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Pour yourself a stiff drink...

I've been studying various gamma aspects over the past year and recently posted these graphics to the Wilcox thread


The point was comparing common cold PP gamma to hot PP Essex and my basic conclusion was that cold PP has the same input and output as hot PP when both PP are equal, but that cold PP would require a larger DP than hot PP (and cautioned that this was a crude conclusion). I then added graphic comparing these 2 models where I subdivided the gas into 2 equal volumes to show gas flow into PP and everything appeared hunkey-dory.

I then expanded my parametric study with cold PP Stirling = hot PP Essex


with 3 thermal ratios where A = 300-450k, B = 300-600k, C = 300-900k, but I made 2 slight Essex errors on frames A E-4 and C E-4 (note corrected callout).

The main takeaway here is that for Stirling, the compression ratio remains constant regardless of the thermal ratio due to DP and PP are same temp during compression. Meanwhile, for Essex, the compression ratio varies with thermal ratio since DP and PP are different temps during compression. Conversely, Essex expansion ratio remains constant regardless of thermal ratio due to DP and PP are same temp during expansion vs Stirling expansion ratio varies with thermal ratio.

Most important, the Essex has a direct relationship between the thermal ratio and the expansion ratio, but the Stirling does not.

Continuing my study, I discovered a major Stirling anomaly...


These are previous graphics considered by gas flow via gas vols A and B (similar lead graphic).

The 300-900k cycle C Stirling appears inline expectations similar lead graphics, despite a higher thermal ratio cycle AND lower volume ratio (300-900k cycle w DP/PP=1.5 vs 300-600k cycle w DP/PP=2). Note that during expansion, 1/3 of gas (10m of 30m) expands in DP while 2/3 of gas (20m of 30m) flows to PP...AT CONSTANT VOLUME just like lead graphic.

Meanwhile, check out 300-450k cycle A where 1/2 of gas expands TWICE despite 1.5 thermal ratio AND 1/2 of gas flows to PP...and EXPANDS. Is everyone getting this...the thermal ratio effects the gas flow in an unexpected manner whereby the "same" engine becomes a "different" engine depending on the thermal ratio !!! In this cycle A example, the gas flow goes waaay below 300k as it moves to PP (under 300k just past 3/4 transfer). So, PVT values may match expectations by the end of transfer, but the volume difference is a dead giveaway that something else happened.

Heads up...there's no math, computer sim or cutting metal that compares with parametric studies !!!

What's going on with cycle A ??? Note vol A = 300cc and vol B = 300cc. We expect that the 300-450k cycle will expand vol B 1.5x from 300cc to 450cc inline 1.5 thermal ratio. Indeed, this happens, but it expands another 1/3 from 450cc to 600cc. Likewise, vol A also expands "another" 1/3 from 300cc to 400cc. It's all about energy balance, but driven by pressure. So, both volumes may "over expand", however vol B is via heater input vs vol A is via ambient cooler "backflow".

Once you get your head around this, consider that this very cycle is the Ian-Hall pitch...

The plot thickens further.

Is everyone getting this...the thermal ratio effects the gas flow in an unexpected manner whereby the "same" engine becomes a "different" engine depending on the thermal ratio !!!

Very well put Matt, and something I should have been able to articulate in some other threads.

In this cycle A example, the gas flow goes waaay below 300k as it moves to PP (under 300k just past 3/4 transfer). So, PVT values may match expectations by the end of transfer, but the volume difference is a dead giveaway that something else happened.

Looking forward to some studies that break this down further. I have some ideas on how to "capture" this low temperature, but need more information from your graphics.

VincentG wrote: ↑Tue Jul 30, 2024 4:52 pm
Looking forward to some studies that break this down further. I have some ideas on how to "capture" this low temperature, but need more information from your graphics.

ice up a cold one...this is gonna shake stuff up big time with a bunch of answers that lead to even more questions !?!

Bump, and hurry up Matt

I need to preface a few things first...

(1) "m" values within earlier posts were sheer accident, merely chosen to compare Ian-Hall gamma vs Essex, but these m values proved very convenient
(2) double dwell simplifies study
(3) sequence from Vmin and Tmin with 1 bar charge

and recently established...

