## 3 Determining the Uncertainty of the Discharge Coefficient  
This chapter presents the results of the conducted analysis for determining the uncertainty [12, 13] of discharge coefficient \(C\) through the tested multi-opening orifice. Fluid volume stream \(q_{v}\) flowing through the tested multi-opening orifice can be described with a known dependence (1) [5] as  
\[q_{v}=\frac{C\cdot\varepsilon\cdot F_{04}}{\sqrt{1-m^{2}}}\cdot\sqrt{\frac{2 \cdot\Delta p}{\rho}} \tag{1}\]  
The orifice constriction cross-section area is composed of four identical openings with diameter \(d_{4}\) (Figure 1), and is presented with a dependence (2):  
\[F_{04}=\frac{4\cdot\pi\cdot d_{4}^{2}}{4}=\pi\cdot d_{4}^{2} \tag{2}\]  
It was assumed that the ratio of constriction from the cross-section area \(F_{04}\) to the pipeline cross-section \(F_{D}\) is module \(m\) of the orifice. In further deliberations, an individual constriction of a single opening of the four-opening orifice was introduced as \(\beta_{4}=d_{4}/D\) (the following diameters were considered in flow cross-sections: diameter of pipeline \(D\) and \(d_{4}\) as one of four identical openings in the multi-opening orifice). The result was a dependence (3):  
\[m=\frac{F_{04}}{F_{D}}=\frac{4\cdot d_{4}^{2}}{D^{2}}=4\cdot\beta_{4}^{2} \tag{3}\]  
Assuming incompressibility of flowing water (\(\varepsilon=1\)) for experimental studies, and considering dependence (2) and (3) in Formula (1), an expression for volume stream was obtained (4):  
\[q_{v}=C\cdot\pi\cdot\frac{d_{4}^{2}}{\sqrt{1-16\cdot\beta_{4}^{4}}}\cdot\sqrt{ \frac{2\cdot\Delta p}{\rho}} \tag{4}\]  
By transforming the designated dependence (4), formula for discharge coefficient \(C\) was determined in the following form (5):  
\[C=\frac{q_{v}\cdot\sqrt{1-16\cdot\beta_{4}^{4}}}{\pi\cdot d_{4}^{2}\cdot\sqrt{ \frac{2\cdot\Delta p}{\rho}}} \tag{5}\]  
Figure 3: Flow characteristics of the tested orifice \(q_{v}=\text{f}(\Delta p)\). For the discharge coefficient defined in (5), the standard relative uncertainty of its determination \(u(C)/C\) was calculated. It was determined from measurements of the values, indirectly according to the propagation of uncertainty rule [12] from Formula (6):  
\[ \begin{array}{c}\frac{u(C)}{C}=\left[\left(\frac{1}{C}\frac{\partial C}{ \partial q_{p}}\cdot u(q_{p})\right)^{2}+\left(\frac{1}{C}\cdot\frac{\partial C }{\partial q}\cdot u(\Delta p)\right)^{2}+\left(\frac{1}{C}\cdot\frac{\partial C }{\partial p}\cdot u(\rho)\right)^{2}+\right.\\ \left.+\left(\frac{1}{C}\cdot\frac{\partial C}{\partial d_{4}}\cdot u(d_{4}) \right)^{2}+\left(\frac{1}{C}\cdot\frac{\partial C}{\partial D}\cdot u(D) \right)^{2}\right]^{0.5}\end{array}  \tag{6}\]  
Next, based on dependence (6), partial derivatives were determined. When multiplied by the converse discharge coefficient (5), they yield components of Equation (6). They are presented in Table 2.  
After substituting components of Equations (7)-(11) in Equation (6) for combined relative standard uncertainty of determining discharge coefficient \(u(C)/C\), the dependence (12) was obtained:  
\[ \begin{array}{c}\frac{u(C)}{C}=\left[\left(\frac{u(q_{p})}{q_{p}}\right)^{2 }+\frac{1}{4}\left(\frac{u(\Delta p)}{\Delta p}\right)^{2}+\frac{1}{4}\left( \frac{u(\rho)}{\rho}\right)^{2}+\left(\frac{2}{1-16\cdot\beta_{4}^{4}}\right)^ {2}\cdot\left(\frac{u(d_{4})}{d_{4}}\right)^{2}+\right.\\ \left.+\left(\frac{32\beta_{4}^{4}}{1-16\beta_{4}^{4}}\right)^{2}\cdot\left( \frac{u(D)}{D}\right)^{2}\right]^{0.5}\end{array}  \tag{12}\]  
Table 3 presents relative standard components of uncertainty, present in Equation (12), necessary for calculating the combined relative standard uncertainty of discharge coefficient \(u(C)/C\).