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Modeling of cross-flow osmotic pressure controlled membrane separation processes under turbulent flow conditions
Journal of Membrane Science 201 (2002) 203–212
Modeling of cross-flow osmotic pressure controlled membrane separation processes under turbulent flow conditions Kosaraju Sreenivas, P. Ragesh, Sunando DasGupta, Sirshendu De∗ Department of Chemical Engineering, Indian Institute of Technology, Kharagpur 721302, India Received 30 March 2001; received in revised form 30 March 2001; accepted 2 November 2001
Abstract A comprehensive predictive model for the osmotic pressure controlled turbulent cross-flow membrane separation process is presented. A thin rectangular channel is selected as the system geometry. The model can handle solutes with wide range of diffusivities (from sodium chloride to dextran) under different modes of membrane separations (RO, UF, etc.). It is not limited to the condition that the mass transfer boundary layer lies within the viscous sublayer of the hydrodynamic boundary layer. The governing equations are solved using an integral method. The performance characteristics of the system in terms of permeate flux and permeate concentration can be predicted simultaneously. The experimental data available in the literature under turbulent conditions [Ind. Eng. Chem. Fundam. 3 (1964) 210; J. Membr. Sci. 22 (1985) 117] are successfully compared with the model predicted results. A parametric study has been carried out to observe the effects of various operating conditions on the permeate flux and permeate concentration (in terms of observed retention). © 2002 Elsevier Science B.V. All rights reserved. Keywords: Cross-flow; Diffusivities; Hydrodynamic boundary layer; Turbulent flow; Permeate flux
1. Introduction One of the major problems associated with membrane separation processes, which restricts the widespread application of this process in the industry is the decline in flux. This occurs due to the build up of the solute concentration near the membrane surface, which is defined as concentration polarization [1]. The transport of species towards the membrane is strongly influenced by the conditions inside the hydrodynamic and mass transfer boundary layers. Therefore, the control of the boundary layers is crucial for the optimization of the process. The conventional method to reduce concentration polarization ∗ Corresponding author. Tel.: +91-3222-83926 (Off)/83927 (Res); fax: +91-3222-55303. E-mail address: [email protected] (S. De).
is to increase turbulence e.g. stirring the solution or cross-flow of the feed. An accurate quantification of concentration polarization as a function of process conditions is necessary to estimate the system performance satisfactorily. Many researchers attempted modeling of UF/RO systems in the past [2–5]. Initial models are based on film theory [1,6] where concentration boundary layer is assumed to be fully developed. This led to the under-prediction of flux. The models which consider developing concentration boundary layer contain assumptions like no axial variation of permeate flux [7,8], surface concentration [9,10], and simplified hydrodynamics, etc. [11,12]. In order to solve the governing mass transfer equation one needs to evaluate the detailed velocity profiles in the particular system. There has been a considerable effort towards developing a detailed transport model
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 7 3 0 - X
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Nomenclature A B1 , B2 , B3 c D de h k l L Lp P Re Ro Rr u uτ u+ uo u v v vw v w + vw x y y+
damping length constant (26ν(τ w /ρ)−1/2 ) viral coefficients solute concentration (kg/m3 ) diffusivity (m2 /s) equivalent diameter (=4 h) (m) Channel half height (m) Von Karman’s constant (0.4) Prandtl mixing length (m) channel length (m) membrane permeability (m/(Pa s)) transmembrane pressure (Pa) Reynolds number (uo de /ν) observed rejection intrinsic rejection axial velocity (m/s) √ friction velocity ( τw /ρ) (m/s) dimensionless axial velocity (u/uτ ) time averaged axial velocity (m/s) time averaged velocity fluctuation (m2 /s2 ) vertical velocity (m/s) permeate flux (m/s) average permeate flux (m/s) dimensionless suction velocity (v w /uτ ) axial coordinate (m) transverse coordinate (m) dimensionless transverse coordinate (yuτ /ν)
Greek letters α1, α2 , α3, α4 coefficients in Eqs. (A.1)–(A.4) δ concentration boundary layer thickness (m) δ+ dimensionless concentration boundary layer thickness, defined as δ + = (δρ/µ)uτ ν kinematic viscosity (m2 /s) Π osmotic pressure (Pa) Π osmotic pressure difference across the membrane (Pa) ρ density (kg/m3 ) τ total shear stress (N/m2 ) wall shear stress (N/m2 ) τw ξ coefficient in Eq. (15)
Subscripts m o p
membrane surface feed bulk permeate
in a variety of channel geometries. Belfort and Kleinstreuer [13] have reviewed much of the analytical and numerical work in this area. These provide insight into the fluid flow problems in the membrane systems. Most of the work in modeling of UF/RO systems has been undertaken in the laminar range [2,3,14,15]. This severely restricts their applicability, as these systems often operate under turbulent flow conditions or employ turbulence promoters. Silva-freire [16], Sucec and Oljaca [17] have undertaken detailed calculation for turbulent transpiration flows. However, these solutions are for the case of external flows and when the free stream velocity is known as a function of the axial distance. The transpiration considered in their solution is injection and it is taken to be constant with axial distance. The most common approach for modeling turbulence in membrane separation processes is to use empirical correlations for the mass transfer coefficients [18]. However, these correlations are system specific and valid in certain ranges of the operating conditions thereby limiting their applicability. Recently, more fundamental modeling of turbulent cross-flow UF was attempted [11,12]. These analyses are valid for low diffusivity (e.g. dextran T70) solutes. For these solutes, the concentration boundary layer lies within the viscous sub-layer (of the hydrodynamic boundary layer). Another simplified assumption regarding the hydrodynamics is that the x-component velocity remains unaffected by membrane permeation. Moreover, the turbulent mass transport terms (Reynolds stresses) are neglected, which may not be true outside the viscous sub-layer (of the hydrodynamic boundary layer). Hence, such analysis cannot account for the filtration of the high diffusivity solutes (sodium chloride, sucrose, etc.) for which the concentration boundary layer may grow beyond the viscous sub-layer in to the buffer zone or even in to the turbulent core. In the present work, most of the above drawbacks for the osmotic pressure controlled turbulent cross-flow UF/RO have been taken care of. The axial
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205
For the fully developed hydrodynamic boundary layer, the Eq. (1) reduces to −ρvw
∂u ∂τ = ∂y ∂y
(4)
Integrating the above equation yields the following expression for the shear stress τ = τw − ρuvw
The turbulent shear stress term can be expressed according to the Prandtl mixing length theory as [22,23] 2 ∂u ∂u (6) τt = ρl ∂y ∂y
Fig. 1. Schematic of the system.
