Hydrogen Yield Enhancement in Biogas Dry Reforming with a Ni/Cr Catalyst: A Numerical Study

Abstract

This numerical study investigates the impact of the reaction temperature, molar ratio of CH₄:CO₂, and catalyst porosity (ε_p) on the H₂ yield and H₂ selectivity during biogas dry reforming over a Ni/Cr catalyst. Using COMSOL Multiphysics, we conducted detailed simulations to elucidate the underlying reaction characteristics. Our findings reveal that increasing ε_p from 0.1 to 0.95 significantly provides a 5 times increase in H₂ production and a 2.3% increase in H₂ selectivity while simultaneously reducing CO selectivity by 2.3%. This effect is attributed to the improved mass transfer within the catalyst bed, leading to more efficient reactant conversion and product formation. Additionally, we observed a strong correlation between higher reaction temperatures and increased H₂ yield and H₂ selectivity. By optimizing these operational parameters, our results suggest that Ni/Cr catalysts can be effectively employed for the sustainable production of H₂ from biogas.

1. Introduction

The production of H₂ as a clean energy carrier has attracted considerable attention in recent years because of its potential to address climate change and decrease reliance on fossil fuels. Among the different methods for generating H₂, biogas dry reforming (BDR) stands out as a promising approach to utilizing a renewable feedstock and creating a valuable product. A biogas is a fuel consisting of CH₄ (55–75 vol%) and CO₂ (25–45 vol%) [1], generally. It is produced from fermentation by the action of anaerobic microorganisms on raw materials, e.g., garbage, livestock excretion, and sewage sludge. In total, 1.46 EJ of produced biogas was obtained in the world in 2020 [2]. It was five times as large as that in 2000 [2]. We can expect the amount of produced biogas to increase more and more in the near future. Consequently, this study thinks biogas is a promising source for producing H₂.

While biogas is commonly utilized as a fuel for gas engines and micro gas turbines [3], its lower heating value due to its significant CO₂ content can limit power generation efficiency. To overcome this limitation, the authors propose a synergistic system integrating a BDR reactor with a solid oxide fuel cell (SOFC) [4,5,6]. CO, which is a by-product from BDR, as well as H₂, can be used as a fuel for SOFC. This approach allows for the valorization of biogas, dry-converting its CO₂ and CH₄ components into valuable syngas, which can then be efficiently utilized as a fuel in the SOFC. Consequently, this study thinks this system can be used for a wider operation application compared to the existing system.

Much research on biogas dry reforming (BDR) has been reported [7,8,9,10,11,12,13,14,15,16,17]. To decide the performance of a BDR reactor, this study thinks the selection of the catalyst is important. According to the literature survey, Ni-based catalysts have emerged as promising candidates due to their high-superiority performance in converting biogas into valuable syngas for BDR [7,8,9,10,11,12,13,14,15,16,17]. Research has investigated various Ni-based catalysts, including Ni/Co/TiO₂, Ni-promoted pyrochlore, Ni/Al/LDH with Mg and Zn, Ni-Ce/TiO₂-ZrO₂, Ni/Ru, etc. to optimize BDR performance. These studies have demonstrated the influence of the catalyst composition, reaction temperature, and molar ratio of CH₄:CO₂ on the conversion rates, H₂ yield, and product selectivity. While the Ni-based catalyst has shown promising results, challenges such as catalyst deactivation, carbon formation, and H₂/CO ratio control remain to be addressed. Recent advancements in catalyst design and process engineering have aimed to mitigate these issues and enhance the overall efficiency of BDR. For example, the incorporation of promoters or supports can improve catalyst stability and selectivity, while advanced reactor configurations can optimize heat and mass transfer.

At 900 °C, the Ni/Co/TiO₂ catalyst performed CH₄ conversion of 87.13%, CO₂ conversion of 92.6% and H₂ production of 41.1% [7]. In this study [7], the optimum ratio of Ni and Co was 7 wt% and 4 wt%, respectively. Ni with hollow silica spheres performed CH₄ conversion of 94.4% and CO₂ conversion of 95% [8]. Ni-promoted pyrochlore showed that the highest CH₄ conversion and CO₂ conversion were obtained in the case of CH₄:CO₂ = 1.1 and 1.85:1, respectively [9]. A BDR experiment was carried out using a Ni/Al/LDH with Mg and Zn in the case of CH₄:CO₂:N₂ = 1.5:1:7.5 at the temperature range from 500 °C to 700 °C [10]. This catalyst performed CH₄ conversion of 73%, CO₂ conversion of 90% and a H₂/CO ratio of approximately 2. Ni-Ce/TiO₂-ZrO₂ performed the highest CH₄ conversion of approximately 98% at 750 °C and the highest H₂/CO ratio of approximately 0.71 and 800 °C [11]. The Ni/Ru catalyst was investigated, exhibiting that CH₄ and CO₂ conversions were 70% and 78%, respectively, at 750 °C [12]. In the case of a CH₄-rich condition, i.e., CH₄:CO₂ = 2:1, low Ni loading (2.5 wt%) modified with Gd, Sc or La showed the highest CH₄ conversion of 49%, the highest CO₂ conversion of 95% and the highest H₂/CO ratio of 1.0 at 750 °C [13]. Kaviani et al. investigated the effect of Ni loading on the performance of the Ni/SiO₂/SiO₂ core-shell for BDR [14]. As a result, when the Ni content increased from 2.5 wt% to 15 wt%, the Ni loading of 15 wt% performed the highest CH₄ conversion of 75%, the highest CO₂ conversion of 90%, and the highest H₂/CO ratio of 0.9. The Ni-SiO₂ sandwiched core-shell catalyst performed the highest CH₄ conversion of 71.5% and CO₂ conversion of 91.5% in the case of CH₄:CO₂ = 1.5:1 at 700 °C [15]. The SiO₂ layer in the sandwiched catalyst prevented the agglomeration of metal particles and minimized carbon deposition. The Ni/LnOx (Ln = La, Ce, Sm or Pr)-type catalyst was developed for the investigation of the characteristics of BDR [16]. As a result, CH₄ and CO₂ conversions increased with the increase in reaction temperature from 600 °C to 750 °C irrespective of catalyst type, while the temperature change of H₂/CO ratio depended on catalyst type. According to the review paper on Ni-based catalyst [17], CH₄ and CO₂ conversions increased with the increase in the reaction temperature from 300 °C up to 1000 °C, irrespective of the molar ratio of CH₄:CO₂. On the other hand, the H₂/CO ratio decreased with the increase in the reaction temperature from 300 °C to 500 °C and kept a constant value, i.e., approximately 0 over 500 °C. According to the other previous study [14], when using a Ni-SiO₂@SiO₂ core-shell catalyst, the H₂/CO ratio increased with the increase in the reaction temperature from 500 °C to 700 °C, attaining approximately 0.8. On the other hand, the H₂/CO ratio kept approximately the same value of 1.5 with the increase in the reaction temperature from 500 °C to 750 °C when using the Mg/F/Ni-Al catalyst [10]. Therefore, the authors think that the impact of the reaction temperature on the H₂/CO ratio is influenced by the catalyst type and experimental conditions.

