Pattern Recognition of Random Corks

Question

Firstly I apologise in case the tags are inaccurate. I simply don't know where this question fits.

Suppose I have a big hollow cylinder with an open top so that I can put things inside it. As a matter of fact I decide to fill it with wine corks. I know the dimensions of this cylinder and of the corks.

The thing is that, once the cylinder is full, I would like to estimate how many corks we have inside. The problem, of course is: How am I supposed to estimate, in a random configuration, the volume the corks are not occupying. How am I supposed to model this random configuration? It seems there is some reseblance with the lattice configuration of cristal, although it is random.

Of course, here, we are not considering gravity (unless it does not make things difficult). I've tried to think about minimizing the energy of the system but couldn't come to a conclusion.

Also, I would prefer an approach that does not involve simulation, if it is possible. Any thoughts?

Answer 1

Clearly you won't expect a precise answer. I can just try to provide a sound estimate in the engineering way.

Let's denote with $R$ the radius of the container, and with $r$ and $h$ the radius and height of the corks, assumed to be cylindrical.

If $r << h$, i.e. if we had nails instead of corks, they will tend to lie horizontally at the bottom. If we "pour" them slowly while constantly and mildly shaking the container, the first nails upon bumping on each other will tend to align almost parallelwise: many of them will not end to be "tip-to-tip"
and so remain a bit inclined.
As the number increases, those being pushed toward the circular wall will tend to be blocked "secant" to it, and induce to be parallel to them other two or three adjacent nails.

Passing to corks , with a much reduced $h/r$ ratio, the situation will be almost the same, except that the "tangential" corks will not induce much other to parallell them, and instead some blocks "head-to-side" ( $T$, $\Pi$, rectangular, ..) will appear.
Such blocks, if the $h/r$ ratio is integral, will be quite compact and won't disrupt much the compactness of the layer.

That said, for a configuration with the corks fully aligned

we can compute that a layer will contain max $$ L = 2\sum\limits_{k = 1}^{\left\lfloor {{R \over {2r}}} \right\rfloor } {\left\lfloor {{{2\sqrt {R^{\,2} - 4k^{\,2} \,r^{\,2} } } \over h}} \right\rfloor } = 2\sum\limits_{k = 1}^{\left\lfloor {{R \over {2r}}} \right\rfloor } {\left\lfloor {{2 \over {h/R}}\sqrt {1 - 4k^{\,2} \,\left( {r/R} \right)^{\,2} } } \right\rfloor } $$ corks.

If we take the corks of the 2nd layer to lie across the first one, and so that a layer be $2r$ thick, we are mitigating for the mis-alignment.

In conclusion for corks , $h/r = 3, \cdots, 5$, and for a container with, say, $R/r= 5, \cdots, 15$ my estimate is that the number of corks per layer ($2r$ thick) be $70 \, - \, 90 $ % of $L$.

Answer 2

I don't think that you can obtain through mathematical analysis an estimate of the number of corks in your cylinder. Such corks have a cylindrical radius of about $11$ mm and a length of about $33$ mm. Throwing them into a container does not create a densest package. For congruent corks I conjecture that the densest package is in layers of vertical corks with densest disc packing in each layer.

I would not suggest a simulation but some experiments with real corks and reasonably large cylindrical containers. You will then see that there is an overall density resulting from "random throwing" of the corks into the container, and this will result in a practical estimate of the number of corks per m$^3$ container volume in a normal setting.