Chemical Engineering Science 58 (2003) 877–885www.elsevier.com/locate/ces

The e�ects of abrupt T -outlets in a riser: 3D simulation using the kinetictheory of granular )ow

Juray De Wilde, Guy B. Marin, Geraldine J. Heynderickx∗

Laboratorium voor Petrochemische Techniek, Ghent University, Krijgslaan 281, Blok S5, 9000 Gent, Belgium

Abstract

Gas–solid )ow in circulating )uidized beds is calculated using the Eulerian–Eulerian approach with the kinetic theory of granular )ow.The usefulness of this approach and the necessity of performing 3D calculations are illustrated by calculating the exit e�ects of singleabrupt outlet con4gurations of di�erent outlet surface area.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Fluidization; Hydrodynamics; Turbulence; Multiphase )ow; Exit e�ects; Kinetic theory of granular )ow

1. Introduction

The hydrodynamics of circulating )uidized beds (CFBs)or risers have been modelled in several ways. The mostpopular approach in industry, is the 1D plug )ow modelwith slip between the phases (Froment & Bischo�, 1990).This idealized model is however not capable of describingthe complex hydrodynamic features of CFBs. The complex-ity of the hydrodynamics is due to the boundary conditions(for example solid walls, the outlet con4guration, etc.). Ex-tra complexity is intrinsically related to the two-phase char-acter of the )ow (Ha�, 1983; Needham & Merkin, 1983).Typical for gas–solid )ow in CFBs are solids segregationnear solid bounding walls (Sinclair & Jackson, 1989; Bader,Findley, & Knowlton, 1988), strong velocity pro4les witheventual down)ow near the solid wall of the tube (Baderet al., 1988), )uctuations in the )ow 4eld (i.e. unsteady be-havior) (Needham & Merkin, 1983), etc. The outlet con4g-uration was observed to sometimes a�ect the )ow patternnot only in the exit region of the riser, but also more up-stream in the middle and bottom part of the riser (Grace,1997). Because of the strong deviation from plug )ow, amore reliable prediction of the hydrodynamics in CFBs isof the utmost importance.

Core-annulus models (Nakamura & Capes, 1973) weredesigned following the experimental observation of a di-lute up)owing core surrounded by a dense down)owing

∗ Corresponding author. Tel.: +32-9-264-45-16; fax: +32-9-264-49-99.E-mail address: geraldine.heynderickx@rug.ac.be

(G. J. Heynderickx).

annulus. These models are however far from general andfail when used in the inlet and exit regions of the riser(Pugsley, Patience, Berruti, & Chaouki, 1992). A more gen-eral approach was taken by Anderson and Jackson (1967):both the gas and the solid phase are treated as continua,entirely mixed with each other. Both phases are describedby Navier–Stokes type equations. A diDculty to overcomein this so-called Eulerian–Eulerian approach, is the calcu-lation of the physical properties appearing in the equations(i.e. the solid phase pressure and viscosity). These prop-erties can be calculated either on an empirical or theoreti-cal basis. The kinetic theory of granular )ow (KTGF) re-lates the particle–particle collisions (which are the immedi-ate origin of the solid phase stress) to the turbulent motionof the solid phase (Gidaspow, 1994; Ha�, 1983; Jenkins &Savage, 1983). The solid phase turbulence is calculated froman extra transport equation. The KTGF however is far frommature and shows a tremendous sensitivity towards one ofthe model parameters, namely the restitution coeDcient forparticle–particle collisions. In case the particle–particle col-lisions are inelastic, the solid phase turbulent kinetic energyis strongly dissipated. As a result, the solid phase pressureand viscosity are low, especially in the core )ow. Hence,)uctuations in the )ow 4eld are less dampened, causing in-stability (slugging (De Wilde et al., 2001, 2002)). Therefore,Pita and Sundaresan (1993) suggest to use the KTGF allow-ing only elastic particle–particle collisions. The cases pre-sented here also have this restriction.

The aim of this paper is to show that, despite the im-perfections of the KTGF, it is possible to calculate certain

0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0009-2509(02)00619-X

878 J. De Wilde et al. / Chemical Engineering Science 58 (2003) 877–885

features of the )ow 4elds in CFBs. This is illustrated withthe calculation of CFBs with an abrupt T-outlet of di�erentsurface area. The characteristics observed with such an outletcon4guration are predicted correctly applying the Eulerian–Eulerian approach with the KTGF.

