Prediction of gas transport across amine mixed matrix membranes with ideal morphologies based on the Maxwell model

Abstract

The incorporation of highly selective molecular sieve such as carbon molecular sieve (CMS) and highly affinitive solvent such as diethanolamine (DEA) into polyethersulfone (PES) have been implemented to synthesize amine mixed matrix membranes (MMMs) with an enhanced gas performance. Synergetic effect of CMS and DEA has caused the improvement of carbon dioxide (CO₂) permeability and ideal CO₂/CH₄ selectivity. While existing theoretical models define the relative permeability well for binary mixed matrix membranes they fail to predict the relative permeability of amine mixed matrix membranes. In fact, the degree of deviation from the simple model predictions provides understanding into the detailed properties of the third component, which has been neglected in previous analyses of these models. Modification of an existing model, namely the Maxwell model, provides an outline to analyze the gas permeation properties of model systems with CMS and DEA in glassy polymer phase. The new model is developed by modifying the basic Maxwell MMMs model. The modification also includes the optimization of λ_dm, which is defined as the ratio of dispersed phase permeability to matrix permeability, and the determination of permeability of the dispersed phase. Furthermore, this Maxwell model has been extended to model the performance of amine mixed matrix membranes by incorporation of combined volume fraction of filler and amine φ^*_ad. The proposed approach can predict the permeability of CO₂ through amine MMMs and also lowers the AARE % value.

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1. Introduction

Membrane technology is an energy efficient process as compared to traditional gas separation processes.^1–5 Polymers are the most attractive materials for the synthesis of membranes, due to their low cost and good intrinsic transport properties. On the other hand, inorganic materials have great potential in separation of gases at industrial operating conditions. However, their applications are still limited due to reproducibility problems in synthesis steps, short life span and high cost.^6,7 A new approach has been developed to overcome these limitations of polymeric and inorganic materials by implementation of mixed matrix membranes (MMMs). These membranes have the properties of inorganic materials and organic materials.^8–11 MMMs are efficient in gas separation such as for O₂/N₂, CO₂/CH₄ etc. These membranes exhibit enhanced separation performance and can preserve the good processibility properties of polymeric membranes.

As mentioned in our recent works^12,13 the third component which is an amine (in this instance diethanolamine, DEA) has been added in polymer (polyethersulfone, PES) and filler (carbon molecular sieve, CMS) sol for the synthesis of amine mixed matrix membranes. The performance of amine MMMs exhibited a good CO₂/CH₄ separation factor, α, in the range of 5.20–51.39 as cited in our recent work.¹² The enhancement in the performance of amine MMMs was due to the presence of the combined effect of carbon molecular sieve (CMS) and amine, in which there is increased solubility of CO₂ and lower solubility of CH₄. The mechanism can be better understood by modeling the transport of these gases using an appropriate model. The transport mechanisms of these gases across these amine MMMs can be described as follows: (a) the diffusion rate of gases depends on the affinity of gases towards the polymeric membrane surface. In the case of CO₂ and CH₄ where polyethersulfone (PES) membrane is used as per the current study, the rate of diffusion of CO₂ toward PES membrane is stronger as compared to CH₄ due to the polarity of CO₂ to the PES surface, moreover, the molecular size of the gases also play an important role in their diffusion. Gas transport through CMS is based on the differences in adsorption kinetics of different gases present in the gaseous mixture. In the separation of CO₂ and CH₄ by CMS molecular sieve, the smaller (3.30 Å) CO₂ molecule adsorbs more rapidly as compared to the larger (3.80 Å) CH₄ molecule. (b) For amine MMMs, the type of amine is important for the high rate of diffusion. For CO₂ and CH₄ separation by diethanolamine (DEA), the diffusion rate of CO₂ is higher as compared to CH₄ due to the high affinity of CO₂ with DEA. Furthermore, the CO₂ transport across amine MMMs occurs by solution diffusion, molecular sieve and DEA interaction.

