Modeling of CO₂/CH₄ gas mixture permeation and CO₂ induced plasticization through an asymmetric cellulose acetate membrane

Abstract

The target of this study is derivation of a mathematical model for permeability and effective diffusivity of mixed gases in glassy polymeric membranes in the presence of plasticization. Diffusion coefficients for all components were assumed to be a function of the plasticizing component. The partial immobilization model was employed to determine the fraction of mobile sorbed gases. The model accurately predicted the mixed gas permeation behavior of CO₂ as a plasticizer and CH₄ as a second component through the asymmetric cellulose acetate membrane in the presence of plasticization. The model parameters were calculated by fitting experimental data from the literature. The plasticization parameter (β) decreased for both CO₂ and CH₄ with increasing fraction of CH₄ in the feed. This means that plasticization of a glassy polymer was suppressed. This decrease was caused by competitive sorption between CO₂ and CH₄. Indeed, CH₄ in the feed acts as an anti-plasticizer. In addition, permeances of the feed gas components were decreased in comparison to those of the pure gases, which might be attributed to reduction of sorption and the occupation of Langmuir sites by the second component. Moreover, the immobilization factor (F) for CO₂ and CH₄ decreased with increase in CH₄ fraction due to reduction of plasticization. D_eff/l for pure CO₂ was significantly pressure dependent. However with increasing fraction of CH₄ in the feed, this dependency almost disappeared. Finally, the model predicted the decreasing trend of the separation factor for CO₂/CH₄ mixed gases with pressure accurately. Therefore, the presented model is capable of being a useful tool with which to enhance our knowledge related with permeation behavior of mixed gas systems through glassy polymeric membranes in the presence of plasticization.

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1. Introduction

Membranes with different organic and inorganic materials, such as polymers, carbon molecular sieves (CMS), zeolites, ceramics, and graphene sheets, are widely used in gas separation processes.^1–5 In addition, nano-structure materials, such as silica nanoparticles, metal oxide nanoparticles, carbon nanotubes (CNTs), and metal–organic frameworks (MOFs), have also been used to produce mixed matrix membranes (MMMs) for gas separation.¹ However, polymers are the dominant membrane materials used in natural gas separation processes.^3,4 In order to preferentially remove carbon dioxide (CO₂), glassy polymeric membranes are often applied rather than rubbery polymeric membranes because of their high CO₂/CH₄ or CO₂/N₂ selectivity.^4,6–9 It is well known that most of the rubbery polymers exhibit high permeability but at the cost of low selectivity.⁹ Although some types of glassy membranes have a good performance in CO₂ separation, at high-pressure CO₂/CH₄ or CO₂/N₂ separation, the performance of these membranes can be hindered by the plasticization phenomenon.^7–11 The polymeric matrix usually swells due to the highly sorbed CO₂, as a condensable gas. Then, interaction between adjacent segments of the polymer chain is reduced, and the glass transition temperature is depressed.^8,10,11 Therefore, it will cause an increase in segmental mobility and free volume of the polymeric matrix.^8,10,11 Furthermore, diffusivity and permeability of sorbed gases increase with pressure, and therefore eventually, the membrane loses its selectivity.^8,10 In the permeability vs. pressure curves, the permeability goes through a minimum, which is known as the “plasticization pressure”. This is the minimum CO₂ pressure necessary to induce plasticization.^10,12–16 In some cases in which the membrane has a thin skin (especially asymmetric membranes with a thin skin layer), there is no minimum pressure and permeance increases with pressure monotonously,^11,17 which means plasticization pressure decreases with decrease in thickness.¹⁸

