Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 13.5.3 - The Sub-grid Hydrodynamic Behavior of Accelerating Gas-solid Flows
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 Material Information
Title: 13.5.3 - The Sub-grid Hydrodynamic Behavior of Accelerating Gas-solid Flows Particle-Laden Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Milioli, C.C.
Milioli, F.E.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: two-fluid model
sub-grid simulation
riser
 Notes
Abstract: Literature presents sub-grid two-fluid simulations (SGS) that are intended to support large-scale simulations (LSS) of gas-solid riser flows. Small periodic domains are considered and an additional gas phase pressure gradient is introduced in the gravitational direction to account for the flow driving force. Such additional term is chosen to exactly match the gravity acting on the gas-solid mixture, so that the simulations give rise to low velocity gas-solid suspensions. The present work aims to verify whether or not the hydrodynamics that prevails in such suspensions is relevant to rapid gas-solid flows. In order to do this, we perform SGS simulations applying an additional gas phase pressure gradient in excess to that required to match the gravity acting on the gas-solid mixture. As a consequence, instead of reaching a suspension statistical steady state regime, the flow keeps accelerating throughout a suitable range of gas phase velocities. We considered a high Stokes number particulate typical of low density risers (520 μm diameter, 2620 kg/m3 density), and run simulations for solid phase average volume fractions between 0.015 and 0.09. We found that the solid phase sub-grid effective stress components increase by up to two orders of magnitude as the gas velocity increases from about 3 to about 9 m/s. This result suggests that suspension hydrodynamics does not suit well rapid gas-solid flows, at least from the present two-fluid modelling point of view.
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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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010



The Sub-grid Hydrodynamic Behavior of Accelerating Gas-solid Flows


Christian Costa Milioli and Fernando Eduardo Milioli


University of S~io Paulo, School of Engineering of Sio Carlos, Department of Mechanical Engineering,
Av. Trabalhador S~io-carlense, 400, 13566-590, S~io Carlos, SP, Brasil
ccosta asc.usp.br and milioli~sc.usp.br


Keywords: two-fluid model, sub-grid simulation, gas-solid flow, riser




Abstract

Literature presents sub-grid two-fluid simulations (SGS) that are intended to support large-scale simulations (LSS) of gas-solid
riser flows. Small periodic domains are considered and an additional gas phase pressure gradient is introduced in the
gravitational direction to account for the flow driving force. Such additional term is chosen to exactly match the gravity acting
on the gas-solid mixture, so that the simulations give rise to low velocity gas-solid suspensions. The present work aims to
verify whether or not the lwdrodynamics that prevails in such suspensions is relevant to rapid gas-solid flows. In order to do
this, we perform SGS simulations applying an additional gas phase pressure gradient in excess to that required to match the
gravity acting on the gas-solid mixture. As a consequence, instead of reaching a suspension statistical steady state regime, the
flow keeps accelerating throughout a suitable range of gas phase velocities. We considered a high Stokes number particulate
typical of low density risers (520 Cpm diameter, 2620 kg/m3 density), and run simulations for solid phase average volume
fractions between 0.015 and 0.09. We found that the solid phase sub-grid effective stress components increase by up to two
orders of magnitude as the gas velocity increases from about 3 to about 9 m/s. This result suggests that suspension
hydrodynamics does not suit well rapid gas-solid flows, at least from the present two-fluid modelling point of view.


Introduction

It is expected that two-fluid modelling under large cluster
simulations (LCS) can accurately predict gas-solid flows in
real risers. This expectation is far from realized mainly due
to modelling shortcomings like the absence of validation for
granular thermodynamic related theories such as the kinetic
theory of granular flows, and for the continuum Newtonian
fluid Navier-Stokes equations applied to particulate phases.
Other very significant unresolved difficulties relate to the
absence of appropriate treatments for solid phase's
turbulence, related to the inexistence of proper closure for
drag at a micro-scale level, and related to the lack of
separation of scales that prevails in gas-solid flows. Of
course, the previous issues must be dealt with and resolved
before any prediction at all can be called realistic. Grace &
Taghipour (21 r 4,i do provide us with a clear insight into the
present capabilities for verification and validation of CFD
based models for granular fluidized bed flows. The authors
stress that verification is virtually impossible, while no
properly validated CFD models are still available. They
observe, in addition, that no current CFD based model can
perform better than simpler mechanistic models. In view of
the above, a warning must be placed regarding the validity,
or the lack of validity of the two-fluid model to be described
in the next section. In spite of the many problematic and
open questions, the model seems to produce a qualitatively
correct topology of a riser flow. Accordingly, only
qualitative behaviors shall be analyzed here, and even so
with much care.


