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 µm.
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]:
where
ni is the number density, Γ
i expresses the flux for the species
i, and
Ri,m are the reaction rates between species
i and species
m.
For a typical reaction between species
the reaction rate depends on the density of each species,
nA and
nB.
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,
σ(
ε), by the following relationship [
17]:
The collisional cross section can be written as follows:
where
Ip is a parameter close (but not always equal) to the ionization or appearance potential for a given ionization channel (expressed in electron-Volts (eV)),
ε is the collision energy (eV),
Bi (
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 Wang [
20] and Kozak [
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*.
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,
μi, and spatial diffusivity,
Di, i.e.,
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]. The mobility of ions was calculated according to the Langevin equation:
where
αi is the polarization of background gas per unit of cubic angstroms; its value for various gases is presented in the existing literature on gaseous discharges [
27]. In this work, the mobilities for CO
2+ and C
+ species were 0.0012 and 0.0009 m
2/Vs, respectively.
The diffusion coefficient of the electrons and ions were instead calculated from the Einstein relation:
with
Te(i) and
qe(i) being the temperature and charge of electrons and ions [
28].
For neutral species, the diffusion coefficients were calculated using the distribution coefficients of Lennard–Jones [
29].
The rate of change of the electron energy density is described by [
1]:
where ω
e is the electron energy density,
is the energy loss or gain due to inelastic collisions, the term
accounts for the ohmic or joule heating of the electrons in the electric field, and
is the electron flux energy, that is described by:
The electron energy loss or gain
is obtained by summing the collisional energy loss or gain over all reactions [
14]:
where
xj is the mole fraction of the target species for reaction
j,
kj is the rate coefficient for reaction
j,
Nn is the total neutral number density and
is the energy loss from reaction
j.
The electron energy density ω
e, the mean electron energy ε, and the electron temperature
Te are correlated with each other through [
30]:
For non-electron species, the following equation was solved for the mass fraction of each species [
30]:
where
is the diffusive flux vector,
is the rate expression for species
k,
is the mass averaged fluid velocity vector,
denotes the density of the mixture and
is the mass fraction of the
kth species.
The diffusive flux vector is defined as [
30]:
with
, being the multicomponent diffusion velocity for species
k.
To initiate discharge in the reactor, electric potential should be applied between the electrodes, thus Poisson’s equation must also be considered in the model [
14]:
where
is the electric potential,
is the vacuum permittivity and
σ is the charge density, that can be written in terms of density of the charged species,
nk, and their charge, e
Zk [
31]:
In this work, 16 different neutral and ionized species were considered in the model (
Table 2). Thus, 16 continuity equations together with Poisson’s equation were solved with the employment of a stabilized FEM.
2.2.2. Boundary Conditions
To obtain a unique solution for the system of coupled equations with the geometry presented in
Figure 3, the boundary conditions (Dirichlet and Neumann boundary conditions) must be imposed. The boundary conditions applied for the AC plasma reactor corresponded to those found in the existing literature [
32]. The following boundary condition was used to account for the particle flux in walls:
where
is the normal vector pointing toward the tube wall and,
is the thermal velocity of particles [
32]:
and the number
ai is defined by:
For electrons, as a special case, the particle flux due to secondary electron emission (SEE) was added to the system and is defined as follows [
32]:
where
are the SEE coefficients, which defines the average number of electrons emitted per impact of ions
p on the tube wall.
Similarly, a boundary condition for electron energy was [
32]:
Here, the second term is the SEE energy flux, being the mean energy of the secondary electrons.
The discharge was driven by a sinusoidal electric potential applied to an electrode and the other electrode was grounded. Then the boundary condition of electric potential in the grounded electrode was:
and the electric potential in the powered electrode was given by:
with
f = 50 Hz and
V0 = 22 kV.