# Improved Adaptive Time Step Method for Natural Gas Pipeline Transient Simulation

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## Abstract

**:**

## 1. Introduction

## 2. Implicit Finite Difference Method

#### 2.1. Governing Equations

#### 2.2. Discretization

#### 2.2.1. Hydraulic Equations

#### 2.2.2. Thermodynamic Equation

#### 2.2.3. Boundary Conditions

## 3. Improved Adaptive Time Step Method

#### 3.1. The Procedures of the Improved Adaptive Time Strategy

#### 3.2. The Energy Number

#### 3.3. Notes

- The value of $\theta $ is always greater than 1, because of the H211b controller. Equation (21) itself has the function of adjusting the time step. When ${\epsilon}_{t}^{j+1}\le TO{L}_{t}$, the values of ${\left(\frac{TO{L}_{t}}{{\epsilon}_{t}^{j+1}}\right)}^{{\beta}_{1}}$ and ${\left(\frac{TO{L}_{t}}{{\epsilon}_{t}^{j}}\right)}^{{\beta}_{2}}$ in Equation (21) are less than 1, and the time step of the next time layer will decrease, which also plays a role in adjusting the time step. Compared with the estimation method of $\Delta {t}^{j+1}=\frac{\Delta {t}^{j+1}}{2}$, the change in time step is relatively mild, which is the first point of improvement.
- Energy number, $\varphi $, is a synthetic parameter that needs to be calculated after the transient simulation of each time layer. It can directly judge whether the time step adjustment is appropriate, rather than changing the pipe network state [2], so as to reduce the calculation of transient simulation in the process of adjusting the time step and improve the efficiency of the time step adjustment, which is the second point of improvement.
- The time step should be in a suitable range to avoid time steps too small or large.
- The tolerable error should also be suitable. Referring to the adaptive simulation of the natural gas pipeline [22], the tolerable errors are set as $TO{L}_{t}^{P}={\Vert p\Vert}_{2}\times {10}^{-3}$, $TO{L}_{t}^{m}={\Vert m\Vert}_{2}\times {10}^{-1}$, and $TO{L}_{t}^{\varphi}=0.001$.

## 4. Results and Discussion

#### 4.1. The Virtual Pipeline

#### 4.1.1. Simulation Case

^{3}/h, 15 °C, and 3 MPa, respectively. The outlet flow rate changes suddenly from 0 Nm

^{3}/h to 1.0 × 10

^{5}Nm

^{3}/h at the beginning, jumps to 0.5 × 10

^{5}Nm

^{3}/h at the 24th hour, and then jumps to 0.1 × 10

^{5}Nm

^{3}/h at the 48th hour; one more time, the inlet temperature jumps from 15 °C to 30 °C at the beginning and jumps to 40 °C at the 24th hour, as shown in Figure 2. The inlet pressure remains 3 MPa during the entire 72 h.

#### 4.1.2. The Computational Accuracy

#### 4.1.3. The Effect

#### 4.1.4. The Efficiency

#### 4.2. The Actual Pipeline

#### 4.2.1. Simulation Case

#### 4.2.2. The Computational Accuracy

#### 4.2.3. The Effect

#### 4.2.4. The Efficiency

#### 4.3. Discussion

## 5. Conclusions

- High accuracy. In the virtual transient case, the accuracy of the results obtained by the improved time step method is almost the same as that in reference [22]. In addition, in the actual transient case, the pressure relative errors of the improved method at the last moment are only 0.184%.
- Acceptable effect and high efficiency. The improved time step method can not only adjust the time step according to the state change of the pipeline but can also consider the change in boundary conditions. When the boundary conditions change rapidly, the time step is adjusted more efficiently.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The result of the simulation. (

