Modeling of flux decline during crossflow ultrafiltration of colloidal suspensions

Modeling of ¯ux decline during cross¯owultra®ltration of colloidal suspensions

Yonghun Lee, Mark M. Clark*

Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 N. Mathews Avenue,

Urbana, IL 61801, USA

Received 7 January 1998; accepted 1 May 1998

Abstract

Mass transfer during cross¯ow ultra®ltration is mathematically expressed using the two-dimensional convective±diffusion

equation. Numerical simulations showed that mass transfer in cross¯ow ®ltration quickly reaches a steady-state for constant

boundary conditions. Hence, the unsteady nature of the permeate ¯ux decline must be caused by changes in the hydraulic

boundary condition at the membrane surface due to cake formation during ®ltration. A step-wise pseudo steady-state model

was developed to predict the ¯ux decline due to concentration polarization during cross¯ow ultra®ltration. An iterative

algorithm was employed to predict the amount of ¯ux decline for each ®nite time interval until the true steady-state permeate

¯ux is established. For model veri®cation, cross¯ow ®ltration of monodisperse polystyrene latex suspensions ranging from

0.064 to 2.16 mm in diameter was studied under constant transmembrane pressure mode. Besides the cross¯ow ®ltration tests,

dead-end ®ltration tests were also carried out to independently determine a model parameter, the speci®c cake resistance.

Another model parameter, the effective diffusion coef®cient, is de®ned as the sum of molecular and shear-induced

hydrodynamic diffusion coef®cients. The step-wise pseudo steady-state model predictions are in good agreement with

experimental results of ¯ux decline during cross¯ow ultra®ltration of colloidal suspensions. Experimental variations in

particle size, feed concentration, and cross¯ow velocity were also effectively modeled. # 1998 Elsevier Science B.V. All

rights reserved.

Keywords: Ultra®ltration; Concentration polarization; Diffusion; Flux decline; Speci®c cake resistance

1. Introduction

In addition to applications in a large number of

industrial processes, membrane technologies have

been receiving increasing interest as an alternative

or add-on process to conventional drinking water

treatment in meeting the current and future demands

for high quality drinking water. Ultra®ltration (UF)

and micro®ltration (MF) have a great potential for

removing particulates, microorganisms, and colloidal

material from potable water supplies and wastewater

streams. A major obstacle to these applications is the

permeate ¯ux decline due to concentration polariza-

tion and fouling. During ultra®ltration of colloidal

suspensions, particles within the feed stream are con-

vectively driven to the membrane surface where they

accumulate and tend to form a cake or gel layer. This

Journal of Membrane Science 149 (1998) 181±202

*Corresponding author. Tel.: +1-217-3333629; fax: +1-217-

3339464; e-mail: [email protected]

0376-7388/98/$ ± see front matter # 1998 Elsevier Science B.V. All rights reserved.

P I I : S 0 3 7 6 - 7 3 8 8 ( 9 8 ) 0 0 1 7 7 - X

particle build-up near the membrane surface is known

as concentration polarization, and results in increasing

hydraulic resistance to permeate ¯ow; as a result the

permeate ¯ux declines with time. The goal of this

research was to develop a numerical model which can

describe the ¯ux decline behavior due to concentration

polarization during cross¯ow UF of colloidal suspen-

sions. This work was motivated by an increasing

interest in employing membrane technologies for

removing particulates, microorganisms, and colloidal

material from water supplies and wastewater streams.

The capital and operational costs of membrane sys-

tems are directly dependent on membrane permeate

¯ux. Therefore, the permeate ¯ux and the factors

affecting it are central considerations in determining

membrane process performance and cost.

2. Literature review on existing modelsfor flux decline

Many different models have been proposed to pre-

dict ¯ux decline during UF and MF. The oldest model

is the resistance model based on the cake ®ltration

theory. When a suspension contains particles which

are too large to enter the membrane pores, then a

sieving mechanism is dominant and a cake layer of

rejected particles forms on the membrane surface. The

cake layer provides an additional resistance to ®ltra-

tion, so the permeate ¯ux declines with time. The cake

layer and membrane may be considered as two resis-

tances in series, and the permeate ¯ux is then

described by Darcy's law:

J � 1

Am

dVp

dt� �P

��Rm � Rc� (1)

where J�permeate ¯ux, Am�membrane ®ltration

area, Vp�total volume of permeate, t�®ltration time,

�P�transmembrane pressure, m�viscosity of the

permeate, Rm�intrinsic membrane resistance, and

Rc�cake resistance. The cake ®ltration theory has

been successful in describing ¯ux decline during

dead-end MF/UF of particulate suspensions. Many

experimental results demonstrate the well-known rela-

tionship drawn from cake ®ltration theory, Vp/t1/2.

The theory for the transient cake build-up and the

associated ¯ux decline for conventional dead-end

®ltration may also apply for the initial cake build-

up in cross¯ow ®ltration, prior to the action of the

tangential ¯ow which causes the cake growth to be

arrested [1]. This model, however, is not appropriate

for application to cross¯ow ®ltration where the feed

solution continuously recirculates. Furthermore,

macromolecules and/or colloidal particles experience

diffusion which is not considered in this model.

The concentration polarization model based on the

®lm theory was developed to describe the back diffu-

sion phenomenon during ®ltration of macromolecules.

