Kroger, F. A., “The Chemistry of Imperfect Crystals,” pp 337405, North-Holland Publishing Co., Amsterdam, 1964. Lamer, J. E., University of hlinnesota, Mineral Resources Research Center, Minneapolis, iMinn,, 1968. Mular, A. L., Trans. A I M E 232, 204 (1965). Plaksin, I. N., Shafeev, R. S., Trans. ZMM 72, 715 (1963). Rao, B. V. P., Lovell, H. L., Simkovich, G., Trans. AZME 241, 328 (1968). Schwab, G. AI., Schmid, H., J . Appl. Phys. Suppl. 33, 426 (1962). Simkovich, G., Trans. AZME 227, 306 (1963). Simkovich, G., Wagner, J. B., Jr., J . Electrochem. SOC.110, 513 (1963). Sun, S. C., Morgan, J. D., Jr., Wesner, R. F., Trans. AZME 187, 369 (1950). Wagner, C.,’ “Atom Movements,” J. H. Holloman, Coordinator, pp 153-173, ASM, Cleveland, Ohio, 1951. Wolkenstein, T., “Advances in Catalysis,” D. D. Eley, el al., Ed., Vol. 12, pp 189-264, Academic Press, New York, 87. Y., 1960.
G. SIMKOVICH Metallurgy Section Department of Mineral Sciences The Pennsylvania State University Cniversity Park, Pa. 16806
F. F. APLAN* R. A. FEKSTERER Department of Mineral Preparation The Pennsylvania State lniversity Cniversity Park, Pa. 16802 RECEIVED for review February 16, 1971 ACCEPTEDNovember 12, 1971
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Observations on Buoyant Convection in Reverse Osmosis Observations are reported on the character of the buoyant convection patterns that develop in the salt diffusion layer adjacent to the membrane for flows encountered in reverse-osmosis water desalination. An unstirred batch cell with the membrane at the top of the cell was studied by the shadowgraph technique, and a fully developed laminar channel flow with the membrane on the top of the channel was investigated with electrical conductivity microprobes. A convection-jet model is suggested for the instability, whose characteristic length and time scales are lo-’ cm and 10’ sec, respectively. On physical grounds, the instability can occur in all reverse-osmosis desalination equipment. It is of practical importance and of potential utility in improving system performance whenever concentrationpolarization is significant.
c o n c e n t r a t i o n polarization, the buildup of a layer containing an excessive concentration of salt adjacent t o t h e membrane during the process of reverse osmosis, is a limiting factor t h a t must be considered in the design of reverse-osmosis desalination systems. Salt is left behind in the solution near the surface of a semipermeable membrane when water is forced through the membrane by the excess of the hydrostatic pressure difference over the osmotic pressure difference across the membrane. Efficient removal of this salt from t h e vicinity of t h e membrane improves the performance of the system. I n studies t h a t were not directed specifically toward the investigation of buoyant convection (Williams, et al., 1970), we noticed a distinct enhancement in the rate of water flow across the membrane when it was situated at the upper boundary of the highpressure brine. Effects of gravity on reverse-osmosis desalination have been observed earlier (e.g., Baldwin, e f al., 1965). We performed a few exploratory studies to develop some tentative ideas concerning t h e character of the buoyant convection t h a t was responsible for flow enhancement. The principal results and conclusions of these studies are described in the present note. Buoyant instabilities have been studied at considerable length in fluid mechanics. The character of the instabilities depends on the value of the Rayleigh number, which can be defined appropriately as Ra
gH4v,co
ff --
VD2
for reverse-osmosis systems. Here g is the acceleration of gravity, H is the height of the cell or channel, vw is t h e velocity 276 Ind.
Eng. Chem. Fundam., Vol. 1 1 , No.
2, 1972
of the flow through the membrane, co is the salt concentration d In p/dc is a salinity exof the bulk of the solution, a pansion coefficient, Y is the kinematic viscosity, and D is the diffusion coefficient. Much of the earlier work has been directed toward studies of peculiar cellular patterns t h a t occur at relatively low Rayleigh numbers. Studies relevant to our lo9) are fewer. Sone has experimental conditions (Ra considered cases in which there is a net upward convection velocity, an aspect t h a t appears t o be critically important in the reverse-osmosis application and to modify substantially the structure of the buoyant convection patterns. We have made shadowgraph observations of convection patterns in an unstirred batch cell and space-time resolved measurements with electrical conductivity microprobes in a two-dimensional channel-flow configuration of a desalination system. These results are summarized in the next two sections. A partially complete model t h a t correlates our batch-cell observations is outlined in t h e final section.
