## 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 H_{2}:CO and CO_{2}: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

_{2}—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

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 CO_{2} 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.

## 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, $Q\u02d9$_{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 (SiO_{2}–Al_{2}O_{3}), 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.

### 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:

where *ρ* is the density, *Y*_{i} is the mass fraction of the species i = {Ar, H_{2}O, H_{2}, CO, CO_{2}}, **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*_{f}*e + *0.5*ρ*_{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.

*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:

where *ρ*_{s} = *f*_{V}*ρ*_{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*_{p}^{3}), 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*

*Radiation—ϕ*

_{radiation}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]

*κ*

_{λ}is the spectral absorption coefficient for wavelength

*λ*and Ω is the solid angle. The rate of change of the spectral radiative intensity

*i*

_{λ}along the path

**s**is given by the equation of radiative heat transfer:

*β*

_{λ}and

*σ*

_{s,λ}are the spectral extinction and scattering coefficients, respectively, and Φ

_{λ}is the scattering phase function from the direction

**s**

_{in}to

**s**. For

*f*

_{V}< 10

^{−3,}independent scattering regime is valid and

*κ*

_{λ},

*σ*

_{s,λ}, and

*β*

_{λ}can be calculated as a function of the corresponding properties for a single sphere using [34]:

*Q*

_{abs,λ},

*Q*

_{sca,λ,}and

*Q*

_{ext,λ}are the spectral absorption, scattering, and extinction efficiency factors. For size parameter

*ξ*=

*πd*

_{p}/

*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. Φ

_{λ}is determined with the Henyey–Greenstein approximation [33,34]:

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 $n\xaf$ = *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(sin*θ*_{s} − *θ*_{s}cos*θ*_{s})/(3π) (*g*_{λ} = −0.889) [33,34]. The gas phase is assumed nonparticipating because *β*_{λ,f} ≪ *β*_{λ,s} [22].

*Chemistry—ϕ*

_{chemistry}accounts for the enthalpy change of the chemical reaction

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.

*Convection—*The convective heat exchange between the two phases is modeled as:

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].

### 2.3 Boundaries and Interfaces.

_{solar}and the absorbed heat flux incident on the cavity walls $Q\u02d9$

_{absorbed}are modeled with an in-house MC module [38,39], while $Q\u02d9$

_{reflected}= $Q\u02d9$

_{solar}− $Q\u02d9$

_{absorbed}. The radiosity (enclosure theory) method is applied for calculating the radiative heat exchange among cavity elements $Q\u02d9$

_{reradiation}The convective heat losses from the cavity walls to ambient $Q\u02d9$

_{convection,cavity}are accounted by using the adjusted Nu correlation for cavity receivers [40,41]

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 ($Q\u02d9$_{convection,shell} and $Q\u02d9$_{convection,front}) and by radiation ($Q\u02d9$_{radiation,shell} and $Q\u02d9$_{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 $n\u02d9H2O$, determined by the H_{2}O: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 $Q\u02d9$_{emitted,volume} = $4V\kappa P,sTs4$, where *V* is the elemental volume and *κ*_{P} 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 (*Ka*_{1}, *Ka*_{2}, *K*_{3}) 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}.

