## 2 Mathematical Model
The pressure differences in the individual branches may be calculated by adjusting Bernoulli's equation for an unsteady flow of a viscous fluid [23]:  
\[\rho\cdot g\cdot z+p+\rho\cdot\frac{v^{2}}{2}+\rho\cdot\int_{\text{f}}\frac{ \partial v}{\partial\tau}ds=const. \tag{5}\]  
\[\Delta p_{1-4}=\rho\cdot g\cdot\frac{0.0827\cdot\lambda_{1-4}}{d_{1-4}^{5}} \cdot L_{\text{ekv},1-4}\cdot Q_{1}^{2}+\rho\cdot g\cdot\Delta H_{1-4}+\rho \frac{4\cdot L_{1-4}}{\pi\cdot d_{1-4}^{2}}\cdot\frac{\Delta Q_{1}}{\Delta\tau} \tag{6}\]  
\[\Delta p_{4-\text{V2}}=\rho\cdot g\cdot\frac{0.0827\cdot\lambda_{4-\text{V2}} }{d_{4-\text{V2}}^{5}}\cdot L_{\text{ekv},4-\text{V2}}\cdot[Q_{1}+Q_{3}]^{2}+ \rho\cdot g\cdot\Delta H_{4-\text{V2}}+\rho\cdot\frac{4\cdot L_{4-\text{V2}} }{\pi\cdot d_{4-\text{V2}}^{2}}\cdot\frac{\Delta Q_{4}}{\Delta\tau} \tag{7}\]  
\[\Delta p_{3-4}=\rho\cdot g\cdot\frac{0.0827\cdot\lambda_{3-4}}{d_{3-4}^{5}} \cdot L_{\text{ekv},3-4}\cdot Q_{3}^{2}+\rho\cdot g\cdot\Delta H_{3-4}+\rho \cdot\frac{4\cdot L_{3-4}}{\pi\cdot d_{3-4}^{2}}\cdot\frac{\Delta Q_{3}}{\Delta\tau} \tag{8}\]  
\[\Delta p_{2-\text{V4}}=\rho\cdot g\cdot\frac{0.0827\cdot\lambda_{2-\text{V4}} }{d_{2-\text{V4}}^{5}}\cdot L_{\text{ekv},2-\text{V4}}\cdot Q_{2}^{2}+\rho \cdot g\cdot\Delta H_{2-\text{V4}}+\rho\cdot\frac{4\cdot L_{2-\text{V4}} }{\pi\cdot d_{2-\text{V4}}^{2}}\cdot\frac{\Delta Q_{2}}{\Delta\tau} \tag{9}\]  
wherein \(\rho\) is the density of the transported water (kg\(\cdot\)m\({}^{-3}\)); \(g\) is the gravitational acceleration (m\(\cdot\)s\({}^{-2}\)); \(\lambda_{\text{x-y}}\) is the coefficient of the pressure loss caused by the pipe length (1); \(d_{\text{x-y}}\) is the pipe diameter (m); \(L_{\text{ekv},\text{x-y}}\) is the equivalent length of the pipe in the x-y Branch (m); \(Q_{i}\) is the water flow rate (m\({}^{3}\cdot\)s\({}^{-1}\)); \(\Delta H_{\text{x-y}}\) is the difference between the elevations of the x-y Branch inlet and outlet (m); \(L_{\text{x-y}}\) is the real length of the pipe (m); \(\Delta Q_{\text{x-y}}\) is the change in the water flow rate in the pipe over the time interval \(\Delta\tau\) (m\({}^{3}\cdot\)s\({}^{-1}\)); and \(\Delta\tau\) is the analysed time interval (s). \[p_{1}-\Delta p_{1\text{-}4}-\Delta p_{4\text{-V2}}=p_{\text{V2}}\;(\text{Pa}) \tag{1}\]0 A diagram of flow rates, pressure differences and relative pressures in the pipeline system. The coefficients of the pressure loss in the pipes caused by the pipe length may be calculated using Moody's equation for the transition zone of a turbulent flow with a Reynolds number range of \(4000<Re<10^{7}\):  
\[p_{1}-\Delta p_{1\text{-}4}-\Delta p_{4\text{-V2}}=p_{\text{V2}}\;(\text{Pa}) \tag{1}\]1  
wherein \(k\) is the absolute roughness of the pipe (m) and \(Re\) is the Reynolds number (1). By substituting \(k_{1}\) with \(k_{8}\) and \(k_{\text{a}}\) with \(k_{\text{d}}\), Equations (6)-(9) may evidently be simplified as follows:  
\[p_{1}-\Delta p_{1\text{-}4}-\Delta p_{4\text{-V2}}=p_{\text{V2}}\;(\text{Pa}) \tag{1}\]2  
\[p_{1}-\Delta p_{1\text{-}4}-\Delta p_{4\text{-V2}}=p_{\text{V2}}\;(\text{Pa}) \tag{1}\]3  
\[p_{1}-\Delta p_{1\text{-}4}-\Delta p_{4\text{-V2}}=p_{\text{V2}}\;(\text{Pa}) \tag{1}\]4  
\[p_{1}-\Delta p_{1\text{-}4}-\Delta p_{4\text{-V2}}=p_{\text{V2}}\;(\text{Pa}) \tag{1}\]5  
In cases where it is necessary to analyse a steady-state system with constant flow rates and pressures, Equations (11)-(14) may be used while neglecting the last term with a time difference in the flow rate. Otherwise, with known boundary conditions of pressures and baseline flow rates, it is possible to identify the changes in flow rates over time.