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Adjustment of drag coefﬁcient correlations in three dimensional
CFD simulation of gas–solid bubbling ﬂuidized bed
Ehsan Esmaili, Nader Mahinpey
⇑
Dept. of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, AB, Canada T2N 1N4
a r t i c l e i n f o
Article history:
Received 13 November 2009
Received in revised form 8 November 2010
Accepted 10 March 2011
Available online 9 April 2011
Keywords:
Multiphase ﬂow
Fluidized bed
Computational Fluid Dyn

Transcript

Adjustment of drag coefﬁcient correlations in three dimensionalCFD simulation of gas–solid bubbling ﬂuidized bed
Ehsan Esmaili, Nader Mahinpey
⇑
Dept. of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, AB, Canada T2N 1N4
a r t i c l e i n f o
Article history:
Received 13 November 2009Received in revised form 8 November 2010Accepted 10 March 2011Available online 9 April 2011
Keywords:
Multiphase ﬂowFluidized bedComputational Fluid DynamicsInter phase drag modelCoefﬁcient of restitutionEulerian–Eulerian model
a b s t r a c t
Fluidized beds have been widely used in power generation and in chemical, biochemical, and petroleumindustries. 3D simulation of commercial scale ﬂuidized beds has been computationally impractical due tothe required memory and processor speeds. In this study, 3D Computational Fluid Dynamics simulationof a gas–solid bubbling ﬂuidized bed is performed to investigate the effect of using different inter-phasedrag models. The drag correlations of Richardon and Zaki, Wen–Yu, Gibilaro, Gidaspow, Syamlal–O’Brien,Arastoopour, RUC, Di Felice, Hill Koch Ladd, Zhang and Reese, and adjusted Syamlal are reviewed using amultiphase Eulerian–Eulerian model to simulate the momentum transfer between phases. Furthermore,a method has been proposed to adjust the Di Felice drag model in a three dimensional domain based onthe experimental value of minimum ﬂuidization velocity as a calibration point. Comparisons are madewith both a 2D Cartesian simulation and experimental data. The experiments are performed on a Plexi-glas rectangular ﬂuidized bed consisting of spherical glass beads and ambient air as the gas phase. Com-parisons were made based on solid volume fractions, expansion height, and pressure drop inside theﬂuidized bed at different superﬁcial gas velocities. The results of the proposed drag model were foundto agree well with experimental data. The effect of restitution coefﬁcient on three dimensional predictionof bed height is also investigated and an optimum value of restitution coefﬁcient for modeling ﬂuidizedbeds in a bubbling regime has been proposed. Finally sensitivity analysis is performed on the grid intervalsize to obtain an optimum mesh size with the objective of accuracy and time efﬁciency.
2011 Elsevier Ltd. All rights reserved.
1. Introduction
Gas–solid ﬂuidized bed reactors are used in many industrialoperations, such as energy production and petrochemical pro-cesses. Some of the distinct advantages of gas–solid ﬂuidized bedreactors over other methods of gas–solid reactors are controlledhandling of solids, isothermal conditions due to good solids mixingand the large thermal inertia of solids, and high heat ﬂow and reac-tion rates between gas and solids due to large gas-particle contactarea. Hence, the ﬂuidized bed reactors are widely used in gasiﬁca-tion, combustion, catalytic cracking and various other chemicaland metallurgical processes. Two approaches are typically usedfor CFD modeling of gas–solid ﬂuidized beds. The ﬁrst one isLagrangian–Eulerian modeling [1–6], which solves the equationsof motion individually for each particle and uses a continuousinterpenetrating model (Eulerian framework) for modeling thegas phase. In large systems of particles, the Lagrangian–Eulerianmodel requires powerful computational resources because of thenumbers of equations that are being solved. Bokkers et al. [5] havestudied the effect of implementing different drag models on simu-lation of gas–solid ﬂuidized bed using Discrete Particle Model(DPM) which assume a Lagrangian–Eulerian model for the multi-phase ﬂuid ﬂow. van Sint Annaland et al. [6] have also studiedthe particle mixing and segregation rates in a bi-disperse freelybubbling ﬂuidized bed with a new multi-ﬂuid model (MFM) basedon the kinetic theory of granular ﬂow for multi-component sys-tems. The second approach is Eulerian–Eulerian modeling [7–13],which assumes that both phases can be considered as ﬂuid andalso take the interpenetrating effect of each phase into consider-ation by using drag models. Therefore, applying a proper dragmodel in Eulerian–Eulerian modeling is of a great importance.Many researchers have applied 2D Cartesian simulations tomodel pseudo-2D beds [1,7,11,13]. Behjat et al. [11] applied a
two-dimensional CFD (Computational Fluid Dynamics) techniqueto the ﬂuidized bed in order to investigate the hydrodynamic andthe heat transfer phenomena. They concluded that the Eulerian–Eulerian model is suitable for modeling industrial ﬂuidized bedreactors. Their results indicate that considering two solid phases,particles with smaller diameters have lower volume fraction atthe bottom of the bed and higher volume fraction at the top of the bed. They also showed that the gas temperature increases asit moves upward in the reactor due to the heat of polymerization
0965-9978/$ - see front matter
2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.advengsoft.2011.03.005
⇑
Corresponding author. Fax: +1 403 284 4852.
