The behavior of suspensions and macromolecular solutions in crossflow microfiltration

journal of MEMBRANE SCIENCE ELSEVIER

Journal of Membrane Science 96 (1994) 1-58

Review

The behavior of suspensions and macromolecular solutions in crossflow microfiltration Georges Belfort”T*, Robert H. Davisb, Andrew L. Zydney” “Howard P. Isermann Department ofChemical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, bDepartment of Chemical Engineering, b’niversity of Colorado at Boulder, Boulder, CO 80309-0424, USA ‘Department of ChemicalEngineering, Ciniuersity of Delaware, Newark, DE 19716-3110, CL!%

USA

Abstract

Although microfiltration is one of the oldest pressure-driven membrane processes, it is probably the least understood when it comes to the filtration of suspensions and macromolecules. Microfiltration is characterized by operation at low pressures, by high permeation fluxes, and by crossflow mode in flat or cylindrical geometries. The major limitation of microfiltration is membrane fouling due to the deposition and intrusion of macromolecules, colloids and particles onto and into the microporous membrane. In this review, we analyze the various components of this problem by focusing on the formation of cakes, the behavior of suspension flows and particle transport in simple geometry ducts, and on the formation and behavior of fouling layers including those resulting from macromolecules, colloids and particles. Some of the work we report on is very recent or is still in progress and needs independent verification. With this understanding, we hope that the reader will be able to use these concepts for analyzing other systems and for investigating new module designs. Kqvwords: Suspensions;

Macromolecular

solutions; Crossflow microfiltration

Contents I. Introduction ........................................................................................................................................................................... 1.1.Background ....................................................................................................................................................................... 1.2. Applications ...................................................................................................................................................................... 1.3. Microporous membranes .................................................................................................................................................. 1.4.Module design ................................................................................................................................................................... I .5. Solutions and suspensions ................................................................................................................................................ I .5.1.Solutions .................................................................................................................................................................... 1.5.2.Suspensions ............................................................................................................................................................... I .6. Mass transfer in membrane modules ................................................................................................................................ 1.6.1.Approach ................................................................................................................................................................... 1.6.2. Concentration polarization and fouling .................................................................................................................... 1.6.3. How to reduce concentration polarization and fouling ............................................................................................. *Corresponding

author.

0376-7388/94/$07.00

0 1994 Elsevier Science B.V. All rights reserved

XTDI0376-7388(94)00119-J

9 9 10 11

12 I .6.4. Cleaning ..................................................................................................................................................................... 13 1.7. FOCUSof this review ..........................................................................................................................................................

13 2. Cake formation ...................................................................................................................................................................... 14 2.1. Darcy’s law ........................................................................................................................................................................ 14 2.2. Cake resistance .................................................................................................................................................................. 15 2.3. Cake growth during dead-end filtration ............................................................................................................................ 16 3. Suspension flow and particle transport ................................................................................................................................. 17 3.1. Laminar flows ................................................................................................................................................................... 19 3.2. Concentration polarization and Brownian diffusion ........................................................................................................ 20 3.3. Shear-induced diffusion .................................................................................................................................................... 21 3.4. Inertial lift ......................................................................................................................................................................... 22 3.5. Flowing cakes and surface transport ................................................................................................................................. 24 3.6. Growth of thick cake layers for crossflow microfiltration ................................................................................................ 25 3.7. Comparison of theory and experiment ............................................................................................................................. 30 3.8. The solids flux model ........................................................................................................................................................ 32 3.9. Unsteady and secondary flows .......................................................................................................................................... 3.9.1. Roughness .................................................................................................................................................................. 34 36 3.9.2. Pulsation .................................................................................................................................................................... 36 3.9.3. Secondary flows (instabilities) ................................................................................................................................. 38 4. Protein fouling ....................................................................................................................................................................... 39 4.1. Protein-membrane interactions ....................................................................................................................................... 39 4.1.1. Protein adsorption ..................................................................................................................................................... 39 4.1.2. Protein deposition ..................................................................................................................................................... 41 4.2. Correlation of flux decline data ........................................................................................................................................ 44 4.3. Fouling mechanisms ......................................................................................................................................................... 45 4.4. Properties of protein deposits ........................................................................................................................................... 48 4.5. Summary of protein fouling .............................................................................................................................................. 49 5. Colloid and particle fouling ................................................................................................................................................... 49 5.1. Colloid stability ................................................................................................................................................................. 5.2. Flocculation: Browman (perikinetic) and velocity gradient (orthokinetic) ................................................................... 50 5.3. Particle capture by a membrane: deposition, intrusion (pore narrowing and constriction, pore plugging and deposition) 50 .............................................................................................................................................................................. 52 6. Summary ................................................................................................................................................................................ 53 Acknowledgements .................................................................................................................................................................... 53 References ..................................................................................................................................................................................

1. Introduction 1.1. Background Pressure-driven membrane processes are being increasingly integrated into existing reaction and recovery schemes for the production of valuable chemical and biological molecules. The properties of these membrane systems that are most often exploited are their operation without a phase change, without a temperature excursion from ambient, without the need for additives, and with relatively low energy consumption. Of these processes, reverse osmosis (RO) and ultrafiltration (UF) have been commercialized over the past 30 years. Microfiltration (MF), a much older process, was first commercialized by

the Sartorius Werke GmbH in Gottingen, Germany in 1929. The original impetus behind this development was the bacteriological assay resulting from the need to rapidly determine the safety of drinking water supplies in bombed-out German cities following World War II. The first patent for microporous membranes was issued to Zsigmondy in 1922 [ 13. As Lonsdale reports in his fascinating review [ 21, the first effort to develop microporous membranes in the United States occurred just after Zsigmondy’s work and was by Professor Goetz of the California Institute of Technology in Pasadena. With a grant from the U.S. Army Chemical Corp., he developed, characterized and tested such membranes for the detection of biological warfare agents. This know-how was then transferred to the Lov-

G. Bdforf

et al. / .Imrrnal of Memhrune

ell Chemical Co. in Watertown, MA which later became the Millipore Corp. Microfiltration is a pressure-driven membrane process in which suspended colloids and particles in the approximate size range 0.1 to 20 mm are retained by microporous membranes. Microfiltration is usually operated at relatively low transmembrane pressures ( < 50 psi or 3.4 bar or 0.35 MPa) and very high permeation fluxes ( 10p4-1 O-* m/s for unfouled membranes), and it is these factors which distinguish MF from both RO and UF. I. 2. Applications Because of their speed of analysis and ability to treat a much larger volume of water than other current methods, microporous membranes were first primarily used for bacteriological analysis of water. Although they are still used for this application, microfiltration membranes are also widely used for depth and surface filtration applications because of their microporosity and large internal surface area. These include bacterial, yeast, and mammalian cell harvesting; clarifrcation of air, other gases, antibiotics solutions, aqueous HPLC samples, water for the electronic industry, and viral containing solutions; and sterilization of additives, alcohol solutions, solutions containing antibiotics, valuable biological molecules and DNA, and tissue culture media [ 3 1. Intravenous and other pharmaceutical solutions are usually sterilized with microfiltration prior to bottle-filling and during administration. Microfiltration processing is also widely used in the food industry such as in wine, juice and beer clarification and in the automobile industry for filtering colloidal (latex) paints. Other applications include battery separators, blood oxygenators, oil-water separations, separation of colloidal oxides and hydroxides for metal recovery, wastewater treatment, and plasma separation from blood for therapeutic and commercial uses. I. 3. Microporous membranes Zsigmondy’s original microporous membrane was solvent cast from cellulose nitrate and cellu-

Scimce

96 (I 994) l-58

3

lose acetate. Depending on the application, newer polymers and inorganic materials with wider pH and temperature ranges are available. They include polypropylene, polyethylene, polycarbonate, ceramic, zirconium oxide, borosilicate glass, stainless steel, silver, poly (tetrafluoroethylene), pure cellulose acetate, poly (vinylidene fluoride ), regenerated cellulose, acrylic polymers, polyamide and polysulfone. Several of these materials are solvent cast (the last six mentioned), while the others are produced by a radiationtrack-etched process (polycarbonate), by hot and cold stretching and heat setting (polypropylene), by an electrochemical deposition process (some ceramic membranes are made this way ), by a thermal quenching procedure (polypropylene), by controlled stretching [ poly (tetrafluoroethylene) 1, by a sintering process (stainless steel, silver and ceramic) and by sol-gel processing (ceramic) [4a]. Typical materials used for microfiltration membranes and their properties are summarized in Table 1. Scanning electron micrographs of a selection of these microporous membranes are shown in Fig. 1. 1.4. Mod& design The main requirements of a membrane module or test cell is to separate the feed suspension from the permeate solution or suspension with a synthetic membrane and to maintain sufficient pressure drop across the membrane so as to obtain the desired permeation flux and selectivity. If the feed suspension flow is directed perpendicular to the membrane, usually with one inlet port, this type of flow pattern is termed “dead-ended flow” or “impact flow” and is most often used for small volume laboratory applications (less than a few liters). If, on the other hand, the feed suspension flow is directed parallel or tangential to the membrane with inlet and outlet ports, this type of flow pattern is termed “crossflow” or “tangential flow” and is most often used for large volume laboratory and process applications (greater than several liters), see Fig. 2. The main advantage of the dead-ended filtration mode is simplicity. The feed suspension is not recycled or passed across the membrane and

4

G. BeIfort et al. /Journal

Table 1 Typical materials used for microfiltration Material

Mixed cellulose nitrate and acetate Polypropylene Poly( tetrafluoroethylene) Bisphenol polycarbonate Borosilicate glass Silver Poly(vinylidene fluoride) (PVDF) Modified PVDF

membranes

of Membrane

Science 96 (1944) I-58

and their properties

[3 ] Max. temp.

Pore size (lum)

Thickness (pm )

Compatibility

0.025-8

150

aqueous

25-80 0.2-3

180-300 175

many organics many organics

100 130

aqueous and some organics aqueous and some organics many organics many organics

140

0.1-10

10

w-2.7

500

0.45, 0.8 0.22-0.45

50 125

0.1-5

125

(“Cl

aqueous and some organics

75

550

Protein binding cap.“. (mg/cm*) 150 high very high 3 60-l 10

200 120

NA. 150

85

4

aBovine serum albumin.

costly exit ports to accomplish this are unnecessary. Intense concentration polarization and membrane fouling can occur under these conditions. The permeation flux drags all solutes, suspended and dissolved, toward the membrane resulting in solute intrusion and adsorption into and/or deposition onto the microporous membrane. It was quickly realized in the 196Os, especially with RO, that moving the suspension tangential to the membrane surface resulted in much higher permeation fluxes [ 41. Reentrainment of polarized and deposited solutes by wall shear stresses have been used to explain this improved performance. Thus, crossj7owfiltration is widely used in nearly all commercial large-scale pressure-driven membrane plants. The most common designs include flat sheet, spiral wound, tubular (with internal diameters greater than 0.635 cm), capillary (with internal diameters between 0.1 and 0.635 cm), and hollow fiber (with internal diameters between 0.025 and 0.1 cm ). In designing the first generation of such modules, the membrane packing densities were maximized without much concern for optimizing the fluid mechanics and hence the permeation flux. Only after the realization that fluid mechanics is very important in maximizing the total capacity of a membrane module, was this

factor considered. Further discussion on this will be presented below in Section 1.6. Since the cost of membrane processing is directly related to the capacity of a particular module and hence its mass-transfer area and permeation flux, high membrane packing densities have been sought to minimize cost [ 5 1. Unfortunately, the situation is more complicated than just maximizing the surface area per unit volume of module (or element ). The main complicating factors have to do with (i) the build-up of dissolved solutes in the solution near the membrane surface (concentration polarization) and the deposition on or in the membrane of dissolved and/ or suspended solutes (fouling), both of which can effect the permeation flux and the retention of desirable and undesirable solutes, (ii) the effectiveness of the feed suspension fluid mechanics in depolarizing and defouling the membrane, (iii) the dependency of membrane and cake resistances to permeate flow on the transmembrane and transcake pressures, (iv) the dependency of the suspension viscosity, osmotic pressure, and solute diffusivity on suspended solids concentration, and (v) the ability of suspended particles to intrude into the membrane thereby decreasing the permeation flux. Many of these complications are negligible with relatively

G. Belfast et al. / .Journal

of.bfembrat~e

Science

96 (I 994) I-38

(b) Polyvinylidene

fluoride (hydrophobic)

(a) Mixed cellulose ester (nitrate+acetate) Hydrophilic

(C) Polytetrafluoroethylene (with polyethylene backing)

(e) Borosilicate glass

(d) Bisphenol polycarbonate

(f) pure silver Fig. 1. Scanning electron micrographs of a selection of microporous roughness (Courtesy of Dr. Yair Egozy, Millipore Corp., MA).

(g) polypropylene membranes

screen

showing different morphology

and surface

G. Belfort et al. / JournaL of Membrane

6

OR GEL) MEMBRANE

V

PERMEATE

a. Impact or Dead Ended Filtration

CENTRATE OUT

b. Cross-Flow

96 (1994) I-SS

up is possible, these designs are often quite expensive because of the need to compensate for their intermediate to very low packing densities. Flat sheet systems can be made to operate without separator screens (often erroneously called “turbulence promoters” and which could catch the suspended material and plug the flow path) and with short path lengths. This can also be an advantage since the mass-transfer boundary layer is renewed at each entrance. For very difficult suspensions containing macromolecules which tend to form compressible cakes, tubular and flat sheet MF designs are often used at very high crossflow velocities in the turbulent flow regime. Domestic wastewater and cell culture media are examples of this type of feed.

Cross Flow vs Dead Ended

CAKE

Scicncr

Filtration

Fig. 2. Schematic showing the feed flowing perpendicular and tangential in (a) dead-ended and (b) crossflow filtration. respectively

“clean” feed solutions, i.e., solutions without suspended material and membrane-adsorbing solutes such as macromolecules (proteins ). Unfortunately, this does not usually occur for MF, where very high permeation fluxes and complicated suspensions containing a distribution of particle sizes with dissolved macromolecules are exposed to the membrane. Because of this, the scale-up and the transfer of module designs from RO and UF to MF have been difficult and sometimes less than successful. A list of desirable module design characteristics is provided in Table 2 for various commercial designs. With suspensions, the tendency to clog hollow-fiber modules with feed flow on the outside of the hollow fibers (extra-capillary space) makes them essentially impractical. With spiral wound modules, the problem is less critical but often can also be debilitating. Flat sheet, tubular, and capillary designs are all preferred for crossflow filtration of suspensions. They have intermediate to very low tendency to clog and exhibit high permeation fluxes. Although scale-

1.5. Solutions and suspensions Although several MF separations are conducted in organic solvents, most applications are with aqueous suspensions containing dissolved solutes. Below we discuss the properties of aqueous solutions and suspensions with special reference to MF. Common examples include the variants of milk (whole, skim, whey), blood (whole, plasma), cell culture broths (bacterial, yeast, mammalian), paints (colloidal lattices) and effluents (domestic and industrial wastewater ) .

1.5.1. Solutions An aqueous feed solution, as defined here, is one in which the solutes are completely dissolved. Salts and small organic molecules such as sugars, small peptides, amino acids, urea, alcohols etc. are characterized by very high osmotic pressures and mutual diffusion coefficients. For an ideal (dilute) solution, the osmotic pressure (in kPa) can be estimated by the van? Hoff equation, II= vcRT/M

(1)

where Y is the number of ions formed if the solute dissociates, c is the solute concentration (kg/ m3),R=8.314x103kgm2/s2Kkgmol, risthe temperature (K), and M is the molecular weight (kg/kg mol ). For small molecules such as su-

G. Bc(fort rt al. /Journal of Membrane Science 96 (I 994) l-58 Table 2 Desired module characteristic9 Desired characteristics

Hollow fiberb

Spiral wound

Flat sheet

Tubular’

Capillary’

1. Membrane area 2. Wall shear rate 3. Scale-up 4. Permeation flux 5. Tendency to clogd

vh

i

i

vl

i

vl

i

i

vh

i

e 1

e i

id h

d h

d h

vh

h

i

vl

1

“Symbols difficult. bWith the ‘With the dClogging

in the table represent: vh, very high; h, high; i, intermediate;

vl, very low; 1, low; e, easy: id, intermediate

difficult; d,

suspension flow on the outside of the hollow fibers (with internal diameters between 0.025 and 0.1 cm ). suspension flow on the inside of the tubes and the capillaries. is defined as obstruction of the feed flow path.

crose (M= 342 kg/kg mol) at c= 50 kg/m3 and 20°C I7=0.356 MPa. For a large colloid (M= 1O7kg/kg mol) at the same concentration and temperature, II= 1.22 x 1O-” MPa. Clearly, the build-up of small solutes at the membranesolution interface cannot be tolerated for MF due to their very high I7 values relative to the operating pressure. Intermediate size solutes such as proteins can also exert relativley high osmotic pressures when present at large concentrations. The build-up of large colloids and particles, on the other hand, exert negligible osmotic effects. Crystallization and deposition of inorganic salts and precipitation and adsorption of (denatured) proteins and polyelectrolytes can occur inside a microporous membrane and can have a significant effect on reducing the permeation flux. Examples include the formation of a colloidal MgNH4P0, inorganic precipitate resulting from sterilization of a bacterial fermentation broth in microporous (0.8 mm nominal pore size ) stainless steel membranes [ 6 1, and aggregation and multilayer deposition of proteins such as bovine serum albumin (BSA ) from mammalian (hybridoma) cell culture medium inside a hydrophilized (sulfonated) polysulfone microporous (0.45 pm nominal pore size) membrane [ 7 1. The effects of protein deposition on both

permeation flux and macromolecular retention are discussed in more detail in Section 4. 1.5.2. Suspensions A colloidal suspension, as defined here, is one in which particles with a linear dimension of between 10e9 and 10 A6 m (nanometers to micrometers) are suspended in an aqueous fluid [ 81. Particles much larger than a micrometer in linear dimension are considered large particles with respect to MF. The range of particles of most interest for MF (0.1-20 pm) is summarized in Table 3 for several relevant industrial fluids. For colloidal particles where M is very large, II is negligible as compared to the transmembrane pressure for MF (dp= 100-300 kPa). The translational diffusion coefficient of suspended colloids and particles (in units m*/s) can be estimated at infinite dilution from the EinsteinSutherland equation, D, = kT/f

(2)

Boltzmann’s constant where k . ( = 1.380~ 10-2’sJ/mol K) andfis the frictional coefficient (kg/s mol). For hydrated flexible macromolecules, obtaining an estimate off is difficult (see Cantor and Schimmel [ lo] ). However, for rigid colloidal particles of dimension much larger than the hydrated layer, fcan

