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Flow Turbulence Combust (2011) 86:477–495DOI 10.1007/s10494-011-9335-2

A Modelling Study of Evolving Particle-ladenTurbulent Pipe-flow

Tobias Strömgren · Geert Brethouwer ·Gustav Amberg · Arne V. Johansson

Received: 15 December 2009 / Accepted: 10 February 2011 / Published online: 5 March 2011© Springer Science+Business Media B.V. 2011

Abstract An Eulerian turbulent two phase flow model using kinetic theory ofgranular flows for the particle phase was developed in order to study evolving upwardturbulent gas particle flows in a pipe. The model takes the feedback of the particlesinto account and its results agree well with experiments. Simulations show that thepipe length required for particle laden turbulent flow to become fully developedis up to five times longer than an unladen flow. To increase the understanding ofthe dependence of the development length on particle diameter a simple model forthe expected development length was derived. It shows that the development lengthbecomes shorter for increasing particle diameters, which agrees with simulations upto a particle diameter of 100 μm. Thereafter the development length becomes longeragain for increasing particle diameters because larger particles need a longer time toadjust to the velocity of the carrier phase.

Keywords Turbulent gas-particle flows · Modelling

1 Introduction

Turbulent gas-particle flows are encountered in a variety of industrial processes suchas cleaning of polluted gases, solid fuel combustion and pharmaceutical- and foodprocessing. In these flows the two phases interact with each other. Particles aretransported by the mean flow but particles can also affect the mean flow [19, 30].

T. Strömgren (B) · G. Brethouwer · G. Amberg · A. V. JohanssonLinné Flow Centre, KTH–Mekanik, 100 44 Stockholm, Swedene-mail: [email protected]

G. Brethouwere-mail: [email protected]

478 Flow Turbulence Combust (2011) 86:477–495

Momentum transfer from the solid phase to the carrier phase can either augmentor attenuate the turbulence intensity of the carrier phase [14]. Collisions betweenparticles can affect the flow even for rather small particle concentrations [32].Taking all those interactions into account make simulations of gas-particle flowscomplex.

As far as we are aware of evolving turbulent particle laden pipe flows have onlybeen studied by Picano et al. [22], Portela et al. [23], Cerbelli et al. [7] and Strömgrenet al. [28]. Nevertheless, it is of significant interest since in many industrial processesthe turbulent particle laden flows will not reach a fully developed state. Picanoet al. [22] studied the spatial development of particle concentration in a pipe usingdirect numerical simulation (DNS) and found that the development length increasedwith the Stokes numbers of the particles. DNS by Portela et al. [23] showed thata very long developing length (∼300 pipe diameters ) is required in order to getstatistically steady particle concentrations. Cerbelli et al. [7] analysed the transientbehaviour of the particle concentration and developed a model that better capturedthe turbophoretic drift of particles.

The effect of the particle phase on an evolving pipe flow was not addressed in theprevious studies. Strömgren et al. [28] studied an evolving particle laden turbulentdownward pipe flow for moderate Stokes numbers (St+ ≤ 70, where St+ is the ratiobetween the particle response time and the viscous time scale) using an Euleriantwo-fluid model and found that the pipe length required for the flow to become fullydeveloped increased with particle loading. The objective of this work is to continuethe study by Strömgren et al. [28] by investigating the effect of the particles on thepipe length required for the flow to become fully developed for a broader range ofStokes numbers.

In order to study gas-particle flow in a pipe the Eulerian two-phase flow modelused in Strömgren et al. [27] taking into account the feedback of the particles isextended to cylindrical coordinates. The benefits of a model compared to DNS isthat cases resembling industrial applications can be studied, i.e. larger Reynoldsnumbers, more complex geometries and higher particle concentration. This studyshows that even for rather low mass loadings the pipe length needed for the flowto become fully developed is five times longer than for an unladen simulation. Theinfluence of the particles on the mean turbulent kinetic energy of the gas phase andthe accumulation of particles in regions with low turbulence intensity are studiedas well.

