Energy and Power En gi neering, 2011, 3, 246-252
doi:10.4236/epe.2011.33031 Published Online July 2011 (
Copyright © 2011 SciRes. EPE
CFD Simulation of Dilute Gas-Solid Flow in 90˚
Square Bend
Tarek A. Mekhail, Walid A. Aissa, Soubhi A. Hassanein, Osama Hamdy
Department of Mechanical Power, Faculty of Energy Engineering (Aswan), South Valley University, Qena, Egypt
Received May 11, 2011; revised June 2, 2011; accepted June 10, 2011
Gas-solid two-phase flow in a 90˚ bend has been studied numerically. The bend geometry is squared cross
section of (0.15 m × 0.15 m) and has a turning radius of 1.5 times the duct’s hydraulic diameter. The solid
phase consists of glass spheres having mean diameter of 77 µm and the spheres are simulated with an air
flowing at bulk velocity of 10 m/s. A computational fluid dynamic code (CFX-TASCflow) has been adopted
for the simulation of the flow field inside the piping and for the simulation of the particle trajectories. Simu-
lation was performed using Lagrangian particle-tracking model, taking into account one-way coupling, com-
bined with a particle-wall collision model. Turbulence was predicted using k-ε model, wherein additional
transport equations are solved to account for the combined gas-particle interactions and turbulence kinetic
energy of the particle phase turbulence. The computational results are compared with the experimental data
present in the literature and they were found to yield good agreement with the measured values.
Keywords: Gas-Solid Two-Phase Flow, Dilute, 90˚ Bend, CFX
1. Introduction
Bends are a common element in any piping system of
gas-solid flow applications such as pneumatic conveyers,
pneumatic dryers, chemical industries and food process-
ing. The gas-solid flow in bends is affected by complex
parameters, such as centrifugal forces, formation and
dispersion of ropes, secondary flows and erosion of bend
outer walls. The gas-solid flow in 90˚ bend has been
studied by many researchers. Yang and Kuan [1] and
Kuan [2] performed a CFD predictions of dilute gas-
solid flow through a curved 90˚ duct bend based on a
Differential Reynolds Stress Model (DRSM) for calcu-
lating turbulent flow quantities and a Lagrangian particle
tracking model for predicting solid velocities. They
found that the more complex DRSM failed to predict the
pressure gradient effects that prevail within the bend; the
predicted turbulence intensity only bears qualitative re-
semblance to the measured distribution. Further, they
stated that the mean streamwise velocities based on
DRSM display good qualitative representation of the
measured profiles while the standard k-ε is able to
achieve better quantitative agreement. Levy and Mason
[3] studied the effect of the bend on the cross-sectional
particle concentration and segregation of solid particles
from the carrier gas. They found that the rope region
increased as the curvature ratio decreased. Mohanaran-
gam et al. [4] and [5] reported a numerical investigation
into the physical characteristics of dilute gas-solid flows
over a square sectioned 90˚ bend. They employed the
modified Eulerian-Eulerian two fluid model to predict
the gas-particle flows and studied a dilute gas-particle
flows over a square sectioned 90˚ bend employing two
approaches to predict the gas-particle flows, namely the
Lagrangian particle tracking model and Eulerian two
fluid model. The computational results are compared
with the LDV results of Kliafas and Holt [6] and were
found to yield good agreement with the measured values
and the Eulerian model provided useful insights into the
particle concentration and turbulence behavior, they
found that both Eularian-Eularian and Eularian-Lagran-
gian approaches provided reasonably good comparison
for gas and particle velocities together with the fluctua-
tion for the gas phase. Further, they stated that despite
the fact that the particle fluctuation using the Eularian
model showed good comparison with the experimental
data. They found that the more computational mesh and
time is required for Lagrangian particle tracking model
in comparison to Eularian model. Ibrahim et al. [7] stud-
ied numerically the behavior of gas-solid flow in 90˚
bend using two different turbulence models and they
found that the total pressure loss for gas-solid flow in 90˚
bend is greater than the corresponding value obtained for
gas only and its value is greatly affected by the fluid and
solid parameters. Bradley [8] gives a review of the
causes of attrition and wear in pneumatic conveying, the
consequences, and the techniques which may be applied
to overcome them in a practical context. Tian et al. [9]
investigated the performance of both the Eulerian La-
grangian model and the Eulerian-Eulerian model to
simulate the turbulent gas-particle flow. The validation
against the measurement for two-phase flow over back-
ward facing step and in a 90˚ bend revealed that both
CFD approaches provide reasonably good prediction for
both the gas and particle phases. Chu and Yu [10] simu-
lated numerically gas solid flow in complex three-di-
mensional (3D) systems by means of Combined Contin-
uum and Discrete Method (CCDM). They compared the
results, quantitatively and qualitatively, with experiment-
tal data and good agreement was noticed. Chen et al. [11]
investigated the relative erosion severity between plug-
ged tees and elbows for dilute gas-solid flow applying a
CFD based erosion prediction model. They conducted
experimental tests to verify the simulation results. The
ratio of erosion at the end of the plugged section to that
in an elbow was found to approach a constant value for a
range of conditions. A correlation was presented that
provides the ratio of erosion of the outer downstream
corner of the plugged tee to that in an elbow. Deng et al.
