Kinetic Modeling for Coke Combustion over a Nickel Catalyst Deactivated under Methane Dry Reforming Conditions

Abstract

This study validates the effectiveness of a composite kinetic processing approach combining the isoconversional method and the master-plot method to determine the kinetic triplet (reaction model, apparent pre-exponential factor, and activation energy) for the nonisothermal combustion at various heating rates of a coked nickel catalyst employed in the dry reforming of methane. First, the average apparent activation energy (164.5 kJ mol^–1) was determined using the Kissinger-Akahira-Sunose (KAS) isoconversional method. Next, the master-plot method was applied to establish the regeneration model function, revealing that the combustion of the carbon nanotubes followed the reaction order model (n = 1.36). The consistency of the regeneration kinetics model was validated by comparing experimental and calculated thermoanalytical curves at a constant heating rate.

1. Introduction

Carbon dioxide (CO₂) and methane (CH₄) are the primary anthropogenic emissions that contribute significantly to the rise in atmospheric greenhouse gases (GHGs) concentrations. While methane is present in lower quantities compared to carbon dioxide, it is 21–25 times more potent as a greenhouse gas. The detrimental effects of GHGs on climate change, including global temperature increases, changes in weather patterns, and more frequent extreme weather events, have led to regulatory initiatives to manage these emissions. Catalytic dry reforming of methane (DRM, CH₄+CO₂→2CO+2H₂) offers a promising approach to mitigating CO₂ and CH₄ emissions by converting them into valuable syngas, which an ideal H₂/CO ratio of approximately 1.^1,2 Although this process offers significant environmental benefits, its main drawback is its highly endothermic nature, which requires operation at temperatures ranging above 650 °C. Under these demanding thermal conditions, uncontrollable growth of the crystalline active phases can occur, resulting in a less efficient catalytic performance.^3,4 More importantly, the simultaneous presence of two carbon sources in the feedstream notably 'favors'catalyst deactivation due to coke formation.⁵⁻⁷ Thus, carbon deposition, which causes pore plugging and active site coverage, occurs primarily due to the decomposition of the methane molecule into carbon and hydrogen (CH₄ decomposition, CH₄ → C + 2H₂) and the CO disproportionation (2CO → CO₂ + C), also known as the Boudouard reaction, which involves the conversion of carbon monoxide into CO₂ and carbon. It is challenging to monitor these two reactions in isolation during DRM, owing to the multitude of concurrent side reactions, such as reverse water gas shift, methanation reactions, and steam reforming among others. Nevertheless, it is reasonable to hypothesize that the balance between carbon formation by CH₄ dissociation and/or CO disproportionation versus surface carbon oxidation by CO₂ determines the rate and extent of carbon deposition.

The catalysts commonly used in the DRM reaction consist of metallic nickel well-dispersed on high surface area supports such as alumina.^8,9 In this sense, the use of nickel aluminate as precursor is well-known for preparing active Ni/Al₂O₃ catalysts with nanometric metallic particles that have a high resistance to sintering.^10,11 However, this type of catalysts still may undergo deactivation by coke deposition, particularly when operating at temperatures below 700 °C, since the formation of carbon is strongly favored from a thermodynamic point of view. Most of the recently published reviews on DRM technology emphasize that regeneration of spent catalysts is critical for industrial implementation and encourage researchers in this field to define appropriate strategies to recover the intrinsic catalytic performance.¹²⁻¹⁴

Restoring catalyst activity through regeneration strategies is therefore essential for industrial operations. This process typically involves the removal of coke in an air or oxygen-rich atmosphere.¹⁵ Other efficient alternative strategies include gasification with steam or hydrogenation.¹⁶ However, to the best of our knowledge, there is a notable lack of research into the regeneration of spent catalysts deactivated under dry methane reforming conditions. Therefore, the development of a reliable and accurate kinetic model to simulate the regeneration process of the spent DRM catalyst is crucial for the design and optimization of industrial catalysts for this application.

Numerous methods have been developed to evaluate the kinetic parameters (kinetic triplet) based on data obtained by thermogravimetric analysis (TGA).¹⁷⁻²⁰ This technique not only provides an easy way to identify different types of coke by the observed peak combustion temperatures, but also generates reliable data on the kinetics of coke combustion. Kinetic analysis methods are generally divided into model-fitting and model-free (isoconversional) approaches. Model-fitting methods are popular for their ability to determine a kinetic triplet at a single heating rate by fitting experimental data to various model functions. However, the resulting kinetic parameters can vary significantly depending on the model function chosen. In contrast, model-free methods do not rely on predefined model functions, thereby avoiding errors from model selection and allowing the determination of activation energy at different reactant conversions. Consequently, model-free methods are often considered superior to model-fitting methods in terms of accuracy and flexibility.

Therefore, the aim of this research is to investigate the regeneration kinetics of a spent Ni-based catalyst employed in the dry reforming of methane, using both isoconversional and master-plot methods. To minimize kinetic uncertainties, thermogravimetric analysis of coke combustion on the DRM catalyst was conducted at various heating rates in an oxidizing atmosphere (21%O₂/N₂). First, the Kissinger–Akahira–Sunose (KAS) isoconversional method was used to determine the apparent activation energy and its possible dependence on conversion. Based on the assumed value of the apparent activation energy, the master-plot method was subsequently applied to define the regeneration model function. The feasibility of the developed kinetic model as a prediction tool was evaluated.

