CFD simulation of dilute phase gas–solid riser reactors: Part I—a new solution method and flow...

Chemical Engineering Science 59 (2004) 167–186www.elsevier.com/locate/ces

CFD simulation of dilute phase gas–solid riser reactors: Part I—a newsolution method and +ow model validation

A.K. Dasa, J. De Wildea, G.J. Heynderickxa, G.B. Marina ;∗, J. Vierendeelsb, E. Dickb

aLaboratorium voor Petrochemische Techniek, Ghent University, Krijgslaan 281, S5, 9000 Gent, BelgiumbDepartment of Flow, Heat and Combustion Mechanics, Ghent University, St. Pietersnieuwstraat 41, 9000 Gent, Belgium

Abstract

A three-dimensional simulation of a dilute phase riser reactor (solid mass +ux: 2 kg m−2 s−1) is performed using a novel density basedsolution algorithm. The model equations consisting of continuity, momentum, energy and species balances for both phases, are formulatedfollowing the Eulerian–Eulerian approach. The kinetic theory of granular +ow is applied. The gas phase turbulence is accounted forvia a k–� model. An extra transport equation describes the correlation between the gas and solid phase +uctuating motion. The solutionalgorithm allows a simultaneous integration of all the model equations in contrast to the sequential multi-loop solution in the conventionalpressure based algorithms, used so far in riser simulations. The simulations show an unsteady behaviour of the +ow, but a core-annulus+ow pattern emerges on a time-averaged basis. The abrupt nature of the T type outlets causes a signi=cant recirculation of gas and solidfrom the top of the riser. The +ow near the outlets is highly non-symmetric and has a three-dimensional character. A signi=cant decreaseof the gas phase turbulence and particle granular temperature across the riser length is attributed to the presence of small particles, whichis qualitatively consistent with the experimental data from literature.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Multiphase +ow; Multiphase reactors; Fluidization; Hydrodynamics; Simulation; Turbulence

1. Introduction

Gas–solid riser reactors are used in many important pro-cesses e.g. +uid catalytic cracking (FCC), +uidised com-bustion of coal, adsorption of SO2–NOx from +ue gases,etc. The main interest for studying the detailed hydrody-namics of gas–solid +ow in risers is the accurate predictionof their performance. A riser usually operates in the tur-bulent regime, with +uctuations in velocity, pressure andconcentration =elds at the corresponding length and timescales. The presence of the solid phase also results in +uc-tuations at other length and time scales. For example, themacro-scale +ow non-uniformities occur at a scale com-parable to the riser diameter/height. Core-annulus +ow(Bader et al., 1988) and density inversion waves (Dry andChristensen, 1988) are examples of such macroscale phe-nomena. Such complex +ow patterns are essentially 3D and

∗ Corresponding author. Tel.: +32-9-264-45-16; fax: +32-9-264-49-99.E-mail addresses: guy.marin@ugent.be, guy.marin@rug.ac.be

(G.B. Marin).

have a signi=cant eFect on the +ow and reaction vari-ables. The complete 3D simulation of riser reactors is thusdesirable.

In this work, a three-dimensional simulation is performedfor a riser used in the simultaneous adsorption of SO2–NOx

from +ue gases over a Na–�-Al2O3 sorbent. Two main fea-tures of this riser are the very low solid +ux, 2 kg m−2 s−1,and a particle diameter of 60 �m. A gas–solid +uctuatingmotion correlation model (Simonin, 1996) and the inter-phase transfer of the +uctuating kinetic energy between thetwo phases, are included in the simulation to account for theattenuation of the gas phase turbulence by such small parti-cles. The simulation is based on a new solution algorithm,discussed along with its convergence behaviour and thesimulation results. It is the aim of this work to demonstratethe applicability of the new density based solution methods(De Wilde et al., 2002) for both the transient and steadysimulation of gas–solid +ow described with the Eulerian–Eulerian approach and the kinetic theory of granular+ow.

The grid sizes being used in this work are certainly not=ne enough to resolve all the +uctuations that occur in ariser, in particular the +uctuations due to meso-scale clusters

168 A.K. Das et al. / Chemical Engineering Science 59 (2004) 167–186

(Agrawal et al., 2001). However, calculations are limited todilute risers (�s ¡ 0:01) in which the eFects of meso-scaleclusters are probably less important.

2. Flow model equations

2.1. Transport equations

Following the Eulerian–Eulerian approach, both gas andsolid are treated as continuous phases. The average numberof particles with a diameter of 60 �m in a typical controlvolume of 1 cm3 for a solid fraction of �s = 0:0006 as inthis work, is about 6× 103. This is a large enough numberto consider the solid phase as continuous.

Table 1 summarizes the transport equations adopted inthis work.

2.1.1. Gas phase equationsThe time smoothed conservation equations for the tur-

bulent gas +ow Eqs. (T1.1)–(T1.4) are obtained from theinstantaneous equations after Reynolds averaging. The in-stantaneous gas velocity u and any scalar variable ’ areexpressed as

u = Ku + u′; ’ = K’ + ’′; (1)

where Ku and K’ are the mean values, resulting from aver-aging over the turbulent time-scale and u′ and ’′ are theinstantaneous deviations from the mean values.

Furthermore, it is assumed that the turbulent +uctuationof the void fraction �g and the gas phase density �g arenegligible as invoked in the Favre averaging (Favre, 1969).

The Reynolds stresses appearing in the equations afterReynolds averaging due to the turbulent +uctuations aremodeled by the relationship proposed by Boussinesq (1877).The turbulent kinetic energy k is de=ned as

(u′21 + u′22 + u′23 ) =12

(u′ · u′) (2)

and the normal Reynolds stress is assumed isotropic.The last term in the RHS of Eqs. (T1.1) and (T1.2) and

the two last terms of Eq. (T1.4) represent the net changeof mass, momentum and energy in the gas phase due to theinterphase transport.

In the energy equation Eq. (T1.4), the total energy, Eg, isgiven by

Eg = eg + qg + k: (3)

The internal energy for an ideal gas is calculated from

eg = cv KT = (cp − R) KT =1

�− 1

KP�g

where � = cp=cv: (5)

In the RHS of Eq. (T1.4), the fourth term and the sixthterm correspond to the interphase energy transfer by dragfor the mean and the +uctuating parts respectively. For theenergy transport by drag, it is assumed that the interfacialsurface between the two phases has a velocity equal to themean velocity of the two phases (De Wilde et al., 2002).Thus, for the gas phase, the interphase transport of the meankinetic energy by drag is

− �( Ku − v) ·(

Ku + v2

)=−�

2( Ku · Ku − v · v): (6)

The last two terms in the RHS of Eq. (T1.4) model theinterphase transport of kinetic/+uctuating kinetic energy andof enthalpy, caused by the interphase mass transfer. Themass transfer results from diFusion over a =lm around theparticle. To de=ne the momentum and energy transport dueto mass transfer, following transport condition is assumed(De Wilde, 2000): a molecule can only diFuse between thegas and the solid phase when its velocity equals that of thesolid phase. Additional momentum and energy transport isentirely attributed to the drag.

The =fth term of the RHS of Eq. (T1.4) corresponds tothe work performed by the gravitational force.

The gas phase turbulence is modelled using the k–� model,modi=ed to account for the presence of the solid phase andthe interphase mass and turbulence transport Eqs. (T1.5)–(T1.6) (De Wilde et al., 2002). The reason to choose thismodel is its wide range applicability using only two diFer-ential equations. For details, reference is made to the workof Bolio et al. (1995). The one but last term in the RHS ofEq. (T1.5) is the net dissipation of the +uctuating energydue to the drag between the +uctuating velocities of eachphase. It is obtained by multiplying the drag force betweenthe +uctuating velocities of the two phases by the +uctuat-ing velocity of the gas phase i.e.,

− �u′ · (u′ − C) =−�(u′ · u′ − u′ · C)

=−�(2k − q12); (7)

where C is the random +uctuation velocity of the particles.By de=nition, the gas solid +uctuating motion correlation

q12 = 〈u′ · C〉 (8)

and calculated using a semi-empirical equation discussedlater.

