The effects of abrupt T-outlets in a riser

Chemical Engineering Science58(2003)877–
885
/locate/ces
The e ects ofabrupt T-outlets in a riser:3D simulation using the kinetic
theory ofgranular ow
Juray De Wilde,Guy B.Marin,Geraldine J.Heynderickx∗
Laboratorium voor Petrochemische Techniek,Ghent University,Krijgslaan281,Blok S5,9000Gent,Belgium
Abstract
Gas–solid ow in circulating uidized beds is calculated using the Eulerian–Eulerian approach with the kinetic theory ofgranular ow. The usef ulness ofthis approach and the necessity ofperf orming3D calculations are illustrated by calculating the exit e ects ofsingle abrupt outlet conÿgurations ofdi erent outlet surf ace area.
?2003Elsevier Science Ltd.All rights reserved.
Keywords:Fluidization;Hydrodynamics;Turbulence;Multiphase ow;Exit e ects;Kinetic theory ofgranular ow
1.Introduction
The hydrodynamics ofcirculating uidized beds(CFBs) or risers have been modelled in several ways.The most popular approach in industry,is the1D plug ow model with slip between the phases(Froment&Bischo ,1990). This idealized model is however not capable ofdescribing the complex hydrodynamic features of CFBs.The complex-ity ofthe hydrodynamics is due to the boundary conditions (for example solid walls,the outlet conÿguration,etc.).Ex-tra complexity is intrinsically related to the two-phase char-acter ofthe ow(Ha ,1983;Needham&Merkin,1983). Typical for gas–solid ow in CFBs are solids segregation near solid bounding walls(Sinclair&Jackson,1989;Bader, Findley,&Knowlton,1988),strong velocity proÿles with eventual down ow near the solid wall ofthe tube(Bader et al.,1988), uctuations in the owÿeld(i.e.unsteady be-havior)(Needham&Merkin,1983),etc.The outlet conÿg-uration was observed to sometimes a ect the ow pattern not only in the exit region ofthe riser,but also more up-stream in the middle and bottom part ofthe riser(Grace, 1997).Because ofthe strong deviation f rom plug ow,a more reliable prediction ofthe hydrodynamics in CFBs is ofthe utmost importance.
Core-annulus models(Nakamura&Capes,1973)were designed 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 and fail 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 described by Navier–Stokes type equations.A di culty to overcome in this so-called Eulerian–Eulerian approach,is the calcu-lation ofthe 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 ofgranular ow(KTGF)re-lates the particle–particle collisions(which are the immedi-ate origin ofthe solid phase stress)to the turbulent motion ofthe solid phase(Gidaspow,1994;Ha ,1983;Jenkins& Savage,1983).The solid phase turbulence is calculated from an extra transport equation.The KTGF however is far from mature and shows a tremendous sensitivity towards one of the model parameters,namely the restitution coe cient for particle–particle collisions.In case the particle–particle col-lisions are inelastic,the solid phase turbulent kinetic energy is strongly dissipated.As a result,the solid phase pressure and viscosity are low,especially in the core ow.Hence, uctuations in the owÿeld 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 ofthis paper is to show that,despite the im-perfections of the KTGF,it is possible to calculate certain
0009-2509/03/$-see front matter?2003Elsevier Science Ltd.All rights reserved. doi:10.1016/S0009-2509(02)00619-X
878J.De Wilde et al./Chemical Engineering Science58(2003)877–885
features of the owÿelds in CFBs.This is illustrated with the calculation ofCFBs with an abrupt T-outlet ofdi erent surface area.The characteristics observed with such an outlet conÿguration are predicted correctly applying the Eulerian–Eulerian approach with the KTGF.
