# Gas-Liquid Mixability Study in a Jet-Stirred Tank for Mineral Flotation
## 2 Methodology
### Multiphase and Turbulence Modelling
While other models, such as the Shear Stress Transport (SST) model, the Reynolds Stress Model (RSM), and Direct Numerical Simulation (DNS), offer higher accuracy, they require significantly more computational resources. Consequently, their practicality is limited given the scale and complexity of the present study. The flow field inside the mixing tank exhibits complex turbulent characteristics, with the standard k \(-\)\(\varepsilon\) turbulence model commonly used in current studies [22; 23; 24; 25; 26]. However, this model has limitations, particularly in accurately predicting shear flow in jet-mixing applications. [PERSON] and [PERSON] [27] conducted mixing experiments in a semi-industrial-scale tank (depicted in Figure 2a) using a side-incidence injector. Mixing profiles were determined by monitoring the homogenization process of a dark blue Nigrosine solution in the tank. The experiments involved varying jet angles of 0\({}^{\circ}\), 22.5\({}^{\circ}\), and 45\({}^{\circ}\), with data recorded using an in-line detector photometer. Comparing experimental data with numerical simulations using various turbulence models, the mixing curves in Figure 2b demonstrate that the RNG model predicts peaks similar to experimental observations with a reasonable time delay. To further quantify the accuracy of the RNG model, the root mean square error (RMSE) was calculated and presented in Table 2, showing that the RNG model yields results closer to experimental measurements compared to other models tested. The root mean square error (RMSE) is a commonly used metric to evaluate the discrepancy between model predictions and actual observed data. It reflects the magnitude of the difference between predicted and actual values [28]. The smaller the RMSE value, the closer the model's predictions are to the actual observations, indicating higher model accuracy. To enhance the validation of the accuracy and reliability of the RNG turbulence model in this study, we drew upon the work of [PERSON] and [PERSON] [29], who combined tracer experiments with computational fluid dynamics (CFDs) simulation techniques to conduct an in-depth investigation into the complex fluid flow and mixing mechanisms within a jet mixer. By employing Residence Time Distribution (RTD), a statistical method for measuring the distribution of fluid or particle residence times within a chemical reactor, they systematically compared the experimentally measured RTD data with the RTD data predicted by CFDs simulations, as shown in Figure 3. We write the root mean square errors for the three different turbulence models in Table 3. The comparison results showed that the RTD curve predicted by the RNG k \(-\)\(\  
The \(\mathrm{k}-\varepsilon\) model based on the Reynolds-averaged Navier-Stokes (RANS) approach is widely applicable for the prediction of turbulent flow fields, in which the turbulent viscosity \(\mathrm{\upmu t}\) is obtained by two additional transport equations for the turbulent kinetic energy \(\mathrm{k}\) and the turbulent dissipation rate \(\varepsilon\)[30]. The RNG \(\mathrm{k}-\varepsilon\) model is based on the standard \(\mathrm{k}-\varepsilon\) model and modified by a mathematical technique called the renormalization group (RNG) method, which can achieve a better response to rapidly strained and low Reynolds number flows [31].