## 2 Mathematical Model
### Description of Mathematical Models of Pump Processes  
The models are combined into a set of differential equations that describe the time-varying behavior of the pump under different operating conditions (Table 1). Solving these equations requires numerical methods (e.g., Runge-Kutta) to simulate the dynamic response of the pump in different scenarios [16; 17]. Fluid flow into the suction chamber of the pump at location 1, Figure 2:  
\[\frac{dQ_{1}}{dt}=\sigma_{1}\cdot\mu_{1}\cdot A_{1}\cdot\sqrt{2\cdot p_{\mathrm{ s}}|p_{u}-p_{\mathrm{s}}|} \tag{1}\]  
where  
\(\sigma_{1}=1\) for \(\mathrm{p_{u}}\geq\mathrm{p_{s}}\), \(\sigma_{1}=-1\) for \(\mathrm{p_{u}}<\mathrm{p_{s}}\);  
\(A_{1}\)--geometrical flow section of the intake pipe;  
\(\mu_{1}\)--flow coefficient of the intake pipe. \begin{table}
\begin{tabular}{c c} \hline
**Symbol** & **Description** \\ \hline \(A_{u}\) & geometric flow cross-section \\ \hline \(\mu_{u}\) & geometric flow cross-section coefficient \\ \hline \(z_{c}\) & cylinder numbers \\ \hline \(E\) & modulus of elasticity \\ \hline \(\varphi\) & the driving shaft angle \\ \hline \(V_{cx}\) & the volume of the pump cylinder achieved by moving the piston \\ \hline \(x_{k}\) & current displacement of the piston \\ \hline \(A_{2}\) & geometrical flow section of the delivery pipeline \\ \hline \(D_{c}\) & cylinder diameter \\ \hline \(\Delta r\) & radial clearance between the piston and the cylinder \\ \hline \(\eta\) & dynamic viscosity \\ \hline \(x_{k}(\varphi)\) & current displacement of the piston \\ \hline \(p_{c}\) & cylinder pressure \\ \hline \(p_{s}\) & suction pressure \\ \hline \(\rho_{c}\) & fluid density in the cylinder \\ \hline \(\mathrm{F_{h}}\) & the hydraulic force \\ \hline \(\mathrm{P}\) & the pressure of the fluid \\ \hline \(\mathrm{A}\) & A is the cross-sectional area of the piston \\ \hline \(\mathrm{F_{m}}\) & spring forces (like a force from a spring) \\ \hline \(\mathrm{F_{f}}\) & friction forces between the piston and the cylinder wall \\ \hline \end{tabular}
\end{table} 
Table 1: The descriptions of parameter symbols. Mass flow of fluid when filling one pump cylinder:  
\[\frac{dQ_{u}}{dt}=\sigma_{u}\cdot\mu_{u}\cdot A_{u}\sqrt{2\cdot p_{s}\cdot|p_{s}- p_{c}|} \tag{2}\]  
where  
\(\sigma_{u}=1\) for \(p_{s}\geq p_{c}\), \(\sigma_{u}=-1\) for \(p_{s}<p_{c}\);  
\(A_{u}\)--geometric flow cross-section;  
\(\mu_{u}\)--geometric flow cross-section coefficient. The mass balance of the suction chamber is as follows:  
\[\frac{dQ_{s}}{dt}=\frac{dQ_{1}}{dt}-\sum_{j=1}^{z}\frac{dQ_{u,j}}{dt} \tag{3}\]  
where  
\(j=1,2,\ldots,z_{c},z_{c}\)--cylinder numbers. Pressure equation in the suction chamber of the pump:  
\[\frac{dp_{s}}{d\varphi}=\frac{E}{V_{s}\cdot\rho_{s}}\cdot\left(\frac{dQ_{1}}{ d\varphi}-\sum_{j=1}^{z_{c}}\frac{dQ_{u,j}}{d\varphi}\right) \tag{4}\]  
\(E\)--modulus of elasticity;  
\(\varphi\)--the driving shaft angle. Pressure in the pump cylinder:  
\[\frac{dp_{c}}{d\varphi}=\frac{E}{V_{c}}\cdot\left[\frac{A_{c}\cdot v_{k}}{ \omega}+\frac{1}{\rho_{c}}\cdot\left(\frac{dQ_{u}}{d\varphi}-\frac{dQ_{i}}{d \varphi}\right)\right] \tag{5}\]  
where  
\(V_{c}=V_{\rm{cmin}}+V_{cx};V_{cx}=A_{c}\cdot x_{k}\);  
\(V_{cx}\)--the volume of the pump cylinder achieved by moving the piston;  
\(\frac{dV_{s}}{dt}=-A_{c}\cdot v_{k}\), \(x_{k}\)--current displacement of the piston. Cylinder volume by moving the piston:  
\[\frac{dQ_{\varphi}}{dt}=\sum_{j=1}^{z_{c}}\frac{dQ_{i,j}}{dt}-\frac{dQ_{2}}{dt} \tag{6}\]  
where \(j=1,2,\ldots,z_{c}\)--the numbers of piston. Fluid flow from the pressure chamber of the pump:  
\[\frac{dQ_{2}}{dt}=\sigma_{2}\cdot\mu_{2}\cdot A_{2}\cdot\sqrt{2\cdot\rho_{t} \cdot|p_{v}-p_{n}|} \tag{7}\]  
where  
\(\sigma_{2}=1\) for \(p_{v}\geq p_{n}\), \(\sigma_{2}=-1\) for \(p_{v}<p_{n}\);  
\(A_{2}\)--geometrical flow section of the delivery pipeline.