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Thanks, HSS leaves a pretty good finish either way. I wonder if running a ball hone down the cylinders would help with lubricating the ringless pistons.

Funny! I actually ordered a 2-3/4" one which should be perfect the 70 mm bore.

Half way through the new set of simulations.

10 bar helium, this one is at 600 RPM (transfer time of 50 ms). The generator is an extruded cordielite cylinder with 0.90 x 0.90 mm square holes through its entire length (8mm), OD is 20mm.

You're still looking at a volume of hot (600K) gas coming out of the heater going through the generator. Initial condition for the solid (regenerator) is ~300K.

Take a look at what happens when the regenerator is basically too small for the job. The end of the regenerator heats up quicker than is bulk length due to greater convection effects - probably due to turbulence.

That's not something we would see without significant pressurization. That's probably due to the fact the pressure and fluid density exiting the regenerator counteracting the turbulence. At 10 bar however, while there is still a zone of lower pressure, the density is still high enough for significant heat transfer to take place.

Anyway, I thought this was interesting.


Oh and also, 10 bar significantly increases the power carried by the fluid. Where this regenerator looked properly sized for non-pressurized conditions, it's pretty obvious that it is quite undersized for 10 bar operation.

Very interesting so pressurizing the engine will have greater real world affects than just the added delta pressure, by way of reducing the consequences of losses to surface contact. But conversely the heat exchanger needs to be more efficient as well.

it's a bit of a trick question I can't answer. Say you got an optimized engine for atmospheric pressure operation and a given/known heat source. Now, you pressurize it and keep the same heat source (same power input), what happens to the RPM? Does it spin faster or slower or stay the same?

By optimized I mean the regerenator is providing a nice temperature gradient across its length.

What I mean is that with a pressurized engine, each stroke can carry more energy; but since the input power is the same, then it means the temperature of the gas will be lower (since it has higher capacity). As such I would be tempted to think that in this scenario the engine RPM would be lower.

The trick is that aside from people using electrical heat source for research/experimentation, you can't really control how much heat you're putting in.

As such, assuming you are just burning fuel, the pressurized engine would be able to utilize the available better a lot better than the non-pressurized engine and as such the RPM would likely increase.

Which one is correct?

Hmm I think it depends on so many things. If the unpressurized engine was utilizing the btu's available and was then pressurized, I suppose the rpm may drop or stay the same.

However I think in practice most of these engines have heat input that far exceeds what is actually needed to affect the temperature of the gas. Again it then comes down to the effectiveness of the heat exchanger in transferring available btu to inside the engine.

Here's the data I gathered on that 20mm diameter regenerator (1mm pitch mesh) and 8 mm long. Material is Cordielite.

The only difference in this data and before, is the fluid (helium) is pressurized to 10 bar.

and of course the point was to run simulation at various RPM.
the RPM changes the mass flow rate but also the duration of the simulation itself, denoted by the "transfer time" in the table below.
the heat yielded by the gas to the regenerator is calculated off analytical calorimetry by using at the mass flow rate, specific heat and the temperature delta between inlet and outlet.

Keep in mind that 10 bar is clearly too much for this regenerator other than at very low RPM. The gas is existing way too hot, which is why we get that secondary heat transfer toward the trailing edge of the generator cause by added the turbulence. This should never happen as ideally we want the gas temperature to be close to Tcold by the time to exists the regenerator medium.

For this reason, I spent the last day or so figuring out what the idea diameter of a regenerator should be. Since I've seen it done so many times, I decided with the typical Gamma style regenerator where the displacer piston "is" the regenerator and as such it often has the dimensions of the power piston. In my case the power piston being 70mm, I modeled a regenerator that is 69mm. Results were bad and it's not surprising. This is simply way to big. The regenerator material has so much thermal mass that the temperature of the gas drops really fast and the temperature gradient in the regenerator is too little. I settled on 30mm diameter and 40mm long. The length is meant to be more than needed but considering the very low thermal resistance of the material this should not affect anything, in fact it help observes what is going on better in my next set of simulations.


