Numerical investigation and assessment of the Langmuir–Hinshelwood–Hougen–Watson model for dry reforming of methane

Abstract

The co-activation of methane (CH₄) and carbon dioxide (CO₂) during dry reforming has led to numerous reduced kinetic models based on detailed surface reaction mechanisms, which are traditionally evaluated through parameter fitting, and their predictive validity is rarely examined. In this work, an alternative reduced Langmuir–Hinshelwood–Hougen–Watson (LHHW) kinetic model based on the literature data set has been tested numerically, without re-calculating any kinetic parameters. A plug-flow reactor (PFR) model is used to simulate conversions from low to high temperatures. Temperature-resolved statistical measures, residual structure analysis, and apparent activation energy were used to assess the model’s performance. This study qualitatively replicates the conversion trends at high and moderate-to-high conversion levels and shows systematic deviations at low temperatures. The apparent activation energy suggests that the high activation energies are due to composite CH₄ activation mechanisms rather than a single elementary step. The study provides a diagnostic reanalysis of a literature-derived LHHW model, highlighting its strengths and illustrating how mechanistic information can be obtained without re-fitting model parameters or independent validation. The present work demonstrates the predictive performance of a literature-derived LHHW kinetic formulation for the catalyst and operating conditions. It identifies the operating regime in which the selected model is applicable for the assessment of reduced kinetic models in reactor-scale simulations.

Introduction

The process of dry reforming of methane has received increasingly more research attention over the past few decades, due to its ability to harness CO₂ as well as to produce synthesis gas with a low H₂/CO molar ratio^1–3. An extensive amount of experimental and theoretical research has been focused on the synthesis of catalysts, reaction mechanisms and kinetic modeling, in an attempt to clarify the complicated chemistry of the surface. The large number of surface intermediates and elementary reaction steps found in detailed mechanistic descriptions results in the construction of extensive microkinetic networks that can contain a large number of elementary reaction steps and surface intermediates^4–6. The overall dry reforming reaction can be written as: CH₄ + CO₂ → 2CO + 2H₂. In one step, two of the greenhouse gases are consumed, and synthesis gas is generated for chemical and fuel production applications. In the literature, reduced kinetic expressions based on extrapolation of the above detailed mechanisms have been proposed by various studies to be used in reactor-scale modeling⁷. The commonest expressions used are LHHW-type rate expressions that also take into account adsorption equilibrium and even inhibition phenomena. In such quasi-empirical models, only a few lumped kinetic parameters such as surface reaction rate constants^8–10 and adsorption equilibrium constants of CH₄, CO₂ and H₂^11,12–14 are used. The values of such parameters are normally obtained through fitting experimental data of conversions under well-defined operating conditions, and the resulting models are then widely used in reactor design and catalyst benchmarking.

Even though simplified kinetic models are widely used in DRM reactor analysis, their applicability is often assessed only within the experimental conditions used for parameter estimation¹⁵. Consequently, uncertainties may arise when such models are applied outside their original operating ranges. The present work evaluates a literature-derived LHHW kinetic formulation using the originally reported parameter set within a plug-flow reactor framework. The analysis is limited to the catalyst system, feed composition, and operating conditions reported in Ref.¹⁶ and therefore should not be interpreted as a test of model transferability to other catalysts or reactor systems. The objective of this study is to perform a diagnostic reanalysis of the selected kinetic model through statistical evaluation, residual analysis, and apparent activation energy assessment. Since the kinetic parameters and experimental data originate from the same literature source, the present work should be viewed as an assessment of model behavior and mechanistic information content rather than an independent predictive validation. By identifying the operating conditions under which the model provides satisfactory agreement with experimental observations, the study aims to clarify the strengths and limitations of the selected reduced kinetic formulation.

A critical evaluation of a scaled-down kinetic model of CH₄ dry reforming is done in this work through independent prediction and systematic diagnostic analysis. The reported expressions of LHHW rate and the corresponding kinetic parameters are used without re-fitting in a plug-flow reactor (PFR) model to predict the conversion of CH₄ and CO₂ over a very large temperature and space–time domain. Validation is based on only experimental conversion data in literature. It is important to note that the current study is not aimed at universal applicability of LHHW kinetic models in dry reforming of methane. The analysis is limited to the kinetic expression, catalyst system and operating conditions of reference^16,17. This means that the findings reported here are specific to this model and its predictive performance and diagnostic utility. The investigated outcomes should be extrapolated only to other catalyst formulations and/or feed compositions, pressures, and/or reactor configurations that have been tested with the same kinetic parameters and experimental data. The model performance is quantitatively evaluated using statistical metrics, residual analysis, and apparent activation energy calculations. By identifying the operating conditions where the reduced model is predictive and clarifying the chemical reasons for its limitations, this work aims to provide a framework for the reliable application of reduced DRM kinetic models in reactor-scale simulations. It should be noted that this study does not carry out independent predictive testing. The kinetic parameters used were obtained from Ref.¹⁶ and the same set of data is used for the current evaluation. It should therefore be viewed as a reanalysis of the existing model with a view to deriving mechanistic understanding; it would not be appropriate to evaluate its predictive power on independent data sets.

