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# PARCEL-BASED APPROACH FOR THE SIMULATION OF GAS-PARTICLE FLOWS

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We focus on a parcel-based approach, similar to the one used by O'Rourke and Snider, 2010, which tracks the motion of a so-called "parcel" of particles. We derive a scaling law for a linear-spring dashpot interaction model that enables tracking of clouds of particles through DEM-based simulation of (scaled) pseudo-particles. This guarantees convergence to a DEM-based simulation of the unscaled system, i.e., a system where all the individual particles are tracked. We use a BGK-like relaxation term to model collisions between particles in dilute regions of the flow field. This combined approach is implemented in an in-house code that runs on GPUs (Radeke et al., 2010), and is used to study a granular jet impinging on a plane surface, as well as a simple shear flow. We find that a BGK-type relaxation model is necessary when using parcel-based approaches for capturing some prominent flow features.
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8th International Conference on CFD in Oil & Gas, Metallurgical and Process Industries
SINTEF/NTNU, Trondheim Norway
21-23 June 2011
CFD11-124
1
PARCEL-BASED APPROACH FOR THE
SIMULATION OF GAS-PARTICLE FLOWS
1
2
, Johannes G. KHINAST
2,3
, Sankaran SUNDARESAN
1
1
Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ
2
Research Center Pharmaceutical Engineering GmbH, Graz, Austria
3
Institute for Process and Particle Engineering, Graz University of Technology, Graz, Austria
ABSTRACT
We focus on a parcel-based approach, similar to the one
used by O'Rourke and Snider, 2010, which tracks the
motion of a so-called “parcel” of particles. We derive a
scaling law for a linear-spring dashpot interaction model
that enables tracking of clouds of particles through
DEM-based simulation of (scaled) pseudo-particles.
This guarantees convergence to a DEM-based
simulation of the unscaled system, i.e., a system where
all the individual particles are tracked. We use a BGK-
like relaxation term to model collisions between
particles in dilute regions of the flow field. This
combined approach is implemented in an in-house code
that runs on GPUs (Radeke et al., 2010), and is used to
study a granular jet impinging on a plane surface, as
well as a simple shear flow. We find that a BGK-type
relaxation model is necessary when using parcel-based
approaches for capturing some prominent flow features.
Keywords: Discrete element method, simulation, granular
impinging jet, shear flow.
NOMENCLATURE
Latin Symbols
c Damping coefficient of primary particles,
[kg/s].
C Constant in the filter function.
d Diameter, [m].
e
p
Coefficient of restitution.
E Young’s modulus, [N/m²].
f Particle distribution function.
F Force, [N].
g
0
G Filter function.
h Switch-off function.
k Stiffness of primary particles, [N/m].
K Constant.
m Mass, [kg].
N Number of particles in a parcel.
p Pressure, [N/m²].
P Probability distribution function.
r
32
Sauter mean radius of primary particles, [m].
t Time, [s].
T Granular temperature, [m²/s²].
T Torque, [N
.
m].
U Initial velocity, [m/s].
v Velocity, [m/s].
y Vertical distance, [m].
Greek Symbols
α
Scaling ratio.
β
Size ratio of colliding particles.
δ
Overlap, [m].
Dimensionless filter length.
φ
Volume fraction.
γ
Shear rate, [1/s].
η
Damping function.
κ
Exponent.
λ
Experimental parameter for the scattering
angle.
µ Friction coefficient.
Π
Dimensionless parameter.
θ
ρ
Particle density, [kg/m
3
].
σ
σσ
σ Stress, [N/m²].
τ
Shear stress [N/m²].
τ
D
Relaxation time, [s].
ω
ωω
Sub/superscripts
*, *’ Dimensionless quantities.
BGK Bhatnagar–Gross–Krook.
CP Close packed.
eff Effective.
el Elastic.
eq Equilibrium.
fluct Fluctuation.
half Half.
impact At impact.
jet Jet.
model Modeled term.
n Normal direction.
t Tangential direction.
off Value at switch-off.
p Parcel.
prim Primary particle.
2
PP Particle-particle.
PW Particle-wall.
roll Rolling.
sample At sample position.
0 Reference value.
INTRODUCTION
Discrete Particle Models (DPM), aim at tracking
individual particles, or parcels thereof, in the flow
domain. By using this approach, the treatment of a
granular assembly made up by particles with different,
size, density, shape, or composition is straightforward.
Particle-based methods offer the possibility to include
complex particle-wall interactions, the handling of rare
events, as well as to extend the method to gas-particle-
droplet systems (Link et al., 2009; O'Rourke et al.,
2009; Zhao et al., 2009). Interest in the latter systems
has been recently motivated by the pharmaceutics
2010), and also by classical applications like coking
(typically performed in fluidized beds, see Darabi et al.,
2010; Radmanesh et al., 2008), or catalyst impregnation.
When using a DPM, two major approaches can be
chosen: First, each particle in the system can be tracked.
In this case, relatively simple models based on
instantaneous (“hard-sphere” model), or enduring
collision dynamics (“soft-sphere” model, recently
summarized by Cleary, 2009) can be employed. The
soft-sphere approach is more general, and allows also
for the simulation of dense regions with enduring
contacts, and is usually referred to as “Discrete Element
Method” (DEM).
Second, the particle population can be represented by
so-called “parcels” of particles, i.e., a cloud of particles,
and hence the name parcel-based approach. Another
way of interpreting such an approach is to think of a
(discrete) approximation of the particle distribution
function by test particles. For gas-solid flows, Andrews
and O'Rourke, 1996, proposed the so-called “Multi-
Phase Particle-In-Cell” (MP-PIC) approach, which is a
parcel-approach. MP-PIC has been applied widely to
fluidized beds, sedimentation, hopper flow, as well as
other dense granular flows (O'Rourke et al., 2009;
O'Rourke and Snider, 2010; Snider, 2001; Snider,
2007). In principle, a parcel-based approach does not
require a second-phase to be present; however, MP-PIC
requires an implicit coupling to the fluid phase due to
stability reasons. The MP-PIC approach does not track
collisions between particles directly, but employs a
simple “particle pressure” model to prevent particles
from becoming close-packed. Instead of modeling
particle interaction forces with a particle pressure,
Patankar and Joseph, 2001, explored the use of a simple
soft-sphere model (without friction) in conjunction with
a parcel-based method for a relatively small system. A
similar approach was taken by Sakai et al., 2010; Sakai
and Koshizuka, 2009, as well as Mokhtar et al., 2011.