G. Bd$ort et al. /Journal

8

qfMembranr

Table 3 Solutes and colloids in industrial fluids most relevant to microfiltration Industrial fluid

Scirnw

96 jIW4)

I-58

[9]

Sizes of solutes, colloids and particles (m) IV9

IO-*

1V5

10-6

10p7

bm)

(nni)

microfiltration ~~~~ 1. Whole milk

lactose salts

2. Whole blood

salts

casein micelles

fat globules

proteins

crythrocytcs

amino acids peptides and proteins

3. Cell culture broth

>

_

platelets

bacteria

amino acids lipids glucose, vitamins, antibiotics salts

mammalian cells proteins

4. Paints

surfactants

organic macromolecules

5. EMuents

salts,

organic macromolecules

latices particles

humic and fulvic acids viruses

be obtained from the classical Stokes law and the Perrin shape factor, F, f= GnquF

(3)

and F=f/&

= ( 1 -p2)

1’2/ (p2’3H)

(4)

where for a prolate ellipsoid with p=b/c, H=ln([l+(l-g’)“‘]/p},andforanoblateellipsoidwithp=c/b,H=tan-‘[ (p’- l)‘/‘],and where c/b is the ratio of the long to the short semiaxes. F has been both plotted and tabulated as a function ofp by Cantor and Schimmel [ 101, For a sphere, F= 1. q is the viscosity (Pa s) and a is the radius (m > of an equivalent sphere with the same volume as the non-spherical solute. The volumes of the three shapes are (4/3 ) m3 (sphere), (4/3)&b (oblate), and (4/3)ncb2 (prolate). From Eq. (2 ) D, is proportional to T/f and from Eq. (3) f is proportional to q, therefore D, K T/q or WQ=W(Krl)

(5)

where D, and D, and q and rs are the solute diffusion coefficients and the viscosities of the fluid at the temperatures T and T, (standard), respectively. Using Eqs. (2) and (3 > for a suspended sphere in a fluid with the viscosity of water ( 1 cP) at 2O”C, D,=2.15~10-‘~/a (with D, in m’/s and a in m ). Hence for a micron size sphere in water, D, = 2.15 x lo- I3 m’/s, some three to four orders of magnitude lower than the diffusivity of glycine or sodium chloride! As will be seen below, this has a significant effect on the Brownian back-diffusion of colloids away from the membrane and on the potential of large colloids and particles to foul MF membranes. The presence of suspended colloids and particles will also affect the flow properties, causing the viscosity of a suspension (q) to differ from that of the pure fluid ( qO, water in our case). Hiemenz [8] gives the following expression for the viscosity of a suspension with a volume fraction of solids of 0, q/?j~~= 1+2.5@+k, where @co.40

@

(6)

and k, has a value of - 10 for

9

r//q0 = 1+2.5@+

k, 9?

(a)

(6)

where Q~0.40 and k, has a value of N 10 for spheres. As @+O, this result reduces to the Einstein linear equation for dilute suspensions (@
FEED SOLUTION/SUSPENSION 1.

t 2. MEMBRANE TYPE t

. MODULEDESIGN I

1

b. PERFORMANCEPREDICTION --

& DESIGN

I -----@I

4.

FLUIDMECHANICS Flow field-U, V, W

4A -

Solute mass balance

Continuity Eq.

(Navier-Stokes

Eq.)

1.6. Mass transfer in membrane modules 1.61. Approach The typical approach to analyzing mass transfer in membrane modules is to simulate the single phase flow field in the feed flow channel of the module of interest and to couple the solution of the flow field to the solute mass-transfer balance through the convective-diffusion equation. Often many simplifying assumptions have made such analyses impractical to use for design or prediction of performance. More sophisticated approaches using modem numerical techniques have yielded valuable insight. However, only very few people can follow these complicated analyses and even fewer companies have used them for actual design or on-line prediction of performance. The generic approach to mass-transfer analysis is summarized in Fig. 3. Thus, the continuity equation, the Navier-Stokes equation, and the initial and boundary conditions are solved for a prechosen channel geometry. See Kleinstreuer and Belfort for a detailed summary of the governing equations and various methods of solution [ 111. For example, Berman in 1953 [ 12 ] and Yuan and Finkelstein in 1956 [ 13 ]

Fig. 3. Generic approach to mass-transfer analyses. (a) Process of analysis, and (b) details of how the flow field and the solute mass balance are combined for a given geometry and used with the convective-diffusion equation to predict performance and obtain new designs.

used a perturbation analysis for 2-D flow in a porous slit and porous tube, respectively. To make the problem tractabIe, they needed to make the following assumptions: steady-state operation, incompressible fluid, laminar flow regime, fully developed parabolic flow, uniform permeation flux along the flow channel (path >, no external forces and constant physical properties (viscosity, diffusivity and density ), Many analyses have been reported in the literature for flow in porous tubes, slits and annuli. Attempts to relax the many restrictive assumptions listed above have also been made. Details have been summarized by Belfort [ 141.

10

G. Brlfrt

el d. i Journul uJ.blrrrh-ane

1.6.2. Concentration polarization and fouling Concentration polarization is the reversible build-up of dissolved or suspended solute in the solution phase near the membrane-solution interface due to a balance between the convective drag toward and through the membrane (resulting from the permeation flux) and back-transport away from the membrane. The presence of concentration polarization usually manifests itself in reduced permeation flux by increasing the osmotic pressure at the upstream face of the membrane and hence reducing the effective transmembrane pressure-driving force. This is most significant for salts and small organic molecules that exhibit large I7values in RO. See discussion above in Section 1.5.1. Solute and particle deposition onto and into the porous membrane is likely to occur when the permeation flux is very high relative to the various backtransport mechanisms. This scenario most often occurs in MF with colloids and particles which have low diffusion coefficients (see Section l-5.2). This phenomenon, together with macromolecular adsorption onto the membrane surface (both on the upstream face and inside the pores), is termed membrane fouling. Much of the experimental data in the literature has been expressed as permeation flux versus time showing the temporal decline of performance and as permeation flux versus logarithm of the bulk concentration (C in mol/l or @in volume fraction). The reason for the latter formulation will be discussed in detail in Section 3. From the experimental literature, several distinct periods of behavior have been identified. Using the schematic in Fig. 4, we will describe thephysiculphenomenon thought to be associated with each period [ 6 1. The concentration of the bulk solution increases with time as the fluid is removed through the membrane (e.g., for cell concentration ) . Period 1 - Fast internal sorption of macromolecules. During the very early phases of the run, the membrane is immediately exposed to the dissolved macromolecules in the culture medium. If the membrane chemistry is such that these dissolved macromolecules sorb onto the membrane

Science 96 (1994) 1-58

surface (most of which is within the membrane structure), then it is reasonable to assume that the permeation rate will decline. The kinetics of macromolecule sorption is thought to be fast and the binding constants high [ 15 1. When all the sorption sites are occupied, a pseudo-steady state is reached. Period 2 - Build-up of$rst sublayer. During this period, the suspended cells begin to deposit onto the membrane slowly increasing the sublayer coverage. Since for most of this period monolayer coverage has not yet been attained, there is little effect on the permeate rate. Isolated particles or clumps offer very little resistance to permeation flow. As monolayer coverage approaches, the permeation flux begins to decline toward that observed in period 3. Period 3 - Build-up of multisubZayers.During this period, the flux of solids towards the membrane remains relatively constant at a maximum. The product of the flux and the bulk concentration is a measure of the solid flux to the wall. The masstransfer coefficient for the permeating fluid is a constant for this period as can be seen by the constant negative slope of the curve in period 3 in Fig. 4. Several sublayers are built up, thereby, affecting both the crossflow and permeation velocities. The cross-sectional area for axial flow is reduced, increasing the wall shear rate and axial pressure gradient for a constant displacement pump. This results in increased back-migration of solids due to Brownian diffusion, shear-induced back-diffusion, inertial lift and/or surface transport as discussed in Section 3. Two competing effects, often difficult to disentangle, influence the permeation velocity. Increased transmembrane pressure provides additional driving force for an increased permeation velocity, but it also compresses the sublayers, thereby reducing the permeation velocity. Clearly, as the sublayers grow and become densified, the constant solids flux rate declines to that of period 4. Period 4 - Densification of sublayers. After the sublayer growth has stabilized, the permeation rate declines rather slowly since the mass-trans-

G. Belfort ef al. /Journal

-

,,,,

l

l

of Membrane Science 96 (1994) 1-58

11

-n//,,

-l-r.,

/ ,,,,,,,,,,

sorption of dissolved/ solutes InternoIly Initial deposat

0

-2 s multiloyer build-up and clogging

l

i Cl

i

i

c2

c3

Logarithm Fig. 4. Periods of different physical phenomena

sublayers reorrongement (compociion)

subloyer blinding -bulk viscosity increase l

(Concentration)

during flux decline with suspended and dissolved solutes in the feed [ 61.

fer coefficient for permeate flow is mainly affected by particle rearrangement rather than the deposition of additional solids within the sublayer. This densification of the sublayers continues while the bulk concentration increases rapidly until the viscosity of the bulk solution becomes sharply in period 5 - _ non-Newtonian

[W. Period 5 - Increase in bulk viscosity. As the concentration of particles in the bulk solution increases and approaches that of the sublayers, axial pumping and lateral permeation of the very viscous non-Newtonian solution becomes difficult. A precipitous drop in permeation velocity is observed. In addition to these fouling effects, there can also be changes in the membrane pore structure arising from physical compaction (and polymer plasticity) of the membrane or even chemical degradation of the polymeric material. These effects (on performance) are generally small compared to the fouling phenomena discussed above, and they are very specific to the particular mem-

brane under investigation; they will not be considered further in this review. In summary, several different phenomena are responsible for the decline in permeation rate during UF or MF of suspensions containing both dissolved macromolecules and suspended particles. These include macromolecular sorption (period 1 ), particle deposition (periods 2 and 3 ), sublayer rearrangement (period 4), and nonNewtonian viscous effects (period 5 ). A comprehensive quantitative description of these interacting phenomena is not yet available; however, attempts to model specific phenomena are reported in the literature. 1.63. How to reduce concentration polarization and fouling To counteract these phenomena, several practical approaches have been pursued. These include modifying the surface chemistry of the membrane so as to reduce the attractive forces or increase the repulsive forces between the solute and the membrane. Direct chemical techniques such as heterogeneous chemical modification [ 17 1, adsorption of hydrophilic polymers

12

G. Bclfort et al. /Journal

ofMembrane

[ 18 1, irradiation methods [ 19 ] and low temperature plasma activation [ 201 have been used commercially. Others have focused on increasing the back-transport of solutes away from the membrane. Blatt et al. [ 2 I] developed a theoretical model for the flux in crossflow filtration based on Brownian back-diffusion (referred to as the gel polarization model). It was soon realized that performance was better than predicted by this model for colloids and particles suggesting that back-migration was higher than expected. Green and Belfort [ 22 ] called this the “flux paradox” and offered inertial lift as an explanation. Belfort and coworkers then analyzed and developed design strategies to minimize particle capture through inertial lift [ 23-271. A few years later, Zydney and Colton [28 ] and Davis and coworkers [29-321 focused on another possible mechanism, shear-induced diffusion, as the reason for the additional back-migration away from the membrane. These models are discussed in more detail in Section 3. Other methods that help reduce concentration polarization and fouling include the use of various modes of crossflow MF. These include (i) diafiltration (the inlet water flow rate to the feed reservoir is set equal to the membrane permeation rate; this flushes low molecular weight components out of the system), (ii) batch (open and closed systems), (iii) feed and bleed (batch and continuous systems ), and (iv ) cascading/ staging. Typical methods to reduce concentration polarization and fouling are illustrated in Fig. 5. Chemical methods are described above while physical methods include the addition of seed particles that attract dissolved macromolecules and drag them away from the membrane and the use of electric fields to move charged molecules away from the membrane [ 33 1. Most of the recent developments in new module design have focused on improving the mass transfer at the membrane surface and the back-migration of retained species away from the membrane-solution interface. Thus membrane modules have been designed to include: turbulent flow, inserts to induce mixing, periodic reverse filtration of permeate back through the membrane (often re-

Scmce

96 (1994) I-58

REDUCE CONCENTRATION POLARIZATION

AND

FOULING

Fig. 5. Typical methods to reduce concentration and fouling.

polarization

ferred to as backflushing or backpulsing), and fluid instabilities to sweep the membrane. Rough surfaces, pulsation of the feed stream, and the production of centrifugal instabilities such as Taylor and Dean vortices have all been used to improve performance (see Winzeler and Belfort [ 341 for a review of these developments). 1.64. Cleaning Effective cleaning methods are crucial for maintaining adequate long-term operation of all pressure-driven membrane processes [ 3 5 1. Frequent manipulation of the operating variables to induce a short-term increase in flux has proved successful. Thus, periodic flow reversal of the feed stream is common practice. Other common techniques include a sudden increase in axial flow rate or a decrease in transmembrane pressure, a change in the feed temperature or composition, the use of cleaning solutions in place of the feed stream, and backflushing from the permeate side of the membrane with filtrate or cleaning solutions. Other have used sponge-ball cleaning for tubular systems, in which a sponge ball larger

G. Bclfort et al. /Journal

ofMembrane

than the tube diameter is forced through the feed flow channel thereby scouring any cake build-up on the membrane. This technique has been mostly used for RO. Using a stainless steel microporous membrane to filter a B. polymyxa fermentation broth, Nagata et. al. [ 6 ] developed a successful cleaning procedure using 1 N NaOH followed by 1 N HN03 for various time periods. Although deionized water was ineffective and long periods of NaOH was only partially effective in cleaning the membrane, short rinses with HN03 were very efficient. This is shown in Fig. 6.

Origmol 5-

0

4-

0 - 4 HI. 45Hr.

Permeoblltly NoOH

two,

0 - 0 H,. AI o15 g2

.

Science

96 (I 994) I-58

13

1.7. Focus of this review Because most process scale MF plants treat solutions and suspensions at low operating pressures, high permeation fluxes, and in crossflow mode in flat or cylindrical geometries, it is important to understand the behavior of these processes. As mentioned above, the major limitation of such processes is membrane fouling due to the deposition and intrusion of macromolecules, colloids and particles onto and into the microporous membrane. Hence, we review the various components of this problem by focusing on the formation of cakes, the behavior of suspension flows and particle transport in simple geometry ducts, and on the formation and behavior of fouling layers including those resulting from macromolecules, colloids and particles. With this understanding, we hope that the reader will be able to use these concepts for analyzing other systems and for investigating new module designs.

2. Cake formation

Pressure (kg/cm’)

(b)

.3-

Original

permeobillty _-

--b

_

.2 -

/ .I _ 1N NoCti

Jo/

1

) 1N HN03

//

I I

I

I

2

3

Time (Hr

, 4

1 5

1

Fig. 6. Recovery of permeation flux as an indication of different cleaning methods. (a) Flux versus transmembrane pressure. (b) Slope of lines in (a) versus time of cleaning. Experimental details are given in the text and in [ 61.

When a suspension contains particles which are too large to enter the membrane pores, then the surface filtration mechanism of sieving occurs. The retained particles accumulate on the membrane surface in a growing cake layer. The cake layer provides an additional increasing resistance to filtration, so that the permeate flux declines with time. For unstirred dead-end filtration, in which the fluid motion is perpendicular to the membrane surface, the cake continues to grow until the process is stopped. For crossflow filtration, however, the fluid motion tangential to the membrane may arrest the cake growth so that extended operation is possible. In this section, we consider cake formation and growth for dead-end filtration or for crossflow filtration under conditions where tangential flow does not affect the cake formation. In Section 3, we consider the arrest of cake growth and the corresponding long-term flux due to the tangential flow in crossflow MF.

14

G. Belfort et al. /Journal of Membrane Science 96 (I 994) 1-58

2. I. DarcyS law When the sieving mechanism of MF is dominant, a cake layer of rejected particles usually forms on the membrane surface, as shown in Fig. 7. The cake layer and membrane may be considered as two resistances in series, and the pressure-driven permeate flux is then described by Darcy’s law [ 36,371:

where J is the permeate flux, V is the total volume of permeate, A is the external membrane

surface area, t is the filtration time, Ap is the pressure drop imposed across the cake and membrane, qO is the viscosity of the permeate, R, is the membrane resistance, and R, is the cake resistance. The membrane resistance clearly depends on the membrane thickness, its nominal pore size, and various morphological features such as the tortuosity, porosity, and pore-size distribution. For a membrane whose pores consist of cylindrical capillaries of uniform radius perpendicular to the face of the membrane, the membrane resistance is obtained from Poiseuille flow as [ 7,361

I+-+& P

(8)

P

where ~1,is the number of pores per unit area, rp is the pore radius, and S, is the membrane thickness, indicating that the membrane resistance increases with increasing membrane thickness and decreases with increasing pore size and number density. Eq. ( 8 ) holds when the entrance and exit pressure drops are negligible and porosity is low. Batch conditions ensure the pressure at the pore mouth is equal to the bulk pressure far from the membrane surface. For a membrane with uniform cylindrical pores, the void fraction is t, = rz,nr,Z and the specific surface area of the pores is S,= 2~r,n,/ ( 1 -em); Eq. (8 ) then becomes R, =2( 1 --S,)2S,&JE;

(9)

For other membranes, this expression may still be used, but with the factor 2 replaced by a constant, K, which depends on the membrane morphology and pore structure [ 381. In practice, the membrane resistance is often determined from Eq. (7) using flux measurements obtained in the absence of a cake layer, and its value may increase with time due to fouling. 2.2. Cake resistance

Fig. 7. Schematics of dead-end MF using (a) a flat membrane with a cake forming on top, (b) a tubular membrane with a cake forming on the inside, and (c) a tubular membrane with a cake forming on the outside.