Notwithstanding all the previously mentioned difficulties,
there is a continuing research effort to provide sub-grid data
for LCS obtained from sub-grid scale simulations (SGS)
with two-fluid modelling (Sundaresan, 2000; Agrawal et al.,
2001, Andrews IV et al., 2005, Igci et al., 2008). Regarding
the solid phase, SGS can also be numerical direct simulation
(NDS). For a typical LCS grid cell (order of centimeters)
taken to be an SGS domain, NDS is computationally
feasible for the solid phase. In that case SGS requires the
micro-scale of the solid flow to be properly described. Here
we call micro-scale the scale free of any solid phase
coherent structure. The current state of the art does not
include micro-scale experimental information of gas-solid
flows, and the so called kinetic theory of granular flows
(KTGF) is widely applied. This theory is an analogy with
the kinetic theory of dense gases, which is modified to
account for the inelastic particle collisions characteristic of
gas-solid flows (Bagnold, 1954, Jenkins & Sarage, 1983,
Lun et al., 1984, Gidaspow, 1994).

Following the above, computational experiments have been
performed by some researchers to provide sub-grid scale
correlations to be used in LCS of gas-solid riser flows
(Agrawal et al., 2001, and then the following works of
Andrews IV et al., 2005, and Igci et al., 2008). In general,
the simulations are performed in small periodic domains
which are thought to repeat themselves throughout the
whole volume of a riser. As periodic boundaries are applied
an additional gas phase pressure gradient is introduced in
the gravitational direction to account for the flow driving






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

representative. We concentrate the current analysis on the
behavior of the solid phase sub-grid effective stresses in
view of their determining effect over the gas-solid flow
topology.


Nomenclature


Paper No


force. Such additional term is chosen to exactly match the
gravity acting on the gas-solid mixture, so that the
simulations give rise to low velocity gas-solid statistical
steady state suspensions. This assumption is brought from
previous studies on the instabilities that develop in unforced
granular materials when the inelasticity of the collisions
among particles is accounted for, which ultimately leads to
the formation of clusters (Goldhirsch et al., 1993: Tan &
Goldhirsch, 1997). Those studies stand for regimes where
particulates arrange themselves in low velocity suspensions.
In spite of that, the clustering mechanism that prevails is
believed to be also relevant to rapid gas-solid flows (Tan &
Goldhirsch, 1997). We will call this statement as Tan &
Goldhirsh s assumption throughout the article.

In addition to the above, Agrawal et al. (2001) imposed
macro-scale shear rates through opposing parallel vertical
boundaries in their small scale periodic domains. By volume
averaging their sub-grid scale predictions the authors
determined meso-scale parameters of the flow that were
analyzed as a function of the imposed macro-scale shear
rates and volume average solid phase volume fractions.
Andrews IV et al. (2005) further extended the analysis of
Agrawal et al. (2001) by actually deriving expressions for
the meso-scale parameters. The imposition of macro-scale
shear rates on those works appears as a first attempt to deal
with the lack of scale separation issue. Extending the
previous works of Agrawal et al. (2001) and Andrews IV et
al. (2005), Igci et al. (2008) evaluated the effects on the
meso-scale results of different LCS filter sizes applied over
the sub-grid scale predictions. They found meso-scale
results quite dependent on filter size.

Following the community effort, we have also performed
some two-fluid sub-grid simulations of gas-solid flows in
periodic domains (Milioli & Milioli, 2007a,b). All the
previous literature works considered a particulate typical of
high density catalytic cracking fluidized bed reactors (75
pLm size, with density of 1500 kg/m ), while our previous
simulations included also a particulate typical of low
density circulating fluidized bed combustors (520 pLm size,
with density of 2620 kg/m ). We have also imposed a flow
driving force through an additional gas phase pressure
gradient chosen to balance the gravity over the mixture and,
therefore, like in the previous literature works, our
predictions provided low velocity gas-solid suspensions.

