Paul - 2004 - Reformulation of the solution-diffusion ... · PDF fileJournal of Membrane Science 241 (2004) 371–386 Reformulation of the solution-diffusion theory of reverse osmosis

Journal of Membrane Science 241 (2004) 371–386

Reformulation of the solution-diffusion theory of reverse osmosisD.R. Paul∗

Department of Chemical Engineering, Texas Materials Institute, The University of Texas at Austin, Austin, TX 78712, USA

Received 12 December 2003; received in revised form 21 April 2004; accepted 3 May 2004

Abstract

Many transport concepts have been proposed for desalination by reverse osmosis; however, one of the most useful has been the solution-diffusion mechanism as articulated mathematically by Lonsdale, Merten, and Riley in 1965. Their simple formulation has proved effective fordescribing desalination by reverse osmosis in spite of themany simplifying assumptions, either explicitly or implicitly, used in its development.The result is a linear model where the flux of solvent increases without limit and solute rejection approaches 100% as the pressure differentialbecomes large. There is good reason to expect that many of the simplifications in the classical theory for the solution-diffusion mechanismwill not be appropriate for separation of organic systems. There are numerous opportunities for reverse osmosis-type separations involvingorganics that might become practical as suitable membrane systems become available. The development of such membranes will require anunderstanding of the issues that affect the transport of both the solute and solvent in the polymer membrane; for this, a suitable theoreticalframework for analysis of experimental data in terms of fundamental parameters is needed. This paper revisits the formulation of the theoryfor pressure-driven processes where both solute and solvent transports occur exclusively by the solution-diffusion mechanism. This morerigorous treatment of the thermodynamic boundary conditions naturally leads to nonlinear responses where the flux of solvent must reach afinite value as the pressure differential becomes very large. The practical consequence of this becomes more important as the solvent molarvolume increases. Likewise, the effects of pressure on solute rejection may not be negligible as its molar volume increases. Finally, theeffects of solute–solvent coupling may become important and this is considered using the Maxwell–Stefan formulation for multi-componentdiffusion. Various approximate forms of the general formulation are given and illustrated by calculations.© 2004 Elsevier B.V. All rights reserved.

Keywords: Solution-diffusion theory; Reverse osmosis; Desalination

1. Introduction

The idea of desalination of sea- and brackish water by re-verse osmosis was seriously proposed 50 years ago [1]. Thefirst step towards realization of this concept was the demon-stration by Reid and Breton [2] and Reid and Kuppers [3]that cellulose acetate (CA) had attractive properties as themembrane material for rejecting salt in this process; how-ever, the observed water fluxes were too low to be practical.This problem was resolved by the far-reaching discoveryby Loeb and Sourirajan [1] of high flux membranes fromcellulose acetate; Riley et al. [4] elucidated the asymmetriccharacter of the Loeb–Sourirajan membranes. Merten et al.[1,5–7] developed a simple solution-diffusion model to de-

∗ Tel.: +1-512-471-5392; fax: +1-512-471-0542.E-mail address: [email protected] (D.R. Paul).

scribe the reverse osmosis process and did extensive experi-ments to characterize the equilibrium and transport behaviorof the CA/water/salt system; this classic theory has provenquite adequate for describing desalination despite its sim-plicity.The technology of reverse osmosis was further advanced

during the 1970s by the introduction of hollow fiber perme-ators, the development of thin-film composite membranesand use of a variety of materials in addition to celluloseacetate. Numerous reviews have appeared over the yearsdescribing the principles and practice of reverse osmosistechnology [8–13]. Today, there is a relatively mature mem-brane industry that serves the markets for desalination andother purifications of water. Increased world demands forpotable and even higher purity water will, no doubt, providesignificant growth opportunities for this industry.In principle, reverse osmosis offers opportunities for

separations other than producing water. Indeed, there have

0376-7388/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.memsci.2004.05.026

372 D.R. Paul / Journal of Membrane Science 241 (2004) 371–386

already been small commercial applications in refineriesfor lube oil processing [14,15]. There is currently academicand industrial research aimed at broader ranges of applica-tions of reverse osmosis for separation of organic mixtureswhere the objective may be to recover a solvent or to con-centrate a solute [16–28]. The success of these efforts willhinge on developing suitable membranes for these pur-poses, and such research is only in its infancy. The classicalsolution-diffusion model developed by Merten et al. maynot prove to be an adequate framework for interpretingdata, guiding the search for new membranes, or designingthe processes needed in this potential growth area owingto the inherent assumptions in this model. The objective ofthis paper is to reformulate this theory in a way that relaxessome of these limitations and to provide a pathway for evenmore complicated models that can be developed as needed.

2. The classical solution-diffusion theory

2.1. Derivation of the theory

Fig. 1 schematically illustrates the system of interest anddefines much of the nomenclature needed. The symbols havethe following meanings: p = pressure, C = concentration, a

Fig. 1. Schematic illustration of reverse osmosis and associated nomen-clature.

= activity, π = osmotic pressure, µ = chemical potential.The superscripts s and m denote the solution and membranephases. The subscripts denote either the component, i.e., 1= solvent, 2 = solute, or m = membrane, or the location,i.e., m = membrane (in the case of pressure), 0 = upstream,and ℓ = downstream in either the solution or membranephases. The basic premise of the solution-diffusion conceptis that permeating species dissolve in the membrane mate-rial and molecularly diffuse through it as a consequence ofa concentration gradient in contrast to the pore flow mecha-nism where the membrane material is not an active partici-pant at a molecular level. It is conceptually easy to visualizehow an imposed pressure difference across the membrane,(p0 − pℓ), can cause transport when there are pores travers-ing the membrane, but it is less intuitive how a pressure dif-ference could induce transport, or diffusion, when there arenot pores. The theory of Merten et al. [5–7] deals with thisissue correctly, within certain simplifying assumptions, butdoes not give a physically satisfying understanding of howthis occurs; a later section deals with this more thoroughly.Merten et al. begin with a familiar but simplified form ofFick’s law to describe the steady-state mass flux of water(or solvent) through the membrane in terms of a diffusioncoefficient and the concentration gradient of solvent in themembrane phase, i.e.

n1 = −D1mdCm1dz

(1)

They assume the membrane–solvent mixture is thermody-namically ideal so that the chemical potential and concen-trations within the membrane are related byµm1 = µo1 + RT lnCm1 (2)

They then write the flux in terms of the gradient of chemicalpotential as suggested by irreversible thermodynamics

n1 = −D1mCm1RT

dµm1dz

∼=D1mCm1RT

#µm1ℓ

(3)

Standard thermodynamics allows the chemical potential dif-ference between the two phases to be written in terms ofthe pressures applied to these phases and their solvent ac-tivity. This difference, #µs1, is the same as that within themembrane; hence, they write

#µs1 = [RT ln as10 + V̄1p0]− [RT ln as1ℓ + V̄1pℓ]= V̄1[#p − #π] = #µm1 (4)

where V̄1 is the partial molar volume of the solvent (seedefinition of π in Fig. 1). Combining Eqs. (3) and (4) gives

n1 =D1mCm1 V̄1(#p − #π)

ℓRT(5)

This implies that the net pressure driving force, (#p − #π),induces a concentration difference within the membrane be-tween its two surfaces given by

(Cm10 − Cm1ℓ) =Cm1 V̄1(#p − #π)

RT(6)

D.R. Paul / Journal of Membrane Science 241 (2004) 371–386 373

However, this point is never explicitly discussed by Mertenet al.To develop a relation for salt flux from Fick’s law, it was

correctly argued that the pressure should have a negligibleeffect for the conditions of interest; thus, they write

n2 = −D2mdCm2dz

∼= D2m#Cm2

ℓ= D2mK2

!