In this model, the rejection of particles gives rise to a

thin fouling layer on the membrane surface, overlaid

by a concentration polarization layer in which parti-

cles diffuse away from the membrane surface. At

steady state, convection of particles towards the mem-

brane surface is balanced by diffusion away from the

membrane [2]. If the solute retention is assumed to be

equal to one, i.e., all particles are assumed to be

rejected by the membrane, then the steady-state

permeate ¯ux can be obtained by integrating the

one-dimensional convective±diffusion equation

across the concentration polarization layer:

Jv � D

�ln�w

�b

� k ln�w

�b

(2)

where D�diffusion coef®cient, ��thickness of con-

centration boundary layer, �w�solids volume fraction

at the wall, �b�solids volume fraction in the bulk

solution, and k�mass transfer coef®cient�D/�. This

model introduces two important parameters ± the mass

transfer coef®cient (k) and the solids volume fraction

at the wall (�w) ± which should be determined either

theoretically or experimentally. One should keep in

mind that although this model starts from the one-

dimensional convective±diffusion equation, the result-

ing equation for steady-state ¯ux includes a parameter

(i.e., the mass transfer coef®cient) with two-dimen-

sional characteristics like shear rate and channel

length. Therefore, this model is inherently weak in

describing two-dimensional mass transport mechan-

isms during cross¯ow ®ltration. Also, this model

results in the well-known ¯ux paradox problem: the

predicted permeate ¯ux can be much less than that

measured during ®ltration of colloidal suspensions

[2].

In order to resolve the ¯ux paradox for colloidal

suspensions, two distinctive models have been devel-

oped: one is the lateral migration model and the other

182 Y. Lee, M.M. Clark / Journal of Membrane Science 149 (1998) 181±202

is the shear-induced hydrodynamic diffusion model.

According to the lateral migration model proposed by

Green and Belfort [3], the permeate ¯ux declines until

the permeation velocity equals the lift velocity eval-

uated at the surface of cake layer. However, during

®ltration of colloidal suspensions, diffusion is another

important mass transport mechanism, and this was not

considered in Green and Belfort's model. Zydney and

Colton [4] proposed to modify the concentration

polarization model by replacing the Brownian diffu-

sion coef®cient with the shear-induced hydrodynamic

diffusion coef®cient. Davis et al. [5,6] developed a

more comprehensive model based on the shear-

induced hydrodynamic diffusion phenomenon. They

deliberately incorporated two-dimensional character-

istics of cross¯ow ®ltration into the one-dimensional

convective±diffusion equation by de®ning the shear-

induced hydrodynamic diffusion coef®cient.

As an alternative to back-transport of particles away

from the membrane by mechanisms such as shear-

induced diffusion and inertial lift, it is possible that the

particles are carried to the membrane surface by

permeate ¯ow and then roll or slide along the mem-

brane surface due to the tangential ¯ow. The rejected

particles are assumed to form a ¯owing cake layer.

Convective-¯ow mathematical models describe the

simultaneous deposition of particles into the cake

layer and the ¯ow of this layer toward the ®lter exit

[7,8]. The fully-developed laminar ¯ow equations

were solved for the velocity pro®les in the bulk

suspension and in the cake layer, and the thickness

and the permeate ¯ux at a steady-state cake can be

determined. In general, the cake layer thickness

increases and the permeate ¯ux decreases with

increasing distance from the ®lter entrance. This sur-

face transport model predicts that the steady-state

permeate ¯ux increases with shear rate and particle

radius.

Recently, many attempts have been made to fully

describe two-dimensional mass transport mechanisms

involved in cross¯ow ®ltration. The most popular one

is the continuum approach. The particle movement

during cross¯ow ®ltration is governed by the two-

dimensional convective±diffusion equation. Many dif-

ferent authors have tried to solve the differential

equation numerically in order to obtain the concentra-

tion pro®les inside the membrane channel; most of

these efforts were limited to the steady-state case.

Although the concentration pro®les can explain the

trends in ¯ux decline, they cannot be directly used to

predict the ¯ux decline. Therefore, it is necessary

to develop a comprehensive model which can predict

the ¯ux decline during cross¯ow ®ltration. The

model presented here can be a powerful tool

for choosing optimal module length and channel

height, and for determining the optimal operating

conditions like transmembrane pressure and cross¯ow

velocity.

3. Experimental

3.1. Materials

All suspensions were prepared using ultrapure

water produced from a Milli-Q system (Millipore,

Bedford, MA). The Milli-Q system consists of four

stages of puri®cation: one activated carbon cartridge,

two mixed-bed ion exchange units, and a ®nal 0.22 mm

micro®ltration ®lter. A buffer solution was added to

the Milli-Q water (10 ml of 0.01 M NaHCO3 per liter),

and the ®nal pH was adjusted to 7.0�0.1 using 0.05 N

HCl or 0.02 N NaOH.

Ideal colloidal suspensions were prepared for ¯ux

tests using monodisperse polystyrene (PS) latex

microspheres (Duke Scienti®c). Particles of ®ve dif-

ferent sizes were used for ¯ux tests, and their proper-

ties are listed in Table 1. Each latex particle

suspension is available in 15 ml bottles of 10% solids.

First, 0.1% stock solutions were prepared by diluting

10% solids with the buffered Milli-Q water. Then feed

suspensions were prepared by diluting the stock solu-

tions to the desired concentration.

Table 1

Properties of latex particles

Material Polystyrene particles (PS)

Color White

Diameter (standard deviation, �) 0.064, 0.121, 0.300, 0.966,

2.16 mm (0.0070, 0.0050,

0.0057, 0.0126, 0.054 mm)

% solids (g/100 g) 10%

Specific gravity 1.05

Refractive index 1.59 at 589 nm at 258CAcid content Low sulfate content

Bacteria/Fungus 0 cfu/ml

Y. Lee, M.M. Clark / Journal of Membrane Science 149 (1998) 181±202 183

A hydrophilic UF membrane with 100 KDa mole-

cular weight cut-off (MWCO) was used in this study,

and its properties are summarized in Table 2. The

membrane was purchased as a sheet (36 in. long-

�12 in. wide), and membranes of the required size

were cut for the experiments. All membranes were

conditioned prior to use by presoaking for 4 h in Milli-

Q water. The water was changed every half hour, up to

2 h, after which no additional water exchanges were

made. Additionally, the membrane was ¯ushed with

enough Milli-Q water at 25 psi until a steady initial

clean water ¯ux was established.

