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Ideal and non-ideal diffusion through polymers
Journal of Membrane Science 191 (2001) 95–102
Ideal and non-ideal diffusion through polymers Application to pervaporation P. Schaetzel a , Z. Bendjama b , C. Vauclair a,∗ , Q.T. Nguyen c a
Laboratoire de Thermodynamique des Procédés, Département Génie Chimique, IUT de Caen bld, Maréchal Juin 14000, CAEN, France b Laboratoire de Valorisation des Substances Naturelles, USTHB, Alger, Algeria c UMR 6522, Polymères, Biopolymères, Membranes, Rouen University, 76 821 Mont-Saint-Aignan, France Received 15 April 1998; received in revised form 16 April 2001; accepted 18 April 2001
Abstract Using the approach developed in a previous paper, we integrate the generalised Stefan–Maxwell (SM) diffusion equations for a unique species in both Lagrangian and Cartesian coordinates. The dusty gas membrane model is considered. Two activity–concentration relationships are used: the Flory–Huggins equation and the Freundlich relationship. The ethyl acetate (EA)/PDMS system follows the Flory–Huggins equation and the computed interaction parameter allows to predict quantitatively the non-dimensional EA flux through a PDMS membrane when the Lagrangian coordinates are utilised. The prediction is less satisfactory when integrating the SM equation in the Cartesian coordinates. The system EA/PDMS can be qualified as an ideal system in diffusion. The water/ethanol/PVA-based membrane at 60◦ C system follows Freundlich’s equation at equilibrium. The bad fitting of the experimental water flux versus volume fraction of water at the feed side of the membrane suggests that this system cannot be described by the dusty gas/Stefan–Maxwell theory even when the Lagrangian coordinates are used. This type of system, defined here as non-ideal, obeys theories of the free volume type, as quoted in a recent paper. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Pervaporation; Diffusion; Stefan–Maxwell; Flory–Huggins; Lagrangian coordinates; Sorption
1. Introduction The permeation of liquids or vapours through dense polymer membranes is generally described by using Fick’s law with a concentration-dependent diffusion coefficient. The concentration-dependent diffusivity equation used is mostly Long’s equation in which the diffusivity depends in an exponential way on the molarity or the volume fraction of the diffusing species in the polymer. Recently, a new approach in which the Stefan–Maxwell (SM) equation was integrated ∗ Corresponding author. E-mail address: [email protected] (C. Vauclair).
through the membrane has been proposed by Schaetzel et al. [1]. The theory was successfully applied to ion-exchange membranes in pervaporation. The membrane model used was the “dusty gas model” [2,3] where the polymer is represented as a suspension of immobile, giant molecules. The integration of the SM equation needs the knowledge of the activity– swelling dependency at polymer–diffusant equilibrium. Schaetzel et al. [1] used the Flory–Huggins and the Freundlich relationships to fit their equilibrium data. In this paper, we use this new approach to treat the experimental results obtained in pervaporation and in sorption equilibrium of two pseudo-binary systems, namely the
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P. Schaetzel et al. / Journal of Membrane Science 191 (2001) 95–102
Nomenclature
2. Theory
List of symbols ai activity of species i Ai Freundlich law constant Bi Freundlich law exponent species i molarity (mol m−3 ) ci mutual diffusion coefficient (m2 s−1 ) Dij molar flux (mol m−2 s−1 ) Ji R ideal gas constant (J mol−1 K−1 ) T absolute temperature (K) Vi i-component molar volume (m3 mol−1 ) z membrane thickness (m) in the Cartesian coordinates Z membrane thickness (m) in the Lagrangian coordinates
2.1. Representation of the polymer
Greek letters χ ij Flory–Huggins interaction parameter i-component volume fraction in φi polymer membrane µi chemical potential (J mol−1 ) Subscripts e membrane downstream side i, j permeating species m membrane o membrane upstream side 1 key component 2 slow and less concentrated species
polydimethylsiloxane (PDMS) membrane/ethyl acetate/water system and the PVA-based membrane/ water/ethanol system, where the PVA-based membrane consists of a blend of polyvinylalcohol (PVA) and a poly(acrylic acid-co-maleic acid) copolymer. For the first system, we will neglect the concentration and the diffusion of water in the membrane; for the second system, we will neglect the flux and the concentration of ethanol. Using the Flory–Huggins/Stefan–Maxwell (FHSM) equation and the Stefan–Maxwell/Freundlich (FSM) equation in the Cartesian and Lagrangian coordinates to fit the experimental curves, we will show that the permeation behaviours of a solvent through a membrane are quite different for an ideal system (EA/PDMS membrane) and for a non-ideal system (water/PVA-based membrane).
The representation of the polymer in the “dusty gas” model, is compatible with that used by Flory. Actually, the “dusty gas” model was used by Spiegler [2] for the modelling of ion-exchange membranes. The name and the precise definition of the representation were given by Mason et al. [3] for gas-diffusion computations. In the “dusty gas” model, the medium (i.e. the matrix of the swollen polymer) is visualised as a suspension of giant molecules held stationary in space. No assumption is made on the geometrical shape of the particles called polymer species. The swollen polymer is modelled as a homogeneous mixture of polymer particles held stationary in space and of moving solvent particles (the permeate species).
