Journal of Colloid and Interface Science231,228–237 (2000)doi:10.1006/jcis.2000.7139, available online at http://www.idealibrary.com on

An Experimental Study of the Kinetics of Particle Deposition in a Wall-JetCell Using Total Internal Reflection Microscopy

Anders Goransson and Christian Tr¨agardh1

Food Engineering, Center for Chemistry and Chemical Engineering, Lund Institute of Technology at Lund University, P.O. Box 124, SE-221 00 Lund, Sweden

Received May 14, 1999; accepted July 31, 2000

The deposition of polystyrene latex particles of 0.46µm diameterwas studied in situ using a wall-jet cell in combination with totalinternal reflection microscopy. The particles were deposited ontoan indium tin oxide surface in a laminar flow field for up to 13 h.The initial particle flux was found to be mass-transfer-controlled.It was shown that one period of the time-dependent deposition rateperiod was of first-order nature. The effective particle transfer co-efficient during this period appeared to be correlated inversely tothe wall shear rate. With the help of three characteristic empiricalconstants, one of them being a mass transfer coefficient, the overalldeposition process was described by a model equation. The concen-tration dependency was elucidated using a Langmuir-type pseudo-isotherm. C© 2000 Academic Press

Key Words: wall-jet cell; deposition; polystyrene particles; to-tal internal reflection microscopy; evanescent wave; first-orderkinetics.

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INTRODUCTION

In the area of cleaning and process hygiene it is importo understand the underlying mechanisms for the deposof proteins, macromolecules, and particles onto surfaces.deposition process can be described in three steps: tranattachment, and detachment. The transport of particles tosurface is followed by attachment, if the particles have enotime to establish contact with the surface and if the adheforce is strong enough. If the hydrodynamic forces are strenough, the particles can be detached from the surface.

Experimental work carried out to date has involved the stof the deposition or reentrainment of bacteria (1), colloidal pticles (2, 3), or macromolecules (4, 5). Experimental setupshave been used include the sedimentation cell (6), the rotadisc cell (7), the impinging jet cell (also called the stagnatipoint flow cell) (8, 9, 10), and the parallel-plate flow chan(11, 12). It is important to have a well-defined flow when stuing the kinetics of the colloidal coating process. With the roing disc cell the possibility of homogeneous particle transpothe collector surface was established. The rotating disc prov

1 To whom correspondence should be addressed. E-mail: christian.tragarlivstek.lth.se.

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220021-9797/00 $35.00Copyright C© 2000 by Academic PressAll rights of reproduction in any form reserved.

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a constant hydrodynamic boundary layer thickness but a mdrawback is that it is not possible to continuously follow the pticle depositionin situ. The collector surface must be rinsed adried before any further analysis can be performed. This plem can be overcome by using an impinging jet, together wdark field illumination. This gives the opportunity to follow thparticle depositionin situ under well-defined flow conditionsColloidal particle deposition was studied in this way in a stucarried out by Dabros and van de Ven (9). It was found that desition showed a decreasing with time flux after a certain crititime, and that this could be explained by particle detachmblocking effects by deposited particles, coagulation of the cloidal particles prior to deposition, and changes in the energinteraction between collector and particle. The number of avable sites also decreases, of course, and this contributes ttime-dependent deposition rate (13). Deposited particles wfound to be able to block an area much bigger than their gmetrical cross section (14). This can be explained by the factthe local flow field was changed due to the deposited partiand thus the trajectories of the flowing particles were modifiAnother explanation of blocking could be that the particlestached to the surface repelled flowing particles and causedto deviate from the surface. Factors that contribute to the inaction energy very close to a wall are the surface roughnesthe collector and particle, the nonuniform distribution of electcharges and chemical factors (9).

In this study, the aim was to adopt a system and methodofor investigating the deposition kinetics of particle transport tglass surface. An attempt was also made to formulate a mamatical description of the kinetics on which deposition analycould be based. The experiments were carried out using a wjet cell in order to achieve a well-developed flow profile, togetwith an evanescent wave technique (TIRM). The data analincluded the determination of the kinetic coefficient and thenetic order.

