Analysis of Reverse Osmosis Data for the System Polyethylene Glycol

414

Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979

multiple species chemical interactions.

Kretschmer, C. B.. Nowakowska, J., Wiebe, R., J. Am. Chem. SOC.,70, 1795

Conclusion The CLAM model (Renon and Prausnitz, 1967) has been shown to be capable of predicting VLE data from hE data with accuracy of between 5 and 10% for binary mixtures of hydrocarbons and alcohols when used in the method of Hanks et al. (1971).

Kretschmer, C. B., Wiebe, R., J . Chem. Phys., 22, 1697 (1954). Kudryavtseva, L. S.,et al., Zh. Prikl. Khim., 36(7), 1471 11963). Lee, L. L., et al., J. Chem. Ens. Data, 12(4).497 (1967). Lee, S . C., J. Phys. Chem., 35(2),3559 (1931). Marina, J. M., Tassbs, D. P., Ind. Eng. Chem. Process Des. Dev., 12,67 (1973). Mevers. R. S . . Clever. H. L.. J . Chem. Thermodvn.. 2. 53 (19701. - ~, Mrazek; R. V.; Van Ness, H. C., AIChE J., 7, 190 (1961). Neau, E., et al. J . Chim. Phys., 70(5),843 (1973). Nguyen, T. H., Ratcliff, G. A., J. Chem. Eng. Data, 20(3), 252 (1975). Niini, A., Ann. Acad. Sci. Fenn., A55, No. 8 (1940). Nitta, T., Katayama, T., J. Chem. Eng. (Jpn.), 6(1),l(1973). Orye, R. V., Prausnitz, J. M., Ind. Eng. Chem., 57, 18 (1965). Pennington, R. H., "Introductory Computer Methods and Numerical Analysis", 2nd ed, Macmillan, London, 1965. Prabhu, P. S., Van Winkle, M., J. Chem. Eng. Data, 8(1),14 (1963a). Prabhu, P. S.,Van Winkle, M., J. Chem. Eng. Data, 8(2).210 (1963b). Prausnitz, J. M., "Molecular Thermodynamics of FluaPhase Equilibria", Chapters 6 and 7, Prentice-Hall, Englewood Cliffs, N.J., 1969. Prigogine, I., Mathot, V., Desmyter, A., Bull. Soc. Chim. Be@.,58,547 (1949). Prigogine, I., "Molecular Theory of Solutions," North-Holland Publishing Co., Amsterdam, 1957. Rao, P. R., Chipanjivi, C., Dasarao, C. J., J. Appl. Chem.. 18(6),166 (1968). Ratcliff, G. A., Chao, K. C., Can. J . Chem. Eng., 47, 151 (1969). Redlich, O.,Kister, A. T., J . Chem. Phys., 15(12).849 (1947). Renon, H., Prausnitz, J. M., Chem. Eng. Sci., 22,299 (1967);Errata, 22, 1891

(1948).

Literature Cited Abrams, D. S., Prausnitz, J. M., AIChE J . , 21(1),116 (1975). Asselineau, L., Renon, H., Chem. Eng. Sci., 25, 121 1 (1970). Barker, J. A., J. Chem. Phys., 20, 794 (1952a);20, 1526 (1952b). Benedict, M., et al., Trans. Am. Inst. Chem. Eng., 41,371 (1945). Bonauguri, E., Bicelli, L.. Spiller, G., Chim. Ind. (Milan), 33, 81 (1951). Brown, I., Fock, W., Aust. J. Chem., 14,387 (1961). Brown, I., Smith, F., Aust. J. Chem., 12, 407 (1959). Brown, I., et al., Aust. J. Chem., 17, 1106 (1965). Byer, S. M., Gibbs. R. E., Van Ness, H. C., AIChE J., 19(2),245 (1973). Christensen, J. J., O'Neill, T. K., Rossiter, B. E.,Jr., Hanks, R. W., J. Chem. Thermcdyn.. IO. 829 (1978). Christensen, J. J., Izatt, R. M., Stitt, B. D., Hanks, R. W., J . Chem. Thermodyn.,

Downloaded by RUTGERS UNIV on September 9, 2015 | http://pubs.acs.org Publication Date: July 1, 1979 | doi: 10.1021/i260071a011

