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Electrolyte transport through amphoteric nanofiltration membranes
Chemical Engineering Science 57 (2002) 2921 – 2931
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Electrolyte transport through amphoteric nano"ltration membranes P. Fievet ∗ , C. Labbez, A. Szymczyk, A. Vidonne, A. Foissy, J. Pagetti Laboratoire de Chimie des Materiaux et Interfaces, 16 route de Gray, 25030 Besancon Cedex, France Received 19 December 2001; received in revised form 16 May 2002; accepted 20 May 2002
Abstract The electrolyte transport through amphoteric nano"ltration membranes is investigated using a model based on the application of the extended Nernst–Planck equation coupled with the electroneutrality condition in the membrane, and the assumption of a Donnan equilibrium at both membrane=solution interfaces. The model gives the possibility to distinguish between the di8erent transport mechanisms, namely, convection, di8usion and electromigration. The in9uence of ion valence, ion di8usion coe:cient and permeate volume 9ux on the di8erent transport mechanisms and electrolyte retention is presented. All calculations are carried out for a membrane with "xed structural parameters (e8ective pore radius of 2 nm and e8ective thickness=porosity ratio of 7 m). ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Donnan partitioning; Mass transfer; Membranes; Nano"ltration; Simulation; Transport processes
1. Introduction Nano"ltration (NF) membranes have intermediate molecular weight cut-o8 (MWCO) between ultra"ltration and reverse osmosis membranes. Their e8ective pore diameters range from ∼ 1 nm to a few nm (Wang, Tsuru, Nakao, & Kimura, 1995). Another feature of these membranes is that most of them are electrically charged in aqueous media due to their materials or adsorption of charged species. Hence, their separation mechanisms involve not only the ordinary “sieve e8ect” but also the “charge e8ect”. Therefore, they can reject charged solutes of much smaller size than the dimensions of the pores. This charge e8ect can be particularly used to remove ions from wastewater (Bowen & Mohammad, 1998a; Katselnik & Morcos, 1998) or whey (Alkhatim et al., 1998), and also separate ions according to their ionic valences (Eriksson, Lien, & Green, 1996). Many types of NF membranes are currently commercially available but all are made of an organic material. Despite of their numerous advantages compared to organic membranes (higher chemical, thermal and mechanical stability), no ceramic nano"lters (MWCO ¡ 1000 Da) can be found in the market now. This situation is due in part to the inherent practical di:culties that need to be overcome in order to prepare lightly sintered granular porous membranes with ∗ Corresponding author. Tel.: +33-381-66-2032; fax: +33-381-66-2033. E-mail address: patrick."[email protected] (P. Fievet).
grains sizes in the 5 –10 nm range and to a highly connected porosity (Palmeri, Blanc, Larbot, & David, 1999, 2000). However, for some years, researchers succeed in producing ceramic NF membranes (Baticle et al., 1997; Benfer, Popp, Richter, Siewert, & Tomandl, 2001; Palmeri et al., 2000). An interesting feature of ceramic membranes is their amphoteric character which allows to control the sign and the charge density of the membrane by pH control. As to the fundamentals of the nano"ltration process, even if the separation mechanisms are not yet well known, major progress has been made since the early 1990s due to the increasing development of new applications. For the mathematical description of the mass transfer process in NF, models that are commonly used are: Theorell–Meyer–Sievers model (Wang et al., 1995), Space-Charge model (Wang et al., 1995), Hybrid model (Bowen & Mukhtar, 1996; Bowen, Mohammad, & Hilal, 1997) and Donnan–Steric partitioning Pore model (DSPM) (Bowen & Mukhtar, 1996; Bowen et al., 1997; Schaep, 1999; Schaep, Vandecasteele, Mohammad, & Bowen, 1999, 2001). This latter has particularly proven to be useful in describing the retention properties of nano"ltration membranes (Bowen & Mohammad, 1998a, b; Schaep, 1999; Schaep et al., 1999, 2001). Moreover, it has proven to be realistic: a very good agreement between predictions and experimental data of retention for mixed NaCl=Na2 SO4 solutions was obtained by Bowen and Mukhtar (1996). It was also successful in predicting the performance of the CA 30 NF membrane for a dye=salt solution (Bowen & Mohammad, 1998a, b).
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 1 8 8 - 4
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In this model, the solute transport is described as occurring through discrete pores and size e8ects are taken into account by incorporating steric hindrance factors for di8usion and convection. This model is based on the application of the Extended Nernst–Planck (ENP) equation to describe the electrolyte transport through the membrane. ENP equation is coupled to the electroneutrality condition inside the membrane, and the assumption of the Donnan equilibrium at both membrane-solution interfaces (Bowen & Mukhtar, 1996; Bowen et al., 1997; Schaep et al., 1999, 2001). In other words, that means that the transfer of charged solutes is controlled by both electrostatic interactions occurring between the charged solute and the charged membrane surface, and transport phenomena in the membrane: convection (pressure gradient), di8usion (concentration gradient) and electromigration (electric potential gradient). The DSPM model provides a description of separation properties in terms of three key parameters: the e8ective pore radius (rp ), the e8ective thickness=porosity ratio (Mx=Ak ) and the e8ective charge density (X ). Thus, it allows to characterise the structural and electrical properties of NF membranes from experimental data of retention (Bowen et al., 1997; Bowen & Mukhtar, 1996; Bowen & Mohammad, 1998a; Schaep, 1999; Schaep et al., 1999, 2001). Some theoretical studies with the DSPM model have been presented in the literature. Most of them deals with the in9uence of the model input parameters (rp ; Mx=Ak and X ), permeate volume 9ux, electrolyte feed concentration, ion valence and ion size on electrolyte retention (Bowen & Mohammad, 1998a, b Schaep, 1999; Schaep et al., 2001), but no studies have been focused on the in9uence of these parameters on the di8erent transport mechanisms—convection, diffusion and electromigration—through nano"ltration membranes. The aim of this paper is then to study the in9uence of some parameters (sign and charge density of the membrane, permeate volume 9ux, valence and di8usion coe:cient of ions) on the three transport mechanisms—convection, diffusion and electromigration—for single "ctitious electrolyte solutions and a membrane with a pore radius "xed at 2 nm. This value corresponds approximatively to the smallest pore radius of currently commercially available ceramic membranes. So, we aim to investigate the electrolyte transport through amphoteric membranes where electrostatic interactions between the membrane material and charged species in solution play important role in separation process in comparison with the size exclusion.
Table 1 Summary of equations used for modeling. Equations for the Donnan– Steric–Pore (DSPM) model Ji = −Ki; d Di Ji (D) =
dx
−
zi cim Ki; d Di RT
dcm −Ki; d Di d xi ,
Ji (EM) = Ji (C) =
dcim
z cm K D − i i RTi; d i
F
d m dx
F
d m dx
+ Ki; c cim Jv ,
(2a) (2b)
,
(2c)
Ki; c cim Jv ,
(2d)
Ki; d = K −1 (; 0),
(3)
Ki; c = (2 − ) G(; 0),
(4)
with K −1 (; 0) = 1:0 − 2:30 + 1:1542 + 0:2243 ,
(5)
G(; 0) = 1:0 + 0:054 − 0:9882 + 0:4413 ,
(6)
= (1 − )2 ,
(7)
= n
rs , rp
(8)
zi ci = 0,
(9)
zi cim = −X , n Ic = i=1 Fzi Ji = 0,
(10)
i=1
n
i=l
dcim dx
=
d m dx
=
i; d i F n i=1 RT
(zi2 cim )
at x =
0−
at x =
Mx+
ci = ci; p ,
= exp
iF − zRT M D
cim ci
(11)
zi cim − K JvD (Ki; c cim − ci; p ) − RT i; d i n zi Jv m i=1 K D (Ki; c ci −ci; p )
R=1−
ci = ci; f ,
ci; p . ci; f
,
F
d m , dx
(12) (13) (14a) (14b)
,
(15) (16)
For the mathematical derivation of the DSPM model, the following assumptions are adopted: • The membrane consists of a bundle of identical straight cylindrical pores of radius rp and length Mx (with Mx rp ). • The e8ective membrane charge density (X ) is constant through the membrane. • The ion size is expressed as a Stokes radius (rs ) and is calculated from the di8usion coe:cient (Di ), with the Stokes–Einstein equation: kT : (1) rs = 6!"Di
2. Theory
• The ion 9uxes (Ji ), ion concentrations (ci ), electric potential ( ) and permeate volume 9ux (Jv ) are all de"ned in terms of radially averaged quantities. • The volume 9ux is fully developed inside the pore and has a parabolic pro"le of the Hagen–Poiseuille type.
The theory of DSPM model (Donnan and Steric partitioning Pore model) has been presented in detail by Bowen et al. (Bowen et al., 1997; Bowen & Mukhtar, 1996) and Schaep et al. (Schaep, 1999; Schaep et al., 1999). That is why only a brief description of the model will be made in this section.
The main equations used for the DSPM model are summarized in Table 1 (All symbols are de"ned in the “Notation”). As mentioned earlier, this model is based on the ENP equation to describe the ionic transport through the membrane (Eq. (2a)). The terms on the right-hand side of
P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
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Charged membrane Jco-ions (C) Jcounter-ions (C) Jco-ions (D) feed
Jcounter-ions (D)
permeate
Jco-ions (EM) Jcounter-ions (EM) Fig. 1. Schematic representation of co-ions and counter-ions 9uxes due to convection (C), di8usion (D) and electromigration (EM) through a charged membrane in the case of a positive retention.
this equation represent transport due to di8usion (D), electromigration (EM) and convection (C), respectively. The hindered nature of ion transport inside pores is taken into account by using steric hindrance factors (Eqs. (3) and (4)). These are a function of , the ratio of the ion radius (rs ) to the e8ective pore radius (rp ) (Eq. (8)). The conditions for electroneutrality in the bulk solution and inside the membrane are given by Eqs. (9) and (10), respectively. The requirement for zero overall electrical current passing through the membrane is expressed by Eq. (11). The concentration and electrical potential gradients (Eqs. (12) and (13), respectively) that can be obtained by rearranging Eq. (2a), are solved by taking the boundary conditions given by Eq. (14) together with the equations for electroneutrality (Eqs. (9) and (10)). Partitioning of ions at the interfaces membrane–feed solution and membrane– permeate is governed by both Donnan and steric e8ects (Eq. (15)). Retention is given by Eq. (16)). The model gives the possibility to distinguish between the di8erent transport mechanisms (C; D and EM). The in9uence of parameters such as surface charge density (X ), permeate volume 9ux (Jv ), di8usion coe:cient (Di ) and ion valence (zi ) on the contribution of each transport mechanism to the overall electrolyte transport can be then computed. Fig. 1 provides a schematic representation of the convective, di8usive and electromigrative 9uxes through a charged membrane for both co-ions and counter-ions when retention is positive. It can be noted that in this case, only the electromigrative 9ux of counter-ions (Jcounter−ions (EM)) is oriented towards the feed solution. For a negative retention (permeate more concentrated than the feed), the two di8usive 9uxes would be additionally oriented towards the feed solution. For any electrolyte, the sum of the three co-ions 9uxes, Jco−ions , is equal to the sum of the three counter-ions 9uxes, Jcounter−ions (if 9uxes are expressed in eq s−1 m−2 ) and represents the overall electrolyte transport through the membrane.
