Modeling of cadmium salts rejection through a nanofiltration membrane: relationships between solute concentration and transport parameters

Journal of Membrane Science 211 (2003) 51–58

Modeling of cadmium salts rejection through a nanofiltration membrane: relationships between solute concentration and transport parameters Y. Garba, S. Taha∗ , J. Cabon, G. Dorange Département Procédés et Analyses, Laboratoire de Chimie des Eaux et de l’Environnement, ENSCR, Avenue du Général Leclerc, 35700 Rennes, France Received 17 April 2001; received in revised form 25 June 2002; accepted 28 June 2002

Abstract Based on the model initially proposed for nanofiltration combining the modified equation of Nernst–Plank and the film theory, expressions were developed to allow accurate characterization of solute transports during nanofiltration. In nanofiltration, the solute concentration in the feed solution is one of the factors which have an effect on solute rejection. So, it was important to study the effect of solute concentration on these transport parameters. In this approach, the model transport parameters, Keff and Φ have been related to the feed solution concentration. While serving to us as the equilibrium equations of Donnan and that of Kirianov, we have established relations between these parameters of transport to the concentration of the feed aqueous solutions. Both the effective transfer coefficient and the transmittance increase when concentration increases. The relations describe our experimental results satisfactorily. They have also allowed us the determination of the order of the charge density of the membrane. It was found to be 8.57 × 10−4 and 49.5 × 10−4 equivalent mol l−1 which correspond to −33 and −191 mC m−2 , respectively for a 0.4 ␮m membrane thickness. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Nanofiltration; Modeling; Transport parameters; Solute concentration; Charge density; Heavy metals

1. Introduction Heavy metals are recognized as one of the serious environment contaminants because of their higher toxicity, accumulation and retention in human body. Cadmium is attracting wide attention from environmentalists as one of the most toxic heavy metals. It is a cumulative toxicant causing progressive chronic poisoning, thus, the maximum permissible level treated waste water is 5 ␮g l−1 . The pollution sources ∗ Corresponding author. Tel.: +33-2-99-87-13-15; fax: +33-2-99-87-13-99. E-mail address: [email protected] (S. Taha).

of cadmium in waters are industrial activities such as electroplating industry, pigment and plastics industries [1]. Therefore, it is in interest to remove such a pollutant from water and nanofiltration appears to be a potential treatment process. This membrane process has attracted increasing attention in water treatment [5,7,14] during recent years; thanks to the development of new and wide application in drinking water [27] and industrial effluents [28]. The transport mechanisms in synthetic membranes have been of interest since the beginning of membrane research [16,22]. Solvent and solute transport mechanisms through a nanofiltration membrane requires to know the characteristics of the membrane, the solution

0376-7388/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 6 - 7 3 8 8 ( 0 2 ) 0 0 3 2 8 - 9

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Y. Garba et al. / Journal of Membrane Science 211 (2003) 51–58

Nomenclature a ai api a0i A1 A2 b B0 B1 B2 C∗ Ci∗ Cmi C0 C0i Cpi Cx D Deff Di F Ji Js Jv Keff Km Kp K0 M Mi M Mi pi R

coefficient ion activity in the bulk solution (mol m−3 ) ion activity in the permeate solution (mol m−3 ) ion activity in the feed solution (mol m−3 ) coefficient (m3/2 mol−1/2 ) coefficient (m3 mol−1 ) coefficient coefficient (m−1 s) coefficient (m1/2 s mol−1/2 ) coefficient (m2 s mol−1 ) solute concentration in the membrane (mol m−3 ) ion i concentration in the membrane (mol m−3 ) ion i concentration in the film (mol m−3 ) solute concentration in the feed solution (mol m−3 ) ion i concentration in the feed solution (mol m−3 ) ion i concentration in the permeate solution (mol m−3 ) membrane charge density (mol m−3 ) solute diffusion coefficient (m2 s−1 ) solute effective diffusion coefficient in the system (m2 s−1 ) ion i diffusion coefficient (m2 s−1 ) Faraday (96,485 C) ion i flux (mol m−2 s−1 ) solute flux (mol m−2 s−1 ) permeate flux (m s−1 ) effective transfer coefficient (m−1 s) constant (V m) constant (V m) constant (V m) solute transport coefficient (m2 s−1 ) ion i transport coefficient (m2 s−1 ) solute transport coefficient ion i transport coefficient ion i permeability (m2 s−1 ) perfect gas constant (8.31 J mol−1 K−1 )