2.1.2. Solid phase equationsUnlike the gas phase, the solid phase consists of dis-

crete particles. The solid phase properties can be trans-ferred via three mechanisms (i) binary collision of par-ticles, (ii) free movement of particles between collisions,(iii) +uctuations arising from collective behaviour of parti-cles e.g. clusters, strands, etc. For the last mechanism, the+uctuations in the locally averaged properties of the solidphase are to be considered, suitable theories for which are

A.K. Das et al. / Chemical Engineering Science 59 (2004) 167–186 169

Table 1Flow model: conservation equations

Gas phaseContinuity equation@@t

(�g�g) +@@x

· (�g�g Ku) =∑jMjkgj av(Y

sj − Yj) (T1.1)

Momentum equations (x; y; z)

(�g�g Ku) +@@x

· (�g�g Ku Ku) = − @(

KP + 23 �gk

@x− @

@x· (�gsg) + �g�gg− �( Ku − v) +

∑jMjkgj av(Y

sj − Yj)v (T1.2)

sg = smg + stij = −[(

#g − 23$g

) (@ · Ku@x

)+ ($g + $tg)

{(@ Ku@x

)+(@ Ku@x

(T1.3)

Energy equation

(�g�gEg) +@@x

· (�g�gEg Ku) =@@x

·(�g(% + %t)

@ KT@x

)− @

@x·((

KP +23�gk)

− @@x

· (�gsg · Ku) − �2

( Ku · Ku − v · v) + �g�gg · Ku − �(k − 3

+∑jMjkgj av(Y

sj − Yj)

(12v · v +

+∑jkgj av(Y

sj − Yj)hsmjs

(T1.4)

Turbulent kinetic energy equation

(�g�gk) +@@x

· (�g�g Kuk) =@@x

·(�g

$g + $tg(k

[�g$tg

[(@ Ku@x

)+(@ Ku@x

)T]]:(@ Ku@x

− �g�g�− �(2k − q12) +∑j

32Mjkgj av(Y

sj − Yj)& (T1.5)

Turbulence dissipation equation

(�g�g�) +@@x

· (�g�g Ku�) =@@x

·(�g

$g + $tg(�

@�@x

)+ C1�

[�g$tg

− C2��g�g�2

k− �(2k − q12)C4�

(T1.6)

Solid phaseContinuity equation for solid phase@(�s�sp)

@(�s�spv)@x

= −∑jMjkgj av(Y

sj − Yj) (T1.7)

Continuity equation for sorbent/catalyst only@(�s�s)

@(�s�sv)@x

= 0 (T1.8)

Momentum equations (x; y; z)

@(�s�spv)@t

+@(�s�spvv)

@x= − @Ps

@x− @(�sss)

@x+ �s�spg + �( Ku − v) −∑

jMjkgj av(Y

sj − Yj)v (T1.9)

where ss = −[(

#s − 23$s

) (@@x

((@v@x

)+(@v@x

(T1.10)

Granular temperature equation

(�s�sp&) + ∇ · (�s�sp&v)]

= −(PsI + �sss) : ∇v + ∇ ·(�s+s

)+ �q12 − 3�&− �−∑

32Mjkgj av(Y

sj − Yj)& (T1.11)

Gas-solid turbulence correlation

(�s�spq12) +@@x

(�s�spq12v) =@@x

·(�s�sp

[�s�sp,12

@@xv)T]]

[�s�sp,12

[(@@xv)

Ku)T]]

− �s3

[�spq12I + �sp,12

· Ku)]

− �s3

[�spq12I + �sp,12

· Ku)

· v)]

· Ku)

− �s�sp�12 − �(q12 +

�s�sp�g�g

q12 − 2k − �s�sp�g�g

(T1.12)

yet to be developed. Fortunately, for moderate solid +uxes(upto 20 kg m−2 s−1), the =rst two mechanisms are dom-inating. Hence, the solid phase conservation equations arederived from the kinetic theory of granular +ow (KTGF)(Sinclair and Jackson, 1989; Gidaspow, 1994) followingthe Chapman–Enskog approach for dense gases (Chapmanand Cowling, 1970). Table 1 (Eqs. (T1.7)–(T1.11)) showsthe solid phase mass, momentum and granular temperaturetransport equations.

Note that the drag term in Eq. (T1.9) is approximated bya time smoothed value of Ku, assuming the single turbulentcorrelation of u′ with the actual particle velocity c i.e., 〈u′〉,known as the drifting velocity, is negligible for a coarse gridsimulation used in the present work. The drifting velocitytakes into account the dispersion eFect on the drag due tothe particle transport by the +uid turbulence and reduces, atleast for small size particles, to single turbulent correlation〈u′〉 between the concentration +uctuation of the solid phaseand turbulent velocity of the gas phase (Simonin, 1996).

In the granular temperature transport equation,Eq. (T1.11), the last term in the RHS is related to inter-phase mass transport. The third and fourth term in the RHSare the production due to the gas–solid +uctuating motioncorrelation and the dissipation due to viscous dampingrespectively (Gidaspow, 1994).

The gas–solid +uctuating motion correlation q12 is calcu-lated using the semi-empirical partial diFerential equationEq. (T1.12) following Simonin (1996), which is based onthe concept of the eddy viscosity. The =rst term in the RHSis for the diFusive transport, the next four terms are produc-tion of q12 by shear, expressed as a function of the meanvelocity gradients, in line to the eddy viscosity concept ofthe +uid +ow (Simonin, 1996). The sixth term in RHS ac-counts for the dissipation of q12 due to the viscous actionof the +uid phase. The dissipation �12 is de=ned in the con-stitutive equation Eq. (T2.13) in Table 2. The last term inEq. (T1.12) represents the eFect of the turbulence transferbetween the two phases and is non-conservative, meaningthat the turbulence loss of one phase does not imply a cor-responding turbulence gain of the other phase.

The solid phase temperature is assumed equal to that ofthe gas phase. Hence, the energy equation for the solid phaseis not required.

The solid phase equations used are not Reynolds av-eraged. Closure relations for the Reynolds averaged solidphase equations are still to be developed. As a result, ex-tremely =ne meshes have to be used for the integrationof the solid phase equations (Agrawal et al., 2001). Thisresults in unrealistic calculation times. In case a coarsemesh is used, Reynolds averaged equations should be usedand a turbulence model for the solid phase is to be in-cluded to account for the eFects of the meso-scale +uctu-ations on the Reynolds-averaged properties. For lack of areliable solid phase turbulence model, calculations in thiswork do not take into account the eFects of meso-scale+uctuations.

Table 2Flow model: constitutive equations

Dissipation of granular temperature by inelastic particle particlecollision

� = −ℵc

(12mC2

)= 3(e2 − 1)�2

s �sg&

−∇ · v]

(T2.1)

Radial distribution function

g = 1 + 4�s

[1 + 2:5�s + 4:5904�2

s + 4:515439�3s

[1 − ( �s�s;max

)3]0:67802

](T2.2)

Solid phase pressure

Ps = [1 + 2(1 + e)�sg]�s�s& (T2.3)

Bulk viscosity solid phase

#s =43�s�sdpg(1 + e)

(T2.4)

Shear viscosity solid phase

$s = $cs +$ks(

1 + 85

(1+e)2 �sg

) (1 + 8

5 �sg)

�sg(T2.5)

$cs =45�s�sdpg(1 + e)

(T2.6)

$ks = 1:016516

(T2.7)

Thermal conductivity solid phase

+s = +cs ++ks(

1 + 125

(1+e)2 �sg

) (1 + 12

5 �sg)

�sg(T2.8)

+cs = 2�s�sdpg(1 + e)

(T2.9)

+ks = 1:025137564

(T2.10)

Turbulent viscosity of gas phase

$tg = C$�gk2

�(T2.11)

Kinematic viscosity for turbulence correlation and dissipation

,12 =C$

2k�q12

[1 + C�

(v − Ku)2

)]−1=2

(T2.12)

�12 =q2

3,12(T2.13)

Interphase momentum transfer coe;cient

if �g ¡ 0:80; � = 150�2s

$g(dp/s)2

+ 1:75�s�gdp/s

| Ku − v| (T2.14)

if �g ¿ 0:80; � =34Cd

�s�gdp/s

�g| Ku − v|�−2:65g (T2.15)

Rep =�g�gdp| Ku − v|

$g(T2.16)

Cd = 0:44 for Rep ¿ 1000

=24Rep

(1 + 0:15Re0:687p ) for Rep ¡ 1000 (T2.17)

In the model presented, the restitution coeOcient e forparticle–particle collisions is a constant. For (nearly) elasticparticle–particle collisions, the particle velocity distribution

is close to isotropic Maxwellian. Brilliantov and PPoschel(2003) derived an impact velocity dependent coeOcientof restitution for visco-elastic particles. Such an approachcould be extended to particles in general. The solid phaseequations used in the present work are subject to somefurther limitations. Particles are assumed spherical. Rota-tional momentum of the particles is not accounted for. Onlyfrictionless collisions between particles are considered.Wylie and Koch (2000) account for rotation of the parti-cles. Jenkins and Zhang (2002) propose a KTGF modelthat accounts for rotational momentum and for frictionalcollisions.

2.2. Constitutive equations

The constitutive equations for the calculation of the gasand solid phase properties are summarized in Table 2. Theequations for the solid phase properties are adopted fromNieuwland et al. (1996) and include both the kinetic andcollisional eFects. Note that for very dilute phase +ow, thekinetic eFects dominate over the collisional eFects. Theturbulent viscosity of the gas phase is obtained from thePrandtl–Kolmogrov equation Eq. (T2.11). The kinematicviscosity for the turbulence correlation ,12 in Eq. (T2.12)and the corresponding dissipation �12 in Eq. (T2.13) aretaken from Simonin (1996).

The model used in the present work (Table 1) accountsfor interactions between the mean motion of the gas andsolid phase via a well-established drag source term only. Theinterphase momentum transfer coeOcient � (Eqs. (T2.14)–(T2.17)) is calculated via the expressions of Wen andYu (1966) (�s ¡ 0:2) and Ergun (1952) (�s ¿ 0:2). Gas–solid interactions are however complex and their modelinghas recently further developed. Hill et al. (2001a,b) andWylie et al. (2003) investigated the role of inertia of the+uid phase making use of lattice-Boltzmann simulations(Ladd, 1994a,b). Hill et al. (2001a,b) examined the tem-poral evolution of the drag force in an accelerating +ow.

