THEORETICAL AND CCMPUTATIONAL MODELS OF REACTING SILANE GAS FLOWS:
LASER DRIVEN PYROLYSIS
OF
SUBSONIC AND SUPERSONIC JETS
by
Ibrahim Sinan AKMANDOR
B.S. ME, Bogazici Universitesi
(June
198Q)
B.S. Math, Bogazici Universitesi
(June
1980)
S.M. A.A., Massachusetts Institute of Technology
(June
1982)
SUBMITTED TO THE DEPARTMENT OF AERONAUTICS
AND ASTRONAUTICS IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June
1985
Copyright @ 1985 Massachusetts Institute of Technology
Signature of Author
Department of Aeronautics and Astronautics
May 3rd,
1985
Certified by
Prof. Julian Szekely
Thesis Supervisor
Certified
by
Prof. Leon Trilling
Thesis Committee Member
Certified
by
Prof. Wesley H. Harris
Thesis Committee Member
Certified
by
Dr. John S. Haggerty
Thesis Committee Member
Accepted by
/r'~'
Prof.
Harold
Y. Wachman
Chairman, Departmental Graduate Committee
Archives
MASACA4SEr; NS1TRUTE
OFIECHMOGY
MAY 3 0 1985
LIBRAIE3
TITLE
i.1
THEORETICAL AND COMPUTATIONAL MODELS OF REACTING SILANE GAS FLOWS:
LASER DRIVEN PYROLYSIS OF SUBSONIC AND SUPERSONIC JETS
by
Ibrahim Sinan AKMANDOR
Submitted to the Department of Aeronautics and Astronautics on 26 May, 1985
in partial fulfillment of the requirements for the degree of Doctor of
Science
in Gas Dynamics.
ABSTRACT
The velocity, temperature, pressure and concentration fields of a
reacting silane gas have been calculated by analytical and computational
techniques. The results have been successfully compared to the experimental
data obtained from the actual powder cell reactor. Silane (SiH4 ) is flowing
coaxially with the annular argon gas. The reaction zone is detached from the
inlet nozzles. In the theoretical approach, the cell is divided into 3
subregions. The governing equations are simplified and then integrated'to
yield the velocity and temperature fields. The reaction flame dimensions are
small compared to the reactor dimensions and the flow is treated as a free
jet flowing at a constant cell pressure. The temperature related density
changes are taken into account by defining a Dorodnitzyn-Howard-Illingworth
transformation. In the analytical calculations, it is assumed that the
reaction occurs spontaneously once the reacting gas is heated up to a
threshold temperature (873.15 K), because the chemical reaction time is
shown to be much smaller than a typical heat conduction or a convection time.
The reaction is weakly exothermic and the absorbed laser beam power is the
main source of heat. The numerical models are based on two finite difference
codes. The dependent variables are the velocity, stagnation enthalpy,
pressure, and mass concentrations of silane and argon. The silicon and
hydrogen concentrations are calculated from the overall chemical equilibrium
equation by using the stoichiometric ratio. The gas properties are also
functions of the temperature. For the subsonic case, the effect of the
reactor cell walls is taken into account by using an elliptic iterative type
algorithm. A fixed grid with a variable mesh size is used with the implicit
algorithm. The computational results show that the mixing of argon with the
reacting gas is significant in lowering the reacting gas temperature,
especially in the reaction zone where the gas expands. Heat loss by
radiation is also found to be important but to a lesser extent. The
supersonic reacting jet is solved by using a marching type algorithm, an
implicit scheme and a variable grid. The radial pressure gradients are also
taken into account by using the SIMPLE model of Patankar and Spalding, thus
predicting the diamond pressure pattern of a supersonic jet. It is shown
that if the laser beam is focused on the shock, the reaction starts and the
downstream flow pattern is substantially affected. A new velocity measuring
technique, has been developed called "velocity measurements by a perturbation
1.2
method". The technique records the time taken by a perturbation front to
travel between two locally predetermined points. Finally, the nature of the
unsteady reaction flame problem has been addressed both theoretically and
experimentally. High speed movie pictures showed that for high cell pressure
and low silane
flow
rates, the
fixed frequency around 20 Hz.
has been solved
as an eigenvalue
reaction flame
flickered
indefinitely
at a
The one dimensional unsteady energy equation
problem
and a critical
gas velocity
has been
deduced beyond which the reaction flame is relatively stable. The
comparison of the theoretical results and the experimental data shows good
agreements.
:
:
Dr. Julian Szekely
Professor of Materials Engineering
Chairman of the Doctoral Thesis Committee
Thesis committee member:
:
Title
Dr. Leon Trilling
Professor of Aeronautics and Astronautics
Member of the Doctoral Thesis Committee
Thesis committee member:
:
Title
Dr. Wesley L Harris
Professor of Aeronautics and Astronautics
Member of the Doctoral Thesis Committee
Thesis Committee member:
:
Title
Dr. John S. Haggerty
Program Director, Advanced Energy Materials
Member of the Doctoral Thesis Committee
Thesis supervisor
Title
i .3
TO MY PARENTS
I!
Dr. Y. Muh. Mehmet Neet AKMANDOR
and
Dog. Dr. Ayten AKMANDOR
in recognition of their infinite love and affection.
"The working iron does also shine ."
(Turkish Proverb)
i.4
ACKNOWLEDGEMENTS
I would
like to express
my sincere
and deepest
gratitude
to my thesis
supervisor Professor Julian Szekely. He provided invaluable insight,
guidance, encouragement and enthusiasm throughout the research. My
deepest gratitude is also due to Dr. Haggerty who has supported me both
technically and financially through the Advanced Energy Materials Program.
I would like to express my most sincere appreciations to Professor Leon
Trilling and Professor Wesley H. Harris who have not only provided
invaluable guidance to this research, but also taught me advanced concepts
of Aerodynamics during my education at MIT.
Thanks are also due to Professor Tau-Yi Toong, John Flint Dr. ElKaddah, Dr. David Casey, Dr. Garry Garvey and Bob Frank. They have
provided vital inputs toward the accomplishment of my reserach. I would
like to take this occasion to thank all my friends and colleagues at the
Mathematical
Modelling
Group
in 8-135
and 4-033
and to my friends
in
various Powder Groups (Energy Lab) in 12-Onm; n,m = 0,1,2,3...
Last
but not least,
I thank
Ali Ozbek
(Doctoral
Candidate
E.E. MIT)
for proofreading the Thesis. Thanks are also due to Cindy Cali and
Patricia Normile for their excellent typing.
TABLE
OF CONTENTS
i.5
PAGE
ABSTRACT
i.1
DEDICATION
i.3
ACKNOWLEDGEMENT
1.4
TABLE OF CONTENTS
1.5
LIST OF FIGURES
1.8
LIST OF SYMBOLS
i.18
1.
1
INTRODUCTION
1.1 DYNAMICS OF LASER DRIVEN SILANE REACTIONS
1
1.2 GEOMETRY OF THE PROBLEM
1.3 GOALS AND MOTIVATIONS
3
5
1.3.1 MAIN PHYSICAL ASPECTS
1.3.2 MAIN MATHEMATICAL ASPECTS
1.4 LITERATURE REVIEW
5
6
8
1.4.1
PYROLYSIS
OF SILANE
1.4.2 DIFFERENCES BFTWEEN A COMBUSTION FLAME
AND A THERMAL DECOMPOSITION FLAME
1.4.3 BACKGROUND
2.
11
12
DETAILED ANALYSIS OF THE STEADY STATE PROCESS
16
2.1. APPROACH TO THE 2-D AXISYMMETRICAL FLAME
2.2. THEORETICAL MODEL OF THE LASER DRIVEN PYROLYSIS
OF A SILANE GAS: PREDICTION OF THE VELOCITY AND
TEMPERATURE FIELD IN A CONTINUOUS POWDER REACTOR
2.2.1 PURPOSE AND MOTIVATION OF THE THEORETICAL MODEL
2.2.2 ASSUMPTIONS AND CALCULATION PROCEDURES
2.2.3 SOLUTION OF THE TWO DIMENSIONAL REACTING
SILANE JET PROBLEM: THE THEORETICAL MODEL
2.2.4 ANALYTICAL RESULTS AND DISCUSSION
A. DETERMINATION OF THE CONSTANTS OF INTERGRATON
16
B.
3.
8
RESULTS
18
18
19
22
27
27
27
VELOCITY MEASUREMENTS IN THE REACTION FLAME
30
3.1 CONVENTIONAL VELOCIMETERS: DIFFICULTIES IN
DATA ACQUISITION
3.2 VELOCITY MEASUREMENTS BY A PERTURBATION METHOD
30
31
3.2.1
PROPERTIES
OF THE NEW METHOD
3.2.2 PHYSICAL CONCEPT BEHIND THE NEW METHOD
3.3 THE EXPERIMENTAL SET-UP
31
31
33
i .6
4.
5.
3.4 EXPERIMENTAL RESULTS AND COMPARISON WITH
THE ANALYTICAL SOLUTION
35
THE COMPUTATIONAL MODEL OF THE STEADY SUBSONIC REACTION FLAME
37
4.1 INTRODUCTORY REMARKS ON THE ALGORITHM
4.2 CHOICE OF THE UPWIND DIFFERENCE SCHEME OVER A
CENTRAL DIFFERENCE SCHEME
4.3 CHOICE OF THE RELAXATION PARAMETER
4.4 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS
4.5 ASSUMPTIONS AND ESTIMATION OF THE DIFFUSION
COEFFICIENTS
4.6 RESULTS OF THE SUBSONIC FLAME AND DISCUSSION
37
THE HYPERBOLIC CODE:
SUPERSO0IC FLOW
6.
43
46
SOLUTION OF THE REACTING
49
5.1 DIFFERENCES AND SIMILARITIES BETWEEN THE
SUPERSONIC JET AND THE BOUNDARY LAYER TYPE ALGORITHMS
5.2 MATHEMATICAL PROCEDURE OF THE SUPERSONIC CODE:
IMPLEMENTATION OF THE SIMPLE ALGORITHM
5.3 TRANSFORMATION
39
41
42
OF THE GOVERNING
EQUATIONS
49
50
FOR
THE HYPERBOLICAL FLOW
5.3.1 THE MOMENTUM EQUATION
5.3.2 THE ENERGY EQUATION
5.3.3 THE CHEMICAL SPECIES EQUATIONS
5.3.4 GENERAL FORM OF THE GOVERNING EQUATIONS
5.4 RADIAL PRESSURE CALCULATION FOR HIGH SPEED FLOW
5.5 CHOCKING CONDITION AND FEASIBILITY OF THE SUPERSONIC RUN
5.6 RESULTS AND DISCUSSION FOR THE SUPERSONIC
REACTING FLOW
51
52
52
55
56
57
59
ANALYSIS OF THE UNSTEADY REACTION FLAME
64
6.1 EXPERIMENTAL INVESTIGATION
6.1.1 INTRODUCTION
6.1.2 DESCRIPTION OF THE OSCILLATIONS
6.1.3 OSCILLATIONS OF THE FIRST TYPE
6.1.4 OSCILLATIONS OF THE SECOND TYPE
6.1.5 THE EXPERIMENTAL SET UP
A. HIGH SPEED MOVIE PICTURES OF OSCILLATIONS
64
64
64
65
67
70
OF TYPE 1
B.
62
70
EXPERIMENTAL SET-UP FOR OSCILLATIONS OF
TYPE
2
71
i.7
6.1.6 EXPERIMENTAL RESULTS AND DISCUSION
A. RESULTS FROM HIGH SPEED MOVIE PICTURES
B. RESULTS FROM THE HOT WIRE OUTPUT
6.1.7 CONCLUSION FOR THE EXPERIMENTAL APPROACH
6.2 REACTION FLAME INSTABILITIES: THEORETICAL APPROACH
6.2.1
ANALYSIS
OF THE UNSTEADY
FLAME
6.2.2 SOLUTION OF THE UNSTEADY ENERGY EQUATION
7.
73
73
76
79
80
80
81
SYNTHESIS OF THE RESEARCH AND GENERAL CONCLUSION
85
7.1 GENERAL RESULTS
85
7.2 ORIGINALITY
OF THE RESEARCH
AND CONTRIBUTIONS
88
7.3 GENERAL CONCLUSION
89
8.
REFERENCES
91
9.
APPENDIX
95
10. BIOGRAPHY
105
11. FIGURES
106
LIST OF FIGURES
i.8
Page
Figure
1:
Approach to the problem
106
Figure
2:
The Flame Shape and Dimensions
107
Figure
3:
Geometry
of the Problem
and
Definition of Subregions
dimensions in mm
Figure
4:
Patching of 2 Velocity Profiles
in a Non-Similar
Figure
5:
Cell
7:
8:
9:
30 cc/min
0.7 Atm.
110
pressure:
0.2 Atm.
111
given
in Figure
5
112
Radial Temperature Profiles
Run Conditions:
Laser power: 180 W unfocused, Gaussian Profile
Argon flow rate: 1000 cc/min
Silane flow rate: 38 cc/min
Cell pressure:
Figure
pressure:
Axial Temperature Profiles
run conditions
Figure
0.2 ATm.
Radial Velocity Profiles
Run Conditions:
Laser power: 180 W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
Silane flow rate: 38 cc/min
Cell
Figure
pressure:
Silane flow rate:
Cell
6:
109
Region
Axial Velocity Distribution
Run Conditions:
Laser power: 180W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
upper curve: Silane flow rate: 38 cc/min
lower curve:
Figure
108
0.2 Atm.
113
Temperature Map of the Reaction Zone:
Flame Boundaries at the Isotherm = 873.15°K
Run Conditions:
Laser power: 180 W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
Silane flow rate: 38 cc/min
Cell
pressure:
0.2 Atm.
114
1.9
Page
Figure
10: Theoretical Axial Distribution of Velocity
Run Conditions:
Laser
power:
Top hat profile
180 W unfocused,
Argon flow rate: 1000 cc/min
Silane flow rate: 38 cc/min
Cell pressure:
0.2 Atm.
1.48 mm
Nozzle diameter:
Figure
115
11: Theoretical and Experimental
Distribution of Temperture
Run Conditions:
Laser
power:
180 W unfocused,
Top hat profile
Argon flow rate: 1000 cc/min
Silane flow rate:
Cell
pressure:
Nozzle diameter:
Figure
38 cc/min
0.2 Atm.
1.48 mm
116
12: Radial Velocity Profiles from Analytical Results
Run Conditions:
Laser power: 180 W unfocused, Top hat profile
Argon flow rate: 1000 cc/min
Silane flow rate: 38 cc/min
Cell pressure:
0.2 Atm.
Nozzle diameter:
Figure
13:
power:
180 W unfocused,
Argon flow rate: 1000 cc/min
Silane flow rate:
Cell
pressure:
Nozzle diameter:
Figure
Figure
14:
15:
16:
117
Radial Temperature Profiles from Analytical Results
Run Conditions:
Laser
Figure
1.48 mm
Top hat profile
38 cc/min
0.2 Atm.
1.48 mm
118
Experimental Set-Up for Velocity Measurements
By a Perturbation Method: Top View
19
Experimental Set-Up for Velocity Measurements
By a Perturbation Method: Side View
120
Signals Displayed by the Photodetectors:
Propagation of the Disturbance Front at
Different Axial Locations
121
Figure
17:
The Basic Structure of the Computational Codes
122
Figure
18:
Flow Chart of the Elliptical Algorithm
123
Figure
19:
The Grid and the Forward Step in the Elliptical
Iterative Computer Code
124
i .10
Page
Figure
20:
Velocity Fields Obtained From
The Elliptical Algorithm
Run Conditions:
Laser power: 180 W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
Upper:
Silane flow rate:
30 cc/min
Cell pressure:
0.7 Atm.
Nozzle diameter: 1.19 mm
Silane flow rate: 38 cc/min
Lower:
Cell pressure:
0.2 Atm.
Nozzle diameter:
dimensions
1.19 mm
in SI units
NO RADIATION
Figure
21:
125
Velocity and Temperature Fields Obtained from
The elliptical Algorithm
Run Conditions:
Laser power: 180 W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
Silane flow rate: 30 cc/min
Cell
pressure:
0.7 Atm.
Nozzle diameter:
dimensions
1.19 mm
in SI units
NO RADIATION, BUOYANCY
Figure
22:
Concentration Fields of Silane and Argon
from the Elliptical Algorithm
same
Figure
23:
24:
run condition
run condition
Figure
25:
26:
21
127
as Figure 21
128
Velocity and Temperature Fields Obtained from
the Elliptical Algorithm
Run Conditions:
Laser power: 180 W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
Silane flow rate: 30 cc/min
Cell pressure: 0.7 Atm.
Nozzle diameter: 1.19 mm
dimensions
Figure
as Figure
Concentration Fields of Hydrogen and Silicon
from the Elliptical Algorithm
same
Figure
126
in SI units
NO RADIATION, NO BUOYANCY
129
Concentration Fields of Silane and Argon
from the Elliptical Algorithm
same run condition as Figure 24
130
Concentration Fields of Hydrogen and Silicon
from the Elliptical Algorithm
same run condition as Figure 24
131
i .11
Page
Figure
27:
Velocity and Temperature Fields Obtained from
the Elliptical Algorithm
Run Conditions:
Laser power: 180 W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
Silane flow rate: 38 cc/min
Cell pressure:
0.2 Atm.
Nozzle diameter:
1.19 mm
dimensions in SI units
NO RADIATION, NO BUOYANCY
Figure
28:
Concentration Fields of Silane and Argon
from the Elliptical Algorithm
same
Figure
29:
30:
run conditions
as Figure
27
run conditions
as Figure
27
in SI units
NO RADIATION, BUOYANCY
31:
32:
i
33:
run conditons
run conditions
34:
30
136
as Figure
30
137
Velocity and Temperature Fields Obtained from
the Elliptical Algorithm
Run Conditions:
Laser power: 180 W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
Silane flow rate: 38 cc/min
Cell pressure: 0.2 Atm.
Nozzle diameter: 1.19 mm
dimensions
Figure
as Figure
Concentration Fields of Hydrogen and Silicon
from the Elliptical Algorithm
same
Figure
135
Concentration Fields of Silane and Argon
from the Elliptical Algorithm
same
Figure
134
Velocity and Temperature Fields Obtained from
the Elliptical Algorithm
Run Conditions:
Laser Power: 180 W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
Silane flow rate: 38 cc/min
Cell pressure: 0.2 Atm.