THEORY

The Flow Field of the Wall-Jet Cell

The case of a submerged jet impinging perpendicularly oflat surface has been described in many papers (15, 16). This

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PARTICLE DEPOSITION I

FIG. 1. The impinging jet.

of flow is called Hiemenz flow, named after the first scientisfind a solution to a differential equation coupled to this typeflow. Four distinct hydrodynamic regions can be distinguishwhich are illustrated in Fig. 1.

The first region is called the potential core region. Thiscone-shaped region with a height of 4.7 to 7.7 jet diametersThe flow then changes from pipe flow to free jet flow, regionThis region extends to a distance of 1.6–2.2 jet diametersthe surface. The third region is the stagnation region or the wpipe region. In this region the jet changes direction and becoradial directed. The radius of this area is 0.6–1.4 jet diameThe flow is axisymmetric inviscid irrotational and the thickneof the hydrodynamic boundary layer is independent of the radistance close to the stagnation point. The final region is the wjet region, which begins at a radial distance greater thandiameters from the stagnation point. Characteristic of this reis that the velocity and the hydrodynamic boundary layer bshow a radial dependency. The velocity decreases and thuboundary layer grows with increasing radial position.

In this investigation, particle deposition was studied in wjet flow, and therefore, all interest is now focused on the walregion. In a study by Glauert (16) the flow of a wall jet winvestigated for the laminar case. Albery (17) gives an expexpression for the radial fluid velocity close to the surface,

ur = C · zr 11/4

, [1]

wherez is the distance to the wall andr is the radial distance fromthe stagnation point. The variableC equals{(5M)3/(216v5)}1/4.The flux of exterior momentum,M , is expressed by

M = k4eV3

f

2π3a2, [2]

whereke is an experimental constant determined to be 0.86

AN IMPINGING-JET CELL 229

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Yamada and Matsuda (18).Vf is the volumetric flow rate andais the diameter of the jet.

Mass Transfer in the Wall-Jet Cell

The mass transfer of particles is described by the FokPlanck equation. For dilute particle suspensions, when thereinteraction between the particles, this equation can be replby the following mass-balance equation:

∂c

∂t+ ∇ · j = 0, [3]

with the convective–diffusive particle flux expressed as

j = c · U − D∇c. [4]

Over a flat surface, where there is no wall-normal velocEq. [4] reduces to

j = k∇c, [5]

wherek = D/δ. δ is the boundary layer thicknessGlauert (16) and Albery (17) have derived the mass-trans

controlled initial deposition step in a wall-jet system. The initiconstant particle flux decreases with increasing distance fromstagnation point:

j0 = 0.165D2/3V3/4f v−5/12a−1/2r−5/4cb. [6]

D is the diffusion coefficient of the particles,Vf is the volumetricflow rate,v is the kinematic viscosity,a is the diameter of the jetr is the radial distance from the stagnation point, andcb is theparticle bulk concentration. The particle transfer coefficientk,is identified comparing Eqs. [5] and [6]. It is under mass trancontrolled conditions assumed that the particle concentratiothe vicinity of the wall is close to zero considering thatk isdivided into one mass transfer step and one attachment ste

1

k= 1

kmass+ 1

katt, [7]

and that the attachment rate is very much faster that the mtransfer rate.

The decreasing with time flux commences when particle cerage exceeds a critical value, implying that the attachment trfer coefficient approaches and eventually gets lower thanmass transfer coefficient. The initial flux equation, Eq. [6], is thno longer valid since it does not consider certain factors, whinfluence the attachment process. Examples of such factorsurface blocking and change in surface potential. The periotime-dependent deposition rate is described by a Langmuir-model (13),

by dt= k(1− θ )n, [8]

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230 GORANSSON A

whereθ is the degree of coverage,k is a net transfer coefficienandn is the kinetic order. This relation is directly proportionto the flux in Eq. [5]. The mass-transfer-controlled initial stis thus of zeroth order, and the nonlinear time-dependent recommences as soon asn> 0. n = 1 represents the Langmumodel or first-order kinetics.

