Study on Hydrogen Production Characteristics by Methanol Steam Reforming in a Fresnel Lens-Tapered Cavity Solar Thermal Concentric-Tube Reactor

Abstract

The endothermic nature of methanol steam reforming (MSR) for hydrogen production induces varying thermal effects along the flow direction, resulting in a non-uniform temperature distribution within the catalytic bed. Optimizing temperature uniformity has been demonstrated to enhance hydrogen production efficiency. In this study, a novel Fresnel lens-driven non-evacuated tapered cavity solar reactor was proposed for methanol steam reforming, which can provide a reference for optimizing hydrogen production using Fresnel lens solar concentrators. The thermal flux distribution on the reactor’s inner walls was determined by Monte Carlo ray-tracing simulations. A three-dimensional CFD model integrating fluid flow, heat and mass transfer, and methanol steam reforming reaction kinetics was developed to investigate the effects of key operational parameters on this novel reactor performance. Multi-objective optimization using response surface methodology revealed that high reactant inlet temperature (T_in > 550 K) and low flow velocity (u_in < 0.2 m/s) conditions significantly improve reactor methanol conversion (99.99%) and hydrogen yield (91.48%), but at the cost of increased CO selectivity (S_CO > 28%). Conversely, low temperature (T_in < 500 K) and high flow velocity (u_in > 0.4 m/s) conditions suppress CO formation (S_CO < 0.03%), although with reduced hydrogen production efficiency.

1. Introduction

To address the increasing challenges of energy supply and environmental pollution, actively promoting renewable and clean energy sources (such as solar, wind, and hydrogen) has become an effective strategy to mitigate these issues. Hydrogen, due to its high calorific value, zero emissions, and pollution-free nature, is regarded as a primary energy carrier for the future [1], but its storage and transportation costs remains high. Methanol, as an ideal hydrogen carrier and source, effectively overcomes the difficulties of hydrogen storage and transportation due to its high hydrogen-to-carbon ratio, liquid state at ambient temperature and pressure, and ease of storage and transport. Methanol steam reforming, with its extensive industrial application experience and relatively low reaction temperature, is considered one of the most promising hydrogen production methods. To achieve the green and sustainable development of methanol steam reforming-based hydrogen production, researchers have actively explored the use of various clean energies as heat sources to drive the methanol steam reforming reaction. Depending on the nature of the heat source, the heating methods for methanol steam reforming can be categorized into waste heat recovery (engine waste heat [2,3], industrial waste heat [4,5], and heat carried by solid particles [6,7], etc.), electric heating [8,9], exothermic chemical reactions [10,11,12,13], and solar energy [1,14,15,16,17,18,19,20,21,22,23].

Liao et al. [2] conducted methanol reforming for hydrogen production experiments using engine exhaust waste heat recovery strategies to verify that exhaust waste heat can replace external heat sources. Tang et al. [3] established a waste heat recovery reformer model for online reforming in methanol–diesel dual-injection engines, analyzing methods to enhance reforming efficiency through flow rate and temperature optimization. Poran A et al. [4] developed a high-pressure thermally regenerative direct-injection internal combustion engine integrated with methanol reforming, comparing power output and emission characteristics with and without the reforming system, demonstrating that waste heat reforming reduces pollutants and improves thermal efficiency. Yao et al. [5] built a compact waste heat recovery reformer to intensify internal transport processes and increase hydrogen production rates. Zheng et al. [6] investigated the impact of catalyst particle size on reforming performance in waste heat-heated reactors, optimizing heat transfer and reaction matching within the bed. Gao et al. [7] analyzed the effect of porosity in the heat exchange wall structure of particulate steam generators on heat transfer in packed beds and overall energy efficiency of the reforming hydrogen generation system. Lian et al. [8] employed insulated plasma catalysis to enhance methanol reforming, enabling efficient hydrogen production under low-temperature waste heat conditions. Guan et al. [9] designed an electromagnetic induction heating tubular catalytic coated reactor, utilizing dual heat sources—electromagnetic heating and waste heat—to improve reforming conversion rates. Andisheh Tadbir M et al. [10] developed a three-dimensional simulation of a coated microreactor, coupling methanol reforming with exothermic oxidation self-heating. Chein R et al. [11] integrated a miniaturized hydrogen generation unit combining fuel vaporization, reforming, and CO removal, completing prototype testing. Wang et al. [12] constructed a self-thermal micro-reformer using SiC honeycomb ceramic carriers, achieving autonomous thermal operation via reaction exothermicity and waste heat. Li J M et al. [13] systematically conducted flow and heat transfer experiments on pin-fin self-heating reforming reactors, quantifying the relationship between hydrogen production performance and fin structure and operating conditions. Hong H et al. [14] provided the earliest proposal of a coupled methanol decomposition and low-temperature solar thermal power cycle and laid the foundation for the idea of solar methanol thermochemical energy storage. The proposal of a low-grade solar heat collection system matched with a methanol reforming hydrogen production system, and the thermodynamic modeling was also completed [15]. In reference to [16], the test platform was built to verify the feasibility of direct solar heating driving reforming hydrogen production at medium temperatures, and it was proven that solar energy could meet the heat demand of the reaction. The works from the same team [1,19,20] established a three-dimensional numerical model of medium-temperature solar energy reception/reactor, optimized the cavity flow path and heat absorption structure, and proposed a completely new integrated heat collection–reaction composite structure. Zheng et al. [18] carried out the numerical optimization of catalyst filling arrangements for the non-uniform heat flow density of parabolic trough receivers to alleviate local overheating/insufficient reaction. Zhao et al. [22] carried out the systematic study of parabolic solar reforming reactors, and comprehensive analyses of heat flow, catalytic layer, and working fluid flow coupling effects. Kumar R et al. [23] investigated the effects of particle size, concentration ratio, and particle shape on the improvement of the system’s energy storage efficiency. This material can be used for solar heat storage and then provide heat for methanol reforming to produce hydrogen. Real D et al. [17] researched pure renewable solar energy-driven methanol reforming, comparing the energy efficiency of photovoltaic electrolysis and photothermal reforming routes. Zhao et al. [21] completed multiple groups of tests of solar thermal driving hydrogen production at medium temperatures, carried out thermodynamic and exergy analyses of the entire system, and identified the key links of system energy loss.

Thermal energy storage is a promising and sustainable solution in the field of long-term energy storage. However, the low thermal conductivity inherent in the storage materials limits their wider application. It has been reported that introducing nanoparticles and mixing enhancement techniques can lead to significant progress in thermal performance [24]. On the other hand, the heat-absorbing methanol reforming reaction can be utilized to enhance the storage and release processes, thereby expanding its application scope and improving its efficiency. When utilizing the low energy density of solar energy, solar concentrating technology can also effectively address this issue by focusing solar radiation into a small area with the help of concentrators, thereby increasing its energy density.

