# Study on the Mathematical Model of Vacuum Breaker Valve for Large Air Mass Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Problems with the Air Valve Mathematical Model

_{P}, B

_{P}, C

_{M}, and B

_{M}= constants;γ is the specific gravity of water, $\mathrm{kg}/\left({\mathrm{m}}^{2}\cdot {\mathrm{s}}^{2}\right)$; $Z$ is constant to be the height of the pipe top at the location of the air valve, m; ${m}_{0}$ is the mass of the gas in the air valve at the beginning of the time step, $\mathrm{kg}$; $\dot{m}$ is the mass flow of the air passing through the air valve, $\mathrm{kg}/\mathrm{s}$; and ${\dot{m}}_{0}$ is the mass flow of the air passing through the air valve at the beginning of the time step, $\mathrm{kg}/\mathrm{s}$.

#### 2.2. Mathematical Model of the Vacuum Breaker Valve

_{P}, B

_{P}, C

_{M}, B

_{M}are constants; ${Q}_{1}$ is the inflow of the section; ${Q}_{2}$ is the discharge flow of the section; $V$ is the volume of gas, γ is the specific gravity of water, $\mathrm{kg}/\left({\mathrm{m}}^{2}\cdot {\mathrm{s}}^{2}\right)$; $p$ is the pressure of the gas in the air valve, Pa; ${p}_{a}$ is the atmospheric pressure, Pa; $Z$ is constant to be the height of the pipe top at the location of the air valve, m; and $A$ is the cross-sectional area of the valve hole, ${\mathrm{m}}^{2}$.

_{P}, B

_{P}, C

_{M}, and B

_{M}= constants; γ is the specific gravity of water, $\mathrm{kg}/\left({\mathrm{m}}^{2}\cdot {\mathrm{s}}^{2}\right)$; $p$ is the pressure of the gas in the air valve, Pa; ${p}_{a}$ is the atmospheric pressure, Pa; $A$ is the cross-sectional area of the valve hole, ${\mathrm{m}}^{2}$; ${Z}_{0}$ is the water level in the vacuum breaker valve at the beginning of the time step, m; $Z$ is constant to be the height of the pipe top at the location of the air valve, m; ${m}_{0}$ is the mass of the gas in the air valve at the beginning of the time step, kg; $\dot{m}$ is the mass flow of the air passing through the air valve, $\mathrm{kg}/\mathrm{s}$; and ${\dot{m}}_{0}$ is the mass flow of the air passing through the air valve at the beginning of the time step, $\mathrm{kg}/\mathrm{s}$.

