Blotechnol. hog. lQQ3,9, 625-634
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Crossflow Microfiltration of Yeast Suspensions in Tubular Filters Sanjeev G. Redkar and Robert H. Davis* Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80309-0424
Crossflow microfiltration experiments were performed on yeast suspensions through 0.2-pm pore size ceramic and polypropylene tubes a t various operating conditions. The initial transient flux decline follows dead-end filtration theory, with the membrane resistance determined from the initial flux and the specific cake resistance determined from the rate of flux decline due to cake buildup. For long times, the observed fluxes reach steady or nearly steady values, presumably as a result of the cake growth being arrested by the shear exerted a t its surface. The steady-state fluxes increase with increasing shear rate and decreasing feed concentration, and they are nearly independent of transmembrane pressure. The steady-state fluxes for unwashed yeast in deionized water or fermentation media are typically 2-4 times lower than those predicted by a model based on the properties of nonadhesive, rigid spheres undergoing shear-induced back-diffusion. In contrast, the steady-state fluxes observed for washed yeast cells in deionized water are only 1*30% below the predicted values. The washed yeast cells also exhibited specific cake resistances that are an order of magnitude lower than those for the unwashed yeast. The differences are due to the presence of extracellular proteins and other macromolecules in the unwashed yeast suspensions. These biopolymers cause higher cell adhesion and resistance in the cake layer, so that the cells a t the top edge are not free to diffuse away. This is manifested as a concentration jump from the edge of the cake layer to the sheared suspension adjacent to it. The shear-induced diffusion model includes a dimensionless parameter (referred to as the crossflow integral) that accounts for the concentration jump as well as the viscosity and diffusivity properties of a given suspension, and this parameter was determined for the unwashed yeast suspensions by fitting the experimental flux data to the model. The resulting values of this parameter are significantlylower than those for rigid spheres but show relatively little variation between experiments a t different operating conditions for a given filter, indicating that the parameter is an intrinsic property of the suspension/membrane system. However, a significant difference in the parameter values was obtained between the ceramic and polypropylene membranes, indicating that the adhesive and fouling properties of the cake depend on the membrane material with which it is in contact. The resulting fluxes obtained from the model using single values for the membrane resistance, specific cake resistance, and crossflow integral for each data set (ceramic and polypropylene filters) are in good agreement with the measured fluxes. Inertial lift theory with no adjustable parameters was also compared to the data. It was found to overpredict the steady-state fluxes a t high shear rates and underpredict the fluxes at low shear rates.
1. Introduction
Crossflow microfiltration is a downstream solid-liquid separation process in which the suspension to be purified is forced tangentially through a filter with microporous walls. It is used in biotechnology applications to harvest cells and to clarify cell lysates. Crossflow microfiltration can be operated on a continuous basis because the flow on the retentate side of the filter exerts a shear at the membrane surface. This shear prevents the continued buildup of a foulingor cake layer on the membrane surface, and a steady or quasi-steadypermeate flux which increases with the shear rate is observed. The mechanism responsible for the steady state reached in crossflow filtration processes is not clearly understood. Brownian diffusion is a likely mechanism for particles less than 0.1pm diameter (Blatt et al., 1970). For larger particles, several possible mechanisms may be responsible for the steady state. These include back-transport mechanisms, such as inertial lift (Altena and Belfort, 1984;Drew et al., 1991) and shear87567938/93/3009-0625$04.00/0
induced diffusion (Zvdnev and Colton. 1986: Davis and Leighton, 1987; Davis and Romero, 19881, which carry particles away from the membrane and surface-transport mechanisms, for which the shear forces and torques on a particle at the cake surface overcome contact and adhesive forces (Lu and Ju, 1989). Many researchers have previously performed crossflow filtration experimentson yeast suspensions. Cheryan and Mehaia (1984) used a hollow-fiber unit attached to a fermentor in order to recycle cells and achieve a productivity of ethanol as high as 100 g/L.h. Hoffman et al. (1987)also studied the use of crossflow microfiltration for increasing cell concentration and ethanol productivity in yeast fermentations. They found that the average flux decreased with increasingcell concentration and that backflushing could be used to help regain the flux. Kavanagh and Brown (1987) also found that the permeate flux decreases with increasing yeast concentration and that periodic back-flushingincreasesthe average filtration rate. Matsumoto et al. (1987)fiitered a yeast suspensionthrough
0 1993 American Chemical Society and American Institute of Chemical Englneers
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Figure 1. Schematic diagram of the experimental setup: 1, yeast feed reservoir; 2, water jacket; 3, magnetic stirrer;4, thermometer; 5, temperature-controlledwater bath; 6, peristaltic pump; 7, three-way valve; 8, pressure gauge; 9, filter module; 10, needle valve; 11, permeate reservoir; 12, electronic balance; 13, microcomputer; 14, water feed reservoir.
a cellulose acetate filter at different flow rates and feed concentrations. Lower concentrations and higher flow rates favored high fluxes. In perhaps the most complete study to date, Ofsthun (1989) measured the filtration properties (resistance and porosity of the cake) of yeast by conducting a variety of experiments. The porosity decreased with increasing pressure below a few psi and then did not change with higher pressure. She also performed crossflowmicrofiltration experiments on yeast suspensions through different filters. A pressureindependent, steady-state flux was observed that showed a weak dependence on the shear rate a t the wall. The flux was seen to be proportional to the logarithm of the inverse of the bulk concentration. Romero and Davis (1991) compared experiments using acrylic and polystyrene beads filtered through a rectangular, glass-walledcrossflowmicrofdter and a ceramictube bundle with their shear-induced diffusion model (Romero and Davis, 1988,1990). The initial rates of particle buildup at the membranesurfaceand of the flux declinewere found to be essentially the same as those predicted by dead-end filtration theory. The measured steady-state fluxes are in good agreement with the theory using independent empirical correlations for the properties (diffusivity, viscosity, and specific cake resistance) for egid, nonadhesive spheres. In the present article, crossflow filtration experiments have been performed on washed and unwashed yeast suspensions through ceramic and polypropylene filters. The feed concentration, flow rate, and transmembrane pressure were varied. The focus is on determining the transient flux decline due to the formation of a cake layer of rejected yeast cells and the steady-state flux due to shear forces arresting the growth of the yeast cake. The transient results are compared to those from dead-end filtration theory, and the steady-state results are compared to theoretical predictions based on shear-induceddiffusion and inertial lift. The goal is to obtain a better qualitative and quantitative understanding of the factors affecting yeast filtration under crossflow conditions.
