Mass Transfer Modeling for Salt Transport in Amphoteric Nanofiltration

1096 Lisboa Codex, Portugal. The mass transfer in the tangential turbulent flow inside a nanofiltration tubular membrane is described by a modified ed...
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Ind. Eng. Chem. Res. 1998, 37, 4118-4127

Mass Transfer Modeling for Salt Transport in Amphoteric Nanofiltration Membranes Maria Dina´ Afonso* and Maria Norberta de Pinho Chemical Engineering Department, Instituto Superior Te´ cnico, Avenida Rovisco Pais, 1096 Lisboa Codex, Portugal

The mass transfer in the tangential turbulent flow inside a nanofiltration tubular membrane is described by a modified eddy diffusivity model, which accounts for the effect of high permeation fluxes on the mass transfer rates through a permeation Reynolds number. The correlation achieved for the polarization modulus in this situation is C ) exp(0.162Rep0.93Re-0.29Sc2/3). The ionic transport in the membrane active layer is described through the extended Nernst-Planck equations and the Donnan equilibrium at the membrane-solution interfaces. The active layer of the nanofiltration membrane is made of an amphoteric polymer containing quaternary amine and sulfonic acid groups. In the experiments of permeation of sodium chloride solutions, the Reynolds number ranges from 9.6 × 103 to 1.1 × 105, the pressure from 10 to 25 bar, and the feed concentration from 1.8 to 193 mol/m3. A good agreement is observed between the experimental and the calculated permeation fluxes and the salt rejections. The membrane effective charge depends on the feed concentration upon the relationship ln CM (mol/m3) ) 4.14 + 0.527 ln CSf (mol/m3). Introduction The early 1980s were characterized by important developments of reverse osmosis membranes for seawater and brackish water desalination. In the later case, the membranes of low sodium chloride rejections (∼96%) were, at first, generally designated by loose reverse osmosis membranes and, only later on, became to be known as nanofiltration (NF) membranes. Their separation characteristics stand between ultrafiltration (UF) and reverse osmosis (RO) and the membrane selectivity was often attributed to the interplay of both molecular sieving mechanisms characteristic of UF and diffusion mechanisms characteristic of RO. The development of NF membranes from polymer materials with acidic/basic functionalities has led to the occurrence of membrane-electrolyte interfaces displaying surface charge distributions of major relevance on the separation performance. The main objective of this work is the prediction of nanofiltration performance by an integrated model describing the concentration polarization and the mass transport mechanisms in the ampholytic active layer of a NF membrane. Extended Nernst-Planck equations should be used to adequately describe mass transfer in the membrane active layer since electromigration has to be considered together with the diffusion-convection mechanisms. In the several works where the Nernst-Planck equations were used for modeling NF mass transfer, two approaches can be identified, whether the equations are coupled with the Poisson equation or with the electroneutrality condition. The study of the ionic transport in charged capillaries of small dimensions, where the double electrical layer width is comparable to the pore dimensions, requires the coupling of the Nernst-Planck equations to the Poisson equation.1,2 The Poisson * To whom correspondence may be addressed: phone, +351 1 8417595; fax, +351 18499242; e-mail, [email protected].

equation can be replaced by the electroneutrality condition, when the local density charge due to the fixed charges and the mobile ions is low. This hypothesis greatly simplifies the integration of the Nernst-Planck equations.3-7 With respect to the boundary conditions used by the referred authors in the interfaces solution-membrane, Menon and Spencer3 used the Nernst-Planck equations coupled to the Poisson equation in both external solutions, Tsuru et al.5 mentioned the Donnan interfacial equilibrium but apparently they did not use it, Bowen and Mukhtar6 and Bowen et al.7 used the Donnan interfacial equilibrium, and Timmer et al.4 only considered the concentration polarization in the interface membrane-feed solution. In the present work, due to the low-density charge on the membrane surface (up to 1 M), electroneutrality is assumed both in the membrane and in the feed and permeate solutions, allowing the application of the Donnan equilibrium in both interfaces solution-membrane. Summing up, the model proposed in this work is based on the application of the extended Nernst-Planck equations coupled to the electroneutrality condition in the membrane and the assumption of the Donnan equilibrium in the interfaces membrane-feed solution and membrane-permeate. A modified eddy-diffusion model takes into account the concentration polarization at the feed solution. The integrated model is validated by nanofiltration experimental results covering a wide range of operating conditions. The salt rejection is predicted as a function of the salt feed concentration, which in turn is correlated to a model parameter corresponding to the membrane effective charge concentration. Theory The diffusion and the convection mechanisms are usually associated to UF, NF, and RO, whereas the electromigration has been traditionally associated to

