Mechanisms responsible for sub-micron particle deposition in a laminar wall-jet

Mechanisms responsible for sub-micron particle deposition ina laminar wall-jet

Anders Goransson, Christian Tragardh *

Food Engineering Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, Lund SE-22100, Sweden

Received 4 December 2001; accepted 2 May 2002

Abstract

An experimental study was performed to investigate the dependency of particle size on the deposition rate.

Polystyrene latex particles of two radii (0.23 and 0.38 mm) were deposited onto a glass surface covered with indium-tin

oxide. Reduced deposition efficiency was observed for the larger particles, although an increase in the deposition

efficiency was seen as that the diffusion boundary layer became thicker. A critical degree of surface coverage, ucrit, was

defined as the fraction of surface coverage at which the linear variation of flux with time ended. When the two particle

sizes were compared, it was found that the value of ucrit depended only on the wall shear stress. This result indicates that

surface shielding is an important factor in the decline in particle flux. The deposition process was divided into two

separate processes, described by a mass-transfer coefficient and an adhesion rate coefficient. It was found that the

process was governed by mass-transfer during the initial period, but for higher degrees of surface coverage adhesion

became the rate-determining factor.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Wall-jet; Particle deposition; Deposition efficiency; Shielding

1. Introduction

It is important to have knowledge about the

steps governing deposition in a wide range of

different types of processes, e.g. in applications

where proteins bind to surfaces, or in adhesion of

bacterial spores in the food and pharmaceutical

industries [1�/5]. The process of deposition of

colloidal particles can be described schematically

in three steps: the first step consists of convective-

diffusive transport of particles from the bulk

solution to the surface; the second step involves

the adhesion of the particles to the surface under

favourable conditions; and finally, the third step

consists of detachment, which occurs if the re-

moval forces are sufficiently strong to overcome

the adhesion forces. At the beginning of the

deposition process, when the surface still offers a

large number of energetically favourable sites, the

attachment of particles is a linear process with

time. As the number of available sites becomes

fewer, the deposition rate declines. The critical

* Corresponding author. Tel.: �/46-46-222-9807; fax: �/46-

46-222-4622

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

Tragardh).

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point at which this non-linear period commences isstrongly flow-dependent, as was shown by Gor-

ansson and Tragardh [6].

The deposition of colloidal particles has been

investigated with different methods. Marshall and

Kitchener [7] employed a rotating-disc apparatus

to investigate the deposition of carbon black

particles under different electrostatic interactions

between particles and surface. In other investiga-tions, parallel-plate channels have been used

[5,8,9]. The stagnation-point flow collector was

developed by Dabros and van de Ven [10], with the

intention of observing particle deposition in situ.

Albery et al. [11] employed a system using a wall-

jet, originally used in the field of electrochemistry.

In both these latter systems, the deposition process

can be monitored without disassembling the appa-ratus. Another advantage of the two systems is

that analytical expressions can be used to calculate

the theoretical flux, flow velocities close to the

surface and the thickness of the diffusion bound-

ary layer, for example. In a study by Albery [12] an

analytical expression for the initial mass-transfer-

controlled deposition flux in a wall-jet was given

as:

j0�0:165D2=3V3=4

f n�5=12a�1=2r�5=4cb; (1)

where D is the diffusion coefficient in the bulk

solution, Vf the volumetric flow rate, n the

kinematic viscosity, a the diameter of the outlet

of the jet, r the radial position and cb the bulk

particle concentration. The validity of Eq. (1) has

been demonstrated in earlier studies [6,11,13,14]for particles with a diameter of about half a

micrometre.

As the deposition process enters the non-linear

phase, different mechanisms control the process

and contribute to the decrease in deposition rate.

One of these mechanisms is the phenomenon of

blocking, in which part of the free surface is no

longer available for nondeposited free-flowingparticles [5,10,14]. The deposited particles occupy

an area that is larger than their cross-sectional

area. Another factor contributing to the fall in

particle flux and which is responsible for the non-

linear temporal behaviour is the detachment of

deposited particles.