An appropriate theoretical approach for the prediction of MMMs permeability is of great interest, especially as such membranes become more important for the separation of gases. The expression of gas transport through MMMs is a complex problem. Some successful studies are made by different researchers to predict the performance of MMMs with the help of various theoretical expressions based on Maxwell, Bruggeman,¹⁴ Lewis Nielsen, Pal and Fleske.^15,16 There are also some other models available in the literature based on Böttcher¹⁷ and Higuchi,¹⁸ Bouma,¹⁹ Hashemifard–Ismail–Matsuura (HIM),²⁰ to predict the performance of MMMs.²¹ The analysis of these analytical models has been reviewed in previous cited reviews.^16,22 The analysis of some models also will be discussed in a later section of this paper. These models are successfully employed to predict the relative permeability of MMMs. Limitations of these models, such as that none of the described models include the effect of a third added component in the polymer/filler matrix, will also be addressed. Consequently, there is a need to develop a theoretical approach that can include the effect of pressure on relative permeability of amine MMM, volume fraction of filler and volume fraction of a third component. It is believed that such a prediction approach can improve the understanding of the various parameters on the performance of amine MMMs.

The objective of this study is to develop a theoretical approach in an idea to predict the performance of amine mixed matrix membranes. Prior to the development of this approach, some of the existing models for MMM are briefly reviewed. From the cited literature it is concluded that there are very few studies which study the inorganic filler permeability with respect to pressure. Therefore, in this contribution a systematic method was employed to calculate the permeabilities of inorganic filler by the optimization of permeability ratios of filler to polymer matrix. Then a new theoretical approach for the prediction of amine MMMs performance is also include the modification of basic MMMs using the Maxwell model by introducing a volume fraction of amine. In addition, the predicted results from the modified model have been compared with the experimental data. To the best of our knowledge, this is the first approach to predict the performance of amine mixed matrix membranes.

2. Analysis of existing mixed matrix membrane models

The available present models for MMMs are adopted from thermal and electrical conductivity models. These models have close relations between electrical and thermal conduction and permeation of species through composite materials.^16,23

The Maxwell model was initially established for the measurement of electrical conductivity of particulate heterogeneous media composite in 1873.²⁴ This model provides the exact solution for the conductivity of random distributed and non-interacting homogeneous solid spheres in a continuous matrix.²⁵

where the relative permeability of gas, P_r is the ratio of P/P_m, P is the permeability of MMMs and P_m is the permeability of polymer matrix, φ_d is the volume fraction of the inorganic dispersed phase and λ_dm is the permeability ratio P_d/P_m while P_d is the permeability of inorganic filler. This model generally described the permeability accurately when φ_d was less than about 0.2. At higher values of φ_d, significant deviations were expected between the predictions of eqn (1) and actual values. Also, the Maxwell model failed to predict the correct behavior at φ_d → φ_m, where φ_m is the maximum packing volume fraction of filler particles. It is important to note that at φ_d = φ_m, the relative permeability P_r was expected to deviate for MMMs with permeability ratio λ_dm → ∞. Moreover, this model did not consider the particle size distribution, particle shape, and aggregation of particles.¹⁶

The Bruggeman model¹⁴ was developed for the estimation of the dielectric constant of particulate composites; this model could be modified for the permeability prediction of MMMs.

The differential effective medium approach was used to propose this model. Although this model was better than the Maxwell model, the model had limitations i.e. it did not give the correct estimation at φ_d → φ_m. Additionally, it did not account for particle size distribution, particle shape, and aggregation of particles. Moreover, the Bruggeman model is an implicit relationship that needs to be solved numerically for the permeability.^16,26,27

Another model was originally proposed for an elastic modulus of particulate composite materials by Lewis–Nielsen.^28,29 The permeability of MMMs could be predicated by this model as follows.

whereHere, φ_m is the maximum packing volume fraction of filler particles (φ_m = 0.64) for random close packing of uniform spheres. The Lewis–Nielsen model, eqn (3), gave a precise prediction at φ_d → φ_m. The relative permeability, P_r at φ_d = φ_m diverges when the permeability ratio λ_dm → ∞. As φ_m was sensitive to particle size distribution, particle shape, and aggregation of particles, this model considers the effects of morphology on the gas permeability. It was interesting to note that when φ_m → 1, the Lewis–Nielsen model reduces to the Maxwell model (eqn (1)).¹⁶

The Pal model²⁶ was proposed for the prediction of thermal conductivity of particulate composite materials. The performance of MMMs can be predicted by this model as:

The Pal model was developed using the differential effective medium approach by taking into account the packing difficulty of filler particles. As φ_m → 1, the Pal model reduces to the Bruggeman model (eqn (2)). The Pal model gave the correct behavior at φ_d → φ_m. It also considered the morphology effect on permeability through the parameter φ_m, which was sensitive to morphology. However, the Pal model is an implicit relationship that needs to be solved numerically for P_r.¹⁶

Applying the Maxwell–Wagner–Sillar (MWS) model, the P_eff of MMM with a dilute dispersion of ellipsoids is described by eqn (6):

where the subscript C indicates the continuous matrix, d indicates the dispersed phase, φ_d is the volume fraction of the filler and n is the shape factor of the dispersed filler. The above equation (eqn (6)) is an analytical solution that can be found by embedding one unit cell of the dispersed phase in the polymer matrix phase. For prolate ellipsoids, i.e. the longest axis of the ellipsoid is directed along the applied pressure gradient, 0 < n < 1/3. For spherical filler particles, n = 1/3. For oblate ellipsoids, i.e. the shortest axis of the ellipsoid is directed along the applied pressure gradient, 1/3 < n < 1.²²

These models have been adopted to predict the effective permeability of a gaseous penetrant through these MMMs as a function of permeability of polymer matrix (P_m), permeability of dispersed phase (P_d), and the volume fraction of filler (φ_d).²² Despite their practicality and advantages, these models have been evaluated to have some limitations as stated below.

(a) The Maxwell model is used for representing electrical conductivity of particulate composites and also for measuring the permeability in composites. It predicts reasonably well for low to moderate values of filler concentrations φ_d ≤ 0.2. Due to its explicit nature, this model is easy to compute. However, the Maxwell model does not consider the effect of size, shape, agglomeration and distribution of particles.^16,30

(b) The Lewis–Nielson model incorporates the effects of morphology on permeability through the parameter φ_m, and is expected to predict the relative permeability satisfactorily at φ_d → φ_m, but deviates at φ_d = φ_m, when the permeability ratio λ_dm → ∞.²¹

(c) The Maxwell–Wagner–Sillar model accounts for the shape of filler by introducing the shape factor n. The incorporation of n helps to use this model for different physical shapes of fillers. Each shape of fillers, such as cube, square, circular, ellipsoid, is associated with a specific value of n. This factor is unity for spheres and between 0 and 1 for all other shapes.³¹ The most challenging part of this model is to determine the shape for non-ideal morphology of MMM especially when agglomeration occurs at high loading of fillers as well as when different shapes of particles are present in MMMs. Hence, when this situation occurs, the actual shape filler need to be determined experimentally.

3. Development of the present model

The model was developed for the steady-state gas permeation across membranes in a flat sheet membrane module. As portrayed in Fig. 1, the synthesized amine MMMs has the concentration of alkanolamine solution φ_a along with the polymer concentration and filler loading φ_d in order to enhance the performance of MMMs. It was also observed that the alkanolamine solution which was embedded in the polymer matrix facilitated transport of CO₂ and retards the transport of CH₄ (Fig. 2).

Fig. 1 Schematic diagram of (a) mixed matrix membrane, (b) amine mixed matrix membrane.¹³

The basic Maxwell model was chosen on the basis of some characteristics such as (i) simple formulation, (ii) able to incorporate all physical components of MMMs, such as matrix and filler permeability as well as filler volume fraction, (iii) easy computation, (iv) explicit nature and (v) less number of assumptions required,^24,32 to model the performance of enhanced MMMs. As our previous published literature^12,13 shows that the presence of amine has enhanced the solubility of CO₂ through amine MMMs while retarding the solubility of CH₄, it is seen that as the amine concentration (φ_a) increased, the CO₂ permeance was also increased until it reaches the maximum saturation point before it starts to decrease. Under the same set of conditions, the permeance of CH₄ shows a contrary behavior and values tend to decrease significantly. Generally, the CO₂ transport across amine MMMs follows three steps, (i) gas transport through polymer, (ii) gas transport through inorganic filler (carbon molecular sieve (CMS)) and (iii) gas transport through amine. The presence of amine and inorganic filler restrict the transport of CH₄ across amine MMMs.