Jin’s team [14,15,25] constructed a medium-temperature solar receiver/reactor based on a single-axis parabolic trough concentrator, utilizing solar heat at approximately 473–573 K to drive the methanol steam reforming reaction. Through experimental and numerical studies, they investigated a 5 kW mid-to-low-temperature solar thermal hydrogen production system and later expanded it to 15 kW [1,24,26]. This demonstrated the feasibility of combining solar heat with alternative fuels for hydrogen production at around 473–573 K, broadening the thermal applications of medium-temperature solar energy. However, compared to conventional methanol steam reforming reactions under uniform heat flux or temperature conditions, the circumferential solar flux density distribution in parabolic trough solar receivers/reactors is non-uniform [27,28]. This can lead to localized hot spots, high temperatures, and significant circumferential temperature gradients [29], resulting in issues such as the degradation of selective absorption coatings [30,31], sintering of catalyst particles [25], and thermal deformation of the receiver [32]. Additionally, the axial light flux distribution in line focusing systems exhibits uniform characteristics, which inherently conflict with the reaction kinetics requirements of methanol steam reforming—high-density heat flux is needed initially to overcome the activation energy barrier, while only heat supply to maintain the reaction temperature is required in later stages. This mismatch between heat supply and endothermic reaction demand limits system energy efficiency. Beyond parabolic trough concentrators integrated with hydrogen production systems, Men et al. [33] developed a linear Fresnel solar absorber/reactor model for photothermal–chemical reactions. By optimizing heat flux distribution, the system’s chemical reaction performance approached ideal conditions, reducing the risk of catalyst overheating and sintering-induced deactivation. Yu et al. [34] developed a system using a Fresnel lens as the concentrator to study methanol-to-hydrogen conversion under combined photothermal and photocatalytic effects outdoors. They prepared a CuO/ZnO/ZrO₂ nanocatalyst with 91.4% full-spectrum solar absorption, achieving a solar-to-chemical energy conversion efficiency of 10.7% at a low temperature of 422 K under 700 W/m² solar irradiance intensity. Zhang et al. [35] employed a Fresnel lens as the concentrator and constructed a hybrid catalyst bed methanol steam reforming reactor, where Pt/CuO facilitated photothermal catalysis and Cu/ZnO/Al₂O₃ enabled thermal catalysis, achieving a maximum solar-to-chemical energy conversion efficiency of 24.1%. Zeng et al. [36] utilized a Fresnel lens to concentrate sunlight onto nanoparticles for solar-driven methanol decomposition, demonstrating that the nanoparticle-based reaction bed exhibited superior solar capture performance with a low concentration ratio of 17.5 and a high energy conversion efficiency of 35.5%. Zhao et al. [37] used a linear Fresnel concentrator with a reflector and secondary homogenizing mirror to drive a vacuum tube-based hydrogen production reactor, comprehensively analyzing the effects of reactor surface heat flux and key operational parameters while highlighting the feasibility and effectiveness of response surface methodology in studying methanol steam reforming performance. Gu et al. [38] pioneered the integration of a compound parabolic concentrator (CPC) with vacuum encapsulation, employing a double-sided TiNOx-coated copper receiver (absorption rate up to 0.94) for methanol steam reforming hydrogen production, offering a viable portable solution for methanol reforming.

Response surface methodology (RSM) is a widely used experimental design and analysis method [39]. Constructing response surface plots helps users intuitively understand how factors in an experimental design interact and influence response variables. Chen et al. [39] investigated the methanol steam reforming process for hydrogen production using a commercial Cu-Zn catalyst, combining Box–Behnken Design (BBD) and Artificial Neural Network (ANN). They systematically studied the effects of the steam-to-methanol ratio (S/C, 1.5–2.5), reaction temperature (180–280 °C), and gas hourly space velocity (GHSV, 15,000–30,000 h⁻¹) on methanol conversion and hydrogen yield. The results indicated that temperature and S/C ratio significantly influenced the target responses (temperature > S/C > GHSV), while GHSV had a limited effect. The optimal operating conditions were identified as high temperature (280 °C) and low S/C (1.5), achieving high methanol conversion (98.56%) and hydrogen yield (2.98 mol/mol). Monyanon et al. [40] employed a 2⁴ full factorial design with four center points to systematically examine the effects of operating temperature, steam-to-methanol ratio liquid feed rate, and catalyst weight-to-helium flow ratio (W/F) on the catalytic performance of methanol steam reforming, aiming to maximize methanol conversion while minimizing carbon monoxide (CO) selectivity. The results showed that the liquid feed rate had a much stronger influence on the response than other factors, overshadowing their significance. Further optimization using Central Composite Rotatable Design (CCRD) under RSM minimized CO selectivity while maintaining high methanol conversion. The optimal conditions were found at 295–307 °C and S/C = 1.82–2.00, achieving nearly a 100% methanol conversion with CO selectivity of around 1.5%. Sarafraz et al. [41] conducted methanol steam reforming in a microreactor and optimized operating parameters using Box–Behnken Design. Under conditions of 773 K, 1.0 g catalyst loading, and a flow rate of 24,000 mL/(g·h), methanol conversion was enhanced to about 100%.

Imtiaz Hussain and Lee [42] conducted a comparative study between point-focus and linear-focus Fresnel lenses, analyzing two collectors with identical areas. They found that the point-focus collector exhibited an average efficiency 7% higher than its linear-focus counterpart. Unlike dish concentrators, Fresnel lenses eliminate the need to account for shadowing effects caused by cavity absorbers on reflective mirrors [43]. Cavity-type solar absorbers, as the core component of point-focus solar concentrating systems, demonstrate significant technical advantages and application potential in medium-to-high-temperature solar energy conversion. These absorbers employ a unique black cavity design, allowing incident light to undergo multiple reflections and absorption processes on the inner walls, thereby minimizing optical spillage losses and effectively suppressing convective and radiative heat losses. Additionally, the cavity absorber, positioned behind the focal plane, uniformly distributes high-density solar radiation across its inner surface and facilitates stable heat transfer through a thermal conduction medium [44]. Moreover, compared to conventional vacuum tube collectors, cavity absorbers adopt a non-vacuum design, eliminating the need for complex sealing processes and significantly reducing production costs and maintenance difficulties. Xie [45] designed and analyzed the optical performance of eight cavity structures, finding that a conical cavity absorber (with a vertex angle of 60°) paired with a point-focus Fresnel lens achieved the highest optical efficiency (89.95%) and the most uniform internal energy distribution. Yuan et al. [46] addressed the dead-zone issue in traditional cavity receivers by designing a concave-bottom cavity absorber system using a Fresnel lens as the concentrator. Experimental results showed that the convex-bottom cavity receiver exhibited a 6.02% higher optical efficiency than conventional designs. Asrori et al. [47] applied a Fresnel lens-based absorber for sensible heat studies, experimentally comparing the thermal efficiency of two conical cavity receivers of different sizes. The results indicated that the smaller receiver, due to its higher concentration capability and lighter thermal load, achieved superior steam generation efficiency and faster response times. During testing, the average focal temperature for the larger receiver was 285.3 °C, while the smaller one reached 478.94 °C. Kaddoura et al. [48] integrated a point-focus Fresnel lens with a sensible heat storage system to investigate monthly thermal energy storage profiles in Lebanon. Their study demonstrated that a lab-scale Fresnel lens could elevate the heat transfer fluid temperature by 200 °C, with projected thermal efficiencies ranging from 93.6% to 97.2%.

Therefore, in this study, a novel methanol steam reforming hydrogen production reactor based on a Fresnel lens cavity absorber was designed, considering the characteristics of the methanol steam reforming reaction in solar energy-driven hydrogen production. This design not only utilizes solar thermal collection for heating but also improves the temperature distribution within the reactor through optimized design, thereby enhancing the reactor performance.

2. Physical Model of Fresnel-Concentrating Solar Cavity Absorber Coupling Reforming Reactor

2.1. Model Description

Based on the literature review of cavity absorbers, most studies indicate that conical cavities exhibit better thermal collection performance. Therefore, this study adopts the principle of equal heat absorption area as a reference to design a conical heat collection cavity [49]. The working principle involves using a Fresnel lens concentrator to focus solar energy onto the walls of the conical cavity absorber to heat methanol steam reforming for hydrogen production. A schematic diagram of the model is shown in Figure 1, with specific dimensions provided in

Table 1. Main design parameters of the concentrator and receiver.