2 Conservation Equations

The averaged equations governing turbulent gas-particle flows are presented here[27]. For details about the derivation see Jackson [16]. The generalised volumefraction weighted averaged continuity equation can be written as

∂

∂t(ρk�k) + ∂

∂xi(ρk�kUk,i) = 0 (1)

Subscript k refers to the phase (k = g for the gas phase and k = p for the particlephase), ρk is the mean density, �k is the mean volume fraction, Uk,i = uk,i is the mean

Flow Turbulence Combust (2011) 86:477–495 479

velocity and uk,i is the instantaneous velocity. Global continuity requires �g + �p =1. The mean momentum equation is written as

∂

∂t

(ρk�kUk,i

) + ∂

∂x j

(ρk�kUk,iUk, j

) = −�k∂ Pg

∂xi− ∂τk,ij

∂x j+ Ik,i + �kρkgi (2)

Here Pg is the gas phase pressure present in both phases, τk,ij is the stress term. Ik,i

is the inter phase momentum exchange and is modelled according to

Ip,i = −Ig,i = (1 + 0.15Re0.687

r

) �pρp

τpUr,i (3)

where the term inside brackets is a correction for larger relative Reynolds numbers

[25], τp = ρp D2p

18ρgνgis the particle response time, Dp is the particle diameter and νg is the

gas phase kinematic viscosity. Rer = |Ur |Dp

νgis the relative Reynolds number, Ur,i =

(U p,i − Ug,i) − Ud,i is the mean relative velocity and Ud is the average of the fluidvelocity fluctuations with respect to the particle distribution [26], and is modelledaccording to Benavides [3].

Ud,i = 1

3Kpgτ

tpg

(1

�p

∂�p

∂xi− 1

�g

∂�g

∂xi

)(4)

where Kpg = u′p,iu

′g,i is the gas-particle velocity correlation and τ t

pg is the correlationtime scale of the fluid turbulence viewed by the particles.

In the present work a two-equation turbulence model with an isotropic eddy-viscosity is used together with a corresponding approach for particle phase. Thistype of modelling was used also by e.g. Gobin et al. [9] although with a somewhatdifferent approach to the modelling of the individual terms, such as the fluid-particlevelocity correlations. This class of models represents a natural first step to improvethe understanding of e.g. the influence of particles on the development length.Anisotropic effects are likely to modify the conclusions to some extent and a naturalnext step will be the development of an explicit algebraic model for the particle phasestresses, to be used in conjunction with an explicit algebraic Reynolds stress modelto account for anisotropy in similar ways for both phases. A further step would be toexplore the possibility to adapt the modelling to that of the sub-grid scale stresses forapplication in large-eddy simulations (see also Riber et al. [24]).

2.1 Gas phase

The stress in the gas phase consists of viscous and turbulent Reynolds stresses.

τg,ij = −2ρg�gνgSg,ij + ρg�gu′g,iu

′g, j (5)

where νg is the kinematic viscosity of the gas and Sk,ij = 12

(∂Uk,i

∂x j+ ∂Uk, j

∂xi

)is the mean

strain rate tensor. The Boussinesq assumption is used to close the gas phase Reynoldsstress

u′g,iu

′g, j = −2νtgSg,ij + 2

3

(Kg + νtg

∂Ug,m

∂xm

)δij (6)


Here νtg is the turbulent viscosity and Kg = 12 u′

giu′gi the mean turbulent kinetic

energy. To model the turbulent viscosity of the gas phase the low Reynolds numberK − ω model by Wilcox [31] is used, i.e. νtg = α∗ Kg

ω, where ω is the inverse time scale

of the gas phase turbulence. The equation for Kg reads

ρg�g∂Kg

∂t+ ρg�gUgj

∂Kg

∂x j= 2ρg�gνtgSg,ijSg,ij − ρg�gCμωKg

+ ∂

∂x j

(ρg�g

(νg + νtg

σK

)∂Kg

∂x j

)

− (1 + 0.15Re0.687

r

) ρp

τp�p

(u′

giu′gi − u′

piu′gi

)(7)

where

Cμ = 9

100

5/18 + (ReT/Rβ)4

1 + (ReT/Rβ)4(8)

and σK = 2. The last term in Eq. 7 is due to the interaction between the gas andthe particle phase and originates from the Stokes drag in the gas phase momentumequation. The equation for ω reads

ρg�g∂ω

∂t+ ρg�gUgj

∂ω

∂x j= 2ρg�gαSg,ijSg,ij − ρg�gβω2

+ ∂

∂x j

(ρg�g

(νg + νtg

σω

)∂ω

∂x j

)

− ω

Kg

(1 + 0.15Re0.687

r

) ρp

τp�p

(u′

giu′gi − u′

piu′gi

)(9)

where the last term represents the influence of the particle phase on ω. Here

α = 5

9

αo + ReT/Rω

1 + ReT/Rω

(α∗)−1 (10)

α∗ = α∗o + ReT/Rk

1 + ReT/Rk(11)

β = 3

40, σω = 2, α∗

o = β/3, αo = 1/10 (12)

Rβ = 8, Rk = 6, Rω = 27/10, ReT = Kg

ωνg. (13)

ω is decomposed according to

ω = ω + ωw (14)


The first part in Eq. 14 is the dominating term far from the boundary, whereas theterm ωw = 6νg

