7th International Conference on Multiphase Flow,

ICMF 2010, Tampa, FL, May 30 June 4, 2010

Developing particle-laden turbulent pipe-flow

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

Linn6 Flow Centre, Department of Mechanics, KTH, SE-100 44 Stockholm, Sweden

tobias@mech.kth.se, geert@mech.kth.se, gustava@mech.kth.se and johansson@mech.kth.se

Keywords: particle-laden gas, turbulent flow, pipe-flow, modelling

Abstract

An Eulerian turbulent two phase flow model using kinetic theory of granular flows for the particle phase was

developed in order to study evolving upward turbulent gas particle pipe flow. The model takes into account the

feedback of the particles. Simulations show that the pipe length required for particle laden turbulent flow to become

fully developed is up to five times longer than an unladen flow. To increase the understanding of the dependence

of the development length on particle diameter a simple model for the expected development length was derived. It

shows that the development length becomes shorter for increasing particle diameters, which agrees with simulations.

The initial slip velocity has also a large influence on the development length.

Introduction

The objective of this paper is to study the development of

turbulent gas-particle flow in a pipe. Studies of evolving

turbulent gas-particle pipe flows are of significant inter-

est since in many industrial processes the turbulent par-

ticle laden flows do not become fully developed. How-

ever, there are only a limited number of studies of these

flows. Picano et al. (2009) studied the spatial develop-

ment of particle concentration in a pipe using direct nu-

merical simulation (DNS) and found that the develop-

ment length increased with the Stokes numbers of the

particles. DNS by Portela et al. (2002) showed that a

very long developing length (~ 300 pipe diameters ) is

required in order to get statistically steady particle con-

centrations. Str6mgren et al. (2009a) also found, using

model simulations that particles substantially increase

the pipe length required for the flow to become fully

developed. Cerbelli et al. (2001) analysed the transient

behaviour of the particle concentration in pipe flow and

developed a model that better captured the turbophoretic

drift of particles.

In this study a two-phase flow model with particle

stresses based on kinetic theory of granular flow is im-

plemented in order to study developing turbulent gas-

particle flows for different particle Stokes numbers.

Conservation Equations

First, the averaged equations governing turbulent gas-

particle flows are presented (Anderson & Jackson 1967).

The generalised volume fraction weighted averaged con-

tinuity equation can be written as

at ai

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

Subscript k refers to the phase (k g for the gas phase

and k p for the particle phase), p, is the mean density,

+< is the mean volume fraction, Uk,i uTti is the mean

velocity and uh,t is the instantaneous velocity. Global

continuity requires b, + ~p 1. The mean momentum

equation is written as

a (PkhkU,i) + (pkhkUk,iUk,j)

a ) +

(9Jh,ij) + Iht + tPhti

Oxj

Pk

Ox,

(2)

Here P, is the gas phase pressure present in both phases,

HI,ij is the stress term representing the viscous and tur-

bulent stress in the gas phase and the kinematic and col-

lisional stress for the particle stress. Iki is the Stokes

drag and is modelled according to

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

Tp

where ,T = pP is the particle response time, Vg is the

gas phase cinematic viscosity and Dp is the particle di-

ameter. Rer u D is the relative Reynolds number,

Ur,i (Up,, U9,i) Ud,i is the mean relative velocity

and Ud is the average of the fluid velocity fluctuations

with respect to the particle distribution (Simonin et al.

1993), and is modelled according to

U 1 ts { 1 K,

U ^, (^;

ud'^ S p M~c

1, D4' N

where 7~g is the correlation time scale of the fluid tur-

bulence viewed by the particles and Kp .' u =

2K93T (Hadinoto & Curtis 2004) is the gas-particle

velocity correlation, K = u'u'i and T = ..' .

2 Yi Yi 3

is the gas phase mean turbulent kinetic energy and the

granular temperature, respectively.

Gas phase. Viscous stresses in the gas phase are given

by

7th International Conference on Multiphase Flow,

ICMF 2010, Tampa, FL, May 30 June 4, 2010

is the radial distribution function where axa = 0.64

is the maximum value of bp for random packing of

spheres.

4 T

A1P 3 Dp1 pppgo(1 + ep)J

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

kin 2 Tp, 7 2 c 1

P 1\ P39 P -1 g

(11)

where T ..' is the granular temperature and

t^ Y Y(1 + C 12 is the time-scale of the fluid

turbulence viewed by the particles where

where Vg is the kinematic viscosity of the gas, Sk,i,

(1 u + ` ) is the mean strain rate tensor and

last term is the Reynolds stress which is closed using

Boussinesq assumption

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

Yx3 0m

Here vt, is the turbulent viscosity. To close this term

low Reynolds number K w model by Wilcox (1S

is used with an additional term due to the interaction

tween the gas and the particle phase both in the K9

the w equation, see Str6mgren et al. (2009a). w is

inverse time scale of the gas phase turbulence.