2.2. The Stefan–Maxwell diffusion equation in the dusty gas model The Stefan–Maxwell diffusion equation was obtained from the statistical physics theory of ideal gases [4] and generalised to non-ideal solution by Scattergoot and Lightfoot [6]. The general expression when only a pressure gradient and a concentration gradient are applied can be written for a multicomponent mixture j =n j =1
cj RT Ji Jj dp dµi − Vi =− − , cDij ci cj dz dz
c=
j =n
cj
j =1
(1) In pervaporation, the pressure gradient term is generally neglected. For a polymer (m) single solvent (i) system, the Stefan–Maxwell Eq. (2) becomes d ln ai Ji cm = dz (cm + ci )Dim ci
(2)
The relationship between ci and cm is Vi c i + V m c m = 1
(3)
P. Schaetzel et al. / Journal of Membrane Science 191 (2001) 95–102
2.3. Integration of the Stefan–Maxwell equation for a single component when the equilibrium follows Flory–Huggin’s theory To link the Flory–Huggins (FH) equation with the SM equation, the permeate volume fraction is expressed as a function of the molarity by φi = Vi ci
The integration [1] of Eq. (2) in which the permeate molarity in the membrane is related to the permeate activity according to Eqs. (4) and (5), gives Vi2 Dim Ji = 2 Vm − 2Vi − 2χim Vi + Vm V i ze 2 − φ2 ) (φio 2 ie 3 3 + χim (Vi − Vm )(φio − φie ) 2 3 1 − φio (6) −Vi ln 1 − φie
×
In this equation, the subscripts ‘o’ and ‘e’ refer to the membrane upstream and downstream sides, respectively, and ze is the thickness of the membrane. 2.3.2. Integration in the Lagrangian coordinates The swelling of a polymer in contact with the solvent at large activities may be high. For example, when the polymer is equilibrated with pure ethyl acetate, the volume fraction of ethyl acetate in the swollen PDMS membrane is ca. 0.60. One cannot neglect the variation of the thickness of the membrane in pervaporation. To take into account the variation in the local swelling of the membrane in the flow direction, we choose the coordinates bound to the dry membrane, the Lagrangian coordinates Z, defined by dZ = (1 − Φi ) dz
side of the membrane. We obtain the expression of a single species permeation flux in the frame of the FHSM model in the Lagrangian coordinates Dim Vi (φio − φie ) Ji = 2 V 1 Ze 2 − φ2 ) (φio ie (Vm − 2Vi − 2χim Vi ) 2 3) (φ 3 − φie + io (2χim Vi − (Vm − Vi )(1 + 2χim )) 3 4 − φ4 ) (φio ie (8) (Vm − Vi )χim + 2
+
(4)
2.3.1. Integration in the Cartesian coordinates The general Flory–Huggins [5] equation for a single species can be written as 1 − Vi ln ai =ln φi + (1−φi )+χim (1 − φi )2 z (5) Vm
(7)
We replace the Cartesian coordinates by the Lagrangian coordinates in differential Eq. (2) and perform the integration from the wet side to the dry
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2.4. Integration of the Stefan–Maxwell equation for a single component when the equilibrium follows Freundlich’s equation 2.4.1. Integration in the Cartesian coordinates Freundlich’s equation fits many experimental equilibrium data in the case of the adsorption of organic vapours onto mineral adsorbents; in the swelling of polymer, this equation has to be considered as an empirical one. For a single component, Freundlich’s equation is written as [1] ai = Ai φiBi
(9)
Ai and Bi are two constants depending on the polymer/solvent system and the temperature. Putting this expression in the Stefan–Maxwell Eq. (3) and integrating the new equation from the membrane upstream side to the downstream one yield [1] Ji =
Dim Bi Vi z e Vm Vm 1 − φio × 1− (φio − φie )− ln Vi Vi 1 − φie (10)
2.4.2. Integration in the Lagrangian coordinates Using Eqs. (2), (4), (7) and (9) and integrating from the membrane upstream side to the downstream side, one obtains the Freundlich–Stefan–Maxwell (FSM) equation in the Lagrangian coordinates 2 2 Φio − Φie Vm Bi Dim Ji = −1 + Φio − Φie Vi Ze Vi 2 (11)
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3. Checking of the models To check the model, we use the pervaporation data concerning the EA/water/PDMS membrane system at 30◦ C (this work), and the recently published data on the water/ethanol/PVA-based membrane system at 60◦ C [7].
3.1.3. Pervaporation experiments The experimental device and procedure have been described in a previous paper [8]. Briefly, the device consists of a stirred pervaporation cell maintained at 30◦ C and a vacuum gauge. In all experiments, the vacuum was maintained at a level better than 2 mmHg. The 70 m thick membrane was used in all experiments.
