Computational modeling of CO₂ conversion by a solar-enhanced microwave plasma reactor

The rate of change of electron number density n_e (m⁻³), considering flow advection, diffusive transport, and plasma reactions, is described by:

$$\begin{equation}{\partial _{\text{t}}}{n_{\text{e}}} + {\text{(}}{\mathbf{u}}{ } \cdot { }\nabla {\text{)}}{n_{\text{e}}} + \nabla { } \cdot { }{{\boldsymbol{\Gamma }}_{\text{e}}} = \,\,{R_{\text{e}}}{{,}}\end{equation} \tag{ 1 }$$

where u (m s⁻¹) is the bulk flow velocity, $\nabla$ is the gradient operator, ${{\boldsymbol{\Gamma }}_{\text{e}}}$ (m⁻² s⁻¹) the electron flux, and R_e (m⁻³ s⁻¹) the net reaction rate for electrons. The electron flux ${{\boldsymbol{\Gamma }}_{\text{e}}}$ is given by:

$$\begin{equation}{{\boldsymbol{\Gamma }}_{\text{e}}} = - {\mu _{\text{e}}}{n_{\text{e}}}{{\mathbf{E}}_{\text{s}}} - {D_{\text{e}}}\nabla {{\text{n}}_{\text{e}}}{{,}}\end{equation} \tag{ 2 }$$

where ${\mu _{\text{e}}} = \frac{e}{{{m_{\text{e}}}{v_{\text{e}}}}}$ (m² V⁻¹ s⁻¹) is the electron mobility, E_s (V m⁻¹) is the electrostatic electric field, and ${D_{\text{e}}} = \frac{{{k_{\text{B}}}{T_{\text{e}}}}}{{{m_{\text{e}}}{v_{\text{e}}}}}$ (m² s⁻¹) the electron diffusion coefficient, with e (C), m_e (kg), k_B (m² kg s⁻² K⁻¹), and v_e (s⁻¹) as the electron charge, electron mass, Boltzmann constant, and electron collision frequency, respectively.

The evolution equation for the energy density of electrons, n (V m⁻³), is given by:

$$\begin{equation}{\partial _t}{n_\varepsilon } + ({\mathbf{u }} \cdot \nabla ){n_\varepsilon } + \nabla { } \cdot { }\boldsymbol{{{\Gamma }}_{{\varepsilon }}} + {\mathbf{E}}_{\text{s}}{ } \cdot { }\boldsymbol{{{\Gamma }}_{\text{e}}} = \,\,{R_{{\varepsilon }}} + {\dot Q_h},\end{equation} \tag{ 3 }$$

where R (V m⁻³ s⁻¹) represents the rate of energy change due to elastic and inelastic collisions, ${{\mathbf{E}}_{\mathbf{s}}}{ } \cdot { }{\boldsymbol{{\Gamma }}_{\mathbf{e}}}$ heating or cooling of electrons depending on whether their drift velocity is aligned or not with the electric field, ${\dot Q_{\text{h}}}$ (J m⁻³ s⁻¹) describes the heating of electrons due to the microwaves, and ${\boldsymbol{{\Gamma }}_\varepsilon }$ (V m⁻² s⁻¹) is the electron energy flux, which is given by:

$$\begin{equation}{{\boldsymbol{\Gamma }}_{{\varepsilon }}} = - {{{\mu }}_{{\varepsilon }}}{{\mathbf{E}}_{\text{s}}}{{{n}}_{{\varepsilon }}} - {{\text{D}}_{{\varepsilon }}}\nabla {{{n}}_{{\varepsilon }}}{{,}}\end{equation} \tag{ 4 }$$

where ${{{\mu }}_\varepsilon } = \frac{{\text{5}}}{{\text{3}}}{{{\mu }}_{\text{e}}}{ }$(m² V⁻¹ s⁻¹) is the electron energy mobility, and ${D_\varepsilon } = \frac{{\text{5}}}{{\text{3}}}{\mu _{\text{e}}}{T_{\text{e}}}$ (m² s⁻¹) is the electron energy diffusivity. Provided that the electron energy distribution function is Maxwellian, the mean electron energy is ${n_\varepsilon }/{n_{\text{e}}} = \frac{{\text{3}}}{{\text{2}}}{k_{\text{B}}}{T_{\text{e}}}$.

The source terms R_e and R follow from collision processes involving electrons. Considering that there are M reactions that contribute to the growth or decay of electron density and P inelastic electron-neutral collisions (in general, P≫ M), the electron source term is given by:

$$\begin{equation}{R_{\text{e}}} = \sum\limits_{r = {\text{ 1}}}^M {x_r}{k_r}{n_{\text{n}}}{n_{\text{e}}}{{,}}\end{equation} \tag{ 5 }$$

where x_r is the mole fraction of the target species for reaction r, k_r is the rate coefficient for reaction r (m³ s⁻¹), and n_n is the total number density of neutral species (m⁻³). Complementarily, the electron energy source term is obtained by summing the collisional energy change over all reactions, namely:

$$\begin{equation}{R_\varepsilon } = \sum\limits_{r = {\text{ 1}}}^P {x_r}{k_r}{n_{\text{n}}}{n_{\text{e}}}{{\Delta }}{\varepsilon _r}{{,}}\end{equation} \tag{ 6 }$$

where ${{\Delta }}{\varepsilon _r}$ (V) is the energy change due to reaction r.

The power transfer from the electromagnetic field to the electron in equation (3) is given by:

$$\begin{equation}{\dot Q_{\text{h}}} = \frac{{\text{1}}}{{\text{2}}}{\text{Re(}\sigma }{\mathbf{E}}{ } \cdot { }{{\mathbf{E}}^{\text{*}}}),\end{equation} \tag{ 7 }$$

where $\sigma$ is the electrical conductivity, E is the microwave field, ^* denotes the complex conjugate, and Re(·) represents the real component of a complex number.

The Ar–CO₂ working fluid is described as consisting of a total of 13 species, these are 7 neutral species, 2 positive ions, 3 negative ions, and 1 electronically-excited argon species, which are listed in table 1. The set of Ar species corresponds to that used by Baeva and collaborators in (Baeva et al 2018), the set of CO₂ species were compiled from the reduced CO₂ chemistry model presented by Aerts and collaborators in (Aerts et al 2015), together with the set used by Bekerom and colleagues (Bekerom et al 2019). Additionally, the carbon atom is included as a species in the model because of its role in Ar–CO₂ interaction reactions and CO dissociation reaction.

Table 1. Species in Ar–CO₂ solar-microwave plasma model.

