Study on the reaction mechanism and kinetics of CO hydrogenation on a fused Fe–Mn catalyst

Abstract

The kinetics of the CO hydrogenation reaction over a Fe–Mn fused catalyst was investigated in a fixed-bed micro-reactor under the following conditions: temperatures of 573–603 K, pressures of 1–15 bar, H₂/CO feed ratios of 0.7–3.4 and a space velocity of 4500 h⁻¹. A reaction rate equation for the supported Fe–Mn catalyst was derived on the basis of the Langmuir–Hinshelwood–Hougen–Watson and Eley–Rideal models. An activation energy of 105 ± 3.7 kJ mol⁻¹ was obtained for the best fitted model. In addition, the power-law equation model was also evaluated for the experimental data. According to the power-law model, the activation energy was obtained as 95.5 ± 2.5 kJ mol⁻¹. Furthermore, the effect of temperature on the reaction partial order was investigated with respect to the reactants, using four simple power law equations. Characterization of the catalyst was carried out using BET and XRD techniques.

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1. Introduction

Fischer–Tropsch synthesis (FTS) has great potential for the production of ultraclean transportation fuels, like diesel and jet fuel, from synthesized gas produced from more abundant resources such as coal, natural gas and biomass. It has been found that several metals, such as nickel (Ni), cobalt (Co), ruthenium (Ru) and iron (Fe), can be activated for the FT reaction.¹ In the high-temperature Fischer–Tropsch (HTFT) process, the company Sasol used a catalyst prepared from fused iron oxides together with chemical and structural promoters.² The catalyst prepared from fused iron oxides was non-porous, so it obviously had a lower surface area compared to other preparation methods.³ However, structural promoters, such as the oxides of aluminum, magnesium, lanthanum or titanium, were added to increase the active surface area of the catalyst.⁴ The most important catalysts prepared by this method are the promoted iron for high-temperature FTS and catalysts for ammonia synthesis.³ For iron-based catalysts, the water-gas shift (WGS) reaction can affect the FTS reaction rate by changing the hydrogen and carbon monoxide partial pressures. The addition of small amounts of manganese to the catalyst enhanced the formation of olefinic products.⁵ Fe–Mn catalysts have attracted much attention due to their high olefin selectivity, lower methane selectivity and excellent stability.^5–9 Under a high reaction temperature, deactivation of the catalyst may occur due to the deposition of carbon on the catalyst surface. The carbon deposits may block the pores and sites of the catalyst, resulting in diffusion limitations and decreasing the activity of the catalyst.¹⁰ Temperature plays an important role in the amount of carbon deposition on the catalyst active sites during exothermic FTS, and this reaction should be performed in a way that maintains the near-isothermal conditions inside the catalyst beds.¹¹ Numerous studies have reported about the FTS kinetics over iron-based catalysts. Most kinetic expressions have been developed empirically, by fitting the data to a simple power-law relationship; it was generally found that the reaction order of hydrogen was positive, whereas that of carbon monoxide was negative.¹² Some researchers derived rate expressions of the reactant consumption based on Langmuir–Hinshelwood–Hougen–Watson- (LHHW) or Eley–Rideal-type mechanisms.^13,14 One of the most popular mechanisms for hydrocarbon formation on iron catalysts is the surface carbide mechanism using CH₂ insertion.^15–19 In particular, iron-based catalysts form stable carbides under the FTS reaction.^20,21 The differences in the rate expressions proposed for the consumption of synthesis gas are mainly because of the effect of adsorbed CO, H₂, and their products (H₂O and CO₂) on the catalyst surface. Carbon monoxide and water adsorb more strongly on the catalyst surface than H₂ and CO₂ do.^22,23 The most evident consequence is the usual assumption that water has a strong inhibiting influence on the reaction rate.¹⁴ The perceived negative influence of water on the reaction rate was ascribed to the competitive adsorption between water and CO on the catalyst surface. With increasing water concentration in the surface of the catalyst, the fraction of CO converted to hydrocarbons decreases due to an increase in the WGS reaction. Therefore, water has essentially an indirect effect on the FTS reaction rate, and increasing the water partial pressure will reduce the amount of surface carbon, which leads to a decrease in the rate of hydrocarbon formation.²⁴ The following simple relationships exist between the rates of the FTS reaction and the WGS reaction:²⁵

r_WGS = r_CO2 (1)

r_FT = −r_CO − r_CO2, (2)

where r_CO2 is the rate of CO₂ formation and r_CO is the rate of CO consumption. Dry reported that the CO₂ inhibition is not as strong as the water inhibition, due to the large difference in adsorption coefficients.²³ The negative effect of CO was ascribed to an extensive coverage of the catalyst surface by adsorbed carbon monoxide, inhibiting the adsorption of hydrogen.^24,26

The main objectives of the present work are to investigate the kinetics of the CO hydrogenation reaction over a Fe–Mn fused catalyst, and also to investigate the power-law equation model and obtain the kinetic parameters using these models. Furthermore, we also attempt to investigate the effect of reaction temperature on the reactant partial orders.

