EnvirOn. Sci. Techno/. 1992, 26, 1611-1621
Particle Transport in Clean Membrane Filters in Laminar Flow Shankararaman Chellam and Mark R. Wlesner'
Department of Environmental Science and Engineering, P.O. Box
rn Particle transport and deposition studies in laminar cross-flow membrane filtration are reported. Particle residence time distributions (RTDs) in experimental conditions typical of ultra- and microfiltration are compared with theoretical predictions incorporating the effects of hydrodynamic, Coulombic, electrodynamic, and external gravity forces. Numerical simulations show that, for a given flow field, mechanisms controlling lateral migration in the far-field region in membrane filters depend primarily on inertial, gravity, and permeation drag forces. The theory accurately predicts first passage times and multimodal RTDs under conditions of high membrane permeability and fast axial flows. Differences between experimental and theoretical RTDs are interpreted as evidence of shear-induced particle resuspension, transport along the membrane surface, and/or unfavorable attachment phenomena in the near-field region. Such an approach may be useful in screening membrane technologies for water and wastewater treatment based on the size distribution of particles in the feed water. Introduction Recent developments in the technology and economics of membrane separation processes have motivated much research in the evaluation of membranes for water and wastewater treatment (1). The accumulation of rejected materials on or near the membrane frequently leads to a decline in the permeate flux over time. Irreversible losses in flux have been termed fouling whereas hydrodynamically or chemically reversible losses are referred to as reversible fouling or as colmatage from the French word for clogging (2). These phenomena play an important role in determining the feasibility of membrane technologies as reductions in membrane flux decrease the effective capacity of membrane installations. Much work has been done to describe colmatage by concentration polarization in perikinetic systems of solutes and small macromolecules where convection and Brownian diffusion are the principal transport phenomena (3, 4). Less is known about the effects of colloidal and macrocolloidal materials on the performance of pressure-driven membrane processes such as ultrafiltration (UF)and microfiltration (MF). Transport mechanisms in addition to convection and Brownian diffusion have been proposed for these larger species. Colloidal materials with a wide variety of physicochemical characteristics are encountered in water and wastewater treatment applications. A theory that accounts for differences in the density, size, and surface chemistry of the contaminants is needed to predict the potential for fouling and colmatage of membranes. The dynamics of the formation and growth of concentration boundary layers and deposited layers that may impede the flow of water across the membrane can be analyzed in two separate steps of transport and attachment. Several transport mechanisms may interact to bring particles close to the membrane surface or to preferentially transport them away from the membrane. Particles that are transported sufficiently close to the membrane may deposit if electrostatic repulsion, London forces, and other near-field factors favor attachment to the membrane or to previously deposited materials. Calculations of the trajectories followed by particles as they move through 0013-936X/92/0926-161 1$03.00/0
1892,
Rice University, Houston, Texas 7725 1
membrane units can incorporate multiple transport mechanisms and may serve as a basis for predicting the colmatage and fouling of membranes by colloidal species. Analysis of particle trajectories in packed-bed filters has proved to be a useful approach in interpreting the performance of filters in water and wastewater treatment (5-8).
