## Abstract

A pressurized solar reactor for effecting the thermochemical gasification of carbonaceous particles driven by concentrated solar energy is modeled by means of a reacting two-phase flow. The governing mass, momentum, and energy conservation equations are formulated and solved numerically by finite volume computational fluid dynamics (CFD) coupled to a Monte Carlo radiation solver for a nongray absorbing, emitting, and scattering participating medium. Implemented are Langmuir–Hinshelwood kinetic rate expressions and size-dependent properties for charcoal particles undergoing shrinkage as gasification progresses. Validation is accomplished by comparing the numerically calculated data with the experimentally measured temperatures in the range 1283–1546 K, chemical conversions in the range 32–94%, and syngas product H2:CO and CO2:CO molar ratios obtained from testing a 3 kW solar reactor prototype with up to 3718 suns concentrated radiation. The simulation model is applied to identify the predominant heat transfer mechanisms and to analyze the effect of the solar rector's geometry and operational parameters (namely: carbon feeding rate, inert gas flowrate, solar concentration ratio, and total pressure) on the solar reactor's performance indicators given by the carbon molar conversion and the solar-to-fuel energy efficiency. Under optimal conditions, these can reach 94% and 40%, respectively.

## 1 Introduction

Solar-driven gasification thermochemically converts carbonaceous feedstock into energy-rich and high-quality syngas—a mixture of CO and H2—using concentrated solar energy as the source of high-temperature process heat [1,2]. The thermochemical conversion involves principally two sequential processes: pyrolysis and steam-based char gasification, with the former occurring typically in the range 500–900 K and the latter becoming dominant at above 1200 K. The simplified net stoichiometric reaction can be represented by
$C(s)+H2O=H2+CO ΔH298K°=131 kJ/mol$
(1)

## 2 Solar Reactor Configuration and Model Formulation

### 2.1 Solar Reactor Configuration.

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 daperture, the internal cavity diameter dcavity, the internal cavity length Lcavity, the cavity wall thickness tcavity, the internal reactor diameter dreactor, the internal reactor length Lreactor, the internal reactor volume Vreactor, and the shell diameter dshell.

Fig. 1
Fig. 1

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.

### 2.2 Governing Conservation Equations.

Fluid phase (subscript f)—The flow field is modeled with the Euler equation (laminar, inviscid, and nonconducting), as justified by Re < 125 and Pe > 12. The mass, momentum, and energy conservation equations for the fluid phase are:
$∂ρfYi∂t+∇⋅(ρfYiU)=ϱi$
(2)
$∂(ρfU)∂t+∇⋅(ρfU×U+pI)=0$
(3)
$∂E∂t+∇⋅((E+p)U)=ϕchemistry,f−ϕconvection$
(4)

where ρ is the density, Yi is the mass fraction of the species i = {Ar, H2O, H2, CO, CO2}, U is the velocity vector of the flow, ϱi 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 = ρfe + 0.5ρf|U|2, e is the specific internal energy, and ϕchemistry and ϕconvection are the source terms associated with the enthalpy change of the chemical reaction and the gas-particle convective heat exchange, respectively.

Solid phase (subscript s)—The particles are assumed to be fully entrained in the fluid phase, as justified by Sk < 0.02. The mass and energy conservation equations for the particle phase are:
$∂ρs∂t+∇⋅(ρsU)=ϱC$
(5)
$∂Γ∂t+∇⋅(ρfeedstockΓU)=0$
(6)
$cp,feedtsockρs∂Ts∂t+∇⋅(ρsTsU)=ϕchemistry,s+ϕradiation+ϕconvection$
(7)

where ρs = fVρfeedstock with fV being the solid volume fraction and ρfeedstock the apparent density of the carbonaceous particles (including the internal pores), cp is the specific heat capacity, T is the temperature, and ϕradiation 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 dp, Eq. (6) yields Г = 6 fV/(π dp3), which is used to calculate dp. A uniform temperature is further assumed within the particle, as justified by Bir < 0.28, where Bir is the Biot number for combined radiation and convection [32].