Because co-ions are excluded from the membrane and counter-ions are attracted (due to the Donnan e8ect), the concentration of co-ions in the membrane is lower than that for counter-ions. Consequently, calculations of the contribution of each transport mechanism (C; D and EM) to the overall electrolyte transport (from the feed side to the permeate side) will be performed with co-ions that are the limiting species to the overall electrolyte transport through the membrane. For example, the contribution of convection to the overall electrolyte transport is given by the ratio of co-ions 9ux due to convection (Jco−ions (C)) to the sum of co-ions 9uxes due to convection, di8usion and electromigration (Jco−ions (C) + Jco−ions (D) + Jco−ions (EM)). It should be noted that for a negative retention, no co-ion would be transferred by di8usion towards the permeate (Jco−ions (D) ¡ 0). In this case, the contribution of di8usion to the overall electrolyte transport (from the feed side to the permeate side) must be taken equal to zero. 3. Numerical calculation procedure Solving the di8erential equations for concentration and electric potential gradients through the membrane (Eqs. (12) and (13)) allows the determination of the retention for given membrane (rp ; Mx=Ak and X ) and solution (cf ; rs ; Di ; T , etc.) parameters according to the following procedure. First, the concentration of cations and anions at the feed-membrane interface (just inside the membrane) is evaluated by solving Eq. (15), taking into account boundary conditions (Eq. (14)) together with electroneutrality requirements (Eqs. (9) and (10)). Next, the potential gradient over the membrane is determined by means of Eq. (13), considering electroneutrality conditions (Eqs. (9) and (10)) again. To this end, an arbitrary value is chosen for the permeate concentration (cp = c+; p =#+ = c−; p =#− ). Then, by solving Eqs. (12) and (13) by accounting for the zero
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P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
electric current condition (Eq. (11)), concentrations of cations and anions across the membrane are calculated and a new value of the permeate concentration is obtained. This latter is compared with the initial one. If the di8erence between initial and new values (Mcp ) is lower than the stop criterion (Mcp =cp ¡ 0:1%), the numerical calculation is then stopped. Otherwise, a new permeate concentration is set by using a function minimization algorithm for the numerical solution of the di8erential equations (Eqs. (12) and (13)). This procedure is repeated until the stop criterion is satis"ed. The numerical integration of Eqs. (12) and (13), and the function minimization are carried out using a fourth-order Runge–Kutta and a quadratic method, respectively. 4. Results of calculations and discussion All numerical simulations presented in this work have been carried out for an e8ective pore radius of 2 nm (value which corresponds approximatively to the smallest pore radius of commercial ceramic membranes) and an e8ective thickness=porosity ratio (Mx=Ak ) of 7 m. This value corresponds approximatively to the mean value obtained by considering an intergranular porosity in the 25 –50% range (Burggraaf, 1996) and a layer thickness in the 1–3 m range (El Marraki, 2001; Labbez et al., 2002). The electrolyte concentration (Cf ) was "xed at the value of 1 eq m−3 and the permeate volume 9ux at 50 m s−1 (value based on data for a “real” membrane with such structural features (Labbez et al., 2002)). Three "ctitious 1–1 electrolytes with D+ = D− = 10−9 m2 s−1 , D+ ¿ D− (3:10−9 and 10−9 m2 s−1 , respectively) and D+ ¡ D− (10−9 and 3:10−9 m2 s−1 , respectively) have been considered to study the in9uence of ionic di8usion coe:cients. Two "ctitious symmetric (1–1 and 2–2) and asymmetric (1–2 et 2–1) electrolytes having identical ionic di8usion coe:cients (D+ = D− = 10−9 m2 s−1 ) have been used for studying the e8ect of ion valence. 4.1. Transport mechanisms 4.1.1. Ion 6uxes Figs. 2a–c show the variations of the convective, diffusive and electromigrative 9uxes of co- and counter-ions as a function of the normalized membrane charge density (% = X=Cf ) for a 1–1 electrolyte with D+ = D− . Figs. 2a and b show the magnitude of the six 9uxes for 0 6 |%| 6 1 and 0 6 |%| 6 20, respectively. Fig. 2c presents the convective, di8usive and electromigrative 9uxes of co-ions as well as the sum of these 9uxes for 0 6 |%| 6 20. First of all, it can be seen from Figs. 2a and b that C; D and EM 9ux curves are symmetrical for both co- and counter-ions about the isoelectric point (IEP) where % = 0. These symmetries are not unexpected since, even if the roles of co-ions and counter-ions are reversed when the membrane charge changes sign, the
type of co- and counter-ions does not change with changing sign of the membrane charge (D+ = D− and z+ = z− ). As mentioned earlier, it can be seen from Figs. 2a and b that only the electromigrative 9ux of counter-ions is negative, i.e. oriented towards the feed side. At the IEP of the membrane, the following observations can be made (Fig. 2a): • The di8usive 9uxes of cations (J+ (D)) and anions (J− (D)) are very low but di8erent from zero due to the presence of a concentration gradient across the membrane. Indeed, as will be shown later, the retention of the membrane is not zero for % = 0 (R = 2%) due to purely steric e8ects. • Ions are not transported by EM through the membrane (J+ (EM) = J− (EM) = 0) due to the absence of an electrical potential gradient across it. Indeed, di8usion of ionic species through an uncharged membrane can not induce an electrical potential gradient across it when their diffusion coe:cients are identical (Fievet, Aoubiza, Szymczyk, & Pagetti, 1999). • The convective 9uxes of cations (J+ (C)) and anions (J− (C)) largely dominate the di8usive 9uxes, which is in agreement with the high value of the PQeclet number (which measures the importance of convective electrolyte transport over di8usion) in the membrane. As can be seen in Figs. 2a and c, when the membrane charge di8ers from zero (% = 0), EM phenomena then occurs (Ji (EM) = 0). The electrical potential gradient induced through the membrane by convection and di8usion phenomena is due to the excess of counter-ions in the pores. For small values of |%| (|%| ¡ 1), it appears from Fig. 2a that the highest 9uxes (in absolute value) are the convective 9uxes for both counter-ions (Jcounter−ions (C)) and co-ions (Jco−ions (C)), and electromigrative 9ux of counter-ions (Jcounter−ions (EM)). The di8usive 9uxes of co-ions (Jco−ions (D)) and counter-ions (Jcounter−ions (D)) are low due to a low-concentration gradient across the membrane (di8usion is directly proportional to the concentration gradient as expressed by Eq. (2b)). It will be shown later that the retention is less than 10% for |%| ¡ 1. The electromigrative 9ux of co-ions (Jco−ions (EM)) is lower than that for counter-ions due to a lower concentration of co-ions in the membrane pores (electromigration is proportional to both electrical potential gradient and ionic concentration inside the membrane as expressed by Eq. (2c)). For high values of |%| (|%| ¿ 10), counter-ions are in large excess inside the membrane pores in comparison to co-ions. As a result, the convective and electromigrative 9uxes of counter-ions, which are proportional to their concentration as expressed by Eqs. (2c) and (d), are dominant (Fig. 2b). It is important to note that these two 9uxes are quasi-identical in absolute value but in the opposite direction. It can also be noted from Figs. 2a and b that the convective 9ux increases with |%| for counter-ions whereas it
P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
2925
0.01
0.0006
0.008 0.006
0.0004 -1
Ji (eq m s )
0.002
-2
0.0002
-2
-1
Ji (eq m s )
0.004
0 -1
-0.5
0
0.5
1
0 -0.002 -20
-15
-10
-5
0
5
10
15
20
-0.004 -0.006
-0.0002
-0.008
-0.0004
(a)
-0.01
ξ
ξ
(b)
0.0003
-2
-1
Jco-ions (eq m s )
0.0004
0.0002
0.0001
0 0
5
10 ξ
(c)
15
20
Fig. 2. (a–c) Variation of co-ions and counter-ions 9uxes due to convection (C), di8usion (D) and electromigration (EM) with normalized membrane charge density (%); Jv = 50 m s−1 . : Jcounter−ions (C); : Jco−ions (C); : Jcounter−ions (D); : Jco−ions (D); •: Jcounter−ions (EM); ◦: Jco−ions (EM); — Fig. 2c): Jco−ions (C) + Jco−ions (D) + Jco−ions (EM).
m co-ions
/cf)/d(x/∆ x)
0.15
0.1
0.05
d(c
decreases for co-ions, which is in agreement with the increase of the counter-ions concentration and decrease of co-ions concentration in the membrane. As to the transport of ions by di8usion, it can also be seen from Fig. 2a that the di8usive 9uxes of co-ions and counter-ions are identical. This equality, observed on the whole range of membrane charge density, is due to the fact that the concentration gradient is the same for both type of ions in order to satisfy the electroneutrality requirements. As can be seen in Fig. 2c, the di8usive 9ux of co-ions (or counter-ions) as a function of normalized membrane charge density (%) passes through a maximum. The decrease in diffusive 9ux of ions with increasing % is a direct consequence of the variation of the driving force for di8usion (concentration gradient across the membrane) with %, as shown in Fig. 3 (where the curve shows also a maximum with respect to % at the same value of % as in Fig. 2c). As for di8usion, the electromigrative 9ux of co-ions "rst increases with the normalized membrane charge density and then decreases. Since the electromigrative 9ux of an ion i depends on both electrical potential gradient and concentration of i inside the membrane (Eq. (2c)), the shape of this curve results then from the % dependence of both concen-
0 0
Fig. 3. Dimensionless %; Jv = 50 m s−1 .
