Phenomenological modeling and analysis of gas transport in polyimide membranes for propylene/propane separation

Abstract

Olefins and paraffins are the main building blocks for many products in the petrochemical industry. Various research studies have demonstrated the viability of polyimide membranes for high performance olefin/paraffin separation. Further advancements in this field require a thorough understanding of both sorptive and diffusive factors of permeation. This research study presents an extensive analysis on using frame of reference/bulk flow and Maxwell–Stefan models in order to elaborate on the transport and prediction of the performance in the case of propylene/propane separation using polyimide membranes. Sorption data of pure gases are utilized to calculate the sorption level of gases in a binary mixture. The contribution of kinetic and thermodynamic coupling effects (TCE) are assessed using the Maxwell–Stefan approach. Moreover, the dual-mode diffusion coefficients are evaluated and optimized for achieving higher accuracy predictions in the case of a binary gas mixture. The results reveal the significant role of thermodynamic compared to kinetic coupling effects in governing the transport properties. Overall, the Maxwell–Stephan model with the contribution of TCEs offers improved predictions compared to the frame of reference/bulk flow model. The findings highlight the inevitable role of taking into account the prominent interactions of feed components in model development for better prediction of performance and evaluation of the propylene/propane separation unit using polyimide membranes.

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1 Introduction

Gas and vapor separation processes have gained much attention in chemical and petrochemical industries due to the growing need for high purity chemicals.^1,2 Nitrogen and helium separation along with oxygen enrichment,^3–6 hydrogen separation and purification,^7–9 CO₂ removal and natural gas processing,^10–12 and hydrocarbon recovery and separation^13–15 comprise the main industrial applications of membrane-based technologies.

Separation of light olefins from paraffin gases is of paramount importance in the petrochemical industry due to their application as raw materials for a wide range of chemicals.^16–19 Compared to the prevalent separation technologies, which are highly energy-intensive, polymeric membranes can offer a low-cost and simple process alternative.^{5,13,16–18,20–22} Among the studied polymers, polyimides have attracted much attention due to their chemical resistance, thermal stability and mechanical strength in many gas and vapor separation applications.^23–27 Extensive research on the separation performance of various polymeric membranes for olefin/paraffin separation has indicated the superior characteristics of polyimides compared to the conventional polymers.^{16,18,28–32}

The separation performance in the polyimide membrane is based on the difference in sorption and diffusion of olefin and paraffin molecules. The dual mode sorption model, based on the superposition of the Langmuir model and Henry's law, has been extensively used to represent transport in glassy membranes.^33–39 This model describes permeation of binary gas mixture using sorption data of pure gases, while in binary systems the flux of each component cannot be considered unaffected by each other which was not accounted for in the dual mode model. Some researchers tried to overcome this shortcoming partly by considering the convective flux via applying the frame of reference/bulk flow model.^40–44 Accordingly, using bulk flow model the permeation of binary gas mixture can be described as a ternary system comprising two penetrants and a membrane. This approach successfully predicted permeation of binary gas mixture through some polyimide membranes while failed in some other cases. Das and Koros^44,45 analyzed the results of propylene/propane separation by 6FDA-6FpDA with both dual mode and frame of reference/bulk flow model. According to the results, calculated selectivity by dual mode model was higher than the experimental selectivity of binary gas mixture while the results of frame of reference/bulk flow model were in good agreement with them. In contrast, frame of reference/bulk flow model was not able to predict the separation performance of 6FDA-DAM for the same binary gas mixture in the research done by Burns and Koros.⁴³ Dual mode model overestimated the permeability of propylene in the mentioned study while underestimated that of propane. Moreover, this model deviated dramatically from the experimental selectivity both in the amount and the trend. One probable reason for this discordance may be coupling of the diffusion coefficients which can be investigated through applying Maxwell–Stefan approach in the mixed gas membrane separation.⁴³

Although the application of the dual mode and frame of reference/bulk flow models were examined in the literature for predicting propylene/propane separation performance in several different membranes, the efficiency of Maxwell–Stefan model has not been assessed yet in the same systems. However, the vast amount of studies on different membrane separation systems utilizing the Maxwell–Stefan model have proved the ability of this approach in overcoming the complications of mass transport in membranes.^41,46–54 In addition, to the best of our knowledge so far no critical investigation has been reported comparing different approaches of mass transport through polyimide membranes for the separation of propylene/propane binary mixture.

