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Modelling for Gas Transport in Enhanced Polymeric Blend Membrane.

Byline: Asim Mushtaq, Hilmi Mukhtar and Azmi Mohd Shariff

Summary: The main aim of this research work is to develop a model of carbon dioxide (CO 2) separation from natural gas by using membrane separation technology. This study includes the transport mechanism of the porous membrane. The fundamental theories of diffusion, poiseuille (viscous) flow, Knudsen diffusion and surface diffusion are used. The developed model of incorporating three diffusion mechanisms to be modified for modeling of polymeric blends towards membrane selectivity and permeability. For the purpose of assessing the gas permeance using the theoretical models, the experimental data taken from CO2 permeance in the PSU/PVAc (85/15) wt. %/DEA enhanced polymeric blend membrane was considered. The results obtained from modified Cho Empirical model of total gas permeance showed the least error as compared to other models.

The modified Cho Empirical mathematical models were extended by blending factor to predict CO2 gas molecule transport in Enhanced Polymeric Blend Membrane (EPBM) to obtain precise theoretical values that are close to the experimental values. The Extended modified Cho Empirical model validation demonstrated the ability to predict CO2 permeance with reasonable accuracy.

Keywords: Carbon dioxide; Methane; Polymeric blend membrane; Porous models; Permeance.

Introduction

Gas separation processes involve both upstream and downstream flows through the membrane. The pressure gradient occurs between these two streams facilitates separation. Permeation is the rate of gas diffuse across the membrane while the degree of separation depends on membrane selectivity under conditions of separation, which include temperature, pressure, flow rate and membrane area [1-3]. The transport of gasses across the membrane can be described using the following schematic diagram.

Fig 1 shows the two major processes sorption and diffusion that play main roles within the overall gas transport. Sorption defines the interactions between gas molecules and therefore the membrane surface, and diffusion refers to the rate of gas passage through the membrane [2, 4]. Quantitative and qualitative analysis of the involvement of these steps is important so as to grasp the gas transport mechanism. As each process will contribute to the total permeation rate, and its importance can vary allowing to such variables as pressure, temperature, and composition. Sorption of gas molecules from the bulk gaseous state to the membrane surface can occur chemically or physically liable on the nature of the force between the surface and the gas molecules [5]. In the following transport process, the adsorbed molecules diffuse through the membrane in a very numerous manner beneath the driving forces such as concentration and pressure.

The process reverses to sorption is known as desorption. This occurs in a system being in the state of sorption equilibrium among an adsorbing surface and bulk phase. Once the pressure or concentration of the substance in the bulk phase is lowered, some of the sorbed substance modified to the bulk state [6].

Table-1: Gas permeability and permeance units.

Expression###Unit###Dimension

###10-

Permeability, PA###Barrer###10 cm3 (STP) . cm / cm2 . sec . cmHg

###10-

Permeance, PA/l###Gas Permeation Unit###6 cm3 (STP) / cm2 . sec . cmHg

###(GPU)

The units of permeability and permeance across the membrane are given in Table-1 [7]. Generally, membranes models can be divided into two categories according to their structural characteristics, dense and porous membranes. The dense membranes are free of discrete structure. The difference between the dense and porous can be conveniently detected by the presence of any pore structure beneath electron microscopy. The efficiency of a membrane strongly depends on the selection of material used, the type of species to be separated and their interactions of species gasses with the membrane [8].

Theory

Gas Transport Mechanism in Porous Membrane

The properties of gas flow in porous media can be determined by the ratio of molecule-molecule collisions and the molecule-wall collisions as shown in Fig. 2.

The membrane dependent on membrane pore size and the sizes of the pore depending the process gasses. The pores characteristics of porous materials have a very complex structure and morphology and many studies have been devoted to describing and characterizing them [6, 8]. Schematic diagram of different types of pores is given in Fig. 3. As can be seen in the diagram, isolated pores, and dead ends do not contribute to the permeation in steady conditions. Dead ends do also subsidize to the porosity as measured by adsorption techniques but do not contribute to the actual porosity in permeation [9]. Pore shapes are channel - like or slit - shaped. Pore constrictions are significant for flow resistance, particularly when surface diffusion and capillary condensation phenomena occur in systems with a quite large internal surface area.

The four basic transport mechanisms across porous membranes are known as Poiseuille flow, Knudsen diffusion, surface diffusion and capillary condensation [7, 11]. For effective separation a mixture of chemical components, a membrane must have a high permeance and its ratio for the species enduring separation [7]. Permeance for a certain species diffusing through a membrane of a known thickness is closely resembling a mass transfer coefficient for the flow rate of that species per unit of cross-sectional area of membrane per unit of driving force [7]. The molar trans-membrane flux of a species i is given by;

(EQUATION)

where P'i is the permeability of gas species i, fd is driving force and tm is membrane thickness. When a mixture on each side of a microporous membrane is gas, Fick's law expresses the rate of a species diffusion. When pressure and temperature on either side of the membrane are equal, and the ideal gas law holds, the trans-membrane flux is expressed in terms of a partial pressure driving force as follows [7]:

(EQUATION)

where Cm is the total concentration of the gas mixture given as P/RT by the ideal gas law. Thus, equation (2) can be written otherwise as [7]:

(EQUATION)

The detail description of the four basic transport mechanism is discussed in the following section.

