Granular flow during hopper discharge.

Granular material freely discharging from a hopper under gravity is one of the oldest and most widely studied problems in granular flow. Despite the apparent simplicity of the system, expressions relating the discharge rate of the hopper to the properties of the individual grains are difficult to determine and typically empirical in nature. A mathematical model for discharge from a hopper is derived based on the dynamics of individual particles just within the outlet. In contrast to previous models, this analysis derives a flow rate from granular dynamics, rather than dimensional arguments. The model, therefore, uses no assumptions about the form of the stress distribution within the hopper, or the addition of empirical factors or fitting parameters. Our model gives a flow rate identical in form to a well-known empirical expression, showing that an experimentally determined constant used in this expression is purely geometry-dependent. Our analysis is also extended to derive an expression for the flow rate incorporating gas drag, which becomes dominant at small grain sizes, significantly reducing the outflow rate. The resulting expression shows excellent agreement with a range of computational simulations using a coupled discrete element and Navier-Stokes method. These simulations also show that the gas flow is much more complex than previously assumed in this region, and simplified assumptions used in prior hopper flow models do not hold.