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The two-dimensional and self-consistent fluid model of the SF
_{6} discharge was established based on the electron and ions continuity transfer equations coupled to Poisson’s equation, and also simultaneously considered the photoionization event, and then the flux corrected transport technique (FCT) was employed to numerically solve the particle flux-continuity equations, and some significant microphenomena were achieved that the dynamic behaviors of the charged particles, the spatio-temporal evolution of the discharge channel and the transformation law of the avalanche-streamer for the SF6 narrow-gap were revealed in this paper.

Usually, the approaches to explore gas discharge phenomenon can be classified into experimental and theoretical methods. Experimental method can directly observe the discharge events, but the discharge mechanisms have not been clearly discovered in detail. However, the theoretical method not only gains key data for the dielectric breakdown of gases, but also some microcosmic parameters on controlling dynamic behaviors of charged particles produced in the time of the gas discharge process are founded [

Despite the fact that the electron and ion densities can attain very steep gradients and make the shock fronts in particule fluids along the discharge channel when the gas discharge is triggered and developed in the overall process under the stimulation of electric field, the FCT technique (flux corrected transport) shows high efficiency and accuracy dealing with the intricacy situation [_{6} is widely used in the eletrical industry, especially in the high voltage circuit breaker technology, although it might be replaced by the environment friendly insulating gases. Nevertheless, it is necessary to accurately find microcosmic mechanisms of the particle dynamical behaviors, the spatio-temporal characteristics of the electric field and the track evolution of the discharge channel along the direction of the specific discharge development for broader application domain.

This paper is organized as following: the mathematical model, the FCT algorithm and constraint conditions are carefully presented in Section 2, and the results of the mathematical simulation are analyzed in Section 3 including the following contents: 3.1, avalanche phase; 3.2, streamer formation phase; 3.3, the discharge channel evolution and the photoionization effect. Some valuable conclusions are given in Section 4.

The model for the narrow-gap with parallel plate electrodes which are filled with SF_{6} gas has been presented in detail [_{6} discharge overall process is mathematically modeled by a set of equations governing the transport of particles, moment and energy for ions and electrons together with the electric field equation as follows [

∂ ( N e ) ∂ t = S p h + N e α | v e | − N e η | v e | − N e N p β − ∂ ( N e v e ) ∂ z + ∂ ∂ z D ∂ N e ∂ z (1)

∂ ( N p ) ∂ t = S p h + N e α | v e | − N e N p β − N n N p β − ∂ ( N p v p ) ∂ z (2)

∂ ( N n ) ∂ t = N e η | v e | − N p N n β − ∂ ( N n v n ) ∂ z (3)

here t is the time, r and z are the radius and axis distances for the calculating subregion; N e , N p and N n are the electron, positive and negative ion densities; v e , v p and v n are respectively the electron, positive and negative ion drift velocity; the symbols α , β , η and D are ionization, adsorption, recombination coefficient and electron diffusion coefficient, respectively, their values have been taken from reference literature [

S p h ( z ) = γ p ∫ 0 d Ω ( z − z ′ ) N e ( z ′ ) α * ( z ′ ) | ν e ( z ′ ) | × exp ( − μ | z − z ′ | ) d z ′ (4)

here γ p , α * and μ are second ionization, excitation and absorption coefficients, and Ω is the solid angle subtended at z ′ by the disk charge at the point z. A detail solution about S p h can be referred to the literature [

Taking into account the distortion of space charge effects on the electric field, the Poisson equation is given by:

∇ 2 φ = ∂ 2 φ ∂ r 2 + 1 r ∂ φ ∂ r + ∂ 2 φ ∂ z 2 = − q ε 0 ( N p − N e − N n ) (5)

where φ is the electric potential; q is the electronic charge; ε 0 is the permittivity of the free space. The current I in the external circuit due to the motion of electrons and ions between the electrodes is calculated by the Sato formula [

I = π r 2 q d ∫ 0 d ( N p v p − N n v n − N e v e ) d z (6)

here, r is the radius of the discharge channel, q is the electronic charge.

The convection term of particle’s continuity Equations (1)-(3) are solved by the FCT technique and other items are used the finite difference directly to solve and the Poisson’s Equation (5) is numerically resolved by the over-relaxation iteration.

