PROCEEDINGS OF THE FILTRATION SOCIETY

Rapid Gravity Filtration - Towards a DeeperUnderstanding

By R I MackieDept of Civil Engineering, The University, Dundee DD14HN

The process of deep-bed filtration has been studiedusing a Coulter Counter to analyse the size distributionof the suspension at various depths in the filter and atvarious times. The results show that an understandingof deep-bed filtration based merely on the mean size ofa suspension is inadequate, and that the size distribu­tion must be taken into account. A mathematical modelto predict the effect of size distribution on removalefficiency has been developed, and a qualitativedescription of the model is given in the paper. Acomparison between the experimental and predictedresults is presented. The model predicts that in certaincircumstances the efficiency of filtration may be im­proved by 'dosing' the feed with larger particles.

where c is the concentration at depth I and time t, I. is the filtercoefficient and u the superficial velocity of the fluid, and a is thespecific deposit (absolute volume of deposit per unit filter volume).Sometimes the bulk specific deposit, a', is used in place of a, wherea' takes account of the fact that the deposited matter has its ownporosity, Ed' a is related to a' by:

Basic EquationsThe filtration process is described by two linked partial differentialequations. The first is the kinetic equation which describes thechange in concentration throughout the depth of the bed, and thesecond is the mass balance, or continuity equation. These equationswere first presented by Iwasaki'?' as:

RAPID GRAVITY, or deep-bed filtration is a well established processused in water purification and, over the last fifty years or so,considerable effort has been put into understanding the process. Oneof the main aims of this work has been to develop models which canaccurately predict the removal efficiency of a filter and the headlossacross the bed, both of which change as the bed becomes progress­ively clogged up. For a long time the work adopted a very empiricalapproach(1), and this route is still pursued by some, but in recentyears a much more analytical approach has been followed!". Thebasic method used in this approach is to calculate the trajectory ofparticles' flowing through the filter, and this has had considerablesuccess in predicting the removal efficiency of clean filters(3. ~).However, until recently little progress had been made in predictingthe change in the removal efficiency of a filter!". This paper discussesthe result of work, both experimental and theoretical, on this veryproblem carried out at Dundee. The description of the theoreticalwork will be in qualitative terms

jthe mathematical details having

been fully described elsewhere'?'. This paper also examines thepractical Implications of the work. However, before embarking onthe main body of the paper, the results of an examination of the basicequations governing removal in deep-bed filtration will be given.

for a layer of fluid to flow through the bed. It has been argued that if tis replaced by filter time, T, using the following equation:

T = t - f~ ~ dz (7)

then the simple form of the equations can be used. But this has beenshown not to be so(l\). Eq (7) would be valid if E varied only withdepth, whereas it is a function of both depth and time, so Eq (7) isinvalid. In addition Eq (7) adds further complications to theequations.

A far better solution is that used by Ivcs'!' who simply arguesintuitively that the additional terms which appear in the exactequations arc negligible, and so it does no harm to use the simpleequations. Mathematical support for this conclusion has beenprovided and a fuller discussion of the basic equations can beIound'!". Therefore the author concludes that it is perfectlylegitimate to use the approximate equations.

Experimental WorkA series of experiments was conducted on a 138mm diameter verticalfilter. Ballotini glass beads were used as the filter medium, and thesuspension was made up of PVC microspheres (Corvic P721757supplied by ICI). Experiments were conducted at different flow rates(the superficial velocity ranging from 2.4 to 6.0mh- I ) , and usingballotini of different sizes (ranging from a mean size of 0.458 to1.095mm). Table 1 shows a schedule of the experiments; experiments7 and 8 were carried out under essentially identical conditions inorder to test the repeatability of the results. Samples of thesuspension were simultaneously taken from various depths in thefilter bed, and a Coulter Counter was used to analyse them. Thisenabled the size analysis of the suspensions to be determined, and itis believed that this is the first time that a detailed size analysisthroughout the history of a filter run has been carried QUI.