At higher pressures than the plasticization pressure, permeability of pure CO₂ in a glassy membrane increases with pressure, whereas for an inert gas, such as N₂ or CH₄, permeability decreases with pressure; therefore, ideal gas selectivity increases with pressure.¹⁹ The mixed gas permeation behavior in glassy polymers, especially in CO₂/CH₄ or CO₂/N₂ separation, is significantly different compared with that in pure gas separation.^19–21 For example, mixed gas CO₂/CH₄ selectivity for polyimide (6FDA-mPD) was reported to be about 4 at a feed pressure of 17.5 atm, whereas the ideal selectivity for this was observed to be about 60.²¹ Moreover, permeation behavior of Matrimid membranes under a mixed gas of CO₂/CH₄ showed that the selectivity of the membranes plasticized by CO₂ dramatically decreased with pressure. Normally, CO₂ swelling and plasticization cause permeability of CH₄ versus pressure to be increased more than CO₂ permeability; therefore, in contrast to that for pure gas, the selectivity of the mixture of gases decreases rapidly.^22,23 Moreover, Donohue et al.²⁰ reported that unlike the ideal selectivity, the mixed gas selectivity of a CO₂/CH₄ mixture by a cellulose acetate membrane decreased with pressure due to plasticization. They interpreted that the presence of CH₄ not only reduces CO₂ solubility, but also lowers the diffusivity of CO₂ for a given partial pressure, which resulted in lowering of the CO₂ permeability in the presence of CH₄. On the other hand, the presence of CO₂ in the feed decreased CH₄ solubility in reference to pure CH₄, whereas due to the membrane plasticization by dissolved CO₂, diffusivity of CH₄ increased as well. They reported that the enhancement in CH₄ diffusivity was much greater than the decrease in solubility, which finally led to an increase in CH₄ permeability.

Furthermore, Visser et al.¹⁹ demonstrated that introducing N₂ or CH₄ to the CO₂ feed mixture apparently suppressed plasticization of asymmetric a polyethersulfone (PES)/polyimide (PI) hollow fiber membrane. Its effect was more pronounced at higher concentrations of inert gases. By introducing N₂ or CH₄ as a second component to the feed, due to lower sorption of CO₂, permeances of CO₂ were fewer than those of the pure gas one.

As a first study, Koros et al.²⁴ developed a model for permeation of mixed gases in polymeric membranes based on the Dual Mode Sorption (DMS) model of a mixed gas system,²³ but they did not consider plasticization. As a result, their model could not predict the permeation of mixed gases when a plasticization phenomenon occurs.¹⁹ Lee et al.²⁶ developed a model for permeation of mixed gases in polymeric membranes in the case of plasticization. Based on their model, the diffusion coefficient of each component is dependent on all of the other components and all of the sorbed gases are considered as mobile molecules.

Although the permeation behavior of mixed gases is significantly different from pure gases due to competitive sorption, less attention has been given to mixed gas permeation. Therefore, a simple and comprehensive model is required to simulate this behavior.

In a previous study,²⁷ a mathematical model for permeation of mixed gases in glassy polymeric membranes in presence of plasticization was developed by us. In this study, a model for effective diffusivity for a gas component in a mixed gas feed based upon Fick's first law was developed. Then, permeation behavior of CO₂/CH₄ mixed gas in an asymmetric cellulose acetate membrane¹⁸ was studied and parameters of the model were calculated and discussed.

2. Theory and modeling

2.1. Sorption

Based on the concept that polymers in a glassy state contain some microvoids or “holes” throughout the polymer matrix, two mechanisms of sorption occur in these polymers: (i) ordinary dissolution based on Henry's law and (ii) “hole-filling” according to Langmuir theory. This type of sorption is known as the Dual Mode Sorption (DMS) model.^28,29 The equilibrium isotherm for a pure gas “A” is expressed as follows:where C is the gas concentration in the polymer (cm³ (STP) per cm³ polymer), C_D is Henry's law solubility, C_H is Langmuir solubility, k_D is Henry's law solubility coefficient (cm³ (STP) per cm³ polymer per kPa), C′_H is the hole saturation constant (cm³ (STP) per cm³ polymer), b is the hole affinity constant (kPa⁻¹), which represents the ratio of the rate constants of gas adsorption and desorption in the microvoids, and p is pressure (kPa). In eqn (1), the first term represents ordinary dissolution, whereas the second term represents sorption in microvoids or holes.²⁷ Solubility of gas “A” in polymeric membranes is defined as follows:^30,31

