DESALINATION BY REVERSE OSMOSIS T . K . S H E R W O O D , P . L . T . B R I A N , A N D R . E. F I S H E R Massachusetts Institute of Technology, Cambridge, Mass.
Experimental studies of salt and water transport through cellulose acetate membranes in both forward and reverse osmosis support the simple but useful model of transport by the parallel processes of diffusion and "pore flow." Data on forward and reverse osmosis using a rotating cylindrical membrane operated over a range of rotational speeds check the theory developed earlier to describe the effect of concentration polarization in the design and operation of equipment for desalination by reverse osmosis.
s
water and inland brackish waters are now being commercially desalinated by electrodialysis, distillation, and freezing. Reverse osmosis, though not yet commercial, is the subject of much current research and development since it shows promise of being economically competitive. Polymeric membranes have been developed which give moderately high water fluxes with high salt rejection (78, 79, 2 5 ) . Cellulose acetate shows particular promise, and techniques have been developed to form and support this material as an exceedingly thin consolidated membrane which can withstand the high pressure differentials required ( 2 6 ) , Present-day membranes suffer from two disadvantages : T h e practical water fluxes are not so great as desired, and the service life is short. Membranes giving both high water flux and high salt rejection might be formulated on a rational basis if the mechanisms of salt and water transport were understood, but this understanding has yet to be developed. I n this situation it is useful to have a gross empirical representation of membrane behavior, based on relatively crude models of the membrane transport processes. The water flux through cellulose acetate and similar membranes has been found (2, 76, 20, 23) to be very nearly proportional to AP - AT, where AP is the applied pressure difference across the membrane and A x is the difference between the osmotic pressures on the two sides. If AP is greater than AT, the flux is from saline to fresh water. Except for the few data of Hodges (76) the upstream salt concentration (on the saline or high pressure side) was held constant while AP was varied. 'The same relation between water flux and (AP - AT) is indicated, whether AP or salt concentration is varied, but in practice the membrane properties are changed by high applied pressure and by contact with salt solutions. The results described were obtained using different salt concentrations a t constant AP. Data obtained for both forward and reverse osmosis are analyzed here in terms of simple models (73, 76) of the membrane transport process. These models are then used to describe the membrane properties in an analysis of the results of a study of boundary-layer concentration polarization. EA
Salt and Water Transport through Cellulose Acetate Desalination Membranes
Theoretical. A simple theory of membrane transport is based on the assumption that water and solute cross the membrane by the parallel processes of diffusion and pore flow. Activated diffusion of solute and water is postulated to involve dissolution into the water-swollen polymer matrix at the upstream membrane surface, diffusion across the membrane 2
l&EC FUNDAMENTALS
as a result of the gradient of chemical potential or concentration, and desorption from the downstream membrane surface. I t is further postulated that upstream solution passes through open channels or pores, with negligible change in solute concentration due to diffusion in the liquid, at a rate which is proportional to AP. I t is the relatively greater diffusional transport of water than of salt that gives rise to the selectivity of the membrane in reverse osmosis. This means that pore flow must be a small fraction of the total flux if the selectivities required for practical desalination are to be achieved. O n the basis of this simple picture of the process, the salt flux, ATs,and the water flux, ATu, are given by the following equations:
- c p ) f ksM,APcf ki(AP - AT) f kzM,APc,
NS = k3M,(c,
(1)
N,
(2)
=
where kl, kz, and k3 are the membrane coefficients for diffusion of water, pore flow, and diffusion of salt, respectively; ci and ,c are the salt and water concentrations a t the upstream membrane surface; c p is the downstream or product salt concentration; and M , and M , are the molecular wcights of salt and water. Coefficient k3 is proportional to the product of the diffusivity of salt in the membrane and the distribution coefficient of salt between solution at the surface and the polymer matrix. There is no allowance for any effect of pressure on salt content of the membrane. The foregoing requires no theory or postulate as to the fundamental mechanism of transport of either water or salt. I t is assumed only that both cross the membrane by two parallel mechanisms, one in which the salt flux is proportional to ci - c p and the water flux to AP - AT, and one by which upstream solution is transferred through the membrane without change in salt concentration, a t a rate proportional to AP. Though the theoretical basis of the three transport coefficients has not been developed, Equations 1 and 2 provide a useful basis for the correlation of data on forward and reverse osmosis. Figure 1 shows the data on reverse osmosis reported by Manjikian (22) using aqueous sodium chloride with a cellulose acetate membrane. The pressure difference, AP, was varied at constant feed composition (0.5 weight % NaC1) with c I - c p remaining essentially constant. The indicated straight line and intercept agree with the form of Equation 1; k, is 1.1 X 10-8 cm./(sec.)(p.s.i.) and k3 is 4.2 X cm./sec. Both are presumably inversely proportional to the thickness of the consolidated polymer skin, which was unknown but constant. REVERSE OSMOSIS. The product salt concentration can be related to the solution concentration at the upstream surface by
dependent. There is no pressure gradient and the pore flow is zero. Diffusion of salt and water may occur to some extent through the solution contained in pores. Coefficient ka describes salt diffusion in both pores and polymer matrix. T h e gradient of water concentration is so small, however, that water diffusion in the pores is negligible.
