7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Effect of Interface Contamination on ParticleBubble Collision Behavior
Z. Huang *t, D. Legendre and P. Guiraud t
University de Toulouse; INPT, UPS; IMFT (Institut de M6canique des Fluides de Toulouse)
All6e Camille Soula, F31400 Toulouse, France
CNRS ; IMFT ; F31400 Toulouse, France
t University de Toulouse; INSA, UPS, INP; LISBP, 135 Avenue de Rangueil, F31077 Toulouse, France
INRA, UMRA792 Ing6nierie des Systemes Biologiques et des Proc6d6s, F31400 Toulouse, France
CNRS, UMR5504, F31400 Toulouse, France
pascal.guiraud@insatoulouse.fr
Keywords:ParticleBubble Collision, Surface contamination, Direct numerical simulation, Inertial forces
Abstract
Particlebubble collision in flotation involves the effects of gravitation, interception, inertia, turbulence, Brownian
diffusion and interfacial forces.Concerning the hydrodynamic aspect of collision, major attention has been paid to two
extreme situations: bubbles with clean surface or fully contaminated bubbles. However depending on the contaminant
concentration, bubble's surface may be partially contaminated: the front of bubble is mobile and the rear is immobile.
The distribution of contaminant on the bubble surface depends on the tangential advection and diffusion along the
interface. When the tangential advection dominates the diffusion, surfactants accumulate at the rear part of the bubble.
Many authors have experimentally, computationally or analytically studied the effects of surfactant on bubbles motion
(Cuenot et al., 1997). The interface mobility has to be considered into the collision modeling (Legendre et al., 2009).
This study focuses on the impact of the interface contamination the on the collision between bubbles and inertial
particles. This situation is observed in most of the flotation processes. The work presented here is limited to spherical
bubbles rising in a suspension composed of small solid particles. The bubble's surface mobility has been integrated
into the collision modelling by using the hydrodynamics stagnantcap model, in which the angle Oc,,,, is introduced
to characterise the interface contamination level. Direct Numerical Simulation has been performed for various
bubble's Reynolds numbers (1 < Reb < 100) and particle's Stokes numbers (0.001 < Stp < 1). The NavierStokes
equations has been firstly solved for the flow field around the bubble. The Lagrangian tracking was performed for
the solid particles by solving the full particle trajectory equation, in order to find the critical grazing trajectory. The
collision efficiency was then calculated, according to its definition, as the ratio of the number of the particles located
in the body of revolution made by the critical grazing trajectory to the particles located in the cylinder formed by
bubble's projection area. The computational results show that particle flux near the surface is controlled by the
tangential velocity for the mobile part of the interface and by the velocity gradient for the immobile part. As a result,
the surface contamination gives an important effect on the behavior of collision efficiency, especially near 0,clea1.
There is a critical angle Ocrit, if Oclean < Ocrit, the collision may occurs both on the mobile and the mobile part of
the surface and only the positive inertial effect is observed. While if Octean > Oc it, the collision occurs only on the
mobile interface, the negative inertial effect due to the nonzero tangential velocity may greatly reduce the collision
probability.
Nomenclature P pressure (N.m 2)
Q flowrate (m3.s1)
r rayon (m)
Roman symbols r rayon (n)
Re Renolds number ()
Eoll collision efficiency ()Re Ros number
Sgravitational constant (m.s St Stokes number ()
g gravitational constant (m.sl )
U fluid velocity (m.s 1)
V particle velocity (m.s1)
Greek symbols
0c collision angle ( )
Ocean clean angle ( )
p dynamic viscosity (Pa.s)
p density (kg.n 3)
2 Vorticity (s 1)
q4 trajectory ()
7 stream function (n2.s 1)
Subscripts
b bubble
f fluid
p particle
Superscipts
max maximum
crit critical
Introduction
The capture of small particles in suspension by bubbles,
called sometimes heterocoagulation, combines the dy
namics of collision with the thermodynamics of interfa
cial forces linking the bubble and particles forming an
aggregate. An overall capture efficiency, usually defined
as the ratio between the number of particles captured by
a bubble and the number of particles in the volume swept
out by the bubble, is generally considered as the product
of the contributions of three successive steps (Schulze,
1989; Ralston etal., 2002): collision, attachment and
particlebubble aggregate stability. Since the governing
forces for each step are independent, they can be treated
separately to simplify the modelling of each process.