In order to model the performance of amine MMMs the, the following assumptions are made:

(i) The CMS and amine are homogenously distributed in the polymer matrix. The amine MMMs has ideal morphology with uniform thickness.

(ii) The active N–H group of DEA is homogenously distributed in the amine MMMs.

(iii) The diffusion of CO₂ towards the membrane surface and the rate of diffusion depend on the concentration of amine and loading of CMS.

(iv) Gas transport through CMS is based on the differences in adsorption kinetics of different gases present in the gaseous mixture. In the separation of CO₂ and CH₄ using CMS, the smaller (3.30 Å) CO₂ molecule is transported more rapidly as compared to the larger (3.80 Å) CH₄ molecule.

All these assumptions (i–iv) are valid for the synthesized amine MMMs as long as the developed model is used within the operating conditions i.e. CMS loading 10–30 wt%, 10 and 15 wt% DEA concentrations and operating pressure at room temperature 25 °C which is far superior to previous approaches, where smaller ranges are used for all parameters except the pressure.

In order to extend the basic Maxwell model with the presence of the third amine component, an additional parameter which combined the volume fraction of CMS and amine as well as the effects of non-ideality and facilitation, defined as φ^*_ad, is introduced in eqn (1). In addition, the permeability ratio of filler to polymer matrix λ^*_dm would also need to be optimized since most permeation experimental data for CMS available from literature is at fixed pressure i.e. 2 bar. In this study, the effect of pressures was also investigated from 2 to 10 bar. The detailed optimization of λ^*_dm is given in a later section.

Therefore, the enhanced modified Maxwell model after incorporating the φ^*_ad and λ^*_dm terms is given as,

The combined volume fraction of the filler and alkanolamine φ^*_ad is given as,

φ^*_ad = (φ_a + φ_d) + F_c (8)

where φ_a is the volume fraction of the amine and φ_d is the volume fraction of the dispersed phase. The term F_c is included in eqn (8) to introduce the effects of non-ideality and facilitation of the membrane. The values of facilitation parameter F_c, which is the function of volume fraction of filler φ_d and volume fraction of amine, φ_a are evaluated by scattered interpolant technique (Delaunay triangulation) in MATLAB® environment.

In the Maxwell model (eqn (1)), λ_dm is related to the permeability of filler P_d and permeability of polymer matrix P_m. As this ratio depends on P_d and P_m, with reference to previous cited literatures the modeling of MMMs is carried out at constant pressure, therefore, λ_dm has one value at a particular pressure. Since, this study focuses to investigate the effect of pressure on relative permeability of amine MMMs, the value of λ_dm needs to be estimated. A least-squares approach was used to determine the λ^*_dm. The fitting and optimization procedures were conducted on the Maxwell model against the experimental data to produce the results for λ^*_dm. It can be expressed as:

λ^*_dm = P^*_d/P_m (9)

P^*_d = λ^*_dmP_m (10)

where λ^*_dm is the ratio of the predicted filler permeability P_d over the permeability of polymer matrix P_m, P^*_d is the estimated permeability of filler against pressure. In practice, the determination of experimental filler permeability P_d is difficult and results are essentially varied due to various factors involved, particularly the brittleness of inorganic membrane which may be unable to withstand high pressure, and the membrane synthesis procedure.³³ A least-square polynomial computational approach using MATLAB® was made to estimate the percentage average absolute relative error (AARE %).^21,30

According to the proposed model (Fig. 2), the relative permeability of gases in amine mixed matrix membranes is a function of six parameters, namely: the volume fraction of filler φ_d, volume fraction of amine φ_a, ratio of dispersed filler to matrix permeance λ^*_dm and combined volume fraction of amine and dispersed filler φ^*_ad, permeability of polymer matrix P_m and permeability of dispersed phase P^*_d as expressed below.

P_r = f(φ_d,φ_a,P^*_d,P_m,λ^*_dm,φ^*_ad) (11)

Fig. 2 Flow chart for modeling of amine MMMs.

4. Estimation of model parameters

The parameters described in eqn (11), including λ^*_dm, φ^*_ad, P^*_d, need to be determined in order to calculate the relative permeability, P_rcal across the amine mixed matrix membrane. The P_rcal will be later compared with the P_rexp, hence through calculation of AARE %, the validity of the present enhanced modified Maxwell model can be validated.

