Znd. Eng. Chem. Res. 1990,29, 259-263
259
SEPARATIONS A Comparison of Solute Rejection Models in Reverse Osmosis Membranes. 2. System Water-Sodium Chloride-Asymmetric Polyamide Vito L. Punzi,* Karen B. H u n t , + and Gregory P. Muldowney* Department of Chemical Engineering, Villanova University, Villanova, Pennsylvania 19085
Experimental separation data collected for the system water-sodium chloride-asymmetric polyamide are used to compare four theoretical solute rejection equations. The general results of this study extend the results obtained earlier for cellulose acetate and thin-film composite polyamide: ideal membrane models provide a reasonable description of membrane behavior; models corrected for solute passage produce significantly better results; and solute separation is accurately correlated over a wide range of pressure, feed rate, and feed concentration by either the corrected diffusive flow model or the corrected viscous flow model. In addition, several mechanistic parameters which characterize homogeneous asymmetric polyamide membranes are estimated. The predicted values of these parameters (particularly the distribution coefficient ratios) are found to be similar for asymmetric polyamide and cellulose acetate membranes of comparable selectivity.
In an earlier investigation (Muldowney and Punzi, 1988), a methodology was developed that facilitates simultaneous study of the following three issues critical to the understanding of the solute rejection mechanism in reverse osmosis (RO): (1)whether solute transport is diffusive or convective, (2) which mechanistic (physical) parameters best characterize a real membrane with respect to an ideal (perfect) separator, and (3) whether ideal and real models can be applied in parallel to quantify nonidealities present in real membranes. In this paper, the results of an investigation involving homogeneous asymmetric polyamide membranes are presented and are used to extend the results of earlier studies involving cellulose acetate (Muldowney and Punzi, 1988, Punzi et al., 1990) and thin-film composite polyamide membranes (Punzi et ai., 1989). In addition, this paper compares the results obtained using homogeneous polyamide membranes to those obtained using cellulose acetate membranes of comparable selectivity. All of the analyses described in this paper are based on a single experimental RO data base collected using the system water-sodium chloride-asymmetric polyamide membranes. Polyamide RO membranes are studied because their use is likely to increase significantly in the future, as they can operate at higher temperatures, are more selective than cellulose acetate membranes, and are not subject to hydrolysis and biological attack (Riley et al., 1976; Soltanieh and Gill, 1981; Kesting, 1985).
and the corrected diffusive flow model CY=---
K,“ Pi Dim K2’ P2 RT
In the classical solution-diffusion equations, solute concentrations in the membrane and the bulk solution are interrelated through the same distribution coefficient (Kz) on both sides of the membrane (Lonsdale, 1966; Merten, 1966; Soltanieh and Gill, 1981; Kesting, 1985). This procedure assumes (K,”/K,’) is unity. As shown previously (Muldowney and Punzi, 1988) and confirmed here, the restriction of equal distribution coefficients severely inhibits the applicability of the diffusive flow equations. The models summarized by eqs 1and 2 accommodate possibly unequal coefficients K,” and K,’. This difference in methodology distingusihes the present approach from previous investigations. Two models based on the convective transport mechanism have been derived: the ideal viscous flow model
and the corrected viscous flow model
Theory and Analysis Two models based on the diffusive transport mechanism have been derived: the ideal diffusive flow model
* Author to whom correspondence should be addressed.
Present address: Sun Company, Marcus Hook, PA 19061. *Present address: Mobil R&D Corp., Paulsboro, NJ 08066.
