Assessment of the impact of water and methanol removal on the kinetics of methanol synthesis over CZA catalysts for sorption-enhanced applications

1 Introduction

The development of accurate kinetic models constitutes a fundamental aspect of chemical reaction engineering, providing the basis for understanding, designing, and optimizing catalytic processes. Kinetic modeling helps predict reaction behavior under varying conditions and improves understanding of the links between reaction mechanisms, transport, and thermodynamics. Particularly in catalytic systems involving competitive reactions, reliable kinetic descriptions are essential for understanding rate-limiting steps and for identifying optimal process conditions to maximize product selectivity and yield (Bisotti et al., 2022).

Methanol synthesis via CO₂ and CO hydrogenation has gained renewed attention as a promising route for chemical energy storage and CO₂ valorization (Goeppert et al., 2018; Olah et al., 2018). This process enables the direct conversion of synthesis gas or captured CO₂ with hydrogen, preferably derived from renewable sources, into methanol. This molecule can be used as a fuel, hydrogen carrier, or feedstock for chemicals. The overall reaction network includes the hydrogenation of CO₂ to methanol, the hydrogenation of CO to methanol, and the reverse water–gas shift (RWGS) reaction. These reactions are typically catalyzed by Cu/ZnO/Al₂O₃ materials (CZA), which remain the industrial benchmark due to their high activity and selectivity under moderate pressures and temperatures (Sehested, 2019). Over the past decades, numerous kinetic models for CZA catalysts have been proposed to describe methanol synthesis from syngas or CO₂-rich feeds (Bussche and Froment, 1996; Graaf et al., 1988; Park et al., 2014; Seidel et al., 2018; Slotboom et al., 2020; Bisotti et al., 2021). Most models are based on Langmuir–Hinshelwood–Hougen–Watson (LHHW) equation formalism that account for surface adsorption, surface reactions, and desorption steps of key intermediates.

Recent interest in sorption-enhanced and membrane reactor concepts introduces new challenges for the kinetic modeling of methanol synthesis (Dehghani et al., 2014; Hamedi et al., 2021; Maksimov et al., 2022; Menéndez et al., 2024; Hauth et al., 2025). In these systems, the in situ removal of reaction products, typically water or methanol, shifts the local reaction equilibrium and modify the concentration of molecules at the catalyst surface. Consequently, the apparent kinetics and rate parameters can differ significantly from those observed in conventional reactors. To describe these effects, kinetic models must explicitly account for product sorption within the catalytic-sorbent bed, as this phenomenon strongly influences the local reaction environment and observed reaction rates. Developing such models is essential for accurately describing the behavior of intensified systems and for enabling predictive reactor simulations to support process design and optimization (Bisotti et al., 2021).

The present work aims to address these challenges by improving the understanding of methanol synthesis kinetics over CZA catalysts under sorption-enhanced conditions. To this end, the objective is to identify a kinetic model that remains applicable in sorption-enhanced reactors by evaluating conventional kinetic formulations using a dedicated experimental setup specifically designed to reproduce product-depletion conditions. Experiments were conducted in a simulated sorption-enhanced reactor, accounting for the selective removal of water and/or methanol to investigate the influence of product depletion on reaction rates. Indeed, to generate data representative of operation under water and/or methanol depletion conditions, the most straightforward and intuitive approach would be to perform experiments combining a catalytic bed with a suitable sorbent. However, several technical challenges must be considered:

• ➢ Such systems inherently operate under transient conditions during the sorption step, which prevents the acquisition of steady-state data, a prerequisite for the establishment of reliable kinetic laws.
• ➢ Under transient sorption-enhanced operation, the reaction mixture composition evolves continuously, which complicates the accurate determination of inlet and outlet molar flows. The characteristic timescales of these transients are typically shorter than the sampling times required for accurate analytical measurements, resulting in time-averaged data that do not accurately reflect the instantaneous system behavior.
• ➢ The variation in the amounts and relative proportions of the species to be trapped, depending on the experimental conditions, would require extensive knowledge of a wide range of sorbents with specific sorption properties, which exceeds the currently available expertise or resources. Moreover, a new catalytic bed would need to be prepared for each experiment, potentially introducing an additional experimental variable.

