a novel approach for modeling concentration polarization in crossflow membrane filtration based on...

A novel approach for modeling concentration polarization incross¯ow membrane ®ltration based on the equivalence

of osmotic pressure model and ®ltration theory

Menachem Elimelech1,*, Subir Bhattacharjee

School of Engineering and Applied Science, 5732 Boelter Hall, University of California, Los Angeles, CA 90095-1593, USA

Received 4 November 1997; received in revised form 18 February 1998; accepted 20 February 1998

Abstract

A theoretical model for prediction of permeate ¯ux during cross¯ow membrane ®ltration of rigid hard spherical solute

particles is developed. The model utilizes the equivalence of the hydrodynamic and thermodynamic principles governing the

equilibrium in a concentration polarization layer. A combination of the two approaches yields an analytical expression for the

permeate ¯ux. The model predicts the local variation of permeate ¯ux in a ®ltration channel, as well as provides a simple

expression for the channel-averaged ¯ux. A criterion for the formation of a ®lter cake is presented and is used to predict the

downstream position in the ®ltration channel where cake layer build-up initiates. The predictions of permeate ¯ux using the

model compare remarkably well with a detailed numerical solution of the convective diffusion equation coupled with the

osmotic pressure model. Based on the model, a novel graphical technique for prediction of the local permeate ¯ux in a

cross¯ow ®ltration channel has also been presented. # 1998 Elsevier Science B.V.

Keywords: Concentration polarization; Cross¯ow ®ltration; Filtration theory; Osmotic pressure

1. Introduction

Reverse osmosis, nano®ltration, and some ultra®l-

tration processes are primarily characterized by use of

membranes having pores that can retain solute parti-

cles ranging from a few angstroms to several nan-

ometers in diameter. Solutes in this size range include

simple molecules, proteins, and macromolecules. Fil-

tration of such molecules from their aqueous solutions

generally does not lead to formation of a cake layer on

the membrane surface. Thus, these membrane ®ltra-

tion processes are traditionally explained using the

concept of concentration polarization, employing

thermodynamic principles governing true solution

behavior [1±3].

An alternative approach for modeling ¯ux decline

during membrane separation processes is based on the

®ltration theory [4±6]. Filtration or hydrodynamic

theories are based on the premise that a separate cake

or a gel layer forms on the membrane surface, which

offers a hydrodynamic resistance to permeate ¯ow.

Journal of Membrane Science 145 (1998) 223±241

*Corresponding author.1Present address: Department of Chemical Engineering, Yale

University, 9 Hillhouse Avenue, New Haven, CT 06520, USA.

0376-7388/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved.

P I I S 0 3 7 6 - 7 3 8 8 ( 9 8 ) 0 0 0 7 8 - 7

Since cakes are rarely formed during membrane

®ltration of very small solutes, application of

®ltration theories to membrane processes where solely

concentration polarization occurs has been questioned

[7,8]. Considerable debate exists regarding the appro-

priate theoretical framework that needs to be invoked

to describe these processes [7±11]. This dichotomy

often leads to confusion regarding the domains of

validity of the thermodynamic and hydrodynamic

models.

A more formal approach towards uni®ed treatment

of membrane separation processes is emerging over

the recent years [10,12±14]. These theories allow for

the fact that beyond a critical solute concentration at

the membrane surface, a ®lter cake or a gel layer

would form on the membrane as a result of some phase

transition. When an assemblage of molecules in a

solution is compressed by extracting the solvent mole-

cules from the assemblage, the solutes lose their

degrees of freedom, and the retained solution tends

to become more structured [15,16]. Beyond a critical

concentration and pressure, the ¯uid forms a separate

phase where the Brownian motion of the solute mole-

cules is `̀ frozen'' [16]. This phase transition point

demarcates between the true solution behavior (dis-

ordered phase) and a cake type behavior (ordered

phase) [16]. In classical studies on ultra®ltration,

the solute concentration at this point was often termed

the `̀ gel concentration'' [1,4,7,17]. In modern the-

ories, the mathematical formulation initiates the for-

mation of a cake once this concentration is attained

[10,12]. Below this concentration, the permeate ¯ux is

considered to be governed by the solution thermo-

dynamics [10,12].

In light of a recently developed cross¯ow ®ltration

theory [12], ®ltration models also seem to account for

the typical physico-chemical phenomena that are

observed during membrane ®ltration in absence of

cake formation. The theory indeed reveals the typical

¯ux ± pressure relationship prior to the onset of cake

formation. The rationale of the theory is to consider

the particulate nature of a solute, instead of following a

continuum approach. In a concentration polarization

layer, we can visualize a solution with spatially vary-

ing solute concentration normal to the membrane as a

bank of stationary solute particles across which the

solvent percolates. Determination of the solvent ¯ux

using such a model requires evaluation of the total

hydrodynamic resistance of the polarized layer to

solvent ¯ow [3,8,9,12]. In other words, this visualiza-

tion leads to a ®ltration model where the speci®c

resistance of the polarized layer (or its porosity) varies

spatially along the transverse (normal to membrane)

direction.

The thermodynamic approach, on the other hand,

uses Darcy's equation where the applied pressure is

modi®ed by introducing an osmotic pressure term

[17±19]. According to this model, the permeate ¯ux

across the membrane arises due to the effective driving

force, which is the difference between the applied

pressure and the transmembrane osmotic pressure.

The approach focuses on determination of the solute

concentration at the membrane surface, and appar-

ently does not consider any resistance to solvent

transport across the polarized layer.

Thus, the above two approaches appear to be in

con¯ict. While the osmotic pressure model apparently

does not concern itself with the transport phenomena

in the polarized layer, the ®ltration theory is based

on the hydrodynamic resistance of the layer. One of

the objectives of this study is to formally resolve

this apparent contradiction and establish the equiva-

lence of the two approaches. Such attempts are

not entirely uncommon. Wijmans et al. [20] showed

that the hydrodynamic resistance of the boundary

layer is equivalent to the osmotic pressure difference

across the membrane. However, their approach

subsequently leads to an empirical resistance in

series model where the resistance of the boundary

layer has to be determined experimentally. A further

source of empiricism in their approach lies in the

use of mass transfer coef®cients, which need to be

speci®cally determined for a given system from

experiments.

In this study, the equivalence of the osmotic pres-

sure and ®ltration approaches has been derived from

the basic governing equations of thermodynamics and

hydrodynamics. Based on this equivalence, we pro-

ceed to develop a model for prediction of permeate

¯ux during steady-state cross¯ow membrane ®ltration

of hard spherical solutes. This model largely incorpo-

rates the theoretical framework of Song and Elimelech

[12]. The developed model leads to an analytical

expression for the local permeate ¯ux in a cross¯ow

membrane ®ltration channel, and yet, relaxes most of

the assumptions present in earlier models based on

224 M. Elimelech, S. Bhattacharjee / Journal of Membrane Science 145 (1998) 223±241

integral or similarity approaches [7,21,22]. Although

the model predicts the permeate ¯ux due to concen-

tration polarization, it is capable of predicting the

point where cake formation is initiated. The model

predictions are compared with the exact numerical

solution of the convective-diffusion equation in a

concentration polarization layer coupled with the

osmotic pressure model.

2. Modeling permeate transport during crossflowfiltration

Cross¯ow membrane ®ltration processes are gov-

erned by two coupled transport phenomena, namely,

the transverse (normal to the membrane) transport of

solvent across the polarized layer and the membrane,

and the axial (parallel to the membrane) transport of

the solution in the polarized layer. These two transport

phenomena are modeled using the convective-diffu-

sion equation along with either the osmotic pressure or

®ltration theories.

