# Quasi-static Flow Model for Predicting the Extreme Values of Air Pocket Pressure in Draining and Filling Operations in Single Water Installations

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

- The water column movement is represented by a steady-state equation [1].

#### 2.1. Draining Process

_{v}= resistance coefficient of a drain valve.

#### 2.2. Filling Process

#### 2.3. Numerical Resolution

## 3. Model Validation

#### 3.1. Experimental Facility

_{1}) was positioned at the upstream end to fill the pipe installation. A pressure transducer was located at the highest point of the hydraulic installation to record air pocket pressure oscillations. In addition, another manual ball valve (MBV

_{2}) was placed at the downstream end, where the draining operation was performed. During all experiments, air pocket sizes of 0.205 m (0.67 ft), 0.340 m (1.11 ft) and 0.450 m (1.48 ft) were configured in the pipeline, as well as pipe slopes of 0.457 rad and 0.515 rad. Different valve travels in the MBV

_{2}were performed, which were defined through marks on a parallel white surface with regard to the position of the MBV

_{2}. Table 1 shows the characteristics of all performed runs, which have been previously discussed by the authors [9].

#### 3.2. Model Validation

## 4. Case Studies

#### 4.1. Draining Process

_{T}= 600 m, D = 0.35 m, k = 1.2, f = 0.018, Δz/L

_{w,t}= sin θ = 0.025 m/m, R

_{v}= 0.06 m/s/m

^{6}and t = 10 s. The initial conditions were described by the following parameters: x

_{0}= 200 m, v(0) = 0 m/s and ${p}_{a}^{*}={p}_{atm}^{*}$= 101,325 Pa. Figure 7 presents the results of water column length, water velocity and air pocket pressure.

- The quasi-static model can predict the evolution of the water column length during the draining process, as shown in Figure 7a. The quasi-static and inertial models show how part of the water column remains inside of the analysed single pipeline with a mean value of 221.2 m (725.7 ft), where the end of the hydraulic event is presented.
- Figure 7b presents the evolution of water velocity. The maximum water velocity using the quasi-static model is 2.71 m/s (8.89 ft/s) at 15 s, while the inertial model presents a maximum value of 2.63 m/s (8.63 ft/s) at 19.9 s. The quasi-static model cannot reproduce negative velocities.
- Regarding the air pocket pressure head (see Figure 7c), the quasi-static model can predict the minimum drop in the sub-atmospheric pressure head of 4.80 m (15.75 ft), which is critical to select the pipe stiffness class. In this hydraulic system, the air pocket pressure head value of 4.80 m (15.75 ft) remains constant from 130 s to the end of the hydraulic event.

#### 4.2. Filling Process

_{T}= 600 m, D = 0.30 m, k = 1.2, f = 0.018, Δz/L

_{w,t}= sin θ = 0.020 m/m, R

_{r}= 0.11 m/s/m

^{6}and t = 10 s. The initial conditions were x

_{0}= 500 m (1640.2 ft), v(0) = 0 m/s (0 ft/s), and ${p}_{a}^{*}={p}_{atm}^{*}$= 101,325 Pa. Figure 9 shows a comparison between the quasi-static and inertial models [7] of the main hydraulic and thermodynamic variables.

- The quasi-static model is not suitable for predicting oscillation patterns of water column position, water velocity, and air pocket pressure because it neglects the internal term (dv/dt = 0); however, extreme values can be predicted by the quasi-static model, which are used to select the pipe class.
- The quasi-static flow and inertial models can predict the final position of the air–water interface; as shown in Figure 9a, the water column approaches 382 m (1253.3 ft) from 120 s to the end of the hydraulic event.
- The peak of water velocity is predicted by the quasi-static model with a value of 5.03 m/s (16.5 ft/s) at 10 s (see Figure 9b); however, water velocity oscillations from 120 to 240 s are not detected by the proposed model. In this range, the quasi-static model reaches a value of 0 m/s (0 ft/s), while the inertial model presents some oscillations between −0.73 m/s (2.40 ft/s) and 0.50 m/s (1.64 ft/s).
- The more important variable during a filling process is the air pocket pressure. The quasi-static model reaches a maximum value of the absolute pressure head of 28.04 m (91.99 ft/s) at 107 s (see Figure 9c), while the inertial model presents a peak value of 30.7 m (100.72 ft). Subsequently, a difference in the maximum air pocket pressure of 2.66 m (8.73 ft) is presented by comparing these models.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$A$ | Cross-sectional area of pipe (m^{2}) |

