Ted Warbrooke's Stirling 1: Question

[Tom Booth]

Tom,

when you do your calculations you will get the same answers as shown in the table at the bottom of that Wikipedia page (assuming you do them correctly!)


https://en.wikipedia.org/wiki/Surface-a ... lume_ratio

The table uses the following sizes of cubes (called "side of cube"): 2, 4, 6, 8, 12, 20, 50 and 1000. The units (feet, inches, cubits) are irrelevant and not given as they apply to any unit of length (provided you don't change units when calculating your ratios).


In other words..... you should get the same answers as Wikipedia for the ratios of surface areas to volumes (3:1, 3:2, 3:3, 3:4, 3:6, 3:10, 3:25 and 3:500 (or, if you prefer to see the ratios expressed as numbers, then 3, 1.5, 1.0, 0.75, 0.5, 0.3, 0.12 and 0.006) if you use those cube sizes.


If you don't get the same answers for the surface area to volume ratios for those cube sizes then you have made a mistake somewhere.

So.... you aren't really arguing with me as a person, but with mathematical reason itself. I'll be interested in seeing your arithmetic steps.

Good luck!

IMO (ahem, not really just opinion IMO but anyway) all that Wiki stuff is nonsensical as it produces inconsistent results. That is, the resulting ratio changes when a different unit of measure is used, as was already demonstrated earlier. You suggest ignoring this discrepancy, but it is evidence of an underlying problem. The actual ratio does not actually change so attempting to get the same WRONG answers as Wikipedia proves what?

My approach is quite different.

We are all, presumably, familiar with the temperature scales and conversion formulas.

For example, to convert Celsius to Fahrenheit:

C=(F-32)x5/9

Fahrenheit to Kelvin:

K=(F-32)x5/9+273.15

Etc.

So, postulating that the ratio between the surface area and the volume of an object; a cube, for example, is actually fixed, it should be possible to work out conversion formulas for converting the surface area into volume and volume to surface area consistently.

So, how to begin?

First, put the scales side by side, begining at zero and look for patterns:

Resize_20220209_075911_1170.jpg (307.36 KiB) Viewed 3499 times

I did this with surface area and volume.

There was indeed, a consistent pattern, as can be found when comparing two different temperature scales.

From there it was possible to work out the conversion formulas:

To convert surface area to volume use:

V=SA(n)/6

Where n is the length of the side of the cube using any unit of measure.

To convert volume to surface area use:

SA=6V/n

This gives consistent results across the board using any unit of measure, including fractional (example: cube length = 6 and 3/4") etc and for any and all size cube.

If there was really some physical progressive divergence in the ratio between smaller and larger cubes, no such consistent conversion formulas would be possible.

As with the various temperature scales, there is the measuring units which produce conflicting results numerically, then there is the actual reality

Water always freezes and boils at the SAME temperatures, though the various temperature scales show different values.

Likewise, the ratio between the volume and Surface area of a large and small object do not actually change when scaling up or down.

[ccspring3021]

Yes, I agree with you. "Thermoacoustic" is amazing but I cannot find any high-power stirling take use of its structure. Maybe the internal air movement is complicated when the machine become larger.
I'm happy to see some geniuses come up with innovative solutions to improve the thermoacoustic engine so as to improve the power, but I know I'm not such a genius.
So my idea is to think about whether it is possible to change the structure of the existing high-power Stirling machine to achieve better purposes, such as simpler, more efficient or other strengthening.
For example, I noticed that some people want to transform the displacer from the piston form to the rotating form.

[Alphax]

Tom,

Firstly..... thank you for taking the time and trouble to do the arithmetic. I appreciate it.

Secondly..... you have not done the last step - which is to divide the Surface Area by the Volume (in each row) to derive the Surface Area to Volume Ratio (SA/V).

Thirdly....... but when anyone looking at the table of numbers in your photograph does that simple series of divisions they get PRECISELY the same numbers as in the Wikipedia table!!!

In other words.... just carry on doing the arithmetic and you will see that the SA/V ratio truly is different for each size of cube in your list - I have done that arithmetic for you and written the answer on your own photograph so that you can double check.

For example pick your working out for Surface Area (SA) when side = 6 and we see that you (correctly) calculate SA=216. Now compare that to Volume (V) when side =6 and you (correctly) calculate V=216. Now divide SA by V and you get 1.0 (which is also correct, but you haven't done it).