In this section, the equivalence between the

osmotic pressure and ®ltration models is formally

established. Based on this equivalence, a governing

equation for the transverse permeate transport across

the polarized layer and the membrane is developed.

Coupling the permeate transport equation with an

integral form of the convective diffusion equation

leads to the complete mathematical model for predic-

tion of permeate ¯ux during cross¯ow membrane

®ltration of small solute particles. Finally, an analy-

tical expression for the permeate ¯ux is obtained using

this model.

2.1. Transport in a polarized layer: Osmotic pressure

model and filtration theory

A typical steady-state cross¯ow ®ltration process is

depicted in Fig. 1(a). The retained solutes at the

membrane surface result in a polarized layer with

spatially variable solute concentration. At steady state,

the thickness of the polarized layer, the solute con-

centration at the membrane surface, and the permeate

¯ux depend on the axial position in the ®ltration

channel. A typical region of the polarized layer and

the membrane at any axial distance downstream from

the channel entrance is shown schematically in

Fig. 1(b). While traveling from the feed solution to

the low pressure side of the membrane, the permeate

encounters the resistance of the polarized layer fol-

lowed by that of the membrane. These two resistances

are viewed differently by the ®ltration and osmotic

pressure models.

When considering the two approaches for modeling

permeate transport, it should be noted that while

®ltration theory is traditionally applied to a ®xed

bed of stationary particles, the osmotic pressure model

is applied to a liquid solution in which the solutes

exhibit Brownian motion. As we are essentially deal-

ing with a liquid solution in a concentration polariza-

tion layer of small solute particles, application of

®ltration theory may be faulty unless the essential

differences between a ®xed bed of stationary particles

and a solution (in liquid state) are incorporated in this

theory. Thus, while the permeate ¯ux in both the

®ltration theory and the osmotic pressure model can

Fig. 1. Schematic representation of the steady-state concentration

polarization phenomenon in a crossflow filtration unit depicting (a)

the buildup of the polarized layer under the influence of axial flow

and permeation drag, and (b) the accumulation of solutes near the

membrane at any axial position of the filtration channel. (b) also

shows the mechanical and osmotic (thermodynamic) pressures at

different locations in the polarized layer. In both figures, CP

denotes the concentration polarization layer.

M. Elimelech, S. Bhattacharjee / Journal of Membrane Science 145 (1998) 223±241 225

be expressed phenomenologically as

v � �Peff

�R(2.1)

the interpretations of the effective pressure

difference, �Peff, and the resistance to permeate

¯ow R, are considerably different in these two

approaches.

In the case of a ®xed bed of stationary particles,

the particles impart a drag force on the ¯owing

solvent. Thus, the solvent encounters a pressure

drop when traversing the particle bed, which is

equal to the cumulative drag force of the stationary

particles. For a solution in the polarized layer,

however, the situation is quite different. As the solu-

tion is in a liquid state, there is no mechanical pressure

drop across the polarized layer. In other words, any

pressure applied to the polarized layer from the bulk

solution is transmitted undiminished to the membrane

surface.

Let us consider the osmotic pressure viewpoint for

permeate transport across a concentration polarization

(CP) boundary layer and a membrane. Referring to

Fig. 1(b), the permeate ¯ux can be expressed in two

equivalent forms, namely [20],

v � �Pÿ��m

�Rm

� �Pÿ��b

��Rm � RCP�(2.2)

where Rm and RCP are the resistances of the membrane

and the polarized layer to solvent ¯ow, respectively.

Here �P�PmÿPp in the ®rst expression and PbÿPp in

the last, with Pb, Pm, and Pp being the pressures at the

bulk solution, membrane surface, and the permeate,

respectively. We note, however, that �P is identical in

both the expressions, as Pm�Pb from the preceding

discussion. The terms ��m and ��b denote the

osmotic pressure difference between the membrane

surface and permeate (�mÿ�p), and between the feed

bulk and permeate (�bÿ�p), respectively.

The ®rst expression for the permeate ¯ux in

Eq. (2.2) considers the transport across the membrane

alone, while the last expression considers the overall

transport across the polarized layer and the membrane.

From the equivalence of the two expressions in

Eq. (2.2), it is apparent that the resistance of the

polarized layer RCP must compensate for the differ-

ence between ��m and ��b.

At steady state, a polarized solution in membrane

®ltration contains two oppositely directed concentra-

tion gradients for the solute and solvent. The increase

in solute concentration near the membrane is accom-

panied by a decrease in the solvent concentration. The

resulting imbalance in solvent chemical potential

causes a solvent ¯ow from the bulk solution towards

the membrane. Thus, denoting the osmotic pressure

difference across the polarized layer as ��CP��bÿ�m, another expression for the permeate ¯ux

can be written as

v � ÿ��CP

�RCP

(2.3)

From the equivalence of Eqs. (2.2) and (2.3), we can

relate the various osmotic pressure drops as

��b��CP��m.

Eq. (2.2) can now be rearranged after eliminating

RCP using Eq. (2.3) as

�Pÿ��b � �Rmvÿ��CP (2.4)

Eq. (2.4) suggests that the total pressure drop across

the composite system comprising the polarized layer

and the membrane (�Pÿ��b) is equal to the sum of

the transmembrane pressure drop (hRmv) and the

osmotic pressure difference across the polarized layer,

��CP.

We can now adopt the viewpoint of ®ltration theory,

and consider the term ��CP in Eq. (2.4) as an effec-

tive pressure drop encountered by the solvent while

traversing the polarized layer. This effective pressure

drop must be an outcome of the frictional drag

between the stationary solute particles and the ¯owing

solvent molecules [12]. Thus, the equivalence of the

®ltration and osmotic pressure models can be estab-

lished if we show that the cumulative drag force

exerted by the stationary solutes in the polarized layer

is equal to the osmotic pressure difference of the

solution across the polarized layer. A force balance

on individual solute molecules in the polarized layer

establishes this equivalence.

2.2. Equivalence of the osmotic pressure gradient

and the hydrodynamic drag

The forces acting on the solute molecules in the

polarized layer comprise a net thermodynamic force


arising due to the oppositely directed concentration

gradients of the solute and solvent, and an external

force arising due to the hydrodynamic drag of

the ¯owing solvent past the solute particles. For a

stationary layer of solute particles, these two forces

will balance each other. In Appendix A, it is

shown that the net thermodynamic force Fth acting

on a solute particle is related to the osmotic pressure

gradient r� as

Fth � ÿ r��nÿ nb� (2.5)

where n is the actual number concentration of the

solute particles in the polarized layer and nb is the

number concentration of the solute particles in

the bulk solution. For ¯ow of solvent past the

stationary solute particles in the polarized layer, the

hydrodynamic drag force acting on a single solute is

given by the modi®ed Stokes±Einstein equation

[12,15]

FD � kT

Dÿ sv (2.6)

where ÿ s is a correction factor to account for the ®nite

size and concentration of the solute, k the Boltzmann

constant, T the absolute temperature, and D is the

solute diffusion coef®cient. Equating the drag

force and the net thermodynamic force on a solute

particle in the polarized layer using Eqs. (2.5) and

(2.6) yields

kT

Dÿ sv�nÿ nb� � ÿr� (2.7)

We note that this force balance is valid locally at every

point in the polarized layer even though the solute

concentration varies spatially along the transverse

direction in this layer.

