Application of Newton–Raphson Method for Computing the Final Air–Water Interface Location in a Pipe Water Filling
Abstract
:1. Introduction
2. Numerical Approach
2.1. Filling Operation
2.2. Governing Equations
- Air–water interface position is given by:
- 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.
2.3. Initial Conditions
2.4. Velocity Field
2.5. Computation of Critical Point
3. Analysis of Results and Discussion
3.1. Case Study
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 (m2) |
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) |
LT | Pipe length (m) |
p0* | Initial absolute pressure supplied by an energy source (Pa) |
patm* | Atmospheric pressure (101,325 Pa) |
p*1 | Air pocket pressure (Pa) |
Rv | Resistance coefficient (ms2m−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 |
Water density (kg m−3) | |
θ | Pipe slope (rad) |
γw | 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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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 |
---|---|---|---|---|---|---|
x0 (m) | 0.46 | 0.96 | 1.36 | 0.46 | 0.96 | 1.36 |
(bar) | 1.75 | 1.75 | 1.75 | 2.25 | 2.25 | 2.25 |
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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
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 StyleBonilla-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