The design procedure for a 3 kWth prototype solar thermochemical reactor to implement isothermal redox cycling of ceria for CO2 splitting is presented. The reactor uses beds of mm-sized porous ceria particles contained in the annulus of concentric alumina tube assemblies that line the cylindrical wall of a solar cavity receiver. The porous particle beds provide high surface area for the heterogeneous reactions, rapid heat and mass transfer, and low pressure drop. Redox cycling is accomplished by alternating flows of inert sweep gas and CO2 through the bed. The gas flow rates and cycle step durations are selected by scaling the results from small-scale experiments. Thermal and thermo-mechanical models of the reactor and reactive element tubes are developed to predict the steady-state temperature and stress distributions for nominal operating conditions. The simulation results indicate that the target temperature of 1773 K will be reached in the prototype reactor and that the Mohr–Coulomb static factor of safety is above two everywhere in the tubes, indicating that thermo-mechanical stresses in the tubes remain acceptably low.

## Introduction

Solar thermochemical redox cycling of metal oxides utilizes the reducing power of partially or fully reduced metal oxides to split H2O and CO2 into syngas (H2 and CO) and oxygen [1,2]. In two-step cycles, the metal oxide is first reduced according to
$1ΔδMxOy-δox→1ΔδMxOy-δred+12O2$
(1)
In reaction (1), $Δδ=δred-δox>0$ is the change in the oxygen nonstoichiometry in the metal oxide, where $δred$ is the nonstoichiometry in the reduced state and $δox$ is that in the oxidized state. In the second step, the reduced metal oxide is oxidized with H2O or CO2
$1ΔδMxOy-δred+H2O→1ΔδMxOy-δox+H2$
(2a)
$1ΔδMxOy-δred+CO2→1ΔδMxOy-δox+CO$
(2b)
The net reactions are splitting of H2O and CO2
$H2O→H2+12O2$
(3a)
$CO2→CO+12O2$
(3b)

Several metal oxides have been proposed for this process, including ferrites (Fe3O4) [3–5], zinc oxide (ZnO) [6–11], ceria (CeO2) [12–16], and mixed metal oxides [17–20]. In recent years, ceria has received increasing attention due to its potential of reaching oxygen nonstoichiometries up to $δ$ = 0.25 without undergoing a phase change, thus maintaining its high oxygen conductivity and avoiding the need for a separation process to recover the metal oxide from the product gas mixture [14]. Porous ceria substrates, which provide high specific surface area, short diffusion length scales, and effective heat and mass transfer have shown the potential of maintaining open architectures even after repeated heating to approx. 1773 K [21].

The thermodynamics of ceria favor operating reaction (1) at a higher temperature than reactions (2) [22]. Consequently, most of the prior work has focused on two-temperature cycles, typically with reduction temperature ≥ 1773 K and oxidation temperature ≤ 1273 K [21,23–26]. However, efficient operation of the two-temperature cycle requires a high degree of solid-phase heat recovery [26,27], which has proven difficult to implement in a solar reactor. In addition, long-term temperature cycling is likely to result in material damage of the structural parts of the solar reactor and the reactive material [26]. Operating the cycle isothermally or with a modest temperature swing on the order of 100 K, the scheme underlying the reactor described in this paper, circumvents these challenges. Assuming conditions expected in the reactor with the ceria reduced in a flow of nitrogen sweep gas (10 ppm O2 impurity) and oxidized in a pure stream of oxidant, and both steps conducted at 1773 K with a gas-phase heat recovery effectiveness of 90%,4 the theoretical solar-to-fuel conversion efficiency without consideration of parasitic energy requirements
$η¯th=·¯nfuel,specHHV¯Q·solar$
(4)

is 10.2% for CO2 splitting5 and 4.28% for H2O splitting [28]. A temperature drop of 100 K from reduction to oxidation has the potential to boost the cycle efficiency to 36.6% and 17.9% due to an increase in the fuel produced per mol of sweep gas [28]. As discussed by Bader et al. [28] and Ermanoski et al. [29], the amount of sweep gas required in the reduction step has a strong influence on the parasitic energy demand of the process, if the sweep gas is N2 produced via air separation.

