This work reports on the development of a transient heat transfer model of a solar receiver–reactor designed for thermochemical redox cycling by temperature and pressure swing of pure cerium dioxide in the form of a reticulated porous ceramic (RPC). In the first, endothermal step, the cerium dioxide RPC is directly heated with concentrated solar radiation to 1500 °C while under vacuum pressure of less than 10 mbar, thereby releasing oxygen from its crystal lattice. In the subsequent, exothermic step, the reactor is repressurized with carbon dioxide as it cools, and at temperatures below 1000 °C, the partially reduced cerium dioxide is re-oxidized with a flow of carbon dioxide. To analyze the performance of the solar reactor and to gain insight into improved design and operational conditions, a transient heat transfer model of the solar reactor for a solar radiative input power of 50 kW during the reduction step was developed and implemented in ANSYS cfx. The numerical model couples the incoming concentrated solar radiation using Monte Carlo ray tracing, incorporates the reduction chemistry by assuming thermodynamic equilibrium, and accounts for internal radiation heat transfer inside the porous ceria by applying effective heat transfer properties. The model was experimentally validated using data acquired in a high-flux solar simulator (HFSS), where temperature evolution and oxygen production results from model and experiment agreed well. The numerical results indicate the prominent influence of solar radiative input power, where increasing it substantially reduces reduction time of the cerium dioxide structure. Consequently, the model predicts a solar-to-fuel energy conversion efficiency of >6% at a solar radiative power input of 50 kW; efficiency >10% can be obtained provided the RPC macroporosity is substantially increased, and better volumetric absorption and uniform heating is achieved. Managing the ceria surface temperature during reduction to avoid sublimation is a critical design consideration for direct absorption solar receiver–reactors.

## Introduction

_{2}, using a solar-driven thermochemical cycle, and the solar-produced syngas can be catalytically processed to produce liquid hydrocarbon fuels [1–4]. Recent efforts have focused on nonstoichiometric metal oxide redox reactions [5], especially the two-step cerium dioxide (ceria) cycle [6]. The two-step ceria cycle is represented by, first, a high-temperature endothermic reduction, typically approaching 1500 °C

_{2}and/or H

_{2}O, typically at temperatures below 1000 °C

where *δ* denotes the nonstoichiometry, which is a measure of the amount of oxygen exchanged during reduction and oxidation. Ceria has emerged as an attractive redox material due to its fast reaction rates and crystallographic stability [7–11]. Various solar reactor concepts have been proposed to affect the ceria redox cycle and other two-step thermochemical cycles, including cavity receiver–reactors with rotating [12–14] or stationary redox materials [15], moving [16,17] and fluidized bed reactors [18], and aerosol flow reactors [19,20].

Previously, we developed a solar cavity-receiver at the 4 kW scale that featured a ceria reticulated porous ceramic (RPC) structure with dual-scale porosity: millimeter-scale pores with struts containing micrometer-scale pores [4]. The millimeter-scale pores enhance volumetric absorption of concentrated solar radiation during the reduction step, while the micrometer-scale pores within the struts increase the specific surface area which enhances reaction kinetics during the oxidation step [21]. The solar receiver–reactor is operated with a temperature and pressure swing, thereby effecting both steps of the redox cycle in a single and stationary reaction vessel. The solar reactor was analyzed numerically using a heat and mass transfer model [22], and its operational parameters were experimentally optimized [23] through which a solar-to-fuel energy conversion efficiency, defined as the ratio of the higher heating value of the fuel produced to the solar energy delivered and accounting for parasitic losses, of *η*_{solar-to-fuel} = 5.25% was demonstrated.

In this work, we present a transient heat transfer model of a scaled-up solar receiver–reactor featuring ceria RPC structures with dual-scale porosity, which was developed in the context of a project aimed at demonstrating a fully integrated solar liquid fuels production facility [24]. The model was validated by comparing numerical results for RPC temperature evolution and oxygen release to experimental results obtained while testing the solar reactor in a high-flux solar simulator (HFSS). The validated numerical model is used to investigate the influence of various operational and design parameters on the performance of the solar reactor.

## Solar Reactor Configuration and Experimental Setup

The solar reactor configuration is shown schematically in Fig. 1. It has a water-cooled aluminum front (region 1) with a circular aperture of 16 cm diameter through which concentrated solar radiation enters. The aperture is sealed with a 12 mm thick circular quartz window that has a diameter of 300 mm. The aluminum front is attached to the reactor shell (region 2), which is made out of stainless steel (316L). The outside of the shell is insulated with a jacket (region 3) made from woven glass fibers and filled with ceramic mat board. The inside of the reactor is insulated with KVS184/400 (Rath, Inc., Newark, DE), which is primarily composed of 80% Al_{2}O_{3} and 20% SiO_{2} (region 4). The reaction cavity (region 6) is assembled with an interlocking structure of RPC bricks (region 5) made of 99.9% pure CeO_{2} (Sigma–Aldrich, Darmstadt, Germany, 1–5 *μ*m powder basis). The ceria bricks have a thickness denoted *t*_{RPC}. Between the RPC and the Al_{2}O_{3}–SiO_{2} insulation, there is a 10 mm gap (region 7) which facilitates gas flow through the reaction cavity.

