Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
A binary kinetic theory of granular flow model for fluidized beds
Zhongxi Chao*,Yuefa Wang* ,Jana P. Jakobsen t,
Maria Fernandino# and Hugo A. Jakobsen*
Department of Chemical Engineering, NTNU, Trondheim, 7049,Norway
T SINTEF Energy AS, Trondheim,7049,Norway
#Department of Energy and Process Engineering, NTNU, Trondheim, 7049,Norway
Email: hugo.atle.j @chemeng.ntnu.no
Keywords: kinetic theory of granular flow, fluidized bed, particle segregation, friction drag
Abstract
The paper presents a multifluid Eulerian model derived from binary kinetic theory of granular flows and an empirical
friction theory. In the process of deriving the constitutive equations, the exact discrete particle velocity
difference c, = Ck + vk is used in this work whereas in previous studies a simplification c,k = Ck is used. As the effects
due to hydrodynamic velocity differences and particleparticle friction are considered, some unconventional terms are
produced. Model validation by use of the data from Goldschmidt et al. (2003) shows that the particleparticle drag models
play an important role and the friction drag must also be included in order to predict the particle segregation phenomena in
dense binary fluidized beds.
Introduction
Fluidized beds are widely used in many industrial processes.
In some applications, there exist 2 types of particles with
completely different physical properties (for example
particle density and size). The particles will segregate due to
the property differences. Previous investigations (Rowe and
Nienow, 1976; Hoffmann and Romp, 1991; Wang and Chou,
1995) show that the small, light particles called flotsam tend
to rise, and the large, heavy particles called jetsam tend to
sink. Some scholars put forth some mathematical models to
describe the flow behavior. Among them, the KTGF (kinetic
theory of granular flow) EulerianEulerian models are often
used. The KTGF is based on an analogy to the classical
kinetic theory of gases. The basic particle billiard ball
collision theory and some statistical methods are adopted to
obtain continuity equations, NavierStokeslike momentum
equations and other transport equations. Lun et al. (1984)
and Gidaspow (1994) successfully applied the theory to a
monoparticle system. Jenkins and Mancini (1987) extended
a monoKTGF model to a binary particle system. In the
model, it is assumed that both particle species share the
same local granular temperature. Gidaspow et al. (1996),
Manger (1996), Mathiesen et al. (2000), Lu et al. (2000,
2003 and 2007), Iddir et al. (2005) applied binary KTGF
models with unequal individual granular temperatures.
Lathouwers (2000) also derived a binary KTGF model with
unequal individual granular temperatures, but with different
constitutive equations. In these models only the transport
effects due to the peculiar particleparticle velocity
differences are considered, and the effects due to the
hydrodynamic particleparticle velocity differences are
neglected. This may be one reason why the model
underpredict the particleparticle momentum coupling. In
dense flows, in which the long term particleparticle sliding
and rolling contacts play important roles, the friction
stresses and friction drag should be considered. Fan and Fox
(2008) and Gera et al. (2'k14) used an empirical friction
stress and a particleparticle drag in order to make the
particleparticle drag sufficiently large. Neglecting these
particleparticle couping terms may be a second reason why
the previous models underpredict the particleparticle
coupling. Hence, in this paper, the authors present a model
considering the effects due to the hydrodynamic velocity
differences and the particleparticle friction. Furthermore,
the data from Goldschmidt et al. (2003) is used to validate
the model, and some discussions are given about the effects
of friction drag and the hydrodynamic particleparticle
velocity differences.
Nomenclature
Cfr, particle friction coeffcient
Ck peculiar particle velocity difference between
particle i and k, Clk= C1 Ck, (ms1)
c,k discrete particle velocity difference between
particle i and k, Ck= C1 Ck, (ms1)
d particle diameter (m)
d,k sum of the radii of particle i and k (m)
e,k coefficient of restitution for a collision between
particle i and k
g gravitational constant (m2s)
g,k radial distribution function for a collision
between particle i and k
mo sum of the masses of particle i and particle k(kg)
h,m initial packed bed height (m)
Kf,, correction coefficient for frictional particle drag
N number of particle species
n particle number density (m3)
p pressure (Nm 2)
Paper No
k,
v
vA
Greek
a
P
At
P,
Eg
Subsripts
g gas phase
i particle i
k particle k
col collisional
fri frictional
kin kinetic
large large particle
small small particle
Supersripts
max maximum
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
some unconventional terms related to the hydrodynamic
velocity difference v,k and conventional terms related to the
gas turbulent kinetic energy (m2s2)
velocity (ms1)
particle hydrodynamic velocity difference
between particle i and k, vl= v, vk, (ms1)
letters
volume fraction of a particle species or gas
density of the particles or gas (kgm 3)
granular temperature (m2s2)
viscosity (Pas)
particleparticle drag coefficient between species
i and k (kgm'3s)
gassolid drag coefficient for particle i (kgm 3s)
gas turbulent kinetic energy dissipation rate per
unit mass(m s 3)
similar to the
(4)
(5)
Z(l+ek),k 5 k=1
The dilute viscosities are:
4p,0,
HAr,dilute N 1 1
Sk= I n di (I +0
k=1
Governing equations
The authors derived a binary kinetic theory of granular flow
model, in which the model equations for the particle phases
are as follows.