2. Materials and Methods 2.1. Experimental Work. A schematic diagram of the setup is given in Figure 1. Experimentswere conducted
with two types of tubular membranes: ceramic and polypropylene. The ceramic membrane is a Norton Ceraflo tube with an average pore size of 0.2 pm. Its internal diameter is 2Ho= 0.24 cm, its outer diameter is 0.48 cm, and the length of the membrane after its enclosure in an acrylic casing is L = 24 cm. This membrane has an asymmetric pore structure and is hydrophilic. The polypropylene membrane is an Accurel filter fiber marketed by Enka Microdyn. The average pore size of the membrane is 0.2 pm. Its internal diameter is 2H0 = 0.18 cm, ita outer diameter is 0.22 cm, and the length of the membrane is L = 24.5 cm. The polypropylene membrane is also asymmetric but is intrinsically hydrophobic. A peristaltic pump marketed by Masterflex is used to pump the feed suspensionthrough the filter, which consists of an individual tube enclosed in an acrylic casing. The retentate is recycled back into the feed reservior through a needle valve that is used to control the pressure inside the filter. Pressure gauges are attacked at the entrance and exit of the filter. Approximately 50 mL of permeate is collected in a single experiment, whereas the feed suspension volume is loo0mL. Hence, the increase in the concentration of the feed due to the removal of permeate isneglected. Thepermeateiscollectedinar~rvoirplaced on an electronic microbalance manufactured by Mettler (Model PE 3600). The balance is interfaced to a personal computer using the RS232 configuration. Software was written to compute the flux rate (volumepermeate formed per time per membrane area) by numerically differentiating the mass of permeate versus time data. The suspension used for filtration is Saccharomyces cereuisiae (Fleischmann’s dry yeast) suspended in deionized water with a resistivity of 18.2 Mfbcm. A Coulter Multisizer was used to measure the size of the cells. The number-average equivalent sphere diameter of the cells was measured to be 4.2 f 0.2 pm. The value was not seen to change after the yeast was suspended in deionizedwater for several hours. No broken cells were observed during this time period. The volume fraction of the suspended yeast cells was determined using a wet cell density of 1.1 gm/cm3 (Szlag, 1988) and a dry yeast cell density of 0.33 times the wet density (Ofsthun, 1989). For most of the experiments, the powdered yeast was not washed after rehydration. Sincethe dried yeast preparations contained
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extracellular proteins and other molecules or colloids, the supernatant for these experiments was not clear. The absorbance of the supernatant for a suspension made with 5 g/L dry yeast was 0.60 at 280 nm. This value was steady for approximately 1 h, and then it gradually increased to 0.80 after 6 h. The increase is presumably due to osmotic cell lysis, in which the cells leak macrosolutes while retaining their shape. Using an absorbanceversus protein concentration calibration curve made in our lab for bovine serum albumin, the 0.60 initial absorbance reading corresponds to 1.0 g/L protein in the supernatant of a 5 g(dry)/L yeast suspension. Thus, the extracellular macromolecules and colloids are expected to affectsignificantly the filtration behavior of the yeast suspensions. For comparison, additional experiments were performed with unwashed yeast cells rehydrated in a rich fermentation medium (20 g of peptone, 10 g of yeast extract, and 10 g of dextrose in 1 L of water) and with washed yeast cells in deionized water. The washing procedure involved suspending the yeast in water, centrifuging,decanting the supernatant, and repeating the procedure three times. The resulting supernatant from a suspension of 5 g/L dry yeast was clear, initially, with no measurable absorbance, and the absorbance gradually increased to 0.03 after 2 h. Each experiment consisted of passing deionized water (which was prefiltered with a reverse-osmosis system to remove colloidalparticles) through the filter for a certain period of time (typically 30-60 min), followed by passing the yeast suspension through the filter for a slightlylonger period of time (typically 60-120 min), until the flux remained nearly steady. The temperature was controlled a t 25 OC for the polypropylene tubes and was monitored in the range 23-25 "Cfor the ceramic tubes. The operating conditions that were varied include the bulk volume fraction of cells in the feed (C#Jb), the transmembrane pressure (AP), and the bulk suspension flow rate (8).Two of the three conditions were held constant, and the third was varied. The parameter ranges studied were 5 IAP $30 psi, 0.003 I C#Jb5 0.06, and 360 IyoI8,700 8, where yo= 4Q/(?rHO3) is the wall shear rate for parabolic laminar flow in a tube of radius H,. The Reynolds number, Re 2poQ/(?rpJlo), was less than 2000 for all experiments except those at the highest flow rate, with po and po being the pure fluid density and viscosity, respectively. Most of the experiments were repeated 2-3 times. Between experiments, back-flushingfollowed by backwashing was used to clean the membrane. For the ceramic filters, a pipe cleaner with a caustic soda solution was used to help remove the cake layer formed.
