As shown in Figure 1, the solar parabolic trough methanol steam reforming reactor (SPTR) in this study mainly comprises three parts: the concentrator system for sunlight collector, the vacuum absorber for solar-to-thermal energy conversion, and the reaction tube for thermal-to-chemical energy conversion, respectively. Our previous work (Tang et al., 2021) showed that a well-designed secondary reflector for PTC can nearly achieve a circumferentially uniform distribution of solar flux on the absorber’s surface. As a result, it is reliable and easy to build a 2-D axisymmetric SPTR model, as shown in Figure 1B by considering uniform circumferential solar flux distribution. In the SPTR, methanol and steam enter through the inlet, absorb solar heat and react on the reaction particle bed consisting of Cu/ZnO/Al₂O₃ catalyst (Peppley et al., 1999), and finally produce fuel products with solar energy stored. The relevant structural and physical parameters of SPTR are listed in Table 1, and a detailed description of the concentrator structure can be found elsewhere (Tang et al., 2021). It should be mentioned that the axial solar flux is not necessarily uniform to boost the reactor performance, which will be optimized in this study.

To build the multi-physics model for SPTR, it is useful and necessary to introduce reasonable assumptions in order to save computational resources while ensuring sufficient realism. The present 2-D multi-physics SPTR model is developed based on the following assumptions: 1) the catalytic reaction bed is an isotropic homogeneous porous medium, 2) the gas mixture flows in a steady laminar flow, 3) the heat loss is only for radiation and convection with the atmosphere, and 4) the local thermal equilibrium assumption in the porous reaction bed is valid (Liu et al., 2016). In the following, we briefly present the governing equations for simulating the mass/species transfer, fluid flow, heat transfer, and chemical reactions of the gas mixture.

The solar flux distribution is optimized to get high-quality energy conversion and improve the overall SPTR performance by achieving a better match between the energy field and the reaction field. The primary performance evaluation parameters of SPTR are the methanol conversion efficiency η _m and the solar thermochemical conversion efficiency η _s, given below: η m = m CH 3 OH, in − m CH 3 OH, out m CH 3 OH, in × 100 % (28) where m CH 3 OH ,  in and m CH 3 OH ,  out indicate the inlet and outlet methanol flow rate, respectively. η s = Q chem Q sun = ∰ S chem d V q sun ( z ) π D ao L × 100 % (29) where Q _chem and Q _sun denote the thermochemical reaction energy and the total solar input energy, respectively.

Typically, one of the most important SPTR performances is the methanol conversion efficiency η _m, and other evaluation parameters are directly or indirectly related to the conversion rate, such as the hydrogen yield m H 2 . Thus, the conversion efficiency η _m is chosen as the maximization target in this study.

For solar flux distribution, it is described by the Bezier curve. This curve is a way that can describe various types of shapes with mathematical expressions, and it has been widely used in scenarios such as airfoil optimization (Wei et al., 2020) and robot motion control (Song et al., 2021). Furthermore, the curve has the property of high-order derivability, which is well suited for the gradient-based optimization algorithm in this study. The corresponding nth-order Bernstein basis function representing the deviation Δq _sun(z) of solar flux relative to the uniform flux q _ave is expressed as: Δ q sun ( z ) = q ave ∑ k = 0 n C k ( z L ) k ( 1 − z L ) n − k (30) where C _k is the control point of the curve and is also the control variable.

Since the sintering temperature of the Cu/ZnO/Al₂O₃ catalyst is at 573.15 K (Yong et al., 2013), a maximum temperature constraint is required. Furthermore, a point-by-point constraint of constant surface solar flux greater than 0 is set to prevent unreasonable negative flux values. Also, in order to accelerate the optimization, an advance constraint of greater than 0 at the control points at both ends, that is, C ₀, C _n ≥ −1 is imposed. To sum up, for a certain total solar input, the optimization problem can be concluded as: { max C 0 ∼ C n :      η m = m CH 3 OH, in − m CH 3 OH, out m CH 3 OH, in s .t . :        T max ≤ 573.15  K               C 0 , C n ≥ − 1               q sun ( z ) ≥ 0               ∯ q sun ( z ) d A = π D ao 2 L q ave 4 (31)

The optimization process of the solar flux distribution is solved by the Sparse Nonlinear Optimizer (SNOPT) based on Sequential Quadratic Programming, an algorithm suitable for solving large-scale constrained optimization problems with smooth non-linear functions in the objective and constraints (Gill et al., 2005). The overall process of SNOPT-based optimization of SPTR solar flux distribution is shown in Figure 2B.

COMSOL is a powerful numerical analysis software for multi-physics fields based on the finite element method, covering various fields such as mechanics, fluids, electromagnetism, heat transfer, chemicals, electrochemistry, and acoustics. It is especially suitable for developing models involving multiple physical fields (Fu et al., 2022; Tang et al., 2022). Calculations were performed on a computer with an 8-core 3.6 GHz CPU and 32 GB RAM, and the entire SPTR multi-physics model was solved iteratively by the full-coupling method in COMSOL (about 10 min), and the solar flux distribution is optimally solved by SNOPT (about 133 min). The convergence criterion for the above two solution processes was determined to be 10^–5 by the tolerance test.

