modeling concentration polarization in reverse osmosis processes.pdf

Desalination 186 (2005) 111128

0011-9164/06/$ See front matter 2005 Elsevier B.V. All rights reserved.

*Corresponding author.

Modeling concentration polarization in reverse osmosisprocesses

Suhan Kim, Eric M.V. Hoek*Department of Civil and Environmental Engineering, 5732-G Boelter Hall, University of California,

Los Angeles, CA 90095-1593, USATel. +1 (310) 206-3735; Fax +1 (310) 206-2222; email: [email protected]

Received 11 April 2005; accepted 23 May 2005

Abstract

Accurate prediction of concentration polarization (CP) phenomena is critical for properly designing reverseosmosis (RO) processes because it enhances trans-membrane osmotic pressure and solute passage, as well assurface fouling and scaling phenomena. The objective of this study was to compare available analytical CP modelsto a more rigorous numerical CP model and experimental CP data. A numerical concentration polarization modelwas developed to enable local description of permeate flux and solute rejection in crossflow reverse osmosisseparations. Predictions of channel averaged water flux and salt rejection by the developed numerical model, theclassical film theory model, and a recently proposed analytical model were compared to well-controlled laboratoryscale experimental data. At operating conditions relevant to practical RO applications, film theory and the numericalmodel accurately predicted channel-averaged experimental permeate flux and salt rejection data, while the morerecent analytical model did not. Predictions of local concentration polarization, permeate flux, and solute rejectionby film theory and the numerical model also agreed well for realistic ranges of RO process operating conditions.

Keywords: Concentration polarization; Reverse osmosis; Film theory; Modeling; Membranes

1. Introduction

Concentration polarization (CP) is one of themost important factors influencing the perform-ance of membrane separation processes [1]. Pre-diction of solute concentration polarization iscrucial for designing reverse osmosis processes,

predicting their performance, and especially forunderstanding surface fouling phenomena [25].For example, accurate description of local varia-tions in permeate velocity, solute rejection, andconcentration polarization are required to predictthe onset of surface scale formation by sparingsoluble minerals [6]. However, some debate lin-gers among membrane scientists and engineers

doi:10.1016/j.desal.2005.05.017

112

as to the most appropriate means of predictingconcen-tration polarization in reverse osmosis andother semi-permeable membrane separations likenano-filtration and ultrafiltration [7].

Concentration polarization is governed bysolute properties, membrane properties, and hyd-rodynamics [810]. It is influenced by both axialand transverse flow fields, which are in turn influ-ence by concentration polarization [1113]. Thecoupling of momentum and mass transport makessimultaneous solution of the NavierStokes, con-tinuity, and convection-diffusion equations com-putationally intensive even for simple systems [13,14]. Although concentration polarization inreverse osmosis processes under laminar flow con-ditions can be rigorously modeled [15,16], theseaccurate numerical solutions have little bearingon real membrane separations in spiral wound ele-ments with complex hydrodynamics due to feedspacers [1719]. However, these relatively idealizedCP models are important for proper interpretationof laboratory scale mechanistic studies, especiallythose focused on elucidating fundamental mecha-nisms governing fouling and scaling phenomena.

Early CP models were limited to semi-em-pirical mass transfer analogies of heat transfercorrelations, which were developed to describestagnant film layer formation in conduits withimpermeable walls [20]. Stagnant film models arefundamentally ill-suited to accurately describe allmechanisms of concentration polarization incrossflow membrane filtration processes becausethese models assume uniform concentration alongthe filtration channel wall and do not account forthe impact of permeate convection on the filmlayer thickness. Despite these limitations, filmtheory is simple, analytical, and (reasonably) accu-rate for most reverse osmosis separations. Further,film theory can be extended to describe CPphenomena in spacer-filled RO modules, whichis of tremendous value in practical process designand evaluation.

Recently, a model was proposed [21], whichattempts to capture the fundamental interplay

between concentration polarization layer develop-ment and a locally varying permeate flux (in aclosed form analytical solution) by making useof the retained solute concept [22]. This modeldescribes two dimensional convective-diffusivesolute transport without complex computationaleffort, and hence, can be used to predict local con-centration polarization and permeate flux inspacer-free reverse osmosis flow channels. Theretained solute model appears an attractive alter-native to film theory and is simpler than numericaltechniques, but it has not been subjected torigorous analysis or comparison to well-controlledexperimental data. A more rigorous numericalconvection-diffusion based concentration polari-zation model was developed for comparison withthe retained solute and film theory models. Predic-tions from all three models were compared toexperimentally-derived permeate flux and ob-served salt rejection data. The accuracy, limita-tions, and applicability of each model for ROprocesses are discussed.

2. Theoretical

2.1. Analytical film theory (FT) model

A film theory approach to describe concen-tration polarization was developed by Michaelsand others [810]. Film theory simplifies a com-plex transport problem to a one-dimensional mass-transfer problem by assuming axial solute convec-tion near the membrane surface is negligible.Integrating the one dimensional (transverse) con-vection-diffusion mass balance from the mem-brane surface out to a finite mass boundary (film)layer thickness, , yields the relationship betweenconcentration polarization and permeate flux. Theresult is

expw p wb p

c c vc c D

= (1)

where cw is concentration at the membrane surface,

S. Kim, E.M.V. Hoek / Desalination 186 (2005) 111128

113

or channel wall, for the rejected salt, cb and cp arethe bulk and permeate solute concentrations, res-pectively, vw is the permeate water velocity at thechannel wall, and D is the solute diffusion coef-ficient.