The aim of the present work was to determinethe deposition of sub-micron-sized polystyrene

latex particles which is governed by different

flow-related mechanisms. The effect of particle

size on deposition rate was also investigated, since

it is known that the bulk diffusion coefficient

decreases at short distances from a surface.

2. Theory

The phases of the particle deposition processhave been extensively described in the literature for

a range of different conditions. The presence of a

surface will change the conditions for particle

transport. As a particle reaches the vicinity of the

wall, it is affected by the surrounding fluid, and at

nanometre distances electrostatic interactions have

an influence on whether or not the particle will

adhere to the surface. When the particle hasadhered to the surface, the balance of force

between FH (the sum of the hydrodynamic inter-

actions) and FA (the sum of the colloidal interac-

tions) determines whether the deposited particle

will remain on the surface or detach [15]. If FH5/

FA no detachment will occur, but if FH�/FA the

hydrodynamic interactions will force the particle

to become detached from the surface. A briefsummary of the hydrodynamic forces that affect a

free-flowing particle close to a surface and a

deposited particle on a collector surface follows.

2.1. Hydrodynamic forces acting on a free-flowing

particle

A particle translating in a viscous fluid close to a

wall is affected by the surrounding liquid. In the

presence of a shear flow an inertial lift force will

affect the particle according to Saffman [16]:

Flift�6:46n1=2r(ur�up)r2pj dur

dz j1=2

; (2)

where r is the density of the fluid, ur and up the

velocity of the fluid in the radial direction and

particle velocity, respectively, rp is the particle

radius and z is the distance from the wall. The lift

force is proportional to r2p; implying a greater lift

A. Goransson, C. Tragardh / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 133�/144134

force for larger particles. The resistance to movingcloser to the wall increases in the direction normal

to the wall, as described by Brenner [17]:

F zdrag�6plmr2

p

duz

dz; (3)

where uz is the particle velocity normal to the walland l is a resistance coefficient expressed by

Dahneke [18] as a function of particle radius and

gap width:

l(rp; h)�1�rp

h; (4)

where h is the gap between particle and wall. The

resistance coefficient describes the increase in

resistance experienced by a particle when moving

towards a wall, due to the presence of the

molecules of the surrounding medium that must

be displaced. For large gaps the function ap-

proaches unity. In an experimental investigation

by Adamczyk et al. [19] the resistance coefficientfor a solid sphere moving towards a surface was

presented.

2.2. Hydrodynamic forces acting on a deposited

particle

A particle in creeping flow (RepB/1) in contact

with a wall is exposed to forces of various natures.One of the hydrodynamic forces involved in the

process is the viscous drag force parallel to the

surface, expressed as Stokes’ drag force

F rdrag�6pf mr2

p

dur

dzjy�rp

�6pf twr2p; (5)

with a factor f which compensates for wall effects

given as 1.7009 according to O’Neill [20]. The

torque acting on the deposited particle at the point

of contact is expressed by Sharma et al. [21] as

Tcontact�1:399F rdragrp�44:85twr3

p; (6)

with the drag force acting at a distance of 1.399

particle radii from the wall [20]. The torque is thus

proportional to twr3p: The lift force acting on a

stationary particle deposited on the wall was

expressed by Hubbe [22] as:

Flift�81:2m�1=2n�1=2t3=2w r3

p; (7)

where the inertial lift force is proportional to

t3=2w r3

p: In the case of particle Reynolds’ numbers

much smaller than unity, as in our case, the effect

of the inertial lift force on a deposited particle is

negligible [22].

Apart from the above-mentioned hydrodynamic

forces, there are also colloidal interactions of

various strengths and ranges, which act both onfree-flowing particles in the vicinity of the surface

and on deposited particles. The attractive

London�/van der Waals’ interaction energy and

the repulsive electrical double-layer energy are

both proportional to the particle radius.