Particle phase. In order to describe the stress:

the particle phase kinetic theory of granular flow

adopted. This model has earlier been used by

example, Peirano & Leckner (1998), Benyahia el

(2005), Benavides & van Wachem (2008), Zhant

Reese (2001) and Str6mgren et al. (2009b). Follow

this approach the stress due to particle-particle collisi

is written as

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

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

where A, is the is the bulk viscosity, Pp T(

240pgo(l + ep)) is the granular pressure, Vkin is

kinematic viscosity, Vp01 is the viscosity due to parti

particle collisions, vkin is the particle viscosity du

kinematic stresses, e, is the coefficient of restitution

1

go= 1+( )1/3

C 1.8 1.35 Upi

SUriI lUp

31U,2 '

T ~) is the Eulerian time-scale of the gas phase tur-

bulence,

Pco1 Dp 12 (14)

S 244pgoT (T

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

1) and a, (1 + ep)(3 ep).

- (3 p, CD -1.7 r)-1

P9 4pp Dp

(1 + 0.15Re0.687)

ing is the characteristic time-scale of gas-particle momen-

ions tum transfer where CD 2 (1 + 0.15Re 687) is the

drag coefficient.

The equation for the granular temperature can be de-

rived from the momentum equation of the particle phase.

(7) In order to close the unclosed terms the kinetic theory of

granular flows is used. The equation for T becomes

3 DT 3 OT

1 2+ PP 2p(Ii 2p j3PJpU

the

cle- 2pp9,p( >in .S, pp T

e to p (16)

and + 0 3 p(, + ,. )T

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

TP

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

09x

-D 1' .1 g ) (5)

where

(3^^ + ,o 3(l (1 e+) (2e

TK ,T go5(1 +eT) (2ev

9 +

is the kinetic contribution to the diffusivity,

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

is the collisional contribution to the diffusivity and p =

10(1 + ep)(49 33ep).

Simulations of evolving gas-particle flow in a

pipe

Simulation setup. An upward turbulent particle laden

flow in a vertical pipe has been simulated. The pipe was

200 D long, which was long enough to achieve a fully

developed flow. The boundary conditions developed by

Johnson & Jackson (1987) were used for the particle

velocity and the granular temperature. The gas phase

velocity has a no-slip condition at the wall whereas the

particle volume fraction and the pressure have a Neu-

mann condition at the wall. There is no flux of particles

in the wall-normal direction. The particle volume frac-

tion was initially homogeneous with o = 5 10-4 and

pp/p, 2000. The inlet conditions were constant for

gas- and particle velocities, K., c and +p. At the outlet

only the pressure has a Dirichlet condition. Two dif-

ferent initial conditions for the slip velocity were used

to investigate its effect on the development length. The

model described above was implemented in cylindrical

coordinates in a finite element code (Do-Quang et al.

2007).

Simulations were made for Stokes numbers between

St+ 34 and 3950. The Stokes number is defined as

where u, = T lpis the friction velocity and T, is

the wall shear stress. The Reynolds number, Re D= u

is 24000 where D is the pipe diameter. An unladen sim-

ulation for this Reynolds number was also performed.

The simulations were run until the flow was steady.

The model has been validated with several sets of

experimental and direct numerical simulation data in

Str6mgren et al. (2009a) and Str6mgren et al. (2009b).

It is shown that the model predicts the gas- and particle

mean- and rms velocities very well.

7th International Conference on Multiphase Flow,

ICMF 2010, Tampa, FL, May 30 June 4, 2010

(a)

08

)6

)4

2 x/D=5

--x/D=25

x/D 50

x/D-150

0 02 04 06 08 1

x/D=5

x/D=25

x/D=50

Figure 1: Mean gas phase velocity profiles normalised

with the gas phase center-line velocity at four different

downstream positions. St+ 34 (a); St+ 1775 (b).

Model results. In order to understand the axial devel-

opment of velocity and particle concentration profiles at

different downstream positions in the pipe are compared

for St = 34 and 1775. Here x is the downstream dis-

tance from the inlet of the pipe.

In figure 1 the gas phase velocity profiles normalised

with the gas phase center-line velocity (Ug,c1) at differ-

ent downstream positions are shown for St+ 34 and

St+ 1775, respectively. For St+ 1775 (figure lb)

the gas phase velocity profile is almost fully developed

at x/D=20 while a much longer development length is

required for St+ 34(figure la).

