## 3 Results and Discussion
### Pressure Drops and Friction Factor Analysis
Near the latter zones, the reduction in the actual passage leads to an increase in velocity magnitude, as illustrated by the contour plot for sample S1 in Figure 15b. Therefore, in the proximity of inlet and outlet, the pressure trend differs from that observed in the middle part. This becomes evident when studying the gradient \(\Delta P/\Delta x\), with \(x\) being the longitudinal coordinate measured from the HP center: for instance, a plot of this quantity, averaged over the cross-sectional areas and scaled relative to the value measured at \(x=0\) (denoted as \(\Delta P_{mid}/\Delta x\)), is depicted for sample S1 in Figure 16, at different Reynolds numbers. With an almost constant \(\Delta P/\Delta x\) in the central region, pressure drops linearly, experiencing a significant increase, which is practically symmetric, where the flow direction changes, before reducing again at the ports. A similar pattern can be extracted by comparing the velocity components along the width, as they grow near the outer zones. Varying the Reynolds number, the situation remains consistent, as the curves are nearly overlapped, leading to an identical estimation of the distribution areas' extent; this was based on a threshold of a 5% rise in \(\Delta P/\Delta x\). Figure 12:0 Channel regions: (**a**) schematic representation; (**b**) velocity contour for sample S1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample** & \begin{tabular}{c} **CAD**, \\ \(\widetilde{D_{h}}\widetilde{D_{h}}\) \\ \end{tabular} & \
\begin{tabular}{c} **Prediction** \\ \(\widetilde{D_{h}}\widetilde{D_{h}}\) \\ \end{tabular} &
\begin{tabular}{c} _Error_ \\ **[\%]** \\ \end{tabular} \\ \hline S1 & 1.21 & 1.19 & \(-2\) \\ S2 & 0.82 & 0.82 & 1 \\ S3 & 1 & 1.03 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of hydraulic diameter values. Table 4 presents the relative lengths and contributions to the global \(\Delta P\) for the three examined channel models. The properties of each individual part change across the geometries, explaining their subsequent influence on the friction factor; in particular, the \(\Delta P\) for sample S3 is primarily determined by the losses along the middle section, as the plate has a high aspect ratio \(L/W\). To account for these variations and improve the friction factor formulation for a better pressure drops prediction, an analysis has been conducted to incorporate the effects illustrated. The pressure drop due to the channel passage can be expressed as follows:  
\[\Delta P=\Delta P_{C}+\Delta P_{DA} \tag{12}\]  
where \(\Delta P_{C}\) is derived by applying Equation (7), with \(f\) obtained in both cases from the matrix behavior. Regarding \(\Delta P_{DA}\), an analogous equation is employed, with the length and average velocity \(U_{DA}\) adjusted to reflect the flow behavior in the external zones. It is assumed to move through a hypothetical stream tube, with one boundary defined by the closed port, which, in conjunction with the open port, directs the flow tangentially to them. This approximation enables the estimation of the reference length \(L_{DA}\), which is used in Equation (7), as well as the longitudinal extent \(L_{DA}^{*}\) of the distribution areas. The concept is depicted in Figure 17.  
By introducing the port spacing \(w_{p}\) along the width, \(L_{DA}^{*}\) is written as follows:  
\[L_{DA}^{*}=\frac{W}{2}tan\bigg{(}arcsin\bigg{(}\frac{D_{p}}{w_{p}}\bigg{)} \bigg{)} \tag{13}\]  
with the argument of the arcsine representing the angle between the tangent to the ports and the line connecting their centers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multicolumn{2}{c}{**Central Region**} & \multicolumn{2}{c}{**Distribution Areas**} \\ \cline{2-5}  & **Extent [\%]** & \(\Delta P\) **[\%]** & **Extent [\%]** & \(\Delta P\) **[\%]** \\ \hline S1 & 59 & 46 & 41 & 54 \\ S2 & 62 & 55 & 38 & 45 \\ S3 & 86 & 95 & 14 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Channel regions characteristics.