3.2. Filtration systems and methods

Two kinds of membrane ®ltration units were used

for this study. One is a dead-end ®ltration unit, and the

other is a cross¯ow ®ltration unit. Both ¯ux tests were

conducted in constant transmembrane pressure mode.

The temperature effects were neglected in calculating

permeate volume and ¯ux. Since all ¯ux tests were

performed in the range of 21�28C, there was less than

0.4% error due to density/viscosity changes.

Dead-end ®ltration tests were performed using an

Amicon model 8200 dead-end ®ltration cell in order to

independently determine the speci®c cake resistance,

which is an important model parameter. All ¯ux tests

were carried out under unstirred conditions unless

otherwise stated. After determining the clean water

¯ux, the cell was ®lled with 200 ml of a prepared latex

suspension, and ®ltration was continued until the total

permeate volume reached 120 ml. The permeate ¯ow

rate was measured using a ¯ow sensor (Model H-

32703-50, Cole-Palmer Instruments, Chicago, IL),

and the readings were continuously logged into a

computer through a LabView data acquisition system

(National Instruments, Austin, TX).

A channel-type cross¯ow UF/MF membrane ®ltra-

tion unit (Model SEPA CF System, Osmonics, Min-

netonka, MN) was used for cross¯ow ®ltration tests.

Instead of the manufacturer's mesh spacer, a special

silicon rubber feed spacer was fabricated to form a

long rectangular channel (3.9 cm wide, 13.8 cm long

and 0.16 cm high). The feed stream supplied by the

pump ¯ows only through this channel, so the effective

membrane surface area is 53.8 cm2 and the cross-

sectional area normal to the ¯ow is 0.62 cm2.

A schematic diagram of the pilot scale cross¯ow

membrane ®ltration system is shown in Fig. 1. A 12 l

glass jar served as a feed tank, and a large magnetic

stirrer was used for mixing the feed tank. A Master¯ex

paristaltic pump (Model H-07017-00, Cole-Palmer

Instruments, Chicago, IL) was used to drive the feed

water to the membrane cell. The pump was driven by

an adjustable speed gearmotor (Model Type

42DSBEPM-E1, Bodine Electric Company, Chicago,

IL) and a permanent magnet control with analog

interface board (Model Type FPM 856, Bodine Elec-

tric, Chicago, IL). Two pulsation dampeners (Model

H-07596-20, Cole-Palmer Instruments, Chicago, IL)

were installed serially close to the pump to reduce the

pulsations created by the peristaltic pump. This pro-

vided almost pulsation-free feed to the channel-type

cross¯ow UF/MF membrane ®ltration unit.

A general-purpose programming system called

LabView was employed to control the system as well

as to continuously log data onto a computer during

®ltration tests. All measurement instruments in the

system generate analog signals, which could be logged

onto a computer. Inlet, outlet, and permeate pressures

were measured by digital pressure transmitters (Model

PG-4/20, PSI-Tronix, Tulare, CA). Permeate and

retentate ¯owrate were measured by 150 mm variable

area ¯owmeters (Model H-03229-31 and H-03229-35,

respectively, Cole-Palmer Instruments, Chicago, IL)

equipped with ¯owmeter electronic conversion mod-

ules (Model H-03298-00, Cole-Palmer Instruments,

Chicago, IL). Temperature was measured by an RTD

probe (Model DD93560-02, Cole-Plamer Instruments,

Chicago, IL), which was inserted into the permeate

tube line. The temperature probe was connected to an

RTD indicator-transmitter (Model H-08099-00, Cole-

Palmer Instruments, Chicago, IL).

Table 2

Properties of membrane used

Membrane ID UF100K

Manufacturer Millipore

Membrane type UF

Nominal pore size 100 K MWCO

Base material Cellulose acetate

Pore structure Anisotropic

Hydrophilicity Hydrophilic

Clean water flux [l/m2/h] 360±600@10 psi, 208CTemperature/pH/pressure limit 1008C/4±11/55 psi


4. Model development

4.1. Steady-state modeling of concentration

polarization

Berman [9] proposed a perturbation technique to

obtain the steady-state velocity pro®les for laminar

¯ow within a porous channel, and Yuan and Finkel-

stein [10] also used a similar technique for laminar

¯ow in a porous tube. These solutions give a

good approximation in the case of small values of

permeation velocity and channel height (or tube

radius), and are presented elsewhere [11,12]. The

particles are assumed to have no effect on the velocity

®eld.

Fig. 2 shows a problem domain of channel-type

cross¯ow UF, and a coordinate system for the numer-

ical model. The feed stream with an inlet concentra-

tion of C0 ¯ows in the x-direction, and the clean

permeate goes out of the membrane in the y-direction.

During cross¯ow UF, particles within the feed

stream are subjected to a force in the y-direction

due to the permeate drag, and they are accumulated

near the membrane surface. On the other hand, par-

ticles within the higher concentration layer near the

membrane surface tend to diffuse away from the

membrane surface to the feed stream due to the

concentration gradient. At the same time, the shear

stresses arising from the transverse velocity gradient

tend to augment the diffusion process. These mass

Fig. 1. Schematic diagram of channel-type crossflow UF/MF membrane pilot system. (a) data acquisition system: input signal; output signal

(b) membrane filtration system: retentate; permeate; feed; membrane cell unit; feed tank; Qf; Qr; Qp; Pin; Pout; Pp.

