Silo Hopper Discharge

What is modelled

Grain in a tall silo does not press on the base like water. Friction against the walls carries much of the weight, so the pressure stops growing with depth (Janssen). The outflow through a hole in the bottom is also unlike water: it depends on the outlet size, to the power two and a half, and not on the height of grain above it (Beverloo). A cohesive powder can bridge a narrow outlet with an arch and stop flowing altogether.

Equations

• Janssen p(z) = ρ g λ (1 − e−z/λ) with λ = Rh / (μw K) and hydraulic radius Rh = D / 4.
• Beverloo W = 0.58 ρ √g (D0 − 1.4 d)5/2.
• Arching limit Bmin = H(α) σc / (ρ g) with H ≈ 2 + α/60.
• Discharge time t = M / W.

Checks run in the page

• The Janssen slice balance integrated numerically against the closed form.
• The local power-law exponent of the discharge rate against the Beverloo form.
• The smallest non-arching outlet found by bisection.
• The level stepped down through the straight wall and cone to the discharge time.

Honest simplifications

The lateral pressure ratio is a fixed 0.4, the wall friction and the cohesion are constants, the cohesion is a single low-stress strength rather than a full flow function, the Beverloo constants are the textbook values for round holes, and the hopper dynamics, rat-holing and dynamic overpressure at start-up are ignored. Use it for the trends, not for designing a silo.

• Press Tall narrow silo and watch the base pressure saturate far below the liquid value.
• Press Small outlet to see the discharge rate collapse with the outlet size.
• Press Cohesive powder arches to see the outlet block, then widen it or steepen the hopper.