Modeling and analytical simulation of rotating disk ultrafiltration module

Journal of Membrane Science 320 (2008) 344–355

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Modeling and analytical simulation of rotating disk ultrafiltration module Debasish Sarkar a , Chiranjib Bhattacharjee b,∗ a b

Department of Chemical Engineering, Calcutta University, Kolkata 700 009, India 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 2007 Received in revised form 2 April 2008 Accepted 9 April 2008 Available online 4 June 2008 Keywords: Ultrafiltration Rotating disk membrane Permeate flux Back transport Mathematical model

a b s t r a c t An unsteady state mass transfer model has been developed for rotating disk ultrafiltration module. Starting from the basic physics of the system, analytical expression of back transport flux generated due to rotation-induced shear field is determined, which is subsequently incorporated in the fundamental material balance equation. In order to get an analytical solution of governing partial differential equation via Laplace transformation, pseudo steady state consideration is imposed both on permeate as well as back transport flux. Once the analytical form of concentration field is obtained using the expression permeate flux, membrane surface concentration are evaluated using polymer solution theory and irreversible thermodynamics. Finally an iterative scheme is designed to simulate the permeate flux and membrane surface concentration under specified set of operating parameters. The prediction from this model is found to be in good agreement with experimental data obtained from PEG-6000/water system using cellulose acetate membrane of 5000 Da molecular weight cut-off. © 2008 Published by Elsevier B.V.

1. Introduction Over the last few decades ultrafiltration (UF) has emerged as a cutting edge technology because of its inherent ability to separate chemical and biochemical compounds up to molecular level. The present day application of UF includes the treatment of industrial effluents, oil emulsion, biological macromolecule, colloidal system and many others. From a fundamental standpoint ultrafiltration is nothing but a rate governed separation process in which pressure acts as a driving force. Generally the operating pressure varies in the range of 10–140 psi, with membrane pore size ranging between 10 and ˚ Despite all its attractive features, the process of ultra100 A. filtration suffers from a serious drawback of continuous solute accumulation on the membrane surface due to their rejection by membrane, characteristics of any pressure driven membrane separation process. Because of this phenomenon, known as concentration polarization, a concentration gradient set in, evidently in the directions opposite to that of permeate flux. As a result the permeate flux decreases, which is not at all desirable for efficient separation. A broad overview of concentration polarization was first reported by Bruin et al. [1]. Later on several studies on different aspects of concentration polarization were reported. 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: [email protected] (C. Bhattacharjee). 0376-7388/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.memsci.2008.04.015

instability both in dead end and cross flow module. Thermodynamic interpretation of concentration polarization was presented by Peppin and Elliott [3]; where as effect of viscosity was studied by Gill et al. [4]. Zaidi and Kumar reported a detail experimental analysis of concentration polarization in dead end ultrafiltration of dextran [5]. Different modules, proposed so far are basically designed with a sole objective to reduce the effect of concentration polarization, though in some of the recent articles it is established that concentration polarization can be reduced by using electric field [6,7], two phase system [8] or by direct gas sparging [9]. One of the popular methods in module design is to induce a high-shear field near the membrane surface so as to homogenize the adverse concentration gradient, thereby minimizing the effect of concentration polarization. In cross flow modules the very structure of the module incorporates a shear field, where as in batch UF cell the primary step is to introduce a stirrer placed in very close proximity of membrane. Several analytical models from different standpoints for single stirred UF cell have been developed over years [10–13], in addition to that neural network models for the same are also available [14,15]. Following the idea of inducing high-shear field, progresses have been made by introducing the concept of rotating disk membrane (RDM), with the same fundamental intention to increase the effect of shear on concentration polarization. In RDM module, membrane and stirrer of practically same diameter are placed face to face with a very small distance of separation between them. Under this condition as membrane and stirrer, both rotates in the direction opposite to each other, a high-shear field