To overcome these limitations, an alternative experimental approach was implemented to represent a sorption-enhanced reactor process using a fictitious bed. This approach combines a sorption-enhanced simulated bed with an effective catalytic bed, in which water and/or methanol depletion is imposed at the reactor inlet, more exhaustively described in the Experimental section. Then, the experimental data were compared with established kinetic models for CZA catalysts reported in the literature, namely, those of Graaf et al. (1988), Park et al. (2014), Seidel et al. (2018), and Slotboom et al. (2020). These models were selected to represent different levels of complexity and mechanistic assumptions commonly used to describe methanol synthesis. The model of Graaf et al. (1988) is one of the most widely referenced formulations and serves as a benchmark for conventional reactor kinetics. The model proposed by Park et al. (2014), derived from Graaf’s approach, introduces additional adsorption sites, offering a more detailed catalyst surface description. Seidel et al. (2018) developed a model available in both steady-state and dynamic formulations, making it particularly relevant for systems involving transient or non-conventional reactor behavior. Finally, the recent model of Slotboom et al. (2020) was included because it does not consider CO hydrogenation, providing an alternative mechanistic viewpoint that is particularly relevant under the operating conditions investigated in this study. Finally, the results are analyzed and discussed to evaluate the applicability of conventional kinetic models under intensified reaction conditions and to identify potential modifications required for accurate representation of sorption-enhanced systems.

2 Kinetic model and operating conditions

2.1 Operating conditions and catalysts

Table 1 summarizes the catalysts used in the literature, the operating conditions, and the number of constants in each model for a given temperature. Each model also corresponds to a distinct reaction mechanism and rate-determining step, which are not discussed in detail in this paper but are illustrated in Figure 1 (adapted from Slotboom et al. (2020)).

TABLE 1. Summary of the kinetic models, CZA catalyst, operating conditions, and number of fitted parameters.

2.2 Kinetic model description

The overall reaction network comprises the exothermic hydrogenations of CO and CO₂, along with the endothermic RWGS reaction, as represented in Equations 1–3. The Slotboom kinetic model’s neglects the CO hydrogenation pathway, considering its minor or negligible effect under typical methanol synthesis conditions.

C O 2 + 3 H 2 ⇌ C H 3 O H + H 2 O Δ r H 298 K 0 = − 49.2 k J / m o l (1)

C O 2 + H 2 ⇌ C O + H 2 O Δ r H 298 K 0 = 41.6 k J / m o l (2)

C O + 2 H 2 ⇌ C H 3 O H Δ r H 298 K 0 = − 90.8 k J / m o l (3)

The thermodynamic equilibrium constants (K_p°) are calculated in this work according to the expressions proposed by Skrzypek et al. (1994) detailed in Equations 4–6.with K p C O 2 in bar^-2 and T in Kwith K p R W G S dimensionlesswith K p C O in bar^-2

log K p C O 2 = 15.0921 + 1571.7 T − 8.7639 × log T + 2.1105 × 10 − 3 × T − 1.9303 × 10 − 7 × T 2 (4)

log K p R W G S = 1.2777 − 2167 T + 0.5194 × log T − 1.037 × 10 − 3 × T − 2331 × 10 − 7 × T 2 (5)

log K p C O = 13.8140 + 3784.4 T − 9.2833 × log T + 3.1475 × 10 − 3 × T − 4.2613 × 10 − 7 × T 2 (6)

2.2.1 Graaf model

Graaf et al. (1988) developed a kinetic model based on the three reactions described in Equations 1–3, which occur on two types of adsorption sites: Θ G A, associated with CO and CO₂ activation, and θ G B, representing the interaction with H₂ and H₂O (Equations 7–11).

r C O 2 = k C O 2 K C O 2 f C O 2 f H 2 3 2 − 1 K p C O 2 T · f C H 3 O H f H 2 O f H 2 3 2 Θ G A θ G B (7)

r R W G S = k R W G S K C O 2 f C O 2 f H 2 − 1 K p R W G S T f H 2 O f C O Θ G A θ G B (8)

r C O = k C O K C O f C O f H 2 3 2 − 1 K p C O T f C H 3 O H f H 2 1 2 Θ G A θ G B (9)

Θ G A = 1 + K C O f C O + K C O 2 f C O 2 − 1 (10)

