12
Modelling the transport of asymmetric electrolytes through nanofiltration membranes Anthony Szymczyk a, *, Yannick Lanteri b , Patrick Fievet b a Université Rennes 1, UMR 6226 CNRS-UR1-ENSCR, équipe Chimie et Ingénierie des Procédés, 263 Avenue du Général Leclerc, Bâtiment 10 A, CS 74205, 35042 Rennes Cedex, France Email: [email protected] b Université de Franche-Comté, Institut UTINAM, UMR CNRS 6213, 16 route de Gray, 25030 Besançon Cedex, France Received 27 June 2008; revised 06 January 2009; accepted 09 February 2009 Abstract In this work, we used the SEDE (Steric, Electric and Dielectric Exclusion) model to investigate the rejection rate of asymmetric electrolytes by nanofiltration membranes. The SEDE model predicts that the rejection rate of asymmetric electrolytes with divalent counter-ions is a non-monotonous function of the volume charge density. Because the Donnan exclusion screens the interaction between ions and their images, it was shown that there is a range of membrane volume charge densities for which the increase in the Donnan exclusion is overcompensated by the strong decrease in the interaction between the ions and their images. As a result, the rejection rate decreases even if the membrane volume charge density increases. At very high membrane charge densities, the image charge interaction vanishes and the rejection rate tends to the value predicted by the Donnan theory. The separation per- formances of two NF polyamide membranes were investigated. It was shown that they cannot be described without taking into account the dielectric exclusion. *Corresponding author. Presented at the conference Engineering with Membranes 2008; Membrane Processes: Development, Monitoring and Modelling From the Nano to the Macro Scale (EWM 2008), May 2528, 2008, Vale do Lobo, Algarve, Portugal. 1. Introduction Nanofiltration (NF) membranes are made of organic or ceramic materials the most of which develop a surface electric charge when brought into contact with a polar medium. The generation of a fixed charge onto the membrane surface results from various phenomena including sur- face site dissociation and/or adsorption of charged species [1,2]. NF membranes have a molecular weight cut- off ranging from a few hundreds to a few thou- sands Dalton, i.e. intermediate between reverse osmosis and ultrafiltration membranes. The Desalination 245 (2009) 3 96 407 doi:10.1016/j.desal.2009.02.003 0011-9164/09/$– See front matter © 2009 Elsevier B.V. All rights reserved.

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Page 1: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

Modelling the transport of asymmetric electrolytes throughnanofiltratio n membranes

Anthony Szymczyka,*, Yannick Lanterib, Patrick Fievetb

aUniversité Rennes 1, UMR 6226 CNRS-UR1-ENSCR, équipe Chimie et Ingénierie des Procédés, 263 Avenuedu Général Leclerc, Bâtiment 10 A, CS 74205, 35042 Rennes Cedex, France

Email: [email protected]é de Franche-Comté, Institut UTINAM, UMR CNRS 6213, 16 route de Gray, 25030 Besançon Cedex,

France

Received 27 June 2008; revised 06 January 2009; accepted 09 February 2009

Abstract

In this work, we used the SEDE (Steric, Electric and Dielectric Exclusion) model to investigate the rejectionrate of asymmetric electrolytes by nanofiltration membranes. The SEDE model predicts that the rejection rate ofasymmetric electrolytes with divalent counter-ions is a non-monotonous function of the volume charge density.Because the Donnan exclusion screens the interaction between ions and their images, it was shown that there is arange of membrane volume charge densities for which the increase in the Donnan exclusion is overcompensatedby the strong decrease in the interaction between the ions and their images. As a result, the rejection rate decreaseseven if the membrane volume charge density increases. At very high membrane charge densities, the image chargeinteraction vanishes and the rejection rate tends to the value predicted by the Donnan theory. The separation per-formances of two NF polyamide membranes were investigated. It was shown that they cannot be described withouttaking into account the dielectric exclusion.

*Corresponding author.

Presented at the conference Engineering with Membranes 2008; Membrane Processes: Development, Monitoring andModelling – From the Nano to the Macro Scale – (EWM 2008), May 25–28, 2008, Vale do Lobo, Algarve, Portugal.

