2004 - jms - jain & gupta

8/8/2019 2004 - JMS - Jain & Gupta

1/17

Journal of Membrane Science 232 (2004) 4561

Analysis of modified surface force pore flow model with concentrationpolarization and comparison with SpieglerKedem

model in reverse osmosis systems

Semant Jain1, Sharad K. Gupta

Department of Chemical Engineering, Indian Institute of Technology, New Delhi 110016, India

Received 3 July 2003; received in revised form 4 November 2003; accepted 24 November 2003

Abstract

The Modified Surface Force Pore Flow (MD-SF-PF) model is used for predicting the performance of the sodium chloridewater andsodium sulphatewater reverse osmosis systems. Unlike the previous analysis available in literature on this model, the present work takesconcentration polarization into account explicitly. The requiredmass transfer coefficient is estimated by relating it to the feed flow rate throughtwo additional parameters. The model equations being non-linear are solved using the orthogonal collocation technique. The model solutioncode is validated by comparing results with those available in literature. Simulation studies indicate observed rejection versus flux has asimilar trend to that predicted by the SpieglerKedem model. From some of the experimental observations of Murthy [Studies on membranetransport models, Ph.D. thesis, I.I.T. Delhi, July 1996], model solution and mass transfer parameters are estimated through a combination ofthe Downhill Simplex method and Monte Carlo search. Simulated results using above parameters are found to be in very good agreementwith the remaining experimental observations. Prediction of the membrane performance and mass transfer coefficient are also found to bevery close to those estimated through the SpieglerKedem model. The membrane specific parameters such as the membrane thickness andpore radius are calculated from experimental data for the two different systems on a similar cellulose acetate membrane. The closeness of

these parameter values again shows the validity of MD-SF-PF model. 2004 Elsevier B.V. All rights reserved.

Keywords: Concentration polarization; Downhill simplex; Orthogonal collocation; Sodium chloride; Sodium sulphate; SpieglerKedem

1. Introduction

When a semi-permeable membrane (permeable to sol-vent, impermeable to solute) is placed between twocompartmentsone containing pure solvent and the othercontaining solutiondue to high osmotic pressure in thesolvent side, solvent tends to go to the solution side. Butif a pressure gradient greater than the osmotic pressure is

Abbreviations: BC, back calculated; CFSK, combined filmtheorySpieglerKedem model; E, Expt., experimental observation; Hr.,hour; IT, irreversible thermodynamics; Lt., litre; P, value as published inliterature; rms, root mean square; S, Sim., MD-SF-PF model simulatedresult; SK, SpieglerKedem model Corresponding author. Tel.: +91-11-26591023;

fax: +91-11-26581120. E-mail addresses: [email protected] (S. Jain),

[email protected] (S.K. Gupta).1 Present address: Department of Chemical Engineering, University of

Michigan, Ann Arbor, MI 48109, USA.

applied across the membrane, against the direction of theosmotic pressure gradient, then this tendency is reversed;i.e., the solution becomes more concentrated while thesolvent in it passes to the solvent rich side.

Reverse Osmosis has become a very popular method forwater purification. The major advantage of the process isits ability to perform direct separation at ambient conditionsand allowing separation of heat sensitive materials. It is cur-rently being used extensively in the production of potableand industrial water, treatment of domestic and industrialwastewater, and in food processing.

Development of a reliable membrane transport model fordescribing mass transport processes in a reverse osmosismembrane is a highly desirable goal since not only doesthis allow the analysis of performance characteristics of amembrane, but this can also assist in the design of newmembranes. Onsager [1,2] used the principles of irreversiblethermodynamics to relate the fluxes with the forces throughphenomenological coefficients. However, a major limitation

0376-7388/$ see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.memsci.2003.11.021

8/8/2019 2004 - JMS - Jain & Gupta

2/17

46 S. Jain, S.K. Gupta/ Journal of Membrane Science 232 (2004) 4561

of irreversible thermodynamics is that it is valid for sys-tems not far from equilibrium. The next step was the de-velopment of KedemKatchalsky model [35] for a dilutetwo-component system, consisting of water and a solute.The three adjustable parameters in this model are simplefunctions of the original phenomenological coefficients. In

the SpieglerKedem model [6], the assumption on the ap-plicability of the linear laws throughout the thickness of themembrane was resolved by rewriting the original linear ITequations in the differential form and then integrating themover the membrane thickness. Although the SpieglerKedemmodel [6] gives quite accurate results, it is not a mechanis-tic based model; i.e., it does not explain the mechanism oftransport or the nature of the membrane structure.

Out of the mechanistic based models, the Solution Diffu-sion model [7,8] has been applied to many different systemsas it is simple and has only two adjustable parameters and itis only appropriate for systems where the rejection is closeto unity.

The three parameter Solution Diffusion Imperfectionmodel [9] allowed for the presence of some imperfections,which permit the salt to pass undiluted, but dodged the dif-ficulty of the explicit inclusion of the convection or viscousflow.

The four parameter Finely Porous model [8,10,11] isbased on a balance of applied and frictional forces in aone-dimensional pore, while the Frictional model [5,6] hasa very simple physical interpretation, which amounts to aheuristic derivation. Both these models keep all the mech-anisms in their equations from the start but Mason andLonsdale [12] felt the treatment of viscous flow was either

misinterpreted or mishandled in both of them.The Diffusion Viscous flow or the Highly Porous model[11] uses the Poisuielles law to describe the total volumeflux and the total solute flux is expressed as the sum of acontribution due to viscous and diffusive flow.

Mason and Lonsdale [12] reviewed all membrane trans-port models described above and compared them with thestatistical mechanical theory of membrane transport. Theyconcluded all these models are equivalent in the sense thatthey end up writing down the same transport equations.However, these models differ in their predictions aboutthe transport coefficients. They finally recommended theSpieglerKedem model for describing and predicting theperformance of reverse osmosis and ultra filtration mem-branes.

As the SpieglerKedem model relates the membrane sur-face concentration to the permeate concentration, it needs tobe combined with concentration polarization if the permeateconcentration is to be related to the bulk feed concentrationwhich results in the combined film theorySpieglerKedemor CFSK model. Murthy [13] also analyzed the predictionsof the many membrane transport models with his experi-mental observations and found the CFSK model to give theclosest results. On plotting the reciprocal of the observed re-jection against volume flux, he observed that the JminV value,

where the observed rejection is maximum, is predicted moreaccurately by the CFSK model as compared to other mod-els [13,14]. Thus, the CFSK model has been selected forcomparing the predictions of our work.

The Surface Force Pore Flow model [15] is a relativelyrecent two-dimensional model for describing the transport

processes in a reverse osmosis membrane. Since the modeltakes into account the membrane structure and variousmembranesolute interactions, it is expected the model willpresent a more accurate description of the mass transferprocess taking place outside the membrane.

Mehdizadeh and Dickson [16] pointed out the SF-PFmodel uses an incorrect form of the material balance, thepotential function in the pore is inconsistent with the cylin-drical geometry and the solute concentration just inside thepore is equal to that of the permeate. These corrections ledto the development of the Modified Surface Force Pore Flowmodel. The model contains four parametersnamely, 1and 2, required for estimating the potential function, RW,

the membrane pore radius, and /, the ratio of the mem-brane tortuosity to its porosity. Due to the complexity of thenon-differential equations of the MD-SF-PF model, numer-ical techniques are required for obtaining the solution suchas the orthogonal collocation scheme [17,18].

The paper in which the MD-SF-PF model was defined[16] assumed the solute bulk concentration is equal to thesolute concentration on the membrane surface and used thisconcentration to compute the theoretically observed rejec-tion. Their later work [19] mentions the concentration po-larization model can also relate the bulk feed concentrationto the membrane surface concentration. However, detailed

analysis of the same was not carried out in their reference[20]. As concentration polarization effects are very impor-tant in reverse osmosis, they must be included for modelprediction and parameter estimation analysis.

