Chemical Engineering Science 60 (2005) 4567 – 4580
www.elsevier.com/locate/ces
The effects of particle and gas properties on the fluidization
of Geldart A particles
M. Ye, M.A. van der Hoef, J.A.M. Kuipers∗
Fundamentals of Chemical Reaction Engineering, Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede,
The Netherlands
Received 17 November 2004; received in revised form 8 March 2005; accepted 8 March 2005
Abstract
We report on 3D computer simulations based on the soft-sphere discrete particle model (DPM) of Geldart A particles in a 3D gas-fluidized
bed. The effects of particle and gas properties on the fluidization behavior of Geldart A particles are studied, with focus on the predictions
of Umf and Umb , which are compared with the classical empirical correlations due to Abrahamsen and Geldart [1980. Powder Technology
26, 35–46]. It is found that the predicted minimum fluidization velocities are consistent with the correlation given by Abrahamsen and
Geldart for all cases that we studied. The overshoot of the pressure drop near the minimum fluidization point is shown to be influenced by
both particle–wall friction and the interparticle van der Waals forces. A qualitative agreement between the correlation and the simulation
data for Umb has been found for different particle–wall friction coefficients, interparticle van der Waals forces, particle densities, particle
sizes, and gas densities. For fine particles with a diameter dp < 40 m, a deviation has been found between the Umb from simulation
and the correlation. This may be due to the fact that the interparticle van der Waals forces are not incorporated in the simulations, where
it is expected that they play an important role in this size range. The simulation results obtained for different gas viscosities, however,
display a different trend when compared with the correlation. We found that with an increasing gas shear viscosity the Umb experiences a
minimum point near 2.0 × 10−5 Pa s, while in the correlation the minimum bubbling velocity decreases monotonously for increasing g .
䉷 2005 Elsevier Ltd. All rights reserved.
Keywords: Discrete particle simulation; Geldart A particles; Fluidized bed; Fluidization
1. Introduction
Geldart A particles are defined as aeratable particles,
which normally have a small particle size (dp < 130 m)
and low particle density (< 1400 kg/m3 ). This kind of particles can be easily fluidized at ambient conditions (Geldart,
1973). The enormous relevance of the fluidization properties of Geldart A particles for industrial applications has
long been recognized in chemical reaction engineering, in
particular in the context of fluidized bed reactors containing FCC powders. A typical property of Geldart A particles
is that they display an interval of non-bubbling expansion
∗ Corresponding author. Tel.: +31 53 489 3000; fax: +31 53 489 2479.
E-mail address: [email protected] (J.A.M. Kuipers).
0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2005.03.017
(homogeneous fluidization) between the minimum fluidization velocity Umf and the minimum bubbling velocity Umb ,
which is absent in the fluidization of large particles (Geldart B and D particles). It is precisely this homogeneous
fluidization which is responsible for many unique features
displayed by these reactors. Notwithstanding the intense
experimental research that has been conducted in the past
30 years (Geldart, 1973; Abrahamsen and Geldart, 1980;
Tsinontides and Jackson, 1993; Menon and Durian, 1997;
Cody et al., 1999; Valverde et al., 2001), there is still no
consensus on the precise mechanism underlying the homogeneous fluidization. Consequently, there exists currently
no comprehensive theoretical approach, which is capable
of describing both the homogeneous fluidization and bubbling behavior on the basis of gas and particle properties.
Foscolo and Gibilaro (1984) suggested that the fluid–particle
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M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
interaction is the dominant factor that controls the stability of the homogeneous fluidization regime. On the other
hand, Rietema and Piepers (1990) and Rietema et al. (1993)
proposed that the interparticle forces are responsible for the
homogeneous fluidization behavior of small particles. Although both viewpoints are partially supported by some experiments (Geldart, 1973; Abrahamsen and Geldart, 1980;
Tsinontides and Jackson, 1993; Menon and Durian, 1997;
Cody et al., 1999; Valverde et al., 2001) and theoretical work
(Koch and Sangani, 1999; Buyevich, 1999; Buyevich and
Kapbasov, 1999; Sergeev et al., 2004), a complete hydrodynamical description, based on either of them, is still not
sufficient to model dense gas–solid flows involving Geldart
A particles. This significantly limits the use of state-of-theart CFD techniques in the design and scale-up of fluidized
bed reactors with Geldart A particles.
Clearly, a detailed study of the particle–particle interactions and particle–fluid interaction at a more fundamental
level is highly desirable. Discrete particle models (DPM)
can play a valuable role in such studies. DPM has been
widely used in the study of gas-fluidized beds, for example,
the hard-sphere approach by Hoomans et al. (1996), Ouyang
and Li (1998), and Zhou et al. (2002), and the soft-sphere
approach by Tsuji et al. (1993), Xu and Yu (1997), Mikami et
al. (1998), and Kafui et al. (2002). The idea of discrete particle simulation is to track the motion of each particle in the
system by solving Newton’s equations of motion. In DPM
the details of the particle–particle (and particle–wall) collisions, including friction, can be readily incorporated. Furthermore, because of the two-way coupling, discrete particle
simulations allows to study the influence of particle properties on the bed dynamics or vice versa (Li and Kuipers,
2003).