11 (1979). Ellis, S. R. M., Spurr, M. J., Brit. Chem. Eng., 6,92 (1961). Ferguson, J. B., J . Phys. Chem., 36, 1123 (1932). Ferguson, J. B., Freed, M., Morris, A. C., J . Phys. Chem., 37, 87 (1933). Flory, P. J., J. Chem. Phys., 12(11),425 (1944). Fredenslund, A., Jones, R. L., Prausnitz, J. M., AIChE J.. 21(6),1089 (1975). Galska-Krajewska, A., Rocz. Chem. 7 , 40(5),863 (1966). Galska-Krajewska, A., Rocz. Chem. 7 , 41(3),609 (1967). Garber, Yu N., et al., J. Appl. Chem. USSR, 35, 392 (1962). Gelbin, D., Chem. Technik, 13,710 (1961). Goates, J. R., et al., J. Phys. Chem., 65,335 (1961). Gurukui, S. M. K. A., et al., J. Chem. Eng. Data, 11(4),501 (1966). Hanks, R. W., Gupta, A. C., Christensen, J. J., Ind. Eng. Chem. fundam., 10,

504 (1971). Hanks, R. W., Tan, R. L., Christensen, J. J., Thermochim. Acta, 23(1),41 (1978a). Hanks, R. W., Tan, R. L., Christensen, J. J., Thermochim. Acta, 27(1),(1978b). Haskell. R. W., Holinger, H. B., Van Ness, H. C., J. Phys. Chem., 72,4534

(1968). Hsu, K. Y., Clever, H. L., J. Chem. Eng. Data, 20(3),268 (1975). Hwa, S.C. P., Ziegler, W. T., J. Phys. Chem., 70, 2572 (1966). Kogan, V. B., Fridman, V. M.. Romanoya, T. G., Russ. J. Phys. Chem., 33(7),

1

~

~~

.

~

(1967). Renon, H., Prausnitz, J. M., AIChE J., 14, 135 (1968). Sadler, L. Y., et ai., J. Chem. Eng. Data, 16(4),446 (1971). Sarolea-Mathot, L., Trans. faraday SOC.,49,8 (1953). Savini, C. G., et al., J. Chem. Eng. Data, 10(2),168 (1965). Seetharamaswamy, et al., J. Appi. Chem., 19,258 (1969). Smirnova. N. A., et al., Zh. f i z . Khim., 43, 1883 (1969). Smith, C. P.. Engei, E. W., J. Am. Chem. Soc.. 51,2660 (1929). Smith, V. C.,et al., J. Chem. Eng. Data, 15(3),391 (1970). Stookey, D. J., Smith, B. D., Ind. Eng. Chem. Process Des. Dev., 12,372

(1973). Tan, R. L., Hanks, R. W., Christensen, J. J., Thermochim.Acta, 21(2) 157 (1977). Tan, R. L., Hanks, R. W.. Christensen, J. J., h e r m h i m . Acta, 23(1),29 (1978). Tompa, H., J. Chem. Phys., 21,250 (1953). Van Ness, H. C., et al., J. Chem. Eng. Data, 12(2),217 (1967a). Van Ness, H. C., et al., J. Chem. Eng. Data, 12(3),346 (1967b). Wiehe, I. A., Bagley, E. B., Ind. Eng. Chem. Fundam., 6,209 (1967). Wilson, G. M., J . Am. Chem. Soc., 88, 127 (1964).

Receiued f o r review February 9, 1978 Accepted March 24, 1979

34 (1959).

Analysis of Reverse Osmosis Data for the System Polyethylene Glycol-Water-Cellulose Acetate Membrane at Low Operating Pressures Fu-Hung Hsieh, Takeshi Matsuura, and S. Sourirajan' Division of Chemistry, National Research Council of Canada, Ottawa, Canada, K7A OR9

Reverse osmosis separations of polyethylene glycol (PEG) solutes PEG6000, PEG9000, PEG15000, and PEG-20000 in aqueous solutions in the concentration range 50 to 5000 ppm of solute have been studied using porous cellulose acetate membranes in the operating pressure range 25 to 100 psig. While the feed solution systems involving PEG-6000, PEG9000, or PEG-15000 behaved normally, the system involving PEG20000 showed the characteristics of pre-gel polarization and retarded flow of pre-gel through membrane pores during reverse osmosis. On the basis of analysis of all experimental results, the necessary physicochemical data and practical techniques have been generated for predicting membrane performance (both solute separation and product rate) for reverse osmosis separations of different PEG solutes in aqueous solutions from only a single set of experimental data for a reference feed solution system such as one involving a dilute solution of PEG-6000.

