Journal of Membrane Science 320 (2008) 344–355

Contents lists available at ScienceDirect

Journal of Membrane Science

journa l homepage: www.e lsev ier .com/ locate /memsci

Modeling and analytical simulation of rotating disk ultrafiltration modulea b,∗

ansfes of tld is drderseudonalytincenrativ

ified sperimlecul

Debasish Sarkar , Chiranjib Bhattacharjeea Department of Chemical Engineering, Calcutta University, Kolkata 700 009, Indiab Department of Chemical Engineering, Jadavpur University, Kolkata 700 032, India

a r t i c l e i n f o

Article history:Received 22 September 2007Received in revised form 2 April 2008Accepted 9 April 2008Available online 4 June 2008

Keywords:UltrafiltrationRotating disk membranePermeate fluxBack transportMathematical model

a b s t r a c t

An unsteady state mass tring from the basic physicrotation-induced shear fierial balance equation. In oLaplace transformation, ptransport flux. Once the aflux, membrane surface comodynamics. Finally an iteconcentration under specin good agreement with exmembrane of 5000 Da mo

1. Introduction

Over the last few decades ultrafiltration (UF) has emerged as acutting edge technology because of its inherent ability to separate

chemical and biochemical compounds up to molecular level. Thepresent day application of UF includes the treatment of industrialeffluents, oil emulsion, biological macromolecule, colloidal systemand many others.

From a fundamental standpoint ultrafiltration is nothing buta rate governed separation process in which pressure acts as adriving force. Generally the operating pressure varies in the rangeof 10–140 psi, with membrane pore size ranging between 10 and100 A. Despite all its attractive features, the process of ultra-filtration suffers from a serious drawback of continuous soluteaccumulation on the membrane surface due to their rejectionby membrane, characteristics of any pressure driven membraneseparation process. Because of this phenomenon, known asconcentration polarization, a concentration gradient set in, evi-dently in the directions opposite to that of permeate flux. Asa result the permeate flux decreases, which is not at all desir-able for efficient separation. A broad overview of concentrationpolarization was first reported by Bruin et al. [1]. Later on sev-eral studies on different aspects of concentration polarization werereported. Youm et al. [2] studied the effects of natural convection

∗ Corresponding author. Tel.: +91 33 2414 6203; fax: +91 33 2414 6203.E-mail address: cbhattacharyya@chemical.jduv.ac.in (C. Bhattacharjee).

0376-7388/$ – see front matter © 2008 Published by Elsevier B.V.doi:10.1016/j.memsci.2008.04.015

r model has been developed for rotating disk ultrafiltration module. Start-he system, analytical expression of back transport flux generated due toetermined, which is subsequently incorporated in the fundamental mate-

to get an analytical solution of governing partial differential equation viasteady state consideration is imposed both on permeate as well as back

cal form of concentration field is obtained using the expression permeatetration are evaluated using polymer solution theory and irreversible ther-e scheme is designed to simulate the permeate flux and membrane surfaceet of operating parameters. The prediction from this model is found to beental data obtained from PEG-6000/water system using cellulose acetate

ar weight cut-off.© 2008 Published by Elsevier B.V.

instability both in dead end and cross flow module. Thermody-namic interpretation of concentration polarization was presentedby Peppin and Elliott [3]; where as effect of viscosity was studiedby Gill et al. [4]. Zaidi and Kumar reported a detail experimentalanalysis of concentration polarization in dead end ultrafiltration ofdextran [5].

Different modules, proposed so far are basically designed with

a sole objective to reduce the effect of concentration polariza-tion, though in some of the recent articles it is established thatconcentration polarization can be reduced by using electric field[6,7], two phase system [8] or by direct gas sparging [9]. One ofthe popular methods in module design is to induce a high-shearfield near the membrane surface so as to homogenize the adverseconcentration gradient, thereby minimizing the effect of concen-tration polarization. In cross flow modules the very structure of themodule incorporates a shear field, where as in batch UF cell theprimary step is to introduce a stirrer placed in very close proximityof membrane. Several analytical models from different standpointsfor single stirred UF cell have been developed over years [10–13],in addition to that neural network models for the same are alsoavailable [14,15]. Following the idea of inducing high-shear field,progresses have been made by introducing the concept of rotat-ing disk membrane (RDM), with the same fundamental intentionto increase the effect of shear on concentration polarization. InRDM module, membrane and stirrer of practically same diame-ter are placed face to face with a very small distance of separationbetween them. Under this condition as membrane and stirrer, bothrotates in the direction opposite to each other, a high-shear field

of Mem

� xt

Assuming Newtonian fluid, existing shear field can be expressedin the following form:

�(r) = �dv�

dx= −

(ω1 + ω2

xt

)r (3)

It may be argued that in dealing with PEG in water, power lawmodel must be used for proper estimation of shear field, but keep-ing in mind the high dilution level of the system, Newton’s law issaid to be fairly valid. Now in order to correlate the axial veloc-ity to shear stress a thin cylinder between r and r + dr, with axial

D. Sarkar, C. Bhattacharjee / Journal

is introduced, which counters the effect of concentration polar-ization. In reality, the primary effect of shear field is to give risea new flux called “Back Transport Flux”, in the direction same asthat of concentration polarization flux. Halstrom and Lopez-Livafirst proposed the basic structure of RDM module [16]. Experi-mental studies using mineral suspension [17] and black liquor [18]are also available. In a recent article computational fluid dynamicmodel and simulation of RDM was presented by Torras et al. [19].In a previous article of single stirred UF cell, Bhattacharjee andDatta reported the analytical simulation of permeate flux by using asemi-empirical parameter called “Back Transport Coefficient” [10].Though the model was good enough to predict the behavior of sin-gle stirred cell, but it cannot be directly extended to RDM moduleas the flow field, hence the proper evaluation of the back transportflux is much complicated in the latter system.

