I n d . Eng. Chem. Res 1988,27, 2341-2352
hf = froth height, mm hL = clear liquid height at the froth-to-spray regime transition, corrected for the effect of weir height on spray regime entrainment, mm h,, Hw = outlet weir height, mm L = liquid flow rate, m3/h m of weir length n = exponent defined by eq 7c p = hole pitch (center-to-center hole spacing), mm S = tray spacing, mm u, = gas superficial velocity, based on tray bubbling area AB, m/s Greek Letters p~ = gas density, kg/m3 p L = liquid density, kg/m3 u = surface tension, dyn/cm x = factor defined by eq 7a [ = correction term for eq 2, defined by eq 3
Literature Cited Bain, J. L.; Van Winkle, M. AIChEJ. 1961, 7, 363. Benke, N. S. B. E. Thesis, School of Chemical Engineering, University of New South Wales, 1974. Bennett, D. L.; Agrawal, R.; Cook, P. J. AIChEJ. 1983, 29, 434. Brook, W. E.; Hannold, D. E.; Cunningham, W. C.; Huntington, R. L. Pet. Eng. 1955, 27, C-32. Calcaterra, R. J.; Nicholls, C. W.; Weber, J. H. Br. Chem. Eng. 1968, 13, 1294. Colwell, C. J. Ind. Eng. Chem. Process Des. Dev. 1981, 20,298. Fair, J. R. PetrolChem. Eng. 1961, 33, 45. Friend, L.; Lemieux, E. J.; Schreiner, W. C. Chem. Eng. 1960, Oct 31, 101. Hsieh, C. L.; McNulty, K. J. Presented at the Annual Meeting of the AIChE, Miami Beach, FL, Nov 2-7, 1986. Hunt, C. d’A.; Hanson, D. N.; Wilke, C. R. AIChEJ. 1955, I, 441. Jeronimo, M. A. da S.; Sawistowski, H. Trans. Inst. Chem. Eng. 1973, 51, 265.
2341
Kister, H. Z.; Haas, J. R. “Distillation and Absorption 1987”. Inst. Chem. Eng. Symp. Ser. 1987,104, A483. Kister, H. Z.; Pinczewski, W. V.; Fell, C. J. D. Ind. Eng. Chem. Process Des. Dev. 1981a, 20, 528. Kister, H. Z.; Pinczewski, W. V.; Fell, C. J. D. Presented at the 90th National Meeting of the AIChE, Houston, TX, April 1981b. Lemieux, E. J.; Scotti, L. J. Chem. Eng. Prog. 1969, 65, 52. Lockett, M. J. Distillation Tray Fundamentals; Cambridge University Press: Cambridge, 1986. Lockett, M. J.; Banik, S. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 561. Lockett, M. J.; Spiller, G. T.; Porter, K. E. Trans. Inst. Chem. Eng. 1976, 54, 202. Muller, R. L.; Prince, R. G. H. Chem. Eng. Sci. 1972, 27, 1583. Nutter, D. E. Chem. Eng. Prog. Symp. Ser. 1973, 68(124), 73. Payne, G. J.; Prince, R. G. H. Trans. Inst. Chem. Eng. 1975,53,209. Pinczewski, W. V.; Fell, C. J. D. Trans. Inst. Chem. Eng. 1972,50, 102. Pinczewski, W. V.; Fell, C. J. D. Inst. Chem. Eng. Symp. Ser. 1982, 73, D1. Porter, K. E.; Jenkins, J. D. “Distillation 1979”. Inst. Chem. Eng. Symp. Ser. 1979, 56, 5.1/1. Priestman, G. H.; Brown, D. J. Trans. Inst. Chem. Eng. 1981,59, 279. Priestman, G. H.; Brown, D. J. Presented at the Annual Meeting of the AIChE, Chicago, IL, Nov 10-15, 1985. Priestman, G. H.; Brown, D. J. “Distillation and Absorption 1987”. Inst. Chem. Eng. Symp. Ser. 1987,104, B407. Shakhov, Yu.A.; Noskov, A. A.; Romankov, P. G. Zh. Prikl. Khim. (Leningrad) 1964, 37, 2074. Stichlmair, J. Grundlagen der Dimensionierung des Gas/ Flussigkeit-Kontakt Apparates-Bodenkolonne; Verlag Chemie: Weinheim, 1978. Thomas, W. J.; Ogboja, 0. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 429.
Received f o r review January 12, 1988 Revised manuscript received July 14, 1988 Accepted July 31, 1988
A Comparison of Solute Rejection Models in Reverse Osmosis Membranes for the System Water-Sodium Chloride-Cellulose Acetate Gregory P. Muldowneyt and Vito L. Punzi* Department of Chemical Engineering, Villanova University, Villanova, Pennsylvania 19085
An investigation is described which models the solute rejection mechanism in reverse osmosis (RO) and develops physical parameters which characterize cellulose acetate membranes. Theoretical rejection equations are derived based on the solution-diffusion mechanism and the convective transport mechanism. Each equation is cast in two forms, one for an ideal membrane and one corrected for solute passage. Four models result which are individually fit to experimental separation data from a commercial RO unit. Ideal models provide a reasonable first approximation to membrane behavior. Corrected models improve significantly on ideal models; comparison of the two forms allows membrane nonideality to be quantified. Solute separation is accurately predicted over a wide range of pressure, feed rate, and feed concentration by either the corrected diffusive flow model or the corrected viscous flow model. Membrane performance is discussed in terms of directly measured variables and fitted rejection parameters, focusing on observed gains in selectivity over time. Reverse osmosis (RO) is a promising separation technology which has reached full-scale application only in the last 25 years. Cellulose acetate membranes were the first to offer a practical balance of selectivity and solvent permeation rate and, though developed originally for water
* Author to whom
correspondence should be addressed. Present address: Department of Chemical Engineering, University of Illinois, Urbana, IL 61801.
desalination, are still used extensively for separation of both organic and inorganic solutes (Kesting, 1985). However, RO process development requires an understanding of membrane semipermeability, which, despite extensive research, remains a controversial topic. Of many proposed models of solute rejection (Soltanieh and Gill, 1981; Punzi and Muldowney, 1987), none may claim either general acceptance versus competitive theories or great utility in applications of reverse osmosis membranes.
