1-s2.0-S0376738806000044-main

Journal of Membrane Science 279 (2006) 466–478

Modeling of a radial flow hollow fiber module and estimationof model parameters for aqueous multi-component mixture

using numerical techniques

S. Senthilmurugan 1, Sharad K. Gupta ∗Chemical Engineering Department, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India

Received 14 April 2005; received in revised form 17 December 2005; accepted 25 December 2005Available online 7 February 2006

Abstract

A mathematical model is developed for the separation of two solutes from aqueous solutions in a hollow fibre module. The model combines theextended Spiegler–Kedem model with the film theory model and takes into account the pressure drops in both feed side and permeate side streams.This model is then used for the simulation of the separation of phenol and NaCl in an aqueous solution at different operating conditions and thertehbc©

Kfi

1

wttbidnbmu

(

cT

0d

ole of the solute–solute interaction parameter appearing in the extended Spiegler–Kedem model is further investigated. To validate this model,he separation data for a ternary NaCl–KBr aqueous solution are obtained in a hollow fiber reverse osmosis (HFRO) module. Using part of thexperimental results for the binary and ternary systems separation, the six unknown membrane parameters of the model, namely the pure waterydrodynamic permeability, the two solute permeabilties, two reflection coefficients and the solute–solute interaction parameter were determinedy using an optimization technique—the simplex search. These values are then used to predict and compare the performances at other operatingonditions. The theoretically predicted and the experimental values are found to be in excellent agreement.

2006 Elsevier B.V. All rights reserved.

eywords: Extended combined film Spiegler–Kedem model; Estimation of solute–solute interaction; Reverse osmosis; Separation of inorganic mixture; Hollowber module

. Introduction

The reverse osmosis is one of the important techniques foraste water treatment, desalination and many industrial separa-

ion processes. In all these applications, the feed consists of morehan one solute. In the last four decades, there has been num-er of studies for understanding the transport processes involvedn the separation of multi-component liquid mixtures by usingifferent membrane separation processes like reverse osmosis,anofiltration and ultrafiltration. For effective use of these mem-rane processes, there is a need to develop better mathematicalodels for design and analysis of various membrane mod-

les. This requires better understanding of the multi-component

∗ Corresponding author. Tel.: +91 11 26591023; fax: +91 11 26581120.E-mail addresses: [email protected], [email protected]

S. Senthilmurugan), [email protected] (S.K. Gupta).1 Present address: Reprocessing Research and Development Division, Repro-

essing Group, IGCAR, Kalpakkam, Tamilnadu 603 102, India.el.: +91 44 274 80126.

mass transport in the membrane phase. Several research work-ers [1–33] have proposed different membrane transport models.However, most of these models have been tested experimentallyin a membrane test cell made of a small piece of a flat sheet mem-brane and only few studies are available where the experimentaldata is actually obtained in a membrane module like hollow fiberor spiral-wound module. For determining the membrane param-eters, the concentration polarization also plays an important role.Some authors have included the concentration polarization forestimating these parameters [2,4,6,10–14,19,24,25,29,32] whileothers have neglected it. First, some of the membrane transportmodels developed in these studies is reviewed. Later, the math-ematical models for various membrane modules for separationof multi-component systems are also reviewed.

Broadly, the membrane transport models available in litera-ture for membrane processes like reverse osmosis, nanofiltrationand ultrafiltration can be divided into two categories: (1) modelsfor neutral membranes and (2) models for charged membranes.The mathematical models like extended preferential sorption-capillary flow model [2,4,6–9], extended solution diffusion

376-7388/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
oi:10.1016/j.memsci.2005.12.041

S. Senthilmurugan, S.K. Gupta / Journal of Membrane Science 279 (2006) 466–478 467

model [13], extended Kedem–Katchalsky model [7,15,19,28],extended Spiegler–Kedem model [8,28,29] and Langmuir-typemodel [28] have been used for neutral membranes or wherethe membrane charge effects may be neglected. In case ofcharged membranes, the extended Nernst–Planck equation[9,16–18,22,24,25,27–29,31,33], electrostatic and steric hin-drance model [23,26] have been tried. Since, the module used inpresent experimental work is B9 HFRO module where the hol-low fibers are made of neutral aromatic polyamide membranes,only the models for neutral membranes and the correspondingexperimental work are discussed in detail.

The preferential sorption-capillary flow model was extended[2,4,6,10,11] for analyzing the performance of multi-componentsystems of mixed electrolytes using cellulose acetate mem-branes. The concentration polarization was determined by usingthe film theory. The mass transfer coefficient of each ion wastaken as the average of the mass transfer coefficient of variousions present in the concentration polarization layer. There is aninconsistency in these works as the number of independent equa-tions in model is less than the number of unknowns. Hodgson[2] overcame this difficulty by assuming equal diffusivities whileRangarajan et al. [4,6,10,11] assumed the electro-neutrality con-dition for calculation of last ion rejection. Brusilovsky andHasson [14] tackled this problem by introducing a correctionfactor for the electrical gradient and with this the model predictsbetter than previously proposed models.

au[tteot[getTfTlc

tpctcfiottctN

the solute permeability were estimated with respect to the con-centration of the second solute.

The separation of a ternary system (NaCl–acetic acid–water)in B9 HFRO module was attempted by Sapienza et al. [15] andthe interaction between the inorganic salt and a weak organicacid were investigated. The acetic acid rejection was enhancedby adding NaCl, but the NaCl rejection was decreased by addingacetic acid. The variations of solute permeability with respectto feed concentration in ternary system were explained by theLangmuir sorption model. However, the pressure, concentrationand velocity variations in both axial and radial directions of themodule were not included and the concentration polarizationwas also neglected.