In the following empirical expression the time-dependent pticle flux is described in a different way than in Eq. [8]. The flis presented as a function of time and degree of surface cove

dθ

dt= k1

t(1− θ ), [9]

wherek1, θ1, and t1 are empirical constants. The solutionEq. [9] for the boundary conditiont = t1 andθ = θ1 is given as

θ (t) = 1− (1− θ1)

[t1t

]k1

. [10]

The Principle of Total Internal Reflection Microscopy

Different techniques can be used to investigate particle dsition in situ. Dabros and van de Ven (9) described a metin which dark field illumination was used to visualize adsorbparticles. In a paper by Alberyet al. (2) an optical techniquetotal internal reflection microscopy (TIRM), was utilized to folow the deposition of carbon black particles. The principleTIRM is described below.

Light is propagated in a denser medium than the surround(n1> n2). Total reflection occurs when the angle of incidengreater than the critical angle for the specific material (Fig.Despite the total reflection a small amount of light energy leinto the surrounding medium. This is the evanescent wave, wpropagates parallel to the surface. Its amplitude falls expotially with increasing distance from the interface. With the hof Maxwell’s equations, the electric field for a linearly polarizplane wave can be expressed as

E = E0te−iωt exp[(i K t sinθt)x] · exp[−Ktuz], [11]

whereE0t is the amplitude of the electric field,ω is the circularfrequency,t is time, Kt is the magnitude of the propagatiovector, andθt is the transmission angle (19). It can be seenthe disturbance at the interface has the same circular frequas the incoming wave. The electric field amplitude decreaexponentially with increasing distance from the interface. Tis expressed as

E = E0(θi ) · e−z/dp, [12]

whereE0(θi ) is the electric field at the interface anddp is the pen-etration depth (Ktu = d−1

p ) defined as the distance over whithe electric field decreases toe−1 of the original electric field

amplitude. Equation [13] shows the dependency of the incidangle,θi , on the penetration depth of the electric field (20), whi

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is illustrated in Fig. 2b:

dp = λ

2πn1(sin2 θi − (n2/n1)2)1/2. [13]

It can be seen how the penetration depth depends not only oincident angle but also on the wavelength of the light, togewith the optical densities of the materials.

MATERIALS AND METHODS

Materials

The experimental wall-jet cell is shown in Fig. 3. It was maof perspex with cell diameter and height of 18 and 14 mrespectively. The diameter of the jet nozzle,d, was 0.5 mm.The distance between the orifice and the opposite surfaglass surface, was 16d (8 mm) in all the experiments. Fluid wapumped through the jet nozzle perpendicular to the surfacethe help of self-pressure in order to achieve a pulse-free flAt the stagnation point, liquid was radially distributed oversurface. The cell was sealed with an O-ring and tighteneda clamp. The cell was drained at the bottom (Fig. 3). It w

entchpossible to move the measuring cell in thex- andy-directionswith two horizontal translators.

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PARTICLE DEPOSITION IN

FIG. 3. Schematic cross section of the wall-jet cell.

A 5-mW linearly polarized HeNe laser (Melles Griot),λ =632.8 nm, was directed at a right angle toward a 70◦, right-angled prism (BK7,n = 1.5151 atλ = 632.8 nm), which waspositioned on top of the glass slide in order to lead the beinto the glass slide. Immersion oil with a refractive indexn = 1.516 was used. The laser was aligned with the help oftwo horizontal translators (x- andy-directions) and an adjustablsupport table in thez-direction.

A neutral density filter (Melles Griot) was positioned betwethe prism and the laser to decrease the intensity of the light

A light microscope (Jena) with an object lens (LeiEF 32/0.40) with a long working distance gave a magnificatof 200×. A head-on photomultiplier tube (Hamamatsu R110in a holder was mounted on top of the ocular. The anasignal was digitized and processed in a computer acquisprogram.