θ G B = f H 2 1 2 + K H 2 O K H 2 1 2 f H 2 O − 1 (11)

2.2.2 Park model

In the kinetic model proposed by Park et al. (2014), the three chemical reactions (Equations 1–3) are also considered, with adsorption assumed to take place on three distinct surface sites. Θ s 1 and Θ s 3 are attributed to the activation of CO and CO₂, respectively, whereas Θ s 2 accounts for the adsorption of H₂ and H₂O species (Equations 12–17).

r C O 2 = k C O 2 K C O 2 f C O 2 f H 2 3 2 − 1 K p C O 2 T * f C H 3 O H f H 2 O f H 2 3 2 Θ s 2 Θ s 3 (12)

r R W G S = k R W G S K C O 2 f C O 2 f H 2 − 1 K p R W G S T f H 2 O f C O Θ s 2 Θ s 3 (13)

r C O = k C O K C O f C O f H 2 3 2 − 1 K p C O T f C H 3 O H f H 2 1 2 Θ s 1 Θ s 2 (14)

Θ s 1 = 1 + K C O f C O − 1 (15)

Θ s 3 = 1 + K C O 2 f C O 2 − 1 (16)

Θ s 2 = 1 + K H 2 f H 2 + K H 2 O f H 2 O − 1 (17)

2.2.3 Seidel model (simplified)

Seidel et al. (2018) proposed a kinetic model with several levels of complexity, which can be adapted to both dynamic and steady-state systems. In the present work, the simplest version of this model was employed. As in the previous models, the three chemical reactions are considered. In addition, the rate expression includes a parameter, ϕ, which accounts for the influence of catalyst surface morphology depending on its reduced or oxidized state. Three types of adsorption sites are defined: Θ ⊙, associated with CO adsorption; Θ *, representing the interaction of H₂O, CO₂, and H₂ with the solid; and Θ ⨂, corresponding to the activation of H₂ alone (Equations 18–26).

r C O 2 = ϕ 2 k C O 2 f C O 2 f H 2 2 1 − 1 K p C O 2 T f C H 3 O H f H 2 O f C O 2 f H 2 3 Θ * 2 Θ ⨂ 4 (18)

r R W G S = ϕ 1 − ϕ k R W G S f C O 2 1 − 1 K p R W G S T * f C O f H 2 O f C O 2 f H 2 Θ * Θ ⊙ (19)

r C O = 1 − ϕ k C O f C O f H 2 2 1 − 1 K p C O T f C H 3 O H f C O f H 2 2 Θ ⊙ Θ ⨂ 4 (20)

Θ ⊙ = 1 + K C O f C O − 1 (21)

Θ * = 1 + K H 2 O K O K H 2 f H 2 O f H 2 + K C O 2 f C O 2 + K H 2 O f H 2 O − 1 (22)

Θ ⨂ = 1 + K H 2 f H 2 − 1 (23)

ϕ = 1 2 1 − 1 − K 1 K 2 f H 2 f C O f H 2 O f C O 2 1 + K 1 K 2 f H 2 f C O f H 2 O f C O 2 – 0 , 1 (24)

K 1 = exp − Δ G 1 R T (25)

K 2 = exp − Δ G 2 R T (26)

2.2.4 Slotboom model (simplified)

The model proposed by Slotboom et al. (2020) considers only the hydrogenation of CO₂ and RWGS reactions. In this simplified approach, a single type of adsorption site (θ *) is assumed to be available for the adsorption of H₂, H₂O, and CH₃OH, whereas the surface coverage of CO and CO₂ is assumed to be negligible (Equations 27–29).

r C O 2 = k C O 2 f C O 2 f H 2 3 2 1 − 1 K p C O 2 T f C H 3 O H f H 2 O f H 2 3 f C O 2 θ * 2 (27)

r R W G S = k R W G S f C O 2 f H 2 1 2 1 − 1 K p R W G S T f C O f H 2 O f C O 2 f H 2 θ * (28)

θ * = f H 2 1 2 K H 2 + f H 2 O K H 2 O + f C H 3 O H − 1 (29)