The main objective of the present study is to evaluate and analyze both frame of reference/bulk flow and Maxwell–Stefan models for predicting the permeation and selectivity in polyimide membranes for propylene/propane separation. Besides, the contribution of kinetic and thermodynamic coupling effects were also investigated via employing Maxwell–Stefan model. Moreover, the influence of diffusion coefficients on the model predictions was also assessed and optimized. Sorption data of pure gases from available literature were employed to calculate sorption levels of binary gas mixture using dual sorption model. Fluxes and subsequently permeability were calculated using the two mentioned models. The governing transport equations were solved using MATLAB codes.

2 Theory and fundamentals

2.1 Dual mode sorption

Sorption in glassy polymers can be described by the dual mode model which presents two types of sorption modes; one includes the molecularly dissolved mode and the other accounts for non-equilibrium pre-existing gaps or excess volume between chains.^34,55 Accordingly, sorption isotherm of component “i” exhibited the following form:^34,44,55,56where k_Di is the Henry's law constant and C^′_Hi is the Langmuir capacity constant which is an indication of the second sorption mode. The affinity constant, b_i, is the measure of polymer attraction for the sorbed molecule.

Fick's law is the most widespread model that presents a simple relation between the flux of the penetrant and the gradient of its concentration, described according to eqn (2).⁵⁵

Dual mode sorption can be extended to represent the behavior of binary gas mixture as follows:

where C_Di is the Henry concentration and C_Hi is the Langmuir concentration.

2.2 Partial immobilization theory

Partial immobilization theory provides a more general form than that of Fick's speculating a finite mobility for the Langmuir sites along with Henry's law sites:^55,57where D_Di is the diffusion coefficient of component “i” in the Henry's sites and D_Hi is the diffusion coefficient of component “i” in the Langmuir sites. In most cases, partial immobilization theory gives an appropriate description of mass transport in membrane. However in some cases, it is unable to show the relation between the concentration within the polymer and flux of penetrants.^46,58,59 This formulation of diffusion is based on the assumption that diffusion of each molecule is independent of other molecules. However, in several situations such assumption fails to provide an appropriate description of the separation performance.⁴⁶ Applying this model along with sorption model leads to the following expression for the permeability of component “i”.⁵⁷

According to the partial immobilization theory, Henry's population is completely mobile and is able to fully participate in performing the diffusive jump while only a fraction of Langmuir's population is able to do so.⁴⁰ Therefore, mobile concentration comprises of C_Di along with a fraction of C_Hi presented according to eqn (6).

C_Mi = C_Di + F_iC_Hi (6)

where F_i is the ratio of diffusion coefficients D_Hi/D_Di.³⁵

According to eqn (4) and (6) the following expression can be written for the flux of component “i”:⁴²

Since molar volumes of the gaseous penetrants are not readily available, it is more convenient to express equations in mass units. Therefore, the mobile concentration (g/g pol.) also known as “sorption level” is defined as follows:

where Mw is the molecular weight of each component and ρ is the density of membrane.

Employing eqn (6) and (8) leads to the following expression for the mobile concentration:⁴⁴

3 Governing transport models

A thorough understanding of gas transport in polymeric membranes is very important from industrial point of view. Moreover, designing new class of polymeric membranes requires a comprehensive knowledge of permeation mechanisms in the membrane. Accordingly, frame of reference/bulk flow model and Maxwell–Stefan were used as the two main models to investigate the transport and separation performance of propylene/propane binary mixture in polyimide membranes.

3.1 Frame of reference/bulk flow model

According to the frame of reference/bulk flow model, the transport of a binary gas mixture through the membrane can be presented as a ternary system comprised of two gases and a membrane using the following equations:⁴⁰where n is the mass flux of components through the membrane and subscript m refers to the membrane. The first term of the right hand side is the convective or frame of reference term. This term is required to be considered when studying permeation in the membranes since diffusion of each molecule in a mixed gas environment is affected by the presence of other molecules which can produce an effective bulk flow.^40,42,44

Considering n_m = 0, rearranging eqn (10)–(12) for flux of each component leads to the following equations:

Boundary conditions are as follows:

z = 0; p₁ = p_1,0; p₂ = p_2,0; w₁ = w_1,0; w₂ = w_2,0 (16)

z = l; p₁ = p_1,l; p₂ = p_2,l; w₁ = w_1,l; w₂ = w_2,l (17)

Mass fluxes of components 1 and 2 are obtained after integrating eqn (13) and (14) and applying the above boundary conditions:⁴⁰