2. Experimental Section

2.1. Preparation of the Nickel Catalysts and Experimental Details for DRM Tests

In this work, a Ni/Al₂O₃ catalyst derived from a stoichiometric NiAl₂O₄ spinel was examined for the dry reforming of methane. Briefly, the synthesis of the oxide precursor (nickel aluminate) was conducted via coprecipitation, initiated from solutions of nickel(II) and aluminum(III) nitrates, adjusted in concentration, with a solution of sodium carbonate (1.2). After aging at 80 °C and a pH of 9, the solid was filtered, dried at 110 °C and calcined at 850 °C for 4 h to obtain the thermally stable spinel phase. Finally, the oxide was subjected to an in situ reduction step at 850 °C for 2 h with a 5%H₂/N₂ mixture to obtain the catalytically active sample (NiAl₂O₄ + H₂ → Ni + Al₂O₃). The physicochemical properties of the precursor and catalyst have been described in detail elsewhere.²¹ The DRM reaction was carried out in a bench-scale fixed-bed reactor at constant temperature (650 °C) for 12 h. The feedstream consisted of a 10%CH₄/10%CO₂/80%N₂ mixture that was admitted into the reactor with a total flow rate of 1200 mL min^–1 giving a gas hourly space velocity of 90,000 h^–1. In the experiment, 0.1 g of the catalyst (particle size 0.25–0.30 mm) was mixed with 0.9 g of inert quartz (particle size 0.5–0.8 mm) and introduced into the reactor. The evolution of CH₄, CO₂, H₂ and CO was monitored by gas chromatography (Agilent 490 microGC, equipped with a thermal conductivity detector). The state of the used catalyst was characterized by X-ray diffraction (XRD), Raman spectroscopy, transmission electron microscopy (TEM), dynamic thermogravimetry coupled to mass spectrometry (TGA-MS), and N₂ physisorption. Experimental details of the characterization techniques employed are given in the Supporting Information.

2.2. Nonisothermal TGA Combustion Tests

The oxidative removal of coke present in the nickel catalyst employed in the DRM reaction was studied by dynamic thermogravimetry using a TGA 550 thermobalance from TA Instruments under atmospheric pressure coupled to an HPR-20 EGA mass spectrometer from Hiden Analytical. The mass loss and temperature of the sample were continuously recorded using a computerized data acquisition system. The thermo-kinetic experiments were performed at constant heating rates of 0.5, 1.5, 3, 5, and 7.5 °C min^–1 from room temperature to 850 °C. Runs were also carried out at three additional heating rates of 4, 6.5, and 10 °C min^–1 to verify the feasibility of the predicted regeneration kinetic model. The oxidant stream was dry synthetic air (40 cm³ min^–1) flowing through the platinum sample holder. A thermocouple was placed next to the crucible to measure the temperature.

The samples for thermogravimetric analysis were prepared by loosely mixing the coked catalyst and silicon carbide with a spatula in a mass ratio of 1/10. The incorporation of SiC reduced the flow resistance of the sample bed and acted as a heat sink, reducing the temperature gradients. Thus, the sample for each run was 40 mg of the mixture (approximately 3.6 mg of coked catalyst). The experimental conditions such as sample size, dilution ratio, atmosphere and gas flow rate were identical for all kinetic experiments. As a large excess of oxygen was supplied, the combustion reaction took place with a negligible change of the oxygen partial pressure. Previous laboratory experiments in which the dilution ratio, sample size and gas flow rate were varied showed that the experimental conditions used in this study (dilution mass ratio of 1:10, sample size of 40 mg, and dry air flow rate of 40 cm³ min^–1) were not affected by mass- and heat-transfer limitations.²²

3. Kinetic Theory

The regeneration of coked catalysts by combustion is a typical gas–solid heterogeneous reaction. Therefore, its kinetic equation (eq 1) can be defined as

1

In this expression dα/dt stands for the regeneration rate, α represents the coke conversion, t is the reaction time, k represents the reaction rate constant expressed as Ae^(−E/RT), where A (the pre-exponential factor) and E (the activation energy) are the Arrhenius parameters, R is the ideal gas constant and T is the regeneration temperature, respectively. The Arrhenius parameters, together with the reaction model, are sometimes referred to as the kinetic triplet.

On the other hand, f(α) represents the regeneration model functions depending on the mechanism and f(P_O2) is a function of the oxygen partial pressure. Since the experiments were conducted under conditions of oxygen excess, the effect of the oxygen partial pressure on the process could be considered constant. A large number of kinetic models exist to describe the most common mechanisms in solid-state reactions. These are the so-called nucleation (A_n) models, geometrical contraction (P_n) models, diffusion (D_n) models, reaction order (R_n) models and exponential nucleation (L_n) models. The corresponding differential (f(α)) and integral (g(α)) forms are given in Table 1.^23,24

Table 1. Kinetic Model Functions f(α) and g(α) Usually Employed for Solid-State Reactions.

