AJUSTE DE LOS COEFICIENTES DE ARRASTRE ESMAILI-MAHINPEY.pdf
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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 Dynamics
Inter phase drag model
Coefficient of restitution
Eulerian–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 to
the required memory and processor speeds. In this study, 3D Computational Fluid Dynamics simulation
of a gas–solid bubbling fluidized bed is performed to investigate the effect of using different inter-phase
drag 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 a
multiphase 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 on
the experimental value of minimum fluidization velocity as a calibration point. Comparisons are made
with 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 the
fluidized bed at different superficial gas velocities. The results of the proposed drag model were found
to agree well with experimental data. The effect of restitution coefficient on three dimensional prediction
of bed height is also investigated and an optimum value of restitution coefficient for modeling fluidized
beds in a bubbling regime has been proposed. Finally sensitivity analysis is performed on the grid interval
size 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 industrial
operations, such as energy production and petrochemical pro-
cesses. Some of the distinct advantages of gas–solid fluidized bed
reactors over other methods of gas–solid reactors are controlled
handling of solids, isothermal conditions due to good solids mixing
and the large thermal inertia of solids, and high heat flow and reac-
tion rates between gas and solids due to large gas-particle contact
area. Hence, the fluidized bed reactors are widely used in gasifica-
tion, combustion, catalytic cracking and various other chemical
and metallurgical processes. Two approaches are typically used
for CFD modeling of gas–solid fluidized beds. The first one is
Lagrangian–Eulerian modeling [1–6], which solves the equations
of motion individually for each particle and uses a continuous
interpenetrating model (Eulerian framework) for modeling the
gas phase. In large systems of particles, the Lagrangian–Eulerian
model requires powerful computational resources because of the
numbers of equations that are being solved. Bokkers et al. [5] have
studied 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 studied
the particle mixing and segregation rates in a bi-disperse freely
bubbling fluidized bed with a new multi-fluid model (MFM) based
on 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 and
also take the interpenetrating effect of each phase into consider-
ation by using drag models. Therefore, applying a proper drag
model in Eulerian–Eulerian modeling is of a great importance.
Many researchers have applied 2D Cartesian simulations to
model pseudo-2D beds [1,7,11,13]. Behjat et al. [11] applied a
two-dimensional CFD (Computational Fluid Dynamics) technique
to the fluidized bed in order to investigate the hydrodynamic and
the heat transfer phenomena. They concluded that the Eulerian–
Eulerian model is suitable for modeling industrial fluidized bed
reactors. Their results indicate that considering two solid phases,
particles with smaller diameters have lower volume fraction at
the bottom of the bed and higher volume fraction at the top of
the bed. They also showed that the gas temperature increases as
it 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: [email protected] (N. Mahinpey).
Advances in Engineering Software 42 (2011) 375–386
Contents lists available at ScienceDirect
Advances in Engineering Software
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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 stationary
bubbling fluidized beds. The results of their simulations show that
two-dimensional simulations should be used with caution and only
for sensitivity analysis, whereas three-dimensional simulations are
able to reproduce both the statics (bed height and spatial distribu-
tion of particles) and the dynamics (power spectrum of pressure
fluctuations) of the bed. In addition, they assumed that the accurate
prediction of the drag force (the force exerted by the gas on a single
particle in a suspension) is of little importance when dealing with
bubbling beds. However, in the present study, it is found that using
a 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 predicted
by 2D and 3D simulations of a rectangular fluidized bed using the
commercial software ANSYS-CFX (a CFD software). The bed expan-
sion, bubble hold-up, and bubble size calculated from the 2D and
3D simulations were compared with the predictions obtained from
the Darton equation [15]. A more realistic model of physical behav-
ior for fluidization was obtained using 3D simulations. They also
indicated 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. They
also investigated the effect of using different coordinate systems.
Their results show that there is a significant difference between
2D and 3D simulations, and only 3D simulations can predict thecorrect bed height and pressure spectra. Li et al. [12] conducted a
three-dimensional numerical simulation of a single horizontal
gas jet into a laboratory-scale cylindrical gas–solid fluidized bed.
They proposed a scaled drag model and implemented it into the
simulation 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 experimental
data, as well as with predictions of empirical correlations. Zhang
et al. [16] suggested a mathematical model based on the two-fluid
theory to simulate both homogeneous fluidization of Geldart A
particles and bubbling fluidization of Geldart B particles in a
three-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 flow
structures in dense gas-fluidized beds with ideal collisional parti-
cles (elastic and frictionless) by employing the discrete particle
method, with special focus on the effect of gas–particle interaction.
They have concluded that gas drag, or gas–solid interaction, plays a
very important role in the formation of heterogeneous flow struc-
tures in dense gas-fluidized beds with ideal and non-ideal particle–
particle collision systems. They discovered that the non-linearity of
gas drag has a ‘‘phase separation’’ function by accelerating particles
in the dense phase and decelerating particles in the dilute phase to
trigger the formation of non-homogeneous flow structures.