Introduction Reverse osmosis systems involving polyethylene glycol (PEG) solutes having the general formula HO(CH2CH,0),CH2CH20Hin the weight-average molecular 0019-7882/79/1118-0414$01.00/0

weight (I$,) range 200 to 6750 in dilute aqueous solutions have been studied using porous cellulose acetate membranes at operating pressures of 100 psig or less (Hsieh et al., 1979). The results showed that in the above systems, 0 1979 American

Chemical Society


415

Table I . Membrane Specifications f r o m Reverse Osmosis Data for t h e Feed Solution System 50 p p m PEG-6000-Water film number

1 operating pressure, a t m pure water permeability constant, A x l o 6 g-mol of H, O / c m 2 s a t m solute separation, % product rate, g/ha operating pressure, a t m pure water permeability constant, A x l o 6 g-mol of H , O / c m Z s a t m solute separation, % p r o d u c t rate, g/ha solute transport parameter for PEG-6000, D,,/Ks X lo6,c m / s

In


a

C*NaCl

4

5

6

2.31 4.80

2.21 8.70

2

2.11 9.32

3

2.01 10.90

1.80 18.80

1.91 30.20

97.0 9.29 7.42 4.53

92.0 16.16 7.32 8.10

91.2 16.56 7.22 8.50

87.4 18.65 7.11 9.74

78.0 28.43 6.91 16.60

62.3 49.25 3.61 29.70

95.1 28.44 2.76

84.6 49.77 12.0

79.4 51.60 12.4

68.9 58.42 19.4

34.2 96.42 45.4

35.4 89.92 79.5

-8.08

-6.61

-6.58

-6.13

-5.28

-4.72

Membrane area = 13.2 e m 2 .

there was no indication of the occurrence of pre-gel or gel-polarization phenomena (Blatt et al., 1970), and the PEG solutes could be treated just as other simple nonionized polar organic solutes in dilute aqueous solutions (Matsuura et al., 1976, 1977) in reverse osmosis systems where water was preferentially sorbed a t the membranesolution interface. On the basis of the above results, numerical values for the parameters representing the polar (interfacial free energy change, -AhG/RT), steric (6*CE,) and nonpolar ( w * x s * ) forces governing reverse osmosis separations of PEG solutes have been generated (Hsieh e t al., 1979). This paper extends the above work and includes PEG solutes in the A&, range 6750 to 23 800, solute concentrations in feed solutions ranging from 50 to 5000 ppm, and reverse osmosis operation in the pressure range 25 to 100 psig. The polar, steric, and nonpolar parameters generated in previous work offer a means of specifying the porous structure of cellulose acetate membranes for which the average pore size on the membrane surface is too big t o give sufficient reverse osmosis separation for NaCl for purposes of precise membrane specification in terms of solute transport parameter (Dm/Kd) for NaCl (Sourirajan, 1970). This work is concerned with such membranes, for = 6750) was the specifications of which PEG-6000 (MW used as the reference solute. Such specifications for the membrane, along with the basic transport equations (Sourirajan, 1970; Sourirajan and Matsuura, 197713) and the other relevant physicochemical data, offer a technique for predicting membrane performance (f and (PR)) in reverse osmosis for aqueous feed solutions involving different PEG solutes at different feed concentrations in the operating pressure range 25 to 100 psig. This technique is illustrated. Experimental Section Four PEG solutes, PEG-6000, PEG-9000, PEG-15000, and PEG-20000 having &fw values of 6750, 9500, 15000, and 23800, respectively, and six flat cellulose acetate membranes of different surface porosities, made in the laboratory by the method described by Kutowy et al. (19781, were used in this work. The PEG materials were obtained from B.D.H. Chemicals, Toronto (PEG-6000), Fluka AG, Buchs, Switzerland (PEG-9000 and PEG15000), and Sigma Chemical Co., St. Louis (PEG-20000). All the membranes were initially subjected to a pure water pressure of 150 psig for 3 h prior to reverse osmosis experiments to stabilize their porous structure. All experiments were carried out at the laboratory temperature (23 to 25 "C) in the apparatus shown in Figure 2 of Sourirajan and Matsuura (1977a). The operating pressure for the reverse osmosis experiments was in the range 25 to 100 psig. The concentrations of PEG solute in the feed so-

5

I n C*N&

IO

15

20

25

30

A X 106 g mol H ~ O / sC a t~m

Figure I. Effect of (a) average pore size on membrane surface and (b) pure water permeability constant on mass transfer coefficient k. Solute, PEG-6000; operating pressure, 100 psig; feed concentration, 50 ppm of solute.

lutions used were in the range 50 to 5000 ppm, and the feed flow rate was kept constant at 530 f 10 cm3/min. In each experiment, the data on (PWP) and (PR) in g/h per given area of membrane surface (13.2 cm2 in this work) were determined, and fraction solute separation f was obtained from the relation solute ppm in feed - solute ppm in product (1) f= solute ppm in feed A Beckman total carbon analyzer Model 915A was used to measure the concentrations of PEG solute in the feed and product solutions; a calibration curve was always prepared just prior to analysis. Results and Discussion Properties of Membranes, PEG Solutes and Their Aqueous Solutions. Table I gives the experimental reverse osmosis data on (PWP) and (PR) (in g/h per 13.2 cm2 of film surface) and f for PEG-6000 at two operating pressures in the range 25 to 100 psig obtained with six membranes of different surface porosities using aqueous feed solutions containing 50 ppm of solute. Using these data and eq lA, 2A, and 3A given in the Appendix, the applicable values of pure water permeability constant A , solute transport parameter DAM/K6for PEG-6000, mass transfer coefficient k for the solute on the high pressure side of the membrane, and the average pore size on the membrane surface as represented by the quantity In C*Nacl (Matsuura et al., 1976) were calculated for each membrane by the methods outlined in the Appendix and the results obtained are included in Table I and Figure la. The values of h obtained from eq 2A (Appendix) for the films and the experimental conditions used in this work cm/s to 5 x cm/s. Such were in the range 2.4 X low values of k are attributable primarily to the relatively

