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Design and Optimization of Hierarchically Ordered Porous Structures for Solar Thermochemical Fuel Production Using a Voxel-Based Monte Carlo Ray-Tracing Algorithm
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Design and Optimization of Hierarchically Ordered Porous Structures for Solar Thermochemical Fuel Production Using a Voxel-Based Monte Carlo Ray-Tracing Algorithm
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ACS Engineering Au

Cite this: ACS Eng. Au 2023, 3, 5, 326–334
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https://doi.org/10.1021/acsengineeringau.3c00013
Published September 13, 2023

Copyright © 2023 The Authors. Published by American Chemical Society. This publication is licensed under

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Abstract

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Porous structures can be favorably used in solar thermochemical reactors for the volumetric absorption of concentrated solar radiation. In contrast to isotropic porous topologies, hierarchically ordered porous topologies with stepwise optical thickness enable more homogeneous radiative absorption within the entire volume, leading to a higher and more uniform temperature distribution and, consequently, a higher solar fuel yield. However, their design and optimization require fast and accurate numerical tools for solving the radiative exchange at the pore level within their complex architectures. Here, we present a novel voxel-based Monte Carlo ray-tracing algorithm that discretizes the pore-level domain into a 3D binary digital representation of solid/void voxels. These are exposed to stochastic rays undergoing reflection, absorption, and re-emission at the ray-solid intersection found by querying the voxel value along the ray path. Temperature distributions are found at radiative equilibrium. The algorithm’s fast execution allows its use in a gradient-free optimization scheme. Three hierarchically ordered topologies with parametrized shapes (square grids, Voronoi cells, and sphere lattices) exposed to 1000 suns radiative flux are optimized for maximum solar fuel production based on the thermodynamics of a ceria-based thermochemical redox cycle for splitting H2O and CO2. The optimized graded-channeled structure with square grids achieves a 4-fold increase in the volume-specific fuel yield compared to the value obtained for an isotropic reticulated porous structure.

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1. Introduction

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Porous structures are of interest in concentrated solar applications. (1) They absorb the incident high-flux solar radiation volumetrically, i.e., sunrays are gradually absorbed as they penetrate and travel across the absorber’s volume, improving heat transfer rates and reducing thermal loads. The so-called volumetric effect is manifested when the hottest zone of the absorber is reached deeper within the structure instead of on the exposed face, further reducing reradiation losses to the surroundings. (1,2) However, the commonly applied isotropic porous structures with homogeneous porosity cause Bouguer’s law exponential attenuation of the incident radiation, which, in turn, leads to undesired temperature gradients along the radiation path. Especially in solar thermochemical reactors for solar fuel production, significant portions of the reacting material cannot reach the reaction temperature, detrimentally affecting the fuel yield and the solar-to-fuel energy efficiency. (3) In principle, higher porosity enhances radiative penetration, but this occurs at the expense of the structure’s effective density ρeff, which again lowers the fuel yield. In contrast to isotropic structures, hierarchically ordered structures have shown great promise to circumvent this problem and can achieve a more uniform heating of the whole volume while maintaining a relatively high ρeff. (2,4,5) Yet, fast and accurate pore-level numerical tools are required to optimize their porous topology for maximizing the production of solar fuel.
Monte Carlo (MC) ray tracing is an appropriate technique for solving the equation of radiative heat transfer within absorbing–emitting–scattering media in complex porous geometries. Typically, the intersection between a solid surface and a generic ray r⃗ is determined by solving a 3D vector equation. However, the computational cost becomes prohibitively high when the goal is to simulate complex porous geometries at the pore level due to the significant number of surfaces required to describe them. Previous modeling studies of hierarchically ordered and multilayered solar volumetric absorbers were limited to the optical analysis of radiation propagation (6,7) but did not solve the equation of radiative transfer, and even that limited scope was computationally costly. (6) To some extent, the computational cost can be alleviated by extracting the effective (homogenized) transport properties from the smallest representative volume for use in volume-averaged simulations of porous media, (8−13) but this approach obviously cannot be applied for pore-level optimization. In contrast, voxel discretization offers a very fast computational alternative. In this work, we present a novel voxel-based MC algorithm for pore-level radiative transfer modeling that can be applied for designing and optimizing complex porous structures, especially those featuring anisotropic graded topologies. We verify this numerical technique by applying it to various geometries exposed to irradiation and comparing its results to those obtained by the analytical radiosity method for a cavity, the exact analytical solution for overlapping spheres, and a standard MC ray tracer for a reticulated porous structure. We further compare the numerical results to experimental measurements performed on a graded-channeled structure exposed to concentrated irradiation. The MC numerical technique is applied to optimize three porous topologies made of ceria, exposed to 1000 suns radiative flux, to maximize the production of solar fuels. As shown in the analysis, the potential for improving the thermal performance of solar reactors is substantial.

