A Pressurized High-Flux Solar Reactor for the Thermochemical Gasification of Charcoal Slurry—Two-Phase Flow and Heat Transfer Analysis

Vis-à-vis conventional autothermal gasification, solar-driven allothermal gasification is free of pollutants and/or combustion by-products, yielding high-quality syngas with low CO₂ intensity because its calorific value is solar-upgraded over that of the original feedstock by an amount equal to the enthalpy change of the endothermic reaction. Furthermore, an upstream air-separation unit is not required as steam is the only oxidant [1]. Solar-driven gasification has been experimentally demonstrated with various solar reactor concepts based on packed beds [3–9], molten salts [10], fluidized beds [11–14], drop tubes [15–17], and entrained flows [18–20]. Of particular interest is the latter featuring a continuous vortex flow of steam laden with carbonaceous particles, which was shown to achieve high conversion rates in short residence times [18,20]. This solar reactor concept can be either directly or indirectly irradiated depending on whether the reactants are directly exposed to the concentrated solar irradiation or there is an opaque absorber wall in between. Although the directly irradiated approach provides efficient heat transfer to the reaction site, it requires a transparent window, which becomes troublesome for scale-up and under pressurized operation. In contrast, the indirectly irradiated approach employs a robust, windowless cavity that enables pressurized operation but at the expense of an additional conductive heat transfer resistance across the solid wall. Despite this inherently less efficient heat transfer mechanism, the indirectly irradiated reactor exhibited experimental carbon conversions and energy efficiencies comparable to those obtained with the directly irradiated counterpart [20]. To elucidate the fundamental transport mechanisms and guide the optimization for an improved process performance, we have developed a numerical heat transfer and fluid flow model of the solar reactor. Previous pertinent modeling studies include heat transfer analysis of a suspension of reacting carbonaceous particles for directly irradiated [21–23] and indirectly irradiated flows [24], and aerodynamic analysis for a directly irradiated vortex flow [25,26]. Still missing is a model that captures in detail both heat transfer and fluid-flow characteristics of an indirectly irradiated solar reactor for the gasification of a particulate carbonaceous flow at high pressures, including coupled radiative transfer and chemical kinetics with shrinking particles. This paper presents the development of such a model, including the formulation of the governing conservation equations and their numerical solution by finite volume computational fluid dynamics (CFD) [27] and Monte Carlo (MC) numerical techniques [28–30]. The numerical results are compared with experimentally measured data obtained using a 3 kW solar reactor prototype tested in a high-flux solar simulator [20]. The validated model is applied to identify the dominant heat transfer mechanisms and to analyze the effect of the solar rector's geometry and operational parameters on its performance indicators given by the carbon molar conversion and solar-to-fuel energy efficiency.

The solar reactor is shown schematically in Fig. 1(a). Its engineering design, fabrication, and testing were described previously in detail [20]; only the main features are highlighted here. It consists of a silicon-carbide (SiC) cylindrical cavity with a hemispherical dome to efficiently absorb the incident concentrated solar radiation, $Q˙$_solar. Absorbed heat is conducted across the cavity walls and transferred by combined conduction, convection, and radiation to the reacting gas-particle vortex flow created by the tangential injection of a carbon/water slurry and supported through the tangential injection of argon. The vortex flow is confined to the annular gap between the absorber cavity and the coaxially aligned cylindrical ceramic lining (SiO₂–Al₂O₃), which is sealed with a stainless-steel shell and an air-cooled Inconel front. The reactor's key dimensions are the aperture diameter d_aperture, the internal cavity diameter d_cavity, the internal cavity length L_cavity, the cavity wall thickness t_cavity, the internal reactor diameter d_reactor, the internal reactor length L_reactor, the internal reactor volume V_reactor, and the shell diameter d_shell.