0888-5S85/S8/2627-2341~~~.50/0 0 1988 American Chemical Society
2342 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988
Understanding of the solute rejection mechanism requires clarification of two points. The first is which type of transport, diffusive or convective, is more deterministic to semipermeability, i.e., whether the membrane acts as a barrier or a conduit. The second is what physical parameters best characterize a real membrane, which provides partial separation of solute from solvent, relative to an “ideal” one achieving perfect separation. Numerous solute rejection models exist which variously address these two points (Soltanieh and Gill, 1981; Punzi and Muldowney, 1987). Unfortunately, the most comprehensive reviews of RO membrane transport theory either do not reconcile model equations with experimental separation data or do so inconsistently. No contemporary investigation has sought to resolve the key issues of transport and membrane characterization by systematic analysis of solute rejection models using a single experimental data base. This study considers four models of membrane selectivity: diffusive transport, convective transport, and the “ideal” version of each. The models are tested against reverse osmosis data for cellulose acetate membranes separating aqueous sodium chloride solutions. Such a comparison has also been performed using data from membranes of thin-film polyamide (Punzi et al., 1988) and asymmetric polyamide (Punzi and Hunt, 1988). In each case the goal is to capture the essential physics of semipermeability in a model equation which offers predictive value for real RO membranes. Theory General. An ideal membrane which permits no solute passage is the starting point for classical treatments of membrane transport (Staverman, l951,1952a,b). A real membrane is then described by a “reflection coefficient” (Staverman, 1951) between 0 and 1, designating the fraction of solute it rejects. Transport models for ideal membranes may be modified by using the reflection coefficient to obtain corrected models which describe solute rejection in real membranes (Muldowney, 1985). Diffusion and convection define the possible extremes of mass transport, and reverse osmosis membranes have microscopic features which suggest either mode could prevail (Kesting, 1985). Existing membrane models based on irreversible thermodynamics cannot clarify the transport mechanism because convective processes violate their assumption of mechanical equilibrium (Soltanieh and Gill, 1981). Also of no assistance are models based on weighted diffusive and convective contributions-these lead to empirical equations without physical significance. The most successful approach is to test the basic diffusive and convective transport models as predictors of membrane performance. If in addition the ideal version of each model is compared to that corrected for solute passage, the nonidealities present in real membranes may be assessed. For highly selective membranes, the most sensitive measure of performance is the solute separation factor, a , defined as the ratio of feed to product solute concentrations (Kesting, 1985): a = C2//C2/1. Solute rejection models relate separation to a solvent-solute flux ratio. A practical reverse osmosis model expresses a in terms of pressure, flow rate, temperature, and parameters specific to the solvent-solute-membrane system. For greatest utility, these parameters should be insensitive to driving forces, i.e., constants or ratios of similar variables. Solution-Diffusion Mechanism. This mechanism is the most common schematic of solute rejection (Laidler and Shuler, 1949; Lonsdale et al., 1965; Lonsdale, 1966). Solvent and solute molecules dissolve into the membrane on the high-pressure side, diffuse through independently,
and emerge on the low-pressure side. A separation results because the solvent and solute diffusion rates are different. Closely related is the hydrogen-bonding model (Breton, 1957; Reid and Breton, 1959) which proposes that solvent molecules migrate through the membrane along a path of successive hydrogen-bonding sites; separation occurs because most solutes hydrogen bond poorly. These models are considered collectively here because both describe mass transport by a diffusive mechanism: the distinction between them is finer than the scope of this study. The essential feature of the solution-diffusion model is that the fluxes of solvent and solute are uncoupled. Each flux is proportional to a chemical potential gradient across the membrane: solvent: solute: Here C, and Mm are the species concentration and mobility in the membrane, p i the chemical potential of species i, and x the direction of the fluxes J . In general, pi varies with activity, pressure, temperature:
Membrane separations are usually carried out isothermally, and heats of dissolution in the membrane are negligible. This justifies omission of the third term in eq 2. Evaluating the remaining terms using dilute-solution laws (i.e., with concentration replacing activity) yields dpi/dx = RT(d In Cim/dn)
+ &(dp/dx)
(3)
where Dim is the partial molar volume of i in the membrane and R is the universal gas constant. For each species, the concentration derivative is rewritten as solvent:
solute:
Here ir denotes the solution osmotic pressure, given by [(-RT/ijlm) In C,,] when ulm is independent of pressure (Merten, 1966). The essential property of the solute which makes reverse osmosis possible is that Uzm is much smaller than (l/Cz,) (Soltanieh and Gill, 1981). Thus, for the solute, the second term in eq 3 is negligible compared to the first. With this simplification the fluxes (eq 1)become solvent: d (54 (P - a) J1 = -ClmMlmUlm solute:
Separation based on differing diffusion rates in the membrane must occur monotonically, through some active layer of thickness A. Across this layer, it is reasonable to ap-
Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2343 proximate p and Czmwith linear profiles, whereupon the x derivatives become differences:
solvent:
Corrected Diffusive Flow Model. A real membrane permits some solute passage; permeate thus exits with nonzero solute concentration, CZm”,and osmotic pressure, a”. To describe this case, eq 9 is recast using distribution coefficients:
solute: Another CY evolves in the bracketed term, leading to the following explicit equation for solute separation: with A denoting the difference from the high-pressure to the low-pressure side. The solution-diffusion mechanism involves no other significant transport processes; thus, the total flux, Ni, of either species is equal to Jiand the solvent-to-solute flux ratio is
N1 Clmulm M1m A@ - a) - = -(7) NZ R T M2m ACZm Equation 7 summarizes a key consequence of the solution-diffusion mechanism: separation is independent of membrane thickness. The physical implications of eq 7 become clearer upon additional refinement. Since the solution of solvent or solute in the membrane is dilute, the mobilities obey the Nernst-Einstein equation Mi, = Dh/RT and are replaced by a diffusivity ratio. Further, since solvent concentration is essentially invariant across the membrane, C1, = Clm”. Defining a permeate-side distribution coefficient K F = Clm”/C1” then eliminates Clm” in favor of KI”C1”, and the flux ratio becomes
and produces in the permeate (per mass conservation) an equivalent concentration ratio, CF/Ci’. This equality is rearranged to obtain the solute separation factor:
The derivation of eq 9 assumes Olm, Dlm,and C1, are independent of pressure, a valid assumption in the pressure range typical of RO processes (Soltanieh and Gill, 1981). For the equation to be of value, the Czmterms must be eliminated and all concentrations collected on the left-hand side. Ideal Diffusive Flow Model. An ideal membrane may permit solute dissolution at the high-pressure side, but no solute diffusion to the low-pressure side. The permeate is thus pure solvent for which CZm”is zero and a“ is negligible. For this ideal case, eq 9 takes the form
using a feed-side solute distribution coefficient, K i = CZm’/Ci. Defining the permeability Pi DimKF for each species then yields cy*
=
K i f Pi Dim -- -(‘p
K i Pz R T
- a’)
Equation 11 summarizes the ideal diffusive flow model. Perfect separation in the solution-diffusion model occurs because Dzmis zero; i.e., solute cannot diffuse through the membrane material. This implies zero solute permeability (P2)and, by eq 11, an infinite separation factor.