2. Voxel-Based MC Ray-Tracing Algorithm

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The voxel-based MC ray-tracing algorithm solves the pore-level radiative heat transfer equation within an absorbing–emitting–scattering porous domain. The methodology consists of following the stochastic paths of a large number of rays carrying the same amount of energy (i.e., equivalent to an energy bundle) as they travel through the interacting porous domain by considering absorption, emission, and reflection at the pores’ inner surfaces. (14) The assumptions are (i) geometric optics regime; (ii) void phase is transparent; and (iii) inner surfaces are opaque-gray-diffuse. The simulation model can be easily extended to treat semitransparent or nongray surfaces and participating gases filling the voids with directional and spectrally dependent optical properties by incorporating the appropriate probability density functions. The porous domain is discretized into a 3D binary digital representation of solid/void voxels. Within a single voxel, all properties are assumed to be constant. Rays are traced and intersected by considering only the transited voxels until a solid one is reached (found by querying the voxel binary value), which is computationally faster than calculating ray-surface intersections in structures with relatively short free path lengths. Similar ray tracing approaches have been applied to increase the speed of computer graphics scene rendering, (15,16) extract effective optical properties of porous structures, (8,12) and calculate radiative exchange factors. (17) To the best of our knowledge, its application to directly solve the radiative exchange is novel.
A voxel traversing algorithm determines which voxels, in order, are traversed by the generic ray. (18) Six-connected voxels ensure that neighboring voxels share a common face and that a ray cannot travel from one voxel to another through an edge or vertex. In addition, solid or void voxels surrounded only by solid voxels are not exposed to rays and, therefore, do not participate in the energy balance to calculate temperatures in radiative equilibrium. Once a traversing ray reaches a solid voxel, it can undergo either reflection or absorption depending on the pore’s inner surface absorptivity α (based on a collision-based MC approach). If absorbed, another ray carrying the same energy is emitted from the same voxel to ensure radiative equilibrium without accumulation, source, or energy sink. Thus, every voxel absorbs precisely as much radiative energy as it re-emits. Both re-emitted and reflected radiation is assumed to be diffuse, simulated by so-called secondary rays. Their direction is obtained from the appropriate probability density function according to Lambert’s cosine law, for which the normal vector n̂ at the intersected solid voxel is needed. n̂ is found using the information on the neighboring voxels, without considering their faces’ orientation, effectively smoothing out the jagged surface created by voxel discretization. (19) Self-intersection is avoided in flat surfaces by relaunching any secondary ray with an incident angle >90°. The history of each ray entering the domain is then a complete random sequence of reflections, absorptions, and re-emissions, which terminates when it escapes the domain. This procedure is repeated for a large number of rays to obtain statistically meaningful results. For each voxel, we keep counters of rays entering the domain, absorbed and re-emitted by the walls, and escaping the domain, and verify energy conservation. The energy balance on each voxel i yields its equilibrium temperature, Ti=(nabs.PrayασA)0.25, where nabs is the number of rays absorbed, Pray is the radiative power carried by a single ray, A is the surface area of the solid voxel’s faces that are exposed to void voxels, and σ is the Stefan–Boltzmann constant. The calculation is performed on voxel neighborhoods to reduce the statistical error such that the neighborhood size determines the resolution of the temperature distribution.
Verification – Three tests were performed to verify the new voxel-based MC model:
(i)

Application of the voxel-based MC ray tracer to a cavity subjected to collimated irradiation and comparison of the temperature distribution to the values obtained by the analytical radiosity (enclosure theory) method.

(ii)

Application of the voxel-based MC ray tracer and a standard MC ray tracer to an Identical Overlapping Opaque Spheres (IOOS) structure subjected to collimated irradiation and comparison of: (a) the attenuation of the radiation intensity to values obtained by the exact analytical solution and (b) the temperature distribution, obtained at radiative equilibrium.

(iii)

Application of the voxel-based MC ray tracer and the standard MC ray tracer to a reticulated porous ceramic (RPC) structure for determining the attenuation of the radiation intensity and comparison with previously published results.

The first test consists of applying the voxel-based MC ray tracer to a cavity exposed to collimated irradiation. The problem is extracted from ref (20). The domain is depicted in Figure 1. It consists of a cavity formed by two surfaces: a spherical cavity of radius R2 (surface 2) containing a top circular opening of radius R1 and merged to a bottom hemispherical cavity of radius R1 (surface 1). All surfaces are assumed to be perfectly insulated and black. One-sun (1 kW/m2) collimated radiative flux is uniformly distributed and normally incident through the top opening. The voxel-based MC ray tracer is applied to compute the equilibrium temperatures of surfaces 1 and 2 for different voxel sizes. The numerical values are compared to those obtained by the analytical solution using the radiosity (enclosure theory) method. As the voxel resolution increases, the numerical temperature values quickly converge to those obtained by the analytical solution (see Table 1). Differences are due to the discretization of smooth surfaces into jagged voxels, and as expected, these differences decrease for smaller voxel sizes. Also note that when the incident radiation on surfaces is not uniformly distributed, the analytical radiosity method does not give an exact solution. We have also solved for the case of diffusely distributed flux incident through the top opening, yielding a 4% difference between the voxel-based MC ray tracer (voxel size = 0.25 mm) and the analytical radiosity method (T1 = T2 = 364.4 K).

Figure 1

Figure 1. (a) Schematic of the domain, consisting of a spherical cavity of radius R2 (surface 2) containing a top circular opening of radius R1, and merged to a bottom hemispherical cavity of radius R1 (surface 1). (b) Temperature distribution obtained by the voxel-based MC ray tracer. Boundary condition: 1 sun collimated radiative flux, uniformly distributed, and normally incident on the top opening. Problem extracted from ref (20).

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Table 1. Temperature of Surfaces 1 and 2 Obtained by the Analytical Radiosity Method and Comparison to Values Obtained by the Voxel-Based MC Ray Tracer for Different Voxel Sizesa
voxel sizeanalytical solution [K]voxel-based MC ray tracer [K, and % difference]
4 mm2 mm1 mm0.67 mm0.5 mm0.25 mm
surface 1429.0323.7 (−25%)344.5 (−20%)400.7 (−7%)411.8 (−4%)414.0 (−4%)411.0 (4%)
surface 2360.7307.4 (−15%)334.7 (−7%)345.5 (−4%)348.5 (−3%)366.7 (−2%)361.8 (−0%)
a

Domain and boundary conditions are defined in Figure 1.