Goldschmidt et al. [13] investigated a two-dimensional multi-fluid
Eulerian 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–O0Brien drag model (–)B constant in RUC-drag model (–)B constant in Syamlal–O0Brien drag model (–)C n drag factor on multi-particle system (–)
ds 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 s2) g 0 the general radial distribution function (–)I the unit tensor (–)I 2D the second invariant of the deviatoric stress tensor (–)K sg drag factor of phase s in phase g (kg m3 s1)kHs
conductivity of granular temperature (kg m1 s1)n coefficient in the Richardson and Zaki drag correlation
(–)P pressure (Pa)P s solids pressure (Pa)
P s, fric frictional pressure (Pa)DP pressure drop (Pa)r qs diffusive flux of fluctuating energy (kg m1 s3)Re the Reynolds number (–)Rem the modified Reynolds number in the Richardson Zaki
correlation (–)Res the particle Reynolds number (–)t time (S)Dt time interval (S)U mf minimum fluidization velocity (m s1)us,i, us, j solid phase velocity in the i and j direction (m s1)~V velocity (m s1)v r the relative velocity correlation (–)w factor in the HKL drag correlation (–)
Greek lettersb angel of internal friction()e g gas phase volume fraction (–)es solid phase volume fraction (–)cHs
dissipation of granular temperature (kg m1 s3)
D change in variable, final–initial (–)r the Dell operator (m(1))Hs granular temperature(m2 s2)ks bulk viscosity (kg m1 s1)l g gas viscosity (kg m1 s1)ls granular viscosity (kg m1 s1)ls,col collisional viscosity (kg m1 s1)ls,kin kinetic viscosity (kg m1 s1)ls, fric frictional viscosity (kg m1 s1)ldil dilute viscosity in Gidaspow kinetic viscosity model
(kg m1 s1)p the irrational number p (–)q g gas density (kg m3)qs solid density (kg m3)s the stress–strain tensor (Pa)
Subscriptscol collisionaldil dilute fr frictional g gas or fluid phasekin kineticmax maximummf minimum fluidization conditionmin minimumq general phase qs solid phase
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beds. They demonstrated that, in order to obtain reasonable bed
dynamics from fundamental hydrodynamic models, it is signifi-
cantly important to take the effect of energy dissipation due to
non-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 to
obtain 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 momentum
transfer between phases and to compare the results of 3D and
2D 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 fluidization
velocity as the calibration point. The effect of restitution coefficient
on the three dimensional prediction of bed height is also investi-
gated and an optimum value of restitution coefficient for modeling
fluidized beds in bubbling regime has been proposed.
2. Experimental setup
Experiments were carried out in the Department of Chemical
and Biological Engineering at the University of British Columbia.The fluid bed is a Plexiglas rectangular shape column consisting
of spherical glass beads with ambient air as the gas phase. The
column dimensions are 0.280 (m) in width, 1.2 (m) in length,
and 0.0254 (m) in depth. Ambient air is uniformly injected into
the column via a gas distributor which is a perforated plate with
a hole to plate cross sectional area ratio of approximately 1.2%.
Pressure drops were measured using three differential pressure
transducers located at the elevations of 0.03, 0.3 and 0.6 (m) above
the gas distributor. Fig. 1 illustrates the shape of the column used
in this research, along with its dimensions and pressure transducer
locations. Spherical, non-porous glass beads, Geldart group B parti-
cles, with a particle size distribution of 250–300 (lm) and density
of 2500 (kg/m3) were used as the granular parts. The static bed
height is 0.4 (m) with a solid volume fraction of approximately
60%. Several experiments were conducted at steady-state bed
operations in order to calculate the void fraction and minimumfluidization velocity. In order to estimate the minimum fluidization
velocity, measurements were carried out at increasing velocity
increments from fixed bed to high inlet velocity (0.6 (m/s)). From
the 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 on
Eulerian–Eulerian approach has been derived. The model solves
sets of transport equation for momentum and continuity of each
phase and granular temperature for the solid phase. These sets of
equations are linked together through pressure and interphase
momentum transfer correlations (drag models). The solid phaseproperties have been obtained using the kinetic theory of granular
flow.
3.1. Continuity equation
The continuity equation in absence of mass transfer between
phases is given for each phase by:
Fig. 1. Geometry of 3D Plexiglas fluidized bed.
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@
@ t ðe g q g Þ þ r ðe g q g ~V g Þ ¼ 0; ð1Þ
@
@ t ðesqsÞ þ r ðesqs
~V sÞ ¼ 0: ð2Þ
And the volume fraction constraint requires e g + es = 1.
where e, q, and ~V are the volume fraction, the density and the
instantaneous velocity, respectively. By considering the mass
transfer between the phases, the term ð _m gs _msg Þ would then beadded to the right hand side of the above equations, where, _m is
the 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 phase
can be expressed as:
@
@ t ðe g q g ~V g Þ þ r ðe g q g ~V g ~V g Þ ¼ r s g e g rP þ e g q g g þ K sg ð~V s ~V g Þ;
ð3Þ
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 explained
in detail in Section 3.5. The gas stress tensor s g is given by:
s g ¼ e g l g r~V g þ ðr~V g ÞT
þ e g k g þ 2
3l g
r ~V g I : ð4Þ
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 solid
phase can be expressed as:
@
@ t ðesqs
~V sÞ þ r ðesqs~V s~V sÞ ¼ r ss rP s þ esrP þ esqs g þ K sg ð~V s ~V g Þ;
ð5Þ@
@ t ðesqs
~V sÞ þ rðesqs~V s~V sÞ ¼ r ss rP s þ esrP þ esqs g þ K sg ð~V s ~V g Þ;
ð6Þ
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 to
particle collisions. In the regions where the particle volume fraction
es is lower than the maximum allowed fraction es,max, the solid pres-
sure is calculated independently and is used in the pressure gradi-
ent term rP s It can be expressed as (Lun et al. [19]):
P s ¼ esqsHs þ 2qsð1 þ eÞe2s g 0Hs; ð7Þ
where Hs 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 on the simulation of bubbling fluidized bed has been
investigated in order to obtain an optimum value for the entire
range of study. The results are presented in Section 5.3. For the ra-
dial distribution function, g 0, the following correlation has been
proposed by Ibdir and Arastoopour [20] and it is well related to
the data from the molecular simulator by Alder and Wainwright
[21].
