Particle penetration depth distribution in deep bed filtration

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1991 J. Phys. D: Appl. Phys. 24 2111

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J. Phys. D: AJI. Phys. 24 (1991) 2111 -2114. Printed in the UK

1 RAPID COMMUNICATION I I Particle penetration depth distribution

1 in deep bed filtration

Claude Ghidaglia, Elisabeth Guazzelli and Luc Oger Laboratoire de Physique et MBcanique des Milieux HBtBrogbnes, URA au CNRS 857, ESPCI, 10 rue Vauquelin. 75231 Paris Cedex 05, France

Received 6 August 1991

Abstract. Small particle penetration depth in a random fixed bed of larger glass spheres has been studied experimentally under laminar flow conditions and in the absence of Brownian motion. The small particles, marked with a thin coating of gold, were tracked in the fixed bed made optically transparent by matching the index of refraction of the suspending fluid to that of the glass spheres. Small particle penetration depths were examined across different ratios of the small particle diameter to the large sphere diameter.

Filtration is the process of separating particles in sus- pension from a carrier fluid by passing the fluid through a permeable material which may be larger particles, granular, porous or fibrous media. The separated solids may be collected as a cake on the surface of the fil- tration medium or retained within the pores of the medium [l, 21. We focus here on the latter case, which is known as deep-bed filtration. The objective of the present work is to characterize the penetration depth of small non-Brownian spherical particles, of diameter d , flowing into a porous medium made by a random packing of larger spherical particles, of diameter D.

In the case of a purely steric particle capture, the small particle penetration depth is essentially deter- mined by the number of constrictions in the packing that the particle is able to cross before being captured. The control parameter of the problem is, therefore, the ratio between the small particle diameter and the large sphere diameter, 8 = d / D . The capture threshold corresponds to the diameter ratio, OC = 2 / ~ - 1 (=0.155), at which the small moving particle fits exactly the hole between three tangent identical spheres [3]. In the case of an infinite medium, if B > 8,, the particle is captured by the medium. Conversely, if 0 < 0, the particle is not retained by the medium and the penetration depth is infinite.

However, particle collection is not a simple matter of geometrical interception. The capture of small sus- pended particles in laminar flow is also usually a conse- quence of hydrodynamic interactions or molecular, electrical or gravitational forces, acting alone or in combination [4]. But since the inclusion of all the forces mentioned above proves to be rather complicated, the

0022-3727/91/112111 + 04 503.50 0 1991 IOP Publishing Ltd

present study is restricted to hydrodynamic and gravi- tational forces without regard to physico-chemical for- ces that enter whenever small particles come in close enough proximity to large particles [5].

The conceptually simple experiment designed to accomplish the objective was one in which small marked particles were visually tracked in a fixed bed of large unmarked glass spheres, made optically trans- parent by matching the index of refraction of the sus- pending fluid to that of the glass spheres. The fluid selected was a mixture of 60% dibutyl-phtalate and 40% butyl-benzyl-phtalate (Santicizer 160 produced by Monsanto) with an index of refraction equal to 1.52, a density pr = 1.07 g and a kinematic viscosity U = 27 CS at 25 "C. The fixed bed consisted of a random packing of glass spheres with an index of refraction equal to 1.52, a density p. = 2.50 g cm-3 and a diameter D = 4 ? 0.1 mm. The packing volume fraction was of the order of 0.604.62. The randomness character of the packing was realized by partially filling the first layer of the bed with larger glass spheres of diameter 10 ? 0.1 mm which induced disorder throughout the bed [6]. The small particles were acrylic beads with a density pp = 1.19g C I I - ~ . These particles were marked by a uniformly smooth thin coating of gold using a metal deposition device. The thin coating has incon- sequential effects on particle density. Three sizes of marked beads were used, one batch with a diameter range of 275 ? 15 pm, one of 430 2 10 pm and one of 655 ? 25 ptn.