Cs20 − Cs2ℓ

ℓ

"

(7)

where K2 is the distribution coefficient for salt (or solute)between the solution and membrane phases. Using the defi-nition of solute rejection given in Fig. 1 and a mass balance,Eqs. (5) and (6) can be combined to obtain the familiar result

R =#

1+D2mK2RTCs1ℓ

D1mCm1 V̄1(#p − #π)

$−1

(8)

For our purposes here, it will prove useful to recast theseresults in another form by defining permeability coefficientsas follows:

P2 = K2D2m, K2 =Cm2Cs2

(9)

P1 = K1D1m, K1 =Cm1Cs1

∼=Cm1ρ1

(10)

where in the latter the solvent concentration in the externalphase is taken to be the pure solvent density ρ1 which isvalid for relatively dilute solutions. We now define a selec-tivity factor α for solute over solvent in analogy with othermembrane terminology

α = P2P1

= αSαD =!

ρ1K2Cm1

" !

D2mD1m

"

(11)

where αS and αD represent the solubility and diffusivitycomponents. To further simplify notation, especially later,we set

x = V̄1#p

RTand #x = V̄1(#p − #π)

RT(12)

Thus, the fluxes can be rewritten as

n1 = ρ1P1#x

ℓand n2 =

P2Cs20

ℓ[1− SP] (13)

and rejection and solute passage as

R = #x

α + #x(14)

SP = α

α + #x(15)

2.2. Application to desalination by cellulose acetatemembranes

Merten et al. determined the parameters in their theorythat describe the solubility and diffusion of water (1) and

Fig. 2. Solute (or salt) rejection vs. pressure for various values of selec-tivity, α, as predicted by the classical theory (Eq. (14)). For the pressurescales shown in psi and atm, V̄1, has been assumed to be 18 cm3/mol.

sodium chloride (2) in various cellulose acetate materialsusing extensive experiments independent of reverse osmosisbehavior [6]. The relatively good prediction of rejection ver-sus pressure from these data compared with that observedin reverse osmosis experiments using asymmetric celluloseacetate membranes may be taken as validation of these con-cepts [1]. Unfortunately, similar data are not available forother reverse osmosis membrane materials; this is quite un-derstandable for the complex thin-film composites. Usingtypical data from Merten et al., we estimate αS ∼= 0.22 andαD ∼= 5.9 × 10−4. Thus, both solubility and diffusion favorrejection of sodium chloride over water. The overall selec-tivity α is 1.3 × 10−4.Fig. 2 shows rejection versus #x as predicted by Eq. (14)

for various values of α. The scales below show the corre-sponding net pressure driving force at 25 ◦C using V̄1 =18 cm3/mol for water. The very small value of α = 1.3 ×10−4 means that good rejection can be obtained at relativelylow pressure driving forces.

2.3. Shortcomings of the classical theory

The results of the classical theory, summarized in Eqs. (5),(7) and (8), while quite appropriate for desalination by cel-lulose acetate membranes, have a number of limitations forbroader application that stem from some simplifying as-sumptions in their derivation; some are obvious and othersare not. The main limitations are summarized here alongwith an indication of the pathway for correcting these short-comings. Subsequent sections will develop most of thesein some detail while others are very system-dependent andonly the pathway is suggested.Eq. (5) predicts solvent flux to increase linearly with driv-

ing pressure without limit, i.e., the flux goes to infinity as thedriving pressure becomes infinite. This makes some sensefor a pore-flow model, but clearly this is untenable for asolution-diffusion process since the concentration gradient


induced by the pressure difference must be bounded. Thisflaw arises because of a subtle simplification implied inEq. (3) which leads to the prediction that the induced con-centration difference within the membrane is linear in thepressure driving force, see Eq. (6), and, hence, unbounded. Afurther consequence of this is that the rejection is predictedto approach unity as the driving pressure increases withoutlimit regardless of the permselectivity characteristics of themembrane. One can easily see why this would be the caseby noting that according to Eq. (7) n2 must always be finitewhile Eq. (5) allows n1 to become infinite; thus, the defini-tion of rejection in Fig. 1 indicates that R → 1 in this limitregardless of the value of α (see also Eq. (14)). A more rig-orous approach for dealing with the thermodynamics of thisproblem was developed more than 30 years ago in our labo-ratory [29–38] and has been widely adopted in the literature[16,26,39,40]. It naturally leads to nonlinear relationshipsthat replace Eqs. (5) and (6) and places an understandablelimit on solvent flux as will be shown in Section 3.The classical theory also predicts that the solute rejec-

tion, R, must always be positive, see Eq. (14), when clearlynegative values seem plausible and have been observed ex-perimentally in some cases [7]. As it turns out, this stemsfrom neglecting the effect of pressure on the solute transportwhich may not be an acceptable approximation particularlyfor large solute molecules. Thus, this effect is included inthe more general theory developed in Sections 4 and 5.The classic theory considers the transport of solute and

solvent to be completely independent without any effect ofone on the other when, in general, they may be coupled byeither frictional or convective effects. The huge literatureon irreversible thermodynamics outlines a formalism for de-scribing such effects and gives reciprocal relationships thatmust be obeyed. The relative success of Eqs. (5) and (7) fordescribing desalination seems to argue that such effects mustbe relatively small in this case. Regardless of their origin, itis reasonable to expect that coupling effects are less signif-icant, the lower the concentrations of solvent and solute inthe membrane. As more diverse applications of reverse os-mosis are contemplated, it seems prudent to build any newtheory around a more rigorous multicomponent diffusionformalism that can account appropriately for coupling aris-ing from both friction and convection. The Maxwell–Stefanequations [41–45] seem to be the most obvious and favoredstarting point for this. This approach to the task of refor-mulating the solution-diffusion theory of reverse osmosis isdeveloped in Section 3.Finally, the classical theory implicitly assumes the solu-

tion part to obey simple rules of thermodynamic idealityand ignores any concentration dependence of the diffusioncoefficients. For many membrane processes, these simpli-fications are not acceptable even though they appear to beadequate approximations for desalination. In what follows,the methodology for dealing with these complications willbe indicated but not dealt with in great detail since each onerequires very system-dependent parameters for modeling.

Section 6 gives example calculations where the assumptionsof solution ideality and concentration-independent diffusioncoefficients are used. These assumptions allow demonstra-tions of some principles and estimates of trends withouthaving system-dependent knowledge that frequently is notavailable at least in the early stages of material selection.The following development will make it clear that consid-

erable care and judgment must be exercised in defining dif-fusion coefficients. For example, the diffusion coefficientsin Eqs. (1) and (3) cannot be the same in general; however,this becomes acceptable when one can assume solution ide-ality and neglect convective issues.

3. Ternary Maxwell–Stefan equations

The Maxwell–Stefan equations were originally developedto describe multicomponent diffusion in gases at low den-sity and can be derived from kinetic theory [41,42]. How-ever, they have been extended with good success to densegases, liquids, and polymers [41]; many review papers andbooks have discussed uses of these equations [41,43]. Thegeneralized Maxwell–Stefan equations for isothermal mul-ticomponent mixtures can be written as

di = −%

j ̸=i

xixj

–Dij(vi − vj) (16)

where the –Dij are multicomponent diffusion coefficients[41], xi is the mole fraction of i in the mixture, and vi thevelocity of i relative to stationary coordinates. The term di isa generalized force for component i that causes it to diffuserelative to other species. Its general form isCRTdi = Ci∇µmi − wi∇pm (17)

where∇µmi = RT∇ ln ami + V̄i∇pm (18)

In Eq. (17), C is the molar density of the mixture and Ci

the molar concentration of i; of course, xi = Ci/C. As wewill see in Section 4, there is no pressure gradient in themembrane so ∇pm = 0; thus, di = xi∇ ln ami and Eq. (16)is replaced by

−%

j ̸=i

xixj

–Dij(vi − vj) = xi∇ ln ami (19)

