An extension of the preconditioned advection upstream splitting method for 3D two-phase flow...

An extension of the preconditioned advection upstreamsplitting method for 3D two-phase flow calculations in

circulating fluidized beds

Juray De Wilde a,1, Geraldine J. Heynderickx a,*, Jan Vierendeels b, Erik Dick b,Guy B. Marin a

a Laboratorium voor Petrochemische Techniek, Ghent University, Krijgslaan 281, Blok S5, 9000 Gent, Belgiumb Department of Flow, Heat and Combustion Mechanics, Ghent University, St. Pietersnieuwstraat 41, 9000 Gent, Belgium

Received 28 September 2001; received in revised form 31 May 2002; accepted 31 May 2002

Abstract

For the calculation of gas�/solid flow in circulating fluidized beds, a Eulerian�/Eulerian approach is taken. An integration scheme

based on dual time stepping and a finite volume technique is developed and implemented in 3D. The inviscid part of the equations is

treated following an extension of the preconditioned advection upstream splitting method (AUSMP) to two-phase flows.

Calculations on an industrial size straight riser are performed. The influence of the inelasticity of particle�/particle collisions on the

stability of the flow is investigated. Further, the effects of a double abrupt side outlet configuration are shown.

# 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Advection upstream splitting method (AUSM); Gas�/solid flow; Circulating fluidized bed (CFB); Kinetic theory of granular flow

(KTGF); Fluid transients

Nomenclature

a linearization matrixA point linearization matrixB explicit information matrixCP heat capacity at constant pressure (J kg�1 K�1)c speed of sound (m s�1)C, D viscous flux matrixdp particle diameter (m)D diagonal of a matrixe restitution coefficientE total energy (J)F flux in x -direction

/g/ gravity (mr s�2)G superficial mass flux (kg m�2 s�1)H enthalpy (J kg�1)

* Corresponding author. Tel.: �/32-9-264-4516; fax: �/32-9-264-4999

E-mail address: [email protected] (G.J. Heynderickx).

Computers and Chemical Engineering 26 (2002) 1677�/1702

www.elsevier.com/locate/compchemeng

0098-1354/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 9 8 - 1 3 5 4 ( 0 2 ) 0 0 1 5 7 - 6

mailto:[email protected]

i i-coordinate

i grid point ii? neighboring grid point i?ii? cell interface between i and i?ii? distance between i and i? (m)j j-coordinate

k turbulent energy gas phase (J kg�1)k k -coordinate

m mass (kg)M Mach numbern pseudo-time/iteration level

/n/ normal directionnn number of nodesP mean total pressure (N m�2)Ps solid phase pressure (M mr

�2)q12 gas�/solid turbulence correlation (m2 s�2)qg/s kinetic energy gas�/solid phase (m2 s�2)

/q/ fluctuation energy flux (kg s�3)Q vector of conservative variables

/r/ position vectorRe Reynolds numbers speed (m s�1)

/s/ viscous stress tensor (kg m�1 s�2)S matrix containing source termst time (s)T temperature (K)

/u/ local mean velocity gas phase (mr s�1)

/v/ local mean velocity solid phase (mr s�1)

V volume (m3)wr maximum connective speed in the flow field (m s�1)x x-position (m)y y-position (m)

y1 distance to the solid wall from node near the solid wall (m)z z-position (m)

Greek notations

b interphase momentum transfer coefficient (kg mr�3 s�1)

b parameter for artificial dissipation (m s�1)/G/ preconditioning matrixg dissipation of kinetic fluctuation energy by inelastic particle�/particle collisions (kg mr

�1 s�3)g ratio of specific heat CP/CV

D differencedad parameter for artificial dissipationo dissipation of turbulent kinetic energy of the gas phase (mr

2 s�3)

o12 dissipation of q12 (mr2 s�3)

og volume fraction gas phase (mg3 mr

�3)

os volume fraction solid phase (ms3 mr

�3)

os,max maximum solids concentration�/0.64356z local axe parallel to interfaceh local axe parallel to interfaceU granular temperature solid phase (J kg�1)k von Karman constant (gas phase wall functions)k conductivity kinetic fluctuation energy solid phase (kg m�1 s�1)l conductivity (W m�1 K�1)mg molecular viscosity gas phase (Pa s)mg

t turbulent viscosity gas phase (Pa s)ms shear viscosity solid phase (kg m�1 s�1)

J. De Wilde et al. / Computers and Chemical Engineering 26 (2002) 1677�/17021678

n kinematic viscosity (m2 s�1)n12

t turbulent correlation viscosityj local axis normal to interfacejs bulk viscosity solid phase (kg m�1 s�2)rs gas density (kgg mg

�3)rsp solid phase density (kgsp msp

�3)s parameter turbulence equationst pseudo time (s)fs sphericity factor particlesf flux in normal directionf matrix of the conservative variables

/f/ flux vectorOther

�� averaged over the bottom inletSubscriptg gas phasei of grid point ii? of grid point i?ii? cell interface between i and i?n normalp particleref references solid phasesp solid phasew wallx x-direction

y y-direction

z z-direction

Superscript

_ vector= tensor? viscous terms? neighbor�/ negative part�/ positive partad artificial dissipation(c) convective termsg gas phasei of grid point ii? of grid point i?ii? cell interface between i and i?inb at the bottom inletn iteration numbernum numerical(P) pressure terms(Ps) solid phase pressures solid phasesp solid phaseT transposedt turbulenttot totaltur turbulent

J. De Wilde et al. / Computers and Chemical Engineering 26 (2002) 1677�/1702 1679

1. Introduction

The circulating fluidized bed (CFB) is characterized

by the unique feature of the fast fluidized state in whichrapidly upflowing solid particles are fluidized in a gas

stream that flows at even higher velocities. In CFB’s, the

mean solids volume fraction is usually less than 10%.

CFB’s are encountered in many chemical and petro-

chemical processes, including fluid catalytic cracking

(FCC) (Avidan & Yerushalmi, 1985), calcination of

alumina trihydrate to high-purity alumina, combustion

of low-grade coal in power generation plants (Wirth,1988), oxidation of n-butane to maleic anhydride

(Contractor, Bergna, Horowitz, Blackstone, Chowdhry

& Sleight, 1988; Contractor, 1999), and simultaneous

removal of SO2 and NOx by adsorption on Na-g-

Alumina (Mortensen, 1995; De Wilde & Marin, 2000).

Despite their widespread application for several

decades, the design and scale-up of CFB’s is still very

difficult. This is mainly due to the complex hydrody-namic behavior. Experimental investigations have

shown a non-uniform solids distribution in both the

axial and radial direction (Tsuji, Morikawa & Shiomi,

1984; Miller & Gidaspow, 1992; Nieuwland, Annaland,

Kuipers & vanSwaaij, 1996; Nieuwland, Meijer, Kuipers

& vanSwaaij, 1996). In their experimental investigation

Bader, Findley and Knowlton (1988) found that only

45% of the total riser area managed 75% of the total gasflow through the riser due to particle segregation. In

some cases, the segregation of particles is found to be

concentrated near the pipe wall, while in others large

regions of high particle concentrations, called streamers,

extend well into the center of the pipe (Pita &

Sundaresan, 1993). Other complex flow phenomena

characteristic for CFB’s are the suppression or enhance-

ment of the gas-phase turbulence by the presence ofparticles, and the multiplicity of pressure gradients and

solids holdup for specified values of gas and solids flow

rates (Pita & Sundaresan, 1991).