Concentrator Parameters	Values (mm)	Cavity Absorber/Reactor Parameters		Values (mm)
Radius (R)	250	Conical Cavity	Height (h)	257
Top Radius (r)	0.5		Top diameter (R1)	80
Thickness (d)	3
Image Distance (f)	750	Catalytic Layer	dr	4
Object Distance	∞

. To reduce thermal losses in the cavity receiver, the inner surface is coated with an absorption layer (absorptivity of 0.9). A detailed cross-sectional view of the heat collection cavity and reactor is presented in Figure 1b. The entire device is constructed using stainless steel as the base material.

2.2. Optical Solution Approach

Based on Monte Carlo Ray Tracing (MCRT) technology, the optical simulation software TracePro 7.4 was adopted to investigate the optical characteristics. As a mature illumination design software, TracePro has been widely applied in numerous studies [45,50,51]. The optical simulation settings include a direct normal irradiance (DNI) of 1000 W/m², with the angular density in the beam settings configured as solar distribution, unpolarized, and an incident light wavelength of 546.1 nm [52]. The light source is a grid source with a circular grid pattern, 200 rings, and 119,401 rays. The non-uniform solar irradiance intensity distribution on the inner wall of the conical heat collection cavity, calculated through ray tracing, is applied as the thermal boundary condition for the reactor.

3. Numerical Methods

3.1. Governing Equations

In this study, commercial software COMSOL 6.0 was used to establish a three-dimensional reactor model and perform calculations. The computational domain of this model primarily includes (1) the solid domain of the stainless steel absorption reactor; (2) the porous medium domain of the Cu/ZnO/Al₂O₃ catalyst bed and multi-component gas mixture; (3) the air domain within the receiver cavity [49]. To simplify the analysis, the following assumptions are used: the mixture is considered an ideal gas; the porous catalyst bed is isotropic with local thermal equilibrium assumed, and thus heat transfer within the catalyst bed is neglected; the heat flux density around the circumference remains consistent at a fixed height (the heat flux density in this study is obtained from Tracepro 7.4 software based on Monte Carlo principles, with data processed to provide heat flux density values along the depth of the cavity wall surface every 0.25 mm). The governing equations are as follows:

Continuity equation:

$$\nabla \stackrel{⃑}{\mathit{u}}=0$$

(1)

The velocity field within a catalytic porous medium is governed by Darcy’s law, where pressure differentials, viscous dissipation, and bed morphology (characterized by permeability tensor k) collectively dictate flow dynamics. This relationship is expressed as:

$$\stackrel{⃑}{\mathit{u}}=-{\displaystyle \frac{k}{\mu }}\nabla P$$

(2)

where $\mu$ is the dynamic viscosity (Pa$·$s); k is the permeability of the porous matrix [53] ($m^2$); P is the pressure (Pa). The permeability k is defined as follows:

$$k={\displaystyle \frac{d_p^2{\epsilon }^3}{150{(1-\epsilon )}^2}}$$

(3)

where $\epsilon$ is the bed porosity; $d_p$ is the equivalent particle diameter.

Momentum equation:

Inside the reactor’s porous catalyst bed, the volume averaging technique is commonly used to assume an even distribution of both the solid catalyst phase—such as Cu/ZnO/Al₂O₃—and the fluid phase, which includes the methanol steam reforming reaction mixture of reactants and products. Based on the assumption of local thermal equilibrium [54], the momentum equation for the porous medium is formulated by adding an additional momentum source term, denoted as $F_i$ in Equations (4) and (5) [55]:

$${\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_i{u}_j\right)=-{\displaystyle \frac{\partial P}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[{\mu }_{eff}\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)-{\displaystyle \frac{2}{3}}{\mu }_{eff}{\displaystyle \frac{\partial u_l}{\partial x_l}}{\delta }_{ij}\right]+\rho g_i+F_i$$

(4)

where the interfacial momentum sink $F_i$ accounts for both viscous and inertial resistances; its definition is as follows:

$$F_i=-({\displaystyle \frac{\mu }{k}}{u}_i+c_F{\displaystyle \frac{1}{2}}\rho |u|u_i)$$

(5)

where $c_F$ is the inertial resistance coefficient, expressed as:

$$c_F={\displaystyle \frac{3.5(1-\epsilon )}{d_p{\epsilon }^3}}$$

(6)

Energy equation:

Assuming the porous medium is homogeneous and isotropic, under quasi-steady-state conditions, the heat transfer equation for methanol steam reforming reaction driven by solar energy is shown in Equation (7) [49]:

$$\nabla ·{(-\lambda }_{eff}\nabla T_{sr}+{\left(\rho C_P\right)}_f{\mathit{u}·\nabla T}_{sr}=\epsilon Q$$

(7)

where $T_{sr}$ (K) is the spatial temperature field in the reaction bed; ${\lambda }_{eff}(\mathrm{W}/(\mathrm{m}·\mathrm{K}))$ is the effective thermal conductivity; $Q(\mathrm{J}/({\mathrm{m}}^3·\mathrm{s}))$ is the volumetric reaction heat source. Effective thermal conductivity ${\lambda }_{eff}$ is defined as follows:

$${\lambda }_{eff}=\epsilon {\lambda }_f+(1-\epsilon ){\lambda }_{pm}$$

(8)

where the subscripts f and pm represent the fluid and porous matrix, respectively. ${\lambda }_f$ and ${\lambda }_{pm}$ represent the thermal conductivities of the gas mixture and the porous matrix, respectively. The volumetric reaction heat source $Q$ is defined as follows:

$$Q=-∆H_{SR}·r_{SR}-∆H_{DE}·r_{DE}+∆H_{WGS}·r_{WGS}$$

(9)

where $H(\mathrm{J}/(\mathrm{m}\mathrm{o}\mathrm{l}·\mathrm{K}))$ represents the reaction enthalpy, and r is the surface reaction rate.

The mass conservation equation represents the steady-state condition of the Maxwell–Stefan diffusion and convection process [1].

$$\nabla \cdot (\rho {\omega }_{i}u-\rho {\omega }_i\sum _{k=1}^n{\tilde{D}}_{e,ik}(\nabla x_k+(x_k-{\omega }_k){\displaystyle \frac{\nabla p}{p}})-D_{e,i}^T{\displaystyle \frac{\nabla T}{1}})=R_i$$

(10)

$${\tilde{D}}_{e,ik}={\displaystyle \frac{0.00143T^{1.75}}{P{M}_{ij}^{1/2}{\left[{({\sum }_v)}_i^{\frac{1}{3}}+{({\sum }_v)}_j^{\frac{1}{3}}\right]}^2}}$$

(11)

$$M_n={(\sum _i{\displaystyle \frac{{\omega }_i}{M_i}})}^{-1}$$

(12)

where ${\omega }_i$ is the mass fraction of species i; $x_k$ is the molar fraction of species k; $D_i^T(\mathrm{k}\mathrm{g}/(\mathrm{m}·\mathrm{s}))$ is the thermal diffusion coefficient; ${\tilde{D}}_{e,ik}({\mathrm{m}}^2/\mathrm{s})$ is the effective Fickian diffusivity tensor; $R_i(\mathrm{k}\mathrm{g}/({\mathrm{m}}^3·\mathrm{s}))$ is the reaction rate; $M_i(\mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l})$ is the molar mass of substance i. M_n is the molar mass of the mixture.

