Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 15.4.4 - Numerical simulation of turbulent gas-particle flow in a riser using a quadrature-based moment method
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 Material Information
Title: 15.4.4 - Numerical simulation of turbulent gas-particle flow in a riser using a quadrature-based moment method Fluidized and Circulating Fluidized Beds
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Passalacqua, A.
Fox, R.O.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: quadrature-based moment method
gas-particle flow
riser flow
 Notes
Abstract: Gas-particle flows are used in many industrial applications in the energy, oil and gas fields, such as coal gasification, production of light hydrocarbons by fluid catalytic cracking, catalytic combustion and different treatments aiming to reduce or eliminate pollutants. The particle phase of a gas-particle flow is described by analogy to a granular gas, by finding an approximate solution of the kinetic equation in the velocity-based number density function. In the recent past, many studies have been published on the mathematical modeling of gas-particle flows using hydrodynamic models (e.g. Enwald et al. 1996), where Navier-Stokes-type equations are solved to describe the particle phase as a continuum, computing its stress tensor using moment closures from kinetic theory (Gidaspow 1994). These closures, however, are obtained assuming that the flow is dominated by collisions and near equilibrium, which corresponds to considering a very small particle-phase Knudsen number. This assumption leads to inconsistencies and erroneous predictions of physical phenomena when these models are applied to dilute fluid-particle flows, where rarefaction effects are not negligible. In these flows, the wall Knudsen layers extend inside the bulk of the fluid, and cannot be accounted for with the simple addition of partial-slip boundary conditions. Recently Desjardin et al. (2008) showed that two-fluid models are unable to correctly capture particle trajectory crossing, seriously compromising their ability to correctly describe any velocity moment for finite Stokes numbers. These authors clarified that the particle segregation captured by two-fluid models for finite Knudsen numbers is artificially high due to their mathematical formulation, which leads to the formation of delta-shocks. In order to overcome these shortcomings, Fox (2008) developed a third-order quadrature-based moment method for dilute gas-particle flows, which has been successfully coupled to a fluid solver to compute dilute and moderately dilute gas-particle flows by Passalacqua et al. (2010) in two dimensions. These authors validated their model against Euler-Lagrange and two-fluid simulations. In this work, the fully coupled quadrature-based fluid-particle code described in Passalacqua et al. (2010) is applied to simulate turbulent gas-particle flow in the riser described by He et al. (2009), using a three-dimensional configuration. This application shows the predictive capabilities and the robustness of the quadrature-based moment method to predict the behavior of gas-particle flows in accordance with experiments (He et al. 2009).
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00379
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 1544-Passalacqua-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Numerical simulation of turbulent gas-particle flow in a riser using a
quadrature-based moment method


A. Passalacqua and R. O. Fox

Department of Chemical and Biological Engineering, Iowa State University, Ames, IA 50011-2230, USA
Salbertop~iastate.edu and rofox~iastate.edu

Keywords: Quadrature-based moment method, gas-particle flow, riser flow




Abstract

Gas-panticle flows are used in many industrial applications in the energy, oil and gas fields, such as coal gasification,
production of light hydrocarbons by fluid catalytic cracking, catalytic combustion and different treatments aiming to
reduce or eliminate pollutants. The particle phase of a gas-particle flow is described by analogy to a granular gas, by
finding an approximate solution of the kinetic equation in the velocity-based number density function. In the recent
past, many studies have been published on the mathematical modeling of gas-particle flows using hydrodynamic
models (e.g. Enwald et al. 1996), where Navier-Stokes-type equations are solved to describe the panticle phase as a
continuum, computing its stress tensor using moment closures from kinetic theory (Gidaspow 1994). These closures,
however, are obtained assuming that the flow is dominated by collisions and near equilibrium, which corresponds to
considering a very small particle-phase Knudsen number. This assumption leads to inconsistencies and erroneous
predictions of physical phenomena when these models are applied to dilute fluid-particle flows, where rarefaction
effects are not negligible. In these flows, the wall Knudsen layers extend inside the bulk of the fluid, and cannot be
accounted for with the simple addition of partial-slip boundary conditions. Recently Desjardin et al. (2008) showed
that two-fluid models are unable to correctly capture particle trajectory crossing, seriously compromising their
ability to correctly describe any velocity moment for finite Stokes numbers. These authors clarified that the particle
segregation captured by two-fluid models for finite Knudsen numbers is artificially high due to their mathematical
formulation, which leads to the formation of delta-shocks. In order to overcome these shortcomings, Fox (2008)
developed a third-order quadrature-based moment method for dilute gas-panticle flows, which has been successfully
coupled to a fluid solver to compute dilute and moderately dilute gas-panticle flows by Passalacqua et al. (2010) in
two dimensions. These authors validated their model against Euler-Lagrange and two-fluid simulations. In this work,
the fully coupled quadrature-based fluid-particle code described in Passalacqua et al. (2010) is applied to simulate
turbulent gas-panticle flow in the riser described by He et al. (2009), using a three-dimensional configuration. This
application shows the predictive capabilities and the robustness of the quadrature-based moment method to predict
the behavior of gas-particle flows in accordance with experiments (He et al. 2009).


