Mathematical modeling of mass transfer in multicomponent gas mixture across the synthesized composite polymeric membrane

Journal of Industrial and Engineering Chemistry 19 (2013) 870–885

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Mathematical modeling of mass transfer in multicomponent gas mixture across the synthesized composite polymeric membrane Abtin Ebadi Amooghin a, Pardis Moradi Shehni b, Ali Ghadimi c, Mohtada Sadrzadeh c, Toraj Mohammadi b,* a b c

Young Researchers and Elites club, Science and Research Branch, Islamic Azad University, Tehran, Iran Department of Chemical Engineering, South Tehran Branch, Islamic Azad University, P.O. Box 11365-4435, Tehran, Iran Research Centre for Membrane Separation Processes, Chemical Engineering Department, Iran University of Science and Technology (IUST), Narmak, Tehran, Iran

A R T I C L E I N F O

Article history: Received 16 September 2012 Received in revised form 28 October 2012 Accepted 3 November 2012 Available online 10 November 2012 Keywords: Mass transfer Ternary gas mixture Mathematical modeling Frisch method Diffusion Membranes

A B S T R A C T

This study presents a new mathematical model to investigate the ternary gas mixture permeation across a synthesized composite PDMS/PA membrane. A novel algorithm is introduced for direct determination of diffusion coefficients. It pertains to study gas permeation through concentration dependent systems and comparing with traditional time lag method confirms the precision of this approach. Feature is that this method does not require physical properties of the membrane. Accordingly, it can be used as a general comprehensive model. In addition, molecular pair and molecular trio interactions were taken into account and in order to investigate the deviation of gas mixture from ideality, fugacities were calculated. The results showed that permeabilites of H2 and CH4 increase with increasing feed temperature and fugacity, while that of C3H8 decreases. Moreover, increasing C3H8 concentration improved permeation properties of all components. The results demonstrated that considering the concentration dependent system (CDS) leads to the small deviation of about less than 10%, while the deviation of 50–100% by the concentration independent system (CIS) was acquired. Additionally the results indicated that permeability of the lighter gases is specially affected by diffusivity, while solubility is dominant on permeability of the heavier gases. ß 2012 The Korean Society of Industrial and Engineering Chemistry. Published by Elsevier B.V. All rights reserved.

1. Introduction Liquefied petroleum gas (LPG) is defined as the C3+ fraction, which can be recovered from various hydrocarbon source streams such as refinery gases, particularly fuel gas streams wherein C3+ constitutes a small portion [1]. Recently, membrane separation has been found to be a cost effective method for recovery of LPG from permanent gases due to inheriting certain advantages compared to other methods such as low energy expenditure, simplicity of operation, compactness and easy maintenance [1,2]. Along with the membrane constructive materials, rubbery polymer membranes preferentially permeate the larger and more condensable gases rather than the smaller, non-condensable gases [3]. High flexibility of the rubbery polymer results in insignificant differences in diffusion rates between smaller and larger molecules. Thus, selective separation is primarily driven by penetrant solubility differences [4]. Polydimethylsiloxane (PDMS) is one of the most commonly rubbery membrane materials used for

* Corresponding author. Tel.: +98 21 77240496; fax: +98 21 77240495. E-mail address: [email protected] (T. Mohammadi).

separation of heavier hydrocarbons from permanent gases. PDMS membranes, which have been synthesized and used in gas separation applications were in the form of a single layer [5– 13], a composite with a microporous support [14–26], a mixed matrix [27–29] and a copolymer [30,31]. Recently, many studies have been accomplished to investigate permeation properties of pure and mixed gas using PDMS membranes [3–8,16– 18,21,22,32,33]. In addition to experimental works, many developments have been carried out on modeling and simulation of gas transport phenomenon through membranes. Some of them are applicable for prediction of sorption, diffusion and permeability of gases, as well as extending the results to unavailable data [34,35]. Hashemifard et al. [36] extensively studied gas permeabilities in mixed matrix membranes using theoretical models such as Maxwell, modified Maxwell, Lewis–Nielsen, modified Lewis–Nielsen and Felske, and compared them with available experimental data. They found that Felske model is the most reliable model for predicting gas permeabilities in mixed matrix membranes. Al-Marzouqi et al. [37] introduced a mathematical model to simulate gas transport in membrane contactors. Their model was considered for nonwetting conditions, taking into account axial and radial diffusion in

1226-086X/$ – see front matter ß 2012 The Korean Society of Industrial and Engineering Chemistry. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jiec.2012.11.003

A. Ebadi Amooghin et al. / Journal of Industrial and Engineering Chemistry 19 (2013) 870–885

Nomenclature ai ami A b B BðiÞ C C C ðiÞ C0 Cb D D¯ DD

DH Ed Ep f DHs J kc kd K Kij l L M N N0 p P Pt Q R S t T u X y y Y Y z Z

v v f u

Strength parameters for various intermolecular forces coefficient for terms in second virial coefficient correlations membrane effective area (cm2) sorption isotherm parameter (kPa1) second Virial coefficient (cm3 mol1) sets of universal functions of reduced temperature concentration (mole/m3) third Virial coefficient (cm6 mol2) sets of universal functions of reduced temperature initial concentration (mole/m3) upstream bulk concentration (mole/m3) diffusion coefficient (cm2/s) concentration-averaged effective diffusivity (cm2/s) diffusion coefficient used for the portion of the gas dissolved in the polymer according to the Henry’s law expression diffusion coefficient used for the portion of the gas contained in the excess free volume activation energy of diffusion and permeation (kJ/ mol) activation energy permeation (kJ/mol) Fugacity (bar) enthalpy of sorption (kJ/mol) steady-state gas flux (cm3 (STP)/cm2s) mass transfer coefficient (m/s) Henry’s law constant (cm3(STP)/cm3 atm) dimensionless partition coefficient binary interaction parameter Membrane thickness (cm) membrane diameter (cm) stiff-spring constant gas flux (mol/m2.s) inward flux (mol/m2.s) pressure (bar) permeability coefficient (Barrer: 1  1010 3 2 cm (STP) cm/cm s cmHg) absolute pressure (N/m2) total flux (cm3(STP)/cm2 s) universal gas constant (6236.56 cm3 cmHg/(mol K)) solubility coefficient (cm3(STP)/cm3 atm) time (s) absolute temperature (K) velocity vector (cm/s) mole fractions in the feed streams mole fraction distance on y axis (cm) mole fractions in the permeate streams model thickness (cm) distance on z axis(cm) compressibility factor penetrant mass fraction in the polymer eccentric factor Fugacity coefficient time lag (s)

871

Subscripts/superscripts downstream or permeate side 1 upstream or feed side 2 gas A and B A, B Bulk b boiling point Bo critical property c anisotropic (effective) eff Local loc component i zone j Membrane m Permeate side P reduced property r Support s characteristic property *