3 Experimental, operating conditions and data exploitation

3.1 Experimental setup

3.1.1 Simulation of sorption-enhanced reactor under steady-state conditions

The setup can be considered as two consecutive fixed beds (Figure 2). The first, a fictitious bed, is simulated by a feed composed of a gas mixture of H₂, CO₂, CO, CH₃OH and H₂O with precisely controlled molar flow rates. The composition of this mixture is selected to represent that of a catalytic-sorbent bed operating in a sorption-enhanced reaction process (SERP), with predefined values of CO₂ conversion, selectivities, and fractions of water and methanol removed. The second bed is a real catalytic bed, consisting of a CZA catalyst (Alfa Aesar, ref. 45776) diluted in silicon carbide, whose catalytic performance depends on the composition of the upstream-generated mixture. The same fixed bed was used throughout the entire experimental campaign to avoid variability associated with changes in catalyst mass, packing, or position within the reactor. Prior to the study, the catalyst was aged under reaction conditions for 48 h to ensure operation at a stable productivity plateau. In the following sections, this second bed will be referred to as the “effective bed”, in contrast to the upstream “fictitious bed”. This configuration enables the evaluation of catalyst performance through the flow rates of the products formed under feed representative of SERP operation (Equation 30).

F p r o d , e f f . = F o u t , e f f . − F i n , e f f . (30)

3.1.2 Experimental apparatus

The experimental setup used in this study is shown in Figure 3. The reactor feed is supplied by three mass flow controllers (Brooks, Delta Smart Mass Flow) connected to gas cylinders containing hydrogen (purity 4.5), carbon monoxide (purity 3.7), and a ternary mixture composed of 65.6%_mol H₂, 22.0%_mol CO₂, and 12.4%_mol N₂. The system is assumed to be without pressure drop throughout the setup, and the pressure is measured using a manometer with an uncertainty of 0.1 bar within the operating range (Keller, Leo2). The incoming gases then pass through a bubbler containing a controlled H₂O/CH₃OH liquid mixture maintained at a precise temperature. Due to the high amount of liquid in the bubbler with respect to the water and methanol gas flows exiting the bubbler, its composition is assumed to be constant over the duration of the test. This configuration allows for the generation of controlled vapor flow rates of water and methanol, depending on the bubbler temperature and the composition of the liquid phase.

The reactor, with an internal diameter of 6.35 mm and a length of 30 cm, is heated using an electrical resistance element. A type-K thermocouple positioned at the center of the reactor and connected to a control interface (Coreci, MiniCore) regulates the temperature at the reactor midpoint. This thermocouple rests above a thin layer of quartz wool that supports the catalytic bed. The system pressure is controlled using a piston-type back-pressure regulator (Tescom, model 26–1700). A three-way valve directs the outlet gas either toward analysis or to a vent. Downstream of the reactor and up to the analyzer, all lines and components are heated at 140 °C to prevent condensation of H₂O or CH₃OH.

3.1.3 Analytical system

Gas analysis at the reactor outlet was performed using modified Shimadzu (Nexus GC-2030) gas chromatography (GC). The instrument configuration, equipped with two parallel columns, enabled the acquisition of chromatograms containing multiple species. To prevent condensation and ensure complete analysis of the outlet stream, the entire sampling system was maintained at an elevated temperature. Two six-way valves, each equipped with a 250 µL injection loop, were used to introduce the outlet gas into two parallel analytical lines housed within the same oven. Each line included a split module to adjust the amount of gas directed into the respective columns. The first column MS5A column (15 m × 0.25 mm) was dedicated to the analysis of light gases, H₂, N₂, CH₄, and CO, using argon as the carrier gas. A pre-column (RTU, 7.5 m × 0.25 mm) combined with a backflush system allowed for the separation of these light gases from water and methanol. The second line, using helium as the carrier gas, was equipped with an RTQ column (15 m × 0.25 mm) for the separation of CO₂, H₂O, and CH₃OH. Chromatograms were recorded using two thermal conductivity detectors (TCDs), positioned downstream of each analytical line. The experimental setup, combined with the analytical system, allows direct quantification of all chemical species.

