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Modeling and simulation of osmotic pressure controlled electro-ultrafiltration in a cross-flow system
Journal of Membrane Science 199 (2002) 29–40
Modeling and simulation of osmotic pressure controlled electro-ultrafiltration in a cross-flow system Venkataraman Karthik, Sunando DasGupta, Sirshendu De∗ Department of Chemical Engineering, Indian Institute of Technology, Kharagpur 721 302, India Received 14 May 2001; received in revised form 12 September 2001; accepted 12 September 2001
Abstract Modeling and subsequent simulation of electric field enhanced ultrafiltration (UF) under osmotic pressure controlled regime in a rectangular channel under laminar flow conditions is presented herein. Bovine serum albumin (BSA) at pH 7.4 in 0.15 M NaCl solution is considered as the system. The analysis is carried out under the framework of boundary layer theory in conjunction with the electro-kinetic phenomena for a colloidal solution. Important effects due to the interaction between charged macro- and micro-particles with the solvent, e.g. viscoelectric effects, variation of diffusivity and Debye length with concentration, etc. have been incorporated in the model. It has been observed that the external dc electric field has a significant effect on the permeate flux and permeate concentration. A parametric study has been carried out to observe the effects of different operating conditions on the system performance. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Membranes; Mathematical modeling; Electrophoresis; Laminar flow; Electro-ultrafiltration; Mobility
1. Introduction Energy efficient membrane-based separation processes are becoming important unit operations in recent years to supplement the conventional separation processes, like distillation, centrifugation, etc. These processes include reverse osmosis (RO), ultrafiltration (UF), microfiltration (MF) etc. and they have numerous applications in the pharmaceutical, biotechnology and many other process industries, like desalination, dye treatment, textile, etc. [1]. The applications of membrane separation processes are mainly, separation of specified solutes, purification of a particular stream and fractionation of the components in a mixed feed. The major limitation of such processes is the decline in permeate flux with time. This is due to the con∗ Corresponding author. Fax: +91-3222-55-303. E-mail address: [email protected] (S. De).
centration polarization effect, i.e. the accumulation of the solute particles on the membrane surface which offers an extra resistance to the solvent flow through the membrane. Concentration polarization contributes to the cost of the process by reducing throughput and results in fouling. This requires frequent cleaning and/or replacement of the membranes. Among several approaches to reduce concentration polarization, following are the most common [2] (i) modification of the membrane material, (ii) change of the hydrodynamic conditions in the flow channel, e.g. introduction of turbulent promoters, (iii) application of external body forces e.g. dc electric field. Out of these methods, application of the external dc electric field for the control of concentration polarization is explored in the present work. However, it must be emphasized that the use of dc electric field will have minimal effects for controlling fouling which is essentially an irreversible process. This method
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 1 ) 0 0 6 7 4 - 3
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Nomenclature a c c1 D De De0 e E h k L Lp Mi Mw n0i Nav P q Q Ro Rr T u u0 ue
v ve vw x x∗ y z zi Zp
particle diameter (m) concentration of the macro-ion (BSA) (kg/m3 ) molar concentration of the electrolyte diffusivity (m2 /s) dielectric constant dielectric constant for the bulk of the solution electronic charge (C) dc electric field strength (V/m) half channel height (m) Boltzmann constant (J/K) length of the channel (m) membrane permeability (m/Pa s) molar concentration of ith micro-ion molecular weight number concentration of ith micro-ion (number/m3 ) Avagadro number pressure difference (Pa) charge density (C/m2 ) charge (C) observed retention real retention temperature (K) axial velocity (m/s) bulk velocity (m/s) electrophoretic mobility (m2 /V s). The subscripts 1–3 are the electorphoretic mobilities incorporating progressive corrections (Figs. 3–6). transverse velocity (m/s) particle velocity due to electrophoresis (m/s) permeate flux (m3 /m2 s) axial distance (m) dimensionless axial distance (x/L) normal distance (m) valency, z is same for both co- and counter-ions for symmetric electrolyte valency of ith micro-ion BSA surface charge number
Greek letters ε0 permittivity in free space (C/V m)
φ κ κm µ π ζ ψ
volume fraction inverse of Debye length (m−1 ) inverse of modified Debye length (m−1 ) viscosity (Pa s) osmotic pressure difference (Pa) zeta potential (V) electric potential (V)
Subscripts m membrane surface 0 bulk p permeate can be developed and implemented independent of the modification of the membrane and/or change in hydrodynamic conditions. An excellent review of the electro-kinetic methods to control reversible membrane fouling (i.e. concentration polarization) specifically by an externally imposed dc electric field is given by Jagannadh and Muralidhara [3]. Beechold [4] utilized a combination of electro-osmosis and electrophoresis to purify colloids in an electrofilter. Later, Bier [5] developed a membrane-based technique using a dc electric field to de-water colloidal suspension. Henry et al. [6] presented a simple mathematical model based on film theory for cross-flow electro-ultrafiltration under the gel-polarization domain for kaolin clay suspension and oil-in-water emulsion. Yukawa et al. [7] investigated cross-flow electro-ultrafiltration in a tubular module for gelatin solution. Mullon et al. [8] have worked on prevention of protein and paint fouling using an electric field. It may be noted here that these works are for gel layer controlled ultrafiltration and the model parameters are estimated from the experimental results. Ultrafiltration of most of the protein solutions is in the osmotic pressure controlled regime [9]. Very few studies have been conducted to model and characterize osmotic pressure controlled electro-ultrafiltration for colloidal solutions. The importance of the development of the design equations of such system and their solution within a proper engineering perspective cannot, therefore, be over-emphasized. In the present work, osmotic pressure controlled electro-ultrafiltration for a protein, bovine serum albumin (BSA) under laminar flow regime is modeled. The developed
V. Karthik et al. / Journal of Membrane Science 199 (2002) 29–40
model takes into account the underlying complications due to the application of dc electric field and property variation due to the solute concentration, e.g. variation of diffusivity with concentration, viscoelectric effects, effects of concentration on zeta potential, etc.
2. Theory In the present study, the protein bovine serum albumin (macro-ion) at pH 7.4 is taken as the solute in 0.15 M NaCl (micro-ion) solution. pH is a major operating condition in such a system. Based on the pH of the solution, protein molecules can be made positively or negatively charged. For BSA, isoelectric pH is 4.7. Therefore, by setting pH at 7.4, protein molecules can be made positively charged. During filtration, solutes will be deposited on the membrane surface. On application of a suitable external dc electric field, the protein molecules will move away from the membrane surface due to the electrostatic attraction and concentration polarization will be reduced. However, the motion of the charged particles in presence of other particles (concentration effects) and its final manifestation in the permeate flux and permeate concentration depends upon the combined analysis of the concentration profile and electro-kinetic effects close to the membrane surface. It needs to be emphasized here that apart from the electro-kinetic phenomena, electrochemical reaction
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and Joule heating may take place specially due to the presence of micro-ions [3]. The effects of the last two are evaluated separately and the contributions of them to the specific separation process studied herein are found to be insignificant. For example, the increase in temperature due to Joule heating for the worst operating condition (highest micro-ion concentration, lowest cross-flow velocity, maximum dc electric field strength) is found to be of the order of 10−3 ◦ C. The calculation of electrochemical reaction similarly shows that the amount of gases liberated is of the order of 10−4 l per unit residence time, whereas, the cell volume is 0.12 l. Thus, the two effects are not considered in the subsequent development of the model. The schematic of the system geometry is presented in Fig. 1. It must be mentioned here that the velocity field in the flow channel is considered under laminar flow regime in the present analysis. The steady state solute mass balance in the concentration boundary layer can be expressed as ∂c ∂c ∂c ∂ ∂c u +v + ve = D(c) (1) ∂x ∂y ∂y ∂y ∂y The assumptions involved in the above equation are (i) flow is steady and fully developed; (ii) diffusive flux in x-direction is small compared to the convective flux in x-direction; and (iii) u v w . The initial condition for Eq. (1) is at x = 0,
Fig. 1. Schematic of the geometry.
