1-s2.0-S0009250907000085-main (1).pdf

Chemical Engineering Science 62 (2007) 2068–2088www.elsevier.com/locate/ces

Dilute gas–solid two-phase flows in a curved 90◦ duct bend: CFD simulationwith experimental validation

B. Kuan∗, W. Yang, M.P. SchwarzCooperative Research Centre for Clean Power from Lignite Division of Minerals, Commonwealth Science and Industrial Research Organisation, Box 312,

Clayton South, Victoria 3169, Australia

Received 31 May 2006; received in revised form 20 October 2006; accepted 21 December 2006Available online 13 January 2007

Abstract

Computational fluid dynamics (CFD) simulations of dilute gas–solid flow through a curved 90◦ duct bend were performed. Non-uniformsized glass spheres with a mean diameter of 77 �m were used as the dispersed phase. The curved bend is square-sectioned (150 mm ×150 mm)

and has a turning radius of 1.5D (D = duct hydraulic diameter). Turbulent flow quantities for Re = 100, 000 were calculated based on adifferential Reynolds stress model. The solids mass loading considered is 0.00206 and hence justifies the application of one-way couplingbetween gas and particles. A Lagrangian particle-tracking algorithm which takes into account the effect of shear-slip lift (SSL) force on particlesand particle-wall interactions (PWIs) has been utilised to predict velocities of the dispersed phase. The predictions were compared against theexperimental data measured using Laser–Doppler Anemometry (LDA). The study found that the predicted gas flow field has a strong influenceover the predicted particle velocities. PWI model considerably affects the prediction of particle velocity and distribution of particles at theinner duct wall within the bend. Inclusion of the SSL force also helps the distribution of the particle tracks towards the duct centre in thevertical duct downstream of the bend. Within the bend, particle velocities near the inner wall have been grossly over-predicted in the simulation,especially at mid-bend. The present study thus highlights the importance of the predicted gas flow field, SSL force and particle-wall collisionsto Lagrangian particle tracking.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Multiphase flow; Mathematical modelling; Simulation; Penumatic conveying; LDA; Turbulence

1. Introduction

Elbows and bends are commonly used in pneumatic convey-ing systems to change flow direction so as to transport the sus-pended material to the desired delivery point within a limitedspace. In the case of coal-fired power plants that operateon a continuous supply of pulverised coal to furnaces, mal-distribution of pulverised fuel often occurs as coal particlesare pneumatically transported from the mill through ductsconsisting of numerous bends and straight sections.

Apart from the duct geometry, the coal pulverisation processis also a strong contributor to mal-distribution of the pulverisedfuel. Field measurements at a lignite-fired power plant carriedout to support this study found that, depending on factors such

∗ Corresponding author. Tel.: +61 3 95458687; fax: +61 3 95628919.E-mail address: [email protected] (B. Kuan).

0009-2509/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2006.12.054

as feed size and mill speed, the fuel pulverisation process pro-duces coal particles ranging between 10 and 1000 �m in size.Fig. 1 shows a typical coal particle size distribution measuredat the mill outlet of a lignite-fired power station (McIntosh andBorthwick, 1984). The extreme non-uniformity in particle sizecombined with a centrifugal effect arising from the duct bendare believed to lead to the formation of a stratified gas–solidflow, known as a particle rope, downstream of the elbow evenat a low solids mass loading, L < 0.1. This invariably createsdifficulties for the plant operators to monitor and control thepulverised fuel supply to individual burners, and hence to main-tain an optimal combustion condition inside the furnace.

There are a large number of documented studies, both nu-merical and experimental, on particle roping in dust conveyingsystems with solids mass loading L > 0.3, but most of themfocus on particles with a size distribution that is either heavilyskewed towards the lower end of the size range (Yilmaz and

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Fig. 1. Typical coal size distribution at the mill outlet.

Levy, 1998; Akilli et al., 2001) or more evenly spread acrossthe entire size range (Huber and Sommerfield, 1994). In bothcases, the tested particle samples contain a considerable amountof fine particles. A comprehensive experimental investigationwas conducted by Huber and Sommerfield (1994) using phase-Doppler anemometry (PDA) in various pipe elements. Theyused glass particles with a mean diameter of 40 �m and theirresults demonstrated the significance of wall-roughness in pre-venting the glass beads from settling in a horizontal pipe.

The research group at Lehigh University has conducted sev-eral physical experiments (Yilmaz and Levy, 1998, 2001; Akilliet al., 2001; Bilirgen and KLevy, 2001) investigating forma-tion and dispersion of the particle ropes involving pulverisedcoal particles. A closed flow loop consisting of two 6.1 m hor-izontal pipes, one 3.4 m vertical pipe and two 90◦ elbows withadjustable turning radii was set up to facilitate measurementsof particle rope characteristics in a vertical pipe following ahorizontal-to-vertical bend. All pipe elements had an internaldiameter of 0.154 m. They identified a number of parametersthat affect the behaviour of the particle rope downstream ofa horizontal-to-vertical duct bend. These included solids massloading, conveying velocity, and duct bend radius.

Levy and Mason (1998) reported a similar study utilisingthree different particle sizes. Their numerical simulations indi-cated a change in particle roping characteristics with particlesize: coarse particles form a concentrated rope which propa-gates more than 2.4D downstream of the bend; for the fineparticles, they only noticed a localised peak in particle con-centration just downstream of the bend. There were howeverno published experimental data at the time to validate theircalculations.