Nozzle diameter: 1.19 mm
dimensions
Figure
133
Concentration Fields of Hydrogen and Silicon
from the Elliptical Algorithm
same
Figure
132
in SI units
RADIATION, BUOYANCY
138
Concentration Fields of Silane and Argon from the
Elliptical Algorithm same run conditions as Figure 33
139
i .12
Page
Figure
35:
Concentration Fields of Hydorgen and Silicon
from the Elliptical Algorithm
same
Figure
36:
run condition
as Figure
140
33
Velocity and Temperature Fields Obtained from
the Elliptical Algorithm, Bigger Scale
Run Conditions:
Laser power:
Argon
flow
180 W unfocused, Gaussian profile
rate:
1000 cc/min
38 cc/min
Silane flow rate:
Cell pressure:
0.2 Atm.
1.19 mm
Nozzle diameter:
dimensions
in SI units
141
NO RADIATION
Figure
37:
Concentration Fields of Silane and Argon
from the Elliptical Algorith, Bigger Scale
same run conditions
Figure
38:
39:
142
36
Concentration Fields of Hydrogen and Silicon
from the Elliptical Algorithm
same
Figure
as Figure
run conditions
as Figure
143
36
Radial Velocity and Temperature Profiles
of Argon
Near Pipe Exit X=0.004
m
from the Parabolical Code
Cell pressure: 0.7 Atm
1000 cc/min
Argon mass flux:
Figure
40:
at S.T.P.
Radial Velocity and Temperature Profiles
of Argon Near Pipe Exit X=0.008 m
from the Parabolical Code
Cell Pressure:
0.7 Atm
1000 cc/min at S.T.P.
Argon mass flux:
Figure
41:
42:
145
Radial Velocity and Temperature Profiles
of Argon Near Pipe Exit X=O.01 m
from the Parabolical Code
Cell pressure: 0.7 Atm
1000 cc/min at S.T.P.
Argon mass flux:
Figure
144
146
Radial Velocity and Temperature Profiles
of Argon After the Pipe Exit X=0.012 m
from the Parabolical Code
0.7 Atm
Cell pressure:
1000 cc/min
Argon mass flux:
at S.T.P.
147
i.13
Page
Figure
43:
Radial Velocity and Temperture Profiles
of the Annular Argon Jet X=0.016 m
from the Parabolical code
0.7 Atm
Cell pressure:
Argon mass flux:
Figure
44:
0.7 Atm
Argon mass flux:
45:
1000 cc/min at S.T,,P.
0.7 Atm
Argon mass flux: 1000 cc/min at S.T.P.
30 cc/min at S.T.P.
Silane mass flux:
46:
149
Radial Velocity and Temperature Profiles
of Silane and the annular Argon Jets
at the Silane Nozzle Exit X=0.02005 m
the Radial Mass Concentration of Silane and Argon
are also Calculated by the Parabolical Code
Cell pressure:
Figure
148
Radial Velocity and Temperature Profiles
of the Annular Argon Jet X=0.02 m
from the Parabolical code
Cell pressure:
Figure
1000 cc/min at S.T.P.
150
Mixing of the Silane and Argon and Jet Before
Reaction:
Radial Velocity and Temperature Profiles
of Silane and the Annular Argon Jets
at X=0.02397
m
The Radial Mass Concentration of Silane and Argon
are also Calculated by the Parabolical Code
Cell
pressure:
0.7 Atm
Argon mass flux: 1000 cc/min at S.T.P.
Silane mass flux:
30 cc/min at S.T.P.
Figure
47:
151
The Reaction Zone:
Radial Velocity and Temperature Profiles
of the Inner Reacting Jet and the Annular
Argon
Jet at X=0.02797
m
The Radial Mass Concentration of Silane, Argon
and Silicon are also Calculated by the
Paraboiical Code
Cell
pressure:
0.7 Atm
Argon mass flux: 1000 cc/min at S.T.P.
30 cc/min at ST.P.
Silane mass flux:
152
i .14
Page
Figure
48:
The End of the Reaction
Zone:
Radial Velocity and Temperature Profiles
of the Inner Reacting Jet and the Annular Argon
m
Jet at X=0.03197
The Radial Mass Concentration of Silane, Argon
and Silicon are also Calculated by the Parabolical
Code
Cell
0.7 Atm
pressure:
Argon mass flux:
Silane
Figure
49:
mass
flux:
1000 cc/min at S.T.P.
30 cc/min
at S.T.P.
The Post-Reaction Zone:
Radial Velocity and Temperture Profiles
of the Inner Reacting Jet and the Annular Argon
Jets at X=0.03597 m
The Radial Mass Concentration of Silane, Argon and
Silicon are also Calculated by the Parabolcial Code
0.7 Atm
Cell pressure:
1000 cc/min at S.T.P.
Argon mass flux:
30 cc/min at S.T.P.
Silane mass flux:
Figure
50:
Figure
51:
52:
154
The Post-Reaction Zone:
Radial Velocity and Temperature Profiles
of the Inner Reacting Jet and the Annular Argon and
Silicon are also Calculated by the Parabolical Code
Cell
Figure
153
pressure:
0.7 Atm
Argon mass flux: 1000 cc/min at S.T.P.
30 cc/min at S.T.P.
Silane mass flux:
155
Axial Distribution of Velocity, Temperture
and Mass Concentration as Calculated by the
Parabolical Code at r=0.0
Run Conditions:
Laser power: 180 W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
Silane flow rate: 30 cc/min
Cell pressure: 0.7 Atm
Nozzle diameter: 1.19 mm
156
Radial Velocity, Temperature and
Concentration Profiles at a Post-Reaction
Position X=0.10
Run Conditions:
Laser power: 180 W unfocused, Gaussian profile
Argon flow rate: 1000 cc/min
Silane flow rate: 30 cc/min
Cell
pressure:
Nozzle diameter:
1.0 ATm.
1.19 mm
157
i.15
Page
Figure
53:
Axial Velocity, Temperature, and Concentration
at r=0.0
Profiles
Run Conditions:
Laser Power: 180 W unfocused, Gaussian
profile
Argon flow rate: 1000 cc/min
Silane flow rate: 30 cc/min
Cell
pressure:
1.0 Atm.
Nozzle diameter:
1.19 mm
158
Figure
54:
Flow Chart of the Supersonic Computer Code
159
Figure
55:
The Grid and the Marching Procedure in the
Supersonic Algorithm
160
Underexpanded Non-Reacting Supersonic Jet
Axial, Distribution (Confined Jet). Comparison
with Kurkov's Results
161
Underexpanded Non-Reacting Supersonic Jet
Radial Distribution (Confined Jet)
162
Overexpanded Non-Reacting Supersonic Jet
Axial Distribution (Confined Jet)
163
Schematic of the Underexpanded
Confined and Free Jets (Non-Reacting)
164
Underexpanded Supersonic Silane-Argon Jet
Axial Distribution (Confined Jet)
165
Figure
Figure
Figure
Figure
Figure
Figure
56:
57:
58:
59:
60:
6i:
Underexpanded Supersonic
Silane-Argon Jet Axial Distribution
166
(Free Jet)
Figure
62:
Underexpanded Reacting Supersonic
Silane-Argon Jet
Axial Distribution
Figure
Figure
Figure
63:
64:
65:
(Free Jet)
167
Diamond Pressure Pattern of
Underexpanded Reacting Supersonic
Silane-Argon Jet
168
Temperature Map of
underexpanded Reacting Supersonic
Silane-Argon Jet
169
Velocity
Map of
Underexpanded Reacting Supersonic
Silane-Argon Jet
170
i.16
Page
Figure
Figure
Figure
Figure
Figure
66:
67:
68:
69:
70:
Reaction Focused on the Shock for an
Underexpanded Silane in a Tube
171
Reaction Focused Before the Shock
for an Underexpanded Silane in a Tube
172
71:
Cell and Gas
Supply Lines and Analog Circuit
173
Experimental Set-Up of High Speed
Movie Camera
174
Fourier Analysis of the Oscillations
of Type
Figure
of the Reaction
Schematic
175
1
Fourier Analysis of the Oscillations
of Type 2
176
Figure
72:
Schematic of the Flame Oscillations
177
Figure
73:
Stable and Unstable Regions for a
Reaction
178
Figure
74:
Flame Velocity vs Time.
179
Figure
75:
Flame Height as a Function of Time
180
Figure
76:
Flame Base Position as a Function of Time
181
Figure
77:
Axial Temperature Distribution
and Flame Propagation Limit
182
Figure
78:
Oscillations of Type 1
183
Figure
79:
Oscillations of Type 2
184
Figure 80:
Experimental Set-Up for the Hot Wire
185
Figure
Disturbances Due to a Full Filter and
the Hot-Wire Response
186
Slow Response of the Pressure
Controlled Valve to an Induced
Disturbance Transducer
187
Figure
Figure
81:
82:
83:
Convergence history of the velocity field
for the
run condition
of Figure
27, 28, 29
188
i.17
Page
Figure 84:
Figure
85:
Comparison between theoretical, computational
and experimental axial velocity distribution
189
Comparison between theoretical, computational
and experimental axial temperature distribution
190
i.17
Page
Figure
84:
Figure 85:
Comparison between theoretical, computational
and experimental axial velocity distribution
189
Comparison between theoretical, computational
and experimental axial temperature distribution
190
i .18
LIST OF SYMBOLS
Symbols
a 0 ,a ,a 2
·..
for Chapters
--- constants in the temperature equation for
Region
C
1
2C,C 3
C4
1 and 2
1
--- integration constants determined by patching
corresponding subregions.
Cp
--- specific
f(x)
--- axial dependence of the temperature field
heat
for Region
g(r)
1 (see Appendix
A)
--- radial dependence of the temperature field
for Region 1 (see Appendix A)
g( )
g(n)
--- non-dimensional stream function (Region 2)
h
--- Planck constant
Gr
--- Grashof Number (buoyancy effect in Region 2)
Jo( ),
Ji/ ( )
in equation
(1.1)
--- Bessel functions (2.11), (A.9)
2
IC
--thermal conductivity
L
--a length scale; average flame width
eo(r),e (r),e2(r) --- radial dependence of the temperature for
Region
1 (2.9)
p,P
--- pressure
q0
--- incident laser powder (2.15)
-- energy source term (included the radiation
effect)
q
r and
---radius and radius corrected for temperature
variation (2.17)
F
--silane nozzle radius
R
du 2
Ro
---
radius at which ul(Ro)=-d- Ir=Ro
T(x,r)
---
tempature profile (Tw
=
wall temperature)
i .19
--- local and average
u and U
silane
velocity
subscripts: 1 referes to Poiseuille profile
2 referes to Squire profile
"ar" referes to the argon velocity
radial velocity component
v----
AL,AL
diffusivity
~~--thermal
~a
C
--- emissivity of the silicon powder and Stefan-Botzmann
constant
and a
'~~n
length of the absorbing volume
~---
non-dimensional independent variable (from the self
(2.2), (2.16)
similarity assumption)
aABS
--- absorption
T
--- Schvab Zeldovich transport variable (1.8)
Mk
--- mass fraction of chemical species k
coefficient
(1.6),
(2.15)
k = SiH 4 , Ar, Si, H 2 .
Additional
Symbols
for Chapters
3, 4, 5, 6, and 7
A
--- Arrhenius rate constant (4.14)
at
--- speed of sound at a stagnation temperature (5.30)
a,b
--- constants (5.20), (5.21)
aw
--- "West" area of a nodal
C
--- specific heat of mixture
C w , Dw
--- computational coefficients
Fx
--- body force
Ji
--- flux of variable t
cell
(4.1)
(4.3), (4.4)
(5.7)
(4.6)
--- stagnation enthalpy
r
w
--- transfer
point
w
coefficient
of the variable
0 at a nodal
i .20
K
reaction rate (3.2)
---
M
--Mach number
Msi
--local mass fraction of silicon (4.8)
also
subscripts
for:
SiHl4, Ar, Si, H 2
Pexp
--preexponential factor (4.14)
Pr
--Prandtl number
RSiH
source term of silane gas (5.9)
---
S
general source termjsubscript:
---
ABS:
RAD:
KE:
Ts
period of high speed movie frames
---
V,V
--radial velocity
Ws
ws
--Shear work (5.9)
absorption
radiation
kinetic energy
--stream function (5.3)
i, nw1
--transformation variables (5.4)
Sx
--(xi+_xi) (5.24)
ratio of specific heats (5.27)
---
Subscript
"t"
stagnation values:
Tt= stagnation temperature
Pt= stagnation pressure
1
1.
INTRODUCTION
1.1
THE DYNAMICS OF LASER DRIVEN SILANE REACTIONS
By a synthesis process developed at M.I.T., submicron silicon
particles are produced by the pyrolysis of the silane (SiH4) gas.
The
present work will focus on the fluid mechanics and the heat transfer
phenomena of this process.
Silane gas is continuously injected into a
reaction chamber where it is heated by absorbing the photon energy emitted
from a C02 laser beam.
After reaching a threshold temperature around
600°C, the gas reacts and decomposes thermally according to the overall
reaction given below:
SiH 4
If a C
2
()
hv
Si
(s) + 2H 2
()
(1.1)
laser is used as the main heat source, the direct synthesis
of silicon particles from a gas phase reactant has the following principal
advantage:
High purity levels can be achieved in the final silicon
product by controlling the purity of the silane gas.
the
initial
gas composition
is not sufficient
But the purity in
if the heat transfer
necessary for the reaction to occur is to be made through a medium having
a high
risk of contamination
such as a heated
tube wall
or an arc plasma.
Laser driven synthesis of silicon powder does not require the presence of
a medium
through
which
the heat
is to be conducted
or convected.
Instead,
the silane gas is optically heated by absorbing the P(20) line at 10.591
Pm emitted in the infrared by a C
2
laser beam.
emitted by a simple untuned laser beam [1].
The P(20) line is usually
The reaction zone is confined
2
to the perimeters of the central inner jet which is bounded by a coflowing
inert
annular
argon gas.
a contaminated surface.
to
sustain
the reaction
room temperature.
is not in contact
with
Inversely, there is no need for a heated surface
and the wall of the powder
reactor
can be kept at
Also it is interesting to note that no laser energy is
used to heat argon,
directly
flow
the reacting
Hence,
emmitted by the laser.
since
gas does not absorb the line
this inert
Therefore, the source of energy is mainly used for
the thermal decomposition of the silane gas, and for the nucleation and
In this aspect, heating with a laser
growth of the silicon particle.
source can be very efficient whenever the emission and absorption lines
coincide.
Although the laser driven synthesis process has some natural
advantages, it is extremely important to understand quantitatively the
different mechanisms of heat, momentum and mass transfer present in the
problem, in order to improve the process and achieve the following
ultimate
goals of an ideal
- to produce
a fine
silicon
formed
as a final composition
synthesis:
size, less than
grain
sintering characteristics.
powder
0.5 micron
for good
With laser driven synthesis, the powder is
and size so that this
criterion
is met.
But
it should be emphasized that, although small silicon particles can be
produced
in the range of 25-150
angstrom,
it has not been possible
to
produce bigger particles and accurately control the size of the silicon
particles.
This thesis will
shed some
light
on the mechanisms
of mass
transport and concentration gradients; but it is essential to understand
the kinetics of the process in order to address the issue in some depth.
- to obtain a narrow particle size distribution:
This can be
achieved if all the nucleating silicon particles have identical time-
3
temperature histories, especially during their residence time in the
reaction zone.
The temperature, velocity and concentration profiles which
will be generated in this thesis will give the direction towards further
macroscopic process refinements, such as deciding whether the silane flow
should be laminar or turbulent.
The present analysis will also present a
ground for comparison different laser beam profiles (specifically gaussian
and top hat profiles).
- to obtain the desired grain structure, amorph or crystalline.
This
will be strongly dependent on both the peak temperatures reached in the
laser beam and the heating and cooling rates in the reaction zone.
laser
heat source
and the radiation
heat sink terms
will
be shown to play
an important role along with convection and conduction terms.
of silane
and argon gases
transfer process.
also plays
an important
The
The mixing
role in the heat
The kinetic heating effect will be shown to be only
relevant at the high Mach number flows.
- to obtain a reproducible silicon powder with identical
characteristics:
the minimum requirement for reproducibility is a steady
reaction flame.
The periodic and random unsteadiness will be covered with
some detail in Chapter 6 of the thesis.
These different types of flame
instability will be experimentally shown and the main parameters affecting
the flame unsteadiness will be discussed.
1.2
THE GEOMETRY
OF THE PROBLEM
The theoretical and computational formulation has assumed a twodimensional axisymmetrical flow configuration [Fig. 3].
Such geometry
closely resembles the actual experimental set-up, although, the laser-flow
.4
configuration is not always axisymmetrical;
especially for focused laser
beams with diameters (2 mm) which are much smaller than the maximum flame
cross section diameter (6 mm).
In these cases, the flame cross-sections
have elliptical shapes (rather than circular ones), being elongated along
the laser axis.
The flow and the typical dimensions are as follows:
The reacting
silane gas emerges vertically from a 1.2 mm diameter nozzle into a
continuous powder reactor kept at constant pressure.
Cell pressure can be
changed across a wide range from 0.2 Atm to 2.0 Atm.
The silane nozzle
diameter can also be varied to accomodate higher or lower mass flow rates.
Typical average silane jet speed varies between 0.65 m/sec - 2.83 m/sec.
Argon gas is injected through a larger annular nozzle (19.7 mm) which
surrounds the inner reacting gas with a typical average speed ranging from
0.07 m/sec to .24 m/sec.
given in [2].
More details on the actual powder reactor is
The purpose of the argon gas flowing coaxially with a
reacting silane gas is two-fold.
Firstly, the argon flows into the
reactor (1000 cc/min STP) at a rate which is 30 times the silane gas flow
rate (around 30 cc/min STP), so that agron fills most of the reaction
chamber.
Hence, the cell pressure is mainly a function of the argon gas
which is inert and steady, and not a strong function of the silane gas
which is continuously reacting whenever the temperature crosses the
threshold temperature.
Secondly, argon presence diffuses the unreacted
silane -if any is left- after the reaction.
The main reason is the
extreme flammability of silane when in contact with oxygen.
releasing into the atmosphere, the flow is diluted.
Thus, before
The presence of
silane in the post-reaction zone can be explained as follows:
Although
5
most of the silane gas reacts, a small fraction
uickly diffuses radially
outward into the argon flow, and is never exposed to the laser beam.
gas reaches the cell outlet without being reacted.
The
The computer models
developed in this research estimate the unreacted silane gas mass fraction
to less than
1.3
.1% of the total
silane flux.
GOALS AND MOTIVATIONS
The main goal is to develop appropriate solutions to the laser driven
reacting flow and to calculate the temperature, velocity and pressure
fields along with the concentration profiles for the chemical species.
Comparison of the theory with the available experimental data will be
carried out.
Experimental temperature and velocity data will be taken and
the "research
loop"
[Fig. 1] will be properly
closed.
Some of the
interesting aspects of the present research have been listed below.
1.3.1- MAIN PHYSICAL ASPECTS
1) Sources
the problem.
(and sinks)
of mass,
energy
and momentum
are present
in
The mass sources in the species equations are present,
because of the chemical reaction.