The present work has been undertaken in an attempt to developa rigorous mathematical model of RDM cell used in the process ofultrafiltration. Starting from the basic physics of the system, theback transport flux is evaluated exploring the probable locus of asolute molecule initially at the membrane surface. Once the ana-lytical expression of back transport flux is at hand, fundamentalmass balance equation is developed under unsteady state condi-tion, which is analytically solved. No analytical solution for RDM orany such high-shear device has been reported in the literature. Inthis context, the proposed model could be used for the simulation ofUF performance in RDM and the approach could also be extended toother high-shear device, like multi-shaft disk (MSD), etc. The modelprediction is validated with experimental data for RDM moduleduring Ultrafiltration of PEG-6000 by cellulose acetate membrane

under unsteady state condition for different transmembrane pres-sure, initial concentration, stirrer speed and membrane speed.

2. Theoretical development

2.1. Development and analytical solution of model equation

The schematic diagram of a rotating disk membrane module hasbeen shown in Fig. 1, in which different flux components have beenhighlighted clearly. As stated earlier, the stirrer and the rotatingmembrane are placed face to face, and rotate in opposite direc-tion with respect to each other with angular speeds, ω1 and ω2,respectively. In order to impart a high-shear field in the vicinity ofmembrane surface, the distance of separation between the stirrerand membrane is made very small compared to the radius of theUF cell, so that a linear velocity profile can be assumed to exist inthe space between the stirrer and membrane.

The primary step of theoretical modeling is to quantify the shearinduced back transport flux. This term is used in order to accountfor the back transport of solute due to eddy back mixing; naturally it

Fig. 1. Mass balance diagram inside rotating disk membrane module.

brane Science 320 (2008) 344–355 345

should have the same dimension of volumetric flux, i.e., LT−1. Froman intuitive point of view it can be inferred, as the membrane andthe stirrer are rotating in opposite direction, any particle releasedtangentially on the membrane surface will follow a helical pathfrom membrane to the stirrer. The tangential velocity versus axialdistance profile for the proposed locus can be depicted qualitativelyas in Fig. 2. Though the helix angle, ˛ will vary from point to point,for simplicity the stated angle is assumed to be constant over theentire path. In accordance with the assumption the axial velocityat any radial location can be related to the tangential velocity at thesame point as

vx(r) = v�(r) tan ˛ (1)

Whereas the tangential velocity profile can be written as

v (r) =(

1 − ω1 + ω2 x)

r (2)

span from the membrane to stirrer surface is considered. The torquebalance equation over any horizontal section of the cylinder yields:

�(r)da′r = � dVr2� (4a)

Simplifying Eq. (4a)

�(r)�

da′

dVr = r� = at = v�

dv�

ds′ (4b)

Fig. 2. Tangential velocity profile vs. longitudinal distance for any solute particleinside RDM module.

of Mem

The flux condition at the membrane surface can be represented

346 D. Sarkar, C. Bhattacharjee / Journal

where at is the tangential acceleration and ds′ is the differentiallength traversed by the particle along the helix. Now

v�dv�

ds′ = vx

tan2 ˛

dvx(r)ds′ (5)

Combining Eqs. (4b) and (5)

�(r)�

da′

dVr = vx(r)

tan2 ˛

dvx(r)ds′ (6)

Noting that da′/dV = 1/xt for the cylinder, Eq. (6) can be simplifiedas

�(r)�

tan2 ˛

xtds = vx(r)dvx(r) (7)

Integrating Eq. (7) from any point between stirrer and mem-brane where s = 0 up to the stirrer surface, where s′ = 2�rn (n isthe number of helix turns necessary to reach the stirrer surfacefrom the stated intermediate position) and vx(r) = 0, incorporatingthe expression of shear stress in Eq. (3) the following relation isobtained:

vx(r) =√

4�n(ω1 + ω2)��

(r

xt

)tan ˛ (8)

Averaging the axial velocity over the radial space, the volumetricexpression of back transport flux at any intermediate point betweenthe stirrer and the membrane can be written as follows:

JBT =√

16�n(ω1 + ω2)�9

(R′

xt

)tan ˛ (9)

Once the basic expression of back transport flux is obtained,an unsteady state solute balance over a differential shell betweendistance x and x + dx from membrane surface yields the followingPDE:

(J − JBT)∂c

∂x+ D

∂2c

∂x2= ∂c

∂t(10)

Eq. (10) assumes constant diffusivity and a pseudo steady stateapproximation on permeate flux, J. The boundary conditions are asfollows:

(i) at t = 0, c = c0 for all x(ii) at x = 0, c = cm for t > 0

(iii) as t → ∞, c remains finite for all x.

The boundary condition also assumes a pseudo steady stateapproximation on membrane surface concentration, cm the timeevolution of which will be solved by separate physical considera-tion.

Introducing the following dimensionless parameters, Eq. (10)can be reduced to a dimensionless form, which is easier to dealwith:

c∗ = c

c0, x∗ = x

xtand t∗ = Dt

x2t

With these parameters Eq. (10) reduces to

∂2c∗

∂x∗2+ A

∂c∗

∂x∗ = ∂c∗

∂t∗ (11)

where A = (J − JBT)x2/D.The dimensionless boundary conditions are as follows:

(i) at t* = 0, c* = 1, for all x*

(ii) at x* = 0, c* = cm/c0 for t* > 0(iii) as t* → ∞, c* remains finite.

brane Science 320 (2008) 344–355

Taking Laplace transform of Eq. (11) and applying boundary con-dition (i)

∂2c∗

∂x∗2+ A

∂c∗

∂x∗ − sc∗ = −1 (12)

Eq. (12) can be solved by solving for CF and PI individually as

I. Complementary function:

CF : c∗(x∗, s) = C1 em1x∗ + C2 em2x∗

where m1, m2 = −A ±√

A2 + 4s/2, C1, C2 are arbitrary con-stants.