Table 3Reaction model: component conservation equations

Component continuity equation in gas phase

@(�gCj)@t

· (�gCj Ku) = kgj (am�s�sp)(RTP

j − Cj) +@@x

[(Dm +

)�g@ KCj

](T3.1)

Component continuity equation for gas in solid phase

@(�s�sgCsj )

· (�s�sgCsj v) = kgj (am�s�sp)

)(Cj − Cs

j ) + Rj�s�sCt +@@x

[(�s�sg

) @Csj

](T3.2)

Surface species continuity equation in solid phase

@(�s�sp&l)@t

· (�s�sp&lv) =@@x

[(�s�sp

)@&l@x

]+ Rl�s�s (T3.3)

Wylie et al. (2003) calculate the viscous dissipation in theinterstitial +uid due to inertia of the +uid. Furthermore, theReynolds stress induced in the interstitial +uid by the randommotion of the particles is determined. Wylie et al. (2003)found that the drag associated with the mean relative motionof the two phases is coupled with the viscous dissipationcaused by the +uctuating motion of the particles. Agrawalet al. (2001) and Zhang and VanderHeyden (2002) show animportant eFect of solid phase meso-scale +uctuations onthe drag. In the present work, the eFects on gas–solid inter-actions of inertia of the +uid phase and meso-scale +uctua-tions of the solid phase are not taken into account.

3. Reaction model equations

The reaction model equations are summarized in Table 3.Following the Eulerian–Eulerian approach, the reaction

model consists of the mass conservation equation for thegaseous components in the bulk gas and inside the solidphase as well as for the surface species in the solid. Sincethe particle size of the solid sorbent/catalyst used in thiswork is very small (¡ 100 �m), the internal pore diFu-sion resistance can be neglected. However, the external =lmresistance may be limiting in very dilute phase +ow, re-quiring a so-called heterogeneous reactor model (Fromentand BischoF, 1990). Hence, it is necessary to formulatethe continuity equation for the component in the bulk gasphase (Cj) (Eq. (T3.1)) as well as inside the solid phase(Cs

j ) (Eq. (T3.2)).

3.1. Component continuity equation in the gas phase

The gas phase concentrations are written as

Cj = Yj�gMg

): (9)

This allows to clearly distinguish +ow and reaction duringthe calculations.

For turbulent +ows, the gas phase component continuityequations are Reynolds averaged. Furthermore, +uctuationsin �g and �g are neglected, as in the Favre averaging appliedearlier. This results in Eq. (T3.1).

Note that there is no source term due to reaction inEq. (T3.1), since all the reactions occur inside the solidphase. The Reynolds +ux due to turbulent mass transfer isexpressed in terms of the mean gradient. Hence

− u′C′j =

2tg(c· �g @Cj

@x; (10)

where 2tg = $tg=�g is the turbulent kinematic viscosity of the

gas phase, calculated via Eq. (T2.11), and (c is a turbulencemodel parameter which relates the turbulent diFusivity andthe turbulent kinematic viscosity of the gas phase.

The second term of the RHS in Eq. (T3.1) clearly distin-guishes the molecular and turbulent diFusion. Whereas theformer is due to the micromixing, the latter results from themesomixing due to the turbulent eddies.

3.2. Component continuity equation in the solid phase

The continuity equation of component j inside thesolid phase Eq. (T3.2) follows from the KTGF. Note thatEq. (T3.2) indeed accounts for the diFusional transport ofmass due to particle +uctuations via the last term in the RHS,expressed in analogy to the gas phase equation Eq. (10).In Eq. (T3.2), the granular diFusivity 2s=(s is introduced,corresponding to a similar term called granular conduc-tivity, used by many authors to express the diFusionaltransport of heat +ux arising from the particle level +uctu-ations (Gidaspow, 1994). The granular diFusivity is equiv-alent to the molecular diFusivity Dm for gas phase and isde=ned as2s(s

(s�sp; (11)

where $s is the granular viscosity given by Eq. (T2.5). In ab-sence of any suitable expression for the parameter (s whichrelates the granular diFusivity and the granular viscosity, avalue of (s = 1 is used, in line with Gidaspow (1994). Thegranular diFusivity has a very small value due to the largevalue of the solid phase density in Eq. (11).

The =rst term in the RHS of Eq. (T3.2) follows from asimilar term in Eq. (T3.1) and represents the mass transferbetween gas and solid phases. The second term in the RHSis due to the chemical reactions taking place inside the solidphase, where Rj is the net production rate of a componentj, calculated from

Rj =Np∑p=1

�j;prp; (12)

where �j;p is the stoichiometric coeOcient of the compo-nent j in the reaction step p; Np is total number of reactionsteps. The individual production rates rp in Eq. (12) for dif-ferent reactions are expressed in terms of concentrations and

coverages. The use of these equations requires the knowl-edge of the kinetic model, as explained in part II.

3.3. Surface species continuity equation in the solid phase

Following the component continuity equation Eq. (T3.2),the instantaneous locally averaged conservation equationsfor the solid phase surface species can be written as shownin Eq. (T3.3). Note that the diFusional transport of solidphase species is accounted for by the =rst term of RHS inEq. (T3.3).

4. Initial and boundary conditions

Appropriate boundary conditions have to be imposed atthe inlet(s), outlet(s) and solid walls. In accordance with aneigenvalue analysis (Tsuo and Gidaspow, 1990), all vari-ables except the gas phase pressure are imposed at the in-let(s) whereas the gas phase pressure is imposed at theoutlet(s).

Regarding the inlet conditions, the solid phase veloc-ity is assumed radially uniform whereas a trapezoidal ra-dial pro=le, corresponding with fully developed turbulentgas +ow, is imposed for the gas phase axial velocity, i.e.u(r) = 1:2245uavg

g (1− r=R)1=7. A turbulence intensity of 5%is imposed for the gas phase:

k inlet(x; y) = (Tuinlet)2 ((uinletx )2 + (uinlet

+ (uinletz )2) =2:0 with Tuinlet = 0:05: (13)

The inlet gas phase turbulence dissipation rate is calculatedfollowing:

�inlet(x; y) = 0:1k inlet(x; y)

×∣∣∣∣∣∣uinlet

[1−(√

x2 + y2

)](−6=7)/(−7R)

∣∣∣∣∣∣: (14)

The particle level +uctuation intensity is taken 50% basedon the inlet solids velocity (Neri and Gidaspow, 2000). Thegas–solid +uctuating motion correlation, q12, is assumednegligible at the inlet.

The no-slip condition is used for the gas phase at solidboundary walls. Near solid walls, the gas phase wall shearstress and the turbulence properties are calculated from wallfunctions. The latter do not account for the presence of par-ticles yet. More details can be found in De Wilde et al.(2002).

The solid phase speci=c wall shear stress (w, the produc-tion of granular temperature due to wall shear and the dis-sipation of granular temperature at the wall due to inelasticparticle–wall collisions are calculated in the present workfrom the relationships proposed by Sinclair and Jackson(1989). These relationships assume some particles sliding

relative to the solid wall and relate the tangential and normalstresses by a so-called Coulomb yield condition. Further-more, spin of the particles at the boundary is not accountedfor. Jenkins (1992) further developed the solid phase bound-ary conditions. An evaluation of the work of Jenkins (1992)by Louge (1994) shows that the behaviour at low friction iswell described, whereas at large friction the boundary con-ditions fail. Louge (1994) suggests that the boundary con-ditions may be improved by accounting for rotations of theparticles at the solid wall. At large friction, the particles mayroll at the solid wall without slipping. Jenkins and Louge(1997) have developed boundary conditions making a dis-tinction between particles that slide upon contact with thewall and particles that stick. Approximate expressions forthe boundary conditions corresponding to intermediatesituations are given. These authors also account for thecorrelation between two orthogonal components of the +uc-tuation velocity at the points of contact of the particles withthe wall.

The gas solid turbulence correlation q12 in the vicinityof the wall is calculated following an algebraic equationproposed by Louge et al. (1991). This relation is applicablefor +ow with a solid fraction less than 0.005.

The solid wall is assumed to be a perfect insulator.The initial =elds in the entire riser are obtained by assum-

ing a uniform solid fraction in all the control volumes andare calculated from the given solid mass +ow rate with ano-slip condition between gas and solid. The pressure =eldsare initiated from the given outlet pressure accounting onlyfor the hydrostatic pressure drop. The velocity =elds are ini-tially assumed identical to those at the inlet. Note that forsteady state calculations, the initial conditions do not haveany eFect on the =nal solution, which only depends on theboundary conditions.

Regarding the initial and boundary conditions for the re-action variables, reference is made to part II on the SO2–NOx process, since it involves a substantial discussion onthe process schematic and reaction steps.

5. Numerical solution methods

Two widely used families of algorithms are available forthe solution of PDEs: (i) pressure based and (ii) densitybased methods. Salient features of these methods are high-lighted below. So far the use of density based methods wasmostly restricted to single phase +ow simulations. In thiswork, we apply the novel density based solution method fortwo phase +ow. To our knowledge, most of the simulationsof gas solid risers so far have used the pressure based solu-tion methods (Neri and Gidaspow, 2000; Nieuwland et al.,1996; Samuelsberg and Hjertager, 1996; Gao et al., 1999).The computational load in the latter method is so highthat most of the two phase +ow simulations are only twodimensional.