3. Materials and methods

Particle deposition was studied in a wall-jet cell,

originally developed by Albery et al. [11]. The

experimental set-up is described in detail by

Goransson and Tragardh [6]. The flow cell, made

of Perspex, is shown schematically in Fig. 1. Fluid

was pumped, by self-pressure to avoid a pulsating

flow, perpendicular to a glass surface covered with

Fig. 1. Schematic illustration of the wall-jet cell (r , radial

position from stagnation point; a , the diameter of the jet

outlet).

A. Goransson, C. Tragardh / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 133�/144 135

indium-tin oxide (Donnelly Applied Films Cor-poration, Boulder, CO, USA). The cell was

mounted on two x -translator stages, which made

it possible to observe the particle deposition at

several positions, exposed to different wall shear

stresses, in one single experiment. The deposition

kinetics was analysed using total internal reflection

microscopy (TIRM). A 5 mW linearly polarized,

He�/Ne laser (Melles-Griot, Irvine, CA, USA) wasused to create the evanescent wave, directed at a

right angle towards a 708, right-angled prism

(BK7, n�/1.5151 at l�/632.8 nm). A head-on

photomultiplier tube (Hamamatsu R1104, Hama-

matsu, Japan), mounted on top of a light micro-

scope (Jena, Jena, Germany), equipped with an

object lens (Leica EF 32/0.40, Wetzlar, Germany)

with a long working distance (total magnification200�/) was used to measure the light intensity

from the deposited particles. The sampled data

were digitalized and transferred to a computer.

Using a calibration procedure, described pre-

viously [6], the intensity was transformed to the

number of particles per surface unit area.

Polystyrene latex particles of two radii, 0.23

(Sigma Latex Beads, Lot 57H1207, Sigma-AldrichCorp., St. Louis, MO, USA) and 0.38 mm (Sigma

Latex Beads, Lot 067H0291, Sigma-Aldrich Corp.,

St. Louis, MO, USA) were used in the deposition

experiments. The bulk particle concentrations

were 4�/109 cm�3 for the 0.23 mm particles and

8.7�/108 cm�3 for the 0.38 mm particles. These

concentrations were chosen to obtain the same

volume fraction in both cases. During an experi-mental run the bulk particle concentration was

determined by measuring the turbidity of the feed

at 700 nm, and the concentration was frequently

adjusted by adding fresh colloid solution to the

feed. The feed vessel was continuously stirred

throughout the whole experiment. All colloid

solutions were prepared with deionized, double-

distilled water (DDDW) from a Milli-Q plant(Millipore Co., Bedford, MA, USA). The electro-

lyte concentration was 50 mM KNO3, and a

citrate�/phosphate buffer solution was used to-

gether with HNO3 and KOH to set the pH to 5.5.

The volumetric flow rates were chosen so as to

obtain the same hydrodynamic drag forces for the

two particle sizes at corresponding radial positions

from the stagnation point. For the smaller parti-cles the flow rates were set to 0.12 and 0.20 cm3

s�1, corresponding to Reynolds’ numbers of 154

and 253 (Re�/ua /(2n ), where u is the velocity of

the fluid in the jet nozzle, a the diameter of the

nozzle and n the kinematic viscosity). For the

larger particles, the flow rates were chosen to be

0.07 and 0.12 cm3 s�1, with Reynolds’ numbers

equal to 94 and 156. All experiments were carriedout at a temperature of 219/1 8C.

4. Results and discussion

4.1. Mass-transport-controlled particle deposition

In Fig. 2(a),(b) the initial particle flux, linearlyincreasing with time, is shown as a function of

r�5/4 for the 0.38 and 0.23 mm-sized particles,

respectively. For the larger particles the volumetric

flow rate was 0.07 cm3 s�1 and for the smaller

particles it was 0.12 cm3 s�1. In both cases the

initial flux followed the r�5/4 dependency accord-

ing to Eq. (1). From the slope of the regression

lines fitted to the data points it is possible tocalculate the particle diffusion coefficients by

employing Eq. (1). The experimentally calculated

diffusion coefficient for the 0.23 mm-sized particles

was found to be 8.6�/10�9 cm2 s�1, based on the

data in Fig. 2(b). This is in good agreement with

the diffusion coefficient according to the Stokes�/

Einstein relation, which is 9.3�/10�9 cm2 s�1.

