The stochastic differential equations for particle motion in a homogeneous fluidized bed have been modified to incorporate a bubble-particle interaction. In a freely bubbling bed this interaction causes an instantaneous random step upwards when a particle collides with a bubble. In addition the particles are subject to a Brownian Motion due to particle-particle interactions and to a vertical drift velocity. The particle motion was modeled by the set of stochastic differential equations
where N and W are Poisson and Wiener processes respectively. U is the velocity vector, X the vertical coordinate, Y and Z the horizontal coordinates. This set of equations has been solved and the form of the probability density function p(x, t) has been obtained. Predictions are in terms of the mean bubble frequency at a point λ the average displacement associated with a particle-bubble collision, and the dense phase diffusion coefficient.
The main result is that the effective diffusion coefficient in the vertical direction is given by
where E is the dense phase diffusion coefficient and 1/α is the average displacement suffered by a particle on collision with bubble.
The model was tested experimentally using random forcing function techniques. All parameters were evaluated independentely; λ by direct measurement and the other two inferred from available published data. The predictions of the theory agreed well with experimental observations of the solids dispersion.