Chemical Engineering Science 66 (2011) 2554–2567

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

n Corr

E-m

chiranji

journal homepage: www.elsevier.com/locate/ces

Simulation of continuous stirred rotating disk-membrane module: Anapproach based on surface renewal theory

Debasish Sarkar a, Diptendu Datta b, Dwaipayan Sen c, Chiranjib Bhattacharjee c,n

a Department of Chemical Engineering, Calcutta University, Kolkata, West Bengal 700009, Indiab Department of Chemical Engineering, Heritage Institute of Technology, Kolkata, West Bengal 700107, Indiac Department Chemical Engineering, Jadavpur University, Kolkata, West Bengal 700032, India

a r t i c l e i n f o

Article history:

Received 11 November 2010

Received in revised form

16 February 2011

Accepted 28 February 2011Available online 21 March 2011

Keywords:

Back transport flux

Dynamic simulation

Membranes

Mathematical modeling

Ultrafiltration

Bovine serum albumin

09/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ces.2011.02.056

esponding author. Tel.: þ91 98364 02118; fa

ail addresses: cbhattacharyya@chemical.jduv.

b_b@yahoo.com (C. Bhattacharjee).

a b s t r a c t

A rigorous surface renewal model has been developed describing the aspects of mass transfer in a

rotating disk-membrane (RDM) ultrafiltration cell. The model takes into consideration of two

distribution functions of random surface elements, one with respect to their point of origin and the

other related to the corresponding residence time on the membrane surface. The back transport flux

and the permeate flux are evaluated at the membrane surface in order to develop a surface component

balance equation. The component balance equation and a flux-rejection relationship arising from

irreversible thermodynamics are solved simultaneously to develop a dynamic simulation. The

simulation elucidates on permeate flux, membrane surface concentration and the permeate concentra-

tion under various operating conditions of transmembrane pressure, bulk concentration, membrane

and stirrer speeds. For validation of the proposed model, experiments were conducted with bovine

serum albumin (BSA)/water as feed in a standard rotating disk membrane module fitted with

polyethersulfone (PES) membrane of 30 kDa molecular weight cut-off (MWCO). The model predicted

flux and permeate concentration was found to be in good agreement with the experimental data, and

the maximum absolute deviation for both cases was found to be well within 75%.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Membrane separation processes are gaining considerableinterests in different segments of chemical industries nowadaysbecause it offers high precision in separation and simple opera-tional procedure. Apart from the various advantages mentionedwith membrane separation technique, one of the primal disad-vantages is the reduction in permeate flux, which is primarilycaused by the well-known phenomenon of concentration polar-ization. In order to address this deleterious effect upon permeateflux, many researchers carried out investigation regarding thesolute transport through membrane based separation processes,particularly emphasizing the region close to the membrane sur-face. A broad perspective of polarization phenomenon was firstreported by Bruin et al. (1980). Later on, several studies wereperformed to explore different aspects of polarization in differentmodules. Youm et al. (1996) had studied the effects of convectionon the degree of polarization both in dead end as well as in crossflow modules. On the other hand, Trettin and Doshi (1980) first

ll rights reserved.

x: þ91 33 2414 6203.

ac.in,

proposed a theoretical model, based on the unification of macro-molecular solution theory with classical filtration theory, whichpointed out the connection between reversible polarization phe-nomenon with that of irreversible membrane fouling, thoughtheir predictions differ significantly compared to that of a similarstudy made by Shen and Probstein (1977). Some of the research-ers, had also proposed that the reduction of permeate flux mighthave been caused by the increased viscosity due to polarization(Gill et al., 1988). In relation to this it may be noted that a detailedthermodynamic interpretation of the polarization phenomenonwas reported by Peppin and Elliott (2001). Different methodolo-gies were suggested by many researchers to alleviate the pro-blems due to concentration polarization. Application of electricfield (Enevoldsen et al., 2007; Mollee et al., 2006) and injectinggas in the feed (Cui and Taha, 2003) to ultrafiltration (UF) modulewere such techniques addressing to the reduction of fouling issuedue to polarization effect.

Designing high sheared membrane devices is another funda-mental approach that could be considered to alleviate the con-centration polarization effect in the vicinity of the membranesurface. This can be accomplished simply by placing a stirrer closeto the static membrane (leading to single stirred/rotating diskmodule) or by mounting the membrane on a hollow shaft(leading to rotating membrane module) or by both i.e., by placing

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–2567 2555

a stirrer close to the counter rotating membrane (leading todouble stirred/rotating disk membrane module). In this referenceit is also to be noted that a more advanced module, known asmulti-shaft disk (MSD) module has been proposed and recentlybeen commercialized, which comprises of circular disk mem-brane-mounted parallel shaft assembly with overlapping mem-branes of adjacent shafts (He et al., 2006). All these different typeof modules are collectively known as ‘‘high-sheared membranemodules’’. High-sheared membrane modules are becomingincreasingly popular, particularly over last few years because oftheir inherent ability not only to produce a high shear field nearthe membrane surface and thereby reducing the effects ofpolarization on permeate flux, but also to decouple the membraneshear rate from the inlet flow rate to the module, which can bevaried independently (Jaffrin, 2008).

So far various theoretical works were reported on the analysisof the limiting flux phenomena in a single stirred/rotating diskmodule (Torras et al., 2009; Bhattacharjee and Datta, 2003, 1996;Bacchin et al., 2002; Bhattacharjee et al., 1996). In addition to theclassical film theory based models, several neural network basedmodels were also available for the same (Cabassud et al., 2002;Teodosiu et al., 2000). On the other hand, different experimentalstudies of rotating disk membrane (RDM) module with mineralsuspension (Bouzerar et al., 2003), black liquor (Bhattacharjee andBhattacharya, 2006), skim milk (Frappart et al., 2008) and manyothers were already reported. Even in a recent article, theexperimental performance analysis of turbulence promoter fittedRDM module was explored by Sen et al. (2010). In spite of severalexperimental studies modeling and simulation of RDM is anascent topic till date. A semi-analytical model based on theevaluation of the back transport flux was reported by Sarkar andBhattacharjee (2008). A CFD based numerical simulation of RDMmodule, mostly concerned with the hydrodynamic characteristicsof the system is also available in literature (Torras et al., 2006).

Surface renewal theory was in application quite successfully toanalyze physical situations of different systems where turbulentstructure predominated. The theory was used to analyze thetransport process characteristics of different systems like turbu-lent pipe flow (Muschenga et al., 1992), three phase fluidized beds(Kim et al., 1999), fixed bed reactor (Lesage et al., 2002), atmo-spheric surface layer dispersion (Katul et al., 2006), enhanced gasabsorption (Kordac and Linek, 2006), direct contact condensation(Gulawani et al., 2009) and simultaneous extraction and strippingusing liquid membrane (Ren et al., 2007).