Calculations of the trajectories of particles in vertical tubular membrane ducts were first undertaken by Hung and Tien (9). They considered the effects of drag, gravitational, and surface forces on particle transport. The torque acting on the particle was assumed to be uniformly zero. Altena and Belfort (10) later developed a particle trajectory model by considering only drag and inertial effects on rotating neutrally buoyant particles. Previous experimental studies to test transport theories have been conducted with very large (113 pm), neutrally buoyant particles in slow flows in a vertical channel with low wall permeability (11). Belfort (12-14) has reviewed the theories of fluid mechanics in describing particle transport in membrane filters. There is a need for more theoretical and experimental investigations of particle transport in membrane channels operating under conditions typical of ultra- and microfiltration applications encountered in water and wastewater treatment. In this paper, we extend the theory developed by Altena and Belfort to include the effects of sedimentation, van der Waals attraction, double-layer repulsion, and added mass of entrained fluid on lateral migration. Calculations of the magnitudes and relative importance to particle transport of each of these mechanisms are discussed. The validity of the transport theory is evaluated by comparing computer-generated trajectories and residence time distributions (RTDs) of rigid, negatively buoyant colloidal particles in the size range 0.24-12.85 pm with experimental data produced under cases of permeation- and inertial-drag-dominated lateral transport in a horizontal membrane channel. (In this paper, particle size refers to particle radius, up.) Theoretical Analysis The experimental investigations of particle transport presented in this paper were carried out in a channel with one porous wall having a rectangular cross section. The following theoretical development also utilizes this geometry. A schematic of the flow channel and the coordinate system used in developing an equation for particle trajectories is described in Figure 1. L and h are the channel length and height, respectively. The origin of the coordinate system 0 is located at the intersection of the channel entrance and the membrane surface. The positive dimensionless transverse distance coordinate, 0,is measured from the porous wall, and the positive dimensional axial distance coordinate, x , is measured from the channel entrance. The path followed by a particle in the membrane channel can be determined by performing a force and torque balance on the particle using classical mechanics (Figure 1). Here, the general case of a negatively buoyant, freely rotating particle suspended in 2-D flow subiect to a nonuniform shear field is considered. The force (9and torque (!f') balances under conditions of dynamic equilib-
0 1992 American Chemical Society
Environ. Sci. Technoi., Vol. 26, No. 8, 1992
1611
T
+
Double-layer repu‘siGn
h
I
x +
LL’I Flgure 1. Schematic representation of a parallel plate membrane filter showing steady 2-D Poiseuille flow in a channel bounded by one porous wall. 0 is the origin of the coordinate system. Also shown are the forces and torques acting on a charged, spherical particle at finite Reynolds number in the proximity of the membrane. The particle velocity has both rotational and translational components which have to be considered in determining its trajectory.
rium of the suspended particle are vector equations and are written in terms of the different components as
-
$drag
-,
4
Tdrag
4
+ p a d d e d ma% + Fpressure gr$dient + Finertia Fdouble layer + F v a n der Waals + Fgravity +
Tpressure glradient
+
+ Tvander
Waals
4
Tadded mys
+
Tdoublelayer
6
(1)
where urn*is the dimensional local maximum axial velocity and f(P) is a function only of the dimensionless transverse coordinate, P, for a given flow field and membrane geometry:
f(P) = fo(P) + Re, fiW + 0(Rew2)
(5)
Tinertia_+
+ Tgravity = 6
(2)
Equations 1 and 2 are linked in that the torque balance is needed to calculate the rate of particle rotation, which in turn is required to calculate the inertial force. It is noted that fluid inertia, pressure gradient, gravity, and added mass do not induce any torque as they are assumed to act through the center of mass of the particle. It is further assumed that any surface charge on the particle is uniformly distributed, and therefore, electrostatic and electrodynamic interactions are also absent from the torque balance. Particles are assumed to exist in a dilute suspension so as to neglect the effect of particle-particle interactions. Given ita stochastic nature, Brownian motion which is the result of fluid-particle interactions, is not incorporated in the trajectory equation. Interactions arising from adsorbed layers and structure of the fluid are also not considered. In the remainder of this section, equations are developed corresponding to each of the forces and torques in eqs 1 and 2. Undisturbed Fluid Flow. Consider isothermal, incompressible, laminar flow in a channel with the geometry as illustrated in Figure 1. Let u, represent the uniform inlet velocity, u, the constant permeation rate, and v the kinematic viscosity of the fluid. Chellam et al. (15) presented a first-order regular perturbation solution for fluid flow in such a module, assuming an arbitrary slip velocity at the membrane surface. For no-slip at the permeable boundary, their solution reduces to the following as given by Green (16):