The convective terms of Equations (1), (2) and (3) are written as ∂ N ∂ t | conv = − ∂ ( N v ) ∂ x , where symbol N shows the density of the particle species and v is their velocities.

Taking ( r N ) as the dependent variable for an axisymmetric cylindrical coordinate system, then

∂ ( r N ) ∂ t | conv = − ∂ f ∂ r − ∂ g ∂ z (7)

where

f = r N v r , g = r N v z (8)

The flux corrected transport algorithm is as follows [

1). Compute F i + 1 2 , j L and G i , j + 1 2 L by a low order monotonic scheme (donor

cell).

2). Compute F i + 1 2 , j H and G i , j + 1 2 H by a high order scheme.

3). Define the anti-diffusive fluxes:

A i + 1 2 , j = F i + 1 2 , j H − F i + 1 2 , j L , A i , j + 1 2 = G i , j + 1 2 H − G i , j + 1 2 L (9)

4). Compute the low order time advanced solution:

N i , j td = N i , j t − 1 Δ V i , j ( F i + 1 2 , j L − F i − 1 2 , j L + G i , j + 1 2 L − G i , j − 1 2 L ) (10)

5). Limit the anti-diffusive fluxes:

A i + 1 2 , j C = A i + 1 2 , j C i + 1 2 , j , 0 ≤ C i + 1 2 , j ≤ 1 (11)

A i , j + 1 2 C = A i , j + 1 2 C i , j + 1 2 , 0 ≤ C i , j + 1 2 ≤ 1 (12)

6). Apply the limited anti-diffusive fluxes:

N i , j t + Δ t = N i , j td − 1 Δ V i , j ( A i + 1 2 , j C − A i − 1 2 , j C + A i , j + 1 2 C − A i , j − 1 2 C ) (13)

where V i , j , r i , j , N i , j and C i , j are the volume, radial distance, density and anti-diffusive coefficient of the ( i , j ) cell, and further details and calculation procedure on the FCT can be found in the literature [

In this paper, schematic diagram and the constraint conditions for parallel-plate electrodes discharge under atmospheric pressure are set as shown in _{6} gas is 5 mm under the pressure 0.1 MPa and temperature

300 K. The spatial mesh chosen to be uniform with 40,000 mesh points, namely, the longitudinal axis ( z -axis) and the radial axis ( r -axis) are all uniformly divided into 200 grids. Then Δ t = 0.05 × 10 − 9 s is taken as time step, which is significantly smaller than that required for stability of the used numerical scheme [

n e | t = 0 = n p | t = 0 = n 0 exp [ − ( r δ r ) 2 − ( z − z 0 δ z ) 2 ] (14)

where r , z is the radial and axial coordinates respectively; the origin of coordinates ( r = z = 0 ) is positioned at the center of cathode surface, the peak value density of particles (seed electrons and positive ions) is n 0 = 10 6 m − 3 , the position of initial plasma z 0 = 0.1 cm , characteristic scales δ r = 2.5 × 10 − 4 m and δ z = 2.5 × 10 − 4 m .

Boundary conditions for electrons and positive ions at the electrodes are as follows:

∂ n e ∂ z | z = 0 = ∂ n e ∂ z | z = d = 0 , ∂ n p ∂ z | z = 0 = ∂ n p ∂ z | z = d = 0

The solution of Poisson's equation is subject to the following boundary conditions:

V | z = 0 = V 0 , V | z = 0 = 0 , ∂ V ∂ r | r = 0 = 0 , V | r = R = V 0 z d

where V 0 is the applied voltage, r is the radius of the computational domain.

The space between parallel-plane electrodes 0.5 cm apart is filled SF_{6} gas with standard atmospheric pressure and commercial purity, and the 46 kV DC is applied on the anode plane, then the cathode grounding. The incepting discharge of the model is trigged by the seed electrons having a Gaussian distribution near the cathode at t = 0 ns (located at 0.1 cm from the cathode as mentioned earlier). Under the electric field, the seed electrons obtain energy to migrate, impact the neutral molecules and produce much more charged particles, so that the current through parallel-plane electrodes is shown in

The electric field stress 92 kV/cm applied to the SF_{6} gap, which slightly larger than the threshold value 89.6 kV/cm, easily renders the discharge happen smoothly. With the electrons migrating to the anode and impacting with neutral molecules, an electron swarm is quickly made up as shown in