The formulation of the reactor model considers an axisymmetric (two-dimensional) two-phase gas/particle flow composed of solid carbonaceous particles entrained in a steam/Ar flow. Although the axisymmetric assumption neglects the flow's helical path, this simplification was necessary to ease the computational complexity with minimum effect on the temperature distributions or the performance indicators, as shown by a preliminary three-dimensional CFD simulation that indicated a maximum relative difference on the Nu value for convective heat transfer between the fluid flow and the inner cavity walls of less than 10% [31]. In terms of radiative exchange—the predominant heat transfer mode at above 1000 K—the particle flow is treated as three-dimensional nongray, absorbing, emitting, and scattering participating medium. As the particles undergo thermochemical gasification (see Eq. (1)), they shrink causing their thermal and optical properties to vary.

where ρ is the density, Y_i is the mass fraction of the species i = {Ar, H₂O, H₂, CO, CO₂}, U is the velocity vector of the flow, ϱ_i is the volumetric production/consumption rate of species i, p is the pressure, I is the identity tensor, E is the total energy per unit volume, E = ρ_fe + 0.5ρ_f|U|², e is the specific internal energy, and ϕ_chemistry and ϕ_convection are the source terms associated with the enthalpy change of the chemical reaction and the gas-particle convective heat exchange, respectively.

where ρ_s = f_Vρ_feedstock with f_V being the solid volume fraction and ρ_feedstock the apparent density of the carbonaceous particles (including the internal pores), c_p is the specific heat capacity, T is the temperature, and ϕ_radiation is the source for radiative heat. Г is the number of particles per unit volume, which is conserved (Eq. (6)) even after complete gasification because particles contain a small amount of inorganic matter which ends as residual ash. Assuming spherical particles of mean diameter d_p, Eq. (6) yields Г = 6 f_V/(π d_p³), which is used to calculate d_p. A uniform temperature is further assumed within the particle, as justified by Bi_r < 0.28, where Bi_r is the Biot number for combined radiation and convection [32].

where k is the thermal conductivity.

Source terms

where g_λ is the asymmetry factor [35] and θ_s is the scattering angle of the incoming ray. Q_abs,λ, Q_ext,λ, and Φ_λ are calculated based on ξ and the complex refractive index of the particle $n¯$ = n−ik. For ξ< 0.3, isotropic scattering is assumed (Φ_λ = 1, g_λ = 0); for 0.3 <ξ< 5, the values are calculated using the routine Bohren and Huffman Mie (BHMIE) [36]; and for ξ > 5, the reflectivity is calculated by applying the Fresnel equation for unpolarized light, while diffuse reflecting sphere is assumed for Φ_λ = 8(sinθ_s − θ_scosθ_s)/(3π) (g_λ = −0.889) [33,34]. The gas phase is assumed nonparticipating because β_λ,f ≪ β_λ,s [22].

where h(T) = c_p(T − T_ref) + h_ref is the specific enthalpy evaluated at temperature T from the originating phase with the reference enthalpy h_ref at T_ref = 273.15 K.

where α_sf = Nu k_f/d_p is the convective heat transfer coefficient between solid and fluid phases, with Nu = 2 for a fully entrained spherical particle [37].

where T_cavity is the mean wall temperature of the cavity and T_ambient is the ambient temperature. The heat losses from the shell and front face by convection (⁠$Q˙$_{convection,shell} and $Q˙$_{convection,front}) and by radiation (⁠$Q˙$_{radiation,shell} and $Q˙$_{radiation,front}) are calculated by using the appropriate Nu correlation [37] and the radiosity method, respectively. The inlet boundary condition for the CFD module at r_in, z_in is defined through the mass flowrate of the feedstock ṁ_feedstock with the initial particle size d_p,in, the molar argon flowrate ṅ_Ar, and the molar water flowrate $n˙H2O$⁠, determined by the H₂O:C molar ratio at the inlet temperature T_in. The slurry (feedstock and water) and the Ar are fed at T_ambient, while the water evaporates before the injection point into the reactor chamber, which is at T_in. The sensible and latent heats required to heat the slurry from T_ambient to T_in are accounted for in the energy balance. At the outlet r_out, z_out, a static pressure p_out is set. The spectral power emitted from the particle phase is calculated by $Q˙$_{emitted,volume} = $4VκP,sTs4$⁠, where V is the elemental volume and κ_P is the Planck mean absorption coefficient.

Activated charcoal particles (Fluka, Sigma-Aldrich 05120) were used as the model feedstock. Their properties are listed in Table 1, where lower heating value (LHV) is the lower heating value and specific surface area. The rates of formation and consumption for each species are predicted by a set of kinetic rate expressions of the Langmuir–Hinshelwood type, with rate constants (Ka₁, Ka₂, K₃) derived for charcoal [11]. The effectiveness factor determining the active surface area of the particles is proportional to the diffusivity, which in turn is inverse proportional to the total pressure [42]. Thus, the kinetic rate constants are proportionally reduced with increasing pressure [43]. Because of the high heating rates and low volatile content of the charcoal, the release of volatiles during pyrolysis is assumed to occur immediately upon entrance of the particles into the reactor. The properties for the gas species and the solid materials are listed in Table 1. The combined k of solid materials is determined with the thermal resistance analogy [37], while k_f of the gas mixture is determined based on the molar fraction of each gas species. The carbon molar conversion is denoted X_C.

