Mathematical model


In the stationary state the calculation is made for a current intensity I of 50 A, the electric field E being constant and uniform. The calculation procedure starts by resolving the energy equation (19). Resolution of this equation allows boundary conditions to be set for the temperature on the surface of the upstream electrode. Equation (19) is resolved by setting a temperature of 3000 K at the edges of the electrode. This corresponds to the temperature of the gas at the inlet (Entry) and that of the walls. The temperature profile determined for x = 0 on the electrode is then used to initialise the calculation range. The model is then resolved in two dimensions to obtain the stationary solution.

Initialisation with this method enables the calculation to be started off stable. On extinction, from t = 0, the electric field is taken as being nil. The initial profiles of temperature and velocity are given by the stationary model and the initial densities of the 19 species are given by the equilibrium composition. The models (hydrodynamic and kinetic) are linked through pressure (20) and mass density (21). The time step Dt is set at 10-10 s, this value is chosen using a kinetic criterion:

Were (Da)Max represents the maximum value of Dai. For the calculation of time Dt, the densities of the molecular SFX (X = 2 to 6) species are not taken into account.

We shall present a few results of the hydrokinetic model during arc decay for an initial pressure of one atmosphere. The kinetic values, the reaction rates, the partition functions and the initial densities are collected for temperatures up to 12000 K. In order for this boundary value not to be outreached in our calculation, the initial temperature was calculated by setting the initial stationary current at 50 A. In figures 4 and 5, we present the temperature fields (in Kelvin) during decay for times. For the initial instant and 20 ms figures 6 and 7 report the corresponding temperature field and velocity field (in m.s-1). Indeed, the input gas is injected in a ring with an axial velocity component. It therefore only very slightly penetrates into the heart of the plasma. The gas outlet, however, is located at the centre of the discharge and has a cross sectional area which is smaller than the inlet. The radial velocities therefore increase towards the outlet causing confinement of the plasma and an increase in temperature. Figures 4 and 5 show that a constriction occurs on the temperature profiles near the upstream electrode. On extinction there is an increase of the radial velocities towards the axis of the arc because of pumping phenomena which tend to compensate for the drop in pressure caused by cooling. We then see the steady inclusion of the cold injection gas which disturbs the plasma by cooling it.


Figure 4: Temperature fields ( t = 0 ms - 60 ms) (movie 3D).

Figure 5: Temperature fields ( t = 0 ms - 50 ms) (movie 2D).

Figure 6: Temperature field (K, t = 20 ms).

Figure 7: Velocity field (m/s, t = 20 ms).


In figure 8 we have plotted the relative electron number density field (the relative density is defined as the ratio of the calculated number density to the equilibrium value of the local pressure and temperature). Our results mainly show an under-population of electron density at the edges of the plasma i.e. in the temperature range 4000 K < T < 6000 K. In figures 9 and 10 we have plotted the relative densities of S2+ and S2 respectively. We can note an overpopulation of S2+ and S2 at the edges of the arc. The under-population of electron density is explained by electron recombination with S2+ molecules, this effect being enhanced by cold gas convection. In fact, it is difficult to compare the results obtained with two different values of the inlet mass flow rate. For a given time, the temperature fields are not identical, but a comparison made between two different times leading to approximately the same temperature field shows that the electron under-population is accentuated by the convection.

Figure 8: Relative density field (electron, t = 20 ms).

Figure 9: Relative density field (S2+, t = 20 ms).

Figure 10: Relative density field (S2, t = 20 ms).

Figure 6: Temperature field (K, t = 20 ms).


In figures 11, 12 and 13 we have plotted the inverse partial relaxation times of species S2 , S2+ and electrons, respectively, versus the temperature for various chemical reactions. Figure 16 shows that charge exchange S+ + S2 => S + S2+ is responsible for the creation of the molecule S2+ when T > 4000 K. Indeed, equation (6) shows that for high values of tA-1 there is strong disappearance of S2 and therefore abundant formation of S2+. Figures 12 and 13 show that electron-ion recombination S2+ + e => S + S is responsible for the disappearance of electrons in the temperature range 4000 K to 6000 K.

The results show that departures from equilibrium created by strong convection can lead to an increase of the plasma resistivity. They show a disappearance of the electron density more quickly than we can deduced from an equilibrium composition on a temperature range 4000-6000K. But as we can found using equilibrium model, it is a critical temperature range for thermal cut-off in SF6 circuit breakers.


Figure 11: Contribution of the reactions on the inverse total relaxation time of the specie S2.

Figure 12: Contribution of the reactions on the inverse total relaxation time of the specie S2+.

Figure 13: Contribution of the reactions on the inverse total relaxation time of the electron.

Figure 1: Variations of densities in SF6 plasma equilibrium.