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Abstract-In the domain for the improvement of acid gas removal technologies from valuable product gas streams by membrane application, this paper discussed gas permeation theoretical models for mixed matrix membranes (MMMs). The models that were considered include Maxwell model, modified Maxwell model, Bruggeman model, Lewis-Nielson model, Pal model, Felske model and modified Felske model. Experimental permeability data on mixed matrix membrane (MMMs) were used to evaluate the selected theoretical models. Comparison of those models based on absolute average relative error percent was conducted. Moreover, present methodology was proposed on modifying Pal model to account for non ideal performance of three phase MMMs and hence minimize the higher deviations obtained by using the original Pal model. The present methodology for the estimation of permeability in MMMs was in a good agreement with experimental literature data.

Key words- mixed matrix membrane (MMMs), CO2 permeability prediction, theoretical models.

I. INTRODUCTION Natural gas is a vital component of the world's supply of

energy. It is one of the cleanest, safest, and most useful types of energy sources. Natural gas produced at the wellhead usually contains impurities such as CO2, H2S, water vapor and higher hydrocarbons, which must be removed to meet the pipe-line quality standard specifications: <2% for CO2, <4 ppm for H2S, and <0.1 g/m3 for H2O [1].

According to reference [2], the annual U.S. production of natural gas was 5.6×1011 m3 (STP), and approximately 20% of this gas contains excess CO2 which must be removed to meet the U.S. pipeline specification, i.e., 2 vol. % or less. Membrane technologies represent an alternative method of acid gas removal without many of the difficulties associated with conventional acid gas separation techniques [3]. Mixed matrix membranes (MMMs) are hybrid membranes that involve the latest membrane morphology emerging with the potential for future applications containing inorganic fillers such as rigid molecular sieving materials of superior gas separation properties embedded in an organic polymer matrix for ease of processability and economics [4].

To make efficient use of the mixed matrix membranes, the variation of permeability of a penetrant with the kind and concentration of filler materials should be known. Knowledge of the permeabilities of different penetrants is required for the design and operation of a mixed matrix membranes (MMMs) separation process.

The proper theoretical description of the permeability of composite polymeric materials such as MMMs in order to model their permeation properties is of great interest, particularly in view of the growing technological importance of these materials [5].

In the open literatures, the existing models for permeation through mixed matrix membranes (MMMs) are reported as adaptations of thermal/electrical conductivity models. Since there exists a close analogy between thermal/electrical conduction in composite materials and permeation of species through such materials, the conductivity models are readily adapted to permeability of species in mixed matrix membranes (MMMs) [6].

The models that were considered include Maxwell model and its modified model, Bruggeman model, Lewis-Nielson model, Pal model, Felske model and its modified model. The existing models for gas permeation through mixed matrix membranes were reviewed, compaired and a new methodology was proposed to modify Pal model.

II. BACKGROUND Mathematically, gas transport through a mixed matrix

medium presents a complex problem. Several theoretical models have been used to predict the permeation properties of mixed matrix (or heterogeneous) membranes as functions of the permeabilities of the continuous and dispersed phases. The Maxwell model [7], originally developed for electrical conductivity of particulate composites, can be adapted to permeability as: 2 1 1 22 1 (1)

Where is the relative permeability of species, is the effective permeability of species in MMMs, is the permeability of species in the matrix (continuous phase), is the volume fraction of the filler particles, and is the permeability ratio / ( is the permeability of species in dispersed phase).

The Bruggeman model [8], originally developed for the dielectric constant of particulate composites, can be adapted to permeability as:

= 1 (2) The Lewis–Nielsen model [9, 10], originally proposed for

the elastic modulus of particulate composites, can be adapted to permeability as:

Gas Permeation Models in Mixed Matrix Membranes

Biruh Shimekit, Hilmi Mukhtar Chemical Engineering Department Universiti Teknologi PETRONAS

Tronoh, Malaysia mail2biruh@yahoo.com, hilmi_mukhtar@petronas.com.my

1 2 1 / 21 1 / 2 (3) where 1 1 (3a)

And is the maximum packing volume fraction of filler particles ( is 0.64 for random close packing of uniform spheres).

