# Application of Newton–Raphson Method for Computing the Final Air–Water Interface Location in a Pipe Water Filling

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Approach

#### 2.1. Filling Operation

#### 2.2. Governing Equations

- Air–water interface position is given by:$$\frac{dL}{dt}=v$$
- Polytropic model: this formulation describes how the air pocket size changes with air pocket pressure pulses. These changes are described by the polytropic law (see Equation (3)), which is applied without injected air into hydraulic installations.

^{2}); $f$ = friction factor; and $D$ = internal pipe diameter; $L$ = length of a water filling column; ${R}_{v}$ = resistance coefficient of a valve (s

^{2}/m

^{5}); and $A$ = cross-sectional area of a single pipe (m

^{2}).

#### 2.3. Initial Conditions

#### 2.4. Velocity Field

#### 2.5. Computation of Critical Point

_{f}). An isothermal process is characterized by a slow transient event; thus, a polytropic coefficient of k = 1.0 can be used to compute the value of ${L}_{f}$ (see Equation (9)):

## 3. Analysis of Results and Discussion

#### 3.1. Case Study

^{2}/m

^{5}, and an initial pressure supplied by a tank of 202,650 Pa. Figure 3 shows the evolution of the main hydraulic and thermodynamic variables over time for the analyzed case study. Figure 3a presents that the maximum water velocity occurs at 10.7 s with a value of 5.34 m/s, while the minimum value reached was −0.76 m/s (at 130.0 s), indicating a reverse water flow. After that, many oscillations occur, since the system trends to a resting position ($v$ = 0) at the end of the hydraulic event. The pipe is filled rapidly by the water column since it takes only 130.0 s to reach a maximum value of 390.9 m. After that, the air–water interface exhibits many pulses moving around its final position (384.42 m). Figure 3b presents the evolution of air pocket pulses for the analyzed case study. The air pocket starts at atmospheric condition (101,325 Pa). The air pocket pressure head reached a peak value of 31.1 m (at 110.6 s) which was provoked by the compression of the filling water column. A similar trend is detected with regard to air pocket pressure oscillations, where the system finished with a value of 28.3 m.

#### 3.2. Final Position of Air–Water Interface

#### 3.3. Sensitivity Analysis

#### 3.4. Experimental Validation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## 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_{0*} | Initial absolute pressure supplied by an energy source (Pa) |

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

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

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 |

$\mathsf{\rho}$ | 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 |

s | Specific experimental test |

Superscripts | |

‘ | Derivative |

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**Figure 1.**A filling process configuration: (

**a**) hydraulic system at rest (t = 0); and (

**b**) hydraulic system at $t\ne 0$.

**Figure 3.**Evolution of hydraulic and thermodynamic variables: (

**a**) water velocity and water column length; and (

**b**) air pocket pressure.

**Figure 6.**Sensitivity analysis for dependent parameters: (

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

**b**) pipe slope ($\theta $ ); (

**c**) air pocket size (${x}_{0}$ ); and (

**d**) initial absolute pressure (${p}_{0}^{*}$ ).

**Figure 7.**Sensitivity analysis for non-dependent parameters: (

**a**) internal pipe diameter ($D$); (

**b**) resistance coefficient of a regulating valve (${R}_{v}$ ); and (

**c**) friction factor ($f$ ).

**Figure 9.**Evolution of the analyzed variables for the experimental facility: (

**a**) air pocket pressure; and (

**b**) water column lengths.

$\mathit{i}$ | ${\mathit{L}}_{\mathit{i}}\left(\mathbf{m}\right)$ | $\mathit{j}\left(\mathit{y}\right)$ | $\mathit{j}\mathbf{\prime}\left(\mathit{y}\right)$ | ${\mathit{L}}_{\mathit{i}+1}\left(\mathbf{m}\right)$ | ${\mathit{L}}_{\mathit{i}+1}-{\mathit{L}}_{\mathit{i}}\left(\mathbf{m}\right)$ |
---|---|---|---|---|---|

0 | 422.58 | −0.15559 | −0.00479 | 390.10 | −32.48 |

1 | 390.10 | −0.02036 | −0.00365 | 384.53 | −5.57 |

2 | 384.53 | −0.00038 | −0.00352 | 384.42 | −0.11 |

3 | 384.42 | 0.00000 | −0.00352 | 384.42 | 0.00 |

Test No. | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

x_{0} (m) | 0.46 | 0.96 | 1.36 | 0.46 | 0.96 | 1.36 |

${p}_{0}^{*}$ (bar) | 1.75 | 1.75 | 1.75 | 2.25 | 2.25 | 2.25 |

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**MDPI and ACS Style**

Bonilla-Correa, D.M.; Coronado-Hernández, Ó.E.; Fuertes-Miquel, V.S.; Besharat, M.; Ramos, H.M.
Application of Newton–Raphson Method for Computing the Final Air–Water Interface Location in a Pipe Water Filling. *Water* **2023**, *15*, 1304.
https://doi.org/10.3390/w15071304

**AMA Style**

Bonilla-Correa DM, Coronado-Hernández ÓE, Fuertes-Miquel VS, Besharat M, Ramos HM.
Application of Newton–Raphson Method for Computing the Final Air–Water Interface Location in a Pipe Water Filling. *Water*. 2023; 15(7):1304.
https://doi.org/10.3390/w15071304

**Chicago/Turabian Style**

Bonilla-Correa, Dalia M., Óscar E. Coronado-Hernández, Vicente S. Fuertes-Miquel, Mohsen Besharat, and Helena M. Ramos.
2023. "Application of Newton–Raphson Method for Computing the Final Air–Water Interface Location in a Pipe Water Filling" *Water* 15, no. 7: 1304.
https://doi.org/10.3390/w15071304