Chemical Engineering Science 64 (2009) 43 -- 51
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Chemical Engineering Science
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Direct numerical simulation of particle clustering in gas–solid flow with a macro-scale
particle method
Jingsen Ma a,b , Wei Ge a,∗ , Qingang Xiong a,b , Junwu Wang a,b , Jinghai Li a
a
b
State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, P. O. Box 353, Beijing 100190, China
Graduate School of Chinese Academy of Sciences, Beijing 100049, China
A R T I C L E
I N F O
Article history:
Received 18 March 2008
Received in revised form 2 September 2008
Accepted 6 September 2008
Available online 20 September 2008
Keywords:
Clustering
Fluidization
Hydrodynamics
Multiphase flow
Simulation
Smoothed particle hydrodynamics
A B S T R A C T
Particle clustering has long been a focus in the study of gas–solid flow. Detailed flow field information
below the particle scale is required to understand the mechanism of its formation and the statistical
properties of its dynamic behavior, but is not easily obtained in both experiments and numerical simulations. In this article, a meshless method is used to reveal such details in the destabilizing of a suspension
with hundreds of particles. During the process, doublets, quadruplet and larger clusters are seen to form
and disintegrate dynamically, showing a tendency to minimize local voidages. At the same time, single
vertical streams, pairs of parallel streams and many irregular streams appear and disappear between particle clusters alternatively, exhibiting a tendency to suffer lowest resistance. Globally, the spatio-temporal
compromise between these two tendencies results in a configuration of large clusters separated by fast
flow streams. In the clustering process, the inter-phase slip velocity is seen to increase long after the forces
on each phase have stabilized, suggesting that inter-phase friction is not a function of local voidage and
Reynolds number only, as commonly considered. The article concludes with prospects on the sub-grid scale
models for continuum description of gas–solid flow that can be established upon such simulation results.
© 2008 Elsevier Ltd. All rights reserved.
1. Introduction
Evidences of natural particle clustering in both gas–solid and
liquid–solid systems can be dated back to more than half a century
ago (Wilhelm and Kwauk, 1948). The heterogeneity thus produced
has significant effect on the transport and reaction properties of the
system, and complicates the sub-grid scale models for its macroscale hydrodynamic description.
To understand the origin of clusters, detailed information on the
motion of individual particles is necessary, since clustering starts
with the formation of doublets (Chen et al., 1991). Furthermore,
according to the energy minimization multi-scale (EMMS) model
(Li and Kwauk, 1994), clustering is shaped by the compromise
between the minimization of particle gravitational potential and
the resistance to gas flow (Li and Kwauk, 2001, 2003). Therefore,
simultaneous visualization of both individual particle motion and
the surrounding fluid flow is favorable to an extensive exploration
on clustering mechanism. Though many experiments on the direct
∗
Corresponding author. Tel.: +86 10 82616050; fax: +86 10 62558065.
E-mail address: wge@home.ipe.ac.cn (W. Ge).
0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2008.09.005
visualization of cluster formation have been reported (Fortes et al.,
1987; Chen et al., 1991; Liu et al., 2005), such pictures were experimentally unattainable until recently (Martin et al., 2005). However,
the number of particles that can be visualized is quite limited (with
8 and 16 particles in the whole system up to now), and accurate
measuring of the flow field in the wakes of the particles was still
considered to be difficult for very few PIV particles were observed
to enter this region (Martin et al., 2005), whereas the wake effect
is usually taken as an important inducement for clustering (Fortes
et al., 1987; Chen et al., 1991).
In comparison, computer simulation can provide a more flexible and exhaustive alternative for this purpose, and the dramatic
leaping in computational capability provided by parallelization has
ranged simulations of fairly large systems into reasonable time. Several direct numerical simulations (DNS) of “large-scale” liquid–solid
systems have been reported (Hu, 1996; Potapov et al., 2001; Pan
et al., 2002; Cho et al., 2005; Nguyen and Ladd, 2005; Derksen and
Sundaresan, 2007). And flow structures below the particle scale in
fixed beds have been obtained also (Cate and Sundaresan, 2006;
Beetstra et al., 2007 and the references therein). However, to our
knowledge, few publications (e.g., Tang et al., 2004; Ma et al., 2006)
have addressed high-resolution simulations on the more heterogeneous gas-solid suspensions where dynamic clustering is inherent.