Speaking of which I am already getting ready to the run the next set. I will be pushing things further by alternating flow, I think I found a way to do this. The idea is to perhaps evidence the hysterisis I theorized in the other thread a few weeks back.

Here are the results for the Diameter 30 mm by 40 mm regenerator.
It has the same 0.9x0.9 mm holes and 0.1mm wall thickness.

Presenting the results in the form of temperature cutouts at the end of the transfer.

Keep in mind that the amount of helium mass transferred is always the same per stroke but depending on the engine RPM, the transfer time varies. In other words the mass flow rates is function of the RPM. As a result the fluid velocity and pressure drop also varies in these 5 different tests.

I'll post the table with numbers in the next post but you'll see that the heat transferred from the gas to the regenerator significantly increases depending on engine RPM. This is completely due to the shorter transfer time. In fact, as you can see from the temperature gradient, the helium exiting the pipe is increasingly higher temperature as engine RPM increase, which means with this design the efficiency of the regenerator decreases with RPM.

At 100 RPM, the regenerator is doing able to get the gas temperature from 600 K to 293.51 K (in 300 ms). At this speed, the renegerator is actually much longer than needed. The temperature of the hot end of the regenerator is really pretty close to the 600 K reference, at about 585 K. At 600 RPM, the regenerator is still doing pretty good. The fluid exits the regenerator at a temperature of 328.25 K (in 50 ms). For an engine this size, I think 600 RPM isn't bad a all. In fact, I can see this particular design being suitable for an actual prototype build. The temperature of the hot end of the regenerator is still pretty close to the 600 K reference, at about 546 K. At 1200 RPM, things are starting to look bad. The fluid exit temperature is 364 K and the temperature of the hot end of the regenerator is down to 527 K. Keep in mind that the temperature of the hot end of the regenerator at the end of the hot to cool transfer will be the upper limit of the next transfer. In other words, when the fluid flows back to the hot end, it will not be able to get preheated higher than 527 K which is IMO too much hysteresis. At 1800 RPM, things continue to get worse. Exit temperature: 390 K and the hot end of the regenerator is down to 514 K. Finally, at 3000 RPM: Exit temperature: 425 K and the hot end of the regenerator is down to 495 K. Obviously, the temperature gradient in the regenerator is a great indication on what's going on (they're hard to see on those screenshots). Its length is no the problem. Rather its surface cross section is the problem. [The reason why the length isn't helping here: increasing length increases surface area. However, due to the aspect ratio of the channels (very long relative to their cross section), we probably have significant boundary layer development causing the downstream fluid close to the walls to insulate heat transfer. This causes the vast majority of the length of the regenerator to be pretty useless. This brings us back to why disjoint regenerators have very good efficiency. But more on that later.]

Even though the velocity increases with RPM, it doesn't increase enough to compensate for the shorter amount of time to transfer the energy. As a result there is a point where there is just not enough time to transfer enough of the energy carried by the fluid and the fluid exits the regenerator with a temperature higher than it should.

Ways of solving this would be:
- to increase the diameter of the regenerator to increase the number of holes - at the cost of lower velocity
- to decrease the size of the holes to increase the surface area without decreasing fluid velocity

The second option is much better but at least with ceramics you quickly face manufacturability issues.

I was really happy with the ceramic (Cordielite) material I found however the boundary layer building up issue might be a deal breaker unless I can find them in a much finer structure (0.1-0.2 mm?). If I can't find them in bit of a finer form, I will revisit the disjoint meshes as with enough gapping between meshes, they should fully address boundary layer build-up at only the cost of dead volume; the only issue with meshes is material properties - if only metals are available, the question also becomes manufacturability to achieve thin-enough wire size. Also, I received some porous glass samples with various porosity levels, they might be a good alternative to either of those regenerator designs. The samples I have are Diameter 32 mm x 4 mm discs, if made is slightly thinner discs, they could be arranged in a disjoint manner as a great alternative to meshes.