A procedure for estimating the convectivediffusion layer thickness in a channel with solubleor rapidly reacting walls forms the basis of mostfilm theory models [23]. For fully developedlaminar flow in a thin rectangular channel, thefilm layer thickness is described by

( ) 1/ 32 /3max

1.475x h D

x x u h = (2)

where x is the longitudinal coordinate, h is thechannel half-height, and umax is the maximumcrossflow velocity at channel center. Assumingconstant diffusivity, a local mass transfer coeffici-ent is described by

( ) ( )1/32

1/ 32

1 31.475 2

0.538 w

D u Dk xx h x

Dx

= = =

(3)

where u is the bulk average crossflow velocity(=Q/2hW), W is the channel width, Q is thevolumetric feed flow rate) and ( 3 / )w u h = is thewall shear rate [24,25]. A key limitation of apply-ing film theory to membrane separations is theassumption that the transverse component of con-vection (permeate flux) does not influence theboundary layer thickness, .

The channel averaged mass transfer coeffici-ent, k , derives from integrating the local flux,vw(x), given by Eqs. (1) and (3) assuming the mem-brane, permeate, and bulk concentrations, as wellas the permeate and crossflow velocities remainconstant along the channel length, L. This assump-tion is unrealistic for large scale RO systems, butreasonable for short membrane channels where

recovery is negligible. The channel average masstransfer coefficient is expressed as

1/ 32

0.807 wDkL

= (4)

which is identical to the mass transfer correlationfor laminar flow in a thin channel commonlyreported in terms of a Sherwood number, e.g.,( )1/ 3Sh / 1.85 ReSc /H Hkd D d L= = [9].

In reverse osmosis processes, the driving forcefor permeation is the difference between theapplied pressure, p, and the transmembraneosmotic pressure, m. Thus, the permeate flux isdescribed by

( )w mv A p= (5)where A is the apparent water permeability of themembrane. The transmembrane osmotic pressureis given as

( )m os w p os i wf c c f R c = = (6)where Ri = (1 cp/cw) is the intrinsic salt rejectionof membrane and fos is a coefficient that convertsmolar salt concentration to an osmotic pressurevia an appropriate expression [26]. vant Hoffsequation gives fos = 2RT for NaCl (R is the uni-versal gas constant and T the absolute tempera-ture), which is accurate for dilute concentrations.

Provided with the system operating parameters(p, u , T), solution and solute characteristics (cb,D), and membrane properties (A, Ri), the wall con-centration, cw, is the only unknown parameterlinking concentration polarization, permeate flux,and observed rejection (Ro = 1 cp/cb). The wallconcentration can be solved for when Ri is knownfrom [5]

1

1 expw wi ib

c vR Rc k

= + (7)

which is called the CP modulus [1].


114

2.2. Analytical retained solute (RS) model

The retained solute model attempts to accountfor the interplay between solvent permeation,solute retention, and concentration polarization.The coupling of water permeation and concentra-tion polarization becomes more complex incrossflow filtration when the local variation of CPis considered along the filtration channel. Ananalytical concentration polarization model waspreviously developed to describe the combinedinfluences of colloidal particle diffusion, cross-flow velocity, and permeate flux in crossflowultrafiltration using the concept of retained solute[21].

At steady state, the concentration of retainedsolute in the CP layer satisfies the following con-dition, assuming negligible axial transport

( ) 0w Cv x C D y+ = (8)

where x and y are the longitudinal and transversecoordinates, respectively, vw(x) is the permeateflux at x, and C indicates the concentration ofretained solute (the actual solute concentration inthe polarization layer is cb + C) [22]. At steadystate, the longitudinal solute flux at any point xalong the channel is equal to the total amount ofsolute rejected by the membrane from the inlet tothe location x, that is

( )0 0

x

w o b wyCdy R c v x dx

= (9)where Ro (= 1 cp/cb) is the observed salt rejectionand x is a dummy integration variable.

Using Eqs. (5), (8) and (9) a closed-formanalytical expression for the local permeatevelocity can be written as

( ) ( )

( )

( )

01/32 3

1/31/ 2

2 3

1/31/ 2

2 3

1 6 /

1 4 21 6 /

1 4 21 6 /

w

s

s

s

A Fv xF X N

F X N

F X N

=+ + + +

+ +

(10a)

with 00

pF = (10b)

and 1/ 32

0

1 1 ws

DNA L

=

(10c)

Here, is the viscosity of water, 0 is the osmoticpressure difference between feed and permeatesides of the membrane (0 = fos (cb cp) = fosRocb),X is the dimensionless longitudinal distance, x/L,and F and Ns are the dimensionless driving forceand dimensionless membrane resistance, respec-tively [21]. The salt concentration on the mem-brane surface can be derived from vw(x) viaEq. (5), that is

( ) ( ) ( ) 0/ 1w os owos

p v x A f R cc x

f + = (11)

This analytical concentration polarizationmodel is attractive because it accounts for thefundamental interplay between concentrationpolarization and permeate flux in a closed formanalytical solution.

