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Chemical Engineering Science 62 (2007) 3397–3409www.elsevier.com/locate/ces
Computation of gas and solid dispersion coefficients in turbulent risers andbubbling beds
Veeraya Jiradilok a, Dimitri Gidaspowa,∗, Ronald W. Breaultba Illinois Institute of Technology, Chicago, IL, USAbUS Department of Energy, Morgantown, WV, USA
Received 1 July 2006; received in revised form 30 January 2007; accepted 30 January 2007
Available online 24 March 2007
Abstract
A literature review shows that dispersion coefficients in fluidized beds differ by more than five orders of magnitude. To understand the
phenomena, two types of hydrodynamics models that compute turbulent and bubbling behavior were used to estimate radial and axial gas
and solid dispersion coefficients. The autocorrelation technique was used to compute the dispersion coefficients from the respective computed
turbulent gas and particle velocities.
The computations show that the gas and the solid dispersion coefficients are close to each other in agreement with measurements. The
simulations show that the radial dispersion coefficients in the riser are two to three orders of magnitude lower than the axial dispersion
coefficients, but less than an order of magnitude lower for the bubbling bed at atmospheric pressure. The dispersion coefficients for the bubbling
bed at 25 atm are much higher than at atmospheric pressure due to the high bed expansion with smaller bubbles.
The computed dispersion coefficients are in reasonable agreement with the experimental measurements reported over the last half century.
2007 Elsevier Ltd. All rights reserved.
Keywords: Fluidization; Gas-particle flow; Computational fluid dynamics; Reynolds stresses
1. Introduction
Traditional design of gasifiers for the FutureGen project and
other reactors requires the knowledge of dispersion coefficients,
as demonstrated by Breault (2006). However, they are known to
vary by 5 orders of magnitudes (Gidaspow et al., 2004; Breault,
2006).
From experimental investigations over the last half-century,
the dispersion coefficients are known to be large for large diam-
eter bubbling beds and small at low gas velocities. Surprisingly,
they differ by two to three orders of magnitudes at the same
gas velocity. Hence, a better understanding of the phenomena
causing such large differences is needed. This study presents a
∗ Corresponding author. Department of Chemical Engineering, Illinois In-stitute of Technology, 10 west 33rd street, Chicago, IL 60616, USA.
Tel.: +1312 5673045; fax: +1312 5678874.
E-mail address: [email protected] (D. Gidaspow).
0009-2509/$- see front matter 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2007.01.084
computational method of determining the gas and solid ax-
ial and radial dispersion coefficients in bubbling beds and
risers.
The physical definition of dispersion coefficients is based
on the kinetic theory of gases (Bird et al., 2002; Chapman
and Cowling, 1970) and granular flow (Gidaspow, 1994). For
diffusion of gases or particles the diffusivity, D, is defined as
the mean free path, L, times the average velocity, C , as shown
below:D = L × C, (1)where the peculiar velocity, C is given by
C = c − v, (2)where c is instantaneous velocity and v is the velocity averaged
over velocity space, as shown below:
v = 1n
cf (c)dc with n =
f(c) dc. (3)
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3398 V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409
Tartan and Gidaspow (2004). measured the instantaneous
particle velocities, c, in a riser and computed the hydrodynamic
velocities, v, as a function of time. From these two types of
velocities the particle stresses, CC, and the Reynolds stresses,
vv, were computed, where v=v−v and v is the time averagedvelocity. In the center of the riser the velocity, v, could be
obtained from Poiseuille flow.The mean free path was obtained from the average velocity
and collision time, :
L = C × , (4)
since C and√ C2 differ by only 10%.
The dispersion matrix can be defined as
DP = CC × . (5)
Similarly for turbulent oscillations, the dispersion matrix is de-
fined as
DT = vv × . (6)
This definition is identical to that used in single phase tur-
bulent flow (Hinze, 1959). For gas–particle flow we have two
types of dispersion coefficients, that for the gas and for the par-
ticles. In this paper we computed only the turbulent dispersion
coefficients.
Hence the dispersion coefficients in the x and y directions
are computed from the normal Reynolds stresses in the x and
y directions. For the riser, the normal Reynolds stresses in the
direction of flow are shown here to be about two orders of
magnitude larger than the radial Reynolds stresses due to thefact that the radial velocities are small compared to the axial
velocities. This provides an explanation for the large anisotropy
of the dispersion coefficients in the riser and not in bubbling
beds, where the velocities in radial and axial directions are
similar.
