Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 2.5.5 - Euler/Lagrange computations of particle-laden gas flow in pneumatic conveying systems
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 Material Information
Title: 2.5.5 - Euler/Lagrange computations of particle-laden gas flow in pneumatic conveying systems Particle-Laden Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Lain, S.
Sommerfeld, M.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: Lagrangian simulations
particle-laden gas
flow structure
elbow
 Notes
Abstract: This paper deals with the transport of solids in pneumatic conveying systems. Particle transport is calculated by considering all the relevant forces (including transversal lift forces), dispersion due to turbulence and two-way coupling. Particle-wall collisions and roughness are modelled according to Sommerfeld and Huber (1999) and inter-particle collisions are described by a stochastic modelling approach (Sommerfeld, 2001). As known from many single-phase studies a secondary flow is developing along the pipe bend. As in the present study rather small glass powder is considered (i.e. 10 6m < Dp < 80 6m) the particle transport is affected by this secondary flow which in turn is modified by the particle phase due to two-way coupling. In particular, due to wall roughness, at the outer wall of the bend two small recirculation areas appear, which affect the secondary flow structure and pressure drop in the bend. The different mechanisms, e.g. wall roughness, inter-particle collisions and mass loading, on the flow structure in the bend and the resulting pressure drop are analysed. Also the effect of the horizontal pipe length, related with the degree of particle gravitational settling, on the resulting structure of the particle laden flow is investigated in this paper. As a result, consideration of inter-particle collisions leads to a more concentrated dust rope at the outer wall of the bend regarding the two-way coupling case. The computations will be compared with the data of Huber and Sommerfeld (1998) in both, the horizontal pipe and the vertical duct.
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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Resource Identifier: 255-Lain-ICMF2010.pdf

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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Euler/Lagrange Computations of Particle-Laden Gas Flow in Pneumatic Conveying
Systems


Santiago Laina and Martin Sommerfeldab


aEnergetics and Mechanics Department, Universidad Autonoma de Occidente (UAO), Cali, Colombia
E-mail: santiago.lain@gmail.com
bZentrum fir Ingenieurwissenschaften, Martin-Luther-Universitit Halle-Wittenberg, D-06099 Halle(Saale) Germany
E-mail: martin.sommerfeld@iiw.uni-halle.de


Keywords: Lagrangian simulations, particle-laden gas, flow structure, elbow


Abstract

This paper deals with the transport of solids in pneumatic conveying systems. Particle transport is calculated by considering all
the relevant forces (including transversal lift forces), dispersion due to turbulence and two-way coupling. Particle-wall collisions
and roughness are modelled according to Sommerfeld and Huber (1999) and inter-particle collisions are described by a
stochastic modelling approach (Sommerfeld, 2001). As known from many single-phase studies a secondary flow is developing
along the pipe bend. As in the present study rather small glass powder is considered (i.e. 10 pmr < Dp < 80 upm) the particle
transport is affected by this secondary flow which in turn is modified by the particle phase due to two-way coupling. In particular,
due to wall roughness, at the outer wall of the bend two small recirculation areas appear, which affect the secondary flow
structure and pressure drop in the bend. The different mechanisms, e.g. wall roughness, inter-particle collisions and mass loading,
on the flow structure in the bend and the resulting pressure drop are analysed. Also the effect of the horizontal pipe length, related
with the degree of particle gravitational settling, on the resulting structure of the particle laden flow is investigated in this paper.
As a result, consideration of inter-particle collisions leads to a more concentrated dust rope at the outer wall of the bend regarding
the two-way coupling case. The computations will be compared with the data of Huber and Sommerfeld (1998) in both, the
horizontal pipe and the vertical duct.


Introduction

Pneumatic conveying is widely used in industry for
transporting fine powders within a production process over
large distances. Depending on the application different
conveying regimes are used, such as dense- and dilute-phase
conveying. Essential for the design of such systems is the
pressure drop as a function of superficial gas velocity. This
pressure drop is composed of the gas phase pressure drop,
which is quite well known already for different pipe elements,
and the additional pressure drop due to particle transport. The
single-phase pressure loss depends on the wall friction
coefficient and hence on the flow Reynolds number and the
wall roughness. For an entire pipe system, the contributions
of the different elements (e.g., straight pipes, pipe bends
and/or constrictions) have to be added. For the additional
pressure loss of the particles also, different contributions may
be identified, namely, pressure loss caused by particle-wall
friction (i.e., resulting from the momentum loss due to wall
collisions), pressure loss due to particle lifting in vertical
conveying and pressure loss due to particle acceleration upon
injection or after a bend, so it depends strongly on the pipe
geometry considered. In addition, the particle phase pressure
drop remarkably depends on pipe material and diameter,
particle size, shape and material, wall roughness (since it is
strongly correlated with the particle-wall collision frequency)
as well as particle phase mass loading, defined as the ratio
between the particle mass flow rate and the gas mass flow