44
J. Ma et al. / Chemical Engineering Science 64 (2009) 43 -- 51
In this article, a gas–solid suspension with 389 solid particles is
simulated with a macro-scale particle method (Ma et al., 2006), based
on smoothed particle hydrodynamics (Gingold and Monaghan, 1977;
Lucy, 1977), macro-scale pseudo-particle modeling (MaPPM, Ge and
Li, 2001, 2003b) and moving particle semi-implicit (Koshizuka et al.,
1995; Koshizuka and Oka, 1996). The whole destabilizing process
from a uniform particle configuration is presented numerically, and
the mechanism of clustering is explored based on qualitative description and quantitative analysis of the flow field around the particles
gradually involved in the clustering and disassembling process. A
correlation between the average slip velocity and clustering degree is
discussed, which are of great interests to higher level simulation
approaches such as Eulerian two-fluid models (TFM, e.g., Murray,
1965; Gidaspow, 1994) and Eulerian–Lagrangian discrete particle
models (DPM, e.g., Tsuji et al., 1993; Hoomans et al., 1996).
2. Simulation
2.1. Methods
This work adopts a Lagrangian–Lagrangian scheme which combines macro-scale particle methods (MaPM) for the gas flow and
Table 1
Simulation and physical parameters.
Dimensional
Dimensionless
Dimensional
Dimensionless
H (m)
W (m)
(kg/m3 )
(m2 /s)
d (m)
h (m)
P (Pa)
F (m/s2 )
7.68e−3
352
7.72e−3
354
1.225
11.11
1.46e−05
5.0e−04
6.54e−6
0.3
7.27e−6
1/3
3.403
1.717e−5
3.617e2
4.39e−9
m (kg)
R1 (m)
R2 (m)
gs (m/s2 )
ms (kg)
s (kg/m3 )
t (s)
e
1.142e−15
1
1.20e−4
5.5
1.09e−4
5
−9.8
−1.19e−10
1.306e−10
114348.47
160.49
1455.93
1.627e−08
1
0.8
Fig. 1. Initial solid distribution in both cases. (a) Initial solid distribution in the “regular” case (the white lines and the repeated parts outside of them represent periodic
boundaries), (b) initial solid distribution in the “random” case (periodic boundaries are omitted for simplicity) and (c) initial solid velocity vector values distribution in the
“random” case.
J. Ma et al. / Chemical Engineering Science 64 (2009) 43 -- 51
hard-sphere (disk) collision for particle motion. In this scheme, the
interactions between solid and fluid particles are converted into that
between fluid particles by discretizing the solid particles into “frozen
particles”, whose properties are the same as fluid particles but their
velocities and displacements are always consistent with the solid
particle they compose. The main formulations are presented here
and more details about the model and its quantitative validation can
be found in Ma et al. (2006).
The incompressible N–S equation is discretized in particle presentation as
dva
= − c2
dt
2
b
a + b
+ 2D
b
∇a Wab
vba Wab
+ Fa
(a + b )/2(Da + Db )/2
with
Da =
b
2 W
rab
ab
(1)
where particle mass m = 1 is assumed and a quintic spline form
of the kernel function W (Morris et al., 1997) is used, and c is the
isothermal sound speed of the fluid, , a and Fa are the dynamic
viscosity, density and body force per mass at particle a, respectively.
The interactions between solid particles are hard-sphere collisions, i.e.,
V1 = V1 +
(1 + )m2 (V1 − V2 )(r1 − r2 )
(r2 − r1 ),
m1 + m2
|r2 − r1 |2
(2)
45
where V, r and m are the velocity, position and mass of the two
particles, 1 and 2, and is their restitution coefficient. The prime
denotes the value after collision.
Parallelization of the numerical scheme on massive parallel processing (MPP) systems running the message-passing interface (MPI)
protocol has been carried out based on our previous work using
space-decomposition, shift-mode communication and dynamic load
balancing (Tang et al., 2004; Wang et al., 2005).
2.2. Simulation setup
Similar to our previous simulations (Ma et al., 2006), the simulated gas–solid suspension is enclosed in a rectangular domain with
periodic boundaries in both directions. The Cartesian coordinate is
built for the system with x-axis paralleling the width direction and yaxis paralleling the height direction. The origin is at the bottom-left
corner. The major parameters are listed in Table 1, where R, s and ms
are the radius, density and mass of a solid particle, and H and W are
the height and width of the flow field, respectively. The dimensional
values are mapped from the dimensionless values actually used in
simulations by taking t = 1.627×10−8 s, m = 1.142×10−15 kg and
3h = 2.181×10−5 m. However, for simplicity, only the dimensionless
values are used hereafter.