The continuity equations are:
(ap,)+V.(apv,) 0
at
The momentum equations are:
(ap,,) + V (av,,)
at
(1)
aVpg V (pk, +
PC, +Pf,)+(, +Pi,(Vgv,)+a,p,g (2)
The granular temperature equations are:
38 3
3 (a,p0,)+ V(a,p,v,0,) =(PCO +P, +
2 at 2
pf,) :Vv V (qco + qk,,) 3,,0, + (3)
In order to close the model equations, the KTGF is used to
obtain the kinetic pressure tensor pk ,collisional pressure
tensor po,, the momentum source term , the kinetic heat
flux vector q, ,the collisional heat flux vector qco, and the
dissipation term for the granular temperature equation In
a dense fluidized bed, the friction pressure tensor pf should
also be included as the particleparticle sliding and rolling
contacts become important.
In calculating the constitutive equations, the assumption
,k = Ck which was used by Manger (1996) is abandoned.
The exact expression for discrete particle velocity difference
c, = C, +v,, is used. So in the equations, there are
0.500, 025)
(7)
The collisional pressure tensor is:
N
Pool = ,kV,kV,k +a,Pcoll a, (2,u,0S, +
k k=l
/,BV v,I+2/Uk,colSk + /k,BV Vk) (8)
Where the standard strains for particle i and k are
S, = I(Vv +Vv,) V.v,I (9a)
Sk ((Vvk) + Vvk) VVkJ (9b)
The collisional pressure which contains both the effects of
hydrodynamic and peculiar velocity differences is:
P akPlPk (1 +ekdkk ( ,k) k
kl 3mo 5
The correction coefficients ,k are:
2 d 3 a,akp, pk (+ek)
7 dk g (1+er)
k 15 kg
The collisional viscosity for particle i is:
7 (1+ ek k akppk
,co 72
S 1 1
302 +402
k I
1.95504E)
The collisional viscosity for particle k is:
(r + ) 4 a^kP0Pk
kc 72 = (1+ ek) gkd m, ak
72 mO
1 1
302 +402
1 k
1 1
1.95504 )4
(12)
(13)
peculiar velocity difference C,k
The form of the kinetic pressure tensor is
expression used by Mathiesen et al. (2000).
Phkn =a, (p,k +2/,,,, S, )
The kinetic pressure for the particle i is:
P,kin = PO,
The kinetic viscosities are:
Paper No
The bulk viscosity for particle i is:
J (l+ek)g", 4 ckP9Pk
,B =36 1+m e ,
1 15
1.95504E 4
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The friction pressure for particle i is considered using a
similar method as used by Lindborg (2008).
ax aCx
1 l
(14)
f Zx (25)
l (25)
The bulk viscosity for particle k is:
k 36 e(1 )g,4 akp'Ok
36 m,
1 1+
302 +402
1.95504
(15)
The form of the kinetic heat flux is similar to the
expression used by Mathiesen et al. (2000).
qkn =ak ,kk,.V (16)
The kinetic heat conductivity for particle i is:
2k 6 N 12
kkn N ,lute ik
1 l ) 5 k=1
N k=1
(17)
Where the dilute heat conductivity for particle i is:
15p,,
N
z d72 (0 + 02
k=l1
S) akco
The collisional flux term is:
N
q1 = k equ Vk V
k=l
0.500; ')
(18)
V0, (19)
The first term is the hydrodynamic heat correction flux and
the second term is the collisional diffusion heat flux. The
hydrodynamic correction coefficient uk is:
2
z mm mnnk (1 32 (3
uk = 1k ( k+)2 gd (20)
10 m,
The equivalent granular temperature Oequ is:
equ = ,k+30, +3k (21)
The collisional granular heat conductivity for particle i is:
2 2xz ak P / k
kco 2 k (1+ ek)g zkd4 (22)
k=l 5 mo
The dissipation term for the granular temperature equation
is:
y, = /1T~1. mmknnk ( i
k k
k 1 8 Mo I I
1.98,8 + 1.98 k d 6 m nk (e (23)
k 1 J 60 WO k
(V .vk (98,+158k)+v.v (98k +158,))
For a dense system, the friction pressure tensor should also
be included.