The non-uniform solids distribution and solids flow

influences the particle residence time distribution and

thus the reactor performance. Therefore, the under-

standing and simulation of the complex hydrodynamics

is of crucial importance for optimal operation, design,and scale-up of CFB’s.

One-dimensional models, like the plug flow model

with slip between the gas and the solid phase (Froment

& Bischoff, 1990), have not been successful for the

simulation of risers without the introduction of empiri-

cism in the form of fitted parameter values which

depend on the nature of the particles and the size of

the pipe (Arastoopour & Gidaspow, 1979; Arastoopour& Cutchin, 1985; Mountziaris & Jackson, 1991; Nieuw-

land et al., 1996).

Core-annulus models have been developed from the

early 70’s (Nakamura & Capes, 1973) and assume a core

region characterized by a dilute gas�/solids mixture

flowing upwards, and a dense annulus with downwards

moving particles. Core-annulus models accounting for

clustering in the annulus have been successful in theprediction of axial pressure profiles in a CFB (Horio &

Takei, 1991) but still involve significant empiricism.

Berruti and Kalogerakis (1989) formulate a core-annu-

lus type model based on the mass and momentum

conservation for the solid phase. Theoretical predictions

of the core diameter, core voidage, and velocity profiles

show reasonable agreement with experimental data. In

core-annulus type models the solids velocity in theannulus is either zero or negative, indicating solids

moving downwards at the wall. While this may be the

case for low flux risers, cold flow data under FCC

operating conditions show no catalyst raining down at

the wall (Fligner, Schipper, Sapre & Krambeck, 1994).

Therefore other models, like the ‘two phase cluster

model’ (Fligner et al., 1994), have been proposed. A

further disadvantage of core-annulus type models is thatthe application is restricted to the fully developed flow

zone of the riser, where the solids and the gas have been

accelerated to their steady-state velocity, and the average

solids holdup remains essentially constant. Thus sepa-

rate modeling is required for the bottom acceleration

zone and the exit region (Pugsley, Patience, Berruti &

Chaouki, 1992).

A more general approach was taken by Anderson andJackson (1967): the gas and solid phase are considered

as two continuous media, fully penetrating each other.

This assumption for the solid phase is justified if the

number of particles in the system of interest is very high.

The present calculations use FCC conditions, with

about 5% particles by volume, justifying this approach.

This so-called ‘Eulerian�/Eulerian’ modeling is based on

the conservation equations of mass, momentum, andenergy for both the phases and was further developed by

several research groups (Nieuwland et al., 1996; Simo-

nin, 1990; Gidaspow, 1994). Certainly with respect to

turbulence modeling, the equations of motion are still

evolving.

The set of partial differential equations describing the

hydrodynamic behavior of both phases in the riser is not

easily integrated and several numerical techniques fromthe domain of computational fluid dynamics (CFD)

have to be used. For one-phase flow calculations, CFD

has known a real break-through and tremendous evolu-

tion is seen in the numerical techniques applied. In

general it can be stated that numerical schemes are

nowadays built on the physical nature of the flow. This

has made the codes faster and more stable. In two-phase

flows, attention has mainly gone to the development ofthe model equations and not much progress was made in

the numerical solution techniques. Almost all the

integration codes used by research groups working on

two-phase flow, are originating from the mid-seventies


(Patankar & Spalding, 1972; Harlow & Amsden, 1975;

Rivard & Torrey, 1977, 1979) and are pressure-based

and using a staggered grid. In this work, a density-based

integration method is developed and recent numerical

techniques used in one-phase flow calculations are

introduced, to enhance the solution accuracy, the

numerical stability, and the convergence speed.

2. Two-fluid model

The gas and solid phase are considered as two

continuous media, fully penetrating each other (Ander-

son & Jackson, 1967). This assumption for the solid

phase is justified for the present calculations, as the

number of particles in the system of interest is high

Table 1

Conservation equations


enough (about 5% particles by volume). The so-called

‘Eulerian�/Eulerian’ modeling is based on the conserva-

tion equations of mass, momentum, and energy for both

the phases and was further developed by several

research groups (Nieuwland et al., 1996; Simonin,

1990; Gidaspow, 1994). Within the solid phase, stress

is transmitted by forces exerted at points of mutual

contact between the particles. The solid phase shear

stress is usually written in a Newtonian form, introdu-

cing a solid phase pressure and solid phase viscosity.

These closure terms can be calculated either on an

empirical base (Tsuo & Gidaspow, 1990; Sun &

Gidaspow, 1999) or by introducing the kinetic theory

of granular flow (KTGF) (Haff, 1983; Jenkins & Savage,

1983; Lun, Savage, Jeffrey & Chepurniy, 1984; Johnson

& Jackson, 1987; Sinclair & Jackson, 1989). In the

KTGF, the collisions between the particles are seen as

interactions of the fluctuating and the mean motion of

the particles. The solid phase pressure and viscosity are

thus directly related to the solid phase turbulence,

commonly quantified by the granular temperature. A

transport equation for the solid phase turbulence is

derived through the KTGF and solved simultaneously

with the mass, momentum, and energy conservation

equations. In the present calculations, the solid phase

temperature is assumed to equal the gas phase tempera-

Table 2

Constitutive equations


ture. As a result, no solid phase total energy equations is

to be solved.

The gas phase turbulent viscosity is estimated follow-

ing the Prandtl�/Kolmogorov relation. A two-equationk �/o model modified for gas�/solids interactions is used

for the calculation of the gas phase turbulent kinetic

energy and dissipation. The k �/o model (Laundar &

Spalding, 1974) has been successfully used in one-phase

flow predictions, reason for attempting to extend this

model to a multi-phase environment. Simulations were

carried out to investigate the importance of the gas

phase turbulence.In this work it was tried to include the mutual

influence of gas�/solids turbulence by gas�/solids drag.

Due to the gas�/solids turbulence correlation, enhance-

ment of the gas�/solids turbulence by gas�/solids drag is

possible. Louge, Mastorakos and Jenkins (1991) were

the first to investigate the influence of the gas phase

turbulence on dilute gas�/solids flow, using a one

equation closure model for the gas phase turbulence.For the gas�/solids turbulence correlation these authors

proposed an algebraic expression based on the func-

tional form by Koch (1990) for a dilute gas�/solids

suspension at very low particle Reynolds number,

extended to higher particle Reynolds numbers. He and

Simonin (1990), Simonin, Deutsch and Minier (1993)

calculated the value of the gas�/solids turbulence corre-

lation from an additional semi-empirical transportequation, an approach adopted in this work. The

covariance tensor components are computed with the

help of the eddy viscosity concept (Simonin, 1990, 1996).

The fluid�/particle turbulent viscosity is written in terms

of the fluid�/particle velocity covariance and an eddy�/

particle interaction time. The transport equation for the

gas�/solid turbulence covariance is derived from the

fluid Langevin equation (Simonin, 1990) and theparticle dynamic equation (accounting only for the

drag force) after cross-multiplication by the velocity

fluctuations (Simonin et al., 1993). The fluid�/particle

covariance equation is written in an approximate form

which underlines the particulate transport. On the left-

hand side (LHS) of the Eq. (11), the first term is the

time-dependent term and the second term models the

convective transport. The first term on the right-handside (RHS) of the equation models the transport of the

covariance by the velocity fluctuations. The next four

terms represent the production by the mean velocity

gradients. The last but one term on the RHS accounts

for the destruction rate due to viscous action in the

continuous phase and the loss of correlation by cross-

ing-trajectory effects. The last term represents the

influence of the turbulent momentum transfer betweenthe phases due to drag.