Ultimate analysis (dry and ash free) | [20] | |
---|---|---|

C | 95.03 wt % | |

H | 0.49 wt % | |

O | 2.13 wt % | |

S | 0.75 wt % | |

N | 0.60 wt % |

Ultimate analysis (dry and ash free) | [20] | |
---|---|---|

C | 95.03 wt % | |

H | 0.49 wt % | |

O | 2.13 wt % | |

S | 0.75 wt % | |

N | 0.60 wt % |

Proximate analysis | [20] | |
---|---|---|

Fixed carbon | 90.16 wt % | |

Volatiles | 2.37 wt % | |

Moisture | 1.62 wt % | |

Ash | 5.83 wt % |

Proximate analysis | [20] | |
---|---|---|

Fixed carbon | 90.16 wt % | |

Volatiles | 2.37 wt % | |

Moisture | 1.62 wt % | |

Ash | 5.83 wt % |

Langmuir–Hinshelwood kinetic parameters [11] | ||
---|---|---|

Frequency factor | Activation energy | |

Ka_{1} | 1017.9 mol/(m^{2} s Pa) | 182 kJ/mol |

Ka_{2} | 5 × 10^{6}mol/(m^{2} s Pa) | 366 kJ/mol |

K_{3} | 7.3 × 10^{−12} 1/Pa | −166 kJ/mol |

Langmuir–Hinshelwood kinetic parameters [11] | ||
---|---|---|

Frequency factor | Activation energy | |

Ka_{1} | 1017.9 mol/(m^{2} s Pa) | 182 kJ/mol |

Ka_{2} | 5 × 10^{6}mol/(m^{2} s Pa) | 366 kJ/mol |

K_{3} | 7.3 × 10^{−12} 1/Pa | −166 kJ/mol |

Property | Value/correlation | Source |
---|---|---|

c_{p,air} | Correlation from: | [44] |

c_{p,Ar} | 20.79 J/(mol K) | [45] |

c_{p,CO} | 34.54 J/(mol K) | [45] |

c_{p,CO2} | 56.53 J/(mol K) | [45] |

c_{p,feedstock} | 1733.3 J/(kg K) | [45] |

c_{p,H2} | 31.08 J/(mol K) | [45] |

c_{p,H2O,steam} | 44.28 J/(mol K) | [45] |

c_{p,H2O,water} | 75.68 J/(mol K) | [45] |

d_{p,in} | 42.6 μm | [20] |

ε_{Al2O3–SiO2} | 0.28 | [46] |

ε_{SiC} | 0.90, average from | [33,47] |

ε_{SiO2–Al2O}_{3} | 0.39, average from | [46,48] |

ε_{steel} | 0.60 | [49] |

h_{ref},_{CO} | −110.50 kJ/mol | [45] |

h_{ref},_{CO2} | −393.50 kJ/mol | [45] |

h_{ref},_{H2O} | −241.82 kJ/mol | [45] |

h_{vap,H2O} | 40.63 kJ/mol | [45] |

k_{air} | Correlation from | [44] |

k_{Al2O3–SiO}_{2} | Exponential fit, values from: | [50] |

k_{Ar}, k_{H2} | Correlations from | [51] |

k_{CO}, k_{CO2} | Kinetic gas theory | [52] |

k_{H2O,steam} | Correlations from | [53] |

k_{inconel} | Linear fit, values from | [54] |

k_{SiC} | Correlation from | [55] |

k_{SiO2–Al2O3} | Fifth order polynomial fit for Al_{2}O_{3} from | [56] |

k_{steel} | Linear fit, values from | [57] |

LHV_{feedstock} | 29.33 MJ/kg | |

ρ_{feedstock} | 450 kg/m^{3} | [58] |

Specific surface area | 689.7±32.1 m²/g | Brunauer–Emmett–Teller method, micromeritics TriStar 3000 |

