124
Ind. Eng. Chem. Prod. Res. Dev. 1984,23, 124-133
[ VR’lwater = retention volume of water represented by that of
DzO, m3 Y(Rb)= normal pore size distribution function, l / m Greek Letters LY = dimensionless solution velocity profile in the pore = dimensionless solution viscosity p2 = dimensionless operating pressure I‘ = surface excess of the solute, mol/m2 6 = length of cylindrical pore, m 7 = solution viscosity, Pa.s 7r = osmotic pressure, Pa p = dimensionless radial distance a = the standard deviation of pore size distribution, m a1 = the standard deviation of the first distribution, m a2 = the standard deviation of the second distribution, m 7 = parameter characterizing the size of colloidal aggregate = dimensionless potential function 4 = potential function of interaction force exerted on the solute from the pore wall, J/mol XAB = DAB/RT Literature Cited Bechhold, H. 2.Phys. Chem. 1008, 64, 328. Blatt, W. F.; Dravld, A.; Michaels, A. S.;Nelson, L. I n “Membrane Science and Technology”, Flinn, J. E., Ed.; Plenum: New York, 1970; p 47. Chan, K.; Matsuura, T.; Sourlrajan, S. Ind. Eng. Chem. prod. Res. Dev. 1082, 27, 605. Chuduk, N. A,; Ettekov, Yu. A.; Kiselev, A. V. J. Colloid Interface Scl. 1081, 8 4 , 149. Grant, W. H.; Dehi, R. S. I n “Adhesion and Adsorption of Polymers”, L.-H. Lee, Ed.; Plenum: New York, 1979; p 827. Green, D. M.: Antwiler, G. D.; Moncrief, J. W.; Decherd, J. F.; Popovich, R. P. Trans. Am. SOC.ArNf. Int. Organs 1076, 22, 627.
Hsieh, F. H.; Matsuura, T.; Sourlrajan. S. J. Sep. Proc. Techno/. 1070, 1 ,
50. Kutowy. 0.; Thayer, W. L.; Sourirajan, S. Desallnatlon 1078, 26, 195. Matsuura, T.; Blais, P.; Sourirajan, S. J . Appl. Powm. Sci. 1076, 20, 1515. Matsuura, T.; Sourlrajan, S. J . Colloid Interface S d . 1078, 66, 589. Matsuura, T.; Sourirajan, S. J . Appl. Poly” Sci. 1073, 77, 3683. Matsuura, T.: Sourlrajan, S. Ind. Eng. Chem. Process Des. Dev. 1081, 2 0 , 273. Matsuura, T.; Taketani, Y.; Sourlrajan, S. “Synthetic Membranes”. Turbak, A. F., Ed.; ACS Symposium Series, No. 154, American Chemical Society: Washington, DC, 1981a: p 315. Matsuura, T.; Taketani, Y.; Sourirajan, S. “Use of Liquid and Gas Chromatog raphy Data for Estimation of Interfacial Forces Governing Reverse Osmcsis Separations”; Proceedings of the 2nd World Congress of Chemical Engineering, Montreal, Oct 4-9, 1981b; Vol. IV, p 182. Matsuura, T.; Tweddle, T. A.; Sourirajan, S. “Predictability of Performance of Reverse Osmosls Membranes from Data on Surface Force Parameters”, paper presentedat the Symposium on Membrane Processes for Industrial Wastewater Treatment in the Summer National AIChE Meeting, Cieva lend, OH, A 4 29-Sept I , 1982. McBain, J. W. “ColloM Science”, Chapter 11; D. C. Heath: Boston, MA, 1950. Michaels, A. S. Sep. Sc/. Techno/. 1080, 75, 1305. Michaels, A. S.; Nelson, L.; Porter, M. C. I n “Membrane Processes in Industry and Biomedicine”, Bier, M., Ed.; Plenum Press: New York, 1971; p 197. Sarboiouki, M. N. Sep. Scl. Technol. 1082, 77, 381. Sourirajan, S.; “Synthetic Membranes”, Turbak, A. F., Ed., ACS Symposium Series No. 153, American Chemical Society: Washington, DC, 1981: p 11. Sourirajan, S.; Matsuura, T. The Chem. E-. (London), 1082, No. 385,359. Taketani, Y.; Matsuura, T.; Sourlrajan, S. Sep. Sci. Technol. 1082, 77, 821.
