## 2 Materials and Methods
### Experimental Flow Tests  
In order to validate that the diameter values obtained from the Sphere method are equivalent to \(d_{h}\), experimental incompressible turbulent fluid flow tests were conducted on all of the parts with water as the fluid medium. Figure 4 illustrates a schematic of the experimental setup. Figure 3: Example of adjusted histogram data and corresponding approximation by B-spline, with red rectangles on the left and right indicating the data ranges of small wall-thickness values that were evaluated to obtain _msth_ and of large wall–thickness values that were evaluated to obtain \(d_{mean}\) and \(d_{eq}\). The specimens were flow-tested in the turbulent flow regime while varying the inlet pressure (\(P_{1}\)) upstream of the channel. As a result, for a given inlet pressure, the corresponding mass flow rate (\(\dot{m}\)) was calculated by collecting and measuring the mass of water discharged through the channel per unit of time. The measurements were performed at five different \(P_{1}\) values on all the specimens. The uncertainty of the entire experimental setup that includes the mass flow and pressure measurements is \(\pm 1\%\). To verify the consistency of the measurements, the flow number was calculated for all five readings. The flow number is a non-dimensional characteristic parameter to evaluate the stability of the fluid flow measurements. The flow measurements are considered consistent and stable when the flow number is found to be constant for all readings. Equation (1) represents the general formulation of the flow number [10]. \[\frac{\dot{m}}{\Delta P^{(0.5)}}=Constant \tag{1}\]  
where:  
\(\dot{m}\)--Fluid mass flow rate;  
\(\Delta P\)--Overall pressure drop. Concerning the pressure measurements, the pressure at the outlet is considered to be equal to the ambient pressure (\(P_{amb}\)) since water is denser than air. Therefore, the difference between the inlet and outlet pressures is the overall pressure drop (\(\Delta P\)) or total head loss (\(H\)) exerted during the fluid flow through the channel. However, it must be noted that \(H\) is the sum of the major and minor losses associated with the flow. In the case of the experimental setup depicted in Figure 4, the loss at the inlet due to the interconnection with the funnel geometry needs to be considered for both specimens. In addition, the minor loss due to the bend section with an inclination of 30\({}^{\circ}\) from the vertical must be regarded for the test parts shown in Figure 1c. The minor losses are calculated by applying a loss coefficient to resolve the general Bernoulli equation. The major loss associated with the flow in all the specimens is the loss due to friction created by the hydrodynamic sand grain roughness (\(k_{s}\)) on the internal surface of the channel [19]. The major loss is calculated using the Darcy-Weisbach and Colebrook-White equation [20]. Equations (2) and (3) represent the Darcy-Weisbach equation in the pressure-loss and head-loss forms, and Equation (4) is the Colebrook-White equation. \[\frac{\Delta P_{friction}}{l}=f_{d}\frac{\rho}{2}\frac{\cdot V^{2}}{d_{h}} \tag{2}\]  
Figure 4: Schematic representation of the experimental flow test setup indicating the direction of the flow and the corresponding pressure measurements. \[\frac{\Delta h}{l}=f_{d}.\frac{V^{2}}{2\cdot g\cdot d_{h}} \tag{3}\]  
\[\frac{1}{\sqrt{f_{d}}}=-2 log\Bigg{[}\frac{\varepsilon}{3\cdot f_{d}}+\frac{2.51 }{R_{ad}\sqrt{f_{d}}}\Bigg{]} \tag{4}\]  
where:  
\(\Delta P_{\it friction}\)--Pressure drop due to friction (Pa);  
\(\Delta h\)--Head loss due to friction (m);  
\(l\)--Length of the channel (m);  
\(g\)--Acceleration due to gravity (m/s\({}^{2}\));  
\(f_{d}\)--Darcy--Weisbach friction factor (no unit);  
\(\rho\)--Density of the fluid (kg/m\({}^{3}\));  
\(V\)--Velocity of the fluid flow (m/s);  
\(d_{h}\)--Hydraulic diameter of the channel (m);  
\(\varepsilon\)--Mean roughness height (m);  
\(R_{ad}\)--Reynold's number (no unit).