Freeman, M. P., Skrivan, J. F., A.Z.Ch.E. J . 8, 450 (1962). Grey, J., ZSA Trans. 4, 102 (1965). John, R. R., Bade, W. L., A m . Rocket Soc. J . 29, 523 (1959). Johnson, J. R., M.S. thesis, Texas A&M University, College Station, Tex., 1966. Kubanek, G. R., Gauvin, W. H., 6lst National Meeting of A.1.Ch.E ., Houston, Tex., February 1967. Moore, L. L., J . Aeron. Sci. 19, 505 (1952). Schlichting, H., “Boundary Layer Theory,” 4th ed., McGraw-Hill, New York, 1960. Skifstad, J . G., Jet Propulsion Center, Purdue University, Lafayette, Ind., Interim Report Contract Nor 1100 (17) (August 1961). Skrivan, J. F., von Jaskowsky, W., IND. ENG.CHEM.PROCESS DESIGN DEVELOP. 4, 371 (1965). Stokes, C. W., Knipe, W. W., Streng, L. A., J . Electrochem. Soc. 107, 35 (1960).
Tankin, R. S., Berry, J. M., Phys. Fluids 7, 1620 (1964). Wethern, R. J., Brodkey, R. S., A.Z.Ch.E. J . 9, 49 (1963). RECEIVED for review April 19, 1967 ACCEPTED September 21, 1967 IYork supported by the National Aeronautics and Space Xdministration through NASA Grant 32550 for Space Technology Project No. 3. Material supplementary to this article has been deposited as Document No. 9655 with ,4DI Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington, D. C. A copy may be secured by citing the document number and by remitting $2.50 for photoprints or $1.75 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
CONCENTRATION POLARIZATION EFFECTS IN R E V E R S E OSMOSIS U S I N G POROUS C E L L U L O S E ACETATE M E M B R A N E S SHOJl
K I M U R A A N D S. S O U R I R A J A N
Diuision of Applied Chemistry, .Vational Research Council of Canada, Ottawa, Canada
Analytical expressions have been developed to predict concentration polarization effects in reverse osmosis from the data on membrane specifications given in terms of the pure water permeability constant, A, and the solute transport parameter, DAM/K6. The results are illustrated for a set of Loeb-Sourirajan type porous cellulose acetate membranes using 0.5M aqeuous sodium chloride solution as the feed at an operating pressure of 102 atm. The case considered here is for turbulent and laminar flow in rectangular channels between flat parallel membranes. Some factors relating to the economic analysis of the process are also indicated. HE practical importance of the problem of concentration Tpolarization in reverse osmosis has been recognized, and several analytical studies have been reported with particular reference to saline water conversion (Brian, 1965a, 1965b, 1965c; Gill et al., 1965, 1966a, 1966b; Johnson et al., 1966; Sherwood et al.,1965; Srinivasan et al.,1967). These studies assume that the membrane exhibits either complete salt rejection or incomplete salt rejection at a constant level. Such an assumption is, in general, invalid, since salt rejection can vary widely depending on feed concentration and feed flow rate, even a t the same operating pressure. A more desirable approach to the subject is given by Sherwood et al. (1967), who have coupled the equations of solute and solvent transport through the membrane to the theory of concentration polarization. A similar approach is offered by the results of the Kimura-Sourirajan analysis (Kimura and Sourirajan, 1967) of the experimental reverse osmosis data obtained with the Loeb-Sourirajan type porous cellulose acetate membranes. The latter analysis, however, is different from that of Sherwood et al. (1967). The Kimura-Sourirajan analysis is based on a generalized pore diffusion model applicable for the entire possible range of solute separation. I t gives rise to a set of basic equations relating the pure water permeability constant, A , the transport of solvent water, iVB, the transport of solute, ArA, the solute transport parameter, DA.M/Kb, and the mass transfer coefficient, k, all of which can be determined from a single set of
