Chemistry reduction of complex CO₂ chemical kinetics: application to a gliding arc plasma

The model used in this work is a 0D model, which solves the time-evolution of the species densities by balance equations, taking into account the various production and loss terms by chemical reactions. We use an existing code ZDPlaskin [64], which features an interface for description of the plasma species and reactions, a solver for the set of differential equations and a Boltzmann equation solver BOLSIG+ [65]. In this 0D model, transport processes are not explicitly considered, since a 0D model only calculates the species densities as a function of time. Nevertheless, the transport of the arc through the GA reactor can be mimicked by translating the temporal behavior, as calculated in the model, into a spatial behavior, corresponding to the position in the reactor, by means of the gas flow rate. This allows us to mimic the typical plasma evolution behavior of a GA used for CO₂ conversion [66].

The comparison of the calculated conversion and energy efficiency with experimental values for the same conditions in our previous work [66] indicates that the detailed chemistry set used in this model is generally reasonable for a GA discharge, although there is no guarantee that the kinetics of all species are correct, as the latter can only be concluded from direct validation of the chemistry of those particular species with experiments. Seventy-two different species are considered in the model. Figure 1 shows the energy diagram of the CO₂ vibrational levels included in the model. The energies of the CO₂ vibrational levels can be calculated using the anharmonic oscillator approximation. We consider the four lowest effective symmetric mode levels (denoted by letters) and 21 asymmetric mode levels (denoted by numbers) up to the dissociation limit of the molecule. The chemical reactions include electron impact reactions, vibrational energy transfer reactions and heavy particle reactions leading to breaking and formation of new species. For more detailed information about the reactions and the rate coefficients used, see [67–70].

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In a GA discharge, the arc is ignited at the shortest interelectrode gap when the electric field is high enough to cause breakdown. Then the arc expands upwards along the surface of the electrodes upon effect of the gas flow, and elongates until it extinguishes [71]. Although a 0D model cannot describe the increase of arc length from the shortest gap to a larger gap, we can use the time evolution of the plasma parameters as input, which are related to the increase of arc length during the arc downstream movement. Indeed, the gas molecules move together with the arc on their way throughout the reactor. This is thus taken into account in the model by translating the spatial variations of the plasma characteristics in the reactor into temporal variations, to be used as input in the model, as described below.

The power density is assumed to be constant with time in the region between the electrodes where the arc is formed. This assumption is justified from our previous calculations [66]. The electric power in the plasma is applied to the electrons by setting a certain value for the reduced electric field (E/N, i.e. ratio of electric field over gas density). The reduced electric field is calculated at each time step, to keep the desired power density due to the changing gas composition as a function of time (upon CO₂ conversion into CO and O₂). We consider a power density of 10 kW cm⁻³. In our model the power density in the arc zone (i.e. the region between the electrodes) is calculated as the plasma power divided by the arc volume. It is reported in literature that the GA is a plasma string with a diameter of about 1 mm, surrounded by a weakly ionized zone [72, 73]. Thus, in our 0D model the arc volume is seen as a cylinder with a diameter of about 1 mm. Therefore, at an applied power on the order of tens of watts [66], the arc column diameter is usually about 1 mm [72, 73], and the arc length is on the order of the interelectrode distance, hence around 1–3 mm [66]. This results in an arc column volume in the order of 10⁻³ cm³, and thus the power density is in the order of 10⁴ W cm⁻³. In practice, the plasma parameters are not uniformly distributed due to the plasma arc core surrounded by a weakly ionized zone. So the power density should be high in the arc core, and low in the arc fringe region. Hence, the magnitude of the power density in our model, which applies to the arc zone, may be achieved in practice. Moreover, it yields an electron density and electron temperature (see figure 2), in agreement with experimental values in the order of 10¹⁸ m⁻³ and 1–2 eV, respectively [74, 75]. In the region beyond the electrodes, i.e. during the post-discharge stage, the power density is set to zero, and thus, both the electron density and electron temperature also drop to zero, as is clear from figure 2. In this study, we set the arc time as 8 ms based on the GA cycle in the experiments, and the total gas residence time is assumed to be 15 ms, which is sufficient for relaxation of the reactive species to equilibrium. In the calculation, the gas pressure is kept constant at one atmosphere.