5
10 ξ co-ions
concentration
15
gradient
20
versus
tration of co-ions inside the membrane and electrical potential gradient across it. Fig. 4 shows that the increase in % leads to a stronger driving force for electromigration (electrical potential gradient). On the contrary, at increasing %
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P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
1.5
0.0004
D+ = DD+ < D-
0.0003
-1
Jco-ions (eq m s )
1
-2
d ψadim /d(x/∆x)
D+ > D-
0.5
0 0
5
10
15
20
ξ Fig. 4. Dimensionless %; Jv = 50 m s−1 .
electrical
potential
gradient
Contribution to electrolyte transport (%)
100
versus
C D EM
80
60
40
20
0 0
5
10
15
20
ξ Fig. 5. Contribution of the di8erent transport mechanisms to the overall electrolyte transport versus normalized membrane charge density (%); Jv = 50 m s−1 .
the concentration of co-ions inside the membrane becomes more and more low. Then, above a critical value of %, the increase of the electrical potential gradient does not allow to compensate the decrease of the number of co-ion in the membrane pores. It can be also seen in Fig. 2c that C mechanism dominates the overall co-ions transport, i.e. the overall electrolyte transport, at low membrane charge density (% ¡ 1) whereas D mechanism becomes predominant in the electrolyte transport for large values of % (% ¿ 10). Moreover, at high normalized membrane charge density, convective and electromigrative 9uxes of co-ions become identical. 4.1.2. Contribution of the three mechanisms to the overall electrolyte transport Fig. 5 shows the contributions of C; D and EM to the overall electrolyte transport (i.e. to the overall co-ions
0.0002
0.0001
0 -20
-15
-10
-5
0 ξ
5
10
15
20
Fig. 6. Variation of co-ions 9ux with normalized membrane charge density (%) for three "ctitious 1–1 electrolytes; Jv = 50 m s−1 . Electrolyte with D+ = D− = 10−9 m2 s−1 ; electrolyte with D+ = 10−9 m2 s−1 and D− = 3:10−9 m2 s−1 and electrolyte with D+ = 3:10−9 m2 s−1 and D− = 10−9 m2 s−1 .
transport) through the membrane as a function of the normalized membrane charge density. In comparison to Fig. 2c, it can be noted that the contribution of D to the overall electrolyte transport monotonously increases with % even if the co-ions 9ux due to di8usion "rstly increases and then decreases with % (as shown in Fig. 2c). This result is justi"ed by the fact that the decrease of co-ions 9ux due to D is less important than that of convective and electromigrative 9uxes of co-ions. The contribution of D tends to reach a limiting value of ∼ 60% at su:ciently high membrane charge density (% ¿ 10). In this case, the contribution of convective and electromigrative mechanisms is ∼ 20%. In summary, at low membrane charge density, the co-ions are only slightly excluded from the membrane and can be, a priori, transported by C or EM since these transport mechanisms are proportional to concentration. However, C is the dominant transport mechanism because the driving force for EM (electrical potential gradient through the membrane) is not strong enough as a result of the low membrane charge density. On the contrary, for large values of %, less co-ions (or electrolyte) can enter the membrane pores and are able to be transported by C or EM. As a result, D becomes the dominant mechanism involved in the overall electrolyte transport. However, it is important to keep in mind that these observations are valuable at permeate volume 9ux low enough (as will be shown later). 4.2. In6uence of parameters 4.2.1. E7ect of ionic di7usion coe8cients Fig. 6 presents the % dependence of the overall co-ions 9ux (i.e. the overall electrolyte 9ux) for three "ctitious 1– 1 electrolytes. As can be seen, the overall 9ux curves are asymmetric about the IEP for electrolytes with D+ = D− .
P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
100
Retention (%)
80 60 40 D+ = D20
D+ < DD+ > D-
0 -20 -20
-15
-10
-5
0
5
10
15
20
ξ
Fig. 7. Retention versus % for the same electrolytes as in Fig. 6; Jv = 50 m s−1 .
These asymmetries are not unexpected since the roles of co-ions and counter-ions are reversed when the membrane charge changes sign. The upshot of this observation is that for the same normalized membrane charge density, |%|, the overall co-ions 9ux will be lower when the membrane will have the same sign as the ion with smaller mobility (or di8usion coe:cient). It should also be noted that the maximum of the overall 9ux curves does not occur at the IEP but at low membrane charge for electrolytes with D+ = D− . The variation of the overall co-ions 9ux as a function of the normalized membrane charge density (%) re9ects how the electrolyte retention varies with %. Indeed, the decrease of the overall co-ions 9ux with increasing |%| leads to a higher retention, as shown in Fig. 7. The most striking result in this "gure is that a negative retention can be found for small values of |%| when Dco−ion ¿ Dcounter−ion . In this case, the retention is a decreasing function of the membrane charge density, |%|, over a certain range of |%| values. It also appears that retention is a strong function of the co-ion diffusion coe:cient. Figs. 8a–c show the contributions of the three transport mechanisms (C; D and EM) to the overall electrolyte transport as a function of the normalized membrane charge density (%) for the same electrolytes as in Figs. 6 and 7. As expected, C; D and EM curves are asymmetric about the IEP for electrolytes with D+ = D− (Figs. 8b and c), the asymmetry being particularly pronounced for EM curve. Indeed, the variation of the electromigrative contribution with % is much more important when the membrane has the same sign as the ion with greater mobility (or di8usion coe:cient). The following observations can also be made: • For an uncharged membrane (%=0); C greatly dominates the electrolyte transport whatever the ionic di8usion coe:cients. As can be seen, ions are able to be transported by EM when D+ = D− due to the di8usion potential induced by the concentration gradient across the membrane
2927
(at % = 0; R ∼ −0:2% due to the steric e8ects). Nevertheless, the contribution of this type of transport is very weak. • For a strongly charged membrane (|%| ¿ 10), di8usive mechanism remains the dominant mechanism involved in the overall electrolyte transport whatever the ionic diffusion coe:cients (provided that the permeate volume 9ux is low enough as will be shown later). Electromigrative transport is more important than convective transport when Dco−ion ¿ Dcounter−ion and conversely, convective transport dominates electromigrative transport when Dcounter−ion ¿ Dco−ion . • For intermediate values of |%| (1 ¡ |%| ¡ 10), either C or D dominates the transport when Dco−ion 6 Dcounter−ion . On the contrary, either EM or D is the dominant mechanism when Dco−ion ¿ Dcounter−ion . 4.2.2. E7ect of ion valence Fig. 9 shows the in9uence of ion valences on Retention-% curves. It must be mentioned that calculations have been performed for electrolyte solutions with the same concentration in eq m−3 and ions have identical di8usion coe:cients. At "rst, it can be seen that di8erences in ion valence are very much in9uencing retention. The e8ect of ion valence on the electrolyte retention is found to be in agreement with the qualitative aspects of the Donnan exclusion mechanism (Schaep, 1999): a higher co-ion valence leads to a higher retention whereas a higher counter-ion valence leads to a lower retention. As it can be seen, retentions calculated for a 1–1 electrolyte and a 2–2 electrolyte are identical due to the fact that both electrolytes have the same concentration expressed in eq m−3 . It also appears that retentions of asymmetric electrolytes are highly asymmetric about the IEP. This behavior is due to the marked change in co- and counter-ion valences when the membrane charge changes sign and the roles of co- and counter-ion are reversed. For example, in the case of the 1–2 electrolyte, we see that for a positive membrane charge, there is a monovalent co-ion that will be weakly rejected by Donnan exclusion and a divalent counter-ion that will be strongly attracted by the membrane, leading to low retention. For the opposite membrane charge polarity, conversely, there will be a divalent co-ion that will be strongly excluded, and a monovalent counter-ion that will only be weakly attracted by the membrane, leading to high retention. The theoretical predictions for electrolyte retention presented in Fig. 9 for both positive and negative membrane charge can be considered as the signature of amphoteric membranes that reject ions principally by electrostatic interactions (as steric e8ects are negligible here). Retention measurements obtained by Palmeri et al. (2000) with hafnia ceramic NF membranes for CaCl2 and Na2 SO4 electrolytes as well as those obtained by El Marraki (2001) with ZnAl2 O4 =TiO2 NF membranes for Na2 SO4 are in quite good agreement with the retention pattern presented in Fig. 9.
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P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931 D
Contribution to electrolyte transport (%)
Contribution to electrolyte transport (% )
100
EM C
80
60
40
20
0 -20
-10
0
10
ξ
(a)
20
100
D EM C
80
60
40
20
0 -20
0
10
20
ξ
100 Contribution to electrolyte transport (%)
-10
(b) D EM C
80
60
40
20
0 -20
-10
0
10
20
ξ
(c)
Fig. 8. (a–c) Contribution of the di8erent transport mechanisms (D, EM and C) to the overall electrolyte transport versus normalized membrane charge density (%); Jv = 50 m s−1 . (a) Electrolyte with D+ = D− = 10−9 m2 s−1 ; (b) electrolyte with D+ = 3:10−9 m2 s−1 and D− = 10−9 m2 s−1 and (c) electrolyte with D+ = 10−9 m2 s−1 and D− = 3:10−9 m2 s−1 .
100
Retention (%)
80
60
40 1-1 2-2 20
1-2 2-1
for the same electrolytes as in Fig. 9. It clearly appears that changes in co- or counter-ions valence strongly in9uence the contribution of the di8erent transport mechanisms. The most striking results in Figs. 10b and c are the 9atness of C; D and EM curves as |%| increases for a membrane whose charge sign corresponds to co-ions with valence greater than that of counter-ions. The contributions of the three mechanisms level o8 because co-ions 9uxes due to C; D and EM tend to reach a constant value as % increases. For large values of |%| (|%| ¿ 10), the following sequences are observed depending on ionic valences:
Fig. 9. Retention versus % for symmetric (1–1 and 2–2) and asymmetric (1–2 et 2–1) electrolytes having identical ionic di8usion coe:cients (D+ = D− = 10−9 m2 s−1 ).
• D contribution ¿ EM contribution ¿ C contribution when zco−ion ¿ zcounter−ion , • D contribution ¿ C contribution ¿ EM contribution when zco−ion ¡ zcounter−ion , • D contribution ¿ C contribution (C contribution = EM contribution) when zco−ion = zcounter−ion .
