Numerical Investigation of Equilibrium and Kinetic Aspects for Hydrogenation of CO₂

Abstract

Even if huge efforts are made to push alternative mobility concepts, such as electric cars and fuel-cell-powered cars, the significance and use of liquid fuels is anticipated to stay high during the 2030s. Biomethane and synthetic natural gas (SNG) might play a major role in this context, as they are raw material for chemical industry that is easy to be stored and distribute via existing infrastructure, and are a versatile energy carrier for power generation and mobile applications. Since biomethane and synthetic natural gas are suitable for power generation and for mobile applications, they can therefore replace natural gas without any infrastructure changes, thus playing a major role.In this paper, we aim to comprehend the direct production of synthetic natural gas from ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2$ in a Sabatier process based on a thermodynamic analysis as well as a multi-step kinetic approach. For this purpose, we thoroughly discuss ${\mathrm{CO}}_2$ methanation to control emissions in order to maximize the methane formation along with minimizing the CO formation and to understand the complex methanation process. We consider an equilibrium and kinetic modeling study on the NiO-${\mathrm{SiO}}_2$ catalyst for methanation focusing on ${\mathrm{CO}}_2$-derived SNG. The thermodynamic analysis of ${\mathrm{CO}}_2$ hydrogenation is preformed to define the optimal process parameters followed by the kinetic simulations for catalyst development. The investigation presented in this paper can also be used for developing machine learning algorithms for methanation processes.

1. Introduction

The primary factor contributing to the rise in atmospheric temperature is believed to be the increase in ${\mathrm{CO}}_2$ concentration. Therefore, it is crucial to capture and reintegrate secondary products like ${\mathrm{CO}}_2$ and recover ${\mathrm{H}}_2$ into the energy supply, ensuring that direct ${\mathrm{CO}}_2$ emissions are minimized [1,2]. Hence, ${\mathrm{CO}}_2$ reduction is a critical aspect of addressing climate change, and there are several methods available to deal with this problem. Some of the most important ones are summarized here along with their advantages and disadvantages. The chemical conversion processes aim to transform ${\mathrm{CO}}_2$ into usable fuels or chemicals, such as methanation, hydrogenation to methanol, Fischer–Tropsch, and electrochemical ${\mathrm{CO}}_2$ reduction. Furthermore, the physical and chemical separation techniques, for instance, absorption, adsorption, membrane separation, and cryogenic separation, play a major role in capturing ${\mathrm{CO}}_2$ from gas streams for storage or utilization. Nevertheless, each of these methods bring its own advantages and challenges based on the specific context and the desired products.

Methanol produced during the hydrogenation of ${\mathrm{CO}}_2$ is a valuable feedstock with multiple industrial uses and is also easier to store and transport when compared to the gases like methane; however, it needs significant energy input in producing hydrogen. Another limitation is the infrastructure needs and catalyst deactivation over time, leading to increased operational costs and reduced efficiency. Fischer–Tropsch synthesis has issues with operational complexity and costs due to complex processes; however, it involves flexible feedstock, producing liquid hydrocarbons that can directly substitute fossil fuels in existing engines and infrastructure. The reverse water–gas shift reaction is well studied and can be optimized for different outputs and to produce CO, which can be used in other chemical processes discussed above, but the reaction requires high temperatures and needs a significant amount of hydrogen. The conversion efficiency is also a concern in the reverse water–gas shift reaction. Electrochemical ${\mathrm{CO}}_2$ reduction is good in terms of the direct use of renewable energy and flexible products depending on catalysts; however, the cost of the catalyst and electrode materials along with low efficiency are major challenges.

Absorption offers a well-established and scalable solution for capturing ${\mathrm{CO}}_2$ from industrial emissions; however, energy consumption, material degradation, and cost are the major challenges. When compared to absorption, adsorption requires less energy, but the capacity, selectivity, adsorbent degradation, and operational complexities are the limitations. The simple operation and scalable and modular approach, along with the lower energy consumption, make the membrane separation approach attractive. Nevertheless, the cost, degradation, and selective permeability of membrane are the major disadvantages in the membrane separation method. In the case of cryogenic separation, pure ${\mathrm{CO}}_2$ is produced, and no chemical solvents are needed, but it requires significant energy for cooling to very low temperatures.

We are interested in methanation for converting ${\mathrm{CO}}_2$ into valuable fuels and chemicals, potentially contributing to energy storage and a circular carbon economy. This can also be energy intensive and may demand significant infrastructure and investment. However, our focus is to produce usable fuel that can be integrated into existing energy systems, and hence, methanation offers clear advantages by creating a versatile and easily stored energy carrier as methane. In methanation, the excess energy produced by renewable energies is converted into chemical energy with the possibility to feed the produced ${\mathrm{CH}}_4$ into the existing network of natural gas referred to as the ’power-to-gas’ approach (PtG) [3,4]. In PtG, hydrogen and methane store energy by utilizing the existing natural gas grid for the storage of synthetic natural gas (SNG) without requiring any investment in storage infrastructure. The stored energy can be used in times of the absence of regenerative production and during dark doldrums.

A sufficient supply of ${\mathrm{H}}_2$ is required for the hydrogenation of ${\mathrm{CO}}_2$, and this hydrogen must be generated by some kind of renewable energy in order to ensure a ${\mathrm{CO}}_2$-neutral process. Alkaline-based or proton exchange membrane (PEM) water electrolysis can be used for hydrogen production [5]. Another promising alternative to the hydrogen source is coke oven gas (COG) [4,6]. To make the SNG production a recycling process, the exhaust ${\mathrm{CO}}_2$ emitted by a lignite power plant, a refinery, or a cement production plant can be used as a ${\mathrm{CO}}_2$ source to fully convert the hydrogen/${\mathrm{CO}}_2$ to methane as performed by the Sabatier reaction:

$${\mathrm{CO}}_2+4{\mathrm{H}}_2\to {\mathrm{CH}}_4+2{\mathrm{H}}_2\mathrm{O},\Delta {\mathrm{H}}_{298}^0=-165.0\mathrm{kJ}/\mathrm{mol}.$$

(1)

For the given educt gas composition, the Sabatier reaction is thermodynamically favored. Nevertheless, the reaction is limited in kinetics, and hence catalysts are needed to achieve acceptable conversion from ${\mathrm{CO}}_2$ into ${\mathrm{CH}}_4$ [7,8,9]. Before entering into the complex multi-step reaction kinetics, we want to comprehend the thermodynamics involved in the processes occurring in such complicated reaction systems. Also, the chemical equilibrium of a reaction is interesting for catalyst research, as the equilibrium finally reveals the maximum amount of a substance that can be converted [10].

It is interesting to understand if the products of a reaction are thermodynamically stable and whether a chemical reaction propagates endothermically or exothermically. Therefore, to investigate these phenomena, thermodynamic equilibrium calculation of complex chemical systems needs to be performed. We can also study the impact of reaction parameters, such as temperature, pressure, and reactant ratios, to understand the methanation process.

