Interactions of Two Settling Spheres: Settling Rates and Collision Efficiency

The behavior of two spheres settling in a low Reynolds' number region is described from the hydrodynamic equations developed by others. Even when they are not close enough to form a floe, neighboring particles increase their settling velocity above that described by Stokes. The collision efficiency factor in differential sedimentation $\left({\alpha }_{\mathrm{DS}}\right)$ is calculated from trajectory analysis that includes the hydrodynamic effects and particle attraction. ${\alpha }_{\mathrm{DS}}$ is a function of many parameters and the sensitivity of ${\alpha }_{\mathrm{DS}}$ to each governing parameter is determined. ${\alpha }_{\mathrm{DS}}$ increases as particle sizes decrease (at constant size ratio), as particle size ratio (small to large) approaches unity, as the gravity acceleration decreases, and as the density of the particle decreases. Viscosity does not affect ${\alpha }_{\mathrm{DS}}\text{.}$ Comparisons to previously published values of ${\alpha }_{\mathrm{DS}}$ are made and the differences are discussed. Ramifications of ${\alpha }_{\mathrm{DS}}$ to the collision frequency function and the modeling of flocculation of heterodisperse suspensions are noted.