## 2 System Modeling
### Mathematical Model
#### 2.2.2 Dynamic Mathematical Model  
Figure 4 is a schematic diagram of the dynamic characteristics of the multi-channel concentrated water supply process. The straight lines and boxes in the figure represent the relevant objects of the concentrated water supply process, including water supply systems, agglomeration structures, valves and pipes. Combined with the mathematical model of each component of the system and the simplified assumptions of the whole system, a dynamic mathematical model of the MCCS process can be obtained.  
Figure 4: Schematic diagram of the dynamic characteristics of the multi-channel clustering process.  
According to the principle of conservation of mass, the dynamic mathematical model of each channel can be obtained as follows:  
\[\left\{\begin{array}{l}P_{i}=K_{1}+K_{2}Q_{i}+K_{3}Q_{i}^{2}\\ Q_{i}=\frac{S_{i}}{\sqrt{{\lambda_{0}}}_{o}}\sqrt{\frac{2(P_{0i}-P_{0i})}{\rho}} =\sqrt{\frac{P_{0i}-P_{0i}}{R_{i}}}\\ R_{i}=\frac{\Lambda_{i}d_{i}+\Lambda_{i}}{2g_{i}d_{i}\Lambda_{i}^{2}}\\ i=1,2,3,4\end{array}\right. \tag{18}\]  
where \(P_{i}\) is the pressure at the outlet of the centrifugal pump of channel \(i\), \(i\) = 1, 2, 3, 4, \(R_{i}\) is the equivalent resistance coefficient of the corresponding pipe in the figure, \(i\) = 1, 2, 3, 4, 5, 6, \(P_{ii}\) is the pressure at the junction of channel \(i\) and the gathering structure pipe, \(i\) = 1, 2, 3, 4, and \(P_{0i}\) is the pressure in front of the control valve on channel \(i\), \(i\) = 1, 2, 3, 4.  
According to the principle of the conservation of mass and energy, the dynamic mathematical model of the clustered water supply structure can be obtained as follows:  
\[2\sum_{i=1}^{4}P_{ii}-8P_{5}=\frac{4\rho+2A_{0}^{2}\rho gR_{5}}{A_{0}^{2}}Q_{0} ^{2}-\frac{\rho}{A^{2}}\sum_{i=1}^{4}{Q_{i}}^{2} \tag{19}\]  
where \(P_{5}\) is the pressure at the connection between the main pipe and the gathering structure pipe.  
The dynamic mathematical model of the main pipe can be obtained as shown in the following formula based on the principle of conservation of mass. The control valve on the main pipe is the main regulating valve, and the valve resistance is much greater than the pipeline resistance, so the pipeline resistance is ignored.  
\[\rho\lambda_{v}Q_{0}^{2}-2S^{2}(P_{5}-P_{0})=0 \tag{20}\]  
The water-consuming system can be equivalent to a resistance element, which is replaced by a pipe load simulation valve. The pressure at the outlet of the water-consuming system is atmospheric, and its dynamic mathematical model is as follows:  
\[P_{0}=R_{7}Q_{0}^{2} \tag{21}\]  
where \(P_{0}\) is the pressure before the load simulation valve, \(Q_{0}\) is the flow before the load simulation valve, and \(R_{7}\) is the equivalent resistance coefficient of the load simulation valve and its front and rear accessories.  
Equations (18)-(21) represent the dynamic mathematical model of the MCCS process. Through this dynamic mathematical model, it can be seen that the main factors affecting the pressure of the main pipe are the pre-aggregate flow rate, the main pipe control valve and the equivalent resistance of the pipe load simulation valve.