Neutral	${\text{Ar}}$, ${\text{C}}{{\text{O}}_2}$, ${\text{CO}}$, ${{\text{O}}_3}$, ${{\text{O}}_2}$, ${\text{O}}$, ${\text{C}}$
Positive ions	${\text{A}}{{\text{r}}^ + }$, ${\text{CO}}_2^ +$
Negative ions	${{\text{e}}^ - }$, ${\text{O}}_2^ -$, ${{\text{O}}^ - }$
Excited states	${\text{A}}{{\text{r}}^*}$

For each ion and neutral species, a transport equation resembling the drift-diffusion equation for electrons is solved for the mass fraction of each species, y_j , i.e.

$$\begin{equation}\rho {\partial _{\text{t}}}{y_j} + \,\,\rho {\mathbf{u}} \cdot { }\nabla {y_j} + \nabla { } \cdot { }{{\mathbf{J}}_j} = \,\,{R_j}{{,}}\end{equation} \tag{ 8 }$$

where the subscript j indicates the j^th species, ρ (kg m⁻³) is the total mass density, R_j (kg m⁻³ s⁻¹) is the net reaction rate, and J _j (kg m⁻² s⁻¹) the mass flux describing mass transport due to migration from the ambipolar field by the electrostatic field E_s and diffusion from concentration gradients.

The net reaction rate for species j is given by:

$$\begin{equation}{{\text{R}}_{\text{j}}} = {{\text{M}}_{\text{j}}}\mathop \sum \limits_{\text{r}} {{{\upsilon }}_{{\text{jr}}}}{{\text{r}}_{\text{r}}}{{,}}\end{equation} \tag{ 9 }$$

where M_j (kg mol⁻¹) is the molar mass of species j, ${\upsilon _{jr}}$ the stoichiometric coefficient for species j in reaction r, and ${r_r} = {k_r}\prod\nolimits_j {c_j}^{{\upsilon _{jr}}}$ the reaction rate for reaction r, where k_r is the reaction rate coefficient and c_j = $\rho {y_j}/{M_j}$ is the molar concentration of species j. The mass flux for species j is defined as:

$$\begin{equation}{{{\mathbf{J}}j = \rho }}{{{y}}_{{j}}}{{\mathbf{V}}_{{j}}}{{,}}\end{equation} \tag{ 10 }$$

where ${{\mathbf{V}}_j}$ is the multicomponent diffusion velocity for species j described by a mixture-averaged mass diffusion model. The species mass fraction y_j is related to its corresponding molar fraction x_j by x_jn_n = ρ(y_j /M_j )N_A with N_A as Avogadro's number.

The Ar–CO₂ chemical kinetics model is composed of plasmachemical reactions consistent with the set of species in table 1. A total of 62 reactions are included in the Ar–CO₂ plasma chemical kinetic model, accounting for 13 elastic and ionization electron impact, 10 electron attachment and recombination, 26 neutral, 4 dissociation, and 9 ion-molecule reactions. These reactions are a compilation of the set of reactions used by Baeva and collaborators (Baeva et al 2018) for argon plasma, the set used by Aerts and collaborators (Aerts et al 2015) for reduced CO₂ plasma chemistry, that by Bekerom and colleagues (Bekerom et al 2019) for CO₂ thermochemistry, and the set by Beuthe and Chang (1997) for Ar–CO₂ interactions. The set of chemical reactions to evaluate R_j and their corresponding rate coefficients are listed in appendix. The only electron impact ionization reactions considered in the model are the ionization of the feedstock gases Ar and CO₂ shown in table A.1, together with elastic electron impact reactions of the neutral species in the model and Ar electron excitation. The rate coefficient of all elastic and ionization electron impact reactions in table A.1 are obtained using cross-section data from the Biagi-v7.1 database for argon-related species and the Morgan database for oxygen and carbon-related species, both within LxCat (Carbone et al 2021, Pitchford et al 2017 and Pancheshnyi et al 2012). The reaction rate coefficients of electron attachment and recombination reactions are listed in table A.2, electron impact dissociation reactions following molecular electronic exictation in table A.3, ion-molecule reactions are presented in table A.4, and chemical reactions involving neutral species in table A.5.

While vibrational excitation has a key role in CO₂ conversion in low-pressure microwave plasmas (Silva et al 2021) and in high temperature microwave discharges (Pietanza et al 2020), species describing vibrationally excited CO₂-related species are not included in the model. Inclusion of vibrational kinetics would involve species describing the vibrationally-excited levels of CO₂ (all 21 levels of the asymmetric stretch mode, together with the symmetric stretch and bending modes (Zhang et al 2020)), and vibrational modes of CO (63 levels (Kozák and Bogaerts 2015)) and of O₂ (three levels (Zhang et al 2020)). Meritorious examples of the modeling of CO₂ vibrational kinetics are given in (Berthelot and Bogaerts 2016) and (Wang et al 2016). Nevertheless, the work of Bekerom et al (2019) indicates that the role of vibrational kinetics is secondary compared to thermal effects at atmospheric pressure. Additionally, no photochemistry is included in the model, despite the important role of photons in plasma chemical kinetics (Capitelli et al 2017). This omission is due to the high complexity and computational cost associated with the description of radiative-collisional kinetics, especially for a broad-enough range of wavelengths suitable to describe solar radiation, and the reliance on the use of radiative properties to describe the effect of radiation on the plasma (section 3.7). Therefore, the solar-plasma kinetics model is able to describe potential CO₂ conversion enhancements due to both thermal and ion interaction effects. The set of species and reactions provide a reasonable description of CO₂ conversion by solar-microwave plasma, particularly of thermal effects, while making 2D fully-coupled computational simulations practicable.

The fluid flow through the discharge tube is described by the Navier–Stokes equations assuming laminar flow. These equations are constituted by the equation of total mass conservation, i.e.

$$\begin{equation}{\partial _{\text{t}}}\rho \, + \nabla { } \cdot {\text{ (}}\rho {\mathbf{u}}{\text{)}} = 0,\end{equation} \tag{ 11 }$$

and the equation of mass-averaged momentum conservation, i.e.

$$\begin{equation}{{\rho }}{\partial _{\text{t}}}{{{\mathbf{u}} + \rho }}\left( {{\mathbf{u}} \cdot { }\nabla } \right) \mathbf{u} = \nabla { } \cdot {{ ( - p}}{\mathbf{I}} + {\boldsymbol{\unicode{x03C4}}}{\text{),}}\end{equation} \tag{ 12 }$$

where p is the pressure, ${\boldsymbol{\unicode{x03C4}}}$ is the viscous stress tensor, and I is the identity tensor. In equation (12), the Lorentz force F_L $= \frac{{\text{1}}}{{\text{2}}}{\mu _{\text{0}}}{\text{Re(}}\sigma {\mathbf{E}} \times {{\mathbf{H}}^{\text{*}}}{\text{)}}$, where ${\mu _0}$ is the free space permeability and H the magnetostatic field, has been neglected.