The effect of particle size on solid- as well as gas-phasemotions in vertical and horizontal two-phase flows has beenextensively studied in the literature. With the aid of a laser-Doppler velocimeter (LDV), Tsuji et al. (1984) tested plasticparticles with diameters ranging from 200 �m to 3 mm in avertical pipe having an inner diameter of 30.5 mm and studiedthe effect of particle size on particle motion and gas turbulence.At high solids mass loading (L > 2.0), they found the meanair velocity profile flattens across the duct cross-section withdecreasing particle size. Fan et al. (1997) performed a similarexperiment with 100 and 300 �m quartz particles in a 100 mm

diameter vertical tube. At L < 2.0, they observed a flattening ofthe mean particle velocity profile with increasing particle size.

Tsuji and Morikawa (1982) has also performed the sameLDV measurements for a horizontal two-phase flow, but onlyfor L > 2.0. They observed a stronger influence of the par-ticles on the background gas turbulence as the particle sizeincreased. Laín et al. (2002) experimentally studied the effect ofparticle size, solids mass loading and wall roughness on a two-phase horizontal channel flow. While all tests were performedat L > 0.2, their results indicate that the flow was increasinglysensitive to particle size distribution as L decreases.

In pneumatic transport, it is commonly accepted that move-ment of large particles is predominantly influenced by particle-wall interactions rather than fluid turbulence. In recognition ofthis, Frank et al. (1993), Kussin and Sommerfeld (2002), andSommerfeld and Huber (1999) conducted series of experimentsto study particle-wall interaction for a large combination of par-ticle size, material and wall roughness in an attempt to createa reliable physical particle-wall collision model.

Compared to the flow regime prevailing at the power plantmill duct (i.e., L < 0.1), all of the above-mentioned studiesexamined gas–solid flow behaviour in the L > 0.2 range andhence their results were not representative of the dilute mill-ductflow considered in the present study. Further, most of the pub-lished conveying duct experiments only consider the scenariowhere a substantial amount of fine particles were present in thesystem. In the present study, a dispersed phase with a wider sizedistribution but a smaller fraction of fine particles is consideredso as to better represent the gas–solid flow system prevailing inthe mill-duct flow. Numerical simulation of such a flow usingLagrangian particle tracking technique requires a large set ofparticle trajectories to be solved. A special discretisation tech-nique has been developed to overcome the need for an exceed-ingly large number of particle tracks. This paper aims to study,both numerically and experimentally, the behaviour of a dilutegas–solid flow which uses non-uniform sized spherical glassbeads as the dispersed phase and at a low solids mass loadingL = 0.00206. Glass spheres having a volume weighted meandiameter of 77 �m were tested in purpose-designed laboratoryexperiments which involve a square-sectioned horizontal-to-vertical 90◦ bend.

Numerical simulations were performed for the gas phase andthen followed by the solution of solids motion. The solution ofparticle motion is based on a Lagrangian particle tracking ap-proach where a single particle track represents the trajectoriesof a group of particles of the same size. The calculation consid-ered various hydrodynamic forces acting on the particles, suchas drag and shear-slip lift (SSL), as well as particle wall inter-actions. The hydrodynamic forces were determined based onthe knowledge of the predicted gas flow. An accurate solutionof the gas flow field is thus crucial for a realistic prediction ofthe particle motion.

One main feature of the square-sectioned duct flow consid-ered in this study is a strong curvature in the streamwise direc-tion along the duct length. This necessitates the application ofa turbulence model that is capable of resolving the extra strainsarising from curvature in the streamlines. There exists a wide

2070 B. Kuan et al. / Chemical Engineering Science 62 (2007) 2068–2088

range of turbulence models in the published literature dealingwith streamline curvature effects. Nallasamy (1987), and Pateland Sotiropoulos (1997) have both provided comprehensivereviews on various approaches to model streamline curvatureeffects in the turbulence model.

Gibson (1978) and Rodi and Scheuerer (1983) have proposeda number of ad hoc modifications to the standard two-equationturbulence model to address this problem. However, the appli-cability of these model corrections has largely been limited toflows with simple geometry where axis of streamline curvatureis fixed relative to the principal flow direction, such as flowover a back-ward step. Richmond and Patel (1991) have testedone of the above models in various curved wall-bounded flows.The model produced satisfactory mean velocity prediction onthe convex wall while under-estimating the mean velocities onthe concave wall.

Choi et al. (1989) have applied an algebraic Reynolds stressmodel in the core flow and mixing-length hypothesis in theviscous sublayer in an attempt resolve streamline curvature ina square-sectioned U-bend. The model performs better than thestandard k.� model and provides a reasonable representationof the primary flow but it tends to over-predict gas velocity atthe convex wall and over-predict it at the concave wall.

Luo and Lakshminarayana (1996) have applied a differentialReynolds stress model and a modified eddy dissipation rate �transport equation proposed by Shih et al. (1995) to accountfor streamline curvature effects. They tested their model in 2D90◦ and 180◦ pipe bends. The results indicated a negligibleimprovement on the predicted mean gas velocities as comparedto the case with unmodified differential Reynolds stress model.However, the new model produced a higher level of turbulenceintensity at two stations inside and shortly downstream of thebend.

Patel and Sotiropoulos (1997) has thus concluded that, forflows containing streamline curvature, a differential Reynoldsstress model without any modifications should provide rea-sonable flow prediction as compared to that based on case-specific two-equation turbulence models. The current studyadopts the differential Reynolds stress model of Speziale et al.(1991) which is tensor-invariant and does not require the addi-tion of wall-reflection terms into its pressure–strain correlationmodel.