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Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4119

To carry out the integration a correlation between the eddy diffusivity, (t), and the dimensionless distance, y+, has to be introduced. We selected one recommended for the region near the wall11 and used in the adjustment of heat and mass transfer data obtained by several authors12

(t)/υ ) 0.00096(y+)3

Figure 1. Scheme of solute concentration profiles in the feed and permeate solutions due to the concentration polarization phenomena. The diffusion and convection result from the solute concentration gradient in the feed solution and the pressure difference across the membrane, respectively.

processes involving ion exchange membranes, such as electrodialysis.8 However, NF membranes in contact with electrolytes acquire a surface charge distribution, and therefore the electromigration becomes an important transport mechanism in the salt separation. Whereas the ion exchange membranes have fixed charges, NF membranes present surface charge distributions induced by the operating conditions (pH, concentration, and composition of the feed solution), their charge densities being lower than those of ion exchange membranes.9 For the mass transfer in nanofiltration to be modeled, the following phenomena are then considered: concentration polarization in the feed solution; Donnan equilibrium in the interfaces feed-membrane active layer and permeate-membrane active layer; diffusion, convection, and electromigration in the membrane active layer, described by the extended Nernst-Planck equations. 1. Concentration Polarization. The preferential solvent permeation occurring in NF membranes leads to a solute concentration profile developing from the membrane surface to the feed bulk (Figure 1). This phenomenon, commonly designated by concentration polarization, is usually described through the film theory. A steady-state mass balance to solute S in the feed polarization layer yields10

NSy ) constant ) vpCSp

NSy ) (DSW + (t))

dCS + vpCS dy

dCS ) vp(CSp - CS) dy

∫CC

Sf

vp dCS ) CS - CSp υ

Scp

∫0δ

dy

1 (t) + Sc υ

CSf - CSp

(3)

(4)

(

) exp 16.66

)

vp Sc2/3 〈u〉 xfF

(6)

The inclusion of this equation in the integrated model leads to predicted rejections showing an evident gap with respect to the corresponding experimental rejections. In fact, the correlation for the eddy diffusivity was obtained for impermeable walls, and it is well-know that the mass transfer rates across the membrane play an important role on the reduction of the concentration polarization. To take this into account, a permeation Reynolds number, Rep, for fully developed turbulent tubular flow, high Schmidt numbers (Sc > 100), and moderate permeation rates in RO membranes (vp > 5 × 10-5 m/s) was introduced.13 In the present work, this parameter should be of major relevance due to the higher permeation rates occurring in NF membranes. The Re and Rep are isolated in eq 7, and their exponents c and b, respectively, together with the constant a, are determined through the algorithm presented in Scheme 1.

CScp - CSp CSf - CSp

(

)

Repb 2/3 ) exp a c Sc Re

(7)

The circulation and the permeation Reynolds numbers are defined, respectively, as

Re )

FfD〈u〉 µf

(8)

FpDvp µp

(9)

Rep )

The effect of the osmotic pressure on the permeation rate is given by

vp )

The integral of both members over the limits of the polarization concentration layer is

-

CScp - CSp

(2)

Introducing this flux equation in eq 1

(DSW + (t))

Introducing this correlation and the friction factor given by the Blasius formula, which is valid for fully developed turbulent flow in long, smooth, circular tubes, the solute concentration in the interface feed solution-membrane is finally obtained

(1)

being the constant determined by the flux boundary condition. Considering that the total flux of solute S is the sum of a diffusive flux and a convective flux, being the former the sum of two contributions, the molecular and the eddy diffusion

