# Stirling Engine Simulator & Guide

The Stirling engine simulator is a simple isothermal cycle simulator for Stirling engines. The guide below should be helpful to those interested in using the simulator. Although it’s a fairly simple simulator, you can learn a lot about Stirling engine design by trying out different designs and operating conditions.

#### What is it?

The Stirling engine simulator is an isothermal cycle simulator for Beta or Gamma Stirling engine configurations. If you aren’t familiar with Beta and Gamma configurations see Stirlilng Engines (Wikipedia). These engines use a displacer to force the operating gas back and forth between hot and cold regions and a power piston to extract work. To use the simulator you enter the key dimensions, dead volumes, hot and cold gas temperatures, and average system pressure. The simulator also accepts inputs for the ideal gas constant for using gases other than air, and lets you specify a cycle rate to generate power output.

This simulator performs simple thermodynamic calculations for ideal gases but does not perform heat transfer computations. That means the simulator can compute the gas pressure versus crankshaft angle if you supply the configuration and temperature information. The simulator can also compute the work done per engine cycle (one revolution). Heat transfer computations would determine the rate at which the gas changes temperature. Those computations are both extremely complex and would require a huge amount of engine design detail.

The simulator can only determine power output at a specific RPM if you specify the hot and cold gas temperatures and the cycle rate. Low cycle rates will give the most reliable information. As cycle rate goes up, heat transfer limitations will decrease the hot gas temperature and increase the cold gas temperature. Those temperature changes are highly dependent on the heat transfer characteristics of your design and will probably be difficult for you to compute or even estimate with any accuracy. Unfortunately, If you want to get accurate results from the simulator, you will have to know the average hot volume gas temperature and the average cold volume gas temperature versus cycle rate.

### Simulator assumptions:

1. The simulator assumes ideal gas characteristics and computes the pressure in the engine based on the volume and temperature changes over one cycle.

2. At any instant the gas pressure is uniform in all regions. This assumption is not satisfied in real engines except when operating at very low speed. To use the simulator properly you need to enter the average hot volume and cold volume temperatures. At higher engine speeds the hot gas temperature will decrease and the cold gas temperature will increase. If you aren’t able to supply this information correctly, this simulator will indicate higher power than the real engine can produce. This information is still useful by providing an upper limit on the possible power output of the engine using the specified values.

3. Related to (2) above, the simulator does not take into account friction of any kind including gas movement, motion of shafts, piston, displacer, and associated seals. You will have to make your own allowances for these friction sources and reduce the work and power outputs accordingly.

4. The simulator assumes the engine is not limited by heat transfer. You control the heat transfer assumptions by the temperatures and cycle rates you input. Producing power from engines at high cycle rates is very dependent on good heat transfer design.

5. Gas in the hot end of the displacer cylinder is always assumed to be at the hot gas temperature. Gas in the cold end of the displacer cylinder and in the power piston cylinder will be at the cold gas temperature. Dead volumes are as you have assigned them. More on dead volumes later.

6. The temperature in the regenerator is computed as Th-Tc/(ln(Th/Tc)) where Th and Tc are in Kelvin. The computed value is close to the average of the hot and cold temperature (Th + Tc)/2, but corrects for the decrease in density of the gas at elevated temperatures.

7. The phase angle between the displacer and the power piston is fixed at 90 degrees.

8. This simulator assumes an ideal gas. At high pressures (over 15 Atm) this simulator will be less accurate but should still be adequate for the estimation purposes for which it is intended.

### Simulation Inputs

Power Piston – The first pair of inputs is for the bore and stroke of the power piston in mm.

Displacer – The second pair of inputs is for the diameter and stroke of the displacer. If there is a significant clearance between the outside diameter of the displacer and the inside diameter of the displacer cylinder as is sometimes the case on simple engines, use the diameter of the displacer and account for the annular volume between the displacer and displacer cylinder as dead volume.

Dead Volumes – The next three inputs are dead volumes. In a Stirling engine the operating gas never leaves the engine but is compressed or expanded by the power piston motion and moved around by the displacer. Any volumes in the engine that are fixed and never have the piston or displacer entering them are dead volumes. Add up the cold and hot dead volumes and put them in their respective inputs. If the engine has a regenerator, include that volume in the regenerator dead volume. Be sure to account for all significant dead volumes because they have a large influence on the pressure variations and power output of the engine. Some examples of dead volumes include the following:

1. The volume between the top of the power piston and the top of the
cylinder when the piston is at TDC. In a normal engine this will be a cold volume.

