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

The use of renewable energy to convert carbon dioxide (CO2) into higher-value products can help meet the demand for fuels and chemicals while reducing CO2 emissions. Solar-Enhanced Microwave Plasma (SEMP) CO2 conversion aims to combine the scalability and sustainability of solar thermochemical methods with the high efficiency and continuous operation of plasmachemical approaches. A computational study of a built SEMP reactor operating with up to 1250 W of microwave power together with up to 525 W of incident solar power at atmospheric pressure is presented. The study is based on a fully-coupled 2D computational model comprising the description of fluid flow, heat transfer, Ar-CO2 chemical kinetics, energy conservation for electrons and heavy-species, electrostatics, and radiative transport in participating media through the discharge tube, together with the description of the microwave electromagnetic field through the waveguide and the discharge tube. Numerical simulations reveal that the plasma is concentrated near the location of incident microwave energy, which is aligned with the radiation focal point, and that CO2 decomposition is highest in that region. The incident solar radiation flux leads to more uniform distributions of heavy-species temperature with moderately greater values throughout most of the discharge tube. Modeling results show that, at 700 W of electric power, conversion efficiency increases from 6.8% to 10.0% with increasing solar power from 0 to 525 W, in good agreement with the experimental findings of 6.4% to 9.2%. The enhanced process performance is a consequence of the greater power density of the microwave plasma due to the absorption of solar radiation.