Finally, the transient heat conduction across solids (cavity, insulation, and front) is given by:
$cpρ∂T∂t−∇⋅(k∇T)=0$
(8)

where k is the thermal conductivity.

Source terms

Radiation—ϕradiation is the difference between the emitted and absorbed radiation of the particle suspension, given by the divergence of the radiative heat flux along the path s [33,34]
$∇⋅q˙radiation(s)=4π∫λ=0∞κλ(s)(iλ,b(s)−14π∫04πiλ(s,Ω)dΩ)dλ$
(9)
where κλ is the spectral absorption coefficient for wavelength λ and Ω is the solid angle. The rate of change of the spectral radiative intensity iλ along the path s is given by the equation of radiative heat transfer:
$diλds=−βλiλ+κλiλ,b+σs,λ4π∫04πiλ(sin)Φλ(sin,s)dΩin$
(10)
where βλ and σs,λ are the spectral extinction and scattering coefficients, respectively, and Φλ is the scattering phase function from the direction sin to s. For fV < 10−3, independent scattering regime is valid and κλ, σs,λ, and βλ can be calculated as a function of the corresponding properties for a single sphere using [34]:
${κλ,σs,λ,βλ}=3fV2dp{Qabs,λ,Qsca,λ,Qext,λ}$
(11)
where Qabs,λ, Qsca,λ, and Qext,λ are the spectral absorption, scattering, and extinction efficiency factors. For size parameter ξ = πdp/λ < 0.3, Rayleigh scattering applies; for the range 0.3 <ξ< 5, Mie scattering is valid; and for ξ > 5, geometrical optics regime applies. Φλ is determined with the Henyey–Greenstein approximation [33,34]:
$Φλ(dp,θs)=1−gλ2(1+gλ2−2gλ⋅ cos θs)3/2$
(12)

where gλ is the asymmetry factor [35] and θs is the scattering angle of the incoming ray. Qabs,λ, Qext,λ, and Φλ are calculated based on ξ and the complex refractive index of the particle $n¯$ = nik. For ξ< 0.3, isotropic scattering is assumed (Φλ = 1, gλ = 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 Φλ = 8(sinθs − θscosθs)/(3π) (gλ = −0.889) [33,34]. The gas phase is assumed nonparticipating because βλ,fβλ,s [22].

Chemistry—ϕchemistry accounts for the enthalpy change of the chemical reaction
$ϕchemistry=∑iϱihi(T)$
(13)

where h(T) = cp(T − Tref) + href is the specific enthalpy evaluated at temperature T from the originating phase with the reference enthalpy href at Tref = 273.15 K.

Convection—The convective heat exchange between the two phases is modeled as:
$ϕconvection=Γπdp2αsf(Tf−Ts)$
(14)

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

### 2.3 Boundaries and Interfaces.

The model domain is subdivided into a solid domain (cavity, insulation, and front) and the two-phase domain. The model domains, the boundary conditions, and the interfaces between the domains are shown schematically in Fig. 1(b). The incoming concentrating radiation providing $Q˙$solar and the absorbed heat flux incident on the cavity walls $Q˙$absorbed are modeled with an in-house MC module [38,39], while $Q˙$reflected = $Q˙$solar − $Q˙$absorbed. The radiosity (enclosure theory) method is applied for calculating the radiative heat exchange among cavity elements $Q˙$reradiation The convective heat losses from the cavity walls to ambient $Q˙$convection,cavity are accounted by using the adjusted Nu correlation for cavity receivers [40,41]
$Nu=0.0051 Gr1/3 (TcavityTambient)0.18 (dapertureLcavity)1.12−0.98(daperture/Lcavity)$
(15)

where Tcavity is the mean wall temperature of the cavity and Tambient 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 rin, zin is defined through the mass flowrate of the feedstock feedstock with the initial particle size dp,in, the molar argon flowrate Ar, and the molar water flowrate $n˙H2O$, determined by the H2O:C molar ratio at the inlet temperature Tin. The slurry (feedstock and water) and the Ar are fed at Tambient, while the water evaporates before the injection point into the reactor chamber, which is at Tin. The sensible and latent heats required to heat the slurry from Tambient to Tin are accounted for in the energy balance. At the outlet rout, zout, a static pressure pout 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 κP is the Planck mean absorption coefficient.