2. Experimental test facility

2.1. Rig configuration

All experiments were conducted in an open-circuithorizontal-to-vertical suction wind tunnel system which wasset up for an earlier study by Yang and Kuan (2006). Aschematic sketch of the entire test facility is provided inFig. 2(a). It consists of an open-circuit suction wind tunnelwhere the airflow is drawn into the system by means of a cen-trifugal fan through an entry piece that consists of an ellipticalbell-mouth inlet and a honeycomb flow straightening section.The air then passes through a converging nozzle to attain ahigher velocity before entering into the test section. Design

of the converging nozzle follows Borger’s contractor profile(Borger, 1973) to ensure flow uniformity within ±0.5%.

The duct geometry and flow coordinate system are shownin Fig. 2(b). The square-sectioned (150 mm × 150 mm) testsection is constructed using 10 mm thick Perspex sheets, and themain components of the test facility include a 3.5 m horizontalstraight duct, a 90◦ bend with a turning radius R of 225 mmand a 1.8 m vertical straight duct. The R/D ratio for the bendis thus 1.5.

Glass spheres with a volume weighted mean diameter of77 �m were released into the gas flow from a fluidised-bedfeeder at a rate of 2 kg h−1 to give a solids mass loading L of0.00206. A digital balance underneath the fluidized-bed feedermonitors the rate at which the particles are released so as toensure a dilute gas–solid flow regime inside the test section.Data on particle size distribution (based on volume fraction)are obtained from a wet analysis using a Malvern particle sizeanalyser and are shown in Fig. 3.

Gas phase measurements were obtained at a bulk gas veloc-ity Ub, of 10 m s−1 in the absence of the glass spheres. A mistof fine sugar particles generated from a jet atomiser using 5%sugar solution have a mean diameter of 1 �m and were used asseeding particles in the single-phase measurement. Turbulenceintensity in the main stream was approximately 1% at the cen-tre of duct cross-section 10D upstream of the 90◦ bend. TheReynolds number, based on the bulk velocity, hydraulic diam-eter of the square-sectioned duct and kinematic viscosity of airwas subsequently 105.

Flow measurements were performed on the duct verticalsymmetry plane. At each streamwise location along the ductlength, flow measurements, including mean gas and solidsvelocities, and turbulence intensities were carried out. In near-wall areas, flow statistics were measured at 1 mm from theduct walls. LDA measurements outside the vertical symmetryplane were not taken because of poor access of the laser beamto the probe locations. Secondary gas flow, which is charac-terised by a pair of counter-rotating vortices along the ductlength downstream of the bend, was not studied in the presentwork. It is assumed to display similar characteristics to thelaboratory flow of Sudo et al. (2001) where R/D = 2.0.

2.2. Dynamic similarity

In order to ensure the flow scenario tested in the laboratorymodel reflects the physical duct operating condition, a dynamicsimilarity analysis was performed prior to the experiment. Ouranalysis is based on the collective experience of Boothroyd(1971) and Fan and Zhu (1998). In general they found the flowsystems that share similar geometric and kinematic boundaryconditions display similar dynamic responses when the dimen-sionless numbers Re, Rep, St , and Fr are the same

Re = �f UbD

�, (1)

Rep = �f UT dp

�, (2)


Fig. 2. Experimental flow system: (a) schematic diagram; (b) duct geometry and flow coordinate system.

Fig. 3. Particle size distributions by volume (— wet analysis data; � modelleddistribution).

St = �p

�f

; �p = �pd2p

18�; �f = D

Ub

, (3)

Fr =√

U2b

Dg, (4)

whereby Re denotes duct Reynolds number; Rep is particleReynolds number; St is particle Stokes number as commonlydefined in the literature (Giddings et al., 2004); Fr is Froudenumber for gas–solid flows; UT denotes particle terminalvelocity; Ub is fluid bulk velocity based on fluid mass flowrate and duct cross-section area.

Stokes number St is a ratio of particle response time to thefluid travelling time; a higher St thus implies more collisionsof the particles with the walls of the bend. The Froude numberFr measures the dominance of inertial effects over gravity onthe airflow.

Dynamic similarity parameters are compared in Table 1. Thetable also lists some of the characteristic flow parameters forboth cases. The smaller duct diameter and particle sizes in thelaboratory system as compared to the industrial flow are largelyresponsible for the lower Re, Rep, and St values at the upperend of the particle size range. A lower Rep implies that the par-ticles in the laboratory flow system are more tightly coupled tothe gas flow than in the industrial counterpart. The power plantmill-duct flow contains a greater proportion of coarse particleswith a very high St, indicating a higher chance of particle-wallcollisions. However, motion of the larger particles is beyond thescope of the present investigation because it is predominantly


Table 1Flow and dynamic similarity parameters

Laboratory rig Power station

Flow parametersD (m) 0.15 1.76� (kg m−1 s−1) 1.8 × 10−5 1.95 × 10−5

�f (kg m−3) 1.18 0.78�p (kg m−3) 2500 1400L 0.00206 0.1dp (�m) 4 77 160 20 80 410

(min.) (mean) (max.) (5%) (50%) (95%)UT (m s−1) 1.21 × 10−3 0.36 1.1 0.0155 0.224 2.32Ub (m s−1) 10.0 28.6Re 1.0 × 105 2.0 × 106

Dynamic similarity parametersRep 3.17 × 10−4 1.82 11.5 0.0124 0.716 38.1St 8.23 × 10−4 3.05 13.2 0.063 1.701 44.7Fr 8.24 6.88S/dp 80.9 21.1

driven by particle’s own inertia as well as collisions with thewall surfaces, instead of gas–solids interactions.

As seen in Table 1, the current experiment covers the lowerend of the industrial particle size range which has lower Rep

and St values. One can thus expect the laboratory rig to realis-tically reproduce the characteristics of the power station mill-duct flow in the absence of very coarse particles.