However, for particles with a radius of 0.38 mm theexperimentally calculated diffusion coefficient was

found to be only 27% of the theoretical diffusion

coefficient, which is calculated to be 5.6�/10�9

cm2 s�1. It can thus be concluded that Eq. (1) does

not describe the particle flux for the larger

particles under the prevailing conditions.

The particle flux onto a flat surface was shown

by Levich to be proportional to the cubic root ofthe wall shear stress. In Fig. 3 the initial flux of the

small particles is shown as a function of the cubic

root of the wall shear stress, t1=3w ; for the two

volumetric flow rates, 0.20 and 0.12 cm3 s�1. It

can be seen that, for both cases, good correlation

was obtained. Similar behaviour was seen by


Parsons et al. [23] when studying the initial flux of

polystyrene particles of 0.13 mm radius.

The initial experimental flux, j0,exp, of the

particles with a 0.38 mm radius is shown as a

Fig. 2. The initial particle flux, linear with time, j0,exp, as a function of r�5/4 with r being the radial distance from the stagnation point.

(a) The flux of 0.38 mm particles, with a bulk particle concentration of 8.7�/108 cm�3 and volumetric flow rate of 0.07 cm3 s�1. (b)

Particle radius of 0.23 mm, with a bulk particle concentration of 4�/109 cm�3 and volumetric flow rate of 0.12 cm3 s�1.


function of wall shear stress, tw, for a volumetric

flow rate of 0.12 cm3 s�1 in Fig. 4(a). The

theoretical flux, j0,theo, calculated from Eq. (1), is

also included and it can be seen that the experi-

mental flux was considerably lower than that

predicted by theory. By introducing the deposition

efficiency, x , defined as the ratio of the experi-

mental initial flux to the theoretical initial flux

(x�/j0,exp/j0,theo), the reduction in the initial parti-

cle flux can clearly be seen. The reason for this

discrepancy between theory and experiment is that

the initial particle flux equation, Eq. (1), is based

on Smoluchovski�/Levich approximation, i.e. the

reduced hydrodynamic mobility of a particle

approaching a surface, as shown by Brenner in

Eq. (3), is assumed to be balanced by attractive

van der Waals’ forces. As was shown in Fig. 2(b),

the Smoluchovski�/Levich approximation is valid

for the smaller particles, but not for the larger

particles. According to Eq. (3) the hydrodynamic

drag force is proportional to the square of the

particle radius. The van der Waals’ interaction, on

the other hand, shows a linear relation to the

particle radius. It thus follows that for larger

particles the hydrodynamic resistance dominates

over the attractive forces, resulting in a reduction

in the particle flux to the deposition surface. The

deposition efficiency is strongly dependent on the

wall shear stress, which can be clearly seen in Fig.4(a), as well as the particle size. The deposition

efficiency in Fig. 4(a) increased with decreasing

wall shear stress, as described by a power-law fit:

x(tw)�0:29t�0:21w : (8)

The deposition efficiency is proportional to t�0:21w ;

which resembles the results of van de Ven andMason [24], who investigated the coagulation

efficiency of spheres exposed to a shear flow. The

coagulation efficiency was proportional to/g�0:18 (/g

being the wall shear rate). This trend was also

found in an investigation by Sjollema and

Busscher [5], in which the predicted deposition

efficiency increased at lower Peclet numbers (low

wall shear stresses). However, they were not ableto see the increase in deposition efficiency in their

experiments at low Re numbers, since the diffusion

boundary layer was too thick (10 mm). In Fig. 4(b)

the deposition efficiency is shown as a function of

the thickness of the diffusion boundary layer for

the data in Fig. 4(a). The diffusion boundary layer

Fig. 3. The initial particle flux, j0,exp, as a function of the cubic root of the wall shear stress, t1=3w ; for the smaller particles at volumetric

flow rates of 0.20 cm3 s�1 (m) and 0.12 cm3 s�1 (j).