A second test consists of applying the voxel-based MC ray tracer and a standard MC ray tracer (21) for simulating radiative equilibrium in an IOOS structure exposed to 1 sun (1 kW/m2) collimated radiation. For this topology, an exact analytical solution of the attenuation of the radiation intensity I/I0 is available. (22) The domain is defined as a 500 × 500 × 500 voxel cuboidal containing 350 spheres of radius R = 31 voxels in the voxel-based ray tracer. The domain’s lateral faces are assumed to be prefectly specularly reflective; top and bottom faces are assumed open to nonparticipating surroundings; spheres are assumed black (α = 1). For the standard MC ray tracer, the analogous problem was defined using a 1 cm/voxel resolution. A total of 105 rays are launched for both MC ray tracers, uniformly distributed and normally incident on the domain’s top face. Figure 2 shows (a) the 3D digital representation of the IOOS structure; (b) radiation intensity I/I0 as a function of the ray path length S obtained by the exact analytic solution and by both MC ray tracers; and (c) the temperature profiles along the z-direction in radiative equilibrium for both MC ray tracers; plotted is the mean temperature of voxels at a given z. The inset in Figure 2c shows the 3D temperature field for the voxel-based MC ray tracer.The maximum temperature is 357 K, while the stagnation temperature of a blackbody exposed normally to 1 sun radiation is 364 K. The root mean square errors (RMSEs) in the computed attenuation by the voxel-based and standard MC ray tracers compared to the exact analytical solution are 0.03 (0.44%) and 0.03 (0.47%), respectively. The RMS deviation (RMSD) between the computed temperature profile by both MC ray tracers is 18.8 K (4.4%), mainly due to the error calculation of normal vectors to surfaces. There is no analytical solution for the temperature distribution.

Figure 2

Figure 2. (a) 3D digital representation of the IOOS porous structure, defined by a 500 × 500 × 500 cm cuboidal domain containing 350 spheres of radius R = 31 cm. (b) Radiation intensity I/I0 as a function of the ray path length S obtained by the exact analytical solution (22) and by both MC ray tracers. (c) Temperature profiles along the z-direction in radiative equilibrium for both MC ray tracers. The inset shows the 3D temperature field obtained by the voxel-based MC ray tracer. Boundary condition: 1 sun collimated radiative flux, uniformly distributed, and normally incident on the top face of the domain.

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A third test consists of applying both MC ray tracers, the voxel-based and the standard one, (21) for determining the attenuation of radiation across a 10-pores-per-inch (ppi) RPC structure. Its exact 3D digital representation was obtained by computer tomography. (8,10) The same boundary conditions are applied as in the second test. Uniformly distributed and normally incident 105 rays are launched toward the top face of the cuboidal domain (1.75 × 1.75 × 1.75 cm, α = 1), which has been meshed for the standard MC ray tracer using 200,000 faces (237 μm mean edge length) and voxelized for the voxel-based MC ray tracer using 204,830 voxels (239 μm voxel-face diagonal). Figure 3a,b shows a photograph of the RPC sample, made of ceria, and its 3D digital representation of the voxelized domain. Figure 3c plots the attenuation of the radiation intensity I/I0 as a function of the ray path length S obtained by the voxel-based and the standard MC ray tracers. The RMSD between the two sets of data points is 0.034 (0.45%). In terms of computational speed, the standard MC and voxel-based MC ray tracers take 12.4 min and 8.4 s, respectively (∼90× faster), without parallelization. Included in Figure 3c are the results by Ackermann et al. (10) and by using the empirical correlation of Hendricks and Howell. (23) The extinction coefficient β is obtained from the cumulative density function G(S), of the ray path length S (according to Bouguer’s law, I/I0 = e–βS = 1 – G(S)). (22) The computed value is 348 m–1, matching the value previously reported. (10)

Figure 3

Figure 3. (a) Photograph of the 10-ppi RPC sample. (b) 3D digital representation of the voxelized domain. (c) Radiation intensity I/I0 as a function of the ray path length S obtained by the voxel-based MC ray tracer, by the standard MC ray tracer, by Ackermann et al., (10) and by using the empirical correlation of Hendricks and Howell. (23)

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2.1. Experimental Comparison

The voxel-based MC ray tracer is used to simulate a graded-channeled structure of square grids exposed to concentrated irradiation. The numerically computed temperatures under radiative equilibrium are compared to those obtained by experimental measurements. The domain is defined as a 25 × 25 × 40 mm cuboid, with top and bottom faces open to nonparticipating surroundings, perfectly and diffusely reflective lateral faces, and surfaces’ absorptivity ∝ = 0.7. The source of radiation is the ETH’s high-flux solar simulator, schematically shown in Figure 4a: a Xe-arc coupled with a truncated ellipsoidal specular reflector, focusing on a 45° mirror which re-directs the concentrated radiative beam toward the top face of the domain and incident at a mean radiative flux of 1000 suns (1290 suns peak). This optical configuration represents the experimental setup used to study porous structures for application in solar reactors, (5) which mimics the radiation characteristics of solar concentrating systems. The standard MC ray tracer (21) simulates the high-flux solar simulator launching a total of 107 rays and delivers the directional distribution of incident radiation (Figure 4a), which sets the top boundary condition on the domain. Figure 4b,c shows a photograph of the graded-channeled sample, made of ceria, and its 3D digital representation of the voxelized domain. Figure 4d shows the temperature profile along the z-direction under radiative equilibrium, obtained by the voxel-based MC ray tracer and by the experimental measurements. (5) z = 0 and z = 40 mm represent the bottom and top faces, respectively. The curve corresponds to the mean temperature of voxels at a given z. The data points correspond to the temperature measurements performed at 2.4, 11.8, and 22 mm from the illuminated surface (top), with indicated error bars due to the uncertainty in the location of thermocouples. The RMSE is 196 K (10.4%). Note that convection and conduction heat losses to the surroundings occurring in the experimental setup (5) were not considered in the MC simulation and resulted in higher computed equilibrium temperatures. The anomalies in the curve at z = 20 and 30 mm are due to the stepwise change of ρeff. The measured temperature profile of the graded structure was higher and more uniform than that of an isotropic RPC, tested under the same radiative flux (5) (not shown in Figure 4d).

