Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 10.7.4 - Evaluation of collision models applied to varying dense particle jets flows
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00271
 Material Information
Title: 10.7.4 - Evaluation of collision models applied to varying dense particle jets flows Collision, Agglomeration and Breakup
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Gumprich, W.
Chrigui, M.
Braun, M.
Sadiki, A.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: particle collision
stochastic model
Lagrange
dispersion
 Notes
Abstract: Standard stochastic inter-particle collision models are derived from the kinetic theory of gases. These models differ from each other in their algorithmic approach and their modification of the collision frequency, accounting for additional effects such as turbulence. In comparison to deterministic models, the major advantage of stochastic models regards their lower computational cost. However, while stochastic models allow the simulation of inter-particle collisions for cases in which a deterministic model would be impracticable, the computational cost is still a critical issue. The Nanbu-Babovsky model achieves a computational expense linearly proportional to the number of parcels and conserves energy, being advantageous over other stochastic models. The model, implemented into the commercial CFD software ANSYS FLUENT, is validated based on results of Discrete Element Method (DEM) simulations. Two cases with different particle volume fractions are considered. The Nanbu-Babovsky model is also evaluated in comparison to the O’Rourke model. These models are evaluated based on their accuracy, statistics and computational cost.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Bibliographic ID: UF00102023
Volume ID: VID00271
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 1074-Gumprich-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Evaluation of collision models applied to varying dense particle jets flows


W. Gumpricht, M. Chriguit, M. Braun* and A. Sadikit

ANSYS Germany GmbH, Birkenweg 14A, 64295 Darmstadt, Germany
t Institute for Energy and Powerplant Technology, TU Darmstadt, Petersenstr. 30, 64287 Darmstadt, Germany
gumprich@ekt.tu-darmstadt.de and markus.braun@ansys.com
Keywords: particle collision, stochastic model, lagrange, dispersion




Abstract

Standard stochastic inter-particle collision models are derived from the kinetic theory of gases. These models
differ from each other in their algorithmic approach and their modification of the collision frequency, accounting for
additional effects such as turbulence. In comparison to deterministic models, the major advantage of stochastic models
regards their lower computational cost. However, while stochastic models allow the simulation of inter-particle
collisions for cases in which a deterministic model would be impracticable, the computational cost is still a critical
issue. The Nanbu-Babovsky model achieves a computational expense linearly proportional to the number of parcels
and conserves energy, being advantageous over other stochastic models. The model, implemented into the commercial
CFD software ANSYS FLUENT, is validated based on results of Discrete Element Method (DEM) simulations.
Two cases with different particle volume fractions are considered. The Nanbu-Babovsky model is also evaluated in
comparison to the O'Rourke model. These models are evaluated based on their accuracy, statistics and computational
cost.


Introduction

Stochastic collision models are straightforward ap-
proaches for the simulation of inter-particle collisions
in dense dispersed two-phase flows. They can be eas-
ily coupled to standard Euler-Lagrange simulations and
the physical phenomena taking place during a collision
process can be properly modeled. These phenomena
depend on the material properties of the particles. In
case of solid-particles, for instance, effects such as cen-
tral collision, friction, rotation or rupture may occur,
while the collision of liquid-droplets may result in co-
alescence, bouncing or breakup.
Some well known models, such as O'Rourke (1981),
Oesterle & Petitjean (1993) and Sommerfeld (1996),
are implemented in commercial CFD software and
widely used in the simulation of dispersed two-phase
flows. In comparison to deterministic models, the ma-
jor advantage of stochastic models is their lower com-
putational cost. They allow the representation of several
real particles by a single simulated parcel, reducing the
number of particles to be computed, while the transient
time-step can be considerably larger.
However, while stochastic models allow the simula-
tion of inter-particle collisions for cases in which a de-


terministic model would be impracticable, the compu-
tational cost is still a critical issue. In order to obtain
reasonably accurate results, the required amount of com-
puted parcels is, in many cases, of the order of hundreds
of thousands. O'Rourke (1981), for instance, computes
collisions for each particle pair in a computational grid
cell, resulting in a computational cost proportional to
the square of the number of parcels. Oesterl6 & Petit-
jean (1993), extended by Sommerfeld (1996), achieve a
computational expense linearly proportional to the num-
ber of parcels by considering a collision partner to be
virtual, with averaged properties of particles in a cell,
and computing collisions for each particle. However,
the conservation of energy cannot be guaranteed if the
collision partner is virtual.
The Nanbu-Babovsky stochastic inter-particle colli-
sion model is derived from the original Nanbu (1980)
model, which was developed to predict molecule colli-
sions in rarefied gas dynamics and to approximate the
solution of the Boltzmann equation. Babovsky (1989)
has modified the original model in order to reduce its
computational cost and to guarantee the conservation of
energy during collisions. The model, which calculates
collisions for each particle in a cell, resulting in a com-
putational cost linearly proportional to the number of