According to Schulze (1989), particulebubble col
lision mechanisms includes interception, gravitational
sedimentation, inertial collision, brownian diffusion, tur
bulent diffusion and cloud effect. Collision may be dom
inated by one or several mechanisms, depending on the
liquid flow around bubble, particle's weight and density.
In most particlebubble collision studies, it is often sup
posed that the particle size is very small compared to
the bubble size. This assumption leads easily to the case
of interception collision where particles' inertia is ne
glected, so the particles trajectory can be simplified to
be assimilated to the liquid streamlines. Based on this
assumption, the first collision models haved been es
tablished, where the collision efficiency was supposed
to be a function of the particles to bubble size ratio
and of bubble's Reynolds number (Sutherland, 1948;
Gaudin, 1957; Yoon & Luttrell, 1989; Nguyen & Kmet,
1992; Heindel & Bloom, 1999). Weber (1981);
Weber& Paddock (1983); Nguyen (1994, 1998) have
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
successivelly proposed efficiency expressions, where the
effect of gravitional sedimentation has been considered.
Inertial effect on collision efficiency was examinated an
alytically by Flint & Howerth (1971) and later numeri
cally by Dobby & Finch (1987) as a function of parti
cles Stokes number St 2pUbr /9trb for Stokes,
potential and intermediate flows respectively. The colli
sion efficiency was shown to be significantly increased
by particles inertia when St > 1. Dukhin et al. (1995)
revealed the negative inertial effect, shown as a "cen
trifugal force" on collision efficiency, which is related
to the tangential component of flow rate at the interface.
However, the overall collision efficiency is not always a
simple summation of these effects. In particular, when
the collision efficiency is controlled by two or more ef
fects, a complex relationship is needed to predict the be
havior of the collision process (Schulze, 1989). Nguyen
(1998) used Taylor expansion to solve liquid flow around
bubble up to Reb = 500 and accounting for inertial
and gravity effects on the trajectories of solid particles.
More recently, in Phan et al. (2003) and Nguyen & V
(2009), the BassetBoussinesqOseen equation for parti
cles is solved for clean and fully contaminated bubble by
taking into consideration the effect of particle density.
The work cited above deal with the collision effi
ciency and major attention have been paid to two ex
treme situations: bubbles with clean surface or fully con
taminated bubbles. However, a bubble rising in a solu
tion containing surfactant or other impurities accumu
lates them on its surface. Depending on the contaminant
and impurities concentration and the adsorption capac
ity, the interface can be totally or partially recovered.
Thanks to the analytical solution in Stokes flow obtained
by Sadhal & Johnson (1983), many authors have experi
mentally, computationally or analytically studied the ef
fects of surfactant on bubbles motion for low to moder
ate Reynolds numbers (Bel Fdhila & Duineveld, 1996;
McLaughlin, 1996; Cuenot et al., 1997). The numer
ical study of Sarrot et al. (2005) clearly show that the
flow field around a bubble is strongly influenced by the
level of contamination resulting in a significant effect
on the behavior of the probability of collision.Based on
numerical simulations and hydrodynamical arguments,
Legendre et al. (2009) give a modelling of the effect of
the partial contamination of the bubble interface. How
ever, these studies are limited to the inertial free parti
cles. For a partially contaminated bubble, probability
behaviors are given by the flux of particles near the sur
face which is controlled by the tangential velocity for
mobile interfaces and by the velocity gradient for im
mobile interfaces. When inertial forces are considered,
their effect may be different from the mobile to the im
mobile interface, the collision probability behavior as a
result maybe also changed.