3.1.4 Verification of kinetic regime

The effective catalytic bed consisted of 67.4 mg of catalyst (ρ_app = 1899 kg/m³) diluted with 271.5 mg of SiC (ρ_app = 961 kg/m³) provided by SiCat. The solids were crushed and sieved to a particle size range of 63–125 µm. The operating conditions were selected to ensure a kinetically controlled regime. According to the correlation proposed by Berger et al. (2002), the deviation in conversion attributable to diffusion effects (Δ) was negligible under the present conditions (Equation 31). For a CO₂ conversion of 5.0%, representing the upper range of the expected values, Δ was estimated at only 0.20%, confirming that mass transfer limitations were insignificant.where b is the volumetric dilution factor, d p the particle diameter (taken as the mean value of the selected particle size range, i.e., 94 µm), L B the bed length, and X C O 2 the CO₂ conversion.

Δ = b 1 − b · d p L B · X C O 2 2 (31)

In addition, the uniformity of flow through the plug-flow reactor was theoretically verified by ensuring compliance with the Mears and Gierman criteria, according to inequalities (Equations 32, 33) (Mears, 1971; Gierman, 1988). These conditions were satisfied, as the left-hand term of the inequality (Equation 32) was equal to 68. For the inequality (Equation 33), the left-hand side is equal to 100, while the right-hand side is equal to 12, thereby validating this inequality. Here, d _*t* denotes the bed diameter, n the reaction order and B o the Bodenstein number.

d t d p > 10 (32)

L B d p > 8 B o n · ⁡ ln 1 1 − X C O 2 (33)

3.2 Operating conditions

The compositions of the mixtures targeted correspond to the feed of the effective bed, assuming an upstream CO₂ conversion of 5%–10% with a selectivity of 50% toward CH₃OH, based on an inlet mixture to the fictitious bed (H₂/CO₂ = 3.0) that may contain CO. Approximately 30% of the total amount of water and methanol produced are removed by fictitious sorption, with a methanol sorption selectivity (noted S r e a c t i o n , C H 3 O H) between 33% and 67%. The ranges of these parameters are reported in Table 2, while detailed information on all experiments is provided in the Supplementary Material.

S s o r p t i o n , C H 3 O H = F s o r b e d , C H 3 O H F s o r b e d , C H 3 O H + F s o r b e d , H 2 O (34)

TABLE 2. Variables and ranges targeted for the different parameters simulated in the fictitious bed.

Due to the difficulty of achieving absolute control over the vapor pressure entrained from the bubbler, the target values were only approximately reached. However, since the flow rates used for the regression correspond to the measured values, these uncertainties do not introduce errors in the regression results.

3.3 Data exploitation

3.3.1 Partial pressure instead of fugacity

In most literature models, reaction rates are expressed in terms of fugacity, as the high operating pressures can cause noticeable deviations from ideal gas behavior (Graaf et al., 1988; Park et al., 2014; Seidel et al., 2018; Slotboom et al., 2020). The use of fugacity is particularly justified in high-pressure studies with significant water and methanol contents, as these species are more sensitive to non-ideal effects.

However, the calculation of fugacity coefficients requires an equation of state with parameters defined for a fixed mixture composition. Since the gas composition varies along the reactor, this approach becomes tedious due to its dependence on conversion. Slotboom et al. (2020) reported fugacity coefficients at 30 bar and higher conversions than those considered here, showing only small deviations from ideality: +2% for H₂, N₂, and CO, 0% for CO₂, and –6% and −4% for H₂O and CH₃OH, respectively. Based on the work of Bennekom et al. (2012), the use of partial pressures instead of fugacities is considered valid up to 50 bar, provided that H₂O and CH₃OH molar fractions remain below 10%. As these conditions are well above those of the present study, the use of partial pressures is deemed acceptable.

3.3.2 Regression method

The regression procedure employed in this study is based on minimizing the deviation between the experimental and modeled molar flow rates of CO and CH₃OH formed in each experiment, as expressed in Equation 35. The use of the experimental flow rate in the denominator allows the deviations to be expressed as a proportion relative to the corresponding experimental product flow rate. This approach provides an accurate representation of flow rate variations while making the model fitting independent of the conversion level in each experiment. Without this normalization, experiments yielding higher product flow rates would exalt a disproportionate influence compared to those with lower conversions, leading to a regression primarily weighted toward high-conversion data and an unbalanced representation of the entire dataset.