Fig. 2. Problem domain and coordinate system: C0; U0; H0; Vw; x; y; L.


transport mechanisms can be mathematically

expressed using the two-dimensional, steady-state

convective±diffusion equation. When the axial ¯uid

velocity is much greater than the transverse ¯uid

velocity, the axial diffusion term becomes negligible

compared to the others [13,14]. Then, the ®nal gov-

erning equation and appropriate boundary conditions

for either the upper or lower half of the problem

domain are as follows:

PDE : u@c

@x� v

@c

@y� Dy

@c2

@y2(3)

BCs : c�x � 0; y� � C0 (4)

@c

@y

� �y�0

� 0 (5)

vcÿ Dy

@c

@y

� �y�H0

� 0 (6)

where u�axial velocity, v�transverse velocity,

c�particle concentration, C0�feed concentration,

Dy�diffusion coef®cient in the transverse direction,

and H0�half channel height. Eq. (4) refers to the inlet

concentration pro®le at the entrance of UF membrane,

which is uniform and equal to C0. Eq. (5) satis®es the

symmetry characteristics of the problem. Since the

speci®c gravity of latex particles used in this study was

1.05, gravity has no in¯uence. Eq. (6) is the boundary

condition at the membrane surface such that there is

no particle accumulation on the membrane surface at

steady-state. This boundary condition is valid for a

totally retained system, and all sizes of latex particles

investigated here were almost 100% rejected by the

membrane used in this study.

Application of the upwind ®nite difference approx-

imation to the convection terms and the central dif-

ference approximation to the remaining diffusion term

of Eq. (3) results in a system of linear algebraic

equations with a tridiagonal matrix. It can be easily

solved by using the Thomas algorithm to give the

concentration pro®les. The numerical scheme was

found to be unconditionally stable through the von

Neumann stability analysis [14]. The behavior of the

steady-state model was thoroughly presented in a

recent article by Lee and Clark [13], and it was shown

that the steady-state model predictions are consistent

with the fundamental mass transport mechanisms

during cross¯ow UF. Section 4.3 describes how the

steady-state model can be adapted to simulate ¯ux

decline with time.

4.2. Nature of flux decline during UF

For the fully developed steady velocity pro®le, the

mathematical problem can be reduced to the solution

of the steady form of the convective±diffusion equa-

tion for the solute concentration [15]. Using a transient

model, the authors examined how fast mass transfer

reaches an equilibrium for a given hydraulic ®eld

condition. The numerical simulations showed that

the mass transfer reaches a steady-state in a very short

period. Thus, it is not necessary to solve the transient

mass transfer equation for a given velocity ®eld in

order to obtain concentration pro®les within the feed

stream. It may be assumed that the mass transfer

reaches steady-state just after a velocity ®eld has been

changed.

Thus, the unsteady nature of the permeate ¯ux

actually results not from unsteady mass transfer but

from changes in the hydraulic boundary condition due

to cake formation. As ®ltration proceeds, a cake layer

is growing at the membrane surface and the permeate

¯ux decreases due to this cake resistance. A reduction

in the permeate ¯ux results in a changed velocity ®eld

within the feed stream, and this affects the mass

transfer. The true steady-state permeate ¯ux will be

established when there is no more particle accumula-

tion in the cake layer. At this point, the particle

deposition rate is in equilibrium with the particle

diffusion rate (molecular and shear-induced diffu-

sion).

4.3. Step-wise pseudo steady-state model of flux

decline

Basic assumptions for the ¯ux decline model are

that the particle concentration at the membrane sur-

face (Cw) will not exceed a limiting value (Cmax)

which depends on cake porosity and particle density

[i.e., Cmax��p(1ÿn)], and that an excessive amount of

particle mass will contribute to build a cake layer at

the membrane surface. As discussed above, the mass

transfer is fast enough to assume a steady-state. Then,

the ¯ux decline can be simulated using a steady-state

model as follows:


First, obtain concentration profiles by solving the

steady-state model using the velocity field at the

previous time step. For example, at time zero the

initial velocity field is determined using the initial

clean water flux and inlet axial velocity.

Second, determine whether flux decline occurs at

that time period based on the above assumption. At

any axial location where the simulated concentra-

tion at the membrane surface is greater than Cmax,

a cake layer will grow and the local permeate flux

will decrease. The cake growth at that axial

location can be calculated from a mass balance

between mass convection rate and mass accumula-

tion rate at the membrane surface, and the

permeate flux decline can be determined using

the resistance model. Therefore, the local permeate

flux changes at each axial location are known.

Next, determine a new velocity field with the

above flux boundary condition, and simulate

concentration profiles again using the steady-state

model.

Then, repeat the 2nd and 3rd steps untill Cw�Cmax

at every axial location.

The main idea of this approach is based on calcula-

tion of the step-wise pseudo steady-state, i.e., for a

given hydraulic condition, the mass transfer within a

membrane channel/tube reaches an equilibrium

(steady-state) very fast, but this steady state is mod-

i®ed by the transient characteristics of the ¯ux bound-

ary condition (¯ux reduction due to cake formation).

In other words, the mass transfer time scale is much

smaller than that for cake formation. The true steady

state can be maintained when there is no more change

in the ¯ux boundary condition. The algorithm for this

iteration is shown in Fig. 3.

4.4. Sensitivity analysis for the step-wise pseudo

steady-state model

Numerical experiments were performed in order to

show the behavior of the step-wise pseudo steady-state

model. The parameters like operating conditions,

Cmax, and rc were arbitrarily chosen to emphasize

or amplify the effect of a speci®c parameter on the

¯ux decline. All the parameter values for these simu-

lations were within reasonable ranges. Figs. 4 and 5

show the model behavior in response to various feed

concentrations and cross¯ow velocities. As expected,

the model simulations show that the permeate ¯ux

becomes smaller as the feed concentration increases.

Fig. 5 shows the effect of cross¯ow velocity on ¯ux

decline behavior. Model simulations show that a

higher cross¯ow velocity results in a higher permeate

¯ux. Note that less time is required to reach steady-

state as the cross¯ow velocity increases.

Fig. 6(a) shows that effect of cake thickness on the

permeate ¯ux decline with and without a correction of

the velocity ®eld for the channel constriction. As

®ltration proceeds, the cake thickness tends to grow

so the effective channel height for feed stream

decreases. This would result in increasing shear rate

at the membrane surface, so it is expected that the ¯ux

prediction will be more accurate if the more realistic

channel height is used. The model simulations in

Fig. 6(a) show the effect of channel height on ¯ux

decline. Fig. 6(b) shows the cake thickness along the

x-axis at steady-state.

