Chemical Engineering Science 59 (2004) 4637–4651www.elsevier.com/locate/ces

Pneumatic transport of granular materials through a 90◦ bend

Lai Yeng Leea, Tai Yong Queka, Rensheng Dengb, Madhumita B. Raya, Chi-Hwa Wanga,b,∗aDepartment of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576, Singapore

bSingapore-MIT Alliance, E4-04-10, 4 Engineering Drive 3, Singapore 117576, Singapore

Received 12 January 2004; received in revised form 26 June 2004; accepted 7 July 2004

Abstract

In the present study, a pneumatic conveying system incorporating a 90◦ bend is investigated. This study employs the use of threenon-invasive instruments to measure solids concentration and velocity distribution determination in the pneumatic conveying system.They are namely the electrical capacitance tomography (ECT), particle image velocimetry and phase doppler particle analyzer. Pressuretransducers were also used to monitor the pressure drop characteristics along the post-bend vertical pipe region. Two different classes ofgranular materials, polypropylene beads (2600�m, Geldart class D) and glass beads (500�m, Geldart class B), were used to investigate thedifferences in the flow characteristics for granular particles of various Geldart classes. The experimental results show a constant frequencypulsating flow for polypropylene beads in the dense-phase flow regime. This is illustrated by the visualization, ECT and pressure dropdata. For dilute-phase flow regime, both polypropylene and glass beads show a continuous annulus flow structure. Numerical simulationusing the Euler–Euler method was also conducted using computational fluid dynamics and the fluid and particle flow characteristics werecompared with the experimental data obtained in the present study.� 2004 Elsevier Ltd. All rights reserved.

Keywords:Pneumatic conveying; Multiphase flow; Bend; Granular materials; Simulation

1. Introduction

Pneumatic conveying is an important process in the foodand pharmaceutical industry for transportation of granularparticles. The transport phenomenon of the conveying pro-cess in gas–solid system is not fully understood despitenumerous studies, both experimental and numerical, havebeen conducted on different pneumatic conveying systemsto characterize the flow profiles of the solids in the pipesof different sizes and for different pipe bends. These stud-ies helped to optimize the pneumatic conveying process andto assess the different methods of monitoring the conveyingsystems. The gas–solid two-phase flow in a vertical pipe isheterogeneous by nature and locally unsteady. As the solidsmass loading increases, particles may come together to formgroups such as sheets, streamers or clusters. Some of theparticle groups may even experience back-flow or slipping

∗ Corresponding author. Tel.: +1-65-6874-5079; fax: +1-65-6779-1936.E-mail address:chewch@nus.edu.sg(C.H. Wang).

0009-2509/$ - see front matter� 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2004.07.007

near the pipe wall (Rautiainen et al., 1999). Several studiesare reported on different kinds of vertical pneumatic con-veying systems (Rautiainen et al., 1999; Yilmaz and Levy,1998, 2001; Dyakowski et al., 2000; Van de Wall and Soo,1994, 1998; Van den Moortel et al., 1997; Plumpe et al.,1993; Zhu et al., 2003). These studies report various fluctu-ations and clustering of particles in the vertical pipe.Table 1summarizes the parameters investigated in some of the re-search. Most of these researches were conducted for smallersized particles falling under class A of Geldart classification.The Geldart classification of particles provides a guidelinedescribing the ease of fluidization of particles and its easeof handling in pneumatic conveying (Klinzing et al., 1997).

Yilmaz and Levy (1998, 2001)measured solids velocityand mass concentration using a fiber optic probe in two dif-ferent 90◦ pipe bends with bend radius to pipe diameter ratioof 1.5 and 3.0. Their investigations suggested a continuousrope-like structure that formed within the elbow and disin-tegrated further downstream into large and discontinuousclusters. A continuous stream of particles was observed near

4638 L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651

Table 1Operating parameters used in pneumatic conveying studies in the literature

Reference dta Solid particles Ug

b Gsc Geldart class

(mm) Material Dp(� m) (m s−1) (kgm−2 s−1)

Rautiainen et al. (1999) 192 Glass beads (spherical) 64 3.5–13 0–141 AYilmaz and Levy (1998, 2001) 154, 203 Pulverized coal 70–76 15–29 — ADyakowski et al. (2000) 150 Powders — ∼ 4.8 30–121 AVan de Wall and Soo (1994, 1998) 127 Glass beads (spherical) 50 10–15 — — A

42–60 20Van den Moortel et al. (1997) 200 Glass beads (spherical) 120 — —- APlumpe et al. (1993) 51 Glass particles 85–105 8, 10 — A

aPipe internal diameter(dt ).bSuperficial air velocity(Ug).cSolids mass loading(Gs).

the post-bend region, while the rope dissolves into loose,agglomerated particles further downstream. This flow phe-nomenon did not seem to change with the radius of curvatureof the bend (Yilmaz and Levy, 2001). The Euler–Lagrangiannumerical model tested byYilmaz and Levy (2001)showedgood agreement with their experimental results. They alsoshowed that secondary flows induced by the 90◦ bendswere responsible for the dispersion of the particles after thebend. However, their simulations did not show any signifi-cant changes in the differences in the rope dispersion withdifferent bend radii, in contrast to the observation that sec-ondary flows are caused by pressure gradients between theinner and outer walls of a bend, and they are different fordifferent curvatures of the bend. Furthermore, particle in-teractions within the bend in terms of particle–wall impactswere shown to be less dependent on bend radius of curva-ture than other factors, such as conveying velocity (Li andShen, 1995). Thus, the phenomena and detailed structure ofroping and clustering induced by the bend in the post-bendregion are exciting areas of research but are not fully inves-tigated in the literature.

Monitoring and control of pneumatic conveying sys-tems are important to ensure reliable transport of materials(Klinzing et al., 1997). One of the common instruments formonitoring gas–solids flow systems is transducer, which isuseful for determining fluctuations and flow condition inthe pipe. Tomography technique has been applied for non-intrusive measurements of solids concentration and velocitycharacterization for pneumatic conveying systems (Raoet al., 2001; Zhu et al., 2003) and is useful in dense-phaseconveying systems. In this study, distributions of solidsconcentration for different flow regimes in a vertical pipefollowing a 90◦ bend were determined and analyzed usingthe electrical capacitance tomography (ECT) technique.Laser techniques such as particle image velocimetry (PIV)and phase doppler particle analyzer (PDPA) were also usedto investigate the velocity and velocity distribution of the500�m glass beads in the post-bend vertical pipe region. Inaddition, numerical simulations using computational fluid

dynamics (CFD) were conducted, and the experimentaldata were compared with the numerical simulations. AnEuler–Euler two-phase flow model was adopted to simulatethe dense- and dilute-phase solids flow in the vertical bend.