Soltanieh and Sahebdelfar [28] confirmed that the multi-component effects can be used to improve membrane sep-aration properties. They used ternary systems of aceticacid–NaCl–water, oxalic acid–NaCl–water in a membrane cellto verify their conclusions. The salt rejection decreased byadding acid whereas the acid rejection increased by adding thesalt. Finally, they concluded that the model based on irreversiblethermodynamics (extended S–K, K–K model) as well as solu-tion diffusion model are adequate for limiting case of small flux,low concentrations and high rejections.

For developing the mathematical model for membrane mod-ules, the membrane transport models discussed above may becombined with the concentration polarization model and thenttiadseptvwt

meoiuffiCscbeitndop

The separation of NaCl, Na2SO4, MgSO4 and MgCl2 fromqueous solution was carried out in both spiral wound mod-le and flat sheet membrane test cell by Marinas and Selleck19]. The extended Kedem–Katchalsky model was used forhe prediction of multi-component fluxes and rejections fromhe single component membrane transport parameters. How-ver, the variation of pressure, concentration and the velocitiesf feed and permeate in the module were neglected. Further,he extended Kedem–Katchalsky model has also been applied7,15,19,28] for understanding the separation of organic, inor-anic and both mixed solutions. Clifton and Fowler [7] used thexperimental data on NaNO3, Na2SO4, urea and glucose mix-ures to investigate the coupling between the various solutes.hey found that the negative coupling coefficient was present

or some systems but the overall influence was rather small.he concentration polarization correction also had relatively

ittle effect on the final values of the solute–solute couplingoefficients.

Vonk and Smit [8] extended the Spiegler–Kedem modelo ternary system by introducing the solute–solute interactionarameter for ternary system. They observed no significanthange in the calcium chloride rejection by adding NaCl. Onhe other hand, the NaCl rejection decreased by adding calciumhloride as the second component. Further, this model was veri-ed by Soltanieh and Sahebdelfar [28] using the separation dataf acetic acid–NaCl–water and oxalic acid–NaCl–water sys-ems. Recently, Ballet et al. [33] has estimated the membraneransport parameters of extended SK model by assuming theoupling between the solute is negligible. The permeate charac-eristics of cadmium ion was analyzed by adding NaCl, Na2SO4,aNO3 and the parameters, such as the reflection coefficient and

he local solute and solvent equations may be integrated overhe whole module. Palanki and Gupta [13] developed an analyt-cal solution for multi-component separation in tubular, spiralnd plate-and-frame reverse osmosis modules. The analyticalesign equations predict membrane area and concentrations ofolutes in the permeate stream for a given solvent recovery. How-ver, the variations in the pressure and velocities in both feed andermeate side of the streams were neglected. Also, the interac-ions between the solutes were also not taken into account and aery simple solution–diffusion type membrane transport modelas used for each solute for describing the mass transport across

he membrane.From the literature [1–33], it is clear that the separation of

ulti-component mixtures in any membrane module has beenxperimentally studied only by Sapienza et al. [15] who workedn a radial flow hollow fiber module and that a very little works available in literature on design and analysis of these mod-les. In the present work, a mathematical model is developedor the separation of a ternary mixture in a radial flow hollowber module (HFRO). The basis for this model is the work ofhatterjee et al. [34] on hollow fiber module for a single solute

ystem. It is assumed that the transport phenomena for the multi-omponent system in the membrane phase may be describedy the extended Spiegler–Kedem model [8]. To understand theffect of various operating conditions and the role of the solutenteraction parameter in the extended Spiegler–Kedem model,he simulation studies are carried out for the separation of phe-ol and NaCl in an aqueous solution. In addition, the separationata for inorganic ternary system of NaCl and KBr were alsobtained in a radial flow HFRO module. The binary transportarameters, such as the pure water hydrodynamic permeability,

468 S. Senthilmurugan, S.K. Gupta / Journal of Membrane Science 279 (2006) 466–478

solute permeabilities and the reflection coefficients were esti-mated from the separation data of the binary systems. Finally,the solute interaction parameter in the extended S–K model isdetermined by using part of the separation data for the ternarysystem by using error minimization-simplex method and thanthese transport parameters were used for prediction and com-parison of the remaining separation data.

2. Theory

2.1. Modeling

A mathematical model is developed for the separation of aternary mixture in a radial flow hollow fiber module. The basisfor this model is the work of Chatterjee et al. [34] on hollowfiber module for a single solute system. Here, it is assumedthat the transport phenomena for the multi-component sys-tem in the membrane phase may be described by the extendedSpiegler–Kedem model [8] and that the concentration polariza-tion for each solute may be calculated by using the film theory.Fig. 1 shows both how the bulk (feed to reject) and permeatestreams flow through a DuPont B9 module and how a grid can becreated for the equations using the method of finite differences.The radial direction (r-axis) and the axial direction (z-axis) aredivided into ‘m’ and ‘n’ segments, respectively. Here, the axialdfl

m

•

•

•

• The permeator is in operation for sufficient time and steadystate has been achieved.

• Solution contains two solutes (ternary aqueous solution).• Film theory is applicable within the membrane module and

there is no solute–solute interaction in the film.• Fluid properties and diffusivities remain constant inside the

module.• Perfect mixing exists in fluids within the finite element strip.