3.1. Experimental 4. Results 3.1.1. Membranes PDMS membranes are obtained by mixing the Rhodorsil 141A and Rhodorsil 141B components manufactured by Rhone–Poulenc Co. 3.1.2. Equilibrium The sorption data were obtained from classical sorption–desorption experiments: the membrane sample was allowed to reach the sorption equilibrium at different compositions of the water/EA mixture in closed flasks maintained at 30◦ C. Then the membrane sample was quickly blotted with a paper sheet, and the sorbed mixture was extracted out of the membrane under vacuum. The desorbed vapour was condensed in a liquid air trap, and the collected liquid was weighed and analysed by gas chromatography.
4.1. FHSM model in pervaporation We use here the experimental results concerning the EA–water–PDMS membrane system at 30◦ C (Figs. 1 and 2). In order to apply the developed FHSM model, we have to check first if the investigated system is a pseudo-binary one, i.e. if the concentration of water at equilibrium in the membrane is low enough to be neglected. Fig. 1 shows that the ratio between the volume fraction of water in the membrane to that of EA does not exceed 1%. It can be also seen in Fig. 2 that the ratio between the molar water flux and the ethanol one is lower than 10%. Thus, the system can be considered to be a pseudo-binary one (EA/PDMS).
Fig. 1. Equilibrium volume fraction of EA (1), water (2) in the polymer vs. the mass fraction of EA in the liquid mixture (EA/water/PDMS membrane system at 30◦ C).
P. Schaetzel et al. / Journal of Membrane Science 191 (2001) 95–102
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Fig. 2. Molar flux of EA (1) and water (2) through the PDMS membrane vs. the mass fraction of EA in the solution (EA/water/PDMS membrane system, 30◦ C, membrane thickness: 70 m).
4.2. Flory–Huggins equation for sorption equilibrium To compute the pervaporation flux, some parameters are to be calculated from experiment or/and from literature data. The interaction parameter χ i m is obtained from the experiment of pure ethyl acetate sorption in PDMS. Assuming Vi /V m = 0 (negligible permeate molar volume compared with the polymer one), we obtain from Eq. (5) χim = −
ln Φi + (1 − Φi ) (1 − Φi )2
(12)
Fig. 3 shows that we can correctly predict the isotherm of ethyl acetate sorption in the PDMS membrane from the EA–water mixtures by using the Flory–Huggins equation for the EA/PDMS system with the interaction parameter χ i m determined from the PDMS–pure EA equilibrium data (χim = 0.70). The activity of ethyl acetate in EA–water mixtures at 30◦ C is calculated with the NRTL equation for vapour–liquid equilibria (Aspen Plus Release 9.2). The data were validated with the values of two experimentally determined compositions of the vapour phase in equilibrium with the EA–water mixtures. It should be emphasised that the four data points in the mixing gap in Fig. 3 merge into one point on the isotherm, as neither the volume fraction nor the activity of ethyl acetate are change.
Fig. 3. Experimental sorption equilibrium values and calculated isotherm (Eq. (5) with χim = 0.7, EA/water/PDMS 30◦ C).
4.3. Validation of the Flory–Huggins/Stefan–Maxwell (FHSM) equation We assume, in these calculations, that the downstream side of the membrane is dry for the pervaporation under high vacuum (φio = 0), and that Vi /V m =0.
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Fig. 4. Experimental values and calculated curves of the EA dimensionless flux using the FHSM theory in the Cartesian and in the Lagrangian coordinates (EA/water/PDMS membrane system, 30◦ C, membrane thickness: 70 m).
The dimensionless ethyl acetate flux can be calculated from Eqs. (6) and (8)
Ji Jio,ref
=
2 /2 − 2/3χ Φ 3 Φ1o im 1o 3 2 Φ1o,ref /2 − 2/3χim Φ1o,ref
(13)
where Ji ,ref is the flux obtained with pure ethyl acetate (in this case, Φio = Φio,ref ) The steps for the calculations are the following: 1. for each feed composition, the EA activity is computed with the NRTL equation; 2. the EA volume fraction in the membrane at the upstream face is calculated with the Flory–Huggins Eq. (5), and the dimensionless flux is obtained with Eq. (10); 3. the predicted and experimental ethyl acetate fluxes are shown in Fig. 4. One can see that the measured EA flux follows well the FHSM equation. However, the scattering of the data do not allow us to conclude that the Lagrangian coordinates lead to a better agreement than the Cartesian coordinates.
Fig. 5. Activity of water vs. volume fraction of water in the membrane (water/ethanol/PVA-based membrane, 60◦ C, membrane thickness: 30 m).
