## 2 Hybrid Numerical Model
### Governing Equations of the Numerical Model  
The macroscopic numerical model described in this study is based on the hydrodynamical model; its governing equations include: the continuity equations for electrons and heavy particles [30], the energy density equation of electrons [31], and the Poisson equation [5]. In order to introduce the microscopic chemical reactions of plasma into the macroscopic model, the Boltzmann equation [32] is used to calculate the electron transport coefficient (such as mobility, diffusion coefficient) and chemical reaction rate. The electronic continuity equation is shown in Equation (1), which contains some source terms introduced by the microscopic chemical reactions of plasma. \[\left\{ \begin{array}{l}\frac{\partial n_{\rm e}}{\partial t}-\
abla\Gamma_{ \rm e}=R_{\alpha}-R_{\rm n}-R_{\rm ep}+R_{\rm det}+S_{\rm ph}+S_{0},\\ \Gamma_{\rm e}=-D_{\rm e}\
abla n_{\rm e}-n_{\rm e}\mu_{\rm e}E,\end{array} \right. \tag{1}\]  
where \(n_{\rm e}\) is electronic density, \(\Gamma_{\rm e}\) is electronic flux, \(D_{\rm e}\) is the electronic diffusion coefficient, and \(\mu_{\rm e}\) is the electronic mobility. The source terms \(R_{\alpha}\), \(R_{\rm n}\), \(R_{\rm ep}\), \(R_{\rm det}\), \(S_{\rm ph}\), and \(S_{0}\) respectively represent the collision ionization, the adsorption reaction, the composite reaction, the electron dissociation, the photoionization, and the initial number of electrons; finally, \(E\) represents the electric field intensity. The continuity equation of heavy particles is a multi-component diffusion and transport equation, which is obtained by simplifying the basic Maxwell-Stefan equation. The expression of the multi-component diffusion and transport equation is shown in Equation (2):  
\[\left\{ \begin{array}{l}\rho\frac{\partial n_{\rm e}}{\partial t}+\rho(u\cdot \
abla)w_{\rm k}=\
abla\cdot j_{\rm k}+S_{\rm k}\\ j_{\rm k}=\rho w_{\rm k}v_{\rm k}\\ v_{\rm k}=D_{\rm kM}\frac{\
abla n_{\rm k}}{w_{\rm k}}+D_{\rm kM}\frac{\
abla M _{\rm n}}{M_{\rm k}}+\frac{D_{\rm kT}}{\rho w_{\rm k}}\frac{\
abla T}{T}-z_{ \rm k}\mu_{\rm kM}E\\ S_{\rm P}=R_{\alpha}-R_{\rm ep}-R_{\rm np}+S_{0}\\ S_{\rm N}=R_{\rm n}-R_{\rm np},\end{array} \right. \tag{2}\]  
where \(\rho\) is the density of the mixture, \(w_{\rm k}\) is the mass fraction of heavy particles (index k is the number of heavy particles), \(u\) is the average velocity vector of fluid, \(j_{\rm k}\) is the diffusion flux vector of heavy particles, \(S_{\rm k}\) is the source term of heavy particles (indexes P and N represent positive and negative ions, respectively), \(v_{\rm k}\) is the diffusion velocity vector of heavy particles, \(D_{\rm kM}\) is the Maxwell diffusion coefficient, \(M_{\rm n}\) is the average molar mass of the mixture, \(D_{\rm k}\) is the thermodynamic diffusion coefficient, \(z_{\rm k}\) is the charge of heavy particles, \(\mu_{\rm KM}\) is the average mobility of heavy particles, and \(R_{\rm np}\) is the compound reaction term of positive and negative ions. In the microscopic chemical reactions of plasma, the average energy density of electrons directly affects the reaction's rate and the energy transfer's rate when electrons collide with other particles [33]. In order to describe the generation and loss mechanism of electron energy in the corona discharge process, considering the influence of temperature, the electron energy conservation equation is shown in Equation (3):  
\[\left\{ \begin{array}{l}\frac{\partial n_{\rm e}}{\partial t}+\
abla\Gamma_{ \rm\varepsilon}+E\cdot T_{\rm e}=S_{\rm\varepsilon}\\ T_{\rm\varepsilon}=-D_{\rm\varepsilon}\
abla n_{\rm\varepsilon}-n_{\rm \varepsilon}\mu_{\rm\varepsilon}E\\ T_{\rm e}=\frac{2n_{\rm\varepsilon}}{3R_{\rm np}},\end{array} \right. \tag{3}\]  
where \(n_{\rm\varepsilon}\) is the energy density of electrons, \(\Gamma_{\rm\varepsilon}\) is the average energy flux of electrons, \(D_{\rm\varepsilon}\) is the electron diffusion coefficient, \(\mu_{\rm\varepsilon}\) is the electron mobility, and \(k_{\rm B}\) is the Boltzmann constant.