# Three-Dimensional Analysis of Air-Admission Orifices in Pipelines during Hydraulic Drainage Events

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Model

## 3. 3D CFD Model

#### 3.1. Governing Equations

#### 3.2. Computational Mesh

#### 3.3. Boundary Conditions

#### 3.4. Numerical Schemes

## 4. Results and Discussion

#### 4.1. Air Inflow Effect

#### 4.2. Influence of Air-Admission Orifice on Water Drainage Velocity

## 5. Conclusions

- The three-dimensional CFD models adequately represent the dynamics of different air-admission orifices in different drainage events with trapped air, showing good numerical results in obtaining vacuum pressures, even in the representation of the air inflow effect using the calibration curves of orifices with diameters of 1.5 and 3.0 mm. The Root Mean Square Error values are adequate (between 2.34 and 6.67%), according to the contribution of Besharat et al. [25].
- The vorticity effect in the trapped air pocket corresponds to a phenomenon that has been scarcely studied in air-admission orifices of hydraulic systems. The 3D CFD model provides the possibility to know the behaviour of the airflow streamlines inside the pipe over time. This information is useful to verify the impact of the airflow turbulence phenomena inside the pipe.
- In similar drainage events, an air-admission orifice with a diameter ${d}_{adm}$ = 3.0 mm guarantees a differential volume fraction of 0.41%, thus allowing a continuity of the airflow over the trapped air pocket and an adequate expansion of the air pocket during the drainage processes. On the other hand, the orifice with ${d}_{adm}$ = 1.5 mm generates a higher differential volume fraction than the orifice ${d}_{adm}$ = 3.0 mm (1.40%). For the maximum time instant analysed (t = 1.50 s), the continuity of the airflow over the air pocket is approximately 3.25 times more effective at the 3.0 mm diameter orifice than at the 1.5 mm diameter orifice. An effective control of the vacuum pressures and hydraulic transients lies significantly in the appropriate sizing of the air-admission orifices. A larger orifice allows the admission of a higher volumetric rate in order to preserve the principle of continuity between the air-admitted volume and the air pocket volume. If a rapid drainage process is considered (high drain valve-opening percentages), it is important to choose to use a larger orifice in order to compensate for the volume differential $\Delta V$ generated between the air pocket volume and the air-admitted volume to the hydraulic system over time.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${B}_{1}$ | blending function (-) |

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

${d}_{adm}$ | air-admission orifice diameter (m) |

${F}_{s}$ | body forces (N) |

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

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

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

M | Mach number (-) |

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

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

${Q}_{adm}$ | air inflow in orifice (m${}^{3}$/s) |

t | time (s) |

${t}_{0}$ | drain valve-opening time (s) |

T | temperature (K) |

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

${u}_{i}$ | velocity component (m/s) |

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

V | volume (m${}^{3}$) |

${\gamma}_{a}$ | air-phase fraction (-) |

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

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

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

${\tau}_{0}$ | drain valve-opening percentage (%) |

${\tau}_{t}$ | shear stress (N/m${}^{2}$) |

$\omega $ | dissipation frequency rate (m${}^{2}$/s${}^{3}$) |

Subscripts | |

The following subscripts are used in this manuscript: | |

a | refers to air phase |

w | refers to water phase |

m | refers to mixture conditions |

t | refers to turbulent conditions |

p | refers to pipeline |

$atm$ | refers to atmospheric conditions |

* | refers to absolute scale |

Coefficients—$\mathit{k}$-$\mathsf{\omega}$ SST model | |

The coefficients of the k-$\omega $ model show the following values: | |

${\alpha}_{1}$ | 0.55 |

${\alpha}_{2}$ | 0.44 |

${\beta}_{1}$ | 0.075 |

${\beta}_{2}$ | 0.0828 |

${\beta}^{*}$ | 0.09 |

${\sigma}_{k,1}$ | 0.85 |

${\sigma}_{k,2}$ | 1.0 |

${\sigma}_{\omega ,1}$ | 0.5 |

${\sigma}_{\omega ,2}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$ | 0.856 |

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**Figure 1.**Detail of experimental model: (

**a**) pressure transducer, (

**b**) drain valve, and (

**c**) air-admission orifice.

**Figure 2.**Characteristic curves of the orifices calibrated at hydraulics laboratory of the Universitat Politècnica de València.