Receioed for review February 15, 1983 Revised manuscript received July 1, 1983 Accepted August 29, 1983
Issued as N.R.C. No. 22902.
Effect of Shrinkage on Pore Size and Pore Size Distribution of Cellulose Acetate Reverse Osmosis Membranes Kam Chan, Llu Tlnghul, Takeshl Matsuura, and S. Sourlrajan’ National Research Council of Canada,
Ottawa, K1A OR9, Canada
The average pore size and the pore size distribution of reverse osmosis membranes have been determined on
the basis of the surface force-pore flow model by use of separation data of reference solutes. The results confirm the correspondence between the pore size distrlbutlon of unshrunk membranes and the shrinkage temperature profile proposed by Kunst and Sourirajan. The change in the pore size of unshrunk membranes to those of membranes shrunk at various temperatures is further related to the change in the thermodynamic properties associated with the movement of polymer molecules. The significant effect of the type of pore size distribution on the fractionation of solute molecules is also illustrated.
Introduction The pore size and the pore size distribution have been the central theme of ultrafiltration since the very early stage of its development. The determination of the membrane pore size goes back to the paper by Gu6rout (1872) in which the average pore size was calculated from the pure water flux by use of the Poiseuille law. Bechhold later applied the Laplace law and developed the fluid permeability and bubble pressure method (Bechhold and Schnurmann, 1929), which permits the calculation of the radius of the largest pores. Combination of the above two methods gives some idea of the broadness of the pore size distribution. Following these early developments, various other methods for the determination of the pore size distribution such as the mercury intrusion method, gas permeability method, and electron microscope method were proposed. While the last method visualizes the pore 0196-432118411223-0124$01.50/0
in the membrane, all other methods are based on certain physical principles on the transport of permeants, either in gas or in liquid form, through the membrane. Probably the latest and the most comprehensive review of the available methods of pore size determination was given by Kamide and Manabe (1980) with their own experimental results. Furthermore, a new technique of thermoporometry, where the differential scanning calorimetry is utilized, was recently developed (Desbrieres et al., 1981). Despite the development of various physical methods described above, the most pragmatic and probably the best method of determining the pore size and pore size distribution is on the basis of the permeation of the solute molecules of known size (van Oss, 1970). In this respect the recent works of the determination of pore size distributions by applying Gaussian normal distribution (Green et al., 1976) and log-normal distribution (Michaels,
Published 1984 by the American Chemical Society
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 1, 1984 125
1980; Schwarz et al., 1982) are of practical interest. The approach was based on the sieving mechanism of ultrafiltration, though the effect of adsorption and desorption characteristics of the solute on the ultrafilter surface has been frequently mentioned (Dejardin et al., 1980; Ingham et al., 1980). Realizing the importance of the pore size and the pore size distribution as well as the preferential sorption characteristics as governing factors of ultrafiltration, Chan et al. (1982) incorporated both the pore size distribution and the interaction force parameters into transport equations based on surface force-pore flow model and determined the pore size distribution of ultrafiltration membranes. In contrast to ultrafiltration membranes, where pores are visible by electron micrography, pores of the size 400 cm3/min. In each experiment the fraction solute separation, f, defined as f = [(ppm of solute in feed) (ppm of solute in product)]/(ppm of solute in feed) and product rate (PR) and pure water permeation rate (PWP) in kg/h for given area of film surface, 13.2 cm2 in this work, were determined under the specified experi-
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Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 1, 1984
Table 111. Membrane SDecificationsa membranes
A X
batch 316-unshrunk batch 316-67 batch 316-77 batch 316-82 batch 602-unshrunk batch 602-62.5 batch 602-67.5 batch 602-70.0 batch 602-80.0 batch 18-unshrunk batch 18-77.5 batch 18-85.0 batch 18-87.5 batch 18-90.0
l o 7'
9.948 2.792 2.114 2.120 13.56 7.052 6.068 3.211 1.485 13.63 7.127 3.452 2.044 0.964
NaCl sepn, %
product rate x lo3, kg/h
36.5 79.6 90.0 96.1 10.6 37.6 53.2 71.2 94.4 7.0 32.5 55.2 71.9 93.2
148.2 35.6 27.4 26.8 205.0 97.8 83.1 42.5 18.6 206.0 92.5 46.8 26.1 12.4
525.2 12.07 4.916 1.684 1345 159.5 93.57 24.20 1.616 1380 111.2 47.27 12.27 1.456
a Operating pressure, 1724 kPag (= 250 psig); feed NaCl concentration, 0.06 m ; k = 22 x kg-mol of H20/mz.s,kPa. Dimension of (D,M/K~),,~~,m/s.