Koltuniewicz and Naworyta (1994) first applied the surfacerenewal theory for predicting the dynamic behavior of a crossflow ultrafiltration module, particularly in terms of the permeateflux. In their approach, the initial concentrations of all the surfaceelements were assumed to be same and equal to the bulkconcentration, which was exactly in accordance with the originalDanckwerts (1951) model. It was also assumed that the local fluxcould be approximated by the flux observed under steady-statecondition. The effective permeate flux was then calculated byaveraging the contribution of each random surface elements withdifferent residence time on the membrane surface. It is to benoted that their model was semi-empirical in a sense that itrequires the experimental values of initial and steady state fluxesto determine the average permeate flux over the processing time.

In the light of the foregoing analysis, an attempt was made inthe present study to develop a rigorous surface renewal model fora standard RDM module. The primary feature of the proposedmodel was the consideration of two separate distribution func-tions related to the kinematics of the random surface elementsbased on their point of origin (in order to account for the effect ofpolarization on the initial concentration of the surface elements)and the other related to their residence time on the membrane

surface. Using these two distribution functions, the volumetricpermeate flux and the Fickian back diffusion coupled backtransport flux was evaluated at the membrane surface. Knowingthe algebraic expressions of these two fluxes a componentbalance equation was developed at the membrane surface witha basic objective to obtain the dynamics of the transient mem-brane surface concentration. The permeate concentration, appear-ing in the component balance equation as well as in theexpression of permeate flux (because osmotic pressure modelwas applied to obtain the flux contribution by individual surfaceelements) was related to the membrane surface concentrationand permeate flux via the well known flux-rejection relationshipobtained from the thermodynamic analysis of membrane trans-port process in general. As a result, the proposed model wascomposed of three equations involving permeate flux, membranesurface concentration and permeate concentration, which wassolved simultaneously leading to the unsteady state simulation ofthree major system variables as mentioned under differentoperating conditions of transmembrane pressure (TMP), feedconcentration, stirrer and membrane speeds. The proposed modelwas validated with respect to experimental data obtained duringthe ultrafiltration (UF) of simulated aqueous solution BovineSerum Albumin (BSA) in a standard RDM module fitted withpolyethersulfone (PES) membrane of 30 kDa molecular weightcut-off.

2. Theoretical development

The schematic diagram of a rotating disk membrane modulehas been shown in Fig. 1, in which different rotating accessorieshave been highlighted clearly. As stated earlier, the stirrer and therotating membrane are placed face to face, and rotate in oppositedirection with respect to each other with angular speeds, O1 andO2, respectively. In order to impart a high-shear field in thevicinity of membrane surface, the distance of separation betweenthe stirrer and membrane is made very small compared to theradius of the UF cell. The basic hypothesis required for the modeldevelopment is elaborated in the following section.

2.1. Model hypothesis

The basic mass transfer picture as shown in Fig. 2 is exactly inaccordance with the original surface renewal theory as proposedby Danckwerts (1951). A random surface element originated at adistance z from the membrane surface reaches the membrane(considered as interface) and stays there for a short while when itdelivers permeate flux that passes through the membrane. At theend of its stay the element moves back into the liquid phasecarrying with it the solute it has picked up during its brief stay atthe membrane surface. The model specific assumptions related tothe mass transfer picture of the proposed module are as follows:

(I)

The average concentration of a surface element is same asthe characteristic concentration of its point of origin (let it bec(z)).

(II)

The probability of a potential surface element reaching themembrane surface decreases monotonically with increasingdistance of its point of origin from the membrane surface (z).

(III)

Residence time of a surface element on the membrane (T) isindependent of its point of origin.

(IV)

Residence time distribution of the surface elements is asym-metric owing to the existence of different hydrodynamiczones on the membrane surface.

(V)

Pseudo steady permeate concentration (cp) over the resi-dence time of a surface element.

Fig. 2. Locus and dynamics of a random surface element residing over the

membrane surface for time T and having a point of origin at a distance z from

the membrane surface.

Fig. 1. Schematic diagram of rotating disk membrane (RDM) module.

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–25672556

Additionally the assumption related to the hydrodynamicpicture is as follows:

(VI)

As the axial distance between the membrane and the stirreris very small compared to the diameter of the module, thecorresponding velocity profile may be assumed to be linear.

Based on the above-mentioned hypothesis the details formu-lation of the proposed model has been illustrated in the followingsections.

2.2. Model formulation

The objective of the proposed model is to obtain analyticalexpressions of three major fluxes present in the system, namely,(i) the pressure driven volumetric permeate flux (J) (bulk phase to

membrane surface), (ii) the stirrer induced back transport flux(membrane surface to bulk phase) and (iii) the concentrationpolarization induced Fickian diffusion flux (membrane surface tobulk phase). Amongst these, the Fickian diffusion flux can beexpressed in terms of existing concentration gradient and thediffusivity of the solute. But the knowledge of inter-phase masstransfer is required for developing analytical expression for othertwo fluxes.

An analytical expression for back transport flux is developedfirst using a modified surface renewal scheme. This flux is thenused to predict the permeate flux (J), membrane surface concen-tration (cm) and permeate concentration (cp). Finally, a time-dependent relationship is put forward as a function of these threeprocess variables developing a component balance equation atthe membrane surface.

2.2.1. Formulation of the back transport flux

It is well known that any random surface element is funda-mentally an eddy owing to stirring action in a rotating diskmembrane module, which can maintain its identity over a systemspecific length, universally known as Prandtl mixing length (w).Therefore, in an attempt to develop the analytical expression forthe back transport flux in terms of a suitable distribution functionof its initial position (z) and its residence time over the membranesurface (T), the initial position must belong to a closed domainz) [0,w], while the residence time belong to a semi-open domainof T) [0,N]. With these rationales, the solute back transport fluxat the membrane surface ðJs

BT9z ¼ 0Þ can be expressed as

JsBT jz ¼ 0 ¼

Z w

0

Z 10

JsBT ,0ðz,TÞjðz,TÞdTdz ð1Þ

In this equation, j(z,T)dTdz is the fraction of the surfaceelements having residence time lying between T and TþdT withits initial coordinate lying between z and zþdz. As the residencetime and the initial position of any random surface element are