088S5885/90/2629-0259$02.50/0
Detailed derivations of eqs 1-4 are presented elsewhere (Muldowney and Punzi, 1988). Development of the models as four equations of similar form is central to this study. Within each pair, the first equation assumes a perfect membrane, while the second includes a correction for solute passage. In all four equations, the dependent variable is the solute separation 0 1990 American Chemical Society
260 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990
factor ( a )defined as a = C2’/C;‘. The critical feature of the equations is that they result from a novel interpretation of familiar transport models. In all analyses involving eqs 1-4, three groups of information must be measured or known: RO process data (a,Ap, V”, T ) which are usually measured experimentally; solvent-solute data (ulm, T,O2Jwhich are usually obtained from the literature; and membrane structure data ( A , A, 7,t) which are usually obtained or derived from discussions of membrane morphology. As indicated above, membrane performance is measured in terms of the solute separation factor, a (or a*). Known properties of the solvent-solute pair, membrane, and RO process are then grouped in an independent variable ( X , ) defined uniquely for each model lz. In each case, a linear relationship results. (For the viscous flow models, a linear form may be obtained either by direct method from eqs 3 and 4 or by substituting a first-order accurate Taylor series expansion for the exponential term-in essence exp(-x) = 1 - x (Thiel, 1989). Typical values of the argument x in this study are of the order of The present analysis uses eqs 3 and 4 directly.) The dominant mode of transport is suggested by the relative capacity of the corrected diffusive flow model (eq 2) and the corrected viscous flow model (eq 4) to correlate RO separation data. A standard linear regression analysis is used to objectively compare the models. In essence, this procedure seeks to establish whether pressure difference or permeate flow rate is the independent variable to which solute separation more strongly responds. Equations 2 and 4 can also be used to determine system-specific mechanistic parameters which summarize the solvent-solute-membrane interaction. Three parameters follow from the data fit and are used to describe membrane behavior in physical terms: (P1/P2)for diffusion, (e/&‘) for convection, and (K,”/K2’)for both modes. All three ratios are insensitive to pressure and flow rate over the ranges tested. Any concentration effects are established by separately analyzing data sets of varying V” and Ap at several feed concentrations. In addition, the ideal forms of these three variables can be determined from eq 1 and 3. The values so obtained can be compared to those obtained via the two corrected models as a means of quantifying membrane nonideality. That is, while the ideal parameters lose their physical significance when applied to data from imperfect membranes, they nonetheless retain value as measures of membrane nonideality when compared to the parameters obtained from the corrected models. In eqs 1-4, a and a* denote the ratio of solute concentrations at the high- and low-pressure membrane surfaces. These may differ from measured feed and permeate (bulk) concentrations due to concentration polarization (the accumulation of solute a t the feed side of the membrane). However, it has been shown (Muldowney and Punzi, 1988; Punzi et al., 1989) that the correction for concentration polarization is naturally absorbed into the mechanistic parameters with negligible effect on their numerical values so that the physical meaning of the parameters is essentially preserved. Membrane Morphology The skin layer thickness (A) in homogeneous asymmetric polyamide is 0.1-0.3 pm (Soltanieh and Gill, 1981; Osmonics, 1988). Electron microscopy studies of pore size and density in this layer suggest a porosity (t) between 0.005 and 0.052 (Kesting, 1985; Osmonics, 1988). Polymer segments in the skin form a random network for which the mean tortuosity (7) is 2.5; a typical range is 2-3 (Soltanieh
and Gill, 1981; Kesting, 1985). Assimilating the three ranges gives ( A 7 / 4 values of 3.8-180 pm. Both viscous flow models are tested here, using, in turn, (AT/€) of 3.8, 180, and 26.2 pm (the geometric mean). However, the fit of the convective transport equations is found to vary negligibly with the quantity ( A 7 / 6 ) , confirming the approximately linear behavior of the exponential term noted earlier. The results discussed below for the two viscous flow models are those at ( A T / E ) of 180 pm. The membrane properties noted above could, in an alternative approach, be fit to the experimental data. However, such a procedure would compromise the model comparison because the convective transport equations would become nonlinear and contain an additional degree of freedom over the diffusive flow models. The present strategy uses known data for the membrane properties, testing the full range of each parameter to guarantee the generality of the results. Experimental Section Separation data are obtained using a commercial RO unit (Osmonics OSMO 3319-SB) representative of industrial systems. The unit features three spiral-wound modules of homogeneous asymmetric polyamide membrane: each has an area of 1.05 m2 (11.3 ft2)and provides 0.002 kg/(m2*s)(4.4 gpd/ft2) maximum product water flux. The homogeneous asymmetric polyamide substructure is cast on a polyester nonwoven substrate and is formed via the classical phase-inversion process (Sourirajan, 1970; Kesting, 1985), which results in the simultaneous formation of a very dense skin