Applying Reaction Kinetics to Pseudohomogeneous Methanation Modeling in Fixed-Bed Reactors

2.1 Kinetic Models

A large variety of kinetic models has been proposed over the past decades. Overviews can be found in ⁴, ⁹, ¹⁴.

In most studies, the set of three rate equations proposed by Xu and Froment ¹⁰ is used for methanation simulations ⁴, ¹⁶, ¹⁷. It was initially created to model methane steam reforming, which takes place at higher temperatures than the methanation reactions. However, by including the backwards reactions, the modeling of methanation is also possible. The temperature range in which the model was established is 573–848 K and the pressure range is 3–15 bar. The chosen catalyst was particularly designed for methane steam reforming.

Klose and Baerns ¹¹ developed a CO methanation model with an 18 wt% Ni/Al₂O₃ catalyst in the temperature range of 453–557 K and the pressure range of 1–25 bar. This model was adapted by Zhang et al. ¹⁵ for a 50 wt% Ni/Al₂O₃ catalyst and complemented with the WGS reaction from the Xu-Froment kinetic model. Rönsch et al. ⁹ complemented CO methanation with a term that characterizes the reverse reaction and corrected the adsorption term of the WGS reaction. The model by Rönsch et al. only considers CO methanation and the WGS reaction.

Model 6b of Kopyscinski et al. ¹⁴ was developed in temperature and pressure ranges of 553–633 K and 1–2 bar. The CO methanation does not include a backwards reaction, as this model was developed for temperatures below 633 K, where the methane steam reforming activity is negligible.

Another possibility to attain a microkinetic model that is fitted to a specific catalyst under certain methanation conditions was presented by Ducamp ¹³. He adapted activity coefficients and adsorption constants of the three kinetic rates proposed by Xu and Froment to his experimental data. The temperature and pressure ranges of the experimental validation data used by Ducamp differ from those examined in ¹⁰. Ducamp fitted the kinetic model at temperatures between 553 and 673 K and pressures between 1 and 10 bar.

The model of Koschany et al. ¹² only considers CO₂ methanation and includes a backwards reaction. The kinetic model was fitted at temperatures ranging from 453 to 613 K and pressures ranging from 1 to 10 bar. An overview of the conditions under which the kinetic models were created is shown in Tab. 1.

2.2 CFD Model

The steady-state model was developed with Ansys Fluent 2019R1. User-defined functions were used to implement the porosity distribution, the mass diffusivity, and the thermal conductivity.

2.2.1 Fixed-Bed Modeling

Pseudohomogeneous, continuous reactor models enable the time-efficient modeling of complex methanation reactor geometries. The equations of state are solved only for one phase, and interfacial heat and mass transport limitations cannot be considered. The influence of the catalyst pellets on the simulation, such as porosity distribution, thermal conductivity, and mass diffusivity, are considered via effective quantities, which consider both the gas phase and the solid catalyst.

The influence of transport limitations can be estimated via dimensionless quantities such as the two Mears criteria for external heat and mass transfer ¹⁸, the Weisz-Prater criterion for intraparticle mass transport ¹⁹, and the Anderson criterion for internal heat transport ²⁰. In general, the influence of internal mass transport and external heat transport must be considered when modeling methanation reactors with large catalyst pellets, small pore radii, and small Reynolds numbers ²¹. To enable reasonable computation times for complex reactor geometries, only pore diffusion effects are considered in this work. The consideration of heat transfer limitations, which was not done here, might lead to superheating of the catalyst pellets and higher reaction rates in the front part of the reactor.

In pseudohomogeneous reactor modeling, pore diffusion effects can either be considered by a fixed effectiveness factor η (e.g., η = 0.1 ⁷, ¹⁶, ¹⁷) or an analytical solution of the reaction-diffusion differential equation based on Thiele moduli ϕ (Eq. 4) (e.g., reaction order n = 1 ²²). A numerical solution of the diffusion-reaction differential equation is computationally more expensive ⁶. The analytical solution (Eq. 6) is used in this work. The effectiveness factors η are calculated for each cell under the assumption that the diffusion of the carbonaceous species is rate-limiting for a first order reaction according to the equations:

(4)

(5)

(6)

where ϕ_K is the Thiele modulus for a spherical particle, r_cat the radius of a catalyst particle, k_v the reaction rate coefficient, c_s the species concentration, D_eff the effective diffusion coefficient, and r_V the volumetric reaction rate.

For large Thiele moduli, Eq. 6 approaches the asymptote:

(7)

The diffusion coefficient D_eff within the catalyst pores is calculated according to the Wilke-Bosanquet formula (Eq. 8) ²³, as a superposition of Knudsen and free molecular diffusion after Fuller's method (Eq. 10) ²⁴. The free molar diffusion coefficient D_mol within the catalyst pores is calculated under the assumption of isolated diffusion of the carbonaceous species in an H₂ bulk.

(8)

(9)

(10)

where D_Kn is the Knudsen diffusion coefficient, d_pore the pore diameter, M_i are molar masses, and v₁ and v₂ are diffusion volumes.