Computational fluid dynamics module—The governing equations for the CFD module, Eqs. (2)–(7), are discretized for an axisymmetric configuration (N_CFD,r × N_CFD,z) with a spatial first-order cell-centered finite volume method and integrated in time with the explicit Euler method [27]. The fluxes for the fluid phase across the cell boundaries are determined by solving the approximated Riemann problem with the method of Roe for multiple species, assuming isentropic gas [61–64]. The steady-state solid heat conduction equation (Eq. (8) with dT/dt = 0) is discretized for the cavity and insulation domain along the axial direction (N_cavity and N_insulation) and for the front domain along the radial direction (N_front).

Monte Carlo module—The pathlength MC ray-tracing method with ray redirection is applied for solving the equation of radiative heat transfer (Eqs. (9) and (10)) [28–30] on a grid discretized in radial and axial direction (N_MC,r × N_MC,z). A given number of rays, N_ray, are initialized from the elemental surface or volume using the probability density functions for direction and wavelength of emission/scattering from particles and emission/reflection from boundary walls. Each ray can undergo three types of interactions: (1) absorption and scattering by the elemental volume; (2) absorption or reflection by the cavity, insulation, or front; and (3) transmission to the adjacent volume element. The ray's history is terminated either when its radiative power is diminished by subsequent absorption events below a set threshold or when it exits the domain. T_s, Г, and f_V are mapped from the CFD module to the MC module by conserving mass and energy.

where N is the total number of unknowns, k is the cell index, and j is the time-step index.

The correctness of the CFD module was verified by comparing the model results to the analytical solution for a fully developed flow in a tubular duct with constant wall temperature (Nu = 3.66) [37,64]. The correctness of the MC module was verified by comparing the model results to two analytical solutions: for a cavity without participating media using the enclosure theory (radiosity method) and for an emitting, absorbing, and scattering media between two infinitely long concentric cylinders [64–66]. The discretization error of the CFD and MC modules was assessed with a grid-refinement study using the boundary conditions and the dimensions of a representative (baseline) solar run, listed in Table 2 (run #25 [20]). The results of a coarse grid with N_CFD,r × N_CFD,z = 15 × 75 and N_MC,r × N_MC,z = 8 × 30, using N_ray = 100,000 were compared to the results of a finer grid with N_CFD,r × N_CFD,z = 30 × 150 and N_MC,r × N_MC,z = 16 × 60, using N_ray = 250,000. The relative discretization errors between these two grid levels amounted to 0.47%, 1.83%, and 0.84% for T_reactor,T_out, and X_C, respectively.

Experimental data were obtained using a 3 kW solar reactor prototype [20]. Experimentation was conducted at the high-flux solar simulator of ETH Zurich, which comprises an array of seven high-pressure Xenon arcs, each close-coupled with truncated ellipsoidal specular reflectors, to provide an external source of intense thermal radiation—mostly in the visible and IR spectra—that mimics the heat transfer characteristics of highly concentrating solar systems. The solar radiative input power $Q˙$_solar was measured optically on a Lambertian target with a calibrated CCD camera and verified with a water calorimeter. Temperatures were measured with type-K thermocouples. The pressure of the reactor chamber was monitored by a pressure transmitter (Keller, 33×). The gas mass flow rates were controlled by electronic flow controllers (Bronkhorst HI-TEC, Aesch, Switzerland) calibrated for an accuracy of ±1%. The composition of gaseous products was analyzed on-line by infrared detectors (Siemens Calomat 6 and Ultramat; frequency 1 Hz) and by gas chromatography (Varian Micro-GC; frequency 8 mHz). A preset flowrate of N₂ (purity: 99.999%) was introduced into the product gas stream as tracer for the determination of the molar flowrates of species. The measurement errors were evaluated based on the standard deviation error propagation. Main source of inaccuracy was in the measurement of the mass flowrate of the slurry because of particle deposition in the feeding system.