rate. As a result of this complexity, universal correlations for
the pressure drop as a function of conveying velocity are not
available and normally experiments are required to develop
such correlations (Siegel, 1991), resulting in an empirical
design of pneumatic conveying systems. Quite often, phase
diagrams are used to correlate the pressure drop with the
superficial gas velocity and the particle mass loading as a
parameter. Such experimental phase diagrams unfortunately
depend on parameters such as pipe diameter, particle size or
size distribution, which implies that a conveying of another
type of particles requires conducting new measurements.
Numerical computations, especially with the Euler/Lagrange
approach, have a large potential in predicting pressure drop
of pneumatic conveying systems (Lain & Sommerfeld,
2008a; Lain & Sommerfeld, 2008b). These studies showed
that a detailed modelling of particle-wall collisions including
wall roughness and inter-particle collisions is required for
correctly predicting pressure drop. The previous studies
concentrated on horizontal pneumatic conveying in channels
and pipes (Sommerfeld & Lain, 2009). Now pneumatic
conveying through a horizontal inlet pipe, a pipe bend and a
connecting vertical pipe is considered. As before, particle
transport is calculated by considering drag, gravity/buoyancy
and transverse lift forces, as well as dispersion due to
turbulence. Particle-wall collisions and roughness are
modelled according to Sommerfeld and Huber (1999) and
inter-particle collisions are described by a stochastic
modelling approach presented in Sommerfeld (2001).





Paper No


Naturally, two-way coupling is accounted for in all
computations, as it is essential for predicting the additional
pressure drop due to the particle phase. As known from many
single-phase studies a secondary flow is developing along the
pipe bend. As in the present study rather small glass powder
is considered (i.e. 10 vtm < Dp < 80 vtm) the particle transport
is affected by secondary flow and in addition the particle
phase modifies the structure of the secondary flow as well as
turbulence due to two-way coupling. The different
mechanisms, e.g. wall roughness, inter-particle collisions
and mass loading, on the flow structure in the bend and the
resulting pressure drop are analysed. The computations are
compared to experimental data of Huber & Sommerfeld
(1998).

Nomenclature

c model constant [-]
CD drag coefficient [-]
D diameter [m]
F force [N]
g gravity [m/s2]
I moment of inertia [Akgin]
k turbulent kinetic energy [m2/s2]
m mass [kg]
P mean pressure [Pa]
R Reynolds stress tensor [m2/s2]
Re Reynolds number [-]
S source term equation
t time [s]
T torque [kgin .]
u instantaneous velocity [m/s]
U mean velocity [m/s]
x position [m]


Greek letters
f parameter slip-shear force [-]
Ay wall roughness parameter [-]
F Diffusion tensor
S Kronecker delta [-]
E dissipation rate [m2/s3]
rq mass loading ratio [-]
lu dynamic viscosity [kg m-1 s-1]
0 generic variable
P production term [kg ,,: ]
p density [kg in"]
Co angular velocity [s- ]
Q relative angular velocity [s-1]
Subscripts
p particle
i tensor subscript
cv control volume
L Lagrangian quantity


Summary of Numerical Approach

The numerical scheme adopted to simulate dispersed
two-phase flow developing in a pipe bend is the fully coupled
stationary and three-dimensional Euler/Lagrange approach
(Lain et al., 2002).
The fluid flow was calculated based on the Euler approach by
solving the Reynolds-averaged conservation equations in


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

connection with the k-s turbulence model equations which
were extended in order to account for modulation by the
dispersed phase, i.e., two-way coupling (Kohnen &
Sommerfeld, 1997). The time-dependent three-dimensional
conservation equations for the fluid may be written in the
general form (using tensorial notation) as:


(p)4,+(pU,),, = (F,k), + S + SO,


where p is the fluid density, U, are the Reynolds-averaged
velocity components, and Fk is an effective transport tensor.
The usual source terms within the continuous phase
equations are summarised in So, while Sp represents the
additional source term due to phase interaction. Table 1
summarises the meaning of these quantities for the different
variables ), where P is the mean pressure, p the gas viscosity
and R,1 = u' u', the components of the Reynolds stress
tensor.


Table 1. Summary of terms in the general conservation
equation for the different variables describing the gas phase
by the k-e turbulence model.

The simulation of the particle phase by the Lagrangian
approach requires the solution of the equation of the motion
for each computational particle. This equation includes the
particle inertia, drag, gravity-buoyancy, slip-hear lift force
and slip-rotational lift force. The Basset history term, the
added mass and the fluid inertia are negligible for high ratios
of particle to gas density. The change of the angular velocity
along the particle trajectory results mainly from wall
collisions but also the viscous interaction with the fluid (i.e.,
the torque T). Hence, the equations of motion for the
particles are given by:


dxp
dt up


SS k S,

1 0 0

U, (+Tj -PJ+(k U,kJ

-(2P k+PT Uk,k j+PgJ

k (+T /okk P-P



P= -p R, Uj

IlT =PC,

p R, =(2pk+PT k,k 6J jT (U, +uJ,

c, = 0.09 c,, = 1.44 2 =1.92

o, =1.0 = 1.3





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


dup 3 p
m mPcD(u, -u,)u-u, +
dt 4 Dp -p + (3)


mg, 1- + F1, + Fi,,
P p

I P = Ti (4)
dt

here, xp are the coordinates of the particle position, upi are its
velocity components, u, = U, + u', is the instantaneous
velocity of the gas, Dp is the particle diameter and pp is the
density of the solids. mp = (7/6) ppDp3 is the particle mass and
Ip = 0.1 mpDp2 is the moment of inertia for a sphere. The drag
coefficient is obtained using the standard correlation:

c 24Re1 (1+0.15 Re68 ) Re, 1000
CD =0.44 Re (5)
D 0.44 Re > 1000


Rep = P DpDU- pu


the particle Reynolds


number.
The slip-shear force is based on the analytical result of
Saffman (1965) and extended for higher particle Reynolds
numbers according to Mei (1992):