The experimental setup is schematically illustrated in Fig. 2. Experiments were performed at the HFSS of the Paul Scherrer Institute (PSI). An array of ten xenon arc lamps, close-coupled to truncated ellipsoidal reflectors, provides an external source of intense thermal radiation, mostly in the visible and infrared spectra, which closely approximates the radiative properties of highly concentrating solar systems such as towers and dishes [25]. The radiative flux distribution at the aperture plane of the solar reactor was optically measured using a calibrated CCD camera focused onto a water-cooled, Al_{2}O_{3} plasma-coated Lambertian target. The total solar radiative power input *P*_{solar} was obtained by integration of the measured flux distribution in the aperture plane, including adjusting for the absorption and reflection losses at the quartz window. The positions of the temperature measurements are indicated in Fig. 1. The temperature of the reacting ceria was monitored at three positions at the back surface of the RPC using B-type thermocouples. The temperature of the lateral Al_{2}O_{3}–SiO_{2} insulation was measured at three different depths using K-type thermocouples. K-type thermocouples were also used to measure the temperatures of the outer lateral surfaces of the reactor shell and the insulating jacket. Gas flow rates were regulated using electronic mass flow controllers (Bronkhorst, Ruurlo, The Netherlands, EL-FLOW Select). The pressure inside the reactor was measured at the gas outlet using a Pirani gauge sensor combined with a capacitance diaphragm vacuum gauge (Leybold, Cologne, Germany, THERMOVAC TTR 101). A dry, multistage roots vacuum pump (Pfeiffer, Annecy, France, ACP 40) was attached to the outlet port of the solar reactor via two parallel evacuation valves. A membrane valve was used to slowly evacuate the reactor at the beginning of the reduction step (path shown by the red arrow in Fig. 2), and a gate valve with bigger nominal diameter was opened once the pressure was sufficiently low (< 50 mbar). During the oxidation step (path shown by the blue arrow in Fig. 2), the vacuum pump was bypassed by use of a manual membrane valve. The composition of the product gas was continuously (frequency 1 Hz) analyzed downstream using an electrochemical sensor for O_{2}, and IR detectors for CO and CO_{2} (Siemens, Berlin, Germany, Ultramat 23). The gas composition was verified by gas chromatography (Agilent, Santa Clara, CA, 490 Micro GC) with a measurement frequency of 0.005 Hz.

During an experimental run, the solar reactor was first slowly preheated with a radiative power input *P*_{solar} between 5 and 10 kW for approximately 75 min, followed by a slow precycle. During a precycle, the reactor was heated with *P*_{solar} = 10–25 kW ramping up, for approximately 30 min until the nominal reduction temperature was reached. The precycle was terminated by closing the shutters of the solar simulator (thereby affecting *P*_{solar} = 0 kW) and letting the reactor naturally cool down to the nominal oxidation temperature. The ceria was fully re-oxidized using CO_{2}. Once the average temperature of the three measurement positions at the back of the RPC (*T*_{RPC,nom}) reached 730 °C, the primary reduction step was initiated with *P*_{solar} = 30.5 kW, while the reactor was evacuated using the vacuum pump. The total pressure in the reactor during the reduction step was in the range of 5–9 mbar, depending on the current rate of oxygen release. To protect the quartz window from deposition of sublimated ceria [26] and to govern the fluid flow field when operating under vacuum conditions, an argon flow rate of 5 L min^{−1} (SLPM; volume flow rate calculated at 273.15 K and 101,325 Pa) was introduced to the reactor directly behind the window. When *T*_{RPC,nom} reached 1485 °C, re-oxidation was initiated by removing input power and re-pressurizing the reactor with CO_{2}. After *T*_{RPC,nom} fell to 1030 °C by natural cooling, 70 L min^{−1} of CO_{2} was flown through the reactor to fully re-oxidize the ceria, producing a mixed flow in the outlet composed of CO and unreacted CO_{2}.

## Heat Transfer Analysis

The solar receiver–reactor, which features inherent axial symmetry, was simulated using a two-dimensional axisymmetric heat transfer model implemented in the commercial CFD software ANSYS cfx (release 17.0). The aluminum front, steel shell, insulating jacket, and Al_{2}O_{3}–SiO_{2} insulation are modeled as solid domains. The cavity and the open region behind the RPC are modeled as fluid domains that are nonparticipating in radiation. During the reduction step, the fluid is assumed to be stationary, as the previous modeling results have shown that the contribution of convection to heat transfer is negligible, especially while operating under vacuum [22]. The ceria RPC is modeled as a homogeneous and radiatively participating porous media.

### Governing Equations.

*ρ*is the density,

*h*is the enthalpy, and

*k*is the thermal conductivity. The energy conservation equation for the solid phase of the porous RPC domain is

where *ϕ*_{dual} is the dual-scale porosity of the RPC, *k*_{eff} is its effective thermal conductivity, *S*_{solar} is the source term accounting for the absorbed incoming solar radiation from the HFSS, and *S*_{reaction} is the energy sink accounting for the endothermic reduction of ceria. The conservation equations for solid and fluid phases are coupled by the source term $Qsf=\u2212Qfs=hfs\u22c5Afs(Tf\u2212Ts)$, where *h*_{fs} is the interfacial heat transfer coefficient, *A*_{fs} is the fluid–solid area density, and *T*_{s} and *T*_{f} are the temperatures of the solid and the fluid, respectively. An artificially high *h*_{fs} of 10,000 W m^{−1} K^{−1} enforces thermal equilibrium between the two phases (*T*_{s} = *T*_{f}). This is reasonable due to the assumption of the fluid being stationary. According to the correlation described in Ref. [27], *A*_{fs} is set to 951.8 m^{−1}.

*S*

_{radiation}is the source term accounting for radiation exchange. Due to a constraint in ANSYS cfx that only allows radiative heat transfer in the fluid and not in the solid phase of porous domains, the radiative properties of the fluid are set to the effective radiative properties of the RPC; local thermal equilibrium between the fluid and the solid phase is enforced. The radiative transfer equation for the RPC, modeled as an isotropic, gray, absorbing–emitting–scattering participating media, is

*s*is the path length,

*β*,

*α*, and

*σ*are the extinction, absorption, and scattering coefficients, respectively,

*I*is the radiation intensity,

*I*

_{b}is the blackbody radiation intensity depending on the local temperature, and

*ω*is the solid angle. For the geometric optics regime, $\alpha =\beta \u22c5(1\u2212r)$ and $\sigma =\beta \u22c5r$, where