Experimental. REVERSE OSMOSIS.“Brackish water” membranes (78) were cast on a 1-inch 0.d. support cylinder with a 2-inch porous section, as shown in Figure 2. The stirring shaft of a 1-gallon autoclave was replaced by the I-inch rotating cylinder supporting the membrane. This was hollow, permitting the withdrawal of the product water passing through the membrane. Figures 3 and 4 show the autoclave and the arrangement of auxiliary equipment. The autoclave was charged with saline solution of known concentration and the noncontaminating piston pump operated to maintain pressure and supply make-up feed as product water discharged through the membrane and the hollow rotating cylinder. Effluent salt concentrations were measured by means of a conductivity cell and product flow rates by weighing timed samples. External boundary-layer polarization was minimized by rotating the cylindrical membrane at 900 r.p.m. Pressure and temperature were maintained a t 600 to 605 p.s.i.g. and 25’ i. 0.3’ C . FORWARD OSMOSIS.Experiments at zero AP were carried out using the dual recirculating apparatus shown in Figure 5, with semicommercial cellulose acetate membranes obtained from the General Atomic Division, General Dynamics Corp. High circulation rates were employed on both sides of the membrane to reduce boundary-layer polarization and to ensure that the solutions in contact with the membranes would be well mixed. The change of salt concentration in recirculating system A was negligible during each test since the liquid holdup volume was large. Recirculating system B had a small liquid holdup, the salt flux being determined from the time rate of change of the measured salt concentration of the small volume of circulating solution. The water flux was obtained from the observed time rate of change of the liquid level in the open buret connected to recirculating system B. Possible leakage was checked by circulating fresh water on both sides of the membrane, after which both systems were charged with water or saline solutions. System A always contained the solution having the greater salt concentration; in most tests system B was initially charged with fresh water. The salt concentration in system B, which was measured by the use of a conductivity cell, increased during the test as salt passed through the membrane from A to B.
AP, psi
Figure 1. Variation of salt flux with applied pressure difference Feed solution 0.5 weight Data of Manjikian (22)
% NaCI;
Ac essentially constant.
first noting that, from a steady-state salt balance,
cp =
NsPT
MJNS f NWJ)
(3)
where p T is the density of the product solution. The osmotic pressure of aqueous sodium chloride is nearly proportional to concentration; as an approximation let T = kbc. Then the simultaneous solution of E:quations 1, 2, and 3 gives
where
k2PT
:=
kiki
- koM,
For the conditions of the present study klk5 is larger than kzMS; so y, K , and h are positive. Equation 4 is useful in the interpretation of data on reverse osmosis, particularly in studies of concentration polarization a t the membrane surface. E,quations 1 and 2 apply to forward FORWARD OSMOSIS. osmosis if the activated diffusion of salt and water is in-
Discussion of Results. REVERSEOSMOSIS. Figures 6, 7, 8, and 9 present the data obtained with a single rotating cylindrical membrane, using both sodium and calcium chlorides. Values of ct were calculated from the upstream bulk Porous Bearing
-/ Figure 2.