CY
K,” P1ulm K,’ Pz R T
= -- -(Ap
- a’) +
Equation 13 represents the corrected diffusive flow model: the second term accounts for nonideality. (Hereafter, the a’ and a” terms are written together as AT.) The diffusive flow models (eq 11and 13) differ from the classical solution-diffusion equation by accommodating possibly unequal distribution coefficients, K,’ and K,”. In the conventional presentation (Lonsdale et al., 1965; Lonsdale, 1966; Merten, 1966; Soltanieh and Gill, 1981; Kesting, 19851, ACzmin eq 9 is equated to K2AC2: this assumes (K,“/K,’) is unity. Convective Transport Mechanism. The convective transport mechanism assumes the existence of discrete membrane pores through which solvent passes in viscous flow, conveying solute with it. A separation is achieved if the solute concentration in the pore liquid differs from that in the solution on the high-pressure side. Such a concentration change on entering a membrane pore is variously explained in the literature (Soltanieh and Gill, 1981; Punzi and Muldowney, 1987),one noteworthy theory being that solvent is preferentially adsorbed on the membrane to the exclusion of solute (Sourirajan, 1963, 1964, 1970). The “finely porous” (Merten, 1966) and “capillary flow” (Sourirajan, 1970) models may be regarded as special cases of the convective transport mechanism and are collectively represented here by a general treatment. The essential feature of the convective transport mechanism is viscous flow of solvent, which couples the solvent and solute fluxes. Pore liquid of solute concentration Cz flowing at mean velocity up constitutes a flux upCZp.Sofute also diffuses in the (moving) solvent due to a concentration gradient along the pore. Total solute flux, Nzp,is the sum of these processes: Nzp =
upczp
+ (-Dzi dC,p/dxp)
(14)
The diffusive flux in eq 14 differs from that of the solution diffusion mechanism in two ways: the solute diffusivity in the solvent, DZ1,is much greater than that in the membrane, Dzm (typically by about lo5 (Sourirajan, 1970; Soltanieh and Gill, 1981));and xp is a coordinate following the (possibly tortuous) pore, whereas x measures straight distance through the membrane. Two physical constants relate pore quantities to macroscopic ones-tortuosity (71, the mean ratio of pore length to membrane thickness, and porosity (e), the mean ratio of pore area to total area: up = U / C xP = 7~ Nzp = N ~ / C (15) Czp = Czm/t The first relation is valid because volumetric flow is conserved as pore liquid enters the permeate. The remaining equalities merely convert distance, flux, and concentration from a pore basis to a membrane basis (of length, area, and volume, respectively). Solute flux at the low-pressure side in macroscopic terms is then (using eq 15 in eq 14)
2344 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988
diffusive flow models: ideal: The solute flux, N2,corresponds to a velocity, u”, of permeate with solute concentration of C c . Permeate velocity is the volumetric flow, V”, divided by the membrane area, A. Following these replacements in eq 16, the differential equation is solved for C2,:
Czm= Ci’t
+ I? ex.[
-9 I)i]
'Jim
where XD* = -(Ap KT
- x ’ ) (21)
where XD = -(Ap L
- AT) (22)
corrected:
(17)
D21
The integration constant B is fixed by the feed-side boundary condition Czm CZm’= K2’Ci at x = 0. This gives the solute concentration profile through the membrane: C2, = C2”t
+ (K2’C,’ - eC2“) exp
[ y(L)L] --
RT
viscous flow models: ideal:
(18)
D21
Thus, the convective transport mechanism yields an exponential profile where the solution-diffusion mechanism leads to a linear one. A form of eq 18 is sought which gives a in terms of quantities other than concentration. Ideal Viscous Flow Model. The ideal membrane delivers a permeate of zero solute content at the low-pressure side. This case is described by eq 18 with C2, = 0 at x = A, which is rearranged to obtain the solute separation factor:
Equation 19 describes the ideal viscous flow model. A membrane governed by convective transport can achieve perfect separation only if no solute enters the pores, since all pore liquid reaches the permeate. The solute rejection mechanism must occur at the high-pressure surface, requiring that CZm‘be zero. This leads to zero K,’ in eq 19 and an infinite separation factor. Corrected Viscous Flow Model. In a real membrane, the concentration profile is nonzero through the pore and reaches a value Ch” at the low-pressure side. The general condition C2, = K,”“” at x = X is imposed on eq 18 to obtain a for this case: a = 1K2’ [1-exp(-~:k)]+
which is the corrected viscous flow model. The correction for nonideality is the second term in eq 20. In contrast to the solution-diffusion equations, both viscous flow models feature a dependence on membrane thickness. When applied to real membranes, A, r , and c refer to a less well-defined geometry than in the conceptual model: h is the mean thickness of the possibly irregular skin layer, c is the fraction of open area between adjacent polymer segments in this layer, and r is the mean number of distances X traveled by a solute particle through random, noncircular, and interconnecting channels thus formed (Kesting, 1985). Analysis. The strategy of this study is to analyze the four transport models in a consistent, parallel format which addresses both the solute rejection mechanism and the quantification of membrane nonideality:
a*=(&)
X,*
(
where X,* = 1 - exp -- - (23)
corrected:
( ”,”: Ll)