2. Hydrodynamics models
The physical principles used are the conservation laws of
mass, momentum and energy for each phase, the fluid phase and
the particulate phases. This approach is similar to that of Soo
(1967) for multiphase flow and of Jackson (1985) for fluidiza-
tion. A Newtonian or power law constitutive equation for the
surface stress of phase “k ” will depend at least on its symmet-
rical gradient of velocity. The kinetic theory of granular flow
provides a physical motivation for such an approach (Gidaspow,
1994). Hence, the general balance laws of mass and momen-
tum for each phase, with phase change, are given by Eqs. (7)
and (8) and the constitutive equation for the stress is given by
Eq. (9).
Continuity equation for phase k :
t
(kk)
+ ∇ (kkvk)
= ˙mk . (7)
Table 1
Hydrodynamic viscosity model
Continuity equations
(gg)
t + ∇ · (ggvg) = 0
(ss)
t + ∇ · (ssvs) = 0
Momentum equations
(ggvg)
t + ∇ · (ggvgvg) = −∇ PI + ∇ · g − B
vg − vs
+ gg(ssvs)
t + ∇ · (ssvsvs ) = − ∇ P sI + ∇ · s + B(vg − vs)
+ sg
s −
N k=g,s
kg
g
Constitutive equations
(1) Definitions
g + s = 1
(2) Gas pressure
P g = g R̃T g
(3) Stress tensor (i = gas or solid)
i = 2iDi + (i − 23i) tr(Di )I
with
Di = 12 [∇ vi + (∇ vi)T ](4) Empirical particulate phase viscosity and stress model
∇ P s = G(g)∇ s
G(g) = 10−8.686g+8.577 dyne/cm2
s = 0.1651/3s g0 poise(5) Fluid-particulate interphase drag coefficients for < 0.8 (based on the
Ergun equation)
= 1502sg
2gd 2p
+ 1.75gsgd p
|g − s |
for > 0.8 (based on the empirical correlation)
= 34Cd
gs |g − s |d p
−2.65g
where
Cd = 24Rep
[1 + 0.15Re0.697p ] for Rep < 1000
Cd = 0.44 for Rep > 1000
Rep =ggd p|vg − vs |
g
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3400 V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409
These equations are similar to Bowen’s (1976) balance laws
for multicomponent mixtures. The principle difference is the
appearance of the volume fraction of phase “k ” denoted by k.
In the case of phases not all the space is occupied at the same
time by all phases, as it is by components due to the negligi-
ble size of molecules. As in the case of the mixture equations
for components, the mixture equations for phases show that thesum of the phase change productions in Eq. (7) is zero and the
sum of the drag forces in Eq. (8) is zero. In convective form,
the phase change momentum in Eq. (8) is zero, insuring in-
variance under a change of frame of reference for translation.
Eq. (9) is the usual Newtonian expression for the stress which
arises from the assumption that the stress is a function of its
own symmetrical gradient of velocity. For the fluid P k is the
fluid pressure. When this form is substituted into the momen-
tum equation, the result is not the usual momentum balance
presented by Gidaspow (1986) and widely used in gas–liquid
two-phase flow. It is a slightly modified version of the momen-
tum balance called model B by Bouillard et al. (1989).
In this study we are applying this model to single size
particle–gas system with no reaction or phase change. The
particle viscosity and the solid stress are input into the vis-
cosity model given in Table 1. In the kinetic theory version,
Table 2, the particle viscosity and solid stress are automatically
System Geometry and System Properties
Riser diameter 0.186 m.
Riser inlet diameter 0.093 m.
Riser height 8 m.
Particle size 54 µm
Particle density 1398 kg/m3
Restitution coefficient, e 0.9
Wall restitution coefficient, ew 0.6
Specularity coefficient, 0.6
Grid size, (∆ x × ∆ y) 0.465 cm × 2.68 cm
Grid number 42 (radial) × 300 (axial)
Time step 5 × 10-5
Outlet
Y
Inlet
X
Fig. 1. System geometry for simulations based on Wei et al. (1998a) experiments.
computed. The restitution coefficient is a fitting parameter. In
the simulation presented here the drag was also modified for
flow of FCC particles as described in Jiradilok et al. (2006).
