Paper No 7t" International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Flowinduced forces in agglomerates
Jos Derksen1 and Dmitry Eskin2
'Chemical & Materials Engineering, University of Alberta, Edmonton, AB, Canada T6G 2V4, jos@ualberta.ca
2Schlumberger DBR Technology Center, Edmonton, AB, Canada T6N 1M9, deskin@slb.com
Keywords: agglomerates, direct simulations, latticeBoltzmann, suspensions
Abstract
Direct simulations of laminar solidliquid flow in microchannels with full resolution of the solidliquid interfaces have been
performed. The solids phase consists of simple agglomerates, assembled of monosized, spherical particles. The flow of the
interstitial liquid is solved with the latticeBoltzmann method. Solids and fluid dynamics are twoway coupled. The
simulations keep track of the flowinduced forces in the agglomerates. The effects of agglomerate type (doublets, triplets, and
quadruplets), solids loading, and channel geometry on (the statistics of the) flow and collisioninduced forces have been
investigated. By comparing these forces with agglomerate strength, we would be able to assess the potential of microchannels
as agglomerate breakage devices.
Introduction
In many processes involving solid particle formation or
solids handling, particles have a tendency to stick together.
In crystallization processes crystals tend to agglomerate due
to the supersaturated environment they are in. Suspension
polymerization processes go through a "stickyphase" with
significant agglomeration levels. In colloidal systems a
variety of interactions can cause agglomeration (related to
Van der Waals forces, electrolyteinduced interactions,
surface chemistry), and stabilization of colloids is a central
issue. In biorelated applications agglomeration plays a role
in such diverse fields as blood flow and biomolecular
crosslinking of particles. The application which is driving
the present research is the behavior of asphaltenes, more
specifically their deposition on rock walls in oil reservoirs
(Boek et al., 2008, and references therein). Asphaltene
agglomeration is a key step in asphaltene deposition since
the agglomerate size and the agglomerate sticking
probability to the wall are intimately related: only relatively
small agglomerates stick to the wall while bigger ones are
removed with the flow. The agglomerate size distribution
evolves as a result of agglomeration and also
deagglomeration (i.e. breakage of agglomerates).
For an agglomeration event to occur, particles (primary
particles and/or agglomerates) need to collide first.
Typically collisions are induced by Brownian motion,
gravity, and velocity gradients in the fluid carrying the
particles since these phenomena bring about relative
velocities between particles. Also particle inertia can be a
source of collisions.
Next to promoting collisions, fluid velocity gradients and
particleparticle interactions are potential reasons for
agglomerate breakage since they cause mechanical loads on
agglomerates. In this paper we focus on the latter aspect: We
investigate the mechanical load on agglomerates due to
deformation of the surrounding liquid and the presence of
other particles/agglomerates. Interactions with other
particles/agglomerates can be either direct (collisions) or
indirect, e.g. transmitted by the interstitial liquid.
In modeling processes involving agglomeration, population
balances are often used to keep track of agglomerate size
distributions (ASD's) (Hounslow & Reynolds, 2006; Aamir
et al., 2009). In order to equip population balances with
adequate agglomeration and breakage physics, rate laws
(usually termed kernel functions) are being developed that
relate numbers of agglomeration and breakage events per
unit volume and time to local (flow and agglomerate)
conditions. With the purpose of devising kernels, there is
extensive literature on agglomeration and breakage as a
result of hydrodynamics for small (though nonBrownian)
agglomerates in turbulent flow (see Babler et al., 2008 and
references therein). Small in this context means that the
agglomerate size is significantly smaller than the smallest
dynamical scale of turbulence, i.e. the Kolmogorov scale.
If this is the case the flow field surrounding the agglomerate
can be assumed to scale with the (local) rate of energy
dissipation and be of some canonical, simple nature, and
disruptive forces, as well as collision probabilities can be
modeled fairly accurately based on Stokes flow principles
(see e.g. Nir & Acrivos, 1973; Babler et al., 2008). In a
previous paper (Derksen, 2008), however, it was argued that
the situation significantly complicates if the agglomerate (in
that paper a doublet of spheres) has a size of the same order
as the Kolmogorov scale, or is larger. It then experiences an
inhomogeneous flow and the (fluctuating) details of the
hydrodynamics around it are crucial for its internal forces
and thus breakage probability. A similar situation occurs in
microreactor equipment where (as an example)
agglomerate slurries are being sent through nozzleshaped
microchannels to facilitate breakage (Zaccone et al., 2009).