When the avalanche volume reaches to the critical value, it instantly changes the avalanche phase into the streamer phase, namely, the streamer phase formation [

From the 8.0 ns moment, the streamer discharge is rapidly development, its volume is promptly expanded and the length is also stretched quickly, and simultaneously the electric field distortion is also exacerbated as shown in

1) The anode-directed streamer

When the transformation avalanche into the streamer phase, the peak value of the electron densities from location z = 0.25 cm at 7.7 ns to the point z = 0.44 cm at t = 9.2 ns is shown in _{6} molecule ionization in this small region is dramatically aggravated, and many more electrons are reproduced and the head radius continuously becomes bigger and bigger, the iterative process finally ends until the head arrives at the anode plate.

2) The cathode-directed streamer

As shown in _{6} molecule is greatly accelerated and much stronger, the velocity of the cathode-directed streamer at 9.2 ns is 0.92 × 10 6 cm / s at z = 0.088 cm location, which is only about 70% of the velocity of the anode-directed streamer in the same moment.

As mentioned above, the discharge process in the narrow-gap of the SF_{6} undergoes transformation from the avalanche to the streamer phase, once the streamer is triggered it soon develops respectively towards the anode and cathode plate, namely, the anode-directed streamer and the cathode-directed streamer, at the same time accompanied by the photoionization appeared, the streamer volume not only grows quickly presenting a near cylindrical shape and but also the streamer length becomes much longer till through the anode and cathode plate, then the discharge path of the both electrode is formed and known as a breakdown channel. The fact has been proved by the experiment results and theoretical demonstration, the photoionization plays an important role and not been overlook when the whole discharge process in the narrow-gap of the SF_{6} undergoes transformation from the avalanche to the streamer phase. The results of the mathematical modeling in this paper shows in

Based on the fluid model of gas discharge and the FCT algorithm, dynamic characteristics of the SF_{6} breakdown process in narrow-gap has mathematically modeled and some important facts has demonstrated that the FCT algorithm is an efficient theoretical way to deal with the troublesome problems having shock fronts in the discharge channel. The results are both shown via easy visualization for the complicated course of the SF_{6} discharge and revealed dynamic characteristics of the charged particles during the SF_{6} discharge process in narrow-gap.

According to the mathematical model and numerical analysis results in this paper, the breakdown process in narrow-gap of SF_{6} still presents two phases at the standard atmospheric pressure, and parallel-plane electrodes 0.5 cm apart and the 46 kV DC applied. Moreover, some facts are indicated that when electrons increase to a certain amount and the avalanche phase changes into the streamer phase. On the one hand, in the electron avalanche phase, the collision ionization is the key rule for producing electrons and making them grow fast until the electron swarms up to the 1.01 × 10 18 cm − 3 . The microcosmic mechanisms of the particle dynamical behaviors, the spatio-temporal characteristics of the electric field and the track evolution of the discharge channel are shown in _{6} molecule and seed electrons is only triggered by the external electric field. When time ranges from t = 7.7 ns to the 8.0 ns, a dramatic change happens in the electron number and electric field of the gas-gap; that is to say, the streamer discharge would be coming. On the other hand, in the streamer phase, from the 8.0 ns moment, the streamer discharge is rapidly developing; its volume is promptly expanded and the length is also stretched quickly, and simultaneously the electric field distortion is also exacerbated as shown in

In a nutshell, the photoionization effect plays a pivotal role in the process of the streamer discharge phase.

Accompanied by the photoionization, the space charges of the gas-gap are multiplicatively increased and led to extremely distorting of the electric field within the discharge channel, and then the streamer dramatically develops toward both anode and cathode plates, and a plasma region is left in the central part of the streamer; meanwhile, the distorted electric field also speeds up the streamer velocity forward to both plates till bridging the anode and cathode plate; finally, the gas-gap is absolutely broken down.

This work was supported financially by National Natural Science Foundation of China, No. 51077032.

The authors declare no conflicts of interest regarding the publication of this paper.

Zheng, Q.P. and Zheng, D.C. (2019) Mathematical Modeling on Dynamic Characteristics of the Breakdown Process in Narrow-Gap of SF_{6} Based on the FCT Algorithm. Applied Mathematics, 10, 769-783. https://doi.org/10.4236/am.2019.109055