3. Explanation of the modeling approaches

#### 3.1 0D chemical kinetic modeling

Most models describing a detailed plasma chemistry apply the 0D chemical kinetic approach, which allows to handle a large number of species and chemical reactions, with limited computational effort. This approach is based on solving balance equations for the various species densities, based on production and loss rates, as defined by chemical reactions:

$$\frac{dn\_i}{dt} = \sum\_j \left\{ \left( a\_{ij}^{(2)} - a\_{ij}^{(1)} \right) k\_j \prod\_l n\_l^{a\_{ij}^{(1)}} \right\} \tag{1}$$

where aij (1) and aij (2) are the stoichiometric coefficients of species i, at the leftand right-hand sides of a reaction j, respectively, nl is the species density at the lefthand side of the reaction, and kj is the rate coefficient of reaction j. For the electron reactions, the energy-dependent rate coefficients are determined from the average electron energy, while the rate coefficients of the chemical reactions between neutral species or ions are adopted from the literature.

The species typically included in such models, for either pure CO2 or pure CH4, as well as the extra species included in CO2/CH4, CO2/H2O, CO2/H2, or CH4/O2 gas mixtures and in CO2/N2 and CH4/N2 mixtures are listed in Table 1. The same species can be included in the CO2/CH4, CO2/H2O, CO2/H2, and CH4/O2 models, because these combinations produce similar molecules. All the species listed in Table 1 might chemically react with each other. Hence, a large number of chemical reactions (typically up to 1000) are incorporated in these models, including electron impact reactions, electron-ion recombination, and ion-ion, ion-neutral, and neutral-neutral reactions. Details of these chemistries for the specific gas mixtures can be found, for example, in [7, 28–30, 61–63, 66, 73–77, 83].

Specifically for CO2 conversion in MW and GA plasmas, the vibrational levels of CO2 are very important, because they allow energy-efficient CO2 conversion [117], so the vibrational kinetics of CO2 must be incorporated and especially the asymmetric stretch mode of CO2, which is the most important channel for dissociation [117]. Likewise, in CO2/N2 mixtures in MW or GA discharges, the N2 vibrational levels must be included, as they can populate the CO2 vibrational levels [76]. Furthermore, also the vibrational levels of CO and O2, and some electronically excited levels, are typically taken into account in such models (see Table 1). These vibrationally and electronically excited levels are indicated in Table 1 with the symbols "V" and "E". Details about their notations can be found in [29, 76] or in Table 1 for the N2 electronically excited levels. Although vibrationally excited levels might also be important for CH4, H2O, and H2 molecules, they are not yet included in the available models, to our knowledge, as these mixtures have only been described up to now for a DBD plasma, where the vibrational levels are of minor importance [117].

the plasma characteristics vary as a function of distance traveled by the gas, in the same way as they would vary in time in a batch reactor. The time in the balance equations thus corresponds to a residence time of the gas in the reactor, and the time variation can be translated into a spatial variation by means of the gas flow

Overview of the species typically included in plasma chemistry models for pure CO2, pure CH4, as well as extra species included in CO2/CH4, CO2/H2O, CO2/H2, and CH4/O2 gas mixtures and in CO2/N2 or CH4/N2

Modeling for a Better Understanding of Plasma-Based CO2 Conversion

DOI: http://dx.doi.org/10.5772/intechopen.80436

Besides balance equations for the species densities, 0D chemical kinetic models typically also apply balance equations for the electron temperature and/or the gas temperature, again based on source and loss terms, defined by the power deposition (or electric field) and the chemical reactions. Alternatively, instead of calculating the electron temperature with a balance equation, 0D models often solve a

Boltzmann equation (e.g., Bolsig+ [118]), to calculate the electron energy distribution function (EEDF) and the rate coefficients of the electron impact reactions as a

rate.

11

Table 1.

mixtures.

Although the above balance equations only account for time variations, thus neglecting spatial variations due to transport in the plasma, spatial variations can be included in such models, by imposing a certain input power or gas temperature as a function of time. For instance, this allows to account for microdischarge filaments in a DBD, through which the gas molecules pass when flowing through the reactor, by applying a number of pulses as a function of time (see, e.g., [28, 48, 62]). In a similar way, this method can account for the power deposition profile in a MW plasma (being at maximum at the position of the waveguide) by means of a temporal profile. Thus, the plasma reactors are considered as plug flow reactors, where

Modeling for a Better Understanding of Plasma-Based CO2 Conversion DOI: http://dx.doi.org/10.5772/intechopen.80436


#### Table 1.

3. Explanation of the modeling approaches

dni

tral species or ions are adopted from the literature.

dt <sup>¼</sup> <sup>∑</sup><sup>j</sup> <sup>a</sup>ð Þ<sup>2</sup>

can be found, for example, in [7, 28–30, 61–63, 66, 73–77, 83].