Apart from dynamic similarity parameters that charac-terise the level of particle–gas interactions in both systems,Table 1 also lists inter-particle spacing S (with respect to dp) asestimated from

S/dp =(

�

6�p

)1/3

, (5)

�p = 1�p

L�f

+ 1(6)

which have been used by Sommerfeld (2000) to classify theimportance of interaction mechanisms in dispersed two-phaseflows. In general, effect of inter-particle collisions can beignored for an inter-particle space, S/dp greater than 10.According to Table 1, both flow systems satisfy this conditionand hence inter-particle collisions were not considered in thisstudy.

3. Mathematical models

3.1. Gas flow

Steady-state, isothermal gas flow properties and turbu-lence quantities are calculated numerically by solving a setof governing partial differential equations (PDE) using acommercial CFD software ANSYS CFX-10. The Reynolds-averaged Navier–Stokes equations and the Reynolds stresstransport equations considered are written in Cartesian tensor

notations below

��Ui

dxi

= 0, (7)

��UiUj

dxj

= −�P

�xi

+ �

�xj

[�

(�Ui

�xj

+ �Uj

�xi

)− �u′

iu′j

], (8)

��Uku′iu

′j

dxk

= �

�xk

[(� + 2

3�Cs

k2

�

) �u′iu

′j

�xl

]

+ Gij − 2

3��ij + ij , (9)

��Uk�

dxk

= �

�xk

[(� + �t

�

)��

�xl

]+ C�1

�

kGkk − C�2

��2

k, (10)

Gij = �u′iu

′k

�Uj

�xk

− �u′j u

′k

�Ui

�xk

, (11)

where Ui and u′i are, respectively, mean and fluctuating veloc-

ities; u′iu

′j denotes the Reynolds stress tensor; Gij is the tur-

bulence production term; ij is the modelled pressure–straincorrelation given by Speziale et al. (1991). Model constants Cs ,C�1, C�2, and �, respectively, have the following values: 0.22,1.45, 1.83, and 1.375. This approach is known as differentialReynolds stress modelling (DRSM).

4. Particle tracks

4.1. Hydrodynamic forces

Instantaneous positions and velocities of the dispersed phaseare solved through a Lagrangian particle tracking method. Mo-tion of individual particles suspended in a continuous fluid isdetermined by numerically integrating the equations of motion


for the dispersed phase in a fluid flow. The equation of particlemotion may be expressed as

mp

dup

dt= FD + Fg + Fpg + FA + Fsl , (12)

dxp

dt= up, (13)

where subscript p represents particle properties and subscriptsD, g, pg, A and sl, respectively, denote force components arisingfrom drag, gravity, flow pressure gradient, added mass effectand slip-shear lift.

The drag force is calculated from

FD = mp

uf − up

�r

(14)

with uf = U + u′ and particle relaxation time �r defined by

�r = �pd2p

18�fD

(15)

and the Schiller–Naumann drag correlation fD for a sphere

fD ={

1 + 0.15Re0.687p , Rep �1000,

0.01833Rep, Rep > 1000,(16)

CD = 24

Rep

fD .

As Rep increases beyond 1000, the corresponding fD leads toa constant drag coefficient CD of 0.44.

The force components due to gravity Fg , added mass FA andpressure gradient Fpg are, respectively, given by

Fg = mp

(1 − �f

�p

)g, (17)

FA = −1

2mp

�f

�p

dup

dt, (18)

Fpg = − 14�d3

p∇P . (19)

The present calculation adopts a SSL model formulated by Mei(1992). In vector form, the model is written as

Fsl = �

8d3p�f Csl((uf − up) × �f ), (20)

where

Csl = 4.1126√Res

f (Rep, Res), (21)

f (Rep, Res) =⎧⎨⎩

(1 − 0.3314�1/2)

×e(−0.1Rep) + 0.3314�1/2, Rep �40,

0.0524(�Rep)1/2, Rep �40,

(22)

Res = �f d2p|�f |�

, (23)

� = 0.5Res

Rep

; �f = ∇ × uf . (24)

Assuming that all force components, except the drag, are con-stant during a time step �t , Eq. (12) can be integrated analyti-cally to yield

up = uf + (u0p − uf )e−�t/�r + �r

mp

(1 − e−�t/�r )

× (Fg + FA + Fsl + Fpg) (25)

with superscript 0 indicating the start of a time step.Similarly, Eq. (13) can be integrated analytically to

xp = x0p + up�t . (26)

Eqs. (25) and (26) were solved within a given cell in the particletracking calculations.

In order to solve the instantaneous particle velocity andlocation using the integrated equations of particle motion,Eqs. (25) and (26), for every particle track in the flow domain,the instantaneous fluid velocity has to be specified at all par-ticle locations. This is made possible through the applicationof a classical stochastic approach of Gosman and Ioannides(1981), which estimates fluctuating gas velocity componentson the basis of isotropic turbulence. Subsequent particle trackintegration which takes into account the turbulent dispersioneffect was thus carried out.

5. Particle-wall interactions

The present simulation adopts a modified version of the PWImodel of Matsumoto and Saito (1970). The base-case modelallows the particles to either slide along the wall surface whenthe angle of incidence is small, or rebound away from the wallafter impact. However, it is based on the assumption of a con-stant restitution coefficient and dynamic friction, both of whichare sensitive to a range of parameters, such as incidence angleand wall material, as found in published experimental investi-gations of Frank et al. (1993), Sommerfeld and Huber (1999).

