Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Turbulent Heat Transfer of GasSolid Flow in a Horizontal Channel
Mahdi Abkar a,b, Zohreh Mansoori b, Majid SaffarAwal a and Goodarz Ahmadi c
a Amirkabir University of Technology, Department of Mechanical Engineering, Energy and Control Center of Excellence,
Tehran, 158754413, Iran
b Amirkabir University of Technology, Energy Research Center, Tehran, 158754413, Iran
Clarkson University, Department of Mechanical and Aeronautical Engineering, Potsdam, NY 136995725, USA
m.abkar@aut.ac.ir, z.mansoori @aut.ac.ir, mavval@aut.ac.ir, ahmadi@clarkson.edu
Keywords: Two phase heat transfer, Gasparticle flow; EulerianLagrangian approach; Turbulence modulation
Abstract
Turbulent heat transfer of gassolid flow in a horizontal channel is simulated numerically using EulerianLagrangian approach.
Low Reynolds Number k1 model and a turbulent Prandtl number model are used for studying the fluid phase. A novel
approach based on a sourceterm formulation derived by Crowe (2000) is presented that can predict turbulence augmentation
due to large particles in the core of the channel. The particle trajectories and velocities are determined by integrating the
particle equations of motion. Particleparticle and particlewall collisions are simulated based on deterministic approach and
coupling terms representing the fluidparticle interactions are also taken into account. In this study, dilute or moderate
suspensions (loading ratio up to 3) are considered. The predicted fluid and particles mean velocity and solid volume fraction
profiles are in good agreement with the available experimental data by Tsuji et al. (1987) involves large particles with mean
diameter 1 mm. Additional numerical results such as the turbulent intensity, the suspension Nusselt number, the wall
temperature and wall heat flux (in constant wall heat flux and wall temperature boundary conditions respectively) are also
investigated for different values of loading ratio by means of a complete fourway coupling description. The results show large
particles (in comparison with integral length scale) augment turbulence in the core of the channel and have great influence on
eddy viscosity and subsequently on thermal eddy diffusivity. This effect increases by interparticle collision and increasing
loading ratio. On the other hand it is demonstrated that increasing loading ratio in the thermally developed region, resulted in
the continuous reduction of the wall temperature and continuous growth of the wall heat flux and in both conditions,
suspension Nusselt number is initially decreases and then increases. The effect of particle specific heat has been studied as
well.
Introduction
Turbulent gassolid flows are frequently found in technical
and industrial processes, such as vertical risers, fluidized
beds, waste treatment, mixing devices, combustion of
pulverized coal particles and in air pollution control. Such
particleladen flows, are affected by various parameters,
such as loading ratio, particle characteristics, interparticle
collisions, particlewall collision and particleturbulence
interactions.
Inter particle collisions largely modify the flow structure
and consequently have a considerable effect on the transport
phenomena (Louge and Yusof, 1993). Mansoori et al.
(2000) showed that the particle interactions and collisions
could markedly influence the particle thermal fluctuation
intensity. An EulerianLagrangian approach is used to
model turbulent gas solid in heated pipe by Chagras et al.
(2005). The particleparticle collision has been found to
increase the overall heat transfer as much as 8% in vertical
pipe gassolid flow. It has been confirmed that direct
thermal exchange during solidsolid contact are negligible
in the range of loading ratio up to 10. Saffar et al. (2007)
showed that profiles of particle concentration and particle
velocity are flattened due to interparticle collisions and this
effect becomes more pronounced with increasing loading
ratio.
An important issue in the development of models for
fluidparticle flows is the turbulence of the carrier phase
that is responsible for mixing, particle dispersion, change in
the eddy viscosity and consequently the thermal eddy
diffusivity and the heat transfer (Crowe ,2000). The
experimental data showed a general trend that small
particles tend to attenuate turbulence and large particles
tend to enhance turbulence (Gore and Crowe 1989).
Kenning and Crowe (1997) have proposed a model for
turbulence modulation for gas particle flows based on the
work done by particle drag and the dissipation based on a
length scale corresponding to the inter particle spacing. The
model showed reasonable agreement with the limited
available data.
