7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Settling of Cuboid Particles Having Various Aspect Ratios
Jinsheng Wang, Haiying Qi and Changfu You
Tsinghua University, Key Lab for Thermal Science and Power Engineering of Ministry of Education
Beijing 100084, P.R. China
wangjinsheng06@mails.tsinghua.edu.cn
Keywords: cuboid particles, aspect ratios, drag, terminal velocity
Abstract
Twophase flow with nonspherical particles plays an important role in both technical and nature processes, such as biomass
combustion, chemical blending, mineral processing and transport of sediment in rivers. Sedimentation is a common method
used to study such flows. The settling of cuboid particles with various aspect ratios was measured experimentally in calm
fluids with the viscosity from 1.523 to 872.8 Pa' s. The terminal velocity, ut, was measured with a high speed camera and then
the drag coefficient, CD, was calculated from a force balance. The results show that the relationship between the terminal
velocity per unit mass ut/mp, and the volumeequivalent spherical diameter, do, was independent of the aspect ratio. The
difference between the drag coefficient on the experimental particles and that of spheres reached a minimum at the transition
point where the particles enter the self excited flow domain. The difference between CD for cubes and that for spheres was
larger than for plates and rods during sedimentation. The prospect to the particle shape study of nonspherical particles is
discussed.
Introduction
Two phase flow are very common in both technical and
natural processes. Modeling of these processes relies on
the assumption of spherical particles. However, particles
are nonspherical in practice, such as biomass combustion,
chemical blending, mineral processing and transport of
sediment in rivers. The particle and fluid reactions are
affected by the particle shape strongly. The characteristics
of settling, which occurs in all particle flows, are different
from those of spherical particles and they are needed and
foundational to fully understanding the interactions
between these particles and the fluid.
There have been many studies of the settling of
nonspherical particles. The terminal velocity and drag
coefficient of a particle strongly depend on its shape
strongly during sedimentation in addition to the fluid
properties. Thus, the research of particle shape is very
important for two phase flow of nonspherical flow.
Wadell proposed the concept of the particle sphericity, 0, to
describe the nonspherical particles in 1934. The sphericity
was defined as the ratio of the surface area of the
equivalentvolumesphere to the actual surface area of the
particle, A (Wadell, 1934):
=. /, A (1)
where dv is the equivalentvolumesphere diameter. For a
true sphere, 0 is 1. For irregular particles, 0 is difficult to
determine because the surface area is not easy to measure
directly. However, this shape factor has been widely used
to characterizing the shape and related to the drag
coefficient model of nonspherical particles (Haider &
Levenspiel, 1989; Hartman, Trnka & Svoboda, 1994;
PettyJohn & Christiansen, 1948).
Reference to the spericity, the particle circularity is defined
as the ratio of the sphere perimeter with the equivalent
projected area to the projected particle perimeter
(TranCong, Gay & Michaelides, 2004):
(O= PA /P (2)
Where PA is the sphere perimeter with the equivalent
projected area and Pp is the projected particle perimeter.
Different from the particle sphericity and particle
circularity, the Stokes' shape factor is not based on the
particle geometry but the terminal velocity, ut, of the
particle PettyJohn & Christiansen, 1948). The terminal
velocity is compared to that of spheres with the same mass
and volume. The Stokes' shape is defined as:
18 /f ,
stokes (p ).g. (3)
where pp and pf are the density of particle and fluid, and wif
is the fluid viscosity. This shape factor can be obtained by
measuring the particle terminal velocity in the Stokes
regime.
However, developing correlation for drag coefficient of
nonspherical particles relies on a large amount of
experiment data. A number of empirical correlations have
been proposed for different shapes including spheroidal
particles (Militzer, Kan, Hamdullahpur, Amyotte &
AlTaweel, 1989; Tripathi, Chhabra & Sundararajan, 1994),
cylinders (Marchildon, Clamen & Gauvin, 1964; Zhu, Lin
& Shao, 2000; Gabitto & Tsouris, 2008),
parallelepipeds(Heiss & Coull, 1952), cones (Jayaweera &
Mason, 1965), fibers (Fan, Yang, Yu & Mao, 2003) and
hemispheres and sphere segments (Wang, Qi &You, 2009).
Although, these studies are accurate for the selected shapes
and orientations, they are not suitable for other shapes.
Haider and Levenspiel (1989) were probably the first to
interest in the development of universally applicable drag
coefficient expressions for various shapes. They proposed
the following correlation for spherical and nonspherical
particles based on more than 500 data points:
24 K
C (1+K Re )+
Re 1+K /Rep
K = 0.0964 + 0.55650 (4)
3
K = exp ,a'2 ( = 2,3,4)
where the values of Kj were found by the minimizing the
root mean square errors in the calculated and experimental
values of the drag coefficient. The values of a, are given in
Table 1.