Most models describing a detailed plasma chemistry apply the 0D chemical kinetic approach, which allows to handle a large number of species and chemical reactions, with limited computational effort. This approach is based on solving balance equations for the various species densities, based on production and loss

> ij � a ð Þ1 ij � �

and right-hand sides of a reaction j, respectively, nl is the species density at the lefthand side of the reaction, and kj is the rate coefficient of reaction j. For the electron reactions, the energy-dependent rate coefficients are determined from the average electron energy, while the rate coefficients of the chemical reactions between neu-

The species typically included in such models, for either pure CO2 or pure CH4, as well as the extra species included in CO2/CH4, CO2/H2O, CO2/H2, or CH4/O2 gas mixtures and in CO2/N2 and CH4/N2 mixtures are listed in Table 1. The same species can be included in the CO2/CH4, CO2/H2O, CO2/H2, and CH4/O2 models, because these combinations produce similar molecules. All the species listed in Table 1 might chemically react with each other. Hence, a large number of chemical reactions (typically up to 1000) are incorporated in these models, including electron impact reactions, electron-ion recombination, and ion-ion, ion-neutral, and neutral-neutral reactions. Details of these chemistries for the specific gas mixtures

Specifically for CO2 conversion in MW and GA plasmas, the vibrational levels of CO2 are very important, because they allow energy-efficient CO2 conversion [117], so the vibrational kinetics of CO2 must be incorporated and especially the asymmetric stretch mode of CO2, which is the most important channel for dissociation [117]. Likewise, in CO2/N2 mixtures in MW or GA discharges, the N2 vibrational levels must be included, as they can populate the CO2 vibrational levels [76]. Furthermore, also the vibrational levels of CO and O2, and some electronically excited levels, are typically taken into account in such models (see Table 1). These vibrationally and electronically excited levels are indicated in Table 1 with the symbols "V" and "E". Details about their notations can be found in [29, 76] or in Table 1 for the N2 electronically excited levels. Although vibrationally excited levels might also be important for CH4, H2O, and H2 molecules, they are not yet included in the available models, to our knowledge, as these mixtures have only been described up to now for

a DBD plasma, where the vibrational levels are of minor importance [117].

Although the above balance equations only account for time variations, thus neglecting spatial variations due to transport in the plasma, spatial variations can be included in such models, by imposing a certain input power or gas temperature as a function of time. For instance, this allows to account for microdischarge filaments in a DBD, through which the gas molecules pass when flowing through the reactor, by applying a number of pulses as a function of time (see, e.g., [28, 48, 62]). In a similar way, this method can account for the power deposition profile in a MW plasma (being at maximum at the position of the waveguide) by means of a temporal profile. Thus, the plasma reactors are considered as plug flow reactors, where

kj Y l n a ð Þ1 lj l

(2) are the stoichiometric coefficients of species i, at the left-

(1)

� �

3.1 0D chemical kinetic modeling

Plasma Chemistry and Gas Conversion

rates, as defined by chemical reactions:

(1) and aij

where aij

10

Overview of the species typically included in plasma chemistry models for pure CO2, pure CH4, as well as extra species included in CO2/CH4, CO2/H2O, CO2/H2, and CH4/O2 gas mixtures and in CO2/N2 or CH4/N2 mixtures.

the plasma characteristics vary as a function of distance traveled by the gas, in the same way as they would vary in time in a batch reactor. The time in the balance equations thus corresponds to a residence time of the gas in the reactor, and the time variation can be translated into a spatial variation by means of the gas flow rate.

Besides balance equations for the species densities, 0D chemical kinetic models typically also apply balance equations for the electron temperature and/or the gas temperature, again based on source and loss terms, defined by the power deposition (or electric field) and the chemical reactions. Alternatively, instead of calculating the electron temperature with a balance equation, 0D models often solve a Boltzmann equation (e.g., Bolsig+ [118]), to calculate the electron energy distribution function (EEDF) and the rate coefficients of the electron impact reactions as a

function of the electron energy. A more detailed description of the free electron kinetics in CO2 plasma is provided in [32–37], where a state-to-state vibrational kinetic model was self-consistently coupled with the time-dependent electron Boltzmann equation.

0D models allow to predict the gas conversion, the product yields, and selectivities, based on the calculated plasma species densities at the beginning and the end of the simulations, corresponding to the inlet and outlet of the plasma reactor. Furthermore, based on the power introduced in the plasma and the gas flow rate, the specific energy input (SEI) can be computed, and from the latter, the energy efficiency (η) can be obtained with the following formulas:

$$\text{SEI}\left(\frac{kJ}{l}\right) = \frac{Plasmapower(kW)}{Flowrate\left(\frac{l}{\text{min}}\right)} \* \text{60}\left(\frac{s}{\text{min}}\right) \tag{2}$$

$$\eta(\text{@}) = \frac{\Delta H\_R \left(\frac{k\bar{l}}{mol}\right) \* X\_{CO\_2}(\text{@})}{\text{SEI}\left(\frac{k\bar{l}}{l}\right) \* 22.4\left(\frac{l}{mol}\right)}\tag{3}$$

for gas flow and gas heating are typically combined into a multiphysics model: the calculated gas velocity is inserted in the transport equations of the plasma species, and the gas temperature determines the gas density profile and thus the chemical

Modeling for a Better Understanding of Plasma-Based CO2 Conversion

0D chemical kinetic models typically provide information about the calculated gas conversion, energy efficiency, and product formation, as a function of specific operating conditions, as well as about the underlying chemistry explaining these results. The latter will be illustrated here, based on the modeling work performed within our group PLASMANT, for pure CO2 splitting, as well as CO2/CH4, CH4/O2, CO2/H2, and CO2/H2O mixtures. For more details about the modeling results in these mixtures, and more specifically the calculated conversions, product yields and energy efficiencies, and comparison with experiments, we refer to the original research papers mentioned below, as well as two recent review papers [84, 116].

