Hourglass flow rate

Why does sand pour through an hourglass at the same steady rate whether the upper bulb is full or nearly empty, when water would not?

A water clock must be built as a carefully tapered vessel, because water drains fast when the tank is full and slowly when it is nearly empty. An hourglass is two plain bulbs and a hole, and keeps time anyway.

That should be surprising. Sand is heavier than water and just as subject to gravity. Pile more above the hole and the weight pressing down goes up. Why does the flow not go up with it?

Two separate things must be established: where the column's weight goes, and what sets the speed at the aperture.

Consider a horizontal slice of sand of thickness dz in a container of area A and wall perimeter P, with σ the vertical stress at depth z. Its own weight is ρ g A dz, but it is also gripped at the wall. Grains push sideways with a horizontal stress some fraction K of the vertical one — the crucial property of a granular pile, that pushing down produces a sideways push, because grains are wedges and not a liquid column — and wall friction μ resists with μKσ per unit wall area:

A dσ = ρ g A dz − μ K σ P dz

which rearranges to dσ/dz = ρg − (μKP/A)σ, and for a cylinder of radius R, P/A = 2/R. Watch what happens as z grows: the stress rises, the friction term grows with it, and the two cancel at dσ/dz = 0. The stress stops increasing. It saturates at

σ∞ = ρ g R / (2μK)

reaching it over a depth λ = R/(2μK). Put numbers in: a neck of radius 5 mm, μ ≈ 0.4, K ≈ 0.5, so λ = 0.005/(2 × 0.4 × 0.5) = 0.0125 m. Twelve millimetres. Within about four centimetres of depth the stress is within a few per cent of its ceiling and stays there ever after.

That is Janssen's result, from the 1890s, and it answers half the question. Sand at the bottom of a full bulb is not carrying the weight above it; the walls are. Adding sand on top adds no load below, because the extra weight is shed sideways into friction before it can arrive.

Now the aperture. Grains do not squeeze through under pressure like toothpaste. Just above the opening is a region — conventionally a free-fall arch, of size comparable to the orifice — where grains lose contact with the load-bearing network and simply drop. Falling from rest through a distance of order D, the orifice diameter, they reach the throat at speed of order √(gD).

That gives the scaling immediately. Flow rate is density times area times velocity:

Q ~ ρ_b · D² · √(gD) = ρ_b √g · D^(5/2)

which is Beverloo's law, usually written Q = C ρ_b √g (D − kd)^(5/2), with d the grain diameter and C and k empirical constants near 0.58 and 1.4 — both quoted from memory and treated as soft. The (D − kd) correction says grains cannot pass within about a grain radius of the rim, so the effective hole is smaller than the drilled one.

Notice what is absent: the head. No depth of sand appears, because the velocity was set by falling a distance of order D, not by pressure from above — and that pressure had saturated anyway. Contrast water: Torricelli gives v = √(2gh), so flow falls as √h and a tank drained to a quarter of its head passes half the flow. An hourglass drained to a quarter of its sand passes the same.

The D^(5/2) exponent is the testable part. Double the orifice and flow should rise by 2^2.5 = 5.66 times — not 4, as a fixed-velocity picture gives, and not 8. That decides the account: if discharge scaled as D², or if adding a tall extension column of sand above a hopper raised the rate, both the Janssen and free-fall arguments would be wrong.

The boundary first. This collection reasons about grains twice elsewhere, and the mechanism here is different. The Brazil-nut effect is vibration opening a jammed bed so a geometric ratchet can work; shear-thickening pastes are stress switching lubricated contacts to frictional ones, moving the jamming threshold. Neither involves what governs here: stress in a static granular column saturates with depth rather than growing linearly. That saturation is the fixed point of difference, and no liquid can do it.

Hold the free-fall arch loosely: it is the standard heuristic and gives the right exponent, but whether a literal persistent arch exists is disputed — recent work derives the same 5/2 power from measured velocity and packing profiles at the outlet without one. The exponent is solid; the picture is a simplification.

The constancy has limits at both ends. Below roughly five or six grain diameters of orifice, the flow does not slow but jams outright — which is why hoppers block. And at the very end of the run, when the sand left is no deeper than the hole is wide, the head genuinely does matter; a real hourglass hides this with a neck small compared with the bulb.

One complication is specific to hourglasses: fine grains drag air with them, and in a sealed instrument that air must flow back up through the neck against the descending sand, which can throttle the rate. So a good hourglass is a little more than dry Beverloo physics — but the near-independence of head is the granular part, and it is real.

Every term the collection defines is gathered in the glossary.