Front. Energy Res. Frontiers in Energy Research Front. Energy Res. 2296-598X Frontiers Media S.A. 1608831 10.3389/fenrg.2025.1608831 Original Research A fixed-bed reactor for methane dry reforming via a ceria-based redox cycle – modeling and experimental validation Zuber et al. 10.3389/fenrg.2025.1608831 Zuber Mario 1 Software Writing - original draft Writing - review and editing Investigation Formal Analysis Methodology Data curation Visualization Conceptualization Validation Ackermann Simon 2 Writing - review and editing Supervision Steinfeld Aldo 1 * Writing - review and editing Supervision Funding acquisition ETH Zurich, Department of Mechanical and Process Engineering, Zurich, Switzerland Synhelion SA, Lugano, Switzerland Correspondence: Aldo Steinfeld, aldo.steinfeld@ethz.ch 05 01 2026 2025 13 1608831 09 04 2025 23 09 2025 03 11 2025 Copyright © 2026 Zuber, Ackermann and Steinfeld. 2026 Zuber, Ackermann and Steinfeld https://creativecommons.org/licenses/by/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This study explores the production of sustainable aviation fuels through the sun-to-liquid pathway, focusing on the thermochemical production of syngas–a mixture of H2 and CO–using concentrated solar energy. One promising route, termed dry redox reforming, involves the dry reforming of CH4 via a two-step redox cyclic process using non-stoichiometric ceria (CeO2). This process consists of: (1) the methanothermal reduction of ceria to form syngas; and (2) the oxidation of reduced ceria with CO2 to form CO. Thermodynamic modeling and fixed-bed tubular reactor simulations of dry redox reforming are presented. Ellingham diagrams and species-temperature diagrams illustrate reaction favourability and equilibrium compositions, detailing how elevated non-stoichiometries improve syngas selectivity. The thermodynamic relations are incorporated into a computational fluid dynamic solver developed within OpenFOAM. The transient solver models a system of reversible heterogeneous reactions and fluid flow over a non-stoichiometric solid oxide as a fixed-bed, accounting for the governing conservation equations in both fluid and solid phases. The model is validated with experiments conducted in a lab-scale tubular reactor (D fxb = 19 mm, l fxb = 30 cm, 5% educts, V ˙ in = 1 Ln·min−1, T sp = 936 °C–1,016 °C, ambient pressure, ceria pellet morphology), showing relative errors within 7% for key metrics such as cumulative and instantaneous conversion and selectivity. The validated model is applied as a design tool. A geometric case study considering 100% CH4 educt, ambient pressure, lfxb = 2 m, and temperatures at around 990 °C, suggests optimal operation of a reactor tube at V ˙ in = 10 Ln·min−1, with tube diameters of 7–10 cm. A novel operating strategy is introduced, priming and cycling, that leverages the δ-gradient in the fixed-bed to enhance syngas selectivity and conversion, indicating the potential production of a high-purity syngas stream.

 solar thermochemistry redox cycles cerium oxide methane thermodynamics tubular reactor experimental computational The authors declare that financial support was received for the research and/or publication of this article. This work was funded by the Swiss Federal Office of Energy (Project HYBREC–Grant No. SI/501854-01). We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [PGSD3-545236-2020]. section-at-acceptance Solar Energy Introduction

“Sun-to-liquid” is a particularly effective pathway to produce sustainable aviation fuels (SAF) through solar thermochemical processes which use concentrated solar energy as the source of high-temperature heat to drive endothermic reactions. The utilization of the entire solar spectrum offers a thermodynamically efficient route to the synthesis of syngas–a mixture of H2 and CO which can be further processed to SAF (Romero and Steinfeld, 2012; Wang et al., 2012; Cabrera and de Sousa, 2022).

One method of producing syngas is via solar-driven methane reforming, which has been extensively investigated (Spiewak et al., 1993; Agrafiotis et al., 2014; Sheu et al., 2015; Simakov et al., 2015; von Storch et al., 2015; Said et al., 2016; von Storch et al., 2016; Gao et al., 2018; Tavasoli and Ozin, 2018; Agrafiotis et al., 2021; Bhattacharjee et al., 2024; Pan et al., 2024; Wang et al., 2024). This process is represented by:

dry reforming: C H 4 + CO 2 ⇌ 2 CO + 2 H 2   Δ H 1273   K o = 256 kJ mol CH 4 wet reforming: C H 4 + H 2 O ⇌ CO + 3 H 2   Δ H 1273   K o = 226 kJ mol C H 4 Unless specified otherwise, species are in the gaseous phase. Industrial reformers typically operate at temperatures around 900 °C and elevated pressures around 15 bar (Bulfin et al., 2021a; Farsi, 2023). Syngas is produced continuously through mixed dry/wet reforming. To improve kinetics, the reaction typically proceeds over a catalyst; however, catalysts require extra costs and often deactivate due to sintering, oxidation, or carbon deposition (Abdulrasheed et al., 2019). To mitigate carbon formation steam is fed in excess, however, this affects the energy efficiency detrimentally (Trimm, 1997; Sehested, 2006). The process is further dependent on the source of CH4, and for the process to be considered sustainable it is crucial that CH4 is sourced biogenically in the form of biogas. Despite challenges, there is potential in dry reforming as it valorizes and converts two major greenhouse gases, CO2 and CH4, to SAFs (Neiva, 2010; Lavoie, 2014; Schwab et al., 2015).

Another method of producing syngas is via a two-step thermochemical redox cycle for splitting H2O and CO2 using metal oxides (Steinfeld, 2012). This route has long-term potential and has undergone significant developments (Chuayboon and Abanades, 2020; Safari and Dincer, 2020; Haussener, 2022; Warren and Weimer, 2022; Abanades, 2023; Budama et al., 2023; Wei et al., 2023; Tran et al., 2024). Early studies identified ceria (cerium (IV) oxide, CeO2) as an effective oxygen carrier, and subsequent research has further highlighted the potential of the material by detailing its rapid kinetics, high oxygen diffusivity, crystallographic stability, and cycling stability (Panlener et al., 1975; Abanades and Flamant, 2006; Zinkevich et al., 2006; Chueh et al., 2010; Chueh and Haile, 2010; Ackermann et al., 2014; Ackermann et al., 2015; Marxer et al., 2015; Marxer et al., 2017). The proven concept of the ceria redox cycle has led to successful on-sun demonstrations of the complete solar-to-fuel chain (Schäppi et al., 2021; Zoller et al., 2022). The two-step H2O/CO2-splitting redox cycle based on ceria is represented by:

First step thermal reduction: 1 Δ δ Ce O 2 − δ ox ⇌ 1 Δ δ Ce O 2 − δ red + 1 2 O 2   Δ H 1273   K , δ < 0.03 o ≈ 466 kJ mol O Second step oxidation with CO2: 1 Δ δ Ce O 2 − δ red + C O 2 ⇌ 1 Δ δ Ce O 2 − δ ox + CO  Δ H 1273   K , δ < 0.03 o ≈ − 185 kJ mol CO 2 Second step oxidation with H2O: 1 Δ δ Ce O 2 − δ red + H 2 O ⇌ 1 Δ δ Ce O 2 − δ ox + H 2   Δ H 1273   K , δ < 0.03 o ≈ − 216 kJ mol CH 4

Unless specified otherwise, metal oxides are in the solid phase. Ceria is not consumed; the net reactions are H 2 O   ⇌ H 2 + 1 2 O 2 and C O 2 ⇌ CO + 1 2 O 2 . δ denotes the non-stoichiometry, and measures the redox extent. Δ δ = δ red − δ ox . Reaction enthalpies are listed at 1,000 °C to align with the range covered in the ceria dataset (Bulfin et al., 2016). The redox cycle is operated under a temperature-swing mode and pressure-swing mode. Typically, the reduction step is performed at around 1,500 °C and 10 mbar to reach δ red ≈ 0.03, and the oxidation step is performed at around 900 °C and ambient pressure. This cycle has been demonstrated in a robust cavity-type reactor design which optimizes radiation heat transfer by featuring ceria in the form of a reticulated porous ceramic (RPC) (Furler et al., 2012; Furler et al., 2014; Marxer et al., 2015; Marxer et al., 2017; Pullar et al., 2019; Schäppi et al., 2021; Zoller et al., 2022). The ceria redox cycle saw significant advancements, yet faces notable inherent challenges. Temperature swings introduce energy losses, and impose mechanical and thermal stresses. High-temperature radiative transfer restrict the ceria porous topology, inhibiting structures with higher packing density or higher specific surface area (SSA) than the RPC. Fuel produced per cycle is limited, as the process fails to exploit the full reduction potential of ceria (Abanades, 2023; Lidor and Bulfin, 2024). Nonetheless, on-going developments on optimized ceria structures (Sas Brunser et al., 2023; Sas Brunser and Steinfeld, 2023) and efforts towards heat recovery during the temperature swings (Lidor et al., 2023) could boost the energy efficiency to competitive levels (CORDIS, 2023).

A hybrid approach to produce syngas combines the aforementioned routes through the methanothermal reduction of a metal oxide. This process is referred as redox reforming and has been extensively studied and reviewed (Sheu and Ghoniem, 2014; Tang et al., 2015; Krenzke et al., 2017; Mao et al., 2020; Agrafiotis et al., 2021; Liu et al., 2024). For this process ceria also emerges as an attractive metal oxide for the characteristics already discussed, along with additional properties critical to redox reforming including its reducibility, oxygen mobility, resistance to sintering, and ability to directly react with (solid) carbonaceous species (Zheng et al., 2017; Das et al., 2018; Stroud et al., 2018; Abdulrasheed et al., 2019; Yadav and Das, 2021). The potential of this process operating with pure ceria has prompted various reactor demonstrations (Jang et al., 2014; Krenzke et al., 2016; Nair and Abanades, 2016; Warren et al., 2017; Welte et al., 2017; Chuayboon et al., 2019; Fosheim et al., 2019; Chuayboon et al., 2020; Haeussler et al., 2020; Warren et al., 2020), including one performed on-sun in a concentrating solar tower (Zuber et al., 2023). The 2-step redox reforming cycle based on ceria is represented by:

First step methanothermal reduction: 1 Δ δ Ce O 2 − δ ox + C H 4 ⇌ 1 Δ δ Ce O 2 − δ red + CO + 2 H 2   Δ H 1273   K , δ = 0.15 o = 466 kJ mol CH 4 Second step dry oxidation with CO2: 1 Δ δ Ce O 2 − δ red + C O 2 ⇌ 1 Δ δ Ce O 2 − δ ox + CO  Δ H 1273   K , δ = 0.15 o = − 140 kJ mol CO 2 Second step wet oxidation with H2O: 1 Δ δ Ce O 2 − δ red + H 2 O ⇌ 1 Δ δ Ce O 2 − δ ox + H 2   Δ H 1273   K , δ = 0.15 o = − 172 kJ mol H 2 O

Ceria is not consumed; the net reactions are Reactions R1 and R2. The reduction step employs methane as a reducing agent, effectively lowering the thermodynamic barrier and temperature for reducing the metal oxide, while simultaneously producing syngas through the partial oxidation of methane. Notably, the process operates isothermally and isobarically, typically at ambient pressure and temperatures between 900 °C and 1,100 °C, and high reduction extents are possible ( δ red < 0.345) (Bulfin et al., 2016). Operation enables selective, tunable syngas composition, and ceria’s oxygen affinity supports full dry reforming due to its coking resistance (Warren et al., 2017; Bulfin et al., 2021a). Although the process is dependent on CH4, it avoids the use of catalysts, produces syngas in both redox steps, and operates isothermally at moderate temperatures, thereby positioning itself as a potential bridge technology to the pure H2O/CO2-splitting redox cycle (Reactions R3–R5). (Chuayboon and Abanades, 2020; Bhosale, 2023). The moderate temperatures of redox reforming also enhance the flexibility in designing ceria morphologies, facilitating optimized micro-structures, higher SSA, and increased packing density (Laosiripojana and Assabumrungrat, 2005; Krenzke et al., 2016; Heya et al., 2020). Moreover, the moderate reactor temperatures together with advancements in cavity-receiver technologies can deliver a gaseous heat transfer fluid (HTF) at temperatures above 1,000 °C, (Karni et al., 1997; Ambrosetti and Good, 2019; Patil et al., 2021; Casati et al., 2022), and thus support the transition towards allothermally heated systems. In these systems, the solar receiver and the chemical reactor are spatially de-coupled and thermally connected via an HTF; advantages of this approach include: enhanced flexibility, precise control of reactor conditions, 24/7 operation through heat storage, high packing density, and independent optimization of components (Tescari et al., 2014; Agrafiotis et al., 2021; Lidor and Bulfin, 2024; Tran et al., 2024). Furthermore, experimental setups that feature the reactive material in externally heated tubes show similarities to indirectly heated reactors (Krenzke et al., 2016; Warren et al., 2020). This configuration not only streamlines the experimental process, enabling quick and straightforward tests, but also decreases the complexity of the domain for computational fluid dynamics (CFD) modeling.

In this study, the dry redox reforming cycle based on ceria is analyzed, i.e., Reactions R6 and R7, with a focus on developing a CFD model that captures its non-stoichiometric behaviour in a fixed-bed configuration. The thermodynamics, together with experimental results, are discussed to provide the necessary background and validation procedure, and an application of the model illustrates a practical use case of the CFD model. We begin by detailing the thermodynamics of dry redox reforming, utilizing published thermodynamic model equations of ceria (Bulfin et al., 2016). Previous thermodynamic studies have investigated redox reforming (Scheffe and Steinfeld, 2012; Krenzke et al., 2017; Warren et al., 2017; Bose et al., 2019; Bayon et al., 2021; Bulfin et al., 2021a). This study focuses on reaction spontaneity, equilibrium species composition, and the dependency on the non-stoichiometry. The detailed thermodynamics are ultimately integrated into a CFD model, developed within the OpenFOAM environment, for the purpose of simulating dry redox reforming in a fixed-bed tubular reactor. Numerically representing this system is challenging due to the intricate reaction mechanisms, kinetic complexity, solid and gaseous phases, equilibrium and reversibility, reaction enthalpies, heat and mass transfer, and the need to account for the porous structure of the fixed-bed (Fabrik et al., 2022; Kaydouh et al., 2024). As a more mature technology, the reforming (i.e., Reactions R1 and R2) processes have been simulated and validated for fixed-bed catalytic reactors (Behnam et al., 2012; Wehinger et al., 2015; Fabrik et al., 2022; Kaydouh et al., 2024). Recent computational studies on reforming have explored biomass as a feedstock, addressed the challenging problem of modeling solid carbon formation, supported simulations with machine learning, and investigated hydrogen-selective membranes (Hajizadeh et al., 2022; Al-Otaibi et al., 2023; Alotaibi et al., 2024; Habibzadeh et al., 2024). Models for redox processes have been developed but are limited to one-dimensional analyses, fail to directly model the solid oxide, do not thoroughly account for kinetics, or lack robust experimental validation (Guene Lougou et al., 2019; Zoller et al., 2019; Bulfin et al., 2023; Hill C. M. et al., 2023). Other studies have extended OpenFOAM’s capabilities to more accurately simulate complex chemistry or heterogeneous reactions (Kwiatkowski et al., 2013; Zhou et al., 2022); for example, porousGasificationFoam was developed to simulate gas flow through a fixed porous media while considering limited irreversible homogeneous and heterogeneous reactions (Żuk et al., 2022). Through modifications to this solver, porousRedoxFoam was developed in this study to accurately model the δ -dependent thermodynamics of dry redox reforming in a tubular reactor, i.e., a system of reversible heterogeneous reactions featuring fluid flow over a non-stoichiometric solid metal oxide as a fixed-bed. This solver is unique as it is developed on an open source platform rather than commercial software, incorporates both thermodynamics and kinetics, introduces the δ variable resulting in a dynamic heterogeneous system, resolves multiple dimensions to capture temperature and species gradients, and, although it does not simulate solid carbon formation, it models solid and gaseous phases that actively participate in the reaction going beyond conventional catalysis–thereby laying the groundwork for future solvers to include reaction mechanisms involving solid carbon. We conclude the study by validating the CFD solver against experiments conducted in an electrically heated tubular reactor. Additional insights obtained from parametric simulations concerning reactor dimensions, flow settings, and operating modes are also discussed.