416

Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 1.025

PEG - 6000 /PEG-9000 /PEG-ZOO00

"

PEG-I5000

7

mY E i'ozo1.015-

22-

E

0

*e

18-

N Ln

+ I.Ol0-

0

'2

>

14-

5

+

1.005-

W

PEG-6000 0 "

6-

PEG-20000

1 / 8 1 ,

I

,

#

#

I

#

,

,

I

,

,

I

I

, , , , , , , , I

0

10.'

10-

I

I

I

5

IO

15

20

SOLUTE CONCENTRATION x IO2, g /cm3

SOLUTE CONCENTRATION, g/cm3

Figure 2. Diffusivities of PEG solutes in aqueous solutions at 25 "C, as a function of concentration.

Figure 4. Densities of aqueous solutions of PEG solutes a t 25 "C, as a function of concentration. \ N

u IO0

1

-

; 0

10-1

t

>

-

+

-

u >

-


m

-

-

0

t

t

PEG-I5000

13-'V1 10-4

,

, , , ,,

,

5

-v//

, , , , ,,,,

10-2-

Y

-

z

, , , , , , , ,1

, , , , , , , , I

.,,,,I

, , , , , , , , I

L

, , , , ,,,,/

IO-'

1

10-1

SOLUTE CONCENTRATION, g/cm3

Figure 3. Osmotic pressures of aqueous solutions of PEG solutes a t 25 "C, as a function of concentration.

low value for the diffusivity of PEG-6000 in water (=0.976 x lo4 cm2/s compared to 16.1 X lo4 cmz/s for NaC1). A plot of In C*NaC1 vs. k for PEG-6000 (Figure la) shows that k increases essentially linearly with increase in In C*NaCl, which is consistent with similar correlations reported in the literature (Rangarajan et al., 1978). Figure l a also gives the straight line correlation of In C*NaCl vs. k obtained by least-squares analysis of all the experimental data; this straight line correlation was used for obtaining the applicable values of k for the other PEG solutes in dilute aqueous feed solutions as discussed later. Both A and In C*NaC1 are functions of average pore size on the membrane surface. Further, A is also a function of the general method of making membranes. When the above method is the same, one can expect a unique correlation between different values of A and In C*NaC1. Since such is the case between the values of A and In C * N a c ~ obtained for the films used in this work, the linear correlation of ln C*NaC1 vs. k given in Figure l a can also be expressed as a unique correlation of A vs. k as given in Figure l b which will be used later in this discussion. Table I1 gives some physicochemical data for PEG-6000, PEG-9000, PEG-15000, and PEG-20000; further, data on diffusivity, osmotic pressure, density, kinematic viscosity, and molar density for the above PEG solutes as a function of concentration are given in Figures 2 to 6, respectively. Table I1 and Figure 7 give the applicable values of the polar, steric, and nonpolar parameters for the above solutes. The methods for determining the above properties

3

B

1 '

4&-J

10-

'"""1

'",','I

' 13- 3

'

'

",','I 0-

10-2

SOLUTE CONCENTRATlON.g/cm'

Figure 6. Molar densities of aqueous solutions of PEG solutes a t 25 "C, as a function of concentration. -I

PEG-20000

-x

x-x-

70-

a 25 p r ~ g A 50 P I # $

60u)

5.1

*s

0

7 5 ps,q

0

100 p r q

50-

-a

PEG- 15000

B B -B --

40 -

301R &"

-8

PEG-9000

W -8-

8-6

-7

-5

en cfN.,cI

Figure 7. Effect of average pore size on membrane surface on nonpolar parameters for PEG solutes at different operating pressures.

of the PEG solutes and their aqueous solutions are outlined in the Appendix. Analysis of Reverse Osmosis Data for PEG-6000 and PEG-15000 at Different Feed Concentrations Reverse osmosis experiments were carried out with four or five membrane samples of different surface porosities a t 100 psig using feed solutions containing PEG-6000 or PEG-15000 solutes in the concentration range 50 to 5000 ppm. The results obtained were analyzed by means of the

Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 417 FILM NO

5x10C5 OPEN SYMBOLS P E G - 6 0 0 0 CLOSED SYMBOLS P E G - I 5 0 0 0

I

lo3

IO'