Table IV. Pore Size Distributions of Membranes -
Rb,l membranes batch 316-unshrunk batch 316-67 batch 316-77 batch 316-82 batch 602-unshrunk' batch 602-62.5 batch 602-67.5 batch 602-70.0 batch 602-80.0 batch 18-unshrunk' batch 18-77.5 batch 18-85.0 batch 18-87.5 batch 18-90.0
Rb.2
x 10",
m 50.9 41.9 39.9 35.9 54.2 48.4 45.9 42.9 37.9 60.8 47.9 44.9 44.2 41.9
ai/Eb.i
0.002 0.002 0.002 0.002 0.002 0.005 0.004 0.003 0.002 0.002 0.004 0.003 0.003 0.002
02
/%.
Dimension of A ,
h2
2
0.196 0.196 0.195 0.195 0.185 0.177 0.137 0.108 0.049 0.160 0.147 0.098 0.098 0.098
0.050 0.005 0.003 0.001 0.060 0.019 0.010 0.006 0.002 0.080 0.030 0.010 0.005 0.001
from data of shrunk membranes.
Comparison of Experimental and Calculated Solute Separationn pure water permeation 1,3-dioxolane rate x lo3, kg/h exptl calcd
batch batch batch batch batch batch batch batch batch batch batch batch
-
lo'',
m 10.37 9.17 8.37 7.47 10.37 9.42 9.07 7.87 6.87 10.37 8.37 7.29 7.20 7.20
' Pore size distributions reconstructed Table V.
X
m/s.
316-unshrunk 316-67 316-77 316-82 602-62.5 602-67.5 602-70.0 602-80.0 18-77.5 18-85.0 18-87.5 18-90.0
156.8 44.0 33.3 33.4 111.1 95.6 50.6 23.4 112.3 54.4 32.2 15.2
0 5.2 9.8 21.0 0 0.1 3.2 10.1 0 8.8 12.0 13.4
' Solute concentration in feed, 100 ppm.
-1.31 3.3 9.4 21.0 0.03 6.3 8.8 17.1 3.0 14.4 20.3 25.2
solute separation, % p-dioxane
12-Crown-4
15-Crown-5
18-Crown-6
exptl
calcd
exptl
calcd
exptl
calcd
exptl
calcd
0 16.1 27.4 38.5 11.2 17.5 24.1 48.3 11.3 31.5 41.8 53.1
-1.31 16.1 27.2 38.7 11.7 18.1 24.5 36.0 10.2 27.0 34.2 42.2
21.5 75.3 78.3 89.2 45.2 61.5 74.7 89.8 35.5 63.8 76.4 89.9
26.0 75.3 80.8 89.6 43.7 56.6 72.5 88.3 33.8 59.5 73.4 88.7
35.2 86.1 92.96.9 50.3 64.8 78.1 94.8 41.1 65.5 77.1 92.5
38.0 88.5 91.4 97.3 53.9 67.8 83.1 97.0 43.9 68.8 82.0 96.2
42.7 90.5 92.4 96.7 55.1 73.0 82.1 96.0 46.6 67.8 80.5 92.6
36.0 88.1 91.1 97.2 52.3 66.3 82.4 96.8 42.1 67.5 81.3 96.0
Operating pressure, 1724 kPag (= 250 psig).