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–2567 2557

independent to each other (assumption iii), the combined dis-tribution function j(z,T) can be expressed as a product of twoseparate distribution functions j1(z) and j2(T) so that j1(z)dz

represents the fraction of the surface elements having its initialcoordinate lying between z and zþdz, while j2(T)dT becomes thecorresponding fraction of same surface elements having theresidence time lying between T and TþdT. The physical signifi-cance of j1(z) reflects from the fact that the probability of asurface element to reach the membrane surface decreases veryrapidly with the increase in its initial distance from the mem-brane surface (assumption ii). Therefore, j1(z) is considered to beexponential and can be expressed as

j1ðzÞ ¼ Ae�Az for 0rzrw

¼ 0 elsewhere ð2Þ

where A(40) is the distribution parameter. On the other hand,the significance of j2(T) can be understood from the flow regime.The flow regime of the RDM module is generally highly turbulent.The rotational Reynolds number which has been defined asRef ¼OR2

0=g is found to be in the range of 105–1.5�105. There-fore, an asymmetric distribution of the corresponding mean maybe considered to be suitable (assumption iv). After careful con-sideration of this physical situation, the two-parameter Weibulldistribution (1951) ascribing this asymmetry is selected for j2(T)as given below

j2ðTÞ ¼ abTb�1e�aTbfor T40

¼ 0 elsewhere ð3Þ

Assuming any arbitrary surface element having a point oforigin at z, with initial concentration c(z) (assumption i) residingover the membrane surface for a time interval of T, the expressionfor the average flux can be given from the well known penetrationtheory (Higbie, 1935) as

JsBT ,0ðz,TÞ ¼ 2

ffiffiffiffiffiffiD

pT

rcm�cðzÞ½ � ð4Þ

Incorporating Eq. (2) and (4) in Eq. (1) the expression for backtransport flux can be given as

JsBT jz ¼ 0 ¼ 2

ffiffiffiffiD

p

ra1=2bG 1�

1

2b

� �cm�A

Z w

0cðzÞe�Azdz

� �ð5Þ

Further, the back transport flux at the membrane surface canalso be expressed in terms of mass transfer coefficient (k) as

JsBT z ¼ 0 ¼ kðcm�c0Þ�� ð6Þ

In order to obtain a correlation for mass transfer coefficient, alocal Sherwood number distribution may be written analogous tothe relation developed by Soong (2003) as

Sh¼ 0:197ðnþ2:6Þ0:2Sc0:6 Refr

R0

� �2" #0:8

ð7Þ

The local radial Sherwood number (Sh¼kr/D) distributionvaries with the known values of n and rotational Reynoldsnumber Ref ¼ R2

0O=g�

. In turbulent flow regime, where rota-tional Reynolds number varies from 105 to 1.5�105 the averagevalues of n is determined from the data published elsewhere(Dorfman, 1963) to be �0.4418.

Using assumption (vi) the local rotational speed at any pointbetween the membrane and the stirrer may be given as,O¼O1�(O1�O2/zT)z. Hence, a correlation for the average masstransfer coefficient can be determined by integrating Eq. (7) withrespect to the both in radial and axial span of the system as

follows:

k¼ 0:00981D

R0Sc0:6Re0:8

fa ð8aÞ

where

Refa ¼O1:8

1 �O1:82

O1�O2

!1:25R2

0

g ð8bÞ

The expression of k given in Eq. (8a) is strictly valid for counterrotating disk-membrane and stirrer module only where O1 andO2 are of opposite sign.

Now, with Eq. (5) and (6) the following equality is as obtained:

kðcm�c0Þ

2ffiffiffiDp

qa1=2bG 1� 1

2b

�cm�A

R w0 cðzÞe�Azdz

� ¼ 1 ð9Þ

Noting the mathematical symmetry between the numeratorand the denominator on the left-hand side of Eq. (9), it can beinferred that

k¼ 2

ffiffiffiffiD

p

ra1=2bG 1�

1

2b

� �ð10Þ

and

c0 ¼ A

Z w

0cðzÞe�Azdz ð11Þ

In fact, Eq. (10) reduces the degrees of freedom about theselection of the adjustable parameters (a, b, and A) to unity, i.e., ifthe value of b is fixed in the proposed model then a becomesunique and at the same time Eq. (11) may be regarded as adeterministic equation for parameter A. Accordingly, the modelbecomes a single adjustable parameter model with b as theadjustable parameter to be determined by optimum match ofthe model prediction with respect to experimental data.

2.2.2. Formulation of volumetric permeate flux

The determination of volumetric permeate flux contributed byany random surface element follows the same distribution ofresidence time and position from the membrane surface. Thisconstitutes a basis of process simulation. The permeate flux insuch a situation can be formulated in a manner similar to thatfollowed for back transport flux (Eq. (1)–(3)) as

J¼

Z 10

Z w

0Jðz,TÞAe�AzabTb�1e�aTb

dzdT ð12Þ

Unlike back transport flux, no straightforward averaging tech-nique exists for permeate flux as the permeate flux contributed byan arbitrary surface element varies over its residence time on themembrane surface. However, the average permeate flux contrib-uted by the same element can be derived by stretching theresidence time span [0,T] with respect to a local time scale t.The permeate flux contribution by each surface element betweent and tþdt i.e., J(z,t) can be expressed using the well-knownosmotic pressure model (Kedem and Katchalsky, 1961) as fol-lows:

Jðz,tÞ ¼ DP�sDpmRm

ð13Þ

For macromolecular solution, generally the well known Flory-Huggins theory (1941) is used for osmotic pressure calculation.But considering the highly non-linear form of Flory-Hugginsequation, the simple Van’t Hoff equation of osmotic pressurehas been used in the present analysis. At the same time, a pseudosteady permeate concentration is assumed over the residence

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–25672558

time of the corresponding surface element, hence

Jðz,tÞ ¼DP�sRy½cðz,tÞ�cp�

mRmð14Þ

So, the average permeate flux contribution by the arbitrarysurface element residing over the membrane surface for a time T

becomes

Jðz,TÞ ¼1

T

Z T

0Jðz,tÞdt¼ DPþsRycp

mRm�

sRymRmT

Z T

0cðz,tÞdt ð15Þ

In order to evaluate the average surface element concentra-tion, a local stretched coordinate (x) spanning over the depth ofthe particular surface element is introduced as shown in Fig. 2.Furthermore, the transient concentration field inside the sameelement has been assumed to be in exact accordance with thatpredicted by the well-known penetration theory. Thus the tran-sient local concentration inside a surface element can be reportedas

cðx,z,tÞ�cðzÞ

cm�cðzÞ¼ 1�erf

1ffiffiffiffiffiffiffiffiffi4Dtp

� �ð16Þ

The average instantaneous concentration c(z,t) can beobtained by considering a dummy parameter l, equal to the depthof the surface element. Considering negligible solute penetrationlength over element’s residence time with respect to the depth ofthe element (l) the space averaged transient concentration of thesurface element can be obtained as

cðz,tÞ ¼lim

l-1

1

l

Z l

0cðx,z,tÞdx ð17Þ

It is to be noted that this criterion is exactly consistent withthe boundary condition of penetration theory. IncorporatingEq. (16) in Eq. (17) the average transient concentration becomes

cðz,tÞ ¼ cm�ffiffiffiffiffiffiffiffiffi4Dtp

½cm�cðzÞ� ð18Þ

Again incorporating the transient average concentration(Eq. (18)) in Eq. (15), the expression for permeate flux contribu-tion by any arbitrarily surface element with residence time T andpoint of origin at Z is as obtained