5.1. Pressure based methods

In the pressure based methods, pressure and velocity aresolved iteratively for each time step in a sequential manner.The following basic steps are performed, although somevariations are possible:

(i) both the pressure and velocity =elds are guessed;(ii) the correction of the velocity =eld is obtained using the

guessed values in the momentum equation;(iii) the continuity equation is used to obtain the correction

for the pressure =eld;(iv) the corrections for other scalar quantities such as con-

centration, temperature etc. are obtained by solving thespecies balance and energy conservation equations;

(v) the above steps (ii)–(iv) are repeated for each timestep until the corrections are within acceptable limits.

These corrections are iteratively brought down to zero, i.e. toconvergence. The SIMPLE method of Patankar and Spald-ing and many of its variations are examples of such a scheme(Versteeg and Malalasekera, 1995). The discretisation of thepressure =eld is the main diOculty of these methods andtherefore heavy under-relaxation to the pressure correctionis often applied. In addition, often a staggered grid approachis used.

Focussing on each of the above-mentioned steps to calcu-late the corrections, the solution of the algebraic equationsresulting from the discretisation involves several iterationloops. For example, in steady state problems, the calcula-tion of the corrections for u and P involves the simulta-neous solution of the momentum and continuity equationsover the entire space domain (Versteeg and Malalasekera,1995). Such a large number of equations are normally solvedusing an iterative method such as Gauss–Seidel or Jacobior line methods based on the Thomas algorithm. Diagonaldominance is essential for assuring convergence of thelatter methods. Therefore the pressure based algorithmshave to iterate not only in an outer loop for the correction ofthe pressure and velocity =elds, but also in an inner loop forthe iterative solution of the momentum equations as well.Such multi-loop iterations result in a signi=cant computa-tional load.

5.2. Density based methods

In the density based methods, all the +ow equations i.e.,the continuity, momentum and energy equations of both thephases, are solved simultaneously. It is also possible to solveboth +ow and reaction equations simultaneously. However,when the coupling of +ow and reaction variables is not verystrong, it is possible to solve the +ow and reaction equationsseparately.

The advantage of a separate solution of the +ow and reac-tion part is that +ow and reaction equations suFer from stiF-ness for diFerent reasons. The integration scheme can thus

be adapted to the speci=c demands of the +ow and reactionpart.

The density based methods employ a time stepping proce-dure, both implicit (Pulliam and Steger, 1985) and explicit(Jameson et al., 1981) to solve the conservation equations.Pseudotime is introduced as a numerical stepper to iter-ate towards the solution. If an explicit or semi-implicit ap-proach is used, the RHS of the discretised equations can beevaluated with values from the previous time step. Hence,a direct solution method such as Gauss elimination can beapplied pointwise for each node, without the requirementto solve for the entire domain. The density based methodshave the following major bene=ts over the pressure basedformulations:

(i) additional convergence of pressure and velocity cor-rection (called outer loop previously) in each timestep is not necessary, since all the equations are solvedsimultaneously. This implies a signi=cant gain in thecomputational speed especially for time dependentcalculations;

(ii) all the variables can be stored in the center of the controlvolumes using a conventional non-staggered grid lead-ing to a considerable ease in programming and mem-ory requirement. This, however, is not a fundamental

diFerence between the pressure and density based meth-ods, although the density based methods generally usethe non-staggered grid approach.

On the other hand, the density based methods have to over-come the following major diOculties, specially for incom-pressible or low Mach number +ow applications, such astwo phase +ow in a riser:

(i) The pressure velocity coupling is insuOcient. As ex-plained earlier, this problem is better handled by thepressure based methods since the correction in eachtime step is interlinked by the additional convergenceloop for u and P.

(ii) Use of a non-staggered grid and the advection up-stream splitting method (Liou and Edwards, 1999;De Wilde et al., 2002), results in a pressure discreti-sation amounting to a central diFerencing method forlow-Mach +ows. The latter is known to be suFeringfrom insuOcient transportiveness.

(iii) As noted earlier, low-Mach +ows suFer from stiFnessdue to the large diFerence in the eigenvalues sinceCg � u (Weiss and Smith, 1995). As a result, theallowable size of the time step is severely restricted,leading to very slow convergence or even breakdownof convergence.

Due to these challenges it is not surprising that the den-sity based methods are more suitable for high Mach numberapplications. In fact, these methods were originally devel-oped for transsonic (Beam and Warming, 1976) compress-ible +ows and later on extended to low Reynolds numberand incompressible +ows (Kwak et al., 1985). The exten-sion of this method to low Mach number +ows is achievedby preconditioning techniques (Weiss and Smith, 1995).

6. The density based solution method

The set of partial diFerential equations constituting themodel (Tables 1 and 3) is integrated using a cell centered=nite volume technique. The complete set of equations forboth +ow and reaction models is written in matrix formula-tion as

+@@x· �= K; (15)

where Q is the vector of conservative variables and � isthe +ux vector constituted of convective, acoustic and vis-cous +uxes. K represents the source terms due to interphasetransport, gravity, reaction, etc.

For the +ow model, the set of conservative variables Q isgiven as

�s�sp �s�s �g�g �s�spvx �s�spvy �s�spvz �g�gux �g�guy ←-

�→ �g�guz �g�gEg32�s�sp& �g�gk �g�g� �s�spq12

: (16)

The set of primary variables is given below

W T = (�g �sp P vx vy vz ux uy uz T & k � q12): (17)

For an arbitrary =nite control volume V , Eq. (15) can beformally integrated as

∫ ∫ ∫Q dV +

∫ ∫ ∫∇ · � dV

=∫ ∫ ∫

K dV = KV: (18)

Applying the divergence theorem, Eq. (18) can beexpressed as

∫ ∫ ∫Q dV +

∫� · n ds = KV: (19)

n being the normal unit vector for each interface, havingsurface area S, of the control volume. The surface integralis discretised over all the interfaces of the =nite volumeresulting in an algebraic +ux summation.

6.1. Preconditioning

To reduce the stiFness of the equation set at low Machnumber, preconditioning of the eigenvalues is essen-tial (Weiss and Smith, 1995). Therefore, the pseudotime

derivative is premultiplied with a preconditioning matrix >:

∫ ∫ ∫Q dV +

∫ ∫ ∫Q dV

∫� · n dS = KV: (20)

> is formulated such that the acoustic speed is replaced by asuitably de=ned reference speed. The latter is usually scaledaccording to the convective speed. Because in the model,the gas phase pressure gradient is distributed over the gasphase only (Eqs. (T1.2) and (T1.9)), preconditioning is onlyaFecting directly the gas phase equations and not the solidphase equations. For more details of the formulation of thepreconditioning matrix for gas–solid +ow, reference is madeto De Wilde et al. (2002).

De=ning Q as the cell averaged value

∫ ∫ ∫Q dV; (21)

assuming the +uxes as constant over each cell interface, andperforming the surface integration piecewise face by face,Eq. (20) is transformed into

∑faces

� · nS = K (22)

for each =nite volume cell.Remark that the pseudotime derivative is pre-multiplied

with the preconditioning matrix. Thus, the time-dependentbehaviour of the equations is not altered. Omitting the tildesin Eq. (22) the discretised equation transforms to the fol-lowing diFerential equation:

∑faces

� · nS = K: (23)

Upon convergence, the pseudotime derivative termapproaches zero and the original conservation equationEq. (19) is revived.

6.2. Discretisation of the time derivative

First order discretisation of the time and pseudotimederivatives transforms Eq. (23) to the following form to besolved for UQ = �Q(m+1)

i − Q(m)i �:(

> + I)(

Q(m+1)i − Q(m)

∑faces

� · nS

i − Q(m)i

)+ Ut K; (24)

where m indicates the iteration level of the pseudotime loopand Qt

i the conservative variables for node i at time level t.The size of the physical time step Ut is based on the

desired temporal accuracy for transient calculations. For agas–solid riser, since the +ow is inherently unsteady, it was

found that a small real time step of say 0:01 s is necessaryto resolve the +uctuations in the +ow. On the other hand,the pseudotime step is limited by numerical stability criteria,discussed later.

For steady state calculations, a large value of Ut (=5000 s)is used in Eq. (24). Eq. (24) is then transformed to a singletime stepping form involving only the pseudotime.

6.3. Flux discretisation and treatment of the source terms

The advection upstream splitting method (AUSM) devel-oped by Liou and Edwards (1999) for single phase +ows wasextended by De Wilde et al. (2002) to gas–solid +ows. Inthis scheme, the convective and acoustic +uxes are handledseparately, recognising the diFerences in their characteristicspeed. A major diOculty of the AUSM scheme is that thesplitting of the acoustic +uxes becomes central at low Machnumbers. Hence, to avoid the formation of wiggles and nu-merical instability, arti=cial dissipation is to be introducedin the scheme. In the present work a =rst order dissipationscheme, as described by De Wilde et al. (2002), is adopted.

After spatial discretization, Eq. (24) can be written in thefollowing form for each node i:(

> + I + Ut

{A(m)i −

(@K−

)(m)})(

Q(m+1)i − Q(m)

i − Q(m)i

)− A(m)

i UtQ(m)i − A(m)

i′ UtQ(m)i′

+ Ut(C(m)A + D(m)

B + (K+i )(m) + (K−

i )(m)): (25)

As shown in Eq. (25), the source terms are split in a posi-tive and a negative part. The latter is handled implicitly toenhance numerical stability (Merci et al., 2000).