### 2.4 Carbonaceous Feedstock and Material Properties.

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 kf of the gas mixture is determined based on the molar fraction of each gas species. The carbon molar conversion is denoted XC.

Table 1

Chemical composition, properties, and kinetic parameters of the activated charcoal particles

Ultimate analysis (dry and ash free)[20]
C95.03 wt %
H0.49 wt %
O2.13 wt %
S0.75 wt %
N0.60 wt %
Ultimate analysis (dry and ash free)[20]
C95.03 wt %
H0.49 wt %
O2.13 wt %
S0.75 wt %
N0.60 wt %
Proximate analysis[20]
Fixed carbon90.16 wt %
Volatiles2.37 wt %
Moisture1.62 wt %
Ash5.83 wt %
Proximate analysis[20]
Fixed carbon90.16 wt %
Volatiles2.37 wt %
Moisture1.62 wt %
Ash5.83 wt %
Langmuir–Hinshelwood kinetic parameters [11]
Frequency factorActivation energy
Ka11017.9 mol/(m2 s Pa)182 kJ/mol
Ka25 × 106mol/(m2 s Pa)366 kJ/mol
K37.3 × 10−12 1/Pa−166 kJ/mol
Langmuir–Hinshelwood kinetic parameters [11]
Frequency factorActivation energy
Ka11017.9 mol/(m2 s Pa)182 kJ/mol
Ka25 × 106mol/(m2 s Pa)366 kJ/mol
K37.3 × 10−12 1/Pa−166 kJ/mol
PropertyValue/correlationSource
cp,airCorrelation from:[44]
cp,Ar20.79 J/(mol K)[45]
cp,CO34.54 J/(mol K)[45]
cp,CO256.53 J/(mol K)[45]
cp,feedstock1733.3 J/(kg K)[45]
cp,H231.08 J/(mol K)[45]
cp,H2O,steam44.28 J/(mol K)[45]
cp,H2O,water75.68 J/(mol K)[45]
dp,in42.6 μm[20]
εAl2O3–SiO20.28[46]
εSiC0.90, average from[33,47]
εSiO2–Al2O30.39, average from[46,48]
εsteel0.60[49]
href,CO−110.50 kJ/mol[45]
href,CO2−393.50 kJ/mol[45]
href,H2O−241.82 kJ/mol[45]
hvap,H2O40.63 kJ/mol[45]
kairCorrelation from[44]
kAl2O3–SiO2Exponential fit, values from:[50]
kAr, kH2Correlations from[51]
kCO, kCO2Kinetic gas theory[52]
kH2O,steamCorrelations from[53]
kinconelLinear fit, values from[54]
kSiCCorrelation from[55]
kSiO2–Al2O3Fifth order polynomial fit for Al2O3 from[56]
ksteelLinear fit, values from[57]
LHVfeedstock29.33 MJ/kg
ρfeedstock450 kg/m3[58]
Specific surface area689.7±32.1 m²/gBrunauer–Emmett–Teller method, micromeritics TriStar 3000
$n¯$ = nikLinear fit, values from[59,60]
PropertyValue/correlationSource
cp,airCorrelation from:[44]
cp,Ar20.79 J/(mol K)[45]
cp,CO34.54 J/(mol K)[45]
cp,CO256.53 J/(mol K)[45]
cp,feedstock1733.3 J/(kg K)[45]
cp,H231.08 J/(mol K)[45]
cp,H2O,steam44.28 J/(mol K)[45]
cp,H2O,water75.68 J/(mol K)[45]
dp,in42.6 μm[20]
εAl2O3–SiO20.28[46]
εSiC0.90, average from[33,47]
εSiO2–Al2O30.39, average from[46,48]
εsteel0.60[49]
href,CO−110.50 kJ/mol[45]
href,CO2−393.50 kJ/mol[45]
href,H2O−241.82 kJ/mol[45]
hvap,H2O40.63 kJ/mol[45]
kairCorrelation from[44]
kAl2O3–SiO2Exponential fit, values from:[50]
kAr, kH2Correlations from[51]
kCO, kCO2Kinetic gas theory[52]
kH2O,steamCorrelations from[53]
kinconelLinear fit, values from[54]
kSiCCorrelation from[55]
kSiO2–Al2O3Fifth order polynomial fit for Al2O3 from[56]
ksteelLinear fit, values from[57]
LHVfeedstock29.33 MJ/kg
ρfeedstock450 kg/m3[58]
Specific surface area689.7±32.1 m²/gBrunauer–Emmett–Teller method, micromeritics TriStar 3000
$n¯$ = nikLinear fit, values from[59,60]