For the calculation of the inter-phase total energy

transport by drag, appearing in the gas phase total

energy equation, it is assumed that the contact plane

between the phases has a velocity equal to the mean

velocity of the two phases.

Table 1 shows all the conservation equations in-

volved. The model parameter values used for thecalculations are found in De Wilde (2001). The con-

stitutive equations (Nieuwland et al., 1996; Nieuwland,

1995) are presented in Table 2 (De Wilde, 2001). The

physical properties of the solid phase result from

particle�/particle collisions caused by the solid phase

turbulence. The expression for the solid phase viscosity

and for the conductivity of the solid phase turbulence

consists of a collisional and a kinetic part. The solid�/

solid interaction is indeed determined by the number of

collisions and by the intensity of a collision. The latter

depends on the mean free path of the particles and

becomes dominant for low particle concentrations.

The inter-phase momentum transfer coefficient b is

modeled as in Gidaspow (1994), Ferschneider and Mege

(1996). For dense regimes (ogB/0.80), b is calculated

using the Ergun-equation. In dilute regimes (og]/0.80),b is determined using the correlation of Wen and Yu

(1966).

The turbulent viscosity for the correlation tensor

between the fluctuating velocities of the two phases

(n12t ) is calculated as in Csanady (1963), Simonin (1990,

1996).

The presented model has been developed and used by

many other two-phase research groups (Simonin, 1990;Nieuwland et al., 1996) but is certainly to be taken with

precaution. Indeed the model still shows several short-

comings and the equations of motion for gas�/solid flow

and even for simple gas flow are still evolving, certainly

with respect to turbulence modeling (Agrawal, Loezos,

Syamlal & Sundaresan, 2001; Zhang & VanderHeyden,

2001).

3. Boundary conditions

Boundary conditions have to be imposed at the inlets,

outlets, and solid bounding walls of the domain.

Following the eigenvalue analysis (De Wilde, Heyn-

derickx & 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 behavior of the gas velocity in the

vicinity of a solid wall is rather complex (Schlichting,

1979; Rodi, 1987). Calculation of the near wall behavior

is possible but demands a lot of extra grid points. In

order to avoid an exponential rise of CPU-time, use is

made of wall functions to calculate the effect of the solid

wall on the bulk flow at this stage of the modeling.Wall functions are applied to the grid point in the

immediate vicinity of the wall. This grid point has to be

positioned in the full turbulent zone at a distance y��/


y1ut /n (ut see Eq. (31)) between 30 and 100. This means

that the external grid points are not positioned at the

wall itself, but at a certain distance.

Wall functions for pure gas flow are applied.Although it is expected that the gas flow field near the

wall will be modified by the presence of solid particles,

no accurate model is available yet.

The value of the wall shear force is calculated from

the well known logarithmic law (Hinze, 1959):

u

ut

�1

kln

�y1ut

nE

�with ut�

ffiffiffiffiffitw

rg

s(31)

Assuming a value y� of 50 results in:

ut�u

(1=k)ln[50E](32)

The k �/o model used in the bulk flow (Fox, 1996) is

not valid in the immediate vicinity of the wall. The gas

phase turbulence variables near the solid wall are

calculated from:

k�u2tffiffiffiffiffiffiCm

p

o�u4t

50kn�

u3t

ky1

(33)

These wall functions do not take into accountinteractions with the solid particles yet. It is assumed

that the particles will disturb the viscous sublayer of the

gas, but this topic still needs a lot of research.

For the solid phase it is also assumed that the mean

flow in the vicinity of the wall is parallel to the wall.

Sinclair and Jackson (1989) calculate the value of the

specific shear stress as:

sw��osms

@nT

@r jw

�f?

ffiffiffi3

pprsposU

1=2i nTi

6osmax[1 � (os=osmax

)1=3](34)

The flux of pseudothermal energy to the wall is

calculated as (Sinclair & Jackson, 1989):

qpU��osk@U

@r�gw�n � sw (35)

In this equation gw is the dissipation of solid phase

turbulent energy due to inelastic collisions between

particles and the wall. The last term models generation

of pseudothermal energy by slip.

The dissipation of solid phase turbulent energy at the

solid wall is modeled as:

gw�

ffiffiffi3

pposrspU

3=2i (1 � e2

w)

4osmax[1 � (os=osmax

)1=3](36)

The gas�/solid turbulence covariance in the vicinity of

the wall is calculated following the formula of Louge et

al. (1991). This approach is only valid for very dilute

particle flow (og�/0.995), somewhat lower than the

expected particle concentration range in our calcula-

tions.

q12�4

p0:5�dp(u � v)2

T12U0:5

(37)

Introducing:

T12�4dprsp

3Ccdsrg½

ffiffiffiffiffiffi2k

p�

ffiffiffiffiffiffiffi3U

p½

(38)

with:

Ccd �

24

Recp

(1�0:15Rec0:687

p ) (39)

valid in the range:

0BRecpB800 : Rec

p�½

ffiffiffiffiffiffi2k

p�

ffiffiffiffiffiffiffi3U

p½ogrgdpfs

mg

(40)

The solid wall does not perform any work. Further-more, it is assumed that the solid wall is adiabatic. Thus

there is no contribution of the solid wall in the

discretized total energy equation of the gas phase.

4. Computational method

Most algorithms for incompressible or low speed flow

and for two-phase flow are pressure-based, which

essentially means that the equations are solved sequen-

tially (Patankar & Spalding, 1972; Harlow & Amsden,

1975; Rivard & Torrey, 1977, 1979). Thus, the pressure

and the velocity field are solved iteratively in a

segregated manner, introducing an additional iteration

loop. For two-phase flow calculations, usually heavyunder-relaxation is imposed and a staggered grid

approach is taken (Harlow & Amsden, 1975; Rivard &

Torrey, 1977, 1979).

In the past years, density based methods are extended

towards low Mach number flows with the precondition-

ing technique (Weiss & Smith, 1995; Liou & Steffen,

1993; Liou & Edwards, 1999), which shows promising

results. In density based methods all the equations aresolved simultaneously, eliminating the pressure�/velocity

correction loop. We extended this method for two-phase

flow (Liou, 2001).

4.1. Preconditioning

To reduce the stiffness of the set of equations due tothe low-Mach gas flow, preconditioning is applied to the

gas phase equations. The preconditioner used in this

work is based on the preconditioner of Weiss and Smith


(1995). However, some modifications for the presence of

the solid phase have been introduced.

The derivation of the preconditioning matrix begins

by defining the transformation matrix @Q=@W: The

vector Q of the conservative variables is:

The matrix W of the primitive variables is:

WT�(og P vx vy vz ux uy uz T U

k o q12) (42)

The transformation matrix is:

(43)

with:

K1�@rg

@P jT

�rg

P

K2�@rg

@T jP

��rg

T

C1�og

��Eg�

P

rg

�K2

��rgCP

�(44)

The preconditioning matrix in terms of the primitive

variables is obtained by replacing K1 by T1 (Weiss &

Smith, 1995):

G ˜W�

@Q

@W(K1 l T1) (45)

with:

T1��

1

u2ref

�1

CP � T

�(46)

and:

uref � ½u½�2(mg � mtur

g )

�1

Dx�

1

Dy�

1

Dz

�rg

if ½u½Bcg

cg if ½u½�cg

(47)

QT�

osrsp ogrg osrspvx osrspvy osrspvz ogrgux ogrguy

� ogrguz ogrgEg

3

2osrspU ogrgk ogrgo osrspq12

264

375 (41)


The form of Eq. (47) ensures that as uref approaches

cg, T1 reduces to g /cg2.