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The time-resolved temperature distribution in figure 5 is based on the combination of experimental measurements and modeling, since the experimental measurements did not provide a time-resolved gas temperature distribution. The maximum temperature 2600 K in figure 5 is obtained according to a measured time-space averaged temperature in the experiment [78]. The gas temperature profile in figure 5 is assumed based on our earlier findings [76]. The gas temperature first increases with time during the arc downstream movement, and then almost stays constant in our previous 3D model results [76], as presented in figure 3. The geometry used in the 3D modeling is presented in figure 4, which shows a cross section through the middle of the anode and the cathode [76]. A pair of semi-ellipsoidal electrodes are 50 mm long and the shortest interelectrode distance is 3.2 mm in the model. The entire geometry, including the region outside the electrodes where the gas can flow without passing through the arc, is a cylinder, with radius of 31.8 mm and length of 150 mm. Note that the gas temperature plotted in figure 3 was obtained for argon, and it is higher in a CO₂ discharge, due to the vibration-translation (VT) relaxation processes [44]. Therefore, in our 0D model for CO₂ we consider a fixed gas temperature profile, increasing from 300 to 1000 K in the first 0.05 ms, due to the short characteristic time (10⁻⁵ s) for VT relaxation processes [67]. Subsequently, until 6 ms, the gas temperature increases further to a maximum value of 2600 K, which is based on experimental measurements from literatures [77, 78], and then it stays constant during the rest of the arc stage (see figure 5). From our previous 3D results, in the post-discharge region, the electron density and electron temperature suddenly drop to negligible values, while the gas temperature gradually decays [76]. Therefore, after the gas leaves the arc, the gas temperature profile is assumed to gradually decrease to 500 K, which corresponds to the outlet gas temperature measured in the experiments [66].

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The theory of DRG is well suited to abstract the coupling among the species. Specifically, each node in a DRG represents a species in the full chemistry, and there exists an edge from vertex A to vertex B, if and only if the removal of species B would directly induce a significant error to the production rate of species A. That is, an edge from A to B means that B has to be kept to correctly evaluate the production rate of A [58–60].

The expression of the production rate of species A is:

$$\begin{eqnarray}&&{R}_{{\rm{A}}}=\displaystyle \displaystyle \sum _{i=1,j}{\nu }_{{\rm{A}},i}{\omega }_{i},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\omega }_{i}={k}_{i}\displaystyle \prod _{l}{n}_{l}^{L},\end{eqnarray} \tag{ 2 }$$

where summation is made over all production reactions, ${\nu }_{{\rm{A}},i}$ is the stoichiometric coefficient of species A in the reaction i, ${\omega }_{i}$ is the production rate, expressed by equation (2), where k_i is the reaction rate constant, and ${n}_{l}^{L}$ is the density of the lth reactant in reaction i.

To quantify the direct influence of one species on another, a normalized contribution of species B to the production rate of species A can be defined as [58]:

$$\begin{eqnarray}&&{r}_{{\rm{A}}{\rm{B}}}=\displaystyle \frac{\displaystyle {\sum }_{i=1,j}\left|{\nu }_{{\rm{A}},i}{\omega }_{i}{\delta }_{{\rm{B}}i}\right|}{\displaystyle {\sum }_{i=1,j}\left|{\nu }_{{\rm{A}},i}{\omega }_{i}\right|}\end{eqnarray} \tag{ 3 }$$

${\delta }_{{\rm{B}}i}=1$ if the ith elementary reaction involves species B, ${\delta }_{{\rm{B}}i}=0$ otherwise.

If the normalized contribution r_AB is sufficiently large, the removal of species B from the chemistry set is expected to induce significant error on the production rate of species A. Consequently, if A has to be kept, B should also be kept. In such a case, we conclude that species A strongly depends on species B.

To quantify the dependence of A on B, we define a small threshold value ε such that, for r_AB < ε, the dependence can be considered negligible. In this paper, a relatively large value ($\varepsilon =0.2$) and a small value ($\varepsilon =0.01$) are used to quantify the dependence of one species on another, for the reason explained in next section. Note that $\varepsilon =0.1$ is a typical value that can be used in most cases [42]. In other words, species A depends on species B if and only if there exists a directed path from A to B in DRG, i.e. B is reachable from A. For each species A, there exists a group of species, which are reachable from A, and this set of species is defined as the dependent set of A, denoted as S_A. If species A is an important species to be kept in the reduced mechanism, its dependent set S_A should be kept as well.