Next, we note some of the thermodynamic and kinetic investigations of the methanation reactions discussed in the open literature. The thermodynamic equilibrium compositions for three CO methanation reactions, $\mathrm{CO}+3{\mathrm{H}}_2\leftrightarrow {\mathrm{CH}}_4+{\mathrm{H}}_2\mathrm{O}$, $\mathrm{CO}+{\mathrm{H}}_2\mathrm{O}\leftrightarrow {\mathrm{CO}}_2+{\mathrm{H}}_2$, and $2\mathrm{CO}\leftrightarrow \mathrm{C}+{\mathrm{CO}}_2$, are given in [11]. The thermodynamics of the hydrogenation of carbon oxides to produce the hydrocarbons and oxygenated organic molecules is studied in [12]. In [13], a detailed thermodynamic equilibrium analysis of methanation reactions of carbon oxides (CO and ${\mathrm{CO}}_2$) using the minimization of the Gibbs free energy is discussed in the temperature range 200–800 °C, and in [14], a thermodynamic investigation is presented focusing on the hydrogenation of ${\mathrm{CO}}_2$ to various products, such as CO, hydrocarbons, alcohols, aldehydes, and carboxylic acids, by using the same approach as that used in [13]. The thermodynamic calculations of ${\mathrm{CO}}_2$ methanation based on the catalyst under the considered reaction conditions are conducted in [15,16], and those for CO methanation over Ni/${\mathrm{Al}}_2{\mathrm{O}}_3$ catalysts are conducted in [17].

Although there are thermodynamic as well as kinetic investigations available in the literature, a deeper understanding of methanation reactions to optimize the processes is still needed. Hence, in the present paper, we thoroughly discuss the thermodynamic impact of ${\mathrm{CO}}_2$ methanation on the formation of products as ${\mathrm{CH}}_4$, CO, and ${\mathrm{H}}_2\mathrm{O}$ in order to maximize the methane formation and minimize the CO formation. We also consider the effect of various initial conditions, for instance, temperature, pressure, and ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio. To optimize the performance of the processes, we also consider the dilution of syngas with species, such as ${\mathrm{H}}_2\mathrm{O}$, ${\mathrm{O}}_2$, CO, and ${\mathrm{CH}}_4$. Further, the kinetic modeling is performed to support the development of the catalyst and also to provide detailed information in order to establish the experimental setup. This study not only provides information for the catalyst and experiment but also aims to make a huge dataset available for developing data-based models and machine learning algorithms.

2. Methodology

In our previous studies, we used the LOGEcat model for catalyst investigations [18,19,20] from the LOGEsoft software suite [21]. This model is used for the kinetic simulations discussed in this paper. Apart from the chemically reacting flow models, there is also an equilibrium reactor model in the software package, and this model is used to perform the equilibrium simulations discussed in the present work. The equilibrium reactor model only requires thermodynamic data for all the species in each phase and allows to determine the chemical state of a mixture under equilibrium conditions, including any number of gas-phase or bulk species for the equilibrium calculations.

The equilibrium composition of a reactive system is calculated using the Gibbs free energy minimization method, which is based on the principle that the total Gibbs energy of the system has its minimum value at the chemical equilibrium, in which the individual equilibrium constants are not considered [22,23,24]. The distribution of the products under a minimum free energy is achieved by utilizing a general mathematical technique without knowledge of the chemistry of the reactions, provided all the species in a reaction system, including the reactants as well as the products, are given. The main quantity here is the Gibbs energy expressed using the Gibbs energy under standard conditions and the law of the mass action as:

$${\mathsf{\Delta }}_r\mathrm{G}={\mathsf{\Delta }}_r{\mathrm{G}}^{\theta }+\mathrm{RT}ln\prod _i{a}_i^{{\nu }_i}$$

(2)

where ${\mathsf{\Delta }}_r\mathrm{G}$ is the change in the Gibbs free energy, ${\mathsf{\Delta }}_r{\mathrm{G}}^{\theta }$ is the standard Gibbs free energy for the reaction r, R is the universal gas constant, T is the temperature, ∏ is the product across all i-indexed variables, a is the activity coefficient, and $\nu$ is the stoichiometric coefficient of species i. The change in free energy at equilibrium is zero; hence, the equilibrium constant is (${\mathrm{K}}_{\mathrm{eq}}$):

$${\mathrm{K}}_{\mathrm{eq}}=exp\left({\displaystyle \frac{-{\mathsf{\Delta }}_r{\mathrm{G}}^{\theta }}{\mathrm{RT}}}\right)$$

(3)

The direction of the reaction from the above expression is determined by the sign of ${\mathsf{\Delta }}_r{\mathrm{G}}^{\theta }$. If ${\mathsf{\Delta }}_r{\mathrm{G}}^{\theta }<$ 0, then the reaction proceeds in the direction of the products (${\mathrm{K}}_{\mathrm{eq}}>$ 1), and if ${\mathsf{\Delta }}_r{\mathrm{G}}^{\theta }>$ 0, then the reaction flows in the direction of the reactants (${\mathrm{K}}_{\mathrm{eq}}<$ 1). The change in the standard free energy ${\mathsf{\Delta }}_r{\mathrm{G}}^{\theta }$ can be calculated using the Gibbs–Helmholtz equation as:

$${\mathsf{\Delta }}_r{\mathrm{G}}^{\theta }=\mathsf{\Delta }{\mathrm{H}}^{\theta }-\mathrm{T}\mathsf{\Delta }{\mathrm{S}}^{\theta }$$

(4)

where ${\mathsf{\Delta }}_r{\mathrm{G}}^{\theta }$ in the above equation is temperature dependent, $\mathsf{\Delta }{\mathrm{H}}^{\theta }$ is the change in the reaction enthalpy, and $\mathsf{\Delta }{\mathrm{S}}^{\theta }$ is the change in the reaction entropy. The equilibrium solver uses the polynomial functions of temperature to determine the state functions and heat capacity. The polynomial coefficients for each species can be provided in the state function input file in a standard format for NASA coefficients.

These polynomials can be used to drive all the other thermodynamic properties listed in

Table 1. Summary of the thermodynamic data for all the species used in the present simulations at 300, 500, 1000, 1500, and 2000 K. The values are calculated from NASA polynomials [25].