(5)

lp [∆Poper - (πcp(CScp) - πp(CSp))] Fp

(10)

2. Donnan Equilibrium at the Solution-Membrane Interfaces. The restrictive character of Donnan equilibrium validity is well documented throughout the scientific literature. The Donnan exclusion principle determines the ionic partition between the membrane and the electrolytic solution, for feed concentrations lower than the membrane fixed charge concentration.14 Some authors pointed out its nonvalidity for high or low electrolyte concentrations and for nonhomogeneous membranes (nonuniform charge distribution).15,16

4120 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 Scheme 1. Schematic Diagram of the Optimization Procedure

tion and equal to unity on the grounds that the membranes are swollen in water.17,19-24 For hydrophilic membranes the ratio of the activity coefficients, γAγC/ γ(2, can be made equal to unity.25 The errors due to these approaches are incorporated in the determination of the membrane charge density.24 The correction of the activity coefficients only appears to be important for concentrations higher than 0.5 M.23 Bearing in mind that the concentrations of the feed and the permeate solutions are not high enough to occur ionic aggregates and also that the membrane under study is hydrophilic, we can safely assume that the ratio of the activity coefficients in both phases, γ(membrane/ γ(solution, is approximately equal to unity. Moreover, for the electrolyte concentrations tested, membrane contamination nor the development of thin electrical double layers surely do not occur; therefore, the hypothesis of small deviations to Donnan equilibrium is assumed. Consider two phases in equilibrium, 1 (feed or permeate solutions) and 2 (charged membrane), containing dissolved cations and anions of valence zC and zA, respectively. In such a case, the electrochemical potentials of each dissolved ion in both phases should be equal, or in terms of the chemical and the electrical potentials

µC1 + zCFψ1 ) µC2 + zCFψ2

(11)

µA1 + zAFψ1 ) µA2 + zAFψ2

(12)

Eliminating ψ1 and ψ2 from eqs 1 and 2, introducing the chemical potential definition, and assuming that the standard chemical potential of each species is equal in both phases, the Donnan equilibrium equation is finally obtained

() () aA1

aA2

zC

)

aC2

-zA

aC1

(13)

The establishment of a Donnan equilibrium requires electroneutrality in both phases, i.e.

For high electrolyte concentrations, the typical distance between the adjacent charged groups in the membrane is comparable or higher than the width of the electrical double layer around these groups, and therefore the role of the electrostatic rejection mechanisms decreases and the nonelectrostatic mechanisms contribution increases.16,17 For low electrolyte concentrations, several authors attributed the deviations with respect to the Donnan model to the membrane contamination and the counterions shield effect, though the nonhomogeneity of the membrane charge distribution should be the main reason for this fact.16 Electrolyte sorption and nonuniform charge distribution in the membrane may be responsible for the deviations to the Donnan equilibrium, and a correction of the membrane electrolyte activity coefficients can be made to take these effects into account.18 In fact, an important assumption often used is to consider the ions activity coefficients independent of the ionic concentra-

zCCC1 + zACA1 ) 0

(14)

zCCC2 + zACA2 - CM ) 0

(15)

where CM is the effective charge concentration in the membrane active layer, which is supposed to be uniform and low (up to 1 M). In a previous work, where the NF membrane tested was characterized, we concluded that the membrane presents a negative surface charge.26 2.1. Feed-Membrane Interface. The Donnan equilibrium and the electroneutrality equation for a uniunivalent salt at the interface membrane active layerfeed solution (interface 1) are, respectively

aC1aA1 ) aCcpaAcp

(16)

CC1 ) CA1 + CM

(17)

Introducing the definition of each ion activity (the activity of species j is defined as the product of the species concentration and its activity coefficient aj ) Cjγj)

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4121

CC1γC1CA1γA1 ) CCcpγCcpCAcpγAcp

(18)

Introducing the electroneutrality condition and the definition of the salt activity coefficient (the activity coefficient of salt Cϑ+Aϑ-, is defined in terms of each ion activity coefficient: γ( ) (γCϑ+γAϑ-)1/(ϑ++ϑ-)) and bearing in mind that CCcp ) CAcp ) CScp