Notwithstanding the efforts, how representative such low
velocity fields are regarding the sub-grid scale of high
velocity real risers, it remains to be shown. In the present
article we address this particular issue. We propose.
differently from all the previous works, to apply an
additional gas phase pressure gradient in excess to that
required to match the gravity acting on the gas-solid mixture.
In this case the flow becomes accelerated, and never reaches
a statistical steady state regime. We present instantaneous
domain averaged predictions at suitable gas phase velocities
inside a range typical of circulating fluidized beds, and then
analyze the results as for the effect of the gas phase velocity
on the flow hydrodynamics. While the predictions at any
particular mesh point should not be regarded as significant
in view of the instability of the flow, the sub-grid scale
domain averaged results, on the other side, are quite


CD
dp
D


go
I
NI
P

Re
t

u v~
UVsrw
xshp


drag coefficient (nd)
particle diameter (m)
strain rate tensor (s- )
restitution coefficient (nd)
gravity acceleration (ms )
radial distribution function (nd)
unit tensor (nd)
interface drag force (Nm )
pressure (Nm -
additional gas pressure gradient (Nm )
Reynolds number (nd)
time (s)
velocity vector (ms ')
Cartesian components of velocity (ms ')
8Xi81 Slipvelocity (F-v,) (ms ')
Cartesian coordinates (m)


Greek letters
volume fraction (nd)
P gas-solid friction coefficient (kgm s ')
o granular temperature (m~s )
n bulk viscosity (Nsm )
i" dynamic viscosity (Nsm )
P density (kgm )
r viscous stress tensor (Nm )
T, effective stress tensor (Nm 2)
u9 particle sphericity (nd)
w driving force factor (nd)

Subscripts
e meso-scale or effective
a gas phase
I interface
n~r maximum
P particle
solid phase
x. L = Cartesian directions

Others
_LCS resolved

(.--) volume average


Mathematical Modelling

Multiphase flow two-fluid models stand on the major
hypothesis of continuum for all of the phases, no matter
fluid or particulate. The phases are treated as
inter-penetrating dispersed continue in thermodynamic
equilibrium. The theory of two-fluid models has been
developed by many researchers. Some classical reference
developments on this matter can be found in the works of
Anderson & Jackson (1967), Ishii (1975), Drew (1983),






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


Paper No


Gidaspow (1994), Enwald et al. (1996), among many others.
The lwdrodynamic two-fluid models comprise a basic set of
mass and momentum averaged conservative equations plus
closure laws for stress tensors, viscosities, pressures and
drag.

We must formulate our two-fluid model to perform SGS,
which remains large eddy simulation alike regarding the gas
phase, but is required to become NDS regarding the solid
phase if all the scales of clusters are to be captured. In this
way, the gas phase would require closures at both the micro
and the meso-scales. Agrawal et al. (2001) found the
turbulence of the gas phase not to significantly affect the
volume averaged solid phase parameters, so that a constant
molecular viscosity approach seems to be good enough for
the gas phase. For our present purposes, turned mamnly to
solid phase lwdrodynamics, tlus assumption shall be good
enough. The micro-scale closure for the solid phase is
established by applying the kinetic theory of granular flows
(KTGF), where solid phase micro-scale properties are
derived as a function of a granular temperature determined
from a pseudo thermal energy balance.

In addition to the conventional formulation, as periodic
boundaries are to be applied, an additional gas phase
pressure gradient must be enforced in the gravitational
direction to account for the flow driving force. We present
next a sub-grid scale lwdrodynamic formulation of the
two-fluid model including closure laws based on the KTGF
(Gidaspow, 1994: Syamlal et al., 1993: Agrawal et al., 2001),
and including gravity compensation for applying periodic
boundary conditions. The formalism involved in the
derivation of the large eddy simulation alike formulation for
the gas phase can be found, for instance, in Sagaut (2001)
and Lesieur et al. (2005).

The continuity and momentum conservative equations for
the gas and the solid phases are, respectively:


Eg +a, = 1


The additional gas phase pressure gradient in the
gravitational direction for exactly matching the gravity
acting on the gas-solid mixture is given by:


V7~= (pa, +p,a,)g


The filtered viscous stress tensor for the gas phase is
disregarded as of lower order in relation to the effective
stress tensor, which is assumed to be given by:


_ pg, ug Yu n (Ae- V-ui


It isalso assumed that u, = constant and n,, = o.

