Estimation of CO2 Separation Performances through CHA-Type Zeolite Membranes Using Molecular Simulation

Chabazite (CHA)-type zeolite membranes are a potential material for CO2 separations because of their small pore aperture, large pore volume, and low aluminum content. In this study, the permeation and separation properties were evaluated using a molecular simulation technique with a focus on improving the CO2 separation performance. The adsorption isotherms of CO2 and CH4 on CHA-type zeolite with Si/Al = 18.2 were predicted by grand canonical Monte Carlo, and the diffusivities in zeolite micropores were simulated by molecular dynamics. The CO2 separation performance of the CHA-type zeolite membrane was estimated by a Maxwell–Stefan equation, accounting for mass transfer through the support tube. The results indicated that the permeances of CO2 and CH4 were influenced mainly by the porosity of the support, with the CO2 permeance reduced due to preferential adsorption with increasing pressure drop. In contrast, it was important for estimation of the CH4 permeance to predict the amounts of adsorbed CH4. Using molecular simulation and the Maxwell–Stefan equation is shown to be a useful technique for estimating the permeation properties of zeolite membranes, although some problems such as predicting accurate adsorption terms remain.


Introduction
Zeolite membranes separate via molecular sieving and selective adsorption, which makes them a promising candidate technology for energy-efficient separations. Geus et al. successfully formed polycrystalline MFI-type zeolite layer on a porous substrate and investigated the permeation properties of hydrocarbons [1][2][3][4][5]. In the 1990s, Kita and coworkers developed a commercially available LTA-type zeolite membrane and applied it to dehydration of ethanol [6][7][8].
There are many reports about the permeation and separation mechanisms in zeolite membranes [3][4][5][14][15][16]. Since CO 2 and hydrocarbon molecules adsorb onto zeolite strongly, the adsorbed molecules move to a neighboring adsorption site according to the concentration gradient across the membrane. The permeation phenomenon due to the surface diffusion is quantitatively described by the Maxwell-Stefan equation [3][4][5]. Many studies have predicted the permeation and separation properties of zeolite membranes from the adsorption and diffusion properties of single gases. Bakker et al. checked that the equation is suitable for expression of the single gas permeation properties through a silicalite-1 membrane [4]. Additionally, van den Broeke et al. applied the equation to separation of binary hydrocarbon mixtures [5], and the gas permeation properties for binary mixtures could be described using the adsorption and diffusion parameters obtained by single component gases.
The permeation and separation properties of zeolite membranes are explained by adsorption of molecules on zeolite and diffusion in zeolite channels. Both the adsorption and diffusion properties can be predicted by molecular simulation such as grand canonical Monte Carlo and molecular dynamics, respectively [29][30][31][32][33][34]. Vujic et al. reported the potential parameters applicable to many zeolites [33].
In this study, the CO 2 separation performances of CHA-type zeolite membranes with Si/Al = 18 were predicted using the molecular simulation technique and Maxwell-Stefan equation to understand the permeation behavior in CHA-type zeolite membranes.

Molecular Simulation
The interaction between adsorbate and adsorbent atoms is described as the sum of interactions between bonded and nonbonded atoms as [33]: The interaction between bonded atoms is calculated as the sum of bond-stretching and angle-bending as: where k b and k θ are the force constants for bond-stretching and angle-bending, respectively. The interaction between nonbonded atom pair is calculated as the sum of van der Waals and coulomb interactions as: where the depth of interaction φ ij and zero-interaction distance σ ij for the pair of different atoms are calculated as: When two atoms are in the same structure and separated by three covalent bonds (known as a 1-4 interaction), the interaction is treated as a nonbonded interaction with scaling factor of 0.5. Nonbonded interactions are ignored for directly bonded atoms (1-2 interaction) and two atoms separated by two bonds (1-3 interaction) since they are included in the bond-stretching and angle-bending interactions.