Figure 4

Figure 4. (a) Scheme of the experimental setup at the ETH’s high-flux solar simulator: a Xe-arc coupled with a truncated ellipsoidal reflector (in blue) and a 45° mirror (in gray), incident on the top face of the cuboidal domain. (b) Photograph of the graded-channeled sample. (c) 3D digital representation of the voxelized domain. (d) Temperature profile along the z-direction under radiative equilibrium, obtained by the voxel-based MC ray tracer and by the experimental measurements. (5) z = 40 mm is the top face exposed to 1000 suns radiative flux delivered by the ETH’s high-flux solar simulator.

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3. Optimization of Hierarchically Ordered Structures

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Thanks to its fast execution speed, the voxel-based MC ray tracer can be exploited for the optimization of complex porous topologies. The numerical algorithm is parallelized in eight cores/workers and coupled with a gradient-free optimization subroutine (MATLAB’s “Surrogate Optimization”) which allows the optimization of non-differentiable and expensive functions. (24) Surrogate optimization is incorporated to optimize the geometries of the porous structures, which are defined as parametrized 3D shapes to reduce the number of decision variables related to their topology. Three parametrized shapes are selected: (i) square grids, (ii) Voronoi cells, and (iii) sphere lattices. The parametrized square grids are defined by 5 layers and 10 decision variables, the height, and the pattern (number of squares) of each layer. This leads to the type of graded-channeled topology shown in Figure 4c. The parametrized Voronoi cells are defined by 11 parameters (also using five layers), the height of each layer, the number of cells of each layer, and the radius of the Voronoi edges. The parametrized sphere lattices are defined by 25 different parameters, the height of each layer, the radius of the spheres forming the lattice in each layer, the spacing between the spheres in each layer, the lattice configuration, and a binary parameter (for each layer) describing whether the spheres are the solid or the void phase (one being the negative image of the other). A porous media generator (25) was adapted to create the Voronoi cells and the sphere lattices.
The optimization algorithm starts by generating 3D architectures with random parameters. It then proceeds to apply the voxel-based MC ray tracer to obtain a set of values of the objective function (one value for each architecture), which is used to construct a surrogate function by interpolating these values. This surrogate function is then minimized using MATLAB’s subroutine “Surrogate Optimization” to yield new parameters for creating better-performing 3D architectures. These improved architectures are in turn re-evaluated by the MC ray tracer to progressively improve the surrogate function fitting and therefore approach the minimum value of the objective function. This procedure is repeated up to a given maximum number of iterations in the range of 8000–25,000, depending on the geometry and number of parameters. The final output of the surrogate optimization is the set of parameters used to create the 3D binary digital representation of the solid/void voxels containing the optimized topology. This approach has been proven to converge to a global solution for continuous objective functions on bounded domains. (24)
The optimization algorithm is applied to porous structures that are used in solar reactors for the thermochemical production of solar fuels. (26) Specifically, we consider the co-splitting of H2O and CO2 via a thermochemical redox cycle for producing syngas─a mixture of H2 and CO that serves as the precursor for synthesizing drop-in fuels for transportation. The two-step redox cycle using ceria as the redox material is represented by
Reduction:CeO2→CeO2−δ+12δO2
(1)
Oxidation:CeO2−δ+δCO2→CeO2+δCO
(2a)
⁣CeO2−δ+δH2O→CeO2+δH2
(2b)
where δ denotes the nonstoichiometry─a measure of the oxygen exchange capacity and, therefore, of the fuel yield per cycle. The reduction, eq 1, is the endothermic step (ΔH° ≈ 475 kJ per 12 mole O2) that uses concentrated solar radiation as the source of high-temperature process heat. During the reduction step performed at temperatures above 1500 K and total pressures below 10 mbar, radiation is the principal mode of heat transfer while conduction and convection within the porous ceramic structure are relatively negligible, i.e., a pure radiative transfer analysis should be satisfactory for design and optimization purposes. This has been corroborated by previous CFD modeling simulations using correlations of the effective thermal conductivity and interfacial convective heat transfer coefficient of porous ceria structures for the relevant ranges of Reynolds and Prandtl numbers and validated versus experimental results. (4,27) Further justification on the dominance of radiative heat transfer for applications involving porous ceramics operating at high temperatures and vacuum pressures is given by a coupled conduction–radiation analysis for open-cell SiC cellular ceramics with porosities in the range 0.8–0.9 and at temperatures in the range 1200–1800 K. (28) Note that our structures have comparable porosities and that the thermal conductivity of solid SiC is about 50 times higher than that of solid CeO2 in the relevant temperature range. The Stark number N, also named the conduction–radiation parameter (29) N=keffβ4σTref3 (where keff is the effective thermal conductivity and β is the extinction coefficient, both obtained from ref (10) for the RPC) is in the range of N = 0.06–0.1, further supporting this assumption.
The complete process chain from H2O and CO2 to methanol and kerosene was demonstrated in a concentrating solar dish (30) and solar tower. (3) In these experimental demonstrations, the solar reactor consisted of a cavity-receiver containing an isotropic RPC structure, made of ceria, directly exposed to the incident concentrated solar radiation. However, because of the exponential-decaying attenuation of the incident radiation (see Figure 3c), about half of the RPC inside the solar cavity receiver did not reach the reduction temperature and, thus, did not contribute to fuel production. Thus, the goal of the optimization is to maximize the solar fuel yield by designing a hierarchically ordered porous structure with high ρeff (i.e., high ceria mass loading inside the cavity), which absorbs the incident solar radiation volumetrically to ensure uniform heating of the entire volume to the reaction temperature.

