A modelling study of surfactant evolution on a film surface coupled to lamellar film drainage has been carried out. This study simulates the surfactant transport onto a lamella in a foam fractionation column with reflux: such columns offer greater control over enrichment and recovery of the surface active material. Insoluble surfactant is assumed during the simulation as such surfactants potentially derive more benefit from a reflux system. There are two terms involved in the equation for surfactant flux which are the Marangoni flow – from the Plateau border to the centre of the lamella – and the film drainage – which is directed from the centre of the lamella to the Plateau border. The extent of film drainage is bounded using two extremal assumptions of mobile or rigid interfaces. On a mobile interface, the film drainage dominates the Marangoni effect, while on a rigid interface, the Marangoni effect is dominant. The numerical simulation was carried out using a material point method followed by a bookkeeping operation to regrid the film. Analytical solutions for the case of no drainage and for a (quasi) steady state in which Marangoni flows and drainage are balanced were used to verify the numerical simulation. From the simulations, it can be concluded that the film drainage obtained using surfactant with a mobile interface is much faster than that determined using surfactant with a rigid interface, meaning that surfactant tends to be washed out of the film in the mobile case. The desirable condition in a foam fractionation column is however where the Marangoni flow dominates the liquid drainage which can be achieved when using surfactant that gives a rigid interface. The (quasi) steady state solution verifies the simulation result at later time for the case of film with a rigid interface. An asymptotic solution in the case of film with a mobile interface gives a prediction of the surface concentration of surfactant at the centre of the film, but also predicts growing surfactant concentration gradients and growing Marangoni stresses near the film's edge, which may then invalidate model assumptions.
- interfacial tension
- mathematical modelling
- Marangoni forces
- film drainage with rigid/mobile film interface