Now if we do the SAME CALCULATION for each value of cube length (the ones in your list) you will get the result that the SA/V ratio varies with the size of the cube!!

[Alphax]

So, Tom,

Just look at the column of figures that I have written on your sheet (photo in my previous post). You will see the numerical value of the ratio SA/V which is calculated using exactly the numbers you have just calculated.

The numerical value of SA/V in my column (column heading is 'RATIO SA/V') is 6, 3, 2, 1.5, 1.2, 1.0, 0.86, 0.75 etc. etc. NOTE the numbers are different to each other! That is the whole point - those numbers are the numerical value of the ratio of the surface area to volume for each cube and they are different because the size of cubes are different.

[Tom Booth]

Well, I guess there's no hope. You don't get the point

If you take a rectangle that is say 5x10 inches

If you make the measurements in centimeters the 1:2 ratio does not change, because it is a real ratio based on real physical dimensions on the same two dimensional plane.

Ratios are not supposed to change just because a different scale of measurement is used

SA:V however does change when different scales of measurement are used, because linear measuring units are being used to compre 2D surface area with 3D volume. This gives rise to weird mathematical anomalies.

It is like trying to measure the "ratio" between Celsius and Fahrenheit and finding that the ratio changes as temperature rises or falls, which is nonsense.

But anyway, I tried

[Alphax]

Tom,

If you make the measurements in centimeters the 1:2 ratio does not change, because it is a real ratio based on real physical dimensions on the same two dimensional plane.

Ratios are not supposed to change just because a different scale of measurement is used

That is only true if both the numerator (the variable above the division line) and the denominator (the variable below the line) are NOT raised to a power. As soon as you have different exponents in the ratio then the numerical value changes and is not constant.

Area has an exponent of 2 (a number raised to the power 2, also known as squared).

Volume has an exponent of 3 (raised to the power 3, or cubed).

Dividing a variable raised to the power 2 by a variable raised to the power 3 always gives a different numerical value when the variable is changed.


Lets leave it there..... as you said earlier when you did a bit of research, the world (and me) have a different interpretation to you!

If you ever get the chance I suggest you ask someone with some understanding of maths - this is basic high school stuff, not difficult, and even your own list of arithmetic values confirms that you get exactly the same result as the rest of the world (and me) but for some reason you just aren't seeing it.
They are not "weird mathematical anomalies" as you put it! It is just that you haven't "got it" yet, but ask a friend to explain it to you and suddenly a whole world of understanding will be there for you, just like someone switching on a lightbulb. Honestly - you'll glad you did!

Also - to be clear - it has absolutely nothing whatsoever to do with "different scale of measurement". Somehow you've gotten confused on different units which is utterly irrelevant to the basic maths.

[Alphax]

Tom, also.... just a small point. You cannot "convert surface area to volume". Volume is volume, surface area is surface area. They do not convert from one to the other.

But I do see what you are getting at because if you know a cube length you automatically know the cube volume and the cube's surface area, but physically there are not convertible from one to the other. The equation that you present allows you to to calculate the volume of a cube (but not any other shape of object) from knowledge of either the cube length or the cube surface area (they both carry the same information value).

Sorry if that sounds "picky" but it is the sort of thing a maths teacher would want to explore with you, so again, I urge you to ask someone to go over this with you because you seem to have both the aptitude and the motivation to "get it right" and you are so close! You won't regret it, I promise.

Or just get a basic maths book that covers ratios with (different) exponents in the numerators and denominators.

[Alphax]

Hi Tom

I thought you might like this explanation (you might have already seen it).

I think it is helpful because it gives a nice example of how the ratio (SA/V) changes when you simply whack a brittle object (like a lump of coal) into pieces with a hammer - you produce vastly more surface area (because it is now a powder) but the mass hasn't changed therefore the volume hasn't changed. But because the surface area has changed the ratio SA/V has changed as a result - and all because you hit it with a hammer!

enjoy ... (please do read it carefully)

https://simple.wikipedia.org/wiki/Surfa ... lume_ratio

[Tom Booth]

Hi Tom

I thought you might like this explanation (you might have already seen it).

I think it is helpful because it gives a nice example of how the ratio (SA/V) changes when you simply whack a brittle object (like a lump of coal) into pieces with a hammer - you produce vastly more surface area (because it is now a powder) but the mass hasn't changed therefore the volume hasn't changed. But because the surface area has changed the ratio SA/V has changed as a result - and all because you hit it with a hammer!

enjoy ... (please do read it carefully)

https://simple.wikipedia.org/wiki/Surfa ... lume_ratio

Again, that is nonsense and inapplicable to "scaling up" or down and the ratio between parts of two different size objects.