II. Particular integral:

PI :1s

Hence the complete solution in Laplace domain can be writtenas

c∗(x∗, s) = C1 em1x∗ + C2 em2x∗ + 1s

(13)

Applying boundary conditions (ii) and (iii) the solutionbecomes

c∗(x∗, s) =(

(cm/c0) − 1s

)exp

(−A −

√A2 + 4s

2x∗

)+ 1

s(14)

Taking inverse Laplace transform using shifting theorem andLaplace transform tables the complete solution in time domainis as obtained:

c∗(x∗, t∗) =(

(cm/c0) − 12

)[erfc

(x∗

2√

t∗ + A√

t∗

2

)

+ A e−Ax∗{

erfc

(x∗

2√

t∗ − A√

t∗

2

)}]+ 1 (15)

The next step is to get the expression of permeate flux.

2.2. Calculation of permeate flux

as

J(cm − cp) = −D∂c

∂x

∣∣∣∣x=0

+ JBT

∣∣x=0

cm ⇒ J

= − D

cm − cp

∂c

∂x

∣∣∣∣x=0

+ JBT

∣∣x=0

cm

cm − cp(16)

Eq. (16) is based on the pseudo steady state assumption of mem-brane surface concentration.

Now

∂c

∂x

∣∣∣∣x=0

= c0

xt

∂c∗

∂x∗

∣∣∣∣x∗=0

(17)

Combining Eqs. (15)–(17) followed by subsequent simplifica-tion, the expression for permeate flux could be obtained as follows:

J = D

cm − cp

(c0

xt

)(1 − (cm/c0)

2

)[2√�t∗ exp

(−At∗2

4

)

+A

{1 + erf

(A√

t∗

2

)}]+ JBT|x=0cm

cm − cp(18)

of Mem

D. Sarkar, C. Bhattacharjee / Journal

where

A = (J − JBT|x=0)xt

D

In the expression of permeate flux, back transport flux, JBT isto be considered separately. Looking into the expression of JBT inEq. (9) it is evident that the form of back transport flux has beenderived by averaging the axial velocity at any particular distancefrom membrane surface over the radial space. But in Eq. (18) backtransport flux is the same evaluated at membrane surface, i.e., atx = 0. In order to get that expression Eq. (7) is to be integrated fromthe membrane to stirrer surface assuming N number of helix turnsis required to reach the stirrer from membrane surface (n ≤ N, as nis the number of helix turns necessary to reach the stirrer from anyintermediate point between the stirrer and the membrane). Thusintegrating Eq. (7) the back transport flux at the membrane surfaceis as obtained:

vx(r)|x=0 =√

4�N(ω1 + ω2)��

(r

xt

)tan ˛ (19)

Before averaging the expression over the radial space total num-ber of helix turns, N can be replaced in terms of radial distance, rand helix angle, ˛ as

p = xt

N= 2�r tan ˛ (20)

where p is the pitch of the traced helix. Incorporating Eq. (20) in(19) and averaging the expression over the radial space the backtransport flux at the membrane surface becomes:

JBT|x=0 =√

32(ω1 + ω2)�R′

25�xttan ˛ (21)

For calculation, solute diffusivity is expressed as a function ofmolecular weight of the same, M following a standard empiricalequation as given below [20]:

D = 2.74 × 10−9M−1/3pol (22)

Now from the expression of permeate flux, as given in Eq. (21) itcan be inferred that in order to get a time evolution of the same timehistory of membrane surface concentration, cm as well as permeateconcentration, cp are to be determined, so the next task of the modelis to explore polymer solution theories in order the evaluate thestated parameters.

2.3. Calculation of permeate and membrane surfaceconcentration

In the process of UF, no matter whether the cell is stirredor unstirred transmembrane pressure drop, P acts as a drivingforce and because of concentration difference on two sides of themembrane surface, osmotic pressure differential, �′ acts as a hin-drance factor to the process objective. Still, because of inherentirreversibility the net driving force is not exactly the difference oftransmembrane and osmotic pressure, but difference of P and� �′, where � is known as reflective coefficient. The permeateflux, which represents the rate of the process can be related tothe stated difference in accordance to a linear equation (osmoticmodel):

J = P − � �′

�Rm(23)

where Rm is the membrane hydraulic resistance, and is determinedby making a series of water runs in UF module, after allowing initialcompaction. Rm is calculated for all the experiments after allowing

(

(

(

brane Science 320 (2008) 344–355 347

for initial compaction, by inducing the membrane to high pres-sure than that of experimental range. The average of all the valuesobtained from different water runs is considered as the accept-able value for Rm. The viscosity of PEG-6000 solution in water wascorrelated with the solution concentration at 30 ◦C as [21,22]:

� =

0.85 + 0.1446c′ + 2.734 × 10−4c′2 − 4.276 × 10−6c′3

+ 2.84 × 10−8c′4

103(24)

where c′ is in g cm−3

In order to express �′ as a function of cm and cp, Flory–Hugginstheory is to be used. The theory states that osmotic pressure of apolymeric solution can be related to solute concentration as

�′ = −RT

V1

[ln(1 − vp) +

(1 − 1

n

)vp + �12v2

p

](25)

where vp = c/�pol, n = Mpol/Mmono.For PEG, �pol = 1125 kg m−3, Mpol = 6000, Mmono = 44, �12,

the Flory–Huggins parameter depends upon the type ofpolymer–solvent interaction and is 0.45 for PEG–water sys-tem. The values of R,T and V1 used are R = 8314 Pa m3 kmol−1 K,T = 303.15 K and V1 = 0.001 m3/kg.

Using Flory’s equation (Eq. (25)) for permeate and membranesurface concentration followed by substitution of osmotic pressuredifferential in Eq. (23) results:

J =

P + �RTV1

[ln(

(�pol − cm)/(�pol − cp))

+{(

1 − (1/n))

+ �12(cm + cp)/�pol

}(cm − cp)/�pol

]�Rm

(26)

Eqs. (18) and (26) constitutes two non-linear equations relat-ing J, cm and cp. In order to determine the time evolution of thesethree parameters another equation relating these three parametersare required. Additionally for the back transport flux at membranesurface, the unknown helix angle (˛) is to be determined separately.