velocity profile is developed using the van Driest mixing length hypothesis. This is then coupled to the mass conservation equation. The numerical solution of the model equations predicts the system performance in terms of permeate flux and permeate concentration. The model predictions are compared with the experimental data available in the literature [19,20]. 1.1. Theory The flow configuration is a rectangular channel as shown in the Fig. 1. The fluid flows axially over the membrane surface. The process is assumed to be in steady state with fully developed turbulent hydrodynamic boundary layer. Physical properties of the components are assumed to be constant. 1.1.1. Hydrodynamic considerations The momentum equation for a two dimensional, incompressible, turbulent boundary layer with zero pressure can be written as u
∂u ∂u 1 ∂τ +v = ∂x ∂y ρ ∂y
(1)
where τ is the total shear stress and is expressed as ∂u − ρu v τ = τl + τt = µ ∂y
(2)
where τ l is the laminar and τ t the turbulent shear stress [21]. The boundary conditions for Eq. (1) are u = 0,
v = −vw ;
τ = τw
at
y=0
(5)
(3)
where l is the Prandtl mixing length. The Prandtl mixing length can be expressed using the van Driest hypothesis in the following way [23]: y (7) l = ky 1 − exp − A where A is the damping length constant, defined as 26ν(τ w /ρ)−1/2 . The modified value of A for suction + ) [23], with is given as A = 26ν(τw /2)−1/2 exp(5.9vw k as Von Karman’s constant (equal to 0.4) [23]. It may be noted that the entire domain of the turbulent hydrodynamic boundary layer is represented by the expression of the mixing length given by Eq. (7). On substitution of Eq. (7) into Eq. (6), the following expression for the turbulent shear stress is obtained: y 2 du 2 τt = ρ ky 1 − exp − (8) dy A Using the above expression of τ t , Eq. (5) may be written as y 2 du + ρ ky 1 − exp − τ = τl + τt = µ dy A 2 du × = τw − ρuvw (9) dy Simplifying the above expression, the velocity gradient may be expressed as du −µ + µ2 + 4(τw − ρuvw )B = (10) dy 2B where y 2 B = ρ ky 1 − exp − A
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The wall shear stress in turbulent flow can be expressed in terms of the time smoothed average velocity (u0 ) [24], 2ν 0.25 2 (11) τw = 0.03325 ρuo de uo Using the above expression for wall shear stress, Eq. (10) can be expressed in an implicit form as du = E(vw , y) dy
(12)
The boundary condition for Eq. (12) is u = 0 at y = 0.
An integral method is employed for solving the governing mass balance equation (Eq. (15)), by assuming the concentration profile is of the form c = a1 + a2 η + a3 η2 where η =
y δ
The coefficients a1 , a2 , a3 are evaluated using the boundary conditions on the mass transfer governing Eqs. (16) and (17) along with, the condition at the edge of the boundary layer, (∂c/∂y)y=δ = 0. Evaluating the constants, the concentration profile becomes c cm y 2 y y = 1− + 2− (19) c0 c0 δ δ δ
1.2. Mass transfer boundary layer
Substituting the above concentration profile into Eq. (15), the following equation is obtained:
The steady state mass balance near the membrane surface (within the concentration boundary layer) is described by the governing species conservation equation: ∂ J¯y ∂c ∂c ∂ 2c u +ν =D 2 − (13) ∂x ∂y ∂y ∂y
f
where J¯y is a term representing turbulent mass transport and is expressed using Prandtl analogy as [21] du ∂c J¯y = −l 2 (14) dy ∂y Assuming ξ = l 2 |du/dy|, Eq. (13) is modified as ∂c ∂ξ ∂c ∂ 2c u (15) + v− = (D + ξ ) 2 ∂x ∂y ∂y ∂y The boundary conditions for Eq. (15) are c(0, y) = c0
(16)
c(x, δ) = c0
(17)
The solute mass balance at the membrane surface, i.e. at y = 0 gives the third boundary condition: ∂c −v(c − cp ) + D =0 (18) ∂y
dcm dδ +g = k1 dx dx
(20)
where
y y y y f = u co 2 2 − 1 + 2cm 1 − δ δ2 δ δ y 2 g = 1− δ (cm − c0 ) ∂ξ k1 = 2(D + ξ ) + v + w ∂y δ2 y 1 − × 2cm 1 − δ δ y 1 y 1 + − + 2− δ δ δ δ 2 ∂ξ du ky − l l 2d u =l + 2l + ∂y dy A y dy 2 The expression for (d2 u/dy2 ) can be obtained from Eq. (10) (given in Appendix A). 1.3. Osmotic pressure model Solvent flow through the membrane is quantified by Darcy’s law for flow through a porous medium. vw = Lp (P − Π )
(21)
Since, the mass transfer boundary layer is very thin compared to the half channel height, the transverse velocity profile may be expressed as [11,12,25]
Osmotic pressure of any macromolecular solution can be expressed in the form of a viral expansion:
v = −vw
Π = B 1 c + B 2 c 2 + B3 c 3
(22)
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Osmotic pressure difference can be estimated if the concentration of the solute at the membrane surface (cm ) and the permeate concentration (cp ) can be evaluated. These two concentrations are related by intrinsic retention (Rr ) which is constant for a specific membrane–solute system [26] defined as
Substituting Eq. (29) in Eq. (20), results in a nonlinear first-order differential equation for the mass transfer boundary layer:
cp Rr = 1 − cm
Integrating the above equation over the concentration boundary layer thickness, the following equation is obtained: δ δ dδ (f + gf1 ) dy = k1 dy (31) dx 0 0
(23)
The quality of permeate stream is characterized by the observed retention (Ro ) Ro = 1 −
cp co
(24)
Using Eqs. (22) and (23), the osmotic pressure difference across the membrane can be expressed in terms of the membrane surface concentration as Π
2 = B1 cm Rr + B2 cm (1 − (1 − Rr )2 ) 3 +B3 cm (1 − (1 − Rr )3 )
(25)
2 vw = Lp [P − B1 cm Rr − B2 cm (1 − (1 − Rr )2 )
(26)
Combining Eqs. (18) and (19) the following expression for flux can be obtained: vw =
2D(cm − co ) cm δRr
(28)
∗ = c /c and α , α , α , α are given in where cm m o 1 2 3 4 Appendix A. Differentiating the above expression with respect to x gives
2(cm − co ) dδ dδ dcm = = f1 dx Fδ dx dx
dδ = k1 dx
dδ =T dx
(30)
(32)
where P and T are the functions of x, v w , δ and cm : δ (f + gf1 ) dy (33) P = 0
T =
δ
k1 dy
(34)
0
Eq. (32) has to be solved along with the Eqs. (12) and (28) to obtain the profile of the concentration boundary layer and in turn the axial variation of cm and v w . The average permeate flux is obtained using
L 1 vw = vw dx (35) L 0 Once the membrane surface concentration profile is determined by the above method, the profile of observed retention is obtained.
(27)
Combining Eqs. (26) and (27), we get an algebraic equation containing surface concentration and concentration boundary layer thickness, which can be given as ∗ − 1) (cm ∗ ∗2 ∗3 ∗4 − α 2 cm − α 3 cm − α 4 cm ] = [α1 cm 2(δ/de )
P
On substituting Eq. (25) into Eq. (21), the permeate flux can be obtained as a function of the membrane surface concentration:
3 −B3 cm (1 − (1 − Rr )3 )]
(f + gf1 )
(29)
where F is a polynomial given in the Appendix A.