the all membrane compartments. Additionally, their predictions were in good agreement with experimental results in different situations of gas and liquid flow rates, gas to liquid ratios, and temperatures. In another work, Faiz and Al-Marzouqi [38] presented a wide-ranging 2D mathematical model for simultaneous absorption of CO2 and H2S using monoethanol amine (MEA) in hollow fiber membrane contactors. This model included physical and chemical absorption considering non-wetting and partial-wetting conditions, respectively. Coroneo et al. examined the CFD approach based on a numerical solution of the Navier– Stokes equations on a three-dimensional domain for separation of a gas mixture through inorganic membranes and the simulation results confirmed the achieved experimental data [39]. Shirazian et al. [40] numerically studied the mass transfer in gas–liquid membrane contactors. They represented that CO2 removal increases by increasing liquid velocity in shell side and confrontation with opposite effects with increasing temperature and gas velocity in tube side. Macchione et al. [41] investigated theoretically and experimentally influence of residual solvent in dense Hyflon AD60X membranes. From analysis of permeability, diffusion and solubility coefficients of six gases, they reported that plasticization by the residual solvent reduces permselectivity and increases permeability. In our previous work, a composite polydimethylsiloxane (PDMS)/polyamide (PA) membrane was synthesized and effect of operating parameters on sorption, diffusion and permeation of C3H8, CH4 and H2 were studied both in pure and mixed gas experiments. The experimental results demonstrated ability of the synthesized PDMS/PA membrane for separation of organic vapors from permanent gases [42]. However, it must be mentioned that direct calculation of diffusion coefficient values in rubbery membranes in laboratory is not feasible. This is due to high solution rate of gas into the polymer matrix. Thus, the unsteady state profile of pressure cannot be determined. In addition computing the unsteady state duration of the gas transport process through the membranes using existing instruments is intricate. Therefore, the values of diffusion coefficient are calculated using well-known solution–diffusion mechanism in polymeric membranes using permeation and sorption tests. Hence, in order to determine the computational error of diffusion coefficient, the errors of the two mentioned tests are multiplied. Consequently, no convincing attempt is conducted for considering the gas transport unsteady state profiles, which is arisen from the intense tendency of gas transport profile to switch into steady state.

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The present study is objected to develop and solve a comprehensive mathematical model for gas permeation behavior of a ternary gas mixture (C3H8, CH4 and H2) through the synthesized composite PDMS/PA membrane. Unsteady state profiles of the ternary gas mixture transport through the membranes and unsteady state durations of the transport are also determined. Since the swelling effect takes place across the PDMS membrane, in order to determine the diffusivities for a concentration dependent system (CDS), the time lag method based on The Frisch method for Fickian diffusion is employed and the outcomes are compared with experimental data. In addition, for emphasizing the importance of swelling effect on the system, diffusion coefficients based on concentration independent system (CIS) are also calculated and compared with the results obtained from CDS system. The solution-diffusion mechanism is taken into account as the governing transport of species system across the membranes. Permeation through the membranes is considered to be conducted by both convection and diffusion contributions in either y or z directions, or feed and permeate sides of the

membranes. The simulation results are then validated with experimental data. 2. Mathematical model equations 2.1. Modeling of gas permeation A material balance was implemented on the polymeric synthesized composite PDMS/PA membrane to expand the main equations for the mathematical model. A crossflow membrane cell made from stainless steel 316L was employed to conduct the experiments and the model was extended to four separate phases of the module: the feed and permeate sides, the PA support and PDMS skin layers, as shown in Fig. 1a and b. The model was proposed for the unsteady state mass transfer in multicomponent gas mixture across the membranes. Fig. 2a demonstrates the model (four zones) with boundaries. Fig. 2b demonstrates the meshgeneration used in this study. The transient two-dimensional material balance was carried out for the domains. The multicomponent gas mixture stream (C3H8, CH4 and H2), was assumed to go

Fig. 1. A schematic diagram for (a) Schematic view of the gas permeation set-up (b) The composite membrane used for modeling (scale in y-coordinate is 1/10).

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873

@C i;P þ r  ðDi;P rC i;P Þ ¼ ui;P  rC i;P ðPermeate sideÞ @t

(4)

where C is concentration, D diffusion coefficient, u velocity vector and t is time. The process is diffusion-driven, that is, components diffuse through a membrane due to concentration gradient between the feed and permeate sides of the membrane. Separation between the components takes place as a result of dissimilarity in diffusion rates across the membrane which is arisen from diversity in molecular size and solubility. A momentum balance using the Navier–Stokes equation was also considered for simultaneous solving with the continuity equation in the feed and permeates sides for obtaining the concentration and velocity distributions [44]. In this case, combining the Navier–Stokes equations with the continuity equation is very common and frequently used by many researchers. In addition, coupling of the mentioned equations for simulation of two-dimensional straight membrane geometry was applied by many [45,46]. In this case, the Navier–Stokes equation is defined as follows [47]:

ri

  @ui;FP þ ri ui;FP rÞui;FP ¼ r pi;FP @t   T  þ rhi rui;FP þ rui;FP þF

(5)

where r denotes the gas density, u the velocity vector in x direction, p the pressure, h the dynamic viscosity and F is a representative of field forces. Additionally, the isotherm gas transport without any chemical reaction was considered in the laboratory. Hence, heat transfer equations were disregarded. The set of boundary conditions considered in model are as follows: (i) Feed side (Ternary gas mixture): @y ¼ 0 Fig. 2. (a) The model four zones and relevant boundaries (y1 = 0.15 cm, y2 = 0.012 cm, y3 = 0.12 cm, y4 = 0.2 cm). (b) Mesh-generation in the studied domains.

C i j ¼ Ci;0 j

ðConcentrationÞ

   n:N ¼ M C i; jþ1  K j C i; j ; N

@y ¼ y1

¼ Di; j rC i; j þ C i; j ui; j in at (y = 0) and move out from (z = L) and each component to diffuse through the membrane (y direction). The mathematical modeling of diffusion is based on the following assumptions: i. The polymer material is homogeneous. ii. The membrane thickness, Y, is significantly smaller than other dimensions. iii. The penetrant sorption/desorption at the membrane feed/ permeate faces is happening much faster than its diffusion in the membrane, which is the rate determining step. In other words, the interfacial sorption/desorption equilibrium is instantaneous and steady. iv. The diffusion process is Fickian and time-dependent.

@z ¼ 0

  @C i;F þ r : Di;F rC i;F ¼ ui;F : rC i;F ðFeed sideÞ @t

(1)

@C i;m þ r  ðDi;m rC i;m Þ ¼ 0 ðMembraneÞ @t

(2)

@C i;s þ r  ðDi;s rC i;s Þ ¼ ui;s  rC i;s ðSupport layerÞ @t

(3)

(7)

ðInsulation=SymmetryÞ

(8)

   n: Di; j rC i; j ¼ 0

ðConvective FluxÞ

(9)

(ii) Membrane (PDMS skin layer):    n:N ¼ M K m C i; j1  C i;m ; N @y ¼ y1 (10)

¼ Di;m rC i;m @y ¼ y2

   n:N ¼ M C i; jþ1  K m C i;m ; N ¼ Di;m rC i;m

@ z ¼ 0; z ¼ L; The continuity equations for each component of the ternary gas mixture for component i in each zone, can be expressed as [43]:

ðStiff-Spring Continuous FluxÞ

n:N ¼ 0; N

¼ Di; j rC i; j þ C i; j ui; j @z ¼ L

(6)

n:N ¼ 0; N ¼ Di;m rC i;m

(11) (12)

(iii) Membrane (PA support layer):   @y ¼ y2  n:N ¼ M K s C i;m  C i;s ; N ¼ Di;s rC i;s þ C i;s ui;s @y ¼ y3

   n:N ¼ M K s C i; jþ1  C i;s ; N

¼ Di;s rC i;s þ C i;s ui;s @z ¼ 0; z ¼ L

(13)

n:N ¼ 0; N ¼ Di;s rC i;s þ C i;s ui;s

(14) (15)

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(iv) Permeate side    n:N ¼ M C i;s  K j C i; jþ1 ; N

@y ¼ y3

¼ Di; j rC i; j þ C i; j ui; j @y ¼ Y

   n: Di; j rC i; j ¼ 0

@z ¼ 0; z ¼ L

n:N ¼ 0; N ¼ Di; j rC i; j þ C i; j ui; j

(16) (17)

2

(18)

where N and Ci;0 j denote gas flux and initial concentration, respectively. Initial concentrations were calculated by dividing inlet pressure by (RT) and multiplying by each component composition. The Navier–Stokes boundary conditions are: (v) Feed side @y ¼ 0; z ¼ L ; p ¼ p0 ðPressureÞ

