Stirling Cycle efficiency via Carnot

[matt brown]

I've been busy this year, so haven't been onboard in months, but recently had a brainstorm that should clarify Carnot limits as it applies to conventional Stirling Cycles. Any engineers that happen by may roll their eyes, but thermo is often hard to relate to laymen. OK, here goes...

The beauty of the Stirling Cycle is its simplicity, but this is also the sucker bait. I'm going to use a conventional Stirling Cycle with 4 processes wherein T1 is the high temperature, T2 is the low temperature, T1 & T2 are in deg K, and an ideal working gas. When T1 & T2 are isotherms, any heat added will require volume expansion and any heat removed will require volume contraction, wherein heat=work and processes are reversible. Thus, all heat equals work, and all work equals heat, but only per process, not per cycle !!!

Interestingly (or not so, depending upon your dogma) both T1 & T2 have a unique CONSTANT internal energy that is directly linear deg K per mole regardless of PV values. So, adding heat at T1 will expand V and decrease P, but the internal energy is unchanged. Likewise, removing heat at T2 will reduce V and increase P, but the internal energy is unchanged.

Now, if T1=600k and T2=300k, we can assign similar values for internal energy U, whereby T1U=600 and T2U=300 (yeah guys, an ideal gas has internal energy exactly this linear K). And a little thinking should conclude that no matter how much heat is added at T1 for work W, heat reduction at T2 will equal W/2 to complete a cycle. Thus, the often touted Carnot eff equation (T1-T2)/T1 remains intact. Note that I left regen out of this, but as an ideal cycle, regen is 100% and changes nothing...

So, Tom, how do you get 100% eff unless you recoup some waste heat at T2 per Daemon's recent post ???

[Tom Booth]

I think, for most Stirling engines, particularly if running, (stop/period) expansion and contraction are mostly adiabatic not isothermal. But even the "Ideal" Carnot cycle includes an adiabatic expansion leg of the cycle

Resize_20221115_092020_0426.jpg (28 KiB) Viewed 17712 times

Now consider the abrupt and unnatural transition depicted from 1-2 isothermal to 2-3 adiabatic

Nevertheless, A very abrupt drop in temperature is depicted between the two isotherms.

What causes this sudden drop in temperature?

Adiabatic expansion.

In a real engine, however, such abrupt changes are not possible. Perfect isothermal expansion is not possible. Adiabatic expansion begins much sooner and extends further. Real engines have weight, velocity, inertia the transitions are smooth or curved and elastic not so angular as depicted in the "ideal" case.

I'm not particularly concerned with impossible, imaginary engines that do not exist and could never exist in the real world. I'm more of an experimentalist than a theorist. What are the actual measurements?

We have this:

Resize_20220309_094457_7057.jpg (63.03 KiB) Viewed 17712 times

Which indicates temperatures dipping above and below the isotherms.

How is that possible?

Well, we have cooling by adiabatic expansion.

Is it reasonable to believe that the transition between 2-3 and 3-4 is so abrupt and angular?

No, pistons have weight, the gas itself has some velocity. When the energy of the gas is expended in the process of expanding and pushing the piston does everything just abruptly stop. The laws of physics would dictate otherwise I think objects in motion tend to stay in motion unless or untill they meet with some opposing force.

So where does the gas get the energy to continue on expanding? Why the temperature drop below the isotherms?

Internal energy is not constant.

[Tom Booth]

To address your specific question:

So, Tom, how do you get 100% eff unless you recoup some waste heat at T2 per Daemon's recent post ???

Not entirely sure what statement made by Daemon you are referring to, other than his general observation that in his experience the "sink" or cold side of a Stirling tends to heat up more when more heat is added.

So, if we take the alleged case of my experiment where there is no waste heat to recoup;

Atmospheric pressure causes the piston to return once the internal temperature and pressure drop low enough, so then atmosphere does work on the gas.

It is again the conversion of work into heat that allows the "recoup" of heat during compression, just as the conversion of heat into work accounts for the "extra" cooling.

Carnot was not aware of the equivalency between work and heat, that one could be converted to the other and back again when he wrote his book.