Methodology

Data source and operating conditions

The kinetic data reported¹⁶ were obtained on supported nickel-based catalysts under conditions which reflect intrinsic kinetic data, with the authors verifying that there is no mass-transfer or heat-transfer limitation. The main goal of¹⁶ was to develop and parameterize an LHHW kinetic model by regression of the experimental rate data and by model discrimination procedures. The present study, however, does not address parameter estimation, model discrimination or kinetic model development. In contrast, the kinetic expression and the values of the parameters reported have not been modified and are used in a plug-flow reactor model. The aim of the present work is to analyse the reactor-scale behaviour of the existing kinetic model, with the aim of recovering the information that is contained in the mechanism. A literature-derived model is used to calculate methane and CO₂ conversions, apparent surface site fractions, apparent activation energies and reaction orders, and to interpret. The novelty of this study is not in the introduction of a new kinetic expression but in the demonstration of the use of an existing kinetic model which has been previously validated as a diagnostic tool to investigate the interplay of adsorption phenomena and intrinsic kinetics with reactor-scale performance. The mechanistic insights are those made based on analysis of the original model predictions, and no re-parameterization of the kinetic constants has been done.

The kinetic parameters obtained in that work consequently reflect intrinsic reaction rates, and the plug-flow reactor (PFR) model employed in the present study will likewise operate within the same intrinsic kinetic regime. This dataset was selected because the authors experimentally verified the absence of external mass transfer limitations, internal diffusion, heat transfer effects, pressure drop, and catalyst deactivation under the studied conditions. Therefore, the reported conversions can be considered kinetically determined rather than influenced by transport limitations. The mathematical expressions used in this analysis are provided in Eqs. (1)–(17), and the corresponding numerical model is illustrated in Fig. 1. The experiments were conducted at atmospheric pressure using an equimolar feed of CH₄ and CO₂, with inlet partial pressures defined in Eqs. (6) and (7). This is the only feed composition for which stable, reproducible operation was achieved. The temperature range of 778–898 K covers the kinetically relevant region for dry reforming while remaining below conditions where significant homogeneous gas-phase reactions or catalyst deactivation become dominant. Space time, defined as the ratio of catalyst mass to the inlet molar flow rate of CH₄, is used as the independent variable to analyze conversion trends in an integral PFR, as expressed in Eq. (5). CH₄ is selected as the reference species because it is consumed exclusively in the dry reforming reaction network. In contrast, CO₂ conversion is also affected by the concurrent reverse water–gas shift (RWGS) reaction. For example, the RWGS reaction, was incorporated through an equilibrium relationship to account for the coupled DRM–RWGS reaction network.

Fig. 1.

Numerical framework of the reactor model. Graphical overview of the plug-flow reactor (PFR) model used in this study, incorporating the Langmuir–Hinshelwood–Hougen–Watson (LHHW) kinetic formulation. The figure illustrates the governing equations, boundary conditions, and numerical solution strategy applied to simulate methane and carbon dioxide conversion over a wide temperature and space–time domain.

Numerical framework

The kinetic model employed in this investigation is an LHHW formulation from the literature¹⁶, selected through systematic discrimination of 27 alternative mechanistic models. This model was chosen because it satisfies criteria for statistical significance and thermodynamic consistency, with physically realistic adsorption enthalpies and activation energy boundaries. The intrinsic CH₄ consumption rate is described by Eq. (1), with the temperature dependence of the kinetic rate constant and adsorption equilibrium constants given by Eqs. (2)–(4). The adsorption equilibrium assumption is commonly employed in reduced LHHW kinetic formulations when adsorption and desorption processes are assumed to occur on a timescale much faster than the surface reaction step¹⁶. It represents the mechanistic assumption that adsorption and desorption processes are sufficiently rapid compared to the surface reaction step under the conditions studied. Based on this, the denominator of the adsorption term in the rate expression accounts only for the major adsorbing reactants, specifically CH₄ and CO₂, which is consistent with the kinetic model identified in reference¹⁶. Dry reforming of CH₄ is amenable to the LHHW framework because, in this case, the reaction occurs by a surface-mediated mechanism where the reactants are competitively adsorbed on catalytically active sites, which can be quantified by the surface coverage⁴. Based on well-established fundamental knowledge, this framework assumes molecular adsorption of CO₂ and dissociative adsorption of CH₄²⁴. The rate-limiting stage is assumed to be the surface reaction between CH₄-derived and CO₂-derived adsorbed species, which are represented by the adsorption and reaction terms appearing in Eq. (1). The reactants and products of the RWGS reaction are assumed to satisfy thermodynamic equilibrium. Therefore, the RWGS reaction is not modeled as a discrete kinetic step in the present model. The effect is accounted for by the equilibrium relation in Eq. (10) relating the partial pressures of CO₂, CO, H₂ and H₂O. This methodological approach follows the original kinetic model and maintains consistency with the experimental conditions under which the model parameters were calibrated. With the assumption of equilibration of the RWGS reaction, the kinetic analysis will focus on the net rate of CH₄ consumption through dry reforming with the auxiliary CO₂ conversion included by considering the RWGS equilibrium, which is expressed in Eqs. (9) and (10). The RWGS equilibrium investigation was introduced as a crude approximation rather than a fully resolved reaction scheme; consequently, the CO₂ balance is expressed in equations. (9)–(10) presenting an adequate explanation of the regime under the investigated numerical model conditions. Although the intrinsic DRM surface reaction is expressed as a forward reaction within the LHHW formulation, the overall reactor behavior is thermodynamically constrained by the reverse water–gas shift reaction, which is treated as an equilibrium process. As a result, the predicted conversion extents are limited by the equilibrium constraints of the coupled DRM-RWGS reaction network. Accordingly, the present numerical study does not attempt to generalize these kinetics to all DRM catalysts. Instead, it aims to examine the predictive behavior and assess the information content inherent in the reported kinetic formulation when applied as an independent input within a reactor-scale model. No modification or re-estimation of the parameters was made, and all were taken directly from Barroso Quiroga and Castro Luna¹⁶, as reported in Table 1. The present study is aimed at assessing the behavior of the published kinetic formulation with the original parameter set.

