## 5 Iterative Process to Obtain the Pressure Loss Coefficients per Unit Length  
The objective of the iterative process is to obtain the values for the pressure loss coefficients for each of the 81 cases introduced in Table 2. As such, the same iterative process is applied to each of the 81 cases. The process for each of the cases starts with 5 initial simulations with randomized and different values for \"\(\varphi_{p}\)\", \"\(\varphi_{z}\)_int\"_ and \"\(\varphi_{z}\)_ext\"_. This way, the response of the model (which could be described as a response surface of the coefficients) to coefficient values changes is captured. From each of the 5 simulations, values are obtained for the area weighted average of the tangential velocity \"\(\overline{\varphi_{t}}\)\", the swirl number \"\(S\)\" and the minimum and the maximum axial velocities \"\(V_{min}\)\" and \"\(V_{max}\)\" from the outlet cross-section (\(z=40\) mm in Figure 2). These magnitudes are the flow magnitudes that [PERSON] et al. [16] defined in order to  
Figure 3: Representation of the mesh for one of the cases simulated of the compact model. correctly define a flow pattern with an annular shape. Therefore, it might be said that the values of the flow magnitudes in each of the 5 simulations correctly define the different flow patterns (as a result of the different pressure loss coefficients) at the outlet cross-section of the model. The next step is to find a function that represents the response surface model. That function can be described by obtaining the values \(A\), \(B\), \(C\) and \(D\) in Equation (10). This equation represents the relationship between the pressure loss coefficients (\(\varphi_{p\ i}\), \(\varphi_{z\ int\ i}\) and \(\varphi_{z\ ext\ i}\)) used for the five initial simulations and the result obtained for \(\phi_{i}\) in the five cases, where \(\phi_{i}\) is one of the flow magnitudes (\(\overline{v_{t}}\), \(S\), \(V_{min}\) and \(V_{max}\)):  
\[\left( \begin{array}{c}\phi_{1}\\ \phi_{2}\\ \phi_{3}\\ \phi_{4}\\ \phi_{5}\end{array} \right)-\left( \begin{array}{cccccccc}\varphi_{p\ 1}&\varphi_{p\ 2}&\varphi_{p\ 3}&\varphi_{p\ 4}&\varphi_{p\ 5}\\ \varphi_{z\ int\ 1}&\varphi_{z\ int\ 2}&\varphi_{z\ int\ 3}&\varphi_{z\ int\ 4}& \varphi_{z\ int\ 5}\\ \varphi_{z\ ext\ 1}&\varphi_{z\ ext\ 2}&\varphi_{z\ ext\ 3}&\varphi_{z\ ext\ 4}& \varphi_{z\ ext\ 5}\end{array} \right)-D=0 \tag{10}\]  
Thus, for each of the flow magnitudes, a group of values for \(A\), \(B\), \(C\) and \(D\) is obtained. With these values, a new step is carried out to search for the values for \(\varphi_{p}\), \(\varphi_{z\ int\ i}\) and \(\varphi_{z\ ext\ i}\) that minimize Equation (11) according to the response surface (\(A\), \(B\), \(C\) and \(D\)) that has been obtained for each of the flow magnitudes. In other words, a combination of pressure loss coefficients that create a flow pattern and pressure drop like that obtained in the detailed model is sought. Equation (11) consists of a summary of the differences (Equations (12)-(15)) obtained when comparing the flow magnitudes obtained with the detailed model from [PERSON] et al. (2016) and those obtained with the compact model whose coefficients are being determined. The solution that is obtained (\(\varphi_{p\ i}\), \(\varphi_{z\ int\ i}\) and \(\varphi_{z\ ext\ i}\)) is the one that minimizes Equation (11) according to the response surface (\(A\), \(B\), \(C\) and \(D\)) obtained in the previous step.