Pfn= a(,, (p f+2/f~S,) (24)
N
0 if a,
t=l
The friction viscosity for particle i is given by:
p,, 2 sin m
f ,_ (26)
2a, S, S + ,/ d
All the parameters in the model are purely empirical, in the
paper, the parameter values proposed by Ocone et al.
(1993) and Lindborg (2008) as listed in Table 1 are used.
Table 1. Empirical parameters for the friction stress and pressure
model proposed by Ocone et al. (1993) and Lindborg (2008)
9( angle of internal friction, 28
F constant, Nmn2 0.5
r constant 2
s constant 3
amm lower volume fraction limit
for friction 0
The gas particle drag for particle i is given by:
150pU(la )a, 1.75a pg vv
a, d2 d .8
3C g, 165 >0.
4d a .8
(27)
Where the coefficient CD is:
if Re, <1000
if Rep, >1000
The particle Reynolds number for particle i is:
Rep,
ad pg v v 
The radial distribution function proposed
(2008) is used:
9k k
Sk ~d g, + ddk
dc +d,
The particleparticle drag force is:
N
k=, l(k v,)
k=
(29)
by Gao et al.
(30)
(31)
In a dense fluidized bed, the particleparticle drag
i,ddlute
1 1
302 +402
Paper No
coefficient includes 3 contributions which consist of the
two particleparticle collision effects due to hydrodynamic
and peculiar velocity differences respectively, and the
particleparticle friction effect.
A = fpec + hyd + (33)
The peculiar drag coefficient which is deduced from the
effects due to the particle peculiar velocity differences is:
,kec Mkmnnk d, (1+ ek )gk
mo
27 + 1 e7 2H0 25 0 25 (34)
The hydrodynamic drag coefficient which is deduced from
the effects due to the particle hydrodynamic velocity
differences is:
1hyd mm k 2 k
k k (1+ ek )gk ( k
mo 2
1.135v5( 025 025 0O.80 125 0125)) (35)
The empirical frictional drag coefficient used in the paper
is:
,fn m'mkn k d(1+el )gkkKfCf, nv (36)
mo 2
For the gas phase, the turbulent k model is used
(Lindborg,2008). The governing equations and constitutive
relations for the gas phase are as follows.
The continuity equation is
a (p) + V .(agpgvg) =0
The momentum equation is
a(ugpgvg) + V (apvv,)
at
(37)
agVpg +
V ((ag a +agRe ) aggg (gk(Vk Vg)(38)
k
The turbulent kinetic energy equation is
(a (pkg)+V(agpgkgvg V ag Vk +
at O kg )
gr Re VVg +aSkg g gpgg (39)
The turbulent energy dissipation rate equation is
0 i Vg +
agC1 Re VV g+ C Skg agpgC2 (40)
k k k
g g g
The viscous stress tensor is modeled as
Tg 2pg1 (Vvg ) V *3 V1 (41)
The turbulent stress tensor is modeled by using the
gradientand Boussinesq hypotheses:
e 3 gkI+2P 1v ) + Vvg 3) I
(42)
The turbulent viscosity is given by:
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
k2
=g = gC (43)
g
The production of the kinetic energy due to the existence
of the particles is modeled by using the equation proposed
by Jakobsen (1993).
S~ichj2 (44)
Skg Ck (Vk g) (44)
k=l
The empirical parameters are shown in Table 2
Table 2 Empirical parameters for the k E model
C kg cg C1 C2 b
0.09 1.00 1.30 1.44 1.92 0.25
Model validations and discussions
In order to validate the models, the experimental data from
Goldschmidt et al. (2003) is used. These experiments were
conducted in a 15cmwide, 70cmhigh, 1.5cmdeep pseudo
2D gas fluidized bed. Dynamic bed heights and segregation
rates were obtained by using digital image analysis
measurements. 2 types of spherical glass beads with the
properties shown in Table 3 were used. The fluidization gas
was mainly operated from Im/s to 1.3m/s. In the
experiments, the whole 2D domain was divided into many
small cells. Goldschmidt et al. (2003) defined the bed
heights for individual species as
C at,cellhcell
(h,) (45)
C a6eN
cell
where the summations are over all grid cells, a,,cei is the
average cell volume fraction for particle i, and hkez is the
cell height. A relative segregation rate is defined as
S1
s = (46)
in which Smax and S are defined like:
S =(2 x,) / (1 Xm) (47)
S = hsma ) / largee
(48)
where xsmaii is the mass fraction of the small particles. The
value range of the relative segregation rate s is between 0
and 1 which means zero segregation and complete
segregation respectively.