2.Modelling
2.1.Conservation equations
The model consists ofthe conservation equations f or mass,momentum and total energy for both the gas phase and the solid phase.Furthermore,a turbulence model is required for each phase.For the gas phase,a k– model, modiÿed for gas–solid interactions was implemented.The solid phase turbulence equation is derived through the KTGF(Gidaspow,1994;Ha ,1983;Jenkins&Savage, 1983).The turbulence ofthe gas and the solid phase are in uencing each other.The interphase transport ofturbu-lent kinetic energy depends on the correlation between the turbulence ofthe gas and the solid phase.In case the cor-relation is complete,turbulent energy lost by one phase by interaction with another phase,will be entirely converted into turbulent energy ofthis other phase.Otherwise,part ofthis turbulent energy is converted into another f orm of energy(internal energy for example).The gas–solid tur-bulence correlation is calculated from an extra transport equation(Simonin,Deutsch,&Minier,1993).
In the present calculations,the solid phase temperature is assumed to equal the gas phase temperature.As a result,no solid phase total energy equations is to be solved.
Table1lists all the conservation and transport equations involved.
2.2.Constitutive equations
The constitutive equations for the solid phase physical properties appearing in the conservation equations,are ob-tained through the KTGF and are taken from Nieuwland (1995).The gas phase turbulent viscosity is calculated from the well-known Kolmogorov equation.The expressions for the calculation ofthe gas–solid turbulence correlation are as proposed by Simonin et al.(1993).
For a summary ofthe constitutive equations and f or the values ofthe turbulence model constants,ref erence is made to 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 ofthe domain. Following the eigenvalue analysis(De Wilde,Heynder-ickx,&Marin,1999),all variables except the gas phase pressure should be prescribed at the inlets.The gas phase pressure is prescribed at the outlets.
For the gas phase the no-slip condition at solid walls is applied.
The f orm ofthe gas velocity proÿle in the vicinity ofa solid wall is rather complex(Schlichting,1979).The k− model used in the bulk ow(Fox,1996)is not valid in the immediate vicinity ofthe wall.Calculation ofthe near wall gas phase behavior is possible but demands a lot ofextra grid points.In order to avoid an exponential rise ofCPU-time, use is made ofwall f unctions to calculate the e ect ofthe solid wall on the bulk ow,at this stage ofthe modeling. Wall functions are applied to the grid point in the imme-diate vicinity ofthe wall.This means that the external grid points are not positioned at the wall itself,but in the full tur-bulent zone at a certain distance ofthe wall.In the present calculations,y+is taken50.
Wall functions for pure gas ow are applied.Although it is expected that the gas owÿeld near the wall will be mod-iÿed by the presence ofsolid particles,no accurate model is available yet.
The value of the wall shear force is calculated from the well-known logarithmic law(Hinze,1959).More details are found in De Wilde(2000)and De Wilde et al.(2001,2002). For the solid phase it is also assumed that the mean ow in the vicinity ofthe wall is parallel to the wall.The values ofthe speciÿc shear stress and the ux ofpseudothermal energy to the wall are calculated as in Sinclair and Jackson (1989).
For the value ofthe speciÿc shear stress:
w=− s s
@ T
@ r
w
=
√
3 sp s 1=2i v T
i
6 s
max
[1−( s= s
max
)1=3]
:(12) For the ux ofpseudothermal energy to the wall:
q p =− sÄ
@
@ r
= w−v· w:(13)
In this equation w is the dissipation ofsolid phase turbu-lent energy due to inelastic collisions between particles and the wall.The last term models generation ofpseudothermal energy by slip.
The dissipation ofsolid phase turbulent energy at the solid wall is modeled as
w=
√
3 s sp 3=2i(1−e2w)
4 s
max
[1−( s= s
max
)1=3]
:(14)
In the present calculations,e w is0.9, has a value of0.5, and E wall is taken8.432.
The gas–solid turbulence covariance in the vicinity ofthe wall is calculated assuming a constant velocity gradient for the solid phase near the solid wall.
The solid wall does not perform any work.Furthermore it is assumed that the solid wall is adiabatic.Thus there is no contribution ofthe solid wall in the discretised total energy equation ofthe gas phase.