The numerical scheme used in this study is the implicit con-
tinuous Eulerian (ICE) approach. The model uses donor cell
differencing. The conservation of momentum and energy equa-
tions are in mixed implicit form. It means that the momentumequations are fully explicit. The continuity equations excluding
mass generation are in implicit form.
3. Simulations
3.1. Flow of FCC particles in the riser
3.1.1. System properties
Jiradilok et al. (2006) have already shown that the kinetic
theory model with the EMMS approach, as shown in Table 2
(Yang et al., 2004), is capable of computing turbulent fluidiza-
tion of FCC particles in agreement with experimental data. The
computed granular temperatures, particles viscosities, solid
pressures and oscillation frequencies agreed with experiments.
The simulations were carried out for the riser section of a
circulating fluidized bed. A two-dimensional Cartesian coordi-
nate system was used. Initially the riser column was empty and
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V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409 3401
0
1
2
3
4
5
6
7
8
H e i g h t , m .
800
700
600
500
400
300
200
100
5 10 15
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.0001
Laminar Granular Temperature m2 /s20.001 0.01 0.11 1
Fig. 2. (a) Snapshot of solid volume fraction, (b) axial laminar granular temperature profile for W s = 132kg/m2 s and U g = 4.57 m/s.
the velocities of both phases were assumed to be zero. At the
outlet, atmospheric pressure was prescribed. Particularly im-
portant is also the specification of appropriate boundary condi-
tions at the wall. For the gas a non-slip boundary condition was
used. For the granular temperature wall boundary condition the
Johnson and Jackson (1987) boundary condition was used. It
was obtained by equating the granular flux to collisional dissi-
pation with a correction for slip.
Fig. 1 gives the system geometry and the system properties
for describing the experiments of Wei et al. (1998a). The main
fluidizing gas is air. The solid phase consisted of FCC particles.
A restitution coefficient of 0.9 was used (Jiradilok et al., 2006).
3.1.2. Flow structureWe have shown that the standard kinetic theory-based CFD
model with a modified drag as suggested by Li group (Yang
et al., 2004) is capable of correctly describing the coexistence
of the dense and dilute regimes for flow of FCC particles in a
riser in the turbulent regime (Jiradilok et al., 2006).
Fig. 2(a) displays the snap shot at 7.2 s for the solid flux of
132kg/m2 s and the gas velocity of 4.25 m/s. The top part of
the riser is dilute and the bottom part is dense. The structure
at the bottom part is core-annular. There is a low concentration
of solid at the center and a high solid volume fraction near the
wall, which approximately agrees with the experimental data.
Using the kinetic theory model, the laminar granular tem-
perature was computed. The axial profile of laminar granular
Table 3
Equations for obtaining the averaged velocity and stresses
The mean velocity
particle
vi(r) = 1m
mk=1
vik (r, t)
The normal Reynolds
stress
vivi =
1
m
mk=1
(vik (r, t) − v̄i (r))(vik (r, t) − v̄i (r))
i represents x or y directions, m is the total number of data over a given time
period.
temperature is shown as Fig. 2(b). The laminar granular tem-
perature increases with increasing bed height due to the oscil-
lation of individual particles.
3.1.3. Reynolds stressesReynolds stresses are produced due to the random velocity
fluctuations of the hydrodynamic velocity. This is the principal
characteristic of turbulent flow. The Reynolds stresses are used
to estimate the turbulent part of granular temperature and the
dispersion coefficients. The Reynolds stress can be calculated
as a function of hydrodynamic velocity and mean velocity, as
shown in Table 3.
The time-average values of normal Reynolds stresses per
unit bulk density in the axial direction of gas and solid phases
with various heights are shown as Fig. 3. The values vary as a
function of the radial position for both phases. The oscillations
on the top, 2 m, and bottom, 6m, parts occur due to the effects
of the inlet and outlet.