1. Introduction

Nanofiltration (NF) membranes are made of

organic or ceramic materials the most of which

develop a surface electric charge when brought

into contact with a polar medium. The generation

of a fixed charge onto the membrane surface

results from various phenomena including sur-

face site dissociation and/or adsorption of

charged species [1,2].

NF membranes have a molecular weight cut-

off ranging from a few hundreds to a few thou-

sands Dalton, i.e. intermediate between reverse

osmosis and ultrafiltration membranes. The

Desalination 245 (2009) 3 –96 407

doi:10.1016/j.desal.2009.02.003

0011-9164/09/$– See front matter © 2009 Elsevier B.V. All rights reserved.

Page 2: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

combination of pore sizes around a few nanome-

ters with electrically charged materials makes

the prediction of NF membrane performances

extremely difficult since the separation of

solutes results from a complex mechanism

including steric hindrance as well as Donnan

and dielectric effects. These dielectric effects

can be split in two contributions. The first one is

connected to the lowering of the dielectric con-

stant of a fluid trapped in nanodimensional cav-

ities [3] and is referred as the Born effect. It

corresponds to the variation of the solvation

energy of an ion when it is transferred from the

bulk solution to the pore inside. If the effective

dielectric constant of the solution confined

inside pores is lower than the dielectric constant

of the bulk solution, the excess solvation energy

is positive and ions are rejected by the mem-

brane pores. The second kind of dielectric effect

arises because of the difference between the

effective dielectric constant of the solution

inside pores and the dielectric constant of the

membrane itself. Because of the dielectric dis-

continuity at the pore surface, the electric field

generated by an ion close to the surface polar-

izes the surface, and the induced polarization

charge interacts with the ion. This phenomenon

is usually referred as the interaction between

ions and their images [4].

The standard theory of NF is based on a

macroscopic description of transport which is

actually a simplified version of the so-called

Space Charge model [5]. The solute separation

process is described in three steps. The first one

is the solute partitioning at the interface between

the feed solution and the pore inlet. Next, the

transport of solutes through the membrane pores

is considered. Finally, the last step is the parti-

tioning of solutes at the interface between the

membrane and the permeate. During the last

decade, several transport models have been

developed which mainly differ from each other

by the equation used to describe the partitioning

of solutes at the membrane/solution interfaces.

We recently proposed the SEDE (Steric, Electric

and Dielectric Exclusion) model, as a tool of

investigation of transport properties of NF mem-

branes [6,7]. Within the scope of the SEDE

model, the partitioning coefficient of an ion is

expressed by means of a modified Donnan equa-

tion that takes into account the steric hindrance,

the Donnan exclusion, i.e. the electrostatic inter-

action between ions and the membrane fixed

charge, and the so-called dielectric exclusion

(DE) which is expressed in terms of Born effect

and image charges. In this work, we used the

SEDE model with the aim of investigating the

contribution of Donnan and non-Donnan exclu-

sion mechanisms in the nanofiltration of an asym-

metric electrolyte (calcium chloride) in aqueous

solution.

2. Theoretical background

The SEDE model has been described in

details in previous works [6,7]. The governing

equations of the model are collected in Tables 1

and 2. The distribution of ions at the feed solu-

tion/membrane (0+|0–) and the membrane/perme-

ate (Δx–|Δx+) interfaces is described by Eqs. (1a)

and (1b), respectively, which are modified Don-

nan relations including steric hindrance and DE.

Within the scope of the SEDE model, the Born

effect is described by Eq. (4) which are modified

Born equations that consider the radius of the

cavity formed by the ion i in the solvent (ri,cav).