-
Batch-Cell Observations
Aqueous solutions of MgSOa were tested in a cylindrical cell 20 em high and 12 em in diameter, with a cellulose acetate membrane fixed to its upper surface. The membrane was rectangular with sides 5 and 40 mm in length. Shadowgraph images were obtained by passing the collimated beam of an He-Ne laser through the test-cell windows and through the solution, in a direction parallel to t h e membrane surface and normal t o the longer side of the rectangle. A representative shadowgraph is shown in Figure 1. From the dark membrane at the top of the figure, a number of bright filaments bounded
Table 1. Comparison of Theoretical and Experimental Growth Times, Filament Spacings, and Filament Lengths.
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Applied pressure, A p , psi Membrane flux, pv,, g/cm2 sec Rayleigh nnmber, R a Number of filaments, N Area fraction,
Figure 1. Shadowgraph of buoyant convection in batch cell: MgSOn; Ap = 100 psi; ca = 0.0045 M; 1-mm scale divisions
Figure
2. Dependence
of concentration profile in channel MgSOd; Ap = 150 psi,
flow on orientation of channel:
30.5
66
2.64 X 3.8 X
log
14
102
6.86 X
1 0 . 4 X 10-5
7 . 8 X log
10.6 X 100
18
20
I 88 X 113 X 125 X 10-4 Observed growth time, tl, sec 240 160 120 Observed filament spacing, XI. cm 0.38 + 0.05 0 . 2 ~ 0.05 Observed filament length, ze, em 0.1 0.1 0.1 Theoretical growth time, t,, sec 580 360 290 Theoretical filament spacing, A,, cm 0.44 0.35 0.31 Theoretical filament length, zt. cm 0.07 0.058 0.05 a co = 0.00833 M (osmotic pressure 5.8 psi), cp = 0.15e0, MgSO,, 20'C. as the time interval between pressurization and the first appearance of filaments. The filament spacing XI was obtained under the assumption that the filaments form a, regular retangular array over the surface of the membrane. Comparison of Figure 1 with the entries in Tahle I corresponding to a hydrosta.tic pressure difference across the membrane of Ap = 102 psi reveals that in less concentrated solutions the filaments are less numerous and more elongated.
Re = 137
Channel-Flow Observations
by darker z o n a are seen to extend vertically downward for a short distance. Although the average fluid motion in the vicinity of the membrane i s upward, the filaments can he interpreted as downward-moving jets (or plumes) of increased salinity. The pattern develops a few minutes after the cell is pressurized, with filaments first appearing a t the membrane and then descending slowly. Configurations like that of Figure 1 remain approximately invariant for periods of 5 to 10 min, as filaments slowly fade in brightness and eventually disappear. At irregu1a.r time intervals a new filament will be horn a t a new position on the membrane surface. At all times the spacing between active filaments possesses a degree of regularity. If the cell is depressurized suddenly, then existing filaments elongate relatively rapidly and eventually extend beyond the lower boundary of the field of observation. The first eight rows in Tahle I contain data t h a t were obtained in connection with the shadowgraph observations. The area fraction f is an estimate of the fraction of the membrane area covered by filaments; this was obtained from the observed number of filaments and from t h e observed filament diameter, typically 0.5 mm. The growth time ti is defined
Traverses in a direction y normal t o the surface of a cellulose acetate membrane were made with electrical conductivity microprobes in a steady-state channel-flow apparatus (Hendricks and Williams, 1971). Profiles of salt concentration c(y) were calculated from conductivities measured at a position x = 180 cm downstream from the leading edge of a memhrane 21 em in width. The Reynolds-number based on the bulk velocity U and on the channel half-spacing h was R e = 4hU/v = 137. For a 500-ppm solution of MgS04, a t Ap = 150 psi and h = 1.6 mm (corresponding t o a nondimensional distance from the leading edge of 4 = AAp(3xh/D2U)"' = 1.1, where A is the membrane constant), concentration profiles t h a t were obtained are shown in Figure 2. The main brine flow was in a horizontal direction, and the channel was rotated through various angles about a n axis parallel to this flow direction. The points marked On, which correspond t o t h e membrane a t the bottom of the channel (buoyantly stable) agree with the points marked go", for which the channel was vertical. No transient effects were apparent in either of these orientations. Therefore it appears t h a t in the vertical orientation the fluid flow tends to stabilize the process, For the points marked Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
277
--
1
I Min. Cmcentmtion Fluctuotlonr on Gmvrty Unrtabolizrd Camgmtim
1
Figure 3. Strip-chart recordings of error voltage of bridge for buoyant convection in channel flow, a t various distances from the membrane; NaN03; Ap = 31 7 psi, Re = 137
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POLARIZATION L A Y E R
MEMBRA NE
\
\
\ \
vw
\
\ \ 2
Figure 4. Diagram of convection-jet model