2. Modelling

2.1. Conservation equations

The model consists of the conservation equations formass, momentum and total energy for both the gas phaseand the solid phase. Furthermore, a turbulence model isrequired for each phase. For the gas phase, a k–� model,modi4ed for gas–solid interactions was implemented. Thesolid phase turbulence equation is derived through theKTGF (Gidaspow, 1994; Ha�, 1983; Jenkins & Savage,1983). The turbulence of the gas and the solid phase arein)uencing each other. The interphase transport of turbu-lent kinetic energy depends on the correlation between theturbulence of the gas and the solid phase. In case the cor-relation is complete, turbulent energy lost by one phase byinteraction with another phase, will be entirely convertedinto turbulent energy of this other phase. Otherwise, partof this turbulent energy is converted into another form ofenergy (internal energy for example). The gas–solid tur-bulence correlation is calculated from an extra transportequation (Simonin, Deutsch, & Minier, 1993).

In the present calculations, the solid phase temperature isassumed to equal the gas phase temperature. As a result, nosolid phase total energy equations is to be solved.

Table 1 lists all the conservation and transport equationsinvolved.

2.2. Constitutive equations

The constitutive equations for the solid phase physicalproperties appearing in the conservation equations, are ob-tained through the KTGF and are taken from Nieuwland(1995). The gas phase turbulent viscosity is calculated fromthe well-known Kolmogorov equation. The expressions forthe calculation of the gas–solid turbulence correlation are asproposed by Simonin et al. (1993).

For a summary of the constitutive equations and for thevalues of the turbulence model constants, reference is madeto De Wilde (2000) and De Wilde et al. (2001, 2002).

2.3. Boundary conditions

Boundary conditions have to be imposed at the inlets,outlets, and solid bounding walls of the domain.

Following the eigenvalue analysis (De Wilde, Heynder-ickx, & Marin, 1999), all variables except the gas phasepressure should be prescribed at the inlets. The gas phasepressure is prescribed at the outlets.

For the gas phase the no-slip condition at solid walls isapplied.

The form of the gas velocity pro4le in the vicinity of asolid wall is rather complex (Schlichting, 1979). The k − �model used in the bulk )ow (Fox, 1996) is not valid in theimmediate vicinity of the wall. Calculation of the near wallgas phase behavior is possible but demands a lot of extra gridpoints. In order to avoid an exponential rise of CPU-time,use is made of wall functions to calculate the e�ect of thesolid wall on the bulk )ow, at this stage of the modeling.

Wall functions are applied to the grid point in the imme-diate vicinity of the wall. This means that the external gridpoints are not positioned at the wall itself, but in the full tur-bulent zone at a certain distance of the wall. In the presentcalculations, y+ is taken 50.

Wall functions for pure gas )ow are applied. Although itis expected that the gas )ow 4eld near the wall will be mod-i4ed by the presence of solid particles, no accurate model isavailable yet.

The value of the wall shear force is calculated from thewell-known logarithmic law (Hinze, 1959). More details arefound in De Wilde (2000) and De Wilde et al. (2001, 2002).

For the solid phase it is also assumed that the mean )owin the vicinity of the wall is parallel to the wall. The valuesof the speci4c shear stress and the )ux of pseudothermalenergy to the wall are calculated as in Sinclair and Jackson(1989).

For the value of the speci4c shear stress:

�w = −�s�s@T@ Kr

∣∣∣∣w

=�′√3 �sp�s�

1=2i vTi

6�smax [1 − (�s=�smax )1=3]: (12)

For the )ux of pseudothermal energy to the wall:

qp� = −�s�@�@ Kr

= �w − v · �w: (13)

In this equation �w is the dissipation of solid phase turbu-lent energy due to inelastic collisions between particles andthe wall. The last term models generation of pseudothermalenergy by slip.

The dissipation of solid phase turbulent energy at the solidwall is modeled as

�w =

√3 �s�sp�

3=2i (1 − e2

w)4�smax [1 − (�s=�smax )1=3]

: (14)

In the present calculations, ew is 0.9, �′ has a value of 0.5,and Ewall is taken 8.432.

The gas–solid turbulence covariance in the vicinity of thewall is calculated assuming a constant velocity gradient forthe solid phase near the solid wall.