The 389 solid particles are initially arranged on a hexagonal lattice with a nearest neighbor distance of 19, except for the pairs of
particles across the boundaries of the domain, so as to speed up the
destabilizing process, which can be seen in Fig. 1a. To establish the
force balance sooner, the particles are assigned initial velocities with
the same magnitude vs 0 = 2.0×10−5 and are parallel to the gravity
gs . The nearest neighbor distance between the fluid particles is d.
Fig. 2. Destabilizing of the gas–solid suspension. The left insets display the whole flow field at a series of times, with the areas around the tracked particles zoomed in the
corresponding right insets. White arrows from particle centers show the magnitudes and directions of solids velocities and the green ones correspond to the inter-phase
forces on particles. Black arrows in the flow field show the directions of the flow at their bigger ends, while the magnitudes of the flow velocities are presented by the color
spectrum (as shown below), white represents velocities higher than the red one. The numbers identify different particles. |v| denotes the magnitude of local flow velocity.
(a) t = 300,000, (b) t = 600,000, (c) t = 900,000, (d) t = 1,300,000, (e) t = 1,700,000, (f) t = 2,200,000, (g) t = 2,250,000, (h) t = 6,000,000.
46
J. Ma et al. / Chemical Engineering Science 64 (2009) 43 -- 51
Fig. 2. (continued).
The particles are driven by a pressure drop P which, as in Morris
et al. (1997), exerts on each fluid particle via an upward body force F.
To understand the clustering process under a more general background, we also carry out a simulation with random initial distribution of the 389 solid particles (hereafter named the “random” case
while the previously described case is named the “regular” case).
This is achieved by giving the solid particles in the “regular” case
with a random initial velocity instead (but of the same magnitude
vs 0 = 2.0×10−5 ), and applying no gravity and F for 3,000,000 time
steps. As shown in Fig. 1b and c, the resulting particle positions are
indeed random and the distribution of the magnitudes of particle
velocities is nearly Maxwellian.
3. Result and discussion
3.1. The clustering process
Fortes et al. (1987) concluded from their studies on liquid–solid
fluidized beds that clustering could be characterized by the stages
of particle drafting, kissing and tumbling (DKT). Similar process has
been observed in our simulation. Fig. 2 shows the snapshots in a
time series for the “regular” case where four particles are colored
differently and their velocities and inter-phase forces are displayed
by the white and green arrows, respectively.
The overall balance between particle drag force and gravity is
reached at t = 300,000 when the initial particle configuration is still
maintained, as seen in Fig. 2a. As particle 1 has a closer neighbor (across the periodic boundary) horizontally, it suffers stronger
resistance when settling at the same velocity as other particles and
is gradually left behind. In the meantime, particle 2 is accelerated
taking advantage of the lower gas velocity in the wake of particle
1, so that the two particles are getting closer, as shown in Fig. 2b
at t = 600,000. Note that these two process are mutually promoted,
presenting the so-called “drafting” period until the two particles
“kiss” each other at about t = 900,000, as shown in Fig. 2c. The same
thing happens to other couple of particles in their columns, which
produces many doublets in the system. The doublets further merge
into a new column which gives way to faster gas flow on its sides, as
shown in Fig. 2d. The faster gas flow then exerts stronger resistance
on particle 3 and a new round of “drafting” and “kissing” happens
to the columns of particles 3 and 4.
While the formation of doublets is largely owing to the wake effect as found in experiments (Fortes et al., 1987; Chen et al., 1991),
the development of fast gas streams between the doublet columns
seems to be the main cause for their further merging into quadruplets. According to the Bernoulli principle, the fast stream has relatively lower pressure and a horizontal pressure drop develops across
the doublet columns on both sides of the fast stream. Moreover, according to Kurose and Komori (1999), particles 1 and 3 also suffer
Magnus lift forces directing to the fast stream as they rotate in different directions under the stronger shearing of the fluid in between.
The stream velocity peaks at t = 1300,000 as shown by Fig. 2d, and
the two columns of doublets are being sucked in until they “kiss”
at around t = 1,700,000 to form the quadruplets (e.g., particles 1,
2, 3 and 4). For each doublet, this process also corresponds to the
“tumbling” of its rear particle as it bypasses the leading particle.