The viscous stress tensor for the solid phase is given by:


The dynamic and bulk viscosities of the solid phase are
obtained by analogy with the kinetic theory of dense gases,
and usually include terms due to kinetic and collisional
effects. Here we exclude the kinetic term since it brings the
solid phase stresses to unrealistically high values at very
low solid fractions: the collisional term, on the other side,
correctly vanishes as expected at lower solid fractions (Lun
et al, 1984). The dynamic and bulk viscosities of the solid
phase are, respectively (as presented in Gidaspow, 1994):


Y = afpdg(+ (9)


(10)


(1) '' -,= a~pXd~o\1+e8-


"6 ) #s lly) o


er(apsp) + V -(asPsus) = o


The radial distribution function is taken form Ding &
(2) Gidaspow (1990), and is given by:


np(O-, VIVPe~)


The solid phase pressure also comes from the analogy with
the kinetic theory of dense gases, and is given by (Gidaspow,
1994):


at (ap,u,) + V -(asppusu) = -a/VP, + grP,+ Vp s
+V*(a,r,)+ asp,g + M,, (4)

As the formulation for the gas phase is for large eddy
simulation, the conserved variables are the filtered
parameters to be resolved. For the solid phase, on the other
hand, the flow field variables are fully resolved as NDS is
assumed.

Closures are required for various parameters that appear in
the conservative equations. The volumetric continuity
imposes that:


P,= asp,O[1+2(1+ e)goas]


The granular temperature required in the previous equations
is determined here from the algebraic equation of Syamlal
et al. (1993), which is derived by assuming that the
pseudo-thermal energy is locally generated by viscous stress
and dissipated by inelastic collisions. It is given by:


r,=p, V.cu,+@usn +(,-yi)/-uji


go
5 a


+ V.- a(r,-rge)] pgagg + NQ,





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


Paper No


application of either free slip, partial slip, or periodic
COnditions to vertical boundaries gives rise to the same flow
(13) topology. In the present work we apply the simpler free slip
condition.

Finally, if averaged sub-grid or meso-scale data are to be used
for solid phase's closures in LCS, then a bridge needs to be
built between the SGS solution and the filtered parameters
(14) required in LCS. Following the large eddy simulation theory,
the components of the solid phase sub-grid effective stress
tensor result (Sagaut, 2001; Lesieur et al., 2005):


Klatr(D,)+ Ki7tr ID


)a, +4K arK tr ID )+2K tr(D 2


where


Ds 1 Yus+ Yusn


K, = 2(1+e)pso


r,~se Ps tsits-Tisis)



Z~ =Z

uzse u~se s 1ss 1ss



rz,se, rese= Ps("sts-\s's)


4dpps(1+e)asgo 2
K, = -K3


d~p~js 8asgo(1+e)
K3 --[1+0.4(1+e)(3e -1)asEOl
2 33-e) 5 ~


_12 1- e PsEO
K,


A stationary drag force is applied to account for
interface momentum transfer between the gas and the
phases, which is given by:

Ms;= -MgI s g-u)

The gas-solid friction coefficient is determined by app]
the procedure frequently referred to as Gidasp
procedure, where Ergun's equation (Ergun, 1952) is us
high solid fractions and Wen & Yu's correlation (Wen &
1966) is used at low solid fractions. The procedure
follows (Gidaspow, 1994):

Pgasl ug -us|
P =150 +1.75 for as >0.2
ag dpa~s) (ps ,


(25)

(26)


the
solid


(19)

lying
Iol's
ed at
iYu,
is as


If a sub-grid filter is assumed that exactly fits the SGS
domain, then the filtered parameters become equal to their
volume averaged sub-grid values, so that:


r ,,= P (NUz) (Ils)(rts) (31)




r = p, (v:v, (~,)(v~,) (32)


(20)
se.~ se Ps 11s s 11 s)~) (34)


(21) r = =p r e)-( )(s (35)


r , = c ,, = p (i, e) (v,)( ty)) (36)

Equations (31) to (36) are the components of the sub-grid
(22) effective stress tensor that are required to be introduced in the
Solid phase's momentum equation, in LCS.