J.De Wilde et al./Chemical Engineering Science 58(2003)877–885
879
Table 1
Conservation equations
Gas phase total mass balance
@
@t ( g g )+@@ r ( g g u
)=0:(1)
Solid phase total mass balance @@t ( s sp )+@
@
r ( s sp v
)=0:(2)
Momentum conservation gas phase @@t ( g g u )+@@ r ( g g u u )=−@@ r P +23 g k −@@ r ( g s g )−ÿ( u − v )+ g g g;(3)where s g =− g −23 g @@ r u I +( g + t g ) @@ r u + @@ r u T
:(4)
Momentum conservation solid phase
@@t ( s sp v )+@@ r ( s sp v v )=−@@ r P s −@@ r ( s s s )+ÿ( u − v )+ s sp g;(5)where s s =− s −23 s @@ r v I +( s )
@@ r v + @@ r v T :(6)
Total energy conservation equation gas phase @@t ( g g (e g +k +q g ))+@@ r ( g g u (e g +k +q g ))−@@ r g ( + t )@T @
r
=−
@@
r
P +23 g k u
−@@ r ( g s g u )−ÿ2(( u u )−( v
v ))−ÿ
k −32
+ g g g u:(7)
Turbulence equations gas phase
k -equation @
@t ( g g k )+@@ r ( g g uk
)=@@ r g g + t g k @k @
r +
g t
g
@@
r u +
@
@
r u T :
@
@
r u
− g g −ÿ(2k −q 12):(8)
-equation
@@t ( g g )+@
@
r ( g g u
)=@@ r
g g + t g @
@
r
+C 1 k
g t g
@@
r u +
@
@
r u T :
@
@
r u
−C 2 g g 2k −ÿ(2k −q 12)C 4
k
:
(9)
Transport equation for the kinetic uctuation energy of the solid phase 32 @@t ( s sp )+@@ r ( s sp v ) =@@ r s Ä@@ r − P s I + s s s : @@ r v − +ÿ(q 12−3 ):
(10)
Transport equation gas–solid turbulence correlation
@@t ( s sp q 12)+@
@
r ( s sp q 12 v
)=@
@ r
s sp t 12 q @q 12@
r
+
s sp t 12
@
@ r u + @
@
r v T :
@
@
r v
+ s sp t 12
@
@
r v
+
@@
r u T :
@@ r u
−
s
3
sp q 12 I + sp t 12
@
@ r v
+
@
@
r u
I :
@
@
r v
− s
3
sp q 12
I + sp t 12
@
@
r u +
@@
r v
I :
@@ r u − s sp 12−ÿ
q 12+ s sp g g q 12−2k − s sp
g g
3
:(11)
880J.De Wilde et al./Chemical Engineering Science58(2003)877–885
3.Calculation method
The integration scheme is based on dual time stepping.
A novelÿnite volume technique was developed and imple-mented in3D.The treatment ofthe inviscid uxes is based on an extension ofthe advection upstream splitting method (AUSM)to two-phase ow(Liou&Ste en,1993;De Wilde,2000;De Wilde et al.,2001,2002).A correct scaling ofthe di erent terms is obtained by applying preconditioning (Weiss&Smith,1995).Pressure–velocity coupling is guar-anteed by an artiÿcial dissipation term(Liou&Edwards, 1999;De Wilde,2000).For details,reference is made to De Wilde(2000)and De Wilde et al.(2001,2002). Calculations start from an initialÿeld using a very large physical time step.Thus,the in uence ofthe initialÿeld is minimized and a steady state is sought.A steady state will however not always exist.In such cases,no convergence is obtained and the owÿeld is seen to oscillate around a mean state.The physical time step is decreased to a value small enough to capture the uctuations in the owÿeld. The ow is then calculated in a time-accurate way.
4.Simulation conditions
All calculations are performed on an industrial scale riser with a diameter of1:56m and a height of14:434m. Both gas and particles are fed from the bottom at di erent velocities.An abrupt side outlet conÿguration is simulated (Fig.1).Table2summarizes the simulation conditions for the di erent case studies.The restitution coe cient for particle–particle collisions e is taken1.0,i.e.the particle–particle collisions are considered to be fully elastic.