Table 1.

The kinetic and adsorption factors used in the LHHW model¹⁶.

Parameter	Value	Unit	Factors description
Ak₁	3.59 × 1021	mol·gcat⁻1·h⁻1·bar⁻1	Pre-exponential factor of surface reaction rate constant
Ea	332.04	kJ·mol⁻1	Activation energy of surface reaction
AkCH₄	2.89 × 10⁻⁸	bar⁻1	Pre-exponential factor for CH₄ adsorption equilibrium constant
ΔHCH₄	− 109.68	kJ·mol⁻1	Adsorption enthalpy of CH₄
AkCO2	3.53 × 10⁻⁸	bar⁻1	Pre-exponential factor for CO2 adsorption equilibrium constant
ΔHCO2	− 125.39	kJ·mol⁻1	Adsorption enthalpy of CO2

Dependence of kinetic parameters

The methodology described above does not assume that the kinetic parameters are universal or transferable to different catalyst formulations or reactor configurations. Instead, its primary objective is to examine the internal consistency and predictive limits of literature-based kinetic forms when used independently, thereby elucidating the extent to which reactor-scale behavior is determined by the intrinsic structure of the model. Since the kinetic parameters employed in this study are not re-estimated but are taken directly from the published literature, the work is oriented toward evaluating model behavior rather than parameter identification. Consequently, no parameter sensitivity or identifiability analyses have been performed. The focus is instead on how the reference parameter set propagates through the reactor-scale model and whether the resulting predictions align with experimentally observed conversion trends. The Arrhenius and van 't Hoff relations describe the temperature dependence of the kinetic rate constant and the adsorption equilibrium constants, respectively, as expressed in Eqs. (2)–(4). All kinetic and adsorption parameters were taken directly from¹⁶, where they were initially determined by global non-linear regression across all temperatures and subsequently verified for thermodynamic consistency. Using these published parameter values without any re-fitting ensures that the model in this work is evaluated for its predictive capacity, rather than its ability to fit data through parameter adjustment.

Model and mass balance formulation

The reactor is modeled as an isothermal plug-flow reactor (PFR), consistent with the intrinsic kinetic regime established in the original experimental study¹⁶. In that work, the catalyst was operated under conditions where axial dispersion, external mass transfer, and heat transfer limitations were experimentally verified to be negligible. Under such intrinsic kinetic conditions, the PFR approximation provides a conventional framework for describing the differential reactor behavior commonly employed in the analysis of heterogeneous catalytic kinetics. The reactor was modeled as an isothermal PFR, consistent with the experimental setup in¹⁶. CH₄ conversion was determined by integrating the differential species balance given in Eq. (5). The partial pressures of CH₄ and CO₂ were expressed as functions of CH₄ conversion using Eqs. (8) and (9), respectively. The kinetic rate expression also depends on hydrogen partial pressure due to a competitive adsorption term associated with CH₄ dissociation. Explicit hydrogen and carbon monoxide mass balances were not solved. The approximation in Eq. (11) assumes that hydrogen production is proportional to methane conversion and does not include a term for the hydrogen consumption in the reverse water–gas shift reaction. Therefore, the values of the partial pressures of hydrogen calculated should be considered as effective partial pressures employed for closure of the reduced kinetic model instead of as predictions for the reactor scale. This approximation may affect model predictions at high conversion levels where hydrogen concentrations are significant due to the presence of hydrogen in the inhibition term of the LHHW expression. The cutoff parameter ε was chosen as 10^–8, and its value only affects the solution very near zero conversion. Calculated hydrogen partial pressures were several orders of magnitude higher than ε throughout the entire experimental conversion range; thus, the deviations observed in the low-temperature regime are not caused by the stabilization procedure used in the calculation. According to the literature examined, hydrogen inhibition has been modeled by incorporating it into the adsorption term of the Langmuir–Hinshelwood–Hougen–Watson (LHHW) rate expression rather than through a separate species balance. Although it would be possible to include explicit balances for hydrogen and carbon monoxide in a more detailed reactor model, the present work focuses on evaluating the predictive capability of the simplified kinetic formulation rather than constructing a complete reaction network model. Parallel reactions commonly associated with dry reforming of methane (DRM), such as methane decomposition (CH₄ → C + 2H₂) and the Boudouard reaction (2CO → C + CO₂), are not explicitly included in the kinetic model under consideration. The experimental study from which the kinetic model was derived reported negligible catalyst deactivation and stable conversion patterns under the conditions investigated, indicating that carbon-forming reactions did not dominate the observed regime. The K_RWGS is the equilibrium constant for the reverse reaction of the water–gas shift reaction and is used as an additional thermodynamic constraint in the reduced model of the reactor. It is only used for approximation of the influence of RWGS on the whole DRM-RWGS system in the present work, and it is not considered as an independently validated kinetic quantity. Consequently, the reduced kinetic expression can be confined to the primary DRM pathway and the equilibrated RWGS reaction, consistent with the assumptions of the original model. The reduced model does not contain explicit species balances for CO and H₂O. Thus, the numbers used in the RWGS equilibrium relationship should be treated as effective variables that are used to close the simplified reactor formulation and not as variables that are solved independently.