Nomenclature K11 Knudsen number
Ma Mach number
Roman symbols
CD drag coefficient in, particle mass (kg)
rl, particle diameter (m1) n, quadrature weight
t: particle-particle restitution coefficient
e, particle-wall restitution coefficient Mf, momentum exchange due to drag (kg/(112 s2))
f distribution function Mi k velocity moment ((11/s)')
F force acting on each particle (N)
p pressure (Pa)
g gravitational acceleration (m1/s )
Pr Prandtl number
go radial distribution function
Re Reynolds number
I identity matrix







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


able to properly predict the behavior of a gas-solid flow
in a riser if all the interactions between the phases are
accounted for, and correlated these interactions to the
averaged and fluctuating components of the phase ve-
locity fields. Sinclair and Jackson (1989) also devel-
oped the first stationary hydrodynamic model able to
predict the particulate phase stresses through the kinetic
theory of the granular flow, as a function of the parti-
cle fluctuating energy (granular energy). Pita and Sun-
daresan (1991) modified the Sinclair and Jackson (1989)
model and validated it against Bader et al. (1988) exper-
imental data, showing its high sensitivity to the value
of the restitution coefficient, whose reduction may lead
to a wrong prediction of the particles segregation pat-
tems inside the duct. Lounge et al. (1991) introduced
a one-equation turbulence model to describe the gas
phase turbulence, and adopted standard wall-functions
for the zone near the wall. Gidaspow and Therdthi-
anwong (1993) adopted a zero-equation closure for the
gas phase turbulence. Ocone et al. (1993) extended the
work of Sinclair and Jackson (1989) to arbitrarily in-
clined ducts, accounting for the effects of particles slid-
ing and rotation. Bolio et al. (1995) adopted a low-
Reynolds two-equation /c-e model for the gas phase, to
eliminate the need for wall functions. Hrenya and Sin-
clair (1997) studied the influence of turbulence both on
the transport equations and on the kinetic theory clo-
sure equations, leading to a reformulation of the dissi-
pation term of the granular energy, which resulted in
a reduced sensitivity of the model to the value of the
particle restitution coefficient. Benyahia et al. (2000);
Arastoopour (2001) simulated a FCC riser using the ki-
netic theory of granular flow, neglecting the gas-phase
turbulence. Peirano and Leckner (1998); Peirano et al.
(2002); DeWilde et al. (2002) adopted a model that cou-
ples a two-equation turbulence model with a set of two
equations for the particulate-phase turbulent kinetic en-
ergy and for the gas-particle velocity correlation Gi-
daspow and Therdthianwong (1993); Samuelsberg and
Hjertager (1996); Mathiesen et al. (2000); Huilin and
Gidaspow (2003); Huilin et al. (2006) introduced the
Smagorinsky (1963) sub-grid stress model to describe
the turbulence of the gas phase following the principles
of large-eddy simulation. Recently Ibsen et al. (2004)
compared discrete methods and multi-fluid models in
circulating fluidised beds. Zeng and Zhou (2006) devel-
oped a two-scale second-order moment particle turbu-
lence model for dense gas-particle flows. Moreau et al.
(2009) proposed a new large-eddy simulation approach
for particle-laden turbulent flows in the framework of the
Eulerian formalism for inertial particles. He et al. (2009)
performed experiments in a riser with mono-dispersed
particles and compared the measurements with the re-
sults of numerical simulations performed using the La-


St Stokes number


t time (s)

U velocity (m/s)

Uai quadrature abscissa (m/s)
v particle velocity (m/s)
x position vector
Greek symbols
volume fraction
agje set of equilibrium moments
8 granular temperature (m /s )
Dynamic viscosity (Pa s)

p density (kg/m )

ogy particle-phase stress tensor components
Ty fluid-phase tensor
7o collision time (s)
Subscripts
f Fluid
p Particle

Superscipts
f Fluid
p Particle

Introduction

Gas-particle flows in risers have been the topic of ex-
tensive research in order to develop reliable computa-
tional models capable of describing their peculiarities,
Generally speaking, two kind of approaches are possi-
ble to describe the particle phase: the Lagrangian ap-
proach, where each particle trajectory is resolved inde-
pendently, applying fundamental laws of mechanics, and
the Eulerian approach, in which the particle phase is de-
scribed by transport equations of moments of the par-
ticle velocity distribution function. The computational
convenience and the absence of statistical noise char-
acteristic of Eulerian models made them very attractive
both for research and applications, and \ignilk~.llll effort
to improve their formulation has been spent in the last
two decades. Syamlal and Gidaspow (1985); Gidaspow
(1986) developed hydrodynamic models for CFB reac-
tors, accounting for heat transfer and introducing a nor-
mal stress modulus for the particulate phase. Gidaspow
et al. (1989); Tsuo and Gidaspow (1990) adopted the
Wen and Yu (1966) drag correlation in their model, and
properly predicted flow regimes typical of the circulat-
ing fluidised bed risers. Sinclair and Jackson (1989);
Sinclair (1997) showed that a hydrodynamic model is







