## 5 CFD Modeling
### Turbulence Model  
A study was carried out on four turbulence models (Standard k-\(\epsilon\), RNG k-\(\epsilon\), MP k-\(\epsilon\), and Realizable k-\(\epsilon\)) to see the effects of the different turbulent models on simulation results. From the steady-state simulations, magnitudes of the average velocity at ten different points in the computational domain for each case were determined, and results were compared. The measurement points were positioned along two straight lines located 1.7 (5.6 ft) and 1.9 m (6.2 ft) above the ground and 12.0 (39.4 ft) and 14.0 m (45.9 ft) from the test gallery inlet, respectively, in the computational domain, as shown in Figure 7. There were five points on the first line spaced 0.5 m (1.6 ft) apart starting 1.5 m (4.9 ft) from the coal face and ending 3.5 m (11.5 ft) from the coal face, while there were five points on the second line spaced 0.5 m (1.6 ft) apart starting 1.8 m (5.9 ft) from the coal face and ending 3.8 m (12.5 ft) from the coal face. The first line was chosen in the high turbulence zone (in front of the scrubber inlet), whereas the second line was chosen in a relatively low turbulence zone (in the middle of the modified shearer). An analysis of steady-state results indicated a maximum deviation of 4.2% among the four turbulence models from their mean velocity, as shown in Figure 8a,b.  
Figure 7: Plan view of the longwall test gallery (not to scale) showing the locations of point of average velocity measurement for the grid independence test.  
Based on the above findings, the most commonly used and well-validated turbulence model in engineering, the standard k-\(\varepsilon\) turbulence model was chosen for this study. The standard k-\(\varepsilon\) turbulence model is a two-equation model, which solves the following two separate transport equations for turbulent kinetic energy, k, and specific dissipation rate, \(\varepsilon\).  
\[\frac{\partial\rho k}{\partial t}+\frac{\partial\rho ku_{i}}{\partial x_{i}}= \frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}} \right)\frac{\partial k}{\partial x_{j}}\right]+G_{k}+G_{b}-\rho\varepsilon \tag{4}\]  
where \(k\) is the turbulent energy in m\({}^{2}\)/s\({}^{2}\), \(\varepsilon\) is the turbulent dissipation rate in m\({}^{2}\)/s\({}^{3}\), \(G_{k}\) is the generation of turbulent kinetic energy that arises due to mean velocity gradient, \(G_{b}\) is the generation of turbulent kinetic energy that arises due to buoyancy, and \(\sigma_{k}\) is the turbulent Prandtl numbers for the turbulent kinetic energy.  
\[\frac{\partial\rho\varepsilon}{\partial t}+\frac{\partial\rho\varepsilon u_{i }}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_ {t}}{\sigma_{k}}\right)\frac{\partial\varepsilon}{\partial x_{j}}\right]+C_{ 1\varepsilon}\frac{\varepsilon}{\bar{k}}(G_{k}+G_{3\varepsilon}G_{b})-C_{2 \varepsilon}\frac{\rho\varepsilon^{2}}{k} \tag{5}\]  
where \(k\) is the turbulent energy in m\({}^{2}\)/s\({}^{2}\), \(\varepsilon\) is the turbulent dissipation rate in m\({}^{2}\)/s\({}^{3}\), \(G_{k}\) is the generation of turbulent kinetic energy that arises due to mean velocity gradient, \(G_{b}\) is the generation of turbulent kinetic energy that arises due to buoyancy, and \(\sigma_{\varepsilon}\) is the turbulent Prandtl numbers for the turbulent specific dissipation rate. The eddy viscosity, \(\mu_{t}\), is given by  
\[\frac{\partial\rho\varepsilon}{\partial t}+\frac{\partial\rho\varepsilon u_{i }}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_ {t}}{\sigma_{k}}\right)\frac{\partial\varepsilon}{\partial x_{j}}\right]+C_{ 1\varepsilon}\frac{\varepsilon}{\bar{k}}(G_{k}+G_{3\varepsilon}G_{b})-C_{2 \varepsilon}\frac{\rho\varepsilon^{2}}{k} \tag{6}\]  
where \(\sigma_{k}\), \(\sigma_{\varepsilon}\), \(C_{1\varepsilon}\), \(C_{2\varepsilon}\), \(C_{3\varepsilon}\), and \(C_{\mu}\) are the empirical constants and their values are 1.0, 1.3, 1.44, 1.92, 0.0, and 0.09, respectively.