6.4. Pointwise solution of linearised algebraic equations

A semi-implicit approach is taken, i.e. information fromnode i is treated at the new iteration level (m+ 1), whereasinformation from neighbouring nodes i′ is treated explic-itly, that is at the current iteration level (m). Consequently,pointwise solution of Eq. (25) is possible.

Eq. (25) is the linearised form of the conservation equa-tions for node i. Finally, the Eqs. (25) for all nodes can berecast as

A(m)UQ = B(m) (26)

from which the correction vector UQi for the conservativevariables of node i can be obtained. Because information re-lated to neighbours is treated explicitly, equation set Eq. (26)can be solved pointwise for each pseudotime step, using adirect solution method. In this work, Gauss elimination isused for the solution of the set of algebraic equations innode i.

6.5. Stability criteria

To achieve numerical stability, the computational domain,de=ned by the grid sizes, has to satisfy the Von Neumanand CFL conditions (Anderson, 1995). As a result of theapplication of preconditioning, the dominating character-istic speed, i.e. the speed of sound for the gas phase, isreplaced by a numerical speed of sound Cnum. For a 3Dmesh, the CFL condition restricts the pseudotime step asfollows:

U? = CFL1

(u1+Cnum1 )

Ux + (u2+Cnum2 )

Uy + (u3+Cnum3 )

; (27)

where 1,2,3 refer to the x; y; z components and

Cnumj =

√u2j + �2

n j = 1; 2; 3 (28)

and �n is de=ned as

�n = |umax|+2($g + $t

g)i�gi |ii′|

: (29)

A CFL value of 0.9 is normally used in this work. Ac-cording to Eq. (29), the numerical speed of sound is of thesame order of the hydrodynamic speed and is much smallerthan the actual speed of sound. Computationally, it allowsa larger pseudotime step as seen in Eq. (27). This is indeedthe important advantage of the preconditioning technique.It can be noted that in the pressure based solution methodsthe imparity of the eigenvalues and the resulting stiFness ofthe system of PDEs is handled by adopting the decoupling

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

(i) (ii) 0

outlet 2 outlet 1 outlet 1

outlet 2

Fig. 1. (i) Distribution of nodes in a horizontal plane; (ii) axial distribution of horizontal planes in the riser. XX and OO cross sections are used inFigs. 4 and 5.

approach: pressure and velocities are solved for separately.In this respect, the density based solution method indeedprovides a signi=cant gain in computational speed whenpreconditioning is applied.

7. Numerical aspects

A structured grid is used for the simulations. The grid de-sign involves the distribution of nodes in an axial cross sec-tion of the riser (Fig. 1(i)). A rectangular–circular approachis taken to guarantee a homogeneous node distribution overthe cross section and to allow grid re=nement near the solidbounding wall. The cross section is then adopted at severalaxial positions, distributed in a non-uniform way to providegrid re=nement in the bottom and top section of the riser(Fig. 1(ii)). Around each node, a octahedral control volumeis constructed. For details, reference is made to De Wilde(2000). For the simulation of the SO2–NOx riser, 188 nodesin each axial cross section and 67 axial layers are used, re-sulting in a total of 12 596 control volumes. The grid size istypical about 0:08 m in the radial direction and 0.08–0:2 min the axial direction.

Computations are made on IBM RS-6000 workstations,normally requiring 90 s CPU time for each pseudotime step.A physical time step of 0:01 s typically requires 100 pseu-dotime steps, so the computational time for covering about20 s real time simulation amounts to about 5000 h of CPUtime. The latter shows the enormity of the computationalload for transient 3D simulations.

8. Simulation of the SO2–NOx riser

Table 4 summarizes the dimensions and operating con-ditions for the SO2–NOx riser being simulated in thiswork. The large scale riser of 1:56 m diameter was usedas pilot unit to demonstrate a process for the simultane-ous adsorption of SO2–NOx from +ue gases (Das et al.,2001).

In Table 4, note that the restitution coeOcient e is as-sumed as unity. Some authors (Neri and Gidaspow, 2000;Samuelsberg and Hjertager, 1996) previously have used evalues slightly diFering from unity (≈ 0:995) during tran-sient simulation studies, although the basis to choose suchvalues is not clear. The over-sensitivity especially of thegranular temperature prediction on the value of e is welldocumented (Pita and Sundaresan, 1991) even for steadystate calculations. However, Bolio et al. (1995) have shownthat the sensitivity of results on the e value is not so sig-ni=cant for solid fractions in the range discussed in thepresent work. Recently, Agrawal et al. (2001) have shownthat the over-sensitivity of e is largely due to the incom-plete resolution of the meso-scale +uctuations if the kinetictheory of granular +ow (KTGF) is applied on a coarsegrid (≈ 5 cm). Using a much =ner grid size (≈ 1 mm),they have shown that the eFect of e on the granular tem-perature is only gradual in nature. Since the present workuses a coarse grid, the eFect of non-elastic collisions can-not be accounted for appropriately. The approximationof elasticity (e = 1) is reasonable since the mass +ux inthe SO2–NOx riser is suOciently low (¡ 5 kg m−2 s−1),for the dissipation of & due to inelastic collisions to beinsigni=cant.

Table 4Simulation conditions SO2–NOx riser

Property/parameter Symbol (unit)

Riser height H (m) 14.43Riser diameter D (m) 1.56Outlet elevation Hout (m) 12.65Total outlet area Sout (m2) 2.5

Particle diameter dp (�m) 60Particle density �p (kg m3) 1550Restitution coeOcient e (dimensionless) 1.0Wall restitution coeOcient ew (dimensionless) 0.9

InletSolid fraction �inb (dimensionless) 0.00137Solid +ux Gs (kg m−2 s−1) 2.12Average gas velocity ug;avg (m s−1) 3.07Gas/solid temperature T (K) 414Granular temperature & (m2 s−2) 1.56Gas turbulence intensity I (dimensionless) 0.05

Outlet pressure P (Pa) 101 300

9. Convergence characteristics

To monitor the convergence, the following residuals arede=ned:

Qr =∑

allcells

|UQ|; (30)

Br =∑

allcells

|B|; (31)

where UQ and B are de=ned in Eq. (26).The novel density based algorithm was initially applied to

study the convergence of single gas phase +ow. It was foundthat both Qr and Br for the gas phase continuity residualdropped monotonously, indicating a smooth convergence(Das et al., 2002).

For the two phase gas–solid +ow simulations at condi-tions shown in Table 4, Fig. 2 shows the convergence be-haviour for the +ow equations of the SO2–NOx riser fordiFerent values of the granular temperature. In the caseshown in Fig. 2, numerical stability restricts the pseudo-time step to roughly 0:001 s. Although the axis in Fig. 2corresponds to the number of pseudotime steps, it is feltthat the observed oscillations in Fig. 2 are comparable, atleast qualitatively, to the real time oscillations. A possi-ble reason is that the solid phase equations are not directlyaFected by preconditioning (De Wilde et al., 2002). Fur-thermore, similar oscillations are indeed found while per-forming a real time transient simulation, as shown laterin Fig. 3.

In Fig. 2, it is seen that the solid phase continuity equa-tion residual Qr (Eq. (31)) for the base simulation oscillatesand that a steady state solution is never reached. The oscil-lation with the lowest frequency has a period of typically6 s in pseudotime and there are many intermittent smalloscillations as well. The minimum values of the residualin Fig. 2 corresponds to the quasi-steady state conditionwhere the +ow variables remain almost unchanged for afew thousands of pseudotime steps before a new +uctu-ation originates. On the other hand, the residual is foundto decrease monotonously when applied to a riser withhigher solid +uxes (50 kg m−2 s−1 (Das et al., 2002);578 kg m−2 s−1 (De Wilde et al., 2002)). This indicatesthat in very high dilute phase risers as in this work, the +owis always oscillatory in nature. The lower damping eFect bythe solid, as will be discussed later, is at the origin of theoscillations.

To verify that the observed oscillations in Fig. 2 are in-deed physical and not numerical, a transient simulation isperformed using a very small time step Ut=0:01 s, in whicheven the high frequency +uctuations are expected to be re-solved. This is further discussed in the next paragraph. Notethat the physical time step Ut is not restricted by any nu-merical consideration but is determined by the time scaleof the model which re+ects the desired temporal accuracy.The pseudotime step on the other hand has to satisfy the

0 2 4 6 8 10 12

(103) Pseudo time steps

10 3 pseudotime steps =1s

quasi-steady

base case simulation22, −sm�

1.6 0.1 0.01

Fig. 2. Steady state convergence behaviour of the solid phase continuity residual: eFect of the granular temperature. Model: Table 1. Conditions:Table 4.–, base case simulation: typical &= 0:01 m2 s−2 at 3:7 m riser height;—, simulation neglecting interphase +uctuating motion interactions: typical& = 0:1 m2 s−2 at 3:7 m riser height; .., simulation neglecting interphase +uctuating motion interactions: typical & = 1:6 m2 s−2 at 3:7 m riser height.

0 2 4 6 8 10 12 14 16 18time, s

H =11 m; r/R=0.9

solid fraction

solid velocity

t = 0 : transient calculation startedafter 21 s of single time stepping

Fig. 3. Local transient +ow =eld behaviour. Model: Table 1. Conditions:Table 4. Position in the riser: r=R = 0:9 and elevation = 11 m.