parcels, is implemented into the commercial CFD soft-
ware ANSYS FLUENT.
The implementation of the stochastic collision model
considers the outcome of solid-particle collisions to be
a central collision with randomly determined collision
point.
In this paper, results of Nanbu-Babovsky, O'Rourke
and Discrete Element Method (DEM) simulations are
compared. Two cases with different particle volume
fraction of about 0.8% and 3%, in the region of higher
concentration, are considered. The aim is to validate the
stochastic models based on results of the deterministic
model and evaluate the limitations of the stochastic mod-
els for denser regimes where the occurence of multiple
particle collisions is relevant. Stochastic models are lim-
ited to the calculation of binary particle collisions.
DEM simulations were performed with an internal
ANSYS DEM code. They were configured to only ac-
count for recoil effects on particle collisions, neglecting
other inter-particle interaction mechanisms. These re-
coil effects are described by spring forces resulting from
particle overlapping, based on the Hooke's Law of elas-
ticity, and are equivalent to a central particle collision.
Another aim of this work is to evaluate the Nanbu-
Babovsky and the O'Rourke models regarding their
computational cost and capability of meeting the ex-
pected statistics for different particle and domain dis-
cretizations.

Nomenclature

Roman symbols
D spring constant (N/m)
Fspring spring force (N)
k restitution coefficient
mi particle mass (kg)
np number of particles in parcel
N number of parcels in a cell
No011 expected number of collisions in a cell
Pi probability of collision
ri particle radius (m)
Ate collision time-step (s)
vi particle velocity (m/s)
vij | particles' relative velocity (m/s)
Vc cell volume (m3)
Greek symbols
vij collision frequency (1/s)
V) uniformly distributed random number
particle volume fraction in a jet
6 length of particle overlapping (m)
Subscripts
i, j particle index
n direction of collision


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Stochastic Inter-Particle Collision Models

The standard stochastic models are derived from the ki-
netic theory of gases. They differ from each other in
their algorithmic approach and in their modification of
the collision frequency, given by

Vij (r + r)2 Vij (1)
Vc
for particles i and j. The collision frequency describes
the probability per unit time that a particle in parcel i
collides with a particle in parcel j. It assumes that par-
ticles are moving in vacuum and that particle velocities
and positions are uncorrelated.
Based on equation 1, the expected number of colli-
sions to occur in a cell over a time-step Ate is computed
with


1 N N
Noi = n jV At
i=1 j=1


where the 1/2 term is a result of symmetry.
While different models modify the collision fre-
quency to account for the influence of the flow on inter-
particle collisions, their base theory is the same and
modifications of the collision frequency can be applied
to any of these models. Sommerfeld (1996), for in-
stance, takes turbulence effects into account by consider-
ing the fluctuating velocity in the relative velocity |vj |.
O'Rourke (1981) introduced a so called collision effi-
ciency to correct vij, accounting for the interaction of
the droplets with the surrounding gas flow.
Moreover, these models are limited to the occurrence
of binary particle collisions and only particles located in
the same computational grid cell are allowed to collide
with each other.

O'Rourke
The O'Rourke model calculates collisions for each
particle pair in a grid cell. It describes the probability
that a particle in parcel i collides with n particles in par-
cel j with a Poisson distribution. Hence, the probability
of no collision between i and j is given by


Pi,o = exp(-VijAtc)


A collision occurs when y > Pij,o, where E [0, 1]
is a uniformly distributed random number.
In case of solid-particles, however, a particle in i
may collide with at most one particle in j. Therefore,
vijAt, < 1 is required in order to achieve the expected
statistics.