Introduction
Global energy demands have risen consistently over the previous few decades, with fossil fuels meeting most of the increase (International Energy Agency 2018). As a result, energy-related emissions of carbon dioxide (CO 2 ), the primary greenhouse gas, have continually increased, exacerbating the need to adopt sustainable energy solutions to avoid the worst impacts of climate change. The sustainable use of CO 2 for the synthesis of value-added products such as organic acids, esters, alcohols, etc, can play an important role in limiting CO 2 emissions (Zhang et al 2017). Particularly, the synthesis of solar fuels, i.e. hydrocarbons from CO 2 , water, or other feedstock using solar energy, is seen as a sustainable way to meet the demand for fuels in difficult-to-decarbonize sectors, such as long-haul aviation and maritime transportation (Detz et al 2018). Solar fuels can be produced via direct or indirect processes. Direct processes harness the energy in solar radiation to produce fuel without an intermediary energy conversion step. Among direct processes, solar thermochemical ones, which make direct use of solar radiation to drive high-temperature endothermic chemical reactions, are very promising due to their potential for scalability and lower costs (Schäppi et al 2022). However, direct processes suffer from the intermittent nature of solar radiation, which limits their economic viability (Roy 2010, Abedini Najafabadi et al 2019. In contrast, indirect processes have solar energy converted to another form of energy, such as biomass or electricity, first, and then use that energy in fuel synthesis. By de-coupling the reception and conversion of solar energy, and moreover, by allowing the use of other forms of renewable energy such as wind or geothermal, indirect processes can operate continuously. Among indirect approaches, plasma-based CO 2 conversion processes have demonstrated high conversion and energy efficiencies, and great potential for industrial-scale deployment (Liu et al 1999, Kozák and Bogaerts 2014, 2015, Lindon and Scime 2014, Snoeckx and Bogaerts 2017. Solar thermochemical processes largely rely on the absorption of radiative energy by a catalytic solid medium to increase its temperature and promote chemical reactions over its surface, given that the most common feedstock gases (CO 2 , H 2 O, CH 4 , etc) are transparent to solar photons (Lapp et al 2012). These processes depend on high-temperature reactors and high concentration of solar radiation to achieve efficient operation (Marxer et al 2017). Typically, solar-driven thermochemical CO 2 decomposition processes operate at temperatures above 1400 K (Buelens et al 2016). Operating temperatures above 2500 K are often needed to achieve reasonable degrees of CO 2 decomposition in single-step reduction or direct thermolysis processes (Smestad and Steinfeld 2012). For instance, thermodynamic and chemical equilibrium considerations indicate that at temperatures above 2700 • C, nearly 50% of CO 2 is decomposed in a single-step process (Abanades and Chambon 2010). In contrast, two-step CO 2 reduction processes, commonly based on the use of metal oxides operating in reduction and oxidation cycles, can operate effectively at significantly lower temperatures (Al-Shankiti et al 2017). A review of the state-of-the-art solar thermochemical approaches for CO 2 decomposition is given in (Pullar et al 2019).
Plasmachemical approaches are generally based on driving the feedstock gas to a plasma state using electrical discharges. Among the different types of plasma sources (arc, glow, corona discharges, inductively coupled, etc (Fridman 2008)), microwave plasma has potentially the highest energy efficiencies for molecular dissociation thanks to effective stepwise vibrational excitation, an effective channel for dissociation; as well as relatively high heavy-species temperature, which promotes thermal dissociation (van Rooij et al 2015, Bongers et al 2017, Silva et al 2017. Such characteristics make microwave plasma sources ideal for gas-phase chemical synthesis, including the decomposition of CO 2 . A significant amount of research has been performed on CO 2 decomposition by microwave plasma. Among these, approaches based on (near) atmospheric pressure operation are particularly appealing for their potentially greater viability, given that the absence of vacuum systems can lead to less expensive installations and to greater compatibility with other unit operations. Mitsingas and collaborators investigated CO 2 conversion by atmospheric pressure microwave plasma, achieving a maximum conversion of 9% with an energy efficiency of 50% at a Specific Energy Input (SEI, amount of energy per molecule) of 0.5 eV mol −1 (Mitsingas et al 2016). Their results showed that CO 2 conversion increases significantly with decreasing flow rate and that the effect of electric power on conversion was not significant. Spencer and Gallimore investigated CO 2 decomposition in an atmospheric pressure microwave plasma-catalyst system for a wide range of SEI, from 2 to 28 eV mol −1 Gallimore 2010, 2012). Their results showed that by increasing the SEI, conversion efficiency increases, but energy efficiency decreases. Their reported highest conversion efficiency of around 45% occurred at a SEI of 28 eV mol −1 , corresponding to their lowest value of energy efficiency of around 5%. Bekerom and collaborators established the importance of thermal conversion in the dissociation of CO 2 in microwave discharges (Bekerom et al 2019). The authors found that thermal processes dominate at higher power densities, favoring thermal dissociation over radial energy transport-a particularly relevant finding in atmospheric pressure processes and in processes that receive additional energy inputs, such as solar radiation in the present work.
Exploiting the advantages of solar thermochemical methods (i.e. direct use of solar radiation as a free form of sustainable energy and process scalability) and plasmachemical approaches (i.e. flexibility and continuous operation afforded by their reliance on electrical energy) into integrated solarplasma processes could lead to more viable and/or sustainable CO 2 conversion (Trelles 2022). Specifically, solar-plasma processes can utilize electricity to compensate for fluctuations in solar radiation (during daytime) or even substitute it (at nighttime) and therefore can operate continuously. Simultaneously, the direct use of solar radiation limits the sustainability penalties associated with the generation, transmission, and storage of electricity. Moreover, a previous study by the authors (Elahi et al 2020Elahi and Trelles 2021a, 2021b showed that CO 2 in nonequilibrium plasma state depicts drastically greater solar radiation absorption than CO 2 in thermodynamic equilibrium. The absorbed radiation can lead to new chemical pathways and enhanced CO 2 conversion. Solar-plasma processes can be divided between Plasma-Enhanced Solar thermochemical (PES) and Solar-Enhanced Plasmachemical (SEP), depending on the dominance of either incident solar power or electric input power, respectively (Trelles 2022). In PES processes, concentrated solar power is greater than the electric power used to sustain the plasma. Therefore, the role of plasma is to enhance solar thermochemistry. An example of a PES process is the decomposition of CO 2 using the solar-gliding arc reactor by Nagassou et al (2019), (2020). The reactor was designed to operate with up to 240 W of electric power and up to 525 W of concentrated solar power, at atmospheric pressure conditions and with feedstock ranging from 100% CO 2 to CO 2 mixtures with water, nitrogen, and methane. In SEP processes, the amount of electric power to sustain the plasma is greater than the solar input power, whose role is to augment plasma-driven chemical reactions. A distinct implementation of a SEP process is the conversion of CO 2 using the Solar-Enhanced Microwave Plasma (SEMP) reactor devised by Mohsenian et al (2019a), (2019b). The SEMP reactor was designed to operate with up to 1250 W of microwave power (MP) at 2.45 GHz and up to 525 W of concentrated solar power at atmospheric pressure conditions with argon-CO 2 and nitrogen-CO 2 mixtures.
The present work constitutes the first computational study of solar-plasma CO 2 conversion in a SEMP reactor. Specifically, this study focuses on unveiling characteristic features of solar-plasma CO 2 conversion, particularly the role of solar radiation, during operation of the SEMP reactor under the conditions experimentally investigated in Mohsenian et al (2019a) and (2019b). The computational solar-plasma reactor model couples a microwave plasma flow model, based on previously-reported models (i.e. Baeva et al 2018, Bekerom et al 2019, Baeva et al 2021 to a novel model of radiative transport in participating nonequilibrium plasma media to describe the effect of solar radiation on CO 2 conversion. The model is validated with experimentally-determined outflow temperature and CO 2 conversion efficiency as function of input electric power and input solar power. Future developments of the model will be aimed at integrating comprehensive descriptions of photon-driven and photon-mediated chemical kinetics and to guide reactor design and process improvements.