Though several Ni alloy catalysts have been investigated, few Ni/Cr catalysts were reported, except for the experimental studies using a membrane reactor, which were carried out by the authors of this paper [5,18]. These previous studies reported that Ni/Cr exhibited the better performance compared to the Ni catalyst due to it preventing a carbon deposition [5,18]. The H₂ yield obtained by the Ni/Cr catalyst was approximately 10–100 times as large as that of the Ni catalyst, which depended on the experimental condition. The H₂ yield increased with the increase in temperature from 400 °C to 600 °C, irrespective of the molar ratio of CH₄:CO₂. The highest H₂ yield, which was 12.8%, was obtained using Ni/Cr in the case of CH₄:CO₂ = 1.5:1 at 600 °C, and the differential pressure between the reaction chamber and the sweep chamber was 0.010 MPa without sweep gas. In addition, the CO selectivity was much larger than H₂ selectivity within the temperature range from 400 °C to 600 °C, which was over 90%. According to the previous studies [5,18], it was thought that not only dry reforming (DR) but also the other reactions were conducted. However, the numerical analysis of the characteristics of the Ni/Cr alloy catalyst used for BDR including the other reactions has not been investigated. It is perceived that only numerical analysis allows for establishing reaction mechanisms. In addition, a catalyst porosity (ε_p) which influences the performance of a catalyst as well as a reactor has not been investigated numerically and experimentally.

While the Ni/Cr catalyst has shown a promising performance in previous experimental studies for BDR [5,18], its reaction mechanisms and the influence of ε_p on performance remain unclear. The aim of this study is to clarify the reaction mechanism of BDR and the other reactions as well as the influence of ε_p on the performance of these reactions through the numerical simulation using the CFD software COMSOL Multiphysics ver. 6.2. This study also aims to reveal the reaction characteristics and optimize the H₂ yield in BDR using the Ni/Cr catalyst by investigating the impact of the reaction temperature, molar ratio of CH₄:CO₂, and ε_p on the characteristics. Since there are more previous studies using COMSOL Multiphysics for the numerical analysis of BDR and reports on the reaction characteristics as well as heat and mass transfer [19,20,21,22], we have adopted COMSOL Multiphysics for the numerical analysis in this study. To investigate the impact of the reaction temperature, molar ratio of CH₄:CO₂, and ε_p on the characteristics of BDR including the other reactions, the following reactions are considered:

<Dry reforming (DR)>

CH₄ + CO₂ ↔ 2H₂ + 2CO + 247 kJ/mol

(1)

<Steam reforming (SR)>

CH₄ + H₂O ↔ CO + 3H₂ + 206 kJ/mol

(2)

<Reverse water gas shift reaction (RWGS)>

CO₂ + H₂ ↔ CO + H₂O + 41 kJ/mol

(3)

<Methanation reaction (MR)>

CO₂ + 4H₂ ↔ CH₄ + 2H₂O − 165.0 kJ/mol

(4)

<Methane decomposition reaction (MDR)>

CH₄ ↔ C + 2H₂ + 74 kJ/mol

(5)

<Boudouard reaction (BD)>

2CO ↔ C + CO₂ − 172 kJ/mol

(6)

2. Numerical Simulation of BDR over the Ni/Cr Catalyst

2.1. A Mathematical Formulation

The numerical simulation was conducted by multi-physics software, COMSOL Multiphysics, ver. 6.2. In COMSOL Multiphysics, users can select or customize various partial differential equations and combine them to achieve direct coupled multi-physics field analysis easily [23]. COMSOL Multiphysics has simulation function codes. The following governing equations are involved in COMSOL Multiphysics.