SOLUTE CONCENTRATION IN FEED, ppm


Figure 8. Effect of feed concentration on solute transport parameters of PEG-6000 and PEG-15000 for different membranes. Operating pressure, 100 psig.

basic Kimura-Sourirajan transport equations (Sourirajan, 1970) to obtain data on solute transport parameter DAM/Kband mass transfer coefficient k in each experiment. The osmotic pressure data given in Figure 3 were used in this analysis. The results showed that in most cases the boundary concentration was considerably higher than the feed concentration, and in several cases the former was about 10 to 20 times the latter, which was not surprising (Kozinski and Lightfoot, 1972) in view of the very low diffusivity of the solutes involved (Figure 2 ) . With respect to each film and each solute, the experimental values of DWIK6 remained essentially constant and hence independent of feed concentration as shown in Figure 8. The experimental values of k showed that they varied as a function of diffusivity D A B of the solute and kinematic viscosity u of the feed solution. A plot of k/DAB2I3vs. v corresponding to each experiment yielded a series of parallel straight lines with a slope of 500 and intercepts different for different membranes, so that

DAB^^^

=

CY

+ 5001,

(2)

where the intercept o( is a constant characteristic of the surface porosity of the membrane. Consequently, for any particular membrane under consideration, the value of CY can be calculated from the values of k and D A B and u(kinematic viscosity of pure water = 0.008963 cm2/s a t 25 "C) applicable for a very dilute aqueous feed solution, for example, one containing 50 ppm of solute. Thus N

= [ k / D ~ ~ * ' ~ soln, ] d i -] ,(500 x 0.008963)

(3)

From eq 2 and 3, with respect to any particular membrane, the value of k/DAB2I3for a PEG feed solution of viscosity u is given by the relation

DAB*/^

= [ k / D ~ ~ ~ ' ~soln, ] d i+i ,5 0 0 ( ~- 0.008963)

(4)

For using eq 4, the value of k for the dilute solution of the PEG solute under consideration (Le., the quantity k in the term [ k / D ~ ~ ~ ~soln, ~ ]on d ithe l . right side of eq 4) can be obtained from the relation (Sourirajan and Matsuura, 197713) =

kref[DAB/(DAB)ref1213

(5)

where the subscript ref refers to the reference solute PEG-6000 for which data on krefand (DAB)ref are given in Figure l a and Table 11, respectively. Figure 9 gives a comparison of experimental k values obtained with different membranes and PEG feed solutions of different viscosities, with corresponding k values calculated from eq 4 using eq 5 as described above. The results show good agreement between the experimental and calculated values

418


Table 111. Reverse Osmosis Separations of PEG-20000. Comparison of Experimental and Calculated Results ~~

solute concn in feed X A

3.657 X

48

3.827 X 10.’

50

lo-’

51

3.896 X


ppm

I

3.816 X

503

3.683 X

4850

a

operating exptl pressure, (PWP),a (PR),‘ atm g/h glh

film no. 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6

2.21 2.11 2.01 1.80 1.91 3.91 3.81 3.71 3.50 3.61 5.61 5.51 5.41 5.20 5.31 2.21 2.11 2.01 1.80 1.91 2.21 2.11 2.01 1.80 1.91

16.29 16.75 18.71 28.56 49.30 28.02 28.76 32.89 50.75 78.44 38.76 39.72 45.59 68.55 98.81 15.83 15.91 18.35 28.59 48.77 15.42 15.51 18.15 28.47 47.39

14.63 14.91 16.64 24.39 34.10 24.68 25.08 28.17 40.13 49.79 34.09 34.55 38.73 52.89 65.17 13.83 13.75 16.28 23.59 32.30 12.44 12.31 14.27 20.87 28.12

model 1

(PRhU f, % 98.8 97.8 96.9 93.5 83.6 97.6 96.1 95.2 90.8 79.8 97.7 95.9 94.4 88.3 78.1 99.0 98.8 98.5 97.3 93.8 99.3 99.1 98.9 97.8 94.9

glh 16.28 16.74 18.70 28.55 49.28 28.01 28.7 5 32.88 50.74 78.42 38.75 39.7 1 45.58 68.54 98.79 15.62 15.69 18.01 28.11 48.05 13.44 13.49 14.56 24.12 39.10

.