For purposes of analysis the suspension was split up into four sizegroups, group 1 containing the smallest particles. Table 2 shows thesize range for each group. A typical size distribution of the feed to thefilter was 13:29:37:21. The results were expressed in the form C/Cn ;,

where C, is the concentration of particles in size group i, and Cn i is theinlet concentration of particles in group i. Fig 1 shows the change inC/Cn i with time at a depth of 37mm for filter run 3, and Fig 2 showsthe change in total concentration at the same depth. It is immediatelyclear that the qualitative behaviour differs with particle size. From(3)

(2)

(I)acaT - - I.C

aa + u Oc = 0at al

and Eq (2) becomes:

(5)

(..J)

Eqs (I) and (4) have the advantage of being simple. However" theyare in fact only approximations to the exact equations. Horner'"? hasshown that the exact mass balance equation (neglecting diffusion) is:

aa' cc ee(I - Ed - c)ai + Eat + uaT = 0

where E is the porosity of the bed, and Mackic'Y'.has shown that theexact kinetic equation is:

Table 1: Experimental Programme

Grain Flow AverageSize Velocity Conc

ExptNo (BS Sieve) (m/hour) -c Porosity (mgfl)

1 22-25 2.4 18 0379 2162 22-25 36 18 0376 1973 22-25 48 19 0376 1704 22-25 60 19 0376 1605 22-25 7.2 19 0376 1006 30-36 48 23 0370 1007 18-22 48 23 0372 1978 18-22 4.8 23 '0373 2129 14-16 48 22 0369 264

cc + ~ oc = _ I.C01 u st

(6) Table 2: Particle Size Groups

Obviously it would be desirable to use the simplified Eqs (I) and(4) in preference to (5) and (6) if it is legitimate to do so. One devicefor doing this, adopted by some(lO.2), has been to introduce theconcept of filter time, ic to take account of the fact that it takes time

Size Group

1234

Size Range (,1m)

063- 1,26126- 252252- 504504-1008

32 January/February 1989 Filtration & Separation

PROCEEDINGS OF THE FILTRATION SOCIETY

Fig 2 one would say that at a depth of 371010 the removal efficiencyincreases for the first 2-3 hours and then goes into steady decline.However, Fig I shows that what is actually happening is that theremoval efficiency for groups I and 2 is increasing all the time, whilethe removal efficiency for particles in group 4 begins to decrease afterI hour, with the behaviour of particles in group 3 being somewherebetween the two extremes. This sort of behaviour was observed in allthe experiments performed. Fig 3 shows a comparison of the resultsfor runs 7 and 8, illustrating the repeatability of the results. It isapparent that observing the. change in total concentration alone doesnot give a complete picture of what is happening. Fuller details of theexperimental procedure and the results can be found(6).

Mathematical ModelA prototype mathematical model has been developed to predict andexplain the experimental results. The model follows the trajectoryanalysis approach, extending the method to predict the completehistory of a filter run. As mentioned in the introduction, only aqualitative description will be given here. The model has three mainelements:o Deposited particles act as collectors;o The effect of deposition is viewed both macroscopically and

microscopically;o The effect of shear forces on deposition is modelled.

Deposited Particles Acting as Collectors. It is well established thatdeposited particles act as collectors!':", and models to predict theeffect of deposited particles on removal efficiency have beendcvelopcd'P'. These models allow chains, or dendrites, of severalparticles to form, and also consider each particle individually, hencethey arc very complex. The current model adopts a simpler approach.First it is assumed. that "dendrites" of one particle only in length areformed. In air filtration, for which dendritic modelling was originallydeveloped!':". this would not be a valid assumption, but in waterfiltration the viscous forces are much greater and It is probably not anunreasonable simplification. Secondly, rather than considering indi­vidual particles, simple probability theory is used to calculate the"average' effect of deposited particles. This greatly simplifies thecalculations and still gives good results.

Macroscoplc/Mlcroscopic Effect of Deposition. Consider a grainwith deposit upon it. If the grain could be viewed from a distance itwould look like Fig -la, but if a close up view of a section of the grainwere taken it would look like Fig -lb, The dome shape of the depositwill affect the general flow field around the grain (the macroscopiceffect), and the particles which protrude above the general profile ofthe deposit will act as collectors in the same way as a depositedparticle on an otherwise clean grain will (this is the microscopiceffect). The Dundee model combines both these effects in calculatingthe change in removal efficiency.