S_A = C_A/p_A (2)

Koros et al.²⁵ extended the DMS model for mixed gas component systems to consider a competitive sorption effect. Based on their assumption, components of a gas mixture in the Henry region of a glassy polymer are sorbed independent of each other, whereas the gas molecules in the mixture compete for sorption on Langmuir sites. The concentration of gas “A” of a binary mixture is obtained by²⁵

Similarly, the sorption isotherm for component “B” is given by²⁵

The total sorbed gas concentration is

Obviously, when either p_A or p_B approaches zero, eqn (5) reduces to the pure gas relation, i.e. eqn (1).

2.2. Permeation

Based on the partial immobilization model, the total concentration of sorbed gas in glassy polymers is divided into a mobile part with a diffusion coefficient D and concentration C_m, whereas the balance (C − C_m) is totally immobilized. This means that all the gas dissolved in the Henry region is mobile, whereas for the Langmuir sites, a fraction (F) of the adsorbed gas molecules, is mobile and the remainder (1 − F) is immobile.^32,33 F is usually called the immobilization factor, which depends on the nature of the penetrant–polymer system as well as the system temperature.^18,34 This factor represents the ratio of the diffusivity through the microvoids to that through the polymeric matrix .³³

Then, the flux (N) of component “A” of a two component system is expressed as follows:^32,35,36

where²⁷

Diffusivity of component “A” in the presence of plasticization is given by^27,35

D_A(C_mA) = D_A0 exp(β_AC_mA) (8)

where D_A0 is the diffusion coefficient of pure gas in the limit C_mA → 0, and β_A is an empirical constant that depends on the nature of the penetrant–polymer system, temperature and membrane thickness, which is known as the plasticization parameter, indicating the penetrant plasticizing capability.^18,34,35

Then, eqn (6)–(8) yield the following expression for the flux of penetrant gas in glassy polymers:

Then, effective diffusivity from eqn (9) is calculated as follows:

Furthermore, under steady state conditions, the permeability “A” (P_A) can be defined as follows:^37,38

where subscripts 2 and 1 represent the upstream and downstream conditions, respectively.

By substituting eqn (8) in eqn (6) and then integrating and combining it with eqn (11), permeability of component “A” in a binary mixture through glassy polymers in the presence of plasticization is expressed as follows:²⁷

It should be noted that negligible downstream pressure of component “A” (p_A1 = 0) was assumed in the derivation of eqn (12).

For component “B” in a binary mixture, diffusivity is given by eqn (13):²⁷

D_B(C_mA) = D_B0 exp(β_BC_mA) (13)

Moreover, concentration of the mobile part of component “B” is calculated as follows:

Combining eqn (6), (13), (14) and (11) yielded the permeability of component “B” as follows:²⁷

Downstream pressure of components was assumed to be zero.

Moreover, by writing eqn (9) for component “B”, effective diffusivity for component “B” is given as follows:

It can be noted that permeability for component “i” in polymeric membranes is defined as the product of diffusivity and solubility:^31,39,40

P_i = D_ave,i × S_i (17)

For a two component system, the selectivity of the membrane is defined as the ratio of their permeability coefficients, which is given by eqn (18):^39,41

3. Results and discussion

3.1. Model validation and mathematical procedure

To validate the model, the predictions of the proposed model were compared against the permeation of CO₂/CH₄ mixtures with different compositions in a cellulose acetate membrane.²⁰ The mathematical procedure to predict permeation of a mixed gas in a cellulose acetate membrane was as follows:

(i) Calculation of parameters of DMS model (eqn (1)) for pure CO₂ and CH₄ by fitting this equation using sorption experimental data.

(ii) Parameters obtained from step (i) were combined with eqn (12) and fitted against experimental data for permeation of CO₂ to compute parameters β, F and D₀/l for CO₂.