In order to incorporate the concentration polarizationmodel into the MD-SF-PF model, the value of the masstransfer coefficient is required. The mass transfer coefficientis dependent on several factors like cell geometry, feed flowrate and temperature. Although its value can be computedfor fully developed flow in well-established geometries likechannels and pipes, in actual experimental set-up for reverseosmosis the exact cell geometry and flow regime are oftenunknown. This introduces the need to relate the value ofthe mass transfer coefficient to the flow rate for a specifiedsolutesolvent system.

The objective of this work is to incorporate concentrationpolarization in the MD-SF-PF model and to develop a nu-merical solution for solving the differential equations of themodel. The mass transfer coefficient may be related to flowrate through the use of two additional parametersa andbincreasing the number of parameters to be estimatedto six. Thus, a numerical scheme is developed for estimat-ing the parameters for a particular solutesolvent system. Toevaluate the accuracy of the MD-SF-PF model predictions,the estimated parameter set is used to predict the membrane

8/8/2019 2004 - JMS - Jain & Gupta

3/17

S. Jain, S.K. Gupta/ Journal of Membrane Science 232 (2004) 4561 47

performance for a given system for the entire range of op-erating conditions with the results being compared with theexperimental observations. Additionally, the mass transfercoefficient parameters and extrapolated trends of the recip-rocal rejection ratio to the volume flux have been comparedwith predictions of the CFSK model.

2. Theory

2.1. Membrane mass transport models

The general purpose of a membrane mass transport modelis to relate the performance (usually expressed in terms offluxes of solute and solvent and observed rejection) to theoperating conditions (usually expressed in terms of pressureand concentration driving forces). In a model, some coeffi-cients emerge that must be determined based on some ex-perimental data. The success of a model can be measured

in terms of its ability to describe mathematically the datawith coefficients that are reasonably constant over the rangeof operating conditions. Here, we briefly discuss two mod-els: IT based SpieglerKedem model and mechanism basedModified-Surface force-pore flow model which are used forcomparison in this study.

2.2. SpieglerKedem model

The SpieglerKedem [6] model predicts the followingequations for volume flux, JV, and actual rejection, RA:

JV = LP(p ) (1)

RA = ePe 1ePe

(2)

where

Pe = exp

JV(1 )PM

(3)

and LP is the hydraulic permeability coefficient of themembrane; PM is the overall permeability coefficient and is the reflection coefficient.

On combining the relationship between RA, the actual re-jection, and RO, the observed rejection, with the film theory,

the following equations are obtained:CA2 CA3

CA1 CA3= exp

JV

k

(4)

Eliminating CA1, CA2, and CA3 in terms of the actual rejec-tion, RA, and observed rejection, RO, we obtain

1 RORO

=1 RA

RAexp

JV

k

(5)

ln

1 RORO

= ln

1

1

1 exp(Pe)

+

JV

k

(6)

The Eqs. (1), (2) and (6) are the basic equations of thecombined filmSpieglerKedem model (CFSK). By usinga non-linear parameter estimation technique, the unknownparameters, PM, and the mass transfer coefficient, k,can be determined from the given experimental data of ob-served rejection ratio versus volume flux.

When the observed rejection ratio is plotted against vol-ume flux for reverse osmosis systems, Murthy [13] foundfrom his experimental data that a maximum in the observedrejection is observed. The volume flux, JminV , where thismaximum occurs can be found from Eq. (6) by differenti-ating it with respect to JV and then setting it to zero. Thefollowing relations are obtained:

JminV = kln(1+ Pe)

Pe(7)

where

Pe =(1 )k

PM(8)

In the above relations, JminV is the value ofJV where a min-imum of (1/RO 1) or a maximum in observed rejectionwould occur at the value ofJV given by Eq. (7).

The initially decreasing and then increasing trend of thereciprocal of observed rejection, (1/RO 1), can be ex-plained as follows. At low volume fluxes, the second termon the right in the last form ofEq. (6) can be neglected. Asthe other terms in the equation are not affected by pressure,(1/RO 1), decreases with an increase in JV. On the otherhand, at high volume fluxes, as the contribution of JV tothe first term becomes smaller, (1/RO 1), is expected to

increase exponentially with JV.2.3. Modified-surface force-pore flow model

This work is based on the equations proposed in theMD-SF-PF model [16]. A few key equations are repro-duced below while their derivation can be found from theabove-mentioned reference.

The differential equation for the velocity profile inside apore can be found by a force balance in the z-direction on afluid element in the annular region between z and z+dz andbetween r and r + dr inside a pore. The detailed derivationtakes into account the net force due to difference in pressure,

viscous shear stresses using Newtons law of viscosity andthe net force due to friction between the solute and the porewall, leading to the following equation:

d2()d2

+1

d()d

+P (1 e())

1

() e()

1

1

1b()

1 +

e() 1

= 0 (9)

where

1 =DAB

R2W2(10)

8/8/2019 2004 - JMS - Jain & Gupta

4/17


P=p

2(11)

=2 3

2(12)

=r

RW

(13)

The boundary conditions for Eq. (9) are:

d()d

= 0 at = 0 (14)

() = 0 at = 1 (15)

The potential function used by SF-PF model had the poten-tial function coming to a peak at the pore centerline insteadof the expected smooth function. In the MD-SF-PF model[16], it was proposed that an empirical equation could be ap-plied to any solute (electrolyte or non-electrolyte) and mem-brane (charged or uncharged):

() =

1

RWexp(2

2), when < 1

very large and positive, when 1 (16)

where

=RS

RW(17)

In the above equations, 1 affects the centerline potentialfunction and 2 affects the radial slope of the potential. Theradial gradient at the pore centerline is zero for this function.

An expression is required in the SF-PF or the MD-SF-PFmodel, which describes the friction between the solute and

the pore wall. This friction is the result of the hydrodynamicdrag for a solute molecule moving in a small pore. It was firstsuggested to use an empirical relationship for the frictionfunction, b [15]. In the later papers [21,22] a modified formof the Faxen equation has been used:

b() =

11 2.104 + 2.093 0.955

, when 0.22

44.57 416.2 + 934.92 + 302.43, when > 0.22

(18)

Although the friction function should be a function of theradial position, the modified Faxen equation is a reasonableapproximation [16,19] and has been used in our analysis.

The solute and solvent fluxes have been averaged over thepore cross-sectional area to obtain:

JA=2

XAB

1

10

()

b()

2 +

2 3

exp(())1

exp(())

(19)

JB =2

XAB

1

CRT1

0() d (20)

In the above integration, the solute molecule has been as-sumed to be no closer than one solute radius from the pore

wall. After integrating solute and solvent fluxes over the areaof a single pore, they have been generalized over the sur-face area of the membrane. In order to relate average soluteand solvent fluxes through a single pore to the flux througha membrane, the fractional pore area on the membrane face,, is used. Thus, one gets:

NA = JA (21)

NB = JB (22)

Then, the total flux is given by:

NT = NA + NB (23)

The permeate concentration, CA3, is related to the solute andsolvent fluxes as:

CA3 = CJA

JA + JB(24)

The analysis in this paper assumes all pores on the membrane

surface have the same radius. However, in case a pore sizedistribution is present, then the solute and solvent fluxesneed to be integrated across all the possible pore sizes.

