## 3 Results and Discussion
### A Comparison of the Experimental Data and the Model  
The analytical and experimental data were compared while using the data of the hydraulic system that was measured in the industrial enterprise with a 60 s interval over the total time of 10 h. Flow rates \(Q_{\rm I}\) through \(Q_{\rm 4}\) and pressure boundary conditions of the system were measured. The measured data was used to specify two sets of the measured data at various flow rates and pressures, while the measurement deviation in the defined 5 min interval corresponded to the accuracy of the used sensors.  
\begin{table}
\begin{tabular}{c c c c c} \hline Branch elevation & \(\Delta H_{\rm I-4}=23.76\) m & \(\Delta H_{\rm 4-V2}=0.8\) m & \(\Delta H_{\rm 3-4}=13.3\) m & \(\Delta H_{\rm 2-V4}=28.46\) m \\ \hline Pipe diameter & \(d_{\rm 1-4}=1.1\) m & \(d_{\rm 4-V2}=1.1\) m & \(d_{\rm 3-4}=0.9\) m & \(d_{\rm 2-V4}=1.1\) m \\ \hline Pipe length & \(L_{\rm 1-4}=7269\) m & \(L_{\rm 4-V2}=792\) m & \(L_{\rm 3-4}=4317\) m & \(L_{\rm 2-V4}=8700\) m \\ \hline \end{tabular}
\end{table}
Table 1: Pipe parameters in the individual system branches.  
The aforesaid sections could be regarded as steady, without any dynamic changes. Subsequently, the coefficients defining the parameters of the pipeline sections listed in Table 2 were calculated using Equations (28) through (32).  
The key advantage of the approach described above is defining the coefficients that take into account the real condition of the pipeline, including the incrustation and hidden local resistances. For the purpose of dynamic calculations, it was necessary to define coefficients \(k_{\text{a}}\) through \(k_{\text{d}}\), and they were identified using the pipeline design parameters. The analytical calculations were made by applying the finite-difference time-domain method with the aim of identifying the changes in flow rates, using Equations (19)-(21). The input data included the previously identified pipeline coefficients and the pressure boundary conditions. The comparisons of the measured data and the resulting flow rate values that were calculated with the use of the mathematical model are presented in Figures 2-4. The time step applied in the calculations was 60 s, which was in concordance with the time step applied in recording the measured data.  
The results of the calculations were in good concordance with the experimental measurements performed in the real facility, while the average deviation of the absolute value of the difference between the measured and the calculated flow rate values was 5.19% for flow rate \(Q_{1}\), 0.99% for flow rate \(Q_{2}\), and 2.7% for flow rate \(Q_{3}\). Figure 5 shows the instantaneous deviations between the calculated and the measured flow rate values.  
Local deviations between the calculated and the measured values were the highest at the flow rate \(Q_{1}\), where the proposed model is more complicated due to the fact that water is supplied from two different sources. The lowest deviations were observed in the simple V4 Branch with the flow rate \(Q_{2}\). The deviations were also significantly affected by the measurement uncertainty, which was used in the calculation in the form of the pressure boundary conditions, and this increases the probability of inaccuracy. Secondary reasons  
Figure 2: The curves of the measured and the calculated flow rate \(Q_{1}\) with the known pressure boundary conditions in the V2 Branch.