# Numerical Modeling of Space–Time Characteristics of Plasma Initialization in a Secondary Arc

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Governing Equations

^{−3}; D is the diffusion coefficient, in m

^{2}/s; f is the net rate of the generation and loss processes, in m

^{−3}s

^{−1}; and t represents time, in s. The main processes usually considered in Equations (1) and (3) are represented by their rates: Electron impact ionization, ${f}_{\mathrm{ion}}=\alpha {N}_{\mathrm{e}}{\mu}_{\mathrm{e}}E$; attachment of electrons to electronegative molecules (CO

_{2}, H

_{2}O, O

_{2}, etc.) present in air, ${f}_{\mathrm{att}}=\eta {N}_{\mathrm{e}}{\mu}_{\mathrm{e}}E$; detachment of electrons from negative ions, ${f}_{\mathrm{det}}={k}_{\mathrm{det}}{N}_{\mathrm{e}}{N}_{\mathrm{n}}$; electron-ion recombination, ${f}_{\mathrm{ep}}={\beta}_{\mathrm{ep}}{N}_{\mathrm{e}}{N}_{\mathrm{p}}$; recombination of positive and negative ions, ${f}_{\mathrm{pn}}={\beta}_{\mathrm{pn}}{N}_{\mathrm{p}}{N}_{\mathrm{n}}$; and natural background ionization, f

_{0}. In the expressions above, α is Townsend’s ionization coefficient, in m

^{−1}; μ is the mobility, in m

^{2}/Vs; E is the electric field strength, in V/m; η is the attachment coefficient, in m

^{−1}; k

_{det}is the detachment coefficient, in m

^{3}/s; and β is each respective recombination coefficient, in m

^{3}/s. Hence, the net rates for different charged species are:

_{0}is the vacuum permittivity, and ε

_{r}is the dielectric constant of the material (unity for air). Equations (1) and (8), with boundary and initial conditions specific to the problem, form a self-consistent model that must be solved numerically because of the strongly non-linear nature of the model.

_{e}= 3.2 × 10

^{3}× (E/N)

^{0.8}m/s and D

_{e}= 7 × 10

^{−2}+ 8 × (E/N)

^{0.8}m

^{2}/s, respectively. The parameters are selected based on analysis of the literature and slightly adjusted according to the convergence of the model.

#### 2.2. Computational Domain and Meshing

^{5}Pa).

#### 2.3. Numerical Modelling of the Short-Circuit Arc

_{e2}, positive ion g

_{p2}, negative ion g

_{n2}) was set on the ignition line to simulate the high-charge-density arc channel generated by the short-circuit combustion. The particle sources were established with the following Gaussian pulse functions (Figure 3) [22]:

#### 2.4. Boundary and Initial Conditions

^{13}/m

^{3}, and 600 kV was loaded on the top electrode.

#### 2.5. Experimental Platform

## 3. Results and Discussion

#### 3.1. Experiment Verification

#### 3.2. Particle Density Distribution and Development Law

^{14}m

^{−3}, and the electron concentration was 1.3 × 10

^{14}m

^{−3}above what it was at the end of the simulation, which proves that the short-circuit discharge increased the concentration of space charge and provided necessary environmental conditions for the generation of a subsequent secondary arc.

#### 3.3. Spatial Distribution of the Electric Field During Discharge

#### 3.4. Particle Reactions in Discharge Process

## 4. Conclusions

- (1)
- The brightness distribution obtained by high-speed cameras of the experimental short-circuit arc was basically consistent with the predicted distribution of electron density, demonstrating that the simulation was effective and supported the subsequent analysis of the plasma interior.
- (2)
- With the short-circuit discharge, the electron density along the ignition line first increased and then decreased, and its distribution was quite different from the general streamer discharge. Over time, the concentration of negative ions rose and then levelled off, and due to the differences of diffusion, convection, and adsorption coefficients between positive ions and negative ions, the changing curves of concentrations of positive ions and negative ions had slight differences despite showing the same trend. Near the end of the simulation time, there was a considerably larger number of charged particles than the initial level, which provided the necessary environmental conditions for subsequent secondary arc generation.
- (3)
- The initial stage of discharge was mainly point discharge in space. The spatial electric field intensity showed an S-shaped upward trend in the discharge process. The end regions were Sssignificantly affected by the high-voltage electrode, whereas the middle area was mainly affected by the particle reaction.
- (4)
- The time correspondence between the detachment reaction and the ion source generated in the short-circuit discharge process was basically consistent, and the detachment reactions were mainly concentrated in the middle area and near the negative electrode. The average recombination reaction rates were consistent with the trend of the detachment reaction rate during the discharge, with differences only in magnitude.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The geometric model and meshing result of the secondary arc simulation: (

**a**) The geometric model; (

**b**) meshing result.

**Figure 4.**Experimental platform for secondary arc reproduction: (

**a**) Experimental circuit; (

**b**) experiment field; (

**c**) high-speed cameras.

**Figure 7.**Electron density distribution along the ignition line at key time nodes during the discharge.

**Figure 8.**The negative ion density at the positive electrode (point 58), the negative electrode surface (point 53), and the central point (point 60) during the discharge.