Late in life however he did become aware of this and abandoned his original hypothesis, unbeknownst to the thermodynamicists following the letter of his published writings.

[Tom Booth]

And, according to that graph, there is approximately 160° of the cycle where the internal gas temperature falls below the temperature of the cold isotherm.

And, of course, heat will naturally have some tendency to flow from a warmer to a colder region.

This would imply then, that atmosphere is not so much a heat "sink" as a secondary heat source.


https://youtu.be/SHyke4hUNOs


The video is a presentation by Andrew Hall of the Stirling Engine Society

[Tom Booth]

So, yes, you do recoup heat at T2, (or during the return compression/contraction stroke, which does not really precisely follow an isotherm).

Just not "waste" heat necessarily.

IMO a goal of efficient engine design should probably be to push adiabatic expansion and cooling as far as possible so as to allow atmospheric pressure to do a substantial portion of the work of driving the engine

Heat addition should be as rapid and as brief as possible in order to convert heat to VELOCITY, as it is the velocity of the piston/gas that maximizes adiabatic cooling. The result being that the atmosphere contributes a greater proportion of the heat and pressure that powers the engine.

[Tom Booth]

So there you have the rationale behind my 100% efficiency (+) plus.

If you get 100% efficiency from the applied heat, or close to it, (in that all the heat is utilized, resulting in temperature and pressure low enough for atmosphere to take over on the return stroke), then you have atmosphere doing roughly 1/2 the work. Well, however that works out exactly, I can't see how it could possibly be the paltry efficiency predicted by the Carnot equation.

The engine has to do work to displace a portion of the atmosphere, however, before atmosphere is able to reclaim that space by pushing the piston back. So that cancels out, but I think you are still looking at very high efficiency.

[Tom Booth]

Put another way;

If ambient temperature is, let's say 20°C and we heat up the hot side to 300°C so the gas expands rapidly and the adiabatic expansion takes the temperature of the working fluid down to below 20°C, which logically I think would be necessary for the pressure to fall below atmospheric pressure

This has used 100% of the heat added to bring the temperature of the gas from 20°C to 300°C

But momentum carries the piston further so that adiabatic cooling, as well as heat being converted to work which results in cooling results in MORE heat being utilized than just ALL the heat that was added to set the piston in motion.

This is not utilization of "all the heat" down to absolute zero. IMO that is absurd, but we hear it all the time, that to get 100% efficiency the temperature would have to go to absolut zero. Why? We didn't start out at absolute zero, we started out at 20°C

[matt brown]

Kudos, Tom, now I'm getting it. In my ideal 300-600k Stirling example, 100% of heat was converted to work during expansion, regen was 100%, but compression backwork reduced this example to Carnot buzz. This is why I often include that Carnot has caveats and restricted mainly to common 4 process cycles with interleaved processes (heating opposite cooling, compression opposite expansion). So, you're right, if you could find a way to remove Wneg of compression from Wpos of expansion (and not take compression force from flywheel) then Carnot could take a hike, and we could have near 100% eff low ratio cycles. Now, I see why you're looking at atmospheric engines, etc.

BTW, the Daemon post I was referring to was a recent response on the long 100% eff thread where he hinted at some way to recoup waste heat (sounds like he's chasing a patent - lol). He somehow reminded me of the old idea of having an open cycle 'steam engine' with an alcohol working fluid (you probably see this coming) where the alcohol is vaporized, then run thru the engine, then fuels the boiler; whereby, the latent heat of vaporization is not lost to ambient, but returned to system during combustion. Yep, a 3am newbie brainfart that shows its folly during morning calcs. Heck, if this was...credible...lots of stuff would be different.

Anyways, you have an interesting proposal in trying to nix typical Stirling Cycle compression which mimes common ICE. Also, good idea looking beyond Stirling, especially exploring adiabatic schemes. Most of my thermo schemes during the past 20 yrs involve adiabatic processes due to speed and chucking laggardly (ie bogus) isothermal processes. However, be advised that adiabatic cycles are harder to analyze than isothermal cycles, and few guys have ever pondered them beyond ICE versions. Nevertheless, welcome aboard...