4.4. FSM model: pervaporation of ethanol–water mixtures through PVA–PAA(co)maleic acid membrane at 60◦ C 4.4.1. Equilibrium Freundlich’s equation for sorption As shown by Schaetzel et al. [7], for water volume fraction in the membrane lower than 0.55, the ratio between the volume fraction of ethanol and that of water in the polymer at equilibrium (at 60◦ C) is constant and equal to 6.6%, and the ratio between the molar flux of ethanol and that of water is constant and equal to 2.3%. Thus, the water/ethanol/PVA-based membrane system can be regarded as a pseudo-binary one, i.e. the ethanol concentration and flux have a negligible influence on the water flux. One can notice in Fig. 5 that the water sorption data for the considered system at 60◦ C are well fitted by Freundlich’s equilibrium relationship. 4.4.2. Checking of the FSM equation The validity of SMF model was checked by least square fitting of experimental data using the Freundlich/Stefan–Maxwell relationships in the Lagrangian (Eq. (10)) and Cartesian (Eq. (11)) coordinates (Fig. 6) [9]. The adjustable parameter was Di m . The poor agreement indicates that these two equations are not suitable to describe the mass transfer through this type of polymer.
P. Schaetzel et al. / Journal of Membrane Science 191 (2001) 95–102
Fig. 6. Molar flux of water vs. volume fraction of water in the membrane at the upstream side (water/ethanol/PVA/PAA-based membrane, 60◦ C, membrane thickness: 30 m).
5. Discussion The EA permeation flux through a PDMS membrane is well predicted with the SMFH equation without any adjustable parameters; for the prediction, only the pure EA flux and the pure EA swelling are required. We would like to remind the readers that in the classical mass transport treatment of pervaporation (when a linear or an exponential relationship holds between diffusivity and concentration), the determination of the plasticization coefficient and the limiting diffusivity needs (for example) the least square fitting of the concentration profile in the membrane. No fitting of an empirical function is needed in our treatment. Another advantage of the model is the compatibility of the Stefan–Maxwell equation with the non-equilibrium thermodynamics: if the model is validated in the case of a single-species transfer, there is no objective reason to doubt its validity for the transfer of two species. This type of behaviour can be qualified as an ideal diffusion behaviour. The fact that the assumption validated for the sorption equilibrium (i.e. Vi /V m = 0) holds for the computation of the diffusion transport in the PDMS membrane suggests that the two models (Flory and Stefan–Maxwell) involve equivalent microscopic molecular picture for their description.
101
The equation integrated in the Lagrangian coordinates would give a better prediction than the one calculated in the Cartesian coordinates, although the scattering of the experimental data did not allow us to prove it clearly. The integration of the SM equation. in the Lagrangian coordinates takes into account the variation in the coordinate dimension, due to the gradient in the membrane swelling, between two boundaries. This approach is reminiscent of the use, by Heintz et al. [9], of the “Hitdorf” frame of reference instead of the “Fick” frame of reference to take into account the swelling of the polymer in the diffusion equation. The diffusion of water through a PVA-based membrane does not follow this ideal model. The main reason of this behaviour may be found in the hypothesis on which is founded the dusty gas model: in this over-simplified model, the polymer is represented by a suspension of immobile, giant molecules. This means that the polymer–solvent mixture is seen as a homogeneous solution. For the water/PVA-based membrane, this hypothesis is not fulfilled. Schaetzel et al. [7] showed that this system follows non-ideal models, like modified solution–diffusion models (the total free-volume theory) and simplified solution–diffusion models (the key component model), where the diffusion of water (and ethanol) can be predicted by the free-volume diffusion theory (Long’s relationship). The non-ideal diffusion of water through the PVA-based membrane could be due to the nature of the membrane. Although, this membrane consists of a miscible blend (based on the criterion of unique glass transition temperature of the blend materials) of two polymers [8], the solvation of the polymer sites and/or the diffusion transport may be not homogeneous on molecular scales.
References [1] P. Schaetzel, E. Favre, Q.T. Nguyen, J. Néel, Mass transfer analysis of pervaporation through an ion exchange membrane, Desalination 90 (1993) 259–276. [2] K.S. Spiegler, Transport processes in ionic membranes, Trans. Faraday Soc. 54 (1958) 1408. [3] E.A. Mason, A.P. Malinauskas, R.B. Evans III, Flow and diffusion of gases in porous media, J. Chem. Phys. 46 (1967) 3199–3208. [4] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954.
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[5] J. Flory, P.J. Rehner, Stastical mechanics of swelling of network structures, J. Chem. Phys. 11 (1943) 512–526. [6] M. Scattergood, E.N. Lightfoot, Diffusional interaction in an ion-exchange membrane, Trans. Faraday Soc. 64 (1968) 1135. [7] P. Schaetzel, C. Vauclair, Q.T. Nguyen, G. Luo, Mass Transfer in Pervaporation: The Total Free Volume Model, Euromembrane, Jérusalem, Israël, 24–27 September 2000.
[8] C. Vauclair, H. Tarjus, P. Schaetzel, Permselective properties of PVA–PAA blended membrane used for dehydration of fusel oil by pervaporation, J. Membr. Sci. 90 (1997) 265–274. [9] A. Heintz, H. Funke, R.N. Lichtenthaler, Sorption and diffusion in pervaporation membranes, in: R.Y.M. Huang (Ed.), Pervaporation Membrane Separation Processes, Elsevier, Amsterdam, 1991, pp. 279–320.
