7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
QLEMMS and QLEMMSn Drag Model for Heterogeneous GasSolid Two Phase Flows in the
Frame of Eulerian Approach
Haiying QI, Fei Li, Jingsheng Wang and Changfu You
Key laboratory for Thermal Science & Power Engineering of Ministry of Education, Tsinghua University
Beijing 100084, P.R. China
hyqi~mail.tsinghua.edu.cn
Keywords: QLEMMS drag model, heterogeneous gassolid two phase flow, Eulerian approach, numerical simulation
Abstract
Some efforts to develop new drag models named QLEMMS and QLEMMSn by coupling the Eulerian approach and the
Energy Minimization MultiScale (EMMS) theon, are reported here. The main focus is on the improvement of existing EMMS
drag models by taking two major measures. The first measure is adding local particle accelerations in force balance of the
heterogeneous flow. The second measure is modifying further the cluster size distribution by inducing a correct factor based on
experimental data. The drag function B used in Eulerian momentum equation is continuously updated by an EMMS subroutine
and fed back to the Eulerian CFD iteration process to renew local flow parameters like gas and particle velocity as well as
particle volume fraction. The simulations for two cases of circulating fluidized bed (CFB) show a good performance of new
drag model and a better agreement with experiments than other models found in literature.
Introduction
As wellknown, drag models play a key role in Eulerian
modeling of dense heterogeneous gassolid twophase flow
with strong clustering effects in fluidized process to get
inner recirculation of particles under high superficial
velocity. The aim to develop new models is to realize an
accurate Eulerian simulation of the flow and enable
numerical description of other complex processes that take
place within the flow like mass and heat transfer,
combustion and chemical reactions (Hou, 2004; Li, 2009).
The main features of fluidized gassolid two phase flow are
cluster formation and frequent fragmentation caused by gas
phase as well as the consequent heterogeneous particle
distribution in concentration, which results in strong
turbulence and a great drag reduction. To characterize drag
behavior in dense heterogeneous flow by means of Eulerian
approach, suitable drag models are needed to close the
momentum equation.
However, the most of drag models found in literature such
as Gidaspow (1994), Wen & Yu (1966), Felice (1994) and
so on are in principle available for homogenous flow and
therefore don't fully success in the simulation (Qi, 1997).
An empirical model (OS model) presented by O'Brien &
Syamlal (1993) is much more capable to describe drag
reduction phenomenon in heterogeneous flow, but is not
universally extended due to limited test conditions. The
model includes some empirical coefficients that are only
valid at two certain values of solid mass flux Gs.
In the past decade, more efforts focus on theoretical
solutions to deal with drag issues in stead of experiments, in
which the Energy Minimization MultiScale Method
(EMMS, Li & Kwauk, 1994) shows its power to describe
fluidized flows. As an independent theory from the Eulerian
approach, EMMS focuses on mechanical energy changes
for suspension transportation in fluidization systems and
indicates that there exits the minimum of the transportation
energy when the system is in stable state. Flow parameters
can be also determined by EMMS method like Eulerian
approach. Therefore, it may be possible to use the energy
minimization as an additional condition to close Eulerian
conservative equation group, in other words, coupling
EMMS and Eulerian CFD process can probably get a new
drag relationship (Xiao, 2001). As a result, one tried to
present a new drag model in term of the coupling idea (Qi,
1997: Xiao et al. 2003). After that, further models based on
EMMS appear in literature (Yang et al. 2003; Wang & Li,
2007; Wang et al. 2008: Wang et al.2009; Hartge et al.2009).
According to the EMMS, a heterogeneous flow is divided
into three scales to signify the particle dense phase, particle
dilute phase and interaction phase that are approximately
regarded as homogeneous and the existing drag models
mentioned above could be still valid. Each phase has its
own flow variables like velocity, volume fraction and so on.
Then, 6 algebraic relations for 8 state variables of three
phases like force balance and continuum equation are
established (see below). State variables including drag could
be determined by solving these equations and searching the
energy minimum (Li & Kwauk, 1994).
Figure 1 shows plots of drag function B that is regarded as
the momentum exchange coefficient vs solid volume
fraction as given by some existing drag models. The dashed
line represents a continuous increasing of B with increasing
solid volume fraction in homogeneous suspensions
(Gidaspow, 1994). Many other models found in literature
give also a similar behavior. The OS model mentioned
above shows a drag reduction especially in the range of es
=0.01~0.35 for a given solid mass flux Gs. However, two
models based on EMMS (Yang et al., 2003; Wang et al.