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


The collision term C is described using the Bhatnagar-
Gross-Krook (BGK) collision operator (Bhatnagar et al.
1954):
C 1 (es ) ,(5)

where 7, is the collision time and fes is the equilibrium
distribution function, extended to account for inelastic
collisions:


grangian discrete particle model (DPM) approach.
In this work we present the results of the simulations
of the same system considered by He et al. (2009) ob-
tained with a fully three-dimensional implementation of
the quadrature-based moment method developed by Fox
(2008) and coupled with a fluid solver by Passalacqua
et al. (2010). Results are compared to experimental data,
statistics of the particle phase are computed and dimen-
sionless parameters such as the particle-phase Mach,
Knudsen and Stokes numbers of the flow are examined.


Model description

Fluid-phase governing equations. The fluid phase,
assumed to be incompressible and isothermal, is de-
scribed by a continuity and a momentum equation as
in traditional two-fluid models (Drew 1971; Gidaspow
1994; Enwald et al. 1996). The fluid continuity equation
has the form


fes =
[det (2i7r)]1/

1 (-(


ct~x 1~?1 tl) -


where A1 is the inverse of the the matrix A, defined by

A = yW 8,I + (yW2 2yw + 1) o (7)

with y = 1/ Pr, w = (1 + e)/2, \Il" the number den-
sity of particles (zero-order moment), U, the mean par-
ticle velocity, e the restitution coefficient, 8, the gran-
ular temperature, and 0- the velocity covariance matrix.
In this work y 1 so that Pr 1 in the standard BGK
model (Struchtrup 2005).
In this work, following Fox (2008); Passalacqua et al.
(2010), a set of twenty moments of f up to third order
is considered. Each moment is defined through integrals
of the distribution function as


i)

and the fluid momentum equation is


(QfPpUf) + V -(QfPpUfUf) =
Of


0, (1)


V (agef) ar~lp + afppg + Mfp, (2)


ii" = fdy,


Afi' = c; lfdy,

Aff = Sv


where at, pc, Uf are the fluid-phase volume fraction,
density and mean velocity, Mf, represent the momen-
tum exchange term due to the drag between the fluid
and particle phases, and g is the gravitational acceler-
ation vector.
The stress tensor 7f is


Transport equations for the moments are obtained by ap-
plying the definition (8) to both sides of the kinetic equa-
tion, leading to (Fox 2008)


-Pr (V Ur) I, (3)


.i = pr VOUr + (VUr)'


a~t itrj


where pr is the dynamic viscosity of the fluid and I the
unit tensor.
Particle-phase governing equations. The particle
phase is described in analogy to a gas made of smooth,
mono-disperse, non-cohesive spheres. Its governing
equation is represented by a kinetic equation for the par-
ticle number density function f (t, x, v) (Chapman and
Cowling 1961; Cercignani et al. 1994; Struchtrup 2005)


VYi)Af i)Af"k
dt itrg


"3 3

Cs A3


where ,4,1, A24@ and 4 p are the source terms due to the
acceleration acting on each particle, and C,' and CL're
are those due to the collision operator.
The set of transport equations (9) is not closed, be-
cause each equation contains the spatial flux of the mo-
ments of order immediately higher, and the source terms
due to the force and to collisions. Gaussian quadrature


O~f O~f i) F
+ v- + f-
DLt i~x i"vin,


where F is the force acting on each particle, including
gravity and drag, and C represents the rate of change
in the number density function due to binary collisions
between the particles.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


fonnulae are used to provide closures for these source
tenns as a function of a set of weights n, and abscissas
U,. Weights and abscissas are computed from the set
of transported moments by means of an inversion algo-
rithm. Following Fox (2008), a set of eight weights and
abscissas per each velocity component is used.
The moments can be computed as a function of the
quadrature weights and abscissas using the definition
(8):




1
(10)