Ideal and non-ideal diffusion through polymers Application to pervaporation P. Schaetzel a , Z. Bendjama b , C. Vauclair a,∗ , Q.T. Nguyen c a
Laboratoire de Thermodynamique des Procédés, Département Génie Chimique, IUT de Caen bld, Maréchal Juin 14000, CAEN, France b Laboratoire de Valorisation des Substances Naturelles, USTHB, Alger, Algeria c UMR 6522, Polymères, Biopolymères, Membranes, Rouen University, 76 821 Mont-Saint-Aignan, France Received 15 April 1998; received in revised form 16 April 2001; accepted 18 April 2001
Abstract Using the approach developed in a previous paper, we integrate the generalised Stefan–Maxwell (SM) diffusion equations for a unique species in both Lagrangian and Cartesian coordinates. The dusty gas membrane model is considered. Two activity–concentration relationships are used: the Flory–Huggins equation and the Freundlich relationship. The ethyl acetate (EA)/PDMS system follows the Flory–Huggins equation and the computed interaction parameter allows to predict quantitatively the non-dimensional EA flux through a PDMS membrane when the Lagrangian coordinates are utilised. The prediction is less satisfactory when integrating the SM equation in the Cartesian coordinates. The system EA/PDMS can be qualified as an ideal system in diffusion. The water/ethanol/PVA-based membrane at 60◦ C system follows Freundlich’s equation at equilibrium. The bad fitting of the experimental water flux versus volume fraction of water at the feed side of the membrane suggests that this system cannot be described by the dusty gas/Stefan–Maxwell theory even when the Lagrangian coordinates are used. This type of system, defined here as non-ideal, obeys theories of the free volume type, as quoted in a recent paper. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Pervaporation; Diffusion; Stefan–Maxwell; Flory–Huggins; Lagrangian coordinates; Sorption
1. Introduction The permeation of liquids or vapours through dense polymer membranes is generally described by using Fick’s law with a concentration-dependent diffusion coefficient. The concentration-dependent diffusivity equation used is mostly Long’s equation in which the diffusivity depends in an exponential way on the molarity or the volume fraction of the diffusing species in the polymer. Recently, a new approach in which the Stefan–Maxwell (SM) equation was integrated ∗ Corresponding author. E-mail address: [email protected] (C. Vauclair).
through the membrane has been proposed by Schaetzel et al. [1]. The theory was successfully applied to ion-exchange membranes in pervaporation. The membrane model used was the “dusty gas model” [2,3] where the polymer is represented as a suspension of immobile, giant molecules. The integration of the SM equation needs the knowledge of the activity– swelling dependency at polymer–diffusant equilibrium. Schaetzel et al. [1] used the Flory–Huggins and the Freundlich relationships to fit their equilibrium data. In this paper, we use this new approach to treat the experimental results obtained in pervaporation and in sorption equilibrium of two pseudo-binary systems, namely the
0376-7388/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 6 - 7 3 8 8 ( 0 1 ) 0 0 4 5 6 - 2
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P. Schaetzel et al. / Journal of Membrane Science 191 (2001) 95–102
Nomenclature
2. Theory
List of symbols ai activity of species i Ai Freundlich law constant Bi Freundlich law exponent species i molarity (mol m−3 ) ci mutual diffusion coefficient (m2 s−1 ) Dij molar flux (mol m−2 s−1 ) Ji R ideal gas constant (J mol−1 K−1 ) T absolute temperature (K) Vi i-component molar volume (m3 mol−1 ) z membrane thickness (m) in the Cartesian coordinates Z membrane thickness (m) in the Lagrangian coordinates
2.1. Representation of the polymer
Greek letters χ ij Flory–Huggins interaction parameter i-component volume fraction in φi polymer membrane µi chemical potential (J mol−1 ) Subscripts e membrane downstream side i, j permeating species m membrane o membrane upstream side 1 key component 2 slow and less concentrated species
polydimethylsiloxane (PDMS) membrane/ethyl acetate/water system and the PVA-based membrane/ water/ethanol system, where the PVA-based membrane consists of a blend of polyvinylalcohol (PVA) and a poly(acrylic acid-co-maleic acid) copolymer. For the first system, we will neglect the concentration and the diffusion of water in the membrane; for the second system, we will neglect the flux and the concentration of ethanol. Using the Flory–Huggins/Stefan–Maxwell (FHSM) equation and the Stefan–Maxwell/Freundlich (FSM) equation in the Cartesian and Lagrangian coordinates to fit the experimental curves, we will show that the permeation behaviours of a solvent through a membrane are quite different for an ideal system (EA/PDMS membrane) and for a non-ideal system (water/PVA-based membrane).
