Hourglass Flow Lab

Why an hourglass keeps time

A liquid drains faster when its level is high (Torricelli: v = √(2 g h)), so a water clock slows down. Dry grains do not: friction between grains and with the walls makes the pressure near the neck saturate with height, a result due to Janssen. For a tube of hydraulic radius Rh and wall friction μ, dP/dz = ρ g - μ K P / Rh, so P(z) = ρ g λ (1 - e-z/λ) with λ = Rh/(μ K) (K is the ratio of horizontal to vertical stress). Because the neck pressure no longer depends on the sand height, the discharge is constant. The empirical Beverloo law gives the mass flow W = C ρb √g (D - k d)5/2 with C ≈ 0.58, k ≈ 1.4, orifice diameter D and grain diameter d. The grains need free fall through an empty annulus of width k d at the rim of the hole, hence the 5/2 exponent: with the hole area scaling as D² and the free-fall speed as √D.

What the simulation does

The upper bulb is a cone of half-angle α topped by a cylinder; the level is found from the sand volume by exact frustum integration. The lower pile is a cone at the angle of repose, which becomes a flat-topped heap when it reaches the glass. The liquid curve integrates dV/dt = -Cd A √(2 g h) with Cd = 0.6 in the same bulb for comparison.

Honest simplifications

• Beverloo's law is empirical and holds for D/d above roughly 3 to 6; the model simply declares an arch jam below 3 grain diameters, although real jams are probabilistic.
• The Janssen parameters use an active-state K for a 35 degree friction angle and a constant bulk density of 1500 kg/m³; air drag on fine grains, humidity and electrostatics are ignored.
• Grain dots in the falling stream are schematic; the amount of sand and the rate are computed, not counted from grains.

• Load Wide neck and note that doubling the orifice multiplies the rate by about 5.6 (2^2.5), not 4.
• Load Near jamming and increase the grain size until the status turns to ARCH JAM.
• Compare the sand and liquid curves in the first graph: the straight line is what makes an hourglass a clock.