$n\xaf$ = n–ik | Linear fit, values from | [59,60] |

Property | Value/correlation | Source |
---|---|---|

c_{p,air} | Correlation from: | [44] |

c_{p,Ar} | 20.79 J/(mol K) | [45] |

c_{p,CO} | 34.54 J/(mol K) | [45] |

c_{p,CO2} | 56.53 J/(mol K) | [45] |

c_{p,feedstock} | 1733.3 J/(kg K) | [45] |

c_{p,H2} | 31.08 J/(mol K) | [45] |

c_{p,H2O,steam} | 44.28 J/(mol K) | [45] |

c_{p,H2O,water} | 75.68 J/(mol K) | [45] |

d_{p,in} | 42.6 μm | [20] |

ε_{Al2O3–SiO2} | 0.28 | [46] |

ε_{SiC} | 0.90, average from | [33,47] |

ε_{SiO2–Al2O}_{3} | 0.39, average from | [46,48] |

ε_{steel} | 0.60 | [49] |

h_{ref},_{CO} | −110.50 kJ/mol | [45] |

h_{ref},_{CO2} | −393.50 kJ/mol | [45] |

h_{ref},_{H2O} | −241.82 kJ/mol | [45] |

h_{vap,H2O} | 40.63 kJ/mol | [45] |

k_{air} | Correlation from | [44] |

k_{Al2O3–SiO}_{2} | Exponential fit, values from: | [50] |

k_{Ar}, k_{H2} | Correlations from | [51] |

k_{CO}, k_{CO2} | Kinetic gas theory | [52] |

k_{H2O,steam} | Correlations from | [53] |

k_{inconel} | Linear fit, values from | [54] |

k_{SiC} | Correlation from | [55] |

k_{SiO2–Al2O3} | Fifth order polynomial fit for Al_{2}O_{3} from | [56] |

k_{steel} | Linear fit, values from | [57] |

LHV_{feedstock} | 29.33 MJ/kg | |

ρ_{feedstock} | 450 kg/m^{3} | [58] |

Specific surface area | 689.7±32.1 m²/g | Brunauer–Emmett–Teller method, micromeritics TriStar 3000 |

$n\xaf$ = n–ik | Linear 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 (*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 d*T*/d*t* = 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.

## 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 [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.

Dimensions | Discretization | Boundary conditions | |||
---|---|---|---|---|---|

d_{a}_{perture} | 0.030 m | N_{cavity} | 38 | $Q\u02d9$_{solar} | 1909 W |

d_{cavity} | 0.040 m | N_{insulation} | 35 | C | 2700 suns |

L_{cavity,0} | 0.095 m | N_{front} | 8 | $m\u02d9$_{feedstock,0} | 9 × 10^{−6} kg/s |

t_{cavity} | 0.005 m | N_{CFD,r} × N_{CFD,z} | 15 × 75 | ṅ_{Ar,0} | 2.975 × 10^{−3} mol/s |

d_{reacto} | 0.100 m | N_{MC,r} × N_{MC,z} | 8 × 30 | ṅ_{H2O} | 1.307 × 10^{−3}mol/s |

L_{reactor} | 0.110 m | N_{ray} | 100000 | H_{2}O:C | 1.98 |

V_{reactor} | 0.778 × 10^{−3} m^{3} | p_{out,0} | 10^{5} Pa | ||

d_{shell} | 0.274 m | T_{in} | 400 K | ||

r_{in}, z_{in} | 0.05, 0.01–0.014 m | T_{ambient} | 288 K | ||

r_{out},z_{out} | 0–0.01, 0.17 m | $m\u02d9$_{air} | 0.95 × 10^{−3} kg/s |

Dimensions | Discretization | Boundary conditions | |||
---|---|---|---|---|---|

d_{a}_{perture} | 0.030 m | N_{cavity} | 38 | $Q\u02d9$_{solar} | 1909 W |

d_{cavity} | 0.040 m | N_{insulation} | 35 | C | 2700 suns |

L_{cavity,0} | 0.095 m | N_{front} | 8 | $m\u02d9$_{feedstock,0} | 9 × 10^{−6} kg/s |

t_{cavity} | 0.005 m | N_{CFD,r} × N_{CFD,z} | 15 × 75 | ṅ_{Ar,0} | 2.975 × 10^{−3} mol/s |

d_{reacto} | 0.100 m | N_{MC,r} × N_{MC,z} | 8 × 30 | ṅ_{H2O} | 1.307 × 10^{−3}mol/s |