Integration of Eq. (2.7) over the polarized layer

results in

kT

Dv

ZCP

ÿ s�nÿ nb� dy � ÿZCP

r� dy � ÿ��CP

(2.8)

Eq. (2.8) shows the equivalence of the cumulative

hydrodynamic drag and the osmotic pressure differ-

ence across the polarized layer. Finally, substitution of

Eq. (2.8) in Eq. (2.4) yields

�Pÿ��b � �Rmv� kT

D

ZCP

ÿ sv�nÿ nb� dy (2.9)

Thus, either Eq. (2.4) or Eq. (2.9) may be used as a

force balance that governs the transport of permeate

across the boundary layer. It is emphasized that the

permeate ¯ux v in Eq. (2.9) is the local permeate ¯ux

in the ®ltration channel and will vary along the axial

direction.

The term ÿ s was introduced in Eq. (2.6) to account

for the non-ideality of the solution. This term depends

on the intermolecular interactions, solute concentra-

tion, and particle size. For ideal solutions, ÿ s assumes

a value of 1. In the ®ltration theory of Song and

Elimelech [12], ÿ s was expressed using Happel's cell

model [15]. In the present study, the equivalence of the

osmotic pressure and the drag force is utilized to

express ÿ s in terms of the osmotic pressure difference

across the polarized layer.

2.3. Material balance in the polarized layer

Theoretical treatment of concentration polarization

involves a coupled solution of the permeate transport

Eq. (2.9) derived above, along with some governing

material balance equations for the cross¯ow channel.

At steady state, neglecting axial concentration gradi-

ents in the polarized layer, the differential material

balance describing the accumulation of solutes along

the y direction in the polarized layer at any axial

position of the ®ltration channel (Fig. 1 (a)) is

Dd2n

dy2� v

dn

dy� 0: (2.10)

The boundary conditions associated with this equation

are

n � nm at y � 0 �membrane surface� (2.11a)

and

n � nb as y!1�feed bulk� (2.11b)

The second boundary condition (2.11b) indicates that

the concentration polarization layer is thin compared

to the channel height. Solution of Eq. (2.10) subject to

the above boundary conditions (2.11) yields the con-


centration pro®le

nÿ nb � �nm ÿ nb� exp ÿ v�x�yD

� �(2.12)

A second integral material balance for the polarized

layer relating the axial (along x direction) and trans-

verse (along y direction) convection of the solute can

be written as [12]Z10

u�nÿ nb� dy � nb

Zx0

v�x0� dx0 (2.13)

Here, v(x) is the local permeate ¯ux at position x and u

is the axial (cross¯ow) velocity in the channel. Assum-

ing a linear velocity pro®le, u� y, where is the shear

rate, we can integrate Eq. (2.13) after substituting

Eq. (2.12) to obtain [12]

nm ÿ nb � nb

D2v�x�2

Zx0

v�x0� dx0 (2.14)

which relates the local membrane surface concentra-

tion to the local permeate ¯ux at any axial position x of

the ®ltration channel.

2.4. Coupling the material balance and permeate

transport equations

Eq. (2.10) can be integrated once using the bound-

ary condition Eq. (2.11b) to yield

Ddn

dy� ÿv�x��nÿ nb� (2.15)

Using Eq. (2.15) in Eq. (2.9) and integrating the

resulting expression from 0 to 1 by applying the

mean value theorem leads to

�Pÿ��b � �Rmv�x� � kTÿ s��n��nm ÿ nb� (2.16)

where �n �nb < �n < nm� is the mean value of the solute

number density. Eqs. (2.4) and (2.16) provide a rela-

tionship between ÿ s��n� and the osmotic pressure at the

membrane surface, given by

ÿ s��n� � ÿ ��CP

kT�nm ÿ nb� ��m ÿ�b

kT�nm ÿ nb� (2.17)

When the concentration dependence of osmotic pres-

sure is known independently, ÿ s��n� can be evaluated

from the knowledge of membrane surface concentra-

tion (Section 2.7).

Eqs. (2.14),(2.16) and (2.17) constitute a coupled

system of equations that implicitly relate the

permeate ¯ux to the membrane surface concentration.

These can be solved simultaneously along with an

appropriate expression for the concentration depen-

dence of osmotic pressure to determine the permeate

¯ux. However, solution of the resulting integral

equation is relatively complicated. In Section 2.5,

we present an alternate methodology to arrive at an

analytical expression for the permeate ¯ux, thus

circumventing a numerical solution of the integral

equation.

2.5. Analytical solution of the crossflow filtration

equations

The analytical solution presented here is based on

the method outlined in [12]. Combining Eqs. (2.14)

and (2.16) we obtain

�Pÿ��b � �Rmv�x�

� kTÿ s��n� nb

D2v�x�2

Zx0

v�x0� dx0

24 35(2.18)

Denoting

� � nbkTÿ s��n� D2

(2.19)

and neglecting its variation with x, the derivative of

Eq. (2.18) with respect to x can be written as

0 � �Rm � 2�

Zx0

v�x0� dx0

24 35 dv�x�dx� �v�x�3 (2.20)

Rearranging Eq. (2.20) after substituting for the inte-

gral term from Eq. (2.18) yields

dv�x�dx� ÿ �v�x�4

2��Pÿ��b� ÿ �Rmv�x� (2.21)

Integration of Eq. (2.21) with the initial condition

v�0� � �Pÿ��b

�Rm

(2.22)


provides a cubic equation in v(x),

�Rm

�Peff

� �3

� 6�

�Peff

x

" #v�x�3 � 3�Rm

�Peff

v�x� ÿ 4 � 0

(2.23)

the real root of which is

v�x� � �Peff

��3m � 6��P2

effx�1=3*�3

m

�3m � 6��P2

effx

� �� 4

� �1=2

�2

( )1=3

ÿ �3

�3m � 6��P2

effx

� �� 4

� �1=2

ÿ2

( )1=3+;

(2.24)

where �Peff��Pÿ��b and �m��Rm. Finally, the

channel averaged permeate ¯ux can be determined

directly from Eq. (2.18) as

V � 1

L

ZL0

v�x� dx � �Peff

L�v�L�2 1ÿ �Rm

�Peff

v�L��

(2.25)

where L is the channel length.

It should be noted that the membrane surface con-

centration increases rapidly near the channel entrance

and subsequent variations of nm are more gradual.

Consequently, since � is a function of the membrane

surface concentration, the assumption of constant �used in the above analysis should be appropriate at

moderate to large distances from the channel entrance.

When the variation of � along the ®ltration channel

must be considered, Eq. (2.18) should be solved

numerically.

The assumption of constant � can, however, be

relaxed considerably by determining the local varia-

tions of the parameter using an iterative solution of the

governing equations. In this study, the permeate ¯ux

was determined from a solution of Eqs. (2.16) and

(2.24), along with an appropriate expression for the

concentration dependence of osmotic pressure. At

each iterative step, the parameter � was updated on

the basis of the local membrane surface concentration.

Further details regarding the procedure are given in

Section 4, where the technique is used to predict the

permeate ¯ux.

2.6. Axial variation of membrane surface

concentration and cake formation

The downstream distance from the channel entrance

x corresponding to a given membrane surface con-

centration nm is obtained by rearranging Eq. (2.23)

and using Eq. (2.16), which yields

x � �Peff

6�

(�3

m

�1� 3kTÿ s��n��nm ÿ nb�=�Peff ��Peff ÿ kTÿ s��n��nm ÿ nb��3

ÿ �m

�Peff

� �3)

(2.26)

The term ÿ s��n� in the above expression is determined

using Eq. (2.17), with�m evaluated for nm. Eq. (2.26)

can be solved for known operating conditions and a

known value of x to yield the local membrane surface

concentration nm. Thus, Eq. (2.26) is an independent

expression for the spatial (along x) variation of the

membrane surface concentration. Note that a prior

knowledge of the permeate ¯ux is not necessary to

obtain the membrane surface concentration using this

expression.

