Topological and hydrodynamic analyses of solar thermochemical reactors for aerodynamic-aided window protection

ABSTRACT Particles and condensing vapors contribute to the contamination of optical components of solar thermochemical reactors including windows and secondary concentrators and result in severe reactor performance deterioration. We propose an impinging-jet solar reactor in a distinctive topology to enhance aerodynamic-aided window protection. The vector fields on the two-dimensional inner surfaces of the impinging-jet reactor and four conventional reactors are analyzed for the first time using differential topology. The topological analysis demonstrates that the application of an impinging jet in the reactor leads to the prevention of the stagnation point in the vicinity of the window, which is inevitable in any axisymmetric flow field of the conventional reactors. The transient two-dimensional flow fields of all reactors are numerically investigated using the finite volume method to understand the effects of the reactor geometry, aspect ratio, and sweep gas mass flow rate on the window protection performance characterized by the contaminant residence time and peak volumetric concentration near the window. The impinging-jet reactors with aspect ratios of one and two are found to be the most effective designs in aerodynamic-aided window protection among all investigated configurations.


Introduction
High-temperature heat obtained from concentrated solar radiation can be used to drive power cycles and thermochemical processes for the production of electricity, fuels, and commodities (Bader & Lipiński, 2017;Chueh et al., 2010;Khorampoor et al., 2020;Lipiński, 2021;Meier et al., 2005;Romero & Steinfeld, 2012;Schäppi et al., 2022). In the solar thermochemical processes, the reactors are typically equipped with optical components such as: (i) a window to confine the thermal and/or chemical processes from the ambient environment (Lougou et al., 2020;Maag et al., 2011;Shuai et al., 2011;Z'Graggen & Steinfeld, 2004) and (ii) a secondary concentrator, e.g. a conical concentrator (Butti et al., 2021;Steinfeld, 1992) or a compound parabolic concentrator (CPC) (Chaves, 2016;Winston et al., 2004), for reducing radiative spillage losses and further increasing concentration ratios (L. Li et al., 2021), and (iii) reflective cavity walls for redirecting solar irradiation to desired locations (Larrouturou et al., 2014;X. Li et al., 2020;Martinek & Weimer, 2013;Sedighi et al., 2021;Steinfeld CONTACT Bo Wang bo.wang@nus.edu.sg & Fletcher, 1988). Particles and condensing vapors generated from chemical reactions or physical abrasion during these processes contaminate the optical components and, consequently, deteriorate the overall performance of a reactor. Figure 1 shows two examples of reactor contamination: (i) vapor condensation on the quartz window of a directly-irradiated solar reactor, and (ii) particle settlement on the CPC of an indirectly-irradiated solar reactor for the reduction of iron-manganese oxide (L. Li et al., 2019;Wang et al., 2020Wang et al., , 2021. Window contamination in solar reactors can be alleviated using gas streams to sweep the window or positioning optical components further from the reactive materials (Alonso & Romero, 2015). Applying sweep gas to the window, as the most commonly adopted method, has the following benefits: (i) reduction of a local contaminant concentration, (ii) increase of the shear stress, and (iii) cooling of the entrained particles and reduction of their adhesion. The effects of using sweep gas were experimentally investigated in numerous studies. Steinfeld et al. used an auxiliary gas flow to actively cool and sweep the  Li et al., 2019;Wang et al., 2020Wang et al., , 2021 window in a solar reactor (Steinfeld et al., 1998). The auxiliary gas flow was then applied in various particle-laden processes (Nikulshina et al., 2006;Schrader et al., 2020;Z'Graggen et al., 2006). Kogan et al. designed a conical solar reactor with a tornado flow field to prevent window contamination (A. Kogan et al., 2004;M. Kogan & Kogan, 2003). The flow field formed by the sweep gas was experimentally visualized using a smoke-charged flow. The particle settlement on the window was reproduced in a laboratory-scale experiment by charging carbon black particles in the flow and applying a thin layer of oil on the inner surface of the window. The majority of the particles settled at the stagnation point in the center and at the edge of the window, indicating that the window contamination primarily occurs in the low-velocity regions. Tornado flows, or cyclones, were widely adopted in solar reactors to avoid particle deposition and clogging Chien et al. (2016); Krishna Ozalp (2013); Ozalp et al. (2013); Shilapuram et al. (2011). Other than the applications involving particles, sweep gas was also employed to prevent zinc vapor condensation on a reactor window (Keunecke et al., 2004;Moller & Palumbo, 2001). Gokon et al. proposed a beam-down fluidized-bed reactor which enables the placement of the window further from the reactive material (Gokon et al., 2008(Gokon et al., , 2011. A convex window was employed to increase the distance between the window and the reactive material, and to sweep the window with specially-designed flow fields (Chambon et al., 2011). In several extreme examples, the window was extended to a glass chamber to confine the entire reactor (Abanades et al., 2007;Chambon et al., 2010).
