Predicting the filtration of noncoagulating particles in depth filters

Pergamon PII:

Chemical EngineerinO Science, Vol. 52, No. 1, pp. 93 105, 1997 Copyright (C 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved oo09 2509/97 $17.00 + 0.00

S0009-2509(96)00367-3

Predicting the filtration of noncoagulating particles in depth filters David D. Putnam and Mark A. Burns* Department of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109, U.S.A. (Received 22 August 1995; accepted 7 May 1996)

Abstract--A predictive model is developed for the filtration of noncoagulating particles in packed-bed depth filters. The model uses the trajectory analysis results of Rajagopalan and Tien (1976, A.I.Ch.E.J. 22, 523-533) to calculate initial collection efficienciesof the granular media. Stochastic simulations of particle deposition are used to predict the decrease in collection efficiency that results from deposited particles not only occupying sites on the collector but also shadowing large deposition areas. A comparison of model predictions with data obtained from batch latex filtration experiments showed qualitative but not quantitative agreement. The observed decrease in collection efficiency could be described by a simple empirical expression, characterized by a single shadowing exponent, that was first suggested by Terranova and Burns (1991, Biotechnol. Bioengng 37, 110~120). By developing a correlation for the shadowing exponent from the stochastic simulations and adjusting the expression for the initial collection efficiency, reasonably accurate model predictions could be made once the model parameters from a single experiment were determined. A study of the effect of dimensionless model parameters on predicted breakthrough curves showed that the optimal operating conditions to filter a given liquid suspension occur when the ratio of the particle-to-collector diameter is maximized. An approximate analytical solution is also developed to predict breakthrough behavior in lieu of a numerical solution. Copyright ~) 1996 Elsevier Science Ltd Keywords: Depth filtration; modeling; deposition; particle adsorption; porous media.

However, just as solutes with different physico-chemical properties can be separated in an adsorption column, particles with different physico-chemical properties can be separated in a depth filter. For example, cell affinity chromatography is essentially a depth filter that separates cell subpopulations based on their differing abilities to bind to the granular media (Sharma and Mahendroo, 1980). An essential component of predicting filter performance is predicting the deposition rate of cells or other particles. The simple rate expressions used to model adsorption columns, where the adsorption rate is uniform around the spherical adsorbent, cannot be used to model depth filters. Particle deposition has been shown to be a strong function of the position on the spherical collector (Mackie et al., 1987; Elimelech and Song, 1992; Song and Elimelech, 1993). Also, developing particle adsorption rate expressions is further complicated by the effect of deposited particles on deposition rates. In addition to occupying sites on the granular media, deposited particles shadow portions of the collector surface (Tien et al., 1977; Wang et al., 1977). Consequently, the deposition rate is observed to decrease with time (Rajagopalan and Chu, 1982; Riley, 1987; Vaidyanathan and Tien, 1989; Terranova and Burns; 1991; Song and Elimelech, 1993). If

INTRODUCTION

The removal of suspended particles from gas or liquid streams has many industrial applications. Drinking water must go through several filtration steps before it meets water quality standards. Similarly, therapeutic drugs must be purified from the cell suspensions from which they are derived. Granular filtration is a typical filtration step used to remove low concentrations (less than 1 wt%) of suspended particles (Tien, 1989). Since the particles deposit throughout the entire filter media, granular filtration is often referred to as deep-bed or depth filtration. The advantage of a depth filter is that it can process suspensions with a wide range of particle sizes at high removal rates. The removal of particulates by a depth filter is similar to the removal of solutes by an adsorption column (particles are transported to a solid surface), but the dominant mechanism of transport is different. In adsorption, solute transport is dominated by diffusion; in depth filtration, cell or particle transport is dominated by gravity and hydrodynamic forces.

*Corresponding author. 93

94

D.D. Putnam and M. A. Burns

particle-particle interactions are favorable, the deposition rate increases with time because deposited particles act as additional deposition sites (Tien and Payatakes, 1979; Vigneswaran and Aim, 1985; Chiang and Tien, 1985a; Vigneswaran and Song, 1986; Adin and Rebhun, 1987; Mackie et a/.,'1987; Vigneswaran and Tulachan, 1988; Chang and Vigneswaran, 1990). Due to these effects and the complicated three-dimensional and stochastic nature of the deposition process, the majority of transient depth filtration models have been of limited predictive value since they still contain model parameters that cannot be predicted a priori.

Initial deposition rates (collection efficiencies) of the granular media can be calculated by analyzing the trajectories of particles as they approach the collector, and several trajectory analysis models are available (Tien, 1989). The models differ primarily in their geometrical representation of the granular media and inclusion of surface forces. However, a study by Tien (1989) of three models (Spielman and FitzPatrick, 1973; Payatakes et al., 1974a,b; Rajagopalan and Tien, 1976) found there is little difference in the accuracy between them; current trajectory analysis models can be expected to predict initial collection efficiencies to within at least an order of magnitude. Trajectory analysis, however, only applies to the initial stage of filtration. To model the filtration process in its entirety, the effect of deposited particles on collection efficiencies must be calculated. The effect of deposited particles on collection efficiencies can readily be calculated with no adjustable parameters using stochastic simulations of particle deposition (Tien, 1984). This simulation approach, pioneered by Tien and coworkers (Tien et al., 1977; Wang et al., 1977), has been useful in calculating the increase in collection efficiency when deposited particles act as additional deposition sites in both liquid (Chiang and Tien, 1985b) and gas systems (Beizaie et al., 1981; Pendse and Tien, 1982; Jung and Tien, 1993), but to our knowledge has not been used to calculate the decrease in collection efficiency when deposited particles do not act as deposition sites (noncoagulating particles). Current rate expressions (Rajagopalan and Chu, 1982; Terranova and Burns, 1991; Song and Elimelech, 1993) used to model noncoagulating particles are not predictive, since they contain parameters that cannot be calculated a priori or do not apply at high surface coverages (Vaidyanathan and Tien, 1991). In this work, we use stochastic simulation methods to calculate the effect of noncoagulating deposited particles on collection efficiencies. Simulation results were compared with batch latex filtration data. Since the decrease in collection efficiency was predicted to be non-linear, the complete model equations had to be solved numerically. However, to facilitate the use of the model, an approximate analytical solution is also developed. Note that, throughout this paper, 'particle' will refer to the suspended particles being

filtered and 'collector' will refer to the larger particles in the bed that adsorb the suspended particles. MODEL DEVELOPMENT

The model consists of differential mass balances on the bed that are coupled by an expression for the total rate of particle adsorption. The total adsorption rate is written in terms of the collection efficiency of individual collectors in the bed. Mass balances

For a filter of length L with uniform cross-sectional area, A, and void fraction, ~:, the differential mass balances on the liquid suspension and granular media at a given time, t, are ~c 0% c')t = Dt (~z2

vt ~c ~ ~Z j

ga/Ot = Ro

R, e

(1)

(2)

where c is the particle concentration in the liquid (mass of particles per unit volume of liquid), a is the specific deposit (mass of particles per unit bed volume), v~ is the superficial liquid velocity through the bed, D~ is an axial dispersion coefficient, and R, is the total rate of adsorption of particles per unit bed volume. The initial and boundary conditions are Initial conditions: e(z, 0) = 0

(3)

~(z, O) = 0.

