The solar-thermal experimental system is schematically illustrated in Figure 1. Concentrated solar irradiation is generated from a custom-made high flux solar simulator, which consists of a 10 kWe xenon short arc bulb (Superior Quartz, SQP-SX100003) and a silver coated aluminum ellipsoidal reflector (Optiforms, E1023F). The power output from the solar simulator is controllable by a power supply. The heat flux distribution at the focal plane inside the reactor is characterized to be Gaussian-Lorentzian (Abuseada et al., 2022a): q s ′ ′ r , I = A s I 1 − α L σ G 2 π exp − r 2 2 σ G 2 + α L π σ L r 2 + σ L 2 (1)

In the Equation 1, A s ( I ) = 0.683 I − 28.52 kW/m is the amplitude parameter, I the power supply input current, and r the radius from the geometric center of the substrate. The weighting factor for balancing the contribution between the Gaussian and Lorentzian distribution is α L = 0.595 . The Gaussian parameter σ G and Lorentzian parameter σ L are 0.00684 m and 0.0222 m, respectively.

The solar-thermal reactor is made of stainless steel and consists of three major sections. The reactant gas with controlled flow rate is supplied into the first section, a full nipple, which is 9.7 cm in inner diameter. Then the reactant flows through a fibrous porous carbon substrate (FuelCellEarth, C100) attached to the second major section, a reducing flange with an inner diameter of 6.86 cm. After passing through the porous reaction zone, the remaining reactant and product mixture flow into the third section, a 4-way cross. An infrared camera (FLIR, A655sc) is mounted on one of the ports of the cross, pointing toward the backside (the side without direct illumination) of the substrate, and the reactant and product gaseous mixture is passed into downstream in-situ laser diagnostics and mass spectrometry systems for species analysis and eventually into the exhaust through the bottom port.

Inverse methods have been widely applied in solving heat transfer problem in various branches of engineering (Alifanov, 2012). We implement an inverse approach for FPM temperature estimation employing the Nelder-Mead method (Gao and Han, 2012). A schematic of the computational domain with thermal and geometric conditions is depicted in Figure 2. The back-side temperature of the substrate is measured using an infrared (IR) camera. The 2D temperature matrix is azimuthally averaged and used as an input to the inverse problem.

A ZEISS Supra 40VP field emission scanning electron microscope (SEM) with a secondary electrons (SE) detector was used to obtain SEM images of graphitic carbon deposition followed by the local fiber diameter measurements. These values are applied for calculating the deposition rate and chemical reaction-induced internal heat sink.

After the inverse problem is solved, we use the results to estimate reaction kinetics. The global methane decomposition reaction is usually considered to be a first order reaction (Abanades and Flamant, 2007; Holmen et al., 1995; Trommer et al., 2004), noting that such a treatment is simplified without considering the complex subreactions network and minor intermediate species. The net rate of the decomposition can thus be expressed as: d n C H 4 d t = − A exp − E a / R T n C H 4 (17) where A is the pre-exponential factor, E a is the activation energy, R is universal gas constant, and n C H 4 is the methane amount in moles. The mass deposition rate per unit length can be calculated as: m ̇ C T L = ρ g π r f 2 − r i 2 Δ t (18) where ρ g = 1650 kg/m³, r i = 4.8 μ m and Δ t = 20 min.

Incorporating all the information and equations above, the activation energy can be estimated from a linear-fit of the following Arrhenius expression: ln m ̇ C T L = ln A M C n C H 4 L − E a R T (19)

The solar methane pyrolysis experiment is conducted at a total solar power of 1.86 kW, system pressure of 3.33 kPa, methane flow of 100 sccm and experimental duration of 20 min. Substrate back-side temperature is measured using an IR camera as shown in Figure 5A. The temperature distribution is clearly symmetric about the substrate’s geometric center. As such, the two-dimensional temperature contour is azimuthally averaged without losing accuracy. The reaction-zone (front-side) temperature without considering chemical reaction, penetration depth, anisotropic effective thermal conductivity, is estimated inversely by matching the experimentally measured back-side and numerically-estimated back-side temperatures. These temperature profiles are summarized in Figure 5B. The backside temperature estimated from the inverse heat transfer model falls into the measurement uncertainty range, showing good agreement with the measured back-side temperature.

In the inverse heat transfer model, several thermal and optical parameters can influence the estimated temperature. A sensitivity analysis is conducted to quantify the effects of these parameters on the inversely-determined peak front-side temperature, T f , and activation energy, E a , as described by sensitivity coefficients S T f and S E a , respectively. These two coefficient are obtained from ± 10 % perturbations around the nominal parameters values using a central difference approach: S T f = ∂ T f ∂ P P o T f , o = T f P 0 + Δ P − T f P 0 − Δ P 2 Δ P P 0 T f , o (20) S E a = ∂ E a ∂ P P o E a , o = E a P 0 + Δ P − E a P 0 − Δ P 2 Δ P P 0 E a , o (21) where P is the parameter to be analyzed including emissivity, absorptivity to solar and surrounding irradiation, and the temperature of the surroundings. P o , T f , o and E a , o are the nominal value of these parameters in Equations 20, 21.

Based on the results summarized in Figure 6, the model is most sensitive to the absorptivity of solar irradiation α s , which directly determines the most dominant input energy captured by the substrate. Approximately 8 % deviation in peak temperature and 6 % in activation energy are observed with ± 10 % perturbation in α s . The second most sensitive parameter is the central region emissivity ε c , which quantifies the rate of energy the substrate can release into the surroundings from the central part of the substrate. Additionally, the estimated results exhibit weak sensitivities to the emissivity of the rest region ε r , surrounding temperature T surr , and absorptivity to the surroundings α surr .

To evaluate the influence of finite penetration depth, anisotropic k and chemical reactions. Each of these factors is considered and incorporated into the code to quantify their effects on the temperature profiles. The temperature profiles and differences relative to the standard case are summarized in Figure 7.

These factors clearly have limited influence on predicted temperatures. The most influential consideration is the anisotropic k , at least potentially due to a slightly improved fitting owing to more floating parameters in k . As for the penetration depth, this study considers only a simplified case with uniform heat distribution, which can be refined with detailed material characterization and optical analysis in future work. Considering the thin penetration depth applied in this work, its effect on predicted temperature is negligible. The chemical reaction-induced temperature difference is the least significant in part due to the low solar-to-chemical efficiency, which is less than 1% for the conditions considered here (Abuseada, 2022). The approximated parameters in k and peak front-side temperature of the four foregoing cases are summarized in Table 1 below.

Spatial reaction-zone temperature of the standard case and fiber diameters measured using SEM at different substrate radii are shown in Figure 8A. The deposition rates derived from fiber diameters are further employed to approximate the activation energy of the global direct solar methane pyrolysis experiment through a linear fit (see Figure 8B) (Xu et al., 2024). The resulting activation energy of E a = 325 kJ/mol lies in the typical range for thermal methane decomposition (Holmen et al., 1995). The activation energy corresponding to inclusion of more detailed processes in the model as discussed above produce very similar predictions for activation energy: 327 kJ/mol when considering chemical reaction effects, 322 kJ/mol when incorporating penetration depth effects, and 336 kJ/mol when including anisotropic thermal conductivity. Clearly, these more detailed considerations have limited influence on the estimated temperature profile and activation energy.