# An Implicit Formulation for Calculating Final Conditions in Drainage Maneuvers in Pressurized Water Installations

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Approach

_{1}), water velocity (v), and water column length (L) during drainage maneuvers in pressurized water installations was developed by the authors in previous publications [4]. A system of 3 × 3 algebraic and ordinary differential equations needs to be solved to obtain the responses of the aforementioned variables. Figure 1 describes the hydraulic and thermodynamic variables for the initial and final conditions during an emptying process without admitted air. Air pocket size is represented by x. The subscript 0 indicates the initial condition of an analyzed variable, and the subscript f is used to represent the final condition. Figure 1a shows the situation when a system is at rest, while Figure 1b presents the final position of an emptying process.

_{T}and an internal pipe diameter D can be analyzed using the formulations as follows:

_{1}= air pocket pressure, p

_{atm}

^{*}= atmospheric pressure, r = water density, g = gravity acceleration, f = the Darcy–Weisbach friction factor, θ = pipe slope (rad), A = cross-sectional area of pipe, R

_{v}= resistance coefficient, and k = polytropic coefficient.

_{T}-x

_{0}, and p

_{1}*(0) = p

_{atm}* = 101,325 Pa.

_{1}*). Commonly, mathematical packages (e.g., Matlab, Scilab, or Octave) are utilized to compute the aforementioned variables. Equations (1)–(3) correspond with the mathematical model developed by the authors in previous works [4,23].

_{T}-x

_{0}, 0) at time t = 0.

_{f}value, the initial value of ${L}_{f,0}$ is found for the method,

_{f}can be found with few iterations. The definition of $j\left({L}_{f,i}\right)$ is given by:

_{f,i}) is not null, when considering seed values ranging from 0 to L

_{T}-x

_{0}. The value L

_{i+1}can be computed by considering the following formulation:

## 3. Analysis of Results

#### 3.1. Practical Application

_{T}= 600 m, f = 0.018, D = 0.35 m, R

_{v}= 0.06 ms

^{2}m

^{−6}, x

_{0}= 200 m, k = 1.2, g = 9.81 ms

^{−2}, r = 1000 kg/m

^{−3}, p*

_{atm}= 101,325 Pa, and θ = 0.025 rad. Figure 3 presents the evolution of length and water velocity for the case study, which was obtained by solving the differential ordinary system composed by Equations (1)–(3). The numerical resolution of these equations was implemented by the authors in previous publications. At the beginning of the transient event, a maximum water velocity value of 2.66 m/s was attained at 25 s. A minimum value of −0.62 m/s was found at 160 s. After that, water velocity pulses were observed. From 3000 to 5000 s, the water velocity tended to be null (v = 0), which corresponded to the final water position during the transient event. The length of the water column rapidly decreased, passing from 400.0 m (at 0 s) to 202.9 m (at 124 s), which implied that almost half of the water column was drained. Then, some oscillations were detected, tending to a final value of 221.2 m (from 3000 to 5000 s).

^{−17}, which is practically null). According to the results, the Newton–Raphson method can converge rapidly (within four iterations) to find the required value. In this sense, the first iteration provided a value of 220.16 m, the second one found a value of 221.19 m, and from the third iteration, the value remained practically constant (221.20 m). The remaining iterations provided the same value. Figure 5 shows the results of the application of the Newton–Raphson method, which confirms how the convergence process is rapidly carried out. It shows how, from iteration three, there was no variation in the computation of L

_{f}.

#### 3.2. Validation

_{p}= 0.2 m). A pressure transducer was installed at the upstream end, and a scour valve (ball valve) was installed at the downstream end to control the variation of water flow.

_{p}represents the vertical pipe length (0.2 m).

