Stochastic theory of particle flow under gravity

Abstract

A discrete theory of cohesionless particle (radius a) flow under gravity is developed, which assumes particle positions are restricted to the sites of a (random) lattice. Downward particle flow toward and through an orifice is represented in terms of an equivalent upward-biased random flight of voids (unoccupied lattice sites) originating at the orifice. The void migration rate is assumed limited by the time required for a particle above a void to fall freely through a distance of 2a into the void. Because of the strong + z (upward) void-jump bias, the theory of random flight yields for the steady-state concentration c of voids the equation ∂2c∂x 2+∂2c∂y2=1α∂c∂z, from which the usual term ∂2c/∂z2 is absent. Since this equation is formally identical to the two-dimensional diffusion or heat flow equation under the substitution (time) → z, steady-state problems in gravitational particle flow, termed stochastic flow on our model, may be converted to boundary value problems in ordinary diffusion theory. Solutions representing a point orifice and a finite orifice (radius R) in the floor of a semi-infinite bed and a point orifice in the bottom of a pipe are given. Particle flux lines and marker motion and exit times are defined and calculated. Time-dependent flow is discussed briefly. Finally, an idealized theory of flow in the immediate vicinity of the orifice is given, which pictures the transition between stochastic flow in the particle bed and bulk free fall in the orifice vicinity as occurring abruptly on a dome-shaped interface surmounting the orifice. Calculated values of the total rate of particle flow through the orifice show the observed particle size effects and agree well with measured values in the literature for 5