### 2.5 Numerical Implementation.

Computational fluid dynamics module—The governing equations for the CFD module, Eqs. (2)(7), are discretized for an axisymmetric configuration (NCFD,r × NCFD,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 [6164]. 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 (Ncavity and Ninsulation) and for the front domain along the radial direction (Nfront).

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)) [2830] on a grid discretized in radial and axial direction (NMC,r × NMC,z). A given number of rays, Nray, 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. Ts, Г, and fV are mapped from the CFD module to the MC module by conserving mass and energy.

The solution is considered converged when the residual of the monitored quantities Ψ
$residual=N∑k=1N(Ψkj+1−Ψkj)2/∑k=1NΨkj<10−6$
(16)

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

## 3 Model Validation

### 3.1 Verification of Implemented Routines.

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 [6466]. 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 NCFD,r × NCFD,z = 15 × 75 and NMC,r × NMC,z = 8 × 30, using Nray = 100,000 were compared to the results of a finer grid with NCFD,r × NCFD,z = 30 × 150 and NMC,r × NMC,z = 16 × 60, using Nray = 250,000. The relative discretization errors between these two grid levels amounted to 0.47%, 1.83%, and 0.84% for Treactor,Tout, and XC, respectively.

Table 2

Baseline model parameters for the baseline solar experimental run (run #25 [20])

DimensionsDiscretizationBoundary conditions
daperture0.030 mNcavity38$Q˙$solar1909 W
dcavity0.040 mNinsulation35C2700 suns
Lcavity,00.095 mNfront8$m˙$feedstock,09 × 10−6 kg/s
tcavity0.005 mNCFD,r × NCFD,z15 × 75Ar,02.975 × 10−3 mol/s
dreacto0.100 mNMC,r × NMC,z8 × 30H2O1.307 × 10−3mol/s
Lreactor0.110 mNray100000H2O:C1.98
Vreactor0.778 × 10−3 m3pout,0105 Pa
dshell0.274 mTin400 K
rin, zin0.05, 0.01–0.014 mTambient288 K
rout,zout0–0.01, 0.17 m$m˙$air0.95 × 10−3 kg/s
DimensionsDiscretizationBoundary conditions
daperture0.030 mNcavity38$Q˙$solar1909 W
dcavity0.040 mNinsulation35C2700 suns
Lcavity,00.095 mNfront8$m˙$feedstock,09 × 10−6 kg/s
tcavity0.005 mNCFD,r × NCFD,z15 × 75Ar,02.975 × 10−3 mol/s
dreacto0.100 mNMC,r × NMC,z8 × 30H2O1.307 × 10−3mol/s
Lreactor0.110 mNray100000H2O:C1.98
Vreactor0.778 × 10−3 m3pout,0105 Pa
dshell0.274 mTin400 K
rin, zin0.05, 0.01–0.014 mTambient288 K
rout,zout0–0.01, 0.17 m$m˙$air0.95 × 10−3 kg/s

### 3.2 Comparison With Experimental Data.

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 Treactor (■), Tcavity (pentagrams), Tout (●), Tshell (hexagrams) (a), carbon molar conversion XC (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 LN/min, $m˙$feedstock = 7 − 21 × 10−6 kg/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 Treactor, Tcavity, and Tout agree well with the experimental measured values (Fig. 2(a)), while Tshell 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 XC, attributed mainly to the error in the measurement of the charcoal feeding rate [20]. XC,numerical lies within the inaccuracy of XC,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].