 Thermodynamics

The chemical system of dry redox reforming extends beyond the two principal reactions (i.e., R6 and R7). Three additional key reactions are used to describe the reduction step:

complete oxidation of CH4: 4 Δ δ Ce O 2 − δ ox + C H 4 ⇌ 4 Δ δ Ce O 2 − δ red + CO 2 + 2 H 2 O   Δ H 1273   K , δ = 0.15 o = 880 kJ mol CH 4 CO as reducing agent: 1 Δ δ Ce O 2 − δ ox + CO ⇌ 1 Δ δ Ce O 2 − δ red + CO 2   Δ H 1273   K , δ = 0.15 o = 140 kJ mol CO 2 H2 as reducing agent: 1 Δ δ Ce O 2 − δ ox + H 2 ⇌ 1 Δ δ Ce O 2 − δ red + H 2 O   Δ H 1273   K , δ = 0.15 o = 172 kJ mol H 2 O All reactions are marked as reversible ( ⇌ ) to emphasize the focus on the thermodynamics. The complete oxidation of CH4 is an undesired reaction and typically occurs at low non-stoichiometries ( δ ≲ 0.05) at the beginning of the reduction step. CO and H2 act as reducing agents, effectively the reverse of Reactions R7 and R8 respectively; the reactions are relevant at low δ and in downstream sections of the reactor where the CO/H2 reduce the ceria to produce undesired products. In our study these reactions are part of the five key reactions (R6, R7, R9, R10, and R11) critical in describing dry redox reforming. A relevant homogeneous side reaction not discussed is the reverse water-gas shift (RWGS: H 2 O + CO ⇌ H 2 + C O 2 ). Solid carbon formation is not considered, as it is kinetically limited and carbon absence is experimentally proven, but the relevant side reactions concerning C(s) include: methane decomposition ( CH 4 ⇌ C s + 2 H 2 ), reverse Boudouard ( C s + CO 2 ⇌ 2 CO ), gasification with H2O ( C s + H 2 O ⇌ CO + 2 H 2 ), and oxidation with ceria 1 Δ δ Ce O 2 − δ ox + C s ⇌ 1 Δ δ Ce O 2 − δ red + CO , 2 Δ δ Ce O 2 − δ ox + C s ⇌ 2 Δ δ Ce O 2 − δ red + CO 2 ) .

The chemical system’s equilibrium and reaction favourability are defined by the Gibbs free energy. It is calculated for the five key reactions by adapting the published model equations of ceria. The thermodynamic model equations of ceria consider the thermal reduction of ceria as per Reaction R3 (where δ ox = 0). The standard enthalpy, entropy, and Gibbs free energy of ceria are given as a function of temperature and non-stoichiometry (Bulfin et al., 2016): Δ h ¯ O , R 3 ∘ = 395000 − 31400 ⁡ log 10 δ Δ s ¯ O , R 3 ∘ = 160 + 1 δ max R ln δ max − δ − ln δ Δ g ¯ O , R 3 ∘ = Δ h ¯ O , R 3 ∘ − T Δ s ¯ O , R 3 ∘ Partial molar values are considered and interpreted as values per mole of elemental oxygen. To determine the standard Gibbs free energy for the key reactions, the reactions are decomposed into two parts; with one part equal to a multiple of Reaction R3. An example of this decomposition applied to Reaction R6 is: R 6 = 1 2 O 2 + C H 4 ⇌ + CO + 2 H 2 + R 3 The standard Gibbs free energy for the gaseous reactions with O2, represented in square brackets (i.e., [rxn]), is calculated through a Van’t Hoff analysis: ln K ¯ eq , rxn T K ¯ eq , f , rxn ∘ = ∫ 298   K T Δ h ¯ rxn ∘ T R T 2 d T Δ g ¯ O , rxn ∘ = − R T ⁡ ln K ¯ eq , rxn Ultimately, the Gibbs free energy for the key reactions is calculated via a superposition of the two decomposed parts: Δ g ¯ O , rxn ∘ = Δ g ¯ O , rxn ∘ + 2 ∑ e n O 2 − ∑ p n O 2 Δ g ¯ O , R 3 ∘ ∑ e n O 2 − ∑ p n O 2 denotes the number of moles of O2 on the educt side minus the product side in [rxn]. This methodology was applied to each of the key reactions and the results are visualized in Figure 1 as an Ellingham diagram where the standard Gibbs free energy of each reaction is plotted as a function of temperature, at a low and high non-stoichiometry. Generally, the reduction reactions are favoured at higher temperatures while the oxidation reactions are favoured at lower temperatures. Ideally, the redox cycle is operated such that only syngas production is favourable (i.e., Δ g ¯ R 6 ∘ < 0 , Δ g ¯ R 7 ∘ < 0 ; Δ g ¯ R 7 , R 8 , R 9 ∘ > 0 ), and undesired products (CO2, H2O) are mitigated. At δ = 0.001 syngas production is favourable at 660 °C < T < 685 °C. At δ = 0.150 syngas production is favourable at 820 °C < T < 1,055 °C. Higher temperatures enhance kinetic properties, thus motivating cycling at higher non-stoichiometries for enhanced syngas selectivity.

Standard Gibbs free energy as a function of temperature, for the five key reactions (R6, R7, R9, R10, R11) describing the chemical system for dry redox reforming. For reference, dry reforming (Reaction R1) and thermal reduction (Reaction R3) are included. Results at low, δ = 0.001 (−), and high, δ = 0.150 (- - -), non-stoichiometries are plotted. Reactions are considered favourable for Δ g ¯ rxn ∘ < 0 .

Graph depicting reaction equilibrium free energy (ΔG) versus temperature (T) for various chemical reactions involving cerium dioxide (CeO₂) and methane (CH₄). Eleven reactions are represented with color-coded lines, both solid and dashed, indicating two different values of δ. Temperature ranges from 600 to 1200 degrees Celsius, and free energy ranges from -200 to 200 kilojoules per mole of educt gas. Lines intersect and diverge, demonstrating the temperature dependence of each reaction's equilibrium.

To determine the equilibrium species composition, the system of equations and minimization of the Gibbs free energy are solved with Cantera using the Gas Research Institute (GRI)-Mechanism 3.0, and coded within MATLAB. In brief, mixtures of O2/CH4 (or O2/CO for oxidation) at various initial ratios are equilibrated. The resulting p O 2 is matched to a δ via Δ g ¯ O , R 3 ∘ + 1 2 R T ⁡ ln p O 2 p ∘ = 0 The equilibrium species composition corresponds to this state (i.e., δ , T ), and the enthalpy of reaction is also extracted. It is assumed that there is an unlimited amount of ceria (at each solved state) available for the reaction. Previous studies have applied similar approaches to solving a redox system (Bulfin, 2019; Bulfin et al., 2021a; Bulfin et al., 2023). This methodology was applied for dry redox reforming and the results are presented in species-temperature diagrams, at a low and high non-stoichiometry, not considering the solid carbon species (C(s)), and determined for an initial educt concentration of 100%. Figure 2A shows the equilibrium composition during the reduction step. At higher δ the oxidizing potential of ceria decreases; the syngas concentration increases, however CH4 conversion decreases, and the enthalpy of reaction per mole of methane also decreases. The effects of C(s) and varying pressures are not shown here but have been explored and are detailed in (Zuber, 2024). In brief, the reduction reaction and conversion of CH4 is favoured at lower pressures, and C(s) formation occurs in regions where low CH4 conversion is observed in Figure 2A. Figure 2B shows the equilibrium composition during the oxidation step. At higher δ there is an increased number of oxygen vacancies in the ceria lattice leading to increased CO2 conversion and CO production. The oxidation reaction is not affected by varying pressure, and C(s) formation occurs (due to the Boudouard reaction) at low temperatures (<700 °C) where high CO production is observed in Figure 2B. Overall, the results indicate that operating at high non-stoichiometries is preferred for syngas production. Video plots of the species-temperature diagrams are made available at: https://github.com/mgzuber-eth/CeO2-CH4-CO2.

Product molar concentrations of species i and reaction enthalpy as a function of temperature at low ( δ = 0.001, ―) and high non-stoichiometry ( δ = 0.150, - - -), not considering the solid carbon species (C(s)), determined for 100% initial educt concentration at 1 atm, during: (A) methanothermal reduction. (B) dry oxidation. The equilibrium compositions were solved with Cantera, assuming unlimited ceria availability at each state.

Graph of methanothermal reduction and dry oxidation processes depicts chemical species concentrations and enthalpy change versus temperature (in degrees Celsius) ranging from 600 to 1200. The top graph shows curves for CH₄, H₂, CO, H₂O, CO₂, and enthalpy change, with variables represented by colored lines. The bottom graph displays curves for CO, CO₂, and enthalpy change. Each graph includes solid and dashed lines for different delta values, with labels showing specific delta values and species. Computational model

The main objective of the computational model is to accurately capture the transient behaviour and physical dynamics of reversible heterogenous reactions involving a non-stoichiometric solid oxide as a fixed-bed. A representative reaction of this type involving generic species is: 1 δ MO + α A ⇌ 1 δ M O 1 − δ + β BO + ∂ D In these reactions the solid actively participates, distinguishing them from catalytic reactions. The computational model solves the reactions using equilibrium thermodynamics, with the capability of incorporating a kinetic model to account for intermediate products. The porous domain is modeled with effective properties and fluid flows are evaluated on either 2-D or 3-D configurations–an important consideration when assessing gradients and determining reactor dimensions. To address these elements, the solver porousRedoxFoam was developed within the OpenFOAM environment (openfoam9, released in July 2021 and distributed by The OpenFOAM Foundation, openfoam.org). The solver is based on two solvers, reactingFoam and porousGasificationFoam, and introduces modifications primarily to accommodate reversible reactions involving multiple educts and products (Żuk et al., 2022). This enables the application of δ -dependent thermodynamic relations (e.g., K ¯ eq , rxn = f δ , T ), as outlined in the previous section. These relations describe the evolution of the chemical species and are represented by a system of ordinary differential equations (ODE) in the form of a Jacobian matrix that is solved within OpenFOAM. porousRedoxFoam is written in the framework of a continuum heterogeneous model with effectiveness factors and considers conservation equations in the fluid (f) and solid phases (s). The governing conservation equations, their boundary conditions, and essential material property definitions are succinctly outlined. Comprehensive information is found in (Zuber, 2024), and porousRedoxFoam is made available under the GNU General Public License version 3 at: https://github.com/mgzuber-eth/porousRedoxFoam.

Governing equations

The foundation of the conservation equations originate from porousGasificationFoam, with further background available in the corresponding article (Żuk et al., 2022). The conservation equations include, solid continuity (Equation 8): ∂ ∂ t 1 − ϵ ρ s transient overall = 1 − ϵ ∑ k R k s source  heterogeneous rxn solid species (k) mass conservation: ∂ ∂ t 1 − ϵ ρ s Y k s transient overall = 1 − ϵ R k s source   heterogeneous rxn

solid energy conservation: ∂ ∂ t 1 − ϵ ρ s c p s T s transient overall − ∇ · 1 − ϵ K k s · ∇ T s Laplacian conduction = − 1 − ϵ ∑ k h f , k ∘ R k s source heat   of   heterogeneous   rxn − 1 − ϵ T s − T f ∑ j c p , j R j s source interphase heating from heterogeneous rxn + h conv a v T f − T s source convection volume conservation: ∂ ∂ t ϵ transient overall = − 1 − ϵ ∑ k 1 ρ k s R k s source heterogeneous rxn fluid continuity: ∂ ∂ t ϵ ρ f transient overall + ∇ · ρ f u   divergence flow = 1 − ϵ ∑ j R j s source heterogeneous rxn fluid species (j) mass conservation: ∂ ∂ t ϵ ρ f Y j f transient overall + ∇ · ρ f Y j f u divergence flow − ∇ · ϵ ρ f D f ∇ Y j f Laplacian diffusion = ϵ ω ˙ j f source homog − eneous   rxn + 1 − ϵ R j s source heterogeneous rxn fluid energy conservation: ∂ ∂ t ϵ ρ f c p f T f transient overall   + ∇ · ρ f c p f T f u divergence advection − ∇ · ∇ ϵ k f T f Laplacian conduction = − ϵ ∑ j h f , j ∘ ω ˙ j f source heat of homogeneous rxn   + 1 − ϵ T s − T f ∑ j c p , j R j s source interphase heating from heterogeneous rxn − h conv a v T f − T s source convection fluid momentum conservation: ∂ ∂ t ρ f u transient overall + ∇ · ρ f u u divergence flux , relative + ϵ ∇ p f gradient pressure − ∇ · μ f ∇ u Laplacian shear   stress − ρ f g scalar gravity = − μ f D · u tensor viscous , Darcy − F · u tensor inertial , Forchheimer

ϵ ( V f / V total ) is the fixed-bed porosity, ρ is the density, R i is the volumetric heterogeneous reaction rate of a species, Y i is the mass fraction, k is the thermal conductivity, h f ∘ is the specific enthalpy of reaction at standard reference state, h conv is the heat transfer coefficient, a v is the volume-specific surface area, u is the velocity vector, D is the mass diffusion coefficient, μ is the dynamic viscosity, and D and F are the Darcy viscous and Forchheimer inertial resistance tensors respectively. The heat of reaction from the heterogeneous reaction is imposed on the solid phase. The interphase heating term represents the energy transfer that occurs when gases, produced via a heterogeneous reaction at T s , equilibrate to the surrounding gas temperature (Żuk et al., 2022). Volume conservation ensures a porosity balance as the solid participates in the reaction. The source term for the heat of homogeneous reaction also manages the enthalpies of the fluid species from the heterogeneous reaction.