The mean particle velocity profile normalised with

U,,c1 at different downstream positions is shown in fig-

ure 2 for St+ 34 and St+ 1775, respectively. For

St = 34(figure 2a) the particle center-line velocity is

nearly equal to the gas velocity at all positions but the

velocity at the wall decreases with x/D, i.e. the profile

becomes less flat. At larger St+ (figure 2b) the parti-

cle phase velocity profile is uniform at all downstream

7th International Conference on Multiphase Flow,

ICMF 2010, Tampa, FL, May 30 June 4, 2010

x/D=5

x/D 25

x/D-50

x/D 15J

14

12

08

06

04 06 08

02 04 06 08

r/R

(b)

-x/D5

x/D 25

x/D-50

x/D-150

14

12

08

02 x/D=5

x/D=25

x/D=50

x/D=150

0 02

0 02

04 06 08

r/R

Figure 2: Mean particle phase velocity profiles nor-

malised with the gas phase center-line velocity at four

different downstream positions. St 34 (a); St

1775 (b).

positions whereas the velocity decreases with increas-

ing x/D. The final velocity at the wall is about the same

for both cases. The more uniform profile at large Stokes

numbers is due to a weaker coupling between the phases.

Since the wall is smooth the particles experience almost

a free slip condition.

Profiles of the mean particle volume fraction, %p, nor-

malised with its initial value, Go, are shown at four

downstream positions in figure 3a and b for St+ 34

and St 1775, respectively. For St = 34(figure 3a)

the particles accumulate in the center of the pipe and in-

creases with x/D. The particle concentration in the cen-

ter of the pipe is almost three times higher than near the

wall when the flow is fully developed. A higher parti-

cle concentration around the pipe axis was also found in

experiments by Tanaka & Tsuji (1991).

In figure 4 the slip velocity (Up U,) normalised with

the center-line velocity is shown at four different down-

stream positions for St+ 1775 and 3950, respectively.

02 04 06 08

r/R

Figure 3: Mean particle concentration profiles nor-

malised with the center-line velocity at four different

downstream positions. St+ 34 (a); St+ 1775 (b).

The development length is seen to increase with increas-

ing St In order to investigate the effect of slip velocity

the same case as in figure 4 is shown in figure 5 with the

difference that the inlet slip velocity is put equal to the

final slip velocity at the outlet in order to neutralise the

effect acceleration or deceleration of the particles on the

development length. The development length is found

to be significantly shorter when the inlet is adjusted to

match the slip velocity for a fully developed flow. Inlet

slip velocity has thus a large influence on the develop-

ment length.

Development length. The influence of the particle di-

ameter and mass loading on the length required for the

flow to become fully developed will be studied in more

detail. First an estimation of the development length

of the particle phase is made by studying the govern-

ing equations in order to understand which parameters

govern this length. In a fully developed flow the convec-

tive term (second term on the L.H.S.) must be smaller

than the most dominating term in equation (2), which is

08

02 x/D=5

x/D=25

x/D=50

x/D=150

0 02

0 02

x/D=5

x/D=25

0 x/D=50

x/D=150

02 04 06 08 1

x/D=5

x/D=25

0 8 x/D50

x/D=150

06-

04-

02-

7th International Conference on Multiphase Flow,

ICMF 2010, Tampa, FL, May 30 June 4, 2010

(a)

x/D=5

--x/D25

0 8 x/D=50

x/D=150

x/D=5

x/D=25

0 8 x/D=50

x/D=150

06

02 04 06 08

r/R

Figure 4: The slip velocity normalised with the the gas

phase center-line velocity at four different downstream

positions. St 1775 (a); St 3950 (b).

the diffusion term (second term on the R.H.S.). Since

vP1 < kin for dilute flows (ip < 1 10-3) this con-

dition becomes

Ups 8 J 8Upi 8Up 0

Up3 < kivp ( ,+ O)U (20)

The derivatives in the convective and the diffusive term

are estimated as U/1 and U/D2, respectively. Here U is

the characteristic mean streamwise velocity and I is the

axial development length. Equation (20) then yields

U2 'in U

< D (21)

After rewriting, the development length can be ex-

pressed as

1 DU

D > in (22)

u v^

Since

kin T )( 21

S- Kpg ` + T) ( + ) 0

3 T 3 pf T,-p + Ipco

Figure 5: The slip velocity normalised with the the gas

phase center-line velocity at four different downstream

positions. St+ 1775 (a); St+ 3950 (b).

In the limit of small Stokes numbers (St+ << 1) and

small (Dp (< 1 10 4) the time between particle-particle

collisions, Tp01 (eqn. 14), will be larger than T7 (eqn.