3.1. Objective Function

The objective function is defined as the amount of ceria mass reaching a targeted temperature, set at 1600 K. A penalization is added for mass heated to below a lower threshold temperature, set at 1000 K, because such unreacted mass does not contribute to fuel production. The optimum topologies were found after 155, 2366, and 12,000 iterations for the square grid, the Voronoi cells, and the sphere lattices, respectively, but 8000, 8000, and 25,000 objective function evaluations were required to ensure convergence to the global optimum, respectively. In addition to the three optimized topologies, we consider the 10-ppi RPC described in the third test (Figure 3) as the reference design because it is the state-of-the-art structure currently applied in solar reactors. (3,30) Its 3D digital geometry was obtained by computer tomography. (8,10)

3.2. Domain and Boundary Conditions

The domain is cuboidal with a 25 × 25 mm square section; the height H is a decision variable of the optimizer. The boundary conditions for all four topologies are the same as the ones applied in the experimental comparison (Figure 4): perfectly and diffusely reflective lateral faces, top and bottom faces are open to nonparticipating surroundings, and the top face is exposed to 1000 suns radiative flux with a directional distribution given by the ETH’s high-flux solar simulator (Figure 4a). The void phase is assumed to be transparent, justified by the vacuum gas pressure during the reduction step. For simplification, the surfaces are assumed opaque-gray-diffuse with α = 0.7 (for reduced ceria with δ = 0.04 (31)), but nongray surfaces can be treated by incorporating the spectral distribution of solar radiation (approximated by Planck’s blackbody emissive power at 5780 K) and the spectral hemispherical reflectivity of ceria. (31) In addition to the aforementioned boundary conditions, the value of H for the RPC’s domain was initially set to H = 35 mm, corresponding to the RPC’s thickness used in the solar reactor tested in the solar tower. (3) However, for a fair comparison of the RPC with the graded topologies in terms of the volume-specific fuel yield, the bottom region of the RPC having temperatures below 1000 K was cut out because such a region does not contribute to fuel generation.

4. Results and Discussion

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Figure 5 shows the attenuation of incident radiation (I/I0) and the temperature distribution under radiative equilibrium as a function of the z-direction perpendicular to the front face for the four topologies: (a) RPC, (b) optimized square grids, (c) optimized Voronoi cells, and (d) optimized sphere lattices. Plotted is the mean temperature of voxels at a given z; z = 0 and z = H represent the bottom and top faces, respectively. Included are the 3D digital representations of the voxelized topologies and their corresponding 3D temperature fields. The geometrical parameters of these optimized topologies are given in Table 2. The convergence of the voxel-based MC model was verified for the square grids by comparing the temperature profiles obtained with 107 and 108 rays, resulting in an average temperature difference of 0.6% over the whole domain (z = 0–34.75 mm, see Table 2).

Figure 5

Figure 5. Attenuation of incident radiation (I/I0) and temperature distribution as a function of the z-direction perpendicular to the front face (z = 0 and z = H represent the bottom and top faces, respectively) for the four topologies: (a) 10-ppi RPC, (b) optimized square grids, (c) optimized Voronoi cells, and (d) optimized sphere lattices. Included are the 3D representations of the voxelized topologies and their corresponding 3D temperature fields. Boundary conditions: perfectly and diffusely reflective lateral faces, top and bottom faces are open to nonparticipating surroundings, and the top face exposed to 1000 suns radiative flux with a directional distribution given by the ETH’s high-flux solar simulator (Figure 4a). The surfaces are opaque-gray-diffuse with α = 0.7.

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Table 2. Geometrical Parameters of the Optimized Topologies of Figure 5a
square grids (voxel size: 0.25 mm, wall thickness: 4 voxels)Voronoi cells (voxel size: 0.25 mm, edge radius: 3 voxels)
layer n°height [voxel]# of squareslayer n°height [voxel]# of cells
151691233
254842245
3662563242
422164246
5411657647
sphere lattices (voxel size: 0.1 mm)
layer n°height [voxel]radius [voxel]center-to-center spacing [voxel]solid (0) or void (1) sphereslattice type
148351face-centered cubic
2855221face-centered cubic
30    
40    
513535351simple cubic
a

Layers are numbered starting from the bottom (z = 0).