The source in the energy equation
contains the laser energy absorption, the radiation and the kinetic
heating terms.
Finally, the momentum equation carries a source term which
accounts for the buoyancy effect.
2) The reaction zone is "suspended" in the flow field [Fig. 3].
The
temperature and velocity fields upstream of the flame are affected by the
reaction.
In other words,
the gas is preheated
in the region 1 [Fig.
and several mechanisms of heat transport can be hypothesized for the
2]
6
preheating.
Heat conduction from the reacting zqne will be shown to play
a crucial role upstream of the flame.
The second mechanism might be a
coupled process involving heat radiation from the reacting flame which
would then heat the steel nozzle and the silane entering the reaction
cell.
The axial
length of the preheated
region
is a very
important
parameter in determining whether periodic flame oscillations will occur or
not.
The greater the distance between the flame base and the nozzle, the
greater the chance of periodic oscillation at high cell pressure.
oscillations
are described
in detail
in Chapter
Such
6 of this thesis.
3) The presence of a free boundary around a reacting thin jet implies
boundary layer type equations and solutions.
equations,
the pressure
are considered.
is not a function
In boundary layer type
of radius
r and no wall effects
This will only be true for unconfined subsonic flows.
For the supersonic case, oblique expansion and compression waves are shown
to form the well known diamond patterns.
Hence for the pyrolysis of a
supersonic silane gas, the radial gradients of the pressure will be taken
into account but the shock-boundary layer interaction will be neglected.
More precisely, the SIMPLE (Semi-Implicit Method for Pressure Linked
Equation) method will be applied to the supersonic case for the evaluation
of the oblique shock position and strength.
1.3.2
MAIN MATHEMATICAL ASPECTS
1) The energy equation carries a source term representing the amount
of laser power absorbed by the reacting gas.
The absorption coefficients
for the silane gas from a CO2 laser have been determined experimentally
and they
are reported
in reference
[3].
In the computational
parabolical
7
case, the exothermic heat release has also been
aken into account.
The
amount of heat release is proportional to the amount of silane gas
consumed.
2) The reaction is suspended in the flow above the silane gas nozzle
and such geometrical characteristics lead to fundamental mathematical
treatments different from the ones covered in the diffusion flame
literature.
In other words, the heat transfer problem in this research is
elliptic, because for low gas flow rates, the axial heat conduction term
cannot be neglected when compared with the radial heat conduction term.
However, results from the
r
abolical and the elliptical computer
algorithms show that, although for low flow rate the axial heat conduction
is very important, this term can be neglected for higher silane flow rates
above 50 cc/min.
This is not a sharp limit, but it has been determined
that 50 cc/min corresponds to an average gas velocity of 3.71 m/sec at the
cell inlet, while, the thermal diffusivity is 2.0 m/sec at the same run
conditions.
For this case, the reaction occurs after the gas enters the
laser exposed reaction zone.
It has been visually observed that at low
silane gas velocities, the flame front comes closer to the inlet nozzle
generating a large amount of heat flux counter-flowing from the laser
heated zone.
3) The box-like shape of the reaction cell creates recirculation
zones around the jet and the corresponding Navier-Stokes equation has to
conserve the axial diffusion terms in order to predict such regions.
An
elliptical algorithm developed in the thesis decribes such separated
regions which are enhanced by the cell walls, downstream of the flame.
the cell reactor is large compared to the reaction, it is possible to
If
8
neglect wall-effects on the flame.
approximately
.6 cm while
The maximum fame
the cell diameter
diameter is
is 7.62 cm.
Because
of this
difference in the order of magnitude, the theoretical model assumes a free
boundary problem and solves for the temperature and velocity values
assuming an infinite rate of silane pyrolysis once the gas reaches the
threshold temperature.
The closed form results are derived as an inverse
to self-similar solutions obtained in the incompressible domain by using a
Dorodnitzyn-Howard-Illingworth transformation of the compressible flow
equations.
With this transformation, a incompressible equation of
continuity and momentum equation correspond respectively to the
compressible equation of mass and momentum.
The solution to the
incompressible problem is obtained; and by using the transformation stated
above, the results in the compressible domain are obtained.
In the
incompressible domain, self similarity of the flow is assumed except in
the region closed to the nozzle, where the jet is still developing.
1.4
LITERATURE REVIEW
1.4.1
THE PYROLYSIS
OF SILANE
This thesis treats a unique case of reaction in the sense that the
problem
to be solved
is not a combustion
process,
but rather a pyrolysis
of a gas which decomposes thermally after absorbing enough energy to break
the Si-H bonds of the silane molecule.
For example, the energy required
for the dissociation of SiH4 molecule into a SiH3 and a hydrogen atom has
been calculated to be 318 kj/mol.
The chemical mechanism governing mass
and heat transfer is different for a combustion and a pyrolysis and we
9
will quickly underline these differences.
This
ill help determine the
degree to which the numerous papers published in the field of combustion
are helpful to this work:
silicon is a member of the group IV elements
appearing directly below carbon on the periodic table.
As a result, the
silane and methane molecules are analogous:
H
H
l
I
H
C
H
H
Si
l
l
H
H
H
This chemical-structural similarity may imply that a similar chemical
kinetic mechanism of the oxidation of these two compounds exist [4].
In
the combustion of methane, the principal initiation reaction is by
hydrogen abstraction:
CH4
+ M = CH3 + H + M
Kf = 2.1017
e
88 4 2
1/RT
(1.2)
Using the chemical-structural similarity, silane combustion has been
proposed
to be as follows
COMBUSTION:
SiH
[6]:
+ M = SiH 3 + H + M
Kf = 2.1017
-59oo0/RT
e
(1.3)
where the activation energy is obtained by scaling according to the ratio
of C-H and Si-H bond energies.
On the other hand, for silane pyrolysis,
10
it has been determined [7] that there are a total, of 120 reaction steps to
the thermal decompostion of silane whose overall equation is given by:
SiH4
(9)
-
> Si
(s)
+
aH = -30 Kj/mol
2 H 2 ()
(1.4)
It is widely acknowledged in the literature that the initial decompostion
step which is given below, also determines the time evolution of all the
species concentration [7,8]:
-221(Kj)
PYROLYSIS:
SiH4 = SiH2
+ H2
with
Kt = 2.13x10
13
e RT
(1.5)
We see a fundamental difference between Equation 1.3 and 1.5.
For
the combustion of silane to start and be sustained, the oxidant must come
in contact with the "fuel" (SiH4
such pre-condition exists.
)
while for the pyrolysis of silane no
Instead, the thermal decomposition or
pyrolysis is more like a premixed gas combustion, where the reaction
starts whenever the gas is heated to the point where the activation energy
level i
overcomed.
In the case of silane decomposition, the activation
energy for Equation 1.5 is given as 226 kj/mol.
Contrary to the
diffusional heat transfer in laminar premixed diffusion flames, the heat
transfer in laser-driven pyrolysis reaction is mainly done through silane
absorption according to the following formula:
q = q(1.0 - e
ABSA
)
(1.6)
11
Where qo:
P:
a:
incident power density,
cell
pressure,
absorbing volume lentgh,
aABS: absorption coefficient.
More specifically, the gas does absorb the P(20) line of the CO2
laser but this is not an absolute requirement for a laser-driven
pyrolysis.
Since
the only
role of the laser
is to be the thermal
source,
even if the reactant itself does not absorb the line emitted by the laser,
any inert gas absorbing the laser power could be used to conduct heat to
the vapor phase reactant.
The characterization of the reaction flame is
another fundamental difference between the combustion and the pyrolysis of
the silane gas.
1.4.2
DIFFERENCES BETWEEN A COMBUSTION FLAME AND A THERMAL DECOMPOSTION
FLAME
In a combustion process, the flame itself is the reaction zone where
the reactant is oxidized.
The color of the flame is given by the energy
which is released from the reaction and radiated in the visible.
On the
other hand, the description of the flame during the thermal decompostion
of the silane can only be done qualitatively.
The flame results from the
later stage of the pyrolysis when the silicon particles, which have
nucleated, begin to thermally radiate their energy content.
Hence, in a
pyrolysis process, the flame front ca,. be defined by the radiating silicon
particles and the reaction front can be described as the location where
the actual thermal decomposition starts.
Although a clear distinction can
be drawn between these two phenomena, it is likely that the two phenomena
(i.e., particle radiation and gas decomposition) occur in regions where
12
both are present because of the non-uniform veloRity and temperature
distribution of the reaction zones.
No analytical attempt has been made
to distinguish between the two fronts.
1.4.3
BACKGROUND
Having established the similarities and the differences that exist
between the combustion and the thermal decomposition processes, it is
useful to outline the successful theoretical treatments of laminar
diffusion flame and heated round jet theories, available in the
literature.
The subject of laminar diffusion flames is a rich pool of
classical papers.
The papers of Burke and Schuman [9], Hottel, Gazley and
Kapp [10], Shvab [11], and Zeldovich [12] are some of the reasons for such
a rich background.
The Schvab-Zeldovich model is a popular technique to
solve the laminar diffusion flame.
The theory is cast within a set of
idealisations which can be summarized as follow:
1.
The flame is laminar (urely
2.
The reaction rates are infinite so that the flame front is infinitely
thin (the Burke-Schumann flame sheet model [13]).
3.
The gas is a "perfect" gas with constant specific heats, and changes
in the molecular weight accross the flame are neglected.
4.
The Schmidt, Prandtl, and Lewis numbers are equal to unity.
5.
The coefficient of viscosity is directly proportional to the
temperature.
6.
The effect of buoyancy is neglected.
7.
The inital
viscous field).
fuel jet has a plug flow profile.
13
Using these assumptions the governing equatipns can be written in the
following general form [14]:
Continuity:
(
a
r u)
+
a
(
r v)
using Schvab - Zeldovich transport
= 0
(1.7)
variables,
T-Ti
(1.8)
TF- T i
Momentum, energy and equation of species become:
P
p.
au +
P
ax
p
a
1
1
au
ar
(rau
(1.9)
Re
riau(
P u aT
p.
a-x
u
a
yk
p
+
p~,ax
p
a)
P v aT
1
1
a
po
Re
r
ar Pr
ar
ayk
1
1
rR e r
a
(1
r
ayk
"
term
Yk = chemical
species
Pr = Prandtl number
p,
(1.11)
)
Tr (Srr T
S = source
(1.10)
Lr)
ar
= reference
density
"k"
14
where we have used the Schvab-Zeldovich transport variables:
T-Ti
=
T Ti
T
=
T
b
= adiabatic
i
=
flame temperatures
(1.8)
initial temperature
= Schvab-Zeldovich
transport
variable
(1.12)
1
TT
b
Cp(
fq
~y
) T +-
f = mass
-
ratio
(of fuel for k=fuel)
q = specific exothermic energy(1.13)
The Schvab-Zeldovich technique cannot be used in our case because the
axial temperature gradients are important, especially in the preheated
region.
The mathematical model developed in the thesis also allows for
the buoyancy effect to be taken into account.
Assumptions 1, 2, 3, given
above, have been adopted as well. The main improvements of the analytical
solutions developed here can be summarized as follows:
1.
The Prandtl
number is not restricted
2.
Axial temperature gradients are taken into account in the preheated
region.
3.
Thescoefficient
4.
Buoyancy is taken into account and the Grashof number if constant in
the reaction region.
5.
The initial jet can have an arbitrary velocity profile.
of viscosity
to 1.
is constant.
These generalizations have been made at the expense of having a
patched solution across the flow field.
Several papers in the field of
heated jet have also been used in the development of the formulation
presented in Chapter 2 of the thesis.
We will briefly review one of the
15
papers relevant to this work.
An elegant solution is presented in the
field of heated round jet by H.B. Squire [16].
This closed form solution
is given for a heated jet where a point heat source is placed at the
origin.
As it is the case for all self-similar solutions, the presented
solutior is not valid near the origin.
But this solution is valid for the
outer flow field where the heat source can be interpreted as a point.
theoretical
model
will
be discussed
in more
detail
in the next chapter.
The
16
DETAILED
2.
2.1
ANALYSIS
APPROACHES
OF THE STEADY
STATE
TO THE 2-D AXISYMMETRICAL
PROCESS,
FLAME
Two main approaches have been adopted to solve the steady flame
problem.
The first one is the analytical approach which gives close form
solutions but in turn, treats a much simpler problem.
The second approach
is the computational modelling which removes most of the limitations of
the theoretical model.
two subsections.
The numerical calculation is further divided into
The elliptical algorithm (Chapter 4) which solves the
flow in the reacting cell and the marching type algorithm (Chapter 5)
which solves the reacting supersonic free jet.
The main goal of the
theoretical model is to obtain the temperature and velocity profiles
within or close to the reaction flame while the aim of the computational
model is threefold:
- To solve the problem to an extent greater than the theoretical model
by solving for the reacting species, and the pressure field without
neglecting the mixing occuring between the annular argon flow and the
inner silane jet.
To compare the parabolical algorithm with the elliptical one and to
demonstrate the extent to which the preheated region and the recirculation
zones are important.
In addition, to find the flow conditions for which
preheating effects are small and where the generated temperature outputs
of both computer codes are similar.
- To solve for the supersonic run condition and to look at the shock
wave - reaction flame interaction.
The supersonic program takes into
account the radial pressure gradient so that the oblique shock wave
17
positions are calculated.
However, the code is
ssentially inviscid in
the sense that the radial momentum equation does not have any viscous
terms, so that mixing of the two coaxial jet is mainly neglected.
The
supersonic run condition have been solved for the following reasons:
Although supersonic flow cannot be achieved with the present laboratory
equipment, this flow condition illustrates the pressure variation and its
effects on the reaction flame.
The absolute pressure is an important
parameter which affects the silane reaction rate.
From this point of
view, any sharp changes in the pressure can substantially affect the
velocity field, the temperature and the concentration profiles of the
reaction.
Although the cell pressure can be changed from one experiment
to another, the absolute pressure within the flow is hardly changed
because the pressure gradients are very small for laminar subsonic thin
flames.
Relative to the absolute pressure, they are on the order of 10- 5 .
Hence, their impact on the reaction flame cannot be truly assessed.
It is
acknowledged that sharp temperature gradients and a narrow reaction zone
decrease the rate of nucleation and increase the silicon particle size,
thus leading to fewer and bigger particles.
If this hypothesis is
correct, then the shock would be an ideal region of reaction.
By solving
for the supersonic case, the feasibility of such a run condition has also
been investigated.
18
2.2
THEORETICAL MODEL OF THE LASER DRIVEN PYROLYSIS OF SILANE GAS:
PREDICTION OF THE VELOCITY AND TEMPERATURE FELDS IN A CONTINUOUS
POWDER REACTOR
2.2.1
PURPOSE AND MOTIVATION OF THE THEORETICAL MODEL
The theoretical model solves a simplified problem where the silane
decomposition is assumed to be instantaneous with an infinite rate of
reaction.
Nevertheless, this does not remove the following numerous
advantages of an analytic solution:
a) Trying to simplify the problem is only possible by identifying the
important terms in the governing equations and by isolating them.
By doing so, a better understanding of the problem is achieved.
b) The closed form solutions give the general behavior of the
dependent variable, for instance, it will be shown that the
temperature distribution in the radial direction has a gaussian
shape in the reaction and the post-reaction zones,
c) The solution also gives the group of physical quantities to which
,
a particular variable depends.
For instance, it will
e shown
that the temperature in the reaction region depends on:
Ur2
T(x,
r) = f(
a qx
)
,
4ax
UK
U:
average axial velocity
a:
thermal diffusivity
K:
thermal
qo:
conductivity
absorbed energy density
while the velocity in the same region depends on
(2.1)
19
u(n, x)
n
where
=
f(
Gr1 /2
-
2
x)
(2.2)
r
x
Gr = Grashof number
By identifying these groups of variables and by varying them
systematically, it would be possible to isolate the conditions which
produce a particular silicon particle size, shape and material
characteristics.
This would ultimately lead to an optimized and precisely
controlled silicon powder process.
2.2.2
A.
ASSUMPTIONS AND CALCULATION PROCEDURES
ASSUMPTIONS
The following simplifying assumptions have been used to generate
closed form solutions for the temperature and velocity fields of a laser
driven pyrolysis of silane gas flow.
- Wall effects have been neglected by assuming a thin reaction flame.
The reaction flame had a maxium diameter of 1 cm and the diameter of the
cell was 7.62 cm.
Because of this difference in the length scale, it was
possible to neglect the wall effects.
treated as a reacting free jet.
This has allowed the problem to be
Furthermore, the thin flame assumption
led to boundary layer type equations in the reaction zone.
This
implicitly establishes that there is no radial pressure gradient in the
region of interest.
This is particularly true at low cell pressure runs
20
(0.2 Atm) where the jet velocities are high and the flame is relatively
thinner
of 6 mm).
(in the order
For this
last assumption,
we used the y-
momentum equation which reduces to the following for a boundary layer type
flow:
BP-
ay
(2.3)
0
- The reaction is weakly exothermic and the heat released from the
reaction
is neglected:
In the case
gas,
of the silane
of the pyrolysis
the main source of energy is the absorbed laser beam.
From the
equation given in (1.6), the amount of energy absorbed can be calculated
to around
given
10 W while
the amount
of heat
0.36 W for a silane
in (1.4) to be around
can be evaluated
released
from AH
flow of 20 cc/min.
- The silane gas have an infinite decomposition rate:
The chemical
reaction time has been found to be much smaller than a typical heat
conduction
or heat convection
time.
As it is shown
in Chapter
3 of this
thesis, a typical reaction time is in the order of a micro-second while
the.heat conduction and the convection times are on the order of a
millisecond.
Hence, the reaction was assumed to occur spontaneously once
the silane gas was heated up to 873.15°K.
The spontaneous reaction
affects the density of the reacting gas directly.
As it can be seen from
the equilibrium equations given in (1.4), one mole of silane gas produces
two moles of hydrogen gas.
This has the direct effect of decreasing the
local density by half, and this assumption was adopted.
In other words,
it is assumed that the sudden volume change affects primarily the density
21
without affecting the cell pressure.
This is a good approximation
especially when the cell volume is much bigger than the flame size.
- The silicon particles follow thieflow.
The assumption allows the
possiblity of treating the problem as a single gas phase, multicomponent
environment.
The particles are aerodynamically shaped (spherical) so that
the drag is small.
Furthermore, the flow is laminar and steady.
As
discussed in [20], such particles will follow the main flow even when
subject to a periodically unsteady flow with low frequency (106 Hz).
B.
CALCULATIONS
The reaction cell is divided into three sub-regions [Fig. 3].
Region
1 is upstream of the laser beam, Region 2 is in the laser beam and Region
3 is downstream of the laser beam.