Again there is a nontrivial flow field around the
71" International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
agglomerate that induces internal forces that can break it.
From the above it may be clear that information regarding
flowinduced and particleinteractioninduced forces in
agglomerates are key in describing their probability to break.
In this paper a computational procedure is presented to
determine these forces from first principles. In the
computations, agglomerates are assembled of primary
spherical particles all having the same size, and released in a
flow field. We directly solve the flow around the
agglomerates, and fully couple flow and agglomerate
motion. The force and torque required to keep a primary
particle attached to the agglomerate follow from this
computational procedure. Comparing that force and/or
torque with a measure of the agglomerate strength allows
for assessing the breakage probability. Usually a primary
sphere has more than one contact point with the other
primary spheres in the agglomerate. This (in general) makes
it impossible to calculate the force and torque at each
contact point from the momentum balance directly.
However, for a few simple agglomerate configurations the
force and torque per contact point can be determined with
minimal assumptions.
It should be noted that during the simulations presented in
this paper the agglomerates keep their integrity; we do not
actually break them. The results of the simulations comprise
detailed representations (time series, probability density
functions) of the flowinduced forces and torques in
agglomerates as a function of process conditions. From this
detailed information breakage probability can be assessed
once data regarding the mechanical strength of the specific
agglomerates at hand is available.
In this paper we first define the flow systems in terms of
their dimensionless parameters, and indicate which part of
the parameter space we will be exploring. Then the
computational framework for calculating the flowinduced
forces in agglomerates is explained. We then apply the
method to flow of agglomerate slurries in microchannels
where we compare forces in different types of simple
agglomerates, viz. doublets, triplets arranged in triangles,
and quadruplets arranged in tetrahedrons, all made of
equally sized spherical particles. By considering a range of
solids volume fractions of the agglomerate slurries, the role
of particleparticle interactions on the mechanical load on
agglomerates is assessed. Furthermore, a few different
microchannel layouts have been compared. Specifically we
compare straight, square channels with channels having
contractions.
Nomenclature
primary sphere radius
deformation tensor
force on sphere i & contact force on sphere i
normal force & dimensionless normal force
body force driving flow
gravitational acceleration
channel width & height
height of obstruction in channel
moment of inertia tensor of agglomerate & of
primary sphere
angular momentum of agglomerate
mass of primary sphere
number of primary spheres in agglomerate
Re,
r, ra, ri
Ti, Te,i
t,t*
Reynolds number based on wall shear velocity
location, agglomerate center & sphere i center
torque on sphere i & contact torque of sphere i
time & dimensionless time
Va, vi velocity of agglomerate & of sphe
u wall shear velocity
Greek letters
r, r, deformation rate & wallshear rate
p dynamic viscosity
v kinematic viscosity
0 solids volume fraction
p, p, liquid & solid density
T wall shear stress
re i
oa ,Coi angular velocity of agglomerate & of sphere i
Flow System & Parameter Space
The basic flow geometry in this study is a square channel;
see Figure 1 for a definition of its dimensions and
coordinate system. The laminar flow in the channel is driven
by a body force f, acting in the x (=streamwise) direction
mimicking a pressure gradient (or, if vertically placed,
gravity). At the four side walls a noslip boundary condition
applies; the system is periodic in streamwise direction. A
Reynolds number characterizing the flow in the channel can
be based on the wall shear velocity u =
SP
uH
Re = u with v the kinematic viscosity of the liquid
v
in the channel, and p its density. The average wall shear
stress r follows from an overall force balance:
1 H3/2 1/2
4H =H2.f so that eventually Re,= 1
2 p1/2v
Inspired by work due to Zaccone et al. (2009), in some of
the simulations the channel has a contraction as defined in
the bottom panels of Figure 1. The contraction is
twodimensional, i.e. the channel is only contracted locally
in the zdirection; the width in the ydirection remains H.