The equations of mass conservation in porous material can be defined as follows:

$$\overrightarrow{u}=-{\displaystyle \frac{\kappa }{\mu }}\left(\nabla p+\rho g\nabla D\right)$$

(7)

$$\nabla \cdot \left(\rho \overrightarrow{u}\right)=Q_m$$

(8)

$$\nabla \cdot \rho \left[-{\displaystyle \frac{\kappa }{\mu }}\left(\nabla p\right)\right]=Q_m$$

(9)

where $\overrightarrow{u}$ is a velocity vector [m/s], κ is permeability [m²], μ is viscosity [Pa·s], p is pressure [Pa], ρ is density [kg/m³], g is gravitational acceleration [m²/s], D is height [m], and Q_m is a mass source term [kg/(m³·s)].

The equation of momentum conservation in porous material can be defined by the Brinkmann equation, as follows:

$$\begin{gathered} {\displaystyle \frac{\rho }{{\epsilon }_p}}\left(\overrightarrow{u}\cdot \nabla \right){\displaystyle \frac{\overrightarrow{u}}{{\epsilon }_p}}= \\ -\nabla p+\nabla \cdot \left[{\displaystyle \frac{1}{{\epsilon }_p}}\left\{\mu \left(\nabla \overrightarrow{u}+{\left(\nabla \overrightarrow{u}\right)}^T\right)-{\displaystyle \frac{2}{3}}\mu \left(\nabla \cdot \overrightarrow{u}\right)\overrightarrow{I}\right\}\right]-\left({\kappa }^{-1}\mu +{\displaystyle \frac{Q_m}{{\epsilon }_p^2}}\right)\overrightarrow{u}+\overrightarrow{F} \end{gathered}$$

(10)

where ε_p is porosity [-], $\overrightarrow{I}$ is a unit vector [-], and $\overrightarrow{F}$ is a force vector [kg/(m³·s)].

The equation of mass transfer in porous material can be defined by the Maxell–Stefan equation, as follows:

$$\nabla \cdot \left(\rho {\omega }_i\overrightarrow{u}-\rho {\omega }_i{\displaystyle \sum _{k=1}^n{\epsilon }_p^{\frac{3}{2}}\widetilde{D_{ik}}\left(\nabla x_k+\left(x_k-{\omega }_k\right)\frac{\nabla p}{p}\right)-{\epsilon }_p^{\frac{3}{2}}{D}_i^T\frac{\nabla T}{T}}\right)=R_{i,tot}$$

(11)

where ${\omega }_i$, ${\omega }_k$ is the mass ratio of chemicals i and k, respectively [-], $\widetilde{D_{ik}}$ is a Fick’s diffusion coefficient of effective multi components [m²/s], $D_i^T$ is a thermal diffusion coefficient [kg/(m·s)], and $R_{i,tot}$ is a reaction rate of chemical species [mol/(m³·s)].

The equations of heat transfer in porous material can be defined as follows:

$${\rho }_f{C}_{p,f}\overrightarrow{u}\cdot \nabla T+\nabla \cdot \overrightarrow{q}=Q+Q_p+Q_{vd}$$

(12)

$$\overrightarrow{q}=-k_{eff}\nabla T$$

(13)

where f is a physical quantity on fluid [-], C_p is the constant pressure specific heat [J/(kg·K)], $\overrightarrow{q}$ is a thermal flux vector [W/m²], Q is a thermal source [W/m³], Q_p is a thermal source due to pressure loss [W/m³], Q_vd is a thermal source due to viscosity dissipation [W/m³], and k_eff is an effective thermal conductivity [W/(m·K)].

2.2. 3D Numerical Modeling of BDR

Figure 1 illustrates the overview of the 3D model. Figure 2 illustrates the front view of the 3D model. The 3D model simulates the experimental apparatus of the reaction chamber used for the previous experimental studies carried out by the authors [5,18]. Though the BDR experiment was carried out using a membrane reactor in the previous studies [5,18], this study focuses on the reaction chamber only without H₂ separation to clarify the reaction characteristics of biogas dry reforming including the other reactions. The dimensions of the 3D model are 40 mm × 40 mm × 100 mm. This study has adopted the dimensions of the 3D model of 40 mm × 40 mm × 100 mm for the numerical simulation following the experimental studies conducted by the authors [5,18]. In this experimental study, the diameters of the inlet and outlet of the reactor were 6 mm. Therefore, this study has set the area of 6 mm × 6 mm as the area of the inlet and outlet of the full-scale model.

In this study, the numerical simulation is carried out by the one-fourth-sized model cut with the dimensions of 10 mm × 10 mm × 100 mm, which are also shown in Figure 1 and Figure 2, since the similarity of the results toward horizontal and vertical directions have been confirmed by this study.

2.3. Kinetic Modeling of BDR

This study considers the reaction shown in Equations (1)–(5) for the numerical simulation. The reaction rates of these equations are defined as follows [19,23,24,25]:

<Dry reforming (DR)>

$$r_1=k_1\cdot \left[{\displaystyle \frac{K_{C{O}_2,1}{K}_{C{H}_4,1}{P}_{C{O}_2}{P}_{C{H}_4}}{{\left(1+K_{C{O}_{2,1}}{P}_{C{O}_2}+K_{C{H}_4,1}{P}_{CH4}\right)}^2}}\right]\cdot \left[1-{\displaystyle \frac{{\left(P_{CO}{P}_{H_2}\right)}^2}{K_1{P}_{C{O}_2}{P}_{C{H}_4}}}\right]$$

(14)

where $k_1=1290\times 5000e^{{\displaystyle \frac{-102065}{RT}}}$, $K_{C{O}_2,1}=2.64\times {10}^{-2}{e}^{{\displaystyle \frac{37641}{RT}}}$, $K_{C{H}_4,1}=2.63\times {10}^{-2}{e}^{{\displaystyle \frac{40684}{RT}}}$, and $K_1=e^{34.011}\cdot e^{{\displaystyle \frac{-258598}{RT}}}$.