mental conditions. The data on (PR) and (PWP) are corrected to 25 OC. Reverse osmosis experiments were conducted with respect to sodium chloride and five reference organic solutes. Concentrations of NaCl were determined by a Beckman total carbon analyzer, Model 915-A.
Results and Discussion Effect of Shrinkage on the Pore Size Distribution of RO Membranes. The pore size distribution determined for each membrane is listed in Table IV and the agreement of calculated solute separation on the basis of the pore size distribution listed in Table IV with the ex-
perimental separation data is shown in Table V. The pore size distribution is further illustrated both in differential form according to eq 1and eq 2 (Figure 3) and in integral form according to eq 3 (Figure 4) with respect to batch 316 membranes using numerical values listed in Table IV. It is clear from Table IV that two pore size distributions are necessary for the RO membranes studied, one with the average radius of 6.9 to 10.4 X m and the other the average radius of 36 to 61 X m. For example, the quantity Sum was calculated to be 0.0232 for both the best fit uniform and single normal distribution, while Sum was 0.0015 for the best fit two normal distributions with respect to the CA-316-67 membrane. The ratio of the total number
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 1, 1984 129 BATCH 316-77
99 -
v)
W
98-
?2 1
a
97-
8 z
96-
0
IO 20 30 40 50 60 W R E RADIUS B E F O R E S H R I N K A G E ,
Figure 6. (Rb)l vs. 0
IO
50 60 PORE RADIUS x 10",m 20
30 40
100
8oc
90
1
I
60
80
90
x dOm
(R& at different shrinkage temperatures.
70 I
Figure 4. Integral pore size distribution of batch 316 membranes.
f
70
1 I 70 80 SHRINKAGE TEMPERATURE~~C
I
1
90
100
Figure 5. Shrinkage temperature profiles of batch 316, batch 602, and batch 18 membranes: membrane material, cellulose acetate Eastman E-398; operating pressure, 1724 P a gauge (= 250 psig); sodium chloride concentration in feed, 0.06 m ;feed flow rate, 400 cm3/min.
of the second distribution to that of the first distribution ranges from 0.1 to 8%, depending on the casting solution composition, casting condition, and shrinkage temperature. The data further indicate that the average radii of both distributions decrease with the increase in shrinkage temperature, though the effect of decreasing h2 is more significant. In other words, more pores are brought from the second distribution to the first distribution at higher shrinkage temperature. Interestingly, no matter how different the pore size distributions of the membranes without shrinkage are, membranes acquire ultimately a very similar pore size distribution, at the highest shrinkage temperature applied for each batch, with an average pore radius of the first distribution from 6.9 to 7.5 X 10-lo m and an average radius of the second distribution from 36 to 42 X 10-lo m. Pore Size Distribution and Shrinkage Temperat u r e Profile. According to Kunst and Sourirajan (1970) and Kunst (1982), a higher shrinkage temperature to give a particular level of solute separation indicates a larger average pore size, and a steeper shrinkage temperature profile (Figure 5) indicates a more uniform pore size distribution in unshrunk membranes. Then from the shrinkage temperature profiles shown in Figure 5, we may
conclude that batch 316-unshrunk membrane possesses the smallest average pore size with the least uniform distribution while batch 18-unshrunk membrane possesses the largest average pore size and the most uniform pore size distribution. Batch 602-unshrunk membrane lies in the intermediate region. When we look at the pore size distribution of unshrunk membranes listed in Table IV, we find that R b , l and U l / R b , l data are the same for all the membranes under consideration, and therefore they do not cause any change in the performance of unshrunk membranes. On the other hand, Rb,2, ~72/Rb,2, and h2 change from membrane to membrane considerably. Furthermore, comparing these data, we find that the increasing order of R , , is batch 316-unshrunk < batch 602-unshrunk < batch 18-unshrunk, while the decreasing order of u2/Rb,2 is batch 316-unshrunk > batch 602-unshrunk > batch 18-unshrunk. Considering that the smaller u2/Rb,2indicates the more uniform pore size distribution, the above orders are precisely the same as those concluded from the shrinkage temperature profiles. Thus, their earlier statement on the significance of shrinkage temperature profiles was confirmed by the application of the new technique of the determination of the pore size distribution to RO membranbs. Effect of Shrinkage on Pore Size and Pore Size Distribution. The pore size distribution of RO membranes given in Table IV can be used to determine the radii of pores which were produced on membranes shrunk at various temperatures and were originated from a given pore radius of unshrunk membranes by the method illustrated in the Appendix. Figure 6 shows the results, where the pore radius of the shrunk membrane designated as (Rb)s is a function of the pore radius of the unshrunk membrane designated as (Rb),, at various shrinkage temperatures. There are several features unique to the figure. (1)When the pore radius is large there is only a moderate decrease in pore size by shrinkage. However, when the pore radius of the unshrunk membrane comes down to a threshold value, the pore radius is drastically reduced to a small pore radius. The latter radius corresponds to the radius of the first distribution (see Figure 3). When the pore radius of the unshrunk membrane is below the threshold value, the pore radius after the shrinkage remains constant. (2) When the temperature increases, the magnitude of the moderate decrease in the large pore region (of the unshrunk membrane) increases. (3) The threshold pore radius also increases with increase in shrinkage temperature. (4) There is a slight decrease in
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Ind. Eng. Chem. Prod. Res. Dev.. Vol. 23. NO. 1. 1984
POLYMER MATERIAL
- A R E A STRETCHED BY THE WORK ‘ZllRbl: - ~ R & I x INTERFACIAL TENSION
Figure 7. Schematic illustration of polymer material stretching at the pore shrinkage.
-
Figure 9. Change in Gihbs free energy as the function of slit distance d.
SHRINKAGE
ASTJ the entropy contribution -TAS becomes greater, as shown in Figure 10. On the other hand, AH contribution if practically independent of the temperature and AG vs. d curve becomes such as shown by a solid line in Figure 10. The energy input at the shrinkage temperature of T2 is also incrtased to Starting again from the pore diameter of (dl),, of the-unshrunk membrane we e_ndup at the pore diameter of (d;), which is smaller than (dl),. The larger magnitude of the moderate decrease in the radius of the pores larger than the threshold value at the higher shrinkage temperature (see Figure 6) is thus understandable. From the functional form of Gibbs free energyjt is also obvious that any pore diameter below (d;), (>(d2),,) is reduced to d, by shrinkage at temperature T2.Thus, the threshold pore diameter at temperature T2is larger than that at TI,which is also in accordance with the results shown in Figure 6. The small pore size d, achieved by the shrinkage at T2should not be very different from that at Tl. Experimentally, there was also only a slight decrease in the small pore radius at the higher temperature (see Figure 6), which is probably due to the small effect of temperature on the enthalpy curve. Thus, all the main features of the pore shrinkage illustrated in Figure 6 are fully understandable on the basis of the form of the free energy function involved in the pore shrinkage. A totally new concept of membrane pores emerges from the above discussion on the effect of membrane shrinkage. The two pore size distributions found in RO membranes probably resulted from the shape of the energy function associated with the distance between two macromolecular walls. A smaller pore distribution is related to the high potential wall at the small molecular distance and may be called the potential pore, while the other distribution of larger pores is related to the lower energy barrier induced by the entropy change and therefore may be called entropy pore. The shrinkage is a process to supply a sufficient amount of