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

is introduced, which counters the effect of concentration polarization. In reality, the primary effect of shear field is to give rise a new flux called “Back Transport Flux”, in the direction same as ¨ and Lopez-Liva that of concentration polarization flux. Halstrom first proposed the basic structure of RDM module [16]. Experimental studies using mineral suspension [17] and black liquor [18] are also available. In a recent article computational fluid dynamic model and simulation of RDM was presented by Torras et al. [19]. In a previous article of single stirred UF cell, Bhattacharjee and Datta reported the analytical simulation of permeate flux by using a semi-empirical parameter called “Back Transport Coefficient” [10]. Though the model was good enough to predict the behavior of single stirred cell, but it cannot be directly extended to RDM module as the flow field, hence the proper evaluation of the back transport flux is much complicated in the latter system. The present work has been undertaken in an attempt to develop a rigorous mathematical model of RDM cell used in the process of ultrafiltration. Starting from the basic physics of the system, the back transport flux is evaluated exploring the probable locus of a solute molecule initially at the membrane surface. Once the analytical expression of back transport flux is at hand, fundamental mass balance equation is developed under unsteady state condition, which is analytically solved. No analytical solution for RDM or any such high-shear device has been reported in the literature. In this context, the proposed model could be used for the simulation of UF performance in RDM and the approach could also be extended to other high-shear device, like multi-shaft disk (MSD), etc. The model prediction is validated with experimental data for RDM module during Ultrafiltration of PEG-6000 by cellulose acetate membrane under unsteady state condition for different transmembrane pressure, initial concentration, stirrer speed and membrane speed.

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should have the same dimension of volumetric flux, i.e., LT−1 . From an intuitive point of view it can be inferred, as the membrane and the stirrer are rotating in opposite direction, any particle released tangentially on the membrane surface will follow a helical path from membrane to the stirrer. The tangential velocity versus axial distance profile for the proposed locus can be depicted qualitatively as in Fig. 2. Though the helix angle, ˛ will vary from point to point, for simplicity the stated angle is assumed to be constant over the entire path. In accordance with the assumption the axial velocity at any radial location can be related to the tangential velocity at the same point as vx (r) = v (r) tan ˛

(1)

Whereas the tangential velocity profile can be written as v (r) =



1−



ω1 + ω2 x r xt

(2)

Assuming Newtonian fluid, existing shear field can be expressed in the following form: (r) = 

dv =− dx

ω + ω  1 2 xt

r

(3)

It may be argued that in dealing with PEG in water, power law model must be used for proper estimation of shear field, but keeping in mind the high dilution level of the system, Newton’s law is said to be fairly valid. Now in order to correlate the axial velocity to shear stress a thin cylinder between r and r + dr, with axial span from the membrane to stirrer surface is considered. The torque balance equation over any horizontal section of the cylinder yields: (r)da r =  dVr 2 ¨

(4a)

2. Theoretical development Simplifying Eq. (4a) 2.1. Development and analytical solution of model equation The schematic diagram of a rotating disk membrane module has been shown in Fig. 1, in which different flux components have been highlighted clearly. As stated earlier, the stirrer and the rotating membrane are placed face to face, and rotate in opposite direction with respect to each other with angular speeds, ω1 and ω2 , respectively. In order to impart a high-shear field in the vicinity of membrane surface, the distance of separation between the stirrer and membrane is made very small compared to the radius of the UF cell, so that a linear velocity profile can be assumed to exist in the space between the stirrer and membrane. The primary step of theoretical modeling is to quantify the shear induced back transport flux. This term is used in order to account for the back transport of solute due to eddy back mixing; naturally it

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

dv (r) da r = r ¨ = at = v   dV ds

(4b)

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

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where at is the tangential acceleration and ds is the differential length traversed by the particle along the helix. Now v

vx dvx (r) dv = ds tan2 ˛ ds

(5)

Taking Laplace transform of Eq. (11) and applying boundary condition (i) ∂c ∗ ∂2 c ∗ + A ∗ − sc ∗ = −1 ∗2 ∂x ∂x

Combining Eqs. (4b) and (5)

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

da

vx (r) dvx (r) (r) r=  dV tan2 ˛ ds

(6)

I. Complementary function:

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

∗

CF : c ∗ (x∗ , s) = C1 em1 x + C2 em2 x

as (r) tan2 ˛ ds = vx (r)dvx (r)  xt

(7)

Integrating Eq. (7) from any point between stirrer and membrane where s = 0 up to the stirrer surface, where s = 2rn (n is the number of helix turns necessary to reach the stirrer surface from the stated intermediate position) and vx (r) = 0, incorporating the expression of shear stress in Eq. (3) the following relation is obtained:



vx (r) =

4n(ω1 + ω2 ) 

r xt

tan ˛

where m1 , m2 = −A ± stants. II. Particular integral: PI :

