Modelling of particle capture by expanding droplets

Froth flotation by small air bubbles has been traditionally used in industry to capture fine minerals and other hydrophobic particles. This method, however, is not efficient for capturing very small particles. The present work is motivated by a new agglomeration process that overcomes this lack of efficiency. It consists of mixing a particle suspension and saltwater-filled droplets covered with semi-permeable oil layers. This paper investigates the two-particle dynamics of a solid particle and a semi-permeable spherical drop that expands due to osmosis in an external, pure extensional flow field. A dimensionless engulfment parameter measures the relative effects of droplet growth and convective flow. The computational results from numerical integration determine a transient collision efficiency, which describes the influence of hydrodynamic interactions and osmotic flow on particle capture. The results show that drop expansion, which decays slowly with time, greatly increases particle capture rates, especially for small particles. Moreover, as the engulfment parameter increases, there is a transition from flow-dominated capture to expansion-dominated capture. For the case of a non-expanding droplet, we provide a numerical solution for the transient pair distribution function, which enables us to explain the transient particle-capture rate in terms of the microstructure of the suspension. Furthermore, we derive an analytical expression for the initial collision efficiency at zero times, which agrees with our numerical data. The numerical results for non-expanding droplets at long times show increasing collision efficiency as the permeability increases and when the size ratio is near unity, in agreement with previous steady-state calculations.