c = c0
(2)
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The two boundary conditions are at y = 0,
D
∂c + (v − ve )(c − cp ) = 0 ∂y
2.3. Partial retention and permeate concentration (3)
The above boundary condition is obtained from the fact that at steady state, net convective flux towards the membrane is equal to the net diffusive flux away from the membrane. Since the thickness of the concentration boundary layer is inversely proportional to the Schmidt number [10], which is very high for the protein solution, thickness of the concentration boundary layer is very small compared to the channel height. Therefore, for mathematical simplicity, the following boundary condition can be used [11] y = ∞,
c = c0
(4)
For fully developed laminar flow, the axial velocity profile in the rectangular channel is given as 3 y−h 2 u(y) = u0 1 − (5) 2 h Within a thin concentration boundary layer, i.e. for y h neglecting the term containing y2 /h2 the velocity can be expressed as 3u0 y h
(6)
For the thin concentration boundary layer, the transverse velocity profile can be approximated as [11,12] v = −vw
(7)
2.2. Transport law for the flow through the porous membrane Permeate flux through the porous membrane can be expressed using Darcy’s law vw = Lp (P − π )
From this equation, cp can be expressed in terms of cm , provided Rr is known. 2.4. Electrophoretic mobility (ue ) The particle velocity is related to the electrophoretic mobility by the following equation ve = ue E
2.1. Velocity profiles
u=
Real retention of the membrane is a parameter which indicates the extent of separation for a specific membrane–solute system. It is constant for a particular membrane–solute–solvent combination [13]. Real retention is defined as cp (9) Rr = 1 − cm
(8)
Osmotic pressure is a function of concentration and the relationship is presented in Appendix A.
(10)
The electrophoretic mobility for a typical colloidal solution can be expressed by Smoulochowski’s equation [14] ue =
ε0 De ζ η
(11)
2.5. Zeta potential (ζ ) When a charged colloidal particle moves in a solution, a layer of counter-ions moves along with it as an envelope. This layer is called the Stern layer and the potential on the outer surface of the stern layer is known as zeta potential. This zeta potential is an important parameter in electrophoresis which indicates the net potential with which the particle is exposed to the external electrolytic solution. Although BSA is an ellipsoidal molecule, it is assumed spherical and zeta potential is expressed [14] in terms of the particle net surface charge Q as ζ =
Q 4π ε0 aDe (1 + κa)
(12)
2.6. Debye length (κ −1 ) Debye length is a parameter that signifies the thickness of the double layer around the charged sphere. It
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is defined as [14] e2 n0i zi2 κ= ε0 De kT
(13)
If the concentration of the electrolyte is expressed in molar, then number concentration of the ith micro-ion is related to the molar concentration as, n0i = 103 Nav Mi 2.7. Solute–solute interaction 2.7.1. Effect of macro-ion concentration on the Debye–Huckel parameter For a concentrated macro-ion solution, the ionic environment around the charged sphere gets changed. The modified Debye–Huckel parameter (modified Debye length) can be expressed as [15] z2 e 2 2nb + (3qφ/zae) 2 κm = (14) ε0 De kT 1−φ where nb = ( i n0i )/2 for a symmetric electrolyte. The volume fraction φ, can be expressed in terms of concentration of the macro-ion in kg/m3 as φ=
c 4 3 π a × 103 × Nav × 3 Mw
(15)
2.7.2. Viscoelectric effect The electric field in the neighborhood of the charged sphere is expected to exert some influence on the structure of the surrounding fluid and the extent to which the viscosity and dielectric constant are modified is known as the viscoelectric effect. The viscosity is considered to vary as [14] ∂ψ 2 η(x) = η0 1 + f (16) ∂x and the expression for mobility can be modified as [14] dψ ε0 De ε ue = (17) η0 0 1 + 2fC2 sinh2 (zψ/kT) where C2 = 14.1 × 1015 c1 V2 /m2 and c1 is the molar concentration of the electrolyte, f the viscoelectric constant which is a characteristic for the solvent and for water its value is 10−15 m2 /V2 . Along
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with the viscosity of the surrounding solution inside the double layer, the solution dielectric constant can be appreciably affected by this phenomenon. The variation of dielectric constant can be expressed as [14] ∂ψ 2 (18) De = De0 1 − B ∂x where B = 4 × 10−18 m2 /V2 for water at 25 ◦ C. The final modified electrophoretic mobility incorporating the variations of the viscosity and the dielectric constant with the potential in the double layer can be expressed as [14] √ 2+ p ε0 De0 ξ ue = (19) √ dψ η0 0 3 + (f/B)(1 − p) where
p = 1 − 12BC2 sinh2
zψ 2kT
In Eq. (19), both De0 and η0 are the values of the dielectric constant and viscosity at the bulk of the solution where there is no effect of the electric field. The bulk viscosity of a protein solution is a function of its concentration and can be expressed as η0 = η0 (c)
(20)
For BSA solution, variation of the viscosity with concentration is presented in Appendix A. Combining Eqs. (19) and (20), the mobility can be obtained as √ 2+ p ε0 De0 ξ ue = (21) √ dψ η0 (c) 0 3 + (f/B)(1 − p) Eqs. (1), (6)–(9) and (21) are solved numerically and the profiles of the permeate flux and permeate concentrations are obtained. The permeate concentration can be expressed in terms of the observed retention as R o = 1 − cp /c0 . The length average permeate flux can be obtained as 1 vw dx ∗ (22) vw = 0
3. Numerical scheme The governing equations and the boundary conditions have been non-dimensionalized and finite differ-
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ence was used to solve them in order to get the profiles of permeate flux and the permeate concentration. In this method, the region of interest is divided in a mesh (number of grid points). The partial derivatives (at any grid point, i, j) can be expressed as ∂ci,j +1 ci+1,j − ci,j ∂ci,j +1 ci,j +1 − ci,j = and = ∂x x ∂y y The second derivatives are expressed using the Crank–Nicholson approximation, which is recommended for solving such initial value, boundary value problems. ∂ 2 ci,j 1 ci,j +1 − 2ci,j + ci,j −1 = 2 ∂y 2 y 2 1 ci+1,j +1 − 2ci+1,j + ci+1,j −1 + 2 y 2 An iterative technique (using matrix inversion) is employed to solve the equation and the concentration profile at every location along the channel is obtained.