When a cake is incompressible, its porosity and resistance are independent of the imposed differential pressure. The specific cake resistance per unit thickness may then be estimated by the Carman-Kozeny equation [ 39 ] :

G. Belfort et al. / Jour-ml qfMer&me

R, =K’

( 1-

E,)2g/E,3

(10)

where E, is the void fraction of the cake and S, is the solids surface area per unit volume of solids in the cake. For rigid spherical particles of radius a, the specific surface area is S,= 3/a, the void fraction of a randomly packed cake is E, = 0.4, and the constant K’ is w 5.0 [ 401. It is often convenient to define K =RI(Ps$%)

(11)

where pS is the mass density of the solids comprising the cake, &= 1 -cc is the solids volume fraction of the cake, and RL is the cake resistance per unit mass deposited per unit surface area. Many cake materials, such as flocculated clays and microbial cells, are highly compressible, exhibiting a decrease in void volume and an increase in the specific resistance as the differential pressure is increased. The effects of cake compressibility are often estimated by assuming that the specific cake resistance is a power-law function of the imposed pressure drop [ 4 1,42 ] : R+x,(dp)”

(12)

where cy, is a constant related primarily to the size and shape of the particles forming the cake, and s is the cake compressibility which varies from zero for an incompressible cake to a value near unity for a highly compressible cake. These quantities are determined by measuring the specific cake resistance at various pressure drops and then plotting the logarithm of RL versus the logarithm of dp. For flat membranes (Fig. 7a), the cake resistance is proportional to the cake thickness, 8,: R, A’,&

(13)

In many MF applications, however, the membrane surfaces are cylindrical. Unless the cake is very thin relative to the radius of curvature of the membrane, Eq. ( 13) must then be modified to take into account the change in cake area with radial position due to the curvature. For filtration from inside a cylinder to outside (Fig. 7b), the cake resistance is [ 43 ] : R, =&Riln

[Ri/ (Ri -8,)

]

(14)

15

Science 96 (1994) 1-58

where Ri is the inside radius of the tube and the flux defined by Eq. ( 7 ) is based on the inside surface area of the tube (A=2@L, where L is the tube length). For filtration from outside a cylinder to inside (Fig. 7c), the cake resistance becomes &=&JnI

(&+W&l

(15)

where R, is the outside radius of the tube and the flux defined by Eq. (7) is based on the outside surface area of the tube (A = 2x&L). 2.3. Cake growth during dead-endJiltration In this subsection, we briefly review the theory for the transient cake build-up and the associated flux decline for a conventional dead-end microfilter employing a membrane which removes the suspended particles by the sieving mechanism of surface filtration. The analysis also applies for the initial cake build-up in a crossflow filter, prior to the action of the tangential flow causing the cake growth to be arrested [ 431. We consider a batch filtration process that starts with a clean membrane on which a cake layer of rejected particles accumulates with time as the filtration proceeds. The rate of cake growth during dead-end filtration using a membrane which completely rejects the particles forming the cake may be determined with the aid of a particle mass balance at the edge of the growing cake layer:

(16) where &, is the solids volume fraction in the suspension being filtered, and & is the solids volume fraction in the cake. Combining Eqs. (7 >, ( 13 ) , and ( 16) yields a first-order ordinary differential equation for the cake-layer thickness on a flat filter:

d& 4J -dt -(&-4) WP =

WC-4bmL

+&%>

(l’)

subject to the initial condition of S, = 0 at t = 0.

16

G. Belfort et al. / Jourrd

of Membrane

Dead-end batch filtration is often carried out with a constant imposed pressure drop. In this case, integration of Eq. ( 17 ) yields 2tl?,q&Ap 1+(4,_&)qoR$

l/2

)

-*

Sciencr 96 (I 994) l-58

:R,[R’

- (Ri -6c)2] +$cRi

1 (18)

In performing the integration, it is assumed that R, is constant (no significant membrane fouling or compaction over time) and that & and R, are constant (no significant changes in cake compression over time). By combining Eqs. ( 17 ) and ( 18 ), the flux expression is

=

WWI

A linear plot of At/V versus V/A then allows the specific cake resistance and the membrane resistance to be determined from the slope and intercept, respectively. Alternatively, if the membrane resistance is measured separately, then the specific cake resistance may be determined from the slope of a linear plot of (JJ J) 2- 1 versus t according to Eq. ( 19 ) . For a cylindrical filter with filtrate passing from the inside to the outside, Eq. ( 17 ) is modified to

Upon integration, this yields an algebraic equation for the cake thickness as a function of time:

APRiht (22 > %($c -9%)

Once the root of this equation is found, the timedependent permeate flux is determined by substituting the result in Eqs. (7) and ( 14). A similar result holds for filtrate passing from the outside to the inside of a tube:

;R,[

(R,+c~,)~-R:]

+;R,R,

2(R,

- (R, +d,)‘+R;

(19) where the initial flux is given by J,= Ap/qJ&,. The permeate flux starts at its initial value for a clean membrane and then decreases linearly with time for short times, J(t) -J,( 1 - [ t&&,Ap/ ((&-GhoR2m) I>, and inversely proportional to the square root of time for long times. Eq. ( 19) may be integrated with respect to time to yield

Rf - (Ri -dc)’

1

ApRo @I,t =%(Qc -4)

(23)

3. Suspension flow and particle transport During the past two decades, crossflow filtration has been increasingly used as an alternative to dead-end filtration. Crossflow MF is similar to that of UF and RO in that the bulk suspension is made to flow tangential to the surface of the membrane. Although this can be accomplished on a small scale using a batch stir cell, the common mode of operation is to pump the suspension through narrow tubes or flat channels having microporous membrane walls. As shown in Fig. 8, the permeate crossflow carries particles to the membrane surface, where they are rejected and form a cake layer which is analogous to the gel layer in UF. This layer initially grows with time, which reduces the permeate flux and constricts the channel. Unlike dead-end filtration, however, this cake layer does not build up indefinitely. Instead, the high shear exerted by the suspension flowing tangential to the membrane surface sweeps the particles toward the filter exit so that the cake layer remains relatively thin. This allows relatively high fluxes to be maintained over prolonged time periods Theoretical research has focused on various mechanisms by which the tangential shear ar-

G. Belfort et al. /Journal

of Membraw Scrmce 96 (1994) I-58

17

Permeate

e

Fig.

8.

Schematic of crossflow MF with cake formation

rests the cake growth, leading to different models for predicting the permeate flux. In this section, recent models which predict the steady-state permeate flux during crossflow membrane MF are reviewed and compared. The focus is on laminar flows with particle motion occurring due to Brownian diffusion, shear-induced diffusion, inertial lift, and surface transport. These models are then compared to available experimental data. The section concludes with a discussion of turbulent and other unsteady or secondary flows which may be employed to reduce cake build-up and flux decline. 3. I. Laminarflows The flow of Newtonian fluids in tubes and other closed channels with nonporous walls is usually laminar when the Reynolds number is less than -2x lo3 [44]. The Reynolds number is defined as Re=2pUH,/q, where p is the fluid density, q is the fluid viscosity, U is the average fluid velocity, and Ho is the channel half-height or tube radius. Typical values for crossflow microfilters are p= lo3 kg/m3, q= 10e3 Pa s, U= 1 m/s and Ho= 10m3 m [45], which yield Re=2 x 1O3 and indicate that the flow may be close to the transition between laminar and turbulent. The behavior of fluid flow in a porous channel with suction is different than that in a nonporous-walled channel. Mellis et al. [ 461 have measured transcartridge (axial) pressure drops (L&.) for water flowing in a porous MF tube under all regimes of flow as a function of wall suction and axial flow rates. At low axial flow rates (Ret l,OOO), low values of wall suction (Re,=pHJ/q<0.25) have a minimal effect on the axial pressure drop. At very high values of axial flow rate (Re> 20,000), all values of suc-

Concentrated

(from [ 50 ] ).

tion have a minimal effect on the pressure drop. Wall suction has its maximum effect on the axial pressure drop at intermediate axial flow rates ( 1,000 < Re-c 15,000). Moreover, wall suction is stabilizing, so that the value of Re, for transition to turbulence increases with Re,. It is in this range that most commercial membrane modules operate. A summary of these results is shown schematically for flow in a porous tube in Fig. 9, where it can be seen how wall flux has a feedback effect on itself through the axial presure drop and, hence, the transmembrane pressure drop. Dilute suspensions of noncolloidal particles exhibit Newtonian behavior, with an effective viscosity which increases with the particle concentration [47]. In the absence of particle-particle interactions, the seminal theory of Einstein [ 48 ] applies: (24) which is a truncated version of Eq. (6). For more concentrated suspensions, Leighton and Acrivos [ 49 ] correlated their measurements of the effective shear viscosity with Euler’s equation:

VW=%

(

;w

l+l_c,4 max

i

G-3)

where [r] is the intrinsic viscosity and &,,, is the maximum particle volume fraction. The bestlit vaIues at a low shear rate are [ TJ]= 3.0 and @,,, = 0.58, with a weak shear-thinning behavior observed [ 49 1, This equation predicts that q/ QY,=l-5,2.0,2.9, and 5.4 for#=0.2,0.3,0.4, and 0.5, respectively, and underpredicts Eq. (6) at low values of @

18

(4

I

s

lo4

lo3

Axial Reynolds Number, ReF

10 -

RCW

.

0.00

0

0 10

0

0.25

+

0.50 0.75 turbulent

I 00 I 2s

I I

regime

\ \

I

ReF

Fig. 9. (a) Schematic and (b) experimental measurements of the effect of axial (I&) and wall (RP~,) Reynolds numbers on a normalized transcartridge pressure drop (also called a friction factor). R stands for the volumetric recovery ratio. The membrane was a 0.5 pm porous stainless steel tube with 0.163 and 6.15 cm inner and outer diameter. respectively [ 461.

For fully-developed laminar flow of Newtonian suspensions, the parabolic Poiseuille velocity profile is described by:

u= Qn,x( 1 -Y21Hz,)

(26)

where y is the distance from the center of a tube of inside radius H,, or a slit of half-height HO. For

flow in a tube, the maximum velocity is twice the average velocity, U,,, = 2 U, whereas U,,, = 3 U/ 2 for a slit. An important quantity in the analysis of crossflow MF is the nominal shear rate at the membrane surface:

teracts the concentration cles near the membrane.

i, = 2 Kl,,XIK

When the particles being filtered are very small or highly compressible, then a thin fouling layer will quickly form (within a few minutes or less) on the membrane surface. This fouling layer will impart a substantial resistance to filtration, so that the permeate flux quickly reaches a steady or quasi-steady value that is significantly lower than the initial clean-membrane flux. The rate at which particles are carried to the membrane surface with the permeate flow is then balanced by back-transport of particles away from the membrane surface, and by the convection of these particles toward the filter exit by the suspension flow tangential to the membrane. It was originally thought that the analogy with UF of macromolecules would allow the traditional concentration-polarization model (often referred to as “film theory”) to predict the steady-state MF flux. In this model, the rejection of particles carried toward the membrane by the permeate flow gives rise to a thin fouling layer on the membrane surface, overlaid with a flowing concentration-polarization layer in which particles diffuse away from the membrane surface, as sketched in Fig. 10. At steady state, the convection of particles toward the membrane is balanced by diffusion away from the membrane and their convection toward the filter exit by tangential flow. Integration of the one-dimensional convective-diffusion equation across the polarization boundary yields [ 5 3 ]

(27)

The above analysis is strictly valid for laminar flow in nonporous tubes and slits. However, it also applies for crossflow filtration for the typical case where the permeate velocity is small compared to the average longitudinal velocity, J-K U, where J is the velocity normal to the membrane surface and is equal to the permeate flux. In this case, the transverse velocity profile is [50]: Slit:

Tube: v=2

y/H,, -;(y/HO)3 JT

I

(28b)

These expressions neglect the distortion of the flow profiles due to concentration polarization. In particular, the polarization layer near the membrane surface will likely have a high viscosity and so will flow with a reduced shear rate. For concentrated suspensions of noncolloidal particles in which the particle volume fraction exceeds -0.1, non-Newtonian effects such as normal stresses and shear thinning are important. The velocity profile for concentrated suspensions is blunt [ 5 I]. This is thought to be due to shear-induced migration of particles from regions of high shear near the channel wall to regions of low shear near the channel center [ 52 1. This migration occurs as a result of hydrodynamic interactions between the particles which cause a particle to be “kicked” down a gradient in shear rate due to a greater frequency of interactions experienced on its side with higher shear. A related phenomenon of shear-induced diffusion of particles down a concentration gradient is important in crossflow MF, because it coun-

polarization

of parti-

3.2. Concentration polarization and Brownian diffusion

J=~nbL/Qd

(29)

SUSPENSION Y 9 = +, ’ P4nMKoEB ’L E ‘MEMBRANE

Fig. 10. Schematic of the concentration-polarization ary layer for crossflow MF (from [ 43 ] ).

bound-

20

G. Belfort et al. / Jownal ofkfembranr

where J is the permeate flux, &, and @,,are the particle volume fraction at the edge of the cake layer and in the bulk suspension, respectively, and K is a mass-transfer coefficient, For laminar flow, the length-averaged mass-transfer coeffrcient is determined by the Leveque solution for thin boundary layers [ 5 3 ] : (30) where L is the tube or channel length and D is the particle diffusivity. Eq. ( 30 ) can also be developed from the complete Graetz solution in the limit of small axial positions. For submicron particles, Brownian diffusion is important, with the Brownian diffusivity, II&,, of an isolated spherical particle in a fluid of viscosity v0 given by Eqs. (2) and (3), the Stokes-Einstein relationship. Combining Eqs. (2 ) , (3 ), (29) and (30) for spherical particles (F= 1)) then yields the following expression for the length-averaged permeate flux:


(31)

Note that the predicted flux for the Brownian diffusion mechanism increases with shear rate or feed velocity to the one-third power, and decreases with particle radius to the two-thirds power. The particle volume fraction, #,, in the boundary layer immediately above the thin fouling layer on the membrane surface may be determined experimentally from a semi-log plot of flux versus bulk particle concentration. Alternatively, if the particles are nonadhesive, then &, will be equal to the maximum random packing density of particles in the adjacent cake layer, and it may then be estimated that &,,= 0.6 for rigid spherical particles of equal size and &Z 0.8-0.9 for compressible or polydisperse particles. The Leveque solution was derived for linear shear flows near nonporous walls and is strictly valid only when the permeate fiux becomes vanishingly small. For porous walls, the concentration dependence of the viscosity causes a nonlinear velocity profile in the concentrationpolarization layer, and the transverse compo-

Screrw 96 (I 994) I-58

nent of the velocity modifies the mass-transfer rate. Trettin and Doshi [ 541 used a similarity solution to derive numerical and asymptotic results which take the transverse result into account (but not the viscosity variation). Their solution asymptotes to that given by Eq. (3 1) for concentrated suspensions (&V-@b+%J, whereas for dilute solutions (&K $,) they showed that: (J) = 1.3 1 ( j.@;,&&,L)

“3

(32)

Wnfortunately, predicted fluxes for micronsized particles using the Brownian diffusivity given by the Stokes-Einstein relationship are found to be one or more orders of magnitude less than those observed in practice [ 2 I,5 3 1. As noted in Section 1.6.2, Green and Belfort [ 221 refer to this discrepancy as the “flux paradox for colloidal suspensions”. It follows from the fact that the Brownian diffusivities of micron-sized particles in water are on the order of 1O- I3 m*/s, which is three orders of magnitude lower than the molecular diffusivities of typical macromolecules (yielding lower predicted fluxes), whereas the membrane and cake permeabilities for MF are higher than the corresponding permeabilities for UF (yielding higher observed fluxes). Moreover, experimental evidence for MF of suspensions shows that the permeate flux increases with particle size and that it increases with shear rate to a power larger than one third [ 551; both of these findings contradict the predictions of the concentration-polarization model with Brownian diffusion. Several alternative mechanisms have been proposed to explain the flux paradox for colloidal suspensions, and the predictions of models based on the primary mechanisms of shear-induced diffusion, inertial lift, and surface transport are summarized in the following sections. 3.3. Shear-induced diffusion As a possible resolution to the flux paradox, Zydney and Colton [ 281 proposed that the concentration-polarization model could be applied to MF provided that the Brownian diffusivity was replaced by the shear-induced hydrodynamic

G. Be~ot-~ et al. /Journal

of Membrane

diffusivity first measured by Eckstein et al. [ 56 1. Shear-induced hydrodynamic diffusion of particles occurs because individual particles undergo random displacements from the streamlines in a shear flow as they interact with and tumble over other particles. Zydney and Colton [ 281 used an approximate relationship for the shear-induced diffusion coefficient measured by Eckstein et al. [56] for 0.2<&,<0.45 Ds =0.3j,a2

(33)

The shear-induced hydrodynamic diffusivity is proportional to the square of the particle size multiplied by the shear rate, whereas the Brownian diffusivity is independent of shear rate and inversely proportional to particle size. As a result, Brownian diffusion is important for submicron particles and low shear rates, whereas it is dominated by shear-induced hydrodynamic diffusion in typical crossflow MF applications involving micron-sized and larger particles. The shear-induced diffusion coefficient of a particle with a radius of one micron at a modest shear rateofj,=1000s-‘is3x10-7cm2/s,whichis more than two orders of magnitude greater than its Brownian diffusivity. Note that the steadystate permeate flux becomes proportional to the shear rate when Ds replaces DsO in Eqs. (3 1) and (32): (J) =0.078+,(a4/L)1/31n(&,/&) &V-$tj
(34) &, +=-K&

(35)

It also increases with particle size. Davis and Sherwood [ 30 ] have performed an exact similarity solution for the convective-diffusion equation governing the steady-state concentration-polarization boundary layer in crossflow MF of fine particles, under conditions where shear-induced diffusion is the dominant mechanism of particle back-transport. Their solution includes the concentration-dependent effective viscosity given by Eq. (25 ) and the shear-induced hydrodynamic diffusivity for sheared suspensions of spherical particles reported by Leighton and Acrivos [ 5738 1:

Science

21

96 (I 994) 1-58

D,=0.33j~‘$~(

1 +0.5e8,86)

(36)

where j=1;,~J~(@) is the local shear rate. The result is similar to Eqs. (34) and (35), except for a slightly different dependence on the particle volume fraction. For dilute suspensions (@,,< 0.1) of monodisperse rigid spheres which are nonadhesive and have a maximum random packing in the boundary layer of #Ww 0.6, Davis and Sherwood [ 301 found that: (J) = O.O6Oj, ( a4/4,L)

L/3

= O.O72j,, ( &,a4/@,,L) “3

(37)

which is identical to Eq. (35 ), except that the value of the leading coefficient is lower in Eq. (37 ). This difference is primarily because the concentration-dependent viscosity empIoyed by Davis and Sherwood [ 301 leads to a lower local shear rate and, hence, decreased shear-induced diffusion in the concentration-polarization boundary layer. As with the concentration-polarization model, the analysis of Davis and Sherwood [ 301 is restricted to apphcations where the cake layer remains thin compared to the channel half-height or tube radius and has a large resistance which dominates the membrane resistance. 3.4. Inertial lift Belfort and coworkers [ 22-271 have proposed that a possible resolution of the flux paradox is that the back-diffusion of particles away from the membrane is supplemented by a lateral migration of particles due to inertial lift. If the conditions are such that the inertial lift velocity is sufficient to offset the opposing permeate velocity, then the particles are not expected to be deposited on the membrane. Inertial lift arises from nonlinear interactions of a particle with the surrounding flow field under conditions where the Reynolds number based on the particle size is not negligible and so the nonlinear inertia terms in the Navier-Stokes equations play a role. The inertial lift velocity of spherical particles under laminar flow conditions in dilute suspensions, where particle-particle interactions are negligible, is of the form:

22

G. Bclfort et al. /Journal

of Meembraw

Scienw

96 (1994) I-58

3.5. Flowing cakes and surface transport

where p0 is the fluid density and b is a dimensionless function of the dimensionless distance from the wall. For Eq. (38) to hold a/ (2H,) << 1 and aU/v K 1 are necessary conditions. In the region near the wall, b is positive, indicating that the inertial lift velocity carries the particles away from the wall. Its maximum value near the wall under slow laminar flow conditions (channel Reynolds numbers small compared to unity) is b=1.6foraslit [59] andb=1.3 foratube [60]. Particle trajectories calculated from the resultant of axial flow and net lateral flow in the direction of the wall compare very well with measurements over nearly two orders of magnitude in wall flux [ 261. However, most crossflow filtration operations are carried out under fast laminar flow conditions (channel Reynolds numbers large compared to unity), for which Drew et al. [ 27 ] have recently shown that the maximum value is given by b=O.577. The inertial lift velocity for fast laminar flow increases with the cube of the particle size, and the square of the tangential shear rate, and so is expected to be significant for large particles and high flow rates. For the typical situation where the permeate flux for a clean membrane exceeds the inertial lift velocity, a concentrated layer of deposited particles forms on the membrane surface. If this fouling layer has a high resistance, then it will reduce the permeate flux until it just balances the inertial lift velocity. For fast laminar flow with thin fouling layers, the steady-state flux predicted by the inertial lift theory is then [27] J= v,,, = 0.036p,a3~~/q,

As an alternative to back-transport of particles away from the membrane by mechanisms such as diffusion and inertial lift, it is possible that the particles are carried to the membrane surface by the permeate flow and then roll or slide along the surface due to the tangential flow. Both continuum and single-particle models have been developed to describe this possibility. In the continuum approach, the rejected particles are assumed to form a flowing cake layer. Convective-flow mathematical models describe the simultaneous deposition of particles into the cake layer and the flow of this layer toward the filter exit [ 50&l 1. The fully-developed laminar flow equations are solved for the velocity profiles in the bulk suspension and in the cake layer, and the steady-state cake thickness and permeate flux are also determined. In general, the cake thickness increases and the permeate flux decreases with increasing distance from the filter entrance and with decreasing axial flow rate, but quantitative predictions to compare with experiments are difficult because the particle concentration, effective viscosity, and specific resistance of the flowing cake layer are not known a priori. In the single-particle models, the basic concept is to consider a spherical particle on the surface of the membrane, or on the surface of a stagnant cake layer, and perform force and torque balances on the particle to determine if it will adhere to the surface or be transported along the surface [ 62,63 1. Fig. 11 shows a spherical particle of radius a on the edge of a cake which has a

(39)

which is inversely proportional to the solution viscosity and independent of the filter length and the concentration of particles in the bulk suspension. For nondilute suspensions, however, it is expected that the inertial lift velocity would need to be modified to account for interactions among particles. For Eq. (39) to hold, a2Re<< 1, where a! = a/ (2H,) and Re= 2H, U/V, is a necessary condition.

Fig. I 1. Schematic of forces acting on a spherical particle on the surface of a cake layer.

G. Bevor& et al. /Journal qfhlemhrme

nonuniform surface morphology. The tangential drag force due to the shear flow may be expressed as: F, = 3n71a2j, Ct

(40)

where C, is a correction factor to the Stokes law due to the presence of the cake. For a sphere resting on a smooth impermeable surface, O’Neill [ 641 found that C,= 1.70, but the value is expected to differ for rough filter cakes. The normal drag force due to the permeate flow may be expressed as: F, = 6qaJC,

t41)

where J is the permeate velocity at the edge of the cake layer and C,, is a hydrodynamic correction factor to the Stokes law due to the presence of the cake layer. Lu and Ju [ 62 ] used a correction factor from Goren [ 65 1: C,= (2R,,,a/3+1.15)‘/’ where Rtot is the total resistance underlying cake and membrane: R,,, =AP/

(roJ)

(42) offered by the (43)

However, the analysis of Goren 165 ] is restricted to porous membranes which are thin compared to the sphere radius, or which only permit flow in the direction normal to the membrane surface. More recently, Sherwood [66] considered the drag force on a sphere touching the surface of an isotropic porous medium which allows flow in any direction and which is thick compared to the sphere radius. This situation is more representative of the present case of a particle at the surface of a cake layer, and the result is C,, =0.36(kp/a2)-‘/5

(44)

where &, is the permeability of the porous medium. If there is no cake, then the permeability is that of the membrane (k,=&,/R,), For the more common situation of a particle at the edge of the cake layer, the permeability is that of the cake (k,= l/&). Under the simplest situation in which other forces, such as gravitational, lift, and adhesive forces, are negligible, then the two components

Science 06 (I 904) l-58

23

of the drag force are balanced by contact forces at the locations where the test sphere is in contact with the cake surface. For the critical condition where the sphere is just able to roll along the surface, the contact forces are zero except at the pivot position (labeled A in Fig. 11). A torque balance about this pivot position is F,asird=F,acod

(45)

where 8 is the angle of repose (alternatively, a ( I- cos 0) is the height of a protuberance over which the sphere must roll). Combining Eqs. (40)- (43) for this critical condition yields the predicted permeate flux after the cake layer has reached its steady-state thickness: J= 2.4aj, ( a2&) 2/5cot0

(46)

where cot 0 depends on the surface morphology. This equation uses the hydrodynamic correction factors from [ 641 and [ 65 ] for smooth surfaces, and so the constant value of 2.4 is expected to differ for surface topologies which are not smooth on the particle length scale. The surface-transport model predicts that the long-term flux increases linearly with shear rate and particle radius (note from Eq. ( 10) that 2, is expected to be inversely proportional to a’ ) . If there is a distribution of particle sizes, then the smaller particles will continue to be captured after the permeate flux is reduced to a value where the larger particles are swept along the surface. As a result, the cake may become enriched in the smaller particles as it grows. Experimental measurements have observed that finer particles selectively deposit in the cake at later times and at higher crossflow velocity [ 67 ] . This causes the specific cake resistance to be crossflow dependent, and may even lead to decreased steady-state flux with increased shear rate [ 67-691. In addition, the cake surface is expected to be inhomogeneous, so that there is a distribution of protuberance heights, resulting in a distribution in 8 values. This leads to the possibility of a particle rolling along the cake surface until reaching an obstacle which is sufficiently high to stop it. This situation has been observed and analyzed by Stamatakis and Tien [ 631.

24

G. Belfort et al. /Journal

of Membrane Science 96 (1994) I-58

3.6. Growth of thick cake layers for crossflow microfiltration The flux predictions presented in the preceding subsections for the mechanisms of Brownian diffusion, shear-induced diffusion, inertial lift, and surface transport, are all based on the assumption that a thin cake layer with a high specific resistance dominates the membrane resistance and controls the flux. This is expected to be valid for a wide variety of suspensions which contain particles which are very small and which are compressible or have a size or shape distribution so that they pack very tightly in the cake layer. On the other hand, suspensions of larger particles which are nearly rigid and monodisperse may form thick cake layers with low specific resistances. To illustrate these situations, Ofsthun and Colton [70] used a freeze-substitution procedure to observe that compressible red blood cells form compact layers of only a few cells thick in hollow-fiber microfilters. In contrast, nearly incompressible yeast cells were observed to form cakes which nearly filled the tube cross section [ 7 11. In crossflow MF experiments using a rectangular filter with glass side walls and monodisperse suspensions of plastic beads of 50 and 80 pm diameter, Romero and Davis [ 72 ] observed cake layers as much as 0.5 cm thick. When suspensions which form thick cakes with low specific resistances are filtered, the approach to steady state may be relatively slow and take minutes or even hours [ 72 1. At the start of filtration, there is a relatively high flux resisted only by the membrane. The particles convected to the membrane surface by the permeate flow begin to form a cake layer on the membrane surface, unless the shear rate is sufficiently high to prevent cake formation. As the cake grows, it reduces the permeate flux. It also constricts the channel so that the shear rate at the cake surface increases because the axial flow must pass through a reduced cross section. These two effects cause a steady-state cake thickness to eventually be reached when the rate of particle convection to the cake surface is balanced by the shear-moderated back-transport or lateral transport of the particles away from the surface.

Transient models of cake growth during crossflow MF have been presented for shear-induced diffusion [ 32 1, inertial lift [ 22 1, and surface transport or capture [63,67]. Reviews of the mathematical developments and numerical results for the inertial lift and shear-induced diffusion mechanisms are also available [ 43,45 1. A key conclusion is that Eqs. ( 18 ) and ( 19 ) from the dead-end filtration theory provide very good approximations for the transient cake growth and flux decline under crossflow conditions, until the shear arrests the cake growth. As noted previously, however, the specific cake resistance (2,) may be higher under crossflow conditions than for dead-end filtration. Romero and Davis [ 3 1 ] developed a general model of crossflow MF to predict the steady-state cake thickness and permeate flux for the shearinduced diffusion mechanism. In dimensionless form, the results depend on two parameters [ 45 ] : P= awcn

(47)

L/G

(48)

=@&J~l(~~~~a4&)

where p is a dimensionless cake resistance and L/x,, is a dimensionless filter length, with x,, being the distance from the filter entrance where a cake first forms (see Fig. 10) +The dimensionless excess particle flux (&) is defined by Davis and Leighton [ 29 ] and depends only on the particle volume fraction in the bulk, 4, It is equal to the dimensionless rate at which particles are convected toward the filter exit in the polarization layer ($ > &, ), minus the corresponding rate in the absence of polarization (@= 4,). Its value may be determined from an integration across the boundary layer of the dimensionless shear viscosity and shear-induced diffusivity 129 1. For dilute monodisperse suspensions of nonadhesive rigid spheres, &.z 1 x 10e4 [ 29 1. Figs. 12 and 13 show the predicted cake thickness and flux at steady state. For small values of the dimensionless cake resistance, very little flux reduction occurs but the highly permeable cakes may be very thick. For large values of the dimensionless cake resistance, the cake layers remain thin and the flux is reduced substantially from the initial

G. Be&outet al. /Journal of Membrane Science 96 (I 994) I-58

25

0.8

0 12

5

102

10

IO3

Jo/\~L,o

103

10*

distance,

x/x,,

Fig. 12. Dimensionless cake thickness versus dimensionless distance from the filter entrance for the steady-state shearinduced diffusion model of crossflow MF: the solid lines are for flat membranes and the dashed lines are for tubular membranes (from [ 431). /3 is the dimensionless cake resistance.

0

-10

103

FltTER

106

LENGTH,

IO9

L/xcr

Fig. 13. Dimensionless average permeate flux versus dimensionless filter length for the steady-state shear-induced diffusion model of crossflow MF; the solid lines are for flat membranes and the dashed lines are for tubular membranes (from [ 43 ] ). p is the dimensionless cake resistance.

Fig. 14. Dimensionless average permeate flux versus the inverse of the dimensionless inertial lift velocity for the steadystate inertial lift model of crossflow MF, the solid lines are for flat membranes and the dashed lines are for tubular membranes (from [ 431 ). p is the dimensionless cake resistance.

value for clean membranes. Fig. 14 shows the results of a similar analysis for the inertial lift mechanism [ 45 1. For a cake composed of incompressible spheres of radius a, the specific cake resistance is given by Eq. (lo), with &=3/a, K=5, and eC=0.4, yielding I?, = 2 5 3 /a 2. For spheres of one micron diameter, the result is I?, z 10’ 5 m-‘. Since typical MF membrane resistances are on the order of R,= 10”-1012 m- ‘, the dimensionless cake resistance, /I, is - I- 10 for micron-sized rigid particles in a 1 mm channel. Thus, very thick cake layers are predicted. In this case, the steady-state flux increases with the shear rate more slowly than linearly for shear-induced diffusion, or more slowly than quadratically for inertial lift. For smaller and compressible particles, however, the values of p may be 100 or greater, so that relatively thin cake layers form. The parameter L/ x,, may take on a wide range of values, but it is typically several orders of magnitude greater than unity [45]. 3.7. Comparison of theory and experiment The relative magnitudes of the particle transport mechanisms of Brownian diffusion, shear-

induced diffusion, inertial lift, and surface transport depend strongly on the shear rate and particle size, and to a lesser extent on the bulk concentration of particles in the feed suspension. When the filtration resistance is controlled by a thin fouling layer, then the steady-state flux is independent of the transmembrane pressure drop and instead is governed by the transport mechanism (s) in the concentration-polarization layer. Table 4 compares the parameter dependence of the four transport mechanisms, It is assumed that the feed suspensions are dilute and composed of nonadhesive spherical particles which form cake layers that dominate the membrane resistance. In this case, Eq. (32 ) for Brownian diffusion, Eq. (37) for shear-induced diffusion, Eq. (39) for inertial lift, and Eq. (46) for surface transport all have the form: (J>=ci)~a’“@Lqq$

mechanism predicts a decrease in flux with increasing particle size, whereas the other three mechanisms predict an increase in flux with increasing particle size, Both diffusion mechanisms show a decrease in the length-averaged flux with increasing particle concentration and filter length, whereas the other two mechanisms are based on single-particle analyses that do not depend on particle concentration and filter length. Only Brownian diffusion and inertial lift depend on the fluid viscosity. Also shown in Table 4 are the predicted values of the steady-state fluxes from the four different mechanisms for particles of 1 and 50 pm radius under the typical conditions i)O= 1000 s- ‘, T= 293 K, qO=O.O1 g/cm s, p_,= 1.O g/cm3, @,=0.6, &,=O.Ol, L=lO cm, R,a2=253, cot I% 1.O and H, = 0.1 cm. Of the three back-transport mechanisms, Brownian diffusion is strongest for very small particles (less than a micron), shear-induced diffusion dominates for small particles (from a micron to - 30-40 pm), while inertial lift is most important for larger particles (greater than - 30-40 pm). The surface-trans-

(491

The values of the exponents, n, m, p, q and Y are listed in Table 4. All four mechanisms predict an increase in the long-term flux with increasing shear rate. The Brownian diffusion

Table 4 Parametric dependence of the long-term flux for various transport mechanismsa

Shear rate (j,, ) Particle size (a) Concentration

(&)

Filter length (L) Suspension viscosity (q,) Predicted flux (cm/s)’ fora=lpm fora=50pm

Brownian diffusionb

Shear-induced diffusion’

Inertial lift*

Surface transporV

increase n=0.33 decrease WI= -0.67 decrease p= -0.33 decrease q= -0.33 decrease r=-1

increase n=l increase m= 1.33 decrease p= -0.33 decrease q= -0.33 decrease r= -0.33

increase n=2 increase m= 3 no effect p=o no effect

increase n=l increase m=l no effect p=o no effect

q=o

q=o

decrease r= - 1

no effect r=o

6.3x 1O-5 4.6 x lO-+j

2.4 X 1V4 4.4x IO-2

4.5x lo-’ 5.6x lo-”

1.1 54.9

“According to (J) =~?,“a “@gL Y&. bFrom Eq. (32). ‘From Eq. (37). dFrom Eq. (39). ‘From Eq. (46). fFor F,,= IO3 s-‘, T=293 K. ~OZO.Ol g/cm s, S,= 1 g/cm3, &=0.6, erg/m01 K, H,=O. 1 cm.

@,,,=O.Ol,L= 10 cm, &a2=253,

cot 0= 1, k= 1.38X 1O-‘6

G. Beljort et al /.lournal(~fMembran~Scienc~

port mechanism predicts a much higher flux, one which is two orders of magnitude greater than typical clean-membrane fluxes, There are several possible explanations for why the surfacetransport model as presently employed greatly overpredicts the permeate flux. One is that the cake surface is likely to have a variable topology on the particle length scale, so that there are numerous pits or protuberances which a particle rolling along the surface can get caught in or behind. Mathematically, these would be described by values of cot f3which are much smaller than unity or even close to zero. For nonspherical particles, such as cells or clays, the particles would likely have their flat or long sides resting on the cake surface, which would increase the normal drag force holding the particles down while decreasing the tangential drag force which would cause the particles to move along the surface. Perhaps most significant is that many particles are adhesive and therefore would not easily move once in contact with the cake. Considering for the moment only the three back-transport mechanisms, inertial lift is the dominant mechanism for large particles and high shear rates, whereas Brownian diffusion is dominant for small particles and low shear rates. Shear-induced diffusion is important for intermediate particle sizes and shear rates. This is illustrated quantitatively in Fig, 15, where the steady-state MF flux of particles in water at 20°C versus particle diameter is plotted for a typical shear rate of j, =2,500 s- ’ for each of the three back-transport mechanisms acting independently. It is assumed that the feed suspension is dilute, and so the predicted steady-state fluxes are given by Eqs. (32), (37), and (39), respectively, for Brownian diffusion, shear-induced diffusion, and inertial lift. From Fig. 15, it is seen that Brownian diffusion is only important for particles smaller than about a few tenths of a micron in diameter, whereas inertial lift is important for particles larger than several tens of microns in diameter. The shear-induced diffusion mechanism is important for particles with diameters in the intermediate range of 0.5 pm < a< 30 pm, although this range will vary slightly with the system parameters.

27

96 (1994) I-58

SHEAR-INDUCED

2a (pm)

Fig. 15. Length-av-eraged permeate flux versus particle diameter for steady--state models of crossflow MF with thin cake layers according to three back-transport mechanisms; the conditions are @,=O.Ol. @,=0.6. T=293 K, qO=O.O1 g/cm s,p,=1.0g/cm3> j,=2,500sP’,andL=30cm (from [43]).