2.3. Numerical convectiondiffusion (CD) model

We have developed a new concentration polari-zation model accounting for the two-dimensional


115

mass transfer occurring in crossflow RO processes.The velocity field is composed of axial (feed) andtransverse (permeate) components. In principle,the velocity profile can be obtained by completesolving the NavierStokes and the continuityequations; however, a Poiseuille flow profile isclassically assumed if the permeation velocity issmall and does not significantly alter the axial flowprofile [15]. An additional complication arisesbecause both velocity components influenceconcentration polarization of rejected salt ions.Simultaneously, concentration polarization reducespermeate flux, which in turn alters the polarizationlayer characteristics. A complete description ofthe process requires solving the fully coupledNavierStokes, continuity, and convectiondif-fusion equations.

In order to simplify the coupled momentumand mass transport problem, the axial velocityprofile is assumed independent of concentrationpolarization in our model. The axial (u) andtransverse (v) components of velocity are ex-pressed as [15]:

( ) ( )( )

2

max 2

2

2

, 1

3 12

ru x y u xh

ru xh

= =

(12a)

( ) ( )( )

2

2, 3 ;2

Re 1

w

ww

r rv x y v xh h

hv x

= =

(12b)

where, r is the distance from center line of arectangular channel (r = y h) and is kinematicviscosity of water. As indicated, Eq. (12b) is validwhen the wall Reynolds number based on thepermeation velocity is much smaller than unity.A further simplification used in our model is that

the CP layer thickness is assumed negligiblecompared to the channel height, which enablesthe transverse velocity within the boundary layerto be held constant and equal to the permeatevelocity at the wall, vw(x).

Conservation of mass is considered in tworegions of the crossflow filtration channel. Oneregion is above the boundary of the CP layerand the other is within the CP layer. Fig. 1 showsthe domain for solving the mass balance problem.At steady state, solute and solvent mass shouldbe conserved everywhere. Since the height ofconcentration polarization layer is much smallerthan the channel height, solvent mass is consideredabove concentration polarization layer, that is

( ) ( ) ( )x x wx

Q x Q x x v x Wdx+

= + + (13)where Q(x) and Q (x + x) are the axial flow rateat x and x + x, respectively, and vw(x) is the per-meate flux at x. The solute mass balance aboveconcentration polarization layer is described as

( ) ( ) ( ) ( )( ) ( )

b b

x x

w bx

Q x c x Q x x c x x

v x Wc x dx+

= + + + (14)

where cb(x) and cb(x + x) are the bulk concen-tration at x and x + x, respectively. By solvingEqs. (13) and (14) simultaneously, cb(x) is provedconstant and denoted as cb.

Solute mass should be conserved within con-centration polarization layer as well, that is,

( ) ( ) ( ) ( ) ( ),,w w pc x yv x c x y D v x c xy+ = (15)

where c(x, y) is the solute concentration at (x, y)within concentration polarization layer. Eq. (15)is a form of convectiondiffusion equation andits solution can be written as


116

Fig. 1. The problem domain used in developing the numerical convection diffusion model for a laboratory scale crossflowmembrane filter without mesh feed spacer. Feed water flows from left to right along the channel length and some portionof water flows through the membrane.

( ) ( ) ( ) ( )( ) ( )

, exp

;

ww p

p

v xc x y c x c x y

D

c x y x

= +

(16)

At steady state, axial and transverse input andoutput should be balanced in the problem domainwithin concentration polarization layer as shownin Fig. 1, that is

( ) ( )( )

( ) ( )( )

( ) ( )

0

0

, ,

, ,

x

at x

x

at x xx x

b p wx

u x y c x y y

u x y c x y y

c c x v x dx

++

=

(17)

The integrations in Eq. (17) are solved num-erically using the trapezoidal rule. The concen-tration profile starting from an arbitrary (initial)wall concentration is solved for by Eq. (16). Asthe transverse coordinate (y) increases, the concen-tration decreases due to solute back transport byconvection and diffusion. The extent of the CPboundary layer thickness is defined as the positionwhen the concentration, calculated from Eq. (16),equals the bulk concentration. Concentrationpolarization layer thickness is defined as thedistance from that position to the channel wall.Above the CP layer, all the concentrations areequal to the bulk concentration.

Since the initial wall concentration is anarbitrary value, an iteration process is required.This is performed by solving Eqs. (12), (16) and(17) simultaneously to obtain the salt concen-tration locally in the channel. Once the local salt


117

concentration on the membrane surface (cw) isdetermined, the local permeate flux can be cal-culated from Eq. (5) and the osmotic pressure fromEq. (6). Provided with the system operating para-meters (T, p, u ), solution and solute charac-teristics (cb, D) and membrane properties (A, Ri),concentration polarization, permeate flux, and per-meate concentration can be predicted. All massconservation equations above assume constant andequal density and viscosity in feed, retentate, andpermeate solutions.

