# Analysis of Sub-Atmospheric Pressures during Emptying of an Irregular Pipeline without an Air Valve Using a 2D CFD Model

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. One-Dimensional Mathematical Model

## 3. Experimental Model

## 4. D–CFD Model with OpenFOAM

## 5. Results

#### 5.1. Analysis of the Variables

#### 5.1.1. Air Pocket Pressure

#### Type A Pattern: Partial Opening of the Ball Valve

#### 5.2. Type B Pattern: Full Opening of the Ball Valve

#### 5.2.1. Water Velocity

#### Type A Pattern: Partial Opening of the Ball Valve

#### Water Velocity Flow for Type B Pattern (Full Opening of the Ball Valve)

#### 5.2.2. Water Column Analysis

_{w}) is analysed, as shown in Figure 7. The water column in case A2 has an initial length of 3.16 m and a length of 3.14 m at the end of the hydraulic process, thus indicating that, by restricting the ball valve opening, only 0.016 m can be drained in 2 s during this process. In contrast, the water column of case B5 begins with a length of 3.16 m (at t = 0 s) and ends with 3.13 m, indicating that 0.21 m can be drained in 2 s during the process. And in the same way, the process is studied for all the following cases which observe how cases A1 and B4 have an air pocket of 0.28 m and a water column of 3.49 m at the initial time; however, by time 2 s it has a minimum water column of 3.48 m and 3.46 m.

## 6. Discussion

## 7. Conclusions

- -
- The type A pattern is the least critical scenario in the pipeline, because the sub-atmospheric pressure pattern reaches values close to atmospheric pressure. The hydraulic event begins with an atmospheric pressure (10.33 m) and then decreases rapidly until reaching sub-atmospheric pressure lows of 9.683 m (at 1.14 s), 9.857 m (at 1.63 s), and 10.045m (at 1.82 s) in runs no. 1 and no. 2 and o. 3, respectively. This pattern shows no oscillations in the first two seconds due to the partial opening of the valve. Subsequently, minimal fluctuations are observed in the pressure stabilisation that occurs in different cases as a function of the air pocket size. The more water left to exit the pipeline, the more its pressure will vary.
- -
- In the type B pattern, the valves are completely open. The sub-atmospheric pressure pattern in the hydraulic event begins under atmospheric conditions and then decreases rapidly until reaching minimum values of 9.387 m (at 0.38 s), 9.624 m (at 0.46 s) and 9.90 m (at 0.53 s) in runs no. 4, no. 5 and no. 6, respectively. After reaching minimum values, oscillations occur throughout the hydraulic event. As the air pocket increases, the pressure fluctuations within the pipeline become increasingly closer to atmospheric pressure, decreasing the amplitude of pressure changes and increasing the time between the different sub-atmospheric drops.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Complete pipeline model with details of lengths for the different behaviours of the study cases.

**Figure 2.**Details of the experimental installation: (

**a**) details of the mesh in the ball valve, (

**b**) details of the reduction mesh, (

**c**) details of the blocks as a function of the pipe symmetry.

**Figure 3.**Comparison of pressure patterns between the experimental model and the 2D CFD OpenFOAM symmetry model for cases A: (

**a**) run A1; (

**b**) run A2; (

**c**) run A3.

**Figure 5.**Comparison of pressure patterns between the experimental model and the 2D CFD OpenFOAM symmetry model for cases B: (

**a**) run B4; (

**b**) run B5; (

**c**) run B6.

**Figure 6.**Comparison of water velocity patterns between the experimental model and the 2D CFD OpenFOAM symmetry model for cases B: (

**a**) run B4; (

**b**) run B5; (

**c**) run B6.

**Figure 7.**Velocity vectors representative of case B5: (

**a**) valve opening at t = 0.50 s; (

**b**) valve opening at t = 2.00 s.

**Figure 8.**Analysis of the water column length in cases, with an initial air pocket length of 0.61 m, 0.28m and 0.98 respectively: (

**a**) case A2; (

**b**) case B5; (

**c**) case A1; (

**d**) case B5; (

**e**) case A3; (

**f**) case B6.

**Figure 9.**Sensitivity analysis of cases of partial valve opening: (

**a**) case A1; (

**b**) case A2; (

**c**) case A3.

**Figure 10.**Comparison of pressure and speed values for the different models: (

**a**) air pocket pressure values for case A1; (

**b**) air pocket pressure values for case B5.

No. | Equations | |
---|---|---|

1. Water phase equation | $\frac{d{V}_{j}}{dt}=\frac{{P}_{i}^{*}-{P}_{atm}^{*}}{{P}_{w}{L}_{w,j}}+g\frac{\u2206{Z}_{j}}{{L}_{w,j}}-{f}_{j}\frac{{V}_{j}\left|{V}_{j}\right|}{2D}-\frac{g{A}^{2}{Q}_{t}\left|{Q}_{t}\right|}{{L}_{w,j}{K}_{s}^{2}}$ | (1) |

2. Air phase equation | ${P}_{i}^{*}{({L}_{j}-{L}_{w,j}+{L}_{j+1}-{L}_{w,j+1})}^{m}={P}_{i,0}^{*}{({L}_{j}-{L}_{w,j,0}+{L}_{j+1}-{L}_{w,j+1,0})}^{m}$ | (2) |

3. Air–water interface equation | $\frac{d{L}_{w,j}}{{d}_{t}}=-{V}_{j}\to {L}_{w,j}={L}_{w,j,0}-{\int}_{0}^{t}{V}_{j}dt$ | (3) |