When the solute concentration at the membrane

surface attains a critical value where phase transition

begins, a cake layer is formed on the membrane

surface. For instance, for a monodisperse suspension

of rigid spherical particles, it is usually believed that

the transition to an ordered solid phase occurs at a

maximumpackingvolumefraction�max(�4�a3nmax/3)

of 0.64 (random close packing) [12,16]. The ¯ux

decline behavior in the cross¯ow ®ltration unit is

osmotic pressure governed as long as the solute

volume fraction � < �max.

Substituting the maximum solute particle concen-

tration nmax corresponding to the maximum packing

density of the solutes in Eq. (2.26), we can also

obtain the axial position xc at which cake formation

starts in the ®ltration channel. The axial position in

the ®ltration channel where cake formation starts

depends on the operating pressure, feed solute

concentration, particle size, shear rate, and membrane

resistance. It may be noted that as long as


�Peff<�(�max), there will be no cake formation. Cake

formation may occur only above a critical pressure

�Pc��(�max). Furthermore, above the critical pres-

sure, a cake layer may form only if the concentration

polarization is severe enough to raise the membrane

surface concentration to �max. Thus, even above the

critical pressure, �max may not be attained when the

®ltration channel is shorter than xc. To ascertain

whether a cake will form anywhere in the ®ltration

unit above the critical pressure, we only need to verify

if xc < L. Determination of the permeate ¯ux using the

present model should be restricted to distances less

than xc.

Another quantity of interest is the local ®ltration

number NF. The ®ltration number can be considered as

the ratio of the energy required to bring a solute

particle from the membrane surface to the bulk solu-

tion to the thermal energy of the particle [12]. It is

de®ned as

NF � 4�a3��Pÿ��b ÿ �Rmv�x��3kT

(2.27a)

Using this de®nition, and rearranging Eq. (2.16),

another expression for the ®ltration number can be

obtained

NF � ÿ s��n��m ÿ �b� (2.27b)

Eqs. (2.27a) and (2.27b) relate the ®ltration number to

the local permeate ¯ux and the local membrane sur-

face concentration, respectively. Using Eq. (2.27b),

the ®ltration number corresponding to a given mem-

brane surface concentration can be determined.

Furthermore, the permeate ¯ux can be evaluated from

Eq. (2.27a) once NF is known.

Eqs. (2.26),(2.27a) and (2.27b) can be solved

simultaneously to yield the local variation of the

permeate ¯ux and the membrane surface concentra-

tion in a ®ltration channel. For a given set of operating

conditions, the membrane surface concentration at a

given axial position can be evaluated by solving the

nonlinear Eq. (2.26). Substituting this membrane sur-

face concentration in Eq. (2.27b) gives NF, which is

then used in Eq. (2.27a) to determine the local perme-

ate ¯ux. Hence, Eqs. (2.26),(2.27a) and (2.27b) are

useful for a priori prediction of permeate ¯ux. These

equations are also amenable to a graphical treatment,

as described later in Section 4.2.

2.7. Osmotic pressure of hard-sphere solute particles

Prediction of permeate ¯ux using the procedure

mentioned above requires an independent relationship

between the osmotic pressure and the membrane sur-

face concentration. The osmotic pressure enters the

model through the expression for ÿ s��n�, which is

given by Eq. (2.17). For hard-spherical solute parti-

cles, the Carnahan±Starling equation can be used for

the osmotic pressure [24]:

��n� � nkT1� �� 2 ÿ �3

�1ÿ ��3 (2.28)

where n�3�/(4�a3) is the solute number density and a

is the radius of the spherical solute particle. Thus, the

osmotic pressure can be determined from the knowl-

edge of solute particle size. Using Eq. (2.28) in

Eq. (2.17) provides ÿ s��n�. Similarly, other known

forms of concentration dependence of osmotic pres-

sure (such as experimental correlations) can also

provide estimates of ÿ s��n�. Once ÿ s��n� is determined,

the parameter � given by Eq. (2.19) can be evaluated,

which enables calculation of the local permeate ¯ux

using Eq. (2.24). Alternatively, ÿ s��n� can be used in

Eqs. (2.26),(2.27a) and (2.27b) to arrive at a graphical

solution as described in Section 4.2.

3. Numerical solution of the steady-stateconvective-diffusion equation

The model for concentration polarization during

cross¯ow ®ltration described in the previous section

was developed using several approximations. In writ-

ing the steady-state solute material balance (2.10), the

axial solute concentration gradients were considered

to be negligible. Furthermore, all integrals in the

transverse direction were evaluated from 0 to 1(which is tantamount to considering a thin polarized

layer compared to the channel height). In order to

study the in¯uence of these approximations, the model

predictions were compared with a detailed numerical

solution of the steady-state convective-diffusion equa-

tion in a cross¯ow ®ltration system.

The steady-state concentration polarization phe-

nomenon in a cross¯ow ®ltration process is described

by the convective-diffusion equation, coupled with the


osmotic pressure model for permeate transport

[25,26,27]. Here we brie¯y summarize the appropriate

governing equations, and the methodology for numeri-

cally solving the system of equations.

3.1. The convective-diffusion equation

The steady-state differential solute material balance

in the concentrated boundary layer may be written for

the cross¯ow geometry as

u@n

@xÿ v

@n

@y� D

@2n

@y2(3.1)

where a linear variation of the axial velocity u� y is

assumed in accordance with the model presented in

the previous section. The boundary conditions used

with this equation are

n � nb as y!1�at the feed bulk� (3.2a)

n � nb at x � 0 for all y �at the channel entrance�(3.2b)

and

ÿ v�x��nm ÿ np� � D@n

@yat y � 0

�at the membrane surface� (3.2c)

The permeate concentration np is zero for a perfectly

rejecting membrane and the membrane surface con-

centration nm varies with x.

An additional relationship between the permeate

velocity and the membrane surface concentration is

required for the solution of the above partial differ-

ential equation. Here we use the expression for the

osmotic pressure governed local permeate ¯ux

v�x� � �Pÿ��nm��Rm

(3.3)

The above model has been used extensively in cross-

¯ow membrane ®ltration of macromolecular systems

where only concentration polarization is encountered

[3,10,11,25±27].

3.2. Solution of the convective-diffusion equation

The partial differential Eq. (3.1) with the boundary

conditions (3.2) can, in principle, be solved numeri-

cally using the ®nite difference technique [25,26,27].

However, a more ef®cient technique described else-

where [25] was employed to solve the above system of

equations. Although this so-called generalized inte-

gral technique requires numerical solution, it is con-

siderably faster than the conventional ®nite difference

scheme, and yields the exact (numerical) solution of

the governing partial differential equation. Use of this

technique facilitated a detailed assessment of the

accuracy of the presently developed model over a

wide range of operating conditions.