## 3. Results

#### 3.1. Project Description

#### 3.2. Numerical Simulation of the Air Valve and Vacuum Breaker Valve

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Lescovich, J.E. Locating and Sizing Air-Release Valves. J. Am. Water Works Ass.
**1972**, 64, 457–461. [Google Scholar] [CrossRef] - McPherson, D.L. Air valve sizing and location: A prospective. In Proceedings of the Pipelines Specialty Conference 2009, San Diego, CA, USA, 15–19 August 2009; pp. 905–919. [Google Scholar]
- Coronado-Hernández, O.E.; Fuertes-Miquel, V.S.; Besharat, M.; Ramos, H.M. Experimental and Numerical Analysis of a Water Emptying Pipeline Using Different Air Valves. Water
**2017**, 9, 98. [Google Scholar] - Manual of Water Supply Practices—M51: Air-Release, Air-Vacuum, and Combination Air Valves, 1st ed.; American Water Works Association: Denver, CO, USA, 2001.
- Bergant, A.; Kruisbrink, A.; Arregui, F. Dynamic behavior of air valves in a large-scale pipeline apparatus. J. Mech. Eng.
**2012**, 58, 225–237. [Google Scholar] [CrossRef] - Lucca, Y.F.L.D.; Aquino, G.A.; Filho, J.G.D. Experimental apparatus to test air trap valves. IOP Conf. Ser. Earth Environ.
**2010**, 12, 1–8. [Google Scholar] [CrossRef] - Arregui, F.J. Air Valves Dynamic Behavior; ITA, Universitat Polytecnica Valencia: Valencia, Spain, 2012. [Google Scholar]
- Lee, T.S.; Leow, L.C. Numerical study on the effects of air valve characteristics on pressure surges during pump trip in pumping systems with air entrainment. Int. J. Numer. Meth. Fl.
**1999**, 29, 645–655. [Google Scholar] [CrossRef] - Lubbers, C.; Clemens, F. Detection of gas pockets in pressurized waste water mains using dynamic system response analysis. Water Sci. Tech.
**2007**, 55, 31–38. [Google Scholar] [CrossRef] - Ramezani, L.; Karney, B.; Malekpour, A. The challenge of air valves: A selective critical literature review. J. Water Resour. Plan. Manag.
**2015**, 141, 04015017. [Google Scholar] [CrossRef] - Bergant, A.; Tijsseling, A.S.; Vitkovsy, J.P.; Covas, D.I.C.; Simpson, A.R.; Lambert, M.F. Parameters affecting water-hammer wave attenuation, shape and timing-Part 2: Case studies. J. Hydraul. Res.
**2008**, 46, 382–391. [Google Scholar] [CrossRef] - De Martino, G.; Fontana, N.; Giugni, M. Transient flow caused by air expulsion through an orifice. J. Hydraul. Eng.
**2008**, 134, 1395–1399. [Google Scholar] [CrossRef] - Fontana, N.; Galdiero, E.; Giugni, M. Pressure surges caused by air release in water pipelines. J. Hydraul. Res.
**2016**, 54, 461–472. [Google Scholar] [CrossRef] [Green Version] - Lingireddy, S.; Wood, D.J.; Zloczower, N. Pressure surges in pipeline systems resulting from air releases. J. Am. Water Works Ass.
**2004**, 96, 88–94. [Google Scholar] [CrossRef] - Li, G.; Baggett, C.C.; Rosario, R.A. Air/vacuum valve breakage caused by pressure surges—Analysis and solution. In Proceedings of the World Environmental and Water Resources Congress 2009, Kansas City, MO, USA, 2009, 17–21 May; pp. 112–121.
- Ramezani, L.; Karney, B. Water column separation and cavity collapse for pipelines protected with air vacuum valves: Understanding the essential wave processes. J. Hydraul. Eng.
**2017**, 143, 04016083. [Google Scholar] [CrossRef] - Cabrera, E.; Fuertes, V.S.; García-Serra, J.; Arregui, F.; Gasc, L.; Palau, V. Reviewing air valves selection. In Proceedings of the Pumps, Electromechanical Devices and Systems Applied to Urban Water Management, Valencia, Spain, 22–25 April 2003; pp. 633–640. [Google Scholar]
- Miao, D.; Zhang, J.; Chen, S.; Yu, X.D. Water hammer suppression for long distance water supply systems by combining the air vessel and valve. J. Water Supply: Res. T.
**2017**, 66, 319–326. [Google Scholar] [CrossRef] - De Aquino, G.A.; De Lucca, Y.D.F.L.; Dalfre Filho, J.G. The importance of experimental tests on air valves for proper choice in a water supply project. J. Braz. Soc. Mech. Sci. Eng.
**2018**, 40, 403. [Google Scholar] [CrossRef] - Bianchi, A.; Mambretti, S.; Pianta, P. Practical formulas for the dimensioning of air valves. J. Hydraul. Eng.
**2007**, 133, 1177–1180. [Google Scholar] [CrossRef] - JÖNSSON, L. Maximun transients’ pressures in a conduit with check valve and air entrainment. In Proceedings of the International Conference on the Hydraulics of Pumping Stations, Manchester, UK, 17–19 September 1985. [Google Scholar]
- Zhou, L.; Liu, D.; Karney, B. Investigation of hydraulic transients of two entrapped air pockets in a water pipeline. J. Hydraul. Eng.
**2013**, 139, 949–959. [Google Scholar] [CrossRef] - Zhou, L.; Liu, D.; Karney, B.; Wang, P. Phenomenon of white mist in pipelines rapidly filling with water with entrapped air pockets. J. Hydraul. Eng.
**2013**, 139, 1041–1051. [Google Scholar] [CrossRef] - Zhou, L.; Liu, D.; Karney, B.; Zhang, Q. Influence of entrapped air pockets on hydraulic transients in water pipelines. J. Hydraul. Eng.
**2011**, 137, 1686–1692. [Google Scholar] [CrossRef] - Tran, P.D. Pressure Transients Caused by Air-Valve Closure while Filling Pipelines. J. Hydraul. Eng.
**2016**, 143, 04016082. [Google Scholar] - Albertson, M.L.; Andrews, J.S. Transients Caused by Air Release. In Control of Flow in Closed Conduits; Colorado State Univ.: Fort Collins, CO, USA, 1971; pp. 315–340. [Google Scholar]
- Carlos, M.; Arregui, F.J.; Cabrera, E.; Palau, C.V. Understanding air release through air valves. J. Hydraul. Eng.
**2011**, 137, 461–469. [Google Scholar] [CrossRef] - Zhou, F.; Hicks, F.E.; Steffler, P.M. Observations of air water interaction in a rapidly filling horizontal pipe. J. Hydraul. Eng.
**2002**, 128, 635–639. [Google Scholar] [CrossRef] - Zhou, F.; Hicks, F.E.; Steffler, P.M. Transient flow in a rapidly filling horizontal pipe containing trapped air. J. Hydraul. Eng.
**2002**, 128, 625–634. [Google Scholar] [CrossRef] - Izquierdo, J.; Fuertes, V.S.; Cabrera, E.; Iglesias, P.L.; Garcia-Serra, J. Pipeline start-up with entrapped air. J. Hydraul. Res.
**1999**, 37, 579–590. [Google Scholar] [CrossRef] - Wylie, E.B.; Streeter, V.L.; Suo, L.S. Fluid Transients in Systems; Prentice-Hall, Inc.: Englewood Cliffs, NJ, USA, 1993. [Google Scholar]
- Chaudhry, M.H. Applied Hydraulic Transients; Springer: New York, NY, USA, 2014. [Google Scholar]