SEMP reactor
SEMP chemical conversion aims to combine the advantages of solar thermochemical and microwave plasma processes. The SEMP CO 2 conversion system developed by Mohsenian et al (2019a) is schematically depicted in figure 1. The reactor was designed to operate with up to 1250 W of electric power from a 2.45 GHz magnetron and up to 525 W of concentrated solar power (effective incident power from a 6.5 kW high-flux solar simulator), using as feedstock CO 2 diluted in argon or nitrogen.
In the SEMP reactor, the input electromagnetic power propagates along the waveguide, which is tapered at the point of plasma formation. A concentrator is used to direct the flux of solar radiation from a high-flux solar simulator into the SEMP reactor. Radiation enters the reactor via an optical aperture at the base of a conical chamber. The conical chamber allows the uninterrupted transmission of the incoming concentrated solar radiation to its focal point inside the reactor, which by design coincides with the location of plasma formation. The feedstock gas enters the conical chamber tangentially near the aperture. The interaction between concentrated solar radiation and microwave plasma, starting at the concentrator's focal point inside the discharge tube and proceeding downstream, leads to the absorption of photons by plasma species. The experimental results obtained by Mohsenian et al showed that up to 20% of incident solar radiation is absorbed by the microwave plasma, and that the absorbed radiation led to an increase from ∼ 6.4% to 9.2% in CO 2 conversion Mohsenian et al (2019a). Further studies focused on the effect of SEI and the use of argon-nitrogen mixtures on CO 2 conversion (Mohsenian et al 2019b). The enhanced conversion seemed to be due to the greater power density of the microwave plasma, which, as indicated by the work of Bekerom et al (2019), favors CO 2 dissociation. The use of solar radiation appears to be effective at depositing power into the plasma, circumventing the skin effect that typically limits electric power deposition in microwave discharges.

Mathematical model
The analysis of solar radiation-plasma interaction requires the concurrent description of physical-chemical-radiative phenomena due to the reception of electric power to sustain the plasma and radiative power from concentrated solar radiation. The SEMP reactor model is based on a fluid flow (continuum) approximation that encompasses the description of chemical kinetics, energy conservation of electrons and heavy-species (molecules, atoms, ions), fluid flow, electrostatics, and radiative transport in participating media through the discharge tube, together with the description of the microwave electromagnetic field through the waveguide and the discharge tube. The reactor operates at atmospheric pressure with an argoncarbon dioxide mixture (Ar:CO 2 ) supplied in a 7:1 ratio by volume as gas feedstock.
The model considers the plasma in a state of chemical nonequilibrium and non-local thermodynamic equilibrium (NLTE), which includes thermal nonequilibrium. Under thermal nonequilibrium, free electrons and heavy-species have different Maxwellian velocity distributions characterized with an electron temperature T e and a heavy-species temperature T h , respectively, with T e > T h . It is to be noted that the relatively high values of T h encountered in microwave discharges at atmospheric (or higher) pressure often makes them to be considered as quasi-thermal (Trelles 2019). This characterization contrasts with thermal plasmas (e.g. high-intensity arcs and radio-frequency discharges at relatively high pressures), which depict large regions where the LTE assumption is valid, and with atmospheric pressure nonthermal plasmas (e.g. glow and corona discharges), which depict thermal nonequilibrium throughout their extent.