R R∗ Ri rm rp r0 Sh x Zi zi Greek αp δ λ µi µ0i νi Ψ

Ψ D Ψm Ψp Ψ0

solute observed rejection (%) solute real rejection (%) ion i observed rejection (%) average distance between two ions in the membrane (m) average distance between two ions in the permeate solution (m) average distance between two ions in the feed solution (m) Sherwood coefficient distance variable (m) ion i charge ion i valence letters proportionality coefficient film thickness (m) membrane thickness (m) ion chemical potential (J mol−1 ) ion standard chemical potential (J mol−1 ) stoichiometric coefficient electrostatical potential (V) Donnan potential (V) electrostatical potential in the membrane (V) electrostatical potential in the permeate solution (V) electrostatical potential in the feed solution (V)

and the operating conditions [9,10]. In general, most commercial membrane characteristics are unknown. The membrane material and manufacture method are most of the time a confidential information. In addition, values of the pores size, the membrane thickness and its charge density are not given by the manufacturer [3,20]. In previous work [8,24], from Nernst–Planck and film theory equations, we have established a linear model characterized by two transport parameters: effective transfer coefficient, Keff , and transmittance, Φ. Both transport parameters depend on M and M/λ which are respectively the coupling coefficient and the transfer coefficient through the membrane. The determination of values of the two latter parameters have been carried to understand transport mechanisms of

Y. Garba et al. / Journal of Membrane Science 211 (2003) 51–58

cadmium salts solutions through a nanofiltration membrane [25]. Moreover, in nanofiltration, the solute concentration in the feed solution is one of the factors which have an effect on solute rejection [18,19]. So, it is important to study the effect of solute concentration on these transport parameters. In this paper, relationships between concentration and transport parameters, Keff and Φ, have been established in order to study the effect of the concentration on the solute rejection and to determine the lack of membrane properties such as the charge density.

2. Theory 2.1. Fundamental model equation The fundamental equation of our simple model was developed in combination of the extended equation of Nernst–Planck proposed by Dresner [2,4,5] with the film theory [6,7] and assuming that the rejection is low [8], the model equation was established as follows: ln(1 − Ri ) =

1 Jv + ln Φi Keff

(1)

All math transformations and expressions of 1/Keff and Φ i have been detailed in our previous work [24]. Assuming that the concentration effects on both parameters do not cancel each other out, both Keff and Φ i are function of solute concentration. It is so important to study the evolution of these parameters when solute concentration in the bulk varies. This approach will be of great practical interest owing to the fact that it will enable us to determine an important characteristic of the membrane, the charge density, by carrying out some tests according to the feed concentration, which can be used to estimate other properties of the system. 2.2. Relationship between concentration and transmittance, Φ During the filtration, it can be considered that a part of solute remains in the membrane due to the electric neutrality and the adsorption phenomena, an other part is rejected due to the membrane physical resistance and the rest passes into the permeate under the transport mechanisms [11,17]. Taking into account the part

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of the solute which passes through the membrane, it can be considered that the rest is rejected. The amount of solute in the membrane is strongly dependent on the solute transmittance, Φ, which is the percentage of solute passing through the membrane when the permeate flux is near to zero. Considering the assumption of Dresner [2], the solute concentration in the membrane can be considered to be constant. Moreover, solute concentrations in the bulk and in the membrane are related by the distribution equation of Boltzmann [10,12] where Donann potential can be written, in dilute solutions cases, as follows [13,14]:
∗  Cj RT

ΨD = ln (2a) zj F vj C 0  ∗  Ci RT (2b) ln

ΨD = zi F vi C 0 C0 is the solute concentration in the feed solution. Ci∗ and Cj∗ are respectively the concentrations of cations i, and anions j, in the membrane. The electroneutrality in the membrane allows to write Eq. (3): Cx = zi Ci∗ − zj Cj∗