2. Numerical solution The coupled differential and algebraic equations (Eqs. (12), (28), (29) and (32)) are solved using a fourth-order Runge–Kutta method. The governing hydrodynamic equation (Eq. (12)) becomes indeterminate at y = 0. This problem is circumvented by the following approach. As the velocity profile very close to the wall is linear in nature and is given by u+ = y + , a very small value of y (10−9 m) and the corresponding value of u (from the viscous sub-layer relation) are chosen as the initial condition for the solution of the velocity profile (Eq. (12)). The lattice spacing in the transverse direction is taken as 10−8 m. No appreciable changes in the calculated values of the dependent
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variables are noticed by changing the y spacing and the starting value of y. The mass transfer governing equation (Eq. (32)) has been coupled with the differential Eq. (12) and is solved using the fourth order Runge–Kutta method. The coefficients P and T (Eq. (32)) are evaluated numerically at each step of integration using Simpson’s one-thirds rule. It may be noted that the algebraic equation (Eq. (28)) is coupled with Eqs. (12) and (32) and is solved using Newton–Raphson’s method. The initial condition for the solution of Eq. (32) is δ = 0 at x = 0. This will make the solution indeterminate. Therefore, an asymptotic solution is sought to the limit x → 0, δ → 0. Under these conditions, the developing concentration boundary layer (near the entrance) lies well within the viscous sub-layer. The form of the asymptotic solution of Eq. (32) will be as follows [11]: dδ 8h3 = dx Zδ 2
(36)
where u2 d 3 Z= τ e 64Dv The solution of Eq. (36) is 24x 1/3 δ=h Z
Fig. 2. Comparison with the experimental data for NaCl [19]. The predictions are the solid lines whereas the experimental data are presented by the symbols.
(37)
(38)
The axial starting point for integration is taken as x = 10−10 m. Any further decrease in the initial point did not result in any appreciable increase in accuracy. The axial lattice spacing is reduced until the solution was effectively independent of spacing. Variable step size is employed with an initial axial spacing of 10−12 m and it has been increased up to 10−4 m at the end of the channel.
3. Results and discussion 3.1. Comparison of model predictions with the experimental data The model Eqs. (12), (28), (29) and (32) are solved as outlined in the numerical solution section. The solution provides the profiles of the permeate flux and the membrane surface concentration along the
channel length. The length averaged permeate flux is obtained using Simpson’s one-thirds rule. The experimental data for two different solutes—sodium chloride (high diffusivity, order of magnitude 10−9 m2 /s, [19]) and dextran-70,000 (low diffusivity, order of magnitude 10−11 m2 /s [20]) are taken for comparison. The comparison of sodium chloride data under turbulent flow conditions with the predicted values is presented in Fig. 2. The predictions are the solid lines whereas the experimental data are presented by the symbols. The experiments were conducted [19] in a cell of dimensions 75 mm × 25 mm × 2.5 mm. The feed concentration was 4.2 wt.% and the operating pressure was 103 atm. Cross-flow velocities were varied from 0.08 to 0.5 m/s. Eastman E-398-3 cellulose acetate membrane was used with a membrane permeability of 7.6 × 10−6 g/(cm2 s atm). The comparison for dextran-70,000 is presented in Fig. 3. The ultra-filtration experiments [20] were carried out in a thin channel of dimensions 100 mm × 60 mm × 5.9 mm. Kalle polysulphone membranes were used with a permeability of 1.44 × 10−10 m3 /(N s). The feed concentrations were varied from 0.43, 0.935 and 1.42 kg/m3 ; the operating pressure differences were 200, 400 and 600 kPa; cross-flow velocities were 1.06, 1.84, 2.75 m/s. It is clear from these two figures that the predicted results are in good agree-
K. Sreenivas et al. / Journal of Membrane Science 201 (2002) 203–212
Fig. 3. Comparison with the experimental flux (×105 m3 /(m2 s)) data for dextran [20].
ment with the experimental data. This indicates that the model presented herein is suitable for solutes of widely varying diffusivities. The major advantage of this model is that it can be successfully used, with the assumptions inherent in the development of the model (as described in the Section 1.1), for high diffusivity solutes for which the concentration boundary layer will definitely cross the viscous sublayer. Fig. 4 represents the dimensionless concentration boundary layer development along the channel length for different solutes, namely, sodium chloride, sucrose and dextran. The solutes are chosen in the decreasing order of diffusivity (orders of magnitudes of 10−9 , 10−10 , 10−11 m2 /s, respectively). Physical properties of these solutes in aqueous solution are presented in the Appendix B. As is evident from the figure, that for the high diffusivity solutes (NaCl and sucrose, curves 1 and 2), the concentration boundary layer develops and goes beyond the viscous sub-layer into the buffer region of the hydrodynamic boundary layer. Only for dextran (curve 3), which is having the smallest diffusivity (the highest Schmidt number) of the three solutes, the concentration boundary layer lies well within the viscous sub-layer. Thus, the present model is more generalized compared to the existing fundamental models of turbulent cross-flow filtration [11,12].
209
Fig. 4. Variation of concentration boundary layer thickness along the channel length. co = 1 kg/m3 , R r = 1, L = 1000 mm, P = 800 kPa, Re = 4600. Curve: 1, NaCl; 2, sucrose; 3, dextran.
The axial velocity profile after incorporation of suction is expressed in Eq. (12). Fig. 5 represents a plot of the x-component of the velocity profile as a function of the distance from the wall when the average velocity in the channel is u0 = 1 m/s. Curve 1 represents the velocity profile when there is no suction (i.e. v w = 0). The effect of suction (v w = 0.001 m/s) on the velocity profile is presented in curve 2. It can be clearly observed from the figure that the velocity decreases across the channel due to suction.
Fig. 5. Variation of x-component of velocity with suction. Curve: 1, v w = 0.0 m/s; 2, v w = 0.001 m/s.
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Fig. 6. Variation of membrane surface concentration along the channel length for NaCl solution. co = 42 kg/m3 , P = 103 atm. Curve: 1, Re = 4600; 2, Re = 9200; 3, Re = 13,800.
Since the major thrust of this work is to develop a model that would be valid even for high diffusivity solutes, a parametric study is undertaken with NaCl as the solute. The aim of the study is to quantify the effects of various operating conditions (e.g. transmembrane pressure, Rr , u0 ) on the permeate flux and the observed retention. Fig. 6 represents the variation of dimensionless membrane surface concentration along the length of the channel for various cross-flow velocities. Curve 1 is for the lowest and curve 3 is for the highest cross-flow velocities. It is clear from the figure that increased turbulence (high cross-flow velocity) impedes the accumulation of the solutes on the membrane surface. This would result in a lower value of membrane surface concentration. The relative decrease in membrane surface concentration results in a reduction in the build up of osmotic pressure, thereby, increasing the effective driving force across the membrane. The consequent increases in the permeate flux are presented in Fig. 7, where curves 1–3 are arranged in increasing order of cross-flow velocity. Fig. 8 presents the variation of quality in the form of observed retention of the permeate along the channel for different cross-flow velocities. Curve 3 is for the highest and curve 1 is for the lowest cross-flow velocity. It can be observed that the quality deteriorates
Fig. 7. Variation of permeate flux along the channel length for NaCl solution. Operating conditions are same as in Fig. 6.
along the channel due to increase in the membrane surface concentration (Fig. 6). As explained earlier, an increase in cross-flow velocity decreases the membrane surface concentration, thereby decreasing the chemical potential difference across the membrane. This would result in less solute transport through the membrane (lower permeate concentration) and hence, the observed retention will increase.