Jðz,TÞ ¼DPþsRycp

mRm�sRymRm

cm�2

3

sRymRm

cm�cðzÞ½ �ffiffiffiffiffiffiffiffiffi4DTp

ð19Þ

Finally, in order to get the total permeate flux, Eq. (19) is to beincorporated in Eq. (12) resulting in

J¼DP�sRy cm 1þ 2

3

ffiffiffiffiffiffiffi4Dp

a�1=bG 1þ 12b

�n o�cp

h imRm

�2

3

sRymRm

ffiffiffiffiffiffiffi4Dp

:a�1=bG 1þ1

2b

� �Z w

0AcðzÞe�Azdz ð20Þ

The integrand in the right-hand side of Eq. (20) can be replacedby c0 using Eq. (11) to obtain the final form of permeate flux as

J¼DP�sRyðcm�cpÞ

mRm�

2

3

sRymRm

ffiffiffiffiffiffiffi4Dp

:a�1=bG 1þ1

2b

� �ðcm�c0Þ ð21Þ

2.2.3. Dynamics of membrane surface concentration

The membrane surface concentration dynamics can be devel-oped by writing a component balance equation at the membranesurface. As mentioned earlier, the effective fluxes present at themembrane surface are (a) permeate flux, (b) back transport fluxand (c) Fickian diffusion flux due to concentration polarization. Infact, the back transport flux derived at the membrane surfaceðJs

BT9z ¼ 0Þ takes into account the Fickian diffusion flux as well. As aresult, the stirrer induced back transport flux and the polarizationinduced Fickian flux are effectively merged into a single one.

Considering the Fickian diffusion coupled with back transportflux, the component balance equation at the membrane surfacebecomes

Jðcm�cpÞ�JsBT9z ¼ 0

� A0 ¼ Vsur

dcm

dtð22Þ

For an average surface layer thickness of d, Eq. (22) changesinto the following form:

Jðcm�cpÞ�JsBT9z ¼ 0

� ¼ d

dcm

dtð23Þ

The surface layer thickness (d) of the system is assumed to besame as that of the root mean square average radius of gyrationffiffiffiffiffiffiffiffiffiffiffi

R2g

D Er� �of the BSA molecule which was found to be

3.05�10�9 m at the isoelectric point in the literature (Panseraet al., 2005). Incorporating Eqs. (5), (11) and (21) into Eq. (23), thefinal form of the membrane surface concentration dynamicsbecomes

DP�sRyðcm�cpÞ

mRm�

2

3

sRymRm

ffiffiffiffiffiffiffi4Dp

a�1=bG 1þ1

2b

� �ðcm�c0Þ

� �ðcm�cpÞ

�2

ffiffiffiffiD

p

ra1=2bG 1�

1

2b

� �ðcm�c0Þ ¼ d

dcm

dtð24Þ

In order to solve Eq. (24), the permeate concentration (cp) is tobe replaced in terms of membrane surface concentration (cm).Moreover, the gross dynamic simulation of three major processvariables necessitates a time-implicit algebraic equation incor-porating J, cm and cp in addition to Eqs. (21) and (24).

2.2.4. Relation of flux and rejection from irreversible

thermodynamics

The algebraic equation comprising J, cm and cp can be directlyobtained from the well established permeate flux to ‘‘real rejec-tion’’ relationship using irreversible thermodynamics (Nakao andKimura, 1981), as follows:

Rr � 1�cp

cm¼sð1�FÞ

1�sF¼

s½1�expf�ð1�sÞJ=Pmg�

1�s½1�expf�ð1�sÞJ=Pmg�ð25Þ

Eqs. (20), (24) and (25) form the set of mathematical equationsby which the dynamic simulation of a rotating disk membrane UFmodule can be accomplished, provided the value of b, the loneadjustable parameter of the model, is supplied externally. Theother parameter of the model a, thus becomes unique owing tofixing b in accordance with Eq. (10). Hence, the model can beregarded as a single adjustable parameter model. The value of theadjustable parameter (b) can be obtained by the closest fitbetween model predicted and experimentally determined perme-ate flux values under fixed operating conditions of transmem-brane pressure drop (DP), bulk concentration (c0) and membraneand stirrer rotation speed (O1 and O2, respectively). The followingobjective function has set forth, minimization of which ensuresthe closest fit:

EkðbÞ ¼Xmk

i ¼ 1

1�Jmodeli

Jexpi

!2

for k¼ 1,2,. . .N ð26Þ

3. Simulation algorithm

The detailed algorithm for performing the simulation is pre-sented through a flow chart as shown in Fig. 3. The flowchartconsists of a single time-loop for which the step size (Dt) wasmaintained to be 1 s for all the runs with different parametricconditions of transmembrane pressure drop, bulk concentrationand stirrer speed. Since analytical solution of Eq. (24) is not

Fig. 3. Flow chart showing the algorithm for the simulation of permeate flux, membrane surface concentration and permeates concentration.

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–2567 2559

possible, 4th order Runge–Kutta (RK) method was employed forsolving it numerically throughout the entire time domain for allruns to obtain cm(tþDt), whereas standard Newton–Raphsonnumerical method was used for solving the simultaneous non-linear equations for J(tþDt) and cp(tþDt) (Eqs. (21) and (25)).

The only adjustable parameter b of the proposed model wasdetermined by minimizing the objective function E(b) asdescribed earlier. With a chosen value of b, the algorithm wassubjected to generate the sum of the normalized square error fora specific run. In the next iteration, b was changed to bþDb andthe entire process was repeated. Considering the characteristics ofWeibull distribution b should be a real positive number rangingbetween b¼1 (onset of exponential distribution provided a is alsoequal to unity) and b¼3.4 (onset of Gaussian distribution pro-vided a is also equal to unity), i.e., bA[1,3.4]. The optimum valueof b at which E(b) attains its minima was located by the standardsingle variable Fibonacci search technique in this analysis.