3.2. Methanol Steam Reforming Comprehensive Kinetic Model

This study employs the three-rate model for the methanol steam reforming reaction. The reaction process on the Cu/ZnO/Al₂O₃ catalyst involves:

Methanol steam reforming reaction (MSRR):

$${CH}_{3}OH+H_{2}O\to {CO}_2+3H_2∆H=+49.5\mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$$

Methanol decomposition reaction (MDR):

$${CH}_{3}OH\to CO+2H_2∆H=+92.0\mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$$

Water–gas shift reaction (WGSR):

$$CO+H_{2}O\to {CO}_2+H_2∆H=-41.1\mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$$

The reaction rates of the three-rate model have been studied by scholars, and the detailed calculations of the reaction rates used in this article can be found in the literature [22].

3.3. Boundary Conditions

(1)

Wall boundary conditions: The heat flux density of the collector cavity wall is determined using Tracepro. The outer wall of the reactor is effectively insulated with thermal insulation material, while the stainless steel shell maintains no heat exchange with the environment.

(2)

Inlet and outlet boundary conditions: The inlet flow rate, inlet temperature, and mass fractions of each component are defined. All outlets are set as atmospheric pressure outlet boundary conditions.

3.4. Parameter Definitions

Key evaluation indicators for methanol steam reforming include the methanol conversion rate [56] ($X_{{CH}_{3}OH}$), hydrogen production rate [57] ($Y_{H_2}$), and CO selectivity [57] ($S_{CO}$). The calculation formulas are as follows.

$$X_{{CH}_{3}OH}={\displaystyle \frac{C_{{CH}_{3}OH,in}-C_{{CH}_{3}OH,out}}{C_{{CH}_{3}OH,in}}}$$

(13)

Here, $C_{{CH}_{3}OH,in}\mathrm{and}{C}_{{CH}_{3}OH,out}$ represent the mass fractions of methanol at the inlet and outlet, respectively.

$$Y_{H_2}={\displaystyle \frac{F_{H_2}}{F_{{CH}_{3}OH,in}}}$$

(14)

$$S_{CO}={\displaystyle \frac{F_{CO}}{F_{CO}+F_{C{O}_2}}}$$

(15)

The solar-to-chemical energy conversion efficiency [20] ${\eta }_{so-ch}$ is defined as the following:

$${\eta }_{so-ch}={\displaystyle \frac{Q_{so-ch}}{Q_{solar}}}={\displaystyle \frac{F_{{CH}_{3}OH,in}{·X}_{{CH}_{3}OH}·∆H_r}{IA{\eta }_c}}$$

(16)

where $F_{{CH}_{3}OH,in}$ is the molar flow rate of methanol at the inlet (mol/s), $∆H_r$ is the reaction enthalpy (J/mol), I is direct normal irradiance (W/m²), A is the concentrator area (m²), and ${\eta }_c$ is optical concentration efficiency [58], calculated using Equation (17).

$${\eta }_c={\displaystyle \frac{Q_{ab}}{IA}}$$

(17)

where $Q_{ab}$ is the solar radiation energy absorbed by the inner wall surface of the cavity (W).

3.5. CFD Model Validation

The validation of the kinetic model for the reaction dynamics was conducted using the same catalyst, reactor geometry, and reaction conditions as those described in Peppley’s literature [59,60]. The methanol conversion rate as a function of W/F (catalyst mass in the reactor (kg)/inlet reactant mass flow rate (mol/s)) was compared, and, according to Figure 2, the methanol conversion rate trends at 513 K and 533 K show good consistency with the experimental results. The largest deviation occurred at a reaction temperature of 533 K and a W/F-value of 5.378 kg/(mol/s), with a maximum deviation of 6.765%. Given that the experimental reaction temperature was measured by a thermocouple placed on the centerline of the reactor rather than the reactor wall, this level of deviation is acceptable considering the differences between numerical simulation and experimental measurements.

The validated kinetic model was applied to investigate and optimize the comprehensive performance of the Fresnel lens-based double-tube reactor. To verify the grid independence of the numerical model, a mesh sensitivity study was conducted under fixed operating conditions: reactant inlet velocity of 0.1 m/s, inlet temperature of 493.15 K, and S/C of 1. As shown in

Table 2. Verification of grid independence for the model.

Number of Grids/Pieces	Methanol Conversion Rate/%	Reactant Outlet Temperature/K
110,002	64.43	549.68
228,916	63.82	549.68
303,312	63.44	549.69
428,723	64.12	549.86
588,054	64.11	549.91

, when the grid number increased from 228,916 to 303,312, the fluctuation range of the methanol conversion rate was only 0.6%, and the maximum temperature deviation in the reaction zone was merely 0.01 K. Considering the balance between computational efficiency and accuracy, the numerical simulations subsequently adopted 228,916 grid elements as the optimal discretization scheme.

4. Results and Discussion

4.1. Optical Solution Results

Since the inner wall of the receiver is curved, while the data obtained from TracePro ray tracing is in the form of a square matrix (representing irradiance projected onto a central plane), the two-dimensional projection was meshed to derive the actual heat flux distribution on the curved surface. After processing by MATLAB 2024, the irradiance distribution along the cavity depth was obtained, as shown in Figure 3. The irradiance initially increases and then decreases along the cavity depth. This trend occurs because the upper portion primarily receives light concentrated by the outer rings of the Fresnel lens, while the lower portion is illuminated by the more central rings.

4.2. Effect of Reactant Inlet Velocity on Hydrogen Production Performance

This section investigates the influence of reactant inlet velocity on reactor performance under fixed conditions (inlet temperature being 493.15 K, S/C being 1). Figure 4 presents the variations in methanol conversion (X_CH3OH), solar-to-chemical efficiency (η_so-ch), CO selectivity (S_CO), hydrogen yield (Y_H2), and maximum reaction zone temperature (T_max) with increasing inlet velocity u_in.

Data analysis indicates that X_CH3OH, Y_H2, S_CO, and T_max all decrease with the increase in u_in, with the largest decline occurring from 0.1 m/s to 0.2 m/s conditions. This is because when u_in increases, the contact time between reactants and catalyst particle surfaces shortens, causing part of reactants to leave the reactor without undergoing effective reactions. When u_in increases from 0.1 m/s to 0.5 m/s, X_CH3OH decreases from 63.82% to 14.3%. η_so-ch increases with the rise in u_in as, although the methanol conversion rate decreases at higher u_in, the total amount of converted methanol increases with the increase in u_in. As shown in Figure 4d, the increase in speed accelerates total methanol conversion amount, increasing the heat absorbed by the reaction, thus reducing the highest temperature within the reaction domain, as shown in Figure 4c.

4.3. Effect of Reactant Inlet Temperature on Hydrogen Production Performance

This section examines how reactant inlet temperature T_in (393.15 K to 493.15 K) affects reactor performance under other constant conditions (S/C = 1, u_in = 0.1 m/s). Figure 5 shows that higher temperatures significantly improve both methanol conversion (X_CH3OH) and solar-to-chemical efficiency (η_so-ch), reaching peak values of 63.82% and 53.33%, respectively. This occurs because both the methanol steam reforming reaction (MSRR) and methanol decomposition reaction (MDR) are heat-absorbing processes—higher temperatures promote forward reactions. In addition, within this reaction temperature range, the CO selectivity is less than 8%, and the maximum temperature in the reaction zone is 560.52 K (below the sintering temperature of 573 K for Cu/ZnO/Al₂O₃ catalyst). Therefore, if the reactant inlet temperature needs to be further increased to improve hydrogen production efficiency, both S_CO and T_max must be considered comprehensively.