Interestingly, the pressure driving force does not enter thepicture by the flux law but through the boundary conditionsused in its integrated form as will be made clear later. Atfirst sight, this may seem a little strange.Next, Eq. (19) will be adapted in ways that make it more

useful for membrane problems. First, velocities should beconverted to fluxes via the following definition:ni = wiρvi (20)

where ρ is the mass density of the membrane, solvent, plussolute mixture. Second, it is important to note that the flux


of the membrane material, relative to stationary coordinates,i.e., nm, is always zero at steady state. Third, it is not useful touse mole fractions when one of the components is a polymer,particularly since its molecular weight may be unknown,or even infinite (when cross-linked); and there is alwayssome distribution. We could convert the mole fractions inEq. (17) to volume fractions which would be convenientsince thermodynamic theories for polymer mixtures, e.g.,the Flory–Huggins theory, generally are expressed in theseterms [45]. However, we choose to use mass fractions heresince this is more compatible with the mass fluxes used andmay be more familiar for some. The interested reader canalways restate what follows in terms of volume fractions ifdesired. The conversion can be accomplished by substitutingthe following:

xi = M

Miwi, M =

&

% wj

Mj

'−1(21)

where Mi is the molecular weight of i and M the numberaverage molecular weight of the mixture. For our ternarymixture, there are three independent diffusion coefficientsthat arise –D1m, –D2m, and –D12. After conversion of molefractions to mass fractions in the Maxwell–Stefan equations,it becomes useful to redefine the diffusion coefficients in thefollowing way [45]:

Dij = –DijMj

M(22)

since they always appear in these combinations. Whilethis may seem strange and artificial, it will be of consid-erable convenience and does preserve the basic form ofthe Maxwell–Stefan equations. This substitution may im-ply something about scaling relationships or concentrationdependence of the transport parameters, but for now it isbest to view the Dij as parameters to be determined by ex-periment. After considerable algebra, the Maxwell–Stefanequations for the 1–2-m ternary can be reduced to thefollowing two equations:

n1 +!

w2n1 − w1n2wm

"

ε1 = −ρw1D1mwm

∇ ln am1 (23)

n2 +!

w1n2 − w2n1wm

"

ε2 = −ρw2D2mwm

∇ ln am2 (24)

where we define the εi terms as

ε1 = D1mD12

and ε2 = M2M1

D2mD12

(25)

These capture the frictional coupling effects and intuitivelywe expect D12 to be larger than either D1m or D2m, butthere is little information one can gain from the currentliterature about the relative magnitudes of these coefficients.The wm (= 1− w1 − w2) in the denominator of two termsin each equation reflects convection or frame of referenceconsiderations inherently included in the Maxwell–Stefanequations.

Alternate forms of Eqs. (23) and (25) can be written interms of gradients of concentrations, or weight fractions,within the membrane rather than activity by making use ofthe following definition:

Dim = Dim

!

∂ ln ami∂ lnwi

"

T,p

(26)

The new forms are

n1 +!

w2n1 − w1n2wm

"

ε1 = −ρD1mwm

∇w1 (27)

n2 +!

w1n2 − w2n1wm

"

ε2 = −ρD2mwm

∇w2 (28)

It is interesting to note that in the absence of frictional cou-pling, i.e., ε1 = ε2 = 0, the Maxwell–Stefan equations forthe unidirectional case become

n1 = −ρD1mwm

dw1dz

(29)

n2 = −ρD2mwm

dw2dz

(30)

wherewm = 1− w1 − w2 (31)

It has been suggested that transport in ternary membrane(note nm = 0) problems can be described by the followingextension of Fick’s law [41,46,47]:

n1 − w1(n1 + n2) = −ρD1mdw1dz

(32)

n2 − w2(n1 + n2) = −ρD2mdw2dz

(33)

These equations may be solved for n1 and n2 to give

n1 = −(1− w2)ρD1mwm

dw1dz

− (w1)ρD2mwm

dw2dz

(34)

n2 = −(w2)ρD1mwm

dw1dz

− (1− w1)ρD2mwm

dw2dz

(35)

This approach introduces a type of coupling because of theway the convection terms in Fick’s law are handled. It isbelieved that the Maxwell–Stefan approach is the more ap-propriate one because in Eq. (16) we can clearly see that themovement of one species relative to another, e.g., vi − vj ,requires some motive force. This physically intuitive con-cept is not apparent in Eq. (34) or (35).It is important to note that the Maxwell–Stefan equations

without coupling, see Eqs. (29) and (30), reduce exactly tothe predictions of Fick’s law, including the convective part,for a binary system, i.e., when we set eitherw1 orw2 to zero.

4. Pressure-induced diffusion

The following sections develop in more clear detail howan applied pressure difference across a dense membrane


leads to a concentration gradient which causes subsequentmolecular diffusion of solvent via Fick’s law through themembrane. The analysis closely follows ideas initially pre-sented by Paul and Ebra-Lima in 1970 and subsequently ex-panded in other publications from this laboratory [29–38].The thermodynamic part described here is a somewhat moregeneral version than originally presented. This section is lim-ited to the case of pure solvent transport to allow a simpleexposition of the concept; however, it proves to be relativelystraightforward to extend this to include solute transport ina general way via the Maxwell–Stefan equations developedin the previous section.

4.1. Fick’s law analysis for a binary system

The general form of Fick’s first law for a binary system(solvent = 1, membrane = m) can be written as follows fora membrane where there is a concentration gradient createdby any means [41]:

n1 = w1(n1 + nm) − ρD1m∇w1 (36)

where the first term on the right is the convective or frameof reference term that is often neglected in many treatments.This is justified when w1 is small but results in a redefini-tion of the binary diffusion coefficient whenw1 is significantcompared to unity. Since in steady state nm = 0, the unidi-rectional form of Fick’s law can be written in either of thefollowing differential or integrated forms when ρ and D1mare independent of concentration

n1 = − ρD1m1− w1

dw1dz

= ρD1mℓ⟨wm⟩

(w10 − w1ℓ) (37)

The convective term in the denominator, (1−w1), of the dif-ferential form means the concentration profile, or w1 versusz, is not strictly linear which complicates the integration. Toretain the expected form of the integration and its associ-ated physical meaning, it is convenient to define an averagevalue of 1−w1 = wm designated as ⟨wm⟩. A more completeanalysis [29,41] shows this quantity to be given by

⟨wm⟩ = w10 − w1ℓln((1− w1ℓ)/(1− w10))

= wmℓ − wm0ln(wm0/wmℓ)

(38)

The thickness in Eq. (37) is the actual thickness of the mem-brane with a solvent gradient which may differ from the drythickness, ld, when there is no solvent in the membrane orthe uniformly swollen thickness lo. These various quantitiesare related by the following [29]:

ℓdℓo

= (φm0)1/3 and φm0ℓo =

( ℓ

0φm dz (39)

where the φ represents volume fractions. Since ρwi = ρiφi,the integral equation can be written in the following form:

ρ0wm0ℓo =( ℓ

0ρwm dz (40)

Fig. 3. Chemical potential, pressure, and solvent concentration profiles ina membrane where pressure-induced diffusion occurs.

When ρ and D1m are constant, these reduce to

ℓ⟨wm⟩ = ℓowm0 (41)

which gives a convenient way to compute the correspondingterms in Eq. (37). These results would also apply to a ternarysystem when w2 ≪ 1.