The representation of the polymer in the “dusty gas” model, is compatible with that used by Flory. Actually, the “dusty gas” model was used by Spiegler [2] for the modelling of ion-exchange membranes. The name and the precise definition of the representation were given by Mason et al. [3] for gas-diffusion computations. In the “dusty gas” model, the medium (i.e. the matrix of the swollen polymer) is visualised as a suspension of giant molecules held stationary in space. No assumption is made on the geometrical shape of the particles called polymer species. The swollen polymer is modelled as a homogeneous mixture of polymer particles held stationary in space and of moving solvent particles (the permeate species).
2.2. The Stefan–Maxwell diffusion equation in the dusty gas model The Stefan–Maxwell diffusion equation was obtained from the statistical physics theory of ideal gases [4] and generalised to non-ideal solution by Scattergoot and Lightfoot [6]. The general expression when only a pressure gradient and a concentration gradient are applied can be written for a multicomponent mixture j =n j =1
cj RT Ji Jj dp dµi − Vi =− − , cDij ci cj dz dz
c=
j =n
cj
j =1
(1) In pervaporation, the pressure gradient term is generally neglected. For a polymer (m) single solvent (i) system, the Stefan–Maxwell Eq. (2) becomes d ln ai Ji cm = dz (cm + ci )Dim ci
(2)
The relationship between ci and cm is Vi c i + V m c m = 1
(3)
P. Schaetzel et al. / Journal of Membrane Science 191 (2001) 95–102
2.3. Integration of the Stefan–Maxwell equation for a single component when the equilibrium follows Flory–Huggin’s theory To link the Flory–Huggins (FH) equation with the SM equation, the permeate volume fraction is expressed as a function of the molarity by φi = Vi ci
The integration [1] of Eq. (2) in which the permeate molarity in the membrane is related to the permeate activity according to Eqs. (4) and (5), gives Vi2 Dim Ji = 2 Vm − 2Vi − 2χim Vi + Vm V i ze 2 − φ2 ) (φio 2 ie 3 3 + χim (Vi − Vm )(φio − φie ) 2 3 1 − φio (6) −Vi ln 1 − φie
×
In this equation, the subscripts ‘o’ and ‘e’ refer to the membrane upstream and downstream sides, respectively, and ze is the thickness of the membrane. 2.3.2. Integration in the Lagrangian coordinates The swelling of a polymer in contact with the solvent at large activities may be high. For example, when the polymer is equilibrated with pure ethyl acetate, the volume fraction of ethyl acetate in the swollen PDMS membrane is ca. 0.60. One cannot neglect the variation of the thickness of the membrane in pervaporation. To take into account the variation in the local swelling of the membrane in the flow direction, we choose the coordinates bound to the dry membrane, the Lagrangian coordinates Z, defined by dZ = (1 − Φi ) dz
side of the membrane. We obtain the expression of a single species permeation flux in the frame of the FHSM model in the Lagrangian coordinates Dim Vi (φio − φie ) Ji = 2 V 1 Ze 2 − φ2 ) (φio ie (Vm − 2Vi − 2χim Vi ) 2 3) (φ 3 − φie + io (2χim Vi − (Vm − Vi )(1 + 2χim )) 3 4 − φ4 ) (φio ie (8) (Vm − Vi )χim + 2
+
(4)
2.3.1. Integration in the Cartesian coordinates The general Flory–Huggins [5] equation for a single species can be written as 1 − Vi ln ai =ln φi + (1−φi )+χim (1 − φi )2 z (5) Vm
(7)
We replace the Cartesian coordinates by the Lagrangian coordinates in differential Eq. (2) and perform the integration from the wet side to the dry
97
2.4. Integration of the Stefan–Maxwell equation for a single component when the equilibrium follows Freundlich’s equation 2.4.1. Integration in the Cartesian coordinates Freundlich’s equation fits many experimental equilibrium data in the case of the adsorption of organic vapours onto mineral adsorbents; in the swelling of polymer, this equation has to be considered as an empirical one. For a single component, Freundlich’s equation is written as [1] ai = Ai φiBi
(9)
Ai and Bi are two constants depending on the polymer/solvent system and the temperature. Putting this expression in the Stefan–Maxwell Eq. (3) and integrating the new equation from the membrane upstream side to the downstream one yield [1] Ji =
Dim Bi Vi z e Vm Vm 1 − φio × 1− (φio − φie )− ln Vi Vi 1 − φie (10)
2.4.2. Integration in the Lagrangian coordinates Using Eqs. (2), (4), (7) and (9) and integrating from the membrane upstream side to the downstream side, one obtains the Freundlich–Stefan–Maxwell (FSM) equation in the Lagrangian coordinates 2 2 Φio − Φie Vm Bi Dim Ji = −1 + Φio − Φie Vi Ze Vi 2 (11)
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3. Checking of the models To check the model, we use the pervaporation data concerning the EA/water/PDMS membrane system at 30◦ C (this work), and the recently published data on the water/ethanol/PVA-based membrane system at 60◦ C [7].
3.1.3. Pervaporation experiments The experimental device and procedure have been described in a previous paper [8]. Briefly, the device consists of a stirred pervaporation cell maintained at 30◦ C and a vacuum gauge. In all experiments, the vacuum was maintained at a level better than 2 mmHg. The 70 m thick membrane was used in all experiments.