Recently, several attempts have been made (Kobayashi et
al., 2002; Xu et al., 2002; Ye et al., 2004a) to study the
fluidization behavior of Geldart A particles by use of 2D
discrete particle simulations. Kobayashi et al. (2002) studied the effect of both the lubrication forces and the van der
Waals forces on the relationship between pressure drop and
the gas velocity for Geldart A particles. They showed the
existence of a non-bubbling (homogeneous) regime, where
it was found that both the cohesive and lubrication forces
affected the profile of pressure drop for a decreasing gas
velocity, but not for an increasing gas velocity. Xu et al.
(2002) investigated the force structure in the homogeneous
fluidization regime of Geldart A particles, where they found
that the van der Waals forces acting on the particles are balanced by the contact forces. They also reported void structures during the “homogeneous” fluidization. In a previous
2D DPM study, we observed many of the typical features
of Geldart A particles in gas-fluidized beds, such as the homogeneous expansion, gross particle circulation in the absence of bubbles, fast bubbles at fluidization velocities beyond Umb (Ye et al., 2004a), and void structures (Ye et al.,
2004b). An analysis of the velocity fluctuation of Geldart
A particles suggests that homogeneous fluidization actually
represents a transition phase resulting from the competition
between three kinds of basic interactions: the fluid–particle
interaction, the particle–particle collisions (and particle–wall
collisions) and the interparticle van der Waals forces (Ye et
al., 2004a,b). However, these DPM simulations were based
on 2D geometries, and focused on the influence of cohesive
forces on the flow patterns or flow structures. No modeling
work has been carried out so far which studies the effect of
the properties of both the particulate phase and gas phase
on fluidization of Geldart A particles, although the classical empirical correlations (Abrahamsen and Geldart, 1980)
have been proposed more than two decades ago. The main
purpose of this paper is, for the first time, to make a comprehensive comparison with the well-known empirical correlation by Abrahamsen and Geldart, 1980 (in particular for
Umf and Umb .), using a full 3D soft-sphere DPM to model
the fluidization of Geldart A particles. In Section 2 the discrete particle model is briefly described. The details of the
simulation procedure are discussed in Section 3, which is
followed by a presentation of the simulation results. The paper ends with conclusions and a discussion.
2. Discrete particle model
In the discrete particle model, the gas-phase hydrodynamics is described by the volume-averaged Navier–stokes
equations, following the approach of Kuipers et al. (1992).
j(εg )
(1)
+ (∇ · ε g u) = 0,
jt
j(εg u)
+ (∇ · ε g uu)
jt
= −ε∇p − Sp − ∇ · (ε) + ε g g.
(2)
No energy equations are considered in our model. This can
be justified since we are studying the fluidization behavior
at ambient conditions where it is anticipated that heat effects
are small, so that the gas and particle flows can be safely assumed as isothermal. The gas flow is treated as compressible
as the local gas pressure and density might be locally different. The gas phase flow field is computed on a Eulerian grid
(with computational cell volume V ) using the well-known
SIMPLE algorithm (Patankar, 1980). The gas phase density
g is calculated via the equation of state of an ideal gas law:
g =
pM g
,
(3)
RT
where R is the universal gas constant (8.314 J/mol K), T
the temperature, and Mg the molar mass of the gas. The
equation of state of the ideal gas can be applied for most
gases at ambient temperature and pressure. The coupling
with the particulate phase is included by means of a source
term Sp , which is formally defined as
1
Fdrag,a (r − ra ) dV ,
Sp =
(4)
V
M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
where Fdrag,a is the drag force acting on particle a, and the
integration is performed over the volume of the computational cell V . To solve the gas-phase hydrodynamical equations, there are two types of boundary conditions that can
in principle be used to account for the sidewalls: no-slip
and free-slip boundary conditions. For the free-slip boundary conditions, the normal components of gas velocity near
the sidewalls are zero and the normal gradient of the tangential components vanishes. In the case of no-slip boundary conditions, both the normal and tangential components
of gas velocity at the side-walls vanish (Kuipers, 1990). In
this research, we use both no-slip and free-slip boundary
conditions for the sidewalls. As we will see in the following
sections, in our particular system, the free-slip wall boundary conditions predict smaller minimum bubbling velocities
(compared to no-slip boundary conditions), which are in better agrement with the values calculated from the empirical
correlations (Abrahamsen and Geldart, 1980). The minimum
fluidization velocity, on the other hand, is hardly affected by
the type of wall boundary condition.
The particulate phase is described by the Newtonian equations of motion for each individual particle in the system.
The equations of motion for a single particle a are given by
ma
Ia
d 2 ra
= Fc,a + Fvdw,a + Fdrag,a − Va ∇p + ma g,
dt 2
da
= Ta .
dt
(5)
(6)
The first and second term on the RHS of Eq. (5) are the
total contact force and the van der Waals force exerted by
neighboring particles, respectively.
The contact force between two particles (or a particle and
a wall) is obtained from a soft-sphere model proposed earlier
by Cundall and Strack (1979). In that model, a linear-spring
and a dashpot are used to formulate the normal contact force,
while a linear-spring, a dashpot and a slider are used to
compute the tangential contact force.
The interaction of particle a with the surrounding fluid
follows from a drag force Fdrag,a , which depends on the
relative velocity of the two phases, and can be written as
Fdrag,a = 3g ε 2 dp (u − vp )f (ε).
