A Useful Excel-Based Program for Kinetic Model Discrimination

Abstract

In the present study, the ANEMONA.XLT tool, an Excel template that was designed for calculation of enzyme kinetic parameters, has been successful adapted to some proposed models for dry reforming reaction, such as Eley-Rideal or Langmuir-Hinshelwood kinetic models. Model discrimination by non-linear regression analysis has been applied to data from the literature; the predicted kinetic parameters that were obtained using ANEMONA.XLT were fully comparable with those already published. Thus, the template can be a helpful and user-friendly alternative tool for researchers who do not have advanced skills in computer programming.

1. Introduction

In the last decade the growing energy demand, furnished primarily from fossil fuels, have created environmental issues due to the generation of greenhouse gases (CO₂ and CH₄). The reduction of CO₂ and CH₄ emissions into the atmosphere and their possible recycling, induced a growing interest in dry reforming (DR) reaction (CO₂ + CH₄ → 2H₂ + 2CO; ΔH°_{298 K} = 247 kJ/mol), able to produce syngas, a building block for the synthesis of different chemicals and oxygenated fuels [1,2,3]. Besides, CH₄ and CO₂ gases are the main constituents of biogas, a fuel that is produced extensively by various anaerobic biological waste treatments. The biogas quality is often adequate to DR reaction but its use can be enhanced through the mentioned process [4,5]. Common catalysts for DR reaction are composed either by noble metals (Pt, Rh, Ru, Ir) or transition metals, such as Ni, Co and Fe. The major problem encountered in the process is the rapid catalyst deactivation by carbon deposition. Although noble metals are very active and more coke-resistant than Ni-based catalysts, the former systems are preferred to noble metal-based samples, as they are cheaper and widely available, making DR reaction a more industrially-viable process. To study the mechanisms of DR reaction and to determine the related kinetic parameters is fundamental for developing effective catalytic systems and consequently for commercializing methane-reforming technologies. Kinetic analysis, performed in order to find the suitable mechanistic model that adequately describes the reaction rate and defines the chemical process, can be used for the optimization of the catalyst design.

Generally, there are three typical kinetic models that are applied to describe reforming reactions: Power law, Eley-Rideal and Langmuir Hinshelwood models [6,7]. The Power law model, in the form of:

$$r_i=A\exp \left(-\frac{E_a}{RT}\right)p_{\mathrm{CH}4}^m{p}_{\mathrm{CO}2}^n$$

(where; r_i, rates of consumption of CH₄ and CO₂; m and n, reaction orders; A, pre-exponential factor; E_a, apparent activation energy; R, universal gas constant; T, reaction temperature; p_CH4 and p_CO2, partial pressure of CH₄ and CO₂, respectively), has the advantage of simplicity in its application and determination, but it concerns only the overall activation energy and reaction orders and, besides, it is valid for only a narrow range of partial pressures. In the Eley-Rideal models, it is assumed that one reactant is associatively adsorbed onto the catalyst surface at thermodynamic equilibrium, while the second reagent remains in the gas phase; the slow reaction between the adsorbed species with the gaseous reactant is the rate determining step (RDS). For the basic reaction $({\mathrm{CH}}_4+{\mathrm{CO}}_2\stackrel{kref}{\leftrightarrow }2\mathrm{CO}+2{\mathrm{H}}_2)$, the generalized rate equation can be represented as:

$$r_{ref}=k_{ref}\left(p_{\mathrm{CH}4}{p}_{\mathrm{CO}2}-\frac{p_{\mathrm{CO}}^2{p}_{\mathrm{H}2}^2}{K_{ref}}\right)$$

where k_ref is the thermodynamic equilibrium constant of the reforming reaction. Alternatively, one or two elementary reactions are assumed as RDS in Langmuir Hinshelwood models, generalized as:

$$r_{ref}=\frac{k_{ref}{K}_{\mathrm{CO}2}{K}_{\mathrm{CH}4}\left(p_{\mathrm{CH}4}{p}_{\mathrm{CO}2}-\frac{p_{\mathrm{H}2}^2{p}_{\mathrm{CO}}^2}{K_{ref}}\right)}{{\left(1+K_{\mathrm{CO}2}{p}_{\mathrm{CO}2}+K_{\mathrm{CH}4}{p}_{\mathrm{CH}4}\right)}^2}$$

while the others are at thermodynamic equilibrium. The elementary reactions steps are generally derived from the dissociative adsorption of both reactants, i.e., CH₄ decomposition and CO₂ decomposition/activation [8].