2. Methodology

2.1. Materials

A schematic diagram of the experimental setup is shownin Fig. 1. Two types of granular solids, polypropylene andglass beads, were used and their physical properties are sum-marized inTable 2. The test section consists of a vertical

Key:

1. Solids feeder 5. Filter tank 2. Acrylic pipe (In blue) 6. Rotameter 3. Glass pipe (in green) 7. Control valve 4. Hose pipe (in red) 8. Vortex blower Pressure transducer

ECT (connected to analyzer)Plane of measurement for PIV and PDPA

5

8

6

3m

2

3

4

2

27

0.75m

1.5m

0.65m

1

1m

Fig. 1. Schematic diagram of the pneumatic conveying facility.

L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651 4639

Table 2Physical properties of particles

Material Mean diameter± SD Geldart class Density(�m) (kg m−3)

Polypropylene 2800± 500 D 1123Glass beads 500± 50 B 1755

glass pipe of 1.5 m (internal diameter= 0.05 m) connectedto two acrylic pipes of lengths 0.65 and 0.75 m (internal di-ameter = 0.05 m) at its top and bottom, respectively. Theglass pipe was specially chosen as the vertical test sectionfor ECT, PDPA and PIV measurements because of its rigid-ity and transparency over acrylic or other ordinary plasticpiping material. ECT measurements are very sensitive toany deformation in the section of the pipe where the ECTsensors are mounted and the higher rigidity of a glass pipeprovides better resistance to deformation. The use of PDPAand PIV requires penetration of laser beam in the pipe, andtransparency of the pipe is necessary for these measure-ments. The upper portion of the vertical pipe is connectedto the vortex blower by a 4.5 m long-reinforced hose (in-ternal diameter= 0.032 m) through a filter tank. The lowerportion is connected to a 3 m horizontal conveying section(internal diameter= 0.050 m) by a 90◦ L-shaped bend withzero bend radius.

2.2. Methods

Air and solids were fed into the pneumatic conveyingsystem through the inlet at the horizontal conveying sec-tion. The solids were conveyed up the vertical test pipethrough the 90◦ bend and are subsequently collected in thefilter tank. The experiments were conducted for conveyingof polypropylene beads with solids mass loading (Gs) of15.8, 31.1 and 46.3 kg m−2 s−1. For each mass flux, exper-iments were conducted at several superficial gas velocities(Ug) ranging from 10.6–17.0 m s−1. For glass beads, exper-iments were conducted with solids mass loading of 15.8 and29.0 kg m−2 s−1 in the same range of air superficial velocity.The system was left to run for a steady stream of air beforethe introduction of solids. Several non-invasive instrumentswere utilized in this study to obtain the solid-phase flowcharacteristics measurements in the post-bend vertical pipesection.

2.2.1. Pressure transducersTwo pressure taps were mounted at locations of 0.65 and

2.35 m, respectively, from the 90◦ bend to install a differ-ential pressure transducer (Model-DT1400-1UD-125 Stel-lar). Differential pressure measurements were obtained be-tween the two pressure taps and the data were acquired usingDataVIEW (Version 1.1, Cumming, Georgia). The samplingrate was set at 200 Hz and the pressure data were acquiredfor 30 s.

2.2.2. ECTTwo 12-electrode ECT sensors were mounted on the

glass pipe at 1.64 and 2.14 m, respectively, from the bend.ECT data were acquired using ECT32 software and a data-acquisition module (Process Tomography Ltd, Wilmslow,Cheshire, UK). The sampling rate was set at 40 Hz and thedata were acquired for 30 s for each experimental run.�s

(x, y, z, t), defined as the local volume fraction of the solidswas obtained from post-processing of ECT data for eachinstant of time(t) using the simultaneous iterative recon-struction technique (SIRT) given bySu et al. (2000). x andy denote the Cartesian coordinates in the cross-sectionalplane and were normalized using the pipe diameter as thecharacteristic length, the axial coordinatez denotes the lo-cation of the ECT electrodes. Particle volume fraction isexpressed as�s , while the (�t ), known as the time-averagedparticle volume fraction values were generated by averag-ing �s (x, y, z, t) over a time periodT (in this case, thefirst 25 s of data acquired for each run):

�t (x, y, z) = 1

T

∫ T

0�s(x, y, z, t) dt. (1)

The instantaneous value of the cross-sectional average par-ticle volume fraction,�s(z, t), is defined as

�s(z, t) = 1

A

∫ ∫�s(x, y, z, t) dx dy (2)

and the time-average value of�s(z, t) is denoted by〈�〉(z)

〈�〉(z) = 1

T

∫ T

0�s(z, t) dt ≡ 1

A

∫ ∫�̄t (x, y, z) dx dy.

(3)

2.2.3. PIVMeasurements using a PowerView 2D PIV system (TSI

Corporation, USA) were conducted at the vertical pipe re-gion 1 m downstream from the bend. The laser light sheetgenerated by the Laser PulseTM Solo Mini Dual Nd:YAGlaser was introduced into the glass pipe and two snapshotsof the lighted particles were taken by a PowerViewTM 4M2K×2K camera for a very small time interval (dT =100�s).The cross-correlation yields the distance traveled by the par-ticles from the first snapshot to the second and the velocitydistribution of particles in the plane can be determined bydividing by the time interval. In this study, the variation ofvelocity in the post-bend region in the axial direction of thehorizontal pipe was measured and the results were comparedwith PDPA measurements and simulated numerical results.

2.2.4. PDPASome measurements using PDPA (XMT204-2.2, TSI Cor-

poration, USA) were taken to compare with the velocitymeasurements obtained by PIV under the same flow condi-tions for dilute-phase flow. The laser wavelength used was514 nm. The phase doppler method is based upon the princi-ples of light-scattering interferometry. The PDPA is able to

4640 L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651

measure the local velocity of each particle and the particlesize distributions accurately. The signal analyzer and post-processing software (Aerometrics Real-Time Signal Ana-lyzer, DataVIEW) could also record the number density ofthe measurement volume online. The PDPA measurementswere taken at three radial points. For each measurement lo-cation, at least 1000 particles were sampled.

2.2.5. Numerical simulation (methods and boundaryconditions)

Commercial CFD package FLUENT 6.0 was used tosimulate the pneumatic conveying system. The air andsolid flows were modeled using conservation equations ofAnderson and Jackson (1967). The volume-averaged ap-proach was used for the conservation equations (refer toAppendix A). The phase-coupled SIMPLE (PC-SIMPLE)algorithm (Vasquez and Ivanov, 2000) was used for thepressure–velocity coupling. The velocities for both solid andgas phases were solved from the coupled differential equa-tions using the block algebraic multi-grid scheme (Weisset al., 1999) with the governing equations solved sequen-tially in an implicit, unsteady fashion. A total of 30,000structured hexagonal grids were used in the geometry withsize ranges from 3.8×10−8 to 3.2×10−7 m3. Grid indepen-dence studies were performed to show that increasing thenumber of grids does not result in noticeable changes to thesimulation results (Quek, 2003). Time steps of 1× 10−6 nd1 × 10−4 s were used for simulation of polypropylene andglass beads, respectively. Details of the model constructioncan be found in Appendix A.

