Indian Journal of Chemical Technology Vol. 8, September 2001, pp. 335-343

Reverse osmosis transport models evaluation: A new approach

Samir Y Vaidya, Amit V Simaria & Z V P Murthy*

Department of Chemical Engineering, Dharmsinh Desai Institute of Technology (Deemed University), P.B.No. 35, College Road, Nadiad 387 001, India

Received 8 February 2001; revised 12 June 2001; accepted 2 July 200/

Various transport models have been developed to describe the transport phenomena through reverse osmosis membrane. Such models g ive relationship amongst the operating variables, such as rejection, flux and driving force and membrane parameters that may include the physical properties of membrane. Knowing membrane parameters, one can predict the performance of membrane with minimum number of experiments, and can be extended to whole range of operating parameters without performing experiments. The available models are based on two concepts. One is on the basis of irreversible thermodynamics, and the others on the transport mechanisms. At present, the membrane transport models are used with the experimental data to estimate the parameters involved assuming that the mass transfer coefficient is available in the literature. The disadvantages of the same were reported recently in the literature. In the present paper an attempt has been made to evaluate the various models available in combination with the film theory .The experimental data obtained on the membrane is used to estimate the membrane parameters and mass transfer coefficient simultaneously, which will have more re levance than the existing methods. With this combined method, the models are compared.

Various membrane transport models are proposed to estimate the performance of reverse osmosis (RO) membranes 1• These models involve different parameters, knowledge of which enables one to predict the membrane performance with minimum number of experiments. In the present paper, various models are compared based on the parameters evaluated from experimental data.

Till date, transport model parameters were evaluated using the mass transfer coefficient data from the literature. Due to discrepancies and limited applicability of theoretical estimations of mass transfer coefficients, errors may creep m the simulation of membrane processes2•

The objective of this work is to develop a method, which eliminates the need for mass transfer coefficient estimated from literature. Using this method, mass transfer coefficient can be obtained simultaneously with membrane parameters, by nonlinear curve fitting. Various transport models are clubbed with film theory, which introduces mass transfer coefficient in the final model as one of the parameters.

The method requires very less amount of experimental data for parameters estimation, using

*For correspondence Present address: Department of Chemical Engineering, S.V. Regional College of Engineering and Technology, Surat 395 007, India. (Fax: 0261 228394; E-mail: znivi @vikram.svrec.ernet.in)

which the process can be simulated for a wider range of operating conditions.

Transport models Membrane transport models have been derived

from two independent general approaches 1• First are the models based on non-equilibrium or irreversible thermodynamics, where the membrane is treated as a black box in which relatively slow processes are taking place near equilibrium. No information is needed on the mechanism of transport. In the second approach, some mechanism of transport is assumed and accordingly, the fl uxes are related to the forces that exist in the system. Thus, physiochemical properties of membrane and system are involved in the model.

Various membrane transport models can be clubbed with film theory. According to this theory, a relation among the membrane true rejection (Ri), observed rejection (R0 ), flux (Jv) and mass transfer coefficient (k) can be written as3

ln(l-Ro)=ln(l-R;)+ l v Ro R, k

(I)

A brief description of each model is given below: Kedem-Katchalsky Mode/1-It is based on the

concept of irreversible thermodynamics, clubbed with phenomenological relations, for transfer of non­electrolytes as well as electrolytes. This model assumes that the Onsager's reciprocal relationships

336 INDIAN J. CHEM. TECHNOL., SEPTEMBER 2001

(ORR), which is valid when the system is close to equilibrium, and the linear laws related to fluxes and forces are valid. But for systems that are far from equilibrium, as is often the case in RO, the ORR may not be correct. The validity of the ORR has been discussed in the literature'. It is a three parameter model. Basic equations correlating the fluxes are

l v = Lp (~p- crMt) (2a)

}A= A~n+(l-a)CA;.Jv (2b)

where

CA'" = CA2- CAJ = - CA2Ri ln(CAJCAJ) ln(l-R;)

(All the nomenclature given at the end.) Substituting Eq.(3) in Eq.(2),

(1- 0' )CMJvR; J A = c AJJ v = A ~7l' - ....:.._---';-----;--

In (1- R;) Assuming n=aCA and simplifying,

00 0 (3)

00 0 (4)

1- R; Aa 1-a ---=-- ) ... (5)

R; Jv ln(l- R;

Substituting Eq.(l) in Eq.(5) will give

1- Ro [ 1- a Aa] -----;;;;-- = In{l- Ro(l- exp(lvlk ))}-ln(l- Ro) + lv

ex{ ~v ) 00 0 (6)

Expanding logarithmic term up to two terms and si mplifying,

1 R;o = [ 1 + 2~a + [ 1 + z~a J -2:~' }x{- ~' ) Here the parameters are

a 1 = 112Aa

G) = Ilk

00 0 (7)