_{0}) of 0.205 m and a pipe slope (θ) of 0.515 rad; and (ii) Test No. 2, where an x

_{0}value of 0.45 m and an θ value of 0.457 rad were considered. Programmed maneuvers were used with maximum resistance coefficients (R

_{v}) of 30.86 × 10

^{6}and 14.79 × 10

^{6}ms

^{2}m

^{−6}for Tests No. 1 and No. 2, respectively, in the ball valve. Each test was repeated twice to confirm the measurements. Figure 7 shows the results of the mathematical model (see Equations (1)–(3)). The final air pocket pressure was computed by considering both Equations (3) and (19). The measurements and the implicit formulation provided air pocket pressure heads of 8.22 and 8.54 m for Test No. 1 and No. 2, respectively, at the end of the transient event (at 6 s). This shows that the implicit formulation is suitable for computing final conditions, since it is based on a physical equation.

## 4. Sensitivity Analysis

_{v})), or the friction factor (f), as shown in Equation (11). In this sense, a sensitivity analysis was performed based on the case study of Section 3. For the analysis, the internal pipe diameter varied from 0.10 to 0.70 m; the resistance coefficient ranged between 0.03 and 1000 ms

^{2}m

^{−6}; and the friction factor varied from 0.010 to 0.026. Figure 6 presents the results of emptying simulations considering the variation of the mentioned parameters. Figure 8a–c shows that the final position of the water column length had no variation, independently of the used initial values of D, R

_{v}, and f. For all cases, a final value of 221.20 m was found.

_{0}, θ, and L

_{T}according to Equation (11). Figure 9 presents the results for the mentioned parameters. Polytropic coefficients from isothermal (k = 1.0) to adiabatic (k = 1.4) were considered for simulations. The greater the polytropic coefficient, the lower the obtained value of the length of the water column (see Figure 9a). Adiabatic behavior always provides lower values of water column length compared to an isothermal evolution. The initial value of air pocket size (x

_{0}) influences the final position of water column length, as shown in Figure 9b. In this sense, considering an initial air pocket size of 100 m, a final value of 302.1 m for the water column length was computed; while, for an initial air pocket size of 500 m, the final position of the water column length was 47.1 m. The pipe slope (θ) is another parameter that is important to analyze for detecting changes regarding the final position of the water column length. The higher the pipe slope, the lower the attained values of the water column length. The total length (L

_{T}) of the pipeline was analyzed considering values between 550 and 650 m. The greater the value of the total length, the greater the reached values of the final water column length.

_{T}-x

_{0}(0 to 400 m). If an isothermal evolution (k = 1.0) is presented, then the final position of the water column length can be computed using Equation (14). For other polytropic coefficients ($1.0<k\le 1.4)$, Equation (14) can be used for a seed value. A sensitivity analysis was performed using seed values from 0 to 400 m (see Figure 10). For all seed values, the Newton–Raphson method is suitable to compute the final position of the water column length (221.20 m). The seed value computed with Equation (14) (SV = 204.33 m) was the best starting point to calculate the final position of the water column length.

## 5. Conclusions

_{T}-x

_{0}. For the case study, the Newton–Raphson method provided excellent results, independently of a selected seed value. Seed values varying from 0 to 400 m were used, obtaining the searched root with a maximum of four iterations.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