4.2. Thermodynamic boundary conditions

For any membrane system, the value of w10 − w1ℓ to beinserted in Eq. (37) stems from the thermodynamic equilib-rium the two membrane interfaces have with the upstreamand downstream external phases. Paul and Ebra-Lima de-scribed an explicit procedure for handling this thermody-namic analysis in reverse osmosis or the so-called hydraulicpermeation [29]. The fundamental premise of this approachis to realize that the conditions of mechanical equilibriumrequire the pressure in a homogeneous, supported membraneto be constant throughout its thickness at the value imposedupstream, p0, as schematically illustrated in Fig. 3. This pointwhich was somewhat controversial when first introduced[29–38] now seems to be well recognized [16,23,39,40].For any species i, or component 1 as shown in Fig. 3, thechemical potentials in the phases on each side of eithermembrane–solution interface are equal as stipulated by ther-modynamics. Inside either the solution or the membranephases, we can write the following expression for the chem-ical potential of i in terms of activity (or concentration) andpressure as follows:

µi = µoi + RT ln ai + V̄i(p − pr) (42)

where pr is an arbitrary reference pressure (we will take thisto be pℓ) and µoi is a corresponding integration constant thatdepends on reference pressure chosen. Some authors preferto choose this reference as the saturation vapor pressure;however, this is not necessary and unnecessarily complicates


expressions given below. Eq. (42) assumes the membraneor solution phases are effectively incompressible; and if thisis not the case, the last term can be replaced by

) ppr

V̄i dp.At the upstream interface (z = 0) where the pressure is p0in both the solution and membrane phases, thermodynamicsconnects the activity in the two phases as follows:

ami0 = asi0 e−(V̄mi0−V̄ si0)(p0−pℓ)/RT (43)

At the downstream surface (z= ℓ), the pressure in the mem-brane phase is p0 while that in the solution phase is pℓ sothere is a pressure discontinuity and the activities on eitherside are related by

amiℓ = asiℓ e−V̄miℓ (p0−pℓ)/RT (44)

In general, the partial molar volume of i will be different inthe two phases and will depend on the concentration in thesephases; thus, we need to designate V̄i by both a superscript(phase) and a subscript for location (z = 0). In many cases,we may regard V̄i as a constant independent of compositionwhich is the same in both phases. In this case, Eqs. (43) and(44) reduce to the following more simple forms [29]:

ami0 = asi0 (45)

amiℓ = asiℓ e−V̄i(p0−pℓ)/RT (46)

By a suitable theory or by experiment, the relationship be-tween the concentration of i and its activity in a given phasecan be established. For example, the Flory–Huggins gives aconvenient framework for polymer systems [29].We see from Eq. (45) that the concentration of solvent

in the membrane at its upstream surface in hydraulic per-meation is the equilibrium swelling, w10, of the polymerin the pure solvent (as10 = 1) or the upstream solution forthe general case. On the other hand, Eq. (46) shows thatthe pressure discontinuity decreases the activity of solvent(hence, its concentration) in the membrane at the down-stream surface, i.e., solvent is “squeezed” out leading to aconcentration gradient within the membrane. This is the ori-gin of the diffusional flux induced by the pressure appliedupstream.

4.3. Limiting cases

It is useful to explore some limiting cases of Eq. (37) andthe associated boundary conditions given by Eqs. (45) and(46) when as10 = as1ℓ = 1, i.e., no solute is present. Whenthe membrane is immersed in pure solvent, it will imbibe anequilibrium amount of solvent which can be expressed as aweight fraction w∗

1 or a volume fraction φ∗1. This becomes

the upstream boundary condition in the membrane since nosolute is present, hence w10 = w∗

1 or Cm10 = ρw10. If weassume solution ideality in the membrane, we can write atthe downstream boundary

w1ℓ = w10am1ℓ = w10 e−V̄1#p/RT (47)

Fig. 4. Schematic illustration of the nonlinear flux–pressure driving forcerelationship in pressure-induced diffusion. Insets show concentration pro-files in the membrane at each state; note that flux reaches a finite plateauwhen the concentration of solvent in the membrane at the downstreamsurface is forced to zero at very high pressures.

With these simplifications and the definition of x in Eqs. (12)and (37) becomes

n1 =Cm10D1mℓ⟨wm⟩

[1− e−x] (48)

If we further assume that w10 is small enough that ⟨wm⟩ ∼ 1and that x is small enough that we can use the approximatione−x ∼= 1− x + · · · , then Eq. (48) reduces to

n1 =D1mCm10V̄1(#p)

ℓRT(49)

which is the classical result in Eq. (5) when #π = 0. Thesimple linear relation between (Cm10−Cm1ℓ) and the net pres-sure driving force given in Eq. (6) only holds for small driv-ing forces. Clearly Cm1ℓ, or wm1ℓ, can, at most, only go tozero and this occurs at infinite pressure. Thus, the relationbetween solvent flux and the pressure driving force must benonlinear in the general case as schematically suggested inFig. 4. The upper limit on flux, when #p → ∞, is given by

n1 →Cm10D1mℓ⟨wm⟩

(50)

Rather than imposing a hydraulic pressure upstream, a fluxof solvent through the membrane will also occur if the down-stream activity is reduced by replacing the downstream so-lution with a vapor phase whose partial pressure p1 is lessthan saturation vapor pressure, p∗

1, i.e.,

am1ℓ = p1p∗1

(51)

and

w1ℓ = w10

!

p1p∗1

"

(52)


Fig. 5. Volumetric flux of hexane through a swollen cross-linked rubbermembrane plotted vs. the difference in hexane volume fractions in themembrane at the upstream (φ10) and downstream (φ1ℓ) surfaces. Solidpoints represent pressure-driven experiments while the open circles rep-resent pervaporation-type experiments.

In this pervaporation mode, the flux is given by

n1 =Cm10D1mℓ⟨w10⟩

&

1− p1p∗1

'

(53)

Thus, the flux goes to the same limit, Eq. (50), when p1 = 0as when #p → ∞. This simple connection between hy-draulic permeation and pervaporation has been experimen-tally demonstrated for hexane permeation in a cross-linkedrubber membrane [37] with the results summarized in Fig. 5.The volumetric flux of hexane is plotted there versus the dif-ference in the volume fractions of hexane in the membranesat the upstream and downstream surfaces; φ10 = constantwhile φiℓ is reduced by imposing a hydraulic pressure up-

Fig. 6. Dimensionless solvent flux vs. x = V̄1#p/RT illustrating the linear and nonlinear models discussed in text.

stream (solid points) or imposing a reduced vapor activitydownstream (open points). As expected, the data from thetwo modes form a single relationship. This relationship isnonlinear owing to the concentration dependence of the dif-fusion coefficient [37] which is not accounted for in thesimple model equations given above.For reverse osmosis or hydraulic permeation, one might

ask when do nonlinearities in flux–pressure relationshipbecome important? First, it is important to recognize threeseparate causes for nonlinearities. The simplest arises fromthe exponential term in Eq. (48) when x is too large touse the truncated series expansion introduced to obtainEq. (49). The graph in Fig. 6 compares the linear and non-linear forms. The two differ by 5% (typical of the level ofexperimental sensitivity) when x = V̄r#p/RT = 0.1. Forwater where V̄1 ∼= 18 cm3/mol, this requires pressures ofthe order of 13.8MPa (2000 psi or 136 atm); such pressuresare usually not reached in reverse osmosis applications sono nonlinearity from this source would never be expected.However, for an organic solvent with V̄1 ∼= 100 cm3/mol,this 5% discrepancy occurs at the much lower pressureof 2.48MPa (360 psi or 24.5 atm); such pressures mighteasily be required in such a membrane operation. AtV̄1 ∼= 300 cm3/mol, the pressure is correspondingly lowerat 0.83MPa (120 psi or 8.2 atm). Thus, high partial molarvolumes bring these effects into a range where they maynot be negligible; water is a special case because of its lowmolecular weight.For highly swollen membranes, the term ⟨wm⟩ cannot

be approximated as unity and will depend on pressure, seeEq. (38), which adds another potential source of nonlinear-ity. The final source of nonlinearity can be concentration de-pendence of the diffusion coefficient. In our prior work onrubbery membranes swollen to various extents by organicsolvents these two effects were rather minor compared tothe one described above [29–38].