The partial differential equations of
mass, momentum and energy are simplified according to the physics of the
problem and then integrated to yield the velocity and temperature fields.
A series solution is assumed for the temperature in the Region 1.
solution is obtained in terms of the Bessel functions.
The
For Region 2 and
3, the simplified 2-dimensional energy equations have been considered.
By
patching the solutions obtained for each subregion, the whole region of
interest has been covered.
mathematical sense.
Patching, here, has been used in the
Two solutions, valid in their respective domains are
equated at their common boundary.
The derivatives of such solutions have
also been equated at the same common boundary.
The temperature and the
velocity profiles are parabolical to first order, especially near the jet
axis where most of the convected mass and momentum are concentrated.
Furthermore, in Region 2, the velocity and temperture profiles are also
22
parabolical to the zeroth order.
Hence, by matching the temperature and
velocity values at the jet centerline, the overall match between the two
velocity profiles and two temperature profiles are assured across the two
regions.
The matching procedure is more valid between Regions 2 and 3
since these two regions have similar solutions.
No attempt has been made
to use the matched assymptotic expansion procedure, because the
computational models which are presented in the following paragraphs
alleviate the problems encountered in obtaining a closed form solutions.
The following procedures has been used to obtain the temperature and
velocity field:
1.
The source terms without radiation are calculated,
2.
The temperature field is obtained,
3.
The energy source term is corrected for radiation,
4.
The new temperature field is calculated,
5.
The velocity field is calculated.
Several iterations can then be made between the second and the fifth steps
given above.
2.2.3
SOLUTION OF THE TWO-DIMENSIONAL REACTING SILANE JET PROBLEM:
THEORETICAL MODEL
THE
The influences of convection, diffusion, radiation, and other source
terms are different upstream of the reaction zone, in the reaction zone
and downstream of it.
Regions 1,2 and 3.
We have respectively labelled these regions as
Hence, by neglecting the lower order terms in these
sub-regions, it is possible to integrate the governing equations of mass,
23
momentum and energy and obtain a closed form solution over the whole
domain.
REGION 1:
UPSTREAM OF THE REACTION ZONE
The axial
nozzle.
length of this
region is about
5-10 diameter
of the silane
The length of this region can be freely changed by bringing the
silane nozzle closer to the laser heated zone.
In Region 1, the
similarity assumption does not hold, since the jet is "developing" and the
core region still persists.
For most of Region 1, the "Poiseuille"
profile at the silane nozzle can be a fair approximation and was adopted
here.
For large radius r, this solution is not valid because the
parabolic velocity profile will go to minus infinity as the radius goes to
infinity.
For large radius, the Poiseuille solution will be patched to
the solution of Squire, developed for a round laminar jet exiting from a
pipe.
In the energy equation, the convection term is balanced by the
axial and radial conduction terms.
A series solution is assumed and the
result is expressed in terms of the Bessel function of the first kind of
the zeroth order.
The axial part of the series expansion in this solution
can be obtained by separation of variables (Appendix A).
It should be
pointed out that the series have been carried to the second term only.
The complete solution would probably be in terms of a summation of Bessel
functions.
Superposition of such functions would then eliminate the
oscillation which is present as r goes to infinity.
Therefore, in this
analysis, we have obtained the leading order solution for the temperature
and velocity.
listed below.
The appropriate equations governing Region 1 have been
24
au_0
a
0
Mass:
Momentum:
v
u(-) =
B.C.:
(2.4)
))
(ar
=
ar
ar;
dP
(2.5)
O.
(2.6)
r
Velocity:
Poiseuille
Squire
Energy:
profile:
pCp TUJ(
ax
p
Assumed
Form:
profile:
T(x,
a
u 2 (r) = C
T
ax
ax
r) = T
u 1 (r) = 2
0
r
a
r
(1- (r)2)
V+ Ur
ar
or
r
1 1
T(O, r) = T,
Temperature;
valid
for (
REGION 2:
T(x, )
=
W
C(x
+
a x2 ).J (-r)+T
T(x, r)
T
(2.7)
-o
(2.8)
)2
T
B.C.:
r > R
T)
(r) + a
+ a
r < Ro
) x + a
0 (r)x 2
2 2
(2.9)
0.
(2.10)
0
(2.10)
(~~~ 0
(2.11)
r < .3)
THE REACTION ZONE
The axial length of this region is determined by the diameter of the
horizontally impinging laser beam.
Usually an unfocused laser beam has 6
mm of diameter and a focused beam 2 mm of diameter.
In this region the
self similarity of the jet is assumed and a transformation similar to the
·LIPflrPUI·IIL?xll··Lr;li;-i-il
II-
25
Illingworth-Stewartson transformation has been defined for axially
symmetrical flows.
This transformation allows the density of the gas to
be dependent on the temperature and at the same time it decouples the
momentum equation from the energy equation.
This approximate formula,
also called the Dorodnitzyn-Howard-Illingworth transformation, gives
relatively good results especially if the radial density gradients are not
large.
Hence, the method is suitable for locating the flame base (which
exhibits large axial density gradient and relatively smaller radial
density gradient).
The momentum equation which has been transformed into
an ordinary differential equation has been solved by direct integration.
Differently from the approach in the available literature, the momentum
equation includes a ouoyancy term.
The Grashof number associated with the
buoyancy is assumed to be constant throughout Region 2.
numbers are around 4300.
Typical Grashof
The energy equation has a source term which is
balanced by convection from the main direction of the flow and radial heat
conduction.
The radiation is also taken into account by substracting the
f;
amount of energy radiated by the silicon particles from the energy amount
absorbed from the CO2 laser beam.
The appropriate equations and solutions
related to Region 2 have been listed below:
Mass:
ax (pur) +
Momentum:
Energy:
a)u
pu a
p C
-aT
- (pvr) =
+ pv
.
~)
-ar=r
1
r(<r
r =- a!(icr
p Tax
r8ra3r
(2.12)
_u
r
au)
T) + q
++
pg(r T
pgaT
(2.13)
(.
(2.14)
26
1
where
- eA
q = q
u
x
(T4 -T4 ref)
L
n
dgl
dii'
11
(2.15)
r
(2.16)
x
- r dr)1 /2
= ( 2 fr
0
(2.17)
p00
Gr
C2 (2
g(rn)
c(-a
Ea
4
g'(n)
'
x
bsP-A2
16)
-
112
=
n2
C2 ( 4
Gr
G
n1
valid
1
Gr
06 ) +
Gr
16
1
1536
1/2
2
for:
.1 7 5
(2.18)
n4
U-r 2
Temperature profile:
REGION 3:
T(x, r)
=
e 4a-x
+
aq x + C 4
U_
(2.19)
DOWNSTREAM OF THE REACTION ZONE
This region extends beyond the upper edge of the laser beam [Fig. 2].
The flow has been assumed to be self-similar.
The Grashof number in this
region has been set equal to zero since the surrounding argon gas has also
been heated (through conduction and radiation).
The source term in the
energy equation has also been set to zero since there is no more
absorption from the laser.
The equations governing this region is similar
They can be written as follows:
to the one developed for Region 2.
u =
x
-'
ni
(2.20)
27
- Ur 2
C5
T =
4ax
e
2x
C 1 , C 2 , C 3 ,-C 4 , C 5 , C 6 , are constants
2.2.4
A.
(2.21)
+ C6
of integration.
ANALYTICAL RESULTS AND DISCUSSION
DETERMINATIONS
OF THE CONSTANTS
OF INTEGRATIONS
The constants of integration which are present in the closed form
solutions have been determined by "patching" the corresponding results of
two sub-regions at their common boundary.
For Region 2, an integral
balance between the amount of energy absorbed and the amount convected has
been performed to find the integration constant of the velocity
expression.
The energy balance has been chosen since the jet does not
have a constant masss (due to entrainment) and the present reacting jet
does
not conserve
the initial
momentum
it had at the exit of the nozzle
due to the buoyant forces created by temperature gradients.
Hence, an
integral energy balance over the absorbing region was adopted instead.
This
balance
is given below:
q
211
vol
pCp(
ax
) u r dr dx
(2.22)
where C2 is defined in the "u" expression.
B.
RESULTS
The theoretical results from the sample run is given in Figures 5-13.
As it can be seen from the centerline temperature [Fig. 7] distribution,
28
the agreement between the experiment and the theory is good downstream of
the flow field.
lower.
In the preheated Region 1, the predicted temperatures are
This is probably because too few terms have been retained in the
series solutions of Region 1.
Two axial temperature distributions are
presented in Figure 7 and Figure 11.
Two parameters have been changed:
The silane nozzle diameter and the laser source profile.
In the lower
curve of Figure 7, the jet velocities are higher because the inlet silane
nozzle is smaller and consequently, the peak temperature is lower due to a
shorter residence time in the laser heated region.
This inverse trend
between the nozzle exit velocity and the peak reaction temperature have
been consistently observed in the analytical and computational models and
has also been verified experimentally.
The radial temperature gradients
given in Figure 8 and 13 and do reflect some differences.
is the result of different laser source profiles.
The difference
The laser has a
gaussian profile in Figure 8 while in Figure 13, the laser source has a
"top hat"
profile
in the radial direction.
The latter
profile
can be
experimentally obtained by using a cylindrical lense to stretch the
gaussian profile.
The axial velocity profiles are given in Figure 5.
The
agreement between the theoretical and experimental results is evident.
The experimental velocity measurement technique will be described in the
next chapter.
The velocity decreases due to a combined effect of the
volume expansion and the presence of viscosity.
as it is heated.
decomposes.
The volume of gas expands
Once the threshold temperature is reached, silane
For each mole of gas which is decomposed, two moles of
hydrogen is produced, thus, further accentuating the gas expansion.
sudden change in density affects the axial velocity profile but not
The
29
drastically [Fig. 5].
trend [Fig. 51].
The computational velocity, profiles show the same
30
VELOCITY MEASUREMENTS IN THE REACTION FLAME
3.
3.1
CONVENTIONAL VELOCIMETERS:
DIFFICULTIES IN DATA ACQUISITION
From the analytical results given in the preceding paragraph
[Fig. 7], it can be seen that the flame base reaches 950°C and that the
axial temperature profile varies around 1000°C.
Because of these high
temperatures, it is difficult -if not impossible- to measure the gas
velocity with a hot wire.
Furthermore, the presence of the silicon powder
produced in the flow is likely to contaminate the probe and to cause
errors in the magnitude of the velocity determined.
At a first glance,
the laser Doppler velocimeter seems to be more suitable but this technique
has also failed to produce results, mainly because of the following
reasons:
a. The flame is relatively thick for a low power (0.005 Watts).
w
_ Ne laser:
The number
density
·
of the particles
.-
He-
is as high as
!
I(
4.0x10 12 particles/cm3 [2].
The presence of large number of
particles in the probe volume decreases the signal to noise ratio
of the photodetector which collects the scattered light from the
probe.
b. Most of the silicon particles have diameters on the order of 300100A, which are smaller than the typical average size of 0.5 pm
for a seeding particle [19].
Even though seeding particles of
this size could be introduced, this would have two major
deficiencies.
Firstly, these "big" particles would themselves be
heated, thus, altering the temperature and the velocity profiles
31
of the reaction.
Secondly, the scattering signal from the probe
volume would still be attenuated by the presence of nucleating and
growing silicon particles.
These conventional velocity measuring
equipment failed to produce data and a new approach suitable to
the physics of the problem was adopted instead.
VELOCITY MEASUREMENTS BY A PERTURBATION METHOD
3.2
3.2.1
PROPERTIES OF THE NEW METHOD
The velocity measurements by the perturbation method which will be
explained in this section, does not have the disadvantages of the
conventional techniques described above:
a. No physical probes are physically inserted into the flow.
This is
important because the probes are inoperational in the high
temperature environment where particle nucleation occurs.
-b. The transmitted He-Ne light is analysed, instead of the scattered
!
light which is much weaker and which constitutes the primary
disadvantage of the Laser Doppler Velocimetry.
THE PHYSICAL CONCEPT BEHIND THE NEW METHOD
3.2.2
The new technique is to introduce a "marker" in the flow which is in
our case, a perturbation front.
Then, the time taken by such a marker
between two predetermined points A and B [Fig. 14] gives the velocity
V
AX
At'
(3.1)
32
But how can such a marker
silane reactor?
be generated
in an oxygen-proofed
continuous
As it can be seen from the different time scales listed
below, it is possible to locally stop the reaction for a very short period
of time (on the order of a micro-second) without affecting the overall
flame structure because the convection times are 1000 times longer
compared to the time of reaction:
-
Convection time scale:
L/U
(1 millisec.),
-
Conduction time scale:
L2 /4a (1 millisec.),
-
Reaction time scale:
1./K
(3.2)
(1 microsec.),
where
L:
width of the reaction zone (or nozzle diameter),
U:
mean flow velocity (at the silane nozzle exit),
a:
thermaldiffusivity,
K:
Arrhenius rate constant for the silane decomposition.
The reaction mechanism of the silane decomposition is still not very
well known.
In the dimensional analysis, the order of the reaction was
assumed to be 1.
The main point of this scale analysis is to show that
the chemical reaction is much faster than the heat transfer by convection
or conduction, and this point was shown experimentally as described in the
next section.
The considerable shortness of the chemical time scale
suggests that a small and local disturbance in the incident laser power
with a duration of 1 vs. would be sufficient to locally stop the pyrolysis
of the silane gas.
In return, the disturbance would be too short to
affect the overall flame structure.
Thus, the thermal decompostion of
silane would be temporarily stopped in the disturbed region and the
nucleation and growth of the silicon particles would be locally
33
diminished:
Thus, a front would be formed across which the number density
of silicon particles would be lower.
As a result, two He-Ne laser beams
(0.0025 Watts each), impinging on the flame at different axial locations
[Fig. 14 and 15) would detect the travelling front at different times.
The time delay
t between the two signals from the detectors are shown in
Figure 16 and it is used in the basic equation given above.
The main
question to be asked is why, in this case, the velocity of propagation of
the disturbance
is equal to the local
gas velocity.
This question
is
important because the perturbation method does not measure the velocity of
the gas flow, but instead, the velocity of the disturbance front.
The
silicon particles which are travelling with the front are very small (300100 A) and their shape is spherical.
Furthermore, the flow is steady.
Hence, in the light of these conditions, it can be safely assumed that
each sub-micron particle will be virtually entrained at the local fluid
velocity.
front,
Hence, the velocities of the particles, of the perturbation
and of the local gas are in equilibrium
and equal
to each other.
So we conclude that tne speed of the perturbation front is equal to that
of the flow.
3.3
THE EXPERIMENTAL SET-UP
The experimental apparatus has been detailed in Figure 14 and 15.
The following equipment was used in the measurement:
-
A 180 W laser source:
Coherent Everlase model 150.
This is the main energy source to be disturbed.
-
A He-Ne 0.015 W laser used as the probing beam.
-
A 50%-50% beam splitter.
34
-
Two high response photodetectors.
-
An axial and radial translation
-
A beam chopper with a variable rpm (0-4000 rpm)
plate.
The incident laser power was chopped as shown in Figure 14.
The
duration of the disturbance pulse was slightly lower than 1 milliseconds,
being limited by the motor speed of the chopper.
But this did not pose
any problems since the magnitude of the disturbance could be easily
changed
the chopper
by moving
blade
up or down
[Fig.
14 and 15].
than 7% of the total incident energy was periodically disturbed.
Less
The
disturbances were introduced from the laser heated point which is closest
to the silane inlet nozzle.
Thus, by introducing the marker from the
bottom of the flame, it was possible to determine the axial velocity
profile of the reaction and post-reaction zones.
as a probing beam.
not necessary).
The He-Ne laser was used
This beam was split into two beams (although this was
Each of these beams was directed towards the flame
centerline, but at different axial locations.
The x-distance between the
two probing beams was 1 mm and was not changed during the experiments.
The He-Ne laser source, the beam splitter, and the receiving
photodetectors were all mounted on a translational plate mounted around
the reactor cell.
This plate was free to move in the axial direction as
well as in the radial direction.
Hence, the distance between the probing
He-Ne beams could be used throughout the experimental run with no need of
resetting it.
Axial and radial traverses were also easy and no optical
realignment was necessary during the runs.
35
3.4
EXPERIMENTAL RESULTS AND COMPARISON WITH THE ANALYTICAL SOLUTION
When the incident laser beam was chopped over a full cross-sectional
area, the whole flame was unsteady.
This was displayed by the
oscilloscope, which was connected to the output of the detectors [Fig.
16].
These photodectors have the high bandwidth of 6 MHz and they
measured the transmitted intensities of the He-Ne laser beams.
When 10%
of the laser was chopped from the lower part of the beam (this corresponds
to the point C on Figure 14), the flame stabilized [Fig. 16] even though a
small disturbance was superposed on the steady reaction every 12
milliseconds.
runs.
This frequency was not changed during the experimental
The duration of the disturbances was 8.0x10- 4 seconds which was
slightly less than the typical
convection or conduction time.
The time
lapse between the two recorded signals [Fig. 16] was obtained as At.
Each
pair of pulses shown in these figures belongs to the perturbation front
passing through points A and B as illustrated in Figure 14.
The flame and
the probe have been magnified in Figures 14 and 15 to show the detail.
From these figures it can be seen that
t increases as the probing laser
beams are located further downstream of the nozzle.
This clearly shows
that the velocity of the gas decreases, because the disturbance takes more
time to travel between the probing beams.
The distance
x between the
probing beams was never changed as the detectors were moved from one axial
location to another.
Both of the beams and their corresponding
photodetectors were carried on the same translational plate moving along
the x axis.
The velocity data is plotted on Figure 5.
The second set of
velocity values are plotted as squares on the graph and they were obtained
by evaluating the diameter of the reaction zone at each axial location
36
(Hence the area A) from a flame photograph.
rate
Given the silane mass flow
and the axial temperature distribution (hence, p), the average
velocities at different axial locations were determined from m
=
pUA.
The centerline velocity was then calculated by assuming a gaussian
velocity profile.
technique.
This simple calculation shows agreement with the new
A major disadvantage of this simple calculation is that the
local density of the reacting gas must be known and this is intimately
related to the complex reaction mechanism of the silane gas decomposition.
Again, the new technique does not involve such complications.
37
4.
THE COMPUTATINAL MODEL OF THE STEADY SUBSONIC REACTION FLAME
THE ELLIPTIC CODE:
In this thesis two computer programs have been consecutively
developed for the subsonic and the supersonic flow conditions.
4.1
INTRODUCTORY REMARKS ON THE ALGORITHM
A computational model of the subsonic reacting laminar flow has been
developed.
Given the incident laser power and given the initial silane
and argon mass fluxes, the computer program solve for the velocity,
temperature and concentration field of the chemical species present inside
the powder reactor.
The program assumed a parabolical inlet profile for
the silane flow and an annular profile for the argon flow.
laser power is also assumed gaussian.