layer with a spongy substructure. All three membranes were produced in one lot. The system allows independent feed rate control to each membrane. A total of 30-40 conditions (combinations of feed flow rate and feed-side gauge pressure at 20 O C ) are studied, a t each of 5 nominal feed chloride concentrations (250, 1000,2500,3750,5000 mg/L). An additional 15 conditions are studied at 7500 mg/L. Flow rates to each membrane are 12.1-27.3 cm3/s (11.5-26 gph) and pressures 345-1723 kPa (50-250 psig). Although broader ranges of flow rate and pressure are encountered in practice (and might test the rejection models more conclusively),the purpose of this study is to parallel the conditions used previously for cellulose acetate (Muldowneyand Punzi, 1988). A study at higher pressures and feed concentrations is reported elsewhere (Punzi et al., 1990). Experimental data collection and analysis procedures are otherwise identical with those reported in the earlier study. General Results and Discussion The solute separation factor ( a )and permeate flow rate ( V ” )are similar but not identical for the three test membranes at each set of operating conditions; typically a varies among the membranes by 30%. This is expected, since all three were manufactured in one lot. The values of a for all three membranes range between 2 and 24 (51-96% rejection); the wide range in a is attributable to the wide range of operating conditions represented by the 197 conditions used to study each membrane. Approximately 50% of the cy values are between 2 and 6, and approximately 85% are between 2 and 12. Flow rates (V”) for each membrane range up to 0.62 cm3/s. In general, both a and V” increase with increasing applied pressure and decrease with increasing feed concentration, consistent with previous findings (Lonsdale, 1966; Muldowney and Punzi, 1988; Punzi et al., 1989, 1990).
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 261
t
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t
101
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0 0
t
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:
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1
2
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5
D i f f L s v e Transpo.-t P a r a m e t e r X c
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Figure 1. Performance of the corrected diffusive flow model in correlating solute separation data for a 2500 mg/L feed.
Figure 2. Performance of the corrected viscous flow model in correlating solute separation data for a 2500 mg/L feed.
Comparison of Solute Rejection Models via Regression Analysis The results of the regression analysis performed by using each of the four solute rejection models are useful in evaluating both membrane nonideality and the mechanism of semipermeability. A comparison of the residual squares (or fraction of squares removed) obtained by using the ideal and corrected model representing a particular solute rejection mechanism is used to quantify membrane nonideality. A comparison of the two corrected models in terms of the same statistical parameter(s) is used to suggest the mechanism of semipermeability. The ideal models produce a significant reduction in the unaccounted variation in a*. Typically, the residual squares are 1 or 2 orders of magnitude smaller than the total squares (Le., 90-99% fraction of squares removed). This indicates that the ideal models provide a reasonable description of solute rejection. The corrected diffusive flow model generally removes 5044% of the total squares. These results, although not as impressive as those obtained for cellulose acetate, are nontheless uniform throughout, with no strong dependence on concentration or membrane. Feed concentration dependence is also well correlated, evident from 50-55’70 squares removed in the pooled data sets. Figure 1 illustrates the typical performance of the corrected diffusive flow model in this study. The corrected viscous flow model removes for 55% to 84% of the total squares, and performance does not depend on the value of ( X T / C ) . The feed concentration dependence is also well correlated, evident from 50% squares removed in the pooled data sets. Figure 2 illustrates the typical performance of the corrected diffusive flow model in this study. Thus, the results of the regression analysis also indicate that using the corrected rejection equations provide a better description of solute rejection than the ideal models. This is not surprising since most of the experimental data are at greater than 90% rejection. From the similar success obtained by using either the corrected diffusive flow model or the corrected viscous flow model, it follows that a two-parameter linear functionality of cy either to the pressure difference (Ap - AT) or to the exponential of permeate flow ( V ” ) can be used to adequately represent the basic mechanism of solute rejection in highly selective homogeneous asymmetric polyamide membranes. This basic conclusion is identical with that of the earlier studies involving cellulose acetate and thin film composite polyamide. While this observation seems to further the persistent disagreement over the solute re-
Table I. Best-Fit Mechanistic Parameters for the Ideal and Corrected Solute Rejection Models diffusive flow viscous flow nominal models models feed concn, mg/L membrane ideala correctedb idealC correctedd 250 1000 2500 3750 5000
1 2 3 1 2 3 1 2 3 1 2 3 1 2
3 7500
1 2
3
2000 1900 2300 2900 2500 2300 1900 1600 1500 1500 1400 1200 1600 1400 1300 2200 2100 1800
1600 1500 1500 2300 1800 1800 1300 960 870 1000 900 610 770 570 450 600 550 400
b[(Pl/P*)(K,”/K,/)I.