2008, 2009) have a great deviation from the plot of OS
model with some unreasonable turning points in the same
region, because they includes a hypothesis on local force
balance for solid phase and some empirical factors.
06
105. Yang et al. (2003)
Gidaspow (1994)
O'Brien & Syamlal (1993)
103l, / /G =98 kg m2
121 Z Wang et al. (2008, 2~009)
10 
0.0 0.1 0.2 0.3 0.4 0.5
as []
Figure 1: Comparison of existing drag models
(Parameter: Ct,=1.81e5 Ns m2, p,=1.205 kg m3, d,=100
Cpm, ur= 1.0 m/s)
To overcome defects in above EMMS models, local particle
accelerations in force balance for heterogeneous
suspensions should be considered through analysis of forces
acting on particle phase and a new model is developed.
After verification by OS model and experimental data,
further modifications to the model are made by inducing a
correct factor for cluster size distribution.
Nomenclature
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
p density (kg m3
9L dynamic viscosity (Pa.s)
volume fraction ()
Subsripts
0 initial condition of simulation
cthe particle dense phase
f the particle dilute phase
g gas phase
i the interaction phase
k universal index, =g, s, r, c, f, i
max maximum
mf minimum fluidization
rrelative parameter of gas and solid phase
S solid phase
Deriving of QLEMMS drag model
1. Revision for existing EMMS model
Basic equations of a typical EMMS model (Li & Kwauk,
1994) as followed.
State variables:
1 = (se,,"gf,Uge ,Ug ,,Usc,U,, f, dal) (1)
Equation for force balance:
* for the particle dense phase
fnF +nqF = j1e jp~p,pg (2)
for the particle dilute phase
(1 f)n,F, = (1 f)(1E,)(p, p,)g (3)
static pressure drop
S(4)
(Ap Alk n~k, k = c, f, i
Continuum equations:
* Gas phase
U, = Ugf (1 f) + Ug,f (5)
Solid phase
Us s (1 f )+ Us J (1)
Cluster diameter dal:
1 6mxm U,,,+ 
de;= a ,a dp (2)
Ps mf U,
Ns, > min
pp 6g 6g (3)
N,, = gU g c2 1 ),
Ps 1 e
In above EMMD model it is assumed that the local drag is
equal to the gravity of particle. This simple force balance
results in solution troubles for drag. In fact, eq.(2) and (3)
are incorrect in a real fluidized flow, because of particle
acceleration. According to EMMS theory, the drag acting on
particles within the dense phase is caused by its inner gas
and the particle dilute phase. Similarly, the drag acting on
particles within the dilute phase is only from its inner gas.
Therefore, two new state variables, the particle acceleration
particle acceleration (m s2)
empirical factor in OS model ()
cluster diameter (m)
particle diameter (m)
volume fraction of the particle dense phase ()
clustering correct factor ()
correct factor for cluster diameter ()
drag force (N)
gravitational acceleration (m s2
solid mass flux (kgm2S1)
Initial bed height (m)
suspension transportation energy (J/kg)
pressure (kPa)
bed pressure drop (kPa)
Reynolds number based on particle diameter ()
total calculation time (s)
time step of iteration (s)
real velocity (m/s)
apparent velocity (m/s), Uk=Uk~k, k=g, s, r
superficial velocity of circulating fluidized bed
(m/s)
letters
drag function (Ns m4)
relative error (%)
a
C3
del
d,
f
fe
fa
F
g
Gs
ha
Nst
p
Apbed
Re,
teal
at
u
U
U o
Greek
13
8
0.5
0.
o 0.3
0.2
0.1
0 0
I I I I .
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
the energy minimization in terms of local flow variables Ug,
U, and as offered by Eulerian CFD module, and replacing
eq.(7) by the function de(e,). Then, all of state variables
including drag function 13 are obtained (see Figure 4).
in the particle dense phase a, and the dilute phase at are
induced in eq. (2) and (3), respectively, to replace particle
gravity g, then we have eq.(9) and (10).
fn~ Fy +n,?( = f(1 see P7 Ps) as (4)
(1f)n,F,=(1f)1s E)p,pZ)a, (5)
Combining these two equations and eq.(4) can deduce
eq.(11) to (13) that determine drag force in three phases Fc,
Ff and F,.
ncE = A~ pg ) gc b~q+l)( t gf f (6)
nF, =( 1eP p )a, (7)
n,~(/7 = p)(1f) f( 1Ege)acr 1 Ean (8)
Besides, an equation that describes the relationship between
the local average voidage e, and variables of the particle
dense phase and the dilute phase should be added in the
equation group with eq.(5) to (8) and eq.(11) to (14). Then,
drag function is able to be calculated by eq.(15). Thus, a
new drag model named QLEMMS is obtained, in which no
empirical factors are introduced.