The incident
Any initial profiles can be
incorporated in the program without difficulty.
the characteristics of the computer code.
The following highlights
This algorithm solves the
governing equations of mass, momentum, stagnation enthalpy and equations
of species for argon and silane.
It incorporates all the diffusional
gradients (i.e., both the axial and radial terms).
This way, the
equations have a general elliptical shape and the entire boundary around
the domain of solution affects the flow field.
therefore be predicted by using this program.
Any recirculating zone can
Also, any counter-flowing
heat flux which preheats Region 1, can be properly modelled.
The reacting
code has been developed from the basic 2/E/FIX program of Pun and Spalding
[24] which solves the laminar non-reacting flow through a pipe.
Because
of the steady state character of the problem, the formulation of the
38
finite differences equations is necessarily of the implicit variety to
achieve unconditional stability.
It is interesting to note that several
researchers have solved the 2-D steady state flow problems as an
assymptotic limit of the time-dependent problems [25-26].
Specifically,
this approach was used by Harlow and Fromm [38] and by Macagno [40].
The
time-dependent procedure may have a better convergence rate but the choice
of the step size is crucial [39].
The main reasons of choosing a fully
implicit steady-state algorithm are given below:
1. the interest is the eventual steady state solution.
2. the convergence of the present code is fast:
is given
in Fig.
83.
It slows
the convergence rate
down by a factor
of 4 when the
chimney downstream of the flow is reduced to a small radius (on
the order of the Argon inlet nozzle).
Even in this case the
relative error of the velocity after 100 iteration is less than
1%.
This iteration number is reached after 43 min of CPU time on
a VAX 730 for a 15x15 grid.
convergence
can be considered
From this point of view, the
as fairly
fast.
3. The algorithm is unconditionally stable.
On the other hand, the
stability of an iterative time-dependent algorithm is linked to
the Courant number (for a forward time, centered space scheme).
UAX
2a
< 1
1,
and Ax for our system varies in a wide range.
stability problem.
This can cause serious
The upwind fully implicit scheme has been chosen so
39
that no convergence limitation is imposed such as the step size condition
(given above) imposed on the time dependent fully explicit scheme (the
Richardson and Jacobi interaction [26]).
The choice of the upwind
difference scheme is outlined in the following paragraph.
4.2
CHOICE OF THE UPWIND DIFFERENCE SCHEME OVER A CENTRAL DIFFERENCE
SCHEME
A grid molecule has been shown below.
The nodal points are refered
as W,N,E,S and P and the internodal points as w,n,e,s.
stand for West, North, East, and South respectively.
These letters
The main flow
direction is from left to right or -using the notation illustrated belowfrom 'West' to 'East'.
hi
E
AREA
AW
Let us consider, the west face of a cell having an area aw. The finite
difference equation for the diffusion flux normal to this area can be
expressed
as:
Diffusion flux:
aw J
w = - aw r,
w (P-tw)/6w
(4.1)
40
where
r~, w
transfer
=
coefficient,
= dependent variable (velocity, stagnation enihalpy),
w
= the "West"
aw
= distance between adjacent nodes w, P,
nodal point,
by using the following notation:
(pua)w
Flux:
if we denote:
Dw
w
(4.2)
C w = (pua) w
(4.3)
rw
aw/6
(4.4)
,
The total flux expression for central difference can then be expressed
as:
awJt,
,
awJtot' i w
(D + (½c
)(w
ww
- wP
-
7
Cw)pP
(4.5)
The negative sign in front of the coefficient of bp imply that
change oppositely to
Ow
w.
p tends to
This is unrealistic because an increase in the
value should also induce an increase in bp.
The upwind-difference
gives the following expression for the total flux J:
Cw>
0
awJtot, , w
(Dw + Cw)w
Dw P
(4.6)
41
Cw < 0
awJtot,
, w
Dw
w
(Dw
Cw)P
(4.7)
In this case no negative coefficients do appear and even a very high -Cw
w to affect the cell surrounding the node P.
permits some influence of
Due to these reasons, the upwind-fully
has been chosen.
The computational
implicit finite difference scheme
grid has a
artesian
fixed width (YN-Y1) independent of the axial position.
AY difference is varying with the radial location.
shape
and has a
the
Nevertheless,
This was adopted in
order to have enough computational nodes in the reaction zone while in the
regions far from the reaction less nodes were used to provide an effective
and adequate storage.
Hence, an efficient code was achieved without
neglecting any regions of the solution domain.
4.3
CHOICE OF THE RELAXATION PARAMETER
Textor (1968) [27] and Tejeira
(1966) [28] studied the parabolical
type governing equations by using different relaxation values and they
both found that the value of 'relax = 1/Re' was required.
This
observation can be applied to our flow configuration because the reactor
dimensions are large compared to the thickness of the jet flame, and it is
possible to state -for the reaction zone and the post reaction zone onlythat the flow will behave like a parabolical flow, especially near the jet
centerline.
Based on this observation, the criteria of Textor and Tejeira
was used as a relaxation parameter of 0.01 was successfully applied
throughout the calculations.
It should be pointed out that the Reynold
number of a typical run condition at 0.2 Atm is R=106 based on the total
mass flux of argon and silane and the outer nozzle diameter (i.e., the
42
argon nozzle).
83.
This
Figures
4.4
The convergence history of the vlocity
rate of convergence
is given in Figure
is for the run conditions
illustrated
in
27, 28, 29.
GOVERNING EQUATIONS AND BOUNDARY CONDITIONS
The equations are expressed in x, r coordinate system.
variables
are the axial
and radial velocity
components
The dependent
u and v, the
stagnation enthalpy h, the mass fractions of argon and silane.
Since the
overall chemical equilibrium equation was considered, the mass fraction of
hydrogen and silicon have been calculated from the mass fraction of silane
and argon and with the use of the stoichiometric ratio 'STOICH':
MS
[1 - M
MAMSH]
* STOICH
Ar
~SiH
Si
MH 2
[1
(4.8)
4
MAr -MSiH]
*
(1 - STOICH)
(4.9)
24
where Mk is the mass fraction of species k.
The governing equations can be cast into a general form given below:
~
18 ~~ax)
~ ~~
+
a (pu)+ 1 a (pv) +
=x(r
r a(r r~~~
r ~~
')~(4.10)
where
= 1 for continuity.
= u, v, h, MSiH , MAr.
(4.10)
The source term
havethefollowing
form for the corresponding equations;
have the following form for the corresponding 4equations;
43
So
= 0
S
= Pref
~ = 1
for
(4.11)
(continuity)
Tref
(1
--
) for
_PAa
So = q
(1
-
0~~~~
ABS
= u
(4,12)
c
(T4-T4
-
(T
T r
for
=
MSiH
frf
h
(4.13)
_A/RT
S
=
Pexp MSiH
* e
Boundary conditions:
(4.14)
The no-slip conditions were used at the walls of the
powder cell and the axial symmetry at the jet centerline implied zero
gradient conditions for all dependent variables.
4.5
ASSUMPTIONS AND ESTIMATION OF THE DIFFUSION COEFFICIENTS
From the definition of the Prandtl number (Pr = CP/K)
the thermal exchange coefficients as r =
gases:
Pr = 0.7.
Prandtl
number
/Pr.
we can write
We note that for most
For the silane gas near the reaction temperature the
has been
found to be Pr = 0.739,
with the generally assumed value of 0.7.
which
is in good agreement
The thermal conductivity near
the reaction temperature is estimated to be
= 0.164 J/ms°K [37].
The
assumptions on the gas properties is listed as follows:
a) The Lewis number is unity throughout the solution domain.
Assuming the Lewis number as unity is a common practice; this
implies that the Prandtl number is equal to the Schmidt number.
In orther words, the thermal diffusivity and the mass diffusivity
44
of the gases are equal.
A calculation of,the diffusion
coefficient between argon and hydrogen is given in this paragraph.
The calculation assumes a Lewis number equal to unity.
The result
is compared with the tabulated value and the agreement is good.
This is one of numerous examples which supports this assumption.
b) The diffusion coefficients are calculated from the following
formula:
r = p D=
Pr
where
Pr
= Sc
(4.15)
It is interesting to note that, in the preheated region, the diffusion is
a binary process between silane and argon.
Similarly, it can be stated
that in the post reaction zone, the diffusion is a binary process between
hydrogen and argon.
The following paragraph briefly justify these
remarks.
As it can be verified from the computational results, only
infinitesimal traces of hydrogen and silicon are present in the preheated
region.
From this point of view, it is safe to assume that the diffusion
coefficient is approximately equal to the binary diffusion coefficient
DSiH
Ar.
Unfortunately, this value is not tabulated in the open
literature and the diffusion coefficient have been estimated from the
viscosity according to the formula given in (4.15).
In the reaction zone,
all of the chemical species have comparable mass concentration and the
diffusion is a multicomponent process.
The reaction takes place in an
narrow region especially for high cell pressures.
The length of this
narrow region where all the chemical species coexist, can be estimated.
45
As it can be seen for the concentration countours of the silane gas [Fig.
22, 25], the depletion of the reactant is complete before reaching the
laser beam centerline at X=0.006 m.
This
suggests that the axial length
of the region where silane is decomposed, is around 2 mm, much less than
half the laser beam diameter (6mm).
The diffusion coefficient for the
reaction zone has been deduced from the local viscosity, according to the
formula given above (4.15).
However, in the post reaction region, it is
possible to estimate the diffusion coefficient from tabulated values [22].
The following paragraph calculates the diffusion coefficient in the
postreaction zone and compares it with the one obtained from the formula
given above (4.15).
In the post reaction zone most of the flow if formed by argon,
silicon and hydrogen.
The silicon powder has a submicron particle size
and does not occupy a significant amount of volume in the flame.
Scatter
extinction measurements have determined silicon particle number density to
be 5x1012 pp/cm3 .
TEM measurements estimate the average size of the
particle to be around 200A.
By using these numbers, it is shown that the
volume of particles amount to only 1/10 for each cm 3 of flow volume.
Hence, in the post reaction the main gas diffusion is between hydrogen and
argon and this region, can be modelled as a binary media (in terms of
exchange coefficients).
In reference [22] the H 2 -Ar diffusivity
coefficient is given as DH
Ar = 0.181 cm2 /sec at S.T.P..
Ar=6.3x10-5 m 2 /sec at 0.7 Atm.
corresponds to DH
This
In the computational
2
models D was varied according to the gas temperature between 5.516x10-5 1.1482x10-5.
This shows that the tabulated value belongs to this range.
Unfortunately there is no experimentally known data for silane-argon
46
binary diffusion and no comparison can be made.
As more diffusional data
becomes available, the present computer code can be easily upgraded to
utilize different diffusion coefficients for different chemical species.
At the actual moment, no theory permits the accurate calculation of the
diffusion coefficient in a multicomponent media and the experimental
measurements are unsually difficult to make.
4.6
RESULTS OF THE SUBSONIC FLAME AND DISCUSSION
The results for the subsonic jet are given in
Figures 83-85.
Figures 17-38 and
The run conditions can be summarized as follows:
- 0.2 or 0.7 atmosphere
cell pressure.
- 30 or 38 cc/min silane (at standard temperture and pressure:
S.T.P.). and 1000 cc/min argon (S.T.P.).
- Laser total power:
180 watts.
- Silane's laser absorpitivity:
- Flame radiation:
- Buoyancy:
0.005 m/N.
taken into account for Figures 33, 34, 35.
taken into account for Figures 21, 22, 23, and 30-38.
In Figure 20, the circulation zones are apparent.
For this part of
the flow, the problem is similar to a 'circular driven cavity' problem.
The center of the vortex is downstream of the reaction flame.
During, the
experimental runs, some silicon powder which was entrained in the
recirculation region, qualitatively certified the location of the annular
vortex.
magnitude
The 'z' marks on most of the velocity vectors show that the
is out of scale
(scale on Fig.
20:
1 cm=0.10
m/sec).
In Figure 21, the circulation zone is no more visible because the
velocities are small.
In Figure 23, it can be seen that the relative size
47
of the arrow decreases considerably as the reacted plume leaves the
reaction zone.
field.
This is mainly due to the temperature drop in the flow
The temperature
contours
in
Figure
describe very well the actual flame shape.
21, 24, 27, 30, 33, and 36,
Maximum temperature was 1340°K
for the run condition given in Figure 27, 28, 29, and 36.
When radiation
and buoyancy effects are included, the peak temperature drops to 1290°K.
The axial heat conduction effect is also apparent in the temperature maps.
The flame influence is noticeable in the last 1/3 portion of the preheated
region.
The following figures show the silane and argon concentration
field Fig. 28, 31, 36, 37.
The effect of the recirculation on these
profiles are apparent especially in the entrance region.
Finally, the
silicon hydrogen field is given in Figures 23, 26, 29, 32, 35, 38.
The
mass fraction of these species were very low at each computational node
and the mass fraction given in these concentration curves must be divided
by 1000.
Hence, the mass fraction of the hydrogen varies between 0.02%-
0.08% while the silicon mass fraction varies between 0.2%-0.55%.
The
results from the elliptic type algorithm will now be compared with the
results obtained from a boundary layer type algorithm Fig. 39-51.
The
latter program assumes that the gas is reacting in a infinite enviroment
whose pressure is held constant at the specified cell pressure 0.7 Atm.
The main limitation of a parabolical code comes from the fact that axial
diffusion terms have been neglected.
For the present flow configuration,
this will lead to unreasonably sharp temperature gradients at the flame
base [Fig. 51].
This shows that the axial heat conduction term in the
energy equation is the main mechanism of heat transfer in Region 1.
parabolical code shows the developement of the annular argon flow
The
48
[Fig. 39-50], which exit from the inlet nozzle. ,At the step 300
[Fig. 41], the argon jet emerges into the powder cell.
It should be
noted, that inner silane nozzle tip is at a higher axial location
[Fig. 44].
At x=0.02
m, the silane
flow emerges
into the cell
[Fig. 45],
and the following figures illustrate the argon-silane jet mixing and the
slowing
down of the reacting
jet [Fig. 44-50].
buoyancy have been taken into account.
Both
radiation
and
Finally, the reacted jet assumes
the Gaussian profile predicted by the analytical calculations [Fig. 50].
The axial profiles of the dependent variables are also presented in Figure
51.
It can be seen that
the peak temperature
is 1330°K which
is in
excellent agreement with the value of 1290°K reported from the elliptic
type code [Fig. 21].
The following Figures 52 and 53 are for a different
run condition at 1 atmosphere cell pressure.
The shape of the radial and
axial profiles are in agreement with the analytical results.
49
5
5.1
THE HYPERBOLIC CODE:
SOLUTION OF THE REACTING SUPERSONIC FLOW
DIFFERENCES AND SIMILARITIES BETWEEN THE SUPERSONIC JET AND THE
BOUNDARY LAYER TYPE SUBSONIC JET
Both codes are marching type algorithms which begin from the gas
inlets where the silane and argon mass flux are specified.
It should be
emphasized that the supersonic jet is hyperbolical in nature.
The
similarities and the differences between the marching type parabolical and
hyperbolical formulations are given below.
MAIN SIMILARITIES:
- There is a single predominant direction of flow.
- Transfers of momentum, heat and mass occur only at right angles to
the predominant direction of flow.
The axial gradients of the
diffusional terms are also neglected.
MAIN DIFFERENCES:
- In the parabolical formulation, a point in the flow is affected by
the whole domain upstream of it; while in the hyperbolical
formulation, only the domain within the Mach cone affects the point
in question.
- In the parabolical boundary layer equation, the pressure is only a
function of x, because the left hand side of the y-momentum
equation vanishes, giving:
dP
dP =0
where
where
>
p = f(x)
p = pressure
p : pressure
(5.1)
50
- In the hyperbolical equation, this is no
onger the case, because
the right hand side of the equation does not vanish and the pressure is
both function of the axial and radial coordinates:
p = f(x,
where
r),
(5.2)
P = pressure.
This program is not iterative and sweeps the domain of solution only once.
Because of its hyperbolical character, a boundary condition located
outside the Mach cone does not have any effect on the points located
within the zone of influence.
The velocity changes across the shock or
the expansion waves are also taken into account.
The use of a marching
type integration is important, because the computation can be diminished
to a single sweep and this reduces the computer storage and time.
The
SIMPLE method outlined by Patankar and Spalding has been used to take into
account the radial pressure gradient by using the y-momentum equation .
Hence, for the supersonic flow condition, it is possible to predict the
diamond shape of the pressure field for under-or over-expanded supersonic
jets.
5.2
MATHEMATICAL PROCEDURE OF THE SUPERSONIC CODE:
SIMPLE ALGORITHM
IMPLEMENTATION OF THE
The supersonic code was developed from the general mixing program of
Spalding and his co-workers.
Changes to this basic program were made in
the following areas:
- The V-momentum equation was added to take into account the radial
pressure gradients.
51
- Radiation and heat absorption terms were incorporated as the source
term in the energy equation.
- Two species equations have been introduced for the silane and argon
gases.
- Supersonic boundary condition were specified whenever the flow
speed exceeded the speed of sound.
- The code was modified so that all gas properties were updated for
the local flow pressure.
5.3
TRANSFORMATION OF THE GOVERNING EQUATIONS FOR THE HYPERBOLIC FLOW
By using the transformation given first by Von Mises, it is possible
to transform the boundary layer type equations into heat conduction type
equations.
function
Instead of using the
artesian coordinates x and y, the stream
will be used as one of the independent coordinates.
x, the
axial length in the problem, will remain as the second independent
variable.
The velocity components u and v can be defined in terms of the
stream function
as:
u
-1
rp
a
ar '
(5.3)
v -1
aT
rp ax
By defining two new variables
found from equation as:
and n, the partial derivatives can be
52
= x,
n =
,
an =rpv,
a,
1,
ax
(5.4)
an = rpu,
a
ar
ar
The partial derivatives
of u with
respect
= 0.
to x and r are obtained
as
follows:
au = au a
ax
a; ax
au
ar
5.3.1
an
an ax
au
a;
+ au
au a + au an
a ar
an ar
rv
au
a
au
0
+rpu
(5.5)
(5.6)
THE MOMENTUM EQUATION
Inserting these partial derivations into the x-momentum equation, we
obtain:
ax
5.3.2
a
(r2
TX ~
puP
a)
a IF
+ 1 (F
aP)
pu
x -x
(5.7)
THE ENERGY EQUATION
The general energy equation has a term defining the work due to shear
forces and a term due to the enthalpy change of the species.
The work due
to shear or kinetic heating is very small in subsonic flow conditions, but
becomes important in supersonic flow.
In order to remain operational in a
53
wide range of flow conditions, the kinetic heatiqg term has been retained
in the code.