5. Results and discussion

5.1. Determination of specific cake resistance from

dead-end filtration

The speci®c cake resistance is an important para-

meter affecting the ¯ux decline rate [see Fig. 8(a)].

Dead-end ®ltration tests were performed to indepen-

dently determine the speci®c cake resistance. Accord-

ing to the resistance model, the permeate ¯ux is

J � �P

��Rm � Rc� (7)

where �P�transmembrane pressure, ��permeate

viscosity, Rm�intrinsic membrane resistance, and

Rc�cake resistance. The values of �P and Rm remain

constant during ®ltration. The ¯ux declines because

the value of Rc increases due to cake formation on the

membrane surface. The speci®c cake resistance is

de®ned as the resistance per unit thickness of cake

layer,

Rc �Z �c

0

rcd� (8)

where rc �speci®c cake resistance and �c�cake layer

thickness. If the cake layer is assumed to be


homogeneous, the above equation becomes

Rc � rc�c (9)

and

J � �P

��Rm � rc�c� (10)

In order to determine the speci®c cake resistance

from the ¯ux test results, the cake thickness should be

known. However, it is hard to physically measure the

cake thickness without disturbing the cake layer. For a

homogeneous cake, the cake thickness is

�c � volume of cake

area of membrane� mp

�p�1ÿ n�Am

(11)

where mp�total dried mass of cake, �p�density of

particles, n�porosity of cake layer, and Am� mem-

brane ®ltration area. Substituting Eq. (11) into

Eq. (9), and combining with the Carmen±Kozeny

equation,

Rc � 180�1ÿ n�2

d2pn3

mp

�1ÿ n�Am�p

� 180�1ÿ n�

d2pn3

mp

Am�p

(12)

Then, Eq. (10) becomes

J � �P

��Rm � r0c�0p�(13)

where

r0c � 180�1ÿ n�

d2pn3

(14)

Fig. 3. Algorithm for step-by-step pseudo steady-state model.


and

�0p �mp

Am�p

(15)

Here, r0c�the resistance per unit solid volume (here-

inafter referred as to the speci®c cake resistance), and

�0p�particle volume per unit membrane area. Now, the

speci®c cake resistance can be determined from the

¯ux test results as follows:

r0c ��P=�J ÿ Rm�

�0p� ��P=�J ÿ Rm�Am�p

mp

(16)

Fig. 4. Model simulations for various feed concentrations (L�0.5 m, H0�0.002 m, U0�0.5 m/s, Cmax�1620 kg/m3, �P�10 psi,

Rm�2E12 mÿ1, a�1.0 mm, rc�1E16 mÿ2). Time [min]; JJ0

; ÐÐÐ C0�25 kg/m3; - - - C0�50 kg/m3; ± - ± C0�100 kg/m3.

Fig. 5. Model simulations for various crossflow velocities (L�0.5 m, H0�0.002 m, C0�25 kg/m3, Cmax�1620 kg/m3, �P�10 psi,

Rm�2E12 mÿ1, a�1.0 mm, rc�1E16 mÿ2). Time [min]; JJ0

; ÐÐÐ Uo�0.5 m/s; - - - U0�0.8 m/s, ± - ± U0�1.0 m/s.


There was no irreversible fouling during ®l-

tration tests with latex suspensions. The initial

clean water ¯ux was always fully recovered after

back¯ushing. The total particle mass of the cake

layer formed during ®ltration can be determined as

follows:

Fig. 6. Model simulations for effective channel height changes (L�0.5 m, H0�0.002 m, U0�0.5 m/s, C0�1 kg/m3, Cmax�600 kg/m3,

�P�10 psi, Rm�5E11 mÿ1, a�4.0 mm). (a) Flux decline versus time: Time [min]; JJ0

; ÐÐÐ rc�1E15 mÿ2 (with correction for channel

restriction); ~ rc�1E15 mÿ2 (without correction for channel restriction); ± ± ± rc�2E15 mÿ2 (without correction for channel restriction); *rc�2E15 mÿ2 (without correction for channel restriction); (b) Cake thickness along x-axis: x[m]; �c

H0Ð Ð Ð rc�5E15 mÿ2; ± ± ±

rc�1E16 mÿ2.


mp � C0V0 ÿ CrVr ÿ CpVp (17)

where C0, Cr, and Cp are concentration of feed,

retentate, and permeate, respectively, and V0, Vr,

and Vp are volume of feed, retentate, and permeate,

respectively. The permeate concentration is nearly

zero because the membrane pore sizes are small

enough to retain all latex particles. The concentrations

were indirectly measured using a turbidimeter (Model

43900, Hach, Ames, IA). There was a linear relation-

ship between concentration of latex suspensions and

turbidity. After ®ltration, the turbidity of the retentate

was measured, and it turned out to be almost identical

to the initial feed concentration. Therefore, the total

particle mass in the cake becomes

mp � C0�V0 ÿ Vr� � C0Vp (18)

where Vp�120 ml for all ¯ux tests.

5.1.1. Effect of particle size

Fig. 7 shows the ¯ux decline curves for various

particle sizes during unstirred dead-end ®ltration. The

smaller the particle size is, the bigger the permeate

¯ux decline. It implies that the cake resistance

increases as the particle size decreases. Fig. 8 shows

the speci®c cake resistance for each particle size. Each

curve remains fairly constant, which implies that the

cake layer is homogeneous as assumed. The unstirred

dead-end ®ltration tests were performed in triplicate

for each particle size, and each data point in Fig. 9

represents an average value (standard deviations were

3% to 7%). According to the Carmen±Kozeny equa-

tion, the speci®c cake resistance is inversely propor-

tional to d2p [see Eq. (14)]. However, the experimental

results did not agree with this. In Fig. 9, the speci®c

cake resistances obtained from the dead-end ®ltration

tests are compared to the theoretical values. The

straight lines in Fig. 9 represent the theoretical pre-

diction from the Carmen±Kozeny equation for a given

cake porosity value. Assuming that the cake porosity

is independent of the particle size (e.g., n�0.4), the

theoretical prediction of Carmen±Kozeny equation

tends to underestimate the speci®c cake resistance

for larger particle sizes whereas the theoretical pre-

diction tends to overestimate for smaller particle sizes.