4.1. Determination of λ^*_dm

There is no data available for dispersed phase permeability at different pressures and different loading of filler. Therefore, a regressive method is adopted to optimize model parameters against experimental data to minimize the predicted errors from the models. The optimization approach has been carried out on the model against the experimental data for 10–30 wt% CMS. Tables 1 and 2 show the optimized values of λ_dm for CO₂ and CH₄, respectively, across various loadings of filler and pressure.

Table 1 Optimized value of λ_dm for CO₂

Pressure/bar	φd	λ^*dm
4	0.05	14.15
6	0.05	32.85
4	0.09	30.59
6	0.09	45.69
4	0.13	17.68
6	0.13	69.87

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Table 2 Optimized value of λ_dm for CH₄

Pressure/bar	φd	λ^*dm
4	0.05	0.27
6	0.05	0.28
4	0.09	0.27
6	0.09	0.20
4	0.13	0.12
6	0.13	0.23

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4.2. Determination of dispersed phase permeability (P_d)

In most of the cited literature, the pressure component is kept constant for model development and thus, permeability of dispersed phase, P_d is deemed constant for correlation. However, this work develops a technique that incorporates the effect of pressure on relative permeability, hence, P_d cannot be held constant. This is attributed to the fact that the permeation of the dispersed phase is affected by the membrane synthesis procedure and rarely equals the experimentally determined value of the isolated filler.³³ The permeability of pure CMS membranes for CO₂ and CH₄ cannot be found from the literature. Therefore, to solve this problem the CMS filler permeability should be an adjustable parameter.³² Consequently, the permeability based model can be used to predict the values of filler permeability from experimental data. The filler permeability P_d can be calculated from experimental data of MMMs. The determination of P_d is carried out by evaluating the PES-CMS MMMs data in order to get accurate values of P_d. The CMS permeability was determined by eqn (10). Based on the regression of the values of estimated CMS permeability P^*_d for CO₂ and CH₄ are tabulated in Tables 3 and 4, respectively.

Table 3 The values of estimated P^*_d for CO₂

P/bar	φd	Pm	P^*d
4	0.05	42.41	600.0
6	0.05	29.71	976.0
4	0.09	42.41	1297.3
6	0.09	29.71	1357.4
4	0.13	42.41	750.0
6	0.13	29.71	2075.8

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Table 4 The values of estimated P^*_d for CH₄

P/bar	φd	Pm	P^*d
4	0.05	42.41	2.43
6	0.05	29.71	1.72
4	0.09	42.41	1.90
6	0.09	29.71	1.22
4	0.13	42.41	1.08
6	0.13	29.71	1.37

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4.3. Determination of φ^*_ad

The determination of φ^*_ad is carried out by using eqn (8), this equation describes the volume fraction of amine φ_a and volume fraction of dispersed phase φ_d. As detailed earlier, the presence of amine has enhanced the performance of amine MMMs. In addition, eqn (8) also includes the facilitation parameter F_c which was determined by Delaunay triangulation in MATLAB® environment. The values of F_c are bounded by 0.13 < F_c < 0.27 for 10 and 15 wt% addition of DEA in PES-CMS-DEA amine MMMs. F_c is the function of volume fraction of filler and amine. The data is tabulated in Table 5 for CO₂ and CH₄.

Table 5 The values of φ^*_ad for CO₂ and CH₄

Pressure/bar	φa	φd	φ^*ad
4	0.1300	0.0500	0.3455
6
4	0.1300	0.0900	0.3637
6
4	0.1300	0.1300	0.5330
6

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5. Modeling results

The estimated parameters i.e. φ^*_ad, P^*_d and λ^*_dm are used to determine P_rcal by using eqn (7) across the operating pressure from 2–10 bar, volume fraction of filler from 10–30 wt% and amine concentration which is 15 wt%. Later on, the calculated relative permeability P_rcal is compared with experimental relative permeability P_rexp.