(4)

Boundary conditions: e ( z = O, t) = co

~c(z = L, t)

Ot

= 0.

(5) (6)

The initial conditions correspond to a bed free of particles, and the boundary condition [eq. (5)] corresponds to a liquid suspension entering the bed at a concentration Co. Equation (6) is a Danckwerts (1952) boundary condition. The total rate of adsorption per unit bed volume that couples the mass balance equations is the product of the number of collectors per unit bed volume and the rate of particle collection for an individual collector (O'Melia and All, 1978): R , = NccvtA~q.

(7)

In this expression, q is the individual collector efficiency and is defined as the fraction of the particles approaching the collector through a projected area, Ap, that are deposited. The projected area for fluid flow that is consistent with Happel's porous media model (Happel, 1958) is ~d2~/4p z. The individual collector efficiency can be calculated by considering two stages of particle deposition: an initial stage when the bed is free of deposited particles

95

Filtration of noncoagulating particles and the collector efficiency is relatively constant, and a later stage where the collector efficiency changes due to the presence of deposited particles (Tien, 1989). Mathematically, this is written as (8)

= qoF

where t/o is the initial collector efficiency, and F is a dimensionless expression used to account for the change in collection efficiency as a function of particle deposition. The initial collection efficiency is calculated using the particle trajectory analysis of Rajagopalan and Tien (1976) and the change in collection efficiency, expressed in terms of F, is calculated using stochastic simulations of particle deposition. Initial collection efficiency tlo

To calculate the initial collection efficiency of the granular media, we have chosen the following expression developed by Rajagopalan and Tien (1976): qo = ~As(1

- - ~:)2/3 N 2 ×

''LO

+ (2.25 × 10-3)N~ 2 N~ 24] + 4(1 - e)2/3 A~/3 N~2/3

(9)

system for this simulation are shown in Fig. 1. The system corresponds to Happel's porous media model and consists of a spherical collector surrounded by a fluid envelope whose radius, b, is chosen to match the overall bed void fraction. Particles originate at random positions on the radius of the fluid envelope in proportion to the volumetric flow rate through the area elements (Jung and Tien, 1993). Note that the trajectory of a particle approaching the collector is assumed to coincide with fluid streamlines; this assumption is valid for most liquid systems where hydrodynamic forces dominate and particle collection is frequently referred to as interception. Deposition can occur only if the area necessary for deposition is neither occupied nor shadowed by a previously deposited particle. The presence of deposited particles is assumed not to affect particle trajectories. The simulation is terminated when all sites on the collector are either occupied or shadowed. The output of the stochastic simulations is the number of deposited particles, m, vs the number of attempts to place particles on the collector, M. The change in collection efficiency as a function of the extent of deposition is calculated using (Beizaie et al., 1981) V = (drfi/dM)/(dtfi/dM)M _ o.

where 2

A s = - - (1 -- pS)

(10)

W

p = (1 - e)1/3

(11)

w = 2 - 3p + 3p 5 - 2p 6.

(12}

This expression is extremely useful since initial collection efficiencies must typically be predicted by solving the particle trajectory equations. This expression was developed by solving the trajectory equations and correlating the effects of dimensionless groups (Na, N~, NLO) on the initial collection efficiency. The last term in the correlation was added separately to include the effect of Brownian diffusion (Rajagopalan and Tien, 1982). The expression has been shown to be accurate to within at least an order of magnitude over a wide range of experimental conditions (Rajagopalan and Tien, 1976; Chiang and Tien, 1985a, b; Tobiason and O'Melia, 1988; Choo and Tien, 1995). Change in collection efficiency with particle deposition, F

For noncoagulating particles, the collection efficiency decreases not only due to deposited particles occupying deposition sites on the collector, but also because these particles can block large areas behind these sites on the collector (shadow effect). The shadow effect is illustrated in Fig. 1 and occurs because deposited particles block particle trajectories that would have otherwise led to contact with the collector. To determine the decrease in collection efficiency with particle deposition, expressed in dimensionless form as F, we have simulated the deposition behavior of individual particles. The collector and coordinate

(13)

Since the simulations are stochastic, a sufficient number of simulations must be run to describe the average deposition behavior. By calculating the change in collection efficiency as a function of the mass of particles deposited per unit bed volume a, the complete form of the deposition rate expression can be determined. Our present simulations allow for the deposition of particles onto a two-dimensional spherical collector. The assumption of two-dimensional deposition simplifies the shadow area calculations. With a two-dimensional collector surface, the shadow area created by a deposited particle corresponds to shadow angle, Os (Wang et al., 1977). In contrast, for a three-dimensional collector surface, a locus of all trajectories that just graze the deposited particle must be calculated (Vaidyanathan and Tien, 1991). To relate the change in collection efficiency predicted by the simulations to an actual three-dimensional filter, the change in collection efficiency is reported in terms of the fraction of deposition sites occupied on the collector ~9. Note that the total number of sites initially available (~/2NR) includes only the upstream half of the collector; when particles follow fluid streamlines, it is impossible for a particle to come into contact with the collector greater than 90 ~' from the front of the collector (see Fig. 1). The model equations were dedimensionalized with the following variables: (~ = C/Co

(14)

~9 = a/am = a/2ppNR(1 -- ~)

(15~

= z/L

(16)

® = t/T~

(17)

96

D. D. Putnam and M. A. Burns 2

90 °

Dat = i(1 -- ~:)1/3qo L dc 9

= ~ A s ( 1 - ~)

Flow Direction Op=0° ~

1so° X

Fig. 1. Shadow effect. An area on the collector behind the deposited particle becomes unavailable for deposition since the particle blocks particle trajectories that would have otherwise led to contact. The angle covering the shadow area, Os = 59.4~ for 0p = 19.9°, NR = 0.05, e = 0.38, is calculated by determining the trajectory of a particle that barely grazes the deposited particle.

L

[ ] ~~rv L'/s Ar15/8 + (2.25 × 10-3)N~-2N~O.* O ~'R

rate of deposition rate of liquid convection a,,

2pp N~(1 - ~)

COg,,

COE

(25)

K=

mass of particles deposited in a monolayer mass of particles in column (26)

where O is the dimensionless concentration of particles in the liquid, ~ is the dimensionless axial distance, and ® is the dimensionless time relative to the residence time of the liquid in the column, q = Le/v, or, equivalently, the number of column void volumes processed by the filter. ~, is the fraction of sites occupied on collector; it is the specific deposit relative to the specific deposit assuming monolayer coverage and no shadowing, a,~ is determined by calculating the number of sites on the collector initially available for deposition and then converting the result to the appropriate units. With these defined dimensionless variables, the differential mass balances and initial and boundary conditions become ~¢ ~O

l c~20 Pet ~(2 ~

~0 8~ Da~

Da~ OF

(18)

MATERIALS

OF.