JBT =

  16n(ω1 + ω2 ) R 9

xt

tan ˛

∂2 c ∂c ∂c +D 2 = ∂x ∂t ∂x

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

1 s

∗

∗

c ∗ (x∗ , s) = C1 em1 x + C2 em2 x +

(8)



c ∗ (x∗ , s) =

(9)

(cm /c0 ) − 1 s





exp

(10) + A e−Ax

∗





erfc

2.2. Calculation of permeate flux

The boundary condition also assumes a pseudo steady state approximation on membrane surface concentration, cm the time evolution of which will be solved by separate physical consideration. Introducing the following dimensionless parameters, Eq. (10) can be reduced to a dimensionless form, which is easier to deal with:

∂c  J(cm − cp ) = −D ∂x 

x Dt x = and t ∗ = 2 xt xt

(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.



A2 + 4s ∗ x 2



+

1 (14) s

√  A t∗ x∗ +1 √ − 2 2 t∗

(15)

x=0



x=0

cm ⇒ J



+ JBT  x=0

cm − cp

(16)

x=0 cm

Eq. (16) is based on the pseudo steady state assumption of membrane surface concentration. Now



(11)



+ JBT 

∂c  D =− cm − cp ∂x 

∂c  ∂x 

With these parameters Eq. (10) reduces to

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



The flux condition at the membrane surface can be represented as

∗

∂c ∗ ∂c ∗ ∂2 c ∗ +A ∗ = ∗ ∂x ∂t ∂x∗2

−A −

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

(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.

c c = , c0

(13)

Taking inverse Laplace transform using shifting theorem and Laplace transform tables the complete solution in time domain is as obtained:  

 √  (cm /c0 ) − 1 x∗ A t∗ ∗ ∗ ∗ c (x , t ) = erfc √ + 2 2 2 t∗

Eq. (10) assumes constant diffusivity and a pseudo steady state approximation on permeate flux, J. The boundary conditions are as follows:

∗

1 s

Applying boundary conditions (ii) and (iii) the solution becomes

Once the basic expression of back transport flux is obtained, an unsteady state solute balance over a differential shell between distance x and x + dx from membrane surface yields the following PDE: (J − JBT )



∗

Hence the complete solution in Laplace domain can be written as

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



(12)



= x=0

c0 ∂c ∗  xt ∂x∗ 

(17) x∗ =0

Combining Eqs. (15)–(17) followed by subsequent simplification, the expression for permeate flux could be obtained as follows: J =

D cm − cp



+A

 c   1 − (c /c )  2 m 0 0 xt

1 + erf

√

2

 √  ∗ A t 2

+

t ∗

JBT |x=0 cm cm − cp



exp

−

At ∗2 4



(18)

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

where (J − JBT |x=0 )xt A= D In the expression of permeate flux, back transport flux, JBT is to be considered separately. Looking into the expression of JBT in Eq. (9) it is evident that the form of back transport flux has been derived by averaging the axial velocity at any particular distance from membrane surface over the radial space. But in Eq. (18) back transport flux is the same evaluated at membrane surface, i.e., at x = 0. In order to get that expression Eq. (7) is to be integrated from the membrane to stirrer surface assuming N number of helix turns is required to reach the stirrer from membrane surface (n ≤ N, as n is the number of helix turns necessary to reach the stirrer from any intermediate point between the stirrer and the membrane). Thus integrating Eq. (7) the back transport flux at the membrane surface is as obtained:



vx (r)|x=0 =

4N(ω1 + ω2 ) 

r xt

tan ˛

(19)

Before averaging the expression over the radial space total number of helix turns, N can be replaced in terms of radial distance, r and helix angle, ˛ as p=

xt = 2r tan ˛ N

(20)

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



JBT |x=0 =

32(ω1 + ω2 )R tan ˛ 25xt

−1/3

for initial compaction, by inducing the membrane to high pressure than that of experimental range. The average of all the values obtained from different water runs is considered as the acceptable value for Rm . The viscosity of PEG-6000 solution in water was correlated with the solution concentration at 30 ◦ C as [21,22]:

=

0.85 + 0.1446c  + 2.734 × 10−4 c  2 − 4.276 × 10−6 c  3 + 2.84 × 10−8 c  4

where c is in g cm−3 In order to express  as a function of cm and cp , Flory–Huggins theory is to be used. The theory states that osmotic pressure of a polymeric solution can be related to solute concentration as  = −

RT V1



ln(1 − vp ) + 1 −

1 n



vp + 12 vp2

 (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 of polymer–solvent interaction and is 0.45 for PEG–water system. 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 membrane surface concentration followed by substitution of osmotic pressure differential in Eq. (23) results:

   RT V1 ln (pol − cm )/(pol − cp ) 

P + J=

+

1 − (1/n) + 12 (cm + cp )/pol (cm − cp )/pol Rm

 (26)

Eqs. (18) and (26) constitutes two non-linear equations relating J, cm and cp . In order to determine the time evolution of these three parameters another equation relating these three parameters are required. Additionally for the back transport flux at membrane surface, the unknown helix angle (˛) is to be determined separately. 2.4. Relation of flux and rejection from irreversible thermodynamics Irreversible thermodynamics is used to express the rejection as [23] Rr ≡ 1 −

2.3. Calculation of permeate and membrane surface concentration In the process of UF, no matter whether the cell is stirred or unstirred transmembrane pressure drop, P acts as a driving force and because of concentration difference on two sides of the membrane surface, osmotic pressure differential,  acts as a hindrance factor to the process objective. Still, because of inherent irreversibility the net driving force is not exactly the difference of transmembrane and osmotic pressure, but difference of P and   , where  is known as reflective coefficient. The permeate flux, which represents the rate of the process can be related to the stated difference in accordance to a linear equation (osmotic model):

P −   Rm



(22)

Now from the expression of permeate flux, as given in Eq. (21) it can be inferred that in order to get a time evolution of the same time history of membrane surface concentration, cm as well as permeate concentration, cp are to be determined, so the next task of the model is to explore polymer solution theories in order the evaluate the stated parameters.

J=

(24)

103

(21)

For calculation, solute diffusivity is expressed as a function of molecular weight of the same, M following a standard empirical equation as given below [20]: D = 2.74 × 10−9 Mpol

347

(23)

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

cp (1 − F) = 1 − F cm

(27)

where F = exp{−(1 − )J/Pm }, Pm is the permeability of the membrane. Now three Eqs. (18), (26) and (27) can be solved simultaneously to predict the time evolution of J, cm and cp , but the prerequisite is the quantitative determination of back transport flux, which is yet not done. The reflection coefficient,  and solute permeability, Pm are determined by a modification of the method outlined by Nakao and Kimura [23]. The iterative technique is briefly outlined as follows: (i) Initially a  value is assumed, using this value membrane surface concentration is temporarily determined for all experimental runs using Eqs. (23)–(25). (ii) Pm is calculated from Eq. (27) for the mentioned assumed value of  and using experimental values of J and cp . The procedure is repeated to calculate all Pm values for different experiments. (iii) Standard deviation in the Pm values is now calculated for the same assumed value of . (iv) In this way different  values are assumed and the procedure is repeated to calculate standard deviation for all of them.

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

Fibonacci search technique is used to locate the  value after minimization of standard deviation. (v) Average of all Pm values for different experiments gives the correct value of solute permeability. For the stated system, the values of Pm and  determined by the 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 membrane surface, 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 of pressure differential, P; stirrer speed, ω1 ; membrane speed, ω2 and bulk concentration, c0 are curve fitted in order to minimize the experimental error and for data smoothening. The best-fit equation in terms of highest correlation coefficient and minimum standard deviation is found to be J = J0 (1 − a + abt )

(28)

In Eq. (28), J0 is the initial flux, time (t) is in minutes, whereas the values of coefficient a and b are different for different experimental conditions. As an example, for an experiment conducted at c0 = 20 kg m−3 , P = 827 kPa, ω1 = 5.2 rad/s and ω2 = 55.5 rad/s, the calculated 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) can be combined for a particular set of c0 , P, ω1 and ω2 as J0 (1 − a + abt ) =

D cm − cp

 c   1 − (c /c )   2 m 0 0

√ t ∗     √  A t∗ At ∗2 exp − + A 1 + erf 4 2 +

xt

(29)

P − Rm J0 

(30)