Here's the data table for this study:


And here's the first successful animation of an alternating flow. I can't simulate the moving pistons, but I was able to apply a time dependent pressure gradient across the pipe which is a decent way of representing what's going on, at least from a thermal standpoint.

This animation is for 6 transfers (3 hot to cold and 3 cold to hot), at 3,000 RPM. This was before I analyzed the data at each RPM. Otherwise I would have done it directly at 600 RPM.

It's not "zoomed in" on the walls of the regenerators but you'll get the picture:
- on the Hot to Cold transfers, only the left side does get to heat up, and the fluid existing the pipe is still really warm
- on the Cold to Hot transfers, the fluid gets barely heated up, for 2 reasons:
- the regenerator did not get very hot from the previous transfer
- the area of the regenerator that is somewhat warm is in the fully developed boundary layer which means significantly reduced heat transfer

I am running that 600 RPM simulation with alternating flow right now, I should have the results either later today or tomorrow. I'll see if I can find a better way to plot, maybe with a split views with zooms on the hot and cold ends of the regenerator.

Regardless, it's pretty clear the boundary layer build up is something that we need avoid.

https://file.io/h6eKUmMngzib

Excellent research Stephenz !!! Many thanks for your sensitivity study and posting your results. Your regen graphics are outstanding and give a rare visual of what's 'really' happening. I have a wide range of hot air interests (various cycles) but you're discovering why I tend to limit my Stirling schemes to 600k and 600 rpm. Unfortunately, even with the best regen most Stirling designs will suffer greatly from their typical out-of-phase dynamics and low volume ratios. I'd suggest doing a simple comparison between Ericsson style isobaric regen vs Stirling style isochoric. A Stirling cycle has 3 main pressure points vs an Ericsson cycle only has 2 main pressure points.

Thank you, I have my mind set on figuring out basic rules on sizing/designing regenerators for stirling engine but I wouldn't mind doing some analysis on other cycles as well. I would need to start doing more research on them as time allows.


The 600 RPM alternative flow simulation is through and put together an animation:

https://file.io/kAHMZTt1cJjT

This is only 3 full cycles, and it might take a bit more than that to get to the final H->C and C->H gradients; more on that later.

The boundary layer issue I was highlighting with the 3000 RPM simulation, is still present at 600 RPM as we already know.

On the H->C transfer, the temperature drops pretty well.
However, on those C->H transfer, the fluid isn't get preheated that much.
This is most likely the result from the fact the fluid starts making contact with the hot part of the regenerator really far inside those channels where heat transfer is less efficient.

As opposed on the H->C transfers where the 600 K fluid hits the 292K at start up, causing great heat transfer (great temperature delta and highly turbulent flow, with no insulating boundary layer).


At time consuming and resource intensive this may sound, I think I need to run this exact simulation on much longer time scale
Right now, each transfer at 600 RPM, each transfer is 50 ms long. Doing 3 full cycles equates to 6 transfers, or 300 ms.

I think I need to run this for 10, maybe 20 times longer to see what this temperature gradient looks like after some time.


Right now I am tempted to thing this boundary layer issue is a problem as the C->H transfer takes place far down the length of the regenerator. But maybe after some time the gradient might spread across the entire length of the regenerator effectively allowing for more efficient C->H transfers.

Considering the model is built and running well, I should just keep running for a few days.

Before I launch this tomorrow, I will do a plot over time of the following average temperature:
- Hot fluid inlet/outlet
- Cold fluid inlet/outlet
- Regenerator Hot End
- Renegerator Cold End
- Regenerator Halfway Section

I was able to extract average temperature data for the Hot and Cold ends of the regenerator.
I had not set the inlet/outlet as goal surfaces in the previous simulation so I couldn't get to plot those without extensive work, but these 2 plots are really good by themselves.