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3402 V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409
1
2
3
4
5
6
7
8
9
10
v ′ y
v ′
y m
2 / s 2
0
1
2
3
4
5
6
7
8
9
10
v ′
y v
′ y m
2 / s 2
0
1
2
3
4
5
6
7
8
9
10
v ′ y
v ′
y m
2 / s 2
0
-1
Gas phase Solid phase
-0.8
r/R
-0.6 -0.4 -0.2 0
-1 -0.8 -0.6 -0.4 -0.2 0
-1 -0.8 -0.6 -0.4 -0.2 0
Fig. 3. Axial normal Reynolds stress per bulk density of gas and solid phases
at (a) 2m, (b) 4m and (c) 6m.
Due to the turbulences in the direction of flow, the normalReynolds stresses per unit bulk density in the axial direction
are higher than those in the radial direction for both phases,
as shown in Fig. 4. We see that the Reynolds stresses per unit
bulk density for the gas and the solid are close to each other.
3.2. Bubbling commercial size fluidized beds
3.2.1. System properties
To show that the model predicts the effects of pressure on
dispersion coefficients, two bubbling fluidizations were simu-
lated. Simulations of bubbling commercial size fluidized beds
from a two-dimensional rectangular bed were performed using
0
0.05
0.1
0.2
0.25
0.15
v ′ x
v ′
x m
2 / s 2
0
0.05
0.1
0.2
0.25
0.15
v ′
x v
′ x m
2 / s 2
0
0.05
0.1
0.2
0.25
0.15
v ′
x v
′ x m
2 / s 2
-1 -0.8 -0.6 -0.4 -0.2
r/R
Gas phase Solid phase
0
-1 -0.8 -0.6 -0.4 -0.2 0
-1 -0.8 -0.6 -0.4 -0.2 0
Fig. 4. Radial normal Reynolds stress per bulk density of gas and solid phases
at (a) 2m, (b) 4m and (c) 6m.
the viscosity model, as given in Table 1. The system geome-
try and the system properties are summarized in Fig. 5. For
boundary conditions, the non-slip boundary condition was used
for the gas phase at the wall. For the solid phase, the free-slip
boundary condition was chosen. Initially, the bedwas filled with
particles at a solid volume fraction of 0.58. In this study we
simulated bubbling beds under two different pressures, a high
pressure, 25 atm, and a low pressure, atmospheric pressure.
3.2.2. Flow structure
Instantaneous snapshots of the solid volume fraction profiles
of the bubbling bed are shown in Fig. 6. Figs. 6(a) and (b) were
obtained with atmospheric pressure and 25 atm, respectively.
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Geometry of the reactors
• Height 12.19 m.
• Reactor Diameter (OD) 1.22 m.
• Reactor Diameter (ID) 1.14 m.
• Angle 75
• Number of inlet cell
• Position of outlet right hand side• Outer diameter of the outlet 0.16 m.
There is one cell of outlet on right hand side.
Solid properties
• Density 2500 kg/m3
• Diameter 500 µm
Fluid properties
• Air
Grid size
• Radial ∆ x = 4.064 cm
• Axial ∆ y = 16.475 cm
Number of Grid
• Radial × Axial 30 × 74 cells
(Including boundary wall cells)Operating Condition
• Gas velocity 0.45 m/s
• Initial solid volume fraction 0.58
• Initial bed height 4 m.
• Temperature 300 K
• Pressure
Case 1 Atmospheric pressure, 1 atm
Case 2 25 atm
4
Fig. 5. System geometry and operating conditions for bubbling bed simulations.
Fig. 6. Instantaneous plots of solid volume fraction field in bubbling bed
simulations: (a) 1 atm, (b) 25 atm.
The bed expansion ratio is calculated based on the fluidized
bed height and the initial bed height,
=H f /H 0. Using David-
son’s bubble-growth model (Darton et al., 1977), the equivalent
Table 4A comparison of expanded height ratio and equivalent bubble diameter at
two pressures
P (atm) Expanded height ratio (H/Ho) Equivalent bubble diameter (m)
1 1.21 0.70 ± 0.1425 2.40 0.58 ± 0.18
bubble diameter can be estimated as a function of the distance
above the distributor in a bubbling fluidized bed, as follows:
DB = 0.54(U 0 − U mf )0.4(h + 4 A0)0.8/g0.2, (10)
where DB is the equivalent bubble diameter, U 0 is the superfi-
cial gas velocity, U mf is the minimum fluidization velocity, h
is the initial bed height, 4√ A0 is 0.03 m for a porous-plate gas
distributor.
