Ind. Eng. Chem. Process Des. Dev. 1981, 20, 273-282
(10) Henke. A. M. OilGas J. 1970, 68(14), 97. (1 1) Hiemenz, W. Discussion, Section 3. Paper 20, Sixth World Petroleum Congress, Frankfurt, Germany, June 1963. (12) Hughes, C. C.; Mann, R. ACS Symp. Ser. 1978, 65, 201. (13) Inoguchl, M.; Kagaya, H.; Daigo, K.; Sakurada, S.; Satomi, Y.; Inaba, K.; Tate, K.; Nishiyama, R.; Onishl, S.; Nagai, T. Bull. Jpn. Pet. Inst. 1971, 13(2), 153. (14) Inoguchi, M.; Sakurada, S.; Satomi, Y.; Inaba, K.; Kagaya, H.; Tate, K.; Mizutori. T.; Nlshiyama, R.; Nagal. T.; Onishi, S. Bull. Jpn. Pet. rnst. 1972, 14(2), 153. (15) Larson, 0. A.; Beuther, H. Prepr., Dlv. Pet. Chem., Am. Chem. SOC. 1968, 11(2), 895. (16) Luss, D.; Amundson, N. R. AIChE J. 1987, 13(4), 759. (17) Mkloux, N.; Charpentier, J. C. Chem. Eng. Sci. 1973, 28, 2108. (18) Newson, E. J. Ind. Eng. Chem. Process Des. Dev. 1975, 14, 27. (19) Nltta, H.; Takatsuka, T.; Kodama, S.; Yokoyama, T. DeactivationModel for Residual Hydrodesulfurlzation Catalysts, 86th National AIChE Meeting, Houston, Texas, April 1979. (20) Ohtsuka, T. Catal. Rev. 1977, 16(2), 291. (21) Oxenreiter, M. F.; Frye, C. G.; Hockstra, G. E.; Sroka. J. M. Fuel 011 Desulfurization Symposium, Japan Petroleum Institute, Tokyo, Japan, Nov 29, 1972. (22) Ozakl, H.; Satomi, Y.; Hisamlsu, T. PD18(4), Ninth World Petroleum Congress, Tokyo, Japan, 1975. (23) Parkin, E. S.; Paraskos, J. S.; Frayer, J. A. Use of Analog Computer Simulation in the Development of a Commercial HDS Process, 74th National AIChE Meeting, New Orleans, La., March 1973. (24) Prasher, B. D.; Ma, Y. H. AIChE J. 1977, 23(3), 303. (25) Prasher, B. D.; Gabriel, G. A.; Ma, Y. H. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 266.
273
(26) Rajagopolan, K.; Luss, D. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 459. (27) Richardson, R. L.; Alley, S. K. Prepr., Dlv. Pet. Chem.. Am. Chem. Soc.1975, 20(2), 554. (28) Rossi, W. J.; Deighton, 8. S.; Mcdonald, A. J. Paper No. 46-77, Fortysecond Mldyear API Meeting, Chicago, Ill., 1977. (29) Sato, M.; Takayama, N.; Kurlta, S.; Kwan, T. Nippon Kagaku Zasshl 1971, 92, 834. (30) Satterfleld, C. N. “Mass Transfer in Heterogeneous Catalysis”, MIT Press: Cambridge, Mass., 1970; p 134. (31) Scott, J. W.; Bridge, A. G.; Christensen, R. I.; Gould, Q. D. Fuel Oil DesulfurizationSymwsium. JaDan Petroleum Institute, Tokyo, . . . Jamn, . March 1970. (32) Scott, J. W.; Bridge, A. G. A&. Chem. Ser. 1971, No. 103, 113. (33) Shah, Y. T.; Paraskos, J. A. Ind. Eng. Chem. Process Des. Dev. Ig7S. - - - - .14. . - ,268. (34) Spry, J. C.; Sawyer, W. H. Paper No. 3OC, Sixty-elghth Annual AIChE Meeting, Lo8 Angeles, Calif., 1975. (35) Todo, N. et al. Kogyo Kagaku Zasshi 1971, 74(4). 563. (36) Thiele, E. W. I d . Eng. Chem. 1939, 31(7), 916. (37) Weekman, V. W. Chem. React. Eng., froc. Int. Symp. 4th, 1978, 615. (38) Welsz, P. B. Chem. Eng. Progr. Symp. Ser. No. 25 1954. 55, 29. (39) Wheeler, A. “Catalysis”, Voi. 11, Reinhold: New York; 1955, pp 105- 158. (40) Wheeler, A.; Robell. A. J. J. Catal. 1989, 13, 299.
Received for review December 19,1979 Accepted October 13,1980
Reverse Osmosis Transport through Capillary Pores under the Influence of Surface Forces Takeshl Matsuura and S. Sourlrajan’ Division of Chemistry, National Research Council of Canada, Ottawa, Canada, K I A QR9
A new approach to analysis of experimental reverse osmosis data is presented. In this analysis, the relative solute-membrane material interactions at the membrane-solution interface are expressed in terms of electrostatic or LennardJones-type surface potential functions, and the transport of solute and solvent through the membrane under the influence of such surface forces Is expressed through appropriate mass transport equations applicable for individual circular cylindrical pores. From such expressions, equatlons for solute separation and ( P R ) / ( fW f ) ratio are derived. The use of the above equations for the analysis of experimental reverse osmosis data is illustrated for four different types of systems involving cellulose acetate as membrane material, water as solvent, and NaCI, p-chlorophenol, cumene, or benzene as solute material. The results show that all the different types of variations in solute separations and ( f R ) / (PWP) ratios observed experimentally are predictable by the above analysis.