In this study the turbulent heat transfer of gas solid flow in a
horizontal channel is presented using a four way coupling
interaction for gassolid flow using EulerianLagrangian
approach. Numerical simulation is based on the
experimental study performed by Tsuji et al (1987) in a
horizontal channel involved massive particles with mean
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
diameter of 1 mm. Inter particle collision is treated by the
deterministic method. In order to obtain turbulence
enhancement for large particles, the formulation proposed
by Crowe (2000) is used which includes the
extraproduction term in the source term for the turbulent
kinetic energy and the extradissipation term by defining
hybrid length scale. So Low Reynolds Number k1 model is
used for studying the fluid phase and for investigating the
thermal behavior, turbulent Prandtl number model is used.
The simulation result is compared with available
experimental data. The influence of turbulence modulation
on the turbulent kinetic energy and the eddy viscosity are
studied. Numerical results pertain to the heat transfer such
as suspension Nusselt number, wall temperature and wall
heat flux in both constant wall heat flux and constant wall
temperature boundary conditions are investigated.
Nomenclature
Ap surface area of particle
c, Specific heat capacity
c, ratio of particle specific heat capacity to gas
specific heat capacity
d, particle diameter
e restitution coefficient
FD drag force
FLS slipshear lift force
FLR rotational lift force
g gravity acceleration
h, heat transfer coefficient around particle
k turbulent kinetic energy
1 initial dissipation length scale
lh hybrid dissipation length scale
im particle mass
mr solid loading ratio
Nu Nusselt number
Nuo pure air Nusselt number
P Pressure
Pr Prandtl number
Pr, turbulent Prandtl number
Q, wall heat flux
Re channel flow Reynolds number
Re, particle Reynolds number
Spk kinetic energy source term
S,, momentum source term
Sp, fluctuation of momentum source term
Sp@ heat source term
T instantaneous temperature
Tm bulk average temperature
T, wall temperature
U instantaneous velocity
U, mean fluid velocity distribution across channel
u velocity fluctuation
u friction velocity
x axial coordinate
Xp particle location
y distance from the wall
Greek letters
a thermal diffusivity
a, thermal eddy diffusivity
e dissipation rate of turbulent kinetic energy
u friction coefficient
channel half width
fluid thermal conductivity
solid volume fraction
mean solid volume fraction
kinematic viscosity
eddy viscosity
density
Subsripts and superscripts
f fluid
p particle
w wall
xm mean quantity across the channel
x time average quantity
x+ quantity in wall unit
Model Formulation
The mathematical model of nonisothermal turbulent
gassolid flow in a horizontal channel is developed by using
an EulerianLagrangian approach. In dynamic formulation
the gas flow is assumed to be incompressible and hydro
dynamically fully developed convectivee terms being
neglected and velocity profiles only altered due to fluid
particle interactions). The solid phase is modeled by
Lagrangian simulation and particles are solid, spherical,
with a constant diameter. The resulting equation system is
closed by means of standard k1 model.
Gas Phase Equation
Momentum equation:
1 dP d dU
(1 p) + p(1 p)(v + v) I
Sdx dyL dy
1
+ sp, = 0.
Turbulent energy equation:
d r dl V(dk1 dU, 2
dy L k I dyj v \ dy 9
1
(1 p)E+ Spk =0.
In Eq. (2) dissipate rate defined as follow:
k3/2 k
E = C (1 eApRk) + Cd2 V
'ft 
where lh is a hybrid length scale which approaches the
interparticle spacing for interparticle spacing less than the
dissipation length scale. Approximated as follow:
1 1 1
with
S= d )1/3 1
Paper No
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Nup = 2 + 0.6Re06Pr33
Af p
0.06(1 )4)
L5 )
where A is the interparticle spacing introduced by Crowe
(2000), 1 the initial dissipation length scale for a channel
(Schilichting, 1982), 6 the channel half width and y the
transverse coordinate measured from the wall of the
channel.