3. Experimental

3.1. Membrane and reagents

A commercial RO membrane designated asXLE (Dow-FilmTec Corp., Edina, MN) was usedin this study. Upon receipt from the manufacturer,the membrane was immersed in de-ionized water(Milli-Q Synthesis, Millipore, Billerica, MA) andstored at 5C. Deionized water was changed dailyfor the first two weeks and subsequently changedabout every two weeks. Salt stock solutions wereprepared using ACS grade NaCl (Fisher Scientific;Pittsburgh, PA) dissolved in de-ionized water.

3.2. Crossflow membrane filter

The crossflow membrane filter (CMF) used infouling experiments was a modified version of acommercially available unit (Sepa CF, Osmonics,Inc.; Minnetonka, MN). A schematic illustrationof the experimental apparatus and a completedescription of modifications are provided else-where [27]. The crossflow membrane filter wasrated for operating pressures up to 6895 kPa(1000 psi), and had channel dimensions of 14.6 cm,9.5 cm, and 1.73 mm for channel length, width,and height, respectively. Membrane surface areawas 1.39102 m2 and cross-sectional flow areawas 1.64104 m2.

3.3. Pure water permeability

At the start of each fouling experiment, de-

ionized water was filtered through the membraneovernight at constant temperature of 25C to allowfor membrane compaction and other unknowncauses of flux decline inherent to laboratory-scalerecirculation systems. After stable flux wasachieved, pure water permeability was determinedby measuring pure water flux over a range ofapplied pressures. The relationship governing thepure water flux is

wv A p= (18)Pure water permeability (A) was determined

from a linear regression of the measured purewater flux and applied pressure data.

3.4. Concentration polarization, salt rejection andosmotic pressure

After membrane permeability was determined,an appropriate volume of stock NaCl solution wasadded to provide the desired feed ionic composi-tion. Flux and crossflow were set at the desiredvalues for each fouling experiment and the systemwas allowed to equilibrate up to 24 h to ensurestable performance. Temperature was maintainedat 25C by a recirculating chiller. Experimentswere conducted with 27 (333) combinationsof cb (10, 20, and 50 mol/m

3 NaCl), p (689, 1034,and 1379 kPa), and u (0.017, 0.042, and 0.068 m/s;290, 590, and 1170 as Reynolds number) atunadjusted pH of 5.8 0.2.

Due to concentration polarization of rejectedionic constituents, the driving force for permeationis the difference between the applied pressure (p)and the transmembrane osmotic pressure (m).During electrolyte equilibration, observed saltrejection (Ro) was determined from feed (f) andpermeate (p) conductivity measurements (Ro = 1 p/f).Conductivity of NaCl solutions wasdetermined linear over the range of ionic strengthsused.

Intrinsic salt rejection (Ri = 1 cp/cw) was deter-mined from the calculated salt concentration atthe wall (cw), and the measured permeate concen-


118

tration (cp). In order to calculate cw, osmoticpressure was first calculated using Eq. (5). Then,cw was calculated using the experimental permeateconcentration (cp) from the conductivity measure-ment as in Sutzkover et al. [28].

4. Results and discussion

4.1. Comparison of CP modulus predicted bymodels

Prediction of observed salt rejection and per-meate water flux at any applied pressure andcrossflow rate are dependent on the concentrationpolarization modulus (=cw/cb), intrinsic salt rejec-tion, and intrinsic water permeability. Once theCP modulus, rejection, and permeability are known,the channel averaged permeate flux may be pre-dicted from Eqs. (5) and (6). Observed salt rejec-tion is then predicted from the intrinsic salt rejec-tion and known feed salt concentration. Hence,the variable CP modulus combined with intrinsicwater permeability and salt rejection determinesRO process performance. Table 1 lists experi-mentally determined intrinsic salt rejections andCP moduli, plus CP moduli predicted by the FT,RS, and CD models.

According to results in Table 1, CP modulusincreases with increasing applied pressure, de-creasing Reynolds number, and decreasing bulkconcentration. All model and experimental resultsare in qualitative agreement. Higher applied pres-sure and lower crossflow rate make the CP mod-ulus larger by increasing permeate convection anddecreasing shear rate (mass transfer), respectively.At the same applied pressure and crossflow rate,the permeate flux decreases with increasing bulkconcentration because the feed solution osmoticpressure is larger. Salt rejection increased withapplied pressure and crossflow, but decreased withincreasing ionic concentration.

All three models underestimated the CP mod-ulus at the lowest operating pressure. At the inter-mediate pressure, the FT and CD models werequite accurate, but the RS model dramatically

underestimated CP. At the highest pressure, theFT and CD models overestimated the CP modulus,whereas the RS model underestimated CP again.All three models were increasingly accurate asionic concentration increased. At a given appliedpressure, the accuracy of all models improved athigher crossflow.

The overall average prediction errors by theFT, RS, and CD models were 4.0 10.4, 37.8 7.9, and 2.1 10.0%, respectively, where the value represents the standard deviation of all per-cent differences. A negative percentage errormeans that the average predicted value is smallerthan the experimental one. The RS model con-sistently underestimated the CP modulus by asmuch as 51%, while the FT and CD models neverdeviated by more than 29 and 26%, respectively.The numerical CD model was most accurate, butthe predicted CP moduli by the FT model werealso quite reasonable.

