Enhancement of Mass Transfer and Coke Resistance in DRM through Hierarchical Porous Catalysts

Abstract

Dry reforming of methane (DRM) is one of the feasible strategies for carbon capture and utilization. However, DRM has a high tendency toward coking, which is restricted to industrial applications. The primary cause of coking in DRM is the limitation of mass transfer inside porous catalysts. To overcome this limitation, optimizing the pore structure of the porous catalyst becomes crucial. Hierarchical pore structure has received considerable attention in recent years due to its superior mass transfer performance. Therefore, this chapter focuses on the structure-performance relationship of hierarchical porous catalysts in DRM. Specifically, two types of porous catalysts, namely porous pellet and open-cell foam, are examined. The impacts of various hierarchical pore structure parameters on the catalytic activity and coke resistance are investigated. The findings offer a theoretical foundation and technical guidance for the design of porous catalysts with hierarchical pore structures.

1. Introduction

Dry reforming of methane (DRM) offers a promising solution by converting two primary greenhouse gases, carbon dioxide (${\mathrm{CO}}_2$) and methane (${\mathrm{CH}}_4$), into value-added synthesis gas (${\mathrm{H}}_2$ and $\mathrm{CO}$), which can be applied to synthesize oxygenated chemicals [1] and hydrocarbons via Fischer-Tropsch synthesis [2]. Due to their remarkable cost-effectiveness and excellent reaction performance, catalysts based on nickel (Ni) have gained widespread utilization in DRM. However, the endothermic nature of the DRM process requires high temperature (>873.15 K) [3], leading to a high tendency toward carbon deposition on Ni-based catalysts, resulting in rapid deactivation and hindering industrial applications [4, 5].

To promote the industrial application of DRM, two effective methods have been pursued: active site design and catalyst pore structure optimization. In recent years, an increasing number of researchers have devoted their efforts to catalyst active site design, leading to notable enhancements in the activity and coke resistance of Ni-based catalysts. Various approaches, such as utilizing different support materials [6], incorporating promoters [7], and optimizing preparation and activation methods [8], have been employed to reinforce the coke resistance of Ni-based catalysts. These studies have made significant progress in improving the catalytic activity of DRM catalysts. Concurrently, the optimization of catalyst pore structure has emerged as a crucial strategy for mitigating catalyst deactivation. Notably, Rao et al. [9] have demonstrated that pore structure optimization can nearly double the lifespan time of catalyst and enhance coke resistance against deactivation. However, it is noteworthy that only limited research has been focused specifically on catalyst pore structure optimization.

In pursuit of enhancing the catalytic activity and resistance to coke formation, the catalyst pore structure has been studied in this work. Porous pellets are extensively employed in the DRM process. However, Baiker et al. [10] highlighted that the catalyst performance of porous pellets in many industrial processes is restricted by intraparticle diffusion despite their high catalytic activity. To overcome this limitation, the concept of hierarchical pore structures has garnered significant attention in recent years due to their superior mass transfer properties [11, 12, 13]. Notably, Lakiss et al. [14] observed that porous pellets with hierarchical pore structures exhibit shorter diffusional paths. As a result, numerous researchers have made efforts to prepare porous pellets with bimodal hierarchical pore structures, involving combinations of micro-mesopores [15], micro-macropores [16], and macro-mesopores [17, 18]. The hierarchical pore structure of porous pellets also demonstrates remarkable coke resistance. Despite these advancements, however, there remains a lack of comprehensive studies associated with the effects of porous pellets with hierarchical pore structures on both catalyst performance and coke resistance for DRM.

Additionally, structured catalysts present another category of porous catalyst worth considering for DRM. In contrast to unstructured catalysts like porous pellets, structured catalysts offer advantages by minimizing regions of restricted mass transfer, thereby mitigating catalyst deactivation [19, 20]. Among the structured catalysts, open-cell foam stands out for its widespread use in heterogeneous reactions, owing to its high porosity, and permeability, as well as excellent heat and mass transfer performance. Notably, Richardson et al. [21] conducted experimental studies confirming that foam catalysts exhibited a higher effectiveness factor than commercially available porous pellets in DRM. Furthermore, investigations have revealed that foam catalysts maintain comparable catalyst performance and selectivity to powdered pellet catalysts. Consequently, many researchers have explored the use of open-cell foam as a catalyst in DRM [22, 23, 24]. Based on the superior mass transfer performance of the hierarchical pore structure, however, a critical knowledge gap remains as the impact of open-cell foam with hierarchical pore structure on coke resistance during the DRM process has not been thoroughly explored yet. Understanding this aspect can elucidate the potential advantages and limitations of employing such structured catalysts, thereby contributing to the advancement of open-cell foam structure catalysts.