where X, = -exp -- - - (24) Membrane performance is measured in terms of a (or a*); known properties of the membrane, solventfsolute pair, and RO process are collected in X (or X *). Each model linearly relates a to X through mechanistic parameters which summarize the solventsolutemembrane interaction corresponding to semipermeability. The dominant mode of transport is established by comparing the capacity of eq 22 and 24 to correlate reverse osmosis separation data. While the adequacy of a correlating equation does not prove the correctness of a solute rejection model, it strongly suggests that the functionality of separation to operating conditions is correctly represented, which is often sufficient to permit a confident statement about the physics. Three groups of information are known: RO process data (CY,A p , VI’, T ) measured experimentally, solvent-solute data (Ulm, T,DZl)obtained from the literature (Sourirajan, 1970), and membrane structure data ( A , A, r , 6 ) reported in discussions of cellulose acetate morphology (Meares, 1966; Soltanieh and Gill, 1981; Kesting, 1985; Osmonics, 1985). Only the mechanistic parameters are unknown. Being specific to the solvent-solute-membrane system, these parameters are fit to the data and used to evaluate membrane behavior in physical terms: ( P I / P 2for ) diffusion, (cfKz’) for convection, and (K,”fK,’) for both modes. All three are ratios of like variables ( E = K1’ for dilute pore liquids (Merten, 1966))expected to be insensitive to pressure and flow rate over the ranges tested. Concentration effects (if any) are established by separately analyzing data sets of variant V ” and A p at several feed concentrations. Both corrected models have two mechanistic parameters, mo and ml, in the dimensionless equation a = mo + mlX. The authors of this paper consider inconsistent the approach of previous treatments (Merten, 1966; Soltanieh and Gill, 1981) which evaluated the one-parameter solution-diffusion equation (i.e., with K2/’/K2/ 1) against convective transport equations having two or three parameters, in particular the viscous flow model (eq 24) with (7X/tDq1) fit also. Fitting this term is additionally unde-
Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2345 sirable because the resulting form a = mo + m, exp(-mzV”) represents at least three other models of membrane transport (Soltanieh and Gill, 1981) in which the parameters m take on different physical meanings and afford less parallelism to the ideal case. Membrane nonideality is described using eq 21 versus eq 22, and/or eq 23 versus eq 24. Although the ideal models are only special cases of the corrected models (valid as cy a),the departure of membrane selectivity from perfect might be significant before the utility of eq 21 and eq 23 as predictors of separation is lost. Ideal forms are included in the data correlation to establish whether the resulting values of (P1/P2)*, (e/K2/)*,and (K2/I/Ki)* bear any relation to those obtained “rigorously” via eq 22 and 24. The ideal parameters lose their physical significance (all three are infinite) when applied to data from imperfect membranes. However, they may retain value as measures of membrane nonideality on comparison to parameters obtained from the corrected models. As predicted by the four models, a (or a*) refers to concentrations at membrane surfaces, which may differ from bulk stream solute levels due to concentration polarization (rejected solute accumulation at the selective surface). A feed-side mass-transfer coefficient, k, relates the surface and bulk concentrations CpI and C2,{ as follows:
-
which results from either film theory or an eddy-diffusion treatment (Brian, 1966). Model equations a* = ml*X * and a = mo mlX are revised to include boundary layer effects using eq 25, yielding respectively a b * = ml,b*X * and f f b = mo,b ml,bX with mo,b= mo exp(-V”/Ak) + [l - exp(-V”/Ak)] (26a)
+
+
ml,b = ml exp(-V”/Ak)
ml,b* = ml* exp(-V”/Ak) (26b) and a b the bulk separation factor (C2,{/C2/1). Conversion to bulk concentrations thus preserves the linear form of the solute rejection equations but intSoduces into the mechanistic parameters an explicit dependence on V ”. This functionality, however, is extremely weak. Turbulent feed-side mass transport is induced in RO ystems by high fluid velocities and mesh obstacles (Osmonics, 1985), leading to coefficients, k, of the order cm/s (Jonsson and Boesen, 1975). A typical flux (V”/A) is 2.5 X lo4 cm/s, and the permeate flow varies under normal conditions by at most a factor of 3, thus yielding exp(-V ”/Ak) values of 0.96-0.99. Because this factor is close to unity, mb differs from m in eq 26 by a very small and nearly invariant amount. Thus, the model equations are fit as ab* = ml*X* and f f b = mo -k mlX, noting that the adjustment for concentration polarization is absorbed into m with negligible effect. This approach is also consistent with using Ap in eq 21 and 22 without subtracting the pressure drop across the boundary layer, an adjustment also commonly found to be negligible (Jonsson and Boesen, 1975). The capacity of any model to predict a from X is determined via linear regression in terms of three statistical quantities: (1) the total sum of squares-the squared deviations of a from the mean a value-(2) the sum of squares removed-the variation in a accounted for by the correlation to X-and (3) the residual sum of squares-the a variation unaccounted for and presumed to represent random deviations of measured a values from the true ones (Volk, 1969; Box et al., 1978). In general, models must be compared in terms of the residual sum of squares, the
smaller residual indicating the better correlation. Models with the same number of fitted parameters, however, will have identical total sums of squares and therefore may be evaluated using the fraction of squares removed, i.e., the squares removed divided by the total sum of squares. The square root of the fraction of squares removed is the familiar “correlation coefficient”.