Electron transport
The rate of change of electron number density n e (m −3 ), considering flow advection, diffusive transport, and plasma reactions, is described by: where u (m s −1 ) is the bulk flow velocity, ∇ is the gradient operator, Γ e (m −2 s −1 ) the electron flux, and R e (m −3 s −1 ) the net reaction rate for electrons. The electron flux Γ e is given by: where µ e = e meve (m 2 V −1 s −1 ) is the electron mobility, E s (V m −1 ) is the electrostatic electric field, and D e = kBTe meve (m 2 s −1 ) the electron diffusion coefficient, with e (C), m e (kg), k B (m 2 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 ε (V m −3 ), is given by: where R ε (V m −3 s −1 ) represents the rate of energy change due to elastic and inelastic collisions, E s · Γ e heating or cooling of electrons depending on whether their drift velocity is aligned or not with the electric field,Q h (J m −3 s −1 ) describes the heating of electrons due to the microwaves, and Γ ε (V m −2 s −1 ) is the electron energy flux, which is given by: where µ ε = 5 3 µ e (m 2 V −1 s −1 ) is the electron energy mobility, and D ε = 5 3 µ e T e (m 2 s −1 ) is the electron energy diffusivity. Provided that the electron energy distribution function is Maxwellian, the mean electron energy is n ε /n e = 3 2 k B T e . The source terms R e and R ε 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: where x r is the mole fraction of the target species for reaction r, k r is the rate coefficient for reaction r (m 3 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: x r k r n n n e ∆ε r , where ∆ε 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: where σ is the electrical conductivity, E is the microwave field, * denotes the complex conjugate, and Re(·) represents the real component of a complex number.

Species transport
The Ar-CO 2 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 CO 2 species were compiled from the reduced CO 2 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-CO 2 interaction reactions and CO dissociation reaction.
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.
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: where M j (kg mol −1 ) is the molar mass of species j, υ jr the stoichiometric coefficient for species j in reaction r, and r r = k r ∏ j c j υjr the reaction rate for reaction r, where k r is the reaction rate coefficient and c j = ρy j /M j is the molar concentration of species j. The mass flux for species j is defined as: where 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 j n n = ρ(y j /M j )N A with N A as Avogadro's number. The Ar-CO 2 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-CO 2 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 CO 2 plasma chemistry, that by Bekerom and colleagues (Bekerom et al 2019) for CO 2 thermochemistry, and the set by Beuthe and Chang (1997) for Ar-CO 2 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 CO 2 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  While vibrational excitation has a key role in CO 2 conversion in low-pressure microwave plasmas (Silva et al 2021) and in high temperature microwave discharges (Pietanza et al 2020), species describing vibrationally excited CO 2 -related species are not included in the model. Inclusion of vibrational kinetics would involve species describing the vibrationallyexcited levels of CO 2 (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 O 2 (three levels (Zhang et al 2020)). Meritorious examples of the modeling of CO 2 vibrational kinetics are given in  and . 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 CO 2 conversion enhancements due to both thermal and ion interaction effects. The set of species and reactions provide a reasonable description of CO 2 conversion by solar-microwave plasma, particularly of thermal effects, while making 2D fullycoupled computational simulations practicable.

Total mass and momentum conservation
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.
and the equation of mass-averaged momentum conservation, i.e.
where p is the pressure, τ is the viscous stress tensor, and I is the identity tensor. In equation (12), the Lorentz force F L = 1 2 µ 0 Re(σE × H * ), where µ 0 is the free space permeability and H the magnetostatic field, has been neglected.

Heavy-species energy conservation
The conservation of the energy of heavy-species is described by: where q is the heat conduction flux and C p the specific heat at constant pressure for the mixture, i.e.
where C p,j denotes the heat capacity of species j. The source termQ el represents the energy gained due to elastic collisions between electrons and heavy-species, and is modeled as:Q where v ej is the collision frequency for elastic collisions with species j. The source termQ n−n describes the heat released due to non-electron collisions, and is given by: with υ ′ jr , υ ′ ′ jr , and h j representing the stoichiometric coefficients for forward and backward reactions, and the enthalpy of the products and reactants, respectively. Finally, the source termQ r describes the net energy transported due to radiation, which is described below.

Electrostatic field
Poisson's equation is used to describe the evolution of the ambipolar electric field E s generated through the discharge tube, i.e.
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 ε 0 the permittivity of free-space.