Figs. 10a–c show the contributions of the three transport mechanisms (C; D and EM) to the overall co-ions transport as a function of the normalized membrane charge density (%)
As expected, convection remains the most important transport mechanism at low membrane charge density (|%| ¡ 1) for both symmetric and asymmetric electrolytes. Retention results presented in this paper clearly show that for a given electrolyte (ionic valences and di8usion
0 -20
-15
-10
-5
0 ξ
5
10
15
20
P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931 C D EM
80 60 40 20 0
100
Contribution to electrolyte transport (%)
Contribution to electrolyte transport (%)
100
2929 C D EM
80 60 40 20 0
-20
-15
-10
-5
0 ξ
(a)
5
10
15
20
-20
-15
100
Contribution to electrolyte transport (%)
-10
-5
(b)
0 ξ
5
10
15
20
C D EM
80 60 40 20 0 -20
-15
-10
-5
(c)
0 ξ
5
10
15
20
Fig. 10. (a–c) Contribution of the di8erent transport mechanisms (D, EM and C) to the overall electrolyte transport versus normalized membrane charge density (%); Jv = 50 m s−1 . D+ = D− = 10−9 m2 s−1 . (a): 1–1 (or 2–2) electrolyte; (b): 1–2 electrolyte and (c): 2–1 electrolyte.
4.2.3. In6uence of permeate volume 6ux Fig. 11 presents the retention-% curves for the 1–1 electrolyte with D+ = D− at two permeate volume 9uxes (Jv ) of 50 and 500 m s−1 . The increase in retention with increasing Jv is a general result due to an important increase of the convective transport. The contributions of the three mechanisms to electrolyte transport for both values of Jv are shown in Fig. 12. It appears that the contribution of both C and EM increases with Jv on the whole range of % values whereas the contribution of D decreases. As shown by Szymczyk et al. (2001), when the permeate volume 9ux increases, the driving forces for convection (Jv ) and electromigration (electrical potential gradient) increase whereas the driving force for di8usion (concentration gradient) tends to reach a limiting value at high permeate volume 9ux.
100
80 Retention (%)
coe:cients), retention will be higher if the membrane is positively or negatively charged. One of the advantages of ceramic membranes, in comparison to most of organic membranes, is their amphoteric behavior which often allows the membrane charge density to go from high positive values at low pH to high negative values at high pH.
60
40
20
0 0
5
10
15
20
ξ Fig. 11. Retention versus % for two permeate volume 9uxes; 1–1 electrolyte with D+ =D− =10−9 m2 s−1 . —: Jv =50 m s−1 ; - - -: Jv =500 m s−1 .
For strongly charged membranes (% ¿ 10), it may be seen that contributions of C and EM become identical whatever the permeate volume 9ux. However, it is important to mention that this equality is observed only for symmetrical electrolytes with D+ = D−. It also appears that C and EM are
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P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
Contribution to electrolyte transport (%)
100
80
60
40
20
0 0
5
10 ξ
15
20
Fig. 12. Contribution of the di8erent transport mechanisms (D, EM and C) to the overall electrolyte transport versus normalized membrane charge density (%) for two permeate volume 9uxes Jv (50 and 500 m s−1 ). : convection, Jv =50 m s−1 ; : convection, Jv =500 m s−1 ; : di8usion, Jv = 50 m s−1 ; : di8usion, Jv = 500 m s−1 and ◦: electromigration, Jv = 50 m s−1 ; •: electromigration, Jv = 500 m s−1 .
the dominant transport mechanisms at high permeate volume 9ux whereas di8usion greatly dominates at low permeate volume 9ux. 5. Conclusion Transport mechanisms and electrolyte retention for amphoteric nano"ltration membranes have been investigated using a model based on the extended Nernst–Planck equation in combination with the Donnan equilibrium. Retention and contribution of each transport mechanism to overall electrolyte transport, namely, convection (C), di8usion (D) and electromigration (EM), were computed for a range of normalized membrane charge densities (%), covering both positive and negative values. The in9uence of ion valence (zi ), ion di8usion coe:cient (Di ) and permeate volume 9ux on both retention and contributions of the three transport mechanisms were investigated. Numerical calculations have been carried out for a membrane with "xed structural parameters (e8ective pore radius of 2 nm and e8ective thickness=porosity ratio of 7 m). As expected, retention increases with membrane charge density (provided retention is positive) whatever the kind of electrolyte and permeate volume 9ux, since less co-ions can enter the membrane pores. It is shown that retention curves are asymmetric about the IEP for electrolytes with z+ = z− or D+ = D− , the asymmetry being much more pronounced for asymmetrical electrolytes. This behavior is due to the interchange of co- and counter-ions when the membrane charge changes sign. For symmetrical electrolytes with D+ = D− , negative retentions are obtained at low membrane charge density when Dco−ion ¿ Dcounter−ion . In this case, retention is a decreasing function of % over a certain
range of % values. Moreover, it appears that for a same value of % (in absolute value), retention is stronger when the membrane has the same sign as ion with greater mobility. As expected, for asymmetrical electrolytes, retention is also stronger when the membrane charge sign is the same as ion with higher valence. Besides, the increase of the permeate volume 9ux leads to a better retention for any electrolyte. As for retentions, contributions of C; D and EM to electrolyte transport are asymmetrical about the IEP for electrolytes with z+ = z− or D+ = D− . The asymmetry of the three contributions is strongly marked for asymmetric electrolytes. The asymmetry of electromigrative contribution is also strongly pronounced for symmetrical electrolytes with D+ = D− . It appears that, for small values of |%|, convection is the dominant transport mechanism whereas for large values of |%|, the electrolyte transport is governed by di8usion mechanism. For large values of |%|, convective contribution is more important than electromigrative contribution when Dco−ion ¡ Dcounter−ion (with zco−ion = zcontre−ion ) or zco−ion ¡ zcontre−ion (with Dco−ion = Dcounter−ion ) and conversely, electromigrative contribution becomes stronger than convective contribution when Dco−ion ¿ Dcounter−ion (with zco−ion = zcontre−ion ) or zco−ion ¿ zcontre−ion (with Dco−ion = Dcounter−ion ). Equality between these two transport mechanisms (C and EM) is observed at high normalized membrane charge density when co- and counter-ions are strictly identical (zco−ion = zcontre−ion and Dco−ion = Dcounter−ion ). For intermediate values of |%|, EM mechanism can be the dominant mechanism involved in the electrolyte transport through the membrane in the case of a symmetrical electrolyte with Dco−ion ¿ Dcounter−ion . It should be kept in mind that above results have been obtained for a permeate volume 9ux of 50 m s−1 (as contributions of the three mechanisms are also a8ected by the volume 9ux). In the case of a symmetrical electrolyte with D+ = D− , it is shown that contributions of C and EM increase with volume 9ux on the whole range of % values whereas the contribution of D decreases. For large values of |%|; C and EM appear to be the dominant transport mechanisms at high permeate volume 9ux whereas di8usion greatly dominates at low permeate volume 9ux. This theoretical study clearly showed that relative contributions of C; D and EM to the ion transport are subject to a much slower or faster change with the membrane charge density, permeate 9ux, ion valence and di8usion coe:cients than the retention. Notation Ak c D F
membrane porosity, dimensionless concentration, mol m−3 di8usion coe:cient, m2 s−1 Faraday’s constant, 96487 C mol−1
P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
G Ic Ji Jv k K Ki; c Ki; d rp rs R T x X zi
lag coe:cient, dimensionless current density, A m−2 solute 9ux, mol s−1 m−2 permeate volume 9ux, m s−1 Boltzmann’s constant, 1:38 × 10−23 J K −1 enhanced drag coe:cient, dimensionless hindrance coe:cient convection, dimensionless hindrance coe:cient di8usion, dimensionless e8ective pore radius, m solute radius, m ideal gas constant, 8:3143 J mol−1 K −1 temperature, K axial coordinate, m membrane charge density, eq m−3 charge number, dimensionless
Greek letters Mx M D " # !
e8ective membrane thickness, m Donnan potential, V dynamic viscosity of the electrolyte, kg m−1 s−1 ratio of the solute radius to the pore radius, dimensionless ionic stoichiometric coe:cient, dimensionless steric partitioning term, dimensionless standard dimensionless constant, dimensionless electrical potential, V
Subscripts i f p + −
ion feed permeate cation anion
Superscripts m
membrane parameter
Acronyms C D DSPM EM MWCO
convection di8usion Donnan–Steric partitioning Pore Model electromigration molecular weight cut-o8
References Alkhatim, H. S., Alcaina, M. I., Soriano, E., Iborra, M. I., Lora, J., & Arnal, J. (1998). Treatment of whey eTuents from dairy industries by nano"ltration membranes. Desalination, 119, 177–183. Baticle, P., Kiefer, C., Lakhchaf, N., Larbot, A., Leclerc, O., Persin, M., & Sarrazin, J. (1997). Salt "ltration on gamma alumina nano"ltration
2931
membranes "red at two di8erent temperatures. Journal of Membrane Science, 135, 1–8. Benfer, S., Popp, U., Richter, H., Siewert, C., & Tomandl, G. (2001). Development and characterization of ceramic nano"ltration membranes. Separation and Puri=cation Technology, 22–23, 231– 237. Bowen, W. R., & Mohammad, A. W. (1998a). A theoretical basis for specifying nano"ltration membranes-Dye=salt=water streams. Desalination, 117, 257–264. Bowen, W. R., & Mohammad, A. W. (1998b). Dia"ltration by nano"ltration: Prediction and optimization AIChE Journal, 44, 1799– 1812. Bowen, W. R., Mohammad, A. W., & Hilal, N. (1997). Characterisation of nano"ltration membranes for predictive purposes—use of salts, uncharged solutes and atomic force microscopy. Journal of Membrane Science, 126, 91–105. Bowen, W. R., & Mukhtar, H. R. (1996). Characterisation and prediction of separation performance of nano"ltration membranes. Journal of Membrane Science, 112, 263–274. El Marraki, Y. (2001). Elaboration et caracterisation de membranes a> base d’oxyde de titane et d’aluminate de zinc pour la nano=ltration et l’ultra=ltration. Thesis, UniversitQe de Montpellier II, France. Eriksson, P. K., Lien, L. A., & Green, D. H. (1996). Membrane technology for treatment of wastes containing dissolved metals. Extr. process. treat. minimization wastes 1996. Proceedings of the Second International Symposium (p. 649). Fievet, P., Aoubiza, B., Szymczyk, A., & Pagetti, J. (1999). Membrane potentials in charged porous membranes. Journal of Membrane Science, 160, 267–275. Katselnik, P., & Morcos, S. Y. (1998). Reduction of nickel in plating operation eTuent with nano"ltration. Plating and Surface Finishing, 85, 46–47. Labbez, C., Fievet, P., Szymczyk, A., Vidonne, A., Foissy, A., & Pagetti, J. (2002). Analysis of the salt retention of a titania membrane using the “DSPM” model: e8ect of pH, salt concentration and nature. Journal of Membrane Science, submitted for publication. Palmeri, J., Blanc, P., Larbot, A., & David, P. (1999). Theory of pressure-driven transport of neutral solutes and ions in porous ceramic nano"ltration membranes. Journal of Membrane Science, 160, 141– 170. Palmeri, J., Blanc, P., Larbot, A., & David, P. (2000). Hafnia ceramic nano"ltration membranes. Part II. Modeling of pressure-driven transport of neutral solutes and ions. Journal of Membrane Science, 179, 243–266. Schaep, J. (1999). Nano=ltration for the removal of ionic components from water. Ph.D. thesis, Katholieke Universiteit Leuven, Heverlee, Belgium. Schaep, J., Vandecasteele, C., Mohammad, A. W., & Bowen, W. R. (1999). Analysis of the salt retention of nano"ltration membranes using the Donnan-steric partitioning pore model. Separation Science and Technology, 34, 3009–3030. Schaep, J., Vandecasteele, C., Mohammad, A. W., & Bowen, W. R. (2001). Modelling the retention of ionic components for di8erent nano"ltration membranes. Separation and Puri=cation Technology, 22–23, 169–179. Szymczyk, A., Labbez, C., Fievet, P., Vidonne, A., Foissy, A., & Pagetti, J. (2001). Contribution of convection, di8usion and migration to electrolyte transport through nano"ltration membranes. Journal of Membrane Science, submitted for publication. Wang, X. L., Tsuru, T., Nakao, S., & Kimura, S. (1995). Electrolyte transport through nano"ltration membranes by the space-charge model and the comparison with Teorell–Meyer–Sievers model. Journal of Membrane Science, 103, 117–133.