βy4 , where y is the distance to the wall, dominates near the wall [10]. Withthis decomposition equation (9) becomes

ρg�g∂ω

∂t+ ρg�gUgj

∂

∂x j(ω + ωw) = 2ρg�gαSg,ijSg,ij − ρg�gβω2

− ρg�g2βωωw + ∂

∂x j

(ρg�g

(νg + νtg

σω

)∂ω

∂x j

)

+ ∂

∂x j

(ρg�g

(νtg

σω

)∂ωw

∂x j

)

− ω + ωw

Kg

(1 + 0.15Re0.687

r

)

× ρp

τp�p

(u′

giu′gi − u′

piu′gi

). (15)

2.2 Particle phase

In order to describe the stresses in the particle phase an approach which is basedon the kinetic theory of granular flows, is adopted in a two-phase flow context. Themodel takes into account the effect of the interstitial fluid. This model (Eqs. 16–27)has earlier been used by for example, Balzer et al. [2], Peirano and Leckner [21],Benyahia et al. [5], Benavides and van Wachem [4], Chan et al. [8], Zhang and Reese[33] and Strömgren et al. [27]. Following this approach the stress due to particle-particle collisions is written as

τp,ij = −2ρp�p

(νcol

p + νkinp

)(Sp,ij − 1

3Sp,kkδij

)+ (

Pp − λpSp,kk)δij (16)

Here νcolp is the viscosity due to particle-particle collisions and νkin

p is the viscositydue to kinetic stresses, λp is the bulk viscosity and Pp = ρp�p

(1 + 2�pg0(1 + ep)

)T

is the granular pressure.

νkinp =

(2

3Kpg

τ tpg

τf

pg

+ T(1 + ζc�pg0)

)(2

τf

pg

+ σc

τ colp

)−1

, (17)

νcolp = 4

5�pg0(1 + ep)

(

νkinp + Dp

(Tπ

)1/2)

, (18)

λp = 4

3Dp�pρpg0(1 + ep)

√Tπ

(19)

T = 13 u′

p,iu′p,i is the granular temperature, ep is the coefficient of restitution,

g0 = 1

1 +(

�p

�maxp

)1/3 (20)


is the radial distribution function and �maxp = 0.64 is the maximum value of �p for

random packing of spheres [20]. Kpg = u′g,iu

′p,i is the correlation between fluctuating

gas and particle velocities and is modelled as Kpg = √2Kg3T [12]. τ t

pg = τ tg

(1 +

Cξ)−1/2 is the time-scale of the fluid turbulence viewed by the particles where

C = 1.8 − 1.35

(UriU pi

|Uri||U pi|)2

(21)

ξ = 3|Uri|22Kg

. (22)

τ tg = 3

2ωis the Eulerian time-scale of the gas phase turbulence,

τ colp = Dp

24�pg0

(π

T

)1/2(23)

is the inter particle collision time, ζc = 25 (1 + ep)(3ep − 1) and σc = 1

5 (1 + ep)(3 − ep).

τ fpg =

(3

4

ρg

ρp

CD

Dp�−1.7

g |Ur|)−1

= τp(1 + 0.15Re0.687

r

) (24)

is the characteristic time-scale of gas-particle momentum transfer where CD =24Rer

(1 + 0.15Re0.687r ) is the drag coefficient.

The equation for the granular temperature can be derived from the momentumequation of the particle phase. In order to close the unclosed terms the kinetic theoryof granular flows is used. The equation for T becomes

3

2ρp�p

∂T∂t

+ 3

2ρp�pU pj

∂T∂x j

= 2ρp�p

(νkin

p + νcolp

)SpijSpij − ρp�p

(1 − e2

p

)

2τ colp

T

+ ∂

∂x j

{3

2ρp�p

(κkin

p + κcolp

) ∂T∂x j

}

+ (1 + 0.15Re0.687

r

) ρp�

τp

(Kpg − 3T

)(25)

where

κkinp =

(3

5

τ tpg

τf

pg

Kpg + T(

1 + �pg03

5

(1 + ep

)2 (2ep − 1

))) (

9

5τf

pg

+ ξp

τ colp

)−1

(26)

is the kinetic contribution to the diffusivity,

κcolp = �pg0(1 + ep)

(6

5κkin

p + 4

3Dp

(T/π

)1/2)

, (27)

is the collisional contribution to the diffusivity and ξp = 1100 (1 + ep)(49 − 33ep).