5. Integrated forms of the ternary Maxwell–Stefanequations

Section 4 has laid the foundation needed for applyingthe more general ternary Maxwell–Stefan equations toreverse osmosis problems. The differential forms of theMaxwell–Stefan equations developed in Section 3 will needto be integrated across the membrane in order to obtainforms useful for comparison with experimental data. Thisraises a question of whether it is better to start from theversions given as Eqs. (23) and (24) or the alternate formsgiven by Eqs. (27) and (28). If we ignore frictional couplingissues for the moment by setting ε1 = 01, the steady-stateunidimensional form of Eq. (23) becomes

n1 = −ρw1D1mwm

d ln am1dz

(54)

Integration of this equation, coupled with Eq. (46) and thedefinition of #p, gives

ln!

am10am1ℓ

"

= V̄1(#p − #π)

RT= n1

( ℓ

0

&

wmρw1D1m

'

dz (55)

Assuming the integrand in Eq. (55) to be constant leads tothe prediction that n1 is always linear in (#p − #π) whichmeans the nonlinear effects predicted in Section 4 are en-tirely missed. Of course, when w1ℓ is significantly less thanw10 at high driving pressure, the term w1 in the denomina-tor obviates such an assumption, regardless of how the otherterms in the integrand may vary across the membrane. In-deed, for thermodynamically ideal systems with a constantdiffusion coefficient, it is the w1 term in this integrand thatis alone responsible for the nonlinearity described earlier.On the other hand, starting from Eq. (27) with the same

assumption that ε1 = 0 leads to

n1 = −ρD1mwm

!d ln am1d lnw1

"

dw1dz

= −ρD1mwm

dw1dz

(56)

This equation can be integrated to give the following:

(w10 − w1ℓ) = n1

( ℓ

0

&

wmρD1m

!d ln am1d lnw1

"'

dz

= n1

( ℓ

0

&

wmρD1m

'

dz (57)

This result more obviously captures the nonlinear effects,even when the integrand is constant, because this is inher-ently included in the term w1ℓ via Eq. (44) or (46). Theformulation involving Dim absorbs the w1 term that appearsin the integrand of Eq. (55) into the thermodynamic termthat connects the two diffusion coefficients (see Eq. (26)).For what follows, the latter approach based on concentrationgradients seems more convenient.The ternary Maxwell–Stefan equations based on concen-

tration gradients, Eqs. (27) and (28), can be rearranged andintegrated for the unidirectional, steady-state case to obtainthe following:

n1

( ℓ

0wm dz + n1

( ℓ

0ε1w2 dz − n2

( ℓ

0ε1w1 dz

=( w10

w1ℓ

ρD1m dw1 (58)

n2

( ℓ

0wm dz − n1

( ℓ

0ε2w2 dz + n2

( ℓ

0ε2w1 dz

=( w20

w2ℓ

ρD2m dw2 (59)

These general results can be simplified in various ways de-pending on what level of assumptions are justified. For ex-ample, if we assume all transport coefficients to be constantsindependent of concentration, we get

n1

( 1

0wm dξ + n1ε1

( 1

0w2 dξ − n2ε1

( 1

0w1 dξ

= ρD1mℓ

(w10 − w1ℓ) (60)

n2

( 1

0wm dξ − n1ε2

( 1

0w2 dξ + n2ε2

( 1

0w1 dξ

= ρD2mℓ

(w20 − w2ℓ) (61)

where ξ = z/ℓ. Often it is satisfactory to assume all the wi

profiles across the membrane are linear, i.e.

wi = (wi0 − wiℓ)ξ + wiℓ (62)

Thus, all the integrals in equations can be simply evaluated,i.e.( 1

0wi dξ = wi0 + wiℓ

2≡ w̄i (63)

Thus, Eqs. (60) and (61) become!

1+ ε1w̄2w̄m

"

n1−!

ε1w̄1w̄m

"

n2 = ρD1mw̄mℓ

(w10 − w1ℓ) (64)

!

1+ ε2w̄1w̄m

"

n2−!

ε2w̄2w̄m

"

n1 = ρD2mw̄mℓ

(w20 − w2ℓ) (65)

Solving Eqs. (64) and (65) simultaneously gives

n1 =

ρD1m(1+ ε1w̄2/w̄m)(w10 − w1ℓ)

+ ρD2m(ε1w̄1/w̄m)(w20 − w2ℓ)

w̄mℓ[1+ ε1w̄2/w̄m + ε1w̄1/w̄m](66)

n2 =

ρD2m(1+ ε1w̄2/w̄m)(w20 − w2ℓ)

+ ρD1m(ε2w̄2/w̄m)(w10 − w1ℓ)

w̄mℓ[1+ ε1w̄2/w̄m + ε2w̄1/w̄m](67)

Eq. (66) reduces to Eq. (37) when ε1 = ε2 = 0 and w̄m ∼=⟨wm⟩; the latter quantities differ only to the extent that theprofile of wm across the membrane differs from linearity,compare the definitions in Eqs. (38) and (63). Eq. (67) isanalogous to Eq. (7) when ε1 = ε2 = 0 and w̄m ∼ 1.


These flux equations may not be as simple to use as theymight appear when ε1 and ε2 are finite for two reasons. First,the composition of the downstream solution in reverse osmo-sis is determined by the ratio of solute to solvent flux; how-ever, these fluxes cannot be calculated until the compositionof the downstream solution is known. Given the complexityof these equations and the fact that they must be combinedwith Eqs. (43) and (44) and a model that connects activitieswith compositions at the two boundaries generally meansan iterative calculation is involved. This, of course, assumesall the transport coefficients are known; generally, the oppo-site is true and one wishes to extract these by comparisonof the flux equations to experimental flux data. Second, itis somewhat doubtful if the assumption of constant trans-port coefficients is generally valid particularly when strongcoupling effects exist (high concentrations of solvent and/orsolute in the membrane).At the current time there are very few data in the literature

indicating the magnitude of frictional coupling effects. Thereare some attempts to analyze pervaporation experiments interms of the Maxwell–Stefan approach [48,49]; however, inthese cases plasticization effects are very strong and tendto compromise evaluation of the coupling terms in theseanalyses.

6. A simplified model and example calculations

In this section, a simplified model is discussed that as-sumes: coupling effects are absent, i.e., ε1 = ε2 = 0, themixtures obey a form of thermodynamic ideality, and thatD1m and D2m are constant. However, the effect of pressureon solute transport is not ignored.

6.1. Model development and calculations

As described earlier, we assume the equilibrium swellingof the membrane in pure solvent is given by w∗

1. With ourassumption of ideality, the upstream boundary condition inthe membrane when solute is present, i.e., as10 < 1, can bewritten as

w10 = w∗1am10 = w∗

1 e−V̄1π0/RT (68)

since the presence of solute upstream reduces the swellingand this effect can be expressed in terms of the osmoticpressure in the adjacent solution, i.e., am10 = e−V̄1π0/RT andam10 = as10. Likewise, we may re-write Eq. (46) at the down-stream surface as

am1ℓ = as1ℓ e−V̄1#p/RT = (e−V̄1πℓ/RT)(e−V̄1#p/RT) (69)

or

w1ℓ = w∗1(e

−V̄1πℓ/RT)(e−V̄1#p/RT) (70)

Thus, the concentration driving force for solvent permeationcan be written in the following forms:

(w10 − w1ℓ) = w∗1(e

−V̄1π0/RT)[1− e−V̄1(#p−#π)/RT]

= w10[1− e−V̄1(#p−#π)/RT] (71)

This can be substituted in the Maxwell–Stefan equation forcomponent 1 that ignores coupling or in Eq. (37) to get

n1 = ρw10D1mℓ⟨wm⟩

[1− e−V̄1(#p−#π)/RT] (72)

It is assumed that solute solubility in the membrane can bedescribed by the distribution coefficient introduced earlierso that at the upstream membrane surface

w20 =K2C

s20

ρ(73)

At the downstream membrane surface, the pressure effect isalso considered

w2ℓ =!