In contrast with Weiss and Smith (1995), in this work

the conservative variables were chosen as the dependentvariables. The preconditioning matrix in terms of the

conservative variables is obtained by back-transforma-

tion:

GQ�GW ��@Q

@W

��1

(48)

It was experienced that the preconditioner obtained

this way does not perform well. The convergence

problems were solved by setting the off-diagonal ele-

ments in the first column of both the matrix @Q=@W

and GW equal to zero. For the solid phase equations, the

elements in the first column of the preconditioner (45),

describing the dependence on the solids volume fraction,

are very large compared with the other values in thatmatrix. This is due to the fact that, in these matrix

elements, the large solid phase density values are no

longer compensated for by the small solid volume

fraction values. Thus, the diagonal dominance of the

system is disturbed. Therefore, in the definition of the

preconditioner, not the full Jacobian is considered and,

except for the solid phase mass balance, the partial

derivatives towards the solids volume fraction areneglected. This implies a decoupling of the gas phase

and the solid phase. Preconditioning is then restricted to

the gas phase, for which it was initially developed.

Remark that the speed of sound is not directly appear-

ing in the elements set to zero (see Eq. (43)). Further-

more, not considering these elements will not affect the

solution because the preconditioning matrix is in the

LHS of the equation to be solved (see paragraph onSection 4.6).

4.2. Spatial discretisation

A finite volume cell centered approach is taken.

Application of the divergence theorema results in:

gVi

9 � f dV �gSi

f � n dS (49)

The surface integral is discretized and an ordinary

summation of fluxes is obtained.

4.2.1. Inviscid fluxes

The inviscid fluxes are treated following an extension

of the preconditioned advection upstream splitting

method (AUSMP) from single phase to multiphase

flow. For each phase, the inviscid fluxes are split intoconvective terms and acoustic flux terms. Convective

flux terms at a cell interface are treated upwind

following the value of the advective velocity at the cell

interface.

�f(c)

g=s n�ii?�fg=s

i=i?(sg=s n)ii?�f

g=s

i=i?sg=snii?

�fg=s

i=i?cgnum=s

ii? Mg=snii?

(50)

with:

fi=i?�fi if Mnii?

]0

fi? if Mnii?B0

(51)

Liou and Edwards (1999) have clearly shown the

improvements of AUSM-family schemes compared with

the flux difference splitting (FDS) and flux vector

splitting (FVS) methods. A distinct feature of the

AUSM-schemes is the formulation of the fluxes in terms

of a Mach number. This allows an easy generalization to

both low Mach number and multiphase applications.Remark that, as a result of the preconditioning, for

the gas phase, the numerical speed of sound cgnum

ii? is to be

used in the formulation Eq. (50) (Liou & Edwards,

1999). The new numerical speed of sound at the cell

interface ii? is calculated according to:

cnumii? �c

gii?

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4M2

refii?� (Mg

nii?)2(1 � M2

ref ii?)2

q1 � M2

ref ii?

(52)

The calculation of the numerical speed of sound at the

cell interface ii? cii?num requires the knowledge of (Mn

ii?

g)2

and Mrefii?

2 which are obtained as follows:

M refi �

urefi

cgii?

M refi? �

urefi?

cgii?

9>>>=>>>;[M2

ref ii?�

(M refi )2 � (Mref

i? )2

2(53)

Mgni�uinii?=c

gii?

Mgni?�ui?nii?=c

gii?

�[(Mg

nii?)2�

(Mgni

)2 � (Mgni?

)2

2(54)

and:

cgii?�

cgi � c

gi?

2(55)

A general formulation for the cell interface advective

velocity, for both subsonic and supersonic flow of gas

and solid phase, combines the wave speeds travellingtowards the cell interface ii? from the adjacent cells. This

is formally written by combining the contributions from

the grid points i and a neighbor i? (Liou & Steffen,

1993):

Mnii?�M ii?

ni�M i?i

ni(56)

A superscript jk stands for movement starting from

grid point j towards grid point k .

The advective velocity for both phases is calculated

applying the Van Leer splitting (Van Leer, 1982).For Mach numbers smaller than 1, the Van Leer

splitting in the external normal direction to the interface

ii? is:


MachB1

M ii?ni��

1

4(Mni

�1)2

Mi?ini?��

1

4(Mni?

�1)2

(57)

For Mach numbers exceeding 1 or equal to 1, the Van

Leer splitting in the external normal direction to the

interface ii? is:

Mach]1

M ii?ni�

1

2(Mni

� ½Mni½)

Mi?in i?�

1

2(Mni?

� ½Mni?½)

(58)

where:

Mni�Mi � nii? Mni?

�Mi? � nii? (59)

As preconditioning is applied to the gas phase, Mgni

is

to be replaced by a newly defined Mnumn

i

and Mgni?

by˜Mnum

ni?in Equations (57) and (58).

˜Mnumni

�[(1 � M2

ref ii?)Mnum

ni� (1 � M2

ref ii?)Mnum

ni?]

2(60)

˜Mnumni?

�[(1 � M2

ref ii?)Mnum

ni?� (1 � M2

ref ii?)Mnum

ni]

2(61)

with:

Mnumni

�ui � nii?

cnumii?

(62)

Mnumni?

�ui? � nii?

cnumii?

(63)

For the formulation of ˜Mnumni

and ˜Mnumni?

reference is

made to Liou and Edwards (1999).

For each phase, the pressure terms at the cell interface

ii? can be written as:

Pg=sii?�Pii?

g=si�Pi?i

g=si?(64)

The superscript jk indicates information travelling

from j towards k .The acoustic flux terms are governed by the acoustic

wave speeds and the pressure splitting is weighted using

the polynomial expansion of the characteristic speeds.

The simplest first order form is:

½Mg=snii?

½B1

Pii?g=si

�Pg=si

2(1�Mg=s

nii?)

Pi?ig=si?

�Pg=si?

2(1�Mg=s

nii?)

(65)

½Mg=snii?

½]1

Pii?g=si

�Pg=si

2

�1�

Mg=snii?

½Mg=snii?

½

�

Pi?ig=si?

�Pg=si?

2

�1�

Mg=snii?

½Mg=snii?

½

� (66)

It can easily be seen that the pressure splitting is

nearly central for low Mach numbers. Therefore,

artificial dissipation should be added to the massbalances and to the total energy equations (Vierendeels,

Riemslagh & Dick, 1999). As in gas�/solid flow the

pressure drop in the tube is mainly due to hydrostatic

effect, i.e. the weight of the gas�/solid mixture in the

gravitational field, the artificial dissipation in terms of a

pressure difference is distributed over both phases,

following the weight fraction of both phases in the

mixture (De Wilde, 2001). Thus the artificial dissipationresults in following contributions to:

. solid phase mass balance:

Xcell interfaces

�dad

�osrsp

ogrg � osrsp

�ii?

(Pi? � Pi)

bn

� Dt � Sii?

Vi

�(67)

Fig. 1. Integration scheme for the flow field calculation based on

pseudo-time stepping coupled to a fourth order Runge�/Kutta scheme.


. gas phase mass balance:

Xcell interfaces

�dad

�ogrg

ogrg � osrsp

�ii?

(Pi? � Pi)

bn

� Dt � Sii?

Vi

�(68)

. gas phase total energy equation:

Xcell interfaces

�dad

�osrg

ogrg � osrsp

�ii?

Hgii?(Pi? � Pi)

bn

� Dt � Sii?

Vi

�(69)

with: wr� ½umax½ the maximum convective speed in the

flow field; dad�/0.5 a parameter that can be chosen;Hg�/Eg�/P /rg the enthalpy;

Hgii?�

(Hgi � H

gi?)