Nanbu-Babovsky
The Nanbu-Babovsky model calculates collisions for
each particle i in a grid cell. The probability that i col-
lides with any other particle in the cell is given by

N
P, = EPi (4)
j-1

where Pij is the 1p' b.ibilii \ ih.l i collides with, defined
as
1
Pij = 2 At (5)

A collision occurs when

> Pij (6)

where y E [0, 1] is a uniformly distributed random num-
ber. On average, this is equivalent to y < Pi. The colli-
sion partner j is determined randomly

j = [Nj + 1 (7)

where [...J describes the integer part of the argument.
Hence, the sum in equation 4 is not solved and the com-
putational cost of the model is linearly proportional to
the number of parcels.
There is a time-step limitation associated with the
Nanbu-Babovsky model. This limitation results from the
condition that NPj < 1. The fulfillment of this condi-
tion can be guaranteed by limiting the collision time-step

2
Atc < -- (8)
VijN

Hence, the maximum allowed collision time-step
At.,max is determined by estimating the maximum col-
lision frequency for all particles in a cell

2
A x max(vij)N


Collision Outcome
The collision outcome is a central collision with ran-
domly determined collision point. The collision point is
the point where the two colliding particles touch each
other. It results from the intersection of the straight line
connecting the centroid of both particles and their sur-
face. The frame of reference of particle i is consid-
ered and the collision direction is described by the ba-
sis vector n, which points in the direction of the straight
line connecting the centroid of i and the collision point.
Hence, derived from the laws of momentum and energy
conservation, the post-collisional velocity components


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


of the particles in n direction is computed with follow-
ing equations:


mnvi,n + ". kmj(vi, j,) (10)
mi + mj


new ivi,n + ..* .. kmi(vj,n vi,n)
n (11)
mi + mnj

where m, and my describe the mass of each particle.
Note that only the velocities in the direction of n are
changed. k is the restitution coefficient. k 1 in ideal
elastic collisions, while in ideal plastic collisions, k = 0.

Solution Strategy
Stochastic inter-particle collision models are executed
after particle trajectories are calculated with the La-
grangian solver. Since only particles located in the same
computational grid cell are allowed to collide with each
other, the collision routine loops over all cells and calcu-
lates collisions for the particles located in each one.
The occurrence of collisions is determined through
a Bernoulli-Experiment. The Nanbu-Babovsky imple-
mentation estimates the collision time-step before cal-
culating collisions for the particles in a cell. If the es-
timated time-step is smaller than the Lagrangian time-
step, the Bernoulli-Experiment is repeated more than
once, until the collision time is equal to the Lagrangian
time-step. The O'Rourke implementation considers the
Lagrangian time-step to be the collision time-step.
In case of O'Rourke, the routine loops over all par-
ticle pairs in the cell and tests each pair for collision.
The Nanbu-Babovsky routine loops over all particles in
a cell, testing each particle for collision. Once a colli-
sion occurs, the direction of collision is randomly deter-
mined and the post-collisional velocities are calculated
with equations 10 and 11.

Discrete Element Method

DEM is a deterministic model and particle collisions
occur only if particles overlap each other. Simulations
were configured to only account for recoil effects on par-
ticle collisions, neglecting other inter-particle interac-
tion mechanisms. These recoil effects are described by
spring forces resulting from particle overlapping, given
by


Fspring D6


based on the Hooke's Law of elasticity, where D is the
spring constant and 6 is the length of overlapping. The
spring force acts in the direction of the straight line con-
necting the centroid of both particles.
This requires a sufficiently small computational time-
step At to accurately predict the outcome of collisions,


SI


























Figure 1: Problem geometry.


depending on the given spring constant. A larger At
results in a larger particle overlapping and, depending
on D, the recoil force may be strongly over-predicted.
Hence, even though momentum is conserved in all coor-
dinate directions, the method does not conserve energy
and the error of the post-collisional kinetic energy de-
pends on the computational time-step.
For the simulations presented in this work, a spring
constant D = le+05 N/m and a At = 2e-08 s were con-
sidered. Investigation has shown that, with this configu-
ration, the expected error is about 1.5% and 4.5% for
the cases with lower and higher particle density, respec-
tivelly. This error is comparable to experimental mea-
surement devices and is sufficient to assess the quality
of the stochastic models.

