Journal of Chemical Technology and Metallurgy, 53, 4,53, 2018, 664-673 Journal of Chemical Technology and Metallurgy, 4, 2018
THE INFLUENCE OF SURFACTANTS ON THE DRAINAGE AND RUPTURE OF MOBILE LIQUID FILMS BETWEEN DROPS: A PARAMETRIC NUMERICAL STUDY Silvia Alexandrova1, Maria Karsheva2, Aleksandar M. Spasic3, Abdellah Saboni4
Université de Pau et des Pays de l’Adour, ENSGTI / LATEP-IPRA rue J. Ferry BP 7511, 64075 Pau cedex, France E-mail:
[email protected] 2 Department of Chemical Engineering University of Chemical Technology and Metallurgy 8 Kliment Ohridski, 1756 Sofia, Bulgaria 3 Institute for Technology of Nuclear and Other Mineral Raw Materials Department of Chemical Engineering, 86 F. d’Esperey St., 11000 Belgrade, Serbia 4 Université de Pau et des Pays de l’Adour, IUT/GTE-Fédération IPRA Avenue de l’Université, 64000 Pau, France 1
Received 14 September 2017 Accepted 16 March 2018
ABSTRACT Film drainage and rupture during coalescence between two drops approaching each other under a constant force in the presence of insoluble surfactants affected by the van der Waals forces is studied numerically. The mathematical problem is based on the coupled equations of flow in each phase and the convection-diffusion equation governing the surfactant distribution at the interface as well as the related Marangoni effects. The latter are subject to the boundary conditions in the limit of gentle interactions (small-deformations) for which the drops are nearly spherical, except in the near-contact region, where a flattened thin film forms. The finite difference method is used to discretize the lubrication equation in the gap and the interfacial diffusion-convection equation while the boundary integral method is employed to solve the flow in the drops. In this work, a parametric study is carried out by a numerical simulation for Peclet numbers (Pe) ranging from 102 to 104, initial dimensionless surfactant concentrations (Γ0*) from 0 to 10 and dimensionless Hamaker parameter (A*) from 10-7 to 10. The results indicate that the critical film thickness (hc) is strongly dependent on the combination of the three parameters which determines the predominance of van der Waals forces or those of Marangoni. Keywords: film drainage and rupture, drops, insoluble surfactant.
Introduction The understanding of coalescence phenomena is important for many liquid-liquid systems. Coalescence or its prevention is an essential element in the preparation and stability of emulsions and foams, liquid-liquid extractions, emulsification, separation, multiphase transportation and other technical problems referring to liquid-liquid dispersions [1]. The design of nuclear reactors, chemical reactors, boiling and condensation
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equipment are some examples of such applications. Most of the time the liquid-liquid dispersions contain surfactants that can strongly influence the hydrodynamics and the interactions between the drops. For pure systems, numerical studies of the coupled processes of drop deformation and film drainage under the action of a constant interaction force with and without van der Waals forces have been carried out by many authors [2 - 8]. The results show that the drop begins to flatten near the axis of symmetry whereas the film thinning rate is larger near the
Silvia Alexandrova, Maria Karsheva, Aleksandar M. Spasic, Abdellah Saboni
edges of the flattened regions. As time progresses, the interface forms a dimple and the film thinning process slows down. The case of constant approach velocity of the drops has also been studied in the partially mobile case, with and without van der Waals forces [9 - 10]. The presence of surfactants or mass transfer can have a significant influence on the coalescence of drops. The effect of a surfactant or interphase mass transfer on film drainage arises from the extra “Marangoni” forces generated by gradients of the interfacial tension, associated with variations of the solute concentration over the interface [1, 10, 11 - 12]. Yeo et al. [10] investigated numerically the drainage of the film between two drops approaching one another at constant velocity in the presence of an insoluble surfactant. Chesters & Bazhlekov [13] have presented several calculations of the film drainage between drops colliding at a constant force in the presence of an insoluble surfactant. The case of tangentially mobile interfaces has been discussed but only for purely convective surfactant distribution and in the absence of van der Waals forces. More recently, Alexandrova [14] has investigated numerically the drainage of the film between two drops approaching one another at a constant force in the presence of an insoluble surfactant. In the absence of a surfactant, the numerical results have been validated against the numerical studies of Rother et al. [4] and Bazhlekov et al. [5]. The results have also confirmed the long-time asymptotic equations of Nemer et al. [7]. In this study, which is an extension of the work of Alexandrova [14], a further detailed parametric study is carried out using a numerical simulation. Particular interest is focused on how the Peclet number, the van der Waals forces and the surfactant concentration affect the film drainage and rupture. Our study is limited to “tangentially-mobile” interfaces, for which the time scale for near contact motion is determined by the drop phase viscosity [2]. In this case, the results apply to most liquid-liquid systems with the exclusion of the extreme viscosity ratios (fully mobile interfaces, immobile interfaces). We solve the mathematical problem consisting of the coupled equations of flow in each phase and the convection-diffusion equation governing the surfactant distribution at the interface as well as the related Marangoni effects. The latter are subjected to the boundary conditions. The following part describes the equations governing the film drainage and rupture in the presence of surface-active species together with the relevant scal-
ing’s, initial and boundary conditions. These equations are then rewritten in terms of dimensionless variables aiming to reduce the number of system parameters to three dimensionless groups: a surface Peclet number (Pes), a transformed surfactant concentration (Γ0*) and a transformed Hamaker parameter (A*). We analyze the results and examine the effects on the drainage process of the dimensionless groups governing the system. Finally, the conclusions are derived. FORMULATION OF THE PROBLEM Two drops of deformable interfaces subjected to a constant interaction force are considered in the presence of van der Waals intermolecular forces and insoluble surfactants. Both the dispersed phase and the continuous phase are assumed immiscible, Newtonian and incompressible. The study examines the class of gentle, axi-symmetrical interactions, for which the equations governing film drainage assume a universal form in the pure liquid case [3 - 5, 14]. By “gentle” it is meant that the film radius is much smaller than that of the drop, i.e. a/R
