Ted Warbrooke's Stirling 1: Question

[Alphax]

Tom, I think I might be able to help you visualise the numerical value of 6(s2)/s3.

Here is the graphical solution for all cubes in the range side = 1 to 10 (any dimension you like, cm or kilometres or inches) showing the value (as a curve) of SA:V = 6(s2)/s3.

(You can ignore the curve for spheres which is 4pir2/(4/3)*pir3 - but it too follows a similar shape to cubes and most other simple 3D shapes)






Sorry - not sure why that image came up twice!!


But you can see that the smaller the cube (or sphere) the larger the ratio of surface area to volume becomes.



Back to the main point at issue here.


The ratio of surface area of the working fluid (ie the contact area between engine surfaces and working fluid) to volume of the working fluid is larger in a small engine than it is in a larger engine of the same design. The significance of that is a smaller engine running in steady state conditions will lose MORE heat (to the ambient environment) relative to its mass of working fluid than a larger one will relative to its own mass of working fluid.


Because most measures of Stirling engine efficiency take into account thermal "leakage" from the engine overall, it will always be true to say that this particular effect means that - in this one particular aspect - smaller engines will be intrinsically and unavoidably less efficient than larger engines of the same design.

BUT.... how much less efficient? We haven't discussed that yet, and I'm guessing there isn't a lot of appetite to do that!

It is an interesting way of looking at the problem of developing higher efficiency Stirling cycle engines, though, because you can apply similar (at least related) logic to the different phases (expansion/contraction/work etc) of the cycle as a whole and determine where the losses are likely to be significant and where they might not be so bad.

But maybe this isn't the time or the place.........?

[Tom Booth]

So, using that logic, are we to believe that the efficiency of our cube shaped engine will actually be different depending on the system of measurement used?

Take a cube 1x1x1 foot.

What is the SA:V if calculated in feet? inches? centimeters?

Is the efficiency of the self same engine actually altered by simply using a different unit of measure?

Measuring our 1 foot cube in feet, the SA:V is 6:1

In inches 1:2

In centimeters 1:5 (approximating 30cm/ft)

Egyptian cubits? Knotted rope?

So our engine will have different efficiencies depending on what country we are in? The historic time period and culture and what system of measurement happens to be in use?

[Alphax]

Tom, take it a step at a time.....

The engine isn't a cube shape. The cube is simply an arbitrary shape that follows the general characteristics of all three dimensional objects, including irregular shapes like engines (or even mice and elephants which don't actually differ all that much in shape but differ enormously in size). Whatever the shape, as a general rule if you make a larger version of the same shape you will always find that its surface area to its volume ratio is less than the smaller version you started with.

That is because objects scale in length, which is a linear quantity and also in (surface) area which is a square (ie length squared) quantity and volume which is a cubic quantity (length cubed).

Imagine you have a 3 dimensional object - say it is a child's wooden building block in the shape of a cube. Say that it is 2 inches along the side. We know that the characteristic length (along any of its 6 sides) is 2 inches. We know that its total surface area (A) is proportional to this length squared and that its volume (V) is proportional to this length cubed. That gives us 3 numbers : 2, 4 and 8 (notice there are 6 sides, but that 6 is a constant for all cubes in the universe irrespective of size, so we generally say "proportional to 4" rather than "equal to 6 x 4" which would be more exact).

Now, let us think what would happen if we wanted to go out into the garage (workshop) and make some extra large blocks for the child. Say we make them into cubes with a 4 inch side. What would the areas and volumes become? Well, if l=4 now, then A is proportional to l squared (ie 16) and V is proportional to (actually exactly equal to) l cubed which is 4 x 4 x 4 which is 64 cubic inches.

Forget the units (inches for length, square inches for area and cubic inches for volume) for just a moment. As long as we stick to one system it doesn't matter. Look at the rapid increase in VOLUME in the 4 inch blocks compared to the original 2 inch blocks. The new, 4 inch blocks have a volume of 64 cubic inches compared to just 8 cubic inches volume of the original 2 inch blocks.