(19)

Initial conditions: 0(~, ® = o) = 0

(20)

~,(~, ® = 0) = o.

(21)

Boundary conditions: 0(~. = 0, ®) = 1 c~¢(~ =

The model equations with the appropriate initial and boundary conditions can be solved numerically using finite difference techniques. The finite difference method used depended on the value of the Peeler number. For Peclet numbers greater than 200, the method of lines (Davis, 1984) was used. This method has the advantage that it does not require the linearization of the deposition rate expression. For Peclet numbers less than I00, the model equations were solved using the Crank-Nicholson formulation of finite differencing and quasilinearization (Ramirez, 1989). The Crank-Nicholson formulation is more applicable for low Peclet numbers since, in this range of Peclet numbers, the method of lines would have required prohibitively small time steps to maintain numerical stability.

1, O)/OO

=

(22) O.

(23)

In addition to the expression for the dimensionless collection efficiency, F, dedimensionalization of the model equations results in the following three dimensionless groups that affect the performance of a depth filter: the liquid phase Peclet number (Pet), a first DamkShler number (Dal), and the mass capacity ratio

Polystyrene latex microspheres with mean diameters of 0.78 and 1.44/tin were obtained from Polysciences, Inc., Warrington, PA. The manufacturer's reported size distributions of the latex suspensions were + 0.02 and 0.04 #m, respectively. The reported density of the latex microspheres was 1.05 g/mE Hiindex glass beads ( - 140 + 200 mesh) were obtained from Potters Industries, Hasbrouck Heights, NJ. The reported density of the glass beads is 4.49 g/ml. Distilled water was obtained from a Barnstead (Boston, MA) glass still, and deionized using a Barnstead Nanopore II deionizer. Alumina sol (Dispal ® 23N4-20) was donated by Vista Chemical, Houston, TX. Chromic-sulfuric acid cleaning solution was obtained from Fisher Scientific, Itasca, IL. All other chemicals were reagent grade and obtained from commercial sources.

(•). METHODS

Lvt Pet = - Dte =

rate of liquid convection rate of liquid dispersion

(24)

Preparation of granular media and particle suspensions The granular media consisted of hi-index glass beads. The beads are washed in a 1:9 ratio of

Filtration of noncoagulating particles chromic-sulfuric cleaning solution to water, and thoroughly rinsed in distilled and deionized water. The beads are then placed in a 3 mM solution of N a O H for 20 min and then washed in distilled and deionized water. Next, the beads are placed in a 20% (wt) aqueous dispersion of Dispal * (23N4-20) colloidal Alumina sol that has a reported average dispersed particle size of 50 nm and stirred occasionally for 20 min. The beads are then washed with distilled and deionized water and stored at 4°C in distilled and deionized water until used. Latex suspensions were diluted with distilled and deionized water or a 0.025 M NaCI solution.

Characterization of granular media The mean diameter of the glass beads 100.8 (+ 9.4) #m was calculated by averaging the projected areas of 100 randomly chosen beads. A representative sample of glass beads were adhered to a microscope slide that was covered with a thin layer of DowCorning clear silicone. The glass beads were then viewed at 100 x magnification under transmitted light illumination with a Zeiss Axioskop microscope. A random field of view was then chosen which contained several beads and a digital image of the field of view was taken using an Oncor Inc. (Rockville, MD) Personal Workstation Real Time Video Frame Grabber/ALU Module, Imaging software (version 1.6), Hamamatsu Newvicon camera with C2400 controller, and a Macintosh IIfx computer. The image analysis software was then used to measure the projected area (pixels) of each bead and converted to a collector diameter. Calibration with a 1 mm square in a Reichert haemocytometer (Buffalo, NY) showed that at 100 × magnification each pixel represented 2.00/~m.

Filter construction The filter consisted of a 1.5 cm i.d. Kontes chromatography column (Rainin Instrument Co., Emeryville, CA). Particle suspensions were pumped using Masterflex ~ peristaltic pumps purchased from ColeParmer Instrument Co., Niles, IL, and entered at the top of the column through a flow adapter obtained from BIO-RAD (Hercules, CA). The flow distributors located at the bottom of the column and in the flow adapter were replaced with a liquid distributor cut from a porous polypropylene sheet (X4900, fine) donated by Porex Technologies (Fairburn, GA). The reported mean pore size is 30 pm. Particle concentration exiting the bed vs time was determined by measuring the absorbance at 610 nm using a HP 8452A diode array spectrophotometer and l ml flow cell with a path length of l0 mm.

Batch .filtration experiments After loading the granular media into the column, the media was fluidized using an equilibrating solution (liquid at the same electrolyte concentration of the feed suspension) at a liquid velocity well above the minimum fluidization velocity of the granular media

97

Table 1. Standard operating conditions for latex filtration experiments Parameter Column height Collector diameter Liquid velocity Particle concentration Salt composition pH Bed void fraction

Value 0.032 m 100.8 ± 9.4 pm 0.00417 m/s 0.005%dry weight 0.025 M NaC1 6.5 0.38

to remove any air bubbles in the bed. The bed was allowed to settle to its packed-bed height, a flow adapter was placed 1 cm above the bed, and the flow through the bed was reversed. At the start of the filtration run, the liquid feed to the column was then switched to the particle suspension by clamping the tubing exiting the reservoir of the equilibrating solution and unclamping the tubing exiting the reservoir containing the particle suspension. Unless otherwise stated, the standard operating conditions are those reported in Table 1. RESULTS AND DISCUSSION

Simulatin9 particle deposition As described in the Model Development section, the prediction of the particle concentration exiting the filter requires calculating how the collection efficiency changes as a function of particle deposition. The change in collection efficiency is expressed in dimensionless form as F [-see eq. (8)]. To predict F, the deposition of individual particles was simulated. Figure 2 shows that the decrease in collection efficiency predicted by the simulations (no adjustable parameters) is primarily linear with the fraction of sites occupied but deviates from linearity when the collection efficiency approaches zero. Initially, there is very little overlapping of shadow areas between deposited particles on the collector, and the collection efficiency decreases in proportion to the fraction of sites occupied. However, as more particles deposit, the overlapping of shadow areas increases and the collection efficiency decreases more slowly and non linearly. A completely linear decrease in collection efficiency was predicted by Vaidyanathan and Tien (1991) since they did not include the effect of overlapping shadow" areas. The difference in the linear rate of decrease in F between our simulations and the results of Vaidyanathan and Tien is due solely to our assumption of two-dimensional particle deposition. Note that in all predictions the final surface coverages are much less than a corresponding monolayer (~9 = 1.0). It is apparent from Fig. 2 that the rate of decrease in collection efficiency increases as N~ (ratio of particle to collector diameter) decreases. A similar trend is shown using the results of Vaidyanathan and Tien (1991). This trend results from shadow angles across the collector that are relatively independent of the ratio of particle-to-collector diameter (see Fig. 3). As