Model validation

Different statistical measures were used to determine the performance of the model. Equations (12)-(14) define the coefficient of determination, mean absolute error, and root-mean-square error. Effective surface site fractions were further estimated by the kinetic model to indicate the distribution of the surface coverage amongst the methane-derived species, carbon-dioxide-derived species, and free sites, as in Eqs. 15–17. The model was validated using a set of quantitative diagnostics, including parity plots comparing predicted and experimental conversions, residual structure analysis across the full temperature and space–time domain, and statistical error measures. These methods enable a precise evaluation of prediction performance by identifying systematic deviations in the model outputs. Model predictions are considered consistent with experimental data when statistical indicators fall within broadly accepted thresholds for kinetic model validation. Specifically, a coefficient of determination (R² > 0.9) and small absolute deviations between predicted and experimental conversions are used as criteria, as reported in Table 2. These benchmarks provide a practical cutoff for assessing the predictive applicability of the reduced LHHW formulation within the temperature and space–time range investigated.

Table 2.

Statistical indicators used for validation of the LHHW model predictions.

Parameter	Value
Number of experimental observations	24
Temperature range (K)	778–898
Space–time range (gcat·h·mol⁻1)	0.10–0.70
Coefficient of determination (R2)	0.9556
Mean absolute error (MAE)	0.0475
Root mean square error (RMSE)	0.0542

Model agreement with experimental data is good as shown by the statistical indicators, within the range of operation investigated. The high coefficient of determination and relatively small error metrics suggest that the selected LHHW model is able to replicate the experimental conversion behavior without the need for parameter re-estimation.

1

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8

9

10

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12

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14

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17

In Eqs. 1–17, rCH₄ refers to the inherent rate of the conversion of methane; XCH₄ refers to the level of conversion, and W/F_CH4 (gcat·h·mol⁻¹) and W/F⁰ are the space–time factors. The surface reaction rate constant is denoted as k₁, where A_k1 and E_a are the pre-exponential factor and the activation energy of this reaction, respectively. The adsorption constants for CH₄ and CO₂ are denoted by k_CH4 and k_CO2, respectively, with Ak_CH4 and Ak_CO2 representing their respective pre-exponential factors. The adsorption enthalpies of CH₄ and CO₂ are expressed as ΔH_CH4 and ΔH_CO2, respectively. The partial pressures of CH₄, CO₂, and H₂ in the reactor are denoted by p_CH4​​, p_CO2​​, and p_H2​​, respectively, while P^oCH₄​ and P^oCO₂​ indicate the inlet partial pressures of CH₄ and CO₂. The inlet mole fractions of CH₄ and CO₂ are denoted by y_CH4 ​​ and y_CO2​​, respectively. P represents the total reactor pressure, R is the ideal gas constant, and T is the absolute temperature. The parameter ε is a small numerical cutoff used to avoid singularities in the hydrogen pressure calculation at very low methane conversion levels. This parameter has no physical meaning and serves only to maintain numerical stability during reactor integration. The fractional surface coverages of adsorbed CH₄ and CO₂ are denoted by θ_CH4 ​​ and θ_CO2​​, respectively, whereas the fraction of vacant catalytic sites is expressed as θ_free ​. Model accuracy is evaluated using the coefficient of determination (R²), mean absolute error (MAE), and root mean square error (RMSE), where N represents the number of experimental data points¹⁶ included in this investigation.

Scope and limitations

The numerical framework is based on a literature-derived LHHW kinetic model specific to a particular catalyst formulation. Consequently, the kinetic parameters obtained cannot be extrapolated to all DRM catalysts, as adsorption and surface reaction constants vary significantly with catalyst composition and metal-support interactions. The model is evaluated within the experimental context from which the parameters were originally derived and does not include balances for all species or account for side reactions such as methane decomposition or the Boudouard reaction. Hydrogen inhibition is implicitly captured through the adsorption term in the rate expression. Literature data¹⁶ are digitized, which could result in some uncertainty when compared to original experimental data. This uncertainty, determined by repeated digitization and cross-checking, is estimated to be about ± 1–2% in conversion, and has been considered in the evaluation of the statistical validation results (R², MAE, RMSE) reported in the Results section. The LHHW model assumes quasi-equilibrated adsorption and a single type of active site. Therefore, the quantities derived such as apparent activation energies, reaction orders, and surface site fractions should be interpreted as effective parameters that describe the simplified kinetic formulation rather than as true microscopic quantities. Another restriction is the simplified consideration of the partial pressure of hydrogen. The RWGS reaction is not considered explicitly in Eq. (11), and thus hydrogen concentrations may be overestimated for certain operating conditions. The approximation may lead to uncertainty in prediction at high conversion because hydrogen is involved in the inhibition term in the kinetic expression. The orders of reaction that are found to apply to the reaction of H₂ are only apparent and should be considered as the effective numbers involved in the reduced kinetic reaction and not as the mechanistic orders of reaction. Since explicit species balances for hydrogen and carbon monoxide are not solved, the partial pressure of hydrogen calculated is only an approximate closure relation that is consistent with the LHHW model used. Accordingly, the present work constitutes a diagnostic examination of a reduced kinetic model rather than a detailed microkinetic analysis of DRM catalysis.

