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THE ROLE OF RETAINED PARTICLES IN DEEP BED FILTRATION
Prog. Wat. Tech. 1978, Vol. 10, Nos 5/6, pp. 167-182. Pergamon Press. Printed in Great Britain.
THE ROLE OF RETAINED PARTICLES IN DEEP BED FILTRATION Charles R. O'Melia and Waris Ali Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill, NC 27514, U.S.A. Summary A model is formulated for the development of head loss and removal efficiency during filtration by packed beds, based on the postulate that some retained particles can act as filter media and thereby improve filtration efficiency. The model is calibrated and tested using data from laboratory filters. Small suspended particles are shown to produce much greater head losses than larger particles. The effects of suspended particle size and concentration, pretreatment by flocculation, and media size on removal efficiency and head loss are discussed. INTRODUCTION Present theories for the filtration of water and wastewater by packed beds describe the performance of clean filters with some success. For example, the head loss at the start of filtration depends upon media size, filtration rate, and bed porosity in a predictable manner (Kozeny, 1927; Fair and Hatch, 1933). Similarly, the removal efficiency of clean filters depends primarily on the size, density, and surface characteristics of the suspended particles, and also on the size and surface characteristics of the filter media, the filtration velocity, and the depth and porosity of the filter bed (0TMelia and Stumm, 1967; Yao et al., 1971; Spielman and FitzPatrick, 1973; Fitzpatrick and Spielman, 1973; Habibian and O'Melia, 1975). During filtration, both head loss and removal efficiency change with time. These changes are caused by particles that are retained in the filter bed. The purpose of this research is to evaluate the effects of particles that are filtered from suspension on the changes of removal efficiency and head loss that occur with time during filtration. The deterioration in effluent quality that occurs late in a filter run when filter capacity is depleted is not considered; emphasis is placed on the ripening period that occurs after filtration begins. A model is formulated in mathematical terms, calibrated with a set of experimental data from laboratory tests, tested (verified) with other experimental data, and then used to simulate the effects of several physical factors on the filtration process. The results provide some new insights into the mechanisms of filtration and suggest directions for filter design.
167
C.R. 0?Melia and W. Ali
168 MODEL DEVELOPMENT
The model developed here is suggested by some experimental observations. First, a suspended particle size with a minimum removal efficiency by a clean filter exists. This size is of the order of 1 ym (Yao et al., 1971). Second, this critical size persists during a filtration run (Habibian and O'Melia, 1975). Third, the effects of suspended particle size on head loss development are dramatic and surprising. For a given mass of material removed, small suspended particles produce much larger head losses than large ones as filtration proceeds. Fourth, the removal of suspended particles from air by fibers has been shown to produce chains or dendrites (Billings, 1966), in which particles removed by the filter fiber serve as sites for the deposition of additional material. The formation of these dendrite deposits has been analysed and modeled by Tien and coworkers (Payatakes and Tien, 1976; Tien, Wang, and Barot, 1977) 0 Habibian and O'Melia (1975) have suggested that particles removed during the early stage of a filter run can serve as collectors or deposition sites for particles reaching the bed at a later time, thereby improving filtration efficiency. This is the basic postulate in the filtration model that follows. A sketch of particle deposition during filtration is shown in Fig. 1, adapted from Tien et al. (1977). The filter media is represented by a large sphere termed a collector. Particles in suspension are depicted as smaller spheres of uniform size. Fluid streamlines for laminar flow are indicated. Particle removal by interception is illustrated. For this case, suspended particles follow the fluid flow exactly and contact a collector because of their finite size. Other mechanisms operative in water and wastewater filtration are Brownian diffusion and sedimentation, due to thermal and gravity effects. These are not illustrated in Fig. 1. Particle 1 is removed on the spherical collector by interception at point A. This has two effects, First, it provides an additional contact site for further deposition by protruding into the original flow field. Particle 2, which would have been carried past the "clean" collector, can then be removed. Second, removal by the media grain in a shadow area downstream from point A is impaired, at least when removal is by direct interception or settling. The result of these effects is the formulation of a deposit consisting of chains or dendrites. Following practice in air and water filtration (Friedlander, 1958; Yao et al., 1971), the single collector efficiency of a clean collector (η) is defined as follows η =
Rate at which particles strike the collector Rate at which particles flow towards the collector
.-v
For the case considered here where the actual collector consists of a filter grain and an associated number (N) of particles attached to it that also serve as collectors,
169
Retained particles in deep bed filtration
. n
c
=
/Rate at which particles \ , N [strike a retained parti- J „ \ cle acting as a collector/ /Rate at which particles \ \flow towards the collector/
)
^
Here n c is the combined contact efficiency of the filter grain and its associated retained particles· The rate at which particles flow towards the collector may be written as n 0 v 0 d c 2(n74), where n 0 and v 0 are the undisturbed or bulk particle concentration and fluid velocity, respectively, and (ïï/4)dc is the cross-sectional area of the collector perpendicular to v 0 . It will be assumed that the diameter of the collecter, d c , is approximately equal to that of the media grain. Rearrangement of Eq. 2 yields
nc=n+Nnp(^)2
(3)
Here η ρ is the contact efficiency of a retained particle, defined as the rate at which particles strike a retained particle acting as a collector divided by n 0 v 0 dp (π/4), and dp is the diameter of a suspended particle. Removal of suspended particles by packed bed filters involves two steps: (1) transport of suspended particles from the bulk fluid in the pores to the solid-liquid interfaces presented by the filter media and the associated retained particles, and (2) attachment of the suspended particles at these interfaces. The first step is primarily physical mass transport for which some quantitative theories exist; it is represented here by the terms n c , η, and η ρ in Eq. 3. The second step is primarily chemical and is not as well understood. It will be described here by two empirical coefficients, a and oip· The fraction of the contacts between a filter grain and suspended particles that result in attachment and removal is termed a. Similarly, otp is the fraction of the contacts between retained particles and suspended particles that result in attachment and removal. Hence, a and otp are particleto-filter grain and particle-to-particle attachment coefficents that can range from 0 to 1. They are not necessarily equal, since the surface properties of the filter grains are usually different from those of the particles to be filtered. With these definitions, Eq. 3 may be rewritten as
n r = αη + ΝαρηΥ^Ρ- J
(4)
in which n r is the single collector removal effrciency of a filter grain and its associated retained particles. The first term on the R.H.S. of Eq. 4 represents the removal efficiency of a single clean filter grain; the second term represents the increase in removal brought about by suspended particles
C.R. 0!Melia and W. Ali
170 that are retained.