2. Experimental

2.1. Catalyst preparation

The Fe–Mn catalyst used in the present work was prepared using a fusion procedure. In order to prepare the fused iron catalyst, the required amounts of Fe(NO₃)₃·9H₂O (99% Merck), Mn(NO₃)₂·4H₂O (99% Merck), La₂O₃ (10 wt% based on the total catalyst weight) and Cs₂O (1 wt% based on the total catalyst weight), with a nominal composition of 50% Fe–50% Mn–10 wt% La₂O₃–1 wt% Cs₂O, were premixed in a crucible. The obtained mixture was heated and dried at 120 °C for 14 h in an oven to give a material denoted as the catalyst precursor. The obtained precursor was fused in an electrical furnace at 1500 °C for 2 h and then cooled slowly. To prevent mass transfer limitations, the obtained catalyst was crushed and screened to collect the catalyst particles of 30–70 mesh (210–590 μm).

2.2. Catalysts characterization

2.2.1. X-ray diffraction (XRD). Powder XRD measurements were performed using a D5000 X-ray diffractometer (Siemens, Germany).

Scans were taken with a 2θ step size of 0.02° and a counting time of 1.0 s, using a CuKα radiation source (λ = 0.15406 nm) generated at 35 kV and 20 mA. Specimens for XRD were prepared by compaction into a glass-backed aluminum sample holder. Data were collected over a 2θ range from 5 to 70°. The line broadening of the Fe₂O₃ and MnO₂ diffraction peaks localized at 33.2° and 39° 2θ values was used to estimate the average particle sizes, according to Scherrer’s equation.

2.2.2. BET measurements. BET surface areas, pore volumes and average pore sizes of the catalyst precursor and calcined samples (before and after the test) were measured by N₂ physisorption using a Quantachrome Nova 2000 automated system (USA). Each catalyst sample was degassed under a nitrogen atmosphere at 300 °C for 3 h. In order to obtain the BET surface areas, pore volumes and average pore sizes, different samples were evacuated at −196 °C for 66 minutes.

2.3. Catalyst testing

The experiments were carried out in a fixed-bed tubular stainless steel micro-reactor. A schematic representation of the experimental set-up is shown in Fig. 1. All gas lines to the reactor bed were made from 1/4′′ stainless steel tubing. Three mass flow controllers (Brooks, Model 5850E), equipped with a four-channel read-out and control equipment (Brooks 0154), were used to automatically adjust the flow rate of the inlet gases (CO, H₂ and N₂, with purities of 99.999%). The mixed gases in the mixing chamber passed into the reactor tube, which was placed inside a tubular furnace (Atbin, model ATU 150-15) capable of producing temperatures up to 1500 °C and controlled by a digital programmable controller (DPC). The reactor tube was constructed from stainless steel tubing; its internal diameter was 20 mm, with the catalyst bed situated in the middle of the reactor. This single tubular micro-reactor was surrounded by an alumina jacket to achieve a uniform wall temperature along the length of the reactor. A preheating zone ahead of the catalyst packing was filled with inert quartz glass beads. External heating was provided by an electrical element wrapped around the alumina jacket and placed in through the firebrick part. The required amount of the catalyst was diluted using inert silica sand with the same particle size range as the catalyst sample, and placed among the inert quartz glass beads. The temperatures of all of the different zones, including the preheating zone, the catalyst bed and the underneath zone of the reactor, were checked using three separate thermocouples placed in different parts of the reactor. The temperature of the catalyst bed was monitored with a thermocouple located exactly in the middle of the catalyst bed. The inlet feed gas arrived from the top of the reactor, and the outlet products exited from the lower part of the reactor. The meshed catalyst (2.0 g) was held in the middle of the reactor using quartz wool. An electronic back pressure regulator was used, which can control the total pressure of the desired process using a remote control via integration with the TESCOM software package, which can improve or modify its efficiency; it is capable of working on pressures ranging from atmospheric pressure to 100 bar. The catalyst was pre-reduced in situ at atmospheric pressure under H₂–N₂ (flow rate of each gas = 50 ml min⁻¹) at 350 °C for 16 h before synthesis gas exposure. The FTS was carried out under reaction conditions of T = 573–603 K, pressures of 1–15 bar, H₂/CO feed ratios of 0.7–3.4 and a space velocity of 4500 h⁻¹. In each test, 2.0 g of catalyst was loaded and the reactor operated for about 24 h to ensure that steady-state operations were attained. Some experiments (reported in Table 1) were repeated three times, and a comparison of the obtained results showed that a steady state was achieved after 24 h. Reactant and product streams were analyzed on-line using a gas chromatograph (Thermo ONIX UNICAM PROGC+) equipped with a sample loop, two thermal conductivity detectors (TCDs) and one flame ionization detector (FID) able to perform the analysis of a wide variety of gaseous hydrocarbon mixtures; one TCD was used for the analysis of hydrogen, and the other one was used for all the permanent gases such as N₂, O₂ and CO. The FID was used for the analysis of hydrocarbons. The system is applicable to the analysis of non-condensable gases, i.e. methane through to C₈ hydrocarbons. The contents of the sample loop were injected automatically into an alumina capillary column (30 m × 0.550 mm). Helium was employed as a carrier gas for optimum sensitivity (flow rate = 30 ml min⁻¹). The GC calibration was carried out using various calibration mixtures and pure compounds obtained from the gas company Matheson (USA).