S im ulations


4 (dpa~s


for a, 50.2


The drag coefficient is given by (Rolve, 1961):


S1 +0.15Re 0.08i for Re, <1000
Ds -
0.44 for Rep 21000

were


The complex set of partial differential non-linear coupled
equations of the two-fluid models can only be solved
through numerical procedures. In this work, the numerical
model available in the software CFX (CFX, 2005) was used.
The simulations were performed for a high Stokes number
particulate typical of low density risers (520 Cpm diameter,
2620 kg/m' density), for solid phase average volume
fractions of 0.015, 0.03, 0.05, 0.07 and 0.09. Accelerating
flows were generated and the results were analysed for


l i, us d, pp
Re, =(23)


Periodic conditions are applied at entrance and exit, i.e. in the
horizontal boundaries normal to the vertical gravitational
direction. This means that the flow at the inlet is exactly the
same that flows at the outlet. Free slip is applied in all of the
vertical boundaries. Agralval et al. (2001) showed that the












































S 6







0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
time (s)

Figure 2: Averaged axial velocities and axial slip velocity
as a function of time, for y = 1 and 1.5, for an average
solid volume fraction of 0.05.


Figure 2 also shows that, while both the phase axial velocities
grow in time, the slip velocity is kept approximately constant.
This result suggests that the clustering topology of the flow is
not significantly affected by the increasing gas axial velocity.
This is an evidence in favor of Tan & Goldhirsch's
assumption. Plots similar to those of Figure 2, for solid
volume fractions from 0.015 to 0.09, show the same behavior.

Figures 3 to 7 present sub-grid effective shear stresses of the
solid phase as a function of the averaged gas phase axial
velocity, for various average solid volume fractions. The
effective shear stresses resulted very oscillating, and seem to


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

becomes accelerated. Notice that the volume averaged solid
phase axial velocity results negative in the statistical steady
state regime, in the stage of the simulation where the gravity
acting on the suspension is exactly matched. This seems to be
a clear consequence of the clustering that develops in the flow,
since the gravity does not simply act on a bunch of individual
particles, but on a relatively heavier bunch of clusters of
particles.


Paper No


increasing gas velocities from about 3 to about 9 m/s.
A 2x2 cm wide and 8 cm tall vertical hexahedral domain was
considered, applying a 1xlx1 mm uniform hexahedral
numerical mesh. The flow enters the domain through the
bottom and exits at the top. The density and viscosity of the
gas phase were, respectively, 1.1614 kg/m3 and 1.82 x 10 5
N.s/m A solid phase volume fraction at maximum packing of
0.38 was applied following Gidaspow and Ettehadieh (1983),
and a restitution coefficient of 0.9 was taken following
Agrawal et al. (2001).

The driving force factor (W) multiplying the additional gas
phase pressure gradient required to match the gravity over the
mixture was set at 1.5. This value allowed the simulations to
go along a range of gas axial velocities typical of circulating
fluidized beds in a reasonable computing time. Initial
conditions for the accelerating runs were obtained by running
previous simulations applying = 1 until statistical steady
state regimes were found. This is done because, if the
accelerating runs are to be realistic, then they must depart
from realistic gas-solid flow conditions. Therefore, the
simulations developed in two stages, the first for 7y = 1, and
the second for y = 1.5 The initial conditions for the first
stage were of uniform quiescent suspensions with fixed
uniform solid volume fractions. It should be noted that the
axial periodic boundary condition that was applied and the
continuity constraint cause the volume average solid volume
fraction to be sustained throughout the whole simulation.

A time step of 5x10-' s was applied which is suitable for solid
phase NDS. The lower characteristic time scale of clusters of
the order of 102 s (Sharma et al., 2000) is expected to be fully
captured. Also, for the present 520 pLm particulate size the
smaller clusters on the flow are expected not to be larger than
5.2 mm (following Agrawal et al., 2001). Therefore, regarding
the solid phase, both the spatial and temporal meshes which
were applied are suitable for NDS. The convergence criterion
for the numerical procedure was a rms of 1 x 10 5


Results

We will now use the results of the present simulations to
collect arguments both in favor and against the assumption of
Tan & Goldhirsch (1997), that the clustering mechanism in
low velocity suspensions also prevails in rapid gas-solid flows.
Figure 1 shows grayscale plots of the solid volume fraction in
an axial section of the domain for increasing volume averaged
gas phase velocities, for the average solid volume fraction of
0.05. It is clearly seen that the flow topology progressively
becomes more and more homogeneous as the gas velocity
increases towards pneumatic transport. This is an evidence
against Tan & Goldhirsch's assumption. Plots similar to those
of Figure 1, for solid volume fractions from 0.015 to 0.09,
show the same behavior.