Fig. 8. Effect of cross-flow velocity on the profile of observed retention along the channel length. Operating conditions are same as in Fig. 6.
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of the operating conditions. The results from the model are successfully compared with the data available from literature for sodium chloride and dextran under turbulent flow conditions.
Appendix A α1 = ΘRr α2 =
Fig. 9. Effect of intrinsic retention on the profile of observed retention along the channel length. co = 42 kg/m3 , P = 103 atm; Re = 4600. Curve: 1, R r = 0.98; 2, R r = 0.95; 3, R r = 0.90.
Fig. 9 represents the effects of Rr on the observed retention. For a fixed Rr , Ro decreases along the channel. This decrease appears less than the previous figure (Fig. 8) due to the length scale for Ro chosen here to accommodate larger variation in Ro . The axial variation (reduction) in Ro is consistent with the relative increases in Cm (Fig. 6) for a fixed value of Rr . At a fixed location in the channel, Ro is lower for smaller values of Rr . Low values of Rr indicate that the membrane is more porous and therefore, the solute concentration in the permeate stream tends to increase. This leads to a decrease in the values of Ro with smaller Rr .
4. Conclusions A transport model for the osmotic pressure controlled turbulent cross-flow membrane filtration process is developed. The limitations of the previous models, specially, for high diffusivity solutes have been overcome in this work. Thus, the model is applicable for most of the solutes typically encountered in membrane separation processes (from RO to UF). An integral method is employed to solve the coupled governing hydrodynamic and mass transfer equations. The solution simultaneously predicts the permeate flux and permeate concentration as a function
(A.1)
ΘRr2 c0 B1
(A.2)
(P )
α3 =
ΘRr co2 B2 [1 − (1 − Rr )2 ] (P )
(A.3)
α4 =
ΘRr co3 B3 [1 − (1 − Rr )3 ] (P )
(A.4)
where Lp de (P ) Θ= 4D
(A.5)
∗ ∗ 2 ) + 3α1 δ ∗ (cm ) F = 2 − α1 δ ∗ + 2α1 δ ∗ (cm ∗ 3 +4α1 δ ∗ (cm )
(A.6)
where ∗ cm =
cm co
and
δ∗ =
δ de
d2 u µ dB ρvw [B + (dB/dy)] M dB = − − MB dy 2 2B 2 dy 2B 2 dy (A.7) where M = µ2 + 4(τw − ρuvw )B y 2 B = ρ ky 1 − exp − A
y dB = ρk 2 y 1 − exp − dy A
y 2 2y exp(−y/A) × + 1 − exp − A A where A = 26ν
τw ρ
−1/2
+ exp(5.9vw )
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Appendix B. Physical properties of the solutes (i) Diffusivity [19,20,27]: NaCl, 1.5 × 10−9 m2 /s; sucrose, 5.24 × 10−10 m2 /s; dextran, 4 × 10−11 m2 /s. (ii) Viscosity: Viscosity of the solutions have been taken as 0.001 Pa s. (iii) Density: Density of the solutions have been taken as 1 g/cm3 . (iv) Osmotic pressure: Osmotic pressure of macromolecular solutions can be expressed in a viral form: Π = B1 C + B2 C 2 + B3 C 3
[10]
[11]
[12]
[13]
[14]
where Π is in Pa and C is in kg/m3 . For dextran T70 [20]: B1 = 37.5; B2 = 0.752; B3 = 0.00764. For NaCl [19]: B1 = 83086.5; B2 = 0.0; B3 = 0.0. For sucrose [27]: B1 = 3770; B2 = 38.79; B3 = −0.04. References [1] W.F. Blatt, A. Dravid, A.S. Michaels, L. Nelson, Solute polarization and cake formation in membrane ultra-filtration: causes, consequences and control techniques, in: J.E. Flinn (Ed.), Membrane Science and Technology, Plenum Press, New York, 1970. [2] S. Srinivasan, C. Tien, W.N. Gill, simultaneous development of velocity and concentration profiles in reverse osmosis systems, Chem. Eng. Sci. 22 (1967) 417. [3] J.S. Shen, R.F. Probstein, On the prediction of limiting flux in laminar ultra-filtration of macromolecular solutes, Ind. Eng. Chem. Fundam. 16 (1977) 459. [4] D.R. Trettin, M.R. Doshi, Ultra-filtration in an unstirred batch cell, Ind. Eng. Chem. Fundam. 19 (1980) 189. [5] J.G. Wijmans, S. Nakao, C.A. Smolders, Flux limitations in ultra-filtration: osmotic pressure model and gel layer model, J. Membr. Sci. 20 (1984) 115. [6] R. Rautenbach, R. Albrecht, Membrane Processes, Academic Press, New York, 1986. [7] T.K. Sherwood, P.L.T. Brian, Salt concentration at phase boundaries in desalination by reverse osmosis, Ind. Eng. Chem. Fundam. 4 (1965) 113. [8] L. Dresner, Boundary layer build-up in the demineralization of salt water by reverse osmosis, Oak Ridge National Laboratory Report, 1964, p. 3621. [9] D.R. Trettin, M.R. Doshi, Pressure independent ultra-filtration —is it gel limited or osmotic pressure limited? in: A.F. Turbak
[15]
[16]
[17]
[18]
[19]
[20]
[21] [22] [23] [24] [25]
[26]
[27]
(Ed.), Synthetic Membranes. Part II. Hyper and Ultra-filtration Uses, ACS Symposium Series, 1981, p. 154. P.L.T. Brian, Concentration polarization in reverse osmosis desalination with variable flux and incomplete salt rejection, Ind. Eng. Chem. Fundam. 4 (1965) 439. R. Prabhakar, S. DasGupta, S. De, Simultaneous prediction of flux and retention for osmotic pressure controlled turbulent cross-flow ultra-filtration, Sep. Purif. Technol. 18 (2000) 13. V.S. Minnikanti, S. DasGupta, S. De, Prediction of mass transfer with suction for turbulent flow in cross-flow ultra-filtration, J. Membr. Sci. 4079 (1999) 1. G. Belfort, C. Kleinstreuer, Mathematical modeling of fluid flow and solute distributions in pressure driven membrane modules, in: G. Belfort (Ed.), Synthetic Membrane Processes: Fundamentals and Water Applications, Academic Press, New York, 1984. S. Ganguly, P.K. Bhattacharya, Development of concentration profile and prediction of flux for ultra-filtration in a radial cross-flow cell, J. Membr. Sci. 97 (1994) 185. L. Huang, M.T. Morrisey, Finite element analysis as a tool for cross-flow membrane filter simulation, J. Membr. Sci. 155 (1999) 19. A.P. Silva-freire, An asymptotic solution for transpired incompressible turbulent boundary layers, Int. J. Heat Mass Transfer 31 (5) (1988) 1011. J. Sucec, M. Oljaca, Calculation of turbulent boundary layers with transpiration and pressure gradient effects, Int. J. Heat Mass Transfer 38 (15) (1995) 2855. V. Gekas, B. Hallstrom, Mass transfer in the membrane concentration polarization layer under turbulent cross-flow. Part I. Critical literature review and adaptation of existing Sherwood correlations to membrane operations, J. Membr. Sci. 80 (1987) 153. U. Merten, H.K. Lonsdale, R.L. Riley, Boundary layer effects in reverse osmosis, Ind. Eng. Chem. Fundam. 3 (1964) 210. J.G. Wijmans, S. Nakao, J.W.A. van den Berg, F.R. Trolestra, C.A. Smolders, Hydrodynamic resistance of concentration polarization boundary layers in ultra-filtration, J. Membr. Sci. 22 (1985) 117. R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960. P. Bradshaw, T. Cebeci, Momentum Transfer in Boundary Layers, Hemisphere, Washington, DC, 1977. T. Cebeci, A.M.O. Smith, Analysis of Turbulent Boundary Layers, Academic Press, New York, 1975. R.W. Fox, A.T. McDonald, Introduction to Fluid Mechanics, Wiley, New York, 1985. S. De, S. Bhattacharya, P.K. Bhattacharya, A. Sharma, Generalized integral and similarity solutions for concentration profiles for ultra-filtration, J. Membr. Sci. 130 (1997) 99. W.S. Opong, A.L. Zydney, Diffusive and convective protein transport through asymmetric membranes, AIChE J. 37 (1991) 1497. S. Sourirajan, Reverse Osmosis, Academic Press, New York, 1970.