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

D | Internal pipe diameter (m) |

f | Friction factor (-) |

g | Gravity acceleration (m s^{−2}) |

k | Polytropic coefficient (-) |

j | Used function in the Newton–Raphson equation |

L | Water column length (m) |

L_{T} | Pipe length (m) |

P | Function based on piston flow model |

p_{atm}^{*} | Atmospheric pressure (101,325 Pa) |

p^{*}_{1} | Air pocket pressure (Pa) |

Q | Function based on both mass oscillation equation and polytropic law |

R_{v} | Resistance coefficient (ms^{2} m^{−6}) |

t | Time (s) |

v | Water velocity (m s^{−1}) |

V | Vector field on a region in the plane (L, v) |

x | Air pocket size (m) |

X | Vector function of t |

r | Water density (kg m^{−3}) |

θ | Pipe slope (rad) |

γ_{w} | Water unit weight (N m^{−3}) |

Subscripts | |

0 | Refers to an initial condition |

f | Refers to a final condition |

i | Iteration number |

Superscripts | |

′ | Derivative |

## References

- Ramezani, L.; Karney, B.; Malekpour, A. The Challenge of Air Valves: A Selective Critical Literature Review. J. Water Resour. Plan. Manag.
**2016**, 141, 04015017. [Google Scholar] [CrossRef] - Fuertes-Miquel, V.S.; Coronado-Hernández, Ó.E.; Mora-Melia, D.; Iglesias-Rey, P.L. Hydraulic Modeling during Filling and Emptying Processes in Pressurized Pipelines: A Literature Review. Urban Water J.
**2019**, 16, 299–311. [Google Scholar] [CrossRef] - AWWA (American Water Works Association). Manual of Water Supply Practices M51—Air Valves: Air Release, Air/Vacuum and Combination; AWWA: Denver, CO, USA, 2016. [Google Scholar]
- Coronado-Hernández, O.E.; Fuertes-Miquel, V.S.; Besharat, M.; Ramos, H.M. Subatmospheric Pressure in a Water Draining Pipeline with an Air Pocket. Urban Water J.
**2018**, 15, 346–352. [Google Scholar] [CrossRef] - Coronado-Hernández, O.E.; Fuertes-Miquel, V.S.; Iglesias-Rey, P.L.; Martínez-Solano, F.J. Rigid Water Column Model for Simulating the Emptying Process in a Pipeline Using Pressurized Air. J. Hydraul. Eng.
**2018**, 144, 06018004. [Google Scholar] [CrossRef] - Coronado-Hernández, Ó.E.; Fuertes-Miquel, V.S.; Quiñones-Bolaños, E.E.; Gatica, G.; Coronado-Hernández, J.R. Simplified Mathematical Model for Computing Draining Operations in Pipelines of Undulating Profiles with Vacuum Air Valves. Water
**2020**, 12, 2544. [Google Scholar] [CrossRef] - Laanearu, J.; Annus, I.; Koppel, T.; Bergant, A.; Vučkovič, S.; Hou, Q.; van’t Westende, J.M.C. Emptying of Large-Scale Pipeline by Pressurized Air. J. Hydraul. Eng.
**2012**, 138, 1090–1100. [Google Scholar] [CrossRef] - Tijsseling, A.; Hou, Q.; Bozku¸s, Z.; Laanearu, J. Improved One-Dimensional Models for Rapid Emptying and Filling of Pipelines. J. Press. Vessel Technol.
**2016**, 138, 031301. [Google Scholar] [CrossRef] - Besharat, M.; Coronado-Hernández, O.E.; Fuertes-Miquel, V.S.; Viseu, M.T.; Ramos, H.M. Backflow Air and Pressure Analysis in Emptying Pipeline Containing Entrapped Air Pocket. Urban Water J.
**2018**, 15, 769–779. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Besharat, M.; Coronado-Hernández, O.E.; Fuertes-Miquel, V.S.; Viseu, M.T.; Ramos, H.M. Computational fluid dynamics for sub-atmospheric pressure analysis in pipe drainage. J. Hydraul. Res.
**2020**, 58, 553–565. [Google Scholar] [CrossRef] - Liou, C.P.; Hunt, W.A. Filling of pipelines with undulating elevation profiles. J. Hydraul. Eng.
**1996**, 122, 534–539. [Google Scholar] [CrossRef] - Izquierdo, J.; Fuertes, V.; Cabrera, E.; Iglesias, P.; Garcia-Serra, J. Pipeline start-up with entrapped air. J. Hydraul. Res.
**1999**, 37, 579–590. [Google Scholar] [CrossRef] - Martins, N.M.; Delgado, J.N.; Ramos, H.M.; Covas, D.I. Maximum transient pressures in a rapidly filling pipeline with entrapped air using a CFD model. J. Hydraul. Res.
**2017**, 55, 506–519. [Google Scholar] [CrossRef] - Zhou, L.; Liu, D. Experimental Investigation of Entrapped Air Pocket in a Partially Full Water Pipe. J. Hydraul. Res.
**2013**, 51, 469–474. [Google Scholar] [CrossRef] - 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] - Fuertes-Miquel, V.S.; López-Jiménez, P.A.; Martínez-Solano, F.J.; López-Patiño, G. Numerical modelling of pipelines with air pockets and air valves. Can. J. Civ. Eng.
**2016**, 43, 1052–1061. [Google Scholar] [CrossRef] - Chapra, S.; Canale, R. Numerical Methods for Engineers, 7th ed.; Mcgraw-Hill Education, Cop: New York, NY, USA, 2015. [Google Scholar]
- Stoer, J.; Bulirsch, R. Introduction to Numerical Analysis; Springer: New York, NY, USA; London, UK, 2011. [Google Scholar]
- Zill, D. Differential Equations with Boundary-Value Problems; Cengage Learning: Melbourne, Australia, 2016. [Google Scholar]
- Suribabu, C.R. Location and Sizing of Scour Valves in Water Distribution Network. J. Hydraul. Eng.
**2009**, 15, 118–130. [Google Scholar] [CrossRef] - Suribabu, C.R. Optimal locations and sizing of scour valves in water distribution networks. J. Pipeline Syst. Eng.
**2020**, 11, 04019056. [Google Scholar] [CrossRef] - Fuertes-Miquel, V.S.; Coronado-Hernández, Ó.E.; Mora-Melia, D.; Iglesias-Rey, P.L. Transient phenomena during the emptying process of a single pipe with water–air interaction. J. Hydraul. Res.
**2018**, 57, 3. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of an emptying process with a trapped air pocket in a single water pipeline: (

