Modelling of sub-micron-sized particle deposition in a laminar wall jet—an investigation of the role of shielding

Colloids and Surfaces

A: Physicochemical and Engineering Aspects 205 (2002) 149–159

Modelling of sub-micron-sized particle deposition in alaminar wall jet—an investigation of the role of shielding

Anders Goransson, Christian Tragardh *Food Engineering, Center for Chemistry and Chemical Engineering, Lund Uni�ersity, P.O. Box 124, SE-22100, Lund, Sweden

Received 22 May 2001; accepted 12 November 2001

Abstract

The deposition of polystyrene latex particles (0.23 �m in radius) onto a glass surface, covered with indium–tinoxide, has been studied using a wall-jet cell. The particle deposition could be followed in a single experiment at severalpositions, exposed to a wide range of wall shear rates. A deposition model including a shielding function was appliedin order to describe the process. Three shielding functions obtained from the literature, which assumed differentshielded areas, were compared. The size of the excluded area was shown to be dependent on the wall shear rate: ahigher shear rate resulted in a larger area being blocked by deposited particles, thus preventing non-deposited,free-flowing particles from reaching the surface. © 2002 Elsevier Science B.V. All rights reserved.

Keywords: Particle deposition; Wall-jet cell; Evanescent wave; Surface blocking; Shielding function

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1. Introduction

The study of particle deposition covers a widefield of applications, such as filtration, paper mak-ing, clean-room applications, fouling in plate-heatexchangers, etc. Marshall and Kitchener [1] used arotating disk system to study the deposition ofcarbon black particles, Bowen and Epstein [2] aparallel-plate channel to follow the initial deposi-tion rate of sub-micron silica particles, Dabrosand van de Ven [3] adopted an impinging-jet cellto follow the deposition in situ, and Albery et al.

[4] used a wall-jet cell in combination with a glasssurface coated with a film of conductive indium–tin oxide to obtain a variable surface potential, inorder to study the adhesion kinetics of carbonblack particles.

The deposition process can be separated inthree processes, the transport of particles to thesurface from the bulk solution, adhesion of theparticles to the surface and re-entrainment if thehydrodynamic removal forces acting on the de-posited particles exceed the attractive forces inmagnitude. In the case of negligible surface cover-age, the particle deposition rate is not dependenton the time. Depositing particles do not have anyproblems finding a free surface area, and, accord-ing to Bowen and Epstein [2] and Albery et al. [4],the transport of particles to the surface can be

* Corresponding author. Tel.: +46-46-222-9807; fax: +46-46-222-4622.

E-mail address: [email protected] (C.Tragardh).

0927-7757/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.

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A. Goransson, C. Tragardh / Colloids and Surfaces A: Physicochem. Eng. Aspects 205 (2002) 149–159150

described by the convective-diffusive equation. Asthe degree of surface coverage becomes higher,the particle deposition rate decreases, due to, forexamples, a change in the surface charge resultingfrom the charge on the deposited particles, re-en-trainment of deposited particles and/or blockingof the surface by already deposited particles.

The blocking phenomenon, or surface shield-ing, was mentioned by Dabros and van de Ven [5]as a reason for the development of a flux decreas-ing with time as the degree of surface coverageincreased. They explained this as being the resultof electrostatic interactions between particles andhydrodynamic effects. A deposited particle forcesa flowing, non-deposited particle further out fromthe wall and it is thus affected by a stronger fluidvelocity. In the 1970s, Bell and Brown [6] investi-gated the shielding phenomenon in the case of gasadsorption. Rajagopalan and Chu [7] applied themodel to colloidal deposition, and Meinders et al.[8] described particle deposition in a parallel-platechannel assuming that the shielding conceptplayed a vital role in the process.

In the present study, particle deposition wasinvestigated both experimentally and theoreti-cally. The experiments were carried out in a wall-jet cell. The construction of the cell made itpossible to follow the deposition process in situfor a wide range of wall shear rates in one singleexperiment. The results of the experiments werecompared with theoretical models describing theparticle deposition. Shielding functions were usedto describe particle deposition for different wallshear rates in the range of 32–236 s−1. The mainpurpose was to quantify the excluded area as afunction of wall shear rate in a laminar wall jet.