Modeling of cross-flow osmotic pressure controlled membrane separation processes under turbulent flow conditions Kosaraju Sreenivas, P. Ragesh, Sunando DasGupta, Sirshendu De∗ Department of Chemical Engineering, Indian Institute of Technology, Kharagpur 721302, India Received 30 March 2001; received in revised form 30 March 2001; accepted 2 November 2001
Abstract A comprehensive predictive model for the osmotic pressure controlled turbulent cross-flow membrane separation process is presented. A thin rectangular channel is selected as the system geometry. The model can handle solutes with wide range of diffusivities (from sodium chloride to dextran) under different modes of membrane separations (RO, UF, etc.). It is not limited to the condition that the mass transfer boundary layer lies within the viscous sublayer of the hydrodynamic boundary layer. The governing equations are solved using an integral method. The performance characteristics of the system in terms of permeate flux and permeate concentration can be predicted simultaneously. The experimental data available in the literature under turbulent conditions [Ind. Eng. Chem. Fundam. 3 (1964) 210; J. Membr. Sci. 22 (1985) 117] are successfully compared with the model predicted results. A parametric study has been carried out to observe the effects of various operating conditions on the permeate flux and permeate concentration (in terms of observed retention). © 2002 Elsevier Science B.V. All rights reserved. Keywords: Cross-flow; Diffusivities; Hydrodynamic boundary layer; Turbulent flow; Permeate flux
1. Introduction One of the major problems associated with membrane separation processes, which restricts the widespread application of this process in the industry is the decline in flux. This occurs due to the build up of the solute concentration near the membrane surface, which is defined as concentration polarization [1]. The transport of species towards the membrane is strongly influenced by the conditions inside the hydrodynamic and mass transfer boundary layers. Therefore, the control of the boundary layers is crucial for the optimization of the process. The conventional method to reduce concentration polarization ∗ Corresponding author. Tel.: +91-3222-83926 (Off)/83927 (Res); fax: +91-3222-55303. E-mail address: [email protected] (S. De).
is to increase turbulence e.g. stirring the solution or cross-flow of the feed. An accurate quantification of concentration polarization as a function of process conditions is necessary to estimate the system performance satisfactorily. Many researchers attempted modeling of UF/RO systems in the past [2–5]. Initial models are based on film theory [1,6] where concentration boundary layer is assumed to be fully developed. This led to the under-prediction of flux. The models which consider developing concentration boundary layer contain assumptions like no axial variation of permeate flux [7,8], surface concentration [9,10], and simplified hydrodynamics, etc. [11,12]. In order to solve the governing mass transfer equation one needs to evaluate the detailed velocity profiles in the particular system. There has been a considerable effort towards developing a detailed transport model
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 7 3 0 - X
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Nomenclature A B1 , B2 , B3 c D de h k l L Lp P Re Ro Rr u uτ u+ uo u v v vw v w + vw x y y+
damping length constant (26ν(τ w /ρ)−1/2 ) viral coefficients solute concentration (kg/m3 ) diffusivity (m2 /s) equivalent diameter (=4 h) (m) Channel half height (m) Von Karman’s constant (0.4) Prandtl mixing length (m) channel length (m) membrane permeability (m/(Pa s)) transmembrane pressure (Pa) Reynolds number (uo de /ν) observed rejection intrinsic rejection axial velocity (m/s) √ friction velocity ( τw /ρ) (m/s) dimensionless axial velocity (u/uτ ) time averaged axial velocity (m/s) time averaged velocity fluctuation (m2 /s2 ) vertical velocity (m/s) permeate flux (m/s) average permeate flux (m/s) dimensionless suction velocity (v w /uτ ) axial coordinate (m) transverse coordinate (m) dimensionless transverse coordinate (yuτ /ν)
Greek letters α1, α2 , α3, α4 coefficients in Eqs. (A.1)–(A.4) δ concentration boundary layer thickness (m) δ+ dimensionless concentration boundary layer thickness, defined as δ + = (δρ/µ)uτ ν kinematic viscosity (m2 /s) Π osmotic pressure (Pa) Π osmotic pressure difference across the membrane (Pa) ρ density (kg/m3 ) τ total shear stress (N/m2 ) wall shear stress (N/m2 ) τw ξ coefficient in Eq. (15)
Subscripts m o p
membrane surface feed bulk permeate
in a variety of channel geometries. Belfort and Kleinstreuer [13] have reviewed much of the analytical and numerical work in this area. These provide insight into the fluid flow problems in the membrane systems. Most of the work in modeling of UF/RO systems has been undertaken in the laminar range [2,3,14,15]. This severely restricts their applicability, as these systems often operate under turbulent flow conditions or employ turbulence promoters. Silva-freire [16], Sucec and Oljaca [17] have undertaken detailed calculation for turbulent transpiration flows. However, these solutions are for the case of external flows and when the free stream velocity is known as a function of the axial distance. The transpiration considered in their solution is injection and it is taken to be constant with axial distance. The most common approach for modeling turbulence in membrane separation processes is to use empirical correlations for the mass transfer coefficients [18]. However, these correlations are system specific and valid in certain ranges of the operating conditions thereby limiting their applicability. Recently, more fundamental modeling of turbulent cross-flow UF was attempted [11,12]. These analyses are valid for low diffusivity (e.g. dextran T70) solutes. For these solutes, the concentration boundary layer lies within the viscous sub-layer (of the hydrodynamic boundary layer). Another simplified assumption regarding the hydrodynamics is that the x-component velocity remains unaffected by membrane permeation. Moreover, the turbulent mass transport terms (Reynolds stresses) are neglected, which may not be true outside the viscous sub-layer (of the hydrodynamic boundary layer). Hence, such analysis cannot account for the filtration of the high diffusivity solutes (sodium chloride, sucrose, etc.) for which the concentration boundary layer may grow beyond the viscous sub-layer in to the buffer zone or even in to the turbulent core. In the present work, most of the above drawbacks for the osmotic pressure controlled turbulent cross-flow UF/RO have been taken care of. The axial
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205
For the fully developed hydrodynamic boundary layer, the Eq. (1) reduces to −ρvw
∂u ∂τ = ∂y ∂y
(4)
Integrating the above equation yields the following expression for the shear stress τ = τw − ρuvw
The turbulent shear stress term can be expressed according to the Prandtl mixing length theory as [22,23] 2 ∂u ∂u (6) τt = ρl ∂y ∂y
Fig. 1. Schematic of the system.