This work aims to investigate the relationship between porous catalysts with hierarchical pore structure and catalyst performance during DRM. To achieve this, a numerical model is employed to analyze the mass transfer and DRM reaction in porous pellet and open-cell foam. Moreover, two artificial numerical algorithms are developed specifically for constructing hierarchical pore structures for porous pellet and open-cell foam, respectively. Based on Benguerba et al. [25], five reactions associated with DRM shown in Table 1 are used in the study, where $R_3$, $R_4$ and $R_5$ are related to coking. The impacts of various hierarchical pore structure parameters on the catalyst performance and coke resistance are investigated.

Index	Reactions	Reaction rates
R1	CO2 + CH4 ↔ 2CO + 2H2	$R_1=\frac{k_1{K}_{{\mathit{CO}}_2,1}{K}_{{\mathit{CH}}_4,1}{P}_{{\mathit{CH}}_4}{P}_{{\mathit{CO}}_2}}{{\left(1+K_{{\mathit{CO}}_2,1}{P}_{{\mathit{CO}}_2}+K_{{\mathit{CH}}_4,1}{P}_{{\mathit{CH}}_4}\right)}^2}\left(1-\frac{{\left(P_{\mathit{CO}}{P}_{H_2}\right)}^2}{K_{P_1}{P}_{{\mathit{CH}}_4}{P}_{{\mathit{CO}}_2}}\right)$
R2	CO2 + H2 ↔ CO + H2O	$R_2=\frac{k_2{K}_{{\mathit{CO}}_2,2}{K}_{H_2,2}{P}_{H_2}{P}_{{\mathit{CO}}_2}}{{\left(1+K_{{\mathit{CO}}_2,2}{P}_{{\mathit{CO}}_2}+K_{H_2,2}{P}_{H_2}\right)}^2}\left(1-\frac{{\left(P_{\mathit{CO}}{P}_{H_{2}O}\right)}^2}{K_{P_2}{P}_{{\mathit{CO}}_2}{P}_{H_2}}\right)$
R3	CH4 ↔ C + 2H2	$R_3=\frac{k_e{K}_{{\mathit{CH}}_4,3}\left(P_{{\mathit{CH}}_4}-\frac{P_{H_2}^2}{K_{P_3}}\right)}{{\left(1+K_{{\mathit{CH}}_4,3}{P}_{{\mathit{CH}}_4}+\frac{P_{H_2}^{1.5}}{K_{H_2,3}}\right)}^2}$
R4	C + H2O ↔ CO + H2	$R_4=\frac{\frac{k_4}{K_{H_{2}O}}\left(\frac{P_{H_{2}O}}{P_{H_2}}-\frac{P_{\mathit{CO}}}{K_{P_4}}\right)}{{\left(1+K_{{\mathit{CH}}_4,4}{P}_{{\mathit{CH}}_4}+\frac{P_{H_{2}O}}{K_{H_{2}O,4}{P}_{H_2}}+\frac{P_{H_2}^{1.5}}{K_{H_2,4}}\right)}^2}$
R5	C + CO2 ↔ 2CO	$R_5=\frac{\frac{k_5}{K_{\mathit{CO},5}{K}_{{\mathit{CO}}_2,5}}\left(\frac{P_{{\mathit{CO}}_2}}{P_{\mathit{CO}}}-\frac{P_{\mathit{CO}}}{K_{P_5}}\right)}{{\left(1+K_{\mathit{CO},5}{P}_{\mathit{CO}}+\frac{P_{{\mathit{CO}}_2}}{K_{{\mathit{CO}}_2,5}{K}_{\mathit{CO},5}{P}_{\mathit{CO}}}\right)}^2}$

Table 1.

Five reactions kinetics during DRM.

2. Porous pellet

The DRM process in a porous pellet is illustrated in Figure 1. In porous pellets, three kinds of intraparticle diffusion behaviors are considered in this work, which are self-diffusion, multicomponent diffusion, and Knudsen diffusion.

Figure 1.

Schematic diagram of the DRM process in a porous pellet.