*r*is the reflectivity of CeO

_{2}. The radiation exchange source term in Eq. (6) is calculated as

### Boundary Conditions and Source Terms.

The boundary conditions and source terms are schematically indicated in Fig. 1. At the outer surface of the insulating jacket, the exposed surface of the reactor shell not covered by the jacket, and the lateral surface of the aluminum front, energy is lost by radiative and convective heat transfer. The convective heat transfer coefficient is conservatively assumed to be 15 W m^{−1} K^{−1}. Due to moderate surface temperatures, the temperature distribution within the solar reactor is insensitive to the value of heat transfer coefficient taken on these surfaces. The ambient air temperature, water cooling channel temperature, and the quartz window temperature are all assumed to be at 293 K. The front surface of the aluminum front of the reactor is assumed to be adiabatic. The mean transmissivity of the 12 mm thick quartz window was experimentally measured to be *τ* = 0.929. Flux maps were acquired by a calibrated CCD camera viewing a Lambertian target while it was irradiated with the HFSS. The 300 mm diameter quartz window was placed in front of the target such that it could intercept the entire light cone produced by the HFSS. By comparing flux maps taken with and without the window intercepting the radiation, the mean transmissivity, thereby averaged over all incident angles of the radiation, could be extracted. The volumetric and surface heat sources *S*_{solar} within the RPC and on the front insulation surface were derived using a decoupled Monte Carlo (MC) ray tracing model. This model yields the absorbed radiative power delivered by the HFSS. For the calculation of the nonstoichiometry $\delta $ of ceria, thermodynamic equilibrium is assumed, as the previous experimental work with similar ceria RPCs has shown that the reduction step is heat transfer limited [26]. The oxygen partial pressure is assumed to be constant at *p*_{O2} = 5 mbar due to operation under vacuum. The energy sink *S*_{reaction}, accounting for the endothermic reduction reaction, is calculated using the two expressions for equilibrium *δ* [28] and reaction enthalpy Δ*H*_{O2} [22], listed in Table 1.

Variable | Correlation | Unit | Ref |
---|---|---|---|

Dual-scale porosity | $\varphi dual=0.78$ | ||

Strut porosity | $\varphi strut=0.3561$ | ||

Single-scale porosity | $\varphi single=\varphi dual\u2212\varphi strut1\u2212\varphi strut=0.6583$ | ||

Number of pores per inch | $nppi=10$ | ||

Mean pore diameter | $dm=(5.302\xd710\u22125\u22c5\varphi single+2.155\xd710\u22125)\u22c5357nppi=2.015\xd710\u22123$ | m | [27] |

Extinction coefficient | $\beta =\u2212630.674\u22c5\varphi single2\u2212120.06\u22c5\varphi single+1229.361000\u22c5dm=435.15$ | m^{−1} | [27] |

Total hemispherical reflectance (at δ = 0.035) | $r=0.41184\u22122.419\xd710\u22125\u22c5T$ | [29] | |

Density CeO_{2} | $\rho CeO2=7220$ | kg m^{−3} | [30] |

Molar mass | $MCeO2=0.1721$ | kg mol^{−1} | [31] |

Specific heat capacity | $cp,CeO2=67.95\u22129.9\xd7105\u22c5T\u22122+0.0125\u22c5TMCeO2$ | J kg^{−1} K^{−1} | [32] |

Thermal conductivity CeO_{2} | $kCeO2=\u22121.723\xd710\u22129\u22c5T3+1.12\xd710\u22125\u22c5T2\u22120.024\xd7T+17.8$ | W m^{−1} K^{−1} | [33] |

Effective thermal conductivity | $keff=kCeO2\u22c5(1\u22120.6223\u22c5\varphi dual)\u22c5(1\u22121.055\u22c5\varphi dual)$ | W m^{−1} K^{−1} | [34] |

Equilibrium thermodynamics (T in °C) | $\delta =10\u2212(2.15\xd710\u22126\u22c5T2\u22129.88\xd710\u22123\u22c5T+12.2)\u22c5(pO2p0)1.25\xd710\u22127\u22c5T2\u22123.1\xd710\u22124\u22c5T\u22121.83\xd710\u22122$ | [28] | |

Reaction enthalpy | $\Delta HO2=969.409\u2212503.739\u22c5\delta 0.5$ | kJ mol^{−1} | [22] |

Variable | Correlation | Unit | Ref |
---|---|---|---|

Dual-scale porosity | $\varphi dual=0.78$ | ||

Strut porosity | $\varphi strut=0.3561$ | ||

Single-scale porosity | $\varphi single=\varphi dual\u2212\varphi strut1\u2212\varphi strut=0.6583$ | ||

Number of pores per inch | $nppi=10$ | ||

Mean pore diameter | $dm=(5.302\xd710\u22125\u22c5\varphi single+2.155\xd710\u22125)\u22c5357nppi=2.015\xd710\u22123$ | m | [27] |

Extinction coefficient | $\beta =\u2212630.674\u22c5\varphi single2\u2212120.06\u22c5\varphi single+1229.361000\u22c5dm=435.15$ | m^{−1} | [27] |

Total hemispherical reflectance (at δ = 0.035) | $r=0.41184\u22122.419\xd710\u22125\u22c5T$ | [29] | |

Density CeO_{2} | $\rho CeO2=7220$ | kg m^{−3} | [30] |

Molar mass | $MCeO2=0.1721$ | kg mol^{−1} | [31] |

Specific heat capacity | $cp,CeO2=67.95\u22129.9\xd7105\u22c5T\u22122+0.0125\u22c5TMCeO2$ | J kg^{−1} K^{−1} | [32] |

Thermal conductivity CeO_{2} | $kCeO2=\u22121.723\xd710\u22129\u22c5T3+1.12\xd710\u22125\u22c5T2\u22120.024\xd7T+17.8$ | W m^{−1} K^{−1} | [33] |

Effective thermal conductivity | $keff=kCeO2\u22c5(1\u22120.6223\u22c5\varphi dual)\u22c5(1\u22121.055\u22c5\varphi dual)$ | W m^{−1} K^{−1} | [34] |

Equilibrium thermodynamics (T in °C) | $\delta =10\u2212(2.15\xd710\u22126\u22c5T2\u22129.88\xd710\u22123\u22c5T+12.2)\u22c5(pO2p0)1.25\xd710\u22127\u22c5T2\u22123.1\xd710\u22124\u22c5T\u22121.83\xd710\u22122$ | [28] | |

Reaction enthalpy | $\Delta HO2=969.409\u2212503.739\u22c5\delta 0.5$ | kJ mol^{−1} | [22] |

### Material Properties.

Material properties of the ceria RPC with dual-scale porosity are listed in Table 1. The dual-scale porosity of the RPC was calculated by measuring its mass and volume. The strut porosity was assessed with a combination of mercury intrusion porosimetry (Quantachrome Poremaster 60-GT) and geometric approximations to calculate the size of the hollow struts. The value for the number of pores per inch was provided by the manufacturer of the polyurethane foams used to manufacture the RPCs. The effective heat and mass transfer properties of the RPC structure were taken from literature. The correlation for the total hemispherical reflectance of CeO_{2} was evaluated for an average reduction state of *δ* = 0.035 and depends on the local temperature in the heat transfer model. To calculate the heat source *S*_{solar} using the MC ray-tracing model, a correlation weighted according to Planck's law for blackbody temperatures of 5780 K, which is a good approximation of solar radiation, was used, resulting in *r *=* *0.2905 [29]. For the calculation of the effective thermal conductivity of the RPC, thermal conductivity of the fluid was set to zero, due to operation under vacuum.