Experimental Section For a solute rejection model to have practical design value, it must be insensitive to variability typical of real RO processes. Separation data are therefore obtained by using a commercial unit (Osmonics OSMO 3319-SB) representative of industrial RO systems. The unit features three spiral-wound modules of cellulose acetate membrane: each has an area of 1.05 m2 (11.3 ft2) and provides 4.7 X g/(cm2.s) (10 gpd/ft2) maximum product water flux. The three membranes are of 89-92% nominal sodium chloride rejection at a test pressure of 27.2 atm (400 psig) (Osmonics, 1983). For experiments, the modules are piped in parallel with valves allowing independent control of feed rate to each. The experiments occur in two phases. In the initial phase, three aqueous sodium chloride feed solutions--1000, 1750, and 2500 mg/L chloride (nominal)-are in turn separated by reverse osmosis at 28 combinations of feed flow rate and feed-side gauge pressure. In the confirmatory phase (performed 5 months later) the feed concentration range is expanded to nine nominal 1eve1s-250,500,1000, 1750,2500,3750,5000,6750, and 7500 mg/L-and the flow rate/pressure combinations reduced to a representative 12. Flow rates are 12-21 cm3/s (11-20 gph) and pressures 6.1-16 atm (90-230 psig). Operation is isothermal a t 20 “C. At every set of operating conditions, the concentrate and permeate from each membrane are sampled twice over a 20-min period. Chloride content in the samples is recorded, along with the pressures on feed and permeate sides and the flow rates of feed, concentrate, and permeate. Solute concentrations (C,l and C2/‘) are determined to at least four significant figures using a chloride selective electrode (Orion Model 94-17, with double-junction reference 90-02) and a pH/mV/ion meter (Fisher Accumet 825MP). Pressures (hence Ap) are measured by Bourdon gauges and flow rates (in particular V ’0 by rotameters, each to two significant figures. More precise instrumentation is not warranted because in commercial RO systems, including the OSMO 3319-SB, pumps impart high-frequence fluctuations of approximately 0.5% to pressures and flow rates even during steady operation. Membrane properties needed in the data analysis are reported as small ranges. The skin layer thickness (A) in cellulose acetate is 0.25-0.50 pm (Meares, 1966; Soltanieh and Gill, 1981; Osmonics, 1988). Electron microscopy studies of pore size and density in this layer fix the fractional open area (e) between 2.9% and 5.2% (Meares, 1966; Kesting, 1985; Osmonics, 1988). (Both the thickness and porosity of the support layer are much larger.) Polymer segments in the skin form a random network for which the mean tortuosity (7) is 2.5 (Kesting, 1985);values of 2-3 are typically used (Jonsson and Boesen, 1975; Soltanieh and Gill, 1981). Assimilating the three ranges gives ( A T / € ) values of 9.6-51.7 pm. Previous studies (Jonsson and Boesen, 1975; Soltanieh and Gill, 1981) have noted “effective skin layer thicknesses” ( A 7 / t ) for cellulose acetate of 26-48 pm. Both viscous flow models are tested here using in turn (AT/€) of 9.6, 51.7, and 22.3 pm (the geometric mean) to investigate the generality of the results. The data base for each experimental phase is analyzed using in turn one of the four solute rejection models. All
2346 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988
measured data values are included. Computations are performed in BASIC on a VAX 11/780 using double precision to minimize subtraction errors in calculating the regression analysis parameters. Final results are reported to two significant figures, consistent with the least precise experimental measurements.
General Results and Discussion The initial-phase experiments investigate pressure and flow rate effects on solute rejection and assess the variability of mechanistic parameters over a small concentration range. The confirmatory experiments clarify concentration effects and test the reproducibility of membrane performance (hence duplication at 1000, 1750, and 2500 mg/L). In what follows, the results of all experiments are presented together. Differences between phases are noted for later discussion on the effects of membrane age. The solute separation factor, a, and permeate flow rate, V ”, are found to differ among the three test membranes at each set of operating conditions. Values of a are 2.0-8.3 (47-87% rejection) in the initial phase and 2.2-12.0 (52-91 % rejection) in the confirmatory phase, with most values between 4.0 and 6.0. Membrane 3 shows the highest selectivity a t all conditions and membrane 2 the lowest. Nominal rejection levels (89-92 %) are approached only at the highest feed-side pressure. Flow rates, V”, span the 0.44-5.2 cm3/s range in both phases. Permeation is without exception greatest for membrane 2 and least for membrane 3, illustrating the familiar trade-off between solute rejection and permeate flow. Both a and V ” are found to increase with increasing applied pressure and decrease with increasing feed concentration, consistent with previous findings in RO systems (Lonsdale, 1966). These trends hold throughout both phases and are discernible even over small ranges of the independent variables. Comparison of Solute Rejection Models Analysis of the four solute rejection models leads to the statistics presented in Table I. These are the basis for subsequent discussion of both membrane nonideality and the mechanism of semipermeability. The following should be noted a t the outset: (1)Data point counts for each nominal feed differ between the initial and confirmatory phases and must be noted when comparing total or residual squares. Smaller data sets occur a t 6750 and 7500 mg/L because some pressures in the standard set of operating conditions fall below the solution osmotic pressure at these chloride levels and are omitted. (2) Since it is impractical to present graphical results for every membrane-feed combination, several representative cases are selected from Table I. Membrane 1 is chosen for its intermediate selectivity among the three. Nominal feeds of 500 and 5000 mg/L are chosen to bracket a realistic range of chloride content. Initial and confirmatory results are plotted for comparison at 2500 mg/L, the highest duplicated feed concentration. (3) The fit of the convective transport equations is found to be unaffected by the quantity ( A T / € ) . Results listed in Table I for the ideal and corrected viscous flow models are those at (AT/€) of 51.7 gm, which agrees most nearly with literature values (Jonsson and Boesen, 1975). Pertinent results using 9.6 and 22.3 pm are mentioned for comparison. Ideal versus Corrected Models. Table I reveals that the ideal models reduce the unaccounted variation in a* from several hundred (total squares) to 25.0 or less (residual squares), This establishes that the ideal case physics
7 ,
, ’
I
Coni rmato-) F Me-broqe
’
5001.gi. o Ideal
A Ca-rec’ed
c
-I
x
itds
,e
z r s c o r t P g v m e er x - *
-r
Y
1 1
Figure 1. Performance of diffusive transport models in correlating solute separation data for a 500 mg/L feed. I
1 ~
6 +&
I
Corflciatorv Phase Membrcve Ij 5000mq/L o Ideal
i
’a
9 Corrected
3
2
B
”
k // O
4 2
I
I
1
34 1
” 5 i-
o
a 2
7
c i i u s ve T r a r s p o r i
/
E
4 Darometer y c * or X~
1
x
2’
Figure 2. Performance of diffusive transport models in correlating solute separation data for a 5000 mg/L feed.