Table 6 shows the comparison of P_rexp and P_rcal of CO₂ across the PES-CMS-DEA 15 wt% amine MMMs which is predicted by the modified Maxwell model along with AARE %. A good agreement between predicted and experimental values of CO₂ relative permeability across all pressure ranges and combined volume fractions for amine MMMs (PES-CMS-15 wt% DEA) was observed. The values of AARE % was found in the range of 5.7 to 11.9% for the selected pressure range. Fig. 3 shows the comparison of calculated permeabilities of CO₂ using the modified Maxwell model with experimental data. As expected, an acceptable agreement is seen between the calculated and experimental data for PES-CMS-15 wt% DEA amine MMMs.

Table 6 The values of the model parameters obtained in the present model in PES-CMS-DEA 15 wt% MMMs system for the relative permeability of CO₂

Pressure/bar	φ^*ad	Pm	P^*d	λ^*dm	Prexp	Prcal	AARE %
4	0.3455	42.41	600.00	14.15	1.95	2.17	11.9
6		29.71	975.98	32.85	2.72	2.38
4	0.3637	42.41	1297.32	30.59	2.37	2.48	5.7
6		29.71	1357.45	45.69	2.39	2.55
4	0.5330	42.41	750.00	17.68	2.92	3.47	9.8
6		29.71	2075.84	69.87	4.16	4.13

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Fig. 3 CO₂ relative permeability: experimental data versus model values predicted by the modified Maxwell model.

The presented results showed that there is a slight difference in predicted and experimental values of CO₂ relative permeability (P_r). This might be due to a number of factors: (a) the modified model (eqn (7)) has been derived from the basic Maxwell model which was originally used to estimate electrical conductivities of composite materials and has been shown to produce good results.²³ However, dependence of electrical proprieties is related to percolation threshold. As this critical value is reached a dramatic increase on conductivity is noticed. However, mass and electrical charge transport in mixed matrix membranes are not completely analogous phenomena.³⁴ (b) In addition, the Maxwell model is proposed for a two-phase system as reported in previous cited literature.²² In this study the model is modified for the incorporation of a third component (DEA) so there are some possible limitations such as complete absorption of amine concentration parameter, φ_a, in the Maxwell model. (c) The synthesized membranes (amine MMMs) obey three transport mechanisms: (i) solution diffusion, (ii) molecular sieving and (iii) facilitated transport, while the Maxwell model is based on MMMs which has only two mechanisms (i) solution diffusion, (ii) molecular sieving, which limits the performance of the developed model. (d) Another reason of deviation is the formation of carbamates and protonated amine due to the reaction of CO₂ with amine. The developed model does not address such formation of carbamates and protonated amine for the prediction of amine MMMs performance. Beyond these limitations of the Maxwell model and newly developed model, the difference of predicted and experiment results is small enough to consider that this model is good enough to predict the performance of amine MMMs at given conditions.

Table 7 shows the comparison between experimental and predicted relative permeability of CH₄ which is predicted by the modified Maxwell model along with AARE %. The values of AARE % were found in the range of 0.0–1.6% for the selected pressure range. An excellent agreement between predicted and experimental values of CO₂ and CH₄ relative permeability across all pressure ranges and combined volume fractions for enhanced MMMs (PES-CMS-15 wt% DEA) was observed.

Table 7 The values of the model parameters obtained in the present model in PES-CMS-DEA 15 wt% MMMs system for the relative permeability of CH₄

Pressure/bar	φ^*ad	Pm	P^*d	λ^*dm	Prexp	Prcal	AARE %
4	0.3455	8.86	2.4	0.27	0.71	0.70	0.7
6		6.08	1.7	0.28	0.71	0.71
4	0.3637	8.86	1.9	0.22	0.65	0.66	1.6
6		6.08	1.2	0.20	0.64	0.65
4	0.5330	8.86	1.1	0.12	0.46	0.46	0.0
6		6.08	1.4	0.23	0.53	0.53

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Fig. 4 shows the comparison of calculated permeabilities of CH₄ using the modified Maxwell model with experimental data. As expected, an excellent agreement was observed between the calculated and experimental data for PES-CMS-15 wt% DEA amine MMMs.

Fig. 4 CH₄ relative permeability: experimental data versus model values predicted by the modified Maxwell model.

