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The role of coupling in pervaporation
Journal of Membrane Science, 41 (1989) 277-284 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
271
THE ROLE OF COUPLING IN PERVAPORATION
ORA KEDEM Department
of Membrane
(Received
October 31,1988;
Research,
Weizmann
Institute
of Science,
Rehovot
(Israel)
accepted in revised form May 17,1989)
Summary In some pervaporation systems coupling between flows may play an important role. Specifically, this appears to apply to the pervaporation of water and alcohol through ion-exchange membranes. Transport equations including a mutual drag coefficient are discussed and a procedure for the measurement of this coefficient is described. It is shown that a concave concentration profile for the less permeable species may be a consequence of coupling. Coupling may explain the close agreement between pervaporation selectivity and sorption selectivity in a range of concentrations observed in ion-exchange membranes.
Introduction
In the literature on pervaporation (PV), the term coupling has often been used for the thermodynamic interaction between components [ 11. Here it denotes a relation between flow i and flow j, so that a substance can flow without or even against its own driving force. In the customary formulation of irreversible thermodynamics this is expressed in the dissipation function and in the phenomenological equations. The entropy production, or dissipation function, per unit area is given by a sum of terms, the product of the flow and its conjugate driving force for each solute and for the heat flow, J,: c$= J,X,
+ JzXz + J,X,
where Ji are flows and Xi driving forces. @ is always positive, but one of the first two terms can be negative when coupling causes a flow to go “uphill”. The heat flow term contributes substantially to the entropy production, but so far no evidence has been presented for coupling between J, and the solute flows. Such coupling is of course well known: the Soret effect in solution and thermo-osmosis through membranes. It may contribute but has not been investigated so far, and certainly is not essential for pervaporation. We shall not discuss here the possible consequences of coupling between heat flow and the flow of matter. In reverse osmosis (RO ) , the coupling between the flow of water (expressed
0376-7388/89/$03.50
0 1989 Elsevier Science Publishers
B.V.
278
in units of volume) and salt flow is represented by the reflection coefficient, o. If the salt is not dragged along by the water, the reflection coefficient is unity. This is very nearly true for good RO membranes. Permeation of water and salt through the selective membrane layer with negligible influence of coupling between them is called in the RO literature the “solution-diffusion model”. This term is logical if it is meant as short for: water is homogeneously distributed, “dissolved”, in the membrane phase; the transport of both water and solute is adequately described by a single term, like Fick’s law in free diffusion. It should not, however, be implied that in free diffusion in solution, there is no coupling between solute and solvent flows and, more generally, that whenever diffusion takes place, there is no coupling between flows. In the pervaporation of two components, both dissolve in the polymer phase at the upstream surface, in contact with the feed solution. They diffuse to the other surface (downstream), where both concentrations are kept very low. Are these two diffusion flows coupled? In the following, an expression of this coupling in the transport equations will be shown, and a possible method of measurement will be indicated. Driving forces in pervaporation
It is generally accepted that the driving force for pervaporation is the difference in vapour pressure pi between the feed solution and the evacuated downstream compartment. The local equations are usually expressed in terms of concentrations, weight fractions or activities [l-3]. The integration of the local transport equations between upstream and downstream boundaries can be more readily carried out if the local driving force is given as a gradient of vapour pressure. One can define the local vapour pressure as the composition of the vapours that would fill a small cavity formed in the membrane at point x during pervaporation. This method, regarding a small equilibrium phase at point X, has been applied previously to RO [ 41. The flux equations
For the description of transport of two mobile components symmetrical flow equations are needed, which give each flow as a function of its own driving force and the other component’s flow. The local flux equations are then of the form
279
The subscript denotes zero gradient for p1 or p2 respectively. p1 and p2 are the local permeability coefficients. In order to correlate the two coefficients, J1/J2 at dp,/dx = 0 and J2/J1at d&d, = 0, we go back to the dissipation function and the Onsager relation. The thermodynamic driving forces are the gradients of the thermodynamic potentials. Assuming ideal behaviour in the vapour phase: d(lnpi) =RTd(lnpi) and the dissipation function is g=-J~RTd'~"_J,Rrd"~)+~,~~
(4)
It is convenient here to write the forces as functions of flows, with J, and J2 not coupled to J,.