www.elsevier.com/locate/ces
Electrolyte transport through amphoteric nano"ltration membranes P. Fievet ∗ , C. Labbez, A. Szymczyk, A. Vidonne, A. Foissy, J. Pagetti Laboratoire de Chimie des Materiaux et Interfaces, 16 route de Gray, 25030 Besancon Cedex, France Received 19 December 2001; received in revised form 16 May 2002; accepted 20 May 2002
Abstract The electrolyte transport through amphoteric nano"ltration membranes is investigated using a model based on the application of the extended Nernst–Planck equation coupled with the electroneutrality condition in the membrane, and the assumption of a Donnan equilibrium at both membrane=solution interfaces. The model gives the possibility to distinguish between the di8erent transport mechanisms, namely, convection, di8usion and electromigration. The in9uence of ion valence, ion di8usion coe:cient and permeate volume 9ux on the di8erent transport mechanisms and electrolyte retention is presented. All calculations are carried out for a membrane with "xed structural parameters (e8ective pore radius of 2 nm and e8ective thickness=porosity ratio of 7 m). ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Donnan partitioning; Mass transfer; Membranes; Nano"ltration; Simulation; Transport processes
1. Introduction Nano"ltration (NF) membranes have intermediate molecular weight cut-o8 (MWCO) between ultra"ltration and reverse osmosis membranes. Their e8ective pore diameters range from ∼ 1 nm to a few nm (Wang, Tsuru, Nakao, & Kimura, 1995). Another feature of these membranes is that most of them are electrically charged in aqueous media due to their materials or adsorption of charged species. Hence, their separation mechanisms involve not only the ordinary “sieve e8ect” but also the “charge e8ect”. Therefore, they can reject charged solutes of much smaller size than the dimensions of the pores. This charge e8ect can be particularly used to remove ions from wastewater (Bowen & Mohammad, 1998a; Katselnik & Morcos, 1998) or whey (Alkhatim et al., 1998), and also separate ions according to their ionic valences (Eriksson, Lien, & Green, 1996). Many types of NF membranes are currently commercially available but all are made of an organic material. Despite of their numerous advantages compared to organic membranes (higher chemical, thermal and mechanical stability), no ceramic nano"lters (MWCO ¡ 1000 Da) can be found in the market now. This situation is due in part to the inherent practical di:culties that need to be overcome in order to prepare lightly sintered granular porous membranes with ∗ Corresponding author. Tel.: +33-381-66-2032; fax: +33-381-66-2033. E-mail address: patrick."[email protected] (P. Fievet).
grains sizes in the 5 –10 nm range and to a highly connected porosity (Palmeri, Blanc, Larbot, & David, 1999, 2000). However, for some years, researchers succeed in producing ceramic NF membranes (Baticle et al., 1997; Benfer, Popp, Richter, Siewert, & Tomandl, 2001; Palmeri et al., 2000). An interesting feature of ceramic membranes is their amphoteric character which allows to control the sign and the charge density of the membrane by pH control. As to the fundamentals of the nano"ltration process, even if the separation mechanisms are not yet well known, major progress has been made since the early 1990s due to the increasing development of new applications. For the mathematical description of the mass transfer process in NF, models that are commonly used are: Theorell–Meyer–Sievers model (Wang et al., 1995), Space-Charge model (Wang et al., 1995), Hybrid model (Bowen & Mukhtar, 1996; Bowen, Mohammad, & Hilal, 1997) and Donnan–Steric partitioning Pore model (DSPM) (Bowen & Mukhtar, 1996; Bowen et al., 1997; Schaep, 1999; Schaep, Vandecasteele, Mohammad, & Bowen, 1999, 2001). This latter has particularly proven to be useful in describing the retention properties of nano"ltration membranes (Bowen & Mohammad, 1998a, b; Schaep, 1999; Schaep et al., 1999, 2001). Moreover, it has proven to be realistic: a very good agreement between predictions and experimental data of retention for mixed NaCl=Na2 SO4 solutions was obtained by Bowen and Mukhtar (1996). It was also successful in predicting the performance of the CA 30 NF membrane for a dye=salt solution (Bowen & Mohammad, 1998a, b).
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 1 8 8 - 4
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P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
In this model, the solute transport is described as occurring through discrete pores and size e8ects are taken into account by incorporating steric hindrance factors for di8usion and convection. This model is based on the application of the Extended Nernst–Planck (ENP) equation to describe the electrolyte transport through the membrane. ENP equation is coupled to the electroneutrality condition inside the membrane, and the assumption of the Donnan equilibrium at both membrane-solution interfaces (Bowen & Mukhtar, 1996; Bowen et al., 1997; Schaep et al., 1999, 2001). In other words, that means that the transfer of charged solutes is controlled by both electrostatic interactions occurring between the charged solute and the charged membrane surface, and transport phenomena in the membrane: convection (pressure gradient), di8usion (concentration gradient) and electromigration (electric potential gradient). The DSPM model provides a description of separation properties in terms of three key parameters: the e8ective pore radius (rp ), the e8ective thickness=porosity ratio (Mx=Ak ) and the e8ective charge density (X ). Thus, it allows to characterise the structural and electrical properties of NF membranes from experimental data of retention (Bowen et al., 1997; Bowen & Mukhtar, 1996; Bowen & Mohammad, 1998a; Schaep, 1999; Schaep et al., 1999, 2001). Some theoretical studies with the DSPM model have been presented in the literature. Most of them deals with the in9uence of the model input parameters (rp ; Mx=Ak and X ), permeate volume 9ux, electrolyte feed concentration, ion valence and ion size on electrolyte retention (Bowen & Mohammad, 1998a, b Schaep, 1999; Schaep et al., 2001), but no studies have been focused on the in9uence of these parameters on the di8erent transport mechanisms—convection, diffusion and electromigration—through nano"ltration membranes. The aim of this paper is then to study the in9uence of some parameters (sign and charge density of the membrane, permeate volume 9ux, valence and di8usion coe:cient of ions) on the three transport mechanisms—convection, diffusion and electromigration—for single "ctitious electrolyte solutions and a membrane with a pore radius "xed at 2 nm. This value corresponds approximatively to the smallest pore radius of currently commercially available ceramic membranes. So, we aim to investigate the electrolyte transport through amphoteric membranes where electrostatic interactions between the membrane material and charged species in solution play important role in separation process in comparison with the size exclusion.
Table 1 Summary of equations used for modeling. Equations for the Donnan– Steric–Pore (DSPM) model Ji = −Ki; d Di Ji (D) =
dx
−
zi cim Ki; d Di RT
dcm −Ki; d Di d xi ,
Ji (EM) = Ji (C) =
dcim
z cm K D − i i RTi; d i
F
d m dx
F
d m dx
+ Ki; c cim Jv ,
(2a) (2b)
,
(2c)
Ki; c cim Jv ,
(2d)
Ki; d = K −1 (; 0),
(3)
Ki; c = (2 − ) G(; 0),
(4)
with K −1 (; 0) = 1:0 − 2:30 + 1:1542 + 0:2243 ,
(5)
G(; 0) = 1:0 + 0:054 − 0:9882 + 0:4413 ,
(6)
= (1 − )2 ,
(7)
= n
rs , rp
(8)
zi ci = 0,
(9)
zi cim = −X , n Ic = i=1 Fzi Ji = 0,
(10)
i=1
n
i=l
dcim dx
=
d m dx
=
i; d i F n i=1 RT
(zi2 cim )
at x =
0−
at x =
Mx+
ci = ci; p ,
= exp
iF − zRT M D
cim ci
(11)
zi cim − K JvD (Ki; c cim − ci; p ) − RT i; d i n zi Jv m i=1 K D (Ki; c ci −ci; p )
R=1−
ci = ci; f ,
ci; p . ci; f
,
F
d m , dx
(12) (13) (14a) (14b)
,
(15) (16)
For the mathematical derivation of the DSPM model, the following assumptions are adopted: • The membrane consists of a bundle of identical straight cylindrical pores of radius rp and length Mx (with Mx rp ). • The e8ective membrane charge density (X ) is constant through the membrane. • The ion size is expressed as a Stokes radius (rs ) and is calculated from the di8usion coe:cient (Di ), with the Stokes–Einstein equation: kT : (1) rs = 6!"Di
2. Theory
• The ion 9uxes (Ji ), ion concentrations (ci ), electric potential ( ) and permeate volume 9ux (Jv ) are all de"ned in terms of radially averaged quantities. • The volume 9ux is fully developed inside the pore and has a parabolic pro"le of the Hagen–Poiseuille type.
