Flotation rates of fine, spherical particles and droplets

Flotation rates of fine, spherical particles or droplets by a spherical, rising bubble or drop have been computed for conditions relevant to microflotation. The buoyancy-driven motion of the rising drop (or bubble) is characterized by small Reynolds, and large Péclet numbers. Particular attention is given to suspended droplets (or particles) much smaller than the rising drop; dilute conditions are assumed. Two complementary regimes are considered: (1) convective capture dominated by the buoyancy-driven, relative motion between a collecting drop and smaller, dispersed droplets, and (2) capture dominated by diffusive transport of small droplets within a boundary layer on the surface of a larger collecting drop. The first regime is relevant for the flotation of micron-size and larger droplets; the second is relevant for submicron sizes. A scaling analysis reveals three distinct mechanisms for droplet capture by a larger rising drop. According to the scaling analysis, flotation rates depend strongly on the size of the dispersed droplets, the size of the collecting drops, and the mobility of the collecting drop interface; a bubble with a free surface is a more efficient collector than a viscous drop, and a collector with an interface immobilized by surfactant is least efficient. The scaling analysis predicts that flotation rates depend weakly on the strength of van der Waals forces, and are insensitive to the viscosity or density of the dispersed droplets; flotation rates of droplets and particles are very similar. The scaling predictions are illustrated by dimensionless flotation rates computed by a trajectory analysis in regime (1), and by mass transport formulae for regime (2). Detailed pairwise, hydrodynamic and van der Waals interactions are incorporated into the analysis of the first regime, but not the second where they are shown to be unimportant by scaling arguments. The droplet size that is most difficult to float and its flotation rate are predicted by scaling and estimated numerically.