2.2. Membrane transport equations

The extended Spiegler–Kedem model [8] is used to describethe mass transfer inside the membrane. The model equations forpermeate flux and solute rejections are given below.

2.2.1. The permeate flux equationJv = A(�P − σ[1]�π[1] − σ[2]�π[2]) (1)

where π = νRT

MwC (2)

and �P = Pb − Pp (3)

2.2.2. Equation for rejection by the membraneThe true rejection of solute “i” at a particular point inside the

membrane is given by

R

w

σ

w

pspot

tion f

irection refers to the axis of the hollow fiber along the permeateow direction.

The following assumptions have been made in the develop-ent of our analysis.

The bulk stream flows radially outward and there is sufficientaxial mixing so that the bulk flow variables are only dependenton r. This allows the partial derivative terms, which appear inthe material balance and the pressure drop equations to bereplaced by ordinary derivatives.The element chosen for finite difference analysis within thepermeator is much larger than the fiber dimensions. Hence,for all practical purposes the shell side of the membrane canbe assumed to be a continuous phase.Membrane structure is uniform throughout the module.

Fig. 1. Finite difference mesh construc

[i] = σ[i,j](1 − F[i,j])

1 − F[i,j]σ[i,j](4)

here F[i,j] = exp

(−Jv(1 − σ[i,j])

Pm[i]

)(5)

[i,j] = σ[i] − B[i,j]C[j]mR[j]

Jv(6)

here i, j = 1 or 2, but i �= j.The pure water hydrodynamic permeability (A), the solute

ermeabilities (Pm[i]), the reflection coefficients (σ[i]) andolute–solute interaction parameter (B[i,j] = B[j,i]) are the sixarameters of the extended Spiegler–Kedem model. The valuef B[i,j] may be either positive or negative [9,28] depending onhe nature of the coupling between the flows of solutes. If it is

or a DuPont B9 Hollow fiber module.


positive, then the more permeable component pulls the othercomponents with it through the membrane and as a result theseparation decreases. If it is negative, the opposite behaviourmay occur and the separation may increase.

2.2.3. Concentration polarization modelAccumulation of the solute on the membrane surface leads

to the development of a concentration polarization layer. Thisphenomenon may be mathematically described using the filmtheory [37] as:

φ[i] = C[i]m − C[i]p

C[i]b − C[i]p= exp

(Jv

k[i]

)(7)

The mass transfer coefficient used in Eq. (7) may be expressedas a function of the Reynolds and Schmidt numbers by the fol-lowing equation.

Sh = aReb Sc1/3 (8)

Equations of the similar form have been used in literature [34,35]for estimating the mass transfer coefficients and the values ofa = 0.048 and b = 0.6 are reported for B9 radial flow HFRO mod-ule. By combining Eqs. (4) and (7), we obtain

C[i]p = C[i]b

1 + ((1 − F[i,j])σ[i,j]/(1 − σ[i,j])φ[i])(9)

2

rttsHse

H

E

Hfid

2

ta

2

2

Hence,dvp

dz= Jv

(ζ

θ

)

such that vp∣∣z=0 = 0,

Di

2< r <

Do

2(12)

Here, θ = d2i N/D2

o − D2i , ζ = 4θdo/d

2i · L/Ls and the

length of a hollow fiber is given as,

L =√

L2s + 4(πrW)2

2.2.5.2. Solute concentration in the bulk stream.

2πrLvr|r+�r − 2πrLvr|r = −2πr�rθvp

Hence,d(rvr)

dr= −θ

rvp

L

∣∣∣∣z=L

(13)

subject to vr|r=Di/2 = vF, for 0 < z < L

Likewise for the solute,d(rvrC[i]b)

dr= −θ

rvpC[i]p

L

∣∣∣∣z=L

(14)

subject to C[i]b∣∣r=Di/2 = C[i]F, for 0 < z < L

Differentiation of Eq. (10) and subsequent substitution intoEq. (12) leads to:

daaarea

lakeiida

�

d

R

.2.4. Equations for the pressure dropThe pressure difference across the membrane which is

equired in Eq. (1) for obtaining the permeate flux varieshroughout the membrane because of the friction losses. Forhe inside hollow fiber (subscript p) and bulk side (sub-cript b) streams, the pressure drops may be estimated usingagen–Poiseuille and the Ergun’s equations, respectively. The

ame equations were also used by Chatterjee et al. [34]. Thesequations are given below

agen–Poiseuille equation :dPp

dz= −32µ

d2i

vp (10)

rgun equation :dPb

dr

= −(

150(1 − ε)2µvr

ε3d2p

+ 1.75(1 − ε)ρv2r

ε3dP

)(11)

ere, dp = 6/av = 1.5do is the mean diameter and vr is the super-cial velocity within the module on the bulk side along the radialirection.

.2.5. Material balanceFollowing Chatterjee et al. [34], the material balance equa-

ions for both solute and solvent streams within the membranere given below.

.2.5.1. Permeate stream.

πr�rθ vp∣∣z=z+�z

− 2πr�rθvp∣∣z=z

= 2πr�r�zζ Jv|z=z

d2Pp

dz2 = −32µ

d2i

ζ

θJv

⎧⎪⎪⎨⎪⎪⎩

dPp

dz

∣∣∣∣z=0

= 0

Pp = Patm − ls32µ

d2i

vp∣∣z=L

(15)

The above Eqs. (1)–(15) are solved numerically by the finiteifference method with each of the variables being expresseds a discrete value. Since the permeate flow variables vary onlylong the z-axis while the bulk flow terms vary along the r-xis, the equations are solved sequentially by proceeding from= Di/2 to Do/2 while solving all the z-axis dependent differencequations at a particular radial grid location. The bulk flow termst r = Di/2 are known: Pb = PF, Cb[i] = CF[i] and vr = vF.