Only recently Bierwisch et al., 2009 have shown that a
parcel-based approach with contact detection, when
using appropriately scaled interaction parameters, yields
simulation results independent of the number of particles
making up the parcel. Bierwisch et al., 2009 showed that
this scaling must be based on identical (i) particle
density, (ii) coefficient of restitution, (iii) friction
coefficients, and (iv) Young’s modulus for their
Hertzian repulsion force model. The scaling proposed
by Bierwisch et al., 2009 yields stresses in the quasi-
static regime, and parcel velocities in all regimes, that
are scale independent. However, the analysis of
Bierwisch et al., 2009, is based on a single collision, and
studies on parcel behavior in dense to moderately dense
systems (i.e., for particle volume fractions between 0.05
and 0.50) are still lacking.
Our first objective is to provide a scaling for the
parameters of a DEM-based (linear spring-dashpot)
model, such that stresses and velocities in the quasi-
static flow regime are scale independent. Our second
objective is to establish a method that takes the effect of
sub-parcel collisions, not directly tracked when using a
parcel-based approach, into account. We base our
method on the ideas of O'Rourke and Snider, 2010.
However, we aim at a strategy that is consistent with
DEM, i.e., that yields a “pure” DEM-based simulation in
a situation where only one particle is in the parcel. Our
third objective is to demonstrate the effect of the
proposed scaling and the sub-parcel collision model on
the dynamics of two granular flows. Our simulations are
based on an in-house code running on graphic
processing units (GPUs).
MODEL DESCRIPTION
Dense Region
When using a parcel-based approach, one has to prevent
over-packing in dense regions either with an (indirect)
particle pressure model (Andrews and O'Rourke, 1996),
or a model based on the direct detection of particle
collisions (e.g., the strategy followed by Patankar and
Joseph, 2001). In our model we have chosen to account
for collisions by using a linear spring-dashpot model
(see Eqns. 1-2), which is well established for granular
dynamics. The details of our implementation are
provided in Radeke et al., 2010, and here we only
provide the most essential model equations. For our
present work we included a spring-dashpot-slider model
in the tangential direction (see Eqn. 2), as well as a
rolling friction model between particles, and particles
and the wall (see Eqns. 3-5;
roll
v
is a “rolling velocity”
as defined by Luding, 2008).
nnF +=
nnnnnij
ck
δδ
&
,
(1)
(
)
ttFF +=
ttttnijij,t
ck
δδµ
&
,min
,
(2)
rollrollnrollroll
vvFF =
µ
(3)
[
]
21
ωnωnv
×
×
=
effroll
R
(4)
rollroll
R
FnT
×
=
1
(5)
The central question when using a DEM-based model
for the parcel-based approach is how to connect particle
interaction parameters with parcel interaction
parameters. Specifically, we ask the question, how
Parcel-Based Approach for the Simulation of Gas-Particle Flows / CFD11-124
3
spring stiffness and damping coefficient must be scaled,
when a parcel with a diameter
d
p
=
α
.
d
prim
is used in the
simulation (
α
is the ratio of the parcel diameter to the
primary particle diameter).
In the following we describe the details on such a
scaling that yields identical stresses for dense granular
flow using the linear spring-dashpot model (Eqns. 1-5).
Scaling of Interaction Parameters
We base our analysis on equal energy densities in the
original, and the coarse-grained system. This means that
the density of the particles, as well as the translational
velocity must be invariant. Also, the total rotational
kinetic energy of the original and coarse-grained
particles must be identical. The following analysis is
valid for a linear spring-dashpot model with frictional
slider, and is similar to the analysis for a Hertzian
interaction model by Bierwisch et al., 2009. Since we
base our analysis on an effective mass and radius for the
collision, it is valid for both particle-wall, as well as for
particle-particle collisions with arbitrary size ratios of
the primary particles (i.e., the particles in the original
system). Our analysis does not include differences in
particle densities; however, an extension to these
systems can be easily done in analogy to particle size
differences.
normal direction from Newton’s equation of motion:
nnnnneff
ckm
δδδ
&&&
+=
.
(6)
Here the effective mass is:
( )
3
3
3
13
4
β
βρ
π
+
=
+
=
pi
ji
ji
eff
R
mm
mm
m.
(7)
Inserting these expressions, and using the dimensionless
variables
inn
R
δ
=
*
,
0
*v
nn
δδ
&& =
, and
(
)
0
*vRtt
i
=
,
as well as approximating Young’s modulus with
in
RkE
yields:
*
*
*
0
22
0
1
n
pi
n
pi
nn
n
vR
c
vR
k
K
δ
ρρ
δ
δ
&&&
+
=,
(9)
with
(
)
[
]
33
1
134
ββπ
+=K
. Thus, our scaling is based
on the dimensionless (normal) overlap for the
translational motion of a particle, with the reference
length being the parcel diameter in the parcel approach
and the particle diameter in the original unscaled
problem, i.e., the relative overlap will remain invariant
when scaling the system. From Eqn. 9 the following
dimensionless parameters can be identified:
β
=
Π
1
,
(
)
2
02
vRk
pin
=Π
ρ
,
(
)
0
2
3
vRc
pin
=Π
ρ
(10)
Π
1
requires a constant ratio of the radii of the colliding
particles or parcels. This ratio will remain constant, as
long as each parcel is made up by the same number of
particles N. Π
2
requires that
constRk
in
=
, since we
require also the density and the reference velocity v
0
to
be invariant. This is in agreement with the simulation
results of Chialvo et al., 2010, which found that the
pressure scales with
Rk
n
in a monodisperse, quasi-
static granular flow.