g 0 ¼3
5 1 es
es;max
13
" #1
: ð8Þ
In momentum equation,
ss is the solid stress tensor and can bewritten as:
ss ¼ esls r~V s þ ðr~V sÞT
þ es ks þ 2
3ls
r ~V sI ; ð9Þ
where ks is the granular bulk viscosity that is the resistance of gran-
ular particles to compression or expansion. The following model is
developed from the kinetic theory of granular flow by Lun et al. [19]
for ks:
ks ¼ 45esqsdsð1 þ eÞ
ffiffiffiffiffiffiHs
p
r ; ð10Þ
where ds is the particle diameter.
In the solid stress tensor equation ls is the granular shear vis-
cosity that consists of a collision term, a kinetic term, and a friction
term:
ls ¼ ls;col þ ls;kin þ ls; fric : ð11ÞThe collisional viscosity is a viscosity contribution due to collisions
between particles and has the highest contribution in the viscous
regime. The corresponding correlation is taken from the kinetic the-
ory of granular flow by Lun et al. [19].
ls;col ¼ 45 esqsdsð1 þ eÞ ffiffiffiffiffiffiHsp
r : ð12Þ
The kinetic viscosity is expressed by Gidaspow model [22,23] as:
ls;kin ¼ 2ldil
g 0ð1 þ eÞ 1 þ 4
5ð1 þ eÞes g 0
2
; ð13Þ
ldil ¼ ðconstantÞ ðbulk densityÞ ðmean free pathÞ ðosccillation velocityÞ
ldil ¼5 ffiffiffiffip
p 96
ðesqsÞ ds
es
ffiffiffiffiffiffiHs
p : ð14Þ
The Schaeffer expression [24] for the frictional viscosity can be
written as
ls; fric ¼P s; fric sin b
2 ffiffiffiffiffiffiI 2D
p ; ð15Þ
where P s, fric is the frictional pressure, the constant b = 28.5 [25] is
the angel of internal friction and I 2D is the second invariant of the
deviatoric stress tensor which can be written as
I 2D ¼ 1
6½ðDs11 Ds22Þ2 þ ðDs22 Ds33Þ2 þ ðDs33 Ds11Þ2
þ D2s12 þ D2
s23 þ D2s31; ð16Þ
Dsij
¼1
2
@ us;i
@ x j þ@ us; j
@ xi :
ð17
Þ Johnson et al. [26] made a simple algebraic expression for the solid
pressure in the frictional region:
P s; fr ¼ Fr ðes es;minÞnðes;max esÞ p
; ð18Þ
Fr ¼ 0:1es: ð19ÞIn which es,min = 0.5, n = 2, and p = 3 are all experimental based
parameters.
3.4. Kinetic theory of granular flow (KTGF)
The transport equation for granular temperature of solid phaseHs can be written as:
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3
2
@
@ t esqsHsð Þ þ r esqs
~V sHs
¼ ss : r~V s r qs cHs
3K sg Hs;
ð20Þwhere ss, q s and cHs are the solid stress tensor, flux of fluctuating
energy and collisional energy dissipation respectively.
qs can be written as:
qs ¼ kHsrHs; ð21Þwhere kHs
is the granular conductivity of granular temperature and
the corresponding correlation based on Gidapow model [22] is
given by:
kHs ¼ 150dsqs
ffiffiffiffiffiffiffiffiffiffiHsp
p
384ð1 þ eÞ g 01 þ 6
5ð1 þ eÞe g g 0
2
þ 2dsqse2s ð1 þ eÞ g 0
ffiffiffiffiffiffiHs
p
r :
ð22ÞThe algebraic equation for the collisional energy dissipation, cHs, is
derived by Lun et al. [19] as follow:
cHs ¼12ð1 e2Þ g 0
ds
ffiffiffiffip
p qse2s
ffiffiffiffiffiffiffiH2
s
q : ð23Þ
When the restitution coefficient, e goes to 1, the dissipation of the
granular temperature goes to zero. This means that the particlesare perfectly elastic [19].
3.5. Drag models
The drag force between the gas phase and the particles is one of
the dominant forces in a fluidized bed. Generally, drag coefficients,
K sg , areobtainedfrom two types of experimental data. The first type
is for the high value of the solid volume fractions or packed-bed
pressure drop data, such as the Ergun drag model [27]. These types
of correlations require a complementary drag model for low values
of the solid volume fractions, like the Gidaspow drag model [22,23].
In the second class of data, the terminal velocity of particles in flu-
idized or settling beds is employed to derive the drag model as a
function of void fraction and Reynolds number. An example for thiscategory is the Richardson and Zaki model [28].
