## Materials and Methods
### Two-Dimensional Fluid Model  
#### 2.2.1 Model Equations  
For modeling purposes, half of the AC-PPP reactor was considered and azimuthal symmetry around the reactor axis was assumed. Thus, the spatial description of the problem was mathematically two-dimensional (with only axial and radial directions). The simulated domain was the discharge gap between the high-voltage (HV) and ground electrodes. This domain was extended into the conductive inlet/outlet pipes that can affect the electric field distribution (see Figure 3). The grid size was 4.5 um. The spatial and temporal macroscopic description of the gas discharge inside the reactor was determined by solving the fluid continuity equations for different species coupled with Poisson's equation. These equations were solved using the finite element method (FEM). The continuity equation for all the formed species inside the AC reactor is expressed as follows [14]:  
\[\frac{\partial n_{i}}{\partial t}+\
abla\cdot\Gamma_{i}=\sum_{m}R_{i,m} \tag{1}\]where \(n_{i}\) is the number density, \(\Gamma_{i}\) expresses the flux for the species \(i\), and \(R_{i,m}\) are the reaction rates between species \(i\) and species \(m\). For a typical reaction between species  
\[A+B\to C+D \tag{2}\]  
the reaction rate depends on the density of each species, \(n_{A}\) and \(n_{B}\). \[R=kn_{A}n_{B} \tag{3}\]  
with \(k\), the reaction constant [14; 15]. In this study, two different approaches were considered to obtain the reaction constants. For some reactions, the experimental data for these reaction rates were available in the literature [16]. In other cases, the reaction rate constants were calculated using the total collision cross sections in terms of the collisional energy, \(\sigma(\epsilon)\), by the following relationship [17]:  
\[k(T)=\frac{1}{k_{B}T}\bigg{(}\frac{8}{\pi\mu k_{B}T}\bigg{)}^{1/2}\int_{0}^{ \infty}\epsilon\sigma(\epsilon)e^{-\epsilon/k_{B}T}d\epsilon \tag{4}\]  
The collisional cross section can be written as follows:  
\[\sigma(\epsilon)=\frac{10^{-13}}{EI_{p}}\Bigg{[}B_{1}\ln\bigg{(}\frac{ \epsilon}{I_{p}}\bigg{)}+\sum_{i=1}^{N-1}B_{i+1}\bigg{(}1-\frac{I_{p}}{ \epsilon}\bigg{)}^{i}\Bigg{]} \tag{5}\]  
where \(I_{p}\) is a parameter close (but not always equal) to the ionization or appearance potential for a given ionization channel (expressed in electron-Volts (eV)), \(\epsilon\) is the collision energy (eV), \(B_{i}\) (\(i\) = 1,..., \(N\)) are fitting coefficients, and \(N\) is determined from those conditions for achieving a standard deviation of the fit from data smaller than 3-5% [18]. In the model, only the reactions that showed the greatest influence on the plasma CO\({}_{2}\) kinetics were considered [19; 20; 21; 22; 23; 24]. These reactions were based mainly on the chemistry of [PERSON] [20] and [PERSON] [23] who provided useful results for gliding arc and microwave plasma, respectively. Table 1 shows the selected reactions inside the AC-PPP reactor. A total of 16 electron collisions and six atomic processes were considered. The excited CO\({}_{2}\) states were simplified according to the scheme explained in a previous paper from our group [25]. The levels Va,..., and Vd of Table 1 represent the two non-asymmetric excited vibrational modes; V1 is the first level of the asymmetric vibrational mode, and the excited electronic states of the CO\({}_{2}\) are grouped in the effective level CO\({}_{2}\)*. Figure 3: The simulated domain for the AC-PPP reactor in the 2-D model. The flux term in the continuity fluid equations for all species (Equation (1)) was based on the momentum conservation of each species. The corresponding flux term for each species \(i\) depends on its electrical mobility, \(\mu_{i}\), and spatial diffusivity, \(D_{i}\), i.e.,  
\[\mathbf{\Gamma}_{i}=\pm\mu_{i}n_{i}\mathbf{E}-D_{i}\
abla n_{i} \tag{6}\]  
The plus or minus sign in this equation corresponds to the sign of the charged particles [1; 26]. Only the mobility coefficients for ions and electrons were included [27].