The dominant reaction pathways for CO2 splitting in a DBD plasma, as predicted

<sup>+</sup> (which recombines with electrons or O2

ions into CO

from the model in [7], are plotted in Figure 1. As a DBD is characterized by relatively highly reduced electric field values (typically above 200 Td), and thus relatively high electron energies (several eV), electron impact reactions with CO2 ground-state molecules dominate the chemistry. The most important reactions are electron impact dissociation into CO and O (which proceeds through electronically excited CO2, that is, the so-called electron impact excitation-dissociation), electron

and O and/or O2), and electron dissociative attachment into CO and O (cf. the thick black arrow lines in Figure 1). These three processes account for about 50%, 25%, and 25%, respectively, to the total CO2 conversion [28]. Because these processes require more energy than strictly needed for breaking the C=O bond (i.e., 5.5 eV), the energy efficiency for CO2 splitting in a DBD plasma is quite limited,

The CO molecules are relatively stable, but at very long residence time, they will recombine with O ions or O atoms, to form again CO2 (cf. thin black arrow lines in Figure 1). This explains why the CO2 conversion typically saturates at long residence times. Furthermore, the O atoms created upon CO2 splitting also recombine

While our calculations predict that ca. 94% of the CO2 splitting in a DBD plasma arises from the ground state, and only 6% occurs from the vibrationally excited levels [28], the situation is completely different in a MW or GA plasma. These plasmas are characterized by much lower reduced electric field values (in the order of 50–100 Td), creating lower electron energies (order of 1 eV), which are most suitable for vibrational excitation of CO2. Therefore, the CO2 splitting in MW and GA discharge is mainly induced by electron impact vibrational excitation of the

that is, up to maximum 10% for a conversion up to 30% [1].

quickly into O2 or O3, based on several processes (see also Figure 1).

reaction rates.

4. Some typical calculation results

DOI: http://dx.doi.org/10.5772/intechopen.80436

4.1 0D chemical kinetic modeling

4.1.1 Pure CO2 splitting

4.1.1.1 DBD conditions

impact ionization into CO2

4.1.1.2 MW and GA conditions

13

where ΔHR is the reaction enthalpy of the reaction under study (e.g., 279.8 kJ/mol for CO2 splitting) and XCO<sup>2</sup> is the CO2 conversion. Note that this formula is only applicable to pure CO2 splitting, but a similar formula can be applied to the other gas mixtures, using another reaction enthalpy and accounting not only for the CO2 conversion but also for the conversion of the other gases in the mixture.

#### 3.2 2D or 3D fluid modeling

Even though some spatial dependences of the plasma reactors can be taken into account in 0D chemical kinetic models, as explained above, they are not really suitable for describing detailed plasma reactor configuration or predict how modifications to the reactor geometry would give rise to better CO2 conversion and energy efficiency. For this purpose, 2D or even 3D models are required, and fluid models are then the most logical choice, because they still allow a reasonable calculation time, in contrast to, for instance, PIC-MCC simulations.

These fluid models solve a number of conservation equations for the densities of the various plasma species and for the average electron energy. The energy of the other plasma species can be assumed in thermal equilibrium with the gas. The conservation equations for the species densities are again based on source and loss terms, defined by the chemical reactions, like in the 0D models. The source of the electron energy is due to heating by the electric field, and the energy loss is again dictated by collisions. In addition, transport is now included in the conservation equations, defined by diffusion and by migration in the electric field (for the charged species) and (in some cases) by convection due to the gas velocity. Furthermore, the conservation equations are coupled with Poisson's equation for a selfconsistent calculation of the electric field distribution from the charged species densities, although more simplified quasi-neutral (QN) models have also been used [113], to further reduce the calculation time. Such a QN model neglects the nearelectrode regions and treats only the quasi-neutral bulk plasma. It does not solve the Poisson equation, but calculates the ambipolar electric field from the ion densities and the electron and ion diffusion coefficients and mobilities.

Finally, in many cases, the gas temperature and gas flow behavior are calculated with a heat transfer equation and the Navier-Stokes equations, respectively, while in GA models, the cathode heat balance can also be accounted for, to properly describe the electron emission processes. The fluid (plasma) model and the models

for gas flow and gas heating are typically combined into a multiphysics model: the calculated gas velocity is inserted in the transport equations of the plasma species, and the gas temperature determines the gas density profile and thus the chemical reaction rates.
