Chapter 4 - Transfer of Particles to Interfaces—Linear Problems

Abstract

Abstract The force and torque balance equations are formulated and their limiting forms are derived. The general mobility relationship, describing particle velocity in terms of the mobility matrix is discussed. This relationship is used to calculate particle migration velocity under the action of external forces such as gravitational or electrostatic forces. Sedimentation rates for particles of various shapes and their migration velocities under electrostatic forces in ion-free media are determined. The inertia-less trajectory analysis of particles moving under various flows near interfaces is performed and interception controlled particle deposition rates are calculated. The force balance equations are extended to incorporate the random force exerted on particles by molecules of the suspending medium. The Langevin stochastic equation is derived and the Smoluchowski equation for particle motion under an external force field. Brownian motion of a single particle in the bulk and near interfaces is analyzed. The general diffusion equation for particle systems is derived using the Einstein method. Various particle transport regimes are considered, in particular the convective diffusion regime, and the pure diffusion regime. Methods for analytical solutions of these equations are discussed with the emphasis on the Laplace transformation method. One-dimensional transport of particles through the thin layer adjacent to interfaces is discussed. This allows one to formulate the surface boundary layer theory and to establish the appropriate boundary conditions for bulk transport equations. Solved problems pertinent to linear transport of particles to interfaces are discussed: the non-stationary diffusional transport to spherical and planar interfaces, the convective diffusion transport to uniformly accessible surfaces and the transport in the impinging-jet cell. Analytical expressions for various transport conditions and interface shapes obtained from the convective diffusion theory are tabulated. The influence of electrostatic interactions and external forces on the limiting flux and mass transfer coefficients for various transport conditions is also discussed.