Table 1: Values of a,
ao al a2 a3
K2 2.3288 6.458 2.4486 0.0
K3 4.905 13.8944 18.42222 10.2599
K4 1.4681 12.2584 20.7322 15.8855
Although this expression is a great improvement for two
phase flow with nonspherical particles, it could not be
applied in practice because the expression is only good for
a few shapes but are not accurate for other shapes.
In addition, there have been many other studies on the drag
coefficient for various shapes (Swamee & Ojha 1991;
Ganser, 1993; Hartman, Trnka & Svoboda, 1994; Chien,
1994; Chhabra, Agarwal, and Sinha, 1999, Holzer, &
Sommerfeld, 2008). However, they all have the same
disadvantage as the expression of Haider and Levenspiel
(1989). As yet, the best accuracy of the universally
correlation for all shapes is 14% in the literature the author
have seen.
All of these studies show that research on sedimentation of
nonspherical particles needs an indeep study of particle
shape and relies on a large number of experimental data
points as a function of the particle geometries.
Due to the various shapes of nonspherical particles, the
particles should be divided into regular and irregular
shapes. Considering the complex particle shapes, it is
feasible to study two phase flow of nonspherical particles
with regular shapes. This study is concerned with cuboids
having different aspect ratios with square bases since they
have regular shapes and there is little experimental data for
cuboids in the literature.
This study presents the experimental data and spheres to
analyze the influence of the aspect ratio. Thus, this study
presents important experimental data for nonspherical
particles. In addition, a prospect to the particle shape study
of nonspherical particles is discussed.
Nomenclature
Particle base length (mm)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Ar Archimedes number
CD Drag coefficient
dv volumeequivalent spherical diameter (m)
g Gravitational acceleration i. i
h Particle height (mm)
n Face number of regular polyhedron
Sphere perimeter with the equivalent projected
A area (2)
Pp Projected particle perimeter (m2)
Re, Particle Reynolds number
Ut Terminal velocity (m/s)
pp Particle density, (kg/m3)
pf Fluid density, (kg/m3)
Wf Fluid dynamic viscosity, (Pas)
0) Sphericity
(? Circularity
Experimental Facility
The experiments used a rectangular parallelepiped test
section 1500x300x300 mm as shown in Figure 1.
S Test S Grid of test region
region
Figure 1: Experimental system
The particles were made of aluminium (2703.2 kg/m3),
PVC (1510.3 kg/m3) and plexiglass (1170.8 kg/m3) with
different base lengths, a, and heights, h. The particles were
divided into four series marked P1 to P4. For P1, the base
length, a, was constant at 10 mm and the height, h varied
from 3 to 30 mm. For P2, P3 and P4, the heights were
constant at 10, 15 and 20 mm, and the base length, varied
from 4 to 20 mm. Each series had only one geometry
variable, and the particle shape changed from plate to rod
regularly for each particle series as in Figure 2. The aspect
ratios, h/a, is from 0.3 to 5. Thus, the maximum and
minimum values of the sphericity for the particles are
0.806 and 0.642.
h
z
a
Figure 2: Regular changes of particle shape with size
The experiments used six glycerinwater solutions having
different glycerin volume concentrations as the test fluids.
The fluid parameters are given in Table 2.
The influence of temperature on the fluid parameters was
neglectable because the temperature fluctuated only 3C
from 18 to 21 C during the experiments.
The terminal velocities were measured in two adjacent
regions as shown in Figure 1, both located far from the end
enough for the end effects to be negligible. However,
because of the unstable behavior of these particles, the
instantaneous velocities always changed. Thus, a particle
was assumed to be at the terminal velocity when the
difference between the two velocities was less than 5%,
with ut as the average of the two values. During the
experiment, each particle was dropped three times with
reproducibility of the results less than 2%. The influence of
the wall was eliminated by only calculating the terminal
velocity for cases with the settling oscillation amplitudes
less than 10% of the cross section length.
Table 2: Glycerinwater parameters where C is the
glycerin concentration (20 C)
Dynamic
Concentration Density viscosit
No. C (Vol. %) pf (kg/m') P
[If 10 P
FO 0 971 1.523
F20 20 1028.8 3.064
F40 40 1086.6 5.203
F60 60 1144.4 14.87
F80 80 1198.4 82.77
F100 100 1260 872.8/
At the terminal velocity, the gravitational force, buoy
and drag for the particle are balanced as:
FD = F F = d3 g(p p )
6
Here dv, pf, and pp can be measured, so CD car
calculated from its definition with dv as the character
length :
ic
y
a.s
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The terminal velocity first increases and then remains
almost constant as the particle size increased. The velocity
becomes constant at the cube shape, a=10 mm because as
the particle size increased, the mass and the projected area
both increased. The increased mass increases the terminal
velocity, while the increased projected area reduced the
terminal velocity. When these opposite effects were
balanced, the terminal velocity remained almost constant.