(3)

Cx is the membrane charge density expressed in concentration unity. At the beginning of the filtration, the negative charges of the membrane are neutralized by protons. During the filtration, as cations have a higher charge and are in higher concentration than protons in the solution ([H+ ] ≈ 10−5 and [Cd2+ ] > 4.10−5 mol l−1 ) they reject protons to the bulk and establish the electroneutrality in the membrane. However, anions are in a low amount in the membrane due to the repulsion of the membrane negative charged group. Assuming that anion concentration is negligible [13] in front of cation concentration, by combination of Eqs. (2a), (2b) and (3), we have obtained Eq. (4): −zj /zi  Cj∗ Cx = (4) vj C0 z i vi C 0 The comparison between Eq. (4) and exponential transforming of Eq. (2a) leads to obtain the following equation:   zj F zj C0 (5)

ΨD = ln vi zi RT zi Cx

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The electrochemical balance between the membrane, permeate and retentate, allows to consider that the solute electrochemical potentials are the same in the three compartments. Consequently, this assumption leads in the case of dilute solutions to the following equation:   Cpj zj F ln = (Ψp − Ψ0 ) (6) C0j RT Otherwise, when pressure is low, the solvent flux can be neglected leading to write the fundamental equation of the model (Eq. (1)) for ion j in the form [24]:   Cpj = lnΦj (7) ln C0j Comparison between Eqs. (6) and (7) gives: ln Φj =

zj F (Ψp − Ψ0 ) RT

(8)

Otherwise, the potentials surrounding an ion, described by the law of Coulomb, can be written respectively in the feed Ψ 0 , the film Ψ m and the permeate solutions Ψ p by equations in the following forms:   Km r0 − K0 rm rp Ψm − Ψ 0 = (9) Ψ0 − Ψ p K 0 rp − K p r0 rm where r0 , rm , and rp are respectively the average distances between two ions in the feed, the film and the permeate solutions. K0 , Km , and Kp are constants which depend on the solution dielectric constant and the ions surrounding charges. Assuming that the second term of the Eq. (9) is a constant and equal to αp−1 (where α p is noted as the proportionality coefficient ) allows to obtain the following equation: Ψ0 − Ψp = αp (Ψm − Ψ0 ) = αp ΨD

(10)

Combination of Eqs. (7), (8) and (10) gives the Eq. (11) which can be written in the linear form and is given by Eq. (12) as follows:   zj C0 ln Φj = − αp ln vi zi (11) zi Cx ln Φj = a ln[C0 ] + b with zj a = − αp zi

  zj vi zi b = − αp ln zi Cx

(12)

Determination of values of constants a and b by linear regression of Eq. (12) allows to calculate the value of Cx . 2.3. Relationship between concentration and Keff In the film and the membrane, the anion is generally accompanied by the cation in order to assume the electroneutrality of the system. In these conditions, they loose considerably their character to move individually. Anion and cation move in the same direction at the same rate. By consequence, there is formation of ion pair. According to the concepts on formation of ion pair, Kirianov cited in Antropov [15] has reviewed the relationship between diffusion coefficient and the solute concentration. He has proposed the fundamental equation:  Deff = D(1 − A1 C0 + A2 C0 ) (13) where D is the diffusion coefficient when the solute concentration is close to zero, Deff is the effective diffusion coefficient, A1 is a coefficient which depends on the ion charge, the dielectric constant of the solution and the temperature, A2 is a coefficient depending on the distance which separates the ions charges. The solute effective transfer coefficient given by our model [24] can be expressed as: 1 δ M =− λ+ Keff M D

(14)

where D/δ is the solute transfer coefficient in the film and −M/(M λ) is the solute transfer coefficient in the membrane. The solute transfer in the system film-membrane can be considered as a transfer through an equivalent homogenous medium [21,23] with a transfer coefficient Keff which can be related to the diffusion coefficient by the Sherwood relationship: Keff =

Sh Deff dh

(15)

Combining Eqs. (13) and (15), leads to write the effective transfer coefficient in the form:  Keff = B2 C0 − B1 C0 + B0 (16) where B0 , B1 and B2 are coefficients which depends on the ion charge, the dielectric constant of the solution and the temperature.