F C H 3 O H , exp p r o d , e f f . − F C H 3 O H , mod p r o d , e f f . F C H 3 O H , exp p r o d , e f f . 2 + F C O , exp p r o d , e f f . − F C O , mod p r o d , e f f . F C O , exp p r o d , e f f . 2 = 0 (35)

The minimization of Equation 36 across all experiments was performed by iteratively adjusting the model parameters using the Excel Solver with a Levenberg–Marquardt-type minimization algorithm. For the regression procedure, the piston-like catalytic bed was modeled as a series of 100 perfectly stirred open reactors. In the first step, the kinetic and sorption constants were fitted individually at each temperature. Subsequently, these values were progressively refined to ensure their consistency with a pseudo-Arrhenius relationship. The optimization procedure followed the rules described in Equations 36–38.

Rule 1 : k i > 0 (36)

Rule 2 : E A > 0 (37)

Rule 3 : Δ H a d s 0 < 0 (38)

The kinetic and adsorption constants were finally expressed in pseudo-Arrhenius form, as shown in Equations 39, 40.

K i = A i ⁡ exp B i R T = exp Δ S a d s 0 R exp − Δ H a d s 0 R T (39)

k i = A i exp ⁡ − E A R T (40)

4 Results and discussion

4.1 Operating conditions

To better illustrate the experimental conditions imposed by the fictitious bed and to highlight the depletion aspect of the experiments, Figure 4 is presented. The molar CH₃OH inlet percentages of experiments conducted at 33 bar (blue circles), and at 23 and 46 bar (orange squares) are plotted as a function of corresponding molar H₂O inlet percentages, which implicitly reflect the sorbent’s sorption selectivity as defined by the Equation 34. The dashed line in Figure 4 illustrates the methanol selectivity that would have been obtained without sorption. Points located above the line therefore correspond to water-depleted conditions, while those below correspond to methanol-depleted conditions. In this representation, experiments located below the dashed line correspond to conditions with an excess of H₂O relative to methanol, while those above the line correspond to H₂O depleted conditions. For a reactor with a reaction selectivity of 50% toward CH₃OH, which is targeted in this study and more representative of realistic SERP operation, nearly all experiments (with the exception of a single case) exhibit higher sorption selectivity toward H₂O than toward CH₃OH. This representation highlights the extent of H₂O depletion experienced by the system for a 50% reaction selectivity.

The fact that most points indicate H₂O depletion does not fully reflect our initial aim of exploring a wide range of inlet compositions, include CH₃OH depletion experiments, for a complete kinetic study. This outcome can be attributed to the difficulty of maintaining higher H₂O partial pressures than CH₃OH using the bubbler. Nevertheless, this representation of the SERP system for methanol synthesis, operating under H₂O depleted conditions, is consistent with the behavior of hydrophilic zeolite-based sorbents reported in the literature (Maksimov et al., 2021; Heracleous et al., 2024).

4.2 Experiments selection

Due to relatively low conversions in the effective bed, particularly at 210 °C, some experiments were excluded from the regression used to evaluate the models. Of the total number of the experiments conducted at 33 bar, only 50 were retained for regression. During the regression procedure, additional experiments were further discarded according to the criteria defined in Equations 41, 42, in order to achieve better convergence toward a clear solution. The number of experiments used for parameter estimation thus varied depending on the model, ranging from 28 to 34. Twenty experiments were included in the regression for all models and were consequently used to illustrate model fitting in the parity plots presented in Figures 5, 6. A summary of the experiment selection is provided in Table 3.

TABLE 3. Summary of the experiment selection.

For experiments at 230; 250 °C and 270 °C:

F C H 3 O H , exp p r o d , e f f . − F C H 3 O H , mod p r o d , e f f . F C H 3 O H , exp p r o d , e f f . 2 + F C O , exp p r o d , e f f . − F C O , mod p r o d , e f f . F C O , exp p r o d , e f f . 2 < 0.2 (41)

For experiments at 210 °C, the criterion is less stringent due to the low conversion, which results in lower product flow rates and consequently higher experimental uncertainties.