 Reversible reactions

A crucial feature to porousRedoxFoam; the thermodynamics and reversibilities are considered via the equilibrium constant for each heterogenous reaction (H), K ¯ eq , H = f δ , T s = exp Δ g ¯ O , H ∘ R T s Δ g ¯ O , H ∘ is analogous to Δ g ¯ O , rxn ∘ from Equation 6. The equilibrium constant can also be expressed with respect to the species partial pressures ( p i ), K ¯ eq , H = ∏ p p p p ∘ ν p , H ∏ e p e p ∘ ν e , H ν is the stoichiometric coefficient (0 for solid species), and e and p denote the index of educt and product species respectively. The architecture of porousRedoxFoam was initially designed for standard total pressures, which simplifies the equation to, K ¯ eq , H = ∏ p X p f ν p , H ∏ e X e f ν e , H X j f is the fluid phase molar fraction. The model was later adapted to operate for various total pressures and is detailed in (Zuber, 2024). OpenFOAM primarily operates with mass fraction Y j = X j M j M mix , hence the equilibrium constant in terms of mass units is defined as, K eq , H = K ¯ eq , H ∏ p M p ν p , H M mix , eq , H ν p , H ∏ e M e ν e , H M mix , eq , H ν e , H M mix , eq , H is the molar mass of the equilibrium mixture for the reaction H . The equilibrium constant relates the forward and reverse rate coefficients through, K eq , H = k f , H k r , H The reaction rate of each heterogeneous reaction is expressed as, Ω H = ρ s k f , H ∏ e Y e f | s ν e , H − k r , H ∏ p Y p f | s ν p , H The volumetric reaction rate of each species is expressed as, R i = ∑ H Ω H ν i , p , H − ν i , e , H A Jacobian matrix is constructed, from a system of ODEs where each row corresponds to Equation 22 for each species. The nominal reaction rate coefficients are defined per kg of solid educt species and follow an Arrhenius form, k 0 f , H = A H ⁡ exp − E a , H R T s The final form of the forward reaction rate coefficient includes a scaling factor, k f , H = f δ , T s · k 0 f , H The scaling factor maintains numerical stability, particularly in cases when low educt conversion is predicted, and the thermodynamic driving force is low relative to k 0 f , H . It also accounts for any additional kinetic dependencies on δ (Hill et al., 2021; Hill C. et al., 2023).

 Model setting definitions

Beyond the governing equations, the computational model incorporates specific assumptions, material properties, and numerical strategies in order to run simulations.

General assumptions

A solid metal oxide must be defined by two species, one in the oxidized state (Mox) and one in the reduced stated (Mred). For the case of ceria, Mox = CeO2 and Mred = CeO. The solid oxide system is normalized by δ = 1, signifying a difference of 1 mol of oxygen between the two states. The non-stoichiometry is defined as, δ = m loss M Mox m sample M O If the solid metal oxide is the sole species in the solid phase the non-stoichiometry is evaluated by, δ = Y Mred s Y Mred s + 1 − Y Mred s M Mox − M O M Mox Radiation is considered within the thermal conductivity via the Rosseland diffusion approximation (RDA). Laminar flow is considered, forced convection dominates, and gravity is considered negligible. The kinetic dependency outlined in Equation 23 represents the apparent kinetics of the reaction.

 Solid morphology properties

Solid species are defined with a constant density. The specific heat capacity is implemented based on a power function fit, c p , k s T s = c p 0 , k s T s T ref , c p , k n c p , k The total thermal conductivity considers both fixed-bed and radiative effects, k k s = k k fxb + k rad . The fixed-bed term imposes Landauer’s correction to account for solid-to-solid contact and fluid properties (Landauer, 1952; Smith et al., 2013; Gigantino, 2021; Wild, 2022). The radiative term imposes the RDA, k rad = 16 3 σ T s 3 β R . The total thermal conductivity is implemented on a power function fit, k k s T s = k 0 , k s T s T ref , k , k n k , k The enthalpy change of the solid species is established on the basis of the pure thermal reduction process, for example: 1 δ MO ⇌ 1 δ M O 1 − δ + 1 2 O 2 The change in enthalpy is based on 1 mol of elemental oxygen and defined per unit mass of solid educt. The enthalpy of formation of Mox is specified as, h f , k ∘ = − Δ h ¯ O , R 13 ∘ − 1 2 h ¯ a , O 2 ∘ ν k M Mox For the case of ceria Δ h ¯ O , R 13 ∘ = Δ h ¯ O , R 3 ∘ . Within the OpenFOAM framework the fluid enthalpies are managed, hence, h ¯ a , O 2 ∘ (the absolute enthalpy of oxygen) is subtracted from the enthalpy of formation. The enthalpy of formation of Mred is set to zero, thus the heat of reaction is represented by the enthalpy of the oxidized solid species.

 Fluid properties

Definitions of fluid species properties are well-established within OpenFOAM and are only briefly discussed here. The fluid species are modeled as ideal gases ρ j = p j M j R T f . The thermophysical properties (specific heat capacity, enthalpy, and entropy) and transport properties (dynamic viscosity, thermal conductivity) are specified for each species.

 Interfacial properties

The linear Darcy viscous tensor ( D = D c I ) is defined in terms of the Darcy resistance coefficient, the reciprocal of the permeability D c = 1 / K and calculated via a Kozeny-Carman expression (Pozzobon et al., 2018). The convective heat transfer coefficient is determined via a Nusselt number correlation by Wakao and Kaguei Nu = 2 + 1.1 ⁡ Pr 1 / 3 ⁡ Re 0.6   , which aims to improve the accuracy at low Reynolds numbers (Wakao et al., 1978; Żuk et al., 2022). The volume-specific surface area is evaluated for a morphology in the form of a sphere or ideal cylinder a v = 6 1 − ϵ D morph (Wild, 2022).

 Numerical strategy

To numerically solve the problem, the finite volume method (FVM) is employed with the Eulerian approach. Discretization schemes, solution control, and simulation execution are all well-established tools within OpenFOAM and are only briefly discussed here (OpenFOAM-Foundation, 2019). The transient solver computes the time derivative using a bounded, first-order implicit Euler method. To solve the system of chemical reactions a typical method is a high-order stiff linearly implicit Euler one-step iterative extrapolation (SEULEX) method. Convergence of the iterative solvers is ensured by the (relative) tolerance on the residual set to approximately (10−3) 10−5. The execution of the solver is defined by a maximum time step. To increase the speed of the simulation, parallel processing is employed whereby the mesh is decomposed into a number of subdomains.

 Thermodynamic validation

To substantiate that the modifications for reversible reactions were implemented correctly a validation case was conducted. The simulation results are compared to those obtained when solving the chemical system using Cantera (as presented in Figure 2). In the computational model, dry redox reforming is modeled using a reaction scheme comprised of the five key reactions (R6, R7, R9, R10, and R11); hence, this validation step is crucial as it compares the results of this simplified reaction scheme (5 gaseous species, 2 solid species, and 5 reactions) to that of the well-established GRI chemical mechanism (53 species and 325 reactions) used by Cantera. To set up the solver to model dry redox reforming, the reaction scheme is imposed. As per Equation 22, OpenFOAM solves a chemical system of equations defined by k f and k r , which are related through the equilibrium constant defined in Equation 19. K ¯ eq , H of each of the five key heterogeneous reactions is defined as per Equation 16, where Δ g ¯ O , H ∘ corresponds to Δ g ¯ O , R 6 R 7 R 9 R 10 R 11 ∘ as calculated in Equation 6. Ultimately, the equilibrium constant of each reaction is implemented within the solver, and k f is defined within the user case files of the simulation. To simplify the simulation, the oxidation step is only modeled with the dry oxidation reaction (R7) and the reduction step is only modeled with the four reduction reactions (R6, R9, R10, and R11). It should be noted that the reduction step can be fully described with Reactions R6, R10, and R11; however, when applying the model in this manner to simulations with fluid flow, the numerical nature of CFD causes false fronts of CO and H2 to form in areas with low non-stoichiometry. Although including reaction R9 results in an over-defined and redundant system, it more accurately describes the direct conversion of CH4 into CO2 and H2O under conditions of low non-stoichiometry, while also remaining numerically stable.

The simulations to validate the thermodynamic implementation were conducted at a specified non-stoichiometry, temperature, and pressure. The non-stoichiometry was specified in the user case files, thus inherently omitting Equation 26. The system temperature was constrained to the user-specified value (1,000 °C). The pressure was defined at the outlet (1 atm). The educt (CH4 during reduction, CO2 during oxidation) concentration (100%) was defined at the inlet of a 2-D axisymmetric mesh, simulating the conditions of a tube. A low flow rate, and high reaction rate constants ( k f ) were set to ensure equilibrium compositions were reached at the outlet. The settings for the material properties are not critical for the validation because the product composition is determined solely by the thermodynamics of the heterogeneous reversible reactions.

The results of the reduction simulations are shown in Figure 3A, which details the product molar composition as a function of non-stoichiometry. The model captures the thermodynamic behaviour at low and high non-stoichiometries as well as in cases of incomplete CH4 conversion. The results of the oxidation simulations are shown in Figure 3B. Overall, the results for both oxidation and reduction show strong agreement between theory (determined using Cantera) and the computational model (porousRedoxFoam), thus reinforcing the validity of each approach, and substantiating the implementation of the thermodynamics within the model. The model was also validated under conditions of higher total pressure and lower educt partial pressure, an important consideration since experimental setups often operate with diluted methane streams (Zuber, 2024).

Product molar concentrations as a function of non-stoichiometry during: (A) methanothermal reduction (Reaction R6). (B) dry oxidation (Reaction R7). The results between theory (Cantera) and the computational model (porousRedoxFoam) are compared at 1,000 °C, 100% educt, and 1 atm.

Two graphs illustrate chemical reactions. A) Methanothermal reduction shows concentration changes of H2O, CO2, H2, CO, CH4 over delta. B) Dry oxidation depicts CO2 and CO conversion over delta. Both include theoretical and model data comparisons with distinct colors for each chemical component. Experimental validation

Following the simulations validating the implementation of the thermodynamics and reversible reactions, the model was validated against experimental data, with the product composition and performance indicators serving as the validation criteria. The experimental setup and tests are described, followed by an overview of the model settings and the simulation results.

Experimental system

A lab-scale tubular reactor setup at ETH Zürich was used to perform the experiments. This setup has been used in past studies to investigate countercurrent chemical looping on the RWGS and methane dry reforming (Bulfin et al., 2023; Bulfin et al., 2024). A photo and schematic of the reactor system are shown in Figure 4. A vertical split tube furnace (Carbolite VST 12/400 Carbolite Gero Ltd.) heats the alumina reactor tube to a temperature set point ( T sp ), and mimics the heating by an HTF in an allothermal system. The alumina reactor tube (ALM2519 – Almath Crucibles Ltd.) has an inner diameter of 19 mm, and houses the fixed-bed of ceria within the 30 cm hot zone. The principal gas direction is from bottom to top to ensure stable gas flow at low Reynolds number. The standard flow rate is 1 Ln·min−1, at an operating temperature of around 1,000 °C, and under ambient pressure conditions; correlating to a residence time approximately equal to 1 s. The fixed-bed was formed with a ceria pellet morphology ( D p ≈ l p ≈ 6 mm). The pellets were fabricated using a “rapid wet casting” technique, which involved simple molds and a high-viscosity slurry containing 27-vol.% of 150 μm cylindrical carbon fibers as a pore former. Each pellet was sintered at 1,100 °C, resulting in an effective density of 2,850 kg·m−3 and SSA of 1.44 m2·g−1 (measured by gas adsorption analysis and a Brunauer–Emmett–Teller measurement). The total mass of the fixed-bed was 110.54 g, corresponding to a fixed-bed porosity ( ϵ ) of 0.544. The pellets achieve a high mass loading, while maintaining a porous fixed-bed that enhances heat transfer through radiation. Tests with various ceria morphologies revealed that the pellet morphology maintains its structural integrity throughout testing and outperforms the baseline RPC (typically used in directly irradiated reactors), with methane conversions 1.8 times higher than those achieved with the RPC. This demonstrates the effectiveness of the pellet in allothermally heated systems. Additionally, the rapid, reliable, and scalable pellet fabrication process has led to its adoption in this study. An analogous agglomerate morphology was used in an on-sun demonstration (Zuber et al., 2023).

Tubular reactor system for dry redox reforming: (left) photo of the vertical split tube furnace with coaxial alumina tube; (right) schematic indicating flow direction. A secondary alumina tube supports the crucible and fixed-bed of ceria. A photo of the ceria pellet morphology is shown (D p ≈ l p ≈ 6 mm). The images are not presented to scale relative to each other.

Diagram and photo of a fixed-bed reactor system. Left: an open furnace showing insulation and a central tube. Right: a schematic with labeled parts like the alumina tube, ceria morphology bed, and crucible. The alumina tube dimensions are specified, and ceria morphology is depicted in an inset.

The complete system design is detailed through a piping and instrumentation diagram (P&ID), as shown in Figure 5. The system is designed to operate only under ambient pressure conditions. Educt gas (5 mol% CH4 in Ar, 5 mol% CO2 in Ar) flows are controlled by mass flow controllers (MFC, EL-FLOW–Bronkhorst AG). MFC calibration was confirmed using a flow verifier (Definer 220 – Mesa Laboratories, Inc.). Two K-type thermocouples (TC) are inserted axially through an adapter at the top of the reactor, and are positioned at the middle ( x = 15 cm) and outlet ( x = 30 cm) of the reactor. The hot product gas stream first flows through a condenser (MAK-10 – AGT Thermotechnik GmbH and Co. KG). It then flows into a gas analysis unit comprised of two units: (1) an infrared (IR) sensor (ULTRAMAT 23 – Siemens AG) to measure the concentrations of CH4, CO, and CO2; and (2) a thermal conductivity sensor (CALOMAT 6 – Siemens AG) to measure the concentration of H2. The calibration of the IR sensor was straightforward. However, the thermal conductivity sensor is cross-sensitive to CO, CO2, and CH4, hence these interfering values were deducted from the nominal H2 signal. Upon temporally aligning the signals, the outlet flow rate for a diluted stream is calculated from the molar balance on Ar, V ˙ vent = X Ar , in V ˙ in X Ar , vent After converting to moles through the ideal gas law, a molar balance on H is applied to determine the molar flow rate of H2O: n ˙ H 2 O , out = 2 n ˙ CH 4 , in − n ˙ CH 4 , out − n ˙ H 2 , out Across a time range, the change in non-stoichiometry is determined from a molar balance on O: Δ n O = n H 2 O , out + n CO , out + 2 n CO 2 , out − 2 n CO 2 , in Δ δ = Δ n O m Ce O 2 / M Ce O 2 To verify the calibration of the reactor system, the molar balance on C ( n C , in = n CH 4 , in + n CO 2 , in , n C , out = n CH 4 , out + n CO , out + n CO 2 , out ) for an experiment consisting of 20 cycles ( T sp = 1,016 °C, 1 Ln·min−1, 5% educts) was conducted. The relative error i . e .   value − reference reference of moles out (0.408,160 mol) versus moles in (0.401,153 mol) for C is 1.7% across the entire testing time (> 4 h), indicating an accurately calibrated system. The molar balance of the system was also verified in two independent experiments (Bulfin et al., 2023; Bulfin et al., 2024).