are a reasonable first approximation to solute rejection behavior in the test membranes. Figures 1-4 show as dashed lines the ideal diffusive flow and ideal viscous flow equations for 500 and 5000 mg/L feeds. The data fall in clusters due to the discrete experimental settings of pressure and feed rate. While most points in Figures 1-4 lie close to the dashed lines, their arrangement suggests a correlation of lesser slope would further reduce the residual squares of a. However, because the line a* versus X * represents a one-parameter model-a direct consequence of ideality-its slope is fixed by the mean data point and the origin. A lesser slope through the same mean would require a nonzero a intercept, hence a two-parameter (nonideal) model. Results presented in Table I using the corrected rejection equations show significant improvement over the ideal models, achieving residual squares in all cases of less than 5.0 and most of order 10-1 or Solid lines in Figures 1-4 denote the corrected diffusive flow and corrected viscous flow models, which are seen to provide excellent correlation of the data compared to the ideal forms at both concentrations. Table I reveals that the improvement in fit upon correcting for nonideality is larger at higher feed concentration and greater for membranes 1and 2 than for membrane 3 (e.g., 2500 mg/L, either phase). These observations are consistent with the theoretical basis of the ideal models, expected to hold most strongly for a highly selective membrane in a highly dilute solution. It is not unexpected that correcting for imperfect separation improves the predictions of a using membranes where solute flow clearly occurs. However, it is useful to quantify the improvement, justify a second correlating parameter, and observe theoretical trends experimentally. Also noteworthy is that the ideal models predict a with
Ind. Eng. Chem. Res., Vol. 27, No. 12,1988
2347
Table I. Performance of Ideal and Corrected Rejection Models in Correlating the Solute Separation Factor ideal rejection models corrected rejection models residual squares after correlation by residual squares (fraction removed) after correlation by total squares total squares diffusive nominal feed membrane (CY*- a*)z model viscous model (CYdiffusive model viscous model concn, mg/L Initial Phase 2.2 (0.92) 1.7 (0.94) 7.7 8.5 28.0 1 480 1000" 12.0 0.96 (0.92) 1.2 (0.90) 7.3 6.9 2 290 53.0 2.8 (0.95) 2.8 (0.95) 13.0 13.0 900 3 0.76 (0.91) 8.3 0.70 (0.92) 19.0 19.0 1 320 1750" 6.3 0.68 (0.89) 0.79 (0.88) 2 14.0 12.0 230 42.0 2.7 (0.94) 1.9 (0.96) 21.0 25.0 3 740 6.1 16.0 12.0 0.36 (0.94) 0.36 (0.94) 1 250 2500" 0.34 (0.90) 0.37 (0.90) 2 3.6 170 13.0 8.6 2.4 (0.92) 1.9 (0.93) 29.0 520 18.0 16.0 3 1 11.0 (0.79) 9.6 (0.83) 56.0 all 2 4.0 (0.84) 3.2 (0.88) 26.0 feeds 140.0 8.6 (0.94) 10.0 (0.93) 3 (pooled)b 25OC 50OC lOOOC
175OC 250OC 375OC 50OOc 6750d 7500d
3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 23 1 2
3 all feeds (pooledY
Confirmatory Phase 1.3 2.6 1.8 2.6 5.2 4.8 3.8 5.5 3.8 4.6 6.2 9.3 2.5 3.1 2.6 2.7 6.3 7.6 4.2 5.5 3.4 4.0 7.5 9.7 2.2 1.7 1.4 1.8 1.8 3.4 2.6 2.4 2.5 1.8 5.8 4.3 8.1 2.0 7.0 15.0 13.0 4.0 13.0 0.14 11.0 0.11 17.0 0.37 15.0 0.53 12.0 0.46 1.1 20.0
530 420 710 470 340 650 410 270 610 360 230 550 240 160 400 220 150 350 160 110 260 37.0 30.0 50.0 59.0 43.0 87.0
1 2
1 2
3
6.6 4.3 22.0 1.1 0.44 1.3 1.8 0.68 1.5 1.3 0.49 1.6 2.5 1.2 8.1 2.3 1.1 4.7 2.1
0.81 4.9 0.099 0.027 0.20 0.46 0.16 0.72 170 130 230
0.73 (0.89) 0.89 (0.79) 4.9 (0.78) 0.085 (0.92) 0.11 (0.76) 0.28 (0.78) 0.10 (0.95) 0.10 (0.85) 0.39 (0.75) 0.28 (0.79) 0.14 (0.72) 0.44 (0.73) 0.19 (0.92) 0.14 (0.88) 1.2 (0.85) 0.093 (0.96) 0.12 (0.88) 1.6 (0.66) 0.14 (0.93) 0.051 (0.94) 1.8 (0.62) 0.023 (0.77) 0.0071 (0.74) 0.14 (0.31) 0.042 (0.91) 0.026 (0.84) 0.32 (0.56) 18.0 (0.90) 25.0 (0.80) 29.0 (0.87)
0.96 (0.85) 1.0 (0.76) 4.8 (0.78) 0.075 (0.93) 0.10 (0.77) 0.25 (0.80) 0.091 (0.95) 0.10 (0.85) 0.42 (0.72) 0.25 (0.81) 0.12 (0.75) 0.50 (0.69) 0.31 (0.88) 0.18 (0.84) 1.4 (0.83) 0.069 (0.97) 0.052 (0.95) 1.3 (0.72) 0.090 (0.96) 0.032 (0.96) 1.5 (0.70) 0.016 (0.84) 0.0058 (0.79) 0.090 (0.55) 0.020 (0.96) 0.0067 (0.96) 0.18 (0.76) 11.0 (0.93) 16.0 (0.87) 26.0 (0.89)
Data set contains 26-28 pressure-feed rate combinations. *Pooled data set contains 82 pressure-feed rate-feed concentration combinations. Data set contains 12 pressure-feed rate combinations. Data set contains 6 pressure-feed rate combinations. e Pooled data set contains 96 pressure-feed rate-feed concentration combinations. C o n v e c t ve TransDor: P z r c r r e t e r . XL
-C 925
- 2 975
-3 903
71
--
_,d E30
i
- 1 000
-0980
-3960
-0940
-0920
5t
-0900 1
/
C o n i rmatory P h a s e Vembrane ' 5000mg/L o lcecl
a
A Corrected
I
YS
X"
0
v? a,
0 080
2 '30
0 120
3 '40
0 160
Convective Transport P a r a m e t e r Xv*
Figure 3. Performance of convective transport models in correlating solute separation data for a 500 mg/L feed.