CC12γ(12 - CMCC1γ(12 - CScp2γ(cp2 ) 0

(19)

Dividing both sides of the equation by (γ(1)2 and assuming that γ(1 = γ(cp

CC12 - CMCC1 - CScp2 ) 0

(20)

The meaningful solution of this equation is

CC1 )

CM +

xC

2 M

+ 4CScp2

2

(21)

To determine CA1 the electroneutrality condition is used. 2.2. Permeate-Membrane Interface. The Donnan equilibrium for a uni-univalent salt at the membrane active layer-permeate solution interface (interface 2) is given by

aC2aA2 ) aCpaAp

(22)

Introducing the definitions of each ion activity and the salt activity coefficient:

CC2CA2γ(22 ) CCpCApγ(p2

(23)

Bearing in mind that CCp ) CAp ) CSp, dividing both sides of the equation by (γ(2)2 and assuming that γ(2 = γ(p, the salt concentration in the permeate solution is finally obtained

CSp ) xCC2CA2

(24)

3. Nernst-Planck Equations. Considering the ions fixed (or adsorbed) to the membrane matrix are uniformly distributed, the membrane can then be modeled as a homogeneous gel phase of fixed and mobile ions in the solvated polymer.27,28 For modeling the mass transfer in the nanofiltration membrane active layer, we assume isotropic membrane active layer, characterized by an uniform dielectric constant and an electrical charge distribution of uniform and low density (up to 1 M), unidirectional ionic flux in the perpendicular direction to the membrane surface, neglecting the end effects, and negligible pressure drop and concentration variation in the tangential feed flow direction. As the highest feed concentration tested is 0.2 M and the membrane charge concentration in a NF membrane is supposedly lower than 1 M, we also assume that both the feed and the permeate solutions and the membrane phase are thermodynamically ideal. For steady-state unidirectional flow of an incompressible fluid, the continuity equation in rectangular coordinates yields

vM ) constant ) vp

(25)

Figure 2. Scheme of ion concentration profiles in the membrane active layer. The diffusion, convection, and electromigration result from the solute concentration gradient, the pressure difference and the electrical potential gradient through the membrane, respectively.

A differential steady-state mass balance for ion j yields after integration10

Njy ) constant ) vpCSpϑj

(26)

being the constant determined by the permeate flux boundary condition. The total flux of ion j consisting in the sum of the diffusion, convection, and electromigration contributions (Figure 2), can be described by the extended Nernst-Planck equations for dilute solutions.29 The literature reviewing the limitations of the NernstPlanck equations is mostly concerned with ion-exchange membranes, which do not present the low charge density, the homogeneity, and the isotropicity, characteristics of the amphoteric active layer of the NF membrane under study. A major concern in the application of these equations is the assumption of the ionic flux independence. The validity of this assumption has been experimentally confirmed for dilute electrolytic solutions, and as a matter of fact, in these conditions the cross terms coefficients tend to be zero. (The more general flux equation for each of two mobile ions in a membrane is, for ions 1 and 2: Njy ) lj1(∂µ˜ 1/∂y) + lj2(∂µ˜ 2/∂y). The socalled cross term coefficient is lji.) On the contrary, for higher concentrations (up to 3 M), the cross terms may be 25% of the main term. Though the cross terms cannot be neglected in conventional ion exchange membranes, with 1 to 10 M of active centers,30,31 this is definitely not the case of NF membranes bearing lower charge densities. Besides the coupling of the mobile ions fluxes, the electro-osmotic contribution resulting from the ion-solvent coupling can also be neglected for weakly charged membranes.32,33 The extended Nernst-Planck equations describing the ionic fluxes can be written as follows:

NCy ) -DCM

DCMzCF dψ dCC + vMCC C ) vpCSpϑC dy RgT dy C (27)

NAy ) -DAM

DAMzAF dψ dCA + vMCA C ) vpCSpϑA dy RgT dy A (28)