model 2

(PRLa f, % 97.3 96.8 95.2 90.1 74.7 96.0 94.8 91.5 78.4 44.8 93.3 90.5 83.8 59.8 28.6 97.3 97.0 95.5 91.1 82.4 97.7 97.4 96.1 92.3 85.7

glh 14.67 14.96 16.52 23.67 34.97 24.37 24.91 28.05 40.07 50.41 32.91 33.54 37.79 51.85 63.72 13.06 13.05 15.45 22.91 30.64 11.82 11.79 13.61 20.83 27.19

f, 5% 97.1 97.1 95.8 90.9 82.0 95.8 95.8 93.5 83.2 66.9 92.8 92.8 88.7 70.5 53.1 97.0 97.0 95.7 91.2 85.1 97.3 97.3 96.1 92.1 86.7

model 3

(pRha g/h

f, 7%

14.79 15.03 16.49 23.74 34.12 24.74 25.21 28.26 41.13 52.20 33.66 34.24 38.45 54.32 64.74 13.45 13.41 15.79 22.97 30.98 12.63 12.61 14.54 21.28 28.57

98.3 97.7 97.0 93.8 86.7 98.0 97.2 96.2 90.7 80.9 97.1 95.9 94.4 84.7 73.9 99.0 98.5 98.1 96.2 93.5 99.4 99.2 98.9 97.9 96.4

Membrane area = 13.2 cm’.

o PEG-6000 + PEG-15000

- 0-

t

kexpt

x IO!

cm/s

Figure 9. Comparison of experimental and calculated data on mass transfer coefficient k for PEG-6000 and PEG-15000. Operating pressure, 100 psig; feed concentration, 50-5000 ppm.

of k and illustrate the practical utility of eq 4 and 5 for predicting K values for different PEG solutes at different concentrations in the feed solutions for any given membrane from only a single experimental k value for a dilute feed solution of PEG-6000 taken as the reference solute. Analysis of Reverse Osmosis Data for Feed Solutions Containing PEG-20000. The experimental values of (PR) were considerably less than those of PWP for feed solutions containing PEG-20000 in the entire concentration range of 50 to 5000 ppm studied in this work. With respect to each film, the fact that the difference between (PWP) and (PR) was significant even a t a feed concentration of 50 ppm (Table 111) showed that the normal concentration polarization effect (i.e., the osmotic pressure effect) was not the only factor governing membrane performance in reverse osmosis. The possibility of pre-gel or gel polarization on the membrane surface during ultrafiltration of

macromolecular solutions has been established in the literature (Blatt et al., 1970; Baker and Strathmann, 1970; Porter, 1972). That such pre-gel or gel polarization is also possible in reverse osmosis has already been pointed out (Hsieh et al., 1979). Consequently, the experimental values of (PWP), (PR), and f could not be used directly in the basic reverse osmosis transport equations in the usual manner (Sourirajan, 1970) for obtaining the value of DAM/K6 for PEG-20000 for each film studied. Some indirect method has to be used for estimating the D-lKb value for PEG-20000 comparable to the DAM/K6 values for the other PEG solutes of lower molecular weight when no pre-gel or gel polarization is involved. Let the above DAM/K6value of PEG-20000 be designated as ( D m / K 6 ) 0 . Analysis of experimental (PWP), (PR), and f data for each film on the basis of the corresponding (DAM/Kb)ovalues may be expected to lead to practical techniques for predicting membrane performance for reverse osmosis separations of PEG-20000 in the concentration range studied in this work. Estimation of (DAMIK6)Ofor Different Films and the Values of w*Cs* for PEG-20000. Experimental data showed that for egch film, the value of DAM/K6decreased with increase in M , of the PEG solute, and the plot of In DAM/K6vs. In M, was a straight line with essentially the same slope for films of different surface porosities, with respect to solutes PEG-6000, PEG-9000, and PEG-15000 (Figure 10). Earlier work (Hsieh et al., 1979) also showed similar correlations with respect to the PEG solutes whose M, values were in the range 200 to 6750. Assuming that the above correction would be valid for PEG-20000 also if its reverse osmosis separations were not governed by any additional factors, the values of In (DAMIK6)o for PEG20000 (comparable to the values of In (DAMIK6) for the other PEG solutes of lower M, values) were obtained simply by extrapolating the linear plots of In (DAMIK6) vs. In A?, for the films tested as shown in Figure 10. Using


-1

I

FILM NO

A

-1.1,

,

---

I

9.0

9.5

---x 10.0

U,

Pn

Figure 10. Effect of weight-average molecular weight on solute transport parameter of PEG solutes at different operating pressures.