Introduction Reverse osmosis transport has been discussed extensively on the basis of preferential sorption-capillary flow mechanism (Sourirajan, 1970b, 1978; Sourirajan and Matauura, 1977b). According to this mechanism, negative or positive adsorption of solute at the membranesolution interface and fluid permeation through the pores on the membrane surface under the operating conditions of the experiment govern solute separation and fluid flux through the membrane during the reverse osmosis process. The negative or positive adsorption of solute at the membrane-solution interface arises from net repulsive or attractive forces acting on the solute from the adjacent membrane surfaces. In the earlier analysis of reverse osmosis transport such surface (interfacial)forces have been quantitatively characterized in terms of appropriate polar, steric, and nonpolar parameters (Matsuura et al., 1974, l975,1976b, 1977; Rangarajan et al., 1976,1978;Sourirajan and Matsuura, 1977a), and the solute and solvent fluxes through the membrane pores have been considered for the entire membrane surface including all pores available for
material transport. This paper offers an alternative approach to the analysis of the same problem. In this approach, the surface forces acting on the solute are expressed by an electrostatic or a Lennard-Jones type potential function, and the solute and solvent transport through the membrane under the influence of such forces is expressed through appropriate mass transport equations for an individual cylindrical pore having an average radius and an average effective pore length, along with an expression for the ratio (PR)/ (PWP)(membrane permeated product rate/pure water permeation rate for a given area of membrane surface); the latter expression makes it needless to specify explicitly the number of pores involved in reverse osmosis transport. This analysis results in general expressions for solute separation and fluid flux which are valid for the case of negative adsorption of solute (preferential sorption of water) as well as positive adsorption of solute (preferential sorption of solute) at the membrane-solution interface. This paper derives the above analytical expressions in detail and illustrates that the form of different correlations of experimental reverse
O1964305f 811 1120-0273$01.25/Q 0 1981 American Chemical Society
274
Ind. Eng. Chem. Process Des. Dev., Vol. 20,No. 2, 1981
Similarly assuming that FAB(r,z)is proportional to the relative velocity of solute and solvent at (r,z) and xAB is the corresponding proportionality constant and noting that solvent velocity is a function of r only, and xAB .. is independent of position FAB(r,z) = -XAB[UA(r,z) - UB(r)l Combining eq 4 and 6 and rearranging
Recall that
Figure 1. Cylindrical coordinates in a membrane pore.
osmosis data already available in the literature are well predicted, both qualitatively and quantitatively, by the above analytical expressions. Consequently the basis of the analysis and the results of such analysis presented in this paper confirm our earlier understanding of reverse osmosis transport and offer a further means of developing both reverse osmosis and ultrafiltration processes (Sourirajan et al., 1979). Analysis Expression for Solute Separation. Consider reverse osmosis transport of solute and solvent through a cylindrical pore of radius R and length 6 under steady-state operating conditions. Let any point (or position) inside the pore be represented by the cylindrical coordinates r (radius) and z (length), and let mass transport be in the direction z as shown in Figure 1. Consider an annular region between two coaxial cylinders of radii r and ( r + dr) inside the pore. Let subscripts A, B, and M represent solute, solvent, and pore wall (or membrane surface), respectively. At any position (r,z) in the pore, the forces acting against the movement of the solute are the friction forces FAB(r,z)and Fm(r,z) between the solute and solvent, and the solute and pore wall, respectively, and the driving force for the movement of solute through the pore is the force of diffusion FA(r,z) due to the gradient in chemical potential of solute (pA). Under steady-state operating conditions, these two forces balance each other so that (1) FA(r,z) = -[FAB(r,z) + FAM(r,Z)I Let uA(F,z) and uA(R) be the velocities of solute in the pore a t position (r,z) and at the pore wall, respectively. Assuming that FAM(T,z) is proportional to the relative velocity of the solute and xA&) is the corresponding proportionality constant, and noting that x,&) is a function of r only (2) FAM(~,Z) = -XAM(r)[UA(r,Z) - UA(R)] Further UA(R) = 0
(3)
and (4) where JA(r)represents the molar flux of solute (which is a function of r only) and cA(~,z)is the molar concentration of solute (which is a function of both r and z). Using eq 3 and 4, eq 2 becomes (5)
Inserting eq 12 into eq 7 and rearranging
(1+ Define a dimensionless parameter b(r) = [XAB + XAMWI+ XAB Equation 13 can then be written as
z)
(13)
(14)
Let d ( r ) be a potential function expressing the force exerted on the solute molecule by the pore wall or the membrane surface. The quantity 4(r)is a function of the distance between the pore wall or membrane surface and solute molecule, and hence r; when d(r) is positive, the force is repulsive, and when $(r) is negative, the force is attractive. Let cA2 represent solute concentration outside the pore at pore inlet, which means that cA2 is solute concentration in the boundary phase during reverse osmosis. Let cA3(r)represent solute concentration outside the pore at the pore outlet, which means that CA3(F) is the local solute concentration in the product solution phase at position (r,6). Following the Maxwell-Boltzmann distribution law CA(T,O) = c&?e-+(r)/RT
(16)
CA(r,8) = CA3(r)e-@(r)'RT
(17)
and Further
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981 275
d2UB(r) + -1- dUB(r) dl.2 r dr
-C A-d d cA2
~ X [UB P (r)~ X A /RTl B
(19) 1+-e-+(r)/RT [ ~ X P ( U B ( ~ ) ~ X A-B1/1R T I b (4
Equation 19 has the same form as eq 10 of Jonsson and Boesen (1975), the essential difference being that the former is applicable only to the annular region between the two coaxial cylinders of radii r and (r + dr) within the pore, whereas the latter is considered as applicable to the entire pore; further, while the former includes the term @(r)explicitly, the latter does not. By averaging the solute concentration at the pore outlet cA3(r)over the entire cross-sectional area of the pore, solute concentration in the product solution (cA3) is obtained. Thus
XAM(r)CA3(r)UB(r)
:( ?IrGr)
Solving the differential eq 15, with eq 16 and 17 as the boundary conditions, and also using eq 18
-
9
71
= 0 (24)
Integrating eq 24 from pore inlet z = 0 to pore outlet z = 6, and then dividing by 6 d2UB(r)+ -1- dUB(r) -dl.2 r dr
