Power output measurement of the PE 2 Stirling engine

This article covers power measurements for the PE 2 3D printed Stirling engine. The net output power (the useful power) plus the frictional losses measured earlier are combined with other test data and simulation results to more fully understand this engine’s performance. The techniques and data should be useful for others designing Stirling engines, especially for obtaining realistic simulation results.

Engine power test setup

Figure 1. PE 2 engine test setup for measuring output power.

Figure 1 shows the PE 2 engine test setup for measuring output power. The data acquisition system records the engine speed plus hot plate, cold plate, and ambient temperatures. The motor provides the load by either operating as a generator driven by the engine, or by energizing the motor as a brake to resist engine rotation. The load of the generator was characterized using the same method that was used to measure the friction of the engine components.

There is one difference in computing the rotational inertial of the motor compared with other engine parts. Because the motor is driven at a higher RPM than the engine RPM (due to the difference in the engine hub and motor wheel diameters), the rotational inertia computation is done differently. For this case the rotational inertia of the generator components (including the wheel and O-ring) must be multiplied by:  (motor-generator RPM/engine RPM)2.

Calibration tests

The engine was tested with the motor-generator in multiple configurations. The first is as shown in Figure 1 with the generator leads disconnected, providing minimal load. This configuration is listed as MHO (= Motor to Hub Open circuit). For the second configuration the generator leads were shorted to provide a higher load (MHS = Motor to Hub Short circuit). The calibration results of these two tests are shown in Figure 2A.

motor-generator friction1

Figure 2A. Calibration test for generator friction.

The calibration tests shown in Figure 2A test each of three configurations twice. The first is the engine configured with just the flywheel spinning on the crankshaft (FW). The second test adds the MHO configuration to the flywheel (FW+MHO) and the third test has the generator leads shorted (FW+MHS). The equations used to to approximate the test results are shown. By subtracting the FW-only equation from the two generator configurations, I can compute the engine’s power output at engine speeds up to 6 Hz.

After testing the engine with the above calibrations, I wanted to increase the loading to test the engine power at lower speed. To do this I added additional loading to the engine by using the motor to provide a braking torque. Figure 2B shows the calibration runs for these two tests.

Calibration for motor load

Figure 2B. Calibration test for generator friction.

The motor was driven to resist engine rotation with 0.3Vdc and 0.4Vdc. These levels were  determined by testing the engine first to see what voltage levels would provide the desired loading, then I did the calibration runs afterward.

Power output tests

The results of testing the engine engine power are shown in Figure 3.
net power vs speed

Figure 3. Engine net power output vs rotational speed

The MHO and MHS configurations were tested over a range of engine speed. When using the motor to apply braking torque to the engine, I only checked specific power levels.
Figure 3 shows the engine running in the MHO configuration will spin up to a higher speed than the other configurations, but because the generator is not loading the engine effectively, the actual power output is low. When the generator is shorted (MHS) the engine is loaded up more and isn’t able to turn as fast, but has a greater power output.

Loading the engine further by using braking torque from the motor produced about the same power  as the MHS configuration but at lower engine speed.

net power vs Tratio

Figure 4. Engine power output vs temperature ratio.

The power output vs temperature ratio for the MHO and MHS configurations is also almost perfectly linear above a certain temperature ratio for a given load. I’ve only plotted temperature ratios above that point to get a useful linear curve fit for the data.

The maximum engine output power measured in these tests was about 14 mW. It is possible that further testing might show higher power, but probably not more than 16 mW. This is not impressive and much less than you might be led to expect if you use an engine simulator such and the one I provide. The problem is that when you use a simulator there are a lot of sources of power loss that you may not be taking into account.

Using the results from the information above plus the engine friction measurements in a previous article, I have created a table with some useful data to understand this engine.

Table 1. Engine data from four test configurations.

 max output power MHOmax ouput power MHSMH + .3VdcMH + .4Vdc
Speed, Hz5.
Temperature ratio1.16401.16651.1681.165
Net power output, mW8.312.913.412.4
Engine Friction @ speed29.822.015.612.8
Gross engine power, mW38.134.929.025.2
Output power as a % of gross engine power22374649

Table 1 uses the previously determined engine friction vs speed to provide the gross engine output power. The gross power is the sum of useful output power plus the frictional sources inside the engine. The gross power probably isn’t useful for anything except for analyzing the engine’s performance. The table shows two interesting pieces of information.

First, the gross power of the MHO configuration is the highest of the configurations tested although it has the lowest output power. This isn’t surprising because the MHO configuration test has the engine running at the highest speed and approximately the same temperature ratio as the other tests. What this means is that although the engine is producing more power, most of the power is being used to overcome various sources of engine friction rather than resulting in useful output power.

Second, net engine output power as a percentage of the gross power never goes above 49% and is as low as 22% at 5.02 Hz. At least half the engine’s gross power is being used to overcoming various source of friction.

