# Modeling Polymer Microencapsulation Processes Using CFD and Population Balance Models
## 3 Results and Discussion
### CFD + PBM Model Performance
#### 3.2.2 Particle Size Distribution  
Figure 7 shows the bin fraction for various particle sizes at different rotational speeds, comparing CFD simulations (blue bars) with experimental data (orange lines). At low agitation speeds (5000/7500 rpm), the experimental results show larger fractions for larger particle sizes, indicating that lower agitation speeds favor the formation of larger particles, consistent with a turbulent viscous emulsification regime. For these low-speed cases, the CFD simulations underestimate these larger particles, probably signaling that the turbulence model is overestimating the turbulent kinetic energy (thus obtaining smaller eddies) under those flow conditions. As the rotational speed increases (10,000 rpm or above), the experimental and CFD results exhibit a better agreement across all particle sizes, reflecting a transition towards a turbulent inertial regime. However, the simulations tend to slightly underestimate the fraction of the smallest particles. The cumulative distribution of particle diameters for various rotational speeds presented in Figure 8 highlights the aforementioned. The CFD model fails to capture the size distribution trend at lower speeds (5000 rpm and 7500 rpm), likely making it unsuitable for turbulent viscous emulsification conditions. The particle size distribution obtained from simulations was analyzed using D10, D50, and D90 metrics, representing the particle diameters at the 10 th, 50 th, and 90 th percentiles, respectively. Figure 9 illustrates these results, providing a clearer view of the trends observed in the cumulative size distributions shown in Figure 8. Greater deviations from the experimental trend are observed for the 10 th and 50 th percentile at low agitator rotational speeds, coinciding with an overprediction of number densities for small particle sizes at these flow regimes. Figure 7: CFD vs. Experimental bin fractions. Figure 10 presents the computed relative errors by bin size and agitation speed, revealing higher errors in particle size distribution at lower agitation speeds. It is worth noticing that the largest relative errors occur for smaller particle sizes. Despite minor discrepancies, the CFD simulations closely align with the experimental data at higher agitation speeds (>10,000 rpm), validating the simulation strategy's accuracy in predicting particle size distributions under turbulent viscous operational conditions, with acceptable errors below 10%. However, the model fails to accurately capture the flow characteristics at lower agitation speeds (<7500 rpm) within the turbulent inertial regime, resulting in significantly higher mean errors, meaning that the simulation is unable to accurately capture the flow field and dynamics under those operating conditions. This behavior was also observed in the validation case results (Figure 4). At lower agitation speeds, as the turbulence level decreases, the flow regime may transition to a turbulent viscous regime, where different mechanisms govern droplet breakup and coalescence. The CFD model may struggle to accurately capture these regime-dependent phenomena at lower flow velocities, likely due to the performance of the turbulence model used [62]. As turbulence intensity increases, the influence of interfacial forces, such as surface tension and viscous forces, becomes less significant compared to the turbulent forces acting on the droplets [63], leading to better agreement between experimental and simulation results at higher speeds. These results highlight the necessity for further research to ensure suitable turbulence model selection for the CFD modeling process. A known issue of the realizable \(k\)-\(\varepsilon\) model is  
Figure 8: Cumulative size distributions. Figure 9: Particle diameters at the 10 th, 50 th, and 90 th percentile. its tendency to overpredict turbulent kinetic energy and underpredict energy dissipation in high mixing flow conditions [64], which may explain its higher inaccuracy at low agitation speeds.