Consider next the number of retained particles that can act as collectors (N), which can vary with time and with location in the bed. The removal efficiency of a clean filter grain is an, so that the rate of removal of suspended particles by the clean grain is ann 0 v 0 d c (ττ/4). Neglecting the change in surface coverage that must occur with time, and considering that some fraction (3) of these particles retained directly on the filter grain can act as additional collectors, one can write
If = n«*OVodc2(i)
(5)
The assumptions made in formulating Eq. 5 are debatable and clearly unrealistic in some cases. They are made to provide an approximate and preliminary estimate of 3N/3t. The removal efficiency of a single filter grain is related to the removal accomplished by a packed bed through a mass balance about a differential volume element of the filter. Results have been presented for a bed of clean cylindrical fibers in air filtration (Friedlander, 1958) and for a bed of clean spheres in water filtration. For the case considered here, in which retained particles act as collectors so that removal varies with time, the result of such a mass balance is given as follows:
iE + v0 *L + 1 (1-f) Vo nn r = 0 3t
° 3L
2
dc
°
r
(6)
Here n is the suspended particle concentration at some time (t) and depth (L) in the filter bed and f is the bed porosity. Equations 4, 5, and 6 comprise a mathematical model to describe the removal efficiency of a filter bed in time and space. Analytical solutions have not been obtained; simplifying assumptions have been made. Details are presented elsewhere (Ali, 1977). In this solution, n r and n are considered as step functions rather than continuous functions of time. The result is _ 3/i_. P > Vr , /L ^(l-f)nri„1(^ 2 I n _ILL = - _ 3 n„a i(l-f)/L_) (^i _ f ) / L U\ ±1 ++ nou ap dDuY£) DBv ηραρ3ν^ρ ^ΓΣη Δί - p0^ -p ^kn ^ Qo0Ate «^ JA ( 7 ) Here n^ is the particle concentration in the depth L at the i time step, and nr· i ^ s t n e single collector removal efficiency for the (i-1) time step0 Equation 7 is a recursion equation. To obtain a solution, the value of ηΓ at time zero must be known from theories or from clean f i l t e r experiments. The development of head loss (hf) is based on the Kozeny equation for a clean f i l t e r :
171
Retained particles in deep bed filtration
H£=kÜVaa^)2(AcV L
p g
f3
\V C J
(8)
Here A c and V c are the surface area and volume of the filter media in the bed, k is an empirical coefficient, μ and p are the dynamic viscosity and density of the fluid, g is the gravity acceleration, and v 0 is the undisturbed superficial velocity above the filter bed. Retention of particles changes the hydraulic characteristics of the filter bed. Some investigators have considered the decrease in porosity that accompanies removal, and attempted to relate head loss development to particle removal through the porosity function with little success. In this research, the change in interfacial surface area (Ac) as dendrites develop is considered, while porosity changes are assumed negligible. The result is the following Jl£.=36 μ vp, (1-f) 2 L dc*P g fJ
1 +
1+
NcUci
(9)
Here N c and N p are the number of filter grains and retained particles in the filter bed with length L,and g1 is an empirical coefficient that represents the fraction of retained particles that are exposed to the flowing fluid and contribute to the additional surface area. Deposition is nonuniform with depth, so that the bed is subdivided into several layers and hf/L is calculated for each layer. MODEL CALIBRATION Use of Equations 7 and 9 to describe filter performance requires evaluation of the terms η, η ρ , α, α ρ , 3, and 3 1 . Data obtained by Habibian (1971) have been used to estimate certain of these * coefficients in calibrating the model. In one experiment, Habibian operated four identical filters (dc = 0.38 mm, f = 0.36, L = 14 cm) at 2 gpm/ft2 (0.136 m / s ) , filtering a suspension of latex spheres (concentration = 11 mg/1, dp = 0.1 urn). Filter media were coated with a cationic polymer prior to use, and polymer was added continuously throughout the runs at a dosage selected to maximize particle-to particle attachment. Coagulation within the pores of the filter bed was not significant. Experimental results of the four runs were similar; these have been averaged and plotted in Fig. 2. Particle removal and head loss are plotted as functions of filtration time. In calibrating the model, the single collector removal efficiency (ηα) is calculated from the experimental data for a "clean" filter. For example, ni/no is (1-0.6) or 0.4 for the filters in Fig. 2. Hence, using Eq. 7 at t = 0, ηα = 0.00259. With this initial condition established, the product 3p3 was varied until the calculated values for removal efficiency is a function of time agreed well with the observed results (Fig. 2 ) . A value
J.P.W.T. S.P. D 2
172
C.R. O'Melia and W. Ali
of 0.009 was obtained for oipß. In a similar manner, 3 1 was estimated using Eq. 9. Calculations of head loss as a function of time were compared with experimental results (Fig. 2 ) . A value for 3 f of 0.15 was selected. The contact efficiency of a retained particle, ηρ, was estimated using the theory of Yao et al. (1971) for transport by diffusion, interception, and settling. The models for transport diffusion and interception were modified using Happel's model for flow through porous media. Because of the empirical and preliminary nature of this approach, corrections for hydrodynamic interactions were not made. MODEL VERIFICATION The model was tested or verified using data from other experiments by Habibian (1971) in which one or more filtration variables was significantly different from conditions in the runs used for calibration. Observed values of the particle removal by the clean filters were used to calculate ηα for each run. Other coefficients were kept constant, save that 3 and ß f were set equal to one for large suspended particles where interception and settling were dominant transport mechanisms. This assumption was based on steric arguments. Data from six filtration experiments using three suspended particle sizes, five influent concentrations, and two bed depths are presented in Figs. 3 to 6. Model predictions are also depicted. Special note is given to the results in Fig. 6. Particles of 1 ym size show the poorest removal; smaller particles (0.1 ym) are transported effectively by diffusion and larger ones (7.6 ym) are transported by interception and settling. Head loss development, however, increases dramatically with decreasing particle size. Influent concentrations were all about 50 mg/1, but head loss development was small for 7.6 ym particles and very rapid for 0.1 ym particles. This set of data provides a useful test for the model. Comparisons between calculations and observations are good to excellent for five of the runs. In one experiment (Fig. 3 ) , the head loss calculations do not fit the data. Reasons for this are not known. Based on these agreements, it was decided that the model provides a useful simplified theory for the development of removal efficiency and head loss during filtration. It is able to simulate the effects of suspended particle size on removal efficiency, and also the surprising effects of suspended particle size on head loss development. Small particles are considered to produce large chains or dendrites that increase the surface in the filter bed while having little effect on porosity. SIMULATIONS Simulations of the effects of suspended particle size and concentration, media size, and bed depth on particle removal and head loss have been made with the calibrated and verified model. Results are presented in Figs. 7 to 10.