Experimental setup diagram of the tubular reactor system, including piping and instrumentation. Open valves (△) indicate the flow path utilized for dry redox reforming tests.

Diagram of a gas analysis setup, featuring a tubular reactor with an alumina tube inside a furnace. The system includes heating elements, a condenser, and a gas analysis unit with infrared and thermal sensors. Gas supply lines for carbon dioxide, methane, and argon are linked to a mass flow controller. Arrows indicate direction of unreacted and reacted gas mixtures. Controls involve a switch cabinet, computer, and sensors for temperature and pressure monitoring. Ventilation paths and data lines for controlling and reading are marked. Experimental procedure and sample run

The system is purged with argon and T sp is set. Once the TC readings have stabilized the dry redox reforming cycle begins. The operator sets the educt flow rates and redox times for the reduction and oxidation steps. An argon purge step distinctly separates each redox step. The experiments discussed here were all performed at 1 Ln·min−1 and with 5% educt concentrations (5 mol% CH4 in Ar, 5 mol% CO2 in Ar). Four cycles for the validation of the computational model were performed, on the same day, sequentially from T sp = 996 °C–936 °C in 20 °C increments; the reduction time was fixed ( t red = 420 s), followed by complete oxidation ( t ox = 2,400 to 1,500 s, X C O 2 < 1%). In addition, cycles at 966 °C and 1,016 °C were performed ( t red = 900 s, t ox = 2,580, 3,360 s until complete oxidation) on a separate day to serve as a unique data point for comparison and to confirm reproducibility. The long reduction times ensured a wide range of non-stoichiometries were considered. A sample cycle from one of these tests is shown in Figure 6, where product molar concentrations, average change in non-stoichiometry of the fixed-bed, and temperature reading at the middle of the fixed-bed are plotted as a function of time for T sp = 976 °C, 1 Ln·min−1, and 5% educts. During reduction, complete conversion is observed until the onset of syngas production. This point is termed as the reduction inflection point ( t ip , red ) and occurs at around 71 s. Beyond this point, syngas production is favourable and the CH4 conversion stabilizes. The decrease in CH4 conversion is attributed to the change in oxygen activity of the ceria. As predicted by thermodynamics the selectivity towards syngas improves at higher δ avg . The endothermicity of the reduction reaction is observed by a decrease in temperature. During oxidation, the CO and CO2 concentration profiles appear to extend from their levels at the end of the reduction step. As the oxidation step progresses there is a marginal increase in CO concentration, which is attributed to the rising non-stoichiometry near the reactor outlet. After reduction there is a δ -gradient within the fixed-bed, with higher non-stoichiometries near the inlet compared to downstream sections. Consequently, during oxidation, CO is produced near the inlet, then as the CO moves downstream it acts as a reducing agent in areas with low non-stoichiometry. Eventually, the fixed-bed nears complete oxidation and X CO = X CO 2 , marking the oxidation inflection point ( t ox ) at around 2050 s (1,510 s from the start of oxidation). The exothermicity of the oxidation reaction is observed by an increase in temperature at around 1,500 s; this Δ T occurs partway through the oxidation step, suggesting that the reaction front of the rapid oxidation reaction has reached the middle of the fixed-bed at this time. Evidence of a reaction front and the impact of the δ -gradient highlight the need for a computational model to accurately represent the multi-dimensional effects of dry redox reforming system.

Experimental redox cycle results plotting product molar concentrations, average change in non-stoichiometry of the fixed-bed, and temperature reading at the middle of the fixed-bed ( x = 15 cm) as a function of time. Conditions: pellet morphology at T sp = 976 °C, 1 Ln·min−1, educts 5 mol% CH4 in Ar and 5 mol% CO2 in Ar (Ar not shown). Reduction and oxidation steps are identified respectively by the action of the CH4 and CO2 MFCs.

A graph depicting various gases' concentrations and temperature over time, labeled with both reductive and oxidative phases. Gases include \( \text{H}_2\text{O} \), \( \text{H}_2 \), \( \text{CH}_4 \), \( \text{CO}_2 \), and \( \text{CO} \). Each gas has a distinct colored line. A dashed brown line shows temperature, while a dotted black line represents the average deviation. The X-axis is time in seconds, and there are dual Y-axes for \( X_j \), \( \delta \), and temperature in degrees Celsius. Model settings

To model dry redox reforming in the tubular reactor using porousRedoxFoam process-specific settings must be defined, including: domain, boundary conditions, kinetics of the reaction scheme, and material properties. The cylindrical reaction domain of the tubular reactor ( D = 19 mm, l = 30 cm) was represented with a 2-D axisymmetric model, defined by a wedge geometry with a mesh density of 2 cells·cm−1 axially and 8 cells·cm−1 radially including an inflation layer. The maximum time step was set to 0.004 s. Mesh refinement and time step sensitivity studies substantiated the spatial and temporal discretization. The solution domain was distributed across 12 partitions for parallel processing on a high-performance computing cluster of ETH Zürich. The transient simulation was monitored using log files that were updated every second.

The inlet was defined by a velocity, temperature, and mass fraction; corresponding to 1 Ln·min−1, 5 mol% CH4/CO2 in Ar, T in = T wall , x = 0 , and ambient pressure. The wall was defined by a no-slip condition and a temperature profile, T wall = f x + T sp   K (where T min = T wall , x = 0 = 968.9 °C, T max = T wall , x = 0.24   m = 1,020.1 °C, T wall , x = 0.3   m = 1,019.6 °C). The outlet was defined with a pressure equal to 1 atm. The reduction step was initialized with a fully oxidized fixed-bed of ceria (i.e., δ ∀ x = 0 ). The oxidation step was initialized with the δ and Y s field variables from the completed reduction step.

The reaction scheme applied for the thermodynamic validation is also imposed here. Kinetic rates are imposed, namely, to capture incomplete methane conversion during experimentation; thermodynamics otherwise suggest complete methane conversion (i.e., X CH 4 ≥ 0.98) at temperatures T ≥ 800 °C for non-stoichiometries δ < 0.20. The parameters defining the nominal reaction rate coefficients (k f,0) for each of the five key reactions are summarized in Table 1. The parameter values are guided though practical insights and literature, and tuned through iterative analyses specific to the solver (Nair and Abanades, 2016; Hill et al., 2021; Hill C. et al., 2023). Further information about the kinetic model settings is detailed in (Zuber, 2024).

The pre-exponential factors and apparent activation energy that define the nominal rate coefficient (Equation 23) for each reaction within the reaction scheme.

Reaction (H) A H [s−1] E a , H [J·mol−1]
R6 3.9775 × 108 246,302
R7 37,428 141,247
R9 2.1148 × 109 264,418
R10 2.9607 × 1010 264,418
R11 0.03 0

The solid properties of CeO2 are defined with a constant density ρ s = 2,850 kg·m−3, constant heat capacity c p s = 444 J·kg−1·K−1, and power fit for the thermal conductivity k s = 0.1725 T s 273 1.2853 W·m−1·K−1 (Touloukian, 1967). The fluid properties account for a mixture of species, each defined according to standard OpenFOAM models and formulae with data taken from the GRI-Mechanism 3.0 and literature (White, 2006). The porosity ( ϵ ) of the fixed-bed is set to 0.544. The pellet morphology is modeled as an ideal cylinder and the Darcy viscous resistance coefficient ( D c ) is set to 9,248,117 m−2 (Li and Park, 1998; Pozzobon et al., 2018). The convective heat transfer coefficient ( h conv ) is evaluated as 64.8 W·m−2·K, and the volume-specific surface area ( a v ) is calculated as 456 m−1.

 Model results

Firstly, the model results are directly compared to the experimental results for the case T sp = 966 °C, which was performed on a separate day to the tests used for the kinetic fitting. The primary model settings for this case are summarized in Table 2.

List of select settings for modelling dry redox reforming in the tubular reactor for T sp = 966 °C. The reaction thermodynamics are outlined in previous sections and are defined primarily through Equations 6, 16 and 19. The reaction kinetics are summarized in Table 1.

Parameter Value/correlation Units
Reactions for reduction R6, R9, R10, R11 —
Reactions for oxidation R7 —
Fixed-bed diameter D fxb = 0.019 m
Fixed-bed length l fxb = 0.3 m
Fixed-bed porosity ϵ = 0.544 —
Wall temperature profile T wall = 2365 x 3 − 2019 x 2 + 562 x − 47 + T sp K
Solid density ρ s = 2850 kg·m−3
Solid heat capacity c p s = 444.3 J·kg−1·K−1
Solid thermal conductivity k s = 0.1725 T s 273 1.2853 W·m−1·K−1
Educt flow rate V ˙ in = 1 Ln·min−1
Educt molar concentration X e = 0.05 —
Educt temperature T in = T wall , x = 0 K
Pressure p out = 101325 Pa
Convective heat transfer coefficient h conv = 64.8 W·m−2·K−1
Volume-specific surface area a v = 456 m−1

For the reduction step, the molar concentrations of the product gases as a function of time for both the simulation and experiment are shown in Figure 7A. The model captures the distinctive trends of the redox cycle, such as the inflection point as S syngas increases and X CH 4 decreases. For the oxidation step, the molar concentrations of the product gases as a function of time for both the simulation and experiment are shown in Figure 7B. The model captures the overall trends of the oxidation step, as X CO 2 marginally increases before sharply decreasing once the fixed-bed nears complete oxidation. However, there is a deviation in the timing of the oxidation inflection point, as that of the model occurs 149 s earlier than that of the experiment. This is due to the difference in X CO 2 at the start of the oxidation step. Note that the oxidation step is strongly dependent on the final conditions and δ -profile from the reduction step; the model likely predicts a higher non-stoichiometry towards the outlet of the fixed-bed resulting in higher CO2 conversions at the start of the oxidation step.

Experimental (○) and model (−) product molar concentrations, and average change in non-stoichiometry of the fixed-bed as a function of time for T sp = 966 °C, 1 Ln·min−1, 5% educts (Ar not shown), during: (A) methanothermal reduction. (B) dry oxidation.

Two graphs showing chemical reactions over time in sections A and B. A) Methanothermal reduction: curves for H2O, CO2, H2, CO, CH4, and deviation with both experimental and model data over 900 seconds. B) Dry oxidation: curves for CO, CO2, and deviation over 2500 seconds, with data comparison. Both graphs use symbols and lines to distinguish data types.

To illustrate the accuracy of the model across a temperature range, performance metrics such as educt conversion ( X e ) and product selectivity ( S p ) across all temperature set points are compared between all test cases. The performance metrics are defined as (Bulfin et al., 2021b): X e = n e , in − n e , out n e , in S p = ν e ν p n p , out n e , in − n e , out For reduction, cumulative values consider the total reduction time, and instantaneous values are taken at the end of the reduction step. Instantaneous values aim to provide insight into how well the transient model captures the evolving δ -profile. Relative errors, e r , with respect to the experimental values are presented. CH4 conversions ( e r = 0.7%) are shown in Figure 8A; instantaneous values are lower than cumulative conversions as Reaction R6 is more favourable at higher δ but has slower kinetics than Reaction R9 at lower δ . CO selectivities ( e r = 6.1%) and H2 selectivities ( e r = −3.3%) are shown in Figures 8B,C respectively; cumulative values decrease with increasing temperature as the CO2 and H2O producing reactions (i.e., Reactions R9, R10, and R11) become more favourable. The instantaneous values also follow this trend for T sp ≳ 970 °C, but exhibit the opposite trend for T sp ≲ 970 °C. This is attributed to decreasing CH4 conversions at lower temperatures, hence longer times are required to reach high non-stoichiometries where syngas selectivity is favoured. Additional metrics are used to measure the transient accuracy of the model. Changes in non-stoichiometry ( e r = −6.7%) are shown in Figure 8D; values increase with temperature due to improved CH4 conversions. Reduction inflection point times ( e r = 2.2%) are shown in Figure 8E; values increase with temperature as the CO2 and H2O producing reactions are more favourable.

Cumulative and instantaneous, experimental (○ and ×) and model (―, - - -), performance metrics as a function of temperature set point, for T sp = 936, 956, 966, 976, 996, and 1,016 °C (1 Ln·min−1, 5% CH4): (A) CH4 conversion. (B) CO selectivity. (C) H2 selectivity. (D) Change in non-stoichiometry. (E) Reduction inflection point time. (F) CO2 conversion. (G) Oxidation inflection point time. For reduction step metrics (A–E) t red = 420 s is considered, and instantaneous values are taken at the end of the reduction step. For oxidation step metrics (F–G) instantaneous values are taken at the start of the oxidation step.

Seven graphs (A-G) depict experimental and model data versus temperature (\(T_{sp}\)) ranging from 940°C to 1020°C. A) Shows \(\chi_{CH4}\) values. B) Illustrates \(\text{S}_{CO}\). C) Displays \(\text{S}_{H2}\). D) Shows \(\Delta \delta_{avg,red}\). E) Represents \(t_{ip,red}\). F) Illustrates \(\chi_{CO2}\). G) Displays \(t_{p,ox}\). Each graph includes cumulative and instantaneous experimental data, and model predictions are represented by solid and dashed lines.

The oxidation step is highly dependent on the preceding reduction step, so to maintain observable trends, only cases with a total reduction time of 420 s were considered (i.e., T sp = 936, 956, 976, 996 °C). Instantaneous values are taken at the beginning of the oxidation step. CO2 conversions ( e r = 5.1%) are shown in Figure 8F; cumulative values are lower as the conversions at the end of the oxidation step approach zero. The instantaneous values are overestimated by the model, indicating that the calculated δ -profile from the reduction step overpredicts δ values near the outlet of the fixed-bed. Oxidation inflection point times ( e r = −4.5%) are shown in Figure 8G. For similar reasons the model underestimates the inflection point time.