There are several factors affecting the performance of solar thermochemical reactors. Because it is impractical to evaluate the diverse configurations through experimental tests, numerical models are used to simulate the complex fluid flows in solar thermochemical reactors. Plug flows carrying small particles (Stokes number 1) in solar reactors were approximated using singlephase models (Z'Graggen et al., 2006;Z'Graggen & Steinfeld, 2008). Schunk et al. developed a three-dimensional transient model to determine the optimal flow field for aerodynamic protection of the window against condensable Zn vapor (Schunk et al., 2008(Schunk et al., , 2009. Kogan et al. focused on the numerical investigation of the tornado effect in a swirl flow field (A. Kogan et al., 2007). Tian et al. compared different turbulent models, including the standard k-ε model, the baseline Reynolds stress model, and the shear-stress-transport (SST) k-ω model, and found that the SST k-ω model predicts the flow with satisfactory accuracy within acceptable computing time (Tian et al., 2015). The SST k-ω model is generally adopted to model turbulent flows in various solar receivers (Menni et al., 2020;Reddy & Satyanarayana, 2008;Wu et al., 2010;Xiao et al., 2019) and reactors (Ma & Martinek, 2019;Ortega et al., 2016;Singh et al., 2017;Tapia et al., 2016).
A literature survey reveals two main challenges to the design of solar thermochemical reactors: (i) the presence of a stagnation point near the center of the window, where severe window contamination occurs, and (ii) the great diversity of reactor geometries, which makes a fair comparison of reactor performance in terms of window protection difficult. To address these challenges, we propose an impinging-jet solar reactor with a novel topology to enhance aerodynamic-aided window protection. The ideal vector fields on the two-dimensional inner surfaces of the impinging-jet reactor and four conventional reactors are analyzed using differential topology. The symmetric flow fields of all reactors are numerically simulated using transient two-dimensional (2D) computational fluid dynamic (CFD) models. The effect of the reactor geometry, aspect ratio (AR), and sweep gas mass flow rate is analyzed on the window protection performance, including the contaminant residence time and peak volumetric concentration near the window.

Reactor design
Stagnation points in the vicinity of the window are crucial areas where the contamination takes place due to the lower fluid velocities in these regions. In order to protect the window, the number of stagnation points near the window should be minimized. From the perspective of differential topology, the solar reactor windows, either flat or spherical in geometry, have the same topology as a disk. A stagnation point in an actual flow field can be regarded as a singular point (a point where the vector field is not continuous) in an ideal vector field. Thus, the problem of minimizing the number of stagnation points near the window can be interpreted as the problem of minimizing the number of singular points in a 2D vector field on a disk.

Flow field from a topological perspective
The relation between the vector field and its manifold (a closed differentiable surface) topology is given by the Poincaré-Hopf theorem (Tu, 2017): For a vector field v with isolated singular points on a 2D manifold M, the sum of the indices of the singular points of the vector  field is equal to the Euler characteristic of the manifold (Gray, 1997), where F(s i ) is the index of singular point s i and χ is the Euler characteristic. If the manifold M has a boundary, the Poincaré-Hopf theorem is valid only if the vectors on the boundary are pointing towards the same side, either inward or outward, of the boundary. Traverse a path surrounding a singular point, the times of 2π the field rotates relative to any convenient direction is the singular point's index. Figure 2 shows several non-exhaustive examples of singular points in 2D vector fields and their corresponding indices. It should be noted that the dipole is valid in ideal vector fields but not in actual flow fields because streamlines cannot intersect. The Euler characteristic is a topological invariant that can be obtained from where g is the genus of a manifold. Table 1 gives the Euler characteristics for some common 2D manifolds (Alexandrov, 2011;Henle, 1994). According to the Poincaré-Hopf theorem, a disk manifold can have any number of any kind of singular points on condition that the sum of the indices equals to one, i.e. n i=1 F(s i ) = 1. Thus, there is at least one singular point in the form of a vortex, sink, or source on a disk manifold. There are two methods to avoid the singular point: (i) changing the vectors on the disk manifold boundary to point to different sides of the boundary, and (ii) alternating the manifold topology from a disk to an annulus. The former breaks the prerequisite of the Poincaré-Hopf theorem and makes Equation (1) no longer valid. The latter adopts an annulus topology with an Euler characteristic of zero, allowing a minimum of zero singular points in the manifold.