M = n ni~a,




The source tenns due to drag and gravity are computed








rnFD ; mP, (1

+ ye~ + gy Use ,

(Fp
m'P (11 UloU

( FD," Se,




Thedrg tme copue gyr Ukach veoiyasis,

(F 1(3


Ther h dragfocle trmsae givesn byr isdfndfre


The drsag tie optdfrec vlct bcsas


and the drag coefficient CD is modeled according to Wen
and Yu (1966):
24 26
CD(Rep, ar) = [1 + 0.15(at Re,)0.687] 2.6
at Rep
(15)
The source terms in the moment transport equations
due to collisions are given by





in which the collision time is defined by

To = ,(17)

with 8, defined in tenns of the moments by






1 1 -1 \


933

The radial distribution function of Carnahan and Starling
(1969) is adopted:

go = 1 30 + (20)
1 p 2 (1 ap)2 2(1 ap)3
The moment spatial fluxes are represented by the sec-
ond tenn on the left-hand side of (9), and are computed
according to their kinetic definition (Perthame 1990;
Desjardin et al. 2008; Fox 2008) in order to ensure the
realizability of the set of moments.
Boundary conditions. Boundary conditions for the
moment transport equations are specified in terms of the
quadrature approximation, as shown in Fox (2008); Pas-
salacqua et al. (2010). The specular reflective condition
at walls is defined as



Uon2 -ewUn2 (1


where ew is the restitution coefficient for collisions be-
tween a particle and the wall, and i = 0 represents the
wall, assumed to be along the second direction in the
considered reference frame. Periodic boundary condi-
tions are imposed by enforcing the periodicity on the set
of weights and abscissas.


Repo


pfd, | Uf U, |













W


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010



Table 1: Dimensions of the riser and flow properties
Variable Value Units
if 0.05 11
H 1.5 11
D 0.015 11
L 0.30 11
d, 335.0 p
p, 2500.0 kg/m13
e, 0.97
Gs 10.0 kg/(In' s)
Pc 1.2 kg/m13
py 1.8 105 Pa s
Uf 2.3 11/s
Ut 2.7 li/s


The fluid-phase mass flux has been imposed to match
the mean particle mass flux Gs, by adapting the pressure
gradient along the height of the computational domain.
The grid density used in the simulations, taken from
the work of He et al. (2009), is 25 grid points along 10,
60 along L and 10 along D. Simulations were performed
with an adaptive time stepping, based on the conver-
gence of the fluid-phase residuals (Syamlal et al. 1993),
on the QMOM CFL condition, on the particle collision
time and the drag time. Simulations are performed for
a total of 10 s of simulation time, while time averages
are computed on the last 5 s of simulation time. Since
the domain is periodic, averages are computed assuming
the z direction to be homogeneous, and considering a
symmetry plane, located at 1C/2, normal to the Jr direc-
tion. The RMS values are found by subtracting the time-
averaged values from the instantaneous fields, and time
averaging the square of the differences. The final RMS
value are then the square root of time-averaged square
differences.
The fluid-phase equations are solved with an itera-
tive procedure based on the SIMPLE (Patankar 1980;
Ferziger and Peric 2002), and coupled with the QMOM
equations using the partial elimination algorithm (Spald-
ing 1980), as shown in Passalacqua et al. (2010). The
second-order scheme with superbee limiter was used to
discretize the fluid-phase equations. As convergence cri-
terion for all the variable of the fluid-phase, a reduction
of the residuals below 104 was required to consider the
solution converged.


Results and discussion

An example of the instantaneous solids volume fraction
field at t = 8 s obtained in the simulation performed
using the quadrature-based moment method is reported


Figure 1: Schematic representation of the riser consid-
ered in He et al. (2009). The hashed area rep-
resents the actual portion of the riser consid-
ered in the simulations.


Test case description

The model presented above has been implemented in the
CFD code MFIX (Syamlal et al. 1993; Syamlal 1998)
as detailed in Passalacqua et al. (2010), and is validated
here against the experimental measurements in a gas-
particle flow riser, with mono-disperse particles, realized
by He et al. (2009). The experimental setup used in this
reference is constituted by a riser column of rectangular
section, as schematically represented in Fig. 1. The di-
mensions of the riser and the properties of the fluid and
particle phases are reported in Table 1


Simulation setup

The computational domain considered in this work rep-
resents only a portion of the whole riser examined in the
experiments of He et al. (2009), in the center of the col-
umn, with a height L, as represented by the dashed vol-
ume in Fig. 1. Periodic boundary conditions were im-
posed at the top and at the bottom of the computational
domain, while no-slip conditions were used for the fluid
phase at the wall, and reflective conditions with resti-
tution coefficient e w were used for the particle phase.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


0.025


0.020



$ 0.015


$ 0.010
s

P~0.005


~0.04


.0.03



0.01
0.006


0.000
0.000


0.005 0.010


0.015 0.020


0.025


x [m]


Figure 3: Time-averaged particle volume fraction.



in acceptable agreement with the experimental measure-
ments of He et al. (2009), and similar to those obtained
by the same authors performing Lagrangian simulations.