The Pal model [11], originally developed for thermal conductivity of particulate composites, can be adapted to permeability as: = 1 (4)

The modified Maxwell model, Eqs. (5) and (6) considered the influence of interfacial layer on the permeability of species in MMMs to determine the permeability of three-phase MMM [12]-[14]. Thus, 2 1 1 2 ⁄2 1 ⁄ (5) Where is the volume fraction of total dispersed phase (core-shell particles) in the whole composite and is the effective permeability of a single core-shell particle given as: 2 1 1 2 ⁄2 1 ⁄ (6) Where s is the volume fraction of filler core particle in the combined volume of core and interfacial shell (in a single core-shell particle), and is the permeability of the interfacial shell. The modified Maxwell model, Eqs. (5) and (6), has the same limitations as observed in the case of the Maxwell model (Eq. (1)).

The Felske model [15] recently developed an exact expression for the thermal conductivity of composites of core-shell particles (core particle covered with interfacial layer). The Felske thermal conductivity model, when adapted to permeability (replacing thermal conductivity by permeability), can be written as:

⁄⁄ (7) Where and are given as: 2 2 1

2 2 1 (8) 1 2 1 ⁄

1 2 1 (9) where is the ratio of outer radius of interfacial shell to core radius, is the volume fraction of core-shell particles (volume fraction of total dispersed phase, filler core particles with interfacial layers), is the permeability in the interfacial shell, is the permeability in filler core particle,

is the permeability in matrix, is the permeability ratio / , is the permeability ratio / , and is the permeability ratio / .

The Felske model gives almost the same predictions as the modified Maxwell model (Eqs. (5) and (6)). The Felske model, although somewhat simpler than the modified Maxwell model, Eqs. (5) and (6) has the same limitations as that of the modified Maxwell model.

To account for the morphology and packing difficulty of particles, the following modified-Felske model was reported: [10]

1 2 / 21 / 2 (10)

Where and are given by Eq. (8) and (9), respectively and is given by Eq. (3a) in terms of (maximum packing volume fraction of core-shell particles).

One of the challenging gas separation problems in process engineering is that of CO2/CH4 system due to its significance in natural gas purification. Moreover, the challenge faced by existing membranes have been known to be the difficulty of increasing the CO2 permeability to a sufficiently high rate to achieve high selectivity under the operating condition of interest so as to fulfill the expected pipeline quality standard [16] - [22].

To be more specific, this paper focused on studying the prediction of CO2 permeability in MMMs. As has been discussed on the background section, one of the models, the Pal model, was reported to be capable of taking into consideration the effects of particle size distribution, particle shape and aggregation of particles through the term . However, as to the open literature surveyed so far, the researchers have found out that only limited publications have been reported that attempted to modify the existing Pal model towards prediction of better CO2 permeability using three phases MMMs. Moreover, it has also been reported that the Pal model is an implicit relationship that needs to be solved numerically for permeability. Therefore, the researchers proposed the modification of the Pal model by taking into account the non-ideal performance of MMMs induced by interfacial rigidified matrix chains and reported its findings on subsequent sections.

III. MODEL DEVELOPMENT Calculating the relative permeability of an ideal MMM (a

two phase system consisting of molecular sieve and polymer) using Pal model is an implicit relation and the model equation need to be solved numerically for ideal morphology (no defects and no distortion of the separation properties of the individual phase). However, factors leading to non ideal performance in MMM, such as the effect of interphases are known to occur for real cases. Therefore, the researchers proposed modification of the Pal model by incorporating one of the non ideal cases, the presence of interfacial rigidified matrix chains while numerically solving for the relative permeability so as to reasonably estimate and in turn to systematically search for methods to offset the deviations reported by the basic Pal model.