The conservation of the energy of heavy-species is described by:

$$\begin{equation}\rho {C_p}({\partial _t}{T_{\text{h}}} + {\mathbf{u}} \cdot \nabla {T_{\text{h}}}) + \nabla { } \cdot { }{\mathbf{q}} = {\dot Q_{{\text{el}}}} + {\dot Q_{n - n}} + {\dot Q_r},\end{equation} \tag{ 13 }$$

where q is the heat conduction flux and C_p the specific heat at constant pressure for the mixture, i.e.

$$\begin{equation}{{{C}}_{{p}}} = \mathop \sum \limits_{{j}} \frac{{{{{y}}_{{j}}}{{{C}}_{{{p,j}}}}}}{{{{{M}}_{{j}}}}}{{,}}\end{equation} \tag{ 14 }$$

where ${C_{p,j}}$ denotes the heat capacity of species j.

The source term ${ }{\dot Q_{el}}$ represents the energy gained due to elastic collisions between electrons and heavy-species, and is modeled as:

$$\begin{equation}{ }{\dot Q_{el}} = \mathop \sum \limits_j 3\frac{{{m_e}}}{{{M_j}}}{n_e}{k_B}({T_e} - {T_h}){v_{ej}},\end{equation} \tag{ 15 }$$

where ${v_{{\text{e}}j}}$ is the collision frequency for elastic collisions with species j.

The source term ${\dot Q_{n - n}}$ describes the heat released due to non-electron collisions, and is given by:

$$\begin{equation}{\dot Q_{n - n}} = \sum\limits_j {{R_j}} ({\upsilon _{jr}}\prime - {\upsilon _{jr}}\prime \prime {\textrm{)}}{{\textrm{h}}_{\textrm{j}}},\end{equation} \tag{ 16 }$$

with $\upsilon _{jr}^{^{\prime}}$, ${\upsilon _{jr}}\prime \prime$, and h_j representing the stoichiometric coefficients for forward and backward reactions, and the enthalpy of the products and reactants, respectively. Finally, the source term ${\dot Q_r}$ describes the net energy transported due to radiation, which is described below.

Poisson's equation is used to describe the evolution of the ambipolar electric field E_s generated through the discharge tube, i.e.

$$\begin{equation}\nabla { } \cdot { }{{\mathbf{E}}_{\text{s}}}{{ = - }}{\nabla ^{\text{2}}}V = \frac{{{\rho _v}}}{{{\varepsilon _{\text{0}}}}}{{,}}\end{equation} \tag{ 17 }$$

where V is the electric potential and ρ_v = e(n⁺ —n_e —n⁻ ) is the space charge density (C m⁻³), where n⁺ is the total number of positive ions, n⁻ the total number of negative ions, and ₀ the permittivity of free-space.

In contrast to the fluid flow-related variables in the model, whose equations are formulated in the time domain, the electromagnetic field evolution model is formulated in the frequency domain. Starting from Maxwell's equations and invoking the assumption of a harmonic time variation of the electric field E (V m⁻¹) and the magnetic field H (A m⁻¹) in a non-magnetic medium, leads to the equations:

$$\begin{equation}\nabla \times {\mathbf{E}} = - i\omega {\mu _0}{\mathbf{H}}{\text{ and }}\nabla \times {\mathbf{H}} = \,\,i\omega {\varepsilon _0}{\varepsilon _{{\text{pl}}}}{\mathbf{E}}{{,}}\end{equation} \tag{ 18 }$$

where i² = −1 is the imaginary unit, μ₀ (kg m s⁻² A⁻²) is the free-space permeability, ω = 2πν_f is the angular field frequency with ν_f = 2.45 GHz as the microwave excitation frequency, and _pl is the relative electrical permittivity of the medium, which is given by the Lorentz formula:

$$\begin{equation}{{{\varepsilon }}_{{\text{pl}}}} = {\text{ 1 }} - { }\frac{{{{\omega }}_p^{\text{2}}}}{{{{\omega (\omega - i}}{{\text{v}}_{\text{m}}})}}{{,}}\end{equation} \tag{ 19 }$$

where ${v_{\text{m}}}$ is electron collision frequency for momentum transfer, and ${{\omega }}_p^{\text{2}} = \frac{{{n_{\text{e}}}{e^{\text{2}}}}}{{{\varepsilon _0}{m_{\text{e}}}}}$ is the plasma frequency. By introducing the electrical conductivity as:

$$\begin{equation}{{\sigma = }}\frac{{{{{n}}_{\text{e}}}{{{e}}^{\text{2}}}}}{{{{{\varepsilon }}_0}{{{m}}_{\text{e}}}}}\frac{{\text{1}}}{{{\text{(}}{{{v}}_{{m}}}\, - {{ i\omega }})}}{{,}}\end{equation} \tag{ 20 }$$

and combining the equations in equation (17) leads to the following equation for the microwave electric field (Baeva et al 2018, 2021):

$$\begin{equation}\nabla {{ \times (}}\nabla \times {\mathbf{E}}{\text{) }} - k_{\text{0}}^{\text{2}}\left({\text{1 }} - { }i\frac{\sigma }{{{\varepsilon _{\text{0}}}\omega }}\right){\mathbf{E}} = {\boldsymbol{0}},\end{equation} \tag{ 21 }$$

in which k₀ = ω/c₀, where c₀ is the speed of light in free space.

As described in section 4.1, a rectangular waveguide is used to connect the plasma tube to the microwave source. The rectangular port is excited by a transverse electric (TE) wave, which is a wave that has no electric field component in the direction of propagation. At the microwave excitation frequency of 2.45 GHz, TE₁₀ is the dominant mode, with the lowest cut-off frequency, depicting a half-cycle variation of the field across the width of the waveguide and no cycle variation of the field along the waveguide. As TE₁₀ is the only propagating mode through the rectangular waveguide, electrons do not experience any change in the high-frequency electric field during the microwave time scale. This means that the phase coherence between the electrons and electromagnetic waves is only destroyed through collisions with the background gas. The loss of phase coherence between the electrons and high-frequency fields is what results in energy gain for the electrons. Therefore, the momentum collision frequency is set as the collision frequency between electrons and neutral species (i.e. ${v_{\text{m}}}$ $= \,\,{{\text{v}}_{\text{e}}}$).