F C H 3 O H , exp p r o d , e f f . − F C H 3 O H , mod p r o d , e f f . F C H 3 O H , exp p r o d , e f f . 2 + F C O , exp p r o d , e f f . − F C O , mod p r o d , e f f . F C O , exp p r o d , e f f . 2 < 0.6 (42)

4.3 Adjustment at 33 bar

The regression procedure was applied to all models, with the CO hydrogenation suppressed for the Graaf and Seidel formulations. All kinetic and sorption constants are reported in the Supplementary Material. The kinetic constant associated with CO hydrogenation (k C O) was intentionally set to zero for these two models, as preliminary regressions showed that this reaction had a negligible influence on the results and could even deteriorate the quality of the fit. This approach contradicts the mathematical principle that an additional degree of freedom should improve the fit. However, the virtually negligible influence of k C O in the modelling highlights the difficulty of demonstrating the contribution of methanol formation from CO under the operating conditions studied. Thus, the presence of r C O may either reflect a real phenomenon occurring in the system or result from an artificial adjustment of the other reaction rates, r C O 2 and r R W G S.

The results of the regressions are presented as parity diagrams in Figure 5, with each color corresponding to one of the models. The CO₂ conversion in most experiments lies within the ±20% interval shown by the dashed lines, and this holds true for all models (Figure 5a).

When each reaction product is examined independently, a clear trend appears at higher conversions: the models tend to slightly underestimate the CO produced at the highest formation rates (Figure 5b). Retaining the CO hydrogenation step would have further accentuated this deviation in the Graaf and Seidel models. For methanol, the model adjustments are acceptable but somewhat less accurate (Figure 5c). However, as with CO, none of the models demonstrates a distinctly superior performance.

The error is evaluated as the mean of the absolute percentage deviations between the modeled and experimentally measured produced methanol and CO flow rates (Equation 43). This approach ensures that each experiment contributes equally to the overall error, regardless of its conversion level, and the results are summarized in Table 4.

% e r r , i ¯ = ∑ F i , exp − F i , mod F i , exp (43)

TABLE 4. Mean absolute deviations obtained during the adjustment step for the four kinetic models studied.

Where % e r r , i is the deviation between the modeled and experimental produced flow rate in percentage, and i = CH₃OH or CO.

Based on the fits obtained at 33 bar (Figure 5) and mean errors (Table 4), and also considering the limited number of experiments, all models are able to reasonably reproduce the observed product formation rates. Consequently, it is not possible to identify any model as being significantly superior to the others. This outcome can be explained by the relatively narrow range of operating conditions explored at the same total pressure, which limits the ability to discriminate between kinetic formulations that share many similarities. Nevertheless, the parameters obtained from this adjustment step provide valuable information for the subsequent prediction stage.

4.4 Prediction at 23 and 46 bar

Once the regressions were performed using the experiments at 33 bar, the resulting parameter values were used to predict the amounts of CO and CH₃OH formed in experiments carried out at 23 and 46 bar (Figures 6a,b). The mean absolute deviations for these predictions are summarized in Table 5.

TABLE 5. Mean absolute deviations obtained during the adjustment step for the four kinetic models studied.

It appears that the Park model performs less accurately when pressure varies. In particular, the amount of CH₃OH produced is significantly overestimated compared to the experimental data (Figure 6b). This trend is especially noteworthy because this model is the only one that includes the r C O reaction. Additional regressions were carried out with this reaction removed. However, excluding it from the kinetic model did not improve the quality of the fits. Consequently, the fact that this model yields the poorest predictions suggests that the r C O reaction plays a very limited, or even negligible, role in CH₃OH formation under the investigated operating conditions.

This observation is consistent with previous studies in which CO hydrogenation was neglected, including the classical model of Van den Bussche and Froment(Bussche and Froment, 1996) as well as more recent formulations, such as that proposed by Van Schagen et al. (2025).

Similarly to the results obtained during parameter adjustment, the prediction quality for the experiments at 23 and 46 bar is comparable for the Graaf model (7 parameters) as well as for the Seidel and Slotboom models. The deviations between predicted and experimental values remain small and can partly be attributed to experimental uncertainties, making it difficult to clearly distinguish the models in terms of predictive performance.

It is nevertheless noteworthy that models with fewer parameters provide predictions of similar quality to those of more sophisticated formulations. The Slotboom model, which contains only six adjustable parameters, achieves results comparable to the Seidel model, despite the latter relying on twelve parameters. This suggests that several of the additional parameters exert only a minor influence on the overall model behavior.