4. Experimental

4.1. Materials

Bovine serum albumin (BSA) with average molecular weight66,000 g mol�1 was obtained from E. Merck, Mumbai, India.Moist PES membrane (asymmetric, molecular weight cut-off:30,000 g mol�1) was obtained from Spectrum Medical Industries(USA). The flat disk membrane operable in pH range of 2–10, hadan actual diameter of 0.076 m whereas the effective diameter was0.056 m.

4.2. Apparatus

The RDM module, made of SS-316, was manufactured byGurpreet Engineering Works, Kanpur, UP (India) as per specifieddesign. The module as shown in Fig. 4 is equipped with two

Fig. 4. Schematic diagram of rotating disk module setup.

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–25672560

motors with speed-controllers to provide independent rotation ofthe stirrer and membrane housing. The module had the facility torotate membrane and the stirrer in opposite direction to providemaximum shear in the vicinity of the membrane. The setup wasequipped with necessary arrangement for recycling of permeateto the feed cell, to run it in continuous mode with constant feedcomposition. Digital tachometer was used to measure the rota-tional speed of both the membrane and the stirrer. Adequatemechanical sealing arrangement was provided to prevent anyleakage from the rotating membrane assembly. The magneticdrive stirrer mechanism prevented any leakage possibility fromthe top stirrer.

4.3. Analysis

BSA concentration was measured using a Hitachi dual beamUV–visible spectrophotometer (Model No U-2800, with UV solu-tion software) by the principle of absorbance assay at 280 nm(Layne, 1957). Density and viscosity of BSA solution at differentconcentration were measured by specific gravity bottle andOstwald viscometer at 30 1C. The pH of solution was monitoredby a digital pH meter purchased from Systronix.

4.4. Experimental procedure

Design of experimental points were made in such a way sothat the effect of four independent variables, namely bulk con-centration (1, 10, 20 and 30 kg m�3), transmembrane pressuredrop (294, 490, 686 and 882 kPa), stirrer speed (5.2, 10.5, 15.7 and20.9 rad s�1) and membrane rotation (5.2, 10.5 and 15.7 rad s�1)could be investigated to address the accuracy of the model inprediction of the permeate flux. Any three of the variables werekept constant while the fourth was varied in order to get theactual nature of dependence. As the reflection coefficient (s) andpermeability (Pm) depend on effective charge of solute particle,the solution pH was always maintained at 4.7 which is very nearto the isoelectric point of BSA that exhibit a neutral behaviorwithin pH range of 4.5–5 (Bohme and Scheler, 2007). The solutionpH was maintained at desired level by adding a very smallquantity of 1(N) HCl solution.

The membrane was placed on a disk shaped porous supportmounted on a hollow shaft through which permeate flows outthrough the cell. The cell was equipped with a flat stirrer havingthe same diameter as that of the membrane. Both the stirrer andthe membrane were placed in front of each other with a verysmall clearance between them. In order to overcome compactioneffect of membrane, the cell was pressurized with distilled waterfor at least 7200 s at 900 kPa, which was higher than that of thehighest operational pressure. Complete compaction of the mem-brane was ensured after getting constant water flux and finallymembrane hydraulic resistance (Rm) was evaluated. Stirrer andmembrane speeds were adjusted to some desired rotational speedby using speed controller fitted with digital rpm tachometer.A metering pump was used to charge the cell with feed solution,as well as for the purpose of recycling permeate intermittently, soas to keep bulk concentration more or less constant. The pressureinside the cell was maintained at a fixed preset value usingpneumatic pressure delivered through compressor, controlled bya digital pressure controller. An intermediate air reservoir wasused for this purpose. After every 300 s interval 10�5 m3 ofpermeate was collected in a measuring cylinder and time for thiscollection was recorded for flux calculation with time. A parti-cular run was continued until two successive flux reading wereequal. Once the run was over, the membrane was thoroughlycleaned with distilled water at least for 7200 s to remove anydeposition. The water flux was then again taken to ensure almost90% regaining of the water flux. The same procedure was repeatedfor each set of operating condition.

5. Result and discussions

The experimental validation of the proposed model requiresnumerical values of different process parameters viz., D, s, Pm, Rm,m, u(¼m/r) and d which has been given in Table 1. As density andviscosity depends on solute concentration, polynomial functionsfor the same have been provided in Table 1. In order to obtain thoserelations, experiments were conducted to determine density andviscosity of BSA solution with different concentration in standardspecific gravity bottle and Ostwald viscometer at 30 1C. The solutediffusivity (D) of BSA was calculated from its functionality with the

Table 1Input parameter for the proposed model taking BSA solution as feed.

Parameter Values

D 6.771�10�11 m2 s�1

m (0.94þ0.215c0 þ1.329�10�4c02)�10�3 Pa s

r (100,021þ2.856�10�4c0)�103 kg m�3

d 3.05�10�9 m

Rm 1.68�1012 m�1

Pm 2.473�10�7 m s�1

s 0.894

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–2567 2561

molecular weight (Sherwood et al., 1975) as given below

D¼ 2:74� 10�9M�1=3pol ð27Þ

For BSA (molecular weight of 67,000 g mol�1) the value of D

reported in Table 1 is close to the one estimated by Opong andZydney (1991) (6.7�10–11 m2 s�1). Reflection coefficient (s) andpermeability (Pm) were calculated from a modified method ofNakao and Kimura (1981). The iterative technique of determinings and Pm was discussed elsewhere (Sarkar and Bhattacharjee,2008). The Weibull distribution parameter a of the residence timedistribution j2(T) was determined for each assumed values of busing Eqs. (3) and (10). Amongst various assumed values of b inthe dynamic simulation of the system, a specific value of b wasdetermined by minimizing the objective function Ek(b) asdescribed in Eq. (26).

In the following sections, the transient behaviors of the modelpredictions has been described under various operating condi-tions and has also been compared with the correspondingexperimental findings.

Fig. 5. Variation of experimental and predicted permeate flux as a function of time

for different transmembrane pressure drop (TMP) and membrane speed (O1) at

constant bulk concentration (c0¼1 kg m�3) and stirrer speed (O2¼5.2 rad s�1).