Radiative transport in an absorbing (i.e. non-transparent) and emitting medium is described by the radiative transport equation (RTE) (Modest 2013), which for steady-state conditions and in the absence of scattering is given by:

$$\begin{equation}{\text{s}} \cdot \nabla I ( {\mathbf{x}}{\text{, }}{\mathbf{s}}) = -\alpha I ( {\mathbf{x}}{\text{, }}{\mathbf{s}})+\varepsilon {{,}}\end{equation} \tag{ 22 }$$

where I is the (gray-)radiation intensity, which depends on the spatial location x and propagation direction s; α is the total absorption coefficient, and represents total radiative emission. The description of radiative transport in non-gray media requires the solution of a set of RTEs, one for each wavelength (or frequency, wavenumber, etc) of interest, each equation with a corresponding spectral absorption coefficient α_λ and an emission _λ . Given the high computational cost of the solution of the RTE and consistent with the species transport model described in section 3.2, a gray-medium description of radiative transport (i.e. equation (22)) is adopted as a first step towards the analysis of solar radiation—plasma interaction in the SEMP reactor.

The radiative energy term in equation (13) is given by (Modest 2013, Sun et al 2016):

$$\begin{equation}{\dot Q_r} = {{4\pi \varepsilon - \alpha G,}}\end{equation} \tag{ 23 }$$

where G is the incident radiation, which for a gray medium is G = $\int\nolimits_{{\text{4}}\pi } I{\text{d}\Omega }$, and Ω is the solid angle at the location given by the direction vector s and spanning the unitary sphere (solid angle 4π).

Given that the microwave plasma is in a state of thermal nonequilibrium, the spectral radiative properties α_λ and _λ are functions of electron and heavy-species temperatures, T_e and T_h, in addition to pressure p and the set of species number densities n_j (all needed to determine the thermodynamic state of the medium). The code SPARK (Simulation Platform for Aerodynamics Radiation and Kinetics) developed by Lino da Silva (2007) has been used to determine α_λ and _λ as a function of T_e and T_h, for p = 1 atm and composition corresponding to that of an Ar–CO₂ (7:1 vol.) mixture in chemical equilibrium. SPARK is an adaptive line-by-line numerical code that calculates spectral emission and absorption of plasma in NLTE (including LTE, by setting T_e = T_h).

The calculations of α_λ and _λ considered 34 atomic and molecular transitions, namely: atomic discrete (bound–bound) transitions for C, C⁺, O, Ar⁺ and O⁺; atomic continuum transitions: C, C⁺, O, Ar⁺, and O⁺ photoionization, O Bremstrahlung, and C⁻ and O⁻ photo-detachment; vaccum ultraviolet (VUV) continuum transitions: CO₂, C₂, CO, and O₂ photoionization; VUV-visible transitions: C₂ Phillips, Mulliken, Deslandres–Azamb, Fox–Herzberg, Ballik–Ramsaw, and Swan transitions; CO Angstrom, Asundi, CO⁴⁺, CO³⁺, CO⁺ Comet tail, CO Triplet; O₂ Shumann–Runge, and O₂ Shumann–Runge Cont; and infrared transitions: CO₂ and CO rovibrational, and O₂ Bremstrahlung. Information about the databases used in the models for the calculation of discrete and continuum radiation is given in (Lino da Silva et al 2013). The calculations involved in the order of O(10⁶ ) transitions.

Representative spectral absorption coefficient α_λ of Ar–CO₂ (7:1 vol.) is presented in figure 2(a) (left) for LTE (T_h = T_e= T) for different values of equilibrium temperature T = 500, 1000, and 1500 K and in figure 2(a) (right) for NLTE (T = T_h and T_e = 1 eV). The results show that, despite the high complexity of the absorption spectra, under both, LTE and NLTE conditions, α_λ increases with increasing temperature. Importantly, α_λ is significantly greater for NLTE compared to LTE. Nevertheless, _λ is also drastically larger for NLTE than for LTE. Determining the net effect of radiative properties of a medium in NLTE compared to LTE requires the solution of the non-gray RTE. The study in (Elahi et al 2020) involved such analysis for a one-dimensional domain with constant composition (i.e. no chemical reactions). For the fully-coupled 2D SEMP reactor model in the present work, the solution of the spectral RTE with complete consideration of the dependency of α_λ and _λ on λ, T_e, T_h, p, and n_j is largely impractical, if not unfeasible. Therefore, the model treats the plasma as a gray medium (i.e. solution of the RTE for a gray medium, equation (22)) within the range of visible wavelengths. The gray medium approximation uses total radiative properties, namely total absorption coefficient α and total emission , which are function of T_e and T_h only (given the assumption of chemical equilibrium at p = 1 atm). The total absorption coefficient α is obtained by integrating the spectral absorption coefficient α_λ over the visible range, i.e.

$$\begin{equation}\alpha = \frac{1}{{\lambda_{{\text{UV}}}{ - \lambda_{{\text{IR}}}}}}\int\limits_{\lambda_{{\text{UV}}}}^{\lambda_{{\text{IR}}}} \alpha_\lambda{{\text{d}}}\lambda, \end{equation} \tag{ 24 }$$

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where $\lambda_{UV} =$380 nm and $\lambda_{IR} =$700 nm. The total emission is determined by using Planck's relation = αI_b , where I_b is the blackbody radiation intensity. Figure 2(b) shows the resulting total absorption coefficient α as function of T_e and T_h. The surface representation of α(T_e,T_h) in figure 2(b) show that α is largely negligible under thermal equilibrium (T_e ∼ T_h) and, as expected given the results in figure 2(a), that α increases with increasing T_h and rapidly increases with increasing T_e.

A sectional view of the SEMP reactor presented in (Mohsenian et al 2019a) is shown in figure 3(a), depicting the regions of incidence of microwave power (MP) and solar power (SP), and the inflow and outflow regions together with the spatial domains for the microwave propagation model and the plasma flow model. To reduce the complexity and computational cost of simulating the complete three-dimensional (3D) reactor geometry, the cylindrical discharge tube is modelled using a 2D Cartesian domain (indicated by the regions enclosed by dashed lines in figure 3(a)). Therefore, the cylindrical discharge tube's circular cross-section in the reactor is approximated as a rectangular channel. Figure 3(b) shows the 2D computational domains (i.e. discharge tube and waveguide), with the notation used to identify the different boundaries.

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The origin of the coordinates system used to define the domains (x–y) is placed at the point of intersection between the discharge tube axis and the centerline along the waveguide. The origin of the coordinates system coincides with the focal point of the incident concentrated radiation (experimentally, the latter is determined by the relative placement of the high-flux solar simulator in front of the optical aperture). The boundary AB is constituted by a parabolic segment whose focal point coincides with the focal point of incident radiation. Therefore, the incident concentrated solar radiation entering the discharge tube can be imposed as a flux normal to the boundary AB. The boundary AB also acts as the inflow boundary, assuming that the swirl component of the flow is negligible (i.e. the flow is along the x-axis only). The boundary DC is used to define the outflow. Electromagnetic energy enters the domain through the boundary LE as a 2.45 GHz wave (i.e. microwave) in the TE₁₀ mode with amplitude corresponding to the imposed input electric power. The cut-off tube, i.e. the metal tube surrounding the reaction chamber, is not included in the computational domain to reduce the complexity and size of the discrete model. The electromagnetic modeling reported in (Mohsenian et al 2019a) indicates that this approximation would result in less power absorbed by the plasma. Therefore, this approximation can be expected to lead to a lower temperature and/or CO₂ conversion than that experimentally obtained. Nevertheless, given the shorter size of the domain and the omission of the outflow constriction (see figure 3(a)), simulation results can also be expected to under-estimate recombination events, and therefore over-estimate CO₂ conversion. The net effect of these modeling approximations can only be determined through computational simulations.

