Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 7.5.1 - Developing particle-laden turbulent pipe-flow
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 Material Information
Title: 7.5.1 - Developing particle-laden turbulent pipe-flow Particle-Laden Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Strömgren, T.
Brethouwer, G.
Amberg, G.
Johansson , A.V.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: particle-laden gas
turbulent flow
pipe-flow
modeling
 Notes
Abstract: An Eulerian turbulent two phase flow model using kinetic theory of granular flows for the particle phase was developed in order to study evolving upward turbulent gas particle pipe flow. The model takes into account the feedback of the particles. Simulations show that the pipe length required for particle laden turbulent flow to become fully developed is up to five times longer than an unladen flow. To increase the understanding of the dependence of the development length on particle diameter a simple model for the expected development length was derived. It shows that the development length becomes shorter for increasing particle diameters, which agrees with simulations. The initial slip velocity has also a large influence on the development length.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00186
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 751-Stromgren-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Developing particle-laden turbulent pipe-flow


T. Str6mgren, G. Brethouwer, G. Amberg and A.V Johansson

Linn6 Flow Centre, Department of Mechanics, KTH, SE-100 44 Stockholm, Sweden
tobias@mech.kth.se, geert@mech.kth.se, gustava@mech.kth.se and johansson@mech.kth.se
Keywords: particle-laden gas, turbulent flow, pipe-flow, modelling




Abstract

An Eulerian turbulent two phase flow model using kinetic theory of granular flows for the particle phase was
developed in order to study evolving upward turbulent gas particle pipe flow. The model takes into account the
feedback of the particles. Simulations show that the pipe length required for particle laden turbulent flow to become
fully developed is up to five times longer than an unladen flow. To increase the understanding of the dependence
of the development length on particle diameter a simple model for the expected development length was derived. It
shows that the development length becomes shorter for increasing particle diameters, which agrees with simulations.
The initial slip velocity has also a large influence on the development length.


Introduction

The objective of this paper is to study the development of
turbulent gas-particle flow in a pipe. Studies of evolving
turbulent gas-particle pipe flows are of significant inter-
est since in many industrial processes the turbulent par-
ticle laden flows do not become fully developed. How-
ever, there are only a limited number of studies of these
flows. Picano et al. (2009) studied the spatial develop-
ment of particle concentration in a pipe using direct nu-
merical simulation (DNS) and found that the develop-
ment length increased with the Stokes numbers of the
particles. DNS by Portela et al. (2002) showed that a
very long developing length (~ 300 pipe diameters ) is
required in order to get statistically steady particle con-
centrations. Str6mgren et al. (2009a) also found, using
model simulations that particles substantially increase
the pipe length required for the flow to become fully
developed. Cerbelli et al. (2001) analysed the transient
behaviour of the particle concentration in pipe flow and
developed a model that better captured the turbophoretic
drift of particles.
In this study a two-phase flow model with particle
stresses based on kinetic theory of granular flow is im-
plemented in order to study developing turbulent gas-
particle flows for different particle Stokes numbers.

Conservation Equations

First, the averaged equations governing turbulent gas-
particle flows are presented (Anderson & Jackson 1967).


The generalised volume fraction weighted averaged con-
tinuity equation can be written as


at ai
a(Ph )+h a c(phhU ,h) 0 (1)

Subscript k refers to the phase (k g for the gas phase
and k p for the particle phase), p, is the mean density,
+< is the mean volume fraction, Uk,i uTti is the mean
velocity and uh,t is the instantaneous velocity. Global
continuity requires b, + ~p 1. The mean momentum
equation is written as


a (PkhkU,i) + (pkhkUk,iUk,j)
a ) +
(9Jh,ij) + Iht + tPhti
Oxj


Pk
Ox,


(2)


Here P, is the gas phase pressure present in both phases,
HI,ij is the stress term representing the viscous and tur-
bulent stress in the gas phase and the kinematic and col-
lisional stress for the particle stress. Iki is the Stokes
drag and is modelled according to

Ip, = -19 = (1 + 0.15Rc -687) P ,, (3)
Tp

where ,T = pP is the particle response time, Vg is the
gas phase cinematic viscosity and Dp is the particle di-
ameter. Rer u D is the relative Reynolds number,
Ur,i (Up,, U9,i) Ud,i is the mean relative velocity











and Ud is the average of the fluid velocity fluctuations
with respect to the particle distribution (Simonin et al.
1993), and is modelled according to


U 1 ts { 1 K,
U ^, (^;
ud'^ S p M~c


1, D4' N


where 7~g is the correlation time scale of the fluid tur-
bulence viewed by the particles and Kp .' u =
2K93T (Hadinoto & Curtis 2004) is the gas-particle
velocity correlation, K = u'u'i and T = ..' .
2 Yi Yi 3
is the gas phase mean turbulent kinetic energy and the
granular temperature, respectively.
Gas phase. Viscous stresses in the gas phase are given
by


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


is the radial distribution function where axa = 0.64
is the maximum value of bp for random packing of
spheres.