(7)
In Eq. (7), the effect of the neighboring particles on the
drag force experienced by particle a is included via the
so-called porosity function f (ε), which depends on the local porosity ε and the particle Reynolds number Rep . The
local porosity, which is calculated for each fluid cell, is
determined on a scale that is much smaller than the computational domain, and can thus reflect the effect of local structures in the fluidized bed. Many attempts have been made
to obtain accurate drag force correlation from either experiments (Ergun, 1952; Wen and Yu, 1966) or lattice Boltzmann (LB) simulations (Koch and Hill, 2001; Van der Hoef
et al., 2004, 2005). One of the most widely used correlations
is the Ergun–Wen–Yu model, where the well-known Ergun
4569
equation (Ergun, 1952) is employed for porosities lower than
0.8, and the Wen and Yu correlation (Wen and Yu, 1966) for
porosities higher than 0.8. In terms of the porosity function,
the Ergun–Wen–Yu drag model can be written as
150(1 − ε) 1.75Rep
+
18ε 3
18ε 3
ε < 0.8
(8)
f (ε) = (1 + 0.15Re0.687
)ε −4.65
p
ε 0.8
(9)
f (ε) =
and
for Rep < 1000. The Ergun–Wen–Yu correlation is based on
the measurement of both pressure drop for stationary beds
and terminal velocities of more dilute assemblies of spheres.
Basically this correlation accounts for the gross effect of
the fluid flow, and is suited to two-phase hydrodynamics at
macro-scale. The LB simulations can capture the details of
the flow field around each particle (Koch and Hill, 2001; Van
der Hoef et al., 2004, 2005), and the drag law based on LB
simulations is expected to give a more accurate representation of the drag force acting on a single particle in an assembly, at least for model conditions (static, homogeneous
systems). The effect of mobility and heterogeneity on the
LB-based drag laws are unclear at present. For this reason,
we have used the Ergun–Wen–Yu correlation in this work.
Although the applicability of that relation to the systems we
study is also questionable, in this way one can make contact
with previous modeling results, where the Ergun–Wen–Yu
correlation has been used almost exclusively.
Note that the drag force relation Eqs. (8) and (9) are based
on the average relative velocity of the particles to the fluid
velocity in the drag force relation. In the simulations, however, we use the individual velocity of each particle relative
to the fluid velocity. This seems the most straightforward
way to take the mobility of the particles into account in
the drag model, although the validity of this should still be
tested by LB methods. The individual particle velocity is in
any case much smaller than the fluid velocity for the systems we study, so the effect is expected to be small for this
particular application.
To calculate the interparticle van der Waals force between
two spheres, we adopt the Hamaker expression (Chu, 1967;
Israelachvili, 1991):
Fvdw,ab (S) =
A 2ra rb (S + ra + rb )
3 [S(S + 2ra + 2rb )]2
2
S(S + 2ra + 2rb )
×
−
1
.
(S + ra + rb )2 − (ra − rb )2
(10)
Note that Eq. (10) exhibits an apparent numerical singularity if the intersurface distance S between two particles
approaches zero. In the present model, we define a cut-off
(maximal) value of the van der Waals force between two
spheres to avoid such a numerical singularity when two particles approach, and start to compress. In practice, an equivalent cut-off value S0 for the intersurface distance is used
instead for the interparticle force (Seville et al., 2000).
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M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
3. Simulation procedures
3.1. Fluidizing the system
In the simulations, the superficial gas velocity U0 is set
to increase linearly in time
U0 = Kt.
(11)
Here the slope K is chosen in such a way that the gas velocity
U0 increases slowly, thereby avoiding sudden changes of the
flow field inside the fluidized bed. This procedure was found
to be more efficient, compared to the “step-wise” procedures
adopted by Rhodes et al. (2001) and Ye et al. (2004a). In
the step-wise procedure, the gas velocity is increased step
by step, and for each gas velocity a sufficiently long computing time is required to ensure that the bed reaches a final dynamical equilibrium since a sudden change of the gas
flow will lead to large fluctuations in the flow conditions.
We stress, however, that the linear-increase approach differs
from the common experimental procedure, and could cause
some systematic errors when compared to the experimental
correlations. Nevertheless, this linear-increase procedure is
expected to be useful for investigating the origin of bubbling
fluidization.
Preliminary simulations showed that the larger the slope
K the higher the predicted minimum bubbling point Umb under the same conditions. A smaller K was found to predicts
a Umb closer to that of the step-wise procedure, which, however, requires a longer computing time. In Fig. 1, we present
the typical results of pressure drop and bed height obtained
from the simulations for two different K values and also
for the step-wise procedure. The value K = 0.03 m/s2 was
found to be optimal in the sense that this yields a reasonable speed-up compared to the step-wise method, where the
deviation in the predicted pressure drop and bed height are
minimal (again compared to the results from the step-wise
method).
Some input parameters that have been used in the simulations are listed in Table 1. Other parameters not indicated
here will be specified for the individual simulations.
3.2. The determination of minimum bubbling point
One of the most important parameters that characterize
the fluidization behavior of Geldart A particles is the minimum bubbling point Umb , which is generally defined as the
instant at which the first obvious bubble appears (Geldart,
1973). However, such a definition is qualitative, and it proves
difficult to describe the minimum bubbling point in a more
quantitative way. It has been found that the change of the
spatial fluctuation of local porosities is the most outstanding
observation (Kobayashi et al., 2002; Ye et al., 2004a), although a temporal fluctuation of pressure drop and granular
temperature can also be observed near the transition from
homogeneous fluidization to bubbling fluidization. The typ-
Fig. 1. The pressure drop and bed height obtained from the simulations
for: K = 0.02 (Dashed line); K = 0.03 (Solid line); and the step-wise
procedure (Squares). Simulations are carried out with no-slip boundary
conditions for the sidewalls. Other parameters not listed in Table 1 are:
particle diameter dp = 75 m; Hamaker constant A = 1.0 × 10−22 J;
particle density p =1290 kg/m3 ; gas shear viscosity g =1.8×10−5 Pa s;
gas molar mass Mg = 2.88 × 10−2 kg/mol; size of the fluidized bed
12.0 × 3.0 × 1.2 mm; initial bed height H0 = 3.68 mm; and particle–wall
friction coefficient f = 0.2.