2.3 Model implementation and boundary conditions

The model described above in cylindrical coordinates is implemented in a finiteelement code [1]. The momentum equation for the gas- and particle phase incylindrical coordinates read

ρg�g∂Ugl

∂t+ ρg�g(Ug · ∇)Ugl = −�g

∂ P∂l

+ ρg∇ ·(�g(νg + νgt)

(∇Ugl + ∇U T

gl

))

− ρp�g(νg + νgt)2Ugl

r2er

− ρp∇(

2

3�gδi, j

(Kg + νtg∇ · Ug

)) + Igl (28)

ρp�p∂U pl

∂t+ρp�p(Up · ∇)U pl = −�p

∂ P∂l

+ρp∇ ·(�p

(νkin

p +νcolp

) (∇U pl + ∇U T

pl

)

−ρp�p

(νkin

p + νcolp

) 2U pl

r2er

− ρp∇(((

3

2ρp�p

(νcol

p + νkinp

)

−λp

)∇ · Up+Pp

)δi, j

)+ Ipl +g(ρp−ρg)ex (29)

Where r is the radial direction, x is the axial direction, ∇ · B = 1r

∂rBr∂r + ∂ Bx

∂x ,B · ∇ = Br

∂∂r + Bx

∂∂x and ∇2 = 1

r∂∂r

(r ∂

∂r + ∂∂x2

).

The gas phase has a no-slip condition whereas the pressure and the particle volumefraction have a Neumann condition at the wall. Both Kg and ω are zero at the wall.At the outlet only the pressure has a Dirichlet condition. Jenkins and Louge [17]present a boundary condition model for the particle phase. However, we apply thesimple-to-use boundary conditions proposed by Johnson and Jackson [18] which canbe seen as a reasonable alternative to the ones developed by Jenkins and Louge [17],see Benyahia et al. [5].

�p

(νkin

p + νcolp

) (∇U px + ∇U T

px

)∣∣∣w

= −πg0ϕ�pU px√

3T6�max

p(30)

�p

(κkin

p + κcolp

)∇T = πg0ϕ�pU2

px

√3T

6�maxp

− πg0(1 − e2

w

)�p(3T)3/2

12�maxp

(31)

where ew is the coefficient of restitution for particle-wall collisions and ϕ is thespecularity coefficient which is zero for smooth walls and one for rough walls. Inthe wall normal direction there is a zero net flux of particles at the wall.

3 Model Validation

To validate the model simulations are compared to the experiment by Tsuji et al. [30]for a fully developed turbulent upward pipe flow (Re = 31,000) laden with 200 μm


Fig. 1 Mean gas- and particlevelocity profiles in thepresence of 200 μm particles

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r/R

Ug/

p/Ug,

cl

Ug, present model

Up, present model

Ug, experiments by Tsuji et al.

Up, experiments by Tsuji et al.

particles and with a mass loading of 2.1. ep and ew were assigned values of 0.9 and 0.7,respectively [6] and ϕ = 0.004. The comparison is shown in Fig. 1 and it can be seenthat the model slightly underpredicts the particle velocity but the overall agreementis good for both particle- and gas velocities. The root mean square (rms) velocityfluctuations of the gas- and particle phase are shown in Fig. 2 for a mass loadingof 3.2 and Re = 22,700. The simulated rms velocity fluctuations of the gas phase(Fig. 2a) overlaps the experiments in the whole region. Also the particle phase rmsvelocity fluctuations (Fig. 2b) have a good agreement with experiments, although aslight under estimation of the particle rms velocity for r/R > 0.5 is seen, where R isthe pipe radius.

Model simulations are also compared to experiments by Hadinoto et al. [13] whoinvestigated a particle-laden downward (fully developed) turbulent pipe flow forRe = 10,000. Figure 3 shows the mean axial slip velocity normalised with the gasphase velocity for particles with Dp = 200μm, ρp = 2500kg/m3 and mass loading 0.7.

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

r/R

Stre

amw

ise

gas

phas

e rm

s–ve

loci

ty

(2/3K)1/2/Ug, cl

, present model

<u´g2>1/2/U

g, cl, experiments by Tsuji et al.

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

r/R

Stre

amw

ise

part

icle

pha

se r

ms–

velo

city

(T)1/2/Ug, cl

, present model

<u´p2>1/2/U

g, cl, experiments by Tsuji et al.

Fig. 2 Mean gas- and particle rms velocity fluctuations in the presence of 200 μm particles. Massloading ρp�p

ρg�g= 3.2


Fig. 3 Mean axial slip velocityfor turbulent particle ladendown flow at Re = 10,000 in apipe in the presence of 200 μmparticles and mass loading 0.7

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

r/R

(Up –

Ug)/

Ug

Present modelExperiments by Hadinoto et al.