Gas permeation through Zeolite Layer
Zeolite membranes are often prepared on porous supports, as shown in Figure 1. The molecules are transferred by the concentration gradient across the membrane.
where the depth of interaction ϕij and zero-interaction distance σij for the pair of different atoms are calculated as: When two atoms are in the same structure and separated by three covalent bonds (known as a 1-4 interaction), the interaction is treated as a nonbonded interaction with scaling factor of 0.5. Nonbonded interactions are ignored for directly bonded atoms (1-2 interaction) and two atoms separated by two bonds (1-3 interaction) since they are included in the bond-stretching and angle-bending interactions.

Gas permeation through Zeolite Layer
Zeolite membranes are often prepared on porous supports, as shown in Figure 1. The molecules are transferred by the concentration gradient across the membrane. In the zeolite layer, molecules adsorbed on the adsorption sites within zeolite channels, and then move to a neighboring site according to the concentration gradient. The permeation flux is described as [3][4][5] The elements of matrix B are calculated by: where the mutual diffusivity can be approximated as: (12) Figure 1. Schematic illustration of concentration gradient across the zeolite membrane supported by porous substrate. C f is the concentration in the feed, C p is the concentration in the permeate, and C i is the concentration at the interface of the zeolite and support layers.
In the zeolite layer, molecules adsorbed on the adsorption sites within zeolite channels, and then move to a neighboring site according to the concentration gradient. The permeation flux is described as [3][4][5]: The elements of matrix B are calculated by: where the mutual diffusivity can be approximated as: When the adsorption isotherm is described by a Langmuir Equation (13), the elements of matrix Γ are described using Equation (14).
where δ ij = 1 for i = j, and δ ij = 0 for i = j. The adsorption and diffusion parameters are summarized in Tables 2 and 3, respectively.

Mass Transfer in the Support Tube
In the porous support tube, the overall permeation flux is: where L is the thickness of the support, and C is the concentration shown by: where ε is the porosity of the support. The diffusivity in the porous support is estimated by the Fuller equation [4,35]: where M i is the molecular mass of component i and V i is the diffusion volume of component i. The diffusion volumes of CO 2 and CH 4 were taken as 26.9 cm 3 and 25.1 cm 3 , respectively [35].

Adsorption on Zeolites
The adsorption isotherms of CO 2 and CH 4 on the CHA-type zeolite were simulated by a grand canonical Monte Carlo (GCMC) technique using software (Biovia, Materials Studio 2021 Sorption). For the GCMC simulation, fugacity was applied to the canonical ensemble, and the number and location of molecules with the lowest potential energy were calculated probabilistically. The cutoff distance of the van der Waals interaction was 1.25 nm, and the Ewald summation method was used for the integration of the coulomb interaction. The total number of Monte Carlo cycles were 10 6 , and the average of the final 10 5 steps were used as the simulation result. The fugacity was assumed to be equal to the pressure in this study since the difference between fugacity and pressure is less than 5% below 1 MPa. Figure 2 shows the atomistic models of CO 2 , CH 4 , and CHA-type zeolite. The model of the CO 2 molecule reported by Harris et al. [36] was used. This model can describe the gas-liquid coexistence curve including the critical point region. The carbon atom was connected to two oxygen atoms by chemical bonds 0.1149 nm long, and the bond-stretching was ignored (k b = 0). The original angle of O=C=O was 180 • , and the force constant was k θ = 1236 kJ mol −1 rad −2 . For CH 4 , the model reported by Siepman et al. [37] was used. The carbon atom was connected to four hydrogen atoms with bond lengths of 0.11 nm, and each H-C-H angle was 109.5 • . Although the bond-stretching and angle-bending are ignored in this model (k b = k θ = 0), the gas-liquid coexistence curve can be expressed. The crystal structure of the CHA-type zeolite was imported from the IZA zeolite database [38]. The CHA-type zeolite model with a composition of Si 91 Al 5 Na 5 O 192 was prepared by substituting Si atoms with Al atoms followed by introduction of Na + cations by GCMC simulation. Table 1 lists the non-bonding interaction parameters for CO 2 , CH 4 , and zeolite. Vujic et al. reported that the adsorption of gases such as CO 2 on CHA-type zeolite can be predicted with high accuracy by using these parameters [33].
Membranes 2023, 13, x FOR PEER REVIEW 5 of 15 The CHA-type zeolite model with a composition of Si91Al5Na5O192 was prepared by substituting Si atoms with Al atoms followed by introduction of Na + cations by GCMC simulation. Table 1 lists the non-bonding interaction parameters for CO2, CH4, and zeolite. Vujic et al. reported that the adsorption of gases such as CO2 on CHA-type zeolite can be predicted with high accuracy by using these parameters [33].