4 T
A1P 3 Dp1 pppgo(1 + ep)J


col = o(1+ ep) (in +D(3) 2), (10)


kin 2 Tp, 7 2 c 1
P 1\ P39 P -1 g
(11)
where T ..' is the granular temperature and
t^ Y Y(1 + C 12 is the time-scale of the fluid
turbulence viewed by the particles where


where Vg is the kinematic viscosity of the gas, Sk,i,
(1 u + ` ) is the mean strain rate tensor and
last term is the Reynolds stress which is closed using
Boussinesq assumption


U',/ ,j = -2vtgS,ij + (Kg + vt ) )ij.
Yx3 0m

Here vt, is the turbulent viscosity. To close this term
low Reynolds number K w model by Wilcox (1S
is used with an additional term due to the interaction
tween the gas and the particle phase both in the K9
the w equation, see Str6mgren et al. (2009a). w is
inverse time scale of the gas phase turbulence.
Particle phase. In order to describe the stress:
the particle phase kinetic theory of granular flow
adopted. This model has earlier been used by
example, Peirano & Leckner (1998), Benyahia el
(2005), Benavides & van Wachem (2008), Zhant
Reese (2001) and Str6mgren et al. (2009b). Follow
this approach the stress due to particle-particle collisi
is written as

Ip,ij p ((Pp pSp,kk)6ij

2pp(V7o1l i+ )(Sp,ij 1 )m )

where A, is the is the bulk viscosity, Pp T(
240pgo(l + ep)) is the granular pressure, Vkin is
kinematic viscosity, Vp01 is the viscosity due to parti
particle collisions, vkin is the particle viscosity du
kinematic stresses, e, is the coefficient of restitution

1
go= 1+( )1/3


C 1.8 1.35 Upi
SUriI lUp


31U,2 '


T ~) is the Eulerian time-scale of the gas phase tur-
bulence,
Pco1 Dp 12 (14)
S 244pgoT (T

is the inter particle collision time, c = (1 + ep)(3ep -
1) and a, (1 + ep)(3 ep).


- (3 p, CD -1.7 r)-1
P9 4pp Dp


(1 + 0.15Re0.687)


ing is the characteristic time-scale of gas-particle momen-
ions tum transfer where CD 2 (1 + 0.15Re 687) is the
drag coefficient.
The equation for the granular temperature can be de-
rived from the momentum equation of the particle phase.
(7) In order to close the unclosed terms the kinetic theory of
granular flows is used. The equation for T becomes

3 DT 3 OT
1 2+ PP 2p(Ii 2p j3PJpU
the
cle- 2pp9,p( >in .S, pp T
e to p (16)
and + 0 3 p(, + ,. )T


(8) (1 + 0.15Re, 687 ( -3T)
TP


'II, = -p-Ya (,IqS,ii
09x


-D 1' .1 g ) (5)











where


(3^^ + ,o 3(l (1 e+) (2e
TK ,T go5(1 +eT) (2ev

9 +


is the kinetic contribution to the diffusivity,

Pg 1o( +e ')()(6 +4Dp(Tl/)1/2), (18)

is the collisional contribution to the diffusivity and p =
10(1 + ep)(49 33ep).

Simulations of evolving gas-particle flow in a
pipe

Simulation setup. An upward turbulent particle laden
flow in a vertical pipe has been simulated. The pipe was
200 D long, which was long enough to achieve a fully
developed flow. The boundary conditions developed by
Johnson & Jackson (1987) were used for the particle
velocity and the granular temperature. The gas phase
velocity has a no-slip condition at the wall whereas the
particle volume fraction and the pressure have a Neu-
mann condition at the wall. There is no flux of particles
in the wall-normal direction. The particle volume frac-
tion was initially homogeneous with o = 5 10-4 and
pp/p, 2000. The inlet conditions were constant for
gas- and particle velocities, K., c and +p. At the outlet
only the pressure has a Dirichlet condition. Two dif-
ferent initial conditions for the slip velocity were used
to investigate its effect on the development length. The
model described above was implemented in cylindrical
coordinates in a finite element code (Do-Quang et al.
2007).
Simulations were made for Stokes numbers between
St+ 34 and 3950. The Stokes number is defined as


where u, = T lpis the friction velocity and T, is
the wall shear stress. The Reynolds number, Re D= u
is 24000 where D is the pipe diameter. An unladen sim-
ulation for this Reynolds number was also performed.
The simulations were run until the flow was steady.
The model has been validated with several sets of
experimental and direct numerical simulation data in
Str6mgren et al. (2009a) and Str6mgren et al. (2009b).
It is shown that the model predicts the gas- and particle
mean- and rms velocities very well.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