**a**) The flow rate at the inlet; (

**b**) The pressure at the outlet; (

**c**) The temperature at the outlet.

U | B | F |
---|---|---|

$\left[\begin{array}{l}p\\ m\end{array}\right]$ | $\left[\begin{array}{cc}0& \frac{1}{A}{\left(\frac{\partial p}{\partial \rho}\right)}_{T}\\ \left[A-\frac{{m}^{2}}{A{\rho}^{2}}{\left(\frac{\partial \rho}{\partial p}\right)}_{T}\right]& \frac{2m}{A\rho}\end{array}\right]$ | $\left[\begin{array}{l}{\left(\frac{\partial p}{\partial T}\right)}_{\rho}\frac{\partial T}{\partial t}-\frac{\lambda}{2}\frac{m\left|m\right|}{dA\rho}\\ -A\rho g\mathrm{sin}\theta +\frac{{m}^{2}}{A{\rho}^{2}}{\left(\frac{\partial \rho}{\partial T}\right)}_{p}\frac{\partial T}{\partial x}\end{array}\right]$ |

$T$ | $m/\left(\rho A\right)$ | $\frac{1}{\rho {c}_{v}}\left[\begin{array}{l}-T{\left(\frac{\partial p}{\partial T}\right)}_{\rho}\frac{\partial \left(m/\left(\rho A\right)\right)}{\partial x}\\ +\frac{\lambda}{2}\frac{\rho {\left|w\right|}^{3}}{d}-\frac{4K\left(T-{T}_{g}\right)}{D}\end{array}\right]$ |

Hydraulic Equations | Thermodynamic Equation | |
---|---|---|

Inlet of the pipe | $p=p\left(t\right)$$\text{}\mathrm{or}\text{}m=m\left(t\right)$ | $T=T\left(t\right)$ |

Outlet of the pipe | $p=p\left(t\right)$$\text{}\mathrm{or}\text{}m=m\left(t\right)$ | None |

Boundary Conditions | Energy Number |
---|---|

Pressure (Pa) | $\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$\rho g$}\right.$ |

Mass flow (kg/s) | $\raisebox{1ex}{${m}^{2}$}\!\left/ \!\raisebox{-1ex}{$2g$}\right.$ |

Temperature (K) | $\raisebox{1ex}{${c}_{v}T$}\!\left/ \!\raisebox{-1ex}{$g$}\right.$ |

Length | Diameter | Thickness | Roughness | Ground Temperature | Total Heat Transfer Coefficient |
---|---|---|---|---|---|

24 km | 323 mm | 8 mm | 0.02286 mm | 15 °C | 0.5 W/(m^{2}·K) |

CH_{4} | C_{2}H_{6} | C_{3}H_{8} | N_{2} | CO_{2} |
---|---|---|---|---|

97.07 | 0.17 | 0.02 | 0.71 | 2.03 |

CH_{4} | C_{2}H_{6} | C_{3}H_{8} | H_{2}S | CO_{2} |
---|---|---|---|---|

96.65 | 1.8 | 0.45 | 0.45 | 0.65 |

Total Number of Executions | The Number of Time Levels | The Difference | |
---|---|---|---|

Adaptive time step method | 2174 | 2008 | 166 |

Improved adaptive time step method | 1587 | 1507 | 80 |

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**MDPI and ACS Style**

Guo, Q.; Liu, Y.; Yang, Y.; Song, T.; Wang, S.
Improved Adaptive Time Step Method for Natural Gas Pipeline Transient Simulation. *Energies* **2022**, *15*, 4961.
https://doi.org/10.3390/en15144961

**AMA Style**

Guo Q, Liu Y, Yang Y, Song T, Wang S.
Improved Adaptive Time Step Method for Natural Gas Pipeline Transient Simulation. *Energies*. 2022; 15(14):4961.
https://doi.org/10.3390/en15144961

**Chicago/Turabian Style**

Guo, Qiao, Yuan Liu, Yunbo Yang, Tao Song, and Shouxi Wang.
2022. "Improved Adaptive Time Step Method for Natural Gas Pipeline Transient Simulation" *Energies* 15, no. 14: 4961.
https://doi.org/10.3390/en15144961