4.2. Accuracy of models for predicting flux

The predictive accuracy of the FT, RS, and CDmodels were tested by comparison to experimentalflux data. Fig. 2 shows channel averaged permeateflux produced by model predictions and experi-mental measurements. For the FT model, thechannel averaged permeate fluxes were predicteddirectly. For the RS and CD models, the channelaverage fluxes were determined by integrating thelocally predicted permeate flux over the entirechannel. All predicted and measured flux dataincreased as applied pressure increased, as Rey-nolds number increased , and as feed concentrationdecreased. Higher applied pressures induce alarger driving force for permeation. Higher Rey-nolds number flows enhance mass transfer, whichcauses a smaller CP modulus and osmotic pressuredrop resulting in a larger relative permeate flux.Increasing bulk salt concentration causes a higherosmotic pressure drop, which results in a smallerpermeate flux.

As shown in Fig. 2, fluxes predicted by the FTand CD model are quite close to the experimental


119

Tabl

e 1

Pred

icte

d an

d ex

perim

enta

l CP

mod

uli

c b (R

e)Ri

EXFT

% e

rrR

S%

err

CD

% e

rrRi

EXFT

% e

rrR

S%

err

CD

% e

rrRi

EXFT

% e

rrR

S%

err

CD

% e

rr

10 (2

90)

96.4

%2.

912.

27-2

2.0%

1.39

-52.

2%2.

34-1

9.6%

96.9

%3.

493.

35-4

.0%

1.95

-44.

1%3.

40-2

.6%

97.6

%4.

684.

894.

5%2.

96-3

6.8%

4.79

2.4%

20 (2

90)

95.7

%2.

311.

98-1

4.3%

1.31

-43.

3%2.

04-1

1.7%

96.8

%2.

972.

85-4

.0%

1.76

-40.

7%2.

89-2

.7%

97.5

%3.

893.

920.

8%2.

51-3

5.5%

3.88

-0.3

%

50 (2

90)

93.0

%1.

621.

54-4

.9%

1.15

-29.

0%1.

57-3

.1%

95.4

%2.

092.

04-2

.4%

1.39

-33.

5%2.

07-1

.0%

96.6

%2.

722.

67-1

.8%

1.82

-33.

1%2.

67-1

.8%

10 (5

90)

96.5

%2.

571.

86-2

7.6%

1.25

-51.

4%1.

96-2

3.7%

97.0

%2.

692.

59-3

.7%

1.51

-43.

9%2.

68-0

.4%

97.4

%3.

213.

5310

.0%

2.00

-37.

7%3.

5911

.8%

20 (5

90)

95.9

%2.

011.

71-1

4.9%

1.20

-40.

3%1.

76-1

2.4%

96.9

%2.

392.

32-2

.9%

1.42

-40.

6%2.

390.

0%97

.5%

2.87

3.07

7.0%

1.82

-36.

6%3.

128.

7%

50 (5

90)

93.7

%1.

481.

41-4

.7%

1.10

-25.

7%1.

44-2

.7%

96.0

%1.

831.

830.

0%1.

23-3

2.8%

1.87

2.2%

97.0

%2.

232.

303.

1%1.

47-3

4.1%

2.33

4.5%

10

(117

0)96

.6%

2.43

1.73

-28.

8%1.

22-4

9.8%

1.80

-25.

9%96

.9%

2.38

2.29

-3.8

%1.

38-4

2.0%

2.39

0.4%

97.3

%2.

763.

0410

.1%

1.71

-38.

0%3.

1313

.4%

20

(117

0)95

.9%

1.89

1.60

-15.

3%1.

18-3

7.6%

1.66

-12.

2%97

.0%

2.17

2.12

-2.3

%1.

33-3

8.7%

2.19

0.9%

97.5

%2.

542.

737.

5%1.

60-3

7.0%

2.80

10.2

%

50

(117

0)94

.1%

1.42

1.37

-3.5

%1.

09-2

3.2%

1.40

-1.4

%96

.1%

1.70

1.72

1.2%

1.18

-30.

6%1.

763.

5%97

.1%

1.99

2.13

7.0%

1.35

-32.

2%2.

179.

0%

p =

690

kPa

p =

103

5 kP

ap

= 1

380

kPa


120

400 600 800 1000 1200 1400 16004

8

12

16

20

24

28

RS model

Experiment,FT model,CD model

Experiment FT model RS model CD model

Per

mea

te fl

ux (

m/s

)

Applied pressure (kPa)

(a) Crossflow Reynolds number = 290

400 600 800 1000 1200 1400 16004

8

12

16

20

24

28



Per

mea

te fl

ux (

m/s

)


(b) Crossflow Reynolds number = 590


121

flux data, while fluxes predicted by the RS modelappear significantly off scale. Although the slopeof pressure-flux predictions by the FT and CDmodels is not identical to the experimental data,the slope of RS model prediction gives a lessaccurate slope. The average prediction errors ofthe FT, RS, and CD models were 2.0 3.6,21.9 12.1 and 0.5 3.6%, respectively. Theseerrors were less than those of the CP moduluspredictions. The CP modulus prediction errorsresult in the deviation of predicted osmotic pres-sure from the real value. The osmotic pressure isused to predict the channel average permeate fluxwith the pure water permeability and transmem-brane pressure using Eq. (5). When channelaverage values are compared, the prediction errorswere significantly reduced. The FT and CDmodels appear to be most accurate for the highest

