Viscous flow in multiparticle systems: Slow motion of fluids relative to beds of spherical particles

Abstract

A mathematical treatment is developed on the basis that two concentric spheres can serve as the model for a random assemblage of spheres moving relative to a fluid. The inner sphere comprises one of the particles in the assemblage and the outer sphere consists of a fluid envelope with a “free surface.” The appropriate boundary conditions resulting from these assumptions enable a closed solution to be obtained satisfying the Stokes‐Navier equations omitting inertia terms. This solution enables rate of sedimentation or alternatively pressure drop to be predicted as a function of fractional void volume. Comparison of the theory is made with other relationships and data reported in the literature. Of special interest is its close agreement with the well known Carman‐Kozeny equation which has been widely used to correlate data on packed beds as well as sedimenting and fluidized systems of particles. This is remarkable in view of the fact that the force on each particle in a packed bed can be up to several hundred times that exerted on a single particle in an undistrubed medium.