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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

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a d v e n g s o f t



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

376 E. Esmaili, N. Mahinpey/ Advances in Engineering Software 42 (2011) 375–386



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.

E. Esmaili, N. Mahinpey / Advances in Engineering Software 42 (2011) 375–386 377



@

@ 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


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:




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


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.




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


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

(


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


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.




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], adjusted 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.




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.




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.




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




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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