velocity profile is developed using the van Driest mixing length hypothesis. This is then coupled to the mass conservation equation. The numerical solution of the model equations predicts the system performance in terms of permeate flux and permeate concentration. The model predictions are compared with the experimental data available in the literature [19,20]. 1.1. Theory The flow configuration is a rectangular channel as shown in the Fig. 1. The fluid flows axially over the membrane surface. The process is assumed to be in steady state with fully developed turbulent hydrodynamic boundary layer. Physical properties of the components are assumed to be constant. 1.1.1. Hydrodynamic considerations The momentum equation for a two dimensional, incompressible, turbulent boundary layer with zero pressure can be written as u
∂u ∂u 1 ∂τ +v = ∂x ∂y ρ ∂y
(1)
where τ is the total shear stress and is expressed as ∂u − ρu v τ = τl + τt = µ ∂y
(2)
where τ l is the laminar and τ t the turbulent shear stress [21]. The boundary conditions for Eq. (1) are u = 0,
v = −vw ;
τ = τw
at
y=0
(5)
(3)
where l is the Prandtl mixing length. The Prandtl mixing length can be expressed using the van Driest hypothesis in the following way [23]: y (7) l = ky 1 − exp − A where A is the damping length constant, defined as 26ν(τ w /ρ)−1/2 . The modified value of A for suction + ) [23], with is given as A = 26ν(τw /2)−1/2 exp(5.9vw k as Von Karman’s constant (equal to 0.4) [23]. It may be noted that the entire domain of the turbulent hydrodynamic boundary layer is represented by the expression of the mixing length given by Eq. (7). On substitution of Eq. (7) into Eq. (6), the following expression for the turbulent shear stress is obtained: y 2 du 2 τt = ρ ky 1 − exp − (8) dy A Using the above expression of τ t , Eq. (5) may be written as y 2 du + ρ ky 1 − exp − τ = τl + τt = µ dy A 2 du × = τw − ρuvw (9) dy Simplifying the above expression, the velocity gradient may be expressed as du −µ + µ2 + 4(τw − ρuvw )B = (10) dy 2B where y 2 B = ρ ky 1 − exp − A
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The wall shear stress in turbulent flow can be expressed in terms of the time smoothed average velocity (u0 ) [24], 2ν 0.25 2 (11) τw = 0.03325 ρuo de uo Using the above expression for wall shear stress, Eq. (10) can be expressed in an implicit form as du = E(vw , y) dy
(12)
The boundary condition for Eq. (12) is u = 0 at y = 0.
An integral method is employed for solving the governing mass balance equation (Eq. (15)), by assuming the concentration profile is of the form c = a1 + a2 η + a3 η2 where η =
y δ
The coefficients a1 , a2 , a3 are evaluated using the boundary conditions on the mass transfer governing Eqs. (16) and (17) along with, the condition at the edge of the boundary layer, (∂c/∂y)y=δ = 0. Evaluating the constants, the concentration profile becomes c cm y 2 y y = 1− + 2− (19) c0 c0 δ δ δ
1.2. Mass transfer boundary layer
Substituting the above concentration profile into Eq. (15), the following equation is obtained:
The steady state mass balance near the membrane surface (within the concentration boundary layer) is described by the governing species conservation equation: ∂ J¯y ∂c ∂c ∂ 2c u +ν =D 2 − (13) ∂x ∂y ∂y ∂y
f
where J¯y is a term representing turbulent mass transport and is expressed using Prandtl analogy as [21] du ∂c J¯y = −l 2 (14) dy ∂y Assuming ξ = l 2 |du/dy|, Eq. (13) is modified as ∂c ∂ξ ∂c ∂ 2c u (15) + v− = (D + ξ ) 2 ∂x ∂y ∂y ∂y The boundary conditions for Eq. (15) are c(0, y) = c0
(16)
c(x, δ) = c0
(17)
The solute mass balance at the membrane surface, i.e. at y = 0 gives the third boundary condition: ∂c −v(c − cp ) + D =0 (18) ∂y
dcm dδ +g = k1 dx dx
(20)
where
y y y y f = u co 2 2 − 1 + 2cm 1 − δ δ2 δ δ y 2 g = 1− δ (cm − c0 ) ∂ξ k1 = 2(D + ξ ) + v + w ∂y δ2 y 1 − × 2cm 1 − δ δ y 1 y 1 + − + 2− δ δ δ δ 2 ∂ξ du ky − l l 2d u =l + 2l + ∂y dy A y dy 2 The expression for (d2 u/dy2 ) can be obtained from Eq. (10) (given in Appendix A). 1.3. Osmotic pressure model Solvent flow through the membrane is quantified by Darcy’s law for flow through a porous medium. vw = Lp (P − Π )
(21)
Since, the mass transfer boundary layer is very thin compared to the half channel height, the transverse velocity profile may be expressed as [11,12,25]
Osmotic pressure of any macromolecular solution can be expressed in the form of a viral expansion:
v = −vw
Π = B 1 c + B 2 c 2 + B3 c 3
(22)
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Osmotic pressure difference can be estimated if the concentration of the solute at the membrane surface (cm ) and the permeate concentration (cp ) can be evaluated. These two concentrations are related by intrinsic retention (Rr ) which is constant for a specific membrane–solute system [26] defined as
Substituting Eq. (29) in Eq. (20), results in a nonlinear first-order differential equation for the mass transfer boundary layer:
cp Rr = 1 − cm
Integrating the above equation over the concentration boundary layer thickness, the following equation is obtained: δ δ dδ (f + gf1 ) dy = k1 dy (31) dx 0 0
(23)
The quality of permeate stream is characterized by the observed retention (Ro ) Ro = 1 −
cp co
(24)
Using Eqs. (22) and (23), the osmotic pressure difference across the membrane can be expressed in terms of the membrane surface concentration as Π
2 = B1 cm Rr + B2 cm (1 − (1 − Rr )2 ) 3 +B3 cm (1 − (1 − Rr )3 )
(25)
2 vw = Lp [P − B1 cm Rr − B2 cm (1 − (1 − Rr )2 )
(26)
Combining Eqs. (18) and (19) the following expression for flux can be obtained: vw =
2D(cm − co ) cm δRr
(28)
∗ = c /c and α , α , α , α are given in where cm m o 1 2 3 4 Appendix A. Differentiating the above expression with respect to x gives
2(cm − co ) dδ dδ dcm = = f1 dx Fδ dx dx
dδ = k1 dx
dδ =T dx
(30)
(32)
where P and T are the functions of x, v w , δ and cm : δ (f + gf1 ) dy (33) P = 0
T =
δ
k1 dy
(34)
0
Eq. (32) has to be solved along with the Eqs. (12) and (28) to obtain the profile of the concentration boundary layer and in turn the axial variation of cm and v w . The average permeate flux is obtained using
L 1 vw = vw dx (35) L 0 Once the membrane surface concentration profile is determined by the above method, the profile of observed retention is obtained.