The Reynolds number definition for the contracted channel
cases is the same as for the uniform channel. All channels
L
considered have lengthoverwidth aspect ratio = 2.0.
H
In the liquid that fills the channel agglomerates are released.
They consist of equally sized spheres with radius a. Three
types of agglomerates will be considered: (1) two touching
spheres forming a doublet; (2) three touching spheres
(triplet) forming a triangle (two contact points per primary
sphere); (3) four touching spheres (quadruplet) forming a
tetrahedron (three contact points per primary sphere). The
introduction of the agglomerates in the channel gives rise to
a
three additional dimensionless numbers: an aspect ratio 
H
a density ratio and a solids volume fraction 0. The
P
density ratio is relevant with a view to inertial effects, e.g.
related to slip velocities, particleparticle and particlewall
Paper No
a
d,
Fi, F,i
fo
g
H
h
la, '0
La
mn
Paper No
collisions. We do not consider gravity in the simulations; it
is assumed that
(p, ,O)g <
fo
SH .
Th
Figure 1: Definition of the flow channel and coordinate
system. Top panels: uniform channel; bottom panels:
channel with contraction. In the streamwise (x) direction
periodic conditions apply.
In this paper only part of the parameter space as identified
above has been explored; Re , and ' have been
H' p
fixed to values 2.6, 0.05 and 2.5 respectively (this density
ratio is e.g. representative of glass beads in a watery liquid).
We consider three solids volume fractions: 0=0.031, 0.062,
and 0.093; three different types of agglomerates: doublets,
triplets, and quadruplets; and three different flow
geometries: a uniform channel ( = 0, see Figure 1), and
H
two channels with contractions having =0.2 and 0.3. In
H
all contracted channels
Simulation Procedure
In the simulations, a number of agglomerates made of
equallysized spheres (primary spheres) are placed in the
liquid filled domain. The motion of the agglomerates and
the liquid are fully coupled, i.e. the fluid flow sets the
agglomerates in motion; the motion of the agglomerates on
its turn induces fluid flow. The fluid flow we solve with the
latticeBoltzmann method (LBM). For flows in complexly
shaped domains and/or with moving boundaries, this
method has proven its usefulness (see e.g. the review article
by Chen & Doolen, 1998). In the LBM, the computational
domain is discretized into a number of lattice nodes residing
on a uniform, cubic grid. Fluid parcels move from each
node to its neighbors according to prescribed rules. It can be
proven by means of a ChapmanEnskog expansion that,
with the proper grid topology and collision rules, this
system obeys, in the low Mach number limit, the
incompressible NavierStokes equations (Chen & Doolen,
1998; Succi, 2001). The specific implementation used in our
simulations has been described by Somers (1993), which is
a variant of the widely used Lattice BGK scheme to handle
the collision integral (e.g., see Qian et al., 1992). We use the
scheme due to Somers. as it manifests a more stable
Sfo
)XI
__*fo
[3
71" International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
behaviour at low viscosities when compared to LBGK. A
latticeBoltzmann fluid is a compressible fluid. In order to
mimic incompressible flow, as is done in this paper, the
Mach number must be sufficiently low. In the simulations
presented here the local Mach number never exceeded 0.05.
The fluid flow and the motion of the agglomerates are
coupled by demanding that at the surface of each primary
sphere the fluid velocity matches the local velocity of the
solid surface (that is the sum of the linear velocity va and
wax(rra) with %a being the angular velocity of the
agglomerate the primary sphere is part of, ra is the
agglomerate's center of mass, and r a point on the primary
sphere's surface); in the forcing scheme that is applied here
this is accomplished by imposing additional forces on the
fluid at the surface of the primary spheres (which are then
distributed to the lattice nodes in the vicinity of the particle
surface). The details of the implementation of the forcing
scheme can be found elsewhere (e.g. Ten Cate et al., 2002).