Molecule
Element

Diffusion in Zeolite
The self-diffusivities of CH4 and CO2 in CHA-type zeolite channels were also simulated by a molecular dynamic technique (Biovia, Materials Studio 2021 Forcite Plus). CH4 and CO2 molecules were adsorbed at 1 MPa by GCMC, and the molecular dynamic simulation was conducted with a time step of 2 fs. The total simulation time was 1 ns, and the mean square displacement every 10 ps was plotted against the simulation time. The selfdiffusivity was calculated using the slope by the Einstein equation. The procedure was repeated 5 times and the average value taken as the diffusivity. Figure 3 shows the adsorption isotherms of CO2 and CH4 on the CHA-type zeolite with Si/Al = 18.2 at 253-473 K. The amounts of adsorbed CO2 at 253 K increased significantly at low pressures and was 6.0 mol kg −1 at 100 kPa. At higher pressures, in contrast, the increment became small, with 7.1 mol kg −1 adsorbed at 1000 kPa. This isotherm is typical for adsorption in micropores, and the relationship is described by the Langmuir equa-

Molecule
Element

Diffusion in Zeolite
The self-diffusivities of CH 4 and CO 2 in CHA-type zeolite channels were also simulated by a molecular dynamic technique (Biovia, Materials Studio 2021 Forcite Plus). CH 4 and CO 2 molecules were adsorbed at 1 MPa by GCMC, and the molecular dynamic simulation was conducted with a time step of 2 fs. The total simulation time was 1 ns, and the mean square displacement every 10 ps was plotted against the simulation time. The self-diffusivity was calculated using the slope by the Einstein equation. The procedure was repeated 5 times and the average value taken as the diffusivity. Figure 3 shows the adsorption isotherms of CO 2 and CH 4 on the CHA-type zeolite with Si/Al = 18.2 at 253-473 K. The amounts of adsorbed CO 2 at 253 K increased significantly at low pressures and was 6.0 mol kg −1 at 100 kPa. At higher pressures, in contrast, the increment became small, with 7.1 mol kg −1 adsorbed at 1000 kPa. This isotherm is typical for adsorption in micropores, and the relationship is described by the Langmuir equation (Equation (13)). The adsorption isotherms of CO 2 became linear as temperature increased. A similar trend was observed for CH 4 . The estimated isotherms were calculated for each temperature using the simulated points by Equation (13) and are shown as lines in Figure 3. The agreement between simulated points and estimated isotherm suggests the Langmuir equation is applicable at 253-473 K. Furthermore, the adsorption isotherms of CO 2 and CH 4 on CHA-type zeolite have been reported by several groups [39,40] and our simulated isotherms agree well with their experimental data, which suggests that the potential parameters are reasonable for simulating the adsorption and diffusion behaviors of CO 2 and CH 4 for CHA-type zeolite.