Eg = fEg +(1 f)Eg, (9)
""
0.006
0.005
0.004
0.003
0.002
0.001
s~m
. 0.000
0.4 0.5
0.0
0.1 0.2 0.3
s, []
Figure 3: Relationship between cluster parameter ee, del
and as given by QLEMMS
(Parameters: d,=54 pLm, p,=930 kg/m3, p,=1.205 kg/m3,
pg=1.815e5 Pa s, mEf0.5, Ugn=1.52 m/s, G,=14.3 kg/m s)
l o6
25 104 E
20 lo3 Z
is lo02 CO
/ lo 0.4o
'7 5 0.2 0.3
oo0 0.1 ,
Figure 4: Drag function profile mn inhomogeneous
suspensions (Parameters: same as in Figure 3)
It can be seen in Figure 3 that particle volume fraction in the
dense phase e,, similar to the density of a cluster or a big
particle increases monotonously with increasing as until it
reaches the value Es,me at the minimum fluidization. In fact, it
is assumed in EMMS theory that as, must be less than or
equal to es,me (Li & Kwauk, 1994)
Figure 3 shows that cluster diameter delis about 1.85.8 mm
and larger than particle size several times to hundred times.
Furthermore, del increases to the maximum at e,=0.365 and
turns to decline. The turning point just corresponds to the
position where as, becomes constant. Nevertheless,
according to the EMMS theory, del should not turn down,
while continues to be infinite. But this is physically
impossible. An explanation to this is that a cluster could not
contain more particles to further enlarge itself with
increasing as after E,,'E,~me. In this case, it is only possible
that clusters must become many smaller one through
fragmentation to meet the assume of E,,E48,me (Wang & Li,
2007).
As shown in Figure 4, the heterogeneous drag function
depends on not only particle volume fraction e,, but also
relative velocity U, of gas and solid phase, and forms a
curved surface.
P= A's +(1
11
f) )nF, + nl,
2. Coupling of QLEMMS model with Eulerian
approach
As known in above analysis, the drag depends on the
operation parameters of a fluidization system, local flow
parameters such as gas and particle velocities as well as the
energy minimization. Therefore, to realize the coupling
QLEMMS model is implemented as a subroutine in the
Eulerian CFD iteration process. The later offers new local
flow parameters as initial input conditions to the former and
at the same time receives continuously from it the updated
drag function. So the whole solving procedure consists of an
Eulerian CFD module and an EMMS analysis module, as
shown in Figure 2.
Local tlow parameters
Iteration calculation E M nls od e
1st step
Eulerian
CFD Module (U,,,, Gs) + mln (Nst)t + s,(s), d (~)
I,:b". I2nd step
closure of (u,, u,, ,) + mmn (Nst)
conservatrtm equation +saevrals+B
Drag function P
Figure 2: Schema of the twostep solving procedure by
CFD/EMMS coupling method
Because a traversal algorithm by means of computers has to
be used for searching the energy minimum, a twostep
analysis method in EMMS module is suggested (Wang & Li,
2008) to avoid those time consuming processes.
The first step is to introduce the system operation
parameters superficial velocity Usn and solid mass flux Gs
 plus the energy minimization to determine the functional
relations of clustering parameter E,, and del with the local
average solid volume fraction as in a mesh cell. The results
are shown in Figure 3.
The second step is to solve EMMS equations together with
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
are treated as 2D models and experimental data are taken
from Li (1987) and Ouyang et al. (1995), respectively.
As seen in Figure 6, both reactors have two outlet and two
inlet for beck feeding of circulating bed material.
Fluidization gas enters the riser from the distributor at
bottom. The riser diameter in Case 2 is two times larger than
that in Case 1 and the operation conditions are also quite
different as seen in Table 2.
Usually, two parameters are needed to determine the reactor
state: superficial velocity Ugn, solid mass flux Gs or bed
inventory that directly influence the bed pressure drop.
Except Ugn, one of other two parameters needs to be
specified as initial condition and another one can be
calculated by the simulation (Qi, 1997: Xiao, 2001; Yang et
al.2003; Wang & Li, 2007). In this paper Gs is selected for
comparison with experiments. Moreover, the amount of
circulating particles is kept consistent with the flux at the
outlet.