This method is practically used as follows: first the reactions in which the target species participated are selected, and then the production rate of target species can be calculated. For each species involved in these reactions, the relative contribution of each species to the target species is calculated. Finally the relative contributions larger than the specified threshold value ε are exported, and the graphs (figures 6, 7, 9 and 10) are plotted manually. For even larger sets of species and reactions, a graph searching method, such as so-called 'depth first search', which features a linear searching time proportional to the number of edges in the graph, can be exploited to find all the vertices reachable from the starting species.

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It is pointed out in [79] that a detailed reaction model is composed of three types of species: critical, nonessential, and marginal species. Critical species participate in those reaction channels that largely determine the simulation results, while nonessential species participate in reaction channels having little to no influence on the simulation results. A successful reduction model retains all critical species, eliminates all nonessential species, and properly deals with the marginal ones.

Based on the DRG method, different species are chosen when adopting different threshold values, and therefore, there exist some nonessential species, which are not involved in these important reaction channels, and also some marginal species depending on the desired accuracy, in our case the size of the threshold value. These nonessential species can be straightforwardly removed from the full reaction set. However, when using the above DRG method to obtain a significant reduction by increasing the threshold value ε, the risk of losing critical marginal species increases. So for these marginal species, a further reduction can be achieved using the more time-consuming sensitivity analysis method to evaluate their effects on the target property. By means of a sensitivity analysis, we can calculate the error on a defined target property, following the removal of each of the marginal species, and ranking them accordingly [63]. In our work, we use several quantities as the investigated target property, i.e. the calculated CO₂ conversion, the electron density and the densities of the major neutral species, as well as the vibrational distribution function (VDF) of CO₂. Starting from the reduced chemistry set obtained through the DRG method, the species to be investigated are detected by fixing different threshold values. Once marginal species are detected, they are ranked according to the error induced by their removal: for the ith species, the error is evaluated as

$$\begin{eqnarray}&&{\varepsilon }_{i}=\displaystyle \frac{\left|{{X}}_{{\rm{R}}{\rm{E}}{\rm{D}},i}-{{X}}_{{\rm{D}}{\rm{E}}{\rm{T}},i}\right|}{{{X}}_{{\rm{D}}{\rm{E}}{\rm{T}},i}}\times 100 \% ,\end{eqnarray} \tag{ 4 }$$

where X_DET,i and X_RED,i are the target property before and after the removal of species i, respectively. In this way, the species are ranked according to the induced error and can then be progressively removed from the chemistry set obtained in section 2.3.

3.1.1. Role of species in CO₂ production and loss

In the following DRGs (figures 6, 7, 9, 10), the blue circles represent the neutral species, and the green squares represent charged particles. The species marked red in blue circles and green squares are the marginal species. The direction of the arrows means a direct dependence of one species on another, measured by equation (3), for example, CO₂ → CO, as shown in figure 6, can be considered as the dependence of species CO₂ on CO, including both source and loss terms. When different species depend on the same species, this species will appear multiple times in the same figure.

We start from CO₂ as the major species, and the other species reachable from CO₂ can be searched. Figure 6 shows the resulting species relation graph, in which each directed edge represents a direct dependence of one species on another. The numbers on the directed edges represent the time-integrated normalized contributions of one species to the production and loss rate of the major species, calculated by equation (3). If this number would be 1, it would mean that this species is involved in every loss and formation reaction of the other (major) species. Usually, this is not the case, so the number on the directed edge is typically lower than 1.

The species relations smaller than a critical value ε = 0.2 are neglected in figure 6. It can be clearly seen that CO₂ strongly depends on CO and O, while CO and O are also strongly coupled, and both depend on the electrons and the CO₂ vibrational states. Here we should note that CO₂(V) includes both four effective symmetric mode levels and the 21 asymmetric mode vibrational states up to the dissociation limit (see details in [67, 68]). Therefore, we can conclude that CO₂, CO, O, CO₂(V) and the electrons are the important species, which should be included in the reduced chemistry set.