Adsorption Isotherms
Membranes 2023, 13, x FOR PEER REVIEW 6 of 15 tion (Equation (13)). The adsorption isotherms of CO2 became linear as temperature increased. A similar trend was observed for CH4. The estimated isotherms were calculated for each temperature using the simulated points by Equation (13) and are shown as lines in Figure 3. The agreement between simulated points and estimated isotherm suggests the Langmuir equation is applicable at 253-473 K. Furthermore, the adsorption isotherms of CO2 and CH4 on CHA-type zeolite have been reported by several groups [39,40] and our simulated isotherms agree well with their experimental data, which suggests that the potential parameters are reasonable for simulating the adsorption and diffusion behaviors of CO2 and CH4 for CHA-type zeolite.  Figure 4 shows the effect of temperature on the adsorption amounts at saturation and Langmuir constants of CO2 and CH4. Both the saturated adsorption amounts, a, and Langmuir constants, b, decreased with increasing temperature. An Arrhenius dependence was observed, as is typical for adsorption isotherms, with the temperature dependencies described by:  (13) and Table 2.  Figure 4 shows the effect of temperature on the adsorption amounts at saturation and Langmuir constants of CO 2 and CH 4 . Both the saturated adsorption amounts, a, and Langmuir constants, b, decreased with increasing temperature. An Arrhenius dependence was observed, as is typical for adsorption isotherms, with the temperature dependencies described by: Membranes 2023, 13, x FOR PEER REVIEW 7 of 15 The pre-exponential factors and activation energies are listed in Table 2. Assuming the heat of adsorption is equal to −(Ea + Eb), the heats of adsorption for CO2 and CH4 are 23.8 kJ mol −1 and 16.6 kJ mol −1 , respectively. Maghsoudi et al. experimentally measured the heats of adsorption of CO2 and CH4 to be 21.0 kJ mol −1 and 17.1 kJ mol −1 , respectively [40], which shows good agreement with the current work and further justifies the proposed methods for simulating the adsorption and diffusion behaviors of CO2 and CH4 in CHA-type zeolite.  The pre-exponential factors and activation energies are listed in Table 2. Assuming the heat of adsorption is equal to −(E a + E b ), the heats of adsorption for CO 2 and CH 4 are 23.8 kJ mol −1 and 16.6 kJ mol −1 , respectively. Maghsoudi et al. experimentally measured the heats of adsorption of CO 2 and CH 4 to be 21.0 kJ mol −1 and 17.1 kJ mol −1 , respectively [40], which shows good agreement with the current work and further justifies the proposed methods for simulating the adsorption and diffusion behaviors of CO 2 and CH 4 in CHA-type zeolite.  Figure 5 shows the time courses in the mean square displacement of CO 2 and CH 4 at 298-473 K. Because the mean square displacements were linearly proportional to simulation time before 1 ns, longer diffusional times were not required. The diffusivities of CO 2 and CH 4 in the CHA-type zeolite were calculated as 1/6 of the slope [33], which resulted in temperature dependencies as reported in Figure 6. The diffusivities of CO 2 and CH 4 at 298 K were 3.9 × 10 −10 m 2 s −1 and 1.2 × 10 −11 m 2 s −1 , respectively. The pore diameter of the CHA-type zeolite is 0.38 nm [38], which is identical to the molecular diameter of CH 4 (0.38 nm [41]). In contrast, the molecular diameter of CO 2 (0.33 nm [41]) is smaller than the pore diameter, which results in molecular sieving behavior with a CO 2 diffusivity nearly an order of magnitude higher than CH 4 .  Figure 5 shows the time courses in the mean square displacement of CO2 and CH4 298-473 K. Because the mean square displacements were linearly proportional to simul tion time before 1 ns, longer diffusional times were not required. The diffusivities of CO and CH4 in the CHA-type zeolite were calculated as 1/6 of the slope [33], which resulte in temperature dependencies as reported in Figure 6. The diffusivities of CO2 and CH4 298 K were 3.9 × 10 −10 m 2 s −1 and 1.2 × 10 −11 m 2 s −1 , respectively. The pore diameter of th CHA-type zeolite is 0.38 nm [38], which is identical to the molecular diameter of CH4 (0.3 nm [41]). In contrast, the molecular diameter of CO2 (0.33 nm [41]) is smaller than the po diameter, which results in molecular sieving behavior with a CO2 diffusivity nearly a order of magnitude higher than CH4.  CH4 in all-silica CHA-type zeolite at 300 K. When the fugacities of CO2 and CH MPa, their diffusivities were ca. 4 × 10 −10 m 2 s −1 for CO2 and 8 × 10 −11 m 2 s −1 for C similarity in diffusivity measurements with the current work suggests aluminum dium do not have a significant effect on gas diffusivity. This is considered reaso cause only one aluminum and sodium atom are incorporated per cavity for a S of 18.2, as shown in Figure 2. The effect of temperature on the diffusivity is also described by the Arrheni tion as follows:

Diffusivities
The diffusivities at infinite temperature and activation energies of CO2 and listed in Table 3. Sladek et al. [43] investigated the relationship between the d and heat of adsorption for physical and chemical adsorption species and conclu the activation energy for diffusion was 0.45 times the heat of adsorption. This c well with the current work wherein the activation energy of CO2 diffusivity is 0 the heat of adsorption.  Figure 7 shows the influence of accounting the material transfer in porous on the calculated permeation properties of CO2 and CH4 for the equimolar mixtu K. When the polycrystalline zeolite layer was not supported by a substrate (selfmembrane), the permeances of CO2 and CH4 were predicted to be 2.3 × 10 −6 and mol m −2 s −1 Pa −1 , respectively. The permeances were reduced to 6.3 × 10 −7 mol m Vujic et al. compared to the simulated CO 2 diffusivities with those obtained by experiments [33], and the simulated diffusivities were twice higher than the experimental values for high silica zeolites. Krishna et al. [42] also simulated the diffusivities of CO 2 and CH 4 in all-silica CHA-type zeolite at 300 K. When the fugacities of CO 2 and CH 4 were 1 MPa, their diffusivities were ca. 4 × 10 −10 m 2 s −1 for CO 2 and 8 × 10 −11 m 2 s −1 for CH 4 . The similarity in diffusivity measurements with the current work suggests aluminum and sodium do not have a significant effect on gas diffusivity. This is considered reasonable because only one aluminum and sodium atom are incorporated per cavity for a Si/Al ratio of 18.2, as shown in Figure 2.

CO2 Separation Performance
The effect of temperature on the diffusivity is also described by the Arrhenius equation as follows: The diffusivities at infinite temperature and activation energies of CO 2 and CH 4 are listed in Table 3. Sladek et al. [43] investigated the relationship between the diffusivity and heat of adsorption for physical and chemical adsorption species and concluded that the activation energy for diffusion was 0.45 times the heat of adsorption. This compares well with the current work wherein the activation energy of CO 2 diffusivity is 0.48 times the heat of adsorption.  Figure 7 shows the influence of accounting the material transfer in porous support on the calculated permeation properties of CO 2 and CH 4 for the equimolar mixture at 323 K. When the polycrystalline zeolite layer was not supported by a substrate (self-standing membrane), the permeances of CO 2 and CH 4 were predicted to be 2.3 × 10 −6 and 1.7 × 10 −8 mol m −2 s −1 Pa −1 , respectively. The permeances were reduced to 6.3 × 10 −7 mol m −2 s −1 Pa −1 for CO 2 and 5.0 × 10 −9 mol m −2 s −1 Pa −1 for CH 4 by supporting with the porous support (porosity = 35% and thickness = 0.3 mm). The reduction of CO 2 permeance could be explained by the porosity of the support tube and pressure drop across the support (3.9 kPa). The calculated permeance of CO 2 was almost identical to the experimental data [28]. However, the calculated CH 4 permeance was higher than the experimental value. Since the amounts of adsorbed CO 2 and CH 4 were calculated using the extended Langmuir Equation (13) in this study, it is considered that the concentration gradient of CH 4 across the polycrystalline zeolite layer was estimated to be high. As a result, the higher CH 4 permeance was obtained compared to the experiment.

