How does sphere diameter affect bulk density in a given volume? 5

[pskvorc]

I am an EE, not an ME, so please accept my apology, (and point me to where it should be), if this is not the correct place to ask this question. I am a retired engineer looking to solve a 'personal' problem.

I have looked extensively for a solution to the below defined problem, and have found none. Let me save "you" time: I am well aware of the "64 to 74%" rules of thumb regarding packing density of spheres. Neither of those 'solutions' address my problem.

My problem defined is:
1) I have a cylinder whose dimensions (and therefore, volume), cannot be modified.
2) I need to add as much weight to that cylinder as I practically can using spheres of pure lead. (Rho_Pb = 11.34g/cc) (Price renders tungsten and gold "impractical".)
3) It is obvious that the smaller the sphere used, the higher the BULK density (weight) will be. However, there are practical reasons for not using spheres smaller than a given size. (At a point, spheres become "dust".)
4) The small-sphere handling issue does not have a threshold. Meaning that regardless of the starting sphere diameter, as sphere diameter gets smaller, the handling problems increase.
5) I want to maximize the weight of the cylinder, but I don't want to have to deal with the handling problems associated with "small" spheres. In other words; there is a point of diminishing returns where the gain realized in increased weight is offset by handling issues.

I want to be able to calculate BULK density in a CYLINDER as a function of sphere DIAMETER, when sphere density is known (and constant). The cylinder diameter and height are >> than the largest sphere diameter.

My problem restated in "simple" language is:
How does sphere diameter affect bulk density in a cylinder?

I have been to half a dozen "math" sites, and got NO help. Every time I have to talk to a mathematician I am reminded of why I am an Engineer.

Thanks!
Paul

 

[GregLocock]

Wood's metal will give you 100% fill, at 9.7 tons/m^3, rather better than your optimum with (tiny) single size lead spheres at 8.4

Cheers

Greg Locock


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[IRstuff]

You are seriously confused. And, you didn't bother to look at any of the literature I hunted down for you.

This is basic freshman physics we're talking here. One of the other poster alluded to this already. BCC-body centered cubic, and FCC- face centered cubic are the ideal packing configurations of spheres. Just as with circles on a plane, they cannot get any closer to each other than what's shown below. Those gaps are what limit the aggregate density of the arrangement. The size and scale is irrelevant. A set of circles with 10 mile diameters will have exactly the same packing density as a set of circles with 10 angstrom diameters. A similar argument applies to spheres. That's why the 74% limit applies to any size of sphere smaller than about 1/10th in ratio to the diameter of the cylinder.

And read the darn papers, or any paper on the subject. You're on a wild goose chase.

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[handleman]

Quote:
2) Assume the maximum "packing" of a liquid and obviously the volume of the liquid equals the volume of the cylinder. Now, stepping back from a liquid ever so slightly, to REALLY SMALL spheres, and I am unconvinced (via theoretical mathematical gymnastics) that the volume occupied by the spheres is ONLY 74% of the volume of the cylinder. Assuming that the ds is "microscopic", the volume occupied by the spheres is going to approach equality with the volume of the cylinder.

This is wrong. It's just wrong. The limit is based on geometry. It will most certainly NOT approach the volume of the cylinder. It only fails when you get into molecule sizes where geometry doesn't govern things. Saying "imagine liquid, then step back and have tiny spheres" is invalid. Liquid and spheres are fundamentally different in their geometry. If you have actual spheres, it doesn't matter how big or small the spheres are. As long as they are equal and they are densely packed WITH NO EDGES then the limit is 74%. It's a ratio of volumes. No matter how tiny your UNIFORMLY SIZED spheres are, the amount of space between them relative to the amount of space they take up is THE SAME. If you allow for different sized spheres, those smaller and smaller spheres can fill the spaces between larger spheres and approach liquid density like a Sirpinski carpet.

The thing that gets you away from the 74% is your CYLINDRICAL VOLUME CONSTRAINT. Your cylindrical volume constraint "breaks up" the theoretical best packing because you don't have those "partial spheres" that would be around the edges of your volume. The bigger your spheres are relative to the cylinder, the more mass of "partial spheres" that would complete the perfect pattern can't fit in there. As your spheres get smaller, they fill the container better and better. If you still MUST consider a liquid, consider the fact that the size of the liquid's "particles" is so small relative to any container we can imagine that we can't tell the difference in how well a 10ml beaker vs a 1ml beaker is "packed". We just have the overall bulk density of the liquid, which already takes into account the unimaginably small spaces between the "particles". If you could manufacture a container that only holds 6 water molecules, we could have this same "packing" discussion about those liquid molecules.

 

[pskvorc]

So, IRStuff, just for the record, you're asserting that there is NO correlation between sphere diameter and bulk density? Please confirm. I don't want to put words in your mouth.

OK handleman - I was just trying make a point, not draw a perfect analogy. No "need" to consider a liquid to make my point. Which,stated again, is "Is bulk density a function of sphere size?"

I'm waiting for IRStuff's commitment.

Paul

 

[IRstuff]

This is high school trig level calculation, so it's hardly "asserting," it's fact.