The algorithm used to simulate the performance of a hol-ow fiber module, when the Spiegler–Kedem model parametersnd the constants appearing in the mass transfer correlation arenown, is described in steps as shown in Appendix A. Thexperimental and theoretical predictions of permeate character-stics for both single and multi-component mixture are comparedn terms of recovery and observed rejection. The recovery isefined as the percentage ratio between the permeate flow ratend feed flow rate. Mathematically, it may be written as

R = Qp

QF× 100 (16)

On the other hand, the observed rejection of solute “i” isefined as

[i]obs =(

1 − C[i]p

C[i]F

)× 100 (17)


2.3. Numerical method for parameter estimation

An initial guesses are made for the model parameters, whichare required to be estimated. The predicted exit permeate flowrates and concentrations obtained based on the initial guesses arecompared with experimental values. In case the values chosenfor the parameters were not correct, the values are refined sothat the predicted and experimental permeate conditions are asclose as possible. Thus, evaluate the error for all experimentaldata points as given below.

Error =h∑

i=1

(1 −

Qpesti

Qpexpi

)2

+(

1 −C[1]pesti

C[1]pexpi

)2

+(

1 −C[2]pesti

C[2]pexpi

)2

(18)

Finally, minimize the error by adjusting the parameter valueswith the help of an optimization technique, such as the simplexsearch method [36], till the error is below a tolerance limit.

3. Results and discussion

3.1. Model analysis

appacwfea

Senthilmurugan and Gupta [35] and are also given in Table 1.The solute–solute interaction parameter, B[1,2], for this ternarysystem is varied from 4 × 10−11 to 8 × 10−11 m4 kg−1 s−1.

Figs. 2–5 explains the permeate characteristics forNaCl–phenol–water system with respect to feed pressure, feedflow rate and feed concentration. Fig. 2 shows the behaviour ofthe recovery and observed rejections of both phenol and NaClwith respect to B[1,2] for different feed pressures. As the feedpressure increases, the observed rejection of NaCl decreases,while the observed rejection of phenol increases and at thesame times the recovery increases. Higher feed pressure leads tohigher permeate fluxes and at the same time higher concentra-tions of both solutes at the feed membrane interface. Therefore,the recovery increases while the observed rejection of NaCldecreases. However, for phenol, the less permeable component,the effect of increasing the pressure is just the opposite of whatis observed for NaCl. As can be seen from Eq. (6), the effectivereflection coefficient for phenol depends on the NaCl concentra-tion, its observed rejection by the membrane and as well as on thepermeate flux. Since the observed rejection for NaCl decreasesand the permeate flux increases as the feed pressure is increased,

Table 1The parameters used in simulation of NaCl–phenol–water system

V

MMMMMBFFF

PF at C[NaCl]F = 4000 ppm, C[phenol]F = 4000 ppm and QF = 80 ml s−1.

To understand the significance of the solute-solute inter-ction parameter in the extended Spiegler–Kedem model, thehenol–NaCl–water system was chosen for the simulation pur-ose. The model behaviour of ternary system for different oper-ting conditions, such as feed pressure, feed flow rate and feedoncentration is studied by keeping two of them constant andhile varying the third. The range of the operating variables used

or the simulations is shown in Table 1. The membrane param-ters for the binary system, such as the reflection coefficientnd the solute permeability for phenol and NaCl are taken from

Fig. 2. The effect of B[1,2] on R[NaCl] and R[phenol] with respect to

ariable name Value

embrane A-valuea (kg m−2 s−1 Pa−1) 9.4015 × 10−10

embrane Pm-valuea (NaCl) (m s−1) 1.4115 × 10−8

embrane σ-valuea (NaCl) 0.901embrane Pm-valuea (phenol) (m s−1) 45.7381 × 10−8

embrane σ-valuea (phenol) 0.430

[1,2] (m4 kg−1 s−1) 4–8 × 10−11

eed pressure (PF) (Pa (bar)) 4 × 105–13 × 105

eed flow rate (QF) (ml s−1) 50–140eed concentration (CF) (ppm) 500–1500

a Fixed parameter in simulation.


Fig. 3. The effect of B[1,2] on R[NaCl] and R[phenol] with respect to QF at C[NaCl]F = 4000 ppm, C[phenol]F = 4000 ppm and QF = 80 ml s−1.

the effective reflection coefficient for the phenol increases, thusleading to higher observed rejections for phenol. The effect ofsolute–solute interaction parameter on phenol rejection is quiteobvious: higher the value of the interaction parameter, loweris the effective reflection coefficient and thus lower observedrejections. In fact at higher values of the interaction parame-ter, the phenol shows negative observed rejections where theconcentration of phenol in the permeate is more than its corre-sponding value in the feed phase. This is due to increased drag onthe weakly rejected phenol by strongly rejected NaCl as can beseen from Eq. (6). On the other hand, the observed rejections forNaCl show a very interesting behaviour. The observed rejectioninitially decreases as the interaction increases, passes through a

minimum and then the observed rejection increase as the valueof the interaction parameter is further increased. This may againbe explained by examining Eq. (6). Initially, when the value ofthe interaction parameter is increased, the effective reflectioncoefficient for NaCl also decreases. However, for higher inter-actions, the phenol starts showing negative observed rejectionsand therefore, as can be seen from Eq. (6), the effective reflec-tion coefficient for NaCl starts increasing and thus the observedrejections for NaCl also increase.

