water mixture

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Chemical Engineering and Processing 47 (2008) 245–250

Model of mass transfer in polyvinyl alcohol membrane for isopropanol/water mixture Meng Han a,∗ , Bo Zhao b , Xue-Mei Zhang a , Wei-Jiang Zhang a a

b

Chemical Engineering Research Center, Tianjin University, Tianjin 300072, China School of Chemical Engineering & Technology, Hebei University of Technology, Tianjin 300130, China Received 13 August 2006; received in revised form 3 January 2007; accepted 16 January 2007 Available online 30 January 2007

Abstract Based on Flory–Huggins theory and Fick’s law, the model of mass transfer in polyvinyl alcohol (PVA) membrane for isopropanol/water mixture was established. The predicted results fit well with the experimental data. The interactional parameter between water and PVA membrane is less than that between isopropanol and PVA membrane, which validates that water is preferentially adsorbed and dissolved in PVA membrane. The plasticizing coefficient and diffusion coefficient at infinite dilution of water are larger than those of isopropanol, which shows that the dissolution and permeation of water are greater than those of isopropanol in PVA membrane. So water permeates preferentially. Both the interactional parameter between water and isopropanol in PVA membrane and that in feed rise with the increase of isopropanol content in feed, which shows that the larger isopropanol content is, the higher selectivity of PVA membrane is and the more remarkable separation effect of pervaporation process is. © 2007 Elsevier B.V. All rights reserved. Keywords: Polyvinyl alcohol membrane; Swelling equilibrium; Permeate flux; Isopropanol/water mixture; Pervaporation

1. Introduction Pervaporation (PV) is an energy-efficient process for the separation of liquid mixtures in the chemical processing industry, especially for separation of azeotropic or close-boiling liquid mixtures [1]. Now dehydration of organic solvents is the bestdeveloped area in PV technology. One vital issue for industrial application of PV processes is the ability to tailor membrane materials with high PV performance. Another fundamental issue is modeling pervaporation transport to optimize PV process. The model of mass transfer through the membrane has been studied quite extensively [2]. Many models were proposed to predict the mass transfer process, such as solution-diffusion model [3], thermodynamics of irreversible process [4], Maxwell–Stefan theory [5], pore flow model [6], pseudo phase change solution-diffusion model [7], resistance-in-series model [8], molecular simulation [9] and so on. Among them, solution-diffusion model is most widely used in describing PV transport including sorption and diffusion steps. Considerable attention has been focused on the mass transfer behavior in recent years. ∗

Corresponding author. Tel.: +86 13820919653; fax: +86 22 27409476. E-mail address: [email protected] (M. Han).

0255-2701/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2007.01.032

For modeling pervaporation transport, the permeation flux, which depends on the solubility and diffusivity of components in the membrane, should be obtained. Usually, the solubility is calculated according to the Flory–Huggins theory [10]. For the diffusivity, the predictive methods of component diffusion in polymer solution have been commonly studied [11]. The solvent transport in the membrane is generally considered to be a molecular diffusion mechanism, so Fick’s law [12,13] is often used to describe the diffusion process of component in membrane, which mainly involves in the calculation of diffusion coefficient. In recent years, many papers [14–16] have described the diffusion coefficient of binary mixtures in pervaporation membrane, where the magnitude of obtained diffusion coefficient is about 10−7 to 10−11 m2 h−1 . Isopropanol, a widely used solvent in chemical and pharmaceutical industries, is known to form an azeotrope with water, a characteristic that creates difficulties in its recovery by the conventional distillation [17], while PV technique, as a economical and safe and clean means, is anticipated to achieve dehydration of isopropanol. So the pilot studies have been carried out by the author of this paper and satisfactory results have been obtained [18]. In this study, polyvinyl alcohol (PVA) membrane is employed to separate isopropanol/water mixture. And based on

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Flory–Huggins theory and Fick’s law, the mass transfer model of this PV process is established to predict the swelling and pervaporation properties.

the definition of chemical potential, the equality of chemical potential can be expressed by activity equilibrium:

2. Experimental

where superscripts s and m are respectively feed phase and membrane phase. In Eq. (2), the activity of component i in feed ais can be described as a product of activity coefficient γis and molar fraction in feed xis :

2.1. Swelling experiments The thickness of the PVA homogeneous membrane used in swelling experiments (provided by Beijing Megavision Membrane Technology & Engineering Co., Ltd.) was 20 ␮m. Average degree of polymerization and degree of alcoholysis was 1750 ± 50 and 98% (mol mol−1 ), respectively. And the experimental feed was isopropanol/water mixture. Samples of dried PVA membrane were weighed and then immersed respectively in pure water, isopropanol and the mixture with different content at specific temperature. While the samples reached steady state, they were taken out and weighed immediately after carefully wiping out the excess liquid. The process was repeated three times and an average value was taken. The degree of swelling DS (g(g membrane)−1 ) is as follows: m2 − m1 DS = × 100% (1) m1 where m2 is the mass of swollen polymer, g; m1 is the mass of dried polymer, g. The extraction of the absorbed product was performed under vacuum. The desorbed vapor was collected in a trap cooled with liquid nitrogen, and the desorbed product was analyzed by gas chromatography. 2.2. Pervaporation experiments PV experiments were performed with a 500 mL filter unit from Nalgene Company (American). The downstream pressure was kept about 2 kPa. And the permeate was collected in a trap cooled with liquid nitrogen, weighed, and then analyzed by gas chromatography. The feed was circulated in the upper chamber of the unit to eliminate the effect of concentration polarization on separation process. The membrane used in PV experiments (provided by Beijing Megavision Membrane Technology & Engineering Co., Ltd.) was composed of PVA as separation layer and polyacrylonitrile (PAN) as support layer. The thickness of the PVA layer and that of the PAN layer were respectively 2 and 80 ␮m. The experimental feed and PVA layer material were the same with those of swelling experiments. 3. Theoretical 3.1. Swelling equilibrium model in PVA membrane for isopropanol/water mixture When swelling equilibrium is reached at the interface between the mixture and the polymeric membrane, thermodynamic phase equilibrium between the mixture and the polymeric membrane is established, which indicates that the chemical potential of component i in each phase is equal. According to

ais = aim

(2)

ais = γis xis

(3)

Flory–Huggins theory is introduced to calculate the activity of component i in membrane [19]: m = ln ΦW + (1 − ΦW ) − ΦI ln aW

VW VW − ΦM VI VM

m + (χWI ΦI + χWM ΦM )(ΦI + ΦM )

− χIM

VW ΦI ΦM VI

ln aIm = ln ΦI + (1 − ΦI ) − ΦW m +(χWI ΦW

− χWM

(4) VI VI − ΦM VW VM

VI + χIM ΦM )(ΦW + ΦM ) VW

VI ΦW ΦM VW

(5)

where V is molar volume, m3 mol−1 ; χ is interactional parameter; and subscripts W, I and M are respectively water, isopropanol and PVA membrane. In Eqs. (3)–(5), γis can be calculated by Wilson equation, and the Wilson equation parameters (ΛWI and ΛIW ) are respectively 0.18865 and 0.79491 [20]. VW /VI = (MW /MI )(ρI /ρW ) = (1.8/6) (0.79/1) = 0.237 [17]. Due to the great larger molar volume of PVA membrane than that of water and isopropanol [21], VW /VM and VI /VM are set to zero in this paper. So swelling equilibrium model in PVA membrane for isopropanol/water mixture can be written as:  s s s xW exp − ln(0.18865 + 0.81135xW ) + (1 − xW )  × = ΦW

 0.18865 0.79491 s − 1 − 0.20509xs 0.18865 + 0.81135xW W  m exp 1 − ΦW − 0.237ΦI + (χWI ΦI + χWM ΦM )

× (1 − ΦW ) − 0.237χIM ΦI ΦM ]  s xI exp − ln(0.79491 + 0.20509xIs ) + (1 − xIs )

(6)



 0.79491 0.18865 × − 0.79491 + 0.20509xIs 1 − 0.81135xIs  m ΦW = ΦI exp 1 − ΦI − 4.219ΦW + (4.219χWI + χIM ΦM )(1 − ΦI ) − 4.219χWM ΦW ΦM ]

(7)

From Eqs. (6) and (7), it is found that swelling equilibrium of component in membrane relates to not only the interactional

M. Han et al. / Chemical Engineering and Processing 47 (2008) 245–250

247

parameter between components but also that between component and membrane.

of the PVA membrane is and the more remarkable separation effect of the PV process is.