2.1 Reconstruction of porous pellet with hierarchical pore structure

In DRM, the utilization of porous pellets containing nickel-based catalysts with macro-mesoporous structures is prevalent [26, 27]. Several artificial algorithms to reconstruct macro-mesopore structures were developed by numerous researchers. Hussain et al. [28] introduced the random generation of macro-meso pores (RGMMP) algorithm, specifically designed for reconstructing porous building materials with interconnected macropores and mesopores in series. Another method proposed by Chen et al. [29] involves the random placement of circular solids to reconstruct hierarchical pore structures. While these algorithms offer promising capabilities, their parameters are mathematical variables that may not directly construct hierarchical pore structures through experimental characterization data. To overcome this, Lin et al. [30] proposed a modified RGMMP algorithm, which can reconstruct hierarchical pore structures based on experimental characterization data. The modified RGMMP algorithm, depicted in Figure 2, is governed by five key variables, which are catalyst porosity ($\epsilon$), the ratio of mesopore volume to macropore volume ($V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$), the ratio of average macropore diameter to average mesopore diameter ($d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$), and pore growth direction and mesopore diameter ($d_{\mathit{\text{meso}}}$).

Figure 2.

Flowsheet of the modified RGMMP algorithm and reconstructed porous pellet with macro-mesopore structure [30].

2.2 Mass transfer in porous pellet with macro-mesopore structure

2.2.1 Effect of the ratio of mesopore volume to macropore volume

This section aims to explore intraparticle diffusivity while maintaining a constant total porosity and varying $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$. This study considered nine different ratios of $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ at values of 1/5, 1/4, 1/3, 1/2, 0.7, 0.9, 1.0, 1.2, and 1.4. Figure 3 illustrates the relationship between intraparticle diffusivity and the variation of $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$. Additionally, three catalyst porosities, namely $\epsilon$ = 0.3, 0.4, and 0.5, are examined. The results reveal an interesting trend that as the mesopore volume increases, the intraparticle diffusivity experiences a slight increase initially, but subsequently starts to decrease. This observation suggests the presence of an optimal value for $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$. The underlying reason for the initial rise in intraparticle diffusivity with increased mesopores is the enhanced interconnection of macropores facilitated by the presence of mesopores, which supports the diffusion process. However, as the mesopore volume continues to increase beyond the optimal point, the intraparticle diffusivity starts to decline. This can be attributed to the fact that while a certain amount of mesopores strengthens the interconnectivity of macropores, an excessive amount of mesopores reduces the number of available macropores, ultimately leading to the decline of intraparticle diffusivity. Furthermore, Figure 3 shows that as the catalyst porosity decreases, the optimal value of $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ increases. This suggests that for lower porosity, a higher portion of mesopores is required to achieve higher intraparticle diffusivity.

Figure 3.

Effect of $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ on intraparticle diffusivity [30].

2.2.2 Effect of the ratio of average macropore diameter to average mesopore diameter

To investigate the effect of $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$, we plotted the intraparticle diffusivity against the variation of $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ for $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ = 0.5, as shown in Figure 4. Eight ratios of $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ =1, 2, 3, 4, 5, 6, 8, and 10 were considered. From the observations in Figure 4, it is evident that the maximum intraparticle diffusivity is achieved when the value of $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ is approximately 4. This result aligns with the findings of Peng et al. [31], who emphasized that optimizing $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ leads to optimal mass transfer performance based on generalized Murray’s law. When $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ is less than 4, increasing the macropore diameter enhances intraparticle diffusivity. This is because larger macropore diameters reduce diffusion resistance, as suggested by Wang et al. [32]. Nevertheless, when contemplating a constant quantity of macropore and mesopore volumes, increasing the average diameter of macropores results in larger voids between them. Consequently, to ensure adequate connectivity between neighboring macropores, an increased number of mesopores is required, which consequently causes a reduction in overall macropore interconnectivity. This, in turn, results in a decline in intraparticle diffusivity as the average macropore diameter increases. These findings present valuable insights for optimizing the design of porous pellets through the manipulation of $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$.

Figure 4.

Effect of $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ on intraparticle diffusivity [30].

2.2.3 Effect of macropore growth direction

In the modified RGMMP algorithm, mesopores are important for connecting macropores in porous pellets. This connection determines the growth direction in macro-mesopore structured pellets, which rely on the macropore structure. Four different types of macropore growth directions were defined in this study, which is isotropic, parallel to mass transfer direction, perpendicular to mass transfer direction in the y-axis, and perpendicular to mass transfer direction in the z-axis. Figure 5 shows how intraparticle diffusivity changes with the ratio of average macropore diameter to average mesopore diameter ($d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$), with $\epsilon$ =0.3 and $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ =0.5. It can also be found that intraparticle diffusivities of parallel to the mass transfer direction and perpendicular to the mass transfer direction are the upper and lower bounds of the intraparticle diffusivity, respectively. Intraparticle diffusivity of perpendicular to the mass transfer direction in the y-axis is close to that of perpendicular to the mass transfer direction in the z-axis. For most cases, porous pellets exhibit the isotropic growth direction of macropore, whose intraparticle diffusivity is closer to the lower bound. It clearly indicates that the pore growth direction significantly affects intraparticle diffusivity under identical conditions.