The calculation domain consists of a horizontal inlet pipe(internal diameter= 0.05 m) of length 1 m, a 90◦ elbow ori-ented vertically, followed by a vertical pipe (internal diame-ter = 0.05 m) of height 2 m (as shown inFig. 2a). For illus-tration, all flow quantities were either averaged over a planeor taken along the centerline in a plane. Qualitative contourresults are presented in the vertical (x–y) symmetry planeof the geometry (as shown inFig. 2b).

3. Results and discussion

There is a marked difference in the flow characteristics ofpolypropylene and glass beads. This can be observed fromexperimental results of system fluctuations, solids concen-tration and velocity distribution in the vertical and horizon-tal lines. Some of the experimental observations were alsoobtained from the simulation conducted for the system.

3.1. Flow regimes and fluctuations in the pneumaticconveying system

The flow characteristics of polypropylene beads differfrom glass beads significantly. The flow characteristics ob-served in the experiments for the polypropylene may be di-vided into two distinct types: dense- and dilute-phase flow

Plane A

Plane centerline y

x

Symmetry planes A &B showing the

qualitative feature ofparticle distributions

z

Plane B

Outlet

2 m

1 m

0.05 m Inlet

y

x

g

Origin, 0

Plane A

Plane B

(a)

(b)

Fig. 2. Geometry for numerical simulation based onFig. 1. Pipe diameter= 0.05 m, and 1 m in the horizontal section before the bend, 2 m inthe vertical section after. All numerical results presented either as across-sectional average or along the centerline of the plane. Qualitativepictures were taken in the vertical symmetry planes A& B.

depending on whether the pressure drop decreased or in-creased with increasing superficial gas velocity (Rao et al.,2001). This can be seen from the differences in pressure gra-dient profile, flow pattern, pressure and solids concentrationfluctuation and solids concentration distribution as describedin the following section.

The pressure gradient profile of the system is useful foridentification of the type of flow in the pneumatic convey-ing system. The Zenz state diagram (Zenz, 1949; Herbreteauand Bouard, 2000) obtained by plotting pressure drop withsuperficial velocity for various solids mass loading is usefulfor identification of flow regime in a pneumatic conveyingsystem. In the region where pressure drop decreases with in-creasing superficial gas velocity, the system is operating inthe dense-phase regime with non-homogenous and unsteadyflow in the conveying lines. As superficial velocity of gas

L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651 4641

0.

(a)

(b)

0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

800.0

8.0 10.0 12.0 14.0 16.0 18.0 20.0

∆P/L

, Pa/

m

Gs = 15.8 kgm-2s-1

Gs = 31.1 kgm-2s-1

Gs = 46.3 kgm-2s-1

60.0

70.0

80.0

90.0

100.0

110.0

120.0

130.0

140.0

150.0

8.0 10.0 12.0 14.0 16.0 18.0 20.0

Ug, ms-1

Ug, ms-1

∆P/L

, Pa/

m

Gs = 15.8 kgm-2s-1

Gs = 29.0 kgm-2s-1

Fig. 3. Pressure gradient(Pa m−1) against superficial gas velocity(m s−1) in the vertical pipe (post-bend section) for two-phase flow (a)air-polypropylene beads; (b) air-glass beads.Gs indicated in the legendrefer to the solid mass flux(kg m−2 s−1).

increases further, the pressure drop begins to increase withgas superficial velocity and the flow becomes steady andhomogenous (dilute-phase flow). The flow characteristics ofthe two classes of solids are illustrated by pressure gradi-ent profile plot for polypropylene beads and glass beads inFigs. 3a and b, respectively.

Fig. 3a shows the pressure gradient in the vertical pipegenerally decreased with increasing superficial gas veloc-ity (a dense-phase conveying characteristics). In this regime,the solids were conveyed in alternating pulses of high andlow solids concentration as shown inFigs. 4a and b. Thesolids formed clusters at the horizontal pipe region and theparticles were conveyed in moving dunes as illustrated inFig. 4c. In this case, cluster formation is determined by aqualitative observation of the differences in solids concen-tration and solids distribution in adjacent sections of theconveying line. The formation of moving dunes in horizon-tal conveying systems for the Geldart class D beads wasalso reported in the study byZhu (2003). Dilute-phase flowfor polypropylene beads were obtained for the lowest solidsmass loading near the maximum superficial gas velocityused for the experiments. In this regime, the solids flow wascontinuous, steady and homogeneous.

Conversely, as shown inFig. 3b, the pressure gradient forglass beads increased with increasing superficial gas veloc-ity for the same range of superficial gas velocity used inpolypropylene beads experiments. This is an indication thatthe system was operating in a dilute phase whereby the flowof particles was continuous. No roping or clustering was ob-served both in the horizontal and post-bend vertical sectionof the pneumatic conveying system. Unlike the conveyingof polypropylene beads, a transition from dense- to dilute-phase flow cannot be achieved. For the range of superficialgas velocity and mass loading of glass beads used in thisstudy, only a dilute-phase flow regime was obtained. Reduc-ing the superficial gas velocity below 10 m s−1 caused theglass beads to accumulate at the inlet of the conveying sec-tion and the conveying rate was less than the feeding rate.Therefore, the experiments for glass beads were conductedwith superficial velocities higher than 10 m s−1 to ensure afully developed steady state flow. In that range, the glassbeads move along the horizontal conveying section in a con-tinuous stream. Small clusters of solids like moving dunesflow were not noticeable. In the vertical post-bend region,the glass beads were conveyed in a homogeneous, contin-uous and dispersed stream. The differences in the pressuregradient profiles for the two classes of particles illustrate themarked differences in the flow characteristics.