2.4. Incorporation of concentration polarization model in

the MD-SF-PF model

During the membrane separation process when the mem-brane rejects the solute, the solute concentration near themembrane surface increases. Hence, there is a differencein the feed concentration in the bulk, CA1, and that at themembrane face, CA2. The MD-SF-PF model assumes CA2 isknown [16,19]. But since it is not possible to measure C

A2,

researchers have used the concentration polarization modelto relate it to CA1. The build up of solute concentration inthe boundary layer region can be described by the film the-ory. Thus, on rewriting the first form of Eq. (8) in terms ofconcentrations, one obtains:

CA2 = CA3 + (CA1 CA3) exp

NA + NB

k C

(25)

Although the concentration polarization can be easily inte-grated with the MD-SF-PF model, it has not been done so far.After its addition, the computations become slightly moreinvolved on account of two convergence loops that are now

needed [23]. The algorithm describing the key steps for in-corporating the concentration polarization in the MD-SF-PFmodel has been described in Jain [23].

2.5. Mass transfer coefficient

The film mass transfer coefficient has been found to bedependent on feed flow rate, cell geometry, temperature andsolute system. A generalized correlation of mass transfersuggests that the Sherwood number is related to the Reynoldsand Schmidt numbers as [24]:

Sh = x1 Rex2 Scx3 (26)

8/8/2019 2004 - JMS - Jain & Gupta

5/17


Table 1Physical properties estimation

S. No. Property Units Solute-1 Solute-2

1. Solute name NaCl Na2SO42. Cationic conductance mho-m2 /equivalent 0.0050 0.00503. Anionic conductance mho-m2 /equivalent 0.0076 0.0080

4. Electrolyte conductance mho-m

2

/equivalent 0.0126 0.01305. Cationic valence 1 16. Anionic valence 1 27. Diffusivity m2 /s 1.61 109 1.23 109

8. Solute radius m 1.3555 1010 1.7743 1010

where x1, x2 and x3 are system and flow regime dependentcoefficients.

In literature, Hwang and Kammermeyer [25] used theabove equation for turbulent flows and Porter [26] in fullydeveloped and laminar regimes. All of them they took x3 =0.33 which makes us believe that x3 can be taken as a con-

stant across flow regimes. With this in mind, Eq. (26) canbe simplified to obtain:

k = a Qb (27)

where aandb are the unknown parameters for a particularsolutesolvent combination.

Murthy [13] has shown the value of a depends on thesolutesolvent system, while the value of b ranges from0.33 to 0.8.

2.6. Physical properties estimation

Some physical properties need to be estimated before theMD-SF-PF model can be used for different solutesolventsystems. These include estimating the solute radius and thediffusivity. While discussing the use of orthogonal colloca-tion for solving the membrane equation [27] and propos-ing the MD-SF-PF model [16], the values of diffusivity andsolute radius for sodium chloride were specified. However,the complete procedure used to obtain these values wasnot stated. Moreover, it was observed the DAB values forNaClH2O in the above two papers also differed slightly.In the present work, the solute molecular radius has beenestimated using the StokesEinstein equation [28]:

RS =

k0T

6DAB (28)

where the value of diffusivity of strong electrolytes at infinitedilution may be calculated from an equation obtained byNernst on the assumption of complete dissociation [29]:

D0A = 8.931 1010T

0+ 0

0+ + 0

z+ + z

z+ z(29)

A useful tabulation of ionic conductance at infinite dilutionin water at 25 C is given in Langes Handbook of Chemistry[30], which has been used in the above equation. Thesevalues are used to predict D0A and then, assuming the solution

to be diluted infinitely, equated to DAB for estimation of thesolute radius. Values of the above parameters for NaClH2Oand Na2SO4H2O systems are shown in Table 1.

At 25 C and 1atm, the density of water is taken as1000kg/m3 (1g/cm3) and viscosity as 1 103 Pas (1 cP)[31]. These values are assumed to be constant throughout the

operating conditions. Mehdizadeh and Dickson [27] used thevalue of viscosity of water as 8.965104 Pa s (0.8965cP).

2.7. Experimental observations

In order to estimate the parameters of any membranetransport model, experimental data are required. For thepresent work, experimental data available in literature[13,32] has been used. This section briefly mentions theexperimental conditions at which the above data was ob-tained. More details of experimental set-up and conditionscan be obtained from the above references.

The experimental data was obtained in sets of inlet con-

centrations of 17.10, 103.18 and 207.62mol/m3 (1000,6000, 12,000 ppm) for NaClH2O and 7.00, 42.51, 85.53,158.42 and 217.80mol/m3 (1000, 6000, 12,000, 22,000and 30,000 ppm) for Na2SO4H2O systems. For each inletconcentration, flow rate is kept at 5.0 106, 1.0 105,1.5 105, 2.0 105 and 2.5 105 m3 /s (300, 600,900, 1200 and 1500 ml/min). For each flow rate, pressureis kept at 2020, 3030, 4040, 6060, 8080 and 10,100 kPa(20, 30, 40, 60, 80 and 100atm). These three conditions,taken together, specify one set of operating conditions. Fora specific solutesolvent system, the concentration is ini-tially kept constant. For each concentration, the flow rate

is gradually increased, and at each flow rate, pressure isincremented. A sample data set collected on experimentalobservations on NaClH2O system using Cellulose Acetatemembrane is shown in Table 2.

3. Solution of membrane transport equations

3.1. Orthogonal collocation method

The SF-PF model used the RungeKutta technique tosolve the nonlinear differential equations [15]. This tech-nique has been found to be slow, inefficient and expensive

8/8/2019 2004 - JMS - Jain & Gupta

6/17


Table 2Sample experimental data

S. No. Pressure (Pa) CA3 (mol/m3) JV (m/s) RO

1. 2.02 106 4.0844 1.62 106 0.76132. 3.03 106 2.8353 2.81 106 0.83433. 4.04 106 2.1355 4.46 106 0.8752

4. 6.06

10

6

1.6598 6.87

10

6

0.90305. 8.08 106 1.5075 9.44 106 0.91196. 1.01 107 1.3980 1.28 105 0.9183

System: NaClH2O; CA1: 17.1111mol/m3; Q: 5.0 106 m3/s.

for solving the SF-PF model equations [27]. The orthogonalcollocation of weighted residuals is used to provide an accu-rate and efficient solution to the nonlinear differential equa-tions in the SF-PF and MD-SF-PF models. A brief mentionof the relevant details is made below and further details canbe obtained elsewhere [17,18].

3.1.1. Interior collocation

In order to estimate a dependent variable, in the presentcasedimensionless velocity, it is approximated by a se-ries of expansion containing n undetermined parameters. Fora symmetric second-order boundary value problem in onevariablein the present case the dimensionless velocityasa function of the dimensionless radial position, a suitableform is:

y = y(1) + (1 x2)n1i=0

aiPi(x2) (30)

In the present case, as (1) = 0, assuming the validity of theno slip boundary condition, the above equation reduces to:

= (1 2)n1i=0

aiPi(2) (31)

Using the values of the polynomials Pi(x2) as given in Table1 [17], for cylindrical geometry and n = 3; i.e. 3 unde-termined constants (a0, a1 and a2), the velocity profile be-comes:

(2)= (1 2)[a0 + a1(1 3 2)

+ a2(1 8 2 + 10 4)] (32)

3.1.2. Interior formulaeThe gradient and laplacian operators, in cylindrical coor-

dinates, for the (2) ofEq. (32) are given by:

d(2)d

=i

=

n+1j=1

Aij(2j ) (33)

1

d

d

1

d(2)d

=i

=

n+1j=1

Bij(2j ) (34)

for i = 1, 2, . . . , n + 1. The coefficients Aij and Bij whichhave been used in this work (for symmetrical cylindrical

geometry) can be obtained from Table 3 of Villadsen andStewart [17].