Poiseuille flow

Viscous (Poiseuille) flow plays an important role in the macroporous substrate(s) supporting the separation layer and can affect the total flow resistance of the membrane system. Gas flow takes place by normal convective flow i.e. r/I>> > 1 if the pores of a microporous membrane are 0.1 microns or larger [12]. The Poiseuille flow is also known as viscous flow. The assumption that the pore resembles a perfect cylinder is necessary to model the viscous flow in the pore [11]. This assumption is practical for a piece of thin membrane with a pore size from 1-7 nm, as the gas molecules will collide more frequently with each other than their collision through the cylinder wall, under this condition [13]. Fig 4 show the viscous diffusion.

(EQUATION)

where J is the flux, l is the pore length, dp is the pore diameter, IP is the pressure difference across pore, u is the solvent viscosity, is the porosity (I dp2 N/4, where N is number of pores per cm2), typical pore diameter: MF - 1micron; UF - 0.01 micron.

The average velocity, Vav of gas molecule is defined as:

(EQUATION)

where ph is the high pressure, pl is a low pressure, ui is the viscosity of gas i, t is the time and rp are the radius of the pore.

For total volumetric flow rate across the whole piece of membrane, the number of pores qp can be calculated as,

(EQUATIONS)

where Lm is the length of the cylindrical pore and Rm is the radius of the cylindrical pore

(EQUATION)

The value of gas permeability can also be determined experimentally as was done by Lee and Hwang in 1985 [14];

(EQUATION)

From equation (8) and (9), the permeability of gas molecule through the membrane pores, due to viscous diffusion, can thus be calculated as such:

(EQUATION)

Equation (10) shows that the permeability of gas molecule does not depend on the pressure of the system. It is only a function of the membrane pore size, tortuosity, porosity and the viscosity of the gas. It is important to note that the permeability of gas is a function of temperature indirectly, as the viscosity of gas varies with system temperature. The viscosity of gas can be computed by using the empirical correlation as established by Bird et al. [15]:

(EQUATION)

Knudsen Diffusion

Mesoporous separation layers are commonly in the transient - regime among Knudsen diffusion and molecular diffusion, with large effects on the selectivity (separation factor). Convective flow will be replaced by Knudsen diffusion in a porous membrane, whose pore sizes are less than the mean free path of the gas molecules [16]. Knudsen diffusion arises when the ratio of the pore radius to the mean free path (I>> ~ 0.1 microns) of a gas molecule is less than 1. Diffusing gas molecules then have further collisions with the pore walls and other gas molecules as shown in Fig 5. Gasses with high DK permeate preferentially [12].

For an equimolar feed, the permeation rate of Knudsen diffusion is inversely proportional to the square root of the molecular weight of the different compounds in the following equation [17]:

(EQUATION)

where DK (m2/s) is the Knudsen diffusion coefficient, rp is the average pore radius, v is the average molecular velocity (m/s) and T is the operating temperature.

According to Seader and Henley [7], the ordinary and Knudsen diffusions, Di and Dk, i of gas species i can be estimated using equation (13) and (15), respectively written as:

(EQUATION)

Gas diffusion through a pore occurs by ordinary diffusion and/or in series via Knudsen diffusion when the pore diameter is very small and/or total pressure is low. In the Knudsen-flow regime, more collisions occur between gas molecules and the pore wall than between gas molecules. In the absence of bulk-flow effect or restrictive diffusion, equation (14) is modified to account for both mechanisms of diffusions:

(EQUATION)

where Dei is the effective diffusivity, DKi is the Knudsen diffusivity, which from the kinetic theory of gases as applied to a straight, cylindrical pore of diameter dp is

(EQUATION)

where vi is the average molecule velocity given by [10]:

(EQUATION)

where M is molecular weight. Combining equation (15) and (16):

(EQUATION)

where DK is cm2/s, dp is cm and T are K.

By integrating the equation (1), (3) and (14), the final equation to calculate the permeability of gas species i through the membrane due to Knudsen and ordinary diffusions can be expressed as [7]:

(EQUATION)

Surface Diffusion

When the temperature of the gas is such that adsorption on pore walls is important, experimental results illustrate that the previous laws for gaseous flow are no longer effective [10].

For comparatively low surface concentrations, the surface flux, Js, for a single gas is generally described by the two-dimensional Fick's law [10]:

(EQUATION)

The surface concentration, Cs, can be correlated with the membrane density Im, and the uptake of the gas molecules by the sorbent material h, which has the unit of [mol.g-1] by the following equation [18].

(EQUATION)

By inserting equation (20) into equation (19) yields

(EQUATION)

The uptake of the gas species by the sorbent material, h, is approximated by Henry's law (monolayer adsorption is assumed to take place) to be directly proportional to the equilibrium loading factor, f, and system pressure as below:

(EQUATION)

Dimensional analysis of equation (22) yields;

(EQUATION)

Keizer et al. showed that f is directly proportional to pressure and inversely proportional to temperature [19]. By inserting h from equation (22) into equation (21) yields:

(EQUATION)

where Ds, can be computed from the following empirical relation as established by Bird et al.:

(EQUATION)

where m is 2 for conductive sorbent and 1 for non-conductive sorbent and IH is the specific enthalpy [7]. The heat of adsorption of the gas species to the sorbent material can be approximated with the assumption that condensation occurs on the surface.

It can be estimated by using Trouton's law and Watson Correlation [20].

(EQUATION)

With gas mixtures, enhancement of the separation factor can be obtained by preferential sorption of mobile species of one of the components of the gas mixture. Adsorption does not always lead to enhanced separation. In a mixture of light non-adsorbing molecules and heavy molecules, the heavy molecules move slower than the lighter ones but in many cases are preferentially adsorbed. Consequently, the flux of the heavier molecules is better enhanced by surface diffusion and the separation factor increases.