Determining forces and torques in agglomerates
The collection of forces acting on the fluid at the surfaces of
the primary spheres forming an agglomerate is used to
determine the hydrodynamic force and torque acting on that
agglomerate (action = reaction). In addition to the
hydrodynamic force and torque stemming from the LBM,
the motion of the agglomerates is controlled by lubrication
forces, and by forces arising from the softsphere
interactions that we use to deal with collisions between
agglomerates. The equations of (linear and rotational)
motion that we solve for an agglomerate consisting of n
primary spheres can be written as
dv n
nmo = Z1F
at 11
d ,= (T,+F x(r, r.) (1)
dt i=\
La = Iao
with v, and Co the linear and angular velocity of the
47ff 3
agglomerate, m0 =p, a the mass of a primary sphere,
3
and I, the moment of inertia tensor of the agglomerate.
The vector ri is the location of the center of sphere i. The
torque T on primary sphere i directly follows from the
LBM / forcing scheme.
As already indicated above, the force on primary sphere i
(F, in Eq. 1) has in principle three contributions: the
hydrodynamic force stemming from the LBM (and forcing
scheme), a radial lubrication force, and a softsphere
interaction force which is also radial. The latter two
contributions are nonzero only if a primary sphere
belonging to another agglomerate is in close proximity of
sphere i. Since they are radial they do not contribute to the
torque on the primary sphere.
The lubrication force is added as a hydrodynamic force in
situations where two primary spheres belonging to different
agglomerates are in close proximity and move relative to
one another. At some stage of proximity typically when the
surfaces of the two spheres involved are less than one grid
spacing apart the (fixed, i.e. nonadaptive) grid cannot
accurately resolve the hydrodynamic interactions anymore
and radial lubrication is explicitly added (Nguyen & Ladd,
Paper No
2002). A softsphere approach has been used to deal \
collisions between primary spheres belonging to diffe
agglomerates. In this approach a linear elastic repul
force is switched on when spheres are in close proximity'
effectively prevents particles from overlapping and gi
rise to fully elastic collisions.
The simulations provide us with the force and torque
each of the primary spheres (Fi and Ti ). From solving
set of Eq's 1 we know the acceleration (linear
rotational) of the agglomerate as a whole so that we
know the acceleration of each primary sphere:
dv. dv d r
di = dat (oa x(ri ra))
dt dt dt a
deo deo a
vith
rent
sive
y. It
yves
on
the
and
also
(2)
dt dt
As a consequence we can determine the force Fi and
torque T,, required to keep each primary sphere attached
to the agglomerate:
Fci = m0 Fi
dt
(3)
Tc,i = I0 d Ti
Sdt
2
(with I, = a m, the moment of inertia of the primary
5
sphere about its center).
If primary sphere i has more than one contact point with
other primary spheres in the agglomerate, F, and T,
are the sum of contact forces and torques respectively. In the
general case of an agglomerate arbitrarily configured of
equally sized, contacting spheres it is not possible to
determine the force and torque per contact point based on
the collection of F, 's and T, 's only: Solving for forces
and torques per contact point poses an ill defined problem.
Additional physics e.g. related to mutual particle
displacement within the agglomerate under mechanical
loading is required to close the system of equations. Still, we
consider the availability of the summed contact forces and
torques per primary sphere to be useful information for
assessing breakage probability.
However, in this paper we will be considering three simple
agglomerates for which the forces and torques per contact
point can be determined directly based on F,. and T,
(i=l...n): doublets, triplets, and quadruplets. The doublet
has a single contact point and the resulting F,. and T,,i
are the force and torque in that single contact. The triplets
are arranged in a triangle with two contact points per
primary sphere; the quadruplets are arranged in a
tetrahedron with three contact points per primary sphere. By
making a few sensible assumptions the force (and torque)
per contact point can be determined based on the set of F,'
and T, per primary sphere.
Results
Flow field impressions
Although the flow in the channel is laminar, the presence of
the agglomerates adds fluctuations and smallscale
structures to the overall flow. We see that if we plot
71" International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
instantaneous realization of the flow in terms of contours of
the velocity magnitude, and (clearer) in terms of the
generalized deformation rate y= 2dd, (with
1 (Du Du
d, 21 I + x' the deformation rate tensor) as is done
in Figure 2. It is also important to see that the agglomerates
do not experience a homogeneous deformation rate around
them so that estimating flowinduced disruptive forces
based on the deformation rate of the undisturbed flow (the
flow without agglomerates) is a coarse approximation at
best.