The solid wall does not perform any work. Furthermore itis assumed that the solid wall is adiabatic. Thus there is nocontribution of the solid wall in the discretised total energyequation of the gas phase.

J. De Wilde et al. / Chemical Engineering Science 58 (2003) 877–885 879

Table 1Conservation equations

Gas phase total mass balance@@t

(�g�g) +@@ Kr

(�g�g Ku) = 0: (1)

Solid phase total mass balance@@t

(�s�sp) +@@ Kr

(�s�sp Kv) = 0: (2)

Momentum conservation gas phase@@t

(�g�g Ku) +@@ Kr

(�g�g Ku Ku) = − @@ Kr

(P +

23�gk

)− @

@ Kr(�g KKsg) − �( Ku− Kv) + �g�g Kg; (3)

where KKsg = −[(

g − 23�g

)(@@ Kr

Ku)

KKI + (�g + �tg)

((@@ Kr

Ku)

+(

@@ Kr

Ku)T)]

: (4)

Momentum conservation solid phase@@t

(�s�sp Kv) +@@ Kr

(�s�sp Kv Kv) = − @@ Kr

Ps − @@ Kr

(�s KKss) + �( Ku− Kv) + �s�sp Kg; (5)

where KKss = −[(

s − 23�s)(

@@ Kr

Kv)

KKI + (�s)

((@@ Kr

Kv)

+(

@@ Kr

Kv)T)]

: (6)

Total energy conservation equation gas phase@@t

(�g�g(eg + k + qg)) +@@ Kr

(�g�g Ku(eg + k + qg)) − @@ Kr

(�g(" + "t)

@T@ Kr

)

= − @@ Kr

((P +

23�gk

)Ku)− @

@ Kr(�g KKsg Ku) − �

2(( Ku Ku) − ( Kv Kv)) − �

(k − 3

2�)

+ �g�g Kg Ku:

(7)

Turbulence equations gas phasek-equation@@t

(�g�gk) +@@ Kr

(�g�g Kuk)=@@ Kr

(�g�g + �t

g

�k

@k@ Kr

)

+

[�g�t

g

[(@@ Kr

Ku)

+(

@@ Kr

Ku)T]]

:(

@@ Kr

Ku)− �g�g�− �(2k − q12):

(8)

�-equation@@t

(�g�g�) +@@ Kr

(�g�g Ku�)=@@ Kr

(�g�g + �t

g

��

@�@ Kr

)

+C1��k

[�g�t

g

[(@@ Kr

Ku)

+(

@@ Kr

Ku)T]

:(

@@ Kr

Ku)]

− C2��g�g�2

k− �(2k − q12)C4�

�k:

(9)

Transport equation for the kinetic 7uctuation energy of the solid phase32

[@@t

(�s�sp�) +@@ Kr

(�s�sp Kv�)]

=@@ Kr

(�s�

@@ Kr

�)−

(Ps

KKI + �s KKss)

:(

@@ Kr

Kv)− � + �(q12 − 3�): (10)

Transport equation gas–solid turbulence correlation@@t

(�s�spq12) +@@ Kr

(�s�spq12 Kv)

=@@ Kr

(�s�sp

t12

�q

@q12

@ Kr

)+

[�s�spt12

[(@@ Kr

Ku)

+(

@@ Kr

Kv)T]]

:(

@@ Kr

Kv)

+

[�s�spt12

[(@@ Kr

Kv)

+(

@@ Kr

Ku)T]]

:(

@@ Kr

Ku)− �s

3

[�spq12

KKI + �spt12

[(@@ Kr

Kv)

+(

@@ Kr

Ku)]

KKI]

:(

@@ Kr

Kv)

− �s3

[�spq12

KKI + �spt12

[(@@ Kr

Ku)

+(

@@ Kr

Kv)]

KKI]

:(

@@ Kr

Ku)− �s�sp�12 − �

(q12 +

�s�sp�g�g

q12 − 2k − �s�sp�g�g

3�):

(11)

880 J. De Wilde et al. / Chemical Engineering Science 58 (2003) 877–885

3. Calculation method

The integration scheme is based on dual time stepping.A novel 4nite volume technique was developed and imple-mented in 3D. The treatment of the inviscid )uxes is basedon an extension of the advection upstream splitting method(AUSM) to two-phase )ow (Liou & Ste�en, 1993; DeWilde, 2000; De Wilde et al., 2001, 2002). A correct scalingof the di�erent terms is obtained by applying preconditioning(Weiss & Smith, 1995). Pressure–velocity coupling is guar-anteed by an arti4cial dissipation term (Liou & Edwards,1999; De Wilde, 2000). For details, reference is made toDe Wilde (2000) and De Wilde et al. (2001, 2002).