Effect of Shear Forces. Many researchers have taken the view that theincrease in interstitial velocity in a filter is responsible for thedecrease in removal efficiency(15.17). However, the mathematicalmodels have used the mean interstitial velocity to. characterise the

6

6

x

234 5Time (hrs)

Size group 2

x

2 :3 4 5Time (hrs)

Size group 4

0.2

1.0

0.8

1.0

0.8

0.2

..og 0.6

cJ 0.4

og 0.6

U 0.4

2 3 4 5 6. Time (hrs)Size group 1

2 3 4 5 6Time (hrs)

Size group 3

Fig 1. Change in C;lC o; for run 3 at depth == 37mm

1.0

0.8

0.2

0.4

0.6og

U

1.0

0.8

1.0

0.8

0.2

og 0.6

o 0.4

sg 0.6cS 0.4

0.2

1 2 345Time (hrs)

6

DEPOSITFig 2. Change in total concentration for run 3, depth == 37mm

1.0• x

1.0•

0.8 • l~ 0.8 x

J ~ •x~ • GRAINs x

~ 0.6 0.6 ~

0 .s ~ ~ ,.x·0.4 x experiment No.7

0.4 x(a)

0.2 • experiment No.8 0.2

2 3 4 5 6 7 2 3 4 5 6 7Time (hrs) Time (hrs}

Size group 1 Size group 2

1.0 1.0 DEPOSITED

0.8 0.8 • PARTICLESc 0~ 0.6 • ~ 0.6 x

xU cJ x • •

0.4 x 0.4 • x• • GRAIN

0.2 I "" >l' ~ x x 0.2 x • x

•• x SURFACE1 2 3 4 5 6 7 2 3 4 5 6 7

Timelhrsl Time (hrslSize group 3 Size group 4 (b)

Fig3. Comparison of the results for runs 7 & 8, depth == 37mm Fig 4. Deposit on a grain - (a) macroscopic view; (b) microscopic view

Filtration & Separation January/February 1989 33

PROCEEDINGS OF THE FILTRATION SOCIETY

1.0 X 1.0 X 50 )( 0-37 mm layer 50

~40

• 37 - 87 mm layer400.8 0.8 - theoretical curve x x

0- X X ...~ 30 'E 300

.g 0.6 X x g 0.6

~;< ;<

u cS 20 200.4 0.4

)( experiment 10 100.2 -theory 0.2

234567ox 10"

Size group 4

)(

234567ox 10"

Size group 3

2 3 4 5 6Time (hrs)

Size group 4

2 3 4 5 6 7 2 3 4 5 6 7

2 3 4 5 6 ox 10" ox 10"

Time (hrs) Size group 1 Size group 2

Size group 2 100 100

90 90x

80 80

70 70

:;- 60 )( z: 60

§.50 §. 50.< .<

40 40

30 30

20 20

10 100.2

1.0

0.8..og 0.6..() 0.4

2 3 4 5 6Time (hrs)

Size group 3

123 4 5 6Time (hrs)

Size group 1

x

'\ -:~)

1.0

0.8

0.2

s.g 0.6

cS 0.4

Fig 5. Comparison of theoretical and experimental relative concentra­tion results (ClCo1) for run 3, depth =37mm

Fig 6. Comparison of theoretical and experimental /. - a curves for run3, depth = 37mm

effect. The Dundee model adopts a more sophisticated approach.The basic hypothesis used is:

For particles of size ap there exists a critical velocity v' suchthat particles of size ap cannot adhere to any point on a grainat which the tangential velocity at a distance ap above thesurface is greater than or equal to v'.

Now the tangential fluid velocity, Ve, will have different values atdifferent points on the grains. Also, Va will be zero at thegrain/deposit surface, increasing to a maximum value at the middle ofthe pore. Larger particles will protrude further out into the flowstream than smaller ones, so they will experience a larger shear force.Therefore particles at different parts of the grain and particles ofdifferent size will experience different shear forces. This means thatat a point in time a part of the grain may be able too Collect particles of any size;o Collect particles of some sizes, but not others;o Be unable to collect any particles.

The model takes full account of all these possibilities.