4. Results and discussions 4.1. Estimation of the mobility of BSA particles under dc electric field 4.1.1. Influence of BSA concentration on Debye length From Section 2, it is clear that Debye length would be affected by the solute–solute interaction, which is strongly influenced by the BSA concentration (macro-ion). This variation is given by Eq. (14). Fig. 2(a) presents the variation of the modified Debye length with BSA volume fraction for different micro-ion concentrations. It can be seen from Fig. 2(a) that the modified Debye length decreases with BSA volume fraction for constant micro-ion concentration. For example, when NaCl concentration is 0.15 M and the BSA volume fraction is 0.4, the modified Debye length is almost 70% of the Debye length without the concentration variations. Thus, the incorporation of the macro-ion concentration in the estimate of Debye length leads to a reduction of the electrophoretic mobility (Eq. (11)) of the BSA particles. Fig. 2(b) represents the variation of the modified Debye length with charge number (i.e. different pH
Fig. 2. (a) Variation of the modified Debye length with BSA volume fraction at pH 7.4 (Z p = 20.4). Curve 1: c1 = 0.001 M; curve 2: 0.01 M; curve 3: 0.1 M; curve 4: 0.15 M. (b) Variation of the modified Debye length with BSA surface charge number at φ = 0.5. Curve 1: c1 = 0.001 M; curve 2: 0.01 M; curve 3: 0.1 M; curve 4: 0.15 M.
conditions) for different micro-ion concentrations at BSA volume fraction of 0.5. The trends indicate that the modified Debye length decreases with charge number for a particular micro-ion concentration. With increase in BSA charge number, the counter-ion concentration increases adjacent to the BSA particle. This
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leads to a compact double layer and thus, Debye length is reduced. For a particular charge number, modified Debye length increases with NaCl concentration. As micro-ion concentration increases, the electric double layer becomes more diffuse which results in an increase in the value of the Debye length. This in turn increases the mobility of the BSA particles, as presented in subsequent sections. 4.1.2. Influence of viscosity in the double layer on mobility (viscoelectric effect) Within the double layer, the solution viscosity varies with the electric potential as described by Eq. (16). Fig. 3(a) provides the variation of mobility with micro-ion concentration for different BSA volume fractions. For the results presented in this figure, the variation of the solution viscosity in the double layer as well as that of Debye length with macro-ion concentration is incorporated in the estimation of mobility. The mobility decreases with NaCl concentration for a fixed BSA volume fraction. On the other hand, for a fixed NaCl concentration, mobility is reduced with BSA volume fraction. With increase in macro- as well as micro-ions concentration, viscosity of the solution increases, leading to a reduction of BSA mobility. The effect of BSA surface charge number on mobility is shown in Fig. 3(b). For a fixed NaCl concentration, mobility increases with surface charge number. This is because zeta potential increases with increase in charge number, which in turn enhances the mobility. Due to viscoelectric effects, both viscosity and dielectric constant may vary together as expressed by Eqs. (16) and (18). Eq. (19) incorporates these two factors in the expression for mobility. From Fig. 4(a), it is evident that the mobility is reduced with increase in both the micro- and macro-ion concentrations. Fig. 4(b) indicates that with increase in charge number, mobility increases (as explained earlier).
Fig. 3. (a) Variation of the mobility (ue1 ) of BSA particles (including the modified Debye length and viscosity variation in the double layer) with NaCl concentration at pH 7.4 (Z p = 20.4). Curve 1: φ = 0.1; curve 2: 0.2; curve 3: 0.3; curve 4: 0.4. (b) Variation of the mobility (ue1 ) of BSA particles (including the modified Debye length and viscosity variation in the double layer) with NaCl concentration for φ = 0.3. Curve 1: Z p = 4.5; curve 2: 9.1; curve 3: 20.4.
4.1.3. Influence of variation of bulk viscosity with concentration in addition to the viscoelectric effect The bulk viscosity is a strong increasing function of the BSA concentration as presented in Appendix A. This would further modify the expression of mobility
as in Eq. (21). Fig. 5(a) reveals that the mobility decreases with both the macro- and micro-ion concentration. On the other hand, the mobility increases with the charge number for a fixed micro-ion concentration (Fig. 5(b)).
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Fig. 4. (a) Variation of the mobility (ue2 ) of BSA particles (including the modified Debye length, viscosity and dielectric constant variation in the double layer) with NaCl concentration at pH 7.4 (Z p = 20.4). Curve 1: φ = 0.1; curve 2: 0.2; curve 3: 0.3; curve 4: 0.4. (b) Variation of the mobility (ue2 ) of BSA particles (including the modified Debye length, viscosity and dielectric constant variation in the double layer) with NaCl concentration for φ = 0.3. Curve 1: Z p = 4.5; curve 2: 9.1; curve 3: 20.4.
Fig. 5. (a) Variation of the mobility (ue3 ) of BSA particles (including the modified Debye length, viscosity and dielectric constant variation in the double layer, as well as bulk viscosity variation with BSA concentration) with NaCl concentration at pH 7.4 (Z p = 20.4). Curve 1: φ = 0.1; curve 2: 0.2; curve 3: 0.3; curve 4: 0.4. (b) Variation of the mobility (ue3 ) of BSA particles (including the modified Debye length, viscosity and dielectric constant variation in the double layer, as well as bulk viscosity variation with BSA concentration) with NaCl concentration for φ = 0.3. Curve 1: Z p = 4.5; curve 2: 9.1; curve 3: 20.4.
Fig. 6 summarizes the variation of the mobility with the micro-ion concentration for a fixed volume fraction and the surface charge number. It describes the effects of the stepwise incorporation of the vari-
ation of electro- and physico-chemical properties on the mobility. Curve 1 indicates the variation of mobility (with micro-ion concentration) incorporating the viscosity variation within the double layer. In curve 2,
V. Karthik et al. / Journal of Membrane Science 199 (2002) 29–40
Fig. 6. Variation of ue1,2,3 with NaCl concentration for φ = 0.3 and Z p = 20.4 (pH = 7.4). Curve 1: ue1 ; curve 2: ue2 ; curve 3: ue3 .
in addition to the above-mentioned effect, the variation of the dielectric constant within the double layer is incorporated. The effects of the variation of bulk viscosity (with macro-ion concentration) on mobility in addition to the viscoelectric effects (curve 2) is presented in curve 3. It can be observed from the figure that incorporation of bulk viscosity correction further reduces the mobility (in comparison to the correction due to the vicsoelectric effects). 4.2. Surface concentration, flux profiles and the parametric results 4.2.1. Variation of the permeate flux profile with the change in electrical and other physicochemical properties Fig. 7 presents the variation of the permeate flux with the channel length (dimensionless) for a stepwise incorporation of different electrical and physicochemical properties. The model Eqs. (1), (6)–(9), and (21) have been solved using finite difference technique. It may be noted here that the governing equation (Eq. (1)) accounts for the concentration variation of the solute diffusivity. The results in curve 1 do not consider any variation of physical properties as well as the effects of dc electric field. Since diffusivity of BSA is an increasing function of concentration (for
37
Fig. 7. Variation of the permeate flux along the channel length for BSA at pH = 7.4, P = 69 kPa, c0 = 17.4 kg/m3 , u0 = 0.345 m/s and c1 = 0.15 M. Curve 1: without electric field, no property variations; curve 2: without electric field, diffusivity variation with BSA concentration; curve 3: with electric field (E = 200 V/m) and variable diffusivity; curve 4: with electric field, variable diffusivity and viscoelectric effect; curve 5: with electric field, variable diffusivity, viscoelectric effect and bulk viscosity variation with BSA concentration; curve 6: with electric field, all variations in curve 5 and using the modified Debye length.