K2Cs2ℓ

ρ

"

e−V̄1#p/RT (74)

Combining the latter with the definition of solute pas-sage, i.e. Cs2ℓ = SPCs20, and the analogous version of theMaxwell–Stefan equation (without coupling) or Fick’s lawgives

n2 =K2C

s20D2m

ℓ⟨wm⟩[1− SP e−V̄2#p/RT] (75)

By combining the two flux equations, the following is ob-tained:

ρ1n2Cs20n1

= SP =!

α

e−V̄1π0/RT

"

1− SP e−rV̄1#p/RT

1− e−V̄1(#p−#π)/RT(76)

where r = V̄2/V̄1. It is useful to simplify these results byrestricting consideration to dilute feed solutions so that π0 ∼#π ∼ 0. Solving Eq. (76) for the solute passage gives

SP = α

1− e−x + α e−rx(77)

where α is the separation factor defined in Eq. (11). Thesolute rejection for this case is given by

R = 1− e−x − α[1− e−rx]1− e−x + α e−rx

(78)

These equations represent a generalization of the “classical”theory of reverse osmosis where the simplifying assumptionsthat pressure does not affect solute flux and that lead tothe linearity of solvent flux have not been made. All othersimplifications are the same. The consequence is that in thelatter model, solvent flux does not go to infinity as#p→ ∞and rejection does not go to unity. The current formulationleads to

SP → α or R = 1− α (79)

as #p → ∞. The limits in Eq. (79) reflect the intrinsic se-lectivity of the membrane as should be the case. For small


Fig. 7. Calculation of how rejection at a constant pressure driving force, x = 0.14, responds to the selectivity of the membrane, α, when the effect ofpressure on solute is included (r = 3) or not included (r = 0).

values of x where a truncated series expansion of the expo-nentials is justified, i.e., e−x = 1− x+· · · , the equation forSP reduces to

SP = α

α + (1− αr)x(80)

In Eq. (77) or (80), setting r = 0 amounts to ignoring theeffect of pressure on solute flux. Some simple calculationswith Eq. (77) serve to illustrate this effect. In Fig. 7, rejectionis plotted versus the selectivity of the membrane for the casesof r = 0 and 3 when x = 0.14 (i.e., nonlinear effects arerelatively small); the latter would correspond, for example,to #p = 3.45MPa (500 psi) when V̄1 = 100 cm3/mol. Asthe membrane selectivity for solute to solvent goes fromvery low to very high values, rejection goes from unity tozero when r = 0 similar to the prediction of the classicaltheory (Eq. (14)). However, when r = 3 the rejection goesto negative values as α becomes larger; the classical theorynever predicts R < 0. Fig. 8 shows the effect of pressureon the solute in another way. Here x is set at 0.5 wherenonlinear effects are quite significant (see Fig. 6). At lowvalues of α, the solute passage is essentially unaffected bythe value of r; however, at much higher α, the effects arestrong. The classical theory fails to describe such effectsowing to assumptions made in its formulation.The above calculations indicate that ignoring the effect of

pressure on solute transport is acceptable when the mem-brane has high selectivity for passage of solvent relative tosolute, i.e. very low α. This is clearly the case for H2O/NaClfor cellulose acetate and probably all other commerciallyimportant desalination membranes. For organic separations,the value of α may not be so favorable so the effect of pres-sure on the solute may be important. Furthermore, since or-ganics will invariably have high molar volumes, comparedto H2O or NaCl, a theory that allows for the nonlinearitiesat higher levels of V̄i#p/RT may also be needed. At this

point, it is difficult to know how important coupling effectsmight be.

6.2. Membranes with reverse selectivity

For conventional reverse osmosis or nanofiltration oper-ations, the product is the permeate or solvent (water typi-cally) which has been significantly depleted of solute. Fororganic systems, the product may also be purified solventor it could be the reject stream which has been enriched insolute relative to the feed. It is likely that both possibili-ties will be used as membranes become a more popular op-tion in organic chemical processing. There is another way touse membranes to good advantage that appears possible, atleast on paper. Membranes that pass larger molecules morerapidly than smaller ones are well known in the field of gasand vapor separations [50]. Generally, the mechanism for

Fig. 8. Calculated effect of the solute molar volume, relative to the solvent,on solute passage, SP, for membranes with different selectivities, α, forfixed x = V̄1#p/RT = 0.5.


Fig. 9. Schematic representation of the solute concentration profile whenreverse osmosis produces a permeate more concentrated than the feed.

this “reverse selectivity” stems from solubility rather thandiffusion considerations. It is of interest here to explore thepossibilities of what one could do with membranes havingα significantly greater than unity. In this case, the productcould be a solute-enriched permeate. In the end, the feasi-bility of this is whether such membrane materials can beidentified for a given feed mixture. Here, we explore whatone could expect for a given postulated set of selectivitycharacteristics.Fig. 9 shows the solute concentration profile that must

exist in a reverse osmosis operation that produces asolute-enriched permeate. Even though Cs20 < Cs2ℓ, it isnecessary that within the membrane that Cm20 > Cm2ℓ if so-lute is to be transported from the feed side to the permeateside. In this situation, the boundary conditions are given by

Cm20 = K2Cs20 (81)

Cm2ℓ = (K2Cs2ℓ)(e

−V̄2#p/RT) = Cm20 SP e−rx (82)

Fig. 10. Solute passage, SP, as a function of membrane selectivity, α, for the reduced pressure, x, levels showing regimes of solute rejection, α < 1, andsolute enrichment, α > 1.

Clearly, for this condition to be established the followingcriteria must be met: SP e−rx < 1.Fig. 10 shows calculations, using Eq. (77), of the solute

passage (SP) versus membrane selectivity α for three op-erating conditions, x, when r = 3. Permeates significantlyenriched in solute over the feed are calculated to occur;however, this requires values of α well above unity andquite high pressures. Note that x = 1 corresponds to #p= 24.7MPa (244 atm) when V̄1 = 100 cm3/mol. Of course,low values of α and high x lead to high levels of solute re-jection since R = 1 − SP. Fig. 11 amplifies on the pressureeffect for membranes with α = 10−2, 1, and 102 for valuesof r = 0 and 3. At low α, there is essentially no effect ofsolute partial molar volume on the predicted solute passage.This clearly shows that Merten et al. were well justified inneglecting this since a successful desalination membranetypically has an α even lower than 10−2. At the high α

shown, there is no selectivity of solute over solvent unlessthe pressure effect is included. For high α, SP rises graduallywith x; whereas, for low α, SP drops rapidly at relativelylow values of x. Interestingly, a membrane with α = 1 maystill preferentially pass one component over the other.It should be noted that while it appears possible to gen-

erate a solute-enriched permeate this may come at a signif-icantly compromised productivity because it is difficult toachieve a high driving force in this mode. The solute fluxcan be written as

n2 = D2mℓ⟨wm⟩

[Cm20 − Cm2ℓ] =K2D2mCs

20ℓ⟨wm⟩

[1− SP e−rx] (83)

This may be rearranged into the following dimensionlessform:

ℓ⟨wm⟩n2K2D2mCs20

= 1− e−x

1− e−x + α e−rx(84)


Fig. 11. Calculated effect of pressure on solute passage for membranes with three levels of selectivity, α, when the ratio of solute to solvent partial molarvolumes r = 0 (no pressure effect) or 3 (pressure effect included).

Fig. 12. Reduced solute flux calculated from Eq. (84) for r = V̄2/V̄1 = 0and 3 at the values of selectivity, α, shown.

Fig. 12 shows plots of the normalized flux, defined as theleft-hand side of Eq. (84), versus the driving force for twolevels of α and r. The normalized flux would be unity if therewere no effect of pressure and the solute concentration inthe permeate were zero. While pressure tends to drive soluteto the permeate side, the high solute content in the permeatetends to do the opposite. The result is a low flux of solutein these cases.