98

D. D. Putnam and M. A. Burns

\

,%:;';:2;2°L j.s.OOOt-I Tien, 1991)

:_

\

o.s

~

] simulation results , 7 (Vaidyauathan and ~ N. = 0.01~

~-\\---

o.s

~

0.4

",2

0.2

i

0

i

0.05

I

f

i

J

0.1

i

i

~L

i

0.15

i

i

i

0.2

i

•

0.25

Fraction of Sites Occupied, ~t

Fig. 2. Effect of particle deposition on particle collection efficiency. The rate of decrease in collection efficiency is predicted to be greater when the ratio of particle-to-collector diameter is decreased. A similar trend was predicted by Vaidyanathan and Tien (1991). 90

N:=:o:1

75

I

~

NR= 0.01 t

60

45

but rather is due to a repulsive double layer surrounding the deposited particle (Rajagopalan and Chu, 1982) that reduces the area for deposition by an amount greater than the projected area. Under appropriate experimental conditions, collection efficiency data expressed in terms of F could be obtained by a simple analysis of the filtration breakthrough curve and compared directly with F calculated from our simulations. When particle breakthrough occurs almost immediately, the specific deposit and, therefore, the collection efficiency throughout the filter is relatively constant. This allows the collection efficiency at any time to be related to the particle concentration exiting the filter, and is compared to qo to obtain F using F

2de ~103L(1--

~h = a _ =

am

0

0

15

30

45

60

75

90

Deposition angle, Op

Fig. 3. Shadow angle vs the angle of deposition on the collector. The shadow angle is relatively independent of the ratio of particle-to-collector diameter. the particle diameter and, therefore, NR is decreased, the shadow angle remains relatively constant because the trajectory of a particle that just barely grazes the deposited particle becomes more parallel and leads to deposition farther down on the collector. The net effect is that an equivalent decrease in collection efficiency occurs at a lower fraction of sites occupied when the ratio of particle-to-collector diameter is smaller. The stochastic simulation results apply only when interception is the dominant transport mechanism, diffusion is negligible, and the shadowing (blocking) of sites is flow induced. The first term in eq. (9) accounts for the contribution of interception to the initial collection efficiency while the last term accounts for Brownian diffusion. When Brownian diffusion is important, deposition takes place in a diffusive layer near the collector where particles deposit randomly. In this case, the blocking of sites is not flow induced

In C/Co(Z = L).

(27)

Equation (27) is derived by neglecting the unsteadystate and liquid dispersion terms in the differential mass balance on the liquid phase [eq. (1)] and integrating the resulting equation across the filter. As we will show later, negligible liquid dispersion is justified under typical experimental conditions. Note that q0 is the value that makes F equal to one as the filtration time tends to zero. The fraction of sites occupied at a given value o f F is found from a simple mass balance on the bed,

30

15

~)1/3

toe

c/co dO . (28)

0 -

2ppNR(l -- ,~:)

Breakthrough curves that are amenable to this type of analysis typically have Damk6hler numbers with breakthrough times (defined when C/Co = 0.05) less than 50 column passes (see Fig. 4). Theoretical predictions of the decrease in collection efficiency as a function of the fraction of sites occupied were compared with results obtained from the analysis of batch filtration experiments. Latex suspensions

0.8 oo°OO° o° o°°°

0.6 ¢9 0.4

0.78 1.44 ~m pan 1 0 0

50

100

__L~ , r, ~ i 150 200

Number of Column Passes, O =

250

300

t~ v I

Fig. 4. Experimental breakthrough curves for the filtration of latex suspensions. The standard operating conditions are shown in Table 1.

Filtration of noncoagulating particles

99

1 data

o

} N~ = 0.0143 - -

theory

o.s

0.8 ,o

¢fl

- -

0.4

\\

data } N R = 0.0077 theory

data - - fit (~ = 15.9)

•

~ ~ [ ~

o , o

0.6

~

o

| N s = 0.0143 J

data -

fit ([3 = 20.0)

} NR = 0.0077

0.6 o c~

o

0.4

¢9 o e~

0.2

0.2

0 i .... 0

~'~k,\ ,',',~I , " , ~ " ~ 0.05

0.1

0.15

, I .... 0,2

0.25

0

0.05

Fraction of Sites Occupied, ~¢

0,1

0.15

0.2

0.25

F r a c t i o n of S i t e s O c c u p i e d , V'

Fig. 5. Experimental and predicted decrease in collection efficiency for the filtration of latex suspensions. The stochastic simulations qualitatively predict the greater rate of decrease in collection efficiency as the ratio of particle to collector diameter is decreased. The collection efficiency data is obtained by a simple analysis of the breakthrough curves in Fig. 4 using eqs (27) and (28).

Fig 6. Explicit expression for the decrease in collection efficiency with particle deposition F = (1 - ~,)~.The decrease in collection efficiency could be fit by a simple expression characterized by a single shadowing exponent.

0.1 N a = 0.0143

were used since they have relatively uniform size distributions and are similar in size and charge to typical particle suspensions. Figure 5 shows that the theoretical predictions follow the same trend as the experimental data; the rate of decrease in collection efficiency is greater when the ratio of particle-to-collector diameter is decreased. However, the experimentally derived F curves are significantly nonlinear, even at low ~ values. The primarily linear decrease in collection efficiency predicted by the stochastic simulations is due to assuming two-dimensional particle deposition which limits the n u m b e r of sites and potential overlap of shadow areas between deposited particles. Assuming three-dimensional particle deposition should result in a more nonlinear but not more quantitative agreement with experiments. Including other effects such as weighting the simulation results by the size distributions of the collectors (the particles were considered monodisperse) were investigated but were found to have a negligible effect on the simulation results.

Explicit rate expression and parameter analysis The dimensionless collection efficiency data obtained from batch latex filtration experiments could be correlated using a simple power-law expression where the dependent variable is the fraction of sites unoccupied (1 - ~) and the exponent, fl, is a measure of the degree of shadowing on the collector: F = (1 - ~)P.