CFL criterion (Eq. (27)), since the semi-implicit approachis adopted in this work.

The eFect of enhancing the turbulence of the gas phaseand the granular temperature on the convergence was ex-plored. The granular temperature and the turbulence wereincreased by neglecting the interphase +uctuating motionexchange terms, i.e. terms involving �, in the granular tem-perature equation and in all the turbulence equations (T1.5,T1.6, T1.11, T1.12) and by providing higher inlet values forthe granular temperature. In Fig. 2, it is seen that there is asigni=cant damping of oscillation as the granular tempera-ture level is enhanced. Note that the viscosity of each phasedepends on its +uctuating kinetic energy (see Eq. (T2.11)for the gas phase and Eqs. (T2.5, T2.6, T2.7) for the solidphase). There is a signi=cant loss of gas phase turbulenceand granular temperature when small particles are used atlower solid +uxes (e.g. in Fig. 6(ii)). This in turn causes alower viscosity of each phase and, hence, a lower dampingof large-scale oscillations. The latter allows the local oscil-

lations to grow leading to an increase of the residuals asseen in Fig. 2 (base case).

10. Time averaged and quasi-steady /ow 0elds

As seen earlier in Fig. 2, a steady state solution couldnot be reached and the +ow oscillates between diFerentquasi steady states. To illustrate this further, a transientsimulation with a physical time step of 0:01 s is per-formed for 17 s real time, starting from a quasi steady statesolution.

Fig. 3 shows the transient behaviour of the solid fractionand the solid axial velocity at r=R = 0:9 and at 11 m ele-vation. Both the solid fraction and the solid axial velocity+uctuate with a typical period of 5–6 s. Usually, a lowersolid velocity is associated with a higher solid fraction. Anincreased solids fraction causes bypassing of the gas and adecrease in the solid phase axial velocity. The oscillationscalculated are more pronounced near the wall than at the cen-ter (not shown), consistent with the experimental observa-tions of Lin et al. (1999). Similar transient behaviour of twophase +ow is reported by Neri and Gidaspow (2000) simu-lating a riser with low solid +ux (20 kg=m2 s) and FCC par-ticles of 75 �m diameter. Their simulation and experimentaldata demonstrate a typical dominating frequency of 0:2 Hzi.e. a period of 5 s. Furthermore, several small scale oscil-lations with higher frequencies are also reported by theseauthors.

To obtain steady state results from these transient +ow=elds, two approaches may be considered (i) time aver-aging of the transient +ow =elds obtained using a smallphysical time step Ut, or, (ii) assuming the quasi-steady=elds obtained using a large physical time step as steadystate. The former is more accurate but requires exten-sive computations. Therefore, the quality of quasi-steady

6. 00E -0 45. 82E -0 45. 64E -0 45. 46E -0 45. 29E -0 45. 11E -0 44. 93E -0 44. 75E -0 44. 57E -0 44. 39E -0 44. 21E -0 44. 04E -0 43. 86E -0 43. 68E -0 43. 50E -0 4

solid frac(-)

out2 out2out1 out1

6. 00E -0 45. 82E -0 45. 64E -0 45. 46E -0 45. 29E -0 45. 11E -0 44. 93E -0 44. 75E -0 44. 57E -0 44. 39E -0 44. 21E -0 44. 04E -0 43. 86E -0 43. 68E -0 43. 50E -0 4

solid frac(-)

solid. velo.5 m/s

out2out1

Quasi-steady Time averaged

Quasi-steady Time-averaged

solid. velo.5 m/s

out2out1

Fig. 4. Comparison of time averaged and quasi-steady +ow =elds. Model:Table 1. Conditions: Table 4. Cross section shown: vertical OO-crosssection through the outlets (see Fig. 1(i)). Time averaged results: basedon 17 s of transient simulation (Fig. 3) with Ut = 0:01 s. Quasi-steadyresults: based on steady-state simulation (Fig. 2) with Ut = 5000 s.

+ow =elds compared to time-averaged +ow =elds isinvestigated.

Fig. 4 shows a comparison of the solid fraction andsolid axial velocity obtained using the time averaged andquasi-steady approach. In both cases, a core annular +owpattern is obtained, although the solid fraction is somewhatlower and the solid velocity is relatively higher for the timeaveraged results. In Fig. 4(ii), note the considerable re+ux-ing of the solid in both time averaged and quasi-steadyconditions. Signi=cant re+uxing of solid due to an abruptoutlet con=guration is experimentally observed (Gupta and

Berruti, 2000), especially when the solid +ux is high. Cal-culations by De Wilde et al. (2003) clearly demonstrate therelation between re+uxing and restrictions at the outlet. Inthe present work, both gas and solid are found to recircu-late and the +ow is not symmetric with respect to the twooutlets. Symmetry breaking was also observed by De Wildeet al. (2002). It is attributed to the loss of granular tempera-ture and turbulence and the correspondingly lower eFectiveviscosity of the system. This is con=rmed from the simu-lation with a higher granular temperature (& = 1:6 m2 s−2)(see Fig. 2), where the +ow to the outlets was foundto be symmetric without any recirculation of gas/solid(not shown).

Fig. 5 allows a further comparison of the time aver-aged and quasi-steady +ow =elds. The quasi-steady ve-locities are almost similar to the time-averaged valuesalthough there is some diFerence at higher elevations. Notethat the time averaged pro=les are somewhat more sym-metric than the quasi-steady pro=les, as expected. Axialasymmetry can occur due to the backmixing induced bythe abrupt T-type outlets as seen in Fig. 4(ii), explainedearlier.

The solid fraction curves in Fig. 5(iii) show that thequasi-steady results are close to the time averaged values, inparticular at higher elevations. At lower elevations, the timeaveraged solid fractions are smaller than the quasi-steadyvalues. This is due to the higher gas velocity in the center,as a result of the recirculation of gas induced by the abruptoutlet. The granular temperature plot also shows that thequasi-steady =eld is quite close to the time averaged =eld.Note the remarkably low values of the granular tempera-ture: typically 0:01 m2 s−2. One reason for the low granu-lar temperature values is the very low solid fraction in theriser.

Fig. 5(i) and (ii) show that the shape of the gas andsolid velocity pro=le changes from a turbulent to a laminarparabolic type with increasing riser elevation. This is clearlyillustrated in the solid velocity vector plots (Fig. 4(ii)). Inaddition, there is an increase in the average velocity alongthe riser height. These eFects are attributed to: (a) a largerannulus thickness at higher elevations (see time averagedsolid fraction plot in Fig. 4(i)) which implies a lower ef-fective +ow area (b) the decrease in the gas phase pres-sure and density along the riser height, although this ef-fect is very small since the net pressure drop is¡ 400 Pa(c) the recirculation of gas and solids from the riser top(Fig. 4(ii)) which decreases the +ow area and increasesthe overall upward +ow rate above 8 m (d) the signi=-cant loss of granular temperature and gas phase turbulence(Fig. 5(iv)).

From the above it can be concluded that for the conditionsused in the present work (Table 4), quasi-steady +ow =eldsare in good agreement with time averaged +ow =elds. Thelatter however require at least an order of magnitude morecalculation time. This can justify the use of quasi-steadycalculations.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

r/R r/R

z=elevation, mModel Az=9.2

time averaged

quasi-steady

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

z=elevation, mModel Az=9.2

time averagedquasi-steady

-0.002

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

z= elevation, mModel A

z=3.2z=9.2

time averaged

quasi-steady

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

z=elevation, mtime averagedquasi steady Model A

z=9.2 z=3.7

(iii) (iv)

Fig. 5. Comparison of time averaged and quasi-steady +ow =elds. Model: Table 1. Conditions: Table 4. Pro=les shown: normalized radial pro=le inthe XX-cross section (see Fig. 1(i)) at two riser elevations. Time averaged results: based on 17 s of transient simulation (Fig. 3) with Ut = 0:01 s.Quasi-steady results: based on steady-state simulation (Fig. 2) with Ut = 5000 s.

11. Interactions of gas phase turbulence with solid phase/uctuating motion

11.1. Analysis of the gas–solid >uctuating motionproperties

Fig. 6 shows a comparison of the normalized gas phasemean and +uctuating velocities in single phase gas +owand in gas–solid +ow at 4:9 m elevation, where the +owis expected to be fully developed. Experimental data ofTsuji et al. (1984) for the single phase gas +ow is alsoshown.

For single phase gas +ow, the k–� turbulence model pre-dicts the gas phase mean velocity accurately (Fig. 6(i)).However, the +uctuating velocity and hence the turbu-lence intensity (u′=ucl) for the gas phase are somewhatover-predicted (Fig. 6(ii)). This could be due to a higherinlet value (5%) for the turbulence intensity and the higherReynolds number in the present work.

For two phase gas–solid +ow, the normalised meangas phase velocity is signi=cantly lower at higher r=Ras compared to the corresponding values for the singlephase +ow (Fig. 6(i)). The mean gas phase velocity pro-=le almost corresponds to a laminar pro=le as con=rmedby the lower turbulent intensity for the two phase +ow(Fig. 6(ii)).

