Limiting Flux in the Ultrafiltration of Macromolecular Solutions - ACS

16

Downloaded by UNIV OF GUELPH LIBRARY on July 2, 2012 | http://pubs.acs.org Publication Date: January 1, 1985 | doi: 10.1021/bk-1985-0281.ch016

Limiting Flux in the Ultrafiltration of Macromolecular Solutions MAHENDRA R. DOSHI Institute of Paper Chemistry, Appleton, WI 54912

Some of the causes for the observed flux declines in the ultrafiltration of macromolecular solutions are (a) Osmotic pressure: the applied pressure is approximately equal to the osmotic pressure of the solution at the upstream side of the membrane. (b) Gel formation: solute concentration at the upstream side of the membrane approaches the solubility limit. (c) Solute-membrane interaction (membrane fouling): adsorption of solute on the membrane surface and consequent pore blockage or deposition of solute within pores, etc., are commonly termed membrane fouling. The first two causes (a,b) are governed by boundary layer diffusion, while the third one (c) obeys a filtration equation. In some cases, both diffusion and filtration resistance may be important. A mathematical model is developed to identify causes of limiting flux in the ultrafiltration of macromolecular solutions. An in-depth study of the ultrafiltration of macromolecular solutions was conducted by Blatt and coworkers ( O in 1970. One of their findings was that as the operating pressure is increased the solvent flux through the membrane first increases but eventually approaches a limiting value which is independent of pressure. In concurrence with Michaels (2), the formation of a gel layer at the membrane upstream surface was proposed to explain the observed pressure independent flux. Since 1970, considerable work has been done to understand the causes of limiting flux in the ultrafiltration of macromolecular solutions. Blatt et al. C O used film theory to model the gel polarized region. Subsequently, Probstein et al. 0_-J>) and Trettin and Doshi (6_~_8) used an integral method and more exact theories to gain insight into the limiting flux behavior. 0097-6156/85/0281-0209S06.00/0 © 1985 American Chemical Society

In Reverse Osmosis and Ultrafiltration; Sourirajan, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.


210

REVERSE OSMOSIS AND ULTRAFILTRATION

In general, the osmotic pressure of the dilute macromolecular solutions could be negligible due to their high molecular weight. However, concentration polarization can increase solute concentration at the membrane upstream surface to such an extent that the contribution from second and higher order virial coefficient terms to osmotic pressure could be quite significant. Goldsmith (9_) pointed out the importance of osmotic pressure but did not use an appropriate model to fit his data. Leung and Probstein (_5) used an integral method analysis to study the effect of osmotic pressure. Vilker et al. (10) and Trettin and Doshi (8) developed an asymptotic solution such that the osmotic pressure of the solution at the membrane surface approaches the applied pressure. Trettin and Doshi also proposed a way to distinguish between the osmotic pressure limited case and the gel polarized region. Recently, many researchers have observed that solute-membrane interaction could be very important in the ultrafiltration of macromolecular solutions. Solute could adsorb on the membrane and offer an additional hydraulic resistance for the solvent flow. In some cases, solute adsorption can be dominating so that diffusion can be neglected and filtration theory can be used to explain the data. For example, Dejmak (1^) has shown that his unstirred ultrafiltration data could, in a large number of cases, be explained by filtration theory. Ingham et al. (12) have convincingly demonstrated that albumin adsorption does reduce the solvent flux through the membrane. Fane et al. (13-14) studied the effect of pH on protein adsorption in ultrafiltration membranes and observed that the protein adsorption was maximum at the isoelectric point (pH = 5). Some interesting experiments were conducted by Reihanian et al. (15). They concluded that the filtration theory is adequate in interpreting their unstirred batch cell data. Howell et al. (16) studied flux decline due to protein adsorption and proposed a way to alleviate the problem. Many other researchers have observed the adsorption of albumin or other solutes on a membrane surface. The reader is referred to a comprehensive review by Matthiasson and Sivik (17). In this paper a generalized mathematical model is developed to account for osmotic pressure and solute adsorption. The model developed for an unstirred batch cell reduces to the osmotic pressure limited cases when adsorption is negligible and to the filtration equation when adsorption is dominating and diffusion is negligible. THEORETICAL DEVELOPMENT Mathematical analysis presented here follows closely that developed by Doshi and Trettin (18). Important differences between the two analyses will be noted as the model development advances. The unsteady state diffusion equation and boundary condition for the ultrafiltration of a macromolecular solution in an unstirred batch cell are:



16. DOSHI

Ultrafiltration of Macromolecular Solutions

211

The solute concentration at the interface of adsorbed layer is not known a priori but is assumed to be constant (boundary condition, Equation 3). For the adsorption dominated process, C w will approach the initial concentration C 0 and the concentration gradient will be negligible so that the diffusion boundary layer will not develop. In this case, solvent flux will be determined by the hydraulic resistance of the adsorbed layer (Figure la). On the other hand if adsorption is negligible, osmotic pressure will limit the solvent flux through the membrane (Figure lb). As shown by the analysis of Trettin and Doshi (8_) and Vilker et al. (10) osmotic pressure of the solution on the upstream side of the membrane will approach the applied pressure. In the general case considered here both adsorption and diffusion are important. The solute concentration in the liquid phase at the interface of the adsorbed layer, C w , will have an intermediate value between C w a (osmotic pressure limited case) and C 0 (adsorption limited case). The actual value of C w is not known, and whether it will be independent of time in the asymptotic case considered here is open to question. Also, in this model we will assume that, asymptotically, permeate rate and solute adsorption flux are inversely proportional to the square root of time. More work needs to be done to understand the adsorption process under ultrafiltration conditions. The present model may have to be modified as more information on adsorption kinetics and equilibrium becomes available. For now we will accept that C w is constant with upper and lower bounds of C w a and C Q , respectively. The coordinate y in the diffusion equation is measured from the adsorbed layer which may increase in thickness. Therefore, u and u w are related by the rate at which the thickness of the adsorbed layer increases:

where \\f is the permeation rate with respect to the stationary membrane. The thickness of the adsorbed layer is denoted by S. The net solute flux, N s , consists of two parts: Ns Total net solute flux

Nsa Adsorptive flux

+

Nsp Solute leakage through membrane

(6)

where

and Finally, the phenomenological solvent flux equation completes the problem statement:



212

Figure 1.


Schematic diagram showing causes of limiting flux in the ultrafiltration of macromolecular solution in an unstirred batch cell. N s p _ 0 assumed here but accounted for in the theory. (a) Adsorption dominating. Solvent flux is independent of solute diffusion coefficient and is determined primarily by the hydraulic resistance of the adsorbed layer. (b) Adsorption is negligible. Solvent flux depends on the diffusion boundary layer thickness. Osmotic pressure of the solution at the membrane approaches the applied pressure. (c) Both adsorption and diffusion are important. The value of C w is intermediate between C 0 and C w a .



16. DOSHI


213

where the term in the square brackets represents the hydraulic resistance of the adsorbed layer. For the large time asymptotic solution considered here, the resistance of the membrane is neglected. This is generally reasonable, since in most cases the solvent flux for macromolecular ultrafiltration is quite low compared to that for pure solvents. The following are the principal differences between the model presented here and the analysis of Doshi and Trettin (18): C w is assumed constant in both the models, but is equal to C g in Ref. ( H O , whereas its value lies between C 0 and C w a here; Solute permeation through the membrane, N s p , is considered here but was neglected by Doshi and Trettin; The osmotic pressure effect was not important in the problem solved by Doshi and Trettin. However, the osmotic pressure is considered in this paper, Equation 9. The large time asymptotic solution is obtained by a procedure similar to that used by Doshi and Trettin (18). The above equations are written in the dimensionless form as:

Equation 10 and the associated boundary conditions can be used to obtain the solution in the following form:

where


214


and

Let us consider some of the limiting forms of this solution.


Case I.

Adsorption dominating and diffusion negligible.

If all of the solute convected to the membrane is adsorbed on the membrane and/or permeates through the membrane, diffusion will be negligible, as C w approaches C 0 , i.e., 6 W + 0. Then from Equation 16:

or

which in essence is the dimensionless form of the filtration equation. In filtration, the osmotic pressure effect and solute leakage are usually not important, so that,

and

Case II.

Adsorption negligible, diffusion dominating.

If solute adsorption on the membrane surface is negligible, the flux will be limited by osmotic pressure.

