## 2 Mathematical Modeling
### Mathematical Model of a Pump Process  
#### 2.2.1 Processes in the Suction Chamber of the Pump  
The mass flow \(m\) through opening, at the point of entry of the suction pipe into the suction chamber of the pump, during quasi-stationary discharge is calculated from the known relation for stationary discharge of an incompressible fluid:  
\[\frac{dm_{1}}{dt}=\sigma_{1}\mu_{1}A_{1}\sqrt{2\rho_{s}|p_{u}-p_{s}|} \tag{8}\]  
where:  
\(p\)--pressure,  
\(\rho\)--fluid density,  
\(A\)--geometric flow section,  
\(\mu\)--flow coefficient constant. The index \"\(1\)\" refers to the opening, the index \"\(u\)\" to the suction, the index \"\(s\)\" to the suction chamber. The magnitudes of the constant \(\sigma_{1}\) are:  
\(\sigma_{1}=1\) for \(p_{u}\geq p_{s}\)  
\(\sigma_{1}=-1\) for \(p_{u}<p_{s}\)  
In a similar way, the mass flow of the fluid \(m\) through the distribution organ of the pump (distribution plate or valves) when filling one cylinder of the pump is calculated:  
\[\frac{dm_{u}}{dt}=\sigma_{u}\mu_{u}A_{u}\sqrt{2\rho_{s}|p_{s}-p_{c}|} \tag{9}\]  
where are:  
\(\sigma_{u}=1\) for \(p_{s}\geq p_{c}\)  
\(\sigma_{u}=-1\) for \(p_{s}<p_{c}\). \(A_{u}\)--geometrical flow section, intake opening of the distribution board or intake valve opening. The index \"\(c\)\" refers to the cylinder. The mass balance of the suction chamber reads:  
\[\frac{dm_{s}}{dt}=\frac{dm_{1}}{dt}-\sum_{j=1}^{z_{c}}\frac{dm_{u,j}}{dt} \tag{10}\]  
where:  
\(j=1,2,...,z_{c}\)--serial number of the cylinder,  
\(z_{c}\)--total number of cylinders. The density of the fluid in the suction chamber of the pump is:  
\[\rho_{s}=\frac{m_{S}}{V_{S}} \tag{11}\]  
\(V_{S}\) is the volume of the suction chamber. Logarithmic differentiation of this equation by time yields:  
\[\frac{1}{\rho_{S}}\frac{d\rho_{S}}{dt}=\frac{1}{m_{s}}\frac{dm_{s}}{dt}-\frac{1}{ V_{S}}\frac{dV_{S}}{dt} \tag{12}\]  
The equation of state of the fluid for pressure \(p\) and temperature \(T\) can be written in the form:  
\[p=p(\rho,T) \tag{13}\]  
The total derivative of pressure with time during an isothermal change of state (\(T=const\)) can be written in the form:  
\[\frac{dp}{dt}=\left(\frac{\partial p}{\partial\rho}\right)_{T}\frac{d\rho}{dt} \tag{14}\]  
For \(m=\rho V\) the following differential can be written:  
\[d(\rho V)=\rho dV+Vd\rho \tag{15}\]  
The isothermal coefficient of compressibility of the fluid \(\varepsilon_{r}\) is defined by the equation:  
\[\varepsilon_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T} \tag{16}\]  
Based on Equations (14)-(16), the following relationship between pressure, density and isothermal compressibility coefficient is obtained:  
\[\left(\frac{dp}{dt}\right)_{T}=\left(\frac{\partial p}{\partial\rho}\right)_{ T}=\frac{1}{\varepsilon_{T}\rho} \tag{17}\]  
In the case of isentropic change (s = const) the equation of state can be written in the form:  
\[\frac{dm_{u}}{dt}=\sigma_{u}\mu_{u}A_{u}\sqrt{2\rho_{s}|p_{s}-p_{c}|} \tag{9}\]0  
The total pressure time derivative in this case is:  
\[\frac{dm_{u}}{dt}=\sigma_{u}\mu_{u}A_{u}\sqrt{2\rho_{s}|p_{s}-p_{c}|} \tag{9}\]1  
The isentropic coefficient of fluid compressibility \(\varepsilon_{s}\) is:  
\[\frac{dm_{u}}{dt}=\sigma_{u}\mu_{u}A_{u}\sqrt{2\rho_{s}|p_{s}-p_{c}|} \tag{9}\]2  
Based on Equations (15), (19) and (20), the relationship between pressure, density and isentropic compressibility coefficient is obtained:  
\[\frac{dm_{u}}{dt}=\sigma_{u}\mu_{u}A_{u}\sqrt{2\rho_{s}|p_{s}-p_{c}|} \tag{9}\]3  
Comparing Equations (17) and (21) shows a complete similarity, so a general equation for the relationship between pressure, density and compressibility coefficient can be written:  
\[\frac{dm_{u}}{dt}=\sigma_{u}\mu_{u}A_{u}\sqrt{2\rho_{s}|p_{s}-p_{c}|} \tag{9}\]4  
i.e., in the form:  
\[\frac{dm_{u}}{dt}=\sigma_{u}\mu_{u}A_{u}\sqrt{2\rho_{s}|p_{s}-p_{c}|} \tag{9}\]5wherein:  
\(\varepsilon=\varepsilon_{T}\)--for isothermal change of state  
\(\varepsilon=\varepsilon_{S}\)--for an isentropic change of state.