In this paper, eleven widely used drag models that have been
reported in the literature are investigated for the modeling of a
3D fluidized bed. The corresponding correlations for each drag
model are summarized in Table 1.
3.5.1. Adjustment of drag coefficient
In all drag correlations, the drag force depends on the local rel-
ative velocity between phases and the void fraction. However, in
deriving such general empirical drag correlations some other fac-
tors, such as particle size distribution and particle shape have
not been considered. Also, void fraction dependency is very diffi-
cult to be determined for any condition other than a packed bed
or infinite dilution (single particle). On the other hand, mostresearchers have information on the minimum fluidization veloc-
ity of their own material. In this respect, Syamlal and O’Brien
[37] introduced a method to modify their original drag law using
minimum fluidization velocity, commonly available experimental
information for the specific material.
The parameter C 2 in Syamlal–O’Brien drag equation is related to
the minimum fluidization velocity through the velocity voidage
correlation and the terminal Reynolds number, Ret [37] and is
changed until the following criterion is met:
Objective function : U experimentmf
Rets e g l g q g ds
( ) !Minimize
0;
m g ¼ Rets e g l g q g ds ¼ U
experiment
mf ; ð24Þ
where
Ret ¼ mr ;sRets; ð25Þ
mr ;s ¼ A þ 0:06B Rets1 þ 0:06Rets
; ð26Þ
Rets ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi23:04 þ 2:52
ffiffiffiffiffi ffi4 Ar
3q 4:8
r
1:26
0BB@
1CCA
2
;
ð27
Þ
Ar ¼ ðqs q g Þd3sq g ~ g
l g
; ð28Þ
Ret , is the Reynolds number under terminal settling conditions for
the multi particle system, v r ,s is the terminal velocity, Rets is the
Reynolds number under terminal settling conditions for the single
particle and Ar is the Archimedes number.
Also, the parameter C 1 in Syamlal–O’Brien equation needs to be
adopted in order to guarantee the continuity of velocity voidage
correlation as follows [37]:
C 1
¼1:28
þ
logðC 2Þ
logð0:85Þ:
ð29
ÞUsing the same concept, a method has been proposed to modify the
drag model presented by Di Felice [18]. At minimum fluidization
condition, neglecting the gas-wall friction and the solid stress trans-
mitted by particles, the momentum balance can be written as:
Buoyancy Force = Drag Force
esðqs q g Þ g ¼K sg e g
j~V s ~V g j: ð30Þ
Considering the fact that at minimum fluidization condition ~V s ¼ 0
and ~V g ¼ U experimentmf
, the Eq. (29) can be reduced to:
es;mf ðqs q g Þ g ¼ K sg e g ;mf
U experimentmf
: ð31Þ
Substituting the Di Felice drag correlation into Eq. (30) and utilizing
the least square method as a non-linear optimization algorithm, the
drag model parameters P and Q in Di Felice drag correlations will be
modified for the system under study using experimental data at
minimum fluidization condition U experimentmf ¼ 0:065 ðm=sÞ. When
adjusting the drag models it should be kept in mind that the adjust-
ment should not alter the behavior of the drag correlation when
voidage approaches one. Most drag correlations are formulated
such that in that limit, the single sphere drag coefficient, C D, can
be recovered.
4. Numerical simulation
Governing equations of mass and momentum conservation aswell as the granular temperature equation are solved using finite
volume method employing the Phase-Coupled Semi Implicit Meth-
od for Pressure Linked Equations (PC-SIMPLE) algorithm, which is
an extension of the SIMPLE algorithm to multiphase flow. A mul-
ti-fluid Eulerian–Eulerian model, which considers the conservation
of mass and momentum for each phase, has been applied. The ki-
netic theory of granular flow, which considers the conservation of
solid fluctuation energy, was used for closure of the solid stress
terms. The three-dimensional (3D) geometry has been meshed
using 336,000 structured rectangular cells. Volume fraction, den-
sity, and pressure are stored at the main grid points that are placed
in the center of each control volume. A staggered grid arrangement
is used, and the velocity components are solved at the control vol-
ume surfaces. Fig. 2 shows a schematic view of the staggered gridcells for velocity components and pressure.
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A pressure correction equation is built based on total volume
continuity. Pressure and velocities are then corrected so as to sat-
isfy the continuity constraint. A grid sensitivity analysis is per-
formed using different mesh sizes and 5 mm mesh interval
spacing was chosen for all the simulation runs. The detailed results
for sensitivity analysis have been discussed in Section 5.4. Second-
order upwind discretization scheme was used for discretizing the
governing equations. An adaptive time-stepping algorithm with
100 iterations per each time step and a minimum value of order
105 for the lower domain of time step was used to ensure a stable
convergence. The adaptive determination of the time step size is
based on the estimation of the truncation error associated with
the time integration scheme (i.e., first-order implicit or second-
order implicit). If the truncation error is smaller than a specified
tolerance, the size of the time step is increased; if the truncation
error is greater, the time step size is decreased. The convergence
criteria for other residual components associated with the relative
error between two successive iterations has been specified in the
order of 105. A detailed study has been carried out on the effect
of restitution coefficient and the results have been presented in
Section 5.3. Including the adjusted drag model cases, 12 different
Table 1
Summary of drag coefficient correlations.