E-mail address:
nader.mahinpey@ucalgary.ca (N. Mahinpey).Advances in Engineering Software 42 (2011) 375–386
Contents lists available at ScienceDirect
Advances in Engineering Software
journal homepage: www.elsevier.com/locate/advengsoft
reaction leading to the higher temperatures at the top of the bed[11]. Peiranoa et al. [14] investigated the importance of three
dimensionality in the Eulerian approach simulations of stationarybubbling ﬂuidized beds. The results of their simulations show thattwo-dimensionalsimulationsshouldbeusedwithcautionandonlyfor sensitivity analysis, whereas three-dimensional simulations areable to reproduce both the statics (bed height and spatial distribu-tion of particles) and the dynamics (power spectrum of pressureﬂuctuations)ofthebed.Inaddition,theyassumedthattheaccuratepredictionof the drag force (the force exerted by the gas on a singleparticle in a suspension) is of little importance when dealing withbubbling beds. However, in the present study, it is found that usinga proper drag model can signiﬁcantly increase the accuracy of results in the 3D simulation of bubbling ﬂuidized beds.Cammarata et al. [8] compared the bubbling behavior predictedby 2D and 3D simulations of a rectangular ﬂuidized bed using thecommercial software ANSYS-CFX (a CFD software). The bed expan-sion, bubble hold-up, and bubble size calculated from the 2D and3D simulations were compared with the predictions obtained fromthe Dartonequation [15]. A more realistic model of physical behav-ior for ﬂuidization was obtained using 3D simulations. They alsoindicated that 2D simulations can be used for sensitivity analyses.Xie et al. [10] compared the results of 2D and 3D simulations of slugging, bubbling, and turbulent gas–solid ﬂuidized beds. Theyalso investigated the effect of using different coordinate systems.Their results show that there is a signiﬁcant difference between2D and 3D simulations, and only 3D simulations can predict thecorrect bed height and pressure spectra. Li et al. [12] conducted athree-dimensional numerical simulation of a single horizontalgas jet into a laboratory-scale cylindrical gas–solid ﬂuidized bed.They proposed a scaled drag model and implemented it into thesimulation of a ﬂuidized bed of FCC (Fluid Catalytic Cracking) par-ticles. They also obtained the jet penetration lengths for different jet velocities and compared them with published experimentaldata, as well as with predictions of empirical correlations. Zhanget al. [16] suggested a mathematical model based on the two-ﬂuidtheory to simulate both homogeneous ﬂuidization of Geldart Aparticles and bubbling ﬂuidization of Geldart B particles in athree-dimensional gas–solid ﬂuidized bed. The usage of their mod-el is easy since it does not include adjustable parameters. It is capa-ble of predicting the ﬂuidization behavior leading to similar resultsas the more complex Eulerian–Eulerian models.Li and Kuipers [17] studied the formation and evolution of ﬂowstructures in dense gas-ﬂuidized beds with ideal collisional parti-cles (elastic and frictionless) by employing the discrete particlemethod, with special focus on the effect of gas–particle interaction.They have concluded that gas drag, or gas–solid interaction, plays avery important role in the formation of heterogeneous ﬂow struc-turesin dense gas-ﬂuidized beds with ideal and non-ideal particle–particle collisionsystems. They discoveredthat the non-linearityof gasdrag hasa ‘‘phase separation’’ functionby acceleratingparticlesin the dense phase and decelerating particles in the dilute phase totrigger the formation of non-homogeneous ﬂow structures.Goldschmidt et al. [13] investigated a two-dimensional multi-ﬂuidEulerian CFD model to study the inﬂuence of the coefﬁcient of restitution on the hydrodynamics of a dense gas–solid ﬂuidized