Also, the stresses will be identical during scaling. This is
because the ratio of the elastic normal forces in the
original and scaled system is given by
(
)
(
)
[
]
2
,,
**'
αδαδα
==
inninn
el
nij
el
nij
RkRkFF
.
(our analysis was based on invariant non-dimensional
overlaps
*
n
δ
). Since the macroscopic contact stress is
given by the sum of the dyadic product of contact forces
and the distances between two particles in a control
volume (see Latzel et al., 2000, for details on the
evaluation of stress tensors), the stresses are invariant
when using this scaling in the dense regime.
Π
3
requires that
constRc
in
=
2
, i.e., c
n
scales with
α
²,
which results in an invariant coefficient of restitution
when scaling the system, as well as a damping force that
scales with
α
². Finally, it is easy to see that the already
dimensionless friction coefficients µ and µ
roll
must be
kept constant when scaling the system.
Dilute Region
Previous Work
To take collisional effects between the particles in a
parcel into account, O'Rourke and Snider, 2010,
proposed a relaxation of the particle distribution
function f to an equilibrium distribution f
eq
in a BGK-
like fashion (in our discrete approximation, f is
represented by parcels having individual velocities v).
Thus, they set:
D
eq
ff
t
f
τ
=
(11)
to take into account particle collisions. O'Rourke and
Snider, 2010, suggested the following correction to the
parcel velocity to take collisions of particles within the
parcel into account:
(
)
(
)
( )( )
D
D
t
t
τδ
τδ
2/1
2/
'
+
+
=
vv
v
(12)
Here,
δ
t is the computational time step, and the mass-
averaged particle velocity
v
is given by a summation
over parcels p near the parcel under consideration:
(
)
(
)
=
p
prim
p
prim
mNmN vv
,
(13)
where m
prim
is the mass of primary particles, and N is the
number of particles in each parcel. O'Rourke and Snider,
2010, proposed Eqns. 14 to 16 for calculating the
damping time
τ
D
.
( )
eps
g
pCPp
CPp
p
+
=
φφ
φ
φ
,
,
0
(14)
(
)
21
p
e
+
=
η
(15)
( )
β
β
+
=
+
=1
i
ji
ji
eff
R
RR
RR
R.
(8)
4
( )
( )
( )
( )
( )
( )
+
+
=
p
prim
p
prim
p
p
D
rdN
rdN
g
r
22
32
24
32
0
3
32
2
2
1
3
281
vv
vv
ηηφ
φ
πτ
(16)
Here eps is a small number (taken to be 10
-5
), r
32
is the
prim
is the (primary)
particle diameter of the particles making up the parcel.
Eqn. 16 is based on the kinetic theory of granular flow
with corrections for particle inelasticity, as well as local
particle volume fraction. Also, one can take frictional
contacts between particles into account by using an
effective coefficient of restitution as proposed by
Jenkins and Berzi, 2010. However, it should be kept in
mind that the velocity relaxation proposed by O'Rourke
and Snider, 2010, applies for an approach where
collisions between parcels are not detected; when such
collisions are detected, as in our present study, one must
modify Eqn. 16 to avoid overdamping.
Modified Relaxation Model for Parcel Velocities
The equation for the collision frequency proposed by
O'Rourke and Snider, 2010, models the effect of
collisions between particles in different parcels.
However, since we are already accounting for collisions
between parcels in our implementation (via the spring-
dashpot model), we need to model only collisions that
occur with a frequency 1/
τ
D,model
, i.e.,
pD,DmodelD,
111
τ
τ
τ
=
.
(17)
Assuming that the latter collision frequency between
parcels follows the expression in Eqn. 16, and by
requiring that the fluctuation velocities of particles and
parcels are identical, one obtains:
(
)
α
τ
τ
1111
DmodelD,
=
.
(18)
Hence, the frequency of collisions 1/
τ
D,model
that
requires modeling in addition to the “resolved”
collisions of the parcels, is somewhat lower in our
model than the one proposed by O'Rourke and Snider,
2010. Note, that Eqn. 18 is consistent with a “full”
simulation of the original system, where all particles are
tracked, i.e.,
α
= 1 and 1/
τ
D,model
equals zero.
Furthermore, we have to consider a correction in the
dense limit, where parcel-parcel collisions become more
frequent, and Eqn. 16 is no longer valid. This is because
in the limit of close packing, contact forces due to
enduring contacts (calculated directly with the spring-
relaxation. Thus, we have to reduce the modeled
collision frequency at a certain volume fraction
φ
p,off
to
avoid the divergence of 1/
τ
D,model
at
φ
p
=
φ
p,CP
. We do
this by multiplying Eqn. 18 with a factor h(
φ
p
) that
approaches zero when
φ
p
equals
φ
p,CP
:
( )
=
off
offpCPp
pCPp
p
h
κ
φφ
φφ
φ
,,
,
,1min
.
(19)
Here
κ
off
is an exponent that controls how fast the
collision frequency model is switched off. We choose
κ
off
to be 8 for a rapid switch off, and
φ
p,off
= 0.60. This
rather abrupt switch off is motivated by the publications
of Silbert et al., 2007, Reddy and Kumaran, 2007, as
well as Chialvo et al., 2011. These authors show that
there is an abrupt switch between an inertial regime
(where the model given by Eqns. 14 and 18 is
appropriate) and a quasi-static regime, for which a more
complex rheological model has to be used (in our case
the DEM-based model shown in Eqns. 1-5).
In summary, our modified relaxation model is consistent
with a DEM-based simulation in the dense regime, and
with a “full” DEM-based simulation in case the original
system is modeled, i.e., a situation where
α
equals 1. In
the dilute limit, and when the parcel size is much larger
than the particle size, our relaxation model is identical to
the model based on kinetic theory shown in O'Rourke
and Snider, 2010.