The influence of the porous pellet is taken into account via the pellet porosity ε_pore and the pellet tortuosity τ.

(11)

where D_par is the parallel diffusion coefficient.

The distribution of porosity ψ in fixed-bed reactor pipes is calculated with the Giese equation ²⁵. However, in this work, the Giese equation was modified so that it can be applied to geometries other than tubes (Eq. 12). This is achieved by replacing the term R – r, which calculates the difference between the tube radius and the radial position, with the cell-wall distance d_w, which can be calculated with Ansys Fluent.

(12)

where d_p is the catalyst particle diameter.

The fixed bed is modeled as a porous zone in Ansys Fluent. The pressure loss along the reaction zone (z direction) in the fixed-bed is calculated via the Ergun equation (Eq. 13) ²⁶, ²⁷.

(13)

where μ is the dynamic viscosity and u the velocity.

The thermal conductivity (including heat transfer by radiation) and the mass diffusivity are calculated on the basis of a stagnant and a dynamic contribution via the Λ_r model as described in ²⁸, ²⁹. Therefore, no wall heat transfer coefficient need be considered. The diffusivity is assumed to equal the diffusivity of CO in an H₂ bulk for all species. As gradients in the radial direction are larger than those in the axial direction, the axial values for the thermal conductivity and the mass diffusivity are set equal to the values in the radial direction. The catalyst thermal conductivity is assumed to be λ_cat = 0.87 W m⁻¹K^{−1 6}. The emissivity coefficient of the catalyst pellets ε_cat is assumed to be unity due to the porous nature of the pellet. The pore diameter was obtained via BET analysis to 3.5 nm. Tab. 2 lists all simulation parameters.

Table 2. Simulation parameters of the CFD model.

Parameter	Value
Pore diameter dpore [nm]	3.5
Catalyst pellet diameter dcat [mm]	4.5
Emissivity coefficient εcat [–]	1
Geometry parameter εpore/τ [–]	0.1
Catalyst thermal conductivity λcat [W m−1K−1]	0.87
Catalyst particle density ρcat [kg m−3]	1535
Infinite bed porosity [–]	0.42
Wall thermal conductivity [W m−1K−1]	19

2.2.2 Applied CFD Model

The modeled dented reactor pipe geometry consists of an inlet zone (void space), an inert zone (filled with inert pellets), and the porous zone (filled with catalyst). The structured mesh has 294 860 elements. Inflation in cell size towards the tube wall ensures the accuracy of the Λ_r model. The independence of the solution from the mesh was shown with a 594 000-cell mesh. The 2.9 mm stainless steel (AISI 314) wall of the reactor is modeled as a shell-conduction layer that only considers radial heat conduction and a constant temperature at the cooled side of the wall.

The mathematical model considers three Ansys Fluent submodels: The mass and energy model, which enables heat and mass transfer, the species model, which enables species transport and reactions, and the laminar viscous model, as laminar flow is assumed due to low Reynolds numbers.

2.3 Experimental Validation

Measurements were conducted to validate the methanation reactor model. The dented reactor pipe was loaded with 265 mL of 25 wt% Ni/γ-Al₂O₃ catalyst on top of an inert material layer. Thermocouples (Type K) at ten axial locations measured the temperature. Mass flow controllers (Bronkhorst EL-Flow) controlled the inlet gas composition. In addition, the outlet gas composition was analyzed by using two online gas analyzers (NDIR, TCD, PMA, SICK S710).

A countercurrent jacket cooling system with Julabo Thermal Oil H350 as cooling fluid was used to cool the reactor walls to 533 or 563 K. The inlet gas mixture was preheated to the same temperature. Before approaching the measuring points, the catalyst was activated with a 90/10 mixture of N₂/H₂ for several hours at 673–693 K. The measuring points were approached and the temperatures and outlet gas compositions were averaged over at least 20 min after stationary conditions were reached.

3.1 Extrapolation with Temperature

Due to the influence of heat and mass transport limitations, kinetic models cannot be obtained at temperatures that resemble the maximum temperatures in production-scale methanation reactors ¹⁵. Therefore, an extrapolation is necessary.

The normalized CO methanation kinetic rates of the presented kinetic models are shown in Fig. 3 for the rich gas composition (see Tab. 5), which is based on ³⁰. All shown kinetic models except the CO methanation model of Kopyscinski et al., which only considers the forward reaction, consider the forward and the reverse reaction. The temperature range reflects typical temperatures for fixed-bed methanation reactors. Normalized reaction rates offer the possibility to compare the temperature dependence of the kinetic models without the influence of the catalyst activity. The solid parts of the lines indicate whether the kinetic model is within its range of validity (see Tab. 1).

The low partial pressures of the examined gas compositions also represent a possible deviation from the conditions under which the kinetic models were created. However, this deviation was not investigated in this work.

The model of Xu and Froment shows low normalized rates at low temperatures. This indicates a conflict between ignition and convergence behavior when using this model for simulations. Therefore, when using this model in simulations, the rate has to be adapted with a high fitting factor to enable ignition. The high reaction rates at higher temperatures then lead to large gradients, which influence the convergence behavior negatively.