$D$ | Pipe diameter (m) |

$f$ | Friction factor (-) |

$k$ | Polytropic coefficient (-) |

$g$ | Gravity acceleration (m/s^{2}) |

${L}_{w}$ | Length of the water column (m) |

${L}_{t}$ | Total length of pipe (m) |

${p}_{atm}^{*}$ | Atmospheric pressure (Pa) |

${p}_{a}^{*}$ | Air pocket pressure (Pa) |

${p}_{1}^{*}$ | Absolute pressure supplied by a pump or tank (Pa) |

${R}_{v}$ | Resistance coefficient of a drain valve (s^{2}/m^{5}) |

${R}_{r}$ | Resistance coefficient of a regulating valve (s^{2}/m^{5}) |

$t$ | Time (s) |

$v$ | Water velocity (m/s) |

$x$ | Length of air pocket (m) |

$z$ | Pipe elevation (m) |

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

$\mathsf{\Delta}t$ | Time step (s) |

$\theta $ | Pipe slope (rad) |

$0$ | Refers to an initial condition (e.g., initial air pocket size) |

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**Figure 5.**Comparison between computed and measure of air pocket pressure patterns: (

**a**) Run No. 1; (

**b**) Run No. 2; (

**c**) Run No. 3; (

**d**) Run No. 4; (

**e**) Run No. 5; (

**f**) Run No. 6; (

**g**) Run No. 7; (

**h**) Run No. 8; (

**i**) Run No. 9; (

**j**) Run No. 10; (

**k**) Run No. 11; (

**l**) Run No. 12.

**Figure 6.**Comparison of air pocket pressure patterns between the quasi-static flow and inertial models: (

**a**) Run No. 1; (

**b**) Run No. 11.

**Figure 7.**Comparison between the internal and quasi-static models for a draining process: (

**a**) Water column evolution; (

**b**) Behaviour of water velocity; (

**c**) Air pocket pressure oscillation.

**Figure 9.**Comparison between internal and quasi-static models for a filling process: (

**a**) Water column evolution; (

**b**) Behaviour of water velocity; (

**c**) Air pocket pressure oscillation.

Run No. | ${\mathit{x}}_{0}$ (m) | $\mathit{\theta}$ (rad) | ${\mathit{R}}_{\mathit{v}}\times {10}^{-6}$ (m/s/m ^{6}) |
---|---|---|---|

1 | 0.205 | 0.457 | 11.89 |

2 | 0.340 | 0.457 | 11.89 |

3 | 0.450 | 0.457 | 11.89 |

4 | 0.205 | 0.457 | 25.00 |

5 | 0.340 | 0.457 | 22.68 |

6 | 0.450 | 0.457 | 30.86 |

7 | 0.205 | 0.515 | 14.79 |

8 | 0.340 | 0.515 | 14.79 |

9 | 0450 | 0.515 | 14.79 |

10 | 0.205 | 0.515 | 135.21 |

11 | 0.340 | 0.515 | 138.41 |

12 | 0.450 | 0.515 | 100.00 |

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**MDPI and ACS Style**

Coronado-Hernández, Ó.E.; Fuertes-Miquel, V.S.; Mora-Meliá, D.; Salgueiro, Y. Quasi-static Flow Model for Predicting the Extreme Values of Air Pocket Pressure in Draining and Filling Operations in Single Water Installations. *Water* **2020**, *12*, 664.
https://doi.org/10.3390/w12030664

**AMA Style**

Coronado-Hernández ÓE, Fuertes-Miquel VS, Mora-Meliá D, Salgueiro Y. Quasi-static Flow Model for Predicting the Extreme Values of Air Pocket Pressure in Draining and Filling Operations in Single Water Installations. *Water*. 2020; 12(3):664.
https://doi.org/10.3390/w12030664

**Chicago/Turabian Style**

Coronado-Hernández, Óscar E., Vicente S. Fuertes-Miquel, Daniel Mora-Meliá, and Yamisleydi Salgueiro. 2020. "Quasi-static Flow Model for Predicting the Extreme Values of Air Pocket Pressure in Draining and Filling Operations in Single Water Installations" *Water* 12, no. 3: 664.
https://doi.org/10.3390/w12030664