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

 

[Compositepro]

The data the OP presented simply shows the effect of loss of perfect packing at the walls as the ball size gets closer to the cylinder diameter.
Sphere size does not affect packing density at all except for secondary effects such as wall interference or surface friction. Most materials will have less spherical particles as the particles get smaller. Smaller particles have more friction and below a certain size will not pack well at all. Compare corn kernels to flour. Flour has a much lower bulk density.

As for packing lead into a cylinder, either melt it and cast it or use a hammer and ram rod to pack it. The above discussion is rather irrelevant to solving the problem statement. The basic math is pretty simple.

 

[Compositepro]

Here is a somewhat relevant video:

https://www.youtube.com/watch?v=SqGRnlXplx0

 

[Gary_321]

Here's the best that I can do in a couple of hours whilst watching telly...

It is only the bottom layer, but I think the maximum density can only be achieved if the bottom layer is as full as possible.

A is a single sphere in a cylinder
B is the next smallest diameter cylinder
C the next smallest
and so on.

I measured the nested height of each circle as 0.86d. There will be an accurate value for this on the internet.

There is definitely a pattern starting to form in the numbers, but I would need to carry on for many more layers to fully define what that pattern is.

 

[pylfrm]

The cylinder area is greater than the total sphere section area in most of the trials making this a 2-D problem.

The calculated cylinder height (approximately 0.159 cm) is less than the sphere diameter in most of the trials, so it seems something was reported incorrectly. Shame about that; I was interested in playing around with the data.

If we stick with the reported cylinder volume of 8 cm^3 but assume the cylinder diameter was actually 1.746 cm (mentioned by OP in an earlier post), the results can be approximated by assuming a packing density of 0.50 in the region within one sphere diameter of a wall, and 0.64 in the central region:

[pre]
sphere packing
diam. density
----- -------
0.305 0.54845
0.292 0.55117
0.254 0.55968
0.203 0.57244
0.178 0.57928
0.127 0.59446[/pre]

Those parameters are just the result of trying a few guesses until something looked decent. I'm sure the model could be improved with more data and proper analysis.


pylfrm

 

[hpaircraft]

One very effective way to add weight to an item of known volume is to send it to a place with a high value for G. Like maybe Jupiter or Saturn.

Oh, wait--we want to add mass, not necessarily weight. Right.

Lead is actually pretty easy to melt. With a garage-sale Coleman stove and some used kitchenware you can reduce the packing factor to the molecular level and avoid the pesky dust issue.

--Bob K.

 

[btrueblood]

But be careful you don't set off the powder charge in the shotshells you are reloading with that method.