Surface Interactions in Osmotic Pressure Controlled Flux Decline

P. Kanagaraj , A. Nagendran , D. Rana , T. Matsuura , S. Neelakandan , and K. ... Subir Bhattacharjee, Ashutosh Sharma, and Prashant K. Bhattacharya...
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Langmuir 1994,10,4710-4720

Surface Interactions in Osmotic Pressure Controlled Flux Decline during Ultrafiltration S. Bhattacharjee, Ashutosh Sharma,*and P. K. Bhattacharya Department of Chemical Engineering, Indian Institute of Technology at Kanpur, Kanpur 20801 6, India Received June 10, 1994. In Final Form: September 16, 1994@ Prediction of osmotic pressure controlled flux decline during ultrafiltrationrequiresan accuratetheoretical determination of the osmotic pressure of macromolecular solutions at high concentrations, where nonvan't Hoffian contributionbecomes significant. A model for osmotic pressure determination is used which is based on facile measurements of surface properties of solutes. The knowledge of the apolar and polar (acid-base) interaction energies of polymeric molecules in water leads to the determination of the FloryHuggins interaction parameter. Estimates of the interaction parameter as a function of the solute concentration lead to the direct theoretical prediction of osmotic pressure of various macromolecular solutions up to high concentrationsapproachingthe gel limit. The osmotic pressure of the macromolecular solutions thus determined is used to predict the osmotic pressure controlled flux decline. A model is developed by solving the coupled velocity and concentrationfields to predict flux during unstirred, stirred, and parallel plate ultrafiltration. For all of these configurations,experimental flux decline and limiting flux data are obtained for PEG and Dextran under various operating conditions. The model predictions are compared to experimental flux data which show a remarkable agreement even in the absence of any adjustable parameter in the model. Results help in the identification and quantification of the solute-solute interactions as a factor affecting permeate flux during osmotic pressure controlled flux decline. The theory of osmotic pressure controlled ultrafiltration can also be used as an efficient tool for osmometry.

Introduction Ultrafiltration (UF) has not yet realized its full potential in the industry because of the severe drawback of flux decline associated with it. Flux decline is caused by three factors manifested individually or simultaneously during an UF operation. These are (i) osmotic pressure buildup in the concentrated layer near the membrane surface, (ii) formation and growth of a gel layer at high concentrations, which acts as a secondary membrane offering additional resistance to permeate flow, and (iii) fouling of the membrane caused by solute adsorption or particulate blocking of membrane pores. Phenomena i and ii are manifestations of solute-solvent-solute interactions, whereas iii is governed by solute-solvent-membrane surface interactions (forces). All of the above processes are greatly aided by concentration polarization, i.e.,buildup of very high concentrations on the membrane surface. Intensive research has been directed toward the modeling and minimization of flux decline over the past 2 de~ades.l-~ These efforts were mostly directed toward the (i)minimization of flux decline and (ii) investigation ofthe causes and modeling of flux decline. These objectives require tailoring of the system hydrodynamics4 and membrane surface^,^ as well as development of predictive models offlux decline. While several models are available t o account for flux decline in various types of UF systems,6-ssuch models are largely phenomenologicaland,

* To whom correspondence should be addressed. Abstract published in Advance ACS Abstracts, November 1, 1994. (1)Nakao, S.; Wijmans, J . G.; Smolders, C. A. J.Membr. Sci. 1986, 26,165. (2)Van den Berg, G . B.; Smolders, C. A. Desalination 1990,77,101. (3) Bhattacharjee, S.; Bhattacharya, P. K. J.Membr. Sci. 1992,72, 149. (4)Bruin, S.; Kikkert, A.; Weldring, - J . A. G.; Hiddink, J.Desalination 1980,35,223. (5)Abdul Mazid, M.Sep. Sci. Technol. 1988,23,2191. (6)Van den Berg, G .B.; Smolders, C. A. J.Membr. Sci. 1989,40,149. ( 7 )Wiimans. J . G.: Nakao,. S.:. Smolders. C. A. J.Membr. Sci. 1984, 20, 115. (8)Trettin, D.R.; Doshi, M. R. Ind. Eng. Chem. Fundam. 1981,20, @

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therefore, cannot provide a proper understanding and quantification of the various effects causing flux decline. When all the three effects of concentration polarization occur simultaneously, it becomes difficult to isolate individual resistances by use of phenomenological models. Predictive models that incorporate the underlying physics of surface interactions are therefore better suited to study the mechanism of osmotic pressure increase, gel layer growth, and solute adsorption individually and to establish the importance of each of these phenomena in controlling flux decline. Osmotic pressure is usually the sole controlling factor in UF of low molecular weight ( M W ) solutes employing low adsorbing membranes. It may also remain significant in UF of higher MW solutes, where it acts in concert with the growth of a gel layer.3 The different flux predictive models which take osmotic pressure control into consideration,6-10mostly use experimental data of osmotic pressure at various concentrations to predict the flux. Here, we formulate a model of osmotic pressure controlled flux decline during UF, which incorporates solute-solute interactions manifested as an increase in the osmotic pressure. Such an approach will lead to an understanding of the osmotic pressure controlled flux decline and for discrimination among various types of resistances encountered during UF. Model predictions are tested against experimental observations of flux decline in osmotic pressure controlled UF for various process configurations, e g . , unstirred and stirred batch cells and parallel plate modules. The solutes employed are PEG (polyethylene glycol) of molecular weight 4000 and 6000 and Dextran T-20, with an average molecular weight of 18 600. Osmotic pressure engendered by the solute-solute interactions in a solvent is the outcome of the interfacial forces acting between the solute mo1ecules.l' So far, the surface interactions between solutes and membrane material have been used to account for (9)Michaels, A. S.Chem. Eng. Prog. 1978,64,31. (10)Trettin, D.R.;Doshi, M. R. In Synthetic Membranes Vol I I Hyper and Ultrafiltration Uses; Turbak, A. F., Ed.; ACS Symp. Ser. No.154; American Chemical Society: Washington, DC, 1981;p 373. (11)Van Oss,C. J . Colloids Surf. 1993,A-78,1.