<Steam reforming (SR)>

$$r_2={\displaystyle \frac{\frac{k_2}{P_{H_2}^{2.5}}\left(P_{C{H}_4}{P}_{H_{2}O}-\frac{P_{CO}{P}_{H_2}^3}{K_2}\right)}{{\left(1+\frac{P_{H_{2}O}{K}_{H_{2}O,2}}{P_{H_2}}+K_{CO,2}{P}_{CO}+K_{H_2,2}{P}_{H_2}+K_{C{H}_4,2}{P}_{C{H}_4}\right)}^2}}$$

(15)

where $k_2=2.636\times {10}^{12}{e}^{{\displaystyle \frac{-240100}{RT}}}$, $K_2=1.198\times {10}^{17}{e}^{{\displaystyle \frac{-26830}{RT}}}$, $K_{C{O}_2,2}=8.23\times {10}^{-5}{e}^{{\displaystyle \frac{70650}{RT}}}$, $K_{C{H}_4,2}=6.65\times {10}^4{e}^{{\displaystyle \frac{38280}{RT}}}$, $K_{H_2,2}=6.12\times {10}^{-9}{e}^{{\displaystyle \frac{82900}{RT}}}$, and $K_{H_{2}O,2}=1.77\times {10}^5{e}^{{\displaystyle \frac{-88680}{RT}}}$.

<Reverse water gas shift reaction (RWGS)>

$$r_3=k_3{P}_{C{O}_2}\left(1-{\displaystyle \frac{P_{CO}{P}_{H_{2}O}}{K_3{P}_{C{O}_2}{P}_{H_2}}}\right)$$

(16)

where $k_3=9.28\times {10}^3{e}^{{\displaystyle \frac{-73105}{RT}}}$ and $K_3=68.78e^{{\displaystyle \frac{-37500}{RT}}}$.

<Methanation reaction (MR)>

$$r_4={\displaystyle \frac{\frac{k_4}{P_{H_2}^{3.5}}\left(P_{C{H}_4}{P}_{H_{2}O}^2-\frac{P_{C{O}_2}{P}_{H_2}^4}{K_4}\right)}{{\left(1+\frac{P_{H_{2}O}{K}_{H_{2}O,4}}{P_{H_2}}+K_{CO,4}{P}_{CO}+K_{H_2,4}{P}_{H_2}+K_{C{H}_4,4}{P}_{C{H}_4}\right)}^2}}$$

(17)

where $k_4=6.364\times {10}^{11}{e}^{{\displaystyle \frac{-24390}{RT}}}$, $K_4=2.117\times {10}^{15}{e}^{{\displaystyle \frac{-22430}{RT}}}$, $K_{C{H}_4,4}=6.65\times {10}^{-4}{e}^{{\displaystyle \frac{38280}{RT}}}$, $K_{CO,4}=8.23\times {10}^{-5}{e}^{{\displaystyle \frac{70650}{RT}}}$, $K_{H_2,4}=6.12\times {10}^{-9}{e}^{{\displaystyle \frac{82900}{RT}}}$, and $K_{H_{2}O,4}=1.77\times {10}^5{e}^{{\displaystyle \frac{-88680}{RT}}}$.

<Methane decomposition reaction (MDR)>

$$r_5={\displaystyle \frac{k_5{K}_{C{H}_4,5}\left(P_{C{H}_4}-\frac{P_{H_2}^2}{K_5}\right)}{{\left(1+K_{C{H}_4,5}{P}_{C{H}_4}+\frac{P_{H_2}^{1.5}}{K_{H_2,5}}\right)}^2}}$$

(18)

where $k_5=6.95\times {10}^3{e}^{{\displaystyle \frac{-58983}{RT}}}$, $K_5=2.95\times {10}^5{e}^{{\displaystyle \frac{-84440}{RT}}}$, $K_{C{H}_4,5}=0.21e^{{\displaystyle \frac{-567}{RT}}}$, and $K_{H_2,5}=5.18\times {10}^7{e}^{{\displaystyle \frac{-133210}{RT}}}$.

<Boudouard reaction (BD)>

$$r_6={\displaystyle \frac{\frac{k_6}{K_{CO,6}{K}_{C{O}_2,6}}\left(\frac{P_{CO}}{K_6}-\frac{P_{C{O}_2}}{P_{CO}}\right)}{{\left(1+K_{CO,6}{P}_{CO}+\frac{1}{K_{CO,6}{K}_{C{O}_2,6}}\frac{P_{C{O}_2}}{P_{CO}}\right)}^2}}$$

(19)

where $k_6=1.34\times {10}^{15}{e}^{{\displaystyle \frac{-243835}{RT}}}$, $K_6=1.9393\times {10}^9{e}^{{\displaystyle \frac{-168527}{RT}}}$, $K_{C{O}_2,6}=2.81\times {10}^7{e}^{{\displaystyle \frac{-104085}{RT}}}$, and $K_{CO,6}=7.34\times {10}^{-6}{e}^{{\displaystyle \frac{100395}{RT}}}$. In addition, $r_n$ is a kinetic rate (n = 1, 2, 3, 4, 4, 6) [mol/(kg·s)], $k_n$ is a kinetic constant (n = 1, 2, 3, 4, 5, 6) [mol/(kg·s)], $P_i$ is a particle pressure of chemical species i [Pa], $K_n$ is an equilibrium constant (n = 1, 2, 3, 4, 5, 6) [-], and $K_i$ is the absorption equilibrium constants of chemical species i [-].