The heat transfer properties of the solid domains are listed in Table 2. They were either taken from literature, or values from the manufacturers were used. For the specific heat capacity of the Al_{2}O_{3}–SiO_{2} insulation, a mass-weighted average of alumina and silica heat capacities was calculated according to the chemical composition, as suggested in Ref. [46]. Fluid domain properties were taken as a modified inert gas for simplicity, as the domain has negligible contribution to heat transfer.

Variable | Correlation | Unit | Ref. |
---|---|---|---|

Al_{2}O_{3}–SiO_{2} insulation (Rath, Inc. KVS184/400) | |||

Density | $\rho ins=400$ | kg m^{−3} | [35] |

Specific heat capacity | $cp,ins=\u22123.09\xd710\u221210\u22c5T4+1.71\xd710\u22126\u22c5T3\u22123.48\xd710\u22123\u22c5T2+3.18\xd7T+101$ | J kg^{−1} K^{−1} | [36,37] |

Thermal conductivity | $kins=\u22122.09\xd710\u221211\u22c5T3+1.06\xd710\u22127\u22c5T2+3.69\xd710\u22125\u22c5T+7.07\xd710\u22122$ | W m^{−1} K^{−1} | [35] |

Hemispherical total emittance | $\epsilon ins=0.28$ | [38] | |

Stainless steel 316 L shell | |||

Density | $\rho shell=8000$ | kg m^{−3} | [39] |

Specific heat capacity | $cp,shell=412+0.2\u22c5T\u22122\xd710\u22125\u22c5T2$ | J kg^{−1} K^{−1} | [40] |

Thermal conductivity | $kshell=0.013\u22c5T+11.45$ | W m^{−1} K^{−1} | [39] |

Hemispherical total emittance | $\epsilon shell=0.57$ | [41] | |

Insulating jacket | |||

Density | $\rho jacket=80$ | kg m^{−3} | [42] |

Specific heat capacity | $cp,jacket=840$ | J kg^{−1} K^{−1} | [42] |

Thermal conductivity | $kjacket=5.319\xd710\u22127\u22c5T2\u22122.487\xd710\u22124\u22c5T+6.433\xd710\u22122$ | W m^{−1} K^{−1} | [42] |

Hemispherical total emittance | $\epsilon jacket=0.89$ | [43] | |

Aluminum front | |||

Density | $\rho Al=2700$ | kg m^{−3} | [44] |

Specific heat capacity | $cp,Al=706.7+0.6\u22c5T\u22121\u22c510\u22124\u22c5T2$ | J kg^{−1} K^{−1} | [40,45] |

Thermal conductivity | $kAl=\u22124.01\xd710\u221210\u22c5T4+1.14\xd710\u22126\u22c5T3\u22121.22\xd710\u22123\u22c5T2+0.53\u22c5T+162$ | W m^{−1} K^{−1} | [46,47] |

Hemispherical total emittance | $\epsilon Al=0.09$ | [41] |

Variable | Correlation | Unit | Ref. |
---|---|---|---|

Al_{2}O_{3}–SiO_{2} insulation (Rath, Inc. KVS184/400) | |||

Density | $\rho ins=400$ | kg m^{−3} | [35] |

Specific heat capacity | $cp,ins=\u22123.09\xd710\u221210\u22c5T4+1.71\xd710\u22126\u22c5T3\u22123.48\xd710\u22123\u22c5T2+3.18\xd7T+101$ | J kg^{−1} K^{−1} | [36,37] |

Thermal conductivity | $kins=\u22122.09\xd710\u221211\u22c5T3+1.06\xd710\u22127\u22c5T2+3.69\xd710\u22125\u22c5T+7.07\xd710\u22122$ | W m^{−1} K^{−1} | [35] |

Hemispherical total emittance | $\epsilon ins=0.28$ | [38] | |

Stainless steel 316 L shell | |||

Density | $\rho shell=8000$ | kg m^{−3} | [39] |

Specific heat capacity | $cp,shell=412+0.2\u22c5T\u22122\xd710\u22125\u22c5T2$ | J kg^{−1} K^{−1} | [40] |

Thermal conductivity | $kshell=0.013\u22c5T+11.45$ | W m^{−1} K^{−1} | [39] |

Hemispherical total emittance | $\epsilon shell=0.57$ | [41] | |

Insulating jacket | |||

Density | $\rho jacket=80$ | kg m^{−3} | [42] |

Specific heat capacity | $cp,jacket=840$ | J kg^{−1} K^{−1} | [42] |

Thermal conductivity | $kjacket=5.319\xd710\u22127\u22c5T2\u22122.487\xd710\u22124\u22c5T+6.433\xd710\u22122$ | W m^{−1} K^{−1} | [42] |

Hemispherical total emittance | $\epsilon jacket=0.89$ | [43] | |

Aluminum front | |||

Density | $\rho Al=2700$ | kg m^{−3} | [44] |

Specific heat capacity | $cp,Al=706.7+0.6\u22c5T\u22121\u22c510\u22124\u22c5T2$ | J kg^{−1} K^{−1} | [40,45] |

Thermal conductivity | $kAl=\u22124.01\xd710\u221210\u22c5T4+1.14\xd710\u22126\u22c5T3\u22121.22\xd710\u22123\u22c5T2+0.53\u22c5T+162$ | W m^{−1} K^{−1} | [46,47] |

Hemispherical total emittance | $\epsilon Al=0.09$ | [41] |

### Initial Condition.

To establish the initial condition, the solar reactor was heated from room temperature for 1 h with a radiative power input, evaluated at the reactor window, of *P*_{solar} = 10 kW. Subsequently, *P*_{solar} was set to zero, and the reactor was allowed to cool naturally to the desired start temperature. The temperature field established with the preheating simulation was then applied as an initial condition for the transient reduction simulation and all subsequent analysis. It was confirmed experimentally that the initial condition is accurate and further established that small variations in the initial condition (temperature field) do not have a significant influence on the final temperature field after reduction.

