Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Modelling particle collisions and agglomeration in gasparticle flows
Martin Sommerfeld
Zentrum fur Ingenieurwissenschaften; MartinLutherUniversitat HalleWittenberg
D06099 Halle (Saale), Germany
Email: martin. sommerfeld @iw.unihalle.de
Keywords: gasparticle flows, EulerLagrange approach, interparticle collision, impact efficiency, agglomeration,
agglomerate structure, agglomerate breakup
Introduction
Interparticle collisions are important in many gassolid
flows, even at relative low overall mass loading resulting in
momentum and heat transfer between the particles,
agglomeration, but also may cause particle breakage. In
pneumatic conveying for example, interparticle collisions
yield a redispersion of dense regions and ropes which often
are caused by inertial forces (Lain & Sommerfeld 2009,
Sommerfeld & Lain 2009). Hence, numerical calculations of
particleladen flows even at rather low overall mass loading
require the consideration and modelling of interparticle
collisions. There are basically two ways to model
interparticle collisions in a Lagrangian approach;
deterministic and stochastic models.
A deterministic collision model requires that all real particles
are tracked simultaneously through the flow field, as for
example mostly done in DNS (direct numerical simulation).
An essential task is the detection of a possible collision pair
which is computationally very demanding. Several efficient
collision detection algorithms were described by Sundaram
& Collins (1996).
In the classical Euler/Lagrange approach combined with
turbulence modelling, not all real particles need to be
simulated, but computational particles, often called parcels,
in order to make the computation less time consuming and to
respect the correct particle mass flow rate. In this situation
statistical or stochastic collision models may be used. One of
the first models of this kind was proposed by O'Rourke
(1981), which describes the probability of collisions between
two parcels, but still requires the search for two possible
collision parcels among all parcels in the computational
domain.
Stochastic collision model
The collision model considered here is purely stochastic,
namely in that a possible fictitious collision partner is
sampled for each tracked parcel at each time step
(Sommerfeld 2001). Therefore, this model does not require
information on the location of the surrounding particles and
simultaneous tracking of all parcels is also not a prerequisite.
Hence this model is also applicable when using sequential
tracking of parcels as often done in the case of stationary
flows. The fictitious collision partner is sampled from local
distributions for particle size, velocity and other relevant
properties. These local properties however have to be
sampled for each control volume in a previous Lagrandian
step. Consequently, the fictitious particle is a representative
of the local particle population. However, normally the
velocities of colliding particles are correlated somehow as
they move in the same turbulence structure once colliding.
The degree of correlation depends on the Stokes number and
is given for the fluctuating velocities by:
Ufit,, = R(Stt)urea, +p, 1 1 R(Stt)2
R(Stt)= exp( 0.55 St 4)
St = Q /TL
The correlation function was calibrated based on LES data
(Lavieville et al. 1995). Possible improvements of this
correlation function are presently evaluated using
pointparticle DNS on the basis of LatticeBoltzmann
Method (Lain et al. 2010). With this, all the properties are
known to calculate the collision probability:
P= (Dp +Dpf)2 u U n, AtL
A random process is used to decide whether a collision takes
place or not. In case of a collision, the collision point on the
fictitious particle surface needs to be determined, which is
done in a coordinate system where the fictitious particle is
stationary. Hence, the relative velocity vector is aligned with
the axis of the collision cylinder (Fig. 1) and a stochastic
approach is used to identify the impact point on the front
hemisphere of the fictitious particle (Sommerfeld 2001), i.e.
the lateral displacement L and the angle P in the
crosssection of the collision cylinder. By solving now the
impulse equations (linear and angular motion) for an oblique
collision in connection with Coulomb's law of friction the
new velocities of the considered particle after collision,
which may be a sliding or nonsliding collision, can be
calculated. After retransforming the particle velocities in
the laboratory frame of reference tracking can be continued.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
U rel
collision cylinder
Figure 1: Illustration of the collision of the real particle 0
with the fictitious particle which is stationary and hence
the collision cylinder axis is aligned with relative velocity
vector.