Several possible hypotheses that are reported to explain deviations of the predicted permeability includes but not limited to the following: poor choice of model, incorrect neat polymer or pure sieve data, poor compatibility of the two phases resulting an interphase, blockage of the sieve pores by residual solvent or adsorbed polymer, rigidified matrix chains or reduced mobility of the polymer chains near the sieve surface, loading of inorganic, and inorganic particle size etc. Moreover, reliable data especially for the inorganic materials cannot be found in the literature because of too many experimental conditions that may influence the separation properties of inorganic membranes, such as the number of defects on the surface, the type of supporting layer, inorganic membrane thickness, etc.

As stated on the previous sections, the factors that will be incorporated in the modification of the Pal model will be limited to the case of the presence of interfacial rigidified

matrix chains. Although most literatures reported that reliable data for intrinsic permeability of molecular sieves will be limited, the researchers limited this study to the available permeation data of the MMMs comprising of Matrimid CMS for CO2/CH4 separation as given on [23]. Thus, the researchers collected data and compared the available theoretical models towards their CO2 relative permeation behavior against volume fraction of filler particles and reported percentages of absolute average relative error using Eqs. 11 [5]. Their results are show on Table I.

% 100

(11)

TABLE I. COMPARISON OF THE AVERAGE ABSOLUTE RELATIVE ERROR PERCENTAGE OF THE THEORETICAL MODELS WITH THE MODIFIED PAL MODEL

Theoretical Models %

Maxwell model[7] 26.04

Modified Maxwell model[12]-[14] 12.92

Bruggman model [8] 28.10

Lewis Nielsen model[9, 10] 30.95

Felske model[15] 4.50

Modified Felske model[10] 3.84

Pal model[11] 33.49 From Table I, one can deduce that except for Felske and

its modified model, all the other models were shown to predict the permeation higher than the actual observations. At this stage, one can critically think of the probable reasons of the deviations of the permeation models from experimental observation and the researcher considered the hypothesis of the effects of the presence of ‘interfacial rigidified matrix chains’.

As has been highlighted on the section II and as to the knowledge of the open literature surveyed so far, the Pal model is an implicit relationship that needs to be solved numerically for permeability. Therefore, the researchers proposed the modification of the Pal model by taking into account the non-ideal performance of MMMs induced by interfacial rigidified matrix chains.

The Pal model, although capable of taking into account the effects of particle size distribution, particle shape, and aggregation of particles, the basic Pal model assumes uniform polymer permeability throughout the matrix and, and, thus, cannot account for the three phase MMMs so that it is expected to exhibit significant deviation from the actual behavior as factors leading to non ideal performance in MMMs were not analyzed. Moreover, the model has a limitation of being an implicit relationship that needs to be solved numerically for permeability. Therefore, to offset the deviation, the researchers proposed modification of the Pal model by taking into account the hypothesis of the presence of the interfacial matrix chain regidification. Here, it is apparent that if one can estimate the interphase volume fraction, or equivalently, the thickness, and the interphase

permeability, the Pal model can easily be extended to these more complicated three phase systems.

Theoretical work on the influence of interface in MMMs was first proposed by [25]. Due to the existence of the interface matrix chain rigidification, a three-phase system includes the molecular sieve phase, the polymer phase and the rigidified interface layer between them. The permeability of this three-phase membrane will be obtained by applying the Pal model twice.

First, the researcher introduced the parameter from Flesk model where is assigned to be the ratio of outer radius of rigidified interfacial layer to core radius [6]. The rigidified interface is assumed to occur due to the inhibition of polymer chain mobility near the polymer-molecular sieve interface; in other words, the presence of molecular sieves or zeolite was assumed to rigidify polymeric chains. This phenomenon is an extension of behavior observed in the semi-crystalline polymer literature. Crystallites appear to inhibit polymer chain mobility within the amorphous phase near the crystal, resulting in higher activation energy for diffusion through the amorphous phase [23]. Thus, The permeability in the interface is assumed to be calculated by β. where is the permeability of the polymer matrix and β is called matrix rigidification or chain immobilization (decline in permeability) factor that is obtained by least square fitting of the intrinsic MMMs permeation data [13], [23-24].

