1-s2.0-S0032591000003193-main

Ž .Powder Technology 114 2001 168–185www.elsevier.comrlocaterpowtec

Formation and dispersion of ropes in pneumatic conveying

Ali Yilmaz), Edward K. LevyEnergy Research Center, Lehigh UniÕersity, 117 ATLSS DriÕe, Bethlehem, PA 18015-1729, USA

Received 10 March 2000; received in revised form 18 May 2000; accepted 23 May 2000

Abstract

In this study, the solid flow nonuniformities which develop in lean phase upward flow in a vertical pneumatic conveying linefollowing a horizontal-to-vertical elbow were investigated. Laboratory experiments were conducted in 154 and 203 mm I.D. test sections

Ž .using pulverized-coal particles 90% less than 75 mm for two different 908 circular elbows having pipe bend radius to pipe diameterratios of 1.5 and 3.0. The experiments covered a range of conveying air velocities and solids mass loadings. Experimental measurementsof time-average local particle velocities, concentrations, and mass fluxes were obtained using a fiber-optic probe which was traversedover the cross-section of the pipe. The measurements indicate a continuous rope-like structure forms within the elbow. The rope maintainsits continuous structure until it disintegrates into large discontinuous clusters at downstream locations. Comparisons of the results of CFDsimulations of turbulent gas-particle flow and time-average experimental data were used to explain rope formation and dispersion. TheCFD simulations, based on the Lagrangian particle-source-in-cell method, predict a denser particle rope as the nondimensional radius of

Ž .curvature RrD is increased, agreeing with trends in experimental data. The individual effects of secondary flows and turbulence onaxial dispersion of the rope were studied computationally and the results show both mechanisms are important. q 2001 Elsevier ScienceS.A. All rights reserved.

Keywords: Pneumatic conveying; Elbows; Rope flow; Clusters; Fiber optic probe; CFD Modeling of turbulent gas-particle flows

1. Introduction

Pneumatic conveying of solids has wide ranges ofapplication in the chemical, food processing, pharmaceuti-cal, cement and power industries. In coal-fired boilers,pulverized coal is pneumatically conveyed in large diame-

Ž .ter pipes 400 to 800 mm I.D. with conveying velocitiesof 20 to 30 mrs and with solids loading ratios of 0.33 to 1.As the mixture of air and pulverized coal make a turnwithin an elbow, pulverized-coal particles form a rope-likestructure because of inertial effects. A particle rope, whichcarries most of the conveyed material in a small portion ofthe pipe cross-section, acts as a third phase in the pneu-matic conveying line, with lower particle velocities andrelatively high particle concentration. Roping causes avariety of operational difficulties in coal-fired boilers andaffects the ability to control NO emissions and limitx

efficiency loss due to unburned carbon in the ash.

) Corresponding author. Tel.: q1-610-758-4090.Ž .E-mail address: [email protected] A. Yilmaz .

Research on roping, which goes back to late 1950s and1960s, includes both field and laboratory measurements.Early experiments relied on flow visualization and isoki-

w xnetic sampling. Patterson 40 recorded details of saltationŽ .profiles in large 203–304 mm diameter pulverized-coal

conveying lines. The minimum conveying air velocity wasfound to depend on pipe bends in the conveying pipework.

w x w xWeintraub 57 , in a discussion of Patterson’s paper 40 ,suggested the use of air jets to break up rope flow and

w xavoid saltation in long horizontal pipes. Whitney 58discussed a new burner design which disperses rope flowand creates better pulverized-coal distribution at the burner

w xnozzle exit. Zipse 65 investigated the possibility of usingorifice plates and single phase flow conditioners to dis-perse the rope flow and obtain homogenous gas–solids

w xflow. Cook and Hurworth 8 reported the principal causeŽ .of settlement was rope flow. Particle Image Velocity PIV

w xmeasurements by McCluskey et al. 35 also support thew xsaltation mechanism described by Cook and Hurworth 8 .

w xMore recently, Huber and Sommerfeld 20 characterizedthe degree of segregation in different pipe elements, usingspherical glass beads with a mean diameter of 45 mm. The

0032-5910r01r$ - see front matter q 2001 Elsevier Science S.A. All rights reserved.Ž .PII: S0032-5910 00 00319-3

( )A. Yilmaz, E.K. LeÕyrPowder Technology 114 2001 168–185 169

information on local particle velocity and concentration byŽ .Phase Doppler Anemometry PDA showed a wall rough-

ness element height comparable to the particle size favorsw xthe dispersion of the rope. Huber and Sommerfeld 20 also

showed the orientation of the elbow plays a strong role inrope dispersion behavior.

Numerical studies of turbulent gas-particle flowsthrough pipe bends have been published by Tsuji and

w x w x w xMorikawa 53 , Nobuhisa et al. 37 , Li and Shen 29 , Tuw x w xand Fletcher 55 , Huber and Sommerfeld 21 . Tu and

w xFletcher 55 , using a two-fluid model developed by Chenw xand Wood 6 , simulated turbulent gas-particle flow through

an elbow with a square cross-section 908 bend, predicting aparticle rope near the outer curve of the bend, similar to

w xthat observed by Kliafas and Holt 26 . The results indi-cated that particle–wall interactions are a controlling factorfor the outer-wall region of the flow.

Computational modeling efforts on gas-particle flowscan be divided into two categories: Lagrangian and multi-

w xfluid models 1,3,4,9,13,14,25,30–32,41,46,47,55 . Bothmodeling approaches make use of turbulence closureschemes to obtain gas phase solutions. However, the La-grangian approach treats the particles as discrete entitiesinteracting with turbulent eddies in a Lagrangian coordi-nate frame while the Eulerian approach treats the particu-

w xlate phase 10,14 as a continuum having conservationequations similar to those of the continuous gas phase. Thedetails and range of applicability of the numerical model-ing approaches have been summarized in the literature

w xquite extensively 10–12 . Comparative studies of the twomodeling approaches performed in fully accelerated gas–solids flow in vertical tubes resulted in close agreement onthe predictions of average gas and particle velocity profilesw x1,13 ; however, the Lagrangian particle tracking approachis better suited for modeling dilute phase pneumatic con-

w xveying 10–12 . Due to the lack of understanding of themutual couplings between the gas and particulate phase,almost all modeling efforts have failed to address the

w xcarrier fluid turbulence modulation due to particles 1 .Irregular particle–wall collisions, particle–particle interac-tions, the effect of Saffman lift forces on particles close tothe wall, and turbulent dispersion of fine particles werefound to be important features of dilute phase pneumatic

w xconveying 25,30,46,47 .The present study includes results on the solid flow

nonuniformities which develop in lean phase upward flowin a vertical pneumatic conveying line following a horizon-tal to vertical elbow. Laboratory experiments were con-ducted in 154 and 203 mm I.D. test sections using pulver-

Ž .ized-coal particles 90% less than 75 mm for two different908 circular elbows having pipe bend radius to pipe diame-ter ratios of 1.5 and 3.0. The test conditions covered solidsloading ratios and conveying air velocities within theranges of the operating conditions in a typical fuel pipelinein a pulverized-coal boiler. Experimental measurements oftime-average local particle velocities, concentrations, and

mass fluxes were obtained using a fiber-optic probe whichw xwas traversed over the cross-section of the pipe 62 .

In addition to the laboratory experiments, numericalsimulations were performed using a commercial CFDpackage, CFX-Flow3D version 4.1c, developed by AEA

w xIndustrial Technology 5 . These used Lagrangian particleŽtracking along with a two-equation turbulence model RNG

.kye to model turbulent gas-particle flows through hori-zontal-to-vertical 908 circular elbows. The objective of thenumerical simulations was to study rope formation anddispersion phenomena within circular elbows and to ex-plain the trends observed in the time-averaged experimen-tal data.