The adhesion surface was made of glass slides coveredan indium tin oxide film (ITO) (Donnelly Applied Films Corporation). The glass slides were cut into pieces 30× 40 mm2. Toinvestigate the surface characteristics, atomic force microsc(AFM) was used. An ITO glass surface was prepared folloing the routine described above. A surface of 3× 3 µm2 wasscanned in surface mode. In Fig. 4 an AFM image of the Isurface is shown, revealing a homogeneous surface.

Methods

Deionized, double-distilled water (DDDW) from a Milli-Qplant was used for all solutions.

AN IMPINGING-JET CELL 231

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A colloid solution was prepared using polystyrene latex pticles (Sigma Latex Beads, Lot 57H1207) with a diameter0.46µm and a density of 1.05 g cm−3. Three particle concentrations were used: 2× 109, 4× 109, and 9× 109 cm−3. Theelectrolyte concentration was 50 mM KNO3, and the pH wasadjusted to 5.5 with a citrate–phosphate buffer solutionHNO3/KOH. All experiments were performed at a tempeture of 21± 1◦C. Two different flow rates were used, 7.3 a12.0 cm3 min−1, with jet Reynolds numbers of 154 and 25respectively (Re= u · r/v, whereu is the velocity of the fluidin the jet nozzle andr is the radius of the nozzle).

The cleaning procedure for a surface was as follows. The swas soaked in 4% Decon90 at 60◦C for 1 h and then carefullyrinsed and stored in DDDW for 30 min. The slide was thstored in air until used. Before mounting the slide in the celwas washed in a 1 : 1 mixture of concentrated HNO3 and H2SO4

at room temperature for 10 min and then carefully rinsed wDDDW and ethanol (95%). Slides were used only once anddiscarded.

The cell was adjusted using knownx- and y-positions forthe stagnation point. Electrolyte solution (50 mM KNO3) waspumped through the cell, and a radial surface scan wasformed to measure the background light of the clean surfThe experiment was then started and the electrolyte soluwas replaced by a colloid solution. A certain volume was dcarded before the colloid solution was recycled to avoid dilutby the electrolyte. For long experiments, the bulk concentrahad to be adjusted by adding more colloid solution to thesolution. The absorbance of the colloid solution was measwith a spectrophotometer in order to determine how muchloid solution should be added. The feed vessel was continuostirred to decrease aggregation of the particles. During the16 minutes the surface was continuously scanned in order tolow the rapid change in surface coverage. The scan frequwas then gradually decreased, and after 4 h radial scans werperformed every 30 min until the end of the experiment. Tsurface was scanned at eight positions beginning at 2.00from the stagnation point in steps of 0.50 mm.

FIG. 4. An AFM image of a bare ITO surface.

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232 GORANSSON A

Calibration of the PMT Signal

To interpret the signal from the PMT to give the numberparticles per unit area, a calibration curve was determinedlow particle coverage on the surface, less than 10%, a lirelationship exists between signal intensity and the numbeparticles. For surface coverage higher than 10% there is a dtion from the linear behavior and a calibration curve is requir

A typical calibration experiment started with a radial scanthe bare surface followed by a period of particle deposition. Inend of the deposition phase a second surface scan was perfoand the colloid solution was replaced by DDDW. The sharp din the PMT signal was due to three factors: disappearance obackground signal, loosely attached particles leaving the suand the increase in distance from the adhered particles to thewhen DDDW is introduced. After rinsing the cell with water f30 min the surface was scanned once again. The flow ratehigh in order to expose the particles to a high shear, to minimthe risk of particles falling off the surface when removing tglass slide from the cell. After the scan the slide was remoand dried in a horizontal position.

The slide was mounted in a scanning electron microsc(SEM) and photographs were collected at specific radial ptions of known PMT intensity. The photographs were digitizand processed with an image analysis program (WinViewPrinceton Instruments) to determine the number of particlethe surface. A comparison between image analysis and macounting gave a discrepancy of±2%. A typical calibration curveis shown in Fig. 5a where the degree of coverage,θ , is shown asa function of PMT intensity.