_RTd(lnpl) dx
=RnJl+R12J2
(5)
_RTd(lnp2) =RzlJl+R22J2 dx with the Onsager relation (7)
Rx = &
Equations (5 ) and (6 ) can be rewritten in similar form to eqns. (2 ) and (3 ) :
(8) dp2
R21
-dn-RT~2J~
(9)
Let us define a mutual drag coefficient B by /J= _$
_g
(10)
The minus sign was introduced because, for R,, ~0, the two flows enhance each other, and for RI2 > 0 they hinder each other. The permeability coefficient Pi may be identified with pi=f$ L
II
and the flux equations in terms of P and Q are
(11) (12) & is the local drag coefficient. the integral drag coefficient Q is defined with the integral permeability coefficient. For constant coefficients:
@= (J,lJ,),, PIP, Q=
(JIIJ& PI
=
,&
x
Pl
P&/Ax
Note that -PQ=PQ
(13)
For concentration-dependent local coefficients, symmetry will not always hold for the integral coefficient, measured with large driving forces across the membrane. This is well known from other transport phenomena, e.g., with transport number measured with either large currents or via membrane potentials at zero current. Integrated fluxequations
With constant coefficients, the effect of coupling on the correlation between flows and vapour pressures is obtained by integration of eqns. (11) and ( 12). From eqn. (12):
1 52 “-QP,J, -- 1 52 “-QP2J,
-=exp( -QJl)
(14)
where the prime and double prime denote the upstream and downstream compartments, respectively. For the customary situation in pervaporation, with both downstream vapour pressures very small: p’l=p;=o
the flow of component 2 becomes
QJ1 J2=P2ph[l-exp(-QJI)]
(15)
281
One can readily see the two limiting cases, with exp ( - QJ1) close to unity, or negligible as compared to unity. ForQJ,<
(16)
In this case the influence of coupling is negligible, either because the mechanism of permeation involves little or no coupling, or because J, is small. In the other extreme, for large QJ1: J2=P,p;QJ,=&p;~Jl
(17)
In this case J2 is proportional to J1. This situation is analogous to R, in reverse osmosis and ultrafiltration. The relations of eqns. (15)- (17) describe the typical balance between diffusion and “convection” in transport processes. This does not mean necessarily that membranes are porous - in fact useful PV membranes are nonporous -but only that there may be interaction between the flows. At high flux, when the coupling overtakes the diffusion for one of the components, its gradient of vapour pressure near the feed surface is very small. The ratio between these gradients in the membrane close to both surfaces is
(b/h)’ (b/h)
,,=ew(-Q&J,)
(18)
For (dp,/d.~)’ +O, the values of Pz and Q in eqn. (17) will be those valid at equilibrium with the feed solution. Equations (14)- (18) describe transport defined by constant coefficients. They show that even with this simple assumption one can not necessarily expect linear concentration profiles. In addition, Q, like the permeabilities, may be, and probably often is, concentration-dependent. Measurement of Q Coupling is in general measured in one of two types of condition: either with one flow zero or with one force zero. In the formalism of coupling these were termed static head and level flow [ 51. Thus the reflection coefficient Qmay be derived from a measurement of effective osmotic pressure, at static head for volume flow, or from maximal rejection [ 91, which is in effect obtained at level flow for the solute (dc,/d.r+O near the feed surface). It is not always possible to realize one of these experimental conditions, and the flow equations may then serve to determine the coefficients in any properly designed experimental condition. This has been done for o in biomembranes [ 61. For static head in pervaporation the two flows need to be separately controlled. If a selective adsorbent for one of the components is available, one may maintain continuous flow into a closed vessel, keeping the vapour pressure p1 low, until pz becomes constant. At this point
283
where the distribution coefficients KP1,2are defined by K$’ =aCJp+ Hence the permeability coefficients have the familiar form -
RTK; p'=fiz+Ln - RTKJl p2=fil +Ln and the coupling coefficient is: f21 Q=m=EEg=RT
fl2
&2
(25)
where
(26) If the mutual friction membrane
fzl is larger than the friction of component 2 with the
In this special case the transport selectivity becomes equal to the sorption selectivity, as observed by Bijddeker in substantial regions with water-alcohol mixtures in ion-exchange membranes [ 81. In general, when selectivity depends on both solubilities and diffusion coefficients, and coefficients are concentration-dependent, mutual drag is one more factor in a complex system.