F1,=1.615 D piRe ci, (u u)x


where ) = V u is the fluid rotation, Re,


p D2; |/ is


the particle Reynolds number of the shear flow and Cis =
F, /Fs,Saff represents the ratio of the extended lift force to the
Saffman force:


(1- 0.331 ,f)e Re,10 +0.3314, Re <40 (7)
C/ 0.0524 ,Rep Rep > 40


and p is a parameter given by P = 0.5Re,/Rep.
The applied slip-rotational lift force is based on the relation
given by Rubinow and Keller (1961), which was extended to
account for the relative motion between particle and fluid.
Moreover, several authors allowed an extension of this lift
force to higher particle Reynolds numbers. Hence, the
following form of the slip-rotation lift force has been used:

r3 Re P
F1, =-Dp Cy, QX -u) (8)
8 Rer
with = 0.5 Vxu ) and the Reynolds number of
p D
particle rotation is given by ReR= P D9 /l The lift
coefficient according to Oesterle and Bui Dinh (1998) is
given for Rep < 2000 by:

ci = 0.45 + Re 0.45 je 05684Re Re' (9)

For the torque acting on a rotating particle the expression of
Rubinow and Keller (1961) was extended to account for the
relative motion between fluid and particle and higher
Reynolds numbers:


= 2 R
2 %


where the coefficient of rotation is obtained from Rubinow
and Keller (1961) and direct numerical simulations of Dennis
et al. (1980) in the following way:
64ff
64;T Re, < 32
Re R2 (11)
R 12.9 128.4
+- 32 ,[ReR ReR
The equations to calculate the particle motion are solved by
integration of the differential equations (Eqs. 2 4). For
sufficiently small time steps and assuming that the forces
remain constant during this time step, the new particle
location, the linear and angular velocities are calculated. The
time step for the particle tracking, AtL, was chosen as the
50 % of the smallest of all local relevant time scales, such as
the particle relaxation time, the integral time scale of
turbulence and the mean inter-particle collision time. This
choice guarantees the stability of the numerical integration
scheme. The instantaneous fluid velocity was obtained by
interpolating the fluid mean velocity from the neighboring
grid points to the particle position and adding a fluctuating
component obtained from a Langevin model (Sommerfeld et
al., 1993).
When a particle collides with a wall, the wall collision model
provides the new particle linear and angular velocities and
the new location in the computational domain after rebound.
The applied wall collision model, accounting for wall
roughness, is described in Sommerfeld & Huber (1999). The
wall roughness seen by the particle is simulated assuming
that the impact angle is composed of the particle trajectory
angle plus a stochastic contribution due to wall roughness,
Ay, which depends on the structure of wall roughness and
particle size. In sampling the instantaneous roughness angle
from a normal distribution with standard deviation Ay, the
so-called shadow effect was accounted for.
Inter-particle collisions are modelled by the stochastic
approach described in detail in Sommerfeld (2001). This
model relies on the generation of a fictitious collision partner
and accounts for a possible correlation of the instantaneous
velocities of colliding particles in turbulent flows. For the
particle-particle collisions the restitution coefficient has been
taken as a constant equal to 0.9 and the static and dynamic
friction coefficients were chosen to be 0.4.


Influence of Particles on the Carrier Flow


The source terms for the momentum equations resulting from
the exchange between particles and fluid are obtained on the
basis of the Particle-Source-In-Cell (PSI Cell) concept.
Hence, the momentum exchange is calculated by averaging
over all parcels traversing a given control volume during one
Lagrangian calculation. Instead of summing up all fluid
dynamic forces acting on the particles, which is quite
cumbersome, the momentum exchange is calculated from the
velocity change of the parcels when traversing the control
volume. In this procedure however, the external forces have
to be subtracted yielding the momentum source in the
following form:


Paper No





Paper No


Sp = ymkNkx
_v k (12)

u 1([ ,] u ] g[ i )I1AtL
n k PP)
where the sum over n indicates averaging along the particle
trajectory (time averaging) and the sum over k is related to
the number of computational particles passing the considered
control volume with the volume Vcv. The mass of an
individual particle is mk and Nk is the number of real particles
in one computational particle. AtL is the Lagrangian time step
which is used in the solution of (2)-(4).
The source term in the conservation equation of the turbulent
kinetic energy, k, are expressed in the Reynolds average
procedure as:

Skp = U, Sp U, Sp (13)
while the source term in the e-equation is modelled in the
standard way:


S = c3Skp (14)

with C13 = 1.8 and the sum is implicit in the repeated
sub-index i.
A converged solution of the coupled two-phase flow system
is obtained by successive solution of the Eulerian and
Lagrangian part, respectively. Initially, the flow field is
calculated without particle phase source terms until a
converged solution is achieved. Thereafter, a large number of
parcels are tracked through the flow field (in this case
240,000) and the source terms are sampled for each control
volume. In this first Lagrangian calculation inter-particle
collisions are not considered, since the required particle
phase properties are not yet available. Hence, for each control
volume the particle concentration, the local particle size
distribution and the size-velocity correlations for the mean
velocities and the rms values are sampled. These properties
are updated each Lagrangian iteration in order to allow
correct calculation of inter-particle collisions. Additional
particle phase properties and profiles may be sampled for
each transverse cell when the computational particle crosses
a pre-defined location. From the second Eulerian calculation,
the source terms of the dispersed phase are introduced using
an under-relaxation procedure (Kohnen et al., 1994). For the
present calculations typically about 25 to 35 coupling
iterations with an under-relaxation factor between 0.5 and 0.1
were necessary in order to yield convergence of the
Euler-Lagrange coupling.

Straight Pipe Computations

The considered configuration of the horizontal straight pipe
has a computational length of 10.6 m. According to the
configuration described in Huber & Sommerfeld (1998), the
pipe diameter is 0.15 m. and the mean conveying velocity is
27 m/s. The multi-block structured grid is composed of 5
blocks with a total of 560,000 hexahedral control volumes
(Fig. 1). Such a resolution was found sufficient to produce
grid-independent results.
At the inlet of the pipe, the fluid is injected with a uniform
velocity of 27 m/s and a turbulence intensity of around 3% of
the bulk flow velocity. At the pipe exit an outlet condition is


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

applied while at the pipe walls no-slip conditions are
adopted.
Regarding the two-phase flow computations, the mass
loading ratio considered in these measurements was rT = 0.7,
with a mean conveying velocity of also 27 m/s. The particle
phase consists of glass beads having a density of pp = 2500
kg/m3. The particle diameter distribution was considered
according to the experiments (see Fig. 2) using 7
computational diameter classes ranging from 20 to 80 pm
with a mean diameter of 40 pm and a standard deviation of
around 30 pm. The Stokesian response time of the considered
particles ranges between 3 and 49 ms, which implies that
they are influenced by the turbulence of the carrier phase.
The pipe material was stainless steel which is characterized
by a pretty high roughness. Therefore, the rms value of the
roughness angle distribution has been chosen to be constant
with Ay = 10.


L


Figure 1: Computational grid used for the simulation of the
3D flow through the horizontal straight pipe.


40 60
Particle Diameter [Am]


Figure 2: Particle size distribution in the experiments of
Huber & Sommerfeld (1998).

The particle injection velocities are sampled from a Gaussian
distribution with fixed mean and rms velocities. The mean
velocity is the bulk gas velocity, i.e. 27 m/s, in the
stream-wise direction and zero in the transverse components,
and the rms value is 3% of the bulk gas velocity for the three
velocity components. The particles are tracked through the
entire flow domain until they reach the exit of the pipe. As it
has been said before, in addition to drag also the transverse
lift forces have been considered in the particle equation of






Paper No


motion. Moreover, after a collision with a wall or with
another particle, the particles acquire high angular velocity
which implies that the particle angular momentum equation
(4) must also be solved.


'0 0.2 0.4 0.6 0.8 1 1.2
Normalised particle velocity [-]
Figure 3: Mean and rms normalised particle velocities after a
conveying distance of 8 m. Numerical computations versus
experiments of Huber & Sommerfeld (1998).


o0 -0.5 1 1.5 2 2.5
Normalised particle mass flux [-]
Figure 4: Normalised particle mass flux after a conveying
distance of 8 m. Numerical computations versus experiments
of Huber & Sommerfeld (1998).


Pipe length [m]
Figure 5: Gas phase pressure drop predicted by the numerics.
Single-phase flow versus two-phase flow with and without
inter-particle collisions (i.e., four-way and two-way
coupling).


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

Huber & Sommerfeld (1998) provide results only for the
properties of the solid phase (velocities, mean and rms, and
normalised particle mass flux) at a cross-section 8 m
downstream the inlet. Figures 3 and 4 show the comparison
between the numerical computations and the experimental
data, for particle velocities (fig. 3) and normalised particle
mass flux (fig. 4).
The quality of the numerical predictions with inter-particle
collisions is comparable to that obtained by Huber &
Sommerfeld (1998), where a deterministic collision model
was employed, instead of the stochastic approach of the
present paper.
As the pipe diameter is relatively large and the particles are
not very inertial, their behaviour is mainly influenced by the
gas turbulence. Therefore, the particle-wall collision
frequency is reduced compared to the cases with smaller
diameter pipe considered in Huber & Sommerfeld (1998).
This results in an effective dispersion of particles by
turbulence reflected in a fairly symmetric particle mean
velocity profile (Fig. 3). For the same reason, the dispersive
effect of wall roughness is reduced so the particles are not
homogeneously distributed in the pipe cross-section showing
a remarkable gravitational settling (Fig. 4). Moreover, the
consideration of inter-particle collisions promotes a better
dispersion of particles when compared with the case where
they are neglected (two-way coupling in Fig.4).