L_{reactor} | 0.110 m | N_{ray} | 100000 | H_{2}O:C | 1.98 |

V_{reactor} | 0.778 × 10^{−3} m^{3} | p_{out,0} | 10^{5} Pa | ||

d_{shell} | 0.274 m | T_{in} | 400 K | ||

r_{in}, z_{in} | 0.05, 0.01–0.014 m | T_{ambient} | 288 K | ||

r_{out},z_{out} | 0–0.01, 0.17 m | $m\u02d9$_{air} | 0.95 × 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 $Q\u02d9$_{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 N_{2} (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 H_{2}:CO (▼) and CO_{2}:CO (◆) (c) for 17 experimental runs [20]. The solar reactor was operated under the following range of parameters: *p *=* *1.0 − 5.9 × 10^{5} Pa, $m\u02d9$_{Ar} = 2 − 15 L_{N}/min, $m\u02d9$_{feedstock} = 7 − 21 × 10^{−6 }kg/s, H_{2}O:C molar ratio = 1.48 − 1.98 (accounting for the moisture content in the charcoal), and $Q\u02d9$_{solar} = 1300–2600 W, which corresponds to mean solar concentration ratios^{2} 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 (H_{2}O + CO = H_{2} + CO_{2}) that occurred downstream of the reactor in steel pipes (>2 m) before the syngas analysis [20].

## 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 *T*_{f} (a), *T*_{s} (b), *f*_{V} (c), *d*_{p} (d), and d*X*_{C}/d*t* (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 $Q\u02d9$_{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 d*X*_{C}/d*t*, 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, d*X*_{C}/d*t* is here more evenly distributed with lower peak rates because *ϕ*_{radiation} ≈ 1.2 × 10^{6} W/m^{3} 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.

### 4.2 Parametric Study.

*X*

_{C}= 1 –

*ṅ*

_{C,out}/

*ṅ*

_{C,in}, and the solar-to-fuel energy conversion efficiency is defined as:

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 $m\u02d9$_{feedstock},$Q\u02d9$_{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 $m\u02d9$_{feedstock} and *ṅ*_{Ar}. Figure 4 shows *X*_{C} and *η*_{solar-to-fuel} as a function of $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0} (normalized to the baseline rate listed in Table 2) for two Ar flow rates *ṅ*_{Ar} = 2.975 × 10^{−3 }mol/s (baseline run) and *ṅ*_{Ar} = 0 mol/s. The H_{2}O:C molar ratio was kept constant at 1.98. *T*_{reactor} decreases gradually with increasing $m\u02d9$_{feedstock}(e.g., for $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0} = 4 and *ṅ*_{Ar} = 2.975 × 10^{−3 }mol/s, *T*_{reactor} = 1248 K) because of the additional sensible heat required to heat the reactants. For *ṅ*_{Ar} = 2.975 × 10^{−3 }mol/s, *X*_{C} = 88.6% is achieved for $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0} = 0.5 at the expense of low *η*_{solar-to-fuel} < 8.8%. Doubling $m\u02d9$_{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 $m\u02d9$_{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 $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0} = 2 for this reactor with these conditions. For *ṅ*_{Ar} = 0 mol/s, complete *X*_{C} is achieved for $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0} = 1, while for $m\u02d9$_{feedstock}/$m\u02d9$_{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 $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0} with *ṅ*_{Ar} = 2.975 × 10^{−3 }mol/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 $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0} = 1 and $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0} = 2. For $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0} = 1, *T*_{reactor} increases monotonically with *C* from 1180 K for 1993 suns ($Q\u02d9$_{solar} = 1409 W) to 1473 K for 3408 suns ($Q\u02d9$_{solar} = 2409 W). As expected, *X*_{C} increases with *C* because of increasing *T*_{reactor}, while full conversion is achieved for *C *>* *3054 ($Q\u02d9$_{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 $Q\u02d9$_{reradiation}. For $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0} = 2, *T*_{reactor} increases monotonically with *C* and is on average 77 K lower than that for $m\u02d9$_{feedstock}/$m\u02d9$_{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 $m\u02d9$_{feedstock}/$m\u02d9$_{feedstock,0}= 1 and $m\u02d9$_{feedstock}/$m\u02d9$_{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 $m\u02d9$_{feedstock}/$m\u02d9$_{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 $m\u02d9$_{feedstock}/$m\u02d9$_{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 $Q\u02d9$_{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 H_{2}: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 H_{2}O: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} m^{3}, 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 $Q\u02d9$_{reradiation} (configuration 1: from 196 to 160 W; configuration 2: from 206 to 146 W) and $Q\u02d9$_{convection,cavity} (configuration 1: from 74 to 53 W; configuration 2: from 76 to 50 W). $Q\u02d9$_{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%.