Experiments were performed in a cell of rectangu- lar cross section with an inside width of 100mm, an inside depth of 39 mm and a height of 550 mm. The

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front and back sides of the cell were made of glass to allow visualization. Fluid was circulated upward through the cell by a pump allowing an easy control of the flow velocity, U. The pump and the cell were part of a closed flow loop. The fixed bed was maintained by two grids inside the cell and had a height of 500 mm. A gap with a height of 40 mm was constructed below the fixed bed. The marked particles were injected with- out fluid circulation through a syringe into the median part of the gap.

Two types of particle injection were used. In the first injection type, only a single particle was injected at a time and then, once the particle was in the cell, fluid was circulated. Particles remained in the bed after capture. We measured the vertical and horizontal coor- dinates of the particle centre when it first contacted the packing, respectively Zi and Xi, and when it found a final stable position, respectively Z, and X,. We should mention that there is an ambiguity in the definition of the initial value Zi, since the first layer of the packing is partially filled with large spheres of diameter 1 0 t 0 . l m m . In the present experiment, Zi cor- responds to the vertical coordinate of the grid which held the fixed bed. We are able, therefore, to calculate the particle penetration depth AZ = Z, - Zi and the particle lateral displacement AX = X, - Xi. In the second injection type, a blob of approximatively 60 particles was injected. We measured the final coor-

penetration depth AZ. For all experiments, the particle Reynolds number, Re = Ud/v, was kept constant at 0.045. The Brownian Pdclet number, 3np,vdz U/2kT, was always very large. Therefore, the Brownian effects were negligible.

For the two smaller particle-diameter batches, corresponding to f3 = 0.069 t 0.005 < 8, and f3 = 0.107 ? 0.006 < B,, particles were not retained by the fixed bed for either injection type. Conversely, for the larger particle diameter batch, corresponding to f3 =

J . . ~ . . ~ . - z ~ . . , . ...A:.,~ .~......-~.AJ.> ..-- 2.L .--- _ : _ I _ r - - - - - - - r -.- . ~ - - - ,>IIIL'LC\ "I q.'I':.LI ,,ill llllr. ,;;cL??Lc ?.??,1 ,.!C,.!L,LC..! ?I?< a,", ,,,.,r

0.164 * 0.010 C Oc, particles were captured by the fixed bed. This behaviour confirms that steric effects and hydrodynamic forces were dominant.

For the larger particle diameter batch, a histogram of the penetration depth for the first injection type for 140 particles is shown in figure 1. This figure represents the frequency distribution of penetration depth against the penetration depth interval. The histogram cor- responds to two sets of experiments; one for 100 par- ticles and another for 40 particles with a second random packing. The results of the second set of experiments replicated those of the first set. The histogram has roughly a gaussian shape with a long tail. The mean penetration depth is (AZ) = 26.0 mm ((AZ)/D = 6.5) with a standard deviation of 17.1mm. We should remark here that the mean penetration depth is larger than the size of the spheres of 10 mm diameter partially filling the first layer. Therefore, the penetration is not only due to the influence of this first layer. Particle trajectories were very vertical, which again confirms that steric effects and hydrodynamic forces were dom- inant. Indeed, the mean particle lateral displacement was (AX) = 2.0" ((AX)/D = 0.5) with a standard deviation of 1.7 mm.

For the larger particle diameter batch, the histo- gram of the penetration depth for the second injection type for 240 particles (4 blobs of 60 particles) is shown in figure 2. The histogram no longer has a gaussian >??",,r. ,1111 \""W> d I .L'?I?I I I I !C"!~ "CLIC*.\C. 111 ?ill\ CC\T,

the particles penetrated deeper inside the bed. The mean penetration depth is (AZ) = 45.5 mm ((AZ)/D = 11.4) with a standard deviation of 30.8mm. The com- parison between the cumulative normalized histograms for the two injection types, displayed in figure 3, clearly shows the difference between the histograms of the two injection types.