In the high-pressure system there is a high expansion and
small bubbles, as shown in Table 4. It is well known in the
fluidization community (Sobreiro and Monteiro, 1982; Rowe
et al., 1983; Piepers et al., 1984; Gidaspow, 1994) that under
high pressure, Geldart’s Type B powders undergo considerable
expansion before bubbling.
At atmospheric pressure, the equivalent bubble diameter ob-
tained from the above equation is 0.58 m. This shows a reason-
able agreement with bubble size in Table 4.
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3404 V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409
0
1
2
3
4
5
20 25 30 35 40 45 50
A x i a l S o l i d V e
l o c i t y , m / s
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
20 25 30 35 40 45 50
R a d i a l V e l o c i t y , m / s
Time, sec
Time, sec
Fig. 7. Typical time series of axial (a) and radial (b) hydrodynamic velocities
(v) for particles in the center region at a bed height of 4.
3.2.3. Fluctuations
Fig. 7 shows typical time series of axial and radial hydro-
dynamic velocities for particles at 25 atm. The hydrodynamic
velocities are obtained directly from the code. Their frequency
distributions of hydrodynamic velocities are shown in Fig. 8(a).
The main frequency for the axial direction is 0.233Hz, and
the main frequency for the radial direction is 0.3Hz. Fig. 8(b)
shows the power spectrum density of bed void at 25 atm. The
dominant frequency (f ) of porosity oscillations in the bub-bling bed can be estimated by an analytical solution, as follows
(Gidaspow et al., 2001; Jung et al., 2005):
f = 12
gH
1/2 (3s/g + 2)ss0
1/2, (11)
where s0 and H 0 are some initial solid volume fraction and
initial bed height.
The time averaged solid volume fraction at the center of
column is approximately 0.80. The initial bed height is 4m. The
calculated main frequency for porosity oscillations obtained
from the above equation is 0.24Hz. This shows a reasonably
good agreement with the main frequency in Fig. 8.
0
10
20
30
40
50
60
70
80
90
100
0 1 2 3 4 5
P o w e r S p e c t r a l M
a g n i t u d e
Axial
Radial
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 1 2 3 4 5
P o w e r S p e c t r a l M a g n i t u d e
Frequency (Hz)
Frequency (Hz)
Fig. 8. Frequency and power spectral magnitude of (a) hydrodynamic ve-
locities and (b) bed void. Main frequency: axial velocity = 0.233Hz; radialvelocity = 0.3 Hz; bed void = 0.3Hz.
4. Dispersion coefficients
There are two kinds of mixing in fluidization: that due to in-
dividual particle oscillations and that due to cluster oscillations.
An order of magnitude estimate of the dispersion coefficient
due to individual particles oscillations can be obtained from
the laminar granular temperature (Jiradilok et al., 2006). Tur-
bulent dispersion coefficients can be obtained as a function of
normal Reynolds stress corresponding the Lagrangian integral
time scale as described below.
4.1. Turbulent dispersion coefficients calculation
The dispersion coefficients in the radial and axial directions
are expressed as in Hinze (1959), as follows:
DT (a) = v(a)2T L, (12)
where v(a)2 is the mean square fluctuating velocity corre-sponding to normal Reynolds stress and T L is the Lagrangian
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V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409 3405
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 A x i a l c o r r e l a t i o n c o e f f i c i e n t
R L ( y,t ′)
0
-0.6
-0.4
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 R a d i a l c o r e l a t i o n C o e f f i c i e n t
-0.2
R L ( x,t ′) 0
Time, sec
0.20.40.6
Time, sec
0.6 0.4 0.2
Fig. 9. Typical autocorrelation functions of solid phase (a) axial; (b) radial
for W s = 98.8 kg/m2
s and U g = 3.25 m/s.
integral time scale of the particle and gas motion, defined by
T L = ∞
0RL(a, t ) dt =
∞0
v(t)v(t + t )v2
dt , (13)
where v here is Lagrangian velocity fluctuations and the auto-correlation function, given by
RL(a,t ) = v
(t)v(t + t )v2
. (14)
Eulerian turbulence characteristics can be obtained from
Lagrangian turbulence characteristic (Hinze, 1959). The rela-
tionship between the Eulerian and the Lagrangian turbulence
characteristics has been given by Hay and Pasquil as
T L = T E , (15)where is the coefficient, T E is the Eulerian integral time scale
of the particle and gas motion.