In order to avoid losing critical species, we can reduce the threshold value ε to include more species. Figure 7 shows the DRG based on a threshold value ε = 0.02. Compared with figure 6, the species marked in red and the arrows marked in orange are newly added in this figure, i.e. O₂, the vibrational states of CO and O₂, the electronically excited state CO₂(e), and negative ions CO₃⁻, O₂⁻, which are considered as marginal species. In order to analyse the effect of the removal of these species on the CO₂ conversion, a sensitivity analysis is performed in the next section.

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Figure 8 presents the relative contributions of the most important reactions responsible for the CO₂ loss and formation. The main loss reaction is electron impact dissociation of vibrationally excited states of CO₂ into CO and O, while the reaction of CO (either in the ground state or in vibrationally excited states) with O atoms and a third body (M) is the predominant production process of CO₂. Moreover, electron impact dissociation of ground state CO₂ and dissociation of vibrationally excited states of CO₂ upon collision with O atoms are also quite important CO₂ loss processes, with a relative contribution of 15% and 13%, respectively, while the recombination of CO with O₂ also contributes to CO₂ formation, with a relative contribution of 9%.

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3.1.2. Role of species in sustaining the arc

From our previous paper [66], we know that charged particles almost have no contribution to the loss and production of CO₂, and this finding is also consistent with our results in figure 8. However, the charged particles are very important to sustain the arc, as revealed by our 2D/3D model, especially the electrons. Thus, we now take the electrons as the target species, to reveal which species are reachable from the electrons. The result is presented in figure 9 for the species with normalized contribution above a threshold value ε = 0.2. The directed edges indicate that the electrons strongly depend on the negative ions O₂⁻, O⁻, and neutral species CO₂, CO, O₂, O, C. The numbers next to the arrows indicate the contribution of each species to the electron production and consumption rates calculated by equation (3). The red arrow to CO₂⁺ indicates that its contribution is larger than 0.2 at the beginning, although the normalized contribution integrated over the entire gas residence time is below 0.2.

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Since there are many ions in the full chemistry set [67, 69], we reduce the threshold value ε in order to avoid losing the critical ions. Figure 10 shows the DRG for the target species reachable from the electrons, based on a threshold value ε = 0.01. The positive ions and neutral species marked in red are newly added, and are considered as marginal species. It can be seen that CO₄⁺ ions have a contribution to the production and loss of electrons at a small threshold value, as shown in figure 10. The CO₄⁺ ions are formed by the O₂⁺ ions conversion reaction (O₂⁺ + CO₂ + M → CO₄⁺ + M), so the relative contribution of O₂⁺ ions to the production of CO₄⁺ ions is calculated. It is found the O₂⁺ ions are strongly coupled to the CO₄⁺ ions although they have a negligible direct contribution to the electrons. So these two ions should be retained together in the reduced chemistry set. The other ions not included here (such as CO⁺, CO₄⁻, O₄⁺, O₄⁻, O₃⁻, O⁺, C₂O₂⁺, C₂O₃⁺, C₂O₄⁺, C₂⁺, C⁺), and their chemical reactions, can safely be removed from the full chemistry set. In the next section we will employ sensitivity analysis to evaluate the effect of removing these marginal species one by one on the CO₂ conversion. In this way, we can assure whether these marginal species should be kept in the final reduced chemistry set or not.