(19)

@y ¼ y1 ; z ¼ 0 ui;F ¼ 0

(20)

ðNo slip conditionÞ

(vi) Permeate side @y ¼ y3 ui;P ¼ 0 @y ¼ Y p ¼ p0

ðNo slip conditionÞ ðPressureÞ

@z ¼ 0; z ¼ L ui;P ¼ 0

ðNo slip conditionÞ

(21) (22) (23)

The equilibrium boundary condition used for calculating fluxes in the membrane boundaries is known as stiff-spring. These conditions account for discontinuities in concentration due to partition equilibrium but at the same time allow for continuous fluxes across the membrane [48–51].The stiffspring method was used, in order to eliminate the discontinuities in the concentration profile and get continuous flux over the boundaries. Instead of defining Dirichlet concentration conditions according to the partition coefficient K, which would destroy the continuity of the flux, continuous flux conditions were defined, and consequently at the same time, the concentrations were forced to the desired values. These equations were defined as below:   N 0 ¼ M C i; jþ1  K j C i; j  (24) Kj ¼

C i; jþ1 C i; j

For zones 1 and 4, the binary gas-gas diffusion coefficient in ternary gas mixture was computed by means of Wilke–Lee equation [53]. Impressive diffusion coefficients of each component in the ternary gas mixture were estimated as well [54]. Due to scaling in y-direction, correlating diffusion coefficients must be considered instead of effective diffusion coefficients. The membrane diameter (3.57 cm) was 270 times longer than membrane thickness (single layer PDMS (12 mm) and PA support (120 mm)) dimension. The problem was scaled in order to avoid extreme amounts of nodes and elements. Thus, a new scaled ycoordinate was introduced: y scale

Di;m ¼ 4 scale2 0

Dm;e ff :

3 0 5

2

Ds;e ff :

Di;s ¼ 4 scale2 0

Zone 2

(28)

Di;m

3 0 5 Di;s

Zone 3

(29)

Gas transport in polymeric membranes is widely modeled using solution-diffusion mechanism and is expressed by a permeability coefficient (see ref. [54] Eqs. (13–19)). The application of Henry’s law for rubbery polymers is well accepted, particularly for low–molecular weight penetrants, but is less accurate for glassy polymers, for which alternative theories have been proposed. In this area, sorption of the penetrant occurs by ordinary dissolution in the polymer chains, as described by Henry’s law, and by Langmuir sorption into the holes or sites between the polymer chains of glassy polymers. Therefore, Fick’s law expression for flux through the membrane can be written as: J ¼ DD

dcD dcH  DH dy dy

(30)

Henry’s law sorption occurs in the equilibrium free volume portion of the polymer. On the other hand Langmuir sorption occurs in the excess free volume between the polymer chains that exists in glassy polymers. Permeation of gases in glassy polymers can also be described in terms of the dual sorption model. One diffusion coefficient (DD) is employed for the portion of the gas dissolved in the polymer according to Henry’s law expression and a second, somewhat larger, diffusion coefficient (DH) for the portion of the gas contained in the excess free volume [55,56]. According to above discussions, in a homogeneous and rubbery membrane such as PDMS, the solubility of a gas obeys Henry’s law, and the sorption isotherm can be represented by Henry’s law, and permeability coefficient may be expressed by P ¼ SD.

(25)

where i and j are representatives of each species and zones, respectively, M is a (non-physical) ‘‘Stiff-spring velocity’’ large enough (in this case, 10,000) to guarantee that the value in parenthesis tends to zero and that the fluxes on both sides of the membrane are equal, C is concentration, and Kj is the dimensionless partition coefficient derived experimentally employing Henry’s law [42,52].

y¯ ¼

This gave the following correlated diffusion-coefficients matrixes for single layer PDMS and PA support [52]: 2 3 Di; j 0 5 Zone 1 and 4 (27) Di j;e ff : ¼ 4 scale2 0 Di; j

(26)

2.2. Determination of permeation time lag & diffusivity by The Frisch method The time lag can be used as a useful method for investigating diffusivities [57–59] even for the complicated diffusional systems [60–62]. The concept of time lag is applied for ideal transport through the membrane as well as non-Fickian systems, specifically for the time, position, or concentration dependent. At this point, the Frisch method was used to derive the time lag for parabolic equations, as the most important characteristic of diffusion process [63–65]. Intercept of the tangent line of the linear part of the pressure– time curve with the x-axis gives the time lag value for each membrane [57]. Fig. 3 shows the unsteady state pressure for C3H8 in the PDMS/PA membrane T = 35 8C, P = 5 bar, XC3H8 = 10%. In order to simulate the multicomponent gas mixture permeation across the synthesized composite PDMS/PA membrane, experimental diffusivity values were employed, in which effect of swelling was enclosed [42], this affected the results indirectly. Therefore, the effect of concentration on diffusivity was considered using the Frisch method for Fickian diffusion. The Frisch method implicitly solves the diffusion equations for achieving time lag, which is applied for systems with concentration dependent diffusion coefficients. Generally, The Frisch method was founded on two approaches:

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polarity with a2 being a function of the dipole moment, m, and B(3) takes account of association with a3, an empirical parameter. Using the substance strength coefficients as:

a0 ¼ 1; a1 ¼ v; a2 ¼ a; a3 ¼ b where v is the eccentric factor, a and b are the constant parameters or variable functions of the dipole moment that may depend upon the ‘‘family’’ of the substance or the substance itself. Hence, it simplifies to [76]: BP C ¼ Bð0Þ þ vBð1Þ þ aBð2Þ þ bBð3Þ RT C

(35)

where

Fig. 3. The unsteady state pressure of C3H8 in the PDMS/PA membrane (with two magnifications for better distinct), at T = 35 8C, P = 5 bar, XC3H8 = 10%.

Bð0Þ ¼ 0:1445 

0:330 0:1385 0:0121 0:000607    Tr Tr2 Tr3 Tr8

(36)

Bð1Þ ¼ 0:0637 þ

0:331 0:423 0:008   Tr2 Tr3 Tr8

(37)

Bð2Þ ¼

1 Tr6

(38)

(1) Concentration independent system; CIS (A type: D is constant). (2) Concentration dependent system; CDS (B type: D is considered as a function of penetrant concentration).

Bð3Þ ¼ 

The Frisch method is acquired by integrating the transport equation twice in space and once in time, and by this, achieving the integrated flux. The method is predominantly used for systems with concentration-dependent diffusion coefficients not including explicitly solving the diffusion equations [66] (see Appendix A).