2(70)

½ii?½�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(xi?�xi)

2�(yi?�yi)2�(zi?�zi)

2

q

the distance between two points;

bn� ½wr½�2(mgii?

� mturgii?

)

rgii?

+ ½ii?½

for convective and viscous contribution, respectively.

4.2.2. Viscous fluxes

The viscous fluxes are calculated following a central

scheme. For complex geometrical structures, it is not

possible to find direct information about the gradientsin the x -, y -, and z -directions. This requires the

introduction of a local complete set of directions (j ,

h , z ) at every cell interface, such that information about

the gradients in these local directions is present in the

grid (Steelant & Dick, 1994). The j direction is chosen

normal to the cell interface and is the only direction in

which the gradient depends only on information from i

and i?. The h and z directions are chosen in the cellinterface and the gradients in these directions depend

also on information from non-neighboring grid points

(De Wilde, 2001). A coordinate transformation results

in the gradients in the x -, y-, and z -directions.

@F

@j

@F

@h

@F

@z

0BBBBBBBB@

1CCCCCCCCA�

@x

@j

@y

@j

@z

@j

@x

@h

@y

@h

@z

@h

@x

@z

@y

@z

@z

@z

0BBBBBBBB@

1CCCCCCCCA

�

@F

@x

@F

@y

@F

@z

0BBBBBBBB@

1CCCCCCCCA

(71)

4.3. Treatment of the source terms

Source terms occur in the Navier�/Stokes and in the

turbulence equations as a result of gas�/solid interac-

tions and of gravity. The turbulence equations contain

extra turbulence source terms.

The source terms are split into a positive and anegative part. The negative part is taken into account at

the iteration level (n�/1), the positive part at level (n)

(Steelant & Dick, 1994). Next, linearization of the

negative part is performed:

S�S�(n)�S�(n�1)

�S�(n)�S�(n)�@S�

@Q� (Q(n�1)�Q(n)) (72)

The separate treatment of the positive and the

negative part of the source terms, allows to increase

the diagonal dominance of the LHS matrix.

4.4. Relaxation method

For the integration in time and space, a dual time

stepping method and a finite volume technique are used,

respectively. The integration scheme is based on semi-

implicit pseudo-time stepping. A numerical stepper, thepseudo-time, is introduced for the solution of the non-

steady state problem:

@Q

@t�spatial terms 0

@Q

@t�

@Q

@t�spatial terms (73)

First order backward discretisation in physical and

pseudo time is performed. Thus, for a node i , equation

(73) becomes:

Dt

Dt(Q(t�Dt)(n�1)

i �Q(t�Dt)(n)

i )�Q(t�Dt)(n�1)

i

�Q(t)i �(Dt � spatial terms) (74)

with: i � /{1, 2, . . ., nn}.

The pseudo-time t is related to the iteration number

n , as each iteration performs a pseudo-time step Dt . Aconverged solution for Q[] at a certain physical time

level is obtained when @Q /@t�/0, i.e. when the differ-

ence in the values of Q[] between iterations vanishes.


Preconditioning is acting on the pseudo-time term, so

that the solution of the original equation set is not

modified.

Spatial discretisation and linearization of the con-servation equations result in the following matrix

formulation of the problem to be solved for Qtnew

[] at

each time level tnew:

a(Q[])Qtnew

[]� b(Q[]) (75)

with:

Q[]�

Q1

Q2

nQi

nQnp

0BBBBBB@

1CCCCCCA

(76)

The LHS of Eq. (75) contains all implicit information,

the RHS all explicit information. Both information from

the node i itself and information required from other

nodes i ? can be treated implicitly. Because of the non-

linearity of the equations, i.e. the fact that a(Q[]) and

b(Q[]) in Eq. (75) do depend on Q[], a solution is seeked

iteratively, starting from an initial guess. To obtain

convergence with an iterative method, it suffices that thematrix a (Eq. (75)) is diagonally dominant:

½a[j][j]½�Xnp

i�1; i"j

½a[j][i]½ (77)

This also guarantees the boundedness property for the

numerical integration scheme. To maximize the stability

of the numerical integration scheme, a point semi-

implicit approach is taken in this work. This means

that for the equations in a node i , information required

from the node i itself was treated implicitly i.e. put in the

LHS, whereas the information from the other nodes i ?required to calculate b was treated in an explicit way i.e.

put in the RHS. This reduces the matrix a(Q[]) in Eq.

(75) to a diagonal matrix:

a � DQ[]�b

with :

a�

A1

A2 0:::

Ai :::0 Anp

266666664

377777775�

::: 0D

0:::

24

35

and : DQ[i]�DQi[15�1] b[i]�Bii?i [15�1]

i � f1; 2; . . . ; nng (78)

This allows a pointwise solution of DQi for each

pseudo-time step Dt . Applying preconditioning, linear-

ization of the spatial derivative terms, and properlinearization of the source terms, finally results in the

following form of the conservation equations for each

node i :�Dt

DtG�I�Ai�

@S�

@Qt�

@K�

@Q

�(n)

(Q(t�Dt)(n�1)

i �Q(t�Dt)(n)

i )

��

Q(t)i �

Xi?

(A?i?(n)Q(t�Dt)(n)

i? �(I�A(n)i )Q(t�Dt)(n)

i

�C (n)hi�D(n)

zi�S(n)�K (n))

�(79)

The first term at the RHS of the equations resultsfrom the time dependent term (accumulation term) in

the equations. Ch and Dz contain information for the

viscous fluxes about the gradients in the local h ,

respectively, z direction. An extra superscript-index is

applied, representing the pseudo-time level of the

variables. Remark that the matrix Ai comprises a

summation over all the neighbors i?. Introducing Ai

and Bi , following simplified notation of Eq. (79) isobtained:

A(n)i DQ(t�Dt)

i �B(n)i (80)

Remark that, as in the Jacobi iteration scheme, theupdates in ALL nodes are calculated using information

on the previous iteration level before updating the

values to the new iteration level. Thus the numerical

method applied does not use the most recently available

Table 3

Simulation conditions

Case 1 Case 2 Case 3 Case 4 Case 5

�uzinb� (m s�1) 12.635

vzinb (m s�1) 6.0

Gs (kg m�2 s�1) 577.8

Uinb (m2 s�2) 20.0 20.0 0.0 20.0 0.0

e 1.0 0.995 0.995 1.0 0.995

rsp (kg m�3) 963.0

dp (mm) 60

Outlet type Straight Double abrupt


information. Although immediate use of the calculated

updates increases the convergence speed, the stability

becomes worse.

To further increase the numerical stability, thepseudo-time stepping is combined with a fourth-order

Runge�/Kutta technique which provides some under-

relaxation.

Q0�QnRK

Q1�Q0�a1 � DQ(Q0)

Q2�Q0�a2 � DQ(Q1)

Q3�Q0�a3 � DQ(Q2)

Q4�Q0�a4 � DQ(Q3)

Qn�1RK �Q4

� (81)

Fig. 1 shows the complete integration scheme for the

calculation of the flow field. Vierendeels (1998) proposes

a1�/1/4; a2�/1/3; a3�/1/2; and a4�/1 values adopted in

this work.

5. Case studies

All calculations were performed for an industrial scale

riser with a diameter of 1.56 m and a height of 14.434 m.

Two different outlet configurations are used (Fig. 2): astraight top outlet (configuration A) and two abrupt

side outlets with a surface area of 0.955 m2 each

(configuration C). For calculations with one abrupt

side outlet, reference is made to (De Wilde, Heynder-

ickx, Vierendeels & Marin, 2001).