Results and discussion

Results

Figure 2a demonstrates the relationship between space time and the conversion of methane at the temperatures 778–898 K. At all the studied temperatures, methane conversion rises gradually with space time, indicating a consecutive consumption of reactants across the catalyst bed under the integral plug flow reactor (PFR) mode of operation (W/F, CH₄). This observation is in line with the outcome of the experiment described by Barroso Quiroga and Castro Luna, who achieved monotonic profiles of conversion in a similar set of experiments and did not record any indication of a mass- or heat-transfer restriction. Conversion of methane also shows a strong temperature dependence as per the Arrhenius dependence of the surface-reaction rate constant used in the LHHW kinetic model, where the surface reaction between adsorbed CH₄-derived species and adsorbed CO₂ species is rate-limiting¹⁸. One of the salient properties of the conversion curves as space times are large is a non-linear shape, which implies that there is no ideal first-order behaviour. This curvature also results from the competition between the species formed in the course of the reaction in the adsorbed state from CH₄ and those formed from CO₂ for adsorption on the catalytic surface, changing the effective surface coverage and hence the observed reaction rate¹⁹. Interestingly, this curvature is also replicated by the model without adjusting parameters to prove that the adsorption terms and surface-reaction model, which have been suggested in the literature, are enough to explain the trends that are observed in the experiment.

Fig. 2.

Conversion profiles and model validation results. (a) Methane (CH₄) conversion as a function of space time at temperatures ranging from 778 to 898 K. (b) Carbon dioxide (CO₂) conversion versus space time under identical operating conditions. (c) Parity plot comparing predicted and experimental CH₄ conversion values across the full operating domain. (d) Residual distribution between predicted and experimental CH₄ conversion, with color scale indicating temperature. The results demonstrate strong agreement between model predictions and experimental data.

The same temperature dependence of the conversion of carbon dioxide in space–time is shown in Fig. 2b. In line with other experimental findings, CO₂ conversion is higher than that of CH₄ in all operating conditions. This disparity is typically explained by the simultaneous reverse water–gas shift (RWGS) reaction, which takes in CO₂ and makes hydrogen, and which has been observed to approach thermodynamic equilibrium when the same is taken under similar conditions. In this analysis, the conversion of CO₂ is not modeled separately, but the conversion of CH₄ is obtained through stoichiometric restrictions, by the same assumptions as in the initial kinetic study. The CO₂ conversion trends from the simplified DRM-RWGS approximation are compared to the experimental observations, indicating a good agreement in terms of overall qualitative behavior of the system. Figure 2c makes comparisons of the predicted and experimental conversion of CH₄ through the entire temperature space time²⁰. The data points are closely clustered about the parity line, and this shows clearly that the kinetic model faithfully recreates not only the low-conversion regimes, but also the high-conversion regimes without any systematic bias. This finding correlates with the initial findings of the original research, where it was demonstrated that there is very good agreement when the same kinetic formulation is used¹⁶. Interestingly, the fit here is obtained with digitized kinetic parameters obtained based on van’t Hoff-type temperature dependencies without re-fitting or recalibration. This shows that the proposed kinetic expression is strong against changes in reaction conditions and experimental data. Arrhenius equation is used to represent the temperature dependence of the rate constants. This equation controls the rate constants, and the adsorption equilibrium constants obey van’t Hoff equation, with A_i and E_a,i the pre-exponential factor and activation energy of the i-th elementary or overall reaction step, respectively, and R is the universal gas constant. Apparent activation energies were obtained from the simulated reaction rates by employing a logarithmic derivative approach as described in Eq. 19, where r_CH4 is the consumption rate of methane predicted by the use of the kinetic model at a given W/F°CH₄. This method can be compared with the apparent activation energies obtained from experiments and can point out the adsorption and surface coverage effect on the apparent temperature-dependent activation energies. The comparison of the predicted and experimental conversions of CH₄ in a temperature and space–time range is shown in Fig. 2c. The central parity line is the line of perfect agreement between predicted and experimental conversions (y = x); the upper and lower dashed lines are the ± 10% relative deviation from the central parity line (y = 1.1 × and y = 0.9x, respectively). These bounds are provided as a visual aid for judging the degree of agreement between model predictions and experimental observations. The parity plot suggests a good agreement between the modeling and experimental data within the operating range studied while the values of the reported kinetic parameters are not re-estimated. While some differences are noted especially in higher conversion values, the overall conversion trends are reasonably well reproduced.

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In the literature^21–24, it is reported that apparent activation energies in multi-step processes of heterogeneous catalysis are not necessarily equal to intrinsic barriers, as effects of pathway coupling and coverage can significantly change the apparent activation energy²¹. The apparent activation energy for CH₄ and CO₂ consumption in dry reforming studies has a wide range of values depending on the catalyst and conditions, providing background information on the temperature-dependent reaction results from the present model²². Specific kinetic studies of Ce_0.7La_0.20Ni_0.10O₂ − δ catalysts yield intrinsic activation energies from both LHHW and power-law models, which can be benchmarked for comparisons with apparent values from simulations. The apparent activation energies obtained from simulated rates and Arrhenius plots in multiscale modelling approaches show that they reflect the overall effect of intrinsic kinetic parameters and adsorption phenomena^23,24. Comparisons with the apparent activation energies obtained from experiments capture the role of adsorption on the apparent activation barriers at different temperatures, and the deviation from the intrinsic barrier confirms the mechanistic interpretation of the model.