The effect of feed flow rates and the interaction parameter onthe recovery and observed rejections of both phenol and NaClare shown in Fig. 3. Since higher feed flow rate increases themass transfer coefficient, the concentrations of both solutes at

F o feed
ig. 4. The permeate characteristics of phenol–NaCl–water system with respect t phenol concentration at C[NaCl]F = 4000 ppm and PF = 13.8 × 105 Pa (13.8 bar).


Fig. 5. The permeate characteristics of phenol–NaCl–water system with respect to feed NaCl concentration at C[phenol]F = 4000 ppm and PF = 13.8 × 105 Pa (13.8 bar).

the feed membrane interface are lower and as a result the recov-ery and the observed rejection of NaCl are higher at higher feedflow rate. On the other hand, for weakly rejected solute, the phe-nol, the effect of increasing flow rates is just the opposite. Athigher feed flow rates the concentration of NaCl in the mem-brane phase decreases and therefore, as can be seen from Eq.(6), the effective reflection coefficient for phenol increases andthus leading to higher observed rejections of phenol at higherfeed flow rates. The effect of solute interaction parameter onrecovery and observed rejections is similar to what is observedin Fig. 2.

Figs. 4 and 5 show the permeate characteristics of the aboveternary system with respect to the concentration of the second

solute. In both the figures, the concentration of the first solute isfixed at 4000 ppm while the concentration of the second soluteis changed from 0 to 4000 ppm. When the concentration of thesecond solute is varied, other variables like effective osmoticpressure, effective reflection coefficients and observed rejec-tions of two solutes and recovery change in a complex manner.Fig. 4 shows that the observed rejection of phenol decreases,while the rejection of NaCl first decreases, reaches a minimumand then further starts increasing as the phenol concentration isincreased. Furthermore, the negative observed rejection of phe-nol occurs for high phenol concentration and low feed flow rates.Fig. 5 shows that the observed rejection of phenol decreaseswhile the observed rejection of NaCl increases as the feed NaCl

Cl] an
Fig. 6. The variation of R[Na d R[phenol] in axial direction.


concentration is increased. As far as the recoveries are con-cerned, the recoveries for the case where the NaCl concentrationis fixed and the phenol concentrations are varied are higher whencompared with the case where the phenol concentration is fixedand the NaCl concentrations are varied. This is because theosmotic pressure as well as the reflection coefficient for purephenol is lower than those for NaCl. In fact, the recovery goesthrough a maximum.

To understand the negative observed rejections, the local vol-ume flux (Jv) and true rejections of both NaCl and phenol are alsoinvestigated along the radial and axial directions of the moduleas shown in Fig. 6. Here, the true rejections are calculated basedon local concentrations in the permeate and the feed–membraneinterface. The flux Jv and the true rejection of both phenol andNaCl are nearly constant along the axial direction. On the otherhand, the flux (Jv) decreases while the true rejection of NaClincreases along the radial direction. The true rejection of phenolshows interesting behaviour. It decreases along the radial direc-tion and changes from positive to negative for larger values of

r, but the overall observed rejection for the module still remainspositive. Thus, even when the overall observed rejection is posi-tive, there may be negative rejections at some local points in themodule.

3.2. Experimental results

To confirm the validity of the present model for multi-component systems, the experimental data were obtained fora ternary system of NaCl and KBr in a radial flow hollowfibre module. The experimental set up used is same as reportedin earlier work [35]. The distilled water with the conductivity0.1 �S/cm has been used for the solution preparation. The exper-imental procedure consists of systematic variation of the feedpressure, feed flow rate and feed concentration at constant tem-perature (30.5 ◦C) and pH (6.0 ± 0.3). The feed pressure, feedflow rate and feed concentration were changed from 4 × 105 to13 × 105 Pa (4 to 13 bar), 50 to 100 ml s−1 and 500 to 1500 ppm,respectively. The separation data were collected by keeping

Table 2Data used for parameter estimation of NaCl–water system

PF (×105 Pa (bar)) QF (ml s−1) C[NaCl]F (ppm) Experimental value Predicted values Error

Qp (ml s−1) C[NaCl]p (ppm) Qp (ml s−1) C[NaCl]p (ppm) Qp (%) C[NaCl]p (%)

8.3 70 495 15.5 71 16.5 75 −6.4 −6.189

1 131 41

45547409

1 491 91

26409048

1 201 03

TD

P ue

C[KB

1

1

1

9.7 70 495 18.51 70 495 21.5 12.4 70 495 24.0 15.5 60 961 8.8 16.9 60 961 11.6 18.3 60 961 14.5 19.7 60 961 17.5 21 60 961 20.0 22.4 60 961 22.5 25.5 50 1396 8.0 36.9 50 1396 11.0 38.3 50 1396 13.5 39.7 50 1396 16.1 41 50 1396 18.4 52.4 50 1396 20.9 6

able 3ata used for parameter estimation of KBr–water system

F (×105 Pa (bar)) QF (ml s−1) C[KBr]F (ppm) Experimental val

Qp (ml s−1)

6.9 70 535 13.4 568.3 70 535 16.5 679.7 70 535 19.5 781 70 535 22.8 1035.5 60 985 9.8 986.9 60 985 13.0 1188.3 60 985 16.1 1539.7 60 985 19.0 1971 60 985 22.1 2485.5 50 1450 10.0 1796.9 50 1450 13.0 2148.3 50 1450 16.3 2789.7 50 1450 19.0 3361 50 1450 22.0 418