4 { -
Detail of membrane cast on 1 -inch 0.d. rotating support cylinder VOL. 6
NO. 1
FEBRUARY 1 9 6 7
3
Water Outlet
Figure 3. Autoclave assembly employed for reverse osmosis with rotating cylindrical membrane
solution concentrations (29) ; at the 900 r.p.m. employed c1 - cp differed from cB - cp by less than 4%. Figure 6 shows water flux to be proportional to AP AT, These tests were made at constant pressure and varying upstream salt concentration: Points at the right side of the graph are for low and those a t the left for large values of ct. As the salt concentration was increased A n approached AP, and the AT increased greatly, estimated error in the difference AP as shown by the flags on the plotted points. Consequently, a
-
-
possible intercept indicating a pore-flow contribution to N , could not be observed. Values of kl for the four tests shown on Figure 6 are 0.398, 0.281, 0.259, and 0.258 X mg./(sec.) (sq. cm.) (p.s.i.). These differences suggest the importance of changes in membrane properties due to different treatments. Run 12/22 a t AP = 605 using pure water was almost exactly reproduced in run 12/23 after 20 hours during which the membrane was pressurized in pure water a t 605 p.s.i.g. Irreversible changes in the membrane are indicated in cases where the membrane was held in contact with salt solutions. The 8% drop in pure water flux between runs 12/21 and 12/22 (Figure 6, points a t AP - AT = 605) followed a period of 18 hours during which the membrane was held under pressure in contact with salt solutions of various concentrations as great as 1.5M. Equation 1 suggests that the salt flux, N,, should be proportional to ci (at constant AP) if transport is pressure-dependent (“pore flow”) and proportional to cI - cp if dependent on concentration gradient. As seen from Figures 7 and 8, neither of these limiting cases is supported by the data. The large difference in the results for sodium and calcium chlorides would not be expected if the “pore-flow’’ contribution were large. The more general possibility that transport occurs by both “pore flow” and “diffusion” is tested by replotting the data as shown in Figure 9. If Equation 1 applies, straight lines should result with intercepts k2AP and slopes k3. Values of kz and k3 obtained in this way from Figure 9 are given in Table I. The relative importance of the two mechanisms is readily apparent from Figure 9: The abscissa is the salt rejection and the ordinate is the sum of the two salt fluxes, the intercept representing “pore flow” and the remainder “diffusion.” Diffusive transport of calcium chloride is evidently very small. The parallel-flux transport of both salt and water, as expressed by Equations 1 and 2, is doubtless an oversimplification of the membrane process. For practical purposes, however, it may be simplified still further by neglecting either “pore flow” or salt transport by “diffusion.” Three cases are considered: Case A, transport of both salt and water by diffusion only (k2 = 0); Case B, transport of water but not salt by diffusion and transport of both by pore flow (k3 = 0); and Case C, transport of both salt and water by both mechanisms. Cases A and B are simplifications of Case C.
rConduct ivity
Meter
Pressure Gauge
P r o d u c t Collection
Variable
Speed
Porous Section
Constant Temperature B a t h
Figure 4. 4
I&EC FUNDAMENTALS
Rotating
Support
Diagram of equipment used for studies of reverse osmosis
Drive
1
Conductivity Meter
Top View of Cell
Buret
Temperature
Volume Lucite Body Rubber Gasket Wire Screen Membrane Exploded View of Section A - A
Figure 5.