Configuration

The problem geometry is illustrated in figure 1. Two
particle jets with 0.015 m of diameter and 45 degrees
of inclination relative to the boundary are injected into a
three-dimensional domain. As both jets cross each other,
particle collisions occur causing the dispersion of parti-
cles over the computational domain.
The injected particles have 5e-04 m of diameter and
15 m/s of initial velocity in the direction of the jet axis.
The problem is configured in such a way that parti-
cle motion is affected only by inter-particle collisions.
Hence, particles are considered to be moving in vac-
uum and drag forces as well as gravitation are set to 0.
Inter-particle collisions are considered to be ideal elastic
(k 1), while other material properties are irrelevant.
Moreover, the boundary conditions are set to escape and
particles leave the domain after reaching its borders.
Overall, two cases with particle volume fraction of
about 0. and 1.5% in each jet were computed, while
in the region near the crossing point of both jets the par-
ticle volume fraction may be up to twice as much. The
particle volume fraction in each jet is described with


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


.In order to have enough parcels for a statistical
application of the model, the case with '. 0. .
was computed with one particle per parcel (n, 1).
The other case, with '. 1.5%, was computed with
np 5. In addition, all stochastic simulations were
computed ti.iiiMnicill with a computational time-step At
= le-04 s, and a grid with 16 x 16 x 40 cells was used.


Results and Discussion

Particle Volume Fraction of 0.4% in each Jet
Since the Nanbu-Babovsky and O'Rourke models
share the same collision frequency and the amount of
simulated parcels is sufficient for a statistical treatment
of the problem, both models delivered practically the
same results.
Qualitatively, the stochastic models predict the same
as DEM. One can see in figure 2 (bottom) that, as both
jets cross each other, some particles collide with each
other as are dispersed over the computational domain,
while other particles that do not collide, or collide only
slightly, keep a path similar to their original one. This is
the same process predicted by DEM, illustrated in figure
2 (top).
In figure 3, this process can be recognized in the di-
mensionless particle concentration profiles over the y-
coordinate, in sampling planes A, B and C. The dimen-
sionless particle concentration is defined as the ratio of
the number of particles sampled in each measurement
interval to the mean number of particles. The mean is
the total number of sampled particles in each plane di-
vided by the number of intervals. The profiles are plotted
separately in order to better analyze the behavior of each


A BC

I .



l
I I I DEM



injection-1

injection-0 ..

Nanbu-Babovsky
Z-x

Figure 2: Particle dispersion colored by particle veloc-
ity magnitude for the '. 0. case.














006

007

006

005

0.04
03
0.03


DEM
-*Nanbu-Babovsky
**O'Rourke


a,.
k ~-
''


a 01 Plane A

0 1 2 3 4 5 6 7 8 9 10
o as
008-

0 07 k

006 "-,
005 ,,
00 as-


A.


0.04

003

0.02

Plane B 001


1 2 3 4 5 6 7 8 9 10


007- -

006 -

005 -

0.04-

003-
0 02

oo01 Plane C

01 234 56789
N/Nm


Figure 3: Dimensionless


0.06

0.05

0.04

003 -

0.02



0 1 2 3 4 5 6 7 8 9 10
0.08 ------

0.07 -

0.06 .,

0.05


06 1 2 3 4 5 6 7 8 9 10
0.08


0.06

0.05

0.04 -

003


0.02

001- '

1 23 4 5678
N/Nm


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010



particles is considered. Again, the profiles are qualita-
tively comparable, while a deviation between the results
of stochastic models and DEM is present.

The effects of inter-particle collisions are also de-
scribed by the change in the original velocities of the
particles. In comparison to DEM, this velocity change
is accurately predicted by the stochastic models. Figure
5 (left) shows the particle axial velocity (i.e. x-velocity)
profiles in each plane. These profiles reflect the averaged
velocities of the particles sampled in each measurement
interval. The average axial velocity is smaller in plane
A than in planes B and C, since the concentration of par-
ticles near the jets crossing point is higher and several
particles with a lower axial velocity and a higher trans-
verse velocity are located in this region. As particles
move away from the crossing point, the particles that re-
main near the domain axis are the ones with a higher ax-
ial velocity and lower transverse velocity. Consequently,
the average particle axial velocity near the center axis is
higher in planes B and C. The averaged velocity near the


9 10


particle concentration of


injection-0 (left) and injection-1 (right), for
the '. z 0. case.




jet, where injection-0 describes the jet on the bottom and
injection-1 the one on the top (see figure 2). Since many
particles do not collide, the concentration of particles
is higher within the particle jets. This is described by
the peak of each profile. These peaks are closer to the
boundaries in planes farther from the jet crossing point,
describing the movement of the jets toward the domain
borders, while the symmetry of both jets can be seen.
Even though the stochastic models qualitatively predicts
the particle concentration with good agreement, a small
deviation between the results of stochastic models and
DEM can be seen. After the crossing point, the stochas-
tic models predict a higher particle concentration within
each jet. This reflects an under-predicted number of col-
lisions, which may be a result of the models' grid depen-
dency.