2. Theory

2.1. The deposition process

For low degrees of surface coverage the initialmass-transfer-controlled particle flux, j0, is pro-portional to the bulk concentration of particles,cb, expressed as:

j0=kmasscb, (1)

in which, for the case of a wall jet, the masstransfer coefficient, kmass, is as described by Al-bery [9]:

kmass=0.165D2/3V f3/4�−5/12a−1/2r−5/4, (2)

where D is the bulk diffusion coefficient of theparticles calculated according to the Stokes–Ein-stein relation, Vf is the volumetric flow rate, � isthe kinematic viscosity, a is the diameter of thejet, and r is the radial distance from the stagna-tion point. According to Eq. (2) the initially con-stant flux decreases with radial distance with anr−5/4 dependency, which has been shown experi-mentally by Albery et al. [4] and in one of ourprevious experiments [10].

After an initial constant flux period, the deposi-tion rate becomes time dependent when the parti-cle surface coverage exceeds a critical level. Thedescription of the initial flux by Eq. (1) alone isno longer valid. Assuming that the particle flux inthe vicinity of the surface per unit time is un-changed, there must be other factors involved,which cause the decrease in deposition rate. Asthe surface becomes more occupied, free-flowingparticles are unable to find an unoccupied en-ergetically stable site and Eq. (1) alone cannotdescribe the deposition kinetics. The depositionprocess is described in Eq. (3) by a general Lang-muir model equation employed by Schaaf andTalbot [11], in which the particle surface concen-tration (particles per unit area), cw, is governed bytwo processes, particle transport to the wall andre-entrainment of deposited particles:

dcw

dt=kmasscb�(�)−krcw, (3)

with kmass and kr, describing the deposition andre-entrainment of particles from the surface. Thedeposition rate coefficient, kmass, is given by Eq.(2) for the wall-jet cell [9].

In this equation, shielding effects are taken intoconsideration by the shielding function, �(�), ex-pressed as a function of degree of coveragedefined as cw�rp

2. As the shielding function isintroduced, the effective mass transfer coefficientdecreases as the degree of coverage increases. InSection 2.2, a description of the shielding functionin Eq. (3) is given, together with the characteriza-

A. Goransson, C. Tragardh / Colloids and Surfaces A: Physicochem. Eng. Aspects 205 (2002) 149–159 151

tion of three shielding functions obtained fromthe literature.

2.2. The shielding functions

The shielding, or blocking phenomenon, hasbeen considered by several authors, [5–8,11,12]who found that a deposited particle occupies anarea which is larger than its cross-sectional area.According to Dabros [13], shielding arises fromthe fact that a deposited particle repels a flowingnon-deposited particle of the same electrostaticcharge. The flowing particle is forced further awayfrom the wall into an area of higher tangentialvelocity. It will thus be transported further down-stream before it is deposited on the wall (Fig. 1).

Eq. (3) shows how the flux at the surfacechanges with time and degree of coverage. Theshielding function, �(�), has the value unity for aclean surface and decreases as the particle concen-tration at the wall increases. The shielding func-tion describes the probability of a free-flowingparticle finding a free site on the surface. Theprobability of finding a free, energetically stablesite for a free-flowing particle in the case of asurface covered with particles, should be smallerthan in the case of an almost clean surface.

Rajagopalan and Chu [7] used a surface-shield-ing function to describe colloidal particle deposi-tion in packed beds. When they comparedexperimental data with different deposition mod-els, they found that the decrease in deposition ratecould be explained to a considerable extent bysurface exclusion effects. The adopted shieldingfunction had originally been derived by Bell andBrown [6] and later adapted by Thakur et al. [14]

to describe gas adsorption. The function forshielding, which was based on geometrical consid-erations, was expressed as:

�(�)= (1−�)e−5�/1−�. (4)

A single deposited particle shields a circular areaof six times its cross-sectional area, i.e. a radius of2.45 particle radii.

Schaaf and Talbot [11] presented a shieldingfunction in a polynomial form for the case whenparticles are allowed to diffuse on the surface:

�=1−4�+6�3

��2+2.4243�3. (5)

Eq. (5) is an approximation of a serial expansionto the third order in � and is accurate to degreesof surface coverage up to about 30%. Theshielded area for a single particle in this case isfour times the cross-sectional area of the particle,in comparison with the function derived by Belland Brown [6], which predicted a shielding area ofsix times the cross-sectional area. The shieldedarea has a circular form with a radius of twoparticle radii.