Overall, the model reasonably captures the trends during both reduction and oxidation. The results reinforce the validity of the model settings including the thermodynamics and kinetic rate parameters. Additional information on the model validation, including temperature profiles and numerical stability, is detailed in (Zuber, 2024).

 Model insights and limitations

Contours of the non-stoichiometry taken at various times in the simulation help visualize the δ -gradient. For the case T sp = 966 °C, δ -contour plots during reduction are shown in Figure 9A; a sharp reaction front is not evident, however a δ -gradient along the length of the fixed-bed is. Prior to the inflection point ( t ip , red = 61 s), the front of the bed is most reduced due to the Reaction R9, which has a low activation energy. As the reduction step proceeds, Reaction R6 is more prevalent and spreads across the fixed-bed. The lower values of δ near the inlet are attributed to the lower wall temperatures. Eventually, the fixed-bed is most reduced at around x = 0.1 m. In addition, δ varies radially; due to the higher temperatures, near-wall regions are more reduced than the centre of the fixed-bed. This highlights the importance of considering multiple dimensions, especially in larger diameter beds where heat transfer becomes increasingly critical. The δ -contour plots during oxidation are shown in Figure 9B; contrary to the reduction step a sharp reaction front is evident. The oxidation reaction proceeds from the inlet until it is locally, fully oxidized. Furthermore, there is no observable radial variation of δ . This is because the oxidation reaction is fast, at nearly thermodynamic rate conditions, without an observable kinetic dependence on temperature (in the studied temperature range).

Model contour plot of δ in the fixed-bed for T sp = 966 °C, 1 Ln·min−1, 5% educts during: (A) methanothermal reduction. (B) dry oxidation. The contour plot is shown at three time intervals for each redox step. The X and Y dimensions of the contour plots do not reflect their real proportions; to better visualize the trends, the Y -coordinate has is scaled up by a factor of 5.

Graphs comparing methanothermal reduction and dry oxidation processes over time. Methanothermal reduction shows changes at 50, 450, and 900 seconds, with color transitions from blue to red. Dry oxidation is depicted at 800, 1600, and 2400 seconds, with similar color shifts. Both use a color bar indicating δ values from blue (0) to red (0.12). X and Y axes represent spatial coordinates in meters.

The axial profile of the δ for each temperature set point of the model validation cases is plotted in Figure 10. Higher temperatures enhance the reaction kinetics leading to improved CH4 conversions, but also results in steeper δ -profiles and non-uniformity along the fixed-bed length. While higher temperatures are typically preferred this requires careful management of the δ -profile. Conversely, lower temperatures result in lower CH4 conversions, but produce more uniform δ profiles, which prevents over-reduction in localized areas of the fixed-bed.

Model results of local non-stoichiometry (taken at the central axis of the fixed-bed at 420 s) during reduction as a function of the axial coordinate, for T sp = 936, 956, 966, 976, 996, 1016 °C (1 Ln·min−1, 5% CH4).

Line graph showing the relationship between delta (δ) and x, ranging from 0 to 0.3 meters. Six curves represent different Tsp temperatures from 936°C to 1016°C, indicated by colors. The curves generally peak around 0.1 to 0.15 meters on the x-axis and then decline.

From a sensitivity study on the various modes of heat transfer, it was determined that the main mode of heat transfer is via conduction, which includes the radiative component as part of the RDA. For example, setting the k rad term to zero causes the reduction inflection point to occur 15% earlier, as heat cannot be transferred from the wall through the fixed-bed as easily.

Although the model captures the trends of both redox steps, there are notable limitations. Firstly, the reaction scheme composed of the five key reactions effectively represents the thermodynamics of dry redox reforming, however, it is simplistic compared to more comprehensive mechanisms. The reaction scheme may fail to accurately account for oxygen vacancies, intermediate reaction steps, and other species (e.g., C(s)), as proposed in other studies (Otsuka et al., 1993; Cheng et al., 2013; Zhao et al., 2016; Liu et al., 2024). Furthermore, only the apparent kinetics are modeled, and the dependency on δ is captured through the reaction scheme and a reaction control term within the scaling factor. Thermogravimetric (TG) studies on ceria have detailed kinetic dependencies on diffusion and non-stoichiometry (Ackermann et al., 2014; Ackermann et al., 2015; Ackermann, 2016; Hill et al., 2021; Hill C. et al., 2023). However, applying the kinetic data from TG studies directly to the model of a fixed-bed reactor presents significant challenges because of different experimental conditions (Gigantino, 2021). This is a major reason for tuning the apparent kinetic parameters based on the fixed-bed results, as done in this work. As previously discussed, an accurate δ -gradient is crucial in determining the product molar concentrations. The δ -gradient was not experimentally validated as doing so is a major challenge. For example, validating the δ -gradient after reduction would require sampling the fixed-bed in an inert environment (to prevent oxidation), and then re-oxidizing the sample in a controlled environment such as in TG analysis. Lastly, the model was experimentally validated only under conditions with low educt partial pressures. Consequently, the model may fail to accurately represent pressure variations or reaction orders.

 Model application

The validated computational model is used as a design tool in the optimization of a scaled-up dry redox reforming system. The study here serves primarily as an example of a practical use case of the model rather than a global or comprehensive optimization effort. This design case study focuses on the allothermally heated tubular reactor and considers two main aspects: (1) the tubular reactor dimensions and flow conditions for optimal syngas production; and (2) various operating strategies that manage the δ-profile and enhance the syngas selectivity.

Tubular reactor configuration

As per Figure 8A, the model accurately captures the trends in methane conversion as a function of temperature, thus enabling its application in this geometric design case study. A reactor-heat exchanger system similar to a shell-and-tube configuration is considered, where an HTF heats up a 7-tube arrangement of reactor tubes. Overall, the case study optimizes for maximum syngas production and considers 100% CH4 educt, ambient pressure, 2 m fixed-bed length, and HTF (steam at ambient pressure) at 990 °C.

The configuration of a single reactor tube (i.e., D fxb ; V ˙ in , C H 4 ) is based on a parametric study on the reduction step of dry redox reforming. This step is endothermic and kinetically controlled, making it sensitive to reactor temperatures. The core of the simulation was configured with the model settings for dry redox reforming as applied for the experimental validation (e.g., Table 2). However, distinct alterations were made such as considering 100% CH4 concentrations. This is an important distinction because the model was not experimentally validated at higher educt partial pressures, highlighting that this simulation serves primarily as an example of case study and may contain inaccuracies. The parametric study varied the reactor diameter ( D fxb = 1.5, 1.9, 2.7, 4, 7, 10, 15, 20 cm) and educt flow rates ( V ˙ in , C H 4 = 0.5, 1, 2, 5, 10, 15, 20, 50 Ln·min−1). To decrease the complexity, the HTF was modeled as a temperature boundary condition defined by a heat flux condition and external heat transfer coefficient ( h ext ). The HTF was represented as a typical annulus ( D o / D i = 1.5), with Nusselt number approximately equal to 6, thus yielding h ext values at around 10–100 W·m−2·K−1, within the typical range for gas-to-gas systems (Shah and London, 1978; Rohsenow et al., 1998; Dirker and Meyer, 2005; Cengel and Ghajar, 2015). The tubular reactor walls were not modeled. The educt gas enters at 990 °C. This case study also assumes the reactor is operated at high- δ resulting in a product stream of syngas. For this reason, the non-stoichiometry was fixed to 0.15 (omitting Equation 26), and the simulation is run (200–1,400 s) until steady state conditions are achieved i . e .   ∂ X C H 4 x , t ∂ t = 0 .

The syngas production rate from a single reactor tube is presented in a contour plot as a function of fixed-bed diameter and inlet flow rate, as shown in Figure 11. Lines of constant methane conversion are superimposed on the plot. Production rates tend to increase with inlet flow rates, however begin to decrease beyond V ˙ in , C H 4 = 15 Ln·min−1. Generally, smaller tubes are preferred due to improved heat transfer and conversions. However, multiple small tubes increase the complexity when designing the reactor-heat exchanger. It is recommended to operate a single tubular reactor at around 10 Ln·min−1, using 7–10 cm diameter tubes in a multi-tubular arrangement. Further increasing the flow rate decreases methane conversion, while further increasing the fixed-bed diameter only results in a marginal increase in syngas production. An additional consideration to bear in mind when choosing the tubular reactor configuration is the reduction time required to completely oxidize the fixed-bed to a δ max . An upper limit may be set so as to prevent certain phase changes (Zinkevich et al., 2006). The reduction time to achieve a uniform δ max as a function of fixed-bed volume and flow rate is defined as: t red , δ max ⁡ min ⁡ = ρ Ce O 2 fxb M Ce O 2 V fxb δ max 22.4   L n · mo l − 1 V ˙ in , C H 4 L n · ⁡ min − 1 Only Reaction R6 is considered, and ρ Ce O 2 fxb is the density of ceria as a fixed-bed ( ρ s · 1 − ϵ ). The recommended configuration of D fxb = 7 cm, and V ˙ in , C H 4 = 10 Ln·min−1, has a maximum reduction time of 33 min for δ max = 0.25.

Contour of syngas production rate during reduction as a function of fixed-bed diameter and inlet flow rate ( l fxb = 2 m). Lines of constant CH4 conversion at 70, 80, and 90% are superimposed on the contour plot. Steady-state results were determined using the porousRedoxFoam solver under high- δ conditions, with educt CH4 concentrations at 100%, and with the HTF at 990 °C modeled as an external wall heat flux condition.

Contour plot showing the relationship between \( V_{\text{in, CH4}} \) in liters per minute and \( D_{\text{fxb}} \) in centimeters, with color gradients representing \( \dot{V}_{\text{syngas}} \) in liters per minute. Contour lines indicate \( X_{CH4} \) values from 0.7 to 0.9, with darker colors representing higher values.

For the following analysis, a single-pass reactor-heat exchanger is considered. A single-pass concept is a relatively simple design and features straight tubes offering greater flexibility in reactor tube material, and provides a predictable axial temperature gradient, which simplifies reactor control. A seven-tube arrangement is considered (e.g., six tubes circling one tube) with each tube operating under the recommended configuration ( l fxb = 2 m, D fxb = 7 cm, and V ˙ in , C H 4 = 10 Ln·min−1), resulting in 63 kg of ceria. The influence of neighbouring tubes in the seven-tube arrangement on a single reactor tube’s performance is neglected. The total syngas production rate is 210 Ln·min−1. To drive the endothermic reaction (of R6), steam (at ambient pressure) is employed as the HTF. The steam is assumed to exit the heat exchanger at 990 °C. If a solar receiver heats up the steam to 1,200 °C, a flow rate of 3,000 Ln·min−1 is required to drive the reaction. Assuming a receiver inlet temperature at around 700 °C, this suggests a solar receiver with a capacity of about 50 kWth is required.

 Operating strategies

Different operating strategies can enhance the syngas selectivity. As visualized in Figure 10, the model captures the δ -profile along the length of the fixed-bed, and cases of high conversion tend to have larger variations of δ . The strategies discussed here focus on methods of exploiting or managing the non-stoichiometry of the fixed-bed in a tubular reactor. Experimental studies have explored related approaches. One investigation proposed a non-steady cycling approach with periodic recharging to sustain an elevated δ at the reactor outlet (Hill C. M. et al., 2023). Another study utilized the chemical potential gradient of the δ -profile, and operated the system in a countercurrent chemical looping configuration with a focus on the complete oxidation of methane and inherent separation of CO2 (Bulfin et al., 2024).

The core of the simulation was configured with the model settings for dry redox reforming as applied for the experimental validation (e.g., Table 2), however, distinct alterations were made. To decrease computational cost and quickly achieve high non-stoichiometries the operating strategies were simulated on a fixed-bed, D fxb = 2.7 cm and l fxb = 1 m, operating at V ˙ in = 2 Ln·min−1, 100% educt concentrations, and ambient pressure. A uniform wall temperature boundary condition was imposed and set to T wall , ∀ x = 1,020 °C. This ensures complete conversion of CH4, as intended, for the study here focuses on optimizing for syngas selectivity.

A baseline, standard strategy is presented, followed by three operating strategies that leverage the non-stoichiometry in the fixed-bed: high- δ , countercurrent, and priming. As observed in Figure 10, the inlet section achieves the highest δ , with the effect becoming more pronounced at elevated temperatures. The primary differences between each operating strategy are the redox step times, and location of the educt inlet. The results are consolidated in Figure 12 which includes plots on the molar concentrations of the product gases as a function of time for each operating strategy. The figure also shows schematics of the flow configuration as well as the axial δ -profile to help understand how the non-stoichiometry is leveraged for each strategy. The strategies are discussed and evaluated based on syngas selectivity during reduction, and CO2 conversion during oxidation. For each metric, two values are presented: “main” isolates the key cycles that best represent the strategy in long-term operation (e.g., excluding initial and final steps of high- δ cycling), and “max” which is the maximum instantaneous value achieved.

A baseline strategy and three operating strategies that exploit the non-stoichiometry are presented: (A) standard (baseline) cycling. (B) High-δ cycling. (C) Countercurrent cycling. (D) Priming and cycling. On the left are plots the product molar concentrations and average non-stoichiometry of the fixed-bed as a function of time. On the right are schematics of the flow configurations, and plots of local non-stoichiometry (taken at the central axis of the fixed-bed) as a function of the axial coordinate at the end of the first, third, and (if applicable) fifth reduction and oxidation step. Results determined using the computational model with educt concentrations at 100%, V ˙ in = 2 Ln·min−1, D fxb = 2.7 cm, l fxb = 1m, and T wall , ∀ x = 1,020 °C.

Graphs showing four types of cycling processes: A) standard, B) high-\(\delta\), C) countercurrent, and D) priming & cycling. Each type includes time series plots of variables like \(X_i\), \(CH_4\), \(CO_2\), \(H_2O\), \(CO\), and \(H_2\) against time, and spatial plots showing profiles of \(\delta\) along the reactor length. Arrows indicate the direction of reduction and oxidation stages with varying line styles representing cycle ends. Colors differentiate between phases and stages.

The standard cycling approach (experimentally demonstrated in Figure 6) involves a reduction step followed by complete oxidation of the fixed-bed. The educts enter at the same inlet (i.e., x = 0   m ). Three consecutive cycles were simulated with t red = 300 s and t ox = 800 s, and the results are shown in Figure 12A. After reduction the δ avg of the fixed-bed reaches approximately 0.215. The axial δ -profiles (Figure 12A right) at the end of the redox steps detail a gradient in δ following the reduction step and complete oxidation is evident as δ ∀ x ≈ 0 at the end of the oxidation step. The metrics concerning this operating mode are: S CO , main = 0.70 (max. of 0.92), S H 2 , main = 0.63 (max. of 0.88), and X C O 2 , main = 0.78 (max. of 0.95).