Figure 4 (left) shows the particle volume fraction pro-
files. The volume fraction describes the dispersion of
particles more accurately, since the amount of sampled


* DEM
*Nanbu-Babovsky
-*O'Rourke


Plane A

0204 06 0.8 1 1.2 14 16





.>
/






Plane B -

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6


Plane C


0.2 0.4 0.6 0.8 1 1.2 1.4
DPM volume fraction [%]


0 008


00 0.5 1.5 2 2.5 3
0 08

0.07,

0.06 i

005

004-

003 '

0.02
4
0.01.

0 0.5 1 1.5 2 2.5 3

0 D7
ooa6,--------
007


A

004

003

002

001*1

DO 0 5 1 15 2 25 3
DPM volume fraction [%]


Figure 4: Particle volume fraction for the '
(left) and 1 -.'. (right) cases.


* --













008 -. i i
S. DEM
007 .. Nanbu-Babovsky
O'Rourke
0.06
005 -
I0.04 ,
0.03 -
0.02-
o.o01 / Plane A
0 10 15 20 25
008 i i '
0.07
0.06-
0.05 -
o0.04- 7
0.03
002 l
0.01- PIno R


004
>. /
0.03 -
002- f
o01- Plane C

0 5 l10 15 20
Particle x velocity [m/s]


0.02
0.01- *

20 15 10 5 0 5 10 15 20
008 1
007 -
006 7
0.05 -
004- -"
0.03
0.02
0.01-

-20 -15 -10 -5 0 5 10 15 20
008
t
007
oo006
005- /'
004
0.03-
002- f
001

20 -15 oI -5 0 5 110 15 20
Particle y velocity [m/s]


Figure 5: Axial (left) and transverse (right) velocity
profiles for the '. 0. case.



domain borders is also higher in planes B and C, in com-
parison to plane A. This is explained by a higher concen-
tration of particles in these regions. In plane A, few par-
ticles are sampled near the borders, since only a few par-
ticles collide in such a way that their ratio of transverse
to axial velocity is large enough for them to approach
the borders before crossing plane A. The velocity oscil-
lations in this region is explained by the small number
of sampled particles. Also, the location of the particle
jets, which contains particles that did not collide, is rec-
ognizable in the axial velocity profiles in planes B and
C. In these locations, the velocity profile is more or less
flat, describing the initial particle axial velocity.
The particle transverse velocity (i.e. y-velocity) pro-
files are shown in figure 5 (right). In the domain center
axis (y 0.4), the transverse velocities are averaged to
0 due to symmetry. Closer to the borders, the flat profiles
describe the location of the particle jets, with unchanged
initial velocities. In planes A and B, the velocity oscil-
lates near the domain boundaries due to the low amount
of sampled particles.


:Ir

i


injection-1

iniection-o -



Z-A


Natrliu-Bao:E-. k,


Figure 6: Particle dispersion colored by particle veloc-
ity magnitude for the '. 1 ".'. case.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010



Particle Volume Fraction of 1.5% in each Jet
In a denser regime, most particles collide and tend
to accumulate near the domain axis before being dis-
persed. Qualitatively, this effect is accurately predicted
by the stochastic models, as illustrated in figure 6 for
the '. 1.5% case. In this case, the average dis-
tance between particle centroids may be as low as 2.6
particle diameters in the jets crossing region. Hence, the
occurence of multiple particle collisions is considerable.
Figure 7 shows the dimensionless particle concentra-
tion profiles of each particle jet. Even though the jet
symmetry is recognizable, all peaks are located near the
center axis, describing the accumulation of particles in
this region. DEM predicts a higher relative concentra-
tion of particles near the center axis. However, it does
not mean that the predicted particle volume fraction is
higher, since the number of sampled particles and the
Nmean is smaller in DEM results. This is a result of a
higher particle dispersion in lateral (i.e. z-coordinate)
direction. According to figure 4 (right), DEM predicts
a lower particle volume fraction in this region. Hence,
the stochastic models predict a higher dispersion in y-
coordinate direction, while DEM predicts a higher dis-
persion in z-coordinate direction. This may be explained
by the high concentration of particles near the crossing
point and the occurrence of multiple particle collisions,
which is not taken into account by stochastic models. As
the particle flow coming from the top and bottom of the
domain is intense, more particles tend to disperse to the
sides. Qualitatively, however, this effect is accurately
predicted by the stochastic models, as seen in figure 8.
The axial and transverse particle velocity profiles in
each sampling plane is illustrated in figure 9. The