Normal fluids contain simple molecules and semiempirical extension with a single additional parameter. Molecules in normal fluids are not strongly polar or hydrogen-bonded [77]. For normal fluids, simpler equations of Van ness and Abbott can be used for B(0) and B(1) [78]:

2.3. Thermodynamically formulation of fugacity

Bð0Þ ¼ 0:083 

0:422 Tr1:6

(40)

Bð1Þ ¼ 0:139 

0:172 Tr4:2

(41)

To estimate the fugacity of ternary gas mixture the following steps were used: f ¼ fP

1 Tr8

(31)

(39)

where P is pressure and f is the fugacity coefficient which can be determined by the following equation: Z P Z1 dP (32) lnf ¼ P 0

The third cross coefficient of pure gases is estimated by [79]:   C ðT Þ X T (42) ¼ ai C ðiÞ  2 T V i

where Z is the compressibility factor obtained from Virial equation of state, which is a polynomial series in pressure or in inverse volume, but for mixtures, the coefficients are functions of both T and {y}:

C ð0Þ ¼ 0:01407 þ



 P  2 P þ  þ C  B2 Z ¼1þB RT RT     B C ¼1þ þ þ  V V2 



(33)

where B and C are pressure series of the Virial coefficients and {y} is the set of n  1 independent mole fractions of the mixture’s n components. In the case of a pure gas, these coefficients are calculated as follows [20,67–74]:   BðT Þ X ð iÞ T ¼ a B (34) i V T i where V* is a characteristic volume computed from V* = RTC/PC, T* = TC [75]. B(0) is for simple substances with a0 being unity, B(1) corrects for non-spherical shape and globularity of normal substances with a1 commonly being, v, B(2) takes account of

0:02432 0:00313  Tr2:8 Tr10:5

C ð1Þ ¼ 0:02676 þ

0:01770 0:040 0:0030 0:00228 þ   Tr2:8 Tr3 Tr6 Tr10:5

(43)

(44)

The second cross Virial coefficient for all pairs of components in the mixture, i and j can be estimated as follows: ! Bi j ðT Þ X T ¼ ami BðmÞ  (45)  Ti j Vi j m in which:  1=2    1  Ki j Ti j ¼ T Ci j ¼ T Cii T C j j

(46)

where Kij is the binary interaction parameter calculated from [74]: 2  1=6 33 2 V Cii V C j j 4 5 Ki j ¼ 1   1=3 1=3 VCii þ VC j j

(47)

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876

Table 1 The system specification used in experiments and numerical simulation. Physical properties & Operating conditions

Solver & Mesh details

Parameter

Value

Parameter

Value

Membrane diameter (mm) PDMS skin layer thickness (mm) PA support layer thickness (mm) Membrane effective area (cm2) Feed flux (m3/s) Feed temperature Transmembrane pressure C3H8 concentration in feed

35.7 12 120 10 100 25, 30, 35, 40, 45 8C 2, 3, 4, 5, 6 atm 2, 4, 6, 8, 10 vol.%

Relative tolerance Absolute tolerance Initial time step Linear system solver Number of meshes Number of degrees

0.01 0.001 0.01 Direct (UMFPACK) Backward Euler 92416 567834

The Physical properties & Operating conditions data are measured in the laboratory.



where Vi; j is a characteristic volume for the pair computed by the following equation: 

Vij

3 1=3 1=3 VCii þ VC j j RT Ci j  ¼ ¼  P Ci j 4 Z Cii þ Z C j j  3 1=3 1=3 R VCii þ VC j j ¼ 4ððPC j j V C j j Þ=T C j j þ ðPCii V Cii Þ=T Cii Þ

(48)

The substance-dependent strength coefficients are considered as:  a1i j ¼ vi j ¼

vii þ v j j



2

(49)

For estimating the third coefficient, the below approach can be used [80]: C i jk ¼ ðC i j C jk C ik Þ1=3

n X n X yi y j Bi j ðT Þ

3.1. Setup

(51)

i¼1 j¼1

Expanding Eq. (51) gives: BðT; f ygÞ ¼ y21 B11 ðT Þ þ 2y1 y2 B12 ðT Þ þ y22 B22 ðT Þ þ 2y1 y3 B13 ðT Þ þ y23 B33 ðT Þ þ 2y2 y3 B23 ðT Þ

(52)

Coefficient C is considered for trio interactions as follows: C ðT; f ygÞ ¼

n X n X n X yi y j yk C i jk ðT Þ

(53)

i¼1 j¼1 K¼1

Expanding Eq. (53) gives: C ðT; f ygÞ ¼ y31 C 111 ðT Þ þ 3y21 y2 C 112 ðT Þ þ 3y1 y22 C 122 ðT Þ þ y32 C 222 ðT Þ þ 3y21 y3 C 113 ðT Þ þ 3y1 y23 C 133 ðT Þ þ y33 C 333 ðT Þ þ 3y22 y3 C 223 ðT Þ þ 3y2 y23 C 233 ðT Þ þ 6y1 y2 y3 C 123 ðT Þ

3. Experiments

(50)

Following equation states functionality of B from the molecular pair interactions [77]: BðT; f ygÞ ¼

92416 meshes and 567834 degrees were created and this confirmed that the mesh size is small enough to be selected, that the computer can be able to converge the problem. The model equations with the boundary conditions were solved numerically using COMSOL Multiphysics software which utilizes the FEM for numerical solution of equations [81]. A computer (Intel1 CoreTM i7 CPU 2620 M @ 2.7 GHz 2.7 GHz and 6 GB Ram) was employed to solve mathematical model. It should be pointed out that the COMSOL generates a triangular mesh that is isotropic in size. Afterwards, a large number of elements are created with particular scaling. Along the walls, no-slip boundary conditions are adopted. Consider that the solving method for an unsteady system is dissimilar from that for a steady system.

(54)

Gas permeation experiments are one of the most fundamental methods for finding the structure and morphology of synthesized membranes. In order to carry out these experiments, a set up was assembled [42] (see Fig. 1). The feed is composed by propane gas with purity of 99.9% (supplied by Air Products and Chemicals, Inc.) and hydrogen and methane gases with purity of 99.5% (supplied by Technical Gas Services, Inc.). Feed flow rate was controlled by Brooks Mass Flow Controllers (MFC), model 5850 E (0–18,000 normal mL/min range). A BPR (Back Pressure Regulator: model 26.60 SCFBXE262C086) was employed to adjust the constant transmembrane pressure. The gases were mixed with a certain ratio and then they sent to the membrane module. The gas temperature was set employing a TCS (P&ID temperature control system) [42]. Permeate gases was sent to a GC (gas chromatography: GC-2001 M, Sanayeh Teif Gostar Co., Iran) with a TCD (thermal conductivity detector) and a chromosorb-102 column. It must be noted that Argon was used as carrier gas. The column temperature was programmed In order to detect the peaks of each gas. Initially, H2 and CH4 were detected at 25 8C. Afterwards, the column temperature was increased to 90 8C with a rate of 7 8C/min, and lastly, C3H8 was detected. The TCD current was 60 mA. Additionally, a BFM (Bubble Flow Meter) was used owing to measure the permeate flow rate. Gas permeability of species i at the steady condition was defined by:

2.4. Numerical solution of equations System specifications and physical conditions for the unsteady state permeation of ternary gas mixture were required to solve the governing equations with the specified boundary conditions as listed in Table 1. In addition to employ stiff-spring equilibrium boundary condition, the subminiature meshes near the PDMS film (see Fig. 2a) were selected in order to ensure the accuracy of the transport flux calculated across the walls. According to Table 1,

Pi ¼

22; 414 l p1 dV x A p2  p1 RT i dt

(55)

where A represents the membrane area, R is the universal gas constant, T is the absolute temperature, xi is the mole fraction of species i in the feed gas, dV/dt is the volumetric displacement rate of soap film in the BFM and 22,414 is the number of cm3 (STP) of penetrant per mole [42].