Several reactors have been developed in the past for solar thermochemical syngas production via temperature-swing partial redox cycling of metal oxides in a single device. Agrafiotis et al. [30] present the design and test results of a cylindrical receiver–reactor for H2O splitting. The reactor contains a monolithic recrystallized SiC honeycomb support structure coated with the reactive material, different MnZn- or NiZn-doped ferrites. Sweep gas and pure steam flow alternately through the honeycomb to reduce the reactive material and produce fuel, respectively. The CR-5 reactor developed and tested at Sandia National Laboratories uses counter-rotating disks which serve as support for the reactive material and as solid–solid heat recuperators [26]. The reactive material (ferrites or ceria) rotates through a sunlit reduction zone where sweep gas is passed through it, and through an off-sun fuel production zone where oxidant is passed through it. A reactor concept developed at the University of Minnesota uses two concentric counter-rotating cylinders [31,32]. The hollow outer cylinder is made from porous monolithic ceria, while the inner cylinder is made from an inert ceramic. The inner cylinder provides a mechanism to recuperate part of the sensible heat released by the outer cylinder as it is cooled from the reduction to the oxidation temperature. A reactor developed by ETH Zurich and Caltech consists of a single cylindrical cavity with the interior lined with porous ceria. Reduction is accomplished on-sun with a flow of sweep gas and fuel production is accomplished by taking the reactor off-sun and switching to oxidant flow. Different material morphologies have been tested in this reactor, including a porous monolith [21], a felt [23], and a reticulate porous ceramic (RPC) [24]. A particle-based reactor concept has been presented by Ermanoski et al. [25]. It consists of two separate reaction chambers for the reduction and fuel production steps. The reduction step is conducted in a vacuum. Particles are fed into the sunlit reduction zone by a conveyor auger, which additionally serves as a solid–solid heat recuperator for the particles. The fuel production step is conducted off-sun.

Only two reactors for splitting CO2 with ceria using an isothermal cycle have been proposed previously. Lipiński et al. [33] conceived a reactor comprised of angular porous ceria elements that form a cylinder which is contained in a cylindrical cavity. The ceria elements are sealed by semitransparent walls and undergo alternate reduction and oxidation by switching the type of gas flowing through the elements. The second reactor, proposed by Lichty et al. [34], consists of drop-down tubular reaction channels located inside an irradiated cavity for continuous production of CO/H2.

The present paper presents a reactor concept for continuous fuel production via the isothermal ceria-based redox cycle. The paper describes the design process for a nominal 3 kW prototype reactor in which reduction and oxidation take place simultaneously and fuel is produced continuously. The use of measured rate data rather than chemical equilibrium projections of fuel production provides a realistic prediction of the reactor output. The process for determining the design, the morphology of the reactive material, and the numerical models developed to study heat and mass transfer in the reactor and mechanical stress distributions in its critical structural parts are presented.

## Reactor Concept

The reactor concept consists of a cylindrical solar cavity which contains an array of $NRE$ tubular reactive elements arranged along the cylindrical cavity wall made of high temperature alumina insulation (Fig. 1(a)). Ceramic insulation extends radially outward and is contained within a stainless steel shell. Concentrated solar radiation enters the cavity through the aperture and is distributed by multiple internal reflections. Each reactive element consists of two concentric alumina tubes. A packed bed of mm-sized porous ceria particles is contained in the annular region between the tubes (Fig. 1(b)). The redox cycle is conducted by alternating the gas flowing through the reactive element between high-purity nitrogen sweep gas to reduce the ceria and pure CO2 reactant gas. The gas enters the reactive element through the inner tube. The flow direction of the gas is reversed at the domed end of the outer tube where the preheated gas enters the packed bed of ceria particles. After passing through the packed bed, the gas leaves the reactor through the annular gap between the inner and outer alumina tubes. The reactive element tubes extend beyond the back of the reactor to connect to a tube-in-tube counterflow heat exchanger, which implements gas-phase heat recovery [35].

Fig. 1
Fig. 1
The reactor can be operated in either (i) batch mode or (ii) quasi-continuous mode. In case (i), all reactive elements synchronously undergo reduction followed by oxidation, with cycle step durations $τred$ and $τox$, respectively, and cycle period $τcycle=τred+τox$. In case (ii), a time shift $ΔτiRE=(iRE-1)τcycle/NRE$ is introduced to the cycling period of the $iRE$-th reactive element in the reactor. Every reactive element is at a different stage of the periodic two-step process, but the numbers of reactive elements undergoing reduction and oxidation, $NRE,red$ and $NRE,ox$, respectively, are constant over time and given by the selected durations for each cycle step, $τred$ and $τox$, according to
$NRE,red=τredτred+τoxNRE$
(5a)
$NRE,ox=NRE-NRE,red$
(5b)

where $NRE,red$ is rounded to the nearest integer. The period after which one reactive element switches from reduction to oxidation, and one reactive element switches from oxidation to reduction, is $Δτswitch=τcycle/NRE$. Quasi-continuous operation is possible because the net heat consumed by the chemical reactions in the reactor is approximately constant over time, resulting in a constant average reactor temperature.