2.4. Numerical Simulation Parameters and Conditions for BDR

In this study, the molar ratio of CH₄:CO₂ is changed by 1.5:1, 1:1, and 1:1.5. The initial reaction temperature is changed by 400 °C, 500 °C, 600 °C, and 700 °C. Ni/Cr alloy is assumed to be the catalyst, which was used for the previous experimental studies [5,18]. The weight ratio of Cr to the whole weight of Ni/Cr alloy is 20 wt%. The porosity, permeability, constant pressure-specific heat, and thermal conductivity of Ni/Cr alloy catalysts are estimated considering the weight ratio of Cr. ε_p is changed by 0.95, 0.7, 0.4, and 0.1. This study has selected ε_p of 0.95 following the experimental studies conducted by the authors [5,18]. The other ε_p, i.e., 0.7, 0.4, and 0.1 have been selected to clarify the impact of ε_p on the characteristics of BDR and the other reactions.

Table 1. Numerical simulation conditions.

Initial reaction temperature [°C]	400, 500, 600
Pressure in reactor [Pa]	1.013 × 105
Inlet flow rate of CH4 [NL/min] (CH4:CO2 = 1.5:1, 1:1, 1:1.5)	1.088, 0.725, 0.725
Inlet flow rate of CO2 [NL/min] (CH4:CO2 = 1.5:1, 1:1, 1:1.5)	0.725, 0.725, 1.088
Outlet pressure [Pa]	1.013 × 105
Density of the catalyst [kg/m3]	8901, 8752, 8709, 8664, 8619 (@20 °C, 400 °C, 500 °C, 600 °C, 700 °C)
Porosity of the catalyst (εp) [-]	0.95, 0.7, 0.4, 0.1
Permeability of the catalyst [m2]	1.9 × 10−8, 1.4 × 10−8, 8.0 × 10−9, 2.0 × 10−9 (εp = 0.95, 0.7, 0.4, 0.1)
Constant pressure-specific heat of the catalyst [J/(kg·K)]	458, 558, 551, 541, 538 (@20 °C, 400 °C, 500 °C, 600 °C, 700 °C)
Thermal conductivity of the catalyst [W/(m·K)]	91.2, 65.3, 66.5, 69.1, 72.2 (@20 °C, 400 °C, 500 °C, 600 °C, 700 °C)
Apparent density of the catalyst bed in the reactor [kg/m3]	418, 2504, 5008, 7511 (εp = 0.95, 0.7, 0.4, 0.1)
Apparent thermal conductivity of the catalyst bed in the reactor [W/(m·K)]	3.5, 20.9, 41.7, 62.5 (εp = 0.95, 0.7, 0.4, 0.1)

lists the main physical properties and parameters for the numerical simulation in this study.

In this study, the following assumptions are set:

(i)

The catalyst is a porous material. The porosity, permeability, constant pressure-specific heat, and thermal conductivity are isotropic.

(ii)

The wall temperature is isothermal.

(iii)

G = The gas is a Newton fluid and an ideal gas.

(iv)

W = The wall of the reactor excluding the inlet and outlet is no-slip.

(v)

The pressure of the outlet is atmosphere (gauge pressure = 0 Pa).

(vi)

The temperature of the inflow gas is the same as the initial reaction temperature.

(vii)

The produced carbon is treated as a gas.

As for (vii), generally speaking, a solid carbon is produced by Equations (5) and (6). However, the numerical simulation software COMSOL Multiphysics, which is used by this study, cannot treat a solid carbon as a gaseous carbon. Therefore, this study has set the assumption (vii).

2.5. Evaluaion Factors and Reaction Pathways in BDR

In this study, the following evaluation factors are considered:

$${\mathrm{CH}}_4\mathrm{conversion}={\displaystyle \frac{C_{C{H}_4,in}-C_{C{H}_4,out}}{C_{C{H}_4,in}}}\times 100\left[\%\right]$$

(20)

where $C_{C{H}_4,in}$ is the molar concentration of CH₄ at the inlet [mol] and $C_{C{H}_4,out}$ is the molar concentration of CH₄ at the outlet [mol].

$${\mathrm{CO}}_2\mathrm{conversion}={\displaystyle \frac{C_{C{O}_2,in}-C_{C{O}_2,out}}{C_{C{O}_2,in}}}\times 100\left[\%\right]$$

(21)

where $C_{C{O}_2,in}$ is the molar concentration of CO₂ at the inlet [mol] and $C_{C{O}_2,out}$ is the molar concentration of CO₂ at the outlet [mol].

$${\mathrm{H}}_2\mathrm{yield}={\displaystyle \frac{\frac{1}{2}{C}_{H_2,out}}{C_{C{H}_4,in}}}\times 100\left[\%\right]$$

(22)

where $C_{H_2,out}$ is the molar concentration of H₂ at the outlet [mol].

$${\mathrm{H}}_2\mathrm{selectivity}={\displaystyle \frac{C_{H_2,out}}{C_{H_2,out}+C_{CO,out}}}\times 100\left[\%\right]$$