The variation in the measured and calculated CH₄ conversion with space time and temperature is shown in the colour-scale representation of Fig. 2d. The residuals are evenly spread around zero, and there is no visible reliance on space, time, or temperature. A residual distribution suggests that there is no predominant trend of dependence on space time over the range of operation studied. But a closer look shows that the residuals at the low temperatures are mostly positive, suggesting a slight systematic error at the adsorption-dominated conditions. It is indicative of the low-temperature deviations from the simplified LHHW model when the surface coverage effects are more significant. Although this is a limitation, the overall residual magnitudes are relatively low, and the model has good predictive capability for the range of temperatures studied. Figure 2 below indicates that the LHHW kinetic model is an effective description of CH₄ and CO₂ conversions over a broad temperature-space time regime²⁵. The model replicates non-linear conversion profiles, exhibits unbiased parity behaviour, and produces residuals that are not structured. These properties suggest that the reaction is not controlled by simple power-law kinetics and, as such, surface-mediated reactions are controlled by competitive adsorption, inhibition, and a rate-limiting surface reaction. Figure 3 extends these observations by deriving mechanistic data based on the same kinetic expression of Fig. 2. This analysis clarifies the underlying reasons for the trends observed in Fig. 2.

Fig. 3.

Mechanistic interpretation of kinetic model behavior. (a) Apparent activation energy as a function of temperature, showing the influence of adsorption and surface reaction kinetics. (b) Temperature-dependent prediction errors for CH₄ conversion. (c) Surface site fractions of CH₄-derived species, CO₂-derived species, and vacant sites as a function of space time. (d) Apparent reaction orders of CH₄, CO₂, and H₂ as a function of temperature, illustrating adsorption effects and hydrogen inhibition.

Figure 3a shows the apparent activation energy as a function of temperature, determined from both the kinetic model and experimental conversion data. The apparent activation energy (Ea) of the model is found to be decreasing from 280.83 to 163.41 kJ mol⁻¹ in the range 778–798 K and 873–898 K respectively with intermediate values of 253.34, 220.43, and 189.37 kJ mol⁻¹. The apparent activation energy values indicated that the apparent activation energy is not the intrinsic value of activation energy of the surface reaction (332.04 kJ mol⁻¹) in the LHHW rate expression but is the combination effect of the intrinsic activation energy of the surface reaction and the adsorption equilibrium. This trend explains the strong temperature dependence of CH₄ activation observed in Fig. 2a and the separation of the conversion curves at higher temperatures. Within the adopted kinetic framework, the intrinsic activation energy for the rate-controlling surface reaction is significant and essentially constant at approximately 332 kJ mol⁻¹. However, the observed reaction rate depends not only on this intrinsic barrier, but also on the adsorption equilibria of CH₄ and CO₂²⁶, which appear in the denominator of the LHHW rate expression. At low temperatures, adsorption is favored, and the catalyst surface exhibits high reactant coverage. Increasing temperature simultaneously accelerates the intrinsic surface reaction kinetics through the Arrhenius dependence of the rate constant and reduces reactant surface coverage through the temperature dependence of the adsorption equilibrium constants. Additionally, standard statistical tools were used to evaluate the quantitative performance of the model. The model obtained an overall MAE of 0.0475, an RMSE of 0.0542 and an overall R² of 0.9556. In this operating range, the model calculations are in good agreement with the experimental observations, as reflected by these values.

At low space times, only a small fraction of the catalyst surface is covered, leaving abundant free sites to sustain a high initial reaction rate. As space time increases, CH₄ conversion rises, and its partial pressure falls, reducing the driving force for dissociative CH₄ adsorption²⁷. Consequently, the coverage of CH₄ -derived species declines, while free sites persist. The continued availability of free sites throughout the space–time range explains why CH₄ conversion does not reach complete saturation in Fig. 2, even at long residence times. The evolving ratio of occupied to vacant sites gradually attenuates the reaction rate, providing a dynamic origin for the experimentally observed non-linear conversion profiles. The calculated site fractions are quantities that are derived from the adopted kinetic framework and hence not truly experimental values. These should be recognized against the assumptions of the model, but they do give a good mechanistic understanding of the relative importance of adsorption and surface reaction processes determining the behaviour on the reactor scale. The apparent reaction orders of CH₄, CO_2, and H₂ vs temperature²⁸ in Fig. 3d are obtained by a local sensitivity analysis of the tested rate expression. The seemingly ordered sequence of CH₄ is fractional with a small temperature dependence and not a first-order gas-phase reaction, implying CH₄ activation at high surface coverage. The observed reaction orders in this study should be interpreted as apparent reaction orders under the operating conditions investigated, rather than as intrinsic mechanistic orders that would require independent kinetic experimentation for determination^29,30. The inverse relationship with hydrogen concentration further supports the hypothesis that hydrogen coverage leads to site blocking at lower temperatures, consistent with the adsorption-controlled kinetic regime inferred from the surface coverage analysis. This finding aligns with a mechanism in which dissociative adsorption of CH₄, followed by a surface reaction, constitutes the rate-limiting process.