CO 2 Separation Performance
Membranes 2023, 13, x FOR PEER REVIEW for CO2 and 5.0 × 10 −9 mol m −2 s −1 Pa −1 for CH4 by supporting with the porous supp rosity = 35% and thickness = 0.3 mm). The reduction of CO2 permeance could be e by the porosity of the support tube and pressure drop across the support (3.9 k calculated permeance of CO2 was almost identical to the experimental data [28]. H the calculated CH4 permeance was higher than the experimental value. Since the of adsorbed CO2 and CH4 were calculated using the extended Langmuir Equatio this study, it is considered that the concentration gradient of CH4 across the pol line zeolite layer was estimated to be high. As a result, the higher CH4 permea obtained compared to the experiment.  [28]. The permeation properties were calculated at CO tration = 50 vol%, total pressure = 300 kPa, and temperature = 323 K. The thickness of zeo was 5 μm, and the pore size, porosity and thickness of the support tube were 150 nm, 35% mm [24]. Figure 8 shows the effect of temperature on the estimated permeation prop CO2 and CH4 for an equimolar mixture. The permeances of CO2 and CH4 at 253 3.3 × 10 −7 and 1.8 × 10 −9 mol m −2 s −1 Pa −1 , respectively, with a resultant CO2/CH4 pe ratio of 190. The CO2 permeance increased with increasing temperature until re maximum at 323 K and then decreasing with further rising temperatures. As a re CO2 permeance decreased to 1.5 × 10 −7 mol m −2 s −1 Pa −1 at 473 K, with the permea also decreasing to 38. Notably, the simulated permeances followed similar trend experimental data and the permeances at 473 K were nearly identical. This conv is because the effect of preferential adsorption was marginal at 473 K compared temperatures, which suggests accurate prediction of permeation properties requi rate estimates of adsorption amounts.  [28]. The permeation properties were calculated at CO 2 concentration = 50 vol%, total pressure = 300 kPa, and temperature = 323 K. The thickness of zeolite layer was 5 µm, and the pore size, porosity and thickness of the support tube were 150 nm, 35% and 0.3 mm [24]. Figure 8 shows the effect of temperature on the estimated permeation properties of CO 2 and CH 4 for an equimolar mixture. The permeances of CO 2 and CH 4 at 253 K were 3.3 × 10 −7 and 1.8 × 10 −9 mol m −2 s −1 Pa −1 , respectively, with a resultant CO 2 /CH 4 permeance ratio of 190. The CO 2 permeance increased with increasing temperature until reaching a maximum at 323 K and then decreasing with further rising temperatures. As a result, the CO 2 permeance decreased to 1.5 × 10 −7 mol m −2 s −1 Pa −1 at 473 K, with the permeance ratio also decreasing to 38. Notably, the simulated permeances followed similar trends as the experimental data and the permeances at 473 K were nearly identical. This convergence is because the effect of preferential adsorption was marginal at 473 K compared to lower temperatures, which suggests accurate prediction of permeation properties requires accurate estimates of adsorption amounts. Figure 8. Effect of temperature on the permeation properties of the CHA-type zeolite membrane with Si/Al = 18.2. The permeation properties were calculated at CO2 concentration = 50 vol% and total pressure = 300 kPa, experimental data were taken from [28]. Figure 9 shows the influence of the CO2 concentration on the permeation properties of CO2 and CH4 at 303 K. The pure gas CO2 permeance was estimated to be 3.8 × 10 −7 mol m −2 s −1 Pa −1 and slightly increased with decreasing CO2 concentration until around 40%. However, below 40% CO2 the CO2 permeance increased significantly. This was due to a relative change in the permeance versus partial pressure difference between feed and permeate streams. At 40% CO2, the partial pressures of CO2 on the feed and permeate sides were 120 kPa and 96 kPa, respectively, with a permeate flux of 1.5 × 10 −2 mol m −2 s −1 . At 30% CO2 concentration, the partial pressures and permeate flux were 90 kPa, 84 kPa, and 6.0 × 10 −3 mol m −2 s −1 , respectively. This means the relative permeate flux at 30% CO2 compared to 40% CO2 was 1/2.5, whereas the partial pressure difference was 1/4. As a result, the CO2 permeance increased below 40% CO2. For all conditions, the estimated CH4 permeance was higher than the experimental data, as discussed in Figures 7 and 8, which is the cause of the lower CO2/CH4 permeance ratio. Figure 9. Influence of the CO2 concentration on the permeation properties of the CHA-type zeolite membrane with Si/Al = 18.2 at 303 K. The permeation