180" Ihe membrane mas a t the top of the channel. I n this case the concentration within the diffusion layer \T as observed t o fluctuate. The open circles represent the minimum concentrations observed during the fluctuations; they differ negligibly from the bulk concentration. The vertical arrow is an indication of the maximum amplitude of the fluctuations. The peak excess concentration during fluctuations was roughly half of the steady excess concentration t h a t was obtained in the stable configuration. A more complete characterization of the nature of the buoyant instability, observed in the channel with the membrane above the brine, was obtained for a S a S 0 3solution a t A p = 317 psi and h = 4 mm. I n this case, the fluctuating error voltage of the AC bridge t h a t was used to measure the conductivity was recorded on a strip-chart recorder. Representative records are shown in Figure 3. Since the error voltage is a nonlinear function of t h e probe resistance, the abscissa of Figure 3 is not exactly linear; the t n o end points, C/CO = 1.0 and C / C ~ = 1.8, were obtained by balancing the bridge. Two features are immediately evident from Figure 3: t h e concentration fluctuations are quasiperiodic in time, and the amplitude peaks a t a position between 100 and 2000 p from the membrane. At 300 p from the membrane, the fluctuation amplitude equals the difference between the wall concentration and the bulk concentration, The fluctuations tend to die out before reaching the center line of the channel and also a t the surface of the membrane. Thus, there are qualitative similarities between thp buoyant convection patterns in the batch cell and in the channel flow, 278
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
Comparison of the results in Figure 3 with coticent'ration profiles measured in the stable orientation under otherwise identical conditions reveals that a t 50 p from the membrane, the peak concentration for the unstable case is only 0.38 times the concentration for the stable case. Thus, the concentration a t the membrane is reduced by more than a factor of 2 by buoyant convection. For this test, the Rayleigh number based on channel half-height is approximately lo9, while that based on the stable characteristic diffusional relaxation length, 280 p , exceeds lo4. The degree of reduction of concentration polarization by the process of buoyant convection will depend on the operating conditions of the desalination system. Convection-Jet Model
I t might be remarked that the growth time tl observed for the batch cell buoyant convection can be compared with a n onset time to for haline convection calculated according to a n earlier theory (Foster, 1968). From the comparison shown in Table I, it is seen that the theoretical and experimental dependences on applied pressure are in agreement, although the observed time is less than predicted. One might furt'her hypothesize t'hat the filament spacing is controlled by the horizontal wavelength A, with the maximum rate of amplification (Foster, 1968). Table I indicates that in fact Ai is in good agreement with A,. The significance of these agreements is debatable on a number of grounds. First, the experimental results are not very accurate. Secondly, the connection between a transient amplification wavelength and a steadystate filament spacing is tenuous physically. Finally, the theory (Fost'er, 1968) does not include the mean vert,ical flow that is present in reverse-osmosis experiments. Kevertheless, clearly some qualitative similarities exist between the transient theory and experiment. It is of interest to attempt to develop a new theoretical model t'hat is capable of describing steady-state aspects of batch-cell observations like that shown in Figure 1. Such a model can be used to calculate the finite filament' length z , which has not been observed in previous experiments. It seems reasonable to postulate that the filaments constitute jets in which salty solution moves downward under a balance of viscous and buoyant forces. I n the presence of a mean upward velocity, the viscous forces on a jet can reduce its velocity in the laboratory frame to zero a t a finite distance from the membrane. Because of the large Schmidt number, a t this point the momentum field of the jet will have spread much farther than the concentration field, and a shadowgraph will continue to exhibit a very narrow filament. A jet of diameter d will then remain stationary and be dissipated by diffusion in a time d 2 / D = 230 sec, which is of the same order as our observed filament lifetime. The overall structure of the postulated convection-jet, sysbem is illustrated in Figure 4, where v, is the upward velocity of the salt solution, U P is the downward velocity in the jet, v, is the velocity through the membrane, CO is the initial concentration of the solution, and C P is the average salt concentration in the jet. Conservation of salt mass and of total mass, applied to the polarization layer, require