Spiegler-Kedem Model1- This model is also based on irreversible thermodynamics concept, and involves three parameters. The main assumption in the Kedem­Katchalsky model is that the linear laws are assumed to apply over the whole thickness of the membrane. This model resolved the problem and starts with a local (or differential) equation for fluxes. The volumetric flux equation will be same as Eq.(2a) and the solute flux is given by

JA = p{ d~A )+ (1-a)cAiv 000 (8)

Putting JA = CA3lv

( dCA) . p s ----;;;- + [ (1 - 0') c A - c A3 ] j v = Q 000 (9)

Integrating Eq.(9) with boundary

CA=CA3 and x=~x, CA=CA2

limits as x=O,

CA2 Ill

f dCA f }vdx O (1 -a )cA- CAJ + P. =

CAJ 0

CAJ- CA2(1 -a) = exj _ lv(l- a)) aCAJ ~ PM

where

PM=.!!:_ Ill

Making R; as subject of equation

000 (10)

000 (I I)

I I a j- Jv(I- a)) 1-R; = 1-a- 1-a ex~ PM 00 0 (1 2)

Substituting Eq.(l) in Eq.(12) will result4

l~;o = l~a[l - ex{- Jv~: a))}x{- ~v) Here the parameters are

a,= crl(l-cr)

a2 = (1-cr)l PM

OJ= 1/k

000 (13)

Solution Diffusion model1 --Thjs model is based on transport mechanism where both solvent and solute dissolves in the homogeneous, nonporous surface layer of the membrane. Hence, coupling is not included in this model. It involves two parameters only. Basic equations for fluxes are

lv=A(~p- ~n)

JA = B(CA2-CAJ)= CAJ}v

B CA3 =---

Jv CA2- CAJ Substituting Ri in Eq.(l5), gives

1+~= CA2 = 1 }v CA2 - CAJ R;

Using Eq.(l ) and Eq.(l6) one can get

~= 1-R; = 1-Ro exj _ Jv) Jv R; Ro ~ k

__!}::___ = }v exj- }v) 1- Ro B ~ k Here, the parameters are

a 1 = liB

(l4a)

(14b)

000 (1 5)

000 (16)

(1 7)

(18)

VAIDY A et al.: REVERSE OSMOSIS TRANSPORT MODELS EVALUATION: A NEW APPROACH 337

az= Ilk

The main restriction of this model is that the separation obtained at infinite flux is always equal to unity. However, this limit is not reached for many solutes.

Highly Porous Model1-This is a three parameter model for which the transport mechanism is given by Hagen-Poiseuille equation. It includes the coupling through viscous flow. The fluxes are given by

lv=A(dp/dx)

Ns=UCA + JA

The boundary conditions are given as

At x=O, CA=ksCA2

at x=TD, CA.;_ksCA3

Also,

(19a)

(19b)

Ns = (Eu)CA3 (20) And }A=-Dsw(dCA/dx) (21)

Integrating Eq.(19) using these boundary conditions and using definition of R; will result in

1-IR; = :s +( k:: e }x{-Jv e~w) (22)

__!!::____ =(~ -III-exj- }v__!!__)]exj- Jv) I - Ro ks 1. eDsw 1. k

Substituting Eq.(l) in Eq.(22), we get Here the parameters are

a1 =(Elks)- I

a2 = TD/(EDsw)

a3= llk

(23)

Finely Porous Mode/1-This is an extension of highly porous model, which includes the friction effect between solid molecules and the membrane pore wall. A factor b is introduced to take into account the friction effect. It is given as

b =I+ J~.Jfsw (24)

Here, the volumetric flux equation is same as Eq.(19a) and the solute flux is given as

dCA CA Ns= -De--+-u

dx b

Based on this, Eq.(22) is modified to give

_I =be +(ks-be)exj -}v~) 1- R; ks ks 1_ ebDe

. . . (25)

(26)

Finally, using Eq. (1) with Eq.(26), one can get

~=(be -111-exj -Jv ~)]exj- }v) 1 - Ro ks 1. ebDe 1. k

Here the parameters are

a1 = (bEiks)- 1

a2=TDIEbDe

a3= Ilk

Experimental Procedure

... (27)