reaction model	symbol	f(α)	g(α)
Nucleation models (An)
Avrami–Erofeev equation	A1.5	1.5(1 – α)(−ln (1 – α))1/3	(−ln (1 – α))1/1.5
	A2	2(1 – α)(−ln (1 – α))1/2	(−ln (1 – α))1/2
	A3	3(1 – α)(−ln (1 – α))2/3	(−ln (1 – α))1/3
	A4	4(1 – α)(−ln (1 – α))3/4	(−ln (1 – α))1/4
Geometrical contraction models (Pn)
contracting linear	P1	1	α
contracting area	P2	2(1 – α)1/2	(1 – (1 – α)1/2)
contracting volume	P3	3(1 – α)2/3	(1 – (1 – α)1/3)
Diffusion models (Dn)
one dimensional	D1	1/(2α)	α2
two dimensional	D2	(−ln (1 – α))−1	(1 – α) ln (1 – α) + α
three dimensional	D3	(3/2)(1 – α)2/3(1 – (1 – α)1/3)−1	(1 – (1 – α)1/3)2
Ginstling–Brounshtein equation (three dimensional)	D4	(3/2)((1 – α)−1/3 – 1)−1	(1 – 2α/3) – (1 – α)2/3
Zhuravlev equation	D5	(2/3)(1 – α)5/3(1 – (1 – α)1/3)−1	((1 – α)−1/3 – 1)2
Reaction order models (Rn)
zero order	R0	1	α
first order	R1	(1 – α)	–ln (1 – α)
second order	R2	(1 – α)2	(1 – α)−1 – 1
third order	R3	(1 – α)3	((1 – α)−2 – 1)/2
Exponential nucleation models (Ln)
power law I	L1	4α3/4	α1/4
power law II	L2	3α2/3	α1/3
power law III	L3	2α1/2	α1/2
power law IV	L4	(2/3)α–1/2	α3/2
exponential law	E1	α	ln(α)

For a constant heating rate, T can be calculated as

2

T₀ and β are the initial temperature and the heating rate, respectively. Thus, eq 1 can be expressed as follows:

3

By rearranging this equation (eq 3) and replacing the term Af(P_O2) by A′ (apparent pre exponential factor) the following expression (eq 4) can be obtained:

4

Given that the exponential integral lacks an analytical solution, a number of approximation methods, series expansions and numerical solution methods have been proposed in the literature.^25,26 In this work the Fischer-Jou-Gokalgandhi approximation p(x) ≅ e^–x/x² was employed, where x = E/RT. Other alternative approximations, such as the Doyle approximation and the Senung and Yang approximations, were also used to analyze their effect on the determination of the reaction model (Supporting Information).

3.1. Estimation of Kinetic Parameters

Kinetic parameters can be obtained by both model-fitting and isoconversional methods. In the model-fitting approach, Arrhenius parameters are derived based on the assumed form of f(α). In nonisothermal experiments, where both temperature and extent of reaction change simultaneously, this approach typically fails to clearly separate the temperature dependence, k(T), from the reaction model, f(α). Consequently, almost any form of f(α) can fit the data at the expense of a significant variability in the Arrhenius parameters. These variations compensate for discrepancies between the assumed and the actual, but unknown, reaction model. As a result, model-fitting methods, such as the Coats–Redfern method, often give highly uncertain Arrhenius parameters.²⁷

The limitations associated with model-fitting techniques can be overcome by the application of isoconversional methods. First, since the reaction rate is only a function of the temperature for a fixed value of α, these methods facilitate the evaluation of the dependence of the activation energy on the degree of conversion. Second, this relationship is established without the need to assume a specific reaction model f(α). Consequently, model-free isoconversional methods mitigate the sources of the aforementioned discrepancies, thereby increasing the likelihood of obtaining consistent kinetic parameters from both isothermal and nonisothermal experiments.²⁴ Another important advantage of the isoconversional methods is that when competitive and independent reactions occur simultaneously in a system, the mechanism of the reaction can be revealed by either increasing or decreasing the heating rate. Finally, an additional advantage of isoconversional methods is that the parameters A′ and g(α), which are assumed to be constant, are not required for the calculation of the activation energy. Therefore, from eq 4 we can obtain eq 5, which is formally defined as the Kissinger–Akahira–Sunose equation.

5

In the KAS method, the left-hand side of eq 5 is plotted against 1/T, giving a straight line whose slope allows the activation energy to be calculated. Once the activation energy has been estimated, it is crucial to select an appropriate regeneration model function. This objective can be achieved by using the previously estimated value of E and the master-plot method. The master-plot method is a technique used in kinetic analysis of solid-state reactions to identify the most suitable kinetic model by comparing experimental data with predefined theoretical curves representing different reaction mechanisms. Prior to making this comparison, the experimental data must first be transformed into the corresponding master-plot. Hence, when applying eq 4 for α = 0.5 as a reference point, the following expression can be obtained (eq 6):

6

where x_0.5 = E/RT_0.5, and T_0.5 is the temperature required to attain 50% conversion. The master-plot method can then be obtained by dividing eq 4 by eq 6.

7

To determine the valid regeneration g(α) model function from Table 1, different values of g(α)/g(0.5) were first calculated for various coke conversion values (α = 0.1–0.9). Then, the corresponding values of p(x)/p(x_0.5) were obtained from the experimental data. According to eq 7, g(α)/g(0.5) will be equal to p(x)/p(x_0.5) when the correct regeneration model function is selected. Therefore, by comparing g(α)/g(0.5) from different model functions with p(x)/p(x_0.5) from the experimental data, the appropriate f(α) for the combustion of the coke present in the nickel catalyst can be identified.