Microwave field
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 nonmagnetic medium, leads to the equations: where i 2 = −1 is the imaginary unit, µ 0 (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 ε pl is the relative electrical permittivity of the medium, which is given by the Lorentz formula: where v m is electron collision frequency for momentum transfer, and ω 2 p = nee 2 ε0me is the plasma frequency. By introducing the electrical conductivity as: and combining the equations in equation (17) leads to the following equation for the microwave electric field (Baeva et al 2018(Baeva et al , 2021: in which k 0 = ω/c 0 , where c 0 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, TE 10 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 TE 10 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 m = v e ).

Radiative energy transport
Radiative transport in an absorbing (i.e. non-transparent) and emitting medium is described by the radiative transport 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 ε 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 ε 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: where G is the incident radiation, which for a gray medium is G =´4 π IdΩ, and Ω is the solid angle at the location given by the direction vector s and spanning the unitary sphere (solid angle 4π).

Radiative properties
Given that the microwave plasma is in a state of thermal nonequilibrium, the spectral radiative properties α λ and ε 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 ε λ as a function of T e and T h , for p = 1 atm and composition corresponding to that of an Ar-CO 2 (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 ε λ considered 34 atomic and molecular transitions, namely: atomic discrete (boundbound) 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: CO 2 , C 2 , CO, and O 2 photoionization; VUV-visible transitions: C 2 Phillips, Mulliken, Deslandres-Azamb, Fox-Herzberg, Ballik-Ramsaw, and Swan transitions; CO Angstrom, Asundi, CO 4+ , CO 3+ , CO + Comet tail, CO Triplet; O 2 Shumann-Runge, and O 2 Shumann-Runge Cont; and infrared transitions: CO 2 and CO rovibrational, and O 2 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(10 6 ) transitions.
Representative spectral absorption coefficient α λ of Ar-CO 2 (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, ε 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 ε 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 ε, 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.
where λ UV =380 nm and λ IR =700 nm. The total emission ε is determined by using Planck's relation ε = αI b , where I b is the blackbody radiation intensity. Figure 2 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.
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 TE 10 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 CO 2 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 CO 2 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: 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 θ = 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 CO 2 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.

Boundary conditions
The boundary conditions needed for solving the set of model equations for the variables: n e in equation (1), n ε 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 e,th = (8k B T e /π m e ) 1 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 p 0 = 1 atm is the operating pressure. The gas temperature at the inlet (AB boundary) is set to the ambient temperature of T 0 = 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 = θRd 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. The simulations in section 5 correspond to an inflow of 8 slpm of an Ar:CO 2 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 CO 2 , 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.

Solution approach
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 heavyspecies 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 = p 0 , u(x, y) = (u in (y), 0), V = 0 V, E = 0 V m −1 , T h = 300 K, n ε = 4 eV, n e = 1.0 ×10 17 m −3 , x CO2 = 0.13, x Ar = 0.87, x j ≈ 0 for all other neural species, and n j = 1.0 ×10 13 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 timestep ∆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 ∼10 16 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 ×10 16 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 interdependent 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 CO 2 concentration in the outlet of the reactor).

Operation with microwave power only
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 = { CO 2 , CO, CO 2 + , Ar, Ar O 3 , and C }. 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 smallamplitude 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 ε . 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 ×10 19 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 + , CO 2 + ) and of CO 2 conversion. Regarding the latter, the minimum value of CO 2 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 CO 2 .
The results in figure 4 show that the distribution of negative molecular oxygen ion (O 2 − ) and ozone (O 3 ) 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).
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 ×10 18 m −3 (compared to ∼2200 K, 3.25 eV, and 2.86 ×10 19 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 CO 2 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.

Effect of solar input power
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.
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 CO 2 decomposition (Bekerom et al 2019). This is corroborated by the results in figure 4, which show significantly lower minimum values of CO 2 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 CO 2 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.
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 Figure 5. Effect of microwave power scaling factor. Modeling results of the SEMP reactor operating with 700 W of microwave power and no input solar power (a) with microwave power scaling factor (SF MP = 24.49) and (b) without scaling factor (SF MP = 1.0). Distribution of heavy-species temperature T h , velocity magnitude ||u||, electron temperature Te, electron number density ne, incident radiation G, mass fraction of CO 2 y CO2 , and mass fraction of CO y CO . 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.

Effect of microwave power
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.
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 interdependence 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).