A BC


I 4


I -A
i O E . .M'


,"1 :









7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


0.08---I-'-I-'-I
DEM
007 *Nanbu-Babovsky
S *.O'Rourke
0 06-
005- ^
0 04
>, _--------''
0.03
002-
0oo- Plane A

0 1 2 3 4 5 6 7 8 9 10
0.08i
0074
006
005 "'
004 ">
0.03
002 T
001o Plane B

S1 2 3 4 5 6 7 8 9 10
008- -
007-
0.06 ,
0.05 -
1004 '*
003 .'
002
0.01' Plane C

0 1 2 3 4 5 6 7 8 9 10
N/N


0.08 ----
0.07
0.06


0.05 3


0.02
001

0 1 2 3 4 5 6 7 8 9 10


0 1 2 3 4 5 6 7 8 9 10
0.081------------


0.06 -
0.05 ,
0.04
003
0.02 '
0.01

0 1 2 3 4 5 6 7 8 9 10
N/N


Figure 7: Dimensionless particle concentration of
injection-0 (left) and injection-1 (right), for
the '. w 1.5% case.



stochastic models predict the post-collisional particle
velocities with good accuracy. Here, the tendency is
similar to results for the lower particle volume fraction
S 0. I'. However, since most particles collide and
the particle dispersion over the domain is higher, the pro-
files are smoother and the presence of the particle jets is
no longer recognizable. Near the domain boundaries, the
amount of particles sampled in plane A is low, resulting
in an oscillation of averaged velocities.


Statistics Investigation

A quick investigation was conducted to verify if the
implemented Nanbu-Babovsky model and the O'Rourke
model are able to meet the expected statistics for differ-
ent particle discretizations and grid resolutions. In order
to ensure enough particles for the application of a statis-
tical treatment even for a coarser particle discretization,
a denser regime with a particle volume fraction close to
6% in each injected particle jet was considered.


I 3M601
2 S5101
2 71 01
2 53&t01
242&-01


DPE.l


Figure 8: Cross section view of particle dispersion be-
hind plane A for the '. 1 ".'. case.




The expected number of collisions to occur in the
computational domain, over a time-step At le-04 s,
is the sum of the expected values of each grid cell, com-
puted with equation 2. Figure 10 compares the number
of computed collisions for different particle discretiza-
tions, ranging from 1 to 80 particles per parcel, and the
averaged expected number of collision events over time,
computed based on the case with 1 particle per parcel.
A computational grid with 16 x 16 x 40 cells was used.
The number of computed collisions with the Nanbu-
Babovsky model oscillates about the expected value for
all cases, while the amplitude of oscillations increases
as particle discretization gets coarser. O'Rourke fails to
meet the expected statistics if the particle discretization
is coarse enough, under-predicting the number of colli-
sions. This is due to the large vij At term in equation 3,
resulted from a large np.

As described in equation 1, the collision frequency of
stochastic models depends on the volume of the cell in
which the particles are located. Therefore, the resolu-
tion of the computational grid may have a substantial
influence on the accuracy of the collision model if par-
ticles are non-uniformly distributed over the computa-
tional domain. This occurs due to wasted cell volume
by a coarser grid, resulting in a smaller calculated col-
lision frequency. Figure 11 illustrates the dependency
of the stochastic models on the grid resolution. Simula-
tions with three different computational grids were per-
formed, where parcels representing one single particle
are injected. For each simulation, both the expected and
computed number of collisions were tracked. The num-
ber of computed collisions decreases for coarser grids,
due to the effect of wasted volume. However, since
this dependency is related to the collision frequency, the
number of expected collisions also decreases for coarser
grids. The Nanbu-Babovsky model meets the expected
statistics for all grid resolutions, while O'Rourke consid-
erably under-predicts inter-particle collisions if the com-
putational grid is fine enough. Again, this is a results of
a large ,j At, term in equation 3.


. '* *.