2. Experimental

2.1. Pneumatic conÕeying test facility

The pneumatic conveying test facility consists of two6.1-m-long horizontal pipes and a 3.35-m-long vertical

Ž .pipe as well as two 908 elbows see Fig. 1 . The pipes areŽ .made of schedule 40 carbon steel pipe 0.154 m I.D. . A

cyclone is used to separate the coal particles from the air.The collected coal particles are fed to the hopper by arotary airlock. A screw feeder feeds the recycled materialinto the pneumatic conveying line. This completes onecycle in the continuous operation of coal flow in thesystem. The air flow leaving the cyclone discharges to theatmosphere after passing through an Aget Model FT-40bag filter assembly, where fine particles down to 0.3 mmare captured. The length of the first horizontal run wasselected by performing an acceleration length calculation

w xaccording to Yang’s unified theory 60 . As a result, a fullyaccelerated gas–solid flow prevailed at the inlet to thehorizontal to vertical elbow for all test conditions. The

Ž .vertical sections 154 and 203 mm I.D. were made ofmodular parts to make it possible to fit different elbowsinto the system. During the experiments for the larger pipe

Fig. 1. Sketch of the pneumatic conveying flow facility.

( )A. Yilmaz, E.K. LeÕyrPowder Technology 114 2001 168–185170

Table 1Particle size distribution of pulverized coal

Ž . Ž .Diameter mm Weight %

)125 1.5106–125 11.090–106 17.975–90 16.763–75 13.145–63 20.1-45 19.7

size, the vertical test section and part of the horizontalsections were changed to 203 mm I.D. pipes using reduc-ers in the horizontal sections. The pipe size change wasmade 2 m away from the vertical section to avoid flowdisturbances. The modular construction also makes it pos-sible to perform flow visualization studies with speciallydesigned pipes having Plexiglas windows. The test facilitywas properly grounded to reduce the effects of electrostaticcharging.

The conveying air mass flow rate is measured by anorifice meter designed according to ASME specifications.An Acrison Model 105Z-N volumetric feeder with a vari-able speed control was used to meter the coal flow rate.Coal flow rates of 0.126 to 1.072 kgrs are achievable withthe current feeder configuration, providing a solids loading

Ž .ratio m of between 0.2 and 1. Forty instrumentationports were placed along the pipes at appropriate angularlocations on the pipe cross-section.

Pulverized-coal particles with a weight mean diameterŽ .of 75 mm see Table 1 and true particle density of 1680

kgrm3 were used as the conveying material. Due tocontinuous circulation of pulverized coal within the loop, itwas necessary to assess the attrition rate of pulverizedcoal. In a series of experiments, the loop was run over 100h without any addition of new material. Pulverized-coalsamples were extracted from the hopper after every 15 h ofoperation, which corresponded to discharge and circulationof the inventory of the material in the hopper approxi-

mately 200 times. Sieve analyses performed on the sam-ples showed no significant attrition for the time periodcovered. The weight mean diameters of the pulverized-coalsamples fluctuated in the range of 70 to 76 mm, mostlikely due to random variations in sampling and sieving.

2.2. Fiber optic probe

A reflective fiber-optic measurement system was usedto simultaneously measure the particle velocity and massconcentration. Fig. 2 shows the instrument, which consistsof two similar optical probes, and the detail of a single-probe configuration used in the present study. Two glassfibers were used in each probe: one of the fibers is used to

Ž .send light from a light emitting diode LED into thegas-particle flow region, while the other fiber transfers thereflected light onto the detector area of a silicon photodi-ode. Local particle velocities are measured using thecross-correlation technique; that is, the flight time of parti-cles from the upstream probe to the downstream probe wasestimated using the cross-correlation function:

1 TC t s u t u tqt d t 1Ž . Ž . Ž . Ž .H 1 2T 0

Ž . Ž .where the terms u t and u t are particle flow signal1 2

waveforms obtained from the two fiber-optic probes, whichare aligned in the main flow direction. The flight time of

Ž . Ž .particles t is the time lag t where the cross-correla-max

tion function becomes maximum. Therefore, local particlevelocity is readily computed from

LU s 2Ž .p

tmax

where the term L is the optical distance between the twofiber-optic probes.

The intensity of the particle flow signal generated byŽ .one of the photodetectors I is used as a measure of thepŽ . w xparticle mass concentration C . As described in Ref. 61 ,p

Fig. 2. Fiber-optic measurement system and detail of fiber-optic probe design.


a calibration procedure was developed to convert the parti-Ž .cle flow signal intensities I into a particle mass concen-p

Ž .tration C as follows:p

C sKI n 3Ž .p p

where K and n are calibration constants.Local particle mass flux is the product of local particle

velocity and local particle mass concentration:

mY sU C 4Ž .˙ p p p

An analysis of the cross correlation algorithm and theerror associated with sampling time gave a predicted un-certainty in the local particle velocity measurements of less

w xthan 2.5% 62 . In addition, the local mass flux measure-ments made by the optical probe were compared to themeasurements by the isokinetic sampling probe. The com-parison showed an agreement within 10% between the twomeasurement techniques.

To avoid having the probe disturb the flow, the probediameter must be sufficiently small. In the present experi-ments, each probe consisted of two 600 mm diameterfibers, placed in a 6-mm internal diameter tube. The pipeblockage, the ratio of the area of the probe surface perpen-dicular to the flow direction and pipe cross-section area,

Žfor the worst case scenario the probe tip located in the. 1center of the pipe was 2.6%. The Stokes number based

Ž .on the particle response time 0.12 s for 45 mm particleand a particle transit time through the measurement regionŽ .12 mm was anywhere from 50 to 300 depending on the

Ž .particle velocity measured 5 mrs to 30 mrs . Stokesnumbers of this magnitude ensure that the particle re-sponse to the drag due to local air velocity changes isnegligible. In another study carried out by the authors, theStokes number was made even larger by decreasing themeasuring region to 3 mm. This was accomplished by

Ždesigning a smaller diameter fiber-optic probe 1.83 mm.O.D. tube and by changing the probe tip design to a

parallel fiber arrangement. The pipe blockage for this casewas 0.75% compared to 2.6% with the 6-mm probe design.The Stokes number increased to between 200 and 1200.Local particle velocity measurements performed by the

Žtwo instruments the 6-mm probe used in this study and.the 1.83-mm probe agreed to within 2%, which is also the

magnitude of the measurement uncertainty associated withthe cross-correlation technique employed in this study.

3. Experimental results

The majority of the experiments were performed tounderstand rope flow dispersion characteristics in the verti-

1 Ž .Note: The Stokes number is defned as: Sts t rt , where the termA tŽŽ 2 . Ž ..t r d r 18m is the particle aerodynamic response time and t is theA p p t

particle transit time through the measuring volume.

cal pipe following two 908 circular, horizontal-to-verticalŽ .elbows RrDs1.5 and 3.0 . The experiments covered the

range of conveying air velocities from 15 to 30 mrs andsolids loading ratios between 0.33 and 1. Some of theexperiments were performed in a larger pipe diameterŽ .Ds0.2027 m using an RrDs1.5 elbow.

Measurements were also performed near the elbow inletto characterize the particle concentration and velocity dis-tribution over the pipe cross-section. Since the fiber-opticprobe measures particle velocity and concentration at apoint, it must be traversed over the pipe cross-section toobtain information on flow nonuniformities. This was doneusing measurement ports placed at various locations alongthe pipe and around its circumference. Fiber optic probemeasurements were performed at six different axial dis-

Žtances from the elbow exit plane zrDs1, 3, 5, 9, 13,.and 17 as shown in Fig. 3.

Ž .The effect of the solids loading ratio m on the particlevelocity and concentration profiles at the elbow inlet planewas studied at a constant conveying air velocity of 20mrs. Fig. 4a shows the inlet particle velocity profiles did

Ž .not change as the solids loading ratio m increased from0.33 to 1.0. However, the particle concentration close to

Ž .the bottom wall increased with solids loading see Fig. 4b .Flow visualization of the flow at this location showed arope-like structure for ms0.5 and 1. No rope-like struc-ture was observed for ms0.33. The formation of a rope ina horizontal pipe, as the solids loading increased, was also

w x w xreported by Cook and Hurworth 8 , Flemmer et al. 16w xand Johnson and Means 24 .