Figure 5.

Effect of pore growth direction on intraparticle diffusivity [30].

2.3 DRM reaction in porous pellet with macro-mesopore structure

In DRM, this part aimed to explore the effects of three hierarchical pore parameters on coke resistance and catalyst performance. The investigation involved calculating several characterization parameters to assess the catalyst performance. These parameters were as follows:

$$R_{\mathit{DRM}}=\frac{{\int }_0^{N_T}{r}_{\mathit{DRM}}d{N}_T}{V}$$E1

$$R_{\mathit{\text{Coking}}}=\frac{{\int }_0^{N_T}{r}_{\mathit{\text{Coking}}}d{N}_T}{V}$$E2

$$N_{\mathit{\text{Active}}}=\frac{N_T}{V}$$E3

The larger the difference between $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$, the higher the catalyst performance. In order to investigate how the hierarchical structure of the catalyst affects its performance, we maintain the constant reaction operating conditions for DRM ($T$ = $923.15\mathrm{K}$, $P$ = $1\mathrm{bar}$, $F_{{\mathrm{CH}}_4}/F_{{\mathrm{CO}}_2}$ = 1:1). These conditions align with Benguerba et al. [25].

2.3.1 Effect of catalyst porosity

The variations in catalyst porosities ($\epsilon$ = 0.3, 0.4, 0.5, 0.6, 0.7, 0.75, 0.8) are depicted in Figure 6 and Figure 7, showing the reaction fluxes of $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$, as well as the corresponding catalyst performance. From Figure 6, it becomes evident that the optimal values of $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$ occur when the porosity is approximately 0.6. This finding is attributed to the positive impact of adding porosity dominating at relatively low porosity levels ($\epsilon$ < 0.6), leading to an increase in $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$. However, when the catalyst porosity exceeds 0.6, both $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$ decrease sharply. These observations suggest that the enhanced intraparticle mass transport cannot make up for the decrease in active sites. Moreover, it can be deduced from Figure 7 that the active sites begin to decline once the porosity exceeds 0.6, indicating the competition between heterogeneous reaction and intraparticle diffusion. The trend observed in the variation of $R_{\mathit{\text{Coking}}}$ aligns with $R_{\mathit{DRM}}$, resulting in the maximum $R_{\mathit{\text{Coking}}}$ when $R_{\mathit{DRM}}$ approaches its peak. As shown in Figure 7, it has been determined that optimal catalyst performance can be achieved by attaining a larger difference between $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$, which occurs when the catalyst porosity is close to 0.7. Therefore, for DRM, the optimal catalyst porosity is found to lie within the range of 0.6 to 0.7. Moreover, these findings are consistent with Liu et al. [26], who pointed out that the optimal porosity of catalysts with a macro-mesopore structure for DRM falls within the range of 0.61 to 0.64.

Figure 6.

Effect of $\epsilon$ on the reaction fluxes [33].

Figure 7.

Effect of $\epsilon$ on the catalyst performance [33].

2.3.2 Effect of the ratio of mesopore volume to macropore volume

To gain a deeper understanding of the reaction-diffusion process in porous pellets with macro-mesopore structure, the pore volume becomes a crucial parameter while keeping the catalyst porosity constant. In this case, seven ratios were considered, namely $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ = 0.2, 0.3, 0.5, 0.8, 1.0, 1.25, and 1.5. As depicted in Figure 8, an interesting trend emerges as the mesopore volume increases. The average reaction fluxes of $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$ gradually increase, reaching a peak when the ratio of mesopore volume to macropore volume ($V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$) is equal to 0.5. However, as the mesopore volume continues to increase beyond this point, $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$ drop dramatically. Furthermore, Figure 9 confirms the observation that the optimal catalyst performance is achieved when the value of $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ approaches 0.5 as it maximizes the difference between $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$. Interestingly, it is evident from Figure 9 that the active sites of the catalyst exhibit minimal variation with changes in $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$, indicating that this parameter has little effect on the active sites. Therefore, in this section, the catalyst performance is primarily limited by intraparticle diffusion. According to Lin et al. [30], the presence of an optimal value of $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ in the intraparticle diffusion process can be attributed to the increased mesopores, which enhance pore connectivity and reduce the number of dead pores. Consequently, the catalyst performance is enhanced. However, if the mesopore volume continues to increase continuously, it leads to a reduction in the number of macropores, eventually weakening the catalyst performance. These findings indicate that an optimal balance of mesopore and macropore volumes is crucial for achieving the best catalyst performance. Too few mesopores limit the intraparticle diffusion process, while too many mesopores at the expense of macropores can reduce the overall catalytic activity. Thus, maintaining an appropriate ratio of mesopore volume to macropore volume is essential for optimizing the catalyst performance for macro-mesopore structure.