Nomenclature
A
constant in RUC-drag model (–)
A
constant in Syamlal–O
0
Brien drag model (–)
B
constant in RUC-drag model (–)
B
constant in Syamlal–O
0
Brien drag model (–)
C
n
drag factor on multi-particle system (–)
d
s
diameter of solid particles (m)
e
restitution coefﬁcient of solid phase (–)
F
drag factor in HKL drag model (–)
Fr
friction factor from Johnson et al. frictional viscosity (–)
F
0
,
F
1
,
F
2
,
F
3
drag constants in the HKL drag function (–)
g
the gravitational acceleration (=9.81) (m s
2
)
g
0
the general radial distribution function (–)
I
the unit tensor (–)
I
2
D
the second invariant of the deviatoric stress tensor (–)
K
sg
drag factor of phase s in phase g (kg m
3
s
1
)
k
H
s
conductivity of granular temperature (kg m
1
s
1
)
n
coefﬁcient in the Richardson and Zaki drag correlation(–)
P
pressure (Pa)
P
s
solids pressure (Pa)
P
s
,
fric
frictional pressure (Pa)
D
P
pressure drop (Pa)
r
q
s
diffusive ﬂux of ﬂuctuating energy (kg m
1
s
3
)
Re
the Reynolds number (–)
Re
m
the modiﬁed Reynolds number in the Richardson Zakicorrelation (–)
Re
s
the particle Reynolds number (–)
t
time (S)
D
t
time interval (S)
U
mf
minimum ﬂuidization velocity (m s
1
)
u
s
,
i
, u
s
,
j
solid phase velocity in the i and j direction (m s
1
)
~
V
velocity (m s
1
)
v
r
the relative velocity correlation (–)
w
factor in the HKL drag correlation (–)
Greek letters
b
angel of internal friction(
)
e
g
gas phase volume fraction (–)
e
s
solid phase volume fraction (–)
c
H
s
dissipation of granular temperature (kg m
1
s
3
)
D
change in variable, ﬁnal–initial (–)
r
the Dell operator (m
(
1)
)
H
s
granular temperature(m
2
s
2
)
k
s
bulk viscosity (kg m
1
s
1
)
l
g
gas viscosity (kg m
1
s
1
)
l
s
granular viscosity (kg m
1
s
1
)
l
s
,
col
collisional viscosity (kg m
1
s
1
)
l
s
,
kin
kinetic viscosity (kg m
1
s
1
)
l
s
,
fric
frictional viscosity (kg m
1
s
1
)
l
dil
dilute viscosity in Gidaspow kinetic viscosity model(kg m
1
s
1
)
p
the irrational number
p
(–)
q
g
gas density (kg m
3
)
q
s
solid density (kg m
3
)
s
the stress–strain tensor (Pa)
Subscriptscol
collisional
dil
dilute
fr
frictional
g
gas or ﬂuid phase
kin
kinetic
max
maximum
mf
minimum ﬂuidization condition
min
minimum
q
general phase
qs
solid phase
376
E. Esmaili, N. Mahinpey/Advances in Engineering Software 42 (2011) 375–386
beds. They demonstrated that, in order to obtain reasonable beddynamics from fundamental hydrodynamic models, it is signiﬁ-cantly important to take the effect of energy dissipation due tonon-ideal particle–particle encounters into account.A few works in the literature have investigated the effect of using different drag models in 3D simulation of ﬂuidized beds toobtain an optimum drag model for simulation of bubbling gas–so-lid ﬂuidized beds. Therefore, the underlying objective of this studyis to present an optimum drag model to simulate the momentumtransfer between phases and to compare the results of 3D and2D simulations of gas–solid bubbling ﬂuidized beds. Furthermore,a method has been proposed to adjust the Di Felice Drag Model[18] based on the experimental value of minimum ﬂuidizationvelocity as the calibration point. The effect of restitution coefﬁcienton the three dimensional prediction of bed height is also investi-gated and an optimum value of restitution coefﬁcient for modelingﬂuidized beds in bubbling regime has been proposed.
