Mathematical model

Hydrokinetics model

The model deals with a 2-D SF6 arc in the transient state and limited by a cylindrical wall. It is based on the following main assumptions: the plasma has a cylindrical symmetry and is in thermal equilibrium (locally we define only one temperature T=Te=Th); we consider that the transport coefficients (electrical conductivity s, thermal conductivity k, specific heat CP, viscosity m [4], net emission coefficient eN [5]) are only dependent on temperature and pressure. The net emission coefficient has been used, assuming a mean plasma radius of 2 mm. Diffusion of particles is ignored.

The calculation domain and boundary conditions are given in figure 3. The dimensions of the calculation domain are 2 cm and 0.5 cm in the axial and radial directions respectively for a grid of 40 x 60 points. On the electrode we made a preliminary study, resolving the 1-D energy equation (see equation 19 hereafter) in the stationary state in order to find the boundary conditions for the resolution of the 2-D stationary model.

Figure 3: Domain and boundary conditions.

The region in figure 1 between 2100 and 3000 K corresponds to a transition zone where diatomic species recombine to form polyatomic species. During this abrupt alteration of the population from one of diatomic species to one of polyatomic species, numerical instability can occur in our model. To overcome this problem, we set the temperature at the wall at 3000 K. Assuming the wall temperature equal to 3000 K leads only to an underestimation of the plasma cooling. Indeed for temperatures lower than 3000 K the thermal conductivity presents peaks corresponding to the dissociation of SF6, SF4 and SF2, and these peaks are not taken into account. In our study, the wall temperature has little influence on the calculation of the electrons number density. At 3000 K the electron density is very low ; moreover, the study reported in section 4 shows that deviations from the equilibrium composition up to 3000 K are very weak owing to the very short relaxation times of the polyatomic species.

In stationary state, the gas entry is situated on line DE where the axial velocity profile u(r) of the inlet flow is assumed to be parabolic (10). The mass flow rate D0 is equal to 0.2 g.s-1. In order to limit the axis temperature in the stationary state (the reaction rates were computed for T £ 12000 K) and to have rather strong blowing during extinction, we imposed an increasing inlet flow in the transient state given by (11), during the first 20 ms. We will give now the conservation equations for stationary and transient states. The resolution of these equations is based on the algorithms of Patankar [6]. The boundaries conditions on the pressure are directly deduced on the densities conditions.

The parameters to be calculated (temperature, velocity and pressure) depend on the local variables r (radial distance) and x (axial distance). In the transient state we also calculate the species densities ; all the unknowns are then dependent on the three variables (space and time). The mass conservation (12) is only used in stationary state : in transient state the mass conservation is satisfied indirectly by assuming the species conservation (Eq.1).

Mass conservation

Axial momentum

Radial momentum

Energy conservation



Ohm's law

Maxwell's Ampere's law

Energy equation near the electrode

Energy equation near the electrode

Coupling equations:

Perfect gas law

Mass density