The values of the coefficient of restitution for the particle-particle collisions, theparticle-wall collisions and specularity coefficient are found in Hadinoto and Curtis[11] and are 0.9, 0.14 and 0.008, respectively. The present model results agreevery well with the normalised slip velocity from the experiments. Even thoughthe flow is downward the particles near the wall lag the gas phase velocity in theexperiment. This counter-intuitive phenomena is captured well by the model andcan possibly be explained by the collisional stresses [19]. The satisfactory predictionof this phenomenon shows that the particle stress is reasonably well captured in themodel.

We can conclude that the model shows overall very good agreement whencompared to experiments for pipe flows. In Strömgren et al. [27] good comparisonis made to experiments and DNS using the same model but for a turbulent channelflow.

4 Simulations of Evolving Gas-particle Flow in a Pipe

In order to study developing turbulent gas-particle flows an upward turbulentparticle 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 particle distributionwas initially homogeneous. The inlet conditions were uniform for gas- and particlevelocities, Kg, ω and �p. The gas- and particle velocities are initially the same. Theturbulent intensity was initially 7% and Uinitω/R = 0.02. The initial conditions werethe same as the inlet conditions. The coefficient of restitution was 0.94 for bothparticle-particle collisions and particle-wall collisions and the specularity coefficientwas 0.004.

Simulations were made for Stokes numbers between St+ = 34 and 3950 and massloading ranging from 0.2 to 2, see Table 1. The Stokes number is defined as St+ =τpu2

τ

νgwhere uτ = √

τw/ρg is the friction velocity and τw is the wall shear stress. The


Table 1 Parameters of the simulated cases

Case Re = 2RUcl

νgSt+ = τpu2

τ

ν�p

ρp

ρgDp m

1 24,000 34 5 · 10−4 2,000 25 μm 12 24,000 130 5 · 10−4 2,000 50 μm 13 24,000 465 5 · 10−4 2,000 100 μm 14 24,000 1,775 5 · 10−4 2,000 200 μm 15 24,000 3,950 5 · 10−4 2,000 300 μm 16 24,000 886 5 · 10−4 1,000 200 μm 0.57 24,000 – – – – –

Here m is the mass loading

Reynolds number, Re = 2RUg

νg, is 24,000. An unladen simulation for this Reynolds

number was also performed. The simulations were run until the flow was steady withregard to both phases.

4.1 Model results

In order to understand the axial development of velocity, turbulence intensity andparticle concentration profiles at different downstream positions in the pipe arecompared for St+ = 34 (case 1) and 1,775 (case 4). x is the downstream distanceto the inlet of the pipe and D is the pipe diameter. At x/D = 180 the flow is fullydeveloped for all variables.

In Fig. 4a and b the gas phase velocity profiles normalised with the gas phasecenter-line velocity (Ug,cl) at different downstream positions are shown for St+ =34 and St+ = 1,775, respectively. For St+ = 1,775 (Fig. 4b) the gas phase velocityprofile is almost fully developed at x/D=20 while a much longer development lengthis required for St+ = 34 (Fig. 4a). The reason is the larger change of the gas phasevelocity profile with respect to the inlet profile for St+ = 34. The more parabolicprofile for St+ = 34 than for St+ = 1,775 is due to the lower turbulence intensity forSt+ = 34.

The mean particle velocity profile normalised with Ug,cl at different downstreampositions is shown in Fig. 4c and d for St+ = 34 and St+ = 1,775, respectively. ForSt+ = 34 (Fig. 4c) the particle center-line velocity is nearly equal to the gas velocityat all positions but the velocity at the wall decreases with x/D, i.e. the profile becomesless flat. At larger St+ (Fig. 4d) the particle phase velocity profile is uniform at alldownstream positions whereas the velocity decreases with increasing x/D. The finalvelocity at the wall is about the same for both cases. The more uniform profile atlarge Stokes numbers is due to a weaker coupling between the phases. Since the wallis smooth (ϕ = 0.004) the particles experience almost a free slip condition there.

Profiles of the mean kinetic energy of the gas phase, Kg, normalised with Ug,cl

at five different downstream positions are shown for St+ = 34 and St+ = 1,775 inFig. 5a and b, respectively. A fully developed profile for an unladen case is shown forcomparison. Kg decreases with x/D for both Stokes numbers. However, the profilefor St+ = 34 (Fig. 5a) becomes fully developed at a larger x/D and the magnitudeof Kg is slightly smaller in the whole region than for St+ = 1,775 (Fig. 5b). Thefully developed Kg profile for St+ = 34 is lower than in the unladen case for all


