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CFD SIMULATION OF TWO-PHASE FLOWS
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lffat Tasneem Shaik Mohammad

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CFD SIMULATION OF TWO-PHASE FLOWS IN A VERTICAL DUCT

By

Iffat Tasneem Shaik Mohammad

A THESIS
Submitted to
Michigan State University

in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2006

ABSTRACT

CFD SIMULATION OF TWO-PHASE FLOWS IN A VERTICAL DUCT
By
Iffat Tasneem Shaik Mohammad
The ﬂow of a suspension of spherical particles in air through a vertical rectangular
channel with equal inlet and outlet cross sectional areas was simulated under'isothermal
conditions using Fluent. An Eulerian approach was used in which the multiphase problem
is represented as two interpenetrating continua. A granular model was employed for the
mixture stress and a realizable “eddy” viscosity model was used for the Reynolds
stress. The purpose of this work was to evaluate the ability of the multiphase model to
capture the ﬂow physics associated with catalytic risers. Experimental ﬁndings of Ibsen
et al. (2001) were used for evaluating the numerical predictions. The numerical
predictions were able to capture the axial and span wise particle velocity in some regions
of the ﬂow but failed to capture the correct pressure drop. In addition, while the
turbulence model provided realizable results for the single-phase ﬂows tested, it became

unrealizable when coupled with the multiphase granular model.

To my husband, daughter and parents.

iii

ACKNOWLEDGEMENTS

First and foremost, I would like to thank Almighty ALLAH for everything in my life. My
heartfelt thanks to my parents for everything they did for me; their generous support,
motivation and trust in me since childhood, which helped me aim higher from time to
time. Great thanks to my wonderful husband and beautiful daughter for all the love and
patience and for being a constant source of inspiration. Thanks to my sister and brother
for their moral support. Thanks also to all my relatives whose conﬁdence in me further
encouraged my performance and betterment.

I sincerely thank my advisor Dr. Andre Benard, who helped me work in the area of my
research interest - Computational Fluid Dynamics and for all his support and guidance
throughout my Master’s. His training, advice and insight into things have been
invaluable.

I would like to thank Dr. Kuochen Tsai from Dow chemicals for suggesting this
interesting and challenging research topic. Thanks to Dr. Charles Petty and Dr. Farhad
J aberi, for being on the committee and for their invaluable suggestions. A special thanks
to all the professors at MSU for being there for me at all times. Thanks to my team mates
Deep and Karuna for their invaluable assistance on our summer workshop project.

For technical assistance I would like to thank Kapil Das Sahu ﬁ'om FLUENT (India) for
his immense help all along my Master’s. I would also like to thank Andrew Keen for
helping me get going smoothly with parallel computing.

Finally, I gratefully acknowledge the ﬁnancial support of this work by the NSF I/UCRC-

MTP.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................................................................. vii
LIST OF FIGURES ............................................................................... viii
NOMENCLATURE ................................................................................ xi
CHAPTER 1 _
INTRODUCTION .................................................................................... 1
1.1 Introduction to Circulating Fluidized Bed ........................................ 1
1.2 Principles of a Circulating Fluidized Bed ......................................... 3
1.3 Previous work in the area of Circulating Fluidized Bed ........................ 4
1.3.1 Experimental work ............................................................ 4
1.3.2 Numerical simulations ........................................................ 6
1.4 Objectives of the thesis .............................................................. 8
1.5 Computational domain considered in this work ................................. 9
1.6 Theoretical and Numerical considerations of this work ........................ 10
1.7 Boundary and Initial Conditions applied in this work ........................... 10
CHAPTER 2
SINGLE PHASE FLOW 12

2.1 VERIFICATION OF FULLY DEVELOPED LAMINAR FLOW IN A DUCT ....12
2.1.1 Governing equations applied to laminar single phase simulations . ....12
2.1.2 Analytical solution for ﬁilly developed single phase duct ﬂow . ....12
2.1.3 Numerical results and verification ............................................................ 13

2.2 COMPARISON OF VARIOUS VISCOUS MODELS IN A TURBULENT DUCT

FLOW ......................................................................................................................... 15

2.2.1 Governing equations applied to turbulent single phase simulations .......... 15

2.2.2 Lumley diagram ........................................................................................ 17

2.2.3 Various viscous models and their approximations ........................... 18

2.2.4 Numerical results and veriﬁcation ............................................... 22
CHAPTER 3

TURBULENT TWO PHASE FLOWS 30

Overview of two phase ﬂows ............................................................................... 30

3.1 COMPARISON OF DRAG INTERPHASE MODELS IN A BUBBLY FLOW ...... 30

3.1.1 Case study problem ................................................................................... 31

3.1.2 Governing equations applied to gas-liquid simulations ............................ 31

3.1.3 Theoretical formulation of various drag interphase models ..................... 32

3.1.4 Numerical results and discussions ............................................................ 35
3.2 NUMERICAL SHVIULATION OF A GAS-SOLID FLOW IN A RISER ................. 41
3.2.1 Assumptions made in the gas-solid simulations ....................................... 41
3.2.2 General guidelines for mathematical modeling in ﬂuidization ................ 42
3.2.3 Governing equations applied to gas-solid simulations ............................. 45
3.2.4 Goveming equations applied in the kinetic theory of granular ﬂow ........ 46
3.2.5 Experimental set up of the 1/9 scale Cold Circulating Fluidized Bed
Boiler .............................................................................. 48
3.2.6 Numerical conﬁguration of the gas-solid simulations ........................ 49
3.2.7 Simulation time ................................................................... 50
3.2.8 Numerical results and discussions ............................................. 50
CHAPTER 4
SUMMARY AND CONCLUSIONS ............................................................ 59
APPENDICES
A. Models and near wall treatment applied in the thesis .................................... 62
B. Comparison of terms in various turbulent viscous models ............................... 63
C. Computed eigenvalues and invariants ......................................................... 64
REFERENCES ..................................................................................... 70

vi

2.1

3.1

B1

B2

C1

C2

C3

C4

C5

C6

LIST OF TABLES

Lumley diagram ....................................................................................................... 17
Parameters for the gas-solid simulations ................................................. 49
Table summarizing the various turbulence models and the respective

near wall treatments applied in the simulations along with their y+ values ........... 62

Table summarizing the various terms for the Spalart-Allmaras model and the
Standard k - a) model studied .............................................................. 63

Table summarizing the various terms for the Realizable k - s and the Reynolds
Stress model studied ......................................................................... 63

Computed eigenvalues and invariants for a single phase turbulent
ﬂow using the Spalart-Allmaras Model; Y/D=0; Z/H=-0.5 ............................. 64

Computed eigenvalues and invariants for a single phase turbulent
ﬂow using the Spalart-Allmaras Model; Y/D=O.48; Z/H=—O.5 ......................... 65

Computed eigenvalues and invariants for the primary phase (water)
in the gas-liquid simulations; Y/D=0; Z/H=-0.5 ........................................ 66

Computed eigenvalues and invariants for the primary phase (water)
in the gas-liquid simulations; Y/D=O.48; Z/H=-0.5 .................................... 67

Computed eigenvalues and invariants for the primary phase (air)
in the gas-solid simulations; Y/D=0; Z/H=-O.5 ........................................... 68

Computed eigenvalues and invariants for the primary phase (air)
in the gas-solid simulations; Y/D=O.48; Z/I-I=-O.5 ........................................ 69

vii

LIST OF FIGURES

1.1 Schematic of a Circulating F luidized Bed ................................................................. 3
1.2 Geometry of 3D-vertical duct considered in the simulation ...................................... 9
2.1 Plot of location of points at which the velocity proﬁles are drawn for Z/H=-0.8....13
2.2 Plot of comparison of numerical and analytical span wise velocities. They predict

almost the same with a slight error. See pg13 for error calculation ......................... 14
2.3 Plot of absolute velocity error vs. dimensionless width. A maximum error of

1.4E-05 was found at the centerline ......................................................................... 14
2.4 Sketch of the boundaries of the realizable states forg ............................................. 18
2.5 Velocity vectors for the Spalart-Allmaras model at Z/H=-0.5 ........................ 22
2.6 Velocity vectors for the Standard k — a) model at Z/H=—0.5 .......................... 22
2.7 Velocity vectors for the Realizable k - a model at Z/H=-0.5 .......................... 23
2.8 Velocity vectors for the Reynolds Stress Model at Z/H=-0.5 .......................... 23
2.9 Location of proﬁles at which the realizability plots for the various viscous

models are drawn .............................................................................. 24
2.10 Realizability plot for the Standard k — a) model at proﬁlel.

It predicted that 1111, z 0 .................................................................... 24
2.11 Realizability plot for the Standard k — a) model at proﬁ1e2.

It predicted that 1111, z 0 .................................................................... 25
2.12 Realizability plot for the Realizable k - a model at proﬁlel.

It predicted that 1111, z 0 .................................................................... 25
2.13 Realizability plot for the Realizable k - a model at proﬁ1e2.

It predicted that IHb z 0 ..................................................................... 26
2.14 Realizability plot for the Reynolds Stress Model at proﬁlel .................................... 24
2.15 Realizability plot for the Reynolds Stress Model at proﬁle2 .................................... 27

viii

2.16 Contours of the turbulent kinetic energy for the Standard k — a) model and the

Realizable k - a model at planes Y/D=0, Y/D=O.48 and Z/H=-0.5 ................... 29
3.1 Plot of comparison of the various drag models for the bubbly flow ........................ 34
3.2 Location of proﬁle at which the various drag models are compared ....................... 35

3.3 Plot of dimensionless axial velocity of water vs. dimensionless height for the
Schiller-Naumann model, the Morsi and Alexander model and the Symmetric
model (sn, ma, sym). All models give similar results ............................................... 35

3.4 Plot of dimensionless axial velocity of air vs. dimensionless height for the
Schiller-Naumann model, the Morsi and Alexander model and the Symmetric
model (sn, ma, sym). All models give similar results ............................................... 36

3.5 Plot of dimensionless axial dynamic pressure of water vs. dimensionless height for
the Schiller-Naumann model, the Morsi and Alexander model and the Symmetric
model (sn, ma, sym). All models give similar results ............................................... 36

3.6 Plot of dimensionless axial dynamic pressure of air vs. dimensionless height for
the Schiller-Naumann model, the Morsi and Alexander model and the Symmetric
model (sn, ma, sym). All models give similar results ............................................... 37

3.7 Plot of dimensionless axial turbulent kinetic energy of water vs. dimensionless
height for the Schiller-Naumann model, the Morsi and Alexander model and the
Symmetric model (sn, ma, sym). All models give similar results ............................ 37

3.8 Plot of dimensionless axial volume fraction of air vs. dimensionless height for

the Schiller-Naumann model, the Morsi and Alexander model and the Symmetric
model (sn, ma, sym). All models give similar results .............................................. 38
3.9 Location of proﬁles at which the realizability plots are drawn for Z/H=-O.5 ........... 38

3.10 Realizability plot for the secondary phase (air) in the gas-liquid simulations at

proﬁlel. The realizable k - a model predicted that 111}, z 0 .................................... 39
3.11 Realizability plot for the secondary phase (air) in the gas-liquid simulations at

proﬁleZ. The realizable k - a model predicted that [111, z 0 .................................... 39
3.12 Contours of the turbulent kinetic energy for the primary phase (water) at planes

Y/D=O, Y/D=O.48 and Z/H=-0.5 ........................................................... 40
3.13 Sketch outlining the actual lines and tendencies for the third type of

mathematical modeling in ﬁuidization .................................................................... 43
3.14 Locations of proﬁles used for plotting the gas-solid results .................................... 50

ix

3.15

3.16

3.17

3.18

3.19

3.20

3.21

3.22

3.23

3.24

3.25

Plot of comparison between the measured (Ibsen et a1. 2001) and computed static
pressure vs. dimensionless height at proﬁlel. It can be seen that a linear pressure

drop is predicted which deviates greatly from the experiments .... ...................... 51
Plot of comparison between the measured (Ibsen et a1. 2001) and computed axial
particle velocities at proﬁ1e2. (X=0 is the center of the duct) .. ............................ 52
Plot of comparison between the measured (Ibsen et al., 2001) and computed span
wise particle velocities at proﬁle2. (X=0 is the center of the duct) ........................ 53
Plot of comparison between the measured (Ibsen et al., 2001) and computed axial
particle velocities at proﬁle3 ................................................................................... 53
Plot of comparison between the measured (Ibsen et al., 2001) and computed span
wise particle velocities at proﬁle3 .......................................................................... 54
Contours of volume fraction of solids at various times for Y/D=0 ......................... 55

Contours of volume fraction of solids at various times for Z/H=-O.25,-0.5 and
-0.7 5 ........................................................................................... 55

"I "I

Realizability plot for the u u associated with the primary phase computed with the

Realizable k - 8 at proﬁle 4 ..................................................................................... 56
Realizability plot for the {7757 associated with the primary phase computed with the
Realizable k - e at proﬁle 5 ..................................................................................... 56
Plot of comparison of the two drag models for the gas-solid ﬂows ....................... 57

Contours of the turbulent kinetic energy for the primary phase (air) at Y/D=0.
Y/D=0.8 and Z/H=-0.5 ........................................................................................... 58

x\.
:<
N

sg<gmu

”U

é‘é E Si <1 :1 fall

NIIZ
:r

 