Pore distribution of the membrane material is normally not uniform and the pores can have very different shape, orientation, and length from each other. Thus, the diffusion of gas molecules through all the pores in a membrane system may not be necessary uniform or successful. Effective diffusion, De, is introduced to cater for this discrepancy between pores, and it can be obtained from Fick's Law as [7]:

(EQUATION)

JT the total flux that comprises the flux via gas diffusion and surface diffusion.

Substitution of C = P / zRT into equation (26) yields,

(EQUATION)

From the true definition of JT, the total flux can be written as,

(EQUATION)

By equating equation (28) to equation (26), De is obtained as such,

(EQUATION)

Gas Permeation Models for Porous Membrane

Permeability is a significant parameter in membrane performance. It gives an overview of permeation behavior of a certain gas through a specific type of membrane, either less permeable or highly permeable. The measurement of gas permeability is usually done on pure gas species. The following literature will illuminate and discuss some of the gas permeability models developed by numerous researchers. The permeability, P'i for a pure gas i can be measured from the Seader and Henley 1998 equation (18) [7].

Cho et al. 1995 developed an empirical model for gas permeability prediction based on the three important transport mechanisms in the membrane, namely viscous flow, Knudsen diffusion and surface diffusion [21]. The model is shown as follows:

(EQUATION)

The first term in the above equation represents the viscous flow. The second term caters for Knudsen diffusion whereas the third term describes the surface diffusion that occurs in the pores. It was illustrious that the second term is not dimensional homogeneous with the first and third term [21]. It should not contain the thickness of the membrane in the equation. Equation (30) has been improved from this error so that it is dimensional homogeneous. The error in the relation as developed when the pore size is so small that it approximates the size of a gas molecule, the effect due to pore refining is not obvious [21]. Moreover, Knudsen diffusion will not occur when the pore size is smaller than the size of the gas molecule. The value of gas permeability can also be determined experimentally as was done by Lee and Hwang in 1985 [14]. It was shown that the gas permeability can be described by the following relation as shown in equation (9).

From equation (9), it is apparent that permeability of the gas through the membrane will increase with respect to rising in qp, if the other parameters stay the same. This is true, as more gas permeates through the membrane surface. However, the permeability of gas should increase, by right, when the thickness of membrane decreases and not the other way around as suggested by equation (9). This is because the distant of travel of the gas molecules will decrease as the thickness reduces and thus the gas molecules need the least time to complete the whole path. But, do not neglect the fact that the actual lengths of travel, t of the gas molecules are generally much longer than the membrane thickness, tm, due to the intrinsic structure of the pores. As seen from equation (18), the permeability of gas is inversely proportional to the tortuosity of the membrane material, which is a ratio of t to tm [7, 22].

Hence, it can be concluded that equation (10) is actually true, as the increase of tm will result in a less tortuous membrane material and consequently shows an increase in permeability.

In the Dusty Gas Model (DGM) as presented by Mason and Malinaukas, all the different contributions to the transport are taken into account [9]. According to the model assumption, the wall of the porous medium is considered as a very heavy component and so contributes to the momentum transfer. The model is schematically represented in Fig 7 for a binary mixture. As can be seen from this electrical network analogy, the flux contributions by Knudsen diffusion Jk, i and molecular diffusion of the mixture Jm,12 are in series and so are coupled. The total flux of component i (i=l,2) due to these contributions is Ji,km* The contribution of the viscous flow Jv, i and of the surface diffusion Js, i are parallel with Ji, km and so are considered independent of each other. There is no transport interaction between gas phase and surface diffusion. The flux expression for single species i in a multi - component mixture with n components according to the DGM model results in [10].

(EQUATION)

where (EQUATION) with Kn as the Knudsen number.

(EQUATION)

Present and De Bethune was the first to develop a model (P - D model) including diffusion, intermolecular momentum transfer, and viscous flow [23]. Based on the P-D model, Eickmann and Werner incorporated two parameters (nk and [beta]) in the P - D equations to account for geometric and reflection characteristics of a real membrane [17]. This extended P - D model is very successful to describe the effect of parameters on permeation and separation. Note that surface diffusion is not incorporated in the model. The flux of component i in a binary mixture is given by:

(EQUATION)

The mole fractions for components 1 and 2 (i = l, 2) given by x and 1-x, respectively. The first term in equation (32) describes the Knudsen diffusion while the second and third term accounts for momentum transfer and viscous flow respectively. The different coefficients in equation (32) can be obtained from Burggraaf and Cot [10].

Modeling in Polymeric Membrane

The permeability of gas molecule in porous material as combined influences by all the three types of transport mechanism such as viscous, Knudsen, and surface diffusion [24]. The total permeability of gas species i, P'i could be obtained by summation of all the three mechanisms permeability.

