Unified Non-equilibrium Modelling of Tungsten-Inert Gas Microarcs in Atmospheric Pressure Argon

Abstract

Analysis of the plasma parameters of a tungsten-inert gas microarc with a length of 0.4 mm is carried out by means of a unified one-dimensional model. The model solves the fluid equations for the particle and energy conservation of the electrons and the heavy species in the plasma, and the heat conduction in the thermionic cathode. The particle transport of the electrons and the ions is coupled with the Poison’s equation. The spatial distributions of the densities of the charged particles, the electric potential and field, the components of the electric current density, the heating mechanisms and the resulting temperatures of the electrons and heavy particles are discussed in detail for an electric current density of 10⁶ A/m². The discharge voltage, estimates of the Debye length, the near-electrode voltage drop, and the thickness of the regions of space charge adjacent to the electrodes are obtained for current densities in the range from 5.3 × 10³ up to 2.3 × 10⁶ A/m².

Appendix: Derivation of the expression of the heat flux Q c to the cathode

For the sake of a simple and a feasible derivation, we consider only the presence of atomic ions in the vicinity of the cathode and follow the procedure described in [16]. The results presented in “Results and Discussion” section show that this assumption is well justified.

At stationary conditions and taking into account the plasma-reaction model in Table 1, the source terms in Eqs. (1) and (3) can be written as

$$S_{e} = n_{e} n_{A} R_{4} + n_{e} n_{A*} R_{5} - n_{e}^{2} n_{i} R_{10} + n_{A*}^{2} R_{11} ,$$

(13)

$$S_{i} = S_{e} ,$$

(14)

where R_j denotes the rate coefficients of the reaction of number j in Table 1. Then, it follows from Eqs. (1), (3), (13) and (14) that

$$\nabla \cdot \left( {\varvec{\varGamma}_{\varvec{e}} -\varvec{\varGamma}_{i} } \right) = 0.$$

(15)

Next, we separate $S_{\varepsilon }$ into two parts—the one resulting from net ionization and the other—from net excitation. The latter is associated with the radiation losses. Adding up the stationary parts of Eqs. (2) and (5), we can write the following relation.

$$\nabla \cdot \left( {\varvec{\varGamma}_{\varvec{\varepsilon}} + eE_{ion}\varvec{\varGamma}_{\varvec{e}} + \varvec{q}} \right) = - e\varvec{E} \cdot\varvec{\varGamma}_{\varvec{e}} + Q_{iJ} - Q_{rad}$$

(16)

After multiplying Eq. (15) by − e(E_ion − W) and adding it up to Eq. (16), we get

$$\nabla \cdot \left[ {\varvec{\varGamma}_{\varvec{\varepsilon}} + eW\varvec{\varGamma}_{\varvec{e}} + e\left( {E_{ion} - W} \right)\varvec{\varGamma}_{i} + \varvec{q}} \right] = - e\varvec{E} \cdot\varvec{\varGamma}_{\varvec{e}} + Q_{iJ} - Q_{rad} .$$

(17)

We consider now the left-hand-side of Eq. (17), representing the divergence of the heat flux Q_c due to the transport of energy of electrons and heavy particles, together with the boundary conditions in Table 2 and accounting for that the positive x-direction from the plasma to the cathode (Fig. 1). Further, we recall that Eqs. (5) and (6) are solved for the variable T and the conductive flux (q) from the plasma to the cathode is already implicitly considered. After some rearrangements and replacing $\epsilon_{th}$ and $\epsilon_{se}$, the result reads

$$Q_{c} = \left. {j_{i} \left( {E_{ion} - W} \right) - j_{se} \left( {E_{ion} - W} \right) - j_{em} \left( {W + \frac{1}{e}2k_{B} T_{c} } \right) + \frac{1}{4}n_{e} v_{th,e} \cdot \left( {2k_{B} T_{e} + eW} \right)} \right]_{{\varvec{x} = \varvec{L}_{{\varvec{gap}}} }} .$$

(18)

In Eq. (17), the particle fluxes of secondary and thermionic electrons are replaced by the corresponding current densities $j_{se}$ and $j_{em}$. Equation (17) includes so far contributions from the plasma. In order to account for a cooling of the cathode by black-body radiation, an additional term $- \epsilon_{W} \sigma_{SB} T_{c}^{4}$ is added to right-hand-side of Eq. (17) to obtain the expression of Q_c in Table 2.