(iii) Parameters obtained from steps (i) and (ii) (F_CO2) were combined with eqn (15) and fitted against experimental data for permeation of CH₄ to compute parameters β, F and D₀/l for CH₄.

It should be noted that the parameters of the DMS model and non-linear proposed models for permeation of CO₂ and CH₄ were obtained by a least squares regression technique.

3.2. Sorption

In order to study permeation of a mixed gas system in a cellulose acetate membrane, it was first necessary to estimate the sorption isotherm of the pure gases in the membrane. Then, by using the results of pure gas sorption, combined with the dual mode sorption model for mixed gas systems (eqn (3) and (4)), the mixed gas sorption in the membrane was predicted. Using experimental data, parameters of the dual sorption model (eqn (1)) for pure CO₂ and CH₄ were calculated, as shown in Table 1.²⁰ Fig. 1a and b show the solubility of pure CO₂ and pure CH₄ and also the concentration of CO₂ and CH₄ in the mixed gas (CO₂/CH₄) with different compositions versus pressure in the membrane. For the sorption of gases in the membrane, at lower pressures, solubility severely decreases; however, for higher pressures, due to occupation of Langmuir sites, decrease in solubility slope occurred. For the sorption of CO₂ in the presence of CH₄, CO₂ was sorbed in the Henry region of the glassy membrane independent of the second component, whereas in Langmuir sites, a competitive sorption occurred and a part of these sites were occupied by CH₄ molecules. Then, the sorption of CH₄ in Langmuir sites caused less CO₂ to be sorbed at these sites at a specific pressure, hence the solubility of CO₂ sorbed in the presence of CH₄ in the polymer decreased relative to that of pure CO₂. By increasing the content of CH₄ in the feed, more CH₄ was sorbed in Langmuir sites, and then more reduction of CO₂ sorption in polymer was observed. Moreover, competitive sorption occurred for CH₄ and with increase in CO₂ fraction in the feed, the solubility of CH₄ was decreased.

Table 1 DMS parameters for CO₂ and CH₄ in cellulose acetate membrane²⁰

Component	kD (cm^3 (STP) per cm^3 per kPa)	C′H (cm^3 (STP) per cm^3)	b (kPa^−1)
CO2	1.43 × 10^−2	37.29	1.32 × 10^−3
CH4	1.51 × 10^−3	37	2.22 × 10^−4

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Fig. 1 CO₂ and CH₄ sorption isotherms in cellulose acetate membrane as a function of pressure.

3.3. Permeation

Fig. 2 shows the effect of feed composition on the CO₂ permeances in a cellulose acetate membrane in binary mixtures of CO₂/CH₄. This figure compares the experimental data of Donohue et al.²⁰ with the predictions of the proposed model, calculated by eqn (12), using the parameters of CO₂ and CH₄ shown in Tables 1 and 2. In the case of pure CO₂, permeance increases with pressure due to plasticization. This trend was due to the thin skin of asymmetric membranes, in which permeance increases with pressure monotonously.^19,20 The presence of CH₄ in the feed decreases sorption of CO₂ due to competitive sorption, and this decrease in solubility lowers the diffusivity of CO₂ for a given pressure and also suppresses plasticization, consequently reducing the CO₂ permeance.²⁰ This depression trend was increased with increase in CH₄ fraction. Moreover, permeance of CO₂ with different amounts of CH₄ in the feed was increased with pressure at a lower slope than that for pure CO₂, which means that by introducing CH₄ into the feed, CO₂-induced plasticization was suppressed dramatically. It should be mentioned that although solubility decreased with pressure, diffusivity increased due to plasticization. This increase overcomes the decrease in solubility, therefore permeance increased with pressure for all cases. It was apparent that the model predictions showed good agreement with respect to the experimental points.