Now the osmotic pressure on the feed side can be obtained as  =   +   . Once the numerical value of   is known memm p m brane surface concentration, cm at that time for the same run can be determined using simple Newton–Raphson method of iterative solution for single variable non-linear equation. The working equation for this calculation is f (cm ) =

RT + V1

  ln

cm 1− cp

 

1 + 1− n

 c m pol

+ 12

2 cm 2 pol

(k)

f (cm )

(k)

= cm −

(k)

f  (cm )

(32)

Once the values of cm and cp are determined at a particular time instant, Eq. (29) can be viewed as a non-linear equation of parameter tan ˛, as all the other variables of the same equation have been reduced to their numerical values for a particular run. Modified form of Eq. (29) that can be expressed in a functional form as f(tan ˛ = 0) is solved again by using single variable Newton–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 instant are known, Eqs. (18), (26) and (27) can be solved simultaneously to get the time evolution of permeate flux, J, membrane surface concentration, cm and permeate concentration, cp . An iterative scheme is outlined below for this purpose: (i) A specific t is selected. (0) (0) (ii) Initialization step: t = 0, cm = 0, cp = 0 and J = ( P −       )/Rm with  = m − p . (iii) Eqs. (18), (26) and (27) are simultaneously solved by multivariable Newton–Raphson method with a specific relaxation parameter 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.

3. Experimental [21,22]

JBT |x=0 cm cm − cp

(1 − a + abt )

 m

(k+1)

cm

The chosen t value must be sufficiently small in order to avoid divergence. In the present analysis the chosen criteria for t selection is t < 5 for all the runs.

2

In order to solve for tan ˛ (through the expression of JBT |x=0 in Eq. (9)) using Eq. (29) values of cm and cp are to be determined from experimental data. cp , i.e., the permeate concentration at different time for a particular run can be determined experimentally. But for cm theoretical equations are to be used in combination with experimental data. For this purpose again the Flory–Huggins equation is used, but in a different form. Eq. (25) is directly used to evaluate the permeate side osmotic  −   ) can be pressure, p . Again using Eqs. (23) and (28)  (= m p determined at different time interval for a particular run as

 =

(0)

 at each time instant and using c Knowing the value of m m = c0 as an initial guess cm can be solved as

 (31)

3.1. Materials Polyethylene glycol (PEG-6000, AR grade) of molecular weight range of 6000–7000 dissolved in water was used as feed solution 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). The membranes showed a hydrophilic property, has no adsorption characteristics and resistant to temperature up to 90◦ . 3.2. Apparatus In order to eliminate the effects of concentration polarization so as to obtain highest possible steady state flux high-shear devices are becoming more and more popular as industrial ultra filtration module instead of its inherent mechanical complexity. In the same line of RDM, multi-shaft disk membrane module, though asymmetric, replicates the basic structure of RDM on compartment basis have been already commercialized by Westfalia Separator Filtration GmbH (previously known as Aaflowsystems), Aalen, Germany. The commercial module consists of ceramic membrane disks mounted on parallel hollow shafts with disks of two successive shafts overlapping each other. So with respect to a particular disk, the adjacent disks functions as stirrer, and hence a MSD unit may be viewed as an asymmetric collection of several RDM units placed in seriesparallel sequence, but the flow rate between membranes in actual

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

MSD unit is periodical and unsteady because of multi-shafts geometry. The largest MSD unit known so far consists of 31 cm diameter disks mounted on up to eight shafts with a total membrane area of 80 m2 . In a very recent article He et al. [24] presented an experimental analysis of MSD that shows the increased performance efficiency of MSD over cross flow and single stirred disks. For the present work the RDM module made of SS316, was manufactured by Gurpreet Engineering Works, Kanpur, UP (India) as per specified design. The module (Fig. 3) was equipped with two motors with speed-controllers to provide rotation of the stirrer and membrane housing. The module has the facility to rotate membrane and the stirrer in opposite direction to provide maximum shear in the vicinity of the membrane. Digital tachometer was used to measure the rotational speed of both the membrane and the stirrer. The setup was equipped with necessary arrangement for recycling of the permeate to the feed cell, to run it in continuous mode with constant feed composition. The later mode was not investigated in this study. Adequate mechanical sealing mechanism was provided to prevent leakage from the rotating membrane assembly. The magnetic drive stirrer mechanism prevents any leakage possibility from the top stirrer. The complete schematic diagram of the rotating disk module setup is given in Fig. 3. The flat disk membrane operable in pH range of 1–14, has an actual diameter of 76 mm whereas the effective diameter was 56 mm.