This latter was estimated according to the proce-

dure proposed by Rashin and Honig [8]. The

interaction between the ions and the induced

polarization charges is described by Eq. (5) that

were first derived by Yaroshchuk [9]. The solute

transport through the membrane is described by

means of the extended Nernst–Planck equation

that considers diffusion, electromigration and

convection (see Table 2, Eq. (10)). This equation

can be rewritten so as to establish the expression

of the concentration gradient inside pores, i.e. for

0+ ≤ x ≤ Δx– (see Eq. (11)). The expression of the

A. Szymczyk et al. / Desalination 24 (2009) –5 396 407 397

Page 3: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

A. Szymczyk et al. / Desalination 245 (2009) 396–407

Table 1Partitioning equations used in the SEDE model

Partitioning equations at the membrane/solution interfaces

(1a)

(1b)

with,

(slit-like pores) (2)

(3a)

(3b)

(4)

(slit-like pores) (5a)

(slit-like pores) (5b)

(6)

(7a)

c

cz Wi

ii

i

ii i

0

0

0

0 0 0

+

+

( )( ) =

( )( ) −( ) −( )+ −φ

γ

γexp exp

( ) ,'Δ ΔΨ Born eexp ', ( )

−( )+ −ΔWi im 0 0

μ κ

φγγ

0 0

2

0 0

0

00

0+ −

+ −

( )−

−−

+ ( )= ( )

( ) ( )( ) − −

rz c z Wi i i

i

ii i

p

exp ' ,Δ ΔΨ BBorn im−⎛

⎝⎜⎞⎠⎟

( )+ −( )

−∑ΔW

I

i

i

', 0 0

2 0

απε εi

iz FRTN r

= ( )2

08 p A p

ΔΔ Δ Δ Δ

Wi x x i x x' ln exp

,im

p m

p m

− + − +( ) ( )= − −−+

⎝⎜⎜

⎠⎟⎟ −⎛

⎝⎜α

ε εε ε

μ1 2⎞⎞⎠⎟

⎣⎢⎢

⎦⎥⎥

ΔWi i' ln exp,im

p m

p m0 0 0 0

1 2+ − + −( ) ( )= − −−+

⎝⎜⎜

⎠⎟⎟ −⎛

⎝⎜⎞⎠⎟

⎡α

ε εε ε

μ⎣⎣⎢⎢

⎦⎥⎥

ΔWz ekTrii

i

' ,

,

Born

cav p b

= ( ) −⎛

⎝⎜⎜

⎠⎟⎟

2

08

1 1

πε ε ε

Δ ΔΔ Δ Δ Δ

Ψx x D x x

ekT− + − +( ) ( )= ψ

Δ ΔΨ0 0 0 0+ − + −( ) ( )= e

kT Dψ

φiir

r= −

⎝⎜⎜

⎠⎟⎟1

,Stokes

p

c x

c x

x

xz Wi

ii

i

ii x x i

Δ

Δ

Δ

ΔΔ Δ

Δ Δ

+

+

( )( ) =

( )( ) −( ) −− +φ

γ

γexp exp '

( ) ,Ψ BBorn im( ) −( )− +exp '

, ( )Δ

Δ ΔW

i x x

398

Page 4: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

electric potential gradient (Eq. (12)) is derived

from Eq. (10) and the condition of zero electric

current flowing through the membrane at the

steady state (Eq. (9)). The electroneutrality con-

dition in the external bulk solutions is given by

Eqs. (8a) and (8b). The electroneutrality inside

pores is expressed by Eq. (8c), where X is the

membrane volume charge density which can be

approximately estimated from tangential stream-

ing potential measurements. Solving transport

equations and partitioning equations allows com-

puting the rejection rate of a solute i (Ri), which

is defined as:

(15)

3. Results and discussion

Fig. 1 shows the variation of the intrinsic

rejection rate of a calcium chloride solution ver-

sus the membrane volume charge density. Rejec-

tion rates were computed out by considering

(solid line) or not considering (dashed curve) the

contribution of image charges. The membrane

was characterized by a dielectric constant εm of 3

which is representative of NF organic mem-

branes. As can be seen, the rejection rates com-

puted from the two approaches differ

significantly over a large range of charge densi-

ties for negatively charged membranes. This

makes electrolytes with divalent counter-ions

very good candidates to put in evidence dielectric

effects in NF. Fig. 2 focuses on these electrolytes

with divalent counter-ions, keeping the example

of calcium chloride. At very low charge densities,

the rejection rate first decreases as more fixed

charges are brought onto the pore walls. This first

decrease is not related to any dielectric effect

since it also occurs when no image charge is

included in the calculations (see dashed line in

Fig. 2). This phenomenon occurs when co-ions

are more mobile than counter-ions. It occurs at

low charge densities because of the interplay

between the co-ion exclusion and the acceleration

of these co-ions by the electric field arising

through the pores to ensure the electroneutrality

R xi = −

+

−10

cci

i

( )

