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Stirling Cycle Machine Analysis
Nice to see this posted. I spent probably too much time on Stirling engines and associated stuff (heat engines in general, the Vuilleumier family of devices and other interesting bits & pieces).

It's so tempting, you think there has to be a way to make it work. But I haven't found one yet and I more or less gave up on it. The most practical ones that I'm aware of are in spacecraft for extreme cooling and in a commercially available generator.

If you've never seen one up close I highly recommend building a model of one and watching it run, it is complete magic compared to internal combustion engines which tend to be noisy and dirty. Stirling engines run so quiet you have to be right up close to realize it is running at all (and even then you probably will subconsciously put your hand on it to verify that it is not just your imagination).

The Philips company, when it still meant something, put together a whole slew of pilots: a boat and a bus, and an endless number of them in the lab. Fortunes have been sunk in these and with preciously little to show for it other than an extreme appreciation for how hard it is to make a really good seal.

Wurm et al's book is the standard in the field, it is very thorough and gives a very good grounding in the theory as well as some appreciation on why this is both a tempting and very hard subject at the same time. Most engines are hard to think of and relatively easy to engineer once you've thought of them. There are two exceptions, Stirling engines and Wankel engines and both have very strongly related problems. Solve the seal issue and the world will pave the way to your door with gold.

Stirling engines are a perfect illustration of what I call Odum's Paradox:

  The closer a heat engine/pump gets to maximum theoretical efficiency, the lower its power density.
This is a simple consequence of thermodynamics. By approaching the Carnot efficiency you're asymptotically approaching an adiabatic cycle, so there's less and less entropy gain to drive the system forward. At the Carnot efficiency the cycle becomes completely time-reversible and all power stops, so as you get closer and closer the power density drops toward zero.

Odum observed[0] that (for certain linear assumptions) the maximum power density is achieved at half the theoretical Carnot efficiency. He noted that both real engines and biological systems tend to cluster around this optimum.

What's interesting, but not terribly surprising, is that this implies a fundamental and unavoidable tradeoff between machinery cost and operating efficiency.

[0] https://en.wikipedia.org/wiki/Maximum_power_principle

One interesting mention I can recall of one was in a airport-kiosk type thriller/science fiction novel where a humanoid robot developed by the military wound up in the jungles of South America assisting a group of disparate individuals --- it made a small Stirling engine using a lathe to power itself from the communal campfire --- always wondered if the physics of that would work out numerically.
Most likely not if it was made from stuff you can find the jungle but it is a very creative bit. I've had one running of a 10' solar concentrator, it was tiny (no more than a few hundred mW) but still, to see sunlight directly converted into motion without any steps in between is quite a neat thing.
I think this is ‘Solo’ by Robert Mason.
The first working model engine I ever had (still have, in parts, in a crate somewhere) was a compact stirling engine, roughly 10 x 5 cm at the base, fantastic gadget for a single digit year old child.

IIRC there's a New Zealand company WhisperGen(?) that use them for power from waste heat from water heaters ...

Checking now, they seem to be used as marine power generation these days, I can't find the original site from 15+ years ago.

WhisperGen is the one I was thinking of.
I went down a rabbit hole on that - apparently the WhisperTech factory got destroyed in the New Zealand earthquake and left little behind other than a cohort of dedicated fans keeping old units alive :(

  How It Works

  The WhisperGen PPS16 is based on a four-cylinder Stirling-cycle (external combustion) engine that repeatedly heats and cools a mass of pressurised nitrogen gas.

  Each time the gas is heated and cooled, the changing gas pressure causes the pistons to move up and down.
  This mechanical motion, via a special mechanism called the “wobble yoke”, rotates an alternator to generate DC electricity which can be used to charge a lead-acid battery bank.

  The nitrogen gas is heated by a continuous-combustion burner, and cooled by coolant circulating through engine cavities. Heat transferred to the coolant can be used to heat domestic water cylinders and for space heating.
~ https://www.victronenergy.com/Manuals/WhisperGen/UserManual/...

There was a European branch in Spain, seems to have withered, and some stories about expansion plans and pivoting that got trashed by the earthquake fallout.

~ https://www.smh.com.au/business/nzs-whispergen-to-move-to-eu...

Damn shame - I had access to an early model looong ago and was really looking forward to seeing where that went.

One reason to be interested in this is distributed long term energy storage. This would be done by storing heat, then converting the heat to power with an externally heated engine.