Examining the model structures reveals a key difference: in contrast to the Graaf and Seidel models, the Slotboom model neglects the adsorption of CO₂ and CO on the catalyst surface, in addition to discarding the CO hydrogenation reaction. This simplification appears corroborated by the regression results obtained here. The fitted adsorption constants for CO and CO₂ in the Graaf and Seidel formulations are indeed found to be significantly smaller than those associated with H₂ and H₂O, indicating a negligible contribution of CO and CO₂-related adsorption to the calculation of free surface sites under the studied conditions.

Finally, the pressure and temperature conditions investigated in this study differ from those commonly reported in the literature (Figure 7). The explored operating range partially overlaps with that of the models developed by Graaf and Slotboom, whereas the Park and Seidel models were originally formulated for higher pressures and broader temperature ranges. Consequently, the parameter values determined in this work are expected to be valid primarily for operating conditions similar to those investigated here, which are representative of the reaction–sorption environment encountered in a SERP.

5 Conclusion

This work aimed to identify kinetic models suitable for describing methanol synthesis in a sorption-enhanced reactor or membrane reactor. To this end, a dedicated experimental setup was designed to emulate SERP operation under steady-state conditions. The resulting dataset was fitted using established kinetic models developed for conventional methanol reactors, namely, those of Graaf, Park, Seidel, and Slotboom. A first dataset obtained at 33 bar was used for parameter estimation, while a second dataset collected at 23 and 46 bar served to evaluate the predictive capabilities of the models.

• All kinetic models evaluated in this work provide a satisfactory fit to the SERP-like experimental conditions investigated.
• The contribution of CO hydrogenation is negligible under the investigated conditions.
• Models with fewer adjustable parameters (7-parameter Graaf and the simplified Slotboom models) achieve fitting performances comparable to those of more complex formulations. In models where the adsorption constants of CO and CO₂ are included, their values are significantly lower than those of H₂ and H₂O, suggesting a limited role of CO and CO₂-related adsorption phenomena in the kinetic representation of the system.
• The Graaf (7-parameter), Seidel (12-parameter), and Slotboom models successfully predict the quantities of CO and CH₃OH formed at 23 and 46 bar, demonstrating their robustness when extrapolated beyond the regression conditions.

Although the development of a dedicated kinetic model for SERP operation is premature due to the limited dataset and pressure range explored, this work demonstrates the potential applicability of conventional methanol-synthesis kinetics to sorption-enhanced systems. Advancing the understanding and modeling of water-depletion reactors remains an important task for the scientific community, as such intensified reactor concepts are expected to play a significant role in the future of chemical process design (Nieminen et al., 2022; Boon, 2023).

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fceng.2026.1802912/full#supplementary-material

Supplementary material Supplementary material at the source

Nomenclature

Δ r H 298 K 0 Standard reaction enthalpy K p Thermodynamic equilibrium constant r Reaction rate f Fugacity k Kinetic rate constant K Sorption equilibrium constant Θ Fraction of free surface sites ϕ Total amount of relative reduced surface sites θ Fugacity-dependent free surface coverage F x i n , e f f . Inlet molar flow rate of species x to the effective reactor F x o u t , e f f . Outlet molar flow rate of species x from the effective reactor F y p r o d , e f f . Molar flow rate of species y produced in the effective reactor ρ _**app** Apparent density Δ Conversion deviation due to dilution b Dilution factor d p Particle diameter L B Bed length X C O 2 CO₂ conversion B o Bodenstein number n Reaction order S s o r p t i o n , C H 3 O H Methanol sorption selectivity S r e a c t i o n , C H 3 O H Methanol reaction selectivity (in fictitious bed) Δ S a d s 0 Standard adsorption entropy − Δ H a d s 0 Standard adsorption enthalpy − E A Activation energy % e r r , i Deviation between the modeled and experimental flow rate of i

Footnotes

Edited by: Sergio Rojas, Spanish National Research Council (CSIC), Spain

Reviewed by: Cristina Peinado, Spanish National Research Council (CSIC), Spain

Jhulimar Castro, University of Canterbury, New Zealand