Figure 2 shows the parity plots of the numerically calculated versus the experimentally measured values of T_reactor (■), T_cavity (pentagrams), T_out (●), T_shell (hexagrams) (a), carbon molar conversion X_C (b), and molar ratios H₂:CO (▼) and CO₂:CO (◆) (c) for 17 experimental runs [20]. The solar reactor was operated under the following range of parameters: p = 1.0 − 5.9 × 10⁵ Pa, $m˙$_Ar = 2 − 15 L_N/min, $m˙$_feedstock = 7 − 21 × 10⁻⁶kg/s, H₂O:C molar ratio = 1.48 − 1.98 (accounting for the moisture content in the charcoal), and $Q˙$_solar = 1300–2600 W, which corresponds to mean solar concentration ratios² over the aperture in the range C = 1794–3718 suns. The key dimensions and the discretization levels of the domains/modules are listed in Table 2, while the boundary conditions were the ones measured for each experimental run. Calculated T_reactor, T_cavity, and T_out agree well with the experimental measured values (Fig. 2(a)), while T_shell is over-predicted by the numerical model because steady-state was not reached in the outer shell. The dashed lines of Fig. 2(b) indicate ±11% the experimental inaccuracy of X_C, attributed mainly to the error in the measurement of the charcoal feeding rate [20]. X_C,numerical lies within the inaccuracy of X_C,_experimental but discrepancies are due to particle deposition. In Fig. 2(c), calculated molar ratios are mostly under-predicted, presumably due to the water-gas shift reaction of syngas with excess water (H₂O + CO = H₂ + CO₂) that occurred downstream of the reactor in steel pipes (>2 m) before the syngas analysis [20].

The baseline parameters and boundary conditions are listed in Table 2. Figure 3 shows the axisymmetric two-dimensional contour maps of T_f (a), T_s (b), f_V (c), d_p (d), and dX_C/dt (normalized) (e). Also shown in Figs. 3(a) and 3(b) is the temperature of the absorber cavity T_SiC, which gradually increases with z because of heat conduction and because of $Q˙$_reradiation, which is obviously higher close to the aperture. The temperature difference across the cavity wall thickness is on average 20 K as a result of the good thermal conductivity of SiC. T_f and T_s steadily increase axially at initial high heating rates of about 1770 K/s for the first 5 cm in axial direction, closely following each other because of the relatively high convective heat transfer area provided by the small d_p, and peaking at 1486 K toward the outlet port. In contrast, for the directly irradiated reactor, T_s exceeds 1750 K close to the inlet plane but decreases axially [22]. As expected, f_V is maximum at the inlet (1.55 × 10⁻⁵) and decreases axially because d_p shrinks as the reaction progresses. The carbon conversion rate, given by dX_C/dt, exceeds 50%/s in the indicated streamlines along z = 0.04–0.11 m (Fig. 3(e)) and peaks at 90%/s. In comparison to the directly irradiated reactor, dX_C/dt is here more evenly distributed with lower peak rates because ϕ_radiation ≈ 1.2 × 10⁶ W/m³ at the entrance region is three times lower [22]. The particle residence time for the baseline run is 1.3 s, determined by temporally integrating the velocity in axial direction. This value is consistent to the experimentally determined residence time of 1.5 s, derived from the measured volumetric gas flow assuming a first order reaction (Eq. (4) in Ref. [20]). The high carbon conversion extents in combination with these short residence times suggest that full carbon conversion can be achieved in a large-scale reactor.

This definition assumes that the heating value of unreacted feedstock exiting the solar reactor is not lost because, in principle, particles can be separated from the syngas and refed into the reactor. Since the feedstock contains a small fraction of volatiles, the LHV of unreacted feedstock does not vary significantly compared to the one freshly fed. The energy balance considers the energy flows in and out, including the chemical energy content of the mass flows in and out. The effect of varying $m˙$_feedstock,$Q˙$_solar, p_out, and L_cavity was examined.