Equation 16 simplifies to:

Equation 21 agrees with the solution for the osmotic pressure limiting case considered by Vilker et al. (10) and by Trettin and Doshi (8_). For a given applied pressure (APT and initial concentration, 0 W can be calculated from Equation 21a and dimensionless flux, V w can be predicted from Equation 21b. Results for bovine


16.

DOS HI


215

serum albumin are presented in Figure 2. Note that V w becomes relatively insensitive to AP for pressures above 600 kPa. For dilute solution and low pressure, V w increases linearly with pressure as in filtration. Case III. When osmotic pressure and solute leakage through the membrane are negligible,


Equations 16-18 reduce to corresponding equations obtained by Doshi and Trettin (18). RESULTS AND DISCUSSION The permeation rate of solvent in the ultrafiltration of macromolecular solutions in an unstirred batch cell can be predicted from Equations 16-18, provided the values of the following parameters are known: C 0 , AP, TT(C), D, e, u, a, p , D p , R, C w . For the case of the bovine serum albumin experiments of Vilker et al. (10), C 0 , AP, TT(C), U, and R are known with reasonable certainty. Dimensionless permeation velocity, V w , calculated from Equations 16-18 for different values of C w ranging from C 0 to C w a is plotted in Figure 3 for pH = 7.4 and in Figure 4 for pH = 4.7. Corresponding values of the diffusion coefficient are taken from Trettin and Doshi (7). Two different values of z and D p , and therefore four different values of hydraulic resistance, a, are considered in these diagrams. Reihanian et al. (15) proposed e = 0.51, p s = 1.2 g/mL, and D p = 62 A for bovine serum albumin. However, Vilker et al. (10) obtained p = 1.34 g/mL, which will give e = 0.63 if the adsorption layer is assumed to be a gel of concentration 0.59 g/mL. Thus, it is not possible to estimate exact values of p , e, and D« from the literature. Selected values, shown on the figures, are considered to represent the reasonable estimate of these parameters. Theoretical results in Figure 3 are obtained for C 0 = 0.072 g/mL and AP = 70 kPa. The corresponding osmotic equivalent concentration ^ s c wa = 0*243 g/mL. The value of V w at C w = C w a represents the osmotic pressure limiting case and is independent of the hydraulic resistance of the adsorbed layer. When C w = C 0 , V w can be predicted from the filtration equation and will therefore depend on the resistance of the adsorbed layer, a. In between these two extreme cases, both adsorption layer resistance and diffusion are important. Depending on the value of a, adsorption may increase or lower the permeation rate compared to that for the osmotic pressure limiting case. The solute adsorption affects permeation rate in two opposing directions. Adsorption can result in lower solute concentration in the liquid phase, C w < C w a , and as a result the driving force, AP-ATT, for the permeation velocity will increase. However, the hydraulic resistance of the adsorbed layer will lower the flux. If a membrane is covered with a protective porous membrane to keep a low even after solute adsorption on the protective layer, it is possible to increase flux compared to that for the osmotic pressure limiting case. This may also explain why addition of colloidal particles to macromolecular solutions will increase the flux (19). The



216


Figure 2.

Dimensionless permeation velocity as a function of applied pressure for different values of the initial concentration, C 0 . The solute considered is bovine serum albumin at pH 7.4.



16. DOSHI

Figure 3.


Dimensionless permeation velocity predicted from theory, Eq. (16)-(18) (solid lines). Vertical dotted line on the left at C w -C 0 corresponds to the filtration case (adsorption dominant), while the dotted line on the right at C w = C w a corresponds to the osmotic pressure limiting case. Experimental dimensionless permeation velocity (assuming D = 6.91 x 1(T 7 cm 2 /s) from the data of Vilker _et__aj_. (10) is also shown on this graph for bovine serum albumin at pH 7.4.


217

218


concept of protecting cover has also been advanced by Belfort et al.