I was to resume the simulation after the first 0.3 seconds, and will continue this one going until 3.0 seconds; or until the goal plots show that I have reached equilibrium.

Here's the average hot end temperature Vs time: And here's the average cold end temperature Vs time: My guess that the hot end temperature was getting hotter slowly after each transfer was correct, probably due to insufficient heat transfer on the C->H transfers.
And the obvious rise in temperature of the cold end is most likely due to the reason, as the thermal conduction in the regenerator eventually makes it way from the hot to the end cold end.

There is no choice, I need to find a way to increase heat transfer inside the regenerator, and not just at the leading edges of the regeneator.

stephenz wrote: ↑Sun Jun 18, 2023 10:08 am
My guess that the hot end temperature was getting hotter slowly after each transfer was correct, probably due to insufficient heat transfer on the C->H transfers.
And the obvious rise in temperature of the cold end is most likely due to the reason, as the thermal conduction in the regenerator eventually makes it way from the hot to the end cold end.

There is no choice, I need to find a way to increase heat transfer inside the regenerator, and not just at the leading edges of the regeneator.

I will use the word "density" of the gas for clarity. PVT values are commonly used to relate thermo, but the gas density is often ignored. If we connect two unregulated 100cc cylinders with 180 deg phasing at the same temperature, pressure and density are equal between cylinders. However, if we heat one cylinder while we cool the other cylinder, only the pressure is equal. As I like to say, cold gas pools.

I never studied regen at the level you are, but merely considered the basic thermo involved which leads me to wonder whether any regen cycle is worth the chase. Assuming Schmidt analysis where pressure is equal, regen density will have a gradient relative temperature gradient when regen has uniform cross section and matrix. Obviously, regen cross section and/or matrix material/porosity could vary across regen, but a density variation will remain between start and finish of each regen 'blow'. IOW viewed as ideal distinct events, when the low pressure blow from cold cylinder to hot cylinder starts, the cold gas will pool in cold cylinder under increased pressure from hot cylinder, whereby density thru regen is relatively low. But as this low pressure blow continues, the density thru regen increases. To me, this suggests that a lower thermal ratio is easier to optimize than a lower volume ratio.

I believe the regenerator is the key to high efficiency and high power.
In fact, carnot efficiency can only be reached at perfect regenerator effiency.

Perfect regenerator efficiency is obtained when the 2 green brackets below tend to 0 (once steady state is achieved that is).
I don't know if it's physically impossible to get those deltas equal to 0, but getting close should be possible. And from what I am seeing, this is mostly a thermal engineering problem, i.e. an engineering problem, i.e. I don't see any real "physics" barrier. Materials and/or manufacturing process may limit how close we can get to it; that's too early for me to say.


I won't say it enough, but keep in mind as much as I am trying to model everything as close to an engine as possible, I will not be able to model it perfectly. Right now, this simulation are only looking at the transfers and ignoring compression and expansion all together.

I would need some measurement data to even try to represent the effect of compression or expansion at the same time as those transfers occur. For the purpose of engineering an efficient regenerator I don't think I need to worry about the compression or expansion effects. But when trying to apply those results to an actual engine it's important to keep in mind that all 4 steps of the stirling engine are not discreet and as such, some compression and expansion occur at the same time as these transfer occur. And of course those would affect the temperature and density of the fluid at the same time as it being transferred.


That being said, the plot above show roughly 5 additional transfers since my last post. I mentioned I didn't have inlet/outlet temperature set as goal. I didn't want to restart the simulation from scratch which is why the Red and Blue curves start at 0.3 s. We're at 0.5 s, and there is still some significant temperature changes from one cycle to the next. We're nowhere near steady state yet.

The "hot" green bracket is getting increasingly smaller and the "cold" green bracket is getting increasingly bigger. Steady state will be reached when the sizes of those respective brackets are no longer changing. I'm hypothesizing they might end up being the same size.