 # Mathematical model

## Introduction

During extinction there is strong arc blowing leading to turbulence phenomena. The mechanisms involved are responsible for the energy transfer necessary for the recovery of dielectric rigidity. So a model based only on thermal phenomena cannot explain the behaviour of the plasma where chemical non-equilibrium exists as a result of turbulence or strong cooling (-10E8 K.s-1 ). All the models based on the hypothesis of LTE lead to a post-arc current, unlike in experimental results where post-arc current is often non-existent after the zero of the alternating current. To interpret this difference, we have to consider that molecular species may be present in the hot regions. So, the plasma column could be cut by a portion of gas with low electrical conductivity hindering the circulation of electric current.

The general aim of this work is to simulate decaying arc behaviour taking non-equilibrium effects into account. The aim of the work is not to modelize exactly the behaviour of a circuit breaker and to represent all the arc life evolution (creation, high current, decrease of the intensity, evolution near current zero, post arc phase and the dielectric phase), but to study the post arc phase of the thermodynamic plasma without an application of an RRRV (Rate of Rise of Recovery Voltage). So, we built a mathematical model which couples, in simplified geometry, a hydrodynamic and a kinetic study for an SF6 gas in a two-dimensional flow in a transient state at a pressure of 10E5 Pa. Coupling between hydrodynamics and kinetics is achieved through the pressure and the mass density.

The work is divided into three parts. The first, presents the SF6 plasma composition in stationnary state. The second part focuses on the study of SF6 kinetics. Here, we study the species involved in the processes leading to electron disappearance and those that may be influenced by the strong convection brought about by arc decay. In the third part, we present the foundations of the hydrokinetic model such as the assumptions made, the geometry and the boundary conditions as well as all the equations that enable us to calculate the profiles of temperature, velocity and density over time; the results are then analysed considering the departures from equilibrium.

## Nomenclature 