:1
s...‘

>‘ U‘VQ

NOMENCLATURE

Hydraulic diameter [m]

Drag coefﬁcient [dimensionless]
Drag function [dimensionless]
Coordinates [m]

Depth of riser [m]

Riser height [m]

Width of riser [m]

Volume [m3]

Variation in volume [m3]
Particle diameter [m]

Dimensions of measurement length [m]
Variation in pressure [Pa]

Restitution coefﬁcient

Solids ﬂux [kgmzs]

Gravitational acceleration [m/sz]
Radial distribution coefﬁcient

Mass of solid [kg]

Number of grid cells [dimensionless]

Pressure [Pa]

Solid phase pressure [Pa]

Time [s]

Reynolds number [dimensionless]
Solid Reynolds number [dimensionless]

Mean rate-of-strain tensor [5"]

Axial velocity component [m/s]

Span wise velocity component [m/s]
Span wise velocity component [m/s]
Dimensionless height [dimensionless]
Dimensionless depth [dimensionless]
Dimensionless width [dimensionless]
Numbers

Identity matrix [dimensionless]
Reynolds stress

Generation
Dissipation
Material derivative
Displacement

xi

Greek symbols

Qb>>§¥§le°ﬁbg“'"b’<"s~<‘x:>:-Qm\<or

Subscripts

mWN'B-Qmw‘m

Superscripts

T

Local derivative

Dissipation of turbulent kinetic energy by collision [kg/ms3]
Kronecker delta [dimensionless]

Volume fraction [dimensionless]

Transport coefﬁcient of turbulent kinetic energy [kg/ms]
Dynamic Viscosity [Pa-s]

Kinematic viscosity [mz/s]

Energy exchange between two phases [J /kg]

Modiﬁed turbulent viscosity [mz/s]
Density [kg/m3]

Shear stress [Pa]
Particle relaxation time [s]

Angular velocity, Mean rate of rotation tensor [5"]
Turbulent dissipation rate [m2 /s3]

Speciﬁc dissipation rate [5"]

Constant [dimensionless]

Eigen values [dimensionless]

Interphase exchange coefﬁcient [dimensionless]
Effective density [kg/m3]

Prandtl number [dimensionless]

Gas

Phase number

Solid

Primary phase

Liquid

Turbulent

Turbulent kinetic energy[kg/ms]
Turbulent dissipation rate [m2 /s3]

Transpose

xii

CHAPTER 1
INTRODUCTION
1.1 Introduction to Circulating Fluidized Bed
The study of multiphase ﬂows represents a challenging and fruitful area of endeavor
since the state of the art for multiphase ﬂows is considerably more primitive where the
correct formulation of the governing equations is still under debate whereas the transport
equations of single phase Newtonian ﬂuids are generally well-accepted and closed form
solutions for speciﬁc cases are well documented.

Numerical simulation of ﬂows associated with risers is the main topic of this
thesis. Gas-solids ﬂuidization has a wide application in chemical, metallurgical and
energy industries with the largest applications being in ﬂuidized bed reactors in coal
combustion for large scale thermoelectric power generation and catalytic cracking of
petroleum to produce gasoline and other fuels. The use of circulating ﬂuidized bed
system is increasing as it can be applied to a wide area of applications such as power
generation, catalytic cracking and drying of wet powders with a good result (low cost to
performance ratio). The circulating ﬂuidized bed combustors are superior to pulverized

coal combustors regarding fuel ﬂexibility and emission performance of NOx and 802.

An advantage of circulating ﬂuidized bed combustor is that it can be ﬁred with
low-grade coal, biomass and waste with reasonable efﬁciency whereas its disadvantage
being the emission of small particle. Hence, it is necessary to put energy into separating
the particles from the outﬂow. Today several hundreds of circulating ﬂuidized bed

combustors is operating with the largest ones delivering 250 MW of power.

One of the possible reasons for the rather modest maximal turn out of power is the
lack of good design tools for scale-up of the process. Computational Fluid Dynamics
(CFD) software has proven to be a relatively efﬁcient tool for design of applications with
single-phase ﬂows. However, for multiphase ﬂow, CFD models still need to be improved
before the same precision in the results can be obtained. This study will focus on
simulating a multiphase problem with the purpose of assessing the accuracy of the
simulations through a realizability diagram.

The structure of multiphase ﬂows in circulating ﬂuidized bed columns is quite
complex, showing great variations on solids volumetric fraction throughout the riser,
involving continuous formation and dissipation of clusters, and a high rate of
recirculation of solids leading to an intense superﬁcial contact between gas and solids
providing high reaction rates. Therefore, the knowledge of the ﬂow hydrodynamics is of
great importance, allowing relevant reactive and mass transport parameters to be
determined.

The mathematical modeling of gas-solids ﬂuidization processes represents an
ancillary tool for minimizing the experimental efforts required for developing industrial
plants. However, mathematical modeling and numerical simulation are in continuous
development, contributing in a growing way for the better understanding of processes and
physical phenomena. Mathematical models require experiments in order to be validated
and, concerning ﬂuidization, the required experiments involve complex measurements.
Hence, mathematical modeling also represents an incentive for the development of new

experimental methods and techniques.

In this work, numerical simulations are performed for the gas-solids ﬂow in a
riser of circulating ﬂuidized bed using the Eddy Viscosity Model proposed by Shih et a1.
(1995).

1.2 Principles of a Circulating Fluidized Bed

A Circulating F luidized Bed schematic is as shown in Figure 1.1.

Gas Outlet

II

 

Cyclone

Ri
ser Return Leg

 

 

 

IIIIIII

Primary Gas Inlet

Figure 1.1: Schematic of a Circulating F luidized Bed
It consists mainly of a riser, a cyclone and a return leg. In the riser, particles that form a
particulate bed in the bottom are kept in free motion by an upward ﬂowing gas (or liquid)
hence the term “ﬂuidized bed”. When the superﬁcial gas velocity is sufﬁciently high the
particles will rise to the exit of the riser and enter the cyclone. A cyclone is used to
separate the particles from the out ﬂowing gas. The separated particles are then

reintroduced into the riser through the return leg, hence the term “circulating”. As the

superﬁcial gas velocity increases from zero the system goes through ﬁxed bed, bubbling
regime, slug ﬂow, turbulent regime and ﬁnally in the fast ﬂuidization regime. The ﬂows
studied in this thesis are associated to a circulating ﬂuidized bed operated in the turbulent
and the fast ﬂuidization regime.

1.3 Previous work in the area of Circulating Fluidized Bed

In the following subsections previous work in the ﬁeld of experimental and numerical
analysis of gas-solid ﬂow will be discussed.

1.3.1 Experimental work

Circulating ﬂuidized beds can roughly be divided into two categories based on the layout
of the riser: The ﬂuidized catalytic cracking riser is tall with a height to width ratio well
above 10 and most often cylindrical; the circulating ﬂuidized bed combustor riser which
is low with a height to width ratio below 10 and mostly based on a square sectioned. The
gas-particle ﬂows experienced in these two types of risers differ as a result of the
difference in the layout.

In the literature, various articles can be found on experimental ﬁndings in
circulating ﬂuidized beds. Two types of non-intrusive measuring devices, based upon
laser light have been used in the past for obtaining experimental results namely the
I Laser/Phase Doppler Anemometry and the Particle Image Velocimetry. Zhang et a1.
(1995), Werther et al. (1996), Samuelsberg et al.(1996), Wang et a1. (1998) and
Mathiesen et a1. (2000) obtained experimental results in circulating ﬂuidized bed of the
ﬁrst category of risers using Laser Doppler Anemometry. Laser Doppler Anemometry
was also used by Berkelmann et a1. (1991), Wang et al. (1993), Van den Moortel et al.

(1997), and Zhou et a1. (2000) to obtain experimental results in circulating ﬂuidized bed

of the second category of risers. The circulating ﬂuidized bed investigated by
Berkelmann et al. (1996) was a 4 MW ﬂuidized bed combustor whereas the remaining
references investigated laboratory scale models. Gidaspow et al. (1992) measured solids
ﬂux with a sampling probe and solids concentration with an X-ray probe. From the solids
ﬂux and the solids concentration they deduced the solids velocity. Werther et a1. (1997)
measured solids volume concentrations and solids velocities with a guarded capacitance
probe in a 109 MW circulating ﬂuidized bed boiler at full load.

All the references mentioned above deal with the properties of the solids in a
circulating ﬂuidized bed. References concerning measurements of both gas and solid
velocities are scarce but existent. Hamdullahpur et a1. (1987) measured gas, and two
different solid phases present in a laboratory scale circulating ﬂuidized bed boiler model.
Yang et al. (1993) measured gas and solids velocities in a ﬂuidized catalytic cracking
riser with a Laser Doppler Anemometry system under relatively dilute operating
conditions. The gas ﬂow was seeded with talcum particles (dp = 1 p m). (Bensalah 1999)
used a Phase Doppler Anemometry system for measuring both phases in a circulating
ﬂuidized bed test rig which was operated under relatively dilute conditions. Glass
particles with two non-overlapping particle size distributions were used for the gas and
solid phase, respectively, thereby distinguishing the two phases by their sizes. The mean
diameter for the seeding glass particles was 9 ,u m and the mean diameter for the solids
was 60 p m. Gillandt et a1. (2001) measured gas and solid properties in a particle laden jet

with a particle range of 1:200 with a Phase Doppler Anemometry system.

1.3.2 Numerical simulations

Numerical predictions of multiphase ﬂow by CFD models have gained a more
widespread use over the past years. When narrowing the multiphase ﬂow ﬁeld into gas-
particle ﬂows there exists two different CFD approaches namely the Eulerian-Lagrangian
and the Eulerian-Eulerian. In this thesis, attention will be put on the latter CFD approach.
When modeling the gas-particle ﬂow with an Eulerian multiphase model it has become
more and more customary to rely on the kinetic theory of granular ﬂow for modeling the
particle phase.

The theory of the model is based on kinetic theory of non-unifonn gasses, as
presented by Chapman et al. (1970). The model was developed and matured through
publications as Jenkins et al.(1983), Lun et al.(1984), Jenkins et al. (1987), Gidaspow et
al.(1990) and Gidaspow (1994). A wide variety of models have been used ranging from
very simple approaches solving only the basic equations to more advanced approaches
solving a large number of equations.

Sundaresan et al. (1991), Nieuwland et al. (1996), Samuelsberg et al.(1996),
Mathiesen et al.(1999) and Neri et al.(2000) made numerical predictions with a two-
dimensional two-phase numerical model of a vertical riser section of a circulating
ﬂuidized bed. Gidaspow et al. (1992), Samuelsberg et al. (1996), Mathiesen et al. (2000)
simulated the entire ﬂuidized bed including cyclone and return leg in two dimensions.
Sun et al. (1999) and Benyahia et al. (2000) made two-dimensional two-phase
simulations of circulating ﬂuidized bed riser ﬂow for the participation in the PSRI

challenge problem (1995) presented at the F luidization XIII conference.

Expanding the numerical model from two to three dimensions has the clear
advantage of including three-dimensional effects in the numerical predictions. A
disadvantage is the considerable increase in the size of the problem, which needs to be
solved. Benyahia et al. (1999) predicted the ﬂow in a tall cylindrical riser in three
dimensions using ﬂuidized catalytic cracking particles for the solid phase. Kuipers et al.
(1999), Mathiesen et al. (1999) predicted gas-particle ﬂow in a circulating ﬂuidized bed
with a square sectioned riser in three dimensions and found the largest average
concentration of solids in the corners of the riser. Balzer et al. (1997) predicted the ﬂow
in an industrial circulating ﬂuidized bed boiler of 125MW with a three-dimensional
model using 350,000 nodes. The model was based on a comprehensive set of equations
related to the kinetic theory of granular ﬂow approach. De Wilde et al. (2001) also used a
comprehensive model with a large number of equations to predict the ﬂow in a
cylindrical ﬂuidized catalytic cracking riser in three dimensions. Zhang et al. (2001)
predicted the ﬂow in a ﬂuidized bed riser with a simple model solving a small number of
equations but used a relatively large number of grid cells for the numerical simulation.
The Euler/Lagrange approach has most often been used for predicting the ﬂow in
bubbling gas-ﬂuidized beds (e.g. Hoomans et al. (1996), Hoomans et al. (2000), van
Wachem et al. (2001), but has also been used for predicting the ﬂow in the riser of a
circulating ﬂuidized bed (e.g. Helland et al. (2000), Helland et al. (2002). (Helland et al.
(2000) simulated ﬂuidized bed riser ﬂow in two dimensions with an Euler/Lagrange
model tracking 250,000 particles.

The numerical modeling of bubbling ﬂuidized beds are related to the modeling of

circulating ﬂuidized beds. In the area of bubbling ﬂuidized beds, interesting work has

appeared which can inspire the improvement of circulating ﬂuidized bed modeling.
Syamlal (1998) modeled a bubbling ﬂuidized bed and compared a ﬁrst order upwind
scheme with second order schemes for discretizing the convection terms and found that
more realistic results were obtained with the second order schemes. Enwald et al. (1999)
evaluated four different closures of the two-dimensional form of the Euler/Euler model,
with and without gas and particulate-phase turbulence models. Additionally, they
evaluated a two-dimensional model with the simplest closure on a reﬁned mesh and
concluded that it was of higher priority to carry out the calculations on a ﬁne mesh than
to use more sophisticated models.

