This post will examine some of the practical and theoretical aspects of a regenerator as it applies to the 3D printed PE 2 Stirling engine. Those who just want to build a regenerator for the engine and get good performance, will find what they need here too.
The above data show my tests of the stock PE 2 engine with no regenerator and 3 different regenerator configurations that I’ll discuss in a minute. The temperature ratio scale is based on any absolute temperature scale, typically Kelvin or Rankine, where the ratio is formed by the hot plate temperature divided by the cold plate temperature. For a cold plate temperature of 70 degF (21 degC) the 1.115 ratio corresponds to a hot plate temperature of 131 degF (55 degC) and the 1.145 ratio corresponds to 147 degF (64 degC).
The important information to notice in the above plot is the small yellow triangle in the lower right that corresponds to the engine running with no regenerator. Even at a relatively high temperature ratio that would result in over 300 rpm with any of the regenerators I tested, the no-regenerator case is barely running at 120 rpm. You really want a regenerator in this Stirling engine to get good performance.
What’s a Regenerator?
The regenerator in a Stirling engine is placed in the path the operating gas follows in going back and forth between the hot and cold ends. When the gas leaving the hot end of the engine flows through the regenerator, it transfers heat gradually to the regenerator, cooling the gas. When the gas leaves the regenerator at the cold end, it will still not be cold enough for engine operation, but the heat rejection (cooling) part of the engine won’t have to cool the gas as much as it would without the regenerator.
When the gas leaves the cold end and enters the regenerator, the regenerator is warmer than the gas and the gas is gradually heated and the regenerator cooled. By the time it leaves the regenerator and enters the heating part of the engine (which in this engine configuration is just the hot plate), it will have been pre-heated, but will still need to get hotter for sustained engine operation.
The above cycle continually repeats. The more efficiently the regenerator stores and transfers heat, the fewer watts have to be transferred to the gas on the hot end and rejected from the gas on the cold end.
The ideal regenerator
If you could make a perfect regenerator and perfectly insulated the engine so that no heat was lost to the environment from the heat source except through the operating gas, you would have an engine that was thermodynamically approaching the Carnot limit (theoretical maximum possible thermodynamic efficiency for a heat engine). For a low temperature engine like this (temperature ratio of 1.145) the Carnot limit is about 13% efficiency. The equation is:
Carnot efficiency = 1-(Tc/Th)
Where Th is the absolute heat source temperature and Tc is the absolute heat rejection temperature.
I say thermodynamically approaching the Carnot limit because frictional losses in the engine, both mechanical and gas frictional losses would still be reducing the engine efficiency.
The above plot is from my simulator with the parameters set for this engine at 1.145 temperature ratio and 300 rpm. You can see the 13% cycle efficiency for a 100% efficient regenerator. My simulator also includes computations for the conduction losses through the displacer and displacer cylinder because they are often sources of large losses in thermal efficiency for my engines and they are easy to compute. The best net efficiency I have measured so far for this engine is 0.14 % with a 0.02 w shaft output and 14 w heating with no attempt to insulate for heat loss. My guess is that I could perhaps achieve an overall efficiency of 0.5% to 2% for this engine if I were to insulate the hot plate to minimize heat loss.
A larger engine running with the same temperature ratio should be able to achieve higher efficiency, but even reaching say 1/4 or 1/3 of the Carnot limit for overall engine efficiency would be a respectable achievement. A high-efficiency Stirling engine requires an effective regenerator and a lot of other attention to design details as well.
Below is a plot that shows how much heating (and cooling) is required for the engine thermodynamically. This is just looking at the heat that needs to go into and out of the operating gas for the PE 2 engine operation at 300 rpm and a temperature ratio of 1.145. Without the regenerator it needs almost 10w of heating and nearly as much cooling. My simulator shows gross power output of the engine for this condition around .08 watts so the cooling is less than the heating by .08 watts.
With a regenerator the engine becomes more efficient because the heating and cooling of the operating gas is partially accomplished internally by the regenerator. The external power required for operation is reduced. Engines that run on a low temperature ratio always have low thermal efficiency which means they have to reject almost as much heat (cooling watts) as they require heat (heating watts) to operate.
Although everyone appreciates the need to have a source of heat to make a heat engine run, people new to Stirling engines often tend to underestimate the problems of cooling the engine. Most of us are accustomed to internal combustion engines where the majority of the heat conveniently blows out the exhaust pipe. Stirling engines have no such easy source of cooling. You have to cool by conduction, convection, and radiation. Regenerator efficiency reduces both the heating and the cooling required on a Stirling engine so you win both ways.
I’ve not yet attempted to make any measurements that would enable me to estimate the regenerator efficiency of the ones I’ve tested. I don’t even think I can make a reasonable guess. I know the regenerator improves the performance a lot. Measuring the regenerator effectiveness is something I plan to do in the future.
In the diagram of the engine and regenerator, you saw a two-part regenerators. Why not make it out of just one piece, a one-part regenerator?