Eqn (18) and (19) should be solved for n₁l and n₂l iteratively in order to calculate the corresponding permeability using the following expressions:⁴⁴

Membrane selectivity can be calculated as follows:⁴²

3.2 Maxwell–Stefan model

In fact selection of a proper diffusion model in multi-component permeation is very crucial in modeling mixed gas transport. In most cases, the Fick's law is utilized assuming that the diffusion of each component is not affected by the presence of other components.⁴⁶ This assumption may not be true for permeation in some binary gas mixtures where the flux of one component can alter the diffusive and convective flux of the other component.^40,44

The most common situation in which the Fick's law can fail to predict the mixed gas performance is when the diffusion coefficient of one component is affected by the diffusion of other components present in the mixture.⁴⁶ It means that D is a function of concentration which results to a more complex model.^59–61 Another situation is that the concentrations of the sorbed penetrants are not small in the polymer matrix. And the last situation is when the flux of one component is dependent on the gradient of concentration of other components.⁴⁶ Maxwell–Stefan model was presented as a logical starting framework to deal with the issue of coupling of fluxes.^41,46 This model, which can be derived from kinetic theory, was developed originally to describe diffusion in gas mixtures and relates the forces acting on the molecules of each species to the friction between these species and any other species.^46,48 For a multi-component mixture Maxwell–Stefan equation can be written as follows:^41,46

where x and v are mole fraction and velocity, respectively. Đ_ij is the binary diffusion coefficient and d is driving force. Using mass fraction is more useful for describing diffusion in membrane systems^40,46 as follows:

n_i = w_iρv_i (24)

Using eqn (24) and (25) a more useful expression for describing mass transport in membranes can be obtained:

Accordingly, for a system comprising two penetrants and a polymeric membrane the following equations are obtained:

The following redefinition of the diffusion coefficient leads to more simplified forms of the transport equations.^41,46

Rearranging eqn (27) and (28) for the flux of each component, the Maxwell–Stefan equations reduce to:

The driving force can be defined via the following expression:⁴⁶

where C_md and C are the molar density of the mixture and the molar concentration and p_m is the pressure in the membrane. Chemical potential, μ, is defined as follows:where a is the activity defined as the ratio of partial fugacity to the standard fugacity. The latter is taken always as that of unity so the activity is numerically the same as the partial fugacity. By assuming the ideal gas law, partial fugacity can be represented by partial pressure. Since the pressure throughout the membrane is constant therefore: dp_m/dz = 0.⁴⁶ Accordingly, driving force can be written as follows:

It can be rewritten in the following form:

Substituting eqn (35) in eqn (30) and (31) the following equations can be obtained:

where ε₁ and ε₂ are frictional coupling effects.⁴¹

Eqn (36) and (37) can be written in the matrix form as follows:

n = −ρBη⁻¹∇w (39)

This allows evaluating the contribution of kinetic coupling effects, B, and thermodynamic coupling effects (TCE), η, independently. The matrices are provided in eqn (40) and (41):

and

The off-diagonal elements of the matrix η are indicators of the thermodynamic coupling of fluxes.⁶² The term dp_i/dC_j should be calculated in order to calculate the elements of η. For instance, η_ij is defined as the degree of influence of flux of component j on the flux of component i in the mixture. After considerable algebraic operations, the following equations are obtained by simultaneous solving of sorption equations (i.e., eqn (3)) for components 1 and 2.

E₁p₁³ + F₁p₁² + G₁p₁ + H₁ = 0 (42)

E₂p₂³ + F₂p₂² + G₂p₂ + H₂ = 0 (43)

where coefficients E, F, G and H are provided in Table 1.

Table 1 Equations derived from sorption equations and their respective coefficients

E1p1^3 + F1p1^2 + G1p1 + H1 = 0	E2p2^3 + F2p2^2 + G2p2 + H2 = 0
E1 = kD1kD2b1^2b2CH1 − kD1^2b1b2^2CH2	E2 = kD1kD2b2^2b1CH2 − k2^2b2b1^2CH1
F1 = 2kD1b1b2^2C1CH2 − kD1^2b2^2CH2 − kD1b1b2^2CH1CH2 + kD1b1b2^2C2CH2 + kD2b1^2b2CH1^2 + kD1k2b1b2CH1 − kD2b1^2b2C1CH1	F2 = 2kD2b2b1^2C2CH1 − kD2^2b1^2CH1 − kD2b2b1^2CH1CH2 + kD2b2b1^2C1CH1 + kD1b2^2b1CH2^2 + kD1kD2b1b2CH2 − kD1b2^2b1C2CH2
G1 = 2kD1b2^2C1CH2 + b1b2^2C1CH1CH2 − b1b2^2C1C2CH1 − b1b2^2C1^2CH2 − kD2b1b2C1CH1	G2 = 2kD2b1^2C2CH1 + b2b1^2C2CH1CH2 − b2b1^2C1C2CH2 − b2b1^2C2^2CH1 − kD1b1b2C2CH2
H1 = −b2^2C1^2CH2	H2 = −b1^2C2^2CH1