1.4 Objectives of the thesis

The overall objective of this work is to evaluate the usage of single phase and two-phase
turbulent models using CFD code FLUENT 6.2. in their ability to capture the important
physical phenomena encountered in a riser by comparing the numerical predictions
against the experimental ﬁndings of Ibsen et al.(2001).

These objectives are achieved through the following four tasks.

1. Veriﬁcation of fully developed single phase laminar ﬂow in a channel.

2. Realizability of various turbulent viscous models.

3. Comparison of various drag interphase models in a bubbly ﬂow.

4. Veriﬁcation and realizability of multiphase models in gas-solid simulations.

1.5 Computational domain considered in this work

The computational domain considered in all the simulations is as shown in Figure 1.2.

W=0. 17m
IHI _j_

 

Top view
D=0. 19m

T

 

 

 

 

Uniform pressure I I “II

 

 

 

 

 

outlet
7? R, = $01
u
Dh = 4 x ‘Area
Perimeter
H
. %% W = 8.8
Front View
=1 .2m
g i Y (v
X (W)
Location of origin in the x-y plane
v

 

 

 

 

 

we

Uniform velocity inlet

Figurel .2: Geometry of 3D-vertical upward channel ﬂow considered in the simulation

As the geometry considered here is rectangular, a three dimensional Cartesian geometry
is chosen with dimensions 0.17 x 1.20 x 0.19m, width(x) x height (y) x depth (2)
respectively, in order to match with the experimental device (Ibsen et al., 2001). A
staggered “hex-cooper” volume mesh was created in GAMBIT 2.2 with a very ﬁne mesh
near the walls and a rather coarse mesh at the center. The grid is made of 146,880 cells

and 156,695 nodes (35, 37 and 121 nodes in x, y and 2 directions respectively).

1.6 Theoretical and Numerical considerations of this work

The numerical simulations involve solving the Reynolds Averaged Navier—Stokes
(RANS-) equation and the Reynolds averaged continuity equation for the mean velocity
and mean pressure ﬁelds. A realizable eddy viscosity model for the Reynolds stress is
used in case of two-phase ﬂows as a closure for the RANS-equation.

All the simulations were performed using a segregated solver. The upwind
scheme is used in all the numerical simulations in which the control-volume based
technique is used to convert the governing equations to algebraic equations consisting of
integrating the governing equations about each control volume, yielding discrete
equations that conserve each quantity on a control-volume basis. The quantities at cell
faces are computed using a multidimensional linear reconstruction approach where the
order of accuracy is achieved at cell faces through a Taylor series expansion of the cell-
centered solution about the cell centroid. The set of algebraic equation is solved using the
Algebraic Multigrid Method Oiutchinson et al., 1986)

In case of two-phase ﬂows, the Eulerian multiphase model was used where the
coupling is achieved through the pressure and interphase exchange coefﬁcients using the
Phase Coupled SIMPLE algorithm for the pressure-velocity coupling, (Patankar, 1980).
1.7 Boundary and Initial Conditions applied in this work
The geometry of the physical domain and the boundary conditions are shown in Figure
1.2.

For single phage simulations: On the bottom side of the riser, an upward uniform velocity
was speciﬁed; on the lower side, the pressure relative to a reference pressure was

speciﬁed as zero. The simulation imposes a no slip condition at the wall.

10

For two-phase simulatio_n_§: In addition to the above conditions, the volume fraction of the
dispersed phase on the inﬂow boundary was speciﬁed. The velocity of the primary phase
and the velocity of the secondary phase were assumed equal and uniform. A zero
backﬂow volume fraction was speciﬁed for the dispersed phase on the pressure patch.
Initial conditions for single-phase simulations: Zero initial velocity was speciﬁed in the
ﬂow domain for all the single-phase simulations.

Initial condition_s for tWO-phﬁﬁ simulLticms: The initial concentration of the secondary

phase was zero in the ﬂow domain for all the two-phase simulations.

ll

CHAPTER 2
SINGLE PHASE FLOW
2.1 VERIFICATION OF FULLY DEVELOPED LAMINAR FLOW IN A DUCT
The objective of this section is to perform the numerical simulation of a fully developed
laminar ﬂow in a channel and to validate the computational results by comparing the span
wise velocities against the analytical solution as given in equation (2.3). The simulations
were performed by considering air as a Newtonian ﬂuid (constant p and constant p ) with
a Reynolds number of 5 based on hydraulic diameter. The steady state grid independent
numerical predictions of span wise velocities are compared with the analytical solution
and the error in the numerical predictions has been computed.
2.1.1 Governing equations applied to laminar single phase simulations
The governing equations used in this simulation are as follows
Continuity Equation
v - r7 = o (2.1)

The conservation of momentum is

(U-V)U=—%+VVZU+§ (2.2)

wherep ,vand (7 are the density, kinematic viscosity and velocity of the ﬂuid
respectively. g is the acceleration due to gravity and p is the static pressure.

2.1.2 Analytical solution for fully developed single phase channel flow
For a fully developed single phase laminar ﬂow in a square cross-section, the analytical

solution is given by Constantinescu V. N. (1995) and in Handbook of Physics (2002).

12

 

16(sz
0 = _ 2 £8 °° (n-1)/2[1- cosh(n7rY/W) ] cos(sz/W) (2.3)

- l
7,3” dz ”€355 ) cosh(nw/ 2W) n3

where W and D are the width and depth of the square channel.

2.1.3 Numerical results and verification

 

0.095

T

0.0475

>4 0 ware-ama--a—--a—-~a—--¢---a—-e-~a—a—a-Aend

-0.0475 r

 

 

-0.095 I I I
-0.085 -0.035 X 0.015 0.065

 

Figure 2.1: Plot of location of points at which
the velocity proﬁles are drawn for Z/H = -0.8.

Figure 2.1 shows the location at which a comparison between the numerical and
analytical span wise velocities is made in Figure 2.2 at an axial location of 1m. above the
bottom of the riser. The numerical predictions compares very well with the analytical
solution except near the centerline which exhibits a small error. Figure 2.3 shows the
absolute error in the numerical results. It can be seen that the maximum error of 1.4E-05
is found at the centerline. This could be due to numerical errors encountered in the
calculations. The error is calculated as

Absolute error calculation: Analyticalvalue — Numerical value

13

u(m/s)

Absolute velocity error (m/s)

0.001

 

 

---&--- Numerical
0.0009 - 0 Analytical

0.0008 ~ ~ ' “ ‘x.

 

 

0.0007 »- '9 “a
0.0006 . ii ‘9.
0.0005 ~ I3 5!

 

 

 

Figure 2.2: Plot of comparison of numerical and analytical span wise
velocities. They predict almost the same with a slight error. See pg 13
for error calculation

1 .60E-05

 

 

 

--t---Analy-Num

 

1.405-05 » ,4---
12013.05 ~

1001—2-05 ~ ,4" a“
8.00E-06 ~ f A
6.00E-06 ~ ," \
4.00E-O6 ~ '3" A

2.00E-O6 ~ 4: h

OOOE+OO L [It 1 l 4 4 r r I r r L g 1 r r r r l r r r r l A At] I
O

-0.6 -0.4 -0.2 0 0.2 0.4 0.6
W

 

 

 

Figure 2.3: Plot of absolute velocity error vs. dimensionless width. A
maximum absolute error of 1.4E-05 was found at the centerline.

14

2.2 COMPARISON OF VARIOUS VISCOUS MODELS IN A TURBULENT
DUCT FLOW

2.2.1 Governing equations applied to turbulent single phase simulations
The actual or instantaneous velocity is decomposed into mean ii and ﬂuctuating

velocities it" as

U.
"I

U = ii + u (2.4)
The continuity equation applied to ii + it" then gives

v - t7 = 0 (2.5)
The conservation of momentum applied to the averaged ﬁeld gives

p(ii - va) = —Vp + ,quii + pg — pV {W} (2.6)

where p , ,u and ii are the density, viscosity and velocity of the ﬂuid respectively. g is

the acceleration due to gravity and p is the static pressure.

Ensemble averaging is used to extract the mean ﬂow properties from the instantaneous

N

ones as ui(.ic',t) = lim 1- ZuS")(5c°,t) (2.7)
N—roo N n=1

UNIV): “i(3,t)+“i(x',t) (2-8)

The Reynolds-averaged momentum equation is

. . 6R~
[,qu =_§E.+_€3_ #% +__'J_+p§ (2.9)

where Ry- = —pii;-z'i'j is called the Reynolds stress. The Reynolds stress components

have additional unknowns introduced by the averaging procedure, hence they must be
modeled (related to the averaged ﬂow quantities) in order to close the equations and this

can be done in one of the following ways:

15

l. Eddy-Viscosity Models (Boussinesq, 1877).

It is a mathematical analogy to the stress-rate-of-strain relation for a Newtonian
ﬂuid. According to this hypothesis, the Reynolds stresses are modeled using an eddy (or
turbulent) viscosity y, as

le = —pu,-uj = ﬂt[-ax—; + a] — §[pk + #t EEJO‘U- (2.10)

2. Reynolds Stress Model (Gibson et al., 1978); (Launder, 1989); (Launder et al., 1975).

It involves calculation of the individual Reynolds stress components, iij-ii'j using

differential transport equations. The individual Reynolds stress components are then used
to obtain closure of the Reynolds-averaged momentum equation. The exact form of the
Reynolds stress transport equations may be derived by taking moments of the exact
momentum equation. This is a process wherein the exact momentum equations are
multiplied by a ﬂuctuating property, the product then being Reynolds-averaged.

The scope of this section is to verify the realizability of various viscous models
used in Fluent 6.2 namely the Spalart-Alhnaras, the Standard k- a) , the Realizable k— s and
the Reynolds Stress Model. The steady state numerical simulations of a turbulent single
phase ﬂow in a channel were performed by considering air as a Newtonian ﬂuid with a
Reynolds number of 10,000 based on hydraulic diameter. The near wall treatment

speciﬁcations can be found in Appendix A. The eigenvalues of the Reynolds stress

term — pul'u'j for each of these models have been computed and the invariants are plotted

using a “Lumley diagram” as shown in Figure 2.4 to verify their realizability.

l6

2.2.2 Lumley diagram (Lumley J., 1978)
The following table shows the equations used for constructing the Lumley diagram.

Table 2.1: Lumley Diagram

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Orientation States Elgenvalues Invariants
Remarks
111 ’12 ’13 11b IIIb
A Uniaxial alignment (ID) 1 0 0 2/3 2/9
B Planar anisotropic (2D) l-x x 0 11b =2/9+21Hb 0 S x S 1/2
C Planar isotropic (2D Random) 1/2 1/2 0 1/6 -1/36
D Planar axisymmetry x x l-2x 2111b =6(IIb/6)l'5 1/3 st 1/2
E Isotropic (3D Random) 1/3 1/3 1/3 0 0
F Axial axisymmetry x x l-2x 2111b =6( 11b /6)‘-5 osxs 1/3
The anisotropic tensor is deﬁned as—
b:—: “:3: —15 (2.11)
= tr u "u 3 =

where g is the identity matrix. Three invariants of 1:) can now be deﬁned as

Ib =tl‘(I=))
IIb=tr(=

1111)" = [IL

)0 (2.12)
“)2

IIO‘ IIO‘

Two nontrivial invariants IIb and 1111, are used to describe the realizability of the tensor,

which is shown in Figure 2.4 in the enclosed region.

"I"!

Let 21, 2.2 , 113 are the three eigenvalues of the tensor—FL). The eigenvalues
tr u 'u

21 , 22 , 23 are independent of the choice of the coordinate system. The invariants Ilb and

IHb can be expressed as eigenvalues of the matrix 2 as

111, = (,11 —1/3)2 + (22 -l/3)2 + (2.3 —1/3)2 (2.13)

111., = (21 —1/3)3 + (2.2 —1/3)3 + (23 - l/3)3 (2.14)

17

 

 

A
A
B
\

Set of Realizable

Orientatio: States

F
C
D
E L
m. = at - 2 - 21

Figure 2.4: Sketch of the boundaries of the realizable states for 2

As the sum of the three eigenvalues is unity, [13 is chosen to be dependent on 21 and 22.

A two-dimensional solution space can be obtained from equations 2.13 and 2.14.
2.2.3 Various viscous models and their approximations
The following are the various turbulence models used in Fluent 6.2

The Spalart-Allmaras Model

It is a single transport equation model proposed by Spalart et al. (1992). It treats that the

production and destruction turbulent terms cancels each other and hence solves directly

for a modiﬁed turbulent viscosity 6 as

— 2

Dv 6v

-— = Gv +— - Y 2.15
D: 0' 6x; —-{(y+pvj——v—}+ Cbzp[5 v] v ( )

18

where Eddy — Viscosity is obtained as

V03
(%) + C31

 

m= p 5f“ . fvl a (2.16)

0,, , Yv are the production and destruction of turbulent viscosity, v is the molecular

kinematic viscosity and 0",, =% , CV] = 7.1 and C b2 = 0.622.