5.1. Effect of transmembrane pressure (TMP) and membrane speed

on permeate flux

Fig. 5 shows the permeate flux profile for the different TMPand membrane speed keeping bulk concentration and stirrerspeed constant. It is seen from the figure that permeate flux firstreduces rapidly then gradually with time for all TMP andcorresponding steady state are reached within 2400 s of experi-mental run. It is also observed from the figure that the permeateflux increases with increasing TMP and membrane speed at anygiven point of time. Fig. 5 also reveals that steady state permeateflux improves significantly on enhancing both TMP and mem-brane speed. It was observed during experiments that steadystate permeate flux rises almost 6.5 times on increase of TMP by3 times and membrane speed 2 times keeping all other parameterconstant. It may be attributed to the fact that adverse effect ofosmotic pressure (Dp) on permeate flux is largely countered byvigorous turbulence generated simultaneously by relative speedbetween membrane and stirrer, which limits the concentrationbuild up to a very small distance from the membrane surface.

It is to be noted that with an arbitrary guess value of b thesystem was simulated using the algorithm as described in Fig. 2and there from the sum of the normalized square error as given inEq. (26) has been evaluated. Now using the standard singlevariable Fibonacci search technique E(b) was minimized and thecorresponding b value leading to minimum E(b) i.e., giving theclosest fit between the experimental data and the model predic-tions has been chosen to be the final one and subsequently usedto determine the residence time distribution. For the present caseof TMP variation with fixed O2¼5.2 rad s�1 and c0¼1 kg m�3 it isfound that a unique value of b (b¼1.01 for O1¼10.5 rad s�1 andb¼1.05 for O1¼5.2 rad s�1) gives the closest fit between theexperimental data and corresponding model predictions within

the range of variable studied. The corresponding E(b) values arefound to be 0.094 and 0.504 respectively. On the other hand, asthe mass transfer coefficient (k) for counter rotating disk mem-brane module according to Eq. (8a) is fixed, the other parameterof residence time distribution a becomes unique too for aparticular value of b (a¼6.97 for O1¼10.5 rad s�1 and a¼4.32for O1¼5.2 rad s�1 as determined from Eq. (10)). This establishesthat residence time distribution function j2(T) remains unin-fluenced with the change of TMP as shown in Fig. 6. Thecorresponding mean residence time of random surface elements(on the membrane surface) has been found to be 0.15 and 0.24 sfor the membrane speed 10.5 and 5.2 rad s�1, respectively. Henceit may be considered an important inference from the developedmodel that increases in the membrane speed is more predomi-nant over TMP in reducing the polarization effect in the vicinity ofthe membrane surface.

5.2. Effect of bulk concentration and membrane speed on permeate

flux

The effect of bulk concentration (c0) coupled with membranespeed on permeate flux is depicted in Fig. 7 under specifiedcondition of stirrer speed and TMP. It is evident from the figurethat the permeate flux decreases with increasing bulk concentra-tion. The decrease in permeate flux due to increase of initialsolute concentration may be largely compensated by increase inmembrane speed. It may be inferred from the fact that onincrease in the bulk concentration solute rejection by membranesurface increases, resulting in an increased resistance offered by

Fig. 7. Variation of experimental and predicted permeate flux as a function of time

for different bulk concentration (c0) and membrane speed (O1) at constant

transmembrane pressure drop (TMP¼294 kPa) and stirrer speed

(O2¼10.5 rad s�1).

Fig. 8. Residence time distribution function for different bulk concentration at

constant transmembrane pressure drop (TMP¼294 kPa), membrane speed

(O1¼5.2 rad s�1) and stirrer speed (O2¼10.5 rad s�1).

Fig. 9. Residence time distribution function for different bulk concentration at

constant transmembrane pressure drop (TMP¼3 kg/cm2), membrane speed

(O1¼10.5 rad s�1) and stirrer speed (O2¼10.5 rad s�1).

Fig. 6. Residence time distribution function for different transmembrane pressure

drop (TMP) at constant bulk concentration (c0¼1 kg m�3) and stirrer speed

(O2¼5.2 rad s�1).

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–25672562

the polarized layer, which ultimately causes the decrease in thepermeate flux. Contrary to the effect, higher membrane speedgenerates vigorous shear at the vicinity of membrane that resultsinto partial disruption of polarized layer contributed by solute

rejection, thereby enhancing permeate flux. It was found by experi-ments that steady state permeate flux decreased by 79.7% onincreasing bulk concentration from 1 to 30 kg m�3 at constantmembrane speed of 5.2 rad s�1. On increasing membrane speed to10.5 rad s�1 almost 80% of permeate flux recovery was achievedunder the other prevailing identical parametric conditions.

The adjustable parameter b has been changed contrary to theprevious analysis of varied TMP, for different bulk concentrationrun in order to get the closest fit between the experimental dataand model prediction. Quantitatively, for a change in bulk con-centration from 1 to 30 kg m�3, b is changed from 1.04 to 1.08and 1.008 to 1.04 for the membrane speed of 5.2 and 10.5 rad s�1,respectively, giving the minima of the corresponding objectivefunction E(b). Hence the other parameter a is also varied from6.04 to 3.2 and 8.5 to 7.05 for the identical range of bulkconcentration and membrane speed, respectively. This leads to adifferent residence time distribution of surface elements asshown in Figs. 8 and 9. It can be clearly observed from the figuresthat the skewness of residence time distribution progressivelydecreases with increase in the bulk concentration leading to amonotonic increase in the corresponding mean residence timefrom 0.17 to 0.28 s and 0.12 to 0.23 s for the membrane speed

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–2567 2563

5.2 to 10.5 rad s�1, respectively, as the bulk concentrationincreases from 1 to 30 kg m�3. The result can be clearly explainedin terms of increasing solution viscosity (as obtained from thepolynomial corresponding to m in Table 1) near the membranesurface with increasing bulk concentration, which offset the effectof the average inertial and lift forces responsible for picking up ofrandom surface elements from the membrane.

5.3. Effect of stirrer and membrane speed on permeate flux

In order to investigate the effects of stirring on the perfor-mance of counter rotating disk membrane ultrafiltration cell, thepermeate flux profile for four different stirrer speed coupled withtwo different membrane speed under fixed TMP and bulk con-centration has been shown in Fig. 10. It is seen from the figurethat enhancement of membrane and stirrer speed impart asimilar positive effect on permeate flux. This may be justifiedfrom the fact that back transport of rejected solutes into the bulkphase increases with an increase in stirrer and membrane speed,which in turn reduces the resistance offered by polarized layerresulting into enhancement of permeate flux. From Fig. 10, thegeneral effectiveness of membrane rotation over stirrer speed canalso be established under fixed condition of TMP and bulkconcentration. Considering two comparable profile, one withstirrer and membrane speed 10.5 and 5.2 rad s�1 and anotherwith 5.2 and 10.5 rad s�1, respectively, it can be inferred thatthough sum of the stirrer and membrane speed remain unaltered