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Elements of thermodynamic matrix can be calculated by differentiation of eqn (42) and (43) with respect to concentration (C).

Accordingly, the elements of TCE matrix can be obtained as follows:

where the gradients dE_i/dC_j, dF_i/dC_j, dG_i/dC_j and dH_i/dC_j are provided in Table 2.

Table 2 The elements of TCE matrix and their respective gradients

η11	η12	η21	η22

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4 Results and discussion

The development of advanced polymeric membranes for propylene/propane separation requires a thorough understanding about the factors affecting the transport of penetrants within the membrane. Therefore, it is important to adopt a proper approach in order to predict the performance of the membrane mathematically. Both frame of reference/bulk flow and Maxwell–Stefan models allow accounting for the coexistence of the penetrating molecules through the membrane and hence the results may be helpful to examine the commercial viability of the polyimides. On the other hand, Maxwell–Stefan model originally accounts for the coupling of fluxes of two penetrants while frame of reference/bulk flow model does not. According to the Maxwell–Stefan model, the flux of one component is dependent on the gradient of concentration of other component in a binary mixture while this issue was not considered in the Fick's law which is the foundation of the frame of reference/bulk flow model. Moreover, matrix form of Maxwell–Stefan model can be employed to assess the contributions of kinetic and thermodynamic coupling effects independently. In the present study, sorption data of pure propylene and propane in 6FDA-6FpDA,^44,45 6FDA-DAM⁴³ and 6FDA-TrMPD³⁶ were used to predict the separation performance of membrane in the case of equimolar (50/50) binary gas mixture. These data are presented in Table 3.

Table 3 Dual mode and partial immobilization parameters of the studied polyimides

Polyimide	T (°C)	Penetrant	DD (m^2 s^−1)	DH (m^2 s^−1)	kD (cm^3 STP/(cm^3 psia))	b (psia^−1)	C^′H (cm^3 STP/cm^3)	bC^′H (cm^3 STP/(cm^3 psia))	F (DH/DD)	Ref.
6FDA-6FpDA	35	Propylene	5.96 × 10^−10	3.92 × 10^−10	0.26	0.24	20.3	4.872	6.58 × 10^−1	44
		Propane	8.40 × 10^−11	8.50 × 10^−12	0.20	0.22	11.35	2.497	1.01 × 10^−1
6FDA-6FpDA	70	Propylene	3.45 × 10^−9	1.86 × 10^−10	0.12	0.32	12.3	3.936	5.39 × 10^−2	44
		Propane	3.11 × 10^−10	3.00 × 10^−12	0.10	0.15	9.87	1.480	9.65 × 10^−3
6FDA-DAM	75	Propylene	7.86 × 10^−8	7.90 × 10^−9	0.15	0.205	17.5	3.587	1.01 × 10^−1	43
		Propane	9.56 × 10^−9	1.53 × 10^−9	0.125	0.266	10.7	2.846	1.60 × 10^−1
6FDA-TrMPD	50	Propylene	3.80 × 10^−8	4.30 × 10^−9	0.265	0.258	26	6.721	1.13 × 10^−1	36
		Propane	2.50 × 10^−9	3.00 × 10^−10	0.347	0.476	17	8.095	1.20 × 10^−1

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4.1 Analysis of sorption for pure and binary gas mixture

Sorption in glassy polymers can be well described using dual mode sorption in the base case. This model describes the sorption using the assumption that two regimes are responsible for the penetrant sorption. One regime is the unrelaxed free volume of the glassy polymer and sorption into these sites can be described by the Langmuir model. The other regime is the sorption in densified polymer chains responsible for Henry's law domains that dominates at high pressures when free volume becomes saturated.^40,43