The theory of DSPM model (Donnan and Steric partitioning Pore model) has been presented in detail by Bowen et al. (Bowen et al., 1997; Bowen & Mukhtar, 1996) and Schaep et al. (Schaep, 1999; Schaep et al., 1999). That is why only a brief description of the model will be made in this section.
The main equations used for the DSPM model are summarized in Table 1 (All symbols are de"ned in the “Notation”). As mentioned earlier, this model is based on the ENP equation to describe the ionic transport through the membrane (Eq. (2a)). The terms on the right-hand side of
P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
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Charged membrane Jco-ions (C) Jcounter-ions (C) Jco-ions (D) feed
Jcounter-ions (D)
permeate
Jco-ions (EM) Jcounter-ions (EM) Fig. 1. Schematic representation of co-ions and counter-ions 9uxes due to convection (C), di8usion (D) and electromigration (EM) through a charged membrane in the case of a positive retention.
this equation represent transport due to di8usion (D), electromigration (EM) and convection (C), respectively. The hindered nature of ion transport inside pores is taken into account by using steric hindrance factors (Eqs. (3) and (4)). These are a function of , the ratio of the ion radius (rs ) to the e8ective pore radius (rp ) (Eq. (8)). The conditions for electroneutrality in the bulk solution and inside the membrane are given by Eqs. (9) and (10), respectively. The requirement for zero overall electrical current passing through the membrane is expressed by Eq. (11). The concentration and electrical potential gradients (Eqs. (12) and (13), respectively) that can be obtained by rearranging Eq. (2a), are solved by taking the boundary conditions given by Eq. (14) together with the equations for electroneutrality (Eqs. (9) and (10)). Partitioning of ions at the interfaces membrane–feed solution and membrane– permeate is governed by both Donnan and steric e8ects (Eq. (15)). Retention is given by Eq. (16)). The model gives the possibility to distinguish between the di8erent transport mechanisms (C; D and EM). The in9uence of parameters such as surface charge density (X ), permeate volume 9ux (Jv ), di8usion coe:cient (Di ) and ion valence (zi ) on the contribution of each transport mechanism to the overall electrolyte transport can be then computed. Fig. 1 provides a schematic representation of the convective, di8usive and electromigrative 9uxes through a charged membrane for both co-ions and counter-ions when retention is positive. It can be noted that in this case, only the electromigrative 9ux of counter-ions (Jcounter−ions (EM)) is oriented towards the feed solution. For a negative retention (permeate more concentrated than the feed), the two di8usive 9uxes would be additionally oriented towards the feed solution. For any electrolyte, the sum of the three co-ions 9uxes, Jco−ions , is equal to the sum of the three counter-ions 9uxes, Jcounter−ions (if 9uxes are expressed in eq s−1 m−2 ) and represents the overall electrolyte transport through the membrane.
Because co-ions are excluded from the membrane and counter-ions are attracted (due to the Donnan e8ect), the concentration of co-ions in the membrane is lower than that for counter-ions. Consequently, calculations of the contribution of each transport mechanism (C; D and EM) to the overall electrolyte transport (from the feed side to the permeate side) will be performed with co-ions that are the limiting species to the overall electrolyte transport through the membrane. For example, the contribution of convection to the overall electrolyte transport is given by the ratio of co-ions 9ux due to convection (Jco−ions (C)) to the sum of co-ions 9uxes due to convection, di8usion and electromigration (Jco−ions (C) + Jco−ions (D) + Jco−ions (EM)). It should be noted that for a negative retention, no co-ion would be transferred by di8usion towards the permeate (Jco−ions (D) ¡ 0). In this case, the contribution of di8usion to the overall electrolyte transport (from the feed side to the permeate side) must be taken equal to zero. 3. Numerical calculation procedure Solving the di8erential equations for concentration and electric potential gradients through the membrane (Eqs. (12) and (13)) allows the determination of the retention for given membrane (rp ; Mx=Ak and X ) and solution (cf ; rs ; Di ; T , etc.) parameters according to the following procedure. First, the concentration of cations and anions at the feed-membrane interface (just inside the membrane) is evaluated by solving Eq. (15), taking into account boundary conditions (Eq. (14)) together with electroneutrality requirements (Eqs. (9) and (10)). Next, the potential gradient over the membrane is determined by means of Eq. (13), considering electroneutrality conditions (Eqs. (9) and (10)) again. To this end, an arbitrary value is chosen for the permeate concentration (cp = c+; p =#+ = c−; p =#− ). Then, by solving Eqs. (12) and (13) by accounting for the zero
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P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
electric current condition (Eq. (11)), concentrations of cations and anions across the membrane are calculated and a new value of the permeate concentration is obtained. This latter is compared with the initial one. If the di8erence between initial and new values (Mcp ) is lower than the stop criterion (Mcp =cp ¡ 0:1%), the numerical calculation is then stopped. Otherwise, a new permeate concentration is set by using a function minimization algorithm for the numerical solution of the di8erential equations (Eqs. (12) and (13)). This procedure is repeated until the stop criterion is satis"ed. The numerical integration of Eqs. (12) and (13), and the function minimization are carried out using a fourth-order Runge–Kutta and a quadratic method, respectively. 4. Results of calculations and discussion All numerical simulations presented in this work have been carried out for an e8ective pore radius of 2 nm (value which corresponds approximatively to the smallest pore radius of commercial ceramic membranes) and an e8ective thickness=porosity ratio (Mx=Ak ) of 7 m. This value corresponds approximatively to the mean value obtained by considering an intergranular porosity in the 25 –50% range (Burggraaf, 1996) and a layer thickness in the 1–3 m range (El Marraki, 2001; Labbez et al., 2002). The electrolyte concentration (Cf ) was "xed at the value of 1 eq m−3 and the permeate volume 9ux at 50 m s−1 (value based on data for a “real” membrane with such structural features (Labbez et al., 2002)). Three "ctitious 1–1 electrolytes with D+ = D− = 10−9 m2 s−1 , D+ ¿ D− (3:10−9 and 10−9 m2 s−1 , respectively) and D+ ¡ D− (10−9 and 3:10−9 m2 s−1 , respectively) have been considered to study the in9uence of ionic di8usion coe:cients. Two "ctitious symmetric (1–1 and 2–2) and asymmetric (1–2 et 2–1) electrolytes having identical ionic di8usion coe:cients (D+ = D− = 10−9 m2 s−1 ) have been used for studying the e8ect of ion valence. 4.1. Transport mechanisms 4.1.1. Ion 6uxes Figs. 2a–c show the variations of the convective, diffusive and electromigrative 9uxes of co- and counter-ions as a function of the normalized membrane charge density (% = X=Cf ) for a 1–1 electrolyte with D+ = D− . Figs. 2a and b show the magnitude of the six 9uxes for 0 6 |%| 6 1 and 0 6 |%| 6 20, respectively. Fig. 2c presents the convective, di8usive and electromigrative 9uxes of co-ions as well as the sum of these 9uxes for 0 6 |%| 6 20. First of all, it can be seen from Figs. 2a and b that C; D and EM 9ux curves are symmetrical for both co- and counter-ions about the isoelectric point (IEP) where % = 0. These symmetries are not unexpected since, even if the roles of co-ions and counter-ions are reversed when the membrane charge changes sign, the
type of co- and counter-ions does not change with changing sign of the membrane charge (D+ = D− and z+ = z− ). As mentioned earlier, it can be seen from Figs. 2a and b that only the electromigrative 9ux of counter-ions is negative, i.e. oriented towards the feed side. At the IEP of the membrane, the following observations can be made (Fig. 2a): • The di8usive 9uxes of cations (J+ (D)) and anions (J− (D)) are very low but di8erent from zero due to the presence of a concentration gradient across the membrane. Indeed, as will be shown later, the retention of the membrane is not zero for % = 0 (R = 2%) due to purely steric e8ects. • Ions are not transported by EM through the membrane (J+ (EM) = J− (EM) = 0) due to the absence of an electrical potential gradient across it. Indeed, di8usion of ionic species through an uncharged membrane can not induce an electrical potential gradient across it when their diffusion coe:cients are identical (Fievet, Aoubiza, Szymczyk, & Pagetti, 1999). • The convective 9uxes of cations (J+ (C)) and anions (J− (C)) largely dominate the di8usive 9uxes, which is in agreement with the high value of the PQeclet number (which measures the importance of convective electrolyte transport over di8usion) in the membrane. As can be seen in Figs. 2a and c, when the membrane charge di8ers from zero (% = 0), EM phenomena then occurs (Ji (EM) = 0). The electrical potential gradient induced through the membrane by convection and di8usion phenomena is due to the excess of counter-ions in the pores. For small values of |%| (|%| ¡ 1), it appears from Fig. 2a that the highest 9uxes (in absolute value) are the convective 9uxes for both counter-ions (Jcounter−ions (C)) and co-ions (Jco−ions (C)), and electromigrative 9ux of counter-ions (Jcounter−ions (EM)). The di8usive 9uxes of co-ions (Jco−ions (D)) and counter-ions (Jcounter−ions (D)) are low due to a low-concentration gradient across the membrane (di8usion is directly proportional to the concentration gradient as expressed by Eq. (2b)). It will be shown later that the retention is less than 10% for |%| ¡ 1. The electromigrative 9ux of co-ions (Jco−ions (EM)) is lower than that for counter-ions due to a lower concentration of co-ions in the membrane pores (electromigration is proportional to both electrical potential gradient and ionic concentration inside the membrane as expressed by Eq. (2c)). For high values of |%| (|%| ¿ 10), counter-ions are in large excess inside the membrane pores in comparison to co-ions. As a result, the convective and electromigrative 9uxes of counter-ions, which are proportional to their concentration as expressed by Eqs. (2c) and (d), are dominant (Fig. 2b). It is important to note that these two 9uxes are quasi-identical in absolute value but in the opposite direction. It can also be noted from Figs. 2a and b that the convective 9ux increases with |%| for counter-ions whereas it
P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
2925
0.01
0.0006
0.008 0.006
0.0004 -1
Ji (eq m s )
0.002
-2
0.0002
-2
-1
Ji (eq m s )
0.004
0 -1
-0.5
0
0.5
1
0 -0.002 -20
-15
-10
-5
0
5
10
15
20
-0.004 -0.006
-0.0002
-0.008
-0.0004
(a)
-0.01
ξ
ξ
(b)
0.0003
-2
-1
Jco-ions (eq m s )
0.0004
0.0002
0.0001
0 0
5
10 ξ
(c)
15
20
Fig. 2. (a–c) Variation of co-ions and counter-ions 9uxes due to convection (C), di8usion (D) and electromigration (EM) with normalized membrane charge density (%); Jv = 50 m s−1 . : Jcounter−ions (C); : Jco−ions (C); : Jcounter−ions (D); : Jco−ions (D); •: Jcounter−ions (EM); ◦: Jco−ions (EM); — Fig. 2c): Jco−ions (C) + Jco−ions (D) + Jco−ions (EM).