The topological analysis reveals that there exists at least one stagnation point near a typical reactor window, in either disk or sphere configuration, swept by gas streams in the radial or swirl pattern. Example reactors are shown in Figure 3(a-c). Method (i) of changing vectors on the manifold boundary can be implemented by sweeping the window from side to side, as shown in Figure 3(d). Method (ii) of alternating the manifold topology can be implemented by opening an orifice in the center of the window. An impinging-jet solar reactor with an annulus window is proposed in this study, as shown in Figure 3(e). The specific flow fields of these five solar reactors are theoretically analyzed in the following section.

Reactor geometry analysis
The conventional reactors in diverse geometries have been classified into groups (a)-(d) in Figure 3. Example solar reactors with spherical domes can be found in Refs. Berro et al. (2021) and Guesdon et al. (2006). The spherical reactor allows high-flux solar radiation is incident from a large acceptance angle. The spherical geometry also grants higher mechanical strength to the window which is beneficial when the reaction is operated under vacuum conditions. The gas inlets are placed on the base plane near the window, generating sweep gas flows tangentially to the window. The outlets are located next to the reactive material for fast removal of the generated contaminants. Due to the symmetry of the reactor, the two stagnation points are found at the apex of the dome and near the reactive material. The stagnation point near the window is a sink in the 2D vector field, while the one near the reactive material is a source.
The conical reactor has a cavity confined by a flat window and a conical wall. The conical wall can reflect part of the spillage radiation on the reactive material, which enhances the optical efficiency. Conical reactors can be operated in a radial or swirl flow field. In the radial flow field, the sweep gas enters the cavity in parallel with the window disk along the radius. This type of flow pattern creates a sink near the window and a source near the reactive material, similar to a spherical reactor. In the swirl flow field, the sweep gas enters and exits the cavity at an angle to the radius, creating the cyclone in the reactor. Two stagnation points in the form of vortices are formed at the same locations as in the radial flow field. Example conical reactors can be found from Refs. Furler and Steinfeld (2015), Bellan et al. (2019), and A. Kogan et al. (2007).
The flat reactor has the simplest geometry composed of a flat window and a flat wall in parallel with each other. The sweep gas enters horizontally from one side of the reactor and exits from the other. There are no stagnation points in a flat reactor. However, since the contaminant diffusion direction is normal to the advection direction, the sweep gas flow cannot effectively suppress the contaminants from contacting the window. As a result, the flat reactor design has been rarely adopted in the literature.
From the four types of reactors discussed above, it can be concluded that the symmetric flow field in these geometries inevitably creates a stagnation point near the center of the window, either a sink formed by colliding sweep gas flows or a vortex formed by swirling flows. We propose changing the topology of the reactor by opening an aperture in the center of the window. Thus, the introduction of sweep gas from this aperture removes the stagnation point near the window and forms an impinging jet at the same time, as shown in Figure 3(e). Using such a design results in the prevention of the stagnation point in the vicinity of the window, leaving only one stagnation point near the reactive material.
The singular points qualitatively analyzed using the differential topology do not necessarily reflect all stagnation points in an actual flow field, due to the additional stagnation points generated by local fluid circulation. The extra stagnation points can be identified using numerical simulations and experiments. Numerical simulation is an effective and convenient approach to studying the fundamentals of physical processes, while experiments are impractical due to the intensive time and cost efforts. Hence, to study the flow fields of the reactors under numerous operation conditions, numerical simulation is performed in this work.