In contrast to the RPC, which displays an exponential attenuation of incident radiation, the hierarchically ordered structures exhibit a stepwise attenuation as a result of the stepwise optical thickness, circumventing Bouguer’s laws. This, in turn, leads to a deeper radiation penetration depth and, consequently, to a more uniform temperature profile despite the higher effective density ρeff. Furthermore, due to the volumetric effect, the hottest area is displaced deeper into the structure, thus reducing reradiation losses to the surroundings. This effect can be clearly observed for the square grids, which display a peak temperature of 2330 K at z = 24.7 mm. As expected, the irregularities in the temperature profiles coincide with the stepwise changes of the radiative attenuation. In general, the magnitude of these irregularities strongly depends on the topology’s anisotropy and on the directional distribution of the radiative flux input, which, for this specific simulation is given by the ETH’s high-flux solar simulator (Figure 4a). For the sphere lattices, the highs and lows of the temperature profile coincide with the highs and lows of solid voxels, and a dashed line is used to interpolate the temperature values in the gap created by the optimizer between z = 0.4 and z = 1.2 mm. Note that re-emitted radiation also significantly affects the temperature distributions and contributes to smoothing out the profiles. The difference between the temperature distributions among the three graded topologies is due in part to the total surface area directly exposed to the incoming radiative flux as provided notably by the optimized square grids, and to a lesser extent by the sphere lattices and by the Voronoi cells. As expected, the isotropic RPC exhibits a significant temperature gradient, but stacking RPCs of different porosities can only achieve marginal improvements. (32) Note that all topologies exhibit interconnected (open cell) porosity and are open at the top and the bottom. These openings are essential to enable the flow of reacting gases H2O and CO2 across the structure during the oxidation step of the redox cycle.
Table 3 summarizes the results, listing the values of the height (H) and the corresponding mass of the ceria (m), the percentage of mass reaching temperatures in the desired range of 1000–1600 K, below 1000 K, and above 1600 K. As anticipated, the optimized hierarchically ordered topologies achieve higher masses in the desired range than the RPC; ranked first is the optimized square grids, second the optimized sphere lattices, third the optimized Voronoi cells, and fourth (last) the RPC. In addition, Table 3 lists the corresponding volume-specific fuel yield (H2 or CO). It is calculated based on the values of the nonstoichiometry δ for each solid voxel of the domain, determined at thermodynamic equilibrium at the voxel’s temperature Ti and at a baseline oxygen partial pressure of 0.1 mbar. (10) According to eq 2a/2b, δ directly gives the number of moles of fuel (H2 or CO) per mole of ceria. The summation of δ for all solid voxels, weighted by the mass of ceria contained in each of these voxels and finally normalized by the domain’s volume, gives the volume-specific fuel yield (noted as Y). Note that chemical thermodynamic equilibrium can be assumed because the reaction kinetics are faster than the heating rates during the reduction step. (27) Therefore, the temperature distribution at the end of the heating phase, which corresponds to the end of the reduction step, determines the total O2 evolution during the reduction (eq 1) and, consequently, the total fuel yield during oxidation (eq 2a/2b). This is true because total selectivity is obtained for the splitting of H2O and CO2 (i.e., H2O splits into H2 and 12 O2, and CO2 splits into CO + 12 O2) with no side-reaction or by-products, as demonstrated experimentally. (3,30) Furthermore, this also justifies the intrinsic steady-state assumption of the MC model, since the transient behavior during heating is not required for determining the total fuel yield. The graded-channeled topology with square grids achieves Y = 1.01 mmol CO or H2/cm3, which corresponds to a remarkable increase in Y by a factor of 4.4 compared to the value obtained for an isotropic RPC with H = 20 mm (Table 3). When compared to the RPC with H = 35 mm used in the solar reactor tested in the solar tower, (3) the graded-channeled topology delivers an increase in Y by a factor of 9.
Table 3. Summary of the Optimization Results for the Reference RPC and the 3 Optimized Topologies: Height (H) of Cuboidal Domain, Corresponding Mass of Ceria (m), Percentage of Mass Reaching the Desired Range 1000–1600 K, below 1000 K, and above 1600 K, and the Corresponding Volume-Specific Fuel Yield (Y)
structureheight H [cm]/volume [cm3]ceria mass, m [g]% mass 1000–1600 K% mass < 1000 K% mass > 1600 Kvolume-specific fuel yield Y [mmol CO or H2/cm3]
RPC2.0/12.516.541%14%44%0.23
optimized square grid3.5/21.861.824%17%59%1.01
optimized Voronoi cells2.1/13.122.837%24%38%0.26
optimized sphere lattices1.5/9.418.722%34%45%0.62

5. Summary and Conclusions

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A novel voxel-based MC ray tracing algorithm has been developed to numerically simulate the radiative exchange within complex porous structures at the pore level. Verification was accomplished by comparing the results to those obtained by the analytical radiosity method for a cavity, the exact analytical solution for overlapping spheres, and a standard MC ray tracer for a reticulated porous structure. The numerical results were further compared to experimental measurements performed for a graded-channeled structure exposed to concentrated irradiation. Its fast execution allowed its application for the optimization of three hierarchically ordered topologies (namely, square grids, Voronoi cells, and sphere lattices) for maximum solar fuel production by maximizing the mass of redox material (ceria) reaching a targeted reaction temperature, set at 1600 K. The optimized topologies enabled volumetric absorption of concentrated solar radiation over the entire volume, leading to a more uniform distribution of temperatures under radiative equilibrium and displacing the hottest area deeper into the structure, thus reducing re-radiation losses to the surroundings. Based on the thermodynamics of the ceria-based thermochemical redox cycle for splitting H2O and CO2, the graded-channeled topology with square grids achieved an increase in the volume-specific fuel yield by a factor of 4.4 compared to that of the current state-of-the-art RPC. The optimized structures developed in the present study can be readily implemented in the solar reactor to boost its fuel yield and, consequently, its solar-to-fuel energy conversion efficiency. In general, the voxel-based MC algorithm can be applied for optimizing designs of porous structures used in high-temperature applications.

Author Information

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  • Corresponding Author
  • Author
  • Funding

    This work was funded by the Swiss Federal Office of Energy (Grant No. SI/502552-01).

  • Notes
    The authors declare no competing financial interest.