A. Ebadi Amooghin et al. / Journal of Industrial and Engineering Chemistry 19 (2013) 870–885

3.2. Module A crossflow membrane cell made from stainless steel (grade 316) was used with an effective area of 10 cm2. The membrane was placed in the cell. Rubber O-rings were employed to provide a pressure-tight seal between the membrane and the holder. 3.3. Membrane preparation In our previous work, a thin polydimethylsiloxane (PDMS) film was synthesized and cast over polyamide (PA) with 0.2 mm pore size as support. Casting solutions were primarily prepared from nhexane solution containing 1.0 wt.% Dehesive 944 silicone (Wacker Silicones Corporation, Adrian, MI) with certain ratios of Cross-linker V24 and Catalyst OL provided by Wacker. As supplied by the manufacturer, Dehesive 944 is a solvent-based addition cross-linkable silicone [42]. Simultaneously, a PA porous membrane (Sartorius AG) was put on water surface in a basin to act as the support. Subsequently the casting solution was cast on the PA distilled water-mediated membrane and the solvent was evaporated at room temperature during 48 h. Afterwards, the membrane was put into an oven at 80 8C for 2 h to remove residual solvent and to fully cross-link the polymer. The thickness of permselective layer was determined with a digital micrometer (Mitutoyo Model MDC-25SB) readable to 1 mm by subtracting the thickness of composite membrane from that of the PA support. Thickness of the PA support and the PDMS skin layers were about 120 and 12 mm, respectively. The cross-linked membranes were housed in the gas permeation module [42]. 3.4. Sorption measurement Mixed gas sorption measurements were performed employing a pressure decay module. Gas sorption apparatus consisted of a stainless steel module with certain volume. Additionally, a vacuum pump and TCS with stainless steel valves were joined to the module. The gas pressure in the module was monitored using sensitive pressure transducers and recorded automatically by a data acquisition system employing LabTech software [42]. TCS was used to adjust the gas temperature at a definite value. A water bath was employed to keep the module at this temperature throughout the sorption process. The vacuum pump was connected to this apparatus to degas the module, as required. The sorbed gas concentration of each species in the polymer (at a certain pressure and temperature) was computed by: Ci ¼

iV 22; 414 h m ð pxi Þb  ð pxi Þ f RT Vp

877

with data collected from literature [8,12,14,87], propane, methane and hydrogen pure gas sorption experiments were carried out at pressures up to 7.1, 20.3 and 22.0 atm, respectively. In the mixed gas experiments, a full factorial design of experiments was applied to investigate effects of feed temperature, transmembrane pressure and C3H8 concentration on diffusivity, solubility and permeability of the gas components. C3H8 was used as a heavy hydrocarbon and in contrary CH4 and H2 were selected as lighter hydrocarbons and permanent gases, respectively. Furthermore, feed flow rate was adjusted at 100 cm3/s. It must be noted that, the sorption isotherms and penetrant solubilities were calculated experimentally and these valid data were used for modeling in the present work [42].

4. Results and discussion 4.1. Effect of feed temperature on permeability in ternary gas mixture The model equations with the boundary conditions were solved for all four zones (multicomponent gas mixture in the synthesized composite PDMS/PA) at different operating conditions. The parameters used in the model are summarized in Table 1. A comparison between the experimental results and the model predictions was performed to validate the model. Fig. 4 depicts effect of temperature on permeability of all components in the ternary gas mixture for the composite PDMS/PA membrane. The model shows a small reduction in C3H8 permeability as temperature increases, although permeabilities of H2 and CH4 increase. As mentioned in our previous work [42], lighter gases such as H2 and CH4 have lower solubility in PDMS, their changes of enthalpy of sorption, DHs, is also scanty. As a result, solubility of the lighter gases decreases as temperature increases. However, when temperature increases, their diffusivity increases considerably, and consequently, their permeability increases. As temperature increases, C3H8 solubility decreases, while its diffusivity increases. Thus, its permeability decreases slightly or remains almost constant. As seen, there is a good conformity between the experimental data and the results achieved by the simulated model.

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where C reperesents the uniform concentration of the dissolved penetrant at equilibrium state; R is the universal gas constant; T is the absolute temperature [K]; p is the module pressure; xi is the mole fraction of species i in the gas; the subscripts b and f denote beginning and final steps of the sorption process, respectively; Vm and Vp are volumes of the module and the polymer sample, respectively and 22,414 is the number of cm3(STP) of penetrant per mole [42]. 3.5. Experimental design In pure gas experiments, OFAT (One factor at a time) experimental design was employed to examine effects of transmembrane pressure ranging from 2 to 6 atm and gas temperature ranging from 25 to 45 8C on gas permeation properties of the synthesized composite membrane. Moreover owing to compare sorption of gases in the synthesized membrane

Fig. 4. Effect of temperature on permeability for each components of the ternary gas mixture for the composite PDMS/PA membrane, at P = 5 bar, XC3H8 = 10% and XH2 = 55%.

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Fig. 5. Effect of fugacity on permeability for each components of the ternary gas mixture for the composite PDMS/PA membrane, at T = 35 8C, XC3H8 = 10% and XH2 = 55%.

4.2. Effect of transmembrane pressure on permeability in pure and ternary gas mixture Since the present work studies a gas mixture consisting C3H8, CH4 and H2, the molecular pair interactions and molecular trio interactions must be taken into consideration. In order to investigate the deviation of the gas mixture from ideality, fugacities were calculated. It must be mentioned that, the investigated mixture in this study comprises 10% C3H8 and the fugacity coefficient is chiefly affected by pair interactions rather than trio interactions. It is evident in Fig. 5, similar to the pressure functionality of permeability that the trend of permeation with fugacity is descending for C3H8 and ascending for CH4 and H2. Furthermore, f/p values decrease at higher feed pressures, which implies that at higher feed pressures, the more deviation from ideality is observed as expected and at higher pressures using fugacity instead of pressure is preferred. Experimental and simulation results indicate when fugacity increases, permeability of the lighter gases (H2 and CH4) increases, but that of the heavier gas (C3H8) decreases [82–84]. In pure gas experiments, solubility of the penetrants with high solubility usually increases when pressure increases [3,10–13,85,86]. However, with increasing pressure, solubility of the low soluble penetrants decreases a little or remains constant. In contrary, a totally different behaviour is perceived in gas mixture [42]. Indeed, as fugacity increases, the probability of collision between the molecules of components and the membrane surface increases due to the premiere mobility of H2 and CH4 (lighter gases) in the bulk of membrane. In mixed gas experiments, it was also observed that, the rate of solubility growth for C3H8 is less than that for H2 and CH4. It is due to the fact that, increasing fugacity increases mobility of H2 and CH4 molecules more effectively than that of C3H8 molecules. Hence, the chance for C3H8 molecules to contact the membrane surface decreases and this consequently affects their sorption. The lighter gases with smaller molecules always have higher diffusivity owing to the more interaction between the bigger and the heavier molecules and the polymer structure. In fact, the lighter gases such as H2 are the winner of the competitive diffusion with increasing the fugacity. The model demonstrates that C3H8 permeability decreases a little as fugacity increases, while permeability of H2 and CH4 increases. As observed, there is a perfect agreement between the experimental data and the simulated model results.

Fig. 6. Pure and mixed gas permeability of the synthesised composite PDMS/PA membrane at 35 8C as a function of transmembrane pressure (experimental and simulation). Data obtained in this work are compared with Raharjo et al. [8], Merkel et al. [12], Prabhakar et al. [14], and Stern et al. [87] results.