where fo(P) = -0
- PY (1 + 2P)
(6)
1
fi(p) = - 7o (-19~2+ 4303 - 3544 + 2165 - 1 4 ~ 6+ 407) (7)
and Re, (=v,h/v) is the dimensionless wall Reynolds number. Viscous Drag. Interactions between spheres give rise to extra drag but can be ignored by assuming that particles exist in a sufficiently dilute suspension (a reasonable assumption for considerations of initial particle transport in clean membrane filters). The following discussion applies only to wall effects on the motion of an isolated sphere. (a) Axial Drag. Let K (=ap/h)denote the dimensionless particle size. It has been shown both theoretically (17,18) and experimentally (19) that a small ( K 10 and the surface potentials are less than *60 m$. Relative Importance of Transport Mechanisms. Estimates for the lateral migration velocities in a typical MF or UF membrane module (30 Iu, I100 cm/s, 5 X Iu, I6 X cm/s) induced by each one of the mechanisms discussed in previous sections are summarized in Table I for particles in the size range 50 nm Iup I 10 pm. Maximum values for each velocity have been used to make comparisons and double-layer and van der Waals interactions have been computed at a surface to surface distance of one particle radius. At this distance, the Stokesian drag is magnified by a factor of 2.12 (see eq 11). Under these conditions, Brownian motion and virtual mass velocities are negligible and therefore can be neglected in trajectory calculations in UF and MF. In the far-field region, permeation drag, inertial lift, and sedimentation are expected to dominate particle motion in the transverse direction. Velocities induced by drag and lift forces may be 1 order of magnitude higher than settling velocities. Lateral migration induced by electrokinetic effects is calculated to be important only at small membrane-particle separations and can therefore be neglected in calculating particle transport in the far-field region. Experience with membrane filtration in industrial water treatment has indicated that membrane charge can be an important variable in controlling colmatage and fouling. These calculations indicate that membrane charge is likely to play a role in membrane fouling as it affects deposition and/or adsorption of materials on the membrane rather than their transport in the far-field region of the membrane element. If physical transport mechanisms bring a particle to the near-field region of the membrane, van der Waals and electrical double-layer interactions are expected to contribute to lateral migration and might be calculated explicitly. Alternatively, it may be useful to lump nearfield interactions into an empirical "sticking coefficient" as has been done in describing particle attachment in packed-bed filters (32). The inclusion of dispersion and
Table 11. Input Values for Different Parameters Used in Theoretical Calculations and Experiments' figures
u, (cm/s)
u, (cm/s)
ap (rm)
channel geometry (membrane type)
2-4
26.61
0.019
3.5
46 cm X 2 cm X 762rm
12.85
(Track-Etch membrane) 46 cm X 2 cm X 762 pm (PVDF membrane)
5-7 ap,
35.30
0.017
= 1 g/ mL, pp = 1.05 g/ mL,
Y
= 0.01 cm2/s, AH = 5
X
repulsive forces in the near-field region and gravity forces at all transverse positions will necessarily provide a more accurate description of particle motion in a membrane module. Particle Trajectories and Residence Time Distributions. The trajectory model developed by Altena and Belfort (IO)does not account for chemical interactions and is therefore valid strictly in the far-field region. The effect of sedimentation for nonneutrally buoyant particles was also excluded. The particle trajectory equation developed below accounts for lateral migration induced by permeation drag, inertial, sedimentation, electrostatic, van der Waals, and virtual mass forces. Let t denote the time elapsed after the particle enters the membrane channel. The particle velocity in the axial direction, u , is obtained by a summation of the undisturbed fluicfvelocity (eq 3), the particle slip velocity (eq 8), and the velocity induced by virtual mass effects (eq 15): dx u p = T t =
Vectorial addition of eqs 4,14,17,20,22, and 25 gives the particle velocity in the transverse direction up.
Dividing eq 27 by eq 26, we eliminate the variable t to obtain an ordinary differential equation which describes the position trajectory, B ( x ) , of the particle in the membrane channel
Let the particle enter the channel at x = xi = 0 , p = pi;the subscript i denotes initial values for the variables x , t, and 0,and the subscript f denotes final values a t the channel exit. Thus, the particles' position trajectory in the closed interval [ x i = 0, x = L] is an initial value problem that can be solved by the Eulers method. The time trajectory, P ( t ) can be derived by integrating the reciprocal of eq 28 (where B is an implicit function of x , as given by the position trajectory) to obtain the functional dependence x ( t ) . For submicron particles, trajectory simulations should include a stochastic variable representing Brownian motion. Such calculations usually start from the Langevin
Re,
Re,
Re
8
4.6 X
0.145
1.4
304
16
0.0169
0.127
6.8
403
K
J, A, = 100 nm, 1 / =~90 ~nm, ionic strength =
0.17
M, $B = -30 mV.