3.1.3. Residual function

For the system under consideration, the residual functionat the ith collocation point becomes:

R|i =

n+1j=1

[Bij(j)] +P [1 exp((i))]

1

(i)exp((i))

1

1

1b

1+

exp(())1

(35)

Theoretically, the value of the above function should be zeroat all collocation points. However, as the values of the pa-rameters (a0, a1 and a2) are not known a priori, the resid-ual function will have some finite value. As the iterationsprogress and the parameters approach their optimal value,the residues tend to zero.

3.1.4. Updating guess

The NewtonRaphson technique is employed to updatethe guess of the parameters used for the velocity profile:

xn+1 = xn F(xn)

J(xn)(36)

In the above equation, x represents the vector set of un-known parameters (a0, a1 and a2), n the iteration number,F the residual function value matrix and J the Jacobianmatrix.

3.1.5. Termination condition

Iterations to obtain the optimal set of parameters are con-tinued until the residues at all the collocation points arewithin the tolerance limit of 1010 and the maximum changein the value of any of the unknown parameters is also withinthe tolerance limit of 1010.

3.2. Analytical solutions

Analytical solutions have been used to test the validityof the MD-SF-PF model [16,27]. In the present work, wetoo used the formulae under the Poiseuille flow and totalrejection conditions to test our velocity profile trends underidentical operating conditions as specified in the above pa-pers and found that our estimates matched those publishedexcellently [23].

4. Parameter estimation

The modified surface force pore flow model along with theconcentration polarization model contains six parameters1, 2, /, RW, a and b. These parameters are determinedby the Downhill Simplex method as described below.

8/8/2019 2004 - JMS - Jain & Gupta

7/17


Table 3Data analysis for NaClH2O system

S. No. Parameter Unit Value

(A) Parameters used for data analysisa

1. 1 m 1.5007 109

2. 2 4.8739

3.R

W m 9.9917

10

10

4. / m 1.2644 104

5. b 0.77866. a (m/s)/(mol/m3)0.7786 1.0733

(B) Data analysis using above parameters(1) Operating condition

1.1. CA1 (mol/m3) 17.11 103.18 207.621.2. Conditionb BC BC P

(2) Error analysis (% rms averaged)c

2.1. CA3 6.17 8.63 10.272.2. JV 17.39 21.76 21.712.3. RO 1.50 2.35 2.572.4. k 3.15 2.18 2.052.5. E 13.05 16.55 16.98

a Note: these parameters are averaged parameters obtained from observations at 17.11 and 103.18mol/m3.b BC means back calculated, implying that the experimental data set is used for parameter estimation. P means predicted, implying that the data

set has not been used in parameter estimation in any way.c Error analysis compares experimental observations to and simulation estimations for CA3, JV and RO. k values are compared with CFSK model

predictions. E is the overall objective error function.

4.1. Optimization problem

Experimental values are available in terms of operatingconditions (inlet concentration, pressure drop and flow rate),permeate solute concentration and flux. The model equationsare solved from initial guesses of the six parameters and the

results compared with the experimental observations. Twooptimization functions are formed, first one, E1, from min-imization of the fractional difference in the calculated andexperimental permeate concentration and the other, E2, fromminimizing the fractional difference between the calculatedand experimental flux:

E1 =

ni=1(1 (C

iA3|Sim/C

iA3|Expt))

2

n(37)

E2 =

ni=1(1 (J

iV|Sim/J

iV|Expt))

2

n(38)

The fractional errors are used instead of absolute differences,as this would make the values independent of the units used,thus preventing a bias towards any one parameter. On com-bining these two functions, the overall objective functionbecomes:

E =

E1 + E2

2(39)

The aim of the optimization procedure is to find out thevalues of1, 2, RW, /, a and b at which E is very small,i.e., the theoretical predictions match the experimental oneswith minimum error.

4.2. Downhill simplex method

At this stage we have a six-parameter non-linear optimiza-tion problem. We could not identify a technique that wouldguarantee the global minimum solution to the problem, sowe combined an approach giving a local minimum and de-

veloped a search technique that would optimize the solution[23]. We selected the Downhill Simplex method [33]. Asstarting with any initial guess, this approach almost alwaysconverges to a local minimum.

The manner in which the optimization operations are car-ried out and estimation of intermediate values has been car-ried out follow the development in literature [33,34].

These following sub-sections briefly describe some of theimportant issues we had to address while using the DownhillSimplex method. More details can be found in Jain [23].

4.2.1. Initial guess

Ideally, an optimization technique should be independentof the initial guess. However, since the Downhill Simplexmethod is a local optimization technique, it is important tohave a very good initial guess. At first, the order of mag-nitude of each of the parameters is determined (1: 109,2 5, RW: 109, /: 104, a 1.0 and b 0.7). Thisis followed by simulation runs in which intermediate stepsare analyzed to check for any physical inconsistency. In casenone is observed, then the set is selected for preliminarysimulation purposes. This initial guess is then updated asmore simulations are carried out in an attempt to improvethe overall error. The automated initial guesses updation al-gorithm can be found in Jain [23].

8/8/2019 2004 - JMS - Jain & Gupta

8/17


4.2.2. Convergence criterion

The convergence of the Downhill Simplex method de-pends on the values of arbitrary constants in the reflection,contraction and expansion operations. It has been observedfast convergence is obtained by taking = 1, = 0.5 and = 2 [33,34]; therefore, we adopted the same values for

the present analysis.The convergence criterion is not applied with respect tothe changes in each optimization parameter but to the valueof the total functions, E1 and E2, represented as a whole in

E. The reasoning behind this approach is that in statisticalproblems where one is concerned with finding the minimumof a negative likelihood surface (or of a sum of a square sur-face) the curvature near the minimum gives the informationavailable on the unknown parameters.

The objective function is considered locally optimizedwhen its value becomes a constant while using the centroidparameters for computing its value. It is observed its valuecomes within 0.2 well before 30 iterations of the Downhill

Simplex loop. Further iterations are not carried out sinceno change is observed in the parameters and the error is aconstant. The above approach is acceptable since using theDownhill Simplex method often leads to local minima.

4.2.3. Interdependent parameters

Another issue to be considered is that all the five parame-ters affect both the error functionsE1 and E2. In case eitherflux or permeate concentration is to be changed, selectingany one parameter will affect both of them. For example, ifflux was to be decreased, one could think of increasing thepotential function, meaning increasing 1 and/or 2. How-

ever, as soon as the initial guess of the parameter is changed,in the orthogonal collocation convergence, the coefficientsa0, a1 and a2 will change. This would imply change in (),thus affecting JB and ultimately CA3. Thus, it is very difficultto predict the actual affect of changing a selected parameter.

4.2.4. Space constraints

In addition to the convergence criterion, there are con-straints on the operational space as well; i.e., all the pa-rameters are non-negative. This is incorporated by writingeach parameter in the objective function as the square of itsroot. It is also ensured the parameters have physical corre-lation throughout the simplex operations. A simple checkis maintaining the values of key physical quantitiesCA2,CA3, NA, NB, NT and JV as non-negative. Additionally, thelower limit of CA2 is fixed at CA1, upper limit of CA2 astwice CA1, upper limit of CA3 as CA1, and in accordancewith experimental observations, value of b was maintainedin the range of 0.330.8.

4.3. Monte Carlo search

It is often observed that the Downhill Simplex methodattained local minima after a few iterations itself. As the pa-rameters do not vary, the objective function E becomes a

constant. In order to explore the nearby surface, with smallvariations in the values of the parameters, an approach simi-lar to that adopted in genetic algorithms has been developedalong with the Downhill Simplex loop [23]. The algorithmof the Monte Carlo search technique has been described inAppendix A.