In order to estimate the order of magnitude of the disper-
sion coefficient, the Eulerian integral time scale approximately
equals Lagrangian integral time scale (Hinze, 1959):
T L ≈ T E . (16)
Eq. (12) is a special case of the second-order tensor disper-
sion matrix, Eq. (6).
Fig. 9 shows typical autocorrelation functions of solid phase
in radial and axial directions for W s = 98.8 kg/m2 s and U g =3.25 m/s. The autocorrelation function decays with time from
the maximum value of one, and goes to zero. For the radial
autocorrelation, the profile dips below zero, then oscillates to astationary value of zero due to the wall limitation of x direction.
For the direction of flow, the autocorrelation coefficient simply
decayed exponentially, corresponding to Roy et al. (2005) in
a liquid–solid system and Godfroy et al. (1999) in a gas–solid
riser.
4.2. Characteristic lengths
Fig. 10 shows the snapshots of solid volume fractions of the
computed clusters at 6.5, 7.5 and 8.5 s. The length and width
of clusters can be approximated from characteristic lengths es-
timated from the relation between the dispersion coefficientsand the oscillating velocity as
Dispersion coefficients (D)
= characteristic length × oscillating velocity. (17)The oscillating velocities are obtained from the square root
of normal Reynolds stress. Fig. 11 shows the radial distribution
of characteristic lengths in the axial and the radial directions.
The length and width of clusters depend on the position cor-
responding to Fig. 10. The lengths and widths of clusters are
approximately 10–100 and 0.5–4 cm, respectively.
4.3. Turbulent dispersion coefficients
The computed radial and axial particle and gas dispersion
coefficients varied as a function of positions. The values of
dispersion coefficients are reported by averaging over the cross
section of a riser or bubbling beds. The standard deviations are
based on these lateral position variations.
For flow of FCC particles in the riser, the experimental solid
flux of 98.8 kg/m2 s and the gas velocity of 3.25m/s (Wei
et al., 1998a) was used for the computation of the gas and solid
dispersion coefficients. The computation of gas dispersion co-
efficients was studied based on Jiradilok et al. (2006) study. A
comparison of radial and axial solid and gas dispersion coeffi-cients at various heights is summarized in Table 5. The solid
and gas dispersions are of same order of magnitudes, because
the Reynolds stresses per unit bulk density do not differ from
each other. Substantially both dispersion coefficients vary from
top to bottom of the riser, as expected. The simulations show
that the radial dispersion coefficients in the riser are two to
three orders of magnitude lower than the axial dispersion co-
efficients.
For bubbling commercial size fluidized bed simulations, the
effects of pressure on dispersion coefficients were studied. A
comparison of radial and axial solid and gas dispersion coef-
ficients at 25 atm and at an atmospheric pressure is given in
Table 6. They were calculated at 4 m above the distributor. The
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Fig. 10. Snapshots of solid volume fraction at 6.5, 7.5 and 8 s for W s = 98.8 kg/m2
s and U g = 3.25 m/s.
dispersion coefficients for the bubbling bed at 25atm are much
higher than at atmospheric pressure due to the high bed expan-
sion with smaller bubbles.
Fig. 12 shows that solid dispersion coefficients increase with
the bed diameter because the bubble diameter increases with
the bed diameter. The computed axial solid dispersion coeffi-
cients of bubbling bed at atmospheric pressure agree with the
measured data. The differences between the simulations and
the experiments are in part due to different definitions of the
dispersion coefficients.
Figs. 13 and 14 show the comparisons of computed axial
and radial gas dispersion coefficients with the literature survey
by Breault (2006). The computed dispersion coefficients are in
the range of the literature data.
The axial solid dispersion coefficients for 850 m cork parti-
cle were generated at the National Energy Technology Labora-
tory (NETL) and measured using the autocorrelation method.
This is the same method used in this study. The data are re-
ported in Fig. 15. A numerical simulation of the NETL riser is
in progress.