3.2. Simulation of the Isothermal Process

From a practical point of view, an accurate prediction of the regeneration of the coked catalyst under isothermal conditions may be of interest. Reliable isothermal simulations could be carried out by using eq 8, which only requires experimental data recorded at a specific heating rate and a proper value of the activation energy. This expression allows to calculate of the time (t_α) required to achieve a specific conversion at any arbitrary temperature (T₀) from the experimental value of the temperature (T_α) corresponding to a given conversion (α) at the heating rate (β). By solving this equation for different conversion levels, the relationship between α and t_α at a given temperature can be reasonably proposed.²⁸

8

4. Results and Discussion

4.1. Performance of the Nickel Catalyst under Methane Dry Reforming Conditions and Postrun Characterization

Figure S1, Supporting Information, includes the evolution of both methane and carbon dioxide conversion with time online (12 h) at 650 °C and 90,000 h^–1. Thus specific temperature was chosen because it is often considered to be the lowest temperature in the operating range of the DRM reaction at relatively high gas hourly space velocities, and thus the most challenging for the development of advanced catalysts for this process. In addition, this thermal level allows a more comprehensive view of the effects of coke formation and subsequent catalyst deactivation, as it represents a thermodynamic compromise between reforming and coking. Regardless of the reactions responsible for coking, the thermodynamic equilibrium clearly dictates that the formation of carbonaceous deposits is favored at low temperatures (Figure S2, Supporting Information).

It was observed that the activity progressively decreased from 71%(CH₄) and 68%(CO₂) to 65 and 62%, respectively. A similar pattern was found for the evolution of the yields of H₂ and CO. The aforementioned catalytic results undoubtedly demonstrated a notable deactivation of the investigated Ni/Al₂O₃ catalyst. As previously stated, it is widely accepted that catalyst deactivation in the DRM process is primarily associated with the formation of carbonaceous deposits and/or sintering of the metallic active phases and/or the chemical transformation of the active phase to less efficient counterparts. In particular, coke formation is promoted at relatively low temperatures, mainly due to the occurrence of the Boudouard reaction and/or methane decomposition. These carbon deposits can restrict the access of reactants to the active metal, which may result in reactor plugging. Indeed, when the reaction time interval is extended (70 h), the continuous coke formation eventually leads to a dramatic decrease in activity. Thus, the conversion decreased continuously from 71% at the beginning of the experiment to 30% after 70 h of reaction time. The experiment was therefore stopped to avoid reactor plugging (Figure S1, Supporting Information).

The state of the used catalyst was characterized by X-ray diffraction, Raman spectroscopy, transmission electron microscopy, dynamic thermogravimetry, and N₂ physisorption. This extensive characterization focuses on the stability of the active nickel phase (sintering) and the extent of coke formation including its morphology, chemical nature and the impact on pore blocking. Details of the results obtained from each analytical technique are given below. A brief overview of the main physicochemical properties of the fresh and postrun samples is shown in Table S1, Supporting Information.

Figure S3, Supporting Information, compares the diffraction patterns of fresh and spent nickel catalysts. Thus, for the fresh Ni/Al₂O₃ sample obtained by the high-temperature reduction of the as-synthesized bulk NiAl₂O₄ catalytic precursor, the presence of metallic nickel was evidenced by several signals located at 2θ = 45.0, 52.0 and 76.7° (ICDD 00-004-0850). Similarly, additional diffraction peaks located at 19.4, 32.0, 37.7, 39.7, 45.8 and 66.9°, assigned to the γ-Al₂O₃ cubic phase (ICDD 01-074-2206), were observed, in agreement with the expected phases owing to the complete reduction of NiAl₂O₄-like oxides. The metallic Ni crystallite size, estimated by the Scherrer equation from the signal located at 2θ = 52.0°, was 15 nm. On the other hand, the XRD pattern of the used catalyst revealed only the presence of metallic Ni and γ-Al₂O₃, thus ruling out the possibility of a massive bulk reoxidation of the catalyst during the DRM reaction. Interestingly, metallic sintering was effectively prevented since the crystallite size remained unchanged after reaction (14 nm). However, carbonaceous deposition was evident as shown by the significantly intense signal of graphitic carbon at 2θ = 26.7° (ICDD 00-056-0159).²⁹ Judging from these results, it was reasonable to consider coking as the main cause of the observed deactivation of the nickel catalyst under investigation.

The postreaction sample was additionally characterized by Raman spectroscopy in the 1100–1800 cm^–1 region (Figure S3, Supporting Information). Two main bands at 1360 and 1580 cm^–1 were distinguished. The first signal 1360 cm^–1 (the so-called D-band) was associated with carbon with structural imperfections while the second band (the so-called G-band) was related to graphite layers.^30,31 After calculating the relative intensity of the D- and G-bands (I_D/I_G = 0.48), it could be deduced that the carbonaceous deposits were characterized by a significantly high defect density.

TEM images of the deposited carbon on the spent catalyst are included in Figure S4, Supporting Information. Carbon filaments with a multiwalled (around 30 walls with an intershell distance of 0.36 nm) nanotube morphology were formed. It is known that whisker carbon is formed as a consequence of concentration gradient of dissolver carbon in the nickel particle, which diffuses across its surface and emerges on the opposite side of the particle, generating elongated graphitic hollow filaments (tubular in shape).^32,33 The average diameter of these chain-like nanotubes, with or without embedded Ni particles, is relatively similar, around 20 nm. Specifically, 60% of the observed nanotubes had a thickness between 10 and 20 nm while the remaining measurements were in the 20–40 nm range. The average inner diameter was around 5 nm. This particular morphology of coke deposits has been documented in a multitude of studies examining DRM over nickel catalysts at the laboratory scale. Moreover, it has been reported in semipilot-scale studies that utilized these particular catalysts.^34,35