Retained particles in deep bed filtration
173
Particle size Predictions of particle removal and head loss as functions of time for three different particle sizes are shown in Fig. 7. One-micron particles show the poorest removal at the onset of filtration, as observed by Yao et al. (1971), and also are the most difficult to remove during the run. Small (0.1 ym) particles produce considerable head loss as they are removed, while larger ones (1 and 7.6 ym) show negligible head loss development for the influent concentration (10 mg/1) and filtration time (3 hrs) examined. The importance of suspended particle size in determining removal efficiency has been predicted and observed previously. Its significance in head loss development has been observed experimentally (Habibian and 0 T Melia, 1975); the results presented here provide a plausible model for this effect. Influent concentration For a clean filter, the rate of removal of particles with respect to depth is proportional to the concentration of particles in suspension, i.e., 3C/9L = -AC. Hence, the removal efficiency of clean filters is independent of influent concentration (Ives, 1967). This is not the case as filtration proceeds. Calculated particle removals and head losses are plotted as functions of filtration time in Fig. 8 for four influent concentrations ranging from 1 to 50 mg/1. As expected, low concentrations produce low head losses. They can also produce low removal efficiencies. This is because removal by a packed bed filter can depend on the number of retained particles which act as collectors. When the influent concentration is low, the rate at which new collectors accumulate in the bed is also low. The ripening process is lengthened, but removal efficiency is impaired. This indicates that filters treating low turbidity waters should be deep, while those treating waters with high suspended solids concentrations can be more shallow. Filters treating low turbidity waters must rely on the filter media to provide collectors; those treating more concentrated suspensions remove solids primarily by contacts with previously retained particles. Media size Calculations illustrating the effects of media size on filtration are presented in Fig. 9. As expected, increasing media size produces lower removals and head losses. The reduction in head loss is more marked than the reduction in removal efficiency. These effects, however, are less significant than those noted for suspended particle size and concentration. The role of media size in filtration may be overstated both in the literature and in practice. Media size can be easily measured and readily controlled, but it is not of major importance. Bed depth The effects of bed depth and filtration time are illustrated by the results presented in Fig. 10, in which particle removal and head loss are plotted as functions of bed depth for five filtration times. The calculations are made for a homogeneous bed with a media size of 0.38 mm. Removal efficiency
C.R. 0TMelia and W. Ali
174
and head loss both increase significantly with time, as expected, but their distribution with time is noteworthy. Removal is distributed throughout the bed at the onset of filtration, but becomes localized in the upper region of the bed as filtration proceeds. Head loss follows a similar distribution. Retained particles accumulate in the upper regions of the bed and lead to the retention of more particles in that area. Filtration is seen to have an autocatalytic character. The use of large filter media in an upper layer of a mixed media filter to permit bed penetration is seen to have limitations, especially for concentrated suspension. This is because removal is accomplished primarily by retained suspended particles that can be quite small. DISCUSSION Observation and theory indicate that the size and concentration of the suspended particulates are the most important physical variables influencing packed bed filtration. Particulate sizes in water and wastewater are not often measured because available techniques are time consuming, expensive, and achieve only limited success. While particle sizes cannot be measured easily, they can be altered by flocculation prior to filtration. When a water or wastewater contains substantial quantities of submicron particles (e.g., a high rate trickling filter effluent), it is plausible that flocculation prior to filtration will be beneficial. Filters can be shallow and contain large media, since retained particles will produce the bulk of the removal. When particle sizes are small and the solids concentration is low (e.g., asbestos particles in Lake Superior), either additional particles must be added for flocculation (sweep floes) or deep filters containing smaller media can be used. The effects of the chemistry of the filtration process, as represented by a and dp, have not been presented here. Preliminary analyses have been made by Ali (1977). In the absence of chemical destabilization (a and otp = 0 ) , removal is zero throughout a run and head loss remains unchanged. Attachment requires chemical destabilization and is essential for the filtration process to perform its function of solid-liquid separation. The model presented here is oversimplified and requires refinement for quantitative application. In addition, it does not treat that portion of the filtration cycle when filter capacity is depleted and breakthrough occurs. Possible reintrainment of retained particles into the bulk flow has also been neglected. CONCLUSIONS The following conclusions are made: 1.