Table 2 Properties of the particles
Small articles Large particles
Particle density, kg / m3 2526 2523
Particle diameter, mm 1.5 2.5
Minimum fluidization
0.78 1.25
velocity, m/s
Friction coefficient Cf 0.1 0.15
In order to implement the mathematical model, an inhouse
code was developed. The finite volume method (Patankar,
1980) is applied to discretize the model equations. The
second order central differential scheme is used to discretize
the diffusion terms. A total variation diminishing (TVD)
scheme (Lindborg, 2008; van Leer, 1974, 1977) is applied to
discretize the convection terms. The SIMPLE algorithm for
multiphase flow (Jakobsen, 2008) is used for the
Paper No
pressurevelocity coupling. The Biconjugate gradient
method is used to solve the discretized algebraic equations
(Lindborg, 2008). A fully implicit coupled solver is used to
solve the coupled momentum equation in one dimension at
the time.
In the model the 2 types of particles are mainly coupled by
the particleparticle drag forces. Hence, the particle
segregation is closely related to the drag force models.
Figure 13 show comparisons of the particle segregation
rates and the particle bed heights predicted by the model
with different particleparticle drag force models. Drag
model 1 only includes the peculiar drag force, drag model 2
includes both the peculiar and hydrodynamic drag forces,
and drag model 3 includes the peuliar, hydrodynamic drag
forces and the friction drag force (in this paperKf, = 800,
the friction coefficient takes the average values for the 2
types of particles Cf = 0.125). Figure 1 shows that the final
particle segregation rates are similar using the 3 different
particleparticle drag forces. But the segregation rate is
overpredicted in the whole dynamic segregation process if
drag model 1 or drag model 2 is applied. Drag model 2 can
give a stronger particleparticle coupling than drag model 1
since an additional hydrodynamic velocity difference effect
is included. The simulation results fit well with the
experimental data if drag model 3 is used. Hence the friction
drag is important for calculating the particle segregation in a
binary dense bed. The bed heights for the small and large
particles calculated by using (45) are shown in Figure 2 and
Figure 3. These results show that the rise of the flotsam and
the sink of the jetsam can be well predicted if drag model 3
is used.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
0,09 
Simulation,dragl
SSiulaton,drag3
o 008 ExpenmentlGoldschmldt et al (2003)
a 0 Expenment2,Goldschmldt et al (2003)
0 07 H=15cm
Sv =1 lm/s
0 g
0 05
0 10 20 30 40 50 60
times]
Figure 3: Comparisons of the bed heights of the large
particles using different drag models
A qualitative comparison of the particle segregation using
different drag models is shown in Figure 4Figure7. The
experiments (Goldschmidt et al., 2003) and the simulations
were run for 60 seconds. Some key snapshots are taken at
time 15s, 30s, 45s and 60s. In the experiments, the small
particles are yellow and the large particles are brown. The
variable used in the isoprofiles is the volume ratio of the
small particles aa,, / (asa,, + lr) All (b) Figures
show a similar segregation process to the experimental data
(shown in (a) Figures) as drag model 3 is used. The small
particles gradually rise to the top of the bed and the large
particles sink to the bottom. The (c) Figures and the (d)
Figures show that the particles are segregated more quickly
than the experimental data if drag model 1 or drag model 2
is used. There are no big differences in the trends for the
snapshots at time 30 s, 45s and 60s.
times]
Figure 1: Comparisons of the segregation rates using
different particleparticle drag models.