The actual (3D) spatial domain of the waveguide together with the geometric approximation for the 2D model are depicted in figure 3(c), with the surface of MP deposition highlighted. Similarly, the incident solar radiation deposition surfaces are highlighted on the actual reactor and the geometric approximation for the 2D model in figure 3(d), with the surface of SP deposition highlighted. The power deposited into the 2D Cartesian domain is matched to the actual power deposited into the SEMP reactor by scaling up the incident SP and the MP with the ratio of the power deposition surface areas of the approximated and actual domains. Specifically, the geometric scaling factor (SF) for the incident SP and the MP are given by:

$$\begin{equation}{\text{S}}{{\text{F}}_{{\text{SP}}}} = \frac{{{{{d}}_{{\text{opt}}}}}}{{{{h }}}}\frac{{{\theta }}}{{{\pi }}}{\text{ and S}}{{\text{F}}_{{\text{MP}}}} = \frac{{{{{d}}_{{\text{opt}}}}}}{{{{\pi a }}}}{{,}}\end{equation} \tag{ 25 }$$

where d_opt is the out-of-plane thickness, which is used to associate the 2D domain of the model to the actual 3D domain of the reactor; a is radius the discharge tube; h is the height of the cap representing the surface of incidence of the influx of concentrated solar radiation; and θ is the angle spanning the cap by the radial distance R from the focal point of incident solar radiation. These geometric factors are depicted in the inserts in figure 3(d). Given the geometry of the reactor, as presented in (Mohsenian et al 2019a), the values of these geometric parameters are: d_opt = 1 m (i.e. arbitrarily large length consistent with the 2D approximation), h = 2.16 mm, a = 13 mm, and $\theta$ = 37.9°. Therefore, the values of the SFs used in the simulations in section 5 are: SF_SP = 48.45 and SF_MP = 24.49.

Despite the drastic approximation of the SEMP reactor geometry and scaling of power inputs, the model captures the main characteristics of the solar-plasma CO₂ conversion process, such as plasma formation, flow development, distribution of temperatures, electric field, radiative intensity, and, especially, chemical species, as shown in the results in section 5. A more appropriate description of the reactor requires full consideration of its 3D geometry, which will be the focus of future work.

The boundary conditions needed for solving the set of model equations for the variables: n_e in equation (1), n in equation (3), set of y_j in equation (8), V in equation (17), p and u in equation (11) and equation (12), T_h in equation (13), E in equation (21), and I in equation (22) are summarized in table 2 for the microwave domain and in table 3 for the plasma domain. In these tables, P_mw is the power of the incident microwave, n is the (outer) normal to the boundary, ${v_{{\text{e,th}}}} = {\left( {{\text{8}}{k_{\text{B}}}{T_{\text{e}}}/\pi {m_{\text{e}}}} \right)^{\frac{{\text{1}}}{{\text{2}}}}}$ is the thermal velocity of electrons, n·J _j is the normal mass flux of species j, u_in (y) = u_inmax (1—(y/a)²) is the inflow velocity component along the axis x, and p₀ = 1 atm is the operating pressure. The gas temperature at the inlet (AB boundary) is set to the ambient temperature of T₀ = 300 K, and h_c is the convective heat transfer coefficient (corresponding to external natural convection in ambient air). I_in is the input radiation intensity (concentrated radiation from the solar simulator), and therefore the incident solar power P_solar = I_in S_s, where S_s = $\theta R{d_{{\text{opt}}}}$ is the area of solar power deposition, I_g is the gray radiation intensity calculated assuming a gray-body at the temperature of the corresponding boundary and with a surface emissivity of 0.01, and I_fix is the outlet (DC boundary) radiation intensity, which is set equal to 0.

Table 2. Boundary conditions for the microwave domain.

Boundary	Variable
	E
EL	${P_{{\text{mw}}}}$
IH, HG, FE, LK, JI, FD, DC, CG, JB, BA, AK	${\mathbf{n}} \times {\mathbf{E}}\, = {\boldsymbol{0}}$

Table 3. Boundary conditions for the plasma domain.

	Variable
Boundary	${n_{\text{e}}}$	${n_\varepsilon }$	yj	V	u	Th	I
AB	$- {\mathbf{n}} {\boldsymbol{\cdot}} {{\boldsymbol{\Gamma }}_{\text{e}}} = \frac{{\text{1}}}{{\text{2}}}{v_{{\text{e,th}}}}{n_{\text{e}}}$	$- {\mathbf{n}} {\boldsymbol{\cdot}} {{\boldsymbol{\Gamma }}_\varepsilon } = \frac{{\text{5}}}{{\text{6}}}{v_{{\text{e,th}}}}{n_{\text{e}}}$	${\mathbf{n}} {\boldsymbol{\cdot}} {{\mathbf{J}}_{\mathbf{j}}} = {\text{ 0}}$ a ${x_{{\text{C}}{{\text{O}}_{\text{2}}}}} = {\text{ 0}}{\text{.13}}$ ${{\text{x}}_{\text{j}}} \approx {\text{ 0}}$ b	${\mathbf{n}} \times {\varepsilon _{\text{0}}}{{\mathbf{E}}_{\text{s}}} = {\text{ 0}}$	u = (uin,0)	T0	Iin
DC	$- {\mathbf{n}} {\boldsymbol{\cdot}} {{\boldsymbol{\Gamma }}_{\text{e}}} = {\text{ 0}}$	$- {\mathbf{n}} {\boldsymbol{\cdot}} {\boldsymbol{{\Gamma }}_\varepsilon } = {\text{ 0}}$	${\mathbf{n}} {\boldsymbol{\cdot}} {{\mathbf{J}}_j} = {\text{ 0}}$	${\mathbf{n}} \times {\varepsilon _{\text{0}}}{{\mathbf{E}}_{\text{s}}} = {\text{ 0}}$	p0	−n $\cdot$ q = 0	Ifix
AD, BC	$- {\mathbf{n}} {\boldsymbol{\cdot}} {\boldsymbol{{\Gamma }}_{\text{e}}} = \frac{{\text{1}}}{{\text{2}}}{v_{{\text{e,th}}}}{n_{\text{e}}}$	$- {\mathbf{n}} {\boldsymbol{\cdot}} {\boldsymbol{{\Gamma }}_\varepsilon } = \frac{{\text{5}}}{{\text{6}}}{v_{{\text{e,th}}}}{n_{\text{e}}}$	${\mathbf{n}} {\boldsymbol{\cdot}} {{\mathbf{J}}_j} = {\text{ 0}}$	V = 0	u = 0	−n $\cdot$ q = hc(T0 -Th)	Ig

^a All ion species. ^b All neutral species apart from Ar and CO₂.