Y





S-9X













wo ^"^


Figure 6: Computational grid used for the simulation of the
3D flow through the horizontal-to-vertical elbow.

Fig. 5 shows the pressure drop in the pipe with and without
inter-particle collisions compared with that of the
single-phase flow. The results are presented by substracting
the static pressure at 2 m location, where the initial particle
acceleration has been already completed to a large extent. As
expected, the pressure loss in the particle-laden flow is higher
than in the single-phase flow. However, when inter-particle
collisions are disregarded, Ap is somewhat higher than in the
four way coupling case even though in this case the number
of particle-wall collisions, responsible for the additional


, 0o
/ 0
0
0

o
o





o
0'
0
0




O;o
S,
\ 9


* Exp.Up
- Num. Up 4-way
- Num. Up 2-way
0 Exp. u'p
S- Num. u'p 4-way


I .






Paper No


pressure loss, is slightly lower. This fact could be due to the
secondary flow generated by the particles, which is higher in
two-way than in four-way coupling, a fact currently being
investigated.

Horizontal-to-Vertical Elbow Computations

The considered configuration of the horizontal to vertical
elbow has an inlet horizontal pipe of 5 m in length and an
outlet vertical pipe of 5 m in length. According to the
configuration described in Huber & Sommerfeld (1998), the
pipe diameter is 0.15 m, the elbow radius is 2.54 times the
pipe diameter, and the mean conveying velocity is 27 m/s.
The multi-block structured grid is composed of 25 blocks
with a total of 568,000 hexahedral control volumes (Fig. 6).
Such a resolution was found sufficient to produce
grid-independent results.
As in the case of the straight pipe, at the inlet the fluid is
injected with a uniform velocity of 27 m/s and a turbulence
intensity of around 3% of the bulk flow velocity. At the pipe
exit an outlet condition is applied while at the pipe walls
no-slip conditions are adopted.
Regarding the two-phase flow computations, the mass
loading ratio considered in these measurements was rl = 0.3,
lower than in the straight pipe, with a mean conveying
velocity of also 27 m/s. The particle phase consists of glass
beads having a density of pp = 2500 kg/m3 with the particle
diameter distribution given according to the experiments (see
Fig. 2). Particle size discretisation as well as injection is the
same as for the straight pipe. As the wall material was the
same than in the straight pipe (stainless steel) the rms value of
the roughness angle distribution has also been chosen as Ay =
100.
The two phase flow in the bend is characterized by a
segregation of the mixture, being the particles accumulated at
the outer wall of the elbow due to inertial effects. As a
consequence, rather dense ropes of solids are formed in that
region leading to localised pretty high particle concentration
which supports the occurrence of inter-particle collisions. It
should be noted that even for the cases without inter-particle
collisions the particle volume fraction did not exceed the
value for maximum packing for mono-sized particles (i.e.
face-centred cubic and hexagonal close-packed with (p, =
0.74).
In order to illustrate some interesting phenomena occurring
in the pipe bend, it is convenient to first simplify the problem
by neglecting inter-particle collisions, i.e. conducting
two-way coupled calculations for the specified conditions.
The degree of accumulation of particles at the bend outer
wall depends strongly on wall roughness. When entering the
bend, particles are first driven towards the bend outer wall
due to secondary flow and mainly inertia (Fig. 7 and 8). After
rebounding from the bend outer wall, particles are farther
reflected back into the core flow of the bend for higher
roughness (Fig. 7). This behaviour is due to the shadow effect
of wall roughness (Sommerfeld, 2003; Lain & Sommerfeld,
2008a) implying that the averaged rebound angle becomes
larger than the impact angle (i.e. resulting in an averaged
transfer of wall parallel particle momentum towards the
transverse components). Hence, although a more or less
dense rope is formed at the outer wall of the bend exit, the
particles are distributed over a large portion of the
cross-section (Fig. 7). In the connecting vertical pipe the wall


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

rope is fragmented due to the secondary flow induced by the
bend, which however is being altered through the action of
the particles, and turbulent dispersion. The secondary flow
structure developing in the bend and the vertical pipe is
shown in Fig. 8, where two small counter-rotating vortices
can be appreciated at the 600 and 90 cross-sections in the
bend, i.e. the two upper ones. This new vortices are a result of
the particle reflection from the outer bend wall to the pipe
core and the associated momentum transfer to the fluid
(Sommerfeld & Lain, 2009). Eventually, these small vortices
disappear in the vertical duct after about 2 m from the elbow
exit where the solids are more homogenously distributed due
to turbulent particle dispersion.


Y


x


cm
I 3.43514
2.26363
1.49165
0.982943
0.647723
0.426825
0.281262
0.185341
0.122133
0.0804812
0.0530342
0.0230291
0,0151754









Y
0.01






1,49165
0,982943
0 647723
0426625
0,281262
0.25 0 185341
0 122133
C,0804812
C,0530342
S00349476
S0,0230291
00151754
001


Figure 7: Cross-sectional distribution of particle
concentration in kg/m3 (conveying velocity 27 m/s, rl = 0.3,
two-way coupling, roughness Ay = 100).

