## 2 Geometric and Operational Parameters for the Numerical Runs
### Mathematical Modeling
#### 3.2.2 Porous Media Modeling  
Both volumes corresponding to the screen (the inner and the outer volumes with \"\(\prime\prime\)\" thickness, as shown in Figure 2) are assumed as porous media volumes by using directional pressure loss coefficients (\(\varphi_{x}\), \(\varphi_{y}\) and \(\varphi_{z}\)) per unit length [33]. The same coefficients (\(\varphi_{x}=\varphi_{y}=\varphi_{p}\)) were used for the \"\(x\)\" and \"\(y\)\" directions, and this value is the same for the inner and outer volumes (\(\varphi_{p\,\,int}=\varphi_{p\,\,ext}=\varphi_{p}\)), resulting in a model with greater simplicity. Therefore, the objective of the work is to obtain the pressure loss coefficients \"\(\varphi_{p}\)\", \"\(\varphi_{z}\) _int_\" and \"\(\varphi_{z}\) _ext_\" per unit length that define a compact model.  
The modeling of a screen through the porous media approximation employs a volumetric hydraulic resistance defined by the previous directional pressure loss coefficients per unit length. In the following expression, the generic strain that must be fixed for both the inner and outer porous media volumes in each of the 81 cases from Table 2 is given:  
\[\varphi_{ij}=\left(\begin{array}{ccc}\varphi_{p}&0&0\\ 0&\varphi_{p}&0\\ 0&0&\varphi_{z}\end{array}\right) \tag{8}\]  
This porous media approach works as a momentum sink in the momentum equation. This sink usually has two different components: the first one is related to the viscous losses (proportional to the velocity), and the second one is due to inertial losses (proportional to the squared velocity). For this study, with the EMC screens, viscous loses are neglected as the inertial losses are far higher. Therefore, the momentum sink might be defined as  
\[S_{i}=-\left(\varphi_{ij}\cdot\frac{1}{2}\,\rho\cdot v_{mag}\cdot v_{j}\right) \tag{9}\]  
where \(v_{mag}\) corresponds to the magnitude of the velocity and \(v_{j}\) to the component of the velocity in \"\(y\)\" direction.  
Due to the fact that the interior walls of the pores are not modeled in the porous media, a special treatment for the velocities is needed. For this reason, by ensuring the continuity of the vectors through the porous media, a superficial velocity based on the volumetric flow is employed. Therefore, the physics inside the compact model are not the same as those that occur in reality or in the detailed models. In fact, compact models are not able to correctly predict the jets produced by the pores (see the velocity field at the outlet of the screen in Figure 1). This is the reason why the compact models introduced in this work will be reliable at a certain distance from the screen once the flow has been homogenized and the jets have disappeared.  
\begin{table}
\begin{tabular}{c c} \hline \hline  & _Standard k-\(\varepsilon\)_ \\ \hline \(\Gamma_{k}\) & \(\mu+\frac{\mu_{t}}{\varphi_{z}}\) \\ \(\Gamma_{\varepsilon}\) & \(\mu+\frac{\mu_{t}}{\varphi_{z}}\) \\ \(P_{k}\) & \(\mu_{t}S^{2}\) \\ \(D_{k}\) & \(\rho\) \\ \(P_{\varepsilon}\) & \(C_{1\varepsilon}\frac{\varepsilon}{P_{k}}\) \\ \(D_{\varepsilon}\) & \(C_{2\rho}\frac{\varepsilon}{\varepsilon}\) \\  & \(\sigma_{k}=1.0\); \(\sigma_{\varepsilon}=1.3\) \\ Coeff. & \(C_{1\varepsilon}=1.44\) \\  & \(C_{2\varepsilon}=1.92\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Terms and coefficients for the standard k-\(\varepsilon\) model.