## 5 Iterative Process to Obtain the Pressure Loss Coefficients per Unit Length
### Reliability of Compact Modeling
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Factor** & \(\beta_{0}\) & \(D\) & \(D_{h}\)/\(D\) & \(S\) & \(V\) & \(a\) & \(t\) \\ \hline
**Value** & \(-771.53\) & \(519.09\) & \(142.53\) & \(-26.18\) & \(26.55\) & \(284.48\) & \(557.85\) \\ \hline
**Factor** & \(\varepsilon\) & \(\xi\) & \(D\)-\(D_{h}\)/\(D\) & \(D\)**-\(t\)** & \(D\)**-\(\xi\)** & \(a\)-\(D_{h}\)/\(D\) & \(t\)-\(D_{h}\)/\(D\) \\ \hline
**Value** & \(-74.12\) & \(175.85\) & \(75.16\) & \(-425.94\) & \(86.75\) & \(-116.31\) & \(-125.81\) \\ \hline
**Factor** & \(S\)**-\(\varepsilon\)** & \(V\)**-\(\varepsilon\)** & \(V\)**-\(\xi\)** & \(a\)**-\(t\)** & \(a\)**-\(\varepsilon\)** & \(t\)**-\(\xi\)** & \(t\)**-\(t\)** \\ \hline
**Value** & \(88.50\) & \(55.91\) & \(-53.75\) & \(-108.91\) & \(-57.81\) & \(-108.34\) & \(-193.59\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Terms of the correlation for \(\varphi_{z,\;ext}\). Figure 8: Flow magnitudes for the outlet cross-section of the domain obtained with the coefficients predicted by the correlations vs. the flow magnitudes obtained via the detailed models. Figure 7: Flow magnitudes for the outlet cross section of the domain obtained with the coefficients predicted by the correlations vs. the flow magnitudes obtained with the coefficients previous to the correlations. There is also interest in comparing how using the compact model improves the results that thermal designers attain without this model. Nowadays, thermal design is a challenge that is tackled by using pressure loss coefficients obtained from different handbooks, such as that by [PERSON] [20]. The problem is that thermal designers use only one pressure loss coefficient. In order to show the improvements provided by the compact model, this pressure loss coefficient was calculated from [PERSON] [20] for each of the 81 cases in Table 2. The geometries for each of the cases in Figure 2 were simulated by imposing the corresponding value of the calculated pressure loss coefficient on \(\varphi_{z\ ext}\) and \(\varphi_{z\ int}\), and \(\varphi_{p}=0\), which is the process used most frequently by thermal designers. In Figure 9, the flow magnitudes obtained via simulation for the outlet cross-section are compared with those obtained from the detailed models in [PERSON] et al. [17]. This figure shows that the predictions for each of the flow magnitudes (\(\overline{v_{t}}\), \(S\), \(V_{min}\) and \(V_{max}\)) made by using handbooks are much worse than the predictions obtained by the compact model presented in this work. In fact, the values for R\({}^{2}\) are clearly lower (unless for the maximum velocity). Furthermore, it is important to highlight that the correlation line is far from the ideal case of \(y=x\), which means that in spite of having a good value for R\({}^{2}\) (as is the case for \(V_{max}\)), the error produced with respect to the detailed models is high for nearly all samples. As the figure indicates, for \(V_{max}\), most of the samples are in the \(\pm 20\%\) range of difference, in spite of having an R\({}^{2}\) of 0.93.