7. Evaluation of coupling effects in reverse osmosis

As mentioned earlier, there is very little information in theliterature from which to evaluate how important frictionalcoupling effects may be in any membrane process; however,experience with gas separations and desalination by reverse

osmosis suggest that such effects must be very small sincemodels that ignore this describe performance relatively well.However, in pervaporation the experiments are usually notgood enough to differentiate true coupling due to cross-termsfrom mutual plasticization effects. At this stage, it is notpossible to know the possible impact of these effects onorganic separations by reverse osmosis. The purpose of thissection is two-fold: the first is to recast the Maxwell–Stefanequations in a form that indicates the possible impact onselectivity and the second is to do hypothetical calculationsto explore the magnitude of these effects on desalination viacellulose acetate membranes.

7.1. Solute passage equations with coupling

Expressions for R or SP as predicted by the ternaryMaxwell–Stefan approach could be obtained from the inte-grated forms developed in Section 5. However, a somewhatdifferent approach is used here. The starting point is thedifferential forms given by Eqs. (27) and (28). We identify,for the unidimensional case, the following terminology forthe right-hand side of each of these equations:

no1 ≡ −ρD1mwm

dw1dz

(85)

no2 ≡ −ρD2mwm

dw2dz

(86)

Physically, noi is essentially the flux that i would have for thesame external boundary conditions if no coupling of trans-port occurred inside the membrane. We can apply Eqs. (27)and (28), with the new nomenclature, locally at z = 0 to fixthe concentration variables at known values and solve for ni

or the flux of i with coupling, i.e.,

n1 =(1+ w10ε1/wm0)n

o1 + (w10ε1/wm0)n

o2

1+ (w20/wm0)ε1 + (w10/wm0)ε2(87)


Table 1Hypothetical calculations of the effects of coupling on desalination bycellulose acetate membranes

Assumed values Calculated

ε2 Ro R SPc/SPo

0.1 0.990 0.987 1.330.1 0.90 0.899 1.0150.5 0.990 0.975 2.540.5 0.900 0.894 1.063

Ro = rejection assuming coupling and R = calculated rejection with theassumed level of coupling.

n2 =(1+ w20ε1/wm0)n

o2 + (w20ε2/wm0)n

o1

1+ (w20/wm0)ε1 + (w10/wm0)ε2(88)

The close similarity of these results with Eqs. (66) and (67)should be clear. By defining solute passage terms with (SPc)and without (SPo) coupling, the above results can be com-bined to obtainSPcSPo

=1+ (K2C

s20/ρwm0)[ε1 + (ρ1/Cs20)(ε2/SPo)]

1+ (w10/wm0)[ε2 + (Cs20/ρ1)SPoε1](89)

Thus, from this equation, the effect of coupling on selectivitycan be evaluated if we know all the physical parametersshown plus, of course, values for ε1 and ε2.

7.2. Coupling effects on desalination by cellulose acetate

It will be interesting to do a parametric analysis of whateffect coupling may have on salt rejection in water desali-nation by cellulose acetate. We assume a feed containing2wt.% NaCl, and use the following typical parameters fromLonsdale et al. [6]: K2 = 0.03, w10 = 0.17, and wm0 =0.83. Substitution of these values into Eq. (89) givesSPcSPo

= 1+ 0.036ε2/SPo1+ 0.21ε2

(90)

As it turns out, terms in ε1 are entirely negligible. Table 1shows samples of the consequences on rejection for twomembrane situations: one with very high rejection of 99%and a more modest value of 90%. The calculations assumea coupling coefficient that is conceivable, ε2 = 0.1, and onemuch higher than the author intuitively expects, ε2 = 0.5.While the calculated effect on salt passage can be sizeable,when this is converted to the usual rejection coefficient,the effects are in the third, or at most the second, decimalplaces. Most reverse osmosis measurements are not madewith enough accuracy to detect such an effect. Thus, cou-pling effects might exist to significant levels in currentlyused membranes, but the consequences are too small to benoticeable.

8. Conclusion

The classical solution-diffusion theory of reverse osmo-sis proposed by Merten et al. [1,4–7] has been re-examined

to relax various assumptions in the original theory. TheMaxwell–Stefan equations have been adapted to considerpotential effects of coupling between solvent and solutetransport within the membrane and to include convectiveeffects. The boundary conditions for the multicomponenttransport equations were developed from a generalizationof the approach introduced by Paul and Ebra-Lima [29].This introduces nonlinearities not captured in the clas-sical theory and includes the effect of pressure on thetransport of solute as well as solute. This formalism per-mits solution non-ideality in all phases to be treated asneeded. Integration of the differential transport equationsis discussed and allows for concentration dependence ofthe transport coefficients. It was concluded that the classi-cal theory for reverse osmosis appropriately and elegantlyaccounts for the effects important in typical desalina-tion applications; however, the more general formulationgiven here may be needed as reverse osmosis membranesare developed for separations of various types of organicsystems.The following are some specific conclusions. Coupling

effects may exist in current desalination membrane systemsbut their consequences on selectivity are judged to be in-significant. The same conclusion may not apply to reverseosmosis for other types of systems. High molar volumesof solvent and solute exacerbate nonlinear effects that arelargely not noticeable in aqueous systems owing to the smallmolar volume of water. Transport laws based on concen-tration gradients rather than activity gradients are more ap-propriate for integration when capturing the nonlinear ef-fects mentioned earlier is important. The Maxwell–Stefanequations deal more appropriately with the convection termsfor multicomponent diffusion than do ad hoc extensions ofFick’s law.It is hoped that the results developed here will stimulate

interest in programs for very careful transport measurementsand associated thermodynamic characterization for a broadrange of membrane/fluid systems that may be of interest fornovel membrane separations. It would be of great value tosort out carefully the effects of true transport coupling (D12in the above treatment) versus apparent coupling caused bymutual plasticization of the polymer membrane caused bythe mere presence of the various species as opposed to effectscaused by their gradients.

Acknowledgements

The author thanks the National Science Foundation andThe Department of Energy for their continued support ofmembrane research in this laboratory. In addition, the authorthanks the following colleagues who have contributed tothe completion of this manuscript in many different ways:Susan Chapman, Shuichi Takahashi, Neil Moe, Steve Kloos,Benny Freeman, and Hal Hopfenberg.


References

[1] U. Merten (Ed.), Desalination by Reverse Osmosis, MIT Press,Cambridge, 1966.

[2] C.E. Reid, Breton, Water and ion flow across cellulosic membranes,J. Appl. Polym. Sci. 1 (1959) 133–143.

[3] C.E. Reid, J.R. Kuppers, Physical characteristics of osmotic mem-branes of organic polymers, J. Appl. Polym. Sci. 2 (1959) 264–272.

[4] R.L. Riley, J.O. Gardner, U. Merten, Cellulose acetate mem-branes: electron microscopy of structure, Science 143 (1964) 801–803.

[5] U. Merten, Flow relationships in reverse osmosis, Ind. Eng. Chem.Fundam. 2 (1963) 229–232.

[6] H.K. Lonsdale, U. Merten, R.L. Riley, Transport properties of cel-lulose acetate osmotic membranes, J. Appl. Polym. Sci. 9 (1965)1341–1362.

[7] H.K. Lonsdale, U. Merten, M. Tagami, Phenol transport in celluloseacetate membranes, J. Appl. Polym. Sci. 11 (1967) 1807–1820.

[8] W.S.W. Ho, K.K. Sirkar (Eds.), Membrane Handbook, van NostrandReinhold, New York, 1992.

[9] M. Soltanieh, W.N. Gill, Review of reverse osmosis membranesand transport models, Chem. Eng. Commun. 12 (1981) 279–363.

[10] V.L. Punzi, G.P. Muldowney, An overview of proposed solute re-jection mechanisms in reverse osmosis, Rev. Chem. Eng. 4 (1987)1–40.