The only reason there's any fluctuation at all is as someone "asserted" that the walls of the bounding space don't fit like spheres would.

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[GregLocock]

Bulk density cannot exceed the Gaussian limit. In the case where the sphere diameter is within several orders of magnitude of the minimum dimension of the container there is no simple calculation that will derive the optimum sphere diameter for a given container. A non obtuse engineer would guess that the smaller the sphere, the better the packing density, in general. However there may well be specific cases where the packing density more nearly approaches the optimum, Gaussian limit.

So I'd expect to see curve of cylinder density vs sphere diameter that generally falls with sphere dia, but has various peaks where there is a particular match.

Cheers

Greg Locock


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[pskvorc]

Haven't got the time to wait for the experts to commit, so I'll post the 'data' and say "TTF".

Just a little Q&D exercise, because I already knew there was a non-random relationship between sphere diameter and bulk density.

On the left are the constants: Measured cylinder volume and measured pellet density: 8 ccs and 11 g/cc respectively.
Just for completeness, column "C" is the ratio of the diameters of the different pellet sizes to the diameter of the cylinder.
The numbers in column "K" (the difference between the actual cumulative weight of the pellets to the calculated cumulative weight of the pellets), verify the density of the pellets - 11.00 g/cc.
The data in the rest of the columns should be self-explanatory. The last column labeled "Ratio of absolute Density to Bulk Density" was what I was after.
The r² value on the graph suggests that a linear model explains 87% of the variability in the data. 89% for a Q&D little exercise isn't bad. I made no effort to have the column of pellets "settle" in the cylinder, so this is "random packing". While I'm fairly confident that the relationship between sphere diameter and bulk density isn't linear, I think most of the remaining variation IN THIS DATA SET is due to 'operator error'. You'll note that the actual results in column "M" don't get to the 64-74% rule of thumb, but are consistent with comments in the publications regarding the results of "random packing" of spheres in cylinders. However, where the publications stopped - probably due to editorial heavy-handedness for "rigor" - was an empirical model of what that relationship was. THAT is precisely why I posted the OP.

I didn't come here to pontificate, but given some of the responses, I don't want other people putting words in my mouth. So I will make myself as clear as I can. For those that want to adhere slavishly to theory and simulations, good luck in your careers. Theory and 'laws' (like gravity) cannot be ignored without peril. However, EVERY successful engineering company (I started and ran one for 23 years), has at least one engineer that the others take their problems to when "It won't work. I don't know why. Theory dictates that is should. All the simulations give the 'right' results." And before I get accused of being an "old fogey that doesn't appreciate "technology", I have been at the forefront of simulation and modelling my whole career. That 40+ years of experience with simulations tells me that simulations are only the starting point. It is only the womb of academia that affords the environment to play as if simulations were the "end". I am reminded of when breadboards were first starting to be used, and "things" weren't working "right". ENGINEERS were keeping their wiring in very neat, straight, organized bundles. Unbeknownst to those that couldn't see beyond the 'rules' were the currents induced by "neatness". A trivial example, but illustrates my point. At least from my perspective. As I pointed out in the OP, I had already read "the papers" on the subject before I came here. All I wanted was what I figured was something someone had already done but was so pedestrian that it didn't deserve "publication": A PRACTICAL model of the relationship between sphere diameter and BULK density. (By the way, there are "real world" applications of PRACTICAL bulk density that have very significant implications: Gun powder. Energy density (joules per gram) is critical to ammunition manufacture. BULK density values are critical to prevent "unpleasant" outcomes when igniting progressive powders in confined spaces, and little circles in square boxes, and simulations, don't "get it".)

I thought I might avoid having to "do the work" to come up with a model that I could use. (I figured somebody already had.) But the arrogance in some of the responses here forced me to get off my butt and prove what I already knew but didn't have the numbers for: That there is in fact a relationship between bulk density and sphere diameter. To suggest otherwise, by an ENGINEER, is just stupid.

To those of you that courteously responded, I thank you for taking your time to offer your help.

The the rest:
TF (Ta Ta Forever)
Paul

PS - Greg Locock: Best answer given. Thanks. However, I wasn't after a "simple calculation that will derive the optimum sphere diameter for a given container" because - as I stated in the OP - "optimum" is decided by me based on my willingness to deal with "handling issues". I simply wanted the curve to which you alluded.

Paul

 

[TheTick]

The “theory” behind this problem is still bleeding edge mathematics. The assertion that this is high school trig is a gross oversimplification.

There are some very smart people working on this exact problem. There is a lot of interest from many quarters: energy & battery developers, defense, aerospace to name a few.

That said, experimentation is probably the best route to useful answer.

A recent published paper explored the effect of different shaped pellets. Turns out the optimal shape is that of a plain M&M candy.

 

[3DDave]

The -maximum- value for sphere packing is still high school mathematics. Except for a few special cases, there are only statistical distributions, which aren't "so pedestrian" they don't deserve publication. There is certainly no 'curve' that the OP was looking for. Even 1-D packing is discontinuous for small ratios of container size to item size. There's no sense in believing there's a better result for 3-D packing.