The removal efficiency of a clean filter bed depends on the size of the particles being filtered. A critical particle size exists, in the
Retained particles in deep bed filtration
175
region of 1 ym. This particle size has the lowest opportunity for contact with the filter media, and subsequent removal from suspension. Smaller particles are effectively transported by Brownian diffusion, larger ones by interception and settling. 2.
This effect of particle size on removal continues into the ripening period; i.e., one-micron particles show the poorest removal throughout the period of effective filtration.
3.
For a given mass of particles removed, head loss increase varies in an inverse manner with suspended particle size. Submicron particles can produce enormous head losses when treated by conventional packed-bed filters.
4.
The removal efficiency of a clean filter is independent of influent concentration.
5.
After the onset of filtration, removal efficiency improves with increasing influent concentration during the ripening period, because retained particles act as collectors for other suspended particles.
6.
Flocculation can be used to adjust the size of the particles in the filter influent. Increasing the suspended particle size above the micron range will improve removal efficiency and reduce head loss.
7.
It is plausible that the advantages of multimedia beds have been overstated. In most filtration processes, the particles in the filter influent provide most of the removal after the run has begun. Hence, head loss and removal tend to be localized in the upper regions of the bed during downflow filtration, regardless of the size of the media.
8.
Effective filtration requires that particle-to-particle and particleto-filter grain contacts be successful in producing attachment. This, in turn, requires chemical control of the process during operation.
9.
The model presented here considers that particles retained within a packed-bed filter during the initial stages of a run can act as collectors (filter media) for other suspended particles applied to the bed at a later time. This gives rise to a ripening of the filter, and permits filters to be for more efficient in particle removal than theory and observations on clean filters would suggest. The form of these deposits is likely to be chains or dendrites which extend into the pores, provide considerable surface for fluid drag, and can increase head loss significantly. This model agrees well with results of laboratory experiments.
176
C.R. O'Melia and W. Ali REFERENCES
Ali, W., 1977, "The Role of Retained Particles in a Filter," unpublished master's report, Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill, N.C·, U.S.A. Billings, C. E., 1966, "Effect of Particle Accumulation in Aerosol Filtration, Ph.D. dissertation, California Institute of Technology, Pasadena, CA., U.S.A. Fair, G. M., and Hatch, L. P., 1933, "Fundamental Factors Governing the Streamline Flow of Water Through Sand," Journal, American Water Works Association, Vol. 25, 1551-1565. FitzPatrick, J., and Spielman, L. A., 1973, "Filtration of Aqueous Latex Suspensions Through Beds of Glass Spheres," Journal of Colloid and Interface Science, Vol 43, 350-369. Friedlander, S. A., 1958, "Theory of Aerosol Filtration," Industrial and Engineering Chemistry, Vol. 50, 1161-1164. Habibian, M. T., 1971, "The Role of Polyelectrolytes in Water Filtration," Ph.D. dissertation, University of North Carolina, Chapel Hill, N . C , U.S.A. Habibian, M. T., and O'Melia, C. R., 1975, "Particles, Polymers, and Water Filtration," Proceedings of the American Society of Civil Engineers, Journal of the Sanitary Engineering Division, Vol. 101, No. EE4, 567-583. Ives, K. J., 1967, "Basic Concepts of Filtration," Proceedings from the Society for Water Treatment and Examination, Vol. 16, 147-169. Kozeny, J., 1927, Sitzungberichte Akademie der Wissenschaften Wien, Mathematisch-Naturwissenschaftlich Klassen Abt H a , Vol. 136, 271-306. O'Melia, C. R., and Stumm, W., 1967, "Theory of Water Filtration," Journal American Water Works Association, Vol. 59, 1393-1412. Payatakes, A. C., and Tien, C., 1976, "Particle Deposition in Fibrous Media with Dendrite-Like Pattern: A Preliminary Model," Journal of Aerosol Science, Vol. 7, 85-100. Spielman, L. A., and FitzPatrick, J. A., 1973, "Theory for Particle Collection Under London and Gravity Forces for Application to Water Filtration," Journal of Colloid and Interface Science, Vol. 42, 607-623.
Retained particles in deep bed filtration
177
Tien, C., Wang, C. S. , and B a r o t , D. T . , 1977, " C h a i n l i k e Formation of P a r t i c l e Deposits i n F l u i d - P a r t i c l e Separation, 1 1 Science, Vol. 196, 983985. Yao, K. M. , Habibian, M. T . , and O'Melia, C. R. , 1971, "Water and Wastewater F i l t r a t i o n : Concepts and A p p l i c a t i o n s , " Environmental Science and Technology, Vol. 5, 1105-1112.
C.R. 0TMelia and W. Ali
178
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Retained particles in deep bed filtration
60
Experimental Results Model Predictions
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100 150 200 TIME (min.)
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181
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THE ROLE OF RETAINED PARTICLES IN DEEP BED FILTRATION Charles R. O'Melia and Waris Ali Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill, NC 27514, U.S.A. Summary A model is formulated for the development of head loss and removal efficiency during filtration by packed beds, based on the postulate that some retained particles can act as filter media and thereby improve filtration efficiency. The model is calibrated and tested using data from laboratory filters. Small suspended particles are shown to produce much greater head losses than larger particles. The effects of suspended particle size and concentration, pretreatment by flocculation, and media size on removal efficiency and head loss are discussed. INTRODUCTION Present theories for the filtration of water and wastewater by packed beds describe the performance of clean filters with some success. For example, the head loss at the start of filtration depends upon media size, filtration rate, and bed porosity in a predictable manner (Kozeny, 1927; Fair and Hatch, 1933). Similarly, the removal efficiency of clean filters depends primarily on the size, density, and surface characteristics of the suspended particles, and also on the size and surface characteristics of the filter media, the filtration velocity, and the depth and porosity of the filter bed (0TMelia and Stumm, 1967; Yao et al., 1971; Spielman and FitzPatrick, 1973; Fitzpatrick and Spielman, 1973; Habibian and O'Melia, 1975). During filtration, both head loss and removal efficiency change with time. These changes are caused by particles that are retained in the filter bed. The purpose of this research is to evaluate the effects of particles that are filtered from suspension on the changes of removal efficiency and head loss that occur with time during filtration. The deterioration in effluent quality that occurs late in a filter run when filter capacity is depleted is not considered; emphasis is placed on the ripening period that occurs after filtration begins. A model is formulated in mathematical terms, calibrated with a set of experimental data from laboratory tests, tested (verified) with other experimental data, and then used to simulate the effects of several physical factors on the filtration process. The results provide some new insights into the mechanisms of filtration and suggest directions for filter design.