References 1 M.H.V. Mulder and C.A. Smolders, On the mechanism of separation of ethanol/water mixtures by pervaporation. 1. Calculation of concentration profiles, J. Membrane Sci. 17 (1984) 289. 2 J. Neel, P. Aptel and R. Clement, Basic aspects of pervaporation, Desalination, 53 (1985) 297. 3 R. Rautenbach and R. Albrecht, The separation potential of pervaporation. Part 1. Discussion of transport equations and comparison with reverse osmosis, J. Membrane Sci., 25 (1985) 1. 4 K.S. Spiegler and 0. Kedem, Thermodynamics of hyperfiltration (reverse osmosis): criteria for efficient membranes, Desalination, 1 (1966) 311.
284 5 0. Kedem and S.R. Caplan, Degree of coupling and its relation to efficiency of energy conversion, Trans. Faraday Sot., 61 (1965) 1897. 6 K.J. Ullrich, Permeability characteristics of the human nephron, Handbook of Physiology, American Physiological Society, Washington, DC, 1968, p. 377. 7 R.J. Bearman and J.G. Kirkwood, Statistical mechanics of transport processes. XI. Equations of transport in multicomponent systems, J. Chem. Phys., 28 (1958) 136. 8 K.W. Biiddeker and A. Wenzlaff, Pervaporation with ion-exchange membranes, in: R. Bakish (Ed.), Proceedings of the First International Conference on Pervaporation Processes in the Chemical Industry, Bakish Materials Corp., New Jersey, NY, 1986.
271
THE ROLE OF COUPLING IN PERVAPORATION
ORA KEDEM Department
of Membrane
(Received
October 31,1988;
Research,
Weizmann
Institute
of Science,
Rehovot
(Israel)
accepted in revised form May 17,1989)
Summary In some pervaporation systems coupling between flows may play an important role. Specifically, this appears to apply to the pervaporation of water and alcohol through ion-exchange membranes. Transport equations including a mutual drag coefficient are discussed and a procedure for the measurement of this coefficient is described. It is shown that a concave concentration profile for the less permeable species may be a consequence of coupling. Coupling may explain the close agreement between pervaporation selectivity and sorption selectivity in a range of concentrations observed in ion-exchange membranes.
Introduction
In the literature on pervaporation (PV), the term coupling has often been used for the thermodynamic interaction between components [ 11. Here it denotes a relation between flow i and flow j, so that a substance can flow without or even against its own driving force. In the customary formulation of irreversible thermodynamics this is expressed in the dissipation function and in the phenomenological equations. The entropy production, or dissipation function, per unit area is given by a sum of terms, the product of the flow and its conjugate driving force for each solute and for the heat flow, J,: c$= J,X,
+ JzXz + J,X,
where Ji are flows and Xi driving forces. @ is always positive, but one of the first two terms can be negative when coupling causes a flow to go “uphill”. The heat flow term contributes substantially to the entropy production, but so far no evidence has been presented for coupling between J, and the solute flows. Such coupling is of course well known: the Soret effect in solution and thermo-osmosis through membranes. It may contribute but has not been investigated so far, and certainly is not essential for pervaporation. We shall not discuss here the possible consequences of coupling between heat flow and the flow of matter. In reverse osmosis (RO ) , the coupling between the flow of water (expressed
0376-7388/89/$03.50
0 1989 Elsevier Science Publishers
B.V.