Most packing interest is in materials that either aren't spherical or aren't uniform in size or both, so are outside this investigation.

There is no relationship between bulk density and sphere diameter - that's an axiom. All packing is based on the unit diameter sphere and scaled as required. There is a relationship between bulk density and packing method, which wasn't distinguished in the simple experiment. Had he started with a 2D analysis, as I suggested, this would have become clear and is clearly demonstrated in papers that discuss the contribution of jamming as the limiting case for packing, but for any non-trivial container there can be thousands to trillions of alternate packings that work.

There is also no relationship between the molecular behavior in liquids that depends on sharing electron clouds between flexible molecules and the behavior of idealized discrete rigid spheres.

As to the spreadsheet - it tells an interesting story:
Notes:
1 Typical density of lead is 11.4 gm/cc; somewhat higher than the material in question
2 The density of the material changes depending on diameter, apparently
3 The cylinder area is greater than the total sphere section area in most of the trials making this a 2-D problem.

 

[MacGyverS2000]

I say make a mold the same diameter as the tube, pour molten lead into it, then slide a properly-lengthened chunk of lead into the tube. Done. Nearly 100% packing efficiency, allowing for shrinkage and friction fit.

Dan - Owner

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[IRstuff]

"The assertion that this is high school trig is a gross oversimplification."

The calculation of FCC or BCC theoretical density is trig.

TTFN (ta ta for now)
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[GregLocock]

I think the OP has lost interest. Shame, as the subject is a complex piece of analysis with real world applications. There are a few examples of high density with large balls that are fairly easy to calculate, but I'm not going to do the hard yards.

The first obvious one uses balls of a diameter of 1, and you can fit floor(L) in. The next is where the balls lie on a single pitch spiral, where you can fit floor(L)+1 in. similarly for floor(L)+2 etc.

Then I suspect there is a solution with the crystal stacking formation 2-1-2-1- etc

Then there's 3-1-3-1-, and 4-1-4-1-. then 3-3-3- and 4-4-4-

each of these will have variations depending on whether you can reduce d slightly and fit an extra layer in as a result. The addition of the L constraint makes things far more complex. Perhaps the thing to do is assume an infinitely long tube and then fiddle around.

I suspect crystallographers would have a bit of insight into approaching this.






Cheers

Greg Locock


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[chicopee]

Looking at your linear regression curve for the range of spheres, you don't have a solution. Then accept the fact that the smaller sphere size(acceptable to your work) is the answer. I would try non linear regression but from the data you'll get the same reasoning. Perhaps, lower the limit of sphere size on your data but I sense the same outcome.

 

[EdStainless]

And as was hinted at earlier, mixing sizes will increase packing density.
This works for sand, powdered metals, and spheres.
The first pass is to assume a second size that will just fit into the void between the larger spheres, but in reality a mixture of sizes over some range packs even better.
Getting the correct range of sizes, and distribution among them is a non-trivial problem.
If this were my issue I would buy the 3 or 4 smallest commercially available shot sizes and begin testing.

= = = = = = = = = = = = = = = = = = = =
P.E. Metallurgy, Plymouth Tube

 

[3DDave]

I just remembered my first introduction to 2D circle packing - the Ideal Toy "Booby Trap" game. Due to jamming there would often be a number of pieces that could be removed without affecting the spring loaded bar.

 

[btrueblood]

I do recall seeing, somewhere, an analysis that assumed a gaussian distribution of sphere sizes, i.e. manufacturing tolerances. I seem to recall that having a fairly coarse distribution of sizes resulted in better packing fractions, both analytically and in experiments. Thus, the graph of the OP's showing improving packing fraction with decreasing pellet size could be at least partially due to the increase in size distribution (harder to control size of shot as the dimension decreases).

 

[IRstuff]

Mixed sizes don't tend to want to stay mixed, and I think they actually tend to segregate themselves into layers. Even if they don't separate, there's nothing preventing a smaller sphere from getting itself inserted into the structure so as to maximize and not minimize the gaps.

TTFN (ta ta for now)
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[TenPenny]

I suggest reading this:

http://mazziotti.uchicago.edu/journal/SJGessner.pdf

 

[1gibson]

Simplifying to 2D, let's say one "row" of X diameter spheres in the cylinder is 10 spheres.

At 2X the sphere size, you can fit 5 spheres, no surprise there.

At 0.5X sphere size, it just so happens that you can fit 21 spheres, more weight. Good job, you got lucky.

At 0.25X sphere size, it's only 39, unlucky, oh well.

Aside from actually "counting" (or modeling) the number of spheres that fit in that exact diameter cylinder, there's no right answer. Certainly not a linear one, and the solution is unique to the diameter. And when you get to 3D, the interaction is obviously more complicated.

 

[Gary_321]

Can you tell us the internal diameter of the cylinder and the smallest sphere diameter that you consider practical?

If the base layer is a full layer of spheres, the answer will be denser than if the base layer is not full.

A full base layer will be a function of sphere diameter to cylinder diameter, but I don't know what this function will be. I'm going to spend the next few hours playing with numbers and circles .