Dispersion in particle velocity resulting from random motion through a spatially-varying fluid velocity field in a pipe

A macro-scale probabilistic model of dilute phase pneumatic transport is developed to analyse the dispersion in the velocity of conveyed particles and to predict their velocity statistics. Fluid drag force, proportional to the relative velocity between the particle and fluid, is taken to be the agent that causes particle motion in the axial direction. The basic premise of the approach is that dispersion in axial particle velocity is a result of subsidiary random motion in the radial direction through the fluid velocity field. Two causes are investigated for this radial motion; gravity and inter-particle collisions. As the local fluid velocity experienced by a particle thus continuously varies in an unpredictable fashion, then the associated drag force fluctuates as does the resulting particle velocity. The non-deterministic nature of the fluid velocity acting on the particle is captured by treating fluid velocity as a stochastic process whose description comes from combining knowledge of the flow field with the nature of the radial motion of the particle. This novel approach allows analytical expressions to be obtained for the mean and variance of particle velocity. The accuracy of these predictions was checked by numerical simulation and found to be good. The analysis demonstrates that dispersion in particle velocity is a function of the magnitude of dispersion in fluid velocity (a fluid property), on the inertial rate constant of the particle (a combined particle/fluid property) and the autoregressive parameter (a property reflecting the type of radial motion).