1. Richardon and Zaki [28] (1954)
K sg ¼ 3q g e g es
4dsv 2r
C Dj~V s ~V g jv r ¼ en1
g
n ¼4:65; Rem < 0:24:4Re0:03
m ; 0:2 > Rem < 1
4:4Re0:1m ; 1 > Rem < 500
2:4; Rem > 500
8>><>>:
Rem ¼ Re sv r
Res ¼ q g dsj~V s~V g jl g
2. Wen–Yu drag model [29] (1966)
K sg ¼ 3q g e g ð1e g Þ4ds
C Dj~V s ~V g je2:65 g
C D ¼ 24e g Res
½1 þ 0:15ðe g ResÞ0:687
Res ¼ q g dsj~V s~V g jl g
3. Gibilaro drag model [30] (1983, 1985)
K sg ¼ 17:3Res
þ 0:336h i
esq g
dsj~V s ~V g je1:8
g
Res ¼ q g e g dsj~V s~V g j2l g
4. Gidaspow drag model [31] (1986)
K sg ¼ ð1 usg ÞK Ergunsg þusg K WenYu
sg
K Ergunsg ¼ 150 e2
s l g
e g d2s
þ 1:75 esq g
dsj~V s ~V g j; e g 6 0:8
K WenYusg ¼ 3
4 C Desq g
dsj~V s ~V g je2:65
g ; e g P 0:8
C D ¼24e g Res
½1 þ 0:15ðe g ResÞ0:687; Res < 1000
0:44; Res P 1000
(
Res ¼ q g dsj~V s~V g jl g
usg ¼ Arctan½1501:75ð0:2esÞp þ 0:5
5. Syamlal–O0Brien drag model [32] (1988)
K sg ¼ 3ese g q g
4dsv 2r
C Dj~V s ~V g j
C D ¼ 0:63 þ 4:8 ffiffiffiRev r
p & ’2
v r ¼ 12 ½ A 0:06Re þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið0:06ReÞ2 þ 0:12Reð2B AÞ þ A2
q
A ¼ e4:14 g
B ¼C 2e1:28 g ; e g < 0:85eC 1
g ; e g P 0:85C 1 ¼ 2:65; C 2 ¼ 0:8
8<:
6. Arastoopour drag model [33] (1990)
K sg ¼ 17:3Res
þ 0:336l m
esq g
dsj~V s ~V g je2:8
g
Res ¼ q g dsj~V s~V g jl g
7. RUC-drag model [34] (1994)
K sg ¼ Al g ð1e g Þ2
e g d2s
þ Bq g ð1e g Þ
dsj~V s ~V g j
A ¼ 26:8e3 g
ð1e g Þ23ð1ð1e g Þ
13Þð1ð1e g Þ
23Þ2
B ¼ e2 g
ð1ð1e g Þ23Þ2
8. Di Felice drag model [18] (1994)
K sg ¼ 34 C D
esq g
dsj~V s ~V g j f ðesÞ
f (es) = (1 es) x
x = P – Q exp ð1:5bÞ2
2
h ib = log(Res)
P = 3.7 and Q = 0.65
9. Hill Koch Ladd drag correlation [35] (2001)
K sg ¼ 3q g e g ð1e g Þ4ds
C Dj~V s ~V g jC D ¼ 12e2
g
ResF
Res ¼ q g dse g j~V s~V g j2l g
F ¼ 1 þ 3=8Res; es 6 0:01 and Res 6 ðF 2 1Þ=ð3=8 F 3ÞF ¼ F 0 þ F 1Re2
s ; es P 0:01 and Res 6 F 3 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi ffi
F 23 4F 1ðF 0 F 2Þq
=ð2F 1ÞF ¼ F 2 þ F 3Res; Otherwise
8><>:F 0 ¼
ð1 wÞ 1þ3 ffiffiffiffiffiffiffies=2
p þð135=64Þes lnðesÞþ17:14es
1þ0:681es 8:4e2s þ8:16e3
s
þ w½10es=ð1 esÞ3; 0:01 < es < 0:4
10es
ð1es
Þ3 ; es > 0:4
8><>:
F 1 ¼ ffiffiffiffiffiffiffiffiffiffi
2=es
p =40; 0:01 < es < 0:1
0:11 þ 0:00051eð11:6esÞ; es > 0:4
(
F 2 ¼ð1 wÞ 1þ3
ffiffiffiffiffiffiffies=2
p þð135=64Þes lnðesÞþ17:89es
1þ0:681es11:03e2s þ15:41e3
s
þ w½10es=ð1 esÞ3; es < 0:4
10es
ð1esÞ3 ; es P 0:4
8><>:
F 3 ¼ 0:9351es þ 0:03667; es < 0:09530:0673 þ 0:212es þ 0:0232=ð1 esÞ5; es P 0:0953
w ¼ eð10ð0:4esÞ=esÞ
10. Zhang and Reese drag model [36] (2003)
K sg ¼150
e2s l g
e g d2s
þ 1:75esq g
ds
~V r ; e g 6 0:8
34 C D
esq g
ds
~V r e2:65
g ; e g P 0:8
8<:
~V r ¼ ð~V s ~V g Þ2 þ 8Hs=ph i0:5
C D ¼ ð0:28 þ 6= ffiffiffiffiffiffiffi
Res
p þ 21=ResÞRes ¼ q g ds
~V r
l g
Fig. 2. Schematic view of staggered grid, volume fractions are stored at the main
grid points (P ) while the velocity components at control volume surfaces.