3. Results and Discussion 3.1. Typical Flux Decline Curves. Typical flux decline curves illustrating the effects of varying the operating perameters are shown in Figures 2-4 for the polypropylene filter. The end of the water filtration period is shown, followed by the flux decline during the yeast filtration period. Pure water flux declines of typically 10-50% were observed during the water filtration period for both the ceramic and polypropylene tubes. Since the water had been prefitered through a reverse-osmosisfilter, it is unlikely that pore-plugging by colloidal particles occurred. A more likely explanation is that offered by Errede (1984),who observed that the water flux decline for hydrophobic membranes occurs due to capillaryeffects from the accumulation of trace impurities at the interface between the water and microscopic bubbles trapped by the pores. To confirm this possibility, we observed that the initial water flux could be reattained by wetting the low-pressure side of the membrane with ethanol.
4 X
2 0.006 Lc
e, +
4 0.004
I
0.000 ~ ' " " " " " " " " " " " " " " " " 1500
3500
5500
7500
Time ( s e c ) 2
Figure 2. Flux decline curves at different transmembrane pressures of AP = 10 psi (A)and AP = 20 psi ( 0 )for unwashed yeast in water and AP = 10 psi (+) for washed yeast,in water, through the polypropylene filter with &, = 0.03 and yo = 4370 8-1,
Qpical flux decline curves for two different transmembrane pressures are shown in Figure 2. For AP = 10 psi, results are shown for both unwashed and washed yeast in water of C#Jb= 0.03 and yo = 4370 s-l. The washed yeast suspension had a higher initial flux because a newer membrane was used; it also had a more gradual flux decline and a higher final flux. For hp = 20 psi, the results for unwashed yeast in water at C#Jb= 0.03 and yo = 4370 s-l are shown. From the data for unwashed yeast in water at the two different transmembrane pressures, it is seen that the initial flux a t the start of yeast filtration is proportional to the transmembrane pressure. Upon introduction of the yeast suspension at t = 1800 a, the flux drops rapidly due to the buildup of a cake layer of rejected cells on the membrane surface, with the decline being more rapid for the higher transmembrane pressure since the higher initial flux causes a more rapid cake accumulation. At t = 7000 s, the flux reaches a near steady state, presumably because the shear exerted by the crossflow arrests the cake growth. The final flux for 20 psi transmembrane pressure is only slightlyhigher than the final fluxfor 10psi transmembrane pressure. Figure 3 illustrates the effect of varying the feed cell concentration on the permeate flux decline for unwashed yeast in water a t AP = 10 psi and yo = 4370 s-l. The permeate flux starts at the same value for two concentrations. For the high feed concentration = 0.06), however, the flux decline is more rapid than for the low feed concentration (C#Jb= 0.003). This is because the rate of cake growth increases with increasing concentration. At t = 7000 a, the nearly steady flux for the low feed concentration is almost 150% greater than that for the high feed concentration. For comparison, data are also shown for washed yeast in water and unwashed yeast in the rich medium at the low feed concentration. The initial fluxes for these two cases are higher than the others because newer membranes were used. The case of washed yeast in water exhibits only modest flux decline, whereas the case of unwashed yeast in the rich medium exhibits a very rapid flux decline and a relatively low steady-state flux. The difference is due to the presence of extracellular proteins and other solutes in the rich medium, which foul
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and membrane) in series:
I
J
where J i s the permeate flux, go is the solvent (in this case, water) viscosity,R m is the membrane resistance, 8, is the specific cake resistance, and 6, is the cake thickness. The flux decreases continuously with time during dead-end filtration because the cake thickness continuouslyincreases as the cells are carried toward and rejected by the membrane. For constant-pressure processes with incompressible membranes and cakes (Rm and 8, constant), the flux decline due to cake buildup in dead-end filtration is described by J(t) = Jo[1 + T
1500
3500
5500
7500
Time ( s e c )
Figure 3. Flux decline curves at different feed cell volume fractionsof &, = 0.003 (A)and = 0.06 (0) for unwashed yeast in water, qjb = 0.003(+) for washed yeast in water, and qjb = 0.003 (*) for unwashed yeast in rich media, through the polypropylene filter with yo = 4370 s-l and AP = 10 psi.
where Jo = A P / P J t m is the initial or pure water flux, and to is the lag time. The lag time is the time required for the switch from a pure water feed to a yeast suspension feed until the yeast suspension actually reaches the filter surface. The time constant, T, is given by
(3) where f#)b is the volume fraction of cells in the bulk suspension, and 4, is the volume fraction of cells in the cake layer. The volume fraction of the yeast cells in the cake layer is taken to be 4, = 0.78 (Ofsthun, 1989). Equations 1 and 2 were derived for flat membranes. They also hold for cylindrical filters, provided that the cake layer is thin compared to the tube radius. For thick cake layers, a curvature correction is required for cylindrical filters such as those employed in the present study. Darcy's law then becomes
J(t)=
0.000 F " " " 1500
" " " " " " " "
3500
5500
'
"
AF
(4)
,u0(Rm+ 8floIn [ H J ( H o- 6,(t))l) Dead-end filtration theory for cylindrical tubes yields an implicit expression for the cake thickness as a function of time (Davis, 1993):
7500
Time ( s e c )
Figure 4. Flux decline curves at different shear rates of {o = 870 s-* (0) and yo = 8700 s-l (A) for unwashed yeast in water, through the polypropylene filter with 4b = 0.03and U = 10 psi. the membrane and cause the cake to be adhesive and to have a larger resistance. Figure 4 illustrates the effect of increasingshear rate on the permeate flux decline for unwashed yeast in water at f#)b = 0.03 and AP = 10 psi. The pure water flux and the initial rate of flux decline are essentially independent of shear rate. However, the final flux for the high shear rate = 8700 s-l) is higher than that for the low shear rate (ro= 870 s-l). This is thought to be the result of the higher shear more effectively arresting the growth of the yeast cake. 3.2. Analysis Method for Transient Flux Decline. A primary finding of transient models of crossflow microfiltration is that the cake buildup and associated flux decline for short times do not depend on the shear rate and may be approximated by dead-end filtration theory (Davis,1993). It is assumed in dead-end filtration theory that the permeate flux at any given time is described by Darcy's law for flow through two porous media (cake
(i
(5)
From the flux reached at the end of the pure water filtration, the membrane resistance was calculated using Darcy's law without a cake layer:
Rm = APIP,J~ (6) For illustrative purposes, the data for one of the experiments with unwashed yeast in water using the ceramic filter operated at a transmembrane pressure of 15 psi are shown in Figure 5. For this experiment, the final pure water flux is Jo = (2.9f 0.2)X 103 cm/s, which corresponds to a membrane resistance of R m = (3.9 f 0.2) x 1O1O cm-l, at the 90% confidence level. Note that the transient flux decline follows the dead-end filtration theory given by eq 2 for short times and then levels off for longer times. When the curvature correction is included in dead-end filtration theory (eqs 4 and 5 ) , the deviation at long times is even greater. This is because the flux decreases more rapidly
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Biotechnd. Rog., 1993, Vol. 9, No. 8
I\
V
v Y
0.002
0.000
..