Determining the Stirling engine gas temperature ratios

When using a simulator to predict engine performance, the simulator requires the average hot gas temperature Th and the average cold gas temperature Tc to accurately predict the engine performance. Without an accurate heat transfer model of the engine, one is usually forced to making an estimate of Th and Tc based on the the hot and cold plate temperatures. Figure 5 shows a section view of the PE 2 engine with the gas and plate regions labeled.  Using the gross power output of this engine I now have the opportunity to work backwards to compute what the average hot and cold gas temperatures are based on the power output of the engine.

plate and gas figure

Figure 5. Section view of engine showing plate and gas regions.

I’ll be using the four engine configurations in Table 1 for the computations. The simulation dimensions I use for the PE 2 engine configuration are:

  • Power piston dia x stroke = 22mm x 9.5mm
  • Displacer dia x stroke = 74.4mm x 12.7mm
  • Dead volumes: hot, cold, regenerator = 12.9, 13.6, 35.7 cm3


Table 2 shows the results of comparing simulated and measured values.

Table 2. Deriving gas temperatures from measured engine performance

configuration No loadMHOMHSMH + .3VdcMH + .4Vdc
MeasuredSpeed, Hz6.605.
MeasuredTemperature ratio (plate temperatures)1.161.1491.1551.1681.165
Measuredgross power, mw44.238352925.2
simulatedTemperature ratio required for gross power (gas temperatures)1.0591.0671.0771.0791.079
comparison(Tratio of gas – 1)/ (Tratio of plate – 1).369.409.444.470.479

I used the following process to create Table 2. First the measured values (from Table 1) are shown in the first three lines of the table data. I also added a new column for the engine running with no load (useful power is zero).  The temperature measurements I make on my engines are of the hot and cold plate temperatures. I use them because they are easily measured and repeatable. What I’d really like to know is the average hot gas temperature and the average cold gas temperature.

Assuming that the simulator can accurately predict the power output of the engine given accurate average gas temperatures, I work backward to find the temperature ratio of the gas that will provide the gross power output I measured on the engine. So in the engine column for the MHS configuration, I’m looking for the temperature ratio that will result in 35 mW gross output power.

The simulator I’m using is quite similar, but not identical to the one I’ve provided online. In addition to the Stirling cycle (ideal isothermal expansion) power output provided online, I also have an output approximately corrected for adiabatic expansion. I’ve discussed this issue in my article on power piston sizing and I’ll cover this in more detail in the future. For this application I’m using the power output mid-way between isothermal and adiabatic expansion. The results will be slightly different from what you might see in the online simulator. The ideal isothermal expansion will show a lower temperature ratio than my results.

Neither the simulator online, nor the one I am using, lets me directly enter power and determine the temperature ratio. Instead I have to test Th and Tc values until I zero in on the temperature ratio that gives the measured power output.

For the last row in Table 2 I’ve compared the ratio of engine gas temperatures with the engine plate temperatures. You can see that the gas has less than half the temperature ratio (.37 to .48) of the plate temperature ratio for the configurations I tested.

Although this engine has a reasonably good regenerator, it has poor heat transfer design for heating or cooling the gas to the temperature of the hot and cold plates. Better heat transfer design would help increase the ratio of gas temperature to plate temperature. On the other hand, engines with neither good heat transfer design nor a regenerator will have an even lower gas temperature swing.

Test and analysis results

I’ve compiled what I think is some of the most important information from the above testing and analysis into the following three figures.

power vs speed

Figure 6. Power vs Speed for the PE 2 engine (gross, friction losses, and net output).

The plot of engine friction and net power in Figure 6 show the two components are about equal at 2.7 Hz. Increasing engine speed increases the friction until it consumes the gross engine power at 6.6 Hz.

torque vs speed

Figure 7. Torque vs speed for the PE 2 engine (gross,and net output).

The gross engine torque shown in Figure 7 will theoretically reach a maximum at zero engine speed when the gas temperature Th reaches its maximum value and Tc reaches its minimum value. These temperatures probably will not be the plate temperatures but some weighted average of all the wall temperatures surrounding the gas.

In a real engine such as this, with imperfect seals,  gas leaks will reduce the torque at very low speed so that peak torque will occur before the engine speed reaches zero. The gross torque may be approaching its peak around 2.7 Hz. It would require more data to be sure.

temperature ratio vs speed

Figure 8. Comparison of gas and plate temperature ratios.

The comparison of gas and plate temperature ratios in Figure 8 show the expected trend of decreasing gas temperature swing as engine speed increases. Higher engine speed provides less time for the gas to heat and cool. The temperature ratio is not dropping too quickly with increasing speed, probably due to the regenerator. Better heat transfer between the operating gas and the hot and cold plates should lift the curve higher and probably provide a slower drop with increasing engine speed.


For an engine configuration, size, and operating temperature range similar to the one I used, you might expect to lose between half and 3/4 of the engine power to frictional losses. On top of that you probably can’t expect to see the gas temperature swings to be more than half the hot and cold plate temperature difference unless you have designed in better heat transfer between your hot and cold plates and the operating gas plus an effective regenerator.

Peak power for the PE 2 engine occurs around 4 Hz (240 rpm) operating speed.