The RO experiments were performed with the cellulose acetate membranes prepared in the laboratory by the phase inversion method of Manjikian5. The conditions and composition of the membrane, and the details of the experimental setup are reported elswhere2.4·6. A disc-shaped flat membrane housing was used for the experiments. Initially the membrane was pressure pretreated at 50 atm over night, and then for about 3 h at II 0 atm using distilled water to avoid membrane compaction during separation operation. After the above stabilization process, pure water permeability is measured at different operating pressures. The separation data was obtained for a sodium chloride­water system between the concentrations I 000 to 30000 ppm brine solutions are prepared from sodium chloride (Merck) and distilled water. The feed solutions, about 12 L, are prepared by taking a calculated quantity of salt dissolved in distilled water. After rinsing the experimental lines with some of the brine, the system is operated initially for about 2 h to reach quasi-steady state. The pressure is regulated using a pressure regulating valve. Two samples of permeate solution, to measure flux rate and concentration, are collected during 45 min for very reading at a certain pressure. The feed and product samples are analyzed by the conductivity method (Global Electronics, Hyderabad) at 25°C. The feed rate varied from 300 to 1500 mUmin, the operating pressure from 20 to 100 atm, and the system temperature maintained at around 25°C using a heat exchanger coil.

Parameters estimation All the above models are complex and non-linear

in nature. Experimental data6 is fitted to these models. One of the simplest and most effective methods of minimizing the sum of squares function is Gauss linearization method. It is attractive because it is relatively simple and because it specifies direction and size of the correction to the parameter vector.

338 INDIAN J. CHEM. TECHNOL., SEJYfEMBER 2001

Table !-Experimental and calculated flux ver~us rejection data for NaCI-water system using cellulose acetate membrane prepared by Manjikian method

Set CAl Q D.P Ro lvx 104 R o l R o2 Ro3 %Error! %Error2 %Error3 ppm mL!min atm crn/s X 103 X 103 X 103

1000 300 20 0.76 13 1.62 0.7614 0.7635 0.7566 14.6 287.6 610.9 1000 300 30 0.8343 2.81 0.8342 0.8327 0.8322 17.9 194.7 253.4 1000 300 40 0.8752 4.46 0.8753 0.8746 0.875 1 8.4 68.1 12.2 1000 300 60 0.9003 6.87 0.9003 0.9006 0.9010 0.8 32.5 81.4 1000 300 80 0.9 11 9 9.44 0.91 19 0.9124 0.9126 0.9 53.7 71.8 1000 300 100 0.9183 12.76 0.9 183 0.9180 0.9178 0.3 32.0 58.2

2 1000 600 20 0.7773 1.78 0.7772 0.7791 0.7727 7.7 237.3 590.4 1000 600 30 0.8408 2.94 0.8408 0.8395 0.8389 1.9 152.4 227.6 1000 600 40 0.883 1 4.82 0.883 1 0.8826 0.8831 2.7 60.9 2.0 1000 600 60 0.904 1 6.98 0.904 1 0.9043 0.9048 2.5 23.4 76.3 1000 600 80 0.9 168 9.81 0.9 168 0.9173 0.9174 1.0 49.5 67.2 1000 600 100 0.9233 12.95 0.9233 0.9230 0.9228 0.2 27.7 54.0

3 1000 900 20 0.7890 1.92 0.7890 0.7909 0.7845 0.1 239.9 573.6 1000 900 30 0.8444 3.02 0.8444 0.8432 0.8423 0.4 143.8 243.9 1000 900 40 0.8863 4.98 0.8863 0.8857 0.8862 3.6 72.7 8.2 1000 900 60 0.9067 7.22 0.9067 0.9070 0.9075 5.1 27.8 87.9 1000 900 80 0.9 191 10. 11 0.9 19 1 0.9195 0.9 197 2.8 46.2 67.3 1000 900 100 0.9256 13.4 0.9256 0.9254 0.9251 0.6 25.9 55.7

4 1000 1200 20 0.809 1 2.22 0.8092 0.8106 0.8053 6.7 184.4 470.9 1000 1200 30 0.8478 3. 11 0.8478 0.8470 0.8458 2.7 100.1 236.4 1000 1200 40 0.8895 5. 19 0.8895 0.8889 0.8895 1.9 69.8 4.2 1000 1200 60 0.9093 7.54 0.9093 0.9095 0.9 10 1 2.7 17.0 87.5 1000 1200 80 0.92 16 10.78 0.92 16 0.9220 0.9222 2.8 46.0 66.1 1000 1200 100 0.9270 13.76 0.9270 0.9268 0.9264 0.9 26.3 59.9

5 1000 1500 20 0.8 178 2.38 0.8 187 0.8200 0.8149 107.2 267.4 357.9 1000 1500 30 0.8539 3.26 0.853 1 0.8523 0.8511 95.2 189.2 332.4 1000 1500 40 0.89 14 5.32 0.89 15 0.8909 0.89 16 15.7 54.4 21.0 1000 1500 60 0.9 105 7.69 0.9 106 0.9107 0.9 114 6.5 23.8 99.9 1000 1500 80 0.9224 10.89 0.9223 0.9228 0.9230 6.2 39.4 64.4 1000 1500 100 0.9279 14. 12 0.9279 0.9277 0.9273 1.7 23.5 60.7