Dabros and van de Ven [5] introduced a parti-cle deposition model in which a blocking phe-nomenon was considered. It was assumed in theirshielding function that the excluded area could beexpressed as ��rp

2, where �, is a blocking factorgoverned by hydrodynamic interactions and elec-trostatic repulsion. For the case of close packing,� is about 1.1 [5] and according to Dabros andvan de Ven [3], �, can be as high as 20–30 forhigh wall shear rates. By applying their formula-tion, a shielding function can be written of theform:

�=1−��. (6)

The shielding function shows a direct inverse pro-portionality to the degree of surface coverage. Theblocking factor is a function of both particle wallconcentration and wall shear rate. In Fig. 2, thethree shielding functions are shown as a functionof degree of coverage. The deposition model wasadopted by Meinders et al. [8] for the study ofparticle deposition in a parallel-plate flowchamber.

Fig. 1. Illustration of the origin of the shielded area mecha-nism. A free-flowing non-deposited particle (position 1) fol-lows a streamline. Due to electrostatic repulsive interactions itis pushed further out from the wall and changes streamline(position 2).


Fig. 2. The three shielding functions given by Eq. (4) (six times the particle cross-sectional area) and Eq. (5) (four times the particlecross-sectional area), and Eq. (6) for three values of � (2.6, 3.2 and 3.8), as a function of the degree of surface coverage.

3. Materials and methods

The deposition experiments were performed ina wall-jet cell, which is illustrated in Fig. 3. It isdescribed in detail elsewhere [10]. The wall-jet cellwas mounted on x–y translators, which made itpossible to perform radial scans of the surface.Measurements were carried out at discrete pointsalong the surface, which were exposed to differentwall shear rates. The wall shear rate was calcu-lated from the expression for the radial velocitypresented by Albery [9], ur=Cz/r11/4, one particleradius from the surface, where C equals {(5M)3/(216�5)}1/4, z is the distance from the wall and r isthe radial distance from the stagnation point.

In order to visualize the deposited particles, atechnique employing total internal reflection mi-croscopy was used. Light from a 5 mW linearlypolarized HeNe laser (Melles Griot), �=632.8nm, was directed towards a 70°, right-angledprism (BK7, n=1.5151 at �=632.8 nm) posi-tioned on top of the glass slide. A neutral-densityfilter (Melles Griot) was positioned between theprism and the laser to decrease the intensity of the

light. Inside the glass slide total internal reflectionoccurred, and at the interface of the glass andwater an evanescent wave was generated with apenetration depth of the electric field of approxi-mately 100 nm [15].

The scattered light from adhered particles wascollected by a light microscope (Jena), on whichan object lens (Leica EF 32/0.40) with a longworking distance produced a total magnificationof ×200. The light intensity was measured by ahead-on photomultiplier tube (HamamatsuR1104). The analogue signal was digitalized andprocessed with a computerized data acquisitionprogram.

Glass slides (Donnelly Applied Films Corpora-tion) covered with indium– tin oxide were used asdeposition surfaces. They were cut into piecesmeasuring 30×40 mm2.

Polystyrene latex beads (Sigma Latex Beads,Lot 57H1207) with a radius of 0.23 �m were used.They were mixed with deionized, double-distilledwater from a Milli-Q plant at particle concentra-tions of 2×109 and 4×109 cm−3. The pH wasset at 5.5 with citrate–phosphate buffer solution


and HNO3/KOH, and the electrolyte concentra-tion was 50 mM KNO3. To achieve a constantbulk concentration of particles during the experi-ment, the turbidity was measured (at 700 nm),and fresh particle solution was added to thestirred feed vessel.

Two flow rates were used, 0.12 and 0.20 cm3

s−1, with corresponding Reynolds numbers of154 and 253, respectively (Re=ua/2�, where urepresents the velocity of the fluid in the jetnozzle, and a is the jet diameter). All experimentswere carried out at a temperature of 21�1 °C.