19.6 89 −6.1 0.322.7 105 −5.5 6.525.6 126 −6.8 11.1

8.8 141 0.4 5.511.7 156 −1.1 6.914.6 179 −0.5 8.317.3 209 1.4 9.719.8 245 1.0 1122.2 286 1.4 12.4

7.6 323 4.4 1.110.2 349 7.2 −2.712.6 393 6.6 −0.814.8 449 7.8 −0.316.9 512 8.0 1.518.9 581 9.5 3.5

Predicted values Error

r]p (ppm) Qp (ml s−1) C[KBr]p (ppm) Qp (%) C[KBr]p (%)

13.9 60 −3.9 −6.217.2 69 −4.5 −2.320.5 82 −5.3 −5.423.7 99 −4.2 4.1

9.9 107 −0.9 −9.413.1 128 −0.4 −8.116.1 155 −0.2 −1.419.1 190 −0.4 3.621.9 234 1.0 5.5

8.9 181 10.9 −1.011.7 226 9.9 −5.514.3 282 12.2 −1.616.7 347 12.0 −3.218.9 420 14.0 −0.4

474S.Senthilm

urugan,S.K.G

upta/JournalofM

embrane

Science279

(2006)466–478

Table 4Separation data of NaCl–KBr–water system at C[NaCl]F = 507 ppm and C[KBr]F = 507 ppm used for parameter estimation

PF

(×105 Pa(bar))

QF (ml s−1) Experimental permeatecharacteristics

Theoretical permeatecharacteristics withsolute–solute interaction

Theoretical permeatecharacteristics withoutsolute–solute interaction

Error1a Error2b

Qp

(ml s−1)C[NaCl](ppm)

C[KBr](ppm)

Qp


C[KBr](ppm)

Qp


C[KBr](ppm)

Qp (%) C[NaCl](%)

C[KBr](%)

Qp

(%)C[NaCl](%)

C[KBr](%)

8.3 70 14.2 100 76 15.7 99.4 82.4 15.6 72.9 61.4 −10.56 0.23 −8.42 −9.86 27.10 19.219.7 70 17 117 96 18.7 114.0 93.0 18.6 83.6 69.4 −10.00 2.82 3.23 −9.41 28.55 27.71

11 70 20 143 114 21.5 132.0 106.5 21.4 96.1 78.7 −7.50 8.00 6.25 −7.00 32.80 30.9612.4 70 22.8 173 136 24.2 152.4 122.2 24.0 109.8 89.1 −6.14 12.00 10.15 −5.26 36.53 34.49

6.9 60 11.5 106 83 12.2 110.1 91.3 12.1 74.8 62.7 −6.09 −4.08 −10.40 −5.22 29.43 24.468.3 60 14.5 127 99 15.0 129.6 105.6 14.9 88.2 72.7 −3.45 −2.24 −6.45 −2.76 30.55 26.579.7 60 17 152 123 17.7 152.8 123.3 17.5 103.4 84.1 −4.12 −0.51 0.00 −2.94 31.97 31.63

11 60 20 181 148 20.1 177.2 143.0 19.9 119.6 96.3 −0.50 2.27 3.25 0.50 33.92 34.9312.4 60 22.5 214 169 22.3 200.9 163.2 22.0 135.7 108.7 0.89 6.27 3.15 2.22 36.59 35.68

5.5 50 8.5 119 101 8.7 123.7 103.0 8.6 75.6 62.9 −2.35 −4.14 −2.01 −1.18 36.47 37.726.9 50 11.8 146 114 11.4 149.4 122.0 11.2 92.2 75.5 3.39 −2.64 −6.97 5.08 36.85 33.778.3 50 14.5 179 133 13.8 178.3 144.8 13.5 110.4 89.2 4.83 0.56 −9.04 6.90 38.32 32.939.7 50 17 215 161 15.9 206.1 168.4 15.6 128.9 103.2 6.47 3.97 −4.60 8.24 40.05 35.90

11 50 20 251 192 17.8 231.0 190.9 17.4 146.8 117.0 11.00 7.96 0.52 13.00 41.51 39.0612.4 50 22.5 280 223 19.5 252.3 211.2 19.0 163.4 130.1 13.33 9.86 5.12 15.56 41.64 41.66

a Error1 = (1 − experimental permeate characteristics/theoretical permeate characteristics with solute–solute interaction) × 100.b Error2 = (1 − experimental permeate characteristics/theoretical permeate characteristics without solute–solute interaction) × 100.

S.Senthilmurugan,S.K

.Gupta

/JournalofMem

braneScience

279(2006)

466–478475

Table 5Separation data of NaCl–KBr–water system at C[NaCl]F = 541 ppm and C[KBr]F = 902 ppm used for predictions

PF (×105 Pa(bar))

QF

(ml s−1)Experimental permeatecharacteristics

Theoretical permeatecharacteristics withsolute–solute interaction

Theoretical permeatecharacteristics withoutsolute–solute interaction

Error1a Error2b

Qp


C[KBr](ppm)

Qp


C[KBr](ppm)

Qp


C[KBr](ppm)

Qp


C[KBr](ppm)

Qp


C[KBr](ppm)

8.3 70 14 106 169 15.1 104.6 170.9 15.0 75.1 105.9 −7.86 1.39 −1.34 −7.14 29.15 37.349.7 70 17 121 195 18.1 118.5 189.2 17.9 85.4 118.8 −6.47 2.42 2.92 −5.29 29.42 39.08