Pressure fluctuations in the conveying of solids are use-ful for flow pattern identification (Dhodapkar and Klinzing,1993). The pulsating flow pattern for the dense-phase con-veying of polypropylene beads was demonstrated by themeasurements of the fluctuation of pressure gradient alongthe vertical pipe with time, as shown inFig. 5a. This pulsat-ing flow pattern was also observed in the cross-section aver-age solids volume ratio measurements (�s) made using ECT(Fig. 5b). AsFigs. 5a and b show, the fluctuations have a reg-ular period of about 2 s (i.e. 0.5 Hz). In order to investigatethe fluctuation frequency for this dense-phase flow, powerspectral density (PSD) functions for pressure drop and vol-ume ratio were used to analyze the pressure and ECT dataand the results are illustrated inFigs. 6a and b, respectively.Sampling rates for pressure and ECT measurements wereset at 200 and 40 Hz, respectively, therefore the peaks ob-tained for power spectra of pressure fluctuations were moredistinct (Fig. 6a). The power spectra obtained from both setsof data are similar and show a dominant peak at approx-imately 0.5 Hz. This is consistent with the fluctuation fre-quency of approximately 2 s as observed fromFigs. 5a andb. The power spectra also indicate secondary peaks presentat approximately 1 and 1.5 Hz. Further experiments showedthat at low superficial velocity, the pressure fluctuation ispredominantly near 0.5 Hz. As superficial velocity increasesand the flow becomes dilute, the peak near 0.5 Hz is lesspredominant and fluctuations extend to 10–15 Hz regime.

The ECT data obtained experimentally were reconstructedto obtain the solids distribution profile for the pneumaticconveying system. The ECT visualization results for denseflow of polypropylene beads and dilute flow of glass beads

4642 L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651

Fig. 4. Photograph of conveying of polypropylene beads forGs = 31.1 kg m−2 s−1, Ug = 10.6 m s−1. (a) Pulse of high concentration in vertical pipe;(b) low solids volume fraction, dispersed flow in vertical pipe after a few seconds and (c) moving dunes formed in horizontal conveying section (clustersof solids as shown in circle drawn).

are shown inFigs. 7a and b, respectively.Figs. 7a and bshow the solids volume fraction distribution with 1 s inter-val between two successive frames.Fig. 7a showed consis-tent results with the fluctuations analysis. As the ECT dataindicate, during the dense-phase flow, the solids show an al-ternating core–annulus/annulus structure. There was a highconcentration of solids in both the core and annular regionof the pipe at one time and the core disappears leaving onlyan annular structure 1 s later. When the system was operat-ing in the dilute phase, the solids tend to be in the annularregion. As shown inFig. 7b, the solids were seen to consis-tently form an annular structure for all the 25 frames shown.

From the numerical simulation, differences in the pneu-matic conveying of the two classes of particles were also ob-served.Fig. 8 shows the solids concentration distribution inthe cross-section of vertical pipe at 0.9 m downstream fromthe bend for conveying of polypropylene beads in the dense-phase flow regime.Fig. 8a shows a time when there was lowconcentration of solids moving through, whileFig. 8b showsa high concentration solids core at a short time followingFigure 8a. The time difference betweenFigs. 8a and b is0.11 s.Figs. 8c and d show the solid concentration profile

at the same location but at a few seconds later than the timefor Figs. 8a and b. The time difference betweenFigs. 8c andd is about 0.17 s. This is in agreement with the experimentalresults of alternating high and low solids concentration withtime. However, the frequency of fluctuation is different fromthe experimental results.Figs. 8e and f show the solids con-centration at a distance of 2 m from the bend, with a timedifference of 0.12 s. The variation in solids flow structuredecreases with the increasing distance from the bend. Onthe other hand, numerical results for the conveying of glassbeads showed a steady consistent solids volume ratio distri-bution in the vertical pipe at a plane 1 m downstream of thepipe bend. This is consistent with the experimental observa-tion of a non-fluctuating system. The cross-sectional volumefraction distribution results at the planesy = 0.9 and 2.0 mfrom the bend are shown inFigs. 9a and b, respectively.

Though the numerical results presented inFigs. 8a–dshowed some qualitative similarity toFig. 7a, the solid fluc-tuation frequency, and thus the cluster formation, which maybe attributed mainly to the particle–particle interaction, ap-pears to be different. The fluctuation period was about 2 s forthe ECT experiments, whereas a period of only about 0.1 s

L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651 4643

0.0

(a)

(b)

100.0

200.0

300.0

400.0

500.0

600.0

700.0

0 5 10 15 20 25t, s

∆P/L

, Pa/

m

0

0.01

0.02

0.03

0.04

0.05

0.06

0 5 10 15 20 25t, s

α s

Fig. 5. Fluctuations in pneumatic conveying of polypropylene beads inthe vertical pipe forGs = 31.1 kg m−2 s−1, Ug = 11.9 m s−1 (a) pressuregradient in vertical pipe(Pa m−1) against time (s) and (b) solids volumefraction (�s ) in a single plane in the vertical pipe against time (s).

was recorded for the simulations as illustrated inFig. 10a.The apparently more rapid succession and less-concentratedclusters could be attributed to their differences in the mech-anism of cluster formation. The fluctuations observed in theECT experiments could be mainly due to the inlet air/solidsflow rate variations, and in the simulation model, the inletair/solids flow rates were assumed to be constant. On theother hand, the simulation results for glass beads indicatedthe absence of high fluctuations of solids concentration asobserved in polypropylene beads simulation, which is shownin Fig. 10b.

The differences in solids distribution for the two typesof conveying system were also observed in the simulationresults.Fig. 11shows the simulation results for solids con-centration distribution of particles from various sections ofthe vertical conveying line. Though the numerical simula-tion was carried out in a time-dependent state, the qualita-tive features of the solid flow did not vary significantly withtime. The corner of the bend showed deposition of solids(Fig. 11a) as observed in the experiments. These solids,which were trapped permanently in the corner, affected theparticle–particle collisions and rebound of the solids mov-ing through the bend. In the solids clusters formed in a bend,such particle–particle collisions are frequently encounteredand they are believed to preserve solids size and shape betterthan collisions between unlike materials, i.e., between solids

Fig. 6. Power spectral density (PSD) analysis of fluctuations for thepneumatic conveying of polypropylene beads in the vertical pipe.Gs =31.1 kg m−2 s−1, Ug =11.9 m s−1 (a) PSD of pressure data and (b)PSD of solids concentration data.

and wall. Some cluster like pattern could be seen directly inthe region above the bend, where the mass of solids startedto form groups with higher concentration. These higher con-centration groups separate into distinct clusters at a furtherdistance from the bend, producing a pulsing effect as shownin Fig. 11b. This can be seen to occur within 1 m (20 diam-eters distance) from the bend and similar pulses were ob-served in the experiments shown inFigs. 4a and b where theactual pulsing flow in the experiments is recorded using aconventional video camera.Fig. 12shows the numerical re-sults for pneumatic conveying of glass beads under the samesuperficial velocity and comparable solids mass loading withpolypropylene. For smaller solids (glass beads), a continu-ous particle flow was observed along the vertical pipe. Therewere no significant variations in the cross-sectional solidsconcentration distribution for different planes in the verticalpipe and at different times. Therefore, there was no ropingor clustering of solids as observed in the experimental andnumerical results for polypropylene. This is consistent withthe experimental observation of continuous steady flow ofglass beads as stated above.Table 3shows the time-averagedparticle concentration at the core for ECT in comparisonwith those obtained from simulations. A large change in thecore concentration was observed for different superficial gas

4644 L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651

Fig. 7. (a) ECT images obtained for pneumatic conveying of polypropylenein the vertical pipe.Gs =31.1 kg m−2 s−1, Ug =11.9 m s−1 and (b) ECTimages obtained for pneumatic conveying of glass beads in the verticalpipe. Gs = 29.0 kg m−2 s−1, Ug = 10.6 m s−1.