Fig. 6 shows experimental and simulation results of pure and mixed gas permeability of penetrants (C3H8, CH4, and H2) at 35 8C versus transmembrane pressure for the synthesised composite PDMS/PA membrane at pressure of 2–6 bar. In addition, the results

A. Ebadi Amooghin et al. / Journal of Industrial and Engineering Chemistry 19 (2013) 870–885

were compared to those of single gas permeability presented by Raharjo et al. [8], Merkel et al. [12], Prabhakar et al. [14], and Stern et al. [87]. As observed, the experimental results prove those of the simulation. Additionally, the simulation results of pure gases are in agreement with the aforementioned experimental results. The simulation results illustrate that C3H8 permeability increases as pressure increases. However, permeabilities of CH4 and H2 decrease a little when pressure increases. The simulation results show the exactly opposite behavior for the ternary gas mixture. Permeability of all components is a linear function of pressure in pure gas experiments. The function depends on the three parameters: penetrant solubility, hydrostatic pressure, and plasticization [42]. Plasticization takes place when concentration of the components increases in the polymer due to an increase in the polymer local segmental motions caused by the presence of diffusive molecules within the polymer structure [7,8,12]. With increasing feed pressure, both of the concentration of the components and the plasticization of the polymer increase. This is much prominent for the high soluble components such as C3H8. Solubility is pressure independent for lighter gases such as H2. Thus, these gases do not affect the plasticization of the polymer [42]. Hence, permeability of the so-called gases decreases. On the contrary, increasing the compressibility at high pressure can lead to entrap the lighter gases, owing to the free volume reduction in the polymer matrix. This phenomenon plays a hindering role in diffusion [88]. 4.3. Effect of feed composition on permeability in ternary gas mixture Fig. 7 depicts effect of C3H8 concentration on permeability of all components in the ternary gas mixture for the composite PDMS/PA membrane. The simulation results indicate the increment in permeability of all components with increasing XC3H8 and this is in conformity with the experimental data. Increasing permeability of the components is due to the polymer swelling caused by the inducing plasticization by C3H8. The polymer swelling increases the chain mobility and this consequently leads to a considerable increase in diffusivity of the lighter components. In accordance with the literature, the swelling of PDMS is the result of hydrocarbon vapours [89–91]. C3H8 solubility increases, when partial pressure of C3H8 increases in the feed. The collisions between C3H8 molecules and the membrane surface increases with increasing C3H8 composition (from 2 to 10%) in the composite PDMS/PA membrane, and this, results in an increase in C3H8

Fig. 7. Effect of C3H8 concentration on permeability for each components of the ternary gas mixture for the composite PDMS/PA membrane, at T = 35 8C, P = 5 bar and XH2 = 55%.

879

solubility. Moreover, as C3H8 composition increases, solubility of H2 and CH4 increases. This fact is in contradiction with the typical assumption, which states the solubility of gases in rubbery polymers is independent of each other and it depends individually on the gas properties [8,92]. High mole fraction of C3H8 in the polymer provides the environment of the dense polymer matrix more similar to that of C3H8, wherein H2 and particularly CH4 are more soluble [8]. In this situation, the presence of C3H8 in the polymer structure may generate a more attractive ambience for CH4 and H2 sorption, and this makes possible an increment in their solubility [4]. Diffusion coefficient of all three components increases as C3H8 composition increases. The increasing rate is higher in the lighter gases such as H2 and CH4 than in the heavier gases such as C3H8. As mentioned before, this is owing to the C3H8-induced swelling of the polymer. 4.4. The u values and the diffusion coefficients As known for the rubbery membranes, due to the high solution rate of the gas into the polymer matrix, direct determining an exact unsteady state period of gas transport across the membranes by means of the accessible apparatus is complicated to be performed. As for the severe tendency of pressure profile for prompt switching from its unsteady state into steady state, experimental evaluation of gas diffusivity and solubility values is intricate with a number of errors. Table 2 compares the experimental values of diffusion coefficient of the penetrants in the ternary gas mixture for the composite PDMS/ PA membrane with their corresponding simulated values, obtained from the model solution algorithm (Fig. A.1). The diffusivities were simulated in two basis of CIS and CDS. The results perceptibly showed that in comparison with the traditional time lag method, taking into consideration the concentration functionality of penetrant diffusivity in the simulation process (CDS), leads to improved conformity between the simulated and experimental results. Considerable deviation (between 50 and 100%) of the traditional method is a strong argument for considering the CDS which showed much less deviation (less than 50%). CIS model showed considerable deviation from the experimental data of diffusivity, due to disregarding the effect of C3H8 concentration on the membrane structure, i.e. plasticization phenomenon. As it is observed, using the mentioned algorithm and after n Loop, the deviation of diffusion coefficients are correlated up to 70% in comparison with Loop 1. In this step, the acceptable deviation (between 4 and 100%) was achieved. Definitely, increasing accuracy of the concentration dependent diffusion coefficient function showed better agreement between the simulated results and experimental data. Values of the diffusion coefficient predicted by the model showed an ascending trend in all cases. Table 2a demonstrates an ascending trend for diffusion coefficient as feed temperature increases. The H2 and CH4 diffusion coefficients in Table 2a–c indicate that increasing temperature, pressure, and XC3H8 in the composite PDMS/PA membrane has a more significant effect on diffusion of the mentioned components in comparison with that of C3H8. The simulation results represents that the diffusion profile of H2 reaches to its steady state more rapidly than those of the other two components. For instance, plasticization takes place when concentration of the components increases in the polymer due to an increase in the polymer local segmental motions caused by the presence of diffusive molecules within the polymer structure [6,7,10,93]. According to Table 2, increasing pressure and XC3H8 increases C3H8 concentration in the polymer. Thus, H2 and CH4 experimental diffusion coefficients increase. Additionally, the simulated results

A. Ebadi Amooghin et al. / Journal of Industrial and Engineering Chemistry 19 (2013) 870–885

880

Table 2 Comparison of experimental values for diffusion coefficient of each component in ternary gas mixture through the composite PDMS/PA membrane with predicted values by the simulated model for both CDS and CIS at different operating conditions. PDMS/PA

Experimental Data

Simulation Results

Temperature (8C)

Temperature (8C)

Penetrants

25

30

35

40

45

25

30

35

40

45

C3H8

1.11e5

1.24e5

1.43e5

1.50e5

1.56e5

CISa CDSb CDSc

1.60e6 9.25e6 1.04e5

2.18e6 1.01e5 1.14e5

3.43e6 1.13e5 1.32e5

6.00e6 1.23e5 1.38e5

1.20e5 1.31e5 1.49e5

CH4

2.86e5

3.11e5

3.38e5

3.66e5

3.92e5

CISa CDSb CDSc

8.00e6 2.44e5 2.67e5

8.57e6 2.57e5 2.97e5

1.20e5 3.02e5 3.18e5

2.40e5 3.13e5 3.43e5

3.00e5 3.42e5 3.74e5

H2

3.04e4

3.13e4

3.08e4

3.15e4

3.21e4

CISa CDSb CDSc

1.85e6 2.57e4 2.86e4

2.82e6 2.65e4 2.92e4

4.80e6 2.70e4 2.90e4

8.00e6 2.75e4 2.98e4

2.40e5 2.81e4 3.09e4

CH4 53.37 13.99 5.59

H2 97.38 13.66 5.52

Standard deviation (%)

PDMS/PA

C3H8 CISa 65.42 CDSb 18.04 CDSc 6.91 (a) P = 5 bar, XC3H8 = 10%

Experimental Data

Simulation Results

Pressure (bar)

Pressure (bar)

Penetrants C3H8

2 1.58e5

3 1.54e5

4 1.52e5

5 1.43e5

6 1.38e5

CISa CDSb CDSc

2 1.85e6 1.07e6 1.45e5

3 2.41e6 1.02e5 1.38e5

4 2.82e6 9.98e6 1.40e5

5 3.43e6 9.89e6 1.35e5

6 4.80e6 9.81e6 1.23e5

CH4

3.10e5

3.13e5

3.25e5

3.38e5

3.43e5

CISa CDSb CDSc

4.80e6 2.65e5 2.95e5

6.00e6 2.71e5 3.01e5

9.60e6 2.85e5 3.12e5

1.20e5 2.94e5 3.21e5

4.80e5 3.02e5 3.28e5

H2

2.49e4

2.67e4

2.88e4

3.08e4

3.33e4

CISa CDSb CDSc

2.40e6 1.83e4 2.28e4

3.00e6 1.97e4 2.42e4

4.00e6 2.04e4 2.64e4

4.80e6 2.17e4 2.92e4

8.00e6 2.30e4 3.11e4

CH4 52.07 13.04 4.41

H2 98.51 28.47 7.59

C3H8 CISa 79.06 b CDS 44.22 c CDS 8.59 (b)T = 35 8C, XC3H8 = 10%

Standard deviation (%)