thickness, which is defined as a region close to the

collector surface where a sudden change in the

concentration profile occurs, has been defined for

a wall-jet [25] according to:

ddbl�5:8sp3=4D1=3a1=2n5=12r5=4V�3=4

f ; (9)

where the constant s equals 0.17, and was

experimentally determined by Yamada and Mat-

suda [26]. By adopting Eq. (9) the diffusion

boundary layer thickness was estimated to vary

from 1 to 3 mm. The deposition efficiency in-

creased linearly as the diffusion boundary layer

Fig. 4. (a) The initial particle flux, j0,exp, as a function of wall shear stress. (m) Data from an experiment carried out at a volumetric

flow rate of 0.12 cm3 s�1, a bulk particle concentration of 8.7�/108 cm�3 and a particle size of 0.38 mm; (j) theoretical particle flux

calculated from Eq. (1); (') the deposition efficiency; and the dashed line represents the linear fit to the deposition efficiency data. (b)

The deposition efficiency versus the diffusion boundary layer thickness at a volumetric flow rate of 0.12 cm3 s�1. The particle radius

was 0.38 mm.


became thicker. This behaviour can be explainedby the fact that when the convective contribution

is greater, i.e. the boundary layer is thin, the

particles do not have time to establish contact on

the collector surface, but are dragged away by the

fluid flow. On the other hand, in the case of an

increasing diffusive contribution in particle trans-

port, the free-flowing particles can more easily

reach the collector surface undisturbed by thesurrounding fluid.

4.2. Particle deposition controlled by adhesion

Fig. 5(a) shows typical deposition with time data

for an experiment carried out with particles of 0.38

mm radius, and a volumetric flow rate of 0.12 cm3

s�1 at a radial position 4.00 mm from the

stagnation point. Particle deposition was moni-

tored for 10 h, reaching a degree of surface

coverage of approximately 0.46, which corre-sponds to a particle wall concentration of 1.0�/

108 cm�2. The linear deposition step discussed in

the previous section, prevailed up to a certain

degree of surface coverage, ucrit (the degree of

coverage is defined as cwp2p; where cw is the particle

wall concentration). This critical degree of cover-

age is typical for the flow conditions at the specific

position shown in the figure, which in this case wasapproximately 0.2. The mass-transfer-controlled

step is followed by a deposition phase that is non-

linear with time. The reason for the fall in the

particle deposition flux may be, for example, that

particles on the surface occupy a given area, thus

preventing free-flowing particles from reaching the

surface. The phenomenon is referred to in the

literature as shielding and is strongly flow-depen-

dent, as was demonstrated by Goransson and

Tragardh [14]. The critical degree of surface

coverage, ucrit, can be interpreted as the degree

of surface coverage by deposited particles and

their ‘‘shadows’’ when the non-linear deposition

phase begins. It is then possible to estimate the

magnitude of the area shadowed by a deposited

particle. In Fig. 5(b) the ratio of the excluded area,

Aexcl, to the particle cross-sectional area, ap, is

shown for a range of wall shear stresses for both

particle radii. The experimentally determined ex-

cluded area ratio increased with increasing wall

shear stress. For the flow conditions prevailing in

Fig. 5(a), with a wall shear stress of 0.08 Nm�2,

the excluded area can be estimated to be seven

cross-sectional areas. For higher wall shear stres-

ses, 0.5�/0.9 Nm�2, the size of the excluded area

for the smaller particles reached as high as 25�/30

particle radii. As the deposition process proceeds

the shielded areas will, to some extent, become

occupied by particles, due to factors such as

Brownian diffusion and collisions between free-

Fig. 5. (a) Experimental data for 0.38 mm particles at a radial distance from the stagnation point of 4.00 mm. The wall shear stress at

this distance is 0.08 Nm�2. The volumetric flow rate was 0.12 cm3 s�1 and the bulk particle concentration was 8.7�/108 cm�3. The

degree of coverage at the onset of non-linear particle flux, ucrit, is indicated in the figure. (b) An experimental estimate of the excluded

area is shown as a function of wall shear stress for both particle sizes, (m) 0.23 mm, (k) 0.38 mm.


flowing particles, and thus the particle wall con-

centration will continuously increase.