Transport Parameters | Expression | Description |
---|---|---|

${\mu}_{\mathrm{p}}$ (m^{2}/Vs) | 2.0 × 10^{−6} | Positive ion mobility |

${D}_{\mathrm{p}}$ (m^{2}/s) | 5.05 | Positive ion diffusivity |

${\mu}_{\mathrm{n}}$ (m^{2}/Vs) | 2.2 × 10^{−6} | Negative ion mobility |

${D}_{\mathrm{n}}$ (m^{2}/s) | 5.56 | Negative ion diffusivity |

${\beta}_{\mathrm{ep}}$ (m^{3}/s) | 5.0 × 10^{−14} | Electron-positive ions recombination rate |

${\beta}_{\mathrm{pn}}$ (m^{3}/s) | 2.07 × 10^{−13} | Positive-negative ions recombination rate |

f_{0} (1/m^{3}s) | 1.7 × 10^{9} | Natural background ionization source item |

k_{det} (m^{3}/s) | 1 × 10^{−18} | Electron detachment coefficient from negative ions |

Application Location | Convection and Diffusion ${\mathit{N}}_{\mathbf{e}}$ | Convection and Diffusion ${\mathit{N}}_{\mathbf{p}}$ | Convection and Diffusion ${\mathit{N}}_{\mathbf{n}}$ |
---|---|---|---|

Axis of symmetry | $\frac{\partial {N}_{\mathrm{e}}}{\partial r}=0$ | $\frac{\partial {N}_{\mathrm{p}}}{\partial r}=0$ | $\frac{\partial {N}_{\mathrm{n}}}{\partial r}=0$ |

Top electrode | ${N}_{\mathrm{e}}=0$ | $-\mathit{n}\cdot ({D}_{\mathrm{p}}\nabla {N}_{\mathrm{p}})={f}_{+}$ | ${N}_{\mathrm{n}}=0$ |

Bottom electrode | $-\mathit{n}\cdot ({D}_{\mathrm{e}}\nabla {N}_{\mathrm{e}})={f}_{-}$ | ${N}_{\mathrm{p}}=0$ | $-\mathit{n}\cdot ({D}_{\mathrm{n}}\nabla {N}_{\mathrm{n}})={{f}^{\prime}}_{-}$ |

Ignition line | $-\mathit{n}\cdot ({D}_{\mathrm{e}}\nabla {N}_{\mathrm{e}})={g}_{\mathrm{e}2}$ | $-\mathit{n}\cdot ({D}_{\mathrm{p}}\nabla {N}_{\mathrm{p}})={g}_{\mathrm{p}2}$ | $-\mathit{n}\cdot ({D}_{\mathrm{n}}\nabla {N}_{\mathrm{n}})={g}_{\mathrm{n}2}$ |

Rest of the boundary | $-\mathit{n}\cdot ({D}_{\mathrm{e}}\nabla {N}_{\mathrm{e}})=0$ | $-\mathit{n}\cdot ({D}_{\mathrm{p}}\nabla {N}_{\mathrm{p}})=0$ | $-\mathit{n}\cdot ({D}_{\mathrm{n}}\nabla {N}_{\mathrm{n}})=0$ |

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**MDPI and ACS Style**

Li, J.; Yu, H.; Jiang, M.; Liu, H.; Li, G.
Numerical Modeling of Space–Time Characteristics of Plasma Initialization in a Secondary Arc. *Energies* **2019**, *12*, 2128.
https://doi.org/10.3390/en12112128

**AMA Style**

Li J, Yu H, Jiang M, Liu H, Li G.
Numerical Modeling of Space–Time Characteristics of Plasma Initialization in a Secondary Arc. *Energies*. 2019; 12(11):2128.
https://doi.org/10.3390/en12112128

**Chicago/Turabian Style**

Li, Jinsong, Hua Yu, Min Jiang, Hong Liu, and Guanliang Li.
2019. "Numerical Modeling of Space–Time Characteristics of Plasma Initialization in a Secondary Arc" *Energies* 12, no. 11: 2128.
https://doi.org/10.3390/en12112128