0 330
3C40
3 252
c Q60
Conve-- v e T r a n s p o r t Fo,a-ete
1 3 z73
c
C8C
XJ'
Figure 4. Performance of convective transport models in correlating solute separation data for a 5000 mg/L feed.
2348 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 i
e l
L 0
----
D ,,/'
%.
1 l
__
-
~
- -~
'. tJs i z T
or s m r t F o Jt-e
p i
Y-
x
e
rorsair
___
-.
Pcrcrct+er i
Figure 5. Performance of the corrected diffusive flow model in correlating solute separation data for a 2500 mg/L feed.
Figure 6. Performance of the corrected viscous flow model in correlating solute separation data for a 2500 mg/L feed.
acceptable accuracy for scoping calculations in reverse osmosis process design. Similar success is not anticipated for less selective (i.e., more nonideal) membranes, but sodium chloride rejection below 90% is uncommon in RO applications. Corrected Diffusive Flow Model versus Corrected Viscous Flow Model. Identification of the solute rejection mechanism is sought in the results of the corrected diffusive flow and corrected viscous flow models, compared in Table I using the fraction of squares accounted for (removed) by the correlation a versus X. The corrected diffusive flow model removes 88-96% of the scatter (i.e., total squares) in a in the initial data and, with two exceptions, 62-96% in the confirmatory data. A very good fit is obtained throughout the initial phase with no strong dependence on concentration or membrane. In the confirmatory phase, the fit is still quite satisfactory, though in most cases better for membranes 1 and 2 than for membrane 3, particularly at 3750 mg/L and above. Typical fractions of squares removed of 0.66-0.85, while smaller than in the initial phase, still give correlation coefficients of 0.81-0.92 and imply that the functionality of a to independent variables is well represented. Much lower fractions for membrane 3 at 6750 and 7500 mg/L occur because at some operating conditions the osmotic pressure of the concentrated feed-side solution approaches the applied pressure, whence a typical measurement error in A p causes a large relative error in ( A p - AT) and XD. Feed concentration dependence is well correlated in both phases, evident from 79-94% squares removed in the pooled data sets, which is comparable to the individualcase results. The solid lines in Figures 1 and 2 illustrate the excellent prediction of a by the corrected diffusive flow model at 500 and 5000 mg/L feed concentrations. Figure 5 presents as two lines the initial and confirmatory results for membrane 1 at 2500 mg/L, which likewise show the performance of the model to be extremely good. (The differing slopes of the two lines indicate membrane age effects discussed later.) From the success of the corrected diffusive flow model, it follows that a two-parameter linear functionality of a to the pressure difference ( A p - A T ) captures the essential physics of semipermeability in the operating window studied. The very small individual-case residuals also confirm that variability due to boundary layer effects on feed-side concentration is satisfactorily absorbed into the (fitted) mechanistic parameters. The corrected viscous flow model accounts for 89-95% of the scatter in cy in the initial phase and, with one exception, 69-97% in the confirmatory phase (Table I). Performance of the model depends insignificantly on the value of ( A T ' e ) . The largest effect throughout all results
is in the initial phase at 1750 mg/L for membrane 3: by use of ( X T / ~ )of 9.6 pm, residual squares are 2.9 and the fraction removed is 0.93, versus respective values at (AT/€) of 51.7 pm of 2.7 and 0.94. Uniformly strong correlation is obtained in the initial data for all membranes and concentrations and is largely retained in the confirmatory data for membranes 1 and 2. Somewhat larger residuals occur in the confirmatory phase for membrane 3 (e.g., 3750, 5000 mg/L), but a strong correlation is still apparent. Excellent performance for all membranes in the pooled data sets-83-93% squares removed-verifies that the model accommodates extremely well the influence of concentration. Figures 3 and 4 show the close fit of the corrected form for membrane 1 at 500 and 5000 mg/L. Figure 6 presents separately the initial and confirmatory results for membrane 1 at 2500 mg/L, confirming the predictive value of the model. These results establish that a two-parameter exponential dependence of a on permeate flow adequately represents the basic mechanism of solute rejection under the given test conditions. Boundary layer effects are again found to be insignificant in correlating separation data. From the foregoing, it is evident that either the corrected diffusive flow or corrected viscous flow model accurately predicts solute separation in highly selective cellulose acetate membranes. Given that a consistent evaluation of all models from a common data base has been heretofore lacking, this likely accounts for much of the persistent disagreement over the solute rejection mechanism in reverse osmosis. The present results demonstrate that the solution-diffusion and convective transport mechanisms, though theoretically different, lead to two rejection model equations either of which accounts for much of the variation in a. It is therefore apparent that some overlap exists in the two models-not in theoretical basis, but in mathematical form. This implies a relationship, separate from the solute rejection mechanism, which couples the "independent" variables, X D and X v , through physical quantities. Such a relationship is noted among the general trends: permeate flow rate increases with increasing applied pressure and decreases with increasing feed concentration. Regression analysis of all confirmatory measurements indicates (with 98% of the squares removed) a correlation of the form where n varies from 1.67 to 1.78 among the membranes tested (Muldowney, 1983). Since the functionality of V to p is complicated by the weakly concentration-dependent quantity [@(C,')], the coupling of XDand Xv is partial. This explains the comparable, though not identical, corI'
Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2349 Table 11. Best-Fit Mechanistic Parameters for the Ideal and Corrected Solute Rejection Models diffusive flow models viscous flow models nominal feed ideal corrected concn, mg/L membrane [(PI/P*) * (K,”/Ki )*I [(PI/P~)(&”/&’)] ideal [ ( c / K i ) * ] corrected Initial Phase 1000 1750 250
1 2 3 1 2 3 1 2 3
680 520 930 620 530 950 670 560 980
1
890 790 1000 860 730 1000 860 700 1000 900 720 1100 820 660 1000 980 810 1200 1200 1000 1500 3000 2700 3400 2000 1700 2600
(e/&’)
480 310 670 270 230 610 280 220 620
47.0 30.0 92.0 41.0 29.0 87.0 48.0 32.0 97.0
33.0 19.0 67.0 19.0 14.0 54.0 22.0 15.0 60.0
690 520 1200 290 170 290 400 230 320 310 180 330 470 320 820 510 330 590 490 300 580 390 180 310 450 240 410
58.0 44.0 97.0 57.0 41.0 96.0 58.0 40.0
40.0 28.0 98.0 22.0 13.0 30.0 29.0 16.0 34.0 23.0 13.0 34.0 31.0 19.0 70.0 35.0 22.0 65.0 38.0 21.0 71.0 44.0 20.0 63.0 38.0 20.0 59.0
Confirmatory Phase 250 500 1000
1750 2500 3750
2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2
3 5000
1
2 3 6750
1
7500
2 3 1 2 3
relation of membrane separation data by two rejection models representing theoretical extremes of solute transport. Interpretation of Mechanistic Parameters Tables I1 and I11 present the fitted mechanistic parameters of the four solute rejection models. Ideal and corrected values of [ (Pl/Pz)(K,”/K,’ ) ] are the slopes of the respective diffusive flow equations; ( e / K i ) *is the slope of the ideal viscous flow equation and (e/&‘ ) the intercept of the corrected form. These four parameters are used to quantify membrane nonideality. Distribution coefficient ratios ( K / / K i ) are calculated from the corrected diffusive flow and corrected viscous flow equations as, respectively, intercept or difference of slope and intercept (cf. eq 22 and 24). Comparison of these ratios allows a consistency check between the corrected models. Table IV presents the physical quantities (Pl/Pz), K i , and K P obtained from the corrected parameters. These are evaluated against literature data. Ideal versus Corrected Parameters. In the initial results, the ideal diffusive parameter [(P1/PJ*(K,”/Ki)*I for a given membrane varies less than 6% and averages 660, 540, and 960 for membranes 1, 2, and 3, respectively (Table 11). To the same concentration limit (2500 mg/L) in the confirmatory results, the variation is 8% or less about mean values of 870,720, and 1020. Across the 3750 and 5000 mg/L levels, the parameter steadily increases for all membranes. Values a t 6750 and 7500 mg/L are less reliable for reasons noted earlier.
100 60.0 41.0 110 56.0 38.0 100 66.0 45.0 120 70.0 46.0 130 95.0 60.0 160 75.0 49.0 130
The ideal convective parameter (e/&’)* at ( X ~ / E )of 51.7 pm averages 45.0,30.0, and 92.0 for the three membranes in the initial phase, while up to 2500 mg/L in the confirmatory phase the means are 58.0, 41.0, and 100.0; in both phases the variation is 9% or less. Like the ideal diffusive parameter, ( s / K i ) * begins a gradual increase a t 3750 mg/L, which continues through 5000 and 6750 mg/L. The results a t 7500 mg/L are inconsistent. In every membrane-feed case, both ideal mechanistic parameters increase in the order 2, 1,3, which parallels the membrane selectivity trend. Corrected diffusive parameters [(Pl/Pz)( K { / K i 11 average 340, 250, and 630 for the three membranes in the initial results, and 410, 260, and 490 over the 500-5000 mg/L range of the confirmatory results. (Isolating this range avoids inconsistencies observed above 5000 mg/L as well as errors at 250 mg/L, especially for membrane 3, caused by permeate concentrations falling below the chloride electrode calibration.) The mean corrected parameters in the confirmatory phase are 36-48% smaller than the corresponding ideal values. Corrected convective parameters (e/&‘) at (AT/€) of 51.7 pm average 25.0,16.0, and 60.0 in the initial phase and 29.0, 17.0, and 51.0 from 500 to 5000 mg/L in the confirmatory phase, the latter set decreasing 41-51 % from the ideal-model mean values. In the confirmatory results, [ (P1/PZ)(K,”/K2/)] and (E/&’ ) show greater variability because the solute rejection models are fit to smaller data sets. However, both parameters follow the familiar order of increase 2, 1,3 in all but one case.
2350 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 Table 111. Solute Distribution Coefficient Ratio Predicted by the Corrected Rejection Models corrected rejection models nominal feed Wz”lK21) concn. me/L membrane diffusive flow viscous flow Initial Phase 1 1.2 1.4 1000 2 1.3 1.4 1.8 3 1.6 2.1 1750 1 1.9 1.5 1.7 2 3 1.8 2.3 1 1.7 1.8 2500 1.6 2 1.5 1.9 3 1.5 Confirmatory Phase 1 1.5 2.1 2 3 (