The quantification of the extent of coking was analyzed out by dynamic thermogravimetry coupled to the analysis of the gas stream at the exit by mass spectrometry. This study was conducted at 3 °C min^–1 and using an oxidizing atmosphere (21%O₂/N₂). Figure S5, Supporting Information, includes the corresponding thermoanalytical curve. Three distinct mass changes could clearly be distinguished. The first mass loss observed from ambient temperature up to 250 °C was attributed to the removal of moisture (m/z = 18) from the sample, corresponding to a total water content of approximately 2.0wt.%. Next, a slight increase (0.9wt.%) in the mass weight of the sample was noticed as the temperature increased between approximately 250 and 450 °C, which could be related to the oxidation of metallic nickel present in the catalyst. To validate this assumption, the experiment was repeated with N₂ as the carrier gas instead of synthetic air. Thus, in the absence of ambient oxygen, no weight increase was detected in this temperature range, thereby suggesting the occurrence of nickel oxidation. Finally, the significant mass loss observed above this temperature and up to 670 °C corresponded to the combustion of the coke present in the catalyst, in fairly good agreement with the presence of CO₂ (m/z = 44) in the effluent stream. The peak combustion temperature was around 570 °C. The coke content in the catalyst was estimated to be about 0.38 ± 0.05 g_C g_cat^–1. It must be pointed out that the eventual formation of CO as an oxidation byproduct could not be excluded since its m/z signal (28) was identical to that of N₂. Expectedly, the diffraction pattern of the sample after the thermogravimetric experiment exhibited the corresponding signals at 2θ = 37.1, 43.3, 62.9 and 75.3°, attributable to the NiO phase (ICDD 01-075-0197). Considering the global results from the characterization of the carbon deposits (highly crystalline carbon nanotubes with peak combustion temperatures around 560 °C), it could be concluded that the coke formation under the investigated experimental conditions originates from methane cracking.³⁶⁻³⁸

The surface area and pore volume of the sample used were 59 m² g^–1 and 0.17 cm³ g^–1. Obviously, the value of these properties account for the textural properties of the nickel catalyst and the bundle of carbon nanotubes formed. It is anticipated that the intrinsic surface area of the catalyst may be adversely affected by the expected clogging of smaller pore sizes, ultimately resulting in a reduction of its accessible surface area. A comparison of the BJH pore size distribution of both fresh and used samples revealed a lower volume of the small pores and a concomitant shift of the mean pore size, from 90 Å for the fresh catalyst and 110 Å for the mixture catalyst+coke (Figure S3, Supporting Information). Conversely, an attempt was made to estimate the intrinsic surface area of the carbonaceous deposits using the measured surface area of the used sample and coke content (0.38 g_C g_cat^–1). Two extreme scenarios were proposed in relation to the contribution of the surface area of the postrun catalyst. Therefore, in one scenario, it was postulated that the textural characteristics of the catalyst remained unaltered by pore obstruction (S_BET = 54 m² g_cat^–1). In the alternative scenario, it was postulated that the catalyst’s pores were entirely obstructed by coke (S_BET = 0 m² g_cat^–1). Accordingly, the intrinsic surface area of the carbon nanotubes was found to be 61 and 81 m² g_C^–1, respectively. This plausible range of surface area agreed with the theoretical calculations by Peigney et al.,³⁹ who predicted a surface area of approximately 75 m² g_C^–1 for carbon nanotubes with a diameter/thickness of 20 nm.

4.2. Calculation of the Apparent Activation Energy by the KAS Method

As detailed earlier, the Kissinger–Akahira–Sunose (KAS) method was employed to calculate the activation energy of the process. The coke conversion-temperature curves recorded at 0.5, 1.5, 3, 5, and 7.5 °C min^–1 are depicted in Figure 1. The oxidation reaction occurred within a temperature range of 380–710 °C. Irrespective of the used heating ramp, the reaction proceeded within a relatively narrow temperature window of about 220–240 °C. It could be observed that the conversion degree attained at a given temperature was dependent on the heating rate. These sigmoidal profiles were characterized by a reduction in time required for the sample to remain in the same temperature range as the ramp increased. This resulted in a higher temperature at which the samples were in the same combustion state. This phenomenon, which is frequently observed in thermally stimulated reactions in solids under nonisothermal conditions, could be attributed to thermal hysteresis. In other words, an insufficient transfer of externally supplied heat from the surface to the interior of the sample led to a shift of the conversion curves at higher temperatures.¹⁹ The first derivative (DTG) of the thermogravimetric curves are also included in Figure 1. The corresponding peak combustion temperatures varied between 515 and 600 °C. It is noteworthy that all the traces exhibited a relatively high degree of symmetry, which could indicate a homogeneous chemical nature of the deposited coke. This suggests the formation of a single type of carbon species under the DRM reaction conditions studied. Given the temperature required for its combustion and the results of XRD, Raman spectroscopy and TEM, the formation of whisker-type carbon with a crystalline structure and high defect density was assessed.^40,41

Figure 1.

Kinetic curves for the combustion of coke deposited on the nickel catalyst obtained at various heating rates (0.5, 1.5, 3, 5, and 7.5 °C min^–1).

The KAS isoconversional method requires the determination of the temperature required for fixed conversion values from the various thermograms recorded at varying heating rates. It is well-known that deviations between experimental data and theoretical values are likely to occur at the extremities of solid-state reactions. Therefore, the kinetic analysis was carried out on the dominant reaction for α values between 0.1 and 0.9.⁴²Figure 2 shows the regression lines (r² > 0.99) obtained when plotting ln(β/T²) against the reciprocal of temperature for the same degrees of conversion, from 0.1 to 0.9 in steps of 0.1. The activation energy of the process was estimated from the slope of this family of straight lines. As shown in Table S2 and Figure S6, Supporting Information, a gradual increase in the activation energy was observed with conversion. This ranged from 154 ± 1.4 kJ mol^–1 for 10% conversion to 174 ± 2.5 kJ mol^–1 for 90% conversion.

