How does sphere packing fraction in a long cylinder change with sphere size?

Question

Earlier I had to cut up some materials into little pieces and fit them in a glass tube, and I wondered if it's better to cut the pieces as small as possible, or if it wouldn't matter. If we think about sphere packing in infinite 3D space, then the optimal packing fraction is ~74% for spheres of any radius, so making the spheres smaller offers no advantage. I assume things aren't as simple in a confined volume, though. Have there been any studies on sphere packing in a cylinder? How does the packing fraction change with sphere radius?

Answer 1

While typing up the question, I had the insight to simply Google "sphere packing in a cylinder" and found this paper, which gives the answer to my question in figure 3. Basically, the packing fraction starts out high (67%) when the spheres and the cylinder have the same diameter, then drops very low (33% at sphere/cylinder radius ratio of 1.6-1.7), then rises to just over 50% for a cylinder/sphere diameter ratio of a little over 2. (Presumably, it eventually gets up to 74% as the spheres get infinitely small.) So, if we model cutting up a material into little pieces and fitting them into a tube as sphere packing, then it's probably best to make the pieces small enough for a cylinder/sphere radius ratio of >2.

I welcome any comments.

Answer 2

In looking at packing, you must look at the upper and lower limits.

In one limit, the radius of the spheres is equal to the radius of the cylinder. Each cylinder segment that had only exactly one sphere in it would have a cylinder volume of (cross sectional area x height ).

The cross sectional area would be 4 ( Pi / 4 ) R^ 2 = Pi * R^2 , and the height would be 2 R, so the volume would be 2R x Pi R^2 = 2 Pi R ^3.

The volume of the sphere would be 4/3 Pi R^3, so the empty space would be the difference ( 2 Pi R^3 - 4/3 Pi R^3 ) = ( 6/3 - 4/3 ) = 2/3 Pi R^3.

The empty space would be 1/2 of the volume of the sphere that just fits inside the cylinder.

So 3/3 Volume is cylinder, 2/3 Volume is sphere, and 1/3 Volume empty space.

In the other limit, the spheres would be extremely small and arranged in offset Hexagonal closest packing. There would be one sheet of all hexagons of spheres overlain by another offset sheet of hexagons of spheres such that every sphere in the upper sheet sat in the valley created by three spheres in the lower sheet.

This arrangement has much less empty space, as each sphere touches 3 + 6 + 3 = 12 spheres in three layers.

In the other limit, R = R , it can only touch two other spheres.

Mike Clark Golden, Colorado, USA