Fig. 1 Schematic representation in a flow diagram of the reactor used: 1 = gas cylinders, 2 = pressure regulators, 3 = needle valves, 4 = ball valves, 5 = mass flow controllers (MFC), 6 = digital pressure controllers, 7 = pressure gauges, 8 = non-return valves, 9 = mixing chamber, 10 = valves, 11 = tubular furnace, 12 = tubular reactor and catalyst bed, 13 = temperature indicators (digital program controller), 14 = resistance temperature detector (RTD), 15 = condenser, 16 = trap air, 17 = back pressure regulator (BPR), 18 = flow meter, 19 = silica gel column, 20 = gas chromatograph (GC) and 21 = hydrogen generator.

Table 1 Experimental conditions for kinetic evaluations at P_tot = 1–15 bar, T = 573–603 K, H₂/CO = 1–3 and GHSV = 4500 h⁻¹ in a fixed-bed reactor (FBR)

No.	T (K)	Ptotal (bar)	H2/CO	PCO (bar)	PH2 (bar)	PH2O (bar)	R (mol min^−1 gcat^−1)
a Experiments repeated three times.
1^a	573.15	1	0.97	0.38	0.37	0.03	8.00 × 10^−6
2	573.15	1	1.95	0.22	0.43	0.04	1.55 × 10^−5
3	573.15	3	1.97	0.64	1.26	0.11	2.22 × 10^−5
4^a	573.15	6	1.02	2.27	2.32	0.17	2.03 × 10^−5
5	573.15	6	2.02	1.25	2.53	0.25	2.94 × 10^−5
6	573.15	9	3.15	1.27	4.00	0.46	3.32 × 10^−5
7^a	573.15	12	3.19	1.66	5.30	0.70	3.35 × 10^−5
8	573.15	15	1.04	5.54	5.76	0.36	3.12 × 10^−5
9	573.15	15	2.03	3.00	6.10	0.80	3.50 × 10^−5
10	573.15	15	3.33	2.00	6.65	0.90	3.49 × 10^−5
11^a	583.15	1	0.95	0.38	0.36	0.03	1.26 × 10^−5
12	583.15	1	1.95	0.22	0.43	0.03	1.76 × 10^−5
13	583.15	6	0.99	2.25	2.22	0.19	2.48 × 10^−5
14	583.15	9	1.03	3.25	3.36	0.21	3.54 × 10^−5
15^a	583.15	9	1.97	1.78	3.51	0.45	3.88 × 10^−5
16	583.15	12	1.97	2.39	4.70	0.60	3.89 × 10^−5
17	583.15	12	3.25	1.54	5.00	0.75	4.04 × 10^−5
18	583.15	15	1.03	5.39	5.55	0.42	3.46 × 10^−5
19^a	583.15	15	2.03	2.93	5.95	0.80	3.69 × 10^−5
20	583.15	15	3.34	1.86	6.22	0.93	4.00 × 10^−5
21	593.15	3	2.00	0.62	1.24	0.14	3.16 × 10^−5
22	593.15	3	3.07	0.43	1.32	0.15	3.24 × 10^−5
23^a	593.15	6	0.91	2.20	2.00	0.23	3.10 × 10^−5
24	593.15	6	3.05	0.80	2.44	0.43	3.60 × 10^−5
25	593.15	9	1.99	1.76	3.50	0.50	3.89 × 10^−5
26	593.15	9	3.28	1.15	3.77	0.60	3.97 × 10^−5
27^a	593.15	12	2.05	2.29	4.70	0.70	3.95 × 10^−5
28	593.15	12	3.36	1.49	5.00	0.86	4.10 × 10^−5
29	593.15	15	1.96	2.84	5.56	1.00	3.97 × 10^−5
30	593.15	15	3.37	1.78	6.00	1.15	3.97 × 10^−5
31^a	603.15	1	0.89	0.37	0.33	0.03	1.98 × 10^−5
32	603.15	1	1.85	0.20	0.37	0.03	2.10 × 10^−5
33	603.15	1	2.87	0.15	0.43	0.04	2.08 × 10^−5
34^a	603.15	3	0.91	1.10	1.00	0.10	3.08 × 10^−5
35	603.15	6	0.88	2.15	1.90	0.27	3.70 × 10^−5
36	603.15	9	0.94	3.20	3.00	0.40	3.67 × 10^−5
37^a	603.15	12	0.74	5.00	3.70	0.80	3.85 × 10^−5
38	603.15	12	1.88	2.23	4.20	0.90	4.00 × 10^−5
39	603.15	15	1.82	2.74	5.00	1.00	3.95 × 10^−5
40^a	603.15	15	3.33	1.80	6.00	1.20	4.05 × 10^−5

