# Effect of Leak Geometry on Water Characteristics Inside Pipes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup & Numerical Model

#### 2.1. Experimental Setup

_{d}is the time of the wave propagating with the flow, and T

_{u}is the time of the wave propagating against the flow. For accurate measurement of the distance (L) between ultrasonic transducers (T), two Cricket (40 kHz) sensors, which work on a combination of radiofrequency and ultrasonic technologies, are used. As depicted in Figure 4, D is Lcosɵ or Lsinɵ. The equipment used for measuring the velocity of flow is designed by “MEMSIC Transducer Systems”.

^{5}Pa. The pressure transducers are also used to make sure that the fluid flow in the pipe is steady.

^{3}cm

^{3}/min measuring range and can operate at a pressure value up to 16 × 10

^{5}Pa. To evaluate the amount of water being leaked, the inlet flow and the outlet flow are compared. A non-zero difference in the volumetric flow rate indicates the presence of a leak between ultrasonic fluid flow meters. To measure the amount of water leaked, a comparison between the inlet flow and the outlet flow in terms of volumetric flow rate, as apparent in (2)

^{3}/s), V is the flow velocity (cm/s), and A is the cross-sectional pipe area (cm

^{2}). Since the cross-sectional area along the pipe is constant, it is sufficient to measure the values of velocity along the pipe rather than the flow rate since velocities can be easily measured at different locations along the pipe using ultrasonic sensors. Thus, the fluid flow rate can be evaluated by two measurements taken at the inlet and outlet of the middle pipe.

#### 2.2. Numerical Model

^{®}is used to analyze fluid velocity distribution inside the pipe. The values of velocities are close to the wall where the Laminar and the Turbulent velocities have close magnitudes [46], as shown in Figure 5. The difference between the Laminar and Turbulent velocities is highest in the middle of the pipe. The Reynolds number for used experimental conditions, which indicates turbulent flow, was calculated. Turbulent flow, K-ϵ, from the COMSOL Multiphysics is considered to analyze the flow in a pipe. K-ϵ supports both compressible and incompressible flows. The continuity equation and Navier-‘Stokes’ equations (for conservation of mass and conservation of momentum) are the equations that are being solved by K-ϵ. Three boundary conditions (velocity inlet 1, velocity outlet 2, and pressure outlet 3) are used in the simulation. The inlet of the pipe is referred to as “velocity inlet 1”. The outlet of the pipe is referred to as “pressure outlet 3”. The leak point is referred to as “velocity outlet 2”. Values assigned in different cases to these conditions are given in Table 1.

#### Geometry and Mesh Creation

## 3. Analysis & Results

#### 3.1. Numerical Simulations

^{3}/s through the leaking valve.

#### 3.2. Comparisons between Experimental Results & Numerical Simulations

^{3}/s. A nearly constant velocity magnitude is observed before and after the leak location due to mass conservation. At the center of the pipe, where the leak is located, an experimental peak value of 25.5 cm/s and a numerical peak value of 24.8 cm/s are observed.

^{3}/s, the % error at 89 cm is 2.3%, whereas at the same location, a % error of 2.9% can be seen for the leak flow rate of 12 cm

^{3}/s. At downstream locations far from the leak, the % error is small, ranging between 1.0 and 3.0%. The magnitude of error can be linked to the fact that leaking cracks causes eddies/instabilities to form in the flow field, which affects velocity measurements and, in turn leads to a higher level of measurement errors.

#### 3.3. Pipe Geometry, Crack Geometry & Flow Conditions

^{2}. Table 3 shows the dimensions of the two pipes under study, and Table 4 shows the fluid flow conditions in each pipe. Table 5 shows the dimensions of the cracks formed at the walls of the pipes.

^{3}/s flow rate & 20 × 10

^{3}Pa pressure) conditions with, Figure 17 and Figure 18.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**(