J. Ma et al. / Chemical Engineering Science 64 (2009) 43 -- 51
The quadruplets thus formed are less stable than the doublets
previously. As new fast stream develops on their right and the axial heterogeneity propagates downstream, the flow field soon gets
disordered after t = 2,200,000 (Fig. 2f), though some larger clusters
with six particle members can be observed temporarily at the early
stage (Fig. 2g at t = 2,250,000). A typical steady state at t = 6,000,000
is shown in Fig. 2h, which is characterized by a few loose clusters
amid fast flow streams. Note that, owing to the horizontal symmetry
of the initial configuration, nearly the same phenomenon also occurs
in the right half of the domain until the system is fully destabilized
at t = 6,000,000.
A similar “DKT” process in the “random” case can be found in
Fig. 3a–e. But it should be mentioned that, due to the initial random
distribution, destabilizing is more quickly than in the “regular” case
and is not as typical also.
3.2. The temporal variation of average slip velocity and drag force
The y components of the velocities for both all the solids and
the fluid flow in the whole domain averaged in every 10,000 time
steps, Vsy and Vfy , together with the average fluid force on each
solid particle in the corresponding periods FDy , are measured in the
47
foregoing process. The temporal variation of the average relative
velocity, Vry = Vfy −Vsy , and FDy for the “regular” case are then
plotted in Fig. 4a, where letters a–h denotes the times of Figs. 2a–h,
respectively. It can be seen in Fig. 4a that FDy reaches a plateau soon
at a with a value of 1.36×10−5 and remains steady thereafter, balancing the average gravity exerted on solid particles. The slip velocity also reaches a plateau value of 4.77×10−5 at a, but it increases
again at b, corresponding to the “drafting” in the formation of the
first batch of doublets. Thereafter, the slip velocity keeps fluctuating
with a tendency to increase until it peaks at h and then fluctuates
around a new plateau value of about 7.49×10−5 . A similar process
can be observed in the “random”, and as shown in Fig. 4b, and the
average slip velocity in the steady state is about 7.41×10−5 , which
seems to suggest that different initial solid distribution will lead to
the same statistically steady state.
The space and time scales upon which Vry and FDy are averaged
compare reasonably with those of the computational elements in
prevailing continuum descriptions of particle–fluid flows, both the
two-fluid models (TFM, e.g., Murray, 1965; Gidaspow, 1994) and
discrete particle models (DPM, e.g., Tsuji et al., 1993; Hoomans et al.,
1996). In these methods, the inter-phase friction, that is, FDy in our
case, is solely represented by the drag force which is correlated to
Fig. 3. Destabilizing of the gas–solid suspension in the “random” case. The descriptions are the same as in Fig. 2. (a) t = 20,000 (drafting), (b) t = 180,000 (kissing), (c)
t = 480,000 (tumbling and drafting), (d) t = 540,000 (kissing) and (e) t = 600,000 (tumbling).
48
J. Ma et al. / Chemical Engineering Science 64 (2009) 43 -- 51
Fig. 3. (continued).
J. Ma et al. / Chemical Engineering Science 64 (2009) 43 -- 51
49
Fig. 4. Temporal evolution of average inter-phase force and axial slip velocity of (a) “regular” case and (b) “random” case.
Fig. 5. Average distance between each particle and its nearest neighbor (left) and the fraction of isolated particles (right): (a) “regular” case, (b) “random” case, (c) “regular”
case and (d) “random” case.
50
J. Ma et al. / Chemical Engineering Science 64 (2009) 43 -- 51
Average Local Number Density (loc)
Average Local Number Density (loc)
0.0039
0.0038
0.0037
0.0036
0.0035
0.0034
0.0033
0.0032
0.0031
0
200
400
600
t(104)
800
1000
1200
0.0038
0.0037
0.0036
0.0035
0.0034
0.0033
0.0032
-100
0
100
200
300 400
t(104)
500
600
700
800
Fig. 6. Time variation of averaged local number density loc of the suspension. (a) “regular” case and (b) “random” case.