Because reduction is endothermic and oxidation is exothermic, the temperature of an individual reactive element may fall during reduction and rise during oxidation. Such a negative temperature swing would be detrimental to the reactor efficiency [28]. Therefore, management of the relative temperature via control of gas flows or the incident radiation on the reactive elements, for example with a moving radiation shield, may be beneficial. Relative temperature management is outside the scope of this paper.

## Reactor Design

In this section, we provide the basis for selecting a bed of porous ceria particles and the methodology for selecting operating parameters and designing a prototype reactor.

### Reactive Material Morphology.

A packed bed of mm-sized porous ceria particles provides large surface area for the heterogeneous chemical kinetics, rapid heat transfer, and low pressure drop. To demonstrate the benefits of this configuration, Fig. 2 compares the pressure drop and the effective thermal conductivity of a bed of porous particles to a packed bed of 5 µm solid particles, a porous ceria monolith [14], and a RPC [24]. Morphological and transport properties are compared in Table 1. The porous particles are assumed to have an internal porosity of ∼70%, an internal surface area of 0.1 m2 g−1 and an interparticle void fraction of ∼40%.

Fig. 2
Fig. 2
Table 1

Comparison of porosity $φ$, volume-specific surface area a, permeability $K$, Forchheimer coefficient $CF$, stagnant thermal conductivity $kcond$, and extinction coefficient $β$ of four alternative ceria morphologies

Configuration$φ$$a$$(m-1)$$K$$(m2)$$CF$$kcond$$(Wm-1K-1)$$β$$(mm-1)$
65% Porous monolith [36]65%4 × 1063 × 10−120.60.205–11
10 ppi RPC [37]92%5 × 1022 × 10−70.060.110.2–0.3
Solid particle packed bed40%7 × 1058 × 10−140.40.34180
Porous particle packed bed80%3 × 1069 × 10−90.90.150.18
Configuration$φ$$a$$(m-1)$$K$$(m2)$$CF$$kcond$$(Wm-1K-1)$$β$$(mm-1)$
65% Porous monolith [36]65%4 × 1063 × 10−120.60.205–11
10 ppi RPC [37]92%5 × 1022 × 10−70.060.110.2–0.3
Solid particle packed bed40%7 × 1058 × 10−140.40.34180
Porous particle packed bed80%3 × 1069 × 10−90.90.150.18

RPC has the lowest pressure drop. However, the pressure drop of the porous particle bed is only slightly higher, while its volume-specific surface area is over 1000 times higher, making it the preferred choice. Pressure drop is estimated using the extended Darcy law and the fluid properties of N2 at 1773 K and 1 bar. The permeability and Forchheimer coefficient are obtained from Ref. [36] for the porous monolith, from Ref. [37] for the 10 ppi RPC, and from the Ergun equation [38] for the beds of powder and particles. The pressure drop across the porous particles is 0.5 bar for an anticipated N2 mass flux of 1.4 kg m−2 s−1, and bed length of 0.35 m.

Another important benefit of the packed bed of porous particles is rapid heat transfer. The stagnant thermal conductivity is estimated by applying the geometric mean model at the particle porosity and bed porosity scales
$kcond=(kCeO21-φkCO2φ)1-φbedkCO2φbed$
(6)

using the thermal conductivity of ceria reported in Ref. [39] and the conductivity of CO2 [40]. The radiative conductivity is computed using the Rosseland approximation [41] with the index of refraction taken from Ref. [42]. The extinction coefficients of the porous monolith and the RPC are obtained from Refs. [43] and [44], respectively, and the extinction coefficients for the packed bed of particles and powder are estimated using Mie theory and the theory of geometric optics [45,46]. The overall conductivity is shown in Fig. 2(b) as a function of temperature. Attenuation of radiation in the particle bed is relatively low for the range of favorable operating temperatures, and heat transfer is rapid compared to the other morphologies with comparable volume-specific surface area.

The porous particle configuration also ensures that the reactive surface is fully accessible to the gaseous reactants. Diffusion in the pores of the particles effectively bathes the internal surfaces with CO2 (or, conversely, removes O2 and CO) so that the gas phase of the porous particle does not exhibit gradients in composition [47]. Composition gradients within the solid phase of the particles are negligible due to the short characteristic oxygen chemical diffusion time of only 0.4 ms through 1 µm [48]. Given the high diffusivity, solid-state oxygen diffusion is not likely to constitute the rate-limiting step of the redox reactions [14–18].