In order to evaluate the correct rate expression that most adequately describes the process, it is needed a discrimination between the different proposed models by fitting the experimental data. Many commercial software packages (Polymath, Mathcad, Matlab) can be used, but they often require some knowledge of mathematics, data management is not easy, they tend to display data-results-analysis in multiple and separate windows that can lead to confusion and, furthermore, they are expensive. To overcome these issues, we have tested the ANEMONA.XLT program [9], a free Excel template, for its suitability to estimate kinetic parameters by fitting literature data to some typical kinetic models that were proposed for the DR reaction.

The ANEMONA.XLT program, as developed by Hernández and Ruiz [9] for the specific calculation of enzyme kinetic parameters, is a collection of mathematical models corresponding to mono- or pseudo-monosubstrate reactions (Michaelis–Menten, double Michaelis–Menten, Hill model, substrate inhibition) and the most common bisubstrate mechanisms (ping-pong, ordered, and random). These models can be easily modified according to the different kinetic models, such as those proposed for DR reactions, as evidenced in this note.

2. Background

A brief description of some representative kinetic models that were proposed for the DR reaction considered in this study are shown below. In the Eley-Rideal mechanism, initially proposed by Mark et al. [10], associatively adsorption of one reactant on the catalyst surface at adsorption equilibrium is assumed. Reaction of the adsorbed species with the remaining reactant in gas phase, that leads to the product formation, is the RDS. Theoretically, both reactants can be adsorbed species [7]. For the bimolecular reaction between CO₂ adsorbed on the catalyst surface and un-adsorbed CH₄ in the gas phase, the rate equation is the following:

$$-rat{e}_{\mathrm{CH}4}=\frac{k{K}_{\mathrm{CO}2}{p}_{\mathrm{CH}4}{p}_{\mathrm{CO}2}}{\left(1+K_{\mathrm{CO}2}{p}_{\mathrm{CO}2}\right)}(\mathrm{Model}1)$$

where k is the rate constant for DR reaction, K_CO2 is the adsorption constant of CO₂, p_CH4 and p_CO2 indicate the partial pressure of CH₄ and CO₂, respectively. In the Eley-Rideal model, the rate depends on the concentration of the un-adsorbed species and the extent of the surface coverage of the adsorbed species. Assuming the CO₂ adsorption on the surface and un-adsorbed CH₄ in the gas phase, the rate equation becomes: $r=k{\theta }_{\mathrm{CO}2}{p}_{\mathrm{CH}4}$. The balance between the fraction of occupied sites θ_CO2 (${\theta }_{\mathrm{CO}2}=K_{\mathrm{CO}2}{p}_{\mathrm{CO}2}{\theta }_x$) and the fraction of vacant sites θ_x $\left({\theta }_x=\frac{1}{1+K_{\mathrm{CO}2}{p}_{\mathrm{CO}2}}\right),$ by application of the MARI (most abundant reaction intermediate) concept, tha rate equation results in the formulated Model 1.

Alternatively, a heterogeneous Eley-Rideal model has been proposed by Olsbye et al. [11]. The model involves the CH₄ adsorption on the catalyst surface and its dissociation into CH_x and H; the reaction between CH_x and CO_2,g became the RDS (at low pressure of CO₂). The related kinetic expression is represented by the following equation:

$$-rat{e}_{\mathrm{CH}4}=\frac{k{K}_{\mathrm{CH}4}{p}_{\mathrm{CH}4}{p}_{\mathrm{CO}2}^m}{\left(1+K_{\mathrm{CH}4}{p}_{\mathrm{CH}4}\right)}(\mathrm{Model}2)$$

where K_CH4 is the rate constant for CH₄ adsorption. Similarly the previous concepts, applied to CH₄ adsorbed on the surface, can be extended to the Model 2.

Langmuir-Hinshelwood rate models are generally proposed for reaction mechanisms that involve the dissociative adsorption of CH₄ and CO₂ followed by the surface reactions of the adsorbed species (rate determining step) to produce H₂ and CO.