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r/R

Ug/U

g,cl

Ug/U

g,cl

Up/U

g,cl

Up/U

g,cl

x/D=5x/D=20x/D=30x/D=40x/D=180

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r/R

x/D=5x/D=20x/D=30x/D=40x/D=180

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r/R

x/D=5x/D=20x/D=30x/D=40x/D=180

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r/R

a b

c d

x/D=5x/D=20x/D=30x/D=40x/D=180

Fig. 4 Mean velocity profiles normalised with the gas phase center-line velocity at five differentdownstream positions. St+ = 34, gas phase (a); St+ = 1775, gas phase (b); St+ = 34, particle phase(c); St+ = 1775, particle phase (d)

radial positions. For St+ = 1,775 the fully developed Kg profile is also lower, exceptnear the wall (r/R > 0.95) where Kg is larger than in the unladen case. Particleswith smaller Stokes numbers thus suppress the turbulence while particles with largeStokes numbers can enhance the turbulence [14]. The reason is the changing signof the drag term in the equation for Kg (last term on the right hand side (R.H.S.)in Eq. 7). The sign changes because Kpg = √

2Kg2Kp becomes larger than 2Kg forlarge Stokes numbers, see Fig. 5d.

In Fig. 5c and d the mean particle kinetic energy, Kp = 32 T, normalised with

Ug,cl at five different downstream positions is shown for St+ = 34 and St+ = 1,775,respectively. Kp decreases with x/D for St+ = 34 (Fig. 5c). At the center-line itreaches nearly the same value as Kg (Fig. 5a), but near the wall Kp is slightly largerthan Kg. We should note that Kp also has a non-zero value at the wall. For largerStokes numbers (Fig. 5d) the particles are less affected by the gas phase and thusthe profile of Kp is almost flat. The peak close to the wall disappears due to theabsence of mean strain. In contrast to the case for St+ = 34, Kp for St+ = 1,775(Fig. 5) increases for x/D up to 20 and then slightly decreases to a level higher than atx/D = 5. For St+ = 1,775, Kp is much larger than for St+ = 34, specially in the centerof the pipe where it is seven times larger.


0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

r/R

Kg/U

g,cl

2K

p/Ug,

cl2

Kp/U

g,cl

2K

g/Ug,

cl2

x/D=5x/D=20x/D=30x/D=40x/D=180Unladen flow

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

r/R

x/D=5x/D=20x/D=30x/D=40x/D=180Unladen flow

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

r/R

x/D=5x/D=20x/D=30x/D=40x/D=180

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

r/R

a

c

b

d

x/D=5x/D=20x/D=30x/D=40x/D=180

Fig. 5 Mean kinetic energy profiles normalised with the gas phase center-line velocity for fivedifferent downstream positions. St+ = 34, gas phase (a); St+ = 1,775, gas phase (b); St+ = 34,particle phase (c); St+ = 1,775, particle phase (d)

Profiles of the mean particle volume fraction, �p, normalised with its initial value,�0, are shown at five downstream positions in Fig. 6a and b for St+ = 34 andSt+ = 1,775, respectively. For St+ = 34 (Fig. 6a) the particles accumulate in thecenter of the pipe which increases with x/D. There are almost three times moreparticles in the center of the pipe than near the wall when the flow is fully developed.A higher particle concentration around the pipe axis was also found in experimentsby Tanaka and Tsuji [29]. It is a well known phenomena that particles tend to clusterin regions of lower turbulence intensity, implying a transport of particles in thedirection of the gradient of the turbulence intensity. This is called the turbophoresiseffect [27]. For St+ = 1765 (Fig. 6b) the distribution of �p is almost homogeneousdue to the small gradient of Kp. It is seen that �p increases with increasing x/Dfor St+ = 1,775. This occurs because the particle velocity reduces with x/D (Fig. 4d)whereas continuity implies a constant mass flux.

4.2 Development length

The influence of the particle diameter and mass loading on the length required forthe flow to become fully developed will be studied in more detail. First an estimationof the development length is made.


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

r/R

Φp/Φ

0

Φp/Φ

0

ax/D=5x/D=20x/D=30x/D=40x/D=180

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

r/R

bx/D=5x/D=20x/D=30x/D=40x/D=180

Fig. 6 Mean particle concentration profiles normalised with the center-line velocity at five differentdownstream positions. St+ = 34 (a); St+ = 1,775 (b)

4.2.1 Estimation of the development length

An estimation of the development length of the particle phase is made by studyingthe governing equations in order to understand which parameters govern this length.In a fully developed flow the convective term (second term on the L.H.S.) must besmaller than the most dominating term in Eq. 29, which is the diffusion term (secondterm on the R.H.S.). Since νcol

p << νkinp for dilute flows (�p < 1 · 10−3) this condition

becomes

U pj∂U pi

∂x j<

∂

∂x j

(νkin

p

(∂U pi

∂x j+ ∂U pj

∂xi

))(32)