So, doubling a characteristic length gives a characteristic area 4 times bigger (than the original) and a characteristic volume 8 times bigger (than the original). This is true for any general shape of 3 dimensional object (unless it has lots of very funny re-entrant shapes or fine hairs) including elephants and mice and Stirling engines.

Put it another way, scaling things up (or down) ALWAYS changes the relative ratios of length to area or length to volume or area to volume. Thats why small things and big things behave differently, especially when it comes to things like heat loss, even though those things are the same basic shape (but different sizes). It is actually a well understood area of study in its own right and has been since Galileo's time. He was the first scientist to observe that nature is not scale invariant - that is to say some things do not scale conveniently or as we might expect.

So, some things are said to be scalable, and others are not. Heat flow, especially heat transfer by conduction fr example does not scale in a simple manner and is a notoriously difficult subject in its own right and thats before you even consider any thermodynamic effects in gasses that may be contributory factors.


Your question about units is a good one, so I'll answer it in the next post (this is getting a bit long!).

[Tom Booth]

Your question about units is a good one, so I'll answer it in the next post (this is getting a bit long!).

Looking forward to that.

But taking feet, inches, centimeters...

The relative size of the units of measure, compared with the actual block, gives rise to the mathematical artifact of apparently different ratios.

Making the block itself bigger has the same effect. The chosen unit of measure becomes smaller, giving rise to the same artifacts, an apparent difference in ratio.

The ratio however, is not actually different.

[Alphax]

Tom

You asked about units.

The answer is that, as long as you stick to a self-consistent set of units it doesn't change the calculated efficiency of an engine, or indeed any other parameter of interest.

You suggested Egyptian Cubits. These are fine as long as you stick with them and use Egyptian Cubits for length (an Egyptian cubit is 18.24 inches) and Square Egyptian Cubits for area (a Square Egyptian Cubit is 332.7 square inches) and Cubic Egyptian Cubits for volume (a Cubic Egyptian Cubit is 6069.01 cubic inches). You will get exactly the same scaling results for the S/V versus size graph for cubes and sphere which is why the graph shows no units (dimensions) at all - it simply doesn't matter as long as you are consistent.


Having said that you need to be aware that you don't mix Egyptian Cubits with Cubits used by other cultures since they will be different. There are literally dozens of cubits and they are all different to the Egyptian Cubit. For example, there is an Assyrian Cubit, a Greek cubit a Roman cubit and many others from different times and different countries. They are internally consistent to their time and country of origin, but outside of that country and that time they differ by anything up to 3 inches (the general cubit being the length of a man's forearm). The typical range of cubits is anywhere from 17 inches to 19 inches depending on the local government of the day (and place).


Which takes care of knotted rope too. The standard British measurement of nautical speed is the knot (one nautical mile per hour) and is based - exactly as the name suggests - on a reel of rope run out by a sailor at the back of the boat to measure its speed. Every nautical mile a knot would run through the sailor's hand and he would call out "knot" while someone else counted how many. As long as the rope was allowed to run free and float on the water, all you needed to do was count the time interval between knots to calculate your speed. We still use them. But they aren't units of length - they are units of speed. So no good to us here !

[Alphax]

Tom, our posts crossed!


You are right, as long as you take care and are consistent in carefully using just one system of units you will always get the same ratios of interest.


Mistakes happen when one person (or a group of people) use one system of units and work with a different person (or group) who ASSUME they are using the same units as themselves.... when they are not!


This leads to expensive misunderstandings.

A good example is the NASA mars Climate Orbiter mission which involved Brits and Yanks working together. The Orbiter - as the name suggests - was supposed to study the surface of Mars from orbit and also act as a communications link for the Mars Polar Lander and for the Deep Space Probes.
However, it suffered a catastrophic navigation error caused by a failure to convert feet and inches into metres.

