Involuted planes within a circle, Klingelnberg-Palloid involute

[artis]

So this "seemingly simple" geometry and idea caught my attention.
See the video in the link from 9:00 minute


They talk about a specially designed nuclear fuel canister/bundle, now there is this geometry where they have a cylinder with smaller diameter and then a cylinder with a larger diameter, the inside between these cylinders is connected by bent planes which are in the shape which is called an involute, this is done in order to achieve the same spacing between the parts of planes closer to the inner diameter as well as those that are closer to the outer diameter, the part that I can't understand is how can one maintain the same size for both the spaces and the metal parts at both ends when the outer diameter clearly has a larger circumference?

I assume this same concept can be applied to spiral bevel gears in car axle differentials,
what is the maximum difference between inner and outer diameter of a cylinder where the involute shaped planes would still be able to have the same spacing difference both at the inner as well as at the outer circumference?

 

[Cutter Ketch]

Picture two flat parallel plates with a gap between them. Along a line perpendicular to the plates the gap and the plate thickness are the expected nominal values. However, consider a line at an angle to that perpendicular. The plate thickness and gap size along this new line are increased by $\frac 1 {cos \theta}$. Now consider their curved plates. Along a line which is locally perpendicular to the plates the gap thickness and plate thickness are the same everywhere. However, as you move further out from the center of the circle, the curve of the plates means that the tangent line is at an ever increasing angle to the local perpendicular. The apparent gap size and plate thickness are growing with $\frac 1 {cos \theta}$ allowing the same number of plates and gaps to fill the ever increasing circumference.

 

[artis]

I think I got the idea, sort of, I guess I want to ask what is the minimum angle at which the involuted plane can connect to the outer circumference in order for this to work, surely connecting straight lines from inner to outer circumference will mean a perpendicular or 90 degree connection but that as we know will result in the gap between planes being larger at outer circumference than inner.Also I wonder does the angle at which the plane touch the outer circumference can stay the same if the overall diameter of the cylinder is increased, or does increasing decreasing outer circumference diameter also means one must adjust the angles of the planes?

 

[Cutter Ketch]

I think I got the idea, sort of, I guess I want to ask what is the minimum angle at which the involuted plane can connect to the outer circumference in order for this to work, surely connecting straight lines from inner to outer circumference will mean a perpendicular or 90 degree connection but that as we know will result in the gap between planes being larger at outer circumference than inner.Also I wonder does the angle at which the plane touch the outer circumference can stay the same if the overall diameter of the cylinder is increased, or does increasing decreasing outer circumference diameter also means one must adjust the angles of the planes?

There isn’t a specific hard limit. As $\theta$ approaches 90 degrees $\frac 1 {cos \theta}$ goes to infinity, so mathematically you can stretch this trick out forever. However, there is a practical limit. What matters is the ratio of the radii of the inner and outer circle that are being connected by the plates. As you push that ratio higher the plates become more and more curved eventually becoming spirals that go all the way around the interior and further wrapping around again and again. Obviously you can’t easily manufacture something like that. Instead they do it in zones, for example, making N plates to connect cylindrical walls at r = 1 to r = 2, and then making 2 N plates to connect r = 2 to r = 4, etc. That’s just an example. Actually, since the angle to the tangent has to increase to 60 degrees (cos 60 = 0.5) I suspect a ratio of 2 is actually pushing too far.

 

[artis]

So I guess as long as the difference between the inner diameter and outer is not very large it works, then if the difference becomes too large the involute turns rather into a spiral but if it turns into a spiral then a guess a different geometry appears because the start of each spiral leg connected to the inner circumference will not be possible simply because of the other legs already passing over and so the gaps between each spiral will tend to be different at start right?

if one can allow for small difference in gap between planes at the start from small circumference center one could have ordinary perpendicular planes which then at some later point (larger circumference) begin to transform into a involuted shape , I guess that is another option of tackling the problem if the difference between inner and outer diameter is too great.

 

[gmax137]

See the video in the link

Thanks, I never knew much about the HFIR. The involute shape of the fuel plates is interesting. I think when they talk about the spacing between the plates they mean the perpendicular between the faces. That would be what the fluid flow depends on; the curvature of the plates is low enough that the flow "thinks" it is between two flat plates. Presumably the design keeps the local power-to-flow ratio relatively uniform.