Calculations start from an initial 4eld using a very largephysical time step. Thus, the in)uence of the initial 4eld isminimized and a steady state is sought. A steady state willhowever not always exist. In such cases, no convergenceis obtained and the )ow 4eld is seen to oscillate around amean state. The physical time step is decreased to a valuesmall enough to capture the )uctuations in the )ow 4eld.The )ow is then calculated in a time-accurate way.

4. Simulation conditions

All calculations are performed on an industrial scaleriser with a diameter of 1:56 m and a height of 14:434 m.Both gas and particles are fed from the bottom at di�erentvelocities. An abrupt side outlet con4guration is simulated(Fig. 1). Table 2 summarizes the simulation conditionsfor the di�erent case studies. The restitution coeDcient forparticle–particle collisions e is taken 1.0, i.e. the particle–particle collisions are considered to be fully elastic.

To investigate the in)uence of the outlet surface area, onesimulation uses an outlet surface area of 0:955 m2, whilea second simulation uses an outlet surface area of 1:91 m2

(Fig. 1). Remark that the inlet surface area and the crosssectional riser surface area amount to 1:91 m2.

5. Results and discussion

A number of axial and radial cross sections are selected forthe presentation of the )ow 4eld. A schematic representationis given in Fig. 2. Axial cross section I goes through themiddle of the riser outlet and through the central axis ofthe riser and spans the whole diameter of the riser. Axialcross section II is perpendicular to Axial cross section I andgoes through the central axis of the riser. Furthermore, radialcross sections III at di�erent axial positions in the riser areused for the presentation of the results.

After two physical time steps Ot of 5000 s, an almoststeady-state solution was obtained in both simulation cases.The solution after the 4rst physical time step hardly dif-fers from the solution after the second physical time step.A real steady-state could not be obtained, but the )uctua-

1.56 m

14.4

34 m

Inlet

Outlet

Top view

Front view

1.56

m

11.9

7 m

1.23

5 m

Out

let75˚

or150˚

B

Fig. 1. Geometrical con4gurations used for the simulations.

Table 2Simulation conditions

Case 1 Case 2

〈uinbz 〉 (m s−1) 3.36

vinbz (m s−1) 1.0Gs (kg m−2 s−1) 2.6�inb (m2 s−2) 1.56e 1.0�sp (kg m−3) 1550.0dp (�m) 60Soutlet (m2) 0.955 1.91

Fig. 2. Projection surfaces used for the presentation of the results.

tions around a mean state are seen to be small. The requiredCPU-time to reach a solution is ±80 000 min on an IBMRS-6000-3CT workstation, after which the residuals havedropped by about three orders of magnitude.

Starting from the instantaneous solution obtained in pre-vious paragraph, time-accurate calculations were carried outwith a small physical time step Ot of 0:1 s. On average,per physical time step Ot, about 1200 min of CPU-time are

J. De Wilde et al. / Chemical Engineering Science 58 (2003) 877–885 881

1.00E-02

1.00E-01

1.00E+00

1.00E+01

0 200 400 600 800 1000 1200

Iteration Number

Res

idu

al

Fig. 3. Convergence behavior for the time-accurate calculations. Theresidual is the maximum absolute value of the update of the viscousvariables.

required to reach convergence. The residual drops by aboutone order of magnitude (Fig. 3). The calculated )uctuationsare small and in general it is seen that an instantaneous )ow4eld does not di�er much from the mean )ow 4eld.

Fig. 4 shows the solids volume fraction �s, respectively inAxial cross section I (Fig. 2) and in radial cross sections IIIat 6.4 and 13:9 m height in the riser. Fig. 5 shows projectionsof the solid phase velocity Kv, respectively, in Axial crosssection I (Fig. 2) and in radial cross sections III at 12.4and 13:7 m height in the riser. Fig. 6 shows the granulartemperature �, respectively, in Axial cross section I (Fig.2) and in radial cross sections III at ±11 and 12:05 m heightin the riser.