278
in units of volume) and salt flow is represented by the reflection coefficient, o. If the salt is not dragged along by the water, the reflection coefficient is unity. This is very nearly true for good RO membranes. Permeation of water and salt through the selective membrane layer with negligible influence of coupling between them is called in the RO literature the “solution-diffusion model”. This term is logical if it is meant as short for: water is homogeneously distributed, “dissolved”, in the membrane phase; the transport of both water and solute is adequately described by a single term, like Fick’s law in free diffusion. It should not, however, be implied that in free diffusion in solution, there is no coupling between solute and solvent flows and, more generally, that whenever diffusion takes place, there is no coupling between flows. In the pervaporation of two components, both dissolve in the polymer phase at the upstream surface, in contact with the feed solution. They diffuse to the other surface (downstream), where both concentrations are kept very low. Are these two diffusion flows coupled? In the following, an expression of this coupling in the transport equations will be shown, and a possible method of measurement will be indicated. Driving forces in pervaporation
It is generally accepted that the driving force for pervaporation is the difference in vapour pressure pi between the feed solution and the evacuated downstream compartment. The local equations are usually expressed in terms of concentrations, weight fractions or activities [l-3]. The integration of the local transport equations between upstream and downstream boundaries can be more readily carried out if the local driving force is given as a gradient of vapour pressure. One can define the local vapour pressure as the composition of the vapours that would fill a small cavity formed in the membrane at point x during pervaporation. This method, regarding a small equilibrium phase at point X, has been applied previously to RO [ 41. The flux equations
For the description of transport of two mobile components symmetrical flow equations are needed, which give each flow as a function of its own driving force and the other component’s flow. The local flux equations are then of the form
279
The subscript denotes zero gradient for p1 or p2 respectively. p1 and p2 are the local permeability coefficients. In order to correlate the two coefficients, J1/J2 at dp,/dx = 0 and J2/J1at d&d, = 0, we go back to the dissipation function and the Onsager relation. The thermodynamic driving forces are the gradients of the thermodynamic potentials. Assuming ideal behaviour in the vapour phase: d(lnpi) =RTd(lnpi) and the dissipation function is g=-J~RTd'~"_J,Rrd"~)+~,~~
(4)
It is convenient here to write the forces as functions of flows, with J, and J2 not coupled to J,.
_RTd(lnpl) dx
=RnJl+R12J2
(5)
_RTd(lnp2) =RzlJl+R22J2 dx with the Onsager relation (7)
Rx = &
Equations (5 ) and (6 ) can be rewritten in similar form to eqns. (2 ) and (3 ) :
(8) dp2
R21
-dn-RT~2J~
(9)
Let us define a mutual drag coefficient B by /J= _$
_g
(10)
The minus sign was introduced because, for R,, ~0, the two flows enhance each other, and for RI2 > 0 they hinder each other. The permeability coefficient Pi may be identified with pi=f$ L
II
and the flux equations in terms of P and Q are
(11) (12) & is the local drag coefficient. the integral drag coefficient Q is defined with the integral permeability coefficient. For constant coefficients:
@= (J,lJ,),, PIP, Q=
(JIIJ& PI
=
,&
x
Pl
P&/Ax
Note that -PQ=PQ
(13)
For concentration-dependent local coefficients, symmetry will not always hold for the integral coefficient, measured with large driving forces across the membrane. This is well known from other transport phenomena, e.g., with transport number measured with either large currents or via membrane potentials at zero current. Integrated fluxequations
With constant coefficients, the effect of coupling on the correlation between flows and vapour pressures is obtained by integration of eqns. (11) and ( 12). From eqn. (12):
1 52 “-QP,J, -- 1 52 “-QP2J,
-=exp( -QJl)
(14)
where the prime and double prime denote the upstream and downstream compartments, respectively. For the customary situation in pervaporation, with both downstream vapour pressures very small: p’l=p;=o
the flow of component 2 becomes
QJ1 J2=P2ph[l-exp(-QJI)]