.,'*,*.., -
', ,* )i


131-
I 3901
* rlOr X









7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


0 08 i i i .
SM *DEM
007, Nanbu-Babovsky
*. *'O'Rourke
006-
005
0.04
003 .
002- -
ool .* Plane A

0 5 10 1 5 20 25
008
007 -
0.06
0.05
. 004
003
002 V
0.01- Plane B

00 5 10 15 20 25
0 o08
0.07 .
006 -
005
0 04
0 03
0.02-
o01 Plane C
0.
0 5 10 15 20 25
Particle x-velocity [m/s]


008
007
006
005
0.04
0.03

"'{
002-
o01 4

020 -15 -10 -5 0 5 10 15 20
0.08
0.07
0.06 -
0.05
0.04
003 3
0.02
0.01 *

20 15 10 -5 0 5 10 15 20
0.08 ,
0.07
0.07 -
0.06 -
0.05
0.04
0 03
0.02
001-

20 -15 -10 -5 0 5 10 15 20
Particle y-velocity [m/s]


Figure 9: Axial (left) and transverse (right) velocity
profiles for the '. z 1.5% case.




Computational Cost

The Discrete Element Method is computationally
very expensive. The serial simulation of the. ;
1.5% case, with about 13,000 particles in the domain,
for instance, required more than a week of computa-
tional time in a 2.3 GHz AMD Opteron CPU, while the
stochastic models delivered results in just a few minutes.

Although the computational cost of the O'Rourke
model is substantially lower in comparison to DEM, the
model is still expensive for cases with a large amount
of simulated parcels. In order to compare the compu-
tational expense of the Nanbu-Babovsky and O'Rourke
approaches, simulations injecting different number of
parcels into the domain were performed with each
model. A denser regime with a particle volume fraction
close to i*'. in each particle jet was considered. First,
1,000 time-steps of 2e-05 s were computed to ensure a
more or less stationary state. Then, 200 additional time-
steps were computed while the elapsed computing time
was measured.


012000
-

S10000
E

a 8000

0 6000

o 4000

- 2000
E
z


O'Rourke


I ~i


0 0.01 0.02 0.03
Time [s]


Figure 10: Statistics investigation for different particle
discretizations.




Figure 12 shows the elapsed computing time of simu-
lations with different number of parcels in the computa-
tional domain. If the amount of parcels is small enough
(i.e. < 29,000), other processes have a higher weight
on the computing time than the collision model, and
the cost increase is not even linearly proportional to N.
Between 29,000 and 290,000 parcels, the cost increase
of Nanbu-Babovsky simulations is more or less linearly
proportional to N, being a bit over-proportional after
92,000 parcels. This may be a result of the time-step
condition 8. The O'Rourke approach does not present
a cost increase proportional to the square of the number
of parcels. This may be explained by the model's ap-
proach of computing collisions for particle pairs in each
cell. Since particles are not necessarily proportionally
distributed over the cells after refining the particle dis-
cretization, the cost increase does not have to be pro-
portional to the increase in the overall number of parti-
cles. However, the computational cost of the O'Rourke
approach increases more rapidly than the one of the
Nanbu-Babovsky model. If 290,000 parcels are com-
puted, for instance, the cost of the O'Rourke model is
about 167% higher.

This investigation reflects the behavior of a particular
case, however, the time-step condition 8 also depends
on the collision frequency vij. In cases with a high dis-
crepancy in particle sizes and non-uniform particle dis-








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


.10000

S8000
0
6000

4000
'5


S2000
E


Figure 11: Statistics investigation for different domain
discretizations.


cretization, as well as a large number of real particles,
max(vij) is high, requiring a considerably small At,.
However, it is important to mention that, if these cases
involve solid-particles, a simulation with the O'Rourke
approach also has to account for a certain time-step con-
dition in order to limit the term uij Atc, whether consid-
ering a Poisson distribution or not, since only one colli-
sion is allowed between particles in each colliding par-
cel. Otherwise, the number of collision events is under-
predicted, as seen in figures 10 and 11.


Conclusions

In this work, the Nanbu-Babovsky stochastic inter-
particle collision model, implemented into the commer-
cial CFD software ANSYS FLUENT, is introduced. The
model is validated based on results of Discrete Element
Method (DEM) simulations. An internal DEM code was
used.
Even though stochastic models are advantageous be-
cause of their lower computational cost in comparison
to deterministic models, the computational cost is still a
critical issue. In many cases, the amount of computed
parcels is of the order of hundreds of thousands, requir-
ing a lot of hardware resources to calculate inter-particle
collisions.