Figure 8.

Effect of $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ on the reaction fluxes [33].

Figure 9.

Effect of $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ on the catalyst performance [33].

In Figure 10, mole fraction distributions of species and the amount distribution of coke are presented for different $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$. Evidently, when $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ =0.5, the macro-mesopore structure exhibits optimal mass transport performance, resulting in reduced coke formation.

Figure 10.

Mole fraction distributions of ${\mathrm{CH}}_4$, ${\mathrm{CO}}_2$, $\mathrm{CO}$, and amount distribution of coke (unit: Mol/m2/s) with three $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ [33]: (a) $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ =0.3; (b) $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ =0.5; (c) $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ =1.5.

2.3.3 Effect of the ratio of average macropore diameter to average mesopore diameter

This section investigated the impact of the ratio between the average diameter of macropores and mesopores ($d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$) on the catalyst performance. As depicted in Figures 11 and 12, six pore diameter ratios were considered, namely $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ = 3, 4, 5, 6, 8, and 10. Observing from Figure 11, it becomes evident that $R_{\mathit{DRM}}$ reaches its optimal value as $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ approaches 5. Furthermore, $R_{\mathit{\text{Coking}}}$ surpasses $R_{\mathit{DRM}}$ when $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ is below 3 and above 8. Additionally, Figure 12 illustrates that the maximum catalyst performance occurs at approximately $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ = 5. When $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ is below 5, increasing the macropore diameter effectively enhances the catalyst performance, as the positive impact of intraparticle diffusion outweighs the effect of diminishing active sites. However, as seen in Figure 12, a further increase in $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ from 5 to 10 results in a decline in macropore interconnectivity and active sites, ultimately leading to the degradation of the catalyst performance. The optimal value of $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ arises from the interplay between competitive heterogeneous reactions and intraparticle diffusion processes.

Figure 11.

Effect of $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ on reaction fluxes [33].

Figure 12.

Effect of $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ on the catalyst performance [33].

Figure 13 displays the mole fraction distributions of species, as well as the amount distribution of coke, corresponding to three different $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ =3, 5 and 10. In Figure 13(b), the reactants ${\mathrm{CH}}_4$ and ${\mathrm{CO}}_2$, as well as the product $\mathrm{CO}$, exhibit more extensive penetration lengths and lesser carbon deposition across the entire simulation domain compared to those shown in Figure 13(a) and Figure 13(c).

Figure 13.

Mole fraction distributions of ${\mathrm{CH}}_4$, ${\mathrm{CO}}_2$, $\mathrm{CO}$, and amount distribution of carbon deposition (unit: Mol/m2/s) with three $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ [33]: (a) $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ =3; (b) $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ =5; (c) $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ =10.

3. Open-cell foam

This study examines the fluid flow, mass transport, and DRM reaction within open-cell foam, featuring hierarchical pore structure. As depicted in Figure 14, the convective-diffusion process only occurs in the pore area, while the DRM reaction and coke formation take place on the surface of the solid matrix.

Figure 14.

Schematic diagram of the DRM process in open-cell foam.

3.1 Reconstruction of open-cell foam with hierarchical pore structure

Several idealized geometric models, including cubic models [34, 35], face-centered models, body-centered models [36], and Kelvin’s tetrakaidecahedral model [37, 38], have been utilized to numerically investigate the pressure drop and heat transport characteristics. To construct hierarchical pore structure for open-cell foam, Lin et al. [39] proposed a novel numerical algorithm capable of constructing both uniform and hierarchical pore structures. Four parameters are used to control the algorithm, which are porosity ($\epsilon$), fine pore size ($d_1$), the ratio of the coarse pore size to the fine pore size ($d_2/d_1$), and the ratio of the coarse pore volume to the fine pore volume ($V_2/V_1$). The generation procedure of the new artificial algorithm is shown in Figure 15.

Figure 15.

Schematic diagram of the generation procedure of open-cell foam [39].

