# Detecting Optimal Leak Locations Using Homotopy Analysis Method for Isothermal Hydrogen-Natural Gas Mixture in an Inclined Pipeline

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

#### 2.1. Governing Equation

- R: is the specific gas constant.
- T: is temperature.

- ${C}_{v}$: is the specific heat at constant volume.
- ${C}_{p}$: is the specific heat at constant pressure.
- R: is the specific gas constant.
- P: is pressure.
- $\gamma $: is the flow process index.

#### 2.2. Hydrogen-Natural Gas Mixture Equation

## 3. Homotopy Analysis Method

#### 3.1. Solving the Steady State Equations by High-Order Deformation HAM

#### 3.2. Solving Isothermal Flow of Hydrogen-Natural Gas Mixture by HAM

#### 3.3. Results and Discussion

#### 3.4. Leak Detection Using Homotopy Analysis Method

#### 3.5. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Comparison between homotopy analysis method of orders $M=5,12$ for $\hslash =-0.1$; with Subani et al., 2017 and Elaoud et al., 2010 methods.

**Figure 5.**Transient pressure of hydrogen natural gas mixture for isothermal flow when leakage occurs at ${X}_{L}=L/3$ in horizontal pipeline when $\varphi =0.25$ and $\varphi =0.5$.

**Figure 6.**Transient pressure of hydrogen natural gas mixture for isothermal flow when leakage occurs at ${X}_{L}=L/3$ in an inclined pipeline when $\theta ={15}^{\circ}$ and $\varphi =0.5$.

**Figure 7.**Transient pressure of hydrogen natural gas mixture with $\varphi =0.5$ for isothermal flow when leakage occurs at ${X}_{L}=L/3$ with different angles $\theta $. HAM with order 12 and $\hslash =-0.5$.

**Figure 8.**Celerity wave of hydrogen natural gas mixture for isothermal flow when leakage occurs at ${X}_{L}=L/3$ in horizontal pipeline with different mass ratio $\varphi $.

**Table 1.**Hydrogen properties in working conditions, $P=35$ bar and $T=15{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C = 288 K (See [7]).

Symbol | Fluid Properties | Values (J/kgK) | |
---|---|---|---|

Hydrogen | Natural Gas | ||

${C}_{p}$ | Specific heat at constant pressure | 14,600 | 1497.5 |

${C}_{v}$ | Specific heat at constant volume | 10,440 | 1056.8 |

R | Gas constant | 4160 | 440.7 |

**Table 2.**Parameters used for the simulation (See [7]).

Symbols | Values | Symbols | Values |
---|---|---|---|

Pipe length | L = 600 m | Mass ratio | $\varphi =0,0.5,1$ |

Time | t = 20 | Angle | $\theta =0,\pi /6,\pi /4,\pi /3$ |

Pipe diameter | D = $0.4$ m | Mass flow | ${Q}_{0}$ = 55 kg/s |

Friction coefficient | $f=0.03$ | Absolute pressure | ${P}_{0}$ = 35 bar |

Temperature | $T=15{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C = 288 K |

**Table 3.**Leak location for the hydrogen-natural gas mixture for isothermal flow at leakage ${X}_{L}=L/3$.

Gas Mixture $\left(\mathit{\varphi}\right)$ | Pipeline’s Angle $\left(\mathit{\theta}\right)$ | Leak Location (m) | |
---|---|---|---|

Subani et al., Method | HAM | ||

0 | ${0}^{\circ}$ | 439.4 | 439.8 |

${15}^{\circ}$ | 439.4 | 439.8 | |

0.25 | ${0}^{\circ}$ | 268.3 | 269.04 |

${15}^{\circ}$ | 268.3 | 269.04 | |

0.5 | ${0}^{\circ}$ | 211.1 | 210.3 |

${15}^{\circ}$ | 211.1 | 210.3 | |

0.75 | ${0}^{\circ}$ | 160.6 | 161.01 |

${15}^{\circ}$ | 160.6 | 161.01 | |

1 | ${0}^{\circ}$ | 95.8 | 96.2 |

${15}^{\circ}$ | 95.8 | 96.2 |

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**MDPI and ACS Style**

S. Chaharborj, S.; Ismail, Z.; Amin, N.
Detecting Optimal Leak Locations Using Homotopy Analysis Method for Isothermal Hydrogen-Natural Gas Mixture in an Inclined Pipeline. *Symmetry* **2020**, *12*, 1769.
https://doi.org/10.3390/sym12111769

**AMA Style**

S. Chaharborj S, Ismail Z, Amin N.
Detecting Optimal Leak Locations Using Homotopy Analysis Method for Isothermal Hydrogen-Natural Gas Mixture in an Inclined Pipeline. *Symmetry*. 2020; 12(11):1769.
https://doi.org/10.3390/sym12111769

**Chicago/Turabian Style**

S. Chaharborj, Sarkhosh, Zuhaila Ismail, and Norsarahaida Amin.
2020. "Detecting Optimal Leak Locations Using Homotopy Analysis Method for Isothermal Hydrogen-Natural Gas Mixture in an Inclined Pipeline" *Symmetry* 12, no. 11: 1769.
https://doi.org/10.3390/sym12111769