1 &,6) - P(r,O) t 6 XAM(r)CA3(r)UB(r) = 0 (25) D In order to integrate eq 25, an expression is needed for the pressure gradient {P(r,6)- P(r,O))as a function of r. This can be obtained as follows. Since radial equilibrium exists within the pore, the Gibbs-Duhem equation can be used to relate pressure and potential gradients inside the pore as a function of r (Anderson and Malone, 1974). Therefore
J R ~ ~ 3 ( r ) ~ ~dr( r ) r cA3 =
R
J u d r b dr
(20)
Defining solute separation based on solute concentration in the boundary phase, as f’
f’=
cA2 - cA3 ~
cA2
From eq 26 and applying Boltzmann equation, we obtain P(r,O) = Pi - RTcA2[1- e-+(r)/RT]
(27)
and P(r,6) = Po - RTcA3(r)[1- e-+(r)/RT]
--I - - cA3
(28)
where Pi and Podenote the external pressures prevailing at the pore inlet and outlet, respectively. Subtracting eq 27 from eq 28 P(r,6) - P(r,O) = -(pi - Po) - RqC~3(r) - C A ~ )x [1 - e-9(r)/RT](29)
cA2
Combining eq 19, 20, and 21 f’= 1 -
Using eq 14 and 29, eq 25 can now be written as d2U~(r) 1 dUB(r) dr2 r dr
+--
cA2)
T + -91 (Pi- Po) + -711R-{cA3(r) 6
x [I - e-+(r)/RT]-
(b(r) - l)XABCAS(r)UB(r) 7
In order to evaluate solute separation f’ from eq 22, information about solvent velocity profile uB(r)as a function of r is required. Radial Velocity Profile, uB(r). The radial velocity profile for solvent flow as a function of r can be established by considering the momentum flux in an annular region between two coaxial cylinders of radii r and (r + dr) limited by distances of z and (z + dz),due to pressure (P),friction ( F m ) ,and viscous (7) shear forces. Thus the momentum balance in the region specified above leads to the expression -2ar dr(
r=r
)
dz- xAM(r)cA3(r)uB(r)2ar dr dz
-
= 0 (30)
Integration of eq 30 gives the radial velocity profile for solvent flow in the pore as a function of r; the boundary conditions for eq 30 are atr=O:
dU&) -dr
-0
a t r = R: uB(r) = 0 (32) Expression for Solute Separation Based on Pore Size Distribution on the Membrane Surface. Assume normal distribution of pores on the membrane surface. Let R represent the mean pore radius and u represent standard deviation. The pore size distribution function Y(R) can then be written as
=O
(23) where the first and second terms express the rates of fluid momentum entering due to difference in pressure and friction force, respectively, and the last two terms together express the rate of change in fluid momentum due to viscous flow. Dividing eq 23 by 2ar dr dz q, and rearranging
With such distribution of pores, solute separation f’defmed by eq 21 can now be expressed as
276 Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981
Using eq 19,eq 34 becomes
Expressions for Solute Separation and Radial Velocity Profile for Solvent Flow in Terms of Dimensionless Quantities. Equations 22,30,31,32,and 35 can be expressed in terms of dimensionless quantities for convenience of computations. For this purpose, the following dimensionless quantities are defined p = r/R (36) cA(P) = cA3(r)/cA2
4 ~= U ) B(~)~XAB/RT Pi
= ~/XAJ$~CA~
PZ = (pi - po)/RTcA2 N P )= W / R T
(37) (38) (40)
= gnR4(Pi - Po)/8~6
QPR
gJRu~(r)2nrdr
(49)
Therefore
(41)
xi[
2 s g 1 f f ( P ) P dP
(“)/(PWP)
= QPR/QPWP
=
p2/8pl
(50)
For the case involving a normal pore size distribution, eq 50 assumes the form
exp(a(p))/l+
and the radial velocity profile expressed by eq 30 may be written as
where CA(P)= exp(a(p))/l
QPWP
(48) With respect to the permeation of product solution through the pore
(39)
In terms of the above dimensionless quantities, solute separation expressed by eq 22 may be written as
f’= 1 -
separation is not a unique function of effective film thickness. Such a conclusion was also reached in the analysis of Bean (1969)for reverse osmosis transport. Ratio of Product Rate to Pure Water Permeation Rate. The membrane permeated product rate (PR)and the pure water permeation rate (PWP)(both in kg/h) are usually expressed for the total available area of membrane surface or permeation through a single pore. Denoting the single pore version of the above permeation rates (PWP) and (PR) as QPwP and QPR,respectively, analytical expressions for the ratio (PR)/(PWP) may be obtained as follows. Consider eq 30;for the case of pure water permeation through the pore, the last two terms in eq 30 can be omitted. For such case, integration of eq 30 with the boundary conditions expressed by eq 31 and 32 leads to the well-known Poiseuille equation
b(p)
+ e_,,p,[exp(a(p))- 1) (44)
from eq 19 and 37. For solving eq 43,the boundary conditions given by eq 31 and 32 may be written as dab) when p = 0: -- 0 (45) dP (46) when p = 1: a ( p ) = 0 For the case involving normal pore size distribution, solute separation expressed by eq 35 may be written as
-
I t should be noted that for determining f’either from eq 42 or 47,whereas precise specification of R (for eq 42) together with its distribution (for eq 47) is necessary, no such specification of 6 is necessary. This means that solute
Results and Discussion Determination of Potential Function and Average Pore Size on the Membrane Surface. A set of experimental data on (PWP), (PR) (both in kg/h per 13.2 X lo4 m2 of film area), and solute separation f (based on feed concentration) are given in Table I. These data were obtained at an operating pressure of 9646 kPa (95.2atm) gauge, with NaC1-H20 feed solutions at solute concentration of 171 g-mol/m3 using six cellulose acetate membranes of different surface porosities. The mass transfer coefficient k on the high-pressure side of the membrane applicable for the experimental conditions used was 22 X lo4 m/sfor all the membranes. These data are used here to illustrate the generation of potential function representing the repulsion of NaCl from the membrane surface (pore wall), and also the average pore size on the membrane surface for each film used. In order to use eq 42,solute separation data are needed in terms off’ which is based on boundary concentration cA2. From the basic reverse osmosis transport equations established earlier (Sourirajan, 1970b), and using the fact that the molar densities of NaC1-H20 solutions remain essentially constant for a wide range of solute concentrations, cA2 may be expressed as cA2 = cA3 + (CAI - cA3) exp(h/k) (52) where u, represents the permeation velocity of product solution through the membrane obtained from the relation us = (PR)/36OOSg (53) where g is the density of the solution. From the experi-
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981
277
Table I. Experimental Reverse Osmosis Data for the System NaCl-H,O Using Porous Cellulose Acetate Membranes, Along with Some Calculated Results" (PWP) x 103, (PR)x 103, CA2, R x lolo, A x lolo, film no. kg/h kg/h f f' g-mol/m E1 m 1
85.60 86.33 121.06 146.41 168.58 202.58
2 3 4 5 6
69.45 69.63 90.19 112.13 133.67 156.94
0.978 0.965 0.885 0.857 0.827 0.679
a Effective area of membrane surface = 13.2 x operating pressure = 9 6 4 6 kPa guage (1400 psig).
