## 2 Materials and Methods
### PWC Technique with the Design of Experiment  
The PWC follows certain rules to produce an output. To understand the behavior of the PWC, an experiment in which five water columns were connected in parallel was conducted. This experimental design is called a \"double-nozzle setup\", in which one nozzle makes the main outlet, which injects the additional water into the existing penstock of the PAT, while each water column has a separate nozzle. Computational fluid dynamics (CFD) simulations were performed using ANSYS software R1 2021 [27]. The dimensions of the PWC and specifications of the FLUENT solver are listed in Table 1.  
The geometrical dimensions of the PWC were composed in a design modeler, and the settings were changed from solid to fluid, while meshing of the PWC was performed using ANSYS meshing. A tetrahedral mesh was created using a patch-confirming algorithm, and the span angle center was set to fine. Meshing of the PWC is shown in Figure 2, the details \begin{table}
\begin{tabular}{c c} \hline \hline
**Dimensions/Settings** & **Values** \\ \hline Head & 10 m \\ Diameter of the outlet nozzle & 0.0635 m \\ Upper diameter of each water column & 0.4572 m \\ Diameter of the nozzle used at the bottom of & 0.2286 m \\ each water column & 0.2286 m \\ Working fluid & Water (liquid) \\ Outlet pressure & 0 Pascals \\ Turbulence model & k-\(\omega\) (SST) \\ Density of water & 1000 kg/m\({}^{3}\) \\
**Boundaries** & \\ Inlet & Velocity inlet \\ Outlet & Pressure outlet \\ Wall function & Standard \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimension of the water columns and solver settings. of which are provided in Table 2, including the meshing method, inflation layers, element size, and number of elements per PWC. A mesh independence test was performed for two PWCs to obtain the optimal element size and number of elements. Element sizes of 20, 30, 40, 50, 60, 70, 80, and 90 mm were applied to create 1,477,215, 655,237, 358,456, 237,308, 174,459, 138,933, 100,099, and 77,206 mesh elements, respectively. The optimal element size was 40 mm, which was selected for all the PWC simulation cases. A size of 40 mm yielded 358,456 elements for the two PWCs, as proven by the grid independence test shown in Figure 3.  
In the experimental design, the dimensions of the PWCs were kept constant, while the number of PWCs was increased individually. The velocity inlet was assigned as the \begin{table}
\begin{tabular}{c c c c c} \hline
**No. of PWC** & **Element Size** & **Method of Meshing** & **Inflation Layers** & **No. of Elements** \\ \hline
1 & 40 & Tetrahedrons & 7 & 274,274 \\ \hline
2 & 40 & Tetrahedrons & 7 & 358,456 \\ \hline
3 & 40 & Tetrahedrons & 7 & 543,020 \\ \hline
4 & 40 & Tetrahedrons & 7 & 736,088 \\ \hline
5 & 40 & Tetrahedrons & 7 & 926,713 \\ \hline \end{tabular}
\end{table} 
Table 2: Mesh report of PWC. Figure 3: Grid independence test of the two PWCs. Figure 2: Meshing of PWCs. boundary condition at the inlet of the PWCs, whereas the pressure outlet was assigned at the outlet. The velocity and pressure produced in each case, which were processed in CFD-Post, are illustrated in (a-e) of Figure 4.  
Figure 4: Velocity and pressure produced by (**a**) one water column, (**b**) two water columns, (**c**) three water columns, (**b**) four water columns, and (**e**) five water columns. Two types of results were obtained from the experimental design study. First, the output behavior of the PWC was determined, and a mathematical model was developed. If the dimensions and output of the water column are known, the optimal number of PWCs can be derived for any flow, velocity, or pressure profile. Second, the optimal number of PWCs was derived for the aforementioned PAT, the water flow of which was variable at its inlet owing to the variations in the head of the upper reservoir during generation.