For the present numerical calculation that involves collisionsbetween the glass spheres and the Perspex duct walls, particlevelocity components as well as particle angle of incidence aredefined in Fig. 4. Impact test data for glass beads on Plexiglass

Fig. 4. Definition of velocities before and after impact and particle angle ofincidence.


plates are utilized to characterize the PWI (Sommerfeld andHuber, 1999). The modified PWI model is given by

up2 = Etup1,

vp2 = −Envp1,

En = max(1.0 − 0.015�′1, 0.73),

�d = max(0.4 − 0.00926�′1, 0.15)

ucrit = 3.5�d(1 + En)|vp1|

Et =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1.0 − �d(1 + En)|vp1/up1|,up1 �ucrit(sliding collision),

5/7,

up1 < ucrit(non-sliding collision).

(27)

The effect of wall roughness is introduced through a semi-empirical approach that modifies the “smooth-wall” incidentangle �1 with a random component characterising the presenceof a rough wall

�′1 = �1 + ��, (28)

where �� represents a random component sampled from aGaussian distribution function. � is a Gaussian random numberwith zero mean and standard deviation of unity, and �� is thestandard deviation of the wall roughness angle. �� is approxi-mately 3.8◦ for a Plexiglass plate.

In our simulations, we have assumed the particles are non-rotating and the effect of particle rotation on particle velocityafter the collision is negligible.

6. Numerical procedure

The PDE (8)–(10), are discretised following a finite vol-ume approach. The advection terms were approximated usinga scheme developed by Barth and Jesperson (1989) whichis more than first-order accurate at mesh discontinuities andprovides higher-order accuracy for smoothly varying meshes.

Table 2Modelled particle size distribution and a discrete representation of the particulate flow by particle tracks

i dp,i (�m) Vol.% Particle vol. flow Ni Particle vol. flow i (1/s)rate (m3/s) rate per track (m3/s)

1 5 2.001E − 5 4.416E − 14 8000 5.520E − 18 8.434E − 22 18 5.202E − 5 1.148E − 13 8000 1.435E − 17 4.700E − 33 30 3.112E − 3 6.868E − 12 8000 8.585E − 16 6.073E − 24 45 2.905E − 1 6.411E − 10 8000 8.014E − 14 1.6805 57 3.260 7.196E − 9 8000 8.994E − 13 9.2766 65 10.35 2.284E − 8 8000 2.855E − 12 19.857 76 38.10 8.409E − 8 8000 1.051E − 11 45.738 89 34.03 7.510E − 8 8000 9.387E − 12 25.439 103 9.982 2.203E − 8 8000 2.754E − 12 4.81310 125 2.500 5.518E − 9 8000 6.897E − 13 6.744E − 111 140 1.000 2.207E − 9 8000 2.759E − 13 1.920E − 112 152 4.887E − 1 1.079E − 9 8000 1.348E − 13 7.332E − 2

Total 100.0 2.207E − 7 96 000

All calculations were performed on a 80×80×135 (y ×z× s)

body-fitted grid and the computed flow domain was constructedusing hexahedral cells. The grid has a minimum wall spacingof y+ < 16 and was found to be sufficiently refined to producea grid-independent solution as shown in an earlier study byKuan (2005).

The current simulations utilise a set of fully turbulent inflowconditions generated from a separate calculation in a straighthorizontal pipe of the same cross-section. The solid particleswere released into the duct at the same velocity as the gasphase, i.e., zero slip, and from random locations on the inletplane.

In a dilute two-phase flow where particle volume fraction isless than 10−6 (i.e., L < 0.00211), it is generally accepted thatthe transfer of particle momentum to the carrier-phase is negli-gible (Sommerfeld, 2000). Thus, at L=0.00206, the gas motioncould be considered independent of the solid-phase (i.e., one-way coupling). When particles are introduced into a turbulentgas flow, they can either enhance or reduce the gas turbulence.This is known as turbulence modulation which is assumed tobe insignificant in our simulation.

In the general dilute gas–solid flows, the quality of thedispersed-phase prediction based on Lagrangian particle track-ing is directly dependent on the number of particle tracks solvedin the simulation. In cases where the particle size distributionis wide, one has to proportionally increase the total numberof the considered particle tracks so as to adequately resolvethe smallest size fraction. This often leads to an exceedinglylarge set of particle tracks which are very computationallydemanding to solve and analyse.

In the present study, we have developed a methodologywhich allows one to overcome this limitation. The particle sizedistribution (by volume) as measured in a wet analysis by theMalvern size analyser (Fig. 3) was first discretised into 12 char-acteristic particle size fractions (Table 2). One can calculatethe associated particle volume flow rate for each size fractionto give a total solids volume flow rate of 2.207E − 7 m3 s−1

such that L = 0.00206. A fixed number of particle tracks were


Fig. 5. Mean longitudinal gas velocity profiles within the bend (— prediction; � data).

then allocated to each size fraction and we have used a totalof 96,000 particle tracks in the present study. Individual par-ticle tracks in each size fraction thus carry a uniform portionof the solids volume flow rate as shown in Table 2. Onecan then determine a particle number flow rate i repre-sented by each track and subsequently apply this to convertnumber of particle tracks into number of “real” particles dur-ing the statistical averaging process as discussed in the nextsection.

7. Collection and averaging of predicted particle statistics

Similar to the data acquisition process performed during thephysical experiment, the predicted particle statistics are col-lected at each measurement station on the duct centre-plane.We have adopted the technique by Uijttewall and Oliemans(1996) for estimating the mean solids flow quantities on themeasurement plane. This involves setting up 27 equally spaced“bins” along the duct diameter to collect and store particle track


Fig. 6. Mean longitudinal gas velocity profiles in the vertical duct (— prediction; � data).

information. The particle statistics stored inside each bin arethen averaged based on Table 2 and Eq. (29) to produce a rep-resentative value for the entire bin.