For grid types, structured grids have the advantages of fast generation, good quality, low memory consumption, and are suitable for regular and simple computational domains (e.g., Figure 1B). Thus, custom meshing in COMSOL is used to generate structured grids of different sizes for comparison, and the comparison results are shown in Figure 3. The grid group of 56 × 1,000 has only a relative error of 10^–5 compared to the group of 76 × 1,500, which can be considered since the grid system of 56 × 1,000 is accurate enough for this study.

To verify the accuracy of the current multi-physics field model, the flow-heat exchange-radiation component is validated with the experimental results of Dudley et al. (1994). As shown in Table 3, the results obtained using the present model under different conditions show great agreement with the experimental results.

For chemical validation, the present model was compared with the experimental study by Peppley et al. (1999) and the simulation study by Cheng et al. (2019c). As shown in Figure 4, the methanol conversion of the present model matches each other with the results in Cheng et al. (2019c) and Peppley et al. (1999), which especially agrees well with the results of Cheng et al. (2019c). The aforementioned validation shows the reliability of the model.

To obtain the optimal concentrator combination for improving the SPTR performance, this section first optimized the axial distribution of solar flux on the SPTR at an inlet reaction gas mass flow rate of 0.012 kg s⁻¹, an inlet steam methanol ratio of 1.1, and an average surface solar flux input of 13,000 W m⁻². In the optimization process, two types of solar flux distributions are considered, which are linear and non-linear. The linear distribution corresponds to the 1st-order Bezier function, that is, n equals 1 in Eq. 30, while the non-linear distribution is set to the 5th-order Bezier function, corresponding to n equals 5.

Figure 5 shows the hydrogen generation rate field in SPTR for uniform solar flux distribution and two optimized distributions, where the relevant control parameters of Bernstein basis functions corresponding to the optimized distributions are listed in Table 4. For the distribution characteristics, the final optimization result of the linear flux distribution is decreasing from inlet to outlet, and this result validates each other with the conclusion of Wang et al. (2017), showing the feasibility of the optimization model. However, the optimal non-linear distribution shows an arch-shaped distribution with the solar flux converging in the middle and less on the sides, as in Figure 5C. For the hydrogen generation rate, both optimized solar flux distributions produce a higher hydrogen reaction zone (r _H2>60 mol m⁻³ m⁻¹) compared to the relatively low uniform generation rate (r _H2<60 mol m⁻³ m⁻¹) under the uniform distribution, while it shifts with the location of the energy concentration. Intuitively, the non-linear optimization has a higher and larger high reaction zone, indicating stronger reaction dynamics.

To clearly observe the changes in comprehensive performance before and after flux optimization, Figure 6 synthesizes the comparative results of SPTR performance under different solar flux distributions. Apparently, both non-uniform optimized solar flux distributions exhibit better SPTR performance than uniform distributions, as reflected by higher methanol conversion, solar thermochemical efficiency, and hydrogen yield. Particularly, the enhancement reaches the maximum under the non-linear flux distribution as shown in Figure 5C, where the methanol conversion, solar thermochemical efficiency, and hydrogen yield are enhanced by 2.5, 3.3, and 2.4%, respectively, relative to the uniform flux. Although there is a relative increase of 19.0% in CO yield, the slow methanol decomposition reaction rate and the CO consumption of the water–gas shift reaction result in a CO yield of only about 1/10 of the H₂ yield. Thus, the discussion later focuses on the primary target product H₂ rather than the by-product CO. Moreover, the maximum catalytic bed temperatures are all below the catalyst sintering temperature of 573 K, which implies a long catalyst life and good stability of SPTR performance (Yong et al., 2013).

To clarify the reason for the performance improvement, Figure 7 shows the average temperature, average hydrogen generation rate, and average methanol concentration distribution along the flow direction for the three solar flux distributions. To facilitate the analysis, the catalytic bed is divided into three zones (Z ₁–Z ₃) using the intersection of hydrogen generation rates as the boundary, as shown in the dashed line in Figure 7. As seen in Figure 7B, both optimized flux distributions have more concentrated fast reaction rate zones due to the concentrated solar energy distribution, which correspondingly appear in the Z ₁ and Z ₂ zone. With the linearly optimal solar flux distribution, more energy is invested in the front zone (Z ₁), which raises the reaction temperature more rapidly and reaches the higher fast reaction zone faster compared to the uniform distribution. However, in the Z ₃ zone, as seen in Figure 7A, only a small amount of reactants are present, where more energy will be used to heat the fluid (e.g., the uniform case in Figure 7A) apart from making the reaction complete. Clearly, neither the linear nor the non-linear optimized distribution allocates much energy input in Z ₃. Under the non-linear optimal solar flux distribution, more energy is distributed in the intermediate zone (Z ₂). Simultaneous temperature increases with progressively increasing solar input in Z ₁ at non-linear flux, at which the temperature level of Z ₂ (480–526 K) surpasses the 423–498 K of the optimal reaction zone Z ₁ at linear flux. Further, 80% of the reactant molar fraction remained when the best reaction zone Z ₂ reached under non-linear flux, thus being sufficient to show higher reaction performance in terms of temperature and reactant-dominated reaction kinetics.