experimental pure water permeability, product rate, and solute separation data obtained from laboratory cells. The parameter A is obtained from the pure water permeability data, and hence is independent of any solute under consideration. The parameter D A w / K 6 depends, of course, on the nature of the solute. Both A and D,,*/’KG are dependent on the porous structure of the membrane surface, and hence they are different for different membranes; and both are functions of operating pressure. Further, at a given operating pressure, DA,/K6 is independent of feed concentration and feed flow rate. Also, the values of k are well correlated by a generalized log-log. plot of .VRe us. .VSh/-1rgo0.33 for the type of apparatus used. The distinguishing feature of the above analysis is the fact that the interconnected parameters A and D A,JKG specify a particular membrane-solution system at the given operating pressure with reference to this separation process. Using the above parameters, the membrane performance-i.e., the variations of solute separation and membrane throughput rate as a function of feed concentration and feed flow rate-can be predicted, provided the applicable mass transfer correlations are available. This has been illustrated for the systems [NaC1-H20], [NaNOa-H20], [Na2S04-H20], [MgC12-H20], [MgS04--H20],and glycerolwater (Kimura and Sourirajan, 1967; Sourirajan and Kimura, 1967). Hence the above analysis and correlations are valid at least for the above system. The object of this paper is to illustrate how the concenVOL. 7
NO. 1
JANUARY 1968
41
tration polarization effects in reverse osmosis can be predicted from the specifications of the membrane given in terms of A and DA,/K6. The system [NaCI-H20] is chosen for illustration because of its practical significance to saline water conversion. The case considered here is for turbulent and laminar flow of brine in rectangular channels between flat parallel membranes. Membrane Specifications
Table I gives the specifications of the membranes studied in this work, in terms of A and D A M / K 6 at the operating pressure of 1500 p.s.i.g. The details of the apparatus, and the experimental procedure employed for obtaining these data, have been reported (Sourirajan, 1964; Sourirajan and Govindan, 1965). Basic Transport Equations
The Kimura-Sourirajan analysis gives rise to the following basic equations for the solvent and solute transport through porous cellulose acetate membranes in the reverse osmosis process using aqueous solutions. Solvent water transport :
Table 1.
Membrane Specifications
Film type. Loeb-Sourirajan type porous cellulose acetate membranes System. Sodium chloride-water Operating pressure. 1500 p.s.i.g. A X 106 Gram M o l e H20 D~~ x 106 c m . / s e c . Film No. Sq. Cm. Sec. Atm. KS 1 0.97 0.9 2 3 5 6
1.46
1.87 2.37 2.93
6.0 20.0 75 .O 140.0
All the above assumptions are reasonably valid for the system [NaCl-HzO] a t least up to a concentration of 1.OM (E 5.52 weight % NaCl). This is evident from the following data at 25' C. The molar density of pure water is 5.535 X 10+ mole per cc., and that of 1.OM solution is 5.530 X mole per cc. The osmotic pressure of 0 . 1 M solution is 67 p s i . , and that of 1.OM solution is 673 p.s.i. The mole fraction ( X A ) of NaCl in 1.OM solution is only 0.0177. The above assumptions do not restrict the scope of the following analysis, but they simplify the equations involved in illustrating the effect of the process and design variables relating to sea water or brackish water conversion. Using the above assumptions, Equations 1, 2, and 3 may be rewritten as follows:
where A is the pure water permeability constant which is obtained from the pure water permeability data. Solute transport :
(3) Also x.43
N.4
= NA
+
NB
(4)
Equations 1 to 4 are applicable for the system [NaCI-H20] whether the brine flow through the channel is turbulent or laminar, provided the appropriate mass transfer coefficients are used. where
Turbulent Flow Case
I n this case, the concentration distribution may be considered uniform along the transverse axis owing to the turbulent mixing so that the usually defined mass transfer coefficient can be used. T o facilitate analysis, the following assumptions are made: The molar density of solution is essentially constant-i.e., Cl
=
c2
=
G3
=
G
=
BXA
XAlo
which is the local salt concentration divided by the salt concentration in the brine feed at the channel inlet. Combining Equations 8, 10, and 13,
(6)
.
or
xA3