[11] W.J. Koros, G.K. Fleming, S.M. Jordan, T.H. Kim, H.H. Hoehn,Polymeric membrane materials for solution-diffusion-based perme-ation separations, Prog. Polym. Sci. 13 (1988) 339–401.

[12] B.S. Parekh, Reverse Osmosis Technology: Applications forHigh-purity-water Production, Marcel-Dekker, New York, 1988.

[13] D. Bhattacharyya, W.C. Mangum, M.E. Williams, Reverse osmosis,in: Kirk–Othmer Encyclo. of Chem. Technol., vol. 21, 4th ed., JohnWiley, New York, 1997, pp. 303–333.

[14] L.S. White, A.R. Nitsch, Solvent recovery from lube oil filtrates witha polyimide membrane, J. Membr. Sci. 179 (2000) 267–274.

[15] L.S. White, Transport properties of a polyimide solvent resis-tant nanofiltration membrane, J. Membr. Sci. 205 (2002) 191–202.

[16] D. Bhanushali, S. Kloos, C. Kurth, D. Bhattacharyya, Performanceof solvent-resistant membranes for non-aqueous systems: solventpermeation results and modeling, J. Membr. Sci. 189 (2001)1–21.

[17] X.J. Yang, A.G. Livingston, L. Freitas dos Santos, Experimentalobservations of nanofiltration with organic solvents, J. Membr. Sci.190 (2001) 45–55.

[18] J.A. Whu, B.D. Baltzis, K.K. Sirkar, Nanofiltration studies of largerorganic microsolutes in methanol solutions, J. Membr. Sci. 170 (2000)159–172.

[19] J.A. Whu, B.C. Baltzis, K.K. Sirkar, Modeling of nanofiltra-tion-assisted organic synthesis, J. Membr. Sci. 163 (1999) 319–331.

[20] D.R. Machado, D. Hasson, R. Semiat, Effect of solvent properties onpermeate flow through nanofiltration membranes. Part I. Investigationof parameters affecting solvent flux, J. Membr. Sci. 163 (1999) 93–102.

[21] D.R. Machado, D. Hasson, R. Semiat, Effect of solvent properties onpermeate flow through nanofiltration membranes. Part II. Transportmodel, J. Membr. Sci. 166 (2000) 63–69.

[22] K. Ebert, F.P. Cuperus, Solvent resistant nanofiltration membranesin edible oil processing, Membr. Technol. 107 (1999) 5–8.

[23] S.S. Luthra, X. Yang, L.M. Freitas dos Santos, L.S. White, A.G.Livingston, Phase-transfer catalyst separation and re-use by solventresistant nanofiltration membranes, Chem. Commun. 16 (2001) 1468–1469.

[24] G.H. Koops, S. Yamada, S.-I. Nakao, Separation of linear hydro-carbons and carboxylic acids from ethanol and hexane solutions byreverse osmosis, J. Membr. Sci. 189 (2001) 241–254.

[25] L.Y. LaFreniere, Reverse osmosis process for recovery of C3–C6aliphatic hydrocarbons from oil, US Patent, 5,133,867 (1992).

[26] R. Rautenbach, R. Albrecht, The separation potential of pervapora-tion. Part 1. Discussion of transport equations and comparison withreverse osmosis, J. Membr. Sci. 25 (1985) 1–23.

[27] E. Gibbins, M. D’Antonio, D. Nair, L.S. White, L.M. Freitas dosSantos, I.F. Vankelecom, A.G. Livingston, Observations on solventflux and solute rejection across solvent resistant nanofiltration mem-branes, Desalination 147 (2002) 307–313.

[28] S.S. Luthra, X. Yang, L.M. Freitas dos Santos, L.S. White, A.G.Livingston, Homogeneous phase transfer catalyst recovery and reuseusing solvent resistant membranes, J. Membr. Sci. 201 (2002) 65–75.

[29] D.R. Paul, O.M. Ebra-Lima, Pressure-induced diffusion of organicliquids through highly swollen polymer membranes, J. Appl. Polym.Sci. 14 (1970) 2201.

[30] D.R. Paul, O.M. Ebra-Lima, The mechanism of liquid transportthrough swollen polymer membranes, J. Appl. Polym. Sci. 15 (1971)2199.

[31] D.R. Paul, The role of membrane pressure in reverse osmosis, J.Appl. Polym. Sci. 16 (1972) 771.

[32] D.R. Paul, Relation between hydraulic permeability and diffusion inhomogeneous swollen membranes, J. Polym. Sci. A-2 11 (1973) 289.

[33] D.R. Paul, Diffusive transport in swollen polymer membranes, in:H.B. Hopfenberg (Ed.), Permeability of Plastic Films and Coatingsto Gases, Vapors, and Liquids, Plenum Press, New York, 1974,p. 35.

[34] D.R. Paul, Further comments on the relation between hydraulic perm-eation and diffusion, J. Polym. Sci.: Polym. Phys. Ed. 12 (1974)1221.

[35] D.R. Paul, J.D. Paciotti, O.M. Ebra-Lima, Hydraulic permeation ofliquids through swollen polymeric networks. II. Liquid mixtures, J.Appl. Polym. Sci. 19 (1975) 1837.

[36] D.R. Paul, O.M. Ebra-Lima, Hydraulic permeation of liquids throughswollen polymeric networks. III. A generalized correlation, J. Appl.Polym. Sci. 19 (1975) 2759.

[37] D.R. Paul, J.D. Paciotti, Driving force for hydraulic and pervaporativetransport in homogeneous membranes, J. Polym. Sci.: Polym. Phys.Ed. 13 (1975) 1201.

[38] D.R. Paul, The solution-diffusion model for swollen membranes,Separation and Purification Meth. 5 (1976) 33.

[39] C.H. Lee, Theory of reverse osmosis and some other membranepermeation operations, J. Appl. Polym. Sci. 19 (1975) 83–95.

[40] J.E. Wijmans, R.W. Baker, The solution-diffusion model: a review,J. Membr. Sci. 107 (1995) 1–21.

[41] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nded., John Wiley, New York, 2002.

[42] C.F. Curtiss, R.B. Bird, Multicomponent diffusion, Ind. Eng. Chem.Res. 38 (1999) 2515–2522.

[43] R. Krishna, J.A. Wesselingh, The Maxwell–Stefan approach to masstransfer, Chem. Eng. Sci. 52 (1997) 861–911.

[44] J.M. Zielinski, B.F. Hanley, Practical friction-based approach to mod-eling multicomponent diffusion, AIChE J. 45 (1999) 1–12.

[45] G. Hoch, A. Chauhan, C.J. Radke, Permeability and diffusivity forwater transport through hydrogel membranes, J. Membr. Sci. 214(2003) 199–209.

[46] H.D. Kamaruddin, W.J. Koros, Some observations about the applica-tion of Fick’s first law for membrane separation of multicomponentmixtures, J. Membr. Sci. 135 (1997) 147–159.

[47] J.S. Vrentas, C.M. Vrentas, Transport in nonporous membranes,Chem. Eng. Sci. 47 (2002) 4199–4208.

[48] P. Izak, L. Bartovska, K. Friess, M. Sipek, P. Uchytil, Comparisonof various models for transport of binary mixtures through densepolymer membrane, Polymer 44 (2003) 2679–2687.


[49] A. Heintz, W. Stephan, A generalized solution-diffusion model ofthe pervaporation process through composite membranes. Part II.Concentration polarization, coupled diffusion and the influence ofthe porous support layer, J. Membr. Sci. 89 (1994) 153–169.

[50] R.W. Baker, J.G. Wijmans, Membrane separation of organic vaporsfrom gas steams, in: D.R. Paul, Y.P. Yampol’skii (Eds.), PolymericGas Separation Membranes, CRC Press, Boca Raton, 1994 (Chapter8).

Documents

Paul - 2004 - Reformulation of the solution-diffusion ... · PDF fileJournal of Membrane Science 241 (2004) 371–386 Reformulation of the solution-diffusion theory of reverse osmosis