The system of eqs 27 and 28 satisfies the condition of

4122 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998

Figure 3. Scheme of NF experimental installation: 1, feed tank; 2, washing water tank; 3, rotary pump; 4, centrifugal pump; 5, heat exchanger; 6, tubular modules; 7, permeate; 8 and 9, flowmeters; 10 and 11, manometers; 12, 13, and 14, needle valves; 15, back pressure valve; 16 and 17, ball valves; 18, pressure regulator; 19, pressure control; 20, municipal water; 21, drain.

and separating the variables

nullelectrical current flux

zCNCy + zANAy ) vpCSp(zCϑC + zAϑA) ) 0

(29)

As the variation of the electrical potential with the position, Ψ(y), is unknown, dΨ/dy is isolated in eqs 27 and 28, and the two expressions obtained are equated

dCC - CCvM dy ) DCMFzCCC RgT

vpCSpϑC + DCM

dCA - CAvM dy (30) DAMFzACA RgT

vpCSpϑA + DAM

Multiplying both sides of eq 30 by RgT/(DSMF) and replacing CA from the membrane electroneutrality equation, as well as its derivative, we obtain

(a1CC + a2)

dCC - a3CC2 - a4CC - a5 ) 0 (31) dy

where

(

a1 ) DSM

a2 ) DSMCM

(

a3 ) vM

)

zC2 - zC zA

zC2 - zC zA

(32) (33)

)

(34)

( )

a4 ) vpCSpzC(ϑC + ϑA) - vMCM a5 ) -vpCSpϑCCM

zC -1 zA

(35) (36)

∫CC

C2

C1

(a1CC + a2) dCC

a3CC2

+ a4CC + a5

)

∫0L dy

(37)

Although being possible to solve analytically, the integral of the left-hand side is easily solved numerically. The upper limit of integration of the right-hand side corresponds to the effective length of the ionic transport path, L, which is a modeling parameter taken as 10-9 m. In the situation of homogeneous isotropic porous membranes, a technique developed for the measurement of the membrane effective electrolyte conductivities and diffusivities yielded, in the range of moderate electrolyte concentrations (0.03-1 M), values of effective transport properties that are independent of electrolyte concentration and only dependent on the geometrical hindrances imposed to transport.34 The effective diffusivity of sodium chloride in the NF membrane was experimentally determined by the previous method, and the value obtained35 was 6.6 × 10-13 m2/s. We further assume that the effective diffusivity of both ions is equal to the salt effective diffusivity, i.e., DCM ) DAM ) DSM. In fact, in the absence of an external electrical field, the average distance between the ions obtained by the dissociation of a single salt should be low to guarantee that the solution remains electrically neutral locally. Besides, the electrical current flux must be null. This condition results from local electrical fields developed between the mobile ions such that the opposite charge ions move at the same velocity.31,36,37 Experimental Section 1. Membrane. The NF composite membrane was manufactured by Nederlandse Organisatie voor Toegepast (TNO). The membrane active layer consists of an ampholytic polymer containing quaternary amine groups and sulfonic acid groups. The active layer was formed by interfacial polycondensation on polysulfone com-

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4123

Figure 4. Experimental and calculated permeation fluxes (J) and rejections (R) of sodium chloride solutions at 40 °C as a function of the operating pressure, the circulation velocity, and the feed solution concentration. The pure water permeation fluxes (Jw) at 40 °C are also presented on the left-hand side figures.

mercial tubes with diameter and length of 14.4 mm and 20 cm, respectively. 2. Installation. The experimental setup is represented in Figure 3. A rotary pump pressurizes the feed and a centrifugal pump makes the concentrate partial

recirculation, to achieve high circulation velocities tangential to the membrane surface. 3. NF Experiments. Before the permeation experiments are started, the membrane is conditioned by circulating deionized water at 25 bar for 2 h.