Fig. 2
Fig. 2

## 4 Model Simulation Results and Discussion

### 4.1 Baseline Run.

The baseline parameters and boundary conditions are listed in Table 2. Figure 3 shows the axisymmetric two-dimensional contour maps of Tf (a), Ts (b), fV (c), dp (d), and dXC/dt (normalized) (e). Also shown in Figs. 3(a) and 3(b) is the temperature of the absorber cavity TSiC, 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. Tf and Ts 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 dp, and peaking at 1486 K toward the outlet port. In contrast, for the directly irradiated reactor, Ts exceeds 1750 K close to the inlet plane but decreases axially [22]. As expected, fV is maximum at the inlet (1.55 × 10−5) and decreases axially because dp shrinks as the reaction progresses. The carbon conversion rate, given by dXC/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, dXC/dt is here more evenly distributed with lower peak rates because ϕradiation ≈ 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.

Fig. 3
Fig. 3

### 4.2 Parametric Study.

The solar reactor's performance is characterized by the carbon conversion, XC = 1 – C,out/C,in, and the solar-to-fuel energy conversion efficiency is defined as:
$ηsolar−to−fuel=m˙syngas,outLHVsyngas,outQ˙solar+(m˙feedstock,in−m˙feedstock,out)LHVfeedstock$
(17)

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, pout, and Lcavity was examined.

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

Fig. 4
Fig. 4

Solar concentration ratio—Fig. 5 shows XC and ηsolar-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, Treactor 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, XC increases with C because of increasing Treactor, while full conversion is achieved for C >3054 ($Q˙$solar>2159 W). ηsolar-to-fuel increases with C up to 15.6% (at Treactor = 1522 K) but a further increase in C causes ηsolar-to-fuel to drop because XC ≈ 100%, i.e., reaction is completed, and because of larger $Q˙$reradiation. For $m˙$feedstock/$m˙$feedstock,0 = 2, Treactor increases monotonically with C and is on average 77 K lower than that for $m˙$feedstock/$m˙$feedstock,0= 1, while ηsolar-to-fuel increases gradually with C up to 24.3%.

Fig. 5
Fig. 5

Pressure—Fig. 6 shows XC and ηsolar-to-fuel as a function of pout/pout,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, Tin was set above the boiling point at 425, 438, 448, 456, 464 K for pout/pout,0 = 5, 7, 9, 11, 13, respectively. For $m˙$feedstock/$m˙$feedstock,0 = 1, XC reaches 100% for pout/pout,0 > 3, consequently ηsolar-to-fuel is only marginally improved from 15.2 to 17.9%. For $m˙$feedstock/$m˙$feedstock,0 = 3 XC increases with increasing pout 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 pout/pout,0 from 1 to 5 leads to an increase in ηsolar-to-fuel from 22.8 to 37.9%, mainly because of the increasing XC while keeping constant $Q˙$solar. A further increase in pout only marginally improves ηsolar-to-fuel (ηsolar-to-fuel = 40.1% for pout/pout,0 = 11) despite the increasing XC, attributed to the exothermic water-gas shift reaction, which has more time to react and increases the H2:CO molar ratio (note $LHVH2$ < LHVCO). Note that this model does not capture all physical effects when changing the pressure. For instance, an increase of pout 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 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 ηsolar-to-fuel. A slurry injection nozzle facilitating particle entrainment at higher pout and without the need for carrier inert gas (Ar = 0 mol/s) would be preferable.