The quantities p1and p2 may now be evaluated from the relations =
7
-
0.8941 X10-3
XmR2C~2 1.540
-- 0.5806
x 1012R2C~2
0:
lo3 (55)
using the values of cA2 given in Table I. The repulsive force between the membrane surface and the NaCl solute is electrostatic as shown by Glueckauf (1965) and Bean (1969) with respect to uncharged cellulose acetate membranes, and by Jacazio et al. (1972) with respect to charged clay membranes. Hence the corresponding potential function @ may be expressed simply as inversely proportional to the distance d between the membrane surface (pore wall) and the solute, so that @ = A/d (56) (57)
Further, because of high solute repulsion from the pore wall, the frictional force between solute and membrane surface becomes relatively insignificant; consequently, one can assume that for the reverse osmosis system under consideration, xm may be assumed to be equal to zero, which means, from eq 14, b ( p ) = 1. From a consideration of eq 42 and 43, it is now clear that solute separation is a function of A and R; the other quantities involved in the above equations are available numerically in terms of the experimental conditions involved. While the pore radius R, of course, depends on the porous structure of the membrane, the force constant A is independent of such porous structure, and dependent only on the chemical nature of the membrane, solute, and solvent materials involved. Therefore, with respect to the reverse osmosis system under consideration, the value of
21 21 21 21 21 21
m/s; CAI = 1 7 1 g-mol/m3;
(PWP) R1.5 (54)
(Pi - Po) - 95.2 X 1.01325 X lo5 - 3.891 X O2 = RTCA~ 2.479 x 1O3cA2 cA2
6.9 8.2 12.7 13.2 13.7 18.2
A is the same for all the membranes used, and the value of R is different for each membrane. Further, for any given value off I, different combination of values of A and R can satisfy eq 42 and 43. This means that the latter equations alone are insufficient to determine the unique combination of values of A and R applicable for the reverse osmosis system under consideration. Some additional information is needed to serve as a basis for such determination. One such basis is offered by Glueckauf (1965), who calculated the average value of R for different cellulose acetate membranes from a consideration of repulsions of ions in aqueous solution in a narrow pore of a material of low dielectric constant; correlating such calculated values of R with experimental data on pure water permeation rates, Glueckauf found (1965) that
X
R2CA2
where A is the relevant force constant. Since R=r+d
329 328 37 8 453 53 7 576
m Z f; = (CAI - c A 3 ) / c A l ; k = 22 x
mental data on CAI, (PR),s,k, and f, and the value of cA2 obtained from eq 52, the value of f defined by eq 21 can be calculated for each film. The values of cA2 and f' so calculated are also given in Table I. In order to use eq 43, numerical values of p1and p2 are needed; they can be obtained using the known values q, DAB,RT, xABgiven as follows: q = viscosity of pure water = 0.8941 X Pa-s;RT = 8.314 X 298.2 = 2.479 X lo3 J/g-mol
P1
0.989 0.982 0.948 0.946 0.945 0.905
(59)
or
In (PWP) = constant + 1.5 In R (60) Though the power term on the right side of eq 59 is different from the correaponding term in eq 48,this difference is not a contradiction in principle. Equation 48 is always valid as far as a single pore is concerned. The dependence of overall (PWP)on average pore radius can be less than the 4th power, when there are a very large number of pores with some type of distribution in pore radius and consequent or simultaneousvariations in effective film thickness. For the purpose of illustration, the validity of eq 60 was assumed in this work, and the applicable Combination of values of A and R was calculated as indicated below. Using the experimental f' and (PWP)data given in Table I for each of the six membranes, the value of A (which was kept the same for all the membranes used) and that of R for each membrane, which satisfied simultaneously eq 42,43, and 60 were calculated. The calculation procedure involves the following steps. Step 1: assume a value of A applicable for all the membranes used; step 2: assume a value of R for each membrane; step 3: obtain the value of a ( p ) using eq 43 and the boundary conditions given by eq 45 and 46; step 4: using the above value of a ( p ) , obtain f'from eq 42; step 5 check the value of f'so obtained with the experimental value given in Table I; if the calculated and experimental values do not coincide, repeat steps 2 to 5 until they coincide, which gives the combination of the value of A and the set of six values of R for the six membranes used, satisfying simultaneously eq 42 and 43; step 6: check by least-squares analysis whether the values of R so obtained satisfy eq 60 using experimental data on (PWP)for each membrane; if eq 60 is not satisfied, repeat steps 1to 6 until eq 60 is satisfied, which gives the required combination of the value of A and the set of six values of R for the six membranes used, satisfying simultaneously eq 42, 43, and 60. The values of A and R so calculated are also given in Table I, which shows that the value of A was 21 X
278
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2 , 1981
1
F I L M AREA 13 2 cm' 2 0 0 1 OPERATING P R E S S U R E 9 6 4 6 k P o q w q e
ot E
I
e
1
I
I
I
I
IO
12
14
16
18
R x IO",
I
m
Figure 2. Effect of pore radius on (PWP): -, calculated; 0,experimental. MEMBRANE MATERIAL CELLULOSE ACETATE
v)
I
SYSTEM CELLULOSE ACETATE (E-3981 MATERIAL-NoCI-WbTER
S Y S T E M , SODIUM CHLORIDE-WATER
20
40
60
P, x IO-',
80
100
kPa gauge
Figure 4. Effect of operating pressure on sodium chloride eeparation. Membrane material, cellulose acetate (E-398); feed concentration, 171 g-mol/m3; mass transfer coefficient k = m. I 10
I 0
5
I5
d x IOIO, m
Figure 3. Potential curve of interfacial force for the system cellulose acetate (E-398) material-NaC1-water.