**Figure 2.**Processes of air passing through the air valve: (

**a**) Gas sucking in; (

**b**) gas discharging out.

Water Level of Suction Sump (m) | 28.0 | Diameter of Vacuum Breaker Valve (m) | 0.3 |
---|---|---|---|

Water level of outlet sump (m) | 38.5 | Intake pressure of vacuum breaker valve (m) | −3.0 |

Quantity of pipes | 1 | Quantity of pumps | 1 |

Elevation of pipe center at outlet sump end (m) | 30.0 | Elevation of pump (m) | 25.0 |

Elevation of pipe center at top of siphon pipe (m) | 37.0 | Rated head (m) | 12 |

Horizontal length of pipe (m) | 105.0 | Rated flow (m^{3}/s) | 10 |

Pipe diameter (m) | 2.4 | Rated rotational speed (r/min) | 250 |

Design flow (m^{3}/s) | 10 | Rated motor power (kW) | 1600 |

Elevation of vacuum breaker valve (m) | 40.0 | Flywheel moment (kg·m^{2}) | 3800 |

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**MDPI and ACS Style**

Zhang, X.-y.; Fan, C.-y.; Yu, X.-d.; Zhang, J.; Lv, J.-w.; Xu, T.-y.
Study on the Mathematical Model of Vacuum Breaker Valve for Large Air Mass Conditions. *Water* **2019**, *11*, 1358.
https://doi.org/10.3390/w11071358

**AMA Style**

Zhang X-y, Fan C-y, Yu X-d, Zhang J, Lv J-w, Xu T-y.
Study on the Mathematical Model of Vacuum Breaker Valve for Large Air Mass Conditions. *Water*. 2019; 11(7):1358.
https://doi.org/10.3390/w11071358

**Chicago/Turabian Style**

Zhang, Xiao-ying, Cheng-yu Fan, Xiao-dong Yu, Jian Zhang, Jia-wen Lv, and Ting-yu Xu.
2019. "Study on the Mathematical Model of Vacuum Breaker Valve for Large Air Mass Conditions" *Water* 11, no. 7: 1358.
https://doi.org/10.3390/w11071358