energy to the membrane in the form of heat energy to overcome the low energy barrier and to bring the entropy pores to potential pores. Both small and large pores in RO membranes are thus related to the thermodynamic properties of polymer materials. In contrast to these pores, there are pores of even larger radii (- 100 X 10-lo m) which were found in UF membranes (Chan et al., 1982). These pores have no relation to the thermodynamics of the polymer. For example, we can make a pore of any arbitrary size by making a large hole in a given membrane. Such pores may be called macroscopic pores. From a microwave investigation, Wasilewski (1966) concluded that almost all of the oxygen atoms of the carbonyl groups in the side chains of cellulose acetate molecule are intramolecularly hydrogen bonded. During the heating process, enough energy is absorbed by such a carbonyl group to break this hydrogen bond and make the group to rotate about a single bond; in that process two segments from different cellulose acetate molecules are
40
I
I
60
80
131
100
SUCROSE SEPARATION, %
Figure 11. Separations of ethyl tert-butyl ether and sucrose by membranes with uniform pore sizes (-) and by membranes with two pore size distributions (- - -). Calculation conducted for dilute feed solutions; operating pressure, 1724 Wag (= 250 psig); mass transfer coeffici_ent,infinity. For two pore size distrib_utions&,I = 6 X = 0.002, Rb,* = 40 x m, uZ/Rb,:!= 0.100 were used m, h2 was changed.
brought close together to cause a physical interaction and form a new intermolecular bond, which, being stronger than the intramolecular one, permanently immobilizes the segments in question (Wasilewski, 1966). According to the new concept of pore shrinkage this is a process of overcoming the low energy barrier induced by the entropy change bringing together the cellulose acetate molecule from one side of the energy barrier (entropy pore side) to the other side (potential pore side). The energy supplied during the shrinkage procedure and stored in the membrane thereafter may be released as time elapses. Thus, the pores on the right side of the entropy barrier (see Figure 10) may slide down the AG slope to the larger side until a stable position is reached. On the other hand, the- energy stored at the steep potential wall of distance d, can be released only when the pore drops to the bottom of the potential well, the position of which may correspond to the lattice structure of the polymer crystal. Thus, the transfer from the right side of the entropy barrier (entropy pore side) to the left side (potential pore side) is irreversible, unless a large energy input to open up the pore mechanically is supplied. Effect of the Pore Size Distribution on RO Membrane Performance. Figure 11 demonstrates the results of the calculation for separations of ethyl tert-butyl ether and sucrose with membranes of different pore sizes. Two different types of pore size distributions were considered. In one case the calculation was performed for membranes of uniform pore radii (solid line) varying the pore radius, and in the other case the porosity change was represented by the change in h2of two normal distributions, while other parameters such as average pore radii and the standard deviations of two normal pore distributions were unchanged (broken line). Both ether and sucrose separations decrease as the pore radius in the uniform distribution and h2value in the two normal distributions increase. In Figure 11 the separation of ether vs. separation of sucrose is plotted with respect to two different types of pore distributions. In both casea the mass transfer coefficient from boundary to feed solution was assumed to be infinity. The dilute solution systems were assumed for each calculation. The figure clearly indicates that ether separation is significantly higher by membranes with two distributions than by membranes of uniform pore size at a given sucrose separation. For example, at a fixed sucrose separation of 85%,the membrane with two distributions and the mem-
132
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23,No. 1, 1984