Radial pro=les at diFerent axial positions in the riserfor the granular temperature, the gas phase turbulence

and their correlation are shown in Fig. 7. There is a sig-ni=cant decrease in granular temperature and gas phaseturbulent energy with riser height. Near the solid wall, kand & decrease by about two orders of magnitude between0.7 and 4:9 m (Fig. 7(i), (ii)). The values of k and & arehigher at the center and near the wall than at intermedi-ate r=R. Similar trends are also observed in the +uctuatingmotion correlation q12. The minimum value for k; & andq12 is found at r=R of about 0.8. The increase in k and& towards the center and the wall is related to the highervalues of the +uctuating motion correlation q12. Thesetrends are opposite to those predicted by Bolio et al. (1995)and Louge et al. (1991) for relatively larger diameterparticles.

The main reason for the loss of gas phase turbulence andthe decrease in the solid phase granular temperature withriser elevation is the presence of negative source terms dueto gas–solid interactions i.e.,−3�& in Eq. (T1.11) and−2�kin Eq. (T1.5). The production term due to the +uctuating mo-tion correlation �q12 is mostly lower than the correspondingdissipation term −2�k in Eq. (T1.5) as seen in Fig. 7(iv).The turbulence dissipation � also causes an additional de-crease of gas phase turbulence, especially in the case of twophase +ow, via the contribution of the last term in Eq. (T1.6).On the other hand, the +uctuating motion correlation q12

itself decreases along the riser height via the source term�s�sp�12, as seen in Fig. 7(iv) where the normalised correla-tion q12=2k is rather low at higher elevation. The normalized

00.10.20.30.40.50.60.70.80.9

0 0.2 0.4 0.6 0.8 1r/R

single phase

two phase

elevation: 4.9m

single phase

two phase

elevation: 4.9m

(i) (ii)

Fig. 6. Comparison of gas–solid +ow and single gas +ow. Model: Table 1. Conditions: Table 4. Rein = 175 000. Pro=les shown: normalized radial pro=lesat 4:9 m elevation. (i) Normalized mean gas velocity; (ii) normalized rms of the gas phase +uctuating velocity. Solid circles: data of Tsuji et al. (1984)for single gas +ow at Re = 21 800.

0 0.2 0.4 0.6 0.8 1r/R

0 0.2 0.4 0.6 0.8 1

0.0005

0.0015

0.0025

0.0005

0.0015

0 0.2 0.4 0.6 0.8 1r/R

, m2 s

(i) (ii)

(iii) (iv)

Fig. 7. Gas and solid phase +uctuating motion properties. Model: Table 1. Conditions: Table 4. Pro=les shown: normalized radial pro=les at diFerentriser elevations. (i) Gas phase turbulent kinetic energy k; (ii) granular temperature &; (iii) gas–solid +uctuating motion correlation q12; (iv) normalizedgas–solid +uctuating motion correlation q12=2k.

correlation is more than unity near the wall due to the con-tribution of a higher normal +uctuating velocity of the parti-cles in this region. This eFect could be attributed to the wallfunctions that are used near the solid wall, as discussed inSection 4.

It is known that the gas phase turbulence is enhanced inthe presence of larger particles and attenuated by smallerparticles (dp ¡ 100 �m) (Zevenhoven and Jarvinen, 2001).At lower ratios of solid/gas mass +ow rates, i.e. at lowermass loading, the main source of turbulence production forlarge particles is the shear which shows a maximum valueat r=R = 0:8. On the other hand, with small particles, theturbulence strongly depends on the +uctuating motion cor-

relation. Since the smaller particles can easily follow the+uctuating gas stream, the correlation is higher in the zonewith a higher gas velocity i.e., in the center of the riser(Bai et al., 1996).

To summarize, the loss of +uctuating motion correlationis found to be the main reason for the signi=cant decreasein granular temperature and gas phase turbulence with riserheight. To ascertain this, a simulation was performed ne-glecting completely the interphase +uctuating motion trans-fer terms in Eqs. (T1.5, T1.6, T1.11, T1.12). The results(not shown here) con=rmed that the granular temperatureand gas phase turbulence increase in the absence of theseterms.

11.1.1. Comparison of di?erent models for the >uctuatingmotion correlation

The main objective here is to illustrate the relative magni-tude of the correlation predicted by diFerent models. Fig. 8shows a comparison of the gas–solid +uctuating motion cor-relation calculated using diFerent models available in liter-ature. Model 1 is the q12 model (Simonin, 1996) used inthis work which makes use of an extra transport equation(Eq. (T1.12)). The other models use an algebraic expressionfor the gas–solid +uctuating motion correlation. In model 2(Gao et al., 1999), the +uctuating motion correlation is pro-portional to & 0:5. In model 3 (Agrawal et al., 2001) and 4(Louge et al., 1991) on the other hand, it is inversely pro-portional to & 0:5. Furthermore, in model 3 and 4, the eFectof the gas phase turbulence on the +uctuating motion corre-lation is completely neglected which is a serious limitationat least for small diameter particles. The curves in Fig. 8 arebased on the local values of slip, solids fraction and granulartemperature obtained using model 1 at 1:2 m. It is seen that,overall, models 2, 3 and 4 predict even lower values thanthe present q12 model 1. Especially the values obtained frommodel 3 and 4 are about two orders of magnitude lower thanthose from model 1 and 2. This might be the reason why therole of the correlation term was found negligible by Agrawalet al. (2001), which unfortunately resulted in much lower &values (0:02 m2 s−2) in their study as compared to the ex-perimental value of about 1 m2 s−2 (Gidaspow and Huilin,1998) for a solid fraction of 0.05. For the same reason, asigni=cant under-prediction of the gas phase turbulence andgranular temperature is observed in the work of Louge et al.(1991), i.e. model 4.

11.2. Attenuation of >uctuations in presence of smallparticles

11.2.1. Gas phase turbulence attenuationThe eFect of the mass loading (i.e. the ratio of solid to

gas mass +ow rates) on the gas phase turbulence stronglydepends on the particle size. Kullick et al. (1994) found aturbulence attenuation of nearly an order of magnitude ata mass loading of 0.5 using 70 �m diameter copper parti-cles. Paris and Eaton (1999) show a drop of k by a factorof 3 with a mass loading ratio of only 0.25. Varaksin andZaichik (1998) proposed a simple model to calculate theloss of turbulence with small particles for fully developed+ow:kk0

1 + 2X=C$; (32)

where X =�s�s

1 + (0:55�s�sk0:5)=(�l): (33)

k0 being the turbulent energy for single phase gas +ow andl the mixing length (=0:14R); C$ = 0:09.

Eq. (32) shows that the loss of turbulence increases whenthe solid density �s�s rises. For an average solid fraction of6× 10−4 and a particle density of 1550 kg m−3, a drop of

0 0.2 0.4 0.6 0.8 1r/R

0.0002

0.0004

0.0006

0.0008

elevation 1.2m

Fig. 8. Gas–solid +uctuating motion correlation q12. Model used: (1)Simonin (1996) (present q12 model); (2) Gao et al. (1999); (3) Agrawalet al. (2001); (4) Louge et al. (1991).

k of about one order of magnitude is predicted by Eq. (32).Such a decrease is validated by the mentioned experimentaldata with small particles. Similarly, Adams (1988) reportedone order of magnitude decrease in the gas phase turbulentdiFusivity (see Eqs. (T3.1 and T2.11)) due to such attenu-ation of turbulence while working with particles of 200 �mdiameter. Yang et al. (1984) reported even lower values ofthe turbulent diFusivity than those of Adams (1988), sincetheir study involved particles of 50 �m diameter. In compar-ison, Fig. 7(i) shows a decrease of k by two orders of mag-nitude at these conditions. The relatively higher turbulenceloss in the present simulation is attributed to the lower meanvelocity of the gas and the larger riser diameter. Note thatmost of the laboratory experimental data on turbulence at-tenuation are obtained in smaller diameter risers (D=5 cm)or channels. The high turbulence loss with the present modelcan also be due to the uncertainty concerning the values ofsome of the model parameters e.g. C4� in Eq. (T1.6). AsRizk and Elgobashi (1989) pointed out, a value of 1.2 forthis parameter, as used in this model, is obtained for turbu-lent jet +ow, but appears to be too small for the riser appli-cations. This value leads to the over-estimation of the tur-bulence dissipation � and, hence, an under-prediction of thegas phase turbulence. A parametric study by Simonin (1996)suggests that C4� may be a function of the local +ow condi-tions, especially the mass loading, and, hence, may requiretuning at varying +ow conditions. Since the turbulence cor-relation strongly depends on the gas phase turbulence (seeEq. (T1.12)), the loss of the granular temperature also takesplace due to the above reason.