11 70 20 143 223 20.9 135.7 212.9 20.7 97.2 133.5 −4.50 5.08 4.59 −3.50 32.03 40.1312.4 70 22.5 171 260 23.5 155.2 240.7 23.3 110.2 149.8 −4.44 9.22 7.57 −3.56 35.56 42.38

6.9 60 11.5 111 174 11.7 114.8 193.3 11.5 76.4 107.2 −1.74 −3.41 −11.27 0.00 31.17 38.398.3 60 14.5 132 210 14.4 133.1 217.2 14.2 89.0 123.1 0.69 −0.84 −3.67 2.07 32.58 41.389.7 60 17 159 243 17.0 155.1 247.7 16.8 103.3 140.9 0.00 2.57 −1.74 1.18 35.03 42.02

11 60 20 187 289 19.4 178.8 281.6 19.1 118.5 160.0 3.00 4.32 2.53 4.50 36.63 44.6412.4 60 22.5 206 335 21.6 202.3 316.0 21.2 133.9 179.5 4.00 1.72 5.73 5.78 35.00 46.42

5.5 50 9 136 193 8.3 127.8 224.2 8.0 76.4 106.6 7.78 6.16 −16.04 11.11 43.82 44.776.9 50 12 151 223 10.8 151.6 254.2 10.6 91.8 126.2 10.00 −0.27 −13.74 11.67 39.21 43.418.3 50 14.8 178 270 13.2 179.1 291.9 12.9 108.8 147.5 10.81 −0.40 −8.25 12.84 38.88 45.379.7 50 17.5 213 321 15.3 206.7 331.1 14.9 126.3 169.5 12.57 2.89 −3.12 14.86 40.70 47.20

11 50 20 242 375 17.2 232.1 368.1 16.7 143.5 191.3 14.00 3.94 1.82 16.50 40.70 48.9912.4 50 22.5 271 430 18.8 254.5 401.5 18.3 159.8 212.3 16.44 6.16 6.52 18.67 41.03 50.63

a Error1 = (1 − experimental permeate characteristics/theoretical permeate characteristics with solute–solute interaction) × 100.b Error2 = (1 − experimental permeate characteristics/theoretical permeate characteristics without solute–solute interaction) × 100.


concentration of one solute at fixed value while varying theconcentration of the second solute. The concentration of bothchloride and bromide were estimated by ion-chromatography(792 Basic IC Metrohm) with the anion column (Metrosep ASupp 5) and sodium carbonate (3.2 mM) and sodium bicarbon-ate (1 mM) as eluents.

Tables 2 and 3 show the separation data for NaCl and water,and KBr and water binary systems. This data was analyzed bySenthilmurugan and Gupta [35] and the solute permeabilities(Pm[1] and Pm[2]) and reflection coefficients (σ[1] and σ[2]) ofboth solutes were calculated by their parameter estimation algo-rithm. The solute permeability and the reflection coefficient forNaCl were found to be a linear function of the feed concentrationas given below.

Pm[NaCl] = [0.0021 × C[NaCl]F − 0.6136]× 10−8 m s−1 (19)

σ[NaCl] = 3 × 10−5 × C[NaCl]F + 0.9299 (20)

In the case of KBr, membrane transport parameters werereported to be constant for low feed concentration. The val-ues of solute permeability and reflection coefficient of KBr are0.0535 × 10−8 m s−1 and 0.9271, respectively. The membraneparameters given above were used for prediction of the ternarys

tamσ

TfbcatbfTe

castwTptNdaflat

4. Conclusion

A mathematical model is developed for the separation of twosolutes aqueous solutions in a radial hollow fibre module. Themodel combines the extended Spiegler–Kedem model with thefilm theory model and takes into account the pressure drops inboth feed side and permeate side streams. This model is thenused for the simulation of the separation of phenol and NaClin an aqueous solution at different operating conditions and therole of the solute–solute interaction parameter appearing in theextended Spiegler–Kedem model is further investigated. It isfound that the solute–solute interaction may be very importantand it must be taken into account while considering the designand analysis of the reverse osmosis module. The separation datafor a ternary NaCl–KBr aqueous solution were also obtained ina hollow fiber reverse osmosis (HFRO) module to validate thismodel. By using part of the experimental results for the binaryand ternary systems separation, the six unknown membraneparameters namely the pure water hydrodynamic permeability,the two solute permeabilties, two reflection coefficients and thesolute–solute interaction parameter appearing in this model weredetermined by using an optimization technique—the simplexsearch. The estimated value of B[1,2] for the NaCl–KBr was esti-mated as 2.0886 × 10−11 m4 kg−1 s−1. These values were thenused to predict and compare the performances at other operatingconditions and the theoretically predicted and the experimen-trmflsuo

As

ystem of these solutes.Tables 4 and 5 show the separation data for the NaCl and KBr

ernary system for a fixed NaCl concentration of about 500 ppmnd varying KBr concentrations from 0 to 1000 ppm. The presentodel has five membrane parameters (A, Pm[1], Pm[2], σ[1] and[2]) and another solute–solute interaction parameter (B[1,2]).he pure water hydrodynamic permeability (A) was estimated

rom pure water permeability data. The remaining four mem-rane transport parameters, such as solute permeabilities of twoompounds (Pm[1] and Pm[2]) and reflection coefficients (σ[1]nd σ[2]) were estimated from binary separation data. Finally,he solute–solute interaction parameter (B[1,2]) was determinedy the parameter estimation technique outlined earlier. The dataor 500 ppm concentration of KBr was used for this purpose.he estimated value of B[1,2] for the NaCl–KBr systems wasstimated as 2.0886 × 10−11 m4 kg−1 s−1.