3.1. Experimental 4. Results 3.1.1. Membranes PDMS membranes are obtained by mixing the Rhodorsil 141A and Rhodorsil 141B components manufactured by Rhone–Poulenc Co. 3.1.2. Equilibrium The sorption data were obtained from classical sorption–desorption experiments: the membrane sample was allowed to reach the sorption equilibrium at different compositions of the water/EA mixture in closed flasks maintained at 30◦ C. Then the membrane sample was quickly blotted with a paper sheet, and the sorbed mixture was extracted out of the membrane under vacuum. The desorbed vapour was condensed in a liquid air trap, and the collected liquid was weighed and analysed by gas chromatography.
4.1. FHSM model in pervaporation We use here the experimental results concerning the EA–water–PDMS membrane system at 30◦ C (Figs. 1 and 2). In order to apply the developed FHSM model, we have to check first if the investigated system is a pseudo-binary one, i.e. if the concentration of water at equilibrium in the membrane is low enough to be neglected. Fig. 1 shows that the ratio between the volume fraction of water in the membrane to that of EA does not exceed 1%. It can be also seen in Fig. 2 that the ratio between the molar water flux and the ethanol one is lower than 10%. Thus, the system can be considered to be a pseudo-binary one (EA/PDMS).
Fig. 1. Equilibrium volume fraction of EA (1), water (2) in the polymer vs. the mass fraction of EA in the liquid mixture (EA/water/PDMS membrane system at 30◦ C).
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Fig. 2. Molar flux of EA (1) and water (2) through the PDMS membrane vs. the mass fraction of EA in the solution (EA/water/PDMS membrane system, 30◦ C, membrane thickness: 70 m).
4.2. Flory–Huggins equation for sorption equilibrium To compute the pervaporation flux, some parameters are to be calculated from experiment or/and from literature data. The interaction parameter χ i m is obtained from the experiment of pure ethyl acetate sorption in PDMS. Assuming Vi /V m = 0 (negligible permeate molar volume compared with the polymer one), we obtain from Eq. (5) χim = −
ln Φi + (1 − Φi ) (1 − Φi )2
(12)
Fig. 3 shows that we can correctly predict the isotherm of ethyl acetate sorption in the PDMS membrane from the EA–water mixtures by using the Flory–Huggins equation for the EA/PDMS system with the interaction parameter χ i m determined from the PDMS–pure EA equilibrium data (χim = 0.70). The activity of ethyl acetate in EA–water mixtures at 30◦ C is calculated with the NRTL equation for vapour–liquid equilibria (Aspen Plus Release 9.2). The data were validated with the values of two experimentally determined compositions of the vapour phase in equilibrium with the EA–water mixtures. It should be emphasised that the four data points in the mixing gap in Fig. 3 merge into one point on the isotherm, as neither the volume fraction nor the activity of ethyl acetate are change.
Fig. 3. Experimental sorption equilibrium values and calculated isotherm (Eq. (5) with χim = 0.7, EA/water/PDMS 30◦ C).
4.3. Validation of the Flory–Huggins/Stefan–Maxwell (FHSM) equation We assume, in these calculations, that the downstream side of the membrane is dry for the pervaporation under high vacuum (φio = 0), and that Vi /V m =0.
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Fig. 4. Experimental values and calculated curves of the EA dimensionless flux using the FHSM theory in the Cartesian and in the Lagrangian coordinates (EA/water/PDMS membrane system, 30◦ C, membrane thickness: 70 m).
The dimensionless ethyl acetate flux can be calculated from Eqs. (6) and (8)
Ji Jio,ref
=
2 /2 − 2/3χ Φ 3 Φ1o im 1o 3 2 Φ1o,ref /2 − 2/3χim Φ1o,ref
(13)
where Ji ,ref is the flux obtained with pure ethyl acetate (in this case, Φio = Φio,ref ) The steps for the calculations are the following: 1. for each feed composition, the EA activity is computed with the NRTL equation; 2. the EA volume fraction in the membrane at the upstream face is calculated with the Flory–Huggins Eq. (5), and the dimensionless flux is obtained with Eq. (10); 3. the predicted and experimental ethyl acetate fluxes are shown in Fig. 4. One can see that the measured EA flux follows well the FHSM equation. However, the scattering of the data do not allow us to conclude that the Lagrangian coordinates lead to a better agreement than the Cartesian coordinates.
Fig. 5. Activity of water vs. volume fraction of water in the membrane (water/ethanol/PVA-based membrane, 60◦ C, membrane thickness: 30 m).