(a) Experiment
10
(b) Drag 3
H =15cm
8 v=8 1m/s
Simulation,dragl
Smulaton,drag 2
Simulation, drag3
a Expenmentl,Goldschmidt et al (2003)
o Experment2,Goldschmldt et al (2003)
S10 20 30 40 50 60
time [s]
(c)Drag 2 (d)Dragl
Figure 4 Particle segregations at time 15s
Figure 2: Comparisons of the bed heights of the small
particles using different drag models
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
(a) Experiment
10 ) D
(b) Drag 3
(c)Drag 2 (d)Dragl
Figure 5 Particle segregations at time 30s
(a) Experiment
(b) Drag 3
(c)Drag 2 (d)Dragl
Figure 6 Particle segregations at time 45s
25
20
10
(c)Drag 2 (d)Dragl
Figure 7 Particle segregations at time 60s
The initial bed heights, gas velocities and the small particle
ratio xsmlare 3 important influential factors determining
the particle segregation. Figure 8FigurelO show
comparisons between the experimental data and the
simulation results with a change of bed heights, gas
velocities and the small particle ratio. In these calculations,
the drag (33) is used. The broad range of comparisons
proves that the mathematical model can predict the
segregation behavior in a binary dense fluidized bed with
reasonable accuracy.
simulation
O Expenmentl,Goldschmidt,et al (2003)
28 Expenment2,Goldschmidt,et al (2003) / 8 0
2 h =30cm 0 O
v =1 m/s
0= 5
x 05
14 O
7 O O O
o
0 10 20 30 40 50 60
time [s]
Figure 8.a Comparisons of the segregation rate
o 0,18

S0,17
E
a)
0,16.
0,15

0 10 20 30 40 50 60
time [sl
Figure 8.b Comparisons of the bed height of the small
particles
Paper No
0
D n
sImulation
O Expenmentl,Goldschmldt et al
S D Expenment2,Goldschmidt et al
h =30cm
v =1 1m/s
x ,,,=0 5
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
10 20 30
time [s]
40 50
60
Figure 8.c Comparisons of the bed height of the large
particles
time [s]
Figure 10.a Comparisons of the segregation rate
^

E
S0,10

0,08
0)
(Dl
I
10 20 30 40
time [s]
I 10 20 30
time [s]
40 50 60
Figure 9.a Comparisons of the segregation rate
E

0,10
E
08
E 0,08
0)
10 20 30
time [s]
40 50 60
Figure 9.b Comparisons of the bed height of the small
particles
0,10
0,09
oi
v
S0,08
S0,07
S
E 0,06
S0.05
0 10 20 30 40 50 60
time [s]
Figure 9.c Comparisons of the bed height of the large
particles
A 0,085
0,080
o
0.075
'
0,070
0) 0,065
n non.
0 10 20 30
time [s]
Figure 10.c Comparisons of the bed height of the large
particles
Conclusions
A novel binary multifluid kinetic theory Eulerian
mathematical model has been presented. The validation and
a broad range of calculations show that the drag models play
an important role and the friction drag must be included for
predicting the particle segregation in a dense binary
fluidized bed.
Paper No
0,180
E
S0,165
0,150

0,135
S0.120
 simulation
A experiment ,Goldschmidt,etal(2003)
S 0 experiment2,Goldschmidt,et a(2003)
o O
h =30cm
v =1 1 m/s
x =05
simulation E
O experiments Goldschmidt et al (2003) 0
5so. experiment 2,Goldschmldt et al (2003)
30 y^
20 ] h.n= 15cm
v=115m/s
S=0 5
h =15cm
v =1 15m/s
xsmL=0 25 O
0
 simulation
O experimentl,Goldschmidtetal (2003)
O experiment 2,Goldschmidt et al (2003)
h,=15cm E B
S=1 15m/s O
0
x ,i =0 5 O
0O
O
 simulation
O experiment ,Goldschmidt et al (2003)
O experiment 2,Goldschmidt et al (2003)
Figure 10.b Comparisons of the bed height of the small
particles
h =15cm
S=1 15m/s
E Xsr ,=0 25
\0
S0
O
O
simulation
O experiment ,Goldschmidt et al (2003)
O experiment 2,Goldschmidt et al (2003)
40 50
60
simulation
O experiment1,Goldschmidt et al (2003)
O experiment 2,Goldschmidt et al (2003)
j 0
o
0
B
h =15cm 0
O
v=1 15m/s
x =s05 0
s___________________
50 60
Paper No
Acknowledgements
The PhD fellowship (Chao, Z.) financed through the
GASSMAKS program (Advanced Reactor Modeling and
Simulation) and the PAFFrx project (ParticleFluid Flow
with Chemical Reaction Multi level models for design
and optimization of fluidized bed processes) of the
Norwegian Research Council are gratefully appreciated.
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