To investigate the in uence ofthe outlet surf ace area,one simulation uses an outlet surface area of0:955m2,while a second simulation uses an outlet surface area of1:91m2 (Fig.1).Remark that the inlet surface area and the cross sectional riser surface area amount to1:91m2.
5.Results and discussion
A number ofaxial and radial cross sections are selected f or the presentation ofthe owÿeld.A schematic representation is given in Fig.2.Axial cross section I goes through the middle ofthe riser outlet and through the central axis of the riser and spans the whole diameter ofthe riser.Axial cross section II is perpendicular to Axial cross section I and goes through the central axis ofthe riser.Furthermore,radial cross sections III at di erent axial positions in the riser are used for the presentation of the results.
After two physical time steps t of5000s,an almost steady-state solution was obtained in both simulation cases. The solution after theÿrst 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-
75˚
or
150
Fig.1.Geometrical conÿgurations used for the simulations.
Table2
Simulation conditions
Case1Case2 u inb z (m s−1) 3.36
v inb z(m s−1) 1.0
G s(kg m−2s−1) 2.6
inb(m2s−2) 1.56
e 1.0
sp(kg m−3)1550.0
d p( m)60
S outlet(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 required CPU-time to reach a solution is±80000min on an IBM RS-6000-3CT workstation,after which the residuals have dropped by about three orders ofmagnitude.
Starting from the instantaneous solution obtained in pre-vious paragraph,time-accurate calculations were carried out with a small physical time step t of0:1s.On average, per physical time step t,about1200min ofCPU-time are
J.De Wilde et al./Chemical Engineering Science 58(2003)877–885
881
1.00E-02
1.00E-01
1.00E+001.00E+01
Iteration Number
R e s i d u a l
Fig.3.Convergence behavior for the time-accurate calculations.The residual is the maximum absolute value ofthe update ofthe viscous variables.
required to reach convergence.The residual drops by about one order ofmagnitude (Fig.3).The calculated uctuations are small and in general it is seen that an instantaneous ow ÿeld does not di er much from the mean ow ÿeld.
Fig.4shows the solids volume fraction s ,respectively in Axial cross section I (Fig.2)and in radial cross sections III at 6.4and 13:9m height in the riser.Fig.5shows projections ofthe solid phase velocity v ,respectively,in Axial cross section I (Fig.2)and in radial cross sections III at 12.4and 13:7m height in the riser.Fig.6shows the granular temperature ,respectively,in Axial cross section I (Fig.2)and in radial cross sections III at ±11and 12:05m height in 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 about 2–5times higher than in the core.For the smaller outlet,it is calculated that the core covers only about 40%ofthe cross sectional surface area,for the broader outlet about 50%.In both cases,only a fraction of the cross sectional surface area ofthe riser is available f or the upward transport ofthe gas–solid mixture.This results in very high axial velocities (compared to the inlet boundary values)and mass uxes in the core ofthe riser.Fig.5shows axial velocities in the core mount to about 10m s −1in Case 1and 8m s −1in Case 2,whereas the imposed value ofthe axial velocity at the inlet is only 3:36m s −1.These results are in line with Bader et al.(1988),who experimentally observed that,under certain conditions,only 45%ofthe total surf ace area ofthe riser manages 75%ofthe total upward ow,as a result ofparticle segregation.
The impact ofinternal velocities much higher than the velocities imposed at the inlet is important.In chemical re-actors for example,the residence time drops with increasing velocities and as a result the conversion drops as well.This again shows the importance ofa correct calculation ofthe ow pattern,in particular for example the annulus thickness.
It should be remarked that this impact can only be quantiÿed correctly in a 3D perspective.