5.1. Fully developed middle part of the riser

Solids segregation is seen to take place near the solid wall(Fig. 4). The solids concentration in the annulus is about2–5 times higher than in the core. For the smaller outlet, it iscalculated that the core covers only about 40% of the crosssectional surface area, for the broader outlet about 50%. Inboth cases, only a fraction of the cross sectional surfacearea of the riser is available for the upward transport of thegas–solid mixture. This results in very high axial velocities(compared to the inlet boundary values) and mass )uxes inthe core of the riser. Fig. 5 shows axial velocities in the coremount to about 10 m s−1 in Case 1 and 8 m s−1 in Case 2,whereas the imposed value of the axial velocity at the inletis only 3:36 m s−1. These results are in line with Bader etal. (1988), who experimentally observed that, under certainconditions, only 45% of the total surface area of the risermanages 75% of the total upward )ow, as a result of particlesegregation.

The impact of internal velocities much higher than thevelocities imposed at the inlet is important. In chemical re-actors for example, the residence time drops with increasingvelocities and as a result the conversion drops as well. Thisagain shows the importance of a correct calculation of the)ow pattern, in particular for example the annulus thickness.

It should be remarked that this impact can only be quanti4edcorrectly in a 3D perspective.

The solids segregation results from collisional stresseswithin the solid phase (Ha�, 1983; Jenkins & Savage, 1983;Gidaspow, 1994). In the present paper, these stresses arecalculated through the KTGF and a correct calculation of thegranular temperature pro4le is required. The latter is not ob-vious, as the pro4le will depend on the imposed inlet values,the boundary conditions at the solid walls, and on certainmodel parameters. The inlet and boundary conditions arenot easily described and should be further investigated. Themost important model parameter is the restitution coeDcientfor particle–particle collisions (Pita & Sundaresan, 1993; DeWilde et al., 2001). The KTGF shows a great sensitivity to-wards this parameter (Pita & Sundaresan, 1993; De Wilde,2000; De Wilde et al., 2001, 2002). A small amount ofinelasticity leads to strong dissipation of the solid phaseturbulence and as a result the importance of the solid phasestress terms is reduced. The latter terms are however respon-sible for many of the features of gas–solid )ow (Ha�, 1983;Needham & Merkin, 1983; Pita & Sundaresan, 1993;Gidaspow, 1994). Therefore, in the present paper, particle–particle collisions are assumed fully elastic, making thedissipation term (�) in Eq. (10) zero.

5.2. Exit e:ects

Remarkably, for the smaller outlet surface area, the solidssegregation in the middle and bottom part of the riser ismuch more pronounced at the side of the riser opposite theoutlet than at the side of the riser at which the outlet ispositioned. As a result, in this case, the dilute core of theriser is not located centrally in the riser, but its location isshifted towards the side of the outlet. At the side of the riseropposite the outlet, the exit con4guration induces down)owof both the gas and the solid phase (Fig. 5). The down)owis maximal near the solid wall right opposite the outlet.This is the immediate origin for the increased solid fractionthere. Rhodes, Sollaart, and Wang (1998) also report thatdown)ow is only measured in the annulus of the upper diluteregion. No relation between the measured asymmetry andthe outlet con4guration is examined by these authors.

For the smaller outlet surface area, the outlet con4gurationis seen to determine the )ow pattern in the whole riser,almost down to the riser bottom inlet. An increase of the riseroutlet surface area drastically reduces the e�ective depth ofthe riser exit con4guration. For the broader outlet, up to 10m height, the dilute core of the riser is positioned centrallyin the riser and no in)uence of the riser exit con4gurationis seen. Higher on, its position shifts towards the side of theoutlet, as expected.