4124 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 Table 1. Optimized Parameters for the Mass Transfer in NF a b c

0.162 0.925 0.289

CM1 (mol/m3) CM2 (mol/m3) CM3 (mol/m3)

94.29 254.8 1116

Each permeation experiment consists of three steps: (1) deionized water permeation at pressures of 10, 15 and 20 bar, to determine the pure water permeation fluxes, Jw, and the membrane hydraulic permeability, lp; (2) permeation of sodium chloride solutions of different concentrations, to determine the permeation fluxes, J, and the salt rejections, R; (3) membrane washing with deionized water at low pressure and high flow rate, to recover at least 90% of the initial Jw. A solution 0.05% of P3 Ultrasil-11 is used to clean the membrane for 10 min whenever the pure water permeation flux reached after a deionized water cleaning is lower than 90% of its initial value. NF experiments are performed in closed circuit to keep the feed concentration constant. The experiments are performed sequentially by increasing the pressure and then by decreasing the circulation velocity, to minimize the cumulative effects of the concentration polarization and membrane fouling. 4. Analytical Methods. The conductivity of the feed and permeate solutions are measured at 25 °C by a CRISON conductimeter, model 525. Results and Discussion The membrane hydraulic permeability, 9.1 kg/(m2 h bar), is typical of NF membranes with high permeation fluxes. As far as the permeation fluxes go, they are not significantly affected by concentration polarization since they are practically equal to the pure water permeation fluxes all over the ranges of variation of pressure, feed circulation velocity, and feed concentration (Figure 4). For the highest concentration, J deviates 9% from Jw. Also in Figure 4, the nanofiltration performance with regard to the apparent rejections is characterized by two asymptotic behaviors: Higher rejections that increase with the increasing pressure for values of feed circulation velocity higher or equal to 1.0 m/s. Lower rejections that decrease with the increasing pressure for the feed circulation velocity of 0.44 m/s. The higher rejections depend on the feed circulation velocity, and therefore a mechanism of salt rejection by electrostatic interactions is not acting by itself but surely there is a synergetic effect between the hydrodynamic and the electrical double layers. For the regimen of low rejections, the membrane is not displaying mechanisms of electrostatic rejection and the process is mainly controlled by concentration polarization. The model parameters, a, b, and c, relative to the description of mass transfer at the membrane-feed interface (eq 7), and CM1, CM2, and CM3, relative to the membrane effective charge displayed for each feed concentration tested, are optimized through adjustment to the experimental data. The parameters optimization is carried out by the subroutine E04CCF from NAG library and the objective function is taken as the sum of the absolute values of the deviations between the experimental and the calculated rejections. The calculation sequence is presented in Scheme 1, and the optimized model parameters are shown in Table 1.

Figure 5. Relationship between the membrane effective charge and the feed concentration.

The order of magnitude of the membrane effective charges, CM1, CM2, and CM3, is comparable to those of weak ion exchange membranes (10-103 mol/m3). The variation of the membrane effective charge with the feed solution concentration is displayed in Figure 5 and is correlated by eq 38 with a fitting parameter, R2, of 0.98.

ln CM (mol/m3) ) 4.14 + 0.527 ln CSf (mol/m3) (38) Petropoulos et al.15 observed that the fixed charge density of an ion exchange membrane depends on the electrolyte concentration, and they obtained a linear variation by plotting log CM vs log CSf. In a subsequent work, Petropoulos et al.16 observed a decrease of the fixed charge density for increasing dilution, and they obtained a linear variation by plotting CM vs log CSf. Takagi and Nakagaki21,22 also noticed a dependency of the effective charge density on the feed concentration, being the effective charge density lower than the membrane fixed charge for low electrolyte concentrations. This fact was attributed to the partial electrolytic dissociation of the membrane groups and to the adsorption of counterions in the membrane surface. A decreasing tendency of the charge density of RO membranes was identified for decreasing electrolyte concentrations.17 Some authors observed that the effective charge density of a NF membrane varies with the feed solution concentration.6,38 The increase of the NF membrane charge density as the bulk concentration increases is caused by co-ion adsorption, which in the limiting case may be the unique charging mechanism.6 The findings of the present work, particularly the quantitative description given by the correlation of the membrane surface charge with the salt feed concentration, are in full agreement with the conclusions of the previous authors. The comparison of the experimental and the calculated permeation fluxes and the sodium chloride rejections is presented in Figure 4. There is a good agreement between the predictions and the experimental data, although some slight deviations are detected for the velocity of 5.1 m/s and the concentration of 0.02 M. The relative error between the experimental and the calculated values ranges from 2 to 10% for the permeation fluxes and from 0 to 21% for the rejections, which indicates that the proposed model generally describes the variation of the permeation flux and the rejection with the operating conditions.