ical fluctuation of local porosities with respect to the gas
velocity is shown in Fig. 2. Here the fluctuation of local
porosities is calculated by
Nsub
Nsub Nsub
1
1
2
ε =
ε −
εk
εk
k=1 k Nsub
k=1
k=1
Nsub −1
(12)
with εk is the local porosity in subdomain k. As can be seen
from Fig. 2, there are two clear transitions occurring for the
fluctuation of local porosities with increasing gas velocity.
These two transition points are very close to the minimum
fluidization point (0.0034 m/s) and minimum bubbling point
(0.0082 m/s) determined from the visualization of the simulation results. The window of homogeneous expansion in
the discrete particle simulations can thus be determined by
the transition points of porosity fluctuation. In this paper, the
minimum bubbling point is determined by visual inspection.
M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
Table 1
Parameters used in the simulations
Parameters
Value
Particle number,
Normal restitution coefficient, en
Tangential restitution coefficient, et
Friction coefficient between particles, f
Normal spring stiffness, kn
Tangential spring stiffness, kt
CFD time step,
Particle dynamics time step,
Minimum interparticle distance, S0
Number of cells
Gas temperature, T
Gas constant, R
36 000
0.9
0.9
0.2
7 or 3.5 N/m
2 or 1 N/m
1.0 × 10−5 or 2.0 × 10−5 s
1.0×10−6 or 2.0 × 10−6 s
0.4 nm
48 × 12 × 5
293 K
8.314 J/(mol K)
Umf
4571
Therefore, in the simulation the minimum fluidization point
is determined as the first instant at which the pressure drop
across the bed equals p0 . The empirical minimum fluidization point Umf is given by (Abrahamsen and Geldart, 1980)
Umf =
9.0 × 10−4 dp1.8 [(p − g )g]0.934
0.066
0.87
g
g
.
(15)
Note that Eq. (15) is a purely empirical correlation directly
obtained from a fit of experimental data of fine particles
(Abrahamsen and Geldart, 1980). On the other hand, several correlations for Umf based on drag force models were
also derived from the balance of the gravitational force and
drag force acting on the bed of particles (Abrahamsen and
Geldart, 1980). However, the latter correlations are valid for
more general type of particles, and are not necessarily accurate for Geldart A particles. In fact, according to Abrahamsen
and Geldart (1980), Eq. (15) gave the best predictions for
fine powders used in 48 different solid–gas systems. Therefore, we prefer to to use Eq. (15) for comparison with the
Umf found in our simulations.
Umb
4. Simulation results
4.1. The effect of the sidewalls
Fig. 2. The spatial fluctuation of the local porosity. Simulations are
carried out with free-slip boundary conditions for the sidewalls. Other
parameters not listed in Table 1 are: particle diameter dp =75 m; Hamaker
constant A = 0.0; particle density p = 1495 kg/m3 ; gas shear viscosity
g = 1.8 × 10−5 Pa s; gas molar mass Mg = 2.88 × 10−2 kg/mol; size of
the fluidized bed 12.0 × 3.0 × 1.2 mm; initial bed height H0 = 3.68 mm;
and particle–wall friction coefficient f = 0.2.
We compare our results for Umb with the correlation derived by Abrahamsen and Geldart (1980), which is given by
Umb =
2.07 dp 0.06
g
0.347
g
exp(0.176W45 ),
(13)
where W45 is the weight fraction of particles having a diameter less than 45 m.
3.3. The determination of minimum fluidization point
The determination of the minimum fluidization velocity
Umf is straightforward. The pressure drop p0 across the
bed will just support the weight of particles at the minimum
fluidization point, hence the following relation should hold:
p0
= ε0 g g + (1 − ε0 )p g.
H0
(14)
Since our simulations have been carried out in a relatively
small fluidized bed, it is essential to check first the effect
of the sidewalls on both the solid and gas phase inside the
fluidized bed.
To investigate the effect on the solid phase, several simulations have been carried out with different particle–wall
friction coefficients under no-slip boundary conditions. The
results are given in Fig. 3. It is shown that the predicted minimum fluidization velocities Umf agree well with the values
calculated from the correlation given by Eq. (15). By contrast, the minimum bubbling velocities are overestimated in
the simulation, compared to correlation (13). However, the
influence of the particle–wall friction on the minimum bubbling point is negligible. In Fig. 4 we show the pressure
drop observed in the simulation for different particle–wall
friction coefficients f . Note that in the initial state the local gas pressure has not been assigned a uniform value,
the real pressure drop is not zero at zero superficial velocity. From Fig. 4 it is found that the overshoot of pressure
drop near the minimum fluidization point obviously depends
on the particle–wall friction coefficients. The bigger the
particle–wall friction coefficient, the higher the overshoot
of the pressure drop cross the bed. Without interparticle van
der Waals forces (A = 0), the overshoot is nearly zero for a
zero particle–wall friction coefficient. These findings are in
accordance with the recent experimental results obtained by
Loezos et al. (2002).