Preliminary measurements at downstream of the elbowwith the fiber-optic probe showed a relatively symmetricparticle concentration distribution within the pipe cross-section. Fig. 5 shows a contour plot of particle concentra-tion in the pipe cross-section at zrDs3 with the RrDs1.5 elbow. This contour plot is based on the fiber-opticprobe measurements performed at 32 equal-area points in

Fig. 3. Sketch of coordinate system used for a 908 circular elbow.


Ž . Ž . Ž .Fig. 4. Effect of solids loading ratio m : a particle velocity profiles, b particle concentration profiles one pipe diameter upstream of elbow inlet plane.

the pipe cross-section. It is apparent from this contour plotthat particles are conveyed within a rope in a small portion

Ž .of the pipe cross-section close to the outer wall xrDs0 .Since the largest variations in particle concentration oc-curred in the x direction, most of the experiments wereperformed in this direction along the pipe diameter. Al-though the rope thickness was fairly constant in the x

Ž .direction, the rope width in the y direction see Fig. 3 didchange with flow conditions. Flow visualization showedthe rope cross-sectional area increased with an increase in

Ž .RrD and m, and with a decrease in U see Fig. 6 .o

Fig. 7a illustrates the radial variations of particle veloc-ity and concentration at different axial locations in the

Ž .vertical pipe U s29 mrs and ArFs1.0 . These pro-o

files, obtained at axial positions zrD ranging from 1 to17, show the variations of concentration and velocity alonga diameter in the x direction. As they move through the

Ž .elbow, the particles are forced to the outer wall xrDs0due to the centrifugal effect of the elbow. The relativelyhigh particle mass concentrations close to the wall at the

Ž .first elevation zrDs1 show the centrifugal effect. Ropes

Ž 3.Fig. 5. Particle mass concentration for ms1, c kgrm , followingp

RrDs1.5 elbow at zrDs3.

Ž .formed within the long radius elbow RrDs3.0 aredenser than the ropes created by the short radius elbowŽ .RrDs1.5 . Particles lose kinetic energy due to directionchange in the elbow, resulting from inelastic particle–walland particle–particle collisions. Therefore, particles withinthe rope have substantially lower velocities than that of theconveying air, at zrDs1. Earlier investigations on roping

w xphenomena 18,20,35 revealed that the rope velocity isabout half the velocity of the conveying air at the elbowexit plane. This result is consistent with the findings fromthe present study for the RrDs1.5 elbow, but not withthe rope velocity in a RrDs3.0 elbow. Lower rope

Ž .velocities about 25% of conveying air velocity for theRrDs3 elbow are associated with the higher frictionallosses which occur along the outer wall of the RrDs3elbow.

As they move downstream of the elbow in the verticaldirection, the particles in the rope region are accelerated.Flow visualization downstream of the elbow showed therope is unable to maintain its continuous structure during

Ž .Fig. 6. Rope width in y direction Ds0.154 m at zrDs1.


Ž . Ž . Ž .Fig. 7. a Effect of elbow radius of curvature and b effect of solids loading ratio m on local particle velocity and concentration profiles along the pipediameter in the x direction at zrDs1, 5, 9, and 17 for two 908 circular elbows. Results obtained using fiber-optic probe.

the acceleration and disintegrates into discontinuous largew xclusters 63 . Breaking of the rope into large clusters took

place at around zrDs5. As can be seen from Fig. 7a, therope dispersed at a relatively slow rate from zrDs1 to 5.In addition, the rope created by the RrDs3.0 elbowremained attached to the outer wall in the vertical pipe andits dispersion rate was low compared to the dispersion rateof the RrDs1.5 elbow.

Fig. 7b compares particle velocity and concentrationprofiles, prevailing in the vertical pipe downstream of theRrDs3.0 elbow, for different solids loading ratios at aconstant conveying velocity of 20 mrs. A denser particlerope was formed for high solids loading ratios, while therope flow dispersed at a faster rate for the low values ofsolids loading ratios. Depending on the solids loading, theeffects of conveying air velocity on rope formation anddispersion characteristics change. When the solids loadingratio m is equal to 1, there seems to be no effect ofconveying air velocity on rope formation and dispersionŽ .see Fig. 8 . Fig. 9 shows the plots of the maximumparticle mass concentration values in the pipe cross-sectionas a function of axial nondimensional distance zrD atdifferent flow conditions and elbow geometries. The plotsin Fig. 9 also show the relatively slow dispersion of therope flow for the RrDs3 elbow.

ŽLimited experiments with a larger pipe diameter I.D.s.203 mm showed that rope flow formation and dispersion

Žbehavior are similar over the range of pipe diameters see.Fig. 10 . However, more data are needed to fully assess the

diameter effect on rope formation and dispersion.

4. Numerical modeling

Lagrangian particle tracking along with a Renormaliza-Ž .tion Group RNG kye turbulence model was used to

simulate turbulent gas-particle flows through the elbows.The objective of the numerical simulations was to studyrope formation and dispersion phenomena and compare thenumerical results with trends observed in the time-aver-aged experimental data.

In the Lagrangian particle tracking approach, the inter-action between the gas phase and particulate phase istreated using the particle-source-in-cell method of Crowe

w xet al. 9 . This method was founded on the idea of treatingparticles as sources of mass, momentum, and energy to the

w xgaseous phase 36 . For the present study, only the momen-tum source term due to the particles is considered. Afterparticles are randomly injected at the inlet with a finitenumber of starting locations and with a finite number ofparticle sizes at each starting location, Newton’s secondlaw of motion is used to find the velocity of every particlein the flow domain. Integration of the particle velocities


Fig. 8. Effect of conveying air velocity on particle velocity and particleconcentration profiles along the pipe diameter in the x direction atzrDs1, 3, 9, and 17 for RrDs1.5, ms1.0.

with respect to time gives the particle trajectories, so theparticle velocities and locations can be used to obtain themomentum source terms for each computational cell.

The mutual interphase coupling is introduced with aniterative process which is known as Atwo-wayB coupling.The influence of fluid turbulence on the particles is mod-eled by a stochastic method proposed by Gosman and

w xIoannides 17 who modeled the turbulent flow by a se-quence of individual eddies which interact with particles.The interactions between particles and the wall are mod-eled using a coefficient of restitution, which is the ratio ofnormal velocities before and after the particle–wall colli-sion. In addition, the tangential particle velocity is as-sumed to be constant during a particle–wall interaction.The influence of the particles on fluid turbulence is notmodeled. Furthermore, it is assumed there are noparticle–particle interactions.

4.1. GoÕerning equations

CFX-Flow3D uses the finite volume approach and ageneralized non-orthogonal body-fitted coordinate system

Žto discretize the governing equations. CFX-Flow3D re-. w xlease 4.1c 5 offers two turbulence models for use with

the Lagrangian particle tracking approach. These turbu-

lence models are based on the eddy viscosity hypothesis,i.e. standard kye and renormalization group based RNG

w xkye turbulence models 59 . The RNG kye model wasused to predict turbulent quantities within the flow fielddue to its better performance over the standard kye

model in predicting the streamwise and radial velocitycomponents and Reynolds shear stresses within a 908

w xcircular pipe bend 64 . The RNG model, which is derivedfrom a renormalization group analysis of the Navier–Stokesequations, differs from the standard model through a modi-fication to the equation for C and the use of a different setof model constants. The RNG theory cannot be extendedto viscosity dominated flow regions like the viscous sub-layer of a turbulent boundary layer. Therefore, CFX-Flow3D implements the wall function approach to bridge

w xthe viscous sublayer 44 .The gas phase flow solution is obtained with a set of six

partial differential equations which consists of ReynoldsŽ .averaged Navier–Stokes equations RANS and two equa-

tions for turbulence modeling. These equations can bewritten in a generic transport form

E E EfrU f s G qS qS 5Ž . Ž .i f f f ,pž /Ex Ex Exi i i

Žwhere the terms U are mean velocity components U, V,i. ŽW . The parameter f represents the variables U, V, W, k,.e . The quantity G is the effective viscosity. The termsf

Ž .S are source terms for the gas phase. Eq. 5 also containsf

the additional source term S which represents the netf,p

efflux of f into the gas phase owing to gas–particlew xinteractions through the viscous drag force 13 . These

source terms are calculated by finding the particle trajecto-ries and particle velocities along the particle path.