3.2 Permeability in open-cell foam with hierarchical pore structure

3.2.1 Effect of hierarchical pore volume ratio

The impact of the hierarchical pore volume ratio ($V_2/V_1$) on Darcy permeability was studied to elucidate the fluid flow characteristics in open-cell foam with dual-porosity. Figure 16 compares four hierarchical pore volume ratios ($V_2/V_1$ = 0, 0.25, 1, and 4) with porosity ranging from 0.76 to 0.95. It is evident that, at the same total porosity, three open-cell foams with hierarchical pore structures ($V_2/V_1$ = 0.25, 1, 4) exhibit higher permeability than that with uniform pore structure ($V_2/V_1$ = 0). This result is in line with the experimental findings by Durmus et al. [41], who developed open-cell foam with bimodal pore size. Their results showed that under almost the same porosity, the hierarchical pore structure formed by adding coarse pores to fine pores exhibits lower fluid flow resistance than open-cell foam with only fine pores, which validates the numerical results. Moreover, the increase in $V_2/V_1$ under the same conditions results in a substantial enhancement of permeability. As shown in Figure 16, permeability in the hierarchical pore structure with $V_2/V_1$ = 4 is more than two times higher than that with $V_2/V_1$ = 0.25 when porosity is 0.95. This can be explained by the fact that fluid flow resistance in the coarse pore is lower than that in the fine pore, and it decreases further with the increase of $V_2/V_1$. Notably, permeability increases sharply when $V_2/V_1$ > 1, indicating that fluid flow in the coarse pore dominates the overall permeability.

Figure 16.

Effect of $V_2/V_1$ on permeability [40]: (a) $d_1$ =1.27 mm, $d_2/d_1$ =2; (b) $d_1$ =2.54 mm, $d_2/d_1$ =2.

3.2.2 Effect of hierarchical pore size ratio

Pore size plays a critical factor in influencing the mass transfer process in porous media. According to the findings emphasized by Ahmad et al. [42], γ-alumina membranes featuring hierarchical pore structures demonstrate reduced transport resistance in comparison to membranes with uniform pore structures. As shown in Figure 17, this section examined three hierarchical pore size ratios (${\mathrm{d}}_2/{\mathrm{d}}_1$ = 1, 2, 4) with porosity ranging from 0.76 to 0.95 to investigate the effect of the hierarchical pore size ratio (${\mathrm{d}}_2/{\mathrm{d}}_1$) on permeability. It has been observed that when the diameter of the fine pores (${\mathrm{d}}_1$) remains constant, permeability in open-cell foam increases as the diameter of the coarse pores (${\mathrm{d}}_2$) increases. Interestingly, the effect of increasing ${\mathrm{d}}_2$ on permeability is not significant when ${\mathrm{V}}_2/{\mathrm{V}}_1$ is not larger than 1. This can be explained by the fact that at this stage, the resistance to fluid flow in $V_2$ is still comparable to that in $V_1$. Consequently, increasing the coarse pore diameter does not necessarily lead to a clear increase in permeability. However, when $V_2/V_1>1$, permeability rises significantly. This occurs because the coarse pores in open-cell foam can effectively connect with each other, and the contribution of the fine pore to permeability becomes negligible. As discussed in the previous subsection, the fluid flow resistance in $V_2$ is lower than that in $V_1$. When $V_1>V_2$, the impact of the fine pore on permeability diminishes, and the increase in permeability becomes more noticeable due to the dominant role of the interconnected coarse pores.

Figure 17.

Effect of $d_2/d_1$ on permeability [40]: (a) $d_1$ =1.27 mm, $V_2/V_1$ =0.25; (b) $d_1$ =1.27 mm, $V_2/V_1$ =1; (c) $d_1$ =1.27 mm, $V_2/V_1$ =4.

3.3 DRM reaction in open-cell foam with hierarchical pore structure

To gain insight into the coke resistance during DRM, an additional investigation was conducted. The following parameters are computed to capture the reaction performance in DRM.

$$R_{\mathit{DRM}}=\frac{{\int }_0^S{r}_1\mathit{dS}}{V}$$E4

$$R_{\mathit{\text{RWGS}}}=\frac{{\int }_0^S{r}_2\mathit{dS}}{V}$$E5

$$R_{\mathit{\text{Coking}}}=\frac{{\int }_0^S{r}_{\mathit{\text{Coking}}}\mathit{dS}}{V}$$E6

$$S_p=\frac{S}{V}$$E7

where $S_p$, $R_{\mathit{DRM}}$, $R_{\mathit{\text{RWGS}}}$, and $R_{\mathit{\text{Coking}}}$ are specific surface area, volume-averaged reaction rates of DRM, reverse water–gas shift, and coke formation, respectively. Within this subsection, fine pore diameter and porosity are defined as 1.27 mm and $\epsilon$ = 0.9, respectively. The impacts of hierarchical pore structures with different $V_2/V_1$ and $d_2/d_1$ on initial coke formation are explored.