0 1994 American Chemical Society

Osmotic Pressure Controlled Flux Decline

Lungmuir, Vol. 10, No. 12, 1994 4711

adsorption and some aspects of gel formation during In the present work, we estimate the importance of such interactions in the osmotic pressure controlled flux decline behavior for macromolecular solutions. To this end we employ a theory11J4which relates the FloryHuggins interaction parameter15 (an estimate of the intermolecular interactions), to independentlymeasurable surface properties. Thus, an understanding of the solute interactions from a surface thermodynamic point of view should broaden the theory of osmotic pressure controlled flux decline during UF. UF.12J3

Theory In UF operations for low MW solutes, flux decline is mostly engendered by osmotic pressure buildup of the feed solution near the membrane surface at the upstream side. For such cases, the permeate flux (velocity)may be evaluated by either of the following equivalent phenomenological expression^^,^

or

where AP is the applied pressure difference, 17, the solvent viscosity, and u, the osmotic reflection coefficient. The resistance due to osmotic pressure R,,, is related to the osmotic pressure difference, An in the following way

(3) The osmotic pressure difference across the membrane is expressed as

An = xm - xp

(4)

where, the subscripts m and p denote the membrane surface and the permeate, respectively. Estimationof Osmotic Pressure. There are several approaches for estimating osmotic pressure. Since the osmotic pressure of a solution is a function of solute concentration, it may be expressed as an empirical virial expansion of the solute concentration

+ ugc + ...

x = up2 + uzc2

3

(5)

provides an interpretation of the virial coefficients in eq 5. In eq 6, R is the univsrsal gas constant and T is the absolute temperature. VZand eP stand for the molar volume of the solvent and the density of the polymer, respectively. The ratio nlM denotes the reciprocal of the molecular weight of the monomer unit for a given polymer. For macromolecular systems of interest, the second virial coefficient is of great importance in determining the osmotic pressure of the solution at higher concentrations.lg From eqs 5 and 6, it can be seen that the Flory-Huggins interaction parameter, x12, is related to the second virial coefficient. Determination of osmotic pressure at high concentrations is therefore dependent upon the determination of xlz. While the Flory-Huggins15 solubility relations can be employed for prediction of osmotic pressure of apolar polymers dissolved in apolar solvents, prediction for polar polymers ( e g .PEG, Dextrans)in polar solvents (eg.water) is not as facile. For polar systems of interest in UF, the Flory-Huggins interaction parameter xlz has to be again determined from the experimental fit of data over the concentration range of interest. This approach then becomes analogous to finding the best experimental fit for eq 5. In arecent study, van Osset uZ.l4proposed an alternative route to the determination ofxlz using surface free energy measurements of the solute species in aqueous solutions. The magnitude and sign of the interaction parameter are a measure of the total interaction energy between the solute molecules in a given solvent. The total free energy of interaction (adhesion), which is composed of the apolar (Lifshitz-van der Waals11~20-22) and polar (acidbase11J4,24-32) interactions, can be readily evaluated from surface properties (eg.,components of interfacial tensions) of the solute and the solvent.11J4,24-30The interfacial tension and its components can be derived from data obtained from contact angle goniometry and are available for a large number of polymer solutes and Here, we use the theory ofvan Oss et u Z . ~for ~ the osmotic pressure, which incorporates the solute-solute interactions to estimate the Flory-Huggins interaction parameter, ~ 1 2 . The essential aspects of the theory to be used in conjunction with the modeling of flux decline are

(19)Goldsmith, R. L.Znd. Eng. Chem. Fundam. 1971,10,113. The virial coefficients a l , u2, etc., are either evaluated (20)Lifshitz, E. M. Zh. Eksp. Tcor. Fiz. 1966,29,94. by fitting experimental data to the virial expression, eq (21)Hamaker, H.C. Physica 1937,4,1058. 5 or calculated by semiempirical or empirical theories, (22)Fowkes, F. M. J. Adhes. 1972,4,155. most of which are accurate for dilute solutions ~ n l y . ~ ~ J ~(23)Fowkes, F. M. J.Phys. Chem. 1963,67,2538. (24)Van Oss, C. J.;Chaudhury, M. K.; Good, R. J. Chem. Rev. 1988, Another common approach for the estimation of osmotic 88,927. pressure of macromolecular solutions is the use of Flory(25)Van Oss,C. J. J. Dispersion Sci. Technol. 1991,12,201. Huggins theory.ls The well-known Flory's equation for (26)Van Oss, C. J.;Good, R. J. J.Mucromol. Sci., Chem. 1989,A-26, 1183. the osmotic pressure of a polymeric solute of molecular (27)Van Oss, C. J. InBiophysics of Cell Surfme; Glases, R., Gringell, weight M in a truncated virial form,ls D.,Eds.;Springer-Verlag: Berlin, 1990;p 131. (12)Rodgers, V. G. J.; Sparks, R. E. AZChE J. 1991,37,1517. (13)Nabetani, H.; Nakajima, M.; Watanabe, A.; Nakao, S.;Kimura, S . AIChE J . 1990,36,907. (14)Van Oss,C. J.; Arnold, K.; Good, R. J.; Gawrisch, K.; Ohki,S. J . Mucrmol. Sci., Chem. 1990,A-27,563. (15)Flory, P. J.Principles ofpolymer Chemistry;Come11University Press: Ithaca, NY, 1953. (16)McMillan, W. G.; Mayers, J. E. J. Chem. Phys. 1946,13,276. (17)Stigter, D.;Hill, T. L. J. Phys. Chem. 1959,63,551. (18)Hermans, J. J. In Colloid Science Vol ZI; Kruyt, H. R., Ed.; Elsevier: Amsterdam, 1949;p 48.

(28)Costanzo, P. M.; Giese, R. F.; Van Oss, C. J.J.Adhes. Sci. Techwl. 1990,4, 267. (29)Sharma, A.J. Dispersion Sci. Technol. 1992,13,459. (30)Van Oss, C. J.; Good, R. J. J. Protein Chem. 1988,7,179. (31)Sharma, A.Langmuir 1993,9,3580. (32)Sharma, A.Lungmuir 1993,9,861. (33)Sharma, A.Biophys. Chem. 1993,47,87. (34)Van Oss, C.J.;Good, R. J.; Chaudhury, M. K. Langmuir 1988, 4,884. (35)Van Oss, C. J. J. Protein Chem. 1989,8, 661. (36)Van Oss, C. J.; Chaudhury, M. K.; Good, R. J. Sep. Sci. Technol. 1987,22,1515.