Moreover, as a consequence of the different behaviour of the
particles in the considered size spectrum with respect to
mean flow, turbulence and wall roughness, size segregation
will be observed in the flow through the pipe bend. This is
illustrated for the two-way coupled calculation in Fig. 9. Due
to inertia and secondary flow, particles tend to be
accumulated close to the bend outer wall with which they
experience a collision. Small particles after the rebound will
respond fairly fast to the carrier phase flow and tend to stay
near the wall where they are conveyed towards the bend inner
wall. The larger particles in the spectrum, on the other hand,
have a higher inertia and rebound farther towards the core of






Paper No


the cross-section almost reaching the inner part of the bend
after rebound (Fig. 9). In the region of the highest particle
concentration (Fig. 7) the number mean diameter
corresponds to that of the injected particle spectrum (i.e. 40
pm) as shown in Fig. 9. Rather strong changes in the pattern
of the mean diameter distribution are observed between the
bend exit and one meter downstream of it, likewise the
particle concentration distribution. Particle interaction with
the flow in the vertical pipe section and to some extent wall
collisions, redistribute the particles over the pipe
cross-section leading to a more homogeneous concentration
as well as mean diameter distribution.


Y VEL


X H24,1611
22.1477

14094
12.0805
10.0671
3 8.05369
6.04027
4.02685
2.01342














200.05
o






I 26,1879
26.1745
24,1611
22,1477
20.1342
18.1208
16.1074



-0.25 6.04027
















roughness Ay 100).
2.01342
0














four-way coupling), the particle rope developing in thof absolute flowbend





is forced to be more concentrated and the particles are
velocdistributed in smaller partreamlines of the cross-section near the outer
wall of the lines) for the particle result with only the benwo-way
(conveying velocity 27 m/s, il = 0.3, two-way coupling,





coupling (compare Fig. 7 and Fig. 10). This is caused by
When also inter-particle collisions are taken into account (i.e.




four-way coupling), thellisions of particle rope developing int the bend with those
is forced to be more concentrated and the particles are
distributed in a smaller part of the cross-section near the outer
wall of the bend compared to the result with only two-way
coupling (compare Fig. 7 and Fig. 10). This is caused by
collisions of particles moving into the bend with those
rebounding from the bend outer wall. Hence, the rebound
particles are "pushed" back towards the bend outer wall and
will again collide with it. Also the re-dispersion of the


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


particles in the vertical pipe section is hindered by
inter-particle collisions. Instead a kind of closed rope
develops in the core region which moves upward and is only
dispersed 3 m downstream the bend exit (Fig. 10).


x
X


Figure 9: Cross-sectional distribution
mean diameter in pm (conveying veloci
two-way coupling, roughness Ay = 100).


dp
7 4702E-05
6,9404E-05
5.4106E-05
5.88079E-05
5.35099E-05
4.82119 05
429139E 05
3 76159E-05
3,23179E-05
2.70199E-05
2.17219E-05
1. 64238E-05
1.11258E-05
5.82781E-06










Y








6.9404E-05
6.4106E-05
5.88079E-05
5.35099E-05
4.82119E-05
4.29139E-05
-0.25 3.76159E-05
3.23179E-05
2.70199E 05
2.17219E-05
1.64238E-05
1.11258E-05
5,82781 E-06
5.29801 E-07




of particle number
ty 27 m/s, Tr = 0.3,


A better perspective of the particle concentration distribution
can be obtained from Fig. 11, where a section of the elbow
mid-plane is presented. Here, the stronger particle "roping"
in the case which considers inter-particle collisions can be
clearly distinguished.
Additionally, the particle size segregation is not as
pronounced as in the two-way coupling case although there is
a slight increase of the particle mean diameter at the elbow
exit from the outer wall towards the pipe centre (Fig. 12).
This fact can be also observed in Huber & Sommerfeld
(1994) and Quek et al. (2005), for instance.
As a consequence of particle motion and behaviour in the
elbow the secondary flow pattern within bend and connecting
vertical pipe is altered as shown in Fig. 13. Inter-particle
collisions prevents that the more inertial particles penetrate
back into the core of the pipe avoiding the transfer of
momentum to the fluid responsible for the appearance of the
two small counter-rotating vortices at the outer cross-sections
of the bend (see Fig. 8). In this respect, in the case of
four-way coupling the secondary flow pattern is qualitatively
similar to the single phase flow, where only two large
recirculation cells appear in the elbow.







