Stirling engines come in different forms. I will try to explain the principle with the simplest version, the alpha Stirling engine. I will do this in several stages, because there are many subtleties that can make it hard to understand.
A basic alpha Stirling engine
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A Stirling engine takes the energy from a temperature difference to generate kinetic energy (motion). Heat passes through the engine, levelling out the temperature difference.
It has a hot cylinder (HC) (top right) and a cold cylinder (CC) (bottom left). You can either heat HC with a heat source (eg a fire) or cool CC with a heat sink (eg water), or both.
The released energy makes the flywheel (top left) spin. This could drive a fan or whatever you want to do with it.
Inside the cylinders are pistons, which are connected to the wheel with piston rods.
The cylinder volumes are connected with a tube, which contains a regenerator. The regenerator amplifies the effect of the tube and is what differentiates a Stirling engine from other hot air engines.
The centrelines of the cylinders can be at different angles to each other, which determines at which points the piston rods should be attached to the wheel. The simplest configuration to explain is one in which the rods are connected at the same point and the cylinders are at an angle of 90°, as shown in the illustration.
Note that as a result the cylinders are 90° out of phase; CC makes the same moves as HC, but lags behind a quarter of a cycle.
The connecting tube allows the air in the cylinders to pass from one cylinder to the other. There is a crucial delay in this, partly determined by its size. (Everything works in a limited space of time. The engine wants to get to an equilibrium, but doesn't get the time to get there because the situation constantly changes. But I'm getting ahead of things.)
If the air in HC is heated it will expand and push its piston outside and move the wheel to the position of 270° (see the images below). But the hot air will also move to CC through the tube, so that piston will also be pushed out and the wheel ends up in the 315° position, where the total volume is at its maximum. And then nothing happens.
To get the engine running (once it has heated up enough) you have to rotate the flywheel 180 ° and give it a push. This is where the fun starts. :)
Volume
135° (first image below): This is where the total volume is at its lowest, so the expanding hot air is pushing hard. It wants the volume to increase, and this can happen with both a clockwise and a counter-clockwise move. So this is a tipping point (still assuming the hot air has spread to CC, so both piston rods push equally hard). If the wheel is pushed just a bit beyond 135°, the wheel will start spinning clockwise. Once the process is up and running, the fly wheel takes care of this step.
180°: The piston of CC has moved down a bit, so the volume there has decreased, but not as much as the rise in volume in HC (it's sinus vs cosinus, so to say, because of the 90° phase shift).
225°: The total volume keeps increasing.
270°: HC is at its maximum volume, but it's sinus vs cosinus again and there is still room for expansion.
315°: Maximum total volume. So here it stops? No.
Impulse
If you give a wheel a spin, it will keep on spinning for a while, and the same happens here. However, it will likely not make it all the way to the other side (135°). This is where the flywheel comes in. This puts a strain on the above process and thus stores energy (it is an energy-buffer). This energy is now released. (Another solution would be to put several Stirling engines on a crankshaft, just like with the internal combustion engine of a car.)
Heat
The above suggests that volume is equally important everywhere. But of course it isn't. In the same volume, hot air will push harder than cold air. And we have a hot and cold cylinder. Two effects are:
135°: The air in HC is hotter, so it pushes harder than the air in CC. This is why the tipping point is not at 135°, but a bit before that. Which is nice.
225° to 315°: The volume in HC is at its maximum here, which means that the air can absorb a lot of heat from the heat source. Which would be a problem when it reaches 315° because it would excert most pressure just when it needs to be compressed. But fortunately, there is a delay in this.
Delay
There are several delays, nothing happens instantly with the flow of heat and air.
A first delay is in the transfer of heat from the heat source to the air in HC.
135° to 315°: The system goes from minimum to maximum volume, which is the (primary) driving force of the engine. The heat energy in the air is transferred into kinetic energy (the turning wheel). In the process, the air cools down. (This btw is instantaneous because it is two sides of the same coin (adiabatic expansion).) This cooler air will also temporarily cool the inner wall of HC, so the air will not be heated by them (as much) just before compression, which will make the compression easier.