Of course if I break or cut a large block of ice into smaller ice cubes COLLECTIVELY all the small cubes together will have more total surface area. That is a vastly different situation from comparing the SA:V ratio between ONE large block and ONE small block.

I don't believe mathematics is all-over-the-place without rhyme or reason, just because of squaring while using different units of measure.

The self same cube cannot have a surface area to volume ratio of 1:5 AND 6:1 or whatever it was I calculated previously SIMULTANEOUSlY just due to using different units of measure.

That would imply that the same engine with those completely opposite ratios would be both physically MORE efficient and also LESS efficient just by measuring it in either feet or centimeters.

It certainly IS possible to convert between two different scales of measure. I'm not actually converting volume into surface area, obviously, just illustrating that it's possible to convert from one scale of measure to another, and that this completely removes all the discrepancies immediately.

Inch, foot, yard, centimeter etc. are all arbitrary and lead to odd ball results.

The only measure that is not arbitrary is that based on the actual object itself, the length of some part.

The actual parts are always proportional in exactly the same way when any actual part is used as the unit of measure and used in relation to the other parts. Then there are no discrepancies and everything works out very consistently.

There is certainly nothing wrong with using part A of an object and comparing it with part B and finding that part A is ALWAYS twice the size of B. That is the true ratio, and there is no inconsistency.

Introduce arbitrary and inconsistent, inapplicable units of measure however and the numbers are all over the place.

I say there is something wrong with this SA:V scaling up ratio change business, and there needs to be some reality checking applied.

The surface area of a "BIG" cube cannot be both larger, the SAME AS, AND smaller than that of the "SMALL" cube, in relation to their respective volumes all at the same time, but this SA:V ratio theory says that's just A-OK.

Sorry, but I don't buy it.

Sure, we can leave it at that for now, but IMO I think I've already provided ample evidence that there are inconsistencies in this SA:V ratio theory that need explaining it just doesn't add up

But, maybe you could explain, what actual problems this presents, or what problems it is supposed to present, when scaling up a model engine to full scale size.

Personally, I don't think it is actually any issue at all, just a numerical quirk of trying to compare 2D and 3D reality with the same linear units.

IMO, using the length of the side of the actual cube as a unit of measure, is a valid approach, that eliminates all inconsistencies. By doing so, the actual ratio of all the parts, including total surface area and volume maintain all the same proportions with mathematical precision.

But, regardless, what exactly is the relevance to engine building, scaling up, efficiency etc. ? Why do you think this is actually an issue in the first place?

[Bumpkin]

Well as I said, I like pancakes. Perhaps I should have said “structure” of pancakes but given the topic —. Anyway, for STIRLING engines, (not THERMAL-LAG engines like the Ted Warbrooke engine,) I think a slightly conical, but otherwise flat-as-a-fritter Beta design is the way to go. Conical to give reciprocating and pressure-bearing strength to the displacer/regenerator and the matching chamber ends; otherwise flat to give the greatest surface area for heating and cooling - both internally and externally. I also THINK that keeping the reciprocating regenerator close to the ends would let it better radiantly absorb/emit as a heater/regenerator/cooler — cooler/regenerator/heater, but I also question whether proximity is really important, since radiation seems to travel any distance until it hits something. Yes the thin perimeter of a pancake is a problem to thermally isolate, but I believe the difficulties are offset by the advantages.

Thermal-Lag engines are entirely different. I don’t see how they could possibly have a regenerator since it would reverse the thermal lag, but I’m always open to new understanding. I think the wool seen in many of these engines is (as in my example above,) a simple way to transfer radiant, and in this case, conductive heat to the air. If some method of flow trickery is used to assist the delay, then the heating needs to come quickly. Typically at the other end, the cooling delay is assisted by no more trickery than the unshrouding of the cylinder by the piston. If you graph it you can see that regardless of those effects, there is still a higher average pressure on the expansion stroke than on the compression stroke, (I believe that’s the premise of the original Tailer patent,) but the effect of the cylinder unshrouding is said to benefit, so a pancake design might need other trickery I can’t quite visualize. I try though — Thermal Lag engine simplicity is enticing.