Configuration 1 | Configuration 2 | |||||
---|---|---|---|---|---|---|

L_{cavity}/L_{cavity,0} | L_{cavity} | L_{reactor} | d_{reactor} | V_{reactor} | d_{reactor} | V_{reactor} |

(m) | (m) | (m) | ×10^{−3} (m^{3}) | (m) | ×10^{−3} (m^{3}) | |

2/3 | 0.05 | 0.085 | 0.110 | 0.78 | 0.1 | 0.63 |

4/3 | 0.1 | 0.135 | 0.093 | 0.78 | 0.1 | 0.93 |

5/3 | 0.125 | 0.16 | 0.088 | 0.78 | 0.1 | 1.07 |

2 | 0.15 | 0.185 | 0.084 | 0.78 | 0.1 | 1.21 |

Configuration 1 | Configuration 2 | |||||
---|---|---|---|---|---|---|

L_{cavity}/L_{cavity,0} | L_{cavity} | L_{reactor} | d_{reactor} | V_{reactor} | d_{reactor} | V_{reactor} |

(m) | (m) | (m) | ×10^{−3} (m^{3}) | (m) | ×10^{−3} (m^{3}) | |

2/3 | 0.05 | 0.085 | 0.110 | 0.78 | 0.1 | 0.63 |

4/3 | 0.1 | 0.135 | 0.093 | 0.78 | 0.1 | 0.93 |

5/3 | 0.125 | 0.16 | 0.088 | 0.78 | 0.1 | 1.07 |

2 | 0.15 | 0.185 | 0.084 | 0.78 | 0.1 | 1.21 |

*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 × 10^{5} Pa for $Q\u02d9$_{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 O_{2} 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 × 10^{5} 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, m

^{2}*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/m

^{3}*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/(m

^{2}sr)=*I*identity tensor

*k*=thermal conductivity, W/(m K)

*K*=kinetic rate, Pa

^{−1}- Ka =
kinetic rate, mol/(m

^{2}s Pa) *L*=length, m

- LHV =
lower heating value, J/kg, J/mol

*m*=mass, kg

- $m\u02d9$ =
mass flow rate, kg/s, L

_{N}/min *n*=molar mass, mol

- $m\u02d9$ =
molar flow rate, mol/s

- $n\xaf$ =
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/m

^{2}- $Q\u02d9$ =
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, m

^{3}*X*_{C}=carbon conversion

*Y*=mass fraction

*Z*=axial coordinate, m

### Greek Symbols

*α*=convective heat transfer coefficient, W/(m

^{2}K)*β*=extinction coefficient, m

^{−1}- Г =
number of particles per unit volume, m

^{−3} - $\Delta H298Kdeg$ =
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/m

^{3}*σ*_{s}=scattering coefficient, m

^{−1}*ϕ*=volumetric source term, W/m

^{3}- Φ =
scattering phase function

- Ψ =
monitored quantity

- Ω =
solid angle, sr

- ∇ =
Nabla operator

*ϱ*=volumetric production/consumption rate, mol/(m

^{3}s)

### Subscripts

### Abbreviations

## Footnotes

The solar concentration ratio *C* is defined as *C* = $Q\u02d9$_{solar} /(*I·A*), where $Q\u02d9$_{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}.

## References

**156**(1), pp.

_{Th}Pressurized-Air Solar Receiver

_{2}O

_{th}Pressurized-Air Solar Receiver for Gas Turbines

**64**, pp.