	T [K]	300	500	1000	1500	2000
	${\mathrm{CO}}_2$	37.22	44.62	54.32	58.22	60.46
Heat	${\mathrm{H}}_2$	28.85	29.30	30.16	32.36	34.20
Capacity	${\mathrm{CH}}_4$	35.76	46.50	73.73	90.48	101.67
$\left[\mathrm{J}/\mathrm{mol}/\mathrm{K}\right]$	${\mathrm{H}}_2\mathrm{O}$	33.60	35.21	41.29	47.33	51.68
	${\mathrm{CO}}_2$	−393,441.22	−385,209.56	−360,112.78	−331,892.88	−302,164.12
Enthalpy	${\mathrm{H}}_2$	53.36	5895.49	20,686.65	36,333.76	52,985.94
$\left[\mathrm{J}/\mathrm{mol}\right]$	${\mathrm{CH}}_4$	−74,533.91	−66,388.03	−35,926.06	5403.61	53,626.74
	${\mathrm{H}}_2\mathrm{O}$	−241,763.86	−234,901.14	−215,823.33	−193,586.42	−168,771.45
	${\mathrm{CO}}_2$	214.02	234.88	269.29	292.12	309.21
Entropy	${\mathrm{H}}_2$	130.86	145.77	166.24	178.90	188.46
$\left[\mathrm{J}/\mathrm{mol}/\mathrm{K}\right]$	${\mathrm{CH}}_4$	186.59	207.16	248.31	281.59	309.26
	${\mathrm{H}}_2\mathrm{O}$	189.04	206.53	232.74	250.69	264.93
	${\mathrm{CO}}_2$	−457,646.44	−502,650.09	−629,400.38	−770,070.50	−920,579.81
Gibbs	${\mathrm{H}}_2$	−39,204.47	−66,990.46	−145,550.02	−232,009.22	−323,941.53
Energy	${\mathrm{CH}}_4$	−130,511.52	−169,969.94	−284,233.56	−416,986.69	−564,897.75
$\left[\mathrm{J}\right]$	${\mathrm{H}}_2\mathrm{O}$	−298,474.94	−338,166.22	−448,559.66	−569,615.69	−698,639.12
$\mathsf{\Delta }{\mathrm{H}}^{\theta }\left[\mathrm{J}/\mathrm{mol}\right]$	Equation (5)	−164,833.85	−174,562.75	−190,206.54	−195,211.39	−193,695.80
$\mathsf{\Delta }{\mathrm{S}}^{\theta }\left[\mathrm{J}/\mathrm{mol}/\mathrm{K}\right]$	Equation (6)	−172.79	−197.74	−220.45	−224.73	−223.93
$\mathsf{\Delta }{\mathrm{G}}^{\theta }\left[\mathrm{J}/\mathrm{mol}\right]$	Equation (7)	−112,997.09	−75,690.45	30,247.56	141,889.44	254,169.86
${\mathrm{K}}_{\mathrm{eq}}[-]$	Equation (3)	4.73 $\times {10}^{19}$	8.08 $\times {10}^7$	2.63 $\times {10}^{-2}$	1.14 $\times {10}^{-5}$	2.30 $\times {10}^{-7}$

. For the Sabatier reaction, Equation (1), the Gibbs free energy can be calculated using the equations given below (t = T[K]/1000):

$$\mathsf{\Delta }{\mathrm{H}}^{\theta }\left(t\right)=\mathsf{\Delta }{\mathrm{H}}_{298.15K}^0+\mathrm{d}{\mathrm{H}}_{C{H}_4}+2\ast \mathrm{d}{\mathrm{H}}_{H_{2}O}-\mathrm{d}{\mathrm{H}}_{C{O}_2}-4\ast \mathrm{d}{\mathrm{H}}_{H_2}$$

(5)

$$\mathsf{\Delta }{\mathrm{S}}^{\theta }\left(t\right)={\mathrm{S}}_{C{H}_4}^0+2\ast {\mathrm{S}}_{H_{2}O}^0-{\mathrm{S}}_{C{O}_2}^0-4\ast {\mathrm{S}}_{H_2}^0$$

(6)

$${\mathsf{\Delta }}_r{\mathrm{G}}^{\theta }\left(T\right)=\mathsf{\Delta }{\mathrm{H}}^{\theta }\left(t\right)-T\ast {\displaystyle \frac{\mathsf{\Delta }{\mathrm{S}}^{\theta }\left(t\right)}{1000}}$$

(7)

The equilibrium constant, ${\mathrm{K}}_{\mathrm{eq}}$ (Table 1), for the Sabatier reaction (Equation (1)) due to its exothermic nature reduces with increasing temperature. Due to the high equilibrium constant at low temperatures, the equation plays a major role, and the influence is also captured on the results. The other reactions, however, also affect the ${\mathrm{CO}}_2$ conversion. Hence, it is of great interest to investigate the product composition in the methanation process.

When the equations above are solved, a system of linear simultaneous equations is achieved with many unknowns, which yields a new composition representing a new approximation of the composition giving minimum free energy. The free energy of the system is minimal when the calculated and the prior compositions are identical. These calculations can be used to develop a catalyst for the methanation process as well as to select the most favorable conditions to perform experiments to reduce cost by neglecting the catalyst, reaction kinetics, and the transport process.

Table 2. Reactions (Ri) involved in the methanation of carbon oxides [26,27].

Ri	Reaction	$\mathsf{\Delta }{\mathbf{H}}_{\mathbf{298}}^{\mathbf{0}}$ $\mathbf{\left[}\mathbf{kJ}\mathbf{/}\mathbf{mol}\mathbf{\right]}$	Reaction Type
R1	${\mathrm{CO}}_2+4{\mathrm{H}}_2\leftrightarrow {\mathrm{CH}}_4+2{\mathrm{H}}_2\mathrm{O}$	−165.0	${\mathrm{CO}}_2$ methanation
R2	CO + $3{\mathrm{H}}_2\leftrightarrow {\mathrm{CH}}_4+{\mathrm{H}}_2\mathrm{O}$	−206.1	CO methanation
R3	2CO + $2{\mathrm{H}}_2\leftrightarrow {\mathrm{CH}}_4+{\mathrm{CO}}_2$	−247.3	inverse dry reforming
R4	CO + ${\mathrm{H}}_2\mathrm{O}\leftrightarrow {\mathrm{CO}}_2+{\mathrm{H}}_2$	−41.2	water–gas shift
R5	2CO↔${\mathrm{CO}}_2$ + C	−172.4	Boudouard reaction
R6	${\mathrm{CH}}_4\leftrightarrow 2{\mathrm{H}}_2$ + C	74.8	methane cracking
R7	CO + ${\mathrm{H}}_2\leftrightarrow {\mathrm{H}}_2\mathrm{O}$ + C	−131.3	carbon monoxide reduction
R8	${\mathrm{CO}}_2+2{\mathrm{H}}_2\leftrightarrow 2{\mathrm{H}}_2\mathrm{O}$ + C	−90.1	carbon dioxide reduction
R9	nCO + (2n+1)${\mathrm{H}}_2\leftrightarrow {\mathrm{C}}_{\mathrm{n}}{\mathrm{H}}_{2\mathrm{n}+2}+{\mathrm{nH}}_2\mathrm{O}$	-	-
R10	nCO + 2n${\mathrm{H}}_2\leftrightarrow {\mathrm{C}}_{\mathrm{n}}{\mathrm{H}}_{2\mathrm{n}}+{\mathrm{nH}}_2\mathrm{O}$	-	-

summarizes the possible reactions involved in the methanation of carbon oxides [26,27]. We use the gaseous compounds CO, ${\mathrm{H}}_2\mathrm{O}$, ${\mathrm{CO}}_2$, ${\mathrm{H}}_2$, ${\mathrm{O}}_2$, and ${\mathrm{CH}}_4$ for the calculations and ignore the high hydrocarbons, solid carbon, and oxygen-containing compounds (methanol, methanoic acid, acetic acid, etc.) due to their tiny amount in the equilibrium gas mixture. All the inlet conditions for the simulations are given in