In the Lagrangian approach implemented by CFX-Flow3D, the particulate phase is represented by computa-tional particles whose trajectories are computed by simul-taneously integrating™d x

™psU 6Ž .pd t

and the equation of particle motion which is generallyw xwritten as 7

™

dU™ ™ ™p

m sF qF qF 7Ž .p D B OTHERSd tŽ .Eq. 7 describes the balance of forces acting on the

particle as it moves along its trajectory. The term on theleft side is the inertia force acting on the particle due to itsacceleration and the right-hand side terms are the externalforces acting on the particle. The most influential force

™

acting on the particle is the viscous drag force F exertedD

by the continuous phase. This force is predicted with theaid of the standard drag coefficient C and relative veloc-D

™

ity between the particle and the carrier fluid U ;R

1™ ™ ™2 < <F s p d rC U U 8Ž .D p D R R8


Ž . Ž .Fig. 9. Axial variation of a peak particle concentration in pipe cross-section and b particle velocity at locations where peak particle concentrationŽ .prevails for RrDs1.5 and 3.0 elbows Ds0.154 m .

where the following standard drag coefficient by Torobinw xand Gauvin 52 is used:

240.687C s 1q0.15Re for Re F1000 9Ž .Ž .D p pRep

Particle Reynolds number is defined by™

< <r U dR pRe s 10Ž .p

m

Here, d is the particle diameter, and r and m are thep™

density and viscosity of the carrier fluid. The force F , theB

™buoyancy force due to gravitational acceleration g, isgiven as follows:

1™

™3F s p d r yr g 11Ž . Ž .B p6The other external forces that can play important roles incalculating trajectories for some gas-particle flows are

w xgiven in Ref. 49 . They are the Basset force, whichaccounts for the history effects of the motion, the added-mass term due to the acceleration of carrier fluid in thevicinity of the particle, and the pressure gradient force. Forgas-particle flows where the density ratio r rr is of thep

Ž . ŽFig. 10. Comparison of rope dispersion behaviors for two different pipe diameters Ds0.154 and 0.2027 m RrDs1.5 and comparable flow.conditions .


order of 103, these forces are negligibly small compared tothe drag force.

The effect of turbulence is included within the particletransport model by using the instantaneous velocity of the

Ž . Ž .carrier fluid in Eqs. 7 and 8 as

™ ™ ™™U s Uqu yU 12Ž .� 4R p

w xFollowing the method of Gosman and Ioannides 17 , themotion of particles is tracked as they interact with asuccession of discrete turbulent eddies. Eddies are assumedto have constant velocity, length and time scales during theinteraction with particles. A particle is assumed to interactwith an eddy for a time which is the smaller of either theeddy life time or the transit time required for the particle tocross the eddy. These times are estimated by assuming thatthe characteristic size of an eddy is the dissipation length

w xscale 17

L sC3r4k 3r2re 13Ž .e m

where C is a turbulence model constant, k is the turbu-m

lent kinetic energy, and e is the energy dissipation rate.The eddy life time is computed in a manner similar to

w xShuen et al. 45 by

1

2t sL r 2kr3 14Ž . Ž .e e

The transit time of a particle is found using the linearizedequation of motion for a particle in a uniform flow

™

< <t syt ln 1yL r t U 15Ž .ž /tr e Rž /where the particle relaxation time t is given by

4 r dp pts 16Ž .

™3 < <r U CR D

4.2. Boundary conditions

Ž .The transport equations RANS are elliptic in spacecoordinates and, hence, require that values or gradients oftransported variables are defined all around the flow geom-etry. Two different flow domains are modeled in thisstudy. For studying rope formation and dispersion within ahorizontal-to-vertical elbow, the flow geometry consistedof a horizontal pipe with a length of 5 pipe diameters, theelbow section, and a vertical pipe with a length of 20 pipediameters. Fully developed turbulent flow was assumed atthe inlet to the horizontal section. The velocity profile forfully-developed turbulent flow was approximated by the1r7th power law relation given by

1r7U 2 rinlets 1y 17Ž .ž /U Dc

Turbulence quantities were calculated using the followingrelations

23 Uinletk s 18Ž .inlet ž /2 10

k 3r2inlet

e s 19Ž .inlet 0.3 D

Ž .Eq. 18 assumes that characteristic turbulent eddy velocityat the inlet is 10% of the mean flow velocity. At the outlet,Neumann boundary conditions were imposed on all trans-port variables except the velocity, which was given aconstant normal gradient. The value of that constant isinternally determined by the code such that the total flowrate out of the domain remains equal to the total flow intothe domain at all stages of the solution. This constant isgenerally close to zero. It is exactly equal to zero for fullydeveloped outlet flows: the vertical pipe length was chosenlong enough to satisfy that approximation. Finally, zero-slipwas prescribed at the wall surfaces for the gas phase.

In the simulations, trajectory calculations were per-formed for 2500 to 5000 computational particles, eachcarrying the same solids mass flow rate. To limit thenumber of trajectory calculations, a single computationalparticle represented on the order of 105 real particles. Theinitial particle velocities were set to the average conveying

Ž .air velocity U . The particle trajectory calculations wereo

begun at 125 locations at the inlet, whose coordinates wererandomly sampled. Therefore, 20 to 40 computationalparticles were tracked at each starting location. The diame-ters of the computational particles at each starting locationwere stochastically sampled from a Rosin Rammler distri-bution function

nd

Prob Dia.)d sexp y 20Ž . Ž .ž /dm

The term d is the mean diameter of the distribution andm

the exponent represents the spread of the data from themean. The parameters d and n were found to be 75 mmm

and 2.33, respectively, from the particle-size distributioninformation of pulverized-coal particles used in the experi-

Ž .ments see Table 1 .Particle–wall collisions were modeled through a coeffi-

cient of restitution, e, which is the ratio of normal veloci-ties of the particle before and after the collision. Thetangential velocity component of the particle during theparticle–wall interaction was assumed to be constant. Thecoefficient of restitution, e, was set equal to 0.9 for all thesimulations performed in this study. The coefficient ofrestitution used in the present study comes from data

w xpresented by Tabakoff and Malak 50 for the particle–wallcollision process between fly ash particles and stainless

w xsteel. Tabakoff and Malak’s data 50 show that for colli-sion angles lower than 158 the normal coefficient of resti-tution changes between 0.95 and 0.8 while the tangential


Ž . Ž .Fig. 11. a Numerical grid for a 908 circular elbow and b detail ofnumerical grid for pipe cross-section.

coefficient of restitution changes between 1 and 0.9. In thepresent study, the motion of relatively small particles iscontrolled by fluid motion and turbulent dispersion; thus,the influence of particle–wall interaction is less importantaway from the wall since the particles promptly follow thecarrier fluid. The influence of improper modeling of parti-cle–wall collision can be seen very close to the wallregion. In fact, the CFD predictions of particle velocitiesand concentrations close to the outer wall of the elbow

Ždiffered from the experimental measurements see Fig. 17.in the present study . This is an indication that the present

particle–wall collision model does not produce realisticresults within the elbow, where heavy particle wall colli-sions occur.