' ' ' ' ' ' ' ' ' - i " " " " " " " " " " " " ' ~
3500
4500
5500
6500
Time ( s e c )
Figure 5. m i d flux decline c w e for an experiment with
unwashed yeast in water through the ceramic filter with &, = 0.03,yo= 1820s-l, and AP = 15psi. The symbolsare experimental data, the solid line is dead-end filtration theory given by eq 2, and the dashed line is dead-end fiitration theory given by eqs 4 and 5 for tubes with the curvature corrections.
30
7
20
0
-10 3500
4000
4500 T i m e (sec)
5000
(J,,/a2
Figure 6. - 1 versus time for the experiment shown in Figure 5. The symbolsare experimental measurements, and the solid line is the best-fit straight line through the data for 3650 I t I4450 8-1.
with increasing cake thickness for a tube than for a rectangular channel due to the reduction in surface are available for filtration. In Figure 6, a graph of ( J o / a 2- 1versus t is plotted for the yeast suspension flux decline data of Figure 5. This graph yields a straight line for short times. This confirms that the transient flux decline occurs due to the formation of a cake layer of the rejected particles, similar to deadend filtration. For longer times, the data deviate from the straight line. This is because the arrest of cake growth due to shear causes the flux for crossflow filtration to be greater than for dead-end filtration after the same time interval, hence, the value of ( J o / a 2- 1 becomes less for the crossflow mode as compared to dead-end filtration at longer time intervals. A linear regression of the data was performed to give an equation for the best-fit line. From the slope and intercept, respectively, T = 28 & 1s and to = 3651 3 s,at the 90%confidence level. Since the switch
*
from a water feed to a yeast feed was made after exactly 1h for this experiment, the value of torepresents a 51-8 lag time for the yeast suspension to reach the fiiter. The specific cake resistance is calculated from eq 3 to be & = (5.9 i 0.2) X 1Ol2 cm-z, with 90% confidence. 3.3. Analysis Method for Steady-State Flux. In thissection,we demonstratehow the steady-statepermeate flux may be compared with model predictions based on the back-transport mechanisms of shear-induceddiffusion and inertial lii. A particle or cell in a sheared suspension experiences a fluctuating motion due to interactions with other particles. This leads to a diffusive flux of particles down the gradients of concentration and shear rate (Leighton and Acrivos, 1987). Zydney and Colton (1986) employed shear-induced diffusion in the concentration polarization model of crossflow filtration based on the Levhue solution for mass transfer in a simple shear flow (Porter, 1972). Using a constant shear-induced diffusvity of D = 0.3i0a2from the data of Eckstein et al. (19771, and assuming a linear velocity profile across the polarization boundary layer, they derived a simple expression for the steady-state, length-averaged permeate flux: ( J , ) = 0.078Yo(a4/L)"3
(7) where a is the particle radius, yo is the shear rate, L is the filter length, & is the particle volume fraction in the boundary layer immediately above a thin fouling layer on the membrane, and 4 b is the particle volume in the bulk suspension. Davis and Leighton (1987) proposed a local model based on nonconstant viscosity and shear-induced diffusivity, where the velocity and concentration profiles at any point are evaluated in the flowing boundary layer. Romero and Davis (1988) extended this to a global model by incorporating axial particle convection in the flowing boundary layer. In general, a numerical solution of their integral model is required. For dilute suspensions with resistance to filtration controlled by a thin cake layer, however,Davis and Shenvood (1990)showed that an exact similarity solution exists: (&/db)
( J , ) = 1.31(D32/&$41'3 (8) where Do = +oa2is a characteristic diffusivity, and Iz is the crossflow integral which accounts for the concentration dependence of diffusivity and viscosity when integrated across the concentration-polarization boundary layer. Using empirical expressions for suspensions of nonadhesive, rigid spheres, Davis and Leighton (1987) estimated that IZ = 1 X 10-4 when the bulk concentration is small. The crossflow integran (12) is defined by Davis (1993),and it is proportional to the dimensionless excess particle flux (Q)defined by Davis and Leighton (1987): I2 = ( p b / l ~ ~ ) ~ Q , where po is the pure fluid viscosity and pb is the bulk suspension viscosity. The two become equal in the dilute limit. Particles may also exhibit a tendency to move away from the wall as a result of inertial lift. This phenomenon was first observed by Rubinow and Keller (1961). Cox and Brenner (1968) quantified the inertial lift velocity of aparticle in slow laminar flow. Recently Drew et 01. (1991) developed an expression for the lift velocity using the "near wall" lateral migration model for dilute suspensions with fast laminar flow. The inertial lift velocity for such cases prior to cake formation is
where
po
is the fluid density. When a thin cake layer
Bbtechnol. Prog., 1993, Vol. 9, No. 8
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provides the controlling resistance, the steady-state flux is simply J , = q0. The above expressions for the steady-state flux are restricted to dilute suspensions which form cakes of rejected particles which have sufficiently large specific resistances such that they remain thin relative to the tube radius. This is generally the case for cakes formed by very small or highly compressible particles, such as cell lysates, but not necessarilytrue of the present experiments with yeast suspensions. The values of the parameters 8, and Rm given in section 3.2 can be used in Darcy's law to estimate the average thickness of the cake layer on the filter. The value of the length-averaged final or steadystate flux for the experiment of Figure 5 is (J,)= 0.38X cmls. From eq 4,the cake thickness, a, which yields this flux is 0.037cm, indicatingthat the final cake thickness is about one-third of the tube radius. Thus, the model predictions must account for cake layers which are not thin. Altena and Belfort (1984)used inertial lift to explain the steady state reached in crossflow microfiltration operation with thick cake layers. As long as the particle convection due to permeate flow dominates over