A structured grid is used for the calculations. The 3D

grid is built up starting from a 2D cross section of the

tube (Fig. 3, top). The latter contains about 200 nodesand 67 cross sections are positioned axially in the tube

(Fig. 3, bottom). Grid refinement can be introduced in

the bottom and top section (Fig. 3, bottom) and at solid

bounding walls. Calculations on a finer grid and a grid

independency study were not performed due to the

extremely long computation times.

Table 3 summarizes the simulation conditions for the

different case studies.

5.1. Straight tube with bottom inlet and top outlet

5.1.1. Elastic particle�/particle collisions and high

granular temperature

The simulation conditions are shown in Table 3 (Case

1). The particle�/particle collisions are assumed elastic.

The solid phase is of Geldart A type.

A steady-state solution was obtained after 70 000

min of CPU time on an IBM RS/6000 397

workstation. Fig. 4 shows the convergence history as

a function ofthe iteration number for the first1200 iterations. The residual is defined as the

maximum value of the update in the field. It is seen

that the residual initially drops fast, but then the

convergence slows down. This could be avoided

by using a line-method instead of a point-method.

Although no cylindrical symmetry was imposed,

the solution is cylindrically symmetrical. Therefore,

the results can be presented for an axial

cross section of the riser through the center of the

tube.

In the axial direction, it is seen that the bottom zone

of the riser is more dense than the fully developed top

region (Fig. 5). Remark, however, that the high inlet

solid fraction of 10% (Table 3) decreases sharply within

the first centimeters of the reactor, due to the fast

acceleration of the particles from the inlet axial velocity

of 6.0 m s�1 to the gas phase velocity of about 12.6 m

s�1 by mixing and drag. The denser bottom entrance/

developing zone reaches to a height of about 3�/4 m.

Higher in the riser, the solids volume fraction os is seen

to decrease slowly, due to the increasing axial velocity

(Fig. 6) and the required conservation of mass. In the

radial direction, segregation of the particulate phase

near the solid wall is visible (Fig. 5) and a core-annulus

type of flow pattern develops. The annulus reaches 9/10

cm from the solid wall, which is 12.8% of the tube

radius, corresponding to 9/25% of the cross sectional

surface area. The annulus usually has a thickness, which

Fig. 2. Geometrical configurations used for the simulations.


is an order of magnitude smaller than the column

diameter (Lim, Zhu & Grace, 1995).

Fig. 6 shows the axial solid phase velocity vz in an

axial cross section of the riser. The linear acceleration

with height is remarkable. This acceleration is mainly

due to the expansion of the gas with height, as the

pressure decreases. In the radial direction, towards the

fully developed zone of the riser (z �/3�/4 m), a typical

turbulent velocity profile develops under the influence of

the viscous forces.

In the fully developed zone of the riser, the slip

between the gas and the solid phase is small, with values

between 0.05 and 0.1 m s�1, corresponding to the

terminal velocity of the particles (Froment & Bischoff,

1990).

To investigate the importance of the gas phase

turbulence, simulations were carried out in which the

gas phase turbulence is ignored. In the latter case, the

granular temperature drops quickly. This is due to the

drag term in the granular temperature transport equa-

tion Eq. (10). On ignoring the gas phase turbulence, the

gas�/solid turbulence correlation q12 becomes zero and

the drag term in Eq. (10) becomes a strong negative

source term. However, in the presented simulations that

take into account the gas phase turbulence, it is seen that

the gas and solid phase turbulence are very well

correlated (Fig. 7, Left Fig. 8: q12:/2k :/3U), making

the drag term in Eqs. (8) and (10) less important. Thus, a

possibility to simplify the equation set is to neglect the

drag term in the turbulence equations Eqs. (8) and (10)

(Nieuwland et al., 1996), omitting the calculation of the

gas�/solid turbulence correlation q12. Under these con-

ditions, the effect of putting the gas phase turbulence k

equal to zero on the value of the granular temperature

U is less pronounced. This should, however, be seen as a

positive result of the used k �/o �/U �/q12 model, resulting

in the prediction of good gas�/solid turbulence correla-

tion. However, in the development of the code, the gas

phase turbulence and the gas�/solid turbulence correla-

tion were not neglected because the code was intended

to run both dense and dilute cases. In the latter case, the

gas phase turbulence is believed to play a more

Fig. 3. Grid generation: * in a 2D cross section of the tube (top); *

axial positioning of the cross sections (bottom); (k: axial numbering of

the cross sections).

Fig. 4. Convergence behavior of the integration scheme. The residual is the maximum absolute value of the update of the viscous variables.


important role. Furthermore, it was observed that even

in some dense case studies, the gas phase turbulent

viscosity cannot be neglected. Also, the gas phase and

solid phase turbulence are not necessarily well corre-

lated.

5.1.2. Inelastic particle�/particle collisions and low

granular temperature

Changing the coefficient of restitution e from 1.0 to

0.995 (Case 2 and 3, see Table 3), results in a drastic

change of the hydrodynamic behavior. This is due to the

Fig. 5. Solids volume fraction (�/). Steady-state flow field Case 1 (conditions see Table 3).

Fig. 6. Solid phase axial velocity (m s�1). Steady-state flow field Case 1 (conditions see Table 3).

Fig. 7. Granular temperature U (m2 s�2): Case 1 left, Case 2 right (conditions see Table 3).


Fig. 8. Steady-state flow field Case 1 (conditions see Table 3).

Fig. 9. Time-dependent calculation of the solids volume fraction (os). Unsteady flow field Case 3 (conditions see Table 3).


strong dissipation of the granular temperature (Fig. 7).

The strong sensitivity towards the coefficient of restitu-

tion was also observed by many other researchers (Pita

& Sundaresan, 1991, 1993; Nieuwland et al., 1996).

With a restitution coefficient of 0.995, a steady-state is

not achieved and oscillating behavior is seen. Never-

theless, the solution remains axisymmetrical. Fig. 9

shows the time-dependent behavior of the solids volume

fraction in an axial cross section of the riser. A

perturbation in the solid volume fraction is seen to

originate at the top of the riser, grows from underneath

towards the bottom of the riser and is then blown out.

Large-scale fluctuations, with dimensions ranging up to

the pipe diameter, have been described by Dasgupta,

Jackson & Sundaresan (1994). A theoretical study on

the propagation of a voidage disturbance in uniformly

fluidized beds (Needham & Merkin, 1983), predicts that

slugging behavior can occur as a result of non-linear

effects. Whereas a linearized theory predicts exponential

growth of a disturbance in case of instability, this

growth is curbed by the non-linearity. This study reveals

that under conditions for which a steady state is

unstable to small-amplitude disturbances, the bed may

restabilize into a quasisteady periodic state. This is

clearly the behavior seen in this case study.

Chen, Gibilaro and Foscolo (1999) have also reported

perturbations in the solids volume fraction to primarily

being transported vertically. These authors further

Fig. 10. Time-dependent calculation of the pressure (P ) at the center line (time as in Fig. 9). Unsteady flow field Case 3 (conditions see Table 3).


reported that hydrodynamic interactions result in hor-

izontal transport of the perturbation. This phenomenon

is also seen in the present calculations.The oscillations are also found in the velocity (not

shown) and pressure fields (Fig. 10). Particle segregation

immediately causes a local increase of the pressure and

decrease of the axial velocity. Essential is the speed at

which the system reacts to correct the perturbation, in

other words the inertia of the system. It is seen that, as

the axial particle transport decreases locally, the inertia

of the system causes the particles feed not to be adapted

immediately. This results in growth of the perturbation.