In addition, the present particle–wall collision modeldoes not account for irregular particle wall collision occur-ring between non-spherical coal particles and a rough wall.Recently, some researchers performed particle–wall colli-sion experiments for a rough wall whose characteristicdimension is comparable to particle diameter. Using Parti-

Ž . w xcle Tracking Velocimetry PTV , Massah et al. 33 pre-sented experimental data on the particle–wall collisions ofFCC particles and a rough CFB reactor wall. Sommerfeldw x48 studied particle wall interaction of glass beads andrough wall. These experimental data show the diffusenature of the particle–wall collision, requiring a MonteCarlo method to realistically model the particle–wall colli-sion process. Rebound angles for a fixed collision anglehad a very broad distribution, indicating the effect ofroughness on particle wall collision when particle diameteris comparable to the wall roughness element size. In both

w x w xthe Massah et al. 33 and Sommerfeld 48 experiments,the normal coefficient of restitution was found to be largerthan 1 for an approach angle of about 258 while the

Ž .tangential coefficient of restitution were lower ;0.7 .The authors believe that further experimental study isneeded to properly model the particle–wall collision pro-

cess occurring between non-spherical coal particles andcarbon steel pipe wall.

4.3. Numerical solution procedures

CFX-Flow3D uses the finite volume approach to obtainnumerical solutions at discrete points of a grid that en-compasses a sequence of elemental control volumes. Thecomputational mesh generated by CFX-Flow3D is charac-terized as boundary fitted, multi-block, structured andnon-staggered. CFX-Flow3D uses the Rhie–Chow algo-

w xrithm 42,43 that allows for the implementation of thew xSIMPLE-based algorithms 38,39 on non-staggered grids,

while avoiding the problems of checker-board oscillationsin pressure and velocity, traditionally associated with theincorrect use of non-staggered grids.

w xAs reported by Holm et al. 19 , a single block H-gridcreates instabilities when using the kye turbulence model,due to highly distorted cells close to the wall region.Therefore, the flow domain was discretized by using 15blocks in order to distribute grids as orthogonally as

Ž .possible see Fig. 11 . A combination of five blocks for thepipe cross-section as shown in Fig. 11b provided grids thatwere near orthogonal close to the wall region and orthogo-nal in the pipe center. Table 2 summarizes the number ofgrids along the flow direction for the simulation domains:horizontal section N , pipe bend N , and vertical sectionH B

N and for the pipe cross-section N .V CRŽ .All terms in Eq. 5 except the convection terms are

discretized in space using second-order centered differenc-ing. A third-order accurate QUICK algorithm was used forthe convection terms. To handle the nonlinearity of theequations, the treatment of all transported variables in-volved two nested levels of iterations referred to as innerand outer iterations. Pressure was handled by a specialprocedure, the velocity–pressure coupling algorithm SIM-

w xPLEC 56 .Outer iterations were repeated until the problem satis-

fied a convergence criterion. For the simulations per-formed in this study, the convergence of outer iterationwas judged by how accurately the continuity equation wassatisfied by the current values of the dependent variables.The solution procedure was considered converged whenthe ratio of the summation of absolute mass source residu-

Table 2Numerical grid resolution

Geometry N N N N NH B V CR TOTAL

Ž .Elbow RrDs1.5 Grid1 20 18 80 224 26,432Grid2 40 22 110 333 57,276

Ž .Elbow RrDs3.0 40 44 110 333 64,602Ž .Elbow RrDs5.0 40 60 110 333 69,930

Vertical pipe – – 110 333 36,630


Ž .Fig. 12. History of changes in mass source residuals for two simulations performed for RrDs1.5 and 3.0 elbows .

als to the total rate of mass inflow fell below a prescribedtolerance.

Ncell

< <RÝ M , iis1 Fd 21Ž .

mair

The parameter, d , was taken equal to 1=10y6 for thesimulations performed in this study.

The convergence of the gas-particle flow solution wasjudged by monitoring the history of the residuals of the U,V, and W momentum equations and of the continuityequation during the two-way coupling process. Fig. 12

Žshows the history of the mass source residuals residuals of.continuity equation for two simulations performed in this

study. As can be seen from Fig. 12, the mass sourceresiduals fluctuate during the two-way coupling processeven though the single-phase solutions converge to a pre-scribed tolerance for every coupling. Momentum sourceresiduals show the same kind of fluctuations. The fluctua-tions or jumps of the residuals during the momentumcoupling of the two phases level out to a constant valueafter about 14 couplings for the simulations performed inthis study. Similar fluctuations in the residuals during the

process of two-way coupling were also observed in thew xwork of Kohnen et al. 27 . One of the reasons for the

fluctuations in the residuals is the stochastic process con-tained within the turbulent particle dispersion model. Thefluctuations might also be due to the limited number ofcomputational particles modeled. The height of this jumpin the residuals depends on the solids loading and on the

Ž .under-relaxation factor for particle source terms g . Theunder-relaxation factor g , which can be given values be-tween 0 and 1, is used to prevent divergence of thegas-particle flow solution and is defined as follows

™ ™ ™iq1 i iq1S sS P 1yg qS Pg 22Ž . Ž .f ,p f ,p f ,pŽcalculated .

™ ™iq1 iwhere the terms S ,S are the particle momentumf, p f , pw xsource terms 13 at coupling iterations iq1 and i, respec-

™iq1tively. The term S is the particle momentumf, pŽcalculat ed .source term evaluated during the particle tracking proce-dure.

The gas-particle flow solution diverged for the case ofan elbow with RrDs3.0 when an under-relaxation factorof 0.5 was used for the particle source terms. However, itwas possible to obtain a converged solution for an under-relaxation factor of 0.35. This indicates that the appropri-

Table 3Geometries and physical properties for the simulations

Case Simulation geometry Particles Air3Ž . Ž . Ž . Ž .Type D m RrD L rD L rD Density kgrm Temp. K Pressure kPaH V

1 908 Elbow 0.154 1.5 5 20 1680 300 2.02 908 Elbow 0.154 3.0 5 20 1680 300 2.03 908 Elbow 0.154 5.0 5 20 1680 300 2.04 Vertical pipe 0.154 – – 20 1680 300 2.0D: Inside pipe diameter L : Length of horizontal pipeH

R: elbow radius of curvature L : Length of vertical pipeV


Ž . Ž . ŽFig. 13. Secondary flow patterns for RrDs1.5 elbow at zrDs0.5: a single-phase flow solution, b gas-particle flow solution U s30 mrs ando.ms0.33 .

ate under-relaxation factor must be used to obtain a con-verged solution. The convergence strategy employed inthis study is a global one and it does not guarantee aconverged solution at all local points in the flow geometry.

4.4. CFD results and discussions

CFD simulations were performed for three differentelbow geometries with nondimensional elbow radii ofcurvature RrD of 1.5, 3.0, and 5.0 and conveying airvelocities U of 20 and 30 mrs. All of the simulationso

Ž .were for the lowest solids loading case ms0.33 forwhich experimental measurements were available for com-parison. The CFD simulations were used to study both therope formation process and the individual effects of sec-ondary flows and flow turbulence on rope dispersiondownstream of the elbow exit. Table 3 summarizes theinformation on the flow geometries and on the air andparticle physical properties.

In curved pipes, the more rapidly flowing central partsof the flow are forced outwards by centrifugal action,while the slower parts along the wall are forced inwardswhere the pressure is less, and a so-called Asecondary

w xflowB develops at right angles to the main flow 23 . TheŽ .single-phase simulations grid size independent solutions

predicted the formation of these vortices. The analysis ofthe equations describing laminar flow through curved pipesshows that two parameters characterize the flow; the radius

D D w xratio, and the Dean number, Des Re 2 . Since( (2 R 2 R

the Dean number is equal to the ratio of the square root ofthe product of the inertia and centrifugal forces to theviscous force, it provides a measure of the intensity of the

w xsecondary flow 22 . Single-phase CFD predictions showedan increased intensity of the secondary flows as the Deannumber increased.