4. Model predictions

Predictions of the developed model were compared

with the solution of the convective-diffusion equation

using the same expressions for the osmotic pressure

dependence on solute concentration, and similar

values of the diffusion coef®cient. As already men-

tioned, the osmotic pressure was determined using the

Carnahan±Starling equation of state for hard spherical

particles (2.28). The diffusion coef®cient was esti-

mated from the Stokes±Einstein equation

D � kT

6��a(4.1)

where � is the solvent viscosity. The use of a constant

diffusion coef®cient is reasonable for a system of hard

spheres, as the gradient diffusion coef®cient does not

vary appreciably from the Stokes±Einstein diffusivity

(4.1) for particle volume fractions as high as 0.4 [16].

The model developed in Section 2 is capable of

simultaneously predicting the variation of the perme-

ate ¯ux and the membrane surface concentration along

the ®ltration channel. The permeate ¯ux is determined

at each axial position in the channel using an iterative

solution of Eqs. (2.16) and (2.24). For an assumed

value of the membrane surface concentration nm, we

determine � from Eqs. (2.17) and (2.19). The perme-

ate ¯ux corresponding to this value of � is obtained

using Eq. (2.24). Using this permeate ¯ux in

Eq. (2.16), the value of nm is updated, which is again

used in Eqs. (2.17) and (2.19) to obtain a new value of

�. The procedure is repeated until two consecutive

estimates of permeate ¯ux or nm converge within some

preset limit of accuracy. In all results presented here,

the calculations were carried out to within a relative

accuracy of 10ÿ8.


Before solving the equations, the critical effective

pressure for cake formation is determined by evaluat-

ing the osmotic pressure corresponding to the max-

imum packing density �max�0.64 using Eq. (2.28).

For operating effective pressures greater than this

critical value, the critical distance for cake formation

xc is determined using Eq. (2.26). If the channel length

L is less than this critical distance, the entire ®ltration

process in the channel is predicted using the developed

model. Otherwise, the model is used only to determine

the permeate ¯ux for x < xc. Once these checks are

performed to ascertain the domain where the model

can be applied, the governing equations are solved

using the procedure outlined above.

4.1. Typical concentration and flux profiles in a

crossflow system

Figs. 2 and 3 depict typical membrane surface

concentration and ¯ux pro®les, respectively, in a

cross¯ow channel generated using the developed

model. While Fig. 2 shows the variation of the scaled

membrane surface concentration �m/�b (or nm/nb)

with scaled axial distance x/L from the channel

entrance, Fig. 3 depicts the corresponding variation

of the local permeate ¯ux v(x) scaled with respect to

the pure solvent ¯ux vw (��P/lm) along the cross¯ow

channel. The simulations were performed for a ®ltra-

tion channel of length L�0.5 m and a particle radius

a�2 nm. The pro®les shown are obtained for different

combinations of operating pressure and shear rate. It is

evident from the ®gures that an increase in operating

pressure increases the extent of concentration build-up

at the membrane surface, while an increase in shear

rate causes the build-up to be less severe. The pre-

dictions in both ®gures obtained from the iterative

calculation of the ¯ux based on the present model

(solid lines) compare well with the detailed numerical

solution of the convective-diffusion equation (sym-

bols). In all cases, the membrane surface concentra-

tions predicted by the model are slightly lower than the

concentrations predicted using the numerical solution.

Consequently, the permeate ¯uxes determined using

the model are slightly larger than the exact permeate

¯ux. The local variations of the permeate ¯ux are

predicted accurately (within 6%) by the model.

Fig. 4 compares the scaled permeate ¯ux v(x)/vw

obtained from the analytical and numerical

Fig. 2. Variation of the scaled solute volume fraction at the

membrane surface (�m/�b) with scaled axial position (x/L) in a

crossflow filtration channel. Solid lines represent the model

predictions, while symbols represent the numerical solution of

the convective diffusion equation. The results are shown for a 0.5 m

long channel and a membrane permeability 1/�m�5�10ÿ11

m Paÿ1sÿ1. Other operating conditions are: �b�10ÿ3 and a�2 nm.

The three sets of curves were obtained for different combinations

of applied pressure and shear rates.

Fig. 3. Variation of the scaled permeate flux (v/vw) with scaled

axial position in the filtration channel corresponding to the

concentration profiles in Fig. 2. Operating conditions are similar

to those used in Fig. 2.


approaches for different particle sizes under a speci-

®ed set of operating conditions (�P�400 kPa,

�b�10ÿ3, L�0.5 M, and �400 sÿ1). As the particle

size is increased, the concentration build-up at the

membrane surface occurs more rapidly, and hence, the

permeate ¯ux decline relative to the pure water ¯ux is

also rapid. In case of the two largest particle sizes, the

¯ux pro®les are truncated at smaller scaled separation

distances. This is because the membrane surface

concentration attains the maximum packing value

of �m�0.64 beyond these points, and the correspond-

ing ¯ux decline becomes cake-layer governed.

The rapid decrease of permeate ¯ux with an

increase in particle size, as depicted in Fig. 4, arises

due to the interplay between the convective and

diffusive transport of solute particles in the polarized

layer. The extent of concentration polarization and

permeate ¯ux decline depends on the osmotic pressure

as well as the diffusion coef®cient. An increase in

particle size causes both these properties to decrease.

A lower osmotic pressure indicates a higher initial

(before attainment of steady state) permeate velocity,

and hence, a higher initial convective transport of

solutes towards the membrane. Furthermore, a

decrease in diffusion coef®cient results in a slower

back-diffusion of solutes away from the membrane,

resulting in a greater accumulation of solutes near the

membrane surface. Thus, with an increase in particle

size, both these factors result in a rapid build-up of

membrane surface concentration. Consequently,

although the permeate ¯ux increases with an increase

in particle size initially, the increased build-up of

membrane surface concentration gives rise to a lower

steady-state permeate ¯ux.

The small disparity (<6%) between the ¯ux predic-

tions using the model and the numerical solution in

Figs. 3 and 4 arises due to two assumptions in the

simpli®ed model which are not used in the exact

numerical solution. In the model, we consider a thin

polarized layer compared to the channel height,

which allows the use of y!1 in the boundary

condition (2.11b), and assume negligible axial con-

centration gradients in the governing mass balance

equation. Both these assumptions govern the extent to

which the model deviates from the exact numerical

solution.

The approximation inherent in consideration of a

thin polarized layer can be determined simply by

replacing the upper limit of the integral over the

polarized layer by a ®nite value. Denoting this ®nite

limit by �CP, and performing the integration in

Eq. (2.13) using this ®nite upper limit reveals that

Eq. (2.14) becomes

nm ÿ nb � nb

D2g��CP� v�x�2

Zx0

v�x0� dx0 (4.2)

where

g��CP� � 1ÿ v�x��CP

D� 1

� �exp ÿ v�x��CP

D

� �(4.3)

This parameter attains a value of �1 when the local

Peclet number v(x)�CP/D is greater than�5. For lower

values of v(x)�CP/D, g(�CP) < 1. Therefore, for a ®nite

value of the upper limit, the membrane surface con-

centration will be larger than that predicted for y!1.

Consequently, towards the end of a ®ltration channel,

where v(x) is much lower, the discrepancy between the

model prediction and the numerical solution may

become signi®cant if v(x)�CP/D < 5.

Fig. 4. Variation of the scaled permeate flux with scaled axial

position for different particle sizes ranging from 2 to 4 nm.

Operating conditions are �P�400 kPa, 1/�m�5�10ÿ11

m Paÿ1sÿ1, channel length�0.5 m, �400 sÿ1, and �b�10ÿ3. For

the two largest particle sizes, the flux profiles are truncated at

scaled separations less than 1, as the filtration process undergoes

transition to the cake layer governed domain.