3.2. Interactional parameters

3.2.3. Mathematical description of diffusion in membrane According to solution-diffusion model, diffusion in membrane can be expressed by Fick’s law, where mass concentration can be described by volume fraction Φi :

3.2.1. Interactional parameter between pure component and membrane According to Flory–Huggins theory, the effect of viscoelasticity on chemical potential can be ignored, so the interactional parameter between component i and membrane χiM is described as [19]: χiM = −

[ln(Φi ) + (1 − Φi )] (1 − Φi )2

(8)

where Φi is volume fraction of pure component in membrane, which can be calculated by the degree of swelling of pure  [22]. component DSi By Eq. (8), interactional parameter between component and membrane χiM at 323 K (optimum operation temperature [18]) is calculated. χWM and χIM is 0.744 and 1.671, respectively. Thus it can be concluded that the interaction between water and the membrane is greater than that between isopropanol and the membrane, so water is preferentially adsorbed and dissolved in PVA membrane. 3.2.2. Interactional parameters between isopropanol and water The interactional parameter between isopropanol and water s can be calculated by [23]: in feed χWI   1 xs xs GE s s xW = s (9) ln W + xIs ln I + χWI x W νI νW νI RT where GE is excess Gibbs free energy, J mol−1 , which can be obtained from Wilson equation; vi is volume fraction of component i in feed. s is expressed by v from the To simplify the calculation, χWI I following equation: s = a + bνI + cνI2 + dνI3 + eνI4 χWI

(10)

The constant (a–e) in Eq. (10) can be calculated by Eq. (9). s at 323 K can be described as: So χWI

Ji = −Di

Under the same conditions, the interactional parameter m can be also between isopropanol and water in membrane χWI expressed by Eq. (11) [24]: m = 0.712 − 0.0175uI + 1.547u2I − 2.147u3I + 1.850u4I χWI

∗ JW = −ρW DW exp(ψW ΦW + ψI ΦI )

JI = −ρI DI∗ exp(ψW ΦW + ψI ΦI )

dΦW dz

dΦI dz

(14) (15)

where ψi is the plasticizing coefficient of component i, dimensionless constant; Di∗ is the diffusion coefficient at infinite dilution, m2 h−1 , which is only dependent on temperature [12]. The boundary conditions of Eqs. (14) and (15) are as follows: z = 0,

ΦW = ΦWu ,

ΦI = ΦIu ;

z = y,

ΦW = ΦWd ,

ΦI = ΦId

where y is the thickness of membrane, ␮m; ΦWu and ΦIu are respectively volume fraction in upstream side membrane of water and that of isopropanol; ΦWd and ΦId are respectively volume fraction in downstream side membrane of water and that of isopropanol. After integral and development, one has: JW =

JI =

∗ (Φ ρW DW Wu − ΦWd ) y (ψ W e ΦWu +ψI ΦIu ) − e(ψW ΦWd +ψI ΦId ) × (ψW ΦWu + ψI ΦIu ) − (ψW ΦWd + ψI ΦId )

ρI DI∗ (ΦIu − ΦId ) y e(ψW ΦWu +ψI ΦIu ) − e(ψW ΦWd +ψI ΦId ) × (ψW ΦWu + ψI ΦIu ) − (ψW ΦWd + ψI ΦId )

(16)

(17)

Because of the very low pressure in downstream side, concentration in downstream side membrane can be neglected compared with that in upstream side membrane. Thus Eqs. (16) and (17) can be reduced into: JW =