Up =∑

i=1,12 upnii∑i=1,12 nii

. (29)

In Eq. (29), ni is the number of particle tracks for size fractioni and i is the corresponding particle number flow rate perparticle track.

8. Results and discussions

8.1. Gas flow predictions and validation

Predicted and measured profiles of mean longitudinal gasvelocities are compared in Figs. 5 and 6. Overall, the numericalprediction provides a good representation of the measured flowfield except in the outer-wall region (r∗ < 0.4) within the bend.The laboratory data indicates a growing layer of slow gas streamnext to the outer duct wall. This is presumably due to an adverse


Fig. 7. Mean longitudinal gas velocity profiles along the outer wall, 1 mmfrom the surface.

pressure gradient on the concave face (outer wall) within thefirst half of the bend. In the second half of the bend, the gasat the outer wall is found to accelerate towards the bend exit.The gas flow near the inner wall, however, displays an oppositetrend: it accelerates steadily between = 0◦ and 45◦, and thenslows down abruptly from = 45◦. This is consistent with thefindings of a similar experiment by Sudo et al. (2001) whichattributed this behaviour to changes in local pressure gradientsat the convex face as well as fluid transport by the secondaryflow.

The DRSM adopted is unable to correctly resolve such apressure–velocity interaction and hence over-predicts the meanlongitudinal gas velocity by as much as 18% at = 60◦ inthe outer-wall region within the bend. This is shown in Fig. 7which compares the predicted and measured near-wall longitu-dinal gas velocities at 1 mm above the outer wall. This possiblypoints to a deficiency in the pressure–strain correlation modelof Speziale et al. (1991) or even the modelled turbulent dissipa-tion rate transport equation, Eq. (10), for resolving streamlinecurvature effects as discussed in Gibson et al. (1981) and Luoand Lakshminarayana (1996). In the inner-wall region, how-ever, the DRSM captures the flow acceleration and decelerationprocess satisfactorily.

Between = 90◦ and s/D = 0.5, the numerical predictionindicates the formation of a recirculation zone at the inner wall(Figs. 5 and 6). Although the measured profiles at these twostations provide no direct evidence as to the existence of thisrecirculation zone, it is highly probable that flow reversal mighthave taken place over a very short distance between =90◦ ands/D = 0.5 in the physical experiment as a result of an adversepressure gradient.

Downstream of the bend, the maximum peak in the measuredprofile has moved from r∗ = 0.36 at = 90◦ to r∗ = 0.3 ats/D =0.5, indicating a movement of the core gas flow towardsthe outer wall under the influence of centrifugal effect. At s/D=1.0, a smaller peak in the measured velocity profile at r∗ =0.86suggests that part of the core flow has been carried to the inner-wall region (Fig. 6). According to Sudo et al. (2001), this isattributable to the existence of a secondary flow which carriesthe core gas flow from the outer wall to the inner wall along

the duct circumference. The DRSM is found to reproduce thecore flow movement correctly in the simulation.

Sudo et al. (2001) also found that the secondary flow im-proves the mixing between the fast and the slow gas streamswhich, respectively, originate from the outer and the inner walls.This helps the mean gas flow stabilise and develop towards afully developed structure downstream of s/D = 1.0. While thecurrent set of centre-plane data is insufficient for a direct valida-tion of the ability of the numerical model to capture secondaryflow motion, one can still apply it to assess the predicted coreflow movement which strongly affects secondary flow motioninside the duct. A comparison of transverse gas velocity profilesin Fig. 8 shows that the core of the flow is initially drawn to theupper inner duct wall at the bend entrance as evidenced from apositive transverse velocity at =0◦. Further into the bend, neg-ative transverse velocities imply that the core flow is graduallymoving towards the outer wall. As the core flow travels fromthe inner wall to the outer wall, it entrains the surrounding fluidon either side of it towards the outer wall and then around theside walls. A pair of counter-rotating vortices (i.e., secondarymotion) is thus formed inside the bend. The speed at which thecore flow approaches the outer wall is therefore a good indicatorof the strength of the induced secondary motion. As seen fromFig. 8, the DRSM predicts an excessive movement of the coreflow towards the outer wall from as early as 60◦. The sametrend persists downstream of the bend (Fig. 9) which indicatesan over-prediction of the transverse velocity and hence of thesecondary flow motion by up to 50%. One would thus expectthe secondary motion to disperse more slowly in the simulationthan in the experiment.

Figs. 10 and 11 compare the measured and predicted tur-

bulence intensity√

u′u′ (normalised by the bulk velocity Ub)inside the duct. On the outer wall, the data indicates a rise innear-wall turbulence intensity between =0◦ and 30◦, followedby a drop to 0.1 at s/D = 0.5. By contrast, the turbulence in-tensity at the inner wall rises considerably in the second halfof the bend to a maximum of 1.5 at s/D = 0.5. This is directlydue to a strong reduction of the local gas flow in the streamwisedirection. The numerical solution provides a qualitative repre-sentation of these trends. Flow fluctuations as represented bythe turbulence intensities, however, have been severely under-predicted in these regions. This is mainly due to the applicationof the eddy dissipation transport equation (10), which ignoresthe contribution arising from the extra strain associated withstreamline curvature.

Turbulence intensity falls rapidly after s/d = 0.5 and gradu-ally recovers back to the same level as that at the bend entrance,buy symmetry in the turbulence structure is not completelyrestored even at s/D = 9.0.

8.2. Validation of the predicted particle tracks

Based on the predicted gas flow field, Lagrangian particletracking was first performed without the implementation ofthe SSL force and PWI models. The calculation utilised auniform restitution coefficient E = 0.73. The predicted mean


Fig. 8. Mean transverse gas velocity profiles within the bend (— prediction; � data).

particle velocity profiles were obtained from the data collec-tion process as outlined previously and they are comparedagainst the measured profiles in Figs. 12 and 13. This casewill be referred to as the baseline simulation in the followingdiscussions.