A problem when interpreting the PMT signal from an expiment is the background intensity from the bulk solution.solve this problem, the difference in the signal from the oveintensity (including intensity from both background noise adeposited particles) and the signal stemming from deposparticles only (Ic–Ie), was plotted against the overall intenssignal, Ic (Fig. 5b), and the difference was added to thebackground intensity signal shown in Fig. 5a. A fit is made frthe calibration curve in Fig. 5a, which compensates for baground intensity. Two alternative fitting equations were usecalculate the amount of deposited particles:

Nt = a1[1− exp(−b1I )] + c1[1− exp(−d1I )] [14]

and

Nt = [a2+ b2I + c2I 2]1/2. [15]

Equation [15] gives the best fit for higher intensities, butlower intensities negative arguments arise and Eq. [14] isused.

RESULTS AND DISCUSSION

The deposition process can be divided into two main pha

the initial phase with constant flux and the phase with timdependent flux. In the following, these two phases are analyz

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FIG. 5. (a) A calibration curve. (b) Compensation for the background sig

Constant Flux Phase

Figure 6a shows the initial, constant particle flux as a funcof r−5/4 together with Eq. [6] (solid line). It can be assumed tthe constant flux is mass transfer controlled since the expmental values agreed well with Eq. [6] (2). From Fig. 6a itpossible to calculate the diffusion coefficient of the particlescalculating the slope of the regression line of the data points.experimentally calculated diffusion coefficient was found to(7.8± 0.8)× 10−9 cm2 s−1. The diffusion coefficient accordinto the Stokes–Einstein equation is 9.28× 10−9 cm2 s−1. The dif-fusion constant calculated from the Stokes–Einstein equatihigher than the experimentally found value, in accordanceAlbery et al. (2). This could be due to the fact that Eq. [6] donot take into account hydrodynamic and van der Waals inactions, since they are supposed to cancel each other. Anexplanation is that the particle size distribution gives rise t

e-ed.range of diffusion coefficients, which differs from the theoreticaldiffusion coefficient.

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PARTICLE DEPOSITION IN

FIG. 6. (a) The initial flux vsr−5/4 together with Eq. [6] (solid line). (b) Theinitial flux at the radial distance 2.50 mm for three particle bulk concentrattogether with Eq. [6] (solid line).

Figure 6b shows the constant particle flux as a functionparticle bulk concentration at the radial position 2.5 mm frthe stagnation point. It is shown that Eq. [6] (solid line) is vafor the three particle concentrations used, indicating an inmass-transfer-controlled deposition process.

In Fig. 7,θcrit is shown as a function of the wall shear streThe θcrit value represents the degree of coverage at the pbetween the phases with constant and time-dependent fluxcan be clearly seen that a higher wall shear stress resultslower degree of coverage when the time-dependent flux pebegins, despite the fact that the particle flux (defined accorto Eq. [6]) is much higher closer to the stagnation point,given in Fig. 6a. The nonlinear relationship indicates thatsingle phenomenon can solely explain this behavior. A likexplanation is that a smaller number of sites are “availabfor free-flowing particles at places exposed to higher wall shstress. This in turn could be explained by the fact that alre

deposited particles block the surface, thus preventing partifrom reaching it (21), that particles do not have enough time

AN IMPINGING-JET CELL 233

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FIG. 7. Characterization of the critical degree of coverage where the tidependent flux period commences, as a function of wall shear stress.

establish adhesion contact with the surface due to the influeof the fluid, or that the hydrodynamic force acting on the partidoes not allow it to be kept at the sites then available for them,they thus reentrain into the bulk solution. By visually studyithe deposition process, it was possible to see particles remfrom sites, which implicates that removal forces are stronger tthe instantaneous adhesion forces, but it does not excludeblocking phenomena also play an active role in the procesthus shows that reentrainment takes place and that the depois a dynamic process.

Time-Dependent Flux Phase

In Fig. 8 a typical deposition pattern is shown as a functof time at eight radial positions from the stagnation point (fro

FIG. 8. Results from a typical particle adhesion experiment showingdegree of coverage vs time at different radial positions starting from 2.00 mincremental steps of 0.50 to 5.50 mm for a volumetric flow,Vf , of 12.0 cm3 s−1.

clestoThe Peclet numbers, defined as Pe= urrp/D, for the eight positions are shownin the figure.