The derivatives in the convective and the diffusive term are estimated as U/ l andU/D2, respectively. Here U is the characteristic mean streamwise velocity and l isthe axial development length. Equation 32 then yields

U2

l<

νkinp U

D2(33)

after rewriting, the development length can be expressed as

lD

>DUνkin

p(34)

Since �p << 1, νkinp (Eq. 17) can be simplified to

νkinp =

(2

3Kpg

τ tpg

τf

pg

+ T

)(2

τf

pg

+ σc

τ colp

)−1

(35)

In the limit of small Stokes numbers (St+ << 1) and small �p (< 1 · 10−4) the timebetween particle-particle collisions, τ col

p (Eq. 23), will be larger than τf

pg (Eq. 24) since


τf

pg ∼ D2p and τ col

p ∼ Dp. For the same reason τ tpg/τ

fpg >> 1. Furthermore, Kpg =

2K = 3T and τ tpg � τ t

g since Ur ∼ 0. With these simplifications νkinp can be esti-

mated as

νkinp = 2

32K

τ tg

τf

pg

τf

pg

2= 2K

3

3

2ω∼ νtg (36)

i.e. the kinematic viscosity of the particles approaches the turbulent viscosity of thegas phase, which is in accordance with Tchen–Hinze’s theory [15]. The developmentlength is then the same as for single phase flow, i.e.

lD

>DUνtg

(37)

For large Stokes numbers (St+ >> 1) τ tpg/τ

fpg < 1 and since Kpg has the same order

of magnitude as T Eq. 35 can be written as

νkinp = T

(2

τf

pg

+ σc

τ colp

)−1

(38)

Since 1 + Re0.687r in Eq. 24 and σc are of order unity we get

νkinp = T

1τp

+ 1τ col

p

= T18ρgνg

ρp D2p

+ 24�p

Dp

√Tπ

(39)

An estimate of the development length for gas-particle flows when St+ >> 1 andparticle-particle collisions have a moderate influence (�p ≤ 1 · 10−3) is thus

lD

>DUT

(18ρgνg

ρp D2p

+ 24�p

Dp

√Tπ

)

(40)

Consequently, the development length becomes shorter for increasing Dp and longerfor increasing �p.

When �p > 1 · 10−3 particle-particle collisions will have a large impact on the flowand τ col

p < τf

pg. The development length can then be estimated as

lD

>DU

Tτ colp

= DUT

24�p

Dp

√Tπ

(41)

Also here the development length becomes shorter for increasing Dp and longer forincreasing �p. However, this parameter range is outside the scope of this study.

A similar estimation as above can be made using the equation for the meangranular temperature (Eq. 25). However, similar results for the development lengthwill be found.

4.2.2 Comparison to model results

The estimation of the development length in Eq. 40 applies for the parameter rangeused in this study, i.e. dilute flows with rather large Stokes numbers, and will becompared to numerical simulations.


Fig. 7 Development length,x/D, as a function of Dp forUg, U p and �p

0 50 100 150 200 250 30020

30

40

50

60

70

80

90

100

110

120

Dp (μm)

x/D

Ug

Up

Φp

The development lengths for Ug, U p and �p are shown as a function of Dp inFig. 7. The development length is here defined as the distance to the inlet wherethe parameters maximum difference between the radial profile and the one at 180

0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

r/R

(Up−U

g)/U

g,cl

(Up−U

g)/U

g,cl

(Up−U

g)/U

g,cl

(Up−U

g)/U

g,cl

ax/D=5x/D=20x/D=30x/D=40x/D=180

0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

r/R

bx/D=5x/D=20x/D=30x/D=40x/D=180

0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r/R

cx/D=5x/D=20x/D=30x/D=40x/D=180

0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r/R

dx/D=5x/D=20x/D=30x/D=40x/D=180

Fig. 8 The slip velocity normalised with the gas phase center-line velocity at five different down-stream positions. Dp = 25 μm (a); Dp = 50 μm (b); Dp = 200 μm (c); Dp = 300 μm (d)


0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r/R

(Up−U

g)/U

g,cl

(Up−U

g)/U

g,cl

ax/D=5x/D=20x/D=30x/D=40x/D=180

0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r/R

bx/D=5x/D=20x/D=30x/D=40x/D=180

Fig. 9 The slip velocity normalised with the gas phase center-line velocity at five different down-stream positions. Dp = 200 μm (a); Dp = 300 μm (b)