The closure equation is:
vt = C,f 1 (1 eApRk) (7)
with Uk = 1., Cd = 0.1664, Cd2 = 0.32, Ai = 0.03,
C, = 0.55 and Rk =
The coupling terms Sp, and Spk as proposed by Crowe
(2000) are given by:
Spi = p ( p, dt g,) (8)
and
Spk = ISpu( Ui Up, )> + (SpuUp,) (9)
Energy balance equation as:
9Tf a r aTf]
(1 P )U ( p)(a + at)
a [ )( d ar] (10)
+ SpT
and with closure relation:
Vt
t Prt
The expression of the turbulent
given by Kays (1994) as:
Prandtl number PT
Prt = 0.5882 + 0.228 (V
S0.0441 (f) [1
V 1
exp 5.165 )
vt V 1
rt is
(12)
The coupling term SpT which represents the heat exchange
between the particulate phase and the continuous phase, is
given by:
6(php _
SpT = T T)
d= (7 >
The heat transfer coefficient hp is computed using the
following standard correlation for the particle Nusselt
number:
1 = (0.140.08(1
dI
dUp
mp = FD+ Fr + FQ S + FR
P dt
mcP = A(T T)
In Eq. (15), Xp are the coordinates of the particle position,
Up are the velocity components. T is the torque acting on
the particle, 3p its the angular velocity, I, is the moment
of inertia for a sphere and cpp is the specific heat capacity
of particle FD is the drag force. FLS represents the
slipshear lift force. The slipshear lift force is important in
strong shear gradient. In the channel flow, this force is
dominated near the wall due to strong shear gradient and has
great influence on particle settlement (in horizontal channel)
and particle concentration across the channel width. FLR and
T represent the rotational lift force and the torque for a
rotating sphere. Particle rotation is mainly induced by
wallparticle and particleparticle collisions. All of the
equations in this section are described in details by
Sommerfld (2003).
All these fluidparticle interactions (force, torque, and heat
exchange) involve the instantaneous velocity (U = U +
u) and temperature (T = T + t') of the fluid at the
particle location, which is predicted by means of a model as
described here. The expressions of the fluctuational
components of gas velocity are obtained by a modified
Gaussian random field model proposed by Kraichnan
(1970) which was extended to nonhomogeneous flows by Li
and Ahmadi (1993).
u+(X+,t) = U,[cos(Kn.X + a)nt+)]
n=1
+ U2[sin(Kn.X+ (19)
+ tt t ttt)]
where X+ denotes the position vector and all quantities are
Paper No
pfDy \Ufy
where Rep = pfDP Ir is the particle Reynolds number
and Pr is the Prandtl number of the gas.
Particle Phase Equation
In Lagrangian approach, a large number of particles are
tracked through the fluid field. The motion of each
individual particle is subjected to Newton's equation of
motion. The forces considered to act on the particle are the
drag force, the gravity, the slip shear lift force, and the lift
force resulting from particle rotation. In order to calculate
their instantaneous position, velocity, rotational velocity and
temperature:
Paper No
nondimensionalized with friction velocity u* and
kinematic viscosity.
U+ =u t*u.2 + Xiu*
Uji= t = = (20)
= qn x Kn, U2 = n K,
The component of vectors n, Fn and frequencies w, are
picked independently from a Gaussian distribution with a
standard deviation of unity. Each component of Kn is also a
Gaussian random number with a standard deviation of 1/2.
In Eq. (4) M is the number of terms in the series. Here M =
100 is used as suggested by Li and Ahmadi (1993). This
equation generates an isotropic homogeneous turbulence
and for application to nonhomogeneous flows a scaling
method is applied. That is up = u(X+,t+)e,(X+) in
which e,(X+) are the shape functions for the axial, vertical
and transverse rms velocities (Kraichnan (1970), Li and
Ahmadi (1993). Similar formulation is also used for
generating the instantaneous gas temperature fluctuations as
proposed by Mansoori et al. (2002). The procedure is as the
generation of the fluctuational components of gas velocity
and the nondimensioned fluctuation of gas temperature
t+(X+, t+) is calculated by:
t+(Xt = T [cos(K.X+ + nt+)]
+ Z T2 [sin(Kn.X+ (21)
+ Lnt+)]
Where, tJ+ is the local gas temperature fluctuation and is
properly nondimensioned. To generate nonhomogeneous
fluctuation field a scaling method is applied. That
is, t+ = t+(X+,t+)e+(y,) in which e+(y,) is shape
functions (Mansoori et al., 2002).