The trans-membrane flux of a gas species, N, can be related as follow,

(EQUATION)

By equating equation (33) with equation (27), which is N = JT,

(EQUATION)

Knowing that I = t / tm, hence,

(EQUATION)

By substituting the relation for De, from equation (29) into the above equation,

(EQUATION)

Permeability of gas as the result of viscous diffusion, as shown in equation (10) can be unified with equation (36) to model the characteristic model of gas permeability as a function of the three important mechanisms of transport in pores,

(EQUATION)

Dimensional analysis of the above equation shows that term 1 [m3.s.kg-1] is not dimensional homogeneous with term 2 and 3 [mol.s.kg-1]. The introduction of ideal gas law into term 1 will yield a dimensional homogeneous equation for gas permeability. From ideal gas law,

(EQUATION)

Substitution of equation (38) into equation (37),

(EQUATION)

It is important to note that the pressure as mentioned in equation (39) is the pressure in the membrane pores. However, the measurement of pressure in the pores will be cumbersome and impractical. Thus, the pressure in the pores can be approximated as the average pressure between the feed and permeate side [18]. The contribution of viscous flow towards the permeability of gas is directly proportional to the average pressure between feed and permeate side. It may lose its entire effect towards permeability at high temperature as the gas viscosity is a strong function of temperature. However, for small and fine pores, the contribution of surface diffusion is more apparent.

P'i represent the total permeability of gas i, is membrane porosity, rp is pore size, I is tortuosity, ui is the viscosity of gas i, z is compressibility factor of gas i depending on pressure, tm is membrane thickness, Im is membrane density and f is equilibrium loading factor. Meanwhile, R in equation (39) above stand for the universal gas constant which is equal to 82.06 cm3.atm/mol.K, P is the operating pressure and T is the operating temperature. Di and Dk,i signify the ordinary and Knudsen diffusion of gas i while Ds is surface diffusion. From the equation above, the membrane properties like porosity (), density (Im), tortuosity (I) and membrane thickness (tm) influence the permeability of gas species i together with operating pressure, P and temperature, T.

The first part of the equation (39) above characterizes the permeability of gas species due to viscous diffusion. The viscosity of a pure monatomic gas of molecular weight Mi using the Lennard-Jones parameters I and a|. The gas viscosity, u is carrying the unit of g/cm.s provided the unit of T in Kelvin and I in m (10-10m). The dimensionless quantity a|u is a slowly variable function of the dimensionless temperature KT/ on the order of magnitude of unity. This accounts for details of the molecular paths taken during a binary collision and is called the collision integral for viscosity. a|u is exactly unity if gasses comprise rigid spheres as an alternative of molecules with attractive and repulsive forces. Hence, this function (a|u) can be interpreted as the deviation from rigid-sphere behavior. Although equation (2) is kinetic theory result of monatomic gasses, it remarkably fits polyatomic gasses as well [14, 25].

The second part of the right-hand side of equation (36) estimates the permeability of gas species i due to ordinary and Knudsen diffusion.

The permeability of gas species i due to the surface diffusion is represented by third part of the equation (39). Surface diffusion will only occur at small pore regions, but it gives the highest selectivity due to membrane material's preferential sorptivity of certain gasses than the others. The surface diffusion, Ds, i for gas species i, could be obtained by using equation (25) as proposed by Seader and Henley [7]. For conducting adsorbent such as carbon, m is equal to 2 and for insulating adsorbents, m equal to 1 is used. Typically, the values of surface diffusivity of light gasses for physical adsorption are in the range of 5 x 10-3 to 10-6 cm2/s. In the case of a low differential heat of adsorption, larger values of Ds are applied.

In order to find out the efficiency of the membrane in separating the desired gas, an ideal separation factor, [alpha] (also known as selectivity) is calculated. The selectivity as the quotient of the permeability of two different gasses given as follow [26].

(EQUATION)

The term of [alpha]ij is representing the selectivity of gas species i to gas species j while P'i and P'j are the permeability of gas species i and j, accordingly. The higher the value of [alpha]ij means the better separation through that particular membrane has occurred.

Results and Discussion

The transport properties of PSU/PVAc blend membranes comprising an amount of amine have been examined and correlated with the morphological structure of the blend system. The porous structure by FESEM evidence, PVAc and amine are dispersed in a PSU matrix, also confirms its compatibility to form miscible blend mixtures.

Gas Transport in Enhanced Polymeric Blend Membrane (EPBM)

The development of enhanced polymeric blend membrane exhibited the good separation factor [alpha] (CO2/CH4) were in the range of 11.16 (base PSU membrane) to 31.30 (PSU/PVAc (95/5) wt. %/DEA) at 10 bar pressure [27]. Therefore, the enhancement in the performance of EPBM was due to the presence of the combined effect of polyvinyl acetate (rubbery polymer) and amine, in which the later attracts more solubility of CO2 and retards the solubility of CH4. The formation of complex mechanism can be better understood by modeling the transport of these gases using appropriate model. The transport mechanisms of these gases across these EPBM can be described as the viscous, Knudsen and surface diffusion.

The transport properties of PSU/PVAc blend membranes containing an amount of amine have been investigated and correlated with the morphological structure of the blend system. The porous structure by FESEM evidence, PVAc and amine are dispersed in a PSU matrix, also confirms its compatibility to form miscible blend mixtures [28]. The different composition of PVAc and amines were blended in PSU, change its pore diameter which effected the permeance rate according to the pressure [27]. Diffusion in PSU polymeric membrane is completely different from PVAc polymer membrane because of the difference in the characteristic scales of the micromotions that occur at a segmental level for the two states. In PSU polymeric membrane the motion of gas molecules is much less broad than PVAc polymeric membrane. It is known that the diffusion coefficient is the primary factor in determining the absolute value of gas permeability in polymers [29].

The diffusivity of gases was shown to decrease promptly as the collision diameter of the gas molecule increases. The diffusion coefficient changed ten orders of magnitude with an order of magnitude change in diameter [18]. Other molecular size parameters proposed include square root of molecular weight, molar volume, and Lennard-Jones or kinetic diameter [24]. The interactional relationship of these quantities gives distinctive results.