inertial lift, particles will accumulte on the membrane. As the particles collect, the channel constricts, increasing the shear rate and thus the inertial lift velocity. The cake buildup also reduces the permeate flux. The increase in the cake thickness continues until the inertial lift velocity equals the permeation velocity. Romero and Davis (1988) developed a similar model for the shear-induced diffusion mechanism, and they showed that a stagnant cake forms on the membrane surface only for distances from the filter exit which exceed a critical value: xer = +,,.D&(&b), where Do = you2 is the characteristic shear-induced diffusivity. Davis (1993)presents generalnumerical results in graphical form for both the shear-induced diffusion and inertial lift models. For the former, the dimensionless steady-state flux, ( J , > Jo, / depends on two parameters: B = RcHoIRm (a dimensionless cake resistance) and Llx,, (a dimensionless filter length). For the latter, ( J,)/Jo depends on L3 and Jolul,o. The shear-induced diffusion theory contains the crossflow parameter, 12, which is known for monodisperse suspensionsof nonadhesive, rigid spheres but which must be determined empirically for more complex suspensions. One way to estimate its value is to parameter fit the model prediction to the measured steady-state flux. For the data in Figure 5, the resulting value of 12 is 6.3 X 10-6. This value is more than 1 order of magnitude smaller than that for dilute suspensionsof rigid spheres (Davisand Leighton, 1987),indicating that the rate of back-transport by shearinduced diffusion is lower for the unwashed yeast in water than for rigid spheres. This difference is due to the proteins and other macrosolutes that presumably make the cake layer more adhesive, resulting in a jump in the particle concentration at the edge of the cake layer and, hence, a reduction in the driving force for diffusion. Further evidence for cake adhesion was obtained by observing a drop in the steady-state flux when the suspension flow rate was decreased, but no increase when the suspension flow rate was subsequently increased to its original value (Figure 8))indicating that cake formation is not readily reversible. For the washed yeast, a slight increase in flux was observed after the tangential shear rate was increased. Using the numerical results of Davis and Leighton (1987) for rigid spheres, the volume fraction of yeast cells in the concentration polarization layer adjacent to the edge of the cake layer corresponding to I 2 = 6.3X 10-6 is estimated
2.OE-3 0
m
\If 1.5E-3 c ' L
\
V v
k \
-1
""'.L \
O.OE+Ol ' 0.001
'
"
""
0.01
'
'
" "
0.1
1
Figure 7. Steady-stateflux versus bulk volume fraction for the unwashed yeast in water. The square symbols ( 0 ) are the measured data and the solid line a,linearfit for the ceramic fiter experiments at AP = 20 psi and yo = 1800 s-l. The triangular symbols (A)are the measured data and the dashed line a linear fit for the polypropylene filter experiments at AP = 10 psi and yo = 4370 8-l.
to be & = 0.14. If cake growth is arrested by a diffusion back-transport mechanism, then we would expect that the steady-state flux would approach zero as q!Jb &. A semilog plot of (J,)versus 4b is given in Figure 7. This figure indicates that the flux would become zero for 4 b = 0.22 f 0.06 and f#Ib = 0.29 f 0.13 for the ceramic and polypropylene tubes, respectively, at the 90% confidence level. 3.4. Comparisonof Experimentswith Theory. The values of the specific cake resistances, the membrane resistances, and the crossflow integral were tabulated for all experimentswith the unwashed yeast cells suspended in water for the ceramic and polypropylene membranes (Redkar, 1992). The average values of membrane resistance for the ceramic and polypropylene tubes are R, = (3.8f 1.2)X 1O1Oand (1.4f 1.2) X 1O1Ocm-l, respectively, at the 90% confidence level. The corresponding average values of the specific cake resistance are 8, = (4.1f 2.2) X 10l2and (1.7f 1.6)X 10l2cm--2,respectively.The average value of loglo 1 2 for the ceramic tubes is -5.5 & 1.0,and for the polypropylene tubes it is -8.7 f 1.4. Totals of 28 and 33 experimentswere performed with unwashed yeast in water for the ceramic and polypropylene tubes, respectively. As evidenced by the reported confidence intervals, considerable variation in the measurementswas observed, even when experiments were repeated under identical conditions. This may have resulted from microscopic nonuniformities in the open membrane pores after cleaning and in the cake layers as they formed. No systematic variation in any of the three parameters (A,, Rm, and I 2 ) was seen with varying operating conditions (yo, hp, and &,). The larger resistance of the ceramic membrane is attributed to its greater wall thickness. The lower values of 8, and I 2 for the polypropylene tube are surprising, as they indicate that the cake properties are affected by the underlying membrane. For the experiments on washed yeast cells suspended in water for the polypropylene fdter, the average membrane resistance is R , = (0.6f 0.6)X 1O1O cm-' and the average specific cake resistance is R, = (0.7f 0.6)X loll cm-2, at the 90% confidence level. The large confidence limits are
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B&NI?&W. Rog., 1993, Vd. 9, No. 8 0.01 6
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0
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flowrate decreased
L
flowrate i j c r e a s e d
\
4 0.000
2000
0
4000 Time (sec)
6000
0
1000
1500
(sec-')
Figure 8. Permeate flux versus time at Ap = 20 psi, +o = 5240 s-l, and &, = 0.003 for unwashed yeast in water through a polypropylene fiiter. A t t = 3600 8, the shear rate was decreased to yo= 1750 s-l, and it was increased back to 6240 s-1 at t = 5400
Figure 10. Comparison of the theoretical and the measured steady-state fluxes at different bulk suspension flow rates for unwashed yeast in water using the ceramic fiiter with AP = 20 psi and 4 b = 0.03. The symbols are the experimental data, the solid line is the shear-induced diffusion theory with the crossflow integral chosen as the average value for this suspension/fitar system, the short-dashedline is the shear-induceddiffusion theory with the crossflow integral for rigid, nonadhesive spheres, and the long-dashed line is the inertial lift theory.