**a**) Velocity contours in the longitudinal direction along the pipe, and (

**b**) near leak area.

**Figure 13.**Percentage of error between experimentally and numerically obtained speeds of flow along the pipe.

**Figure 15.**Pressure (

**a**) and Velocity (

**b**) distributions along the 5.08 cm diameter pipe, with flow condition 1.

**Figure 16.**Pressure (

**a**) and Velocity (

**b**) distributions along the 5.08 cm diameter pipe, with flow condition 2.

**Figure 17.**Pressure (

**a**) and Velocity (

**b**) distributions along the 10.16 cm diameter pipe, with flow condition 1.

**Figure 18.**Pressure (

**a**) and Velocity (

**b**) distributions along the 10.16 cm diameter pipe, with flow condition 2.

**Figure 19.**Pressure (

**a**) and Velocity (

**b**) distributions along the pipe, with radius of “0.708 cm, 0.508 cm, and 0.308 cm”.

Sample No. | Flow Rate (cm^{3}/s) |
---|---|

1 | 1.5 |

2 | 3 |

3 | 5.4 |

4 | 1.2 |

5 | 1 |

The Flow Rate through Leak (dia 1.016 cm) (cm ^{3}/s) | Point 1 (50 cm) | Point 2 (83 cm) | Point 3 (89 cm) | Point 4 (96 cm) | Point 5 (127 cm) | |||||
---|---|---|---|---|---|---|---|---|---|---|

Test. (cm/s) | Num. (cm/s) | Test. (cm/s) | Num. (cm/s) | Test. (cm/s) | Num. (cm/s) | Test. (cm/s) | Num. (cm/s) | Test. (cm/s) | Num. (cm/s) | |

1.5 | 23.1 | 23.4 | 23.5 | 23.9 | 25.5 | 24.9 | 23.4 | 23.2 | 23.6 | 23.9 |

3 | 21.6 | 22.0 | 22.4 | 21.9 | 25.1 | 25.9 | 21.1 | 20.8 | 20.1 | 19.9 |

5.4 | 21.7 | 21.1 | 25.1 | 24.3 | 28.1 | 29.2 | 24.6 | 23.6 | 19.2 | 18.7 |

12 | 22.4 | 23.0 | 24.4 | 25.5 | 28.9 | 28.1 | 25.0 | 24.5 | 22.7 | 23.3 |

Length (cm) | Diameter (cm) | Sample No. |
---|---|---|

177 | 5.08 | 1 |

177 | 10.16 | 2 |

Condition 2 | Condition 1 |
---|---|

Flow Rate = 5 × 10^{2} cm^{3}/s | Velocity = 2 × 10^{2} cm/s |

Pressure = 20 × 10^{3} Pa | Pressure = 20 × 10^{4} Pa |

Dimensions | Crack Shape | Sample No. |
---|---|---|

0.508 cm Radius | Circular | 1 |

0.9 cm × 0.9 cm | Square | 2 |

2.54 cm × 0.32 cm | Slot | 3 |

Flow Conditions 1 (Pipe Diameter 5.08 cm) | Flow Conditions 2 (Pipe Diameter 5.08 cm) | Flow Conditions 1 (Pipe Diameter 10.16 cm) | Flow Conditions 2 (Pipe Diameter 10.16 cm) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Pressure (Pa) | Velocity (m/s) | Pressure (Pa) | Velocity (m/s) | Pressure (Pa) | Velocity (m/s) | Pressure (Pa) | Velocity (m/s) | |||||

Min | Max | Max | Min | Max | Max | Min | Max | Max | Min | Max | Max | |

Circular | 200459.3 | 200639 | 25.808 | 20015.91 | 20009.37 | 27.808 | 200167.9 | 200333.9 | 25.08 | 199798.4 | 200151.1 | 28.079 |

Slot | 200490.7 | 200619 | 25.385 | 20015.85 | 20009.41 | 27.385 | 200152.1 | 200344.9 | 25.01 | 199835.5 | 200090 | 28.009 |

Square | 200509.5 | 200621 | 26.4 | 20015.15 | 20013.44 | 28.4 | 200235.5 | 200295.7 | 24.36 | 199956.5 | 20016.7 | 27.364 |

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**MDPI and ACS Style**

Ali, S.; Hawwa, M.A.; Baroudi, U.
Effect of Leak Geometry on Water Characteristics Inside Pipes. *Sustainability* **2022**, *14*, 5224.
https://doi.org/10.3390/su14095224

**AMA Style**

Ali S, Hawwa MA, Baroudi U.
Effect of Leak Geometry on Water Characteristics Inside Pipes. *Sustainability*. 2022; 14(9):5224.
https://doi.org/10.3390/su14095224

**Chicago/Turabian Style**

Ali, Sajid, Muhammad A. Hawwa, and Uthman Baroudi.
2022. "Effect of Leak Geometry on Water Characteristics Inside Pipes" *Sustainability* 14, no. 9: 5224.
https://doi.org/10.3390/su14095224