6 6000 300 20 0.7 127 1.22 0.7 126 0.7153 0.7066 16.9 368.4 858.7 6000 300 30 0.8061 2.22 0.8061 0.8036 0.8032 3.8 304.7 364.9 6000 300 40 0.8703 4.19 0.8703 0.8696 0.8700 1.2 82.6 36.4 6000 300 60 0.8962 6.36 0.8963 0.8966 0.8969 6.5 44.6 79.1 6000 300 80 0.9101 9.01 0.9 10 1 0.9106 0.9107 4.5 57.7 69.4 6000 300 100 0.9 173 12.38 0.9173 0.9170 0.9 168 1.0 34.9 52.0

7 6000 600 20 0.7546 1.54 0.7545 0.7712 0.7550 6.8 2 196.9 58.1 6000 600 30 0.8310 2.31 0.8 130 0.8 180 0.8143 2163.7 1567.5 2013.8 6000 600 40 0.8762 4.38 0.8762 0.8769 0.8781 2.7 77.9 220.1 6000 600 60 0.9022 6.74 0.9022 0.9029 0.9040 3.9 74.0 203.4 6000 600 80 0.9147 9.23 0.9 147 0.9152 0.9157 2.3 56.1 109.6 6000 600 100 0.9227 12.77 0.9227 0.9223 0.9217 0.4 43.0 109.4

8 6000 900 20 0.7761 1.75 0.7757 0.7777 0.7708 57.3 2 10.0 678.9 6000 900 30 0.8 140 2.3 1 0.8140 0.8131 0.8108 1.8 11 6.4 398.5 6000 900 40 0.8835 4.79 0.8836 0.8828 0.8835 10.3 77.7 0.7 6000 900 60 0.9050 6.98 0.9050 0.9052 0.9058 3.6 18.5 83.9 6000 900 80 0.9 182 9.89 0.9 181 0.9186 0.9 188 6.2 4 1.6 62.7 6000 900 100 0.9248 13.1 1 0.9248 0.9246 0.9243 1.9 2 1.9 53.4

Contd.

VAIDYA eta/.: REVERSE OSMOSIS TRANSPORT MODELS EVALUATION: A NEW APPROACH 339

Table ! -Experimental and calculated flux versus rejection data for NaCI-water system using cellulose acetate membrane prepared by Manjikian method-Con/d.

Set CAl Q t:J.P Ro lvx l04 Rol Ro2 Ro3 %Error! %Error2 %Error3 ppm mL!min atm cm/s X 103 X 103 X 103

9 6000 1200 20 0.7839 1.85 0.7839 0.7856 0.7794 6.1 210.7 576.4 6000 1200 30 0.8179 2.38 0.8 179 0.8170 0.8 147 3.0 113.2 388.5 6000 1200 40 0.8859 4.92 0.8859 0.8852 0.8859 1.0 82.4 4.7 6000 1200 60 0.9070 7. 18 0.9070 0.9072 0.9077 0.8 18.7 82.6 6000 1200 80 0.9 197 10.11 0.9 197 0.9201 0.9203 0.3 45.0 64.4 6000 1200 100 0.926 1 13.21 0.9261 0.9259 0.9256 0.0 24. 1 55.1

10 6000 1500 20 0.7895 1.92 0.7894 0.7910 0.7847 16.9 194.6 604.0 6000 1500 30 0.82 10 2.44 0.8210 0.8202 0.8 178 3.0 94.6 392.7 6000 1500 40 0.8875 5.02 0.8875 0.8867 0.8875 2.3 85.2 4.0 6000 1500 60 0.9088 7.41 0.9088 0.9090 0.9096 1.4 17.9 85.0 6000 1500 80 0.9206 10.24 0.9206 0.9210 0.9212 2.0 44.0 66.6 6000 1500 100 0.9272 13.55 0.9272 0.9270 0.9267 0.6 22.7 56.4

I I 12000 300 20 0.6802 1.03 0.6803 0.6824 0.6737 12.0 329.8 948.4 12000 300 30 0.7880 1.94 0.7880 0.7848 0.7846 1.8 411.1 426.0 12000 300 40 0.8644 3.89 0.8644 0.8637 0.8640 1.5 85.1 51.4 12000 300 60 0.8942 6.11 0.8942 0.8946 0.8948 3.9 43.7 65.1 12000 300 80 0.9094 8.86 0.9094 0.9100 0.9 10 1 2.5 68.2 74.4 12000 300 100 0.9 168 12. 11 0.9 168 0.9 164 0.9 163 0.6 41.1 51.0