-----------I
F
500
0.004
n0.003
c
0
-t:
t
E0
I'
-0.002 7 V
0.001 0
10
20
30
AP (psi)
Figure 9. Comparison of the theoretical and the measured steady-state fluxes at different transmembrane prequres for unwashed yeast in water using the ceramic filter with yo= 1820 s-l and &, = 0.03. The symbols are the experimental data, the solid line is the shear-induceddiffusion theory with the crossflow integral chosen as the average value for this suspension/fiiter system, the short-dashed line is the shear-induceddiffusion theory with the crossflow integral for rigid, nonadhesive spheres, and the long-dashed line is the inertial lift theory.
due to the variability in the experiments and the fact that only nine experimentswere performed with washed yeast. The average specific cake resistance for the washed yeast cells experiments is more than an order of magnitude lower than that for the unwashed yeast cells. This indicates that the existing proteins and other macrosolutes in the unwashed yeast suspensions significantly increase the specific cake resistance. Figures 9-11 and 12-14 illustrate the effects of transmembrane pressure, bulk suspension flow rate, and feed particle volume fraction on the steady-state flux for the ceramicmembranes and for the polypropylenemembranes, respectively. The solid lines are the steady-state fluxes predicted by the shear-induced diffusion model for the unwashed yeast in water, with I2 set equal to the respective
0.000
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*
0.04
0.02
0.06
@b
Figure 11. Comparison of the theoretical and the measured steady-state fluxes at different feed volume fractions of cella for unwashed yeast in water using the ceramic fiiter with AP = 20 psi and yo = 1820 s-l. The symbols are the experimental data, the solid line is the shear-induced diffusion theory with the croesflow integral chosen asthe averagevalue for this suspension/ fiiter system, the short-dashedline is the shear-induced diffusion theory with the crossflow integral for rigid, nonadhesive spheres, and the long-dashed line is the inertial lift theory.
average values cited above. The short-dashed lines are the steady-state fluxes predicted by the shear-induced diffusion theory using I2 = 1X 1Vfor dilute monodisperse suspensions of rigid, nonadhesive spheres. The longdashed lines are the corresponding steady-state fluxes predicted by the inertial lift theory with no adjustable parameters. In all of these theoretical predictions, the respective average values of the membrane resistance and the specific cake resistance cited above were used. The symhls are the measured steady-state fluxes,and the error bars represent standard deviations for repeated measure-
Bbtechnol. Rog., lQQ3,Vol. 0,
692
menta. Channelconstriction due to the formation of thick cake layers is accountedfor in the theoretical predictions. Figures 9 and 12 show the effect of increasing transmembrane pressure for the ceramic and polypropylene filters, respectively. The models predict a proportional increase in the steady-state flux for low values of the transmembrane pressure, corresponding to conditions where a cake does not form and the resistance is membrane controlled. For pressures beyond a few psi, the predicted increase in flux is more gradual, corresponding to conditions where both the membrane and cake resistance are important. Then, an increase in pressure leads to an increase in cake thickness, which partially compensates for the flux increase predicted by eq 1. A pressureindependent or limiting flux is only predicted when a thin cake or fouling layer with a high specific resistance is controlling, and this condition is not met for the suspensions of whole yeast cells. Nevertheless, the measured steady-state fluxes for unwashed yeast in water are nearly independent of pressure. This finding is consistent with that of Ofsthun (1989). Figures 10 and 13 show the effect of increasing shear rate for the ceramicand polypropylenefilters,respectively. The models predict a rapid increase in the steady-state permeate flux with shear rate at low and moderate shear rates, due to an increase in the back-transport rate by inertial lift or shear-induced diffusion, whereas cake formation is prevented at sufficiently high shear rates and the flux then becomes independent of shear rate. The flux data for unwashed yeast in water for the ceramic tubes show a gradual increase with shear rate, whereas the trend is less clear due to scatter for the polypropylene tubes. From Figure 13, the data for washed yeast in water for the polypropylene membrane show a string increase in the steady-state flux with increasing shear. The values are only 10-30 9% below the theoretical predictions using the value of the crossflowintegral for rigid spheres in the shearinduced diffusion theory, whereas the corresponding values for the unwashed yeast are nearly an order of magnitude lower and do not show a strong increase with shear. This difference is due to the presence of extracellular macromolecules with the unwashed yeast; these molecules cause the cake to have a higher resistance and to be adhesive so that increasing the shear is not as effective in limiting the cake formation as it is for the washed yeast. Figures 11 and 14 show the effect of increasing the bulk suspension concentration of unwashed yeast in water for the ceramic and polypropylene tubes, respectively. The experimentaldata and the shear-induceddiffusion models show a monotonic decrease in the steady-state flux with increased cell concentration in the feed. This is because thicker cake layers form when the concentration is higher. In contrast, the inertial lift theory is independent of concentration. The steady-state flux shown in Figure 14 for an experiment with unwashed yeast in a rich medium is not significantly different than that for unwashed yeast in water for the same operating parameters. The shear-induced diffusion theory without cake adhesion (short-dashed lines) and the inertial lift theory (long-dashedlines)overpredict the flux for the experiments with unwashed yeast using the ceramicmembrane (Figures 9-11). For the polypropylene membrane, the shearinduced diffusion theory without cake adhesion overpredicta the steady-state flux for unwashed yeast, but the inertial lift theory underpredicts it (Figures 12-14). The primary difference in operating conditions for the ceramic and polypropylene membrane experiments is that higher
0.006
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A P (psi)
Figure 12. Comparison of the theoretical and the measured steady-state fluxes at different transmembrane pressures for unwashed yeast in water using the polypropylene fiter with yo = 4370 8-l and &, = 0.03. The symbols are the experimental data, the solid line is the shear-induced diffusiontheory with the crossflow integral chosen asthe average value for this suspension/ filter system,the shorbdashedline is the shear-induceddiffusion theory with the crossflow integral for rigid, nonadhesive spheres, and the long-dashed line is the inertial l i t theory.