Impact Efficiency
The above model assumes that the impact efficiency is
100 %, i.e. all particle centroids arriving from within the
collision cylinder will collide with the fictitious particle.
However, in most practical cases a particle size distribution
exists, which implies that a small particle might interacts
with a larger collector particle (Fig. 2). Hence, as the small
particle is able to move with the relative flow around the
collector only a fraction of them coming from a circular area
limited by the socalled boundary particle trajectory are
captured by the collector (Ho & Sommerfeld 2002). The
impact efficiency is defined as the ratio of this collection
crosssection to that of the collector particle:
2 7 2
TL = f2Y,Y)
The radius of the collection circle is depending on the small
particle response and hence a relative or impact Stokes
number, given by:
Stl p,, p,Ps D up Up1
Tpass 18 L Dpc
Based on simple calculations of the boundary particle
trajectory Schuch & Loffler (1978) developed a correlation
for the impact efficiency given by:
i = St
S St, +a)
The constants a and b depend on the Reynolds number of the
collector particle based on the relative velocity (Sommerfeld
& Lain 2009). Using the impact efficiency, now a collision
will only take place when the sampled lateral displacement
L is smaller than the value of Y, (Fig. 2). This implies that in
the model, the trajectories of the small particle are still
straight lines up to the point they hit the collector. Hence, no
direct fluid dynamic interaction between the particles is
considered in the collision process. It was shown by Ho &
Sommerfeld (2005) that the consideration of the impact
efficiency can remarkably reduce the collision rate of small
particles.
Stream Lines
Boundary Particle
.. .... ..........
Urel Par
Separated Particle
Collector Particle
Collector Particle
Figure 2: Illustration of the impact efficiency for the
collision of a small particle with a larger collector particle.
Agglomeration Model
The stochastic collision model was also extended to account
for possible agglomeration of dry particles considering only
van der Waals attraction. Comparing the kinetic energy of
the small particle before impact with the dissipated energy
(due to irreversible deformation) and the van der Waals
energy, one can determine a critical velocity below which
the small particle sticks to the collector (Ho & Sommerfeld
2002, 2004). This critical velocity includes additional
parameters which depend on the particle properties, such as,
the restitution ratio, the Hamacker constant, the minimum
contact distance and the material limiting contact pressure.
1 (1k 2 A
2n 2 
Dps kl 71 Z0 V Ppl Pps
These particle properties are depending on the particle
material and have to be obtained experimentally. After the
occurrence of agglomeration, i.e. the normal component of
the collision velocity is smaller than the critical velocity, it is
assumed that the new particle has the diameter of a volume
equivalent sphere in the subsequent tracking, including the
calculation of the fluid dynamic forces. In order to ensure
mass balance, the number of real particles in the considered
parcel has to be reduced (Sommerfeld & Lain 2009), since a
fictitious particle has been captured by the real particle (Blei
& Sommerfeld 2004). In case the normal component of the
relative velocity is larger than the critical velocity a rebound
is calculated using the impulse equations in connection with
Coulomb's law of friction. This agglomeration model has
been tested and validated for a homogeneous isotropic
turbulence (Ho & Sommerfeld 2002), a plane shear layer
(Ho & Sommerfeld 2004) and a gas cyclone (Ho &
Sommerfeld 2005). For the cyclone separator it could be
shown that the consideration of particle agglomeration
considerably improves the separation efficiency for particles
below about 3 im. Larger particles are more inertial and
have higher relative velocities whereby the rebound
probability is high.