:i----_=1 0.8
I
0
10
20
30
40
Axial distancefrom thechannelentrance(crn)
Figure 2. Positlon trajectories of small particles undergoing lateral transport domlnated by permeation drag.
equation and the inclusion of relevant external forces (33). Each term in the numerator of eq 28 corresponds to a specific lateral migration mechanism. Calculations similar to those made in Table I could be made to determine the relative importance of various lateral migration velocities and find out which ones are important for the system under consideration. For example, the ratio 6 = u,/ U,RepK2 describes the relative importance of permeation drag to inertial lift. 6 >> 1 corresponds to permeation drag dominated transport, whereas if 6 = 0(1), both drag and inertial lift should be considered in calculating particle trajectories. Theoretical particle trajectories were calculated for a range of particle sizes. In these calculations, the slip velocity of the particle in the axial direction (eq 9) was incorporated only in the region 0.1 5 0 I0.9. All numerical computations were performed on a Sun Sparc-2 system with double-precision arithmetic using FORTRAN as the programming language. Table I1 gives input values for computer simulations and corresponding experimental parameters. In addition to the Reynolds numbers based on particle size, Re and the wall permeation rate, Re,, Table I1 gives the cfannel Reynolds number, Re, and the value of 6. Permeation Drag Dominated Transport. For high values of 6, inertial effects are negligible and lateral migration is controlled by permeation velocity. Computergenerated trajectories of 3.5-gm particles undergoing permeation drag dominated transport in the channel are shown in Figure 2 (6 = 16). If particles accumulate in a cake and/or concentration polarization layer near the membrane, a reduction in permeate flux may occur due to the additional resistance to permeate flow presented by these deposited materials. Under the assumed hydrodynamic conditions, 3.5-gm particles are predicted to follow fluid streamlines closely except at lateral positions very close to the solid wall. Particles entering the membrane channel at a distance of p = 0.45 or less follow trajectories that appear to intersect the membrane. More accurately, these particles follow trajectories that transport them to the near-field region of the membrane. Brownian diffusion, electrokineticinteractions, and other near-field phenomena determine whether particles attach to the membrane, roll Envlron. Sci. Technol., Vol. 26, No. 8, 1992
1615
1 .o 0.9
3
3
E %
0.8
/I
1
1
id
I
Theoretical RTD
0.8
0’7
o.6 0.5
i
0.4
P
3 0.3 ..e E
I-
L=46cm h = 762pm ap=3,5~m Uo=26.61 cmtsec Vw=0.019 c m m x Membrane: Track-Etch
L=46cm h=762cm Uo =26.61cm/ssc Vw =0.019crn/scc sp=3.5prn
0.2
o. 1 0.0 0.0
2.0
4.0
6.0
Time to traverse the channel length (sec) Figure 3. Residence time of 3.5-pm particles as a function of initial lateral position. Trajectories of particles with initial lateral position higher than 0.45 do not intersect the plane of the membrane.
along the surface, or are resuspended. A baseline for evaluating the importance of near-field phenomena can be obtained by considering the case where all particles entering the near-field region attach to the membrane and are “removed” from the system. This baseline or maximum theoretical removal of particles in the near-field region is obtained by identifying the limiting trajectory, defined as the trajectory intersecting the plane of the membrane at a distance x = L + up associated with a particle introduced at the {argest initial lateral position, Pcrit. Particles entering at higher Piwill escape any possibility of capture and will exit along with the reject (concentrate) stream whereas trajectories of all particles entering at smaller values of & are calculated to intersect the membrane surface. Figure 3 shows the passage times for 3.5-1m particles introduced uniformly at t = 0 across the channel entrance. Only 55% of these particles are predicted to exit the channel (Pcrit = 0.45) without depositing on the membrane. In the baseline case, trajectories that intersect the membrane are assumed to be captured irreversibly without any possibility of reentrainment and their residence time is therefore assumed to be infinite. Also, due to insignificant inertial lift force, particles introduced near the solid wall continue to be associated with the slower moving fluid streamlines and their transit times are predicted to be very long. For example, a 3.5-pm particle released at 0 = 0.99 will be pushed only to slightly faster moving fluid elements. Therefore, the transit time for this particle is predicted to be =7 s. Figure 3 can be used to generate theoretical residence time distributions for a Dirac pulse of particles into the channel at (xi= 0, ti = 0) by counting the fraction of trajectories exiting the channel at predetermined time intervals. The theoretical residence time distribution for the 3.5pm particles is unimodal under the assumed conditions and is shown in Figure 4 superposed on the experimental RTD. (The experimental RTD is discussed in the following section on Experimental Work.) The single peak of the theoretical RTD is associated with fast-moving particles close to the channel centerline that exit the channel first. Particles entering the channel close to the solid wall constitute the tail that skews the RTD toward the larger residence times. Particles entering near the membrane are assumed to be captured by the membrane and therefore do not contribute to the area under the theoretical RTD. As particle size decreases, particle transport is increasingly dominated by permeation drag. Thus, particles smaller than the 3.5 pm simulated here also tend to follow fluid streamlines, producing trajectories and RTDs nearly 1616
Environ. Scl. Technoi., Vol. 26, No. 8, 1992
5
0
10
Time elapsed after injection (sec) Figure 4. RTD of 3.5-pm particles in the membrane filter. Particles undergo transport dominated by permeation drag, and a unimcdai peak is observed/predicted.