The convergence of the averages is considered satis-
factory, in spite of the relatively short averaging time,
since the values of the time-averaged velocity along the
x direction (Fig. 5), are approximately zero, as expected
when the flow reaches the steady-state condition. The
root-mean-square (RMS) of the vertical component of
the velocity of the particle phase is reported in Fig. 6,
and is in qualitative agreement with the experiments of
He et al. (2009), but not in quantitative agreement. Both
the experimental and the computational profiles show a
minimum at the riser centerline, and the maximum value
in proximity of the wall, but at a certain distance from
it. The RMS velocity in the x: direction, reported in
Fig. 7 is in quantitative agreement with the experimental
data. Differences in the numerical predictions compared
to the experimental data might depend on the simplified
numerical configuration used in the simulation, where
only one portion of the system is considered, and peri-
odicity is assumed in the axial direction. Further reasons
that could explain the differences are the choice of the
drag law, as well as other sub-models used in the numer-
ical model, and the systematic errors in the experimental
measurements (He et al. 2009).
The time-averaged granular temperature is reported in
Fig. 8. (Note that the granular temperature is not re-
lated to the RMS velocity statistics!) The values of 8,


Figure 2: Instantaneous solids volume fraction field at
t 8 s across the centerline (D/2).


in Fig. 2. It can be seen that particles segregated at the
walls of the riser, forming ensembles at higher volume
fraction, which tends to move downward, since their
weight wins the resistance exerted by the fluid. As ex-
plained in Passalacqua et al. (2010), the flow evolves
from the uniform initial conditions through an interme-
diate state where particles, due to the reflective condi-
tions at the walls, tend to originate two vertical stripes
parallel to the walls, with lower particle concentration.
In these stripes the fluid accelerates, and the difference in
shear causes the flow to become unstable. The instabil-
ity quickly propagates to the whole flow, leading to the
segregation of particles typically observed in riser flows.
The time-averaged volume fraction, shown in Fig. 3 (ex-
perimental data is not available for this quantity), is low-
est in the center of the flow and highest at the walls,
which is typical of core-annular flow.
Results of the time-averaged vertical component of
the velocity obtained in the simulation performed us-
ing the quadrature-based moment method are reported
in Fig. 4 and compared with the experiments of He et al.
(2009). In all the reported plots, x: 0 represents the
channel centerline, and x: = 0.025 m indicates the chan-
nel wall. The predicted values of the axial velocity show
a core-annular behavior, with negative values at the wall,
indicating particles fall in that region of the system, and
positive values in the center of the riser. Results are







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


m
E o.
x


QMOM
a Exp, Ug= 2
m/s
0.03
SExp, Ug =2.3
m/s

0.02-






0.00-


-0.01
0.000 0.005 0.010 0.015 0.020 0.025

x (m)


Figure 5: Time-averaged x: component of the particle
velocity.


cording to the region where it is computed, following
Kogan (1969), so that


0.000 0.005 0.010 0.015 0.020 0.025

x (m)


Figure 4: Time-averaged z component of the particle
velocity.



present a minimum at the wall of the channel, where
the collision frequency is higher due to higher particle
concentration, due to particle segregation, and a maxi-
mum in the core of the riser, where the flow is dilute
and collisions are not predominant, as shown in Fig. 10.
Before proceeding to consider the behavior of the Knud-
sen number in detail, it is worth noticing that the flow
is in transonic conditions, meaning that there are parts
of the computational domain where the particle veloc-
ity is above or equal to the local particle-phase 'speed of
sound' (Ma > 1), and other parts of the system where
the particle velocity is below the the local value of the
speed of sound (Ma < 1). This is evident from the
values of the time-averaged local particle Mach num-
ber Ma | Up|/0,1/ in Fig. 9, which shows the Ma
is approximately between 6 x 103 and 1.8. Under
these conditions, two regimes are present in the system.
Where Ma < 1, the flow is dominated by diffusive pro-
cesses, regulated by the local value of the granular tem-
perature, which has to be used to compute the character-
istic velocity in this regime. When Ma > 1, the flow is
dominated by convective phenomena, meaning that the
transport of properties is mainly due to the convective
transport of particles more than to diffusive phenomena,
and the granular temperature has to be replaced by the
local mean velocity magnitude in the definition of the
characteristic velocity of the flow. The two zones are
separated by a dashed line in Fig. 9. Since the flow un-
dergoes a transition between two different regimes, the
definition of the Knudsen number has to be modified ac-


K~n =~V L Ol/


Ma< 1
Ma> 1


where the collision time is given by

To = 12goap 8/ (23)

The values of the Knudsen number, computed as-
suming L = 2WD/(W + D), according to the def-
inition of hydraulic diameter of the riser, are reported
in Fig. 10. The diagram shows the flow transitions
from the slip regime, where 0.01 < K~n < 0.1 (Bird
1994), in the region adjacent to the wall, to a more rar-
efied regime (transitional regime) in the center of the
riser, where non-equilibrium phenomena are expected
to happen. The two regions are separated by the dot-
ted line. The lower value of K~n at the wall is justi-
fied by the higher particle concentration in that region
of the system, which leads to higher values of the col-
lision frequency, making the flow locally dominated by
collisions. It is worth noticing that if an hydrodynamic
model (Gidaspow 1994; Enwald et al. 1996), derived in
the by! polhesc'i\ of nearly zero Knudsen number (Contin-
uum regime, K~n < 0.01) were used to perform the sim-
ulation, the adoption of partial slip boundary conditions
(Johnson and Jackson 1987) would have been necessary
to describe the behavior of the flow in the zone adjacent