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Bhattacharjee et al.

summarized next. Detailed theory and experimental techniques for estimation of apolar and polar components of free energy and osmotic pressure may be found el~ewhere.’~J~~~~-~~ Solute-Solute Interactions. The apolar Lifshitzvan der Waals (dispersion, Keesom and Debye) intera c t i o n ~ ~are ~ always , ~ ~ - attractive ~~ between two molecules (particles) of the same material regardless of the nature of the solvent. The shorter ranged polar interactions due to electrodproton exchange are variously described as hydration pressure, hydrogen bonding, hydrophobic attraction, etc. The polar AB interactions (which may be repulsive or attractive) play a crucial role in understanding of the mechanisms of the following (see, e g . , r e v i e w ~ ~ ~ hydrophobic , ~ ~ , ~ ~ ) : attraction; hydration pressure; stability and phase separation of polymers in aqueous media; stability of aqueous dispersions and nonionic micelles; reversed phase liquid chromatography; stability ofthin aqueous films;31equilibrium contact angles of polar liquids ( e g . , water).11,24,25,32 The total free energy of interaction (per unit area) is the sum of the apolar (Lifshitz-van der Waals, LW) and polar (acid-base, AB) interactions. AG12, = AGk;

+ AGEl

(7)

In the case of solute molecules possessing large electrical potentials and relatively weak polar interactions, the electrostatic double layer interaction, AGfkl also makes a significant contribution to the total energy. However, in the present work, electrostatic interaction has not been considered since the solutes used in the experiments (PEG and Dextran) are largely charge neutral, and the “hydrophilic” (AB)repulsion is the sole cause of the non-van’t Hoflian (nonideal) component of the osmotic pressure.14 The apolar and the polar components of the interaction energy (per unit area) in the flat plate approximation decay with intermolecular separation, d, according to the following relations11,24,25,27,31,32

the integral of the form

(10) For spheres, the Derjaguin a p p r ~ x i m a t i o n may ~ ~ be used

to obtain the integral when AG121is the interaction energy per unit area for the flat plate (eqs 8 and 9). However, in the present work, only linear chain polymers (PEG and Dextran) have been treated, for which the total free energy can be obtained from the flat plate approximation.14 The Flory-Huggins interaction parameter is a dimensionless energy of the f o r d 4

where, k is Boltzmann’s constant. Thus, determination of the total interaction energy between two molecules helps us to obtain x12. It is an estimate of the binary interactions in a macromolecular solution.38 Once x 1 2 is estimated, the Flory-Huggins relati~nship,’~ eq 6, can be used to obtain the osmotic pressure of a solution. At higher values of solute concentration, the second term in eq 6 containing x12 becomes important and contributes significantly to the osmotic pressure of the solution. It should be noted that AG121 being a function of intermolecular separation, ~ 1 2 is, also a function of d and, hence, of solute concentration. Further, a positive value of AG121due to polar hydrophilic repulsion gives a negative interaction parameter. Thus, high negative values of interaction parameter indicate a strong hydration repulsion11,14,24,25 between the polymer species. Elementary considerations suggest that for long linear polymeric strands, end-on interactions will be far fewer than the interactions in randomly oriented crossed positions. The mean intermolecular separation distance should therefore decrease almost linearly with the concentration. Thus, whenever AB interactions are dominant, lnlxlzl should vary linearly with concentration (from eqs 9 and 11).This indeed is also shown by an excellent agreement of the theory to experimental osmotic pressure data of PEG.14 The total free energy of adhesion (at d = do) between two particles of material 1 in a solvent 2 is11r24

(12)

AG121(do) = -2712 where do is the minimum equilibrium distance where the extremely short range Born repulsion may be replaced by a vertical rise in the potential to infinity (hard sphere approximation). The best value for do for a vast variety of condensed phase materials is 0.158 nm.11924 The parameter 2 is a characteristic correlation length (decay length), which for “hydrophilic repulsions” in water has a value of about 0.6 nm.11’25,27An exponential decay of the polar (AB) interactions renders them short range as compared to the apolar (LW)interactions, which show an algebraic decay. The LW component of the energy of adhesion, AGkE(do),is always attractive (negative),” whereas the AB component (polar) may be repulsive (positive) for intensely hydrophilic solutes in water.11z24,25 The interaction energy AG121 between two linear chain molecules can be expressed as a function of separation distance, d. This interaction energy per unit surface area, when multiplied with the contactable area, S, gives the total energy of adhesion (AGT,,) between two molecules. However, for molecules with shapes ( e g . ,spherical) where the interaction zone cannot be approximated by flat plates, the total interaction energy can be obtained by evaluating

where y denotes the surface tension. The LW and AB components, AGkE(do)and AGgl(do),of the free energy of adhesion can be evaluated from contact angle goniometry by the following relation^^^*^^-^^

and AGEl(d0) = -4(-Jy:y,+

-

lir:rz -

(14)

where ykw is the apolar component of the surface tension of species i, and y; and y: are the electron donor (proton acceptor) and electron acceptor (proton donor) parameters (37)Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (38) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: London, 1983. (39) hobstein, R. F.; Shen, J. S.; Leung, W. F. Desalination 1978, 24, 1.

Osmotic Pressure Controlled Flux Decline

Langmuir, Vol. 10,No. 12, 1994 4713

of material i , respectively, both of which determine the polar (AB) component of the surface tension as

Modeling of the Batch Cell and Parallel Plate UF Processes. UnstirredlStirred Batch Cell. Neglecting and effects, and assuming l-D flow normal to the membrane surface (u = 0, u = -uw), eq 17 can be written in this case as

The total surface tension of a material is LW

AB

yi = Yi + Yi

(16)

Different surface parameters cy:", y l , and 7): needed for evaluation of free energy of adhesion are readily determined by the measurements of contact angles of diagnostic liquids on a flat surface of the s o l ~ t e . " JThe ~~~~ surface properties and energies of adhesion of many polymershiopolymers and other materials have been determined by this method.11J4,24-36 Osmotic PressureControlled Flux Decline in UF. Once we have a reliable estimate of the osmotic pressure of the solution at relatively high solute concentrations, we can use eq 1to obtain the permeate flux, provided we know the membrane surface concentration and the concentration profiles in the concentrated layer close to the membrane. The problem of predicting permeate flux is coupled with the estimation of the concentration profile. The transient state diffusion problem associated with UF in different setups is expressed in a general form by the following mass balance equation at

+ v - v c = !qzV2C

(17)

where V is the velocity, with its x and y components denoted by u and u , respectively. Equation 17 is the mass balance for transfer of a solute species across a porous barrier without reaction and adsorption. It is also assumed that the diffusivity, &, and density are independent of solute concentration, c. The steady-state counterpart of eq 17 which is more suitable for analysis of flux profiles in a stirred batch cell or a parallel plate module is given by v-vc = q z v 2 c

(18)

The incorporation of the cell geometry and the hydrodynamic parameters make eqs 17 and 18 applicable to specificUF systems. The appropriateinitial and boundary conditions associated with eq 17 are c(xy,O)= co(constant) C(X,",

t )= co

a2c ac ac- q2+ uw at ay2 ay

(19) (20)

and (21) where, co and cpare the feed and permeate concentrations of the solute respectively. Equation 21 is the solute mass balance at the membrane surface, where y is normal to the membrane surface. The permeate flux uw is given by eq 1 in conjunction with the osmotic pressure model discussed earlier. Finally, the velocity components u and u are in general obtained from the continuity equation, V-V = 0, together with the Navier-Stokes equations describing the fluid flow. However, as is shown below, further simplifications are possible for the membrane moduleskonfigurations used in this study.