The numerical predictions of turbulent gas-particle flowthrough an elbow show the single-phase solutions aremodified by the momentum transfer between the gas phase

2Ž . Ž . Ž . Ž .Fig. 14. a Secondary velocities and b rms fluid velocities k in the x direction: comparison of CFD results for two elbows RrDs1.5 and 3.0 .'3


and particulate phase. The secondary flow patterns are alsomodified, resulting in four vortices in the pipe cross-sec-tion instead of the two large vortices present in single-phaseflow. Fig. 13 compares these two different secondary flowpatterns for an RrDs1.5 elbow at zrDs0.5. The sec-ondary flow patterns exhibit a flow current away from theouter wall towards the pipe center. This flow patternprovides the mechanism for removal of particles fromwithin the rope to particle-free regions in the pipe cross-section. At the elbow exit plane, the secondary flows are

Ž .stronger for the short radius bend RrDs1.5 as shown2Ž .in Fig. 14a. The turbulence fluctuation velocities k are(3

Ž .also shown in Fig. 14b at the elbow exit plane zrDs0.0for two elbows with RrDs1.5 and 3.0.

4.5. Rope formation

The formation of a rope is closely related to the migra-tion of particles towards the outer wall of the elbow due tocentrifugal forces and to particle–wall and particle–par-ticle interactions. The predicted particle concentration vari-ation along the outer wall of the three elbows are shown inFig. 15 using the coordinate system defined in Fig. 3. Thecentrifugal force, which is proportional to U 2rR, does noto

seem to be the controlling parameter for the rope forma-tion process for the range of flow conditions and elbowgeometries considered. Instead, the average residence timewithin the elbow plays a stronger role. The longer theparticles experience the inertial effect within the elbow, the

Ž .stronger the rope i.e. larger peak rope concentration is atthe bend exit. Fig. 16 shows peak concentration values inthe pipe cross-section at zrDs1.0 as a function of

Žaverage particle residence time in the elbow t sRŽ ..p Rr2U and compares CFD predictions with experi-o

mental data. The CFD results overpredict peak particleconcentration within the rope at elbow exit, most likelydue to the absence of particle–particle interactions in the

Fig. 15. Numerical simulation results showing the rope formation withinthree 908 circular elbows with RrDs1.5, 3.0, and 5.0.

Fig. 16. Rope peak concentration at zrDs1 as a function of averageŽ .residence time of particles t within elbow.R

w xCFD model 21 . Fig. 17 compares CFD results and exper-imental data on particle concentration profiles in the xdirection at different angular positions from the elbowinlet. The particle concentration progressively increases atthe outer wall with increasing u for the case of CFDresults. However, experimental data show that peak parti-cle concentration reaches a maximum somewhere aroundus678, agreeing LDV measurements performed by Kli-

w xafas and Holt 26 .

4.6. Rope dispersion

A particle rope created by an elbow starts dispersingonce it exits the elbow due to secondary flows and turbu-lence and possibly due to gas-particle flow instabilities.CFD predictions showed rather rapid rope dispersion withinthree to four pipe diameters downstream of the elbow exit

Fig. 17. Particle concentration profiles within RrDs1.5 elbow.


Fig. 18. Comparison of CFD simulation results with experimental data onrope formation and dispersion for RrDs1.5 elbow.

Ž .plane zrDs0 . This dispersion characteristic is accom-panied by a rapid decrease in the intensity of secondaryflows and turbulence levels between zrDs0 and 5. Fig.18 compares CFD predictions with experimental data onthe rope formation and dispersion processes for the RrDs1.5 elbow. Fig. 18 also shows the effects of grid resolu-tion and number of computational particles modeled on theCFD results. Increasing the grid resolution and number ofcomputational particles did not change the overall ropedispersion behavior.

Rope dispersion characteristics were found to be verydifferent for the RrDs1.5 and 3.0 elbows as discussed inthe experimental part of this study. Ropes formed withinthe RrDs1.5 elbow moved towards the center of pipe

and dispersed at a faster rate compared to the cases for theRrDs3.0 elbow. The CFD predictions did not show thissame behavior. For all values of RrD, the dispersionprocess predicted by the CFD simulations was similar tothe one experimentally observed for the RrDs3.0 elbow:the rope flow stayed close to the outer wall and its rate ofdispersion was relatively slow.

Fig. 19 compares the CFD results and experimental dataon rope dispersion behavior of the RrDs1.5 and 3.0elbows, showing particle velocity and particle concentra-tion profiles in the x direction at different zrD distances.The predicted particle concentration contours are alsoshown in Fig. 20 at zrDs1 and 9 for two elbowsŽ .RrDs1.5 and 3.0 . CFD simulations performed for

Ž .lower conveying velocities U s20 mrs did not showo

any effect of conveying air velocity on the rope dispersionprocess.

4.7. Mechanism of rope dispersion

The actual particle dispersion phenomenon in a turbu-lent flow is a complicated process due to the very largenumber of time and length scales. The individual effects ofsecondary flows and flow turbulence on particle ropedispersion in the vertical pipe was studied using CFX-Flow3D. Two simulations were performed in a verticalpipe assuming a particle rope was present at the inlet. Thisparticle rope was the one created by the short radius elbowŽ .RrDs1.5 . The transverse particle velocities within thisrope were set to zero. Using the secondary flow patternsfrom the single-phase solution for the short radius elbow

Ž . Ž .Fig. 19. Comparison of CFD results and experimental data on rope dispersion for a RrDs1.5 and b RrDs3.0 elbows at different zrD distances.


Ž .Fig. 20. Predicted particle concentration contours for two elbows: aŽ .RrDs1.5 and b RrDs3.0 at zrDs1.0 and 9.0 for U s30 mrso

and ms0.33.

Ž .RrDs1.5 as an inlet condition and turning off theturbulent dispersion model in CFX-Flow3D made it possi-ble to study the effect of secondary flows on rope disper-sion. In the second case, a uniform air velocity wasassumed at the inlet of the vertical pipe and the simulationwas run with the particle dispersion model turned on. Thesame rope was used as an inlet condition, but with notransverse velocities present in either the gas or solidphase. The gas phase turbulence quantities at the inlet weretaken from the single-phase solution for the RrDs1.5elbow.

The predicted particle concentration distributions in thepipe cross-section at different axial distances from the inletare shown in Fig. 21. These contours show the secondaryflows carry the particles around the pipe circumferencethrough the particle-free regions, resulting in a spreadingof the particles from within the rope. The results also showthat turbulence does not spread the particles, but insteadcreates a more homogenous distribution by localized parti-cle mixing into the region immediately adjacent to therope. In addition, relatively large peak particle concentra-tion values in the pipe cross-section prevail when theturbulent particle dispersion model was not implementedŽ .see Fig. 21b .

The present study demonstrates that organized largescale fluid motions such as secondary flows created inelbows contribute to the dispersion of particle ropes. Asdiscussed in the review paper written by Eaton and Fesslerw x15 , particles interact strongly with organized fluid struc-tures when the Stokes number based on characteristic timescale of organized large scale motion and the particleresponse time is on the order of unity. The Eaton and

w xFessler paper 15 focused on the centrifuging effect causedby turbulence structures in the flow. In contrast, the pre-sent study shows that both turbulence and secondary flowsdisperse the particle ropes created in the elbow. The Stokesnumber calculated from the turnover time of a secondaryflow structure and particle response time is on the order ofunity. Furthermore, the CFD simulation results show theStokes number calculated from the eddy life time predictedby the kye turbulence model is also on the order ofunity.

Ž . Ž . ŽFig. 21. Particle concentration contours showing the effects of a turbulence and b secondary flows on particle dispersion RrDs1.5, U s30 mrs ando.ms0.33 .


5. Summary and conclusions

Elbows commonly used in pneumatic conveying sys-tems create solids flow nonuniformities leading to a struc-ture referred to as a rope. The present paper combinesexperimental and numerical results to describe the mecha-nisms involved in rope formation and dispersion in verticalpneumatic conveying line following horizontal-to-verticalelbows. The formation and dispersion of a particle ropeflow is strongly dependent on the pipe bend radius and to alesser extent on the conveying air velocity and solidsloading.

The experiments showed that ropes formed within theRrDs1.5 elbow moved towards the center of pipe anddispersed at a faster rate compared to the cases for theRrDs3.0 elbow. The CFD predictions did not show thissame behavior. For all values of RrD, the dispersionprocess predicted by the CFD simulations was similar tothe one experimentally observed for the RrDs3.0 elbow:the rope flow stayed close to the outer wall and its rate ofdispersion was relatively slow. The CFD results overpre-dict peak particle concentrations within the rope at theelbow exit, most likely due to the absence of particle–par-ticle interactions in the CFD model.