m (= 21 A) and the values of R were in the range 6.9 8,(for film 1, f' = 0.989) to 18.2 A (for film 6, f' = 0.905). Extending the above calculations, it was also found that the value of R for a cellulose acetate membrane capable of giving 99.9% solute separation for NaCl for the case cA2 = C A I (which corresponds to k = m) was 5.5 A,which is the same as the value (5.5 A) estimated by Glueckauf and Sammon (1970) for a cellulose acetate membrane capable of excluding NaCl from the pores of a cellulose acetate membrane during reverse osmosis. Using the R values obtained above for the six membranes listed in Table I (for the case A = 21 X 10-lo m (= 21 A)), the corresponding (PWP)data were calculated and the results obtained were in good agreement with the experimental data as shown in Figure 2, which confirmed the vddity of the calculation procedure outlined above. From the known value of A, characteristic of the material of the membrane surface, one can now generate the potential function @ as a function of d using eq 56. This is shown in Figure 3 for A = 21 X 10-lom corresponding to the set of values of R given in Table I. Since @ appears in eq 42 and 43 as exp(-@), and any value of CP more than 10 reduces exp(-@) to nearly equal to zero, the effect of changes in 9 for values of 9 greater than 10 was neglected in Figure 3. The general form of the potential function @ shown in Figure 3 is a definite one from a priori considerations, and the particular form of @ shown in Figure 3 for A = 21 X 10-lom is a unique representation of the electrostatic repulsive force acting on the solute from the membrane surface, and this representation was used in the rest of this paper. Effect of Pressure on Sodium Chloride Separation. Using the potential function CP given in Figure 3, and the corresponding values of R given in Table I, NaCl separations a t different operating pressures were calculated for the six membranes listed in Table I using eq 42 and 43. The results, given in Figure 4, showed that for each membrane, solute separation increased with increase in operating pressure, which confirm the results already well known experimentally for a long time (Sourirajan, 1970a). Reverse Osmosis Separation of p-Chlorophenol in Dilute Aqueous Solutions Using Porous Cellulose Acetate Membranes. Previous experimental work on the subject (Matsuura and Sourirajan, 1972; Dickson et al.,
1979) has shown that the system p-chlorophenol-watercellulose acetate membrane is an example in a class of reverse osmosis systems where solute is preferentially sorbed at the membrane-solution interface because of the greater acidity of the solute relative to that of water and the basic nature of the cellulose acetate membrane material. In such a system, solute separations are usually ratios are negative (and sometimes, positive), (PR)/(PWP) significantly less than unity even when osmotic pressure of the feed solution is negligible, and solute separation decreases with increase in operating pressure. With reference to p-chlorophenol-water-cellulose acetate membrane system, on the basis that the solute is preferentially attracted towards the membrane surface, the mobility of the solute decreases as the membrane surface is approached, and the preferentially sorbed layer in general consists of both immobile and mobile solute molecules, one can now generate the approproate potential function for the system under consideration and study the variations ratios obtainable in solute separation, and (PR)/(PWP) as a function of average pore radius on membrane surface, operating pressure, and pore size distribution on the membrane surface, and then compare such variations with those obtained experimentally. This was done as follows for the films listed in Table I and very dilute feed solutions (solute concentration = 1 g-mol/m3) in the operating pressure range 1724 to 10342 kPa gauge (250 to 1500 psig). The overall interaction force over^) working between a nonionized organic solute and the polymer membrane material can be expressed as the sum of the short range and the still van der Waals attractive force (4attractive) shorter range repulsive force arising from the overlap of electron clouds of interacting atoms, as given by the Lennard-Jones potential function
where B and C are the attractive and repulsive force constants, respectively. For the attractive force between the solute and the membrane surface assumed to be flat (Israelachvili and Tabor, 1973), eq 61 may be modified as
At the membrane surface, the solute molecule is predominantly subject to the repulsive force which practically vanishes as the molecule recedes away from the surface; further, since 4 appears as exp(-4/R7') in eq 35, and the latter quantity becomes nearly equal to zero for values of
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981 279
4/RT > 10, the change in f 'due to change in 4/RT > 10 is negligible. Consequently, eq 62 can be approximated as 4overd= lORT when d I D (634
SYSTEM' CELLULOSE ACETATE (E-398) MATERIALp-CHLOROPHENOL-WATER
5
= -- RT when d > D (631) d3 where D is the distance between the polymer surface and the solute molecule, at which 4overd becomes very large. In terms of nondimensional quantities, eq 63 can be rewritten as CP(p)overd= 10 when 1 - p I D/R (64a)
3
)I
B/R3
= -when 1 - p > D/R (64b) (1 - d 3 Equations 63 and 64 may be considered as the approximate forms of Lennard-Jones equation applicable for the pchlorophenol-water-cellulose acetate membrane reverse osmosis system under consideration. The decrease in the mobility of the solute as the membrane surface is approached (due to attraction of solute by the membrane surface) has to be represented by an appropriate frictional funtion ($); the form of this function was arbitrarily chosen to be as follows for the system under consideration. t) = 10 when d ID (65a) =E/d whend>D (65b)
Figure 6. Potential curves of interfacial forces for the system cellulose acetate (E-398) material-p-chlorophenol-water.
-
M E M B R A N E M A T E R I A L : C E L L U L O S E ACETATE ( E - 3 9 8 ) S Y S T E M ' p - C H L O R O P H E N O L - WATER
1.01
kPa gauge
10342
LL
a
where E is a constant. In nondimensional form, eq 65 can be written as $ ( p ) = 10 when 1 - p I D/R (66a) = -E/R 1-P
when 1 - p
> D/R
(66b)
Further, it was assumed that the dimensionless parameter b ( p ) could be expressed as b ( p ) = eJ.(p)= el0 when 1 - p I D/R (67a) = ex.(
-)
E/R 1-P
when 1 - p > D/R
(67b)
According to eq 67, b(p) is largest at the polymer surface (d = 0, or p = 1) and diminishes rapidly to the center of the pore (d = R, or p = 0). As the magnitude of pore radius becomes large, b(p) approaches unity, and at this point friction force is contributed by solute-solvent interaction (xm) only. Equations 64 and 67 express clearly the interaction force between polymer surface and solute molecule and the frictional force acting against the movement of solute through the membrane pore, respectively. By specifying the numerical values of the constants B, D, and E, the functions CP(p) and b(p) as represented by eq 64 and 67, respectively, are precisely defied. The latter equations in conjunction with eq 42, 43, and 50 offer a means of calculating f'and (PR)/(PWP)ratio for the reverse osmosis system under consideration using the membranes listed in Table I under the experimental conditions specified earlier. These calculations were made as follows. Using the diffusivity data for p-chlorophenol at infinite m2/s), the value of xAB for dilution (Dm = 0.947 X the system was found to be 2.617 X 10l2J.s/m2 g-mol, and the applicable value of k for each membrane was obtained using the 2/3-powerrelation (Sourirajan, 1978). Equation 52 was again used for calculating cA2 values. The values of P1 and P2 were calculated from eq 39 and 40, respec-
R x IO",
I
1
15
20
m
Figure 6. Effect of pore radius on p-chlorophenol separation and (PR)/(PWP)ratio. Membrane material, cellulose acetate (E-398); feed concentration, 1 g-mol/m3; mass transfer coefficient k = 15.5 X lo4 m/s; -, calculated; 0,A, experimental.