330 310 340 380 310 250 230 200 140 180 160 110 190 180 100 210 140 100
170 160 160 220 180 160 130 99 78 96 89 56 99 90 57 120 54 48
‘(e/Kd)*.
d(c/
jection mechanism in reverse osmosis, it is nevertheless significant that the same conclusion is reached for the three different membrane materials that predominate in current industrial applications. The similarity in results, combined with the traditional experimental observation that permeate flow rate increases with increasing applied pressure and decreases with increasing feed concentration, suggests a relationship that partially couples the independent variables of the two corrected equations. A regression analysis of experimental data obtained using cellulose acetate confirmed partial coupling between pressure and permeate flow rate (Muldowney, 1983). This would explain the comparable,but not identical, correlation of separation data by two rejection models representing theoretical extremes of solute transport.
Interpretation of Mechanistic Parameters Tables I and I1 present the fitted mechanistic parameters determined from the four solute rejection modelsparameters that quantify membrane nonideality. Distribution coefficient ratios (K,”/K,’) are calculated from each corrected equation and compared. Table I11 presents values of the physical quantities ( P 1 / P 2 )K, i , and K,“ obtained from the corrected parameters. In general, since the polyamide membranes used in this study and the cellulose acetate membranes used in the
262 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 Table 11. Solute Distribution Coefficient Ratio by the Corrected Reiection Models nominal corrected rejection models feed concn, mg/L membrane diffusive flow‘ viscous f l o e 250 1 2.3 6.2 2 2.5 5.9 3 4.6 7.6 1000 1 2.8 6.7 2 2.9 5.9 2.4 4.5 3 2500 1 2.1 4.0 2 2.3 3.7 2.0 3.0 3 3750 1 1.2 2.9 2 1.0 2.5 3 1.5 2.2 5000 1 1.4 2.1 2 1.6 2.0 3 1.6 1.6 7500 1 2.0 1.9 2 1.9 2.5 1.8 1.8 3
“(K,”/K,’).
earlier study are of comparable selectivity, a certain degree of similarity and agreement of results is expected. Thus, a comparison of the results of the two studies provides an opportunity to assess the consistency of the results obtained by using this methodology. Ideal vs Corrected Parameters. The ideal diffusive parameter [ (P,/P,)*(K,”/K,’)*] averages 2200,1900, and 2000 for the three membranes tested, values that are about 3 times the comparable values obtained for cellulose acetate. The values of this parameter show no apparent dependence on concentration. The corrected diffusive parameter [ (P,/P,)(K,”/K,’ )] averages 1400,1200, and 1100 for the three membranes tested, values that are about 3 times those obtained using cellulose acetate. In general, mean corrected parameters are 55-65% of the corresponding ideal values. However, it appears that the relationship between the corrected and ideal values is more concentration dependent than that observed for the other membrane materials. Specifically, a t low concentrations, corrected parameters are approximately 80% of the ideal values; a t 2500 and 3750 mg/L, corrected parameters are approximately 65% of the ideal values; and at the two high concentrations, corrected parameters are approximately 35% of the ideal values. While greater deviation from ideality is expected as the concentration increases, the effect is more pronounced than that observed for cellulose acetate, where the one-third to one-half fraction was appropriate throughout. Although all three membranes were manufactured in one lot, the mean value of the ideal diffusive parameter is 16% larger for membrane 1 than for membrane 2 and 10% larger for membrane 1 than for membrane 3. The mean value of the corrected diffusive parameter for membrane 1 is 17% and 27% larger, respectively, than the values obtained for membranes 2 and 3.