2. Experimental setup
Experiments were carried out in the Department of Chemicaland Biological Engineering at the University of British Columbia.The ﬂuid bed is a Plexiglas rectangular shape column consistingof spherical glass beads with ambient air as the gas phase. Thecolumn dimensions are 0.280 (m) in width, 1.2 (m) in length,and 0.0254 (m) in depth. Ambient air is uniformly injected intothe column via a gas distributor which is a perforated plate witha hole to plate cross sectional area ratio of approximately 1.2%.Pressure drops were measured using three differential pressuretransducers located at the elevations of 0.03, 0.3 and 0.6 (m) abovethe gas distributor. Fig. 1 illustrates the shape of the column usedin this research, along with its dimensions and pressure transducerlocations. Spherical, non-porous glass beads, Geldart group B parti-cles, with a particle size distribution of 250–300 (
l
m) and densityof 2500 (kg/m
3
) were used as the granular parts. The static bedheight is 0.4 (m) with a solid volume fraction of approximately60%. Several experiments were conducted at steady-state bedoperations in order to calculate the void fraction and minimumﬂuidization velocity. In order to estimate the minimumﬂuidizationvelocity, measurements were carried out at increasing velocityincrements from ﬁxed bed to high inlet velocity (0.6 (m/s)). Fromthe data obtained, minimum ﬂuidization velocity is estimated as
U
mf
= 0.065 (m/s).
3. Hydrodynamic model
In this study the general model of multiphase ﬂow based onEulerian–Eulerian approach has been derived. The model solvessets of transport equation for momentum and continuity of eachphase and granular temperature for the solid phase. These sets of equations are linked together through pressure and interphasemomentum transfer correlations (drag models). The solid phaseproperties have been obtained using the kinetic theory of granularﬂow.
3.1. Continuity equation
The continuity equation in absence of mass transfer betweenphases is given for each phase by:
Fig. 1.
Geometry of 3D Plexiglas ﬂuidized bed.
E. Esmaili, N. Mahinpey/Advances in Engineering Software 42 (2011) 375–386
377
@ @
t
ð
e
g
q
g
Þþ
r
ð
e
g
q
g
~
V
g
Þ¼
0
;
ð
1
Þ
@ @
t
ð
e
s
q
s
Þþ
r
ð
e
s
q
s
~
V
s
Þ¼
0
:
ð
2
Þ
And the volume fraction constraint requires
e
g
+
e
s
= 1.
where
e
,
q
, and
~
V
are the volume fraction, the density and theinstantaneous velocity, respectively. By considering the masstransfer between the phases, the term
ð
_
m
gs
_
m
sg
Þ
would then beadded to the right hand side of the above equations, where,
_
m
isthe rate of mass transfer between phases.
3.2. Gas phase momentum equation
Assuming no mass transfer between phases and no lift and vir-tual mass forces, the conservation of momentum for the gas phasecan be expressed as:
@ @
t
ð
e
g
q
g
~
V
g
Þþ
r
ð
e
g
q
g
~
V
g
~
V
g
Þ¼
r
s
g
e
g
r
P
þ
e
g
q
g
g
þ
K
sg
ð
~
V
s
~
V
g
Þ
;
ð
3
Þ
where
P
is the pressure,
g
is the gravity and
K
sg
is the drag coefﬁ-cient between the gas and the solid phase which will be explainedin detail in Section 3.5. The gas stress tensor
s
g
is given by:
s
g
¼
e
g
l
g
r
~
V
g
þð
r
~
V
g
Þ
T
þ
e
g
k
g
þ
23
l
g
r
~
V
g
I
:
ð
4
Þ
3.3. Solid phase momentum equation
Assuming no mass transfer between phases and no lift and vir-tual mass forces, the conservation of momentum for the solidphase can be expressed as:
@ @
t
ð
e
s
q
s
~
V
s
Þþ
r
ð
e
s
q
s
~
V
s
~
V
s
Þ¼
r
s
s
r
P
s
þ
e
s
r
P
þ
e
s
q
s
g
þ
K
sg
ð
~
V
s
~
V
g
Þ
;
ð
5
Þ
@ @
t
ð
e
s
q
s
~
V
s
Þþ
r
ð
e
s
q
s
~
V
s
~
V
s
Þ¼
r
s
s
r
P
s
þ
e
s
r
P
þ
e
s
q
s
g
þ
K
sg
ð
~
V
s
~
V
g
Þ
;
ð
6
Þ
where
P
s
is the granular pressure, derived from the kinetic theory of granular ﬂow, and is composed of a kinetic term and a term due toparticle collisions. In the regions where the particle volume fraction
e
s
is lower than the maximum allowed fraction
e
s
,
max
, the solid pres-sure is calculated independently and is used in the pressure gradi-ent term
r
P
s
It can be expressed as (Lun et al. [19]):
P
s
¼
e
s
q
s
H
s
þ
2
q
s
ð
1
þ
e
Þ
e
2
s
g
0
H
s
;
ð
7
Þ
where
H
s
is the granular temperature;
e
is the restitution coefﬁ-cient of granular particles and
g
0
is the radial distribution function.Different values for the coefﬁcient of restitution, from 0.73 to 1,have been proposed in literature. In this study the effect of restitu-tion coefﬁcient onthe simulation of bubblingﬂuidized bed has beeninvestigated in order to obtain an optimum value for the entirerange of study. The results are presented in Section 5.3. For the ra-dial distribution function,
g
0
, the following correlation has beenproposed by Ibdir and Arastoopour [20] and it is well related tothe data from the molecular simulator by Alder and Wainwright[21].