In this paper, we focus on the collision aspect between
a spherical bubble and inertial particles in suspension,
with the emphasis on the effects of the inertial force and
the gravitational sedimentation on the collision proba
bility for different bubble surface contamination levels.
Numerical methodology
Statement of problem. The study of Moruzzi & Reali
(2010) show that in the contact zone of DAF (Dissolved
air flotation), the distance between bubbles is about ten
to twenty times bubble diameter, so that the problem
may be simplified to (i) an isolated spherical bubble of
radius rb rising straightly at a steady velocity Ub and
to (ii) small spherical particles at their settling veloc
ity, both in a liquid at rest and of infinite extent, as it is
schematically shown in Fig. 1. The bubble can be consid
ered as a nondeformable sphere and its trajectory keeps
rectilinear, as long as the air bubble diameter doesn't ex
ceed 0.9 mm in water (Duineveld, 1995).
C,
r  K ^ \
Figure 1: Schematic view of Particlebubble collision
Governing Equations. The fluid is Newtonian and its
local velocity and pressure are denoted by U and P, re
spectively. The incompressible flow is governed by the
NavierStokes equations:
VU p (D + UVU)
V (PI + T) (1)
with pf and pf denoting the density and the viscos
ity of the fluid, respectively. This problem can be
characterized by Reynolds numbers defined as Rep
2r. ,, ITf Vpl/pf for solid particles and Reb
2rbpfUb/1f for the bubble. If we consider a bubble
rising in a suspension of particles, the ratio between
the particle settling velocity Us and the rising bubble
velocity Ub scales as Us/Ub (rp/rb)2, so that in
the limit rp/rb 0, the velocity perturbation im
posed to the sedimentation of each particle is small com
pared to the velocity field imposed by the rising bubble.
Bloom & Heindel (1997) show that under the condition
rp/rb < 0.05, bubble's motion is not affected by the
presence of particles.
A particle moving in a fluid experiences several
forces: gravity, buoyancy, drag, inertia and added mass,
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
shear lift and BassetBoussinesq history. Usually the
equation of Martin & Riley (1983) is used to describe
the particle's trajectory. Whereas, at a short separation
distance (about submicro), the interfacial forces as the
electrostatic force, Van der Waals forces and other non
DLVO force may be involved in the interaction between
the particle and the bubble surface. So the full particle
trajectory equation can be described as below:
47rr3 dV
3 dt
47T 4r7T
+cp ,., 
L
4f0, D4 D7(U dV)
+ C 
3 Dtp dV
p
4Pr3 DU
3 DL
"CiP 3(U7) x Q
+ 67prp o K(t
) ds
(2)
where Cd, CM et Ci are respectively the drag, the
added mass and the lift coefficients, K is the ker
nel of the Basset history force. E F,,,f is the sum
of the surface forces that depend only on the parti
cles physicochemical properties, particles size and the
separation distance. In this expression, d/dt and D/Dt
denote the time derivatives taken along the particle path
and the continuous phase path, subscript I/" denotes
continuousphase conditions evaluated at the particle lo
cation in the absence of the particle. However, the lift
force and the history force are usually neglected, be
cause they are expected to be of a secondaryorder effect
comparing with other forces (Nguyen, 2003), which was
confirmed by our numerical simulations.Considering a
bubble of dp = 1 mm and rp/rb 0.01, interfacial
forces are insignificant except for a short separation dis
tance h < 0.001rb. This distance corresponds to about
100 nm. We noted that under this distance, the interfa
cial forces dominate the interaction between the bubble
and the particles, corresponding to the attachment pro
cess. Finally, the simplified dimensionless form of the
particle trajectorie (2) is written as:
Stp (Sp ( D
(u v) + Us (3)
with T Ubt/rb, p PplPf, U = U/Ub, v V/Vb
and = Us/Ub. In Eq.3, the first term on the left
hand side characterises the particle's inertia, while the
second represents the inertial force of the liquid flow.