Fig. 2. Channel averaged permeate fluxes determined experimentally (circles) and predicted (lines) by film theory (FT),retained solute (RS), and convection diffusion (CD) models plotted as a function of applied pressure and salt concentra-tion for crossflow Re of (a) 290 (b) 590 (c) 1170. Constant experimental and simulation conditions employed weretemperature and pure water permeability of 298 K and 2.151011 mPa1s1. Solution pH was 5.80.2.

400 600 800 1000 1200 1400 16004

8

12

16

20

24

28



Per

mea

te fl

ux (

m/s

)


(c) Crossflow Reynolds number = 1170

ionic strength and lowest pressure, which are themost realistic set of conditions for practical ROmembrane desalination processes. It appears thatthe FT and CD models reasonably predicted per-meate flux of the experiments performed.

4.3. Accuracy of models for predicting rejection

Observed salt rejection cannot be predicted bythe RS model because it is an input parameter forthe model. The observed salt rejection may be deter-mined experimentally or by a non-equilibriumthermodynamic model [29]. Since the bulk con-centration (cb) is an input for all models, observedsalt rejection (Ro = 1 cp/cb) is determined by pre-dicting permeate concentration (cp). The permeateconcentration is determined from the product ofsalt passage and the salt concentration on themembrane surface, cp = (1 Ri) cw.


122

Intrinsic salt rejection is an input for the FTand CD models and was determined experiment-ally for every set of operation conditions simu-lated. Since the intrinsic rejection is known, theobserved salt rejection predictions are solely de-pendent on the accuracy of predictions of CPmodulus. In Figs. 3a and 3b the predictions ofobserved salt rejection by the FT and CD modelsare plotted against experimental data. The averageprediction errors of the FT and CD models were0.2 0.1 and 0.3 0.9%, respectively. Hence,

Fig. 3. The comparison between experimental and ob-served salt rejection (Ro) by (a) film theory (FT) modeland (b) numerical convection-diffusion (CD) model. Theexperimental and simulation conditions are the same asthose for Fig. 2.

0.6 0.7 0.8 0.9 1.00.6

0.7

0.8

0.9

1.0

FT m

odel

Experiment

(a)

0.6 0.7 0.8 0.9 1.00.6

0.7

0.8

0.9

1.0

CD

mod

el

Experiment

(b)

it can be concluded that the FT and CD modelsgive reasonably accurate prediction of salt rejec-tion, as well as CP modulus and permeate flux.

4.4. Sources of errors in modeling CP, rejectionand flux

Potential sources for deviations between modelpredictions and experimental results include en-trance, exit, and side wall effects as well as densitydifferences for high salinity feed and low salinitypermeate solutions. For example, the entrancelength, over which a full developed parabolic flowprofile forms, is predicted by Lu = 0.08Reh,where ( )Re 2 /u h= [30]. For Reynolds numbersof 38, 191, 380 and 1910, Lu/L is 0.02, 0.09, 0.18and 0.91 for the crossflow channel geometry usedin this study. Hence, the assumption of a fullydeveloped parabolic profile at the inlet is not satis-fied for any experimentally employed conditions,and is a source of common error for predictionsby all three models.

The analytical retained solute model is basedon a concept introduced by Song and Elimelech[22] to describe particle concentration polarizationin membrane filtration processes. Use of the re-tained solute concept enabled analytical solutionof a two-dimensional convection-diffusion massbalance applicable for crossflow membrane filtra-tion. This approach worked well to predict thelimiting flux during ultrafiltration of nanoparticlesuspensions [31]. However, the RS model did notaccurately predict concentration polarizationphenomena for the reverse osmosis experimentsreported in this study. A possible explanation forthe inaccuracy of the retained solute model inpredicting concentration polarization is providedbelow.

At the membrane surface, net solute masstransfer by convection and diffusion is equal tothe solute flux through membrane, that is,

0w w w p

y

cv c D v cy =

+ = (19)


123

Here, y = 0 means at the membrane surface.Eq. (19) can be transformed by substituting c = cb+ C, where C is the excess solute concentration atthe membrane surface or the retained solute con-centration, into Eq. (19) yielding

( ) ( )0

bw b w w p

y

c Cv c C D v c

y =

+ + + = (20)

Here Cw is the excess concentration at the mem-brane surface. Since the bulk concentration (cb)is constant, the derivative is zero.

Rearranging the left hand side of Eq. (20) andapplying the definition of retained solute, i.e.,Eq. (8), yields

0w b w w p

y

Cv c v C D v cy =

+ + = (21a)

or

0w w p w b

y

Cv C D v c v cy =

+ = (21b)

The term in parentheses is the retained solutemass balance for the concentration polarizationdomain. According to Eq. (21b), this term doesnot equal zero at the membrane surface (y = 0), asspecified in the development of Eq. (8), becausethe bulk and permeate concentrations are sig-nificantly different and non-negligible in ROseparations. This means that Eq. (8) cannot be usedas a boundary condition at the membrane surface.