The contribution of the second approximation to the

discrepancy cannot be ascertained in a straightforward

manner. Although the permeate ¯ux expression (2.24)

was obtained neglecting any axial concentration gra-

dient (i.e., assuming a constant �), the iterative solu-

tion of Eqs. (2.16) and (2.24) ensures that � varies

locally in the axial direction. Thus, the resulting

solution most likely emulates a slow variation in solute

particle concentration in the polarized layer along the

axial direction. This assumption should hold for most

cross¯ow ®ltration units of moderate length.

Despite these approximations, the model provides

remarkably accurate predictions of the membrane

surface concentration and the permeate ¯ux along

the ®ltration channel. It may be noted that the tradi-

tional similarity solutions of the convective-diffusion

equation are based on the consideration of a constant

wall ¯ux (or concentration) [22,25], which render

these solutions unsuitable for determination of the

local variation of the ¯ux. Similarly, analytical expres-

sions for the permeate ¯ux based on integral solutions

also require assumption of constant membrane surface

concentration [21,25]. In both instances, the above

mentioned approaches underpredict (similarity solu-

tions) or overpredict (analytical expressions based on

integral solution) the ¯ux considerably. Furthermore,

applications of these integral and similarity solutions

are often restricted to small domains of operating

conditions, beyond which they yield unreasonable

results. In contrast, the present model is capable of

predicting the permeate ¯ux far more accurately over a

wide range of operating conditions.

4.2. Graphical evaluation of local variation in

permeate flux

It was shown in Section 2.6 that the present model

decouples the transverse hydrodynamics and thermo-

dynamics in a polarized layer from the axial hydro-

dynamics, and can lead to independent determination

of the membrane surface concentration without a prior

knowledge of the permeate ¯ux using Eq. (2.26).

Once the membrane surface concentration is deter-

mined from Eq. (2.26), it can be substituted in

Eqs. (2.27a) and (2.27b) to predict the permeate ¯ux.

The entire procedure of using these two equations to

predict the permeate ¯ux can be implemented graphi-

cally as described below.

The key element of the graphical technique is the

®ltration number de®ned in Eqs. (2.27a) and (2.27b),

which relates the permeate ¯ux and membrane surface

concentration in a ®ltration channel to the osmotic

pressure governing relationship. From Eq. (2.27a), we

observe that plots of the ®ltration number against the

scaled permeate ¯ux v(x)/vw will be straight lines with

a negative slope:

NF � 4�a3�P

3kT1ÿ��b

�P

� �ÿ v�x�

vw

� �(4.4)

The slope of the straight lines will depend on the

particle size and the applied pressure, while the inter-

cept will additionally depend on the feed bulk con-

centration. For low feed concentrations, neglecting the

bulk solution osmotic pressure ��b will render the

intercept equal to the slope of the straight lines. Noting

that the maximum value of the quantity v(x)/vw is 1,

we can obtain several straight lines for different

combinations of operating pressure and particle size.

These lines represent the characteristic ¯ux lines in a

®ltration process.

Eq. (2.27b) provides the equilibrium line for the

®ltration process for a given concentration depen-

dence of the osmotic pressure

NF � ÿ s��n��m ÿ �b� � 4�a3��m ÿ�b�3kT

(4.5)

where Eq. (2.17) was used to obtain the ®nal expres-

sion. Thus, a plot of NF against the membrane surface

concentration �m yields the equilibrium line for the

®ltration process. The equilibrium line is related

solely to the osmotic pressure governing relationship.

Clearly this line does not depend on any of the

operating conditions.

For a known value of �m, the ®ltration number can

now be evaluated from Eq. (4.5), which is substituted

in Eq. (4.4) to yield the scaled permeate ¯ux corre-

sponding to a given applied pressure. This process can

be represented graphically by plotting Eqs. (4.4) and

(4.5) together. Fig. 5 depicts the relevant plots. Here,

the vertical axis represents the ®ltration number, the

lower horizontal axis represents the solute volume

fraction at the membrane surface, and the upper

horizontal axis represents the scaled permeate ¯ux.

In the ®gure, the straight lines represent the plots of

Eq. (4.4) obtained under different operating pressures,


while the single curved line is obtained from Eq. (4.5).

For a given operating pressure and membrane

resistance, the ¯ux corresponding to a given

membrane surface concentration can be easily

determined from the ®gure. The procedure involves

determining NF corresponding to a given �m, followed

by determination of the scaled ¯ux corresponding

to this NF. The procedure is illustrated through the

dotted construction lines abcd in Fig. 5. Clearly,

for the same membrane surface concentration, we

obtain different scaled ¯uxes under different operating

pressures.

Fig. 5 also depicts the critical condition at which

the cake formation will initiate. The dashed horizontal

line PQ depicts the ®ltration number at which the

membrane surface concentration attains its maximum

value �max (0.64). Below this value of NF, the process

is osmotic pressure governed. Therefore, the critical

operating pressure can be determined from the inter-

cept of Eq. (4.4) as

�Pc � NFc3kT

4�a3(4.6)

where NFc is the critical ®ltration number correspond-

ing to the maximum membrane surface concentration.

Below this critical pressure, a cake will never form in

the ®ltration unit.

Although Fig. 5 can relate the scaled permeate ¯ux

with the membrane surface concentration, we still

cannot predict the ¯ux as a function of the axial

position in the ®ltration channel. However, the present

model provides a second independent relationship

between the membrane surface concentration and

the axial position in the ®ltration channel when the

shear rate is known. The axial distance from the

channel entrance at which a speci®c membrane sur-

face concentration is attained can be determined

directly from Eq. (2.26). Since the term � in

Eq. (2.26) depends on the shear rate , different

operating lines corresponding to different shear rates

can be obtained by plotting the distance x against the

membrane surface concentration nm (or �m). These

operating lines will also depend on the particle size,

feed solute concentration, applied pressure, and mem-

brane hydraulic resistance. Note that these curves can

be obtained independently from the operating condi-

tions and the osmotic pressure governing relationship

without prior knowledge of the permeate ¯ux.

Fig. 6 shows the plots of x against the solute volume

fraction at the membrane surface �m. For a given set of

operating conditions, the local membrane surface

concentration at any axial position of a ®ltration

channel can be determined from the ®gure. The four

curves in Fig. 6 are obtained for various combinations

of operating pressure, particle size, and shear rate. It is

evident from the ®gure that increasing the shear rate,

and reducing the pressure and particle size, result in a

lower build-up of membrane surface concentration.

Variation of the permeate ¯ux with the axial dis-

tance in the ®ltration channel can now be determined

Fig. 5. A graphical technique for prediction of the permeate flux

during concentration polarization. The figure was obtained by

superimposing the plots of NF versus v(x)/vw, obtained from

Eq. (4.4), and NF versus �m, obtained using Eq. (4.5). Eq. (4.4)

yields the characteristic flux lines, the slopes and intercepts of

which depend on the operating pressure and particle size. Eq. (4.5),

on the other hand, yields a unique equilibrium line, which is

governed by the osmotic pressure dependence on solute volume

fraction. The horizontal dashed line PQ is the critical line above

which the membrane surface concentration attains a constant

maximum value and the filtration behavior is governed by a cake

layer growth. When a characteristic flux line intersects the critical

line, the filtration process becomes cake layer governed. Following

the example shown by the dotted construction lines abcd, the

scaled permeate flux corresponding to a given membrane surface

concentration and applied pressure can be determined. Other

conditions used to obtain the plots were �b�10ÿ3, a�3 nm, and 1/

�m�2.5�10ÿ11 m Paÿ1sÿ1.


by combining the information in Figs. 5 and 6. From

Fig. 6, the membrane surface concentration at a given

axial position is obtained for a speci®ed shear rate and

other operating conditions. Using the equilibrium line

in Fig. 5, we can then obtain the ®ltration number

corresponding to this membrane surface concentra-

tion. Finally, the scaled permeate ¯ux corresponding

to this ®ltration number is determined from the char-

acteristic line in Fig. 5 at a given operating pressure.