(23)

where $C_{CO,out}$ is the molar concentration of CO at the outlet [mol].

$$\mathrm{CO}\mathrm{selectivity}={\displaystyle \frac{C_{CO,out}}{C_{H_2,out}+C_{CO,out}}}\times 100\left[\%\right]$$

(24)

3. Results and Discussion

3.1. Impact of Catalyst Porosity on Reaction, Heat, and Mass Transfer Phenomena in BDR

Figure 3, Figure 4, Figure 5 and Figure 6 illustrate the 3D distributions of the molar concentrations (CH₄, CO₂, H₂, CO₂, H₂O, and C), temperature, gas pressure, and gas velocity within the reactor. To exemplify the numerical simulation results, Figure 3 and Figure 5 present the data for the CH₄:CO₂ ratio of 1.5:1 at 600 °C with ε_p of 0.95, while Figure 4 and Figure 6 show the corresponding data for the same conditions but with a lower ε_p of 0.1.

As depicted in Figure 3 and Figure 4, the molar concentrations of CH₄ and CO₂ (reactants) decrease progressively along the gas flow from the inlet to the outlet, regardless of ε_p. In the initial section (0 mm ≤ y ≤ 20 mm), the molar concentrations of CH₄ and CO₂ exhibit a radial gradient, with higher values near the center and lower values near the wall. However, this radial distribution becomes less pronounced or disappears in the downstream region (30 mm ≤ y ≤ 100 mm). This behavior can be attributed to the high gas velocity near the inlet, which promotes preferential axial flow and limits radial diffusion. As the gas progresses through the reactor, the diffusion becomes more significant, leading to a more uniform distribution. In addition, the molar concentrations of H₂, CO, H₂O, and C, which are products, increase along the gas flow direction from the inlet to the outlet, regardless of ε_p. Conversely, the temperature profiles depicted in Figure 5 and Figure 6 demonstrate a decrease from the initial reaction temperature of 600 °C, except near the inlet and the outlet regions. This temperature reduction is attributed to the endothermic nature of the reactions described by Equations (1)–(6). These observations validate the accuracy of the simulated reaction model. Furthermore, the gas velocity is notably higher near the inlet and the outlet compared to the central reaction of the reactor. This is due to the larger cross-sectional area in these regions. However, away from the inlet and the outlet, the gas velocity stabilizes as a result of sufficient gas diffusion towards the reactor walls.

As illustrated in Figure 3 and Figure 4, a higher ε_p leads to lower molar concentrations of CH₄ and CO₂ and, conversely, higher molar concentrations of H₂, CO, H₂O, and C. This enhancement in product formation and reactant consumption can be attributed to the improved mass transfer within the porous catalyst structure at higher ε_p values. The enhanced diffusivity within the catalyst facilitates reactant diffusion to active sites and product diffusion away from the catalyst surface, thereby promoting the reaction kinetics. Further analysis of Figure 5 and Figure 6 reveals that the temperature for ε_p = 0.95 is lower than that for ε_p = 0.1. This observation can be attributed to the enhanced diffusion within the catalyst bed at higher porosities, facilitating the progress of the endothermic reactions (Equations (1)–(3) and (5)). Additionally, the pressure near the inlet is lower for ε_p = 0.95 compared to that for ε_p = 0.1, which is directly related to the increased catalyst permeability at higher porosities, as indicated in Table 1.

Figure 7, Figure 8, Figure 9 and Figure 10 illustrate the influence of ε_p on the distribution of molar concentrations along the reactor centerline for CH₄:CO₂ = 1.5:1 at various reaction temperatures (400 °C, 500 °C, 600 °C, and 700 °C). ε_p is changed by 0.95, 0.7, 0.4, and 0.1. The results demonstrate that increasing ε_p leads to a decrease in the molar concentrations of CH₄ and CO₂ while simultaneously enhancing the concentrations of H₂, CO, H₂O, and C. This enhancement is attributed to the improved mass diffusion within the porous catalyst structure, facilitating the progress of the reforming reactions. Moreover, elevating the reaction temperature promotes the formation of H₂ and CO, as evidenced by the increased molar concentrations of these species. The enhanced performance of DR, SR, RWGS, and MR, as described by Equations (1)–(4), at a higher reaction temperature contributes to the observed H₂ and CO production. In addition, according to Figure 7, Figure 8, Figure 9 and Figure 10, the molar concentration of C for ε_p increases along the y-direction at 400 °C and 500 °C. On the other hand, we cannot see the production of C for ε_p along the y-direction at 600 °C and 700 °C. The authors think BD (Equation (6)), which is an exothermic reaction, produces C from CO at a lower reaction temperature after CO is produced by DR (Equation (1)), SR (Equation (2)), and RWGS (Equation (3)).

Figure 11 quantitatively evaluates the influence of ε_p on the distribution of the molar concentration ratios for each gas along the reactor centerline under the condition of CH₄:CO₂ = 1.5:1 at 600 °C. The analysis reveals that the impact of ε_p on the molar concentration distribution becomes more pronounced as ε_p increases, regardless of the gas species. Notably, the effect of ε_p on the products species (H₂, CO, H₂O, and C) is more significant near the inlet (y = 20 m). In addition, it can be seen that the effect of ε_p on the reactant species (CH₄ and CO₂) is more pronounced near the inlet (y = 20 m), e.g., in the case of 0.1/0.95, which is similar to the tendency of products species. This observation can be attributed to the gradual consumption of CH₄ and CO₂ through the various reactions (Equations (1)–(5)) along the gas flow. The enhanced diffusivity within the porous catalyst at higher ε_p values contributes to the more significant impact on reactant concentrations near the outlet. Moreover, it is revealed in Figure 11 that increasing ε_p from 0.1 to 0.95 significantly provides a five times increase in H₂ production.