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2.4. Heat and mass transfer limitations

Heat and mass transfer limitations are two important factors which influence the reaction rate when heterogeneous catalysts are employed, especially at high temperatures. Measuring the reaction rate should not be influenced by deactivation of the catalyst or by heat and mass transfer limitations. When the mass transfer rate is smaller than the reaction rate, it results in a significant effect of mass transfer on the total observed rate, or it even controls and limits the reactant transfer from the gas phase to the catalyst surface.^26,27 In the presence of mass transfer limitations, the apparent activation energy for the reaction will be approximately one-half the true activation energy for the surface reaction.²⁸ Gas space velocity, catalyst particle size and catalyst amount are three important factors influencing heat and mass transfer in heterogeneous catalytic systems. With using a fixed-bed reactor, the heat and mass transfer limitation problem could be overcome by increasing the reaction temperature, and effective heat removal should be accomplished from catalytic active sites during exothermic FTS.²⁹ Before the kinetic experiments, mass transfer limitations in the fixed-bed reactor were investigated by changing gas space velocity and catalyst particle size. These conditions require the elimination of both pore diffusion and intra-particle (film resistance) mass transfer resistances. Preliminary experiments were performed to test the pore diffusion via decreasing the catalyst particle size. The fresh catalyst was crushed and sieved to particles with diameters of 210–590 μm (30–70 ASTM mesh), then isometric catalysts were loaded under the same operating conditions. As shown in Fig. 2, no pore diffusion limitation was observed for particles with sizes lower than 250 μm. As the particle size became smaller, the reaction rate remained constant. In the second set of experiments, intra-particle mass transfer limitation (film resistance) for the reaction was investigated by variation of the space velocity (resident time) with the upstream addition of N₂ to the flow. As shown in Fig. 3, over the range of space velocities between 1500 h⁻¹ and 6500 h⁻¹, as the space velocity increased, the reaction rate remained constant and film resistance was negligible. In addition, by using a small amount of high-density catalyst (low volume) prepared by the fusion method, the heat and mass transfer limitations are minimized.

Fig. 2 Variation of reaction rate as a function of particle size; conditions: T = 603 K, P = 1 bar and GHSV = 3600 h⁻¹.

Fig. 3 Variation of reaction rate as a function of GHSV value; conditions: T = 603 K, P = 1 bar.

3. Kinetic experiments

The FTS kinetic experiments were carried out with a mixture of H₂, CO and N₂ at the temperature range of 573–603 K, P = 1–15 bar, H₂/CO = 0.7–3.4 and GHSV = 4500 h⁻¹. The required amount of the catalyst (2.0 g) was diluted using 10 g of inert silica sand with the same particle size as the catalyst sample and placed among the inert quartz glass beads. For the kinetic measurement tests, the reactor was operated for 24 hours until the measurements became stable. The experimental conditions and obtained data are presented in Table 1. To avoid the effect of deactivation, fresh catalysts were loaded in each experiment series. To achieve isothermal conditions in a catalytic bed, the catalyst was diluted with inert materials (quartz and asbestos); axial temperature distribution was ensured using Mear’s criterion.^30,31 In order to avoid channelization phenomena, the following simplified relation between catalyst bed length (L_b) and mean catalyst particle diameter (d_p) was fulfilled: L_b/d_p > 50. We have a differential flow reactor when we choose to consider the rate to be constant at all points within the reactor. Since rates are concentration-dependent, this assumption is usually reasonable only for small conversions or for shallow small reactors. For each run in a differential reactor, the plug flow performance equation becomes as follows:Therefore, in brief:and hence:The molar flow rate of carbon monoxide in the feed is calculated as follows:

4. Results and discussion

4.1. Kinetic models and rate equations

In order to obtain the rate equation, firstly a reaction mechanism should be considered. For determination of kinetic models, four different mechanisms are presented, on the basis of different monomer formation and carbon chain repartition rate. The elementary reactions of these four offered mechanisms are summarized in Table 2. For the derivation of each kinetic model, firstly one of the elementary reaction steps was considered as the rate-determining step (RDS) and all the other steps were assumed to be at equilibrium. With consideration of different RDSs for the proposed models, 14 different rate expressions were obtained (presented in Table 3). Finally, all of the resulting rate expressions were fitted separately against experimental data. According to the obtained results, FT-I-1 was chosen as the best fitted model (the first step of the elementary reaction of the FT-I model was considered as the RDS). The elementary reaction steps of the FT-I model are presented in Table 2. The reaction rate of the RDS 1 for FT-I is as follows:

−r_CO = k₁P_COθ_s² (mol_CO g_cat⁻¹ min⁻¹), (8)

where θ_s refers to the fraction of free sites. As can be observed in Table 2, the adsorbed species in the FT-I model are C, H and O. The terms θ_C, θ_H and θ_O, respectively, refer to the surface fractions occupied by carbon, hydrogen and oxygen, which can be calculated via the site balance, with the preceding reaction steps being at quasi-equilibrium:

CO + 2s ⇌ C_s + O_s; (9)

k₁P_COθ_s² = k₋₁θ_Cθ_O; (10)

θ_C = θ_O = (αP_CO)^0.5θ_s, (13)

where α is the equilibrium constant of the CO adsorption step. θ_H is obtained from the below steps:

H₂ + 2s ⇌ 2H_s; (14)

k₂P_H2θ_s² = k₋₂θ_H²; (15)

θ_H = (βP_CO)^0.5θ_s. (18)

Table 2 Reaction schemes of CO hydrogenation

Model	Number	Elementary reaction
FT-I	1	CO + 2s ↔ Cs + Os
	2	H2 + 2s ↔ 2Hs
	3	Cs + Hs ↔ HCs + s
	4	HCs + Hs ↔ H2Cs + s
	5	Os + Hs → HOs + s
	6	HOs + Hs → H2O + 2s
FT-II	1	CO + s ↔ COs
	2	H2 + 2s ↔ 2Hs
	3	COs + Hs ↔ HCOs + s
	4	HCOs + Hs ↔ Cs + H2O + s
	5	Cs + Hs ↔ CHs + s
	6	CHs + Hs ↔ CH2s + s
	7	Os + Hs ↔ HOs + s
	8	HOs + Hs → H2O + 2s
FT-III	1	CO + s ↔ COs
	2	COs + H2 ↔ H2COs
	3	H2COs + H2 ↔ CH2s + H2O
FT-IV	1	CO + s ↔ COs
	2	H2 + s ↔ H2s
	3	COs + H2s ↔ H2COs + s
	4	H2COs + H2s ↔ H2O + s

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Table 3 Reaction rate expressions for CO hydrogenation

Number of proposed model	Rate equation	Parameters	MARR (%)
FT-I-1		K = k1	5.50
FT-I-2		K = k2	10.62
FT-I-3		K = k3K1^0.5K2^0.5	8.15
FT-I-4		K = k4K3K2K1^0.5	15.54
FT-II-1		K = k1	12.14
FT-II-2		K = k2	17.67
FT-II-3		K = k3K1K2	22.43
FT-II-4		K = k4K2K3K4	21.32
FT-III-1		K = k1	29.36
FT-III-2		K = k2K1	15.32
FT-III-3		K = k3K1K2	23.32
FT-IV-1		K = k1	18.30
FT-IV-2		K = k2	28.65
FT-IV-3		K = k3K1K2	32.12

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The free sites fraction (θ_s) is calculated from the site balance:

where θ_is refers to the surface fraction occupied with adsorbed species (C, H and O here). The site balance thus becomes:

θ_s + θ_C + θ_O + θ_H = 1. (20)

Substituting eqn (13) and (18) into eqn (20), the free active sites fraction is obtained as follows:

θ_s + 2(αP_CO)^0.5θ_s + (βP_H2)^0.5θ_s = 1; (21)

θ_s(1 + 2(αP_CO)^0.5 + (βP_H2)^0.5) = 1; (22)

By substituting eqn (23) into eqn (8), the following rate expression is obtained:

where K = k₁.