However, the assumption that Eq. (8) is truejust at the membrane-solution interface is the keystep in deriving an analytical solution to theintegral in Eq. (9) and, ultimately, arriving at theclosed form analytical solution. The assumptionis acceptable if the bulk and permeate concentra-tions are approximately equal and if both arenegligible compared to the concentration at themembrane surface (wall) as is the case for

ultrafiltration of dilute particle suspensions.However, in RO separations the solute wall con-centration is only slightly elevated over the bulk;hence, the bulk concentration is not negligible.The lack of mass conservation is likely the primarysource of error in predictions by the RS model;however, the retained solute model does morerealistically simulate the fundamental interplaybetween permeation, crossflow, rejection, andconcentration polarization than does film theory.Thus, the retained solute model offers significantconceptual insight into concentration polarizationphenomena in all crossflow membrane processes.

4.5. Applicability of film theory for RO processes

Film theory models are widely preferred byprocess engineers for their simplicity and reason-able performance predictions. The data of Figs. 2and 3 demonstrates that the film theory modelpredicts permeate water flux and salt rejectionnearly as well as a more rigorous numerical con-vectiondiffusion model. For the experimentalconditions tested, the FT model was less accuratethan the CD model at low fluxes, but slightly moreaccurate at the higher fluxes examined for allcrossflows and salt concentrations. Although thisis counter-intuitive, the film theory neglects theeffect of solute permeation through the membrane,which may suppress the boundary layer thicknessat high fluxes.

Since film theory has been shown to givereasonable predictions of CP, flux, and rejection,it is of interest to further compare the FT and CDmodels across an array of operating conditions,membrane properties, and solution chemistrieslikely to be encountered in real applications. In anew set of simulations, constant conditions assum-ed were temperature, water permeability (A), andsalt rejection (Ri) of 298 K, 1.1210

11 mPa1s1,and 97%. Table 2 summarizes the simulation con-ditions that were varied and Fig. 4 compares chan-nel averaged permeate flux and observed saltrejection data predicted by both models.


124

Crossflow hydrodynamics were varied withinthe laminar range (Re < 2100). Permeate fluxesemployed were in the range frequently used inreal operations and higher. Salt concentrationsranged from fresh water conditions to about twicethe molarity of seawater. Permeate flux and ob-served salt rejection data predicted by both modelsare well-matched according to the data in Fig. 4.Overall there was a difference of 3.5 2.8%, wherethe FT model consistently over predicted the per-meate flux and salt rejection because it under pre-dicted the CP modulus.

4.6. Using film theory to predict local concentra-tion polarization

Accurate prediction of local concentration

Table 2Simulation conditions for the comparison of FT and CD models

# 1 2 3 4 5 6 7 8 9 10 11 Re 38 191 380 1910 191 191 191 191 191 191 191 p, kPa 912 912 912 912 576 1580 2930 912 912 912 912 cb, mol/m3 50 50 50 50 50 50 50 10 100 500 1000

Fig. 4. The comparison between the film theory (FT) and the numerical convection-diffusion (CD) model predictions for(a) channel averaged permeate flux and (b) observed salt rejection. Constant simulation conditions employed were tem-perature, water permeability, and salt rejection of 298 K, 1.121011 mPa1s1, and 97%. Table 2 summarizes the simula-tion conditions that were varied. Crossflow hydrodynamics were varied within the laminar range (Re < 2100).

0 5 10 15 20 25 300

5

10

15

20

25

30

FT m

odel

(m

/s)

CD model (m/s)

(a)

0.6 0.7 0.8 0.9 1.00.6

0.7

0.8

0.9

1.0

FT m

odel

CD model

(b)

polarization phenomena is very important forreverse osmosis processes because, for example,it enables prediction of the onset of scaling bysparing soluble minerals. The FT model may beused to predict the local concentration polarizationmodulus, cw/cb, via

( )( )

( )( )

1

1 expw wi ib

c x v xR R

c x k x

= + (22)

where vw(x) is the local permeate flux predictedby the CD model, k(x) is the local mass transfercoefficient predicted by Eq. (3), and cw(x) and cp(x)denote local wall and permeate concentration,respectively.


125

Note the local permeate flux predicted by theCD model was used as an input for CP moduluspredictions by the FT model because film theorycan not predict the local flux and CP modulus in-dependently. It is more common to assume a con-stant permeation velocity when applying filmtheory to RO processes; however, this has no effecton the local film layer thickness, but does subtlyinfluence the local solute rejection. Also, simula-tions were performed for the geometry of the labscale filter used in the experiments of this study.Hence, the assumption of a constant bulk concen-tration introduces little error because the recoveryin this system ranges from about 0.11.0%.Clearly, a separate mass balance for the axiallyvarying retentate concentration must be performedfor systems with significant recovery [26].