A stepwise mechanism, where in the RDS the CH₄ is decomposed to hydrogen and active carbon followed by the direct and fast conversion of this active carbon to CO₂ and 2CO, is represented as [11,12]:

$$-rat{e}_{\mathrm{CH}4}=\frac{k{p}_{\mathrm{CH}4}{p}_{\mathrm{CO}2}}{\left(1+K_{\mathrm{CH}4}{p}_{\mathrm{CH}4}\right)\left(1+K_{\mathrm{CO}2}{p}_{\mathrm{CO}2}\right)}(\mathrm{Model}3)$$

while the associative adsorption of CH₄ and CO₂ molecules on a single site can be described as [6]:

$$-rat{e}_{\mathrm{CH}4}=\frac{k{K}_{\mathrm{CO}2}{K}_{\mathrm{CH}4}{p}_{\mathrm{CH}4}{p}_{\mathrm{CO}2}}{{\left(1+K_{\mathrm{CO}2}{p}_{\mathrm{CO}2}+K_{\mathrm{CH}4}{p}_{\mathrm{CH}4}\right)}^2}(\mathrm{Model}4).$$

The acid or basic functionalities of the catalyst support can influence the reaction mechanism with a bi-functional pathway where CH₄ and CO₂ are activated on different sites and the reaction intermediates react at the metal-support interfacial sites. Basic supports, like La₂O₃, enhance the non-dissociative CO₂ adsorption in the form of carbonates or bicarbonates [13,14]; this mechanism, with a two-step single site as RDS, can be represented by the following reactions:

$${\mathrm{CH}}_4+S\stackrel{K_1}{\leftrightarrow }S-{\mathrm{CH}}_4 \left(equilibrium\right)$$

(1)

$$S-{\mathrm{CH}}_4\stackrel{k_2}{\leftrightarrow }S-\mathrm{C}+2{\mathrm{H}}_2 \left(slow step\right)$$

(2)

$${\mathrm{CO}}_2+{\mathrm{La}}_2{\mathrm{O}}_3\stackrel{K_3}{\leftrightarrow } {\mathrm{La}}_2{\mathrm{O}}_2{\mathrm{CO}}_3 \left(equilibrium\right)$$

(3)

$${\mathrm{La}}_2{\mathrm{O}}_2{\mathrm{CO}}_3+\mathrm{C}-S\stackrel{k_4}{\leftrightarrow } {\mathrm{La}}_2{\mathrm{O}}_3+2\mathrm{CO}+S \left(slow step\right)$$

(4)

These include the CH₄ activation on metal catalyst (S) and the reaction of La₂O₂CO₃ with C deposited on Ni particles at the interface between Ni and La₂O₂CO₃ as rate determining steps. Hence, the related rate equation can be written as:

$$-rat{e}_{\mathrm{CH}4}=\frac{K_1{k}_2{K}_3{k}_4{p}_{\mathrm{CH}4}{p}_{\mathrm{CO}2}}{K_1{K}_3{k}_4{p}_{\mathrm{CH}4}{p}_{\mathrm{CO}2}+K_1{k}_2{p}_{\mathrm{CH}4}+K_3{k}_4{p}_{\mathrm{CO}2}}(\mathrm{Model}5)$$

where K₁ and K₃ denote the adsorption equilibrium constant for CH₄ and CO₂, respectively; k₂ is the rate constant of methane decomposition (craking) on the metallic surface, while k₄ is the rate constant of the reaction between the surface carbonate species and the C deposited on the catalytic surface [15,16].

Sierra Gallego et al. [17] for the Ni/La₂O₃ catalyst obtained from LaNiNO₃ perovskites structure, proposed an additional dual active site mechanism involving analogous reaction steps:

$${\mathrm{CH}}_4+S_1\stackrel{K_1}{\leftrightarrow }{S}_1-{\mathrm{CH}}_4 \left(equilibrium\right)$$

(5)

$$S_1-{\mathrm{CH}}_4\stackrel{k_2}{\leftrightarrow }{S}_1-\mathrm{C}+2{\mathrm{H}}_2 \left(slow step\right)$$

(6)

$${\mathrm{CO}}_2+S_2\stackrel{K_3}{\leftrightarrow } S_2-{\mathrm{CO}}_2\left(equilibrium\right)$$