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


S0.04-





a Exp.
0.00
0.000 0.005 0.010 0.015 0.020 0.025

x (m)


Figure 7: RMS of the Jr component of the particle ve-
locity.


mulation of particles (delta-shocks), as shown in Des-
jardin et al. (2008), who simulated the behavior of par-
ticles in Taylor-Green flow at different Stokes numbers,
and compared the solutions obtained with Lagrangian
simulations, Eulerian quadrature-based simulations, and
hydrodynamic models.

Conclusions

A portion of a riser of a circulating fluidized bed have
been simulated using a third-order quadrature-based mo-
ment method in a fully three-dimensional numerical
setup, showing the robustness and the capability of the
quadrature-based moment method for this kind of ap-
plications. Results for the time-averaged mean particle
velocity were found in satisfactory agreement with the
experimental results in spite of the simplifying assump-
tions made in the simulation about the periodicity of the
computational domain. RMS velocities were found in
qualitative agreement with experiments, but not in quan-
titative agreement, for what concerns the vertical com-
ponent of the velocity.
Characteristic dimensionless parameters of the parti-
cle flow were examined. The Mach number showed that
the panticle flow is in transonic conditions, with the sub-
sonic region adjacent to the wall. The values of the local
Knudsen number pointed out the flow is in a condition
across two regimes: the slip regime (zone adjacent to the
wall) and the transition regime (center of the channel),
indicating that the adoption of hydrodynamic models in


-QMOM
1.3 -a Exp.

1.1-

0.9-

S0.7-

S0.5

0.3- a"

0.1.

-0.1
0.000 0.005 0.010 0.015 0.020 0.025

x (m)


Figure 6: RMS of the z component of the particle ve-
locity.


to the walls, where K~n < 0.1, however the model would
have deviated from the correct behavior in the center of
the riser, where higher values of the Knudsen number are
present, and higher-order approximations of the kinetic
equation than the hydrodynamic are necessary.
The Stokes number profile is reported in Fig. 11, and
reports values between 66.1 and 123.5, which indicate
particles reaction to changes in the local conditions of
the fluid flow are not instantaneous, but delayed and,
since the flow is dilute, might lead to panticle trajec-
tory crossing. As pointed out in Desjardin et al. (2008),
such phenomenon cannot be predicted by hydrodynamic
models, since they only consider moments up to the sec-
ond order, and define only one local velocity in each
computational cell, whereas multiple local velocities are
necessary to be able to capture the discontinuous veloc-
ity field that originates when panticle trajectory cross-
ing occurs. This becomes clear considering the panti-
cle velocity distribution function. The hydrodynamic
models are obtained assuming equilibrium or nearly-
equilibrium conditions, imposing that the velocity distri-
bution is Maxwellian. When particle trajectory crossing
occurs faster particles pass slower particles, locally the
velocity becomes multi-valued and the distribution func-
tion strongly deviates from the equilibrium condition,
originating discontinuities in the velocity field, which
cannot be described under the hydrodynamic hy! po lhes\is
since single-valued velocity is assumed in these models.
Hydrodynamic models hypothesize the flow is always
dominated by collisions, and predict unphysical accu-








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


0.060


E~0.055


a 0.050


S0.045-


0.040-


S0.035


0.030
0.000 0.005 0.010 0.015 0.020 0.025

x [m]



Figure 8: Time-averaged granular temperature profile.


0.000 0.005 0.010 0.015 0.020 0.025

x (m)



Figure 10: Time-averaged Knudsen number profile.


2.03-

1.83-

1.63-

1.43-

1.23-

cj 1.03-

0.83-

0.63-

0.43-

0.23-

0.03-
0.000


120-


110-


100-


90


80-


70-


60-
0.000


0.005 0.010 0.015 0.020 0.025

x [m]


0.005 0.010 0.015

x (m)


0.020 0.025


Figure 9: Time-averaged Mach number profile.


Figure 11: Time-averaged Stokes number profile.












such a regime would be inappropriate. This was further
confirmed by considering the local values of the Stokes
number, which clarified that particles do not immedi-
ately react to the fluid flow, and can originate particle
trajectory crossing wherever the flow is not dominated
by collisions (high K~n).
Future work will involve the simulation of systems
with more realistic boundary conditions and geometries
than those considered in the present work and the exami-
nation of denser flow conditions. Additionally, idealized
simulations in domains with periodic conditions in all
the spatial directions will be performed to further inves-
tigate the formation of clusters and the conditions that
lead to flow instabilities at the base of the particle segre-
gation.