Fig. 10. Variation of experimental and predicted permeate flux as a function of

time for different stirrer speed (O2) and membrane speed (O1) at constant

transmembrane pressure drop (TMP¼294 kPa) and bulk concentration

(c0¼1 kg m�3).

but the steady-state value of permeate flux for the second casei.e., the case with higher membrane speed was found to be 26.05%higher than the first. This results clarifies the fact the membranerotation is much more effective from the standpoint of permeateflux enhancement than the stirrer speed. As depicted in Fig. 10, itwas found from experiment that an increase in stirrer speed from5.2 to 10.5 rad s�1 (lower side values) steady state permeate fluxincreased by 91% for the membrane speed 5.2 rad s�1 and 36.4%for membrane speed 10.5 rad s�1. While an increase in stirrerspeed from 15.7 to 20.9 rad s�1 (higher side values) steadypermeate flux recorded an increase of 16.95% and 20.6% for themembrane speed 5.2 and 10.5 rad s�1, respectively. Hence, it canbe inferred that effect of stirrer speed is more intense onpermeate flux while membrane speed is low and the effect ofstirrer speed is gradually diminishes with the increase in bothstirrer and membrane speed. This can be further elaborated byconsidering the first order derivative of the mass transfer coeffi-cient (k) with respect to the stirrer speed (O2) that can be directlyobtained from Eq. (8a) and (8b) as follows:

@k

@O2pO�0:2

2

1:8 1�O1O2

�� 1� O1

O2

n o1:8� �� �

1�O1O2

�2

Considering (O1/O2) is of the order �1, then @k=@O2 �O�0:22 .

This clearly follows that the rate of change of k with O2 woulddecrease with increase in O2.

The impulse, drag and lift forces acting on the random surfaceelements located at the membrane surface are bound to changesharply with change in stirrer speed, leading to different values ofb in order to obtain the closest fit. Figs. 11 and 12 show thevariation of residence time distribution function with stirrerspeed while keeping TMP, bulk concentration and membranespeed unchanged. It can be seen from the figures that the peaks ofthe distribution function increases with increase in stirrer speedand the variations amongst these different distribution functionsare more than those obtained in changing the bulk concentration.As a result, a widespread variation in the mean residence time isobserved as shown in Table 2. The data clearly reveals that, withincrease in stirrer and membrane speed mean residence timedecreases on account of vigorous turbulence created by therotation of stirrer and membrane in opposite direction enablingthe random surface elements to be promptly replaced by a freshone on the membrane surface.

Fig. 11. Residence time distribution function for different stirrer speed (O2) at

constant transmembrane pressure drop (TMP¼294 kPa), membrane speed

(O1¼5.2 rad s�1) and bulk concentration (c0¼1 kg m�3).

Fig. 12. Residence time distribution function for different stirrer speed (O2) at

constant transmembrane pressure drop (TMP¼294 kPa), membrane speed

(O1¼10.5 rad s�1) and bulk concentration (c0¼1 kg m�3).

Table 2Variation of mean residence time with membrane and stirrer speed at TMP¼294 kPa

and c0¼1 kg m�3.

Membrane speed(rad s�1)

Stirrer speed(rad s�1)

Mean residencetime (s)

5.2 5.2 0.243

5.2 10.5 0.17

5.2 15.7 0.14

5.2 20.9 0.13

10.5 5.2 0.15

10.5 10.5 0.12

10.5 15.7 0.103

10.5 20.9 0.091Fig. 13. Variation of experimental and predicted steady state permeate flux as a

function of transmembrane pressure drop (TMP) for different Stirrer speed (O2)

and membrane speed (O1) at constant bulk concentration (c0¼1 kg m�3).

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–25672564

In order to be more informative about the trend of steady stateflux with respect to TMP, steady state permeate flux profile fordifferent stirrer and membrane speed at fixed bulk concentrationhas been shown in Fig. 13. It can be seen from the figure thatsteady permeate flux increases almost linearly with TMP. Increasein stirrer and membrane speed recorded a sharp increase insteady permeate flux, especially when increase was madebetween two lower side values of stirrer speed. It was notedfrom experiments that the steady permeate flux increased by87.5% with increase of stirrer speed from 5.2 to 10.5 rad s�1

(lower side values), while it increased by only 19% with theincrease of stirrer speed from 15.7 to 20.9 rad s�1 (higher sidevalues) at the membrane speed 5.2 rad s�1 with TMP 294 kPa. Itwas also observed that the increase in TMP the gradient for thesteady state permeate flux with respect to stirrer speed reducesfor both values of membrane speed.

5.4. Effect of parametric condition on membrane surface

concentration and steady state permeate concentration

Variations of membrane surface concentration with time undervarious parametric conditions have been shown in Fig. 14 and 15.The figures indicate a sharp increase in the membrane surfaceconcentration initially almost like a step function and thereafter itgradually reaches a constant steady state value almost beyond1200 s. It is seen from the figures that the effect of TMP and stirrerspeed over the membrane surface concentration are mutuallyopposite. Membrane rotation and stirrer speed impart a similarreducing effect on membrane surface concentration. It is observed

from the figures that comparative effectiveness in reducing mem-brane surface concentration is higher for membrane rotation thanstirrer speed. In fact, the applied TMP distributes itself instanta-neously over the entire bulk feed resulting in an instantaneousbuilt up of concentration profile as well as membrane surfaceconcentration, but an increase in both stirrer and membrane speedproduce a convective back transport, which causes reduction inmembrane surface concentration. Though it was not studiedseparately, the increase in bulk concentration is likely to increasemembrane surface concentration.

Permeate concentration is one of the very important processvariables so as to get an idea about the performance of any UFoperation. Moreover, knowing the membrane surface concentration,in addition to permeate concentration, the ‘‘real’’ rejection (1�(cp/cm)) of the membrane can be well predicted. In this context, theeffect of TMP on steady state permeate concentration has beenshown in Fig. 16 as a function of membrane and stirrer speed. It canbe seen from the figure that the permeate concentration increaseswith increase in TMP. It may be attributed due to the fact that thevolume of BSA macromolecule gets reduced owing to increase inTMP allowing the chain to pass through the pores of the membrane.It was also observed from experiment and model prediction thatwith the increase in stirrer and membrane rotation speed permeateconcentration decreased. This is probably because higher turbulencein UF cell reduces the mean residence time of random surfaceelement, thereby decreasing the time for mass transfer.

Model predicted results with the same values of b as obtainedby inducing the closest fit for the permeate flux (b¼1.01 for

Fig. 14. Variation of membrane surface concentration with time for different

transmembrane pressure drop (TMP) and membrane speed (O1) at constant bulk

concentration (c0¼1 kg m�3) and stirrer speed (O2¼5.2 rad s�1).