(12) where uI is the relative volume fraction of isopropanol in membrane, uI = ΦI /(ΦW + ΦI ). s and χm both From Eqs. (11) and (12) it can be found that χWI WI rise with the increase of isopropanol content, which shows that the larger isopropanol content is, the higher swelling selectivity

(13)

where an exponential relationship of the diffusion coefficient Di with Φi seems to be more appropriate [13,25,26], so Eq. (13) can be written as:

s χWI = 0.712 − 0.0175νI + 1.547νI2 − 2.147νI3 + 1.850νI4

(11)

dCi dΦi = −ρi Di dz dz

JI =

∗ Φ (ψW ΦWu +ψI ΦIu ) − 1 ρW DW Wu e y ψW ΦWu + ψI ΦIu

ρI DI∗ ΦIu e(ψW ΦWu +ψI ΦIu ) − 1 y ψW ΦWu + ψI ΦIu

(18) (19)

Neglecting the influence of concentration polarization, swelling equilibrium is reached at the interface between the feed mixture and PVA membrane in PV process according to

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solution-diffusion model. Thus, ΦIu in Eqs. (18) and (19) can be calculated by Eqs. (6) and (7). And the membrane thickness y is 2 ␮m. 3.3. Model of mass transfer 3.3.1. Estimation of model parameter Inserting Eqs. (6) and (7) into Eqs. (18) and (19) gives: JW

e(ψW ΦWu +ψI ΦIu ) − 1 ∗ = 0.500 × 109 DW ΦWu ψW ΦWu + ψI ΦIu

JI = 0.395 × 10

9

∗ DW = 4.74 × 10−9 m2 h−1 ,

(20)

(ψW ΦWu +ψI ΦIu )

−1 ψW ΦWu + ψI ΦIu

e DI∗ ΦIu

(21)

where:

 s s s exp −ln(0.18865 + 0.81135xW ) + (1 − xW ) xW  ×

0.18865 0.79491 − s s 0.18865 + 0.81135xW 1 − 0.20509xW



m )ΦWu ΦIu + 0.744Φ2Wu + 0.396Φ2Iu ] + (1.140 − χWI

(22)  xIs exp −ln(0.79491 + 0.20509xIs ) + (1 − xIs ) ×

0.79491 0.18865 − s 0.79491 + 0.20509xI 1 − 0.81135xIs

DI∗

= 1.11 × 10

−9

2 −1

m h

,

ψW = 0.525, ψI = 0.116

It can be found that the plasticizing coefficient and diffusion coefficient at infinite dilution of water are both larger than those of isopropanol, which shows that the dissolution and permeation of water are both greater than those of isopropanol in PVA membrane. So water permeates preferentially, which agrees with the result concluded from the interactional parameters. 3.3.2. Model of mass transfer in PVA membrane for isopropanol/water mixture Inserting the value of Di∗ and ψi into Eqs. (20) and (21), model of mass transfer in PVA membrane for isopropanol/water mixture at 323 K can be expressed by:

m = ΦWu exp[1.744 − 2.488ΦWu + (χWI − 1.377)ΦIu



Eqs. (20) and (21) form a system of non-linear quaternary equations with model parameters ψi and Di∗ . And under definite system and temperature, these parameters are constant. Defining target function by maximum likelihood estimate rule [27], model parameters are estimated by combining model equations with experimental data. This process is carried out by fmincon function of Matlab, and the final results are as follows:

JW = 2.415ΦWu JI = 0.454ΦIu



e(0.531ΦWu +0.112ΦIu ) − 1 0.531ΦWu + 0.112ΦIu

e(0.531ΦWu +0.112ΦIu ) − 1 0.531ΦWu + 0.112ΦIu

(25) (26)

m in Eqs. (25) and (26) can be attained by Eqs. ΦWu , ΦIu and χWI (22)–(24).

m = ΦIu exp[2.671 − 4.342ΦIu + (4.219χWI − 9.029)ΦWu

4. Results and discussion

m )ΦWu ΦIu + 1.671Φ2Iu + (4.810 − 4.219χWI

+ 3.139Φ2Wu ]