The ideal convective parameter (c/K,’)* at ( A T / € ) of 180 pm averages 290, 250, and 200 for the three membranes tested and generally shows no dependence on concentration. These values are about 4 times the comparable values obtained for cellulose acetate. The corrected convective parameter ( t / K i )at ( A T / € ) of 180 pm averages 200, 180, and 140,values that are about 5 times those obtained using cellulose acetate. In general, mean corrected parameters are 70% of the corresponding ideal values, although many of the corrected parameters obtained at individual concentrations are 50-55% of the corresponding ideal values. The variation between the corrected and ideal values as a function of concentration observed using the diffusive parameters is not evident with the convective parameter. Although all three membranes were manufactured in one lot, the mean value of the ideal convective parameter is 15% larger for membrane 1than for membrane 2 and 45% larger for membrane 1 than for membrane 3. The mean value of the corrected convective parameter is also 15% larger for membrane 1 than for membrane 2 and 45% larger for membrane 1 than for membrane 3. Thus, for each corrected rejection model, one mechanistic parameter-[(Pl/Pz)(K,”/K,’)] or ( E / K)-may ~ be reliably estimated as 50-70% of the corresponding ideal value for this type of highly selected membrane material. It is expected that as selectivity increases the mechanistic parameters obtained from the corrected models would more closely approximate the ideal parameters. This trend is confirmed in the two studies performed thus far using three different types of membrane material. Corrected Diffusive Flow vs Corrected Viscous Flow Parameter. Table I1 presents the solute distribution coefficient ratio (K,“/K,’ ) calculated from each corrected model. Ratios calculated from the corrected viscous flow model, which range from 1.6 to 7.6, are generally higher than those based on the corrected diffusive flow model, which typically range between 1.0 and 4.6. The similarity among the predicted values is of particular interest because of the very different theoretical bases used to arrive at these estimates. It should be noted that the values of (K,”/K,’ ) obtained using the polyamide membranes are also quite similar to those obtained for cellulose acetate membranes of comparable selectivity. For cellulose acetate, ratios calculated from the corrected viscous flow model ranged from 1.4 to 5.1; ratios based on the corrected diffusive flow model ranged between 0.94 and 5.2. It is clear that (K,”/K,’) differs significantlyfrom unity. In physical terms, this suggests that the membrane structure is different on the high-pressure and low-pressure sides, which challenges the conventional assumption of a uniform selective layer. Thus, it is again shown that assuming K 2 constant across the membrane severely compromises any comparisons of diffusive and convective models. Predicted Physical Quantities. Table I11 presents the mean permeability ratios (P,/P,) and the mean distribution coefficients ( K i and K,”) calculated in this study. The permeability ratios indicate that the test membranes
Table 111. Physical Quantities Predicted by the Corrected Rejec:tion Models corrected diffusive flow model membrane 1 2 3 a
Solvent-to-solute permeability ratio.
P1IP2,”
dimensionless 630 520 400
corrected viscous flow model KiVb KZff,b dimensionless dimensionless 1.4 X lo4 3.9 x 10-5 1.7 x 10-4 5.2 x 10-5 6.8 X 1.8 x 10-4
Solute distribution coefficieints, based on a porosity
I
= 0.005.
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 263 are 400-630 times more permeable to solvent than to solute. Although no literature data are available at the rejection levels observed here, values of (Pl/P2)reported for "aromatic polyamide membranes" separating aqueous sodium chloride solutions are on the order of lo3 (Soltanieh and Gill, 1981). By use of = 0.005, calculated feed-side solute distriinbution coefficients, K,', are approximately 5 X dicating that the solute is 20 000 times less concentrated in the high-pressure side of the membrane than in the feed solution. Values of the permeate-side distribution coefficient, KF,are approximately 1.6 X lo"', indicating a 6000-fold partitioning of solute. Corresponding K,' and K/ values for cellulose acetate are 10-20 times larger. The smaller coefficients, K,' and K F , obtained for polyamide versus cellulose acetate indicate that, despite the similar rejection levels observed for the two materials, the solute concentration will be lower in a polyamide membrane than in a cellulose acetate membrane contacting the same solution. This finding disagrees with other experiments (Frommer et al., 1973; Matsuura et al., 1983) which concluded that sodium chloride is more concentrated in aromatic polyamide than in cellulose acetate. The studies cited, however, used an immersion method and a liquid chromatography method as opposed to the dynamic measurements made in the present investigation. It is likely that solute partitioning changes in the presence of a strong solvent flow because a true equilibrium in the static sense is never established. The greater solute partitioning in polyamide found here is more consistent with structural differences between the two materials, such as a porosity in polyamide that is 8-10 times smaller than in cellulose acetate (noted, in fact, in the cited studies). Such differences also manifest in a permeate flow rate lower by a factor of 8-9 for polyamide than for cellulose acetate at identical operating conditions.