2.4. Relation of flux and rejection from irreversiblethermodynamics

Irreversible thermodynamics is used to express the rejection as[23]

cp �(1 − F)

Rr ≡ 1 −

cm=

1 − �F(27)

where F = exp{−(1 − �)J/Pm}, Pm is the permeability of the mem-brane. Now three Eqs. (18), (26) and (27) can be solvedsimultaneously to predict the time evolution of J, cm and cp, butthe prerequisite is the quantitative determination of back transportflux, which is yet not done.

The reflection coefficient, � and solute permeability, Pm aredetermined by a modification of the method outlined by Nakao andKimura [23]. The iterative technique is briefly outlined as follows:

(i) Initially a � value is assumed, using this value membranesurface concentration is temporarily determined for all experi-mental runs using Eqs. (23)–(25).

ii) Pm is calculated from Eq. (27) for the mentioned assumed valueof � and using experimental values of J and cp. The procedureis repeated to calculate all Pm values for different experiments.

iii) Standard deviation in the Pm values is now calculated for thesame assumed value of �.

iv) In this way different � values are assumed and the proce-dure is repeated to calculate standard deviation for all of them.

of Mem

(

(

(

348 D. Sarkar, C. Bhattacharjee / Journal

Fibonacci search technique is used to locate the � value afterminimization of standard deviation.

(v) Average of all Pm values for different experiments gives thecorrect value of solute permeability.

For the stated system, the values of Pm and � determined bythe above-mentioned method are 1.8275 × 10−6 ms−1 and 0.98322,respectively.

2.5. Calculation of back transport flux

Considering the expression of back transport flux at membranesurface, the only unknown parameter left is the helix angle (˛),which is necessary for final simulation of RDM. For this purpose,first experimental data of flux versus time for different values ofpressure differential, P; stirrer speed, ω1; membrane speed, ω2and bulk concentration, c0 are curve fitted in order to minimize theexperimental error and for data smoothening. The best-fit equationin terms of highest correlation coefficient and minimum standarddeviation is found to be

J = J0(1 − a + abt) (28)

In Eq. (28), J0 is the initial flux, time (t) is in minutes, whereasthe values of coefficient a and b are different for different experi-mental conditions. As an example, for an experiment conducted atc0 = 20 kg m−3, P = 827 kPa, ω1 = 5.2 rad/s and ω2 = 55.5 rad/s, thecalculated values of J0, a and b are 3.0482492 × 10−5 m3 m−2 s−1,0.11459235 and 0.974425, respectively. Now Eqs. (28) and (18) canbe combined for a particular set of c0, P, ω1 and ω2 as

J0(1 − a + abt) = D

cm − cp

(c0

xt

)(1 − (cm/c0)

2

)[2√�t∗

exp

(−At∗2

4

)+ A

{1 + erf

(A√

t∗

2

)}]

+ JBT|x=0cm

cm − cp(29)

In order to solve for tan ˛ (through the expression of JBT|x=0 inEq. (9)) using Eq. (29) values of cm and cp are to be determined from

experimental data. cp, i.e., the permeate concentration at differenttime for a particular run can be determined experimentally. But forcm theoretical equations are to be used in combination with exper-imental data. For this purpose again the Flory–Huggins equation isused, but in a different form.

Eq. (25) is directly used to evaluate the permeate side osmoticpressure, �′

p. Again using Eqs. (23) and (28) �′(= �′m − �′

p) can bedetermined at different time interval for a particular run as

�′ = P − �RmJ0(1 − a + abt)�

(30)

Now the osmotic pressure on the feed side can be obtained as�′

m = �′ + �′p. Once the numerical value of �′

m is known mem-brane surface concentration, cm at that time for the same run canbe determined using simple Newton–Raphson method of iterativesolution for single variable non-linear equation. The working equa-tion for this calculation is

f (cm) = �′m + RT

V1

[ln

(1 − cm

cp

)+(

1 − 1n

)cm

�pol+ �12

c2m

�2pol

](31)

brane Science 320 (2008) 344–355

Knowing the value of �′m at each time instant and using c(0)

m = c0as an initial guess cm can be solved as

c(k+1)m = c(k)

m − f (c(k)m )

f ′(c(k)m )

(32)

Once the values of cm and cp are determined at a particu-lar time instant, Eq. (29) can be viewed as a non-linear equationof parameter tan ˛, as all the other variables of the same equa-tion have been reduced to their numerical values for a particularrun. Modified form of Eq. (29) that can be expressed in a func-tional form as f(tan ˛ = 0) is solved again by using single variableNewton–Raphson method with an initial guess of ˛ = 0.1 (in degree)and a tolerance of 0.001◦.

2.6. Simulation for J, cm and cp

Once the values of tan ˛, hence JBT at a particular time instantare known, Eqs. (18), (26) and (27) can be solved simultaneously toget the time evolution of permeate flux, J, membrane surface con-centration, cm and permeate concentration, cp. An iterative schemeis outlined below for this purpose:

(i) A specific t is selected.ii) Initialization step: t = 0, c(0)

m = 0, c(0)p = 0 and J = (P −

��′)/�Rm with �′ = �′m − �′

p.iii) Eqs. (18), (26) and (27) are simultaneously solved by multi-

variable Newton–Raphson method with a specific relaxationparameter to enhance the speed of convergence.

iv) t = t + t(v) Steps (iii) and (iv) are repeated up to the desired time. At time

t, the previous step values are used as an initial guess.

The chosen t value must be sufficiently small in order to avoiddivergence. In the present analysis the chosen criteria for t selec-tion is t < 5 for all the runs.

3. Experimental [21,22]

3.1. Materials

Polyethylene glycol (PEG-6000, AR grade) of molecular weightrange of 6000–7000 dissolved in water was used as feed solu-tion and was obtained from Fluka, England. Moist ’Spectra-Por

C5’ asymmetric cellulose acetate complex membrane (cut-off size:5000) was obtained from Spectrum Medical Industries (USA). Themembranes showed a hydrophilic property, has no adsorption char-acteristics and resistant to temperature up to 90◦.