m co-ions
/cf)/d(x/∆ x)
0.15
0.1
0.05
d(c
decreases for co-ions, which is in agreement with the increase of the counter-ions concentration and decrease of co-ions concentration in the membrane. As to the transport of ions by di8usion, it can also be seen from Fig. 2a that the di8usive 9uxes of co-ions and counter-ions are identical. This equality, observed on the whole range of membrane charge density, is due to the fact that the concentration gradient is the same for both type of ions in order to satisfy the electroneutrality requirements. As can be seen in Fig. 2c, the di8usive 9ux of co-ions (or counter-ions) as a function of normalized membrane charge density (%) passes through a maximum. The decrease in diffusive 9ux of ions with increasing % is a direct consequence of the variation of the driving force for di8usion (concentration gradient across the membrane) with %, as shown in Fig. 3 (where the curve shows also a maximum with respect to % at the same value of % as in Fig. 2c). As for di8usion, the electromigrative 9ux of co-ions "rst increases with the normalized membrane charge density and then decreases. Since the electromigrative 9ux of an ion i depends on both electrical potential gradient and concentration of i inside the membrane (Eq. (2c)), the shape of this curve results then from the % dependence of both concen-
0 0
Fig. 3. Dimensionless %; Jv = 50 m s−1 .
5
10 ξ co-ions
concentration
15
gradient
20
versus
tration of co-ions inside the membrane and electrical potential gradient across it. Fig. 4 shows that the increase in % leads to a stronger driving force for electromigration (electrical potential gradient). On the contrary, at increasing %
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P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
1.5
0.0004
D+ = DD+ < D-
0.0003
-1
Jco-ions (eq m s )
1
-2
d ψadim /d(x/∆x)
D+ > D-
0.5
0 0
5
10
15
20
ξ Fig. 4. Dimensionless %; Jv = 50 m s−1 .
electrical
potential
gradient
Contribution to electrolyte transport (%)
100
versus
C D EM
80
60
40
20
0 0
5
10
15
20
ξ Fig. 5. Contribution of the di8erent transport mechanisms to the overall electrolyte transport versus normalized membrane charge density (%); Jv = 50 m s−1 .
the concentration of co-ions inside the membrane becomes more and more low. Then, above a critical value of %, the increase of the electrical potential gradient does not allow to compensate the decrease of the number of co-ion in the membrane pores. It can be also seen in Fig. 2c that C mechanism dominates the overall co-ions transport, i.e. the overall electrolyte transport, at low membrane charge density (% ¡ 1) whereas D mechanism becomes predominant in the electrolyte transport for large values of % (% ¿ 10). Moreover, at high normalized membrane charge density, convective and electromigrative 9uxes of co-ions become identical. 4.1.2. Contribution of the three mechanisms to the overall electrolyte transport Fig. 5 shows the contributions of C; D and EM to the overall electrolyte transport (i.e. to the overall co-ions
0.0002
0.0001
0 -20
-15
-10
-5
0 ξ
5
10
15
20
Fig. 6. Variation of co-ions 9ux with normalized membrane charge density (%) for three "ctitious 1–1 electrolytes; Jv = 50 m s−1 . Electrolyte with D+ = D− = 10−9 m2 s−1 ; electrolyte with D+ = 10−9 m2 s−1 and D− = 3:10−9 m2 s−1 and electrolyte with D+ = 3:10−9 m2 s−1 and D− = 10−9 m2 s−1 .
transport) through the membrane as a function of the normalized membrane charge density. In comparison to Fig. 2c, it can be noted that the contribution of D to the overall electrolyte transport monotonously increases with % even if the co-ions 9ux due to di8usion "rstly increases and then decreases with % (as shown in Fig. 2c). This result is justi"ed by the fact that the decrease of co-ions 9ux due to D is less important than that of convective and electromigrative 9uxes of co-ions. The contribution of D tends to reach a limiting value of ∼ 60% at su:ciently high membrane charge density (% ¿ 10). In this case, the contribution of convective and electromigrative mechanisms is ∼ 20%. In summary, at low membrane charge density, the co-ions are only slightly excluded from the membrane and can be, a priori, transported by C or EM since these transport mechanisms are proportional to concentration. However, C is the dominant transport mechanism because the driving force for EM (electrical potential gradient through the membrane) is not strong enough as a result of the low membrane charge density. On the contrary, for large values of %, less co-ions (or electrolyte) can enter the membrane pores and are able to be transported by C or EM. As a result, D becomes the dominant mechanism involved in the overall electrolyte transport. However, it is important to keep in mind that these observations are valuable at permeate volume 9ux low enough (as will be shown later). 4.2. In6uence of parameters 4.2.1. E7ect of ionic di7usion coe8cients Fig. 6 presents the % dependence of the overall co-ions 9ux (i.e. the overall electrolyte 9ux) for three "ctitious 1– 1 electrolytes. As can be seen, the overall 9ux curves are asymmetric about the IEP for electrolytes with D+ = D− .
P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
100
Retention (%)
80 60 40 D+ = D20
D+ < DD+ > D-
0 -20 -20
-15
-10
-5
0
5
10
15
20
ξ
Fig. 7. Retention versus % for the same electrolytes as in Fig. 6; Jv = 50 m s−1 .
These asymmetries are not unexpected since the roles of co-ions and counter-ions are reversed when the membrane charge changes sign. The upshot of this observation is that for the same normalized membrane charge density, |%|, the overall co-ions 9ux will be lower when the membrane will have the same sign as the ion with smaller mobility (or di8usion coe:cient). It should also be noted that the maximum of the overall 9ux curves does not occur at the IEP but at low membrane charge for electrolytes with D+ = D− . The variation of the overall co-ions 9ux as a function of the normalized membrane charge density (%) re9ects how the electrolyte retention varies with %. Indeed, the decrease of the overall co-ions 9ux with increasing |%| leads to a higher retention, as shown in Fig. 7. The most striking result in this "gure is that a negative retention can be found for small values of |%| when Dco−ion ¿ Dcounter−ion . In this case, the retention is a decreasing function of the membrane charge density, |%|, over a certain range of |%| values. It also appears that retention is a strong function of the co-ion diffusion coe:cient. Figs. 8a–c show the contributions of the three transport mechanisms (C; D and EM) to the overall electrolyte transport as a function of the normalized membrane charge density (%) for the same electrolytes as in Figs. 6 and 7. As expected, C; D and EM curves are asymmetric about the IEP for electrolytes with D+ = D− (Figs. 8b and c), the asymmetry being particularly pronounced for EM curve. Indeed, the variation of the electromigrative contribution with % is much more important when the membrane has the same sign as the ion with greater mobility (or di8usion coe:cient). The following observations can also be made: • For an uncharged membrane (%=0); C greatly dominates the electrolyte transport whatever the ionic di8usion coe:cients. As can be seen, ions are able to be transported by EM when D+ = D− due to the di8usion potential induced by the concentration gradient across the membrane
2927
(at % = 0; R ∼ −0:2% due to the steric e8ects). Nevertheless, the contribution of this type of transport is very weak. • For a strongly charged membrane (|%| ¿ 10), di8usive mechanism remains the dominant mechanism involved in the overall electrolyte transport whatever the ionic diffusion coe:cients (provided that the permeate volume 9ux is low enough as will be shown later). Electromigrative transport is more important than convective transport when Dco−ion ¿ Dcounter−ion and conversely, convective transport dominates electromigrative transport when Dcounter−ion ¿ Dco−ion . • For intermediate values of |%| (1 ¡ |%| ¡ 10), either C or D dominates the transport when Dco−ion 6 Dcounter−ion . On the contrary, either EM or D is the dominant mechanism when Dco−ion ¿ Dcounter−ion . 4.2.2. E7ect of ion valence Fig. 9 shows the in9uence of ion valences on Retention-% curves. It must be mentioned that calculations have been performed for electrolyte solutions with the same concentration in eq m−3 and ions have identical di8usion coe:cients. At "rst, it can be seen that di8erences in ion valence are very much in9uencing retention. The e8ect of ion valence on the electrolyte retention is found to be in agreement with the qualitative aspects of the Donnan exclusion mechanism (Schaep, 1999): a higher co-ion valence leads to a higher retention whereas a higher counter-ion valence leads to a lower retention. As it can be seen, retentions calculated for a 1–1 electrolyte and a 2–2 electrolyte are identical due to the fact that both electrolytes have the same concentration expressed in eq m−3 . It also appears that retentions of asymmetric electrolytes are highly asymmetric about the IEP. This behavior is due to the marked change in co- and counter-ion valences when the membrane charge changes sign and the roles of co- and counter-ion are reversed. For example, in the case of the 1–2 electrolyte, we see that for a positive membrane charge, there is a monovalent co-ion that will be weakly rejected by Donnan exclusion and a divalent counter-ion that will be strongly attracted by the membrane, leading to low retention. For the opposite membrane charge polarity, conversely, there will be a divalent co-ion that will be strongly excluded, and a monovalent counter-ion that will only be weakly attracted by the membrane, leading to high retention. The theoretical predictions for electrolyte retention presented in Fig. 9 for both positive and negative membrane charge can be considered as the signature of amphoteric membranes that reject ions principally by electrostatic interactions (as steric e8ects are negligible here). Retention measurements obtained by Palmeri et al. (2000) with hafnia ceramic NF membranes for CaCl2 and Na2 SO4 electrolytes as well as those obtained by El Marraki (2001) with ZnAl2 O4 =TiO2 NF membranes for Na2 SO4 are in quite good agreement with the retention pattern presented in Fig. 9.