Fig. 7. Correlation between the local average clustering degree and average drag coefficient (where Cn is the local clustering degree defined as loc divided by that at
t = 6,00,000). (a) “regular” case and (b) “random” case.
local average voidage and slip velocity only (e.g., based on Ergun,
1952; Wen and Yu, 1966). Therefore, the results in Fig. 4 propose
a serious challenge in that the average voidage is constant for the
closed system here but the slip velocity changes dramatically after
the inter-phase friction has balanced the fixed particle weight precisely. Our finding gives definite support to the EMMS model (Li
et al., 1988; Li and Kwauk, 1994) and the numerical results of Agrawal
et al. (2001) with fine-grid TFM; both confirm the role of meso-scale
structures on the effective slip velocity. It also justifies the necessity
of incorporating structural factors into the drag coefficient for the
simulation of particle–fluid flows, as has been attempted by Yang
et al. (2004), Xu et al. (2007), Wang and Li (2007) and Wang et al.
(2008) through the application of the EMMS model under the TFM
and DPM frameworks.
3.3. Characterizing the meso-structure-clustering
Wylie and Koch (2000) have proposed an easy-to-calculate measure to characterize the clustering tendency by finding the nearest
neighbor for each particle and then averaging the distances over all
particles. The temporal variation of this quantity for the “regular”
case is plotted Fig. 5a. Starting from t = 600,000 when particle disorder begins, the value decreases sharply and then slows down. Finally, it fluctuates about the steady value of 12.5 from t = 4,000,000
on. Accordingly, the fraction of isolated particles (with a nearest distance larger than 13) plotted in Fig. 5b displays a significant decrease.
Both indexes give quantitative and definite evidence of clustering as
we have observed phenomenologically.
Local number density (loc ) is also used to characterize the clustering, which is calculated by searching its neighbors in a certain
area and then by a weighted averaging method using the quintic
spline, as for the fluid particles in Section 2. Different radii (r) of
the sampled region are tested for the homogeneous initial configuration and r = 63 is selected for it can already give an error smaller
than 0.1% of the theoretical value. loc is then averaged over all
particles and its temporal evolution is shown in Fig. 6a. Similar to
Fig. 4, the value loc increases with time, reflecting the process of
clustering. However, the increase lasts even beyond t = 4,000,000,
until t = 5,000,000 roughly. As can be found in Fig. 4a, it corresponds
J. Ma et al. / Chemical Engineering Science 64 (2009) 43 -- 51
to a sharp increase of slip velocity in that period, during which the
clusters keep growing from quadruplets and clusters with six particle members to even larger ones (see Fig. 2f–h).
Therefore, loc seems to be a better index for the long range
structures in the system, and the difference between Figs. 5 and 6
reveals quantitatively the different stages in clustering: single particles come closer to form doublets or small clusters first, and then
they aggregate to form larger clusters. These stages have also been
elucidated by experimental results (Chen et al., 1991).
The “random” case shows similar variation tendency of the average nearest neighbor distance, the fraction of isolated particles and
the loc in Fig. 5b and d and Fig. 6b. Maybe the only difference is
that the statistical state is reached more quickly.
4. Conclusion and prospects
In this study, using macro-scale particle methods, the clustering process from a uniform gas–solid suspension in a periodic domain with 389 particles is simulated with flow details around each
particle. The typical “drafting–kissing–tumbling” process and the formation of particle doublets and quadruplets are observed. The interphase friction is found to balance particle gravity immediately and
vary very little thereafter, but the slip velocity increases in a fluctuating manner for a much longer time before it reaches a statistically
plateau value, visualizing the compromise between the two dominant mechanisms involved: for the particle to reach minimum local
voidage and the gas to suffer least resistance, as assumed by the
EMMS model.
Owing to the limited number of particles and the conditions
tested, establishing a quantitative relationship between the drag coefficient = FDy /Vry and loc is not yet possible based on these simulation, though Fig. 7a and b does exhibit a tendency of negative
correlation qualitatively. We expect that on larger scales, more pronounced heterogeneity and more distinct clustering structures can
be observed, which may give more definite relations between these
variables. In fact, the dependence of cluster velocity on both cluster
size and alignment has been reported in liquid–solid systems already
(Chen et al., 1991) and it is highly possible that such dependence
will be more significant in more heterogeneous gas–solid systems.
Acknowledgments
This study is financially supported by National Nature Science
Foundation of China under the Grants nos. 20336040, 20490201 and
20221603 and the Chinese Academy of Science (CAS) under the Grant
KJCX-SW-L08. The simulations are partly carried out on the Lenovo
DeepComp 6800 system at the Supercomputing Center of CAS.
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