3.2. Apparatus

In order to eliminate the effects of concentration polarization soas to obtain highest possible steady state flux high-shear devicesare becoming more and more popular as industrial ultra filtrationmodule instead of its inherent mechanical complexity. In the sameline of RDM, multi-shaft disk membrane module, though asym-metric, replicates the basic structure of RDM on compartment basishave been already commercialized by Westfalia Separator FiltrationGmbH (previously known as Aaflowsystems), Aalen, Germany. Thecommercial module consists of ceramic membrane disks mountedon parallel hollow shafts with disks of two successive shafts over-lapping each other. So with respect to a particular disk, the adjacentdisks functions as stirrer, and hence a MSD unit may be viewed asan asymmetric collection of several RDM units placed in series-parallel sequence, but the flow rate between membranes in actual

of Mem

D. Sarkar, C. Bhattacharjee / Journal

MSD unit is periodical and unsteady because of multi-shafts geom-etry. The largest MSD unit known so far consists of 31 cm diameterdisks mounted on up to eight shafts with a total membrane area of80 m2. In a very recent article He et al. [24] presented an experimen-tal analysis of MSD that shows the increased performance efficiencyof MSD over cross flow and single stirred disks.

For the present work the RDM module made of SS316, was man-ufactured by Gurpreet Engineering Works, Kanpur, UP (India) as perspecified design. The module (Fig. 3) was equipped with two motorswith speed-controllers to provide rotation of the stirrer and mem-brane housing. The module has the facility to rotate membrane andthe stirrer in opposite direction to provide maximum shear in thevicinity of the membrane. Digital tachometer was used to mea-sure the rotational speed of both the membrane and the stirrer.

The setup was equipped with necessary arrangement for recyclingof the permeate to the feed cell, to run it in continuous mode withconstant feed composition. The later mode was not investigated inthis study. Adequate mechanical sealing mechanism was providedto prevent leakage from the rotating membrane assembly. The mag-netic drive stirrer mechanism prevents any leakage possibility fromthe top stirrer. The complete schematic diagram of the rotating diskmodule setup is given in Fig. 3. The flat disk membrane operable inpH range of 1–14, has an actual diameter of 76 mm whereas theeffective diameter was 56 mm.

3.3. Analysis

Solution concentrations were measured with a refractometer(model P70, Warsaw, Poland). The density and viscosity were deter-mined by solution concentration at 30 ◦C.

3.4. Design of experiment

In order to validate the model with respect to permeate flux,experiments were designed in such a way so that the effect of

Fig. 3. Schematic diagram of rot

brane Science 320 (2008) 344–355 349

four independent variables, namely bulk concentration (20, 50, 70and 90 kg m−3), transmembrane pressure drop (965, 827, 689 and552 kPa) and stirrer speed (63.3, 55.5, 47.1 and 34.0 rad/s) could beinvestigated. Any three of the variables were kept constant whilethe fourth was varied in order to get the actual nature of depen-dence. The effect of membrane rotation (5.2, 31.2 and 62.8 rad/s)was studied in conjugation with the variable parameter, as anexample while studying the effect of transmembrane pressure onflux profile the membrane speed was changed to the next level(from 5.2 to 31.2 rad/s or from 31.2 to 62.8 rad/s) after exploringthe effect of transmembrane pressure for a particular membranespeed. In order to compare the performance of RDM module withthat of corresponding single stirred cell, all the different runs wererepeated with static membrane (ω1 = 0), with the same paramet-

ric conditions. Additionally for the single stirred cell three differenttypes of stirrer, namely propeller agitator, turbine and flat disk wereused separately in order to compare the general characteristics ofdifferent single stirred cell with that of RDM. This comparison wasnecessary in order to establish the gross effectiveness of introduc-ing membrane stirring effect over different single stirred mode.But as the develop model can handle only the flat disk fitted stir-rer, all the single stirred cell data reported in the study was purelyexperimental.

3.5. Procedure

The membrane was placed on a disk shaped porous supportmounted on a hollow shaft through which permeate flows outthrough the cell, and the cell was assembled which contains a flatstirrer having the same diameter as that of the membrane, placedface to face. In order to overcome compaction effect of membrane,the cell was pressurized with distilled water for at least 2 h at900 kPa, which was higher than the highest operational pressure.After getting constant water flux membrane hydraulic resistance(Rm) was determined. This was followed by actual experimen-

ating disk module set up.

of Membrane Science 320 (2008) 344–355

350 D. Sarkar, C. Bhattacharjee / Journal

tal run. The stirrer and membrane speeds were adjusted first todesired rpm by the use of speed controller fitted with digital rpmtachometer. A metering pump was used to charge the cell withfeed solution, as well as for the purpose of recycling the perme-ate intermittently, so as to keep bulk concentration more or lessconstant. The pressure inside the cell was maintained at a fixedpreset value using pneumatic pressure delivered through compres-sion, controlled by a digital pressure controller. An intermediateair reservoir was used for this purpose. To determine current valueof permeate flux, 10 cm3 of permeate was collected in a measur-ing cylinder and time for this collection was recorded. Flux valueswere recorded every 5 min. A particular run was continued untiltwo successive flux reading were equal.

Once the run was over, the membrane was thoroughly cleanedwith distilled water at least for 2 h to remove any deposition. Thewater flux then again checked to detect any variation in the mem-brane hydraulic resistance. The same procedure was repeated foreach set of operating condition.