Early experimental work on crossflow MF of micron-sized particles in water is summarized by Blatt et al. [21], Henry [73], and Porter [53 1. These authors (and results from more recent work [ 741) made similar findings in that the steady-state permeate fluxes are -5X 10-4-5X10-3cm/s, . are one or two orders of magnitude larger than predicted by the film theory with the Brownian diffusion mechanism, . increase linearly with transmembrane pressure for small transmembrane pressures and then become pressure-independent, decrease with increasing concentration (and hence viscosity), although not always with the logarithmic dependence predicted by the film theory, . increase with increasing shear rate under laminar conditions with an exponent greater than the value of n=0.33 predicted by the Brownian diffusion mechanism. Table 5 summarizes measured exponents for the dependence of the steady-state flux on shear l

l

Table 5 Experimental measurements of the shear-rate dependence of the permeate flux for laminar flow Suspension

Styrene-butadiene polymers latex (5-50% solids by weight) Electrocoat paint ( 15%solids by weight) Whole plasma Whole blood Bacteria ( 1% solids by weight ) Colloidal impurities (5-lOhum) Yeast Yeast Bovine blood

Shear-rate dependence, n

Ref.

0.8-0.85

1531

1.33

[531

0.33 0.6 0.5-0.8

[531 [531 I731

0.49-0.86

L751

0.4-0.7 1.1 0.9

[761 1771 [781

into the mass balance (with back-transport to diffusion and inertial lift ): J(z)~=D(~)d~ldy'+v,,,~

vL,o =

Ww/4+~bL-4)

+p’~Q~-~~~2+~P3(~~-4~3+ 3.3!

2*2!

rate for colloidal suspensions filtered under laminar conditions. A broad range of exponents is observed (0.4 < rz< 1.33 ), with many of the values close to n = 1 or slightly less. These shear-rate dependencies provide support for the shear-induced diffusion and surface-transport mechanisms. An alternate explanation could be the one presented by Belfort et al [ 75 1, in which two back-transport mechanisms (e.g., Brownian diffusion and inertial lift, for which 0.33 < n < 2.0) operate simultaneously. Ignoring the shear-dependence of the diffusion coefficient, Nagata et al. [ 6 ] inserted the following functional relationship W#) =&exp(P@)

J-

... I

(52)

where K is the local mass-transfer coefficient between the particle suspension and the membrane or sublayer on the membrane. For constant diffusivities (&&K 1) and for the case where the particle drag to the membrane is much larger than the inertial lift away from the membrane (JB uL,+),Eq. (52) reduces to Eq. (29). This is an example of the combined effect of more than one back-transport mechanism operating simultaneously. Quantitative comparison between a modified polarization model based on shear-induced diffusion and the measured flux for plasmapheresis of whole blood was made by Zydney and Colton [ 791. The conventional polarization model was modified to account for the concentration-dependent blood viscosity. The results shown in Fig. 16 indicate very good agreement between theory and experiment, including verification of the dependence of the flux on shear rate, cell concentration, and filter length over a wide range of operating conditions. Close observation of the data, however, shows that each set of measurements

PARAMETER VARIED 0 n

Yo L

l

Ob

0 A m

r

(50)

m -

a

/

due l

(51)

where y’ is the distance away from the membrane. They then integrated from y’ =0 and @=&, to y’ =6 and $=Q& to obtain a concentration-dependent expression for the gel-polarization model [see Eq. (29 ) 1, viz.

/

I 0.001

PREDICTED

FILTRATE

FLUX,

Fig. 16. Comparison of predicted and measured flux for crossflow MF of blood using a modified polarization model based on shear-induced diffusion (from [ 79 ] ) .

has a shallower slope than the diagonal! This suggests that the mean of the data follows the trend but that the individual data do not. Zydney and Colton [ 28 ] also compared the modified concentration-polarization model with data from twelve different studies of crossflow MF of suspensions including blood, bacteria, latex, paint, platelets, and clay [ 28 1. Good agreement was found over wide ranges of concentrations, particle sizes, filter lengths, and shear rates, with the average error being - 40% with no adjustable parameters. In contrast, using the Brownian diffusivity in the polarization model predicted fluxes which are two orders of magnitude less than the data and which show a weaker dependence on shear rate, whereas the inertial lift mechanism has a stronger dependence on shear rate than was observed in these experimental studies. Lojkine et al. [ 55 ] recently provided a nice review of models and experimental results for crossflow MF and the effects of operating conditions, membrane materials and pore sizes, and suspension composition and concentration. Their data summary indicates that MF fluxes generally increase with increasing shear rate and particle size, decrease with increasing particle concentration, and may depend on particlemembrane interactions. They concluded that the shear-induced diffusion models appear to be the most accurate predictors, but that there are many complicating features that need further study. These features include the effects of particle adhesion, compressibility, shape, and size distribution on crossflow MF behavior. Further support of the shear-induced diffusion model has been provided for monodisperse latex beads and for yeast cells by Romero and Davis [ 72 ] and Redkar and Davis [ 80 1. They showed that the transient flux decline after the start of filtration is well described by the deadend filtration theory. The dimensionless cake resistances were found to be on the order of p= 10 in both cases, and so thick cake layers were predicted. The long-term fluxes for latex particles of radius 0.32, 0.47, and 0.70 pm agree with the complete shear-induced diffusion theory within an average error of 27%, whereas the Brownian

diffusion and inertial lift models under-predict the observed fluxes by one to two orders of magnitude. For unwashed yeast cells with an average radius of a= 2.1 pm suspended in water or fermentation media, the long-term fluxes were found to be 2-4 times lower than the predictions of the shear-induced diffusion model for nonadhesive rigid spheres. Redkar and Davis [80] attribute the discrepancy to extracellular proteins which cause cell adhesion and higher resistance in the cake layer, so that the cells at the top edge are not free to diffuse away. As further evidence of this, they found that increasing the shear rate after a steady state has been reached does not increase the flux. In contrast, washed yeast cells suspended in water in the absence of extracellular biopolymers exhibited specific cake resistances which are an order of magnitude lower than those for the unwashed yeast, and the steady-state fluxes for the washed yeast are only lo-30% below the predictions of the shear-induced diffusion model for nonadhesive rigid spheres. Because shear-induced diffusion models and surface-transport models have similar flux dependencies on shear rate and particle size, they may provide similar agreement with the data (provided that cot 0 and/or the leading constant in Eq. (46) is chosen to fit the data). Indeed, Stamatakis and Tien [ 63 ] and Mackley and Sherman [ 671 showed that their flux decline could be described by surface-transport models. Mackley and Sherman [ 67 ] observed particles rolling along the surface of the cake, with no evidence of back-diffusion. However, this is not surprising, as it may be shown from the analysis of Davis and Leighton [29] that the polarization boundary layer thickness for the relative large particle sizes (2~ = 150 pm ), high fluxes (Jz 0.3 cm/s), and low shear rates Ij, ~30 s-l) used by Mackley and Sherman [ 67 ] for optical observations is only a fraction of the diameter of a single particle! The surface-transport and the inertial lift mechanisms predict a length-independent flux, and there is some evidence for this [67,8 1 ] as well as counter-evidence which supports the L - li3 dependence of the diffusion models [ 79 1.

30

G. B~lfort et al. /Journal

of Membrane

Since most studies are carried out with a fixed apparatus, data on length dependence are limited. We note that Futselaar [ 821 has recently demonstrated that transverse-flow modules, in which the feed flow passes outside hollow-fiber membranes in a direction perpendicular to their axis, provide for significant flux enhancement. It is thought that this is related to a reduced cake layer of retained particles on the membrane surface because of the short membrane length (one half of the fiber circumference) in the direction of flow. Most of these studies have incorrectly evaluated the inertial lift model by using Eq. (38 ) (with b= 1.3-1.6) which unfortunately is limited to axial Reynolds numbers less than one [ 23,261. Clearly, the expression given by Eq. (39) (with b=0.58) is preferable since it covers all laminar flows below the onset of turbulence and predicts lift velocities above those due to Brownian and shear-induced diffusion for particles larger than -20-40 pm depending on the shear-rate conditions [ 27 1. The surface-transport and inertial lift models do not predict a dependence of the long-term flux on the particle concentration, whereas virtually all investigations have observed decreasing flux with increasing concentration. A need for future study is the incorporation of particle-particle interactions and concentration effects in the surface-transport and inertial lift models. As a crude first approximation, the concentration effects could be introduced into the inertial lift model by replacing the fluid viscosity by the suspension viscosity expressed by Eq. ( 6 ) , ( 24) or (25 ), depending on the suspension concentration range. Given the predictions in Table 4 and Fig. 15, and the very limited testing of these models in the literature, none of the back-transport or surface-transport mechanisms could at present be considered dominant for all MF applications. It is likely, moreover, that more than one of these mechanisms operate simultaneously in different ranges of particle size and physical parameters. 3.8. The solidsflux model [6/ For the mechanisms described above to effectively transport the suspended particles away

Science

96 (1994) 1-58

from the cake or membrane surface toward the bulk flowing solution, it has been assumed that the particles are freely suspended in solution and do not become immobilized on the membrane or cake. Once they leave the solution and attach to the solid substrate, they can only be transferred by surface transport via the moving cake or be reentrained into the suspension. Clearly, the intermolecular forces between the particles and between the particles and the membrane are critical in determining the stability of the cake. In some cases, it is conceivable that particle drag to the membrane followed by strong attachment (attractive) forces will result in cake thickening and ensuing flux decline. In this section, we consider the case for high permeability membranes, such as MF membranes, large channel widths, and relatively low axial velocities. Under these conditions, reentrainment due to Brownian diffusion, shear-induced diffusion and inertial lift is small for sticky particles and particle drag due to the high permeation flux dominates. Clearly, as particles arrive and build up at the membrane interface, they effect permeation by feedback inhibition. Expressing this feedback inhibition as first order by assuming that the rate of decline of permeation velocity with the concentration of suspension solids is proportional to the permeation velocity itself, i.e., $=-biJ

(53)

b

gives upon integration J=J,eXp

( -

bi@b)

(54)

where J, is the pure water permeation rate and b, is a constant describing the characteristics of the sublayer. In contrast to the usual graphical representation of Jversus In @; ’ from Eq. (29)) Eq. ( 54 ) suggests that In J versus &, gives a linear relationship. The slope gives a value for bi and the intercept at infinite dilution gives the pure water flux, J,. Using the same data for crossflow MF of a phosphate buffer solution containing 3.5 ml/l of 0.5 M HEPES buffer, we compare a plot for the gel-polarization model [ Eq. (29) ] with that of the solids flux model [ Eq. (54) ] in Fig. 17. At least the first three periods in Fig. 4 are

31

G. Beffort et al. /Journal of Membrane Scieme 46 (1994) I-58

a

AP - 12 kg/cm* Re -

40.000

Temp

-

a

30°C

Concenrrotlon

IO

100 SOLIDS

RelotweCone (%I

CONCENTRATION,

Cb

b

01

0

1

I

1

20

40

60

I 80

Solrds Concentration

I 100

I 120

I 140

160

WI

evident in Fig. 17a. The data are described well by the solids flux model in Fig. 17b. Also, the flux is not recovered by dilution (replacement of feed by water ) , illustrating irreversibility. Using a graphical method similar to settling tank design and operation [ 83 1, the solids flux approach for membrane fouling of non-back-migrating sticky particles can be obtained from Eq. ( 54 ) as follows,

-biq&,)

20

1

I

/

40

60

80

Sollds

Fig. 17. Permeation velocity (wall flux) versus relative concentration of phosphate buffer solution containing 3.5 ml/l of 0.5 M HEPES buffer according to (a) the gel-polarization model [Eq. (29) ] and (b) the solids flux model [Eq. (55)] [ 61. The measured concentration is normalized by the initial bulk concentration of the feed and is reported on the abscissa as a percentage.

S,=&J=&,J,exp(

0

(55)

where S, is defined as the flux of solids toward the membrane. A diagrammatic representation of Eq. (55) is depicted in Fig. 1Ba. From

I

I00

Concentrollon

,

I20

I

140

/

160

W)

Fig. 18. (a) Theoretical and (b) experimental verification of the solids flux model according to Eq. (56) using the same data as in Fig. 17 161.The measured concentration is normalized by the initial bulk concentration of the feed and is reported on the abscissa as a percentage in (b).

the turning point is at &, = b, ’ and at a solids flux J,/b,e where ez2.7 18. Differentiating Eq. (56), and substituting b; ’ for &, indicates that the coordinates (bi ’ ; J,/bie) are at a maximum at point D. Setting this second derivative to zero gives the coordinates for the inflection point (E) on the solid flux curve. These are (2/bi; 2J,/ bie2). If the permeation velocity J varies with axial length, X, then curves similar to those shown in Fig. 18 are needed as a function of path length. On either side and at the turning point D, the

32

(4

POROUS WALL

,

/

VORTEX FORMATION

(4

(d INSERTS7

1L

POROUS WALL

PARABOLIC

ki_ POROUS

FLOW 1

WALL WITH PULSATIONS

Fig. 19. Different methods for inducing flow instabilities: (a) placing objects (protuberances) onto the membrane surface to form a rough surface, (b) a furrowed channel with flow reversal, (c) placing objects into the flow channel away from the membrane surface, (d) flow oscillations in a smooth-walled duct, (e) centrifugal flow with Taylor vortices in an annulus with a rotating center tube/rod, and (f) centrifugal flow with Dean vortices in a spiral half-tube.

flow of solids to the surface is highest (i.e., region 3 in Fig. 4). Fig. 18a is useful because it explicitly gives the solids flux as a function of increasing feed concentration during the run. Using the data from Fig. 17, a solids flux plot is shown in Fig. 18b, where we notice the similarity with Fig. 18a. Since the solids flux model ignores back-diffusion of solids from the membrane to the bulk solution and assumes that particles stick to the sublayer upon contact, a plot of In J versus &, can be used to check whether Eq. ( 55 ) best de-

scribes the process or whether gel polarization with back-diffusion [ Eq. (32) or its variants such as Eq. (37) or (39) ] is more appropriate. Also, the possibility exists that different models operate at different periods of the process. 3.9. Unsteady and secondaryflows As mentioned

[34J

in the introduction

(Section

1.6.2), one of the main reasons for trying to un-

derstand the factors that limit performance of pressure-driven membrane processes (concen-

33

G. Bclforr et al. / Jourrzal of .bfernbrane Scierm 96 (1994) 1-58

Surface Roughness

a

Flux

(lOa

m/s)

. 1” r/m . 19““7

PROTUBERENCES

,

PERMEATE

b

VOR

UFR milmlrvm2

VORTEX

FORMATION

CORRUGATIONS (Furrowed surface)

Fig. 20. Improved permeation fluxes with (a) corrugations on the membrane surface and on the opposite face to the membrane in a slit channel of different channel heights [ 861. and (b) a furrowed channel with turbulent flow reversal (8 Hz) of whole blood (squares, 28% hematocrit and 83 g/l of total protein) and for saline (circles) with a UF membrane [ 851.

tration polarization and fouling) is to search for methods to alleviate them. Surface treatment has little effect on the behavior of suspended particles once a secondary cake has been established. In this case, fluid mechanical approaches such as steady and unsteady flows are paramount. Steady

flows often require crossflow velocities in the turbulent regime, while unsteady flows can be effective in both the laminar (TayJor and Dean vortex flows) and the turbulent regimes [ 841. Rough channels have been used to induce fluid mixing at the membrane-solution interface

G. BelJh et al. /Journal of Membrarre Science 96 (1994) I-58

34

Inserts

INSERTS1

kfi

POROUS

WALL

Reynolds

Number

(

!$

1

Pulsatile Flow

Steady llow. no mselt (Re 200)

Transmembrane

Fig. 2 1. Improved permeation pulsatile flow [ 99,102 1.

pressure bar

fluxes with objects (insert?) placed in the flow channel away from the membrane

[ 85,861. Reversal of axial crossflow and permeate flow have also been used to increase fluxes [ 87,881. Fluid instabilities due to flow in curved ducts (Taylor and Dean flows) have been used to disturb the flux-limiting effects of concentration polarization and fouling [ 34,89-931. See an earlier review on many of these topics [ 94 1. Although steady crossflow is effective in depolarizing dissolved and non-dissolved solutes and reentraining suspended particles from the viscous sublayer and secondary cake near the membrane [ 95-971, unsteady flow conditions can be even more effective in accomplishing these goals. Various approaches to inducing instabilities in bulk flow across a membrane surface exist [ 341. These include designing membrane surfaces with

surface with

organized roughness, pulsation of axial and lateral flow, and the use of curvilinear flow under conditions that promote instabilities or vortices, see Fig. 19. 3.9.1. Roughness Placing protuberances directly onto the membrane surface at defined separation distances induces periodic unsteady flows in the mass-transfer boundary layer. Instabilities are produced where they are most needed - at the solutionmembrane interface region - to depolarize the solute build-up [98 1. However, since the axial flow is smallest in this interfacial region (laminar sublayer), the intensity of the vortices is relatively low. To overcome this, three approaches

35

Annulus with Rotating Cylinder (Taylor Flow)

;~~-%__s

f ‘I ‘rn

t

a

i

L

rotating (3000 rpm)

02

0

04

06

08

1.0

1.2

1.4

APtm

Vortex Flow vs. Tangential Flow Filtration: E. Coli Harvesting

b

m TFF -Plate & Frame 110 lG0

0

TFF-Hollow Fiber

A

Pacesetter-VFF

90 F

ao

z‘r

70 60

5

50

zi ii

160 g/l dry cell weight

40 30 20 IO 0

0

25

50

75

100

125

150

175

200

225

250

275

300

325

350

375

400

Time (minutes)

Fig. 22. Improved permeation fluxes with centrifugal flow with Taylor vortices. (a ) Diafiltration of disrupted C. bondini cells for the recovery of formate dehydrogenase with 0.1-0.2 pm membranes at 3000 rpm versus a crossflow tube and hollow-fiber membrane modules [ 1041, and (b) Taylor vortices (VFF) versus a crossflow plate and frame (TFF) and a hollow fiber (TFF) for an E. co/i cell broth [ 1061,

have been taken: (i) extend the protuberances further into the faster flowing regions of the fluid to increase the intensity and size of the vortices, (ii ) replace the flat uniform membrane profile with a well-defined rough surface such as a furrowed profile [ 85 1, and (iii) placing protuberances at some defined distances away from the membrane surface with pulsatile bulk flow [ 991.