Acknowledgements


This work has been supported by the National Energy
Technology Laboratory of the U.S. Department of En-
ergy under the award number DE-FC26-07NT43098.


References


H. Arastoopour. Numerical simulation and experimental
analysis of gas/solid flow systems: 1999 Fluor-Daniel
plenary lecture. Powder Technol., 119:59-67, 2001.

R. Bader, J. Findlay, and T. Knowlton. Gas-solids flow
patterns in a 30,5 cm diameter fluidized bed. In P. Basu
and J. F. Large, editors, Circulating Fluidized Bed Tech-
,. 1. . \ II, pages 123 137. Pergamon Press, 1988.

S. Benyahia, H. Arastoopour, T. M. Knowlton, and
H. Massah. Simulation of particles and gas flow behav-
ior in the riser section of a circulating fluidized bed us-
ing the kinetic theory approach for the particulate phase.
Powder Technol., 112:24-33, 2000.

P. L. Bhatnagar, E. P. Gross, and M. Krook. A model for
collisional processes in gases. I. Small amplitude pro-
cesses in charged and neutral one-component systems.
Phys. Rev., 94:511-525, 1954.

G. A. Bird. Molecular Gas Dynamics and the Direct
Simulation of Gas Flows. Oxford University Press,
1994.

E. J. Bolio, J. A. Yasuna, and J. L. Sinclair. Dilute tur-
bulent gas-solid flow with particle-particle interaction.
AIChE J., 41:1375 1388, 1995.

N. F. Carnahan and K. E. Starling. Equation of state for
nonattracting rigid spheres. J. Chein. Phys., 51(2):635 -
636, 1969.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


C. Cereignani, R. Illner, and M. Pulvirenti. The Mathe-
inatical Theory of Dilate Gases. Springer-Verlag, 1994.

S. Chapman and T. G. Cowling. The Mathematical The-
ory ofNon-uniforin Gases. Cambridge University Press,
2 edition, 1961.

O. Desjardin, R. O. Fox, and P. Villedieu. A quadrature-
based moment method for dilute fluid-particle flows. J.
Comput. Phys., 227:2524-2539, 2008.

J. DeWilde, G. J. Heynderickx, J. Vierendeels, E. Dick,
and G. B. Marin. An extension of the preconditioned
advection upstream splitting method for 3d two-phase
flow calculations in circulating fluidized beds. Comput.
Chein. Eng., 26:1677 1702, 2002.

D. A. Drew. Averaged equations for two-phase flows.
Stud. Appl. Math., L(3):205 231, 1971.

H. Enwald, E. Peirano, and A. E. Almstedt. Eulerian
two-phase flow theory applied to fluidization. Int. J.
Multiphase Flow, 22:21-66, 1996.

J. H Ferziger and M. Peric. Computational Methods for
Fluid Dynamics. Springer, 2002.

R. O. Fox. A quadrature-based third-order moment
method for dilute gas-particle flows. J. Comput. Phys.,
227:6313-6350, 2008.

D. Gidaspow. Hydrodynamics of fluidization and heat
transfer: supercomputer modeling. Appl. Mech. Rev.,
39:1-22, 1986.

D. Gidaspow. Multiphase Flow and Fluidization. Aca-
demic Press, 1994.

D. Gidaspow and A. Therdthianwong. Hydrodynamics
and SO, sorption in a CFB loop. In A. A. Avidan, editor,
Circulating fluidized bed -,. Im,. 1. . \ IV, pages 32 39.
AIChE, New York, 1993.

D. Gidaspow, Y. P. Tsuo, and K. M. Luo. Computed
and experimental cluster formation and velocity profiles
in circulating fluidized beds. In J. C. Grace, L. W.
Schemilt, and G. Sun, editors, Fluidization IV, pages 81
- 88. Engineering Foundation, New York, 1989.

Y. He, N. Deen, M. Van Sint Annaland, and J. A. M.
Kuipers. Gas-solid turbulent flow in a circulating flu-
idized bed riser: Experimental and numerical study of
monodisperse particle systems. Ind. Eng. Chein. Res.,
(48):8091 8097, 2009.

C. M. Hrenya and J. L. Sinclair. Effects of particle-phase
turbulence in gas-solid flows. AIChE J., 43(4):853-869,
1997.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


B Perthame. Boltzmann type schemes for compressible
Euler equations in one and two space dimensions. SIAM
J. Num. Anal., 29(1):1-19, 1990.

J. A. Pita and S. Sundaresan. Gas-solid flow in vertical
tubes. AIChE J., 37:1009 1018, 1991.

A. Samuelsberg and B. H. Hjertager. An experimental
and numerical study of flow patterns in a circulating flu-
idized bed reactor. Int. J. Multiphase Flow, 22(3):575 -
591, 1996.