167
C.R. 0?Melia and W. Ali
168 MODEL DEVELOPMENT
The model developed here is suggested by some experimental observations. First, a suspended particle size with a minimum removal efficiency by a clean filter exists. This size is of the order of 1 ym (Yao et al., 1971). Second, this critical size persists during a filtration run (Habibian and O'Melia, 1975). Third, the effects of suspended particle size on head loss development are dramatic and surprising. For a given mass of material removed, small suspended particles produce much larger head losses than large ones as filtration proceeds. Fourth, the removal of suspended particles from air by fibers has been shown to produce chains or dendrites (Billings, 1966), in which particles removed by the filter fiber serve as sites for the deposition of additional material. The formation of these dendrite deposits has been analysed and modeled by Tien and coworkers (Payatakes and Tien, 1976; Tien, Wang, and Barot, 1977) 0 Habibian and O'Melia (1975) have suggested that particles removed during the early stage of a filter run can serve as collectors or deposition sites for particles reaching the bed at a later time, thereby improving filtration efficiency. This is the basic postulate in the filtration model that follows. A sketch of particle deposition during filtration is shown in Fig. 1, adapted from Tien et al. (1977). The filter media is represented by a large sphere termed a collector. Particles in suspension are depicted as smaller spheres of uniform size. Fluid streamlines for laminar flow are indicated. Particle removal by interception is illustrated. For this case, suspended particles follow the fluid flow exactly and contact a collector because of their finite size. Other mechanisms operative in water and wastewater filtration are Brownian diffusion and sedimentation, due to thermal and gravity effects. These are not illustrated in Fig. 1. Particle 1 is removed on the spherical collector by interception at point A. This has two effects, First, it provides an additional contact site for further deposition by protruding into the original flow field. Particle 2, which would have been carried past the "clean" collector, can then be removed. Second, removal by the media grain in a shadow area downstream from point A is impaired, at least when removal is by direct interception or settling. The result of these effects is the formulation of a deposit consisting of chains or dendrites. Following practice in air and water filtration (Friedlander, 1958; Yao et al., 1971), the single collector efficiency of a clean collector (η) is defined as follows η =
Rate at which particles strike the collector Rate at which particles flow towards the collector
.-v
For the case considered here where the actual collector consists of a filter grain and an associated number (N) of particles attached to it that also serve as collectors,
169
Retained particles in deep bed filtration
. n
c
=
/Rate at which particles \ , N [strike a retained parti- J „ \ cle acting as a collector/ /Rate at which particles \ \flow towards the collector/
)
^
Here n c is the combined contact efficiency of the filter grain and its associated retained particles· The rate at which particles flow towards the collector may be written as n 0 v 0 d c 2(n74), where n 0 and v 0 are the undisturbed or bulk particle concentration and fluid velocity, respectively, and (ïï/4)dc is the cross-sectional area of the collector perpendicular to v 0 . It will be assumed that the diameter of the collecter, d c , is approximately equal to that of the media grain. Rearrangement of Eq. 2 yields
nc=n+Nnp(^)2
(3)
Here η ρ is the contact efficiency of a retained particle, defined as the rate at which particles strike a retained particle acting as a collector divided by n 0 v 0 dp (π/4), and dp is the diameter of a suspended particle. Removal of suspended particles by packed bed filters involves two steps: (1) transport of suspended particles from the bulk fluid in the pores to the solid-liquid interfaces presented by the filter media and the associated retained particles, and (2) attachment of the suspended particles at these interfaces. The first step is primarily physical mass transport for which some quantitative theories exist; it is represented here by the terms n c , η, and η ρ in Eq. 3. The second step is primarily chemical and is not as well understood. It will be described here by two empirical coefficients, a and oip· The fraction of the contacts between a filter grain and suspended particles that result in attachment and removal is termed a. Similarly, otp is the fraction of the contacts between retained particles and suspended particles that result in attachment and removal. Hence, a and otp are particleto-filter grain and particle-to-particle attachment coefficents that can range from 0 to 1. They are not necessarily equal, since the surface properties of the filter grains are usually different from those of the particles to be filtered. With these definitions, Eq. 3 may be rewritten as
n r = αη + ΝαρηΥ^Ρ- J
(4)
in which n r is the single collector removal effrciency of a filter grain and its associated retained particles. The first term on the R.H.S. of Eq. 4 represents the removal efficiency of a single clean filter grain; the second term represents the increase in removal brought about by suspended particles
C.R. 0!Melia and W. Ali
170 that are retained.