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Combining the above analysis in sections 3.1.1 and 3.1.2, we obtain a reduced chemistry set, including the following neutral and charged species: CO₂, CO₂(V), CO₂(e), CO, CO(V), O₂, O₂(V), O₃, O, C, CO₂⁺, CO₄⁺, CO₃⁻, O₂⁺, O₂⁻, O⁻, and the electrons. Since different threshold values result in different species in the reduced set, we apply sensitivity analysis to evaluate the effect of removing these marginal species (marked in red in figures 7 and 10) one by one on the CO₂ conversion. Figure 11 shows the results for removing the marginal species one by one, for seven different cases (see legend). We observe in this figure a double knee in the post-discharge stage. In the time range from 8 to about 9 ms, the CO₂ conversion drops faster, while the drop becomes slow between 9 and 10 ms. This phenomenon is a physical effect related to the variations of gas temperature. As shown in figure 5, the gas temperature is below 1500 K at about 9 ms, so the three-body recombination rate of CO with O atoms and a third body M decreases, resulting in a slow drop in conversion value after about 9 ms. From this figure we can deduce that the CO₂ conversion predicted by the magenta and green lines greatly deviates from the conversion obtained by the full reaction set, which means that O₂⁺ and CO₂(e) should be kept in the reduced chemistry set. In order to check whether these species are responsible for the large changes seen when they are removed additionally to other species, figure 12 below shows the effect of removing only O₂⁺ species on the CO₂ conversion. The result is the same as in figure 11 (i.e. no CO₃⁻, CO₄⁺ and O₂⁺). Therefore, it can be inferred that the O₂⁺ species is the one responsible for the large changes seen when it is removed additionally to other species. It is clear that for the other cases the CO₂ conversion obtained with the reduced chemistry sets agrees well with the full reaction set, for the entire gas residence time. The maximum error is about 7% in the arc stage and 1% in the relaxation stage. Thus, the obtained reduced chemistry set, including only CO₂, CO₂(V), CO₂(e), CO, O₂, O, C, CO₂⁺, O₂⁺, O₂⁻, O⁻, and the electrons, is suitable to predict the CO₂ conversion in our GA, at the discharge condition investigated.

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The CO₂ conversion as the sole target property is not a guarantee for the reliability of this chemistry set and for the effect of the chemistry reduction on all other plasma properties. Indeed, the effect on other important properties, like the densities of the various plasma species, should also be evaluated.

Comparison of the densities of the various neutral species predicted by both the full chemistry set and the reduced set is shown in figure 13. The densities of the neutral species obtained with the reduced chemistry set are in good agreement with those obtained with the full chemistry set in the arc stage. In the post-discharge stage, the densities of CO₂, CO and O₂ molecules are the same for the reduced and full chemistry sets, while the density of O is different from that obtained with the full set, but its density is low, and it has no influence on the CO₂ conversion. In general, the densities of the neutral species obtained with the reduced set are basically in good agreement with those obtained with the full chemistry set.

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For the density of charged species, the positive ions density predicted by the full set is higher than that obtained by the reduced set at the beginning, and the positive ions density becomes close to that obtained by the reduced set with the increase of time. The ions have very little direct effect on the CO₂ conversion and are therefore only important for determination of the electron density. So the comparison of electrons density between the reduced and full chemistry set is shown in this study.

Figure 14 presents the comparison of the CO₂ asymmetric VDF predicted by the full set and the reduced set both in the arc discharge and in the post-discharge. The time for the arc discharge is at the end of the discharge, i.e. t = 7.6 ms, and the time for the post-discharge is at t = 8.5 ms. The reduced set predicts a slightly underestimated density for the first eight vibrational levels, but in general the agreement is very good, both in the arc stage and in the post-discharge stage. As the reduced chemistry set does not account for the CO vibrational levels, while the full chemistry set considers 10 CO vibrational levels [67], this shows that, under this condition, the CO vibrational levels do not affect the CO₂ VDF.

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Figure 14 suggests that we can use an additional reduction method to further reduce our chemistry set. Indeed, in [43], a level lumping method was proposed to treat the asymmetric mode vibrational levels of CO₂ in order to reduce the calculation time. In [43] the VDF was not in thermal equilibrium and showed a plateau for the intermediate vibrational levels; it could be represented by three groups in the lumped level method, but figure 14 illustrates that for the conditions investigated here, the CO₂ vibrational levels follow almost a Boltzmann distribution, so that one group is sufficient to describe the 21 CO₂ asymmetric mode vibrational levels. This is indeed what we will do in the 2D model explained in section 5 below. Note also that the VDF, and thus the vibrational temperature (being a measure for the population of the first vibrational level of CO₂) is higher in the arc discharge than in the post-discharge, indicating that the vibrational levels are more populated in the arc discharge than in the post-discharge, which is indeed like expected.