(22)

The initial and boundary conditions for the unstirred cell are given by eqs 19 to 21. The permeate flux is given by eq 1. For a stirred batch cell, the steady state is attained within a very short time (usually less than a minute). Within this interval, the concentration build up is primarily restricted to the membrane surface. Once the steady state is attained, the mass transfer boundary layer adjacent to the membrane can be assumed to be of a steady thickness 6 according to the film theory. The concentration gradient is confined within the boundary layer, and the stirring maintains a uniform concentration for y > 6. The may therefore be replaced by the condition at y condition of uniform bulk concentration at the end of boundary layer

-

QO

c = c o at y = 6

(23)

As is discussed later, the mass transfer boundary layer thickness (which is related to the mass transfer coefficient) depends inversely on the stirrer speed, w , and in the limit of w 0, the case of the unstirred cell (6 w) is recovered. Parallel Plate System. The general mass balance equation within the mass transfer boundary layer (0 5 y I6 ) for this system can be expressed as

-

-

-a+ uc - +a u c- = a q , -c- ; ;

at

ay

a% ay

(24)

where x: and y are the axial (parallel to membrane) and the transverse (perpendicular to membrane) coordinates, respectively. The fully developed axial velocity component within the mass transfer boundary layer of the laminar channel flow may be expressed aslo 6ii

u=-y

h

where u is the local velocity, ii is the average axial velocity, and h is the channel height. In our experiments, the total permeate flow is small compared to the axial feed rate, and therefore ii is approximately constant along the channel length. The transverse velocity component, u , equals the permeate velocity, uiz. u = -u,

(26)

The boundary conditions for eq 24 are given by eq 21 and c(xy,O)= c(Oy,t) = c(x,d,t)= co

(27)

Analytical solution of this model for the gel controlled case using the similarity method is available.1° However, analytical solution of the osmotic pressure controlled case poses difficulty as the osmotic pressure and hence the permeate flux are both complicated functions of solute concentration. It is known that the steady-state profile in a parallel plate system is attained rather fast, and the analysis of

4714 Langmuir,

Bhattacharjee et al.

Vol. 10,No.12, 1994

Table 1. Surface Tension Components of Different Solutes and Water PEG Dextran water

YLW 43.0 42.0 21.8

Y+

Y-

0 0 25.5

64.014 55.014 25.514

the steady-state operation is more important for this system. The mass balance equation (24) is solved for the steady-state case by discarding the time derivative. An appropriate nondimensionalization of these equations and numerical solution techniques are discussed in the Appendix. Estimation of the Resistance Due to Osmotic Pressure. The discussion above leads to the determination of the concentration profile and the membrane surface concentration for a UF process. The evaluation of the membrane surface concentration leads to the estimation of the permeate flux through eq 1. Thus, we can make predictions about the flux decline during an osmotic pressure controlled UF. However, eq 2 leads to another significant insight into the process, namely, the estimation of R,,, and its correlation with the solute-solute interactions in the concentrated layer through eq 3. The approach incorporates solute-solute interactions as the cause of flux decline.

Experiments UF experiments were performed in a batch cell under both unstirred and stirred conditions and in a parallel plate cross flow module. The permeate flux was measured under many different operating conditions encompassing feasible ranges of pressure differentials, stirrer speeds, and cross flow velocities. The initial feed concentration was kept constant. The batch cell and the cross flow module were both developed in our laboratory. The batch cell has a volume of 600 mL and is fitted with a motor-driven stirrer and a pressure gauge. The parallel plate module consists of a flat rectangular channel (0.273 m x 0.038 m x 0.001 m) with adjustments made for the entrance length effects, so that a fully developed cross flow is ensured in the effective area of the cell under the flow conditions used in the experiments. Both the modules can handle a pressure up to 1000 kPa. The stirrer can attain a maximum speed of 600 rpm and the maximum volumetric flow rate recommended in the parallel plate cell is 120 L/h. Pressurization in the batch cell is done by compressed nitrogen or an air compressor. The cross flow cell is connected to ahigh-pressure pump, feed tank (capacity 5 L), pressure gauges, and flow meters. "Spectra Por" cellulose acetate disk membranes, supplied by Spectrum Medical Industries Inc., were used in the batch cell experiments. The membranes were of molecular cut-off 5K and 10K, highly rejecting (298%) and widely pH compatible (pH 2-10). A different cellulosic membrane (PS1544), supplied by Hydronautics, Ltd. (India), was used for the cross-flow experiments. This membrane had a molecular cut-off size of 20K. The solutes used were polyethylene glycols of M W 4000 and 6000 (obtained from Fluka, Germany) and Dextran T-20, M W 18 600 (obtained from Sigma Chemicals). The properties ofthese solutes as stated in the literature are cited in Tables 1 and 2. Feed solutions (1%(w/v))were prepared and subjected to UF for various lengths of time under different settings of operating variables. The experiments were carried out a t three different pressures, namely, 450,600, and 750 kPa. For the stirred cell experiments, three stirrer speeds (150,300,450 rpm) were used. For the crossflow experiments, the cross-flow velocity was maintained at 50, 80, and 120 L/h. The permeate flux was computed from the cumulative volume permeation data measured against time. The concentrations ofthe feed and permeate solutions were measured by a refractometer (Bausch and Lomb) t o obtain the rejection. Membrane Hydraulic Resistance. The experiments were designed for the study ofosmoticpressure controlledflux decline. Due to possibilities of pore blocking and adsorption, the experiments cannot lead t o the estimation of the osmotic pressure

-30

LO

2.4

5.2

3.8

6.6

8.0

dld.