(29)

This simple expression, which gives a complete explicit form of the deposition rate expression, was first put forth by Terranova and Burns (1991) when correlating batch latex filtration data in magnetically stabilized fluidized bed depth filters. The validity of this simple expression to describe the experimental F curves is demonstrated in Fig. 6. Experimental

N R = 0.0077 vl0 = 0.0164 0.01

o~ ,~

0.001

0.0001 1

0.9

0.8

F r a c t i o n of S i t e s U n o c c u p i e d , I-V /

Fig. 7. Determination of experimental shadowing exponents and initial collection efficiencies. The shadowing exponent, fl, is the slope of the best fit line and the initial collection efficiency is the y-intercept of that line. Note that both scales are logarithmic.

values of the shadowing exponent and initial collection efficiency for these curves can be obtained by plotting the logarithm of the collection efficiency data vs the logarithm of the fraction of sites unoccupied (see Fig. 7). The slope of the line is equal to the shadowing exponent, and the initial collection efficiency is equal to the y-intercept. With the explicit expression for the change in collection efficiency determined, the model eqs (18)-(26) can be solved to predict the particle concentration exiting the filter with time (breakthrough curve). Figure 8 shows that the model fits the experimental breakthrough curves very well. Note that the values of the initial collection efficiencies and shadowing exponents used in these predictions were obtained from the plots in Fig. 7. With the development of an explicit expression for the dimensionless collection efficiency completed, the

100

D. D. Putnam and M. A. Burns 1

r , , i i , i~l

0.8

0.8

/ i

,(,"

0.6

0.6 0.4

"!

i i i i i i b , , , i r , , ,

0.4

0.2 0.2

, ~

~,~_o¢~ ¢~,

~ ,"

I // /

, ,(fit I .........

,:

__

:

0 0

50

100

150

200

250

300 0

100

200

300

400

500

N u m b e r of C o l u m n Passes, O =t/r I N u m b e r of C o l u m n Passes, O= t/vz

Fig. 8. Model fit to experimental breakthrough curves for the filtration of latex suspensions. The model parameters used are: Pe=190, Da~=3.77, K=530, fl=20.0 and Pe = 190, Dal = 6.64, x = 979, fl = 15.9 for the 0.78 and 1.44/~m latex suspensions, respectively•

Fig. 10. Effect of first Damk6hler number (Dat) on filtration breakthrough curves. As Dal is increased, the breakthrough time, defined at C/Co= 0.1, increases and the breakthrough curves become sharper• At very high DamkShler numbers, the breakthrough time is limited only by the effective bed capacity. Model parameters used: Pe = 200, x = 500, fl=10.

0.8 0.8 i

,'

J

-

-

0.6 i

0.6

0.4 0.4 0.2 0.2 0

0

50

100

150

200

250

300

350

N u m b e r of C o h t m n Passes, O=t/VL

Fig. 9. Effect of the Peclet number (Pe) on filtration breakthrough curves. Under typical filtering conditions (Pe > 100), the effects of liquid dispersion can be neglected. Model parameters used: Da~ = 15, s: = 667, fl = 12.

effect of dimensionless model parameters on the performance of the filter was analyzed. Analysis of the effect of the Peclet number on filtration breakthrough curves (Fig. 9) shows that except for Peclet numbers less than 100, liquid dispersion is found to have very little effect on filter performance. As the Peclet number is increased from 100 to 1000, the breakthrough time (defined as the number of column passes when C/Co = 0.1) increases only slightly. The Peclet number for the experiments in this study, based on dispersion coefficient data in packed beds (Sherwood et al., 1975), was 190. F r o m Fig. 10 it is apparent that optimal breakthrough curves, characterized by long breakthrough times, can be obtained by increasing the Damk6hler number. When the Damk6hler number is too low, particle breakthrough is observed to occur almost

0 0

200

400

600

800

1000

N u m b e r of C o l u m n Passes, O=t/rL

Fig. 11. Effect of mass capacity ratio (r) on filtration breakthrough curves. When x is increased, the breakthrough time increases proportionately. Model parameters used: Pe = 200, Dal = 15, fl = 10. immediately. At very high Damk6hler numbers, the breakthrough time is limited only by the effective bed capacity. Typical values of the DamkShler number that are sufficient to prevent immediate breakthrough are 10 or greater. Whenever possible, the Damk6hler number should be increased by decreasing the collector diameter rather than increasing the bed length. Decreasing the collector diameter not only increases the Damk6hler number but increases the filtering capacity of the bed. Analysis of the effect of the mass capacity ratio on breakthrough curves (Fig. 11) shows that as the mass capacity ratio is increased, the breakthrough time increases proportionately. An increase in the mass capacity ratio could result from a lower particle concentration entering the filter or a decrease in collector diameter.

Filtration of noncoagulating particles Figure 12 shows that an increase in the shadowing exponent decreases the breakthrough time by decreasing the effective filtering capacity of the bed. Experimentally, the shadowing exponent was observed to decrease as the ratio of particle-to-collector diameter was increased. Consequently, filter performance can be increased by decreasing the collector diameter. Optimal filtering conditions occur when the collector diameter is decreased since this also increases the Damk6hler number and mass capacity ratio. When calculating breakthrough times over a range of parameter values, the breakthrough time was found to scale with the mass capacity ratio divided by the shadowing exponent, K/ft. Figure 13 shows that three

0.8

/ / /

0.6

~

0.4

~

0.2

I

--

13=25 ,8=5

0 0

500

I000

1500

Number of Column Passes, O= t l r z

Fig. 12. Effect of shadowing exponent (fl) on filtration breakthrough curves. As fl is increased, the effectivefiltering capacity of the bed is decreased. This decreased effective capacity has a similar effect as decreasing the mass capacity ratio x; the breakthrough time is decreased. Model parameters used: Pe = 200, Da x = 15, h"= 1000.

4 C.)

Z

. ~y/Ca rio,

,c/a

Fig. 13. The effectivemass capacity ratio, K/ft. By scaling the mass capacity ratio x with the shadowing exponent fl, three surfaces corresponding to different shadowing exponents (5, 9, and 20) can be collapsed to an almost singular universal surface.