Shielding is not the only phenomenon believed

to be responsible for the decline in particle flux.

Another reason for the flux decrease is the

detachment of deposited particles. This detach-

ment may be caused by free-flowing particles

colliding with deposited ones, or removal by the

shear flow acting on the deposited particles,

according to Eq. (5). In Section 2.2 the various

forces affecting particles deposited on a flat sur-

face were presented. By plotting the critical degree

of coverage, ucrit, against twr2p (drag force), twr3

p

(torque), t3=2w r3

p (lift force) and tw (wall shear stress)

the dependency of ucrit was investigated for both

particle sizes (Fig. 6(a�/d)).

Fig. 6(a) shows the critical degree of coverage,

ucrit, as a function of twr2p; i.e. the hydrodynamic

drag force acting parallel to the collector surface

on the deposited particle, according to Eq. (5). The

data shown in the figure are the mean values of

three experiments each at volumetric flow rates of

0.20 cm3 s�1 (rp�/0.23 mm) and 0.12 cm3 s�1

(rp�/0.38 mm). A mutual linear regression fit to

the data for the two particle sizes resulted in a

correlation coefficient of �/0.84. It can be seen that

Fig. 6. The dependency of the end of the initial period of linear flux with time, ucrit, on different phenomena is shown. Mean values are

given for three experiments each, carried out at volumetric flow rates of 0.20 cm3 s�1 (0.23 mm) and 0.12 cm3 s�1 (0.38 mm). (a) ucrit

versus twr2p (drag force), (m) 0.23 mm; (j) 0.38 mm. (b) ucrit versus twr3

p (torque), (m) 0.23 mm; (j) 0.38 mm. (c) ucrit versus twr3=2p (lift

force), (m) 0.23 mm; (j) 0.38 mm. (d) ucrit versus tw (wall shear stress), (m) 0.23 mm; (j) 0.38 mm.


the data for the two particle sizes converge as thedrag force increases. Fig. 6(b) shows the signifi-

cance of the torque on ucrit. The mutual regression

line resulted in a correlation coefficient of �/0.63.

In Fig. 6(c), ucrit is plotted against the lift force,

with a correlation coefficient of �/0.84. The

dependencies of ucrit on the lift force and drag

force are thus higher than that on the torque

acting on the deposited particles. In Fig. 6(d), ucrit

is plotted against the wall shear stress. The data

for both particle sizes are gathered on one line,

suggesting that ucrit is solely dependent on the wall

shear stress. The correlation coefficient was �/

0.97. It also follows that the degree of coverage

when the process becomes non-linear with time

does not depend on the particle size. It might also

indicate that the hydrodynamic forces discussed inSection 2.2 do not play significant roles in the

onset of non-linear deposition phase. Instead, it is

indicated that the shielding phenomenon plays an

important role, being responsible for the decrease

in the particle deposition flux, confirming earlier

works in the area such as Sjollema and Busscher

[5], Dabros and van de Ven [10], Adamczyk et al.

[27], among others.The introduction of a dimensionless particle

wall concentration, cw;crit� ; makes it possible to

compare the four different experimental cases

with volumetric flow rates ranging from 0.07 to

0.20 cm3 s�1. The dimensionless concentration is

expressed with the aid of the bulk particle con-

centration, the bulk particle diffusion coefficient

and the volumetric flow rate:

c+w;crit�cw;critD

cbVf

: (10)

In Fig. 7 the dimensionless particle wall concen-

tration at the end of the linear deposition period is

shown as a function of the wall shear stress for the

experimental conditions studied. From the relation

that describes the wall shear stress dependency,expressed as a logarithmic expression:

log c+w;crit� (�9:86�0:85)log tw; (11)

it is possible to estimate the particle wall concen-

tration at which non-linear deposition commences

for a wide range of wall shear stresses.