Figure 1. Variation of the free energy of interaction of PEG

and Dextran with dimensionless separation distance dido. LW and AB components of the total free energy for PEG are also and do = 1.58 shown. I = 6.0 %1.11p24

control unless the membranes are subjected to proper pretreatment. The pretreatment was done by compacting the membranes initially with distilled water a t 900-1000 kPa for 3 h followed by UF with the feed solution for 1 h a t 750 kPa, prior to the actual experimental run. This resulted in a constant R, value of the membrane and effectivelydecoupled the effect of adsorption and fouling from osmotic pressure rise in the boundary layer. Measurement of water fluxes before and after the actual experiments proved the membrane resistance, R,, to be a constant.

Results and Discussion Estimation of Osmotic Pressure. Predictions of osmotic pressure for PEG and Dextran are done using a procedure which is similar to that used by van Oss et aZ.14 for the prediction of osmotic pressure of PEG solutions of different molecular weights. Thus, only a brief discussion of the method is included here. The total interaction energy or the interaction parameter, ~ 1 2 ,between two molecules can be obtained as a function of intermolecular separation, eq 11, provided AGFE(do) and AGgl(d,) are known in eqs 8 and 9. The surface parameters (yLW, y+, and y-) of different solutes being used are summarized in Table 1. On the basis of these data, the polar and apolar components of the free energies of adhesion in water are obtained from eqs 13 and 14. The variation of the free energy components as well as the total free energywith dimensionless separation distance (dldo),as obtained from eqs 7 to 9, are reported in Figure 1 for PEG and Dextran. The LW components of the free energy are slightly negative (weakly attractive) while the AB components are large and positive (repulsive), indicating strong hydrophilic repulsion in water for these solutes. The free energy behavior of Dextran resembles PEG. The total interaction energy between two molecules of PEG and Dextran were evaluated by multiplying the free energy per unit area with the contactable surface area, assuming the molecules to be thin strands; the strand width being 4.6 A for PEG and 7.5 A for Dextran.14 Variation of the interaction parameter, ~ 1 2 with , intermolecular separation is obtained from eq 11 and is shown in Figure 2 for PEG and Dextran. Finally, the osmotic pressure may be evaluated if the mean molecular distance is known as a function of solute concentration. For long strands ofPEG and Dextran, only the lateral interactions in randomly oriented crossed

Langmuir, Vol. 10, No. 12, 1994 4715

Osmotic Pressure Controlled Flux Decline

Table 2. Properties of the Solutes Required for Modeling Osmotic Pressure Behavior and Flux Decline mol w t nlM density, kg/m3 shape and size diffusivity,mz/s solute

PEG

1050 1114

1.51 x 6.0 x

Dextran

rectangular strands width 4.6 A14 strands with strand width 7.5A14

0

IO'

~

'

"

'

'

4000,6000 18600 '

'

'

'

"

'

'

1/44 11180 '

-i L

*

m

'

-

f

i

k -2 A . -

h-

2

$ -3 I 1 I

2

-4 t

-5 -6

1.00 2.17 3.33 4.50 5.67 6.83 8.00

0.0

0.2

Figure 2. Decay in magnitude of the Flory-Huggins interaction parameter xlz (-AGTzlkZ') with dimensionless separation distance for PEG and Dextran.

0.4

0.6

0.8

Concentration (gm/cc)

d/dO

Figure 3. Concentration dependence of lxlzl for rectangular strands of PEG and Dextran.

positions are significant. For such molecules, the lateral distance, d , should decline linearly with the solute concentration, as discussed in the Theory section and e1~ewhere.l~ For intensely hydrophilic solutes considered here, the total free energy is completely dominated by the polar repulsion as shown already in Figure 1. Therefore, from eqs 9 and 11,In 1x121 is largely proportional to do - d. This also implies that In lx12l is proportional to concentration (for linear molecules), uiz.

where a is a material constant. Thus the free energy profiles can be used to determine the nature of the dependence of x12 on concentration by the above equation. For linear molecules, as the variation of In 1x121 with concentration has been ascertained to be linear, we need to determine two values of x12 a t two different concentrations in order to characterize the system. For PEG and Dextran, the total free energy is known a t concentrations where the hydration may be considered to be negligible (at d = d0).14236 For PEG, the concentration at which hydration is negligible was assumed to be 60%,14 while for Dextran, it was assumed to be close to the gel concentration, ~ 7 0 % . ~ O The maximum concentration corresponds to a single layer of hydration of water molecules. At this maximum concentration, the interaction energy should correspond to the contact energy value and the value of ~ 1 is2 determined from this contact energy. It is also known that in an aqueous solution of PEG and Dextran, phase separation starts at 4% PEG and 4% Dextran.36 For linear macromolecules, the interaction energy at phase separation point has a magnitude of 0.5kT. Following the method described earlier,14xlz for PEG and Dextran are then computed at 8% solute concentration. (40) Wijmans, J. G.; Nakao, S.; Van den Berg, J. W. A.; Troelstra, F. R.; Smolders, C . A. J. Membr. Sci. 1986,22, 117.

0.0

0.2

0.4

0.6

0.8

1.0

Concentration (gm/cc)

Figure 4. Comparison of predicted and experimental osmotic pressure of PEG and Dextran solutions. The symbols W and

represent experimental values for PEG14 and D e ~ t r a n , ~ ~ ~ ~ respectively.

0

Thus, two points can be obtained for each solute on the In 1x121 vs concentration plot, one at the maximum possible concentration and the other a t a more dilute side corresponding to the phase separation point. Figure 3 plots the variation of In 1x121 with concentration for PEG and Dextran as obtained by eq 28 and the above data. The osmotic pressure a t different concentrations can now be evaluated from eq 6 by the use ofx12 values obtained from Figure 3. The predictions are compared with the experimentally determined osmotic pressures for PEG14 and Dextran (data obtained from an experimental correlation)40in Figure 4. An interesting observation concerns the relative importance of ~ 1 and 2 nlM in determining the magnitude of the second virial coefficient of the osmotic pressure. For PEG, nlM is of a much larger magnitude than that for Dextran. Thus, although the magnitude ofX12 for Dextran is higher than that of PEG, its second virial coefficient is much smaller. Osmotic pressure predictions are in good agreement with the data, which is specially remarkable

Bhattacharjee et al.