(a)



08

)6

)4

2 x/D=5
--x/D=25
x/D 50
x/D-150
0 02 04 06 08 1


x/D=5
x/D=25
x/D=50


Figure 1: Mean gas phase velocity profiles normalised
with the gas phase center-line velocity at four different
downstream positions. St+ 34 (a); St+ 1775 (b).


Model results. In order to understand the axial devel-
opment of velocity and particle concentration profiles at
different downstream positions in the pipe are compared
for St = 34 and 1775. Here x is the downstream dis-
tance from the inlet of the pipe.
In figure 1 the gas phase velocity profiles normalised
with the gas phase center-line velocity (Ug,c1) at differ-
ent downstream positions are shown for St+ 34 and
St+ 1775, respectively. For St+ 1775 (figure lb)
the gas phase velocity profile is almost fully developed
at x/D=20 while a much longer development length is
required for St+ 34(figure la).
The mean particle velocity profile normalised with
U,,c1 at different downstream positions is shown in fig-
ure 2 for St+ 34 and St+ 1775, respectively. For
St = 34(figure 2a) the particle center-line velocity is
nearly equal to the gas velocity at all positions but the
velocity at the wall decreases with x/D, i.e. the profile
becomes less flat. At larger St+ (figure 2b) the parti-
cle phase velocity profile is uniform at all downstream








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


x/D=5
x/D 25
x/D-50
x/D 15J


14
12


08
06


04 06 08


02 04 06 08
r/R
(b)


-x/D5
x/D 25
x/D-50
x/D-150


14
12


08


02 x/D=5
x/D=25
x/D=50
x/D=150
0 02
0 02


04 06 08
r/R


Figure 2: Mean particle phase velocity profiles nor-
malised with the gas phase center-line velocity at four
different downstream positions. St 34 (a); St
1775 (b).



positions whereas the velocity decreases with increas-
ing x/D. The final velocity at the wall is about the same
for both cases. The more uniform profile at large Stokes
numbers is due to a weaker coupling between the phases.
Since the wall is smooth the particles experience almost
a free slip condition.
Profiles of the mean particle volume fraction, %p, nor-
malised with its initial value, Go, are shown at four
downstream positions in figure 3a and b for St+ 34
and St 1775, respectively. For St = 34(figure 3a)
the particles accumulate in the center of the pipe and in-
creases with x/D. The particle concentration in the cen-
ter of the pipe is almost three times higher than near the
wall when the flow is fully developed. A higher parti-
cle concentration around the pipe axis was also found in
experiments by Tanaka & Tsuji (1991).
In figure 4 the slip velocity (Up U,) normalised with
the center-line velocity is shown at four different down-
stream positions for St+ 1775 and 3950, respectively.


02 04 06 08
r/R


Figure 3: Mean particle concentration profiles nor-
malised with the center-line velocity at four different
downstream positions. St+ 34 (a); St+ 1775 (b).



The development length is seen to increase with increas-
ing St In order to investigate the effect of slip velocity
the same case as in figure 4 is shown in figure 5 with the
difference that the inlet slip velocity is put equal to the
final slip velocity at the outlet in order to neutralise the
effect acceleration or deceleration of the particles on the
development length. The development length is found
to be significantly shorter when the inlet is adjusted to
match the slip velocity for a fully developed flow. Inlet
slip velocity has thus a large influence on the develop-
ment length.
Development length. The influence of the particle di-
ameter and mass loading on the length required for the
flow to become fully developed will be studied in more
detail. First an estimation of the development length
of the particle phase is made by studying the govern-
ing equations in order to understand which parameters
govern this length. In a fully developed flow the convec-
tive term (second term on the L.H.S.) must be smaller
than the most dominating term in equation (2), which is


08


02 x/D=5
x/D=25
x/D=50
x/D=150
0 02
0 02













x/D=5
x/D=25
0 x/D=50
x/D=150


02 04 06 08 1


x/D=5
x/D=25
0 8 x/D50
x/D=150
06-

04-

02-


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


(a)
x/D=5
--x/D25
0 8 x/D=50
x/D=150


x/D=5
x/D=25
0 8 x/D=50
x/D=150
06


02 04 06 08
r/R


Figure 4: The slip velocity normalised with the the gas
phase center-line velocity at four different downstream
positions. St 1775 (a); St 3950 (b).