### Selection of Operating Parameters.

To determine the specific flow rates of sweep gas and oxidant as well as durations of reduction and oxidation that maximize the cycle-averaged thermal efficiency of the reactor, an optimization procedure was applied to CO production rates obtained for 1 g of porous ceria particles cycled in an IR imaging furnace at 1773 K over a broad range of N2 (O2 impurity of 10 ppm) and CO2 (99.99% purity) flow rates. This procedure was performed without consideration of the impact of the energy requirement for pumping or gas separation. For reduction over 100 s at $n·N2,specific$ = $1.1×10-4$ mol s−1g−1 and oxidation for 155 s at $n·CO2,specific$ = $3.6×10-5$ mol s−1g−1, the predicted cycle-averaged reactor efficiency is 3.7% (Eq. (4)). The prediction is based on a measured cycle-averaged continuous fuel production rate of 7.8 × 10−8 molCO s−1g−1 of ceria at 1773 K, and assumes 20% conductive losses from the reactor [10,49–51], a solar concentration of 3000, and gas-phase heat recovery effectiveness for sweep gas and CO2 of 90%. A tube-in-tube counterflow heat exchanger filled with reticulated porous alumina is under development for gas phase heat recovery and has been tested at the prototype scale [35,52]. The reasonableness of scaling the data from the IR imaging furnace to a larger scale was validated through tests of a reactive element containing 22.4 g of ceria particles in a solar cavity (100 mm long and 120 mm in diameter) operated for 21 cycles in the University of Minnesota high-flux solar simulator [53,54]. The measured average rates of CO production in the IR furnace and solar simulator are consistent within measurement uncertainty.

### Prototype Reactor Design.

Based on the optimization procedure described in Sec. 3.2, the 3 kW prototype reactor was sized to contain nominally 5030 g of ceria. The geometric parameters for the reactor are listed in Table 2.

Table 2

Prototype reactor parameters

 Number of reactive elements, $NRE$ 7 Cavity length (mm) 347 Cavity radius (mm) 150 Outer/inner radius of outer tube (mm) 34.9/31.8 Outer/inner radius of inner tube (mm) 22.2/19.1 Packed bed thickness (mm) 9.5
 Number of reactive elements, $NRE$ 7 Cavity length (mm) 347 Cavity radius (mm) 150 Outer/inner radius of outer tube (mm) 34.9/31.8 Outer/inner radius of inner tube (mm) 22.2/19.1 Packed bed thickness (mm) 9.5

The configuration in Table 2 comprises a well-balanced tradeoff between several practical considerations. A length-to-radius aspect ratio of the cavity near two is selected, which results in an apparent absorptivity close to unity [55]. Gaps are included between the reactive elements to allow the solar radiation to distribute over the entire tube perimeter. The tube radii are selected from standard tube sizes, while also considering the resulting number of reactive elements and concomitant complexity of the reactor, the flexibility to select the cycle step durations, the resulting size of the cavity, and the gap size between the concentric tubes required for the packed bed of ceria particles. Additional considerations include the pressure drop in the particle bed and an estimation of the mechanical stresses in the reactive element tubes. The diameter of the cavity aperture is selected to accommodate a solar power input of 3 kW at an average flux of 3 $MW m-2$. For $NRE=7$, the cycle step durations are adjusted to $τred$ = 100 s and $τox$ = 133 s. The results of the cycle duration optimization show that the impact of this small change in the duration of the oxidation cycle on the average fuel production rate is negligible.

Hence, the 3 kW reactor with approximately 5000 g of ceria is projected to produce fuel at an average continuous rate of 3.9 × 10−4 $mol s-1$ at an efficiency of 3.7% based on the definition given in Eq. (4). This efficiency does not include parasitic energy requirements, which can be significant, as pointed out by Bader et al. [28] and Ermanoski et al. [29]. Based on state-of-the-art industrial processes for gas separation, production of N2 with 1 ppm oxygen impurity at 8 bar from air by cryogenic rectification consumes 0.15 $kWh mN2-3$ [56]. Using this value and a reasonable assumption for the conversion of solar heat to electrical energy (25%), the parasitic energy requirement for the sweep gas production is 29 $MJ molCO-1$ (or approx. 11 kW for the 3 kW reactor). Adding this parasitic energy requirement to the denominator in Eq. (4), the overall process efficiency drops to 0.8%. On the other hand, the theoretical separation work to purify the sweep gas leaving the reactor back to an inlet O2 impurity of 1 ppm is estimated to be 16 $J molN2-1$, or $1.0×104$$J molCO-1$. The theoretical work to separate the CO product from the outlet stream is estimated to be $1.6×104$$J molCO-1$. These values are more than two orders of magnitude smaller than the solar energy required to produce 1 mol of CO, $q¯solar$ = $7.4×106$$J molCO-1$. The isentropic pumping power required to pump the sweep gas through the reactive elements of the 3 kW solar reactor is 60 W, or $1.5×105$$J molCO-1$. These theoretical energy penalties for auxiliary processes are small compared to the solar energy required to produce 1 mol of CO.