(27)
Combining Eqs. (26) and (27), we get an algebraic equation containing surface concentration and concentration boundary layer thickness, which can be given as ∗ − 1) (cm ∗ ∗2 ∗3 ∗4 − α 2 cm − α 3 cm − α 4 cm ] = [α1 cm 2(δ/de )
P
On substituting Eq. (25) into Eq. (21), the permeate flux can be obtained as a function of the membrane surface concentration:
3 −B3 cm (1 − (1 − Rr )3 )]
(f + gf1 )
(29)
where F is a polynomial given in the Appendix A.
2. Numerical solution The coupled differential and algebraic equations (Eqs. (12), (28), (29) and (32)) are solved using a fourth-order Runge–Kutta method. The governing hydrodynamic equation (Eq. (12)) becomes indeterminate at y = 0. This problem is circumvented by the following approach. As the velocity profile very close to the wall is linear in nature and is given by u+ = y + , a very small value of y (10−9 m) and the corresponding value of u (from the viscous sub-layer relation) are chosen as the initial condition for the solution of the velocity profile (Eq. (12)). The lattice spacing in the transverse direction is taken as 10−8 m. No appreciable changes in the calculated values of the dependent
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variables are noticed by changing the y spacing and the starting value of y. The mass transfer governing equation (Eq. (32)) has been coupled with the differential Eq. (12) and is solved using the fourth order Runge–Kutta method. The coefficients P and T (Eq. (32)) are evaluated numerically at each step of integration using Simpson’s one-thirds rule. It may be noted that the algebraic equation (Eq. (28)) is coupled with Eqs. (12) and (32) and is solved using Newton–Raphson’s method. The initial condition for the solution of Eq. (32) is δ = 0 at x = 0. This will make the solution indeterminate. Therefore, an asymptotic solution is sought to the limit x → 0, δ → 0. Under these conditions, the developing concentration boundary layer (near the entrance) lies well within the viscous sub-layer. The form of the asymptotic solution of Eq. (32) will be as follows [11]: dδ 8h3 = dx Zδ 2
(36)
where u2 d 3 Z= τ e 64Dv The solution of Eq. (36) is 24x 1/3 δ=h Z
Fig. 2. Comparison with the experimental data for NaCl [19]. The predictions are the solid lines whereas the experimental data are presented by the symbols.
(37)
(38)
The axial starting point for integration is taken as x = 10−10 m. Any further decrease in the initial point did not result in any appreciable increase in accuracy. The axial lattice spacing is reduced until the solution was effectively independent of spacing. Variable step size is employed with an initial axial spacing of 10−12 m and it has been increased up to 10−4 m at the end of the channel.
3. Results and discussion 3.1. Comparison of model predictions with the experimental data The model Eqs. (12), (28), (29) and (32) are solved as outlined in the numerical solution section. The solution provides the profiles of the permeate flux and the membrane surface concentration along the
channel length. The length averaged permeate flux is obtained using Simpson’s one-thirds rule. The experimental data for two different solutes—sodium chloride (high diffusivity, order of magnitude 10−9 m2 /s, [19]) and dextran-70,000 (low diffusivity, order of magnitude 10−11 m2 /s [20]) are taken for comparison. The comparison of sodium chloride data under turbulent flow conditions with the predicted values is presented in Fig. 2. The predictions are the solid lines whereas the experimental data are presented by the symbols. The experiments were conducted [19] in a cell of dimensions 75 mm × 25 mm × 2.5 mm. The feed concentration was 4.2 wt.% and the operating pressure was 103 atm. Cross-flow velocities were varied from 0.08 to 0.5 m/s. Eastman E-398-3 cellulose acetate membrane was used with a membrane permeability of 7.6 × 10−6 g/(cm2 s atm). The comparison for dextran-70,000 is presented in Fig. 3. The ultra-filtration experiments [20] were carried out in a thin channel of dimensions 100 mm × 60 mm × 5.9 mm. Kalle polysulphone membranes were used with a permeability of 1.44 × 10−10 m3 /(N s). The feed concentrations were varied from 0.43, 0.935 and 1.42 kg/m3 ; the operating pressure differences were 200, 400 and 600 kPa; cross-flow velocities were 1.06, 1.84, 2.75 m/s. It is clear from these two figures that the predicted results are in good agree-
K. Sreenivas et al. / Journal of Membrane Science 201 (2002) 203–212
Fig. 3. Comparison with the experimental flux (×105 m3 /(m2 s)) data for dextran [20].
ment with the experimental data. This indicates that the model presented herein is suitable for solutes of widely varying diffusivities. The major advantage of this model is that it can be successfully used, with the assumptions inherent in the development of the model (as described in the Section 1.1), for high diffusivity solutes for which the concentration boundary layer will definitely cross the viscous sublayer. Fig. 4 represents the dimensionless concentration boundary layer development along the channel length for different solutes, namely, sodium chloride, sucrose and dextran. The solutes are chosen in the decreasing order of diffusivity (orders of magnitudes of 10−9 , 10−10 , 10−11 m2 /s, respectively). Physical properties of these solutes in aqueous solution are presented in the Appendix B. As is evident from the figure, that for the high diffusivity solutes (NaCl and sucrose, curves 1 and 2), the concentration boundary layer develops and goes beyond the viscous sub-layer into the buffer region of the hydrodynamic boundary layer. Only for dextran (curve 3), which is having the smallest diffusivity (the highest Schmidt number) of the three solutes, the concentration boundary layer lies well within the viscous sub-layer. Thus, the present model is more generalized compared to the existing fundamental models of turbulent cross-flow filtration [11,12].
209
Fig. 4. Variation of concentration boundary layer thickness along the channel length. co = 1 kg/m3 , R r = 1, L = 1000 mm, P = 800 kPa, Re = 4600. Curve: 1, NaCl; 2, sucrose; 3, dextran.
The axial velocity profile after incorporation of suction is expressed in Eq. (12). Fig. 5 represents a plot of the x-component of the velocity profile as a function of the distance from the wall when the average velocity in the channel is u0 = 1 m/s. Curve 1 represents the velocity profile when there is no suction (i.e. v w = 0). The effect of suction (v w = 0.001 m/s) on the velocity profile is presented in curve 2. It can be clearly observed from the figure that the velocity decreases across the channel due to suction.