Diagram of equipment used for studies of forward osmosis AP = 0
0
100
500
300
400
500
600
700
A p - A n , psi
Figure 6. Water flux in reverse osmosis with cylindrical membrane rotating at 900 r.p.m. Salt Concentration varied a t constant AP of 600 ta 605 p.s.i. Runs 1 2 / 1 0 and 12/21 with NciCl; no solute used in run 12/22. CaClz used in run 1 2 / 2 3 . Horizontal tlags indicate estimated precision of AT. measurement and calculation of AP
-
0
1.o
2 .o
c i , mg Figure 10 shows the data of run 12/21 plotted as cp us. cI for comparison with the 1:hree cases described. The solid lines represent values of c p calculated by the use of Equation 4 with taken as 700 (p.s.i.)(cc.)/(mg. mole). Values of kl, kz, and ka were obtained from Figure 6 and Figure 8, Figure 7, or Table I for Cases A, B, and C, respectively. T h e three cases fit the data about equally well for the moderately low salt concentrations used. The simpler Cases A and B are used in evaluating membrane properties in the study of concentration polarization with a rotating membrane described below. FORWARD OSMOSIS. Water and salt fluxes measured in the forward-osmosis study are shown in Figures 11 and 12. These data were obtained with the dual-recirculating system (Figure 5) using semicommercial membranes supplied by General Atomic. As in reverse osmosis, fluxes from the high salinity side t o the low salinity side are considered positive; the water fluxes in forward osmosis are inherently negative.
3.0
moles/cm3
Figure 7. Molal salt flux in reverse osmosis with cylindrical membrane rotating at 900 r.p.m. and AP = 600 to 650 p.s.i. Table 1.
Reverse Osmosis Membrane Constants
(From slopes and intercepts of lines on Figure 9 ) kp (Pore Flow), kv (Dgusion), Run No. Solute Cm./ (Sec.) (P.S.Z.) Cm./Sec. 12/10 NaCl 0.0 8 . 0 x 10-6 12/21 NaCl 1 . 3 X lo-* 8 . 4 X IO-' 12/23 CaClg 3 . 1 X 10-8 0.5 x I n each test the salt concentration on the high salinity side of the membrane (recirculating system A) was held constant; the several data points correspond to different salt concentrations on the low salinity side (recirculating system B). ExperiVOL. 6
NO.
1
FEBRUARY 1 9 6 7
5
08
06
04
02
0
0
0.2
0.4
0.6
1.0
0.8
c . , mg
1.2
1.4
1.6
rno~eicm~
Figure 10. Variation of downstream product salt concentration with upstream interfacial salt concentration in reverse osrriosis Three theoretical cases.
Data points from run 12/21
Figure 8. Molal salt flux in reverse osmosis with cylindrical membrane rotating a t 900 r.p.m. and AP = 600 to 605 p.s.i. 70
6.0
50 In
9 4.0 U
0
-..
:
0
2.0 200
100 (AP
1.c
Figure 1 1.
- A" 1,
300
400
psi
Water flux in forward osmosis
Salt concentration varied a t constasit AP = 0. CaClz used in run 13, all other runs with NaCI. Runs 1 to 5 with membrane GA-1, runs 8 A - 1 0 with membrane GA-2, and run 13 with membrane GA-3
0
ci-cp Ci
Figure 9. Test of Equation 1 for salt flux in reverse osmosis with cylindrical membrane rotating a t 900 r.p.m. and AP = 600 to 605 p.s.i.