4. Results and discussion

4.1. Shielding effects at different wall shear rates

In Fig. 4, results obtained at a particle bulkconcentration of 4×109 cm−3 and a volumetricflow rate of 0.20 cm3 s−1 (Re=253) are shownfor three different radial positions, 2.50, 4.00 and5.50 mm, with the corresponding wall shear ratesof 860, 236 and 98 s−1, respectively. The time-de-pendent flux began at a lower degree of surface

coverage for higher shear rates than with lowerones, and a higher wall shear rate resulted in alower degree of coverage for longer times of depo-sition, as found previously [10]. The model de-scribed by Eq. (3) is included in Fig. 4(a–c), withthe shielding function given by Bell and Brown [6]in Eq. (4). The mass transfer coefficient, kmass, isgiven by Eq. (2) and kr is used as the only fittingparameter, interpreted as a re-entrainment ratecoefficient. The deposition kinetics is poorly de-scribed by the model at a radial distance of 2.50mm from the stagnation point, exposed to a wallshear rate of �� =860 s−1 (Fig. 4(a)). The overpre-diction of the particle concentration at the wallmay be explained by the fact that the area ofshielding is larger than that predicted by theshielding function, or that not only shielding isresponsible for the sharp drop in the depositionrate. The deposition process was best described bythe deposition model of Eqs. (3) and (4) at theposition exposed to a wall shear rate of �� =236 s−1 at a radial distance of 4.00 mm from thestagnation point (Fig. 4(b)). If the assumptionthat the shielding effect is one of the mechanismsgoverning the deposition process under the pre-vailing conditions is correct, it may then be con-cluded that the excluded area reached an averageof six times the cross-sectional area of a singledeposited particle. Fig. 4(c) shows the results fromthe same deposition model applied to a particledeposition trace performed at a radial distance of5.50 mm from the stagnation point, exposed to awall shear rate of 98 s−1, i.e. further away fromthe stagnation point. The model underestimatedthe experimental data, since it predicted that asingle, deposited particle would shield an area sixtimes its cross-sectional area. In this case asmaller area was excluded, resulting in a higherdegree of coverage, since particles were depositedmore easily on the surface.

The shielding functions presented in the theorysection Eqs. (4)– (6), assume different sizes of theexcluded area. The shielded area according to Eq.(4) for a single, deposited particle, is six times itscross-sectional area, while the analogous area inEq. (5) is four times its cross-sectional area. Eq.(6) contains the fitting parameter, �, which allowsthe size of the shielded area to be varied. ThisFig. 3. Schematic illustration of the wall-jet cell.


Fig. 4. Particle concentration at the wall (�) vs. time. The volumetric flow rate was 0.20 cm3 s−1 (Re=253) and the bulkconcentration of particles 4×109 cm−3. Solid line represents the deposition model of Eq. (3) together with the shielding functionof Eq. (4), having an excluded area of six times the cross-sectional area of a particle. (a) Deposition at 2.50 mm with �� =860 s−1.The pseudo-steady-state degree of coverage, �pss, reached 0.20 after 13 h. The re-entrainment rate coefficient, kr, was 3×10−4 s−1.(b) Deposition at 4.00 mm with �� =236 s−1, �pss=0.34 and kr=2.5×10−5 s−1. (c) Deposition at 5.50 mm with �� =98 s−1,�pss=0.42 and kr=9.0×10−7 s−1.


Fig. 4. (Continued)

difference in the predicted size of the shielded areabetween the shielding functions can be used toquantify the degree of surface area that is notavailable for flowing non-deposited particles atdifferent shear rates; thus a higher wall shear ratewould result in a larger shielded area. The resultsfrom an experiment carried out at a particle bulkconcentration of 2×109 cm−3 at a volumetricflow rate of 0.12 cm3 s−1 (Re=154) are shown inFig. 5. The particle wall concentration is shown asa function of time for three radial positions, 3.50,4.00 and 5.50 mm from the stagnation point, withcorresponding wall shear rates of 111, 77 and 32s−1, respectively. Fig. 5(a) shows the depositionat a radial position 5.50 mm from the stagnationpoint with a wall shear rate of 32 s−1. Thepseudo-steady-state degree of coverage reached0.38 after 13 h of deposition. The shielding func-tion of Eq. (5) (solid line) was not able to exactlydescribe the deposition process, but underpre-dicted the particle wall concentration, indicatingthat the area which is not available for the deposi-tion of free-flowing particles is smaller than fourcross-sectional particle areas. By introducing theshielding function of Eq. (6) into the deposition