By using
the definitions
Mf
->
at +
P
u
N
* vh = v-(rhvh) +
h,
(5.8)
where
Sh
= a;)+
at
ABS
+
RAD
+ v.(r
u2
h
v
+ v.[(rh C -
+
--
W )
s
(5.9)
r, Mz C)vT
(r -rh)h, vMX] ,
X
and the approximation,
Lewis
number
r Qt =rr
h
= 1,
(5.10)
=Pr
the energy equation can be written as:
ai
+PU->h
=
v.(rhh)+ SRAD + SABSORPTION
+ SKE+ v-{(rhC - r M C )T}.
(5.11)
54
Transforming into the x;
coordinates, the steady energy can be written
in its new form as
Bh ~(r2 ~PUrh
~
ax=)
i)
+ [("-rh)r2
pu
______h
(5.12)
(u/2)]
aM
+
{
f(CL-C)dT
+ (r~ - rh)H}r
pu
+ SRAD + SABS.
The thermal diffusion coefficient is written as
mass exchange
coefficient
is defined
as r
mass diffusion coefficient of the species
= pD
.
= Crh
[Kg/m.s]
[J/m.sK] and the
where
D
is the
The mass diffusion
coefficients for a multicomponent medium such as argon-silane or argonsilane-silicon-hydrogen are not tabulated in the open literature [1,2].
So the first task is to estimate the diffusion coefficients involved in
the problem.
An evaluation of the diffusion coefficients is given in
Equation (5.10).
With these assumptions we obtain
ax =- B ( r 2 purh ah) +
{(
-
rh)r2pu a(U/2)
+ SRAD + SABs.
(5.13)
55
5.3.3
CHEMICAL-SPECIES EQUATIONS
We have two more equations for the argon and silane mass fractions.
Argon is an inert gas so that the associated equation does not carry any
source terms.
The equation can be written on vector form as:
p u
p u
*
*
VMAr = - VJAr,
=
V MSiH
VJSiH
-
(5.14)
(5.15)
+ RSiH,
-221(KJ)
where RSiH
is a source term.
MSiH x 2x101 3x e
RSiH
44
4
T
Using the transformation of Von Mises, this equation is transformed to:
aMAr_
aAr=
ax
a
aMSil,
x
:
-
Ar
(r
2
u r
Ar
=aTv
(5.16)
1
) + uRSiH
SiH
(5.17)
aMsi,
2
(r2
Ar
u
4
DT
The equations for the silane has a "sink" term which assures the
silane depletion when local temperature
873.15°K.
is above the threshold temperature
This source term is set equal to zero whenever the local
temperature drops below the threshold temperature.
For this reason, in
the post reaction region the nonreacted silane concentations are higher
near mid-radius locations and decrease at the jet centerline and at the
56
cell walls.
silane
But it should be emphasized that the amount of unreacted
is minimal
and well
below
1% (of total
initial
silane
mass flux)
except at the mixing region of the two jets.
5.3.4
GENERAL FORM OF THE GOVERNING EQUATIONS
Except for the V-momentum equation which will be studied in detail in
the following paragraph, the governing equations derived above have the
following general form:
x +ax (a + b)
were x and
~as
aaw ( cCaw) +d,
are the independent variables.
X
(5.18)
is the normalized stream
function according to
-
-
YE -
TI
I
IFE: external streamline
I: internal streamline
(5.19)
Constants a,b are functions which are closely related to the mass flux.
Specifically a + b
is the amount of mass flow between the streamline and
the jet centerline divided by the total mass flow.
per unit increment of x.
These mass flows are
Hence, these constants are determined in terms
of the incoming mass fluxes,
a
-1
dI
Ei
' dx
(5.20)
57
b=-1
d
E-
(5.21)
-)I.
d-
This set of simultaneous equations were solved using the finite
difference method by reducing them into a set of algebraic equations, in
the form
Di
iD
= Ai i+1,D + Bii-
1
+
,D
(5.22)
Ci'
where
is the dependent variable thus forming a sparse matrix illustrated
below.
This matrix is solved by the method of Gaussian elimination giving
the desired value of the dependent variable
D2
-B3
2
-
A2t
- B4
:= C2+B2I
3
2 + D3 4b3
3
-
A3
1
= C3
4
+ D4 t4 - A4
= C4
5
- B4 + D5 5 - A6
- B6 i 5 + D6
-
5.4
:
B7
= C5
6
= C6
6
= C7
(5.23)
RADIAL PRESSURE GRADIENT CALCULATION FOR HIGH SPEED FLOWS:
All supersonic flows are hyperbolic in characters as the influence
from an upstream point can only affect points inside the Mach cone.
The
SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) have been used
in this research [23] in order to take into account the radial variation
of the pressure field, thus predicting the diamond pattern formed by
58
shocks and expansion waves.
The important operations in the order of
their execution are:
- The downstream pressure is guessed by imputing the upstream
pressure field as a first estimate.
- The x-momentum, the stagnation enthalpy and the equation of species
*
*
are solved, thus giving U , P field.
- The lateral velocity is adjusted by simple extrapolations along
streamlines and then the change in v from the lateral pressure variation
is computed using an explicit formula (i.e., the y-momentum equation
without the viscous terms)
6x
_
Vi+1/2,D
Vi-1/2, U
P.-Pi
__ __ __ _
_
1+
_
1
(5.24)
i+
(Pu)i+/2,D
~~i+l-r
i2)D
=
where "D" stands for Downstream,
"U" stands for Upstream.
- Having found the velocity field, the streamlines locations and
their slope can be evaluated.
This is used in the pressure correction
The derivation of the pressure correction
equation which is given below.
equation
is given in Appendix
,
I
A.*
-C~
,~~~ ~ }C
Ci+1/2
Pi+1-Ci-1/2
B,
Pi-
+
P
{Ci+1/2
+
Ci-1/2
1
+
p*
*
*
=A. -a. Pi
1
ai1
A.*
~--,i (5.25)
11
- * U*2
Pi Ui
P
59
- The pressure correction equation gives the,corrected velocity and
pressure
field,
U, V and P.
- The temperature and density field are then deduced from the
stagnation
enthalpy
and the ideal gas law.
- The newly found pressure value is treated a a new guessed pressure
*
P
for the downstream point and the above steps are repeated until the
whole solution domain is covered.
5.5
CHOCKING CONDITION AND FEASIBILITY OF THE SUPERSONIC RUN
The flow is assumed to be supersonic at the tip of the inlet nozzles.
Therefore, the speed of sound must be exceeded at some location upstream
of the nozzles.
The "throat" where the gas reaches M=1 is usually a flow
controlling valve and is referred as chocked.
This location can also be a
properly designed convergent-divergent nozzle.
In this paragraph, we will
calculate the mass flow rate which is required to chock this area.
The
pressure ratio between the supply tank and the chocked location will also
be evaluated.
The mass flow rate per unit is given as follows:
pu,
pu
:
P
) M (yRT) 1 / 2
(5.26)
(5.27
60
the stagnation pressure and temperatures are givqn below:
M2 )
(5.28)
( 21 ) M2)Y-1
(5.29)
Tt= T(1+ (-l)
Pt
P(1+
By using the definition for the speed of sound and the ideal gas law, we
evaluate the corresponding stagnation values for the sonic velocity and
the density:
at= (yRTt)l/2
(5.30)
where a t is the stagnation speed of sound based on stagnation temperature
Pt
Pt/RTt
(5.31)
.
By inserting (5.26), (5.28), (5.29), (5.30), (5.31) into (5.27), we
obtain
T
AA t
t
2
1t/
R 1/2
(-)Y
-
=
M
(1+ T-1M
M
(+1)
2 )
this is the flow parameter where
= total incoming mass flux of argon and silane
A = flow area
TtPt=
stagnation temperature and pressure respectively
(5.32)
61
R,y = the gas constant and the specifiq heat ratio respectively
M = the Mach number.
For a chocked nozzle, M=1.
If the chocking area is assumed having a
radius of 0.00051 m and if the inlet static temperature i
taken to be
800°K (i.e. silane is preheated before entering the laser beam) then the
amount
of gas flow
at 0.7 atm can be evaluated
from
(5.32) to be
0.000216 m3 /sec (13t/min) this value can be easily reached achieved in our
laboratory scale and a supersonic experimental run is feasible from that
point of view.
tank pressure P
The pressure ratio P /P
ol1 (i.e.
the ratio of the supply
to the pressure P1 just ahead of the chocked area)
establishes the pressure gradients which drive the flow.
important to evaluate it.
Hence it is
If the flow is assumed to flow from a large
reservoir then the initial Mach number is zero and at the chocked area
M=1, hence from
(5.29) we have:
¥
Po/P1
=
(1b 1)
-
'
= 1.89
M=1
(5.33)
This ratio increases considerably with an increasing mach number
Po/P1 = 7.82 for M=2.0,
P /P1= 64.33 for M=3.381 .
But these numbers are also feasible on a laboratory scale especially for
short run conditions on the order of minutes.
62
It should be emphasized that the pressure rise across the chocked
nozzle also increases (see Shock Tables), thus reducing the effective
pressure drop established by P/P
5.6
1
ratio, across the nozzle.
RESULTS AND DISCUSSION FOR THE SUPERSONIC REACTING FLOW
The results relating to the supersonic are given in Figures 56-67.
Figures 56 and 58 report the axial velocity and pressure distribution for
underexpanded and overexpanded nonreacting jets respectively.
for the underexpanded
jet [Fig.
56], was compared
for the same run conditions [29].
The result
to the results
The agreement is very good.
of Kurkov
By
uniquely changing the pressure values of the inner and outer flow, an
overexpanded inner jet was also simulated [Fig. 58].
By comparing with
the previous Figure 56, it becomes apparent that the axial position of the
expansion wave and the shock wave are virtually interchanged.
This is a
characteristic in the flow structure of supersonic jets which expand and
compress to finally accomodate to the pressure of the surrounding [30].
Figure 57, shows the radial distribution of the velocity and pressure for
the nonreacting coaxial jet confined in a cylinder.
between
a confined
jet and a free jet is schematically
The difference
drawn
in Figure
The corresponding computational results are given in Figures 60-62.
59.
As
depicted in the schematic diagram, the shock wave precede the second
expansion wave when confined in a cylinder.
But for the free jet their
order has been reversed (due to the wave reflection from the free jet
boundary).
At this stage, it should be emphasized that the boundary
layer-shock interaction cannot be accurately modeled by using the
computational "SIMPLE" procedure outlined above, because although the x-
63
momentum equation contains the viscous terms, the y-momentum equation does
not.
The latter equation is basically used for radial pressure
correction.
Hence, very little diffusion effect is present in the
solutions presented here.
When the laser is focused on the shock, the
pyrolysis of silane starts [Fig. 62].
The pressure map in Figure 63
reveals the diamond shape pattern of the pressure field.
The temperature
map shows the dimension and the shape of the reacting supersonic region
[Fig. 64].
The velocity field [Fig. 65] shows the expansion of the
reacting field into the argon field.
The last two figures related to the
supersonic jet [Fig. 66-67], illustrate the pressure, velocity, and
temperature field of a confined jet configuration.
By comparing these two
figures, it can be seen that if the laser is focused slightly ahead of the
shock, the flow structure is substantially affected because in the latter
case the reaction is completed to a greater extent and therefore, a
greater chemical expansion of the reacted flow has occured.
64
6.
ANALYSIS OF THE UNSTEADY REACTION FLAME
EXPERIMENTAL INVESIGATION
6.1
6.1.1
INTRODUCTION
During the laser driven silicon production runs, we encountered
oscillations in the reaction region.
Since vital chemical processes such
as nucleation and growth do occur within or close to the flame, the
instability of the reaction zone may have a direct consequence on the
final shape, size and properties of the silicon powder.
This chapter
presents the research done on the mechanisms of the flame instability.
The thorough understanding of the disturbances is the first and probably
the most important step toward eliminating if not, preventing the flame
flickering problem.
In this chapter, we will examine the different
aspects of the flame instability.
We will describe the phenomena
qualitatively and will determine quantitatively the parameters that relate
to this problem.
6.1.2
DESCRIPTION OF THE OSCILLATIONS
As the 180-Watt laser beam impinges horizontally on the silane jet, a
bright yellow flame develops in the reaction region.
The flame dimensions
(i.e., the maximum width and height of the luminous zone shown in
Figure 3) are usually bigger than the incident laser beam width.
The
unfocused laser beam diameter has been determined to be 6 mm and the
maximum
flame
diameter
to be on the order
of 1 cm.
The flame
brightnes
the result of the energy radiated by newly formed silicon particles.
is
The
65
details of the radiation mechanism and the structure of the reaction flame
were given in the previous sections.
It has been visually observed that
for low silane flow rates and high cell pressures, the intensity of the
reaction flame oscillates at a fixed frequency.
If the flow rate, the
laser power or the cell pressure are not reset, the flickering can be
sustained
indefinitively
in time.
as oscillations of the first type.
We will call
this mode of oscillations
This problem is not a transient
phenomenon, but a self-sustaining one.
The oscillations of the second
type are small amplitude disturbances which are spread over a wider range
of frequencies (0-220 Hz).
These oscillations are not periodic in time
and their amplitudes are small but varying, without any pronounced
relation to a specific frequency.
Qualitatively, the amplitude of these
oscillations are estimated to be less than a millimeter.
Two different
experimental set-ups have been used to investigate these two modes of
oscillations.
High speed movie pictures have been used for the
oscillations of the first type and hot wire anemometry, combined with a
He-Ne laser light detector used for the investigation of the oscillations
of the second type.
recorder.
Data was recorded in real time on a 3-pen chart
More details of the experimental apparatus will be given in
Section 6.1.5.
6.1.3
OSCILLATIONS OF THE FIRST TYPE
In this problem, we can distinguish two different time variables.
These are the reaction time and the travel time of a silane volume to
reach the reaction front.
following velocities:
Equivalently, we can think in terms of the
66
a) A local
burning
velocity
(i.e., the velocity
at which
the flame
is
propagating into the unreacted silane gas).
b) A mean flow velocity.
II
The ratio of these velocities are related to the well known Damkohler
dimensionless
Group
1 given as
r L
u C
chemical reaction rate
bulk mass flow rate
(6.1)
where:
u
= fluid
velocity,
L = characteristic length dimension,
r
= reaction
rate,
C = concentration
of reacting
species.
A classical explanation of the periodic flame-outs encountered during
experiments, is as follows.
When the absolute flame propagation speed is
bigger than the silane flow speed, the flame front propagates towards the
unreacted silane volume, consuming the unburned gas at a rate faster than
the supply rate which is, in our case, the silane mass flow rate.
After a
fixed period of time the reaction stops due to the lack of a silane rich
region.
The reaction will restart only when a finite amount of silane
volume has absorbed a finite amount of energy from the laser source to
overcome the activation energy.
The time lapse between two consecutive
67
flame-outs will be defined as the period of the sustained oscillations of
Type
1.
6.1.4
OSCILLATIONS OF THE SECOND TYPE
As it will be shown later in this thesis, the oscillations of the
first type [Fig. 70] only occur within a region which falls between
specified values of cell pressure and silane mass flow rate (Fig. 73).
The oscillations of the second type [Fig. 71] do not have this
characteristic.
During the experimental runs of 4-5 hours, the reaction
flame was not steady.
It was oscillating randomly.
The fluctuations in
the outer argon gas velocity were measured with a hot wire.
Because of
high temperatures in the flame (around 1400°K), the hot wire could not be
inserted into the silane gas stream.
instantaneous local
The probe was measuring the
argon velocity at the annulus exit.
It was assumed
that the disturbances causing the flame to oscillate would propagate in
the cell and the hot wire would detect any
amplitude disturbances.
high frequency, small
The upper operational limit of the signal
processor was around 400 Hz.
The hot wire sensitivity was such that it
was possible to measure low velocities as small as a fraction of a percent
of
he mean velocity value.
The amplitude o
the velocity oscillations
ranged between 15% to less than 1% of the mean velocity.
This variation
in the amplitude suggested that the cause of the flame fluctuation was not
unique, but rather a combined effect of the different disturbance sources.
We can distinguish between three major sources of disturbances.
as follow:
These are
68
1) Disturbances due to the geometry of the set-up.
The probable
causes of such disturbances are detailed below:
a) Excessive powder accumulation in the collecting filter, causing
backpressure waves to travel upstream in the cell and disturb the
reaction flame;
b) Disturbance caused by the proximity fo the controlling values to
the reaction zone;
c) Mismatch between pipes of different diameters causing an internal
flow separation and a mixing zone upstream of the inlet nozzles;
d) Presence of a natural frequency which would cause the overall flow
geometry
to act as a Helmholtz
or as an organ
pipe
resonator
[35].
2) Disturbances created and magnified by the flow:
a) Amplification of self excited disturbances due to an unstable
silane gas flow profile.
First, Rayleigh showed that for parallel
flow to be unstable, the velocity distribution must show an
inflection point.
Later (1935) Tollmien showed that this
condition is also sufficient for velocity distributions of general
types [31].
Since the axisymmetric silane jet profile has a
gaussian shape, the profile has an inflection point and for
sufficiently high Reynolds number, such a profile is unstable.
As
69
it is shown
amplified)
in [36], the Reynolds
number
(for which
can be as low as 4. for a 2 dimensional
instability
is
jet.
b) Formation of vortex rings generated by the silane nozzle tip with
an unsteady
supply
of silane
gas.
As these
vortices
travel
downstream, they brake-up and disturb the flow.
3) Disturbances due to the laser heat source:
Oscillations in the laser mode of the CO2 laser that acts as the
major heat source.
This might affect directly the reaction and the flame.
The nucleation time of silicon has been estimated to be much less than .01
seconds.
Also, the residence time of a particle (which is a scale for the
growth time of a silicon particle), is on the order of a millisecond.
Hence, a laser power fluctuation above 1000 Hz might affect each single
silicon particle characterisitics and any laser power fluctuations below
1000 Hz would affect the overall powder characteristics.
However, further
experimental work should be done on that area since silane absorbs only
less than 10% of the 180-Watt incident laser power; and although certain
mechanisms can be proposed, it is not clear how an incident laser power of
continuous mode, fluctuating between 180-140 Ywattswould affect a process
absorbing approximately 10 Watts.
have been highlighted above.
Most of the sources of oscillations
series
of experiments
have
been carried
out to find out which
actually occuring during the powder runs.
A
ones are
As it will be discussed in
paragraph 6.1.6, the relevant sources of disturbances were found to be the
ones given in 1-a, 2-a, and 3-a (Page 60-61).
Although the hot wire did
not show any Helmholtz type resonance of the cell, a sample calculation is
70
included in the Appendix C as a future reference to the scale-up process
of the silicon powder facility.
6.1.5
A.
THE EXPERIMENTAL SET-UP
HIGH SPEED MOVIE PICTURES OF OSCILLATIONS OF TYPE
.:
Figure 72 is the schematic drawing of the experimental set-up.