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drag models are studied in this work to simulate the momentum
transfer between the phases (Richardon and Zaki [28], Wen–Yu
drag model [29], Gibilaro drag model [30], Gidaspow drag model
[31], Syamlal–O’Brien drag model [32], Arastoopour drag model
[33], RUC-drag model [34], Di Felice drag model [18], Hill Koch
Ladd drag correlation [35], Zhang and Reese drag model [36], ad- justed Syamlal [37], adjusted Di Felice drag model). The drag mod-
els available in Fluent 6.3 suited for a fluidized bed simulation is
the Gidaspow model, the Syamlal–O0Brien model and the Wen–
Yu drag model. For the other nine drag models, specific User De-
fined Functions (UDF) in C++ have been implemented and up-
loaded into the software. FLUENT 6.3 on a 20 AMD/Opteron 64bit
processor Sun Grid Microsystems workstation W2100Z with 4 GB
RAM is employed to solve the governing equations. Computational
model parameters are listed in Table 2.
5. Result and discussion
CFD modeling has been performed using FLUENT 6.3. Simula-tions have been carried out on a 3D fluidized bed using a transient
Eulerian–Eulerian model. Several superficial gas velocities, 0.11,
0.21, 0.38, and 0:46 ðm=sÞ, that correspond to 1.6, 3.2, 5.8, and
7U mf , respectively, have been studied. In the following section,
the simulation results have been compared with the experimental
data in order to validate the model. As previously discussed, sev-
eral drag models have been proposed in the literature to model
the momentum transfer between the phases. In the present work,
a complete study has been performed on all those drag models and
finally a method has been developed to adjust the drag model pro-
posed by Di Felice [18]. The adjustment of the two drag models
with the methods discussed earlier was taken into account by opti-
mizing the parameters C 1 and C 2 to 11.772 and 0.182 for the Syam-
lal–O’Brien model and the parameters P and Q to 5.2 and 0.31 for
the Di Felice model. The associated parameters of the models were
estimated by adopting the model to experimental data using
non-linear parameter estimation analysis.
Fig. 3 shows a snapshot of solid volume fraction contours for the
twelve drag models studied in this work at a superficial gas veloc-
ity of 0:21 ðm=sÞ and after 10 (s) real-time simulations. In this fig-
ure, comparison between all drag models and experimentalsnapshot has been made in terms of bed height and bubble size
and shape. It can be readily observed that the two adjusted models
(i.e., Di Felice adjusted model and Syamlal–O’Brien adjusted model
[37]) show the best results simulating the bed height. The adjusted
Di Felice model is more accurate in the prediction of the bubble
shapes and fluctuating behavior of the free surface of the bed. It
can be seen that the original Syamlal–O’Brien model [32] repre-
sents the lowest bed expansion and gas void fraction. This fact
could have been foreseen from the minimum fluidization velocity
prediction by this model, which is almost six times larger than
experimental data [38]. Expansion of the bed started with forma-
tion of bubbles for all the models and eventually reached a statis-
tically steady-state bed height. After this point, an unsteady
chaotic generation of bubbles was observed after almost 3 (s) of
real-time simulation. Disregarding the two adjusted drag model,
Fig. 3 shows that the original Di Felice [18] and Gibilaro [30] drag
models have produced better results in predicting the bed expan-
sion among other drag models. The drag model proposed by
Richardson–Zaki [28] has given the worst results with respect to
bubble shapes since it shows symmetry in contours of solid vol-
ume fraction after 10 (s) real time which is not reasonable. The rest
of the models showed approximately the same range of bed expan-
sion. There also exists a more recent correlation which is based on
extensive lattice Boltzmann simulations by van der Hoef et al. [39],
and Beetstra et al. [40]. They have proposed expressions for nor-
malized drag force for both mono-dispersed and poly-dispersed
systems. Their results found to be in excellent agreement (devia-
tion smaller than 3%) with the simulation data of several models
proposed in literatures.
5.1. Pressure drop
Fig. 4 shows the time average pressure drop inside the bed be-
tween two specific elevations (i.e. 0.03 (m) and 0.3 (m) as demon-
strated in Fig. 1) for different studied cases and experimental
results. In order to calculate the average pressure at each pressure
sensor (i.e. y = 0.03 (m)), both spatial and time averaging have been
applied. At first, the spatial averaging, which is the average value of
pressure for all nodes in the plane of first pressure sensor (plane
y = 0.03 (m)) has been utilized. Subsequently, the time averaging
of spatial-averaged pressure values in the period of 3–10 (s) real
Table 2
Computational model parameters.