0 5 10 15 20 25
a (mm)
Figure. 3: Relationship between ut and ut/m and the
geometry for rods and plates
(F60, pp=2703.2 kg/m3, h=10 mm)
The influence of the particle mass was eliminated by
plotting the terminal velocity per unit mass, ut/m, versus
the base length, a as shown in Figure. 3. Using the terminal
velocity per unit mass, the curve no longer had a transition
point where the velocity became constant. Thus, data point
could be represented by a onetoone functional
ancy relationship over the entire range of the particle geometry.
The relationship between ut/mp and the base length, a, in
(5) Figure. 3 can be represented as (R2=0.993):
ut /mp= 7743* a72 (7)
n be
ristic
2FD 4 d,(p p)g
D (6)
SApu2 3 pf 2
where A is the projected area and the terminal velocity.
Results and Discussion
Settling behaviors
During the experiment, various complex behaviors such as
swing and rotation were observed. The complex behavior
is an important character of the particle sedimentation and
is closely related with the terminal velocity and drag.
However, the behaviors were found to be independent of
the initial particle orientation.
Different particle shapes resulted in different settling
behaviors. The plates followed a spiraled track while the
rods zigzagged. The cube rotated almost continually
following a complex helical path. In some extreme cases
with the larger particles and low viscosity fluid, the
particles even ran into the wall.
Terminal velocity per unit mass
The terminal velocity, ut, was measured for the oscillation
smaller than 10% of the column width. The relationship
between ut and the particle geometry is shown in Figure. 3.
where, R is the correlation coefficient.
Interpolation of the data Gave the relationship between
ut/mp and the volumeequivalent spherical diameter, dv, for
h/a=0.3 and 5 as shown in Figure 4. Though the aspect
ratios, h/a, differed greatly, the relationship between ut/mp
and dv for h/a=0.3 and 5 are quite similar. This indicates
that ut/mp is independent of the aspect ratio and the
volumeequivalent spherical diameter, d,, is the most
important parameter for ut/mp.
104
 2
E 102
J 101
1001
0.00 0.02 0.04 0.06
d (m)
Figure 4: Relationship between ut/mo and dv
(h/a=5,0.3,F60)
Comparison with spherical CD
The drag coefficient, CD, was calculated with dv as the
characteristic length using Eq. (6). Figure 5 compares the
drag coefficient CD for the experimental particles and a
pp=2703.2 kg/m3
h/a
o0.3
* 5
sphere. As Rep increased, the difference between CD for the
experimental particles and the sphere first decreased and
then increased.
104 "
101 1
102
101 100 101 102 103 104
Re ()
Figure 5: Drag coefficient for all particles
The dimensionless Archimedes number, Ar, was then used
to obtain a more distinct functional relationship between
the terminal velocity and the particle and fluid
characteristics. The definition of Ar is as:
Ar Pf (p Pf)g
Ar =
The relationship between Ar and Re, was illustrated in
Figure 6 represents all of the data quite well.
104 4
1 Sphere "
1002
100 101
102 103 104 105 106 10 108
Ar()
Figure 6: Relationship between Rep and Ar
This curve can be correlated as
Rep =exp(17.3Ar 079 +12.64) (9)
The difference between the curve fit for Rep for the
particles is shown in Figure 7 with the relationship for CD
shown in Figure 8 where 5 and c were defined
Kep D
I Re Re, sphere
Rep Re
p,sphere
c c
CD CD D,sphere
CD r
D,sphere
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
which strongly influence the flow field. Vortex shedding
occurred more easily at these sharp covers than at the
spherical surface. This caused oscillations of the particles,
thus, the drag coefficient changed little as Rep increased.
For spheres at the same Rep, the drag coefficient still
decreased as Rep increased. Thus, the differences increased
as the Rep increased. The effect of the particle shape in the
low Rep region on the flow field was not as clear. Thus, the
reason for the increased differences as the Re, decreased in
this region needs more indepth study.