brane of uniform distribution exhibit ether separations of 64 and 36%, respectively. These results illustrate that the pore size distribution affects the fractionation of these two solutes considerably. Conclusion Applying the transport equations based on the surface force-pore flow model to the RO experimental separation data and calculating the average pore size and the pore size distribution of RO membrane, the following conclusions emerge. (1) Membrane pores and their distributions can be classified into three groups which are distinct in their origin. The first two distributions of smaller pore radii are closely related to the thermodynamic properties of the membrane polymer material, and the radii of such pores can be changed only in a limited range for a given polymer. The third distribution of the largest pore radii has no relation to the thermodynamic property of the polymer and can be changed arbitrarily. The distribution of pores of any given membrane consists of the combination of these three classes of pores. (2) The pore size distribution affects the fractionation of two solutes considerably. Appendix The reconstruction of the pore size distributions of unshrunk batch 602 and batch 18 membranes requires the following three assumptions: (1) During the shrinkage process the total number of pores does not change. Only the size of the pore changes. (2) The pore radius after shrinkage is a unique function of the original pore radius (before shrinkage) and the shrinkage temperature. This function is common for all batches of cellulose acetate membranes involved in this study. (3) When there are two pores of different sizes in the unshrunk membrane the larger pore remains as the larger pore after the shrinkage. In other words, the order in size of pores in the unshrunk membrane is maintained in the shrunk membrane. Since pore size distributions of batch 316-unshrunk, batch 316-67, and batch 316-77 membranes are given in Table IV, integral distribution curves can be drawn as shown in Figure 4 by applying eq 3. Using these curves, the radii of pores on the membranes shrunk at 67 and 77 OC which originated from a given pore radius of the unshrunk batch 316 membrane are determined in the following way. Figure 4 indicates that 99.76% of pores are with radii less than 65.9 X 10-lom with respect to batch 316-unshrunk membrane, while the same percentage of pores belong to pores with radii less than 42.9 X m and 34.9 X 10-lo m on batch 316-67 and batch 316-77 membranes, respectively. Applying assumptions (1)and (3), it is concluded that the pore with the radius of 65.9 X 10-lo m in the unshrunk membrane is reduced to the pores with radii of 42.9 X 10-lo m and 34.9 X 10-lo m at the shrinkage temperature of 67 and 77 "C, respectively. By this means we are able to produce a list of the pore radii of the unshrunk membrane and the corresponding pore radii at the shrinkage temperatures of 67 and 77 "C, as shown in Table VI, and this table is assumed to be valid not only for batch 316 membranes but also for batch 602 and batch 18 membranes according to assumption (2). A note here is in order about the influence of pores of largest radii on the overall transport. At a first glance 99.76% considered above seems to be 80 close to 100% that 100 - 99.76 = 0.24% of pores (of largest radii) can be neglected. However, it is not true. Since the solution flux of individual pores is proportional to the 4th power of the pore radius, as indicated by eq 4, the contribution of largest pores to the overall transport is significant though the
Table VI. Shrinkage of Pores on Batch 316 Membranes membrane batch 316
316
radius of shrunk membranes. (Rb)s x 1010,'m radius of unshrunk membrane, x 1O1O,m
shrinkage temp, oc
70.9 69.9 68.9 61.9 66.9 65.9 64.9 63.9 62.9 61.9 60.9 59.9 58.9 57.9 50.9 40.9 30.9 20.9 10.9 9.9
67
17
58.3 56.9 49.9 41.7 44.9 42.9 40.2 36.0 25.4 9.2 9.2 9.2 9.2 9.2 9.2 9.2 9.2 9.2 9.2 9.2
55.5 50.5 46.5 43.5 40.3 34.9 24.3 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4
/
I1
SOL 0
111
10
I
20
I
30
I
40
PORE RADIUS x
I
50
I
60
I
70
1
80
IOY m
Figure 12. Integral pore size distribution of batch 602 and batch 18 membranes.