**a**) initial condition and (

**b**) final condition.

**Figure 7.**Comparison between measured and computed (final condition) air pocket pressure: (

**a**) Test No. 1 and (

**b**) Test No. 2.

**Figure 8.**Sensitivity analysis for non-dependent parameters: (

**a**) internal pipe diameter; (

**b**) resistance coefficient; and (

**c**) friction factor.

**Figure 9.**Sensitivity analysis for dependent parameters: (

**a**) polytropic coefficient (k); (

**b**) initial air pocket size (x

_{0}); (

**c**) slope pipe (θ); and (

**d**) total pipe length (L

_{T}).

i | L_{f,i} (m) | J (L_{f,i}) | j′ (L_{f,i}) | L_{f,i}_{+1} (m) | L_{f,i}_{+1} − L_{f,i} (m) |
---|---|---|---|---|---|

0 | 204.33 | −0.03197 | 0.00202 | 220.16 | 15.83 |

1 | 220.16 | −0.00185 | 0.00180 | 221.19 | 1.03 |

2 | 221.19 | −0.00001 | 0.00178 | 221.20 | 0.00 |

3 | 221.20 | 0.00000 | 0.00178 | 221.20 | 0.00 |

… | … | … | … | … | … |

18 | 221.20 | 0.00000 | 0.00178 | 221.20 | 0.00 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Coronado-Hernández, O.E.; Bonilla-Correa, D.M.; Lovo, A.; Fuertes-Miquel, V.S.; Gatica, G.; Linfati, R.; Coronado-Hernández, J.R. An Implicit Formulation for Calculating Final Conditions in Drainage Maneuvers in Pressurized Water Installations. *Water* **2022**, *14*, 3364.
https://doi.org/10.3390/w14213364

**AMA Style**

Coronado-Hernández OE, Bonilla-Correa DM, Lovo A, Fuertes-Miquel VS, Gatica G, Linfati R, Coronado-Hernández JR. An Implicit Formulation for Calculating Final Conditions in Drainage Maneuvers in Pressurized Water Installations. *Water*. 2022; 14(21):3364.
https://doi.org/10.3390/w14213364

**Chicago/Turabian Style**

Coronado-Hernández, Oscar E., Dalia M. Bonilla-Correa, Aldo Lovo, Vicente S. Fuertes-Miquel, Gustavo Gatica, Rodrigo Linfati, and Jairo R. Coronado-Hernández. 2022. "An Implicit Formulation for Calculating Final Conditions in Drainage Maneuvers in Pressurized Water Installations" *Water* 14, no. 21: 3364.
https://doi.org/10.3390/w14213364