the pH used in this study), the incorporation of diffusivity as a function of BSA concentration, results in an increase in flux (curve 2). This is due to the back diffusion of molecules from the membrane surface with enhanced BSA concentration. If a dc electric field is applied (curve 3), the particles are attracted towards the oppositely charged (positive in case of BSA at pH 7.4) electrode, thereby reducing the membrane surface concentration and hence increasing the flux. The results in this curve do not account for any variation of the transport properties inside the double layer which may get affected by the application of dc electric field. If viscoelectric effects are considered (curve 4), which reduces the electrophoretic mobility of the molecules, a decrease in flux results. If the variation of bulk viscosity with concentration is taken into account in addition to the viscoelectric effect, flux reduces even further (curve 5). The Debye length decreases with BSA concentration leading to a decrease in zeta potential. This would reduce the
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electrophoretic mobility of the solute (Eq. (11)) and thus, will reduce the flux further (curve 6). Thus, curve 6 is a complete simulation of the physical phenomena associated with cross-flow electro-ultrafiltration. For the operating conditions considered herein, the predicted increase in permeate flux on application of dc electric field is about 50% (between curves 1 and 6). 4.2.2. Variation of the membrane surface concentration and permeate flux along the length of the channel The variation of the membrane surface concentration and permeate flux with the channel length is presented in Fig. 8. It is observed from this figure that the membrane surface concentration decreases with the dc electric field strength (curves 1–4) with an associated increase in flux (curves 5–8). This is due to the enhanced mobility of the particles away from the membrane surface under the stronger dc electric fields. For high value of field strength (curve 4, E = 400 V/m), the membrane surface concentration remains at almost a constant value throughout the channel. It is interesting to note that for high dc electric fields, and under low polarization conditions (low pressure), the membrane surface concentration remains close to its
Fig. 8. Variation of the membrane surface concentration and the permeate flux along the channel length at P = 69 kPa, c0 = 17.4 kg/m3 , u0 = 0.345 m/s and c1 = 0.15 M. Curves 1–4 are for the membrane surface concentration and curves 5–8 are for the permeate flux. Curves 1 and 5: E = 0 V/m; curves 2 and 6: 125 V/m; curves 3 and 7: 200 V/m; curves 4 and 8: 400 V/m.
bulk value throughout the channel length. This effect is also manifested by the constancy of the flux profile for the conditions depicted in curve 8. 4.2.3. Influence of dc electric field strength on the average flux Influence of the dc electric field strength on the average flux is presented in Fig. 9. It is observed that the flux increases with dc electric field strength almost linearly for high operating pressures (curves 2 and 3). For low pressure, the flux increases rapidly and attains a constant value (curve 1). As seen from Fig. 8, curve 4, for high values of the dc electric fields, under low polarization (low pressure) conditions, the membrane surface concentration attains a constant value and remains almost invariant with position. This would imply a constancy in permeate flux beyond a certain field strength (300 V/m) for the conditions of curve 1. For fixed field strength, flux increases with pressure. 4.2.4. Influence of the cross-flow velocity on the permeate flux The effect of cross-flow velocity on permeate flux is described in Fig. 10. For fixed field strength, flux increases with the cross-flow velocity due to enhanced forced convection. This leads to a decrease
Fig. 9. Variation of the average flux with the electrical field strength at c0 = 17.4 kg/m3 , u0 = 0.345 m/s and c1 = 0.15 M. Curve 1: P = 69 kPa; curve 2: 138 kPa; curve 3: 207 kPa.
V. Karthik et al. / Journal of Membrane Science 199 (2002) 29–40
Fig. 10. Variation of the average permeate flux with electric field strength at P = 69 kPa, c0 = 17.4 kg/m3 and c1 = 0.15 M. Curve 1: u0 = 0.23 m/s; curve 2: 0.345 m/s and curve 3: 0.45 m/s.
in the membrane surface concentration, and thereby an increase in flux. For higher electric field strength (beyond 400 V/m), flux becomes independent of the cross-flow velocity. The high electric field strength enhances the mobility of the particles, which reduces the membrane surface concentration to almost the level of the bulk concentration. For example, at ∗ (equal to E = 400 V/m, and for u0 = 0.345 m/s, cm c/c0 ) has a value approximately equal to 1.1 (Fig. 8). For the operating conditions considered, the invariant flux value is 4.75 × 10−6 m/s, which is close to the pure water flux (5 × 10−6 m/s). 4.2.5. Variation of the observed retention along the channel length The mathematical model developed herein, can also be applied to partially retentive membranes. This is illustrated in Fig. 11. The value of Rr is taken as 0.98. For such a membrane, it is possible to predict permeate quality (Ro ) as well. It is evident from the figure that for a constant dc electric field, Ro decreases along the channel length as expected. With increase in electric field strength, the value of Ro at any location along the channel will increase. This is due to the fact that the membrane surface concentration will be lowered for higher field strength and this would result in less permeation of the solute through the membrane.
39
Fig. 11. Variation of the observed retention along the channel length at P = 69 kPa, c0 = 17.4 kg/m3 , u0 = 0.345 m/s, R r = 0.98 and c1 = 0.15 M. Curve 1: E = 0 V/m; curve 2: 125 V/m; curve 3: 200 V/m.
5. Conclusions A theoretical approach for predicting the performance of a laminar cross-flow electro-ultrafiltration system is presented here. All possible electorphoretic as well as physicochemical property variations with concentration are included in the governing equations of the proposed model. A finite difference scheme was used to solve the model equations. The relative changes due to the incorporation of each of these variations in the electrical and physicochemical properties have been quantified separately and then combined together to estimate the overall effect. The final solution indicates an appreciable increase in flux (as compared to the no field value) on application of an dc electric field. Finally, a parametric study of the process is accomplished. The comprehensive model would be very useful in the design and optimization of an electro-ultrafiltration system.
Appendix A Molecular weight of BSA: 69,000. Equivalent spherical radius of BSA: 31.3 Å [16]. Z p = −20.4 at pH 7.4 [16].
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Viscosity: η = 9.086 × 10−4 exp(2.44 × 10−5 c2 ) [17]. Diffusivity: D = 6.762 × 10−11 (1 − 0.27 exp (−0.05c)) [18]. Osmotic pressure: π = 204.78c − 2.0c2 + 8.44 × 10−3 c3 [16]. Membrane permeability: 7.25 × 10−11 m/Pa s. UF cell: 43 cm × 7.62 cm × 0.38 cm. References [1] P.N. Bungay, H.K. Lonsdale, M.N. De Pinho, Synthetic Membranes: Science, Engineering and Applications, Reidel, Dordrechet, 1983. [2] G. Belfort, R.H. Davis, A.L. Zydney, The behavior of suspensions and macromolecular solutions in cross flow microfiltration, J. Membrane Sci. 96 (1994) 1. [3] S.N. Jagannadh, H.S. Muralidhara, Electrokinetics method to control membrane fouling, Ind. Eng. Chem. Res. 35 (1996) 1133. [4] H. Beechold, Ultrafiltration and electroultrafiltration, in: J. Alexander (Ed.), Colloid Chemistry, The American Catalog Company, 1926. [5] M. Bier, Electrophoresis, Vol. 1, Academic Press, New York, 1959, p. 263. [6] D.J. Henry, L.F. Lawler, C.H.A. Quoch, A solid/liquid separation process based on cross-flow and electrofiltration, AIChE J. 23 (6) (1977) 851. [7] H. Yukawa, K. Shimura, A. Suda, A. Maniwa, Cross flow electro-ultrafiltration for colloidal solution of protein, J. Chem. Eng. Jpn. 16 (4) (1983) 305.