model, and using a value of � of 2.6 (dashed line),i.e. a shielded area of 2.6 cross-sectional areas, thecomplete deposition process was reproduced well.It was found that re-entrainment played a minorrole at this low wall shear rate, thus, in thedeposition model the re-entrainment term wasexcluded by setting kr=0. Fig. 5(b) shows theexperimental data at a radial position of 4.00 mmfrom the stagnation point. The wall shear ratewas 77 s−1 and the degree of coverage reached0.35 at the end of the experiment. The depositionprocess was readily described by adopting thedeposition model of Eq. (3) together with theshielding function of Eq. (5). To successfully de-scribe the entire deposition kinetics both theshielding and the re-entrainment mechanisms hadto be included in the model. The size of theshielded area under the prevailing experimentalconditions was estimated to be four cross-sec-tional particle areas. In Fig. 5(c) where the wallshear rate was 111 s−1 (3.50 mm from the stagna-tion point), the pseudo-steady-state degree of cov-erage reached 0.34 after 13 h of deposition. Theshielding function of Bell and Brown [6], whichconsiders a shielded area of six times the particle


Fig. 5. Data from an experiment carried out at a volumetric flow rate of 0.12 cm3 s−1 (Re=154) and a bulk concentration ofparticles of 2×109 cm−3 at three wall shear rates: (a) Deposition at 5.50 mm with �� =32 s−1 and �pss=0.38 after 13 h. Dashedline represents the deposition model of Eq. (3) together with shielding function of Eq. (6) with � value of 2.6 and kr=0. Solid linerepresents Eq. (3) together with Eq. (5), having an excluded area of four times the cross-sectional area of a particle and kr=0. (b)Deposition at 4.00 mm with �� =77 s−1 and �pss=0.35. Solid line represents Eq. (3) together with Eq. (5) with kr=1.4×10−5 s−1.The dashed line shows Eq. (3) together with Eq. (6) with an � value of 3.2 and kr=5.0×10−7 s−1. (c) Deposition at 3.50 mm with�� =111 s−1 and �pss=0.34. Solid line describes Eq. (3) in combination with Eq. (4), having an excluded area of six times thecross-sectional area of a particle. kr=5.5×10−6 s−1. Dashed line shows Eq. (3) in combination with Eq. (6) using an � value of3.8 and kr=5.0×10−7 s−1.


cross-sectional area, gave a good prediction of theprocess. The results shown support the assump-tion that shielding determines the deposition ofsub-micron particles to a large extent and it canbe concluded that a higher wall shear rate willresult in a larger excluded area, which could beunderstood intuitively.

The deposition kinetics was readily describedfor �� =32 s−1 by adopting the shielding functionof Dabros and van de Ven (Eq. (6)), applying an� value of 2.6 as was seen in Fig. 5(a). In order toinvestigate the applicability of this general func-tion, Eq. (6) was used to describe the depositiondata at �� =77 and 111 s−1 (Fig. 5(b) and (c)). Atthe position exposed to a wall shear rate of 77 s−1

(Fig. 5(b)), the best fit was obtained by adoptingan � value of 3.2 with a lower particle re-entrain-ment rate than was the case when the shieldingfunction of Schaaf and Talbot (Eq. (5)), whichhad a shielded area of four cross-sectional areas,was used. The function of Dabros and van de Venslightly overpredicted the particle concentration atthe wall at the beginning of the deposition pro-cess, followed by underprediction of the particletransport in the later part of the experiment.