Y. Garba et al. / Journal of Membrane Science 211 (2003) 51–58

Applying Eqs. (12) and (16) to our nanofiltration results for CdCl2 and Cd(NO3 )2 solutions enables us to determine the values of the transport parameters in function of feed solute concentration and the charge density of the membrane. 3. Experimental 3.1. Apparatus The experiments were performed on a MILLIPORE PROSCALE pilot operating in the batch circulation mode, which means that both permeate and concentrate were carried back to the vessel. A schematic diagram of the experimental system was given in reference [24]. The nanofiltration module is equipped with a NANOMAX 50 membrane which is a composite polyamide negatively charged membrane in spiral form with a 0.37 m2 area. The hydraulic diameter and the hydraulic permeability were calculated to be in order of 6 mm and 2.5 × 10−6 m s−1 , respectively. It has a macroporous polyester mechanical support (120 ␮m), a microporous polysulfone intermediate structure (40 ␮m) and an active layer in polybenzamide (0.4 ␮m). The data of the manufacturer announce a cut-off about 300 Da for uncharged solutes and a pore diameter of 0.5 nm. The feed flow rate was fixed at 70 × 10−6 m3 s−1 which gives a value of 1780 as estimated Reynolds number. 3.2. Experimental conditions Solutions were prepared using PROLABO cadmium salts (3CdSO4 ·8H2 O; CdCl2 ; Cd(NO3 )2 ·4H2 O) in the concentration range 1–10 mg l−1 for cadmium. The experiments were performed at the 293 K over pressure range of 0.12–1.5 MPa. The pH was 5.8 for all experiments. Permeate and concentrate were sampled after 1 h of filtration. After removing the feed solution from the module, the system was rinsed with HCl 10−2 M. It was then rinsed with water until the conductivity and the pure water fluxes were restored in order to ensure an efficient cleaning.

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interval. Observed rejection was calculated by the relation: R =1−

Cp C0

where Cp and C0 are respectively the salt concentrations in the permeate and in the feed solution. The cations concentrations were measured by atomic absorption spectrophotometry (VARIAN AA-1275) and the anions concentrations by ionic chromatography using WATERS 431 conductivity detector, IC-PAK Anion HR-26765 column, WATERS 501 pump, eluent containing for 1 l: 0.32 g sodium gluconate, 0.36 g boric acid, 0.50 g sodium tetraborate decahydrate, 5 ml glycerin, 20 ml n-butanol, 120 ml acetonitrile and Milli-Q water.

4. Results and discussion 4.1. Validation of the model Transport parameters have been calculated by regression of Eq. (1) using experimental results. Figs. 1–3 show that calculated and experimental results for CdCl2 , Cd(NO3 )2 ·4H2 O and CdSO4 ·8H2 O at 1, 5 and 10 mg l−1 are in good agreement. It also shows that sulfates rejection is higher than chlorides and nitrates rejections. This observation can be explained by the double negative charge of sulfate ions which are strongly repelled by membrane negative charge. The cations are rejected according to Donnan effects and in order to assume the bulk electroneutrality.

3.3. Analytical methods The volumetric flux Jv was determined by measuring the volume of permeate collected in a given time

Fig. 1. Comparison between experimental and calculated rejection rates in function of flow permeate for CdCl2 at 1, 5 and 10 mg l−1 .

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Y. Garba et al. / Journal of Membrane Science 211 (2003) 51–58

Fig. 2. Comparison between experimental and calculated rejection rates in function of flow permeate for Cd(NO3 )2 ·4H2 O at 1, 5 and 10 mg l−1 .