Oo_, n_). In Figures 3 and 4, experimental values of V w reported by Vilker et al. (10) are also shown. It appears from these figures that adsorption of albumin on a cellulose acetate membrane was probably significant at pH 4.7 (close to the isoelectric point) rather than at pH 7.4. But Vilker's concentration profile data indicate that C w is close to C w a in both the cases. More experimental' data are needed to test the theory. Specifically, independent measurements of adsorbed layer porosity and resistance will be quite useful. Also one would like to know more about the nature of the adsorption process under ultrafiltration conditions and the equilibrium solute concentration in the liquid phase. Dimensionless permeation velocity, V w , is shown as a function of the applied pressure in Figure 5 for the following limiting cases: (a) Osmotic pressure limiting, C w = C w a (b) Adsorption limiting (filtration theory applies): two values of a are considered to get the estimate on lower and upper bound. (c) Gel format ion, C w — Co ( V W is independent of pressure in this case). These theoretical results are obtained for bovine serum albumin at pH = 7.4 and C 0 = 0.08 g/mL. Note that it is difficult to distinguish between the lower bound of filtration theory result and the osmotic pressure limiting case for operating pressure below 300 kPa. Hence at C 0 = 0.08 g/mL of bovine serum albumin, experiments in an unstirred batch cell should be conducted at pressures above 300 to 400 kPa so as to be able to identify the cause of limiting flux. For other values of C 0 , pressures above which experiments should be conducted can be represented by a straight line as shown in Figure 6. Experimental conditions should be in the upper portion of the line so that the difference between the filtration theory prediction and the osmotic pressure limiting case is greater than 25%. CONCLUSION A generalized theoretical model is developed to account for the adsorption of solute and diffusion of solute in the ultrafiltration of macromolecular solutions in an unstirred batch cell. The theory reduces to the osmotic pressure limiting case when adsorption is negligible and to the filtration equation when adsorption is dominating. In the intermediate case when both adsorption and diffusion are important, C w (solute concentration in the liquid at the interface of the adsorbed layer) and hydraulic resistance of the adsorbed layer will determine the value of the solvent flux. It is interesting to note that solute adsorption can result in higher flux compared to that for the osmotic pressure limiting case with no adsorption, provided the hydraulic resistance of the adsorbed layer is not too high. Finally, we have shown that experiments conducted at low AP and C 0 are not adequate to identify the cause or causes of limiting flux in the ultrafiltration of macromolecular solutions in an unstirred batch cell. For example, experiments with BSA at pH 7.4 should be conducted under the following guidelines:



16. DOS HI

Figure 4.


Dimensionless permeation velocity predicted from theory, Eq. (16)-(18) (solid lines). Vertical dotted line on the left at C w -C 0 corresponds to the filtration case (adsorption dominant), while the dotted line on the right at C w = C w a corresponds to the osmotic pressure limiting case. Experimental dimensionless permeation velocity (assuming D = 6.79 x 10" 7 cm 2 /s) from the data of Vilker _et_£JL. (10) is also shown on this graph for bovine serum albumin at pH 4.7.


219


220


Figure 5.

Limiting flux in the ultrafiltration of bovine serum albumin (pH 7.4) in an unstirred batch cell. Flux could be limited by adsorption (filtration equation applies) or osmotic pressure or gel formation. Initial concentration, C 0 = 0.08 g/cc.



16.

DOS HI

Figure 6.


221

Threshold value of o p e r a t i n g p r e s s u r e above which experiments should be conducted to d i s t i n g u i s h b e tween osmotic p r e s s u r e l i m i t e d and a d s o r p t i o n l i m i t e d cases.


222

REVERSE OSMOSIS AND ULTRAFILTRATION for 0.01 < C 0 < 0.15 g/mL, AP (kPa) > 655-3170 (Figure 6) and for C 0 > 0.15 g/mL, AP (kPa) > 10 TT0


Nomenclature C Cg C0 C« Cw Cwa

= = = = = =

D Dp K N s Nsa Nsp AP S S+ R t u

= = =

uw V Vw y

=

= = = = = = = = = = = =

solute concentration, g/mL gel concentration, g/mL initial solute concentration, g/mL solute concentration in permeate, g/mL solute concentration at y = o, g/mL asymptotic solute concentration at y = o so that its osmotic pressure equals the applied pressure, g/mL diffusion coefficient, cm^/s particle diameter, A or cm a.u.D.C0/AP, dimensionless N sa + N sp = n e t s °l u t e flux to the membrane, g/cm2 s adsorptive flux, g/cm2 s solute leakage through the membrane, g/cm2 s applied pressure drop, g/cm s 2 or kPa thickness of the adsorbed layer, cm S//4DF 1 - Cp/C w , membrane rejection constant time, s transverse velocity in the cell w.r.t. the coordinate y, cm/s permeate velocity w.r.t. stationary membrane, cm/s - u /4t/D - i^ A t / D transverse distance measured from the adsorbed layer, cm