Fig. 6
Fig. 6

Cavity geometry—Table 3 lists the varied dimensions. Figure 7 shows Treactor, Treceiver, Treceiver,peak the peak of Treceiver (a), XC, and ηsolar-to-fuel (b) as a function of Lcavity/Lcavity,0 (normalized to the baseline length) for two configurations: (1) with constant reactor volume, Vreactor,const= 0.778 × 10−3 m3, and (2) with constant reactor diameter, dreactor,const= 0.1 m. For both configurations, Treactor and Treceiver decrease with increasing Lcavity 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 Lcavity because of the reduced diameter-to-length ratio, enhancing the cavity effect. Overall, these competing effects lead to only marginal variation of XC and ηsolar-to-fuel. For configuration 2 and Lcavity> 0.12 m, XC is slightly higher (factor 1.09) than the values obtained with configuration one because of the larger Vreactor which results on average in a 1 s longer residence time. Treceiver,peak reaches a minimum at 1660 K for Lcavity = 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 Treceiver,peak for the considered parameters is Lcavity = 0.125 m for configuration two (dreactor,const = 0.1 m) with Treactor= 1409 K, resulting in XC = 91.5% and ηsolar-to-fuel = 16.0%.

Fig. 7
Fig. 7
Table 3

Key dimensions for configurations 1 and 2 for different Lcavity and Lreactor

Configuration 1Configuration 2
Lcavity/Lcavity,0LcavityLreactordreactorVreactordreactorVreactor
(m)(m)(m)×10−3 (m3)(m)×10−3 (m3)
2/30.050.0850.1100.780.10.63
4/30.10.1350.0930.780.10.93
5/30.1250.160.0880.780.11.07
20.150.1850.0840.780.11.21
Configuration 1Configuration 2
Lcavity/Lcavity,0LcavityLreactordreactorVreactordreactorVreactor
(m)(m)(m)×10−3 (m3)(m)×10−3 (m3)
2/30.050.0850.1100.780.10.63
4/30.10.1350.0930.780.10.93
5/30.1250.160.0880.780.11.07
20.150.1850.0840.780.11.21

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].

## 5 Summary and Conclusions

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 × 10−6 kg/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.

## Funding Data

• 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).

## Nomenclature

• A =

area, m2

•
• C =

solar concentration ratio

•
• cp =

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

•
• fV =

solid volume fraction

•
• gλ =

asymmetry factor

•
• Gr =

Grashof number

•
• h =

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

•
• i =

•
• 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, LN/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

•
• =

heat flux vector, W/m2

•
• $Q˙$ =

power/heat rate, W

•
• Qabs =

absorption efficiency factor

•
• Qext =

extinction efficiency factor

•
• Qsca =

scattering efficiency factor

•
• Re =

Reynolds number

•
• r =

•
• s =

unit vector

•
• Sk =

Stokes number

•
• t =

time, s

•
• T =

temperature, K

•
• tcavity =

cavity wall thickness, m

•
• U =

velocity vector, m/s

•
• V =

volume, m3

•
• XC =

carbon conversion

•
• Y =

mass fraction

•
• Z =

axial coordinate, m

### Greek Symbols

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

•
• ηsolar-to-fue =

solar-to-fuel energy conversion efficiency, %

•
• θs =

scattering angle, deg

•
• κ =

absorption coefficient, m−1

•
• κP =

Planck mean absorption coefficient, m−1

•
• λ =

wavelength, m

•
• ξ =

size parameter

•
• ρ =

density, kg/m3

•
• σs =

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

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

Abbreviations

• CFD =

computational fluid dynamics

•
• CPC =

compound parabolic concentrator

•
• daf =

dry and ash free

•
• MC =

Monte Carlo

## Footnotes

2

The solar concentration ratio C is defined as C = $Q˙$solar /(I·A), where $Q˙$solar is the solar radiative power intercepted by the cavity aperture of area A, normalized to the direct normal solar irradiation I. C is often expressed in units of “suns” for I = 1000 W m−2.

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