Gt

\

I

419

three transport models were tested for membrane performance for reverse osmosis separations of PEG-20000. In all cases it was assumed that the values of (DAM/K& for PEG-20000 Gust as those for the other PEG solutes studied) were independent of solute concentration and operating pressure. Each model was tested with experimental data obtained with five membranes of different surface porosities (Table 111). The constraints of each model and its adequacy for predicting membrane performance are described below. Model 1. Normal Concentration Polarization Model. This is the usual model where concentration polarization normally controls membrane performance in reverse osmosis, so that product rate and solute separation can be predicted from a knowledge of the applicable values A, D,M/K6, and h under the experimental conditions used. This model was tested using the experimental values of A (Table I), ( D A M I K 6 ) O values obtained from Figure 10, the values of h calculated from eq 4 and 5 (using PEG-6000 as the reference solute), and the following set of basic equations (Sourirajan, 1970)

NB = A [ P - T ( x A 2 ) + K ( x A 3 ) I

(64

\ i -I 102

, ,

8 8 1 ,

lo3 -

>

1

1 1 - 8 8

o4

91

w

Figure 11. Effect of weight-average molecular weight on summation of polar, steric, and nonpolar parameters of PEG solutes.

these values of In ( D A M / K 6 ) 0 , the values of w * C s * (nonpolar parameter) for PEG-20000 were calculated from eq 3A in the Appendix following the procedure identical with that used before for the solutes PEG-9000 and PEG-15000. The values of w * C s * so obtained are also plotted in Figure 7, which shows that the above values are independent of porosity of the membrane surface. The average of the above values of w * C s * for PEG-20000 was found to be 74.70 which may also be considered to be independent of operating pressure as is the case for all other PEG solutes tested. It may be recalled that the values of the polar free energy parameter (-AAG/RT) and nonpolar parameter ( w * C s * ) for the PEG solutes given in Table I1 and also in Table I1 of Hsieh et al. (1979) are independent of the porous structure of the membrane surface. On the other hand, values of the steric parameter (6*CE,) depend on the porous structure of the membrane surface, and those given in the above tables are applicable only to cellulose acetate membranes whose In C*Nacl values are greater than -9.6. Considering only the latter membranes, it is interesting to note that the sum of the polar, steric, and n_onpolar parameters decreases steadily with increase in M , of the PEG solute as shown in Figure 11. This means that the net solute repulsion a t the membrane-solution interface increases with increase in A?fwof PEG solute. The actual solute separation of PEG solutes in reverse osmosis depends not only on the magnitude of the above sum, but also on experimental conditions. T r a n s p o r t Models for Predicting Membrane P e r formance i n Reverse Osmosis. Using the basic Kimura-Sourirajan transport equations (Sourirajan, 1970)

Using the above equations together with the known values of A , P, and X A l (Table 1111, ( D A M I K 6 ) O (Figure lo), h (Figure la, and eq 4 and 5 ) , T ( X A ) (Figure 31, and c (Figure 61, the values of NBand XA3and the corresponding values of product rate and solute separation were calculated (Sourirajan, 1970). The results, given in Table 111, showed that the calculated values were too high for product rate and too low for solute separations compared to the corresponding experimental values, thereby confirming the inadequacy of transport model 1 for predicting membrane performance. Model 2. Pre-Gel Polarization Model. The experimental data (Table 111) showed that solute separation decreased and product rate increased with increase in operating pressure; the increase in product rate, however, was less than proportional to the increase in effective driving pressure for fluid flow. These results are qualitatively consistent with pre-gel polarization on the high pressure side of the membrane as described by Blatt et al. (1970). In this model, following Blatt et al. (1970), it is assumed that the pre-gel offers additional resistance to water flux through the membrane, so that

From the point of view of the analysis of experimental reverse osmosis data, the additional hydraulic resistance offered by the pre-gel is equivalent to the condition that the magnitude of NBis controlled by an apparent reduced pure water permeability constant A , (instead of A ) so that A, =

1 ~

Rm + Rp

(8)

and a correspondingly reduced mass transfer coefficient h, (instead of h ) on the high pressure side of the membrane. Since the porosity of the membrane surface does not change, no change in solute transport parameter is as-

420


sumed. Consequently, in the transport model 2, A = A,, 1 k,, and (DM/KcVO remains constant. Hence the basic transport equations eq 6a and 6c assume the following forms

0.5

a

k

NB = A,[P

-

dx~z) -k dx~3)]

FILM NO

n

w h

(9a)

\

From the experimental solute separation and product rate data, one can calculate N B from the relation (Sourirajan, 1970)

-.-

0.1

0.5

5.0

1.0

xg2 x

IO

IO6

Figure 12. Effects of pre-gel polarization and retarded permeation of pre-gel through membrane pores during reverse osmosis separations of PEG-20000.