349

four independent variables, namely bulk concentration (20, 50, 70 and 90 kg m−3 ), transmembrane pressure drop (965, 827, 689 and 552 kPa) and stirrer speed (63.3, 55.5, 47.1 and 34.0 rad/s) could be investigated. Any three of the variables were kept constant while the fourth was varied in order to get the actual nature of dependence. The effect of membrane rotation (5.2, 31.2 and 62.8 rad/s) was studied in conjugation with the variable parameter, as an example while studying the effect of transmembrane pressure on flux 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 exploring the effect of transmembrane pressure for a particular membrane speed. In order to compare the performance of RDM module with that of corresponding single stirred cell, all the different runs were repeated with static membrane (ω1 = 0), with the same parametric conditions. Additionally for the single stirred cell three different types of stirrer, namely propeller agitator, turbine and flat disk were used separately in order to compare the general characteristics of different single stirred cell with that of RDM. This comparison was necessary in order to establish the gross effectiveness of introducing membrane stirring effect over different single stirred mode. But as the develop model can handle only the flat disk fitted stirrer, all the single stirred cell data reported in the study was purely experimental. 3.5. Procedure

3.3. Analysis Solution concentrations were measured with a refractometer (model P70, Warsaw, Poland). The density and viscosity were determined 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

The membrane was placed on a disk shaped porous support mounted on a hollow shaft through which permeate flows out through the cell, and the cell was assembled which contains a flat stirrer having the same diameter as that of the membrane, placed face to face. In order to overcome compaction effect of membrane, the cell was pressurized with distilled water for at least 2 h at 900 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-

Fig. 3. Schematic diagram of rotating disk module set up.

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tal run. The stirrer and membrane speeds were adjusted first to desired rpm by the use of 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 the permeate intermittently, so as to keep bulk concentration more or less constant. The pressure inside the cell was maintained at a fixed preset value using pneumatic pressure delivered through compression, controlled by a digital pressure controller. An intermediate air reservoir was used for this purpose. To determine current value of permeate flux, 10 cm3 of permeate was collected in a measuring cylinder and time for this collection was recorded. Flux values were recorded every 5 min. A particular run was continued until two successive flux reading were equal. Once the run was over, the membrane was thoroughly cleaned with distilled water at least for 2 h to remove any deposition. The water flux then again checked to detect any variation in the membrane hydraulic resistance. The same procedure was repeated for each set of operating condition.

4. Results and discussion Once the membrane has been compacted, several water runs were taken at different pressures as mentioned before to determine membrane hydraulic resistance (Rm ). Out of all the generated sets of experimental data 24 sets were used for the purpose of comparison with the trend predicted by analytical model. Fig. 4a–c shows the comparative plots of permeate flux profile under different 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 permeate flux reaches its steady state value practically within 5–7 min, where as for a single stirred cell (ω1 = 0) of similar geometry the corresponding time is in the order of 1 h [10]. This shows an extra advantage of RDM module over single stirred cell. The permeate flux clearly shows an increasing trend both with the increase of transmembrane pressure as well as membrane speed. It is observed that for a transmembrane pressure change from 552 to 965 kPa the steady state flux increases by 60–67% under three different membrane speeds. Where as for the change of membrane speed from 5.2 to 62.8 rad/s, the increase of steady state permeate flux is more than 100% for the same operating pressure. In addition to the general flux profile, a comparative bar chart representing the steady state 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 of membrane stirring in addition to stirrer is always much more effective than the most efficient single stirred module (single stirred cell fitted with propeller agitator) reflected by the fact that the steady state flux of RDM was at least 10% higher that of propeller fitted single stirred cell, though the RDM itself is fitted with flat disk stirrer, which is the least effective one in single stirred mode. Moreover with the increase of membrane speed from 5.2 to 62.8 rad/s the permeate flux increases practically by 90%. In order to establish the general effectiveness of membrane rotation, permeate flux profile for different stirrer and membrane speeds under fixed conditions of transmembrane pressure and bulk concentration is shown in Fig. 6. Considering two comparable profiles, one with stirrer and membrane speeds of 63.3 and 31.2 rad/s and another with 34.0 and 62.8 rad/s, respectively it can be inferred that though the sum of the stirrer and membrane speeds remains practically constant but the steady state value of permeate flux for the second case, i.e., the case with higher membrane speed (stirrer: 34.0 rad/s and membrane: 62.8 rad/s) is 12% higher than the first. 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 bulk concentration (c0 = 20 kg m−3 ) and stirrer speed (ω2 = 55.5 rad/s).