However, for higher surface coverage the modelfailed to follow the deposition kinetics and didnot behave as the shielding function of Schaafand Talbot [11]. The same observation was madefor �� =111 s−1 (Fig. 5(c)), where an � value of3.8 gave the best results, compared with theshielding function of Bell and Brown [6], whichpredicts an excluded area of six particle cross-sec-tional areas. Also in this case, overprediction ofthe particle transport was seen at the beginning ofthe deposition process. At approximately �=0.25the model failed to follow the deposition kineticsas the particle concentration at the wall continuedto increase. In Fig. 2, the shielding function of Eq.(6) is shown for the different � values used as afunction of the degree of surface coverage. Forthe case of �=3.2 the straight line coincides fairlywell with the function of Schaaf and Talbot (Eq.(5)) up to �=0.25. This is also the case for�=3.8, but � becomes 0.22. To conclude, thegeneral shielding function of Dabros and van deVen [5] fails to describe the particle depositionkinetics for high particle concentrations at thewall when high wall shear rates dominate. Onepossible explanation of this could be that the

Fig. 5. (Continued)


shielding function does not consider the fact thatdeposited particles share shielded surfaces witheach other, as the shielding functions of Schaafand Talbot [11] and Bell and Brown [6] do.

4.2. Particle bulk concentration effects

The shielding function of Eq. (4) was comparedwith the data from the two experiments given inFig. 4(b) and Fig. 5(c) to describe the depositionkinetics. The particle bulk concentrations were4×109 and 2×109 cm−3. The radial position forthe higher particle bulk concentration was 4.00mm from the stagnation point with a correspond-ing wall shear rate of 236 s−1 and for the lowerconcentration the radial position was 3.50 mmwith a wall shear rate of 111 s−1. The excludedareas in both experiments seem to be of the samemagnitude, although there is a considerable differ-ence between the wall shear rates. At the positionexposed to the higher wall shear rate the excludedarea should be larger. This observation supportsthe hypothesis that at a higher particle bulk con-centration, more particle collisions are likely tooccur, resulting in a smaller excluded area, sinceparticles are pushed down behind deposited parti-cles. The results of Meinders and Busscher [16]indicated a similar concentration dependency ofthe size of the excluded area, although in theirinvestigation, an asymptotic value of the excludedarea was observed for particle bulk concentrationsexceeding 2×108 cm−3.

Considering the re-entrainment rates, quantifiedby the fitting parameter kr, there is a discrepancybetween the two experiments. When the bulk con-centration of particles was 4×109 cm−3 with awall shear rate of 236 s−1 (Fig. 4(b)), the re-en-trainment rate was 2.5×10−5 s−1. When theconcentration was 2×109 cm−3 and the wallshear rate was 111 s−1 (Fig. 5(c)) kr was 5.5×10−6 s−1. The explanation of the difference inre-entrainment rate is twofold. An increase inparticle concentration leads to a higher collisionfrequency between free-flowing, non-depositedparticles and deposited particles, resulting in anincrease in the number of particles leaving thesurface. The second explanation is that a free-flowing particle has a higher linear momentum

and when it collides with a deposited particle theprobability of re-entrainment will be greater.

5. Conclusions

In this study, the deposition of polystyrenelatex particles on a glass surface covered withindium– tin oxide was studied in a wall-jet cell.The cell was constructed in such a way that thedeposition kinetics could be followed at variouswall shear rates in a single experiment.

A deposition model was used to describe thenon-linear deposition as a function of time. Themodel consisted of a transport term, a shieldingfunction and an expression accounting for re-en-trainment of particles. Three shielding functions,which assumed different shielded areas, were com-pared. It was shown that a higher wall shear rateresulted in a larger shielded area. This behaviorwas explained by the fact that a particle beingpushed further out from the wall due to repulsiveforces between the free-flowing particle and thedeposited particle was transported further down-stream before returning to the collector surface.The deposition model, which successfully de-scribed the process at lower wall shear rates failedto characterize the deposition at positions exposedto the higher wall shear rates. The model overpre-dicted the non-linear particle flux with time,which resulted in too rapid an increase in theparticle concentration at the wall.

A higher bulk concentration of particles re-sulted in a smaller shielded area due to a highercollision frequency, since particles are pusheddown behind deposited particles.

An increase in particle bulk concentration alsoresulted in higher re-entrainment rates due to anincrease in the number of particle collisions be-tween free-flowing particles and deposited ones.

Acknowledgements

The Swedish Research Council for EngineeringSciences provided funding for this research.


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Documents

Modelling of sub-micron-sized particle deposition in a laminar wall jet—an investigation of the role of shielding