[Goofy]

I´m not sure if this have been posted before, but here it does belong :

https://www.ohio.edu/mechanical/thermo/ ... index.html

And the use of the term "Exergy" could simplify things a bit. https://en.wikipedia.org/wiki/Exergy

Quote:
"In thermodynamics, the exergy of a system is the maximum useful work possible during a process that brings the system into equilibrium with a heat reservoir, reaching maximum entropy."

And remember : In theory, theory and praxis are the same. In praxis they are not . . .

\Petter

[Tom Booth]

...

BTW, the Daemon post I was referring to was a recent response on the long 100% eff thread where he hinted at some way to recoup waste heat..

viewtopic.php?f=1&t=5410&start=60#p18215

Have you read through the thread on the thermoacoustic "rice" engine?

viewtopic.php?f=1&t=5483

Unfortunately some posts and videos were accidentally deleted, but some really interesting developments.

I'm anxious to see Daemon's slow motion video.

The idea of a kind of condensing steam engine where the usual evaporation and condensation cycle is reduced to an instant or single cycle in a very compact powerful engine is very compelling.

[Tom Booth]

Personally I don't think that the rice or glass beads or whatever condensation medium causes condensation by acting as a heat sink.

I think that the mass of small, whatever, beads, rice, aquarium gravel, just give the water vapor, that cools as a result of adiabatic expansion and work output a convenient landing to condense on.

Condensation is easier and more rapid when there is something to condense on and the matrix of rice or beads or pebbles or whatever ensures the vapor does not have to travel far to find something to condense on.

In other words, I don't think the glass beads or whatever cause condensation because they are cold, but rather the vapor expands and does work and cools as a result and is then ready to condense.

I'm not sure why, but I've always heard it said that clouds form because water droplets condense on dust particles in the air. Apparently the water cannot condense without something to condense on.

But with clouds as well, the dust particles don't cause condensation. It is when warm moist air rises and expands adiabatically and cools that the cloud condenses on whatever particles happen to be available.


https://youtu.be/a0AgF_ysyZc

The rice or whatever simply provides the surface on which the vapor can conveniently condense. The so-called "condensation nuclei".

IMO the vapor probably does not condense because of conducting heat to the (cold) rice.

[Tom Booth]

I had read something somewhere about this a long time ago, someone had claimed that they found that their Stirling engine would run much better and had more power if they put water and smoke in the displacer chamber.

viewtopic.php?f=1&t=77&start=15#p174

Actually, re-reading that post now, it looks like putting smoke in the chamber may have been my idea, to prevent the "problem" of too much condensation rusting the tin can or whatever. It was a long time ago.

Also the end of this post:

viewtopic.php?f=1&t=77#p160

Possibly that kind of force could be generated with the addition of some appropriate gas other than plain air, but I'm pretty well convinced that rapid heating and cooling is the way to get more juice out of one of these things. A little water would probably work if the thing could be kept at an operating temperature somewhere above the boiling point.

I read on one of the Stirling engine websites I was looking at somewhere that adding a little water to the chamber will make a Stirling engine run better.

The response to the idea of adding water to the engine is interesting.

As for putting water in the system... liquid water is less compressible than air and also has a higher degree of thermal retention ... both are properties directly opposed to what you want. Let's not even talk about what water, heat, and air do to metal parts.

Water in displacer chamber...bad, very bad

[Tom Booth]

A difference though, between the rice engine and a cloud formation is that rising air that might form a cloud has more than just water vapor, so when the water vapor condenses it doesn't create a vacuum.

With the rice engine, though, by the time it starts up, the boiling water has created enough water vapor to displace all the air, or at least a good portion of it, so when the water vapor condenses a nearly complete vacuum results.

Likely the sudden vacuum, lowers the boiling point while pulling down the membrane, it also allows the condensed water droplets to suddenly re-evaporate (or "explode"?)

I think Daemon intended to build a glass rice engine to see what was actually happening and took a slow motion video, which should be quite interesting.