Primary assumptions necessary for the validity of

the Carmen±Kozeny equation are (1) uniform particle

size, (2) laminar ¯ow through the pores, (3) validity of

Darcy's law, and (4) absence of long and short-range

forces of interaction. The ®rst assumption is already

satis®ed since the latex particles used for this study are

very uniform and spherical. The second and third

assumptions are reasonable for the ¯ow through the

cake layer during ®ltration. The fourth assumption is

the most questionable.

Fig. 7. Effect of particle size on flux decline (C0�100 mg/l, �P�15 psi, and no stirring). permeate volume, Vp [ml]; JJ0

; dp�2.160 mm; ^dp�0.966 mm; & dp�0.300 mm; ~ dp�0.121 mm; * dp�0.064 mm


It is not yet clear how the particle±water and

particle±particle interaction forces affect the ¯ow

through the cake layer and membrane pores. However,

it is likely that the presence of these interaction forces

in¯uences the permeate migration through the pores.

The presence of these forces creates immobilized

Fig. 8. Effect of particle size on specific cake resistance (C0�100 mg/l, �P�15 psi, and no stirring). permeate volume, Vp [ml]; r0c (1E16) [1/

m2]; dp�2.160 mm; ^ dp�0.966 mm; & dp�0.300 mm; ~ dp�0.121 mm; * dp�0.064 mm.

Fig. 9. Comparison of experimental results with Carmen±Kozeny equation particle size. 1d2

p[1/mm2]; r0 (1E16) [1/m2]; ÐÐÐ r0c from

Carmen±Kozeny equation, * r0c from experimental results.


hydrodynamic layers surrounding each particle (dou-

ble layer or diffuse ion layer). The thickness of these

immobilized hydrodynamic layers is dependent on the

interaction characteristics. For example, increasing

the ionic strength results in compressing these layers,

and correspondingly affects the ¯ow rate for the same

porosity [16]. Faibish et al. [17] studied the effect of

double layer interaction on ¯ux decline during cross-

¯ow ultra®ltration of colloidal silica suspensions, and

they found that ¯ux decline is faster for higher ionic

strength conditions because the Debye screening

length decreases with increasing ionic strength result-

ing in a denser cake layer. Harment and Aimar [18]

have included effects of permeation drag and short-

and long-range colloidal forces on the cake structure

during unstirred dead-end UF of 0.22 mm latex parti-

cles. They showed that for suf®ciently high ¯ux, ionic

strength, and cake mass, the cake layer could be

inhomogeneous, and become dense and less reversible

near the membrane surface. The cake porosity

depends on separation distance between adjacent par-

ticles in the cake layer as well as particle size.

Theoretical porosity values consistent with the

experimental data were calculated using the Car-

men-Kozeny equation, Eq. (14). As shown in

Fig. 10, the theoretical porosity tends to increase as

particle size decreases. Note that the calculated por-

osity for larger particles becomes less than the theo-

retical minimum for close-packed, hard, and uniform

spheres (n�0.26 for tetrahedral packing [19]). This

anomaly implies that the Carmen±Kozeny equation

[Eq. (14)] tends to overestimate speci®c cake resis-

tance for large particles, possibly because the depen-

dency of the speci®c cake resistance on the 2nd power

of particle size may not be correct for large particle

sizes. The long and short-range forces of interaction

are also not considered in the Carmen±Kozeny equa-

tion. The following relationship between theoretical

porosity and particle size was found through regres-

sion of the experimental data:

n � 0:110� 1��dp

p � 0:053 (19)

where dp is the particle diameter in mm. Eq. (19)

implies that the cake layer becomes more porous as

particle size decreases, i.e., the speci®c cake resistance

becomes smaller than the theoretical value estimated

by Carmen±Kozeny equation with a maximum pack-

ing density. There could be two possible scenarios

consistent with the above observations:

(1) Assuming that the double-layer thickness is

independent of particle size, the ratio of this thickness

to particle diameter becomes bigger as particle size

decreases. Since ionic strength was about 2�10ÿ4 M

for all colloidal suspensions, the double-layer thick-

ness should be invariant according to the DLVO

Fig. 10. Theoretical porosity versus particle size. 1��dp

p , dp in [mm] n. * theoretical porosity calculated from Carmen±Kozeny equation; ÐÐÐ

regression curve.


theory, about 30 nm. Therefore, the in¯uence of this

layer becomes more signi®cant as particle size

decreases. At the same time, the volume occupied

by this layer becomes bigger because of larger surface

area as particle size decreases. As a result, the porosity

increases as particle size decreases.

(2) The maximum repulsion for two approaching

colloids, which is a measure of resistance to aggrega-

tion, decreases as particle size decreases [20]. So,

small colloids are more likely to aggregate with each

other than large colloids. If small colloids or primary

particles aggregate into doublets, triplets, and larger

aggregates, it is possible these aggregates can effec-

tively behave as larger primary particles. Thus, a cake

formed of aggregates of small particles could be more

like a cake formed by larger primary particles. Similar

concepts were used by Clark and Flora [21] in a study

of ¯oc restructuring. Therefore, the measured speci®c

cake resistance tends to be smaller than the theoretical

value as particle size decreases because the aggrega-

tion of primary particles results in an increased effec-

tive particle size [see Eq. (14)].

5.1.2. Effect of feed concentration

The effect of feed concentration was investigated

with latex suspensions of dp�0.121 and 0.966 mm at

15psi in the unstirred dead-end ®ltration mode.