315° to 135°: While the air in the cylinders is being compressed, heat from the heat source will move through the cylinder walls and start heating up the air. Of course, this last bit should preferably start to happen at the right time, roughly at 135°. When it does this is determined by the thickness and other qualities of the cylinder walls. This is one of the determining factors in how fast the engine will spin.
Effectively, the heat of the heat source is held back temporarily and let back in at just the right moment. If the engine is designed well, that is.
Another delay takes place in the connecting tube, both in heat transmission and air displacement. This is where it starts to get a bit complicated.
315° to 45°: The air in HC is being compressed (driven by the flywheel). This heats it up (adiabatic compression). But it is also pushed through the tube (plenty room in CC, with a roughly constant volume). The tube now starts absorbing some of the heat, cooling the air, making it easier to compress it.
45°: Most of the volume is now in CC, where the air is at its coolest, and the heated air that has been added from HC has been cooled down by the tube. (Of course it is still warm, but it is the temperature differences that count.)
45° to 135°: This is the major compression stroke of CC. But where does the air go? There isn't much room in HC (at 90° its piston might even block the entrance), so much of the air goes into the tube, which has a much smaller volume, so the air heats up even more by adiabatic compression. And because the tube was previously heated up by the air passing through it from HC, the air will nog lose that heat.
135° to 180°: CC pushes its last bit of air into the tube, which displaces the now heated air that was there, which then enters HC, where it will already start to be heated up further by the now heated wall of HC. The air is hot and compressed and HC is hot too, so all is set for the major power stroke.
180° to 225°: The first half of the major power stroke, largely done by HC. The air cools down as a result (and consequently the cylinder wall too, but with a delay).
225° to 270°: The second half of the major power stroke, largely done by CC. The air cools down even further, and passes through the tube, which consequently also starts cooling down.
270° to 315°: More air, now at its coolest, passes through the tube, cooling it down even further. Which prepares it for absorbing heat at the next stage (the first stage above).
The tube thus acts as another energy-buffer, just like the flywheel. Because it absorbs and gives off heat at just the right moments (again, if the engine is well designed and built!) a Stirling Engine is more efficient than any other hot air engine. Also, the tube and the flywheel work nicely in tandem. During compression the flywheel gives off kinetic energy from outside, while the tube absorbs heat energy on the inside, both of which make compression easier. During the power stroke the tube gives off heat energy internally, while the flywheel absorbs it externally.
This is all very nice. Can we have more of it? Yes:
Regenerator
This basically does the same as the effect of the tube described above. The regenerator is really just an amplification of that process. This is what Stirling added to hot air engines, what makes it a Stirling engine.
It comes in many forms, for example a wire mesh or some tubes. The material should preferably have a high heat capacity (so it can take up much heat) and low conductivity (so the heat stays put). But also a large surface area and low friction. The latter two are at odds. More surface area generally means more friction. If you don't get this right, the friction-disadvantage may outweigh the buffer-advantage.
It's complicated
If you don't understand all of this, don't worry. To quote the Stirling cycle article: "The Stirling cycle is a highly advanced subject that has defied analysis by many experts for over 190 years." And: "The analytical problem of the regenerator [...] is judged by Jakob to rank "among the most difficult and involved that are encountered in engineering"."
Probably the most complicating thing is that there aren't distinct stages. Several processes take place at the same time, which is furter complicated by the fact that they influence each other. And then there is turbulence, one of the least understood physical phenomena. This can be a good thing (mixing of the gas) or a bad thing (friction, although it can also reduce that). And even small details in the design can greatly influence turbulence.
This means that many things have to be found out experimentally.
What exactly happens when is not as straightforward as I described it here. It depends on many factors, such as:
- the friction in the tube and regenerator
- the heat capacity and conductivity of the material the various parts are made of
- the dead space (the unswept volume of the cylinders, at 90° for HC and 180° for CC)
- the volume of the tube - this can be regarded as part of the dead space, also depending on the size and location of the regenerator
- the temperature difference between the heat source (at HC) and the heat sink (at CC)
- the phase difference. I chose 90° for ease of explanation, but that may cause the engine to lock.