The observed in)uence of the outlet con4guration on the)ow pattern is certainly enhanced by the presence of thesolid phase. To explain this, Fig. 5 shows projections of thesolid phase velocity in Axial cross section I and in radial

882 J. De Wilde et al. / Chemical Engineering Science 58 (2003) 877–885

0

2

4

6

8

10

12

14

z(m

)

out let

cross section I

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y (m

)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8x (m)

z = 13.9 m

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y (m

)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8x (m)

z = 6.4 m

cross section III:

1. 57E-031. 53E-031. 49E-031. 45E-031. 41E-031. 37E-031. 33E-031. 29E-031. 25E-031. 21E-031. 17E-031. 13E-031. 09E-031. 05E-031. 01E-039. 70E-049. 30E-048. 90E-048. 50E-048. 10E-047. 70E-047. 30E-046. 90E-046. 50E-046. 10E-045. 70E-045. 30E-044. 90E-044. 50E-044. 10E-043. 70E-043. 30E-042. 90E-042. 50E-042. 10E-041. 70E-041. 30E-04

0

2

4

6

8

10

12

14

z (m

)

outlet

cross section I

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y(m

)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8x (m)

z = 13.9 m

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y (m

)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8x (m)

z = 6.4 m

cross section III:

8.42E-048.20E-047.99E-047.78E-047.56E-047.35E-047.14E-046.92E-046.71E-046.50E-046.29E-046.07E-045.86E-045.65E-045.43E-045.22E-045.01E-044.79E-044.58E-044.37E-044.15E-043.94E-043.73E-043.51E-043.30E-043.09E-042.87E-042.66E-042.45E-042.23E-042.02E-041.81E-041.59E-041.38E-041.17E-049.54E-057.41E-05

2955.0 mSout = 291.1 mSout =

Fig. 4. Solids volume fraction (�s) (conditions see Table 2).

Fig. 5. Solids velocity ( Kv) (conditions see Table 2).

cross sections III positioned at 12.4 and 13:7 m height inthe riser (see Fig. 2). Near the riser outlet, the )ow de)ectstowards the outlet as a result of the pressure gradient. How-ever, as a result of inertia and the dimensions of the riser

top con4guration, part of the de)ected )ow does not reachthe outlet. It is re)ected by the solid walls of the riser anda vortex is formed in the top of the riser. At the side oppo-site to the outlet, the down)ow in the top creates additional

J. De Wilde et al. / Chemical Engineering Science 58 (2003) 877–885 883

0

2

4

6

8

10

12

14

z (m

)outlet

cross section I

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y(m

)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8x (m)

outlet

z = 12.05 m

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y(m

)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8x (m)

z = 10.4 m

cross section III:

1.2E+00

5.0E- 024.2E- 023.5E- 023.0E- 022.5E- 022.1E- 021.8E- 021.5E- 021.3E- 021.1E- 028.9E- 037.5E- 036.3E- 035.3E- 034.5E- 033.8E- 033.2E- 032.7E- 032.2E- 031.9E- 031.6E- 031.3E- 031.1E- 039.4E- 047.9E- 046.7E- 045.6E- 044.7E- 044.0E- 043.3E- 042.8E- 042.4E- 042.0E- 041.7E- 041.4E- 041.2E- 041.0E- 04

0

2

4

6

8

10

12

14

z(m

)

outlet

cross section I

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y(m

)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

x (m)

outlet

z = 12.05 m

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y(m

)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

x (m)

z = 11.1 m

cross section III:

5.0E-023.7E-022.7E-022.0E-021.5E-021.1E-028.2E-036.1E-034.5E-033.3E-032.5E-031.8E-031.4E-031.0E-037.4E-045.5E-044.1E-043.0E-042.2E-041.7E-041.2E-049.1E-056.7E-055.0E-053.7E-052.7E-052.0E-051.5E-051.1E-058.2E-066.1E-064.5E-063.3E-062.5E-061.8E-061.4E-061.0E-06

1.1E+00

2955.0 mSout = 291.1 mSout =

Fig. 6. Granular temperature (�) (m2 s−2) (conditions see Table 2).

resistance for the up)owing gas–solid mixture. As the solidphase su�ers more from inertia than the gas phase, the topsection of the riser (above the outlet) is in general moredense. Also, a zone of increased solids volume fraction isseen right above the outlet and the external boundaries ofthe vortex are more dense than the eye of the vortex. Thelatter is a typical centrifugal e�ect.

Remark that the vortexes formed have a 3D character, asclearly seen in the radial pro4les in Fig. 5.

The calculated pressure drop over the riser is 249 Pa forthe broader outlet and 329 Pa for the smaller outlet. This isdue to the increased solids hold-up in case of the smalleroutlet.