**Figure 3.**Three-dimensional mesh of single pipeline: supplementary section with 90${}^{\circ}$ elbow and drain valve (Lower zone), and air-admission orifice (Upper zone).

**Figure 4.**Comparison of vacuum pressure patterns of air pocket entrapped (CFD model vs. experimental measurements): (

**a**) Test 1, (

**b**) Test 2, (

**c**) Test 3, and (

**d**) Test 4.

**Figure 5.**Comparison of the physical characteristics of air-admission orifices: CFD model vs. experimental curves: (

**a**) ${d}_{adm}$ = 1.5 mm, and (

**b**) ${d}_{adm}$ = 3.0 mm.

**Figure 6.**Air vortices during air inflow in drainage events (${\tau}_{0}$ = 17.5%, ${t}_{0}$ = 0.40 s, and ${L}_{iap}$ = 0.205 m): (

**a**) ${d}_{adm}$ = 1.5 mm and (

**b**) ${}_{adm}$ = 3.0 mm.

**Figure 7.**Air–water interface location during drainage events with ${d}_{adm}$ = 1.5 and 3.0 mm (${\tau}_{0}$ = 17.5%, ${t}_{0}$ = 0.40 s, and ${L}_{iap}$ = 0.205 m).

**Figure 8.**Ratio between admitted air volume and air pocket volume over pipeline volumetric capacity (${\tau}_{0}$ = 17.5%, ${t}_{0}$ = 0.40 s, and ${L}_{iap}$ = 0.205 m): (

**a**) ${d}_{adm}$ = 1.5 mm and (

**b**) ${}_{adm}$ = 3.0 mm.

Test | ${\mathit{L}}_{\mathbf{iap}}$ (m) | ${\mathit{\tau}}_{0}$ (%) | ${\mathit{t}}_{0}$ (s) | ${\mathit{d}}_{\mathbf{adm}}$ (mm) |
---|---|---|---|---|

1 | 0.205 | 12 | 0.40 | 1.5 |

2 | 0.450 | 8.2 | 0.25 | 1.5 |

3 | 0.205 | 24.5 | 0.50 | 3.0 |

4 | 0.450 | 13.4 | 0.45 | 3.0 |

**Table 2.**Root Mean Square Error obtained in the vacuum pressure patterns of the CFD model tests with experimental results.

Test | RMSE (%) |
---|---|

1 | 6.67% |

2 | 6.01% |

3 | 2.74% |

4 | 2.34% |

Test | M (max.) |
---|---|

1 | 0.30 |

2 | 0.27 |

3 | 0.23 |

4 | 0.18 |

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**MDPI and ACS Style**

Paternina-Verona, D.A.; Coronado-Hernández, O.E.; Espinoza-Román, H.G.; Besharat, M.; Fuertes-Miquel, V.S.; Ramos, H.M.
Three-Dimensional Analysis of Air-Admission Orifices in Pipelines during Hydraulic Drainage Events. *Sustainability* **2022**, *14*, 14600.
https://doi.org/10.3390/su142114600

**AMA Style**

Paternina-Verona DA, Coronado-Hernández OE, Espinoza-Román HG, Besharat M, Fuertes-Miquel VS, Ramos HM.
Three-Dimensional Analysis of Air-Admission Orifices in Pipelines during Hydraulic Drainage Events. *Sustainability*. 2022; 14(21):14600.
https://doi.org/10.3390/su142114600

**Chicago/Turabian Style**

Paternina-Verona, Duban A., Oscar E. Coronado-Hernández, Hector G. Espinoza-Román, Mohsen Besharat, Vicente S. Fuertes-Miquel, and Helena M. Ramos.
2022. "Three-Dimensional Analysis of Air-Admission Orifices in Pipelines during Hydraulic Drainage Events" *Sustainability* 14, no. 21: 14600.
https://doi.org/10.3390/su142114600