The relationship between the variable distributions in Figure 7 shows that there is a degree of similarity under the non-linear case, so Figure 8 compares the synergy between the fields by the correlation index described by the Pearson correlation coefficient P _r of Eq. 32: P r = ∑ k = 1 n ( x k − x ¯ ) ( y k − y ¯ ) ∑ k = 1 n ( x k − x ¯ ) 2 ∑ k = 1 n ( y k − y ¯ ) 2 (32) where x and y denote the two discrete variables for comparison, n is the number of discrete variables, and ‾ represents the average of the corresponding variables.

Usually, the closer P _r is to 1, the higher the positive correlation is, especially when the sample of variables is constant, P _r is 0 (e.g., the solar flux in the uniform case in Figure 8 is constant). As seen in Figure 8, the correlations under the non-uniform optimal distribution are all the highest, showing the strongest synergistic effect to enhance the SPTR performance. Here, the solar flux as an external variable directly affects the temperature change, while the temperature and the reaction rate interact with each other, which ultimately manifests as the effect of the solar flux and the reaction rate, that is, the relationship between the solar energy supply and the reaction energy demand. It should be noted that due to the dominant role of MSR in the SPTR reaction system, the hydrogen reaction rate was found to almost represent the reaction heat consumption in this study.

Overall, the rational distribution of non-linear optimization achieves hysteresis matching of temperature distribution, which in turn achieves energy flow matching of energy input and consumption (i.e., reaction rate) through the synergistic relationship between reaction rate and temperature with P _r up to 0.99.

Considering the engineering reality, the current PTC cannot achieve continuous concentrated solar flux, so it is necessary to convert the continuous solar flux distribution into a step-like one in order to get a concentrator group with different apertures and finally get the design guidance of concentrator in SPTR. Since the step-like distribution can be viewed as having an infinite gradient at each junction, the gradient-based segmentation method is used in this study to segment the continuous solar flux distribution to ensure that the features of the distribution are fully preserved.

After comparative consideration, 100 and 75% of the maximum gradient are selected as the splitting points to split the non-linear flux into a step-like flux, as in Figure 9A.

Figure 9B summarizes the corresponding field variable changes and performance diagrams of SPTR under the step-like solar flux distribution. The characteristic of variable change can be observed, where after each flux change, the temperature and reaction rate first have a large rate of change and then gradually level off. In Figure 9B, there is an extremely high correlation of over 0.99 for the results of variable changes between the corresponding step-like and continuous type flux. Also, the SPTR performance is only slightly degraded by about 0.2% due to the loss of some continuous curve properties by the step-like shape, which also shows the effectiveness of the gradient-based segmentation method in this study.

To check the performance effect of the optimized solar flux distribution form (i.e., the step-like solar flux) under different conditions, this section explores the SPTR performance at different reactant mass flow rates and DNIs, which map the demand side and supply side in SPTR, respectively. Different DNIs correspond to different solar average fluxes, where DNI = 1 kW m⁻² corresponds to a solar average flux of 13 kW m⁻², implying an average concentration ratio of 13. Here, the solar average flux of 4–14 kW m⁻² and the reactant mass flow rate of 0.012 kg s⁻¹ are applied.

The performance comparison for different solar average fluxes is shown in Figure 10, where the methanol conversion η _m, hydrogen generation rate m _H2, and maximum catalytic bed temperature T _max all increase with increasing input solar energy, due to the increased total input allowing more chemical and thermal energy conversion. However, the solar thermochemical conversion efficiency η _s decreases as the input energy increases when it reaches a high conversion efficiency η _m, mainly because the chemical energy that can be converted reaches the limit. Furthermore, at different solar flux, the optimized flux all have better performance improvement compared to uniform flux, with the increase ratio (relative percentage improvement of corresponding performance indicator in present figure) increasing as the solar flux decreases. Together with the temperature variation in Figure 10B, where the reaction is weak when the energy input is small, the central concentration feature of the optimized solar flux distribution results in a higher temperature level and thus a more significant performance improvement. However, when the energy input is sufficient, the reaction proceeds completely and the energy input at the end flows to the thermal energy; thus, the optimized solar flux with less end distribution allows more energy to be used in the high reaction zone and reduces the temperature level.

For different mass flow rates, which represent the energy consumed on the demand side, 0.009–0.016 kg s⁻¹ are chosen for the analysis, with a solar average flux of 10 kW m⁻². The performance comparison summarized in Figure 11 shows that as the reactant mass flow rate increases, η _m and T _max decrease due to the increase in the flow rate and the decrease in the reaction residence time. However, m _H2 and η _s are essentially stable because the energy input determines the hydrogen production when the total amount of reactants is greater than the energy input. Similarly, the optimized solar flux is enhanced at different flow rates, and the increase ratio increases with increasing flow rate.

Overall, the aforementioned results indicate that the form of optimal solar flux distribution with spatial energy supply and demand matching can improve SPTR performance regardless of the relationship between solar input and total reactants, and thus the present study is considered valid and general.