0
I
I
10
20
I
30
40
Axial distance from the channel entrance (crn) Figure 5. Position trajectories of large particles whose transport in the lateral direction is dominated by inertial forces.
identical to those shown in Figure 2. Trajectory simulations of these relatively small particles might represent bacterial transport in membrane systems. Since the transport of such bacteria-sized particles is dominated by permeation drag, transport to the membrane surface is predicted. If deposited, bacteria may secrete polymers that may result in irreversible adhesion and/or adsorptive fouling of the membrane. Fouling of membranes by bacteria can be an important operational problem in some membrane installations (34). Inertia-Dominated Transport. Smaller values of 6 correspond to larger influence of inertial lift on particle transport. Trajectories obtained for 12.85-pmparticles are given in Figure 5 (6 = 0.17). For these hydrodynamic conditions, particles starting close to the membrane experience an inertial lift force toward the longitudinal axis of the channel which counteracts the effects of permeation drag, gravity, and attractive forces and results in particle migration across fluid streamlines. Thus, lateral transport is inertia-dominated and particles should not be transported to the membrane. The passage times of particles uniformly introduced into the channel at t = 0 undergoing inertia-dominated transport are plotted as a function of initial lateral position in Figure 6. The particles that exit first are associated with the fastest moving fluid elements near the channel centerline. Particles starting initially from positions close to the solid wall (large Pi) experience smaller permeation drag forces than do particles starting at similar lateral positions from the membrane (small P i ) and are moved to fluid elements having higher axial ve-
Particle free electrolyte
Uo=35.29cm/su:
9
0.8
-
vw=o.ol7cm/su: ap=12.85pm L=46cm
SOIUtIO.
Tctloohtbing Pulseless
gearpump
3 .-* 5
I
t 0.2
t
First
passage 0.0
0.5
I
\
1
1.o
I
\
i
Monodisperse
Time to cross the channel length (sec)
Syringe pump
Figure 8. Residence time of 12.85-pm particles as a function of initial lateral position. All trajectories are predicted to cross the membrane module without intersecting the plane of the membrane. 1.0
Turbulsnce
2.5
2.0
1.5
P
I
Flow meter
r
1
I
n
UVdetrctar Pressuregage
(to data acquisition)
Cancentrats
Membraozfiltratmaunit regulating valve
+Psrmeats(to + +balanceand 4 + dataa quisitionsystem) Concentrationdata
Uo=35.29 cdsrc Vw=0.017 c d s e f ap=12.85 pm
Expcnmental RTD
L=46 cm h=762 pm
Membrane: PVDF
n " n " 0
2
4
6
8
Time elapsed after particles entered the channel (sed
Figure 7. RTD of particles whose transport in the lateral direction is dominated by inertial effects. The RTD derived from theoretical analysis is superposed on the experimentally determined RTD.