### Numerical Solution.

The heat sources *S*_{solar} were calculated by applying an in-house MC ray-tracing code [48] with 10^{9} rays. The heat transfer simulations were performed with ANSYS cfx (version 17.0). To discretize the governing equations in space, between 35,115 and 54,980 hexahedral cell elements were used. Due to a limitation in ANSYS cfx, a single cell had to be extruded in the third direction around the symmetry axis. For the discretization in time, a constant time-step of 1 s was used. The finite volume method was applied with a second-order backward Euler scheme. To solve the radiative transfer equation (Eq. (7)), the discrete transfer model was used, transforming the equation into a set of transport equations for *I* and solving for discrete solid angles along *s*. The simulations were performed using the high-performance cluster Euler of ETH Zurich.

## Experimental Validation

The heat transfer model was validated by comparing the calculated temperature and oxygen evolution to the experimentally determined values measured during testing of the solar reactor in the HFSS [49]. Figure 3(a) shows the numerically calculated (solid lines) and the experimentally measured (dashed lines) temperatures at different thermocouple positions as indicated in Fig. 1. The agreement between simulation and experiment is reasonably good for all thermocouple positions, most importantly the B-type thermocouples in contact with the back of the RPC (standard deviation between experimental and numerical *T*_{RPC,nom} during reduction was 9.4 °C). For both the simulation and the experiment, the RPC temperature at the front position (*T*_{B,3}) is significantly lower than the temperatures toward the back of the RPC (*T*_{B,1} and *T*_{B,2}). The temperature of the Al_{2}O_{3}–SiO_{2} insulation at the innermost position (*T*_{K,1}) is slightly overestimated in the simulation. This is because the thermal conductivity of the porous insulation is assumed constant, whereas in reality it changes between the reduction step, which is operated under vacuum, and the oxidation step, which is operated at atmospheric pressure. Temperatures of the reactor shell (*T*_{K,4}) and the insulating jacket (*T*_{K,5}) are slightly underestimated in the simulation, due to a lower initial condition for the external surfaces, however, the curvature still matches experimental results. In Fig. 3(b), *T*_{RPC,nom} and the O_{2} release rate are shown for the simulation (solid lines) and the experiment (dashed lines). The two curves for *T*_{RPC,nom} match well, with the maximum temperature being 1470 °C for the simulation and 1489 °C for the experiment. The O_{2} release at low temperatures is slightly overestimated in the simulation, however, the integrated value of 31.12 L matches well with the experimentally measured integrated amount of 29.16 L (6% difference).

Note that *Q*_{solar} is only delivered during the endothermic reduction step. Assuming complete re-oxidation using CO_{2}, the energy content of the fuel (CO) produced is calculated as $Qfuel=\Delta Hfuel\u22c52\u222brO2dt$, where Δ*H*_{fuel} is the heating value of CO (Δ*H*_{fuel} = 283 kJ mol^{−1}) and $\u222brO2dt$ is the rate of released O_{2} integrated over the reduction step. *Q*_{solar} is the total solar energy input integrated over the reduction step. *Q*_{pump} and *Q*_{inert} are the energy penalties associated with vacuum pumping and the consumption of the inert gas Ar during the reduction step, respectively, and are calculated as described in Ref. [23]. Note that *η*_{solar-to-fuel} is weakly dependent on the assumptions used for the calculation of these two energy penalties, because *Q*_{solar} is roughly 2 orders of magnitude larger than *Q*_{pump} and *Q*_{inert}. An efficiency of *η*_{solar-to-fuel} = 3.38% was predicted by the simulation, which is comparable to the experimentally determined efficiency *η*_{solar-to-fuel} = 3.17%. Heat recovery was not applied. The slight overestimation is correlated directly to the slight overestimation in total O_{2} yield from the simulation.

## Modeling Results and Discussion

The validated numerical model is a useful tool not only to better understand the performance of the current solar reactor, but also to assess the influence of various design and operational changes on the performance of the reactor. In the subsequent analysis, a base case simulation representing the experimental validation case (as described in Sec. 4) is used to perform a parametric study of several crucial design variables of the ceria RPC. The critical parameters of the base case simulation are summarized in Table 3.