Consider next the number of retained particles that can act as collectors (N), which can vary with time and with location in the bed. The removal efficiency of a clean filter grain is an, so that the rate of removal of suspended particles by the clean grain is ann 0 v 0 d c (ττ/4). Neglecting the change in surface coverage that must occur with time, and considering that some fraction (3) of these particles retained directly on the filter grain can act as additional collectors, one can write
If = n«*OVodc2(i)
(5)
The assumptions made in formulating Eq. 5 are debatable and clearly unrealistic in some cases. They are made to provide an approximate and preliminary estimate of 3N/3t. The removal efficiency of a single filter grain is related to the removal accomplished by a packed bed through a mass balance about a differential volume element of the filter. Results have been presented for a bed of clean cylindrical fibers in air filtration (Friedlander, 1958) and for a bed of clean spheres in water filtration. For the case considered here, in which retained particles act as collectors so that removal varies with time, the result of such a mass balance is given as follows:
iE + v0 *L + 1 (1-f) Vo nn r = 0 3t
° 3L
2
dc
°
r
(6)
Here n is the suspended particle concentration at some time (t) and depth (L) in the filter bed and f is the bed porosity. Equations 4, 5, and 6 comprise a mathematical model to describe the removal efficiency of a filter bed in time and space. Analytical solutions have not been obtained; simplifying assumptions have been made. Details are presented elsewhere (Ali, 1977). In this solution, n r and n are considered as step functions rather than continuous functions of time. The result is _ 3/i_. P > Vr , /L ^(l-f)nri„1(^ 2 I n _ILL = - _ 3 n„a i(l-f)/L_) (^i _ f ) / L U\ ±1 ++ nou ap dDuY£) DBv ηραρ3ν^ρ ^ΓΣη Δί - p0^ -p ^kn ^ Qo0Ate «^ JA ( 7 ) Here n^ is the particle concentration in the depth L at the i time step, and nr· i ^ s t n e single collector removal efficiency for the (i-1) time step0 Equation 7 is a recursion equation. To obtain a solution, the value of ηΓ at time zero must be known from theories or from clean f i l t e r experiments. The development of head loss (hf) is based on the Kozeny equation for a clean f i l t e r :
171
Retained particles in deep bed filtration
H£=kÜVaa^)2(AcV L
p g
f3
\V C J
(8)
Here A c and V c are the surface area and volume of the filter media in the bed, k is an empirical coefficient, μ and p are the dynamic viscosity and density of the fluid, g is the gravity acceleration, and v 0 is the undisturbed superficial velocity above the filter bed. Retention of particles changes the hydraulic characteristics of the filter bed. Some investigators have considered the decrease in porosity that accompanies removal, and attempted to relate head loss development to particle removal through the porosity function with little success. In this research, the change in interfacial surface area (Ac) as dendrites develop is considered, while porosity changes are assumed negligible. The result is the following Jl£.=36 μ vp, (1-f) 2 L dc*P g fJ
1 +
1+
NcUci
(9)
Here N c and N p are the number of filter grains and retained particles in the filter bed with length L,and g1 is an empirical coefficient that represents the fraction of retained particles that are exposed to the flowing fluid and contribute to the additional surface area. Deposition is nonuniform with depth, so that the bed is subdivided into several layers and hf/L is calculated for each layer. MODEL CALIBRATION Use of Equations 7 and 9 to describe filter performance requires evaluation of the terms η, η ρ , α, α ρ , 3, and 3 1 . Data obtained by Habibian (1971) have been used to estimate certain of these * coefficients in calibrating the model. In one experiment, Habibian operated four identical filters (dc = 0.38 mm, f = 0.36, L = 14 cm) at 2 gpm/ft2 (0.136 m / s ) , filtering a suspension of latex spheres (concentration = 11 mg/1, dp = 0.1 urn). Filter media were coated with a cationic polymer prior to use, and polymer was added continuously throughout the runs at a dosage selected to maximize particle-to particle attachment. Coagulation within the pores of the filter bed was not significant. Experimental results of the four runs were similar; these have been averaged and plotted in Fig. 2. Particle removal and head loss are plotted as functions of filtration time. In calibrating the model, the single collector removal efficiency (ηα) is calculated from the experimental data for a "clean" filter. For example, ni/no is (1-0.6) or 0.4 for the filters in Fig. 2. Hence, using Eq. 7 at t = 0, ηα = 0.00259. With this initial condition established, the product 3p3 was varied until the calculated values for removal efficiency is a function of time agreed well with the observed results (Fig. 2 ) . A value
J.P.W.T. S.P. D 2
172
C.R. O'Melia and W. Ali
of 0.009 was obtained for oipß. In a similar manner, 3 1 was estimated using Eq. 9. Calculations of head loss as a function of time were compared with experimental results (Fig. 2 ) . A value for 3 f of 0.15 was selected. The contact efficiency of a retained particle, ηρ, was estimated using the theory of Yao et al. (1971) for transport by diffusion, interception, and settling. The models for transport diffusion and interception were modified using Happel's model for flow through porous media. Because of the empirical and preliminary nature of this approach, corrections for hydrodynamic interactions were not made. MODEL VERIFICATION The model was tested or verified using data from other experiments by Habibian (1971) in which one or more filtration variables was significantly different from conditions in the runs used for calibration. Observed values of the particle removal by the clean filters were used to calculate ηα for each run. Other coefficients were kept constant, save that 3 and ß f were set equal to one for large suspended particles where interception and settling were dominant transport mechanisms. This assumption was based on steric arguments. Data from six filtration experiments using three suspended particle sizes, five influent concentrations, and two bed depths are presented in Figs. 3 to 6. Model predictions are also depicted. Special note is given to the results in Fig. 6. Particles of 1 ym size show the poorest removal; smaller particles (0.1 ym) are transported effectively by diffusion and larger ones (7.6 ym) are transported by interception and settling. Head loss development, however, increases dramatically with decreasing particle size. Influent concentrations were all about 50 mg/1, but head loss development was small for 7.6 ym particles and very rapid for 0.1 ym particles. This set of data provides a useful test for the model. Comparisons between calculations and observations are good to excellent for five of the runs. In one experiment (Fig. 3 ) , the head loss calculations do not fit the data. Reasons for this are not known. Based on these agreements, it was decided that the model provides a useful simplified theory for the development of removal efficiency and head loss during filtration. It is able to simulate the effects of suspended particle size on removal efficiency, and also the surprising effects of suspended particle size on head loss development. Small particles are considered to produce large chains or dendrites that increase the surface in the filter bed while having little effect on porosity. SIMULATIONS Simulations of the effects of suspended particle size and concentration, media size, and bed depth on particle removal and head loss have been made with the calibrated and verified model. Results are presented in Figs. 7 to 10.