The Stokes number is defined as the particle's relaxation
time Tp to the characteristic time of the disturbed flow
Tb 2rb/Ub. For Stokes conditions (Re,  0), the
drag coefficient is Cd 24/Rep that yields:
2(pf pp)r4g
2 2
2St (p + Cpf)r (6)
p PIfrb
For bubbles and particles that are of interest in flotation
(0.1 < db < 1 mm and dp = 1 ~ 100 pm), Us and
Stp roughly remain between 10 4 and 1, especailly in
water treatment and mineral recovery. So, in this work,
the direct numerical simulation are performed for 1 <
Reb < 100, 0 < 01ea < 180, 104 < Stp < 1 and
103 < u < 101.
Surfactant adsorbed at the bubble interface or cap
tured particles migrate along the interface to the rear
stagnation point due to the liquid motion. In the stag
nant cap model (Sadhal & Johnson, 1983), the resulting
bubble surface contamination is characterized via the
angle 0cl,,, limiting contaminated and clean areas (see
Fig. 1): the forward part of bubble surface (0 > Ocea,1),
free of contaminants, move with the liquid ; while the
backward bubble surface (0 < 0 l,,,), covered by the
contaminants, behaves as "Stagnant cap". For solid
spheres (fully contaminated bubbles), the wake presents
a steady axissymmetric vortex for Reb < 20, loses its
axisysmmetry at Reb = 210 where two vortex fila
ments appears. For a clean spherical bubble, no vortex
appears and the wake is steady and axisymmetric even
at large to infinite Reynolds numbers. Indeed path
instability and vortex shedding behind a spherical bulle
is due to the contamination of bubble surface.
Computational method. Numerical computation re
ported below were realized by solving the NavierStokes
and particle trajectory equations for incompressible liq
uid. Note that in this study we only consider axi
symmetrical simulations. The computational domain at
tached to the bubble is a polar domain (r, 0). Differ
ent boundary conditions are imposed on the computa
tional domain. A sysmmetry condition is imposed on
the zaxis. On the outer boundary, the inflow veloc
ity Ub is imposed upstream (0 < 0 < 90) and a
parabolic approximation of the governing equation al
lowing the flow to leave freeely the domain without in
ducing significant perturbations is imposed downstream
(90 < 0 < 120 ): 02p/t,. "1 et 02U/On2.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
At the bubble surface, the Stagnant Cap Model
(Sadhal & Johnson, 1983) is used to model the surface
contamination. For the mobile part of the bubble surface
(Oclean < 0 < 180), a zero normal velocity and zero
tangential stress condition are imposed: U.n 0 and
n x (r.n) 0; while for the immobile part (0 < 0 <
Ocle,,), a noslip condition is imposed on the interface:
U 0. Validation of the hydrodynamic simulations has
been performed via comparisons with literature results
concerning the drag coefficient calculation as a function
of 0cl,,, that demonstrate the accuracy of hydrodynam
ics simulations. Particle trajectory equation is integrated
in time. The fluid velocity and gradient are interpolated
at the particle location using a second order accuracy in
terpolation.
If the concentration of particles is uniform in the liq
uid, the collision efficiency EC011 canbe calculated as the
ratio of the flux of particles which collide with the bub
ble surface and the flux passing through a cylinder of
section r : Ecoll Q(c/7rTUb. The first term is calcu
lated by searching the "grazing trajectory" Ic that sep
arates the trajectories which encounter the bubble from
those that do not (Schulze, 1989). Only the particles lo
cated in the body of revolution made by the grazing tra
jectories can encounter the bubble. So Qc corresponds
to the flow rate passing through the cross section area of
radius r, of the grazing trajectories upstream far from
the bubble: Qc = rUb. The collision efficiency Eco11
can be then written as follows:
Ecol (7)
rb}
This grazing trajectory can be obtained by searching the
contact point between the bubble surface and particle
trajectory. This point is found by trialanderror, vary
ing the initial particle position (ro, 0o) far away from
the bubble, where the particle trajectories are parallel to
the bubble symmetrical axis and not influenced by the
bubble's motion. As this initial position was moving to
the symmetric axis of the bubble, the numerical calcu
lation for obtaining particle trajectories and the check
of the minimal distance between the bubble surface and
the particles center was repeated, until this distance is
less than the particle radius rp.
