## 2 Methodology
### Theoretical Background and Modeling  
In examining the whole pressure drops across a PHE device, three main components [24] can be identified, namely: those occurring in the inlet and outlet manifolds and ports, within the HP passages, and due to elevation changes. Although the pressure loss term associated with the manifolds can be calculated by using an empirical equation [24], its uncertainty significantly impacts the performance evaluation, particularly at high mass flow rates. Because of the mentioned limitations in pre-processing preventing the estimation of that contribution and in pursuit of a well-established procedure, the analysis has been focused on single channels supplied with water at ambient temperature, tested in the same rig as described by [PERSON] et al. [13]. The present activity extends that study, which considered only the central portion of the HP (referred to as the matrix), by investigating both complete and partial HPs. A scheme of the system is depicted in Figure 4a. In summary, the test bench consists of two large stainless-steel vessels to maintain uniform pressure conditions at both ends of the sample--highlighted in red in Figure 4a--which is clamped between them. Detailed information about the test rig is provided in [13]. To replicate the setup, a CAD model of only the fluid domain, comprising the volumes and the channel, was created, as shown in Figure 5. The geometry was modeled using SpaceClaim, part of the Ansys software suite 22 R2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sample** & \(h/h^{*}\) & \(p/p^{*}\) & \(W/W^{*}\) & \(L/L^{*}\) & \(D_{p}/D_{p}^{*}\) & \(L_{p}/L_{p}^{*}\) \\ \hline S1 & 1.11 & 1.21 & 1.25 & 0.58 & 1.78 & 0.53 \\ S2 & 0.74 & 0.92 & 1 & 0.65 & 1 & 0.61 \\ S3 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of investigated plate geometries. Figure 3: Heat plate surface: (**a**) resultant geometry from the forming simulation; (**b**) schematic representation of its overall dimensions. The editing of the HP includes adding the brazed joints, which appear as copper material bridges, connecting contact points between the assembled plates, along with the walls around the closed ports that prevent fluid entry on that side. The brazed material is highlighted in orange in Figure 2b. To simplify their complex shapes, especially when blending with the HP surface without discontinuities, they were approximated as cylinders between the dimples, in the middle part; the diameters were assigned based on the averaged values resulting from optical measurements. The working fluid, water, is modeled as incompressible and single-phase, with constant properties imposed by knowing the operating temperature and pressure. The following equations govern the fluid dynamics problem under investigation:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\overset{\rightarrow}{u} \right)=S_{m}, \tag{1}\]  
\[\frac{\partial\left(\rho\overset{\rightarrow}{u}\right)}{\partial t}+\
abla \cdot\left(\rho\overset{\rightarrow}{u}\overset{\rightarrow}{u}\right)=- \
abla p+\
abla\cdot\left(\overset{\leftarrow}{\tau}\right)+\rho\overset{ \rightarrow}{g}+\overset{\rightarrow}{F}. \tag{2}\]  
representing the laws of conservation of mass (1) and momentum (2), where \(\overset{\rightarrow}{u}\) is the velocity vector; \(S_{m}\) is a general mass source term; \(p\) is the pressure; and \(\overset{\rightarrow}{F}\) denotes the resultant body force. In the present study, \(S_{m}=0\), \(\overset{\rightarrow}{F}=0\), and adiabatic wall conditions were assumed, with no heat transfer to the surroundings. Under these assumptions, Equations (1) and (2) form a decoupled system, with \(\rho=const\). Steady-state analyses were performed using the commercial software Ansys Fluent 22 R2 [25]. Figure 4: Test bench: (**a**) schematic; (**b**) full heat plate; (**c**) matrix obtained from central-portion cut.