then pick a volume ratio that looks appropriate based on the sample engine values I have charted. I would not expect operation above the 35% adiabatic compression line. You can see my low-temperature engines don’t even make the 25% line.

For example, If I want to build an engine that operates at a temperature ratio of 1.6 and doesn’t merely run but puts out some reasonable level of power at that temperature ratio, I would choose a volume ratio between 1.4 and 1.5. I would expect minimum operation around a temperature ratio of 1.4.

This plot will not help you estimate the power of the engine you build unless you build one with the same dimensions as the engines I’ve listed. Even then your engine could be much better or worse depending mostly on friction, the quality of the seals and compression, and your heat transfer design details. What it will do is give you a much better likelihood of an engine that runs.

I can just about guarantee if you choose a volume ratio above and to the left of the engines I have built that you will have an engine that doesn’t run unless you are able to push your temperature ratio higher. Novices often have the urge to use a larger power piston displacement to increase the compression in an effort to make the engine more powerful. I’ll explain in a minute why that won’t work.

If you use a volume ratio below and to the right of what I show as good power, you will probably be giving up power and may also end up with an engine that doesn’t run. A smaller power piston displacement than you need will make the engine less powerful. In small engines the reduced power may not be enough power to overcome the friction.

Once you have picked out the volume ratio to use on your engine, then you can size the power piston (for gamma and beta configurations) displacement volume.

The equation is:

Power piston displacement volume = (volume ratio – 1) x minimum volume

Remember that minimum volume is the number you determined earlier. The relationship between the displacement volume and the piston bore and stroke is:

displacement volume = pi x bore^2 x stroke/4

Those designing alpha type engine will need to do their own math, or perhaps I’ll post a future article specifically for alpha engines. Just note that on alpha engines you’ll need to adjust the phase angle between the two pistons to get the desired volume ratio.

For reference here is a the relevant data on the four engines used in the above plots:

The adiabatic curves

Why you can’t make power pistons arbitrarily large

Stirling engines ideally are reversible. Instead of heating and cooling the appropriate ends to make an engine run you can turn the engine with a motor and the ends of the displacer will get hot and cold. This works with real engines too. You can actually turn even a low-temperature engine with a motor and measure a few degrees of heating and cooling.

If you operate your engine with too much compression for the temperature ratio you’re using, you won’t measure any power out, you’ll have to put power in because it’s still operating as a heat pump, not an engine.

Ideal Stirling engines use isothermal compression and expansion. They also use constant volume heating and cooling. Real Stirling engines use a sloppy approximation of these processes. Compression and expansion have a significant adiabatic (isentropic) component. Heating and cooling happen while volumes are changing.

When you take a fixed volume of gas and compress it while insulating it so that no heat transfer takes place, adiabatic heating occurs. In a Stirling engine we don’t want adiabatic heating, we want isothermal heating. Unfortunately we get quite a bit of adiabatic heating in a typical real Stirling engine.

An ideal Stirling engine could operate above and to the left of the 100% adiabatic heating curve. I doubt if any real Stirling engines do so. I use percentage of adiabatic adiabatic heating as a way to help me gauge how efficient a design is working.

The curves in the above plot correspond with different percentages of adiabatic compression that result in the specified volume and temperature ratios. The plot is computed for normal air. I could make plots corresponding to other gases, but I think anyone building a Stirling engine using helium should be way beyond piston sizing problems.