Collision Model
The collision model used in this study follows mainly the
methodology by Wang and Mason (1992) and explained in
detail by Zhou et al. (2002). In the formulation of collision
problem it is assumed that particles are sphere and rigid,
collisions are binary, instantaneous and inelastic and both the
restitution and friction coefficient are constant. The
simulation of the dispersion phase follows the Lagrangian
code presented by Saffar et al. (2007) with some
improvements which are based on deterministic approach for
considering interparticle collisions.
Numerical Scheme
Equations are numerically solved using a finitedifference
scheme. The geometry of the herestudied channel is
divided into two parts, the adiabatic inlet section and the
heated section. At the end of the first part the flow is
assumed to be dynamically fully developed. At the second
part the flow is involved in the heat transfer and is lead to
thermally developing flow. An iterative scheme is applied
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
in order to allow fourway coupling to be taken into account.
In order to accurately handle the problem, it is necessary
that the mesh width be small enough near the wall. A
logarithmic scheme is used for the transverse coordinate.
All the calculations are performed with meshes
guaranteeing gridindependent results with 100 nods in the
axial direction and 100 nods in the lateral direction. The
total number of time steps is chosen to be N, = 10000.
Boulet et al. (1998) have shown that low number of grid
nodes in axial direction has no significant effects on the
numerical results. The equations of particle motion are
calculated by the fourthorder RungeKutta scheme. The
Lagrangian time step for solving the particle equations is
chosen as described by Lain et al. (1999). The particles are
initially distributed randomly across the flow domain to
achieve a uniform initial concentration distribution. The
initial particle velocity and temperature is chosen as 70%
and 90% of gas mean velocity and local temperature
respectively. The deterministic process is used for
simulating interparticle collisions. In the hard sphere model,
only binary collision is taken into account. If the distance
between particle centers is less than the particle diameter
then collisions are assumed to be occurred. Whenever
particlewall collision is detected, the velocities after wall
collision velocities are obtained. Any particle leaving the
calculation domain (the end of channel length) would be
replaced by a new particle entering in with the same
velocity and inlet temperature. This is done to keep the
number of particles constant in calculation domain.
Additionally, it has to be mentioned that due to the strong
grid tightening associated with the use of a lowReynolds
number model, the particle diameter may exceed the mesh
width in near the wall region. In this case, the source terms
are distributed over several cells according to the particle
surface area pertaining to each cell. All the details
concerning the numerical processing may be found in
Mansoori et al. (2002). Finally, the appropriate thermal
boundary conditions are:
At the channel inlet: Tf = Ti
At the wall:
constant wall heat flux: Q, =
constant wall temperature: T,
Af8
 T at y = 0 and y = 28
=Tw aty = 0 andy = 28
Results and Discussion
Experimental data of Tsuji et al. (1987) in the horizontal
channel experiment is used to validate the dynamic
characteristics. A twodimensional channel made of acrylic
plate comprised the test section. The channel height H was
25 mm. The bulk air velocities tested in experiments were
Um =15 m/s which yielded the channel Reynolds numbers
(ReH) of 25,000. The solid phase was composed of
polystyrene beads with 1 mm diameter and mass density of
1000 kg/m3. The mass loading ratio, m, were within 1 to 3.
The friction coefficient and the restitution coefficient
suggested by Tsuji et al. were 0.4 and 0.8 respectively. But
in this study the appropriate values suitable for the flow
system are employed. The values of ep = 0.8, e, = 0.95,
lp = 0.47 and p, = 0.2 are used in the computations.
Paper No
Therefore, the thermal characteristics of the turbulent
airsolid flows in the horizontal channel were studied.
Unfortunately, no experimental data are available for the
thermal behaviour of such flows. Consequently, validation
of the thermal data may not be as complete as for the
dynamics and just qualitative comparisons with the
available experimental data and the general trend of the
gassolid flow in the heat transfer area are investigated.
Hydrodynamic Characteristics
Velocity profiles are plotted in Fig. 1 for loading ratios of 1,
2 and 3. The numerical results and the experimental
measurements for the dimensionless mean air velocity
distribution ( ) are in good agreement.