4.3. The overall deposition process

The overall deposition rate constant, k , can be

divided into two separate processes, which are

governed by a mass-transfer rate coefficient, kmass,

and an adhesion rate coefficient, katt, expressed as:

1

k�

1

kmass

�1

katt

: (12)

The mass-transfer rate coefficient for the wall-jet is

Fig. 7. The dimensionless particle wall concentration (cw,critD /

(cbVf)) as a function of the wall shear stress*/(m) 0.23 mm,

Vf�/0.20 cm3 s�1; (k) 0.38 mm, Vf�/0.12 cm3 s�1; (j) 0.23

mm, Vf�/0.12 cm3 s�1; (I) 0.38 mm, Vf�/0.07 cm3 s�1.

Fig. 8. Rate coefficients from an experiment carried out with

the smaller particles at a volumetric flow rate of 0.20 cm3 s�1 at

3.50 mm from the stagnation point. The overall rate coefficient,

k , (m), the adhesion rate coefficient, katt, (k) and the mass-

transfer coefficient, kmass (dashed line).


expressed by Eq. (1). In Fig. 8 the three rate

constants are given as a function of the degree of

coverage from an experiment carried out at a

volumetric flow rate of 0.20 cm3 s�1, with a

particle radius of 0.23 mm. The radial position

was 3.50 mm from the stagnation point. It can be

seen that at the beginning of the deposition process

the rate-determining factor is the mass-transfer

rate, but as the particle surface coverage exceeds a

critical level, adhesion becomes the rate-determin-

ing factor.

Fig. 9(a) shows a first-order kinetics plot for the

data shown in Fig. 8. It can be seen that particle

deposition starts to follow first-order kinetics after

approximately 4 h of deposition, at which time the

degree of coverage has reached 0.26. It then

continues until the end of the experiment. Thegradient can be interpreted as the transfer coeffi-

cient in Eq. (13):

du

dt�k(1�u): (13)

Fig. 9. (a) A first-order kinetics plot for the particles of rp�/0.23 mm at a radial distance of 3.50 mm and wall shear stress�/0.34

Nm�2. The first-order period did not commence until approximately 4 h after the start of the experiment and lasted to the end of the

experiment. (b) A first-order kinetics plot of the data shown in Fig. 5(a) at a radial distance of 4.00 mm from the stagnation point and a

wall shear stress of 0.08 Nm�2. The first-order kinetics period lasted for approximately 5.5 h. (c) The dependence of the degree of

coverage at which the first-order kinetics period ends on the wall shear stress for 0.38 mm particles. Experimental conditions are given

in Fig. 5(a).


In Goransson and Tragardh [6] it was shown thatk decreases with increased Peclet number (convec-

tion/diffusion ratio).

A first-order kinetics plot for the larger particles

(Fig. 9(b)) shows different behaviour from that of

the smaller particles. The first-order kinetics of the

experimental data presented in Fig. 5(a) is shown.

It can be seen that the deposition process began

with first-order deposition kinetics and lasted forapproximately 5.5 h. The degree of coverage at

which first-order kinetics ceases to control the

deposition is shown in Fig. 9(c) as a function of

wall shear stress for the same experiment. The

period of first-order kinetics lasts longer for those

parts of the surface that are exposed to lower wall

shear stresses.

5. Conclusions

It was demonstrated in this study that the

deposition efficiency was strongly dependent on

wall shear stress and particle size. It was concluded

that hydrodynamic resistance dominates over

attractive forces, resulting in a reduction in mass-transfer for the larger particles.