(4) any gamma has a constant compression ratio despite various thermal ratios
(5) any Essex has a constant expansion ratio despite various thermal ratios

While studying gamma vs Essex, I saw an opportunity to verify work vs Carnot via online calculator where I would simply calc Essex expansion work and gamma compression work then 'reverse engineer' both gamma expansion work and Essex compression work. And down the rabbit hole I went...


Here's the previous 300-450k cycle A with some graphic mods

(1) added yellow for dead volumes beyond regen
(2) nixed "monkey bar" conduit which was an early graphic evolution between hot and cold PP
(3) subdivided DP and PP volumes into 100cc volumes
(4) added enough add'l notation to choke a horse
(5) 2nd sequence includes 2 add'l frames between A-2 and A-3

Frame A-Z is Vincent's "zero point" where pressure after expansion equals pressure after compression. I found this parity relationship last year and have mentioned it several times, such as here

viewtopic.php?p=20376#p20376

viewtopic.php?p=21789#p21789

but I have expressed this the way I found it via

(Thigh-Tlow)/Thigh = PPvol/DPvol

which is rather hard to apply vs

Thigh/(Thigh-Tlow) = DPvol/PPvol

which results in the same values, but is much easier to grasp (try it and you'll see). And just so we're all on the same page, Vincent's zero point is this


where pts 1 and 3 (on this graphic) define an isobar, but said isobar does not need to be ambient P. However, the Ian-Hall model appears to use my PVT values (note pressure swing)

https://www.youtube.com/watch?v=SHyke4hUNOs

where video graphic is just as fuzzy as post here, but values can be deciphered.

I'm still working on a credible explanation for PP values during expansion. For now, I'm thinking that previous over expansion has some backflow heat input during transfer and as this approaches an adiabatic expansion (prior any 'isothermal' input) then less work is produced during expansion than an ideal PV plot would indicate.

matt brown wrote: ↑Sat Aug 03, 2024 4:20 pm
I'm still working on a credible explanation for PP values during expansion. For now, I'm thinking that previous over expansion has some backflow heat input during transfer and as this approaches an adiabatic expansion (prior any 'isothermal' input) then less work is produced during expansion than an ideal PV plot would indicate.

edit...in lower sequence, all DP temp values should read 450k (I was a tad too fast with cut and paste). Hopefully, I didn't botch anything else...

OK Vincent, here's something that will rekindle your Essex fascination. Chasing down this gamma expansion anomaly via 300-450k cycle A was getting nowhere (fast enough) so I decided to try similar 300-900k cycle C. Bingo, I had what I was looking for and only had to modify 30m gas to 36m to nix fuzzy fractional values. Behold the anomaly...

where a simple value table looks like this...
but obscures the anomaly. Take a close look at the graphic and notice

(1) A2-AX is 12m in 200cc of DP becoming 12m in 100cc of PP or 2:1 vol reduction
(2) AX-AY is 6m in 150cc of DP becoming 6m in 100cc of PP or 1.5:1 vol reduction
(3) AY-A3 is 6m in 200cc of DP becoming 6m in 200cc of PP or 1:1 no vol reduction

Exactly what is going on since there's only a few thermal processes and none of them make any sense...

Jeez, a couple minor errors in previous post and I should have included 300-900k cycle C as reference (top sequence)


hopefully, I didn't botch anything here...

Is everyone lost or just speechless ? Here's another view of this 300-900k cycle during expansion...


The DP gas in light red is mass that transfers to PP in next frame. In effect, DP gas transfers first at constant volume, then isothermally compresses. This is easiest seen when comparing CY-C3 transfer where 12m in 200cc at 900k and 1.5 bar equates to an isochoric conversion to 12m in 200cc at 300k and .5 bar then an isothermal 2:1 compression to 12m in 100cc at 300k and 1 bar. I cross-checked this fuzzy isochoric-isothermal pattern between all 4 frames to verify this pitch. Thus, PVT values in PP follow expectation AFTER transfer (during PP expansion) and PVT values in DP follow expectation during expansion, but the juggernaut remains as to how the actual transfer occurs. My best guess is that the gas flow is being BOTH isochorically cooled and isothermally compressed simultaneously during PP expansion. If so, then isothermal compression during regen likely taxes cold PP gamma beyond reprieve.