Results and discussions
Clean or fully contaminated bubble
Fine particles with a density close to that of the liquid
(Stp 0, us 0), follow totally the liquid stream
lines (V = U), so the collision takes place only by
interception. The collision efficiencies increases with
Reb, which is a direct consequence of the streamlines
contraction near the bubble surface. E,,11 is found to
be a quadratic function of bubble size ratio rp/rb for a
fully contaminated bubble and rather a linear function of
rp/rb for a clean bubble Sarrot et al. (2005).
100
101
102
102 Stp 101
(a) clean bubble
106
10
102 Stp 101
(b) fully contaminated bubble
Figure 2: Collision efficiency (Eco11) v.s. Stokes num
ber (Stp) at Reb = 100. : rp/rb 0.02,
S: rp/rb 0.01, *: rp/rb 0.005, o:
rp/rb 0.002, +: rp/rb 0.001.
Inertial forces are usually neglected because of the tra
ditionally used assumption of small particles size and
bubbles with immobile surface where both liquid and
particle movement is largely reduced. The inertial forces
have been generally overlooked for the medium size par
ticles and bubbles with high velocity (Yoon & Luttrell,
1989; Dobby & Finch, 1987). By taking into consid
eration the bubble surface mobility, Dai et al. (1998);
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Nguyen (1999) have shown successively the negative in
fluence of the inertial forces of the liquid flow at a clean
bubble surface on the particlebubble interaction, that
causes a decrease in the collision efficiency. However, as
shown in Eq. 3, when the particle's density increases, its
inertia overcomes the liquid inertia. A different behavior
of Eco,, is expected. In this section, we try to examine
the global influence of inertial forces on particlebulle
collision by varying the Stokes number. The behavior of
the collision efficiency Eco,, is firstly analysed without
taking into account the gravitational forces (us 0).
The evolution of the collision efficiency E,,o1 and the
collision angle Oc versus the Stokes number Stp is plot
ted in Fig. 2a and 2b for a clean and a fully contaminated
bubble, respectively with the bubble's Reynolds number
Reb = 100. The particle to bubble size ratio varies from
0.001 to 0.02.
It can be noted tat for both cases that the collision
efficiency Eo11 increases with rp/rb. When Stp 0,
Eo11 leads to the value corresponding to the inertialfree
particlebubble collision E oi obtained by Sarrot et al.
(2005). The collision angle 06  00, as Stp 0, which
are equal to 84 and 66 for a clean bubble and a fully
contaminated bubble, respectively. Note that Eo11 for
a fully contaminated bubble are two order of magnitude
lower than those for a clean bubble.
It is interesting to note that for a clean bubble Eo11
experiences a reduction of I r' of its value for Stp 0
until a critical value Sth. For all the size ratio rp/rb
tested, the same evolution is observed with a critical
value of St, near 0.1. Meanwhile, this diminution
in Eco11 does not appear when a fully contaminated
bubble is concerned. The same behavior of collision
efficiency haven been observed for Reynolds number
equal to 1 and 10. Meanwhile, at higher Stokes number
Stp > St1", Eoll increases rapidly with Stp and the
dependence on the size ratio rp/rb is largely reduced.
Considering the collision angle, for both clean bubble
and fully contaminated bubble, 06 decreases as the
Stokes number grows to Stp'" and it then go back
up. In the case of a fully contaminated bubble, 0c
is independent of the particle to bubble size ratio,
while for a clean bubble, 0c grows with rp/rb. In
fact, this behavior depends on the balance between the
negative and the positive effects of the inertial forces.
When the particle approaches the bubble surface, at
the forward part of the bubble (0 < 45 ), particle's
inertia tends to move straighlty towards the interface
rather than following the streamlines (cf. Fig.3a). This
phenomena called in the following as posli\ c inertial
effect" tends to increase the probability of collision.
