# Current Limitations for Predicting Liquid Dispersion in Continuous Flow Bubble Columns Using CFD
## 3 Results
### Liquid-Phase Dispersion in Air-Water Two-Phase System  
Two-phase flow in bubble columns has highly turbulent and anisotropic characteristics [25]. For batch bubble columns, it has been shown that RANS models are suitable for simulating flow when the objective is to obtain a stationary solution and the usual assumption of isotropic turbulence applies. The two-phase systems in bubble columns do not conform to the isotropic turbulence requirement because of the bubbles. In addition, the pulse tracer mixing processes are transient, and therefore the use of RANS models for such systems requires caution [26]. Therefore, for the continuous bubble column, based on the liquid-single-phase dispersion study, the SAS and SST-SAS models were selected for simulating the air-water (two-phase) system. It was found that stable two-phase simulations could not be performed using the SST-SAS model as the solution diverged after a few time steps, irrespective of the time step and the under-relaxation coefficients. Although the SST is a comprehensive model, instabilities develop, resulting in divergence issues in complex flows [20]. Therefore, the SAS turbulence model was adopted, as it had also demonstrated a good fit to the experimental data (Figure 4B-D). In Fluent(r) (Canonsburg, PA, USA), only the VOF two-phase model is compatible with the SAS turbulence model and requires surface tension for accurate simulations [28]. Both the low (\(Q_{w}\) = 1.98 m\({}^{3}\) h\({}^{-1}\), \(Q_{a}\) = 0.954 m\({}^{3}\) h\({}^{-1}\)) and high flow (\(Q_{w}\) = 3.2 m\({}^{3}\) h\({}^{-1}\), \(Q_{a}\) = 4.5 m\({}^{3}\) h\({}^{-1}\)) scenarios were simulated using SAS-VOF. Compared to other LES approaches, SAS considers the ratio of the turbulent length scale to the von Karman length scale as a dimensionless measure of non-homogeneity. Therefore, the model can provide good quantitative results for many unsteady flows, even when relatively coarse meshes are used. LES and DES are known to experience the modeled-stress depletion (MSD) phenomenon, resulting in an incorrect application of LES in areas where RANS should be used. For example, incorrect behavior might occur within confined flows [35]. In addition, LES models need an initial estimate of the velocity field, which is usually obtained with RANS models. Indeed, it is advised that a solution based on two-equation models be used as a starting point (Ansys User Guide). For all models based on LES, including SAS, it is important to know the size of the eddies that the model is attempting to solve. In standard LES models, eddy size is an input parameter, whereas in SAS, it is determined within the simulation. At least two cells are required in each direction for a model to resolve a vortex with a given scale length \(l\) (Ansys Quick Guide). The larger eddies, corresponding to the integral scale length, \(l_{0}\), can be calculated using:  
\[l_{0}=k^{3/2}/\varepsilon \tag{1}\]The mesh resolution determines the fraction of turbulent kinetic energy that can be solved directly. The ratio of the integral scale length to the cell size, i.e., \(l_{0}/\Delta\) (\(\Delta\) = cell volume\({}^{1/3}\)), relates to the fraction of turbulent kinetic energy that can be solved as a function of the size (\(\Delta\)) of the cells. The higher the ratio, the better the mesh (as there are more cells for a given scale length), and the higher the percentage of energy that can be calculated directly. For ratios between 1.25 and 4.8, eddies accounting for 50-80% of the total mechanical energy are calculated. The values of \(l_{0}/\Delta\) in various parts of the bubble column are shown in Figure 54 (Supplementary Materials) to illustrate the resolution capacity of the chosen mesh. For the two-phase simulations, the values of \(l_{0}/\Delta\) oscillated between 1.4 and 2.7, increasing to 3.5 in some regions (Figure S4). This meant that the selected mesh resolved eddies that dissipated 65-70% of the energy.