Compared to the measured profiles, the baseline simulationhas over-predicted longitudinal particle velocities at the outerwall within the bend (Fig. 12) and up to 3D downstream of thebend (Fig. 13). This is partly attributable to an over-prediction

of the near-wall gas velocities (see Figs. 5 and 6). Particle-wallcollisions also strongly affect particle velocity prediction inthis area and this is illustrated in Fig. 14 which plots predictedparticle velocities (un-averaged) for three particle sizes at =75◦. One can see from the figure that the predicted velocityprofiles for fine (5 �m) and coarse (> 76 �m) particles displaydistinctively different characteristics. The fine particles show astrong coupling with the gas phase while the coarse particlesare moving much more slowly within r∗ < 0.4 of the outer wall.


Fig. 9. Mean transverse gas velocity profiles in the vertical duct (— prediction; � data).

This has led us to believe that the coarse particles have alreadyseparated from the carrier fluid and they are moving under theinfluence of their own inertia as well as particle-wall collisionsin the simulation. This subsequently implies that particle sizedistribution also affects the prediction of particle velocity in theouter wall region of the duct bend.

Away from the outer wall, the predicted values follow themeasured velocity profiles with a maximum error of 20%

towards the inner wall, except at = 75◦ where large discrep-ancies between the measured and predicted particle velocitiesare found at r∗ > 0.8 (Fig. 12). With reference to Fig. 14, thisarises from a small group of fine particles possessing negativelongitudinal velocities and is a direct result of a recirculationzone in the predicted gas flow field at = 90◦ (see Fig. 5).The fine particles that have been entrained by the recirculat-ing gas flow at = 90◦ are thrown back towards = 75◦.


Fig. 10. Turbulence intensity (√

u′u′/Ub) profiles within the bend (— prediction; � data).

Velocity statistics pertaining to particles travelling upstreamthus cancel out those travelling downstream in the statisticalaveraging process. Hence, one can conclude that the predictedparticle motion in the inner-wall region of the duct bend isvery sensitive to the background gas flow field.

Mean transverse velocity profiles for the particles are pre-sented in Figs. 15 and 16. The baseline calculation is able to

correctly capture the transverse particle motion on the verti-cal centre-plane of the duct bend (Fig. 15). Both the experi-mental measurement and the calculation suggest a thin layerof particles having very small transverse velocity componentsat r∗ < 0.2 between = 45◦ and 75◦. This phenomenon arisesas some of the particles are moving towards the outer wall andsome are moving (rebounding) away from the wall within the


Fig. 11. Turbulence intensity (√

u′u′/Ub) profiles in the vertical duct (— prediction; � data).

LDA probe volume. As a result, the two types of motion canceleach other out during the averaging process. This observationis consistent with the findings of a number of published studieson solid saltation, e.g. Tanière et al. (1997) and Ciccone et al.(1990).

In the vertical duct, the particle tracks tend to possess a largertransverse velocity component than that in the physical flow

(Fig. 16). This directly relates to a considerable discrepancybetween the measured and predicted gas-phase velocities asseen in Fig. 9. Hence, one should expect the predicted transversevelocity profile to approach the measured distribution as thegas-phase solution improves.

Following the baseline calculation, Lagrangian particletracking was repeated with the inclusion of SSL force and


Fig. 12. Mean longitudinal particle velocity profiles within the bend (— baseline prediction; � prediction considering SSL and PWI models; � data).

sliding/non-sliding PWI in the numerical model. Figs. 12,13, 15 and 16 indicate that they have no major effect on thepredicted mean particle velocities. However, they do contributeconsiderably to the distribution of particles in the duct as illus-trated in Fig. 17. The figure plots the normalised distributionof particle tracks in the inner- and outer-wall regions along theduct, and it is based on the number of particle tracks passingthrough a 5.0 mm layer next to the duct walls. At each station,

particle tracks in the near-wall layer are counted and thenthe sum is normalised by the total number of particle trackspassing through the symmetry axis of that station. Accordingto the figure, the application of SSL force and PWI modelshas helped distribute a larger portion of the particle tracks to-wards the outer duct wall within the bend as compared to thebaseline result. In this region, the baseline model only allowsthe particles to rebound away from the wall after particle-wall


Fig. 13. Mean longitudinal particle velocity profiles in the vertical duct (— baseline prediction; � prediction considering SSL and PWI models; � data).

collisions. By comparison, the PWI model offers the particlesa second option which is to slide along the wall if their angleof incidence is below a critical limit. This directly contributesto a stronger presence of particle tracks at the outer wall of theduct bend.

In the vertical duct, slip velocities (i.e., uf – up) are smallbut positive for a large majority of the particle tracks in theinner-wall layer. The SSL force, Eq. (20), thus distributes more

particle tracks away from the inner wall, resulting in a weakerpresence of particle tracks at the duct inner wall between s/D=3.0 and 9.0 as compared to the baseline case.

Streamwise variation of the mean longitudinal velocity forthe particles within a 1.5 mm layer next to the outer wall isshown in Fig. 18. In the early part of the bend, the data showa rather rapid decrease in particle velocity as the majorityof the particles collide with the outer wall. The wall particle


Fig. 14. Predicted particle velocities by particle size (un-averaged) at = 75◦.

Fig. 15. Mean transverse particle velocity profiles within the bend (— baseline prediction; � prediction considering SSL and PWI models; � data).

velocity then gradually recovers to about 0.56Ub by 90◦ withthe help of the gas flow entraining and resuspending therebounded particles.