3.3.1 Effect of hierarchical pore volume ratio

According to Lakiss et al. [14], hierarchical pore structures have the potential to reduce coke formation significantly by providing shorter pathways for mass transport. To gain a more profound understanding of the impact of hierarchical pore structures on coke formation, the distribution of hierarchical pore volume in open-cell foam was taken into account during the investigation. Within this section, the porosity is held constant ($\epsilon$ = 0.9), seven hierarchical pore volume ratios ($V_2/V_1$) were explored, which are $V_2/V_1$ = 0, 0.25, 0.5,1, 2, 3, and 4. It is important to note that the hierarchical pore size ratio was kept constant ($d_2/d_1$ = 2) for this part focusing on hierarchical pore structure ($V_2/V_1\ne$ 0). As depicted in Figure 18, the coke formation rates ($R_{\mathit{\text{Coking}}}$) in hierarchical pore structures ($V_2/V_1\ne$ 0) are consistently lower than those in uniform pore structures ($V_2/V_1$ = 0). Moreover, the coke formation rates ($R_{\mathit{\text{Coking}}}$) exhibit a noticeable downward trend with the increase of $V_2/V_1$, while the RWGS reaction rate ($R_{\mathit{\text{RWGS}}}$) shows a slight increase, and the DRM reaction rate ($R_{\mathit{DRM}}$) remains almost stable. This phenomenon can be ascribed to the fact that the resistance to mass transport in $V_2$ is comparatively smaller than that in $V_1$. Consequently, increasing $V_2/V_1$ significantly enhances mass transport performance. From the results shown in Figure 18, it becomes evident that the value of $R_{\mathit{\text{Coking}}}$ decreases by approximately 33.6% when $V_2/V_1$ =4, in comparison to the uniform pore structure ($V_2/V_1$ = 0). Table 2 reveals that the specific surface area ($S_p$) follows a decreasing trend with the increase of $V_2/V_1$. This decrease in active sites accounts for the substantial inhibition of coke formation. However, the progressively diminishing quantity of active sites has only a negligible impact on $R_{\mathit{DRM}}$, indicating that increasing $V_2/V_1$ not only improves coke resistance but also enhances the catalyst performance.

Figure 18.

Effect of $V_2/V_1$ on reaction rate for DRM [40].

d2/d1	1	2	2	2	2	2	2
V2/V1	0	0.25	0.5	1	2	3	4
Sp (m^2/m^3)	1984.51	1920.73	1933.73	1671.48	1476.20	1337.77	1300.20

Table 2.

The value of $S_p$ for open-cell foam with different $V_2/V_1$ [40].

The characterization of mass transport efficiency across various open-cell foam structures is achieved by visually analyzing the distribution of components along the primary transportation axis. In Figure 19, the distributions of mole fractions for ${\mathrm{CH}}_4$, ${\mathrm{CO}}_2$, and $\mathrm{CO}$ are displayed for three different open-cell foam structures, namely $V_2/V_1$ =0, $V_2/V_1$ = 0.25, and $V_2/V_1$ = 4. Notably, hierarchical pore structures ($V_2/V_1$ = 0.25 and $V_2/V_1$ = 4) exhibit enhanced mass transport efficiency compared to the uniform pore structure ($V_2/V_1$ = 0). Furthermore, as the hierarchical pore volume ratio ($V_2/V_1$) increases, the mole fractions of ${\mathrm{CH}}_4$, ${\mathrm{CO}}_2$, and $\mathrm{CO}$ distribute more uniformly throughout the computational domain, which promotes the coke resistance.

Figure 19.

${\mathrm{CH}}_4$, ${\mathrm{CO}}_2$, and $\mathrm{CO}$ mole fraction distributions with three $V_2/V_1$ [40]: (a) $V_2/V_1$ =0, ${\mathrm{d}}_2/{\mathrm{d}}_1$ =1; (b) $V_2/V_1$ =0.25, ${\mathrm{d}}_2/{\mathrm{d}}_1$ =2; (c) $V_2/V_1$ =4, ${\mathrm{d}}_2/{\mathrm{d}}_1$ =2.