The entire graphical process can be summarized in a

single diagram as follows.

Fig. 7 depicts the relevant analysis for an operating

pressure of 400 kPa, a particle radius a�3 nm, and

three different shear rates ranging between 100 and

400 sÿ1. While the upper half of Fig. 7 is similar to

Fig. 5, the lower half is similar to Fig. 6. The upper

segment of the ®gure depicts the variation of the

®ltration number with permeate ¯ux as a single

straight line, which implies that the relationship

between these quantities is independent of the shear

rate. This can be readily veri®ed from Eq. (4.4).

Similarly, the equilibrium curve also remains unaf-

fected by the shear rate. The lower segment of the

®gure, which plots the relationship between x and �m,

depicts the in¯uence of the axial hydrodynamics. In

this example, three curves are obtained for the three

different shear rates employed.

The composite ®gure is now used to determine the

permeate ¯ux at a speci®c position in the ®ltration

channel (x�0.5 m). The ®rst step of the procedure

involves determination of the membrane surface con-

centration �m corresponding to this axial distance

from the lower segment of Fig. 7. Using the equili-

brium line in the upper segment of the ®gure, the value

of NF corresponding to �m is determined. Finally, the

Fig. 6. The relationship between the axial distance X and the

membrane surface concentration as depicted by Eq. (2.26). The

rate of concentration build-up will depend on the axial shear rate,

particle size, operating pressure and membrane resistance. The

curves represent the operating lines of the filtration process. All the

curves were obtained for �b�10ÿ3 and 1/�m�2.5�10ÿ11 m Paÿ1sÿ1.

The different curves were obtained under the following sets of

conditions: curve 1: �P�400 kPa, �100 sÿ1, a�2 nm; curve 2:

�P�400 kPa, �200 sÿ1, a�3 nm; curve 3: �P�400 kPa,

�100 sÿ1, a�3 nm; and curve 4: �P�600 kPa, �200 sÿ1,

a�3 nm.

Fig. 7. Combination of the two types of plots shown in Figs. 5 and

6. The straight and the curved lines in the upper half of the figure

represent the characteristic flux line and the equilibrium line,

respectively. For a given applied pressure, these two lines are

independent of the shear rate. The lower segment of the figure is

similar to Fig. 6. Here, the three curves correspond to three

different shear rates of 100, 200, and 400 sÿ1. All the curves were

generated for �b�10ÿ3, a�3 nm, 1/�m�2.5�10ÿ11 m Paÿ1 sÿ1,

and �P�400 kPa. The permeate flux at a given axial position in

the filtration channel can be obtained by following the dotted

construction lines abcde.


scaled ¯ux v(x)/vw corresponding to this value of NF is

obtained from the characteristic ¯ux line (at 400 kPa).

The entire graphical technique is illustrated by the

dotted construction lines abcde.

To summarize, the graphical technique described

above serves as a simple tool for a priori prediction of

the permeate ¯ux for a given set of operating condi-

tions and a given solute size. Such a technique can

obviate the use of detailed computational schemes for

predicting the local variations of the permeate ¯ux in a

cross¯ow ®ltration channel. The entire procedure is

based on the osmotic pressure governing relationship,

for which we employ the Carnahan±Starling

Eq. (2.28). Thus, the procedure described above is

applicable only for hard spherical solutes. In¯uence of

intermolecular interactions can be incorporated in the

model quite easily through separate governing equa-

tions for the concentration dependence of osmotic

pressure [16,24], which will modify the equilibrium

curves shown in Figs. 5 and 7. In absence of accurate

theoretical understanding of the osmotic pressure

variations with solute concentrations, empirical cor-

relations for speci®c solutes based on experiments [7±

10] may also be used in the model

4.3. The channel-averaged permeate flux

Once the local variation in the permeate ¯ux is

determined, the channel-averaged permeate ¯ux can

be evaluated using either forms of Eq. (2.25). As the

expression for the local permeate ¯ux Eq. (2.24) is

quite cumbersome, its direct incorporation in the

integral form of Eq. (2.25) requires numerical inte-

gration to determine the average ¯ux. In contrast, the

second expression in Eq. (2.25) directly yields the

average ¯ux based on a single evaluation of the local

permeate ¯ux at the channel endpoint (i.e., x�L).

Fig. 8 depicts the variation of the channel-averaged

permeate ¯ux with operating pressure for three dif-

ferent shear rates. The solid line represents the exact

numerical predictions of the permeate ¯ux based on

solution of the convective diffusion equation, while

the dashed and dotted lines represent the model results

based on numerical evaluation of the integral in

Eq. (2.25), and the analytical expression in

Eq. (2.25), respectively. All computations were per-

formed for a feed bulk concentration �b�10ÿ3 and a

®ltration channel length of 0.5 m. It is observed that

the ¯ux predicted using the numerical integration of

Eq. (2.25) is remarkably close to the exact numerical

predictions, although it slightly overpredicts the ¯ux at

very high pressures. The analytical result, on the other

hand, underpredicts the permeate ¯ux at high pres-

sures, and deviates from the exact results to a larger

extent compared to the numerical integration results.

The curves corresponding to the shear rate of 400 sÿ1

were truncated at a lower pressure compared to the

other results, since the maximum packing density for

the solutes (0.64) was attained at the membrane sur-

face, and cake formation started beyond this pressure.

As shown above, the analytical result in Eq. (2.25)

deviates from the exact numerical solution to a greater

extent compared to the result obtained when

Eq. (2.25) is integrated numerically. This behavior

becomes apparent when inspecting the predictions

of the local permeate ¯ux shown in Figs. 3 and 4.

In both the ®gures, it is observed that the local ¯ux

predicted by the model deviates from the exact numer-

ical results to a greater extent towards the end of the

Fig. 8. Variation of the channel-averaged permeate flux with

operating pressure for three different shear rates ranging from 400

to 1200 sÿ1. The permeate flux was determined for L�0.5 m and

a�2 nm. Other operating conditions are �P�400 kPa, 1/

�m�5�10ÿ11 m Paÿ1 sÿ1, and �b�10ÿ3. The solid lines represent

the exact numerical solution of the convective diffusion

equation. The dashed lines show the flux predicted by numerically

evaluating the integral in Eq. (2.25), and the dotted lines represent

the predictions obtained using the analytical expression in

Eq. (2.25).


channel. Thus, numerically performing the averaging

in Eq. (2.25) results in an even distribution of the error

over the entire channel. The analytical result, on the

other hand, relies only on the ¯ux at the endpoint of the

channel v(L), which introduces a greater error in the

average ¯ux. This also becomes apparent from

Eq. (2.25), which contains the term v(L)2 in the

denominator. The permeate ¯ux being a small quan-

tity, even a small error in v(L) introduces a relatively

signi®cant error in the estimate of the average ¯ux.

