Z surface area of packing volume of column occupied by packing

Note: it's the volume of the column that's packed, not the volume of the packing material itself, because we're interested in the free space left for vapor and liquid flow.

We know D, and we know V, so all we need to know is N.

This may seem simple. After all, if we know the size of each marble it should be easy to calculate how many can occupy a volume V, shouldn't it? Unfortunately, the answer is no!

The problem of random particle packing has been fascinating mathematicians since ancient times, and is still not completely understood. Experiments have even been done on the space shuttle to study how packing is affected by micro-gravity. Given enough shaking, spheres will always settle to a maximum density, a state known as random close packing (RCP), and this density will be about 0.64 (volume of spheres / volume they occupy).

However, if you carefully stack the spheres, you can improve on this. This was predicted by Johannes Kepler (1571-1630), who calculated that a packing density of K / 181/2 =0.74 could be achieved. As a demonstration of how difficult these mathematical problems are, it wasn't until 2000 that Sal Torquato, a professor at Princeton University proved this to be true!

We should apply the RCP factor of 0.64 to calculate N (it is extremely difficult to achieve 0.74), but because the average marble's diameter is relatively close to the diameter of a small column and this causes interference, we will round that down to 0.6.

The total volume of marbles is NTCD3 / 6 = 0.6V, and Z = NTCD2 / V = 3.6 / D

For marbles with D = 1, then Z = 3.6. If you use marbles or ball bearings only 0.5cm diameter, then Z increases to 7.2

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