Comparisons between computed solid dispersion coefficients
and the literature survey for both directions, axial and radial,
are shown in Figs. 15 and 16, respectively. The computations
show that the gas and the solid dispersion coefficients are close
to each other in agreement with measurements. The simulations
show that the radial dispersion coefficients in the riser are two
to three orders of magnitude lower than the axial dispersion
coefficients, but less than an order of magnitude lower for the
bubbling bed at atmospheric pressure.
5. Conclusions
• We have shown how to compute radial and axial particle andgas dispersion coefficients in the turbulent regime of a riserwith flow of FCC particles and in bubbling commercial size
fluidized beds at low and high pressures.
• The dispersion coefficients were computed from the turbulentvelocity oscillations of the gas and the particles obtained
by direct numerical solutions of the coupled Navier–Stokes’
equations for gas–particle flow in the two fluid model.
• The computed dispersion coefficients are in reasonable agree-ment with the experimental measurements reported over the
last half century. The CFD computations suggest that the re-
ported differences in the dispersion coefficients may be due
to geometrical effects of the risers and the bubbling beds,
since the geometry strongly affects the local gas and particle
velocities from which the dispersion coefficients are derived.
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V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409 3407
• The computed dispersion coefficients and the normal stressesallow the computation of characteristic lengths of clusters.
The lengths and widths agree with snapshots of volume frac-
tion of solid.
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8
A x i a l C h a r e c t e r i s t i c L e n g t h ( c m )
0
1
2
3
4
0 0.2 0.4 0.6 0.8
r/R
R a d i a l C h a r t e r i s t i c L e n g t h ( c m )
200 cm 400 cm 600 cm
r/R
1
1
Fig. 11. Radial distributions of characteristic lengths (a) axial; (b) radial for
W s = 98.8 kg/m2 s and U g = 3.25 m/s.
0.0001
0.001
0.01
0.1
1
0.01 0.1
Bed Diameter (m)
S o l i d s D i s p e r s i o n C
o e f f i c i e n t s ( m 2 / s )
10
May (1959)
Thiel and Potter (1978)Avidan et al (1985)Morooka et al (1972)Du et al (2002)
Jung et al (2005)Lewis et al (1962)de Groot (1967)Liu and Gidaspow (1981)
Lee and Kim (1990)Mostoufi et al (2001)Jung et al (2005)
This study (Computation)
Axial,
Bubbling, 1 atm
Computation
A
B
1
Fig. 12. Effect of the bed diameter on experimental and computed solid
dispersion coefficients for bubbling and turbulent fluidized beds for Geldart
A and B particles (Avidan and Yerushalmi, 1985; Du et al., 2002; de Groot,
1967; Jung et al., 2005; Lee and Kim, 1990; Lewis et al., 1962; Liu andGidaspow, 1981; May, 1959; Morooka et al., 1972; Mostoufi and Chaouki,
2001; Thiel and Potter, 1978).
1
10
0 1 2 3 4 5 6 7 8 9 10
A x i a l G
a s D i s p e r s i o n C o e f f i c i e n t ( m 2 / s )
Dry, 1989
Kim, 1998 & 1999
Wei, 2001
CFD-Riser-FCC particles
Li , 1989
4 m.
2 m.6 m.
CFD,1 atm(Bubbling)
Gas Velocity (m/s)
0.1
0.01
0.001
0.0001
Fig. 13. Effect of the gas velocity on experimental and computed axial gas
dispersion coefficients (Dry and White, 1989; Kim and Namkung, 1998,
1999; Li and Weinstein, 1989; Wei et al., 2001).
Table 5
A comparison of computed radial and axial solid and gas dispersion coefficients at various heights for FCC particles in a riser
Height (m) s Solid dispersion coefficient (m2 /s) Gas dispersion coefficient (m2 /s)
Axial Radial Axial Radial
2 0.25 0.347 ± 0.120 0.002 ± 0.001 0.614 ± 0.323 0.004 ± 0.0044 0.14 1.221 ± 0.289 0.001 ± 0.001 2.032 ± 0.927 0.002 ± 0.0026 0.04 1.331 ± 0.757 0.009 ± 0.005 1.134 ± 0.821 0.004 ± 0.003
Table 6
A comparison of computed radial and axial solid and gas dispersion coefficients at two pressures for the bubbling beds
P (atm) s Solids dispersion coefficient (m2 /s) Gas dispersion coefficient (m2 /s)
Axial Radial Axial Radial
1 0.41 0.069 ± 0.042 0.012 ± 0.005 0.06 ± 0.027 0.019 ± 0.01125 0.20 0.791 ± 0.373 0.030 ± 0.009 0.891 ± 0.464 0.040 ± 0.018
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3408 V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409
1
10
0 1 2 3 4 5 6 7 8 9 10
R a d i a l G a s D i s p e r s i o
n C o e f f i c i e n t ( m 2 / s )
Adanez, 1997Leckner, 2002
Werther, 1992
Rhodes,1993
Wei, 2001Leckner,(hot) 2000
Leckner,(cold) 2000
4 m.2 m.6 m.