Mathematical model and numerical solution
Transient 2D models are developed for the five types of reactors elaborated above to explore their flow fields and their performance in window protection. A gas stream is injected from the reactive material surface as a representative of gaseous or small particle (Stokes number 1) contaminants. In this section, the governing equations, initial and boundary conditions, parameters and properties, and numerical solutions are introduced. The transient Reynolds-averaged continuity equation reads where ρ is density, t is time, andū i is the average velocity component in the direction of x i . Einstein summation convention applies to repeated indices. The mass balance of species c is given by whereω c is average mass fraction of species c and j is diffusive mass flux obtained from the Fick's law considering only the ordinary concentration diffusion. The Reynoldsaverage Navier-Stokes equation is given by Tannehill et al. (2020) wherep is the average pressure. The stress tensor σ ij , due to molecular viscosity, is defined as where μ is the dynamic viscosity and δ ij is the Kronecker delta function. The Reynolds stress is approximated using where μ t is turbulent viscosity. The two-equation SST k-ω model is employed as the turbulence model: where k is turbulence kinetic energy, ω is specific turbulence dissipation rate, G k is the generation of turbulence kinetic energy due to mean velocity gradient, G ω is the generation of ω, k and ω are the effective diffusivity of k and ω, Y k and Y ω are the dissipation of k and ω due to turbulence, and D ω is the cross-diffusion term. Detailed calculation of these terms is omitted for brevity. Further information can be found in the paper by Menter (1994).
The geometrical parameters of all reactors and the operating conditions are given in Tables 2 and 3, respectively. The simulation can be divided into three sequential phases: a stabilizing phase of the flow time of 10 s, a discharging phase of 1 s, and a tracing phase of 50 s. A quasisteady-state is reached by the end of the stabilizing phase. Then the contaminant that enters the reactor during the discharging phase is monitored in the tracing phase. The transient contaminant concentration profile is obtained by post-processing the numerical results. The residence time is defined as the time required to purge 99% of contaminant that enters the reactor within the injecting period of 1 s. The contaminant peak volumetric concentration is the maximum volume fraction of the contaminant during the discharging and tracing phases. The physical properties of the sweep and contaminant gases, provided in Table 4, are representative values used in the current study and should be modified in future studies.
The model is implemented using ANSYS Fluent v21.1. The pressure and velocity of the flow are solved using the Coupled algorithm. The time and space are discretized  using the second-order upwind scheme and the secondorder implicit scheme, respectively.

Verification and validation
The grid independence analysis is conducted for all model cases. As a case in point, the result of grid independence analysis for a spherical reactor with AR = 1 is shown in Figure 4. The quasi-steady-state velocities of seven locations in the reactor are plotted as a function of the number of meshing elements. The mesh with 7680 elements is selected as it provides satisfactory precision with an acceptable computing time. The numbers of mesh elements for the five solar reactors are selected based on the grid independence study and summarized in Table 5. A thorough validation of the model is challenging due to the diverse geometries of the solar reactors explored in this study. The experimentally measured velocity  field of a conical reactor with a swirl pattern is used for the model validation as an example. The details of the conical reactor including its geometry and operation conditions are given in Ref. Chinnici et al. (2017).  The numerically predicted tangential velocity using the model presented in this study is compared against the experimental results, as shown in Figure 5. The numerical predictions agree well with the experimental results across the entire radius span, corroborating the validity of the model.