101

surfaces corresponding to typical shadowing exponents over a range of Damk6hler numbers and mass capacity ratios (the Peclet number is usually unimportant) collapse to an almost singular surface when scaling the mass capacity ratio with the shadowing exponent for each surface (x/fl). If the middle surface (fl = 9) is used as a universal surface, the error in the breakthrough time is 15% or less for fl values from 5-20. The ability to scale the mass capacity ratio with the shadowing exponent occurs since at low surface coverages (1 - ~k)~ ~ 1 -fltp. Predicting batch filtration In this section we use two approaches to predicting batch filtration. First, a priori predictions with no adjustable model parameters are presented. These predictions were found to be in qualitative agreement with experiments. Correlations were then developed based on theoretical calculations and experimental data. These correlations allow reasonably accurate model predictions to be made at other experimental conditions from model parameters obtained from a single experiment. An approximate analytical solution to the model equations that can be used to predict batch filtration in lieu of a numerical solution is also developed. The a priori prediction of the particle concentration exiting a depth filter with time requires a knowledge of how the particle collection efficiency changes as particles deposit throughout the filter. The initial particle collection efficiency, qo, is predicted using the results of Rajagopalan and Tien [eq. (9)]. As described in the Model Development section, the decrease in collection efficiency, expressed in dimensionless form as F, has been predicted by simulating the deposition behavior of individual particles. To solve the dedimensionalized model equations, the predicted numerical values of F as a function of fraction of sites occupied (Fig. 5) had to be fit to an analytical expression (see Appendix for equations). Figure 14 shows a comparison of a priori model predictions with data obtained for latex filtration in a packed-bed depth filter. The model predictions are shown to be in qualitative agreement with the experimental data; the model correctly predicts an earlier breakthrough of the smaller diameter latex suspension. The absence of quantitative agreement between theory and experiments is twofold: the initial collection efficiencies are underpredicted by the expression developed by Rajagopalan and Tien (1976) and the decrease in collection efficiency is overpredicted by the stochastic simulations (see Fig. 5). The expression for the initial collection efficiency [eq. (9)] has been shown to be capable of predicting qo to within at least an order of magnitude, and the predicted initial collection efficiencies show a similar degree of accuracy; they are underpredicted by a factor of 5.6 and 3.6 for the 0.78 and 1.44 pm particles, respectively. The consistent underprediction of initial collection efficiencies may be due to non-negligible, attractive double layer forces between the particle and collector.

D. D. Putnam and M. A. Burns

102

d

0.8

We have developed the following correlations that can be used to predict the shadowing exponent and initial collection efficiency at other experimental conditions once the best fit parameters r/o, and fll are calculated from a single experiment:

// °

0.6

)/'

~/

":f"

0.4

°°°°°°°°°

,

i

data

,:

} 0 . 7 8 lira theory

/ o 0.2

o

• i;_ ^ . . . .

0

data

o°

}

~o°

L44 ~

-t

o O~ . . . . . . . . . 50

100

]

I ....

. . . .

150

of Column

200

250

Passes,

O

i

I ....

300

=t/z l

Fig. 14. Comparison of a priori model predictions with experimental breakthrough curves for the filtration of latex suspensions. Model predictions are in qualitative agreement with experiments. The model parameters used are: Pe = 190, Dal = 0.67, x = 530 and Pe = 190, Dar = 1.82, x = 979 for the 0.78 and 1.44 pm latex suspensions respectively. Expressions for F were obtained from fitting the simulation results in Fig. 5 (see Appendix for equations).

, r , , ~ - l ~ r oo

~ , I ' ' ' r IT,

Deionized W a t e r °••

0.8

, , I~FI

, ,

•. ......

°°"°°*

oooooOO°°°°

0.6

o°

o

o

0 . 0 2 5 M NaG1

o

0.4 o •

L

• • go

°

(31)

f12 = (NR~/NR:) ill.

theory

Number

0.2

(30)

o,, (eq. (9)

o°

' "

0

.o, )] r/o,(eq. (9))

qo2 =

L 1 J

o o

50 Number

o

o

100

150

of Column

200 Passes,

250

The expression for r/o has the same functional from as the expression of Rajagopalan and Tien since their expression has shown good agreement with experiments over a range of experimental condition and was in qualitative agreement with our experiments. Since the current stochastic simulations are able to qualitatively predict the increase in the rate of the decrease in collection efficiency as NR is decreased (see Fig. 5), the correlation for fl was developed by fitting shadowing exponents to the simulation results for different NR values. The method of correlating a single experiment and predicting filtration breakthrough at a different ratio of particle-to-collector diameter is shown in Fig. 16. Equations (30) and (31) were used to predict new values of initial collection efficiency and the shadowing exponent using the best fit values of these parameters for the 1.44 pm suspension, The corresponding DamkShler numbers and mass capacity ratios were calculated using eq. (25) and (26). With this method, reasonably accurate model predictions are obtained. We have developed an approximate analytical solution that can be used to predict breakthrough curves in lieu of a numerical solution. To obtain the approximate analytical solution, the explicit expression for the decrease in collection efficiency, eq. (29), was

300

O=t/v l

Fig. 15. Effect of electrolyte concentration on latex filtration breakthrough curves• When the electrolyte concentration is increased, the breakthrough time, defined at C/Co = 0.l, increases slightly. In addition to a decrease in the shadowing exponent, the initial collection efficiency also decreases.

1 o .~

0.8

0.6 /

0.4 /

As pointed out by Rajagopalan (1974), the contribution due to surface forces should be proportional to the increase in collection due to interception. This is demonstrated in Fig. 15 where analysis of the breakthrough curves showed that the initial collection efficiency increased from 0.0164 to 0.0280 when switching from 0.025 M NaC1 to deionized water. To our knowledge, a theoretical study on the effect of double layer forces when interception is a dominant filtration mechanism has never been done. The shadowing exponent is also sensitive to electrolyte concentration, increasing from 15.9 to 32.9 when using deionized water.

/

•

o of/

o°;~

° /

•

°~~ •/* / °"

0.2

• .....

o **" "/+

(data) ( c o r r e l a t e d } 0 . 7 8 Isrn theory)

(data) - - - (fit)

} 1.44 I~m

0 0

50 Number

100

150

of Colunm

200 Passes,

250

300

O =t / z I

Fig. 16. Correlated model predictions for latex filtration. The model predicts reasonably accurately the filtration of the smaller 0.78pm suspension (Pe = 190, Dar= 2.44, = 530, fl = 29.4) after correlating the initial collection efficiency and shadowing exponent with the best-fit model parameters for the 1.44 pm suspension (Pe = 190, Dal = 6.64, K = 979, fl = 15.9).

103

Filtration of noncoagulating particles 1000

25

20

-

~o

soo --

H

r~

10 r/l

0

i i i 0

i i i i

10

20

600

--

-

Daft20

- - Daf.30

.

/

ii

200

J i i i I.~

i

40

50

50

100

150

Effective Mass Capacity Exponent,

~w~T

~'

o.s

, '~1 ~ ' ~ 1

I!

~ ' ' 7 1 :~-~ ' ~ 1 7 =

'~

r:-' ' w - a - ~

/

o

0.4

_ _ _

0.2

-

.)

-----

0 0

50

100 Number

150

200

of Column

250 Passes,

200

250

300

R a t i o , r//]

fl

Fig. 17. Approximate decrease in collection efficiency. To obtain an approximate analytical solution, the nonlinear expression for the change in collection efficiency observed experimentally, F = (1 - ~b)~,is approximated with the linear expression, F = 1 - Fs~b. For a given shadowing exponent,/L the shadowing factor, Fs, is found by minimizing the error for all 0 between the two expressions.

1

J

400

30

Shadowing

/

.