Effect of solar power on species distributions
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.
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 CO 2 species within the plasma core (region −0.008 < x < 0.008). The enhancement of CO 2 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 CO 2 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 CO 2 conversion. Moreover, this result indicates that the SEMP modeling results are expected to over-predict CO 2 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, O 3 , 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 O 2 − ) 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 O 3 in that region. Figures 9(b) and (c) show the distributions of negativelycharged 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 CO 2 + . This dependency is attributed to the role of heavyspecies 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 O 2 − is significantly larger when solar power is used, a result that is consistent with the increased population of O 3 , as discussed above.

CO 2 conversion
The efficacy of CO 2 conversion by the SEMP reactor is assessed by the conversion efficiency η c , i.e.
where x in CO2 and x out CO2 are the input and average output molar fractions of CO 2 , respectively. Figure 10 shows comparisons between the experimentallyobtained CO 2 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, x out CO2 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, x out CO2 corresponds to the average value x CO2 over the outflow boundary (DC in figure 3(b)).
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 CO 2 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 CO 2 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 photondriven reactions, particularly given the high intensity of radiative fluxes, can have a significant effect in the prediction of CO 2 decomposition by the model.

Conclusion
Solar-Enhanced Microwave Plasma (SEMP) conversion aims to the scalability and sustainability of solar thermochemical methods with the high efficiency and continuous operation of plasmachemical approaches. This paper presents the first computational study of a built and experimentally characterized SEMP reactor for the conversion CO 2 . The study is based on a fully-coupled 2D model of SEMP reactor developed and characterized by Mohsenian et al (2019a), (2019b). The model encompasses the description of fluid flow, heat transfer, energy conservation for electrons, energy conservation for heavy-species, electrostatics, and radiative transport in participating media through the discharge tube, together with the description of the microwave electromagnetic field through the waveguide and the discharge tube. The model is based on a 2D description of the actual reactor, together with the incorporation of geometric scaling factor to scale-up the incident microwave power and incident solar power. The model is used to simulate the operation of the SEMP reactor at atmospheric pressure conditions, with 8 slpm of an Ar-CO 2 mixture (1:7 by volume), powered with 500-900 W of microwave power and from 0 to 525 W of solar power from a high-flux solar simulator used in the experiments.
Modeling results show that absorption of solar power leads to greater temperatures throughout the discharge tube. The maximum incident radiation inside the reactor is comparable with maximum radiation emitted by the plasma (at the plasma core, near the location of incident microwave power) when the reactor operates at the maximum solar power of 525 and 700 W of microwave power. The results indicate that, with 325 W of incident solar power, by increasing microwave power from 500 to 900 W, the conversion efficiency increases from ∼9% to ∼10%. Importantly, conversion efficiency increases from 6.8% to 10.0% with increasing solar power from 0 to 525 W, in good agreement with the experimental findings of 6.4% to 9.2%. The incorporation of solar radiation appears to be an effective means to increase the power density of the plasma circumventing the skin effect that typically limits the operation of microwave discharges at high power levels.
Given the absence of photon-driven kinetics, the model describes solar enhancement due to thermal effects only (i.e. heating due to interaction with solar radiation). Consistent with this simplification, the observed enhancement appears to be a consequence of the greater power density of the microwave plasma (i.e. greater heavy-species temperature) due to the absorption of solar radiation. Future efforts will be aimed at unveiling specific mechanisms of CO 2 conversion enhancement, particularly the incorporation of photon-driven kinetics (collisional-radiative effects), and on adopting a more accurate geometric description of the SEMP reactor (i.e. 3D model geometry encompassing the rectangular tapered waveguide and the cylindrical discharge tube).

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).  (Beuthe and Chang 1997). c Reaction rate coefficient expression taken from (Ionin et al 2007). d Reaction rate coefficient expression taken from (Hokazono et al 1998). Notes: N1-N17 reaction rates are of the form ki (T h ) = k 0,i exp (−Ea,i/kBT h ) where k 0,i is the reaction rate coefficients and Ea,i the activation energy. kB is the Boltzmann constant, and T h is the heavy-species temperature as expressed by Bekerom et al (2019). All other rate coefficients are in (m 3 · s) for the two-body reactions and in (m 6 · s) for the three-body reactions. a Reaction rate coefficient expression taken from (Fridman 2008). b Reaction rate coefficient expression taken from . c Reaction rate coefficient expression taken from (Hadj-Ziane et al 1992). d Reaction rate coefficient expression taken from (Beuthe and Chang 1997). e Reaction rate coefficient expression taken from (Aerts et al 2015).