Obviously, with the same body force acting on the liquid,
the flow rates in the channels reduce if a contraction is
present; see Figure 3. Therefore, compared to uniform
channel flows at the same Reynolds number (and thus the
same body force) contracted channel flows not necessarily
have higher overall (i.e. volume averaged) deformation rates.
Deformation tends to increase locally as a result of the more
complicated flow structure in a contracted channel, and
tends to decrease as a result of decreased flow rate and
quiescent parts of the flow away from the contraction.
Cleary the contractions have strong impact on the spatial
distribution of liquid deformation over the channel; compare
the deformation fields of Figure 3 with the deformation field
in Figure 2. We expect this to have impact on the forces
experienced at the contact points of the primary spheres in
the agglomerates.
a oq
m
oI a ' L N
y o
0
U,
Figure 2: Contours of velocity magnitude (top) and
deformation rate y in the center plane of the uniform
channel. The circular disks represent the cross sections of
the spherical particles in the same center plane. Quadruplets,
S=0.093.
Flow and collisioninduced forces in agglomerates
During the simulations we keep track of the entire force and
torque history of every agglomerate so that we can
reconstruct the contact forces and torques as a function of
time. The discussion of the results in this paper focuses on
the normal contact force. This is the projection of the
contact force on the line connecting the centers of the two
Paper No
spheres sharing the contact point under consideration. A
tensile normal contact force is positive; a compressive
normal contact force is negative. Note that the simulations
provide us with the full (threedimensional) contact force
vectors and contact torque vectors for every contact point in
every agglomerate at every moment in time. Which
components and/or projections of forces and torques are
critical for agglomerate breakage depends on the physics
and/or chemistry of the bonds between the primary spheres
forming the agglomerates. This is a subject beyond the
scope of the present paper. The discussions regarding the
normal contact force allow us to show the detailed level of
information gathered by the simulations, and to indicate
some major trends in parameter space while not overloading
the reader with the full (vectorial) contact force and torque
information.
r
,,:.' T
u 0
x io~
LI
0) 0
ra
71" International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
simulations with quadruplets at three solids volume fractions,
the middle panel compares three channel layouts (again for
quadruplets), and the bottom panel compares doublets,
triplets and quadruplets.
The smooth parts of the time series relate to the agglomerate
translating and rotating through the laminar flow with a
continuous change in the direct laminar hydrodynamic
environment resulting in a gradual change in the normal
force. The time scale of these smooth fluctuations is of the
10
order of . The spikes and apparent discontinuities in the
force signal are due to collisions with other agglomerates in
the channel.
H_ mO 00 c
Figure 3: Contours of velocity magnitude (top two panels)
and deformation rate y for quadruplets in channels with
h
contractions =0.3 and 0.2 respectively. 0=0.062.
H
In Figure 4 we show samples of time series of the normal
force. It has been nondimensionalized according to
F w
F,* = F," with y = with dimensionless time
'kwa JU
being t* = t? Each time series corresponds to one contact
point in one agglomerate. The top panel compares
Figure 4: Time series of the normal force in a single point
of contact. Force has been normalized according to
F .
F,* = time according to t* =ty, with =
/. a2 //
the average wall shear rate. Top panel: quadruplets in a
uniform channel with solids volume fraction 0=0.093 (blue
curve), 0.062 (red), 0.031 (green). Middle panel:
quadruplets at 0=0.062 in a uniform channel (blue), a
channel with contraction
h
 =0.2 (red), a channel with
H
h
contraction =0.3 (green). Bottom panel: doublets (blue),
H
triplets (red), quadruplets (green) in a uniform channel at
0=0.062.
Comparing the different time series obtained under different
conditions and in different channels hints at a few trends. In
J
Xq4t=
Paper No
the denser suspension the agglomerate collides more
frequently leading to a normalforce signal with more spikes
(top panel of Figure 4). Also the flow seen by the
agglomerate is more complex in the denser suspension. This
leads to the smooth fluctuations of the signal in the denser
suspension having higher amplitude and higher frequencies.