For the PSU membrane, surface diffusion is the dominant contributor to the total permeance of CO2 at small pores and it decreases with increasing pore size [30]. Gas transport through PVAc is based on the differences in adsorption kinetics of different gases present in the gaseous mixture. In the separation of CO2 and CH4 by PVAc, smaller (3.3oA) CO2 molecule adsorb more rapidly as compared to larger (3.8oA) CH4 molecule. For amine EPBM, the type of amines is important for the high rate of diffusion. For CO2 and CH4 separation by di-ethanolamine (DEA), the diffusion rate of CO2 is higher as compared to CH4 due to the high affinity of CO2 with DEA.

For the blend membranes PSU/PVAc/amines, at small pore sizes, the movement of the gas molecules are impeded by the narrow pathways of travel. Under this state, the gas molecules have higher tendency to diffuse from the bulk stagnant gas film to the pore surface due to the concentration gradient between bulk gas phase and pore surface. At the pore surface, adsorption of highly adsorbing CO2 gas molecules takes place and thus, contributes to the high total permeability of CO2. Due to the hindered pathways of travel, viscous diffusion and Knudsen diffusion are not apparent at very small pore sizes (<2 nm) [10]. A porous membrane, higher permeability indicates that the membrane has high porosity [2]. With reference to section 2.1 to 2.3, several models were discussed for the prediction of permeability of EPBM as Modified Cho empirical model, Cho empirical model, Lee and Hwang model, Seader and Henley model and other models [7, 10, 14, 21, 31].

The performance analysis of three existing models was carried out in section 3.2 to select the best working model for the prediction of enhanced polymeric blend membranes (EPBM).

Gas Permeance Analysis Using Modeling Approach in EPBM

When evaluating gas permeance using theoretical models, the experimental data was used from CO2 permeance in the base PSU, PSU/PVAc and PSU/PVAc/DEA blends membranes. The principal mechanisms of gas permeation in porous material consist of viscous diffusion, Knudsen diffusion and surface diffusion. Along the simulation of the models, it was assumed that the surface diffusion, which comprises the adsorption of gas molecules on the surface of the pores and then glides along the pores upon the pressure gradient, would behave as ideally as predicted by Henry's law [18, 24]. The basic models for gas permeation of porous polymeric membrane are given above. Basically, three models have been used for the performance analysis of current synthesized polymeric membrane which includes modified Cho empirical model (Eq. 39), Cho empirical model (Eq. 30), and Lee and Hwang model (Eq. 10).

These models are basically developed from transport phenomena including viscous, Knudsen and surface diffusion. Despite their practicality and advantages, these models have been evaluated to have some limitations as stated below.

The Lee and Hwang model in 1985 proposed the viscous (Poiseuille) flow of gas permeability. The permeability of gas molecule does not depend on the pressure of the system as shown in equation (10). It is only a function of the membrane porosity, pore size, tortuosity and the viscosity of the gas [14]. According to Seader and Henley model (1998), gas diffusion through a pore occurs by ordinary diffusion or Knudsen diffusion only when the pore diameter is very small and total pressure is low as shown in equation (18) [7]. Cho et al. 1995 developed an empirical model for gas permeability. It was illustrious that the second term is not dimensional homogeneous with the first and third term. It should not contain the thickness of the membrane in the equation (30) [21].

Modified Cho empirical model (2004) stated that, the permeability of gas molecule in porous material as combined influences by all the three types of transport mechanism such as viscous diffusion, Knudsen diffusion and surface diffusion [31-33]. The total permeability of gas species could be obtained by summation of all the three mechanisms permeability. It is important to note that the pressure as mentioned in equation (39) is the pressure in the membrane pores. However, the measurement of pressure in the pores will be cumbersome and impractical. Thus, the pressure in the pores can be approximated as the average pressure between the feed and permeate side.

To assess the gas permeance using the theoretical model, the experimental data are taken from CO2 permeance in the base PSU polymeric membrane, PSU/PVAc polymeric blend membranes and PSU/PVAc/DEA enhanced polymeric blend membranes was considered. The AARE% values were calculated by the following equation [34];

(EQUATION)

Evaluation of Existing Models using Base PSU Membrane

Fig 8 shows the comparison between the existing models and the experimental data for base PSU membrane. It was found that the modified Cho empirical model is closet to the experimental results as compared to other models.

Table-2: Variation of the different existing models with the experimental data for CO2 permeance of base PSU membrane

###Average Absolute Relative Error

Theoretical models###(AARE %)

###of base PSU membrane

Modified Cho empirical

model###0.66

(2004) Eq. 39

Cho empirical model

###1.65

(1995) Eq. 30

Lee and Hwang model

###68.16

(1985) Eq. 10

Table-2 shows the average absolute relative error (AARE%) between the CO2 experimental and calculated permeance determined by different models for base PSU membrane. Lee and Hwang model has a greater error as compared to modified Cho empirical model and Cho empirical model. This is due to Lee and Hwang model only shows viscous flow of gas permeability. However, in modified Cho empirical and Cho empirical model, the gas permeance occurs in three phenomena's viscous, Knudsen and surface diffusion. From the table it was found that the modified Cho empirical model of base PSU membrane, calculated permeance for CO2 are in good agreement with experimental permeance having AARE % value of 0.66 %.