The simulations in section 5 correspond to an inflow of 8 slpm of an Ar:CO₂ mixture in the ratio 7:1 by volume, leading to a maximum inflow velocity u_inmax = 0.2 m s⁻¹ and to molar fractions equal to 0.87 and 0.13 for Ar and CO₂, respectively. Surface reactions of neutralization and de-excitation are defined over the sidewalls of the discharge tube, as well as zero mass flux for all the other species in the model.

The SEMP reactor model is implemented in Comsol Multiphysics, version 5.4 (Comsol 2022). Comsol uses a stabilized finite element method for the discretization of the set of coupled partial differential equations defining the model. The spatial domain is discretized using 6148 triangular elements. This relatively small number of elements was used due to the high computational cost of the simulations (in terms of both, CPU time and required memory), particularly due to the inclusion of radiative transport in participating media. To appropriately resolve the severe property gradients near the discharge tube walls, eight layers of boundary layer elements with a stretching factor of 1.2, with the first layer element size of 0.1 mm, are used near the AD and BC boundaries. The Debye length near the AD and BC boundaries is approximately 0.5 mm, and therefore the discretization mesh near the wall is capable to resolve the Debye length scale, as needed to resolve the charge separation implied by equation (17).

The Navier–Stokes equations that describe conservation of mass and momentum, the equation for conservation of heavy-species energy, the microwave field equation, the equations for electron and species transport, and the RTE equation are solved in the frameworks of laminar flow, heat transfer in fluids, RF, plasma, and radiation in participating media modules in Comsol, respectively.

Temporal advancement is done with a fully-implicit variable-order (1st or 2nd order) Backward Differences method with automatic step-size selection (between ∼10⁻¹⁶ and 10⁻⁵ s, with and average time-step size of ∼10⁻⁴ s), together with a highly nonlinear Newton method. A frequency-transient solution technique is used for the temporal evolution of the electromagnetic field in equation (21). Solution of the numerical model is accomplished using a segregated approach, in which the discrete linearized problem in each step is solved using the MUltifrontal Massively Parallel sparse direct Solver.

As initial conditions, uniform distribution of variables consistent with the boundary values are used, namely, p = p₀, u(x, y) = (u_in(y), 0), V = 0 V, E = 0 V m⁻¹, T_h = 300 K, n = 4 eV, n_e= 1.0 ×10¹⁷ m⁻³, x_CO2 = 0.13, x_Ar = 0.87, x_j ≈ 0 for all other neural species, and n_j = 1.0 ×10¹³ m⁻³ for all other ion species are set throughout the discharge tube. For the waveguide, E = 0 V m⁻¹ and H = 0 A-m⁻¹ are used. To initiate the solution process, the initial computational time-step $\Delta$ t is equal to 10⁻¹⁶ s. Such a small value is used to facilitate numerical convergence. Moreover, the highly nonlinear Newton method is configured with an initial damping of 10⁻⁴ and minimum damping of 10⁻⁸. Changes in the solution smaller than the relative tolerance of 0.01 define convergence in each time step.

The problem is initially solved without radiative transport (equation (22)), i.e. keeping the 12 radiative intensity variables fixed until plasma ignition is achieved (maximum value of n_e reaches ∼10¹⁶ m⁻³). Despite the omission of the radiative transport model (RTE), the fully-coupled set of model equations is numerically stiff and, therefore, still very difficult to solve. Particularly, the resolution of discharge ignition involves strong electromagnetic field-plasma interaction. During the early stages of discharge evolution, the electric field distribution rapidly changes as the electron density grows, especially when the cut-off density n_ec = 7.6 ×10¹⁶ m⁻³ at 2.45 GHz is achieved. Moreover, in the plasma-microwave interaction region, steep gradients in the spatial distribution of properties such as electron number density, electron temperature, and electric potential are observed, necessitating smaller maximum timesteps (∼10⁻⁸ s).

Once plasma ignition is achieved, the solution of the RTE is included in the solution approach. Solution of the RTE requires the discretization angular domain (i.e. the unit sphere spanned by the vector s in equation (22)), which is accomplished using the discrete ordinates method (DOM) with the S4 discretization with level symmetric even quadrature set (i.e. 12 directions). This S4 method, involving 12 inter-dependent variables to discretely describe the radiative intensity I(x,s), has been shown to be sufficient for many applications (Comsol 2022). The DOM has significant advantages over other approaches (e.g. spherical harmonics), particularly regarding solution accuracy in spatial domains with arbitrary configurations and in problems with widely varying radiative properties. However, this method is computationally expensive, and the required memory for problems with complicated geometries can rapidly exceed the available memory capacity on typical workstations. Therefore, the solver divides these 12 directional intensity variables into three groups (each includes four intensity directions). Each group is computed in a single iterative step before moving on to the next one, which results in the required memory remaining relatively low.

With the above solver and simulation set-up, each simulation was run using eight shared-memory cores (single processor) with 64 GB of memory and required on the order of 10⁻⁵ time-steps to achieve steady-state (defined as a change of less than 0.05% in the average CO₂ concentration in the outlet of the reactor).

The SEMP reactor is first simulated operating with 700 W of microwave power only (i.e. P_mw = 700 W and P_solar= 0 W by setting I_in = 0 in table 3). Figure 4 shows the distribution of all the model variables through the microwave and plasma domains, i.e. magnitude of electric field ||E|| along the waveguide and discharge tube, magnitude of electrostatic field ||E_s|| through the discharge tube, pressure p, velocity magnitude ||u||, heavy-species temperature T_h, electron temperature T_e, incident radiative intensity G (due to radiative heat transfer within the reactor chamber), electron number density n_e, and mass fractions of all participating species y_j with j = { CO₂, CO, CO₂ ⁺, Ar, Ar^*, Ar⁺, O₂, O, O⁻, O₂ ⁻, O₃, and C }.

