## 2 Mathematical Model
### A Model for Calculating the Pipeline Parameters through an Experiment  
Due to the fact that the previous calculations were based on the design parameters of the pipeline that no longer exhibits good concordance between the measured data and the experiments due to the effects of incrustation, a different approach was applied. In that approach, coefficients \(k_{1}\), \(k_{3}\) and \(k_{5}\) for the V2 Branch and coefficients \(k_{7}\) and \(k_{8}\) for the V4 Branch were identified by analysing the measured values of flow rates and pressures in a real and steady-state water flow. The steady state may be achieved with a constant water withdrawal and constant pressures, which often occur in a real facility. In this case, mathematical procedures according to Equations (11)-(14) may be used, neglecting the last terms of the equations that describe the dynamic changes in the system. By deducting Equations (1) and (2), an equation describing the pressure equilibrium at the point where the transported water mixes in point 4 is obtained:  
\[p_{1}-\Delta p_{1-4}=p_{3}-\Delta p_{3-4} \tag{25}\]  
By substituting Equations (11) and (13), while neglecting the terms of equations that describe the changes in the flow rate (the steady state), in Equation (25), the following set of equations for the two sets of measured data is obtained:  
\[p_{1}^{\prime}-k_{1}\cdot{Q^{\prime}}_{1}^{2}-k_{2}=p_{3}^{\prime}-k_{5}\cdot{ Q^{\prime}}_{3}^{2}-k_{6} \tag{26}\]  
\[p_{1}^{{}^{\prime\prime}}-k_{1}\cdot{Q^{\prime\prime}}_{1}^{2}-k_{2}=p_{3}^{{} ^{\prime\prime}}-k_{5}\cdot{Q^{\prime\prime}}_{3}^{2}-k_{6} \tag{27}\]  
wherein the pressures and flow rates with one apostrophe represent the data obtained in Measurement 1, while the values with two apostrophes represent the data obtained in Measurement 2. Achieving the required calculation accuracy is conditioned by conducting analyses in two steady states with significantly different flow rates and pressures in the system during the measurement. Coefficients \(k_{2}\) and \(k_{6}\) represent the hydrostatic pressure of the water column in the pipeline, and they may be simply calculated using the known differences between the pipe inlet and outlet elevations. The only unknown parameters in the set of equations are \(k_{1}\) and \(k_{5}\), and they may be calculated as follows:  
\[ffk_{5}=\frac{p_{1}^{\prime}\cdot{Q^{\prime\prime}}_{1}^{2}-p_{1}^{{}^{\prime \prime}}\cdot{Q^{\prime}}_{1}^{2}+k_{2}\Big{(}{Q^{\prime}}_{1}^{2}-{Q^{\prime \prime}}_{1}^{2}\Big{)}-p_{3}^{\prime}\cdot{Q^{\prime\prime}}_{1}^{2}+p_{3}^{ \prime\prime}\cdot{Q^{\prime}}_{1}^{2}-k_{6}\Big{(}{Q^{\prime}}_{1}^{2}-{Q^{ \prime\prime}}_{1}^{2}\Big{)}}{Q^{\prime}}_{1}^{2}\cdot{Q^{\prime\prime}}_{3}^{ 2}-{Q^{\prime}}_{3}^{2}\cdot{Q^{\prime\prime}}_{1}^{2}} \tag{28}\]\[k_{1}=\frac{p_{1}^{{}^{\prime\prime}}-k_{2}-p_{3}^{{}^{\prime\prime}}+k_{5}\cdot Q ^{{}^{\prime}2}_{\ 3}+k_{6}}{Q^{{}^{\prime\prime}2}_{\ 1}} \tag{29}\]  
Coefficient \(k_{3}\) is identified from Measurement 1 by adjusting Equation (1):  
\[k_{3}=\frac{p_{1}^{\prime}-k_{1}\cdot Q^{{}^{\prime}2}_{\ 1}-k_{2}-k_{4}-p_{ \mathrm{V2}}}{\left(Q_{1}^{\prime}+Q_{3}^{{}^{\prime}2}\right)^{2}} \tag{30}\]  
In order to identify coefficients k\({}_{7}\) and k\({}_{8}\), Equation (3) was used in the two measurements of the steady-state flow:  
\[k_{7}=\frac{\left(p_{\mathrm{V4}}^{\prime}-p_{\mathrm{V4}}^{\prime\prime} \right)-\left(p_{2}^{\prime}-p_{2}^{\prime\prime}\right)}{Q^{{}^{\prime\prime}2 }_{\ 2}-Q^{{}^{\prime}2}_{\ 2}} \tag{31}\]  
\[k_{8}=p_{2}^{\prime}-k_{7}\cdot Q^{{}^{\prime}2}_{\ 2}-p_{\mathrm{V4}}^{ \prime} \tag{32}\]  
In order to identify coefficients k\({}_{7}\) and k\({}_{8}\), only one measurement can be used in a steady water flow in the V4 Branch, while coefficient k\({}_{8}\) is deducted from the known value of the difference between the pipeline inlet and outlet elevations.