4716 Langmuir, Vol. 10, No. 12, 1994 in the absence of any adjustable parameters. The predictions are independent of molecular weight of a solute for large molecular weights in excess of about 1000.14 Therefore, the predictions for PEG-6000 should be the same as those for PEG-4000. Similarly, the predictions for Dextran T-20 are equally good for Dextran T-70. In summary, the interaction parameter and the osmotic pressure depend on the total free energy of interaction between solute molecules in a solvent. For intensely hydrophilic polymers, the polar repulsion, often disguised as the hydration pressure or structural forces, dominates the total energy of interaction. This is true even for charged molecules with zeta potential I25 mV and is certainly applicable for largely uncharged neutral molecules like PEG and Dextran.24 The surface properties, free energies, and therefore the osmotic pressure all depend on the physicochemical environment, e g . , pH, nature of solvent, impurities, etc. The effect of all these parameters may be conveniently quantified by contact angle goniometry.26,28 A further independent test of the osmotic pressure model is provided in the next section by predictions ofthe osmotic pressure controlled UF flux and its comparison with experiments. Osmotic Pressure ControlledFlux Decline in UF: Solution of the Model Equations. The experimental and theoretical membrane surface concentrations in this study were always less than the gel concentrations for the solutes studied-PEG3 and Dextran.’ Thus, for the solutes considered in this study, the UF is entirely osmotic pressure controlled, as the fouling during the UF is minimized (as discussed earlier). The mass balance, eq 17, subject to the boundary conditions, eqs 19 to 21, was solved numerically using the finite difference technique. The model equations were made dimensionless using nondimensional variables, as described in the Appendix. The two dimensional parabolic partial differential equations for the case of unstirred and stirred batch cells were solved to get the concentration profile c b , t ) . In the case of a parallel plate module, the equations were integrated to get the steady-state profile c ( x y ) . In either case, both the dimensions ( x , y ) or b, t ) were discretized using a form of Chevychev polynomial which gradually increases the step size as distance from 0) increases. This technique is the origin (2, y , t computationally more efficient than working with a uniform grid size since the concentration decreases much more rapidly near the origin (membrane surface). The solution of the differential eq 17 in conjunction with the coupled eqs 19to 21 provide the concentration profiles, osmotic pressure difference across the membrane, and permeate flux. A typical plot for the concentration profile is shown in Figure 5. The solution techniques for the parallel plate configuration, operated at the steady state, are analogous to the unsteady operation of the stirred and unstirred batch cell configurations; the difference being that in the former, time is replaced by axial distance. Thus, for a steadystate solution in a parallel plate configuration, the flux is obtained as a function of the downstream distance from the module entrance. As the variation of flux with the axial distance could not be determined experimentally, the average flux for the entire module was computed from the model results by using the expression

-

(29) where L is the channel length. Comparison of this predicted average flux was made with the experimental flux.

0.0’ 0

”

”

’

”

25

”

’

50

y’ (= J ~ Y / ’ D I ? )

Figure 5. Theoretical predictions of concentration profiles obtained during unstirred batch cell UF of PEG-6000 with a 5K MWCO cellulose acetate membrane: AI’ = 450 Wa; L, = 2.43 x d e s ; cmf= 140.3 kg/m3;(curve 1)0 min, (2) 0.2 min, (3) 0.73 min, (4) 4.6 min, (5) 37 min, (6) 60 min.

The results obtained for three different UF configurations are summarized below. Umtirred Batch Cell. The concentrationbuildup is most severe in unstirred batch cell UF. Hence, the applicability of the model to high concentrations is most rigorously tested for this type of UF using different membranes and solutes. The model was tested for PEG-4000 and 6000 using a 5K MWCO membrane, while with Dextran T-20, a 10K MWCO membrane was used. Both PEG and Dextran showed high rejections (PEG-6000 and Dextran 2 98%, while PEG-4000 2 86%). The high rejection indicates that the reflection coefficient, u, in eq 1should have a value close to 1. Hence, in all the computations, the value of u was assumed to be 1. The concentration dependence of flux, obtained from eqs 1,4, and 6, is used in eq 21. We can then solve the partial differential equation (22) to obtain the concentration profile, c(y,t)and the permeate flux, u,. In Figure 5, typical plots for the concentration profiles at different time intervals are shown for unstirred batch cell UF of PEG6000 a t AP = 450 kPa. The solute concentration at the membrane surface was found to rise rapidly and reach a limiting value in about 2 min. This limiting concentration is dependent upon the applied pressure; the higher the pressure, the greater the limiting concentration. After this steep initial concentration buildup at the membrane surface, the growth of the concentrated layer predominates, and the thickness of the concentrated layer increases. This secondary concentration buildup is, however, much slower. This behavior suggests a steep initial decline in permeate flux followed by a more gradual secondary decline, which was indeed observed in the experiments. The concentration buildup is expressed as the osmotic pressure resistance, R,,,, using eq 3. Figure 6 shows the dimensionless osmotic pressure resistance, RasnJR,, against time for PEG-6000 at different applied pressures. This osmotic resistance is a function of the osmotic pressure in the concentrated boundary layer. The osmotic pressure buildup is rapid in the earlier stages and slows down as the membrane surface concentration reaches a limiting value. Data of R,,, generated in this manner for specific solute-membrane pairs, are useful for prediction of flux in a given UF system a t different operating pressures through eq 2. From eq 1it is obvious that as A x equals AP,the flux

Langmuir, Vol. 10, No. 12, 1994 4717

Osmotic Pressure Controlled Flux Decline

I

2

3 -

0

35

70

0

Figure 6. Dimensionless osmotic pressure resistance, R,,d R,, where R, = l/(&), for unstirred batch cell UF of PEG 6000at different applied pressures, computedusingeq 3: Curve 1, AP = 450 kPa; 2,600 kPa; 3,750 kPa. Symbols represent the experimentally obtained values.

4

i

0

35

70

Time (min.)

Time (inin.)

Figure 8. Variation of dimensionless flux,v,JJw,against time during unstirredbatch cell UF'of PEG-6000: Model predictions and experimental fluxes are denoted by lines and symbols, respectively. Feed concentration: 10 kg/m3;membrane: 5K MWCO; permeability: 2.43 x m/Pa.s. Curve 1, AP = 450 kPa; 2,600 kPa; 3,750 kPa. The values of c d are 140.3,161, and 178 kg/m3for the three operating pressures, respectively.

3

i3

35

70

la'

Time (mia.)