-
Ruxdata
locities. Hence, particles entering near the nonporous wall (pi 1) exit the channel sooner than particles entering near the membrane (Pi 0), and the transit times are asymmetric about the longitudinal axis (even though the axial velocity profile is symmetric about the same axis). Again, this plot can be transformed into a RTD which is compared with an experimentally observed RTD in Figure 7. Discussion of the experimentally observed RTD is postponed until the following section on Experimental Work. The asymmetry inherent in the system produces triple peaks in the theoretical residence time distribution of particles with inertia-dominated transport. Particles that spend time toward the channel center (6 = 0.5) contribute to the first peak, the second peak is due to particle trajectories in the upper part of the channel near the solid wall @ 1 0.7),and the last peak is associated with particles closer the membrane boundary (0 50.3). The redistribution of particles initially close to the walls at ,6 = 0.79 and /3 = 0.13 as seen in Figure 5 causes a sharp tail in the RTD.
-
Experimental Work Experiments were designed to determine the RTDs of dilute suspensions undergoing laminar flow in a membrane filter with the goal of comparing theoretical and observed RTDs. A membrane filtration cell similar to that described by Granger et al. (35)was constructed to measure particle RTDs. Particle injections were made at the inlet of the membrane module and particle concentrations monitored at the reject side. Thus, only particles that escape capture by the membrane contribute to the RTD. A schematic of the experimental setup is given in Figure 8. A more detailed discussion of the experimental apparatus, accu-
racy, and protocols is presented elsewhere (36). Numerical simulations of momentum boundary layer profiles indicate that the flow becomes fully developed very early in the filter and that the entrance region is 1% of the channel length. Materials and Procedures. The filtration apparatus consisted of an accurately machined Teflon sheet (46 cm X 2 cm X 762 pm) between two Plexiglass blocks held together by 24 fasteners, all tightened to a uniform torque. The upper block constituted the solid wall while the membrane was housed in the bottom portion. A polyethylene porous sheet (thickness 1/16 in., pore diameter 70 pm, CMS Inc., Houston, TX) provided the mechanical support for the membrane. The membrane support was made hydrophilic by soaking it in a detergent. A Spectra 100 UV detector (Spectra Physics Inc., San Jose, CA) installed at the filter concentrate side was used to determine particle RTDs. The detector was auto-zeroed before each injection. Flux data were acquired using a FX-3000 electronic balance (A&D Co., Milpitas, CA) coupled with a SER420 serial 4-20-mA converter (Rice Lake Weighing Systems, Rice Lake, WI). Five different sizes (0.24,0.47, 1.6, 3.5, and 12.8 pm) of rigid, spherical polystyrene latex particles (Seradyn Corp., Indianapolis, IN) were used as model colloids. All experiments were performed with colloids larger than the membrane pores to avoid effects of pore penetration and possible fouling. Experiments were performed using two types of hydrophilic membranes: a modified polyvinylidene difluoride (PVDF) membrane having a nominal pore diameter of 0.1 pm (Durapore, Millipore Corp., Bedford, MA) and a Track-Etch polycarbonate membrane (0.1-pm average pore diameter, Poretics Corp., Livermore, CA). All experiments were conducted using ultrapure, organic free water (Milli-Qsystem, Millipore Corp., Bedford, MA). The water was typically near pH 6.3 and had a resistivity near 18 Mi2 cm. The electrophoretic mobilities (measured using a Malvern Zeta Sizer 2c, Malvern, England) of the latices were typically near -2.27 pm/s V/cm, corresponding to a {potential of
-
Environ. Sci. Technol., Vol. 26, No. 8, 1992
1817
6.0
t
4.0
1
a .3 8'o 2 u
o o v 00
.
Dab from PVDF membrame Data from Track-Etch membrane
0 I
20
I
60 Theoretical first passage time (sec) 40
I
80
Figure 9. Comparison of theoretical and experimental observations of the first passage time of 0.24, 0.47-, 1.6-, 3.5-, and 12.85-pm particles.