Evaluation of Existing Models using Polymeric blend Membrane

Fig 9 (a, b, c and d) represents the comparison between the existing models and the experimental data for PSU/PVAc blend membranes at different feed pressure. It was found that with the addition of PVAc with different composition 5-20 wt. % in PSU, the deviation had increased in theoretical permeance as compared to experimental permeance. Modified Cho empirical and Cho empirical models show the gaps increased of calculated gas permeance of CO2 as compared to experimental permeance. The calculated permeance from Lee and Hwang model obtained is far from the experimental permeance as discussed in the previous section.

A comparative summary of the deviations between the models is listed in Table-3. When 5-20 wt. % of PVAc was blended in PSU; the AARE% increased from 0.66 % to 48.92 % in modified Cho empirical model. It was also observed that Cho empirical and Lee and Hwang model increased AARE % for CO2 permeance as obtained in the modified Cho empirical model for all polymeric blend membranes.

This deviation in theoretical permeance was due to the absence of the PVAc content in the blend membrane. The PVAc addition caused the model below predicted because the existing models were describing the transport of gases through single polymers. Hence, an additional parameter that account for the effect of blending need to be included in the modified Cho empirical model in order to be used for modeling the synthesis PBM.

Fig 10 portrays the order of the deviation based on the AARE% with increasing composition of PVAc in PSU matrix. It was found that AARE% deviation is in the increasing order as modified Cho empirical model < Cho empirical model < Lee and Hwang model. The results show that the modified Cho empirical model provided the least deviation from the other models.

Table-3: Variation of the different existing models with the experimental data for CO2 permeance of polymeric blend membranes

###Average Absolute Relative Error (AARE %) of polymeric blend membranes

Theoretical models###PSU/PVAc (95/5) wt.###PSU/PVAc (90/10)###PSU/PVAc (85/15) wt.

###Base PSU###PSU/PVAc (80/20) wt. %

###%###wt. %###%

Modified Cho

empirical model###0.66###36.17###43.56###46.46###48.92

(2004) Eq. 39

Cho Empirical model

###1.65###39.25###41.63###50.20###55.95

(1995) Eq. 30

Lee and Hwang model

###68.16###96.88###97.28###98.06###98.70

(1985) Eq. 10

Evaluation of Existing Models using Enhanced Polymeric blend Membrane

Fig 11 (a, b, c and d) portrays the comparison between the existing models and the experimental data for PSU/PVAc/DEA enhanced polymeric blend membranes at 2 to 10 bar feed pressure. It was found that with the addition DEA in PSU/PVAc blend membrane the deviation was further increased in theoretical permeance as compared to experimental permeance. This deviation shows the absence of DEA content in the enhanced polymeric blend membrane.

Table-4 shows the comparative summary of the deviations between the different models for PSU/PVAc/DEA enhanced polymeric blend membranes. However, the agreement between calculated permeance and experimental permeance for CO2 changed when DEA 10 wt. % was added in a different composition of PSU/PVAc blend membrane; the AARE% also increased from 54.99 % to 73.40 % in modified Cho empirical model. This table also shows the comparison of calculated permeance and experimental permeance values of CO2 using Cho empirical model and Lee and Hwang model. It was observed that the Cho empirical and Lee and Hwang model gave more error of calculated permeance with increased AARE% for CO2 permeance. These might be due to the same reasons as stated for CO2 permeance as the model fail to incorporate the effect of amine.

Table-4: Variation of the different existing models with the experimental data for CO2 permeance of enhanced polymeric blend membranes

###Average Absolute Relative Error (AARE %) of enhanced polymeric blend membranes

Theoretical models###PSU/PVAc (95/5) wt. %###PSU/PVAc (90/10) wt. %###PSU/PVAc (85/15) wt. %###PSU/PVAc (80/20) wt. %

###/DEA###/DEA###/DEA###/DEA

Modified Cho empirical model

###54.99###59.67###63.42###73.40

(2004) Eq. 39

Cho Empirical model (1995) Eq. 30###64.06###64.73###66.97###75.98

Lee and Hwang model (1985)

###97.31###98.53###98.67###99.19

Eq. 10

Fig 12 shows the comparison of AARE% between existing models with increasing composition of PVAc in PSU with DEA of enhanced polymeric blend membranes. This Fig shows that the modified Cho empirical model provided the least deviation as compared to the other models. For CO2 the AARE% between calculated permeance and experimental permeance was less at adding of PVAc content and high at adding of PVAc/DEA content as compared to base PSU membrane.

Selecting Suitable Model for Enhanced Polymeric Blend Membrane

The order of the deviation based on the AARE% equation (40) was found in the increasing order as modified Cho empirical model < Cho empirical model < Lee and Hwang, respectively. The Lee and Hwang model represent only single phenomena (viscous diffusion). Thus this model did not predict whole phenomena's occurs in the porous membrane. The results show that the modified Cho empirical model provided the least deviation from the experimental data. This provides a rationale to use modified Cho empirical model as the base equation. However, significant deviations were observed between the calculated data from the theoretical models and the published experimental results which trigger a need for an improved model. Thus, the analysis on the range of the results obtained from the theoretical models seems to point towards the importance of morphology factors that need to be considered as well.

Observation from the FESEM cross-sectional view of the EPBM is described, indicates that the pores are uniform and a packed bed of spheres is a perfect sphere as assumed in the theoretical model. The blend (PVAc and amine) incorporate a spherical shape in the packed bed of sphere [28, 35]. The assumptions are in good agreement with the morphology of membrane cross section. In order to account for the blend factor, modified Cho empirical model was used for the follow-up calculations.