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P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931 D
Contribution to electrolyte transport (%)
Contribution to electrolyte transport (% )
100
EM C
80
60
40
20
0 -20
-10
0
10
ξ
(a)
20
100
D EM C
80
60
40
20
0 -20
0
10
20
ξ
100 Contribution to electrolyte transport (%)
-10
(b) D EM C
80
60
40
20
0 -20
-10
0
10
20
ξ
(c)
Fig. 8. (a–c) Contribution of the di8erent transport mechanisms (D, EM and C) to the overall electrolyte transport versus normalized membrane charge density (%); Jv = 50 m s−1 . (a) Electrolyte with D+ = D− = 10−9 m2 s−1 ; (b) electrolyte with D+ = 3:10−9 m2 s−1 and D− = 10−9 m2 s−1 and (c) electrolyte with D+ = 10−9 m2 s−1 and D− = 3:10−9 m2 s−1 .
100
Retention (%)
80
60
40 1-1 2-2 20
1-2 2-1
for the same electrolytes as in Fig. 9. It clearly appears that changes in co- or counter-ions valence strongly in9uence the contribution of the di8erent transport mechanisms. The most striking results in Figs. 10b and c are the 9atness of C; D and EM curves as |%| increases for a membrane whose charge sign corresponds to co-ions with valence greater than that of counter-ions. The contributions of the three mechanisms level o8 because co-ions 9uxes due to C; D and EM tend to reach a constant value as % increases. For large values of |%| (|%| ¿ 10), the following sequences are observed depending on ionic valences:
Fig. 9. Retention versus % for symmetric (1–1 and 2–2) and asymmetric (1–2 et 2–1) electrolytes having identical ionic di8usion coe:cients (D+ = D− = 10−9 m2 s−1 ).
• D contribution ¿ EM contribution ¿ C contribution when zco−ion ¿ zcounter−ion , • D contribution ¿ C contribution ¿ EM contribution when zco−ion ¡ zcounter−ion , • D contribution ¿ C contribution (C contribution = EM contribution) when zco−ion = zcounter−ion .
Figs. 10a–c show the contributions of the three transport mechanisms (C; D and EM) to the overall co-ions transport as a function of the normalized membrane charge density (%)
As expected, convection remains the most important transport mechanism at low membrane charge density (|%| ¡ 1) for both symmetric and asymmetric electrolytes. Retention results presented in this paper clearly show that for a given electrolyte (ionic valences and di8usion
0 -20
-15
-10
-5
0 ξ
5
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20
P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931 C D EM
80 60 40 20 0
100
Contribution to electrolyte transport (%)
Contribution to electrolyte transport (%)
100
2929 C D EM
80 60 40 20 0
-20
-15
-10
-5
0 ξ
(a)
5
10
15
20
-20
-15
100
Contribution to electrolyte transport (%)
-10
-5
(b)
0 ξ
5
10
15
20
C D EM
80 60 40 20 0 -20
-15
-10
-5
(c)
0 ξ
5
10
15
20
Fig. 10. (a–c) Contribution of the di8erent transport mechanisms (D, EM and C) to the overall electrolyte transport versus normalized membrane charge density (%); Jv = 50 m s−1 . D+ = D− = 10−9 m2 s−1 . (a): 1–1 (or 2–2) electrolyte; (b): 1–2 electrolyte and (c): 2–1 electrolyte.
4.2.3. In6uence of permeate volume 6ux Fig. 11 presents the retention-% curves for the 1–1 electrolyte with D+ = D− at two permeate volume 9uxes (Jv ) of 50 and 500 m s−1 . The increase in retention with increasing Jv is a general result due to an important increase of the convective transport. The contributions of the three mechanisms to electrolyte transport for both values of Jv are shown in Fig. 12. It appears that the contribution of both C and EM increases with Jv on the whole range of % values whereas the contribution of D decreases. As shown by Szymczyk et al. (2001), when the permeate volume 9ux increases, the driving forces for convection (Jv ) and electromigration (electrical potential gradient) increase whereas the driving force for di8usion (concentration gradient) tends to reach a limiting value at high permeate volume 9ux.
100
80 Retention (%)
coe:cients), retention will be higher if the membrane is positively or negatively charged. One of the advantages of ceramic membranes, in comparison to most of organic membranes, is their amphoteric behavior which often allows the membrane charge density to go from high positive values at low pH to high negative values at high pH.
60
40
20
0 0
5
10
15
20
ξ Fig. 11. Retention versus % for two permeate volume 9uxes; 1–1 electrolyte with D+ =D− =10−9 m2 s−1 . —: Jv =50 m s−1 ; - - -: Jv =500 m s−1 .
For strongly charged membranes (% ¿ 10), it may be seen that contributions of C and EM become identical whatever the permeate volume 9ux. However, it is important to mention that this equality is observed only for symmetrical electrolytes with D+ = D−. It also appears that C and EM are
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P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
Contribution to electrolyte transport (%)
100
80
60
40
20
0 0
5
10 ξ
15
20
Fig. 12. Contribution of the di8erent transport mechanisms (D, EM and C) to the overall electrolyte transport versus normalized membrane charge density (%) for two permeate volume 9uxes Jv (50 and 500 m s−1 ). : convection, Jv =50 m s−1 ; : convection, Jv =500 m s−1 ; : di8usion, Jv = 50 m s−1 ; : di8usion, Jv = 500 m s−1 and ◦: electromigration, Jv = 50 m s−1 ; •: electromigration, Jv = 500 m s−1 .
the dominant transport mechanisms at high permeate volume 9ux whereas di8usion greatly dominates at low permeate volume 9ux. 5. Conclusion Transport mechanisms and electrolyte retention for amphoteric nano"ltration membranes have been investigated using a model based on the extended Nernst–Planck equation in combination with the Donnan equilibrium. Retention and contribution of each transport mechanism to overall electrolyte transport, namely, convection (C), di8usion (D) and electromigration (EM), were computed for a range of normalized membrane charge densities (%), covering both positive and negative values. The in9uence of ion valence (zi ), ion di8usion coe:cient (Di ) and permeate volume 9ux on both retention and contributions of the three transport mechanisms were investigated. Numerical calculations have been carried out for a membrane with "xed structural parameters (e8ective pore radius of 2 nm and e8ective thickness=porosity ratio of 7 m). As expected, retention increases with membrane charge density (provided retention is positive) whatever the kind of electrolyte and permeate volume 9ux, since less co-ions can enter the membrane pores. It is shown that retention curves are asymmetric about the IEP for electrolytes with z+ = z− or D+ = D− , the asymmetry being much more pronounced for asymmetrical electrolytes. This behavior is due to the interchange of co- and counter-ions when the membrane charge changes sign. For symmetrical electrolytes with D+ = D− , negative retentions are obtained at low membrane charge density when Dco−ion ¿ Dcounter−ion . In this case, retention is a decreasing function of % over a certain
range of % values. Moreover, it appears that for a same value of % (in absolute value), retention is stronger when the membrane has the same sign as ion with greater mobility. As expected, for asymmetrical electrolytes, retention is also stronger when the membrane charge sign is the same as ion with higher valence. Besides, the increase of the permeate volume 9ux leads to a better retention for any electrolyte. As for retentions, contributions of C; D and EM to electrolyte transport are asymmetrical about the IEP for electrolytes with z+ = z− or D+ = D− . The asymmetry of the three contributions is strongly marked for asymmetric electrolytes. The asymmetry of electromigrative contribution is also strongly pronounced for symmetrical electrolytes with D+ = D− . It appears that, for small values of |%|, convection is the dominant transport mechanism whereas for large values of |%|, the electrolyte transport is governed by di8usion mechanism. For large values of |%|, convective contribution is more important than electromigrative contribution when Dco−ion ¡ Dcounter−ion (with zco−ion = zcontre−ion ) or zco−ion ¡ zcontre−ion (with Dco−ion = Dcounter−ion ) and conversely, electromigrative contribution becomes stronger than convective contribution when Dco−ion ¿ Dcounter−ion (with zco−ion = zcontre−ion ) or zco−ion ¿ zcontre−ion (with Dco−ion = Dcounter−ion ). Equality between these two transport mechanisms (C and EM) is observed at high normalized membrane charge density when co- and counter-ions are strictly identical (zco−ion = zcontre−ion and Dco−ion = Dcounter−ion ). For intermediate values of |%|, EM mechanism can be the dominant mechanism involved in the electrolyte transport through the membrane in the case of a symmetrical electrolyte with Dco−ion ¿ Dcounter−ion . It should be kept in mind that above results have been obtained for a permeate volume 9ux of 50 m s−1 (as contributions of the three mechanisms are also a8ected by the volume 9ux). In the case of a symmetrical electrolyte with D+ = D− , it is shown that contributions of C and EM increase with volume 9ux on the whole range of % values whereas the contribution of D decreases. For large values of |%|; C and EM appear to be the dominant transport mechanisms at high permeate volume 9ux whereas di8usion greatly dominates at low permeate volume 9ux. This theoretical study clearly showed that relative contributions of C; D and EM to the ion transport are subject to a much slower or faster change with the membrane charge density, permeate 9ux, ion valence and di8usion coe:cients than the retention. Notation Ak c D F
membrane porosity, dimensionless concentration, mol m−3 di8usion coe:cient, m2 s−1 Faraday’s constant, 96487 C mol−1
P. Fievet et al. / Chemical Engineering Science 57 (2002) 2921–2931
G Ic Ji Jv k K Ki; c Ki; d rp rs R T x X zi
lag coe:cient, dimensionless current density, A m−2 solute 9ux, mol s−1 m−2 permeate volume 9ux, m s−1 Boltzmann’s constant, 1:38 × 10−23 J K −1 enhanced drag coe:cient, dimensionless hindrance coe:cient convection, dimensionless hindrance coe:cient di8usion, dimensionless e8ective pore radius, m solute radius, m ideal gas constant, 8:3143 J mol−1 K −1 temperature, K axial coordinate, m membrane charge density, eq m−3 charge number, dimensionless
Greek letters Mx M D " # !
e8ective membrane thickness, m Donnan potential, V dynamic viscosity of the electrolyte, kg m−1 s−1 ratio of the solute radius to the pore radius, dimensionless ionic stoichiometric coe:cient, dimensionless steric partitioning term, dimensionless standard dimensionless constant, dimensionless electrical potential, V
Subscripts i f p + −
ion feed permeate cation anion
Superscripts m
membrane parameter
Acronyms C D DSPM EM MWCO
convection di8usion Donnan–Steric partitioning Pore Model electromigration molecular weight cut-o8
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