Unfortunately, each of these approaches have distinct disadvantages. In the case of (i), increased axial pressure drops are produced and diminished active surface area results. High frequency flow reversal is usually used in (ii). These techniques are difficult to scale up and also exhibit high axial pressure drops. As shown in Fig. 20, rough membrane surfaces do indeed give im-

36

G. Bdjbrt

et al. / Jowlal

ofMembrane

Science 96 (I 994) l-58

CONCENTRATE ( RETENTATE )

FILTRATE ( PERMEATE)

OLID STAINLESS STEEL TATING DISC

F =Feed C = Concenfrate

J

0

P = Permeate

Ultrafiltration

Fig. 23. A laboratory unit with a single membrane rotating disc (Pall Corp., NY ).

proved performance [85,86].

over

a smooth

of whole

milk

above a

surface

3.9.2. Pulsation Flow oscillations in smooth porous-walled ducts have been tested for RO of sucrose solutions in laminar and turbulent flow [ 100 1, and for UF and MF of whey and whole blood in laminar flows [ 1011. Illias and Govind [ 87 ] have solved the complete mass-transfer boundary value problem for oscillating flow of a Newtonian fluid in a tubular membrane using a finite difference method and they tested the model with the results of Kennedy et al. [ 1001. They conclude that the advantage of using pulsed axial flow supercedes the penalty of increased power consumption. Besides energy dissipation, flow reversal also results in reduced net crossflow and hence filtering capacity. As shown in Figs. 20b and 2 1, pulsatile flow can be used to good advantage in rough walled ducts and those with inserts [ 99,102 1. Additional discussion is given in [ 943. 3.9.3. Secondaryjhws (instabilities) Lopez-Leiva [ 901 and Lieberherr [ 9 1 ] separately suggested the use of Taylor vortex filtration to help depolarize the solute build-up on membranes lining the annulus of a rotating cylinder membrane device [ 103 1. Kroner and coworkers [ 104,105 ] and Rolchigo et al. [ 106 ]

1

2

3 Fk,

L

Cont.

5

6

7

I

Factor

Fig. 24. Results on filtering whole milk with an early multiple rotating disc filter module (RDF) and an early Taylor vortex filter (RS) [ 1081.

have extensively tested different devices based on this concept and have shown convincingly that both volumetric and solute permeation rates are higher with this device than with flat sheet or tubular crossflow devices (Fig. 22). The advantages of this type of device are excellent bulk fluid mixing, high wall shear rates, and weakly decoupled crossflow with transmembrane flux. The limitations are high energy consumption to rotate the equipment, difficulty to repair, possible sealing and membrane replacement difficulties and most importantly difficulties in scaling up the capacity of the modules. Backwashing the membrane is also not possible. Following the

(a)

o4

10

t

0

IO

20

30 Ttia,uu?

50

60

70

55

6.0

7.0

6.5

Whey Concenfration

7.5

8.0

85

(g/l OOml)

lb)

1

t "0

20

40

60

80

loo

Timc.mu Fig. 25. Improved permeation fluxes with centrifugal flow with Dean vortices for (a) 0.15 wt% suspension of I 1.9pm latex spheres and a 2.0pm membrane, and (b ) 0.30 wt% suspension of baker’s yeast and a 0.2 ,um membrane. Results are shown for flow in a slit with {circles) and without (squares) Dean vortices at 3.8 times the critical Dean number (modified Reynolds number) and @,=75.2 kPa. The diamond symbols are for deionized water [ 741.

early work of Murkes and Carlsson [ 107 1, several companies have recently commercialized a rotating disc system. By rotating a disc between two flat sheet membranes at high enough rotational speed, spiral vortices are formed above a critical rpm. In the early nineties, ABB Flootek, Malmo, Sweden, produced and sold a series of such filters for RO and UF. In the US, similar designs using rotating discs have been recently commercialized by Pall Corp., NY (Fig. 23 ) and by Membrex Inc., NJ. As early as 1980, LopezLeiva [ 1081 built a similar design and showed its excellent performance (Fig. 24). Another system uses a torsional spring to continually rotate back and forth a multiple flat sheet module less

10

100

Time (min) Fig. 26. Improved permeation fluxes comparing centrifugal flow with Dean vortices (circles) for a spiral half-tube with crossflow filtration (squares). (a) Dairy whey, (b) baker’s yeast with a 2.0 pm polypropylene membrane [ 34,l 12 1.

than 360” in a rocking motion at frequencies of 60 Hz, and this device has successfully filtered difficult solutions and suspensions [ 109 1. In order to overcome the limitations associated with rotating Taylor vortex filter devices, Winzeler [ 921 and Belfort and coworkers [ 74,93,110,1 111 have separately suggested using Dean vortices rather than Taylor vortices. In fact, Winzeler and Schomberg [ 1121 have designed a laboratory membrane module based on a spiral half tube to utilize these vortices above a

38

G. Brlfort PIal. /Journal of h’errtbr-ant Science 96 (I 994) I-58

flat membrane, while Brewster et al. [ 113 ] have designed a nested spiral wound module based on continuously maintaining the streamwise flow above the critical Dean number. When a fluid is forced to flow in a curved channel at a modified Reynolds number called a Dean number above the critical value for incipient production of vortices, the flow changes from laminar flow to unstable laminar flow [ 114,115 3. Vortices are produced that twist and spiral in the streamwise direction. It is these vortices that can be used to depolarize the solute build-up near the membrane-solution interface. Besides having advantages of the rotating filter, variants of this type of design are easy to scale up and can be equipped with any type of membrane. In other words, it could be as efficient as the Taylor vortex filter in depolarizing solutes from the membrane-solution interface, and have none of its disadvantages. Dean vortices in the half spiral tube are visualized using optical and magnetic resonance imaging methods. With respect to Dean vortices, Chung et al. [ 1 10,l 1 1 ] have used optical and magnetic resonance imaging to confirm the existence of the instabilities. Also, they have compared filtration performance with and without Dean vortex instabilities for well-defined colloids and baker’s yeast broth with a MF membrane (Fig. 25) [ 741 and dairy whey with an UF membrane (Fig. 26 ) [ 112 1. In summary, secondary flows or flow instabilities in general are very effective in depolarizing solute build-up at the membrane-solution interface, resulting in significantly improved performance for pressure-driven membrane filtration. Although we have classified methods to induce unsteady flows or instabilities into roughness, pulsation and secondary flows (instabilities), several recent publications report on the simultaneous use of more than one of these methods [ 89,99,110,116]. The results reviewed above show that unsteady flow in the presence of vortices gives better performance in terms of permeation flux than steady flow across a membrane. Thus, factors of improvement for the cases mentioned above were as follows: (i) with roughness and surface protuberances ( x 2.5), with inserts without pulsation ( x 7.5 ) , and with

inserts and pulsation ( x 3.3 ), (ii) with roughness and flow reversal ( x 9)) (iii) with a smooth channel with pulsations ( x 1.6 ), and (iv) with centri&guE instabilities with annular (Taylor) flow ( x 3.5-7.8)) or with annular disc impeller ( x 8 ), or with curved channel (Dean) flow (X6).

4. Protein fouling MF is used extensively in the purification of a variety of bioprocess streams that contain large amounts of proteins, e.g., the separation of plasma proteins from blood cells, the sterile filtration of therapeutic proteins prior to final formulation, and the harvesting of bacterial, yeast, or mammalian cells from protein-containing culture media. As discussed in the previous sections, most theoretical and experimental studies of the filtrate flux during MF have focused on the effects of particles or cells. There is, however, considerable experimental evidence that the proteins present in these bioprocess streams can play a critical role in MF, even though the pore sizes of these MF membranes are generally over an order of magnitude larger than the characteristic size of the proteins. For example, Ofsthun et al. [ 1171 obtained data for the filtrate flux as a function of time during the crossflow filtration of 0.65% yeast cells suspended in 6% bovine serum albumin (BSA) through polypropylene, polyamide, and modified polysulfone hollow-fiber MF membranes. In each case, the flux declined dramatically, attaining a steady-state value after -40 min of filtration that was more than an order of magnitude less than the initial flux. Photomicrographs of rapidly frozen fibers showed that significant cakes of yeast cells were formed on the membrane surface during these filtration experiments. However, the flux decline during the yeast cell filtration through the polypropylene membrane was essentially the same as that obtained during filtration of a cell-free BSA solution under identical conditions, suggesting that much of the flux decline observed during the filtration was in fact due to some type of protein-membrane interaction as opposed to any

effect associated with the yeast cells. A significant flux decline was also observed duri ~ltratio~ through the polyamide and m polysu~fo~~ mernbr~n~s~ although the flux decline through these membranes for the BSA alone was much less pronounced than that obtained with the yeast cell suspensions.

The flux decline c,aused by the proteins present in these bioprocess streams can be attributed to one or more of the following phenomena: (a) protein adsorption, which involves a specific interaction between the proteins and the membrane polymer that occ in the rrbsence of any convective flow throu the mernbr~~e, (b) protein deposition, which refers to any additional protein that becomes associated with the on that is over and be a~sorbcd to the -~o~~~g > ~~~~~~~and limitations (often referred to as concentration polarization or boundary layer effects), which refers to the ac-

f the ~ite~~t~re on protein adsorption a tion and their effects on IX’ and MF. ITIaccumulation at the membrane surface can reduce the filtrate flux by increasing the hydraulic resistance to flow or by reducing the thermodynamic driving force (due to the osmotic pressure ofthe retained proteins ). Both of these effects should be negligible for the filtrati f (cell-free) protein solutions through clean membranes since these large pore membranes are essentially nonretentive proteins. However, these portant far ““fouled”’ retention by the cell cake [ 1191 or protein deposit [ 120,12 f ] that can form on the membrane surface during filtration of these protein-cell rn~xt~r~~. 4.1. I. Alt multilayer protein adsorption does occur on some polymeric and inorganic materials, most quantitative studies of protein ad-

sorption on MF membranes (in the absence of Row} have found approximately monolayer adout the i~tcr~a~ pore area of

aluminum oxide [ 122 ], and 0.2 jlrn alumina membranes [ 123 1, hemoglobin adsorption on 0.2 pm alumina membranes [ 1241 1yeast alcohol d~hydrogenase (YADH f ads~~~t~o~ on alumianes [ 125 ] ) and ~~lactaglob~num oxide me nylon, cellulose acetate, and vinylidene fluoride ) (PVDF) membranes [ 126 1, Protein adsorption dots appear to he reduced on more hydrophiIic membranes, with only a fraction of a monolayer ad[122] and s~~~t~o~ for I35 [ I 26 ] on hydrophilic PVDF me dition, protein adsorption can be limited by steric interactiuns in very narrow pores, but this effect t unless the memdoes not appear to brane pores are nn ger than the char~~~er~st~csize of the proteins [ 1271. This type of monolayer protein adsorption will generally have only a small effect on the flux durMF. This can be most easily seen by modelthe membrane as an array ~~par~~~~~unifo ~dr~cal pores, with the flux given by Eqs. ( and ( 8 > with R,= 0. For a typical MF membrane (with rp = 0.1-0.5 jlrn >, the flux reduction caused by monolayer adsorption of a 60 A protein (typical of BSA ) is unly 2-l 2%. L d~et~ons in flux would be cxpec adsorption of very large macromolecules, although these lar proteins may be present in sufficiently dilute concentrations that equilibrium adsorption remains well below a monolayer.

The flux declare during process titration can uch more dramatic than that caused by simple adsorption. Typical experimental data for the flux decline during the stirred cell filtration of 5 g/L BSA so~~t~o~s through ~o~ycarbonate, ES), and ~~~~~t~t~~~~roethyle~e ) (lp E) rnemb~~cs at a constant pressure of 35 kPa (5 psi) are shown in Fig. 27. In each case, the membranes were first preadsorbed with BSA so that the flux decline seen

G. Belfart et al. /Journal of Membrane Sritwre 96 (1994) l-58

40

-

5 s/L 35 kPa pH 7.0

loo

. 0

’ 2

’

’ 4

’

’ 6

.

’ 6

’

’ 10

’ 12

Time, t (s x 10m3)

Fig. 27. Flux decline during the constant pressure stirred cell filtration of 5 g/l solutions of BS.4 through Nuclepore polycarbonate, PES, and PTFE membranes at 35 kPa and pH 7.0 [128].

in Fig. 27 is due to protein deposition; protein concentration polarization (boundary layer accumulation) should be minimal at least over the initial stages of filtration due to the absence of any protein retention at short times [ 128 1. The flux through the PES and PTFE membranes were nearly identical over the entire time course of these experiments, with the flux declining by almost two orders of magnitude after less than 2 h of filtration. A much smaller flux decline was seen with the Nuclepore polycarbonate membrane, reflecting the much lower initial permeability for this track-etched membrane (which is associated with the smaller porosity and larger effective thickness of the Nuclepore membrane). However, the steady-state fluxes through these three membranes are nearly identical, suggesting that the long-time behavior may be only weakly influenced by the physical or chemical characteristics of the membrane under these conditions. Large flux declines have also been observed for BSA filtration through 0.05 and 0.2 pm polycarbonate membranes [ 1291 and through 0.22 pm aluminum oxide membranes [ 122 1. Hlavacek and Bouchet [ 1301 have reported over an order of magnitude increase in transmembrane pressure required to maintain a constant filtrate flux during the dead-end filtration of BSA through

polycarbonate and cellulosic, PVDF, membranes. Although much of the previous experimental work on protein deposition has examined the behavior of BSA, this type of dramatic flux decline has also been seen with a range of other proteins [ 1281. For example, data for the stirred cell filtration of 5 g/l solutions of BSA, immunoglobulin G, hemoglobin, ribonuclease A, and lysozyme in 0.15 M NaCl at pH 7 are shown in Fig. 28. In each case, the flux through the 0.16 pm PES membranes declined by well over an order of magnitude within the first I h of filtration. The steady-state flux for the three proteins that were filtered at a pH near their isoelectric points (hemoglobin, ribonuclease, and immunoglobulin G) are nearly identical and significantly below those for the positively charged lysozyme and the negatively charged BSA. Bansal et al. [ 1241 found that the flux reduction during the MF of hemoglobin solutions is greatest at the protein isoelectric point, with similar results obtained by Palecek and Zydney [ 128 ] for the stirred cell filtration of BSA. In addition, Palecek et al. [ 13 1 ] showed that the steady-state flux for BSA filtration increased monotonically as the salt concentration was reduced; the origin of these effects is discussed in more detail subsequently.

Protein

PI 4.7 6.6 7.1

w Albumin 0 lmmunoglobulins b Hemoglobin

loo 0

2

4

6

8

10

12

Time, t (s x 10M3)

Fig. 28. Flux decline during the constant pressure stirred cell filtration of 5 g/l solutions of albumin, immunoglobulins, hemoglobin, ribonuclease A, and lysozyme through 0. I6 ,um PES membranes at 69 kPa and pH 7.0 [ 1B].

G. Be&w et al. /Journnl ofMembrane

The flux decline during crossflow MF of cellfree protein solutions can also be dramatic. For example, Cbandavarkar [ 1321 and McGettigan et al. [ 1331 reported a sharp decline in flux during crossflow MF of BSA solutions, with the steady-state flux again being almost two orders of magnitude less than the initial flux [ 133 ]. Bansal et al. [ 1241 found a large flux decline during the crossflow filtration of hemoglobin, with a minimum in the flux observed at the protein isoelectric point. Belfort et al. [ 71 observed a significant reduction in flux during the filtration of cell culture media ( 10% fetal calf serum) through 0.45 pm hydrophilic polysulfone membranes in a rotating annular filter. Scanning electron micrographs of MF membranes obtained after protein filtration clearly show that a protein deposit forms on the upper surface of, and in some cases penetrating into, the porous structure of the membrane [ 129,134]. These protein layers are generally on the order of a micron in thickness [ 120,130], although Glover and Brooker [ 135 ] obtained scanning electron micrographs of whey deposits on UF membranes that were as much as 30 pm thick. 4.2. Correlation ofjlux decline data Previous models for the flux decline that occurs during protein MF have generally been based on one or more of the following phenomena: (a) pore blockage, (b) pore constriction, and (c) cake formation. These three cases are shown schematically in Fig. 29 for a membrane with a typical pore size distribution [ 71. Consider a protein (or particle) with diameter d and a pore of diameter dp. When d<< dp, the protein can enter most pores, deposit on the pore walls, and thus reduce the effective pore radius as well as the cross-sectional area available for flow. This is shown schematically as a loss of pores from the pore size distribution and a reduction in the slope of the flux versus transmembrane pressure plot for the solution as compared to the pure water flux. When d z d,,, pore plugging or blockage becomes more significant. When d> d,, the particles are unable to enter most pores and hence a deposit or cake forms on the upper surface of the

Scienm 96 0

41

994) l-58

membrane. The effective pore size distribution for this cake will likely change with filtration time and transmembrane pressure due to particle compaction, particle rearrangement, and/or deposition of smaller particles in the pores of the cake. In this case, the slope of the flux versus transmembrane pressure curve will decrease with increasing pressure due to the increase in cake resistance at high pressure. The mathematical analysis of the flux decline caused by pore blockage, pore constriction, or cake formation is generally performed assuming that the membrane can be described by a single pore size, with the details of these analyses discussed by Hermia [ 1361. The starting point in the development of these models is Darcy’s law which can be written as: J_

AP-a,An

-Jonl

(57)

+m

where R, is the resistance of the protein deposit (cake) and q, and An are the osmotic (Staverman) reflection coefficient and the osmotic pressure difference across the membrane, respectively. The osmotic reflection coefficient is a measure of the leakiness of the membrane to the protein of interest; it varies from one for a fully retentive membrane to zero for a nonretentive membrane. The osmotic reflection coefficient for an unfouled MF membrane is essentially zero due to the very small ratio of the protein to pore size, thus the osmotic pressure term in Eq. (57) is negligible during the initial stages of protein filtration. This term may become important for a fouled membrane, although it has generally been neglected in previous models for the flux decline during protein MF. Chandavarkar [ 132 ] described the early stages of BSA fouling during MF with a pore blockage model in which the number of blocked pores was assumed to be proportional to the volume of solution filtered through the membrane. The filtrate flux was evaluated as a function of time as: J=J,,exp(

- ublotJoAl)

where the constant ablockiS

(58) equal

to the number

42

G. lh(fort

f f (11. /Journal

of Memhratw

Scienw

FOULING SCHEMATICS

DlSTRtl3UTlON

# CASE A:

d i< d,

CASE B:

1PORE NARROWlNGlCONSTRlCTlON

Adsorption

d

CASE c: d >> d,

1

pure water

In E

solution

“w

pore size

r

pore size

r

AP

~”

[PORE PLUGGING] d4

d - d,

96 (I 994) I-58

t-

d’~

#

Blockage

[ GEUCAKE

LAYER FORMATION

PW

LA- E

s

“w

pore size

AP

r

Deposition

&

VW

pore size

r

PW

I

S

: I

1

AD ’ independent of pressure

Fig. 29. Fouling schematics for: Case A - pore narrowing and constriction.