Fig. 1 shows the sorption values as a function of pressure using dual mode sorption model calculated based on sorption data of pure gases. It is evident that considering pure components, sorption of propylene is higher than that of propane in all polyimides indicating higher solubility of propylene than that of propane. According to the data in Table 3, this can mainly be owing to the larger Langmuir capacity constant (C^′_H) of the polymers for propylene than propane, indicating the dominating role of Langmuir sites in the sorption process. Interestingly, a limited difference between the sorption of propylene and propane was observed in case of 6FDA-TrMPD (Fig. 1(d)) compared to the rest of polyimides. It was also noted that, sorption values for propylene and propane in the binary gas mixture were lower than those of respective pure components. This can be attributed to the contribution of additional b_jp_j term in the denominator of Langmuir component provided in eqn (5). In addition, results reveal a generally more pronounced drop in the sorption of propylene when coexists with propane in a binary gas mixture. Larger drops observed for 6FDA-DAM and 6FDA-TrMPD (Fig. 1(c) and (d)) compared to other polyimides can be attributed to the dominating role of Langmuir affinity constants in these polymers for Propane. Therefore, the larger decrease in the sorption of the faster component in the binary gas mixture can be attributed to the competition effect and/or contribution of bulk flow that are not taken into account in dual mode sorption model.^40,44

Fig. 1 Pure and mixed gas sorption isotherms in (a) 6FDA-6FpDA at 35 °C (b) 6FDA-6FpDA at 70 °C (c) 6FDA-DAM at 75 °C and (d) 6FDA-TrMPD at 50 °C.

Fig. 2 demonstrates the corresponding values of sorption level for the investigated polymers. A similar trend of higher sorption level of propylene than that of propane can be seen for all polymers except for 6FDA-TrMPD in pure state. This can be attributed to the larger Henry constant (k_D) of propane compared to that of propylene in 6FDA-TrMPD (Table 3) compared to other polymers.

Fig. 2 Pure and mixed gas sorption levels in (a) 6FDA-6FpDA at 35 °C (b) 6FDA-6FpDA at 70 °C (c) 6FDA-DAM at 75 °C and (d) 6FDA-TrMPD at 50 °C.

In addition, sorption levels for propylene and propane in the binary gas mixture were lower than those of respective pure components as explained earlier. The exceptionally lower sorption level of propylene in binary gas mixture than that of propane in 6FDA-TrMPD can be explained by data presented in Fig. 3 indicating a larger sorption level of propane in Henry sites (W_D2) arising from the larger contribution of Henry constants of 6FDA-TrMPD for this gas.

Fig. 3 Parts of mixed gas sorption level in 6FDA-TrMPD at 50 °C.

Moreover, sorption level of propane in 6FDA-TrMPD is the highest among other polyimides due to the highest k_D and bC^′_H while sorption level of propylene is higher in 6FDA-6FpDA at 35 °C despite lower values of k_D and bC^′_H compared to that of 6FDA-TrMPD. Sorption levels for propylene and propane are presented in Fig. 4.

Fig. 4 Sorption levels of (a) propylene (b) propane in different polyimides.

The highest values of propylene sorption level in 6FDA-6FpDA at 35 °C compared to other polyimides can arise from its highest value of F among other polyimides (Table 3). In order to prove this hypothesis, mobile fraction of Langmuir population in all the studied polyimides were depicted (Fig. 5) and it was observed that this value in 6FDA-6FpDA at 35 °C is the highest among other polyimides. In other words, mobile fraction of Langmuir population plays an important role in determining the sorption level even in the case of small values for k_D and bC^′_H.

Fig. 5 Mobile fraction of Langmuir population for propylene sorption in different polyimides.

4.2 Analysis of frame of reference/bulk flow model

According to the frame of reference/bulk flow model, the transport process is considered as a combination of both bulk and diffusive fluxes. Bulk flux is a function of sorption level in permeation of pure components; however in the case of binary gas mixtures, bulk flux is a function of both sorption level and flux of mobile components.⁴⁰ The contributions of bulk flux of propylene and propane in different polyimides were calculated according to the method described by Kamaruddin and Koros⁴⁰ and are presented in Fig. 6.

Fig. 6 Fraction of bulk flux contribution of propane and propylene in binary mixture compared to the pure gas in (a) 6FDA-6FpDA at 35 °C (b) 6FDA-6FpDA at 70 °C (c) 6FDA-DAM at 75 °C and (d) 6FDA-TrMPD at 50 °C.