The key observation of the present experimental study is the existence of a critical ratio of the small particle diameter to the large sphere diameter below

.L___ L...L.~.~. . ,~ . I ~ . . ~~.~ r - . - . - - - -

0 t 1 3 I 5 6 7 8 9 1 0 1 1 1 1 1 3 1 4

Penetration depth intervals (cm)

Figure 1. Histogram of particle penetration depth for 9 = 0.164 ? 0.010 > Oc (the horizontal axis graduations correspond to the lowest values of the intervals) when particles were injected one by one and for 140 Darticles.

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F r 0.7

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Penetration depth intervals (cm) Flgure 2. Histogram of particle penetration depth for 8 = 0.164 t- 0.006 > Oc (the horizontal axis graduations correspond to the lowest values of the intervals) when a blob of approximately 60 particles were injected at a time for 240 particles (4 blobs of 60 panicles).

which the small particles were not retained by the bed. Above this critical ratio, the particles were captured by the packed bed. This finding shows that, in the present experiment, steric capture was dominant since the physico-chemical forces were negligible.

When particles were injected one by one, the pen- etration depth depended on the geometrical structure of the packing. The moving particles could be captured only when pore size was smaller than the size of the

particles. The penetration depth distribution was linked to the probability P(d) of a moving particle finding a pore size smaller than its diameter. The long tail of the histogram, observed in the present exper- iment above the critical diameter ratio, was also observed in recent numerical simulations [7, 81. In these 3D numerical computations [SI, the large spheres were of the same size (within a relative error of and the small particles could only be captured due to

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Rapid communication

steric effects. It should be also noted that, in these numerical computations, the small particles did not remain in the packing after capture. These numerical simulations showed that the penetration depth histo- gram decreased exponentially with a rate of decay pro- portional to the probability P ( d ) . In the present experiment, size distributions of both moving particles and bed spheres weighted the probability P(d) but did not drastically change the histogram tail decay. When blobs of 60 particles were injected, the penetration depth was modified above the critical diameter ratio and particles penetrated deeper inside the packed bed. This change in behaviour might be due to collective effects which we are presently investigating.

Finally, we should mention that although the par- ticles remained in the packing after capture, their pres- ence did not significantly affect the penetration depth distribution. Indeed, in the first layer of the packing, approximately 500 bed pores were available and only 140 or 240 moving particles were injected.

This work is based upon the DEA research of C Ghidaglia, sponsored by the Centre de Recherche de Voreppe (Pechiney). We wish to acknowledge the stimulus for this work provided through discussions

with J P Desmoulins. Partial support also came from an ATP CNRS ‘Materiaux Heterogknes’, a DRET con- tract no 89/1399 and an ‘Action Specifique’ CNRS. We also thank Monsanto for donating a generous quantity of Santicizer 160 and Du Pont de Nemours and Orkem Norsolor for providing the acrylic beads. We thank F Lacour for help in coating the acrylic beads. We also thank C Baradel, J C Guibert, R Porchet, M Mayeux and D Vallet for technical assistance in constructing the experimental set-up. We also benefited from helpful discussions with C Allain, L de Arcangelis, D Bideau, R Jullien and P Tabeling. ,

References

[I] Dodds J A, Daluais G and Leclerc D 1988 Disarder and Miring ed E Guyon et a1 (Dordrecht: Kluwer) p 163

[2] Houi D 1990 Hydrodynamics of Dispersed Media ed J P H u h et a/ (Amsterdam: Elsevier) p 155

[3] Dodds J A 1980 J. Coll. Inf. Sci. 77 317 [4] Spielman L A 1977 Ann. Rev. Nuid Mech. 9 297 [5 ] Israelachvili J N 1985 Infermolecular and Surface Forces

161 Bideau D. Troadec J P and Oeer L 1983 CR Acad. Sci. (London: Academic)

. I Y

Paris 297 219; 319 171 Imdakm A 0 and Sahimi M 1987 Phvs. Rev. A 36 5304 [8j Meakin P and Jullien R 1990 J. PhyGque 51 2673

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