Mass transfer for droplets and bubbles with a stagnant surfactant cap - an analytical solution of the advection-diffusion equation

Abstract

The mass transfer rate to a liquid droplet or bubble, in motion in another immiscible fluid, is studied when the interface is partly covered by a stagnant surfactant cap. We present an analytical solution by solving the equations for advection diffusion under conditions of creeping flow and a thin concentration boundary layer. The empirical relation between the Sherwood number and drag coefficient for partly coated bubbles proposed in literature that accurately describes both experiments and numerical simulations of CO${}_2$${}_2$-release from a rising bubble follows from our analytical solution. Mass transfer and drag force are interrelated as demonstrated mathematically.This study improves the understanding of mass transfer to partly covered droplets and bubbles at time scales where these are created such as in processing equipment or microfluidic devices as well as for semi-stationary situations where partly covered bubbles or droplets exist as in froth flotation, bubble columns or extraction processes.