So... as long as we are consistent and stick to a choice of units, it doesn't matter what those units are (generally speaking).

[Tom Booth]

I think you're still missing/ignoring the point.

Measuring our 1 foot cube in feet, the SA:V is 6:1

In inches 1:2

In centimeters 1:5 (approximating 30cm/ft)

If the cube is enlarged from one foot to twelve feet to 30 feet we see the exact same progression.

6:1, 1:2, 1:5

Certainly the ACTUAL ratio, represented by the algebraic expression for the SAME engine/cube is not actually different just because we change our unit of measure.

That most engines are not cube shaped is irrelevant. I'm quite sure a cube shaped Stirling is possible.

[Alphax]

Time for bed!

[Alphax]

OK then.... here's a clue.

You are already mixing units in your example. The whole point is that we want to consider scaling up (or down) in size - do we agree?

On the assumption that we do agree (it really is bed time here, so I hope we do agree) we are categorically NOT interested in comparing a cubic foot measured in cubic feet with the same cubic foot measured in cubic inches or cubic centimetres. FORGET COMPARING THE SAME SIZE OF OBJECT IN DIFFERENT SYSTEMS!!!

That is why you are continually getting confused I think.

Instead just pick ONE system of units (doesn't matter which). Now change the size of your cube and recalculate SA:V IN THE SAME UNITS.

Now start again with A DIFFERENT SET OF UNITS (again it doesn't matter which as long as it is different). Now change the size of your cube (it has to be the same size as with your first choice of units so you will have to convert from one to the other) and recalculate the SA:V IN YOUR NEWLY SELECTED UNITS.


You will always get the same answer (ratio SA:V) for the same actual physical size of cube no matter what units you use as long as (a) you don't mix units when going from small cube to big cube and (b) the actual small cube is always the SAME PHYSICAL SIZE and the BIG CUBE is always the same physical size which means you have to know how to convert between your chosen units and ensure you don't vary what you mean by "big cube" and "small cube".


The ratios ALWAYS come out the same as long as you don't mix units up or compare cubes that are ACTUALLY different physical sizes when they are supposed to be the same!

Keep plugging away....... I think you've got it, really!

[Tom Booth]

If one side of a cubical engine is used for heat input the volume of that engine will always be the length of the heated side cubed.

The heat input surface area will always be the length of the side squared.

A consistent unit of measure for calculating proportionality is the length of one side of the given cube. That length will always be 1 cube side length. The heated area will always be 1 squared and the area will always be 1 cubed.

Now we can start slicing and dicing the cube into arbitrary units of measure and come up with seemingly different results, but that IMO, is just so much number juggling.

In other words, I don't think the size of an engine of a given design is relevant in regard to efficiency.

I've given my 2 cents, for what it's worth, as far as what I THINK MIGHT actually make for some improved performance, like using a non-heat conducting ceramic for the heat transfer tube instead of a heat conducting/dissipating metal with cooling fins. It seems to me using metal may be just throwing away heat and degrading efficiency. But that is untested, theoretical opinionated speculation.

The size of the engine may or may not be relevant. Such questions and issues are best settled by actual experiment.

I'm pragmatic. I have a similar issue with so-called "Carnot efficiency". It is a mathematical artifact. By that I mean it has no actual basis in reality.

Similarly, I believe the apparent Surface to Volume ratio difference between a big cube and a small cube is a mathematical artifact. The actual Real SV ratio of a cube is always 6(s2)/s3 regardless of size or measuring unit.

A cube is a cube is a cube.

All cubes are proportionately identical if a consistent self referencing unit of measure is used, i.e. the actual length of the side of the cube in question.

Take away all arbitrary systems of measurement, equiped with only a stick. I can cut the stick to the exact length of the side of the cube and using that method, the proportions of any cube relative to the length of the side of the cube, for that cube will always be the same.

Similarly with any given object of any shape.

Enlarging an object does not change it's proportions relative to itself.

Mathematical abstractions are often misleading and don't actually represent reality. An experiment can sort out fact from fiction.