Local Particle Volume Fraction and Velocity
For the relaxation of parcel velocities, one needs to
know the local mean particle volume fraction
p
φ
, as
well as the mean velocity of the parcels v. Previous
work (e.g., O'Rourke and Snider, 2010) used a Cartesian
grid for this purpose. This requires an interpolation of
particle volume and velocity from and to this grid. We,
however, reconstruct
p
φ
and v from a spatial filtering
on a spherical domain surrounding each parcel. Our
filter function has the form:
( )
(
)
(
)
( )
+
+<+
=
BGKji
BGKji
RRR
RRRRRC
RG 0
2
02
, (20)
with parameters C
2
, and R
0
chosen such that the integral
of G(R) in a spherical domain is unity. This filter
guarantees that the mean quantities are evaluated at the
center of the parcel. For our simulations we choose
BGK
= 3, i.e., filtering was performed in a sphere with a
diameter three times the parcel diameter.
RESULTS
Granular Jet
Cheng et al., 2007, have experimentally investigated the
normal impact of 0.050 mm to 2.1 mm glass and brass
particles (monodispers and spherical) on a circular disc.
They showed that the characteristic scattering angle
θ
half
of the rebounding particles is influenced by the ratio of
particle and jet diameter d
p
/d
jet
:
(
)
jetphalf
dd
=
2atan
λ
θ
(21)
with
λ
= 1.05 ± 0.05. Thus, for a small particle-to-jet
ratio the granular jet behaves like a liquid without a
scattering of particle velocities. When using our parcel-
approach, the goal is that only the size of the primary
particles should affect the scattering angle, and not the
parcel diameter itself.
We performed simulations using (i) an unscaled system
(i.e., the interaction parameters in Eqns. 1-5 were not
Parcel-Based Approach for the Simulation of Gas-Particle Flows / CFD11-124
5
changed with parcel size); (ii) a scaled system (scaling
based on the dimensionless parameters displayed in Eqn.
10), but without relaxation model; as well as (iii) a
scaled system with the relaxation model shown in Eqn.
18. Also, a simulation of the original system, for which
the motion of the primary particles was calculated
directly, was performed. The simulation parameters are
listed in Table 1. The damping coefficient c
n
was
calculated from the coefficient of restitution (see
Luding, 2008), and we set c
t
= c
n
.
Figure 1: Particle velocities near the impact region of a
granular jet. The jet (which has a circular cross section with
diameter d
jet
) impacts the planar surface under an angle of
θ
impact
= π/2.
The scattering patterns of these three systems are shown
in Figure 1a-c; a magnified version of a slightly smaller
region near the impact point for the original system is
shown in Figure 1d. Here we show front views of the
particle jet, which approaches the planar surface from
the top. For the results displayed in Figure 1 we have
chosen the scaling factor
α
as 6.3, i.e., we have grouped
250 single particles into one parcel.
As can be easily seen from Figure 1a, the unscaled
system represents a case with a much larger (effective)
particle diameter, and the characteristic scattering angle
is much larger than that observed in the original system
(see Figure 1d). This behavior is expected, since we do
not perform any scaling, and the scattering angle is in
good agreement with the experimental results of Cheng
et al., 2007 (see the comparison of simulation results for
the unscaled system indicated by dots with the
correlation indicated by the bold line in Figure 3).
Table 1: Parameters for the granular jet simulations.
Parameter Value
d
jet
7.3 [mm]
y
jet
25 [mm]
d
jet
/ d
p
3.5; 14.6; 29.2; 73
d
p
0.10 - 2.19 [mm]
U
p,0
1 - 8 [m/s]
φ
p
0.60
ρ
p
2500 (glass)
e
p
0.75, 0.95
µ
PP
0.10
µ
PW
0.15
µ
roll
0.01
k
n
306 [N/m]
k
t
88 [N/m]
R
sample
63.5 [mm], 20 [mm]
Appropriate scaling of the system significantly lowers
the scattering angle (see Figure 1b), even in the case no
relaxation is employed. However, some scattering of
parcels still occurs. Employing the BGK-like relaxation
(Figure 1c) finally yields a parcel behavior similar to the
one observed in the original system with 250-times more
particles.
Figure 2: The scattering angle distribution - comparison of
the original system (red line), with results using the parcel
approach (symbols).
To quantify the effect of the BGK-like relaxation, we
compare the distribution of particle scattering angles in
Figure 2. Specifically, we plot the number density
distribution P(
θ)
of the scattering angle
θ
(i.e., the angle
measured between the impact surface and the particle
position as illustrated in Figure 1a). To calculate P(
θ)
,
we only consider particles located at a radial distance of
R
sample
= 63.5 ± 2.5 mm from the origin, as per Cheng et
(a) unscaled system
d
prim
= 100 µm, d
p
= 630 µm
(c) scaled system
with relaxation
d
prim
= 100 µm, d
p
= 630 µm
(d) original system
d
prim
= d
p
= 100 µm
0,p
U
r
θ
θθ
θ
d
jet
(b) scaled system
without relaxation
d
prim
= 100 µm, d
p
= 630 µm
R
sample
origin
6
al., 2007; for the original system, though, we set R
sample
= 20 ± 2.5 mm, since we were unable to simulate the full
system. As can be seen from Figure 2, the unscaled and
scaled systems give a wide distribution of scattering
angles, with scattering half angles
θ
half
(defined as
( )
=
half
dP
θ
θθ
0
5.0
) of 0.28 and 0.14, respectively. This
result is in distinct contrast to the very narrow
distribution of the original system (shown by the red line
in Figure 2). However, when we employ the relaxation
model, all simulations using a larger parcel diameter
(see the symbols for different values of d
p
/d
jet
in Figure
2) agree reasonably well with behavior of the original
system. The improved agreement of the BGK cases
indicates the importance of the relaxation model in
replicating the behavior of the much smaller particles
within a parcel.
Figure 3: The scattering half angle
θ
half
as a function of
dimensionless parcel size (symbols represent simulation
results, lines are theoretical predictions).
In Figure 3 we show a summary of our results for the
granular jet for various scalings, expressed as d
p
/d
jet
.
