Hopper Orifice Clogging

What is modelled

When grains leave a hopper through a hole only a few times larger than a grain, a stable arch can form over the hole and stop the flow. Experiments show that the number of grains that pass before the next clog follows an exponential (geometric) law, which means each grain has the same small chance of clogging and the hopper has no memory. That chance falls very steeply as the orifice grows, so the mean avalanche grows roughly like an exponential of the squared size ratio.

Equations

• Mean avalanche ⟨s⟩ = s0 exp[(R / R0)2] with R = D / d and R0 = 2.6 − 1.2 μ.
• Clog chance per grain p = 1 / (1 + ⟨s⟩), size law P(s) = (1 − p)s p, survival (1 − p)s+1.
• Variance (1 − p) / p2; chance to empty N grains without a clog (1 − p)N.

Checks run in the page

• The mean of 20000 sampled avalanches against the geometric mean.
• The sampled survival share against the closed form.
• The sample variance against the geometric variance.
• The size ratio found by bisection at which the mean reaches 1000 grains against its closed form.

Honest simplifications

The exponential growth law and its constants are a phenomenological fit shape, not a measurement: the friction dependence of R0 is a simple linear guess, there is no vibration or tapping, the flow rate is steady and independent of the clogs, and grains are treated as identical. Use it for the statistics of rare events, not for sizing an outlet.

• Press Clogs often and watch short avalanches follow each other.
• Press Rare clogs and compare the mean avalanche with the first preset.
• Move the friction slider to see the safe size ratio shift.