Figure 15 compares the electron density predicted by the full set and the reduced set. The electron density predicted by the reduced set is slightly lower in the arc stage, but in the post-discharge stage, the electron density quickly drops to zero for both chemistry sets. The maximum error in the electron density is about 12% in the arc stage. Because many positive and negative ions are included in the full set, it is difficult to obtain exactly the same electron density with the reduced and full sets. On the other hand, this difference almost has no influence on the CO₂ conversion, so the agreement between the full set and reduced set is very good.

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Finally, we compare in figure 16 the reduced electric field obtained by both the full chemistry set and the reduced set. The reduced electric field predicted by the full set remains almost constant at about 67 Td in the arc stage, while the value obtained by the reduced set is slightly higher, and decreases to about 67 Td at the end of the arc stage. In the post-discharge stage, the reduced electric field quickly drops to zero for both chemistry sets. In general, the reduced electric field obtained with the reduced set is in good agreement with that obtained with the full chemistry set.

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In previous section we have validated the reduced chemistry set for CO₂ conversion in a GA by comparing various calculation results with the full chemistry set. Table 1 lists the species included in the reduced chemistry set, and tables 2–5 present the corresponding chemical reactions. In this section, this reduced chemistry set is applied to a 2D model for a GA.

A reduced non-equilibrium CO₂ plasma chemistry set is also proposed in [43]. The difference is that CO₃⁻ and O₂(V) are included in [43] and O₂⁺ is included in our reduced chemistry set. This can be explained by the fact that at our conditions, the electron temperature is higher, resulting in more ionization, so the O₂⁺ ions cannot be neglected, which also can be seen from its effect on the CO₂ conversion in figure 11. In figure 11 the role of the CO₃⁻ ions and O₂(V) species in the actual CO₂ conversion is minor, so these two species could be removed from the chemical reaction set. Reference [53] presents a dimension reduction method for a CO₂ plasma based on PCA. Two principal components were able to predict the CO₂ to CO conversion at varying ionization degrees, and the densities of other species could be recovered from the principal components by linear interpolation from 2D lookup tables. Compared with our model, the model in [53] greatly reduces the number of species and thus continuity equations, thus reducing the computation time. However, the charged species are not considered in [53], and although they have no direct influence on the conversion of CO₂, they indeed affect the electron density and arc discharge characteristics. In our model, the reduction of both charged particles and neutral species is considered, so this reduced model is more suitable for 2D or 3D numerical simulations of a GA discharge (and probably also other plasma types).

The GA geometry used in the 2D model is presented in figure 20. The cathode and anode curvature are the same in the geometry. The shortest interelectrode distance is 3.2 mm in this model, which is the same as in our previous calculations for an argon GA [80].

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The plasma discharge equations are the same as in our previous paper [80]. They describe the plasma density, electron and gas temperature, and the electric field in the GA. As illustrated in figure 14, the CO₂ vibrational states follow almost a Boltzmann distribution, so we describe the 21 asymmetric mode vibrational states with only one group. The level lumping method is the same as in [43].

Here the total number density of all the levels j within one group is n_g:

$$\begin{eqnarray}&&{n}_{g}=\displaystyle \displaystyle \sum _{j\in g}{n}_{j}.\end{eqnarray} \tag{ 5 }$$

After obtaining the total number density of the group, in order to obtain the number density of each level j in the group, the distribution function f(E_j, T) of the levels within the group needs to be known, where E_j is the energy of the jth level within group, and T is the temperature associated to this group. In the case of a Maxwellian internal vibrational distribution, where all the levels have the same degeneracy, this gives:

$$\begin{eqnarray}&&{n}_{j\in g}=\displaystyle \frac{{n}_{g}\exp \left(-\displaystyle \frac{{E}_{j}}{{k}_{b}T}\right)}{\displaystyle {\sum }_{j\in g}\exp \left(-\displaystyle \frac{{E}_{j}}{{k}_{b}T}\right)}.\end{eqnarray} \tag{ 6 }$$

In order to solve the total density of the group instead of the density of each individual level, the species density equation is as follows [80]:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {n}_{g}}{\partial t}+{\rm{\nabla }}\cdot {\vec{G}}_{g}+({\vec{u}}_{g}\cdot {\rm{\nabla }}){n}_{g}=\displaystyle \displaystyle \sum _{j\in g}{S}_{j},\end{eqnarray} \tag{ 7 }$$

where S_j is the chemical reaction source term for each individual level j.