Clearly, when the system is unscaled (filled circles and
crosses), the parcels behave like large particles. In this
systems the scattering half angle is close to the
correlation established by Cheng et al., 2007, and given
in Eqn. 21. For the unscaled system we find that the
coefficient of restitution has no significant effect at large
dimensionless parcel diameters. Only for the smallest
system studied (i.e., d
p
/d
jet
= 0.04) there is a large
relative difference for our results involving e
p
= 0.75
and e
p
= 0.95. The cause for this behavior is unclear, as
previous computational studies (e.g., that of Huang et
al., 2010) could not afford to simulate that large systems
(the simulations for d
p
/d
jet
= 0.04 required us to track
0.97 million particles) and the experiments of Cheng et
al., 2007, only included particles that had a coefficient
of restitution equal to 0.75.
The scaled system using the BGK-like relaxation model
(blue filled squares in Figure 3) shows a more realistic
behavior, as we obtained the small scattering angles
typical for a system made up by particles of 100 µ m (the
scattering angle for 100 µm particles as expected from
Eqn. 21 is shown as green dashed line Figure 3). Our
simulation results for the original system are indicated
by the leftmost black circle at d
p
/d
jet
= 0.0137. These
simulations involved already 2.7 million particles in a
domain half the size as the one used for the other cases.
Our results when using the relaxation model (blue filled
squares in Figure 3) are somewhat below the expected
results from Eqn. 21, as well as our simulations of the
original system. This behavior is also visible in Figure 2,
where the simulation results using the BGK relaxation
do not match perfectly with the original system (red line
in Figure 2). It seems that the chosen expression for the
relaxation time slightly over-predicts the collision
frequency, even though we already subtract the collision
frequency between parcels. More work, and possibly a
more suitable model, is required to explain this
difference, and to make the scattering angles in both
systems collapse. Another explanation could be that the
switch off particle volume fraction of
φ
p,off
= 0.60 in our
simulations is somewhat too high. Indeed, simulations
with lower values for
φ
p,off
showed that such a parameter
adjustment could be used to improve the collapse.
However, there is no rationale for choosing a different
value for
φ
p,off
, and hence we suggest to first improve the
collision frequency model, rather than to perform a
parameter fit.
Simple Shear
We performed simple shear flow simulations using a
cubic box of particles. Lees-Edwards boundary
conditions (LEBCs, Lees and Edwards, 1972) were
employed on two sides of the box, whereas on the other
four sides periodic boundaries were set. The LEBCs
impose a steady shear motion on the particles. Stresses,
fabric tensors, velocities, as well as particle rotation
rates were recorded during the simulation, and then
averaged over the box. Stresses were probed by
calculating the contact and streaming stress tensor for
each particle, and then summing up the contributions
from each particle. A similar procedure was employed
for the fabric tensor (results not shown). Both
procedures (i.e., for the stress and the fabric) follow the
work of Latzel et al., 2000. The granular temperature in
the system was estimated by calculating the velocity
fluctuation tensor
2
,fluctp
v
, and then setting
3
2
,
=
fluctp
trT v
.
(22)
The velocity fluctuations were calculated from the
instantaneous particle velocity and the local average
velocity. The latter was calculated under the assumption
of a linear velocity profile.
Similar to the granular jet, we performed shear flow
simulations in (i) an unscaled system (i.e., the
interaction parameters for the contact model were not
changed with parcel size); (ii) a scaled system (scaling
based on Eqn. 10), but without relaxation model; as well
as (iii) a scaled system with relaxation model. For our
simulations we used the same particle interaction
parameters as shown in Table 1, except for µ
roll
, which
was set to zero. Simulations were performed with
φ
P
=
Parcel-Based Approach for the Simulation of Gas-Particle Flows / CFD11-124
7
0.55, e
p
= 0.75, and the shear rate was chosen such that
(
)
4
10
==
primpprim
dkd
ργγ
was constant.
Systems with different parcel sizes where investigated
by holding the box size (0.015 m) and the primary
particle diameter (d
prim
= 100 µ m) constant, and
grouping between 4 and 8192 particles into one parcel.
Also, a limited number of simulations were performed
with
φ
P
= 0.62 to show the system behavior in the quasi-
static flow regime. Each of these simulations involved
approximately 4,000 parcels in a box of variable size to
investigate the effect of the parcels size.
Granular Temperature
Already previous work (Benyahia and Galvin, 2010)
showed that the granular temperature in a wall-bounded
shear flow dramatically increases with parcel size. This
finding is expected, since a simple calculation based on
the kinetic theory of granular flow would predict that
(see, e.g., Sangani et al., 1996):
( )
(
)
[
]
( )
p
p
p
e
g
d
T
++
=
115
851121
2
0
2
φπ
γ
(23)
Here g
0
is the radial distribution function at contact, for
which we use the expression in Eqn. 15. Thus, by just
scaling the particle diameter, and performing a naive
DEM-based simulation leads to a significant
overestimation of the granular temperature in the
system.
Figure 4: Granular temperature scaled by the velocity
fluctuation in the original system (d
prim
.
γ
)² (symbols represent
simulation results, lines are theoretical predictions,
φ
P
= 0.55
unless otherwise stated).
The results for the granular temperature (made
dimensionless with the shear rate and the primary
particle diameter d
prim
) are shown in Figure 4. The red
bold line represents the expected granular temperature
of the original system for
φ
P
= 0.55 estimated from Eqn.
23, which should remain constant when using the
relaxation model. Also, we have included a line
representing the theoretical prediction using Eqn. 23 for
the increase in granular temperature in the simulations
without relaxation model (i.e., the scaled and unscaled
system should behave like a system consisting of large
particles, and the granular temperature should increase
proportional to (d
p
/d
prim
)²). Clearly, the results for the
scaled system and
φ
P
= 0.55 agree reasonably well with
the theory of Sangani et al., 1996, which slightly
overpredicts the granular temperature. The unscaled
system, as well as the scaled system with
φ
P
= 0.62
shows a higher granular temperature, but the same
principal trend as the theoretical prediction of Sangani et
al., 1996. This is because the unscaled system represents
a case with a higher coefficient of restitution - this
increase can be easily understood by inspection of Eqn.
23. The scaled system with
φ
P
= 0.62 is in the quasi-
static regime, for which the theory of Sangani et al.,
1996, is no longer applicable.