Variable | Value | Unit | |||
---|---|---|---|---|---|

RPC morphology | |||||

Dual-scale porosity | $\varphi dual=0.78$ | ||||

Number of pores per inch | $nppi=10$ | ||||

Extinction coefficient | $\beta =435.15$ | m^{−1} | |||

Thickness of RPC | $tRPC=25$ | mm | |||

Ceria mass loading | $mRPC=18.38$ | kg | |||

Operational parameters | |||||

Solar radiative power input | $Psolar=30.5$ | kW | |||

Partial pressure of oxygen | $pO2=5$ | mbar | |||

Reduction start temperature | $Tred,start=730$ | °C | |||

Reduction end temperature | $Tred,end=1466$ | °C |

Variable | Value | Unit | |||
---|---|---|---|---|---|

RPC morphology | |||||

Dual-scale porosity | $\varphi dual=0.78$ | ||||

Number of pores per inch | $nppi=10$ | ||||

Extinction coefficient | $\beta =435.15$ | m^{−1} | |||

Thickness of RPC | $tRPC=25$ | mm | |||

Ceria mass loading | $mRPC=18.38$ | kg | |||

Operational parameters | |||||

Solar radiative power input | $Psolar=30.5$ | kW | |||

Partial pressure of oxygen | $pO2=5$ | mbar | |||

Reduction start temperature | $Tred,start=730$ | °C | |||

Reduction end temperature | $Tred,end=1466$ | °C |

### Incident Solar Radiation and Temperature Distribution.

A contour plot of absorbed incoming solar radiation from the HFSS, *S*_{solar}, is shown in Fig. 4(a). *S*_{solar} is constant during the reduction step. Due to the relatively large optical thickness of the RPC ($\tau RPC=\beta \u22c5tRPC=10.9$), more than 90% of the incoming radiation is absorbed within the first five millimeters of the RPC structure, which can clearly be seen in the figure. Due to the uneven distribution of the incoming solar radiation, caused by the discrete nature of the HFSS radiation source, *S*_{solar} is high toward the back corner of the RPC structure and relatively low at the center of the back. Figure 4(b) shows the temperature distribution within the solid and the RPC domain of the reactor at the end of the reduction step, and of the RPC domain only (enlarged). The hottest regions in the temperature profile within the RPC correspond to the areas of highest *S*_{solar}; the front, directly irradiated surface of the RPC reaches the highest temperatures, while the back of the RPC and areas which are less directly irradiated remain at lower temperatures. This nonuniformity of temperature within the RPC limits the efficiency that can be achieved with the solar reactor, as the nonstoichiometry *δ* (a measure of oxygen released during reduction) is directly correlated to the ceria temperature which is achieved. To achieve a more uniform temperature distribution using highly concentrated sunlight, and do so quickly enough to reach a high solar-to-fuel energy conversion efficiency, the macroporosity (millimeter-scale) of the absorber material (in this case an RPC) must be substantially increased.

### Energy Flows.

The instantaneous energy balance for the reduction step is illustrated as a function of time in Fig. 5. Note that heat recovery was not applied. Losses by reradiation from the hot cavity, the change in sensible heat content of the RPC, the remaining reactor components (Al_{2}O_{3}–SiO_{2} insulation, aluminum front, reactor shell, and insulating jacket), the energy consumed by the endothermic reduction reaction, the conductive heat loss to the water-cooled reactor front, and other heat losses are indicated. Other heat losses include reflection of incoming solar radiation inside the reactor cavity and at the quartz window, absorption of incoming radiation at the window, and convection and radiation at the outer reactor surfaces. Initially, sensible heating of the RPC dominates energy consumption, consuming 87% of *P*_{solar}, while on average it consumes 33%. By the end of the reduction step, reradiation dominates heat loss, accounting for 31% of *P*_{solar} on average and 45% at the peak. Reradiation losses could be lowered by decreasing the size of the aperture, provided that solar radiation can be delivered with higher concentration. A selective coating with high transmissivity in the visible region of the solar spectrum, but high reflectivity in the IR region of the radiation emitted by the hot cavity, could be considered for the quartz window, provided that the coating can withstand very high temperatures (>500 °C). Sensible heating of the bulk materials consumes 21% of *P*_{solar} on average, but levels off early in the reduction cycle, with the Al_{2}O_{3}–SiO_{2} insulation being the dominant consumer, while the aluminum front, reactor shell, and insulating jacket consume 1.2% or less each. Energy loss through sensible heating of the bulk materials could be lowered if insulation materials with lower specific heat capacity were used. The energy fraction driving the endothermic reduction reaction of ceria quickly increases with time, and on average, accounts for 5.6% of *P*_{solar}. The conduction heat losses to the water-cooled reactor front are significant, with an average consumption of 2.7% of *P*_{solar}. The losses by convection and radiation at the outer reactor surfaces, as well as the energy lost by reflection of the incoming solar radiation inside the reactor cavity, account for less than 0.3% of *P*_{solar} each. The remaining 7.1% of *P*_{solar} is lost by absorption and reflection at the quartz window (*τ* = 0.929). Although not considered in the simulation, convective losses associated with gases exiting the solar reactor during the reduction step are also negligible (less than 0.3% of the input power). The share of energy used to drive the reduction reaction, and therefore also representative of the solar-to-fuel energy conversion efficiency, could potentially be increased by using doped ceria to increase the reduction extent [50,51], or by minimizing the temperature swing with near isothermal operation [15,52,53], although this does not necessarily increase the efficiency due to other limitations introduced with a lower temperature swing.

### Parameter Study.

Operational and design parameters of the solar reactor can be optimized using the numerical model. The most critical parameters are the level of input power and the structure of the ceria RPC. These two parameters are coupled, as higher power is only beneficial if it can be more uniformly absorbed within the RPC structure. If solar radiation is only absorbed within the first small fraction of RPC depth, performance becomes limited by the maximum sustainable surface temperature of the ceria RPC. A parametric study was conducted using the parameters listed in Table 3 as the base case. The following parameters were varied in the study: RPC thickness *t*_{RPC}, RPC dual-scale porosity *ϕ*_{dual}, and the radiative power input *P*_{solar}. All of the simulations were initialized with *T*_{RPC,nom} = 730 °C, and the duration of the reduction step *t*_{red} was controlled by setting *P*_{solar} to zero once *T*_{RPC,nom} = 1466 °C was reached. The results of the parameter study are shown in Figs. 6(a)–6(f). In the left column (Figs. 6(a), 6(c), and 6(e)), the nominal RPC temperature *T*_{RPC,nom} and the oxygen release rate are plotted as a function of time. In the right column (Figs. 6(b), 6(d), and 6(f)), the variable parameters are plotted versus *η*_{solar-to-fuel}, *t*_{red}, the reduction time required to reach *T*_{RPC,nom} = 1466 °C, and *T*_{RPC,max}, the maximum temperature of the RPC reached at the end of the reduction step, which is a critical value for the mechanical stability of the RPCs.

The effect of RPC thickness *t*_{RPC} is shown in Figs. 6(a) and 6(b). The inner, directly irradiated surface area of the RPC as well as the thickness of the separating gap between the RPC and the Al_{2}O_{3}–SiO_{2} insulation was kept constant, while the thickness of the insulation was adapted slightly (and with negligible effect). For both higher and lower *t*_{RPC} values compared to the base case, *η*_{solar-to-fuel} slightly decreases, while *t*_{red} increases with increasing RPC thickness. This is due to the increasing ceria mass loading of the reactor, and consequently, longer duration of the reduction step, which yields a higher total amount of O_{2} released. This can be seen in Fig. 6(a). Due to the increased thickness of the RPC, *T*_{RPC,max} increases as the end of the reduction step is controlled by the temperature at the back surface of the RPC. It is important to note the scale of the efficiency metric, which shows that large variation in the RPC thickness parameter, while yielding a trend, only impacts the efficiency by a fraction of a percent.