[8] C. Mullon, J.M. Radovich, B. Behnam, A semiempirical model for electro-ultrafiltration–diafiltration, Sep. Sci. Technol. 20 (1) (1985) 63. [9] P. Aimar, V. Sanchez, A novel approach to transfer limiting phenomena during ultrafiltration of macromolecules, Ind. Eng. Chem. Fundam. 25 (1986) 789. [10] R. Bird, E. Stewart, E.N. Lightfoot, Transport Phenomena, McGraw-Hill, New York, 1960. [11] V.S. Minnikanti, S. DasGupta, S. De, Prediction of mass transfer coefficient with suction for turbulent flow in cross flow ultrafiltration, J. Membrane Sci. 157 (2) (1999) 227. [12] S. De, S. Bhattacharjee, A. Sharma, P.K. Bhattacharya, Generalized integral and similarity solutions of the concentration profiles for osmotic pressure controlled ultrafiltration, J. Membrane Sci. 130 (1997) 99. [13] W.S. Opong, A.L. Zydney, Diffusive and convective protein transport through asymmetric membranes, AIChE J. 37 (1991) 1497. [14] R.J. Hunter, Zeta Potential in Colloid Science: Principles and Applications, Academic Press, London, 1981. [15] W.B. Russel, D.A. Saville, W.R. Schowaltar, Colloidal Dispersions, Cambridge University Press, New York, 1988. [16] V.L. Vilker, C.K. Colton, K.A. Smith, The osmotic pressure of concentrated protein solutions: effect of concentration and pH in saline solutions of bovine serum albumin, J. Coll. Interf. Sci. 79 (2) (1981) 548. [17] W.N. Gill, D.E. Wiley, C.J.D. Fell, A.G. Fane, Effects of viscosity on concentration polarization in ultrafiltration, AIChE J. 34 (1988) 1563. [18] G.D. Philles, G.B. Benedek, N.A. Mazer, Diffusion in protein solutions at high concentrations: a study by quasi-elastic light scattering spectroscopy, J. Chem. Phys. 65 (5) (1976) 1883.
Modeling and simulation of osmotic pressure controlled electro-ultrafiltration in a cross-flow system Venkataraman Karthik, Sunando DasGupta, Sirshendu De∗ Department of Chemical Engineering, Indian Institute of Technology, Kharagpur 721 302, India Received 14 May 2001; received in revised form 12 September 2001; accepted 12 September 2001
Abstract Modeling and subsequent simulation of electric field enhanced ultrafiltration (UF) under osmotic pressure controlled regime in a rectangular channel under laminar flow conditions is presented herein. Bovine serum albumin (BSA) at pH 7.4 in 0.15 M NaCl solution is considered as the system. The analysis is carried out under the framework of boundary layer theory in conjunction with the electro-kinetic phenomena for a colloidal solution. Important effects due to the interaction between charged macro- and micro-particles with the solvent, e.g. viscoelectric effects, variation of diffusivity and Debye length with concentration, etc. have been incorporated in the model. It has been observed that the external dc electric field has a significant effect on the permeate flux and permeate concentration. A parametric study has been carried out to observe the effects of different operating conditions on the system performance. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Membranes; Mathematical modeling; Electrophoresis; Laminar flow; Electro-ultrafiltration; Mobility
1. Introduction Energy efficient membrane-based separation processes are becoming important unit operations in recent years to supplement the conventional separation processes, like distillation, centrifugation, etc. These processes include reverse osmosis (RO), ultrafiltration (UF), microfiltration (MF) etc. and they have numerous applications in the pharmaceutical, biotechnology and many other process industries, like desalination, dye treatment, textile, etc. [1]. The applications of membrane separation processes are mainly, separation of specified solutes, purification of a particular stream and fractionation of the components in a mixed feed. The major limitation of such processes is the decline in permeate flux with time. This is due to the con∗ Corresponding author. Fax: +91-3222-55-303. E-mail address: [email protected] (S. De).
centration polarization effect, i.e. the accumulation of the solute particles on the membrane surface which offers an extra resistance to the solvent flow through the membrane. Concentration polarization contributes to the cost of the process by reducing throughput and results in fouling. This requires frequent cleaning and/or replacement of the membranes. Among several approaches to reduce concentration polarization, following are the most common [2] (i) modification of the membrane material, (ii) change of the hydrodynamic conditions in the flow channel, e.g. introduction of turbulent promoters, (iii) application of external body forces e.g. dc electric field. Out of these methods, application of the external dc electric field for the control of concentration polarization is explored in the present work. However, it must be emphasized that the use of dc electric field will have minimal effects for controlling fouling which is essentially an irreversible process. This method
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 1 ) 0 0 6 7 4 - 3
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Nomenclature a c c1 D De De0 e E h k L Lp Mi Mw n0i Nav P q Q Ro Rr T u u0 ue
v ve vw x x∗ y z zi Zp
particle diameter (m) concentration of the macro-ion (BSA) (kg/m3 ) molar concentration of the electrolyte diffusivity (m2 /s) dielectric constant dielectric constant for the bulk of the solution electronic charge (C) dc electric field strength (V/m) half channel height (m) Boltzmann constant (J/K) length of the channel (m) membrane permeability (m/Pa s) molar concentration of ith micro-ion molecular weight number concentration of ith micro-ion (number/m3 ) Avagadro number pressure difference (Pa) charge density (C/m2 ) charge (C) observed retention real retention temperature (K) axial velocity (m/s) bulk velocity (m/s) electrophoretic mobility (m2 /V s). The subscripts 1–3 are the electorphoretic mobilities incorporating progressive corrections (Figs. 3–6). transverse velocity (m/s) particle velocity due to electrophoresis (m/s) permeate flux (m3 /m2 s) axial distance (m) dimensionless axial distance (x/L) normal distance (m) valency, z is same for both co- and counter-ions for symmetric electrolyte valency of ith micro-ion BSA surface charge number
Greek letters ε0 permittivity in free space (C/V m)
φ κ κm µ π ζ ψ
volume fraction inverse of Debye length (m−1 ) inverse of modified Debye length (m−1 ) viscosity (Pa s) osmotic pressure difference (Pa) zeta potential (V) electric potential (V)
Subscripts m membrane surface 0 bulk p permeate can be developed and implemented independent of the modification of the membrane and/or change in hydrodynamic conditions. An excellent review of the electro-kinetic methods to control reversible membrane fouling (i.e. concentration polarization) specifically by an externally imposed dc electric field is given by Jagannadh and Muralidhara [3]. Beechold [4] utilized a combination of electro-osmosis and electrophoresis to purify colloids in an electrofilter. Later, Bier [5] developed a membrane-based technique using a dc electric field to de-water colloidal suspension. Henry et al. [6] presented a simple mathematical model based on film theory for cross-flow electro-ultrafiltration under the gel-polarization domain for kaolin clay suspension and oil-in-water emulsion. Yukawa et al. [7] investigated cross-flow electro-ultrafiltration in a tubular module for gelatin solution. Mullon et al. [8] have worked on prevention of protein and paint fouling using an electric field. It may be noted here that these works are for gel layer controlled ultrafiltration and the model parameters are estimated from the experimental results. Ultrafiltration of most of the protein solutions is in the osmotic pressure controlled regime [9]. Very few studies have been conducted to model and characterize osmotic pressure controlled electro-ultrafiltration for colloidal solutions. The importance of the development of the design equations of such system and their solution within a proper engineering perspective cannot, therefore, be over-emphasized. In the present work, osmotic pressure controlled electro-ultrafiltration for a protein, bovine serum albumin (BSA) under laminar flow regime is modeled. The developed
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model takes into account the underlying complications due to the application of dc electric field and property variation due to the solute concentration, e.g. variation of diffusivity with concentration, viscoelectric effects, effects of concentration on zeta potential, etc.