Standard k — a) Model
The standard k — a) model in FLUENT is based on the Wilcox k — a) model proposed
by Wilcox (1998). It belongs to the general 2-equation Eddy Viscosity Model family.

The exact transport equations for k and a are

Dk— _a[_ u1(k'+£)+v—a—]—m%-s (2.17)
l

 

Dt 6x p 6x, 6x

 

—_ —'_ . —'—_' can’-
2§=_6__8,u;_23221_6£_+v_6_5_2v6u, aulaul+aut J
Dt 6x1 p 6x j ax}- 6x1 6x j ax, 6xj 6x1 6x1

 

 

 

 

—2vu

2
r 2 r r all, 2 r
,6“ 6“ -2v— 6" 6“ ——2[v a “' ] (2.18)

’axjaxmxj 6xj6x1 6x1 6x16): 1-

The modeled transport equations for the turbulence kinetic energy k and the speciﬁc

dissipation rate a) in the Standard k - a) model are

Dk BU, a 6k
—= k +— + 2.19
pm i15’0‘ pﬂ fﬂ w (ix/“K” ELI—j] ( )

19

p-‘E= a9 "a—UI-pﬂfrswz +i[(p+i’—]93] (2.20)

D kryax ax ax
‘ 1 j aw j

6‘ I t k
where (oz—oc— and ,u,=a p—
r a)

Realizable k - a Model
The term "realizable" means that the model satisﬁes certain mathematical constraints on
the Reynolds stresses, consistent with the physics of turbulent ﬂows. The realizable k - a

model proposed by Shih et al. (1995) address the following:

1. A new eddy-viscosity formula involving a variable C ﬂoriginally proposed by

Reynolds.
2. A new model equation for dissipations based on the dynamic equation of the
mean-square vorticity ﬂuctuation.
Transport Equations for the Realizable k - a Model

The modeled transport equations for k and a in the realizable k - a model are

6 6 a ,u, 6k

—-(0k +—Ioku- =— +—— — +6 +0 — s—Y 2.21
a: ) axj 1) ax]. III! Ukjaxj] k b P M ( )
a 6 a p, 66 £2 a

— £+— au-=—— +—-—-+ Sa— -————+C —C G
at ) axj(p j) ax] Hi“ 0‘8)ij pCI pCZk-I-x/V; 18k 38 b

(2.22)

h - k2 C — 043 ’7 -Sk S- 25 S

w erey, —pC#—;—, l—max . ,ﬁ , r)— E, _ ij ij

In these equations, G k represents. the generation of turbulence kinetic energy due to the

mean velocity gradients. G b is the generation of turbulence kinetic energy due to

20

buoyancy. Y M represents the contribution of the ﬂuctuating dilatation in compressible
turbulence to the overall dissipation rate. C2 =1.9 and C15 =1 .44. ck =1.0 and 08 =1.2.
where the ranges are C2 :1.8-2; C18 :1-1.8; 0k :0.9-2; 05 :0.95-3 (Ching and Jaw, 1998).
Reynolds Stress Model

The Reynolds stress model proposed by Gibson et al. 91978); Launder, (1989); Launder
et al. (1975) attempts to address the deﬁciencies of the eddy viscosity model but it is

computationally more expensive.

The Reynolds Stress Transport Equations
The exact transport equations for the transport of the Reynolds stresses p u "u- may be

written as follows

 

 

 

a —) 52410 66
"’."' U ﬁlii')=— —Xk—[pufiij “'17 +p(-5u’- +514 1
. u - J
Local Time Derivative ac: .=.Convection D“). :Turbulent Dzﬁiision

 

a a T 7761?} 27-7617,:
— —u'-'- u'u' +u'-u '— u'0+ u—'-t9

 

 

 

 

 

ﬂ v Gy- z-Buoyancy Production
D L, y- EMoIecular Diﬁ‘itsion Pij sStress Production
5,7. 617'- W
+ p— +——L -— 2p—’-———- - 2,00,, u' u' maﬁa" + u til-maﬁa") (2.23)
axj- 6x,- 6x k 6x k k 1
1 B—W——’ F g- =Pr0duction by System Rotation

¢ij‘ =Pressure Strain 5:] =DiSSipatton

21

2.2.4 Numerical results and veriﬁcation

 

X

Figure 2.5: Velocity vectors for the Spalart-Alhnaras model at Z/H——0.5.

 

 

 

 

Figure 2.6: Velocity vectors for the Standard k — 60 model at Z/H-—0.5.

22

 

‘ , “\\\\\l.

 

 

 

 

Figure 2.8: Velocity vectors for the Reynolds Stress Model at Z/H=-O.5.

23

Figures 2.5 to 2.8 shows the velocity vectors for various turbulent viscous models and it
shows that all the models predict the same proﬁles except the Reynolds Stress Model.

The proﬁles at which the realizability plots for the various viscous models are plotted are

 

 

 

 

 

 

 

 

 

 

shown in Figure 2.9.
Y/D=0 Y/D=0.48
Z/H=—0.5 Z/H=-0.5
Y Proﬁlel Y
Proﬁle2
X X

Figure 2.9: Location of proﬁles at which the realizability plots for the various
viscous models are drawn.

0.05
0.045 ~
0.04 »—
0.035 ~

9 0.03 ~

“0025 .
0.02 L
0.015 —
0.01 —

 

 

0.005 * Walls. Center

0 l l _ 1 #

-0.005 -0.003 -0.001 IIIb 0.001 0.003 0.005

 

 

 

Figure 2.10: Realizability plot for the Standard k - a) model at proﬁlel.
It predicted thatIIIp, z 0.

24

IIb

0.08
0.07
0.06

0.05 -

0.04
0.03
0.02
0.01

0

 

  
    

Center

 

Walls

 

 

l l l A

 

-0.0045 -0.0025 -0.0005 0.0015 0.0035

III b

Figure 2.11: Realizability plot for the Standard k — (0 model at proﬁle 2.
It predicted that 1111, z 0 .

0.04
0.035 -
0.03 ‘
0.025
.9

0005 Center

 

 

 

 

O 1 L l l l I
-0.002 -0.002 -0.001 -5E-04 0 0.0005 0.001 0.0015 0.002
IIIb

 

Figure 2.12: Realizability plot for the Realizable k - 3 model at proﬁle 1
It predicted that 1111, z 0 .

25

0.05 - ~
0.045 I
0.04 +—
0035 ~

9 0.03 ~

... 0.025 +-
0.02 —
0.015 ~
0.01 —
0.005 a

0 .

-0.005 -0.003 -0.001 0.001 0.003 0.005
III b

Figure 2.13: Realizability plot for the Realizable k - a model at proﬁle 2.
It predicted that IIIb z 0 .

 

    

Center

 

 

 

0.8

 

0.7 r
0.6 ~
0.5 r

IIb

0.4 ~

0.3 ~

0.2 —

l

0.1

 

‘ Center
0 I L l l 1

 

 

 

-0.05 0 0.05 0.1 0.15 0.2 0.25

111 b
Figure 2.14: Realizability plot for the Reynolds stress model at proﬁle 1

26

0.8

 

0.7 _
0.6 ~
0.5 -

Kb

0.4 —
0.3 —
0.2 —
0.1

T

 

 

 

 

O l l 1 l

-0.05 0 0.05 0.1 0.15 0.2 0.25
III b

Figure 2.15: Realizability plot for the Reynolds stress model at proﬁle 2

Figure 2.10 to 2.15 shows the realizability plots for the Standard k—(o model, the
Realizable k -6 model and the Reynolds stress model. It was found that all the viscous
models proved to be realizable except the Spalart-Alhnaras. The Spalart-Alhnaras is
found to be highly unrealizable and the computed eigenvalues and invariants for this
model can be found in Table C1 and C2 of Appendix C.

Also it can be seen from Figures 2.10 to 2.13 that the Standard k —a) model and
Realizable k -5 model which uses the Boussinesq hypothesis predicts that the III

invariant of the anisotropic tensor 2 is almost zero which can be explained as follows

For a fully developed steady ﬂow,

a = uz(x, y)a, (2.24)
bu _, 2 au _. _.
Vuz = 1372er + jeyez (2.25)

27

 

 

 

 

 

 

 

 

 

 

 

 

Since .1 = avg + VET) (2.26)
S = l au, axe, + au, “,5, + au, 5,5, + au, 1;,
= 2 6x 6y ﬁx by

r ~ \

O 0 auz
S=— O 0 z or S=—- 0 0 b
= 2 0y = 2 b 0
6172 61:2 0 a

tax 5)’ 2

. 27., 2
Also srnce — pu u = 221,; - §pk£ (2.29)
whichgives "W“ =1 -3”—‘ =11+b (2.30)

- 2pk 3= 2pk= 3= =

hence 2 is related to g as 1:) = _ ”t g

pk

The eigenvalues of the symmetric matrix :2 are {0, W12 + b2, — Vaz + b2 }
Therefore, the eigenvalues of l; are {0, _ 51’ \Iaz + b2, \Ia2 + b2}

&
Pk Pk

 

2
which results in Ib = 0; I11, = 2[%] (32 + b2) and IIIb = 0.

Since in the simulations, the ﬂow is not fully developed, the ml, is predicted as nearly
zero as seen in Figures 2.10 to 2.13 whereas the Reynolds Stress Model does not make

the above assumption and use a different model for — p uj-u’j .

28

(2.27)

(2.28)

(2.31)

 

1]

UN
[IDES
[IDS
01155
UB5
IIIIIﬁ '
004

- Uﬂﬁ Z w
I133
0025
[II]?
0115
Bill
01103

I

]

IA]

 

 

 

 

 

 

 

 

 

 

X X
Y/D=0 Y/D=O.48 Y/D=0 Y/D=0.48

The Standard k — a) model The Realizable k - a model

 

 

 

 

Figure 2.16: Contours of the turbulent kinetic energy for the Standard k - a) model and
the Realizable k — a model at planes Y/D=0, Y/D=0.48 and Z/H=-0.5.

Figure 2.16 predicts a variation in the turbulent kinetic energy for the Standard k - a) model
and the Realizable k — a model at various cross-sections in the riser.

29

CHAPTER 3
TURBULENT TWO PHASE FLOWS
Overview of two phase flows

Signiﬁcant efforts have been made to describe the ﬂow phenomena and dynamic
behavior of the two-phase ﬂows where the transfer of momentum, mass, and energy
across the phase interface is of crucial importance.

Also, numerical simulations should satisfy strong demands in relation to
approximation, especially near a wall, stability, keeping some important integral
properties of governing equations and others. The Boussinesq's hypothesis between
stress tensor and strain velocity tensor forms a rather simple model which enables to
conduct numerical calculation up to bounded walls including a laminar sub layer. The
succeeding chapters deal with the numerical simulation of turbulent bubbly ﬂow and gas—
solid ﬂows.

3.1 COMPARISON OF DRAG INTERPHASE MODELS IN A BUBBLY FLOW
Bubbly ﬂows play a signiﬁcant role in a wide range of geophysical and industrial
processes like oil transportation, mixing in chemical reactors, and elaboration of alloys,
cooling of nuclear reactors, two-phase heat exchangers, aeration processes, ship
hydrodynamics, and atmosphere-ocean exchanges. In a two-phase bubbly ﬂow, the mass,
momentum and energy transfer processes involved are inherently complicated and
closely linked to phase distribution proﬁles through the strong interaction at the gas—
liquid interface. Numerical simulations can help to predict two-phase dispersed ﬂows but
the capabilities of these codes are, to a large extent, limited by the closure models used to

represent the momentum exchange between the dispersed and the continuous phase.

30

Owing to the very weak relative density of bubbles compared to that of the liquid, almost
all the inertia is contained in the liquid, making inertia induced hydrodynamic forces
particularly important in the prediction of bubble motion. Hence, in this section,
predictions of various drag interaction models have been compared for a bubbly ﬂow
problem.

3.1.1 Case study Problem

The scope of this section is to perform steady state numerical simulations of bubbly ﬂow
in a channel using an Eulerian approach in order to compare and validate the various drag
interphase models used in Fluent 6.2 namely the Schiller and Naumann model (Schiller,
1935), the Morsi and Alexander model (Morsi et al., 1972) and the symmetric model.
Two Newtonian ﬂuids, air dispersed in water is assumed with a uniform velocity of
0.3m/s and a 10 vol. % dispersed (air) phase. The air particles were assumed to be
spherical with a diameter of 1mm. The simulations were performed using a realizable
eddy viscosity model. The near wall treatment speciﬁcation for this case can be found in
Appendix A.

3.1.2 Governing equations applied to gas-liquid simulations

The governing equations used in these simulations are based on a separate treatment for
each phase and are given below

The volume fraction of each phase is calculated from a continuity equation as

1 6 ..
_(E(a,pq).v.(a,p,.,)) =0 (3.1)
p r q

Where iiq is the velocity of the primary phase q and prq is the phase reference density of

the phase q in the solution domain. The volume of phase q, is deﬁned by

31

Vq = [61qu (3.2)

V
n
where Z aq =1 (3.3)
q=l .
The effective density of phase q is pq = aqpq (3.4)

Equations (3.2) and (3.3) calculate the primary-phase volume fraction.