The results showed that the CH₄ predicted relative permeability is closer to the experimental relative permeability. As CH₄ has zero affinity with DEA, CH₄ undergoes transport through a solution diffusion mechanism across amine MMMs. In addition CMS also hindered the transport of CH₄ due to the kinetic size of CH₄ which is equal to the kinetic size of CMS. Therefore, the newly developed model is mostly behaving like the basic Maxwell model for CH₄. Thus, the CH₄ transport prediction across amine MMMs is quite close to the experimental results.

Additionally, the optimized permeability ratio λ^*_dm for CO₂ is much larger than the CH₄ optimized permeability ratio. Similarly, the new P^*_d values are exhibit the same trend. The difference in the values is due to the sizes of gas molecules which are transported though amine MMMs. In the amine MMMs the large values of λ^*_dm and P^*_d for CO₂ is due to the high affinity of CO₂ with DEA and small kinetic diameter of CO₂ (3.30 Å) relative to CMS kinetic diameter (3.80 Å), which enhances the transport of CO₂ through amine MMMs. The small values of λ^*_dm and P^*_d for CH₄ is due to the lower affinity of CH₄ with DEA and equal kinetic diameter of CH₄ with CMS, which offered resistance in the transport of CH₄ through amine MMMs.

As also summarized in Tables 6 and 7, the AARE % for different pressures and loading of CMS for CO₂ and CH₄ using the new approach for PES-CMS-15 wt% DEA amine MMMs are significantly low. This shows that modified Maxwell model successfully predicts the CO₂ and CH₄ permeability. The low values of AARE % is due to the use of more accurate values of pure CMS permeability³³ for CO₂ and CH₄, which is estimated by using the stated approach.

The developed approach in this study is different from literature because it is predicting the behaviour of three-phase MMMs with ideal morphology. However, current literature mostly considers the non-idealities of membranes. Recently, Gheimasi et al. (2015) proposed a new model to investigate the effects of particle shape and undesirable defects (polymer chain rigidification, partial pore blockage and void formation). The results showed low average absolute relative error (AARE) of the model for the MMMs of about 0.73–31.53%.³⁵ Similarly, Bakhtiari et al. (2015) proposed an analytical model to predict the MMM permeation performance by incorporating impermeable filler particles and used this to investigate the permeation and concentration distribution through MMMs with ideal morphological structure. A very good agreement with the current model result with the experimental data was found (e.g. AAREs were reduced from 52.65 to 31.74%, from 51.12 to 28.22% and from 55.33 to 31.50% for the Maxwell, the Bruggeman and the Pal models, respectively, in their proposed model). However, while these studies are good enough to predict the performance of MMMs with undesirable defects, as far as a third component is concerned; to our best knowledge, there is no recent study available to predict the third component MMMs performance via existing models. Furthermore, these cited approaches focused on MMMs only; therefore the comparison of current study results with cited literature is not possible. However, we can evaluate the performance of the current approach by that AARE % of the developed approach via Maxwell is lower than cited literature.

It is concluded that the accuracy of the current approach is sensitive to the values of λ^*_dm and P_d. The determination of P_d is essential because the permeation property of pure filler membranes is likely different from that in MMMs. The next section highlights the model validation to test this approach for CO₂ permeation though PES-CMS-DEA amine MMMs.

6. Validation of models

Normally, the developed models are validated upon external experimental data sources. However, due to the unavailability of data for amine MMMs, another set of experiments using 10 wt% DEA in MMMs was used. The values of λ^*_dm and P^*_d are as previously, and eqn (7) and (8) were extended for the validation of the model. By using the optimized λ^*_dm and estimated P^*_d, the AARE % of all fittings using new approach for PES-CMS-10 wt% DEA MMMs are summarized in Table 9 for CO₂. Table 8 shows the values of φ^*_ad for PES-CMS-10 wt% DEA MMMs. Fig. 5 portrays the comparison of P_rcal of CH₄ using the modified Maxwell model with P_rexp. As expected, an excellent agreement between the calculated and experimental data is observed for PES-CMS-10 wt% DEA amine MMMs.

