Sand grains pour through a funnel orifice following the Beverloo discharge law. Narrow the orifice with the slider to watch probabilistic arch formation block the flow — the classic industrial jamming problem.
Beverloo: Q = C · ρ · √g · (D − k·d)^2.5
Normal force: Fn = k_n · δ − γ · v_n
Friction: Ft ≤ μ · |Fn|
Clog prob: P_clog ∝ exp(−α · (D/d − 1))
The Beverloo equation was derived in 1961 by W.A. Beverloo and colleagues studying seed discharge from silos. Despite its empirical origins, it remains the standard engineering formula for hopper design, applied to everything from grain elevators to pharmaceutical tablet presses.
What is the Beverloo law for granular flow?
The Beverloo law states that the mass flow rate through a hopper orifice scales as Q ~ C·ρ·√g·(D − k·d)2.5, where D is the orifice diameter, d is the grain diameter, and k ≈ 1.5 accounts for an empty annular zone near the wall. Unlike liquids, the rate is height-independent due to the Janssen effect.
What causes arch formation in hoppers?
Arch (dome) formation occurs when grains near the orifice arrange into a mechanically stable arch that bridges the gap and blocks flow. This is stochastic: each grain has a finite probability of locking in place. Narrower orifices (lower D/d ratio) dramatically increase clogging probability, following an exponential law P ∝ exp(−α(D/d − 1)).
How does the Discrete Element Method (DEM) work?
DEM treats each grain as a soft sphere. When two grains overlap by δ, a normal repulsive force Fn = k_n·δ − γ·v_n is applied (spring-dashpot). A tangential friction force (bounded by Coulomb's law) is also computed. Newton's second law is integrated with a small timestep to advance positions and velocities.
The jamming transition is the point at which a granular material changes from fluid-like (flowing) to solid-like (clogged) behaviour. In hoppers it is controlled by D/d: below roughly D/d ≈ 4–5, clogging probability becomes very high and flow stops almost immediately after starting.
Unlike liquids, where hydrostatic pressure increases with depth, granular materials transfer stress to container walls via force chains (the Janssen effect). Pressure at the orifice saturates quickly and becomes independent of pile height, so the Beverloo flow rate is also height-independent.
The constant k ≈ 1.5 represents the number of grain diameters of an empty annulus at the orifice edge where grains cannot centre themselves. It accounts for excluded volume near the wall, making the effective flow width (D − k·d) rather than D.
Force chains are quasi-linear networks of particles carrying most of the compressive stress in a granular medium. A stable arch at the orifice is a curved force chain that redirects the weight of overlying grains laterally to the hopper walls, preventing downward force from pushing grains through the gap.
Non-spherical (angular or elongated) grains clog much more readily than spheres because their interlocking geometry creates stronger arch structures. Rice and lentils jam at larger D/d ratios than glass beads. This simulation uses circular discs (2D spheres) for computational simplicity.
Yes. External vibration (tapping, shaking) perturbs grain positions enough to destabilise the arch and restart flow. This is widely used in industrial silos. In this simulation, clicking on the canvas applies a local impulse that may break a clog and resume discharge.
Hopper discharge models are critical for designing pharmaceutical tablet machines, grain silos, coal bunkers, cement plants, food-processing hoppers, and powder-bed 3D printers. Avoiding unexpected clogs prevents production downtime, equipment damage, and safety hazards from sudden avalanche discharge after manual unclogging.
This simulation models granular flow through a hopper orifice using a soft-sphere Discrete Element Method (DEM): each grain is a disc that exerts a spring–dashpot normal force and a Coulomb-limited friction force on its neighbours whenever it overlaps them. Gravity pulls grains down through a funnel, and the discharge rate is governed by the empirical Beverloo law, Q ~ C·√g·(D − k·d)^2.5, where D is the orifice width and d is the grain diameter. Narrow the orifice far enough and the flow does not simply slow down — it can jam completely when grains lock into a self-supporting arch across the gap, a stochastic transition rather than a smooth one. Colour on each grain encodes its instantaneous speed, so you can watch force chains and stagnant zones emerge as the pile empties.
Coloured discs stream from a hopper through a narrow orifice, occasionally freezing into an arch that halts flow entirely. Live stats track grain count, measured flow rate Q, the D/d ratio and how many times the hopper has clogged, while the canvas overlay compares the actual discharge against the theoretical Beverloo prediction.
Drag Orifice width (D) from 2.5d to 10d to control how easily grains escape; below roughly 4–5d clogging becomes frequent. Friction μ (0–0.9) makes arches more or less stable, Grain size d (0.6–1.6) rescales the discs, and Gravity g (0.3–3.0) speeds up or slows down discharge. Use Pause/Reset to restart a run, and click or tap the canvas to shake the hopper and break a jam.
Granular jamming does not need cohesion or glue — pure geometry and friction are enough. A handful of grains bridging an orifice can support the weight of an entire silo above it, which is why real hoppers use vibrators, air jets or mechanical stirrers rather than just gravity to guarantee reliable discharge.
The Beverloo law describes how the mass flow rate Q through a hopper orifice scales with orifice size: Q ~ C·ρ·√g·(D − k·d)^2.5, where D is orifice width, d is grain diameter, g is gravity and k ≈ 1.5 accounts for an empty zone near the wall where grains cannot centre themselves. The simulation overlays this theoretical curve next to the measured flow rate.
Clogging is a stochastic, not a gradual, process. As grains approach the orifice they sometimes lock into a mechanically stable arch that bridges the gap and carries the weight of the grains above sideways into the hopper walls. Once that arch forms, flow halts abruptly rather than tapering off — this is why the "Clogs" counter jumps in discrete steps.
It sets the gap at the bottom of the hopper in units of grain diameter d, from 2.5d up to 10d. Below a ratio of roughly 4–5d, arches form so readily that the hopper clogs almost immediately; above about 6–7d, flow is essentially continuous and follows the Beverloo prediction closely.
Clicking applies a local impulse to nearby grains, mimicking the vibration or tapping used on real industrial silos to dislodge a jam. If the hopper is clogged, a well-placed click perturbs the arch enough to collapse it and let flow resume, which is exactly how the simulation's shake-to-unclog mechanic works.
Friction μ (0–0.9) sets how strongly grains resist sliding past each other; higher friction makes arches more stable and clogging more frequent even at wider orifices. Gravity g (0.3–3.0) scales the driving force pulling grains down, so higher gravity increases discharge speed but does not by itself prevent arch formation, since jamming is governed by geometry as much as by force.