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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 stirrer speed. Moreover it can be observed that the effect of stirrer speed increase is more pronounced at higher side values of the membrane speed 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 ).

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

increases with the increase of membrane speed. A comparative bar chart of steady state permeate flux with respect to stirrer speed under fixed conditions of transmembrane pressure, concentration and membrane speed is shown in Fig. 7. Here once again the steady permeate flux is compared with single stirred cell with different types of stirrer. Effect of bulk concentration on flux profile is depicted in Fig. 8a and b under fixed conditions of stirrer speed and transmembrane pressure. Here the effect of membrane speed was studied in conjugation with bulk concentration. Fig. 8a represents the flux profile at membrane speeds of 5.2 and 31.2 rad/s where as Fig. 8b represents the same variation at 31.2 and 62.8 rad/s. It is very evident that as bulk concentration increases rejection by the membrane surface increases resulting an increased resistance in polarized layer and thereby flux gets reduced. But with the increase of membrane speed the increased effect of back transport flux reduces the influence of concentration polarization; hence permeate flux increases practically by 35–50% for all different concentrations. Fig. 9 shows the variation of helix angle of the solute particle trajectory due to back transport flux at the membrane surface with different stirrer and membrane speed, all the other process parameters remains constant. The helix angle (˛) shows a decrease with increasing stirrer and membrane speed, which is expected because of the fact that increased stirrer as well as membrane speed, induces a tendency in solute to follow horizontally oriented circular path with very little vertical displacement under each turn, exactly consistent with the flow field produced by combined rotation of stirrer and membrane. As the ratio of vertical to circular displacement decreases the helix angle must decrease with increase in stirrer speed. Alternatively it can be argued that in high turbulent regime, flow structure is practically independent of system parameters, as a result the back transport flux that occurs in high turbulent regime becomes constant. Since the flux at membrane surface (JBT |x=0 ) is directly proportional to (ω1 + ω2 ) tan ˛, increase in ω2 or ω1 , i.e., in (ω1 + ω2 ) tan ˛ and therefore ˛ must decrease to keep the entire term constant. For the increase of transmembrane pressure helix angle decreases insignificantly. It is observed that ˛ decreases very sharply with time and attains its steady state value within 5–10 min, consistent with the trend of permeates flux. No significant trend of helix angle with bulk concentration is observed.

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

after a short span of time it becomes almost flat. This is due to the fact that as time increases membrane surface concentration shows a positive trend resulting in increase in the osmotic pressure differential, which opposes the applied pressure differential. This results in decrease of the effective driving force giving rise to a retarded volumetric flux and hence a lower solute transportation rate at the membrane surface. In addition to the above effect, concentration polarization increases with membrane surface concentration and the phenomena of back-diffusion becomes more pronounced and hence the rate of increase in membrane surface concentration becomes small. With the increase in transmembrane pressure that’s why the surface concentration seems to be increasing. 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 transmembrane pressure ( P = 827 kPa) and stirrer speed (ω2 = 55.5 rad/s).

Variation of membrane surface concentration (cm ) with time at different transmembrane pressure and stirrer speed with two distinct membrane speeds are shown in Figs. 10 and 11, respectively. Initially the membrane surface concentration increases very sharply but the rate of increase in cm diminishes with time and