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The distribution of ||E|| shows the propagation of microwaves, from the boundary of incident MP (boundary EL, also see insert in figure 4), along the tapered waveguide, and their absorption within the discharge tube with small-amplitude waves leaving the waveguide domain (through boundary IH). The interaction of the microwave field E with the plasma within the discharge tube leads to a distribution of the electrostatic field E_s. The distribution of E_s directly affects the distribution of electron number density n_e and especially, electron energy density ${n_\varepsilon }$. This dependency is seen in the distribution of T_e in figure 4, resembling that of ||E_s|| and depicting separate high-temperature zones indicative of standing-wave phenomena. Given the coupling between the heavy-species energy and the electron energy due to elastic collisions, the distribution of heavy-species temperature T_h appears correlated to the distribution of T_e.

The plasma region is elongated in the downstream direction due to advection by the inflow stream. The relatively high heavy-species temperature in the plasma leads to a significant reduction in mass density, and consequently, to significant acceleration of flow (i.e. the maximum velocity within the plasma is ∼ 0.84 m s⁻¹, whereas the maximum inflow velocity is 0.2 m s⁻¹) and variations in pressure.

The maximum values of T_h (∼2200 K), T_e (∼3.25 eV), and of n_e (2.86 ×10¹⁹ m⁻³) occur near the region of incidence of microwave energy (KF boundary in figure 3(b)). The distribution of incident radiation G shows that the plasma acts as a radiative source that heats the inner walls of the discharge chamber. The high values of T_h, T_e, and n_e lead to the highest degree of ionization (mass fraction of Ar⁺, CO₂ ⁺) and of CO₂ conversion. Regarding the latter, the minimum value of CO₂ mass fraction y_CO2 ∼ 0.07 and maximum mass fraction of CO y_CO ∼ 0.05 occur within the plasma core, near the region of incident MP, representing ∼6.8% conversion of CO₂.

The results in figure 4 show that the distribution of negative molecular oxygen ion (O₂ ⁻) and ozone (O₃) is highest in the region immediately upstream of the plasma core. This is attributed to the high reactivity of these species, which leads to their decomposition at high temperatures.

The maximum value of T_h of ∼2200 K is relatively low for a microwave discharge operating at atmospheric pressure (Bekerom et al 2019). This is attributed to the geometric approximations used in the 2D model, as discussed in section 4.1. The effect of the scaling factor for MP (SF_MP) used in the model is clearly appreciated in the results in figure 5. Figure 5 shows a sub-set of the results in figure 4 together with the corresponding results when no SF is applied (i.e. SF_MP = 1).

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The results in figure 5 show, as expected, that the use of the geometric SF has a dramatic effect on the results. Particularly, if no SF is used, the maximum heavy-species temperature is just above 900 K, the maximum electron temperature ∼1.35 eV, and the maximum number density ∼3.15 ×10¹⁸ m⁻³ (compared to ∼2200 K, 3.25 eV, and 2.86 ×10¹⁹ m⁻³, respectively when the scaling factor SF_MP = 24.49 is used). These relatively low values of temperatures and electron number density lead to negligible CO₂ conversion, as expected. These results show that the use of a direct 2D model (without a geometric SF) to describe the waveguide-discharge tube assembly results in poor approximation of the microwave plasma system. This result also emphasizes the need to embrace a full 3D description of the reactor geometry to describe the microwave plasma reactor operation more accurately.

To determine the effects of solar input power, the SEMP reactor is simulated operating with 700 W of microwave power and the maximum solar power of 525 W. Representative results are presented in figure 6. These results show that the incident radiation leads to significant heating of the gas, with the maximum heavy-species temperature ∼2200 K without solar power compared to ∼3040 K with solar power. Such intense heating causes minor increases in electron temperature T_e and number density n_e. The incident radiation G near the optical aperture (boundary AB) is ∼1.4 MW m⁻² when solar input power is used (and it is near 0 when it is not), which is comparable to the maximum incident radiation emitted by the plasma core (near the boundary KF). Moreover, part of the incoming solar radiation is absorbed by the discharge tube, resulting in additional heating of the walls.

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As observed in figure 2(b), the total radiative absorption coefficient α increases with increasing heavy-species temperature and/or electron temperature, leading to further increases in the absorption of solar radiation. This result suggests that the incorporation of solar radiation should lead to enhanced thermochemical reactions (i.e. among heavy-species), which as demonstrated by Bekerom and collaborators, plays a major role in CO₂ decomposition (Bekerom et al 2019). This is corroborated by the results in figure 4, which show significantly lower minimum values of CO₂ mass fraction y_CO2 and higher maximum values of CO mass fraction y_CO when input solar power is used than when it is not. The net effect of the incorporation of solar power on CO₂ conversion is discussed in section 5.5.

To observe the effect of increasing solar power, figure 7 shows the distributions of heavy-species temperature T_h and incident radiation G in the discharge tube for increasing values of solar input power, i.e. 0 (i.e. no solar power), 350, 440, and 525 W. As discussed in relation to the results in figure 6, the modeling results show that the discharge tube absorbs a significant part of incoming solar radiation, causing significant heating throughout the discharge tube, and given the increase in absorption coefficient with T_h and/or T_e, the plasma absorbs solar radiation directly as well (i.e. volumetric heating). This leads to additional heating of the plasma. The heating produced by the absorption of solar radiation is clearly observed in the results in figure 7, which show that T_h reaches a maximum value of ∼2200, 2800, 2900, and 3000 K for input solar power of 0, 350, 440, and 525 W, respectively. In all cases, the maximum temperature is found near location of incident microwave power (boundary KF). The maximum incident radiation G is ∼3.8, 12.9, 15.6, and 17.9 MW m⁻² for input solar power of 0, 350, 440, and 525 W, respectively. The location of maximum G occurs near the location of maximum T_h, although for the case of maximum solar power (525 W) the influx of radiative energy (along the boundary AB) is comparable to the amount of radiative energy emitted by the plasma.

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As an initial validation of the model, figure 7(b) shows the comparison of the experimentally-measured gas temperature at the flow outlet and the average heavy-species temperature obtained with the model at the outflow boundary (DC in figure 3(b)) as a function of solar input power and 700 W of MP (i.e. conditions used in the results in figure 7(a)). The experimental results correspond to those reported by Mohsenian et al in (2019a) of the exhaust gas temperature as measured by a K-type thermocouple. The experimental results showed an increase in exhaust temperature from 450 to 504 K with increasing solar power from 0 to 525 W. These results are contrasted against the simulation results, which show an increase in heavy-species temperature from ∼1150 to 1620 K with increasing solar power. The significantly higher temperatures obtained by the model can be a consequence of two main factors. The first one is the geometric approximation of the reactor model (i.e. 2D with geometric power SF versus 3D). And the second one is that, experimentally, the temperature is measured in the outflow region, which is not included in the computational domain and is far downstream the outflow boundary (boundary DC). Therefore, the modeling results under-predict the cooling of gas products downstream.