To establish the effect of wall boundary conditions on the
gas phase inside the fluidized bed, we mutually compare
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M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
Fig. 3. The effect of particle–wall friction on Umf and Umb . The symbols
indicate the simulation results for Umb (circles) and Umf (triangles). The
lines correspond to the prediction for Umb from Eq. (13) (dashed line)
and Umf from Eq. (15) (solid line). Simulations carried out with no-slip
boundary conditions for the sidewalls. Other parameters not listed in
Table 1 are: particle diameter dp = 75 m; Hamaker constant
A = 1.0 × 10−22 J; particle density p = 1495 kg/m3 ; gas shear viscosity
g = 1.8 × 10−5 Pa s; gas molar mass Mg = 2.88 × 10−2 kg/mol; size of
the fluidized bed 12.0 × 3.0 × 1.2 mm; initial bed height H0 = 3.68 mm.
the simulation results with no-slip and free-slip boundary
conditions. In Fig. 5 we show the fluctuation of local porosity
obtained in the simulations. In both cases the first transition
occurs at nearly the same gas velocity, which implies that
the minimum fluidization point is not influenced by the wall
boundary conditions. This is not surprising since the onset
of fluidization only depends on the weight of particles inside
the bed. As can be seen, however, in the case of no-slip
boundary conditions the second transition occurs at a much
later stage (Umb =0.0128 m/s), compared to the results from
(a)
(b)
Fig. 5. The effect of the wall boundary conditions on the fluctuation of
the local porosity. Circles: free-slip condition; crosses: no-slip conditions.
Simulations are carried out under the same conditions as in Fig. 2.
free-slip boundary condition (Umb = 0.0082 m/s). According to the correlation by Abrahamsen and Geldart (1980),
a Umb = 0.0070 is expected for the specified system indicated in Fig. 5. Thus the simulation results for Umb are 17%
and 82% over-predicted under free-slip and no-slip boundary conditions, respectively. On the other hand, the pressure
drop and bed height do not show large deviations for these
two different wall boundary conditions, as can be seen in
Fig. 6.
In fact, we also carried out some parallel simulations with
either no-slip or free-slip boundary conditions for some other
conditions. The general qualitative agreements have been
found, except that higher values for Umb are observed in
case of no-slip boundary conditions.
(c)
Fig. 4. The effect of particle–wall friction on the pressure drop. Hamaker constant A = 0. Other simulation conditions are the same as in Fig. 3. The
(wp)
particle–wall friction coefficient f
is (a) 0.3; (b) 0.2; (c) 0.0.
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Umf & Umb, [m/s]
M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
Granular Bond number Bo, [-]
Fig. 7. The effect of the interparticle van der Waals forces on Umf and
Umb . The symbols indicate the simulation results for Umb (circles) and
Umf (triangles). The solid line corresponds to the prediction for Umf from
Eq. (15). Simulations carried out under free-slip boundary conditions.
Other parameters not listed in Table 1 are: particle diameter dp = 75 m;
particle density p =1495 kg/m3 ; gas shear viscosity g =1.8×10−5 Pa s;
gas molar mass Mg = 2.88 × 10−2 kg/mol; size of the fluidized bed
12.0 × 3.0 × 1.2 mm; initial bed height H0 = 3.68 mm; the particle–wall
friction coefficient f = 0.2.
Fig. 6. The effect of the wall boundary conditions on pressure drop and
bed height. Solid line: no-slip conditions; dotted line: free-slip conditions.
The simulations are carried out under the same conditions as in Fig. 2.
4.2. The effect of interparticle van der Waals force
In Fig. 7, we show the effect of the strength of the van
der Waals forces on the minimum bubbling point and the
minimum fluidization point. The strength of the van der
Waals forces can be quantified by a granular Bond number
Bo , which is defined as the ratio of the interparticle van der
Waals force between two identical spheres to the weight of a
single particle. The simulations have been conducted under
free-slip boundary conditions. It is found that the influence
of interparticle van der Waals forces on Umf is negligible,
and that the predicted minimum fluidization velocities Umf
again agree well with the value obtained from Eq. (15).
On the other hand, the predicted Umb increases with an
increasing Bond number Bo , as shown in Fig. 7. This seems
to imply that the cohesive interactions between particles will
delay the minimum bubbling point in the gas-fluidized bed,
which is in accordance with previous experimental work
(Rosensweig, 1979). In case of relatively strong cohesive
forces, e.g., Bo 10, the bed behaves effectively as Geldart
C particles, where obvious bubbles have not been identified,
not even at a very high gas velocity.
In Fig. 8 the profiles of pressure drop for different
Hamaker constants are shown. It is found that the overshoot
is also affected by the interparticle van der Waals force: the
stronger the interparticle van der Waals force, the higher
the overshoot of the pressure drop near the minimum fluidization point. So on the basis of our discrete particle simulations, we conclude that the generation of the overshoot
of pressure drop in the fluidization of Geldart A particles
is due to both the particle–wall friction and the interparticle van der Waals forces. This confirms the conclusion of
Rietema and Piepers (1990).