4.4. Influence of the Steam-to-Methanol Ratio on Hydrogen Production Performance

Figure 6 shows the variation trends of various parameters when the S/C increases from 0.7 to 2.1 under the conditions of a reactant inlet temperature of 493.15 K and an inlet velocity of 0.1 m/s. X_CH3OH increases from 62.23% to 69.33% with the increase of S/C, Y_H2 increases from 58.76% to 72.6%, but η_so-ch, S_CO, and hydrogen mole fraction (F_H2) show gradual decreasing trends. Zhao et al. [61] also observed the same pattern when investigating the effect of S/C. In conclusion, although increasing water content helps improve methanol conversion rate, excessive water consumes more heat, which not only fails to promote further conversion of reactants but also leads to energy waste.

As the S/C ratio gradually increases, the concentration of steam rises, promoting the forward progression of both the methanol steam reforming reaction and methanol decomposition reaction, thereby increasing methanol conversion rate and hydrogen yield. However, with more steam under unchanged heating conditions, most of the heat is absorbed by the steam and, due to water’s high heat capacity, the solar-to-chemical efficiency decreases (peaking at 61.2%). Additionally, the proportion of steam in the product/reactant mixture increases, which conversely dilutes the hydrogen concentration in the mixture. Consequently, the hydrogen mole fraction decreases with increasing S/C, while S_CO reaches its maximum value under these conditions.

Zhao et al. [61] proposed the following calculation formulas for methanol consumption rate and hydrogen generation rate:

$$r_{C{H}_{3}OH}=-r_{MSR}-r_{MDR}$$

(18)

$$r_{H_2}=3r_{MSR}+2r_{MDR}+r_{WGS}$$

(19)

As can be seen from Figure 7, as the S/C gradually increases, the relative concentration of methanol decreases, reducing the probability of methanol–water collisions per unit time. Consequently, the reaction rate of methanol declines. Since hydrogen is primarily generated through the methanol steam reforming reaction, the decrease in methanol reaction rate directly leads to a corresponding reduction in hydrogen production rate.

4.5. Influence of Catalyst Layer Thickness on Hydrogen Production Performance

To study the effect of catalyst layer thickness (d_c) on hydrogen production performance, Figure 8 shows the variation patterns of various parameters when the catalyst layer thickness increases from 4 mm to 10 mm under the conditions of methanol inlet flow rate of 8.57 × 10⁻⁵ kg/s, T_in of 493.15 K, u_in of 0.1 m/s, and S/C ratio of 1.

As the catalyst layer thickness increases, both the methanol conversion rate and solar-to-chemical efficiency decrease. This phenomenon can be analyzed from both mass transfer and heat transfer perspectives. With increasing d_c, the diffusion range of methanol and steam expands, preventing reactants from distributing uniformly across all catalyst particle surfaces. Consequently, some active sites remain inert, and the overall catalyst activity cannot be fully utilized, making it impossible to improve methanol conversion. Figure 8c shows the methanol mass fraction distribution in the reaction zone under different d_c values. As d_c increases, the methanol mass fraction at the outlet becomes progressively higher, indicating increased amounts of unreacted methanol and thus lower methanol conversion. When the reactant inlet flow rate remains constant, the variation trend of solar-to-chemical efficiency aligns with that of methanol conversion. Additionally, with a constant heat source, methanol steam reforming as an endothermic reaction leads to decreased temperatures in the reaction zone as d_c increases, as shown in Figure 8d.

4.6. Effect of Two Heat Flux Distributions on Hydrogen Production Performance

This section examines the influence of different heat source conditions, the actual non-uniform heat flux distribution in the conical cavity (as can be seen in Figure 3) versus a hypothetical uniform heat flux (Q_mean = 5448.8 W/m²), on reactor performance. Figure 9a compares the two heat flux distributions at T_in = 493.15 K and S/C = 1 across different velocities, showing that methanol conversion under uniform heat flux consistently exceeds that under non-uniform conditions as inlet velocity increases, though the difference diminishes at higher velocities. Notably, at lower velocities, the maximum temperature under uniform heat flux exceeds the catalyst’s sintering limit. Among the tested conditions, the non-uniform heat flux at u_in = 0.1 m/s yields the optimal hydrogen production efficiency (X_CH3OH = 63.82%) while maintaining at safe temperatures (T_max = 560.52 K). Figure 9b demonstrates that while both heat flux conditions show increasing X_CH3OH with rising T_in, the uniform condition produces higher conversions but exceeds the sintering temperature above 473.15 K. Figure 9c reveals that although uniform heat flux delivers higher solar-to-chemical efficiency with increasing S/C, it simultaneously maintains CO selectivity above 10%. Crucially, the non-uniform heat flux effectively controls reaction zone temperatures, as evidenced in Figure 9d, comparing temperature distributions along the cavity depth at u_in = 0.1 m/s, S/C = 1, T_in = 493.15 K, and d_c = 4 mm. Under non-uniform heat flux conditions, the overall temperature within the reaction zone initially increases and then stabilizes, whereas under uniform heat flux conditions, it continuously increases and exceeds the catalyst sintering temperature after the 140 mm depth of the cavity.

4.7. Effect of Different Irradiance Levels on Hydrogen Production Performance

The preceding sections examined the reactor’s hydrogen production performance under quasi-steady-state conditions at a solar irradiance intensity of 1000 W/m². However, in actual operation, solar irradiance intensity fluctuates frequently. This section investigates the combined effects of varying irradiance levels with other parameters on hydrogen production performance, enabling the real-time adjustment of operating parameters based on irradiance changes to improve hydrogen production efficiency. Figure 10 shows the heat flux density distributions formed on the inner cavity wall at solar irradiance intensities of 1000 W/m², 800 W/m², and 600 W/m². Results all show a trend of increasing gradually from the inlet to the peak and then gradually decreasing along the cavity depth.

Table 3. Solar radiation energy absorbed by the inner wall of the conical heat collection cavity and the concentrating efficiency under different irradiance levels.

DNI (W/m2)	Incident Luminous Flux (W)	Qab (W)	Concentration Efficiency (ηc)
600	117.75	107.07	0.91
800	157	142.76	0.91
1000	196.25	178.26	0.91

lists the solar radiation energy absorbed by the inner wall surface of the conical solar collector cavity under different irradiance conditions, and the concentration efficiency was calculated using Equation (17).

Figure 11 shows that under the constant conditions of T_in = 493.15 K and S/C = 1, the methanol conversion rate reaches its highest value when the inlet flow velocity is smaller and the irradiance is greater. However, the trend of solar-to-chemical efficiency is completely opposite; as velocity increases, the solar-to-chemical efficiency becomes higher when irradiance is smaller. This indicates that the additional heat from increased irradiance is not effectively converted into chemical energy because, at lower velocities, the reactants cannot be fully mixed, resulting in lower reaction rates, as shown in Figure 11c. Figure 11b demonstrates that the maximum temperature in the reaction zone remains below the catalyst sintering temperature under all operating conditions. Therefore, under lower irradiance conditions, optimization can be achieved by either reducing the inlet velocity or increasing the reaction temperature to improve hydrogen production efficiency.

Figure 12 shows that as both irradiance and S/C increase, the methanol conversion rate exhibits an upward trend while the CO selectivity gradually decreases. This phenomenon can be attributed to the increased S/C promoting the forward progress of the methanol steam reforming reaction and favoring the water–gas shift reaction toward producing CO₂ and H₂, thereby reducing CO selectivity. However, it should be noted that a higher S/C is not always better, as excessively high ratios cause steam to absorb more heat, consequently lowering the reaction temperature and adversely affecting the reaction process.