Comparison of Theory and ExperimentOne of the problems WIth the empirical models was that theycontained a number of empirical parameters, most of which had nodirect physical meaning. Also the parameter values varied signifi­cantly from situation to situation. The model developed at Dundeehas three sets of parameters: v' the critical velocity used in thehypothesis stated above; Ed the self porosity of the deposit; and X' aparameter to account for the effect of deposited particles on the localflow field. All these have a clear physical meaning, and in time itshould be possible to establish clear experimental or mathematicaltechniques to determine their values. For the present, values werechosen in order to give a reasonable fit to experiment No.3, and thenthe model was used with these same parameter values to predict theresults of the other experiments. Figs 5 and 6 show comparisons ofthe experimental and predicted C/Co i curves and A - 0 curves forrun 3. Fig 7 shows A - 0 curves for run 7. Com-Rarison betweentheory and experiment for all the runs can be found' .

The model accurately predicts all the qualitative effects. ie the

1 2 3 4 5 6TIME (hrs)

1 2. 3 4 5 6TIl'E (hrs )

SIZEGROUP2

Coincludes size group 41nbolhcases

TOTAL

CONCENTRATION

0.2

0.8

0.6

0.4

1.0

0.2

0.8

0.6

0.4.... ---.. - --...

1 2 3 4 5 6TIME (nrs)

123 1+ 5·61'IME (hrs)

SIZEGROUP3

SIZEGROUP1

Fig 8. Effect of size distribution on relative concentration, depth =37mm: -- suspension consists of size groups 1, 2, 3 & 4.

- - - - suspension consists of size groups 1, 2 & 3.

0.2

0.2.

0.8

0.6

0.4

1..0

0.6

0.4

0.8

2345678a x 10']

Size group 2100

90

80

70

::-60

!. 50 x.<

60

50

::-40

!.30 x )(.<

20 x

10

2345678ox 10"

Sizogroup 1

x

)( 0-33 mm layer• 33·83 mm layor

- theoretical curve

x•

10 10

2 3 4 5 6 7 8 2 3 4 5 6 7 8ox 10" ox 10"

Size group 3 Size group 4

100

90

80

70

z: 60

~50

Fig 7. Comparison of theoretical and experimental;' - a curves for run 7

60

50

::-40

!. 30-c

20

10

34 January/February 1989 Filtration & Separation

PROCEEDINGS OF THE FILTRATION SOCIETY

difference in behaviour for particles of different sizes, and the effectsof grain size and superficial velocity. The quantitat ive agreement iscertainly good enough to validate the principles behind the model.

Implications and ApplicationsThe way the mod el works is as follo ws: the initi al increase in removalefficiency is caused by deposited particles protruding into the flowfield and act ing as collectors; the eventual decrease is cau sed by theincrease in she ar forces acting on deposited part icles. Now largeparticles protrude further into the flow field and so experience largershear forces, therefore removal efficiency for larger particlesdecreases while that of smaller particles is still increasing. as observedexperimentally. The Ives' type models explained the initial increasein terms of the increase in specific surface area. It should be notedthat this concept plays absolutely no part in the current model, andthe current model gives a better and fuller description of the removalprocess. In fact it is the author's opinion that changes in specificsurface area have no direct effect on the removal efficiency. Ideallyan experimental te st of the sur face area hypothesis should be carriedout.

It has been seen that the changes in removal efficiency differ fromone particle size to another, and the model predicts this effect.However, the importance of size distribution runs deeper than this ,as can be seen by considering the follo wing situ ation.

Since deposited particles actin g as collectors is the primary cause ofthe increase in removal efficiency, it follows that the size distribu­tions, not just the particle sizes, will influence the removal efficiency,For suppose that all particles in size group 4 were removed from thesuspension prior to it being filtered . This would mean that particles ofsize group 4 could no longer act as collectors, and so it would beexpected to change the removal efficiency of particles in size I, 2 and3. Accordingly model simulations for the case where particles of sizegroup 4 were and were not pr esent were carried out. Th e results areshown in Fig 8. As can be seen, the effect of removing the particles ofsize group 4 is sign ificantly to reduce the removal efficiency for theother particles. So far no exp eriments to confirm the predicted effectof removing the particles of size group 4 have been performed, but itis planned to do so . If the mod el results are confirmed there would beimportant implications for filtrat ion pract ice. For it would mean thatin some circumstances it may be worthwhile 'dosing' a suspension offine particles with some larger ones in order to increase removalefficiency. This would be of particular benefit as fine particles areusually the most difficult to remove whatever the sep aration process

used . Conversely, in situ ations where particl e deposition 'h as adetrimental effect, the predictions of the model emphasise theimportance of removing the larger particles from the suspension, asleaving them in would also increase the deposition of smallerparticles, which would otherwise have only a limited effect.