PDMS/PA

Experimental Data

Simulation Results

XC3H8 (%) Penetrants

2

XC3H8 (%) 4

6

8

10 a

2

4

6

8

10

C3H8

7.34e6

9.96e6

1.15e5

1.30e5

1.43e5

CIS CDSb CDSc

9.60e7 6.55e6 7.15e6

1.14e6 8.47e6 9.47e6

1.50e6 9.89e6 1.04e5

2.00e6 1.01e5 1.24e5

3.43e6 1.09e5 1.35e5

CH4

2.80e5

2.88e5

2.98e5

3.15e5

3.38e5

CISa CDSb CDSc

3.43e6 2.03e5 2.62e5

4.14e6 2.11e5 2.69e5

6.00e6 2.25e5 2.81e5

8.00e6 2.42e5 2.98e5

1.20e5 2.65e5 3.18e5

H2

1.74e4

2.07e4

2.39e4

2.66e4

3.08e4

CISa CDSb CDSc

1.41e6 1.12e4 1.67e4

1.71e6 1.25e4 1.81e4

2.67e6 1.46e4 2.13e4

3.42e6 1.68e4 2.48e4

4.80e6 1.92e4 2.88e4

CH4 78.47 24.70 6.01

H2 98.88 37.73 8.14

Standard deviation (%)

a b c

C3H8 CISa 84.61 CDSb 17.16 CDSc 5.46 (c) T = 35 8C, P = 5 bar

Concentration Independent System [Traditional time lag method]. Concentration Dependent System [Frisch method (Loop 1)]. Concentration Dependent System [Frisch method (Loop n)].

exhibit an ascending trend. The chain mobility increases due to the polymer swelling and this consequently leads to extensive increasing in diffusivity of the lighter components [4]. Diffusion coefficient of all three components showed an ascending trend with increasing C3H8 composition. In comparison with the heavier gas, C3H8, the lighter gases, H2 and CH4, exhibit higher increasing rates.

Table 3 presents the time lag values acquired from CDS and CIS models for the composite PDMS/PA membrane at various operating conditions. The values in the first row were acquired from the P–t curve (traditional method) and in the second one, by modifying the diffusion coefficient via the Frisch method. The unsteady state time lag values for gas permeation are perceptibly shorter than those observed in the experiments (approximately

A. Ebadi Amooghin et al. / Journal of Industrial and Engineering Chemistry 19 (2013) 870–885 Table 3 Time lag values acquired from CDS and CIS for the composite PDMS/PA membrane at various operating conditions. Penetrants

Method

Composite PDMS/PA

25

30

35

40

45

(a)

(b)

(a)

(b)

(a)

(b)

1.67 1.76 2.82

0.60 0.91 0.52

4.82 4.02 4.71

3.97 6.71 5.77

3.41 2.83 2.95

2.93 3.92 3.94

P–t curve Frisch method

15 11

11 9

7 5

4 2.8

2 1

CH4

P–t curve Frisch method

13 10.5

8.5 6.7

5 3

3 1.5

1 0.9

H2

P–t curve Frisch method

3 2

2.8 1.8

2 1.5

1 0.9

0.8 0.6

(a)P = 5 bar, XC3H8 = 10% Method

Composite PDMS/PA Pressure (atm) 2

3

4

5

6

C3H8

P–t curve Frisch method

13 10

10 7.5

8.5 6

7 5.2

5 3.5

CH4

P–t curve Frisch method

10 9

8 6.2

6 4.8

5 4

3 1.5

H2

P–t curve Frisch method

5 4

4 3

2.5 2

2 1.5

0.5 0.4

Method

Temperature test Pressure test XC3H8 (%) test

CH4

H2

permeation properties [42]. According to the simulation results, pressure profile in the support zone as a function of time, shows no delay for reaching to the steadiness. Actually, due to pore size of 0.2 mm, convective flux does not confront with resistance throughout the transport across the membrane. Diffusion coefficients were obtained based on single layer PDMS membrane and the diffusivity of each component in PA support layer was assumed to be large enough. This argue confirms the above mentioned experimental assumption which states the ‘‘support layer has no effect on gas permeation properties’’. 4.5. Ternary gas mixture permeation flux

(b)T = 35 8C, XC3H8 = 10% Penetrants

Standard deviation (%) C3H8

C3H8

Penetrants

Table 4 Standard deviation (%) of the model from the experimental results at different operating conditions. (a) gas permeation flux (b) mole fractions of each component in the permeate side. Test

Temperature (8C)

881

Composite PDMS/PA XC3H8 (%) 2

4

6

8

10

C3H8

P–t curve Frisch method

25 22.8

21 19

16 14.5

12 11

7 5.5

CH4

P–t curve Frisch method

17 15

14 12.5

9 8

7 5.5

5 4

H2

P–t curve Frisch method

7 6

5.8 4.5

4 3.5

3 2.2

2 1

(b)T = 35 8C, XC3H8 = 10%

500 s). In the experiments, after 500 s the trend of gas transport through the membranes reached to its steady state and the permeability values were calculated at this point. This study was accomplished in order to simulate the trend of gas transport from the initial unsteady state to the steady state of 1500 s. The permeability values were obtained for the steady state duration. It was shown that the duration of unsteady state process obtained by the simulation is considerably shorter in comparison with steady state period. Likewise, during 1–25 s, the permeation flux tends towards the steadiness. After modifying the diffusion coefficients, the shorter time lag values (according to Eq. (A.15)) were attained. This definitely showed that considering the concentration-dependent diffusion coefficient and consequently location across the membrane results in obtaining the exact unsteady state period of transport. Table 3a shows that the values of u for H2 at 45 8C are equal to 0.8 and 0.6 s, which are considerably fewer than those of the two other components. Indeed, the simulation results verify the fact that diffusivity is the key parameter of lighter gases permeabilities (H2 and CH4) whereas solubility is the impressive parameter of heavier gases permeability (C3H8). In the experiment, diffusion coefficients were calculated indirectly in accordance with the following method: Single layer PDMS was placed into the absorption chamber and solubility values were determined. Afterwards, by using permeability values, the diffusion coefficients were calculated. In fact, in that work it was assumed that PA as the support layer does not affect

Comparison between the simulation predictions and experimental results of ternary gas permeation flux across the composite PDMS/PA membrane at different operating conditions, are given in Table 4. The simulation results show the good conformity with the reported experimental values for permeation flux. Increasing pressure and XC3H8 increases permeation flux of all three components. However, permeation flux of the components is significantly affected by increasing pressure. Increasing temperature decreases permeation flux of C3H8, while higher H2 and CH4 permeation flux is observed. 4.6. Effect of operating conditions on mole fraction of each component in permeate side One of the fundamental parts of membrane separation processes is the membrane capability for selective separation of components. From the membrane material point of view, most of the existing industrial polymers have been used in the membrane preparation. Selecting a material for the membrane fabrication is not arbitrary, but it is carried out considering various physical and chemical properties [94]. Since permeability is a quantitative criteria of membrane throughput and selectivity is a qualitative criteria, both of them must be appreciated together at the membrane design [95]. Up to now, many studies have been carried out to increase the performance of polymeric membranes. According to these researches, the most important methods for increasing the performance of polymeric membranes are as follows [96–101]: 1. Incorporation of flexible and polar groups such as amines, carboxyles, . . . 2. Mixing with a carrier (fixed carrier membranes) such as type one amino groups as a CO2 carrier. 3. Using a soft segment such as PDMS. 4. Addition of a compatibilizer such as PS-b-PMMA in PMMA/poly methyl ether blend. 5. Polymer blending and interpenetrating polymer networks (IPN). 6. Chemical cross-linking and load-bearing network creation via covalent linkages. 7. Structural modification of block copolymers by block copolymerization with a polymer having specific mechanical