Table 3. Summary of the inlet gas mixtures on which the investigation is performed. The temperature used to perform the simulations is varied in the range 200–1200 °C for each case given below. For all the cases (C1–C15), the pressure is considered 1, 30, and 100 atm. Note that for one case, C2, the pressure is varied in a wide range as 1–300 atm. The inlet composition of the species is given in volume %.

Case	Fuel Composition	${\mathbf{H}}_{\mathbf{2}}$ (vol. %)	${\mathbf{CO}}_{\mathbf{2}}$ (vol. %)	CO (vol. %)	${\mathbf{CH}}_{\mathbf{4}}$ (vol. %)	${\mathbf{H}}_{\mathbf{2}}\mathbf{O}$ (vol. %)	${\mathbf{O}}_{\mathbf{2}}$ (vol. %)
	${\mathrm{H}}_2/{\mathrm{CO}}_2$
C1	2	66.6	33.4	-	-	-	-
C2	4	80.0	20.0	-	-	-	-
C3	6	85.7	14.3	-	-	-	-
	${\mathrm{H}}_2/{\mathrm{CO}}_2/\mathrm{CO}$
C4	4/1/0	80.0	20.0	-	-	-	-
C5	4/1/1	66.7	16.6	16.7	-	-	-
C6	4/1/3	50.0	12.5	37.5	-	-	-
	${\mathrm{H}}_2/{\mathrm{CO}}_2/{\mathrm{CH}}_4$
C7	4/1/0	80.0	20.0	-	-	-	-
C8	4/1/1	66.7	16.6	-	16.7	-	-
C9	4/1/3	50.0	12.5	-	37.5	-	-
	${\mathrm{H}}_2/{\mathrm{CO}}_2/{\mathrm{H}}_2\mathrm{O}$
C10	4/1/0	80.0	20.0	-	-	-	-
C11	4/1/0.2	76.9	19.2	-	-	3.9	-
C12	4/1/0.5	72.7	18.2	-	-	9.1	-
	${\mathrm{H}}_2/{\mathrm{CO}}_2/{\mathrm{O}}_2$
C13	4/1/0	80.0	20.0	-	-	-	-
C14	4/1/0.2	76.9	19.2	-	-	-	3.9
C15	4/1/0.5	72.7	18.2	-	-	-	9.1

.

3. Validation

Before applying the equilibrium solver to new conditions, we validate it against the literature results [13] for the product faction of ${\mathrm{CO}}_2$ methanation at equilibrium for one initial condition, i.e., ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ = 4 and 1 atm in a temperature range of 200–800 °C. Note that the literature results are also simulations using the CHEMCAD solver, and we are comparing our simulation results against literature simulations. The concentration for all the species, ${\mathrm{CO}}_2$, ${\mathrm{H}}_2$, ${\mathrm{CH}}_4$, ${\mathrm{H}}_2\mathrm{O}$, and CO (Figure 1), are on top of the reference results for the considered temperature range. Hence, the solver can be used for equilibrium calculations at various unexplored conditions to optimize the methanation process.

At low temperatures, 200–450 °C, the main products are ${\mathrm{CH}}_4$ and ${\mathrm{H}}_2\mathrm{O}$, and with increasing temperature, the species CO, ${\mathrm{CO}}_2$, and ${\mathrm{H}}_2$ also increase. The ${\mathrm{CO}}_2$ mole fraction reaches its maximum around 550 °C. The reversed water gas shift reaction is dominating at higher temperatures, which leads to an increase in the CO concentration along with an increase in unreacted ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2$ and a decrease in methane. The ${\mathrm{CO}}_2$ methanation is most favorable at low temperatures, 200–400 °C, as the reaction is exothermic and would be unfavored at high temperatures.

4. Results: Equilibrium Calculations

In this section, we first explore several conditions for the thermodynamic equilibrium, for instance temperature, pressure, fuel composition, and fuel dilution. This aims to compile the thermodynamic investigation in one document and also to provide the most favorable regime for performing the kinetic simulations and experiments. In the second part, we focus on the kinetic modeling for the methanation process under various simulations conditions to make better choices in catalyst and parameter selection. Note that we only discuss the species mole fraction at the reactor outlet; however, we provide all the thermodynamic and kinetic data in case the reader is interested in other quantities, such as conversion, selectivity, or yield.

4.1. Pressure and $H_2$/${CO}_2$ Ratio Variation

The effect of ${\mathrm{CO}}_2$ methanation on the product composition in a wide range of temperature for a varying pressure and fuel (${\mathrm{H}}_2$/${\mathrm{CO}}_2$) ratio is shown in Figure 2. The pressure is varied in a range from 1 to 300 atm for fixed fuel composition, i.e., ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ = 4 and the fuel ratio ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ is varied from 2 to 6 for 1, 30, and 100 atm in a temperature range of 200–1200 °C.

The species ${\mathrm{CO}}_2$ (Figure 2a) at the outlet increases with the increase in temperature and attains its peak value around 500 °C for 1 atm. This behavior is shown due to the exothermic nature of the methanation reaction. The peak for ${\mathrm{CO}}_2$ shifts towards higher temperatures with an increase in pressure; for example, at 300 atm, the peak for this species is observed to be near 900 °C. Once the peak is attained, the unreacted ${\mathrm{CO}}_2$ again starts to decrease for temperatures above 500 °C at 1 atm, indicating the reverse water–gas shift reaction consuming ${\mathrm{CO}}_2$. If we check the ${\mathrm{CO}}_2$ mole fraction at lower temperatures, we note that at higher pressures, the ${\mathrm{CO}}_2$ is fully converted to methane and steam. A similar behavior is observed for the ${\mathrm{H}}_2$ mole fraction (Figure 2b). Hence, low temperatures and high pressure are most favorable for the Sabatier reactions.