5. Results

5.1. Validation

The first aspect of the project is to solve the differentialequation for the velocity profile using the orthogonal collo-cation method. The present model solution code is validatedby comparing results with those presented by Mehdizadehand Dickson [27] under identical conditions. For this sec-tion, the same values of physical properties; i.e., diffusivity,solute radius and viscosity are taken which were used in the

original work [27]. Since, in the present code, the concen-tration polarization is taken into account while Mehdizadehand Dickson [27] had neglected it, CA2 is forced to be nearlyequal to CA1 by making mass transfer coefficient quite large.This is achieved by fixing a at 1000, about 108 times thevalue normally observed when predicting the mass trans-fer coefficient from experimental data, and b as 0, whichmakes the mass transfer coefficient independent of flow rateand its value sufficiently high for it to be considered infinite.

Our computed velocity profiles, actual rejection ratios andflux ratios are found to be in excellent agreement [23] withthose published by Mehdizadeh and Dickson [27] validating

the present orthogonal collocation code.

5.2. Concentration polarization model

The effect of introducing concentration polarization to re-late CA1 and CA2 is seen in Figs. 1 and 2. Parameters usedare 1: 5.4434109 m, 2: 0.4910, /: 1.0104 m, RW:1.3 109 m, a: 1000 m/s and b: 0. The operating condi-tions include temperature at 298 K, pressure as 4000 kPa andthe bulk feed concentration as 75 mol/m3. For these simula-tion runs, viscosity of water is taken as 0.001kg/ms (1 cP),diffusivity as 1.566 109 m/s and solute (NaCl) radius as1.555 1010 m (1.555 ).

It is desired to represent a range of values for the masstransfer coefficient for all flow regimesturbulent, laminarand intermediate. This is achieved by maintaining parameterb as 0, making mass transfer coefficient equal to a. Thevalue of a is changed from 103 to 5.0 106 m/s whichcovered the above flow regimes. Figs. 1 and 2 depict theeffect of increasing bulk feed concentration, with the masstransfer coefficient as a parameter, on rejection ratio andvolume flux respectively. The decreasing trends of rejectionratio and volume flux with increasing feed concentration arealong the expected lines. On the other hand, at a particu-lar value of the bulk feed concentration, both rejection ra-

8/8/2019 2004 - JMS - Jain & Gupta

9/17


Fig. 1. Effect of bulk feed concentration on observed rejection ratio with the mass transfer coefficient as a parameter. Operating conditions: T = 298K,P = 4000kPa and CA1 = 75mol/m3 for NaClH2O system. Model parameters: 1 = 5.4434 109 m, 2 = 0.4910, / = 1.0 104 m andRW = 1.3 109 m.

tio and volume flux increase with increasing mass transfercoefficient, which obviously is due to the reduced concen-tration polarization at higher mass transfer coefficients. Theabove graphs reflect the effect of mass transfer coefficientfrom 5 106 to 101 m/s. For higher values of the mass

transfer coefficients than 101 m/s the concentration polar-

Fig. 2. Effect of bulk feed concentration on flux with the mass transfer coefficient as a parameter. Operating conditions: T = 298K, P= 4000 kPa andCA1 = 75mol/m3 for NaClH2O system. Model parameters: 1 = 5.4434 109 m, 2 = 0.4910, / = 1.0 104 m and RW = 1.3 109 m.

ization has already become insignificant and lower valuesthan 5 106 m/s are not present in real reverse osmosisoperations.

When using the SpieglerKedem model [6], on increasingJV, 1/RO1 initially decreases, attains a minimum, and then

starts increasing (Section 2.2). To test this trend, the model

8/8/2019 2004 - JMS - Jain & Gupta

10/17


Fig. 3. Effect of concentration polarization on reciprocal of rejection ratio vs. flux with the mass transfer coefficient as a parameter. Operating conditions:T = 298 K and CA1 = 75mol/m3 for NaClH2O system. Pressure is incremented in steps of 1000kPa from a minimum of 1000kPa to a maximum of4000 kPa. Model parameters: 1 = 5.4434 109 m, 2 = 0.4910, / = 1.0 104 m and RW = 1.3 109 m.

solution code is run at a temperature of 298 K for a bulk feedconcentration of 75 mol/m3 of the NaClH2O solution withpressure being incremented in steps of 1000 kPa from 1000to 4000 kPa. The model parameters used are: 1: 5.4434109 m, 2: 0.4910, /: 1.0104 m,RW: 1.3109 mwiththe mass transfer coefficient having been varied from 5.0106 to 103 m/s. Observations at mass transfer coefficient of

greater than 2.5105

m/s show no discernible change fromthe trend at 2.5 105 m/s and so have not been includedin the graph (Fig. 3).

Looking at the above graphs, it can be said the concentra-tion polarization becomes significant when the mass trans-fer coefficient lies in 106104 m/s rangeessentially theregime in which most of the experimental data sets lie.

5.3. Model evaluation

Parameter estimation has been done in two steps. In thefirst step, all the experimental observations at some selectedbulk feed concentrations are used to estimate the parame-ters. These parameters are then used to back calculate theexperimental observations from which they are produced inorder to study the consistency of the original experimentalobservation.

In the second step, the above estimated parameters areaveraged to obtain the average parameters. These param-eters are then used to predict the membrane performance ofall available experimental conditions including those, whichwere not used in any way for parameter estimation.

For data analysis, we have preferred to examine errorfunctions E1, E2 and E, rather than the parametersindividually. E has been selected as the single point error

function encompassing the effects on both observed rejec-tion ratio and the volume flux. This approach is followedas parameters like 1 and 2 must be examined together,and not individually, for potential function evaluation. Asthe optimization process is fairly random, while 1 may beaffected in some iterations, others times 2 may be changed.With just 2 or 3 data sets being used for computing aver-

aged parameter sets, we felt carrying out an error analysiswould make more sense.For the case of NaClH2O system, all experimental data

at 17.11 and 103.18mol/m3 have been used separately forparameter estimation. When these parameters are used foranalysis of the same set of experimental data, the backcalculation errors (E) are 13.05 and 16.55% for 17.11 and103.18 mol/m3, respectively. The averaged parameters forthese two concentrations are given in Table 3. As shownin this table, using these averaged parameters to estimatethe membrane performance at 207.62 mol/m3, the resultsare found to be in excellent agreement with the experi-mental observations with E being 16.98%. Subsequently,these parameters are used to further predict the membraneperformance and the detailed results are discussed below(Section 5.4).

When a similar procedure is repeated for Na2SO4H2Osystem, with the parameters being estimated at 42.51 and158.42 mol/m3, the back calculation values errors (E) are15.51 and 14.24%, respectively. The averaged parameters forthese two concentrations are shown in Table 4. Using theseparameters, the errors (E) between the predicted and the ex-perimental observations for concentrations 7.00, 85.53 and217.80 mol/m3 are estimated and found to be 18.74, 12.90and 16.98% respectively. The above values in E clearly in-

8/8/2019 2004 - JMS - Jain & Gupta

11/17


Table 4Data analysis for Na2SO4H2O system

S. no. Parameter Unit Value

(A) Parameters used for data analysisa

1. 1 m 2.6144 109

2. 2 7.3283

3.R

W m 1.2090

10

9

4. / m 1.0404 104

5. b 0.77696. a (m/s)/(mol/m3)0.7769 0.8620

(B) Data analysis using above parameters(1) Operating condition

1.1. CA1 (mol/m3) 42.51 158.42 7.00 85.53 217.801.2. Conditionb BC BC P P P

(2) Error analysis (% rms averaged)c

2.1. CA3 10.69 10.60 13.60 8.27 15.332.2. JV 19.15 17.13 22.75 16.26 18.492.3. RO 0.47 0.64 0.53 0.44 1.342.4. k 2.00 8.27 5.81 5.81 3.852.5. E 15.51 14.24 18.74 12.90 16.98

a Note: these parameters are averaged parameters obtained from observations at 42.51 and 158.42mol/m3.b BC means back calculated, implying that the experimental data set is used for parameter estimation. P means predicted, implying that the data

set has not been used in parameter estimation in any way.c Error analysis compares experimental observations to and simulation estimations for CA3, JV and RO. k values are compared with CFSK model

predictions. E is the overall objective error function.

dicate the parameter estimation program has estimated mem-brane parameters, which predict the membrane performancefor unknown conditions quite accurately.

