## 5 Iterative Process to Obtain the Pressure Loss Coefficients per Unit Length
### Correlations for the Coefficients as a Function of the Geometric and Operational Parameters  
The previous section explained the different trends among the cases. Because the trends establish different behaviors for the parameters, obtaining correlations for the pressure loss coefficients as a function of the geometric and operational parameters is not easy. In order to obtain the correlations, certain restrictions were imposed on the coefficient values, in spite of knowing that the quality of the adjustment was going to be slightly reduced. Thermal designers will be interested in obtaining the values for the pressure loss coefficients for their geometric parameters on their own, which is why the correlations are more imperative than the values obtained by the iterative process. The first restriction is to impose the value of zero on \(\varphi_{z\ ext}\) for the cases that had a value of \(\varphi_{z\ ext}\) greater than zero after the iterative process. As explained above, only two of the 81 cases turn out to have a positive value for \(\varphi_{z\ ext}\). These two cases with a positive  
Figure 6: Axial and tangential velocity patterns (m/s) for the outlet cross-section obtained for case 2 in Table 2 via compact modeling and detailed modeling. coefficient reduce the adjustment of the coefficient values predicted by the correlation to those obtained with the iterative process. The second restriction is related to the value of \(\varphi_{z\;int}\). It was verified that for the cases where \(\varphi_{z\;ext}\) is smaller than zero (nearly all cases), the influence of the value for \(\varphi_{z\;int}\) is strongly reduced, which makes it difficult for the iterative process to achieve good results for both coefficients. However, the shape of the flow pattern depends on the relation between \(\varphi_{z\;ext}\) and \(\varphi_{z\;int}\). For this reason, if one of the coefficients is fixed, it is possible to obtain a correct adjustment for the flow pattern through an adequate value for the other coefficient. Therefore, the value of \(\varphi_{z\;int}\) is fixed to 100,000 (which is the value that this coefficient had for the two cases where \(\varphi_{z\;ext}\) was originally greater than zero). This way, once the value of \(\varphi_{z\;int}\) is fixed, it is easier for the iterative process to achieve a correct value for \(\varphi_{z\;ext}\), and at the same time, the error for those two cases is minimized, as they have their original values for \(\varphi_{z\;int}\). Eventually, the complexity of the model is also reduced from a compact model with three unknown pressure loss coefficients to a model with only two unknown coefficients. With the above restrictions imposed on the values, the iterative process was applied again, this time only looking for \(\varphi_{z\;ext}\) and \(\varphi_{p}\), such that expression 10 was reduced to the following:  
\[\left( \begin{array}{c}\phi_{1}\\ \phi_{2}\\ \phi_{3}\\ \phi_{4}\\ \phi_{5}\end{array} \right)-\left(A\;\;B\right)\left( \begin{array}{cccccccc} \varphi_{p\;1}&\varphi_{p\;2}&\varphi_{p\;3}&\varphi_{p\;4}&\varphi_{p\;5}\\ \varphi_{z\;ext\;1}&\varphi_{z\;ext\;2}&\varphi_{z\;ext\;3}&\varphi_{z\;ext\;4 }&\varphi_{z\;ext\;5}\end{array} \right)-D=0 \tag{16}\]  
With the new pressure loss coefficients, two correlations (one for \(\varphi_{p}\) and the other for \(\varphi_{z\;ext}\)) were obtained using DoE techniques [18]. Before they are presented, it is necessary to explain how they were used. The values of the parameters for the models had to be within the range shown in Table 1, and at the same time, their values had to be entered in a coded way.