Parametric study
The contaminant residence time and peak volumetric concentration near the window in spherical reactors with different ARs and sweep gas mass flow rates are shown in Figure 6(a,b), respectively. The residence time decreases with an increasing the sweep gas mass flow rate and a decreasing AR. This observation indicates that the contaminants are more effectively removed from a reactor with a higher sweep gas flow rate and a smaller reactor cavity volume. The peak volumetric concentration of contaminants is not monotonically related to the variants. For example, the lowest peak volumetric concentration in a spherical reactor with AR = 0.5 is found at a sweep gas mass flow rates of 3 × 10 −4 kg s −1 . The quasisteady-state flow fields visualized in Figure 7 indicates that a new vortex (marked with red arrows) emerges in the reactor as the sweep gas mass flow rates increases from 3 × 10 −4 kg s −1 to 9 × 10 −4 kg s −1 . As the size of the vortex grows, the probability of the contaminants being entrained by the vortex and brought to the window vicinity increases, resulting in a higher peak volumetric concentration of contaminants. In comparison, reactors with AR = 1 and 2 remain the same flow patterns at different sweep gas mass flow rates. The direction of gas velocity in the vicinity of the window is independent from the sweep gas mass flow rates. As a result, the contaminant peak volumetric concentrations of these reactors barely change with the varying sweep gas mass flow rates. The performance and flow fields of conical reactors with radial inlets for sweep gas are shown in Figures 8  and 9, respectively. For the conical reactor of AR = 0.5, the sweep gas is able to effectively cover the window at the mass flow rate of 1 × 10 −4 kg s −1 . As the mass flow rate increases to 5 × 10 −4 kg s −1 , three vortices are developed along the window, as shown with arrows in Figure 9. The growth of the middle vortex is the primary cause for carrying over contaminants to the window, leading to an increase in both the residence time and peak volumetric concentration. The two vortices closer to the axis are destructed at higher sweep gas mass flow rates -causing direct contact between the contaminant flow and the window and an abrupt uptake in the contaminant peak volumetric concentration. In the case of AR = 1, it is observed that three vortices emerge at the sweep gas mass flow rate of 3 × 10 −4 kg s −1 . Further increasing the mass flow rate leads to the expansion of the middle vortex and subsequently longer residence time and higher peak volumetric concentration. The flow field for the reactor of AR = 2 does not change significantly. Thus, the contaminant residence time varies monotonically with the sweep gas mass flow rate -approaching lower values by an increase in the sweep gas mass flow rate. It should be noted that the reactor is modeled for a maximum flow time of 50 s. The residence time for the reactor of AR = 2 at a mass flow rate of 1 × 10 −4 kg s −1 exceeds 50 s and is only noted as 50 s in Figure 8(a).
The conical reactors in the swirl flow pattern resemble those in the radial flow pattern, except that the sweep  gas enters the reactor tangentially to the radii of the window. Such inlet arrangement creates a cyclone in the reactor that could potentially suppress the circulation of contaminant to the window. The contaminant residence time and peak volumetric concentration near the window are shown in Figure 10. Since the generated cyclone is coaxial with the reactor, the streamline cannot be displayed in the 2D symmetry plane. Instead, the profiles of the contaminant volumetric concentration are shown in Figure 11. Contrary to the design objective of suppressing contaminant circulation, the created cyclones in reactors with AR = 1 and 2 drain the contaminant near the reactive material and elevates it to the window. This funnel effect becomes more significant at high sweep gas mass flow rates. For the reactor of AR = 2, the sweep gas velocity is not high enough to create a cyclone funnel effect. The contaminant is gently entrained by the sweep gas, resulting in a longer residence time but lower peak volumetric concentration compared to reactors of AR = 0.5 and 1.  The contaminant residence time, peak volumetric concentration near the window, and streamlines for the flat reactors are shown in Figures 12 and 13, respectively. The contaminant diffuses from the ablated sample located at the bottom of the reactor to the window at the top of the reactor. The sweep gas flows horizontally from one side of the reactor to the other. Unlike spherical and conical reactors where downward flows are formed along their axes, the horizontal sweep gas in flat reactors cannot suppress the movement of the contaminants to the window. Adjusting the sweep gas mass flow rate only indirectly affects the contaminant peak volumetric concentration by influencing the residence time. For flat reactors with AR = 1 and 2, no strong vortices are observed under different operating and geometrical conditions. Thus, both the residence time and peak volumetric concentration vary monotonically with the sweep gas mass flow rates. The mild vortices spotted for sweep gas mass flow rates from 5 × 10 −4 to 9 × 10 −4 kg s −1 induce a slight increase in the residence time.  The contaminant residence time, peak volumetric concentration near the window, and streamline plots for the impinging-jet reactors are shown in Figures 14  and 15, respectively. The residence time decreases monotonically with increasing sweep gas mass flow rate and increasing AR. The contaminant peak volumetric concentration near the window for the reactor of AR = 0.5 is high due to the short distance between the window and the reactive material. For reactors of AR = 1 and 2, the peak volumetric concentration is less than 10 −5 for the entire range of the sweep gas mass flow rate. The peak volumetric concentrations of contaminants in impinging-jet reactors are significantly lower than the spherical, conical, and flat reactors. The flow fields of impinging-jet reactors with different ARs and sweep gas mass flow rates show high consistency. A single large vortex is observed in the reactor with minor changes under different ARs and sweep gas mass flow rates. Due to the consistency in the flow pattern, the effects of AR and sweep gas mass flow rate are highly monotonic. The results also reveal  that raising the AR above 1 and the sweep gas mass flow rate over 3 × 10 −4 kg s −1 has merely a minor contribution to reducing the residence time and the peak volumetric concentration of contaminants in the reactor.