Compared to the measured profile, our predictions basedon the SSL and particle-wall collision models are only ableto quantitatively capture the particle deceleration/acceleration


Fig. 16. Mean transverse particle velocity profiles in the vertical duct (— baseline prediction; � prediction considering SSL and PWI models; � data).

process inside the bend. Universally applying a lower resti-tution coefficient to replace the particle-wall collision modeldid not cause a further reduction in particle velocity between = 15◦ and 45◦ where particle-wall collisions are expected tobe the most frequent. Therefore, particle-wall collision is notthe dominating mechanism through which the particles losetheir momentum.

Although we are not able to directly verify the location andsize of particle rope in the duct with the measured data, a higherconcentration of particle tracks in the outer wall layer between = 0◦ and s/D = 1.0 (Fig. 17) does suggest the presence ofa particle rope inside the duct. It is thus very likely that the

particle lose much of their momentum as a result of roping.Within the rope, the particles are strongly affected by inter-particle collisions. One can therefore further improve particlevelocity predictions in this region by considering inter-particlecollisions in the numerical simulation.

9. Conclusion

Numerical simulations have been performed for a curved 90◦duct bend. The gas-phase simulation was based on a DRSMapproach and the solids flow was solved by Lagrangian parti-cle tracking. Our particle tracking calculations involved 96,000


Fig. 17. Normalised distribution of particle tracks within a 5.0 mm layer next to the walls (— baseline prediction; —�— prediction considering SSL and PWImodels).

Fig. 18. Streamwise variation of mean longitudinal particle velocities along the outer wall, 1.5 mm from the surface.

particle tracks and considered a range of hydrodynamic forcesand a particle-wall interaction (PWI) model, but ignored theeffects of particle rotation, turbulence modulation and inter-particle collisions. The CFD solutions have then been validatedagainst the experimental data in an effort to identify areas wherefurther work is necessary to improve the accuracy of numericalprediction for flows with dilute solid suspension. The presentstudy demonstrated the importance of a range of issues that oneneeds to consider when performing a numerical simulation ofgas–solid flow inside a power station mill duct, including:

• Turbulent gas flow solution based on the current DRSM canprovide a reasonable representation of the mean gas flow atthe duct vertical centre-plane, except in regions next to theconcave wall and more than 3D downstream of the bend. Theformer is a deficiency, common to all differential Reynoldsstress models, which has not yet been fully resolved by thefluids modelling community. The turbulence model has con-siderably under-predicted the decay of secondary motion inthe vertical duct.

• In the inner-wall region and also more than 3D downstreamof the bend, the predicted particle motion critically dependson the quality of the gas flow solution. Near the outer wallwhere the particles are more likely to collide with the wall,

PWI model can also considerably affect the prediction ofparticle velocities and distribution of particles near wall.

• Implementation of the shear-slip lift (SSL) force modeldid not directly contribute to a more accurate prediction ofsolids motion. However, it acts to redistribute particle tracksacross the flow depending on local gas velocity gradientsand slip velocity, particularly near the inner duct wall down-stream of the bend. This makes it an important hydrody-namic force component to consider in shear or curved flowcalculations.

• The PWI model adopted in this study does not allow near-wall particles to lose as much momentum as the data suggestsat the outer wall, though it was able to reproduce a similartrend. We thus expect the particles to attain a much lowernear-wall velocity through mechanisms other than PWI inthe physical flow.

Notation

C model constantsCD drag coefficientdp particle diameter, mD hydraulic diameter of the duct, m


E restitution coefficientEn restitution coefficient normal to a surfaceEt restitution coefficient tangential to a surfaceF force vector, NFr Froude numberfD Schiller–Naumann drag correlationG turbulence production, kg m−2 s−2

g gravity vector, m s−2

k turbulence kinetic energy, m2 s−2

L solids mass loading = mp/mf

m mass, kgm mass flow rate, kg s−1

Ni number of particle tracks allocated to size frac-tion i

n number of particle tracksR duct turning radiusRe duct Reynolds numberRes particle Reynolds number based on fluid rotationS inter-particle distance, mSt particle Stokes numbers, r∗ curvilinear coordinate system on the duct

plane of symmetry; r∗ = 0 atouter wall; r∗ = 1 at inner wall, m

u instantaneous velocity vector, m s−1

u′ fluctuating velocity vector, m s−1

u′, v′ fluctuating longitudinal andtransversevelocity components, m s−1

U mean gas velocity vector, m s−1

U, V mean longitudinal and transversevelocity components, m s−1

Ub bulk gas velocity, m s−1

UT particle terminal velocity, m s−1

Greek letters

�p particle volume fraction�1 smooth wall incidence angle, deg�ij Kronecker delta�� standard deviation of the wall

roughness angle; �� = 3.8 for aplexiglass plate, deg

� eddy dissipation rate, m2 s−3

duct turning angle, deg� gas dynamic viscosity, kg m−1 s−1

�0 static friction coefficient�d dynamic friction coefficient� gas kinematic viscosity, m2 s−1

� Gaussian random number with zeromean andstandard deviation of 1

� density, kg m−3

� time scale, s particle rate, s−1

�f fluid rotation vector,s−1

Subscripts

A added massD drag

f fluidg gravityi size fraction indexi, j, k tensor indexp particpg pressure gradientsl slip-shear lift1 pre-impact state2 post-impact state

Acknowledgements

The authors gratefully acknowledge the financial and othersupport received for this research from the Cooperative Re-search Centre (CRC) for Clean Power from Lignite, which isestablished and supported under the Australian Government’sCooperative Research Centres program.

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