The flotation rates of fine spherical particles under Brownian and convective motion

The flotation of spherical colloidal particles by small spherical bubbles is considered. The model accounts for the effects of buoyancy motion, Brownian motion, van der Waals attractive forces, and hydrodynamic interactions. Conditions are such that the fluid has negligible inertia. The suspension is sufficiently dilute that the analysis is restricted to pairwise bubble-particle interactions. The quasi-steady formulation of the Fokker-Planck equation for the pair-distribution function is simplified for negligible transversal diffusion and solved numerically. Allowance is made for bubbles with freely mobile or totally immobile interfaces. For size ratios of the captured particle to capturing bubble of 0.1 and higher, and for bubble Peclet numbers greater than approximately 10(5), convective capture dominates. For these conditions, the collision efficiencies calculated through the more complete Fokker-Planck formulation are in good agreement with those predicted by a particle trajectory analysis, both far free and rigid interfaces. For more extreme size ratios of 0.01 and lower, and bubble Peclet numbers less than approximately 10(5), capture is dominated by diffusion of the small particles within the convective flow field created by the rising bubble; however, it is found that the classical mass-transport formula is not entirely accurate, due to the effects of finite particle size and hydrodynamic interactions when the particles are large enough for boundary-layer mass transfer with high Peclet number to be dominant. A minimum flotation efficiency is observed for a given collecting bubble size, while, for a fixed suspended particle diameter, it is always more effective to utilize smaller bubbles. Bubbles with a rigid interface exhibit lower collection efficiencies than those with mobile interfaces, especially in the regime of convective capture. In all instances, the simple additivity approximation for diffusive and convective capture is shown to overpredict the collision efficiencies, in some cases by up to two-fold. (C) 1998 Elsevier Science Ltd. All rights reserved.