1.0
0.9
0.8
S0.7
0.6
0.5
0.4
100 101 10 10 102 10 6 10 7 10 108
Ar ()
Figure 7: Difference for Rep between the experimental
particles and spheres
5
4
S3
2
1
0
102 101 100 101
Re ()
102 103 104
Figure 8: Difference for CD between the experimental
particles and spheres
Correlation for different aspect ratios
The correlation of u/mp and particle size in Eq. (7) was
used to calculate data for Rep versus Ar. The effects of
different shapes were based on plates, cubes and rods with
aspect ratios h/a=0.3, 1 and 5. The resulting relationship
between Rep and Ar is shown in Figure 9. In the figure, the
differences between the three aspect ratios are not obvious,
but some small differences exist. The data for the three
aspect ratios can be correlated as:
Rep = exp(17.67Aro 075 +13.24)
p ,h/a=5
Re, =exp(18.28Ar0074 +13.78)ha0.3 (11)
Rep =exp(16.78Ar0091 +11.6)
h/a=1
The difference reached a minimum at about Ar=5000 or
Rep=30, where was the turning point for the particle
coming into the self excited flow domain. The
experimental particles had sharp covers in the surface
/y
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
10 4
o3 "
10 3
102
r101
r Sphere h/a p
100 A 0.3 0.677
0 1 0.806
101 a
10 1 5 0.642
1021
102 100 102 104 106 108
Ar ()
Figure 9: Relationship between Rep and Ar
(h/a=0.3, 1, 5)
The correlation in Eq. (11) was then used to calculate the
differences between Rep and CD for the three types of
particles and spheres as in Figure 10 Figure 11.
1.0
0.0
102 101 100 101 102 103 104 105 106 107
Ar ()
Figure 10: Differences of Rep for the particles relative to
spheres
(h/a=0.3, 1, 5)
5
o" 2
1
0
102 101 100 101 102 103 104
Rep ()
Figure 11: Differences of CD of the particles compared
with spheres
(h/a=0.3, 1, 5)
Rep for the minimum difference varied slightly for the
three types of particles. Thus, the transition point for the
three types of particles into the self excited flow domain is
approximate. Even though the shape characteristics were
very different, the sharp covers on the surfaces were
similar, thus the transition points were similar.
1.0
0 0.5
0.0
2 h/a 4
h/a
Figure 12: Relationship between 6c and aspect ratio
Being the most similar to the sphere, the cube had the
largest difference as shown in Figure 12. One reason was
the more irregular settling behavior of the cube compared
with the plates and rods especially the rotation of the cube.
This indicates that the settling behaviors are very important
for two phase flow with nonspherical particles and may
strongly influence the particle and fluid reactions.
Discussion of the particle shapes
According to the above, different particle shapes cause the
different settling characteristics. Due to the various shapes
of nonspherical particles, it is unpractical to experiment
with particles of each type of shapes. Thus, the study of
particle shapes is very significant for two phase flow with
nonspherical particles.
As the most widely used factor, the particle sphericity have
the disadvantage that different particle shapes may have
the same value of sphericity. For example, particles in this
study with different aspect ratios may have the same value
as shown in Figure 13.
1.0
0.8
0.6
0.4
0.4
5 10
h/a ()
15 20
Figure 13: Sphericity of cuboids with different aspect
ratios
Despite this, the sphericity has a great value in description
of particle shapes. The reason of two particle shape having
the same value is that the shape characteristics changes. As
shown in Figure 13, the shape is from plates to rods.
However, for the regular polyhedrons, the value of
sphericity match the face number, n, of the regular
polyhedrons as shown in Figure 14. This indicates that for
this rule of transmutation, the sphericity is powerful.
Re =100
SP
h/a=1
5
1.0
0.9
0.8
0.7
0.7
5 10 15 20 25
Figure 14: Sphericity of the regular polyhedrons
Thus, for the different rules of transmutation, there are
different appropriate shape descriptions. Based on the rules
of transmutation, there are different particle series. For the
regular polyhedron series, the sphericity is useful, but for
the platerod series, it is disabled. In this study, the results
indicate that the particle series having the same base but
different height have the similar relationship between Re,
and Ar because of the same sharp covers around the
surface. According to these, the particle shape diagram for
regular particles is then proposed as Figure 15. Studying of
the typical shape is very important for each series. The
typical shape represents the main difference from spheres
including CD and Rep. However, in this diagram, the
platerod series and the regular polyhedron series have the
crossing at cube. Thus, the inherent relationship between
any two particle shapes in the diagram has been obtained.
Furthermore, other complex shapes can be obtained from
combination or transmutation of the shapes in the diagram.
Consequently, the diagram is an important reference for
two phase flow with nonspherical particles in the further
study.
o0'
Cube
te
Figure 15: Particle shape diagram
Conclusions
The settling characteristics of particles with different
aspect ratios were investigated experimentally. The results
show that:
1) The terminal velocity per unit mass, ut/mp, was an
important parameter with the relationship between ut/mp
and dv being independent of the aspect ratio, h/a.
2) The difference between CD for the experimental
particles and for spheres reached a minimum at a transition
point where the particle enters the self excited flow
domain.
3) The differences between CD for cubes and spheres were
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
larger than that for plates and rods during sedimentation.
4) The particle shape diagram according to the inherent
relationship of different shapes was proposed.
Acknowledgements
This work was supported by the Major Program of the
National Natural Science Foundation of China with Grant
No. 10632070.
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