The solids segregation results from collisional stresses within the solid phase (Ha ,1983;Jenkins &Savage,1983;Gidaspow,1994).In the present paper,these stresses are calculated through the KTGF and a correct calculation ofthe granular temperature proÿle is required.The latter is not ob-vious,as the proÿle will depend on the imposed inlet values,the boundary conditions at the solid walls,and on certain model parameters.The inlet and boundary conditions are not easily described and should be further investigated.The most important model parameter is the restitution coe cient for particle–particle collisions (Pita &Sundaresan,1993;De Wilde 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 of inelasticity leads to strong dissipation ofthe solid phase turbulence and as a result the importance ofthe solid phase stress 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 the dissipation term ( )in Eq.(10)zero.5.2.Exit e ects
Remarkably,for the smaller outlet surface area,the solids segregation in the middle and bottom part ofthe riser is much more pronounced at the side ofthe riser opposite the outlet than at the side ofthe riser at which the outlet is positioned.As a result,in this case,the dilute core ofthe riser is not located centrally in the riser,but its location is shif ted towards the side ofthe outlet.At the side ofthe riser opposite the outlet,the exit conÿguration induces down ow ofboth the gas and the solid phase (Fig.5).The down ow is maximal near the solid wall right opposite the outlet.This is the immediate origin for the increased solid fraction there.Rhodes,Sollaart,and Wang (1998)also report that down ow is only measured in the annulus ofthe upper dilute region.No relation between the measured asymmetry and the outlet conÿguration is examined by these authors.
For the smaller outlet surface area,the outlet conÿguration is seen to determine the ow pattern in the whole riser,almost down to the riser bottom inlet.An increase ofthe riser outlet surface area drastically reduces the e ective depth of the riser exit conÿguration.For the broader outlet,up to 10m height,the dilute core ofthe riser is positioned centrally in the riser and no in uence ofthe riser exit conÿguration is seen.Higher on,its position shifts towards the side of the outlet,as expected.
The observed in uence ofthe outlet conÿguration on the ow pattern is certainly enhanced by the presence ofthe solid phase.To explain this,Fig.5shows projections ofthe solid phase velocity in Axial cross section I and in radial
882J.De Wilde et al./Chemical Engineering Science 58(2003)877–885
2
4
6
8
10
12
14
z (m
)
out cross section I -0.8
-0.6-0.4
-0.200.2
0.40.6
0.8y (m )
x (m)
z = 13.9 m
y (m )
x (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-04
2.90E-042.50E-042.10E-041.70E-041.30E-04
2
4
6
8
10
12
14
z (m
)
cross section I
y (m )
x
(m)
z = 13.9 m
y (m )
x (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
2
955.0m S out =2
91.1m S out =Fig.4.Solids volume fraction ( s )(conditions see Table 2).
Fig.5.Solids velocity ( v )(conditions see Table 2).
cross sections III positioned at 12.4and 13:7m height in
the riser (see Fig.2).Near the riser outlet,the ow de ects towards the outlet as a result ofthe pressure gradient.How-ever,as a result ofinertia and the dimensions ofthe riser top conÿguration,part ofthe de ected ow does not reach the outlet.It is re ected by the solid walls ofthe riser and a 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)877–885883
2
4
6
8
10
12
14
z (m
)
y (m )
x
(m)
z = 12.05 m
y (m )
x (m)
z = 10.4 m
cross section III:
1.2E+005.0E-024.2E-023.5E-023.0E-02
2.5E-022.1E-021.8E-021.5E-021.3E-021.1E-028.9E-037.5E-036.3E-035.3E-034.5E-03
3.8E-033.2E-032.7E-032.2E-031.9E-031.6E-031.3E-031.1E-039.4E-047.9E-046.7E-045.6E-04
4.7E-044.0E-043.3E-042.8E-042.4E-042.0E-041.7E-041.4E-041.2E-041.0E-04
2
4
6
8
10
12
14
z (m
)
-0.8
-0.6-0.4
-0.200.20.40.60.8
y (m )
x
(m)
z = 12.05 m
-0.8
-0.6
-0.4-0.200.2
0.40.60.8
y (m )
x (m)
z = 11.1 m
cross section III:
5.0E-023.7E-022.7E-022.0E-021.5E-021.1E-028.2E-03
6.1E-034.5E-033.3E-032.5E-031.8E-031.4E-031.0E-03
7.4E-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-05
8.2E-066.1E-064.5E-063.3E-062.5E-061.8E-061.4E-061.0E-06
1.1E+002955.0m S out =2
91.1m S out =Fig.6.Granular temperature ( )(m 2s −2)(conditions see Table 2).
resistance for the up owing gas–solid mixture.As the solid phase su ers more from inertia than the gas phase,the top section ofthe riser (above the outlet)is in general more dense.Also,a zone ofincreased solids volume f raction is seen right above the outlet and the external boundaries of the vortex are more dense than the eye ofthe vortex.The latter is a typical centrifugal e ect.