Figs. 5 and 6 describe local CP modulus (cw(x)/cb) predicted by FT and CD model for variousinlet permeate fluxes and crossflow Reynoldsnumbers. The difference between each predictedfactor increases as the initial flux at the membrane

0.0 0.2 0.4 0.6 0.8 1.01

2

3

4

5

6

7

8

9

10

11

12 p FT model CD model 576 kPa 912 kPa 1580 kPa 2930 kPa

CP

mod

ulus

(cw/c

b)

x/LFig. 5. Prediction of local CP moduli depending on different applied pressures using the film theory (FT) and the numeri-cal convection-diffusion (CD) models. Constant simulation conditions employed were temperature, water permeability,and salt rejection of 298 K, 1.121011 mPa1s1, and 97%. Simulation conditions employed were combinations #2, 5, 6,and 7 in Table 2. Crossflow hydrodynamics were varied within the laminar range (Re < 2100).

inlet becomes larger (Fig. 5) and Reynolds numberdecreases (Fig. 6). Film theory works well whenCP factor is small (e.g., higher crossflow Reynoldsnumber and lower operating pressure). At per-meate fluxes below 7.54106 m/s (~16 gfd,gallons per square foot per day), the FT modelaccurately predicts local concentration polariza-tion. The key source of error at higher fluxesrelates to independence of the boundary layerthickness from permeation. Considering most ROapplications employ fluxes smaller than 7.54106 m/s, it appears reasonable to apply film theorybased concentration polarization models.

5. Conclusions

A two dimensional, numerical convection-diffusion mass balance model was developed toenable more fundamental description of concen-tration polarization phenomena in reverse osmosisprocesses. Predictions by two analytical concen-tration polarization models and the numerical


126

model were compared to experimental RO separa-tion performance data. The three models allowedthe prediction of local solute concentration polari-zation, permeate water flux, and permeate salt con-centration given inputs of operating parameters,filter geometry, and membrane properties. Predic-tions by film theory and the numerical convec-tiondiffusion model compared favorably to ex-perimental data and to each other, providing accu-rate and consistent predictions of observed waterflux and salt rejection. An important result of thisstudy is the validation of film theory as a simpleand reliable indicator of concentration polarizationphenomena in reverse osmosis separations, par-ticularly for mechanistic studies of fouling andscaling conducted in laboratory scale crossflowmembrane filters. Further, the use of film theorybased concentration polarization models in realRO processes appears justified given the specificoperating conditions of high crossflow and lowflux used in most practical applications.

0.0 0.2 0.4 0.6 0.8 1.01

2

3 Re FT model CD model 38 190 380 1900

CP

mod

ulus

(cw/c

b)

x/L

Fig. 6. Predicted local CP moduli depending on different crossflow Reynolds numbers using the film theory (FT) and thenumerical convectiondiffusion (CD) models. Constant simulation conditions employed were temperature, water perme-ability, and salt rejection of 298 K, 1.121011 mPa1s1, and 97%. Simulation conditions employed were conditions 14in Table 2.

Acknowlegements

The research described above was performedwhile Dr. Suhan Kim was a postdoctoral fellowat University of California, Riverside. Partialfinancial support for this work was obtainedthrough a post-doctoral fellowship from the KoreaScience and Engineering Foundation (KOSEF) forDr. Kim. Additional support was obtained fromthe California Department of Water Resourcesthrough the Desalination Research InnovationPartnership (DRIP), which is managed by theMetropolitan Water District of Southern California.

Symbols

A Pure water permeability, mPa1s1C Concentration of retained solute, molm3cb Solute concentration in the bulk, molm

3

cp Solute concentration in the permeate,molm3


127

cw Wall concentration for the rejecting salt,molm3

D Solute diffusion coefficient, m2s1dH Channel hydrodynamic parameter, mF Dimensionless driving forcefos Osmotic coefficient, Pamol1h Channel half-height, mk Mass transfer coefficient, ms1k Channel averaged mass transfer coeffi-

cient, ms1L Channel length, mNs Dimensionless membrane resistancep Pressure, PaQ Axial flow rate, m3s1R Universal gas constant,

= 8.314 Jmol1K1Ri Intrinsic salt rejection of membrane

(e.g., Ri = 1 cp/cb)Ro Observed salt rejection (e.g., Ro = 1

cp/cb)r Distance from center line of a rectan-

gular channel, mRe Reynolds number (e.g., Re /Hud= )Sc Schmidt number (e.g., Sc = v/D)Sh Sherwood number [e.g.,Sh = /Hkd D =

1.85 (ReScdH/L)1/3]

T Absolute temperature, Ku Axial components of velocity filed, ms1u Bulk average crossflow velocity, ms1umax Maximum crossflow velocity at channel

center, ms1v Transverse components of velocity filed,

ms1vw Permeate water flux, ms

1

X Dimensionless longitudinal distance,xL1

x Axial coordinate, mx Dummy integration variabley Transverse coordinates, m

Greek

Mass boundary (film) layer thickness,m

f Feed conductivity, Sm1

p Permeate conductivity, Sm1

w Wall shear rate, s1 (e.g., 3 /w u h = ) Viscosity of water, Nsm2 Kinematic viscosity of water, m2s10 Osmotic pressure difference between

feed and permeate sides of the membrane,Pa (e.g., 0 = fos (cb cp) = fosRocb)

m Transmembrane osmotic pressure, Pa

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