A schematic representation of the CO hydrogenation reaction over the Fe–Mn fused catalyst, for production of different hydrocarbons according to the best fitted model, is illustrated in Fig. 4.

Fig. 4 Schematic description of the FT mechanism according to the FT-I-1 model.

4.2. Kinetic parameters estimation

Model parameters were calculated from the experimental data and optimized with statistical indicators. Various plots, provided by the software Polymath 6.0, were used to assess the quality of the regression models and to compare the various models. The parameters that were used in the software Polymath 6.0 consist of the following: graph, residual plot, confidence interval, R², R_adj², variance and R_msd. These are defined as follows:

4.2.1. Graph. Graph is a plot on the basis of the calculated and measured values of R_CO for each proposed model. An inappropriate model shows differing trends. Fig. 5 compares the experimental R_CO with that calculated from the expression which was obtained for FT-I-1 (Table 3), with the assumption that step 1 is the rate-controlling step.

Fig. 5 Comparison between experimental and calculated reaction rates using eqn (8). Reaction conditions: T = 573–603 K, P = 1–15 bar, H₂/CO = 1/1–3/1 and GHSV = 4500 h⁻¹.

4.2.2. Residual plot. The residual plot is a plot that shows the difference between the calculated and measured values of the dependent variable as a function of the measured values. The residuals between the proposed model and the experimental values should be normally distributed, with a zero average line. A comparison between calculated and experimental CO conversion is presented in Fig. 6. This figure shows that the residual relative errors (RRs) between model and experiment are mostly distributed around the zero line.

Fig. 6 The relative residuals for the CO consumption rate (• R_exp − R_cal).

4.2.3. Confidence interval. If the confidence interval is smaller than (or at least equal to) the respective parameter values (in absolute values), then the regression model is stable and statistically valid.

4.2.4. R² and R_adj². The correlation coefficients were used to judge whether the model correctly represents the data. These parameters are defined as shown in eqn (25)–(27):

In the above formulas, the notation n, y_i, exp and calc denotes the number of experimental data points, specific observations, observed data and calculated data, respectively.

4.2.5. Variance and R_msd. Variance and R_msd are defined as shown in eqn (28) and (29):

Some statistical indicators that were used to assess the quality of the proposed model (expression FT-I, RDS 1) in Table 3 are summarized in Table 4.

Table 4 Obtained values of kinetic parameters for the fitted model (FT-I-1) and statistical criteria

Parameter	Value	Dimension
Ea	105.00 ± 3.70	kJ mol^−1
K0	5.93 × 10^6	mol gcat^−1 min^−1 bar^−1
k(573)	1.30 × 10^4	mol gcat^−1 min^−1 bar^−1
k(583)	1.96 × 10^4	mol gcat^−1 min^−1 bar^−1
k(593)	2.95 × 10^4	mol gcat^−1 min^−1 bar^−1
k(603)	4.35 × 10^4	mol gcat^−1 min^−1 bar^−1
α0	1.28 × 10^8	bar^−1
ΔHCO	−68.00 ± 4.50	kJ mol^−1
β0	2.98 × 10^7	bar^−1
ΔHH2	−48.00 ± 3.20	kJ mol^−1
R^2	0.94	—
Radj^2	0.91	—
Rmsd	3.918 × 10^−7	—
Variance	7.23 × 10^−12	—
MARR (%)	5.50	—

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For estimation of the best kinetic model, we made the following assumptions: (1) the mass transport limitations and pressure drop are negligible, and (2) the suitable values for all parameters must be positive and all offered models with negative values of parameters will be refused.

Estimation of parameters and model distinction have been accomplished using a nonlinear regression model and the software Polymath 6.0. Fig. 6 displays the residuals between the offered model and the experimental data, distributed randomly around the zero line. This figure confirms that the offered model is in good agreement with the experimental data.