Parameter Value
Particle density 2500 (kg/m3)
Gas density 1.225 (kg/m3)
Mean particle diameter 275lm
Initial solid packing 0.6
Superficial gas velocity 11.7, 21, 38, 46(cm/s)
Bed dimension 0.28 (m) 1.2 (m) 0.025 (m)Static bed height 0.4 (m)
Grid interval spacing 0.002 (m)
Inlet boundary condition type Inlet – velocity
Outlet boundary condition type Pressure – outlet
Under-relaxation factors Pressure 0.6
Momentum 0.4
Volume fraction 0.3
Granular temperature 0.2
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m)
Fig. 3. Contours of solid volume fraction (U = 0.21 (m/s) t = 10 (s)): (a) experiment; (b) Syamlal–O’Brien adjusted; (c) Syamlal–O’Brien; (d) Arastoopour; (e) Gibilaro; (f) HillKoch Ladd; (g) Zhang–Reese; (h) Richardson–Zaki; (i) RUC; (j) Di Felice adjusted; (k) Di Felice; (l) Wen–Yu and (m) Gidaspow.
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time has been incorporated. As indicated in Fig. 4, the pressure
drop for all the models showed a declining trend with increase of
the superficial gas velocity, providing good qualitative agreement
with the experimental data. It can be seen that the adjusted Di
Felice model gives closer result to experimental data and for all
four superficial gas velocity shows significant improve in result
compared to original Di Felice model [18]. As shown in Fig. 4 the
adjusted model based on Syamlal–O’brien [37] study also gives
acceptable results especially in lower superficial gas velocities.
Fig. 5 shows a comparison between all drag models in predic-
tion of overall bed expansion ratio (i.e., DP 2 as indicated in
Fig. 1) with respect to superficial gas velocity. It can be seen that
the overall pressure drop for all drag models does not change toomuch by increasing the superficial gas velocity. This is in good
agreement with both theoretical and experimental predictions, ex-
cept for the highest velocity, where the deviation may be due to
the fact that at high velocities of gas, the elevation 0.6 m above
the distributor is actually inside the bed for the case of the adjusted
Syamlal–O’Brien drag model [37]. Hence, it, in fact, it represents
the pressure drop between two elevations inside the bed [38].
Fig. 6 shows a comparison between 2D vs. 3D simulation of flu-
idized bed using three different drag models (adjusted Di Felice
and Syamlal–O’Brien [37] drag model and original Di Felice [18]
drag model). It can be seen that the pressure drop for both 2D
and 3D simulation shows a declining trend by increasing the
superficial gas velocity which is in good qualitative agreement
with the experimental data. However, 3D simulations show their
superiority in predicting the pressure drop inside the bed com-
pared to 2D simulations. The reason can be the effect of participat-ing governing equations of the z direction (depth of the bed) in
Navier Stokes equation of multiphase flow. It can be concluded that
although three-dimensional simulation takes more time and
Fig. 4. Pressure drop inside the bed
ðDP 1
¼ P z
¼0:03m
P z
¼0:6m
Þ.
Fig. 5. Overall Pressure drop ðDP 2 ¼ P z ¼0:03m P z ¼0:6mÞ.
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computing processors than two-dimensional simulation, it gives
more accurate results when the models are compared with exper-
imental data.
5.2. Bed expansion ratio
The experimental data of the time-average bed expansion ratio
were compared with corresponding values predicted by different
drag models for various superficial gas velocities as depicted in
Fig. 7. For this series of simulations, a static bed height of
H 0 = 0.4 (m) over a range of superficial velocities 11.7, 21, 38,
and 46 ðcm=s was used. All drag models demonstrate a consistent
increase in bed expansion with gas velocity and predict the bedexpansion reasonably well. Fig. 7 shows the considerable relative
increase in bed expansion as the fluidizing velocity increases; a
5% increase was obtained at 0.11 m/s, a 20% increase at 0.21 m/s,
42% at 0.38 m/s, and up to a 50% increase in bed height was
measured at 0.46 m/s, the highest fluidized velocity investigated.
It can be seen that using adjusted Di Felice drag model, the bed
expansion ratio can be predicted fairly accurately over a whole
range of superficial gas velocities compared to experimental data.
All the available drag correlations with the exception of two ad-
justed drag models (i.e., Di Felice and Syamlal–O’Brien [37]) at high
superficial gas velocity (0.46 m/s), underestimate the bed expan-
sion. The adjusted Syamlal–O’Brien drag model showed good
agreement with experimental results only up to a moderate range
of gas velocity. Fig. 7 shows that for higher superficial gas veloci-
ties, the adjusted Syamlal–O’Brien drag model comparatively over-
estimated the bed expansion ratio. Fig. 7 also reveals that even the
original Di Felice [18] drag model gives the best result for the pre-diction of bed expansion ratio among all the other conventional
drag laws. This fact vindicates the claim that this drag model
was opted for adjustment based on minimum fluidization velocity
[38].
Fig. 6. 2D vs. 3D simulation of pressure drop inside the bed
ðDP 1
¼ P z
¼0:03m
P z
¼0:3m
Þ.
Fig. 7. Comparison of simulated bed expansion ratio with experimental data.
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Fig. 8 shows the predicted contours of solid volume fraction at
t = 10 (s) using adjusted Di Felice drag model for four different
superficial gas velocities. It can be easily seen that by increasing
the gas velocity the bigger bubbles will be generated inside the
bed and as a result the bed height will increase significantly. By
further increasing the superficial gas velocity, the hydrodynamic
regime of the fluid flow inside the bed will transfer from bubbling
regime to slugging regime.