Fig. 5. Variation of x-component of velocity with suction. Curve: 1, v w = 0.0 m/s; 2, v w = 0.001 m/s.
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Fig. 6. Variation of membrane surface concentration along the channel length for NaCl solution. co = 42 kg/m3 , P = 103 atm. Curve: 1, Re = 4600; 2, Re = 9200; 3, Re = 13,800.
Since the major thrust of this work is to develop a model that would be valid even for high diffusivity solutes, a parametric study is undertaken with NaCl as the solute. The aim of the study is to quantify the effects of various operating conditions (e.g. transmembrane pressure, Rr , u0 ) on the permeate flux and the observed retention. Fig. 6 represents the variation of dimensionless membrane surface concentration along the length of the channel for various cross-flow velocities. Curve 1 is for the lowest and curve 3 is for the highest cross-flow velocities. It is clear from the figure that increased turbulence (high cross-flow velocity) impedes the accumulation of the solutes on the membrane surface. This would result in a lower value of membrane surface concentration. The relative decrease in membrane surface concentration results in a reduction in the build up of osmotic pressure, thereby, increasing the effective driving force across the membrane. The consequent increases in the permeate flux are presented in Fig. 7, where curves 1–3 are arranged in increasing order of cross-flow velocity. Fig. 8 presents the variation of quality in the form of observed retention of the permeate along the channel for different cross-flow velocities. Curve 3 is for the highest and curve 1 is for the lowest cross-flow velocity. It can be observed that the quality deteriorates
Fig. 7. Variation of permeate flux along the channel length for NaCl solution. Operating conditions are same as in Fig. 6.
along the channel due to increase in the membrane surface concentration (Fig. 6). As explained earlier, an increase in cross-flow velocity decreases the membrane surface concentration, thereby decreasing the chemical potential difference across the membrane. This would result in less solute transport through the membrane (lower permeate concentration) and hence, the observed retention will increase.
Fig. 8. Effect of cross-flow velocity on the profile of observed retention along the channel length. Operating conditions are same as in Fig. 6.
K. Sreenivas et al. / Journal of Membrane Science 201 (2002) 203–212
211
of the operating conditions. The results from the model are successfully compared with the data available from literature for sodium chloride and dextran under turbulent flow conditions.
Appendix A α1 = ΘRr α2 =
Fig. 9. Effect of intrinsic retention on the profile of observed retention along the channel length. co = 42 kg/m3 , P = 103 atm; Re = 4600. Curve: 1, R r = 0.98; 2, R r = 0.95; 3, R r = 0.90.
Fig. 9 represents the effects of Rr on the observed retention. For a fixed Rr , Ro decreases along the channel. This decrease appears less than the previous figure (Fig. 8) due to the length scale for Ro chosen here to accommodate larger variation in Ro . The axial variation (reduction) in Ro is consistent with the relative increases in Cm (Fig. 6) for a fixed value of Rr . At a fixed location in the channel, Ro is lower for smaller values of Rr . Low values of Rr indicate that the membrane is more porous and therefore, the solute concentration in the permeate stream tends to increase. This leads to a decrease in the values of Ro with smaller Rr .
4. Conclusions A transport model for the osmotic pressure controlled turbulent cross-flow membrane filtration process is developed. The limitations of the previous models, specially, for high diffusivity solutes have been overcome in this work. Thus, the model is applicable for most of the solutes typically encountered in membrane separation processes (from RO to UF). An integral method is employed to solve the coupled governing hydrodynamic and mass transfer equations. The solution simultaneously predicts the permeate flux and permeate concentration as a function
(A.1)
ΘRr2 c0 B1
(A.2)
(P )
α3 =
ΘRr co2 B2 [1 − (1 − Rr )2 ] (P )
(A.3)
α4 =
ΘRr co3 B3 [1 − (1 − Rr )3 ] (P )
(A.4)
where Lp de (P ) Θ= 4D
(A.5)
∗ ∗ 2 ) + 3α1 δ ∗ (cm ) F = 2 − α1 δ ∗ + 2α1 δ ∗ (cm ∗ 3 +4α1 δ ∗ (cm )
(A.6)
where ∗ cm =
cm co
and
δ∗ =
δ de
d2 u µ dB ρvw [B + (dB/dy)] M dB = − − MB dy 2 2B 2 dy 2B 2 dy (A.7) where M = µ2 + 4(τw − ρuvw )B y 2 B = ρ ky 1 − exp − A
y dB = ρk 2 y 1 − exp − dy A
y 2 2y exp(−y/A) × + 1 − exp − A A where A = 26ν
τw ρ
−1/2
+ exp(5.9vw )
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Appendix B. Physical properties of the solutes (i) Diffusivity [19,20,27]: NaCl, 1.5 × 10−9 m2 /s; sucrose, 5.24 × 10−10 m2 /s; dextran, 4 × 10−11 m2 /s. (ii) Viscosity: Viscosity of the solutions have been taken as 0.001 Pa s. (iii) Density: Density of the solutions have been taken as 1 g/cm3 . (iv) Osmotic pressure: Osmotic pressure of macromolecular solutions can be expressed in a viral form: Π = B1 C + B2 C 2 + B3 C 3
[10]
[11]
[12]
[13]
[14]
where Π is in Pa and C is in kg/m3 . For dextran T70 [20]: B1 = 37.5; B2 = 0.752; B3 = 0.00764. For NaCl [19]: B1 = 83086.5; B2 = 0.0; B3 = 0.0. For sucrose [27]: B1 = 3770; B2 = 38.79; B3 = −0.04. References [1] W.F. Blatt, A. Dravid, A.S. Michaels, L. Nelson, Solute polarization and cake formation in membrane ultra-filtration: causes, consequences and control techniques, in: J.E. Flinn (Ed.), Membrane Science and Technology, Plenum Press, New York, 1970. [2] S. Srinivasan, C. Tien, W.N. Gill, simultaneous development of velocity and concentration profiles in reverse osmosis systems, Chem. Eng. Sci. 22 (1967) 417. [3] J.S. Shen, R.F. Probstein, On the prediction of limiting flux in laminar ultra-filtration of macromolecular solutes, Ind. Eng. Chem. Fundam. 16 (1977) 459. [4] D.R. Trettin, M.R. Doshi, Ultra-filtration in an unstirred batch cell, Ind. Eng. Chem. Fundam. 19 (1980) 189. [5] J.G. Wijmans, S. Nakao, C.A. Smolders, Flux limitations in ultra-filtration: osmotic pressure model and gel layer model, J. Membr. Sci. 20 (1984) 115. [6] R. Rautenbach, R. Albrecht, Membrane Processes, Academic Press, New York, 1986. [7] T.K. Sherwood, P.L.T. Brian, Salt concentration at phase boundaries in desalination by reverse osmosis, Ind. Eng. Chem. Fundam. 4 (1965) 113. [8] L. Dresner, Boundary layer build-up in the demineralization of salt water by reverse osmosis, Oak Ridge National Laboratory Report, 1964, p. 3621. [9] D.R. Trettin, M.R. Doshi, Pressure independent ultra-filtration —is it gel limited or osmotic pressure limited? in: A.F. Turbak
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