The effect of the contraction appears to be a reduction in the
amplitude of the normal force (middle panel of Figure 4).
The agglomerates passing through the contraction can be
reconstructed from these time series. A lowfrequency
variation of the force signal is from time to time followed by
faster fluctuations when the agglomerate passes through the
contraction. Comparing the force time series in doublets,
triplets, and quadruplets does not indicate a clear trend. For
this specific time series, triplets show somewhat stronger
force fluctuations than doublets and quadruplets.
PDF
(au)
10
105
50 0 F 50
Figure 5: Probability density functions (PDF's) of the
normal contact force. From left to right: doublets, triplets,
and quadruplets. From top to bottom: uniform channel,
h
contracted channel with =0.2, and contracted channel
H
h
with =0.3. Blue curves have =0.093, red curves have
H
=0.062, green curves have 0=0.031.
In order to more quantitatively analyze the normal force
data based on the full amount of information contained in
the simulations we condensed the normal forces in all
contact points in all agglomerates at each moment in time in
force probability density functions (PDF's), and organized
them in Figure 5. A number of interesting dependencies can
be observed in the various panels of Figure 5, and by
comparing the various panels presented there. An apparent
trend relates to the solids volume fraction. The more
agglomerates, the higher the chances for collisions and thus
the higher the chances for high normal force levels. As a
results, the tails (high and lowend) of the PDF's for the
higher solids volume fractions lie systematically above
71" International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
those for the lower solids volume fractions. The exception is
the lower left panel of Figure 5 (doublets in a strongly
contracted channel) where the compressive forces in the
0=0.031 slurry are higher than those in the ==0.062
agglomerate slurry.
The force PDF's of the different agglomerate types
(doublets, triplets, quadruplets) in the uniform (i.e.
noncontracted) channels (top row of Figure 5) differ
significantly. The (smaller) doublets have narrower PDF's
than the triplets and quadruplets. The difference not only
relates to size but also to shape. The doublets (at least the
ones in the center portion of the channel) tend to align with
the flow, specifically if the solids volume fraction is low so
that collisions do not scatter the orientations. These effects
reduce the chance of collisions and also reduce the (absolute
value of) the contact force since the spheres in the aligned
doublet see more or less the same hydrodynamic
environment. There are minor differences between the force
distributions of triplets and quadruplets.
Changing the channel geometry by placing contractions
influences the flowinduced forces. For the quadruplets and
the triplets the trend is from more Gaussianshaped
distributions (quadratic on the linlogscale of Figure 5) for
uniform channels towards exponential distributions (linear
on linlogscale) for the strongest contractions. For the
doublets the trends as a result of contractions are less
systematic; we do observe a widening of the normal force
PDF as a result of placing a contraction.
F,
Dimensionless force peak levels = are of the
Pya2
order of 50 (see Figure 5). The wall shear rate yk = is
at the high end of the shear rates that would be present in
the undisturbed (single phase) laminar flow. Based on
undisturbed channel flow with yk we would estimate
normal forces as high as Fn 20 (e.g. see Nir & Acrivos,
1973). The complexity of the direct hydrodynamicc)
environment of the agglomerates and collisions thus
significantly add to the normal force peaklevels.
Conclusions
Motivated by the potential role of liquid deformation in the
breakage of agglomerates we have set up a computational
procedure for determining flowinduced forces in
agglomerates and applied it for three simple agglomerate
configurations (doublets, triplets and quadruplets all made
of equally sized spheres) in laminar channel flow. The
simulations fully resolve the liquid flow which is twoway
coupled with the motion of the agglomerates. The flow
simulations are based on latticeBoltzmann discretization;
collisions between agglomerates are based on a softsphere
approach. During the simulations, the agglomerates keep
their integrity and we keep track of the forces at the contact
points required to keep the spheres the agglomerates are
made of attached.
