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.

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