Feeding rate—Both ϕ_radiation and ϕ_convection can be improved by increasing f_V, which in turn is controlled by $m˙$_feedstock and ṅ_Ar. Figure 4 shows X_C and η_{solar-to-fuel} as a function of $m˙$_feedstock/$m˙$_feedstock,0 (normalized to the baseline rate listed in Table 2) for two Ar flow rates ṅ_Ar = 2.975 × 10⁻³mol/s (baseline run) and ṅ_Ar = 0 mol/s. The H₂O:C molar ratio was kept constant at 1.98. T_reactor decreases gradually with increasing $m˙$_feedstock(e.g., for $m˙$_feedstock/$m˙$_feedstock,0 = 4 and ṅ_Ar = 2.975 × 10⁻³mol/s, T_reactor = 1248 K) because of the additional sensible heat required to heat the reactants. For ṅ_Ar = 2.975 × 10⁻³mol/s, X_C = 88.6% is achieved for $m˙$_feedstock/$m˙$_feedstock,0 = 0.5 at the expense of low η_{solar-to-fuel} < 8.8%. Doubling $m˙$_feedstock results in a 141% improvement in η_{solar-to-fuel} from 15.2 to 21.5%, coupled to a decrease in X_C from 80 to 58%. A further increase of $m˙$_feedstock only improves η_{solar-to-fuel} marginally because of the low T_reactor, while X_C drops further. Thus, the optimal feed rate is around $m˙$_feedstock/$m˙$_feedstock,0 = 2 for this reactor with these conditions. For ṅ_Ar = 0 mol/s, complete X_C is achieved for $m˙$_feedstock/$m˙$_feedstock,0 = 1, while for $m˙$_feedstock/$m˙$_feedstock,0 > 2, η_{solar-to-fuel} is improved on average by a factor 1.4. A reduction in ṅ_Ar increases besides f_V the particle residence time and reduces the sensible heat losses, both helping to increase X_C and η_{solar-to-fuel}. T_reactor was on average 57 K higher compared to the same $m˙$_feedstock/$m˙$_feedstock,0 with ṅ_Ar = 2.975 × 10⁻³mol/s. Note that for ṅ_Ar = 0 mol/s, particle deposition might occur and a redesign of the slurry injection nozzle is required without the support of Ar.

Solar concentration ratio—Fig. 5 shows X_C and η_{solar-to-fuel} as a function of the solar concentration ratio C for two feeding rates $m˙$_feedstock/$m˙$_feedstock,0  = 1 and $m˙$_feedstock/$m˙$_feedstock,0  = 2. For $m˙$_feedstock/$m˙$_feedstock,0  = 1, T_reactor increases monotonically with C from 1180 K for 1993 suns (⁠$Q˙$_solar = 1409 W) to 1473 K for 3408 suns (⁠$Q˙$_solar = 2409 W). As expected, X_C increases with C because of increasing T_reactor, while full conversion is achieved for C > 3054 (⁠$Q˙$_solar>2159 W). η_{solar-to-fuel} increases with C up to 15.6% (at T_reactor = 1522 K) but a further increase in C causes η_{solar-to-fuel} to drop because X_C ≈ 100%, i.e., reaction is completed, and because of larger $Q˙$_reradiation. For $m˙$_feedstock/$m˙$_feedstock,0 = 2, T_reactor increases monotonically with C and is on average 77 K lower than that for $m˙$_feedstock/$m˙$_feedstock,0= 1, while η_{solar-to-fuel} increases gradually with C up to 24.3%.

Pressure—Fig. 6 shows X_C and η_{solar-to-fuel} as a function of p_out/p_out,0 (normalized to the baseline pressure) for two feeding rates $m˙$_feedstock/$m˙$_feedstock,0= 1 and $m˙$_feedstock/$m˙$_feedstock,0= 3. Because of the model's limitations, T_in was set above the boiling point at 425, 438, 448, 456, 464 K for p_out/p_out,0 = 5, 7, 9, 11, 13, respectively. For $m˙$_feedstock/$m˙$_feedstock,0 = 1, X_C reaches 100% for p_out/p_out,0 > 3, consequently η_{solar-to-fuel} is only marginally improved from 15.2 to 17.9%. For $m˙$_feedstock/$m˙$_feedstock,0 = 3 X_C increases with increasing p_out because of the longer residence time, which increases from 0.2 to 12.3 s, and because of the improved kinetic rates due to the higher partial steam pressure. An increase in p_out/p_out,0 from 1 to 5 leads to an increase in η_{solar-to-fuel} from 22.8 to 37.9%, mainly because of the increasing X_C while keeping constant $Q˙$_solar. A further increase in p_out only marginally improves η_{solar-to-fuel} (η_{solar-to-fuel} = 40.1% for p_out/p_out,0 = 11) despite the increasing X_C, attributed to the exothermic water-gas shift reaction, which has more time to react and increases the H₂:CO molar ratio (note $LHVH2$ < LHV_CO). Note that this model does not capture all physical effects when changing the pressure. For instance, an increase of p_out reduces the fluid velocity, leading to insufficient particle entrainment. The particle deposition can be mitigated to some extent by increasing the H₂O:C molar ratio and ṅ_Ar, but at the expense of shorter residence times, larger sensible heat losses due to heating inert gas and/or excess water, and consequently lower η_{solar-to-fuel}. A slurry injection nozzle facilitating particle entrainment at higher p_out and without the need for carrier inert gas (ṅ_Ar = 0 mol/s) would be preferable.