Figure 2 shows the time development of the phase's volume
averaged axial velocities and the axial slip velocity for the
simulation with an average solid volume fraction of 0.05.
Departing from a unitary driving force factor, the flow
develops and reaches a statistical steady state regime. Then
the driving force coefficient is turned to 1.5, and the flow


Figure 1: Grayscale plots of solid volume fraction in an
axial section of the domain for increasing volume averaged
gas phase velocities (3, 4, 6, 8 and 9 m/s from the left), for an
average solid volume fraction of 0.05.


rre


























VV I I I I r
3 4 56 789

=V, (nu s)


Figure 5: Sub-grid effective shear stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.05.


Figure 6: Sub-grid effective shear stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.07.



100-












0.01-
3456789




Figure 7: Sub-grid effective shear stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.09.


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


Paper No


vary randomly at the same range, no mater the averaged gas
phase axial velocity. This pattern is observed for all the
average solid phase volume fractions, except for 0.015. In this
case there is a rising tendency of up to two orders of
magnitude as the gas phase axial velocity increases. Here we
have arguments both in favor and against Tan & Goldhirsch's
assumption.

Figures 8 to 12 show sub-grid effective normal stresses of the
solid phase as a function of the averaged gas phase axial
velocity, for various average solid volume fractions. The
effective normal stresses resulted much less oscillating than
the effective shear stresses. In most of the cases there is a
decreasing tendency as the averaged gas phase axial velocity
increases. Variations of up to one order of magnitude are
observed. Again, the above pattern is observed for all of the
average solid phase volume fractions, except for 0.015. In this
case a clear rising tendency is not seen. Here we also have
arguments both in favor and against Tan & Goldhirsch's
assumption.


- o -I I I I I
vs
"A h
-'y 8 A


~ ~vse
use
yrse


iv


i-




1-


i', (r; s)


3456789

~v~ pn s)


Figure 3: Sub-grid effective shear stresses of the solid
phase as a function of the averaged gas phase axial velocity
for an average solid volume fraction of 0.015.


l i e l l i e l


3456789



Figure 4: Sub-grid effective shear stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.03.


























1456789



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













10 -




3456789

nms

Figure 11: Sub-grid effective normal stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.07.












10 --



::456789



Figure 12: Sub-grid effective normal stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.09.


Conclusion

We developed a two-fluid sub-grid scale simulation of an
accelerated gas-solid flow covering a range of gas velocities
typical of risers. Our main objective was to verify the
conunonly applied Tan and Goldhirsch's assumption that
the clustering mechanism in low velocity suspensions also
prevails in rapid gas-solid flows. We considered a high
Stokes number particulate (520 Cpm diameter, 2620 kg/m3
density). The analysis was concentrated on the behavior of
the sub-grid effective stresses of the solid phase. From the
predictions, we have collected a number of evidences both
in favor and against the concerning assumption.

For increasing averaged gas phase velocities, we found that:
(i). slip velocities are kept approximately constant; (ii) the
sub-grid effective shear stresses of the solid phase seem to
be mostly unaffected, except for a solid volume fraction of
0.015, where a variation of up to two orders of magnitude is


I e I I I I e I


Paper No












lo-


-At ~se


Figure 8: Sub-grid effective normal stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.015.


345678 im


Figure 9: Sub-grid effective normal stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.03.


~-100-




lo-






3456789


Figure 10: Sub-grid effective normal stresses of the solid
phase as a function of the averaged gas phase axial velocity,
for an average solid volume fraction of 0.05.






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

Gidaspow, D. Multiphase flow and fluidization. San Diego.
Academic Press (1994).


Paper No


observed; and (iii) the sub-grid effective normal stresses of
the solid phase decrease by up to one order of magnitude for
most of the cases. Here we have arguments both in favour
and against Tan and Goldhirsch's assumption.

Notwithstanding the above conclusions, all we can say, for
the moment, is that Tan & Goldhirsh's assumption seems
not to hold, at least for the particulate that was considered,
and from the present two-fluid modelling point of view. Of
course, the current predictions need much improvement
before better conclusions may be drawn. It is wise to keep
in mind that even though two-fluid modelling can provide
apparently correct qualitative predictions of riser flows, the
same can not be said regarding quantitative predictions.


Acknowledgements

This work was supported by The State of Sio Paulo
Research Foundation (FAPESP), The National Council for
Scientific and Technological Development (CNPq), and The
Coordination for the Improvement of Higher Level
Personnel (CAPES).


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