( )

Δ

A. Szymczyk et al. / Desalination 24 (2009) –

Table 1 (continued)

(7b)

Electroneutrality conditions

(8a)

(8b)

for 0+ ≤ x ≤ Δx– (8c)

μ κ

φγγ

Δ Δ

Δ Δ

Δ

ΔΔ

ΔΔ

x x

i i ii

ii x x

x rz c x

xx

z

− +

− +

( )+

++

= ( )( ) ( )

( ) −

p

2 exp Ψ(( ) ( )+

− −⎛⎝⎜

⎞⎠⎟

( )− +

∑Δ Δ

Δ

Δ ΔW W

I x

i i x x

i

' ', ,Born im

2

z c x Xi ii

( ) + =∑ 0

z c xi ii

Δ +( ) =∑ 0

z ci ii

0 0−( ) =∑

5 396 407 399

Page 5: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

[11,12]. The phenomenon disappears as the mem-

brane charge increases and the rejection rate

increases with the membrane volume charge den-

sity. For higher charge densities the rejection rate

decreases again. This behaviour is brought about

by the combination between the Donnan exclu-

sion and the interaction between ions and their

images. This can be approximately shown in Fig.

3 which presents the variation of the co-ion par-

titioning coefficient due to image charges versus

the co-ion partitioning coefficient due to the Don-

nan exclusion. In this figure, the membrane vol-

ume charge density increases from the right to the

left. It can be seen that the interaction with image

charges decreases (i.e. the co-ion partitioning

coefficient due to image charges increases) as the

Donnan exclusion increases (i.e. the co-ion parti-

tioning coefficient due to the Donnan exclusion

decreases). Otherwise stated, the Donnan exclu-

sion screens the image forces. In Fig. 3, the

dashed line corresponds to the hypothetic case for

which the increase in the Donnan exclusion

would be exactly balanced by the decrease in the

DE, so that the rejection rate would be constant

whatever the membrane charge. Let us compare

this hypothetic case with the real one which is

A. Szymczyk et al. / Desalination 24 (2009) –

Table 2Transport equations used in the SEDE model

Zero electric current condition (steady state)

(9)

Transport equations

(10)

Concentration gradients inside pores

(11)

Electrical potential gradient inside pores

(12)

with [10],

(slit-like pores) (13)

(slit-like pores) (14)

F z ji ii

∑ = 0

Ki ii i i

i,

. . .

. .c =

− + −

+ −−φ

λ λ λ

λ λ1

2 3 4

5

1 3 02 5 776 12 3675

18 9775 15 2185 ii i6 74 8525+

⎝⎜⎜

⎠⎟⎟. λ

Ki ii i i

i i

,

ln .

. . .d =

+ −

+ − +

−φλ λ λ

λ λ

1

3 4

19

161 19358

0 4285 0 3192 0 08428λλi5

⎜⎜⎜

⎟⎟⎟

ddx

z K D dcdx

JA

z K c

FRT

z c K D

i i d ii V

ki

ii c i

i

i i i d ii

ψ =− +∞

∑∑

, , ,

, ,

2

dcdx

JK D A

K c c x z FcRT

ddx

i V

i d i ki c i i

i i= −( ) −∞

+

, ,

, ( )Δ ψ

j K D dcdx

z c K D FRT

ddx

K cVJ c x

Ai i d ii i i i d i

i c iV i

k

= − − + =( )

∞∞

+

, ,

, ,

,

ψ Δ

5 396 407400

Page 6: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

A. Szymczyk et al. / Desalination 24 (2009) –

0.0

0.2

0.4

0.6

0.8

1.0

−500 −375 −250 −125 0 125 250

X (eq/m3)

R intWith images

Without images

Fig. 1. Intrinsic rejection rate (Rint

) vs. membrane volume charge density (X); 0.0005 M CaCl2

solution; rp

= 0.5 nm;

Δx/Ak = 5 μm; εm = 3; JV = 10–5 m/s; no Born effect.

