Analytical description of the unsteady settling of spherical particles in Stokes and Newton regimes

Abstract

In most solid processes, the separation of particles is important and designed based on the terminal settling velocity as the characteristically distributed particle property. This settling velocity is determined by distributed particle properties like size, density, and shape. Although the settling of particles driven by an external force (gravitation or centrifugal field) is an unsteady process, the temporal changes of the particle settling velocity are relevant for the microscopic process design. These temporal changes correspond to spatial changes along the path-line. Generally, the particle state of motion is described by solving numerically the force balance equations in fluid dynamic simulations which are highly complex. In this work, algebraic solutions are presented instead, suitable for describing the temporal changes of the settling velocity and the displacement at settling for laminar and turbulent flows around the particle. These equations solely depend upon particle properties like size, density, and shape and are valid independently from the initial velocity. Thus, and at the difference to known publications, it is not assumed that the particles are initially at rest. Finally, the relaxation time at which terminal settling velocity is approximately reached can be clearly identified.