Fig. 8. Simulation results for polypropylene beads conveying. (a)–(d)Solid phase volume fraction contours in the vertical post-bend region,in the horizontal plane aty = 0.9 m from the bend, taken at differenttimesGs =31.1 kg m−2 s−1, Ug =11.9 m s−1; (e)–(f) Solid-phase volumefraction contours in the horizontal plane aty=2.0 m from the bend, takenat different timesGs = 31.1 kg m−2 s−1, Ug = 11.9 m s−1.

velocities in ECT measurements, but simulation resultsshowed only small changes. This may be attributed to theinability of the model to capture the moving dunes forma-tion at constant frequency in the horizontal pipe section ofthe system.

Fig. 9. Solid-phase volume fraction contours at (a)y=0.9 m; (b) y=2.0 mfrom the bend at timet = 8 s. Gs = 29.0 kg m−2 s−1, Ug = 10.6 m s−1.A similar profile is observed which indicates a steady flow stream in thevertical pipe region.

α S

0.004.5 5.0 5.5 6.0

0.01

0.02

0.03

0.04

x-sectional average αs at 0.9m

x-sectional averageαs at 2.0m

t, s

05 5.5 6 6.5 7 7.5 8

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

α S

x-sectional average αs at 0.9 m

x-sectional average αs at 2.0 m

t, s

(a)

(b)

Fig. 10. Numerical solutions illustrating the differences in fluctuations forthe two systems. Solids volume fraction plane-averaged values(�s ), in theplanes at distances of 0.9 and 2.0 m in the vertical section from the bendfor (a) polypropylene beads,Gs =31.1 kg m−2 s−1, Ug =11.9 m s−1; (b)glass-beads,Gs = 29.0 kg m−2 s−1, Ug = 10.6 m s−1.

3.2. Distribution of solids velocity in the vertical pipe

The volume ratio data obtained simultaneously for the twoECT planes may be processed using the cross-correlationfunction to determine the pattern velocity of the solids in

L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651 4645

Fig. 11. Predicted polypropylene solids volume fraction (�s ) in the sym-metry plane of the entire geometry. High solid concentration in the sharpbend corner, with solid clusters above it and disperse rope at the topof the vertical section. Enlarged sections show the clustering and ropingregions of the vertical pipeGs = 31.1 kg m−2 s−1, Ug = 11.9 m s−1.

Table 3Comparison of experimental time-averaged solid volume fraction ofpolypropylene at the core and polypropylene pattern velocity from ECTwith simulation results

Superficial gas ECT Simulation ECT pattern Simulationvelocity (m s−1) �s �s velocity (m s−1) us(m s−1)

10.6 0.136 0.0719 3.08 1.1111.9 0.048 0.0590 3.33 1.6316.0 < 0.01 0.0300 3.33 3.48

Gs = 31.1 kg s−1.

the pipe. The cross-correlation coefficient,C(d), whereddenotes the delay time, was then computed as

C(d)= 1

T

∫ T

0(�s(z1, t) − 〈�〉(z1))(�s(z2, t + d)

−〈�〉(z2)) dt. (4)

Here,z1 andz2 refer to upstream and downstream planes,respectively, andd was taken to be positive. The domi-nant pattern propagation velocity,V ∗, was estimated from

Fig. 12. Predicted glass beads solids volume fraction (�s ) in the symme-try plane of the entire geometry. Accumulation of solids observed in thesharp bend corner. Continuous flow in the horizontal and vertical sec-tions. No clustering or roping was observed for the vertical pipe region.Gs = 29.0 kg m−2 s−1, Ug = 10.6 m s−1.

V ∗ = L/D , whereL is the axial distance between the twoECT sensors andD is the value ofd at which C(d) as-sumes the largest value. In this study, the cross-sectionalaveraged volume ratios obtained from the two ECT planeswere processed using the cross-correlation function and re-sulting C(d) values were normalized with the maximumC(d) value. The time at which the peak value for a plotC(d) with time occurs is taken as the time delay for estimateof pattern velocity.V ∗ estimated in this manner is clearlybased on cross-sectional averages.

Table 3shows the pattern velocity obtained from ECT datafor polypropylene in comparison with simulated solid-phasevelocities with varying superficial gas velocities. The ex-perimental solid-phase velocities were calculated by cross-correlating the temporal signals about the cross-section aver-aged solids concentrations at the planesy =1.64 and 2.14 mafter the pipe bend. The experimental values and the numer-ical results were within the same order of magnitude, show-ing qualitative consistency of the measurements. However,the ECT data indicate insignificant change in pattern veloc-ity with increasing inlet velocity, while the simulation modelshows higher sensitivity to varying inlet gas velocities.

4646 L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651

0

1

2

3

4

5

6

-25 -20 -15 -10 -5 0 5 10 15 20 25

Position from pipe center (mm)

Sol

ids

velo

city

,us (

m/s

)

10.6 m/s11.9 m/s13.2 m/s17.0 m/s

Pipe wallPipe wall

(a)

(b)

Fig. 13. Experimental results from PIV measurements: (a) Example ofa snapshot taken using the PIV system and (b) velocity(us) profile ofthe glass beads at position 1 m from the pipe bend for four differentsuperficial gas velocities.

Table 4Time-averaged solids velocity of glass beads at different radial positionsin the vertical pipe

r/R PIV, us PDPA, us Simulation,us

(m s−1) (m s−1) (m s−1)

0.6 3.064 2.737 4.8060.25 3.132 2.426 5.2150 3.264 4.336 5.704

Gs = 29.0 kg s−1, Ug = 10.6 m s−1. R is the radius of the pipe.