ConclusionsA theoretical and experimental study of the effect of deposition onremoval efficiency in deep-bed filtration has illustrated the irnpor­tance of size distribution on filter performance. In particular, themodel has indicated that in certain circumstances the operation ofdeep-bed filters may be improved by dosing the suspension withlarger particles (or alternatively not removing all the larger particlesprior to filtration). The work also demonstrated that it is perfectlylegitimate to use the approximate filtration equations, without using'filter time ' .

REFERE.'iCESI. Ives, K J . 'Rapid Filtration'. Water Research 4 201(1970),2, Tien, C and Payatakcs, A C. 'Advances in Deep-Bed Filtration'. AIChEJ . 25. 737

(1979),3. Rajagopalan Rand Tien C. 'Tra jectory Analysis of Deep-Bed Filtration with the

Sphere-in-Cell Porous Media Model'. AIChEJ. 22. 523 (1976).4. Payatakes, A C. Ticn , C. Turian , R M and Rajagopalan. R. 'Trajectory Calculation of

Pan icle Deposition in Deep-Bed Filtrat ion. I: Model Formulation: II: Case Study of theEHeet of Dimensionless Groups and Comparison with Experimental Data'. AIChEJ. 20,889 (J97-lb).

5, Chiang, H W and Ticn, C. 'Tra nsient Behaviour of Deep-Bed Filters. Symposium onAdvances in Solids-Liquid Separation. Univ College. London (1983),

6. Mackie. R I. Horner, R M W and Jarvis. R J. 'Dyn amic Modelling of Deep-BedFiltration', AIChEJ.33. 1761(t 9871'

7. Iwasaki, T. 'Some Notes on Sand Fi tration·.JAWWA 29.1591. (1937).8. Horner. R M W. 'Water Clarification and Aquifer Recharge' . PhD thesis, Univ College.

London (1968)• .9_~lh~~:i~n~ ~j,~l~~\~~b~~;0'i~~~3Je?~fs'1.it ion on Removal Efficiency in Deep·Bed

10. Herzig. J P, LeCferc, D M and uGoff. P, 'Flowof Suspensions through Porous media ­j\pplications to Deep-Bed Filters', Ind Eng Chem 62. (5) 8 (1970).

II. Homer. R M W. Jarvis. RJ and Mackie. R I. 'Deep-Bed Filtration-A New Look at theBasic Equations'. Water Research. 20. 215. (t 986).

12. Payatakes, A C. Parks. H Y and Petrie. J. 'A Visual Study of Particle Deposition and :Re-entrainment dur ing Depth Filtration of Hydrcsorb with a Polyelectrolyte'. ChemEng Sei.36 . 1319, (1981),

13. Wang. C S. Beizaie, M and Tien, C. 'Deposition of Solid Pan icles on a Collector:Formulation of a New Theory' , AIChEJ , 23. 879. (19n) . . .

J-l, Payatakes, A C and Tien. C. ' Panicle Deposition on Fibrous Media with Dentrite-likePanern-A Preliminary Model'.J. Aerosol Sci. 7. 85. (1976),

IS. Stein, P C, 'A Study of the Theory of Rapid Filtration of Water Through Sand' . DSeDiss. Mass Inst Tech. Carnbridae USA (19.10).

16. Ives, KJ . 'Rational Design of f'iltecs·. Proc Inst Civ Eng. 16. 189(J960).17. Tien, C. Turian, R M and Pendse, H. 'Simulation of the Dynamic Behaviour of

Deep-Bed Filters'. AIChEJ. 25. 385(1979).

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Filtration & Separation January/February 1989 35