The decrease in turbulence in the present simulation isalso supported by the fact that the ratio of the particle re-laxation time (?p = �s�s=�) to the Kolmogorov time scaleof turbulence (?k = k=�) amounts to about 100 and that thesolid fraction is in the range of 0.0006, both correspondingto a region of strong attenuation of turbulence (Sirignano,1999). The particle relaxation time in the Stokes regime

Table 5Granular temperature in low solid fraction risers (a: experimental; b:simulated)

Parameter Symbol (unit) Simonin (1996) Gidaspow and Huilin Neri and Gidaspow Bolio et al. Presenta and b (1998) a (2000) b (1995) a and b work b

Solid fraction �s (dimensionless) 0.0001 0.0042 0.005 0.004 0.0006Avg. gas velocity ug;avg (m s−1) 16.0 2.89 2.61 18.0 3.07Particle diameter dp (�m) 406 75 75 200 60Particle density �s (kg m−3) — 1654 1654 1020 1550Granular temperature & (m2 s−2) 0.03 0.13 0.2 0.12 0.002–0.08

can be expressed as ?p = �sd2p=(18$g) which shows that ?p

could be signi=cantly lower for small particle diameters. Asmall ?p and, hence, a higher � implies that the particles willquickly respond to the gas phase +uctuations which amountsto a higher attenuation of k according to Eq. (32). There areseveral mechanisms for the attenuation of turbulence withsmall particles. It appears that small particles, having verylittle inertia, strongly follow the gas phase turbulent +uctua-tions. A considerable turbulent energy from the gas phase islost to accelerate the particles in these +uctuations (Varaksinand Zaichik, 1998). However, according to Paris and Eaton(1999), the distortion of eddies by the presence of parti-cles is in some way responsible for the large reduction inthe turbulence. On the other hand, if the particle diameter ishigher, the turbulence is enhanced by completely diFerentmechanisms: vortex shedding and turbulent wake formation(Hetsroni, 1989; Tsuji et al., 1984).

11.2.2. Solid phase granular temperature attenuationAs shown in Fig. 7(ii), the granular temperature also

drops signi=cantly with riser height. The predicted values of& near the wall are in the range of 0.002–0:08 m2 s−2.Table 5 summarises the measured/simulated granular tem-perature for low solid +ux risers. All the data correspond to aposition close to the wall. It is seen that the predicted & valuesin the present work are well in the ranges given in Table 5.The lower granular temperature is explained by the lowersolid fraction in the riser and by the dissipation of & dueto the lower turbulence correlation values. Gidaspow andHuilin (1998) established a simple empirical relation & =13:2�2=3

s which also predicts a decrease of & as the solid frac-tion is lowered.

Finally, to illustrate the necessity to account for the gas–solid turbulence correlation in simulations, especially at lowsolid +uxes, a comparison is made in Fig. 9 between thepresent model and a model where all the interphase turbu-lence transfer terms i.e., terms involving � are neglected.The riser has a low solid +ux of 10 kg m−2 s−1. It is seenthat the q12 model used in this work gives a closer predictionthan the model where the interphase turbulence transfer isneglected. Despite the success of the q12 model at low solid+uxes, its applicability for higher solid +uxes is not recom-mended since other phenomena e.g. clustering inter-particlecollisions, etc. will have a more dominating eFect.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

z = 1.6 m

Data pointsGao et al.(1999)

Fig. 9. Comparison of gas–solid +uctuating motion interaction models.(1) present model. (2) gas–solid +uctuating motion interaction ne-glected. Solid circles: experimental data of Yang et al. (1984) atGs = 10:0 kg m−2 s−1, ug = 4:33 m s−1, dp = 54 �m, D = 0:15 m.

12. Conclusions

The complex hydrodynamics of two phase gas–solid +owin risers poses serious challenges in the design and scale upof such reactors. A three-dimensional transient simulationbased on =rst principles can provide a deeper understandingof the interaction of +ow and reaction variables. However,the accuracy of such fundamental simulation largely dependson the reliability of the turbulence models used for thesecalculations.

A 3D transient simulation of a low solid +ux riser reactorused for the simultaneous removal of SO2 and NOx is per-formed using a novel simultaneous solution algorithm. The+ow is largely of the core annular type with a signi=cant re-circulation of gas and solids from the riser top determinedby the outlet geometry. There is a major decrease in the gasphase turbulence and particle granular temperature along theriser height, resulting from the loss of gas–solid turbulencecorrelation. This attenuation of turbulence in the presence ofsmall particles is qualitatively consistent with experimentaldata taken from literature. The presently predicted attenua-tion is somewhat high, which is partly attributed to the largerdiameter of the simulated riser and partly due to the sensi-tivity of the turbulence parameters to the solid mass loading.

The loss of gas phase turbulence and particle granulartemperature leads to laminar parabolic velocity pro=les anda signi=cant decrease of the solid fraction in the core re-gion of the +ow. It also reduces the eFective viscosity al-lowing +uctuations to grow and thereby making the +owpattern oscillatory. A comparison of the time averaged andquasi-steady +ow =elds shows that the deviations from thetime averaged =eld are only small. Time averaged results aremore symmetric than the quasi-steady =elds, as expected.

Notation

am external particle surface area per kg solid,m2

p kg−1solid

A left-hand side matrix for each nodeB right side vector (dimension: number of

equations in each node)Br sum of residuals of the conservative variables

over all the nodes in the domainc instantaneous velocity of individual particle,

m s−1

C random +uctuating velocity of individualparticle, m s−1

Cd drag coeOcient for an individual particle,dimensionless

C1�;:4�;$;�;s turbulence model parameters, dimensionlessCj gas phase concentration of the component j,

mol m−3gas

Csj concentration of the gaseous component j,

inside solid phase, mol m−3gas

Cnum numerical speed of sound, m s−1

Ct site capacity of the sorbent, molsite kg−1sorbent

dp particle diameter, mD riser diameter, mDm mean molecular diFusivity of the gas mixture,

m2 s−1

e restitution coeOcient of particles, dimensionlesseg gas phase internal energy, J kg−1

ej unit vector along the axis,ew restitution coeOcient for particle–wall

collisions, dimensionlessEg gas phase total energy, J kg−1

g gravity, m s−2

g radial distribution function,Gs solid mass +ux, kg m−2 s−1

hmj molar enthalpy of component j, J kmol−1j

H riser height, mHout outlet elevation from riser bottom, mi grid point i,i′ neighbouring grid point i′,I unit matrix,k turbulent kinetic energy of gas phase, m2 s−2

kgj mass transfer coeOcient of component j,mol m−2 s−1

K source terms,m pseudotime stepping iteration number,m mass of each particle, kgn normal unit vector to the surface, dimensionlessP gas phase pressure, N m−2

Ps solid phase pressure, N m−2

q12 gas–solid turbulence correlation, m2 s−2

qg mean kinetic energy of gas phase = 12 (u2

x + u2y

+u2z ), m2 s−2

Q set of conservative variablesQr sum of correction of each of the conservative

variables over all the nodes in the domainRj net production rate of the component j in the

gas phase, mol mol−1site s−1

Rl net production rate of the species 1,mol mol−1

site s−1

smg viscous stress tensor for the gas phase,kg m−1 s−2

ss viscous stress tensor for solid phase,kg m−1 s−2

S surface area, m2

t time, sT gas/solid phase temperature, Ku instantaneous, locally averaged velocity of the

gas phase, m s−1

u′ turbulent +uctuation velocity of the gas phase,m s−1

Ku time smoothed hydrodynamic velocity of the gasphase, m s−1

Ur reference velocity for preconditioning, m s−1

v instantaneous, locally averaged hydrodynamicvelocity of the solid phase, m s−1

V cell volume, m3

W set of primary variablesYj mol fraction of component j, dimensionless

Greek letters

� interphase momentum transfer coeOcient,kg m−3

reactor s−1

�n numerical velocity scale, see Eq. (29), m s−1

� dissipation of turbulent kinetic energy of the gasphase, m2 s−3

�g gas volume fraction, dimensionless�s solid volume fraction = (1− �g), dimensionless�sg porosity of the particle, dimensionless�12 dissipation of turbulence correlation, m2 s−3

� dissipation of kinetic +uctuation energy ofthe solid phase by inelastic particle–particlecollision, kg m−1 s−3

� ratio of speci=c heat of the gas = cp=cv,dimensionless

> preconditioning matrix+s granular conductivity of the solid phase,

kg m−1 s−1

$g gas phase shear viscosity, kg m−1 s−1

$s solid phase shear viscosity, kg m−1 s−1

2tg kinematic turbulent viscosity of gas phasem2 s−1

’′ +uctuating component of a scalar�g density of the gas, kg m−3

�s particle density of the solid, kg m−3

(k;�;q;s eddy parameters, dimensionless? pseudotime, s& granular temperature, m2 s−2

&l fractional coverage by the species l,molspecies mol−1

site,12 kinematic viscosity of turbulence correlation,

m2 s−1

#s bulk viscosity of solid phase, kg m−1 s−1

#g bulk viscosity correction of gas phase (=0 forNewtonian +uid), kg m−1 s−1

Additional notations

− time smoothed mean′ +uctuating part〈〉 stochastically averaged meanbold letters, e.g. C vector

Subscripts/superscripts

avg averageb bottomc collisionaleF eFectiveg gas phasein inletj component in gas phasek kineticl species in solidm pseudotime levelout outlets solid sorbent/catalystsp solid phase including speciest turbulentT transpose matrix

Acknowledgements

This work was =nanced by the European Commis-sion within the frame of the Non-Nuclear Energy Pro-gram, Thermie project under contract no SF 243/98DK/BE/UK (project partners: Laboratorium voor Petro-chemische Techniek-Universiteit Gent, FLS miljo a/s—Denmark & Howden Air & Gas Division—UK). Theauthors are grateful to the ‘Fonds voor Wetenschap-pelijk Onderzoek-Vlaanderen’ and the ‘InterUniversitaireAttractie Polen’ of DWTC for =nancial support. Oneof the authors (JDW) would like to acknowledge the

=nancial support of the Instituut voor de aanmoedig-ing van Innovatie door Wetenschap en Technologie inVlaanderen (IWT-Vlaanderen) under contract numberIWT/OZM/020059.

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