# Current Limitations for Predicting Liquid Dispersion in Continuous Flow Bubble Columns Using CFD
## 3 Results
### Liquid-Phase Dispersion in Air-Water Two-Phase System
The experimental variance values confirmed a high level of axial mixing, suggesting that for the low-flow settings, the system approached a perfectly mixed state (Table 4). For all combinations of air and water flow rates, the simulations suggested a system that approximated plug flow. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline \multirow{2}{*}{**Flow Settings**} & \multirow{2}{*}{\(z\) **(m)**} & \multirow{2}{*}{\(r\) **(m)**} & \multicolumn{2}{c}{**Experiments**} & \multicolumn{4}{c}{**Simulations (SAS + VOF)**} \\ \cline{5-10}  & & & \(t_{\text{rm}}\) **(s)** & \({\sigma_{t}}^{2}\) **(s\({}^{2}\))** & \({\sigma_{\theta}}^{2}\) **(-)** & \(t_{\text{rm}}\) **(s)** & \({\sigma_{t}}^{2}\) **(s\({}^{2}\))** & \({\sigma_{\theta}}^{2}\) **(-)** \\ \hline  & 1.05 & 0 & 221.8 & 36,843 & 0.75 & 59.2 & 1210 & 0.35 \\ \(Q_{w}=1.98\) m\({}^{3}\) h\({}^{-1}\); & 1.05 & 0.1 & 218.1 & 35,530 & 0.75 & 60.7 & 1209 & 0.33 \\ \(Q_{u}=0.954\) m\({}^{3}\) h\({}^{-1}\) & 1.84 & 0 & 227.9 & 34,679 & 0.67 & 82.9 & 1603 & 0.23 \\  & 1.84 & 0.1 & 244.4 & 37,890 & 0.63 & 80.6 & 1600 & 0.25 \\ \hline  & 1.05 & 0 & 77.0 & 4409 & 0.74 & 34.8 & 514 & 0.43 \\ \(Q_{w}=3.2\) m\({}^{3}\) h\({}^{-1}\); & 1.05 & 0.1 & 76.1 & 4193 & 0.72 & 35.8 & 523 & 0.41 \\ \(Q_{u}=4.5\) m\({}^{3}\) h\({}^{-1}\) & 1.84 & 0 & 83.5 & 4078 & 0.59 & 44.6 & 540 & 0.28 \\  & 1.84 & 0.1 & 90.7 & 4444 & 0.54 & 44.0 & 538 & 0.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of measured and simulated tracer residence times (\(t_{m}\)) and variance. Figure 5: Comparison of the relative acid concentration versus time in two-phase continuous flow bubble column. Measured and simulated data are compared. Simulations used the SAS-VOF two-phase model. (**A**) Centerline tracer concentration at \(z=1.84\) m (\(Q_{w}=1.98\) m\({}^{3}\) h\({}^{-1}\), \(Q_{a}=0.954\) m\({}^{3}\) h\({}^{-1}\)); (**B**) centerline tracer concentration at \(z=1.84\) m (\(Q_{w}=3.2\) m\({}^{3}\) h\({}^{-1}\), \(Q_{a}=4.5\) m\({}^{3}\) h\({}^{-1}\)); (**C**) centerline tracer concentration at \(z=0.105\) m (\(Q_{w}=1.98\) m\({}^{3}\) h\({}^{-1}\), \(Q_{a}=0.954\) m\({}^{3}\) h\({}^{-1}\)); (**D**) centerline tracer concentration at \(z=1.05\) m (\(Q_{w}=3.2\) m\({}^{3}\) h\({}^{-1}\), \(Q_{a}=4.5\) m\({}^{3}\) h\({}^{-1}\)). Mesh of \(6.9\times 10^{5}\) elements. Irrespective of the axial and radial locations of the measurement points, the values of the mixing parameters were essentially identical for any given combination of the air and water flow conditions (Table 4). This confirmed good radial dispersion. In the axial direction, the lower part of the column (z = 1.05 m) located close to the entrance points of the air and water phases had higher values of the liquid phase dispersion (greater variances \(\sigma_{\theta}\)2) compared to the zone higher up (z = 1.84 m) (Table 4). The temporal evolution of the contours of the tracer mole fraction is shown in Figure 6, 10, 20, and 30 s after tracer injection. Contours for longer elapsed times are included in Figures S5 and S6 (Supplementary Materials). The vertical sections shown represent planes at the center of the column, parallel to the flow (Figure 6). The circular horizontal sections shown represent planes at three different heights across the direction of the flow. For the high flow rates of the two phases (Figure 6A), axial dispersion was evident in the bottom third of the column and in the central plane (elapsed time = 10 s). In the upper sections, the tracer was almost completely dispersed axially. Contours for elapsed times of 20 and 30 s showed no differences in the axial direction. For low-flow combinations (Figure 6B), the tracer was not axially dispersed by the elapsed times of 10 or 20 s after injection. For an elapsed time of 30 s, the horizontal circular plane halfway up the column showed a pattern comparable to that of the corresponding high-flow condition (Figure 6A). This was logical, as the variances for the relevant relative residence times were of a similar order (Table 4).