2. Similar dead volumes remaining in the displacer cylinder when the displacer is at its extreme travel positions. These dead volumes would be hot on the hot end of the displacer and cold on the cold end.

3. If the engine has a regenerator then the gas volume in the regenerator would be used for the regenerator dead volume. The volume taken up by the regenerator material can be subtracted out of the dead volume if it is significant. This is often computed for metal regenerators by dividing the mass of the regenerator material by the density of the material.

4. The volume contained in gas heating or cooling tubes.

5. If no regenerator is provided, the path the gas takes between hot and cold regions of the displacer could be considered the regenerator volume depending on the engine design.

6. In the special case of a Beta engine design it is common to have a negative cold dead volume where the power piston and displacer overlap the same volume in the cylinder (but not at the same time of course). You can design a gamma engines with a negative cold dead volume too, but it is less common. Use a negative dead volume for any gas volume of the engine that is entered by both the displacer and the power piston (at different times).

Engine cycles/sec – The cycle rate input is provided so you can conveniently compute the power output in watts at a given cycle rate. If you expect your engine to operate at 300 rpm you can enter 5 cycles/sec and see the power it would generate (before subtracting friction losses).

Hot and cold gas temperatures – are the average values in the hot and cold regions respectively. You have to provide these values from your computations or estimates. A few things you should keep in mind. The hot gas temperature will always be colder than the highest external temperature measured. If you just heat the end of a displacer cylinder you should expect the gas temperature to be a lot less than you measure externally when the engine is running. Similarly the cold gas temperature will be hotter than you measure externally. As you increase the cycle rate of the engine, Th will decrease and Tc will increase for the same external temperatures.

Average pressure – is the operating pressure for the engine in atmospheres. An unpressurized engine will operate at the default one atmosphere unless it will be operating at an altitude above sea level in which case you would want to enter the actual pressure. For a pressurized engine you would use the average pressure.

Gas constant – The default gas constant is set for air (.287kJ/kg-K). You can change the gas constant for other operating gases. Its only affect on the simulation will be to change the mass of operating gas required. It will not change the power output of the engine. The improvement in efficiency of using a gas such as helium will be due to less heat being used to heat the lower (mass x heat capacity) of gas (and less cooling too). In a real engine it may also result in higher power at a given RPM because the heat transfer should result in a higher Th and a lower Tc at any RPM.

Input errors – I have not put any error trapping on the simulator. All inputs must be numbers. If you see results of “Nan” (means “not a number”) then a non-numeric value has been entered. The simulator should recover if you correct the offending input. If not, reload the page. Allowable entries include numbers like the following 3, 300, 1.23, .004, 1.3e3, 1.3e-3. You can also enter negative numbers but they would only make sense for the temperatures (degrees C) and for the previously mentioned possible negative cold dead volume for some engine configurations. Avoid using Th=Tc as that will cause a divide by zero error.

## Results

• The results table compiles some useful information on the engine cycle. The work is always shown for a single engine cycle. The power is (net work x cycle/sec). If you prefer horsepower the conversion is 1 hp = 746 W.

• I’ll go into more details about the results and how to use them in a future posts, but I want to point out one general rule for successful Stirling engine design. Keep the pressure ratio less than or equal to the temperature ratio. Even though the simulator will show increasing power with increasing pressure ratio, a pressure ratio higher than the temperature ratio will push your design into an area where the compression will heat the operating gas so high that little heat can be transferred to the gas from the heat source. The reverse problem happens after expansion. I’ll go into more detail on this in a future post.

• The simulator also generates a normalized plot of the engine cycle pressure-volume curve. This curve provides a lot of information visually. I’ll go into it more in a later post. The table below the plot lists the pressure and volume for all the plotted data points. The table also shows the work increment for each cycle step. The work is computed assuming the back side of the piston is at the average pressure. This results typically in two power pulses per engine cycle (as it does in a real engine). The first when the pressure is highest and the volume is expanding. The second power pulse is when the pressure is below the average and the volume is decreasing. In the second case the pressure on the back of the piston is pushing the piston up into the lower pressure in front of the piston.