mental errors for the data in which the water flux is greater than about 0.05 mg./(sq. cm.) (sec.) are believed to have been so small as to affect neither ordinate nor abscissa by more than 3 to 4%. As the water flux decreases, the per cent error in the ordinate increases. The fact that the points fall along single straight lines for each membrane on the two graphs indicates that the water flux is independent of the level of salt concentration but proportional to osmotic pressure (AP = O), and that 6
I&EC FUNDAMENTALS
the salt flux is independent of salt concentration but proportional to h(=c i - c p ) . Table I1 lists the values of the membrane constants k l and ks for reverse osmosis, as reported by the membrane manufacturer, and the corresponding values of the same constants obtained from Figures 11 and 12 in the present study offonvard osmosis. The water-flux constant, kl, is seen to be 1 to 4 times as great in fonvard as in reverse osmosis, and the salt-flux constant, k3, is 18 to 23 times as great (for NaCl). Banks and Sharples ( Z ) , who also noted higher membrane constants in fonvard osmosis, attributed the effect to compaction of the membrane under the high applied pressure used in reverse osmosis. The data on reverse osmosis using calcium chloride, discussed earlier, showed that the transport of this salt was
osmotic pressure and water flux independently of the salt concentrations. T h e salt-flux data points scatter appreciably but generally agree with the other data on the same membrane, though water flux N , varied from -0.08 to +0.04. These results support the assumption that salt and water flux across the membrane proceed independently. Water transport through the polymer matrix may be expressed in terms of a diffusion coefficient, D,, as follows (6, 77,
12.0
11.0
IQO
9.0
76,20, 27):
0.0
where cWpis the water concentration in the polymer of thickness t. Thus the over-all coefficient, kl, is proportional to some average value of D,, which may be presumed to vary with cWp. Lonsdale, Merten, and Riley (20) report that a 19% decrease in water content caused a 60 to 80% decrease in D,, based on data for forward osmosis using a homogeneous cellulose acetate membrane. This marked variation of kl with water concentration is also indicated by data on forward osmosis obtained in a study of concentration polarization using a rotating membrane, where kl was found to decrease with increase in upstream salt concentration. Figure 13 shows these results (corrected for a small concentration polarization) plotted as z/t us. activity of water in the membrane. The ordinate z/tis calculated from Equation 5 assuming Henry’s law to apply to water absorption by the polymer, the equilibrium concentration in the polymer being 0.138 times that in the solution a t the membrane surface (76). The mean activity, ii, is the arithmetic mean of the activities of the water a t the two membrane surfaces. These results confirm the reports that D, and hence kl may be expected to vary widely with membrane water content.
7.C
3
a-
*^
6.0
5.c
\
P
-
4.c
2-
3.c
2.c
IS
a AC, rng rnolr/crn’
Figure 12.
Salt flux in forward osmosis
Salt concentration varied at a constant AP = 0. CaCIz was used in run 13, all other runs were with NaCI. Runs 1 1 and 12 had dextrose added to solution in recirculating system B. Runs 1 to 5 with membrane GA-1, runs 8 A - 1 2 with membrane GA-2, and run 1 3 with membrane GA-3
almost entirely by “pore flow.” The value of kl of 1.6 X 10-5 may be presumed, therefore, to represent diRusion through solution in the pores. Assuming the mass flux by diffusion through pores to be proportional to the product of molecular weight and molecular diffusivity, it is estimated that for the first two membranes no more than 3% of the total flux of sodium chloride was by diffusion through solution in the pores. I t would appear, therefore, that the compaction due to applied pressure must have greartly reduced the ability of the salt to diffuse through the polymer matrix. Two runs (11 and 12) were made using membrane GA-2 with dextrose added to the low salinity system so as to vary the
I
Run 6 0
0
R v l 76
0.04
0.03
f
aw
Table II. Values of kl anid k3 for Cellulose Acetate Membranes Membrane NO.
kl,
Solute
M,./(Sec.)
(Sq. Cm.)(P.S.I.)
ka, Cm./Sec.
Values Reported by Manufacturer for Reverse Osmosis 0.31 X 10-3 1 . 0 X 10-6 GA-1 NaCl GA-2 NaC1 0.68 X 10-3 1 . 8 X 10-6 3.7 x 10-3 3.0 X GA-3 NaCl Values Obtained in Forward Osmosis (AP = 0) GA-1 NaCl 1 . 2 x 10-3 2 . 3 x 10-4 GA-2 NaCl 1 . 9 x 10-3 3.3 x 10-4 GA-3 CaClz 3.5 x 10-3 1.6 X
Qo1
~~
a, Figure 13. Variation of water diffusivity in cellulose acetate with water activity Data from (76)
VOL. 6
NO.
