## 2 Methodology
### Numerical Method
#### 2.2.2 Three-Dimensional Gas Flow Model (3D GFM)
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01 & \(e+He\to e+He\) & [PERSON]+ (d) & 0 & 0 & [29] \\
02 & \(e+He\to e+He_{m}\) & [PERSON]+ & 19.8 & 0 & [29] \\
03 & \(e+He\to e+He_{m}\) & [PERSON]+ & 20.6 & 0 & [29] \\
04 & \(e+He\to 2e+He^{+}\) & [PERSON]+ & 24.6 & 0 & [29] \\
05 & \(e+He_{m}\to 2e+He^{+}\) & [PERSON]+ & 4.8 & 0 & [29] \\
06 & \(e+He_{m}\to e+He\) & \(2.9\times 10^{-15}\) & 0.0 & 0 & [29] \\
07 & \(e+He^{\prime}_{2}\to e+2 He\) & \(3.8\times 10^{-15}\) & 0.0 & 0 & [29] \\
08 & \(He^{+}+e\to He_{m}\) & \(1.4\times 10^{-15}\) & 0.0 & \(-4.8\) & [30] \\
09 & \(2e+He^{+}_{2}\to e+He+He_{m}\) & \(2.8\times 10^{-32}\) & 0.0 & 0 & [29] \\
10 & \(e+He+He^{\prime}_{2}\to 2 He+He_{m}\) & \(3.5\times 10^{-39}\) & 0.0 & 0 & [29] \\
11 & \(2e+He^{\prime}_{2}\to He+He^{\prime}_{2}\) & \(1.2\times 10^{-33}\) & 0.0 & 0 & [29] \\
12 & \(e+He+He^{\prime}_{2}\to He+He^{\prime}_{2}\) & \(1.5\times 10^{-39}\) & 0.0 & 0 & [29] \\
13 & \(He_{m}+He\to e+He+He^{+}\) & \(8.7\times 10^{-16}\) & 0.0 & 0 & [29] \\
14 & \(He_{m}+He^{\prime}_{2}\to e+2 He+He^{+}\) & \(5.0\times 10^{-16}\) & 0.0 & 0 & [29] \\
15 & \(He^{\prime}_{2}+He^{\prime}_{2}\to e+3 He^{\prime}\) & \(3.0\times 10^{-16}\) & 0.0 & 0 & [29] \\
16 & \(He_{m}+He_{m}\to e+He^{+}_{2}\) & \(2.0\times 10^{-15}\) & 0.0 & 0 & [29] \\
17 & \(2 He+He^{\prime}\to He+He^{\prime}_{2}\) & \(6.5\times 10^{-44}\) & 0.0 & 0 & [29] \\
18 & \(He^{\prime}_{2}+He^{\prime}_{2}\to e+2 He+He^{+}_{2}\) & \(1.2\times 10^{-15}\) & 0.0 & 0 & [29] \\
19 & \(He_{m}+He^{\prime}_{2}\to e+He+He^{\prime}_{2}\) & \(2.0\times 10^{-15}\) & 0.0 & 0 & [29] \\
20 & \(He_{m}+2 He\to He+He^{\prime}_{2}\) & \(1.9\times 10^{-46}\) & 0.0 & 0 & [29] \\
21 & \(He^{\prime}_{2}+He\to 3 He\) & \(4.9\times 10^{-22}\) & 0.0 & 0 & [29] \\
22 & \(e+N_{2}\to e+N_{2}\) & Bolsig+ & 0.0 & 0 & [31] \\
23 & \(e+N_{2}\to e+N_{2}\) (\(A^{3}\Sigma^{+}_{u}\)) & Bolsig+ & 6.2 & 0 & [31] \\
24 & \(e+N_{2}\to e+N_{2}\) (\(B^{3}\Pi_{S}\)) & Bolsig+ & 7.4 & 0 & [31] \\
25 & \(e+N_{2}\to e+N_{2}\) (\(C^{3}\Pi_{u}\)) & [PERSON]+ & 11.0 & 0 & [31] \\
26 & \(e+N_{2}\to e+N_{2}\) (\(a^{1}\Sigma^{-}_{u}\)) & [PERSON]+ & 8.4 & 0 & [31] \\
27 & \(e+N_{2}\to 2e+N_{2}^{+}\) & [PERSON]+ & 15.6 & 0 & [31] \\
28 & \(2e+N_{2}^{+}\to e+N_{2}\) & \(5.7\times 10^{-39}\)\(T_{e}^{-0.8}\) & 0.0 & 0 & [26] \\
29 & \(e+N_{4}^{+}\to 2N  
The steady-state gas temperature of the reactor is modeled by solving the energy equation (total enthalpy) as:  
\[\
abla\cdot\left(\rho\overset{\rightarrow}{V}h_{T}\right)=\
abla\cdot\left( \lambda\
abla T\right)+\
abla\cdot\left(\overset{\rightarrow}{V}\cdot\tau\right)+S _{GFM} \tag{9}\]  
where \(h_{T}\) (\(=h+\frac{1}{2}\times\sum\limits_{x=1}^{3}u_{x}^{2}\), \(h\) is the enthalpy and \(u\) represents the velocity component) is the total enthalpy, \(\lambda\) is the thermal conductivity of the background gas, and \(S_{GFM}\) is the average volumetric heating source obtained from the 1D PFM. The \(S_{GFM}\) is applied to the reactive zone as the average volumetric heating source evaluated as:  
\[S_{GFM}=S_{SCA-PFM}=\frac{\sum\limits_{m=1}^{N_{cell}}(S_{CA-PFM,m}\times w_{m} )}{d_{gap}} \tag{10}\]  
where the subscript \"\(SCA\)\" represents the spatial cyclic average quantity calculated, \(N_{cell}\) is the cell number used in the 1D PFM for the discharge gap, \(w\) is the cell width, \(d_{gap}\) is the discharge gap as 5 mm, and the subscript \(m\) represents the \(m\)th cell. In the discharge gap, the helium flow is provided as the inlet boundary condition from the top of the reactor. We test whether the small amount of N\({}_{2}\) addition had only a minor effect on the simulated gas temperature; therefore, the small amount of N\({}_{2}\) addition is neglected in the 3D GFM. The outlet boundary condition is assigned on boundaries of ambient air around the reactor to allow air flows in and out of the boundaries as the natural convection is built.