4. Results and discussion

Once the membrane has been compacted, several water runswere taken at different pressures as mentioned before to determinemembrane hydraulic resistance (Rm). Out of all the generated setsof experimental data 24 sets were used for the purpose of com-parison with the trend predicted by analytical model. Fig. 4a–cshows the comparative plots of permeate flux profile under dif-ferent transmembrane pressure (P) and membrane speed (ω1),while keeping the bulk concentration (c0) and stirrer speed (ω2)unchanged. From these profiles it becomes clear that the perme-ate flux reaches its steady state value practically within 5–7 min,where as for a single stirred cell (ω1 = 0) of similar geometry thecorresponding time is in the order of 1 h [10]. This shows an extraadvantage of RDM module over single stirred cell. The permeateflux clearly shows an increasing trend both with the increase oftransmembrane pressure as well as membrane speed. It is observedthat for a transmembrane pressure change from 552 to 965 kPa thesteady state flux increases by 60–67% under three different mem-brane speeds. Where as for the change of membrane speed from5.2 to 62.8 rad/s, the increase of steady state permeate flux is morethan 100% for the same operating pressure. In addition to the gen-eral flux profile, a comparative bar chart representing the steadystate flux both for RDM and single stirred cell fitted with different

stirrers (propeller agitator, turbine and flat disk) is shown in Fig. 5.From the bar chart it becomes very evident that the introduction ofmembrane stirring in addition to stirrer is always much more effec-tive than the most efficient single stirred module (single stirred cellfitted with propeller agitator) reflected by the fact that the steadystate flux of RDM was at least 10% higher that of propeller fitted sin-gle stirred cell, though the RDM itself is fitted with flat disk stirrer,which is the least effective one in single stirred mode. Moreoverwith the increase of membrane speed from 5.2 to 62.8 rad/s thepermeate flux increases practically by 90%.

In order to establish the general effectiveness of membranerotation, permeate flux profile for different stirrer and membranespeeds under fixed conditions of transmembrane pressure and bulkconcentration is shown in Fig. 6. Considering two comparable pro-files, one with stirrer and membrane speeds of 63.3 and 31.2 rad/sand another with 34.0 and 62.8 rad/s, respectively it can be inferredthat though the sum of the stirrer and membrane speeds remainspractically constant but the steady state value of permeate flux forthe second case, i.e., the case with higher membrane speed (stir-rer: 34.0 rad/s and membrane: 62.8 rad/s) is 12% higher than thefirst. This result clarifies the fact that membrane rotation is much

Fig. 4. (a–c) Variation of experimental and predicted flux as a function of time (min)at different transmembrane pressure and membrane speed but at constant bulkconcentration (c0 = 20 kg m−3) and stirrer speed (ω2 = 55.5 rad/s).

D. Sarkar, C. Bhattacharjee / Journal of Membrane Science 320 (2008) 344–355 351

Fig. 5. Variation of steady state flux as a function of transmembrane pressure(TMP, in kPa) at different membrane speed, but at constant bulk concentration(c0 = 20 kg m−3) and stirrer speed (ω2 = 55.5 rad/s).

more effective from the standpoint of permeate flux than the stirrerspeed. Moreover it can be observed that the effect of stirrer speedincrease is more pronounced at higher side values of the membranespeed itself as the individual band width of different flux profile sets

Fig. 6. Variation of experimental and predicted flux as a function of time (min)at different stirrer and membrane speed, but at constant transmembrane pressure(P = 552 kPa) and bulk concentration (c0 = 70 kg m−3).

Fig. 7. Variation of steady state flux as a function of stirrer speed at different mem-brane speed, but at constant bulk concentration (c0 = 70 kg m−3) and transmembranepressure (P = 552 kPa).

increases with the increase of membrane speed. A comparative barchart of steady state permeate flux with respect to stirrer speedunder fixed conditions of transmembrane pressure, concentrationand membrane speed is shown in Fig. 7. Here once again the steadypermeate flux is compared with single stirred cell with differenttypes of stirrer.

Effect of bulk concentration on flux profile is depicted in Fig. 8aand b under fixed conditions of stirrer speed and transmembranepressure. Here the effect of membrane speed was studied in conju-gation with bulk concentration. Fig. 8a represents the flux profile atmembrane speeds of 5.2 and 31.2 rad/s where as Fig. 8b representsthe same variation at 31.2 and 62.8 rad/s. It is very evident thatas bulk concentration increases rejection by the membrane sur-face increases resulting an increased resistance in polarized layerand thereby flux gets reduced. But with the increase of membranespeed the increased effect of back transport flux reduces the influ-ence of concentration polarization; hence permeate flux increasespractically by 35–50% for all different concentrations.

Fig. 9 shows the variation of helix angle of the solute particletrajectory due to back transport flux at the membrane surface withdifferent stirrer and membrane speed, all the other process param-

eters remains constant. The helix angle (˛) shows a decrease withincreasing stirrer and membrane speed, which is expected becauseof the fact that increased stirrer as well as membrane speed, inducesa tendency in solute to follow horizontally oriented circular pathwith very little vertical displacement under each turn, exactly con-sistent with the flow field produced by combined rotation of stirrerand membrane. As the ratio of vertical to circular displacementdecreases the helix angle must decrease with increase in stirrerspeed. Alternatively it can be argued that in high turbulent regime,flow structure is practically independent of system parameters, asa result the back transport flux that occurs in high turbulent regimebecomes constant. Since the flux at membrane surface (JBT|x=0) isdirectly proportional to

√(ω1 + ω2) tan ˛, increase in ω2 or ω1, i.e.,

in (ω1 + ω2) tan ˛ and therefore ˛ must decrease to keep the entireterm constant. For the increase of transmembrane pressure helixangle decreases insignificantly. It is observed that ˛ decreases verysharply with time and attains its steady state value within 5–10 min,consistent with the trend of permeates flux. No significant trend ofhelix angle with bulk concentration is observed.

352 D. Sarkar, C. Bhattacharjee / Journal of Membrane Science 320 (2008) 344–355

Fig. 9. Variation of helix angle (˛) in degree, with time at different stirrer andmembrane speeds (c0 = 20 kg m−3 and P = 827 kPa).

after a short span of time it becomes almost flat. This is due tothe fact that as time increases membrane surface concentrationshows a positive trend resulting in increase in the osmotic pres-sure differential, which opposes the applied pressure differential.This results in decrease of the effective driving force giving rise toa retarded volumetric flux and hence a lower solute transporta-tion rate at the membrane surface. In addition to the above effect,concentration polarization increases with membrane surface con-centration and the phenomena of back-diffusion becomes morepronounced and hence the rate of increase in membrane surfaceconcentration becomes small. With the increase in transmembranepressure that’s why the surface concentration seems to be increas-ing. Further it can be concluded from these figures that turbulence

Fig. 8. (a and b) Variation of experimental and predicted flux as a function of time(min) at different bulk concentration and membrane speed, but at constant trans-membrane pressure (P = 827 kPa) and stirrer speed (ω2 = 55.5 rad/s).

