Nonsteady-State Particle Flow Under Gravity—An Extension of the Stochastic Theory

The steady-state (ss) stochastic theory of convergent, cohesionless particle flow under gravity toward an orifice in the floor of a semi-infinite bed, based on the statistics of random flight and assuming instantaneous propagation of flow disturbances throughout the bed, is extended to nonsteady-state flow and time lag effects. The new theory, of which the ss theory is a special case, assumes flow to be restricted to an expanding zone, surmounting the orifice (opened at t = 0), of particle density ρ_ss, separated from the rest of the bed of the original particle density ρ₀ = ρ_ss + Δρ (Δρ > 0) by a boundary whose elements advance with a velocity v_n = −(1/Δρ)J_n where J_n is the normal component of the particle flux on the inside of the boundary due to flow (assumed to be ss) within the zone. Detailed equations describing the flow zone boundary as a function of time and the flow within the zone are developed; the equations depend on two material parameters (Δρ/ρ_ss, and α of ss theory) and on the quantity of material drained out. Corrections are derived for the analysis of the z² and z^3/2 plots of layer data previously made on the basis of the ss theory. A comparison of the new predictions with one piece of flow data shows the theory capable of accounting for lag effects and for details of the flow pattern in that case. Values of Δρ/ρ_ss and α are deduced, the latter being the order of the particle size in conformity to the expectations of the statistical theory.