1
FEBRUARY
1967
7
The water activity changes so slowly with changes in salt concentration in water, however, that kl should vary but little in the range of salt concentrations normally encountered in processes for desalination by reverse osmosis. Conclusions. Experimental measurements of salt and water transport through cellulose acetate membranes for both forward and reverse osmosis are consistent with a simple model of the over-all process. Both salt and water are transported by a process equivalent to leakage through pores, with upstream solution passing through the membrane at a rate proportional to the pressure difference, AP. A second process, occurring in parallel with the first, transports water a t a rate proportional to AP AT, and salt a t a rate proportional to the difference ci - c9 between the salt concentrations a t the membrane surfaces. For low salt concentrations the model can be simplified by allowing for but one salt transport mechanism. The model described is useful for practical purposes, though it only hints a t the basic mechanisms on the molecular level.
-
Experimental Study of Concentration Polarization in Forward and Reverse Osmosis
Desalination by reverse osmosis may be expected to be economical if all of the irreversibilities inherent in the process can be minimized. One such irreversibility is due to the chemical potential or concentration gradient necessary to cause transport between saline solution and membrane surface. Water flw to and through the membrane results in a buildup of salt in the solution adjacent to the membrane, with a consequent increase in the osmotic pressure difference which must be exceeded by the applied pressure to cause water to pass through the membrane. The irreversibility represented by this “concentration polarization” leads to increased operating pressure and the requirement of greater work input to the saline water feed pump. The result is that the operating costs are increased because the electrical work input per gallon of potable product water is increased. Previous experimental studies (7, 75, 24) have shown the importance of boundary layer resistances in membrane processes and demonstrated that these can be reduced by increasing the rate of flow of liquid over the surface. They have not, however, developed the quantitative connection between membrane performance and liquid flow patterns in brine feed passages of typical geometries. Several theoretical studies of concentration polarization ifi reverse osmosis have treated turbulent flow in round tubes (28, 29), laminar flow between parallel flat membranes (8, 28, 29), laminar flow in round tubes (72), and variable membrane water flux along the membrane in the direction of flow (5, 74). Brian (4) has shown how these theoretical analyses might be employed to calculate the economically optimum equipment design and operating conditions for reverse osmosis. I t has been hoped that experimental confirmation of these theoretical studies might be forthcoming. The present investigation provides this experimental confirmation for one particular flow geometry and for one membrane type. The support given the theory is implicitly broad. EXPERIMENTAL PROGRAM.Both reverse and forward osmosis were studied using cylindrical Loeb-type (78, 79) cellulose acetate membranes rotated in saline solutions. For reverse osmosis the solution was contained in a high pressure autoclave and the product water removed through the hollow rotating shaft. Forward osmosis was studied using a similar membrane rotated in solution contained in an open vessel. The rotating-cylinder geometry was chosen for several reasons: Good correlations of data on mass, heat, and momen8
I&EC FUNDAMENTALS
tum transport between liquids and rotating cylinders are available ; the symmetrical system provides a surface which is “uniformly accessible,” so that diffusion near the surface can be treated as radially one-dimensional; and the batch operation of the saline water supply is simple and relatively trouble-free, THEORY.The general theory of salt buildup a t the membrane surface in reverse osmosis has been presented (29). If allowance is made for salt transport through the membrane, the salt flux is given by
where N , and N , are the mass salt and water fluxes into the membrane and c is the salt concentration in the solution a t distance y from the membrane surface. D, is the constant molecular diffusion coefficient of salt in water and eD is the corresponding eddy diffusivity which varies with y and with flow conditions. Using Equation 3 and the boundary conditions c = ci a t y = 0, c = CB a t y = m, Equation 6 is integrated for constant ( N , Nw) to give