The kinetic model parameterization by Ducamp shows a decrease in reaction rates above 750 K. Therefore, this model is not suitable to model methanation at high temperatures.

In contrast, the kinetic models by Rönsch et al. and Kopyscinski et al. show a moderate increase in reaction rates with temperature over the examined temperature range. Therefore, these models are suitable for modeling methanation reactors that operate over a broad temperature range.

Similar analyses can be conducted for the WGS reaction and CO₂ methanation (Fig. 4). Considering CO₂ methanation, the kinetic model by Koschany et al. shows the most favorable behavior for an extrapolation with temperature due to a well-defined increase in reaction rates and significant reaction rates at low temperatures.

3.2 Applicability of First-Order Effectiveness Factors

Depending on the kinetic model and the reaction conditions, CO methanation reactions proceed with a reaction order in the range of –1 to 1 with respect to CO. Negative reaction orders are obtained for high CO concentrations and low temperatures, whereas for low CO concentrations and high temperatures positive reaction orders are observed ¹¹, ³¹. Negative reaction orders in high-temperature methanation applications can lead to effectiveness factors exceeding unity ³². Under reaction conditions that are strongly limited by pore diffusion phenomena, effectiveness factors based on Thiele moduli can be calculated for arbitrary non-negative reaction orders, by multiplying the effectiveness factor for first-order reactions with a constant factor. This is deduced from the constant vertical distance between the asymptotes in the double-logarithmic plot in Fig. 5. The error resulting from assuming a reaction order of 1 instead of 0 is ca. for strongly limited operating conditions. However, the error that is made when assuming a positive reaction order instead of a negative reaction order can be much larger ³².

Therefore, a kinetic model that is suitable for pseudohomogeneous modeling should have positive reaction orders over the relevant range of operating conditions.

Fig. 6 shows the normalized CO methanation kinetic rates versus the CO partial pressure. The bulk is assumed to consist of H₂ and the overall pressure of H₂ and CO combined is set to be 1 bar. Negative reaction orders with respect to CO are indicated by an increase in reaction rates with a decrease in partial pressure. At a relatively low temperature of 533 K (Fig. 6a), significantly negative reaction orders can be observed for the kinetic models of Xu and Froment, Ducamp, and Rönsch et al. The kinetic model of Kopyscinski et al. does not show reaction rates far above unity. At a temperature of 890 K (Fig. 6b) significantly higher reaction orders are observed for all kinetic models, except for the kinetic model of Kopyscinski et al., which shows similar behavior compared to low temperatures.

Fig. 7 shows the normalized CO₂ methanation kinetic rates as a function of the partial pressure. The normalized kinetic rate by Xu and Froment equals the normalized kinetic rate by Ducamp and has a reaction order of unity. The model of Koschany et al. has reaction orders larger than zero over the whole range of partial pressures, as the rate monotonously decreases. The model of Rönsch et al. does not consider CO₂ methanation explicitly.

3.3 Kinetic Model Adaption

The CO methanation rate by Rönsch et al. and the CO₂ methanation rate by Koschany show suitable rates over a wide range of temperatures and no or only slightly negative reaction orders. Therefore, these two kinetic rate equations (Eq. 15, Eq. 19) are combined with the WGS reaction by Rönsch et al. (Eq. 16) to conduct the CFD simulations. It was not possible to reach sufficient CO₂ conversion by only considering the CO₂ methanation as a combination of CO methanation and the WGS reaction. The kinetic model of Kopyscinski et al. is suitable for the CO-methanation, as well. However, due to the lack of a reverse CO methanation reaction, the model of Rönsch et al. was chosen.

Inhibition of CO₂ methanation by the presence of CO, as discussed in the literature, is not considered ³⁵. This inhibition is assumed to be not present, due to the strong pore diffusion limitation (). Therefore, the local CO content within the catalyst pore is low over the major part of the pore.

The implementation of the unmodified kinetic models leads to too low reaction rates in the methanation reactor. Therefore, the kinetic rates are fitted with a constant factor s_F.

(14)

where r_int is the internal reaction rate.

The s_F factors are chosen in a manner that the maximum temperature in the reactor is slightly above the experimentally determined temperature and the CH₄ conversion rates are reasonable.

Therefore, the CO methanation rate and the WGS rate by Rönsch et al. are multiplied by a factor of s_F,Rönsch = 3.5, and the CO₂ methanation rate from the model by Koschany et al. by a factor of s_F,Koschany = 4.

The equilibrium constants are calculated with Eqs. (17), (18) ³⁴, and (20) ¹². The kinetic parameters for the CO methanation and the WGS reaction are listed in Tab. 3, and the kinetic parameters for the CO₂ methanation in Tab. 4. The temperature dependence of the adsorption constants A and the rate constants k are given by the Arrhenius and the Van't Hoff equations, as shown in the original works ⁹, ¹².

(15)

(16)

(17)

(18)

(19)

(20)

where A_i are adsorption constants and R is the universal gas constant.