(7)

$$S_2-{\mathrm{CO}}_2+S_1-C\stackrel{k_4}{\leftrightarrow } S_2+2\mathrm{CO}+S_1 \left(slow step\right)$$

(8)

where S₁ and S₂ represent metallic nickel and support La₂O₃ active sites, respectively. The related rate of methane conversion assumes the following form:

$$-rat{e}_{\mathrm{CH}4}=\frac{K_1{k}_2{K}_3{k}_4{p}_{\mathrm{CH}4}{p}_{\mathrm{CO}2}}{K_3{k}_4{p}_{\mathrm{CO}2}+K_1{K}_3{k}_4{p}_{\mathrm{CH}4}{p}_{\mathrm{CO}2}+K_1{k}_2{p}_{\mathrm{CH}4}+K_1{k}_2 K_3{p}_{\mathrm{CO}2}}(\mathrm{Model}6)$$

All of these selected models, which can be considered as representatives of the process, have been introduced into the ANEMONA.XLT tool in order to verify the suitability of the program. The existing models have been properly modified; the experimental data (rate_CH4, reagents partial pressure and reaction temperature) employed for the fitting procedure have been derived from the related literature [15,17,18,19,20].

3. Results

Table 1. Model parameters predicted by ANEMONA.XLT template compared with analogous literature data.

Model No.	Parameters from ANEMONA.XLT			Parameters from Literature			Refs.
	Rate Parameters	σ2 (mmol g−1 s−1)	Rmsd	Rate Parameters	σ2 (smmol g−1 s−1)	Rmsd
1	k = 14.34 mmol g−1 s−1 atm−1 KCO2 = 3.00 atm−1	1.6 × 10−6		k = 14.34 mmol g−1 s−1 atm−1 KCO2 = 2.99 atm−1	2.29 × 10−6		[18]
2	k = 2020.0 mmol g−1 s−1 KCH4 = 0.076 atm−1 m = 1.35 atm−1	9.69 × 10−5		k = 2020 mmol g−1 s−1 KCH4 = 0.078 atm−1 m = 1.36 atm−1	8.88·10−5		[18]
3	k = 60.00 mmol g−1 s−1 atm−2 KCH4 = 2.00 atm−1 KCO2 = 6.82 atm−1	3.17 × 10−5		k = 68.18 mmol g−1 s−1 atm−2 KCH4 = 1.55 atm−1 KCO2 = 10.57 atm−1	2.86 × 10−6		[18]
3	k = 2.67 × 10−4 mmol s−1 kPa−(α+β) KCH4 = −5.94 × 10−3 1 kPa−1 KCO2 = 1.96 × 10−2 kPa−1		0.012 (R2 = 0.970)	k = 1.02 × 10−4 mmol s−1 kPa−(α+β) KCH4 = −1.15 × 10−3 1 kPa−1 KCO2 = 3.06 × 10−2 kPa−1		0.028 (R2 = 0.965)	[19]
4	k = 50.0 mmol g−1 s−1 KCO2 = 1.76 atm−1 KCH4 = 0.54 atm−1	5.68 × 10−6		k = 49.9 mmol g−1 s−1 KCO2 = 1.74 atm−1 KCH4 = 0.55 atm−1	9.98 × 10−6		[18]
4	TReaction = 923 K k = 1.80 × 10−4 mmol min −1kPa–(α+β) KCH4 = -6.13 × 10−3 kPa−1 KCO2 = -3.2 × 10−3 kPa−1		0.0058 (R2 = 0.970)	k = 1.56 × 10−4 mmol min−1 kPa–(α+β) KCH4 = −6.29 × 10−3 kPa−1 KCO2 = 3.6 × 10−3 kPa−1		0.028 (R2 = 0.973)	[19]
4	TReaction = 973 K k = 3.69 10−4 mmol min−1 kPa–(α+β) KCH4 = −7.07 × 10−3 kPa−1 KCO2 = −1.53 × 10−3 kPa−1		0.040 (R2 = 0.98)	k = 4.32 × 10−4 mmol min−1 kPa–(α+β) KCH4 = −6.36 × 10−3 kPa−1 KCO2 = 9.19 × 10−4 kPa−1		0.036 (R2 = 0.98)	[19]
4	TReaction = 1023 K k = 7.96 × 10−4 mmol min−1 kPa–(α+β) KCH4 = −4.36 × 10−3 kPa−1 KCO2 = −1.09 × 10−3 kPa−1		0.032 (R2 = 0.985)	k = 1.56 × 10−4 mmol min−1 kPa–(α+β) KCH4 = −6.29 × 10−3 kPa−1 KCO2 = 3.6 × 10−3 kPa−1		0.028 (R2 = 0.973)	[19]