+
where Y is the kinematic viscosity of the solution and Ns0 is the Schmidt number, v/D,. The solution density, p T , varies little in practice and has been taken as constant. Solute transfer across a boundary layer with small solute fluxes and no net solvent flux is given by an integration of Equation 6 with the first term on the right-hand side omitted:
where k, is the usual mass transfer coefficient, having units of gram moles/(sec.) (sq. cm.) (g. mole/cc.), or cm./sec. Substitution of this result in Equation 7 gives
(9) Correlations of experimentally determined values of k , are available for many geometries and conditions of fluid flow over the surface to or from which mass transfer takes place. Many of these take the form of graphs of j D us. NRe,j D being defined by
where uB is the bulk linear velocity of the fluid over the surface. Line B-B of Figure 14 represents the data of Eisenberg, Tobias, and Wilke (70) for mass transfer from a rotating cylinder to a pool of liquid. Their results have been confirmed by the later investigations of Sherwood and Ryan (30) and others (3, 7) and may be used to obtain k, for the conditions of the present study. Alternatively, since jD agrees closely with the corresponding group j H for heat transfer, and also with f / 2 , correlations of j H or f / 2 in terms of NRe may be employed for the purpose. The line F-F for f / 2 was taken to represent j D and so obtain k, in analyzing the data to be reported. In the case of a cylinder rotating in a pool of liquid uB represents the peripheral speed of the cylindrical surface. Taylor vortices are known to develop in the annular space between a rotating cylinder and the cylindrical confining wall
KEY
Curve A-A 8-0 C-C D-D €-E
n
F-F
P X - 0 .-
Investigators
;r j, jH ,j f/2
Dropkin and Carmi
(3)
Pr Sc
=
0.74
= 835- 11,490 Pr = 0.74 Pr = 190 - 670 Pf = 3 . 5 6 - 5 . 5 8
Eisenberg, Tobias and Wilke (Q) Kays and Bjorklund (17) Sebon and Johnson (21) Seban and Johnson(=) Theadorsen and R a g i e r ( 3 )
In-
E! X
.-
f
In’
P X
-IN
*%R, Re= y Figure 14.
Heat, mass, and momentum transfer correlations for a rotating cylinder
of the vessel if the gap between the two is small. These could cause k, to vary with position on the vertical surface and complicate the analysis. Dyc tests with water in a glass scale model of the autoclave demonirtrated that above rotational rates of about 40 r.p.m. the voriex motion could no longer be distinguished. Since the gap between the concentric cylinders was larger in the forward-osimosis equipment than in the reverseosmosis equipment, the transition to fully developed turbulence would occur a t a lower rotational rate. Thus most of the data were taken in the fully developed turbulent regime where the above correlations apply. Both N , and N , are positive in reverse osmosis, and the salt concentration, ct, at the upstream surface is greater than the bulk concentration, cB. No concentration polarization occurs a t the downstream surface of the membrane. In forward osmosis water passes from the membrane on the rotating cylinder into the surrounding saline solution. Both c p and N8 are taken to be zero and N , is inherently negative, so concentration polarization a t the outer surface causes salt concentration in this region to be lower than in the bulk liquid. Bulk flow of water out from the membrane carries salt which diffuses back through the boundary layer. Experimental. The autoclave and rotating cylinder employed for reverse osmosis are described above. The method of operation, however, was different. Rotational speed of the cylindrical membrane w a varied over a wide range to determine the manner in which salt and water fluxes and product composition (N,, N,, c p ) varied with membrane surface velocity. Different membranes prepared in a similar way were used with NaC1 and CaClz solutions. The apparatus used for forward osmosis is shown diagrammatically in Figure 15. Cylindrical “sea water” membranes (79) 2 inches long were attached to the 2-inch 0.d. hollow metal cylinder rotated in a 12 x 12 inch glass jar containing saline
solution. Rotational speed was varied from 10 to 1400 r.p.m. Water flux was measured by timing changes of distilled water level in the pipet to which fresh water was added from time to time. Water flux was corrected for small leakage, using measurements of leakage into pure water as a function of r.p.m. Occasional checks showed the salt concentration inside the rotating cylinder to be negligibly small under operating conditions.