4.3. The effects of particle density
In Fig. 9 we show the simulation results of Umf and Umb
for different particle densities. Again the predicted minimum
fluidization points Umf agree well with the correlation given
by Eq. (15). When the particle density gets higher, Umf
increases rapidly. By contrast, only a weak dependence of
Umb on the particle density is found. The predicted Umb
changes slightly from 0.0082 to 0.0094 m/s by increasing the
particle density from 900 to 4195 kg/m3 . Hence the window
of homogeneous fluidization is decreased for heavy particles,
but this is mainly due to the increase in Umf . At a very high
particle density (p =4195 kg/m3 ), Umf almost equals Umb ,
which indicates a transition from Geldart A to B fluidization
behavior. This transition is clearly shown in Fig. 9. Note
that the correlation of Abrahamsen and Geldart (1980), as
shown in Eq. (13), does not include any information about
the particle density, which suggests a negligible effect of
4574
M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
(a)
(b)
(c)
Fig. 8. The effect of the interparticle van der Waals forces on the pressure drop. Simulation conditions are the same as in Fig. 7. The Hamaker constant
A is (a) 10−21 J; (b) A = 10−22 J; (c) 0.
particle density on Umb . Our simulation results support this
conclusion.
4.4. The effects of particle size
In Fig. 10 the results of Umf and Umb for different particle diameters are shown. The predicted minimum fluidization points Umf agree well with the correlation given by
Eq. (15). A general qualitative agreement is found for Umb
when the particle diameter dp is bigger than 40 m. The
values of Umb are typically over-predicted by 15–25% in
comparison with the correlation by Abrahamsen and Geldart (1980). For particles having a diameter dp = 180 m, a
transition of Geldart A to B fluidization behavior can be distinguished. For fine particles with a diameter dp < 40 m,
the simulation results clearly deviate from the correlation.
For instance, for a particle diameter dp = 37.5 m, a much
lower Umb = 0.0022 m/s is obtained, compared to the prediction from the correlation (Umb = 0.006m/s). Note that
the interparticle van der Waals forces are absent in this simulation, i.e., the Hamaker constant A = 0. As was mentioned
previously, the incorporation of interparticle van der Waals
forces can delay the minimum bubbling point and extend
the interval of homogeneous fluidization. For fine particles
with a diameter dp less than 40 m, the interparticle van der
Waals forces may become more stronger, and will make a
shift of Umb to a higher value that are normally observed in
the experiments. Thus it can be argued that for fine particles
the interparticle van der Waals forces are playing a critical
role in the formation of homogeneous fluidization.
4.5. The effects of the gas density
To change the gas density, we can change either the molar mass Mg or gas pressure p, as shown in Eq. (3). Here
we will only consider the change of the gas molar mass,
and the effect of the gas pressure on the fluidization of Geldart A particles will be the subject of a future research. We
should mention, however, that the gas density is not uniform
inside the fluidized bed since the gas pressure is spatially
heterogeneous. Since the pressure drop across the fluidized
bed is quite small compared to the absolute gas pressure,
the gas density can be determined solely by the inlet gas
pressure. We change the molar mass Mg of the gas phase
from 1.44 × 10−2 to 5.04 × 10−2 kg/mol. As can be seen in
Fig. 11, the influence of the gas density on the porosity
fluctuation is negligible (except in the bubbling regime),
which implies that the effect of gas density on both Umf
and Umb are negligible. Incidentally, in the correlations of
Abrahamsen and Geldart (1980), as shown in Eqs. (15) and
(13), only a weak dependence of Umb and Umf on gas density is found, i.e., Umb ∼ 0.06
and Umf ∼ −0.066
.
g
g
4.6. The effect of the gas viscosity
Figs. 12 and 13 show Umf and Umb for different gas shear
viscosities. For simplicity, the interparticle van der Waals
forces are switched off by setting the Hamaker constant
A=0. Again, the minimum fluidization velocities agree well
with the values calculated from Eq. (15). Umf experiences a
continuous decrease as g increases. The minimum bubbling
M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
0.011
4575
0.024
0.010
0.020
Umf and Umb, [m/s]
Umf and Umb, [m/s]
0.009
0.008
0.007
0.006
0.005
0.004
0.016
0.012
0.008
0.004
0.003
0.002
0.000
0.001
500 1000 1500 2000 2500 3000 3500 4000 4500
0
40
80
120
160
200
160
200
Particle diameter dp, [m]
Particle density p, [kg/m3]
4.5
4
3.5
Umb /Umf, [-]
Umb /Umf, [-]
4.0
3.0
2.5
3
2
2.0
1.5
1
1.0
1000
2000
3000
Particle density p, [kg/m3]
4000
Fig. 9. The effect of particle density on Umf and Umb . The symbols
indicate the simulation results for Umb (circles) and Umf (triangles). The
lines correspond to the prediction for Umb from Eq. (13) (dashed line)
and Umf from Eq. (15) (solid line). Simulations are carried out under
free-slip boundary conditions. Other parameters not listed in Table 1 are:
particle diameter dp = 75 m; gas shear viscosity g = 1.8 × 10−5 Pa s;
gas molar mass Mg = 2.88 × 10−2 kg/mol; size of the fluidized bed
12.0×3.0×1.2 mm; initial bed height H0 =3.68 mm; and the particle–wall
friction coefficient f = 0.2.
velocities, however, manifest a systematic deviation from the
empirical correlation by Abrahamsen and Geldart (1980). As
illustrated in the Fig. 13, Umb first drops and subsequently
increases for an increasing shear viscosity, passing a minimum point at g = 2.0 × 10−5 Pa s. This is clearly in contradiction with Eq. (13), where the minimum bubbling velocity
decreases monotonously for increasing g . At present, we
have no explanation why a minimum in Umb is observed in
our simulations. It is worthwhile to mention, however, that
the correlation of Abrahamsen and Geldart (1980) was actually obtained from gas shear viscosities ranging from 0.9 to
2.0 × 10−5 Pa s. In this regime the predicted Umb also experiences a continuous decrease with an increasing gas shear
viscosity. It would therefore be interesting to perform experiments with gas shear viscosities larger than 2.0 × 10−5 Pa s,
where our simulations predict an increase in Umb .