0.0

0.2

0.4

0.6

0.8

1.0

1 × 10–3 1 × 10–2 1 × 10–1 0 1 × 101 1 × 102 1 × 103

−X (eq/m3)

Rin

t

With image charges - no BorneffectNo dielectric exclusion

Fig. 2. Variation of the intrinsic rejection rate (Rint

) of negatively charged membranes with the membrane volume charge

density (X); 0.0005 M CaCl2

solution; rp

= 0.5 nm; Δx/Ak = 5 μm; εm = 3; JV = 10–5 m/s.

5 396 407 401

Page 7: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

A. Szymczyk et al. / Desalination 245 (2009) 396–407

0

1 × 10–1

4 × 10–1

6 × 10–1

8 × 10–1

1 × 100

0 5 × 10–2 1 × 10–1 2 × 10–1

exp(−zCl-ΔΨD)

exp(

–ΔW

’ im, C

l–)

Increasing Donnan exclusion

Uncharged membraneDecreasing

image exclusion

X (eq/m3) −1000 −60 −0.6 −0.03 0

Fig. 3. Image charge partitioning coefficient vs. Donnan partitioning coefficient for co-ions at the feed/membrane inter-

face; rp

= 0.5 nm; εm = 3; no Born effect; solid line: 0.0005 M CaCl2

solution; dashed line: hypothetic case for which the

increase in the Donnan exclusion would be exactly balanced by the decrease in the image charge contribution.

represented by the solid line. For low charge den-

sities, the solid line is under the dashed line,

which means that the increase in the Donnan

exclusion is not balanced by the decrease in the

DE. This partitioning effect therefore tends to

increase the membrane rejection rate. It must be

stressed that Fig. 3 cannot explain the first

decrease of the rejection rate observed in Fig. 2 at

very low membrane charge densities. The reason

is that the argument based on the results shown

in Fig. 3 takes into account only the partitioning

effect at the membrane/solution interface and it

totally disregards the transport phenomena

through the membrane (i.e. the influence of the

induced transmembrane electric field on the

rejection of charged solutes). Nevertheless, Fig. 3

is helpful to explain the behaviour of more

strongly charged membranes. Indeed, for higher

charge densities, it can be seen that there is a

sharp increase of the solid line which goes over

the dashed line. It means that there is a range of

charge densities for which the increase in the

Donnan exclusion is overcompensated by the

decrease in the DE and so, the rejection rate

decreases even if the membrane charge increases

(as can be seen in Fig. 2). For even more strongly

charged membranes, the situation is reversed (i.e.

the increase in the Donnan exclusion is no more

balanced by the decrease in the DE) and the rejec-

tion rate is expected to be an increasing function

of the membrane charge. This is in agreement

with the results shown in Fig. 2. It is worth men-

tioning that this situation is characterized by par-

titioning coefficients due to image charges close

402

Page 8: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

A. Szymczyk et al. / Desalination 245 (2009) 396–407

0

0.2

0.4

0.6

0.8

1.0

−X (eq/m3)

Rin

t

1 × 10–3 1 × 10–2 1 × 10–1 1 × 100 1 × 101 1 × 102 1 × 103

Fig. 4. Influence of the Born effect on the intrinsic rejection rate (Rint

); 0.0005 M CaCl2

solution; rp

= 1 nm; Δx/Ak = 5

μm; εm = 3; JV = 10–5 m/s; (–): εp = 40; (×): εp = 50; (○): εp

= 60; (▲): εp = 70; (�♦): no Born effect.

to 1 (see Fig. 3). Otherwise stated, the interaction

with image charges is totally screened by the

Donnan exclusion and so, the rejection rate is

expected to tend to the value that is predicted by

the standard Donnan theory (see Fig. 2).

