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

The flotation of spherical colloidal particles by small spherical bubbles i s considered. The model accounts for the effects of buoyancy motion, Browni an motion, van der Waals attractive forces, and hydrodynamic interactions. Conditions are such that the fluid has negligible inertia. The suspension i s sufficiently dilute that the analysis is restricted to pairwise bubble-pa rticle interactions. The quasi-steady formulation of the Fokker-Planck equa tion for the pair-distribution function is simplified for negligible transv ersal diffusion and solved numerically. Allowance is made for bubbles with freely mobile or totally immobile interfaces. For size ratios of the captur ed particle to capturing bubble of 0.1 and higher, and for bubble Peclet nu mbers greater than approximately 10(5), convective capture dominates. For t hese conditions, the collision efficiencies calculated through the more com plete 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 pa rticles within the convective flow field created by the rising bubble; howe ver, it is found that the classical mass-transport formula is not entirely accurate, due to the effects of finite particle size and hydrodynamic inter actions when the particles are large enough for boundary-layer mass transfe r 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 t hose with mobile interfaces, especially in the regime of convective capture . In all instances, the simple additivity approximation for diffusive and c onvective capture is shown to overpredict the collision efficiencies, in so me cases by up to two-fold. (C) 1998 Elsevier Science Ltd. All rights reser ved.