and gel/cake layer formation

Case B - pore plugging, and Case C - solute deposition

[ 71.

of pores that are blocked per unit filtrate volume and A is the area of the upper surface of the membrane. Hlavacek and Bouchet [ 1301 found that a modified form of the pore blockage model, referred to as the intermediate blocking law [ 1361, was in good agreement with their data for the increase in pressure required to maintain a constant flux during dead-end filtration of BSA solutions through four different types of 0.2 pm MF membranes. Bowen and Gan used the pore constriction model, also referred to as the standard blocking model [ 136 1, to describe the flux decline during stirred cell filtration of BSA [ 122 ] and YADH [ 1251 through 0.2 pm aluminum oxide membranes. The reduction in pore radius was assumed to be proportional to the filtrate volume, with the flux given as:

Jo

J= I

+

apore Jo t

2

>

(59)

n, 71Yp2&

where aaorCis the volume of protein that deposits on the interior pore walls per volume of filtrate. Bowen and Gan [ 1221 found that aporeincreased with increasing bulk protein concentration and with increasing applied pressure. The pore constriction model has also been used by Tracey and Davis [ 129 ] in their analysis of the initial stages of BSA filtration through 0.2 pm polycarbonate membranes, although they indicate that their flux decline data could also be effectively described by the pore blockage model. Belfort et al. [ 7 ] used a pore constriction model to describe the reduction in flux during filtration of 10% fetal bovine serum through 0.45 pm hydrophilic polysulfone membranes in an annular rotating filter. They extended the pore constriction model

G. Be@& et al. /Journal

of Memhranr

to include a log-normal distribution of cylindrical pores with mean pore radius r* and variance o2 [ 137 1, with the filtration velocity evaluated by direct integration of the Poiseuille flow through each pore over the pore size distribution. The results for the flux reduction suggested that there were -21 layers of protein in the membrane pores. The cake filtration model has been most widely used to describe the flux decline during protein filtration, with the proteins assumed to deposit in a “cake” on the upper surface of the membrane. The hydraulic resistance of this cake layer is assumed to be proportional to the mass of the deposit anagolous to the approach used to develop Eq. ( 19). In the classical cake filtration model, the mass of the deposit is assumed to be proportional to the amount of protein convected towards the membrane yielding: J=Jo

l

-“’ +2adepositR~GAAPt r10Rl12

(60)

where C,, is the mass concentration of protein in the bulk solution and R L is the specific resistance of the protein cake (or deposit). Eq. (60) is equivalent to Eq. ( 19) except for the parameter adepositwhich is the fraction of protein convected towards the membrane that actually adds to the deposit. The cake filtration model predicts that a plot of total resistance (R,,, = dP/Ar,1,.7) versus filtration time is concave down, in contrast to the internal fouling models (pore blockage and pore constriction) which predict that the total resistance versus time curve is concave up [ 1291. Tracey and Davis [ 1291 used this difference to study the mechanism of BSA fouling during deadend filtration through 0.05 and 0.2 pm track-etch polycarbonate membranes. Their data indicate that the initial flux decline is generally associated with internal fouling (either pore blockage or pore constriction) while the long-term flux decline was due to cake formation, with the transition between these two fouling mechanisms a function of the membrane pore size and bulk protein concentration. Chandavarkar [ 1321 successfully used a cake filtration model to de-

Scirnw

96 (I 994) l-58

43

scribe the long-time flux decline observed during crossflow MF of BSA. Howell and Velicangil [ 138 ] used a cake filtration model to describe the flux decline during BSA ultrafiltration through fully retentive membranes, but the cake growth was assumed to occur via a polymerization reaction at the membrane surface of the form

(61) where md is the cake mass, and C, is the protein concentration immediately adjacent to the membrane surface. The order of the polymerization reaction (n) was evaluated by comparison of the model calculations with their experimental data, with the best fit found using n=2. This type of model has apparently not been applied to protein MF. Suki et al. [ 139 ] used a cake filtration model to describe protein deposition during BSA UF with the growth of the cake layer evaluated using an empirical expression of the form

df”b

-=K”(m,*-m,) dt

(62)

where m z is the maximum value of the cake mass at steady-state and K” is an empirical constant describing the rate of cake formation. Unlike the other filtration models [e.g., Eqs. (58)-(61) 1, this model predicts that the flux attains a steadystate value at long filtration times, which is in agreement with much of the available experimental data for both protein UF and MF. Suki et al. [ 1401 subsequently developed a more mechanistic model for the long-term flux decline based on the aggregation of proteins due to intermolecular interactions occurring at or near the membrane surface. However, neither of these models has yet been used to analyze the flux decline during protein MF. Although these mathematical models can provide useful descriptions of the decline in flux observed during protein MF, it is not currently possible to obtain a priori estimates of the key parameters in these filtration equations (e.g., ablock, apore, or adeposit). In addition, these models

44

G. Belfort et al. /Journal

ofMembrane

do not provide a quantitative basis for understanding the effects of solution properties or membrane characteristics on the flux decline. These models also provide little insight into the actual physical and/or chemical mechanisms by which the proteins become associated with the membrane during these filtration processes. 4.3. Fouling mechanisms Most of the recent fundamental work on the initial stages of protein fouling during membrane MF have focused on the effects of denatured and/or aggregated protein. These protein aggregates can be present in the bulk protein solutions (generated during the initial processing of the proteins) or they may be formed by the pumping of the protein through the MF system or by the high local shear rates that can exist near the membrane surface or in the membrane pores. Kelly et al. [ 1411 have shown that different commercial preparations of BSA can have dramatically different fouling characteristics, with the rate of flux decline for these different preparations, all of which were reported to be greater than 96% pure BSA, differing by well over an order of magnitude. These differences in protein fouling appeared to be due to the differences in protein denaturation associated with the different techniques used to separate the albumin from bovine serum or to further process the protein prior to lyophilization, with the results being very well correlated with the concentration of higher molecular weight BSA oligomers present in these solutions. Chandavarkar [ 1321 used quasi-elastic light scattering to show that pumping of BSA solutions during MF resulted in the formation of large protein aggregates, with the initial fouling behavior during MF caused by pore blockage associated with the deposition of these aggregates on the membrane surface. Chandavarkar suggested that this protein aggregation was caused by both shear and interfacial denaturation of the BSA molecules during pumping, with these denatured proteins aggregating through strong intermolecular interactions. Similar effects were observed by Meireles et al. [ 1421, with the rate and extent

Science 96 (1994) 1-58

of protein denaturation, and in turn aggregation and fouling, increasing with increasing temperature and crossflow velocity. Kim et al. [ 143 ] used scanning electron microscopy to examine the structure of the albumin deposits formed during stirred cell MF (in the absence of pumping). The protein deposits were composed of large BSA aggregates, which were assumed to form from the conformational changes in the BSA molecules associated with the high shear rates that exist near the membrane surface. Kim et al. hypothesized that these conformational changes exposed hydrophobic regions on the BSA molecules which then interacted to form large protein aggregates at the membrane surface. Bowen and Gan [ 122 ] hypothesized that it was the high shear rates in the membrane pores that caused a shear-induced denaturation with subsequent deposition of these denatured proteins on the interior of the pore walls. Kelly et al. [ 1411 showed that BSA fouling could be dramatically reduced by prefiltration of the protein solutions through smaller molecular weight cut-off membranes, with this prefiltration step removing any large protein aggregates that were present in the bulk solutions. Kelly et al. proposed that protein fouling during MF is a two-step process: the initial flux decline is due to the deposition of BSA aggregates present in the bulk protein solution either on or in the membrane pores, with these aggregated proteins then serving as nucleation or initiation sites for the subsequent deposition of nonaggregated BSA. Recent work by Kelly and Zydney [ 134 ] has demonstrated that the aggregation of BSA in bulk solution is governed by the intermolecular thioldisulfide interchange reaction in which the free sulfhydryl group on a given BSA attacks an existing internal disulfide linkage in a separate albumin molecule, This aggregation reaction, and in turn protein fouling, could be completely eliminated by capping the free sulfhydryl group on BSA with either a carboxymethyl or cysteinyl group. In addition, BSA aggregation could be reduced by the addition of metal chelators like citrate or EDTA, both of which reduce the catalytic activity of any divalent metal cations for these

G. Be/j&-r er al. / Journai

qfkfembrane

sulfhydryl-mediated reactions. The rate and extent of protein aggregation also increased at alkaline pH due to the ionization of the free sulfbydryl group. These studies thus suggest that it may be possible to substantially reduce the extent of protein fouling during MF by appropriate control of the intermolecular chemical reactions that lead to protein aggregation in these systems, although considerable additional work is required to demonstrate the generality of these results. These studies of protein aggregation provide a mechanistic basis for understanding the initial stages of protein fouling during membrane MF. Much less work has been done on the long-term fouling behavior and the apparent approach to a steady-state flux at very long filtration times. Palecek and Zydney [ 128 ] showed that the quasisteady fluxes in a stirred device for a wide range of proteins (lysozyme, ribonuclease A, hemoglobin, BSA, and immunoglobulin G) are essentially identical when the proteins are filtered at their individual isoelectric pH, with the flux increasing at pH both above and below the PI. Palecek and Zydney [ 128 ] developed a simple physical model to describe this behavior in which protein deposition continues to occur, and thus the flux continues to decline, until the drag force on the proteins associated with the filtrate flow is no longer able to overcome the intermolecular repulsive interactions between the proteins in the bulk solution and those already in the protein deposit. The steady flux was given as: J=J,,+

(S2K-‘)exp(

-rcd,)/(3E)

(63)

where K- ’ is the Debye length (as defined later in Section 5.1>, s is the protein surface charge density (in units of C/m’), 6 is the permittivity of the solvent (~=6.95~10-‘~ C*/J m for water), and d, is the characteristic separation between the bulk proteins and the deposit at which the proteins are just able to overcome the repulsive interactions and add to the deposit. JPr is the steady-state flux at the protein isoelectric point, which was evaluated as Jp1x6 pm/s for the stirred cell filtration of 5 g/l solutions of BSA, lysozyme, IgG, hemoglobin, or ribonuclease A at 69 kPa [ 128 j . Eq. (63 ) predicts that the steady-

Sciencr

96 (I 994) I-58

45

state flux increases linearly with the square of the protein surface charge density (which is a function of the solution pH for any given protein), with this behavior shown to be in good agreement with experimental data for BSA, lysozyme, hemoglobin, and ribonuclease A. This model is also consistent with the observed increase in the steady-state flux with decreasing salt concentration (i.e., increasing Debye length), with this increase in flux arising from the decrease in the electrostatic shielding between the proteins in the bulk solution and those in the deposit at low salt concentrations. 4.4. Properties ofprotein deposits There have also been a number of investigations of the physical characteristics of the protein deposits that form during membrane Ml?. Lee and Merson [ 1441 examined the structure of protein deposits on 0.4 pm Nuclepore polycarbonate membranes using scanning electron microscopy, The BSA and P-lactoglobulin formed sheet-like deposits on these membranes, while immunoglobulin G formed granules which stacked into layers creating a porous matrix. Kelly and Zydney [ 134 ] and Tracey and Davis [ 1291 have shown that BSA deposits formed during stirred cell MF consist of protein aggregates in an amorphous protein matrix that fully covers many of the pores, with similar results reported by Kim et al. [ 143 ] for BSA deposits formed during protein UF. Opong and Zydney [ 1201 evaluated the hydraulic resistance provided by BSA deposits from data for the saline flux (in the absence of proteins) through protein deposits formed during a stirred cell MF. The data clearly demonstrate that these BSA deposits behave as compressible porous media, with the hydraulic permeability a function of the applied pressure, Typical results are shown in Fig. 30 for the flux of 0.15 M NaCl through deposits formed by filtration of a 10 g/l BSA solution through a 0.16 pm PES membrane at a constant pressure of 104 kPa ( 15 psi) for 4 h. The saline flux through the BSA deposit was nearly two orders of magnitude less than the flux through the membrane prior to the BSA filtra-

al

2 s

2

IA 0.

.-E it

01

I

0

,

100

I

,

I

200

300

400

Time (min)

Fig. 30. Saline flux at pH 7.4 through a BSA deposit formed by a 4 h filtration of a 10 g/l BSA solution at 104 kPa. The transmembrane pressure was rapidly changed from 2 1 to 69 kPa at t = 65 min and then returned to 2 1 kPa at I75 min.

tion (but after BSA adsorption), indicating that the bulk of the resistance to flow for this heavily fouled membrane was in fact associated with the protein deposit. The saline flux at 21 kPa declined slightly with time, reflecting the slow compression of the protein deposit in response to the applied pressure. The saline flux rapidly increased when the pressure was increased to 69 kPa (indicated by the dashed vertical line in Fig. 30), but then decayed to a new steady-state value after -, 100 min of filtration. The steady-state llux at 69 kPa was less than a factor of two greater than that at 21 kPa, corresponding to a 45% reduction in the hydraulic permeability of the deposit. These data were fit to an expression similar to Eq. (12) with the exponent s=O.5 [ 1201 reflecting the compressibility of the BSA deposit. The flux declined sharply when the pressure was reduced back to 2 1 kPa, but in this case the flux slowly increased with time due to the relaxation of the protein deposit at this lower pressure. The steady-state flux during this second filtration at 2 1 kPa was essentially identical to that obtained during the initial 2 1 kPa filtration, indicating that these changes in permeability were completely reversible at least under these experimental conditions. Opong and Zydney [ 1201 hypothesized that the long transient response of the protein deposit to changes in applied pres-

sure was due to the slow rearrangement of the proteins in the very closely packed deposit in response to these pressure changes. The hydraulic permeability of these protein deposits is also a function of the solution pH and ionic strength. Palecek and Zydney [ 145 ] have shown that the solution environment can effect the properties of the protein deposit by: ( 1) altering the protein charge, (2 ) shielding the electrostatic repulsion between adjacent proteins within the deposit, ( 3 ) modifying the electro-osmotic counterflow that is generated by the solvent flow through the charged “pores” in the protein deposit, and (4) altering the protein conformation. The electro-osmotic counterflow arises from the streaming potential that is caused by the solvent flow, with this streaming potential reducing the overall solvent flux through its interactions with the counterions in the diffuse part of the double layer surrounding the individual proteins in the deposit [ 145 ]. The first three phenomena are of greatest importance for stable proteins like ribonuclease A and lysozyme (and for BSA at pH > 4). Under these conditions, the steady-state hydraulic permeability of the protein deposits was minimum at the protein isoelectric point and decreased with increasing solution ionic strength for pH#pI [ 131,145]. These effects were primarily due to the alteration in the protein packing within the deposit, with this packing density determined by the balance between the compressive pressure associated with the filtration and the electrostatic repulsion between adjacent proteins within the deposit (which is a function of both the protein charge and the magnitude of the electrostatic shielding provided by the electrolytes). The transient response of these protein deposits to changes in solution environment is quite complex due to the slow rearrangement of the proteins in response to the changes in electrostatic interactions in combination with the much more rapid alteration in the magnitude of the electroosmotic counterflow [ 145 1. Conformational alterations in the proteins may also be important for less stable macromolecules like hemoglobin [ 1451. Mochizuki and Zydney [ 12 1 ] studied the

G. Bclforf ct al. /Journal ofhfembrane

sieving characteristics of BSA deposits using polydisperse dextrans, with the dextran molecular weight distribution determined using gel permeation chromatography. The actual sieving coefficient (S,) is defined as the ratio of the solute concentration in the permeate (filtrate) to that in the solution immediately adjacent to the upstream surface of the membrane (including the protein deposit); the actual sieving coefficient is thus equal to the fractional transmission of the solute through the protein layer or to one minus the actual rejection coefficient (R,). Typical data for the actual dextran sieving coefficients through a BSA deposit (formed by filtration of a BSA solution through an 0.16 ,um PES membrane) at applied pressures of 35 and 69 kPa ( 5 and 10 psi) are shown in Fig. 31, with the curves representing experimental data obtained with the polydisperse dextrans. The actual sieving coeflicients of the clean 0.16 pm PES membrane are essentially equal to one over the entire range of dextran molecular weights examined in Fig. 31, The BSA deposit is able to retain significant amounts of dextran, with the dextran retention increasing as the dextran molecular weight increases. Also shown for comparison are the dextran sieving profiles for several different molecular weight

cn” E 2

E % 0 F

‘5 .P

rn

7 E

4

lo4

lo5

IO6

10'

Dextran Molecular Weight

Fig. 3 1. Actual sieving coefficients as a function of dextran molecular weight at 35 and 69 kPa for a BSA deposit formed by filtration of a 5 g/l BSA solution for 3 h at 69 kPa. The dashed lines represent the actual dextran sieving coefficients for 50, 100, and 300 kDa PES UF membranes [ 12 11.