(23)

2  ΦIu ΦIu + 1.547 ΦWu + ΦIu ΦWu + ΦIu 3 4   ΦIu ΦIu − 2.147 + 1.850 (24) ΦWu +ΦIu ΦWu + ΦIu

m χWI = 0.712 − 0.0175

Based on Eqs. (22)–(26), the predicted volume fraction in membrane i and permeate flux Ji are obtained. The changes of Φi with the increase of molar fraction of isopropanol in mixture xIs are demonstrated in Fig. 1. The average relative error of volume fraction in membrane is 4.91% and 5.80%, respectively. As shown in Fig. 1, the concentration of water in membrane is much higher than that of isopropanol, which shows

Fig. 1. Changes of volume fraction of water ΦW (a) and that of isopropanol ΦI (b) in membrane with the increase of molar fraction of isopropanol in mixture χIs at infusion temperature 323 K.

M. Han et al. / Chemical Engineering and Processing 47 (2008) 245–250

249

Fig. 2. Changes of permeate flux of water JW (a) and that of isopropanol JI (b) with the increase of molar fraction of isopropanol in mixture χIs at feed temperature 323 K and downstream pressure 2 kPa.

that PVA membrane is preferentially permselectivity to water. Volume fraction of water ΦW gradually descends with the rise of isopropanol content (Fig. 1(a)), while that of isopropanol ΦI increases and then begins to level off when xIs is larger than 0.1 (Fig. 1(b)). The reason is that the effect of isopropanol as a poor solvent on membrane is very small at low isopropanol content in feed. With the increase of isopropanol concentration, membrane structure contracts more remarkably. So volume fraction of isopropanol also begins to decrease. Fig. 2 illustrates the changes of Ji with the increase of xIs . The average relative error of permeate flux is 4.09% and 3.53%, respectively. The curve in Fig. 2 is similar with that in Fig. 1. This is a consequence of polymer membrane swelling, which leads to the same trend of permeate flux compared with that of volume fraction in membrane. The predicted volume fraction and permeate flux fit well with the experimental data, which shows that this model of mass transfer can be used to predict the swelling and PV properties and provide a basis for exhaustive study of separation of isopropanol/water mixture by PV technology. 5. Conclusions (1) Based on Flory–Huggins theory and Fick’s law, the model of mass transfer in PVA membrane for isopropanol/water mixture is established. The predicted volume fraction in membrane and permeate flux are in good agreement with the experimental data. (2) The interactional parameter between water and PVA membrane is less than that between isopropanol and PVA membrane, which validates that water is preferentially adsorbed and dissolved in PVA membrane. (3) The plasticizing coefficient and diffusion coefficient at infinite dilution of water are larger than those of isopropanol, which shows that the dissolution and permeation of water are greater than those of isopropanol in PVA membrane. So water permeates preferentially. (4) Both the interactional parameter between water and isopropanol in the membrane and that in the mixture rise with the increase of isopropanol content in mixture, which shows that the larger isopropanol content is, the higher selectivity

of the membrane is and the more remarkable separation effect of PV process is. Appendix A. Nomenclature

a D DS J m M T u V x y

activity diffusion coefficient (m2 h−1 ) degree of swelling g(g membrane)−1 permeate flux (kg m−2 h−1 ) mass (g) molar mass (kg mol−1 ) temperature (K) relative volume fraction molar volume (m3 mol−1 ) molar fraction membrane thickness (␮m)

Greek symbols γ activity coefficient Λ Wilson equation parameter ρ density (kg m−3 ) ν volume fraction in feed χ interactional parameter Φ volume fraction in membrane ψ plasticizing coefficient Superscripts and subscripts s feed m membrane ’ single * infinite dilution i component d downstream u upstream I isopropanol M membrane W water References [1] M. Han, C.L. Li, B. Zhao, Development of pervaporation for separating VOCs from water, in: Proceedings of 1st Chinese National Chemical

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