It can be observed that the flux of propane in the binary mixture of propylene/propane is much higher than that of pure propane while the flux of propylene remained almost unchanged. It can be inferred that propane molecules swept by the faster penetrant, propylene, and this led to an effective bulk flow of propane.

In addition, Fig. 7 shows that the contribution of bulk flux for propane is much higher in the 6FDA-TrMPD compared to other polyimides. This may be due to the higher sorption level of propane in this polymer as illustrated in Fig. 4(b).

Fig. 7 Fraction of bulk flux contribution of propane in different polyimides.

It is obvious that bulk flux of propane, which was not accounted for in the dual mode model, constitutes a great portion of the total mass flux. This consideration led to more accurate predictions of the separation performance in the case of 6FDA-6FpDA. Nevertheless, this model failed to predict permeability and selectivity in 6FDA-DAM and 6FDA-TrMPD. Even more, applying this model led to the wrong predictions for the trend of selectivity in the case of 6FDA-DAM. This may be attributed to the fact that the flux of each component is dependent on the gradient of concentration of other component as explained by Paul⁴⁶ and can be accounted for by the Maxwell–Stefan model. It should also be noted that different testing temperatures as well as the chemical structure of the studied polymers may paly parts in the obtained trends.

4.3 Analysis of Maxwell–Stefan model

According to Maxwell–Stefan model the flux of each component is related to the driving force of both components. In fact, driving force is a combination of the material fluxes which in turn can be expressed in terms of composition gradients.^49,63 Accordingly, coupling of fluxes is inherently accounted for in this model which was not considered in the bulk flow model.

Propylene/propane selectivity for 6FDA-6FpDA at different upstream pressures calculated using both frame of reference/bulk flow and Maxwell–Stefan models is presented in Fig. 8. It can be observed that predictions of both models are in agreement with the experimental data of permeation in binary gas mixture presented by Das and Koros.⁴⁴ In addition, predicted values of the frame of reference/bulk flow model are almost the same as those calculated by the Maxwell–Stefan model. This can be due to the negligible cross-terms of the kinetic coupling matrix resulted from the infinite small values of frictional coupling effects. This in turn arises from markedly small values of diffusion coefficients of penetrants, D_1m and D_2m, compared to the coupling diffusion coefficient, D₁₂. Experimental dual mode diffusion coefficients (provided in Table 3) were used as D_1m and D_2m and D₁₂ were calculated according to the Chapman–Enskog Wilke–Lee model.⁶⁴

Fig. 8 Comparison of experimental and calculated selectivity for 6FDA-6FpDA at 35 °C (gas composition 50/50 mol%).

Besides, the weight fractions of the sorbed species in the polymer matrix are small enough to assume w_m ≈ 1. The average values of frictional coupling effects (ε₁ and ε₂), sorption level in membrane (w_m) and the calculated kinetic coupling effects (B_average) are listed in Table 4. Consequently, predictions of Maxwell–Stefan model approached to that of bulk flow model which can also be observed in Fig. 9 and 10.

Table 4 The average amounts of frictional coupling effects, membrane sorption levels and kinetic coupling effects

Polyimide	T (°C)	ε1	ε2	wm	Baverage
6FDA-6FpDA	35	2.99 × 10^−7	4.82 × 10^−8	0.97
6FDA-6FpDA	70	7.72 × 10^−8	2.64 × 10^−9	0.98
6FDA-DAM	75	3.35 × 10^−5	3.89 × 10^−6	0.99
6FDA-TrMPD	50	2.09 × 10^−5	1.20 × 10^−6	0.97

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Fig. 9 Comparison of experimental and calculated permeability of (a) propylene and (b) propane for 6FDA-DAM at 75 °C (gas composition 50/50 mol%).

Fig. 10 Comparison of experimental and calculated permeability of (a) propylene and (b) propane for 6FDA-TrMPD at 50 °C (gas composition 50/50 mol%).

Due to the nature of the propane and propylene which are often separated at relatively low pressures and high temperatures (i.e. at low activities), the observed effect of pressure on selectivity (Fig. 8) and permeability (Fig. 9 and 10) in this case is quite modest. More pronounced effects are expected for other gas pairs permeating at higher pressures and is the subject of our ongoing investigations.

However, it was found that the predictions of the aforementioned models deviated from experimental data in Fig. 9 and 10 reported respectively by Burns and Koros⁴³ and Tanaka et al.³⁶ in the case of propane permeability. Whereas, Maxwell–Stefan model with TCE contribution led to a better estimation of permeability of propane according to Fig. 9 and 10.