Greek Letters a e n

= = =

specific resistance of the adsorbed layer, cm/g adsorbed layer porosity y//4Dt

e

=

(c-c0)/c0

8W TT

= =

(C w -C 0 )/C 0 osmotic pressure of the solution, g/cm s z or kPa

AlT

=

TTW -

TTW

=

TT

*p y • ps

= = =

ir (C p ) solvent viscosity, poise 1 - C 0 /[p s (l-e)] solute density, g/mL

(C

TTp w

)

Literature Cited 1.

2.

Blatt, W. F.; Dravid, A.; Michaels, A. S.; Nelsen, L. "Solute polarization and cake formation in membrane ultrafiltration: Causes, consequences, and control techniques" in Membrane Science and Technology, J. E. Flinn, ed., p. 47, Plenum Press, NY, 1970. Michaels, A. S., Chem. Eng. Prog. 1968, 64, 31.


16. DOSHI 3. 4. 5. 6. 7.


8.

9. 10. 11.

12.

13. 14. 15. 16.

17. 18. 19. 20. 21.


223

Shen, J. S.; Probstein, R. F., Ind. Eng. Chem. Fundam. 1977, 16 (4), 459. Probstein, R. F.; Shen, J. S.; Leung, W. F., Desalination 1978, 24, 1. Leung, W. F.; Probstein, R. F., Ind. Eng. Chem. Fundam., 1979, 18 (3), 274. Trettin, D. R.; Doshi, M. R., Chem. Eng. Commun., 1980, 4, 507. Trettin, D. R.; Doshi, M. R., Ind. Eng. Chem. Fundam., 1980, 19, 189. Trettin, D. R.; Doshi, M. R. "Pressure independent ultrafiltration - is it gel limited or osmotic pressure limited?" in Synthetic Membranes, A. F. Turbak (ed.), ACS Symp. Series No. 154, Vol. II, p. 373 (1981). Goldsmith, R. L., Ind. Eng. Chem. Fundam. 1971, 10, 113. Vilker, V. L.; Colton, C. K.; Smith, K. A., AIChE J. 1981, 27 (4) 637. Dejmek, P. "Concentration polarization in ultrafiltration of macromolecules." Ph.D. Thesis, Lund Institute of Technology, Lund, Sweden, 1975. Ingham, K. C.; Busby, T. F.; Sahlestrom, Y.; Castino, F. "Separation of macromolecules by ultrafiltration: Influence of protein adsorption, protein-protein interactions, and concentration polarization," in Ultrafiltration Membranes and Applications, A. R. Cooper, ed., p. 141, Plenum Press, NY, 1980. Fane, A. G.; Fell, C. J. D.; Suki, A., J_. Memb. Sci. 1983, 16, 195. Fane, A. G.; Fell, C. J. D.; Waters, A. G., J_. Memb. Sci. 1983, 16, 211. Reihanian, H.; Robertson, C. R.; Michaels, A. S., J. Memb. Sci. 1983, 16, 237. Howell, J.; Velicangil, O. "Protein ultrafiltration: Theory of membrane fouling and its treatment with immobilized proteases," in Ultrafiltration Membranes and Applications, A. R. Cooper, ed., p. 217, Plenum Press, NY, 1980. Matthiasson, E.; Sivik, B., Desalination 1980, 35, 59. Doshi, M. R.; Trettin, D. R., Ind. Eng. Chem. Fundam. 1981, 20, 221. Bixler, H. J.; Rappe, G. C., U.S. Patent 3 541 006, Nov., 1970. Belfort, G.; Marx, B., Desalination 1979, 28, 13. Green, G.; Belfort, G., Desalination 1980, 35, 129.

RECEIVED February 22, 1985


Limiting Flux in the Ultrafiltration of Macromolecular Solutions - ACS

Recommend Documents