J. L. Sinclair. Hydrodynamic modeling. In J. R. Grace,
A. A. Avidan, and T. M. Knowlton, editors, Circulating
Fluidized Beds, chapter 5, pages 149-180. Blackie Aca-
demic & Professional, London, la edition, 1997.

J. L. Sinclair and R. Jackson. Gas-particle flow in a ver-
tical pipe with particle-particle interaction. AIChE J.,
35:1473-1486, 1989.

J. Smagorinsky. General circulation experiments with
the primitive equations. Mon. Weather Rev., 91:99 164,
1963.

D.B. Spalding. Numerical computation of multi-phase
fluid flow and heat transfer. In C. Taylor, editor, Re-
cent Advances in Numerical Methods in Fluids. Piner-
idge Press, 1980.

H. Struchtrup. Macroscopic Transport Equations for
Rarefied Gas Flows. Springer, Berlin, 2005.

M. Syamlal. MFIX Documentation Numerical Tech-
nique. U.S. Department of Energy, Office of Fossil En-
ergy, Federal Energy Technology Center, Morgantown,
West Virginia, 1998.

M. Syamlal and D. Gidaspow. Hydrodynamics of flu-
idization: prediction of wall to bed heat transfer coeffi-
cient. AIChE J., 31:127 135, 1985.

M. Syamlal, W. Rogers, and T. J. O'Brien. MFIX Doc-
umentation Theory Guide. U.S. Department of Energy,
Office of Fossil Energy, Morgantown Energy Technol-
ogy Center, Morgantown, West Virginia, 1993.

Y. P. Tsuo and D. Gidaspow. Computation of flow pat-
terns in circulating fluidized beds. AIChE J., 86:886 -
896, 1990.

C. Y. Wen and Y. H. Yu. Mechanics of fluidization.
Chem. Eng. Prog. S. Ser, 62:100-111, 1966.

Zh. X. Zeng and L. X. Zhou. A two-scale second-
order moment particle turbulence model and simulation
of dense gas-particle flows in a riser. Powder Technol.,
162(1):27 32, 2006.


L. Huilin and D. Gidaspow. Hydrodynamics of binary
fluidization in a riser: CFD simulation using two gran-
ular temperatures. Chem. Eng. Sci., 58:3777 3792,
2003.

L. Huilin, Z. Yunhua, S. Zhiheng, J. Ding, and J. Juying.
Numerical simulation of gas-solid flow in tapered risers.
Powder Technol., 169:89 98, 2006.

C. H. Ibsen, E. Helland, B. H. Hjertager, T. Solberg,
L. Tadrist, and R. Occelli. Comparison of multifluid and
discrete particle modelling in numerical predictions of
gas particle flow in circulating fluidised beds. Powder
Technol., 149(1):29 41, 2004.

P. C. Johnson and R. Jackson. Frictional-collisional
constitutive relations for granular materials, with appli-
cations to plane shearing. J. Fluid Mech., 176:67-93,
1987.

M. N. Kogan. Rarified Gas Dynamics. Plenum Press,
New York, 1969.

M. Lounge, E. Mastorakos, and J. T. Jenkins. The role of
particle collision in pneumatic transport. J. Fluid Mech.,
231:345 359, 1991.

V. Mathiesen, T. Solberg, and B. H. Hjertager. An ex-
perimental and computational study of multiphase flow
behavior in a circulating fluidized bed. Int. J. Multiphase
Flow, (26):387 419, 2000.

M. Moreau, O. Simonin, and B. Bedat. Development of
gas-particle Euler-Euler LES approach: a priori analy-
sis of particle sub-grid models in homogeneous isotropic
turbulence. Flow Turbul. Combust., 84(2):295 324,
2009.

R. Ocone, S. Sundaresan, and R. Jackson. Gas-particle
flow in a duct of arbitrary inclination with particle-
particle interactions. AIChE J., 39:1261 1271, 1993.

A. Passalacqua, R. O. Fox, R. Garg, and S. Subra-
maniam. A fully coupled quadrature-based moment
method for dilute to moderately dilute fluid-particle
flows. Chem. Eng. Sci., 65(7):2267 2283, 2010. DOI:
10.1016/j.ces.2009.09.002.

S. Patankar. Numerical Heat Transfer and Fluid Flow.
Taylor & Francis, 1980.

E. Peirano and B. Leckner. Fundamentals of turbulent
gas-solid flows applied to circulating fluidized bed com-
bustion. Prog. F,, r Combust. Sci., 24:259-296, 1998.

E. Peirano, V. Delloume, F. Johnsson, B. Leckner, and
O. Simonin. Numerical simulation of the fluid dynamics
of a freely bubbling fludized bed: influence of the air
supply system. Powder Technol., 122:69-82, 2002.




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