Fig. 2 CO₂ permeance in cellulose acetate membrane as a function of pressure with different compositions of the feed.²⁰

Table 2 Parameters of eqn (12) for permeation CO₂ in a cellulose acetate membrane

Composition	βCO2	FCO2	D0,CO2/l	R^2
Pure CO2	0.086	0.06	0.00253528	0.987
70.6% CO2	0.055	0.039	0.0027093	0.967
30.6% CO2	0.036	0.028	0.00293148	0.842

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As mentioned above, parameters of eqn (12) (β, D₀/l and F) were calculated using sorption parameters of pure CO₂ and CH₄ and also the experimental data²⁰ for CO₂ permeances with different fractions in the feed in a cellulose acetate membrane, as shown in Table 2. As can be seen, these parameters are strongly dependent on the feed composition. It can be noted that the experimental data are for an asymmetric membrane and the reported results for permeation are permeability per thickness (permeance in GPU, in which 1 GPU = 10⁻⁶ cm³ (STP)/(cm² s cmHg)). Then, D₀/l and D/l were determined in the present study.

For β_CO2 with increasing CH₄ fraction in the feed, competitive sorption caused less CO₂ to be sorbed in the polymer at a specific pressure, hence β_CO2 showed decreased plasticization ability. Moreover, with increasing fraction of CH₄ in the feed, sorption of CO₂ in the membrane and also plasticization decreases.

The diffusion coefficient of CO₂ at zero penetrant concentration per unit membrane thickness (D_0,CO2/l) decreased with CO₂ fraction. With increasing CH₄ fraction in the feed and decreasing plasticization, D₀/l for CO₂ increases. This trend is consistent with the work of Duthie et al.³⁴ and Okamoto et al.,⁴² which reported that D₀ increases with decreasing plasticization.

The immobilization factor for CO₂ (F_CO2) was decreased with increasing CH₄ (decreasing CO₂ fraction) in the feed, as shown in Table 2. This means that the diffusivity of CO₂ through the microvoids (Langmuir sites) decreased in comparison to the diffusion through the polymer matrix (Henry region). This occurred due to the occupation of a portion of the Langmuir sites by CH₄ molecules, therefore CO₂ molecules had fewer sites for sorption, whereas sorption of CO₂ through the polymer matrix was independent of CH₄. The second major reason for the decrease in F_CO2 was reduction of the mobility of CO₂ molecules due to suppression of plasticization.

The effect of feed composition on the CH₄ permeances in a cellulose acetate membrane in a binary mixture of CO₂/CH₄ is shown in Fig. 3. This figure compares the experimental data of Donohue et al.²⁰ with the predictions of the proposed model, calculated by eqn (15), using the parameters of CO₂ and CH₄ shown in Tables 1–3. Permeance of CH₄ with different fractions of CO₂ increased with pressure, and at higher fraction of CH₄, the increasing trend had a lower slope. This resulted from higher sorption of CH₄ in the membrane, which led to decrease in plasticization. Moreover, at specific pressures, permeances of higher fractions of CH₄ in the feed, due to higher sorption of CH₄, were higher than those of the lower fractions.

Fig. 3 CH₄ permeance in cellulose acetate membrane as a function of pressure with different compositions of the feed.²⁰

Table 3 Parameters of eqn (15) for permeation of CH₄ in a cellulose acetate membrane

Composition	βCH4	FCH4	D0,CH4/l	R^2
29.4% CH4	0.043	0.24	0.0004369	0.978
69.4% CH4	0.036	0.13	0.0007152	0.942

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Parameters of eqn (15) (β, D₀/l and F) were calculated using sorption parameters of pure CO₂ and CH₄, immobilization factor for CO₂ (F_CO2) and also the experimental data²⁰ for CH₄ permeances with different fractions in the feed in a cellulose acetate membrane, as shown in Table 3. As seen in Table 3, similar to the plasticization ability of CO₂, β_CH4 decreased as its fraction increased due to suppression in plasticization. D_0,CH4/l also increases with CH₄ fraction.

Moreover, the immobilization factor for CH₄ (F_CH4) decreases with fraction of CH₄ in the feed. In this case, with an increase in CH₄ fraction in the feed, Langmuir sites occupied by CH₄ was enhanced; however, plasticization and then mobility of CH₄ molecules reduced. This in turn caused a decrease in F_CH4.