12 12000 600 20 0.7 140 1.23 0.7 150 0.7169 0.7090 138.9 409.0 706.6 12000 600 30 0.8009 2. 11 0.8009 0.7984 0.7977 3.7 3 18.2 398.1 12000 600 40 0.87 18 4. 14 0.87 17 0.8709 0.87 13 13.7 101.0 58.4 12000 600 60 0.9008 6.55 0.9007 0.9011 0.90 14 7.9 33.0 63. 1 12000 600 80 0.9 14 1 9. 11 0.9 142 0.9147 0.9 148 9.6 67.8 78.2 12000 600 100 0.9222 12.43 0.9222 0.92 19 0.9217 2.5 36.7 51. 1

13 12000 900 20 0.755 1 1.54 0.7554 0.7576 0.7500 43.6 326.7 675.6 12000 900 30 0.8100 2.24 0.8 100 0.8083 0.8067 1.5 207.0 411.5 12000 900 40 0.88 10 4.62 0.88 10 0.8802 0.8808 3.2 86.8 22.1 12000 900 60 0.9039 6.82 0.9038 0.9041 0.9045 5.8 19.2 70.8 12000 900 80 0.9 17 1 9.60 0.9 172 0.9176 0.9178 5.6 58.0 75.2 12000 900 100 0.9244 12.87 0.9244 0.9241 0.9239 1.5 29.6 54.4

14 12000 1200 20 0.778 1 1.78 0.78 16 0.7801 0.7769 449.2 258.0 158.7 12000 1200 30 0.8198 2.42 0.8226 0.8205 0.8 196 337.9 80.3 26.2 12000 1200 40 0.8854 4.60 0.8826 0.8821 0.8824 315.9 369.6 333.3 12000 1200 60 0.906 1 7.04 0.9068 0.9073 0.9076 78.0 134.8 165.5 12000 1200 80 0.9 193 9.98 0.9 195 0.920 1 0.9202 25.8 86.5 96. 1 12000 1200 100 0.9260 13. 14 0.9259 0.9255 0.9253 14.2 58.4 73.1

15 12000 1500 20 0.7872 1.89 0.79 16 0.7895 0.7869 563.5 289.1 37.3 12000 1500 30 0.8245 2.5 1 0.8280 0.8257 0.8250 426.7 150.0 58.7 12000 1500 40 0.8870 4.60 0.8834 0.8829 0.8832 408.3 463.3 431.9 12000 1500 60 0.9083 7.32 0.909 1 0.9098 0.9 100 91.2 160.3 187.1 12000 1500 80 0.9202 10.12 0.9205 0.9211 0.92 12 32.3 99.2 108.3 12000 1500 100 0.9270 13.42 0.9268 0.9264 0.9262 18.0 69.4 82.9

16 22000 300 20 0.6572 0.92 0.6573 0.6580 0.6504 9.0 117.4 1032.7 22000 300 40 0.8635 3.85 0.8635 0.8629 0.863 1 0.2 71.4 50.6 22000 300 60 0.894 1 6. 11 0.8941 0.8946 0.8947 0.7 58.0 68.4 22000 300 80 0.9092 8.82 0.9092 0.9098 0.9098 0.1 44.2 49.3 22000 300 100 0.9 164 11 .88 0.9164 0.9160 0.9159 0.1 44.2 49.3

17 22000 600 20 0.6871 1.05 0.6860 0.6892 0.6793 166.2 308.8 1140.4 22000 600 40 0.8707 4.08 0.8708 0.8701 0.8704 12.7 63.9 29.0 22000 600 60 0.8990 6.33 0.8990 0.8995 0.8997 4.6 51.9 74.4

Contd

340 INDIAN J. CHEM. TECHNOL., SEPTEMBER 2001

Table 1-Experimental and calculated flux versus rejection data for NaCl-water system using cellulose acetate membrane prepared by Manjikian method-Contd.

Set CAl Q t.P Ro lvxl04 Rol R o2 R o) %Error I %Error2 %Error3 ppm mL!min atm cm/s X 103 X 1.03 X 103

22000 600 80 0.9140 9.08 0.9139 0.9145 0.9145 7.1 52.2 58.4 22000 600 100 0.9214 12.13 0.9214 0.9211 0.9210 2.2 34.8 45.2

18 22000 900 20 0.6983 1.12 0.6983 0.7008 0.6914 6.6 360.6 987.4 22000 900 40 0.8777 4.42 0.8777 0.8771 0.8774 0.8 63.8 28.6 22000 900 60 0.9028 6.67 0.9028 0.9032 0.9034 0.8 48.0 71.4 22000 900 80 0.9169 9.49 0.9169 0.9174 0.9175 0.8 55.5 61.7 22000 900 100 0.9240 12.52 0.9240 0.9237 0.9236 0.2 36.4 47.5

