## 2 System Modeling
### Mathematical Model  
Multi-channel concentrated water supply means that multiple channels are connected in parallel in the same aggregate main pipe, and the switching process is its typical working condition. In fact, the switching process of the two channels is as follows: under the parallel operation of the water supply pump of each channel, the opening degree of the control valve in the operating branch gradually decreases from 100% to zero, while the control valve in the standby branch gradually increases from closed to 100%. If the action between the operating channel control valve and the standby channel control valve is not coordinated during the switching process, the pressure and flow rate in the collecting main pipe will change, resulting in a big sudden change in the pressure in the main pipe. Due to the complex internal mechanism of the MCCS process, the mathematical model cannot be precisely established, and only an approximate model based on experience and test data for approximate processing can be established. The established model makes the following simplifications and assumptions: the system medium is water at room temperature and pressure; the fluid is in a single-phase flow state; the transient flow in the pipeline is one-dimensional and homogeneous. The numerical models of the key components of the whole system are established respectively. The pressure and flow rate of the MCCS system will change during the switching process, which belongs to the transient flow state of the pipeline. The basic equation is composed of a mass conservation equation (continuity equation), a momentum conservation equation (motion equation) and an energy conservation equation (Bernoulli equation) in the transient process [21]. Continuity equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v_{x})}{\partial x}+\frac {\partial(\rho v_{y})}{\partial y}+\frac{\partial(\rho v_{z})}{\partial z}=0 \tag{1}\]  
where \(\rho\) is density, and \(v_{x}\), \(v_{y}\) and \(v_{z}\) are the mean velocity in the \(x\), \(y\) and \(z\) directions, respectively. According to the model simplification, the pipeline flow is a one-dimensional single-phase incompressible transient flow, and the amount of flow in and out of a certain section of the control body along the pipeline is equal, namely:  
\[\frac{\partial v_{x}}{\partial x}+\frac{1}{\rho A}\frac{d(\rho A)}{dt}=0 \tag{2}\]  
where \(A\) is the cross-sectional area of a pipe. Motion equation:  
\[\left\{ \begin{array}{l}f_{x}-\frac{1}{\rho}\frac{\partial\eta}{\partial x}= \frac{\partial v_{x}}{\partial t}+v_{x}\frac{\partial v_{y}}{\partial x}+v_{y} \frac{\partial v_{y}}{\partial y}+v_{z}\frac{\partial v_{z}}{\partial z}=\frac {dv_{x}}{dt}\\ f_{y}-\frac{1}{\rho}\frac{\partial\eta}{\partial y}=\frac{dv_{y}}{\partial t}+v _{x}\frac{\partial v_{y}}{\partial x}+v_{y}\frac{\partial v_{y}}{\partial y}+ v_{z}\frac{\partial v_{y}}{\partial z}=\frac{dv_{y}}{dt}\\ f_{z}-\frac{1}{\rho}\frac{\partial\eta}{\partial x}=\frac{dv_{z}}{dt}+v_{x} \frac{\partial v_{y}}{\partial x}+v_{y}\frac{\partial v_{z}}{\partial y}+v_{z }\frac{\partial v_{z}}{\partial z}=\frac{dv_{z}}{dt}\end{array} \right. \tag{3}\] \begin{table}
\begin{tabular}{c c} \hline
**Device** & **Main Parameters** \\ \hline Centrifugal pump & Model KSL32-160, flow rate 5 m\({}^{3}\)/h, head 32 m, power 1.5 kW \\ Channel control valve & Input signal 4–20 mA, PN1.0 \\ Main control valve & Input signal 4–20 mA, PN1.0 \\ Main flowmeter & Flow range 2–20 m\({}^{3}\)/h, accuracy 0.5\% FS, output signal 4–20 mA \\ Channel flowmeter & Flow range 0.8–8 m\({}^{3}\)/h, accuracy 0.5\% FS, output signal 4–20 mA \\ Pressure transducer & Range 0–0.6 MPa, accuracy 0.5\% FS, output signal 4–20 mA \\  & Acquisition card NI9203 eight input channels, input signal 0–20 mA, sampling \\ Acquisition & rate 200 KS/s \\ control system & Output card NI9266 eight output channels, output signal 0–20 mA, rate 24 KS/s \\  & chChassis cDAQ9185 four slots, rate 1.6 MS/s \\ \hline \end{tabular}
\end{table}