According to the obtained results, the best expression that describes the experimental results for the FT reaction over the Fe–Mn fused catalyst is as follows (eqn (24)):

Kinetic and adsorption parameters which depend on temperature were described and calculated according to the Arrhenius and van’t Hoff equations, respectively (eqn (30) and (31)):

k_i(T) = k_i,0 exp(−E_i)/RT; (30)

α(T) = α₀ exp(−ΔH_ads)/RT. (31)

In these equations, E and ΔH refer to the activation energy and heat of adsorption, respectively; by substituting eqn (30) and (31) in the best fitted model (FT-I-1), we have:

The relation between the temperature and the reaction rate constant, according to the kinetic parameters obtained from the FT-I-1 model, is shown in Fig. 7. According to the Arrhenius-type equation (eqn (30)), a plot of ln(k) versus 1/T should give a straight line with a negative slope of –E_i/R. The relation between the temperature and the adsorption enthalpy (van’t Hoff plot) for the FT-I-1 model is shown in Fig. 8.

Fig. 7 Arrhenius plot of the rate constant (k) according to the FT-I-1 model results.

Fig. 8 Van’t Hoff plots of the adsorption coefficients of (A) a CO molecule and (B) a H₂ molecule, according to the results obtained from the FT-I-1 model.

The mean absolute relative residual (MARR%) between the experimental and the calculated consumption rates of CO is defined as follows:

where N_exp is the number of experimental points. Eqn (24) shows the best fit to the experimental data. A comparison of the calculated and experimental consumption rates of CO for the FT-I-1 model is shown in Fig. 9; the MARR% of this model was obtained as 5.50%. This value is reasonable and shows that the predicted values are 5.50% different from the observed values. The MARR% values of the other obtained kinetic models are presented in Table 3; as can be seen in this table, the FT-I-1 model has the lowest MARR% value. The obtained activation energy for FT-I-1 was found to be 105 ± 3.7 kJ mol⁻¹.

Fig. 9 The calculated CO consumption rate versus the experimental CO consumption rate for the FT-I-1 model.

4.3. Kinetic investigation using a power law model

The effect of reactants on the reaction rate has been investigated by many researchers.^{12,22,24–26,32–34} In the present work, we attempted to investigate the relationship between the partial pressures of the reactants and temperature changes. By using the power law equation (eqn (34)), the order of reaction was obtained at four temperatures for the FTS. To investigate the effect of water on the reaction rate, water pressure was entered in the power law equation:

r_FT = k_FTP_CO^aP_H2^bP_H2O^c. (34)

The obtained results are summarized in Table 5. The obtained results showed that, with increasing temperature from 573 to 603 K, the partial orders of CO and H₂O increased and the partial order of hydrogen decreased. The effect of temperature on the reaction order is plotted in Fig. 10. Previous research showed that the reaction order of H₂ [b] was positive but the reaction orders of CO [a] and H₂O [c] were negative.^23,24,35 It could be argued that the higher CO partial pressure leads to higher coverage of the catalyst surface by adsorbed CO. As is shown in Fig. 10, upon increasing the temperature, the reaction order was changed for all reactants. With increasing temperature, adsorbed CO molecules are consumed more rapidly, and so the number of free active sites for adsorption of H₂ molecules is increased. An increase in the number of free active sites leads to easier adsorption of hydrogen molecules on the catalyst surface, so that the positive effect of higher H₂ partial pressure on the reaction rate is decreased. The calculated activation energy for the CO hydrogenation reaction according to the power law model was found to be 95.54 ± 2.5 kJ mol⁻¹; the high activation energy for hydrocarbon formation suggests that the diffusion interference is not significant in experiments.^36,37 As with intra-particle diffusion limitations, the presence of external mass transfer limitations could be detected via measuring the apparent activation energy. An external mass transfer control regime could lead to an apparent activation energy of just a few kJ mol⁻¹.³⁸

Table 5 The obtained results for the reaction order, using the power law equation, at different temperatures (573–603 K)

No.	T (K)	Power law equation
1	573	R573 = 9.13 × 10^−6PCO^−0.27PH2^0.84PH2O^−0.23
2	583	R583 = 1.44 × 10^−5PCO^−0.18PH2^0.65PH2O^−0.15
3	593	R593 = 2.23 × 10^−5PCO^−0.09PH2^0.39PH2O^−0.11
4	603	R603 = 2.38 × 10^−5PCO^−0.08PH2^0.28PH2O^−0.08

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Fig. 10 Effect of reaction temperature on the reaction partial orders.

According to the power law equation (eqn (34)), four rate constants (k) were obtained for the four different temperatures (Table 6). In order to show the relationship between the inverse of the temperature and the logarithm of the rate constant obtained from this model, the Arrhenius equation (eqn (30)) was used. The obtained plot, displayed in Fig. 11, illustrates this relationship, showing a straight line with the negative slope (−E_a/R). The value of −E/R was found to be −11 492, which yields an activation energy of 95.54 kJ mol⁻¹.

