Modelling W/O/W double emulsions preparation in static mixers: Accounting for the shear-thinning behaviour of the dispersed phase

The dispersion of a shear-thinning non-Newtonian fluid using in-line mixers to prepare double emulsions is investigated experimentally and numerically in this study. A two-step method is adopted for the preparation of double emulsions, comprising in a first step the preparation of a reverse water-in-oil (W/O) emulsion using a high-shear rotor -stator device. This inner emulsion is dispersed in a second step into an outer aqueous phase by continuous parallel pumping of both phases through a set of static mixers. The second preparation step is modelled using a population balance equation (PBE) accounting for droplet breakage inside the mixers. Since the inner emulsion behaves as a shear-thinning fluid, the breakage efficiency depends on the local shear rate. Using classical turbulent breakage kernels is therefore inappropriate, hence there is a need for a kernel accounting for the shear-thinning character of the dispersed phase. To tackle this issue, single -phase CFD simulations of the fluid flow through the mixers are carried out showing a high shear rate magnitude near the walls of the pipe and the mixer crossbars. To avoid CFD-PBE direct coupling, the probability density function (PDF) of the shear rate is extracted from the CFD simulations. This PDF is then used to develop a volume-averaged PBE, where a classical Ostwald -de Waele model is employed to relate the apparent viscosity of the inner emulsion to the shear rate. Different experimental parameters influencing the quality of the dispersion are investigated, including the fraction of the inner emulsion phase, the oil viscosity and the inner and outer aqueous phase viscosities. In the different cases, the model captures the transient evolution of the droplet size distribution of the W/O/W double emulsions at the outlet of the mixers while the computational time remains very low.