# A Quick Survey of the Most Vulnerable Areas of a Water Distribution Network Due to Transients Generated in a Service Line: A Lagrangian Model Based on Laboratory Tests

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Experimental Setup

#### 2.2. Preliminary Tests for the End-User and Service Line Characterization

#### 2.3. Laboratory Transient Tests

#### 2.4. The Lagrangian Model (LM)

## 3. The Effect of the Network Topology

## 4. The Effect of the Transient Generation Point

## 5. Maps of Vulnerability by the Lagrangian Model (LM)

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Water Distribution Network (WDN) at the Water Engineering Laboratory of the University of Perugia: (

**a**) general layout with the indication of the pipe length and principal measurement sections and the service line installed at node 5 (test #1 configuration); (

**b**) end-user 5u; (

**c**) location of the measurement sections around junction 5; and (

**d**) junction 5 (as an example of the service line connection to the WDN).

**Figure 2.**(

**a**) Comparison of the geometric characteristics of the ball valve obtained at WEL and by [40]; (

**b**) hydraulic characteristics of the end-user.

**Figure 3.**Test #1—pressure signals acquired at measurement sections: (

**a**) 5u; (

**b**) 5, 5${}_{4}$, 5${}_{6}$, and 5${}_{8}$; (

**c**) 4, 6, and 8; (

**d**) 7, 32, and 1. Note that to highlight the pressure variations into the network, the y-axis of (

**b**–

**d**) is significantly reduced with respect to the one at 5u (

**a**).

**Figure 4.**Test #1—one of the paths of the generated pressure wave of Figure 3 in the network—pressure signals at measurement sections: (

**a**) 5u; (

**b**) 5; (

**c**) 5${}_{4}$; (

**d**) 45; (

**e**) 4; (

**f**) 47; and (

**g**) 7.

**Figure 5.**Test #1—experimental pressure disturbances, $H-{H}_{0}$, vs. the impulse response function, $\Delta {H}_{n}$ (blue stems), and numerical reconstruction, ${H}_{n}-{H}_{n,0}$ (red line), carried out by the Lagrangian model (LM) at measurement sections: (

**a**) 5u; (

**b**) 5; (

**c**) 4; (

**d**) 6; (

**e**) 8; (

**f**) 7; and (

**g**) 32.

**Figure 6.**Pressure signals acquired during tests #1 (blue lines), #2 (black lines), and #3 (red lines) with ${Q}_{0,5u}$ = ${Q}_{0,6u}$ = ${Q}_{0,7u}$ = 0.12 L/s at measurement sections: (

**a**) end-user (5u/6u/7u); (

**b**) 32; (

**c**) 4; (

**d**) 5; (

**e**) 6; (

**f**) 7; and (

**g**) 8.

**Figure 7.**Tests #2, and #3—experimental pressure disturbances, $H-{H}_{0}$, vs. the impulse response function, $\Delta {H}_{n}$, (blue stems) and numerical reconstruction, ${H}_{n}-{H}_{n,0}$, (red line) carried out by the Lagrangian model (LM) at measurement sections: (

**a**) 6u; (

**b**) 6; (

**c**) 5; (

**d**) 4; (

**e**) 7; (

**f**) 8; and (

**g**) 32, for test #2, and (

**h**) 7u; (

**i**) 7; (

**j**) 5; (

**k**) 4; (

**l**) 6; (

**m**) 8; and (

**n**) 32, for test #3.

**Figure 8.**Frequency distribution of the relative amplitude of the pressure waves given by the LM for test #1 at measurement sections: (

**a**) 5u and 5; (

**b**) 32; (

**c**) 4; (

**d**) 6; (

**e**) 7; (

**f**) 8.

**Figure 9.**Map of vulnerability evaluated by the LM for tests: (

**a**) #1; (

**b**) #2; and (

**c**) #3; respectively (numbers at nodes indicate the value of the index of vulnerability, V).

**Figure 10.**Map of vulnerability evaluated by the LM for a numerical test equivalent to test #1, but with a diameter distribution reversed with respect to the laboratory layout (numbers at nodes indicate the value of the index of vulnerability, V).

Test No. (#) | Layout | Maneuver Type | Discharge [L/s] |
---|---|---|---|

1 | service line connected at node 5 | Total closure of 5u | ${Q}_{0,5u}$ = 0.12 |

2 | service line connected at node 6 | Total closure of 6u | ${Q}_{0,6u}$ = 0.12 |

3 | service line connected at node 7 | Total closure of 7u | ${Q}_{0,7u}$ = 0.12 |

**Table 2.**The reflection and transmission coefficients given by the Lagrangian model (LM) for the singularities of the network.

Singularity | Reflection Coefficient, ${\mathit{C}}_{\mathit{R}}$ | Transmission Coefficient, ${\mathit{C}}_{\mathit{T}}$ |
---|---|---|

constant head reservoir | −1 | 0 |

closed valve/dead end | 1 | 0 |

junction | $\frac{{A}_{j}/{a}_{j}-\left({\sum}_{\begin{array}{c}i=1\\ i\ne j\end{array}}^{n}{A}_{i}/{a}_{i}\right)}{{\sum}_{i=1}^{n}{A}_{i}/{a}_{i}}$ | $\frac{2{A}_{j}/{a}_{j}}{{\sum}_{i=1}^{n}{A}_{i}/{a}_{i}}$ |

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**MDPI and ACS Style**

Meniconi, S.; Maietta, F.; Alvisi, S.; Capponi, C.; Marsili, V.; Franchini, M.; Brunone, B. A Quick Survey of the Most Vulnerable Areas of a Water Distribution Network Due to Transients Generated in a Service Line: A Lagrangian Model Based on Laboratory Tests. *Water* **2022**, *14*, 2741.
https://doi.org/10.3390/w14172741

**AMA Style**

Meniconi S, Maietta F, Alvisi S, Capponi C, Marsili V, Franchini M, Brunone B. A Quick Survey of the Most Vulnerable Areas of a Water Distribution Network Due to Transients Generated in a Service Line: A Lagrangian Model Based on Laboratory Tests. *Water*. 2022; 14(17):2741.
https://doi.org/10.3390/w14172741

**Chicago/Turabian Style**

Meniconi, Silvia, Filomena Maietta, Stefano Alvisi, Caterina Capponi, Valentina Marsili, Marco Franchini, and Bruno Brunone. 2022. "A Quick Survey of the Most Vulnerable Areas of a Water Distribution Network Due to Transients Generated in a Service Line: A Lagrangian Model Based on Laboratory Tests" *Water* 14, no. 17: 2741.
https://doi.org/10.3390/w14172741