σ² = variance of the experimental error; Rmsd = Root mean square deviation.

summarizes the results obtained by fitting the experimental data reported by Özkara-Aydinoğlu and Aksoylu [18] and Ayodele et al. [19] to models from 1 to 4; the goodness of fit is evaluated by the variance of experimental error (σ²) and/or Root mean square deviation (Rmsd), in accordance to what the authors reported in the original articles.

When compared side by side, the estimations of the parameters obtained using the ANEMONA tool were in accordance with those reported by the authors, taking into account the possible inaccuracy of our starting data. A screen capture of the sheet fitting Model 4 after convergence is presented in Figure 1. It is interesting to note that the presence of graph plots (parity plot, analysis of residuals) represents a simple way to visualize the effects on the fit of any variation in the values of the parameter estimated.

The related parity plot depicted in Figure 2, where the experimental reaction rate (at increasing reaction temperatures) is related to the calculated rate, demonstrates the appreciable quality of the fitting with R² ranging from 0.970 to 0.985.

Models 5 and 6 involving the prediction of four parameters (K₁, k₂, K₃, and k₄) become a more complex mathematical model. Model 5 has been applied by Múnera et al. [15] to catalytic systems, including a catalytic active phase (Rh) supported on La₂O₃; the authors derive the equilibrium constant K₃, related to the adsorption equilibrium of CO₂ on the support (Equation (3)), from thermodynamic data reported in literature by Shirsat et al. [21]:

$$K_3=5.817\times {10}^{-9}\exp \left(\frac{1750.2}{T}\right)\left({\mathrm{kPa}}^{-1}\right)$$

(9)

On this basis, the predicted parameter values by application of the ANEMONA tools are summarized in

Table 2. Model parameters predicted by ANEMONA.XLT template from Models 5 and 6.

Model No.	Kinetic Parameters from ANEMONA.XLT			Kinetic Parameters from Literature		Refs.
	Treaction (K)	Rate Parameters	Rmsd
5	823	k1 = 2.82 × 10−2 kPa−1 k2 = 8.00 × 10−4 mol·g−1·s−1 k3 = 10.02 kPa−1 k4 = 6.61 × 10−6 mol·g−1·s−1	5.29 × 10−6	k1 = (48 ± 4.27) × 10−3 kPa−1 k2 = (6.0 ± 0.48) × 10−4 mol·g−1·s−1 k3 = (10.02 ± 0.11) kPa−1 k4 = (7.28 ± 0.073) × 10−6 mol·g−1·s−1	(*)	[15]
5	863	k1 = 1.93 × 10−2 kPa−1 k2 = 1.58 × 10−3 mol·g−1·s−1 k3 = 3.74 kPa−1 k4 = 2.32 × 10−5 mol·g−1·s−1	1.25 × 10−5	k1 = (38 ± 4.34) × 10−3 kPa−1 k2 = (9.9 ± 1.37) × 10−4 mol·g−1·s−1 k3 = (3.74 ± 0.017) kPa−1 k4 = (2.32 ± 0.0043) × 10−5 mol·g−1·s−1		[15]
5	903	k1 = 1.61 × 10−2 kPa−1 k2 = 3.00 × 10−3 mol·g−1·s−1 k3 = 1.85 kPa−1 k4 = 3.21 × 10−5 mol·g−1·s−1	1.95 × 10−5	k1 = (30 ± 3.18) × 10−3 kPa−1 k2 = (1.9 ± 0.31) × 10−3 mol·g−1·s−1 k3 = (1.85 ± 0.003) kPa−1 k4 = (4.05 ± 0.0025) × 10−5 mol·g−1·s−1		[15]
5	973	k1 = 1.600 kPa−1 k2 = 0.0035 mol·g−1·s−1 k3 = 0.017 kPa−1 k4 = 0.020 mol·g−1·s−1	1.10 × 10−4 (R2 = 0.9555)	k1 = 1.817 kPa−1 k2 = 0.0031 mol·g−1·s−1 k3 = 0.0062 kPa−1 k4 = 0.028 mol·g−1·s−1	(R2 = 0.9392)	[20]
5	1023	k1 = 0.890 kpa−1 k2 = 0.0051 mol·g−1·s−1 k3 = 0.0052 kPa−1 k4 = 0.0300 mol·g−1·s−1	7.61 × 10−6 (R2 = 0.9825)	k1 = 0.814 kPa−1 k2 = 0.0061 mol·g−1·s−1 k3 = 0.0041 kPa−1 k4 = 0.0350 mol·g−1·s−1	(R2 = 0.9721)	[20]
5	1073	k1 = 0.495 kPa−1 k2 = 0.0059 mol·g−1·s−1 k3 = 0.0035 kPa−1 k4 = 0.066 mol·g−1·s−1	8.21 × 10−6 (R2 = 0.984)	k1 = 0.188 kPa−1 k2 = 0.0082 mol·g−1·s−1 k3 = 0.0014 kPa−1 k4 = 0.049 mol·g−1·s−1	(R2 = 0.9483)	[20]
6	973	k1 = 200 × 10−3 kPa−1 k2 = 0.223 × 10−3 mol·g−1·s−1 k3 = 6.9 × 10−3 kPa−1 k4 = 11.03 × 10−3 mol·g−1·s−1	1.57 × 10−6	k1 = 141 × 10−3 kPa−1 k2 = 0.223 × 10−3 mol·g−1·s−1 k3 = 15.98 × 10−3 kPa−1 k4 = 13.22 × 10−3 mol·g−1·s−1	(**)	[17]

