## 2 Numerical Modelling in Irregular Pipelines
### Mathematical Modelling in Irregular Pipelines  
The mathematical model for an irregular pipeline associated with the experimental installation was proposed by [PERSON] et al. [17], which considers the following assumptions: (i) rigid column model considering an air-water interaction perpendicular to  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameter** & **Test 1-1** & **Test 1-2** & **Test 1-3** & **Test 1-4** \\ \hline \(L_{\text{i}up}\) (m) & null & 0.54 & 0.92 & 1.32 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Initial conditions of the experimental tests of the irregular pipe.  
Figure 1: Experimental facility of an irregular pipeline with an air valve.  
the pipe axis (flow-piston model); (ii) constant inner diameter and roughness coefficient; (iii) ball valves located at the downstream end; (iv) consideration of friction losses by means of the Darcy-Weisbach equation. Since the pipeline is branched in two sections, the analysis is considered for the left branch (Emptying Column No. 1), and the right branch (Emptying Column No. 2); additionally, an air valve with cross section (\(A_{adm,1}\)) is considered with an admission coefficient \(C_{adm,1}\). Table 2 shows the mass oscillation and emptying column velocity of Emptying Columns No. 1 and 2.  
In Table 2, \(u_{e}\) = water column velocity, \(p_{a}\) = air pocket pressure, \(p_{atm}\) = initial pressure of the air pocket (atmospheric), \(L_{e}\) = water column length, \(f\) = friction factor, \(\Delta z_{e}\) = height difference, \(R_{v}\) = ball valve resistance coefficient, \(A\) = pipe cross section, \(\phi\) = inner pipe diameter and \(g\) = gravitational acceleration. The subscripts 1 and 2 refers to branches 1 and 2, respectively. The equations defined in the above table correspond to the formulations associated with the water phase. For the air phase simulation, the authors of this mathematical model defined three (3) additional equations. To study the continuity of the air during expansion and admission process through the air valve, the authors of the mathematical model use the following differential equation:  
\[\frac{d\rho_{a,1}}{dt}=\frac{\rho_{a,nc}u_{a,nc,1}A_{adm,1}}{A_{2}(L_{2}-L_{e,2})+A_{1}(L_{1}-L_{e,1})}\]  
where \(\rho_{a}\) = air density, \(A_{adm,1}\) = air valve cross section and \(L\) = branch length. The subscripts 1 = actual conditions and \(nc\) = normal conditions. The representation of the evolution of the air pocket during its expansion process is expressed by the following equation:  
\[\frac{dp_{a}}{dt}=-\kappa\frac{p_{a}(u_{e,1}A_{1}+u_{e,2}A_{2})}{A_{2}(L_{2}- L_{e,2})+A_{1}(L_{1}-L_{e,1})}+\frac{p_{a}p_{a,nc}u_{a,nc,1}A_{adm,1}}{A_{2}(L_{2}- L_{e,2})+A_{1}(L_{1}-L_{e,1})}\frac{\kappa}{\rho_{a,1}}\]  
where \(\kappa\) = polytropic coefficient. For the analysis of the air inflow in irregular pipe, the following expression was used:  
\[Q_{a}=C_{adm,1}A_{adm,1}\sqrt{7p_{atm}\rho_{a,nc}\left[\left(\frac{p_{a}}{p_{ atm}}\right)^{1.4286}-\left(\frac{p_{a}}{p_{atm}}\right)^{1.714}\right]}\]  
where \(Q_{a}\) = air inflow and \(C_{adm,1}\) = admission coefficient.  
The mathematical model of [PERSON] et al. [17] was performed in the Simulink-Matlab software R2021b Update 4 of Mathworks (Natick, MA, USA), where the Ordinary Differential Equation 23s (ODE23s) was applied and supported by a modified Rosenbrock formula with a resolution of second order. In addition, a variable time step was applied, with a maximum value of 0.01 s.