The order of reaction of CO₂ rises with temperature, which suggests a change in CO₂ kinetics, from adsorption at low temperatures to the surface reaction taking pre-eminence³¹. This tendency justifies the high temperature sensitivity of CO₂ conversion as noticed in Fig. 2b. In contrast, the apparent order for H₂ is approximately 1 and nearly independent of temperature, confirming consistent inhibition by hydrogen. This suppression arises because hydrogen shifts the dissociative adsorption equilibrium of CH₄, reducing the activity of CHₓ surface intermediates as hydrogen partial pressure increases. The temperature independence of this negative order indicates that hydrogen inhibition is an inherent feature of the reaction mechanism. Collectively, Fig. 3 provides the mechanistic foundation for the reactor-scale observations in Fig. 2. It explains the origins of temperature-dependent apparent activation energies, evolving surface coverages, and non-integer reaction orders within a single LHHW kinetic model. This coherent interpretation demonstrates that the chosen kinetic model offers a physically meaningful and self-consistent representation of dry reforming kinetics under the studied conditions.

Among the available studies, the kinetic investigation by Barroso Quiroga and Castro Luna¹⁶ is among the most comprehensive LHHW analyses of the dry reforming of CH₄. It involved discriminating among several mechanistic models and identifying a rate expression that is both statistically and thermodynamically consistent. Their work focused primarily on model discrimination, parameter identification, and validation against experimental conversion data, with an emphasis on the statistical significance and physicochemical plausibility of the fitted parameters. However, as is typical in kinetic studies centered on parameter determination, the analysis in the literature is primarily forward-looking; it applies a kinetic model to simulate experimental conversions but does not necessarily extract all the secondary kinetic information embedded within the validated rate equation³². For instance, the reference article does not present temperature-dependent apparent activation energies, evolving surface site occupancies, apparent reaction orders, or sensitivity coefficients, even though these are implicitly contained within the LHHW formulation used. This work addresses that gap by shifting the focus from parameter identifiability to information extraction. Using the reported kinetic parameters from the literature, obtained via independent digitization, the same selected LHHW model is re-implemented without any re-fitting or recalibration. This makes sure that any following analysis stems only from the intrinsic structure of the reported or literature framework instead of tuned process parameters. Further investigation of prior assumptions indicates that the apparent activation energy is not constant, even when the intrinsic activation energy of the rate-determining step is held at a fixed, high value (≈332 kJ mol⁻¹). Although the original work observes one intrinsic activation energy and comments on the variety of apparent activation energies reported in the literature on dry reforming, it does not explicitly indicate how this variation arises out of the validated kinetic model. In this case, the relationship between temperature-dependent apparent activation energies and the interaction between the kinetics of surface reactions and adsorption equilibria in the LHHW framework is demonstrated to occur naturally. This strategy explains why a large spectrum of observed apparent activation energies can be experimentally observed, despite no catalyst activity and reaction mechanism changes, because of the multiplicative nature of the relationship that is characterized by the model structure³³.

Moreover, even though the expression of the original rate includes adsorption that is competitive, the occupancies of the surface sites are not reported or rebuilt in the original article. In contrast, this study explicitly calculates apparent surface site fractions along the reactor coordinate. The results indicate that, under the investigated conditions, catalyst surfaces remain only partially covered, with a significant fraction of free sites persisting. This observation explains why CH₄ conversion does not saturate with increasing residence time and provides a physical origin for the experimentally observed non-linear conversion profiles. Such insight is crucial for catalyst and reactor design but has been absent from the literature^34–39. Additionally, while the original work identifies the surface reaction between adsorbed CH₄ fragments and adsorbed CO₂ species as rate-limiting, it does not report reaction orders or their temperature dependence. This study demonstrates how fractional, non-integer, and temperature-dependent apparent reaction orders for CH₄, CO₂, and H₂ are derived directly from the established LHHW rate expression. In particular, the consistently negative reaction order for hydrogen provides quantitative evidence of hydrogen inhibition on the CH₄ dissociation equilibria, a phenomenon often discussed qualitatively but seldom derived from an experimentally validated kinetic model. The conclusions that can be drawn from this study should therefore be considered under the conditions of the catalyst and the operating conditions of the original kinetic model. The quantitative trends observed for apparent activation energy, reaction order and surface coverage as reported for the various literature-derived kinetic formulations in the present work, however, are not assumed to be transferable to other DRM catalysts. Further testing of the catalyst systems and operating conditions would be required to ensure general applicability. More generally, the current research shows that a literature-based LHHW model, which has already been validated on its own at this point, can serve as a source of deeper mechanistic information beyond mere data fitting. Apparent activation energies, time-varying surface coverages, reaction orders, and sensitivity parameters derived from the rate expression give a consistent connection between the reactor-scale phenomena and kinetic processes on a microscopic scale. This methodology supplements the traditional kinetic validation and provides a methodology for re-analysis of the existing kinetic models to achieve a more mechanistic insight without new mechanisms suggested or requiring further experimentation.