Besides the species density equation, another energy conservation equation is required to describe the inner-distribution within this group. In this paper, we have chosen to solve for the mean group vibrational energy:

$$\begin{eqnarray}&&\overline{{E}_{g}}=\displaystyle \frac{1}{{n}_{g}}\displaystyle \displaystyle \sum _{j\in g}{E}_{j}{n}_{j}.\end{eqnarray} \tag{ 8 }$$

In order to obtain the distribution of the levels within the group, we need to know the temperatures T, as shown in equation (6). The relation between T and $\overline{{E}_{g}}$ is given in [43], and the lookup tables can be used to give T as a function of $\overline{{E}_{g}}.$ Then the number density of each individual level in the group can be obtained.

The time and spatial evolution of the plasma characteristics in the CO₂ GA are presented in figures 21–24. The electron temperature (figure 21) in the arc center (core) almost stays constant at 0.8 eV, which is very suitable for vibrational excitation of CO₂ molecules. At 1.2 ms, the electron temperature slightly decreases to 0.7 eV in the arc center. The electron density (figure 22) shows a similar evolution as the electron temperature, decreasing from 2.3 × 10¹⁸ m⁻³ at 0.2 ms to 9.0 × 10¹⁷ m⁻³ at 1.2 ms in the arc center. The calculated electron density and temperature are in agreement with the electron density in the range of 10¹¹−10¹⁴ cm⁻³ and electron temperature in the range of 1–1.5 eV shown in [75], respectively.

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Both the gas temperature and vibrational temperature increase with time, as illustrated in figures 23 and 24. At 0.2 ms, the vibrational temperature (2200 K) is higher than the gas temperature (1900 K), but after 0.4 ms, they reach the same value, rising to 2600 K at 1.2 ms, indicating indeed that the vibrational distribution becomes in thermal equilibrium with the gas temperature.

The gas temperature is a very important parameter for CO₂ conversion: a high gas temperature enhances the vibrational-translational relaxation rate, resulting in a low density of the high vibrational states, and thus also a low CO₂ conversion. Moreover, the rate of the reverse reaction, i.e. recombination of CO with O atoms, also increases at high gas temperature, thus reducing the CO₂ conversion. Therefore, our calculations indicate that a low gas temperature, an appropriate electron temperature (around 1 eV) and high electron density are essential to produce high densities of vibrational states, thus improving the CO₂ conversion in a GA discharge.

In general, the electron density in the 2D model is in the same order of magnitude (∼10¹⁸ m⁻³) as the electron density obtained with the 0D model. The electron temperature of the 2D model is lower than that in the 0D model, but it is still in the same order of about 1 eV. The vibrational and gas temperature increase to a maximum value of 2600 K, which is in agreement with the input values of the gas temperature. Based on these plasma characteristics, the 2D modeling indicates that the reduced chemistry set is applicable for 2D GA simulation. We cannot predict the CO₂ conversion with the current 2D discharge model, because a 3D GA model would be needed to obtain the accurate CO₂ conversion without making any assumptions. The latter is however not yet feasible within a reasonable calculation time. Nevertheless, we may expect that the CO₂ conversion obtained in such multi-dimensional model should be similar to that in the 0D model, as the basic plasma characteristics, such as electron density and temperature, gas and vibrational temperature, are very similar.

In figure 25 we compare the CO₂ VDF obtained with the state-specific vibrational kinetics and the one group model. It is clear from this figure that the one group model can reproduce well the populations of the low vibrational levels [V₁–V₇], but it results in an under-estimation of the populations of the high vibrational levels [V₈–V₂₁]. This indicates that an extra group is needed to describe the distribution of the high vibrational levels. It should be noted that each extra group adds two extra equations to be solved: one for the total number density of that group and one for the mean vibrational temperature of that group. Therefore, we need to make a balance between the accuracy of the model predictions and the computational cost. In our model, the populations of the high vibrational levels are low due to the strong vibrational-translational relaxation processes, and they contribute little to the CO₂ dissociation. Therefore, it is not necessary to obtain the populations of the high vibrational levels with the highest accuracy, i.e. using the model with two groups, given the higher computational cost.

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