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234 GORANSSON A

FIG. 9. (a) A first-order plot from the experiment shown in Fig. 8 for thradial position 3.50 mm. (b) Characterization of the degree of coverage wfirst-order kinetics starts as a function of the logarithm of wall shear str(c) The dependency of the first-order constant,k, on the P´eclet number.

2.00 to 5.50 mm at intervals of 0.50 mm). The wall shear strfalls from 1.57 Pa at 2.00 mm to 0.1 Pa at 5.50 mm from

stagnation point. The volumetric flow rate is 12.0 cm3 min−1

with a particle concentration of 4× 109 cm−3. The deposition

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process was followed for a period of 13 h, and it was seen tthe degree of surface coverage was higher for positions wsmaller Peclet numbers (Pe= urrp/D). At a position 5.50 mmfrom the stagnation point, the surface coverage reachescompared with 0.15 for a position 2.00 mm from the stagnatpoint.

Figure 9a shows a first-order plot of one of the experimentraces (3.50 mm from the stagnation point) from the experimshown in Fig. 8 with ln(1− θ ) given as a function of time. Itcan be seen that a period beginning after around 4 h and lastingalmost to the end of the experiment is of first order, with the gdient interpreted as the transfer coefficient in Eq. [8]. In Fig.the degree of coverage is shown where the first-order periodgins as a function of wall shear stress. A similar pattern tocase where the nonlinear period commences can be seen, wa higher wall shear stress results in an earlier start of the fiorder period. The particle flux decreases with increasing P´ecletnumber, which can be shown by plotting the transfer coefficiek, for different Peclet numbers (Fig. 9c). It is likely thatk hereequalskatt and thus that the deposition rate is controlled by tattachment rate and mentioned phenomena.

Figure 10 shows experimental traces at two radial positio(2.50 and 5.00 mm) for the two volumetric flow rates, 7.3 a12.0 cm3 min−1. For the higher flow rate the values of Pe areand 7.3 for the 2.50- and 5.00-mm positions and for the lowflow rate 16 and 2.4. A comparison between identical positioand a different convection/diffusion ratio shows a higher degof coverage for the position with lower flow intensity. The resuwhich are shown in Fig. 10 also give support to the hypothethat the rate of deposition is controlled, to a large extent, byconvection/diffusion ratio as shown in Fig. 9c. Here,k⊂Pe−0.4

in contrast to the constant rate period whenk⊂Pe3/4 in accor-dance with the approach made in Eqs. [5] to [7]. The relationsshown in Fig. 9c, between particle transfer coefficient and P´eclet

FIG. 10. Particle traces from two experiments with Re= 154 and 253 at

two radial positions (2.50 and 5.00 mm). (s) Re= 154, Pe= 2.4; (d) Re= 253,Pe= 7.3; (u) Re= 154, Pe= 16; (j) Re= 253, Pe= 49.

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PARTICLE DEPOSITION IN

FIG. 11. (a) In the figure the empirical constants used in Eq. [10] are defifor an experimental trace at Re= 253 with a particle bulk concentration o2× 109 cm−3 at a radial position of 2.50 mm. (b) The particle trace together wthe prediction of Eq. [10] (solid line) with the empirical constantsk1= 0.038,t1= 500 s,θ1= 0.04. A first-order kinetic model given by Eq. [8] forn = 1 isgiven (dotted line) for the time period where first-order kinetics is valid.

number (or shear rate), indicates that blocking effects and/orticle drag force effects controls the deposition rate. Both of thscale linear to the shear rate.

In Fig. 9a it was shown that one period of the depositprocess follows a first-order kinetics equation. However, in orto describe the overall process, another formulation musused, expressed in Eq. [9]. Figure 11a shows the interpretaof k1, θ1, andt1, which are used in Eq. [10] from an experimeperformed at a volumetric flow rate of 12.0 cm3 min−1 witha particle concentration of 2× 109 cm−3 at a radial distance o2.50 mm. They-axis shows ln(1− θ ) and the time is representeon thex-axis. The time constant,t1, gives the time at which thelinear period commences at a degree of coverage ofθ1. The slopeof the line isk1. The degree of coverage is shown as a functiontime in Fig. 11b together with the predictions of Eq. [10] (so

line) and the integrated equation, Eq. [8] (dotted line) for tcase whenn = 1.