For glass beads, the solids velocity in the vertical post-bend section could not be accurately determined from cross-correlation of ECT volume ratio data due to the very diluteflow of the system. Therefore, laser techniques namely thePDPA and PIV were employed to determine the solids ve-locity of glass beads.Figs. 13a and b show a snapshot ofthe image taken using PIV for a superficial gas velocity of10.6 m s−1 and the cross-section distribution of solids ve-locity for four different air flow rates, respectively.Table 4shows the comparison of solids velocity for glass beads us-ing PIV, PDPA and simulation. The qualitative results for thevelocity distribution were quite consistent with a very low(close to zero) solids velocity just next to the pipe wall anda maximum velocity close to the center of the pipe. The av-erage solids velocities obtained were also fairly consistent.The normalized velocity profiles for both PIV measurementsand numerical simulation are shown inFig. 14. Simulation

0

0.2

0.4

0.6

0.8

1

1.2

-25 -20 -15 -10 -5 0 5 10 15 20 25Position from pipe center (mm)

Nor

mal

ized

sol

ids

velo

city

(u s

/us,m

ax)

Numerical Results

PIV results

Pipe wall Pipe wall

Fig. 14. Comparison of the normalized velocity profile for glass beadsfrom PIV measurements and numerical simulation.

results showed a more distinct maximum velocity off thepipe center while the measurements from PIV demonstratea flatter velocity profile along the radius of the pipe. Theasymmetric nature of the velocity profile in the numericalresults indicates more significant sensitivity of the simula-tion to the effects of the pipe bend then actual experiments.In the actual experimental runs, the particle–particle inter-actions may enhance the mixing of solids in the post-bendregion which may lead to a more symmetrical flow as ob-served in ECT and PIV measurements of solids concentra-tion and velocity.

3.3. Numerical results for the horizontal conveying section

Simulation data for pneumatic conveying of the class Dparticles (polypropylene beads) were compared with the ex-perimental data obtained from the literature.Fig. 15a showsthe simulated gas-phase axial velocity, plotted along thecenterline of the planes at distances of 12, 16 and 18 diame-ters from the pipe inlet (Fig. 2b), respectively (lines A–C inFig. 15(a)). The simulated results were compared with theexperimental data ofTsuji and Morikawa (1982), conductedunder the experimental conditions similar to those usedin the present numerical simulation (shown by line D inFig. 15(a)). The almost identical velocity profiles at differ-ent distances indicate that the airflow is fully developed be-fore reaching the bend. This is consistent withCarpinliogluand Gundogdu (1998), who measured fully developed two-phase flow for 3< x/D < 4, using wheat particles with amean diameter of 825�m. It was noted that the particle size,shape and the Reynolds number have significant effect onthis development distance.

Fig. 15a also shows that the maximum velocity valueis found closer to the top of the pipe, and this asymmet-ric profile is also qualitatively similar to those found byprevious investigators (Tsuji and Morikawa, 1982; Huber

L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651 4647

ug/Ug

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

y/D

0.0

0.2

0.4

0.6

0.8

1.0

y/D

0.0

0.2

0.4

0.6

0.8

1.0

ABCD

Solid volume fraction, αs

0.00 0.01 0.02 0.03 0.04 0.05 0.06

ABC

(a)

(b)

Fig. 15. (a) Normalized gas-phase axial velocity (centerline values) fromnumerical simulations, at distances of 12 (A), 16 (B) and 18 (C) diame-ters(x =0.5, 0.8&0.9 m) measured from the inlet.Gs =31.1 kg m−2 s−1,Ug =11.9 m s−1. Curve D presents values taken fromTsuji and Morikawa(1982), with 0.2 mm diameter particles through a 0.0305 m diameter pipe,Gs =18.4 kg m−2 s−1, Ug =6 m s−1 and (b) predicted solid volume frac-tion (�s ) for 2.8 mm polypropylene beads in the horizontal pipe section,along the centerline in the plane located at the different distances frominlet, x/D=10 (A), 16 (B) and 18 (C),Gs =kg m−2 s−1, Ug=11.9 m s−1.

and Sommerfeld, 1994, 1998; Zhu et al., 2003; Levy andMason, 1998; Bilirgen and Levy, 2001). This is attributed tothe loss of gas-phase momentum because of the higher par-ticle concentration near the bottom of the pipe than the top(Fig. 15b). Distortion by the bend in front was also cited asa reason for the parabolic profiles (Huber and Sommerfeld,1994, 1998).

The high solid volume fraction(�s) at the bottom of thehorizontal pipe wall, shown inFig. 15(b), was caused mainlyby gravitational settling of solids (Tsuji and Morikawa,1982; Huber and Sommerfeld, 1994, 1998; Zhu et al., 2003;Levy and Mason, 1998; Bilirgen and Levy, 2001). However,this was aggravated by the more intense particle–particleand particle–wall collisions at the bottom of the pipe, result-ing in greater loss of solids momentum as compared to theupper portion of the pipe (Huber and Sommerfeld, 1994). Inaddition to inelastic particle collisions, viscous dissipationof the gas phase also played a role in the momentum loss(Huber and Sommerfeld, 1994).

(u'g2)1/2/Ug, (u's

2)1/2/Ug, us/Ug

0.0 0.1 0.2 0.3 0.4

y/D

0.0

0.2

0.4

0.6

0.8

1.0

ABC

Fig. 16. Predicted (A) axial solid velocity(us/Ug), (B) solid-phase

axial turbulence ((u′2s )0.5/Ug) and (C) gas-phase axial turbulence

((u′2g )0.5/Ug) values. Values taken at the vertical line through the center of

the pipe at a distance of 0.9 m from the pipe inlet,Gs =31.1 kg m−2 s−1,Ug = 11.9 m s−1.

Similar parabolic profile could be seen in the simulationresults of particle velocity (filled circles) data as shown inFig. 16. Fig. 16also shows the axial velocity fluctuations ofthe both gas and solids phases (open circle for solids velocity,and the filled triangles for gas velocity). More fluctuationswere seen in the gas velocity. Such similarities in velocityfluctuations were also observed for the fully developed two-phase flow by previous studies (Tsuji and Morikawa, 1982;Huber and Sommerfeld, 1994, 1998; Zhu et al., 2003). Min-imum velocity fluctuations in both phases coincide at anoff center region, as observed experimentally byHuber andSommerfeld (1994, 1998)while high values were observedat the wall regions. This was attributed to the response ofthe solids to the high gas turbulence near the walls (Huberand Sommerfeld, 1994). Another reason for these observedprofiles was due to the particle–wall collisions occurringin the near wall regions (Tsuji and Morikawa, 1982; Huberand Sommerfeld, 1994, 1998). However, gas-phase turbu-lence at the pipe bottom wall was relatively lower than thatat the pipe top wall. This was due to the higher solids con-centration at the bottom wall (Fig. 15b), leading to highergas-phase turbulence suppression (Huber and Sommerfeld,1994). Numerical simulation scheme adopted in this workseemed to predict the gas and solid velocity profilesreasonably well.