Conclusions The solute rejection mechanism in homogeneous asymmetric polyamide RO membranes is accurately described over a wide range of pressure, feed rate, and feed concentration by either the corrected diffusive flow or the corrected viscous flow model. Two membrane-specific parameters are required for either model and may be obtained from minimal separation data. Because the independent variables in the two models are coupled by a dependence of permeate flow on applied pressure, the solute rejection mechanism cannot be further clarified by analysis of model equations alone. However, the models offer great utility in predicting both solute separation and physical quantities characterizing a solvent-solute-membrane system. Ideal membrane models provide acceptable descriptions of semipermeability at high rejection. Corrected mechanistic parameters are 50-70% of the corresponding ideal parameters and permit membrane nonideality to be quantified. Both ideal and corrected parameters generally parallel selectivity when compared among similar membranes. Distribution coefficient ratios (K2"/K2')predicted by the two corrected models range from 1.0 to 7.6 and are strikingly similar to the values obtained using cellulose acetate membranes of comparable selectivity. Acknowledgment This paper is based on research supported by the National Science Foundation, under Grant CBT-8519698.
Nomenclature A = membrane cross-sectional area C; = concentration of species i at membrane surface s D ,= diffusivity of speecies i in medium j K t = distribution coefficient of species i at membrane surface S
Pi = permeability of species i p = pressure R = universal gas constant T = absolute temperature V 3 = volumetric flow rate at surface s uim = partial molar volume of species i in membrane Xk = independent variable defined uniquely for each model k Greek S y m b o l s a = solute separation factor e, A, T = membrane active layer
porosity, thickness, and
tortuosity a = solution osmotic pressure Superscripts ', " = feed-side surface and permeate-side surface of membrane * = ideal-case model Subscripts 1 = solvent
2 = solute m = membrane Registry No. NaC1, 7647-14-5.
Literature Cited Frommer, M. A.; Murday, J. S.; Messalem, R. M. Solubility and Diffusivity of Water and of Salts in an Aromatic Polyamide Film. Eur. Polym. J . 1973, 9, 367. Kesting, R. Synthetic Polymeric Membranes; Wiley: New York, 1985. Lonsdale, H. K. Properties of Cellulose Acetate Membranes. In Desalination by Reverse Osmosis; Merten, U., Ed.; MIT Press: Cambridge, MA, 1966. Matsuura, T.; Taketani, Y.; Sourirajan, S. Interfacial Parameters Governing Reverse Osmosis for Different Polymer Material-Solution Systems through Gas and Liquid Chromatography Data. J. Colloid Interface Sci. 1983, 95, 10. Merten, U. Transport Properties of Osmotic Membranes. In Desalination by Reverse Osmosis; Merten, U., Ed.; MIT Press: Cambridge, MA, 1966. Muldowney, G. P. A Theoretical and Experimental Investigation to Characterize and Optimize the Reverse Osmosis Separation Process. M.Ch.E. Thesis, Villanova University, 1983. Muldowney, G. P.; Punzi, V. L. A Comparison of Solute Rejection Models in Reverse Osmosis Membranes for the System WaterSodium Chloride-Cellulose Acetate. Ind. Eng. Chem. Res. 1988, 12, 2341. Osmonics, Inc. Verbal communications and product literature. Osmonics: Minnetonka, MN, 1988. Punzi, V. L.; Muldowney, G. P.; Hull, T. J. An Evaluation of RO Solute Rejection Model Performance a t Elevated Pressure and High Feed Concentration. Ind. Eng. Chem. Res. 1990, another paper in this issue. Punzi, V. L.; Muldowney, G. P.; Hunt, K. B. Study of Solute Rejection Models for Thin Film Composite Polyamide RO Membranes. J. Membrane Sci. 1989, in review. Riley, R. L.; Fox, R. L.; Lyons, C. R.; Milstead, C. E.; Seroy, W. E.; Tagami, M. Spiral-Wound Poly(Ether/Amide) Thin-Film Composite Membrane Systems. Desalination 1976, 19, 113. Soltanieh, M.; Gill, W. N. Review of Reverse Osmosis Membranes and Transport Models. Chem. Eng. Commun. 1981,12, 279. Sourirajan, S. Reverse Osmosis; Academic: New York, 1970. Thiel, S. W. Linear Relationships between Rejection and Flux in Pressure-Driven Membrane Separation Processes. J. Membrane Sci. 1989, 43, 307.
Received for review March 8, 1989 Revised manuscript received October 6, 1989 Accepted October 20, 1989