Pattern Type | Case No | Drain Valve Opening (%) * | ${\mathit{X}}_{1,1}\left(\mathbf{m}\right)$ | ${\mathit{X}}_{1,2}\left(\mathbf{m}\right)$ | ${\mathit{X}}_{1}\left(\mathbf{m}\right)\text{}**$ |
---|---|---|---|---|---|

A | 1 | Partial (6%) | 0.28 | 0.28 | 0.60 |

A | 2 | Partial (6%) | 0.61 | 0.61 | 1.26 |

A | 3 | Partial (6%) | 0.98 | 0.98 | 2.00 |

B | 4 | Full (100%) | 0.28 | 0.28 | 0.60 |

B | 5 | Full (100%) | 0.61 | 0.61 | 1.26 |

B | 6 | Full (100%) | 0.98 | 0.98 | 2.00 |

_{1}corresponds to the total air pocket size (X

_{1}= X

_{1,1}+ X

_{1,2}+ 0.04).

No. | Formulation | Equation |
---|---|---|

1. Moment conservation equation | $\frac{\partial \left({\rho}_{m}V\right)}{\partial t}+\nabla \times \left({\rho}_{m}VV\right)=-\nabla P+{\rho}_{m}\overrightarrow{g}+\nabla \times \left[{\mu}_{m}\left(\nabla V+\nabla {V}^{T}\right)\right]$ | (4) |

2. Air–water interface equation | $\frac{\partial \left({\rho}_{m}V\right)}{\partial t}+\nabla \times \left({\rho}_{m}Vu\right)=0$ | (5) |

Variables | Outlet | Walls |
---|---|---|

$T\left(\xb0\mathrm{K}\right)$ | inletOutlet | zeroGradient |

$P(\mathrm{N}/{\mathrm{m}}^{2})$ | prghTotalPressure | fixedFluxPressure |

$V(\mathrm{m}/\mathrm{s})$ | pressureInletOutletVelocity | noSlip |

$\omega ({\mathrm{m}}^{2}/{\mathrm{s}}^{3})$ | inletOutlet | omegaWallFuntion |

$k({\mathrm{m}}^{2}/{\mathrm{s}}^{2})$ | inletOutlet | KqRWallFunction |

Formulation | Equation |
---|---|

$\frac{\partial \left(\rho k\right)}{\partial t}+\frac{\partial \left(\rho {u}_{i}k\right)}{\partial t}={P}_{k}-{\beta}^{*}\rho k\omega +\frac{\partial}{\partial {x}_{i}}\left[\left(\mu +{\sigma}_{k}{\mu}_{t}\right)\frac{\partial k}{\partial {x}_{i}}\right]$ | (6) |

$\frac{\partial \left({\rho}_{\omega}\right)}{\partial t}+\frac{\partial \left(\rho {u}_{i}\omega \right)}{\partial t}=\alpha \frac{1}{{v}_{t}}{P}_{k}-\beta \rho {\omega}^{2}+\frac{\partial}{\partial {x}_{i}}\left[\left(\mu +{\sigma}_{k}{\mu}_{t}\right)\frac{\partial k}{\partial {x}_{i}}\right]+2\left(1-{F}_{1}\right)\rho {\sigma}_{\omega 2}\frac{1}{\omega}\frac{\partial k}{\partial {x}_{i}}\frac{\partial \omega}{\partial {x}_{i}}$ | (7) |

${P}_{k}=\mu \frac{\partial {u}_{i}}{\partial {x}_{j}}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)$ | (8) |

_{1}= 0.075 and β

_{2}= 0.0828. Table 6 shows the variables added in the initial and boundary conditions due to the turbulent model of the system (${\mathsf{\alpha}}_{t}$, k, ω y ${\mathsf{\nu}}_{t}$).

Boundaries | ${\mathsf{\alpha}}_{\mathit{t}}\text{}$ | k | ω | ${\mathsf{\nu}}_{\mathit{t}}$ |
---|---|---|---|---|

Outlet | inletOutlet | inletOutlet | inletOutlet | calculated |

Walls | fixedValue | KqRWallFunction | omegaWallFunction | nutkWallFunction |

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**MDPI and ACS Style**

Hurtado-Misal, A.D.; Hernández-Sanjuan, D.; Coronado-Hernández, O.E.; Espinoza-Román, H.; Fuertes-Miquel, V.S. Analysis of Sub-Atmospheric Pressures during Emptying of an Irregular Pipeline without an Air Valve Using a 2D CFD Model. *Water* **2021**, *13*, 2526.
https://doi.org/10.3390/w13182526

**AMA Style**

Hurtado-Misal AD, Hernández-Sanjuan D, Coronado-Hernández OE, Espinoza-Román H, Fuertes-Miquel VS. Analysis of Sub-Atmospheric Pressures during Emptying of an Irregular Pipeline without an Air Valve Using a 2D CFD Model. *Water*. 2021; 13(18):2526.
https://doi.org/10.3390/w13182526

**Chicago/Turabian Style**

Hurtado-Misal, Aris D., Daniela Hernández-Sanjuan, Oscar E. Coronado-Hernández, Héctor Espinoza-Román, and Vicente S. Fuertes-Miquel. 2021. "Analysis of Sub-Atmospheric Pressures during Emptying of an Irregular Pipeline without an Air Valve Using a 2D CFD Model" *Water* 13, no. 18: 2526.
https://doi.org/10.3390/w13182526