90
o outlet
9.... .utlet 'r
S254
L$ Y
Gas Inlet Gas Inlet
(a) Case 1 (b) Case 2
Figure 6: Geometric models of two CFB reactor
Table 1: Calculation conditions used in two cases
Parameters Case 1 Case 2
Particle diameter d, (Cpm) 54 65
Particle density p, (kg m ) 930 1380
Gas density p, (kg m ) 1.205 1.212
Gas viscosity p (Pas) 1.815e5 1.908e5
Maximal solid volume fraction 06
es~max (
Maximal voidage smax () 0.9997
Initial bed density esn () 0.5
Initial bed height hn (m) 1.17, 1.6 1.782
Superficial velocity Ugn (m s ) 1.52 3.8
Solid mass flux G, (kg m~s ) 14.3 106
Bed pressure drop Apbed (kPa) 5.34, 7.3 12.6
Number of mesh cells 30 x150 30 x154
Time step of iteration at (s) 5e4
Total calculation time teal (s) 70 40
Figure 7 compares Gs calculated from the homogeneous
WenYu model (Wen & Yu, 1966) and heterogeneous
QLEMMS drag model. Gs fluctuate strongly at the
beginning, but tend to be stable after a certain time (10 s).
Both Figure 7 and Table 2 show that results from
QLEMMS model agree with experiments much better than
WenYu model. This implies that the homogeneous model is
totally not applicable.
Table 2 shows quantitative comparison between two models.
The relative error 8 from QLEMMS is 10~25%, while
WenYu model has 8 of 400~600%. The comparison
Results and Discussion
1. Verification of QLEMMS model
It is necessary to evaluate the accuracy and performance of
the model through experimental data. In our point of view,
there should be two ways to verify the model: one is the
direct way and another one indirect.
The direct verification means comparison with OS model
under same conditions. This is the most important and
effective step before the simulation process (Qi, 1997:
O'Brien & Syamlal, 1993: Tanner 1994), because through
this step one can make sure if the model is qualitatively and
quantitatively accurate and reasonable, i.e. if it has a good
agreement with OS model and a same change tendency vs
solid mass flux Gs. Furthermore, one can even predict the
quality of subsequent simulations for fluidized flows
through observing the deviation from OS model and clarify
the mechanism.
The indirect verification is to implement the new model in
simulations and compare the results with experimental data
measured from the same test facilities. If the model shows
its power to get good agreements with experiments in any
cases, it can be regarded as a universally applicable model.
(1) Direct verification
As shown in Figure 5, in one hand, (1) QLEMMS model
reflects drag reduction caused by clustering: (2) It has no
difference from OS model both at very lean region (E,00)
and dense region (Ede,mEf); (3) It has a same tendency with
solid mass flux Gs. In other hand, there are still obvious
differences both in shape and order of magnitude: the drag
reduction in the range of e,=0~0.3 is smaller, while much
larger than OS model where e,>0.3.
106
1051 Gidaspow (1994) .._,"
 14.3
E 1041 . 2.64
,' 147
2cc 103J~~ QLEMMs
12~ OS Model (1993)
G = 147 kgm s
10
0.0 0.1 0.2 0.3 0.4 0.5
Es [
Figure 5: Comparison of QLEMMS with OS model
(Parameters: u,=1.0 ms', the rest is same as in Figure 3)
Anyhow, QLEMMS model originates from theoretical
analysis and predicts reasonably the drag reduction in
heterogeneous twophase flows, which is a great progress in
comparison with other homogenous models. It's also more
reasonable than other EMMS models shown in Figure 1. It
can be foreseen that simulations for heterogeneous flows in
circulating fluidized beds by using QLEMMS model will
approach the practical situation. Of cause, further
improvements are expected.
(2) Indirect verification
To further verify the performance of the new model, two
CFB reactors as two cases with fully different conditions are
selected for Eulerian simulation. The geometric models that
QLEMMS mo
0 10 20 30 40 50 60 70
t [s]
Figure 7: Solid mass flux Gs calculated by WenYu model
and QLEMMS model '
(Condition: Case 1, Apbed=7.3 kPa)
Table 2: Comparison of simulation results given by
diffrent drag models with experimental data
Apbed Gs 8
Drag mdl (kP~a) (kg m2S1) o
Experimental data 14.3
QLEMMS 5.4 12.53 12.4
WenYu 70.46 392.7
SQLEMMS 7.017.95 25.5
WenYu 101.76 611.6
Experimental data 106
SQLEMMS 133 25.5
EMMS/matrix
U 198 86.6
(Wang & Li 2007)
Gidaspow (1994) 444 319
Table 2 further compares the results in Case 2 by using
homogeneous Gidaspow model (Gidaspow 1994) and
heterogeneous EMMS/matrix model reported in the work of
Wang & Li, 2007 and Dong et al. (2008). The precision of
QLEMMS is the highest and the heterogeneous drag
models have better performance than homogeneous models.