number of such pores is as small as indicated above. In order to reproduce the pore size distribution of unshrunk batch 602 and batch 18 membranes, we proceed exactly in the opposite direction. First, the integral distribution curves are drawn for batch 602-67.5 and batch 18-77.5 membranes using pore size distribution parameters listed in Table IV,as illustrated in Figure 12. Then, using the columns corresponding to unshrunk, 67 and 77 O C in Table VI and approximating that the effect of shrinkage at 67 and 77 "C is identical with that at 67.5 and 77.5 "C, respectively, the integral distribution curve for batch 602-unshrunk and batch 18-unshrunk membranes are reproduced. For example, according to Table VI, the pore radius of 44.9 x m on the membrane shrunk at 67 "C corresponds to the pore radius of 66.9 X 10-lom on the unshrunk membrane; therefore, starting from point A on
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 1, 1984 133
the integral distribution curve of batch 602-67.5, we obtain point A’, which point should be on the integral distribution curve on batch 602-unshrunk membrane. Similarly, points B’ and C’ are obtained. Using these three points A’, B’, and C’, we are able to calculate three parameters which describe the second normal distribution of the pore size, i.e., I&, u2,and b.As for the parameters Ral and u1 which represent the first normal distribution the values corresponding to batch 316-unshrunk membrane are adopted. ul, Knowing all the pore she distribution parameters Rb,2, u2, and h2, we are able to draw the entire integral distribution curve for batch 602-unshrunk membrane as illustrated in Figure 12. Likewise, the integral distribution curve for batch l&unshrunk membrane is generated and is also shown in Figure 12. All the parameters describing the first and second normal distributions of batch 602 and batch 18 unshrunk membranes are included in Table IV. A method similar to that used for generating Table VI can be applied for determining the radii of pores which are produced on the membranes shrunk at temperatures other than 67 and 77 “C and correspond to a given pore radius of unshrunk membranes, as shown by the results given in Figure 6. Nomenclature A = pure water permeability constant, kg-mol/m2.8.kPa A, = the surface area of polymer powder in the chromatography column, m2 B = constant characterizing the van der Waals attraction force, m3 b = frictional function given in Figure 4 cA2= molar concentration of the solute at the pore inlet, mol/m3 CA&) = molar concentration of the solute at the pore outlet (a function of radial distance r), moi/m3 CA,b = bulk solute concentration, mol/m3 D = constant characterizing the steric repulsion at the interface, m = molecular radius of water (0.87 X 10-lom) Dwater. d = distance between polymer material surface and the center of solute molecule, m d = slit distance, equivalent to pore diameter, m (Dm/Kb)NaCI= transport parameter of reference sodium chloride in water (treated as single quantity), m/s f = fraction solute separation based on the feed concentration f’ = fraction solute separation based on the solute concentration in the boundary phase AG = change in Gibbs free energy, J/mol g = density of solution, kg/m3 AH = change in enthalpy, J/mol hi = ratio defined by eq 2 k = mass transfer coefficient for solute on the high pressure side of the membrane, m/s ni = number of pores belonging to the ith normal distribution n, = total number of pores AI’ = operating pressure, kPa gauge (PR) = product rate though given area of membrane surface, kg/h (PWP) = pure water permeation rate through given area of membrane surface, kg/h & = pore radius, m Rb = average pore radius, m &,,i = average pore radius of the ith distribution, m R = gas constant r = radial distance in the cylindrical coordinate of the pore, m AS = change in entropy, J/mol-K Sum = quantity defined by eq 16 T = absolute temperature, K
= velocity of water in the pore as a function of r, m/s [VR’]A = chromatography retention volume of solute A, m3 u, = permeation velocity of product solution, m/s Yi(Rb) = normal pore size distribution function, l / m u&)
Greek Letters r = surface excess of the solute, mol/m2 b = length of cylindrical pore, m 0 = solution viscosity, Pa.s Xf = quantity defined by eq 9 r q = standard deviation of ith normal pore size distribution,
m 4 = potential function of interaction force exerted on the solute from the pore wall, J/mol xm = proportionality constant between solute velocity relative to water and the friction force working on solute, J.s/m2.mol Registry No. Cellulose acetate, 9004-35-7. Literature Cited
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Received for review April 19,1983 Reoised manuscript received September 12, 1983 Accepted September 26, 1983
Issued as N.R.C. No. 22903.