There are some limitations with the theoretical equation as experimental values differ significantly with the theoretical values obtained from the equation (39) when PVAc and amines were blended in PSU. The experimental values have been repeated and re-evaluated to obtain reliability and assurance. Therefore, no significant change had been found with the experimental value. Since the main difference between EPBM and base membrane is caused by blending, a 'blending factor' can be introduced to incorporate the effects of blending in the EPBM.

Algorithm of Gas Permeance Modelling

To evaluate the gas permeance using the theoretical permeation models, experimental data was taken from CO2 permeance in EPBM.

Extended Modified Cho Empirical Model

As discussed, modified Cho empirical model was used as the base model in this study. The basic modified Cho empirical model equation (39) is chosen on the basis of some characteristics such as the summation of all three diffusion models (viscous, Knudsen and surface diffusions). It has simple formulation, incorporate all parameters of the porous membrane and less number of assumptions required to model the performance of EPBM. However, the model needs modifications in order to predict the permeance and selectivity of blended membranes. The permeance of a membrane depends on its plasticization and intrinsic absorptivity. When PSU blends with PVAc, enhances the permeance as PVAc chains contain polar carbonyl groups that can collaborate with CO2 polar gas and increase the CO2 solubility in the PSU/PVAc blend membranes. When the MDEA, DEA or MEA were added, they got incorporated into the pores of the membrane and thus increase the intrinsic absorptivity of the membrane.

It was also observed that alkanolamine solution was embedded in the polymer matrix which offered the facilitated transport to CO2 and retards the transport of CH4. Assumptions are important for the models developed to be meaningful. Below are the few assumptions made in this modeling work;

a. The membrane is assumed to be operated isothermally with in the feed and retentate side.

b. The PVAc and DEA are homogenously distributed in the sphere of enhanced polymeric blend membrane.

c. No reaction takes place in the membrane separation space available.

d. No capillary condensation (multilayer adsorption) in the pores.

e. Complete mixing occurs in both the feed and permeates chamber and that the bulk gas phase is moving in a plug flow manner.

When the pressure is increased, the effect is distributed between the blending material and the resulting pores. Small pore sizes provide narrow pathways of travel. Therefore the movement of the gas molecules is obstructed. Increasing operative pressure would increase the collision as well as the interaction between the gas molecules and membrane surface which makes surface diffusion more favourable. Surface diffusion increases primarily because of adsorption processes are favoured at high pressure due to increased molecular density. As a result, the tendency of filler material to absorb CO2 as compared to CH4 increases. It can be concluded that the blending effect manifests itself as a strong function of pressure.

Estimation of PVAc factor in Blending factor

Modified Cho empirical model predicts the permeance of a porous membrane. When a polymer is blended, there is a significant change in its morphology and therefore, the deviation from experimental results is inevitable. Hence, an additional parameter that account for the effect of blending need to be included in the modified Cho empirical model in order to be used for modeling the synthesis PBM. Modelling can be done by considering PBM first, as it provokes the need of blending factor. To extend the basic modified Cho empirical model with the presence of the second component, PVAc an additional parameter is introduced which is the blend factor as a composition of 'x' of PVAc is introduced in equation (39).

This concept of adding a correction factor for improvement of prediction of parameters is not novel. In the case of the Virial equation of state, in which deviation of gas from ideal gas behaviour is predicted by defining a correction factor (named "compression factor") [36]. Another example from the equations for permeability prediction can be observed in Pal's model which can be obtained by adding a factor into the original Bruggmen's model [37].

To determine the blending factor theoretically is cumbersome. Nevertheless, the blending factor could be determined through data fitting and optimization of predicted parameters against experimental data to minimize the model prediction errors. Defining the blending factor as:

(EQUATION)

or

(EQUATION)

The blend composition 'x' represents the fraction of PVAc in this study. Its value varies between 0 and 1, where 0 represents the pure polymer. The parameter are function of pressure 'P', defined as:

(EQUATION)

where; P is the pressure, [alpha], [beta], I3, I, and I are blending constants.

The blend factor is included into the equation (39) to obtain precise theoretical values that are close to the experimental values. The new extended modified Cho empirical model equation to address the deviation will provide an accurate estimation of the theoretical values.

P'i = Base equation x blending factor of PVAc

(EQUATION)

The calculated Bf,p,cal was determined by using equation (42). On the other hand, the experimental values of the blending factor Bf,exp can be determined using the following expression:

(EQUATION)

The values of the model parameter constants, [alpha], [beta], I3, I, and o for a1, a2 and a3 can be found from Table-5. This table shows the optimize blending parameter constant from the fitting process when PVAc was added in the PSU matrix. At this optimized Bf,p,cal, all the parameters contact should satisfy all fitted experimental conditions and reach at a unique solutions. In order to find the values which can provide sufficient approximation of the blending factor Bf,p,cal, the MATLABA(r) curve-fitting tool was used for selecting initial guesses of these values. It should be noted that to avoid overfitting, only low order equations were used in accordance with the trends of the experimentally observed values of the blending factor for compositions PSU/PVAc (80/20) wt. %, PSU/PVAc (90/10) wt. % and PSU/PVAc (95/5) wt. %. Thus the parameter constants were optimized at which the calculated blending factor Bf,p,cal, for these compositions would approach the experimental blending factor Bf,exp.

Table-5: Blending factor parameters for PSU/PVAc or blends.