The
experimental goal was to take high speed movie pictures of the flickering
flame.
Once this was obtained, the phenomenon was "played-back" at a
slower speed, giving the opportunity to visually judge the problem.
By
comparing the flame front position with respect to the silane nozzle, we
have also deduced the flame speed.
The experimental set-up consisted of
the following apparatus:
a) A laser
source:
a Coherent
Everlase
model
150 CO2 laser
with a
maximum output of 180 Watts.
b) A high speed movie camera:
Hycam, Red Lake Lab, Inc., model:
k-
200- 4E-115, with a high speed rotating prism.
c) Two light meters:
The first one was a Honeywell-Pentax light
meter which was used to set the frame rate of the high speed
camera with respect to the available light source (i.e., the
reaction flame).
The second one was a silicon photodiode which
measured the intensity of the flame.
the oscilloscope.
It's output was recorded on
The response of the second detector was much
faster than the oscillation frequency.
71
A Tektronix 434 storage oscilloscope which was
d) An oscilloscope:
used to record the flame oscillations independent of the high
speed movie camera.
B.
EXPERIMENTAL
SET-UP
FOR OSCILLATIONS
OF TYPE
The schematic view is given in Figure 69.
2
The probe consisting of a
He-Ne laser of 5 milliwatts was directed on the reaction, barely touching
the base of the flame.
Since the flame contained the radiating silicon
particles, any motion of the flame in the vertical direction caused the
amount of transmitted and scattered laser probe light to change.
Hence, a
light detector collecting the transmitted laser light could follow flame
fluctuations.
The photodetector then produced an analog imput for the
oscilloscope.
Since the photodetector was a high response device, it did
not inhibit the flame oscillations.
The time history of the detector was
also recorded on a low response chart-recorder.
probe was inserted in the reaction cell.
In addition, a hot wire
We will briefly underline the
A very thin (.006" dia.)
operation principle of the hot wire probe.
Platinum-Irridium wire is inserted into the flow.
This wire is a small
resistance (12.47 Ohms) element which is heated and controlled at 320°C.
The wire is connected to a control circuit (a Wheat-Stone bridge) which
keeps the wire temperature constant by changing the amount of energy
supplied to the resistance.
This energy is dissipated by the cooling
Thus, the voltage output of the
effect of the fluid past the heated wire.
anemometer is related to the fluid flow as,
72
E2 R
E2 R
1
=
(A+B(pV)
/n)(t
(R + R 3 )
-t
s
),
(6.2)
e
where:
A,B = Constants depending on fluid and type of sensor,
p
= density
V
=
n
of gas or liquid,
velocity,
exponent,
ts
=
sensor operating temperature,
te
=
fluid temperature,
R3 = sensor operating resistance,
R
= resistor in series with the sensor,
E
= bridge
voltage.
Using hot wire calibration curves for a probe in the argon flow, we
determined the local, instantaneous velocity values in the cell.
The
components of the anemometer used is listed below:
a) A DISA type
55D05 signal
processor
with
1-1 bridge
ratio and a 16
Ohms external resistance,
b) A hot wire
resistance
probe TSI model
1220-PI
of 16 Ohms an internal
2.5 with
resistance
an operating
of 12.47
Ohms.
73
6.1.6
A.
EXPERIMENTAL RESULTS AND DISCUSSION
RESULTS FROM HIGH SPEED MOVIE PICTURES
A schematic of the high speed movie pictures is displayed in Figure
72.
The time interval between each frame in this diagram is T
1/25
=
seconds since the shutter speed of the camera was set at 25 frames per
second.
From the figure, it is possible to determine the frequency "T" of
the flame oscillations of Type 1.
This is calculated as follows:
flame intensity and axial location vary periodically with time.
The
The
period T of oscillations can be defined as the time lapse between two
consecutive bright flames.
We see that one cycle of flame oscillation is
roughly spread to 7 picture frames (including the picture frame with no
flame).
In other words, there are 7 frames between two consecutives
flame-outs.
In each photograph, the flame position (and intensity) is
displaced with respect to a fixed reference frame because T
T.
Since
every 7 pictures, the flame is at the same position, we can write the
following equality:
7(T-TS ) = T
>
7
T = ().(1/25)
= 47msec.
(6.3)
On the upper right corner of the same Figure 72, a typical analog
output of the light meter is displayed.
The light meter recorded the
intensity of the flame, independent of the high speed camera.
The
lightmeter has determined an oscillation period of 50 msec which is in
good agreement with the data from the camera.
In the film, the flame
74
seems to move upstream toward the nozzle.
propagates downstream.
Actually, the flame front
This optical biasing comes from the fact that the
shutter period (40 msec) is slightly lower than the flame oscillation
period (50 msec).
To determine the direction of propagation of the flame,
high speed pictures have been taken at 100 frames/sec and 150 frames/sec.
These movies revealed that the flame front is propagating downstream away
from the nozzle.
Based on the high speed movie, we have determined the
instantaneous position of the flame front with respect to the silane
nozzle.
The minimum distance between the laser heated zone and the silane
nozzle is approximately 3 mm.
From Figures 76 and 77, we see that the
flame front comes as close as 1.80 mm to the silane nozzle.
This is not
surprising since the chemical reaction will start at any silane rich
region having a minimum temperature of 873.15°K.
The temperature profile
of the region between the laser heated zone and the silane nozzle have
been analytically calculated.
The centerline (r=0, x-axial coordinate)
temperature in this region is given by the following formula:
pCpU
T(x,
0)
=
g(0)
(e
K
1) + T
(6.4)
p CpU
This formula is plotted on Figure 77.
This plot shows that temperatures
as high as 873.15°K can be reached after travelling 1.5 mm downstream of
the nozzle.
Hence, from this figure, we conclude that the flame front can
be as close as 1.50 mm to the nozzle.
This value agrees with the
experimental value of 1.80 mm reported above.
The calculations related
with Figure 77 have been carried out in detail in the Appendix A.
75 plots the height of the rection flame versus time.
Figure
The height of the
75
flame is defined as the maximum height of the bright area seen in the
movie pictures [Fig. 2].
It is interesting to note that Figure 75 and
This is expected since the
Figure 76 are out of phase by 180 degrees.
flame dimensions are minimum when it is very close to the nozzle, in a
region relatively "cold" where the reaction flame is "quenched".
interesting result is given in Figure 74.
(Fig. 76),
it is easy to plot the flame
combining Figures 74-76.
peculiar trend.
In this figure, the flame speed
Since the flame position is also plotted versus
is plotted versus time.
time
Another
speed
position
versus
by
The flame speed reported in Figure 74 has a
The flame front travels downstream with an initial
velocity of .30 m/sec.
Then the velocity is rapidly decreased to .07
m/sec within 6 milliseconds.
Within the next 6 milliseconds the velocity
increased back to .19 m/sec.
A quantitative explanation would be that the
local flame velocity and the silane gas velocity try to equilibrate each
It is interesting to note that the flame speed fluctuates around a
other.
The flame spped
value-close to .18 m/sec which is the silane flow speed.
has also been evaluated from the approximate solution presented in
Reference [32].
A value of .23 m/sec is obtained form the following
formula:
1
2ic
flame speed = Su = (p__)1/2
P
where
(
= adsorbed laser energy,
[
T -
bu1/2
Tb
fT
u
OU
PwhereCp(TbU)
1/2
,
(6.5)
76
Tb =
T
u
temperature
= temperature
Pu = density
of burned
gas,
of unburned
of unburned
gas,
gas.
This result was obtained by replacing the chemical energy source term
present in the original equation [32] by the energy density
the silane from the laser.
hw
absorbed by
The main reason for this replacement is that
the exothermic energy released by the chemical reaction is negligible
compared to the laser input.
The maximum absorbed laser energy is about
10 watts while the exothermic energy is on the order of 1 watt.
B.
RESULTS FROM HOT WIRE OUTPUT
The related experimental set-up has already been described in Section
6.1.5.
A schematic of the experimental laser driven reaction cell and the
analog circuit is presented in Figure 68.
The purpose of the hot wire
velocity measurements was to determine the cause of the small amplitude
flame fluctuations.
The experiment has been designed in such a way that,
for each run, one parameter was varied.
In the following paragraphs, we
present the major result:
a) The gas velocity at the inlet of the cell was very steady and
undisturbed:
If the silane or the argon gas supplies were not steady,
this would have directly affected the reaction flame.
The main valves
were placed upstream of the cell and this might have also introduced some
disturbances in the unreacted gas flow.
But this is a highly unlikely
event since the flow controllers and the valves are located 380 pipe
diameters upstream of the cell.
This distance is largely sufficient for
77
the disturbance to damp out and the velocity profiles to assume a smooth
parabolical shape.
A second cause for the flame disturbance was the argon
flow detachment from the annulus.
Thirty hours of hot wire histogram
related to a probe placed at the exit of the agron nozzle did not show any
signal of flow disturbances.
Hence, no flame disturbances are expected to
be caused by fluctuations of the argon and silane gas supplies.
b) The response of the adjusting valve was found to be too slow to
drive the flame disturbances:
Since the automatic control valve (valve on Figure 68) is the only
mechanical
piece
moving
during
the experiment,
it was
natural
to focus on
the flow disturbances caused by an adjusting valve downstream of the
reaction cell.
Half of the experiments were conducted with the automatic
valve and half of them with a manual valve which was kept fixed during the
runs.
When experimenting with the manual valve, the automatic valve was
paralysed by disconnecting it from the DC power supply.
Neither the
oscilloscope nor the chart recorder ever displayed any flow disturbance in
the range of 1-220 Hertz.
But it must be pointed out that over a period
of 80 minutes, the pressure history varied approximately like a sinusoid,
with a total amplitude of 0.3% of the cell pressure.
When the automatic
valve was on, the pressure change was barely noticeable, changing from
5.165 torr to 5.164 torr which corresponds to .02% of the cell pressure.
Since the fluctuations are further damped by the automatic valve, this
adjusting valve is less likely to induce flame oscillations.
Another
experiment which supported this idea was the evaluation of the
synchronized time response characteristics of the hot wire and of the
pressure transducer (Fig. 82).
Since the cell pressure is controlled by
78
the automatic valve, looking at the cell pressure history is an indirect
way to record the valve response.
pulse.
The disturbance was a Dirac delta type
Figure 82 shows that the pressure settling time is about 45
seconds for a high amplitude velocity disturbance.
But it should be noted
that the induced disturbance pulse was still long enough for the slow
response chart recorder to print the hot wire response.
Hence, none of
the pressure transducer responses could have been inhibited by the chart
recorder.
Based on these observations, we concluded that the automatic
valve has a slow response which cannot drive small amplitude disturbances
with high frequencies.
c) A filter almost filled with silicon powder does induce flame
oscillations:
The spikes shown in Figure 81 occurred more and more often toward the
end of a silicon production run.
fluctuations increased.
Also the amplitude of the velocity
In this velocity histogram, it is shown that
after 4 hours of experimental run, the disturbances occurred more
frequently and finally the back-pressures from the powder collecting
filter were so high that the reaction flame in the cell was put out
violently when 13 gr of silicon powder had accumulated in the filter.
Hence, the increase in the occurrence of disturbance spikes in the
velocity-time or pressure-time plots are mainly due to the filling of the
powder filter and the coupling effect of the automatic valve readjusting
to keep the cell pressure constant.
But this does not explain why the
flame oscillates at the beginning of a run.
given below.
The most probable cause is
79
d) Flame oscillations due to self excited flow disturbances:
Since
the gas flow entering
the reaction
cell has been shown
to be
smooth and steady, the oscillations of the flame can only be caused by
disturbances
that grow
in the cell.
This
mechanism
can be explained
if it
is assumed that the flame oscillation is caused by the instability of the
laminar
jet flow
[33].
The instability
of a 2 dimensional
coordinates) jet can occur as low as Re=4.
flow is on the range
signal
output
of 100.
is done.
This
A Fourier
is given
(in
artesian
The Reynolds number is our
analysis
in Figures
of the photodetector
78 and 79.
These figures
show that the small amplitude disturbances, also called oscillations of
Type 2, have a wide frequency range which spreads between 26 Hz and 216
Hz.
6.1.7
CONCLUSION FOR THE EXPERIMENTAL APPROACH
After comparing Figures 78 and 79, we have concluded that two types
of flame fluctuations can be distinguished:
Fixed frequency, large
amplitude oscillations called oscillations of Type
and small amplitude
oscillations with a broad frequency spectrum (0-225 Hz) called flame
oscillations of Type 2.
The Fourier analysis of the oscillations of the
first type is given in Figure 78.
and a narrow frequency range.
The oscillations have a large amplitude
The Fourier analysis of oscillations of the
second type is given in Figure 79.
The fluctuations have a small
amplitude and a wide frequency range.
The cause of the fluctuations of
the first type is linked to the propagation of the reaction front into the
silane rich low velocity volume and to the depletion of the unreacted gas.
The flame instability of the second type has been found to have two major
80
causes.
One is the amplification of the self excited disturbances because
the velocity profile is unstable (due to a presence of an inflection
point), and the other is the filling of the collecting filter.
mechanisms of flame stabilization exist.
Several
We will report three of them:
The reaction flame can be stabilized by:
- Using a bigger powder collecting filter;
- Creating a swirling annular argon flow [34];
- Increasing the argon velocity by a factor of 2 to 10 so that the
shear between the inner silane flow and the annular argon flow is
diminished.
The following parameters affect the instability:
a) The silane flow speed;
b) The laser intensity;
c) The cell pressure;
d) The position of the silane nozzle with respect to the laser beam:
The further away is the nozzle, the more probable is the flame
instability;
e) The diameter of the nozzle.
The bigger is the nozzle, the slower
is the silane gas, the more probable is the instability.
6.2
REACTION FLAME INSTABILITIES:
6.2.1
THEORETICAL APPROACH
ANALYSIS OF THE UNSTEADY FLAME
The stability of the flame was also studied as an eigenvalue problem.
The oscillations of Type 1 discussed in the previous section exhibit a
fixed characteristic frequency.
The unsteady nature of the reaction zone
81
is so strong
as the mass
flow rate
is decreased
for a given
cell
pressure) that it cannot be caused by a velocity profile which becomes
more unstable as the velocity increases.
Nevertheless, these velocity
profiles which have an inflection point, have been shown to be unstable by
Lord Rayleigh and are likely to induce flame oscillations of Type 2.
Because of these reasons, the influences of unstable temperature and
velocity profiles on a flame oscillation of Type 1 have been neglected.
The effects of these disturbances are assumed to be small compared to the
periodical unsteadiness of the reaction flame which is induced by the
chemical depletion of the silane volume.
6.2.2
SOLUTION OF THE UNSTEADY ENERGY EQUATION
The axial convection and conduction terms have been retained in the
energy equation along with the source term and the unsteady term.
equation
can be written
This
as:
2
+l
r) a 2T + q,
aT
+ aT=
+q
at
ax
' axa)T
a
where
qo
pp(6.7)
qe a
r
n
p
qo= absorbed energy density.
The source term has been linearized as
(6.6)
82
q
R
q
C T
(6.8)
(6.8)
ABS RAp
where
at
= axial length of the laser exposed region,
aABS = absorption
R
coefficient,
= thermodynamic
constant.
Since the equation is linear, the general solution of this problem
can be obtained with a separation of variables:
f(x)-g(t) [for the temperature].
We assume the form T =
Inserting into the energy equation given
above:
I
II
= K
g'(t)
g
f
f
(t)
(x)
(x)
+ C.
If the reaction is unsteady, the temperature
preheated
steady
region will
not increase
flow, but rather
it will
exponentially
be a contant
(6.9)
distribution in the
as
it is the case for the
TC close
temperature with an oscillatory time dependent part.
to the room
The temperature at
each location will be periodically increased as the reacting flame front
sweeps the region back and forth.
With this assumption the boundary
condition in the space domain can be written as
83
f(o)
= 0,
(6.10)
f(Q) = 0,
where
= distance between the nozzle and the laser beam lowest point,
TNew = Told - Tc.
and where the variable is written as :
By solving for
f(x) and applying the boundary conditions we obtain
K
n
= C
aj2fn
2
=
-(6
, where
n
(6.11)
= 0,1,2,.....
For a stable flame, the constant "K" in the exponential of the time
dependent part should decay or at least be equal to zero.
This will set
the stability condition on the velocity:
K
o
= C -
4a
<0
(6.12)
or
U > 2 (Ca)
1/2
(6.13)
By linearizing the source term given below, we obtained an expression
for the critical velocity beyond which the flame is relatively stable.
84
The results from this simple one dimensional analysis are plotted as a
solid curve in Figure 73.
The agreement between the analytical model and
the experimental points is good.
STABLE
The stability criterion is
FLOW:
U > 2
(aC)
1 /2
where
(6.14)
C
=
qo
0 aABS t-p
Further extension to the one dimensional analysis is nevertheless,
necessary because this simplified theory fails to predict any frequency of
oscillations.
85
7.
7.1
SYNTHESIS OF THE RESEARCH AND GENERAL CONCLUSIONS
GENERAL RESULTS
The velocity, temperature, pressure and concentration fields in a
powder cells reactor have been calculated, for the first time, for laser
driven reactions.
The main goal which has been achieved through this
thesis, is a better understanding of the heat, mass and momentum transfer
phenomena in a gas phase pyrolysis.
The results obtained can be grouped
into two main categories.
A) RESULTS OF THE STEADY REACTING FLOW
have been obtained from the following:
*Analytical
calculations,
*Computational
subsonic flow calculations,
* Computational supersonic flow calculations,
* Experimental measurements.
B) RESULTS OF THE UNSTEADY REACTING FLOW
have been obtained from the following:
* Experimental investigation,
*Analytical
calculations.
One of the major results obtained in this research is a relatively
important mixing between argon and silane (preheated region) and between
argon and hydrogen (reaction region).
The mixing starts in the preheated
region where 100% (by mass fraction) of silane is diluted to 50% (average
local mass fraction within 3 mm radius) when entering the laser heated
region.
The mixing is even more pronounced in the reaction zone where the
gas expands substantially.
From the engineering point of view, the mixing
86
in the preheated
zone
and in the very early stage
decomposition is not desirable.
particle
is by aggregation
of teh silane
Whether the growth of the silicon
of silicon
nuclei or by the growth
of a single
particle surrounded by silicon vapor, the argon dilution is unwanted at
this stage.
Intuitively, a large mixing between argon and silane before
the reaction would yield a small-size particle with a high number density.
On the other hand, the argon dilution immediately after the reaction, may
have beneficial effects.
Firstly, it would lower the gas temperature and
the newly formed silicon particles would be cooled by the surrounding gas.
This would reduce the "necking" of product particles which collide and
stick together.
Secondly, argon dilution would reduce the collision rate
between particles, futher improving the beneficial effect mentioned above.