Remark that the vortexes formed have a 3D character,as clearly seen in the radial proÿles in Fig.5.
The calculated pressure drop over the riser is 249Pa for the broader outlet and 329Pa for the smaller outlet.This is due to the increased solids hold-up in case ofthe smaller outlet.
Gupta and Berruti (2000)investigated the exit e ects in circulating 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 the exit 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 of the riser (Grace,1997).This is clearly in agreement with the present calculations.It is also observed that,as a result of the re ection,an area ofdensiÿcation is created around the exit and,depending on the operating conditions and sever-ity ofthe exit restrictions,may extend a long way down the length ofthe riser.The calculations with the smaller outlet clearly illustrate this possibility.
Pugsley,Lapointe,Hirschberg,and Werther (1997)de-veloped a model that provides a reasonable estimate for the densiÿcation observed at the top ofan abrupt exit riser and that focuses on the distance the densiÿcation from the exit propagates down the riser.The present calculations reveal that the densiÿcation 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 violent turbulence (Fig.6)and signiÿcant solids backmixing in the riser 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 in the exit region is not a result ofthe turbulence,but ofthe convective and the acoustic terms.
Fig.6shows a picture ofthe granular temperature in the risers.Four phenomena are clearly visible:the slow dissi-pation ofthe 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 the turbulence quantities near the solid bounding walls,the in-creased value ofthe turbulence variables at the boundaries ofthe outlet,and the increased values ofthe turbulence vari-ables due to recirculation.With respect to the latter,it is
884J.De Wilde et al./Chemical Engineering Science58(2003)877–885
clear that the zone of increased turbulence follows the front where the up ow and the down ow encounter.Steep veloc-ity gradients in this zone are responsible for the increased turbulence.
The calculations have also shown that the gas phase and the solid phase turbulence are well correlated,except near the solid bounding walls.Near solid bounding walls, the gas phase turbulence(not shown)—calculated via wall functions—is higher than the solid phase turbulence.Due to inertia,the solid phase is not able to follow the uctuat-ing motion ofthe gas phase.Just the opposite is seen at the bottom inlet,where the solid phase is imposed to enter with much more turbulence than the gas phase.The gas phase immediately responds to the turbulent motion ofthe solid phase.Therefore,it could be stated that in the bulk ow,it is the solid phase that determines the overall turbulence in a gas–solid mixture.This again stresses the importance ofa correct calculation ofthe solid phase turbulence and ofthe use ofcorrect wall f unctions.
6.Conclusions
A3D simulation ofgas–solid ow using the Eulerian–Eulerian approach with the kinetic theory ofgranular ow is able to give insight in the complex hydrodynamics ofcir-culating uidized beds.This is illustrated by calculating the e ect ofan abrupt T-outlet on the ow pattern.Such an out-let conÿguration leads to vortex formation in the top,which in turn induces recirculation at the wall opposite the outlet. Ifthe outlet surf ace area is decreased,the outlet becomes more restrictive and the recirculation is seen to increase,re-sulting in down ow down to the bottom ofthe riser.
Acknowledgements
This work wasÿnanced by the European Commission within the frame of the Non-Nuclear Energy Program,Joule III project under contract JOF3-CT95-0012and Thermie project under contract no.SF243/98DK/BE/UK(project partners: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)forÿnancial support of the CFD Research.Geral-dine Heynderickx and Guy Marin are grateful to the“Bij-zonder Onderzoeksfonds”BOF-RUG forÿnancial support ofthe gas–solid two-phase ow research.
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