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

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

near the vicinity of membrane surface have an important effect in ascertaining the concentration polarization phenomena. An interesting observation that can be noticed from both of Figs. 10 and 11 that membrane rotation is more effective in reducing the membrane surface concentration compared to the stirring effect. In fact the concentration build up is reduced by maximum 1.1% per unit increase of membrane speed where as it is reduced by 0.75% per unit increase of stirrer speed. Though not separately studied but it is also quite obvious that increase in bulk concentration must give rise to increased membrane surface concentration. In order to be more informative about the trend of steady state flux with respect to transmembrane pressure, a plot of steady flux versus pressure at different bulk concentration and membrane speed is represented in Fig. 12. It is evident from the figure that permeate flux increases almost linearly with pressure, where as with increasing concentration permeate flux by significant amount as expected because increased concentration results in severe polarization thereby reducing the steady flux. The effect of concentration increase is countered more effectively by the increase in membrane rotation. It is to be noted that as the membrane speed increases from 5.2 to 31.2 rad/s there is practically 75–115% increase in permeate flux under different transmembrane pressure and bulk concentration. The effect of stirrer speed increase is separately shown in Fig. 13 in the form of a bar chart. Though the steady flux seems to be increasing with stirrer speed but it is limited to maximum 50%, much lower than the corresponding enhancement with the increase in membrane speed. The effect of transmembrane pressure and membrane rotation on % rejection is depicted in Fig. 14. At lower pressure lower rejection is observed. As pressure increases more liquid permeates through the membrane having more solute to retain thereby constricting the pore opening and subsequently increasing rejection. Turbulence created by membrane rotation has a moderate influence on rejection, though not as high as pressure, but stirrer speed does not seem to have any appreciable effect on rejection.

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 and model prediction may be inherent in the fact that the formulated model is strictly one-dimensional considering variation of different system variables along axial direction only. Though not very signif-

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 ).

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Nomenclature

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

icant but due to high-speed rotational field there must be certain radial variation of the same, which were not included because of mathematical complexity. Secondly, the equation of velocity field (Eqs. (1) and (2)) was simplified in order to get the analytical expression of back transport flux, otherwise it was not possible to get the trend of proposed helix angle with respect to different operating parameters. Instead of these two simplifications on an average the deviation of predicted result from that of experiment was well within ±7%, which establishes the general usability of the model for any standard RDM module.

5. Conclusion An analytical model based on solution of PDE arising from fundamental mass balance incorporating the expression of back transport flux for rotating disk membrane is proposed in this study. Unsteady state membrane surface and permeate concentration are also evaluated through Flory’s equation and related to permeate flux via irreversible thermodynamics, whereas the permeate flux is related to concentration field by balance equation developed at the membrane surface. Finally an iterative scheme is developed to simulate permeate flux and rejection under any operating condition. The model prerequisite is accurate estimation of four system parameters, i.e., Rm , ˛, Pm and , in this point of view the model can be termed as a four-parameter model of RDM. The proposed model is validated with experimental data for PEG-6000 in water treated with cellulose acetate membrane in a standard RDM cell. Low value of deviation (within ±7%) both for permeate flux and rejection establishes that model could be used for accurate simulation of permeate flux and rejection for any system subjected to ultrafiltration in a standard RDM module.

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

a,b at a A c cm cp c0 c c* C1 , C2 D F J JBT Mmono Mpol n N P PEG Pm r R Rm R s t t* T vx v V V1 x xt x*

constants used in Eq. (28) tangential acceleration (ms−2 ) area element defined in Eq. (4a) parameter defined in Eq. (11) solute concentration (kg m−3 ) solute concentration at membrane surface (kg m−3 ) solute concentration in the permeate (kg m−3 ) bulk solute concentration (kg m−3 ) solute concentration (g cm−3 ) dimensionless concentration = c/c0 constants used in Eq. (13) solute diffusivity (m2 s−1 ) parameter defined in Eq. (27) volumetric permeate flux (m3 m−2 s−1 ) back transport flux (m3 m−2 s−1 ) molecular weight of repeat unit (kg kmol−1 ) molecular weight of polymer (kg kmol−1 ) number of helix turns required to stirrer surface from any intermediate point, used in Eq. (8) total number of helix turns used in Eq. (19) hydraulic pressure (Pa) polyethylene glycol solute permeability (ms−1 ) radial coordinate (m) gas constant membrane hydraulic resistance (m−1 ) radius of stirrer and membrane (m) displacement along helix path (m) time (s) dimensionless time = Dt/xt2 absolute temperature in Eq. (25) axial velocity (ms−1 ) tangential velocity (ms−1 ) volume element used in Eq. (4a) specific volume of the solvent axial distance from membrane surface (m) distance of stirrer from membrane surface (m) 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 parameter ω1 , ω2 angular velocity of membrane and stirrer, respectively

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