--33.5 mV. Near-pulse-free flow of water was achieved using a micropump head (Cole-Palmer Instrument Co., Chicago, IL, Catalog No. 07002-27) fitted on a Ismatec digital variable-speed drive. Particle transport was studied with feed flow rates between 2.04 and 5.38 mL/s, corresponding to average inlet axial velocities of 13.4 and 35.3 cm/s. Permeate flow rates were in the range 0.68-2.97 mL/s (0.008 5 u, 5 0.034 cm/s), corresponding to transmembrane pressures of 3-10 psi. Digital instrument control and remote signal monitoring were accomplished using a data acquisition system for the Macintosh computer (MacPacq, Biopac Systems, Goleta, CA). Samples were acquired at a rate of 100 s-l giving a resolution of 10 ms for the first transit time. The resolution of the data acquisition system in all experiments was 20 mV. Pulse inputs of short durations from 200 to 400 ms were made using TTL control of a syringe infusion pump (Harvard Apparatus Inc., Model 22, South Natick, MA). Residence time distributions were obtained as voltsecond plots. Typically each experiment was repeated six times, and the standard deviation for the first passage time was no more than 60 ms. After each experiment, the area under the curve was calculated and used to normalize the data for comparison with theoretical expectations. Impulse inputs of an aqueous solution of sodium dichromate were used to conduct tracer studies for system calibration of the first passage time to account for any nonidealities in the apparatus. A linear regression (r2= 0.999) was obtained from the tracer study, correlating experimental observations of the first response and theoretical predictions of the first passage time. This correlation was later used to adjust experimental observations of first responses obtained from particle injections for comparison with theoretical predictions of the first passage times of particles.
Results and Discussion First Passage Time Studies. A number of measures may be used to compare experimental and theoretical RTDs including the first passage time, modality, mean, variance, and skew. The first passage time is defined as the shortest time for particles uniformly introduced into a membrane module to cross the entire length of the module. It is associated with the fastest moving particles introduced close to the module longitudinal axis and therefore is a reflection of the physical transport mechanisms in the far-field region of the membrane. A comparison of theoretically anticipated fiit passage times and experimental first responses for the five particle sizes investigated using both membranes is shown in Figure 9. It is seen that there is very good correspondence between predicted and observed values for the first passage times 1618 Environ. Sci. Technot., Vol. 26, No. 8, 1992
of particles through the porous channel. Thus, the transport of particles that are relatively unaffected by the action of the near-field forces in the membrane channel is accurately predicted by eq 28. Comparison of experimental and theoretical RTDs by other measures reveals both points of agreement and divergence. Modality and Skew. (a) Permeation Drag Dominated Transport. The residence time distribution for a 3.5-pm particle undergoing lateral transport dominated by permeation drag is given in Figure 4 along with the theoretical prediction. As predicted by theory, the experimentally observed RTDs are unimodal. In contrast with theoretical predictions, long trailing edges were obtained in all the experimental RTDs skewing the distribution toward larger residence times. As discussed earlier, a fraction (55%) of the trajectories of the influent 3.5-pm particles are expected to intersect the membrane. Theoretical RTDs are derived with the assumption that nearfield phenomena favor irreversible deposition of particles transported to the membrane. The long trailing edges of the experimental RTDs are interpreted as artifacts of sticking and transport phenomena near the membrane. Orthokinetic transport mechanisms leading to particle transport along the membrane surface (37, 38), viscous resuspension from the membrane toward the bulk flow (37, 39) due to shear-induced particleparticle interactions, and electrostatic repulsion may reduce particle deposition and produce long tails in experimental RTDs. In addition, some deposition may have occurred at the channel inlet. The feed stream enters the channel in a direction perpendicular to the membrane before assuming a cross-flow configuration. This may have enhanced particle deposition at the channel inlet. The deposited particles may eventually exit the channel due to prevailing orthokinetic mechanisms contributing to the long tails of experimental RTDs. The leading edges in theoretical and experimental RTDs also show differences. Whereas from theoretical calculations of RTDs we expect a steep rising edge, experimental observations of rising edges in particle RTDs exhibit a slower increase in signal due to dispersion in the tubing leading to the detector from the channel outlet. Finite pulse width in experiments conducted (as opposed to