Rotor-Stator Emulsification in the Turbulent Inertial Regime: Experiments toward a Robust Correlation for the Droplet Size

The Sauter mean diameter, d32, is a representative parameter in emulsions that indicates the average size of the oil droplets once the emulsion becomes stable. Several mathematical and physical approaches have been employed in the literature to seek expressions for d32 under different conditions. The present work sheds light on this rich literature and emphasizes that the characterization of emulsions is still a fertile field for investigation. In this paper, a new Π-theorem-based model to predict the normalized Sauter mean diameter for the specific case of rotor-stator emulsification is sought by applying a multiple regression analysis on experimental data of oil-in-water (O-W) emulsions produced using three different oils: paraffin, soybean oil, and isopropyl myristate, at different oil-to-water (O/W) ratios and rotor speeds. The proposed model quantifies the roles of the viscous, inertial, and interfacial tension forces, besides the O/W ratio, in the emulsification process within the turbulent inertial subrange. The developed empirical correlation is then contrasted with relevant literature models for reliability assessment; predictions of the present explicit model are proven to be more accurate for the fluid properties and the experimental conditions under study.


Figure S1 :
Figure S1: Representative image of the rotor-stator emulsifier and the used workhead.

Figure S2 :
Figure S2: Sketch of Richardson regime and energy spectrum of turbulent flow in log-log scale.

Figure S3 :
Figure S3: Representative optical microscope images of emulsions obtained with paraffin, soybean oil, and isopropyl myristate as disperse phases, as a function of the oil concentration at a fixed rotor speed of 5000 rpm.Scale bar 100 µm.

Figure S4 :Figure S5 :
Figure S4: Droplet size distributions of emulsions obtained with (a) paraffin oil, (b) soybean oil, and (c) isopropyl myristate; at different rotor speeds with fixed representative oil concentration of ϕ m = 0.2.

Table S1 :
Models available in the literature for droplet diameter of O-W emulsions in turbulence inertial sub-regime.
µ is the dynamic viscosity.The suffix c and d denote properties for water and oil, respectively.ϵ is the turbulence energy dissipation rate.γ is the interfacial tension.tR is the total residence time for coalescence.*N(rpm) is the rotation speed of the rotor-stator mixer.D is the characteristic diameter, which is the diameter of the rotor head for a rotor-stator mixer.L is the outer rotor diameter.N p is the power number.S7 p 1 , p 2 , and α are constants.* ϕ v is the dispersion volume concentration.ρ is the density.* The dimensionless quantities: Re

Table S2 :
Experimental values of mean and representative diameters of emulsion drops.
m m m

Table S3 :
Dimensionless numbers for all the experimental conditions.

Table S4 :
List of tested models with optimized coefficients/exponents

Table S5 :
Estimation of normalized Sauter averaged diameter for all oils.

Table S6 :
Water properties with different surfactants, in addition to corresponding values of interfacial tension for isopropyl myristate in water.

Table S7 :
Estimation of d 32 /D for sunflower oil and isopropyl myristate (ϕ m = 0.2) with different surfactants in comparison with experiments.