Figure 12 illustrates the influence of ε_p on the distribution of temperature for CH₄:CO₂ = 1.5:1 at 600 °C. As depicted in the figure, a larger temperature drop is observed near the inlet (y = 20 m) with increasing ε_p. This phenomenon is attributed to the dominance of the endothermic reactions represented by Equations (1), (2), (3) and (5), which absorb heat from the surrounding, leading to a localized temperature decrease.

3.2. Influence of the Molar Ratio of CH₄:CO₂ on BDR over the Ni/Cr Catalyst

Figure 13 and Figure 14 compare the distributions of the molar concentration for different molar ratios of CH₄:CO₂ at 400 °C and 700 °C, respectively, with ε_p = 0.95. The results indicate that the highest molar concentrations of CH₄ and CO₂ are achieved at CH₄:CO₂ = 1.5:1 and 1:1.5, respectively, regardless of the reaction temperature. This is primarily attributed to the higher inflow rates associated with these ratios. Additionally, the highest molar concentrations of H₂ and CO₂ are consistently observed at CH₄:CO₂ = 1:1, irrespective of the reaction temperature, aligning with the theoretical molar ratio for DR.

Regarding H₂O production, the highest molar concentration of H₂O is obtained for CH₄:CO₂ = 1:1.5 at 700 °C. This is due to the favorable conditions for MR, which consumes CO₂ to produce H₂O. In contrast, at 400 °C, the highest H₂O concentration occurs at CH₄:CO₂ = 1:1 and 1:1.5. Under these conditions, RWGS, which also produces H₂O, is more favored due to its exothermic nature. The lower molar concentration of H₂O at 400 °C for CH₄:CO₂ = 1.5:1 can be attributed to the reduced availability of CO₂ for RWGS. Moreover, the molar concentration of C is lower at 700 °C compared to that at 400 °C, suggesting a decreased contribution of BD at higher temperatures.

3.3. Comparison of Numerical Simulation and Experimental Results for BDR

To comprehensively evaluate the performance of BDR including the other reactions, this study employs the evaluation merits defined by Equations (20)–(24).

Table 2. Comparison of CH₄ conversion, CO₂ conversion, H₂ yield, H₂ selectivity, and CO selectivity among different ε_p values in the case of CH₄:CO₂ = 1.5:1 at 600 °C.

εp [-]	CH4 Conversion [%]	CO2 Conversion [%]	H2 Yield [%]	H2 Selectivity [%]	CO Selectivity [%]
0.95	49.9	63.5	21.3	48.2	51.8
0.7	45.6	58.2	19.5	48.1	51.9
0.4	33.8	43.3	14.3	47.8	52.2
0.1	11.1	14.8	4.6	45.9	54.1

presents a comparative analysis of the CH₄ conversion, CO₂ conversion, H₂ yield, H₂ selectivity and CO selectivity for various ε_p values at CH₄:CO₂ = 1.5:1 and the reaction temperature of 600 °C.

As shown in Table 2, increasing ε_p leads to enhanced CH₄ conversion, CO₂ conversion, H₂ yield, and H₂ selectivity. Conversely, CO selectivity decreases with rising ε_p. The improved performance can be attributed to enhanced mass diffusion within the porous catalyst structure. This facilitates the progress of the reaction, particularly those consuming CH₄ and CO₂ as reactants. In addition, it is revealed in Table 2 that increasing ε_p from 0.1 to 0.95 provides the increase in H₂ selectivity by 2.3% while simultaneously reducing CO selectivity by 2.3%.

Table 3. Comparison of CH₄ conversion, CO₂ conversion, H₂ yield, H₂ selectivity, and CO selectivity among different temperatures in the case of CH₄:CO₂ = 1.5:1 (based on numerical simulation data).

Reaction Temperature [°C]	CH4 Conversion [%]	CO2 Conversion [%]	H2 Yield [%]	H2 Selectivity [%]	CO Selectivity [%]
400	2.1	2.7	0.8	48.0	52.0
500	16.2	21.0	6.5	47.1	52.9
600	49.9	63.5	21.3	48.2	51.8
700	74.4	92.1	33.2	49.4	50.6

presents a comparative analysis of CH₄ conversion, CO₂ conversion, H₂ yield, H₂ selectivity, and CO selectivity among various reaction temperatures for CH₄:CO₂ = 1.5:1 and ε_p = 0.95.

Table 4. Comparison of CH₄ conversion, CO₂ conversion, H₂ yield, H₂ selectivity, and CO selectivity among different temperatures in the case of CH₄:CO₂ = 1.5:1 (based on experimental data).

Reaction Temperature [°C]	CH4 Conversion [%]	CO2 Conversion [%]	H2 Yield [%]	H2 Selectivity [%]	CO Selectivity [%]
400	24.1	−29.7	0.09	0.16	99.8
500	8.24	−3.08	1.01	2.09	97.9
600	12.7	0.08	4.48	7.24	92.8

provides a similar comparison based on experimental data for ε_p 0.95 obtained from the previous experimental study conducted by the authors using the membrane reactor [5]. Due to the experimental limitations, data for 700 °C were not available in the experimental study [5]. The data presented in Table 4 correspond to the specific condition of 0 MPa of the pressure difference between the reaction chamber and the sweep chamber, without the use of a sweep gas. This condition aligns with the numerical simulation setup, which does not consider H₂ separation.