The conservation of momentum for a ﬂuid phase q is

£9—(aqpqiiq)+ V - (aqpqiiqiiq) = —aqu + V -: + aqpqg + §(qu(up - uq»

at p=1
(3.5)
Here g is the acceleration due to gravity and P is the pressure shared by all phases.
= .. _. 2 .. = .
rq = aqpq (qu + quT)+ aq(kq — qu )V ~ qu (3.6)

Here ,uq and kq are the shear and bulk viscosity of phase q. .

3.1.3 Theoretical formulation of various drag interphase models
Momentum exchange between the phases is based on the value of the ﬂuid-ﬂuid

exchange coefﬁcient K pq. It is the drag function that differs among the exchange-

coefﬁcient models. The following are the three different drag interphase momentum
theories found in FLUENT for the gas-liquid ﬂow.

The exchange coefﬁcient can be written in the following general form as

zaqapppf

(3.7)
T

Pq
P

where q is the primary phase and p is the secondary phase. The particle relaxation time is

given as

32

2
_. '0de

 

 

 

 

r — —— (3.8)
P
1821‘]
where drag ﬁlnction,
CD Re
= 3.9
f 24 ( )
and d p is the particle diameter
The drag coefﬁcient given by various models is as follows
The Schiller-Naumann Model (Schiller, 1935)
0.687
D ' 0-44 Re > 1000 '
The Morsi and Alexander Model (Morsi et al., 1972)
“2 “3
C = a + — + — 3.11
D 1 Re R62 ( )
' 024,0 0 < Re < 0.1
3.690,22.73,0.0903 0.1 < Re <1
1.222,29.1667,—3.8889 1 < Re <10
O.61617,46.50,—116.67 10 < Re < 100
01,02,03 = i (3.12)
0.3644,98.33,—2778 100 < Re <1000
0.357,l48.62,—47500 1000 < Re < 5000
0.46,—490.546,578700 5000 < Re < 10000
L0.5191,—1662.5,5416700 Re 2 10000
The Symmetric Model
2
=aP(aPpP+aqpq)f 1' _(aHpP+aqpq}1p. (3.13)

[’4 ’ P‘I —
rpq 18(appp+aqqu

33

0.687
f _ CD RC Where C - 24[1+ 0.15RC J/RC R6 51000
24 D - 0-44 Re >1000

 

(3.14)

 

 

,l I v I —;—s-n&sym 3
1.75] — m‘a ‘

 

 

 

’ I
1.5 ; :

 

1.25 3 .1

Cr)

0.75 1

0.5 :

 

 

0.25E 222
0 2000 4000 6000 8000 100001200014000
Re

 

Figure 3.1: Plot of comparison of the various drag models for the bubbly ﬂow.

34

3.1.4 Numerical results and discussions
In Figure 3.2, the locations at which the points extracted to produce Figures 3.3 to Figure

3.8 are shown. X/W=0
Y/D=0

 

 

 

 

 

Figure 3.2: Location of proﬁle at which the various drag models are compared.

 

 

1.04 ~

 

 

 

1.02 r

u/<u>

0.98 -

0.96 r

 

 

 

0.94
-0.6 -0.4 -0.2 0 0.2 0.4 0.6

z/H

 

Figure 3.3: Plot of dimensionless axial velocity of water vs. dimensionless height for the
Schiller-Naumann model, the Morsi and Alexander model and the Symmetric model
(sn, ma, sym). All models give similar results.

35

 

1.04

 

1.03 i

 

 

 

1.02 I
1.01 —

u/<u>

0.99 r
0.98 ~
0.97 r

 

0.96 ~

 

 

 

0.95 1 l J I l
-0.6 -0.4 -O.2 0 0.2 0.4 0.6

Figure 3.4: Plot of dimensionless axial velocity of air vs. dimensionless height for the
Schiller-Naumann model, the Morsi and Alexander model and the Symmetric model
(sn, ma, sym). All models give similar results.

 

 

 

 

 

 

 

 

1.2
-*-sn
coo-arm
/\
Q.
X 1 .
Q.
I.
0.8 1 A l l l l I 1 l 1 l l ‘ 1 J 1 1 44 1 1 1 1 1 l a 1 1 1
-0.1 0.1 0.3 0.5 0.7 0.9 1.1

z/H

Figure 3.5: Plot of dimensionless axial dynamic pressure of water vs. dimensionless
height for the Schiller-Naumann model, the Morsi and Alexander model and the
Symmetric model (sn, ma, sym). All models give similar results.

36

 

 

1.06 —
1.04 —
1.02 L

 

 

 

p/<p>

0.98 —
0.96 ~
0.94 ~
0.92 ~

09 111L1111411411L11L1111111111
I

-O.1 0.1 0.3 0.5 0.7 0.9 1.1
z/H
Figure 3.6: Plot of dimensionless axial dynamic pressure of air vs. dimensionless height

for the Schiller-Naumann model, the Morsi and Alexander model and the Symmetric
model (sn, ma, sym). All models give similar results.

 

 

 

 

1.5
1.4 ~
1.3 _
1.2 ~
1.1 L

 

0.9 ~
0.8 -
07 . “’5"
' Momma
0.6 t -O-sym

05 IllillllJlllL4111111114111111
U

-0.1 0.1 0.3 0.5 0.7 0.9 1.]
z/H

Figure 3.7: Plot of dimensionless axial turbulent kinetic energy of water vs.
dimensionless height for the Schiller-Naumann model, the Morsi and Alexander model
and the Symmetric model (sn, ma, sym). All models give similar results.

ke/<ke>

 

 

 

 

 

 

 

37

 

0.998

 

 

 

 

 

 

 

0.9975 . ......“
-€-sym
0.997 _ ‘\\
4; 0.9965 ~
X
‘5‘ 0.996 ~
0.9955 —
0.995 —
0.9945 ”Ullwllliw111111411134...
-0.6 -0.4 -0.2 0 0.2 0.4 0.

z/H

Figure 3.8: Plot of dimensionless axial volume fraction of air vs. dimensionless height
for the Schiller-Naumann model, the Morsi and Alexander model and the Symmetric
model (sn, ma, sym). All models give similar results with a very slight difference.

From Figures 3.2 to Figure 3.8, it can be seen that all the correlations studied predict
similar values of global quantities such as the velocity, pressure, turbulent kinetic energy
and concentration of the dispersed phase for the steady state solution. From the Figure 3.1

it can be seen that for the Reynolds number of 10,000 the C D values given by the various

drag interphase models is about the same.

The proﬁles at which the realizability plots are drawn for the secondary phase is shown in

 

 

 

 

 

 

 

 

 

 

Figure 3.9.
Y/D=0 Y/D=0.4
Proﬁle
Y Y
Proﬁle
X X

Figure 3.9: Location of proﬁles at which the realizability plots are drawn for Z/H = -0.5

38

0.0035

 

0.003 —

0.0025 ~

0.002

IIb

0.0015

0.001

0.0005

 

O l

 

Center

Walls

1 I l

 

 

-0.00005 -0.00003

-0.00001 0.00001 0.00003 0.00005
III b

Figure 3.10: Realizability plot for the secondary phase (air) in the gas-liquid
simulations at proﬁle 1. The realizable k - 6 model predicted thatIIIb z 0 .

 

0.004 .
0.0035
0.003
0.0025
.0
3 0.002
0.0015
0.001

0.0005

 

0

 

J

 

 

-0.0001 -0.00005

0 0.00005 0.0001
III b

Figure 3.11: Realizability plot for the secondary phase (air) in the gas-liquid
simulations at proﬁle 2. The realizable k - 5 model predicted that IIIb z 0 .

39

The realizability plots for the secondary phase (air) using the Schiller-Naumann drag
interphase model is as shown in Figures 3.10 and 3.11. In the two-phase simulations of
gas-liquid ﬂows, the realizable k -8 model proves to be realizable for the secondary
phase (air) while it shows to be highly unrealizable for the primary phase (water) and the
computed eigenvalues and invariants for this unrealizable case can be found in Table C3
and C4 of Appendix C. Again, it can be seen ﬁ'om the above ﬁgure that the IH invariant

of the anisotrOpic tensor b is almost zero.

Also, all the various drag models studied in this case predicts exactly the same
realizability values with a negligible difference. For the sake of brevity; the realizability
plots for all the other drag models are not presented. Figure 3.12 shows a variation in the

turbulent kinetic energy in the riser.

 

3 011021

 

 

 

 

 

 

 

 

x x
Y/D=0 Y/D=0.48

 

 

 

Figure 3.12: Contours of the turbulent kinetic energy for the primary phase (water) at
planes Y/D=0, Y/D=0.48 and Z/H=-0.5.

40

3.2 NUMERICAL SIMULATION OF A GAS-SOLID FLOW IN A RISER

The sc0pe of this section is to perform the transient numerical simulations for the gas-
solid ﬂow under isothermal conditions in a riser using a realizable eddy viscosity model.
The purpose is to evaluate the ability of the multiphase model to capture important
physical phenomena encountered in a catalytic riser. Further, the experimental ﬁndings of
Ibsen et a1. (2001) were used here for evaluating the quality of the numerical predictions.

Transient simulations of gas-solid ﬂows in a riser have been performed using a
realizable “eddy” viscosity model proposed by Shih et al. (1995) to account for the
transport of mean momentum by turbulent ﬂuctuations. The dispersed turbulence model
and a granular model were used to model the gas and solid phase respectively. Air is
considered as a primary phase with constant density and viscosity. The solid particles are
assumed to be spherical in shape with a uniform density. The various parameters used in
this simulation are shown in Table 3.1. The near wall treatment speciﬁcations can be
found in Appendix A.
3.2.1 Assumptions made in the gas-solid simulations
Due to the complex nature of the ﬂows associated with the riser, the following
simpliﬁcations in the geometry and particle size were introduced for the purpose of this
exploratory study.
Particle size: Since the simulations presented here were based on three dimensional
modeling of the gas-solid ﬂow, the numerical modeling was restricted to two-phase
simulations with one solid phase with a representative particle diameter for the particle

size distribution in the experiments in order to keep the computational time down.

41

CirculaMuidized Bed Geometry: The inlet is located at the bottom and the outlet is
located at the top, thereby neglecting the effects of the inlet and the exit locations, which
are placed at the side on the actual riser.

3.2.2 General guidelines for mathematical modeling in fluidization

Harris et a1. (1996) present a classiﬁcation of models for simulating circulating ﬂuidized
beds. According to the authors there are three types of mathematical models:

1. Models that predict the axial variation of the density of solids and disregard its lateral
variations.

2. Models that predict the span wise variation of the density of solids and the high
average slipping velocities, accounting for two or more regions of different ﬂow
characteristics.

3. Models that apply the fundamental conservative equations of the ﬂuid dynamics for
predicting the two-phase gas-solids ﬂow.

The ﬁrst two types of models are mostly used for preliminary design, mainly for
investigating the effects on the process of geometry and operational conditions. These
models can easily include chemical kinetics for simulating the performance of reactors.
The models of the third type, as for example a two-ﬂuid model, are more suitable for
research allowing, for instance, the behavior of ﬂow local structures and the effects of
local geometry to be studied. Figure 3.13 presents actual lines and tendencies for the third
type of mathematical modeling. There are two major tendencies for modeling, following
a treatment either Eulerian for both phases (Eulerian formulation) or Eulerian for the

ﬂuid phase and Lagrangian for the particulate phase (Eulerian-Lagrangian formulation).

42

 

Paths in the mathematical modeling for
ﬂuidization processes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Eulerian formulation for both phases Eulerian formulation for the gas
phase and Lagrangian for the
solid phase

1
' Traditional model With laminar - Discrete particle approach-particle
ﬂow for both phases. dynamic method.
- KTFG with laminar or turbulent . Dynamics of Stokes
110‘” for the 835 phase - Pseudo-particle approach

 

 

 

 

 

 

Figure 3.13: Sketch outlining the actual lines and tendencies for the third type of
mathematical modeling in ﬂuidization.

In the Eulerian formulation both phases are treated as a continuum. Each phase is
modeled separately in terms of a system of conservative equations for mass, momentum
and energy. The conservative equations present terms accounting for interface
interaction, which are related to mass, momentum and energy exchanges through the
interface. In their traditional formulation, those models require the dynamic viscosity of
the solid phase to be speciﬁed. A constant value for the solids viscosity can be obtained
through momentum balances applied to experimental data (Gidaspow et al., 1992).

The traditional Eulerian formulation has been extensively applied to ﬂuidization
(Gidaspow et al., 1983; Gidaspow, 1986; Bouillard et al., 1989). A new approach has
been developed by a number of researchers for dealing with solids phase viscosity
(Jenkins et al., 1983; Lun et al., 1984; Jenkins et al., 1985). This is the kinetic theory of
granular ﬂow, which is based on the kinetic theory of dense gases (Chapman et al.,

1970).

43

Bagnold, (1954) and Gidaspow, (1994) is generally credited with starting the
kinetic theory approach of granular ﬂow. The kinetic theory of granular ﬂow is based on
an analogy between the ﬂow of granular materials and the motion of gas molecules
(Peirano, 1996). In spite of allowing the direct determination of the solids viscosity, the
kinetic theory of granular ﬂow implies in more complex numerical procedures and higher
computation times. Gidaspow, (1994) was the ﬁrst to present a detailed theoretical
derivation of Kinetic Theory of Granular Flow and to consider its application to
particulate ﬂows.