According to Eq. (4), the Born effect (i.e. the

work required to transfer an ion from the bulk

solution to the pore inside) is stronger as the

effective dielectric constant inside pores (εp)

decreases. This leads to higher rejection rates (see

Fig. 4) because of the stronger exclusion of both

co-ions and counter-ions from the membrane

pores (as shown by Eq. (4), the Born effect varies

with the square of the ion charge and so, it expels

both cations and anions from the membrane pores

unlike the Donnan exclusion).

Fig. 5 shows the rejection rate of a 0.0005 M

calcium chloride solution by a polyamide mem-

brane, labelled AFC 40 (PCI Membrane Sys-

tems). The mean pore size of the membrane as

well as its thickness to porosity ratio were esti-

mated by fitting the experimental rejection of

neutral solutes according to the conventional pro-

cedure described in the literature [13,14]. Con-

sidering slit-like pores, the pore half-width (rp)

and the thickness to porosity ratio (Δx/Ak) was

found to be 0.54 nm and 7.8 μm, respectively.

The membrane volume charge density, X, was

inferred from tangential streaming potential

measurements by Szymczyk et al. [15]. This elec-

trokinetic characterization indicated that the

membrane is negatively charged in the calcium

chloride solution (X = –41.3 eq/m3). Both struc-

tural and electrical properties of the membrane

were first used to compute the theoretical rejec-

tion rate by neglecting dielectric effects (i.e. using

403

Page 9: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

A. Szymczyk et al. / Desalination 245 (2009) 396–407

0

0.25

0.5

0.75

1

0 5 × 10–6 1 × 10–5 1.5 × 10–5 2 × 10–5 2.5 × 10–5 3 × 10–5

Jv (m s−1)

Rin

tt Experimental

SEDE model

Model withoutDE

Model: best fitwith X < 0 andwithout DE

Model: best fitwith X > 0 andwithout DE

Xfit = + 2 eq/m3

Xfit = − 650 eq/m3

Fig. 5. Experimental and theoretical intrinsic rejection rates of the AFC 40 membrane vs. permeate volume flux (JV);

0.0005 M CaCl2

solution; rp

= 0.54 nm; Δx/Ak = 7.8 = μm; εm = 3; X = –41.3 eq/m3 [15].

the standard Donnan theory). As can be seen in

Fig. 5, the rejection rate of calcium chloride is

highly underestimated when dielectric effects are

disregarded whereas a satisfying description of

the experimental data is obtained from the SEDE

model (i.e. by taking into account both image

charges and Born effect). The results of the SEDE

model shown in Fig. 5 were obtained by decreas-

ing the dielectric constant inside pores down to a

value of 69. We also tried to fit the experimental

data by neglecting dielectric effects but using the

volume charge density as a fitting parameter

(instead of using the approximate value inferred

from electrokinetic measurements). It can be

noted from Fig. 5 that a fair description of the

experimental data can be obtained by using either

a positive or a negative membrane charge density.

Of course, the positive value of X is not relevant

since it is in contradiction with the experimental

measurements. The best fit obtained by consider-

ing a negatively charged membrane led to X = –

650 eq/m3. Tangential streaming potential

measurements led to a much smaller value (X = –

41.3 eq/m3). It is true that tangential streaming

potential measurements can only provide a rough

estimate of the real membrane volume charge

density inside pores. But because of phenomena

like the so-called charge regulation process [16],

the true charge density inside pores is expected

to be lower than the value inferred from tangen-

tial streaming potential measurements, which can

be viewed as the upper limit of the real membrane

charge density.

Similar qualitative results were obtained with

another polyamide membrane (AFC 30 from PCI

Membrane Systems) in a more concentrate (0.01

M) calcium chloride solution. At this concentra-

tion, the concentration polarization was found to

404

Page 10: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

be non-negligible as shown in Fig. 6. It was taken

into account within the scope of the standard film

theory according to the procedure described in

[17]. Structural features (i.e. rp

and Δx/Ak) of the

membrane sample were determined by Bouranene

et al. [18] and a negative volume charge density (X= –50 eq/m3) was determined from tangential

streaming potential measurements. The results of

the SEDE model shown in Fig. 6 were obtained

from the experimental membrane volume charge

density and a fitted value of the dielectric constant

inside pores equal to 70. It can be noted that the

description of the experimental data without tak-

ing into account the DE requires a huge and phys-

ically unrealistic (fitted) membrane charge density

(around –9000 eq/m3) unless a positive charge is

considered (but this latter is in contradiction with

the negative charge density obtained from electro-

kinetic measurements).