In Figure 2, (c) ${\mathrm{CH}}_4$ and (d) ${\mathrm{H}}_2\mathrm{O}$ is produced at a maximum for low temperatures (200 °C) at 1 atm and decreases with increasing temperature; nevertheless, for higher pressures, the production increases for higher temperatures. The CO formation (Figure 2e) starts at 400 °C at 1 atm and 600 °C at 300 atm. Since the water formation at high pressures (300 atm) is more up to approximately 600 °C, the CO formation for the same operating conditions is shifted to higher temperature (600 °C), suppressing the CO production. Therefore, the CO formation could be alleviated at high pressures. The methanation reactions are favored at low temperatures and high pressures for the fuel ratio investigated in this section. Next, we vary the fuel ratio to understand the methanation process better, and the variation is performed for three pressures in the considered temperature range.

The effect of the ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio on the product fraction is shown in Figure 2f–j. Note that the ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio affects all the species at the outlet remarkably. For ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ = 2, there is a lot of unreacted ${\mathrm{CO}}_2$; nonetheless, the ${\mathrm{H}}_2$ is fully converted for this fuel composition. So, the ${\mathrm{CO}}_2$ conversion at a low ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio is much less. The ${\mathrm{CO}}_2$ conversion is high for a high ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio, for instance, for ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ = 6, the ${\mathrm{CO}}_2$ is 100% converted at 100 atm from 200 to 600 °C. However, for a high ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio, there is a lot of unreacted hydrogen.

Also, at ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ = 6, the methane formation is minimal compared to other two fuel compositions (see Figure 2h). Methane formation is maximal for the case with ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ = 6. The CO formation for the low ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio in the desired temperature range is an issue at low pressures. For ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ = 2 at 1 atm, the formation of CO starts around 380 °C, and it shifts to higher temperatures with increasing pressures. When ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ = 6, the CO formation starts around 700 °C at 100 atm.

The ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio variation study suggests that for the low ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio, i.e., two, the hydrogen is fully reacted with a lot of unreacted ${\mathrm{CO}}_2$, and CO formation becomes an issue. However, for the high ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio in the temperature range 200–400 °C, ${\mathrm{CO}}_2$ is fully converted, leaving a lot of unreacted hydrogen with less methane and CO formation. The most favorable condition turns out to be ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ = 4, where ${\mathrm{CO}}_2$ and hydrogen are converted with maximum methane in the outlet.

4.2. $H_2$/${CO}_2$ Diluted with ${CH}_4$ and $CO$

Since the syngas often contains a certain amount of methane, the effect of ${\mathrm{CH}}_4$ on the methanation reaction is shown in Figure 3a–e. It is also important to recycle the methanation gas product containing ${\mathrm{H}}_2\mathrm{O}$ and ${\mathrm{CH}}_4$ to dilute the feed gas to control the reaction temperature in the industrial methanation process [28,29]. The introduction of ${\mathrm{CH}}_4$ in the feed gas affects the formation of ${\mathrm{CO}}_2$, ${\mathrm{CH}}_4$, and ${\mathrm{H}}_2\mathrm{O}$ the most.

The ${\mathrm{CO}}_2$ conversion at low temperatures is unaffected with additional ${\mathrm{CH}}_4$, and the conversion improves with increasing methane feed from 400 °C for the the pressures considered for the investigation. Note that the additional methane does not lead to a large difference in hydrogen production in the temperature range of our interest for the methanation reaction. The ${\mathrm{H}}_2$ mole fraction is influenced by adding ${\mathrm{CH}}_4$ only for temperatures above 700 °C (1 atm), where the reforming reactions are expected to be dominant, considering that ${\mathrm{CH}}_4$ is a reactant for reforming processes, which further helps to increase ${\mathrm{H}}_2$ production.

The methane formation, Figure 3c, is at a maximum with more methane in the feed (${\mathrm{H}}_2$/${\mathrm{CO}}_2$/${\mathrm{CH}}_4$ = 4/1/3) and is at a minimum with no additional methane. Since ${\mathrm{CH}}_4$ is a product in the methanation reaction, it is expected that the addition of methane would further increase its production. The water production (Figure 3d), however, shows the opposite behavior when compared to methane, and it decreases with increasing methane in the feed. Some minor influence on CO formation is also noted. Although the CO formation is not affected by ${\mathrm{CH}}_4$ dilution in the low temperature range (200–400 °C at 1 atm), at temperatures above 500 °C, there is a sharp increase in the formation of CO. This might also lead to carbon deposition due to the methane cracking occurring at high temperatures, which also explains more ${\mathrm{H}}_2$ at high temperatures.

Additional methane allows more ${\mathrm{CH}}_4$ formation and inhibits ${\mathrm{H}}_2\mathrm{O}$ formation. It can be helpful to optimize the methane production, but at the same time, if there is carbon formation due to a reduction in ${\mathrm{H}}_2\mathrm{O}$, then we need to avoid ${\mathrm{CH}}_4$ from the feed. Therefore, a large recycling of the product gas is not recommended, and ${\mathrm{CH}}_4$ in the feed gas needs to be controlled at a low level to optimize the energy efficiency to obtain high ${\mathrm{CH}}_4$ and low carbon.

The simultaneous methanation of carbon oxides is often seen due to the presence of carbon oxides with CO and ${\mathrm{CO}}_2$ existing in syngas. Therefore, we next investigate the methanation situation with additional CO in the feed. It is expected that the reversed water gas shift reaction and inverse CO methanation play an important role at high temperature.

The influence of the CO co-feed with ${\mathrm{H}}_2$ and ${\mathrm{CO}}_2$ is very strong and shown in Figure 3f–j. The ${\mathrm{CO}}_2$ product fraction is at a minimum without CO dilution because of the CO and ${\mathrm{CO}}_2$ methanation reactions. Unreacted ${\mathrm{CO}}_2$ and methane increase with increasing the amount of CO dilution due to the inverse methane ${\mathrm{CO}}_2$ reforming reaction and reaches a maximum value for ${\mathrm{H}}_2$/${\mathrm{CO}}_2$/CO = 4/1/3 in the temperature range most favorable for Sabatier reactions (200–400 °C). However, at higher temperatures, above 400 °C, the inverse water–gas shift reaction and inverse CO methanation are dominant, leading to a decrease in the ${\mathrm{CO}}_2$ production with both dilution cases. Consider the ${\mathrm{H}}_2$ production at 1 atm; it starts to increase from 200 to 600 °C with as well as without CO dilution. Note that the ${\mathrm{H}}_2$ profile remains qualitatively the same with additional CO.

Methane and water (Figure 3h,i) are formed at low temperatures for all the pressures considered for simulations because of the ${\mathrm{CO}}_2$ methanation reaction without the CO dilution. In the case of additional CO in the feed, other reactions, for instance, CO methanation and inverse methane ${\mathrm{CO}}_2$ reforming reactions, favor the products and lead to an increase in the methane and ${\mathrm{CO}}_2$ along with a decrease in water. The impact of these reactions in also seen on the CO formation (Figure 3j). The CO formation starts around 300 °C at 1 atm for ${\mathrm{H}}_2$/${\mathrm{CO}}_2$/CO = 4/1/3. For this case, the water formation is reported to be minimal, which can allow carbon formation. Hence, additional CO in the feed in unfavorable for the methanation reactions and should be avoided wherever possible.