Table 6 Values of kinetic parameters for the power law model (eqn (34))

Parameter	Value	Dimension
Ea	95.54 ± 2.5	kJ mol^−1
K0	5.01 × 10^3	mol gcat^−1 min^−1 bar^−1
k(573)	9.13 × 10^−6	mol gcat^−1 min^−1 bar^−1
k(583)	1.44 × 10^−5	mol gcat^−1 min^−1 bar^−1
k(593)	2.23 × 10^−5	mol gcat^−1 min^−1 bar^−1
k(603)	2.38 × 10^−5	mol gcat^−1 min^−1 bar^−1
R^2	0.90	—
MARR (%)	8.86	—

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Fig. 11 Arrhenius plot according to the power law equation results.

The kinetic parameters and activation energy of the power law model (eqn (34)) were calculated, and the obtained results are presented in Table 6. A comparison of the calculated and experimental consumption rates of CO (for eqn (34)) is shown in Fig. 12; the MARR% of this model was obtained as 8.86%. This value is reasonable and shows that the predicted values are 8.86% different from the observed values. Fig. 13 shows a comparison between the experimental and calculated intrinsic reaction rates using the power law equation.

Fig. 12 The calculated CO consumption rate versus the experimental CO consumption rate, using the power law model.

Fig. 13 Comparison between experimental and calculated reaction rates for the power law equation (eqn (34)).

4.4. Catalysts characterization

Characterization of the fresh catalyst (sample before the test) and the used sample (catalyst after the test) was carried out using powder X-ray diffraction, and the obtained patterns are illustrated in Fig. 14. The actual phases identified in the fresh catalyst were Fe₂O₃ (hexagonal), MnO₂ (tetragonal), Fe₃O₄ (cubic) and Mn₂O₃ (orthorhombic). In order to identify the changes in the catalyst during the reaction and to detect the phases formed, this catalyst was characterized by XRD after the test and its phases were found to be MnO (cubic), FeO (cubic), Fe (cubic) and Fe₂C (monoclinic). As can be seen, in the tested catalyst there are oxidic and iron carbide phases, which are both active for FTS. Zhang and Schrader³⁹ concluded that two active sites operate simultaneously on the surface of iron catalysts: Fe0/Fe-carbides and magnetite (Fe₃O₄). The carbide phase is active towards the dissociation of CO and formation of hydrocarbons, while the oxide phase adsorbs CO associatively and produces predominantly oxygenated products. The crystallite size of the fresh catalyst was calculated by the Debye–Scherrer equation from the XRD data. The average crystallite sizes for Fe₂O₃ and MnO₂ were calculated to be 91 and 83 nm, respectively.

Fig. 14 XRD patterns of calcined catalysts (before and after the test).

Characterization of the fresh catalyst was also carried out using BET measurements; the obtained results showed that the surface area of this sample was 14 m² g⁻¹. The surface area of the catalyst could be partially improved during the reduction process.^3,5 The addition of Mn appeared to increase the BET surface area of the Fe catalysts.⁴ Dry has suggested that an alkali promoter can decrease the surface area of the Fe catalyst by increasing the Fe crystallite size.⁴⁰

5. Conclusion

The kinetics of the CO hydrogenation reaction were investigated over a fused Fe–Mn catalyst in a fixed-bed micro-reactor over a range of operating conditions. Four different mechanisms, according to the carbide mechanism using the Langmuir–Hinshelwood–Hougen–Watson and Eley–Rideal mechanisms, were derived for CO hydrogenation. The unknown kinetic parameters were estimated from experimental data using a nonlinear regression (Levenberg–Marquardt) method. The reaction rate of CO hydrogenation is determined by the formation of the methylene monomer. In the best fitted model (FT-I-1), both reactants (CO and H₂) were dissociated and adsorbed on the catalyst surface. Furthermore, the power law model was also proposed and evaluated. The results in the present work also showed that upon increasing the reaction temperature, the reaction partial orders of all reactants were changed.

Nomenclature

rFT	Rate of reaction
rWGS	Rate of water gas shift reaction
rCO	Rate of CO consumption
rCO2	Rate of CO2 production
F^°CO	Inlet molar flow of CO
ν°	Volumetric flow rate of input gas
CCO	Concentration of CO
PCO	Partial pressure of CO
PH2	Partial pressure of H2
PH2O	Partial pressure of H2O
T	Gas temperature
R	Universal gas constant
k	Rate constant of reaction
Ea	Activation energy
α	Adsorption coefficient of CO
β	Adsorption coefficient of H2
ΔHH2	Adsorption enthalpy of H2
ΔHCO	Adsorption enthalpy of CO
XCO	The conversion of CO
W	The catalyst weight

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