[12] studied experimentally the effect of particle concen-
tration on the erosion rate of pipe bends in pneumatic
conveyors using different bend radii. Results show that
there was a significant reduction of the specific erosion
rate for high particle concentration. This reduction was
considered to be as a result of the shielding effect during
the particle impacts.
2. Present Study
2.1. Mathematical Model
The purpose of this paper is to provide deeper under-
standing of the parameters which may have an effect on
the erosion and pipeline wear especially at bends. A CFD
simulation of the dilute gas-solid flow in a square-sec-
tioned 90˚ bend (0.15 m × 0.15 m) using a Lagrangian
particle tracking model is presented, considering that all
the particles have been introduced in the flow with ap-
proximately the same bulk velocity of the fluid. The par-
ticulate phase consists of glass particles that assumed to
be spherical with diameter of 77 μm. The Finnie’s ero-
sion prediction model [13] and the standard k-ε were
applied to numerically predict erosion and turbulence in
elbow respectively. The mixture composition and phase
velocities were defined at the inlet boundary. The system
pressure was fixed at the outlet boundary. The aforemen-
tioned models were implemented into the CFX-TASC
flow V 2.9.0 via user-defined subroutines. Using user-
defined subroutines allows the flexibility in extending
the collision model to handle complex engineering flows.
To gain confidence in this numerical study, the predicted
mean velocities for both gas phase and solid phase were
validated against experimental data of Yang and Kuan
2.2. Assumptions
Assumptions made in formulating the tracking model
have introduced some limitations on the model. These
Particle/particle interactions are not included in the
model. Particle interactions may be important in
flows where the discrete phase volumetric concentra-
tion is greater than 1% [14]. This assumption implies
that the model is designed for dilute systems.
There are no particle source terms to the turbulence
equations, and therefore, turbulence is not modulated
by the discrete phase.
The viscous stress and the pressure of the particulate
phase are negligible.
Only inert, spherical particles are considered.
There is no mass transfer over the surface of the par-
ticles due to particle-wall collision.
The flow field is isothermal.
2.3. Gas Phase
The first step is to solve the continuous carrier fluid flow
equations. The continuity and momentum equations em-
ployed by the CFX-2.9.0 [15] are given in Equations (1)
and (2), respectively:
 (1)
 
gg ggg
uieffeff ij
ijji l
xxxx x
 
 
where, i
u represents the gas mass-averaged velocities
in the xi coordinate directions, P*g is a time-average
pressure, ρ is a time-average density, μeff is the effective
viscosity, and the S term is an additional time-average
source term.
Copyright © 2011 SciRes. EPE
2.4. Solid Phase
After obtaining the flow field, the particle trajectories are
simulated. In the current model, the particles are as-
sumed not to affect the flow field, one-way coupling
between the sand particles and the carrier fluid, the fluid
is allowed to influence the trajectories but the particles
do not affect the fluid. It is noted that the one-way cou-
pling method is suitable only for low solid loading. The
particles are introduced at a finite number of starting
locations. In every given time step, their positions and
velocities are calculated according to the forces acting on
the particle and using Newton’s second law. The equa-
tion of motion for a par- ticle [15] can be written as:
12 dd
tgg p
 
where, mp is particle mass, d is particle diameter, u is
velocity, ρ is density, μ is fluid dynamic viscosity, Ccor is
drag coefficient and t0 is the starting time. The subscript
g and p refers to the fluid gas and the particle respec-
The term on the left-hand side is a summation of all
of the forces acting on the particle expressed in terms of
the particle acceleration. The first term on the right hand
side is the viscous drag of fluid over the particle surface
according to Stokes law. The second term is the force
applied on the particle due to the pressure gradient in the
fluid surrounding the particle caused by fluid accelera-
tion. The third term is the force to accelerate the virtual
mass of the fluid in the volume occupied by the particle.