Comparison of the data related to the TCE matrix presented in Fig. 11, reveals that the cross terms are not negligible versus the main terms suggesting that the diffusive fluxes of propane and propylene are affected to some extent when present in the mixture. On the other hand, higher values of η₁₂ compared to η₂₁ may suggest that diffusive flux of propylene is strongly coupled with that of propane. This can be a justifiable reason for less accurate prediction of permeability of propane than that of propylene by Maxwell–Stefan model without TCE contribution.

Fig. 11 Thermodynamic coupling effects of (a) propylene and (b) propane (gas composition 50/50 mol%).

Therefore, the assumption that the gradient of concentration of each component can affect the flux of other component seems to be valid when studying the transport of propylene/propane binary mixture through polyimide membranes. The average values of thermodynamic coupling effects are presented in Table 5.

Table 5 Average values of thermodynamic coupling effects (TCE)s

Polyimide	T (°C)	ηaverage
6FDA-6FpDA	35
6FDA-6FpDA	70
6FDA-DAM	75
6FDA-TrMPD	50

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Despite the importance of TCEs and their effective role in providing a better estimation of the transport behavior in the mentioned systems, it is obvious that TCEs could not be the only decisive parameters in determining the transport properties of binary gas mixture. As it was mentioned before, sorption data of pure gases were used to estimate the behavior of binary gas mixture while it was proved that the presence of penetrants in the binary gas mixture led to an effective bulk flow for each component. Accordingly, it can be speculated that diffusion of each component can also be affected in their binary gas mixture. This was investigated by calculating the diffusion coefficients using experimental permeability and selectivity of binary gas mixture employing Maxwell–Stefan model both without and with contribution of TCEs. The calculated values are presented in Table 4 and used for analysis of the separation performance of 6FDA-DAM and 6FDA-TrMPD. Comparing the calculated diffusion coefficients and the diffusion coefficients of pure gases in Table 6 it can be observed that the optimized diffusion coefficients of propylene in the binary gas system in both 6FDA-DAM and 6FDA-TrMPD reduced to almost half of their original values. However, the optimized diffusion coefficient of propane stayed almost unchanged in 6FDA-DAM while increased in 6FDA-TrMPD.

Table 6 Experimental and optimized diffusion coefficients

Polyimide	T (°C)	Experimental diffusion coefficients^36,43 (m^2 s^−1)		Optimized diffusion coefficients using Maxwell–Stefan model without TCE contribution (m^2 s^−1)		Optimized diffusion coefficients using Maxwell–Stefan model with TCE contribution (m^2 s^−1)
		Propylene	Propane	Propylene	Propane	Propylene	Propane
6FDA-DAM	75	7.86 × 10^−8	9.56 × 10^−9	6.329 × 10^−7	1.25 × 10^−8	3.973 × 10^−8	9.1 × 10^−9
6FDA-TrMPD	50	3.8 × 10^−8	2.5 × 10^−9	3.279 × 10^−8	4.36 × 10^−9	2.033 × 10^−8	3.16 × 10^−9

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The reduced diffusion coefficient of propylene could be ascribed to the competitive sorption of propane and propylene in the binary gas mixture. In addition, using the results illustrated in Fig. 11, it can be inferred that the flux of larger molecule, propane, probably hindered propylene molecules to diffuse unlike the case of pure gas. This phenomenon did not disturb the diffusion of propane in the binary gas environment through 6FDA-DAM. Therefore, another phenomenon might act to increase the diffusion coefficient of propane in 6FDA-TrMPD. According to Fig. 2(d), propane sorption level is higher than that of propylene both in pure and binary gas systems which makes the polyimide susceptible to plasticization. Therefore, one probable reason for the increase in the propane diffusion coefficient in the case of 6FDA-TrMPD could be the plasticization phenomena. The trend in the permeability of propane could be also a clue to the presence of plasticization in the mentioned polyimide since a minimum permeability can be observed in Fig. 10.

Fig. 12 and 13 show the separation performance of both 6FDA-DAM⁴³ and 6FDA-TrMPD³⁶ predicted by Maxwell–Stefan model without and with TCE contribution using data in Table 4. These figures indicate that diffusion coefficients that obtained via performing optimization using the Maxwell–Stefan model with TCE contribution resulted in better estimation of separation performance compared to the Maxwell–Stefan model in which TCEs were not taken into account. The optimized diffusion coefficients can also be used to estimate the propylene/propane separation performance of the polyimides in pressures other than those used for the optimization.