## Numerical Analysis

Thermal and thermo-mechanical models are developed to obtain steady-state temperature and static stress distributions in the reactive elements. The steady-state temperatures are obtained in batch mode operation, i.e., with all reactive elements operated equally. This approach is conservative for evaluating the static mechanical stresses in the tubes for the purpose of reactor design.

### Heat Transfer in the Solar Cavity.

The Monte Carlo ray-tracing method for surface radiative exchange [57] is applied to establish the solar radiation distribution at the cavity walls. The incident radiation is assumed to be uniformly distributed spatially over the aperture and over solid angles within a 37.7 deg half-angle that matches the rim angle of the high-flux solar simulator at the University of Minnesota [53,54]. The cavity walls and the reactive element tubes are modeled as opaque gray-diffuse surfaces with reflectivity of 0.9. The boundary heat flux at the cavity surfaces consists of contributions by convection and radiation. The net radiative heat flux at the inner surfaces of the reactor, $q·rad,in$, is calculated by discretizing the cavity and reactive element surfaces into $Nsurfseg,cavity$ = 10 × 10 and $Nsurfseg,RE$ = 10 × 10 surface segments, respectively, and solving the set of net radiation equations for gray-diffuse surfaces [57]
$∑j=1Nsurfseg,in11-Rj(δkj-RjFk-j)q·rad,in,j =∑j=1Nsurfseg,in(δkj-Fk-j)σTin,j4-q·i,solar,k, k=1,2,...,Nsurfseg,in$
(7)

where $Nsurfseg,in=Nsurfseg,cavity+NRENsurfseg,RE$. The view factor matrix, $Fk-j$, is determined with the Monte Carlo ray-tracing method. Correlations for natural convection between an array of eccentric inner cylinders and an outer cylinder [58] are used to evaluate convective heat transfer coefficients on the cavity and reactive element surfaces. The heat loss through the aperture is estimated by using the correlations for convective heat transfer in an empty, hemispherical, sideward-facing cavity [59]. The bulk mean air temperature in the cavity is determined by considering convective heat transfer between surfaces in addition to the energy lost through the aperture [59]. The heat loss through the alumina insulation is taken into account using a one-dimensional conduction model with temperature-dependent thermal conductivity [60]. The convective heat transfer at the outer reactor surfaces is modeled using Churchill and Chu correlations for free convection on a horizontal cylinder [61] and on a vertical plate [62]. The net radiative heat flux at the outer reactor surfaces is modeled assuming Tsky = 280 K.

### Heat and Mass Transfer in the Reactive Element.

Because all reactive elements undergo the same process, heat and mass transfer is calculated in a single reactive element. This simplification is justified for a uniform incident radiative source. The volume-averaged governing equations for 3D steady-state conservation of mass, momentum and energy are solved to obtain the temperature and flow velocity distribution in the reactive element. The bed of porous particles is treated as an isotropic continuum of 4-mm-dia. solid spheres with a packed bed interparticle porosity of 0.45. This morphology well matches that in the actual prototype reactor. Due to the large difference in pore size and hence in permeability between the voids of the packed bed and the internal pores of the particles, the fluid flow is assumed to occur only in the voids between the particles. The extinction coefficient of the particle bed is evaluated using geometric optics [45,46]. The particles are assumed to be opaque due to high attenuation of radiation in the porous microstructure of ceria [43,63]. The chemical energy heat sink/source is neglected because the rate of energy removal by the gases is tenfold that of the change in enthalpy due to the production of O2 and CO.

All relevant effective transport properties of the porous medium and their dependence on pore-scale morphology are derived using correlations for packed beds [64,65]. The temperature dependent thermophysical properties for the gases and ceria are evaluated from Refs. [39],[40], and [66]. The radiative thermal conductivity of the particle bed is evaluated using the Rosseland diffusion approximation. The flow rates of N2 and CO2 at the inlet of the reactive element are specified as 0.08 mol s−1 and 0.03 mol s−1. Assuming a gas-phase heat recovery effectiveness of 90%, the gases enter the reactive element at 1625 K. The gases are stipulated to leave the reactive element at 1 atm.

