## 3 Numerical Modelling in Single Pipelines
### 3D CFD Model vs. Mathematical Model
Differences between oscillations in water flow and pressure patterns in the CFD and mathematical models were also due to the effect of backflow air, which influences the hydraulic conditions of drainage, a phenomenon that was not considered in the physical equations of the mathematical model. This delay in the drainage pressure and flow rate oscillations was significantly evidenced in Test C1-1 (see Figures 12 and 13), which is influenced by the presence of air bubbles entering upstream. It could be observed that the 3D CFD model was able to adequately represent the behaviour of the air pocket, i.e., the movement of the backflow air, forming an appropriate stratification of air pockets, wherein the mixing of these fluids is rarely generated. In contrast, there were no air bubbles in Test C1-2. This is evident in the good agreement between the pressure oscillations and water discharge flow rate of the CFD model with the recorded oscillations of the mathematical model (see Figure 11d). On the other hand, the thermodynamic behaviour between air and water corresponds to a phenomenon that can be captured by 3D CFD models, which has been rarely studied in the literature. [PERSON] et al. [33] performed a study of the heat transfer between the air and water phases in a rapid filling process and compared with experimental results through the development of a 3D CFD model, which observed an increase in the temperature of the entrapped air pocket to values of more than 100 \({}^{\circ}\)C, which could be observed during the experimental measurements performed. Currently, there are no detailed studies of the thermodynamic behaviour between the air and water phases in pressurised pipe emptying processes. The water phase is characterised by a lower compressibility than the air phase, so its temperature variations are insignificant during transient events. However, the water phase plays an important role in the heat transfer process, as shown in Figure 14, where the temperature gradients that occur from the water phase to the air phase in the emptying processes of Tests C1-1 and C1-2, respectively, are visualised. Additionally, this information is complemented with the air-water interface location during the thermodynamic interaction. Figure 12: Air intrusion due to backflow air in 3D CFD Model (Tests C1-1 and C1-2). Figure 15 shows the variation in the temperature patterns in the air pockets of Tests C1-1 and C1-2, where it is observed that the temperature of the entrapped air pockets decreases during the expansion processes, proceeding from 20 \({}^{\circ}\)C to minimum values of 1.5 \({}^{\circ}\)C and 2.3 \({}^{\circ}\)C, respectively, at \(t\) = 0.5 and 1.0 s. In Test C1-1, temperature oscillations occur due to the smooth oscillations in the water column velocity, which cause compression of the air pocket, accumulating compression energy that is subsequently transformed into heat energy. In Test C1-2, there is a temperature drop at \(t\) = 1.0 s, and, thereafter, the temperature increases progressively, causing the compression and expansion energy to be present in smaller proportions in the air pocket, facilitating its return to ambient temperature by heat transfer between water and air. Figure 14: Heat transfer between the water phase and the air phase. (**a**) Test C1-1 and (**b**) Test C1-2. Figure 13: Air intrusion in Test C1-1 (Experimental Model). Comparison of 2D/3D CFD Models  
The authors of [23] used an experimental setup similar to the single pipe to represent the scenario associated with Case 2 (emptying process with an air admission orifice), where Test C2-1 was compared with the simulation performed by these authors in their research. An analysis of the physical phenomena in Test C2-1 was performed between the results obtained by the 3D CFD model of [PERSON] et al. [24] and the 2D CFD model of [PERSON] et al. [23], where the behaviour of the air-water interface and the velocity distribution of the air flow admitted to the hydraulic system through the orifice were compared.