Pareto optimal solutions
A comparison of the five types of reactors with different ARs and sweep gas mass flow rates indicates that the impinging-jet reactor has a better performance in window protection than the other designs. All Pareto efficient solutions are impinging-jet reactors for the entire range of sweep gas mass flow rate investigated. The contaminant residence time in the impinging-jet reactor can be less than 1.5 s, around only one-fifth of that for the second-best reactors. The contaminant peak volumetric concentration is also several orders of magnitude lower than its counterparts. The impinging-jet reactor manages to achieve short residence time and low peak volumetric concentration simultaneously, unlike other reactors that Figure 15. Streamline plots for impinging-jet reactors with aspect ratios, AR = 0.5, 1, and 2, and sweep gas mass flow rates of 1 × 10 −4 kg s −1 , 3 × 10 −4 kg s −1 , 5 × 10 −4 kg s −1 , 7 × 10 −4 kg s −1 , and 9 × 10 −4 kg s −1 . The color represents the contaminant mole fraction. A schematic of the computational domain is shown at the bottom right corner.
have satisfactory performance in one metric but poor in the other such as the conical reactor in the swirl pattern (AR = 1) and the flat reactor (AR = 0.5). In addition, both metrics are preferred to be a monotonically decreasing function of the sweep gas mass flow rate to allow convenient dynamic control of the reactor: one can simply increase the sweep gas mass flow rate to reduce the residence time and peak volumetric concentration. From the considered configurations, the impinging-jet reactor is the only design that fulfils such a requirement. A quantitative evaluation of the performance of all five reactor types is shown in Figure 16. The scattered plot manifests the contaminant residence time of all five reactor types as a function of the sweep gas mass flow rates. Colours are used to mark the reactor geometry types. The radius of the data points is correlated to the peak volumetric concentration by radius ∝ 1 − log 0.1 log peak vol. conc. .
The larger the data point, the lower the peak volumetric concentration it has. Inferior cases with residence time over 20 s and peak volumetric concentration over 0.1 are eliminated from Figure 16. The figure reveals that among the five types of reactors, the impinging-jet reactor has the shortest residence time for all sweep gas mass flow rates analyzed in this research. Other than the impingingjet reactor, the spherical reactor is a reasonable choice with a slightly longer residence time and higher peak volumetric concentration. The conical reactor in the swirl flow pattern outperforms the radial pattern in most cases.
In the case of the flat reactor, there is no ideal solution that meets the above-mentioned criteria.

Conclusions
Vector fields on two-dimensional inner surfaces of selected solar thermochemical reactor geometries were analyzed using the Poincaré-Hopf theorem. The topological analysis showed the existence of a stagnation point near the window in all the axisymmetric flow fields of conventional reactors. Contaminant accumulation near the stagnation point is the primary cause of window contamination and reactor performance deterioration. To alleviate this concern, a conceptual impinging-jet solar reactor was proposed to enhance the aerodynamic-aided window protection from particle settling and vapor condensation. The two-dimensional flow fields of the reactors, including the impinging-jet reactor and four types of conventional reactors, were simulated using transient two-dimensional numerical models. The models are used to quantitatively explore the effect of the reactor geometry, aspect ratio, and sweep gas mass flow rate on the window protection performance, which is characterized by the contaminant residence time and peak volumetric concentration near the window. It was observed that the impinging-jet reactors with aspect ratios of 1 and 2 are the most effective designs for aerodynamic-aided window protection among all investigated cases.
This study demonstrated a new methodology to compare the performance of solar reactors in various geometrical configurations in terms of contaminant removal. Future work on the proposed impinging-jet solar reactor includes evaluating the heat transfer and optical characteristics in addition to the hydrodynamic characteristics presented in this work. This will provide a comprehensive evaluation of the performance of the proposed impinging-jet solar reactor.