The simulations using the BGK-like relaxation in Figure
4 show a lower granular temperature, i.e., the system
behavior is closer to the theoretical prediction of the
original system (bold red line). Still the granular
temperature is significantly overpredicted using the
relaxation model given by Eqns. 14 and 18. Thus, the
relaxation to the local mean velocity is too week to
dampen the velocity fluctuations of the parcels. Also,
the scaled granular temperature increases with parcel
size, indicating that the dependency of the current
relaxation model on the parcel size is too weak.
Stresses
Figure 5: Pressure (top) and shear stress (bottom) for different
parcel diameters and filter sizes in a scaled and unscaled
system, as well as a in a system with BGK-like relaxation
(symbols represent simulation results, lines are theoretical
predictions,
φ
P
= 0.55 unless otherwise stated).
8
The results for the sum of contact and streaming stresses
are shown in Figure 5. Here we have defined the
pressure as p = (
σ
xx
+
σ
yy
+
σ
zz
) / 3, and the shear stress
τ
is the stress component pointing in the shearing
direction and acting on the surface normal to the
gradient direction (the other shear stress components are
much smaller).
As can be seen from Figure 5, with increasing parcel
size both pressure and shear stress dramatically increase
for the systems with
φ
P
= 0.55. This is explained with
the fact that these systems correspond to a system with a
higher dimensionless shear rate
γ
*’, in case we define
this dimensionless shear rate with
(
)
psp
dkd =
ργγ
, i.e., we use the parcel
diameter instead of the primary particle diameter. Thus,
γ
*’ increases with (d
p
/d
prim
)
3/2
in the unscaled system,
and with (d
p
/d
prim
) in the scaled system since our scaling
is based on k/d
p
= const. As our simulations were
performed with µ
PP
= 0.1, and
γ
*’< 10
-2
, the systems
with
φ
p
= 0.55 were in the inertial regime, and the
systems with
φ
p
= 0.62 were in the quasi-static regime
(for an exact regime definition refer to Chialvo et al.,
2011). Previous studies, and simple theory tells us that
pressure and shear stress scale with the dimensionless
shear rate squared in the inertial regime (at constant
volume fraction and friction coefficient). In the quasi-
static regime the stresses are expected to remain
constant, and only a moderate increase for
γ
*’> 10
-3
(which corresponds to d
p
/d
prim
> 10 in our scaled system)
is anticipated due to a regime transition into an
intermediate flow regime (see the findings of Chialvo et
al., 2011). We have included lines showing the expected
trends for the stresses in the quasi-static and inertial flow
regime in Figure 5 (see the red solid lines).
As can be seen, the simulation results for the scaled
system agree well with the expected scalings in the
inertial and quasi-static regime. For the latter regime our
simulation results for the stresses (black diamonds in
Figure 5) slightly increase with parcel size above d
p
/d
prim
= 10 due to the transition to an intermediate flow
regime. This increase has also been observed by Chialvo
et al., 2011, for flows at different dimensionless shear
rates.
Also the results for the unscaled system with
φ
p
= 0.55
follow the theoretical predictions for flow in the inertial
flow regime; however, they show somewhat higher
stresses. This is explained by the fact that for the
unscaled system also the coefficient of restitution
increases with parcel size, which results in higher
granular temperature and stresses.
In case the BGK-like relaxation is applied, the sum of
contact and streaming stress stays nearly unaffected (see
the filled blue boxes in Figure 5), and only for d
p
/d
prim
>
15 a rather small drop in the stresses is observed. Thus,
even in case we perform a BGK-like relaxation, the
stresses do not significantly change compared to a
scaled system.
CONCLUSION
Even though parcel-based approaches have been in use
for more than fifteen years since the pioneering work of
Andrews and O'Rourke, 1996, there is still a strong need
to analyze the basic features of this simulation approach.
Recent literature highlights this need (see the work of
Benyahia and Galvin, 2010; Benyahia and Sundaresan,
2011).
In this work we performed a relaxation of parcel
velocities to a local mean velocity to account for
collisions of particles making up the parcels. We claim
that such a relaxation is necessary in any DEM-based
simulation that uses bigger (pseudo) particles, and has
the ambition to model the original system consisting of a
much higher number of primary particles. Even in cases
where a proper scaling of particle interaction parameters
is performed (i.e., a scaling like the one proposed by
Bierwisch et al., 2009, or the one presented in this work
for the linear spring-dashpot model), a relaxation seems
necessary. Thus, our simulations involving the
relaxation model gave a nearly parcel-sized independent
behavior for a granular jet impinging on a planar
surface, while simulations without relaxation did not.
The challenge for parcel-based methods is to correctly
compute the collision rate, the granular temperature and
the inter-parcel stress in a dilute to moderately dense
region. Our simulations for a simple shear flow in the
inertial regime agree with the kinetic theory of granular
flow that predicts a massive increase in granular
temperature and stress with parcel size when no
relaxation is used. For the stress this is explained with
the higher dimensionless shear rate when parcels are
used, i.e., the shear rate increases with (d
p
/d
prim
)
3/2
in an
unscaled system, and with d
p
/d
prim
in a system with
scaled interaction parameters to yield identical behavior
in quasi-static granular flow. Thus, there is no easy way
of scaling parcel interaction parameters to yield identical
flow behavior in the inertial flow regime. For the quasi-
static flow regime, however, scaling is less problematic
and we observe a nearly constant stress in our shear flow
simulations. Only the transition to an intermediate flow
regime, as defined by Chialvo et al., 2011, will lead to
an increase in the stresses in a correctly scaled system.
For simple shear flow in the inertial regime the use of
the relaxation model based on O'Rourke and Snider,
2010, used in a slightly modified form in our work,
leads to a granular temperature for the parcel-based
approach closer to the original system, but this is not
true for the stress. It is still unclear how one can obtain
an identical stress for flows in the inertial regime when
using a parcel-based approach. Clearly, more work is
required, especially in connection with the relaxation
model, which is subject of ongoing work.