D is less than 1%. For Dp ≤ 50 μm the development length decreases for increasingparticle diameters (St) for all three variables in agreement with the estimation. Thedevelopment length for Ug increases for Dp between 50 and 100 μm but decreasesfor larger Dp-values until it reaches the development length of an unladen flow(x/D = 21) because the coupling between the two phases weakens. In contrast, thedevelopment length of U p and �p start to increase with Dp for Dp >100 μm. ForDp = 300 μm it is about five times that of an unladen flow. However, according toEq. 40 the development length for the particle phase should decrease with Dp. Thisdiscrepancy for large Dp is likely related to the increasing slip velocity with particlediameter. In Fig. 8a–d the slip velocity for a fully developed flow normalised withits center-line velocity is shown at five different downstream positions for Dp = 25,50, 200 and 300 μm, respectively. The figure shows that the length required to reachthe final slip velocity is much shorter and its magnitude is smaller for Dp < 100 μm

Fig. 10 Development length,x/D, as a function of Dp forUg, U p and �p

0 50 100 150 200 250 3000

20

40

60

80

100

120

140

Dp (μ m)

x/D

Ug

Up

Φp

Eqn. (40)


0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r/R

(Up−U

g)/U

g,cl

(Up−U

g)/U

g,cl

x/D=5x/D=20x/D=30x/D=40x/D=180

0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r/R

a bx/D=5x/D=20x/D=30x/D=40x/D=180

Fig. 11 The slip velocity normalised with the gas phase center-line velocity at five differentdownstream positions. ρp/ρg = 1,000 (a); ρp/ρg = 2,000 (b)

(Fig. 8a and b) than for Dp > 100 μm (Fig. 8c and d), which shows that the slipvelocity only will affect the development length for large Dp. This effect has notbeen taken into account in the previous analysis.

In order to investigate the influence of the inlet slip velocity on the developmentlength the cases in Fig. 9c and d are recomputed with slip inlet conditions equal to themagnitude of the fully developed slip velocity in Fig. 9c and d while keeping the massflux constant. The slip velocity inlet conditions are homogeneous as in the previoussimulations. The results are shown in Fig. 9a and b, respectively. A significantlyshorter development length is found when the slip velocity at the inlet is adjustedto match the slip velocity for a fully developed flow.

All cases in Fig. 7 are recalculated with inlet slip velocities adjusted to match themagnitude of the fully developed slip velocities and the results are shown togetherwith the estimation of the development length in Eq. 40 in Fig. 10. The bulk valuesare chosen for the velocity and granular temperature in Eq. 40. Compared to Fig. 7where the initial slip velocity is zero the development length is much shorter for largeDp (> 100 μm). With these initial conditions the development length decreases evenfor large Dp which agrees qualitatively and quantitatively with the derived estimationfor the development length in Eq. 40.

In Fig. 11a and b the slip velocity profiles normalised with Ug,cl at five differentdownstream positions are shown for ρp/ρg = 1,000 (case 6) and 2,000 (case 4),respectively. The magnitude of the slip velocity and the development length increasewith ρp/ρg, i.e. mass loading. The development length for the two different massloadings are listed in Table 2. It becomes almost twice as long for U p and �p as

Table 2 Development length for different ρp/ρg and flow variables

Variable x/D(

ρpρg

= 1,000)

x/D(

ρpρg

= 2,000)

Ug 31 31U p 36 71�p 41 75

The values are from model simulations


ρp/ρg is increased. For Ug the development length does not change with ρp/ρg but isabout 50% longer than for the unladen case.

5 Conclusions

A two-fluid model using the kinetic theory of granular flow for the particle phasewas developed to study an evolving upward particle laden turbulent pipe flow.Good agreement was found with two different sets of experimental data. We haveinvestigated the influence of the particle diameter (Stokes number) on the pipelength required for the flow to become fully developed. It was found that the particlescan make the development length for the gas velocity up to three times longer thanthat of an unladen flow whereas the development length of the particle velocitycan be up to five times longer than that of the gas velocity of an unladen flow.To understand what governs the development length a simple estimation for thislength was derived from the particle momentum equation. This estimation showedthat the development length decreases with increasing Dp. This was confirmed bymodel simulations for Dp ≤ 100 μm. For larger Dp the development length is verysensitive to the initial slip velocity between the phases. When the initial slip velocityis close to the final slip velocity the development length is much shorter than for caseswith a zero initial slip velocity.

The development of the particle concentration profiles were also studied and anet flow of particles to the center of the pipe was observed. This drift was largestfor small particle diameters. The velocity and kinetic energy profiles of the particlephase were found to become flatter for increasing Stokes number. The gas phasekinetic energy increased slightly close to the wall for large Stokes numbers, whereasfor small Stokes numbers Kg decreased in the whole region.

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