the diffusion term (second term on the R.H.S.). Since
vP1 < kin for dilute flows (ip < 1 10-3) this con-
dition becomes

Ups 8 J 8Upi 8Up 0
Up3 < kivp ( ,+ O)U (20)


The derivatives in the convective and the diffusive term
are estimated as U/1 and U/D2, respectively. Here U is
the characteristic mean streamwise velocity and I is the
axial development length. Equation (20) then yields

U2 'in U
< D (21)

After rewriting, the development length can be ex-
pressed as
1 DU
D > in (22)
u v^


Since


kin T )( 21
S- Kpg ` + T) ( + ) 0
3 T 3 pf T,-p + Ipco


Figure 5: The slip velocity normalised with the the gas
phase center-line velocity at four different downstream
positions. St+ 1775 (a); St+ 3950 (b).


In the limit of small Stokes numbers (St+ << 1) and
small (Dp (< 1 10 4) the time between particle-particle
collisions, Tp01 (eqn. 14), will be larger than T7 (eqn.
15) since T ~ D2 and Teo1 ~ Dp. For the same reason
Ti-g/- >> 1. Furthermore, K,, 2K 3T and
Tpg T. since U,. 0. With these simplifications vkin
can be estimated as


kin 2 Tt Tf
S -" = -2K
Vp 3 f1 2


2K 3
3 2w


Vta (24)


i.e. the kinematic viscosity of the particles approaches
the turbulent viscosity of the gas phase, which is in ac-
cordance with Tchen-Hinze's theory (Hinze 1959). The
development length is then the same as for single phase
flow, i.e.
1 DU
S> -(25)
D Vt9


) For large Stokes numbers (St+ >> 1) Tpg/7T < 1
23) and since has the same order of magnitude as T
and since K, has the same order of magnitude as T







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


equation (23) can be written as

-T( )1 (26)
Tpg Ip

Since ac and 1 + Re 687 in equation (15) are of order
unity we get


140
U
U
120

100

80


+p 1-+1 18pgg 24tp IF
P ppD D- V prD

An estimate of the development length for gas-pal
flows when St+ >> 1 and particle-particle colli,
are not important (p < 1 10 3) is thus

I DU 18pyg 241)p T
D T ppD D, 7
V P


I..... ,0oo


(28) Figure 6: Development length, x/D,
St+ for U, Up and bp.


as a function of


Consequently, the development length becomes shorter
for increasing Dp and longer for increasing bp.
When 4p > 1 10-3 particle-particle collisions will
have a large impact on the flow and Tp7, < 7,7. The
development length can then be estimated as


1 DU
D TT0_1


DU 24p T
T D, w


Also here the development length becomes shorter for
increasing Dp and longer for increasing %p. However,
this parameter range is outside the scope of this study.
An estimation as above can also be made using the
equation for the mean granular temperature (eqn. 16),
but, similar results for the development length will be
found.
Comparison to model results. The estimation of the
development length in equation (28) applies for the pa-
rameter range used in this study and will be compared to
numerical simulations.
Using equation (28) the development lengths for U.,
Up and bp are shown as a function of St+ in figure 6.
The development length is here defined as the distance
from the inlet where the maximum difference between
the radial profile and the one at 180 D is less than 1%.
For St+ < 500 the development length decreases for in-
creasing Stokes numbers (Dp) for all three variables in
agreement with the estimation. The development length
for U. keeps on decreasing until it reaches the devel-
opment length of an unladen flow (x/D=21) because the
coupling between the two phases weakens. In contrast,
the development length of Up and bp start to increase
with St+ for St+ > 500. For St+ = 3950 it is about
five times that of an unladen flow. However, according
to equation (28) the development length for the particle
phase should decrease with St+. This discrepancy for
large Stokes numbers is likely related to the increasing
slip velocity with particle diameter.


S ...... ..... .. oo
st+

Figure 7: Development length, x/D, as a function of
St+ for U, Up and bp.


To examine the influence of the slip velocity all cases
in figure 6 are recalculated with inlet slip velocities ad-
justed to match the magnitude of the fully developed slip
velocities. The results are shown together with the es-
timation of the development length in equation (28) in
figure 7. The bulk values are chosen for the velocity
and granular temperature in equation (28). Compared
to figure 6 where the initial slip velocity is zero the de-
velopment length is much shorter for St+ > 500. With
these initial conditions the development length decreases
even for large Stokes numbers (Dp) which agrees qual-
itatively and quantitatively with the derived estimation
for the development length in equation (28).