The individual effects of secondary flows and turbu-lence on rope dispersion were identified and explainedwith the aid of CFD simulations. The results show sec-ondary flows disperse the rope by carrying particles aroundthe pipe circumference while turbulence disperses the ropeby localized mixing of particles.

List of symbolsA Ž 2 .Pipe cross-section area mŽ .C t Ž 2 .Cross correlation function V

CD Ž .Standard drag coefficient –cp Ž Ž ..Particle mass concentration c s r 1 y e ,p p

Ž 3.kgrmcp,max Max. particle concentration within pipe cross-sec-

Ž 3.tion kgrmcpo Ž Y .Average particle concentration c sm rU ,˙po p o o

Ž 3.kgrmdp Ž .Weight mean particle diameter mD Ž .Inside pipe diameter mDe DŽ . Ž .Dean number Des Re , –(2 R

e Ž .Coefficient of restitution –f Ž .Sampling frequency HzIo Ž .Mean signal intensity without particles VIm Ž .Mean signal intensity with particles VIp Ž . Ž .Mean signal intensity difference I y I , Vm o

k Ž 2 2 .Turbulence kinetic energy m rsK , n ŽParticle concentration calibration constants c sp

Ž .n.K Ip

L Ž .Optical distance between fiber-optic probes mLrD Axial dimensionless distance to the inlet of elbow

Ž .–

Le Ž .Characteristic size of an eddy mma Ž .Air mass flow rate kgrsmp Ž .Solids mass flow rate kgrsmY˙ p Ž 2 .Local solids mass flux kgrm smY˙ p o Ž Y . Ž 2Average solids mass flux m sm rA , kgrm˙ ˙p o p

.sN Number of samplesNcell Number of computational cellP Ž .Pressure PaR Ž .Pipe bend radius mRM, i Absolute mass source residual for computational

Ž .cell i kgrsRep

™Ž < < .Particle Reynolds number Re sr U d rmp R p

s Distance along the outer wall of the elbow fromŽ .the inlet m

™

Sf Ž 3.Gas phase momentum source terms Nrm™

Sf, p Ž 3.Particle momentum source terms NrmSt Ž Ž . Ž .Stokes number Sts t rt , –A t

te Ž .Eddy life time stR ŽParticle residence time in the elbow t sR

Ž .. Ž .p Rr2U , so

ttr Ž .Transit time of a particle through an eddy sT Ž .Sampling period suX Ž .Fluctuating velocity mrs

2X X X 2(< < Ž < < . Ž .u rms fluid velocity u s u s k , mrs(

3X X X 2< < Ž < < . Ž .u rms particle velocity u s u , mrs(p p p

u , u1 2 Voltage wave forms at two locations in the gas-Ž .solid flow V

U Ž .Local fluid velocity in x direction mrsUp Ž .Particle velocity mrsUo Ž .Conveying air velocity mrs™

UR

™ ™Ž . Ž .Relative velocity UyU , mrsp

V Ž .Local fluid velocity in y direction mrsW Ž .Local fluid velocity in z direction mrsxrD Dimensionless transverse distance from outer wall

Ž .of elbow mx, y Ž .Transverse distances in pipe cross-section mz Ž .Axial distance mzrD Axial dimensionless distance downstream of the

Ž .elbow exit m

Greek Lettersd Ž . Ž .Convergence tolerance Eq. 21 , –e Ž 2 3.Turbulence dissipation rate m rs and Voidage

Ž . Ž .es1yV rV , –p TŽ .1ye Ž Ž ..Particle volume concentration c sr 1ye ,p p

Ž .–g Under-relaxation factor for particle source terms

Ž .–m Ž .Fluid viscosity kgrm s and solids loading ratio

Ž .–mt Ž .Turbulence viscosity kgrm sr Ž 3.Gas phase density kgrmra Ž 3.Air density kgrm


rp Ž . Ž 3.Apparent true particle density kgrms Standard deviations 2 Variancet Ž .Time lag in cross-correlation function s and

Particle relaxation time™Ž Ž . Ž < < .. Ž .ts 4r3 r d r r U C , sp p R D

tA Particle aerodynamic response timeŽ Ž 2 .. Ž .t s r d r18m , sA p p

tmax Time lag where cross-correlation function has aŽ .maximum s

t t Particle transit time through the optical probeŽ .measuring volume s

u Ž .Angular position from the elbow inlet 8

References

w x1 A. Adeniji-Fashola, C.P. Chen, Modeling of confined turbulentfluid-particle flows using Eulerian and Lagrangian schemes, Int. J.

Ž .Heat Mass Transfer 33 1990 691–701.w x2 S.A. Berger, L. Talbot, L.S. Yao, Flow in curved pipes, Ann. Rev.

Ž .Fluid Mech. 15 1983 461–512.w x3 A. Berlemont et al., Particle tracking in turbulent flows, ASME FED

Ž .166 1993 121–129.w x4 E.J. Bolio, J.L. Sinclair, Predicting the turbulent flow behavior of

gas–solid mixtures, Prepared for presentation at the November 1–6,1992 AIChE Meeting Session on Turbulence in Multiphase Phenom-ena, Paper 118b, 1992, unpublished.

w x5 CFX FLow3D 4.1c, User Manual, Computational Fluid DynamicsServices, AEA Technology, Harwell, England, 1996.

w x6 C.P. Chen, P.E. Wood, A turbulence closure model for diluteŽ .gas-particle flows, Can. J. Chem. Eng. 63 1985 349–360.

w x7 R. Clift, J.R. Grace, W.E. Weber, Bubbles, Drops, and Particles,Academic Press, New York, 1978.

w x8 D. Cook, N.R. Hurworth, Recent research on pulverized fuel settle-ment in power station pipelines, and the significance of roping,Proceedings Pneumatransport 5. Fifth International Conference onthe Pneumatic Transport of Solids in Pipes. London, England pub-lished by BHRA, Cranfield, England, 1980.

w x Ž9 C.T. Crowe, M.P. Sharma, D.E. Stock, Particle-source-in cell PSI-. Ž .cell model for gas-droplet flows, J. Fluids Eng. 99 1977 325–332.

w x10 C.T. Crowe, REVIEW: numerical models for dilute gas-particleŽ .flows, J. Fluids Eng. 104 1982 297–303.

w x11 C.T. Crowe, Two-fluid vs. trajectory models; range of applicability,Ž .ASME FED 35 1986 91–96.

w x12 C.T. Crowe, T.R. Troutt, J.N. Chung, Numerical models for two-Ž .phase turbulent flows, Annu. Rev. Fluid Mech. 28 1996 11–43.

w x13 F. Durst, D. Milojevic, B. Schonung, Eulerian and Lagrangianpredictions of particulate two-phase flows: a numerical study, Appl.