The fourth term is an external force which may directly
affect the particle such as gravity or an electric field.
The fifth term is the Basset force or history term which
accounts for the deviation in flow pattern from steady
A drag coefficient, Ccor, is introduced to account for
experimental results on the viscous drag of a solid sphere.
For a moderate particle Reynolds number 0.01 < Rep <
260, the drag correction in Equation (8) [15] is:
0.82 0.05
10.1935(Re)Re 20
10.1315ReRe 20
log Re
where, the particle Reynolds number, Rep, is calculated
 (6)
In turbulent tracking, the instantaneous fluid velocity
is decomposed into mean,
, and fluctuating,
2.5. The Particle-Wall Collision
When a particle impacts on the pipe wall, it reflects at an
angle related to the coefficient of restitution, er. The
value of er determines the component of velocity normal
to the surface after impact, , given the incident nor-
mal component,
 (7)
The coefficient of restitution is taken to equal unity.
Figure 1 shows the elastic collision for er equals to unity,
2.6. Particle Erosion Model
Before conducting the flow calculations, it is necessary
to modify source code to calculate particle erosion on the
duct and pipe walls. To calculate the erosion a simplified
Finnie's erosion model [13] is applied. Finnie proposed
that erosive wear is a direct consequence of the cutting of
surfaces by impacting particles. This model assumes that
the erosion rate on a surface; ER may be described by:
where, V is the impact velocity of the particle on the sur-
face, K and n are erosion parameters, and f(θ) is a func-
tion relating wear to the impact angle (angle relative to
the surface normal), θ:
1sin0 0.4π
sin 23cos0.4π
 
Figure 1. Elastic collision (coefficient of restitution = 1).
Copyright © 2011 SciRes. EPE
The parameter K includes the mass flow represented
by the particle. The erosion rate is defined as the mass of
surface removed per unit area per unit time. As such, the
units of K are adjusted depending on the value of the
exponent n.
2.7. Solution Procedure
A three-dimensional pipe system consisting of two
straight ducts of (0.15 m × 0.15 m) cross section and
0.15 m length and a 90˚ bend of the same cross section
and 0.15 m diameter was chosen as the calculation do-
The numerical procedure for solving the governing
equations is based on the finite-volume formulation of
the conservation equations for mass, momentum and
energy for the two phases. The three-dimensional nu-
merical solver and grid generator employs a Multigrid
linear solver to solve the discrete finite volume equations
that result from the discretization process using upwind
difference. This linear solver is usually very reliable. The
solution procedure of finite-volume discretization scheme,
is solved over one grid system that has a cross- sectional
cell density of (30 × 30 × 30) shown in Figure 2.
3. Results and Discussion
3.1. Velocity Vectors
Figures 3-4 show the gas velocity vectors at the bend sec-
Figure 2. Duct geometry and computational grid.
Figure 3. Gas velocity vectors.
Figure 4. Mean velocity vectors and turbulent kinetic en-
ergy contours for gas phase inside the curved 90˚ bend duct
system [1].
tion compared with the results, shown in Figure 5,
measured by Yang and Kuan [1]. As shown in Figure 4,
in the curved 90˚ bend, velocity vectors were calculated
at seven locations from 0˚ to 90˚ at 15˚ intervals. At the
entrance of the straight duct the flow of the gas is fairly
evenly distributed in the upstream pipe bend, when the
gas enters the bend section it is already affected by the
presence of the bend; this can be easily seen as the flow
starts to accelerate near to the inner wall due to the fa-
vorable pressure gradients while at the outer wall of the
bend the flow decelerates due to the unfavorable pressure
After the bend section, the flow begins to decelerate
and the velocity gradient is uneven. This is because of
the separation that has occurred in the inner section of
Copyright © 2011 SciRes. EPE
(a) (b)
(c) (d)
Figure 5. Particle trajectories of different particle diame-
ters. (a) Dp = 10 µm; (b) Dp = 40 µm; (c) Dp = 77 µm; (d) Dp
= 120 µm.
the bend due to the adverse pressure gradient.