0
40
80
120
Particle diameter dp, [m]
Fig. 10. As in Fig. 9, but now for varying particle diameter. The particle
density is equal to p = 1495 kg/m3 .
Fig. 11. The effect of gas density on Umf and Umb . The inlet gas densities
used in the simulations are 0.5990 (squares), 1.1979 (crosses), 1.4974 (triangles) and 2.2461 (circles) kg/m3 . The simulations are carried out under
free-slip boundary conditions. Other parameters not listed in Table 1 are:
particle diameter dp =75 m; particle density p =1495 kg/m3 ; gas shear
viscosity g =1.8×10−5 Pa s; size of the fluidized bed 12.0×3.0×1.2 mm;
initial bed height H0 = 3.68 mm; and the particle–wall friction coefficient
f = 0.2.
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M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
Fig. 12. The effect of gas shear viscosity on Umf . The triangles denote
the simulation results while solid line represents Eq. (15). The simulations
are carried out under free-slip boundary conditions. Other parameters
not listed in Table 1 are: particle diameter dp = 75 m; particle density
p =1495 kg/m3 ; size of the fluidized bed 12.0×3.0×1.2 mm; initial bed
height H0 = 3.68 mm; and the particle–wall friction coefficient f = 0.2.
Fig. 13. The effect of gas shear viscosity on Umb . The circles represent the
simulation results while dashed line represents Eq. (13). The simulations
are carried out under the same conditions as in Fig. 12.
To further compare with the correlation of Abrahamsen
and Geldart (1980), in Fig. 14 we show Umb vs g on a
log–log scale. The best linear fit to our data has a slope
−0.267, compared to the power −0.347 in Eq. (13). If we
only include data up to 1.8 × 10−5 Pa s, which corresponds
to the shear viscosity of air under normal condition (T =
293 K), a slope of −0.354 is obtained.
Fig. 14. Same as Fig. 13, but now in a log–log scale, and where only the
first 8 points of Fig. 13 are shown. The solid line is the best linear fit to
all 8 data points, the dashed line for the first 5 data points.
work, some typical values are dp =75 m, p =1495 kg/m3 ,
and f = 1.8 × 10−5 Pa s. The terminal settling velocity for
this kind of particle is Ut = 0.254 m/s, and thus the corresponding dimensionless numbers are F r t = Ut /gd p = 87.9,
St t = p Ut dp /f = 1583.2, and Ret = f Ut dp /f = 1.269,
respectively.
In Fig. 15 we show the dimensionless numbers based on
the minimum bubbling velocity Umb . The results are obtained for systems with different particle and gas properties. It is interesting to note that for the Froude number a
minimum value can be observed, if we neglect the point
that corresponds to a particle size dp = 37.5 m. In fact, for
dp = 37.5 m a very low Umb has been obtained, which is
probably due to the fact that we have not included the cohesive forces in the simulation for this type of particles. Since
the cohesive forces are relatively large for particles sizes below 40 m, they are expected to give rise to a larger Umb
for such a fine powder. For particles with a diameter larger
than 50 m, a minimum value F r m = 0.09 can be distinguished, which means that the bed will manifest homogeneous fluidization if F r m 0.09. In an early experimental
study, Wilhelm and Kwauk (1948) found that a fluidized bed
would display homogeneous fluidization if F r m 1. We
argue, however, that the Froude number is still too rough
to offer a criterion for the transition from homogeneous fluidization to bubbling fluidization. The typical Stokes number and Reynolds number are St m = O(101 ∼ 102 ) and
Rem = O(10−2 ∼ 10−1 ).
4.7. The analysis of dimensionless numbers
5. Discussion and conclusions
The analysis of the independent dimensionless parameters is of theoretical interest in the study of fluidization of
Geldart A particles (Sundaresan, 2003). Of particular interest are Froude number F r = U 2 /gd p , Reynolds number
Re=f U d p /f , and Stokes number St =p ud p /f . In this
In this research, computer simulations based on the softsphere DPM have been used to investigate the fluidization
behavior of Geldart A particles. The simulations have been
carried out in a 3D fluidized bed, with interparticle van der
M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
0.24
0.20
Frmb
0.16
0.12
0.08
0.04
0.00
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.50
0.55
0.60
0.50
0.55
0.60
1-mb
160
140
120
Stmb
100
80
60
40
20
0
0.30
0.35
0.40
0.45
1-mb
0.14
0.12
Remb
0.10
0.08
0.06
0.04
0.02
0.00
0.30
0.35
0.40
0.45
1-mb
Fig. 15. The dimensionless numbers with respect to the solid volume
fraction at the minimum bubbling point for particle size larger than 40 m.
The results are obtained from Fig. 9 (squares), Fig. 10 (crosses), and
Fig. 12 (triangles).
Waals forces that follow from the Hamaker theory. We first
studied the effects of the sidewalls on the hydrodynamics
inside fluidized beds. It has been found that the generation
4577
of the overshoot of the pressure drop near the minimum fluidization point is affected by both the particle–wall friction
and the interparticle van der Waals forces, which confirm the
experimental results by Loezos et al. (2002) and Rietema
and Piepers (1990).