15) since T ~ D2 and Teo1 ~ Dp. For the same reason

Ti-g/- >> 1. Furthermore, K,, 2K 3T and

Tpg T. since U,. 0. With these simplifications vkin

can be estimated as

kin 2 Tt Tf

S -" = -2K

Vp 3 f1 2

2K 3

3 2w

Vta (24)

i.e. the kinematic viscosity of the particles approaches

the turbulent viscosity of the gas phase, which is in ac-

cordance with Tchen-Hinze's theory (Hinze 1959). The

development length is then the same as for single phase

flow, i.e.

1 DU

S> -(25)

D Vt9

) For large Stokes numbers (St+ >> 1) Tpg/7T < 1

23) and since has the same order of magnitude as T

and since K, has the same order of magnitude as T

7th International Conference on Multiphase Flow,

ICMF 2010, Tampa, FL, May 30 June 4, 2010

equation (23) can be written as

-T( )1 (26)

Tpg Ip

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

unity we get

140

U

U

120

100

80

+p 1-+1 18pgg 24tp IF

P ppD D- V prD

An estimate of the development length for gas-pal

flows when St+ >> 1 and particle-particle colli,

are not important (p < 1 10 3) is thus

I DU 18pyg 241)p T

D T ppD D, 7

V P

I..... ,0oo

(28) Figure 6: Development length, x/D,

St+ for U, Up and bp.

as a function of

Consequently, the development length becomes shorter

for increasing Dp and longer for increasing bp.

When 4p > 1 10-3 particle-particle collisions will

have a large impact on the flow and Tp7, < 7,7. The

development length can then be estimated as

1 DU

D TT0_1

DU 24p T

T D, w

Also here the development length becomes shorter for

increasing Dp and longer for increasing %p. However,

this parameter range is outside the scope of this study.

An estimation as above can also be made using the

equation for the mean granular temperature (eqn. 16),

but, similar results for the development length will be

found.

Comparison to model results. The estimation of the

development length in equation (28) applies for the pa-

rameter range used in this study and will be compared to

numerical simulations.

Using equation (28) the development lengths for U.,

Up and bp are shown as a function of St+ in figure 6.

The development length is here defined as the distance

from the inlet where the maximum difference between

the radial profile and the one at 180 D is less than 1%.

For St+ < 500 the development length decreases for in-

creasing Stokes numbers (Dp) for all three variables in

agreement with the estimation. The development length

for U. keeps on decreasing until it reaches the devel-

opment length of an unladen flow (x/D=21) because the

coupling between the two phases weakens. In contrast,

the development length of Up and bp start to increase

with St+ for St+ > 500. For St+ = 3950 it is about

five times that of an unladen flow. However, according

to equation (28) the development length for the particle

phase should decrease with St+. This discrepancy for

large Stokes numbers is likely related to the increasing

slip velocity with particle diameter.

S ...... ..... .. oo

st+

Figure 7: Development length, x/D, as a function of

St+ for U, Up and bp.

To examine the influence of the slip velocity all cases

in figure 6 are recalculated with inlet slip velocities ad-

justed to match the magnitude of the fully developed slip

velocities. The results are shown together with the es-

timation of the development length in equation (28) in

figure 7. The bulk values are chosen for the velocity

and granular temperature in equation (28). Compared

to figure 6 where the initial slip velocity is zero the de-

velopment length is much shorter for St+ > 500. With

these initial conditions the development length decreases

even for large Stokes numbers (Dp) which agrees qual-

itatively and quantitatively with the derived estimation

for the development length in equation (28).

Conclusions

A two-fluid model using the kinetic theory of granu-

lar flow for the particle phase was developed to study

Equation (28)

. . I .... I .. .

an evolving upward particle laden turbulent pipe flow.

We have investigated the influence of the Stokes num-

ber on the pipe length required for the flow to become

fully developed. It was found that the particles can make

the development length for the gas velocity up to three

times longer than that of an unladen flow whereas the

development length of the particle velocity can be up to

five times longer than that of the gas velocity of an un-

laden flow. To understand what governs the development

length a simple estimation for this length was derived

from the particle momentum equation. This estimation

showed that the development length decreases with in-

creasing St This was confirmed by model simulations

for St+ < 500. For larger St+ the development length

is very sensitive to the initial slip velocity between the

phases. When the initial slip velocity is close to the fi-

nal slip velocity the development length is much shorter

than for cases with a zero initial slip velocity.

The development of the particle concentration profiles

were also studied and a net flow of particles to the center

of the pipe was observed. This drift was largest for small

particle diameters.

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