In terms of probability density functions (PDF's), the forces
in the agglomerates depend on the solids volume fraction in
the agglomerate slurry with increased chances for high force
levels at high solids loading. This is largely due to collisions
Paper No
probabilities increasing in denser suspensions, but also due
to the more complex hydrodynamic environment of an
agglomerate in a dense suspension. Sphere doublets
experience weaker normal forces compared to triplets and
quadruplets.
We also investigated the impact of a contraction placed in
the channel on the flowinduced force levels. Sharp
contractions have been used in experiments to promote
agglomerate breakage (Zaccone et al., 2009). In our study
contracted channels have been compared with uniform
channels on the basis of equal pressure drop. This implies
that the volumetric flow rate in the contracted channel is
smaller than in uniform channel. Still (in general) the
contracted channels widen the normal force PDF's hinting at
their usefulness for promoting breakage.
Future work will be in applying the procedures for
determining flowinduced forces in agglomerates to more
generic flows, first and foremost homogeneous, isotropic
turbulence. Such simulations will allow us to relate
turbulence characteristics such as energy dissipation rate
and Kolmogorov scale (relative to agglomerate size) with
force and torque levels in agglomerates. This information
would be useful for assessing breakage probabilities in
turbulent, largescale process equipment flows. Performing
simulations with finitestrength agglomerates is another
direction for future work. In such simulations we actually
would break agglomerates if certain threshold forces,
torques, and/or stresses are exceeded. By specifying direct
interparticle force fields we could even study the interplay
between agglomerate morphology and fluid flow.
Acknowledgements
Support of the Schlumberger DBR Technology Center is
gratefully acknowledged.
References
Aamir, E., Nagy, Z.K., Rielly, C.D., Kleinert, T. & Judat, B.
Combined quadrature method of moments and method of
characteristics approach for efficient solution of population
balance models for dynamic modeling and crystal size
distribution control of crystallization processes. Ind. &
Engng. Chem. Res., Vol. 48, 8575 (2009)
Bibler M.U., Morbidelli M. & Baldyga J. Modelling the
breakup of solid aggregates in turbulent flows. J. Fluid
Mech., Vol. 612, 261 (2008)
Boek, E.S., Ladva, H.K., Crawshaw, J.P & Padding, J.T.
Deposition of Colloidal Asphaltene in Capillary Flow:
Experiments and Mesoscopic Simulation. Energy & Fuels,
22, Vol. 805 (2008)
Chen, S., Doolen, G.D. Lattice Boltzmann method for fluid
flows. Annual Rev. Fluid Mech., Vol. 30, 329 (1998)
Derksen, J.J. Flow induced forces in sphere doublets. J.
Fluid Mech., Vol. 608, 337 (2008)
Hounslow, M.J. & Reynolds, G.K. Product engineering for
crystal size distribution. AIChE J., Vol. 52, 2507 (2006)
71" International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Nir, A. & Acrivos, A. On the creeping motion of two
arbitrarysized touching spheres in a linear shear field. J.
Fluid Mech., Vol. 59, 209 (1973)
Nguyen, N.Q. & Ladd, A.J.C. Lubrication corrections for
latticeBoltzmann simulations of particle suspensions. Phys.
Rev. E, Vol. 66, 046708 (2002)
Qian, YH., d'Humieres, D. & Lallemand, P Lattice BGK
for the NavierStokes equations. Europhys. Lett., Vol. 17,
479 (1992)
Somers, J. A. Direct simulation of fluid flow with cellular
automata and the latticeBoltzmann equation. App. Sci. Res.,
Vol. 51, 127 (1993)
Succi, S. The lattice Boltzmann equation for fluid dynamics
and beyond, Clarendon Press, Oxford (2001)
Ten Cate, A., Nieuwstad, C.H., Derksen, J.J. & Van den
Akker, H.E.A. PIV experiments and latticeBoltzmann
simulations on a single sphere settling under gravity. Phys.
Fluids, Vol. 14, 4012 (2002)
Zaccone A., Soos M., Lattuada M., Wu H.. Bibler M.U. &
Morbidelli M. Breakup of dense colloidal aggregates under
hydrodynamic stresses. Phys. Rev. E, Vol. 79, 061401
(2009)