In my experience, I started playing with engines as a very young boy with my older brothers, begining with tiny Cox engines on model radio controlled cars and airplanes, then spent the rest of my life working as an engine mechanic.

IMO a tiny Cox engine operates and performs just the same as a larger 10 horsepower engine of similar design.

I think if a tiny model engine works efficiently, the same engine scaled up will work just as efficiently.

That's just my opinion. You can take it or leave it.

[airpower]

......

That most engines are not cube shaped is irrelevant. I'm quite sure a cube shaped Stirling is possible.

Displacer can be any shape, as for the power side that would have to be more like a fire blower
https://www.youtube.com/watch?v=sSVx_QBx5j8

or
https://youtu.be/P0JWvEQPUJs?list=PLq1e ... fDraF54miB

there are also several flat solar setups
Stirling engines once again will rock the world.
No shape limitations can be operated with any fuel, even moonshine if someone likes to use green fuel.

[Alphax]

Hi Tom

In other words, I don't think the size of an engine of a given design is relevant in regard to efficiency.

I know! I can tell.

I suspect that size is one relevant factor, but you know that already. Happy to agree to disagree.

[Alphax]

Hi Airpower

Stirling engines once again will rock the world.

Well, I certainly hope so!

I really like the Stirling-Flachplattenmotor W1 - Waeller1967 has some very nice Stirling engine models on his Youtube channel.

[Tom Booth]

Just to give a response to your earlier question, not to be confrontational or "prove" anything or anybody right or wrong necessarily.

So, Tom, can you give a convincing argument as to why you think that a small engine is not somehow automatically less efficient just because it is smaller than an otherwise identical, but larger, version?

I ask because I think that a smaller engine is automatically less efficient just because it is smaller than an otherwise identical, but larger, version. And I have given my arguments (previous post) to explain why I think that.

I am prepared to be shown I am wrong on this point, but need to understand the reason if that be the case. Thanks.

Simply put, I have never seen anything that would indicate otherwise. Just direct observation

Just for example, there are numerous Youtube videos of tiny coin size LTD Stirling engines. They appear to run just the same as the largest versions ever built that I've ever seen.

https://youtu.be/dAV_nWTwWR4

Now admittedly, "appearing to operate just the same" is just my own subjective observation, not any kind of careful objective measurement of efficiency.

Like you, I'm certainly open to evidence to the contrary, but to date, I don't think I've seen any. A modification of a small model that improves mechanical efficiency will do the same for a full scale engine and vice versa.

At any rate, modifying and testing small models is economically more viable for me as I don't have unlimited funds to build full scale engines to try out every idea that pops into my head. I'm counting on the "fact" that successful modifications to my models will translate to full scale engines. That may not always be true for one reason or another, but, so far, I've seen nothing to contradict it.

I think if a ceramic heat transfer tube improves the efficiency of a small model, the same modification to a full scale engine will do likewise. I've not seen anything, ever, to indicate to me that a small model is inherently inefficient simply by virtue of being small.

By "efficiency" I mean, simply, available fuel utilization. Of course there is more heat available to a big Stirling with a proportionately big heat exchanger, so the actual power output will be greater, but horsepower is one thing, efficiency is another.

[Bumpkin]

Hey Alphax and Tom, this looks like fun so I’ll jump in and disagree with both of you. Firstly take a cube one unit high — six sides = six square units. Three dimensions is 1X1X1 units =1 cubic unit. Ratio of six to one area to volume.
Now compare to a cube two units high — each side is four square units X six sides = 24 square units. Three dimensions is 2X2X2 units = 8 cubic units. Ratio of three to one area to volume. And on and on.
But I disagree that for a given shape a greater surface to volume ratio is a disadvantage. As much thermal transfer area as possible (in the right places) is just what we want. That “in the right places” is where I reckon we could all agree, though I’m sure there could be many different “right’ approaches. I like pancakes myself.
Bumpkin