4.3. $H_2$/${CO}_2$ Diluted with $H_{2}O$ and $O_2$

It is essential to see the impact of ${\mathrm{H}}_2\mathrm{O}$ added to ${\mathrm{H}}_2$ and ${\mathrm{CO}}_2$ because of the moisture in the mixture. In addition, it is shown by the industrial experience that water steam is added into the reactants to suppress carbon formation to a great extent on the methanation catalysts [1]. Therefore, for the catalyst development, we thermodynamically investigate the dilution of ${\mathrm{H}}_2\mathrm{O}$ in the methanation reaction shown in Figure 4a–e.

Figure 4a,b indicate that additional steam with different ratios does not influence the ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2$ at low temperatures. However, at higher temperatures, the addition of steam leads to slight decrease in ${\mathrm{CO}}_2$ conversion. Since ${\mathrm{H}}_2\mathrm{O}$ is one of the products in the ${\mathrm{CO}}_2$ methanation reaction, it is expected that additional ${\mathrm{H}}_2\mathrm{O}$ will inhibit the production of ${\mathrm{CH}}_4$ as observed in Figure 4c. The increased ${\mathrm{H}}_2\mathrm{O}$ production, Figure 4d, helps in reducing the carbon deposition by influencing the carbon dioxide reduction reaction significantly, hence making additional water favorable for the methanation reaction. The production of CO (Figure 4e) remains unaffected for all pressures up to approximately 600 °C with different ratios of steam addition. So, the species production in the temperature range required for the Sabatier reaction is not changed. Only for the higher temperatures does the CO production reduce with increasing the steam in the feed.

The analysis of the methanation reaction with ${\mathrm{O}}_2$ in the feed is important due to the trace amount of ${\mathrm{O}}_2$ in the syngas during the gasification process. Hence, the effect of ${\mathrm{O}}_2$ on CO methanation is considered in [13]; however, in this section, we focus on ${\mathrm{CO}}_2$ methanation with an increasing amount of ${\mathrm{O}}_2$ in the entire temperature range considered for the simulations at 1, 30, and 100 atm shown in Figure 4f–j.

The increase in ${\mathrm{CO}}_2$ production with increasing ${\mathrm{O}}_2$ indicates the decrease in ${\mathrm{CO}}_2$ conversion in the complete temperature range even by adding some trace of ${\mathrm{O}}_2$. The qualitative approach for the ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2$ profile (Figure 4f,g) is similar for all the pressures. The ${\mathrm{H}}_2$ production is not affected at low temperature by adding ${\mathrm{O}}_2$. The impact on ${\mathrm{H}}_2$ production is evident only from approximately 600 °C. Even with a some trace of ${\mathrm{O}}_2$ in the feed, the ${\mathrm{CH}}_4$ production (Figure 4h) is highly influenced, and the ${\mathrm{CH}}_4$ production is reduced by more than 50% from no ${\mathrm{O}}_2$ dilution case (${\mathrm{H}}_2$/${\mathrm{CO}}_2$/${\mathrm{O}}_2$ = 4/1/0) to ${\mathrm{H}}_2$/${\mathrm{CO}}_2$/${\mathrm{O}}_2$ = 4/1/0.5. Although adding ${\mathrm{O}}_2$ leads to large reduction in carbon deposition, it is necessary to remove the ${\mathrm{O}}_2$ in the feed gas for the methanation process when considering the ${\mathrm{CH}}_4$ production.

The production of ${\mathrm{H}}_2\mathrm{O}$ (Figure 4i) is slightly affected by additional ${\mathrm{O}}_2$ in the temperature range 200–600 °C, whereas, at higher temperatures, the influence on the ${\mathrm{H}}_2\mathrm{O}$ profile is noticeable due to the reactions of ${\mathrm{H}}_2$ and ${\mathrm{CO}}_2$ with ${\mathrm{O}}_2$. Similarly, the CO production (Figure 4j) changes only at higher temperatures with an increasing amount of ${\mathrm{O}}_2$ in the feed, owing to the partial oxygenation of CO to ${\mathrm{CO}}_2$, and remains unaffected in the temperature range needed for the Sabatier reaction, i.e., 200–400 °C.

To summarize the thermodynamic equilibrium calculations discussed in this section, we give the species mole fraction of ${\mathrm{CH}}_4$ and CO for all the cases, C1–C15, considered for the investigation in

Table 4. Summary of the species mole fraction of ${\mathrm{CH}}_4$ and CO measured at the reactor outlet for all the cases, C1–C15, considered for the thermodynamic equilibrium investigation at 1 atm and 350 °C as inlet temperature.

Case	${\mathbf{CH}}_{\mathbf{4}}$	CO
C1	2.18 $\times {10}^{-1}$	3.12 $\times {10}^{-3}$
C2	2.94 $\times {10}^{-1}$	1.39 $\times {10}^{-4}$
C3	1.99 $\times {10}^{-1}$	5.09 $\times {10}^{-6}$
C4	2.94 $\times {10}^{-1}$	1.39 $\times {10}^{-4}$
C5	3.18 $\times {10}^{-1}$	3.56 $\times {10}^{-3}$
C6	3.75 $\times {10}^{-1}$	5.72 $\times {10}^{-3}$
C7	2.94 $\times {10}^{-1}$	1.39 $\times {10}^{-4}$
C8	4.36 $\times {10}^{-1}$	5.60 $\times {10}^{-4}$
C9	6.26 $\times {10}^{-1}$	1.25 $\times {10}^{-4}$
C10	2.94 $\times {10}^{-1}$	1.39 $\times {10}^{-4}$
C11	2.78 $\times {10}^{-1}$	9.87 $\times {10}^{-5}$
C12	2.56 $\times {10}^{-1}$	6.44 $\times {10}^{-5}$
C13	2.94 $\times {10}^{-1}$	1.39 $\times {10}^{-4}$
C14	2.57 $\times {10}^{-1}$	1.31 $\times {10}^{-4}$
C15	1.96 $\times {10}^{-1}$	1.75 $\times {10}^{-4}$

. The species mole fractions are given at the reactor outlet for 1 atm and 350 °C as the inlet temperature. We select the 350 °C temperature, as this is the most favorable for methanation reactions. Note that the outlet temperature for 350 °C as the inlet temperature would be different for each case. It would be difficult to select the same outlet temperature for each case; therefore, we consider the inlet temperature for reporting the species mole fraction in the table. For maximal methane formation, C2, C6, C9, C10, and C13 are the most favorable cases, and for minimal CO at the reactor outlet, C3, C4, C9, C12, and C14 are the most supportive cases.