2. Theory In the present study, the protein bovine serum albumin (macro-ion) at pH 7.4 is taken as the solute in 0.15 M NaCl (micro-ion) solution. pH is a major operating condition in such a system. Based on the pH of the solution, protein molecules can be made positively or negatively charged. For BSA, isoelectric pH is 4.7. Therefore, by setting pH at 7.4, protein molecules can be made positively charged. During filtration, solutes will be deposited on the membrane surface. On application of a suitable external dc electric field, the protein molecules will move away from the membrane surface due to the electrostatic attraction and concentration polarization will be reduced. However, the motion of the charged particles in presence of other particles (concentration effects) and its final manifestation in the permeate flux and permeate concentration depends upon the combined analysis of the concentration profile and electro-kinetic effects close to the membrane surface. It needs to be emphasized here that apart from the electro-kinetic phenomena, electrochemical reaction
31
and Joule heating may take place specially due to the presence of micro-ions [3]. The effects of the last two are evaluated separately and the contributions of them to the specific separation process studied herein are found to be insignificant. For example, the increase in temperature due to Joule heating for the worst operating condition (highest micro-ion concentration, lowest cross-flow velocity, maximum dc electric field strength) is found to be of the order of 10−3 ◦ C. The calculation of electrochemical reaction similarly shows that the amount of gases liberated is of the order of 10−4 l per unit residence time, whereas, the cell volume is 0.12 l. Thus, the two effects are not considered in the subsequent development of the model. The schematic of the system geometry is presented in Fig. 1. It must be mentioned here that the velocity field in the flow channel is considered under laminar flow regime in the present analysis. The steady state solute mass balance in the concentration boundary layer can be expressed as ∂c ∂c ∂c ∂ ∂c u +v + ve = D(c) (1) ∂x ∂y ∂y ∂y ∂y The assumptions involved in the above equation are (i) flow is steady and fully developed; (ii) diffusive flux in x-direction is small compared to the convective flux in x-direction; and (iii) u v w . The initial condition for Eq. (1) is at x = 0,
Fig. 1. Schematic of the geometry.
c = c0
(2)
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The two boundary conditions are at y = 0,
D
∂c + (v − ve )(c − cp ) = 0 ∂y
2.3. Partial retention and permeate concentration (3)
The above boundary condition is obtained from the fact that at steady state, net convective flux towards the membrane is equal to the net diffusive flux away from the membrane. Since the thickness of the concentration boundary layer is inversely proportional to the Schmidt number [10], which is very high for the protein solution, thickness of the concentration boundary layer is very small compared to the channel height. Therefore, for mathematical simplicity, the following boundary condition can be used [11] y = ∞,
c = c0
(4)
For fully developed laminar flow, the axial velocity profile in the rectangular channel is given as 3 y−h 2 u(y) = u0 1 − (5) 2 h Within a thin concentration boundary layer, i.e. for y h neglecting the term containing y2 /h2 the velocity can be expressed as 3u0 y h
(6)
For the thin concentration boundary layer, the transverse velocity profile can be approximated as [11,12] v = −vw
(7)
2.2. Transport law for the flow through the porous membrane Permeate flux through the porous membrane can be expressed using Darcy’s law vw = Lp (P − π )
From this equation, cp can be expressed in terms of cm , provided Rr is known. 2.4. Electrophoretic mobility (ue ) The particle velocity is related to the electrophoretic mobility by the following equation ve = ue E
2.1. Velocity profiles
u=
Real retention of the membrane is a parameter which indicates the extent of separation for a specific membrane–solute system. It is constant for a particular membrane–solute–solvent combination [13]. Real retention is defined as cp (9) Rr = 1 − cm
(8)
Osmotic pressure is a function of concentration and the relationship is presented in Appendix A.
(10)
The electrophoretic mobility for a typical colloidal solution can be expressed by Smoulochowski’s equation [14] ue =
ε0 De ζ η
(11)
2.5. Zeta potential (ζ ) When a charged colloidal particle moves in a solution, a layer of counter-ions moves along with it as an envelope. This layer is called the Stern layer and the potential on the outer surface of the stern layer is known as zeta potential. This zeta potential is an important parameter in electrophoresis which indicates the net potential with which the particle is exposed to the external electrolytic solution. Although BSA is an ellipsoidal molecule, it is assumed spherical and zeta potential is expressed [14] in terms of the particle net surface charge Q as ζ =
Q 4π ε0 aDe (1 + κa)
(12)
2.6. Debye length (κ −1 ) Debye length is a parameter that signifies the thickness of the double layer around the charged sphere. It
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is defined as [14] e2 n0i zi2 κ= ε0 De kT
(13)
If the concentration of the electrolyte is expressed in molar, then number concentration of the ith micro-ion is related to the molar concentration as, n0i = 103 Nav Mi 2.7. Solute–solute interaction 2.7.1. Effect of macro-ion concentration on the Debye–Huckel parameter For a concentrated macro-ion solution, the ionic environment around the charged sphere gets changed. The modified Debye–Huckel parameter (modified Debye length) can be expressed as [15] z2 e 2 2nb + (3qφ/zae) 2 κm = (14) ε0 De kT 1−φ where nb = ( i n0i )/2 for a symmetric electrolyte. The volume fraction φ, can be expressed in terms of concentration of the macro-ion in kg/m3 as φ=
c 4 3 π a × 103 × Nav × 3 Mw
(15)
2.7.2. Viscoelectric effect The electric field in the neighborhood of the charged sphere is expected to exert some influence on the structure of the surrounding fluid and the extent to which the viscosity and dielectric constant are modified is known as the viscoelectric effect. The viscosity is considered to vary as [14] ∂ψ 2 η(x) = η0 1 + f (16) ∂x and the expression for mobility can be modified as [14] dψ ε0 De ε ue = (17) η0 0 1 + 2fC2 sinh2 (zψ/kT) where C2 = 14.1 × 1015 c1 V2 /m2 and c1 is the molar concentration of the electrolyte, f the viscoelectric constant which is a characteristic for the solvent and for water its value is 10−15 m2 /V2 . Along
33
with the viscosity of the surrounding solution inside the double layer, the solution dielectric constant can be appreciably affected by this phenomenon. The variation of dielectric constant can be expressed as [14] ∂ψ 2 (18) De = De0 1 − B ∂x where B = 4 × 10−18 m2 /V2 for water at 25 ◦ C. The final modified electrophoretic mobility incorporating the variations of the viscosity and the dielectric constant with the potential in the double layer can be expressed as [14] √ 2+ p ε0 De0 ξ ue = (19) √ dψ η0 0 3 + (f/B)(1 − p) where
p = 1 − 12BC2 sinh2
zψ 2kT
In Eq. (19), both De0 and η0 are the values of the dielectric constant and viscosity at the bulk of the solution where there is no effect of the electric field. The bulk viscosity of a protein solution is a function of its concentration and can be expressed as η0 = η0 (c)
(20)
For BSA solution, variation of the viscosity with concentration is presented in Appendix A. Combining Eqs. (19) and (20), the mobility can be obtained as √ 2+ p ε0 De0 ξ ue = (21) √ dψ η0 (c) 0 3 + (f/B)(1 − p) Eqs. (1), (6)–(9) and (21) are solved numerically and the profiles of the permeate flux and permeate concentrations are obtained. The permeate concentration can be expressed in terms of the observed retention as R o = 1 − cp /c0 . The length average permeate flux can be obtained as 1 vw dx ∗ (22) vw = 0
3. Numerical scheme The governing equations and the boundary conditions have been non-dimensionalized and finite differ-
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ence was used to solve them in order to get the profiles of permeate flux and the permeate concentration. In this method, the region of interest is divided in a mesh (number of grid points). The partial derivatives (at any grid point, i, j) can be expressed as ∂ci,j +1 ci+1,j − ci,j ∂ci,j +1 ci,j +1 − ci,j = and = ∂x x ∂y y The second derivatives are expressed using the Crank–Nicholson approximation, which is recommended for solving such initial value, boundary value problems. ∂ 2 ci,j 1 ci,j +1 − 2ci,j + ci,j −1 = 2 ∂y 2 y 2 1 ci+1,j +1 − 2ci+1,j + ci+1,j −1 + 2 y 2 An iterative technique (using matrix inversion) is employed to solve the equation and the concentration profile at every location along the channel is obtained.