Consequently, it can be concluded that the

DE plays a significant role in the separation

performances of both the AFC 30 and AFC 40

membranes.

4. Conclusion

The SEDE (Steric, Electric and Dielectric

Exclusion) model was used to investigate the

contribution of Donnan and non-Donnan exclu-

sion mechanisms in the nanofiltration of asym-

metric electrolytes. The SEDE model is a 1-D

transport model that considers the separation of

solutes as being the result of both interfacial phe-

nomena (including steric hindrance, Donnan

exclusion and dielectric exclusion through both

the Born effect and the image charges) and trans-

port effects described by means of the so-called

extended Nernst–Planck equations. In the case of

asymmetric electrolytes with divalent counter-

ions, the SEDE model predicts that the rejection

rate is a non-monotonous function of the volume

charge density. It was shown that there is a range

A. Szymczyk et al. / Desalination 245 (2009) 396–407

0

0.25

0.5

0.75

1

0 5 × 10–6 1 × 10–5 1.5 × 10–5 2 × 10–5 2.5 × 10–5 3 × 10–5

Jv (m s-1)

Rej

ecti

on

Experimental: observed rejection rate

Experimental: intrinsic rejection rate

SEDE model

Model: best fit with X < 0 and without DE

Model: best fit with X > 0 and without DEXfit = +34 eq/m3

Xfit = -9020 eq/m3

Fig. 6. Experimental and theoretical intrinsic rejection rates of the AFC 30 membrane vs. permeate volume flux (JV);

0.01 M CaCl2

solution; rp

= 0.43 nm [17]; Δx/Ak

= 4.2 μm [17]; εm

= 3; X = –50 eq/m3.

405

Page 11: Modelling the transport of asymmetric electrolytes through nanofiltration membranes

of membrane volume charge densities for which

the increase in the Donnan exclusion is overcom-

pensated by the strong decrease in the interaction

between ions and their images, which leads to a

decrease in the rejection rate although more fixed

charges are brought onto the pore walls. It was

also shown that the separation performances of

two NF polyamide membranes (AFC 30 and AFC

40) cannot be described by neglecting the dielec-

tric exclusion.

Nomenclature

Ak porosity of the membrane active layer

ci concentration of ion i

Di,∞ bulk diffusion coefficient of ion i at

infinite dilution

e elementary charge

F Faraday constant

I ionic strength

ji molar flux density of ion i

JV permeate volume flux

k Boltzmann constant

Ki,c hydrodynamic coefficient accounting

for the effect of pore walls on convec-

tive transport

Ki,d hydrodynamic coefficient for hindered

diffusion inside pores

NA

Avogadro number

ri, cavcavity radius of ion i

ri, StokesStokes radius of ion i

rp

pore half-width

R ideal gas constant

Ri rejection rate of ion i

T temperature

V fluid velocity inside pores

x axial coordinate

X effective volume charge density inside

the membrane pores

zi charge number of ion i

Greek symbols

ΔΨ dimensionless Donnan potential

Δx effective thickness of the active layer

ΔW’i,Borndimensionless excess solvation energy

due to Born effect for ion i

ΔW’i,im dimensionless excess solvation energy

due to “image charges” for ion i

ε0

vacuum permittivity

εb dielectric constant of the bulk solution

outside pores

εm dielectric constant of the membrane

active layer

εp effective dielectric constant inside

pores

φi steric partitioning coefficient for ion i

γi activity coefficient for ion i

�κ: Debye parameter

μ: effective dimensionless reciprocal

screening length for interaction

between ions and induced polarization

charges

ψ local electrical potential inside pore

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