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Figure 13. Comparison of the theoretical and the measured steady-state fluxes at different bulk suspension flow rates for unwashed yeast (A)and washed yeast ( 0 ) in water using the polypropylene filter experimentswith hp = 10 psi and & = 0.03. The symbols are the experimental data,the solid line is the shearinduced diffusion theory with the crossflow integral chosen as the average value for this suspension/filter system, the shortdashed line is the shear-induced diffusion theory with the croseflow integral for rigid, nonadhesive spheres, and the longdashed line is the inertial lift theory. shear rates were used with the polypropylene filter, and this gives relatively higher inertial lift velocities due to the square dependence in eq 9. The theory with cake adhesion (solid lines) is in better agreement with the experimentaldata for unwashed yeast, but unlike the other models it does contain an adjustable parameter (12). For the washed yeast, using the polypropylene membrane,the shear-induced diffusion theory for nonadhesive, rigid particles is in good agreement with the measured data (Figure 13).
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Figure 14. Comparison of the theoretical and the measured steady-state fluxes at different feed volume fraction of cells for unwashed yeast in water (A)and a rich medium (*) using the polypropylene fiiter with M = 10 psi and ro= 4370 s-l. The symbols are the experimental data, the solid line is the shearinduced diffusion theory with the crossflow integral chosen as the average value for this suspenison/filter system, the shortdashed line is the shear-induced diffusion theory with the crossflow integral for rigid, nonadhesive spheres, and the longdashed line is the inertial lift theory.
In Figures 15 and 16, the dimensionless steady-state flux ((Ja)/Jo) versus the dimensionless filter length (L/ xcr,where x cr = {:a41dJi) is plotted for the unwashed yeast in water with the ceramic and polypropylene tubes, respectively. The dimensionless filter length for each experiment was computed using the average value of IZ accounting for cake adhesion. The solid line is based on the theory of Romero and Davis (1988). The theory contains a family a curves for different values of the dimensionless cake resistance, @ = &Ho/R,. From the average values of 8, and R , reported earlier, the values used for this parameter are @ =I 13 and /? = 11,respectively, for the ceramic and polypropylene tubes. The symbols are the measured data from the experiments. The experimental data follow the predicted trend for all values of the dimensionless filter length, with moderate scatter observed. 4. Concluding Remarks Experiments with yeast suspensions filtered through ceramic and polypropylene tubes under crossflow conditions confirm that the steady-state flux increases with increasing wall shear rate and decreasing feed concentration. These results for washed yeast in water are 1030% below theoretical predictions using a shear-induced diffusion model with diffusivity and viscosity correlations for nonadhesive rigid spheres. For unwashed yeast, however,extracellular macromoleculescause a severalfold reduction in the steady-state flux. The model may still be employed, provided that a parameter referred to as the crossflow integral is determined by fitting the data. Since this parameter is lower for unwashed yeast than for nonadhesive rigid spheres, it is proposed that the cake formed on the membrane is partially adhesive, which implies that the cells in the cake are not free to diffuse away. The measured steady-state fluxes show no significant dependenceon transmembrane pressure in the range 5 Ihp I35psi, whereas the theory predicts a weak increase in flux with pressure. Inertial lift theory was found to
106
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Figure 15. Dimensionlesssteady-stateflux versus dimensionleas fiiter length for unwashed yeast in water using the ceramic filter. The solid line is the theory based on shear-induced diffusion with cake adhesion and the symbols are the measured data.
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Figure 16. Dimensionlesssteady-stateflux versus dimensionleas fiiter length for unwashed yeast in water using the polypropylene fiiter. The solid line is the theory based on shear-induced diffusion with cake adhesion and the symbols are the measured data.