BTW no possibility of isobaric anything due to PVT constraints.

matt brown wrote: ↑Sun Aug 04, 2024 4:16 pm
This is easiest seen when comparing CY-C3 transfer where 12m in 200cc at 900k and 1.5 bar equates to an isochoric conversion to 12m in 200cc at 300k and .5 bar then an isothermal 2:1 compression to 12m in 100cc at 300k and 1 bar.

Yikes, another minor correction...everything correct above except than it's 6m not 12m (3 errors).

Also notice that during this CY-C3 transfer that the "previous" 18m in 200cc of PP at C-Y expands to 18m in 300cc of PP at C-3. Yep, a tad heady keeping track of various volumes thruout this sequence.

And to complicate stuff further, I still think previous "over expansion" is just minor PP expansion beyond zero point with isothermal backflow from cooler.

PV diagrams?. indicator diagrams? Simulations? Isn't it the path that is important too?

I managed to clarify this "anomaly" a tad more via unbundling gas flow more accurately.

Here's previous 300-900k cycle C with Ian-Hall volume ratio where DP/PP=1.5


Vincent's "zero point" relationship is hard to see in above diagram, but quite obvious here


The juggernaut with this buzz is that the volume ratio for this is NOT constant, but related to the thermal ratio (as previously mentioned). In this table, we can clearly see that the transfer volume remains constant thru this 'zero point' relationship. The thing to remember is that the DP gas will 'compress' in transit to PP proportional to the thermal ratio THEN expand until PP pressure equalizes with DP pressure. No doubt, a simple integral across entire transfer, but i'm bogged down with other issues first (integral appears same as isothermal expansion). Now, for anyone thinking that DP gas flow will simply 'fill the void' in PP pressure, a sharp eye (and simple wit) will notice that since both DP and PP have same pressure, that both will expand at the the same rate while each has their own input. Yep, the DP expansion is fed via source while the PP expansion is fed via (wait for it)...sink (aka thermal backflow).

For anyone doubting this Vr/Tr sensitivity, just feast your Vulcan squinties on this...


which I'll call cycle D, but the same as cycle C except that PP volume is reduced 1/4 to 300cc (vs cycle C 400cc) whereby DP/PP=2 now. Furthermore, the gas mass is reduced 1/6 (30m vs cycle C 36m) and the PP TDC is realigned. Now, check out the compression in PP gas vs previous cycle C gas flow expansion...(sorry about the graphic 'inversion' here, but I'm too lazy to redraw this one)

I've mentioned several times that any 600k expansion has only 2x work of same 300k expansion and nobody commented. The point to remember is that this is proportional expansion when isothermal such that ANY 1:2 600k expansion has 'only' 2x the work of ANY 1:2 300k expansion, so a 600k 100-200cc expansion has only 2x work of 300k 100-200cc expansion or 300k 500-1000cc expansion. Returning to 300-450k cycle A, the Ian-Hall model in very interesting...


Within this 300-450k thermal ratio, the zero point volume ratio is 3:1 where if we consider DP volume constant 600cc, then PP=200cc has DP/PP=3 vs Ian-Hall PP=400 has DP/PP=1.5 (common DIY ratio).

The Pmin values are after displacer reciprocates just prior compression. I added the 800cc PP as a lofty extreme, but a sharp eye will catch that DP and PP pressures are 'quickly' converging.

Nobody comments on Hall video where his 300-450k cycle with 400cc PP produce 30w at ~200rpm from 1 bar charge. Are you guys all asleep ??? Wake up and reread the Andy Ross book for his chase of 100w from 100cc. The funny thing about the Hall video is Hall himself distracted by engine running without any flywheel (eyes roll) meanwhile missing heat in vs power out.


Here's any interesting comp on gas flow where I compare DP expansion vs PP expansion on the extreme 800cc PP model. The yellow groups are equal except that DP is 1.5 greater than PP due to 450 vs 300k. Likewise, the orange groups are equal except that DP is 1.5 greater than PP due to 450 vs 300k. Indeed, vol E in DP is sucking heat big time, but if the sink input during expansion can equal the work loss (and heat loss) during compression, then this cycle has 100% eff on paper. IOW when the ambient 300k sink input during PP expansion equals 300k sink output during PP compression then Carnot looses. However, even when the sink input (backflow) during expansion is substantially less than the sink output during compression, this cycle appears to easily be Super-Carnot where Carnot=.33

OK Fool, there you have it, no ice cubes req'd except to chill another drink...