properties were calculated at the total pressure of 300 kPa, and experimental data were taken from [28]. The permeation properties were calculated at CO 2 concentration = 50 vol% and total pressure = 300 kPa, experimental data were taken from [28]. Figure 9 shows the influence of the CO 2 concentration on the permeation properties of CO 2 and CH 4 at 303 K. The pure gas CO 2 permeance was estimated to be 3.8 × 10 −7 mol m −2 s −1 Pa −1 and slightly increased with decreasing CO 2 concentration until around 40%. However, below 40% CO 2 the CO 2 permeance increased significantly. This was due to a relative change in the permeance versus partial pressure difference between feed and permeate streams. At 40% CO 2 , the partial pressures of CO 2 on the feed and permeate sides were 120 kPa and 96 kPa, respectively, with a permeate flux of 1.5 × 10 −2 mol m −2 s −1 . At 30% CO 2 concentration, the partial pressures and permeate flux were 90 kPa, 84 kPa, and 6.0 × 10 −3 mol m −2 s −1 , respectively. This means the relative permeate flux at 30% CO 2 compared to 40% CO 2 was 1/2.5, whereas the partial pressure difference was 1/4. As a result, the CO 2 permeance increased below 40% CO 2 . For all conditions, the estimated CH 4 permeance was higher than the experimental data, as discussed in Figures 7 and 8, which is the cause of the lower CO 2 /CH 4 permeance ratio.  The permeation properties were calculated at CO2 concentration = 50 vol% and total pressure = 300 kPa, experimental data were taken from [28]. Figure 9 shows the influence of the CO2 concentration on the permeation properties of CO2 and CH4 at 303 K. The pure gas CO2 permeance was estimated to be 3.8 × 10 −7 mol m −2 s −1 Pa −1 and slightly increased with decreasing CO2 concentration until around 40%. However, below 40% CO2 the CO2 permeance increased significantly. This was due to a relative change in the permeance versus partial pressure difference between feed and permeate streams. At 40% CO2, the partial pressures of CO2 on the feed and permeate sides were 120 kPa and 96 kPa, respectively, with a permeate flux of 1.5 × 10 −2 mol m −2 s −1 . At 30% CO2 concentration, the partial pressures and permeate flux were 90 kPa, 84 kPa, and 6.0 × 10 −3 mol m −2 s −1 , respectively. This means the relative permeate flux at 30% CO2 compared to 40% CO2 was 1/2.5, whereas the partial pressure difference was 1/4. As a result, the CO2 permeance increased below 40% CO2. For all conditions, the estimated CH4 permeance was higher than the experimental data, as discussed in Figures 7 and 8, which is the cause of the lower CO2/CH4 permeance ratio. Figure 9. Influence of the CO2 concentration on the permeation properties of the CHA-type zeolite membrane with Si/Al = 18.2 at 303 K. The permeation properties were calculated at the total pressure of 300 kPa, and experimental data were taken from [28]. Figure 9. Influence of the CO 2 concentration on the permeation properties of the CHA-type zeolite membrane with Si/Al = 18.2 at 303 K. The permeation properties were calculated at the total pressure of 300 kPa, and experimental data were taken from [28]. Figure 10 shows the influence of the total pressure on the permeation properties of CO 2 and CH 4 at 303 K. When the total pressure was 200 kPa, the estimated permeances of CO 2 and CH 4 were 6.8 × 10 −7 and 5.8 × 10 −9 mol m −2 s −1 Pa −1 , respectively, with a CO 2 /CH 4 permeance ratio of 120. The permeances decreased with increasing total pressure and at 1000 kPa were 2.9 × 10 −7 for CO 2 and 2.4 × 10 −9 mol m −2 s −1 Pa −1 for CH 4 . The decrease was calculated to be similar for both gases so the permeance ratio was nearly independent of total pressure. Membranes 2023, 13, x FOR PEER REVIEW 12 of 1 Figure 10 shows the influence of the total pressure on the permeation properties o CO2 and CH4 at 303 K. When the total pressure was 200 kPa, the estimated permeances o CO2 and CH4 were 6.8 × 10 −7 and 5.8 × 10 −9 mol m −2 s −1 Pa −1 , respectively, with a CO2/CH permeance ratio of 120. The permeances decreased with increasing total pressure and a 1000 kPa were 2.9 × 10 −7 for CO2 and 2.4 × 10 −9 mol m −2 s −1 Pa −1 for CH4. The decrease wa calculated to be similar for both gases so the permeance ratio was nearly independent o total pressure. Figure 10. Influence of total pressure on the permeation properties through the CHA-type zeolit membrane with Si/Al = 18.2 at 303 K. The permeation properties were calculated at a CO2 concen tration of 50 vol%, and experimental data were taken from [28].