In Eulerian-Lagrangian formulation only the gas phase is assumed as a
continuum. This approach allows a better understanding of the particle—particle
interactions. However, the Eulerian-Lagrangian formulation requires a complete set of
equations (Newton’s second and third laws) to be written for each particle in the ﬂow
ﬁeld, difﬁculting its application to ﬂuidization in view of computational limitations.
Otherwise it is a very useful tool for the development of new rheological models for
ﬂuidized suspensions, and to enhance the formulation of closing laws required by two-
ﬂuids Eulerian models.

The analysis of gas-solid ﬂows is complex because of the strong coupling
between the solid and gas phases. The gas ﬂows through the interstitial spaces or voids
created by the particles, moving the particles and re-arranging the gas ﬂow paths. The gas
phase exerts a drag force on the solids; the solids exert an equal and opposite force drag
on the gas. Furthermore, the pressure gradients created in the gas ﬂow give rise to
pressure forces on the particulate phase. Density differences between the two phases

cause buoyancy driven ﬂows. Thus the two phases exchange momentum and energy.

44

3.2.3 Governing equations applied to gas-solid simulations

The mathematical model is based on a three-dimensional Eulerian realizable k — a
multiphase model. The gas phase calculations are done using a continuum approach. The
conservation equations for the solid phases are based on the kinetic theory for granular
ﬂow. The governing equations for the gas solid model may be summarized as follows.

The volume fraction of each phase k is calculated from a continuity equation
0 a
5(akpk)+ V ' (akpkuk) = 0 (3-15)

where a k , pk and 17 k are the volume fraction, density and velocity of phase k respectively.

The conservation of momentum for a ﬂuid phase q is

%(aqpq5q)+ V ' (aqpqﬁqﬁq) = ‘0‘qu + V ' : + aqpqg + Eleq(17p - at!)
q:

(3.16)
The solid-phase stresses are derived by making an analogy between the random particle
motion arising from particle-particle collisions and the thermal motion of molecules in a
gas, taking into account the inelasticity of the granular phase. As is the case for a gas, the
intensity of the particle velocity ﬂuctuations determines the stresses, viscosity, and
pressure of the solid phase. The kinetic energy associated with the particle velocity
ﬂuctuations is represented by a "pseudo thermal" or granular temperature which is
proportional to the mean square of the random motion of particles.

The conservation of momentum for the solid phase s is given as

a = n
5(aspsiis) + V - (aspsiisiis) = -asz - VPs + V . rs + aspsg + Z; qu(1'iq — it's)

q—l
(3.17)

45

The stress-strain tensor is given as

.. ..7‘ 2 ..
rq = aqpq(qu +qu )+aq[/1q --§,uq)V-uq1 (3.18)
where p and g are pressure and gravitational acceleration respectively. pq and liq is the

shear and bulk viscosity of phase q; K sq is the drag coefﬁcient between the phase 3 and

q. The solid phase pressure ps, the bulk viscosity is and the shear viscosity #3 are

derived from the kinetic theory for granular ﬂow.
The interphase momentum transfer coefﬁcient between gas and solid is modeled

as proposed by Gidaspow, (1986) which is

3 asaqpq ifs " l7q _
qu = 2CD dl laq2‘65 (3.19)
S

 

where CD = 24 [l+0.15(aq Re,)°~687] (3.20)
aq Re 5

 

3.2.4 Governing equations applied in the kinetic theory of granular flow

The following are the equations applied in the kinetic theory of granular ﬂow

Granular Temperature

The viscosities need the speciﬁcation of the granular temperature for the solids phase for

which an algebraic equation proposed by Syamlal et al. (1993) is
0: (- 153749;): V17, -79. +¢qs (3.21)

where (— p SI + 2?): V173 is the generation of energy by the solid stress tenor, 765 is the
collisional dissipation of energy and gigs is the energy exchange between the ﬂuid or solid

phase and the solid phase.

46

The collisional dissipation of energy 79, represents the rate of energy dissipation

within the solids phase due to collisions between particles which is given by Lun et al.

(1984) as

2
76 =12(1-ess k0,“ ,0 a
s dsx/Z s

 

2 3/2
s 9. (3.22)

The transfer of the kinetic energy of random ﬂuctuations in particle velocity from
the solids phase to the ﬂuid or solid phase is represented by ¢qp and is given by
Gidaspow et al. (1972).
¢qs ='3 qu9s ' . (3.23)
Solids phase shear viscosity (neglecting frictional viscosity)

It is given as the sum of kinetic and collisional shear viscosities as

 

lopsds 19,7: 4 2 4 a “2
= 1+- 1+ + —a d 1+ -—‘—
#3 96613 (1 + ess )g0,ss 5 g0,ss as( es) 5 5:03 ng,ss( 933 7:

J J

v

”akin ”3:601!
(3.24)
Granular Bulk Viscosity
The solids bulk viscosity accounts for the resistance of the granular particles to

compression and expansion. The expression for this is given by Lun et al. (1984).

4 6 1/2
ks = gaspsdsgo,” (1 + e“ (7;) (3.25)

For granular ﬂows in the regime where the solids volume fraction is less than its

maximum allowed value, a solids pressure is calculated independently and used for the

pressure gradient term, Vps , in the granular-phase momentum equation. Because a

47

Maxwellian velocity distribution (which has a temperature term) is used for the particles,
a granular temperature is introduced into the model, and appears in the expression for the
solids pressure and viscosities.

The solids pressure is composed of a kinetic term and a second term due to

particle collisions and is given by Lun et.al. (1984) as

2
Ps = aspsgs + 2103(1 + 335 )as gO,ss63 (326)
where cm is the coefﬁcient of restitution for particle collisions, go,” is the radial

distribution ﬁlnction, and 63 is the granular temperature.

Radial distribution Function
It speciﬁes a correction factor that modiﬁes the probability of collisions between grains

when the solid granular phase becomes dense. The expression for this is given by Sinclair

et al. (1989)

 

l
go=1—[ a, T (3.27)

3.2.5 Experimental set up of the 1/9 scale Cold Circulating Fluidized Bed Boiler

The dimensions of the experimental set up of Ibsen et al. (2001) were 1.5 m x 0.19 m x
0.17 m corresponding to Height(x) x Depth(y) x width (z) respectively. A cyclone was
located at the rear of the riser, 1.2 m above the primary air distributor to separate the
solids, which passed a particle seal designed as a bubbling bed, before being reintroduced
in the lower part of the riser. No secondary air was used. The amount of solids

recirculated was adjusted to give a pressure drop across the riser equal to 2.7 KPa

48

(corresponding to 8KPa in the full-scale boiler). A detailed description of the model is
provided by J ohnsson et al. (1999).
3.2.6 Numerical conﬁguration of the gas-solid simulations

The numerical ﬂow parameters applied in the gas-solid simulations are tabulated in Table

 

 

 

 

 

 

 

 

 

 

 

3.1.

Table 3.1: Parameters for the gas-solid simulations
5;“???ng the gammy XxYxZ 1.2 x 0.17 x 0.19
Number of Mesh Cells N cell 146,880
Gas Density Pg 1.2 kg/m3
Gas Velocity Vg 1.0 m/s
Gas Viscosity #1 g 1.8 E-05 kg/ms
Particle diameter dp 45 ,u m
Particle Density P p 7800 kg/m3
Amount of Solids Used ms 9 kg
Volume fraction of solid as 0.03 572
Restitution coefﬁcient e 0.95
Maximum Solid Packing es ,max 0.64

 

 

 

 

 

A uniform plug ﬂow is assumed for the gas phase at the inlet with a superﬁcial velocity
of 1.0 m/s (Reynolds number=12,156). The inlet ﬂux of the solid phase is assumed to be
equal to the outlet ﬂux. The simulation imposes a no-slip condition at the wall for all
phases. Zero initial velocity was speciﬁed for the solid phase in the ﬂow domain with the
solids being evenly distributed in the lower half of the riser. Due to the large number of
cells in three dimensions, the simulations were terminated after 208 of real time and the

averaged results were obtained which was found to be adequate. A mean volume length

diameter of d p= 45 p m. is used for the solid phase (Peirano et al., 2000). About 9 kg. of

49

solids were used in the simulations to give a pressure drop over the riser height which is
comparable to the experiments.

3.2.7 Simulation time

Simulations of ﬂuidized beds with about 9 kg of solids took about 8 days to simulate 20
seconds of real time. The simulations were run on 6 nodes with a 2.2 GHz processor. In
comparison Zhang et al. (2001) used 101 days to simulate 21 s with their ﬁne grid on a
SGI Origin 200.

3.2.8 Numerical results and discussions

The numerical predictions are plotted against the experimental ﬁndings along proﬁles 1,

2, 3, 4 and 5 are shown in Figure 3.14. The Hexp=l .5m.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Proﬁle 1 Proﬁle 2 Proﬁle 3 Proﬁle 4
Proﬁle 5
x/w=.0,5, Z/Hexp= -0.5 Z/Hexp= -0.5 Z/chp= -0.4 Z/Hexp= -0.4
Y/D=0 Y/D = 0 Y/D = -0.3 Y/D = 0 Y/D = 0.48

Figure 3.14: Locations of proﬁles used for plotting the gas-solid results

50

3300

 

 

0 Exp
2800 4 ' * ‘ Nu’“

 

 

2300 TD
1800 a
1300 ~ 0

800 +53

300 — ‘\..

P(Pa)

-200 — "x ,.,-'

 

 

 

-700 r 1t— ’ r I

Figure 3.15: Plot of comparison between the measured (Ibsen et al., 2001) and
computed static pressure vs. dimensionless height at proﬁle 1. It can be seen that a
linear pressure drop is predicted which deviates greatly from the experiments.
Static pressure
Figure 3.15 shows the computed and measured static pressure drop as a function of riser
height at proﬁle 1. The numerical results fail to predict the right order of pressure drop.
The simulations showed a linear pressure drop which deviates strongly from the non-
linear ﬁndings of the experiments. Also the pressure drop at the inlet is predicted as 5kPa
which was only 2.7kPa in the experiments.
Particle velocities
The particle velocities at proﬁle 2 and 3 are shown in Figures 3.16 to 3.19. It shows that

the mean volume-length diameter does not give a good agreement with the experimental

ﬁndings when considering particle velocity proﬁles.

51

1.5

 

 

 

 

 

 

 

 

D Exp
-*-Num [
l— _ ..... 4 """""
1 nru'Tn“ D a a
."r 1* 4.0 n l:
A a
0.5 «' o
a I f“
E i x a
:3 ,
0 It
a
n
-0.5 ~
13
_1 L l 1 1 I
-0.5 -O.4 -0.3 -0.2 -0.1 0
W

Figure 3.16: Plot of comparison between the measured (Ibsen et al., 2001) and
computed axial particle velocities at proﬁle 2. (x = 0 is the center of the duct).

The experimental ﬁndings in Figures 3.16 and 3.19 indicate a constant decrease in the
axial and span wise velocities at the centerline whereas in the numerical model this trend
for the particle velocities was not captured. Also in Figure 3.16 the predicted solid
velocities near the wall deviate much from the experimental ﬁndings which are an
indication that wall boundary conditions for the solid particles is quite complex.

In Figure 3.18, the particle velocities are not symmetric about the centre line which may
indicate that the averaging time has been short. Also in Figure 3.19, the numerical

predictions exhibit a rather poor match with the experiments.

52

0.2

0.15

0.1

 

I

I

 

 

13 Exp
—--e~-Num

 

 

 

 

l l 1

 

-0.1 0

Figure 3.17: Plot of comparison between the measured (Ibsen et al., 2001) and
computed span wise particle velocities at proﬁle 2. (x = 0 is the center of the duct).

1.5

 

 

 

 

 

 

 

D Exp
---A---Num
_ :1 "1:1 0 c1
1:1 1:1 :1 0 :1 :1 5k ’6‘. “an c1
:1 a ----- _¢. """ ‘~. :1
& ,zA'“~cr- - A. A“
K ~‘--t-‘£ \
_ ‘5 a £51“
I? 5‘1. A I:
‘5 A
k A
g L 4 4
0 0.5

Figure 3.18: Plot of comparison between the measured (Ibsen et al., 2001) and
computed axial particle velocities at proﬁle 3.

53

0.2

 

 

 

 

 

 

 

 

0 Exp
0.15 ~ -"b"-Num
0.1 [
0.05 - 4-
’0? K ,5 but“: a
E 0 a IK”,—'&' """ 'ﬁ' ----- *"E'k , u a
‘3’ A//' a D D D
’ D
~ 5 n .- Ch
005 m k g a
U a
-0.1 r-
-0.15 —
'0.2 1 1 1 l I
-0.5 0 0.5
y/D

Figure 3.19: Plot of comparison between the measured (Ibsen et al., 2001) and
computed span wise particle velocities at proﬁle 3.