Figure 7. Variation of inverse square of permeate flux with time for unstirred batch cell UF of PEG-6000 at different

operating pressures computed from the model equations: curve 1, AP = 450 kPa; 2,600 kPa; 3, 750 kPa. Symbols represent the experimentally obtained values.

becomes zero. This fact was utilized in the calculations to make the concentration dimensionless. The concentration corresponding to the osmotic pressure at which the flux becomes zero, termed as the limiting membrane surface concentration, c d , was determined a t each operating pressure. The concentration terms were divided by this limiting concentration to render them nondimensional. The membrane surface concentration can never exceed the limiting concentration during an UF operation. In all the cases it was indeed observed that the membrane surface concentration attained values ranging from 58% to 95% of the limiting values after considerable periods of time. Another behavior observed for this case was the linear nature of the variation of 1/u%with time, as shown in Figure 7. This shows that even in osmotic pressure controlled case with varying membrane surface concenholds. The dimentration, the proportionality uw = sionless experimental fluxes (u&,AP) obtained for PEG6000 and Dextran T-20 in comparison with the corresponding model predictions are shown in Figures 8 and 9, respectively.

-

0

35

70

Time (min.)

Figure 9. Comparison of the experimental and predicted dimensionless fluxes as functions of time for unstirred batch cell UF of Dextran T-20. Feed concentration: 10 kg/m3;membrane: 10KMWCO; permeability: 5.556 x lo-" m/Pa.s. Curve 1, AP = 450 kPa; 2, 600 kPa; 3, 750 kPa.

Stirred Batch Cell. In a stirred batch cell, the permeate flux is much higher compared to an unstirred batch cell, and attains a steady-state value within a very short interval. Due to stirring, the concentration buildup in the polarized layer is reduced, resulting in a much lower osmotic pressure resistance. Further, the growth of the concentrated layer is limited and its thickness is assumed to be constant, equal to a thin hydrodynamic boundary layer of thickness 6 (corresponding to a steady-state operation). The model equations in this case were modified to obtain dependence of flux on stirrer speed. For analysis, the estimation of the mass transfer coefficient for the stirred cell was required and the following expressions were sed.^^,^^ (41) Belluci, F.; Drioli, E.; Scardi, V. J. Appl. Polym. Sci. 1976,19, 1639. (42) Colton, C . IC;Frizdman, S.; Wilson, D. E.; Lees, R. S. J.Clin. Invest. 1972, 51, 2472.

4718 Langmuir, Vol. 10, No. 12, 1994

for8000

L

Bhattacharjee et al.

- '32000 (ol")

(9

and for - L 32000

t

In eq 30, w is the stirrer speed in radiandsecond, r is the radius of the stirred cell, v is the kinematic viscosity, and (wr2/v)is the Reynold's number. The mass transfer coefficient can also be estimated experimentally by the velocity variation technique43if the properties of the solute used in eq 30 are not known. In the case of stirred cell UF, the limiting flux phenomenon was most perfectly observed. As the stirrer speed is increased, the back transport of solutes from the membrane surface is enhanced, resulting in a lower membrane surface concentration. If the feed concentration is constant, then the membrane surface concentration attains a steady value after a few seconds. Solving the transient equations for the stirred cell yields results which indicate the attainment of a steady state within 1 min of operation in the case of PEG and Dextran. The steady state is attained quickly enough, enabling the use of the steady state counterpart of the model equations (by dropping the time derivative in eq 17). The flux also becomes constant after about 1 min of operation. However, in long-term operation, there is a secondary decline in flux owing to the increase in the bulk concentration. But, in practically all cases of stirred cell UF, a steady flux can be observed during a period from 1 min to 10 min during which time, increase in co is negligible because there is a little change in the initial (large) volume of the feed solution in the batch cell. Stirring reduces the membrane surface concentration drastically. The membrane surface concentration after the initial flux decline period was found to be much lower than the corresponding unstirred batch cell values. For example, after 10 min of UF of a 10 kg/m3 feed solution of Dextran a t a pressure differential of 600 kPa and a stirrer speed of 200 rpm, cm was found to be 247 kg/m3, while a t 400 rpm, the value was only 37 kg/m3 which was less than 10%of the corresponding limiting concentration, cmf (404 kg/m3). Consequently there is a drastic improvement in permeate flux at high stirrer speeds. The ratio of the predicted limiting flux to the experimental flux for stirred cell UF of PEG-6000 with a 5K membrane, and Dextran T-20 with a 10K membrane, are plotted against the corresponding experimental fluxes in Figure 10. The plot shows a remarkable closeness of the predicted and experimental fluxes. In more than 99% of the cases, the deviation between the two fluxes is less than 4~3%.The limiting flux values were higher at higher stirring velocities. There was a very limited opportunity to collect flux data during the initial transient period which lasted for a few minutes at the most (-1-2 min). A limiting value of the flux and the membrane surface concentration are established quite fast. In the case of a stirred batch cell, the steady-state limiting flux can be predicted analytically. The solution of the steady-state mass balance equation for the stirred ~_____

(43) Bhattachajee, C.; Bhattacharya,P.K. J.Membr. Sei. 1992,72,

137.

t 0.9

8

16

24

32

40

vw(Exp.) ( x 1 0 5 )m/s

Figure 10. Ratio of the predicted and experimental limiting fluxes plotted against the experimental flux for stirred cell UF of PEG and Dextran: A, PEG, 0 , Dextran. cell yields the well-known film theory expression2which can be equated to the phenomenological expression for the flux, eq 1

where A n is a function of the membrane surface concentration, cm, from eq 6 . Osmometry Using Batch Cell Ultrafiltration. If the permeate flux is measured at some known stirrer speed and feed bulk concentration, then it is possible to estimate the osmotic pressure of the solution at the corresponding membrane surface concentration using eq 31. The stirred batch cell UF with a nonfouling membrane can therefore be used for osmometry, provided the mass transfer coefficientcan be determined accurately for the batch cell. The accurate and rapid determination of osmotic pressure using this method can lead to significant developments in the experimental verification of several theories of osmotic pressure based on solute-solute interactions. For instance, using the relationship for the osmotic pressure, eq 6 up to the second virial term, as well as eq 28 for 212, we obtain an equation for the membrane surface concentration C,-C

k In(-)

-

hp RTc, R T B c ~ p--[ 1 - h p -~ h p 0 . 5 -

exp(ac,))

1

= 0 (32)

where (33)