Fig. 11 shows the effect of feed concentration on ¯ux

decline. As expected, the higher the feed concentra-

tion, the more the permeate ¯ux declines. The total

particle mass in the cake layer is directly proportional

to the permeate volume passing through the mem-

brane, and the cake thickness is proportional to the

total particle mass of the cake layer [see Eq. (11)]. So,

it is obvious that the cake resistance increases with the

feed concentration.

As shown in Fig. 12, the cake resistance increases

linearly with the feed concentration, while the speci®c

cake resistance remains constant regardless of feed

concentrations. This supports the idea that the feed

concentration only in¯uences the total particle mass of

the cake layer and the cake layer thickness (i.e., a

higher feed concentration only results in a thicker cake

layer). However, the feed concentration does not affect

the speci®c cake resistance. In conclusion, the feed

concentration would not in¯uence the properties of the

cake layer formed during unstirred dead-end ®ltration

of latex suspensions.

5.1.3. Effect of transmembrane pressure

The permeate ¯ux would be expected to increase

with the transmembrane pressure. However, there is a

negative effect of a higher transmembrane pressure:

the cake layer may become more compact as the

transmembrane pressure increases, leading to a greater

¯ux reduction.

Fig. 13 shows the variations in speci®c cake resis-

tance with permeate volume for various transmem-

brane pressures obtained from the unstirred dead-end

®ltration tests. For a given transmembrane pressure,

the speci®c cake resistance values stay fairly constant

during ®ltration. It suggests that the cake layer is

homogeneous during ®ltration, as assumed. However,

the speci®c cake resistance increases with transmem-

brane pressure. As shown in Fig. 14, both the cake

resistance and speci®c cake resistance are linearly

proportional to the transmembrane pressure.

The theoretical porosity values which agree with the

speci®c cake resistance values were calculated for

different transmembrane pressures using Eqs. (14)

and (16). The porosity tends to decrease with increas-

ing transmembrane pressure. It indicates that a higher

transmembrane pressure results in a denser and more

compact cake layer. The cake porosity can be

expressed as follows:

n � n�p�10 � 10

�P

� ��(20)

where� is the compressibility factor, which was found

to be 0.17 from regression analysis.

5.1.4. Effect of stirring

The effect of stirring was investigated with two

monodisperse latex suspensions of different particle

size. The experimental results are shown in Figs. 15

and 16. Regardless of particle size, there was no

evidence of the in¯uence of stirring on the ¯ux

decline, cake resistance, and speci®c cake resistance

during ®ltration.

For polydisperse colloidal suspensions, it has been

found that stirring or increasing the cross¯ow velocity

can result in a lowering of the permeate ¯ux [22±25].

Electron microscopy has shown that stirred conditions

result in a more ®nely dispersed cake than unstirred

conditions [22]. Under stirred conditions, the propor-

tion of small particles in the cake layer could be


increased due to preferential removal of larger parti-

cles by shear-induced hydrodynamic diffusion, and it

is more likely that small particles will ®ll the pores

between larger particles remaining in the cake. As a

result, stirring may produce a higher speci®c resis-

tance. However, for the monodisperse colloidal sus-

pensions studied here, this negative in¯uence of

stirring was not observed.

Fig. 11. Effect of feed concentration on flux decline (�P�15 psi and no stirring). (a) dp�0.121 mm permeate volume, Vp [ml] JJ0

*C0�50 mg/l; ~ C0�100 mg/l; & C0�150 mg/l; (b) dp�0.966 mm permeate volume, Vp [ml] J

J0; * C0�50 mg/l; ~ C0�100 mg/l; &

C0�150 mg/l.


5.2. Comparison of crossflow filtration test results

with model simulations

It is necessary to determine two parameters for

cross¯ow model simulations; one is the speci®c cake

resistance and the other is the diffusion coef®cient.

The speci®c cake resistance was independently mea-

sured in the dead-end ®ltration tests. The diffusion

coef®cient is another parameter affecting ¯ux decline

behavior, so it is important to determine which diffu-

sion coef®cient is appropriate. As the particle size

decreases, the molecular diffusion coef®cient (Dm)

increases, whereas the shear-induced diffusion coef®-

cient (Ds) decreases. The sum of those two diffusion

Fig. 12. Effect of feed concentration on cake properties at Vp�120 ml (�P�15 psi and no stirring). (a) dp�0.121 mm; C0�50 mg/l;

C0�100 mg/l; C0�150 mg/l; Rc (1E16) [1/m2]; r0c (1E16) [1/m2]; (b) dp�0.966 mm; C0�50 mg/l, C0�100 mg/l; C0�150 mg/l; Rc (1E11) [1/

m]; r0c (1E16) [1/m2].


coef®cients was used as an effective diffusion coef®-

cient (Deff) for the numerical simulations [13,26],

Dy � Deff � Dm � Ds � �T

6��a� 0:03a2 (21)

where ��Boltzman constant, T�absolute tempera-

ture, ��dynamic viscosity, a�particle radius, and

�shear rate.

The cross¯ow ®ltration results for various particle

sizes are compared with the model simulations in

Fig. 17. Although the model tends to predict a

slightly higher ¯ux than the experimental results,

Fig. 13. Effect of transmembrane pressure on specific cake resistance (dp�0.300 mm, C0�150 mg/l, and no stirring). permeate volume, Vp

[mL]; r0c (1E16) [1/m2]; * �P�10 psi; ~ �P�15 psi; & �P�20 psi.