Conclusions

A two-fluid model using the kinetic theory of granu-
lar flow for the particle phase was developed to study


Equation (28)


. . I .... I .. .











an evolving upward particle laden turbulent pipe flow.
We have investigated the influence of the Stokes num-
ber on the pipe length required for the flow to become
fully developed. It was found that the particles can make
the development length for the gas velocity up to three
times longer than that of an unladen flow whereas the
development length of the particle velocity can be up to
five times longer than that of the gas velocity of an un-
laden flow. To understand what governs the development
length a simple estimation for this length was derived
from the particle momentum equation. This estimation
showed that the development length decreases with in-
creasing St This was confirmed by model simulations
for St+ < 500. For larger St+ the development length
is very sensitive to the initial slip velocity between the
phases. When the initial slip velocity is close to the fi-
nal slip velocity the development length is much shorter
than for cases with a zero initial slip velocity.
The development of the particle concentration profiles
were also studied and a net flow of particles to the center
of the pipe was observed. This drift was largest for small
particle diameters.


References

Anderson T and Jackson R., S fluid mechanical descrip-
tion of fluidized beds, Ind. Eng. Chem. Fundamentals,
vol. 6, pp. 527-539, 1967

Benavides A. and van Wachem B., Numerical sim-
ulation and validation of dilute turbulent gas-particle
flow with inelastic collisions and turbulence modulation,
Powder Technology, vol. 24, pp. 294-306, 2008

Benyahia S., Syamlal M. and O'Brien T, Evaluation
of boundary conditions used to model dilute turbulent
gas/solid flows in a pipe, Powder Technology, vol. 156,
pp. 62-72, 2005

Cerbelli S., Giusti A. and Soldati A., ADE approach to
predicting dispersion of heavy particles in wall-bounded
turbulence, Int. J. Multiphase Flow, vol. 27, pp. 1861-
1879, 2001

Do-Quang M., Villanueva W., Amberg G. and Logi-
nova I., Parallel adaptive computation of some time-
dependent materials-related microstructural problems,
Bulletin of Polish Acad. Sci., vol. 55, pp. 229-237, 2007

Hadinoto K. and Curtis J.S., Effect of interstitial fluid on
particle-particle interactions in kinetic theory approach
of dilute turbulent fluid-particle flow, Ind. Eng. Chem.
Res., vol. 43, pp. 3604-3615, 2004

Hinze J., Turbulence, McGraw-Hill, 1959


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Johnson P. and Jackson R., Frictional-collisional consti-
tutive relations for granular materials with application to
plane shearing, J. Fluid Mech., vol. 176, pp. 67-93, 1987

Peirano E. and Leckner B., Fundamentals of turbulent
gas-solid flows applied to circulating fluidized bed com-
bustion, Prog. Energy Combust. Sci., vol. 24, pp. 259-
296, 1998

Picano F., Sardina G. and Casciola C., Spatial develop-
ment of particle-laden turbulent pipe flow, Phys. Fluids,
Vol. 21,093305, 2009

Portela L., Cota P and Oliemans R., Numerical study
of the near-wall behaviour of particles in turbulent pipe
flows, vol. 125, pp. 149-157, 2002

Simonin O., Deutsch E. and Minier J., Eulerian predic-
tion of the fluid/particle correlated motion in turbulent
two-phase flows, Applied Scientific Res., vol. 51, pp.
275-283, 1993

Str6mgren T, Brethouwer G., Amberg G. and Johans-
son A.V, A modelling study of evolving particle-laden
turbulent pipe-flow, submitted to Flow, Turbulence and
Combustion, 2009

Str6mgren T., Brethouwer G., Amberg G. and Johansson
A.V, Modelling of turbulent gas-particle flows with fo-
cus on two-way coupling effects on turbophoresis, sub-
mitted to AIChE J., 2008

Tanaka T. and Tsuji Y, Numerical simulation of gas-
solid two-phase flow in a vertical pipe: On the effect
of inter-particle collision, 4th Symposium on Gas-Solid
Flows, ASME FED, vol. 121, pp. 123-128, 1991

Wilcox D.C., Turbulence Modelling for CFD, DCW In-
dustries, Inc. La Cafiada, California, 1993

Zhang Y. and Reese J.M., Particle turbulence interac-
tions in a kinetic theory approach to granular flows, Int.
J. Multiphase Flows, vol. 27, pp. 1945-1964, 2001




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