3.2. Effect of Particle Diameter on the Particle
Figure 5 shows the trajectories of particles tracked in a
wide range of particle diameters. It can be seen that the
particles with small diameters (case (a) Dp = 10 µm) tend
to follow the flow and turning before reaching the outer
wall while the particles with large diameters (cases (b,c
and d) Dp = 40, 77 and 120 µm respectively) don't follow
the flow and hit the outer wall of the bend. With the in-
crease of Dp the concentration of particles is further in-
creased near the outer wall and further decreased near the
inner wall. Further progresses of the flow into the bend, a
particle free region starts to be identified close to the
inner wall, the thickness of this particle free region
gradually increases until the bend exit with the increase
of the particle diameter.
3.3. Effect of Particles on Gas Speed
The velocity field with and without particles injected is
compared to determine the relative influence of the glass
particles on the fluid flow. Two separate calculations
were made: with and without particles. The speed in the
duct is compared through the relation:
dif_speed = speed – speed_wop (11)
where, dif_speed is the difference in velocity, speed is
the gas velocity with particles and speed_wop is the gas
velocity without particles.
Figure 6 shows the velocity vectors for the gas phase
with and without particles. The difference in velocity for
the flow with and without particles is shown in Figure 7.
The velocity field has been substantially changed by the
particles. As shown in Figure 7 there is no observed
speed difference in the entrance of the duct, as the parti-
cles supposed to be entered with the same gas velocity
(10 m/s). The speed starts to decrease slightly at the be-
ginning of the bend section near to the outer wall, this
may be due to the increasing in particle concentration at
this region. The difference in speed downstream of the
bend is negative, speed is further decreasing especially at
the inner wall of the duct, where particles coming from
the first impact at the bend hit again the inner wall of the
downstream duct. Particles may move faster than the gas
phase or may lag behind the gas phase. Depending on the
movement of the particles they setup slip velocities.
These slip velocities in turn give rise to particle drag.
The gas phase which is embodied with these particles,
lose some of its velocity trying to overcome this drag.
This helps to explain why the gas flows with particles lag
behind the clean gas which has no particles.
3.4. Erosion
The erosion information was generated based on Finnie’s
model which described earlier. Erosion was calculated
on all duct surfaces and plotted as shown in Figure 8.
The current model shows that, any part of the pipe sys-
tem that experiences high flow velocities or sudden
changes in flow direction is subjected to erosion. The
rate of particle erosion is highly dependent on the flow
velocity. As stated in Equation (9). the erosion rate is a
function of (Vn), then any small increases in velocity can
therefore cause substantial increases in erosion.
The peak erosion rate occurs at the outer wall of the
bend section; where the particle impacts, are concen-
trated. Particle size mostly influences erosion by deter-
mining how many particles impact on a surface. Very
small particles (~10 microns) are carried with the fluid
and rarely hit walls, see Figure 5(a). As they are lighter,
small particles more readily follow the flow of the car-
Copyright © 2011 SciRes. EPE
Copyright © 2011 SciRes. EPE
(a) (b)
Figure 6. Velocity vectors of gas phase with and without particles. (a) Velocity vectors of gas phase only; (b) Velocity vectors
of gas phase with particles.
Figure 7. Change in velocity field due to particle flow. Figure 8. Erosion pattern on the duct wall.
rying fluid rather than impacting on the walls. Also,
when they impact they tend to do so at low angles and
they cause less damage. Larger particles tend to travel in
straight lines and bounce off surfaces. So, the higher the
free stream velocity and the larger the particle size, the
more the erosion will be.
ent diameters, also the erosion pattern is predicted. The
mean velocity vectors of the predicted model shows
good agreement with the experimental data obtained by
Yang and Kuan [1]. The current numerical data can be
used to further enhance CFD models, to aid better pre-
diction near the inner wall of the bend by establishing an
effective two-way coupling between the gas and the par-
ticulate phases.
4. Conclusions
The current study simulates gas and solid phases in a
dilute two-phase flow system inside a square sectioned
90˚ bend using CFD. The computational results obtained
by Lagrangian tracking model explain how the flow in-
side the duct would be and show the shape of the gas
velocity vectors and the particle trajectories with differ-
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