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


Y


x


Y


X


I 3,43514
2.26363
1.49165
0.982943
0 647723
0 426625
0,281262
0.185341
0.122133
0.0804812
0.0530342
0 0349476
0 0230291
00151754
go,










Y

Z





cm
I 3.43514
2.26363
1.49165
0.982943
0.647723
0.426825
0.281262
>.25 0.185341
0.122133
0.0804812
0.0530342
0.0349476
0.0230291
0.0151754
0.01


dp
I 7.4702E-0 5
6 9404E-05
5.4106E-05
5.88079 E-05
5.35099 E-05
482119E-05
4.29139E-05
3.76159E-05
323179E-05
270199E-05
2.17219E-05
1.64238E-05
1.11258E-05
5 82781E-06
5 2901 E-07










Y







dp
I 7.4702E-05
6.9404E-05
6.4106E-05
5.88079E-05
5.35099E-05
4.82119E-05
4.29139E-05
>.25 3.76159E-05
3.23179E-05
2.70199E-05
2.17219E-05
1.64238E-05
1,11258E-05
5.82781 E-06
5.29801 E-07


Figure 10: Cross-sectional distribution
concentration in kg/m3 (conveying velocity 27
four-way coupling, roughness Ay = 100).


cm
I 343514
226363
1 49165
0982943
0 547723
0425825
0281262
0 185341
S122133
0 004812
0 0530342
S00349476
00230291
00151754
001








1 343514
226363
1 49165
0 982943
0647723
0426825
0281262
0185341
0 122133
00804812
00530342
00349471
0 0230291
00151754
001


9


of particle
m/s, rl = 0.3,


Figure 11: Particle concentration in kg/m3 in the elbow
mid-plane. Tr = 0.3, four-way coupling, Ay= 100. Two-way
coupling (top) and four-way coupling (bottom).


Figure 12: Cross-sectional distributions of particle number
mean diameter in pm (conveying velocity 27 m/s, rl = 0.3,
four-way coupling, roughness Ay = 100).


An additional effect of inter-particle collisions within the
bend is the increase of particle-wall collisions with the outer
wall compared to the two-way coupling case. This fact is
illustrated in Fig. 14, where the normalised number of
particle-wall collisions per unit area is shown in the unfolded
outer bend for two-way (top) as well as four-way coupling. In
Fig. 14, the theta coordinate represents the angle along the
bend, taking the value of 0 in the end of the horizontal pipe
and the value of 90 in the beginning of the vertical pipe. On
the other hand, the phi coordinate is the angle measured from
the pipe centre towards the outer wall of the bend starting at x
= D/2 (phi = -90) and finishing at x = D/2 (phi = 90). From
this figure it can be readily seen that the majority of
particle-wall collisions occurs between -200 in the most outer part of the elbow. Also, the highest density
of particle-wall collisions appears between 30 < theta < 60,
in the two-way coupling, but it is extended towards the bend
exit for the four-way coupling case.
As it has been discussed in Lain & Sommerfeld (2009), the
pressure loss in the elbow rises as mass loading increases.
However, in this work we are interested in the pressure loss
and secondary flow intensity modification by considering
inter-particle collisions. Secondary flow intensity is defined
as the quotient of the average cross-sectional gas velocity,

Ucrossav, in the bend and the bulk gas velocity (i.e., 27 m/s),
with:


Paper No






Paper No


VEL
281879
261745
24 511
221477
201342
181208
161074
14094
120805
100571
605359
S6,04027
4.02585
201342
0


0.0071 0 069 0.1657 0.2466 0.3264 0.4062 0.4860 0.5558 0.6457 0.7255


Figure 13: Cross-sectional distribution of absolute flow
velocity in m/s (colour) and streamlines of the cross-sectional
flow (white lines) for the particle flow through the bend
(conveying velocity 27 m/s, rl = 0.3, four-way coupling,
roughness, Ay = 100).


U2 crossav -


, ZvC(U ++U
VCVbnd =CV+
Vbend Z=CV


where Vbend is the volume of the bend and Ut is the gas
velocity component being in the particular cross-section of
the bend which is also perpendicular to x-direction.
The results are presented in Table 2 for a mass loading ratio
of r = 0.3 and rms wall roughness angle Ay= 10.


Two-way Four-way Single-phase
Ap [Pa] 828 802 555

Secondary flow 10.2 11.0 11.4
intensity [%]


Table 2: Pressure loss and secondary flow intensity at the
bend for the case rl = 0.3 and Ay = 10.

As it can be observed, the pressure loss in case of neglecting
inter-particle collisions is slightly higher than in the four-way
coupling case, as it also happened in the straight pipe. On the
other hand, the secondary flow intensity in the bend is
reduced by particles, being lower in the two-way than in the


Figure 14. Normalized number of particle-wall collisions per
unit area maps for the two-way (top) and four-way (bottom)
cases in the unfolded bend (see text for details).


Finally, with the objective of providing an idea about the
quality of the predictions, the results obtained with four-way
coupling and a rms value of the roughness angle distribution
Ay = 100 are compared with the experiments of Huber &
Sommerfeld (1998). The comparison of the calculations of
particle mean velocity, normalised with the bulk flow
velocity, with the measurements at three cross-sections
located in the vertical pipe downstream of the elbow (i.e. y =
0, 1 m and 4 m) is shown in Fig. 15. Here the inner wall of the
elbow is on the left side. It can be seen that the tendencies
shown by the experiments are reproduced by the calculations
accounting for wall roughness and inter-particle collisions
although some quantitative differences can be identified
close to the elbow inner wall (left side of the plots).
Fig. 16 presents the comparison for the normalised particle
mass flux at the same cross-sections in the vertical pipe.
Although the agreement at the elbow exit is reasonable,
capturing the magnitude of the particle mass flux peak, larger
discrepancies between calculations and measurements are
observed for the upper cross-sections. In the intermediate


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

four-way coupling case. This damping of the secondary flow
intensity could be expected as particles extract momentum
from the carrier phase. This mechanism is more efficient
when particles are better distributed across the pipe section (a
fact that happens in two-way coupling, compare Figs. 7 and
10).