19 22000 1200 20 0.7177 1.25 0.7189 0.7207 0.7122 171.2 421.5 764.9 22000 1200 40 0.8822 4.66 0.8820 0.8814 0.8818 22.1 85.5 46.4 22000 1200 60 0.9056 6.98 0.9056 0.9060 0.9063 1.6 47.0 74.3 22000 1200 80 0.9182 9.66 0.9183 0.9188 0.9189 6.1 62.7 71.2 22000 1200 100 0.9256 12.89 0.9256 0.9252 0.9251 2.1 38.1 51.9

20 22000 1500 20 0.7292 1.32 0.7292 0.7321 0.7225 5.0 401.3 913.3 22000 1500 40 0.8847 4.82 0.8847 0.8841 0.8846 2.6 63.2 13.5 22000 1500 60 0.9069 7.11 0.9069 0.9072 0.9076 3.33 36.4 72.2 22000 1500 80 0.9197 9.98 0.9197 0.9202 0.9203 1.8 54.6 64.6 22000 1500 100 0.9264 13.07 0.9264 0.9261 0.9259 0.4 34.0 51.7

2 1 30000 300 40 0.8628 3.82 0.8628 0.8630 0.8618 3.7 26.5 115.4 30000 300 50 0.8839 5.11 0.8839 0.8837 0.8838 2.1 23.8 11.2 30000 300 60 0.8932 6.01 0.8932 0.8931 0.8935 1.7 14.2 30.4 30000 300 80 0.9089 8.76 0.8932 0.8931 0.8935 1.7 14.2 30.4 30000 300 100 0.9159 11.61 0.9159 0.9158 0.9156 0.4 7.7 31.9

22 30000 600 40 0.8691 4.01 0.8691 0.8693 0.8681 0.1 24.6 11 6.9 30000 600 50 0.8884 5.26 0.8884 0.8882 0.8883 1.0 20.0 11.4 30000 600 60 0.8980 6.22 0.8980 0.8979 0.8983 1.1 14.6 30.1 30000 600 80 0.9 137 9.02 0.9137 0.9139 0.9142 0.3 19. 1 56.0 30000 600 100 0.92 13 12.01 0.9213 0.9212 0.9210 0.1 6.2 29.8

23 30000 900 40 0.8743 4.23 0.8743 0.8745 0.8732 5.5 20.2 124.4 30000 900 50 0.8941 5.67 0.8940 0.8938 0.8940 8.7 29.1 14.3 30000 900 60 0.9019 6.58 0.9021 0.9019 0.9023 16.8 4.0 49.0 30000 900 80 0.9167 9.43 0.9166 0.9168 0.9171 6.6 11.0 46.4 30000 900 100 0.9237 12.36 0.9237 0.9237 0.9234 1.5" 4.1 28.5

24 30000 1200 40 0.8786 4.44 0.8787 0.8789 0.8777 10.9 28.7 99.3 30000 1200 50 0.8960 5.78 0.8959 0.8957 0.8958 11.2 29.5 18.7 30000 1200 60 0.9040 6.74 0.9040 0.9039 0.9043 0.4 10.2 32.7 30000 1200 80 0.9179 9.57 0.9179 0.9181 0.9184 3.1 21.1 56.1 30000 1200 100 0.9249 12.52 0.9249 0.9248 0.9246 0.9 7.1 30.6

R0 ~o Error I = Rejection and error(%) for data - fitting using Spiegler-Kedem, Finely & Highly porous models R 02, Error 2 = Rejection and error(%) for data - fitting using Kedem-Katchalsky models R 03, Error 3 =Rejection and error(%) for data -fitting using Solution-Diffusion models

This method is effective in seeking minima that are Kanemasu method is favoured for nonlinear reasonably well-defined provided the initial estimates parameter estimation because of its advantages with are in the general region of the minimum. For difficult respect to the relative computer time needed, power to

cases (i.e., those with indistinct minima) solve difficult cases and probability of convergence7

modifications to the Gauss method are used. Some of In all the above models, the two variables are the widely used modifications to the Gauss method

Y = Ro f(l-R0 ) are Levenberg, Marquardt and Box-Kanemasu methods. Out of the above methods, the Box- x=lv

VAIDYA et al.: REVERSE OSMOSIS TRANSPORT MODELS EVALUATION: A NEW APPROACH 34 1

0.95

0.9

0.85

0 0.8 a:

0.75

0.7

·o.65

0 2 4 6 8 10 12 14

Jv x to• c:JD/1

Fig. 1- Result for Finely Porous model for 6'h set of data of NaCI-Water system.

0.95=----- ---------------1

0.9

0.85

~ • 0.8

0.75

0.7

0 2 4 6 10 12 14

Fig. 2-Result for Spiegler-Kedem model for 7'h set of data of NaCI-Water system.