Gupta and Berruti (2000) investigated the exit e�ects incirculating )uidized beds and also found an increasing re-)ection with decreasing exit surface area. These authors ob-served that, for Geldart group A particles, the strongest in-)uence on the re)ection comes from the exit angle. If theexit is at a sharp right angle to the vertical up)owing sus-pension, the particles cannot take the sharp turn, re)ect o�the top and recirculate internally down along the walls ofthe riser (Grace, 1997). This is clearly in agreement with thepresent calculations. It is also observed that, as a result ofthe re)ection, an area of densi4cation is created around theexit and, depending on the operating conditions and sever-ity of the exit restrictions, may extend a long way down the

length of the riser. The calculations with the smaller outletclearly illustrate this possibility.

Pugsley, Lapointe, Hirschberg, and Werther (1997) de-veloped a model that provides a reasonable estimate for thedensi4cation observed at the top of an abrupt exit riser andthat focuses on the distance the densi4cation from the exitpropagates down the riser. The present calculations revealthat the densi4cation is not radially uniform and that 3D ef-fects should be taken into account.

A T-shape abrupt exit is a very strong restrictive exit(Cheng, Wei, Yang, & Jin, 1998). This results in violentturbulence (Fig. 6) and signi4cant solids backmixing in theriser exit region (Pugsley et al., 1997; Cheng et al., 1998;Jin, Yu, Qi, & Bai, 1988; Brereton & Grace, 1994; Zheng& Zhang, 1994). Remark however that the backmixing inthe exit region is not a result of the turbulence, but of theconvective and the acoustic terms.

Fig. 6 shows a picture of the granular temperature in therisers. Four phenomena are clearly visible: the slow dissi-pation of the inlet boundary granular temperature (no dis-sipation by inelastic particle–particle collisions, as e = 1:0)starting from the riser bottom inlet, the high value of theturbulence quantities near the solid bounding walls, the in-creased value of the turbulence variables at the boundariesof the outlet, and the increased values of the turbulence vari-ables due to recirculation. With respect to the latter, it is

884 J. De Wilde et al. / Chemical Engineering Science 58 (2003) 877–885

clear that the zone of increased turbulence follows the frontwhere the up)ow and the down)ow encounter. Steep veloc-ity gradients in this zone are responsible for the increasedturbulence.

The calculations have also shown that the gas phaseand the solid phase turbulence are well correlated, exceptnear the solid bounding walls. Near solid bounding walls,the gas phase turbulence (not shown)—calculated via wallfunctions—is higher than the solid phase turbulence. Dueto inertia, the solid phase is not able to follow the )uctuat-ing motion of the gas phase. Just the opposite is seen at thebottom inlet, where the solid phase is imposed to enter withmuch more turbulence than the gas phase. The gas phaseimmediately responds to the turbulent motion of the solidphase. Therefore, it could be stated that in the bulk )ow, itis the solid phase that determines the overall turbulence ina gas–solid mixture. This again stresses the importance of acorrect calculation of the solid phase turbulence and of theuse of correct wall functions.

6. Conclusions

A 3D simulation of gas–solid )ow using the Eulerian–Eulerian approach with the kinetic theory of granular )owis able to give insight in the complex hydrodynamics of cir-culating )uidized beds. This is illustrated by calculating thee�ect of an abrupt T-outlet on the )ow pattern. Such an out-let con4guration leads to vortex formation in the top, whichin turn induces recirculation at the wall opposite the outlet.If the outlet surface area is decreased, the outlet becomesmore restrictive and the recirculation is seen to increase, re-sulting in down)ow down to the bottom of the riser.

Acknowledgements

This work was 4nanced by the European Commissionwithin the frame of the Non-Nuclear Energy Program, JouleIII project under contract JOF3-CT95-0012 and Thermieproject under contract no. SF 243/98 DK/BE/UK (projectpartners: Laboratorium voor Petrochemische Techniek—Universiteit Gent, FLS miljo a/s—Denmark & Howden Air& Gas Division—UK).

Geraldine Heynderickx and Guy Marin are grateful to the“Fonds voor Wetenschappelijk Onderzoek—Vlaanderen”(FWO-N) for 4nancial support of the CFD Research. Geral-dine Heynderickx and Guy Marin are grateful to the “Bij-zonder Onderzoeksfonds” BOF-RUG for 4nancial supportof the gas–solid two-phase )ow research.

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