tively. The numerical values of the constants B, D, and E were changed by trial and error, until the values off' and (PR)/(PWP)ratio calculated from eq 42 and 50, respectively, were in reasonable agreement with the experimental values. The values of B, D, and E so determined m3, 0.6 X 10-lom, and 0.05 X were 13.5 X m, respectively. The values of CP and $ above 10 and below -13 were considered as 10 and -13, respectively, in order to make the computer calculation possible. On the basis of the values of B, D, and E given above, Figure 5 gives the polymer-solute interaction potential function (a, solid line) and the frictional force function ($, dotted line). Using these functions, Figure 6 shows the calculated values (solid lines) of (PR)/(PWP)ratio and f' for the six membranes listed in Table I as a function of pore radius R, and Figure 7 shows similar performance data as a function of operating pressure for two membranes. These results show: (i) p-chlorophenol is negatively separated in all the experiments; (ii) the (PR)/ (PWP)ratio is less than unity in all the experiments in spite of the fact that the overall osmotic pressure effects are negligible; (iii) decrease in pore radius decreases (PR)/(PWP)ratio; (iv) operating pressure does not affect
280
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981 6895 LPo Pauge (1000p l l q l
MEMBRANE MATERIAL CELLULOSE ACETATE I E - 3 9 8 1 SYSTEM p-CHLOROPHENOL-WATER PORE RADIUS x IO’? m
I
-
I
0.6
I
I
I
8.2
E
690 kPa aauoa
I
,
I
I
I
I
I 100 m a l
I
,
I
I I
I o a 0.3
rNoCl
1.0
I
0 B 0.3
f NaCI
0
v) W
I
-151
0
3 1
1
I
I
I
10
15
20
25
R x IOio
- 20 0
1
5
20
I 40
PI x IO-‘,
1 60
I 80
I n
I00
1
kPo gauge
Figure 7. Effect of operating pressure on p-chlorophenol separation and (PR)/(PWP)ratio. Membrane material, cellulose acetate (E398); feed concentration, 1 g-mol/m3; mass transfer coefficient k = 15.5 X 10” m/s; -, calculated; 0, A, experimental.
(PR)/(PWP) ratio significantly; (v) increase in operating pressure decreases solute separation; and (vi) at a low operating pressure, solute separation passes through a minimum with change in pore radius, and at a relatively high operating pressure, solute separation decreases with decrease in pore radius. These results are not only remarkably similar but also reasonably close to the actual experimental results which are shown in Figures 6 and 7 for comparison. It might be possible to obtain even closer agreement between the calculated and experimental results through better choice of numerical values of the constants B, D, and E,and/or appropriate assumptions on pore size distribution. In any case, the fact that the trends in the variations in the values of f’and (PR)/(PWP)ratio expressed by the results calculated from the set of eq 42, 43, 50, and 52 are exactly the same as those obtained experimentally, and the fact that the calculated and experimental results are as close as they are even without further refinement in calculations shows that eq 42,43, 50, and 52 are fundamentally valid for the analysis of reverse osmosis transport for systems of the type p-chlorophenol-water-cellulose acetate membrane where the solute is preferentially sorbed at the membrane-solution interface. Effect of Pore Size Distribution on Membrane Performance. The set of eq 42,43, 50, and 52 and the set of eq 43, 47, 51, and 52 together offer a means of computing the effect of pore size distribution on solute separation and (PR)/(PWP)ratio as a function of average pore size on the membrane surface at any chosen operating pressure, or as a function of operating pressure for any chosen membrane. Such computations were made assuming dilute aqueous feed solutions for the systems NaC1-H20 and p-chlorophenol-water a t the operating pressures of 6895 kPa gauge (lo00 psig) and 690 kPa gauge (100 psig) for different films as a function of average pore radius; two pore size distributions corresponding to a = 0 and u = 0.3 were considered for illustration. For the latter case, the integrations of eq 43 and 51 with respect to the pore size distribution function Y(R)were carried out from the lower limit of -3u to the upper limit of +3a, so that 99.7 % of the total number of pores involved was included in the pore size distribution function. The results obtained by the foregoing computations are shown in Figure 8. Figure 8 shows that for the chosen assumption of normal pore size distribution, the differences in solute separations and (PR)/(PWP)ratios for the cases of u = 0 and Q = 0.3
I
0
I
I
I
I
5
IO
15
20
/I
25
m
MEMBRANE MATERIAL CELLULOSE ACETATE ( E - 3 9 8 ) SYSTEMS NaCI- WATER AND p-CHLOROPHENOL WATER
-
Figure 8. Effect of pore size distribution on solute separation and (PR)/(PWP)ratio. Membrane material, cellulose acetate (E-398); feed concentration, 1 g-mol/m3; mass transfer coefficient k = -.