ACKNOWLEDGEMENT
We thank Sebastian Chialvo for reviewing the
manuscript. SR acknowledges the financial support of
the Austrian Science Foundation and Princeton
University through the Erwin-Schrödinger fellowship J-
3072.
Parcel-Based Approach for the Simulation of Gas-Particle Flows / CFD11-124
9
APPENDIX
List of animation files:
(A1) “1000k_boxLEBC_PrincetonLogoMix.avi“
(simple shear of 10
6
particles in a box with Lees-
Edwards boundary conditions)
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The Discrete Element Method (DEM) is commonly used for modeling the flow of particulate materials. Unfortunately, such detailed simulations are computationally very demanding, restricting its use for industrially-scaled processes. The number of particles in a simulation can be reduced by introducing parcels (i.e., “coarse graining”), which – in essence – relies on the increase of the particle diameter for interaction calculations. However, sophisticated models are necessary to preserve the original behavior of the material when using such an approach. Our present contribution extends available coarse-graining concepts by introducing models for (i) particle–fluid mass transfer and (ii) the deposition rate of spray droplets on particles. Our mass transfer model is based on an existing model for heat transfer. For the spray deposition model, we introduce an effective particle diameter to compute the correct amount of droplets that impact particles. We show that these models can be used with confidence up to a coarse-graining level of 5, which we demonstrate for a simple-shaped fluidized bed. The models proposed by us are critical for detailed simulations of spray coating processes since they enable precise particle-droplet-air interaction modeling at low computational cost.
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Recent improvements in computing capabilities have made numerical simulation of the behavior and the heat transfer of many particles possible via the Discrete Element Method (DEM). However, even with current computational capabilities, it is still difficult to simulate particle behavior and the heat transfer for manufacturing equipment as rotary kilns, because the large scale and enormous number of particles would be required. To address this issue, we developed the coarse-graining method called modified coarse-grained method for granular shear flow (M-CGSF). In this study, we performed the verification tests for M-CGSF targeting manufacturing scale as a rotary kiln reactor. It was confirmed that the particle behavior and heat transfer with DEM simulations by applying M-CGSF were consistent with the original case.
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Railway operation is faced with windblown sand hazards in desert areas. In order to study the influence of sand intrusion on dynamic mechanical behavior of railway ballast bed, a multi-scale discrete element model of sleeper-ballast bed-surface layer of subgrade has been established with the method of “step filling-particle replacement”. The vibration acceleration of the ballast bed and the dynamic displacement of sleeper field tests in the section with severe sandstorm have carried out to verify the correctness of discrete element analysis model. The scale invariance of the linear contact model and the rationality of the scaling of the sand size are explained by the “dimensional analysis method”. The results show that the sand intrusion reduces the vibration of ballast bed. When the sand content reaches 30%, the vibration and elastic deformation of ballast bed decrease by 94.2% and 63.2% respectively. The invasion of sand can reduce the deformation of the surface layer of subgrade, and the maximum reduction is 53.3%. Therefore, it is necessary to pay attention to the service state of ballast bed with sand content of 30% in windblown sand area, and make the maintenance plan reasonably.
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A simplified mathematical model is proposed to determine the agglomeration tendency of bitumen-coated coke particles in fluid cokers. The model calculates a theoretical critical velocity that depends on key parameters such as the particle size, bitumen viscosity, and bitumen thickness; it also accounts for the temperature- and reaction-dependent variations of the bitumen thickness and viscosity. A peak theoretical critical velocity at the intermediate reaction times for all coking temperatures is predicted. By comparing this peak critical velocity with the estimated inter-particle collision velocities within an industrial-scale reactor, the agglomeration tendency of coke particles is determined within fluid cokers. The results show that at low temperature regions (T=400 °C), there is no agglomeration tendency; however, at high coking temperatures (T=503 and 530 °C), substantial agglomeration tendency is expected. It is also found that the number of coke particles constituting an agglomerate could be as high as a few hundreds.
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Granular flow of three experiments are predicted by a computational particle fluid dynamic CPFD numerical scheme in three-dimension using the true particle size distribution. The experiments are simple which show the distinct characteristics of particle flow which differs from fluid flow. The experiments are flow of particles in sedimentation, a U-tube and from a hopper. The CPFD method models the fluid as a fluid and models the particles as discrete particles (material description). The CPFD method is a form of discrete element method, where each particle has three dimensional forces from fluid drag, gravity, static-dynamic friction, particle collision and possibly other forces. However, unlike DEM models which calculate particle-to-particle force by a spring-damper model and direct particle contact, the CPFD method models collision force on each particle as a spatial gradient. The CPFD numerical method predicts all three experiments.
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We present an Eulerian-Lagrangian numerical simulation (LNS) scheme for particulate flows. The overall algorithm in the present approach is a variation of the scheme presented earlier. In this numerical scheme we solve the fluid phase continuity and momentum equations on an Eulerian grid. The particle motion is governed by Newton’s law thus following the Lagrangian approach. Momentum exchange from the particle to fluid is modeled in the fluid phase momentum equation. Forces acting on the particles include drag from the fluid, body force and the interparticle force that prevents the particle volume fraction from exceeding the close-packing limit. There is freedom to use different models for these forces and to introduce other forces. In this paper we have used two types of interparticle forces. The effect of viscous stresses are included in the fluid phase equations. The volume fraction of the particles appear in the fluid phase continuity and momentum equations. The fluid and particle momentum equations are coupled in the solution procedure unlike an earlier approach. A finite volume method is used to solve these equations on an Eulerian grid. Particle positions are updated explicitly. This numerical scheme can handle a range of particle loadings and particle types. We solve the fluid phase continuity and momentum equations using a Chorin-type fractional-step method. The numerical scheme is verified by comparing results with test cases and experiments.
