## 5 Modeling Charge Transport in Dielectric Gas  
The HVDC-GIS components are always subjected to irradiation fields coming from the surrounding environment. This radiation may be due to natural environmental radiation or may occur more severely in the vicinity of MCFs, as described in Section 3. In both cases, the irradiation field ionizes the dielectric gas with the consequent production of ion pairs. As described in Section 3, this fact cannot be neglected when dealing with nuclear fusion devices and should be taken carefully into account for accurate modeling due to the highly radioactive environment they are forced to operate. Accurately modeling the ionization effect in the presence of high radiation levels, without using experimentally based formulas, is of great interest for the development of HVDC-GISs for nuclear fusion applications [1,21]. In this section, a drift-diffusion reaction (DDR) model is described to couple the radiation field with the production of free ions in the gas. The procedure for the latter case is described in [30]. The DDR model deals with positive and negative ion species, neglecting the presence of free electrons. This assumption is certainly valid for common dielectric gases used in HVDC-GIS apparatus, such as sulfur hexafluoride (SF\({}_{6}\)). Indeed, SF\({}_{6}\) is characterized by a high electron affinity (or electronegativity). Ideally, due to this property, all the free electrons in the gas are captured. Even in the case of a discharge, SF\({}_{6}\) represents a good electric-arc extinction medium, precisely because of its high electron affinity, making it so widespread and difficult to substitute in HVDC applications despite its high environmental impact. Positive and negative ions are described by their number densities [m\({}^{-3}\)], \(n^{+}\) and \(n^{-}\). The source of ion pairs due to the external ionizing radiation field is described by the parameter \(S\) [m\({}^{-3}\) s\({}^{-1}\)]. Two mechanisms of migration occur in the gas. The first one is due to the electric force acting on the ions with mobility coefficients \(\mu^{+}\) and \(\mu^{-}\) [m\({}^{2}\)/(V\(\cdot\)s)]. The second mechanism is the diffusion of ions with diffusion coefficients (\(D^{+}\), \(D^{-}\)) that are defined, e.g., through the Einstein relation [31]:  
\[D^{\pm}=\mu^{\pm}\frac{k_{B}T}{e}, \tag{11}\]  
where \(k_{B}\) is the Boltzmann constant and \(e\) is the elementary charge. Recombination of ion pairs also occurs and is described by the coefficient \(R\) [m\({}^{3}\)/s]. It is worth mentioning that the transport parameters of the gas are generally a function of the electric field, pressure, and temperature [32; 33]. By combining these mechanisms, the following partial differential equations (PDEs) can be written for the local time variation of the ion densities:  
\[\frac{\partial n^{+}}{\partial t} =S-Rn^{+}n^{-}-\
abla\cdot(n^{+}\mu^{+}\mathbf{E}_{G})+D^{+} \Delta n^{+}, \tag{12}\] \[\frac{\partial n^{-}}{\partial t} =S-Rn^{+}n^{-}+\
abla\cdot(n^{-}\mu^{-}\mathbf{E}_{G})+D^{-} \Delta n^{-}. \tag{13}\]  
where the electric field in the gas \(\mathbf{E}_{G}\) is evaluated in a self-consistent way through the solution of the [PERSON]'s law:  
\[\
abla\cdot\mathbf{E}_{G}=\frac{e(n^{+}-n^{-})}{\varepsilon_{G}}. \tag{14}\]  
The boundary and initial conditions for the solution of Equations (12) and (13) are described in [34], as follows:  
* A Dirichlet condition for \(n^{+}\) is imposed in the boundaries, where \(\mathbf{n}\cdot\mathbf{E}_{G}<0\), i.e., \(n^{+}\)= 0, while a natural Neumann condition is imposed to \(n^{-}\) therein, i.e., \(-\mathbf{n}\cdot D^{-}\
abla n^{-}=0\). * A Dirichlet condition for \(n^{-}\) is imposed in the boundaries, where \(\mathbf{n}\cdot\mathbf{E}_{G}>0\), i.e., \(n^{-}=0\), while a natural Neumann condition is imposed to \(n^{+}\) therein, i.e., \(-\mathbf{n}\cdot D^{+}\
abla n^{+}=0\). Here, \(\mathbf{n}\) denotes the unit vector normal to the gas domain.