with respect to the two solution systems considered are not spectacular, but the trends shown appear significant. At a given level of NaCl separation, the corresponding separation for p-chlorophenol is higher when u = 0.3 than when a = 0; this result is consistent with the experimental observations reported earlier (Matsuura and Sourirajan, 1972; Dickson et al., 1979). Further, while (PR)/(PWP) ratio is unity and remains unaffected by a for the system NaCl-H20, the above ratio is less than unity and it is affected by u significantly for the system p-chlorophenol-water. The general conclusion that one can draw from Figure 8 is that variations in pore size distrubiton do affect both solute separation and (PR)/(PWP)ratio, and such variations may be significant for different reverse osmosis systems and different types of pore size distributions, and hence such variations should be studied in detail. Analysis of Reverse Osmosis Systems CumeneWater-Cellulose Acetate Membrane and BenzeneWater-Cellulose Acetate Membrane. Previous work has shown (Matsuura and Sourirajan, 1973) that these systems are examples in a class of reverse osmosis systems where solute is preferentially sorbed at the membranesolution interface due to solute-polymer attraction because of their hydrophobic (nonpolar) character (Matsuura et al., 1976a). The functions CP (representing the solutepolymer interaction) and b (representing the friction force acting against the movement of solute through the membrane pores) were generated for the above two reverse osmosis systems. For this purpose, average pore radii for a set of cellulose acetate membranes were calculated using reverse osmcrsis data for NaC1-H20 feed solutions at 10342 kPa gauge (1500 psig) pressure, and the potential function for the system given in Figure 3. Using the above membranes, experimental data for reverse osmosis separations of cumene and benzene were obtained at the operating pressures of 1724 kPa gauge (250 psig) and 10342 kPa gauge (1500 psig) using very dilute (0.19 and 0.81 gmol/m3, respectively) feed solutions. Equation 52 was again used to calculate CM values in each experiment. The parameters B, D, and E for the cumene-water-cellulose acetate and benzene-water-cellulose acetate systems were chosen by trial and error (just as indicated earlier for the p-chlorophenol-waterllulose acetate membrane system) so that the values of f’and (PR)/(PWP)ratio calculated from eq 42 and 50 were in reasonable agreement with the experimental values. The values of B, D, and E so dem3, 7.0 X m, and 49 X termined were 2916 x 10-lom, respectively for the cumene-water-cellulose acetate membrane system and 13.5 X m3, 1.5 X m,
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981 281 MEMBRANE MATERIAL.CELLUL0SE ACETATE (E- 398)
a
-
SYSTEWCUMENE-WATER
-
r - 0
0
I
p 0.8
n
A
'
LI:
A
I
I
b
10342
I
100-
1
8 90r
0
h
e
-5
2 Lo
SYSTEM CELLULOSE ACETATE ( E - 3 9 8 1 MATERIALCUMENE-WITER
CELLULOSE ACETATE (E-3981 MbTERIALBENZENE-WATER
501
t
B
1
I 15
20
25
30
r
10 .
6
0 I
0.9 -10
A
I 10
10342
b
1724
SYSTEM:BEN~ENEI WATER
ug70-
a
b
Figure 9. Potential curves of interfacial forces for systems (a) cellulose acetate (E-398) material-cumene-water, and (b) cellulose acetate (E-398) material-benzene-water.
and 10.5 X m, respectively for the benzene-watercellulose acetate membrane system. On the basis of the above values those of and b functions were calculated using eq 64 and 67 and those of f'and (PR)/(PWP)ratio were calculated from eq 42 and 50. The results obtained are given in Figures 9 and 10. Figure 9a shows a thick repulsive potential barrier extending to 7 A followed by a deep potential well (attractive force) and friction force for cumene. The former contributes to positive solute separation in reverse osmosis, and the latter forces contribute to strong adsorption and possible agglomeration of cumene at the membraneaolution interface. The corresponding potential and friction forces are similar, but lesser in magnitude for benzene as shown in Figure 9b. The latter result is understandable on the basis that cumene-polymer interaction is stronger than benzene-polymer interaction, since cumene is more nonpolar than benzene as indicated by their respective modified Small's numbers (Zs* for cumene = 791 and Zs* for benzene = 425). A comparison of the potential barrier and friction curves for cumene and benzene on the one hand (Figures 9a and 9b) and those for p-chlorophenol on the other hand (Figure 5) are particularly enlightening. For p-chlorophenol, the repulsive force and the friction force are much less, and the strength of the attractive force diminishes far more rapidly with increase in distance from the membrane surface. This is understandable on the basis of the physical nature of solute adsorption involved in the two cases. In the case of p-chlorophenol, the solute is adsorbed on the polar sites of the membrane surface and does not allow competitive adsorption of water molecule on the same polar site; once a monolayer of p-chlorophenol is formed on the polar sites of the membrane surface, further adsorption of solute on the top of the monolayer becomes rapidly weak. Therefore, solute-polymer attraction decreases rapidly with increase in distance from the membrane surface. In the case of cumene and benzene, the solute is adsorbed on the nonpolar sites of the membrane surface, and the adsorption of water on the polar sites of the membrane surface is unaffected; further, the solute molecules can attract each other because of nonpolar attractive forces working among themselves in the polar environment so that such attractive forces can reach relatively farther into the solution. Figures 10a and 10b give data (solid lines) on f' and (PR)/(PWP)ratio €or the cumene and benzene systems, respectively, calculated on the basis of their respective @
56 % %6 0 Lob
50
10
15 R x IO",
20
25
30
m
Figure 10. Effect of pore radius on solute separation and (Pk)/ (PWP)ratio in systems (a) cumene-water and (b) benzene-water. Membrane material, cellulose acetate (E-398); feed concentrations, cumene 0.19 g-mol/m3 and benzene 0.81 g-mol/mg; mass transfer coefficient, cumene 13.8 X lo4 m/s and benzene 15.6 X lo4 m/s; -, calculated; 0, A, experimental.