The high- δ cycling strategy aims to take benefit of higher syngas selectivities observed at higher non-stoichiometries, as alluded to in Figure 2. Five consecutive cycles were simulated with an initial reduction step time of 300 s. To prevent complete oxidation, the succeeding, main redox times were set to t red = t ox = 100 s. A final oxidation step time of 900 s fully oxidizes the fixed-bed. The educts enter at the same inlet (i.e., x = 0   m ). The results are shown in Figure 12B. During the main cycles (2–4) the δ avg cycles between 0.183 and 0.252. The benefit of high- δ cycling is observed in the improved metrics compared to standard cycling: S CO , main = 0.95 (max. of 0.98), S H 2 , main = 0.93 (max. of 0.97), and X C O 2 , main = 0.97 (max. of 0.99). Due to shorter oxidation times during high- δ cycling only the near-inlet region becomes oxidized (shown in Figure 12B right), thus maintaining elevated levels of non-stoichiometry towards the outlet.

The countercurrent cycling strategy mimics that of a countercurrent heat exchanger, where the inlet of one flow is adjacent to the outlet of the other flow. By leveraging the chemical potential difference between streams, the strategy can improve species transfer by 1–2 times (Bulfin, 2019). An experimental demonstration into the purely thermal reduction of ceria (i.e., Reaction R3) used a countercurrent aerosol reactor to increase the non-stoichiometry beyond the limits of co-current operation (Scheffe et al., 2014). Another means of realizing the countercurrent arrangement for a solid-gas heterogeneous system is through the utilization of a chemical looping regenerative reactor, whereby the educt flows for each redox step enter at opposite ends of the reactor. This concept has been applied to the RWGS process as well as for dry redox reforming with a focus on complete oxidation of methane, where results showed nearly twice the CO2 conversion rate to (standard) co-current operation (Bulfin et al., 2023; Bulfin et al., 2024). In the context of dry redox reforming, countercurrent operation focusses primarily on CO2 conversion, but was nevertheless considered in this study. The flow configuration (Figure 12C right) is such that CH4 enters at x = 1 m during the reduction step, and CO2 enters at x = 0 m during the oxidation step. Three consecutive cycles were simulated with t red = 300 s and t ox = 700 s, and the results are shown in Figure 12C. The improvement in CO2 conversions is evident in the shorter oxidation times and improved metrics relative to standard cycling: S CO , main = 0.70 (max. of 0.92), S H 2 , main = 0.63 (max. of 0.88), and X C O 2 , main = 0.89 (max. of 1.00). This improvement is attributed to the fact that the CO2 educt stream experiences higher non-stoichiometries at its outlet (shown in Figure 12C right) as caused by the previous reduction step.

The priming and cycling strategy is designed such that both redox steps operate at an elevated non-stoichiometry at the outlet of the fixed-bed. As the name suggests, the cycles are primed by the first reduction cycle where the CH4 enters from the opposite end of the reactor (i.e., x = 1 m). All following redox steps have the educts enter at the forepart of the reactor (i.e., x = 0 m). The flow configuration is visualized at the right of Figure 12D. Five consecutive cycles were simulated with t red = t ox = 300 s, and the final oxidation step was set to 900 s. The results are shown in Figure 12D. During the main cycles (2–4) the δ avg cycles between 0.115 and 0.261. The benefit of priming and cycling is observed in the improved metrics: S CO , main = 0.96 (max. of 1.00), S H 2 , main = 0.96 (max. of 1.00), and X C O 2 , main = 0.97 (max. of 1.00). The “max” values of each metric reach unity, indicating the potential to achieve a pure syngas stream. As depicted at the right of Figure 12D, the first reduction step increases the non-stoichiometry at the aft of the fixed-bed (i.e., x = 1 m). The subsequent redox steps reduce and oxidize the forepart (i.e., x = 0 m) of the fixed-bed, therefore maintaining a high non-stoichiometry at the aft and promoting high syngas selectivities. In addition, this strategy directly targets high non-stoichiometries at the aft of the reactor, thereby mitigating excessively high local non-stoichiometries as observed during high- δ cycling (right of Figure 12B) which may lead to phase changes (Zinkevich et al., 2006). The priming and cycling strategy was also experimentally demonstrated using the previously described tubular reactor setup, with the additional use of valves to control the flow direction (Figure 5). Educt conversions and syngas selectivity reached unity under conditions of T sp = 1,016 °C, V ˙ in = 1 Ln·min−1, and 5% educts.

 Conclusion and outlook

This study examined dry redox reforming, i.e., methane dry reforming using a ceria-based redox cycle. To represent the process, a reaction scheme consisting of five key reactions was proposed: (1) methanothermal reduction; (2) dry oxidation; (3) complete oxidation of CH4; (4) CO as reducing agent; and (5) H2 as reducing agent. The thermodynamics of the reaction system were outlined in Ellingham diagrams and species-temperature plots, outlining the dependency of the system on temperature and non-stoichiometry. Results indicated that operating at high non-stoichiometries is preferred for syngas production.

The transient CFD model of the fixed-bed tubular reactor was developed in OpenFOAM and was designed to model a system of reversible heterogenous reactions and fluid flow over a non-stoichiometric solid oxide as a fixed-bed. The thermodynamic implementation was validated. Experiments were then carried out in a tubular reactor ( D fxb = 19 mm, l fxb = 30 cm) for the purpose of validating the model across a wide range of temperatures ( T sp = 936 °C–1,016 °C) and non-stoichiometries (0.001 ∼ 0.12). Typical conditions included 5% educt concentrations of CH4 and CO2 in Ar, T ≈ 1,000 °C, and V ˙ in = 1 Ln·min−1. The experimental conditions were simulated and the model results were compared against the experiments. The model captured the distinctive trends of the redox cycle, such as changes in species concentrations as well as multi-dimensional effects. When comparing performance metrics (conversion, selectivity, Δ δ , t ip ), the model had relative errors within 7% of the experimental values. The validated model was then used as a design tool in the optimization of a dry redox reforming system. First, a geometric design case study considered a tubular reactor operating with 100% CH4 educt, ambient pressure, 2 m fixed-bed length, and HTF at 990 °C. Through a parametric study the reactor diameter and flow rate were optimized for syngas production, with a final recommendation to operate at around V ˙ in = 10 Ln·min−1 (per reactor rube), using 7–10 cm diameter tubes in a multi-tubular arrangement. In a separate application of the model, various operating strategies were investigated and a priming and cycling strategy is suggested. This strategy exploits the gradient of the non-stoichiometry in the fixed-bed, resulting in values of syngas selectivity and conversion approaching unity, indicating the potential to achieve a pure syngas product stream.

This work helps advance the field of SAF forward by addressing multiple key areas identified by the community (Sheu et al., 2015; Bhosale, 2023). The dry redox reforming cycle in itself is advantageous as it utilizes CH4 rather than an inert gas during reduction, thereby decreasing operating temperatures and producing syngas in both redox steps. The experimental investigation reported on a high-conversion porous pellet ceria morphology. The computational model was developed to simulate reversible heterogenous reactions involving a non-stoichiometric solid oxide as a fixed-bed, applied here to the ceria-based dry redox reforming cycle. The model was analytically and experimentally validated at 5% educt concentrations, and further used to identify a novel operating strategy, priming and cycling, that exploits non-stoichiometry gradients yield a pure syngas stream. A fully validated computational model can support future scale-up efforts, as exemplified in the practical case study. Beyond this work, the model provides a platform that can be extended to other redox systems, higher educt partial pressures, and more complex reaction mechanisms involving solid carbon. The study investigated dry redox reforming performed in a cyclic manner. Looking ahead and serving as an outlook, there emerges another promising pathway that warrants further investigation. Through co-feeding a mixture of the educts (i.e., CH4 and CO2), the dry redox reforming process can be operated in a continuous manner.

co-feeding: C H 4 + C O 2 ↔ Ce O 2 − δ 2 CO + 2 H 2   Δ H 1273   K , δ eq o = 256 kJ mol CH 4

This enables the continuous production of syngas at lower energy demands, as it mitigates the separate endothermic reduction step (Bulfin et al., 2021a). A preliminary investigation into co-feeding, as discussed throughout this study, has been conducted (Zuber, 2024). Experiments demonstrated that high SSA ceria morphologies act as an oxygen buffer, mitigate carbon formation, and resist deactivation. This enables co-feeding with a stoichiometric ratio of educts (i.e., 1CH4: 1CO2), giving it a competitive advantage over traditional catalytic reforming which requires an excess of oxidizing educts (Trimm, 1997; Sehested, 2006). Co-feeding is governed by the same thermodynamic system as dry redox reforming, enabling it to be modeled using the code developed in the present study (porousRedoxFoam), with preliminary simulations showing promising results.

 Data availability statement

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found below: https://github.com/mgzuber-eth/CeO2-CH4-CO2_thermoPlots, https://github.com/mgzuber-eth/porousRedoxFoam.

 Author contributions

MZ: Software, Writing – original draft, Writing – review and editing, Investigation, Formal Analysis, Methodology, Data curation, Visualization, Conceptualization, Validation. SA: Writing – review and editing, Supervision. AS: Writing – review and editing, Supervision, Funding acquisition.

 Acknowledgements

The numerical simulations were executed on the Euler cluster operated by the High Performance Computing group at ETH Zurich. We thank Brendan Bulfin for the scientific advice. The manuscript includes adapted segments from the doctoral thesis of the first author (Zuber, 2024).

 Conflict of interest

Author SA was employed by Synhelion SA.

Author AS has financial interests in Synhelion SA.

The remaining author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Generative AI statement

The authors declare that no Generative AI was used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

 Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Edited by: Michael Folsom Toney, University of Colorado Boulder, United States