The mass, species, momentum, and energy transport equations in the fluid phase [64] are listed in the Appendix. Heat transfer in the dense alumina tubes of the reactive elements is modeled using steady-state conduction, while radiative contributions are included via an increase of the thermal conductivity. The spectral data for the absorption coefficient of dense, single-crystal alumina is found in Ref. [67]. The Rayleigh–Gans–Debye approximation is applied to determine the spectral scattering coefficient based on the arguments of suppression of light refraction in dense polycrystalline alumina proposed in Ref. [68]; a mean grain size of 6 μm [69] is used
$σsλ=3Δn2π2rgrainλ02$
(8)

The radiative conductivity is then determined using the Rosseland diffusion approximation with the mean extinction coefficient calculated using the emission spectrum at 1773 K.

Within the packed bed, local thermal equilibrium is assumed and the energy conservation equation is given by
$∇·(〈ρf〉f〈v〉∑i〈Yi〉f〈hi〉f) =∇·(keff∇〈T〉)+∇·(〈ρf〉f∑i〈hi〉fDi,eff∇〈Yi〉f)-〈∇·q·rad〉$
(9)
where the effective conductivity of the medium is
$keff=φkf+(1-φ)ks$
(10)
and the radiative source term is obtained using the radiative conductivity as
$〈∇·q·rad〉=krad∇〈T〉$
(11)

### Numerical Solution of the Heat and Mass Transfer Model.

The cavity walls are divided into $Nlayer,ins$ = 5 radial layers of equal thickness. The 1D heat conduction equation for the cavity walls is discretized using the finite-volume technique, and the discretized equations are solved with the tridiagonal matrix algorithm [70], using fortran 95 and the LAPACK/BLAS libraries. Equation (7) is formulated in matrix notation and solved by matrix inversion [57]. The temperature distribution in the cavity walls is determined by simultaneous solution of Eq. (7) and the energy equation for the cavity walls, using the following convergence criterion (i and i+1 denote consecutive iteration steps)
$1Nsurfseg,cavity∑j=1Nsurfseg,cavity[(Tin,ji+1-Tin,ji)/Tin,ji]2<10-3$
(12)

The transport equations inside the reactive element are solved in AnsysFluent 14 [71]. Temperature and heat flux continuity are applied to all solid–fluid interfaces in the tubes. The SIMPLE algorithm is used to couple the pressure and the velocity fields with the second order upwinding scheme for the advection terms in the momentum and energy conservation equations. The governing equations are solved sequentially with the global residual values set to $10-8$ for continuity and momentum equations, and $10-10$ for species transport and energy equations.

A user defined function is implemented in Fluent to couple the cavity and the reactive element models [72]. At iteration step n, the net surface heat flux ($q·wn$) on the reactive element tubes is obtained from Eq. (7) and the corresponding tube wall temperature ($Twn$), evaluated using Fluent, is used by the cavity model to provide an updated flux distribution ($q·wn+1$) for the subsequent iteration. The iterations progress until the scaled residual for the energy equation is less than $10-10$. Convergence is also monitored by tracking the change in the outlet fluid temperature between successive iterations
$|Tfn+1-Tfn|<0.5$
(13)

### Thermo-Mechanics.

A 3D thermo-mechanical linear elastic finite element model of the outer reactive element tube was created to ensure that the element would not fracture due to stresses attributable to thermal gradients. The analysis focuses on the outer tube because it is subjected to higher stresses than the inner tube. While a 1D analysis, based on assuming an outer surface temperature of 1773 K and a uniform heat flux through the tube, was used to obtain crude factor of safety estimates for the initial selection of tubes, the analysis described here instead utilizes the detailed temperature profile (see Fig. 3(a)) obtained from the heat and mass transfer model.

Fig. 3
Fig. 3
The boundary conditions allow for free expansion of the tube. The tube position is set by specifying zero axial displacement on the face of the tube at its open end. A standard linearly elastic continuum mechanics model is used, where thermal stresses and deformations are accommodated by augmenting the constitutive equations with a temperature-dependent term
$ɛij=12G(σij-ν1+νσkkδij)+αTδij$
(14)

where $T$ is the temperature rise above the strain-free reference temperature. Thermo-viscoelastic stress relaxation is neglected. Because alumina is brittle, Mohr–Coulomb theory [73] is used to estimate the static factor of safety.

The mechanical model was implemented in ANSYS [74]. The temperature dependent material properties of polycrystalline alumina are used. The main driver for potential failure is the marked decrease in tensile strength of alumina at temperatures above 1273 K [75].