Fig. 12 Predictions of the Maxwell–Stefan model without and with TCE contribution using optimized diffusion coefficients for 6FDA-DAM at 75 °C. (a) Propylene permeability and (b) propane permeability (c) propylene/propane selectivity (gas composition 50/50 mol%).

Fig. 13 Predictions of the Maxwell–Stefan model without and with TCE contribution using optimized diffusion coefficients for 6FDA-TrMPD at 50 °C. (a) Propylene permeability and (b) propane permeability (c) propylene/propane selectivity (gas composition 50/50 mol%).

5 Conclusions

Transport of propylene and propane through polyimide membranes were investigated and analyzed by applying two major phenomenological models of frame of reference/bulk flow and Maxwell–Stefan. Results revealed that sorption level of the more soluble gas, propylene, calculated using the dual sorption model, was generally higher than that of propane and is highly affected by the values of k_D, bC^′_H and F. Frame of reference/bulk flow model led to good estimations of the membrane performance in the case of 6FDA-6FpDA while could not provide appropriate predictions for the separation performance of binary gas mixture in the case of 6FDA-DAM and 6FDA-TrMPD. This was attributed to the dependence of the flux of each component to the gradient of concentration of other component. The validity of this speculation was assessed via applying the Maxwell–Stefan model that initially accounted for the co-existence of penetrants. According to the results, Maxwell–Stefan estimations approached to those of frame of reference/bulk flow model due to the small values of the cross terms in the matrix of the kinetic coupling effects. Better predictions were obtained using Maxwell–Stefan model with TCE contribution. Calculating optimized diffusion coefficients by the aid of experimental permeability showed that diffusion coefficients were also affected due to the presence of another component in the binary gas mixture. Diffusion coefficients of all components decreased in the binary gas mixture while that of propane in 6FDA-TrMPD increased. This could be probably due to the plasticization phenomena since the sorption level of propane in 6FDA-TrMPD was higher compared to other polyimides. In addition, the optimized diffusion coefficients could be utilized to predict the behavior of propylene/propane separation in other pressures than those used for the optimization. The findings can be effectively used by researchers for selecting the best strategies in order to represent the best possible predictions about the membrane separation performance for olefin/paraffin separation applications. Nevertheless, more investigations are required to reach to more reliable conclusions which necessitate more experimentations and analysis.

Nomenclature

a	Activity
b	Langmuir affinity constant [1/psia]
B	Matrix of kinetic coupling effects
C	Gas sorption [cm^3 (STP)/cm^3 (polymer)]
CD	Concentration of Henry's law sites [cm^3 (STP)/cm^3 (polymer)]
CH	Concentration of Langmuir sites [cm^3 (STP)/cm^3 (polymer)]
C^′H	Langmuir capacity constant [cm^3 (STP)/cm^3 (polymer)]
d	Driving force
DD	Diffusion coefficient in the Henry's law environment [m^2 s^−1]
DH	Diffusion coefficient in the Langmuir domain [m^2 s^−1]
Dij	Binary diffusion coefficient [m^2 s^−1]
J	Molar flux through membrane [cm^3 (STP)/cm^2 s]
kD	Henry's law constant [cm^3 (STP)/cm^3 (polymer) psia]
l	Membrane thickness [m]
Mw	Molecular weight [g mol^−1]
n	Trans-membrane mass flux [kg s^−1 m^−2]
p	Pressure [psia]
P	Permeability [Barrer] = [ cm^3 (STP) cm/cm^2 s cmHg]
r	Ratio of mass flux of propylene to the mass flux of propane
R	Universal gas constant [J mol^−1 K^−1]
T	Temperature [K]
w	Sorption level (g/g)
wD	Sorption level in Henry's law sites (g/g)
wH	Sorption level in Langmuir sites (g/g)
x	Mole fraction
z	Membrane length coordinate [m]

Greek letters

α	Membrane selectivity
ε	Frictional coupling effects
η	Matrix of thermodynamic coupling effects
μ	Chemical potential [J mol^−1]
ν	Velocity [m s^−1]
ρ	Average density of the polymer [kg m^−3]

Subscript

i, j	Index of components
ij	Interaction of component i and j
m	Membrane
0	Upstream side of the membrane
l	Downstream side of the membrane
1	Propylene
2	Propane

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