3.4. Numerical results near the bend

The characteristics of the flow phenomena of the granularsolids in the vertical section are mainly due to the presenceof the 90◦ bend upstream of the vertical pipe. The clusterformation could be attributed to the inelastic collision withinthe bend, as well as the impact angle. Particles travelingin the horizontal pipe towards the bend can be expected to

4648 L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651

collide at or near an angle of 90◦. This would severely dis-sipate some of the kinetic energy while redirecting particlesfrom the horizontal to the vertical pipe. Particles would thenhave to decelerate, rebound and change flow direction, andre-accelerate upwards. Particles colliding within the benddo not transfer its kinetic energy upwards immediately, butwould rebound backwards and collide with solids arrivinglater. This would cause a net loss of momentum due to in-elastic collisions for the whole group of solids arriving atthe bend within a short period of time. These groups wouldthen be accelerated upwards as a cluster due to the dissipa-tion of granular kinetic energy.

At a further distance, near the outlet of the vertical pipe,a continuous stream of solids of lower concentration can beseen moving rapidly up (Fig. 11c). The solids flow structureevolution in the vertical section can be understood as fourcontinuous stages: cluster formation (Fig. 11a), cluster flow(Fig. 11b), dispersion/rope formation (Fig. 11c) and ropeflow (Fig. 11d).

One point to note is that a sharp bend has been usedin this work as compared to the smooth bend used in thework ofYilmaz and Levy (1998, 2001). The rope formationfrom the disintegration of the clusters was not seen in theexperiments ofYilmaz and Levy (1998, 2001), where anopposite phenomenon of dissipation of a continuous ropeinto discontinuous clusters occurred for smooth bends.

Previous studies (Yilmaz and Levy, 1998, 2001; Huber andSommerfeld, 1994, 1998; Levy and Mason, 1998; Bilirgenand Levy, 2001) reveal that secondary velocities in the gasphase and turbulent flow are responsible for the dispersionof particles in the vertical post-bend region. For the presentstudy, only the turbulent local mixing can be used to explainparticle dispersion, as secondary velocities induced by thebend does not seem to have observable effect on particledispersion, which usually occurs at a short distance after thebend. However, solid clusters are found to be formed within1 m from the bend, and the dispersion of the clusters in arope occur only after about 20 diameter distances from thebend(> 1 m).

4. Conclusions

Distinct differences in the flow characteristics of differentgranular materials were observed in this study. The single-plane ECT data at various axial locations of the conveyingpipe determine the non-uniformities in the cross-sectionalsolids concentration distribution for the vertical bend. ECTimages for the vertical conveying of polypropylene in adense-phase flow show a low-frequency pulsing flow withan alternating core–annulus and annulus structure. This pul-sating flow pattern was also observed in the simulation re-sults. The core–annulus structure is a result of the immaturemoving dunes formed at the horizontal conveying section.At higher superficial gas velocities, the flow becomes moredilute and the time-averaged solids concentration at the core

is found to decrease. In dilute phase, no pulsing flow wasobserved in the vertical conveying section, and no particlesare found in the core and the flow has an annulus structure.The formation of moving dunes in the horizontal convey-ing section is an inevitable phenomenon for conveying ofgranular materials with similar properties as polypropylenebeads. This type of flow behavior in the horizontal pipe haseffects on the flow pattern in the vertical post-bend region.

The conveying of class B particles (glass beads) showsa continuous flow with less fluctuation in solids concentra-tion in the vertical pipe region. This is reported by both thesimulated results and experimental ECT measurements. It isimportant to note that granular materials with similar phys-ical properties to the glass beads used (Class B) have to beconveyed in the dilute phase. Conveying at low superficialgas velocities will lead to unsteady flow of materials withfeeding rate higher than conveying rate and problems suchas choking of pipes may arise.

Numerical solutions for the flow characteristics of thefluid phase in the horizontal section compares well withthe experimental results obtained from the literature reports.The Euler–Euler model for the two-phase flow simulates thecluster formation in the vertical pipe after the bend reason-ably well, although dune formation in the horizontal sectioncould not be predicted without considering fluctuations inflow conditions at the inlet. While core structures in the solidflow were seen in the post-bend region in experiments andsimulations, their fluctuation intervals differ by nearly 20times (0.5 vs.10 Hz). The cross-section averages of the solidvolume fraction in the clusters predicted by the numericalsimulation are of the same order as ECT measurements.

Notations

CD interphase drag coefficientds diameter of solid particle, lengthess interparticle collision coefficient, energy after

collision and before collisiong0,ss radial distribution functionGs solids mass loading, kg m−2 s−1

kg gas-phase turbulent kinetic energy, m2 s−2

ks solid-phase turbulent kinetic energy, m2 s−2

Ksg interphase momentum exchange, from solid-to-gas phase

n number density of solid particles, number of par-ticles per unit volume

Res Reynolds numberug gas-phase velocity, m s−1

us solid-phase velocity, m s−1

Ug superficial gas velocity, m s−1

�Vg phase-weighted gas velocity, m s−1

�Vs phase-weighted solid velocity, m s−1

V ∗ solids pattern velocity, m s−1

L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651 4649

Greek letters

�g gas-phase volume fraction�s solid-phase volume fraction�s solid bulkviscosity, kg m−1 s−1

�s solid shear viscosity, kg m−1 s−1

�g gas-viscosity, kg m−1 s−1

�g gas-phase density, kg m−3

�s solid-phase density, kg m−3

� angle between mean particle velocity and meangas velocity

�s granular temperature, m2 s−2

Acknowledgements

This study has been supported by the National Universityof Singapore under the Grant No. R-279-000-095-112. Wethank Prof. Reginald Beng Hee Tan and Dr. S. MadhusudanaRao for many helpful discussions on the project.