The predictions for the permeate flow rate and permeate con-entrations of both NaCl and KBr were made by using thebove estimated parameters and the results are shown in theame Table 2. As can be observed from this table the predic-ions and the experimental results are in excellent agreementith maximum errors less then 15% for all three predictions.o understand the importance of the solute–solute interactionarameter, the errors were also calculated for the case wherehe solute–solute interaction parameter was assumed to be zero.ow, the errors between the experimental results and the pre-iction are much more then 15% and were found to be as highs 50%. This shows that for the multi-component systems, theow coupling between various solutes may be very importantnd it must be taken into account in any mathematical model forhese types of systems.

al values were found to be in excellent agreement. From theseesults, it may be concluded that the present mathematical modelay be used with confidence for design and analysis of a radialow hollow fiber module for separation of two solutes aqueousystems. It is also suggested that the present approach may besed for other multi-component systems involving more numberf solutes.

ppendix A. Algorithm for finite difference method toolve model equations

Step 1: Obtain the extended Spiegler–Kedem model parame-ters, dimensions of the hollow fiber, the input feed flowrate, concentration and inlet feed pressure.

Step 2: Form the grid for the finite differences method.Step 3: Repeat steps (4–9) for “i” (along the radial direction)

going from 1 to m (the radial grid point r1 = Di/2,rm = Do/2).

Step 4: As an initial estimate for Jv, calculate Jvi,j for j = [1,n], by assuming the feed is pure water and fiber borepressure is in atmospheric. Calculate vp from Eq. (12)using the numerical integration.

Step 5: Solve for the system of linear equations for Ppi,j

by applying central difference method to Eq.(15).

Step 6: Estimate Jvi,j and R[i]i,j for j = [1, n] by solving non-linear equations from (1) to (9), and calculate vpi,n

asmentioned in step 4. Now, Pbi,j

, C[i]bi,jand vri,j can be

replaced by Pbi, C[i]bi

and vri , respectively, since theyonly vary radially.


Step 7: Repeat steps 5 and 6 till the values of Pp and Jvi,j

converge at all points at radial index “i”.Step 8: Obtain C[i]pi,j

from Eq. (12).Step 9: Solve vri+1 and C[i]bi+1 by using forward difference

representation of Eqs. (13) and (14), unless i + 1 > m.For solving C[i]bi+1 the value of C[i]p in Eq. (14) maybe calculated from the ratio between the total soluteflux and solution flux through the membrane along theaxial direction of the finite element “i”.

Step 10: Calculate the values Qp, C[i]p from the following equa-tions.

Qp = 2π�rθ

[0.5r1vp1,n

+m−1∑i=2

rivpi,n+ 0.5rmvpm,n

]

(A.1)

C[i]p = 2π�rθ

Qp

[0.5r1vp1,n

C[i]p1,n+

m−1∑i=2

rivpi,nC[i]pi,n

+ 0.5rmvpm,nC[i]pm,n

](A.2)

Qp Permeate flow rate (ml s−1)Qpexp experimental permeate flow rate (ml s−1)QpThe

theoretically predicted permeate flow rate(ml s−1)

r radial directionR true rejection of solute in Eq. (2), universal gas

constant in Eq. (1) (8314 J kmol−1 K−1)Robs observed rejection of solute in Eq. (19)Re Reynolds numberSc Schmidt numberSh Sherwood numberT temperature (K)Vr radial superficial velocity of bulk fluid (m s−1)vp velocity of permeate in axial direction (m s−1)W number of woundsz axial direction

Symbol∆ difference across the membraneπ osmotic pressure in Eq. (1) and value 3.14 in Eqs.

(12) and (13) (Pa)ε porosity of hollow fiber bundleµ viscosity (×10−3 Pa s (cP))ν Van’t Hoff factor

R

Nomenclature

a constant used in correlation for mass transfercoefficient

A pure water hydrodynamic permeability(kg m−2 s−1)

b constant used in correlation for mass transfercoefficient

B[1,2] solute–solute interaction parameter(m4 kg−1 s−1)

Cb concentration of solute at bulk/reject stream (kgsolute per m3 of solution)

C[i] concentration of solute i (kg solute per m3 of solu-tion)

dp equivalent diameter of hollow fiber (m)di inside diameter of hollow fiber (m)Di inside diameter of hollow fiber module (m)do outside diameter of hollow fiber (m)Do outside diameter of hollow fiber module (m)F[i,j] parameter used in extended Spiegler–Kedem

model defined by Eq. (5)ls thickness of epoxy seal (m)Jv permeate flux (m3 m−2 s−1)k mass transfer coefficient (m s−1)L length of hollow fiber involved in mass transfer

(m)Mw molecular weight (kg kmol−1)N number of hollow fibers in moduleP pressure (Pa)Pm[i] solute permeability of solute i (m s−1)QF feed flow rate (ml s−1)

σ[i] reflection coefficient of solute iσ[i,j] effective reflection coefficient of solute i and j

defined by Eq. (6)

Subscriptsatm atmosphereb bulk sideF feed[i] component ii axial coordinatej radial coordinatem membrane surfacep permeate side

AbbreviationsHFRO hollow fiber reverse osmosisKBr potassium bromideNaCl sodium chloride

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