## 3 Results and Discussion
### Pressure Drops and Friction Factor Analysis  
Leveraging the detailed information from the CFD data, an additional investigation was conducted to explore potential enhancements to correlations of the friction factor \(f\), thereby deriving a predictive expression suited to the studied BPHE technology. For the matrix models, Figure 13 presents a comparison between two sets of \(f\) lines, derived from CFD simulations applying the distinct definitions of the hydraulic diameter described in Section 2.2, with the coefficient \(f\) calculated from the Darcy-Weisbach equation:  
\[f=2\Delta P\frac{D_{h}}{L}\frac{1}{\rho U^{2}} \tag{7}\]  
where the characteristic velocity \(U\) was determined based on the same cross-section employed in the derivation of Equation (3). Figure 12: Wall shear stress contour for sample S1, with fluid flowing from left to right. Figure 13: Comparison of friction factor trends as a function of Reynolds number, with hydraulic diameter defined by (**a**) Equation (3) and (**b**) Equation (4). Figure 13b refers to the formula in Equation (4) and shows lower deviations, alongside a greater consistency, suggesting an effective characterization of the hydraulic diameter attributed to its ability to capture the spatial variation of the \(A/\Gamma\) ratio. The primary challenge in using Equation (4) lies in the need to extract \(V_{u}\) and \(A_{w,u}\), which are difficult to predict accurately without relying on 3D geometry, a method used in generating the friction factor data in Figure 13. To create a tool capable of estimating \(\widetilde{D}_{h}\) without depending on CAD models, the HP unit surface could be approximated as the product of cosine waves:  
\[Z=\frac{1}{2}h\left[cos\left(\frac{2\pi}{p}x\right)cos\left(\frac{2\pi}{p}y \right)+1\right] \tag{8}\]  
The plot of \(Z\) function is generically illustrated in Figure 14.  
Due to the symmetry of the plate, the corresponding unit volume is obtained by mirroring \(Z\) across the longitudinal plane, therefore \(V_{u}\) is calculated as follows:  
\[V_{u}=2\iint_{-\frac{p}{2}}^{\frac{p}{2}}Z\;dxdy \tag{9}\]  
The infinitesimal area of a surface \(Z(x,y)\) in space is expressed as \(\sqrt{1+(\partial Z/\partial x)^{2}+(\partial Z/\partial y)^{2}}\;dxdy\); hence, for function \(Z\):  
\[A_{w,u}=2\iint_{-\frac{p}{2}}^{\frac{p}{2}}\sqrt{1+\left(\frac{\pi h}{p} \right)^{2}}Bdxdy \tag{10}\]  
with  
\[B=(sin(ax)cos(ay))^{2}+(sin(ay)cos(ax))^{2} \tag{11}\]  
and \(a=2\pi/p\). The calculation of both \(V_{u}\) and \(A_{w,u}\) through Equations (9) and (10) was performed using the Gauss-Kronrod quadrature formulae [33]. To achieve higher accuracy, \(A_{w,u}\) should be adjusted by subtracting the area relative to the brazed points, whose impact on \(V_{u}\) is negligible due to their minimal thickness; since their measured diameters across the selected geometries are similar, an averaged value was adopted. Table 3 shows the comparison between the predicted \(\widetilde{D}_{h}\) values and those obtained from the 3D models. The relatively small errors highlight the potential of the aforementioned approach to be integrated into a more comprehensive description of the friction factor as a function of the Reynolds number. Nevertheless, extending it to a broader range of geometries and  
Figure 14: Approximation of the heat plate unit surface using Equation (8). dimple sizes will require the development of a formulation accounting for the variation in brazed points, which future research must focus on. When considering the full plate pressure drops, a greater discrepancy in the \(f\) coefficients is observed. This is presumed to arise from the effects of the HP outer regions, which are not fully included in the area-averaged velocity definition. Generally speaking, and as confirmed by CFD data, a channel can be virtually divided into two main zones, as shown in Figure 15a: a central region (C), where the flow is predominantly aligned along the port-to-port distance, and distribution areas (DAs), where it begins to deviate as it approaches or exits the ports.