3.3.2 Effect of hierarchical pore size ratio

In this section, the effect of the hierarchical pore size ratio ($d_2/d_1$) on the coke formation rate in DRM was thoroughly investigated. Based on our prior study, it was found that increasing $V_2/V_1$ significantly reduces the fluid flow resistance and coke formation. Consequently, the value of $V_2/V_1$ was set to 4 for the hierarchical pore structure in this analysis. Figure 20 illustrates the exploration of seven hierarchical pore size ratios, namely $d_2/d_1$ = 1, 1.5, 2, 2.5, 3, 3.5, and 4. Observing from Figure 20, it becomes evident that both $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$ gradually decrease as $d_2/d_1$ increases. However, the decrease rate of $R_{\mathit{\text{Coking}}}$ is notably more significant than that of $R_{\mathit{DRM}}$. Additionally, $R_{\mathit{\text{RWGS}}}$ exhibits a slight increase due to the reduction in $R_{\mathit{DRM}}$, potentially diminishing the selectivity of the DRM reaction. Reviewing the data in Table 3, it is observed that the specific surface area ($S_p$) experiences a sharp decline with the increase of $d_2/d_1$. Specifically, the specific surface area ($S_p$) decreases nearly twice from $d_2/d_1$ = 1 (uniform pore structure) to $d_2/d_1$ = 4. This substantial reduction in $S_p$ can explain the decreases in $R_{\mathit{DRM}}$ and $R_{\mathit{\text{Coking}}}$ to a significant extent. However, it is essential to highlight that $R_{\mathit{\text{Coking}}}$ is more sensitive to changes. In Figure 20, the transition from $d_2/d_1$ =1 (uniform pore structure) to $d_2/d_1$ = 4 results in approximately 13.90% decrease in $R_{\mathit{DRM}}$ but a significant 57.49% decrease in $R_{\mathit{\text{Coking}}}$. This exemplifies that increasing the hierarchical pore size ratio is a potent strategy for promoting coke resistance within in DRM.

Figure 20.

The effect of $d_2/d_1$ on reaction rate for DRM [40].

d2/d1	1	1.5	2	2.5	3	3.5	4
V2/V1	0	4	4	4	4	4	4
Sp (m^2/m^3)	1984.51	1508.59	1300.20	1175.52	1112.11	1001.25	897.44

Table 3.

The value of $S_p$ for open-cell foam with different $d_2/d_1$ [40].

In Figure 21, the mole fraction distributions for ${\mathrm{CH}}_4$, ${\mathrm{CO}}_2$ and $\mathrm{CO}$ are presented for three different hierarchical pore size ratios, namely $d_2/d_1$ = 1, $d_2/d_1$ = 1.5, and $d_2/d_1$ = 4. Upon examining Figure 21, it becomes evident that, under the constraint of constant $V_2/V_1$, the component distribution in the open-cell foam structure with $d_2/d_1$ = 4 is the most uniform throughout the computational domain when compared to the other two structures. This enhanced uniform distribution implies better mass transfer performance, which can effectively promote coke resistance.

Figure 21.

Mole fraction distributions of ${\mathrm{CH}}_4$, ${\mathrm{CO}}_2$, $\mathrm{CO}$ with three $d_2/d_1$ [40]: (a) $d_2/d_1$ =1, $V_2/V_1$ =0; (b) $d_2/d_1$ =1.5, $V_2/V_1$ =4; (c) $d_2/d_1$ =4, $V_2/V_1$ =4.

4. Conclusion

To promote the industrial application of DRM, hierarchical pore structure of two porous catalysts, namely porous pellet and open-cell foam, were explored, and two artificial algorithms were proposed to construct hierarchical pore structures for porous pellet and open-cell foam, respectively. The impacts of various hierarchical pore structure parameters on the catalyst performance and coke resistance were investigated.

For porous pellet, the influence of the macro-mesopore structure on intraparticle diffusivity and coke resistance was explored. Under constant reaction conditions (using $\mathrm{Ni}/{\mathrm{Al}}_2{\mathrm{O}}_3$ catalyst, $T$ = $923.15\mathrm{K}$, $P$ = $1\mathrm{bar}$ and $F_{{\mathrm{CH}}_4}/F_{{\mathrm{CO}}_2}$ = 1:1), the catalyst performance is superior when $\epsilon$ = 0.7, $V_{\mathit{\text{meso}}}/V_{\mathit{\text{macro}}}$ = 0.5, and $d_{\mathit{\text{macro}}}/d_{\mathit{\text{meso}}}$ = 5.

For open-cell foam, the impact of the hierarchical pore structure on the fluid flow behavior and coke formation characteristics was examined. The results revealed that increasing the coarse pore volume ($V_2$) and coarse pore size ($d_2$) can decrease the fluid flow resistance. Under the restriction of constant hierarchical pore volume ratio ($V_2/V_1$ = 4), increasing $d_2/d_1$ from 1 (uniform pore structure) to 4 results in a 57.49% decrease in the coke formation rate.

Acknowledgments

The authors would like to express their sincere thanks to the National Natural Science Foundation of China (Nos. 52176062 and 22308058).