Sensitivity analysis showed that the numerical inte-

gration results based on the model are remarkably

accurate over a wide range of applied pressure and

shear rates. The resulting predictions of the average

permeate ¯ux never deviate beyond 2% of the exact

value. In contrast, the analytical expression under-

predicts the average permeate ¯ux by about 6% for

high applied pressures, low shear rates, and long

channels (>1 m). In all cases, it was observed that

the model predictions become more accurate when the

concentration polarization is less severe, a condition

achieved by maintaining low applied pressures and

high shear rates.

5. Summary and conclusion

A comprehensive model for the concentration

polarization phenomenon during cross¯ow membrane

®ltration of small hard spherical solute particles is

developed using a unique combination of hydrody-

namic (®ltration) and thermodynamic (osmotic pres-

sure) approaches. The theoretical development is

based on the equivalence of the two approaches, which

is shown here in unambiguous and general terms. The

model yields an analytical expression for the permeate

¯ux. Predictions of permeate ¯ux using the model

compare remarkably well with the numerical solution

of the convective-diffusion equation employing the

osmotic pressure model. Based on the model, a simple

graphical approach for prediction of the local variation

of the permeate ¯ux in a cross¯ow ®ltration channel is

developed. Another remarkable feature of the model is

the simplicity with which it incorporates the onset of

cake formation.

The generality of the approach facilitates the study

of in¯uence of solute properties and inter-particle

interactions on the permeate ¯ux. Although the pre-

sent study re¯ects only the effects of hard-sphere

interactions, long-range interactions (like electrostatic

double layer) can also be incorporated in the theore-

tical framework. Similarly, although constant values

of the solute diffusivity were used in the model,

incorporation of concentration dependent diffusion

coef®cients will not signi®cantly complicate the tech-

nique, except for requirement of another independent

expression for such concentration dependence. The

simplicity of the model and its accuracy facilitate a

clearer understanding of the physico-chemical phe-

nomena underlying cross¯ow membrane ®ltration of

solute particles in the concentration polarization

regime.

Acknowledgements

The research reported in this paper was supported

by the Center for Environmental Risk Reduction

(CERR) at UCLA.

Appendix A. Thermodynamic force on a solutein a polarized solution

In Section 2.1, it was stated in conjunction with

Eq. (2.5) that the net thermodynamic force on the

solute is related to the osmotic pressure gradient. Here,

we present the arguments that lead to the derivation of

Eq. (2.5).

Consider a volume element of the concentrated

solution in the polarized layer. A solute molecule in

this volume element experiences a thermodynamic

force due to its own concentration (chemical potential)

gradient

Fs � ÿr�s (A.1)

Likewise, each solvent molecule experiences a ther-

modynamic force owing to the solvent chemical

potential gradient

F0 � ÿr�0 (A.2)

The two chemical potential gradients are not indepen-

dent. At equilibrium, they are related by the Gibbs±

Duhem relationship

nsr�s � n0r�0 � 0 (A.3)


where ns and n0 represent the number concentrations

of the solute and solvent molecules in the volume

element, respectively. Eq. (A.3) implies that even

when there is no mechanical pressure gradient across

the polarized system, the solvent molecules can move

across the system due to the solvent chemical potential

gradient.

Let us now consider a hypothetical polarized layer

in absence of any external force. This represents a non-

equilibrium system where the solute and solvent

molecules will tend to diffuse in opposite directions

and attain a uniform concentration in the entire layer.

Two forces will govern the diffusion of the solute

molecules. While the chemical potential gradient of

the solute, Eq. (A.1), will give rise to the primary

diffusive force, there will be an opposing force due to

the countercurrent motion of the solvent molecules

(due to solvent chemical potential gradient). Thus, the

net force for diffusion of a solute molecule in a

polarized system is obtained by subtracting the force

exerted by the solvent molecules from the primary

diffusive force. This net force acting on the solute

molecules can be evaluated by rendering the solvent

force free [16,23].

The solvent can be made force free by subtracting a

uniform body force (force/volume) from both solute

and solvent molecules without inducing any additional

relative motion between them. This uniform body

force per unit volume of the solution is given by

F0/v0, where F0 is the force acting on each solvent

molecule and v0 is the speci®c volume of the solvent

[16,23]. Subtraction of this body force from the pri-

mary diffusive force acting on each solute molecule

will result in the net thermodynamic force on a solute

molecule in the polarized layer. Thus, the actual ther-

modynamic force acting on each solute molecule is

Fth � Fs ÿ vs

v0

F0 (A.4)

where vs is the volume of a single solute molecule.

Using the Gibbs±Duhem relationship, Eq. (A.3), and

the volume constraint

vsns � v0n0 � 1 (A.5)

we obtain

Fth � ÿr�s � vs

v0

r�0 � 1

nsv0

r�0 � ÿr�ns

(A.6)

where ÿr�0/v0 is de®ned as the osmotic

pressure gradient r� [16,23]. Thus, the net thermo-

dynamic force acting on each solute particle in the

polarized layer represents the difference between

the total diffusive force acting on the solute (ÿr�s)

and a second force arising due to the solvent ¯ow

(ÿvsr�0/v0). In absence of external forces, the

solute molecules will diffuse towards the bulk

solution from the membrane surface owing to this

net thermodynamic force.

We now note that the chemical potential at any point

in the polarized layer is determined with respect to a

reference chemical potential corresponding to the bulk

solution, and not the pure solvent. In the bulk solution,

both the chemical potential gradients and any relative

motion between the solute and solvent molecules

disappear. Thus, the concentration ns does not re¯ect

the actual particle concentration, but that of only the

accumulated solutes in excess of the feed bulk con-

centration. Consequently, in Eq. (A.6), ns represents

the net concentration of the stationary solute particles

(generally de®ned as excess particle concentration

[12]), which in the present context is the difference

between the actual particle number concentration in

the polarized layer, n, and the bulk particle number

concentration, nb (i.e. nÿnb). This leads to the rela-

tionship between the net thermodynamic force on a

solute molecule in a polarized layer and the osmotic

pressure

Fth � ÿ r��nÿ nb� (A.7)

Appendix B. Nomenclature

a solute radius

D diffusion coefficient calculated from

Eq. (4.1)

FD drag force

Fth net thermodynamic force on a solute

molecule

g(�CP) parameter defined by Eq. (4.3)

k Boltzmann's constant

L channel length

NF filtration number defined by Eqs. (2.27a)

and (2.27b)

n particle number concentration


nmax particle number concentration at random

close packing density

�n value of n between nb and nm

P pressure

R resistance

RCP resistance of concentration polarization

layer

Rm membrane hydraulic resistance

T absolute temperature

u crossflow (axial) velocity

V channel average permeate flux

v, v(x) local permeate flux in a crossflow filtration

channel

vw pure solvent flux (��P/lm)

x axial direction

y transverse direction

Greek symbols

� parameter defined by Eq. (2.19)

ÿ s parameter accounting for non-ideality of a

solution

shear rate

�CP thickness of concentrated boundary layer

� solvent viscosity

� solute volume fraction (�4�a3n/3)

�m membrane resistance defined as lm��Rm

� chemical potential

r� chemical potential gradient

vs,v0 specific volume of solute and solvent,

respectively.

� osmotic pressure

r� osmotic pressure gradient

�� osmotic pressure difference

��b osmotic pressure difference between feed

bulk and permeate solutions

�P applied pressure difference

�Pc critical pressure above which cake forma-

tion occurs

�Peff effective pressure drop; �Peff��Pÿ��b

Subscripts

0 pertaining to solvent

b bulk

CP concentration polarization layer

m membrane surface

p permeate

s pertaining to solute

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a novel approach for modeling concentration polarization in crossflow membrane filtration based on...

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