Variation of membrane surface concentration (cm) with timeat different transmembrane pressure and stirrer speed with twodistinct membrane speeds are shown in Figs. 10 and 11, respec-tively. Initially the membrane surface concentration increases verysharply but the rate of increase in cm diminishes with time and

Fig. 10. Variation of membrane surface concentration with time at different pres-sures and membrane speeds (c0 = 70 kg m−3 and ω2 = 55.5 rad/s).

D. Sarkar, C. Bhattacharjee / Journal of Membrane Science 320 (2008) 344–355 353

Fig. 12. Variation of steady state flux with transmembrane pressure (TMP, in kPa)at different bulk concentration and membrane speed (ω2 = 63.3 rad/s).

The sources of deviation between experimental result andmodel prediction may be inherent in the fact that the formulatedmodel is strictly one-dimensional considering variation of differentsystem variables along axial direction only. Though not very signif-

Fig. 11. Variation of membrane surface concentration with time at different stirrerand membrane speed (c0 = 20 kg m−3 and P = 827 kPa).

near the vicinity of membrane surface have an important effect inascertaining the concentration polarization phenomena. An inter-esting observation that can be noticed from both of Figs. 10 and 11that membrane rotation is more effective in reducing the mem-brane surface concentration compared to the stirring effect. In factthe concentration build up is reduced by maximum 1.1% per unitincrease of membrane speed where as it is reduced by 0.75% perunit increase of stirrer speed. Though not separately studied but itis also quite obvious that increase in bulk concentration must giverise to increased membrane surface concentration.

In order to be more informative about the trend of steady stateflux with respect to transmembrane pressure, a plot of steady fluxversus pressure at different bulk concentration and membranespeed is represented in Fig. 12. It is evident from the figure that per-meate flux increases almost linearly with pressure, where as with

increasing concentration permeate flux by significant amount asexpected because increased concentration results in severe polar-ization thereby reducing the steady flux. The effect of concentrationincrease is countered more effectively by the increase in membranerotation. It is to be noted that as the membrane speed increases from5.2 to 31.2 rad/s there is practically 75–115% increase in permeateflux under different transmembrane pressure and bulk concentra-tion. The effect of stirrer speed increase is separately shown inFig. 13 in the form of a bar chart. Though the steady flux seemsto be increasing with stirrer speed but it is limited to maximum50%, much lower than the corresponding enhancement with theincrease in membrane speed.

The effect of transmembrane pressure and membrane rota-tion on % rejection is depicted in Fig. 14. At lower pressure lowerrejection is observed. As pressure increases more liquid permeatesthrough the membrane having more solute to retain thereby con-stricting the pore opening and subsequently increasing rejection.Turbulence created by membrane rotation has a moderate influ-ence on rejection, though not as high as pressure, but stirrer speeddoes not seem to have any appreciable effect on rejection.

Fig. 13. Variation of steady state flux with transmembrane pressure (TMP, in kPa)at different stirrer speed (ω1 = 31.2 rad/s and c0 = 20 kg m−3).

354 D. Sarkar, C. Bhattacharjee / Journal of Membrane Science 320 (2008) 344–355

Nomenclature

a,b constants used in Eq. (28)at tangential acceleration (ms−2)a′ area element defined in Eq. (4a)A parameter defined in Eq. (11)c solute concentration (kg m−3)cm solute concentration at membrane surface (kg m−3)cp solute concentration in the permeate (kg m−3)c0 bulk solute concentration (kg m−3)c′ solute concentration (g cm−3)c* dimensionless concentration = c/c0C1, C2 constants used in Eq. (13)D solute diffusivity (m2 s−1)F parameter defined in Eq. (27)J volumetric permeate flux (m3 m−2 s−1)JBT back transport flux (m3 m−2 s−1)Mmono molecular weight of repeat unit (kg kmol−1)Mpol molecular weight of polymer (kg kmol−1)n number of helix turns required to stirrer surface

from any intermediate point, used in Eq. (8)N total number of helix turns used in Eq. (19)P hydraulic pressure (Pa)PEG polyethylene glycolPm solute permeability (ms−1)r radial coordinate (m)R gas constantRm membrane hydraulic resistance (m−1)R′ radius of stirrer and membrane (m)s′ displacement along helix path (m)t time (s)t* dimensionless time = Dt/x2

tT absolute temperature in Eq. (25)vx axial velocity (ms−1)v tangential velocity (ms−1)

Fig. 14. Variation % rejection with transmembrane pressure (TMP, in kPa) at differ-ent membrane speed (ω2 = 63.3 rad/s and c0 = 20 kg m−3).

icant but due to high-speed rotational field there must be certainradial variation of the same, which were not included because ofmathematical complexity. Secondly, the equation of velocity field(Eqs. (1) and (2)) was simplified in order to get the analytical expres-sion of back transport flux, otherwise it was not possible to getthe trend of proposed helix angle with respect to different operat-ing parameters. Instead of these two simplifications on an averagethe deviation of predicted result from that of experiment was wellwithin ±7%, which establishes the general usability of the modelfor any standard RDM module.

5. Conclusion

An analytical model based on solution of PDE arising from funda-mental mass balance incorporating the expression of back transportflux for rotating disk membrane is proposed in this study. Unsteadystate membrane surface and permeate concentration are also eval-uated through Flory’s equation and related to permeate flux viairreversible thermodynamics, whereas the permeate flux is relatedto concentration field by balance equation developed at the mem-brane surface. Finally an iterative scheme is developed to simulatepermeate flux and rejection under any operating condition. The

model prerequisite is accurate estimation of four system parame-ters, i.e., Rm, ˛, Pm and �, in this point of view the model can betermed as a four-parameter model of RDM. The proposed modelis validated with experimental data for PEG-6000 in water treatedwith cellulose acetate membrane in a standard RDM cell. Low valueof deviation (within ±7%) both for permeate flux and rejectionestablishes that model could be used for accurate simulation of per-meate flux and rejection for any system subjected to ultrafiltrationin a standard RDM module.