For all cases we studied in this research, the predicted
Umf was found to be in good agreement with the correlation by Abrahamsen and Geldart (1980). The minimum
bubbling velocity Umb , in general, shows a qualitative agreement with the correlation. First, the wall boundary conditions are found to have influences on the predicted Umb . The
free-slip boundary conditions predicts a lower Umb than the
no-slip boundary conditions in our small-scale simulations.
The predicted Umb under free-slip boundary conditions is
found 15–25% higher than value calculated by the correlation, while under no-slip boundary conditions this accounts
to more than 80%. Second, the action of interparticle van
der Waals is found to delay the origin of bubbles and extend the interval of homogeneous fluidization. The higher
the granular Bond number, the higher Umb , until a transition to Geldart C behavior where no Umb can be determined.
Third, the particle density and gas density are shown to have
a weak effect on Umb . For heavy particles, the window of
homogeneous fluidization is decreased mainly due to the increase in Umf , where a transition from Geldart A to B behavior (no homogeneous fluidization) is found at a density
of 4195 kg/m3 for dp = 75 m. Also, it has been found that
the particle size has a strong effect on Umb . The predicted
Umb with different particle diameter agrees with the correlation except for fine particles with a diameter dp < 40 m.
This may due to the fact that we turn off the interparticle
van der Waals forces in the simulations. It can be argued
that for larger particles with a diameter dp > 40 m the interparticle van der Waals forces may have a negligible effect on the formation of homogeneous fluidization. For fine
particles, however, a proper incorporation of interparticle
van der Waals forces is highly desired. One of the problems is that there are currently no reliable estimates for the
magnitude of the cohesive forces, since it has proved not
possible to directly measure these forces in experiments.
Eq. (15) might implicitly reflect the effect of cohesive interaction since for finer particles the cohesive forces might give
rise to some clustering of the particles which would affect the
drag force. However, we are not expecting an apparent increase of Umf for slightly cohesive particles as the clustering
effect will be minimal, where the onset of fluidization is only
due to the balance of gravity and gas–particle interaction
(drag force) for homogeneous systems. Finally, the effect of
gas viscosity has been examined. The minimum bubbling
velocities predicted with different gas viscosity, however,
manifest a systematic deviation from the empirical correlation by Abrahamsen and Geldart (1980). We found that with
an increasing gas shear viscosity the Umb experiences a minimum point near 2.0 × 10−5 Pa s, while in the correlation by
Abrahamsen and Geldart (1980) the minimum bubbling velocity decreases monotonously for increasing g . If we fit the
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M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
Fig. 16. The snapshots of the simulation results of the homogeneous fluidization of Geldart A particles. The far left graph shows the fluidized bed in
3D. Graphs 1–5 show the cross sections of the bed (cutting through the width direction are shown. The simulation conditions are the same as in Fig. 2.
Fig. 17. As in Fig. 16, but for the bubbling fluidization.
data up to 2.0 × 10−5 Pa s, a power of −0.267 has been obtained, which is not far away from the value (−0.347) given
by Eq. (13). Clearly, a more elaborate study of the effect
of viscosity on Umb is required, both from experiment and
simulation.
With respect to the latter, however, we note that it is a
non-trivial task to determine the minimum bubbling velocity
Umb , since there is no unique, quantitative formalism to
relate Umb to parameters that can be directly measured in
the discrete particle simulations. This is directly related to
the fact that the minimum bubbling point is rather loosely
defined, namely, as the instant at which “the first obvious
bubble” appears in the fluidized bed. To illustrate this point,
in Figs. 16 and 17, we show some typical snapshots for
both homogeneous fluidization and bubbling fluidization. As
can be seen, even during the homogeneous fluidization, we
can still find some void structures. It would be extremely
difficult to define a formalism (i.e., a computer code) which
could discriminate the voids and cavities of homogeneous
fluidization from the first obvious bubble, just on the basis
of the particle coordinates.
Notation
A
d
F, F
Fr
g, g
H
I
K
m
n
Nsub
Npart
p
r
r
Re
S
St
t
Hamaker constant, J
particle diameter, m
force, N
Froude number, −
gravitational acceleration, m/s2
bed height, m
moment of inertia, kg m2
Slop, m/s2
mass, kg
normal unit vector, dimensionless
number of sub-domains, dimensionless
number of particles, dimensionless
gas pressure, Pa
particle radius, m
particle position, displacement, m
Reynolds number, dimensionless
Intersurface distance between spheres, m
Stokes number, dimensionless
time, s
M. Ye et al. / Chemical Engineering Science 60 (2005) 4567 – 4580
T
u
U
v
V
W
torque, N m
local gas velocity, m/s
superficial gas velocity, m/s
particle velocity, m/s
volume, m3
the weight fraction of fine particles, dimensionless
Greek letters
ε
f
g
porosity, dimensionless
coefficient of friction, dimensionless
gas shear viscosity, Pa s
density, kg/s3
viscous stress tensor, kg/m s2
angular velocity, 1/s
Subscripts
0
a, b
c
drag
g
mf
mb
p
t
vdw
initial state
particle index
contact force
drag force
gas phase
minimum fluidization point
minimum bubbling point
particlephase
terminal setting velocity
van der Waals force
Superscripts
n
t
normal direction
tangential direction
Acknowledgements
This work is part of the research program of the Stichting
voor Fundamenteel Onderzoek der Materie (FOM), which is
financially supported by the Nederlandse Organisatie voor
Wetenschappelijk Onderzoek (NWO).
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