Ž .Math. Model. 8 1984 101–115.w x14 S.E. Elghobashi, T.W. Abou-Arab, A two-equation turbulence model

Ž .for two-phase flows, Phys. Fluids 26 1983 931–938.w x15 J.K. Eaton, J.R. Fessler, Preferential concentration of particles by

Ž .turbulence, Int. J. Multiphase Flow 20 1994 169–209, Suppl.w x16 C.L. Flemmer, R.L.C. Flemmer, E.K. Johnson, K.H. Means, Roping

Ž .paths in elbows, ASME PVP 193 1990 105–109.w x17 A.D. Gossman, E. Ionnides, Aspects of computer simulation of

liquid-fueled combusters, AIAA Paper, No. 81-0323, 1981.w x18 D. Hoadley, The small scale modeling of deposits in p.f. pipes,

CEGB-Report TPRDrMr1393N84, 1984.w x19 M. Holm, J. Stang, J. Delsing, Comparison of two different grids in

a single elbow flow simulation for the application to flow meter

modeling, Second CFDS International User Conference, 1995, pp.273–285.

w x20 N. Huber, M. Sommerfeld, Characterization of the cross-sectionalparticle concentration distribution in pneumatic conveying systems,

Ž .Powder Technol. 79 1994 191–210.w x21 N. Huber, M. Sommerfeld, Modeling and numerical calculation of

dilute-phase pneumatic conveying in pipe systems, Powder Technol.Ž .99 1998 90–101.

w x22 J.A.C. Humphrey, H. Iacovides, B.E. Launder, Some numericalexperiments on developing laminar flow in circular-sectioned bends,

Ž .J. Fluids Mech. 154 1985 357–375.w x Ž .23 H. Ito, Flow in curved pipes, JSME Int. J. 30 1987 543–552.w x24 E.K. Johnson, K.H. Means, Erosion studies in pneumatic transport

bends, Presented at Pneumatic and Hydraulic Conveying SystemsConference, Florida, April 21–25, 1996.

w x25 G.A. Kallio, D.E. Stock, Turbulent particle dispersion: a comparisonbetween Lagrangian and Eulerian model approaches, ASME FED 35Ž .1986 23–34.

w x26 Y. Kliafas, M. Holt, LDV measurements of a turbulent air-solidŽ .two-phase flow in a 908 bend, Exp. Fluids 5 1987 73–85.

w x27 G. Kohnen, M. Ruger, M. Sommerfeld, Convergence behavior fornumerical calculations by the EulerrLagrange method for stronglycoupled phases, ASMErFED, Numer. Methods Multiphase Flows

Ž .185 1994 191–202.w x29 X. Li, H.H. Shen, Dilute phase pneumatic transport of granular

Ž .materials in a pipe bend, ASME FED 228 1995 127–133.w x30 M.Y. Louge, E. Mastorakos, J.T. Jenkins, The role of particle

Ž .collisions in pneumatic transport, J. Fluid Mech. 231 1991 345–357.

w x31 D. Ma, G. Ahmadi, A thermodynamical formulation for dispersedmultiphase turbulent flows: I. Basic theory, Int. J. Multiphase Flow

Ž .16 1990 323–340.w x32 D. Ma, G. Ahmadi, A thermodynamical formulation for dispersed

multiphase turbulent flows: II. Simple shear flow for dense mixtures,Ž .Int. J. Multiphase Flow 16 1990 341–351.

w x33 H. Massah, F.D. Shaffer, J. Sinclair, M. Shahnam, Measurements ofdiffuse and specular particle–wall collision properties, Proceedingsof International Symposium of the Engineering Foundation, Flu-idization III; Toullouse, France, May, 1995.

w x35 D.R. McCluskey, W.J. Easson, C.A. Greated, D.H. Glass, The useof particle image velocimetry to study roping in pneumatic con-

Ž .veyance, Part. Part. Syst. Charact. 6 1989 129–132.w x36 D. Migdal, V.D. Agosta, A source flow model for continuum

Ž .gas-particle flow, Trans. ASME, J. Appl. Mech. 1967 860–865.w x37 S. Nobuhisa et al., Numerical analysis of particle dynamics in a bend

of a rectangular duct by the direct simulation Monte Carlo method,Ž .ASME FED 166 1993 145–152.

w x38 S.V. Patankar, D.B. Spalding, A calculation procedure for heat,mass, and momentum transfer in three dimensional parabolic flows,

Ž .Int. J. Heat Mass Transfer 15 1972 1787.w x39 S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemi-

sphere, New York, 1980.w x40 R.C. Patterson, Pulverized coal transport through pipes, J. Eng.

Ž .Power 1959 43–54, Jan.w x41 J.A. Pita, S. Sundaresan, Gas–solid flow in vertical tubes, AIChE J.

Ž .37 1991 1009–1018.w x42 C.M. Rhie, A numerical study of the flow past an isolated airfoil

with separation, Ph.D. Thesis, Dept. of Mech. and Indust. Eng.,Univ. of Illinois at Urbana-Champaign, 1981.

w x43 C.M. Rhie, W.L. Chow, Numerical study of the turbulent flow pastŽ .an airfoil with trailing edge separation, AIAA J. 21 1983 1525–

1532.w x44 W. Rodi, Examples of turbulence models for incompressible flows,

Ž .AIAA J. 20 1982 872.w x45 J.S. Shuen et al., Evaluation of a stochastic model of particle

Ž .dispersion in a turbulent round jet, AIChE J. 29 1983 167–170.w x46 M. Sommerfeld, Numerical simulation of the particle dispersion in


turbulent flow: the importance of lift forces and particlerwallcollision models, ASME FED, Numer. Models Multiphase Flows 91Ž .1990 11–18.

w x47 M. Sommerfeld, G. Zivkovic, Recent advances in the numericalsimulation of pneumatic conveying trough pipe systems, in: Ch.

Ž .Hirsch Ed. , Computational Methods in Applied Sciences, Elsevier,Amsterdam, 1992, pp. 201–212.

w x48 M. Sommerfeld, Application of optical non-intrusive measurementtechniques for studies of gas–solid flows, ASME Fluids EngineeringMeeting, Gas–Solid Flow Symposium, June, 1993.

w x49 D.E. Stock, Particle dispersion in flowing gases — 1994 FreemanŽ .Scholar Lecture, Trans. ASME, J. Fluids Eng. 118 1996 4–17.

w x50 W. Tabakoff, M.F. Malak, Laser measurement of fly ash reboundparameters for use in trajectory calculations, J. Turbomachinery 109Ž .1987 535–540.

w x52 L.B. Torobin, W.H. Gauvin, Fundamental aspects of solids–gasflow: Part V. The effects of fluid turbulence on the particle drag

Ž .coefficient, Can. J. Chem. Eng. 1960 189, Dec.w x53 Y. Tsuji, Y. Morikawa, Computer simulation for the pneumatic

transport in pipes with bends, Proceedings Pneumatransport 4. FourthInternational Conference on the Pneumatic Transport of Solids inPipes, California, USA, 1978.

w x55 J.Y. Tu, C.A.J. Fletcher, Numerical computation of turbulent gas–Ž .solid particle flow in a 908 bend, AIChE J. 41 1995 2187–2197.

w x56 J.P. Van Doormal, G.D. Raithby, Enhancements of the SIMPLE

method for predicting incompressible fluid flows, Numer. HeatŽ .Transfer 7 1984 147–163.

w x Ž .57 M. Weintraub, Discussion, J. Eng. Power 1959 52–53, Jan.w x58 G.C. Whitney, Use of models for studying pulverized coal burner

Ž .performance, Trans. ASME 81 1959 380–382.w x Ž .59 V. Yakhot, S.A. Orszag, J. Sci. Comput. 1 1986 .w x60 W.C. Yang, A unified theory on dilute phase pneumatic transport, J.

Ž .Powder Bulk Solids Tech. 1 1977 89–95.w x61 A. Yilmaz, E.K. Levy, Roping phenomena in pulverized coal con-

Ž .veying lines, Powder Technol. 95 1998 43–48.w x62 A. Yilmaz, E.K. Levy, Particle velocity and concentration measure-

ments in lean phase pneumatic conveying using a fiber optic probe,Proceedings of 1998 ASME Fluids Engineering Division SummerMeeting, Paper No. FEDSM98-5282, Washington, DC, 1998.

w x63 A. Yilmaz, E.K. Levy, Dynamics of rope flow in lean phasepneumatic conveying, Proceedings of 1998 ASME Fluids Engineer-ing Division Summer Meeting, Paper No. FEDSM98-4820, Wash-ington, DC, 1998.

w x64 M. Yin, F. Shi, Z. Xu, Renormalization group based C turbulencemodel for flows in a duct with strong curvature, Int. J. Eng. Sci. 34Ž .1996 243–248.

w x65 G. Zipse, The mass flow density distribution in pneumatic convey-ing and the influence on it of inserts in the conveyer duct, Fortschritt

Ž .Berichte VDI-Z 1966 Series 13, No. 3.

Documents

1-s2.0-S0032591000003193-main