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4125

Conclusions The prediction of the nanofiltration performance in terms of permeation fluxes and salt rejections is achieved through an integrated model, considering the mass transfer mechanisms both in the feed solution adjacent to the membrane and in the membrane phase. The mass transfer in the fully developed turbulent flow at the feed stream is described by an eddy diffusion model that incorporates wall permeation and provides a new correlation for the concentration polarization modulus: C ) exp(0.162Rep0.93Re-0.29Sc2/3). The mass transfer by diffusion, convection, and electromigration through the NF membrane is related to the mass transfer rates across the feed solutionmembrane interface by the continuity equation and the Donnan equilibrium. That yields a correlation between the membrane effective charge and the feed salt concentration: ln CM (mol/m3) ) 4.14 + 0.527 ln CSf (mol/ m3). The calculated permeation fluxes and the salt rejections are in good agreement with the corresponding experimental values. The integrated model renders explicit parameters directly related to the hydrodynamics and the NF productivity and selectivity, and therefore it constitutes a major tool for the design and optimization of nanofiltration equipment. The solution permeation fluxes are practically the same as the pure water permeation fluxes all over the ranges of variation of the operating conditions. Even for the highest concentration, J only deviates 9% from Jw. The nanofiltration rejections present two asymptotic behaviors: higher rejections that increase with the increasing pressure for values of feed circulation velocity higher or equal to 1.0 m/s; lower rejections that decrease with the increasing pressure for the feed circulation velocity of 0.44 m/s. The higher rejections depend on the feed circulation velocity and therefore a mechanism of salt rejection by electrostatic interactions is not acting by itself, but surely there is a synergetic effect between the hydrodynamic and the electrical double layers. For the regimen of low rejections, the membrane is not displaying mechanisms of electrostatic rejection and the process is mainly controlled by concentration polarization. These asymptotic behaviors result from the interplay of the NF operating conditions in wide ranges of variation and at high permeation rates. Acknowledgment Part of this work was supported by the Commission of the European Communities under the BRITE/ EURAM program, Contract Breu CT90-0353-(SMA). Nomenclature a ) parameter of eq 7 aA ) anion activity (mol/m3) aC ) cation activity (mol/m3) b ) parameter of eq 7 c ) parameter of eq 7 C ) polarization modulus ((CScp - CSp)/(CSf - CSp)) CA ) anion concentration in the membrane active layer (mol/m3) CC ) cation concentration in the membrane active layer (mol/m3)