The effect of changing RPC porosity *ϕ*_{dual} is illustrated in Figs. 6(c) and 6(d). The only variable adjusted is *ϕ*_{dual}, while *ϕ*_{single} and *n*_{ppi} are kept constant. The chosen values of *ϕ*_{dual} correspond to a change in ceria mass loading of ±25% compared to the base case. *η*_{solar-to-fuel} decreases slightly from 3.54% at *ϕ*_{dual} = 0.725 to 3.14% at *ϕ*_{dual} = 0.835. The influence on *t*_{red} is higher, with a decrease from 16.1 min (*ϕ*_{dual} = 0.725) to 9.15 min (*ϕ*_{dual} = 0.835), mainly caused by the significant difference in ceria mass loading. Similar to the impact of changing RPC thickness, the effect of decreasing reduction time is counteracted by a decrease in total O_{2} released, and therefore, the efficiency only changes slightly. With increasing *ϕ*_{dual}, the optical thickness of the RPC decreases, leading to a slightly lower *T*_{RPC,max}.

The most influential variable is the solar radiative power input, as can be seen in Figs. 6(e)–6(f). Increasing *P*_{solar} drastically decreases *t*_{red} and increases the efficiency *η*_{solar-to-fuel}. Roughly doubling *P*_{solar} from 30.5 kW to 60 kW cuts *t*_{red} by more than half (12.3 min to 5.0 min) and more than doubles *η*_{solar-to-fuel} (from 3.38% to 7.34%). This is attributed primarily to two phenomena: first, heat losses, especially by reradiation, decrease due to a shorter reduction time, and second, higher RPC temperatures toward the irradiated front surface directly lead to higher oxygen nonstoichiometry *δ*. *T*_{RPC,max} increases from 1719 °C at *P*_{solar} = 30.5 kW to 1914 °C at *P*_{solar} = 60 kW.

### Advanced Reactor Design.

An additional case was considered to assess the possibility of designing a solar receiver–reactor with parameters optimized beyond the current means of production. This advanced reactor design features much larger pores (*n*_{ppi} = 3; average macropore diameter ≅ 7 mm) to dramatically enhance volumetric absorption, but the same porosity and thereby mass loading as in the base case. In Fig. 7(a), the absorbed solar radiation *S*_{solar} as well as the local RPC temperature is shown as a function of the penetration depth for the advanced RPC design. The location of extraction of these variables is indicated in Fig. 4. For comparison, the results for the case with *n*_{ppi} = 10 (average macropore diameter ≅ 2 mm) are also shown. In both cases, *P*_{solar} was set to 60 kW and the values correspond to a simulation time of 298 s, which is the time when the reduction step ends in the case of *n*_{ppi} = 10. In the case of *n*_{ppi} = 3, *S*_{solar} is more uniformly distributed, leading to a more uniform distribution of temperature within the RPC. The temperature difference between the front and the back of the RPC equals 113 °C, compared to 377 °C for *n*_{ppi} = 10. The more uniform distribution of temperature within the RPC directly results in higher performance of the solar reactor when it is properly operated. Due to the lower temperature difference between the front and the back of the RPC, *t*_{red} can be extended without exceeding the critical value for the maximum RPC temperature. This is illustrated in Fig. 7(b), which shows the nominal RPC temperature and the rate of released oxygen as a function of time for *n*_{ppi} = 3 (solid lines) and for *n*_{ppi} = 10 (dashed lines). For *n*_{ppi} = 3, *t*_{red} is extended to 423 s. Due to the more uniform temperature distribution, this results in the same critical value of *T*_{RPC,max} = 1914 °C at the end of the reduction step as in the case of *n*_{ppi} = 10. As a consequence, the total amount of O_{2} released drastically increases from 53.55 L (*n*_{ppi} = 10) to 106.5 L (*n*_{ppi} = 3), which ultimately results in a better performance of the solar reactor (*η*_{solar-to-fuel} = 10.2% compared to *η*_{solar-to-fuel} = 7.34% for *n*_{ppi} = 10). However, the path to realizing a ceria structure with the physical parameters required to obtain this level of performance is ongoing research and development [54,55]. Note that this analysis does not consider heat recovery.

It is important to consider the impact of volumetric absorption and uniform heating when analyzing and scaling solar reactors. With typical chemical reactors, for example continuously stirred thermal reactors, scaling up results in significantly increased thermal performance because of the increased ratio of active volume to external surface area [56]. For solar RPC reactor technology, however, this is not the case because the active volume is limited to the ceria RPC. The solar reactor analyzed in this study is more than 12 times larger than its precursor technology, where an efficiency of 5.25% was experimentally demonstrated [23], and yet an efficiency of only 6.12% is predicted here for the nominal 50 kW case and otherwise similar operating conditions. This is directly due to a decreasing active volume fraction which results from scaling an RPC solar reactor; at the 4 kW scale, the ceria RPC represented 60% of the chemical reactor volume, while at the 50 kW scale it is only 30%. Assuming a constant apparent mass density inside the ceria RPC, the total mass loading of the reactor is limited in the same way. In 2012, Furler et al. [26] determined that ceria RPCs outperformed ceria blocks and felts because of the structure's relatively enhanced radiation heat transfer properties, although direct absorption of solar radiation was still limited. In this study, we show that for solar reactors of this type to operate efficiently, increased utilization of the cavity volume by achieving higher volumetric absorption of the incoming solar radiation, and thus volumetric heating, is necessary.

The impact that sensible heat recovery could have on the performance of a solar reactor is evident from the energy balance presented in Fig. 5. Previous studies have considered various forms of heat recovery and the implication on both reactor and system level efficiency [52,57–59]. For the reactor technology discussed here, consisting of stationary redox ceramics which are directly irradiated in temperature and pressure swing operation, heat recovery options are limited. The possibility to actively recover heat during the cooling step after reduction exists, but its impact is limited by the need to utilize an inert gas heat transfer fluid for multiple stages of solid–gas heat exchange [60].