(*) = ±95% confidence interval; (**) = deviation from the experimental data below 5%.

, along with the parameter estimations by the cited authors [15].

The slight differences between our predicted data and those that were reported earlier can be ascribed to different factors (fitting procedure and/or inaccuracy of our starting data); while a good fitting, with R² = 0.99, emerges from the parity plot that is depicted in Figure 3a.

Similarly, the prediction of all the parameters present in Model 5 has been carried out by fitting the experimental data reported by Fan et al. [20] with a bimetallic Ni-Co/MgO-ZrO₂ catalyst. The fitting results, as reported in Table 2 and depicted in Figure 3b, confirm the appreciable reliability of the ANEMONA.

The complexity of Model 6 is solved by Sierra Gallego et al. [17] introducing in the model the value K₁k₂ ($K_1{k}_2=2.61\times {10}^{-3}\exp \left(\frac{-4300}{T}\right), {\mathrm{kPa}}^{-1})$ from previous literature [13]; the individually derived kinetic constants are reported in Table 2. Clearly, the estimations evaluated by ANEMONA were in accordance with those that were reported by the authors in the original article.

4. Advantages and Limitations

The reliability of the ANEMONA tool is dependent on the input parameters; indeed, if the initial parameters are inappropriate, the iteration process can result in a wrong solution. Similar published data, or alternatively, educated guesses can be used. The standard procedure consists in the variation of these data up and down, the repetition of the non-linear estimation, checking and discarding both estimations and initial guesses that produce absurd fitting and focusing on the estimated parameters with good fittings (i.e., little differences between the estimations and the experimental data).

The next step will be to check the physical significance of the fitted parameters; for example, if the adsorption equilibrium constant is one of the derived parameters, the fitted value must decrease with increasing temperature, because the adsorption is an exothermic process. As we have shown in this report, ANEMONA.XLT is flexible enough to be useful in the estimation of kinetic parameters, not only for biochemical reactions, but also for purely chemical ones. The only limitation is the availability of a reasonable mathematical model.

5. Conclusions

In this study, the suitability and adaptability of the Excel template, ANEMONA.XLT, with the purpose of fitting experimental data to some kinetic models proposed for dry reforming reactions, has been evidenced; the reliability of the tools was demonstrated by successful comparison of the results of the analysis using ANEMONA.XLT with those that were reported in the literature obtained with other software packages. The estimated parameters were considered satisfactory and, therefore, ANEMONA.XLT can be a reliable tool for the routine model discrimination pertaining reforming reactions as well as a fast and easy one to use.