\UE/ ili/UUl~lll C
0.4
0.2
Uf / Ume
Figure 1: Mean fluid velocity ()
MU,
The variation of turbulence intensities with mass loading is
shown in Fig. 2. The model following the Crowe
formulation gives satisfactory results. With the presence of
relatively large solid particles, the air turbulence intensities
are increased in the core of the channel. As mention before
it is clear that the turbulence enhancement due to large
particles. This figure shows that turbulence increased in
core of the channel and attenuation in another (far from the
center of the channel). Unfortunately, Tsuji et al. (1987) did
not measure the air turbulence intensities in their channel
experiment. Previously, Tsuji and Morikawa (1982)
measured the streamwise turbulence intensities of airsolid
flows in a horizontal pipe. They found that in general the
presence of relatively large plastic particles with 3.4 mm
diameter increases the turbulence markedly, while small
plastic particles with 0.2 mm diameter reduced it. The
present numerical model simulates particles with 1 mm
diameter. Even though Tsuji et al.(1984) in the experimental
study in a vertical pipe observed similar result for particles
with 500 gm mean diameters. So it can be inferred that this
model has reasonable qualitative agreement with the
measurements and correlation proposed by Crowe (2000)
for the streamwise fluid turbulence intensities for large
particles. It should be mention that the ratio of particle
diameter to integral length scale of the fluid (D/L) is equal
to 0.57 approximately in the channel canter that is greater
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
than 0.1. This value proposed by Gore Crowe (1996) as a
criterion for turbulence modulation.
0.8
0 I
0.6 m 0
  m=
0.4
,.\
0.2 .
0 0.02 0.04 0.06 0.08
Uf/U U.
Figure 2: Air turbulent intensity
0.1 0.12
The effect of particles and turbulence modulation on eddy
viscosity is plotted in Fig. 3. In clear fluid eddy viscosity
increased from zero at the wall and near the channel center,
is started to decreased. But it is observed that with
increasing loading ratio eddy viscosity at the core of the
channel start to increase. This trend can be the result of
increasing the turbulent intensity in the center of channel
because these two parameters according to Eq. (7) directly
related to each other. This parameter investigated here
because of according to turbulent Prandtl relation, eddy
viscosity is tied together eddy thermal diffusivity that has
the great influence on thermal behavior of any kinds of fluid
flow. It has to be stated that, unfortunately there is no
available data related to this parameter.
0.8
0.6
S m o I "\ \
0.4 ' " m = 3
I i
0.2 "
0 20 40 60 80
Vt/V
Figure 3: Normalized eddy viscosity
Thermal characteristics
In this section numerical result for two basic thermal
boundary conditions (constant wall heat flux and constant
Paper No
wall temperature) are presented and general behavior of
such gasparticle flow discussed. In this study suspension
Nusselt numbers have been calculated using the following
definition:
26Qw
Nu Q(22)
A,(T, Tm)
where
f p (1 op) cUTrdy + f pppc,,T,% dy
fPIr(1 (p)cf pUrdy + S pppc ,Updy
The T, and Tm are the wall temperature and the bulk
average temperature, respectively. In case of horizontal
channel because of asymmetric flow due to gravity effects
there is difference between upper and lower walls
temperature and consequently Nusselt number. To overcome
this issue, mean Nusselt numbers, wall temperature and wall
heat flux are selected. Although the differences between
lower and upper walls parameters such as the wall
temperature and the wall heat flux are described in detail.
For investigating the effect of particle specific heat capacity
on the thermal behavior, one terms defined as c, that
represent the ratio of particle specific heat capacity to gas
specific heat capacity.
Figure 4 illustrate the Nusselt number at the end of the
channel ( = 190) as a function of the loading ratio for
both boundary conditions. Some decreasement in the
Nusselt number is first observed, then the results show that
Nusselt number increases with increasing loading ratio.
Similar trends are observed in experimental study performed
by Depew and Farbar (1963) and Jepson et al. (1963). In
this figures it is shown that increasing cp decrease the value
of the suspension Nusselt number. It is obvious that with
increasing heat capacity of particles, they can absorb the heat
from the hot fluid in the near the wall and refund it to cold
fluid in the core of the channel. When heat capacity of
particle is small, the fluid and the particle are almost in the
same temperature. And particles could not transfer the heat
from the region near the wall to area far from the wall.
1.1
0.9
0.8
0.7
0.6
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
In Fig. 5, local Nusselt numbers are plotted versus the axial
location along the channel. This figure shows that the
thermal entrance length is increased due to the existence of
particles (in both mentioned thermal boundary condition).