Figure 2.

Isoconversional plots at various conversion degrees, in the 10–90% range, for the combustion of coke deposited on the nickel catalyst.

This trend may be consistent with the higher reactivity toward combustion of structural imperfections within the graphite. Consequently, lower activation energies were found at low conversion levels. As the process progressed, the combustion of the remaining coke, which is characterized by more stable graphitic layers, required a slightly higher activation energy. In this sense, Contescu et al.⁴³ reported that the oxidation of graphitic materials occurred preferentially over the more disordered domains, as evidenced by the gradual decrease in the I_D/I_G ratio (Raman spectroscopy) with conversion. Thus, given that the estimated relative variation between 10 and 90% conversion was relatively reduced (around 13%), and even smaller for the values found in the 20–80% range (8%), it was then assumed that the mean value (164.5 kJ mol^–1) would be a representative apparent activation energy for the process. In other words, coke combustion was considered as a single-step reaction for the purpose of further kinetic calculations. Unfortunately, to the best of our knowledge, no values for the activation energy of the combustion of coke generated under dry reforming conditions have been reported in the literature. Interestingly, our estimate fell within the typical range (135–180 kJ mol^–1) reported in numerous studies examining the thermal stability and resistance to oxidation shown by carbon nanotubes.⁴⁴⁻⁴⁹ The calculation procedure in these works is based on the prior assumption of a given kinetic model (typically nucleation models, reaction order, diffusion models, and geometric contraction models) and the subsequent fitting of the experimental results to estimate the values of the apparent activation energy.

4.3. Determination of the Kinetic Model for Coke Combustion

After proposing a representative value for the apparent activation energy of the whole combustion process (164.5 kJ mol^–1), the subsequent step was to select an appropriate kinetic model for describing the regeneration process of the coked DRM catalyst. First, the p(x)/p(x_0.5) function was calculated from the thermogravimetric data obtained for each heating rate (0.5, 1.5, 3, 5, and 7.5 °C min^–1) using the master-plot method (Figure 3). It was found that all experimental master-plots constructed under different thermal conditions (at various ramps) exhibited relatively good consistency. Nevertheless, a closer inspection revealed a slight deviation at high coke conversions (α > 0.8), as a decrease in the heating rate resulted in a slight increase in the p(x)/p(x_0.5) ratio.

Figure 3.

Experimental masterplots (p(x)/p(x_0.5) vs α) for the combustion of coke deposited on the nickel catalyst at various heating rates (0.5, 1.5, 3, 5, and 7.5 °C min^–1).

These experimental master-plots were then compared with the theoretical g(α)/g(0.5) values obtained for each conversion model listed in Table 1 (21 kinetic models). It is worth pointing out that these graphs coincide with at α = 0 and at α = 0.5. Most of the large differences between the various g(α) function are apparent within the range α > 0.5. The objective of this graphical representation was to preliminarily discriminate a cluster of models from which further refinement could be undertaken in order to select the most appropriate g(α). Figure S7, Supporting Information shows the comparison between the experimental values of p(x)/p(x_0.5) and the values of g(α)/g(0.5) derived from the nucleation (A_n) models, geometrical contraction (P_n) models and exponential nucleation (L_n) models. It is evident that none of these models were sufficient to accurately describe the kinetics of coke combustion, given the significant discrepancy between the experimental and predicted data.

In contrast, the results included in Figure 4 suggested a relatively high degree of concordance between the experimental data and the R₁ (first order model), D₁ (one-dimensional diffusion model) and D₂ (two-dimensional diffusion model) models. In order to more accurately discriminate between these three possible models, the parity plots of p(x)/p(x_0.5) vs g(α)/g(0.5) were analyzed for each model. Hence, Figure S8, Supporting Information presents the results when using the experimental data recorded at 3 and 5 °C min^–1. The parity plots for both D₁ and D₂ models indicate, however, a notable discrepancy over the whole conversion range (0.1–0.9), which were mutually overcompensated to give relatively high r² (0.978 and 0.988, respectively). However, the R₁ model yielded more representative and statistically valuable results, with r² = 0.993. Thus, based on the results also shown in Figure 4, it was evident that the chemical reaction model f(α) = (1 – α)ⁿ with a reaction order (n) between 1 and 2, was the regeneration model that best fitted the experimental data. It should be noted that order-based models are the simplest, as they are similar to those used in homogeneous kinetics. In these models, the reaction rate is proportional to concentration, amount or fraction remaining reactant raised to a particular power (integral or fractional), which is the reaction order. Although without prior validation of the consistency of kinetic data with respect to this kinetic model, the chemical reaction model is often adopted for the combustion of a wide range of carbonaceous materials, including carbon nanatubes.⁵⁰⁻⁵³

Figure 4.

Theoretical masterplots (g(α)/g(0.5) vs α) for various Rn and Dn models, and experimental masterplots (p(x)/p(x_0.5) vs α) for the combustion of coke deposited on the nickel catalyst at various heating rates (0.5, 1.5, 3, 5, and 7.5 °C min^–1).

