## 2 Physical Modelling and Numerical Methodology
### Numerical Simulation
#### 2.1.1 Equations for Fluid Phase and Flow Solver  
The fluid phase is described using a Eulerian approach in which the RANS equations are solved. When considering a transient flow in the present study, strictly speaking, URANS equations are solved. Equation (2) is the continuity equation and Equation (3) is the Reynolds averaged quantity of motion equation. Here, \(\mathbf{F}_{i}\) is the external body force that arises due to interaction with the dispersed phase, and \(g_{i}\) is the gravitational acceleration vector.  
\[\frac{\partial U_{i}}{\partial x_{i}}=0 \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **D** & \(\mathbf{L}_{1}\) **[mm]** & \(\mathbf{L}_{2}\) **[mm]** & \(\mathbf{R}_{c}\) **[mm]** & \(\mathbf{\theta}\) [\({}^{\circ}\)] \\ \hline Zhang & 104 & 30D & 50D & 2D & 90 \\ Guda \({}^{\star}\) & 76.2 & 5D & 30D & 1.8D & 90 \\ \hline \hline \end{tabular} \({}^{\star}\)Guda case maintains original geometry and is solved for a mass load of 0.35 and 0.42.  
\end{table}
Table 1: Dimensions pipeline validation cases.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & \(\mathbf{\mu}\) **[Pa-s]** & \(\mathbf{\rho}\) **[kg/m\({}^{3}\)]** & \(\mathbf{T}\) **[K]** & **Shape** &
\begin{tabular}{c} **Shape** \\ **Factor [-]** \\ \end{tabular} & **Sizes [\(\mathbf{\mu}\)m]** \\ \hline Air & 1.802 \(\times\) 10\({}^{-5}\) & 1.225 & 288.73 & - & - & - \\ Particles & - & 2710 & - & Non-spherical & 0.755 & d1 = 50 & d2 = 150 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Fluid phase and dispersed phase properties.  
Figure 1: Definition of the geometric parameters in the models.  
\[\frac{\partial U_{i}}{\partial t}+U_{j}\ \frac{\partial U_{i}}{\partial x_{j}}=-\frac{1}{ \rho}\frac{\partial P}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left(v \frac{\partial U_{i}}{\partial x_{j}}\right)-\frac{\partial}{\partial x_{j}} \langle u_{i}^{\prime}u_{j}^{\prime}\rangle+g_{i}+F_{i} \tag{3}\]  
The model describing the transport of mean properties of turbulent flows for the scales being modeled will be the RNG \(k\)-\(\epsilon\) model. The model offer balance and good \"hit rates\" when compared to others, as shown by [PERSON] et al. (2018). The RNG \(k\)-\(\epsilon\) model has governing equations like the standard \(k\)-\(\epsilon\) model, expressed in a turbulence energy transport equation \"\(k\)\" (Equation (4)) and a dissipation rate \"\(\epsilon\)\" (Equation (5)).  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho U_{i}k)}{\partial x_{ i}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\alpha_{k}\mu_{eff}\frac{\partial k}{ \partial x_{j}}\Bigg{]}-\rho\epsilon+G_{k}+G_{b}-Y_{M}+S_{k} \tag{4}\]  
\[\frac{\partial(\rho\epsilon)}{\partial t}+\frac{\partial(\rho U_{i}\epsilon )}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\alpha_{\epsilon} \ \mu_{eff}\frac{\partial\epsilon}{\partial x_{j}}\Bigg{]}+C_{1\epsilon}\frac{ \epsilon}{k}\big{(}\ G_{k}+C_{1\epsilon}G_{b}\big{)}-C_{2\epsilon}\rho\frac{ \epsilon^{2}}{k}-R_{\epsilon}+S_{\epsilon} \tag{5}\]  
The RNG \(k\)-\(\epsilon\) model is obtained using a statistical technique called renormalization group theory that is derived from the instantaneous Navier-Stokes equations, from which analytical derivative results are derived in a model with constants \(C_{\mu}=0.0845\), \(C_{1\epsilon}=1.42\), \(C_{2\epsilon}=1.68\).