Amongst all the operating conditions examined, the maximum temperature in the reaction zone occurred under the conditions of DNI = 800 W/m², u_in = 0.1 m/s, T_in = 493.15 K, and S/C = 1, where the peak temperature T_max reached 559.52 K, below the sintering temperature of the Cu/ZnO/Al₂O₃ catalyst. Consequently, the following investigation focuses on hydrogen production performance at elevated reactant inlet temperatures, specifically within the range of 503.15 K to 573.15 K. As shown in Figure 13, when both inlet temperature and direct normal irradiance (DNI) increase simultaneously, methanol conversion rate can exceed 99%. However, this condition also results in relatively high CO selectivity, reaching up to 30.82%. Therefore, for practical applications, both hydrogen yield and hydrogen purity must be considered comprehensively. For prioritized hydrogen yield condition, higher reactant inlet temperatures are recommended; for prioritized hydrogen purity condition, lower inlet temperatures should be maintained.

Figure 14a shows the temperature variation along the middle section of the reaction zone at DNI = 600 W/m² with increasing reactant inlet temperature. When the inlet temperature is below 533.15 K, the initial reaction rate is relatively low, and the heat absorption is less than the heat supply from the higher heat flux density in the front section, resulting in temperature rise. However, near the outlet, the reduced heat flux density leads to insufficient heat supply coupled with decreased reactant concentration, causing temperature drop. When the inlet temperature exceeds 543.15 K, intense endothermic reactions occur immediately after reactants enter the reactor. In this case, despite the higher heat flux density in the front section, the reaction’s heat absorption exceeds the heat supply, leading to the temperature decrease.

Figure 14b demonstrates that higher T_in corresponds to higher maximum temperatures in the reaction zone as DNI increases, which aligns with the trend observed by Wang et al. [20] regarding maximum temperature variations under different heat flux densities. At DNI = 1000 W/m², to maintain optimal catalyst activity, the inlet temperature should preferably remain below 533.15 K. Higher inlet temperatures lead to greater increases in the maximum reaction zone temperature, which may intensify side reactions of CO and CO₂ methanation. Further, these two highly exothermic reactions can cause uncontrolled temperature rise in the reaction zone, as shown in other studies [62].

5. Multi-Objective Optimization Process Based on Response Surface Analysis

This section focuses on the optimization of a conical cavity-driven methanol steam reforming reactor for hydrogen production. During the methanol steam reforming process, operating parameters such as the S/C, reactant inlet temperature, and inlet flow rate exhibit complex interrelationships and mutual influences on hydrogen production performance [63,64]. The response surface methodology (RSM) is an empirical modeling approach used to predict the relationships between system variables and response values with relatively high accuracy. The Box–Behnken Design (BBD) within RSM was employed to analyze the interactions among these operating parameters, facilitating further optimization of the reaction process.

5.1. Design of Methanol Steam Reforming Process

This study considers three key operating parameters that directly influence the chemical reaction, the S/C, reactant inlet temperature T_in, and inlet flow rate u_in. Using the Box–Behnken Design (BBD) in Design-Expert 13 software, an optimization scheme was developed for the conical cavity methanol steam reforming hydrogen production reactor. The design comprised 15 runs, with all three factors set within consistent ranges, as shown in

Table 4. Factors and corresponding levels for Box–Behnken design.

	Parameters	Unit	Low Level (−1)	Zero Level (0)	High Level (1)
A	S/C	1	0.7	1.4	2.1
B	Tin	K	493.15	533.15	573.15
C	uin	m/s	0.1	0.3	0.5

.

The upper and lower bounds of each factor are referred to as the high and low levels, respectively. The midpoint between these levels is termed the zero level. The average distance from the high or low level to the zero level, calculated as (high level—low level)/2, is defined as the standard deviation. After standardization (subtracting the mean and dividing by the standard deviation), the high, zero, and low levels correspond to coded values of +1, 0, and −1, respectively [65].

5.2. Results Analysis

A three-factor, three-level Box–Behnken design was implemented using Design-Expert 13 software to generate numerical combinations of the three operating parameters listed in Table 4. This experimental design yielded 15 sets of independent variable combinations.

Table 5. Calculations and response results of different combinational designs.

Serial Number	Factor 1 S/C	Factor 2 Tin (K)	Factor 3 uin (m/s)	XCH3OH/%	YH2/%	SCO/%
1	1.4	493.15	0.5	16.99	16.18	2.65
2	1.4	573.15	0.1	100.00	91.48	28.20
3	1.4	533.15	0.3	39.83	38.92	2.98
4	2.1	493.15	0.3	33.71	33.14	2.10
5	2.1	533.15	0.5	30.47	29.84	2.54
6	0.7	493.15	0.3	21.56	20.53	2.58
7	1.4	533.15	0.3	38.60	38.06	3.00
8	0.7	533.15	0.1	73.90	69.34	16.94
9	2.1	573.15	0.3	66.71	65.72	5.05
10	1.4	533.15	0.3	39.66	37.28	3.01
11	1.4	573.15	0.5	37.37	36.57	3.73
12	1.4	493.15	0.1	73.69	71.64	7.29
13	0.7	573.15	0.3	43.94	42.58	6.61
14	0.7	533.15	0.5	19.84	18.78	3.15
15	2.1	533.15	0.1	100.00	94.80	16.09

presents the corresponding methanol conversion rate, hydrogen yield, and carbon monoxide selectivity for each set of independent variables, as calculated via COMSOL 6.0 simulations.

Based on the analysis results, three candidate models were evaluated, the Linear model, the Two-Factor Interaction (2FI) model, and the Quadratic model [63]. As demonstrated by the ANOVA results presented in

Table 6. Statistical summary of the conical sleeve reactor model.

Response Variable	Source	p-Value	Lack of Fit	Adjusted R2	Predicted R2
XCH3OH	Linear	<0.0001	0.0038	0.8689	0.8066
	2FI	0.8487	0.0028	0.8360	0.6153
	Quadratic	<0.0001	0.1012	0.9965	0.9811	Significant
YH2	Linear	<0.0001	0.0070	0.8779	0.8209
	2FI	0.8452	0.0051	0.8476	0.6468
	Quadratic	0.0006	0.0651	0.9905	0.9477	Significant
SCO	Linear	0.0085	0.0023	0.5418	0.2893
	2FI	0.3067	0.0024	0.5889	0.0024
	Quadratic	0.0183	0.0077	0.8978	0.4187	Significant

, the Quadratic model exhibited superior performance for the response variables X_CH3OH and Y_H2 in terms of model significance, goodness-of-fit, and predictive capability, making it the most suitable model for the conical cavity hydrogen production reactor.

However, for the response variable S_CO, although the Quadratic model was recommended, its predictive R² value of only 0.42 indicated insufficient predictive accuracy. To enhance model precision, cubic terms were subsequently incorporated for further optimization, as detailed in later sections.