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properties that forms a nanostructure which has physical crosslinkages with favorite properties. 8. Free volume increasing by adding (nano) particles in polymer matrices. Mole fraction of each component in permeate side were calculated so as to evaluate the ability of the simulated model for separation of each component, as well as comparing experimental data. The deviation of the model prediction with the experimental results was calculated by the following formula:

 Permeability of all components increases through the membrane, as XC3H8 increases. This behavior is due to the polymer swelling occurred by the presence of the heavier component such as C3H8. The polymer swelling increases the chain mobility which consequently leads to a remarkable increase in diffusivity of the lighter components. The model predictions were in acceptable conformity with the experimental results for mixed gas permeation properties, through the self-synthesized composite PDMS/PA membrane. Acknowledgements

100  jexperimental  theoreticalj deviation ¼ experimental

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Deviation values of the model from the experimental results for mole fraction of each component in the permeate side for different operating conditions, are given in Table 4. YC3H8 in the permeate side of the membrane is much higher than mole fractions of the two other components. Additionally, higher values of YC3H8 in the permeate side, show the capability of membrane for C3H8 separation process. The simulation results confirmed that the mole fraction of each component in the permeate side would stand in close proximity with the experimental data. The CH4 deviation was larger than those of the other two components. The simulation results confirmed the model qualifications, in order to predict the area in which no experimental data have been reported. 5. Conclusions A comprehensive model was presented for direct determination of diffusion coefficients. At first, the time lag method was taken into account, and then it was improved to obtain the precise values of diffusion coefficient by the Frisch method. Moreover, the model was developed to simulate gas mixture permeation through composite PDMS/PA membrane. In the regard, convection and diffusion was taken into account along with the solution-diffusion as the governing transport mechanism. The entire results were acquired at various operating conditions (upstream temperatures, pressures and feed compositions). The results are summarized as below:  The results achieved for diffusion coefficient demonstrated that considering the concentration-dependent diffusion coefficient and consequently location across the membrane results in obtaining the exact unsteady state period of transport. The simulation procedure, resulted in achieving better conformity between the simulated and experimental results, comparing to the traditional time lag method. Using this method, the deviation of diffusion coefficients was correlated up to 70%.  As feed temperature increases, a slightly decrease in C3H8 permeability is observed through the membrane, whereas permeabilities of H2 and CH4 increase. Solubility of the lighter gases decreases as temperature increases, while their diffusivity increases considerably, which results in higher permeability. As temperature increases, solubility of C3H8 decreases, but its diffusivity increases simultaneously. Thus, permeability decreases a little or remains constant.  The simulation results of permeability in pure gas experiments indicated that as fugacity increases, C3H8 permeability increases, while those of the lighter gases (H2 and CH4) slightly decrease. In contrary, the exactly opposite behaviour was observed for the ternary gas mixture. That is to say, with increasing fugacity, the lighter gases permeabilitied increase, while that of the heavier gas decreases.

The authors would like to thank Dr. Hamidreza Sanaeepur for his valuable comments.

Appendix A. Model solution procedure Quasi-linear equation used for the multi-component diffusivity is defined by:

@C i;m @ ¼ ½ J ðy; t Þ @y @t

(A.1)

where C and J represent the concentration and the flux density profiles of penetrant, functions of y (perpendicular coordinate as a function of the plane film) and time t. Using the following initial and boundary conditions (Note: in Eqs. (A.1)–(A.15), Y denotes the membrane thickness and y = 0 is considered at the membrane feed face): C i; j ðy; 0Þ ¼ 0; 0 < y < Y

(A.2)

C i; j ð0; tÞ ¼ C 0;i;m ; t > 0

(A.3)

C i; j ðY; tÞ ¼ 0; t > 0

(A.4)

the non-steady rate of flow through the right boundary y = Y is denoted by:

@C i;m J ðY; t Þ ¼  Di;m ðC Þ (A.5) @y y¼Y and the total flow through this boundary at time t is obtained by: Z t Q ðY; t Þ ¼ J ðY; t Þ dt (A.6) o

The first step is to integrate both sides of the diffusion Eq. (A.1) over y from y to Y: Z

Y y

@C i;m dy ¼ Jðy; t Þ  J ðY; t Þ @t

(A.7)

The second one is to integrate over y from 0 to Y. After rearranging, the formula for flux at the right boundary is given by the following equation [62]: (Z ) Z YZ Y C 0;i @C i;M ðx; t Þ 1 dx dy Di;m ðCÞ dC  (A.8) J ðY; t Þ ¼ Y @t 0 0 y Finally, integrating over t from 0 to t and changing the order of integration yield to: ( Z ) Z YZ Y C 0;i 1 t Di;m ðCÞ dC  C i;m ðx; t Þ dx dy Q ðY; t Þ ¼ (A.9) Y 0 0 y

A. Ebadi Amooghin et al. / Journal of Industrial and Engineering Chemistry 19 (2013) 870–885

by [4,32,42,86,102]:

The asymptote Q s ðY; t Þ to Q ðY; t Þ is: Q s ðY; t Þ ¼ JðY; t  uÞ

(A.10)

where its gradient is the steady-state flow rate, J, and the time lag, u, is its intercept on the t-axis. The general time lag relation with re-arranging pervious formulas is given by [66]: RY RY

ui;m ¼

0

y C s;i;m ðxÞ dx dy R C 0;i Di;m ðC ÞdC 0

Y2 6Di;m

D¯ i;m ¼

(A.11)

Z

1 C 0;i;m

C 0;i;m 0

D0;i;m ðC Þ dC

(A.13)

Substitution of Eq. (A.13) into Eq. (A.11) gives: RY RY D¯ i;m ¼

where Cs(x) denotes the steady state concentration distribution of the penetrant molecule. When diffusion coefficient is constant (A type), Eq. (A.11) simplifies to the traditional time lag form:

ui;m ¼

883

0

y

C s;i;m ðxÞ dx dy

ui;m C 0;i;m

(A.14)

Parabolic concentration profile through the membrane, C s;m ðxÞ, is obtained by solving Eq. (2), and using the interrelated boundary conditions. The numerator of Eq. (A.14) is computed by indefinite integration from y to Y, afterwards, specific integration over the obtained function from 0 to Y.

(A.12) A.1. Model solution algorithm

In case of concentration dependent diffusion coefficient (B type), the mean value of diffusion coefficient can be obtained

The model development process is demonstrated in Fig. A.1. In the first step, data source 1 (Ternary gas permeation properties),

Fig. A.1. Solution algorithm applied for numerical analysis.

884

A. Ebadi Amooghin et al. / Journal of Industrial and Engineering Chemistry 19 (2013) 870–885

physical properties and operating conditions were used to solve the model, which consequently provided the simulation values of permeation at different operating conditions. In the second step, for assessing time lag and diffusion coefficients values, the Frisch method was applied based on two general approaches; CIS and CDS. Taking into consideration the Frisch method, parabola concentration profile was acquired entirely the membrane thickness (Loop 1). Afterwards, using Eq. (A.14), and the time lag values acquired from P–t curves, the diffusion coefficients were determined. Subsequently, the new diffusion coefficient was again placed in the next step of model solution to attain a new concentration distribution (Loop 2). This procedure was repeated again and again to satisfy the specific constrains (n Loops). It should be noted that this procedure was performed for the membrane at all operating conditions (upstream temperatures, pressures and feed compositions). Moreover, the results were acquired after about 15 iterations. As a final point, having the correlated concentration distribution functions, u values were obtained from the Frisch method using the following equation [93]: RY RY

ui;m ¼

0

y

C s;i;m ðxÞ dxdy RY J 0 dy

(A.15)

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