AN IMPINGING-JET CELL 235

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In Fig. 12a the rate constantk1 is depicted as a function of Pfor two experiments performed at the two flow rates. Enhanparticle transport is observed as the influence of convectioncomes less pronounced (decreasing Pe). This behavior isurprising as explained above. A comparison between thevolumetric flow rates clearly shows that positions exposed tosame convection/diffusion ratio have similar transport chateristics. Figure 12b shows an analysis of the dependenceθ1

on the wall shear stress. As expected, a higher wall shear sgives rise to a lower degree of coverage when the log-linperiod starts.

Concentration Dependency

In order to illustrate the concentration dependency on pcle deposition, three deposition courses at 3.00 mm are sho

FIG. 12. (a) The rate constant,k1, from Eq. [10] shown as a function oPeclet number for two Reynolds numbers (s) Re= 154 and (d) Re= 253.(b) Characterization of the degree of coverage where the logarithmic p

hestarts as a function of the logarithm of wall shear stress for Re= 253 (from theexperiment shown in Fig. 11).

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236 GORANSSON A

FIG. 13. (a) The particle concentration dependency on the degree of cage for three particle concentrations (d) 2× 109 cm−3, (s) 4× 109 cm−3, and(.) 9× 109 cm−3 at the same Reynolds number at the radial distance 3.00(b) A Langmuir-type pseudoisotherm with experimental data from (a) a13 h.

Fig. 13a. The particle bulk concentrations were 2× 109, 4× 109,and 9× 109 cm−3, with a volumetric flow rate of 7.3 cm3 min−1.For higher particle concentrations the driving force for depotion is much stronger and therefore a higher degree of coveis obtained. Let us define the level after 13 h deposition aspseudo-steady-state degree of coverage,θpss. In Fig. 13bθpss

from Fig. 13a is shown as a function of the particle bulk ccentration given as a Langmuir-type pseudoisotherm.

CONCLUSIONS

A method has been adopted for studying particle deposin situas a function of time. By radially scanning the surfaceis possible to obtain data on the deposition behavior at locat

exposed to different wall shear stress almost simultaneouslyin one experiment.

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ionit

ons

It was found that the initial and constant particle fluxmass-transfer-controlled under prevailing experimental cotions which was proved by that the particle diffusion constderived agreed reasonable well with that based on the StoEinstein relation and measured particle size. It was shownthe critical degree of coverage, where the time-dependentperiod commences, has a linear log-dependency of the wall srate. This kind dependency indicates that no single phenomcan explain this behavior.

The time-dependent deposition course shows a shear ratdegree of coverage dependency. The higher shear rate thedegree of coverage and the slower deposition rate was achiIn addition, it can be concluded that the final phase of the desition process follows first-order kinetics. The first-order mtransfer coefficient,k, was found to have a linear dependencyshear rate, suggesting blocking effects and particle drag-feffects as a limiting factor. The transport coefficient decreawith increasing shear rate. It was also shown that re-entrainmphenomena take place during the deposition course.

An empirical equation was developed to describe the deof particle coverage. The model contains a mass transfer cocient,k1, which relates to the particle deposition to the collecsurface. The mass transfer toward the collector surface decrethe more the flow is dominated by convection. This indicatesalthough the flux of particles to the vicinity of the surface is higthe conditions for the particles to deposit get more difficult. Tcould be due to increased blocking of the surface at higherues of Pe, that the particles do not have enough time to estacontact with the surface, or that the hydrodynamicaly controremoval forces become a dominant factor.

A higher particle bulk concentration causes the driving fofor deposition to become higher. Defining a pseudo-steady-sdegree of coverage shows a Langmuir-type pseudoisotherm

ACKNOWLEDGMENT

The Swedish Research Council for Engineering Sciences has gratefullyvided funding for this research.

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