Abbreviations

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3D

three dimensional

MC

Monte Carlo

IOOS

identical overlapping opaque spheres

ppi

pores per inch

RMSE

root mean square error

RMSD

root mean square deviation

RPC

reticulated porous ceramic

References

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    Nature (London, United Kingdom) (2022), 601 (7891), 63-68CODEN: NATUAS; ISSN:1476-4687. (Nature Portfolio)
    Aviation and shipping currently contribute approx. 8% of total anthropogenic CO2 emissions, with growth in tourism and global trade projected to increase this contribution further1-3. Carbon-neutral transportation is feasible with elec. motors powered by rechargeable batteries, but is challenging, if not impossible, for long-haul com. travel, particularly air travel4. A promising soln. are drop-in fuels (synthetic alternatives for petroleum-derived liq. hydrocarbon fuels such as kerosene, gasoline or diesel) made from H2O and CO2 by solar-driven processes5-7. Among the many possible approaches, the thermochem. path using concd. solar radiation as the source of high-temp. process heat offers potentially high prodn. rates and efficiencies8, and can deliver truly carbon-neutral fuels if the required CO2 is obtained directly from atm. air9. If H2O is also extd. from air10, feedstock sourcing and fuel prodn. can be colocated in desert regions with high solar irradn. and limited access to water resources. While individual steps of such a scheme have been implemented, here we demonstrate the operation of the entire thermochem. solar fuel prodn. chain, from H2O and CO2 captured directly from ambient air to the synthesis of drop-in transportation fuels (for example, methanol and kerosene), with a modular 5 kWthermal pilot-scale solar system operated under field conditions. We further identify the research and development efforts and discuss the economic viability and policies required to bring these solar fuels to market.
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  2. Da Xu, Lei Zhao, Meng Lin. Optimization of porous structures via machine learning for solar thermochemical fuel production. Progress in Natural Science: Materials International 2024, 34 (5) , 895-906. https://doi.org/10.1016/j.pnsc.2024.07.024
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  4. Justin T. Tran, Kent J. Warren, Steven A. Wilson, Christopher L. Muhich, Charles B. Musgrave, Alan W. Weimer. An updated review and perspective on efficient hydrogen generation via solar thermal water splitting. WIREs Energy and Environment 2024, 13 (4) https://doi.org/10.1002/wene.528
  5. Maha A. Alshubramy, Khalid A. Alamry, Mahmoud A. Hussein, Hajar S. Alorfi, Mohamed Reda Aouad, Nadjet Rezki, Abeer M. Alosaimi. Effective synergy of ionic capacity and C3-symmetry porosity in porous ionic polymers based imidazolium/pyridinium-benzene and/or triazine molecular hybrids: A comprehensive review. Results in Chemistry 2024, 9 , 101673. https://doi.org/10.1016/j.rechem.2024.101673
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  • Abstract

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    Figure 1

    Figure 1. (a) Schematic of the domain, consisting of a spherical cavity of radius R2 (surface 2) containing a top circular opening of radius R1, and merged to a bottom hemispherical cavity of radius R1 (surface 1). (b) Temperature distribution obtained by the voxel-based MC ray tracer. Boundary condition: 1 sun collimated radiative flux, uniformly distributed, and normally incident on the top opening. Problem extracted from ref (20).

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    Figure 2

    Figure 2. (a) 3D digital representation of the IOOS porous structure, defined by a 500 × 500 × 500 cm cuboidal domain containing 350 spheres of radius R = 31 cm. (b) Radiation intensity I/I0 as a function of the ray path length S obtained by the exact analytical solution (22) and by both MC ray tracers. (c) Temperature profiles along the z-direction in radiative equilibrium for both MC ray tracers. The inset shows the 3D temperature field obtained by the voxel-based MC ray tracer. Boundary condition: 1 sun collimated radiative flux, uniformly distributed, and normally incident on the top face of the domain.

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    Figure 3

    Figure 3. (a) Photograph of the 10-ppi RPC sample. (b) 3D digital representation of the voxelized domain. (c) Radiation intensity I/I0 as a function of the ray path length S obtained by the voxel-based MC ray tracer, by the standard MC ray tracer, by Ackermann et al., (10) and by using the empirical correlation of Hendricks and Howell. (23)

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    Figure 4

    Figure 4. (a) Scheme of the experimental setup at the ETH’s high-flux solar simulator: a Xe-arc coupled with a truncated ellipsoidal reflector (in blue) and a 45° mirror (in gray), incident on the top face of the cuboidal domain. (b) Photograph of the graded-channeled sample. (c) 3D digital representation of the voxelized domain. (d) Temperature profile along the z-direction under radiative equilibrium, obtained by the voxel-based MC ray tracer and by the experimental measurements. (5) z = 40 mm is the top face exposed to 1000 suns radiative flux delivered by the ETH’s high-flux solar simulator.

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    Figure 5

    Figure 5. Attenuation of incident radiation (I/I0) and temperature distribution as a function of the z-direction perpendicular to the front face (z = 0 and z = H represent the bottom and top faces, respectively) for the four topologies: (a) 10-ppi RPC, (b) optimized square grids, (c) optimized Voronoi cells, and (d) optimized sphere lattices. Included are the 3D representations of the voxelized topologies and their corresponding 3D temperature fields. Boundary conditions: perfectly and diffusely reflective lateral faces, top and bottom faces are open to nonparticipating surroundings, and the top face exposed to 1000 suns radiative flux with a directional distribution given by the ETH’s high-flux solar simulator (Figure 4a). The surfaces are opaque-gray-diffuse with α = 0.7.

