## 5 Conclusions
CFD models provide useful and relevant numerical results for a detailed analysis of hydraulic phenomena resulting from air-water interaction during transient events in pressurised pipelines, such as (i) deformation and real-time location of the air-water interface, (ii) velocity contours, (iii) backflow air intrusion, (iv) air pocket pressure patterns as well as water flow patterns at different points in the pipes and (v) heat transfer between water and air. Based on the development of mathematical and CFD models, the following items can be concluded:  
* Mathematical models based on the rigid column model are a fundamental tool for understanding the physical evolution of several hydraulic and thermodynamic variables based on differential equations that can be manipulated by the user according to the hydraulic scenario that is to be studied. These models have been validated in different investigations, showing good agreement with experimental measurements, and, additionally, they have been applied in large-scale hydraulic scenarios [19; 36];
* A 2D CFD model allows study of the emptying process of an irregular pipeline with an air valve. The presence of air valves facilitates the air admission process during the water drainage process and mitigates subatmospheric pressures. In this sense, 2D CFD models are suitable for a simplified analysis of the interaction between air and water, with acceptable numerical and spatial resolution. The use of 2D CFD models is useful in cases where numerical information associated with physical parameters, such as velocity, pressure and heat transfer at different points of the pipeline, and visual information about the interaction between air and water are needed. These models require less computational time than a 3D CFD model and must be used with restrictions for the study of the emptying of pipes with air valves, when section changes occur (contractions, reductions), with the purpose of adjusting the mass flow conditions by means of a geometric aspect ratio defined in the literature for the sizing of orifices in pipes;
* A 3D CFD model allows simulation of the drainage of a single pipe in two cases (without air valves and with air valves). In Case 1, adverse subatmospheric pressures are generated due to entrapped air pockets and water flow oscillations. On the other hand, in Case 2, fewer critical pressure oscillations are observed compared to Case 1, due to the influence of the air admission orifices. All CFD model results in both cases were compared with the patterns recorded by the mathematical model, generally proposed by [PERSON] et al. [18]. It was observed that 3D CFD models showed hydraulic-thermodynamic phenomena that cannot be obtained in 2D CFD models, such as velocity distribution in pipe cross-sections, transient flows in pipe cross-sections, small air pockets created due to backflow air effect and their influence on water drainage flow transitions. In addition, there is the presence of a phenomenon that has not been studied in the literature, which is the temperature gradients between water and air phases, considering that the air pockets present a decrease in temperature due to the subatmospheric pressures and the water phase influences in heat transfer.