This effect is in agreement by experiment performed by
Depew and Farbar (1963).
Constant wall heat flux
SPr= 0.71
2I I i I i
0 50 100 150 200
x/(26)
Figure 5: Nusselt number as a function of the axial
location.
Until now the effect of presence of particles on Nusselt
number are studied. But it is obvious that Nusselt number
can not describe the thermal behavior of such gas solid flow
and give a comprehensive conclusion about these flows. It is
assumed that the wall temperature in constant heat flux and
wall heat flux in constant wall temperature can effectively
describe the effect of particles motion in the fluid. As it is
described below, according to this parameters,
comprehensive conclusion is possible for researcher.
In Fig. 6 the wall temperature (in constant wall heat flux
thermal boundary condition) is plotted as a function of the
axial location. It is known that in thermally developed
region, wall temperature is increased with the constant rate.
These figure shows that this rate continuously is decreased
with increasing loading ratio and this effect becomes more
pronounce with increasing the specific heat capacity of
particles. In other words at the initial channel length
(thermal entry length) the wall temperature increases with
increasing loading ratio, but in thermally developed region
(that is of importance) the wall temperature is decreased and
this effect intensifies with increasing loading ratio and has a
general trend: In thermally developed region, increasing
loading ratio causes decreasing wall temperature
continuously without exception. Increasing particle heat
capacity augments this effect. It should be say the results
show that lower wall temperature is less that upper wall
temperature in this condition. This effect probably is due to
higher particle concentration in the lower region.
0.5
0 1 2 3
Loading ratio (mr)
Figure 4: Nusselt number as a function of the loading ratio
  Nu Exp. 0.022Reo8Pro
clear gas
mn= 1, Cp =0.8
 m= 1, Cp = 2.
m,= 3, Cp = 0.8
 m,= 3, Cp = 2.
S  c = 0.8 (Constant wall heat flux)
 c =2. (Constantwall heatflux)
** c =0.8 (Constantwall temperature)
 l  c = 2. (Constant wall temperature)
\
\< c.  '
\ .  .  *. 
% A A A
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
0 50 100 150 200
x/(26)
Figure 6: Wall temperature (constant wall heat flux)
Figure 7 presents the wall heat flux (in constant wall
temperature thermal boundary condition) in this case. It is
clear that with increasing loading ratio wall heat flux
increases, and this trend is observed continuously. Some as
previous result, increasing particle heat capacity augments
this effect. So again a general trend is concluded: In
thermally developed region, increasing loading ratio causes
increasing the wall heat flux continuously without exception.
It should be said that in this condition the results show the
lower wall heat flux is more than upper wall heat flux. This
effect as mentioned before is probably related to higher
particle concentration in the lower region.
100
Pr= 0.71
clear gas
1m= 1, Cp 0.8
 m= 1, Cp'= 2.
80  m= 2, Cp =0.8
. m= 2, Cp = 2.
m= 3, Cp =0.8
 mn= 3, Cp 2.
60
00
40
20
0
0 50 100 150 200
x/(26)
Figure 7: Wall heat flux. (constant wall temperature)
It is helpful to point out that according to the bulk average
temperature definition (i.e. Eq. (23)), this value decreases
continuously with increasing loading ratio and particle heat
capacity (Fig. 8).
x /(26)
Figure 8: Bulk average temperature as a function of the
axial location
Conclusions
Simulation of gassolid turbulent flow in a horizontal
channel was performed by using k1 turbulence modeling
including coupling terms introduced by Crowe to simulate
the fluid particle interaction. This new model is presented to
predict the heat transfer between turbulent gas solid flow
and the wall in a horizontal channel being involved with two
basic thermal boundary conditions. Numerical calculations
were carried out for different loading ratios. The results
indicate that kl model with coupling terms introduced by
Crow yields very interesting results that usual or standard
source terms could not predict. The effects of this source
term on turbulent intensity and eddy viscosity were
examined. It is also demonstrated that in the thermal
investigation, wall temperature and wall heat flux are more
useful quantities than Nusselt number, because these
parameters showed continues trend with increasing loading
ratio in contrast with Nusselt Number.
Although this model showed remarkable results, further
study is really necessary in order to extend and validate this
model for higher range of lading ratio, other particle size
and other configuration such as vertical channel.
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