## Numerical Results

The temperature distributions on the cavity walls and inside the reactive elements during oxidation are shown in Figs. 3 and 4. The temperature of the cylindrical cavity wall increases along the length from 1763 to 1793 K. The regions of the back wall that are subjected to higher incident flux reach temperatures of 1793–1813 K. The gases are heated inside the inner tube to 1708 K, before they enter the packed bed. Figure 5 shows the axial temperature distribution in the packed bed of ceria particles. The average temperature of the particle bed is 1767 K for CO2 flow. The particles and the gas are nearly at the same temperature as the outer tube. This result is attributed to the high effective thermal conductivity of the particle bed. The radial variation of temperature within the solid tubes and within the bed is 2–4 K. The maximum circumferential temperature variation is about 8 K in the reactive element tubes close to the midaxial plane (z = 0.1735 m). The low values for pressure drop (0.15 atm for sweep gas and 0.06 atm for the oxidizer) and the relatively uniform temperature distribution over the cross section of the particle bed support this choice of morphology, as well as the selection of the overall reactor configuration.

Fig. 4
Fig. 4
Fig. 5
Fig. 5

Figure 6 illustrates the static factor of safety due to stresses in the reactive element outer tube corresponding to the temperature distribution shown in Fig. 3(a). Static factors of safety lower than unity constitute an expected static failure of the material. Since cyclic stresses attributable to start-up and shut-down have been neglected, and differences between the model and experiments are to be expected, factors of safety substantially higher than one are preferred. Ignoring the end effects, the factor of safety remains above 2.5 throughout the tube, resulting in a nonfailure condition.

Fig. 6
Fig. 6

## Conclusion

A novel solar thermochemical reactor concept for continuous fuel production via isothermal partial redox cycling of ceria and the process for specifying its geometry and operating conditions are presented. The reactor is based on tubular reactive elements, which contain packed beds of mm-sized porous ceria particles, inside a blackbody cavity receiver. The particles are reduced in a N2 sweep gas flow, and re-oxidized with a flow of pure CO2, producing CO. This reactor design provides high specific surface area of $~106 m-1$ for the heterogeneous chemical reactions, rapid heat and mass transfer, and acceptably low pressure gradients in the gas flow. A 3 kW prototype reactor has been designed, based on experimentally determined gas flow rates and cycle step durations.

A heat and mass transfer model of the reactor and a thermo-mechanical model of the reactive element tube have been formulated and solved numerically to predict temperature and stress distributions under static conditions in the reactor. At the design operating point of the reactor, the predicted average temperature of the packed bed of ceria particles is close to the target operating temperature of 1773 K. The largest temperature variations occur in the axial direction, while the radial and circumferential temperature variations in the particle bed are less than 4 and 10 K, respectively. The Mohr–Coulomb static factor of safety in the reactive element tube is 2.5 or higher, indicating that the temperature gradients in the tubes are acceptably low for structural stability. While these results will vary depending on assumptions made during modeling, they are sufficiently encouraging to justify construction of a prototype of the proposed reactor design, which will in turn enable refining the assumptions.

The design process presented here has broad application to the design of reactors for solar fuel production via nonstoichiometric redox cycles. The use of measured rate data rather than chemical equilibrium projections of fuel production provides a realistic prediction of reactor output. It also demonstrates the importance of considering mechanical design as well as thermal design. The results presented reinforce the challenge of reaching high solar to fuel efficiencies for the isothermal redox cycle, particularly if sweep gas is used for reduction. Looking forward, it is imperative to consider the possibility of implementing a modest temperature swing in the “isothermal” cycle and a less energy intensive means of producing the low oxygen partial pressure for reduction. One of the benefits of the presented reactor concept is the flexibility to change operating conditions, including the use of vacuum pumping, gas flow rates, cycle times, and mode of operation (batch or continuous) to boost the efficiency.

## Acknowledgment

The financial support by the U.S. Department of Energy's Advanced Research Projects Agency—Energy (Award No. DE-AR0000182) to the University of Minnesota, and the University of Minnesota Initiative for Renewable Energy and the Environment (Grant No. RM-0001-12) is gratefully acknowledged. The authors acknowledge the computing facilities provided by the Minnesota Supercomputing Institute (MSI) at the University of Minnesota.