4.4. Determination of the Reaction Order and Pre-Exponential Factor

If the regeneration model of the DRM catalyst was considered to obey the reaction order model, the apparent pre-exponential factor A′ and the reaction order could be determined from the integral form of the reaction order (R_n) model (eq 9):

9

In the ideal cases where n = 0, 1, or 2 (R₀, R₁, and R₂ models), the corresponding expressions for g(α) are included in Table 1. According to this expression, the correct reaction order could be determined from the linear fit of the plot of (1 – (1 – α)^1–n)/(1 – n) versus Ep(x)/βR. In other words, the most accurate value of the reaction order would be that which would exhibit the highest correlation coefficient, r². Figure S9, Supporting Information, illustrates the evolution of r² with the reaction order (ranging from 1 to 2 in steps of 0.01) for the experimental results obtained with all the investigated heating rates. It could be concluded that the optimum linear fitting was obtained for n = 1.36. Accordingly, the plot of (1 – (1 – α)^1–n)/(1 – n) against Ep(x)/βR with this value for the reaction order and different heating rates are shown in Figure 5. The apparent pre-exponential factor (A′) of the process was calculated from the slope. As a result, kinetic parameters of the combustion of the coke present in the used DRM catalyst, that is to say, the reaction model, the apparent pre-exponential factor and the reaction order were determined to be as follows: f(α) = (1 – α)ⁿ, A′ = 1.635·10⁹ min^–1 and n = 1.36. The values of the reaction order were also estimated using alternative approximations of the p(x) function to solve the temperature integral (eq 4), namely the Doyle approximation and the Senum and Yang approximation. In addition to confirming that the appropriate regeneration model was the chemical reaction model, the calculated values for n were 1.32 and 1.36, respectively. These comparable values pointed out that the influence of the approximation used for the kinetic calculations was relatively small.

Figure 5.

vs (E/βR)p(x) at n = 1.36 at several heating rates (0.5, 1.5, 3, 5, and 7.5 °C min^–1).

4.5. Verification of the Estimated Kinetic Model for Coke Combustion

Having determined the kinetic triplet, namely the apparent activation energy, the apparent pre-exponential factor and the regeneration model function, the feasibility of the developed kinetic model for predictive purposes was assessed. Hence, a series of new thermo-kinetic runs was performed at three additional heating rates (namely 4, 6.5, and 10 °C min^–1). Two runs were conducted within the range used for the developed kinetic analysis (4 and 6.5 °C min^–1), while the third run was carried out with a heating rate outside this range (10 °C min^–1). According to eq 3, the kinetic model of coke combustion on the DRM catalyst can be rewritten (eq 10) as follows:

10

Figure 6 presents the model predictions in conjunction with the experimental results for these three additional heating rates (E = 164.5 kJ mol^–1, n = 1.36, and A′ = 1.635·10⁹ min^–1). A relatively high degree of agreement with the experimental data was observed, indicating that the proposed kinetic analysis was suitable for modeling the combustion process under linear nonisothermal conditions. While some deviation of the theoretical values from the experimental data was noted at low α values, the dominant reaction was successfully predicted.

Figure 6.

Comparison between experimental (dotted lines) and simulated (solid lines) thermoanalytical curves at several heating rates (4, 6.5, and 10 °C min^–1).

Combustion of the coke present on the DRM catalyst was also simulated under isothermal conditions. This prediction was constructed from experimental data collected at a specific reaction rate and the estimated apparent activation energy, as dictated by eq 8. Figure S10, Supporting Information, shows the evolution of the conversion with reaction time at 625 °C. It was found that regardless of the experimental data chosen for the simulation, the resulting profiles were very similar. Furthermore, using the kinetic data derived from the run at 7.5 °C min^–1, the conversion profiles were calculated under isothermal conditions, between 625 and 725 °C in steps of 25 °C (Figure 7). It could be observed that full coke conversion (>99%) would be achieved at 625 °C/23 min, 650 °C/13 min, 675 °C/7 min, 700 °C/4 min, and 725 °C/3 min. A reasonably good prediction was also found for simulations generated using the full kinetic triplet (eq 1).

Figure 7.

Simulated evolution of conversion with time under isothermal conditions in the 625–725 °C range, calculated from eq 8 when using experimental data taken at 7.5 °C min^–1 (solid lines), and eq 1 (dashed lines).

As a final remark, it must be recalled that the coke combustion mechanism is influenced by multiple factors, such as its surface area and pore structure, morphology, and the degree of crystallinity and defects, which in turn are related to the DRM operating conditions and the type of reforming catalyst employed. All these factors can impact the oxidation rate. Therefore, the results obtained in this study—activation energy, pre-exponential factor, kinetic function model—may not necessarily be valid for other types of coke. However, the present study validates the utility of a composite kinetic processing technique, which integrates the complementary utilization of both isoconversional methods and master-plot methods. This comprehensive and pragmatic approach enables the determination of the kinetic triplet (conversion model, pre-exponential factor, and activation energy), and in principle could be applied for the modeling of the combustion of any type of coke deposits for the purpose of catalyst regeneration.

4.6. Reusability of the Regenerated Catalyst

Finally, the catalytic performance of the regenerated catalyst was examined. As predicted by the kinetic simulations included in Figure 7, the postrun sample was submitted to a combustion step at 650 °C for 15 min with synthetic air to ensure a full removal of coke deposits. Due to the unavoidable oxidation of nickel particles to NiO during this process, the coke-free catalyst was reduced at 850 °C for 2 h. This activation protocol was the same as that used for the fresh catalytic precursor. The complete conversion of NiO to metallic nickel was evidenced by XRD analysis.

The XRD patterns corresponding to the catalyst’s state after calcination (650 °C/15 min) and subsequent reduction (850 °C/2 h) are presented in Figure S11, Supporting Information. The absence of carbon and the formation of NiO were confirmed. Following the high-temperature reduction, the diffraction signals attributed to metallic nickel were observed, consistent with the absence of nickel oxide. The average crystallite size of nickel was 17 nm, as estimated by the Scherrer equation from the signal located at 2θ = 52.0°, thereby suggesting minimal sintering during the regeneration process (15 nm for the fresh catalyst). It is noteworthy that these two diffractograms were collected using the same sample employed in the thermogravimetric analysis, which explains the presence of signals from SiC (ICDD 00-049-1429; 2θ = 34.1, 34.9, 35.6, 38.1, 41.4, 60.0, 65.6 and 71.8°), utilized as a diluent.