CM ) effective concentration of electrically charged groups on the membrane surface (mol/m3) CS ) concentration of solute S (mol/m3) D ) tubular membrane diameter (m) DAM ) anion effective diffusivity in the membrane active layer (m2/s) DCM ) cation effective diffusivity in the membrane active layer (m2/s) DSM ) effective diffusivity of solute S in the membrane active layer (m2/s) DSW ) diffusivity of solute S in water (m2/s) F ) Faraday constant (9.64867 × 104 (C/equiv)) fF ) Fanning friction factor (ratio of the drag force due to friction and the characteristic area and the characteristic kinetic energy per unit volume) J ) solution permeation flux (kg/(m2 s)) Jw ) pure water permeation flux (kg/(m2 s)) L ) effective length of ionic transport path (m) lji, ljj ) phenomenological coefficients (mol2/(J m s)) lp ) membrane hydraulic permeability (kg/(m2 h bar)) NAy ) anion flux in the y direction (mol/(m2 s)) NCy ) cation flux in the y direction (mol/(m2 s)) Njy ) flux of species j in the y direction (mol/(m2 s)) NSy ) flux of solute S in the y direction (mol/(m2 s)) R ) observed rejection ((1- CSp/CSf ) × 100 %) Rg ) gas constant (8.31434 J/(mol K)) Re ) Reynolds number in the feed solution (FfD〈u〉/µf) Rep ) Reynolds number in the permeate solution (FpDvp/ µp) Sc ) Schmidt number in the feed solution (µf/(FfDSW)) T ) absolute temperature (K) 〈u〉 ) feed circulation velocity (m/s) u* ) reference velocity (τ0/F)1/2 m/s) vM ) solution velocity across the membrane (m/s) vp ) permeate solution velocity (m/s) y ) rectangular coordinate perpendicular to the membrane surface (m) y+ ) dimensionless distance from the membrane surface into fluid (y u*/υ) zA ) anion valence (equiv/mol) zC ) cation valence (equiv/mol) Greek Symbols δ ) width of the polarization concentration layer adjacent to the membrane surface (m) ∆Poper ) pressure difference across the membrane (bar) (t) ) eddy diffusivity (m2/s) γ( ) salt activity coefficient γA ) anion activity coefficient γC ) cation activity coefficient ϑA ) moles of anions per mole of dissociated salt (mol/mol) ϑC ) moles of cations per mole of dissociated salt (mol/mol) ϑj ) moles of ions j per mole of dissociated salt (mol/mol) µ ) solution viscosity (kg/(m s)) µ˜ ) electrochemical potential (J/mol) µA ) anion chemical potential (J/mol) µC ) cation chemical potential (J/mol) π ) solution osmotic pressure (bar) F ) solution density (kg/m3) τ0 ) shear stress on the membrane surface (kg/(m s2)) υ ) solution kinematic viscosity (µf/Ff) (m2/s) ψ ) membrane electrical potential at the position y (V) Subscripts cp ) feed solution adjacent to the membrane surface f ) feed solution p ) permeate solution 1 ) interface membrane active layer-feed solution 2 ) interface membrane active layer-permeate solution

4126 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998

Appendix 1. Derivation of Equation 6

(

ln

CSf - CSp

( ) { [

ln

)

CSp - CSp

CSp - CSp

)

CSf - CSp

)

vp u*

+

Sc dy ∫051 + 0.00096Sc(y + 3 )

vp Sc × u* 3(0.00096Sc)1/3

(A1)

]

[(0.00096Sc)-1/3 + y+]2 1 + ln 2 2 (0.0096Sc)-2/3 - (0.00096Sc)-1/3y+ + y+

[

x3 arctg

ln

(

)

CScp - CSp CSf - CSp

)

( (

ln

x3 (0.00096Sc)-1/3

{

vp Sc2/3 1 - ln 1 u* 0.296 2

( )

x3 arctg ln

]}|

2y+ - (0.00096Sc)-1/3

5

}

1 1 + ln 1 + x3 arctg ∞ 2 x3

) )

CScp - CSp CSf - CSp

CScp - CSp CSf - CSp

vpSc2/3 u*

) 11.780

) 16.66

(A2)

0

vpSc2/3 〈u〉xfF

(A3)

(A4)

(A5)

The Fanning friction factor is given by the Blasius formula, which is valid for fully developed turbulent flow (104 < Re < 105) in long, smooth, circular tubes:

fF ) 0.0791/Re1/4

(A6)

Appendix 2. Physical Properties of Sodium Chloride Solutions The correlations between the density, the viscosity, and the osmotic pressure of sodium chloride solutions at 40 °C, and the concentration are obtained by the fitting of experimental data.39-42

F (kg/m3) ) 992.2 + 40.4110CS (M)

(A7)

µ (kg/(m s)) ) 6.536 × 10-4(1 + 0.08216CS (M) + 0.01838CS2 (M)) (A8) π (bar) ) 48.6754CS (M)

(A9)

The sodium chloride diffusivity is assumed to be independent of the concentration40 and the temperature effect is taken into account by the equation:

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Received for review February 16, 1998 Revised manuscript received June 29, 1998 Accepted July 7, 1998 IE980092P