A simple energy balance analysis can be performed to determine the impact of extracting heat from the stationary ceria mass between the reduction and oxidation steps and providing it back to the solar reactor. Considering the case of 50 kW of solar input power for the base RPC parameters listed in Table 3 (total ceria mass of 18.4 kg covering 30% of the reactor volume), a solar-to-fuel energy conversion efficiency of 6.12% was determined. 7813 kJ of energy is contained as sensible heat in the ceria solid (42% of the solar energy input during the reduction step). For the purpose of discussing the potential of minimizing this irreversibility, 100% of this sensible heat is considered to be recoverable. Accounting for this recoverable heat as a subtraction from the denominator of Eq. (9), presumably representing the fact that less solar energy would be required to heat the solid, and further accounting for less energy lost by reradiation because of the resulting shorter reduction time (212 s versus 368 s), an efficiency of *η*_{solar-to-fuel} = 12.75% is determined. It is important to note that removing and reusing even 50% of the sensible heat contained in the ceria RPC represents a major engineering challenge.

## Summary and Conclusions

We have reported on the development and use of an experimentally validated transient heat transfer model of a scaled-up ceria RPC solar reactor designed for pressure and temperature swing thermochemical redox cycling. The performance of the solar reactor was analyzed using the model by considering, among other metrics, the solar-to-fuel energy conversion efficiency. The numerical results indicate the prominent influence of solar radiative input power, and therefore, the solar concentration ratio at the aperture, where increasing power substantially reduces reduction time. For *P*_{solar} = 50 kW, the model predicts *η*_{solar-to-fuel} = 6.12%. For this case, if 100% of the sensible heat is recovered from the ceria RPC mass between reduction and oxidation steps, the cycle efficiency can be increased to 12.75%. Further measures to boost *η*_{solar-to-fuel} include increasing the millimeter-scale porosity of the RPC structure to allow for more volumetric absorption of incoming solar radiation, resulting in a more uniform temperature distribution within the RPC, which ultimately improves the performance of the solar reactor. For example, an increase in macropore diameter from roughly 2 mm to 7 mm (*n*_{ppi} 10 to 3) resulted in an increase of *η*_{solar-to-fuel} from 7.34% to 10.2%. If volumetric absorption and uniform heating is achieved inside the ceria RPC, mass loading could also be increased to obtain higher efficiencies, provided the latter criterion of uniform heating is not compromised in the process. While the numerical model indicates the potential of this solar receiver–reactor technology to achieve high efficiency, critical issues remain: (i) stable ceria structures with optimized volumetric absorption characteristics (i.e., ordered structures) must be fabricated and demonstrated to survive in the solar reactor environment, and (ii) with increased power, the directly irradiated surface area of the redox active material will always be at risk of sublimation; the search for new redox active materials which can be reduced at lower temperatures while maintaining favorable oxidation properties is critically important [51,61,62].

## Acknowledgment

The authors acknowledge the support of the Paul Scherrer Institute for use of their high-flux solar simulator during validation experiments. The contributions by Philipp Haueter and Adriano Patané on the design and fabrication of the solar reactor system are also gratefully acknowledged.

## Funding Data

The EU's Horizon 2020 research and innovation program (Project SUN-to-LIQUID – Grant No. 654408, Funder ID. 10.13039/100010661).

Swiss State Secretariat for Education, Research and Innovation (Grant No. 15.0330).

## Nomenclature

*A*_{fs}=fluid–solid area density (m

^{−1})*c*_{p}=heat capacity (J mol

^{−1}K^{−1})*d*_{m}=mean pore diameter (m)

*h*=enthalpy (J)

*h*_{fs}=interfacial heat transfer coefficient (W m

^{−2}K^{−1})*I*=radiation intensity (W m

^{−2})*I*_{b}=blackbody radiation emission intensity (W m

^{−2})*k*=thermal conductivity (W m

^{−1}K^{−1})*k*_{eff}=effective thermal conductivity of ceria RPC (W m

^{−1}K^{−1})*M*=molar mass (kg mol

^{−1})*m*_{RPC}=mass of ceria RPC (kg)

*n*_{ppi}=number of pores per inch

*p*=pressure (Pa)

*P*_{solar}=solar radiative power input (kW)

*p*_{O2}=oxygen partial pressure (Pa)

*Q*_{fs}=fluid–solid heat source (W)

*Q*_{fuel}=integrated heating value of the fuel produced (J)

*Q*_{inert}=heat equivalent of work required for inert gas separation (J)

*Q*_{pump}=heat equivalent of work required for vacuum pumping (J)

*Q*_{sf}=solid–fluid heat source (W)

*Q*_{solar}=solar radiative energy input (J)

*r*=total hemispherical reflectance of ceria

- $r\xaf$ =
position vector

*r*_{O2}=rate of O

_{2}released (mol s^{−1})*s*=path length (m)

- $s\xaf$ =
direction vector

*S*_{radiation}=radiation exchange source (W m

^{−3})*S*_{reaction}=reaction energy source (W m

^{−3})*S*_{solar}=absorbed solar radiation source (W m

^{−2}/W m^{−3})*t*=time (s)

*T*=temperature (K)

*t*_{red}=duration of the reduction step (min)

*t*_{RPC}=RPC thickness (mm)

*T*_{f}=fluid temperature (K)

*T*_{red,end}=end temperature of reduction step (°C)

*T*_{red,start}=start temperature of reduction step (°C)

*T*_{RPC,max}=maximum temperature of RPC (°C)

*T*_{RPC,nom}=nominal temperature of RPC, measured at back surface (°C)

*T*_{s}=solid temperature (K)

*x*=depth within RPC (mm)

- Δ
*H*_{fuel}= heating value of the fuel CO (J mol

^{−1})- Δ
*H*_{O2}= reaction enthalpy (kJ mol

^{−1})

*α*=absorption coefficient (m

^{−1})*β*=extinction coefficient (m

^{−1})*δ*=nonstoichiometry of ceria

*ε*=total hemispherical emittance

*η*_{solar-to-fuel}=solar-to-fuel energy conversion efficiency

*ρ*=density (kg m

^{−3})*σ*=scattering coefficient (m

^{−1})*τ*=transmissivity of quartz window

*τ*_{RPC}=optical thickness of ceria RPC

*ϕ*_{dual}=dual-scale porosity of ceria RPC

*ϕ*_{single}=single-scale porosity of ceria RPC

*ϕ*_{strut}=strut porosity of ceria RPC

*ω*=solid angle (deg)