A critical degree of surface coverage at which

the linear variation in flux with time ended, ucrit,

was introduced. The reason for the change from

linear to non-linear flux was investigated by

studying the effects of different hydrodynamic

forces on the onset of the non-linear flux phase.

It was concluded that ucrit was strongly dependenton wall shear stress, but not on the hydrodynamic

forces investigated, which may indicate that sur-

face shielding plays an important role in the

decline in particle flux.

By introducing a dimensionless particle concen-

tration it was possible to estimate the particle wall

concentration at which the non-linear deposition

commences for a wide range of wall shear stresses.By separating the deposition process into two

consecutive steps, a mass-transfer step and an

adhesion step, it was possible to demonstrate

how the process changed from being governed by

mass-transfer to becoming controlled by the

adhesion rate.

Acknowledgements

The Swedish Research Council for Engineering

Sciences has gratefully provided funding for this

research.

References

[1] R.B. Dickinson, S.L. Cooper, AIChE J. 41 (9) (1995) 2160.

[2] D. Kim, W. Cha, R.L. Beissinger, J. Colloid Interface Sci.

159 (1993) 1.

[3] C.A.-C. Karlsson, M.C. Wahlgren, A.C. Tragardh, J.

Food Eng. 30 (1996) 43.

[4] R.B. Dickinson, J. Colloid Interface Sci. 190 (1997) 142.

[5] J. Sjollema, H.J. Busscher, J. Colloid Interface Sci. 132 (2)

(1989) 382.

[6] A. Goransson, C. Tragardh, J. Colloid Interface Sci. 231

(2000) 228.

[7] J.K. Marshall, J.A. Kitchener, J. Colloid Interface Sci. 22

(1966) 342.

[8] F. Vasak, B.D. Bowen, C.Y. Chen, F. Kastanek, N.

Epstein, Can. J. Chem. Eng. 73 (1995) 785.

[9] Z. Adamczyk, T.G.M. van de Ven, J. Colloid Interface Sci.

80 (2) (1981) 340.

[10] T. Dabros, T.G.M. van de Ven, Colloid Polym. Sci. 261

(1983) 694.

[11] W.J. Albery, G.R. Kneebone, A.W. Foulds, J. Colloid

Interface Sci. 108 (1985) 193.

[12] W.J. Albery, J. Electroanal. Chem. 191 (1985) 1.

[13] S. Joscelyne, C. Tragardh, J. Colloid Interface Sci. 192

(1997) 294.

[14] A. Goransson, C. Tragardh, Colloid Surf. A (submitted

for publication).

[15] J. Visser, J. Colloid Interface Sci. 55 (3) (1976) 664.

[16] P.G. Saffman, J. Fluid Mech. 22 (2) (1965) 385.

[17] H. Brenner, Chem. Eng. Sci. 16 (1961) 242.

[18] B. Dahneke, J. Colloid Interface Sci. 48 (1974) 520.

[19] Z. Adamczyk, M. Adamczyk, T.G.M. van de Ven, J.

Colloid Interface Sci. 96 (1) (1983) 204.

[20] M.E. O’Neill, Chem. Eng. Sci. 23 (1968) 1293.

[21] M.M. Sharma, H. Chamoun, D.S.H.S.R. Sarma, R.S.

Schechter, J. Colloid Interface Sci. 149 (1) (1992) 121.

[22] M.A. Hubbe, Colloids Surf. 12 (1984) 151.

[23] D. Parsons, T. Harrop, E.G. Mahers, Colloids Surf. 64

(1992) 151.

[24] T.G.M. van de Ven, S.G. Mason, Colloid Polym. Sci. 255

(1977) 468.

[25] H. Gunasingham, B. Fleet, Anal. Chem. 55 (8) (1983)

1409.

[26] J. Yamada, H. Matsuda, J. Electroanal. Chem. 44 (1973)

189.

[27] Z. Adamczyk, B. Siwek, L. Szyk, J. Colloid Interface Sci.

174 (1995) 130.


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Mechanisms responsible for sub-micron particle deposition in a laminar wall-jet