Since the experiments on both the systems are conductedon similar CelluloseAcetate membrane, the membrane spe-cific parameters, RW and /, are expected to remain largely

constant. Their values and the error analysis of these param-eters are compared for both the systems in Table 5. Whilemaking these membranes, Murthy [13] made several batchesof membranes. Although the same preparation techniqueand conditions were kept while making these membranes,some small differences between batches may have remained.This may be the reason for differences of 17.36% for RWand 21.53% for / when the estimations from NaClH2Oand Na2SO4H2O systems were compared as shown inTable 5.

5.4. Membrane performance prediction

In addition to the objective error function E, the mem-brane performance predictions are also carried out for thevolume flux and permeate concentration against pressure

Table 5Comparison of membrane specific parameters

S. no. System RW (m) / (m)

1. NaClH2O 9.9917 1010 1.2644 104

2. Na2SO4H2O 1.2090 109 1.0404 104

3. Error (%)a 17.36 21.53

a %Error = 100[1 parameter(NaCl)/parameter(Na2SO4)].

with flow rate as a parameter (Figs. 4 and 5) and the volumeflux and permeate concentration against bulk feed concen-tration with flow rate as a parameter (Figs. 6 and 7).

From Fig. 4, it can be seen the predictions match the ex-perimental observations very closely. However, the rejectionratios are predicted slightly better than the fluxes. In Fig. 5,

the simulation results excellently match the experimental ob-servations. Slightly higher errors of rejection ratio at lowerpressures are possible because of a lower than expected ex-perimental observation of rejection ratio at 3000kPa.

Figs. 6 and 7 have been prepared in order to show themembrane performance across the entire range of exper-imental conditions. For both the systems, the trend ofpermeates concentration against bulk feed concentrationmatches very closely with the experimental observations.For NaClH2O system, the maximum error at a flow rateof 5.0 106 m3/s and concentration of 207.62 mol/m3 isobserved to be 4.5%. The corresponding figure for flux atthe same operating conditions is observed to be 8.9%.

In Fig. 7, for Na2SO4H2O system, the permeate con-centration is plotted against bulk feed concentrations at aflow rate of 2.0105 m3/s. Errors between the MD-SF-PFmodel predictions and the experimental observations in per-meate concentration are just 2.6, 4.0 and 9.1% for bulk feedconcentrations of 7.00, 85.53 and 217.80 mol/m3, respec-tively. The corresponding values of errors for the flux at thesame operating conditions are 9.3, 0.9 and 22.5%, respec-tively. Figs. 47 and the above analysis clearly show theMD-SF-PF model has been successful in estimating the pa-rameters and in predicting the membrane performance veryclosely.

8/8/2019 2004 - JMS - Jain & Gupta

12/17


Fig. 4. Effect of pressure on rejection ratio and flux for NaClH 2O system with flow rate (m3 /s) as a parameter. S represents simulated resultsand E the experimental observations of Murthy [13]. Operating conditions: T = 298K and CA1 = 207.62 mol/m3 (1000ppm). Model parameters:1 = 1.5007 109 m, 2 = 4.8739, / = 1.2644 104 m and RW = 9.9917 1010 m, b = 0.7786 and a = 1.0733 (m/s)/(m3/s)0.7786.

5.5. Mass transfer coefficient

Mass transfer coefficient and its parameters a andb are computed from the present parameters estimationanalysis and compared with the values obtained from thecombined film theorySpieglerKedem (CFSK) model.Murthy [13] estimated the mass transfer coefficient and

Fig. 5. Effect of pressure on rejection ratio and flux for Na2SO4H2O system with flow rate (m3 /s) as a parameter. S represents simulated resultsand E the experimental observations of Murthy [13]. Operating conditions: T = 298K and CA1 = 85.53mol/m3 (1000ppm). Model parameters:1 = 2.6144 109 m, 2 = 7.3283, / = 1.0404 104 m, RW = 1.2090 109 m, b = 0.7769 and a = 0.8620 (m/s)/(m3/s)0.7769.

the coefficient parameters from the CFSK model usingthe Box-Kanemasu method [35]. When the mass transfercoefficient parameters from the two different methods arecompared for both NaClH2O and Na2SO4H2O systems inTable 6, the maximum error in any parameter, for any sys-tem, is around 3% implying the predicted values from MD-SF-PF and the CFSK models are in excellent agreement.

8/8/2019 2004 - JMS - Jain & Gupta

13/17


Fig. 6. Effect of bulk feed concentration on permeate concentration and flux for NaClH 2O system with flow rate (m3/s) as a parameter. S represents

simulated results and E the experimental observations of Murthy [13]. Operating conditions: T = 298K and P = 4,040kPa. Model parameters:1 = 1.5007 109 m, 2 = 4.8739, / = 1.2644 104 m, RW = 9.9917 1010 m, b = 0.7786 and a = 0.0733 (m/s)/(m3/s)0.7786.

5.6. Extrapolated results

When the reciprocal of the observed rejection is plottedagainst volume flux for both NaClH2O and Na2SO4H2Osystems under consideration, minima is not found within theexperimental region. This necessitated the need to study thesystem in an extrapolated region; i.e., beyond the experimen-tal region. As the experimental data are limited to 10,100 kPa(100 atm), the simulations are carried out to cover the pres-

Fig. 7. Effect of bulk feed concentration on permeate concentration and flux for Na2SO4H2O system with flow rate (m3/s) as a parameter. S representssimulated results and E the experimental observations of Murthy [13]. Operating conditions: T = 298K and P = 10,100kPa. Model parameters:1 = 2.6144 109 m, 2 = 7.3283, / = 1.0404 104 m, RW = 1.2090 109 m, b = 0.7769 and a = 0.8620 (m/s)/(m3/s)0.7769.

sure range of 2,00040,000 kPa (19.8396.0atm). Plots forthe systems under consideration are shown in Figs. 8 and9. Our results indicate the JminV would occur at 23,500 kPa(232.67 atm) for NaClH2O and at 11,500 kPa (113.86 atm)for Na2SO4H2O.

The value of JminV estimated through the MD-SF-PFmodel for NaClH2O system is 2.41 105 m/s and forthe Na2SO4H2O system is 2.01 105 m/s. Correspond-ing values for these systems estimated using the CFSK

8/8/2019 2004 - JMS - Jain & Gupta

14/17


Fig. 8. Effect of flux on rejection ratio for NaClH 2O system. S represents simulated results using MD-SF-PF model, CFSK are values obtained throughcombined filmSpieglerKedem model and E the experimental observations of Murthy [13]. Operating conditions: T = 298K, CA1 = 103.18mol/m3,Q = 1.5 106 m3 /s and P= 2,00040,000 kPa. MD-SF-PF model parameters: 1 = 1.5007 109 m, 2 = 4.8739, / = 1.2644 104 m,RW = 9.9917 1010 m, b = 0.7786 and a = 1.0733 (m/s)/(m3/s)0.7786. SpieglerKedem model parameters: = 0.9395, PM = 4.168 107 m/s,b = 0.7763 and a = 1.0355 (m/s)/(m3/s)0.7763 [13].

model are 2.29 105 and 2.19 105 m/s, respectively.Assuming the CFSK model estimated values as the base,this means an error of5.36% for the NaClH2O systemand an error of 8.08% for the Na2SO4H2O system. Look-

Fig. 9. Effect of flux on rejection ratio for Na2SO4H2O system. S represents simulated results using MD-SF-PF model, CFSK are values obtained throughcombined filmSpieglerKedem model and E the experimental observations of Murthy [13]. Operating conditions: T = 298K, CA1 = 103.18mol/m3,Q = 1.5 106 m3 /s and P = 200040,000 kPa. MD-SF-PF model parameters: 1 = 2.6144 109 m, 2 = 7.3283, / = 1.0404 104 m,RW = 1.2090 109 m, b = 0.7769 and a = 0.8620 (m/s)/(m3/s)0.7769. SpieglerKedem model parameters: = 0.9982, PM = 1.508 107 m/s,b = 0.7769 and a = 0.8395 (m/s)/(m3/s)0.7769 [13].

ing at the above error calculations and the overall trendsdisplayed in the figures, the MD-SF-PF model predictionsof JminV are in excellent agreement with the CFSK estima-tions.

