## 4 Determination of Pump Characteristics and Mechanical Power during the Operation Cycle
The working point of the pump with a non-adjustable drive is determined by the intersection of its \(Q\)-\(H\) characteristics and vertical line \(Q_{req}\)/2. Table 2 represents the calculation results for these pumps operating in parallel for ten different modes of the cycle with the selected control method \(Q_{1}=Q_{2}\). Figure 7 also shows the calculated working points of these parallel pumps in terms of \(Q\)-\(H\) axes. Figure 6: (**a**) Q-H characteristics of pump performance and system curve; (**b**) hydraulic power curves of the pump at different rotational speeds. In Table 2 the following notation is used: \(P_{mech1}\) is the mechanical power of the variable speed pump; \(P_{mech2}\) is the mechanical power of the non-adjustable pump. As can be seen from Figure 7 and Table 2, when using the two pumps, the second parallel pump is used only at the required flow rate of 70% or more. Flow rates in the range of 0-60% can be provided by one of the lower power pumps. When both pumps are used, the equal distribution of the required flow rate \(Q_{1}=Q_{2}=Q_{req}/2\) between them is adopted. As mentioned above, this control law provides minimum power consumption in this case. The head values of the two pumps are not equal to each other, \(H_{1}\
eq H_{2}\), which is achieved by throttling the output of the second pump. The speed of the first pump \(n_{1}\) is adjusted by the VSD. The rotational speed of the second pump powered directly from the mains is assumed to be constant, \(n_{2}=2900\) rpm. The \(H_{1}\), \(H_{2}\), and \(n_{1}\) values are calculated according to the method described in Section 3 (see Figure 5). The mechanical powers \(P_{mech1}\) and \(P_{mech2}\) are evaluated using the \(P_{mech}(Q)\) dependence shown in Figure 2b. Figure 8 represents graphical dependencies of the total mechanical power of the two configurations of a pumping system on the water flowrate according to Tables 1 and 2. It can be seen that during the variation of the flowrate up to \(Q_{req}=14.4\) m\({}^{3}\)/h (60%), the multi-pump system demands less mechanical power. At higher \(Q_{req}\), however, it requires more mechanical power than a system with a single pump. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(Q_{req}\), & \(Q_{req}\), & \(Q_{1}\), & \(Q_{2}\), & \(H_{1}\), & \(H_{2}\), & \(P_{mech1}\), & \(P_{mech2}\), & \(P_{mech}=P_{mech1}\) + & \(n_{1}\), & \(n_{2}\), \\ \% & **m\({}^{3}\)/h** & **m\({}^{3}\)/h** & **m\({}^{3}\)/h** & **m\({}^{3}\)/h** & **m** & **m** & **W** & **W** & \(P_{mech2}\), **W** & **rpm** & **rpm** \\ \hline
10 & 2.4 & 2.4 & - & 8.1 & - & 136 & - & 136 & 1918 & - \\
20 & 4.8 & 4.8 & - & 8.3 & - & 181 & - & 181 & 1975 & - \\
30 & 7.2 & 7.2 & - & 8.7 & - & 247 & - & 247 & 2081 & - \\
40 & 9.6 & 9.6 & - & 9.3 & - & 338 & - & 338 & 2229 & - \\
50 & 12 & 12 & - & 10.0 & - & 463 & - & 463 & 2409 & - \\
60 & 14.4 & 14.4 & - & 10.9 & - & 628 & - & 628 & 2615 & - \\
70 & 16.8 & 8.4 & 8.4 & 11.9 & 17.6 & 394 & 616 & 1010 & 2433 & 2900 \\
80 & 19.2 & 9.6 & 9.6 & 13.1 & 17.1 & 486 & 654 & 1140 & 2579 & 2900 \\
90 & 21.6 & 10.8 & 10.8 & 14.5 & 16.6 & 598 & 691 & 1289 & 2736 & 2900 \\
100 & 24 & 12 & 12 & 16.0 & 16.0 & 730 & 729 & 1460 & 2901 & 2900 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System cycle data for two pumps operating in parallel mode. Figure 7: Working points of pumps operating in parallel.