 

[Cutter Ketch]

Also note that they don’t start with the plates perpendicular to the inner ring. The involuted shape has a lot of curvature starting from perpendicular. If you imagine starting from perpendicular and going from r = 1 to r = 2, you can chop off the part from r = 1 to r = 1.1 and make plates that work from r = 1.1 to r = 2. This effectively skips the part with the highest curvature and makes a more gentle easily manufacturable shape.

 

[artis]

@Cutter Ketch I'm not entirely sure I understood the idea you just said in your last post.

How about if one has a large difference between inner and outer diameter , one can just have the planes involuted to on side then in about halfway from center to rim start the involution angle to the other side?

 

[Cutter Ketch]

@Cutter Ketch I'm not entirely sure I understood the idea you just said in your last post.

How about if one has a large difference between inner and outer diameter , one can just have the planes involuted to on side then in about halfway from center to rim start the involution angle to the other side? View attachment 255855

RE my last point, yes, I can see where that may not have been 100% clear. I really need to draw a diagram.

RE your idea of stopping at some radius and beginning a new curve, there are two problems. First, since those are all continuous plates, the number of plates around the circumference doesn’t change at your discontinuity. That being the case, there is only one angle that fills the circumference without changing the gap size. Basically, you can’t do that. Secondly, a plate with a kink like that would be difficult to manufacture.

 

[artis]

well difficult to manufacture is not a problem here because I am trying to understand just in theory.
But why should the gap size change in this regard, at the point where the planes change curve the new plane that extends from midpoint to outermost radius could in theory have a different curvature if needed?

See the reactor canister, it basically has two tubes so this midpoint is nothing more than just a wall and after it the next set of planes begin and if my vision serves me correctly they have slightly different curvature and angles , but my point is this, what would change if we discarded the mid wall and simply changed the geometry of the planes midway? In theory I can't see a difference, if I am wrong please point this out to me.

 

[Cutter Ketch]

well difficult to manufacture is not a problem here because I am trying to understand just in theory.
But why should the gap size change in this regard, at the point where the planes change curve the new plane that extends from midpoint to outermost radius could in theory have a different curvature if needed?

See the reactor canister, it basically has two tubes so this midpoint is nothing more than just a wall and after it the next set of planes begin and if my vision serves me correctly they have slightly different curvature and angles , but my point is this, what would change if we discarded the mid wall and simply changed the geometry of the planes midway? In theory I can't see a difference, if I am wrong please point this out to me.

Outside the mid wall there are more plates. The number changes. They aren’t just continuations of what’s inside. They did that because they had gone as far as they could go with the inner number. The curvature and length were getting to be too much to manufacture. Adding a wall and starting again they get to start over. The reason the angle goes back to close to perpendicular to the wall is because they used many more plates in the outer zone.

 

[artis]

Ok now I understand, that makes sense.

 

[artis]

@Cutter Ketch, or anyone else for that matter, see the link and page 10
https://neutrons.ornl.gov/sites/default/files/High Flux Isotope Reactor User Guide 2.0.pdf

it seems that in the inner annulus they have 171 involuted plates while in the outer annulus they have 369.
I was thinking that as you increase radius by 2, say from 2 to 4 the number of evenly spaced involutes can be doubled but here it seems they bit more than doubled but I presume that is because there is a layer of separation between the inner and outer annulus so the radius increase not by 2 but by 2.100... something, is that true?

So if I wanted to make in each next annulus exactly twice more involuted layers than either I would need a transition that doesn't take up extra space or each next annulus would need to haev a bit larger space between each involuted plane?

 

[Cutter Ketch]

@Cutter Ketch, or anyone else for that matter, see the link and page 10
https://neutrons.ornl.gov/sites/default/files/High Flux Isotope Reactor User Guide 2.0.pdf

it seems that in the inner annulus they have 171 involuted plates while in the outer annulus they have 369.
I was thinking that as you increase radius by 2, say from 2 to 4 the number of evenly spaced involutes can be doubled but here it seems they bit more than doubled but I presume that is because there is a layer of separation between the inner and outer annulus so the radius increase not by 2 but by 2.100... something, is that true?

So if I wanted to make in each next annulus exactly twice more involuted layers than either I would need a transition that doesn't take up extra space or each next annulus would need to haev a bit larger space between each involuted plane?

Yes, I think that’s right.