(10) The applicable value of k, can be determined as a function of A, and XAl (f(A,,XAl))using Figure l b (where A is set equal to A,) and eq 4 and 5. Thus the applicable relationship between h, and A, and XAlcan simply be stated as

kr

= f(ArrXA1)

(11)

0.05~

I

By solving eq 9a, 9c, and 11 simultaneously, the quantities XA2,A,, and k, can be determined for each set of experimental reverse osmosis data. Using the values of A, and k, so obtained, together with the values of (DM/K6)ofor each film, eq 9a, 6b, and 9c can now be used to calculate NB and XA3, and hence product rate and solute separation as in model 1. These calculated values, also given in Table 111, show that while the product rate data compared well with the corresponding experimental data (which is only to be expected), the solute separation data were too low compared to the experimental data especially for films giving relatively lower levels of solute separation. Thus the transport model 2 also is inadequate for predicting membrane performance for the reverse osmosis separations of PEG-20000. Model 3. Pre-Gel Polarization a n d Retarded P e r meation of Pre-Gel t h r o u g h Membrane Pores. From the foregoing analysis, it is evident that the assumption of ( D - / K C ~ for ) ~ the applicable solute transport parameter in the pre-gel polarization model is not valid. This is not surprising in view of the physical nature of pre-gels. Concentrated polymer solutions become pre-gels by ramifying aggregation of polymer segments in the colloidal state, and the loose framework of such aggregates enmesh and immobilize a considerable amount of solvent material (McBain, 1950). Because of the high solidity of the above colloidal aggregates, the movement of the pre-gel through the membrane pores under reverse osmosis operating conditions is retarded. This means that pre-gel polarization in reverse osmosis reduces both water flux and solute flux. From the point of view of the analysis of experimental reverse osmosis data on the basis of the above considerations (model 3), the resistances offered by the pre-gel to solvent and solute transport are equivalent to the condition that the magnitudes of NB and x.43 (and hence product rate and solute separation) are controlled by reduced values of pure water permeability constant and mass transfer coefficient (A, and h,, respectively, as in model 2), and also solute transport parameter. Designating the applicable reduced value of solute transport parameter

- 6.5

-5.5

-6.0 PI

-5.0

-4.5

c&

Figure 13. Effect of average pore size on membrane surface on the coefficients a and b in eq 12; solute, PEG-20000.

as (DM/K6),,the basic reverse osmosis transport equations for model 3 can be written as follows

NB = Ar[P - ~ ( X A -IZ )a(XA3)I

(94

Further, eq 11 holds good for model 3 also. Consequently, the values of XA2,A,, and k, obtained in the analysis of model 2 are also applicable for model 3. Using the above values of XA2and those of NBobtained earlier from eq 10, the values of (DAM/K6), were calculated from eq 9b for each set of experimental reverse osmosis data given in Table 111. The results showed that the log-log plots of X A.vs. ~ (DAM/K6)r/(DAM/K6)0exhibited reasonable straight line correlations as shown in Figure 12. These correlations can be expressed as follows

and

where a, b, a', and b' are constants characteristic of the pre-gel. By least-squares analysis of the experimental data, numerical values for the above constants were evaluated. In the operating pressure range tested (25 to 75 psig), all of the four constants were independent of operating

Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 421

pressure. The values for the constants a and b were functions of surface porosity of the membranes used; the correlations of a and b with the corresponding values of In C*NaCIare shown in Figure 13. The constants a’and b’were evaluated to be 0.0216 and 4 . 2 2 , respectively; these values were independent of membrane porosity so that eq 13 can be rewritten as


(DAM / K6),/ (DAM/ KS), = O.O216X~2-~.~~ ( 14) The correlations shown in Figure 12 are consistent with pre-gel polarization and retarded permeation of pre-gel through membrane pores resulting in decreased solute and solvent transport through the membrane. The increase in the value of 1- ( A , / A )(Figure 12a) and hence a decrease in the ratio A J A , as well as a similar decrease in the ratio (DAM/K6)r/(DAM/K6)0(Figure 12b) with increase in XA2 illustrate that the effects of pre-gel polarization on both solvent and solute transport increase with increase in solute concentration in the pre-gel ( X A ~ )Further, . for any given value of X A 2 ,the ratio A , / A decreases with increase in the average pore size on the membrane surface (Figure E a ) , which illustrates increased pre-gel effect on solvent transport with increased pre-gel permeation through the membrane pores. Figure 12b shows that for any given value of XA2, as the average pore size on the membrane surface increases, both (Dm/K6), and (Dm/K6)0 increase to the same extent, so that their ratio remains constant, which also means that all solute transport is through pre-gel permeation through the pores. On the basis of the foregoing analysis, one can predict membrane performance (product rate and solute separation) for the reverse osmosis separation of PEG-20000 for any film from data on A , (DAM/K6)0 for PEG-20000, and k for a dilute solution of PEG-6000 (- 50 ppm of solute). Using the above data in conjunction with eq 4, 5, 12, and 14, and Figures l b and 13, one can calculate the applicable values of A,, (DAM/K6),,and k, which can then be used in the set of eq 9a, 9b, and 9c to calculate NBand XA3, and hence product rate and solute separation. Table I11 also illustrates that the data on product rate and solute separation so calculated are in excellent agreement with the corresponding experimental values.

Conclusions For low pressure (

Analysis of Reverse Osmosis Data for the System Polyethylene Glycol

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