4. Results and discussions 4.1. Estimation of the mobility of BSA particles under dc electric field 4.1.1. Influence of BSA concentration on Debye length From Section 2, it is clear that Debye length would be affected by the solute–solute interaction, which is strongly influenced by the BSA concentration (macro-ion). This variation is given by Eq. (14). Fig. 2(a) presents the variation of the modified Debye length with BSA volume fraction for different micro-ion concentrations. It can be seen from Fig. 2(a) that the modified Debye length decreases with BSA volume fraction for constant micro-ion concentration. For example, when NaCl concentration is 0.15 M and the BSA volume fraction is 0.4, the modified Debye length is almost 70% of the Debye length without the concentration variations. Thus, the incorporation of the macro-ion concentration in the estimate of Debye length leads to a reduction of the electrophoretic mobility (Eq. (11)) of the BSA particles. Fig. 2(b) represents the variation of the modified Debye length with charge number (i.e. different pH
Fig. 2. (a) Variation of the modified Debye length with BSA volume fraction at pH 7.4 (Z p = 20.4). Curve 1: c1 = 0.001 M; curve 2: 0.01 M; curve 3: 0.1 M; curve 4: 0.15 M. (b) Variation of the modified Debye length with BSA surface charge number at φ = 0.5. Curve 1: c1 = 0.001 M; curve 2: 0.01 M; curve 3: 0.1 M; curve 4: 0.15 M.
conditions) for different micro-ion concentrations at BSA volume fraction of 0.5. The trends indicate that the modified Debye length decreases with charge number for a particular micro-ion concentration. With increase in BSA charge number, the counter-ion concentration increases adjacent to the BSA particle. This
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leads to a compact double layer and thus, Debye length is reduced. For a particular charge number, modified Debye length increases with NaCl concentration. As micro-ion concentration increases, the electric double layer becomes more diffuse which results in an increase in the value of the Debye length. This in turn increases the mobility of the BSA particles, as presented in subsequent sections. 4.1.2. Influence of viscosity in the double layer on mobility (viscoelectric effect) Within the double layer, the solution viscosity varies with the electric potential as described by Eq. (16). Fig. 3(a) provides the variation of mobility with micro-ion concentration for different BSA volume fractions. For the results presented in this figure, the variation of the solution viscosity in the double layer as well as that of Debye length with macro-ion concentration is incorporated in the estimation of mobility. The mobility decreases with NaCl concentration for a fixed BSA volume fraction. On the other hand, for a fixed NaCl concentration, mobility is reduced with BSA volume fraction. With increase in macro- as well as micro-ions concentration, viscosity of the solution increases, leading to a reduction of BSA mobility. The effect of BSA surface charge number on mobility is shown in Fig. 3(b). For a fixed NaCl concentration, mobility increases with surface charge number. This is because zeta potential increases with increase in charge number, which in turn enhances the mobility. Due to viscoelectric effects, both viscosity and dielectric constant may vary together as expressed by Eqs. (16) and (18). Eq. (19) incorporates these two factors in the expression for mobility. From Fig. 4(a), it is evident that the mobility is reduced with increase in both the micro- and macro-ion concentrations. Fig. 4(b) indicates that with increase in charge number, mobility increases (as explained earlier).
Fig. 3. (a) Variation of the mobility (ue1 ) of BSA particles (including the modified Debye length and viscosity variation in the double layer) with NaCl concentration at pH 7.4 (Z p = 20.4). Curve 1: φ = 0.1; curve 2: 0.2; curve 3: 0.3; curve 4: 0.4. (b) Variation of the mobility (ue1 ) of BSA particles (including the modified Debye length and viscosity variation in the double layer) with NaCl concentration for φ = 0.3. Curve 1: Z p = 4.5; curve 2: 9.1; curve 3: 20.4.
4.1.3. Influence of variation of bulk viscosity with concentration in addition to the viscoelectric effect The bulk viscosity is a strong increasing function of the BSA concentration as presented in Appendix A. This would further modify the expression of mobility
as in Eq. (21). Fig. 5(a) reveals that the mobility decreases with both the macro- and micro-ion concentration. On the other hand, the mobility increases with the charge number for a fixed micro-ion concentration (Fig. 5(b)).
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Fig. 4. (a) Variation of the mobility (ue2 ) of BSA particles (including the modified Debye length, viscosity and dielectric constant variation in the double layer) with NaCl concentration at pH 7.4 (Z p = 20.4). Curve 1: φ = 0.1; curve 2: 0.2; curve 3: 0.3; curve 4: 0.4. (b) Variation of the mobility (ue2 ) of BSA particles (including the modified Debye length, viscosity and dielectric constant variation in the double layer) with NaCl concentration for φ = 0.3. Curve 1: Z p = 4.5; curve 2: 9.1; curve 3: 20.4.
Fig. 5. (a) Variation of the mobility (ue3 ) of BSA particles (including the modified Debye length, viscosity and dielectric constant variation in the double layer, as well as bulk viscosity variation with BSA concentration) with NaCl concentration at pH 7.4 (Z p = 20.4). Curve 1: φ = 0.1; curve 2: 0.2; curve 3: 0.3; curve 4: 0.4. (b) Variation of the mobility (ue3 ) of BSA particles (including the modified Debye length, viscosity and dielectric constant variation in the double layer, as well as bulk viscosity variation with BSA concentration) with NaCl concentration for φ = 0.3. Curve 1: Z p = 4.5; curve 2: 9.1; curve 3: 20.4.
Fig. 6 summarizes the variation of the mobility with the micro-ion concentration for a fixed volume fraction and the surface charge number. It describes the effects of the stepwise incorporation of the vari-
ation of electro- and physico-chemical properties on the mobility. Curve 1 indicates the variation of mobility (with micro-ion concentration) incorporating the viscosity variation within the double layer. In curve 2,
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Fig. 6. Variation of ue1,2,3 with NaCl concentration for φ = 0.3 and Z p = 20.4 (pH = 7.4). Curve 1: ue1 ; curve 2: ue2 ; curve 3: ue3 .
in addition to the above-mentioned effect, the variation of the dielectric constant within the double layer is incorporated. The effects of the variation of bulk viscosity (with macro-ion concentration) on mobility in addition to the viscoelectric effects (curve 2) is presented in curve 3. It can be observed from the figure that incorporation of bulk viscosity correction further reduces the mobility (in comparison to the correction due to the vicsoelectric effects). 4.2. Surface concentration, flux profiles and the parametric results 4.2.1. Variation of the permeate flux profile with the change in electrical and other physicochemical properties Fig. 7 presents the variation of the permeate flux with the channel length (dimensionless) for a stepwise incorporation of different electrical and physicochemical properties. The model Eqs. (1), (6)–(9), and (21) have been solved using finite difference technique. It may be noted here that the governing equation (Eq. (1)) accounts for the concentration variation of the solute diffusivity. The results in curve 1 do not consider any variation of physical properties as well as the effects of dc electric field. Since diffusivity of BSA is an increasing function of concentration (for
37
Fig. 7. Variation of the permeate flux along the channel length for BSA at pH = 7.4, P = 69 kPa, c0 = 17.4 kg/m3 , u0 = 0.345 m/s and c1 = 0.15 M. Curve 1: without electric field, no property variations; curve 2: without electric field, diffusivity variation with BSA concentration; curve 3: with electric field (E = 200 V/m) and variable diffusivity; curve 4: with electric field, variable diffusivity and viscoelectric effect; curve 5: with electric field, variable diffusivity, viscoelectric effect and bulk viscosity variation with BSA concentration; curve 6: with electric field, all variations in curve 5 and using the modified Debye length.
