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

TLDR

The study develops a mathematical model using concentric spheres to represent a random assemblage of particles moving relative to a fluid. By treating one particle as the inner sphere and the surrounding fluid as the outer sphere, the authors impose appropriate boundary conditions to solve the Stokes‑Navier equations analytically, yielding a closed‑form solution that is then compared with existing theories and experimental data. The closed‑form solution predicts sedimentation rate or pressure drop as a function of void fraction and shows close agreement with the Carman‑Kozeny equation, despite the much larger forces experienced by particles in packed beds.

Abstract

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.