a Dirac pulse input in obtaining theoretical RTDs) as well as nonidealities in concentrations at the channel inlet also decrease the slope of the leading edge. Under similar hydrodynamic conditions imposed on the simulation and experiments of transport for the 3.5-pm particles (Figures 2-4), smaller particles are also expected to undergo permeation drag dominated transport. As anticipated, experimental RTDs obtained in response to impulse inputs of 0.24-, 0.47-, and 1.6-pm latex particles are also observed to be unimodal (Figure 10). While significant differences exist between theoretical and experimental RTDs, in all cases where transport is dominated by permeation drag, RTDs are unimodal, have a steep leading edge, and exhibit positive skewness with long trailing edges as predicted by theory. Low cross-flow velocities, high permeation rates, and/or small particle size result in drag-dominated lateral migration and are likely to lead to greater accumulation of colloidal materials on membranes. (b) Inertia-Dominated Transport. In laminar flows, nonlinear phenomena arising from fluid inertia induce a lateral force on particles which opposes particle deposition onto membranes. Equation 17 predicts that the inertiainduced lateral migration velocity should increase as the cube of particle size and the square of the cross-flow ve-
" Z
I
U.J
* 0.4 -
3
I
L d 6 cm
0.3
I
0.2
-
0.1
-
Membrane: Track-Etch
1
8
0.6
.I
*
L=46 cm h = 762pm
8
* +
0.4
ap = 12.85pm Uo=30.71 cudsec Vw=O.O13cdsec Membrane: PVDF
a
-8
.e
i
0.2
z"
0.0
0
2
4
6
Time elapsed after injection (sec)
a
Figure 12. Experlmental RTD of 12.85-pm particles when lateral migration is dominated by permeation drag. 5 10 Time elapsed after particles entered the channel (sec)
0
15
Figure 10. Experimental RTDs obtained for particles whose lateral transport is calculated to be dominated by permeation drag.
o'8
d L
-I
* 3 0.6 a L.
*
3
8
L=46 cm h=762 pm ap = 12.85pm Uo=32.94 cudsec Vw=0.015 c d s e c Membraos: PVDF
0.4
.--1
t
z"
0.2
0.0 0
I, 2
4
6
8
10
12
Time elapsed after particles entered the channel (sec)
Figure 11. Experimental RTD of 12.85-pm particles when lateral migration is inertiadominated.
locity. Transport in the inertial regime was observed by conducting experiments with 12.85-pm particles a t faster flows and lower permeation rates than those used in the drag-dominated regime. The results of one such experiment (u, = 35.3 cm/s, u, = 0.017 cm/s) are shown in Figure 7; only two distinct peaks are observed in this case. Under these hydrodynamic conditions, theoretical RTDs are trimodal. Bimodal RTDs with a tallfirst peak followed by a shorter second peak were also obtained in other experiments where inertial effects were expected to play a significant role in particle transport. For example, results from a second experiment with 12.85-pm particles conducted at a slightly lower cross-flow velocity are given in Figure 11. Under the conditions imposed in the experiments summarized in Figures 7 and 11,inertia-dominated transport is predicted to produce three peaks in the RTD whereas all experimental RTDs produced in the inertial regime were observed to be bimodal. This divergence between theory and experiment is discussed below. However, the question remains as to whether or not the shift from unimodal to bimodal RTDs is attributable to a shift from drag- to inertia-dominated lateral migration. If so, does theory accurately predict this shift? For a given particle size and module geometry, the contribution of inertial lift to particle motion is predicted to decrease with a reduction in the cross-flow velocity. At a slow enough cross-flow velocity, the transport of 12.85-pm particles should revert to the permeation drag dominated regime
and RTDs should be unimodal. Figure 12 shows the RTD of 12.85-pm particles at an axial velocity lower than that used in either of the experiments summarized in Figures 7 and 11 (u,= 30.7 cm/s, u, = 0.013 cm/s). Under these conditions, particle transport is calculated to be dominated by drag rather than inertial effects. In contrast to Figures 7 and 9 and in agreement with theory, the experimental RTD is unimodal. It may be concluded that, in the inertial regime, expectations of multimodal RTDs based on trajectory calculations are confirmed. Nonetheless, important differences between theoretical and experimental RTDs are evident. As in the case of drag-dominated transport, these differences may be explained by variance of the experimental system from the assumption of complete and irreversible deposition of particles transported to the near-field region of the membrane and in part based on nonidealities of the experimental setup. Electrostatic repulsion, orthokinetic back-transport, and transport along the membrane may all contribute to a blurring of peaks in the experimental RTDs and larger trailing edges. However, limitations of the theoretical expressions used to compute the inertial lift velocity should also be recognized in explaining differences between theoretical calculations and experimental observations. Inertial lift theory requires Re,