Results and Discussion. REVERSE OSMOSIS. Figures 16 and 17 show representative data on the variation of total product flux and product salt concentration as a function of r.p.m. for reverse osmosis tests using NaCl and CaC12. Each quantity was measured with a precision of approximately +1%. The graphs are for two different cellulose acetate membranes of the Loeb “brackish water” type (78). Total flux is seen to vary somewhat less than 15% in the range of speeds studied. This variation is small because the experimental values of Air were small compared to A P - Air, so though ct varied considerably, the water flux did not. Variation in flux with r.p.m. would be expected to be much greater with membranes of higher permeabilities. Small changes in the membrane properties with use are indicated by the three points a t 250 r.p.m. which represent first, last, and intermediate tests in each series. The product salt concentration is seen to vary twofold with changes in r.p.m. This is evidently a more sensitive measure of the magnitude of concentration polarization than is the variation in total flux. The solid lines shown on Figures 16 and 17 represent values of ( N , 4-Arm)and c p calculated by combining Equation 9 with the equations for N , and N , developed on the basis of the membrane transport models described above, with membrane constants evaluated from the data obtained a t 250 r.p.m. The procedure is illustrated by the sample calculation given below for NaCl and 1000 r.p.m. VOL. 6
NO. 1
FEBRUARY 1 9 6 7
9
r
Pipet
SUDDOrt Frame
Variable Speed Pulley
D,C,MotorFresh Soluti
Support Cylinder
l ~ ' x 1 2 ' 'Cylindrical Jar
+Speed
Control
Figure 15. Diagram of equipment used for studies of forward osmosis (AP = rotating cylindrical membrane.
0)using
-..I . .i
i
0.50 I
0.40
I
I
200
0
400
600
RPM
800 R. P.M.
1000
1200
1400
1600
2s I
t
9
i
i
I.€
.*
D
0
1.2
E
P ;o.e
u
0.4
I
I
1
200
400
600
I
I
I
I
I
800
1000
1200
1400
1600
R. P.M.
RPM
Figure 16. Water flux and product salt concentrations in reverse osmosis as a function of rotational speed
Figure 17. Water flux and product salt concentrations in reverse osmosis as a function of rotational rate
1 -inch o d. rotating cylindrical membrane.
1-inch 0.d. rotating cylindrical membrane. = 605 p.s.i.
CB
= 0.1 65M NaCI, AP = 605
p.r.i.
SAMPLE CALCULATION. The physical properties of the NaCl solution are: cB = 0.165 mg. mole/cc.; M , = 58.5; = 0.00905 poise; D, = 1.48 X 10-5 sq. cm./sec.; pT = 1006 mg./cc.; v = O.GO90 sq. cm./sec.; .rr = 680 c ; Ns, = 0.0090/ = 608; membrane diameter, 2.54 cm.; area = 1.48 X 40.5 sq. cm.
NR= ~
2.54 X (2.54 X 3.14) (r.p,m.) = 37.4 60 X 0.0090
Substituting in Equation 9 with j
- cp 0.165 - cP Cf
10
= exp
[,
D
x
(r.p.m.)
+
I&EC FUNDAMENTALS
= 0.0545M CaC12, AP
At 250 r.p.m. the measured values of N,, N,, and cp were 0.00252 and 0.605 mg./(sec.) (sq. cm.) and 0.071 mg. mole/cc. The Reynolds number is 9350 and f is found from line F-F of Figure 14 to be 0.0095. Substitution in Equation 10 gives (cf - c P ) / ( c B c p ) = 1.31 and so cf is 0.195 mg. mole/cc. Referring to Equations 1 and 2, with k3 = 0 (Case B, no salt diffusion in the polymer),
-
k -
-
taken equal to f/2,
2 X 6082/3(N, N,)60 1006 (2.54 X 3.14) (r.p.m.)f
CB
kl =
0.00252 = 3.65 X lo-' cm./(sec.)(p.s.i.) 58.5 X 0.195 X 605 0.605
-
3.65 X I O ' X 18 X 605 X 55.5 605 680 (0.195 - 0.071) 0.00074 mg./(sec.) (sq. cm.) (p.s.i.)
-
-
Substitution of these values (with kh = 680) in the defining equations gives y = 0.00220, K = 0, and X = 0.00073. Equation 4 reduces to
2cp
=
-(1.33
-
+ [(1.33 - cJ* + 1.76 c,]"'
G()
(11)
Finally, the model equations for N , and N,, with k3 = 0, give
N,
+ Nu = 0.00074 [605 - 680(c