Figure 8 shows the axial distribution of solid concentration
that also agrees with experiments. The same trendcy at the
dense bottom and dilute upper part of the riser is well
predicted. The calculated value is lower than the
experimental value, that surely because the drag force
predict by the model is too high in the corresponding solid
concentration range (seen Figure 5), which leads the solid
concentration approach to uniform distribution along the
height. The homogeneous Gidaspow drag model predicts a
nearly total uniform distribution in whole bed (Dong et al.,
2008), because it does not take in account of clustering
effcts, so that no internal circulation is given.
QLEMMSn drag model further improvement
The effcts of particle cluster contribute most to the drag
force (Li, 2009) according to the previous investigation.
And the most important character of OS model is taking in
the clustering effects. So the precision of QLEMMS model
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
can probably be improved by introducing the OS model to
correct the del expression.
indicates that QLEMMS model describes the
heterogeneous gassolid twophase interaction well, with
high prediction precision and very good applicability to
different conditions.
d
ile
0.00 0.05 0.10 0.15 0.20 0.25
Figure 8: Comparison of axial distribution of solid
volume fraction with experimental data
(Conditions: Case 2, QLEMMS model)
In OS model, clustering correct factor f, (O'Brien &
Syamlam, 1993) is the key parameter to represent drag
reduction (the second term in eq. (16)). If f, is 0, then the
drag model reduces to a homogeneous model. C3 cnf iS an
empirical factor that is determined by experiments, but fal
not. As seen in Figure 9, fel has a single peak distribution in
the range e,=0~0.3, which just causes the drag reduction in
OS model (Figure 5). While in the dilute and dense region,
fai90. This implies that the drag tends to homogeneous
condition.
0.6
~0.4
0.2
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6
,[1
Figure 9: Profile of correct factor fax
For the del distribution (Figure 3) in QLEMMS model, the
turning point locates out of e,>0.3, which is hard to be
explained physically. If the location of the del turning point
can be modified by fel, then the drag distribution can be
ultimately modified.
In this paper, fel that is normalized by fel~,ma is used as a
correction factor to multiply del in eq. (2). Since clustering
effcts disappear in dilute and dense region, drag force
approaches the homogeneous condition. Taking in condition
of these effcts, del is set to d, when E,00 and 98s,m,f then
we get eq.(17) that modifies the del distribution substantially
as shown in Figure 10
f, = 1+ C3 1 '
for = Rey t, exp 0.005 Re 5)2 90 eB 0.92)2]
WenYu mo
Exp riment
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The simulation results agrees qualitatively well with the
experiment, but there is still a error lies between
10~25%.
4) QLEMMSn model combines the reasonable core of
OS model and QLEMMS model, so the predicted
drag force accord with the physical reality much better.
This model needs further numerical verification.
Acknowledgements
This work is supported by the National Basic Research
Program of China (973 Program) No. 2006CB200305.
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Figure 10 shows that, the distribution of the corrected
cluster diameter del is more reasonable than the origin model.
And that greatly improve the drag curve, despite there is still
some difference in the quantity, the corrected drag possesses
all the major characters of OS model (Figure 11). The new
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120
100( QLEMMS, 's
80
0 60
Figure 10: Cluster diameter profiles before and after
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102 ,
0.0 0.1
0.2 0.3 0.4 0.5
Figure 11: QLEMMSn drag model
Conclusions
1) Heterogeneous QLEMMS drag model has been
proposed. The model combines the Energy
Minimization MultiScale theory (EMMS) and the
Eulerian CFD approach. The local particles force
balance is improved by introducing particle
acceleration.
2) Compared with other EMMS models, QLEMMS
model is totally deduced based on theoretical analysis
without any assumption. In comparison with OS model
that summarized from numerous experimental data, the
present model not only describes the drag reduction due
to clustering effects in heterogeneous flows, but also
autoadaptive to different fluidization conditions.
3) Flows in two CFB reactors with different conditions are
successfully simulated by using QLEMMS drag model.
The results imply that the model has a good universe
applicability since there are no empirical parameters.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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