300 O=

350

400

t/v L

Fig. 19. Approximate breakthrough times. This plot can be used to predict breakthrough times to within an error of 15% for fl values 5-20. however, that the analytical solution is still useful since it can be used to determine Damk6hler numbers that are sufficient to prevent immediate breakthrough. A more accurate estimate of breakthrough times can be obtained from Fig. 19. This plot can be used to estimate breakthrough times, defined at C/Co = 0.1, to within an error of 15% or less. Note that similar plots for breakthrough times defined at different exit concentrations can be easily developed. Figure 19 can also be used to get an approximate a priori breakthrough time. Dal and ~c are calculated using eq. (25) and (26). fl is calculated from Fig. 17 using the value of Fs predicted by the stochastic simulations. Fs is equal to the slope of the linear portion of the predicted decrease in collection efficiency in Fig. 2. Over a typical range of ratios of particle-to-collector diameters (0.01 < NR < 0.1), Fs and can be described by the equation: Fs = 0.24/NR. (33)

Fig. 18. Approximate analytical solution. The approximate solution approximates the numerical solution very well when the DamkShler number is low but underpredicts the breakthrough curve at higher Damk6hler numbers since the linear expression for the decrease in collection underpredicts the true capacity of the filter. Values of the other parameters are: Pe = 200, K = 500, fl = 10.

This equation was obtained by fitting the linear portion of the F curves from the stochastic simulations. Obviously, correlated values oft/0 and fl [eqs (30) and (31)] based on a single filtration experiment can also be used in Fig. 19 to determine the breakthrough times of additional experiments.

approximated with the following linear expression:

A predictive model for the filtration of noncoagulating particles in depth filters was developed. The ability to accurately predict batch filtration depends on the ability to predict the change in collection efficiency as particles are deposited in the filter. The change in collection efficiency can readily be predicted without any adjustable parameters by using stochastic simulations of particle deposition. Two-dimensional simulations of particle deposition showed qualitative agreement with collection efficiency data from latex filtration experiments, but failed to quantitatively predict the magnitude of the nonlinear decrease in collection. The most likely reason for the

CONCLUSIONS

F = 1 - Fs~b.

(32)

For a given shadowing exponent, the shadowing factor, Fs, is calculated by minimizing the error ~b between the two expressions. The shadowing factor for a range of exponents is plotted in Fig. 17. Figure 18 shows that the analytical solution (see Appendix for equations) approximates the numerical solution very well at low DamkOhler numbers but underpredicts the breakthrough curve at higher DamkOhler numbers. At high Damk6hler numbers, eq. (32) underpredicts the true capacity of the collector. Note,

104

D. D. Putnam and M. A. Burns

primarily linear decrease in collection efficiency predicted by the simulations is the simplifying assumption of two-dimensional deposition. A comparison with the work of Vaidyanathan and Tien (1991) suggests that a similar but more complete three-dimensional simulation would result in more nonlinear but not more quantitative agreement with experiments. The decrease in collection efficiency could be described by a simple empirical expression first suggested by Terranova and Burns (1991) for the filtration of noncoagulating latex suspensions in magnetically stabilized depth filters. This expression, characterized by a single shadowing rate exponent, could describe all our filtration data. By developing a correlation for the shadowing exponent from theoretical calculations and adjusting the expression for the initial collection efficiency developed by Rajagopalan and Tien (1976), reasonably accurate model predictions of batch filtration could be made once the model parameters from a single experiment were determined. An analysis of model parameters showed that conditions which result in sharp breakthrough curves and long breakthrough times occur when the ratio of the particle-to-collector diameter is maximized. In addition to increasing the initial collection efficiency of the granular media, decreasing the collector diameter relative to the particle diameter increases the effective capacity of the filter by decreasing the degree of shadowing. An approximate analytical solution which can be used to predict the shape of the breakthrough curve was developed, and a simple plot (Fig. 19) was constructed to estimate breakthrough times in lieu of a numerical solution. Acknowledgements

The authors would like to acknowledge support of the National Science Foundation and the Donors of the Petroleum Research Fund tadministered by the American Chemical Society). This work was also supported in part by funding from the Cellular Biotechnology Training Program, National Institutes of Health grant # 5 R01 GM08353.

Ap

As b e

co dc d, DBM Ol Dat

NOTATION projected area for fluid flow (/rb 2 = gdac/4p 2) parameter defined by eq. (10) radius of fluid envelope for Happel's model [ = d,./2(1 - e,)1/3] mass of particles per unit volume of liquid particle concentration entering the filter collector diameter particle diameter particle diffusion coefficient due to Brownian motion (= kT/3~l~dp) liquid phase axial dispersion coefficient first Damk6hler number [ = 3/2 L(1 - e ) 1/3

qold,,] F Fs g H k L

dimensionless collection efficiency, (= q/qo) shadowing factor [defined by eq. (32)] gravitational acceleration Hamaker constant Boltzmann constant filter length

m M NB NLo Nk N~; p Pe~ r R, t T vj w z

number of particles deposited on the collector number of attempts to place particles on the collector Brownian diffusion parameter (= dpvt/D~M) London force parameter (= 4H/9gl~d 2vl) interception parameter (= dp/dc) gravitational parameter [ = 2d~ g(pp -- p f)~ 91w~] defined by eq. (11) liquid phase Peclet number (= Lvt/Dz~:) radial coordinate rate of adsorption per unit bed volume defined by eq. (7) time absolute temperature superficial liquid velocity parameter defined by eq. (12) axial distance along the bed

Greek letters /~ shadowing exponent ~ bed void fraction f/ collection efficiency qo initial collection efficiency [eq. (9)] 0 angular coordinate 0p particle deposition angle Os shadow angle ® dimensionless time [ = t/(Le/vO] ~,~ mass capacity ratio (= am/coe) t~ fluid viscosity dimensionless axial distance along the bed (= z/L) Pl fluid density pp particle density a specific deposit (mass of particles per bed volume) am specific deposit assuming monolayer coverage [ = 2ppNR(1 -- e)] liquid-phase residence time (= Le/vz) ,p dimensionless liquid concentration (= Uco) fraction of sites occupied by deposited par4, ticles (= a/am) REFERENCES

Adin, A. and Rebhun, M., 1987, Deep-bed filtration: accumulation-detachment model parameters. Chem. Enyng Sci. 42, 1213 1219. Beizaie, M., Wang, C. and Tien, C., 1981, A simulation model of particle deposition on single collectors. Chem. Engng Commun. 13, 153 180. Chang, J. S. and Vigneswaran, S., 1990, Ionic strength in deep bed filtration. Water Res. 24, 1425-1430. Chiang, H.-W. and Tien, C., 1985a, Dynamics of deep-bed filtration: analysis of two limiting situations. A.I.Ch.E.J. 31, 1349 1359. Chiang, H.-W. and Tien, C., 1985b, Dynamics of deep-bed filtration: experiment. A.LCh.E.J. 31, 1360-1371. Choo, C.-H. and Tien, C., 1995, Simulation of hydrosol deposition in granular media. A.I.Ch.E.J. 41, 142(~1442. Danckwerts, P. V., 1952, Continuous flow systems: distribution of residence times. Chem. Engng Sci. 2, I 13. Davis, M. E., 1984, Numerical Methods and Modeling for Chemical Engineers. Wiley, New York, U.S.A.