and b functions shown in Figures 9a and 9b; the experimental results on f' and (PR)/(PWP)are also shown in Figures loa and lob for comparison. The calculated results (solid lines) show that: (i) solute separations are positive in all cases; (ii) cumene separation passes through maximum and minimum as the average pore radius on the membrane surface is varied, while benzene separation decreases with increase in average pore radius; (iii) cumem separation decreases with increase in operating pressure; (iv) (PR)/(PWP)ratio is less than unity in all cases even though the overall osmotic pressure effects are negligible; and (v) (PR)/(PWP)ratio decreases with increase in operating pressure. The above results are of extraordinary significance. They are not only qualitatively identical with the experimental results reported earlier (Matsuura and Sourirajan, 19731,they are also quantitatively in reasonable agreement with the experimental results, confirming the essential validity of the analytical approach developed in this work for reverse osmosis transport. Conclusions This paper offers a new analytical approach to reverse osmosis separations. The transport equations developed in this analysis are applicable whether solute or solvent is preferentially sorbed at the membrane-solution interface. The agreement between the results predicted by this analysis and those obtained experimentally, as illustrated in this paper, confirm the essential validity of the mechanism of reverse osmosis based on solute and solvent transport through capillary pores under the influence of surface forces. The analytical approach presented in this paper is not limited either to polymer membrane materials or to aqueous feed solution systems. In this analysis, four quantities (A, B, D, and E)emerge as interfacial parameters governing reverse osmosis separations. The quantity A represents the electrostatic repulsive force between membrane material and ionized solute; the quantities B and D represent solute-membrane material interactions governed by the Lennard-Jones type potential functions; and the quantity E represents the frictional force on the solute molecule decreasing its mobility through the membrane pores. If the values for the above four quantities can be estimated either through appropriate experimen-
282
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981
tation and/or through molecular and physicochemical basis, then those values can be used in the transport equations derived in this paper to predict possible reverse osmosis separations for a wide variety of membrane material-solute-solvent systems, and operating conditions. Such predictive capability would constitute a major advance in reverse osmosis process design and development. Nomenclature A = constant defined by eq 56, m B = constant defined by eq 62, m3 b = dimensionless parameter defined by eq 14 C = constant defined by eq 62, m12 CA = dimensionless solute concentration defined by eq 37 CA = molar concentration of solute, g-mol/m3 D = distance between pore wall and solute molecule, at which &vera becomes very large, m D m = diffusivity of solute in water, mz/s d = distance between pore wall and center of solute molecule, m E = constant defined by eq 65, m FA = diffusion force F a = friction force between solute and solvent FAM= friction force between solute and pore wall f = fraction solute separation based on the feed concentration f‘ = fraction solute separation based on the solute concentration in the boundary phase as defined by eq 21 g = density of solution, kg/m3 J A = molar flux of solute, g-mol/m2.s k = mass transfer coefficient of solute on the high pressure side of membrane, m/s P = pressure, kPa Pi = operating pressure applied at pore inlet, kPa or Pa when used in eq 54 Po= external pressure prevailing at pore outlet, kPa and Pa when used in eq 54 (PR) = membrane permeated product rate for given area of membrane surface, kg/h (PWP) = pure water permeation rate for given area of membrane surface, kg/h Qpwp = permeation rate of pure water through a single pore, kg/s QPR = permeation rate of product solution through a single pore, kg/s R = pore radius, m R = average pore radius, m R = gas constant r = radial distance in cylindrical coordinate, m S = effective area of membrane surface, m2 T = absolute temperature, K uA= velocity of solute in the pore, m/s U B = velocity of solvent in the pore, m/s v , = permeation velocity of product solution, m/s Y ( R ) = pore size distribution function z = axial distance in cylindrical coordinate, m Greek Letters a = dimensionless quantity defined by eq 38
p1 = dimensionless quantity defined by eq 39
p2 = dimensionless quantity defined by eq 40 6 = length of cylindrical pore, m 7 = solution viscosity, Pa-s p A = chemical potential of solute J/g-mol p ’ ~= chemical potential of solute at a reference concentration, J/g-mol p = dimensionless quantity defined by eq 36 u = standard deviation in pore size distribution = dimensionless quantity defined by eq 41 4 = potential function of force exerted on solute molecule by pore wall, J/g-mol xm = proportionality constant defined by eq 6, J.s/m2.g-mol xw = proportionality constant defied by eq 2, J.s/m2.g-mol $ = frictional function of force restricting the movement of solute molecule Subscripts 1 = bulk feed solution on the high-pressure side of membrane 2 = concentrated boundary solution on the high-pressure side of membrane 3 = membrane permeated product solution on the atmospheric side of the membrane Literature Cited Anderson, J. L.; Maione, D. M. Bbphys. J . 1974, 74. 957. Bean, C. P., Research and Development Progress Report No. 465, Office of Saline Water, U.S. Department of the Interlor, Washlngton, D.C., 1969. Mckson. J. M.. Matsuura, T., Sourirajan, S. Ind. Eng. Chem. Process Des. Dev. 1979, 78, 641. Glueckauf, E. Proceedings, First Intematiinai Symposium on Water Desaiination, Washington, D.C., 1965 (Published by US. Department of the InterC or, Offlce of Saline Water: Washington, D.C., 1965; Voi. I, pp 143-156). Israeiachviii, J. N., Tabor, D. In “Progress in Surface and Membrane Science”, Danieili, J. F.. Rosenberg, M. D., Cadenhead, D. A., Ed.; Voi. 7, Academic: New Ywk, 1973; pp 1-55. Jacazio, G.; Probstein. R. F.; Sonin, A. A,; Yung. D. J. Phys. Cbem. 1972, 76, 4015. Jonsson, G.; Boesen, C. E. Desallnatbn 1975, 77, 145. Matsuura, T.; Baxter, A. G., Souriraian, S. Ind. €ng. Chem. Process Des. Dev. 1977, 16, 82. Matsuura, T.; Bednas, M. E.; Dickson, J. M.; Sourirajan, S. J . Appl. Po!vm. Scl. 1874. 18. 2824. MaGuura, T.;’Blais,P~~Sowirajan, S. J. Appl. P0l)lm. Scl. 1976a, 20, 1515. Matsuura, T.; Dckson, J. M.; Sourirajan, S. I d . Eng. Cbem. Process Des. Dev. 1976b, 75, 149. Matsuwa, T.; Pageau, L.; Sourirajan, S. J. Appl. P N m . Sci. 1975, 19, 179. Matsuura, T.; Sourirajan, S. J . Appl. Polym. Scl. 1972, 76, 2531. Matsuura, T.; Sourirajan, S. J . Appl. Polym. Scl. 1973, 17, 3883. E. C.; Sourirajan, S. Ind. Eng. Rangarajan, R.; Matsuura, T.; -hue, Chem. Process Des. D e v . 1976. 75, 529. Rangarajan, R.; Matsuura, T.; Goodhue, E. C.; Sourirajan, S. Ind. Eng. Chem. Process Des. D e v . 1978, 77, 71. Sourirajan, S. “Reverse Osmosis”; (a) Chapter 2, (b) Chapter 3, Academic Press: New York, 1970. Sourlrajan, S. Pure Appl. Chem. 1978. 50, 593. Sourirajan, S.; Matsuura, T. In “Reverse Osmosis and Synthetic Membranes”, S. Sourirajan, Ed.; (a) Chapter 2, (b) Chapter 3, National Research Council Canada: Ottawa, 1977. Sourirajan, S.; Matsuura, T.; Hsieh, F. H.; Gilded, G. R. In “Ultrafiltration Membranes and Applications”, Cooper, A. C., Ed.; Plenum: New York, 1980; pp 21-43.
Received for review December 26, 1979 Accepted November 12, 1980
Issued as NRC No. 19084.