### Nomenclature

Nomenclature

• $a$ =

specific surface area ($m-1$)

•
• $CF$ =

Forchheimer coefficient ($m-1$)

•
• $D$ =

diffusion coefficient ($m2s-1$)

•
• $Fk-j$ =

view factor

•
• G =

shear modulus (Pa)

•
• $h$ =

enthalpy ($J kg-1$)

•
• $HHV¯$ =

higher heating value ($J mol-1$)

•
• $i,j,k$ =

indices

•
• $k$ =

thermal conductivity ($W m-1K-1$)

•
• $K$ =

permeability ($m2$)

•
• $n$ =

number of moles (mol); index of refraction; iteration step

•
• $N$ =

number

•
• $p$ =

pressure (Pa)

•
• $q·$ =

heat flux ($Wm-2$)

•
• $Q·$ =

heat rate (W)

•
• $rgrain$ =

•
• $R$ =

reflectance

•
• $T$ =

temperature ($K$)

•
• $Tsky$ =

apparent sky temperature (K)

•
• $v$ =

velocity vector

•
• $x,y,z$ =

Cartesian coordinate

•
• $Yi$ =

species concentration ($m-3$)

### Greek Symbols

Greek Symbols

• $α$ =

coefficient of thermal expansion

•
• $β$ =

extinction coefficient ($m-1$)

•
• $δ$ =

nonstoichiometry; Kronecker delta

•
• $Δn$ =

difference in index of refraction

•
• $Δδ$ =

change in nonstoichiometry

•
• $ΔτiRE$ =

time shift of reactive element $iRE$ (s)

•
• $Δτswitch$ =

switching period (s)

•
• $ɛ$ =

emissivity

•
• $ɛij$ =

mechanical strain tensor

•
• $ηth$ =

solar-to-fuel efficiency

•
• $λ$ =

wavelength ($m$)

•
• $λ0$ =

wavelength in air (m)

•
• $μ$ =

dynamic viscosity ($Pa s$)

•
• $ν$ =

Poisson's ratio

•
• $νf$ =

kinematic viscosity ($m2 s-1$)

•
• $φ$ =

porosity

•
• $ρ$ =

density ($kg m-3$)

•
• $σ$ =

Stefan–Boltzmann constant ($W m-2 K-4$); mechanical stress (Pa)

•
• $σs$ =

scattering coefficient ($m-1$)

•
• $τ$ =

duration (s)

•
• $τij$ =

stress tensor (Pa)

### Subscripts

Subscripts

• bed =

packed bed

•
• cond =

conduction

•
• eff =

effective

•
• f =

fluid

•
• i =

incident

•
• in =

inlet; referring to inside surfaces of cavity

•
• ins =

insulation

•
• m =

mean; molar

•
• out =

referring to outside surfaces of cavity

•
• ox =

oxidation

•

•
• red =

reduction

•
• RE =

reactive element

•
• s =

solid

•
• seg =

segment

•
• spec =

specific

•
• surf =

surface

•
• w =

wall

•
• $λ$ =

spectral

### Superscripts

Superscripts

• $"$ =

per unit area

•
• $·$ =

per unit time

•
• - =

average over one fuel production cycle

### Abbreviations

Abbreviations

• RPC =

reticulate porous ceramic

### Operators

Operators

• $〈〉$ =

volume-average

•
• $〈〉f$ =

intrinsic average over the fluid phase

### Appendix

The mass, species, and momentum conservation equations that are solved in the fluid phase in the reactive element are given below.

Mass conservation—Conservation of mass for the fluid phase yields
$∇·(〈ρf〉f〈v〉)=0$
(A1)
Species conservation—In the absence of chemical reactions, all species in the system are conserved, and the conservation equation for species $i$ reduces to the advective and diffusive terms
$∇·(〈ρf〉f〈v〉〈Yi〉f)=∇·(Di,eff〈ρf〉f∇〈Yi〉f)$
(A2)

where $〈ρf〉f$ is the intrinsic average fluid phase density and $v$ is the velocity field of the gases.

Momentum conservation—The extended Darcy–Brinkman–Forchheimer momentum transport formulation is adopted [64]
$(1vf2)〈v〉·∇(〈ρf〉f〈v〉)=-∇〈pf〉f+1vf∇〈τ¯¯〉-μKbed〈v〉 -ρfCF|〈v〉|〈v〉$
(A3)

The term on the left-hand side predicts entrance length effects. The terms on the right hand side account for pressure drop in the fluid phase, bulk viscous shear stresses due to the presence of rigid bounding surfaces, pressure drop due to the viscous stresses within the porous medium, and inertial pressure losses (from left to right). Note that, for the free fluid regions, $vf$ = 1, and, in addition, the last two terms on the right-hand side of Eq. (A3) are removed.

Energy conservation—For the free fluid domain, the energy transport equation becomes
$∇·(vρfhf)=∇·(kf∇Tf)+∇·(ρf∑ihiDi∇Yi)$
(A4)

where the enthalpy flux due to species transport has been taken into account in addition to the advective and conduction terms.

4

These assumptions are consistent with the experiments described in Section 3.2.

5

The higher heating value of CO is 283 kJ/mol.

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