As the particles are fed from the bottom and move

upwards, the zone of increased particle fraction grows

from underneath and thus shows a movement down-

ward, in the sense opposite to the convective direction.

Fig. 11. Time-dependent calculation of the granular temperature (U ) (time as in Fig. 9). Unsteady flow field Case 3 (conditions see Table 3).

Fig. 12. The solids volume fraction (os). Instantaneous flow field Case 4 (conditions see Table 3).


Furthermore, transfer in the radial direction occurs.

This phenomenon, the so-called ‘gravity wave’, is seen to

be a pressure wave. The low propagation speed for the

pressure is typical for two-phase flow.Buyevich (1999) states that in an assemblage of

particles supported by a flowing fluid, random fluctua-

tions of the local assemblage concentration appear. As

the drag force and other constituents of the interphase

interaction force are nonlinear functions of the concen-

tration, the concentrational fluctuations induce a fluc-

tuating random force exerted by the fluid on the

particles, thus violating the balance of gravity, buoy-ancy, and drag forces. Moreover, the concentration

fluctuations bring about fluctuations in the effective

two-phase mixture density, so that an additional fluc-

tuating force arises by the action of gravity. Buyevich

(1999) further states that, in a vertical fluidized bed

supported by an upward flowing fluid, the above

mentioned fluctuating forces accelerate the particles

either upward or downward, depending on the sign ofthe concentration fluctuations, so that primarily vertical

fluctuations occur. This is clearly demonstrated from the

calculation results in the present case and was confirmed

in other calculations.

Fluctuations and instabilities in the flow field result

from the convective, the acoustic, and the source terms.

The viscous terms on the other hand, are dampening the

fluctuations. This was clearly demonstrated by Muddeand Simonin (1999), who showed that when the

turbulent viscosity is high, all oscillations are dampened

and a steady solution is obtained. When the turbulent

viscosity becomes smaller, the flow becomes transient.

Needham and Merkin (1983) have also shown that the

solid phase viscosity gives rise just to dispersive effects.

Using a simple hyperbolic model (i.e. no solids viscosity

ms), Christie, Ganser and Sanzserna (1991), Christie andPalencia (1991) have been capable of reproducing

oscillatory behavior corresponding to slugging. Need-

ham and Merkin (1983) have demonstrated that both

type of models, containing and excluding the solids

viscosity ms, can exhibit smooth transition solutions

connecting two stable states.

A theoretical study by Needham and Merkin (1983)

has clearly shown the strong stabilizing influence of thesolid phase collisional pressure on gas�/solid flow. In

fact, neglecting the particle collisional pressure, the

fluidized bed is predicted to be always unstable to small

amplitude disturbances (Murray, 1965). The study by

Needham and Merkin (1983) has also revealed that the

exponential growth of disturbances in case of instability,

as predicted by the linearized theory, is curbed by the

non-linearity. Thus, in case the fluidized bed is unstableto small-amplitude disturbances, the bed may restabilize

into a quasisteady periodic state. Here again, the solid

phase collisional pressure is seen to play an important

role. Needham and Merkin (1983) conclude that, even

when the solid phase collisional pressure is not suffi-

ciently strong to stabilize the uniform state completely,

it must still be included to give the flow the possibility of

reaching a new, quasisteady state, as it is one of the maindissipative mechanisms in the system curbing the

unbounded growth of voidage perturbations.

Earlier, Fanucci, Ness and Yen (1979) revealed the

effect of the solid phase pressure function on the

possibility of shock formation in gas�/solid fluidized

beds. As the dependence of the solid phase pressure on

the solids volume fraction increases (i.e. @Ps/@os in-

creases), in other words, as the bed becomes more‘incompressible’, there is less chance for shock forma-

tion to occur and the gas�/solid flow becomes more

stable.

Using the KTGF, the solid phase pressure is strongly

related to the value of the granular temperature (Table

2) (Gidaspow, 1994; Nieuwland, 1995):

Ps� [1�2(1�e)osg]osrspU (15)

Looking at the calculational results, it is clear that thesolid phase pressure is much higher for elastic particle�/

particle collisions than for inelastic particle�/particle

collisions, due to the granular temperature (Fig. 7).

Furthermore, from Eq. (15), a higher value of the

granular temperature U results in an increased depen-

dence of the solid phase pressure Ps to the solids volume

fraction os (i.e. @Ps/@os is higher). According to Fanucci

et al. (1979), this leads to an improved stability, asindeed seen from the calculations. In case of oscillating

flow, it is also seen that the termination of the voidage

perturbation growth (Dt�/3.2 s) (Fig. 9) coincides with a

local increase of the granular temperature (Fig. 11) and

thus a local increase of the solid phase pressure and the

dependence of the solid phase pressure to the solids

volume fraction, most pronounced at the central axis of

the tube.Srivastava, Agrawal, Sundaresan, Karri and Knowl-

ton (1998) have experimentally shown that the onset of

unstable flow in CFBs roughly coincides with conditions

where the granular stresses and wall friction become

unimportant. In view of the preceding discussion, this

seems an explainable observation. Indeed, when the

viscous terms and the solid phase pressure become

unimportant, unstable flow develops.For the frequency of the porosity and pressure

fluctuations, the value of 0.15 Hz calculated in this

work, is in good agreement with the power spectrum

diagrams reported in literature (Tsuo & Gidaspow,

1990; Gidaspow, Huilin & Therdthianwong, 1995; Liu,

Grace, Bi, Morikawa & Zhu, 1999). The fact that the

frequency is somewhat lower may be due to the higher

length of the riser simulated in this work. The frequencyof large-scale oscillations produced by gravity decreases

with increasing riser height, as seen from the basic

incompressible frequency relation equal to the square


root of the gravity divided by the riser length (Sun &

Gidaspow, 1999). Applying this relation to the riser

simulated in this work, a typical oscillation frequency of

0.21 Hz should be seen, in good agreement with thecomputations.

Remark that the sensitivity towards the coefficient of

restitution for particle�/particle collisions can be ex-

plained as this parameter describes the energy dissipa-

tion resulting from one collision. As a result, the value of

the restitution coefficient has to be very accurate, which

is not the case. If many collisions take place, the energy

dissipation will follow an exponential rule. Introducinga parameter that describes the energy dissipation result-

ing from a number of collisions would reduce the

sensitivity.

Recent studies (Agrawal et al., 2001; Zheng, Wan,

Qian, Wei & Jin, 2001; Zhang & VanderHeyden, 2001,

2002) give a strong indication that the strong depen-

dency towards the coefficient of restitution e is due to

the inappropriate use of the KTGF. These studiesindicate that to obtain good results with the KTGF,

sufficient resolution is to be used, with grid sizes of only

several particle diameters. For industrial scale risers this

is impossible with the current computational resources.

If rough grids are used, sub grid corrections, accounting

for the turbulent motion of the solid phase, should be

introduced. Sub-grid models for the solid phase are still

in the early developing stage. An approach for devel-oping a sub-grid model is proposed by Agrawal et al.

(2001). Zheng et al. (2001) take into account the meso-

scale structures by using a k �/o �/kp�/op�/U model, in

which the kp�/op model is used to describe the turbulent

motion of the solid phase, whereas the KTGF is used to

take into account the collisions between the particles.