Retained particles in deep bed filtration
173
Particle size Predictions of particle removal and head loss as functions of time for three different particle sizes are shown in Fig. 7. One-micron particles show the poorest removal at the onset of filtration, as observed by Yao et al. (1971), and also are the most difficult to remove during the run. Small (0.1 ym) particles produce considerable head loss as they are removed, while larger ones (1 and 7.6 ym) show negligible head loss development for the influent concentration (10 mg/1) and filtration time (3 hrs) examined. The importance of suspended particle size in determining removal efficiency has been predicted and observed previously. Its significance in head loss development has been observed experimentally (Habibian and 0 T Melia, 1975); the results presented here provide a plausible model for this effect. Influent concentration For a clean filter, the rate of removal of particles with respect to depth is proportional to the concentration of particles in suspension, i.e., 3C/9L = -AC. Hence, the removal efficiency of clean filters is independent of influent concentration (Ives, 1967). This is not the case as filtration proceeds. Calculated particle removals and head losses are plotted as functions of filtration time in Fig. 8 for four influent concentrations ranging from 1 to 50 mg/1. As expected, low concentrations produce low head losses. They can also produce low removal efficiencies. This is because removal by a packed bed filter can depend on the number of retained particles which act as collectors. When the influent concentration is low, the rate at which new collectors accumulate in the bed is also low. The ripening process is lengthened, but removal efficiency is impaired. This indicates that filters treating low turbidity waters should be deep, while those treating waters with high suspended solids concentrations can be more shallow. Filters treating low turbidity waters must rely on the filter media to provide collectors; those treating more concentrated suspensions remove solids primarily by contacts with previously retained particles. Media size Calculations illustrating the effects of media size on filtration are presented in Fig. 9. As expected, increasing media size produces lower removals and head losses. The reduction in head loss is more marked than the reduction in removal efficiency. These effects, however, are less significant than those noted for suspended particle size and concentration. The role of media size in filtration may be overstated both in the literature and in practice. Media size can be easily measured and readily controlled, but it is not of major importance. Bed depth The effects of bed depth and filtration time are illustrated by the results presented in Fig. 10, in which particle removal and head loss are plotted as functions of bed depth for five filtration times. The calculations are made for a homogeneous bed with a media size of 0.38 mm. Removal efficiency
C.R. 0TMelia and W. Ali
174
and head loss both increase significantly with time, as expected, but their distribution with time is noteworthy. Removal is distributed throughout the bed at the onset of filtration, but becomes localized in the upper region of the bed as filtration proceeds. Head loss follows a similar distribution. Retained particles accumulate in the upper regions of the bed and lead to the retention of more particles in that area. Filtration is seen to have an autocatalytic character. The use of large filter media in an upper layer of a mixed media filter to permit bed penetration is seen to have limitations, especially for concentrated suspension. This is because removal is accomplished primarily by retained suspended particles that can be quite small. DISCUSSION Observation and theory indicate that the size and concentration of the suspended particulates are the most important physical variables influencing packed bed filtration. Particulate sizes in water and wastewater are not often measured because available techniques are time consuming, expensive, and achieve only limited success. While particle sizes cannot be measured easily, they can be altered by flocculation prior to filtration. When a water or wastewater contains substantial quantities of submicron particles (e.g., a high rate trickling filter effluent), it is plausible that flocculation prior to filtration will be beneficial. Filters can be shallow and contain large media, since retained particles will produce the bulk of the removal. When particle sizes are small and the solids concentration is low (e.g., asbestos particles in Lake Superior), either additional particles must be added for flocculation (sweep floes) or deep filters containing smaller media can be used. The effects of the chemistry of the filtration process, as represented by a and dp, have not been presented here. Preliminary analyses have been made by Ali (1977). In the absence of chemical destabilization (a and otp = 0 ) , removal is zero throughout a run and head loss remains unchanged. Attachment requires chemical destabilization and is essential for the filtration process to perform its function of solid-liquid separation. The model presented here is oversimplified and requires refinement for quantitative application. In addition, it does not treat that portion of the filtration cycle when filter capacity is depleted and breakthrough occurs. Possible reintrainment of retained particles into the bulk flow has also been neglected. CONCLUSIONS The following conclusions are made: 1.
The removal efficiency of a clean filter bed depends on the size of the particles being filtered. A critical particle size exists, in the
Retained particles in deep bed filtration
175
region of 1 ym. This particle size has the lowest opportunity for contact with the filter media, and subsequent removal from suspension. Smaller particles are effectively transported by Brownian diffusion, larger ones by interception and settling. 2.
This effect of particle size on removal continues into the ripening period; i.e., one-micron particles show the poorest removal throughout the period of effective filtration.
3.
For a given mass of particles removed, head loss increase varies in an inverse manner with suspended particle size. Submicron particles can produce enormous head losses when treated by conventional packed-bed filters.
4.
The removal efficiency of a clean filter is independent of influent concentration.
5.
After the onset of filtration, removal efficiency improves with increasing influent concentration during the ripening period, because retained particles act as collectors for other suspended particles.
6.
Flocculation can be used to adjust the size of the particles in the filter influent. Increasing the suspended particle size above the micron range will improve removal efficiency and reduce head loss.
7.
It is plausible that the advantages of multimedia beds have been overstated. In most filtration processes, the particles in the filter influent provide most of the removal after the run has begun. Hence, head loss and removal tend to be localized in the upper regions of the bed during downflow filtration, regardless of the size of the media.
8.
Effective filtration requires that particle-to-particle and particleto-filter grain contacts be successful in producing attachment. This, in turn, requires chemical control of the process during operation.
9.
The model presented here considers that particles retained within a packed-bed filter during the initial stages of a run can act as collectors (filter media) for other suspended particles applied to the bed at a later time. This gives rise to a ripening of the filter, and permits filters to be for more efficient in particle removal than theory and observations on clean filters would suggest. The form of these deposits is likely to be chains or dendrites which extend into the pores, provide considerable surface for fluid drag, and can increase head loss significantly. This model agrees well with results of laboratory experiments.