If each regenerator channel (there are 12 regenerator channels in this engine) used a single piece of a metal with good thermal conductivity, such as the aluminum I used, then the regenerator should reach an equilibrium temperature about halfway between the hot and cold plate temperatures. There would be small fluctuations of the hot gas heating the metal slightly (depending on the mass of metal) in one direction and then cooling when the gas flowed in the other direction. That would still be a big improvement over no regenerator, but it should be possible to improve this further by cutting the piece of metal used in the regenerator in half. A two-part regenerator.
Cutting the metal such that the gas flowed across one half of the metal first would allow the two pieces of metal to reach different equilibrium temperatures. The half adjacent to the hot end would have a higher temperature than the half adjacent to the cold end. These two plates should allow the gas to heat up slightly higher before leaving the regenerator on the hot end and cooler on the cold end. The thermal independence of a multipart regenerator should improve the regenerator efficiency. The more independent parts of the regenerator in the gas path the better.
With the one-part regenerator I can assume the regenerator would be close to the average temperature of the hot and cold plates. For the two-part regenerator I can’t assume the temperatures would be split into even thirds between the hot and cold end temperatures because it would depend on the a variety of factors in the heat transfer process.
More heat transfer area
Another factor for improving heat transfer is to have the gas flow over a greater surface area. The photo below shows what I refer to as two-fold and four-fold regenerator configurations. The two-fold regenerator piece has 2 folds with 3 straight sections and the four-fold has 5 straight sections. So the four-fold regenerator has 5/3 = 1.67 times as much surface area as the two-fold regenerator.
regenerator surface area
The following table shows the total surface area of the various regenerator configurations and also the interior surface area of the hot or cold plates for comparison. You can see the regenerator provides many times the surface area of the hot or cold plates.
exposed surface area, square inches
|hot or cold plate||9.62|
|2-part, 2-fold regenerator||54|
|4-part, 2-fold regenerator||54|
|4-part, 4-fold regenerator||90|
Although regenerators can make a Stirling engine run more efficiently, there are negative aspects to regenerators too. They usually add what is called dead volume to the engine, volume that is not used for the power piston or the displacer. Dead volume in an engine has two main effects: It increases the amount of operating gas in the engine, requiring more gas to be heated and cooled (and therefore more input power). It also reduces the maximum compression ratio or requires a larger power piston to maintain the same compression ratio. For this low-compression engine, the increase in dead volume is not a serious drawback. The PE 2 engine has the dead volume for the regenerator built-in so there is no volume penalty for adding the regenerator.
The more important potential drawback is that the regenerator always adds some restriction to airflow, requiring the engine to do more work forcing the displacer back and forth to push the operating gas back and forth through the regenerator.
If you tightly pack the regenerator volume of this engine with steel wool for example, it might do a good job as a regenerator of storing and transferring heat, but the resistance of pushing the gas through packed steel wool will take more power than the engine can provide. A very light packing of steel wool would work.
Because the regenerators I tested all fit within the same volume in this engine, the 4-fold regenerator forces the gas to flow through smaller openings and a slightly smaller total cross-sectional area. I wanted to see if the regenerator surface area increase of the four-fold regenerator would increase unloaded engine rpm over the two-fold regenerator or if the increased flow resistance would reduce performance.
My conclusion from the data taken so far (the plot at the top of this post) is that all the regenerators I tested were a huge improvement over no regenerator. I think there was too much noise in my data to make any firm conclusions to compare the effectiveness of the different regenerators. I need to repeat these tests with more precise measurements and hopefully get more precise and repeatable data. It looks like the four-part, four-fold regenerator is performing best although there is some question that it is starting to drop at the highest rpm.
Here’s how I made the four-part, four-fold regenerator:
1. Cut apart an aluminum drink can (soft drink or beer can) to obtain sheets of aluminum about 0.002 to .005 inches thick. I have not found any aluminum foil stiff enough to use. The four-fold regenerator required two cans.
2. Cut the aluminum into strips about 1/4 inch wide. (1/2 inch wide for the two-part regenerator).
3. Bend the strips using a simple jig shown in the photo. The jig will set the length uniformly and accurately without the need to measure each fold. I set the wood piece in the folding jig about 0.70 inches from the end so that the folded piece ends up about 0.75 inches long. Test fit one to make sure it will work before making them all.
4. After the last fold, cut the regenerator piece to length. In the photo you can see I’ve made two pieces from one strip before cutting them.
5. Push the accordion-folded regenerator pieces into the 12 regenerator slots using a thin piece of wood or other tool. Ideally each piece should not actually touch the other pieces for thermal isolation to minimize conduction. I used depth marks on the stick to help set the depth for each regenerator piece.
You want the regenerator pieces to still have some spring in them to help stabilize them in position so they don’t slide around. Don’t squash them too flat. Also try to make the gaps for the gas to flow through somewhat uniform.
Although I would encourage experimentation with the regenerator, you might want to start with one of the ones I made first for a baseline to get reliable performance. I’d probably suggest the four-part, two-fold version. I’m curious how well a one-part, two-fold version would work and will probably have to make one.
If you come up with a regenerator that is very efficient, especially a simple one, let me know and I’ll post it so others can use it.