When 0 increases to 7r/2, the radial component of the
fluid velocity U, (which force the particles to move
towards the bubble) decreases to zero and the tangential
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
component of the fluid velocity Uo becomes important,
so a "centrifugal force" is induced near the bubble's
equator (45 < 0 < 90 ). This effect pulls particles
away from the bubble surface. So in this region, the
probability of collision is reduced and this effect is
called "negative inertial effect" (cf Fig.3b). The greater
the Stokes number, the more important is the deviation.
rp/rb
(a) Positive effect Stp = 0.2 (b) Negative effect Stp = 0.02
Figure 3: Inertial effect on particlebubble collision po
sition. Reb 100, rp/rb 0.01 critical
trajectory for an inertial free particle, crit
ical trajectory for an inertial particle
When Stp increases, not only inertial effects but also
the gravitational effect becomes important. The evo
lution of collision efficiency E,,o1 for different Stokes
numbers Stp as function of size ratio rp/rb is plotted in
Fig.4 for a clean bubble at Reb 100. The numerical
solution for inertialfree particles based on the calcula
tion of the streamline function (Sarrot et al. (2005)) is
presented in solid line.
It can be seen that at low Stokes number, the collision
efficiency increases with rp/rb and is close to the value
for the inertialfree particlebubble collision (Stp 0,
presented by symbols < ). For a given size ratio rp/rb,
when increasing Stp, interception collision is replaced
by inertialgravitational collision, leading to an increase
in collision efficiency as expected. This gravitational ef
fect is more significant for small particles than for larger
ones with the same value of Stp. Moreover, when Stp
exceeds 0.1, Eco,, depends no longer on rp/rb. Collision
becomes totally dominated by gravitational sedimenta
tion.
In order to analyse the effect of gravitational force,
the evolution of the collision efficiency E,,o1 is plotted
for different values of us, for a clean (Fig.5a) and
a fully contaminated bubble (Fig.5b), respectively.
In the case of a clean bubble, there is no significant
effect of the gravitational force for us < 0.001. Since
Figure 4: Collision efficiencies v.s. diameter ratio at
Re 100. : Sarrot et al. (2005), <: Stp
O, o: Stp 0.002, *: Stp 0.005, o:
Stp 0.01, x: Stp 0.02, A : Stp 0.1, o
: Stp 0.2, +: Stp = 0.5, V: Stp = 1.0
the gravitational sedimentation is always in favor of
particlebubble collision, i.e. when us increases, its
positive effect dominate the negative effects of the
inertial forces. For us 0.1, these two adverse effects
become of same intensity. For all the Reynolds numbers
Reb examinated, the same evolution has been observed.
For a fully contaminated bubble, the gravitational
effect is obviously similar to that observed for a clean
bubble. When us increases from 0.001 to 0.1, Eco11
is two orders of magnitude greater than Eco11 obtained
without counting the gravitational effect (us = 0).
This difference comes from the different bubble surface
conditions. As we have explained above, both the fluid
and the particles velocities are significantly reduced near
an immobile interface. Therefore, it takes more time
for a particle to move around a contaminated surface
than a clean surface. As a result, particles have more
time to settle down instead of being taken away by the
inertial forces of the fluid. However, the gravitational
force has only an important positive effect for small Stp.
Collision with a partially contaminated bubble
The study of Sarrot et al. (2005) have shown that
particlebulle collision may be significantly influenced
by the bubble's surface property, because it modifies
the flow field around a bubble. In the next paragraph,
the case of a partially contaminated bubble will be dis
cussed. Firstly, the gravitational effect is not considered
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
in the calculation (us
analysed.
0), only inertial effects being
(a) clean bubble
(b) fully contaminated bubble
Figure 5: Gravitational effect for a clean bubble with
rp/rb 0.01 and Reb 100. E : U, 0; A
Us 103, *: Us 10 2, o: us 10 1.
Figure 6 presents the collision efficiency as function
of Stokes number Stp for a bubble Reynolds number
(Reb 1) and a particle to bubble size ratio (rp/rb
0.01). The contamination angle 0ean, varies from
cl,,, 0 (fully contaminated bubble) to 0,1,an
180 (clean bubble).