8/8/2019 2004 - JMS - Jain & Gupta

15/17


Table 6Comparison of mass transfer coefficient parameters

S. No. Property NaClH2O Na2SO4H2O

1. Parameters a b a b2. Units a a 3. Predictionb 1.0683 0.7786 0.8620 0.7769

4. CFSK

c

1.0355 0.7763 0.8395 0.77695. Error (%)d 3.16 0.29 2.68 0.00

a Units of a: (m/s)/(m3/s)bb Values from simulation based on the MD-SF-PF model in the present

work.c Values from the CFSK model parameters have been obtained from

Murthy [13].d %Error = 100[1 k(Sim.)/k(CFSK)].

6. Conclusions

The addition of concentration polarization to the MD-

SF-PF model resulted in a six-parameter estimation prob-lem which has been solved by using Monte Carlo searchand automated guess updation combined with the Down-hill Simplex technique. The estimated parameters are usedto predict the membrane performance over a range of op-erating conditions, and the simulated results compare verywell with the experimental observations. The membrane spe-cific parameters of NaClH2O and Na2SO4H2O systemson a similar cellulose acetate membrane have been shownto be very close to each other. The rejection ratio versus theflux trend and the predicted JminV values for the system un-der consideration show agreement with the predictions ofSpieglerKedem model. All other observations indicate the

success of MD-SF-PF model as its predictions are foundto be in agreement with the SpieglerKedem modelthemodel recommended by Mason and Lonsdale [12] for re-verse osmosis membranes. Being a mechanism based model,the advantage of the MD-SF-PF model when combinedwith concentration polarization over the SpieglerKedemmodel is that its parameters can be related to the mem-brane structure, the solutesolvent and solutemembraneinteractions.

Acknowledgements

The authors would like to thank Mr. J.K. Jain, Sys-tems Programmer, Department of Chemical Engineering,IIT-Delhi, Mr. Om Vir Singh from the Department of Chem-ical Engineering and Mr. Deepak Garg from the Departmentof Computer Science & Engineering who assisted us withthe coding aspects. Assistance and support extended by Dr.Rajesh Khanna and Dr. Kamal K. Pant, faculty members inthe Chemical Engineering department, is also gratefully ac-knowledged. The authors would also like to thank the twoanonymous reviewers whose numerous suggestions helpedimprove the quality of the paper significantly.

Nomenclature

a unknown in film mass transfer equation(Eq. (27)), (m/s)/(m3/s)b

a0 unknown parameter for velocity profile(Eq. (32))

a1 unknown parameter for velocity profile(Eq. (32))a2 unknown parameter for velocity profile

(Eq. (32))ai unknown parameter for the estimation of the

velocity profile (Eq. (30))Aij first derivative coefficient matrix for

orthogonal collocation (Eq. (33))b unknown in film mass transfer equation

(Eq. (27))b() frictional parameter (Eq. (18))

Bij second derivative coefficient matrix fororthogonal collocation (Eq. (34))

C molar density of solution (mol/m3)CAi solute concentration at ith position (mol/m3)CiA3 solute concentration in permeate solution

during ith observation (mol/m3)D0A solute diffusivity in infinitely diluted solution

(m2/s)DAB solute diffusivity in solution (m2/s)E objective error function (Eq. (56))E1 error in permeate concentration (Eq. (54))E2 error in flux (Eq. (55))F (xn) residual function value matrix (Eq. (36))

JA radially averaged solute flux through a singlepore (mol/m2 s)JB radially averaged solvent flux through a

single pore (mol/m2 s)J (xn) Jacobian matrix (Eq. (36))JV volume flux (m/s)JiV volume flux at ith observation (m/s)JminV value of flux, JV, when 1/RO 1 is

minimum when plotted against JV (m/s)k film mass transfer coefficient (m/s)kO Boltzmanns constant (J/K)l0+ cationic conductance at infinite dilution

(mho/equivalent)l0 anionic conductance at infinite dilution

(mho/equivalent)l0+ + l

0 electrolyte conductance at infinite dilution

(mho/equivalent)LP hydraulic permeability coefficient of the

membrane (m2 s/kg)n number of experimental observations

Ni flux of component i through membrane(mol/m3)

NiT total flux during ith observation (mol/s)

8/8/2019 2004 - JMS - Jain & Gupta

16/17


p pressure drop across the membrane (Pa)P ratio of pressure drop across the membrane to

osmotic pressure at boundary layer (Eq. (11))PA solute permeabilityCFSK

phenomenological parameter (m2/s)PB water permeabilityCFSK

phenomenological parameter (m3 s/kg)Pe Peclet numberPe value ofPe, for which 1/RO 1 is minimum

when plotted against JV (m/s)Pi polynomial for evaluation of the

dimensionless velocity profile (Eq. (30))PM overall membrane permeability coefficient

(m/s)Q flow rate (m3/s)r radial coordinate (m)

R gas constant (J/mol K)RA actual rejection ratio, difference of ratio of

membrane surface concentration to thepermeate concentration from 1

RO observed rejection ratioRe Reynolds numberR|i residual function evaluated at ith

dimensionless radial position (Eq. (30))RS solute radius (Eq. (28)) (m)RW membrane pore radius (m)Sc Schmidt numberSh Sherwood numberT temperature of solution (K)

x independent variable for orthogonalcollocation equations

x1 experimentally determined coefficient forestimating Sh (Eq. (26))



XAB ratio of the product of gas constant andsolution temperature to solute diffusivity insolution (kg/mols)

xn vector set of parameters (a0, a1 and a2) at nth

iteration (Eq. (36))xn+1 vector set of parameters (a0, a1 and a2) at

(n + 1)th iteration (Eq. (36))x membrane thickness (m)

y dependent variable for orthogonal collocationequations

z+ absolute value of cation valencez absolute value of anion valence

Greek letters

parameter used in downhill simplex method() solute velocity

parameter used in downhill simplex method1 parameter used in model solution (Eq. (9))ij frictional constant between i and j (Js/m2 mol) fractional pore area of membrane() dimensionless potential function parameter used in Downhill Simplex method solution viscosity (Pa s) ratio of solute radius to the membrane pore

radiusi osmotic pressure at location i (Pa) osmotic pressure across the membrane (Pa) ratio of difference of osmotic pressure at

boundary layer and permeate to the osmoticpressure at boundary

1 unknown in surface potential function (m)2 unknown parameter in surface potential

function ratio of radial position to pore radiusi radial position of ith interior collocation point

(Eq. (33)) reflection coefficient average pore length taking tortuosity into

account (m)

Subscripts

A soluteB solventE experimental observationS simulated resultT total solutionW pore wall1 feed solution2 boundary layer solution3 permeate solution

Appendix A. Monte Carlo search algorithm

The Downhill Simplex method converges to a minimum,which is most likely, a local one. In an attempt to identify the

global minima of the problem, the nearby surface solutionneeds to be explored. This is accomplished by adding aMonte Carlo search algorithm [23]. The integration of theMonte Carlo search with the Downhill Simplex routine hasbeen done in the following manner:

1. This procedure is activated at every third iteration of thesimplex loop to start after four iterations of the simplexloop have been completed. The objective of this step isto examine the nearby area of a potential local minimaalready identified by the downhill simplex method.