Reviewed by: Sindhu Subramanian, Amrita Vishwa Vidyapeetham University, India

Sohrab Zendehboudi, Memorial University of Newfoundland, Canada

 References Abanades S. (2023). A review of oxygen carrier materials and related thermochemical redox processes for concentrating solar thermal applications. Mater. (Basel) 16 (9), 3582. 10.3390/ma16093582 37176464 Abanades S. Flamant G. (2006). Thermochemical hydrogen production from a two-step solar-driven water-splitting cycle based on cerium oxides. Sol. Energy 80 (12), 1611–1623. 10.1016/j.solener.2005.12.005 Abdulrasheed A. Jalil A. A. Gambo Y. Ibrahim M. Hambali H. U. Shahul Hamid M. Y. (2019). A review on catalyst development for dry reforming of methane to syngas: recent advances. Renew. Sustain. Energy Rev. 108, 175–193. 10.1016/j.rser.2019.03.054 Ackermann S. (2016). Redox kinetics and transport phenomena of reticulated porous ceria. Zürich: ETH Zürich. Doctoral Thesis No. 23339. 10.3929/ethz-a-010788214 Ackermann S. Scheffe J. R. Steinfeld A. (2014). Diffusion of oxygen in ceria at elevated temperatures and its application to H2O/CO2 splitting thermochemical redox cycles. J. Phys. Chem. C 118 (10), 5216–5225. 10.1021/jp500755t Ackermann S. Sauvin L. Castiglioni R. Rupp J. L. Scheffe J. R. Steinfeld A. (2015). Kinetics of CO2 reduction over nonstoichiometric Ceria. J. Phys. Chem. C 119 (29), 16452–16461. 10.1021/acs.jpcc.5b03464 26693270 Agrafiotis C. von Storch H. Roeb M. Sattler C. (2014). Solar thermal reforming of methane feedstocks for hydrogen and syngas production—A review. Renew. Sustain. Energy Rev. 29, 656–682. 10.1016/j.rser.2013.08.050 Agrafiotis C. Wullenkord M. Roeb M. Sattler C. (2021). Solar thermochemistry – solar thermal methane reforming. Adv. Chem. Eng. 58, 91–130. 10.1016/bs.ache.2021.10.002 Al-Otaibi F. Xiao H. Berrouk A. S. Polychronopoulou K. (2023). Numerical study of dry reforming of methane in packed and fluidized beds: effects of key operating parameters. ChemEngineering 7 (3), 57. 10.3390/chemengineering7030057 Alotaibi F. N. Berrouk A. S. Salim I. M. (2024). Scaling up dry methane reforming: integrating computational fluid dynamics and machine learning for enhanced hydrogen production in industrial-scale fluidized bed reactors. Fuel 376, 132673. 10.1016/j.fuel.2024.132673 Ambrosetti G. Good P. (2019). A novel approach to high temperature solar receivers with an absorbing gas as heat transfer fluid and reduced radiative losses. Sol. Energy 183, 521–531. 10.1016/j.solener.2019.03.004 Bayon A. de la Calle A. Stechel E. B. Muhich C. (2021). Operational limits of redox metal oxides performing thermochemical water splitting. Energy Technol. 10 (1), 2100222. 10.1002/ente.202100222 Behnam M. Dixon A. G. Wright P. M. Nijemeisland M. Stitt E. H. (2012). Comparison of CFD simulations to experiment under methane steam reforming reacting conditions. Chem. Eng. J. 207-208, 690–700. 10.1016/j.cej.2012.07.038 Bhattacharjee S. Linley S. Reisner E. (2024). Solar reforming as an emerging technology for circular chemical industries. Nat. Rev. Chem. 8 (2), 87–105. 10.1038/s41570-023-00567-x 38291132 Bhosale R. R. (2023). Recent developments in ceria-driven solar thermochemical water and carbon dioxide splitting redox cycle. Energies 16 (16), 5949. 10.3390/en16165949 Bose A. Farooqui A. Ferrero D. Santarelli M. Llorca J. (2019). Thermodynamic assessment of non-catalytic Ceria for syngas production by methane reduction and CO2+H2O oxidation. Mater. Renew. Sustain. Energy 8 (1), 5. 10.1007/s40243-019-0142-3 Budama V. K. Rincon Duarte J. P. Roeb M. Sattler C. (2023). Potential of solar thermochemical water-splitting cycles: a review. Sol. Energy 249, 353–366. 10.1016/j.solener.2022.11.001 Bulfin B. (2019). Thermodynamic limits of countercurrent reactor systems, with examples in membrane reactors and the ceria redox cycle. Phys. Chem. Chem. Phys. 21 (4), 2186–2195. 10.1039/c8cp07077f 30644473 Bulfin B. Hoffmann L. de Oliveira L. Knoblauch N. Call F. Roeb M. (2016). Statistical thermodynamics of non-stoichiometric ceria and ceria zirconia solid solutions. Phys. Chem. Chem. Phys. 18 (33), 23147–23154. 10.1039/c6cp03158g 27494765 Bulfin B. Ackermann S. Furler P. Steinfeld A. (2021a). Thermodynamic comparison of solar methane reforming via catalytic and redox cycle routes. Sol. Energy 215, 169–178. 10.1016/j.solener.2020.11.076 Bulfin B. Miranda M. Steinfeld A. (2021b). Performance indicators for benchmarking solar thermochemical fuel processes and reactors. Front. Energy Res. 9, 677980. 10.3389/fenrg.2021.677980 Bulfin B. Zuber M. Gräub O. Steinfeld A. (2023). Intensification of the reverse water–gas shift process using a countercurrent chemical looping regenerative reactor. Chem. Eng. J. 461, 141896. 10.1016/j.cej.2023.141896 Bulfin B. Zuber M. Steinfeld A. (2024). Countercurrent chemical looping for enhanced methane reforming with complete conversion and inherent CO2 separation. Chem. Eng. J. 488, 150513. 10.1016/j.cej.2024.150513 Cabrera E. de Sousa J. M. M. (2022). Use of sustainable fuels in Aviation—A review. Energies 15 (7), 2440. 10.3390/en15072440 Casati E. Tièche A. Hartner A. Steinfeld A. (2022). “A solar receiver delivering heat at 1500C and above: experimental results,” in 28th SolarPACES conference, Albuquerque, NM. 10.3929/ethz-b-000573794 Cengel Y. Ghajar A. (2015). Heat and mass transfer fundamentals and applications. New York, NY: McGraw Hill Education. Cheng Z. Sherman B. J. Lo C. S. (2013). Carbon dioxide activation and dissociation on ceria (110): a density functional theory study. J. Chem. Phys. 138 (1), 014702. 10.1063/1.4773248 23298052 Chuayboon S. Abanades S. (2020). An overview of solar decarbonization processes, reacting oxide materials, and thermochemical reactors for hydrogen and syngas production. Int. J. Hydrogen Energy 45 (48), 25783–25810. 10.1016/j.ijhydene.2020.04.098 Chuayboon S. Abanades S. Rodat S. (2019). Syngas production via solar-driven chemical looping methane reforming from redox cycling of ceria porous foam in a volumetric solar reactor. Chem. Eng. J. 356, 756–770. 10.1016/j.cej.2018.09.072 Chuayboon S. Abanades S. Rodat S. (2020). Solar chemical looping reforming of methane combined with isothermal H2O/CO2 splitting using ceria oxygen carrier for syngas production. J. Energy Chem. 41, 60–72. 10.1016/j.jechem.2019.05.004 Chueh W. C. Haile S. M. (2010). A thermochemical study of ceria: exploiting an old material for new modes of energy conversion and CO2 mitigation. Philosophical Trans. R. Soc. A 368 (1923), 3269–3294. 10.1098/rsta.2010.0114 20566511 Chueh W. C. Falter C. Abbot M. Scipio D. Furler P. Haile S. M. (2010). High-flux solar-driven thermochemical dissociation of CO2 and H2O using nonstoichiomeetric ceria. Science 330, 1797–1801. 10.1126/science.1197834 21205663 CORDIS (2023). SUN-to-LIQUID II. Grant agreement ID: 101122206. Taufkirchen, DE: Bauhaus Luftfahrt EV. 10.3030/101122206 Das S. Ashok J. Bian Z. Dewangan N. Wai M. H. Du Y. (2018). Silica–Ceria sandwiched Ni core–shell catalyst for low temperature dry reforming of biogas: coke resistance and mechanistic insights. Appl. Catal. B Environ. 230, 220–236. 10.1016/j.apcatb.2018.02.041 Dirker J. Meyer J. P. (2005). Convective heat transfer coefficients in concentric Annuli. Heat. Transf. Eng. 26 (2), 38–44. 10.1080/01457630590897097 Fabrik M. Salama A. Ibrahim H. (2022). State-of-the-art in methane-reforming reactor modeling: challenges and new insights. Rev. Chem. Eng. 38 (8), 991–1022. 10.1515/revce-2020-0038 Farsi M. (2023). Reforming process design and modeling: steam, dry, and autothermal reforming. Amsterdam, NL: Elsevier, 123–140. 10.1016/b978-0-323-91879-4.00003-5 Fosheim J. R. Hathaway B. J. Davidson J. H. (2019). High efficiency solar chemical-looping methane reforming with ceria in a fixed-bed reactor. Energy 169, 597–612. 10.1016/j.energy.2018.12.037 Furler P. Scheffe J. Gorbar M. Moes L. Vogt U. Steinfeld A. (2012). Solar thermochemical CO2 splitting utilizing a reticulated porous ceria redox system. Energy Fuels 26 (11), 7051–7059. 10.1021/ef3013757 Furler P. Scheffe J. Marxer D. Gorbar M. Bonk A. Vogt U. (2014). Thermochemical CO2 splitting via redox cycling of ceria reticulated foam structures with dual-scale porosities. Phys. Chem. Chem. Phys. 16 (22), 10503–10511. 10.1039/c4cp01172d 24736455 Gao Y. Jiang J. Meng Y. Yan F. Aihemaiti A. (2018). A review of recent developments in hydrogen production via biogas dry reforming. Energy Convers. Manag. 171, 133–155. 10.1016/j.enconman.2018.05.083 Gigantino M. (2021). Materials and reactor engineering for thermochemical energy storage. Zürich: ETH Zürich. Doctoral Thesis No. 28008. 10.3929/ethz-b-000544090 Guene Lougou B. Shuai Y. Chaffa G. Ahouannou C. Pan R. Zhang H. (2019). Analysis of two‐step solar thermochemical looping reforming of Fe3O4 redox cycles for synthesis gas production. Energy Technol. 7 (3), 1800588. 10.1002/ente.201800588 Habibzadeh M. Madadi Avargani V. Zendehboudi S. (2024). Performance assessment of a novel porous catalytic reactor with a hydrogen-selective membrane for enhanced methane dry reforming process – CFD investigation. Int. J. Hydrogen Energy 64, 870–888. 10.1016/j.ijhydene.2024.03.174 Haeussler A. Chuayboon S. Abanades S. (2020). Solar redox cycling of ceria in a monolithic reactor for two-step H2O/CO2 splitting: isothermal methane-induced reduction versus temperature-swing cycle. AIP Conf. Proc. 2303, 170009. 10.1063/5.0028582 Hajizadeh A. Mohamadi-Baghmolaei M. Cata Saady N. M. Zendehboudi S. (2022). Hydrogen production from biomass through integration of anaerobic digestion and biogas dry reforming. Appl. Energy 309, 118442. 10.1016/j.apenergy.2021.118442 Haussener S. (2022). Solar fuel processing: comparative mini-review on research, technology development, and scaling. Sol. Energy 246, 294–300. 10.1016/j.solener.2022.09.019 Heya M. Gao X. Tricoli A. Lipiński W. (2020). Effect of specific surface area on syngas production performance of pure ceria in high-temperature thermochemical redox cycling coupled to methane partial oxidation. RSC Adv. 10 (60), 36617–36626. 10.1039/d0ra06280d 35517936 Hill C. M. Hernaiz E. A. Furler P. Ackermann S. Scheffe J. R. (2021). Characterization of Zr‐doped Ceria and sr‐doped La-Mn perovskites as redox intermediates for solar chemical‐looping reforming of methane. Energy Technol. 10 (1), 2100473. 10.1002/ente.202100473 Hill C. Robbins R. Furler P. Ackermann S. Scheffe J. (2023a). Kinetic investigation of solar chemical looping reforming of methane over Ni–CeO2 at low temperature. Sustain. Energy Fuels 7 (2), 574–584. 10.1039/d2se01452a Hill C. M. Ackermann S. Trimm K. G. McCord D. C. Furler P. Scheffe J. R. (2023b). Enhanced syngas selectivity and carbon utilization during chemical looping reforming of methane via a non-steady redox cycling strategy. Sustain. Energy Fuels 7, 5669–5680. 10.1039/d3se01213a Jang J. T. Yoon K. J. Bae J. W. Han G. Y. (2014). Cyclic production of syngas and hydrogen through methane-reforming and water-splitting by using ceria–zirconia solid solutions in a solar volumetric receiver–reactor. Sol. Energy 109, 70–81. 10.1016/j.solener.2014.08.024 Karni J. Kribus A. Doron P. Rubin R. Fiterman A. Sagie D. (1997). The DIAPR: a high-pressure, high-temperature solar receiver. J. Sol. Energy Eng. 119 (1), 74–78. 10.1115/1.2871853 Kaydouh M.-N. El Hassan N. Chalhoub E. Bukharin N. El Hassan M. (2024). A review of computational fluid dynamics modeling of dry reforming of methane for improved catalyst and reactor design. Fuel 366, 131290. 10.1016/j.fuel.2024.131290 Krenzke P. T. Fosheim J. R. Zheng J. Davidson J. H. (2016). Synthesis gas production via the solar partial oxidation of methane-ceria redox cycle: conversion, selectivity, and efficiency. Int. J. Hydrogen Energy 41 (30), 12799–12811. Krenzke P. T. Fosheim J. R. Davidson J. H. (2017). Solar fuels via chemical-looping reforming. Sol. Energy 156, 48–72. 10.1016/j.solener.2017.05.095 Kwiatkowski K. Bajer K. Żuk P. J. Dudyński M. (2013). Biomass gasificaiton solver based on OpenFOAM. Comput. Phys. Commun 278. 10.1016/j.cpc.2022.108407 Landauer R. (1952). The electrical resistance of binary metallic mixtures. J. Appl. Phys. 23 (7), 779–784. 10.1063/1.1702301 Laosiripojana N. Assabumrungrat S. (2005). Catalytic dry reforming of methane over high surface area ceria. Appl. Catal. B Environ. 60 (1-2), 107–116. 10.1016/j.apcatb.2005.03.001 Lavoie J. M. (2014). Review on dry reforming of methane, a potentially more environmentally-friendly approach to the increasing natural gas exploitation. Front. Chem. 2, 81. 10.3389/fchem.2014.00081 25426488 Li Y. Park C. W. (1998). Permeability of packed beds filled with polydisperse spherical particles. Industrial Eng. Chem. Res. 37 (5), 2005–2011. 10.1021/ie970603s Lidor A. Bulfin B. (2024). A critical perspective and analysis of two-step thermochemical fuel production cycles. Sol. Compass 11, 100077. 10.1016/j.solcom.2024.100077 Lidor A. Aschwanden Y. Häseli J. Reckinger P. Haueter P. Steinfeld A. (2023). High-temperature heat recovery from a solar reactor for the thermochemical redox splitting of H2O and CO2 . Appl. Energy 329, 120211. 10.1016/j.apenergy.2022.120211 Liu T. Zheng H. Wang H. Wang J. Wang Z. Kong H. (2024). A review of methane-driven two-step thermochemical cycle hydrogen production. J. Clean. Prod. 466, 142668. 10.1016/j.jclepro.2024.142668 Mao Y. Gao Y. Dong W. Wu H. Song Z. Zhao X. (2020). Hydrogen production via a two-step water splitting thermochemical cycle based on metal oxide – a review. Appl. Energy 267, 114860. 10.1016/j.apenergy.2020.114860 Marxer D. Furler P. Scheffe J. Geerlings H. Falter C. Batteiger V. (2015). Demonstration of the entire production chain to renewable kerosene via solar thermochemical splitting of H2O and CO2 . Energy Fuels 29 (5), 3241–3250. 10.1021/acs.energyfuels.5b00351 Marxer D. Furler P. Takacs M. Steinfeld A. (2017). Solar thermochemical splitting of CO2 into separate streams of CO and O2 with high selectivity, stability, conversion, and efficiency. Energy Environ. Sci. 10 (5), 1142–1149. 10.1039/c6ee03776c Nair M. M. Abanades S. (2016). Tailoring hybrid nonstoichiometric Ceria redox cycle for combined solar methane reforming and thermochemical conversion of H2O/CO2 . Energy Fuels 30 (7), 6050–6058. 10.1021/acs.energyfuels.6b01063 Neiva L. S. (2010). A study on the characteristics of the reforming of methane: a review. Braz. J. Petroleum Gas, 119–127. 10.5419/bjpg2010-0013 OpenFOAM-Foundation (2019). User Guide - version 7. London, United Kingdom: The OpenFOAM Foundation Ltd Incorporated in England. Otsuka K. Ushiyama T. Yamanaka I. (1993). Partial oxidation of methane using the redox of cerium oxide. Chem. Lett. 22, 1517–1520. 10.1246/cl.1993.1517 Pan H. Li J. Wang Y. Xia Q. Qiu L. Zhou B. (2024). Solar-driven biomass reforming for hydrogen generation: principles, advances, and challenges. Adv. Sci. 11, 2402651. 10.1002/advs.202402651 38816938 Panlener R. Blumenthal R. Garnier J. (1975). A thermodynamic study of nonstoichiometric cerium dioxide. J. Chem. Phys. 36, 1213–1222. 10.1016/0022-3697(75)90192-4 Patil V. R. Kiener F. Grylka A. Steinfeld A. (2021). Experimental testing of a solar air cavity-receiver with reticulated porous ceramic absorbers for thermal processing at above 1000C. Sol. Energy 214, 72–85. 10.1016/j.solener.2020.11.045 Pozzobon V. Colin J. Perré P. (2018). Hydrodynamics of a packed bed of non-spherical polydisperse particles: a fully virtual approach validated by experiments. Chem. Eng. J. 354, 126–136. 10.1016/j.cej.2018.07.214 Pullar R. C. Novais R. M. Caetano A. P. F. Barreiros M. A. Abanades S. Oliveira F. A. C. (2019). A review of solar thermochemical CO2 splitting using Ceria-based ceramics with designed morphologies and microstructures. Front. Chem. 7, 601. 10.3389/fchem.2019.00601 31552219 Rohsenow W. Hartnett J. P. Cho Y. I. (1998). Handbook of heat transfer. New York, NY: McGraw-Hill. Romero M. Steinfeld A. (2012). Concentrating solar thermal power and thermochemical fuels. Energy Environ. Sci. 5 (11), 9234–9245. 10.1039/c2ee21275g Safari F. Dincer I. (2020). A review and comparative evaluation of thermochemical water splitting cycles for hydrogen production. Energy Convers. Manag. 205, 112182. 10.1016/j.enconman.2019.112182 Said S. A. M. Waseeuddin M. Simakov D. S. A. (2016). A review on solar reforming systems. Renew. Sustain. Energy Rev. 59, 149–159. 10.1016/j.rser.2015.12.072 Sas Brunser S. Steinfeld A. (2023). Design and optimization of hierarchically ordered porous structures for solar thermochemical fuel production using a voxel-based monte carlo Ray-Tracing algorithm. ACS Eng. Au 3 (5), 326–334. 10.1021/acsengineeringau.3c00013 Sas Brunser S. Bargardi F. L. Libanori R. Kaufmann N. Braun H. Steinfeld A. (2023). Solar‐driven redox splitting of CO2 using 3D‐Printed hierarchically channeled Ceria structures. Adv. Mater. Interfaces 10 (30), 2300452. 10.1002/admi.202300452 Schäppi R. Rutz D. Dahler F. Muroyama A. Haueter P. Lilliestam J. (2021). Drop-in fuels from sunlight and air. Nature 601, 63–68. 10.1038/s41586-021-04174-y 34732875 Scheffe J. R. Steinfeld A. (2012). Thermodynamic analysis of cerium-based oxides for solar thermochemical fuel production. Energy Fuels 26 (3), 1928–1936. 10.1021/ef201875v Scheffe J. R. Welte M. Steinfeld A. (2014). Thermal reduction of Ceria within an aerosol reactor for H2O and CO2 splitting. Industrial Eng. Chem. Res. 53 (6), 2175–2182. 10.1021/ie402620k Schwab E. Milanov A. Schunk S. A. Behrens A. Schödel N. (2015). Dry reforming and reverse water gas shift: alternatives for syngas production? Chem. Ing. Tech. 87 (4), 347–353. 10.1002/cite.201400111 Sehested J. (2006). Four challenges for nickel steam-reforming catalysts. Catal. Today 111 (1-2), 103–110. 10.1016/j.cattod.2005.10.002 Shah R. K. London A. L. (1978). Laminar forced flow convection in ducts. New York, NY: Academic Press. 10.1016/C2013-0-06152-X Sheu E. J. Ghoniem A. F. (2014). Redox reforming based, integrated solar-natural gas plants: reforming and thermodynamic cycle efficiency. Int. J. Hydrogen Energy 39 (27), 14817–14833. 10.1016/j.ijhydene.2014.07.086 Sheu E. J. Mokheimer E. M. A. Ghoniem A. F. (2015). A review of solar methane reforming systems. Int. J. Hydrogen Energy 40 (38), 12929–12955. 10.1016/j.ijhydene.2015.08.005 Simakov D. S. A. Wright M. M. Ahmed S. Mokheimer E. M. A. Román-Leshkov Y. (2015). Solar thermal catalytic reforming of natural gas: a review on chemistry, catalysis and system design. Catal. Sci. Technol. 5 (4), 1991–2016. 10.1039/c4cy01333f Smith D. S. Alzina A. Bourret J. Nait-Ali B. Pennec F. Tessier-Doyen N. (2013). Thermal conductivity of porous materials. J. Mater. Res. 28 (17), 2260–2272. 10.1557/jmr.2013.179 Spiewak I. Tyner C. Langnickel U. (1993). “Applications of solar reforming technology,” in Sandia report (Springfield, VA: Sandia National Laboratories). 10.2172/10131314 Steinfeld A. (2012). Thermochemical production of syngas using concentrated solar energy. Annual Review of Heat Transfer 15, 255–275. 10.1615/AnnualRevHeatTransfer.2012004537 Stroud T. Smith T. J. Le Saché E. Santos J. L. Centeno M. A. Arellano-Garcia H. (2018). Chemical CO2 recycling via dry and bi reforming of methane using Ni-Sn/Al2O3 and Ni-Sn/CeO2-Al2O3 catalysts. Appl. Catal. B Environ. 224, 125–135. 10.1016/j.apcatb.2017.10.047 Tang M. Xu L. Fan M. (2015). Progress in oxygen carrier development of methane-based chemical-looping reforming: a review. Appl. Energy 151, 143–156. 10.1016/j.apenergy.2015.04.017 Tavasoli A. Ozin G. (2018). Green syngas by solar dry reforming. Joule 2 (4), 571–575. 10.1016/j.joule.2018.02.017 Tescari S. Agrafiotis C. Breuer S. de Oliveira L. Puttkamer M.N.-v. Roeb M. (2014). Thermochemical solar energy storage via Redox oxides: materials and Reactor/Heat exchanger concepts. Energy Procedia 49, 1034–1043. 10.1016/j.egypro.2014.03.111 Touloukian Y. S. (1967). Thermophysical properties of high temperature solid materials. New York, NY: The Macmillan Company. Tran J. T. Warren K. J. Wilson S. A. Muhich C. L. Musgrave C. B. Weimer A. W. (2024). An updated review and perspective on efficient hydrogen generation via solar thermal water splitting. WIREs Energy Environ. 13 (4), e528. 10.1002/wene.528 Trimm D. L. (1997). Coke formation and minimisation during steam reforming reactions. Catal. Today 37, 233–238. 10.1016/S0920-5861(97)00014-X von Storch H. Stadler H. Roeb M. Hoffschmidt B. (2015). Efficiency potential of indirectly heated solar reforming with open volumetric solar receiver. Appl. Therm. Eng. 87, 297–304. 10.1016/j.applthermaleng.2015.05.026 von Storch H. Roeb M. Stadler H. Sattler C. Hoffschmidt B. (2016). Available online efficiency potential of indirectly heated solar reforming with different types of solar air receivers. Appl. Therm. Eng. 92, 202–209. 10.1016/j.applthermaleng.2015.09.065 Wakao N. Kaguei S. Funazkri T. (1978). Effect of fluid dispersion coefficients on particle-to-fluid heat transfer coefficients in packed beds: correlation of nusselt numbers. Chem. Eng. Sci. 34, 325–336. 10.1016/0009-2509(79)85064-2 Wang Z. Roberts R. R. Naterer G. F. Gabriel K. S. (2012). Comparison of thermochemical, electrolytic, photoelectrolytic and photochemical solar-to-hydrogen production technologies. Int. J. Hydrogen Energy 37 (21), 16287–16301. 10.1016/j.ijhydene.2012.03.057 Wang Y. Su B. Yang X. Chen Z. Wang S. (2024). A green hydrogen production system based on solar-assisted biogas steam reforming. Appl. Therm. Eng. 248, 123067. 10.1016/j.applthermaleng.2024.123067 Warren K. J. Weimer A. W. (2022). Solar thermochemical fuels: present status and future prospects. Sol. Compass 1, 100010. 10.1016/j.solcom.2022.100010 Warren K. J. Reim J. Randhir K. Greek B. Carrillo R. Hahn D. W. (2017). Theoretical and experimental investigation of solar methane reforming through the nonstoichiometric Ceria redox cycle. Energy Technol. 5 (11), 2138–2149. 10.1002/ente.201700083 Warren K. J. Carrillo R. J. Greek B. Hill C. M. Scheffe J. R. (2020). Solar reactor demonstration of efficient and selective syngas production via chemical‐looping dry reforming of methane over Ceria. Energy Technol. 8 (6), 2000053. 10.1002/ente.202000053 Wehinger G. D. Eppinger T. Kraume M. (2015). Evaluating catalytic fixed‐bed reactors for dry reforming of methane with detailed CFD. Chem. Ing. Tech. 87 (6), 734–745. 10.1002/cite.201400153 Wei L. Pan Z. Shi X. Esan O. C. Li G. Qi H. (2023). Solar-driven thermochemical conversion of H2O and CO2 into sustainable fuels. iScience 26 (11), 108127. 10.1016/j.isci.2023.108127 37876816 Welte M. Warren K. Scheffe J. R. Steinfeld A. (2017). Combined Ceria reduction and methane reforming in a solar-driven particle-transport reactor. Industrial Eng. Chem. Res. 56 (37), 10300–10308. 10.1021/acs.iecr.7b02738 28966440 White F. (2006). Viscous fluid flow. New York, NY: McGraw-Hill. Wild M. (2022). Design, modelling, experimental demonstration, and investigation of a gas-solid thermochemical heat storage reactor. Zürich, CH: ETH Zürich. Doctoral Thesis No. 28283. 10.3929/ethz-b-000548557 Yadav P. K. Das T. (2021). Study of the perovskite-type catalysts 40LaNi0.75Fe0.25-X MXO3/SiO2 (M=Ce, Zr) for the dry reforming of methane. React. Kinet. Mech. Catal. 132 (1), 279–300. 10.1007/s11144-021-01926-9 Zhao Z. Uddi M. Tsvetkov N. Yildiz B. Ghoniem A. F. (2016). Redox Kinetics study of fuel reduced ceria for chemical-looping water splitting. J. Phys. Chem. C 120 (30), 16271–16289. 10.1021/acs.jpcc.6b01847 Zheng Y. Li K. Wang H. Tian D. Wang Y. Zhu X. (2017). Designed oxygen carriers from macroporous LaFeO3 supported CeO2 for chemical-looping reforming of methane. Appl. Catal. B Environ. 202, 51–63. 10.1016/j.apcatb.2016.08.024 Zhou D. Zhang H. Yang S. (2022). A robust reacting flow solver with computational diagnostics based on OpenFOAM and cantera. Aerospace 9 (2), 102. 10.3390/aerospace9020102 Zinkevich M. Djurovic D. Aldinger F. (2006). Thermodynamic modelling of the cerium–oxygen system. Solid State Ionics 177 (11-12), 989–1001. 10.1016/j.ssi.2006.02.044 Zoller S. Koepf E. Roos P. Steinfeld A. (2019). Heat transfer model of a 50 kW solar receiver–reactor for thermochemical redox cycling using cerium dioxide. J. Sol. Energy Eng. 141 (2), 021014. 10.1115/1.4042059 Zoller S. Koepf E. Nizamian D. Stephan M. Patané A. Haueter P. (2022). A solar tower fuel plant for the thermochemical production of kerosene from H2O and CO2 . Joule 6, 1606–1616. 10.1016/j.joule.2022.06.012 35915707 Zuber M. (2024). Dry redox reforming with concentrated solar energy. Zürich, CH: ETH Zürich. Doctoral Thesis No. 30755. 10.3929/ethz-b-000710527 Zuber M. Patriarca M. Ackermann S. Furler P. Conceição R. Gonzalez-Aguilar J. (2023). Methane dry reforming via a ceria-based redox cycle in a concentrating solar tower. Sustain. Energy Fuels 7 (8), 1804–1817. 10.1039/d2se01726a Żuk P. J. Tużnik B. Rymarz T. Kwiatkowski K. Dudyński M. Galeazzo F. C. C. (2022). OpenFOAM solver for thermal and chemical conversion in porous media. Comput. Phys. Commun. 278. 10.1016/j.cpc.2022.108407 Glossary 2-D