Y

Z


S0071 0.0869 0.1667 02466 03254 0.4062 0.4860 05656 06457 0.7255


Y

z
X
S28.1879
26.1745
24.1611
22.1477
20.1342
18.1208
16.1074
14.094
12.0805
10.0671
8.05369
).265 6.04027
4.02585
2.01342
0






Paper No


cross-section the calculated maximum in the particle mass
flux is lower than the measured one but it is located in the
same area, close to the bend outer wall. Moreover, a
reasonable prediction of the particle mass flux in the highest
cross-section is observed, getting a nearly uniform
distribution similar to the experimental one. Therefore, the
present computations capture the essential quantitative
characteristics of the particle rope dispersion within the
vertical pipe after the bend.
2 1 I I I I I I I I I I I
Exp. y=0m
3 o Exp. y= I m
A Exp. y =4 m
S1,5 -- Numerical y=0 m
S ... Numerical y = m
- Numerical y = 4 m




A-1 A N




-1 0,75 -0,5 -0,25 0 0,25 0,5 0,75 1
z/R[-]
Figure 15: Calculated normalised particle mean velocity
compared with experimental data of Huber and Sommerfeld
(1998) in the vertical pipe downstream of the bend exit
(conveying velocity 27 m/s, r = 0.3, four-way coupling,
roughness Ay= 100).

10 I 1 I 1
Exp. y=Om -
0 Exp. y= m
S A Exp. y= 4 m
C -- Numerical y = 0 m
S- Numerical y =m
6 - Numerical y = 4 m


4-

2013
5 9
2*- 1... ....


0 -0,5 0 0,5 1
z/R [-]
Figure 16: Calculated normalised particle mass flux
compared with experimental data of Huber and Sommerfeld
(1998) in the vertical pipe downstream of the bend exit
(conveying velocity 27 m/s, r = 0.3, four-way coupling,
roughness Ay = 100).

Conclusions

The structure of the particle-laden gas flow through different
conveying system elements has been analised numerically. A
straight pipe as wall as a horizontal-to-vertical elbow have
been investigated applying the fully coupled Euler/Lagrange
approach. Essential for numerically predicting such denser
confined two-phase flows is the consideration of particle
rough-wall collisions as well as inter-particle collisions. As a
result, in both configurations the pressure drop is higher
when inter-particle collisions are neglected than when they


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

are taken into account. This fact is due to the higher gas
momentum extraction by the particles in the two-way
coupling case. Moreover, inter-particle collisions imply a
denser particle rope developing within the elbow as they
prevent larger particles to be bounced back towards the core
of the pipe. The secondary flow evolving in the pipe bend,
known from single phase flows, is altered by the particles due
to trajectory focussing and momentum exchange between the
phases. Consequently, two small recirculation cells are
developing within the bend and in the connecting vertical
pipe in the two-way coupling case. On the contrary, in the
four-way coupling the structure of the secondary flow is
similar to the single phase one, but with reduced intensity.
Finally, the computational results with inter-particle
collisions have been compared with the experiments of
Huber & Sommerfeld (1998) showing a good enough
quantitative agreement in all the particle variables available.

Acknowledgements

The fruitful support by Engineer Leonard Duefias is
gratefully acknowledged.

References

Dennis, S.C.R., Singh, S.N. & Ingham, D.B. The steady flow
due to a rotating sphere at low and moderate Reynolds
numbers. J. Fluid Mech., vol. 101, 257-279 (1980).

Huber, N. & Sommerfeld, M. Modelling and numerical
calculation of dilute-phase pneumatic conveying in pipe
systems. Powder Technology, vol. 99, 90-101 (1998).

Huber, N. & Sommerfeld, M. Characterization of the
cross-sectional particle concentration distribution in
pneumatic conveying systems. Powder Technology, vol. 79,
191-210 (1994).

Kohnen, G., Rager, M. & Sommerfeld, M. Convergence
behaviour for numerical calculations by the Euler/Lagrange
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Kohnen, G. & Sommerfeld, M. The effect of turbulence
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Lain, S., Sommerfeld, M. & Kussin, J. Experimental studies
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Lain, S. & Sommerfeld, M. Euler/Lagrange computations of
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Lain, S. & Sommerfeld, M. Structure and pressure drop in
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Mei, R. An approximate expression for the shear lift force on
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spinning sphere in a range of intermediate Reynolds numbers.
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in horizontal and vertical bends. Ind. Eng. Chem. Res., vol.
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Rubinow, S.I. & Keller, J.B. The transverse force on a
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