5.3. Effect of restitution coefficient
The restitution coefficient, e specifies the coefficient of restitu-
tion for collisions between solid particles. The restitution coeffi-
cient compensates for the collisions to be inelastic. In a
completely elastic collision the restitution coefficient will be equal
to one. Fig. 9 shows a snapshot of solid volume fraction contours at
the superficial gas velocity of U = 0.21(m/s) and t = 10 (s) using ad-
justed Di Felice drag model for seven different restitution coeffi-
cients proposed for simulation of fluidized beds in literature. A
comparison between different values of restitution coefficient
and experiment in terms of bed height and bubble size and shape
is shown in Fig. 9. It can be seen that as collisions become less ideal
(and more energy is dissipated due to inelastic collisions) particles
become closely packed in the densest regions of the bed, resulting
in sharper porosity contours and larger bubbles [13]. In this study
the value of e = 0.92 for the coefficient of restitution has been used
for the whole simulation which seems to be in good agreement
with experiment in terms of bubble shape and bed height.
5.4. Mesh size sensitivity analysis
Wang et al. [41] concluded that in order to obtain correct bed
expansion characteristics, the grid size should be of the order of
three particle diameters which requires smaller grid size and high-
er computer resources.
A mesh size sensitivity analysis has been carried out to study
the effect of grid size resolution on the results predicted by numer-
ical simulation. In this respect, the geometry of the fluidized bed
has been meshed using three distinctive grid intervals of 2, 4,
and 5 (mm) to simulate the hydrodynamic behavior of the bed.
The adjusted Di Felice drag model has been chosen for modeling
the momentum transfer between the phases in sensitivity analysis
simulations. All the simulations performed at superficial gas veloc-ity of U ¼ 0:21 ðm=sÞ. Table 3 shows the predication of pressure
drop inside and across the bed, DP 1 and DP 2, respectively. Predic-
tion of time mean average solid volume fraction at bed elevation of
Z = 0.2 (m) also was checked. The time mean average was calcu-
lated on the real-time simulation interval of 2–10 s to ensure that
statistical steady state behavior inside the bed was attained [38]. It
can be easily observed that the results did not show any notewor-
thy dissimilarity in fluid dynamics behavior of the beds. Table 2
also compares the time required for 10 s of real-time simulation.
(a) (b) (c) (d)
Fig. 8. Contours of solid volume fraction (t = 10 (s), Di Felice adjusted drag model):
(a) U = 0.117 (m/s); (b) U = 0.21 (m/s); (c) U = 0.38 (m/s) and (d) U = 0.46 (m/s).
Experiment
Fig. 9. Comparison between experiment and simulated bed height for various values of the coefficient of restitution ( U = 0.21 (m/s) t = 10 (s), adjusted Di Felice).
Table 3
Grid size sensitivity results.
Mesh
spacing
(mm)
DP 1(kPa)
DP 2(kPa)
Mean solid volume
fraction at z = 0.2 (m)
Simulation time for
10 (s) real time (h)
2 2.945 5.18 0.55 300
4 2.964 5.10 0.54 148
5 2.975 5.05 0.54 52
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It can be seen that the required time for simulating 10 s of 3D flu-
idized bed drastically increases from 52 h to almost 2 weeks for a
decrease in grid interval spacing from 5 to 2 mm, respectively.
Therefore, the mesh interval size of 5 mm has been chosen forthe rest of simulation to obtain reasonable time efficiency without
losing the accuracy of results. Fig. 10 also shows the contours of so-
lid volume fraction for three different mesh size resolutions. Here-
in, similarities of bed expansion and bubble shapes among the
simulations can be easily appreciated. The above results indicate
that the grid size spacing selected for simulation in this work
(i.e., 5 mm) was adequate for satisfactory prediction of the hydro-
dynamics in computational geometry.
6. Conclusion
Numerical simulation of a bubbling gas–solid fluidized bed
were performed in a three dimensional solution domain using
the Eulerian–Eulerian approach to investigate the effect of usingdifferent drag correlations for modeling the momentum transfer
between phases. The drag models of Richardon and Zaki,
Wen–Yu, Gibilaro, Gidaspow, Syamlal–O’Brien, Arastoopour, RUC,
Di Felice, Hill Koch Ladd, Zhang and Reese, and adjusted Syamlal
were reviewed and a method proposed for adjusting the original
Di Felice darg model in a three dimensional domain based on the
experimental minimum fluidization conditions. In this respect,
FLUENT 6.3 was used to perform the calculations while the drag
correlations have been implemented in C++ and uploaded in
FLUENT as User Defined Functions (UDF).The results have been
compared to experimental data in terms of pressure drop and
bed expansion ratio. It is concluded that the adjusted Di Felice
model predicts the hydrodynamic behavior of fluidized bed more
accurately that all other drag models. The effect of using three-dimensional analysis vs. two-dimensional simulation of fluidized
beds is also investigated. The results show that although
three-dimensional simulation takes more time and computing pro-
cessors than two-dimensional simulation, it gives more accurate
results when the models are compared with experimental data.
Finally, sensitivity analysis was carried out to investigate the effect
of using various restitution coefficients as well as the different grid
interval spacing on the results. Further modeling efforts are
required to study the influence of other parameters such as gasdistributors, and also, the effect of particle size distribution which
has been underestimated using the mean particle diameter. More-
over, new experimental studies should be carried out using recent
advancements in instrumentation engineering in order to resolve
the available experimental discrepancies reported in the literature
such as void fraction measurements, and bed expansion ratio.
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