2. Values of E for the past three iterations are compared.If error becomes constant, or consistently increases, it in-

8/8/2019 2004 - JMS - Jain & Gupta

17/17


dicates possible local minima might have been attained.For all other cases the Downhill Simplex routine is per-mitted to continue without any deviations.

3. Only the first six simplex vertices of the total seven areexamined as a six-parameter combination is being han-dled. This is done to prevent any undue weightage being

assigned to any parameter. In case the parameter serialnumber is equal to the vertex serial number, it is furthertested (serial order of parameters: 1, 2, RW, /, a andb).

4. The variation probability of a parameters value has beenfixed at 0.5. Thus, if the randomly generated probabilitychange, between 0.0 and 1.0, for the parameter is greaterthan the requirement of 0.5, the parameters value changefactor is computed.

5. A multiplication change factor is randomly generated ina user-defined range, normally 0.71.3, for the parameterunder consideration. The new value of the parameter isdetermined by multiplying the old one by this factor.

6. The resulting values of simplex vertices and the set ofparameters is accepted only if the error after the abovedescribed change is less than what it was before it car-rying out the algorithm.

References

[1] L. Onsager, Reciprocal relations in irreversible processes. Part I,Phys. Rev. 37 (1931) 405.

[2] L. Onsager, Reciprocal relations in irreversible processes. Part II,Phys. Rev. 38 (1931) 2265.

[3] O. Kedem, A. Katchalsky, Thermodynamic analysis of the perme-

ability of biological membranes to non-electrolytes, Biochem. Bio-phys. Acta 27 (1958) 229246.

[4] O. Kedem, A. Katchalsky, Permeability of composite membranes.Part 3. Series array of elements, Trans. Faraday Soc. 59 (1963)19411953.

[5] A. Katchalsky, P.F. Curran, Non-Equilibrium Thermodynamics inBiophysics, Harvard University Press, Cambridge, MA, 1975.

[6] K.S. Spiegler, O. Kedem, Thermodynamics of hyperfiltration (reverseosmosis): criteria for efficient membranes, Desalination 1 (1966) 311.

[7] H.K. Lonsdale, U. Merten, R.L. Riley, Transport properties of cel-lulose acetate osmotic membranes, J. Appl. Polym. Sci. 9 (1965)13411362.

[8] U. Merten (Ed.), Desalination by Reverse Osmosis, MIT Press,Cambridge, MA, 1966 (Chapter 2).

[9] T.K. Sherwood, P.L.T. Brian, R.E. Fisher, Desalination by reverse

osmosis, Ind. Eng. Chem. Fund. 6 (1) (1967) 212.[10] G. Jonsson, C.E. Boesen, Water and solute transport through cellulose

acetate reverse osmosis membranes, Desalination 17 (2) (1975) 145165.

[11] M. Soltanieh, W.N. Gill, Review of reverse osmosis membranes andtransport models, Chem. Eng. Comm. 12 (1981) 279.

[12] E.A. Mason, H.K. Lonsdale, Statistical mechanical theory of mem-brane transport, J. Membr. Sci. 51 (1990) 1.

[13] Z.V.P. Murthy, Studies on membrane transport models, Ph.D. thesis,I.I.T. Delhi, July 1996.

[14] Z.V.P. Murthy, S.K. Gupta, Sodium cyanide separation and param-eter estimation for reverse osmosis thin film composite polyamidemembrane, J. Membr. Sci. 154 (1999) 89103.

[15] T. Matsuura, S. Sourirajan, Reverse osmosis transport through cap-illary pores under the influence of surface forces, Ind. Eng. Chem.20 (2) (1981) 273282.

[16] H. Mehdizadeh, J.M. Dickson, Theoretical modification of the Sur-face Force Pore Flow model for reverse osmosis transport, J. Membr.Sci. 42 (1989) 119145.

[17] J.V. Villadsen, W.E. Stewart, Solution of boundary value problemsby orthogonal collocation, Chem. Eng. Sci. 22 (1967) 14831501.

[18] J.V. Villadsen, M.L. Michelsen, Solution of Differential EquationModels by Polynomial Approximation, Prentice-Hall, EnglevoodCliffs, 1978.

[19] H. Mehdizadeh, J.M. Dickson, Evaluation of Surface Force PoreFlow and Modified Surface Force Pore Flow models for reverseosmosis transport, Chem. Eng. Comm. 103 (1991) 6582.

[20] H. Mehdizadeh, J.M. Dickson, P.K. Eriksson, Temperature effects onthe performance of thin-film composite aromatic polyamide mem-branes, I & EC Res. 28 (1989) 814824.

[21] T. Matsuura, Y. Taketani, S. Sourirajan, Estimation of interfacialforces governing the reverse-osmosis system: nonionized polar or-ganic solute-water-cellulose acetate membrane, in: A.F. Turbak (Ed.),Synthetic Membranes, ACS Symposium Series, American ChemicalSociety, Washington, DC, 1981 (Chapter 19).

[22] K. Chan, T. Matsuura, S. Sourirajan, Interfacial forces, average poresize and pore size distribution of ultrafiltration membranes, Ind. Eng.Chem. Prod. Res. Dev. 21 (4) (1982) 605612.

[23] S. Jain, Analysis of Modified Surface Force Pore Flow model withconcentration polarization and comparison with SpieglerKedemmodel in reverse osmosis systems, M. Tech. thesis, I.I.T. Delhi, May2003.

[24] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, Wi-ley, New York, 1960.

[25] S.T. Hwang, K. Kammermeyer, Membranes in Separation, Wiley,

New York, 1975.[26] M.C. Porter, Concentration polarization with membrane ultrafilter-ation, Ind. Eng. Chem. Prod. Res. Dev. 11 (3) (1972) 234248.

[27] H. Mehdizadeh, J.M. Dickson, Solving non-linear differential equa-tions of membrane transport by orthogonal collocation, Comput.Chem. Eng. 14 (2) (1990) 157160.

[28] E.L. Cussler, Diffusion: Mass Transfer in Fluid Systems, CambridgeUniversity Press, Cambridge, 1984.

[29] A.H.P. Skelland, Diffusional Mass Transfer, Wiley/Interscience, USA,1974.

[30] J.A. Dean (Ed.), Langes Handbook of Chemistry, 15/e, McGrawHill, 1999, pp. 8.1578.160.

[31] W.L. McCabe, J.C. Smith, P. Harriott, Unit Operations of ChemicalEngineering, McGraw-Hill, 5th(Intl.)/e, 1993, pp. 10941102.

[32] Z.V.P. Murthy, S.K. Gupta, Estimation of mass transfer coefficient

using a combined nonlinear membrane transport and film theorymodel, Desalination 109 (1997) 3949.[33] J.A. Nelder, R. Mead, Simplex method for function minimization,

Comput. J. 7 (1965) 308313.[34] J.L. Dellis, J.L. Carpentier, Nelder and Mead algorithm in impedance

spectra fitting, Solid State Ionics 62 (1993) 119123.[35] J.V. Beck, K.J. Arnold, Parameter Estimation in Engineering and

Science, Wiley, NY, 1977.

2004 - jms - jain & gupta

Documents