4.4. FSM model: pervaporation of ethanol–water mixtures through PVA–PAA(co)maleic acid membrane at 60◦ C 4.4.1. Equilibrium Freundlich’s equation for sorption As shown by Schaetzel et al. [7], for water volume fraction in the membrane lower than 0.55, the ratio between the volume fraction of ethanol and that of water in the polymer at equilibrium (at 60◦ C) is constant and equal to 6.6%, and the ratio between the molar flux of ethanol and that of water is constant and equal to 2.3%. Thus, the water/ethanol/PVA-based membrane system can be regarded as a pseudo-binary one, i.e. the ethanol concentration and flux have a negligible influence on the water flux. One can notice in Fig. 5 that the water sorption data for the considered system at 60◦ C are well fitted by Freundlich’s equilibrium relationship. 4.4.2. Checking of the FSM equation The validity of SMF model was checked by least square fitting of experimental data using the Freundlich/Stefan–Maxwell relationships in the Lagrangian (Eq. (10)) and Cartesian (Eq. (11)) coordinates (Fig. 6) [9]. The adjustable parameter was Di m . The poor agreement indicates that these two equations are not suitable to describe the mass transfer through this type of polymer.
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Fig. 6. Molar flux of water vs. volume fraction of water in the membrane at the upstream side (water/ethanol/PVA/PAA-based membrane, 60◦ C, membrane thickness: 30 m).
5. Discussion The EA permeation flux through a PDMS membrane is well predicted with the SMFH equation without any adjustable parameters; for the prediction, only the pure EA flux and the pure EA swelling are required. We would like to remind the readers that in the classical mass transport treatment of pervaporation (when a linear or an exponential relationship holds between diffusivity and concentration), the determination of the plasticization coefficient and the limiting diffusivity needs (for example) the least square fitting of the concentration profile in the membrane. No fitting of an empirical function is needed in our treatment. Another advantage of the model is the compatibility of the Stefan–Maxwell equation with the non-equilibrium thermodynamics: if the model is validated in the case of a single-species transfer, there is no objective reason to doubt its validity for the transfer of two species. This type of behaviour can be qualified as an ideal diffusion behaviour. The fact that the assumption validated for the sorption equilibrium (i.e. Vi /V m = 0) holds for the computation of the diffusion transport in the PDMS membrane suggests that the two models (Flory and Stefan–Maxwell) involve equivalent microscopic molecular picture for their description.
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The equation integrated in the Lagrangian coordinates would give a better prediction than the one calculated in the Cartesian coordinates, although the scattering of the experimental data did not allow us to prove it clearly. The integration of the SM equation. in the Lagrangian coordinates takes into account the variation in the coordinate dimension, due to the gradient in the membrane swelling, between two boundaries. This approach is reminiscent of the use, by Heintz et al. [9], of the “Hitdorf” frame of reference instead of the “Fick” frame of reference to take into account the swelling of the polymer in the diffusion equation. The diffusion of water through a PVA-based membrane does not follow this ideal model. The main reason of this behaviour may be found in the hypothesis on which is founded the dusty gas model: in this over-simplified model, the polymer is represented by a suspension of immobile, giant molecules. This means that the polymer–solvent mixture is seen as a homogeneous solution. For the water/PVA-based membrane, this hypothesis is not fulfilled. Schaetzel et al. [7] showed that this system follows non-ideal models, like modified solution–diffusion models (the total free-volume theory) and simplified solution–diffusion models (the key component model), where the diffusion of water (and ethanol) can be predicted by the free-volume diffusion theory (Long’s relationship). The non-ideal diffusion of water through the PVA-based membrane could be due to the nature of the membrane. Although, this membrane consists of a miscible blend (based on the criterion of unique glass transition temperature of the blend materials) of two polymers [8], the solvation of the polymer sites and/or the diffusion transport may be not homogeneous on molecular scales.
References [1] P. Schaetzel, E. Favre, Q.T. Nguyen, J. Néel, Mass transfer analysis of pervaporation through an ion exchange membrane, Desalination 90 (1993) 259–276. [2] K.S. Spiegler, Transport processes in ionic membranes, Trans. Faraday Soc. 54 (1958) 1408. [3] E.A. Mason, A.P. Malinauskas, R.B. Evans III, Flow and diffusion of gases in porous media, J. Chem. Phys. 46 (1967) 3199–3208. [4] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954.
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[5] J. Flory, P.J. Rehner, Stastical mechanics of swelling of network structures, J. Chem. Phys. 11 (1943) 512–526. [6] M. Scattergood, E.N. Lightfoot, Diffusional interaction in an ion-exchange membrane, Trans. Faraday Soc. 64 (1968) 1135. [7] P. Schaetzel, C. Vauclair, Q.T. Nguyen, G. Luo, Mass Transfer in Pervaporation: The Total Free Volume Model, Euromembrane, Jérusalem, Israël, 24–27 September 2000.
[8] C. Vauclair, H. Tarjus, P. Schaetzel, Permselective properties of PVA–PAA blended membrane used for dehydration of fusel oil by pervaporation, J. Membr. Sci. 90 (1997) 265–274. [9] A. Heintz, H. Funke, R.N. Lichtenthaler, Sorption and diffusion in pervaporation membranes, in: R.Y.M. Huang (Ed.), Pervaporation Membrane Separation Processes, Elsevier, Amsterdam, 1991, pp. 279–320.