10000


SNanbu-Babovsky
*--*O'Rourke


1000

a,
E


WO 100
E


Nanbu-Babovsky

- ., A(V




S- 8x8x20 cells
... 8x8x20 cells (expected)
16x16x40 cells
-16x16x40 cells (expected)
32x32x80 cells
--- 32x32x80 cells (expected)
0.01 0.02 0.03 0.04 0.


le+06


Figure 12: Elapsed computing time of Nanbu-
Babovsky and O'Rourke simulations.


The main advantage of the Nanbu-Babovsky model
in comparison to other stochastic models is that it is
able to compromise the conservation of energy with a
lower computational cost. While some models consider
a particle's collision partner to be virtual in order to re-
duce their computational expense, the Nanbu-Babovsky
model considers a real collision partner, conserving en-
ergy during collision processes. For certain computa-
tional time-steps, the model has a cost linearly propor-
tional to the number of parcels and, although this lin-
earity does not hold for all time-steps, the model's ex-
pense is considerably lower in comparison to the ap-
proach of the O'Rourke model, which computes colli-
sions for each particle pair is a cell.
Results of both Nanbu-Babovsky and O'Rourke sim-
ulations are compared to DEM results. If the amount
of simulated parcels is high enough for applying a sta-
tistical treatment of the problem, and the collision time-
step is small enough, both models deliver practically the
same results, since they are based on the same theory. In
comparison to DEM, both stochastic models are able to
predict inter-particle collisions with reasonable accuracy
and low computational cost. A DEM simulation that re-
quired more than a week to be concluded was performed
with both stochastic models in just a few minutes. The
accuracy of the models is investigated for particle vol-
ume fractions up to :'. in regions of higher particle con-
centration.
The collision statistics and, consequently, the results
of stochastic models depend on the computational grid
resolution. The Nanbu-Babovsky model is robust re-
garding its capability of meeting the expected statis-
tics. The number of computed collisions always oscil-
lates about the expected values, independent on the par-
ticle discretization and grid resolution. If the number
of simulated particles is high enough, the expected col-
lision frequency is met with good accuracy. This is a
result of the collision time-step estimation in the im-


le+04 le+05
Number of parcels


~ I T T r











plemented model. The Nanbu-Babovsky model con-
siders the collision time-step apart from the Lagrangian
time-step, such that collisions may be computed more
than once per Lagrangian time-step. This is just a rep-
etition of the Bernoulli-Experiment and particle trajec-
tories are not calculated between consecutive collision
time-steps. The O'Rourke model, however, considers
the Lagrangian time-step and, in some cases, the num-
ber of calculated collisions is under-predicted due to a
large expected number of collisions between two parti-
cles over a single time-step.
However, both stochastic models have their limita-
tions. In contrast to deterministic models, the stochastic
models do not take particle positions into account, being
appropriate to the simulation of free flow regimes, where
accumulation of particles does not occur. These mod-
els are unable to control the concentration of particles in
cases where particles tend to accumulate in certain re-
gions, such that the bulk density of the dispersed phase
may exceed the maximum allowed in some grid cells.
Also, stochastic models are limited to the occurrence
of binary particle collisions, being suitable to cases in
which the average distance between particles is much
larger than their diameters.
In order to accurately meet the expected statistics over
each computational time-step, the amount of simulated
parcels needs to be sufficiently large. Even though the
Nanbu-Babovsky model is robust and the number of
computed collision events always oscillates about the
expected values, the amplitude of these oscillations in-
creases as the amount of computed parcels decreases,
possibly resulting in a higher simulation error.
Moreover, the implemented Nanbu-Babovsky model
assumes that particles are moving in a vacuum, not con-
sidering the influence of the continuous phase on inter-
particle collisions and not taking turbulence into ac-
count, being more suitable to flow regimes with St > 1.
In order to achieve accurate results for cases in which
the flow is influential to the occurrence of inter-particle
collisions, the model has to be extended. This can be
easily achieved by modifying the collision frequency.
O'Rourke, for instance, defined a collision efficiency
used to modify the collision frequency, accounting for
the interaction of the droplets with the surrounding gas
flow. In highly turbulent cases, with St < 1, an ap-
proach similar to the one of Sommerfeld, considering
the fluctuating velocities, could be used.


Acknowledgements

The authors appreciate the technical support of Ralf
Kroeger in configuring the Discrete Element Method
simulations.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


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