Fig. 14. Effect of transmembrane pressure on cake properties at Vp�120 ml (dp�0.300 mm, C0�150 mg/l, and no stirring). �P�10 psi,

�P�15 psi, �P�20 psi; Rc (1E11) [1/m]; r'c (1E16) [1/m2].


the model predictions are quite consistent with

the experimental results. Fig. 18 shows the effect of

feed concentration during cross¯ow ®ltration. As

expected, the higher the feed concentration, the

greater the ¯ux decline. The model predictions

for different feed concentrations agree well with

Fig. 15. Effect of stirring on flux decline (C0�300 mg/l and �P�10 psi). permeate volume, Vp [ml]; JJ0

; dp�0.300 mm; dp�0.966 mm; ÐÐÐ

without stirring; - - - with stirring.

Fig. 16. Effect of stirring on cake properties at Vp�120 ml (C0�300 mg/l and �P�10 psi) (a) dp�0.300 mm: without stir; with stir (b)

dp�0.966 mm: without stir; with stir; Rc (1E11) [1/m]; r'c (1E16) [1/m2].


the cross¯ow test results. Fig. 19 shows the effect

of cross¯ow velocity on the ¯ux decline. The ¯ux

is expected to increase with the cross¯ow velocity

because the higher cross¯ow velocity increases

shearing and enhances the shear-induced hydrody-

namic diffusion [see Eq. (21)]. Again, the model

predictions are consistent with the cross¯ow test

results.

Fig. 17. Comparison of simulations with crossflow test results (effect particle size) (C0�100 mg/l, U0�0.3 m/s and �15 psi) time, t [min]; JJ0

;

* exp. with dp�2.16 mm - - - sim. with dp�2.16 mm; ~ exp. with dp�0.966 mm ± - - ± sim. with dp�0.966 mm; & exp. with dp�0.300 mm

Ð - Ð sim. with dp�0.300 mm; ^ exp. with dp�0.121 mm ± ± ± sim. with dp�0.121 mm; exp. with dp�0.064 mm ÐÐÐ sim. with

dp�0.064 mm.


Fig. 18. Comparison of simulations with crossflow test results (effect of feed concentration) (dp�0.300 mm, U0�0.3 m/s and �P�15 psi).

time, t [min]; JJ0

; ~ exp. with C0�50 mg/l; * exp. with C0�100 mg/l; & exp. with C0�200 mg/l; ± ± ± sim. with C0�50 mg/l; Ð - Ð sim.

with C0�100 mg/l; ÐÐÐ sim. with C0�200 mg/l.

Fig. 19. Comparison of simulations with crossflow test results (effect of crossflow velocity) (dp�0.300 mm, C0�100 mg/l and �P�15 psi);

time, t [min]; JJ0

; ~ exp. with U0�0.1 m/s; * exp. with U0�0.3 m/s; & exp. with U0�0.5 m/s; ± ± ± sim. with U0�0.1 m/s; Ð - Ð sim. with

U0�0.3 m/s; ÐÐÐ sim. with U0�0.5 m/s.


6. Conclusions

From the study of dead-end ®ltration of monodis-

perse colloidal suspensions, the following conclusions

can be drawn:

1. As particle size decreases, the speci®c cake

resistance increases. However, it does not increase

quadratically as predicted by the Carmen±Kozeny

equation.

2. A higher transmembrane pressure results in a

denser cake layer, which increases the specific

cake resistance. A general power law can be

applied to describe the effect of transmembrane

pressure on the specific cake resistance.

3. There is no influence of feed concentration and

stirring on the specific cake resistance.

The numerical model of crossflow filtration developed

here successfully explains the fundamental mechan-

isms involved in flux decline during crossflow UF of

colloidal suspensions. The model provides a helpful

tool for investigating the effect of various operating

parameters such as the particle size, feed concentra-

tion, axial velocity, and membrane dimensions. The

model requires a parameter, the specific cake resis-

tance which was independently obtained from the

dead-end filtration tests. The simulations show that

the model predictions are in good agreement with the

crossflow experimental results. A future modification

of the model might involve incorporating the effect of

polydisperse suspensions. Physico-chemical interac-

tions at the particle-to-particle and membrane-to-par-

ticle levels also need to be further investigated.

7. List of symbols

Am membrane filtration area (m2)

a particle radius (m)

c particle concentration (kg/m3)

C0 feed concentration (kg/m3)

Cp permeate concentration (kg/m3)

Cr retentate concentration (kg/m3)

Cw concentration at membrane surface (kg/m3)

D diffusion coefficient (m2/s)

Deff effective diffusion coefficient (m2/s)

Dm molecular diffusion coefficient (m2/s)

Ds shear-induced hydrodynamic diffusion coeffi-

cient (m2/s)

Dy transverse diffusion coefficient (m2/s)

dp particle diameter (m)

H0 half channel height (m)

J permeate flux (m/s)

Jv steady-state permeate flux (m/s)

k mass transfer coefficient (�Dm/�)mp total dried mass of cake (kg)

n porosity of cake layer

Rc cake layer resistance (mÿ1)

Rm intrinsic membrane resistance (mÿ1)

rc specific cake resistance (mÿ2)

r0c resistance per unit solid volume (m/m3)

T temperature (C)

t filtration time (s)

U0 average inlet axial velocity (m/s)

u axial velocity (m/s)

V0 volume of feed (m3)

Vp volume of permeate (m3)

Vr volume of retentate (m3)

v transverse velocity (m/s)

�0p particle volume per unit area (m3/m2)

� compressibility factor

�P transmembrane pressure (Pa)

� thickness of concentration boundary layer (m)

�c thickness of cake layer (m)

%p density of particles (kg/m3)

� solids volume fraction

�b solids volume fraction in the bulk solution

�p solids volume fraction in the permeate

�w solids volume fraction in the wall

shear rate (sÿ1)

� Boltzman constant

� dynamic viscosity (Pa s)

Acknowledgements

The authors wish to thank the National Science

Foundation under Grant No. BCS 90-57387, and U.S.

Army Construction Engineering Research Labora-

tories under Contract No. DACA88-93-D-0010 and

DACA88-93-D-0023 for providing ®nancial support.

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Modeling of flux decline during crossflow ultrafiltration of colloidal suspensions

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