Bumpkin

[Alphax]

Bumpkin:

Thermal Lag engine simplicity is enticing.

Which is why it is the topic of the thread.

I'm not saying there is anything wrong with "pancakes". But the ones I'm aware of are all LTD design and therefore of extremely limited power.

They also have more moving parts (displacer).

If we have to pick a name for the type to which Warbrooke's Stirling-1 belongs I think it would be Thermal Lag, as you suggest.

[Alphax]

Bumpkin:

Thermal-Lag engines are entirely different. I don’t see how they could possibly have a regenerator since it would reverse the thermal lag, but I’m always open to new understanding.

I don't think they are entirely different, but they are different.

One particular Thermal Lag Stirling engine has been designed, built, instrumented and extensively tested by Calcoen and Vandermeersch. This perhaps still to this day the only Thermal Lag engine which has been built based on understanding harvested from previously published literature. The engine was also instrumented to measure what is going on within the engine.

So - obviously - they had to read, process and understand that literature before getting started on their build. Calcoen and Vandermeersch have published the main findings of that literature review and report that much of the existing literature at that time was contradictory. In simple terms it could not be deduced how the Thermal Lag engine actually works with any degree of scientific consensus.

In particular, two world experts (Tailer - the inventor of the Thermal Lag engine and Allan Organ - arguably the world's greatest expert on the workings of Stirling engines) - fundamentally disagreed on how they work.

Moving forward in time....

It is called a Thermal Lag engine by Tailer because he realised that the limiting factor of Stirling engines’ power output is the time it takes for gas to exchange heat. This so called ’thermal lag’ is what inspired Tailer to build an engine that, instead of being limited by it, uses it as a driving phenomenon.

The implications of this idea (of thermal lag) are profound because the lag can be thought of as the phase lag usually managed by the displacer at the hot end. Therefor (the idea is) you no longer need a displacer cylinder, a displacer piston, displacer links and fluid pathways or the associated metalwork if you change the design to exploit the inherent lag instead of working against it (with a displacer and all the friction losses that entails).

So.... does a Thermal lag engine actually have a regenerator? I think it probably is doing two jobs - it is the heat source and it has large thermal gradients across the "wad" of wire because it has such a colossal surface area to volume ratio. If you subscribe to that view, and the idea that the "lag" is equivalent to the traditional displacer piston phase lag, then yes - it has a regenerator.

[Alphax]

So..... back to the original question: does the essence of the basic Thermal Lag engine design scale up?

Is bigger better - if so why?

And if bigger is worse - then why?

I'd be very interested in hearing any opinions whether theoretical, practical or just gut-feel.

[Alphax]

While you are scratching your heads about that (is bigger better/worse/why), consider this:


- it is entirely possible that the wire wool at the hot end is not merely a "regenerator" but is actually controlling the magnitude of the lag itself. In other words, much of the lag (most perhaps?) occurs largely within the wire wool mass itself.

That is an interesting idea because the wool has a colossal surface area to volume ratio, and the finer the grade of wool, the larger that ratio becomes in any given volume of pack. Changing the grade of wool (fine, coarse etc) but keeping its mass the same will change the performance of the engine if this is even partly true.

Scaling the whole engine up but keeping the grade of wool the same has the same effect.

This is why the question at the start of this thread was written the way it was..... the implication (perhaps a little too deeply buried) being that if you scale up a Thermal Lag engine, you may need to pay particular attention to the surface area to volume ratio of the regenerator material if it is controlling both the phase lag (by transient gradients) AND the hot end heat transfer.

Looked at another way.... one of the design parameters which is already recognised is the issue of how much wool you require and where exactly to pack it within the tube. But it seems that the wool may have at least one more trick up its sleeve - it may matter what grade you use (and, of course, how tightly or loosely you pack it).

[Nobody]

It is very clear now to me why Tom won't ever understand the simple concept of the Carnot limit.

Clearly the function of SA/V ratio is not constant. It is a function of Side Length. And therefore his subterfuge is nothing more than a straw man argument, at best.

Changing units does change ratios. The ratio between freezing and boiling changes, when comparing Celsius to Fahrenheit. Tom brought that up and didn't even notice.

Thank you very much Alphax.

It is very clear that as engines get smaller the hot side gets closer to the cold side, the straight across wasted heat loss will be more and more difficult to prevent, higher. Smaller engines will have lower efficiency. For the same design, size being the only only change.