Simulation results of the SEMP reactor operating with different amounts of microwave power, namely 500, 700, and 900 W, for constant input solar power of 320 W are presented in figure 8.

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The distributions of heavy-species temperature T_h and incident radiation G show that absorption of microwave power causes additional heating in the plasma region leading to higher maximum temperatures. Specifically, the heavy-species temperature T_h reaches a maximum value of 2631, 2760, and 2894 K for incident microwave power of 500, 700, and 900 W, respectively. Correspondingly, the incident radiation reaches a maximum of 10.3, 11.8, and 14.2 MW m⁻², respectively. As expected, given the strong temperature-dependence of radiative emission with temperature, the location of maximum G occurs near the location of maximum T_h (i.e. near the boundary KF). Given that the radiative absorption coefficient increases with increasing T_h and/or T_e, the higher temperature within the plasma leads to more intense radiative energy exchange within the discharge tube. This inter-dependence limits further heating, partially explaining why, despite an 80% increase in microwave power (from 500 to 900 W) and almost 50% increase in total (microwave plus solar) power (from 820 to 1220 W), the maximum temperature within the discharge tube increases by only ∼10% (from 2631 to 2894 K).

The changes in species number density along the main axis of the reactor (i.e. along the x-axis), for all the species considered in the model, are depicted in figure 9, for 700 W of microwave power together with either 0 W or 525 W of solar input power. It is to be noted that the focal point for the incident solar radiation is located at x = 0 cm.

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The distributions of neutral species in figure 9(a) shows that the incorporation of solar input power leads to significantly more rapid conversion of Ar and CO₂ species within the plasma core (region −0.008 < x < 0.008). The enhancement of CO₂ conversion by the incorporation of plasma can be observed in the values of species number densities at the outflow boundary (x = 0.12 m). The fact that the CO₂ number density n_CO2 increases, while the CO number density n_CO decreases, both from x ∼ 0.02 m to the end of the plasma domain (x = 0.12 m) is indicative of the significant role of recombination. This result suggests that rapid quenching of products, soon after the core plasma region, can significantly enhance CO₂ conversion. Moreover, this result indicates that the SEMP modeling results are expected to over-predict CO₂ conversion. This aspect of the model is quantitatively addressed in section 5.5.

The results in figure 9(a) also show that the enhanced heating due to the incorporation of solar power has a minor effect on excited argon species Ar^*, but leads to significant increases in O, O₃, and C species at the outlet, with the increase more pronounced for ozone. As already observed in the results in figure 4, ozone (as well as O₂ ⁻) present very high reactivity at the intermediate temperatures found in the upstream region immediately adjacent to the plasma code (i.e. −0.02 < x < 0). This leads to an abrupt and high increase in the number density of O₃ in that region.

Figures 9(b) and (c) show the distributions of negatively-charged species and positively-charged species, respectively, along the main axis of the reactor. The results show that the maximum degree of ionization of the plasma (primarily given by the number densities of e⁻ and Ar⁺ species) remains relatively unaffected by the incorporation of input solar power. Nevertheless, it is interesting to note that the incorporation of solar power leads to higher densities of O⁻ and lower densities of CO₂ ⁺. This dependency is attributed to the role of heavy-species temperature (which increases with the incorporation of solar radiation) on the reaction rate constants associated with these species (e.g. reactions A1 and A10 in table A.1 in appendix). Additionally, the density of O₂ ⁻ is significantly larger when solar power is used, a result that is consistent with the increased population of O₃, as discussed above.

The efficacy of CO₂ conversion by the SEMP reactor is assessed by the conversion efficiency η_c, i.e.

$$\begin{equation}\eta {\text{c}} = \frac{{x_{{\text{C}}{{\text{O}}_{\text{2}}}}^{{\text{in}}} - \overline {x_{{\text{C}}{{\text{O}}_{\text{2}}}}^{{\text{out}}}} }}{{x_{{\text{C}}{{\text{O}}_{\text{2}}}}^{{\text{in}}}}}{{,}}\end{equation} \tag{ 26 }$$

where $x_{{\text{C}}{{\text{O}}_{\text{2}}}}^{{\text{in}}}$ and $\overline {x_{{\text{C}}{{\text{O}}_{\text{2}}}}^{{\text{out}}}}$ are the input and average output molar fractions of CO₂, respectively.

Figure 10 shows comparisons between the experimentally-obtained CO₂ conversion efficiency and that obtained by the SEMP reactor model. Figure 10(a) shows η_c as a function of solar input power, i.e. from zero power to the maximum 525 W delivered by the high-flux solar simulator, while the microwave power is fixed to 700 W, and figure 10(b) shows η_c as a function of microwave power, from 500 to 900 W, for 320 W of solar power. In the experiments, $\overline {x_{{\text{C}}{{\text{O}}_{\text{2}}}}^{{\text{out}}}}$ is measured right after the constriction following the exhaust port of the discharge tube (see figure 3(a)) by gas chromatography. In the simulation results, $\overline {x_{{\text{C}}{{\text{O}}_{\text{2}}}}^{{\text{out}}}}$ corresponds to the average value x_CO2 over the outflow boundary (DC in figure 3(b)).

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The experimental results from (Mohsenian et al 2019a) in figure 10(a) show an increase in η_c from 6.2% to 9.0% by adding 525 W of solar power to 700 W of microwave power. In contrast, the simulation results show an increase in conversion efficiency from 6.8% to 10.0% with 525 W of solar input power. These results can be contrasted against the results of the variation in outlet gas temperature in figure 7(b), which show the increase in temperature with input solar power and that the experimental temperatures are significantly lower than those obtained with the model. Together, these results indicate that the model over-predicts the effect solar radiation on CO₂ conversion. This may be due to over-prediction of the absorption coefficient leading to relatively greater temperatures within the plasma but lower temperatures near the exhaust port, as well as by the geometric approximation implied in the 2D reactor model.

Results of the effect of microwave power on CO₂ conversion efficiency in figure 10(b) show that the agreement between the experimental results in (Mohsenian et al 2019b) and the computational modeling results improves with increasing microwave power. Specifically, the experimental results show η_c increasing from ∼6% to ∼9% and the computational results η_c increasing from ∼9% to ∼10% with increasing microwave power from 500 to 900 W. The discrepancy between results is mainly attributed to the under-prediction of recombination reactions by the model. Specifically, given the limited extent of the plasma domain in the model, increasing microwave power leads to increased temperatures throughout the domain, including the near-outlet region, and consequently to lower rates of recombination reactions. Moreover, given the omission of photon-driven chemical kinetics in the model together with the use of a gray absorption coefficient, the model is only capable to describe thermal effects driven by the influx of solar radiation. The incorporation of photon-driven reactions, particularly given the high intensity of radiative fluxes, can have a significant effect in the prediction of CO₂ decomposition by the model.