The Eulerian�/Eulerian approach and the KTGF

have, however, been used by many research groupsover the years without sub-grid scale corrections (Lun et

al., 1984; Sinclair & Jackson, 1989; Gidaspow, 1994;

Nieuwland et al., 1996), showing interesting results. The

use of the presented model allows to compare the

qualitative results and the model behavior from our

calculations with literature to get confidence in the

mathematical method implemented. The aim of the

present work is to develop a novel density-basedintegration technique for gas�/solid flow calculations

(Liou, 2001). The inclusion of a sub-grid model will only

result in changes in the constitutive equations and the

addition of some turbulence transport equations. The

mathematical treatment presented in this paper remains

valid.

5.2. Influence of the exit geometry: straight tube with

bottom inlet and two opposed abrupt side outlets

Calculations were performed making use of the

geometrical configuration with two side outlets at a

distance of 1.23 m from the top of the riser (configura-

tion C in Fig. 2). The simulation conditions are shown

in Table 3, Case 4 and 5. Both calculations with a

restitution coefficient for particle�/particle collisions

equal to 1.0 (elastic particle�/particle collisions) and

equal to 0.995 have been carried out.

There are two geometrical symmetry planes: a sym-

metry plane across the two outlets (cross section I) and

another symmetry plane between the two outlets (cross

section II) (Fig. 2). These symmetries were not imposed

and, surprisingly, are not found in the solution.

The flow structure in the top outlet region of the riser

with one recirculation eye was found to be stable. Thus,

to obtain a stable state, symmetry breaking has to take

place. This is clearly seen in Fig. 12, showing the solids

volume fraction os in cross sections I and II of the riser

and in Fig. 13 showing projections of the solid phase

velocity vectors in the same cross sections with focus on

the exit and top zone.

A symmetrical solution is not stable. Small perturba-

tions in the solids volume fraction always occur in a

gas�/solid system. In case of a stable solution, such

perturbations are dampened out and disappear, not

affecting the solution. However, if the symmetrical field

is unstable, small perturbations may cause the system to

evolve towards a stable non-symmetrical solution or to

oscillate. If a small perturbation in the solids volume

fraction occurs just beneath the outlets, the flow is

preferentially directed to one side of the riser, with

recirculation to the opposite side of the riser (Fig. 13).

Both sides have an equal probability of becoming the

preferential side. As a result, multiple solutions exist.

Switch-over during operation from one flow field to

another can take place due to large perturbations in the

riser or any other impulse in the system, as was seen in

case of inelastic particle�/particle collisions.

In the bottom and fully developed zone of the riser,

the profiles are very similar to those obtained with a

straight top outlet (Figs. 5 and 6), with only little effect

of the exit configuration seen. This is probably because

the total outlet surface area equals the horizontal riser

surface area and thus the restrictive strength of the exit

is small (Gupta & Berruti, 2000; De Wilde et al., 2001).Particle segregation is seen to occur near the solid

wall, at the external boundary of the vortex in the top

region of the riser, and where the upflow and recycle

downflow encounter (Fig. 12). The eye of the vortex

shows a low solids volume fraction, as expected from the

centrifugal effect due to the inertia of the solid phase.

In the bottom and middle part of the riser, the typical

velocity profiles develop as a result of the viscous and

the gravitational forces. As the flow approaches the

outlets, where the upflowing and downflowing recycled

suspension encounter, a zone of increased solids volume

fraction exists. As explained before, it originates from a


perturbation. It is explained next how this zone of

increased solids volume fraction is sustained.

Since the flow resistance is higher at the side suffering

the zone of increased solids volume fraction, the flow

is directed towards the opposite side which becomes

the preferential side for outflow. The side suffering

the zone of increased solids volume fraction becomes

the recycle side. The increased flow rate at one side of

the riser lowers the probability of solids segregation

to occur at that side of the riser. This is the

first stabilization mechanism for the asymmetric solu-

tion. Another reason for sustaining the zone of

increased solids volume fraction once it is formed, is

the fact that it is positioned where the gas�/

solids suspension flowing upwards from the bottom

inlet and the recycled gas�/solids suspension

flowing downwards from the top of the riser meet.

The inertia of the solid phase then results in

retaining the zone of increased solids volume fraction

at that location. This is a second stabilization mechan-

ism for the asymmetric solution obtained in the calcula-

tions.

Comparing the radial outflow at both outlets shows

that almost equal parts of the total flow are removed

from the riser through both the outlets. Thus, the

internal asymmetry of the riser is not easily detected

externally.

For inelastic particle�/particle collisions, an oscillating

solution, as in case of a straight top outlet, was

found. The period of the oscillation is 9/5 s, which is

somewhat smaller compared with the straight tube.

This is probably due to the shorter distance between

inlet and outlet, as the outlets are situated at 12.6 m

in this case. The frequency of large-scale oscillations

produced by gravity increases with decreasing

riser height, as seen from the basic incompressible

frequency relation equal to the square root of the

gravity divided by the riser length (Sun & Gidaspow,

1999).

The flow pattern in the top exit region of the riser

is influenced by the oscillations occurring in the

middle and bottom part of the riser. Fig. 14 focuses

on the time-dependency of the flow in the top exit

region of the riser, showing the solids volume

fraction and the cross sectional projections of the

solid phase velocity in cross sections I and II of the

riser.

At any time, an increased solids volume fraction at

the top of the riser and at the external boundary of the

vortex is calculated. This is the result of the inertia of the

particles.

The flow field at Dt�/3.6 s shows that the flow is

preferentially directed towards one outlet, with recircu-

lation to the other outlet. Again it is seen that both

outlets are responsible for approximately the same

outflux, one from upflow, the other from downflow.

Thus, again, the internal asymmetry of the riser is

externally not easily detected.

In the next 1.3 s, the orientation of the central vortex

is seen to turn from axial cross section I towards axial

cross section II (Fig. 14). This spiraling motion is

believed to persist in combination with the oscillation

in the middle and bottom section of the riser. The flow

field at Dt�/4.9 s is seen to be symmetrical in cross

section I.

Fig. 13. The cross sectional projection of the solids velocity (/v) (m s�1). Instantaneous flow field Case 4 (conditions see Table 3).


Fig. 14. The solids volume fraction (os) (color) and the cross sectional projection of the solids velocity (/v) (m s�1). Time-dependent calculation Case 5

(conditions see Table 3).


6. Conclusions

A novel integration scheme is developed for the

calculation of gas�/solid flow in circulating fluidizedbeds with the Eulerian�/Eulerian approach. The integra-

tion scheme is based on dual time stepping and a finite

volume technique. The advection upstream splitting

method AUSM is extended to two-phase flows. Pre-

conditioning is applied to accelerate the convergence.

Calculations on a simple straight tube geometry show

that the model behavior, reported in literature, is well

calculated. In particular with respect to the influence ofinelastic particle�/particle collisions, it is shown that the

latter result in a fluctuating flow field.

Calculations with a double abrupt side outlet config-

uration show that the geometrical symmetries are not

always followed by the flow field. Symmetry breaking

occurs, resulting in a stable asymmetric solution.

Acknowledgements

This work was financed by the European Commission

within the frame of the Non-Nuclear Energy Program,

Joule III project under contract JOF3-CT95-0012 and

Thermie project under contract no SF 243/98 DK/BE/

UK (project partners: Laboratorium voor Petrochem-

ische Techniek-Universiteit Gent, FLS miljo a/s-Den-

mark & Howden Air & Gas Division, UK). Geraldine

Heynderickx and Guy Marin are grateful to the ‘Fondsvoor Wetenschappelijk Onderzoek-Vlaanderen’ (FWO-

N) for financial support of the CFD Research. Ger-

aldine Heynderickx and Guy Marin are grateful to the

‘Bijzonder Onderzoeksfonds’ BOF-RUG for financial

support of the gas�/solid two-phase flow research.

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