overpredict the observed fluxes for yeast in polypropylene tubes at high shear rates and to underpredict the fluxes for ceramic tubes at lower shear rates. The transient flux decline under crossflow conditions was observed to follow dead-end filtration theory, independent of the shear rate, for short times. For long times, however, the measured fluxes were observed to level out and reach a near-steady-state value. This results from the hydrodynamicshear imposed by the crossflowarresting the cake growth. The initial flux and the rata of flux decline for short times were used to determine the membrane resistance and the specific cake resistance, respectively. The specific cake resistances for the unwashed yeast are on the order of (2-4) X 10l2 cm-z. The specific cake resistances for the washed yeast are much lower, on the order of (0.5-1.0)X lo1' cm-2. Ofsthun (1989)&observed a large reduction in the cake resistance for freshly rehydrated yeast cells after washing. For pressures in the
634
range employed here, she measured the flow rate through a stagnant yeast bed and found 8, = (5-6)X 10" cm-2 for unwashed yeast, and 8, = (4-6) X 1O1O cm-2 for washed yeast. A possible reason for the lower resistances observed by Ofsthun is that she used larger cells (6.4 versus 4.2 p m average diameter). In addition, it is likely that cakes formed under crossflow conditionsare more compact that those formed dead-end or stagnant conditions. The specific cake resistances for the unwashed yeast are considerably higher than those predicted by the BlakeKozeny equation for laminar flow though incompressible packed beds of monodispersed, rigid spheres (Bird et al., 1960):
Bbtechnol. hog., 1993, Vol. 9,
No. 8
Literature Cited
Althena, F. W.; Belfort, G. Lateral migration of spherical particles in porous flow channels: Application to membrane fiitration. Chem. Eng. Sci. 1984,39,343-355. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960;p 199. Blatt, W. F.; Dravid, A.; Michaels, A. S.; Nelson, L. Solute polarization and cake formation in membrane ultrafiltration: causes, consequences and control techniques. Membrane Science and Technology; Flinn, J. E., Eds.; Plenum Press: New York, 1970;pp 47-97. Cheryan, M.; Mehaia, M. A. Ethanol production in a membrane recycle reactor. Conversion of glucose using Saccharomyces cereuisiae. Process Biochem. 1984 (Dec), 204-208. Cox, R. G.; Brenner, H. The lateral migration of solid particles in Poiseuille flow-I theory. Chem. Eng. Sci. 1968,23,147173. Darcy, H. P. G. Les Fontaines Publiques de la Ville de Dijon; from which 8, = 5 XlOlO cm-2 for 4c = 0.78 and 2a = 4.2 Dalmont: Paris, 1856. pm. This prediction is in good agreement, however, with Davis, R. H. Modeling of fouling of crossflow microfiitration the specific cake resistance that we observed for the washed membranes. Sep. Purif. Methods 1993,21 (2),75-126. yeast cells. Davis, R. H.; Leighton, D. T. Shear induced transport of a particle layer along a porous wall. Chem. Eng. Sci. 1987,42,275-281. Drew, D. A.; Schonberg,J. A.; Belfort,G. Lateral inertial migration Notation of a small sphere in fast laminar flow through a membrane a particle radius duct. Chem. Eng. Sci. 1991,46,3219-3224. D shear-induced diffusivity Eckstein, E. C.; Bailey, P. G.; Shapiro, A. H. Self diffusion of particles in shear flow of a suspension. J. Fluid Mech. 1977, characteristic shear-induced diffusivity, +,a2 Do 79,191-208. tube radius Ho Errede, L. A. Effect of organic ion adsorption on water permedimensionless crossflow integral I2 ability of microporous membranes. J. Colloid Interface Sci. J permeate flux 1984,100,414-422. pure water flux Jo Hoffman, H.; Scheper, T.; Schugerl, K. Use of membranes to improve bioreactor performance. Chem.Eng. J.1987,34,B13steady-state permeate flux J, B19. L fiiter tube length Kavanagh, P. R.; Brown, D. E. Crossflow separation of yeast cell AP transmembrane pressure suspensions using sintered stainless steel fiiter tubes. J.Chem. feed flow rate Technol. Biotechnol. 1987,38,187-200. dimensionless excess particle flux Leighton, D. T.; Acrivos, A. Measurement of the shear-induced coefficient of self-diffusion in concentrated suspension of cake resistance per unit depth 8, spheres. J. Fluid Mech. 1987,177,109-131. Re channel Reynolds number, 2pOQIu~oHo Matsumoto, K.; Katsuyama, S.; Ohya, H. Separation of yeast by membrane resistance Rm cross-flow fiitration with backwashing. J.Ferment. Technol. t time from start of water filtration 1987,65,77-83. time at start of yeast filtration Ofsthun, N. J. Crossflow membrane fiitration of cell suspensions. to Ph.D. Thesis, Massachusetts Institute of Technology, Caminertial lift velocity (eq 9) Ul.0 bridge, MA, 1989. xcr critical distance for cake formation, +&I2/ Jt4b Porter, M. C. Ultrafiltration of colloidal suspensions. Recent dimensionless cake resistance, 8 J l J R m B Advances in Separation Techniques,AIChE Symp. Ser. 1972, cake thickness 68 (NO. 120), 21-30. 6, Redkar, S. G. Crossflow microfiltration of complex mixtures. YO nominal shear rate, 4QlwHi M.S. Thesis, University of Colorado, Boulder, CO, 1992. pure fluid viscosity P O Romero, C. A.;Davis, R. H. Global model of crossflow microbulk volume fraction of cells 4b fiitration based on hydrodynamic particle diffusion. J.Membr. Sci. 1988,39,157-185. volume fraction of cells in cake layer 4c Romero, C. A.; Davis, R. H. Experimental verification of the volume fraction of cells just above cake layer 4w shear-induced hydrodynamic diffusion model of cross-flow pure fluid density Po microfiitration. J. Membr. Sci. 1991,62,249-273. 7 time constant defined by eq 3 Rubinow, S. I.; Keller, J. B. The transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech. 1961,11, 447-459. Acknowledgment Szlag, D. C. Factors affecting yeast flocculation. M.S. Thesis, University of Colorado, Boulder, CO, 1988. Zydney, A. L.; Colton, C. K. A concentration polarization model This research was supported by the Center for Sepafor the filtrate flux in crossflow mirofiltration of particulate rations Using Thin Films at the University of Colorado at suspensions. Chem. Eng. Commun. 1986,47,1-27. Boulder and by Grant CTS-9107703 from t h e National Science Foundation. The authors thank Dawn Downey, Accepted June 11, 1993.' Annetta Razatos, and members of the University of Colorado's Undergraduate Research Opportunities Pro'Abstract published in Advance ACS Abstracts, September 1, 1993. gram for help in the experiments.
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