two-dimensional

 3-D

three-dimensional

 CG

conjugate gradient

 GNU

GNU’s Not Unix!

 e,p

index of an educt species, product species

 ext

external

 fxb

fixed-bed

 H

denotes (index of) heterogeneous reaction

 HTF

heat transfer fluid

 IR

infrared

 i

inner

 i, j, k

index of a species, fluid species, solid species

 in

inlet

 ip

inflection point

 MFC

mass flow controller

 Mox, Mred

solid metal oxide in oxidized, reduced state

 morph

morphology

 Nu

Nusselt number [-]

 ODE

ordinary differential equation

 o

outer

 out

outlet

 ox

oxidation

 P&ID

piping and instrumentation diagram

 PDE

partial differential equation

 Pr

Prandtl number [-]

 rad

radiation

 RDA

Rosseland diffusion approximation

 Re

Reynolds number [-]

 RPC

reticulated porous ceramic

 RWGS

reverse water-gas shift

 red

reduction

 rxn

reaction

 [rxn]

reaction represented with gaseous O2

SEULEX

stiff linearly implicit Euler one-step iterative extrapolation

 SSA

specific surface area [m2·g−1]

 TC

thermocouple

 TG

thermogravimetric

 vent

ventilation

 A H

pre-exponential factor [s−1]

 a v

volume-specific surface area [m−1]

 c p

specific heat capacity [J·kg−1·K−1]

 D

Darcy viscous resistance tensor [m−2]

 D

diameter [m]

 D

mass diffusion coefficient [m2·s−1]

 D c

Darcy resistance coefficient [m−2]

 E a , H

apparent activation energy [J·mol−1]

 e r

relative error [-]

 F

Forchheimer inertial resistance tensor [kg·m−3·s−1]

 g

gravitational constant, 9.81 m s−2]

 h

specific enthalpy [J·kg−1]

 h conv

convection heat transfer coefficient [W·m−2·K−1]

 I

identity matrix

 K

anisotropic thermal conductivity tensor [W·m−1·K−1]

 K

permeability [m2]

 K eq

equilibrium constant in terms of mass fraction

 K ¯ eq

molar equilibrium constant

 k

thermal conductivity [W·m−1·K−1]

 k f , k r

forward, reverse rate coefficient [s−1]

 l

length [m]

 M i

molar mass [g·mol−1]

 M Ce O 2

molar mass of ceria, 172.115 [g·mol−1]

 M O

molar mass of elemental oxygen, 16 [g·mol−1]

 m i

mass [g]

 n

power function fitting parameter

 n i

moles [mol]

 n ˙ i

molar flow rate [mol·s−1]

 p

pressure [Pa]

 p i

partial pressure [Pa]

 R

universal gas constant, 8.314 J mol−1·K−1

R i

volumetric heterogeneous reaction rate [kg·m−3·s−1]

 S i

selectivity [-]

 T

temperature [°C][K]

 t

time [s]

 u

velocity vector [m·s−1]

 V

volume [m3]

 V ˙ i

volumetric flow rate [Ln·min−1]

 X i

conversion [-]

 X i

mol fraction [-]

 X r

molar ratio [-]

 x

axial coordinate [m]

 Y i

mass fraction [-]

 β R

extinction coefficient [m−1]

 Δ g ¯

molar Gibbs free energy of reaction [J·mol−1]

 Δ h ¯

molar enthalpy of reaction [J·mol−1]

 Δ s ¯

molar entropy of reaction [J·mol−1·K−1]

 Δ δ

change in non-stoichiometry [-]

 δ

non-stoichiometry [-]

 δ avg

average non-stoichiometry [-]

 δ ox

non-stoichiometry in oxidized state [-]

 δ red

non-stoichiometry in reduced state [-]

 ϵ

fixed-bed porosity [-]

 μ

dynamic viscosity [Pa·s]

 ν i

stoichiometric coefficient [-]

 ρ i

density [kg·m−3]

 σ

Stefan-Boltzmann constant, 5.67·10−8 W ·m−2·K−4

Ω H

heterogeneous reaction rate [kg·m−3·s−1]

 ω ˙

volumetric homogeneous reaction rate [kg·m−3·s−1]

 · 0

denotes nominal

 · a

denotes absolute

 · eq

denotes equilibrium

 · f

denotes of formation

 · g

denotes gauge

 · n

denotes normal temperature (273.15 K) and pressure (1 atm)

 · O

denotes partial molar values

 · p

denotes pellet

 · sp

denotes set point

 · th

denotes thermal

 · ∘

denotes standard reference state, 101,325 Pa

 · f

denotes the fluid phase

 · s

denotes the solid phase

 · ¯

denotes molar values

   ·  

denotes integral over control volume