(15)
281
One can readily see the two limiting cases, with exp ( - QJ1) close to unity, or negligible as compared to unity. ForQJ,<
(16)
In this case the influence of coupling is negligible, either because the mechanism of permeation involves little or no coupling, or because J, is small. In the other extreme, for large QJ1: J2=P,p;QJ,=&p;~Jl
(17)
In this case J2 is proportional to J1. This situation is analogous to R, in reverse osmosis and ultrafiltration. The relations of eqns. (15)- (17) describe the typical balance between diffusion and “convection” in transport processes. This does not mean necessarily that membranes are porous - in fact useful PV membranes are nonporous -but only that there may be interaction between the flows. At high flux, when the coupling overtakes the diffusion for one of the components, its gradient of vapour pressure near the feed surface is very small. The ratio between these gradients in the membrane close to both surfaces is
(b/h)’ (b/h)
,,=ew(-Q&J,)
(18)
For (dp,/d.~)’ +O, the values of Pz and Q in eqn. (17) will be those valid at equilibrium with the feed solution. Equations (14)- (18) describe transport defined by constant coefficients. They show that even with this simple assumption one can not necessarily expect linear concentration profiles. In addition, Q, like the permeabilities, may be, and probably often is, concentration-dependent. Measurement of Q Coupling is in general measured in one of two types of condition: either with one flow zero or with one force zero. In the formalism of coupling these were termed static head and level flow [ 51. Thus the reflection coefficient Qmay be derived from a measurement of effective osmotic pressure, at static head for volume flow, or from maximal rejection [ 91, which is in effect obtained at level flow for the solute (dc,/d.r+O near the feed surface). It is not always possible to realize one of these experimental conditions, and the flow equations may then serve to determine the coefficients in any properly designed experimental condition. This has been done for o in biomembranes [ 61. For static head in pervaporation the two flows need to be separately controlled. If a selective adsorbent for one of the components is available, one may maintain continuous flow into a closed vessel, keeping the vapour pressure p1 low, until pz becomes constant. At this point
283
where the distribution coefficients KP1,2are defined by K$’ =aCJp+ Hence the permeability coefficients have the familiar form -
RTK; p'=fiz+Ln - RTKJl p2=fil +Ln and the coupling coefficient is: f21 Q=m=EEg=RT
fl2
&2
(25)
where
(26) If the mutual friction membrane
fzl is larger than the friction of component 2 with the
In this special case the transport selectivity becomes equal to the sorption selectivity, as observed by Bijddeker in substantial regions with water-alcohol mixtures in ion-exchange membranes [ 81. In general, when selectivity depends on both solubilities and diffusion coefficients, and coefficients are concentration-dependent, mutual drag is one more factor in a complex system.
References 1 M.H.V. Mulder and C.A. Smolders, On the mechanism of separation of ethanol/water mixtures by pervaporation. 1. Calculation of concentration profiles, J. Membrane Sci. 17 (1984) 289. 2 J. Neel, P. Aptel and R. Clement, Basic aspects of pervaporation, Desalination, 53 (1985) 297. 3 R. Rautenbach and R. Albrecht, The separation potential of pervaporation. Part 1. Discussion of transport equations and comparison with reverse osmosis, J. Membrane Sci., 25 (1985) 1. 4 K.S. Spiegler and 0. Kedem, Thermodynamics of hyperfiltration (reverse osmosis): criteria for efficient membranes, Desalination, 1 (1966) 311.
284 5 0. Kedem and S.R. Caplan, Degree of coupling and its relation to efficiency of energy conversion, Trans. Faraday Sot., 61 (1965) 1897. 6 K.J. Ullrich, Permeability characteristics of the human nephron, Handbook of Physiology, American Physiological Society, Washington, DC, 1968, p. 377. 7 R.J. Bearman and J.G. Kirkwood, Statistical mechanics of transport processes. XI. Equations of transport in multicomponent systems, J. Chem. Phys., 28 (1958) 136. 8 K.W. Biiddeker and A. Wenzlaff, Pervaporation with ion-exchange membranes, in: R. Bakish (Ed.), Proceedings of the First International Conference on Pervaporation Processes in the Chemical Industry, Bakish Materials Corp., New Jersey, NY, 1986.