Discussion

This study is not intended to develop a new intrinsic kinetic model for the dry reforming process. Rather, its purpose is to evaluate the behavior of an existing literature-based LHHW formulation when applied independently within a reactor-scale system. The analysis is directed toward extracting diagnostic information from the mathematical model rather than drawing new mechanistic conclusions about DRM catalysis. The originality of this work therefore lies in demonstrating how an existing LHHW formulation can be interrogated to reveal mechanistic insights that might otherwise remain obscured within simplified kinetic schemes. This approach repurposes literature-based kinetic representations typically used as single-curve fitting tools as diagnostic instruments for interpreting reactor-scale catalytic behavior. A key finding is the explanation for temperature-dependent apparent activation energies, which emerge from the coupling between intrinsic surface reaction kinetics and adsorption equilibria. This result provides a mechanistic rationale for the wide range of apparent activation energies reported in the dry reforming literature, often attributed to different mechanisms or catalyst states. The current discussion shows that this variability can arise even when the intrinsic activation barrier associated with the rate-limiting step remains unchanged. Likewise, the re-creation of apparent surface site fractions shows the catalyst surface as only partially occupied, with a large percentage of free sites remaining along the reactor. This observation explains the absence of full kinetic saturation of the conversion profiles and provides a physical interpretation of the non-linear space–time behaviour, an aspect rarely discussed explicitly in reactor-kinetic investigations despite its relevance to catalyst utilization and reactor design. Estimation of apparent reaction orders also indicates that non-integer, temperature-dependent kinetic orders are an intrinsic property of Langmuir–Hinshelwood kinetics rather than arbitrary fitting parameters. In particular, the consistently negative hydrogen reaction order provides a quantitative measure of H₂ inhibition, a phenomenon that has long been recognized qualitatively in DRM systems. The investigation demonstrates that literature-based kinetic models can be reused as mechanistic diagnostic tools, bridging the gap between observed reactor performance and the underlying surface processes without introducing new mechanistic pathways or re-parameterizing the original kinetic expression.

Conclusion

In this study, the literature-based LHHW of dry reforming of CH₄ was applied and later compared to the experimental data retrieved in the literature, despite the fact that no parameter manipulation was performed. The verified model was used to derive secondary kinetic and mechanistic understanding, which was not outlined previously. The consequent analyses show that apparent activation energies that rely on temperature, fractional reaction orders, hydrogen inhibition, and dynamic site occupancies on the surface become inherent to the kinetic formulation when the model is completely worked out. These results not only give a mechanistic explanation of the heterogeneity of apparent kinetic parameters, which are often observed in the dry reforming literature, but also give support to the fact that such heterogeneity does not necessarily reflect changes in the fundamental reaction mechanism. The way in which adsorption and surface reaction kinetics compete to control the behaviour of the reactor at the reactor scale is demonstrated by the reconstruction of apparent surface coverages and reaction orders. Notably, the research shows that validated kinetic models of such a type can serve as constitutive mechanistic explanatory tools, as opposed to being demoted to the position of a curve-fitting tool. The originality of the current article lies in the organized relocation and interpretation of mechanistic information in an existing model of literature, and, as such, fills a critical gap in knowledge in the dry reforming kinetics. The methodology demonstrated here could be used for other catalytic systems and kinetic models; however, the quantitative results presented in this study are specifically for the catalyst system and operation conditions of the original literature model¹⁶. For more widespread application, the results would have to be confirmed with other sets of data on other catalysts, different feed compositions, pressures, and reactor designs.

Abbreviations

DRM

Dry reforming of methane

LHHW

Langmuir–Hinshelwood–Hougen–Watson

PFR

Plug-flow reactor

RWGS

Reverse water–gas shift

CH₄

Methane

CO₂

Carbon dioxide

CO

Carbon monoxide

H₂

Hydrogen

H₂O

Water

H₂/CO

Hydrogen to carbon monoxide molar ratio

R²

Coefficient of determination

MAE

Mean absolute error

RMSE

Root mean square error

W/F

Space time (catalyst weight to molar flow rate ratio)

W/FCH₄

Space time based on methane feed

W/F₀

Reference space time

XCH₄

Methane conversion

rCH₄

Rate of methane consumption

k₁

Surface reaction rate constant

Aₖ₁

Pre-exponential factor for surface reaction

Eₐ

Activation energy

kCH₄

Adsorption equilibrium constant of methane

kCO₂

Adsorption equilibrium constant of carbon dioxide

k_RWGS

Equilibrium constant of the RWGS reaction

Ak_CH4

Pre-exponential factor for CH₄ adsorption

Ak_CO2

Pre-exponential factor for CO₂ adsorption

ΔH_CH4

Adsorption enthalpy of methane

ΔH_CO2

Adsorption enthalpy of carbon dioxide

p_CH4

Partial pressure of methane

p_CO2

Partial pressure of carbon dioxide

p_H2

Partial pressure of hydrogen

p_CH4⁰

Inlet partial pressure of methane

p_CO2⁰

Inlet partial pressure of carbon dioxide

P_CO

Partial pressure of carbon monoxide

P_H2O

Partial pressure of water

P

Total reactor pressure

y_CH4

Mole fraction of methane

y_CO2

Mole fraction of carbon dioxide

θ_CH4

Surface coverage of methane-derived species

θ_CO2

Surface coverage of carbon dioxide-derived species

θ_free

Fraction of vacant catalytic sites

R

Universal gas constant

T

Absolute temperature

ε

Numerical cutoff parameter

N

Number of experimental data points

x_exp

Experimental conversion

x_model

Model-predicted conversion

Author contributions

Abdulaziz A. M. Abahussain: Investigation, Writing – original draft. Norah Alwadai: Data analysis, Review. Rao Adeel Un Nabi: Conceptualization, Supervision, Writing – review & editing. Abdulrahman Faraj Alharbi: Methodology, Validation. Fekri Abdulraqeb Ahmed Ali: Data curation. Ahmed S. Al-Fatesh: Supervision, Funding acquisition. Hassan Abbas Khawaja: Review and editing. Ahmed E. Abasaeed: Project administration.

Funding

Researchers Supporting Project (PNURSP2026R11). Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Data availability

The datasets used during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Ethics and consent to participate

Not applicable.

Consent for publication

Not applicable.