From x and y data, parameters a 1, a 2 are estimated. Using these values, various quantities like mass transfer coefficient. diffusivity, etc. can be determined.

Results and Discussion The data supplied to the nonlinear parameter

estimation program, based on the Box-Kanemasu modification to the Gauss method, are Ro vs. l v taken at different operating pressures keeping feed rate and feed concentration constant for each set of data given in Table I. The parameters estimated for various models from Eqs.(7), (13), (18), (23) and (27) (i.e., a" a 2 and a3) are used to find the membrane transport parameters and mass transfer coefficient from the respective relations and they are shown in Table 2. These parameters are in turn used to calculate

0.94

0.92

0.9

0.88

0.86 0 ~

0.84

0.82 • E:q>erimeWI

0.8 Theoretical

0.78

0.76

0 2 4 6 8 10 12 14

Jv sto' cm/s

Fig. 3-Result for Kedem-Katchalsky model for 9'h set of data of NaCI-Water system.

0 . 95 ,------------------.

0 .9

0 .85

0 .8

0 .75 • ExperimeoW

Theoretical

0 . 7 +---.---...-----.,----..-~--.------1

0 2 4 8 8 10 12 14

Fig. 4-Result for Solution-Diffusion model for 12'h set of data of NaCI-Water system.

observed rejection coefficient (R0 ) of the membrane for different values of flux (Jv). with respect to various models, and reported in Table 1.

The results in Table 1 show that the calculated R0

from various models have different error ranges. The calculated Ro from Spiegler-Kedem (SK)

model show very negligible error (% Errorl) when compared with experimental values. In case of other models, even though error is very less with respect to experimental values, errors in calculated R0 from Kedem-Katchalsky (KK) model (%Error2) are approximately 10 to 100 times more than that by SK model. Similarly, for Solution Diffusion (SO) model , errors (%Error3) are 20 to 200 times more than that by SK model. The significance of these errors may be

342 INDIAN J. CHEM. TECHNOL., SEPTEMBER 2001

Table 2- Parameters estimated using data-fitting method for various models, for NaCl-water system

Set Spiegler-Kedem Porous .

Kedem-Katchalsk~ Solution-Di ffusion cr PMx l 05 kx l04 (b )Elk, (b)EDABftO cr Aax105 kxl 04 Bx l05 kxl04

I 2 3 4 5 6 7 8 9

10 II 12 13 14 15 16 17 18 19 20 21 22 23 24

0.940 0.940 0.940 0.940 0.939 0.940 0.940 0.939 0.940 0.940 0.940 0.940 0.939 0.939 0.939 0.939 0.939 0.940 0.940 0.940 0.940 0.940 0.940 0.939

cm/s

4.171 4.175 4.170 4.168 4.147 4.174 4.174 4.157 4.168 4.175 4.170 4.185 4.159 4.069 4.029 4. 169 4.153 4.173 4.179 4.173 4.167 4. 178 4. 179 4.160

cm/s

81.91 8 140.82

197.283 244.140 288.639 79.440 132.532 183.683 237.440 275.302 7~13

133:209 •186.106 206.977 265.200 79.325 133.230 183.851 227.825 271 .058 79.043 134.193 186.846 228.198

*bin bracket is included for Finely Porous model

16.608 16.609 16.546 16.549 16.486 16.559 16.625 16.506 16.534 16.580 16.536 16.628 16.459 16.496 16.313 16.476 16.469 16.554 16.591 16.552 16.456 16.531 16.533 16.5 17

more pronounced when the calculated Ra values from different models are used to predict volume flux and other membrane parameters. A graphical comparison is also made through Figs 1 to 4 showing that all the sets are equally fitting.

The values of transport parameters involved in each model and mass transfer coefficients are shown in Table 2. All the models show that the parameters are reasonably constant over the range of operating variables, except the mass transfer coefficient (k). The solute and solvent transport parameters of SK and SD models are more accurate than that of KK model. The mass transfer coefficients estimated through the models differ considerably. The k values from SK model show considerable variations with feed flow rate, which is what one would expect theoretically. But the k values estimated from KK and SD models show negligible variations. It is expected that the k values from KK model should be nearer to that of k values by SK model. But that are, nearer to that of SD model. This discrepancy may be due to the shortcomings of model formulation 1 and approximating the logarithmic quantity involved. The reason for negligible variation in k values of SD model were reported elsewhere2.4. The k values