The strong influence of surface contamination is then
put in evidence: for all Reb examinated, collision effi
ciency for clean bubbles is much greater than that for
Figure 6: Collision efficiency (Eco,,) v.s. Stokes num
ber (Stp) for a partially contaminated bubble
(rp/rb 0.01) at Reb 1. : Oclean 0,
A: Ocea, 20, *: Oclean 30 0:
clean = 45 +: 0ci an = 60 o: clean
90 x: Ocean = 120 V: 0cean = 180.
a fully contaminated bubble. Since the collision oc
curs always on the forward part of bubble surface, for
0clan > 90 , the point of contact between particles
and bubble is still remaining on the mobile part of the
interface. As a result, Eo11 behavior is thus as same
as that for a clean bubble E~l, for 0c,1an > 90 : a
decrease of Ecol with Stp is observed due to the neg
ative inertial effect. When 0clea, decreases under 90,
Eco11 deviates from E'~, and approaches the efficiency
for a fully contaminated bubble It is noted that for
08clan < 30 no more decrease of collision efficiency
is observed as Stp increases. This can be explained by
the decrease of the tangential velocity of the local fluid
near the interface due to the reduction of the surface mo
bility. Legendre et al. (2009) have calculated and given
an approximation of the maximum interfacial velocity
Ur as a function of Oclean. Ur normalized by Ub is
plotted in Fig.7 versus clan, for different levels of sur
face contamination and for Reynolds numbers Re = 1,
10, and 100. As shown in this figure, when Ocican is
above 120 U(Oclean) Ur(180 ); meanwhile when
clean < 90 Ur decreases rapidly with clean and the
negative inertial effect become insignificant. The evolu
tion of Eo11 in the later case is close to that of a totally
contaminated bubble. However the positive inertial ef
fect at the front of the bubble always exists for large Stp.
Now we consider all the effects on the collision effi
ciency: interception, inertia and gravitational sedimen
104
103
6 Al
+
10 14
10'3
104
10'
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
100
10_
30 60 90 120 150 180
Ocean
Figure 7: Maximum tangential velocity Ur/Ub v.s.
Ocean. : Re 1, o: Re = 10, A
:Re 100, : 3/2sin(Oclea) and:
1/2 sin2 (Oan/2) (Legendre et al., 2009)
station. Figure 8 reports the evolution of the collision
efficiency and the collision angle as a function of the an
gle 0clan for a bubble Reynolds number (Reb 100)
and a particle to bubble size ratio (rp/rb 0.01). The
angle 0clan varies from 0c,1an 0 (fully contaminated
bubble) to Ocan = 180 (clean bubble).
It can be noted that when the gravitation forces are
considered, the collision efficiency is significantly in
creased. For Stp > 0.4 (for rp/rb 0.01), Eo11 is
independent of 0,ea1, and mainly depends on Stp. For
Stp < 0.4, Eo11 for a partially contaminated bubble is
between the collision efficiency for a fully contaminated
bubble F and for a perfectly clean bubble E~11. Con
sidering the collision angle 0c, three different behaviors
have been observed:
For 0c,1an > 120 in the range of Reb consid
ered, the evolution of the collision angle with Stp
is the same as that of a clean bubble. 06 is firstly re
duced by the negative effect of "centrifuge forces"
and then increased under the influence of combined
effects of gravity and inertial forces.
For Ocrit < clean < 90 the evolution of 0, is ob
served to experience three stages as it is shown in
Fig.9 for 0,clan 45 In this figure, critical parti
cles trajectories are plotted for the particles of size
rp/rb 0.01 and Stokes number varying between
0.002 and 1.0. The dash line represents the location
of the bubble's surface.