Cavity geometry—Table 3 lists the varied dimensions. Figure 7 shows T_reactor, T_receiver, T_{receiver,peak} the peak of T_receiver (a), X_C, and η_{solar-to-fuel} (b) as a function of L_cavity/L_cavity,0 (normalized to the baseline length) for two configurations: (1) with constant reactor volume, V_{reactor,const}= 0.778 × 10⁻³ m³, and (2) with constant reactor diameter, d_{reactor,const}= 0.1 m. For both configurations, T_reactor and T_receiver decrease with increasing L_cavity because of the lower radiative heat flux incident on the cavity wall as a result of the larger absorber surface area, which in turn improves the heat transfer to the reactor chamber and reduces $Q˙$_reradiation (configuration 1: from 196 to 160 W; configuration 2: from 206 to 146 W) and $Q˙$_{convection,cavity} (configuration 1: from 74 to 53 W; configuration 2: from 76 to 50 W). $Q˙$_reflected slightly decreases with increasing L_cavity because of the reduced diameter-to-length ratio, enhancing the cavity effect. Overall, these competing effects lead to only marginal variation of X_C and η_{solar-to-fuel}. For configuration 2 and L_cavity> 0.12 m, X_C is slightly higher (factor 1.09) than the values obtained with configuration one because of the larger V_reactor which results on average in a 1 s longer residence time. T_{receiver,peak} reaches a minimum at 1660 K for L_cavity = 0.125 m, reducing thermal stresses on the SiC, which is stable up to 1923 K [67]. Thus, the optimal cavity length for high performance and low T_{receiver,peak} for the considered parameters is L_cavity = 0.125 m for configuration two (d_{reactor,const} = 0.1 m) with T_reactor= 1409 K, resulting in X_C = 91.5% and η_{solar-to-fuel} = 16.0%.

Scale-up—The scaling up of the reactor technology for a solar tower foresees an array of solar reactors modules with their cavities arranged side-by-side, each aperture attached to hexagon-shaped compound parabolic concentrators (CPC) in a honeycomb-type structure [68,69]. Cavities made of sintered α-SiC with integrated CPC have been experimentally demonstrated in on-sun solar tower testing at nominal temperatures up to 1600 K and pressures up to 6 × 10⁵ Pa for $Q˙$_solar = 35000–47000 W and C = 2000–2500 suns [70,71]. Such a scale-up concept would also enable the capture of spilled radiation by the concentrating heliostat field and may eliminate conductive losses between adjacent solar reactors. Hybridization with internal combustion (autothermal) by cofeeding O₂ further enables continuous operation under intermittent solar radiation [43,72].

We have developed a solar reactor model for a steam flow laden with carbonaceous particles undergoing gasification at high pressures. Energy and mass conservation equations were formulated for each phase and species and solved by CFD and Monte Carlo techniques. Concentrated solar radiation is absorbed by a SiC cavity, conducted and transferred to the steam-particle flow by combined radiation and convection, which distinguishes this windowless solar reactor concept from a windowed solar reactor featuring directly irradiated particle suspension directly exposed to concentrated solar radiation. The geometry of the dome-type cavity distributes the absorbed heat more evenly across the reactor, enabling high carbon conversion extents within short residence times. Increasing the cavity length reduces its peak temperature while improving heat transfer because of the larger surface area and the lower reradiation losses. For the lab-scale reactor, almost complete carbon conversion is achieved for pressure levels above 3 × 10⁵ Pa at a feedstock feeding rate of 9 × 10⁻⁶kg/s and a solar concentration ratio above 2700 suns. The solar-to-fuel energy efficiency can reach up to 40% by operating the reactor at high pressures, optimal feedstock feeding rate, and without argon, balancing the competing effects of absorption efficiency and sensible heat sink.