5. Results: Kinetic Modeling

After performing the thermodynamic analysis, we further investigate the influence of the detailed surface chemistry on some of the most sensitive conditions along with other parameters that can help in the catalyst development and to set up the experiments. The model used for the simulations is described in our previous work [18,19,20] and is not discussed here to avoid repetition. The validation of the detailed surface reaction mechanism is given in [30]. The thermodynamic data and the kinetic parameters are taken from [25] as the base, and then the surface chemistry is trained to achieve the results comparable to the experiments discussed in [30]. The kinetics used for the present investigations have been developed based on the transient experiments discussed in [30]. The kinetics is developed for the catalyst used for the experiments based on the manufacturer’s specifications as nickel oxide on a silica substrate (NiO/Si${\mathrm{O}}_2$) with a weight proportion of 60 wt% NiO. According to the experiments, the nickel content of 44.8–47.6 wt% is used. The trained surface mechanism can capture the desired species concentration for different conditions considered for the experiments. The kinetics developed holds for the different amount of catalyst distribution on support. Therefore, we also show the results of variations in the amount of catalyst on the support. In total, 52 surface reactions containing 6 gas phase species and 13 surface species are used to perform the simulations.

5.1. Amount of Catalyst

Not only the distribution of the catalyst on the support but also the size of the catalyst particles play important roles in determining the efficiency, selectivity, and the overall performance of catalytic reactions. We consider a uniform distribution of the catalyst across the support to maximize the utilization of the catalyst material by making the active sites more evenly available for reactants for this configuration, leading to higher catalytic activity, better overall efficiency, and consistent reaction conditions across the catalyst. However, a uniform distribution can be technically challenging. Also, we consider a surface reaction mechanism that consists of all the direct reactions, ignoring the thermochemistry of the intermediate species.

The catalyst particle size is crucial, as the smaller catalyst particles have a higher surface area-to-volume ratio, providing more active sites for the reaction, enhancing the catalytic activity by providing more reactants to interact with on the catalyst surface. This increased surface area leads to higher reaction rates and selectivity in the case of a surface reaction. The main challenge with a small particle size is sintering at high temperatures, leading to the loss of the active surface area and reduced catalytic performance, which can also cause desorption resistance by creating the stronger adsorption of reactants or products.

The catalyst distribution on the support and the particle size are critical parameters which influence the performance of catalytic reactions significantly, and therefore, a uniform distribution along with optimal particle size can improve the activity, selectivity, and stability. On the other hand, the poor distribution or inappropriate particle size can cause mass transfer limitations and reduced catalyst lifetime. Hence, a thorough investigation is suggested to comprehend the specific impacts depending on the type of reaction, the properties of the catalyst, and the conditions under which the reaction is conducted. Nevertheless, in our simplified model, the catalyst distribution on the support and the catalyst particle size are dependent on each other; hence, it can be sufficient to study one of the parameters.

Since the main aim is always to obtain a good conversion of ${\mathrm{H}}_2$ and ${\mathrm{CO}}_2$ while using a minimum amount of catalyst, we first check the influence of the amount of catalyst on all the species concentration and discuss them in this section. We consider a wide temperature range for this particular case. Note that the amount of catalyst in our model is represented by the parameter surface area per length, i.e., SA/L.

At low temperatures (Figure 5), the trend for all the species concentrations calculated with the kinetic model is away from the equilibrium calculations. This is because of the fact that the catalyst is not yet activated at these temperatures. However, the kinetic results are moving towards the equilibrium calculations at higher temperatures where the catalyst is fully activated.

The species ${\mathrm{CO}}_2$, Figure 5a, measured at the reactor outlet at low temperatures, 200–300 °C, is maximal, as the reactions are not happening yet, and after 300 °C, the species starts to be used. Hence, it decreases with increasing temperature in the temperature range 300–500 °C. A similar behavior is noted with the kinetic calculations as observed in the equilibrium calculations, i.e., the ${\mathrm{CO}}_2$ increases from around 500 °C and attains a maximum value due to the exothermic nature of the methanation reactions. The peak attained with the kinetic model is shifted to higher temperatures when compared with the equilibrium calculations. The effect of the reverse water–gas shift reaction is also captured, indicating the decrease in the unreacted ${\mathrm{CO}}_2$ with increasing temperature.

Most of the ${\mathrm{H}}_2$, Figure 5b, is reacted in the temperature range 400–600 °C, and above 600 °C there is a lot of unreacted ${\mathrm{H}}_2$ due to the dominance of the reforming reactions at higher temperatures which form the ${\mathrm{H}}_2$. Note that the conversion of ${\mathrm{H}}_2$ is minimal for minimum catalyst loading, and it increases with increasing the amount of catalyst, achieving maximum conversion for highest catalyst loading. For the maximum catalyst available on the surface, the ${\mathrm{CO}}_2$ conversion is more than 90%, starting from 400 °C; however, for ${\mathrm{H}}_2$, the conversion is more than 80% for 400–600 °C, reducing around 15% above 1000 °C.

The qualitative behavior of ${\mathrm{CH}}_4$, ${\mathrm{H}}_2\mathrm{O}$, and CO (Figure 5c–e) measured with the kinetic model is similar to the equilibrium calculations starting from 400 °C. The formation of methane as well as water is greater when the amount of catalyst is large. However, the formation of CO can be challenging with more catalyst loading, as the CO starts to increase from 500 °C for the maximum catalyst amount. This investigation suggests that a higher catalyst amount is favorable for the conversion of reactants as well as a maximum methane amount in the temperature range of 400–600 °C. However, the amount of catalyst needs to be reduced to produce less CO with the kinetic model.

Note that the overall reaction rate in catalytic reactions on the surface can be greatly influenced by the internal and external resistances that appear in a kinetic process. If the surface has limited active sites, this can lead to difficulty in the adsorption of reactant molecules onto the catalyst surface. Another resistance is due to the surface reactions on the catalyst converting the adsorbed reactant into products due to the intrinsic activity of the catalyst and the activation energy for the reaction. Sometimes, there is a difficulty in desorbing the product molecules from the catalyst surface after the reaction has occurred, leading to desorption resistance. For porous catalysts, internal diffusion resistance occurs as reactants diffuse through the pores to reach active sites inside the catalyst particles. Further investigations are needed also for the external resistances, such as heat and mass transfer resistance.

5.2. Mass Flow and Pressure Variation

The above analysis indicates that the temperatures between 400 and 600 °C are favorable for the Sabatier reaction and a lesser amount of catalyst is preferred for CO formation. Therefore, we next consider the effect of mass flow and pressure variation in the temperature range of 200–1000 °C and for the catalyst loading of around 0.1 g.