Table 7. Variance analysis of the regression equation model for methanol conversion rate.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value
Model	10,359.29	6	1726.55	441.01	<0.0001	Significant
A-S/C	641.64	1	641.64	163.89	<0.0001
B-Tin	1301.77	1	1301.77	332.51	<0.0001
C-uin	7376.68	1	7376.68	1884.21	<0.0001
AB	28.16	1	28.16	7.19	0.0278
AC	59.87	1	59.87	15.29	0.0045
C2	951.18	1	951.18	242.96	<0.0001
Lack of Fit	30.43	6	5.07	11.35	0.0832	Not significant
R2	0.9970
Adjusted R2	0.9947
Predicted R2	0.9877
Adeq Precision	63.8564

presents the analysis of variance (ANOVA) results for the reduced quadratic model with methanol conversion rate as the response variable, where non-significant terms (BC, A², B²) have been eliminated. The model demonstrates a sum of squares of 10,359.29 with six degrees of freedom, yielding a mean square of 1726.55. The exceptionally high F-value of 441.01 (p < 0.0001) provides strong evidence for the model’s statistical significance, indicating only a 0.01% probability that such a large F-value could result from noise. Furthermore, the lack-of-fit term shows an F-value of 11.35 with p > 0.05, confirming its non-significance and suggesting minimal influence from unknown experimental factors. The model exhibits excellent predictive capability, as evidenced by an R² and an adjusted R² of 0.9947, indicating that 99.47% of the variation in methanol conversion can be explained by the model, with only 0.53% remaining unexplained. The adequate precision ratio of 63.86 (substantially >4) further confirms the model’s robustness in characterizing spatial variations within the response surface design. The model demonstrates exceptional explanatory power, with an R² value of 0.9970 and an adjusted R² of 0.9947. This indicates that the model accounts for 99.47% of the observed variation in methanol conversion rate, leaving only 0.53% of the variation unexplained. Furthermore, the signal-to-noise ratio of 63.86 (significantly exceeding the threshold value of 4) provides robust evidence that the model effectively captures the spatial variability inherent in the response surface design.

For the optimization of operating parameters in methanol steam reforming, a quadratic model was established through least squares regression analysis to quantify the effects of three operational parameters on X_CH3OH.

$$\begin{gathered} X_{C{H}_{3}OH}=40.5712+8.95576A+12.7562B-30.3658C \\ +2.65327AB+-3.86874AC+15.9618C^2 \end{gathered}$$

(20)

where A represents the S/C; B denotes the inlet temperature; C signifies the inlet flow rate. Positive coefficients indicate a positive correlation between the factor and methanol conversion rate, while negative coefficients represent negative correlations. The absolute magnitude of each coefficient reflects its relative influence on the conversion rate. The equation reveals that inlet flow rate (C) exerts the most significant (negative) effect on methanol conversion, inlet temperature (B) shows the second-strongest influence, and S/C (A) demonstrates the weakest effect among the three parameters.

Table 8. Variance analysis of regression equation model for hydrogen yield.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value
Model	8979.75	6	1496.62	286.10	<0.0001	Significant
A-S/C	652.77	1	652.77	124.79	<0.0001
B-Tin	1124.82	1	1124.82	215.03	<0.0001
C-uin	6378.46	1	6378.46	1219.35	<0.0001
AB	27.75	1	27.75	5.31	0.0502
AC	51.82	1	51.82	9.91	0.0137
C2	744.12	1	744.12	142.25	<0.0001
Lack of Fit	40.50	6	6.75	10.03	0.0934	Not significant
R2	0.9954
Adjusted R2	0.9919
Predicted R2	0.9778
Adeq Precision	51.8191

presents the variance analysis results of three operational parameters affecting hydrogen yield. The model’s F-value is 286.10, with p < 0.0001, indicating that this regression model is highly significant and has high credibility. The p-value for the lack-of-fit term is greater than 0.05, which is insignificant, suggesting that the model error is small. Additionally, the p-values for A, B, C, and C2 are all less than 0.01, indicating that these factors significantly influence Y_H2, with the order of their impact being inlet speed, inlet temperature, and S/C.

The model’s variance (R²) is 0.9954 and the adjusted R² is 0.9919, indicating that only 0.35% of the response variable falls outside the prediction range. The difference between the adjusted R² and the predicted R² is less than 0.2; the precision is greater than 4, confirming the correctness of the model’s response.

The multiple regression fitting equation for hydrogen yield and the three operational parameters is

$$\begin{gathered} Y_{H_2}=39.4588+9.03307A+11.8576B-28.2366C \\ +2.63412AB-3.59919AC+14.118C^2 \end{gathered}$$

(21)

Hydrogen yield is positively correlated with the steam–methanol ratio and inlet temperature, but negatively correlated with inlet velocity, with inlet temperature having the greatest impact.

As previously mentioned, the accuracy of the carbon monoxide selectivity’s second-order prediction model was insufficient. By adding the cubic term A²B and removing insignificant terms, the accuracy of the prediction model was improved, with R² increasing from 0.4187 to 0.9780, as shown in

Table 9. Variance analysis of regression equation model for carbon monoxide selectivity.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value
Model	787.86	7	112.55	864.32	<0.0001	Significant
A-S/C	1.52	1	1.52	11.70	0.0111
B-Tin	120.93	1	120.93	928.67	<0.0001
C-uin	398.43	1	398.43	3059.65	<0.0001
BC	98.36	1	98.36	755.38	<0.0001
B2	2.90	1	2.90	22.28	0.0022
C2	155.80	1	155.80	1196.47	<0.0001
A2B	28.21	1	28.21	216.63	<0.0001
Lack of Fit	0.7632	5	0.1526	2.06	0.3585	Not significant
R2	0.9988
Adjusted R2	0.9977
Predicted R2	0.9889
Adeq Precision	99.3052

. The total sum of squares for the model is 788.20, with degrees of freedom at 10, mean square at 78.82, and an F-value reaching 549.12, with a p-value less than 0.0001 indicating that the model has high significance, with only a 0.01% probability of such a large F-value being due to noise. Among the factors, the p-values for A, B, C, BC, B², C², and A²B are all less than 0.05, making them significant terms in the model. In terms of lack-of-fit testing, the lack-of-fit F-value is 2.06, with a p-value of 0.3585, indicating that the lack-of-fit is not significant relative to pure error, with a 35.85% probability of such a lack-of-fit F-value being due to noise. The non-significant lack-of-fit result is ideal, suggesting that the model fits well. An R² value of 0.9988 means the model can explain 99.88% of the variation in the response variable, demonstrating excellent fit quality. The adjusted R² is 0.9977, close to R², indicating that the model maintains good fit even after considering the number of independent variables. The difference between the predicted R² and the adjusted R² is less than 0.2, further proving that the model’s predictive ability aligns highly with its fitting capability, showing good generalization performance. Additionally, the model’s precision is 99.3052, which measures the signal-to-noise ratio; ideally, this ratio should exceed 4. This model’s ample precision far surpasses 4, indicating a strong signal strength capable of effectively distinguishing different response levels. In summary, these statistical indicators show that the model has excellent fitting effect, predictive ability, and stability, suitable for guiding the exploration of design space.

The multivariate regression equation for the correlation between carbon monoxide selectivity and three operational parameters is

$$\begin{gathered} S_{CO}=3.15525-0.436328A+5.49843B-7.05714C-4.95894BC \\ +0.883739B^2+6.47666C^2-3.7556A^{2}B \end{gathered}$$

(22)

The equation above shows that the S/C and the reactants inlet velocity are negatively correlated with carbon monoxide selectivity, while the feedstock inlet temperature is positively correlated with carbon monoxide selectivity.

5.3. Optimized Results

The multi-objective optimization results for hydrogen production via methanol steam reforming driven by solar energy are illustrated in Figure 15. This study aims to maximize methanol conversion and hydrogen yield while minimizing carbon monoxide selectivity, presenting the interaction effects of three key operational parameters, S/C, T_in, and u_in, on system performance through a two-dimensional contour plot. The yellow areas in the figure indicate the recommended operating range for multi-objective optimization.