- cz,~,
(1)
- f ) - VPOf
( 21
r2u20j= rOvl(1 - f ) and 21,v
= Ul(1
where c, is the salt concentration in the product solution, UP' is the initial velocity of the jet, and f is the ratio of the total
jet initial cross section to the total membrane area. The dependence of the jet velocity vZ on the distance y from the membrane is governed by the fluid mechanics of the jet; as a rough approximation we use the formula for a laminar jet without buoyancy (3)
where S is the total membrane area and N is the tot'al number of filaments. The filament length is defined by setting vz = V I a t y = z in eq 3. Using eq 1 and 2 to eliminate 01 and v ~ O ,we obtain
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z = -
3S(I - f ) ~ , ( c o- c , ) ~ 8 r v Y j ( c Z- co)(c? - e,)
(4)
I n eq 4, c p = pco and c, = yeo, where fi > 1 and y < 1 depend on the structure of the polarization layer. If it is assumed that the st,ructure of the polarization layer changes negligibly after t,ime tl, then this struct,ure can be estimated by applying the batch-cell theory (WiIliams, et al., 1970). The measured values of tl, vu-, f , N , and S t'hen enable one to calculate z from eq 4. The results, shown in the last row of Table I, agree with the experiment,ally measured filament lengths within the combined accuracies of the measurement and the calculation. Also in agreement with the calculation, observed filament lengths decreased slightly with increasing pressure, although the changes were not great enough t'o be recorded in Table I, within the accuracy of the measurement. The convection-jet model suggested here is subject to improvement in many W E L ~ SFor . example, eq 3 is highly quest'ionable, and methods are currently available for calculating the jet structure with much greater accuracy, taking into consideration both buoyancy and viscosity at high Schmidt numbers. I n addition, t'he agreement between the current theoretical and experirnent'al results can be termed qualitative a t best. lIoreover, the model ignores the interesting possibilit'y of long-term buildup of salt a t the bottom of the filaments, with secondary convection induced below. However, it appears that the convection-jet model is likely to constitute one of the essential elements in any complete description of buoyant convection in reverse osmosis. Concluding Comments
It is relevant to emphasize that the buoyant convection investigated here differs from the large-scale buoyant convection patterns for which theoretical analyses have been published by other investigators (e.g., Johnson and Acrivos,
1969) in connection with desa1inat)ion st'udies. This earlier work is concerned not with instabilities but with stable buoyancy-driven flows that may be of interest for use in certain reverse-osmosis systems. I n unpublished work, Johnson, dcrivos, and Bershader completed interesting laser-interferometric observat'ions of large-scale buoyant convection patterns in reverse osmosis t h a t tend to confirm their theoretical calculations. These studies are concerned with flows that are different from those that we have studied, in the sense that the configurations are not unstable to the type instability that we have investigated. I n unpublished work, Davis and Acrivos have shown t h a t in the absence of buoyancy the reverse-osmosis process itself is a n intrinsically unstable flow. Our experimental observations tend to suggest that this intrinsic instability is overpowered by buoyancy, which stabilizes the flow in gravitationally stable configurations and produces the buoyant instability considered herein in gravitationally unstable configurations. The buoyant instabilities that we have studied occur in a very narrow layer near the surface of the menibrane. They are not strongly influenced by the main flow, and in principle they should occur both for turbulent flow in large channels (in the laminar sublayer) and for laminar flow through very narrow channels. The anticipated ubiquity of the phenomenon motivates further study of it, directed toward developing a more thorough quantitative description that can be emoloyed in developing design concepts. literature Cited
Baldwin, W. H., Halcomb, I). L., Johnson, J . S., J . Polynl. Sci. A3. 838 1196.5).
Foster; T . b:, j: Geophys. Res. 73, 1933 (1968). Desalination 9, 155 (1971). Hendricks, T. J., Williams, F. .4,, Johnson, .4.R., Acrivos, A,, IND.ESG.CI-IEJI., FusD.nsr. 8, 359 (1969). Williams, F. A., Hendricks, T. J., Liu, 31. K., "Boundary Layer Flow Problems in Desalination by Reverse Osmosis," llesearch and Development Progress Report S o . 622, Office of Saline Water, U. S. Department of the Interior, U. S. Government Printing Office, Washington, I1.C. 20402, Ilec 1970. T E R R Y J. HEXIIRICKS
J E , l S F. lI.\C0171S FORhLIX A. JTILI,IA-hIS* Department of Aerospace and Jfechanical Engineering Sciences I-niversity of California, Sun Diego La Jolla, Calif. 92037 This work was slipported in part by a grant from the Office of Saline m'ater, U.S.Department of the Interior. I