Science 56 (I 554) 1-S

47

cut-off polyethersulfone UF membranes [ 146 1. The dextran sieving coefficients at 69 kPa are uniformly smaller than those at 35 kPa due to the tighter packing of the proteins within the deposit, and thus the smaller effective pore size, at the higher pressures. The molecular weight dependence of the actual dextran sieving coefficients through the BSA deposit is significantly weaker than that seen for the commercial PES membranes; the sieving coefficients for dextrans with molecular weights less than lo5 are similar to those for a 100,000 molecular weight cut-off UF membrane while those for dextran molecular weights greater than 1O6approach the values obtained with a 300,000 molecular weight cut-off membrane. In addition, the sieving characteristics of the BSA deposit are a function of the solution pH and ionic strength [ 12 I], similar to the behavior decribed previously for the hydraulic permeability. The sieving characteristics of these protein deposits can also affect the filtrate flux through the accumulation of rejected protein and the generation of an osmotic pressure difference across the fouled membrane. Experimental data for the steady-state flux of a 5 g/l BSA solution through an 0.16 pm PES membrane are shown as a function of the applied pressure in Fig. 32. These data were obtained after formation of a protein deposit (formed by filtering a 10 g/l BSA solution at 20 psi for 5 h). The dashed curve in Fig. 32 represents the flux through the clean (pre-adsorbed) membrane, evaluated using Eq. (57 ) with the deposit resistance and osmotic pressure terms set equal to zero and with the membrane resistance evaluated from saline flux data obtained with the 0.16 pm membranes after preadsorption with BSA. The flux through the clean membrane varies linearly with applied pressure, reflecting the constant resistance of the PES membrane under these conditions. The thin solid line represents the flux determined from Eq. (57) with a&=0, i.e., without osmotic pressure effects. In this case, the deposit resistance was evaluated from the data for the steady-state saline flux through the protein deposit in the absence of any proteins in the bulk solution (analogous to the data in Fig. 30). The darker curve

flux both by altering the extent and rate of protein fouling, as well by retaining protein at the cake surface (which can lead to the development of a protein polarization boundary layer above the yeast cake). 4.5. Summary of protein fouling

0

5

10

15

20

25

30

Applied Pressure, AP (psi)

Fig. 32. Ultrafiltrate flux as a function of applied pressure during filtration of a 5 g/l BSA solution through an 0.16 pm PES membrane. Data were obtained after completion of a 3 h filtration at a constant pressure of 10 psi (69 kPa). Model predictions are described in the text.

in Fig. 32 is the predicted flux using Eq. ( 57 ) including both the deposit resistance and the osmotic pressure effect. In this case, the protein concentration at the membrane surface was evaluated using a stagnant film model with the protein osmotic pressure determined from available literature correlations as described by Opong and Zydney [ 1201. The model calculations including both the deposit resistance and the osmotic pressure effect are in good agreement with the experimental data over the entire pressure range. At low pressures, the deposit provides the dominant resistance to flow, with the osmotic pressure effect being negligible due to the small protein retention and the low protein concentration at the membrane surface. The osmotic pressure effect becomes more important at higher pressures due to the increase in protein retention (analogous to the increase in dextran retention seen in Fig. 31) and the increase in the protein concentration at the membrane surface due to the high degree of concentration polarization [ 1201. Arora and Davis [ 119 ] have recently demonstrated that yeast cakes, formed by either gravity sedimentation or vacuum filtration, are able to retain BSA during a subsequent dead-end filtration. These yeast cakes can apparently affect the

Although a complete understanding of the complex interactions between proteins and MF membranes is still lacking, a basic understanding of some of the key phenomena governing these processes has been developed in the last few years. In most cases, protein adsorption (in the absence of filtration) is thought to attain only monolayer values, with this adsorption having a relatively minor impact on the filtrate flux or protein retention due to the large pore size of the membranes relative to the size of the proteins. However, in other cases, especially in the presence of flow and under conditions where the proteins can denature, protein multilayers can form in the pores causing a dramatic decline in performance. A theoretical formalism to describe multilayer adsorption in a membrane with a lognormal pore size distribution has been developed (this is discussed further in Section 5.3). Recent experimental studies of protein fouling indicate that the initial flux decline during the MF of protein-containing solutions is primarily due to the deposition of protein aggregates, with these aggregates either blocking or constricting the membrane pores. These protein aggregates can be formed during the initial preparation of the protein solutions, or they may be generated during the filtration process, either by the pumps employed in most crossflow filtration systems or by the high shear rates that exist in the vicinity of the membrane surface or in the membrane pores. Several studies have demonstrated that any processing steps typically associated with an increase in protein denaturation will also cause an increase in aggregation and in turn protein deposition. In addition, it appears that intermolecular sulfhydryl-mediated interactions between proteins may be the molecular basis for the aggregation of these proteins, and this may provide an appropriate chemical framework for the

G. Bdjhrt ef al. /Journal

oj~hfemhrune Science 96 (I 994) l-58

analysis, and possible control, of protein fouling during MF. Protein deposition will ultimately result in the formation of a protein deposit (or cake) on the upper surface of the membrane. This protein deposit provides an additional hydraulic resistance to flow and can also retain significant amounts of other proteins present in the bulk solution. In addition, these protein deposits are compressible, with the hydraulic permeability and sieving characteristics a function of the applied pressure, solution pH, and salt concentration. This protein deposit appears to continue to grow until the drag force on the proteins associated with the filtrate flow is no longer able to overcome the intermolecular repulsive interactions between the proteins in the bulk solution and those already present in the protein deposit. Protein masstransfer limitations (concentration-polarization effects) may also become important during MF due to the accumulation of bulk proteins that are retained either by a cell cake or protein deposit on the membrane surface.

49

branes containing a fixed negative charge so as to repel the suspended colloids and particles. The size, size distribution (see discussion below) and the charge of suspended material are crucial to their ability to foul MF membranes. It is important to realize that flocculation of colloidal material and hence its stability can be strongly influenced by transport processes (diffusion and/ or convection), ionic strength of the solution in which the colloids are suspended, the concentration of the colloids or particles, and the intermolecular forces between the suspended particles and between the particles and the membranes. The first of these effects will be discussed in the next section while the others will be briefly reviewed below. Consider two identical spherical particles of radius a and wall potential Y.. interacting through a distance h, 240 mV), one obtains the total interaction potential V on adding the repulsive and attractive potentials [ 1471, v= V, + V, = 27rar [ 4RTy/ (29) ] *

5. Colloid and particle fouling In addition to proteins and dissolved macromolecules, there are a range of other colloidal and particulate matter which can foul MF membranes. This includes a variety of inorganic materials, lipids and other fatty or oily particles, latex and other organic particles, and polysaccharides (Table 3 ). In order to understand the fouling behavior of these colloidal materials, it is necessary to examine the intermolecular forces that govern colloidal stability and the kinetics of such aggregation processes. 5.1. Colloid stability Most suspended colloids and particles in nature are negatively charged. This has been used to advantage by some membrane manufacturers for the treatment of surface waters that contain negative colloids. These companies have treated negatively charged suspended particles such as colloids and latex colloidal paints with mem-

xexp(

-h,/K-‘)-aA/(

(64)

12h,)

where y= tanh [ zF!PJ (4RT) ] z 1.O for large wall potential lu,, and the Debye length K- ’ = [ 2000e2N,Il&T] - ‘j2, I=O.SCZ’Mi is the ionic strength, ~=~~,=78.54~8.85~ lo-‘* (C’/J m), k= 1.38x 1O-23 (J/K), e= 1.60x lo-l9 (C) and N,=6.02~ 1O23(molecules/mol). At 25°C -‘=4.31 x 10-10(21)-‘/’ (m). For rapidly iocculating colloids, V< 0, while for a stabilized system, v> 0. In the latter case, one obtains the distance of the primary maximum total interaction potential by differentiating Eq. (64) and setting it to zero, and then finding that hqmax= IC- ’ (critical). Inserting this result into Eq. (64) gives ~~‘(critical)

%:Az,*/y*

(65)

and from the definition of the Debye length, -‘c (zfMi)-lI* where M is the molar conEentration of ions.‘Combinini this with Eq. (65) we obtain the criterion for the critical flocculating electrolyte molar concentration,

G. Be/fort et al. / Journul of Merrrbrune Science 96 (I 994) I-55

50

Mi(critical)==:‘/(A2Zi6)

(66)

coefficient of the proportionality with 3.38~ 1O-36 J2 mol/m3 at 25°C Mi in mol/m3 and the Hamaker coefficient A in J. Eqs. (65 ) and (66) are only valid (in their current forms) for symmetric electrolytes ( 1: 1,2 : 2, or 3 : 3, but not 1:2,etc.).ForlargeYW (>240mV),y=l.O, Eq. (66) predicts that the critical flocculating concentration of indifferent electrolytes such as K+ , Ca2+ and A13+, containing counter-ions with charge numbers z= 1,2 and 3, will be in the ratio of 1:2-6:3-6 or 1000: 15.6: 1.37. This is called the Schulze-Hardy rule. The above analysis is known as the DLVO theory after Derjaguin, Landau, Verwey and Overbeek [ 1481. 5.2. Flocculation: Brownian (perikinetic) and velocity gradient (orthokinetic) Flocculation is a two-step process: transport and attachment. The first effect has to do with transport of two colloids to each other or of one colloid to the membrane surface while the second involves intermolecular forces. The two main transport mechanisms, diffusion and convection, are discussed here. The random motion of colloids due to the bombardment of fluid molecules is measured by a Brownian diffusion coefficient and termed perikinetic flocculation. Assuming a bimolecular reaction for the decrease of the total concentration of particles, N (mm3), at time 1 (s) in a suspension, viz. dN/dt=

- (4W/3rl)w

(67)

where 5 is a collision association factor representing the fraction of the total number of collisions that remain attached, k is the Boltzmann constant, T is the absolute temperature, and q is the viscosity of the fluid. Integrating from the initial concentration N, to N over time t gives

N/Ni=11[1+(tltI/2)l where t1,2 = 3~/ (4ckTNi) and is the time sary to reduce Ni to Ni/2. At 25 “C, 1.6 x 10’ ‘/ ( CNi). For dilute solutions and T,,~ can be very long. Note that the size

VW necestlp =

low [, of the

colloids are absent from these expressions. The likely location for perikinetic flocculation to occur in a pressure-driven membrane flow loop is in the feed reservoir. A more effective transport mechanism is orthokinetic flocculation which is the result of convective transport due to the spatial distribution of the velocity. Such velocity gradients are imparted into a fluid by agitation due to an impeller, a pump, or Bow in a duct. For an initial uniform particle size suspension of diameter di, the rate of orthokinetic flocculation is given by dN/dt = - (2iGd13/3 )w

(69)

where G (m/s) is the mean velocity gradient. Defining the volume fraction of colloidal particles as 52= Ed: Ni and assuming it remains fairly constant, the following is obtained by substitution into Eq. (69), dN/dt=

- (4[GS/n)N

and integrated give,

(70)

from Ni to N and from 0 to t to

N/N, =exp [ - (4[GSZ/n)t]

(71)

Increasing G has a strong effect on destabilizing suspended colloids. The effect of flocculation on increasing particle size influences the mean particle size and hence the distribution fed to the MF process. This is especially important with inertial lift of particles away from the membrane interface since this process is very sensitive to the size (i.e., a’), see Section 3.4. 5.3. Particle capture by a membrane: deposition, intrusion (pore narrowing and constriction, pore plugging and deposition) [7] Suspended particles and colloids are known to foul pressure-driven membranes by depositing onto the upstream face or entering the pores. Membrane fouling can be dissected into pore narrowing and constriction, pore plugging and deposition of a gel/cake onto the upstream face of a membrane. These three cases were described previously in Fig. 29. The drop in the slope of the flux versus transmembrane pressure curves has been attributed to the increased deposit thick-

G. BeFort et al. /Journal ofMembrane Science 96 (I 994) I-58

ness [ 149 J and/or osmotic effects [ 1501. Most analyses of these phenomena have used a model in which the membrane is treated as an array of uniform cylindrical pores of a single pore radius. Belfort et al. [ 7 ] have recently extended this analysis to explicitly account for the presence of a pore size distribution on the flux decline associated with pore constriction effects. Following the approach of Meireles et al. [ 15 11, assuming the correct log-normal pore size distribution, n(r), and assuming that the local flow rate through a pore, j(r), is given by Poiseuille flow, then the solvent flux is given by M VW =

JArMrW

(72)

=

(fq4[ 1+ (fJ/P)2]6

51

(75)

Assuming that k= i. t is the thickness of an adsorbed multilayer inside a pore and that t is the thickness of an adsorbed monolayer (a dimension that can be estimated from the molecular structure of a protein or macromolecule), then i is the number of monolayers adsorbed in the pore. To account for the loss in cross-sectional area for flow through a pore, r is now substituted for by r-k in Eq. (74). Since the number of pores per unit membrane area remains the same, Eq. (73) is unaltered. Explicit expressions [Eq. (75) ] for the modified fluxes without and with adsorbed solute in the pores are presented by Belfort et al. [ 7 1. Their ratio is

0

where the number of pores per unit area is n(r) =pr “;, r x

[log(l+ -

exp

(a/r*)2)]-1’2

[M~~mFT

(73)

>/r*12

21og( 1+(a/fy2>

i

I

and j(r)=velocityxarea=

(&?/8~1)7~r

(74)

where n, is the probability density of radius r, r* is the mean radius, 0 is the standard deviation of the distribution, p is the kinematic viscosity and Eq. (74) is called the Hagen-Poiseuille law. Unfortunately, r* and LT’as used by others are not the mean and variance of their expression, respectively, as evaluated by moment analysis. Eq. (73), on the other hand, does give the mean r* and variance 0’ by moment analysis [ 137 1. Substituting Eqs. (73) and (74) into (72) and rearranging gives a modified flux,

where q= 1 + (~/r*)~. Eq. (76) was conveniently used to obtain an estimate of k assuming that 0 is the same for the membrane with and without adsorbed solute [ 71. The model assumes that on an average the adsorbed layer is uniform throughout. At present little is known on how reasonable this assumption is. In Fig. 33 the integrand of Eq. (75 ) with r sub-

“E 0.6 3

vw -

s, = no

(-H

RAP -

1

8P

[ 27clog( 1+

0.0

( 6/fy2)

] - “2

Xmexp - h%(~JGJm/r”)

JI 0

21og( 1+ (cJ/fy2)

I'

tid

I

r

Fig. 33. Plot of the integrand of Eq. (75) (called the modified frequency,f) with r substituted by r-it in Eq. (74) versus pore radius r for different numbers of adsorbed layers i assuming r”=0.62pm, a=0.48pm and t=72.2 A [ 71.

G. Belfort et al. /Journal

(4 ’.-.

0

’ ’ ’ ’ ’ 0 l.ycr

ofMembrane

’ +

IO

---

Z”

.-a..

1”

IO 20 30 40 Pressure difference. APi,,, x I O-4, (Pa)

O.h

0.4 (I

s

IO

15 No BSA layers,

I

20

25

30

Fig. 34. (a) Flux versus transmembrane pressure difference for different numbers of adsorbed layers i, assuming r* = 0.62 pm, a=0.48 pm and I= 72.2 8, for BSA. The initial slope for i=O is obtained experimentally (choose 0.783E-7 cm/s Pa from Fig. 7 in [ 71) and determines n,/l from Eq. (75). Assuming 0 to be the same for the clean (i=O) and the surfacecovered (i>O) membranes, Eq. (76) is used to determine the decrease in the slope resulting from adsorption of one or more monolayers. (b) Slopes from (a) as a function of i for o= 0.48 pm according to Eq. (76 ) [ 7 1.

stituted by r-it in the flow component [Eq. (74) ] is plotted as a function of the pore radius r for different numbers of monolayers (i= 0, IO, 20, 30) with the molecular diameter t=72.2 8, for BSA, the geometric standard deviation, a=0.48 pm (chosen from Belfort et al. [ 89]), and the mean radius of the membrane pores r*=O.62 pm. Substituting 0 as a function of r* and a known value of the mode rrn= 0.305 pm (from measurements) into ytn= VV4/ [ (P)2+ (a)2]3’2 and then into Eq. ( 76 >, allows one to calculate S as a function of r* only. Sub-

Sctencc~ 96 (1994) I-58

tracting the measured S values from the calculated values of S and squaring the difference for two cases, they obtained the best-fit r* and ~7values [ 891. For the hydrophilic MF membrane, they obtained best-fit values of r* =0.62 pm and a=0.48 pm which were used in preparing Figs. 33 and 34. As the number of adsorbed layers increases, both the peak of the modified frequency, f; and the area under the curves decrease as expected. Knowing the slope of the flux-transmembrane pressure curve for clean water (“no fouling” case) and for the case with protein solution from experiments, an estimate of the number of layers i of protein adsorbed in the pores can be obtained from Eq. (76) knowing r*, 0 and t (Fig. 34). Using a 0.45 pm hydrophilized polysulfone microporous membrane with a feed containing 10% fetal bovine serum and a control of deionized water, linear permeation flux versus transmembrane pressure curves gave slopes of 6.23~ lo-‘and 7.8~ IO-‘cm/s Pa, respectively (slope ratio, S=O.798) [ 71. From Fig. 34b, we obtain an estimate of the mean number of BSA layers that line the pores of the 0.45 nominal pore size membrane as 2 1.

6. Summary

Membrane MF is increasingly used as a unit operation in a wide variety of chemical, environmental, food, and biotechnological applications. As discussed in this review, considerable progress has been made in the last few years in identifying and characterizing the key physical phenomena governing the performance of MF processes in these industries. These phenomena include ( 1) the rate of particle cake formation and the proper description of particle mass transport to the growing cake, (2 ) the effects of protein adsorption and deposition, and (3) the importance of colloidal and particulate fouling and the proper description of colloid-membrane interactions. These studies have also provided important insights into the effects of the membrane properties (both chemical and physical) and the module configuration/operating condi-

tions on these different phenomena, thereby providing a framework that can be effectively used to analyze the design and operation of MF processes for these different applications. Although these studies have provided important insights into the behavior of both suspensions and macromolecular solutions in both dead-end and crossflow MF, it must be noted that most of the qualitative experimental and theoretical investigations have focused on each phenomena separately, be it shear-induced diffusion, inertial particle migration, protein fouling, etc. There is thus a critical need for future work directed at examining the possible coupling or interactions that can occur between these different phenomenon in actual commercial MF processes. This would include, for example, extending available theoretical analyses of inertial migration to explicitly account for the effects of particle-particle interactions (which are the basis for the shear-induced particle diffusion observed in concentrated suspensions). Studies of this kind could ultimately lead to the development of a unified framework for the analysis of particle transport in bulk suspensions that accounts for the combined (and simultaneous} effects of Brownian diffusion, shear-induced particle diffusion, and inertial migration. There is also a need to more effectively integrate the existing models for particle transport in bulk EW pensions (i.e., the concentration-polarizationtype models) with appropriate descriptions of particle capture and cake formation kinetics (which have generally been based on a description of the motion of a single particle and thus largely ignore the effects of bulk mass transport). It is also critically important to develop a more quantitative understanding of the possible interactions that can occur between different proteins, colloids, and particles present in complex process streams. For example, shear-induced particle diffusion can lead to an enhancement in macromolecular transport, although the effects of this increased transport on protein fouling is uncertain. Recent experimental studies have shown that preformed particle cakes can alter the rate of protein deposition during filtration, but there is currently no qualitative understanding of the

effects that the simultaneous deposition of particles and proteins can have on the flux and selectivity during membrane MF. There is also a critical need to extend existing studies on protein fouling to a wider range of proteins with different physicochemical characteristics and to examine in more detail the effects of the solution environment (including the presence of particles, cells, or other colloids), the device fluid mechanics, and the physical/chemical properties of the membrane on the fouling characteristics of these macromolecular solutions. Finally, understanding the behavior of suspensions and macromolecular solutions during MF should lead not only to reduced concentration polarization and fouling, but also to the development of better modes of operation and possibly to more efficient membrane elements. Recent advances in integrating mass and momentum transport, as described in this review, will need to be extended to unsteady non-Newtonian behavior at higher solute concentrations than previously considered. Any advances in the formation of microporous membranes, such as narrowing of the pore size distribution, will also have a direct impact on the performance of MF.

Acknowledgements The authors thank their respective collaborators, students and families.

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