As demonstrated in Figure 8, the catalytic performance of the catalyst that was eventually regenerated is evident. It was substantiated that the proposed route for reactivating the NiAl₂O₄-derived nickel catalyst was efficient, as the initial catalytic activity (methane conversion of 71%) was virtually fully recovered. Furthermore, a similar pattern for the evolution of conversion with time on stream (12 h) was identified.

Figure 8.

Evolution of methane conversion with time on stream over the regenerated Ni/Al₂O₃ catalyst.

5. Conclusions

In this study, multiple dynamic (0.5–7.5 °C min^–1) thermogravimetric analyses were performed under an oxidizing atmosphere (21%O₂/N₂) to investigate the combustion kinetics of coke in a spent nickel catalyst (0.38 g_C g_cat^–1) used for dry reforming of methane. The carbonaceous deposits were essentially filaments with a nanotube morphology (20 nm in diameter), characterized by a crystalline structure and a high density of defects.

The apparent activation energy was determined using the Kissinger–Akahira–Sunose integral isoconversional method without any prior assumptions regarding the conversion model. The activation energy exhibited a slight increase with conversion, which was consistent with the preferential combustion of the defective structure of the coke followed by the combustion of the graphitic domain. The average apparent activation energy for coke combustion was found to be 164.5 kJ mol^–1. This value was used for the application of the master-plot method to identify the model function, which indicated that the coke combustion process in the DRM catalyst can be described by the R_n reaction order model with n = 1.36. The apparent pre-exponential factor was calculated to be 1.635·10⁹ min^–1. The kinetic triplet was accurately estimated, allowing an reliable simulation of the process. A comparison between model predictions and experimental results for three different heating rates, which were not used for the kinetic calculations, showed a good agreement, suggesting that the kinetic triplet could be effectively applied for modeling the regeneration of the spent DRM catalyst. Furthermore, the combustion of coke under isothermal conditions could be evaluated.

Finally, since the effective management of carbon deposition is critical for the successful implementation of industrial DRM, this study suggests that ex situ regeneration of Ni-based catalysts via controlled oxidation in fixed-bed reactors is a viable approach. To prevent structural degradation, coke removal should be conducted at low temperatures, ideally matching the DRM operating conditions. Our findings indicated that complete coke combustion occurred at 650 °C within 15 min, making this a suitable regeneration temperature. Subsequently, the catalyst should be submitted to a reduction treatment in order to convert the NiO generated in the oxidation stage into metallic particles analogous to those found in the freshly activated catalyst. In this sense, postregeneration characterization confirmed minimal Ni sintering (17 vs 15 nm in the fresh catalyst), validating the effectiveness of oxidation with dry air (21%O₂/N₂) followed by reduction with H₂. However, to further mitigate sintering, strategies such as staged oxidation or lower O₂ concentrations could be explored. Implementing periodic regeneration cycles will extend catalyst lifespan, enhancing the commercial feasibility of DRM.

Glossary

Abbreviations

DRM

methane dry reforming

KAS

Kissinger–Akahira–Sunose

GHGs

greenhouse gases

XRD

X-ray diffraction

TEM

transmission electron microscopy

TGA

thermogravimetric analysis

MS

mass spectrometry

BET

Brunauer–Emmett–Teller

BJH

Barrett–Joyner–Halenda

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.5c00824.

• Main physicochemical properties of both fresh and used nickel catalysts; apparent activation energy as a function of the degree of conversion (α) calculated by means of the KAS isoconversional method; evolution of reactants conversion and H₂ and CO yields with time on stream over the Ni/Al₂O₃ catalyst (12 and 70 h); thermodynamic equilibrium of DRM under the investigated reaction conditions and details on the calculation of equilibrium data; characterization of the spent Ni/Al₂O₃ catalyst; XRD pattern of the fresh and used Ni/Al₂O₃ catalyst; Raman spectrum; TEM micrographs; TEM images of the spent Ni/Al₂O₃ catalyst; thermogravimetric analysis for the combustion of coke deposited on the DRM catalyst with a heating rate of 3 °C min^–1; TG curve, dTG curve, and evolution of the composition of the gas stream at the exit of the thermobalance by mass spectrometry; evolution of the activation energy with the degree of conversion (α) calculated by means of the KAS isoconversional method; comparison of (p(x)/p(x_0.5) and g(α)/g(0.5)) at different conversions of coke; nucleation models (An), geometrical contraction models (Pn), and exponential nucleation models (Ln + E1); parity plots (p(x)/p(x_0.5) vs g(α)/g(0.5)) for the R₁, D₁, and D₂ conversion models generated from experimental data recorded at 3 and 5 °C min^–1; variation of the correlation coefficient (r²) with the reaction order from the linear fit versus [E/βR]p(x) (eq 9); simulated evolution of conversion with time under isothermal conditions (625 °C), calculated from eq 8 when using experimental data taken at a several heating rates (0.5, 1.5, 3, 5, and 7.5 °C min^–1); and XRD patterns corresponding to the regenerated catalyst’s state after calcination (650 °C/15 min) and subsequent reduction (850 °C/2 h) (PDF)

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. All authors contributed equally.

The authors declare no competing financial interest.