Acknowledgements

Experimental part of this work was carried out utilizingthe infrastructures developed under Indo-Australian Project,entitled “Milk nutraceuticals: A biotechnology opportunity forAustralian and Indian Dairy Producers”, funded by DBT under Indo-Australian Biotechnology Fund (IABF) (vide sanction letter no.BT/PR9547/ICD/16/754/2006 of DBT/Indo-Aus/01/35/06 dated 02July 2007). The contribution of IABF is gratefully acknowledged.

�

V volume element used in Eq. (4a)V1 specific volume of the solventx axial distance from membrane surface (m)xt distance of stirrer from membrane surface (m)x* dimensionless distance = x/xt

Greek letters˛ helix angle� angular displacement, used in Eq. (4a)

′

� osmotic pressure (Pa)� density of solution (kg m−3)�pol density of polymer (kg m−3)� reflection coefficient� shear stress (Pa)�12 Flory–Huggins interaction parameterω1, ω2 angular velocity of membrane and stirrer, respec-

tively

References

[1] S. Bruin, A. Kikkert, J.A.G. Weldering, Overview of concentration polarizationin ultrafiltration, Desalination 35 (1980) 223.

[2] K.H. Youm, A.G. Fane, D.E. Wiley, Effect of natural convection instability onmembrane performance in dead end and cross flow ultrafiltration, J. Membr.Sci. 116 (1996) 229.

[3] S.S.L. Peppin, J.A.W. Elliott, Non-equilibrium thermodynamics of concentrationpolarization, Adv. Colloid Interface Sci. 92 (2001) 1.

[4] W.N. Gill, D.E. Wiley, C.J.D. Fell, A.G. Fane, Viscosity effect on concentrationpolarization, AIChE J. 34 (2004) 1563.

of Membrane Science 320 (2008) 344–355 355

[

[

[

[

D. Sarkar, C. Bhattacharjee / Journal

[5] S.K. Zaidi, A. Kumar, Experimental studies in the dead end ultrafiltration ofdextran: analysis of concentration polarization, Sep. Purif. Technol. 36 (2004)115.

[6] T.R. Mollee, Y.G. Anissimov, M.S. Roberts, Periodic electric field enhanced trans-port through membrane, J. Membr. Sci. 278 (2006) 28.

[7] A.D. Enevoldsen, E.B. Hansen, G. Jonsson, Electro ultrafiltration of industrialenzyme solution, J. Membr. Sci. 299 (2007) 28.

[8] H.M. Wang, C.Y. Li, S.J. Chen, T.N. Cheng, T.L. Chan, Abatement of concentra-tion polarization in ultrafiltration using n-hexadecane/water two-phase flow,J. Membr. Sci. 238 (2004) 1.

[9] Z. Cui, T. Taha, Enhancement of ultrafiltration using gas sparging: a comparisonof different membrane module, J. Chem. Technol. Biotechnol. 78 (2003) 249.

10] C. Bhattacharjee, S. Datta, Simulation of continuous stirred ultrafiltration pro-cess: an approach based on analytical solution couples with turbulent backtransport, J. Chem. Technol. Biotechnol. 78 (2003) 1135–1141.

[11] C. Bhattacharjee, S. Datta, Analysis of mass transfer during ultrafiltration ofPEG-6000 in a continuous stirred cell: effect of back transport, J. Membr. Sci.119 (1996) 39–46.

12] S. Bhattacharjee, A. Sharma, P.K. Bhattacharya, A unified model for predictionof flux in stirred and unstirred batch ultrafiltration, J. Membr. Sci. 111 (1996)243.

13] P. Bacchin, M. Meireles, P. Aimar, Modeling of filtration: from polarized layer todeposit formation and compaction, Desalination 145 (2002) 139.

14] M. Cabassud, N.D. Vincent, C. Cabassud, L.D. Bourlier, J.M. Laine, Neural net-works: a tool to improve ultrafiltration plant productivity, Desalination 145(2002) 223.

[

[

[

[

[

[

[

[

[

[

15] C. Teodosiu, O. Pastravanu, M. Macoveanu, Neural network models for ultrafil-tration and backwashing, Water Res. 34 (2000) 4371.

16] B. Halstrom, M. Lopez-Liva, Description of rotating ultrafiltration module,Desalination 24 (1977) 39.

17] R. Bouzerar, P. Paullier, M.Y. Jaffrin, Concentration of mineral suspensions andindustrial effluent using a rotating disk dynamic filtration module, Desalination158 (2003) 79.

18] C. Bhattacharjee, P.K. Bhattacharya, Ultrafiltration of black liquorusing rotating disk membrane module, Sep. Purif. Technol. 49 (2006)281.

19] C. Torras, J. Pallares, R.G. Valls, M.Y. Jaffrin, CFD simulation of rotating disk flatmembrane module, Desalination 200 (2006) 453.

20] T.K. Sherwood, R.L. Pigford, C.R. Wile, Mass Transfer, McGraw Hill, New York,1980.

21] C. Bhattacharjee, P.K. Bhattacharya, Prediction of limiting flux in ultrafiltrationof Kraft black liquor, J. Membr. Sci. 72 (1992) 137.

22] S. Bhattacharjee, S. Ghosh, S. Datta, C. Bhattacharjee, Studies on ultrafiltrationof Casein hey using a rotating disk module: effect of pH and membrane diskrotation, Desalination 195 (2006) 95–108.

23] S. Nakao, S. Kimura, Analysis of solute rejection in ultrafiltration, J. Chem. Eng.Jpn. 14 (1981) 32.

24] G. He, L.H. Ding, P. Paullier, M.Y. Jaffrin, Experimental study of a dynamic fil-tration system with overlapping ceramic membrane and non permeating disksrotating at independent speed, J. Membr. Sci. 300 (2007) 63.