Agglomerate Structure Model
So far the agglomeration model does not provide any
Paper No
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
information on the structure of the agglomerates, such as,
number of primary particles, effective surface area, porosity,
sphericity or fractal dimension of the agglomerate. Such
information is however necessary for the layout of a number
of technological processes, as for example spray drying and
flame synthesis. Additionally, such structure information is
also required for considering the real size of the agglomerate
in the particle tracking (i.e. drag coefficient of an
agglomerate) and modelling a possible breakup of
agglomerates (Lipowsky & Sommerfeld 2008). Therefore, a
structure model in the frame of the pointparticle
approximation was developed. This was realized by storing
pointer vectors from a reference primary particle to all the
other primary particles collected in the agglomerate as
illustrated in Fig. 3. For all the agglomerates the location
vectors are stored in a multiple linked list. In addition also
the contact forces between the primary particles can be
stored which is the basis for modelling the breakup of
agglomerates. In the collision treatment, only collisions of
new primary particles with agglomerates are allowed. Since
the number density of agglomerates in the twophase system
is generally very low, collisions between agglomerates can
be neglected. This means, that if the tracked particle is an
agglomerate and the fictitious particle sampled from the
local particle size distribution is also an agglomerate, this
collision is discarded, since it is quite difficult to model.
For the determination of the collision crosssection the
sphere diameter enclosing the entire agglomerate is used.
Once a collision of a primary particle with an agglomerate is
detected based on the collision probability and accounting
for impact efficiency, it has to be decided which primary
particle in the agglomerate is hit by the new primary particle.
For this purpose, first the agglomerate is randomly rotated
and the lateral displacement of the impact point on the
sphere enclosing the agglomerate is sampled. The random
rotation is done since the rotation of the primary particles
and agglomerates in the flow is not considered at the
moment. Then the primary particle in the agglomerate being
hit by a new primary particle has to be found. This is done by
searching the closest neighbour for the primary particle
moving straight with the relative velocity towards the
agglomerate structure. Once a collision occurs with a
primary particle in the agglomerate, the new particle might
stick or be reflected depending on the strength of the
interaction forces.
The structure model was first tested for particles moving in a
box with homogeneous isotropic turbulence. A typical result
of such a simulation is shown in Fig. 4, where the
agglomerate consists of 98 monosized primary particles. In
order to characterise the agglomerate structure, different
properties may be considered as mentioned above. The
porosity of the agglomerate is one important property also
affecting particle motion through the modification of the
drag coefficient and other fluid dynamic forces (Dietzel &
Sommerfeld 2009). For allowing the determination of the
porosity of the agglomerate, the convex hull is used which is
a surface wrapped around the agglomerate as illustrated in
Fig. 5. The agglomerate in Fig. 4 has for example a porosity
of s = 0.73. Other means for agglomerate characterisation
which can be easily determined are the sphericity or the
fractal dimension.
Figure 3: Sketch of the structure model for agglomerating
primary particles (left) and the storage of the position
vectors in a linked list (right).
Figure 4: Simulated structure of an agglomerate obtained
in homogeneous isotropic turbulence (98 monosized
primary particles, porosity s = 0.73, fractal dimension df =
2.2).
ami LX
Figure 5: Sketch of the convex hull approach to allow the
characterisation of agglomerates.
The performance of the agglomeration and structure model
will be demonstrated for polydisperse dry particles moving
in homogeneous isotropic turbulence with defined properties.
Different particle Stokes numbers, shapes of the size
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distribution and initial particle volume fraction will be
considered.
Conclusions and Outlook
The developed stochastic interparticle collision model is
numerically very efficient so that a Lagrangian calculation
also can be effectively parallelised since the model does not
require information on the location of neighboring particles.
The model was also extended to allow the consideration of
dry particle agglomeration through van der Waals attraction.
Other particle interaction forces (e.g. electrical forces and
liquid bridges) could be also considered in the modelling.
The developed structure model allows the estimation of
important agglomerate properties, such as porosity, free
surface area, sphericity and fractal dimension. An extension
of the stochastic collision and structure model was recently
also done for viscous particle, where the involved primary
particles might partly penetrate each other (Stabing &
Sommerfeld 2010). The further validation of the
agglomeration and structure model based on detailed
experiments is in preparation.
Acknowledgements
The contributions of Dr. Ho, Mr. Lipowsky and Mr. Stabing
are gratefully acknowledged.
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ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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