Volume fraction

Figures 3.20 and 3.21 present instantaneous contour plots of solids at various times and at
various sections. The predicted time variation of the solids volumetric fraction through
the riser of Circulating Fluidized Beds is indicative of a quite complex hydrodynamic
behavior. There are steep variations on both span wise and axial solids concentrations,
together with ﬁ'equent formation and dissociation of clusters ﬂowing both upwards and
downwards. The numerical results show that fewer heavier simulated particles
accumulate in the region very close to the wall due to larger inertia. Also, the simulations
did not predict the dense bottom bed which was present in the experimental ﬁndings. The
simulations predicted a mean solids volume fraction in the lowest part of the riser of 5-10
%. The solids volume fraction should have been 2-3 times higher. In addition, the

simulations fail to capture the ﬁne particles rather it shows the formation of clusters.

54

 

 

 

 

 

 

 

 

 

Z=0.9m ]

 

ﬁ‘rA-_ " I- .

3;."

9.?

 

. r
. l
. I ‘

 

 

 

 

 

 

 

 

[l‘ " .' . -- .1? 5" '1' :.: trot.
Z=0.6m 3i 1
1 I

 

 

 

 

 

 

--....a
“I . ‘ I
Z=0.3m . f I 1
I ' I
t = 53 t = 105

 

 

 

 

 

 

t=153

 

 

 

i

 

t=208

 

 

 

“1.1 -I-!._

t=203

Figure 3.21: Contours of volume fraction of solids at various times
for Z/H = -0.25, -0.5 and -0.75.

 

 

03

" 026
" 024
'1 022
" 02
d 0.10
‘ 0.16
" 0.14
" 0.12
'7 0.1

00 000

006
l 004
0.02

 

 

I
a
K.)
00

 

 

012

 

"020 z[

X

0.8

 

0.7 * I' /'
0.6

0.5

IIb

0.4

0.3

0.2

  

 

 

Center-out of plot area

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
111 b

"I"!

Figure 3.22: Realizability plot for the u u associated with the primary phase computed
with the Realizable k - s at proﬁle 4.

0.7

 

0.6 r

 

 

 

 

-0.04 0.01 0.06 0.11 0.16 0.21

"I"!

Figure 3.23: Realizability plot for the u u associated with the primary phase computed
with the Realizable k - a at proﬁle 5.

56

The Realizability plots for the W associated with the primary phase computed with the

Realizable k - a is shown in Figures 3.22 and 3.23. In the two-phase simulations of gas-
solid ﬂows, the realizable k -6 model proves to be highly unrealizable for the primary
phase (air) near the center while it is realizable near the walls. The computed eigenvalues
and invariants can be found in Tables C5 and C6 of Appendix C. Since the dispersed
phase turbulence model is used in the simulations, the calculation of the eigenvalues and

hence the realizability plot for the secondary phase (solids) is not possible.

 

 

 

 

 

 

 

 

 

6 a ........................
“r — — Syamlal-O'Brien I
5 : : —- Gidaspow . 1
4 E]
6 3 E I
2 E
i
1 I
0 5000 10000 1 5000 20000 25000

Re

Figure 3.24: Plot of comparison of the two drag models for the gas-solid ﬂows.

57

 

‘ 0.015
c 001
- 0005
- 0.06
" 0056
r7 0.05
0045
0.04
0.00
0.03
0.025
002
0.015
0.01
0006

 

 

Y I
X
Z/H=-0.5

 

X X
Y/D=0 Y/D=0.48

 

 

 

Figure 3.25: Contours of the turbulent kinetic energy for the primary
phase (air) at Y/D=0, Y/D=0.8 and Z/H=—0.5.

Figure 3.25 shows that the contours of turbulent kinetic energy for the gas-solid
simulations are the same at any cross-section in the riser.

58

CHAPTER 4
SUMMARY AND CONCLUSIONS

The objective of this work was to assess the performance of single phase ﬂow and two-
phase turbulent ﬂow models implemented in FLUENT 6.2 using experiments and the
invariant diagram for the Reynolds’ stress. The Reynolds’ stress must have positive or
zero eigenvalues to be physical or realizable, and the invariant diagram bounds the region
corresponding such eigenvalues.

Weakness observed in the models

The velocity vectors for the various turbulence models applied in the single phase
ﬂow simulations are the same except for the Reynolds Stress Model which predicted a
great deviation from the other three viscous models. In addition, the various turbulence
models proved to be realizable except for the Spalart-Allmaras model (Spalart et al.,
1992) which showed to be unrealizable. Moreover, the Realizable k -6 model (Shih et al.,
1995) which was realizable for single phase ﬂows became very unrealizable for the
primary phase in the two-phase ﬂow simulations.

The gas-solid ﬂow model was not able to predict the right order of pressure drop
across the riser. Also, the predicted time averaged proﬁles of the axial and span wise
particle velocities show good agreement with the experimental ﬁndings (Ibsen et al.,
2001) only in some regions of the ﬂow.

The gas-solid ﬂow model is not capable of predicting the correct interaction of the
turbulent gas phase and particles; the dispersion of the secondary phase is under
predicted. The numerical results did not predict a dense bottom bed as seen in the

experimental set-up. The simulations predicted a mean solids volume fraction in the

59

lowest part of the riser of 5-10 %. The solids volume fraction should have been 2-3 times
higher.
Recommendations

The solid viscosity parameter and the interface momentum transfer applied should
be rigorously modeled through a suitable theory which can accurately characterize the
two-phase interactions and the physical phenomena involved. The kinetic theory of the
granular ﬂow is a promising theory for this purpose.

Turbulence effects must be accounted properly in gas-solid ﬂows through
Reynolds stress terms. According to Enwald et al. (1996), in bubbling ﬂuidized beds
which are characterized by high solids concentrations, the particulate inertia attenuates
the gas phase turbulent parameters. However, for circulating ﬂuidized beds which are
mostly characterized by regions of low solids concentrations, turbulence becomes
signiﬁcant.

Since the gas-solid ﬂows in ﬂuidized beds are characterized by three-dimensional
effects due to non-unifonn shapes and sizes distributions of the particles, asymmetric
solids feeding, asymmetric geometry at the exit of the riser, and the presence of solids
separators such as cyclones, the whole installation should be simulated considering the
above factors. Such simulations demand a very large number of grid cells in order to
obtain grid independent solutions, which increase the computational time considerably.
On the other hand, the uncertainty on the experimental measurements which are
characteristic of gas-solids multiphase ﬂows must be taken into account.

In summary,,the results of simulation presented here strongly suggest that many

features related to two-phase Eulerian approach needs improvement if accuracy is to be

60

achieved. Special attention should be directed towards interface momentum transfer,
turbulence in two phases, pressure and viscosity of the secondary phase, and realistic
boundary conditions. On the numerical side it becomes evident that more reﬁned grids

are required which unfortunately becomes prohibitive in view of the computational time.

61

APPENDICES
A. Models and near wall treatment applied in the thesis

Table A: Table summarizing the various turbulence models and the respective near wall

treatments applied in the simulations along with their y+ values.

 

 

 

 

 

 

 

 

 

 

 

 

 

Case Turbulence Near Wall +
Models Treatment y
Spalart-Allmaras _ 2.75
Standardk — a) - 2.6
Single Phase
Turbulent Flow Realizablek — a Standard wa" 2.75
Treatment
Reynolds Stress Enhanced Wall 2 6
Model Treatment '
. Enhanced Wall Air 0.66
Bubbly Flow Realrzablek — a Treatment Water 925
. . Enhanced Wall Air 9.8
Gas-solld Flow Realrzablek — a Treatment Solid 18

 

 

 

62

B: Comparison of terms for the various turbulent viscous models.
Table B1: Table summarizing the various terms for the Spalart-Allmaras
model and the Standard k - a) model studied.

 

 

 

 

 

 

 

 

 

Turbulence Closure Models

Terms Spalart-Allmaras Standard k — a)

Local Time Derivative g; I») _6_ at(pk); 2 at(par)

a _
Convection ax_(Pv“ j ) 6:jo )6: (I‘M—“mi )
J ax]
Turbulent Viscosity ,u, = pIval pt: apk

a)

- " 2—”—‘ s ---s 2— 88--
Production terms G, = CblpSv O'k lj ; k mil: 1] 1]
Dissipation terms - pﬂ’fﬂ. kw; pﬂfﬂmz

- 2

. v

Destructron terms Yv = Cw1 ”‘42) -

 

 

 

 

 

Table B2: Table summarizing the various terms for the Realizable k - a
model and the Reynolds Stress model studied.

 

 

 

 

 

 

 

 

 

 

Turbulence Closure Models

Terms Realizable k - 19 Reynolds Stress Model
Local Time 6 6 6 :7,-
Derivative 510’"), 5005') 5; (611,111.)
Convection 6 6 6 .. _.

_ . ; __ gu. U '. '

:jlam.>,,<p .1 We. —.1
Turbulent k2
Viscosity =pC,,-— a -
Production 517'. -

2_,u, $.- 1 =77 014i .
terms 0k S," jSIi’ pC181/ZSl-‘l'sij " p[u,- 11k ax—k + u juk é; ,

W60 6 + gju. 6-)
- 2Wk(171}17'm81km + 7151763}ka

Dissipation pC 6.2 6u,’- —a—§_'j
t rms 6‘; ——
e p 2 k + J; 6xk 6xk
Destruction 8 -
terms D6 = C16 IC3£ Gb

 

 

 

63

C. Computed eigenvalues and invariants

Table Cl: Computed eigenvalues and invariants for a single phase turbulent ﬂow
using the Spalart-Allmaras model; Y/D=O; Z/H=-O.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Location Eigenvamgs Invariants
X Al 12 I13 III) 1111,
0085 -14456.1 0057 14457.12 4.18E+08 2.45E+08
-0.0845 -59086.6 0.635 59087 6.98E+09 -3.2E+09
-0.08388 -17465.2 0.586 17465.59 6.1 E+08 -2.3E+08
008309 -50871 .3 -4.2 50876.46 5.18E+09 3.52E+10
008212 -119145 -23.62 1191693 2.84E+10 1.02E+12
-0.0809 -18093.2 -8.5 18102.7 6.55E+08 8.67E+09
007937 -123432 127.45 1233057 3.04E+10 -5.8E+12
-0.07746 -24271.1 -49.84 24321.95 1 .18E+09 8.89E+10
-0.07508 -5700.57 21.84 5679.73 64756219 -2.1E+09
-0.0721 -3055.86 21.55 3035.31 18551858 -5.9E+08
-0.06837 -1033.24 -14.7 1048.93 2168052 48864675
006231 -1070.94 63.76 1008.17 2167373 -2.1E+08
-0.0553 41.1 141 -37.3 79.4 9383.897 3698012
-0.0472 -363.45 154.5 209.9 200040 -3.5E+07
003783 -989.754 -571 .25 1562 3745782 2.65E+09
-0.027 -2164.92 888.26 1277.6 7108310 -7.4E+09
-0.01448 -1586.92 -1352.79 2940.7 12996163 1 .89E+10
2.22E-18 -1748.73 844.1941 905.5 4590694 -4E+09
0.014479 -1586.92 -1352.79 2940.7 12996163 1 .89E+10
0.027002 -2170.92 890.717 1281.2 7147756 -7.4E+09
0.037834 -989.754 -571 .25 1562 3745782 2.65E+09
0.047202 -363.45 1 54.54 209.9 200040 -3.5E+07
0.055304 -41 .1 141 -37.28 79.4 9383.897 3698012
0.062312 -1070.94 63.76 1008.17 2167373 -2.1E+08
0.068374 -1033.24 -14.7 1048.9 2168052 48864675
0.072099 -3053.71 21.54 3033.2 18525687 -5.9E+08
0.075079 -5700.57 21.84 5679.73 64756219 -2.1E+09
0.077463 -24271.1 4984 24321.95 1.18E+09 8.89E+10
0.079371 -1 23432 127.45 1233057 3.04E+10 -5.8E+12
0.080897 -1 8093.2 -8.49 18102.7 6.55E+08 8.67 E+09
0.082117 -119145 -23.62 1191693 2.84E+10 1.02E+12
0.083094 -50871 .3 -4.2 50876.46 5.18E+09 3.52 B1 0
0.083875 ~17465.2 0.58 17465.6 6.1E+08 -2.3E+08
0.0845 -59086.6 0.63 59087 6.98E+09 -3.2E+09
0.085 -1 4456.1 0056 14457.12 4.18E+08 2.45E+08

 

 

 

 

 

64

 