• Swiss National Science Foundation (Grant No. IZLIZ2_156474; Funder ID: 10.13039/501100001711).

• Swiss State Secretariat for Education, Research and Innovation (Grant No. 16.0183; Funder ID: 10.13039/501100007352).

• EU's Horizon 2020 Research and Innovation Program (Project INSHIP) (Grant No. 731287; Funder ID: 10.13039/100010661).

 
• A =

  area, m²

 
• C =

  solar concentration ratio

 
• c_p =

  specific heat capacity, J/(kg K), J/(mol K)

 
• d =

  diameter, m, μm

 
• e =

  specific internal energy, J/kg, J/mol

 
• E =

  total volumetric energy, J/m³

 
• f_V =

  solid volume fraction

 
• g_λ =

  asymmetry factor

 
• Gr =

  Grashof number

 
• h =

  specific enthalpy, J/(kg K), J/(mol K)

 
• i =

  radiation intensity, W/(m²sr)

 
• I =

  identity tensor

 
• k =

  thermal conductivity, W/(m K)

 
• K =

  kinetic rate, Pa⁻¹

 
• Ka =

  kinetic rate, mol/(m² s Pa)

 
• L =

  length, m

 
• LHV =

  lower heating value, J/kg, J/mol

 
• m =

  mass, kg

 
• $m˙$ =

  mass flow rate, kg/s, L_N/min

 
• n =

  molar mass, mol

 
• $m˙$ =

  molar flow rate, mol/s

 
• $n¯$ =

  complex refractive index

 
• N =

  number of elements/rays

 
• Nu =

  Nusselt number

 
• Pe =

  Péclet number

 
• p =

  pressure, Pa

 
• Q =

  energy/heat, J

 
• q̇ =

  heat flux vector, W/m²

 
• $Q˙$ =

  power/heat rate, W

 
• Q_abs =

  absorption efficiency factor

 
• Q_ext =

  extinction efficiency factor

 
• Q_sca =

  scattering efficiency factor

 
• Re =

  Reynolds number

 
• r =

  radial coordinate, m

 
• s =

  unit vector

 
• Sk =

  Stokes number

 
• t =

  time, s

 
• T =

  temperature, K

 
• t_cavity =

  cavity wall thickness, m

 
• U =

  velocity vector, m/s

 
• V =

  volume, m³

 
• X_C =

  carbon conversion

 
• Y =

  mass fraction

 
• Z =

  axial coordinate, m

Greek Symbols
 
• α =

  convective heat transfer coefficient, W/(m²K)

 
• β =

  extinction coefficient, m⁻¹

 
• Г =

  number of particles per unit volume, m⁻³

 
• $ΔH298Kdeg$ =

  standard enthalpy change at 298 K, kJ/mol

 
• ε =

  emissivity

 
• η_solar-to-fue =

  solar-to-fuel energy conversion efficiency, %

 
• θ_s =

  scattering angle, deg

 
• κ =

  absorption coefficient, m⁻¹

 
• κ_P =

  Planck mean absorption coefficient, m⁻¹

 
• λ =

  wavelength, m

 
• ξ =

  size parameter

 
• ρ =

  density, kg/m³

 
• σ_s =

  scattering coefficient, m⁻¹

 
• ϕ =

  volumetric source term, W/m³

 
• Φ =

  scattering phase function

 
• Ψ =

  monitored quantity

 
• Ω =

  solid angle, sr

 
• ∇ =

  Nabla operator

 
• ϱ =

  volumetric production/consumption rate, mol/(m³ s)

Subscripts
 
• b =

  blackbody

 
• CFD =

  fluid/particle domain

 
• const =

  constant parameter

 
• f =

  fluid/gas phase

 
• i =

  index of chemical component

 
• in =

  inlet

 
• out =

  outside/outlet

 
• p =

  particle

 
• ref =

  reference

 
• s =

  solid/particle phase

 
• sf =

  solid-to-fluid

 
• vap =

  evaporation

 
• λ =

  spectral

 
• 0 =

  baseline parameter

Abbreviations
 
• CFD =

  computational fluid dynamics

 
• CPC =

  compound parabolic concentrator

 
• daf =

  dry and ash free

 
• MC =

  Monte Carlo