Stochastic sedimentation and hydrodynamic diffusion

Molecular collisions with very small particles induce Brownian motion. Cons equently, such particles exhibit classical diffusion during their sedimenta tion. However, identical particles too large to be affected by Brownian mot ion also change their relative positions. This phenomenon is called hydrody namic diffusion. Long before this term was coined, the variability of indiv idual particle trajectories had been recognized and a stochastic model had been formulated. In general, stochastic and diffusion approaches are formal ly equivalent. The convective and diffusive terms in a diffusion equation c orrespond formally to the drift and diffusion terms of a Fokker-Planck equa tion (FPE). This FPE can be cast in the form of a stochastic differential e quation (SDE) that is much easier to solve numerically. The solution of the associated SDE, via a large number of stochastic paths, yields the solutio n of the original equation. The three-parameter Markov model, formulated a decade before hydrodynamic diffusion became fashionable, describes one-dime nsional sedimentation as a simple SDE for the velocity process {V(t)}. It p redicts correctly that the steady-state distribution of particle velocities is Gaussian and that the autocorrelation of velocities decays exponentiall y. The corresponding position process {X(t)} is not Markov, but the bivaria te process {X(t), V(t)} is both Gaussian and Markov. The SDE pair yields co ntinuous velocities and sample paths. The other approach does not use the d iffusion process corresponding to the FPE for the three-parameter model; ra ther, it uses an analogy to Fickian diffusion of molecules. By focusing on velocity rather than position, the stochastic model has several advantages. It subsumes Kynch's theory as a first approximation, but corresponds to th e reality that particle velocities are, in fact, continuous. It also profit s from powerful theorems about stochastic processes in general and Markov p rocesses in particular. It allows transient phenomena to be modeled by usin g parameters determined from the steady-state. It is very simple and effici ent to simulate, but the three parameters must be determined experimentally or computationally. Relevant data are still sparse. but recent experimenta l and computational work is beginning to determine values of the three para meters and even the additional two parameters needed to simulate three-dime nsional motion. If the dependence of the parameters on solids concentration is known, this model can simulate the sedimentation of the entire slurry, including the packed bed and the slurry-supernate interface. Simulations us ing half a million particles are already feasible with a desktop computer. (C) 2000 Elsevier Science B.V. All rights reserved.