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Suppose I have an hourglass that takes 1 full hour on average to drain. The grains of sand are, say, $1 \pm 0.1\ {\rm mm}$ in diameter.

If I replace this with very finely-grained sand $0.1 \pm 0.01\ {\rm mm}$ in diameter but keep the hourglass otherwise the same, how long a time will the "hourglass" now measure? Does this depend on the size of the funnel, or should all one-hour hourglasses change in roughly the same way?

Bonus: is the new hourglass more or less precise? (precision defined at $\sigma_t /t$, with $t$ the time to drain)

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I recently saw a colloquium that included some discussion of jammed granular solids. There are phase changes under some conditions, so I suspect the behavior could be highly non-linear under some condition. –  dmckee Apr 16 '11 at 18:38
It is, in fact. The flow in (on) the cones of a hourglas shows tiny avalanches. This is a very popular experiment in chaos theory. –  Georg Apr 17 '11 at 12:23

2 Answers 2

up vote 8 down vote accepted

I have at least an answer :

Does this depend on the size of the funnel ?

Yes, as it was expected (for a very large funnel, all sand falls in the same time, whereas for a very small funnel it doesn't fall at all. More precisely ... the complete answer is probably extremely complex, as one has to take into account the shape of the hourglass, the dynamics of the grains, and so on (see http://arxiv.org/abs/0707.4550 for example). However we can have a rough idea with some dimensional analysis.

Let us consider a cylinder of diameter $D$, with a circular hole punched on the bottom side with radius $a$. We fill the cylinder with a height $H$ of sand. If we look at the speed of sand grains going out the bucket, we observe (experimentally) that it does not depend of the height of sand $H$, if $H$ is big enough (compared to the diameter $D$ - because the constraint saturates). We are left with two parameters : the diameter of the hole $a$ and the gravity field $g$ that makes it fall, so the output speed $v$ has to be proportional to $\sqrt{g a}$. The flow rate is the speed times the section, thus it is $Q \propto v \, a^2$, so it is of order $Q \propto g^{1/2} a^{5/2}$ (this is the Beverloo law).

However we have to take into account the size of grains ; let us suppose they are spherical and let be $d$ their diameter. All grains that are less than half on the hole won't fall, so there is a ring-shaped exclusion region at the border of the hole. The effective diameter of the hole, were grains will actually fall, is thus reduced by $d/2 + d/2 = d$, and is thus $a-d$. We thus replace $a$ with $a-d$ in the Beverloo law and get

$Q \propto g^{1/2} (a-d)^{5/2}$

Now, this becomes invalid when $H$ and $D$ are of the same order, but we can get an idea of the time to drain $T$ by saying that as the flow rate does not depend of $H$, $Q T = V = 4 \pi (D/2)^2 H = \pi D^2 H$ so, dropping the $\pi$ we get

$T \propto g^{1/2} \, \frac{(a-d)^{5/2}}{D^2 \, H}$

and thus for everything but the size of sand grains $d$ constant

$\frac{T(d_1)}{T(d_2)} = \left( \frac{a - d_1}{a - d_2} \right)^{5/2}$

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A link to a paper is always good, but can you please point to the abstract page rather than directly to the PDF. That way I can decide if I want the paper. –  dmckee Apr 17 '11 at 0:17
You're right. Done. –  Georg Sievelson Apr 17 '11 at 0:59
Very good answer! That this Beverloo law is from as late as 1961, strange! –  Georg Apr 17 '11 at 10:16

As a first guess, I would think the strain rate under a fixed amount of shear stress for a unit cube of material should be expected to be proportional to the surface area of the grains. That would imply that the strain rate, and hence the flow rate for fixed geometry should be inversely proportional to grain size. So I think your fine grained hourglass would be much faster. I suspect in reality things would get more complicated then that. What shape are the grains? Spherical is probably faster than angular. What about grain on grain sliding coefficient of friction? Is there some stress level below which no flow/creep takes place?

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