Equation 32 can be used as a useful tool for several purposes. First, it can be used to predict the membrane surface concentration directly, if the values of a and K are known. Equation 32 can be solved by a simple Newton Raphson technique for this purpose. Secondly, from experimental steady flux data for a stirred cell a t known stirrer speeds, the value of a can be obtained by direct substitution of known parameters in eq 32. The determination of a leads to the direct establishment of the dependence of the interaction parameter on solute con-

Langmuir, Vol. 10, No. 12, 1994 4719

Osmotic Pressure Controlled Flux Decline

Table 3. Comparison of Predicted Flux in a Parallel Plate Module with Experimental Dataa hp, Wa

Co, kg/m3

ii, c d s

400 450

10 10

38.9 38.9 58.6 87.7 38.9 58.6 87.7 38.9 58.6 87.7 58.6 58.6

20 600

10

750

10 20

a,(th)(x

lo6),d

s

1.072 1.132 1.237 1.357 0.874 0.952 1.012 1.349 1.433 1.539 1.674 1.338

D,(exp)(x

lo6),d

a Solute: Dextran T-20. Membrane: PS-1544, permeability 5.7 x lo-” ”ass. at different axial distances averaged over the length of the module using eq 29.

centration, which in turn leads to the determination of osmotic pressure dependence on solute concentration. The unstirred batch cell can also be used as an efficient osmometer. The time dependent flux measurements can be used to determine the osmotic pressure dependence on the membrane surface concentration. However, an analytical expression as in the case of the stirred batch cell cannot be obtained in this case. The osmotic pressure dependence on concentration can only be obtained by numerical solution of the eqs 1 , 6 , and 28. However, the osmotic pressure measured from the unstirred batch cell flux data should be more reliable compared to the stirred cell data, as the uncertainties related to the determination of the convective mass transfer coeficient are absent. In addition, a single set of flux us time data can give the dependence of the osmotic pressure on concentration over a wide range of concentrations (CO < C m < Cmf). Parallel Plate System. The model eq 24 subject to appropriate boundary conditions was solved for the steady state concentration profiles in the parallel plate configuration. The appropriately nondimensionalized forms of the model equations, given in the Appendix, were solved to relate the flux decline and the concentration buildup to the various operating variables (pressure differential, feed concentration, cross-flowvelocity, and feed solution properties). A direct comparison of the flux profile with experimental data is not possible as only the average permeate flux, 8, could be measured. The average flux predicted by the model by averaging over the length of the flat plate module (eq 29)) was compared to the experimental average flux for the case of Dextran T-20 a t different operating conditions of pressure and cross-flowvelocity of the feed. The osmotic pressure model describes the flux behavior for this system quite accurately as can be seen from Table 3. The percent deviation between the model predictions and the experimental values are within 4% for the data reported. Thus, it can be stated that the model can be used to predict the osmotic pressure controlled flux decline in a parallel plate system quite accurately. The matching of experimental data with the predicted flux was found to be excellentin all the three configurations used. One feature of the present model is the assumption that the membrane surface concentration does not attain the limiting value instantaneously. While an instantaneous attainment of steady state at the membrane surface is often a s s ~ m e d , this ~ , ~ assumption ~ leads to overprediction, particularly at low pressures and low axial velocities. The present model uses no such approximation for the attainment of steady state concentration profiles and also uses a more accurate expression for the osmotic pressure dependence on solute concentration. These factors result in a better correlation of the model predictions with the experimental data.

s

% dev = I[lOO(a,(th)

1.04 1.13 1.23 1.35 0.877 0.95 1.02 1.3 1.44 1.54 1.7 1.34

- B,(exp))Yfi,(exp)l

3.07 0.177 0.56 0.52 0.34 0.16 0.784 3.83 0.49 0.05 1.529 0.15 The flux is computed from the theoretical predictions

Conclusion A predictive model for the osmotic pressure controlled flux decline is formulated and tested for unstirredstirred cell and parallel plate modules. The non-van’t Hoffian (nonideal)contribution to the osmotic pressure dominates a t high solute concentrations which are encountered near the membrane surface in ultrafiltration. The osmotic pressure engendered by the hydrophilicrepulsions for PEG and Dextran in water is predicted based on their polar “acid-base” surface properties. The coupled differential equations for the permeate velocity and concentration fields were solved numerically for the prediction of osmotic pressure of the polarized layer and the permeate fluxes. Experiments were performed in unstirred and stirred batch cells and in parallel plate module for studying the flux decline with time. Predictions ofthe osmotic pressure and membrane fluxes match the experimental values without any adjustable parameters. An excellent match between the theoretical and experimental fluxes supports the soundness of the theory of the osmotic pressure proposed e 1 s e ~ h e r e . IThe ~ model should be useful for osmotic pressure controlled UF where additional resistances due to gel formation and adsorption are insignificant. When additional resistances are present, the model may be used for assessing their importance from the flux data. It also paves the way for the use of laboratory scale ultrafiltration for osmometry. Appendix Solutionof the Model Equations. For the unstirred batch cell, the governing equations were made dimensionless by using the following dimensionless variables,

8=c/cm,; z=-;

Jw2t @I2

JULY

y’=-; @I2

and V=uIJw (Ai)

where cmf is the final membrane surface concentration and is determined from the phenomenological expression for permeate flux, eq 1,as the concentration at which AP = An or uw = 0. The pure water flux, J, is given by Darcy’s law (=L,AP). For the stirred cell case, the choice of dimensionless variables is slightly different,

The nondimensional set of equations with appropriate boundary conditions are then solved using a finite difference scheme. The solution of the parallel plate system is performed with the assumption of a linear axial velocity profile given

Bhattacharjee et al.

4720 Langmuir, Vol. 10, No. 12, 1994

The constant K is given by

bY

u=-

6iix h

(A3)

The model equations are made dimensionless using the following dimensionless variables.

The nondimensional forms of the equations for the parallel plate system are

The term V is obtained from the osmotic pressure governing relation

Equation A5 can be written as a pair of coupled firstorder nonlinear ODE’Sand solved subject to the boundary conditions of eqs A6 and A7 for a given value of z which denotes the position in the axial direction of the cell. The solution was performed by a backward difference technique using Chebychev type mesh spacing in the 5 as well as the z directions to cluster points at the boundary near the membrane surface. While approximate solutions, eg., the similarity approach of Trettin and Doshi,1°may be applied, they require assuming a constant membrane surface concentration. However, from the direct numerical approach, we find that cm varies considerably along the axial distance for low cross flow velocities and low applied pressures.