Fig. 15. Variation of membrane surface concentration with time for different

stirrer speed (O2) and membrane speed (O1) at constant transmembrane pressure

drop (TMP¼294 kPa) and bulk concentration (c0¼1 kg m�3).

Fig. 16. Variation of steady state permeate side concentration with transmem-

brane pressure drop for different stirrer speed (O2) and membrane speed (O1) at

constant bulk concentration (c0¼1 kg m�3).

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–2567 2565

O1¼10.5 rad s�1 and b¼1.05 for O1¼5.2 rad s�1) were in wellagreement with the experimental data, showing the applicabilityof the proposed model in predicting the permeate concentrationand hence the rejection. The average deviation of around 75%was noticed at steady state.

6. Conclusion

Optimizing any membrane separation process after identifyingthe affecting process parameters is always a difficult scenarioespecially when the experiments were carried out in a high-sheareddevice like RDM module. Developed model in the present studyassists to identify these process parameters and also their extent ofeffect on the permeate flux and rejection. It was found from thesimulation that the mean residence time of random surface ele-ments was independent from the change in TMP whereas enhance-ment of bulk concentration causes an increase in residence time.The effectiveness in reducing polarization effect was found to behigher for membrane speed than TMP. On the other hand, counterrotation of membrane and stirrer reduces the residence time of thesurface elements drastically. It was also found that comparativeeffectiveness in reducing mean residence time was higher formembrane rotation speed than that of stirrer. The maximumabsolute deviations between the experimental and predicted valuesfrom the proposed model throughout the ranges of variables studiedwere found to be well within 5% for permeate flux and concentra-tion. Clearly, it demonstrates that the model predictions may beused for accurate simulation of the major process variables sub-jected to ultrafiltration of protein solution in a standard RDMmodule. The proposed model in this particular study provides aninsight of the process from the microscopic level based on thesurface renewal theory, and thus it truly represents a generalizedapproach for the prediction of membrane throughput. It is a well-known fact that the efficacy of the membrane separation lies in

D. Sarkar et al. / Chemical Engineering Science 66 (2011) 2554–25672566

obtaining a moderate flux during the process, which is also anindicative value to understand the accumulation of solutes on themembrane surface, an important issue with a commercial mem-brane separation process. This particular flux predictive modelillustrates the process from the point of accumulation of soluteson the membrane surface. Therefore the applicability of this modelcan be generalized for different membrane modules after a little bitmodification in the mass transfer coefficient’s expression. Addition-ally because of this generalization, the simulation program, used topredict flux from the model, can also be adjoined with a controlsystem to develop a strategy device in order to adjust all parametricconditions to obtain a maximum flux.

Nomenclature

A parameter of exponential distribution defined in Eq. (2)A0 area of the membrane surface (m2)BSA bovine serum albuminc solute concentration (kg m�3)c0 solute concentration (gm cm�3)cm solute concentration at membrane surface (kg m�3)cm(t) solute membrane surface concentration after time t

(kg m�3)cp solute concentration in the permeate (kg m�3)cp(t) solute concentration in the permeate after time t

(kg m�3)c0 bulk solute concentration (kg m�3)c(z) local concentration at a distance z from the membrane

surface (kg m�3)c(x,z,t) concentration of solute at a depth x within a random

surface element with a point of origin at z with localstretched time scale coordinate t counted from its timeof arrival at the membrane surface (kg m�3)

c(z,t) average solute concentration over the depth of a randomsurface element with a point origin at z with localstretched time scale coordinate t counted from its timeof arrival at the membrane surface (0rtrT), (kg m�3)

D solute diffusivity (m2 s�1)Ek(b) sum of normalized square error between the model

prediction and experimental data for kth runF parameter defined in Eq. (25)J volumetric permeate flux (m3 m�2 s�1)Jexpi experimental permeate flux data at ith experimental

data point of kth runJmodeli model predicted permeate flux corresponding to ith

experimental data point of kth runJ(z,T) volumetric permeate flux contributed by a random sur-

face element having point of origin at z residence time T

on the membrane surfaceJsBT9z ¼ 0 solute back transport flux at the membrane surface

(m3 m�2 s�1)JsBT ,0ðz,TÞ solute back transport flux at the membrane surface

contributed by a random surface element having pointof origin at z residence time T at the membrane surface

k mass transfer coefficient (m s�1)l dummy parameter same as the depth of a random

surface element (m)Mpol molecular weight of polymer (kg kmol�1)n parameter defined in Eq. (7).N total number of experimental runs for different para-

metric conditionsDP transmembrane pressure drop (Pa)Pm solute permeability (m s�1)r radial axis of RDM module (m)R universal gas constant (kJ kmol�1 K�1)

R0 radius of stirrer and membrane (m)Rg radius of gyration of a polymer chain (m)Rm membrane hydraulic resistance (m�1)Rr ‘‘real’’ rejectionRef rotational Reynolds numberRefa Reynolds number defined in Eq. (8b)Sc Schmidt number ðSc¼ m=rDÞ

Sh local radial Sherwood number ðSh¼ kr=DÞ

t time (s)tut time up to which simulation is required (s)Dt time increment in the dynamic simulation (s)T residence time of a random surface element on the

membrane surface (s)TMP transmembrane pressure dropUF ultrafiltrationVsur average volume of the surface phase (m3)x local stretched space coordinate spanned over the depth

of a surface element (m)z distance from the membrane surface, global space

coordinate (m)zT axial length of separation between membrane and

stirrer (m)

Greek letters

a real positive parameter of Weibull distribution(dimensionless)

b real positive parameter of Weibull distribution, the onlyadjustable parameter of the model (dimensionless)

r density of solution (kg m�3)t local stretched time coordinate spanned over the resi-

dence time of a surface element on the membranesurface (s)

p osmotic pressure (Pa)pm membrane side osmotic pressure (Pa)pp permeate side osmotic pressure (Pa)Dp osmotic pressure differential across the membrane

(¼pm�pp) (Pa)y temperature (K)s reflection coefficient (dimensionless)O local angular velocity of fluid at any point between

membrane and stirrer (rad s�1)O1 angular velocity of membrane (rad s�1)O2 angular velocity of stirrer (rad s�1)u kinematic viscosity of the solution (m2 s�1)m viscosity of the solution (kg m�1 s�1)d average thickness of the surface phase (m)j combined distribution function of the point of origin and

the residence time over membrane surface for surfaceelements

j1 distribution function of the coordinate of origin of sur-face elements

j2 distribution function of the residence time of surfaceelements over the membrane Surface

w Prandtl mixing length (m)

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