Conclusions
In this study, the permeation and separation properties of a CHA-type zeolite mem brane were evaluated for improving the CO2 separation performance. The adsorption iso therms of CO2 and CH4 on CHA-type zeolite with Si/Al = 18.2 were predicted by gran canonical Monte Carlo, and the diffusivities in zeolite micropores were simulated by mo lecular dynamics. The CO2 separation performance of the CHA-type zeolite membran was estimated by a Maxwell-Stefan equation, accounting for mass transfer through th support tube. In this study, the influences of the support tube, temperature, CO2 concen tration, and total pressure on the permeation properties were calculated, and the est mated permeation properties were compared with experimental data [24]. The estimate CO2 permeance agreed well with the experimental results due to the inclusion of the effec of the support tube. However, the estimated CH4 permeance was slightly overestimated suggesting that better predictions of the amount of adsorbed CH4 on both the sides of th membrane must be made to obtain more accurate results.

Conflicts of Interest:
The authors declare no conflicts of interest. Figure 10. Influence of total pressure on the permeation properties through the CHA-type zeolite membrane with Si/Al = 18.2 at 303 K. The permeation properties were calculated at a CO 2 concentration of 50 vol%, and experimental data were taken from [28].

Conclusions
In this study, the permeation and separation properties of a CHA-type zeolite membrane were evaluated for improving the CO 2 separation performance. The adsorption isotherms of CO 2 and CH 4 on CHA-type zeolite with Si/Al = 18.2 were predicted by grand canonical Monte Carlo, and the diffusivities in zeolite micropores were simulated by molecular dynamics. The CO 2 separation performance of the CHA-type zeolite membrane was estimated by a Maxwell-Stefan equation, accounting for mass transfer through the support tube. In this study, the influences of the support tube, temperature, CO 2 concentration, and total pressure on the permeation properties were calculated, and the estimated permeation properties were compared with experimental data [24]. The estimated CO 2 permeance agreed well with the experimental results due to the inclusion of the effect of the support tube. However, the estimated CH 4 permeance was slightly overestimated, suggesting that better predictions of the amount of adsorbed CH 4 on both the sides of the membrane must be made to obtain more accurate results. σ Distance at zero-potential energy (m)