As expected from the endothermic nature of H₂ production reactions (Equations (1), (2) and (5)), Table 3 and Table 4 reveal a positive correlation between the reaction temperature and both the H₂ yield and H₂ selectivity. However, a discrepancy exists between the numerical simulation results (Table 3) and the experimental data (Table 4). The experimental data were obtained using a membrane reactor [5]. On the other hand, the 3D model used for the numerical simulation in this study considers the reactions in only the reaction chamber without a separation membrane and a sweep chamber. As the first step in this study, the numerical simulation by a 3D model considering several reactions without H₂ separation has been conducted. As a result, each evaluation factor shown in Table 3 and Table 4 indicates the discrepancy between the numerical simulation results and the experimental results. This study suggests that the impact of H₂ separation might influence the progress of each reaction considered in this study.

According to the literature review by the authors, the numerical studies on biogas dry reforming using a membrane reactor have been mainly conducted for the double circular tube type [26,27,28,29]. On the other hand, this study has conducted the rectangular reactor type since the authors’ previous experimental studies [4,5] have used the rectangular membrane reactor consisting of a reaction chamber and sweep chamber. In this paper, the numerical study using the 3D model for a rectangular reactor has been conducted to compare the results obtained by the authors’ experimental studies using the rectangular membrane reactor. As a result, this study has clarified the characteristics of biogas dry reforming including the other reactions using the rectangular reactor without H₂ separation. This is the first step. In the next step of this study, the numerical study on the rectangular membrane reactor consisting of a reaction chamber and a sweep chamber is going to be carried out to understand the effect of H₂ separation on the biogas dry reforming including the other reactions considered in this study, which is being prepared now. This study would report the results obtained by the numerical study on the rectangular membrane reactor consisting of a reaction chamber and a sweep chamber compared to the results obtained by the authors’ previous experimental studies [4,5] in the near future.

4. Conclusions

This study employed numerical simulations using COMSOL Multiphysics to elucidate the reaction characteristics of a Ni/Cr catalyst employed for BDR. By systematically varying the reaction temperature, molar ratio of CH₄:CO₂, and ε_p, the study aimed to comprehensively understand the impact of these parameters on the BDR process. The findings of this investigation are summarized as follows:

i

The results indicate that increasing ε_p leads to a decrease in the molar concentrations of CH₄ and CO₂ while simultaneously enhancing the molar concentrations of H₂, CO, H₂O, and C. This enhancement is attributed to the improved mass diffusion within the porous catalyst structure, while it facilitates the progress of the reforming reactions.

ii

A positive correlation is observed between the reaction temperature and the molar concentrations of H₂ and CO. The enhancement of H₂ and CO production is attributed to the promotion of DR, SR, RWGS, and MR at elevated reaction temperatures.

iii

The results indicate that the influence of ε_p on the distribution of the molar concentration of all gases increases with rising ε_p. Notably, the impact of ε_p on the product species (H₂, CO, H₂O, and C) is more pronounced near the inlet (y = 20 m), which is similar to the tendency of reactant species (CH₄ and CO₂).

iv

A more pronounced temperature drop near the inlet (y = 20 m) is observed with increasing ε_p. This phenomenon is attributed to the dominance of the endothermic reactions in this region, which absorb heat from the surroundings, leading to a localized temperature decrease.

v

The highest molar concentrations of H₂ and CO₂ are consistently obtained for CH₄:CO₂ = 1:1, regardless of the reaction temperature. This finding aligns with the theoretical stoichiometry of DR, where equal molar amounts of CH₄ and CO₂ are converted to H₂ and CO₂. Therefore, the optimal CH₄:CO₂ for maximizing product yields in BDR is 1:1.

vi

The highest molar concentration of H₂O is observed at 700 °C for CH₄:CO₂ = 1:1.5, whereas the highest molar concentration of H₂O at 400 °C occurred with CH₄:CO₂ = 1:1 and 1:1.5. Additionally, the molar concentration of C is found to be lower at 700 °C compared to 400 °C, suggesting a reduced contribution of BD at higher reaction temperatures.

vii

The results indicate that increasing ε_p leads to higher CH₄ conversion, CO₂ conversion, H₂ yield, and H₂ selectivity. Conversely, CO selectivity decreases with increasing ε_p. Furthermore, both the numerical simulations and experiments’ data demonstrate a positive correlation between the reaction temperature and H₂ yield and H₂ selectivity. This enhancement is attributed to the endothermic nature of the reactions producing H₂, which are favored at higher reaction temperatures.

viii

This study has revealed the reaction mechanism of BDR and the other reactions as well as the influence of ε_p on the performance of these reactions by numerical simulation. This study has also revealed the impact of the reaction temperature and molar ratio of CH₄:CO₂ on the reaction characteristics. The results obtained by this study, e.g., which reaction occurred in the reactor under the different parameters, are novel findings according to the scientific viewpoint. In addition, the optimum molar ratio and reaction temperature have been clarified in this study. These optimum operation conditions are meaningful for practical applications. On the other hand, this study has not conducted a numerical simulation on the BDR membrane reactor, and we would like to conduct this in the near future for a more accurate comparison with the experimental data.