Second, from Eq. (4), a revised version of the Pal model was proposed using Eq. (12-16) to obtain the permeability of the combined interfacial rigidified matrix chains and molecular sieve phase with the interfacial layer as the continuous phase and the molecular sieve phase as the disperse phase:

where is the permeability ratio ( is the permeability of species in dispersed phase). is the permeability of the combined sieve and interfacial rigidified matrix chain layer and it’s the unknown variable and has implicit relation that need to be solved by numerical methods. is the permeability of the dispersed or sieve phase. is the permeability in the rigidified interface layer, and the volume fraction of the sieve phase in the combined phase, given by

(14)

where is the volume fraction of the sieve phase. is the volume fraction of the interfacial rigidified matrix chains.

is the maximum packing volume fraction of filler particles ( is 0.64 for random close packing of uniform spheres) [6]. is the radius of the dispersed molecular seive. is the outer radius of the rigidified interfacial layer and is assumed to be half way the distance between the sieve

/ 1 1 (12)

/ 1 1

(13)

and the polymer obtained by FESEM cross sectional view inspection [24].

Finally, after solving for the value of the permeability of the combined sieve and interfacial rigidified matrix chains,

, the value can then be used along with the continuous polymer phase permeability, , to obtain a predicted permeability for the three-phase mixed matrix membrane by applying the Pal model a second time: / 1 1 (15)

where

� � rR (16)

� is the combined volume fraction of the sieve phase & the interfacial rigidified matrix chains in the whole system. R is the distance from the center of the sieve to boundary of the polymer surface.

In the process of modifying the existing modeling equation of Pal to find a reasonable agreement as that of experimental observation, the researcher used six set of published experimental data of Matrimid CMS MMMs for model development and three other data of Matrimid CMS MMMs for model validation. It was also assumed that the CMS particles were spherical shape with 1μm diameter (based on SEM inspection) [6], [23], [25, 26]. A value of β =3 could be assumed for matrix rigidification or chain immobilization factor β based on the approximate value in the range of typical gas penetrants in semi-crystalline polymer as reported on [23] and previous related studies [13], [24]. However, for rigorousness of the model, the researchers used the values that were obtained by least square fittings from published modified Fleske model as it was reported that the predictions weren’t sensitive to the values used for β justifying that the trend is more important than the actual fit [6], [13].

Thus, using the above input data from the open literatures, the researchers introduced the present methodology (Eq. (12-16)) on the Pal model to evaluate the value of the in the three phase mixed matrix membrane and reported the comparison results of the using the present model and experimental values with respect to volume fraction of filler particles on Table II and Fig 1.

TABLE II. COMPARISONS OF RESULTS OF USING PRESENT MODEL AND EXPERIMENTAL VALUES WITH RESPECT TO VOLUME FRACTION OF FILLER

PARTICLES FOR MATRIMID CMS MMMs.

Relative Permeability

Ref. No. Volume

Fraction of Filler

Particles Experimental Present Model

% Absolute Average Relative

Error

[23] 0.00 1.00 1.00 0.00

[23] 0.17 1.03 1.04 0.97

[23] 0.19 1.06 1. 06 0.00

[23] 0.33 1.15 1.13 1.74

[23] 0.36 1.26 1.25 0.79

Figure 1. Comparison of the present model with the different theoretical model.

From the comparison, it can be shown that the present methodology used to modify the Pal model has minimized the error of its original model by 32.79%.

IV. CONCLUSIONS The present methodology that used for modifying the Pal

model was developed to account for the presence of non ideal performance in MMMs, the interfacial rigidified matrix chains effect. Thus, the method was introduced to the original Pal model and examined for such an effect by developing three phases MMMs system to minimize the higher deviations obtained by using the original Pal model. Therefore, by applying the modified Pal model along with the known intrinsic separation properties of bulk polymer and porous filler, factors leading to non ideal performance in MMMs, such as interfacial rigidified matrix chains can be used to estimate quantitatively the overall relative permeability. Hence, the developed approach can also be used to systematically search for methods to offset the negative effect and further will be useful for the design of MMMs for commercial applications.

ACKNOWLEDGMENT

The authors would like to acknowledge the assistance provided by Universiti Teknologi PETRONAS.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.8

1

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Volume Fraction of Molecular Sieves (Φ)

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