# Rapid Filling Analysis with an Entrapped Air Pocket in Water Pipelines Using a 3D CFD Model

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## Abstract

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## 1. Introduction

## 2. Experimental Facility

## 3. Three-Dimensional Model

#### 3.1. Governing Equations

#### 3.1.1. Partial Vof Equations

#### 3.1.2. Navier–Stokes Equations

#### 3.1.3. Thermodynamic Equations

#### 3.1.4. SST k-$\omega $ Turbulence Model

#### 3.2. Numerical Approach

#### 3.3. Mesh Computational and Boundary Conditions

## 4. Results and Discussion

#### 4.1. Air–Water Interaction

#### 4.2. Water Velocity

#### 4.3. Air Pocket Behaviour

## 5. Conclusions

- The entrapped air pocket exhibits constant volume changes over time. The water column compresses the air pocket to a limit state and then it tends to expand to release the compression energy that was accumulated during the filling event. This compression–expansion phenomenon of the air pocket is cyclic over time. The dynamic behaviour of the trapped air during the volume changes can be visualised in detail through streamlines and velocity contours, where such events manifest their volume reduction and expansion through vortices located at the air–water interface zones that can physically reach maximum velocities of 1.2 m/s (in Test No. 2).
- Transient flows can result in backflows towards the pumping source, which can lead to a loss of the hydraulic efficiency during filling events. During the first few seconds, the velocity transitions are abrupt, and then they gradually dissipate over time due to the damping pressure of the entrapped air pocket. These backflows are more critical in scenarios with higher inlet gauge pressures. In addition, the CFD model predicts in detail the vector field of the water flow by showing the changes in the water trajectory at different time instants.
- Temperature changes are inevitable during filling processes, considering that the compression of the trapped air causes a change in the thermodynamic conditions of the air phase. In such phenomena, the trapped air pocket undergoes adiabatic behaviour, which has been reproduced in previous research on filling processes. The 3D CFD model in the analysed tests shows a non-uniform temperature distribution, showing that away from the air–water interface the highest temperatures occur (up to 120 °C in Test No. 2), whereas in the air pocket near the air–water interface, temperatures between 20 °C and 30 °C occur.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notation

${a}_{\mathrm{eff}}$ | thermal conductivity (W/(m·K)) |

${C}_{p}$ | specific heat (J/(kg K)) |

${D}_{k}$ | diffusion term for k (m${}^{2}$/s) |

${D}_{\omega}$ | diffusion term for $\omega $ (m${}^{2}$/s) |

${F}_{1}$ | blending function (–) |

${F}_{s}$ | surface tension (kg m/s${}^{2}$) |

$\overrightarrow{g}$ | gravitational acceleration vector (m/s${}^{2}$) |

G | turbulent kinetic energy generation (m${}^{2}$/s${}^{3}$) |

k | turbulent kinetic energy (m${}^{2}$/s${}^{2}$) |

${L}_{iap}$ | initial air pocket (m) |

p | static pressure (N/m${}^{2}$) |

${p}_{0}$ | initial absolute pressure (N/m${}^{2}$) |

$\overrightarrow{q}$ | heat flux vector (W/m${}^{2}$) |

R | universal gas constant (J/(K·mol)) |

T | temperature (°C) |

t | time (s) |

$\overrightarrow{u}$ | velocity vector (m/s) |

${u}_{r}$ | velocity field (m/s) |

${y}^{+}$ | distance function (–) |

${\alpha}_{w}$ | phase of fraction (water) (–) |

$\mu $ | dynamic viscosity (Ns/m${}^{2}$) |

${\nu}_{t}$ | turbulent kinematic viscosity (m${}^{2}$/s) |

$\rho $ | density (kg/m${}^{3}$) |

$\omega $ | dissipation frequency (1/s) |

Subscripts | |

a | refers to air (e.g., air density) |

w | refers to water (e.g., water density) |

m | refers to the mixture between air and water (e.g., mixed density) |

t | refers to a turbulent condition (e.g., turbulent dynamic viscosity) |

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**Figure 2.**Geometric domain with detail in distribution of cells and boundary conditions: (

**a**) inlet and electro-pneumatic ball valve, (

**b**) upstream end, and (

**c**) detail of walls in right pipe branch.

**Figure 3.**Analysis of air pocket pressure pulses: experimental measurements versus 3D CFD model. (

**a**) Test No. 1, and (

**b**) Test No. 2.

**Figure 4.**Dynamic behaviour of the entrapped air pocket during the analysed filling process. (

**a**) Test No. 1, and (

**b**) Test No. 2.

Test | ${\mathit{p}}_{0}$ (Pa) | ${\mathit{L}}_{\mathbf{iap}}$ (m) |
---|---|---|

1 | 121,325 | 0.46 |

2 | 176,325 | 0.46 |

3 | 151,325 | 0.46 |

4 | 121,325 | 0.96 |

5 | 176,325 | 0.96 |

6 | 151,325 | 0.96 |

Boundary | Characteristics |
---|---|

Inlet | Corresponds to the boundary where water inflow is generated during filling events. |

Walls | This boundary corresponds to the walls of the existing pipeline and accessories. |

VSI (Valve–Sliding Interface) | Corresponds to the sliding interface that ensures continuity of flow over the electro-pneumatic ball valve. |

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**MDPI and ACS Style**

Paternina-Verona, D.A.; Coronado-Hernández, O.E.; Espinoza-Román, H.G.; Fuertes-Miquel, V.S.; Ramos, H.M. Rapid Filling Analysis with an Entrapped Air Pocket in Water Pipelines Using a 3D CFD Model. *Water* **2023**, *15*, 834.
https://doi.org/10.3390/w15050834

**AMA Style**

Paternina-Verona DA, Coronado-Hernández OE, Espinoza-Román HG, Fuertes-Miquel VS, Ramos HM. Rapid Filling Analysis with an Entrapped Air Pocket in Water Pipelines Using a 3D CFD Model. *Water*. 2023; 15(5):834.
https://doi.org/10.3390/w15050834

**Chicago/Turabian Style**

Paternina-Verona, Duban A., Oscar E. Coronado-Hernández, Hector G. Espinoza-Román, Vicente S. Fuertes-Miquel, and Helena M. Ramos. 2023. "Rapid Filling Analysis with an Entrapped Air Pocket in Water Pipelines Using a 3D CFD Model" *Water* 15, no. 5: 834.
https://doi.org/10.3390/w15050834