Otherwise, sulfate ions have higher hydration energy, 1047 kJ mol−1 , than chlorides and nitrate ions which have 325 and 310 kJ mol−1 of energy, respectively [16]. It is well known that solute transport is favored by a high solvation of solute in the membrane matrix or a low hydration [17]. So, the higher the solute hydration is, the better the solute is rejected. In relation to the decrease of the retention when the concentration increases in the case of chlorides or nitrates salts, the results have shown that when the Cl− or NO3 − ions concentration was increased, Cd2+ retention was decreased. In fact, soluble complex like CdCl+ and CdCl0 could be formed. These latters will be less hydrated and passed easier through the membrane than the free Cd2+ . Bhattacharya et al. [29] have shown in the same manner that the presence of neutral complex CdCl0 allowed a decrease in the retention. As our studied solute concentration was low, the concentrations of the formed complex was relatively too

Fig. 3. Comparison between experimental and calculated rejection rates in function of flow permeate for CdSO4 ·8H2 O at 1, 5 and 10 mg l−1 .

Fig. 4. Evolution of the transmittance parameter against the concentration.

low. So, we could not attribute this decrease of Cd2+ retention to the complex formation. It seems too difficult and unreasonable to let them responsible for the decrease of Cd2+ retention. 4.2. Concentration effects on transport parameters Table 1 show the concentration effect on transport parameters values in the cases of CdCl2 and Cd(NO3 )2 solutions. Figs. 4 and 5 show evolution of these parameters against the concentration. Both the effective transfer coefficient and the transmittance increase when concentration increases. This evolution can be explained by the increasing of the transport by diffusion and by convection [21] when solute concentration increases. According to the Eqs. (12) and (16), values of a, b, B0 , B1 and B2 were determined and shown in Table 2. Values of a and b allow to estimate the membrane charge density Cx values for CdCl2 and Cd(NO3 )2 solutions. It was found to be 8.57 × 10−4 and 49.5 × 10−4 equivalent mol l−1 which correspond to −33 and −191 mC m−2 , respectively for a 0.4 ␮m membrane thickness. The difference observed between these two values can be explained by adsorption phenomena of ions on the membrane which modifies the charge density. A similar result has been obtained by a recent work [26] where the retention of ionic components was analyzed by the DSPM model to evaluate the membrane charge density. The authors showed that the charge density is not constant but depends very much on the salt nature and its concentration. Moreover, it could lead to a change in the sign of the membrane charge from a negative to a positive value. Otherwise, pressure induced potential

Y. Garba et al. / Journal of Membrane Science 211 (2003) 51–58

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Table 1 Transport parameters Keff and Φ values of CdCl2 and Cd(NO3 )2 solutions Cd2+ (mg l−1 )

CdCl2 Keff

1 5 10

Cd(NO3 )2

106

(m s−1 )

50.2 110.7 179.6

Φ (%)

R2

Keff 106 (m s−1 )

Φ (%)

R2

55.8 71.9 78.5

0.99 0.98 1.0

49.9 161.1 470.8

58.6 69.1 72.1

1.0 1.0 1.0

Fig. 5. Evolution of effective transfer coefficient against the concentration.

Table 2 Coefficients values of CdCl2 and Cd(NO3 )2 solutions Salts

a

b

B0

B1

B2

CdCl2 Cd(NO3 )2

0.1499 0.0943

1.1625 0.5659

27.566 212.69

3.6445 92.508

1.3236 12.715

across a composite polyamide/polysulfone reverse osmosis membrane were measured with NaNO3 , NaCl and MgCl2 solutions at different feed concentrations [30,31]. Results have shown that the zeta potential, which is reliable to the charge density, was practically independent of concentration, but it depends on the electrolyte. This is agreed with our obtained results.

the rejection rate according to the concentration and the nature of the associated anion. In addition, we observed that the two parameters of transport, the effective coefficient of transfer and transmittance, characterizing the model, increase when the concentration of the solutions grows. This evolution is dependent, on one hand, the increase in the coefficient of diffusion and the permeability and, on the other hand, the effect shielding of the cation facilitating the passage of the aqueous solution. This model finally led to the qualitative determination of an approximate value of the charge density of the membrane which is about 10−3 equivalent mol l−1 . References

5. Conclusion In spite of the difficulties encountered by the ignorance of a certain number of the characteristics of the used membrane, this study has allowed the modeling of our experimental results. The results obtained show that there is a good agreement between the theoretical and experimental values based on the evolution of

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