Filtration of noncoagulating particles Elimelech, M. and Song, L., 1992, Theoretical investigation of colloid separation from dilute aqueous suspensions by oppositely charged granular media. Separat. Sci. TechnoL 2, 2-12. Happel, J., 1958, Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. A.I.Ch.E.J. 4(2), 197-201. Jung, Y. and Tien, C., 1993, Simulation of aerosol deposition in granular media. Aerosol Sci. Technol. 18, 418~,40. Mackie, R. I., Horner, R. M. W. and Jarvis, R. J, 1987, Dynamic modeling of deep-bed filtration. A.1.Ch.E.J. 33, 1761 1775. O'Melia, C. R. and All, W., 1978, The role of retained particles in deep-bed filtration. Progress Water Technol. 10, 167-182. Payatakes, A. C., Tiem C. and Turian, R. M., 1974a, Trajectory calculation of particle deposition in deep bed filtration: model formulation. A.1.Ch.E.J. 20, 889 899. Payatakes, A. C., Tien, C. and Turian, R. M., 1974b, Trajectory calculation of particle deposition in deep bed filtration: case study of the effect of the dimensionless groups and comparison with experimental data. A.1.Ch.E.J. 20, 900-905. Pendse, H. and Tien, C., 1982, A simulation model of aerosol collection in granular media. J. Colloid Interface Sci. 87(1), 225-241. Rajagopalan, R., 1974, Stochastic modeling and experimental analysis of particle transport in water filtration. Ph.D. Dissertation, Syracuse University. Rajagopalan, R. and Chu, R. Q., 1982, Dynamics of adsorption of colloidal particles in packed beds. J. Colloid Interface Sci. 86, 299-317. Rajagopalan, R. and Tien, C., 1976, Trajectory analysis of deep-bed filtration with the sphere-in-cell porous media model. A.I.Ch.E.J. 22, 523 533. Rajagopalan, R. and Tien, C., 1982, Letter to the editor, A.I.Ch.E.J., 28, 871 872. Ramirez, W. F., 1989, Computational Methods for Process Simulation. Butterworths, Boston, MA, U.S.A. Riley, R. J., 1987, The magnetically stabilized fluidized bed as a solid/liquid separator. M.S. Thesis, University of Michigan. Sharma, S. K. and Mahendroo, P. P., 1980, Affinity chromatography of cells and cell membranes. J. Chromat. 184, 471~,99. Sherwood, T. K., Pigford, R, L. and Wilke, C. R., 1975, Mass Transfer. McGraw-Hill, New York, U.S.A. Song, L. and Elimelech, M., 1993, Dynamics of colloid deposition in porous media: modeling the role of retained particles. Colloids Surf 73, 49-63. Spiehnan, L. A. and FitzPatrick, J. A., 1973, Theory for particle collection under London and gravity forces. J. Colloid lnterJace Sci. 42, 607-623. Terranova, B. E. and Burns, M. A., 1991, Continuous cell suspension processing using magnetically stabilized fluidized beds. Biotechnol. Bioengng 37, 110-120. Thomas, H., 1944, Heterogeneous ion exchange in a flowing system. J. Am. Chem. Soc. 66, 1664-1666. Tien, C., 1984, Stochastic modeling of particle deposition on collectors. Math. Comput. Sire. XXVI, 355-356. Tien, C., 1989, Granular Filtration of Aerosols and Hydrosols. Butterworths, Boston, U.S.A. Tien, C. and Payatakes, C., 1979 Advances in deep bed filtration. A.I.Ch.E, J. 25, 737-759. Tien, C., Wang, C.-S. and Barot, D. T., 1977, Chainlike formation of particle deposits in fluid-particle separation. Science 196, 983 985. Tobiason, J. E. and O'Mdia, C. R, 1988, Physicachemical aspects of particle removal in depth filtration. J. A W W A 80, 54-64. Vaidyanathan, R. and Tien, C., 1989, Hydrosol deposition in granular beds--an experimental study. Chem. Engng Commun. 81, 123-144. Vaidyanathan, R. and Tien, C., 1991, Hydrosol deposition in granular media under unfavorable surface conditions. (?hem. Engng Sci. 46, 967-983.

105

Vigneswaran, S. and Aim, B. R., 1985, The influence of suspended particle size distribution in deep-bed filtration. A.I.Ch.E.J. 312, 321-324. Vigneswaran, S. and Song, C. J., 1986, Mathematical modeling of the entire cycle of deep bed filtration. Water Air Soil Pollut. 29, 155 164. Vigneswaran, S. and Tulachan, R. K., 1988, Mathematical modeling of transient behavior of deep bed filtration. Water Res. 22, 1093-1100. Wang, C.-S., Beizaie, M. and Tien, C., 1977, Deposition of solid particles on a collector: formulation of a new theory. A.I.Ch.E.J. 23, 879-889. APPENDIX

Change in collection efficiency with particle deposition, F The a priori predictions shown in Fig. 14 used the following set of expressions for the 1.44 Ftm suspension:

0 ~ 0.0471: F = 1

15.70

(A1)

0.0471 < 0 ~ 0.0968: F = 1.94 - 6.39 x 1010 + 7.37 x 10202 - 2.92 x 10303

(A2)

0 > 0.0968: F = 0.

(A3)

For 0.78 ~tm suspensions the expressions used are 0 ~< 0.0234: F = 1 - 28.80

(A4)

0.0234 < 0 ~< 0.0533: F = - 2.317 + 4.814 x 1020 - 3.017 x 10402 + 8,523 x 10503 -- 1.138 x 10704 + 5,857 x 1070 s

(A5)

0 > 0.0533: F = O.

(A6)

A higher-order polynomial [eq. (A5)] and greater number of significant figures were necessary to accurately fit the simulation data for the 0.78 ,urn latex suspension. Approximate analytical solution The approximate analytical solutions to the model eqs (18)-(26) with F defined by eq. (32) are

4) = ,'/Co =

((K/rsl ( 3 DaI exp(Dal~ ) + exp~ (® - ~)~ -

3

((~/F0

1

(A7) ( Da~

1 0 = o/am -

Fs

exp - - ( 0 - 0 ~(g/Fs)

}

-1

( DaI ) exp(Da,0 + exp ~ (® - ~)~ - 1

(A8) and were developed using the results of Thomas (1944), who originally solved the mass balances without dispersion and a more general rate expression. The relationship between the shadowing factor and the shadowing exponent can be approximated with the following relationship: k~ = 0.558fl + 0.442 (see Fig. 17).