g
0
¼
35 1
e
s
e
s
;
max
13
#
1
:
ð
8
Þ
In momentum equation,
s
s
is the solid stress tensor and can bewritten as:
s
s
¼
e
s
l
s
r
~
V
s
þð
r
~
V
s
Þ
T
þ
e
s
k
s
þ
23
l
s
r
~
V
s
I
;
ð
9
Þ
where
k
s
is the granular bulk viscosity that is the resistance of gran-ular particles to compression or expansion. The following model isdeveloped from the kinetic theory of granular ﬂow by Lun et al. [19]for
k
s
:
k
s
¼
45
e
s
q
s
d
s
ð
1
þ
e
Þ
ﬃﬃﬃﬃﬃﬃ
H
s
p
r
;
ð
10
Þ
where
d
s
is the particle diameter.
In the solid stress tensor equation
l
s
is the granular shear vis-cosity that consists of a collision term, a kinetic term, and a frictionterm:
l
s
¼
l
s
;
col
þ
l
s
;
kin
þ
l
s
;
fric
:
ð
11
Þ
The collisional viscosity is a viscosity contribution due to collisionsbetween particles and has the highest contribution in the viscousregime. The corresponding correlation is taken from the kinetic the-ory of granular ﬂow by Lun et al. [19].
l
s
;
col
¼
45
e
s
q
s
d
s
ð
1
þ
e
Þ
ﬃﬃﬃﬃﬃﬃ
H
s
p
r
:
ð
12
Þ
The kinetic viscosity is expressed by Gidaspow model [22,23] as:
l
s
;
kin
¼
2
l
dil
g
0
ð
1
þ
e
Þ
1
þ
45
ð
1
þ
e
Þ
e
s
g
0
2
;
ð
13
Þ
l
dil
¼ð
constant
Þð
bulk density
Þð
mean free path
Þð
osccillation velocity
Þ
l
dil
¼
5
ﬃﬃﬃﬃ
p
p
96
ð
e
s
q
s
Þ
d
s
e
s
ﬃﬃﬃﬃﬃﬃ
H
s
p
:
ð
14
Þ
The Schaeffer expression [24] for the frictional viscosity can bewritten as
l
s
;
fric
¼
P
s
;
fric
sin
b
2
ﬃﬃﬃﬃﬃﬃ
I
2
D
p
;
ð
15
Þ
where
P
s
,
fric
is the frictional pressure, the constant
b
= 28.5
[25] isthe angel of internal friction and
I
2
D
is the second invariant of thedeviatoric stress tensor which can be written as
I
2
D
¼
16
½ð
D
s
11
D
s
22
Þ
2
þð
D
s
22
D
s
33
Þ
2
þð
D
s
33
D
s
11
Þ
2
þ
D
2
s
12
þ
D
2
s
23
þ
D
2
s
31
;
ð
16
Þ
D
sij
¼
12
@
u
s
;
i
@
x
j
þ
@
u
s
;
j
@
x
i
:
ð
17
Þ
Johnson et al. [26] made a simple algebraic expression for the solidpressure in the frictional region:
P
s
;
fr
¼
Fr
ð
e
s
e
s
;
min
Þ
n
ð
e
s
;
max
e
s
Þ
p
;
ð
18
Þ
Fr
¼
0
:
1
e
s
:
ð
19
Þ
In which
e
s
,
min
= 0.5,
n
= 2, and
p
= 3 are all experimental basedparameters.
3.4. Kinetic theory of granular ﬂow (KTGF)
The transport equation for granular temperature of solid phase
H
s
can be written as:
378
E. Esmaili, N. Mahinpey/Advances in Engineering Software 42 (2011) 375–386

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