CFD, 1atm
(Bubbling)
CFD-riser(FCC particles)
0.1
0.01
0.001
0.0001
Gas Velocity (m/s)
Fig. 14. Effect of the gas velocity on experimental and computed radial
gas dispersion coefficients (Adanez et al., 1997; Leckner et al., 2000, 2002;
Rhodes et al., 1993; Wei et al., 2001; Werther et al., 1992).
0.001
0.01
0.1
1
10
100
0.01 0.1
A x i a l S o l i d s D i s p e r s i o n ( m 2 / s e c )
Duet al. (2002)Thiel and Potter (1978)Aviden and Yerushalmi (1985)Weiet al. (1998)Weiet al. (1995)Gidaspow et al. (2004), IITRiserNETL unit,850µm Cork particlesJiradiloket al. (2006), FCC particles2 m.4 m.6 m.This study (Bubbling Bed, 500 µm)
Experiment
Computation
Bubble
Computation, 1 atm
Single particleoscillation
FCCparticles-Cluster
Computation
NETL ,
Cork particles
1 10
Gas Velocity (m/sec)
Fig. 15. Effect of the gas velocity on experimental and computed radial solid
dispersion coefficients (Du et al., 2002; Thiel and Potter, 1978; Avidan and
Yerushalmi, 1985; Wei et al., 1995, 1998a, b; Gidaspow et al., 2004; Jiradilok
et al., 2006).
1
R a d i a l S o l i d s D i s p e r s i o
n ( m 2 / s e c )
Du et al. (2002)Koenigdorff and Werther (1995)Wei et al. (1998)We i et al. (1995)Jiradilok et al. (2006), FCCparticles
2 m.4 m.6 m.
This study (Bubbling Bed, 500 µm)
Experiment
Computation
BubbleComputation, 1 atm
FCC particles-clusterComputation
0.1
0.01
0.001
0.1 1
Gas Velocity (m/sec)
10
Fig. 16. Effect of the gas velocity on experimental and computed radial solid
dispersion coefficients (Du et al., 2002; Koenigsdorff and Werther, 1995; Wei
et al., 1995, 1998a, b; Jiradilok et al., 2006).
• The dispersion coefficients for the bubbling bed at 25 atm aremuch higher than at atmospheric pressure due to the high bed
expansion with smaller bubbles.
• The computations show that the gas and the solid dispersioncoefficients are close to each other in agreement with mea-
surements. The simulations show that the radial dispersion
coefficients in the riser are two to three orders of magnitudelower than the axial dispersion coefficients, but less than an
order of magnitude lower for the bubbling bed at atmospheric
pressure.
Notation
a x or y directions
c instantaneous velocity
Cd drag coefficient
d k characteristic particulate phase diameter
DP particle dispersion coefficients
DT turbulent dispersion coefficients, due to cluster os-
cillationse coefficient of restitution
g gravity
g0 radial distribution function at contact
L mean free path
P continuous phase pressure
P k dispersed (particulate) phase pressure
vi hydrodynamic velocity in i direction
vi mean particle velocity in i direction
vivj Reynolds stress (i = j normal Reynolds stress; i =
j shear Reynolds stress)
Greek letters
B interphase momentum transfer coefficient
energy dissipation due to inelastic particle collision
k volume fraction of phase k
granular temperature
granular conductivity
bulk viscosity of phase k
k shear viscosity of phase k
k density of phase k
k stress of phase k
specularity coefficient
Acknowledgment
This study was supported by the US Department of Energy
Grant (DE-FG26-06NT42736).
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