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      Concentrating solar thermal power and thermochemical fuels
      Romero, Manuel; Steinfeld, Aldo
      Energy & Environmental Science (2012), 5 (11), 9234-9245CODEN: EESNBY; ISSN:1754-5706. (Royal Society of Chemistry)
      A review. Concd. solar energy provides a virtually unlimited source of clean, non-polluting, high-temp. heat. This article reviews the underlying principles of concg. solar radiation and describes the latest technol. advances and future prospects of solar thermal power and thermochem. fuel prodn.
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    27. 27
      Furler, P.; Steinfeld, A. Heat transfer and fluid flow analysis of a 4kW solar thermochemical reactor for ceria redox cycling. Chem. Eng. Sci. 2015, 137, 373– 383,  DOI: 10.1016/j.ces.2015.05.056
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      Heat transfer and fluid flow analysis of a 4 kW solar thermochemical reactor for ceria redox cycling
      Furler, Philipp; Steinfeld, Aldo
      Chemical Engineering Science (2015), 137 (), 373-383CODEN: CESCAC; ISSN:0009-2509. (Elsevier Ltd.)
      A solar reactor consisting of a cavity-receiver contg. a reticulated porous ceramic (RPC) foam made of CeO2 is considered for affecting the splitting of H2O and CO2 via a thermochem. redox cycle. A transient 3D heat and mass transfer model of the redn. step is formulated and solved using Monte-Carlo ray-tracing coupled to computational fluid dynamics. Exptl. validation is accomplished in terms of measured temps. and O2 evolution rates obtained with a solar reactor prototype tested under high-flux radiative power inputs in the range 2.8-3.8 kW and mean solar concn. ratios up to 3024 suns. Crit. temps. of up to 2250 K induced CeO2 sublimation, which in turn affected detrimentally the solar reactor performance. The model is applied to analyze an improved geometrical design with alternative flow configuration, enabling more uniform radiative absorption and temp. distributions, and resulting in a higher solar-to-fuel energy conversion efficiency.
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    28. 28
      Badri, M. A.; Favennec, Y.; Jolivet, P.; Rousseau, B. Conductive-radiative heat transfer within SiC-based cellular ceramics at high-temperatures: A discrete-scale finite element analysis. Finite Elem. Anal. Des. 2020, 178, 103410  DOI: 10.1016/j.finel.2020.103410
      There is no corresponding record for this reference.
    29. 29
      Howell, J. R.; Menguc, M. P.; Daun, K.; Siegel, R. Chapter 15 – Conjugate Heat Transfer in Participating Media. In Thermal Radiation Heat Transfer, 7th ed.; 2021; pp 686– 687.
      There is no corresponding record for this reference.
    30. 30
      Schappi, R.; Rutz, D.; Dahler, F.; Muroyama, A.; Haueter, P.; Lilliestam, J.; Patt, A.; Furler, P.; Steinfeld, A. Drop-in fuels from sunlight and air. Nature 2022, 601, 63– 68,  DOI: 10.1038/s41586-021-04174-y
      30
      Drop-in fuels from sunlight and air
      Schappi, Remo; Rutz, David; Dahler, Fabian; Muroyama, Alexander; Haueter, Philipp; Lilliestam, Johan; Patt, Anthony; Furler, Philipp; Steinfeld, Aldo
      Nature (London, United Kingdom) (2022), 601 (7891), 63-68CODEN: NATUAS; ISSN:1476-4687. (Nature Portfolio)
      Aviation and shipping currently contribute approx. 8% of total anthropogenic CO2 emissions, with growth in tourism and global trade projected to increase this contribution further1-3. Carbon-neutral transportation is feasible with elec. motors powered by rechargeable batteries, but is challenging, if not impossible, for long-haul com. travel, particularly air travel4. A promising soln. are drop-in fuels (synthetic alternatives for petroleum-derived liq. hydrocarbon fuels such as kerosene, gasoline or diesel) made from H2O and CO2 by solar-driven processes5-7. Among the many possible approaches, the thermochem. path using concd. solar radiation as the source of high-temp. process heat offers potentially high prodn. rates and efficiencies8, and can deliver truly carbon-neutral fuels if the required CO2 is obtained directly from atm. air9. If H2O is also extd. from air10, feedstock sourcing and fuel prodn. can be colocated in desert regions with high solar irradn. and limited access to water resources. While individual steps of such a scheme have been implemented, here we demonstrate the operation of the entire thermochem. solar fuel prodn. chain, from H2O and CO2 captured directly from ambient air to the synthesis of drop-in transportation fuels (for example, methanol and kerosene), with a modular 5 kWthermal pilot-scale solar system operated under field conditions. We further identify the research and development efforts and discuss the economic viability and policies required to bring these solar fuels to market.
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    31. 31
      Ackermann, S.; Steinfeld, A. Spectral hemispherical reflectivity of nonstoichiometric cerium dioxide. Sol. Energy Mater. Sol. Cells 2017, 159, 167– 171,  DOI: 10.1016/j.solmat.2016.08.036
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      Spectral hemispherical reflectivity of nonstoichiometric cerium dioxide
      Ackermann, Simon; Steinfeld, Aldo
      Solar Energy Materials & Solar Cells (2017), 159 (), 167-171CODEN: SEMCEQ; ISSN:0927-0248. (Elsevier B.V.)
      Nonstoichiometric ceria, CeO2-δ, has emerged as a promising redox material for thermochem. splitting H2O and CO2 using concd. solar energy. Knowledge of its radiative properties is crucial for the design of efficient solar reactors. Samples of various nonstoichiometries (0≤δ≤0.0377) were prepd. by thermal redn. in a thermogravimetric analyzer at high temps. (T≥1473 K) and under low oxygen partial pressures (pO2≤2.5·10-4 atm). The spectral hemispherical reflectivity was measured using a spectroscopic goniometry system in the spectral range 300-2800 nm. A porous ceria sample with interconnected μm-sized pores showed comparable selectivity because of its high optical thickness. The total hemispherical reflectivity was computed for emission temps. in the range 900-6000 K relevant to solar reactors.
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    32. 32
      Xie, T.; Xu, K.; Yang, B.; He, Y. Effect of pore size and porosity distribution on radiation absorption and thermal performance of porous solar energy absorber. Sci. China: Technol. Sci. 2019, 62, 2213– 2225,  DOI: 10.1007/s11431-018-9440-8
      There is no corresponding record for this reference.