The qualitative trend for all the species with varying mass flow, Figure 6a–e, is the same. In the temperature range between 400 and 600 °C, the maximum consumption of ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2$ and the maximum formation of the ${\mathrm{CH}}_4$, ${\mathrm{H}}_2\mathrm{O}$, and CO mole fraction is achieved with the kinetic model. The mass flow can be further reduced to reduce unreacted ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2$ and to increase the methane at the outlet of the reactor. However, we limit our investigation at 7.17 $\times {10}^{-7}$ kg/s due to increase in the formation of CO; hence, the simulations for varying pressure are performed with this mass flow.

To check the effect of pressure in Figure 6f–j, we consider 0.1 g catalyst (SA/L = 2.90 $\times {10}^{-3}$ m), ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ = 4, and mass flow of 7.17 $\times {10}^{-7}$ kg/s. We investigate 1, 10, and 30 atm in the temperature range of 200–1000 °C for the kinetic simulations. Similar to equilibrium calculations, the catalytic hydrogenation of carbon dioxide is also favored at higher pressures in the temperature range suitable for the Sabatier reaction.

The species ${\mathrm{CO}}_2$ is fully consumed already at 400 °C for 10 atm itself, and that is why the species measurements for 30 atm are on top of the 10 atm calculations. This indicates that the reverse water–gas shift reaction is not influencing the species prediction at higher temperatures for the catalytic hydrogenation of carbon dioxide. The computed species then remains the same for all temperatures above 400 °C for higher pressures. However, the behavior of ${\mathrm{H}}_2$ is different for higher pressures. Unlike ${\mathrm{CO}}_2$, the ${\mathrm{H}}_2$ mole fraction decreases from 200 to 400 °C and then starts to increase for temperatures higher than 600 °C. The measurements indicate that, in the case that the unreacted hydrogen needs to be reduced, then the best method is to operate the reactor at higher pressures, for example, at 30 atm in the temperature range 400–600 °C. Further, the product species mole fraction, i.e., ${\mathrm{CH}}_4$, ${\mathrm{H}}_2\mathrm{O}$, and CO, illustrates that the higher pressures support our requirements in the temperature range 400–600 °C. In this range, we have maximum methane yield and minimum CO formation.

5.3. Dilution of the Inlet Gases with ${CH}_4$ and $O_2$

With the help of the equilibrium calculations, it is noted that the addition of methane in the feed improves the ${\mathrm{CO}}_2$ conversion and ${\mathrm{CH}}_4$ selectivity along with an increase in the CO mole fraction. Hence, we select the dilution of ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2$ with ${\mathrm{CH}}_4$ (Figure 7a–e) while performing the simulations with a catalyst. We also vary the catalyst mass to see the influence of the presence of more catalyst.

Unlike equilibrium measurements, the kinetic model shows very good conversion of ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2$ in the entire temperature range even with the lowest dilution. The methane formation at the lowest temperature remains unaffected due to the inactive catalyst. However, when the catalyst is active, the ${\mathrm{CH}}_4$ mole fraction at the reactor outlet increases with increasing temperature and starts to decrease after attaining the peak. So, for methane optimization, additional ${\mathrm{CH}}_4$ in the feed is favorable, and also, the greater amount of catalyst is supportive. However, for the CO formation, a lower catalyst amount is more favorable along with additional ${\mathrm{CH}}_4$.

Next, we check the additional ${\mathrm{O}}_2$, Figure 7f–j, in the feed and find that similar to the equilibrium calculations, kinetic modeling also illustrates that even a small amount of ${\mathrm{O}}_2$ is unfavorable and needs to be avoided for both the catalysts considered for the investigation. With the ${\mathrm{O}}_2$ added in the feed, the unreacted ${\mathrm{CO}}_2$ increases, and the ${\mathrm{CH}}_4$ mole fraction reduces, indicating low selectivity. The CO formation also starts at lower temperatures with oxygen in the feed.

6. Conclusions

A comprehensive discussion for thermodynamic equilibrium calculations for selecting the optimal conditions and kinetic modeling for the catalyst development of ${\mathrm{CO}}_2$ methanation is presented in this paper. All the species considered, ${\mathrm{CO}}_2$, ${\mathrm{H}}_2$, ${\mathrm{CH}}_4$, ${\mathrm{H}}_2\mathrm{O}$, and CO, are shown at the outlet for various inlet conditions, for instance, temperature, pressure, ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio, catalyst loading, mass flow, and different fuel compositions. Some of the major findings from the study are summarized below.

In the case of thermodynamic investigation, we find that the low temperatures and high pressures are favorable for methanation conditions. A low ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio leaves some of the ${\mathrm{CO}}_2$ unused at the outlet, while a high ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio leaves ${\mathrm{H}}_2$ unused along with low ${\mathrm{CH}}_4$ formation. The ${\mathrm{H}}_2$/${\mathrm{CO}}_2$ ratio of four is favorable. Traces of ${\mathrm{H}}_2\mathrm{O}$ in the feed slightly affect the ${\mathrm{CH}}_4$ formation at low temperatures and also reduce carbon deposition, making the additional water favorable for methanation reaction. Traces of ${\mathrm{O}}_2$ in the ${\mathrm{H}}_2$ and ${\mathrm{CO}}_2$ mixture are highly unfavorable for the methanation process and hence need to be avoided wherever possible. Adding CO in the mixture along with ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2$ influences the ${\mathrm{CO}}_2$ and ${\mathrm{H}}_2\mathrm{O}$ mole fraction at the outlet and is unfavorable for the methanation reaction. Putting additional ${\mathrm{CH}}_4$ in the mixture produces more ${\mathrm{CH}}_4$ and reduces ${\mathrm{H}}_2\mathrm{O}$ with increasing the amount of methane in the feed, and hence, its addition needs to be controlled at a low level to optimize the energy efficiency.

For the kinetic modeling, a large amount of catalyst is favorable for good consumption of the reactants and maximum methane formation in the temperature range of 400–600 °C. However, the amount of catalyst needs to be selected carefully considering the formation of CO. High pressures are favorable for kinetic modeling, and high temperatures are needed for the catalyst activation. Low mass flow can be considered to achieve the desirable outcome; however, we limit our investigation based on the formation of CO. Additional ${\mathrm{CH}}_4$ in the feed is favorable for methane optimization, which further improves if a large amount of catalyst is used. Also, even a small amount of ${\mathrm{O}}_2$ is unfavorable for kinetic modeling and needs to be avoided in the methanation process.

The present study aims to summarize the optimal parameters from thermodynamic analysis and also to support the development of catalysts by understanding the kinetics involved in the multi-step detailed surface reaction mechanism. This will also help us to select the best conditions under which to perform further simulations with detailed chemistry as well as experiments. Also, the data produced from the simulations can be used to develop the machine learning algorithms for methanation process, as the major limitation for such processes is the availability of data that can be used to train the data-based models.