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Spout fluidized beds are frequently used for the production of granules or particles through granulation. The products find application in a large variety of applications, for example detergents, fertilizers, pharmaceuticals and food. Spout fluidized beds have a number of advantageous properties, such as a high mobility of the particles, which prevents undesired agglomeration and yields excellent heat transfer properties.A discrete element model is used describing the dynamics of the continuous gas phase and the discrete droplets and particles. For each element momentum balances are solved. The momentum transfer among each of the three phases is described in detail at the level of individual elements.The results from the discrete element model simulations are compared with local measurements of time time-averaged particle volume fractions as well as particle velocities by using a novel fibre optical probe in a fluidized bed of 400 mm I.D. Simulations and experiments were carried out for three different cases using Geldart B type aluminium oxide particles: a freely bubbling fluidized bed; a spout fluidized bed without the presence of droplets and a spout fluidized bed with the presence of droplets. It is found that the experimental and numerical results agree in a qualitative manner.It is demonstrated how the discrete element model can be used to obtain information about the interaction of the discrete phases, i.e. the growth zone in a spout fluidized bed. Additional analysis of the numerical results indicates that liquid breakthrough does not take place for the studied test case.
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Granular flows are extremely important for the pharmaceutical and chemical industry, as well as for other scientific areas. Thus, the understanding of the impact of particle size and related effects on the mean, as well as on the fluctuating flow field, in granular flows is critical for design and optimization of powder processing operations.We use a specialized simulation tool written in C and CUDA (Compute Unified Device Architecture), a massive parallelization technique which runs on the Graphics Processing Unit (GPU). We focus on both, a new implementation approach using CUDA/GPU, as well as on the flow fields and mixing properties obtained in the million-particlerange.We show that using CUDA and GPUs, we are able to simulate granular flows involving several millions of particles significantly faster than using currently available software. Our simulation results are intended as a basis for enhanced DEM simulations, where fluid spraying, wetting and fluid spreading inside the powder bed is considered.
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The kinetics of thermal cracking of films of vacuum residue from Athabasca bitumen in the temperature range of 457–530∘C was modelled with liquid-phase mass transfer, reaction-dependent fluid properties, and coke formation by reaction of cracked products in the liquid phase. Previous investigations on the thermal cracking of vacuum residue in thin films showed that at low film thickness (∼20±2μm) the coke yield was insensitive to the temperature and heating rate for thin films of bitumen. The coke yield increased with the thickness of the initial film, in the range from 20 to 80μm(±2μm). At the same time, the viscosity of the reacting liquid increased rapidly with time, which would slow down the diffusion of products inside the film. This coupling of transport and reaction would enhance the formation of coke by increasing the rate of recombination reactions. The concept of intrinsic coke is used in a new kinetic model to account for the minimum observed coke formation in thin films. With increasing film thickness, the increasing yield of extrinsic coke is modelled through the change in fluid properties as a function of extent of reaction, which reduces the rate of diffusion in the reacting liquid phase. The model was able to properly account for the insensitivity of coke yield in thin films to reaction temperature and the dependence of coke yield on the thickness of the liquid film.
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Purpose The purpose of this paper is to show how particle scale simulation of industrial particle flows using DEM (discrete element method) offers the opportunity for better understanding of the flow dynamics leading to improvements in equipment design and operation. Design/methodology/approach The paper explores the breadth of industrial applications that are now possible with a series of case studies. Findings The paper finds that the inclusion of cohesion, coupling to other physics such fluids, and its use in bubbly and reacting flows are becoming increasingly viable. Challenges remain in developing models that balance the depth of the physics with the computational expense that is affordable and in the development of measurement and characterization processes to provide this expanding array of input data required. Steadily increasing computer power has seen model sizes grow from thousands of particles to many millions over the last decade, which steadily increases the range of applications that can be modelled and the complexity of the physics that can be well represented. Originality/value The paper shows how better understanding of the flow dynamics leading to improvements in equipment design and operation can potentially lead to large increases in equipment and process efficiency, throughput and/or product quality. Industrial applications can be characterised as large, involving complex particulate behaviour in typically complex geometries. The critical importance of particle shape on the behaviour of granular systems is demonstrated. Shape needs to be adequately represented in order to obtain quantitative predictive accuracy for these systems.
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We examine the problem of determining the particle-phase velocity variance and rhe-ology of sheared gas-solid suspensions at small Reynolds numbers and finite Stokes numbers. Our numerical simulations take into account the Stokes flow interactions among particles except for pairs of particles with a minimum gap width comparable to or smaller than the mean free path of the gas molecules for which the usual lubrication approximation breaks down and particle collisions occur in a finite time. The simulation results are compared to the predictions of two theories. The first is an asymptotic theory for large Stokes number St and nearly elastic collisions, i.e. St [dbl greater-than sign] 1 and 0 [less-than-or-equal] 1 - e [double less-than sign] 1, e being the coefficient of restitution. In this limit, the particle velocity distribution is close to an isotropic Maxwellian and the velocity variance is determined by equating the energy input in shearing the suspension to the energy dissipation by inelastic collisions and viscous effects. The latter are estimated by solving the Stokes equations of motion in suspensions with the hard-sphere equilibrium spatial and velocity distribution while the shear energy input and energy dissipation by inelastic effects are estimated using the standard granular flow theory (i.e. St = [infty infinity]). The second is an approximate theory based on Grad's moments method for which St and 1 – e are O(1). The two theories agree well with each other at higher values of volume fraction φ of particles over a surprisingly large range of values of St. For smaller φ however, the two theories deviate significantly except at sufficiently large St. A detailed comparison shows that the predictions of the approximate theory based on Grad's method are in excellent agreement with the results of numerical simulations.
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Discrete particle methods that track the motion of individual particles and their collisions are computationally very expensive. To accelerate these numerical simulations, some basic assumptions have been introduced and reported in the literature. This study investigates two of these common assumptions: (1) the use of computational parcels, or clouds, wherein many particles are lumped together so that only parcels and their collisions are tracked, and (2) the multiphase particle-in-cell, or MP-PIC, wherein the collision forces are replaced by a solids pressure term with the main purpose to avoid exceeding the maximum packing of the granular assembly. Using several cases relevant to the fluidization community, errors associated with these assumptions are computed. For these cases the magnitude of error in the time-averaged flow variables indicates that further research on the validity of these assumptions is warranted.