Appendix A. Equations applied in FLUENT simulationof the pneumatic conveying system

The equations of mass and momentum in general formsare:

For the gas phase:

��t

(�g�g) + ∇ · (�g�g �ug) = 0, (A.1)

��t

(�g�g �ug) + ∇ · (�g�g �ug �ug)

= − �g∇pg + ∇ · �g + Ksg(�us − �ug)

+�g�g�Flift ,g + �g�g �g. (A.2)

For the solid phase:

��t

(�s�s) + ∇ · (�s�s �us) = 0, (A.3)

��t

(�s�s �us) + ∇ · (�s�s �us �us)

= − �s∇pg − ∇ps + ∇ · �s + Kgs(�ug − �us)

+�s�s�Flift ,s + �s�s �g, (A.4)

wheret is the time,�g and�s are gas and solids densities,respectively, the�g and�s are the volume fractions of theair and solid phases, respectively,�Flift ,g and �Flift ,s are thelift forces ( �Flift ,g = − �Flift ,s), �g is the gravitational accelera-tion, ∇pg is the pressure gradient of the gas phase,∇ps isthe solid pressure gradient,Kgs andKsg are the interphaseexchange coefficients,�g and�s are the stress-strain tensorsof the gas and solid phases, respectively. For the gas it iswritten as

�g = �g�g(∇�ug + ∇�uTg ) − 2

3�g�g∇ · �ugI (A.5)

and for the solid, it is:

�s = �s�s(∇�us + ∇�uTs ) + �s(�s − 2

3�s)∇ · �usI . (A.6)

Here�g is the gas viscosity,�s and�s are the solid shear and

bulk viscosities andI is the unit tensor. The solids pressureis composed of a kinetic term and a second term due toparticle collisions:

ps = �s�s�s + 2�s(1 + ess)�2s g0,ss�s , (A.7)

whereess is the coefficient of restitution for particle colli-sions, set at 0.9,g0,ss is the radial distribution function, and�s is the granular temperature. The radial distribution func-tion is modeled as proposed byOgawa et al. (1980).

g0,ss =[

1 −(

�s

�s,max

) 13]−1

(A.8)

The maximum packing limit,�s,max has been set to 0.65 inthis simulation.

The interphase exchange model follows that ofYang andYu (1966),

Ksg = 3

4CD

�s�g�g

∣∣�us − �ug

∣∣ds

�−2.65g , (A.9)

where

CD = 24

�gRes

[1 + 0.15(�gRes)0.687] (A.10)

and the relative Reynolds number,Res is defined by

Res = �gds |�us − �ug|�g

(A.11)

with Ksg = Kgs .The lift force is computed fromDrew and Lahey (1993).

�Flift ,s = −0.5�s�g|�us − �ug| × (∇ × �us). (A.12)

The granular temperature is proportional to the kineticenergy of the random motion of the particles. The transportequation derived from kinetic theory (Ding and Gidaspow,1990) can be written as:

3

2

[��t

(�s�s�s) + ∇ · (�s�s �us�s)

]

= (−psI + �s) : ∇ �us + ∇ · (k�s∇�s)

−12(1 − e2ss)g0,ss

ds

√�

�s�2s�

3/2s − 3Kgs�s , (A.13)

where k�sis the thermal conductivity of pseudo-thermal

energy.The solids bulk viscosity,�s accounts for the resistance

of the solid particles to compression and expansion. It takesthe following form according toLun et al. (1984),

�s = 4

3�s�sdsg0,ss(1 + ess)

(�s

�

)1/2

. (A.14)

4650 L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651

The shear viscosity of the particles is calculated from theparticle momentum exchange due to collisions (Gidaspowet al., 1992; Syamlal et al., 1993) and translation (Gidaspowet al., 1992). These two are added to give the solid shearviscosity as

�s = 4

5�s�sdsg0,ss(1 + ess)

(�s

�

)1/2

+ 10�sds

√�s�

96�s (1 + ess) g0,ss

×[1 + 4

5g0,ss�s(1 + ess)

]2

. (A.15)

The model used also accounts for the interphase turbu-lence transfer. Additional transport equations for the turbu-lence and its effects on the respective phases are calculated.Phase weighted averaging is used in averaging the fluctuat-ing turbulence quantities.

For each phase, two additional transport equations aresolved. For the solid phase, for example:

��t

(�s�sks) + ∇ · (�s�s�Vsks)

=∇ ·(�s

�t,s

�k

∇ks

)+ (�sGk,s − �s�ss)

+Kgs(2kg − Csgks) − Kgs(−→V g − −→

V s) · �t,g

�g�g

∇�g

+Kgs(−→V g − −→

V s) · �t,s

�s�s

∇�s (A.16)

��t

(�s�ss) + ∇ · (�s�s�Vss)

=∇ ·(�s

�t,s

�∇s

)+ s

ks

(1.42�sGk,s − 1.68�s�ss)

+1.2s

ks

(2kg − Csgks)

−1.2s

ks

Kgs( �Vg − �Vs) · �t,g

�g�g

∇�g

+1.2s

ks

Kgs( �Vg − �Vs) · �t,s

�s�s

∇�s (A.17)

Here �Vg and �Vs are the phase-weighted velocities, andGk,s is the generation of turbulent kinetic energy, and isexpressed as

Gk,s = −�su′s,iu

′s,j

�us,j

�xi

(i andj represent thex, y directions, respectively)

(A.18)

and

�t,s = 0.09�s

k2s

s

(A.19)

andCsg can be approximated as

Csg = 2

(gs

1 + gs

)(A.20)

with

gs = 0.135Kgsks

�g�gs(�g/�s + 0.5)

√1 + �2(1.8 − 1.35 cos2�)

(A.21)

and

� =√

2

3· 0.135|�ug − �us |

k12s

. (A.22)

For the interphase momentum transfer, the turbulent dragterms in Eqs. (A.3) and (A.5) are modeled as follows:

Ksg(�us − �ug)=Ksg(−→V s − −→

V g) + Ksg

×(

Ds

0.67�s

∇�s − Dg

0.67�g

∇�g

), (A.23)

where Dg and Ds are diffusivities and are calculated di-rectly from the expressions given below (Simonin andViollet, 1990):

Dg = 2

3ks

(b + gs

1 + gs

)gs�F,gs

+(

2

3kg − 2

3b

(b + gs

1 + gs

)ks

)�F,gs, (A.24)

b = 1.5

(�g

�s

+ 0.5

)−1

, (A.24a)

�F,gs = �g�gK−1gs

(�g

�s

+ 0.5

), (A.25)

Ds can be expressed as above with the terms whose sub-scripts are interchanged accordingly.

A no-slip condition is applied to both phases at the wallsurface, with wall roughness height (Hs) of 1 × 10−5 m. Amodified wall function is used to account for roughness:

0.548upk1/2

�w/�= 1

0.42ln

(5.373

�ypk1/2

�

)− �B, (A.26)

whereup andyp are the mean velocity and the distance fromthe wall, respectively.Hs is the roughness height describedbelow.

The roughness function�B is based on the formulas pro-posed byCebeci and Bradshaw (1977), and its computationdepends on the roughness regime(H+

s ).

H+s = 0.548�

Hsk1/2

�, (A.27)

For H+s < 3, �B = 0, (A.28)

L.Y. Lee et al. / Chemical Engineering Science 59 (2004) 4637–4651 4651

For 3< H+s < 70,

�B = 1

0.42

[H+

s − 2.25

87.75+ 0.5H+

s

]· sin

{0.4258

(ln H+

s − 0.811)}

, (A.29)

For H+s > 70, �B = 1

0.42ln(1 + 0.5H+

s ). (A.30)

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