Table C2: Computed eigenvalues and invariants for a single phase turbulent ﬂow

using the Spalart-Allmaras model; Y/D=O.48; Z/H=-O.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Location Eigenva|ues Invariants
X 111 112 I13 Uh [Uh
-0.085 -1 1377.2 0.543 11377.65 258891356 -81437229
00845 -73150.8 -7.171 73159.01 1.07E+10 1.2E+11
-0.08388 -17592.8 -3.828 17597.59 619180587 3.86E+09
-0.08309 -68104.8 23.31 68082.51 9.273E+09 -3.2E+11
-0.08212 49683.2 -21.29 49705.47 4.939E+09 1.6E+11
00809 -106722 -51.75 106774.9 2.279E+10 1 .78E+12
-0.07937 -95947.7 -45.28 95993.98 1 .842E+10 1.26E+12
-0.07746 -671307 -265.1 6715733 9.017E+11 3.59E+14
007508 -373809 91.76 3737179 2.794E+1 1 - -3.83E+13
00721 -107809 6.227 1078034 2.324E+10 -2.05E+11
-0.06837 44903.7 -4.903 44909.57 4.033E+09 3.17E+10
-0.06231 -70731.2 -14.15 70746.37 1.001 E+10 2.17E+11
-0.0553 -80329.6 -14.57 80345.19 1.291 E+1 0 2.89E+11
00472 -98622 -17.45 98640.47 1 .946E+10 5.19E+11
-0.03783 -93456.7 -17.1 93474.76 1 .747E+10 4.57E+11
-0.027 -113695 -20.63 113717.1 2.586E+10 8.13E+11
001448 -114072 -21.13 1140942 2.603E+10 8.38E+11
2.22E-18 -109972 -20.68 1099937 2.419E+10 7.63E+11
0.014479 -114072 -21.13 1140942 2.603E+10 8.38E+11
0.027002 -113717 -20.63 1137384 2.587E+10 8.14E+11
0.037834 -93456.7 -17.1 93474.76 1 .747E+10 4.57E+11
0.047202 -98622 -17.45 98640.47 1 .946E+10 5.19E+1 1
0.055304 -80329.6 -14.57 80345.19 1 .291E+10 2.89E+11
0.062312 ~70731.2 -14.15 70746.37 1 .001E+10 2.17E+11
0.068374 -44903.7 -4.903 44909.57 4.033E+09 3.17E+10
0.072099 -107809 6.227 1078034 2.324E+10 -2.05E+11
0.075079 -373809 91.76 3737179 2.794E+1 1 -3.83E+13
0.077463 -671307 -265.1 6715733 9.017E+11 3.59E+14
0.079371 -95947.7 4528 95993.98 1 .842E+10 1 .26E+12
0.080897 -106722 -51.75 106774.9 2.27QE+10 1 .78E+12
0.082117 49683.2 -21.29 49705.47 4.939E+09 1.6E+11
0.083094 -68104.8 23.31 68082.51 9.273E+09 -3.2E+1 1
0.083875 -17592.8 -3.828 17597.59 619180587 3.86E+09
0.0845 -73150.8 -7.171 73159.01 1.07E+10 1.2E+11
0.085 -11377.2 0.543 11377.65 258891356 -81437229

 

65

 

in the gas-liquid simulations; Y/D=0; Z/H=-O.5

Table C3: Computed eigenvalues and invariants for the primary phase (water)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Location Eigenvalues Invariants
X 3.1 12 13 Uh Uh,
-0.085 -15.314 0.333 15.981 489.71 0.100727
-0.0845 -78.501 0.331 79.17 12430.05 37.72916
-0.08388 -84.027 0.321 84.706 14235.27 261 .2749
-0.08309 -83.944 0.292 84.652 14212.32 883.6041
-0.08212 -91 .626 0.246 92.379 16928.98 2205.522
-0.0809 -97.869 0.172 98.697 19319.07 4666.574
007937 -102.23 0.057 103.17 21093.31 8747.825
007746 -102.14 -0.105 103.25 21092.16 13874.94
-0.07508 -96.931 -0.282 98.214 19041.36 17578.25
-0.0721 ~89.106 -0.469 90.574 16143.41 19416.09
006837 -88.54 -0.937 90.478 16026.1 1 30537.17
-0.06231 -73.806 -2.816 77.622 1 1480.15 54144.5
-0.0553 -33.228 -4.739 38.967 2644.666 19730.94
-0.0472 -20.715 ~11.24 32.957 1641 .359 23846.03
003783 -14.21 -11.94 27.151 1081.379 14362.18
-0.027 -18.418 -18.12 37.541 2076.644 38630.16
-0.01448 -80.805 -76.15 157.95 37277.1 1 2934399
2.22E-18 -349.24 166.9 183.31 1834354 -3.2E+07
0.014479 -80.805 -76.15 157.95 37276.86 2934369
0.027002 -18.418 -18.12 37.541 2076.633 38629.85
0.037834 -14.21 -11.94 27.151 1081.366 14361.94
0.047202 -20.714 -11.24 32.955 1641.089 23839.91
0.055304 -33.227 -4.739 38.966 2644.532 19729.05
0.062312 -73.805 -2.816 77.621 11479.7 54139.79
0.068374 -88.54 -0.937 90.477 16026.04 30536.01
0.072099 -89.105 -0.469 90.574 16143.19 19415.11
0.075079 -96.931 -0.282 98.214 19041 .36 17577.63
0.077463 -102.14 -0.105 103.25 21092.16 13874.41
0.079371 -102.22 0.057 103.17 21093.26 8747.404
0.080897 -97.869 0.172 98.697 19319.07 4666.31
0.0821 17 -91 .626 0.246 92.379 16928.96 2205.372
0.083094 -83.944 0.292 84.652 14212.31 883.5298
0.083875 -84.027 0.321 84.705 14235.25 261.2384
0.0845 -78.501 0.331 79.17 12430.05 37.71271
0.085 -15.314 0.333 15.981 489.71 0.100622

 

66

 

in the gas-liquid simulations; Y/D=O.48; Z/H=-0.5

Table C4: Computed eigenvalues and invariants for the primary phase (water)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Location Eigenvalues Invariants
X 1.1 12 2.3 Hb IHb
-0.085 -10.07 0.33 10.7 216.36 -0.23
-0.0845 -59.8 0.34 60.47 7234.09 -107.13
008388 -72.67 0.38 73.28 10649.64 -741.163
-0.08309 -83.76 0.47 84.3 14121 .98 -2840.333
-0.08212 -9594 0.6 96.36 18490 -6826.123
-0.0809 -95.86 0.6 96.22 I 18449.4 -8452.243
-0.07937 -88.062 0.65 88.4 1 5572 -7364.19
-0.07746 -85.38 0.65 85.7 14639.8 694207
007508 -86.43 0.64 86.8 1 5004.78 -6806.9
-0.0721 -87.73 0.6 88.13 15463 -6115.5
-0.06837 -89.6 0.52 90.07 16137.8 -4517.27
006231 -92.26 0.4 92.9 17138.43 -1239.33
-0.0553 -92.7 0.28 93.47 17339.98 1437.67
-0.0472 -93.9 0.23 94.68 17784.64 2850.82
-0.03783 -93.7 0.22 94.51 17718.58 2972.55
-0.027 -93.58 0.23 94.35 17659.59 2795.52
-0.01448 -93.9 0.22 94.67 17778.62 2871.7
2.22 E1 8 -93.9 0.22 94.73 17800.57 2979.48
0.014479 -93.9 0.22 94.67 17778.53 2871 .57
0.027002 -93.58 0.23 94.35 17659.45 2795.08
0.037834 -93.7 0.22 94.51 17718.62 2971 .63
0.047202 -93.9 0.23 94.67 17780.68 2844.21
0.055304 -92.77 0.28 93.49 17345.67 1432.34
0.062312 -92.24 0.38 92.86 17132.51 -1222.6
0.068374 -89.6 0.52 90.09 16146.75 -4511.6
0.072099 -87.75 0.6 88.15 15470.52 -6134.7
0.075079 -86.42 0.6 86.78 14999.34 -6818.45
0.077463 -85.37 0.65 85.73 14638.62 -6949
0.079371 -88.06 0.65 88.4 15572.18 -7370.8
0.080897 -95.87 0.64 96.23 18450.72 -8460
0.082117 -95.95 0.58 96.36524 18491.7 -6833.8
0.083094 -83.77 0.47 84.3012 14123.8 -2845
0.083875 -72.67 0.38 73.28319 10650 -741
0.0845 -59.8 0.34 60.471 15 7234.3 -107.4
0.085 -10.07 0.33 10.73 216.3 -0.23

 

 

 

 

 

 

67

 

the gas-solid simulations; Y/D=O; Z/H=-O.5.

Table C5: Computed eigenvalues and invariants for the primary phase (air) in

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Location Eigenvalues Invariants
X 2.1 2.2 13 11b 111 b
-0.085 0.33 0.33 0.34 4.86E-05 -2.1E-09
-0.0845 0.33 0.33 0.34 2.74E-05 -9.5E-10
-0.08388 0.31 0.33 0.354 0.000998 1 .28E-07
-0.08309 0.27 0.33 0.44 0.009 8.02E-06
-0.08212 0.28 0.328 0.4 0.0067 4.93E-05
-0.0809 0.15 0.35 0.54 0.0632 -0.00172
-0.07937 -0.13 0.35 0.774 0.4 -0.0126
-0.07746 0.15 0.38 0.464 0.05 -0.00345
-0.07508 -0.14 0.323 0.84 0.45 0.007
-0.0721 -0.57 0.375 1.1954 1.56 —0.098
-0.06837 -0.99 0.475 1.517 3.18 —0.67
-0.06231 -0.6 0.58 1.02 ‘ 1.41 -0.48
-0.0553 -0.125 0.42 0.7 0.35 -0.047
-0.0472 0.078 0.175 0.7 0.26 0.05
003783 -0.82 0.487 1 .33 2.36 -0.53
-0.027 -1.99 0.458 2.54 10.34 -1.93
-0.01448 -7.62 0.315 8.3 126.68 3.37
2.22E-18 -6.6 -1.03 8.6 119.6 236.7
0.014479 -3.12 -0.71 4.8 33.24 48.79
0.027002 -2.32 0.11 3.2 15.3 4.998
0.037834 -0.67 0.43 1 .24 1 .85 -0.26
0.047202 -0.8 0.24 1 .55 2.78 0.4
0.055304 -0.8 0.205 1.59 2.86 0.54
0.062312 0.014 0.24 0.74 0.27 0.034
0.068374 0.103 0.3 0.6 0.13 0.008
0.072099 0.18 0.3 0.52 0.06 0.002749
0.075079 0.25 0.3 0.45 0.02 0.001082
0.077463 0.15 0.32 0.52 0.07 0.001068
0.079371 -0.00179 0.33 0.671 0.23 0.000976
0.080897 -0.035 0.33 0.7 0.27 0.000129
0.082117 0.218 0.33 0.45 0.03 —8.4E-06
0.083094 0.32 0.33 0.34 0.000135 5E-08
0.083875 0.314 0.33 0.35 0.000758 2.13E-07
0.0845 0.33 0.33 0.33 1.21 E-05 4.02E-10
0.085 0.33 0.33 0.34 7.89E—05 2.34E-09

 

 

 

 

68

 

the gas-solid simulations; Y/D=0.48; Z/H=-O.5.

Table C6: Computed eigenvalues and invariants for the primary phase (air) in

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Location Eigenvalues Invariants
X Al ’12 ’13 11b IIIb
—0.085 0.33 0.333 0.336 1.7E-05 1 .24E-10
-0.0845 0.333 0.333 0.334 7.54E-08 -8.4E-13
-0.08388 0.291 0.333 0.376 0.003595 -6.8E~07
-0.08309 0.224 0.335 0.441 0.023593 -4.9E-05
-0.08212 0.173 0.329 0.499 0.053186 0.000362
-0.0809 -0.014 0.316 0.698 0.254016 0.006552
00793? 0.072 0.318 0.61 0.145081 0.00332
-0.07746 0.139 0.335 0.527 0.07531 -0.00014
-0.07508 0.293 0.34 0.367 0.002817 -2.6E-05
-0.0721 0.236 0.306 0.458 0.025795 0.000996
-0.06837 0.1 15 0.326 0.559 0.098595 0.001086
-0.06231 0.04 0.335 0.625 0.171139 -0.00037
-0.0553 0.147 0.334 0.519 0.069209 -9.2E—05
-0.0472 0.207 0.333 0.46 0.031912 3.08E-05
-0.03783 0.223 0.333 0.444 0.024254 8.41 E-06
-0.027 0.231 0.335 0.435 0.020864 -4.1E-05
-0.01448 0.246 0.337 0.418 0.014756 -7.2E-05
2.22E-18 0.25 0.34 0.41 0.012965 -0.00013
0.014479 0.22 0.338 0.442 0.024735 -0.00019
0.027002 0.184 0.332 0.485 0.045323 0.000121
0.037834 0.258 0.325 0.417 0.012773 0.000156
0.047202 0.269 0.333 0.398 0.008434 2.89E-06
0.055304 0.285 0.332 0.383 0.004827 6.23E-06
0.062312 0.229 0.325 0.446 0.023592 0.000278
0.068374 -0.035 0.338 0.697 0.267441 -0.00188
0.072099 01 0.348 0.752 0.362615 -0.00795
0.075079 -0.06 0.353 0.708 0.295388 -0.00863
0.077463 -0.02 0.355 0.666 0.235889 ~0.00753
0.079371 0.032 0.354 0.614 0.17037 -0.00526
0.080897 0.079 0.346 0.575 0.123601 -0.0024
0.082117 0.176 0.334 0.489 0.049053 -7.9E-05
0.083094 0.27 0.329 0.401 0.008565 5.35E-05
0.083875 0.293 0.333 0.374 0.003315 2.55E-06
0.0845 0.332 0.333 0.334 2.52E-06 7.1E-11
0.085 0.331 0.333 0.336 1 .18E-05 3.91 E10

 

 

 

 

 

 

69

 

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