From Figure 15a, it is evident that when the S/C > 1.5 and T_in > 550 K are high, hydrogen yield significantly increases. This is due to the higher steam-to-methanol ratio enhancing the concentration of steam molecules, promoting complete methanol conversion (CH₃OH + H₂O → CO₂ + 3H₂), and the elevated temperature accelerating the kinetics of endothermic reactions. However, high temperatures also intensify the MDR reaction (CH₃OH → CO + 2H₂), leading to increased CO selectivity, which must be mitigated through gas purification processes such as PSA to reduce CO content. When the S/C is between 1.2 and 1.8 and the inlet temperature is below 500 K, CO selectivity drops significantly (S_CO < 0.05). Lower temperatures inhibit methanol decomposition reactions, while an intermediate S/C optimizes the main reaction pathway by balancing reactant concentrations. However, it should be noted that under low-temperature conditions, hydrogen yield decreases by more than 30% compared to hydrogen-rich conditions, necessitating a trade-off between hydrogen purity and yield based on downstream application requirements.

Figure 15b shows that when the S/C > 1.2 is high and the inlet velocity (u_in < 0.2 m/s) is low, hydrogen yield exceeds 90%. The lower inlet velocity prolongs the residence time of reactants within the catalyst bed, facilitating deep methanol conversion; the high S/C enhances hydrogen yield by suppressing the reverse water–gas shift reaction. At a S/C of 1.4 and an inlet velocity of 0.4 m/s, there is a significant turning point in CO selectivity. This phenomenon is attributed to the high flow rate shortening the residence time, inhibiting CO formation; additionally, increased flow rate improves mass transfer efficiency, encouraging CO’s further participation in the water–gas shift reaction, thereby reducing its final concentration.

As shown in Figure 15c, when the inlet temperature (T_in > 550 K) is high and the inlet velocity (u_in < 0.2 m/s) is low, the hydrogen yield exceeds 90%. High temperatures significantly enhance the reaction rate constant (k∝e^−Ea/RT), while low flow rates ensure adequate contact between reactants and the catalyst. It is important to note that under these conditions, the CO selectivity (S_CO > 15%) is relatively high, necessitating subsequent purification processes. In contrast, at lower temperatures (T_in < 500 K) combined with higher inlet velocities (u_in > 0.4 m/s), CO selectivity can be reduced to below 3%. Lower temperatures inhibit the activation energy requirements for CO formation pathways, while higher flow rates reduce the likelihood of side reactions by diluting reactant concentrations. This mode is suitable for scenarios requiring stringent hydrogen purity (such as hydrogen supply for fuel cells), though it results in a lower hydrogen yield.

When S_CO is less than 15%,

Table 10. CO concentrations corresponding to operating conditions with S_CO less than 10%.

Tin	S/C	uin	SCO	CCO
493.15	1.4	0.5	2.65	0.65
533.15	1.4	0.3	2.98	0.74
493.15	2.1	0.3	2.10	0.52
533.15	2.1	0.5	2.53	0.62
493.15	0.7	0.3	2.58	0.64
533.15	1.4	0.3	3.00	0.75
533.15	1.4	0.3	3.00	0.75
573.15	1.4	0.5	3.73	0.93
533.15	0.7	0.5	3.15	0.78

shows that the CO concentration in the dry reforming gas is less than 1%. High-temperature fuel cells can tolerate CO levels in methanol reformate (less than 1%) and accept a performance decline because, at low current densities, the presence of CH₃OH and CO in the fuel gas is beneficial for fuel cell performance [66].

Response surface analysis revealed that under conditions where methanol conversion exceeds 80% and carbon monoxide selectivity is less than 10%, the optimal operating parameter combination within the range of analysis is S/C = 1.9, T_in = 494 K, u_in = 0.1 m/s. The corresponding predicted values are X_CH3OH at 82.203%, Y_H2 at 77.8495%, and S_CO at 9.0895%. To verify the reliability of the predictive model, repeated simulations based on these optimal conditions were conducted, yielding X_CH3OH at 84.3%, Y_H2 at 82.12%, and S_CO at 8.65%, with errors of 2.49%, 5.19%, and 5.07%, respectively, from the predicted results, indicating the reliability of the predictive model.

For applications aiming for high hydrogen yield, corresponding recommended ranges are also provided, as shown in the yellow area of Figure 16. Under high S/C, higher inlet temperatures, and lower flow rates, the system can simultaneously achieve near-limit methanol conversion and high hydrogen yield (above 94%), as high temperatures accelerate endothermic reaction kinetics, low flow rates extend the residence time of reactants, and high S/C inhibit side reactions. However, this region is accompanied by higher CO selectivity. In practical applications, Chen et al. [67] injected the reformate obtained from methanol steam reforming into an engine without further purification, directly performing hydrogen-enhanced combustion, and demonstrated that reformed hydrogen provides performance equivalent to high-purity hydrogen from compressed gas cylinders or a mixture of 75% H₂ and 25% CO₂.

In scenarios with high hydrogen yield, the optimal operating parameter combination is S/C = 2, u_in = 0.128 m/s, T_in = 565.6 K, predicting X_CH3OH = 100%, Y_H2 = 96.185%, S_CO = 19.93%. Repeated simulations based on these optimal conditions yielded X_CH3OH = 100%, Y_H2 = 94.44%, S_CO = 18.48%, indicating that the prediction model has an error of 1.814% for hydrogen yield and 7.275% for carbon monoxide selectivity, thus demonstrating the reliability of the predictive model.

6. Conclusions

Based on the endothermic nature of methanol steam reforming for hydrogen production process, a novel reactor was developed, which uses a Fresnel lens cavity for heat collection to drive the MSR reaction. Through numerical analysis, the effects of reactant inlet velocity, temperature, S/C, catalyst bed thickness, two types of heat flux density, and irradiance on the reactor performance were discussed. By using response surface methodology, the synergistic impact of three key factors directly influencing MSR reaction on hydrogen production performance was explored, and optimization goals were set to determine the optimal operating parameter range.

The results showed that increasing reactant inlet velocity shortens the residence time of reactants within the reaction zone, leading to a decrease in methanol conversion rate, although the thermal chemical conversion efficiency improves due to an increased total amount of methanol conversion. Higher inlet temperature increases both methanol conversion rate and thermal chemical efficiency, but high temperatures promote reverse WGSR and MDR reactions, resulting in higher CO production. As the S/C increases from 0.7 to 2.1, hydrogen yield rises from 58.76% to 72.6%, although the mole fraction of hydrogen decreases due to dilution by steam. Increasing the thickness of the catalytic layer weakens mass, and heat transfer resulted in methanol conversion rate and thermal chemical conversion efficiency decrement, with the maximum radial temperature difference being only 3.554 K. Under other identical operating conditions, non-uniform heat flux density conditions can more effectively control the temperature distribution within the reaction zone compared to uniform heat flux density. Although methanol conversion rate is slightly lower under non-uniform heat flux density condition, under uniform heat flux density, when the inlet temperature of the reaction zone exceeds 473.15 K, localized temperatures within the reaction zone surpass the sintering temperature of copper-based catalysts, and this may significantly reduce catalyst lifespan. When irradiance conditions interact with other operational parameters, the highest methanol conversion rate occurs at lower inlet flow rates and higher irradiance condition, while the trend for thermal chemical conversion efficiency is opposite. Methanol conversion rate increases and carbon monoxide selectivity decreases as irradiance and S/C rise.

Through a multi-objective optimization study, the optimal conditions for achieving a CO selectivity less than 10% and a methanol conversion rate greater than 80% were determined to be S/C = 1.9, T_in = 494 K, u_in = 0.1 m/s, with a maximum prediction error of 5.19% compared to simulation validation results. This reformed gas can be applied in high-temperature proton exchange membrane fuel cells. Additionally, the optimal conditions for achieving a methanol conversion rate greater than 95% were found to be S/C = 2, T_in = 565.6 K and u_in = 0.128 m/s, with a maximum prediction error of 7.275% compared to simulation validation results.