Table 8 The values of φ^*_ad for CO₂ and CH₄

Pressure/bar	φa	φd	φ^*ad
4	0.1000	0.0500	0.3025
6
4	0.1000	0.0900	0.3205
6
4	0.1000	0.1300	0.4658
6

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Table 9 Comparison of the predicted values of the relative permeability of CO₂ in PES-CMS-10 wt% DEA amine MMMs and experimental data

Pressure/bar	φ^*ad	Prexp	Prcal	AARE %
4	0.3025	1.85	1.98	4.65
6	0.3025	2.10	2.15
4	0.3205	1.88	2.23	11.54
6	0.3205	2.19	2.29
4	0.4658	2.68	2.96	9.78
6	0.4658	3.76	3.42

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Fig. 5 CO₂ relative permeability: experimental data versus model values predicted by the modified Maxwell model.

Table 9 shows the satisfactory agreement between predicted and experimental results with R² = 0.92 and small AARE % is observed, which shows the validity of developed approach. The small value of AARE % is obtainable by the use of new values of CO₂ permeability of pure CMS particles, which are estimated by using a regressive algorithm. Table 9 shows that the predicted values and experimental values are in good agreement. On the other hand the AARE % shows an increase at higher loading. This can be attributed to the fitting of model parameters at higher amine concentration than that used for validation. Such effect can be mitigated by introducing an amine concentration correction factor. However, this introduction adversely affects the simplicity and efficiency of the model.

In conclusion, an effective theoretical and experimental approach has been developed for fitting gas permeation data of MMMs and amine MMMs with ideal morphologies. This strategy can be applied with no limit on the range of filler volume fraction φ_d. The reasonable values of gas permeance of filler P_d are easily predicted and extended to a range of parametric conditions. The value of error is for all amine MMMs is <15%. The proposed model will be useful to enhance the knowledge related to gas diffusion across amine MMMs.

Some of the predicted data from this work cannot be qualitatively and quantitatively explained by the model presented in this study. The simpler Maxwell analytical model of MMMs highlights the fact that at this stage experimental membranes of this study differ from the idealized models in many ways. Therefore, there is need to do more work on both fronts. Theoretical models need to include more of the relevant parameters (as we try to contribute by incorporation of amine concentration φ_a and the effect of facilitation factor and non-idealities F_c) and a more accurate picture of experimental membrane geometry, true mechanism, perhaps including formation of ionic species. Meanwhile, systematic studies that link well with ionic species (carbamates and protonated amines) formation to the experimental parameters will allow the rational design of membranes with the potential for exceptional performance.

7. Conclusions

The modeling of amine MMMs was done using a modified Maxwell model. A new concept using the Maxwell model as the base model was developed by considering the presence of alkanolamine. A three-phase system; containing inorganic phase, polymer phase and alkanolamine concentration accounted for all the experimental variables. The equation proposed by Maxwell was modified by introducing the φ^*_ad, combined volume fraction of CMS and amine as well as the effects facilitation (F_c). In addition, λ^*_dm which was defined as the optimized ratio of dispersed phase permeance to polymer matrix permeance was also incorporated in the modified Maxwell model. The results showed good agreement between experimental and predicted values of CO₂ and CH₄ relative permeability across all pressure ranges with minimum AARE %. Similarly, when these values have been plotted against P_rexp and P_rcal, satisfactory R² values has been observed. In addition, this research is seen to be very useful to provide better understanding of the amine MMM separation performance enhancement by incorporation of the third component (DEA).

Nomenclature

AARE %	Average absolute relative error (%)
Pd	Permeability of dispersed phase
Pm	Permeability of polymer matrix
NDP	Number of data points
Pr	Relative permeability
PC	Permeability of continuous phase (polymer matrix)
P^*d	Estimated permeability of dispersed phase
Fc	Facilitation parameter
MMM	Mixed matrix membranes
PES	Polyethersulfone
CMS	Carbon molecular sieve
DEA	Diethanolamine

Greek letters

φd	Volume fraction of dispersed phase
λdm	Permeability ratio of dispersed phase to polymer matrix
φm	Maximum packing volume fraction of filler particles
φ^*ad	Combined volume fraction of filler and alkanolamine
φa	Volume fraction of alkanolamine solution

Superscripts

exp	Experimental
cal	Calculated

Subscripts

m	Matrix
d	Dispersed phase
a	Amine
C	Continuous phase
ad	Amine and dispersed phase

Acknowledgements

The authors acknowledge the financial and technical support provided by Universiti Teknologi PETRONAS Malaysia.

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