Modeling of phase inversion phenomenon

Abstract

Turbulent liquid–liquid dispersions are frequently encountered in the chemical industry in various operations such as liquid–liquid extraction, liquid–liquid reactions, polymerization, and direct contact heat transfer. Operational efficiency of these operations is a function of drop size distribution and state of mixing of the dispersed phase. If holdup of dispersed phase is increased, at a critical holdup of dispersed phase, dispersion inverts - dispersed phase becomes continuous phase and vice versa. This phenomenon is known as phase inversion and the critical holdup at which it occurs is called inversion holdup. Although the phenomenon has been studied experimentally over the years, the mechanism of phase inversion is not very clear and there is no physical model which can comprehensively describe the various features of phase inversion. One of the interesting features of phase inversion is existence of ambivalent region; in this region either of the phases can be continuous depending on the path history and starting conditions. Since two different types of dispersions show the opposite behavior as interfacial area after inversion increases in case of w/o and decreases in case of o/w dispersion, interfacial energy minimization criteria cannot explain the ambivalent region, an important characteristic of phase inversion. One of the features that these dispersions show is that as agitation rate increases at a very high value, inversion holdup reaches a constant asymptotic value. This asymptotic value has been found to be independent of the way turbulence has been generated. Phase inversion occurs because of imbalance between breakup and coalescence of drops. If the system at equilibrium is given a slight change in holdup, it reaches another equilibrium drop size distribution. If a step change in dispersed phase holdup is given to the system, it can take it to inversion catastrophically. Some of the models proposed for the breakup and coalescence in conjunction with population balance framework were not able to predict phase inversion. As number of drops decreases, the coalescence frequency decreases drastically. Thus most of the models which can adequately describe drop size distribution fail to predict phase inversion. A promising concept to explain phase inversion phenomenon is that breakage and coalescence no longer are independent events. Near inversion point, at high holdup, a drop which has to deform in order to break collides with another drop, resulting in coalescence. The present work aims to model and undertake Monte Carlo type simulation on a two-dimensional lattice for breakup and coalescence of drops at high holdup, including the possibility of breakup-induced coalescence of drops to predict phase inversion. Motion of the drops has been modeled from analogy to kinetic theory of gases, assuming them to be moving randomly. Diffusivities and the breakup frequency depending on the size of the drops are assigned to drops. Event selection is based on the relative probability ratio and for drop selection, the cumulative probability graph is used. If a drop collides with another, depending on the coalescence efficiency, coalescence is permitted or rejected. Proposed model is advantageous in the sense that it can handle multiple and breakup-induced coalescence and system evolution is tracked naturally to predict the catastrophic phase inversion. When coalescence rate dominates breakup rate, phase inversion occurs. Phase inversion in our simulations is realized as formation of infinitely big drop. Simulations show that at high number density of drops, there is no universal scaling exponent for coalescence. They are sensitive to initial conditions and holdup of drops. Simulation scheme is very powerful in terms of explaining various unexplained features of phase inversion: like ambivalent region, constant asymptotic value of inversion holdup, high agitation and wide range in which phase inversion can occur.