176
C.R. O'Melia and W. Ali REFERENCES
Ali, W., 1977, "The Role of Retained Particles in a Filter," unpublished master's report, Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill, N.C·, U.S.A. Billings, C. E., 1966, "Effect of Particle Accumulation in Aerosol Filtration, Ph.D. dissertation, California Institute of Technology, Pasadena, CA., U.S.A. Fair, G. M., and Hatch, L. P., 1933, "Fundamental Factors Governing the Streamline Flow of Water Through Sand," Journal, American Water Works Association, Vol. 25, 1551-1565. FitzPatrick, J., and Spielman, L. A., 1973, "Filtration of Aqueous Latex Suspensions Through Beds of Glass Spheres," Journal of Colloid and Interface Science, Vol 43, 350-369. Friedlander, S. A., 1958, "Theory of Aerosol Filtration," Industrial and Engineering Chemistry, Vol. 50, 1161-1164. Habibian, M. T., 1971, "The Role of Polyelectrolytes in Water Filtration," Ph.D. dissertation, University of North Carolina, Chapel Hill, N . C , U.S.A. Habibian, M. T., and O'Melia, C. R., 1975, "Particles, Polymers, and Water Filtration," Proceedings of the American Society of Civil Engineers, Journal of the Sanitary Engineering Division, Vol. 101, No. EE4, 567-583. Ives, K. J., 1967, "Basic Concepts of Filtration," Proceedings from the Society for Water Treatment and Examination, Vol. 16, 147-169. Kozeny, J., 1927, Sitzungberichte Akademie der Wissenschaften Wien, Mathematisch-Naturwissenschaftlich Klassen Abt H a , Vol. 136, 271-306. O'Melia, C. R., and Stumm, W., 1967, "Theory of Water Filtration," Journal American Water Works Association, Vol. 59, 1393-1412. Payatakes, A. C., and Tien, C., 1976, "Particle Deposition in Fibrous Media with Dendrite-Like Pattern: A Preliminary Model," Journal of Aerosol Science, Vol. 7, 85-100. Spielman, L. A., and FitzPatrick, J. A., 1973, "Theory for Particle Collection Under London and Gravity Forces for Application to Water Filtration," Journal of Colloid and Interface Science, Vol. 42, 607-623.
Retained particles in deep bed filtration
177
Tien, C., Wang, C. S. , and B a r o t , D. T . , 1977, " C h a i n l i k e Formation of P a r t i c l e Deposits i n F l u i d - P a r t i c l e Separation, 1 1 Science, Vol. 196, 983985. Yao, K. M. , Habibian, M. T . , and O'Melia, C. R. , 1971, "Water and Wastewater F i l t r a t i o n : Concepts and A p p l i c a t i o n s , " Environmental Science and Technology, Vol. 5, 1105-1112.
C.R. 0TMelia and W. Ali
178
lo
Formation of particle chains or dendrites during filtration (after Tien et al., 1977).
-i
2
60
(/) (/) O
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_ι
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i
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80
- Fitted Model
-
'
-
Q
<
-
w 20 i
n
50
l
100
1
1
150 200 TIME (min.)
1
250
300
2. Model Calibration. Model fitted to experimental results. L = 14 cm, f = 0o36, dc = 0.38 mm, v 0 = 2 gpm/ft2, dp = 0.1 ym, C0 = 11.0 mg.l.
Retained particles in deep bed filtration
60
Experimental Results Model Predictions
z~
50
3.
100 150 200 TIME (min.)
250
Model Verification. Comparison of experimental results with model predictions. L = 14 cm, f - 0.36, d c = 0.38 mm, v 0 = 2 gpm/ft 2 , d p 0.1 urn, C 0 = 9.7 mg/1.
40 - Experimental Results - Model Predictions
30
0
4.
50
100
150 200 TIME (min.)
250
300
Model Verification. Comparison of experimental results and model predictions. L = 14 cm, f = 0.36, d c = 0.38 mm, v 0 = 2 gpm/ft 2 , dp = 1.0 um, C 0 = 37.2 mg/1.
179
C.R. 0 ! M e l i a and W. A l i
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Experimental Results Model Predictions
c
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r
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.--
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300
5. Model Verification. Comparison of experimental results and model predictions. L = 14 cm, f = 0.36, dc = 0.38 mm, v 0 = 2 gpm/ft2, dp = 7.6 ym, C 0 = 14.7 mg/1.
50
100 150 TIME (min.)
200
250
6. Model Verification. Comparison of experimental results and model predictions. L = 2 cm, f = 0.36, dc = 0.38 mm, v 0 = 2 gpm/ft2. For dp = 0.1 ym, C 0 = 50 mg/1; for dp = 1.0 ym, C 0 = 48 mg/1; for dp = 7.6 ym, C0 = 52 mg/1.
181
Retained particles in deep bed filtration
o 80
30
60 90 120 TIME (min.)
150
180
7. Effects of suspended particle size on filtration. C0 = 10 mg/1, L = 14 cm, f = 0 . 3 6 , d c = 0.38 mm, a = 0 . 5 , 0.06,
O 40
60 90 120 TIME (min.)
8.
150
180
Effects of influent concentration on filter performance. f = 0.36, dc = 0.38 mm, dp = 0.1 ym, a = 0.5, a p = 0.06.
L = 14 cm,
0 T M e l i a and W.
C.R.
182
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Effects of media size on filter performance. dp = 0.1 ym, a = 0.5, a p = 0.06, f = 0.36.
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C a l c u l a t e d d i s t r i b u t i o n s of p a r t i c l e removal. C0 = 10 mg/1, d c = 0.38 mm, dp = 0 . 1 ym, f = 0 . 3 6 , a = 0 . 5 , a p = 0.06.