3.4. Diffusion

Fig. 4 and 5, respectively, illustrate the estimated effective diffusivity per unit membrane thickness (D_eff/l) versus pressure for CO₂ and CH₄ derived from eqn (10) and (16) utilizing parameters from Tables 1–3. Although D₀/l increased as a function of CH₄ fraction, variation of effective diffusivity for CO₂ with pressure is rapidly overwhelmed by the higher degree of plasticization at lower fractions of CH₄, so that the effective diffusivity at higher pressures decreases significantly with increasing CH₄ fraction. For pure CO₂, stronger dependency of D_eff/l on pressure was observed and D_eff/l was increased with increasing pressure due to plasticization. In general, for feeds containing different fractions of CH₄, the effect of plasticization decreased and the influence of pressure on D_eff/l for CO₂ became negligible in comparison to the case with pure CO₂. By increasing CH₄ in the feed, because of less sorption of CO₂ and decrease in plasticization, dependency of D_eff/l for CO₂ on pressure, was more reduced. As can be seen in Fig. 5, D_eff/l for CH₄ in the feed was enhanced as a result of an increase in pressure due to plasticization. Moreover, for feeds with higher fractions of CH₄, because of higher sorption of CH₄, D_eff/l was higher than in the cases with lower fractions.

Fig. 4 Effective diffusivity per unit membrane thickness (D_eff/l) for CO₂ versus pressure.

Fig. 5 Effective diffusivity per unit membrane thickness (D_eff/l) for CH₄ versus pressure.

3.5. Separation factor

Based on eqn (18), the separation factor for a binary gas mixture is the ratio of the permeability coefficients of the components. In Fig. 6, the experimental separation factor for CO₂/CH₄ was compared with the predictions of the model using permeances of CO₂ and CH₄ calculated in Section 3.3. According to this figure, the separation factor of the mixed CO₂/CH₄ with different compositions decreased with increasing pressure and the model showed this trend accurately. As is observed in Fig. 2 and 3, permeances of CO₂ and CH₄ increased with pressure, but increase in CH₄ permeances was more than that of CO₂, and the separation factor decreased with pressure. Moreover, the separation factor for the mixed gas feed at specific pressures was decreased with increasing CH₄ fraction. The presence of CH₄ led to CO₂ permeance reduction with increasing CH₄ fraction in the feed at a specific pressure. On the other hand, permeance of CH₄ increased with increasing CH₄ fraction in the feed. Therefore, with increase in CH₄ fraction at a specific pressure, the CO₂/CH₄ separation factor decreased.

Fig. 6 Separation factor for CO₂/CH₄ mixed gas with different compositions versus pressure.

4. Conclusion

In the current study, a mathematical model was developed to predict permeation behavior of mixed gases through glassy polymeric membranes in the presence of plasticization. Parameters of the model (β, F, D₀/l) were obtained using the experimental data for permeation of a CO₂/CH₄ mixture feed with different compositions through an asymmetric cellulose acetate membrane. It was shown that these parameters were strongly dependent on the feed composition. The observations revealed that β and F for CO₂ and CH₄ declined with increasing CH₄ fraction in the feed due to reduction in plasticization. D₀/l for CO₂ and CH₄ rose with increasing CH₄ fraction. D_eff/l for pure CO₂ was significantly pressure dependent; however with increasing fraction of CH₄ in the feed, this dependency almost disappeared. Moreover, D_eff/l for CH₄ increased with pressure due to plasticization. Separation factor for CO₂/CH₄ with different fractions decreased with pressure, and the model showed this trend accurately. In addition, with increase in CH₄ fraction in the feed, the separation factor decreased at a specified pressure. In conclusion, the presence of the second component along with CO₂ resulted in reduction of sorption of CO₂ due to competitive sorption, which eventually led to decrease in plasticization. The presented model was capable of giving a useful tool to enhance our knowledge related to permeation behavior of mixed gas systems through glassy polymeric membranes in the presence of plasticization.

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