Blend Parameter Constants###[alpha]###[beta]###I3###I,###I

###a1###-29.202###11.653###-1.3156###0.0725###0

###a2###288.6###-104.96###15.619###-0.9005###0

###a3###-823.56###296.61###-48.424###2.8094###0

Table-6: Calculated and experimental values of blending factor for CO2 permeance at different pressures of polymeric blend membranes

###Pressure###"Bf" for CO2 permeance

Membranes

###'bar'###Bf,exp###Bf,p,cal###AARE%

###2###0.7085###0.76

###4###1.1127###1.17

PSU/PVAc (95/5) wt. %###6###1.4691###1.49###3.11

###8###1.7802###1.80

###10###2.1663###2.18

###2###0.949###0.880

###4###1.563###1.478

PSU/PVAc (90/10) wt. %###6###2.029###1.987###3.23

###8###2.482###2.458

###10###2.953###2.940

###2###1.0326###1.029

###4###1.8013###1.785

PSU/PVAc (80/20) wt. %###6###2.4140###2.425###0.42

###8###3.0023###2.997

###10###3.5377###3.546

Subsequently, using optimized blending parameters, the calculated blending factor Bf,p,cal, was determined using equation (42). The comparison between Bf,exp, calculated using equations (45) and (42) are tabulated in Table-6. The AARE% for all conditions fit very well with less than 3%.

The experimental data of composition PSU/PVAc (85/15) wt. % has been with-held for cross-validation of the resulting model in order to avoid overfitting. To validate these blending constants, a new set of experimental data was used at PSU/PVAc (85/15) wt. %. The results are presented in Table-9 with AARE% < 2%. The blending factors Bf,p,cal, are then multiplied in equation (39) when PVAc is blended in PSU to normalize the surface diffusion, Knudsen diffusion and viscous diffusion in total permeance using modeling from 2 to 10 bar pressure.

Estimation of amine factor in Blending factor

In order to extend the basic modified Cho empirical model with the presence of the third component that is an amine, an additional parameter is introduced which is the combined blend factor as a composition of 'y' of amine is introduced in equation (41) as:

(EQUATION)

or

(EQUATION)

The amine composition y represents the fraction of amine in the membrane. Its value varies between 0 and 1, where 0 represents the absence of amine. The parameter ai a function of pressure P, defined as:

(EQUATION)

where P is the pressure, [alpha], [beta], I3, I, and o are blending constants.

The blend factor is included into the equation (39) to obtain precise theoretical values that are close to the experimental values. The extended modified Cho empirical equation to evaluate the error percentage that will provide an accurate estimation of the theoretical values is as follows:

P'i = Base equation x blending factor of PVAc and amine

(EQUATION)

The values of [alpha], [beta], I3, I, and o for a1, a2,... a8 can be found from Table-7. It should be noted that the values of a1, a2 and a3 were previously determined. Hence, only the values of a4, a5,..., a8 are now obtained by minimizing the prediction error of compositions PSU/PVAc (80/20) wt. %/DEA, PSU/PVAc (90/10) wt. %/DEA and PSU/PVAc (95/5) wt. %/DEA. MATLABA(r) curve-fitting tool was used for selecting appropriate initial guesses, whereas fine-tuning of the parameter was obtained by hit-and-trial in a spreadsheet, as was previously done for the determination of a1, a2 and a3 in above section.

Table-7: Blending factor parameters for PSU/PVAc/DEA.

Blend Parameter

###[alpha]###[beta]###I3###I,###I

Constants

###a1###-29.202###11.653###-1.3156###0.0725###-

###a2###288.6###-104.96###15.619###-0.9005###-

###a3###-823.56###296.61###-48.424###2.8094###-

###a4###-0.253###0.237###-0.072###0.009###-

###a5###-1144.9###1060.9###-315.72###39.64###-1.72

###a6###0.01###-0.04###-0.03###-0.004###-

###a7###1224.3###-471.23###-69.893###5.745###-

###a8###-155.41###-25.77###87.396###-5.8892###-

The calculated was intended by using equation (46), whereas the experimental was determined by using equation (45). The calculated and experimental values of the blending factor for different blending compositions and pressures are shown in Table-8. The AARE% for all conditions fit very well with less than 8%.

Table-8: Blending Calculated and experimental values of blending factor for CO2 permeance at different pressures of enhanced polymeric blend membranes.

###Pressure###"Bf" for CO2 permeance

Membranes

###'bar'###Bf,exp###Bf,pa,cal###AARE%

###2###0.9262###0.840

###4###1.8729###2.108

PSU/PVAc (95/5) wt. % /DEA###6###2.8694###2.373###8.96

###8###3.9459###4.345

###10###5.1547###5.126

###2###1.0725###1.060

###4###2.2478###2.279

PSU/PVAc (90/10) wt. % /DEA###6###3.5448###3.600###5.56

###8###4.9388###4.625

###10###6.4745###5.351

###2###1.5810###1.545

###4###3.4289###3.463

PSU/PVAc (80/20) wt. % /DEA###6###5.5718###5.852###3.07

###8###7.8499###7.736

###10###10.101###9.535

As in the previous section, the experimental data of composition PSU/PVAc (85/15) wt. %/DEA has been withheld for cross-validation of the resulting model to avoid overfitting. To validate these blending constants, a new set of experimental data was used at PSU/PVAc (85/15) wt.% /DEA. The results are presented in Table 9 with AARE% ###Mean free path of travel of gas molecules###[m]

###in the pore

I###Membrane tortuosity###[-]

Im###Membrane density###[kg.m-3]

a|u###Lennard-Jones parameters as available in###[-]

###Bird et al.(1960)

I###Kinetic diameter of gas molecule###[oA]

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