x l04 cm/s cm/s cm/s

6.927 0.9492 4.776 14.601 4.647 14.174 6.935 0.9346 4.816 15.937 4.663 15.362 6.901 0.91 63 4.860 16.631 4.675 15.915 6.898 0.8869 4.917 17.149 4.682 16.239 6.837 0.8634 4.944 17.422 4.676 16.396 6.9 12 0.9789 4.713 14.261 4.644 14.018 6.939 0.8564 4.837 15.749 4.492 14.482 6.862 0.9296 4.829 16.378 4.653 15.681 6.892 0.9249 4.849 16.815 4.667 16.068 6.992 0.9141 4.885 17.195 4.684 16.362 6.895 0.9926 4.666 14.045 4.632 13.9 17 6.959 0.9805 4.721 15.451 4.657 15. 190 6.845 0.958 4.780 16.147 4.652 15.623 6.7 12 0.9660 4.641 15.889 4.560 15.571 6.573 0.9668 4.61 1 15.950 4.536 15.658 6.870 0.9990 4.64 1 13.920 4.627 13.864 6.839 0.9895 4.663 15.118 4.621 14.943 6.907 0.9857 4.703 15.853 4.653 15.639 6.934 0.9780 4.738 16.331 4.672 16.046 6.907 0.9675 4.764 16.621 4.673 16.227 6.857 0.8114 4.980 14.897 4.660 13.964 6.906 0.7810 5.042 16.439 4.684 15.247 6.910 0.7430 5.107 17.272 4.706 15.881 6.870 0.7243 5.1 12 17.583 4.697 16.129

estimated from all the models are used to obtain a Dittus-Boelter type relation by non-l inear fitting, which can be represented by simpler relation shown below.

Here n = 0.76 for SK model n = 0.42 for KK model n = 0.41 for SD model

Conclusion

kocQ"

There are many models described in the literature to represent/predict membrane performance. Estimation of membrane transport parameters and mass transfer coefficients were made separately in the literature. In the present work, membrane transport model and film theory are combined to estimate the parameters involved and mass transfer coefficient, simultaneously, from the same expelimental data. In the present case, the transport parameters of each model are reasonably constant over the given range of operating variables. Even though the %errors with respect to the experimental values are small for all the models, the magnitude of %errors of KK and SD models over SK model are very high. These errors

V AIDY A et al.: REVERSE OSMOSIS TRANSPORT MODELS EVALUATION: A NEW APPROACH 343

may propagate when the calculated Ro values are used for predicting volume flux and other membrane parameters. In case of k values too, values from SK model are nearer to that of real values. In the absence of any other reliable method for the prediction/evaluation of membrane performance and mass transfer coefficient, one may use the SK model clubbed with film theory.

Nomenclature

A = Solvent permeability coefficient a = van't Hoff constant in osmotic pressure equation b = Factor for measure of friction between the solute

molecules and the membrane pore wall 8 Solute permeability coefficient CA Local solute concentration per unit membrane volume CA 1 Solute concentration in the feed CA2 Solute concentration at membrane surface on feed side

CA3 Solute concentration in permeate

Logarithmic average of solute and solvent concentration across membrane

Effective diffusion coefficient of the solute in a micropore

Diffusion coefficient of solute in water

Friction coefficient between solute and membrane

Friction coefficient between solute and water (solvent)

Solute flux through membrane Total volumetric flux membrane Mass transfer coefficient

Partition coefficient of solute in membrane with respect to total membrane volume

Hydraulic permeability constant

Ps Q R;

Solute flux with respect to membrane

Solvent permeability coefficient

Local solute permeability per unit membrane thickness Local solute permeability in the membrane Feed flow rate in mL!min.

True rejection coefficient of membrane

= (CA2-CA3)/CA2 Observed rejection coefficient of membrane

= (CAI-CA3)/CAI

u Local centre of mass velocity of the pore fluid

tu 8

l'lP

Mt E

(J

t

Total membrane thickness Skin layer thickness of asymmetric membrane

Pressure difference across the membrane

Osmotic pressure difference across membrane Void fraction of the membrane Reflection coefficient; 0 for no rejection ; I for total rejection Tortuosity of the membrane

References I Soltanieh M & Gill W N, Chem Eng Commun, 12 (1981) 279. 2 Murthy Z V P & Gupta S K, Sep Sci Techno/, 31 (1996) 77. 3 Lonsdale H K, in Synthetic Menzbranes: Science, Engineering

and Applications, edited by Bungay P M, Lonsdale H K & de Pinho M N, NATO-Advanced Study Institute Series, vol.no. l 81 (D.Reidel Publishing Company, Dordrecht, Holland), 1986, 307.

4 Murthy Z V P & Gupta S K, Desalination, 109 (1997) 39. 5 Manjikian S, Ind. Eng Chem Prod Res Dev, 6 ( 1967) 23. 6 Murthy Z V P, Ph D Thesis, Studies on reverse osmosis

membrane transport models, Chemical Engineering Department, I IT, Delhi, India, 1996.

7 Beck J V & Arnold K J, Parameter Estimation in Engineering and Science (John Wiley and Sons, New York), 1977, Chapter 7.