## 3 Results and Discussion
### A Regulation of the Hydraulic System through the Defined Mathematical Model  
The primary purpose of the proposed model was to implement the system of regulating the hydraulic system in the industrial enterprise with the requirement to stabilise discharge pressures \(p_{\rm V2}\) and \(p_{\rm V4}\) while maintaining the required flow rate of water from Source 2. The regulation is based on the knowledge described in Equations (34)-(37). In order to demonstrate the regulation model, three model examples were created for the V2 Branch:  
* Model Example 1: constant increasing of the flow rate, subsequent stabilising and then reducing the flow rate to the initial value;
* Model Example 2: a sine curve of the output flow rate;
* Model Example 3: a stochastic curve of the output flow rate. The analysed pipe dimensions are listed in Table 1 and the measured pipe coefficients are listed in Table 2. Table 3 contains the long-term averages that were taken into account when determining the baseline flow rates at the beginning of the simulation. The total time of the numerical simulation of the flow was 2000 s, with a time step of 1 s. The pressure boundary conditions were calculated using Equations (34) through (37). The density and dynamic viscosity of the flowing water was defined at a temperature of 12 \({}^{\circ}\)C. The required value of output pressures \(p_{\rm V2}\) and \(p_{\rm V4}\) was 0.26 MPa. Flow rate \(Q_{3}\) was regulated with the aim of achieving a constant value of 0.314 m\({}^{3}\).s\({}^{-1}\). In order to facilitate the use of numerical simulation of water withdrawal in the industrial enterprise, pressure \(p_{\rm V2}\) was calculated using an equation in which the sum of all length-induced and local losses in the enterprise may be substituted with the local loss factor \(\zeta_{\rm eq}\):  
\[p_{\rm V2}=\zeta_{\rm ekv}\cdot\frac{8\cdot Q_{4}^{2}}{\pi^{2}\cdot d_{4-V2}^ {4}}\cdot\rho \tag{38}\]  
Figure 5: Deviations between the calculated and the measured flow rate values. \begin{table}
\begin{tabular}{c c c} \hline \(Q_{1}\) (m\({}^{3}\) s\({}^{-1}\)) & \(Q_{2}\) (m\({}^{3}\) s\({}^{-1}\)) & \(Q_{3}\) (m\({}^{3}\).s\({}^{-1}\)) \\ \hline
0.294 & 0.53 & 0.314 \\ \hline \end{tabular}
\end{table} 
Table 3: Baseline flow rates at the pipe inlets. In all the model situations, the considered baseline flow rate was 0.607 m\({}^{3}\)-s\({}^{-1}\) at the required pressure \(p_{\rm V2}\). This corresponded to the value of the equivalent local loss of 1275. In Model Example 1, the value of the local loss factor was reduced over a 4 min interval by 37.25% down to the value of 800; this simulated an increase in the water flow rate at the outlet from the V2 Branch. After the flow rates and pressures were stabilised, the \(\zeta_{\rm eq}\) value was increased again to the baseline value of 1275 over a 2 min interval. Subsequently, the flow rates and pressures were left to stabilise again. Figure 6 shows the curves of pressures and flow rates with coefficient \(\varepsilon\) = 0, at which the regulation was carried out without applying the difference between the measured and the required value of pressure \(p_{\rm V2}\). The calculated pressures \(p_{1}\) and \(p_{3}\) in the V2 Branch were used in the dynamic model with a 10 s delay, which is the mean time of change in the rotational speeds of the pumps within the selected range of pressures. The required pressure at the outlet from the V2 Branch was set to 0.26 MPa. At the time of 600 s, \(\zeta_{\rm eq}\) decreased, which resulted in an increase in flow rate \(Q_{1}\) and a decrease in pressure \(p_{\rm V2}\). Based on the proposed model, pressures \(p_{1}\) and \(p_{3}\) gradually increased and, consequently, the water flow rate in the V2 Branch increased. After 1000 s, the flow rates were stabilised with an unchanged regulated pressure \(p_{\rm V2}\) and flow rate \(Q_{3}\). During the 26.27% increase in flow rate \(Q_{1}\), there was a short-term change in the regulated pressure \(p_{\rm V2}\) by \(-4.38\%\) and a 3.3% change in flow rate \(Q_{3}\).