It can be noted that, at small Stokes number (Stp <
Figure 8: Collision efficiency (Eco11) v.s. Stokes num
ber (Stp) for a partially contaminated bub
ble (rp/rb 0.01). : 8 ,1, 0,A
clean = 20 *: 0,1ean 30 0: Ocan,
45 +: 0cian = 60 o: 0can = 90 x:
Oc0lan = 120 V: clean 180.
0.01), collision occurs only on the mobile part of
the interface, the collision angle being controlled
by the surface contamination level Ocea ,. Because
there is a local strong decrease of the flow rate near
the bubble surface induced by the change of inter
face condition from the zero tangential stress to the
noslip condition resulting in a strong increase of
local vorticity. Consequently, for Ocean > Ocrit,
the flow rate close to the clean part is always larger
than the flow rate close to the contaminated sur
face. Ocrit has been shown to be related to both
bubble's Reynolds number Reb and size ratio rp/rb
(Legendre et al., 2009).
Morever, the effect of gravitational sedimentation
makes the particles easier to collide with bubble
surface at the location of the first minimum distance
ri, as it is shown in Fig.10. As Stp increases, the
gravity also increases and gradually overcomes the
jump effect of the streamline near 0 ,1an. It makes
the collision on the immobile part of the interface
possible. Consequently, collision behavior enter
into another regime, where two minimum distances
between the bubble and the particle trajectory have
been observed, as that for a fully contaminated bub
ble. Particles collide bubble surface at the loca
tion of the second minimum distance r2. At large
Stp, posiLm inertial effect" as well as the grav
itational forces become significant and collision is
S10'1
102
0
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
S0.05
2 0.04
0.03
0.02
0.01
0.01
0
10 20 30 40 50 60 70 80 90
Figure 9: Particles (rp/rb 0.01) critical trajectories in the neighbourhood of a partially contaminated bubble
(Oci,ea 45, Reb 100). o: Stp 0.002, >: St, 0.005, *: Stp 0.01, +: Stp 0.02, o:
Stp 0.1, A: Stp 0.2, D: Stp 0.5, x: Stp 1.0
then dominated by the gravitational sedimentation.
clean part
r,
Oc 45*
Contaminated part
Figure 10: Shape of streamline for partially contami
nated bubble (Legendre et al., 2009).
For Oclean < Ocrit, the velocity gradient reaches
locally a maximum close to the value obtained for
a fully contaminated bubble. Consequently, parti
cles bubble collision is then independent on the sur
face contamination level and the collision behavior
is the same as that of a fully contaminated bubble
and governed by the poi il\ c inertial effect" and
gravitational forces.
Conclusions
In this paper, the inertial particle bubble collision ef
ficiency has been evaluated by solving the full parti
cle trajectory equation. DNS have been realized for
1 < Reb < 100 by taking into consideration the in
terface contamination level. The effect of inertia on the
collision efficiency has been examined at first. It was
found that Eco11 may be strongly influenced by inertial
forces. For all the cases considered, particle inertia has
a positive effect at large Stp, which results in a sharp
increase in the value of Eco11. At small Stp, a negative
effect has been observed near the bubble equator, where
the tangential component of the surface velocity reaches
its maximum for clean or nearly clean bubbles. This
tangential velocity creates a "centrifugal force" which
pulls particles away from the interface and makes col
lision impossible above a certain angle. As a result,
Eco11 decreases as Stp increases. For a partially con
taminated bubble (with 0 can < 90), the flow field
around the surface concerned by collision is significantly
modified and the collision efficiency depends strongly
on clean If Oclean > Ocrit, the collision occurs only on
the mobile interface, as the surface velocity decreases
with 0,,ean, the negative effect being greatly reduced. If
Ocean < Ocrit, particle bubble collision may take place
in both mobile and immobile part of the interface. Eco11
behaves as that of a fully contaminated bubble and only
the positive inertial effect is observed. Secondly, the in
fluence of the gravitational sedimentation on the colli
sion behavior has been analysed. Its contribution to the
collision efficiency becomes important, when us > 0.01,
which should not be neglected.
Acknowledgements
The authors would like to especially thank the CNRS
Fedration FERMAT for the financial and technical
support.
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