# Modeling Polymer Microencapsulation Processes Using CFD and Population Balance Models
###### Abstract
Other authors employed a similar modeling approach to estimate the droplet size evolution in an oil-in-water emulsion [24]. In both instances, the researchers reported the mean particle or droplet size evolution over time but did not consider the size distribution of the particles, focusing only on particle growth. A discrete PBM method coupled with a Eulerian-Eulerian approach for multiphase flow and a \(k\)-\(\epsilon\) turbulence model was validated for a gas-liquid (air in water) dispersion in a tank [25], using the moving reference frame (MRF) technique and [PERSON]'s model for bubble breakage and coalescence [4]. In another study, a discrete PBM method was solved using a sequential solution approach; first, simulating the turbulent flow field and then solving the population balance equations. Due to the extensive computational time required for calculating hydrodynamic variables coupled with population balance equations, the authors recommended parallelizing the model in future work to reduce computational costs [26]. A mixture multiphase flow model in combination with a \(k\)-\(\epsilon\) turbulence model, along with [PERSON]'s model for droplet breakage and coalescence [27], was also tested for liquid-liquid separation in hydro cyclones [28]. Several researchers have coupled CFD flow solutions with a Direct Quadrature Method of Moments (DQMOM) population balance model to study the aggregation and breakage processes in one-dimensional domains. These studies mainly reported on the computation times, numerical methods, and benchmarks of the different CFD packages [29; 30; 31]. The effect of oil phase viscosity in droplet formation frequency in an oil-in-water microchannel emulsification system has also been studied by means of CFD [32]. In this study, the authors compared drop size distributions obtained through CFD modeling against experimental results; unfortunately, no relevant details about the simulation strategy adopted were facilitated. Many studies have focused on modeling droplet breakage [33; 34; 35; 36] and the coalescence [37; 38; 39; 40] phenomena, which are significant in the design of dispersion processes. However, predicting particle sizes in inhomogeneous flows with spatially and temporally varying velocity magnitudes and directions remains a challenge, especially at the microencapsulation particle length scales. The breakup and coalescence processes determine source terms in the population balance equation, while velocity vectors from the transport equations determine the convective term. An important computational issue is the solution method as follows: sequential (first calculating the flow field and turbulence, then solving the population balance) versus simultaneous (solving all equations together from the start). The accuracy of the predictions is influenced by the chosen solution method [32]. Several authors have combined CFD and PBM for particle size distribution prediction in emulsion processes. A study on laboratory-scale mixing tanks investigated the particle size distribution in water-in-oil emulsions using Conroe and Troika oil, examining the effects of different impeller speeds (300, 400, 600 rpm) on water droplet breakage and coalescence [41]. The study presents a computational model using Fluent 6.3.26. A \(k\)-\(\epsilon\) turbulence model, a Euler-Euler multiphase model, and PBM were selected to simulate the mixing process. Validation through comparison with experimental data showed satisfactory similarities. Other studies extended this work by coupling the PBM model with a Euler-Lagrange multiphase model to see the effect of the droplet's trajectories on the size distribution and [42; 43]. Another study compared experimental and simulation data for a baffled mixing tank, using a Nexbase oil mixture and 3.5% NaCl brine at various agitator speeds [44]. By altering impeller speeds and introducing surfactant SPAN 80, the authors obtained experimental particle size distributions using particle vison and measurement (PVM) probes.