the pH used in this study), the incorporation of diffusivity as a function of BSA concentration, results in an increase in flux (curve 2). This is due to the back diffusion of molecules from the membrane surface with enhanced BSA concentration. If a dc electric field is applied (curve 3), the particles are attracted towards the oppositely charged (positive in case of BSA at pH 7.4) electrode, thereby reducing the membrane surface concentration and hence increasing the flux. The results in this curve do not account for any variation of the transport properties inside the double layer which may get affected by the application of dc electric field. If viscoelectric effects are considered (curve 4), which reduces the electrophoretic mobility of the molecules, a decrease in flux results. If the variation of bulk viscosity with concentration is taken into account in addition to the viscoelectric effect, flux reduces even further (curve 5). The Debye length decreases with BSA concentration leading to a decrease in zeta potential. This would reduce the
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electrophoretic mobility of the solute (Eq. (11)) and thus, will reduce the flux further (curve 6). Thus, curve 6 is a complete simulation of the physical phenomena associated with cross-flow electro-ultrafiltration. For the operating conditions considered herein, the predicted increase in permeate flux on application of dc electric field is about 50% (between curves 1 and 6). 4.2.2. Variation of the membrane surface concentration and permeate flux along the length of the channel The variation of the membrane surface concentration and permeate flux with the channel length is presented in Fig. 8. It is observed from this figure that the membrane surface concentration decreases with the dc electric field strength (curves 1–4) with an associated increase in flux (curves 5–8). This is due to the enhanced mobility of the particles away from the membrane surface under the stronger dc electric fields. For high value of field strength (curve 4, E = 400 V/m), the membrane surface concentration remains at almost a constant value throughout the channel. It is interesting to note that for high dc electric fields, and under low polarization conditions (low pressure), the membrane surface concentration remains close to its
Fig. 8. Variation of the membrane surface concentration and the permeate flux along the channel length at P = 69 kPa, c0 = 17.4 kg/m3 , u0 = 0.345 m/s and c1 = 0.15 M. Curves 1–4 are for the membrane surface concentration and curves 5–8 are for the permeate flux. Curves 1 and 5: E = 0 V/m; curves 2 and 6: 125 V/m; curves 3 and 7: 200 V/m; curves 4 and 8: 400 V/m.
bulk value throughout the channel length. This effect is also manifested by the constancy of the flux profile for the conditions depicted in curve 8. 4.2.3. Influence of dc electric field strength on the average flux Influence of the dc electric field strength on the average flux is presented in Fig. 9. It is observed that the flux increases with dc electric field strength almost linearly for high operating pressures (curves 2 and 3). For low pressure, the flux increases rapidly and attains a constant value (curve 1). As seen from Fig. 8, curve 4, for high values of the dc electric fields, under low polarization (low pressure) conditions, the membrane surface concentration attains a constant value and remains almost invariant with position. This would imply a constancy in permeate flux beyond a certain field strength (300 V/m) for the conditions of curve 1. For fixed field strength, flux increases with pressure. 4.2.4. Influence of the cross-flow velocity on the permeate flux The effect of cross-flow velocity on permeate flux is described in Fig. 10. For fixed field strength, flux increases with the cross-flow velocity due to enhanced forced convection. This leads to a decrease
Fig. 9. Variation of the average flux with the electrical field strength at c0 = 17.4 kg/m3 , u0 = 0.345 m/s and c1 = 0.15 M. Curve 1: P = 69 kPa; curve 2: 138 kPa; curve 3: 207 kPa.
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Fig. 10. Variation of the average permeate flux with electric field strength at P = 69 kPa, c0 = 17.4 kg/m3 and c1 = 0.15 M. Curve 1: u0 = 0.23 m/s; curve 2: 0.345 m/s and curve 3: 0.45 m/s.
in the membrane surface concentration, and thereby an increase in flux. For higher electric field strength (beyond 400 V/m), flux becomes independent of the cross-flow velocity. The high electric field strength enhances the mobility of the particles, which reduces the membrane surface concentration to almost the level of the bulk concentration. For example, at ∗ (equal to E = 400 V/m, and for u0 = 0.345 m/s, cm c/c0 ) has a value approximately equal to 1.1 (Fig. 8). For the operating conditions considered, the invariant flux value is 4.75 × 10−6 m/s, which is close to the pure water flux (5 × 10−6 m/s). 4.2.5. Variation of the observed retention along the channel length The mathematical model developed herein, can also be applied to partially retentive membranes. This is illustrated in Fig. 11. The value of Rr is taken as 0.98. For such a membrane, it is possible to predict permeate quality (Ro ) as well. It is evident from the figure that for a constant dc electric field, Ro decreases along the channel length as expected. With increase in electric field strength, the value of Ro at any location along the channel will increase. This is due to the fact that the membrane surface concentration will be lowered for higher field strength and this would result in less permeation of the solute through the membrane.
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Fig. 11. Variation of the observed retention along the channel length at P = 69 kPa, c0 = 17.4 kg/m3 , u0 = 0.345 m/s, R r = 0.98 and c1 = 0.15 M. Curve 1: E = 0 V/m; curve 2: 125 V/m; curve 3: 200 V/m.
5. Conclusions A theoretical approach for predicting the performance of a laminar cross-flow electro-ultrafiltration system is presented here. All possible electorphoretic as well as physicochemical property variations with concentration are included in the governing equations of the proposed model. A finite difference scheme was used to solve the model equations. The relative changes due to the incorporation of each of these variations in the electrical and physicochemical properties have been quantified separately and then combined together to estimate the overall effect. The final solution indicates an appreciable increase in flux (as compared to the no field value) on application of an dc electric field. Finally, a parametric study of the process is accomplished. The comprehensive model would be very useful in the design and optimization of an electro-ultrafiltration system.
Appendix A Molecular weight of BSA: 69,000. Equivalent spherical radius of BSA: 31.3 Å [16]. Z p = −20.4 at pH 7.4 [16].
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Viscosity: η = 9.086 × 10−4 exp(2.44 × 10−5 c2 ) [17]. Diffusivity: D = 6.762 × 10−11 (1 − 0.27 exp (−0.05c)) [18]. Osmotic pressure: π = 204.78c − 2.0c2 + 8.44 × 10−3 c3 [16]. Membrane permeability: 7.25 × 10−11 m/Pa s. UF cell: 43 cm × 7.62 cm × 0.38 cm. References [1] P.N. Bungay, H.K. Lonsdale, M.N. De Pinho, Synthetic Membranes: Science, Engineering and Applications, Reidel, Dordrechet, 1983. [2] G. Belfort, R.H. Davis, A.L. Zydney, The behavior of suspensions and macromolecular solutions in cross flow microfiltration, J. Membrane Sci. 96 (1994) 1. [3] S.N. Jagannadh, H.S. Muralidhara, Electrokinetics method to control membrane fouling, Ind. Eng. Chem. Res. 35 (1996) 1133. [4] H. Beechold, Ultrafiltration and electroultrafiltration, in: J. Alexander (Ed.), Colloid Chemistry, The American Catalog Company, 1926. [5] M. Bier, Electrophoresis, Vol. 1, Academic Press, New York, 1959, p. 263. [6] D.J. Henry, L.F. Lawler, C.H.A. Quoch, A solid/liquid separation process based on cross-flow and electrofiltration, AIChE J. 23 (6) (1977) 851. [7] H. Yukawa, K. Shimura, A. Suda, A. Maniwa, Cross flow electro-ultrafiltration for colloidal solution of protein, J. Chem. Eng. Jpn. 16 (4) (1983) 305.
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