Argon-silane mixing will be reduced if silane is injected at higher
velocities.
The idea is to reduce the radial spreading of the jet before
reaching the laser heated region.
Higher annular argon velocities will
also improve the situation by reducing the shear region between the two
coaxial jets.
if no argon
it is possible to completely eliminate argon-silane mixing
gas is injected
in the reaction
cell.
But this
is not an
attractive solution because the reactant gas will be in contact with the
walls and the problem of contamination will arise.
The solution is to
have turbulent jet flowing coaxially with a high velocity argon flow.
Radiation was determined to be of second importance in cooling the
reacting
mechanism
gas flow.
should
Still,
it should
not be overlooked.
be emphasized
Within
3 mm
that this heat
(in the vertical
transfer
x
direction), the temperature drop (due to radiation) is estimated to be
around 70°-50°C.
This drop in the reacted gas temperature is desirable
87
(to prevent silicon particle aggregation).
Thin ,flame configurations (2-
3 mm flame radius) favors such heat radiation so that from an engineering
point of view, thin reaction flames are recommended.
From
Figures
84 and 85 it can be seen that there
are still
substantial differences between the calculated (theoretical and
computational) results and the experimentally measured values.
This is
especially true for the axial temperature distribution in the preheated
region [Fig. 85]:
There is a better agreement between analytical and
computational results and less agreement between calculated and measured
values.
This suggests two things:
- Firstly, some heating mechanisms might have been overlooked during
the calculations (especially for the preheated region).
- Secondly, more experimental measurements should be done with a
better determination of the data location.
SUGGESTION FOR FUTURE WORK:
- The computational code should be extended to turbulent reacting
jets.
- A better source model for laser absorption and heat radiation
should be incorporated.
- Argon should be introduced from two axial locations:
a) a small mass
flux
of argon
from the inlet nozzles.
should
be introduced
at a high velocity
Thus preventing the spreading of the
silane jet.
b) A large argon mass flux should be introduced at a low velocity il
the post reaction zone to dilute the reacted gas.
88
7.2
ORIGINALITY OF THE RESEARCH AND CONTRIBUTIONS
Before this research was initiated, several questions remained
unanswered.
The most fundamental ones are listed below:
- What is the velocity distribution, especially in and around the
reaction zone?
- What is the effect of heat conduction?
- By how much does the temperature drop due to heat radiation?
- What is the effect argon-silane jet mixing?
- What are the main parameters causing the reaction flame
flickering?
All these questions have been answered quantitatively for different
run conditions.
The results have been tested by widely varying parameters
such as the pressure, the silane mass flow rate and the incident laser
profile.
The agreements between the experiments and the calculations were
good despite the complexity of the problem which involves reaction, laser
absorption, gas mixing, radiation, and chemical gas expansion.
Enumerated
here are some of the contributions made by this work to laser driven
pyrolysis of mixing jets:
1) An analytical closed form solution has been developed for laser
driven reacting jets.
It was shown that, the velocity and the temperature
depend on specific groups of variables such as
Ur 2
q0
Gr1/ 2 n 2
;_8 '
4ax
(7.1)
UK
2) An elliptic computational model revealed the presence and the
89
strength
of a recirculation
region in the powder
cell
reactor.
It has
been determined that the recirculation had no major effects on the flame
shape.
Silane and argon concentrations were found to be slightly affected
especially near the nozzle inlets.
3) Argon-silane
mixing
reaction flame temperature.
was found
to be very important
in lowering
the
Radiation was shown to play a secondary
role.
4) A new velocity measuring technique was developed illustrating the
different time scales present in the problem.
The advantages of the new
technique over conventional velocimeters where also shown.
5) A flame stability criterion has been defined beyond which the
reaction flame does not flicker.
The overall phenomenon has also been
II
described in terms of the Dahmkohler group I parameters.
Major sources of
the flame instability have been indentified.
6) The feasibility of a supersonic laser driven pyrolysis run is
demonstrated by developing an appropriate computer code.
Several aspects
of the laser beam-shock wave interactions have been investigated revealing
the importance of the choice of the axial location of the laser flow.
7.3
GENERAL CONCLUSION
The following conclusions have been achieved from the research
results:
- The axial conduction is the main heat transfer mechanism i
preheated region.
the
90
- The velocities in the reaction zone decrease sharply because of the
combined effect of heated gas expansion and the production of 2 moles of
hydrogen gas for each mole of silane gas.
- The argon mixing in the reaction zone causes a temperature drop of
150°-100°K within 3-4 mm.
- The radiation plays a slightly less important role by lowering the
peak temperature by 10°-70'K within 3-4 mm.
- The effects of radiation and mixing are more pronounced for high
pressure low velocity flows.
- In the supersonic flow, it is possible to start a reaction if the
laser is focused on the shock without significantly altering the basic
flow structure.
- If the laser is focused slightly beforee the shock there is a
substantial change in the flow structure.
- For the subsonic laser driven flows, there is a critical velocity
beyond which the reaction flame oscillates indefinitely at a fixed
frequency.
91
8.
REFERENCES
[1] Haggerty, J.S.
"Growth of Precisely Controlled Powders from Laser Heated Gases"
Proceedings of the International Conference of Ultrastructure.
Processing of Ceramics, Glasses and Composites. February 13-17, 1983
Gainsville.
[2] Flint,
J.H.
"Powder Temperatures in Laser Driven Reactions" M.S. Thesis M.i.T. pp
22-27 (1982).
[3] Haggerty,
J.S.
and Cannon
W.R.
"Sinterable Powders from Laser Driven Reactions".
Report MIT-EL 79-047 pp 24-28 (July 1979).
[4] Chinitz,
Energy Laboratory
W.
"Theoretical Studies of the Ignition and Combusion of Silane-HydrogenAir Mixtures" NASA CR 3876 (February 1985).
[5] Edelman, R.B., and Harscha, P.T.
"Some Observations on Turbulent Mixing With Chemcial Reactions"
Turbulent Comutsion, Lawrence A. Kennedy, ed., AIAA Progres in
Astronautics
[6] Arthur,
and Aeronautics,
N.L., and Bell,
vol. 58 pp. 55-102
(1978).
T.N.
"An Evaluation of the Kinetic Data for Hydrogen Abstraction from Silanes
in the Gas Phase"
Rev. of Chem.
Intermediates
vol. 2 pp. 37-74
(1978).
[7] Coltrin, M.E., Kee, R.J., and Miller, J.A.
"A Mathermatical Model of the Coupled Fluid Mechanics and Chemical
Kinetics in a Chemical Vapor Depostion Reactor", J. Electrochem. Soc.,
Solid State Science and Technology (February 1984).
[8] Marra,
R.A.
"Homogeneous Nucleation and Growth of Silicon Powder from Laser.
Gas Phase Reactants" Ph.D. Thesis M.I.T. (February 1983).
Heated
[9] Burke, S.P., and Schumann, T.E.W.
"Diffusion Flames" First Symposium on Combustion 2-11, Swampscott, MA
(1928). (Reprint of Proceedings published by The Combustion Institute
in 1965).
[10] Hottel,
H.C.,
and Hawthorne,
H.R.
"Diffusion in Laminar Flame Jets" Third Symposium on Combustion and
Flame and Exposion Phenomena, pp 254-266. The Williams and Wilkins
Company, Beltimore, Maryland, (1949).
[11] Wohl,
K., Gazley,
C., and Kapp,
N.
"Diffusion Flames" Third Symposium on Combustion and Flame and Explosion
Phenomena, pp 288-300. The Williams and Wilkins Company, Baltimore,
Maryland,
(1949).
92
[12] Shvab, V.A.
"Relationship Between the Temperature and the Velocity Field of a
Gaseous Flame" Journal of Technical Physics 11, 5 pp 431-442 (1941).
[13] Zeldovich, Ya.B
"On the Theory of Initially Unmixed Gases" Journal of Technical Physics
19, 10 pp 1199-1210
N.A.C.A.
(transl.
TN 1296,
(1951).
[14] Clarke, J.F.
"The Laminar Diffusion Flame in Oseen Flow: The Stoichiometric BurkeSchumann Flame and Frozen Flow" Proc. Roy. Soc. A296, pp 519-545
(1967).
[15] Klajn, M.,
"Influence
Nineteenth
Combustion
Oppenheim, A.K.
of the Exhothermicity on the Shape of a Diffusion Flame"
International Symposium on Combustion, Published by The
Institute pp 223-235 (1982).
[16] Squire, H.B.
"The Round Laminar Jet" Quar. Jour. Mech. and Appl. Math. pp 321-329
(1950).
[17] Pai, S.I.
"Fluid Dynamics
of Jets"
pp. 75-95.
Nostrand
(1954).
[18] Pai, S.I.
"Axially Symmetrical Jet Mixing of a Compressible Fluid" Quar. Appl.
Math.
10, No. 2, pp 141-148
(July 1952).
[19] T.S.I. Thermal System Inc. Laser Velocity Systems pp. 96-109 (1982).
I.S.
[20] Akmandor,
"Laser Doppler Velocimetry: Measurements in Plane Poiseuille Flow" M.S.
pp 39-43 Thesis M.I.T. (1982).
[21] Smith, J.M.
"Chemical Engineering Kinetics" pp 364 Second Edition Tosko printing
Co., Tokyo-Japan.
[22] Cussler,
E.L.
Diffusion; Mass Transfer in Fluid Systems
[23] Patankar,
Suhas.
V.
Numerical Heat Transfer and Fluid Flow. Series in Computational Methods
in Mechanics and Thermal Sciences, pp 126-131 (1980).
[24] Pun, W.M.,
and Spalding,
Brian
D.
A General Computer Program For Two-Dimensional Elliptic Flows HTS/76/2
Imperial College of Science and Technology (1976).
[25] Heywood, J.C.
On the Stationary Solutions of the Navier-Stokes Equations as Limits of
Nonstationary Solutions. Archive for Rational Mechanics and Analysis,
Vol 37, No. 1, pp. 48-60.
93
[26] Roache, P.J.
Computational Fluid Dynamics pp. 106-107 Hermosa Publishers (1976).
[27] Textor, R.E.
A Numerical Investigation of a Confined Vortex Problem Rep. No K-1732,
Union Carbide Corporation, Computing Technology Center, Oak Ridge,
Tennessee (1968).
[28] Tejeira,
E.J.
Numerical and Experimental Investigation of a Two-Dimenational Laminar
Flow With Non-Regular Boundaries. Rep. EM-66-8-1, Department of
Engineering Mechanics, The University of Tennessee, Knoxville,
Tennessee, (August 1966).
[29] Kurkov, P.A.
Mixing o Supersonic Jets Incluidng the Effects of the Transverse
Pressure Gradient Using Different Methods, Lewis Research Center,
Cleveland, Ohio NASA TN D-6592 (December 1971).
[30] Kerrebrock, J.L.
Aircraft Engines and Gas Turbines, pp. 95 M.I.T. Press (1977).
[31] Lin, C.C.
The Theory of Hydrodynamic Stability, pp. 47, Cambridge University Press
(1966)
[32] Toong, Tau-Yi
Combustion Dynamics, The ynamics of Chemically Reacting Fluids", pp.
107-144, Mc-Graw Hill (1983).
[33] Itsuro, Kimura
Stability of Laminar Jet Flames, 10th Combustion Symposium, pp. 12951300
[34] Gupta
(1965).
and al.
Swirl Flows, pp. 103-117, Paragraph 2.7, Swirl Stabilized Flames
(1984).
[35] Braddick
Vibrations, Waves and Diffractions
[36] Kaplan, R.E.
The Stability of Laminar Incompressible Boundary Layers in The Presence
of Compliant Boundaries, M.I.T. ASRL TR 116-1 pp. 147.
[37] Svehla, R.A.
Estimated Viscosities and Thermal Conductivities of Gases at High
Temperatures, pp. 31, 74, 106-107 NASA TR-132 (1962).
[38] Harlow, F.H and Fromm, J.E.
Computer Experiments in Fluid Dynamics, Scientific American, Vol. 212
No.3
(1965).
94
[39] Bush, H.R.
Prediction of Complex, Viscous, Compressible, nternal Flows Using
Implicit Finite Difference Methods, pp. 80, Ph.D. thesis, M.I.T.
(1983).
[40] Macagno, E.0.
Some New Aspects of Similarity in Hydraulics, La Houille Blanche,
Vol.
20, No. 8, pp. 751-759.
[41] Jennions,
I.K.,
MA
A.S.C.,
Spalding
D.B.
A Prediction Procedure for 2-D, Steady, Supersonic Flows, HTS/77/24,
Imperial College of Science and Technology, pp. 5-13 (1977).
95
9.
APPENDIX
1.
DERIVATION OF THE TEMPERATURE PROFILE OF THE PREHEATED REGION:
A
The calculations in this appendix are done by using separation of
variables.
Let
T(x,r) = f(x)-g(r)
(A.1)
By inserting this expression into the energy Equations (2.8) and by using
the following boundary conditions (2.10), we obtain the following
expression.
PCU
T(x, o) -
g(o
pCp U
)
T(ePw
(A.2)
a
for small pCp
U
x, the exponential factor may be expanded and we recover
the axial part of the temperature solution (2.11).
2.
INTEGRATION OF THE MOMENTUM EQUATION FOR REGION 2:
In the imcompressible domain, the momentum equation is
u a
+ v
r ar r
)
+
Tg
(A.3)
96
By using
= vxg(n),
(A.4)
and
r
x
we obtain,
-_
U -=T)T
-
_ (l)2
n
=
g
_
+
+
'n
Gr
(A.5)
Multiplying by n and integrating once, we have
2
(A.6)
2
'n
n
by multiplying by n and integrating another time:
-Ug
-2
g
+
g92 = _ Gr
n
(A.7)
.
This is a Riccati equation, which can be transformed in the following
form:
x2 y" - x y'
+-
16
X4 y = 0
where
and
x
(A.8)
= n
y
g(n)
This ordinary differential equation can be solved either by the Frobenius
method (thus leading to the expression given in (2.18) or, it can be
97
directly inegrated.
It should be remarked that the above equation with
constant Grashof number is nothing more than a Bessel equation which solution
is given below:
y
=
x [C1 J1/2
Grl/2xZ2
8
x )
+2
+ C2 J 1
Equation A.9 can be expanded to yield exactly (2.18)
for
Gr/2
8
x
<
.175
j(Gr1
( Gr 1/22
(A.9)
98
APPENDIX
B
DERIVATION OF THE PRESSURE-CORRECTION EQUATION:[41]
The equation is formed in three steps.
A. relationship is required
between the momentum area and its change with pressure.
continuity area and its change with pressure.
Also between the
Finally, the requirement
that the downstream cell areas have to be equal irrespective of derivation
has to be met.
Let the downstream
quantities,
u, v, p, p, A, a, and r be
composed of two parts, representing the part due to the guessed presure
and the part due to the pressure correction:
P
+
(B.1)
*
u
=
u
+
u
(B.2)
v =
v
+
v'
(B.3)
p
p
+
p'
(B.4)
A = A
+ A'
(B.5)
a =
a
+
a'
(B.6)
r =
r
+
r'
(B.7)
The compatibility of cell areas is expressed as:
A
+ A' = a
+ a'
It is now required to find the equations for a' and A'.
above values into truncated y-momentum equation gives;
(B.8)
Substituting the
99
v'
ii -
)
pu
'+/2
From Figure
X
=(.
i+'1/2
p
i+l
i+.
1+ -r
r
1.B the area a i,D can be calculated
ai,D
the inclination
ai,U + (ri+1/2 a+1/2
angle
of the streamline
(B.9)
i
from:
- ri1/
(B.10)
ai-1/2 ) 6x
(a) being
given
by:
- /)
+1/
2i+/
where
a,
r,
" is the mass flow rate across constant w lines.
u,
v,
v'
from
equations
B.11,
7, 2,
3,
9 are
(B.11)
If the values of
substituted
into
equation B.10, the equation thus formed is:
a = Ci+l/2 (P'i- Pi+1)
(6X2/6r*+
where C+1/
-
C+-1/2
(B.12)
pgi,)
euto
u*
has the value ri+1/ (6x) 2 /(6rPi+/Ui+l/2).
The A,
equation
is formed by considering the mass flow rate throug a downstream cell
face:
piuiAi
= (E
-
'I) (i+l/2
-
i-1/2
(B.13)
*
=
*
u
*
A.
Substituting
+
of p, u, A from equations
the values
p'
p
U'
*
i
100
A.'
.-
+
u
.
B.2, 4, 5, we obtain:
(B.14)
=0
A.
~~~1
1
where the terms containing the cross products p'u', p'A',
The variations of p and u with pressure are now
have been neglected.
needed
'A', p'u'A',
for substitution
into Equation
(B.14).
These
can be obtained
by
assuming pressure adjustments to be isentropic, and from Euler's equation.
They
give:
_ Pi
Y
u'.
(B.15)
i
1
pm=
P
i
1
-
Pi
(B.16)
p
Ui
Substitution of Equations B.15 and B.16 into B.15 gives:
A.i
__
(
1 A
YPi
) Ai
A1
*
Pi
Pi
pj'
~~~~~~~~~~(B.17)
(B.17)
ui
Finally, combining Equations (B.8, 12) and (B.17) and the pressurecorrection equation is obtained:
101
*
-Ci+l/
2 P+l - C1/2
Pi-
+
i+l/
C _l/2
*
* ,}P
i
= A.*
1
(B.18)
- a.*
1
BOUNDARY CONDITIONS
For a supersonic boundary, consider the effect of a pressure wave of
angle
V
Utot,
striking the boundary, as in the following sketch:
I
U2
Figure 2.B
6x
.w= constant
streamlines
rL
area ai,D = Ai,
x
ure 1.B
LI-
- 6x
-
-*
D
102
The truncated lateral momentum equation gives:
av
1
ap
ax
pu
ar
(B.19)
Applying this to the above figure gives:
p1 u1 6r(v1 - v 2 )
=
(P1
- P 2 ) 6x
(B.20)
From geometry:
ar
ax
(+a)
= tan
- tan a
(B.21)
and from the definitions:
1
-sin
(1/
8
)
(B.22)
2 + v1 2 ) / (yp 1/pl)
1
(B.23)
tot,1
M2
1(u
tot'il:
I
the v boundary condition is derived by substituting Equations (B.23) and
(B.22) into Equation (B.20):
) + a]
Vl-V2 = (-P 2) / {Pl U1(tan[sin
-1( /
tot, 1
(B.24)
-
tan a) }
Both u and p are taken, for this boundary condition, to be the same as
they are at the near boundary node.
103
APPENDIX
C
FREQUENCY OF THE HELMHOLTZ RESONATOR
The resonance occurs in geometries similar to the one shown below in
Figure 1.C.
The flow in the tube is adiabatic and the flow is slow so
that the problem can be considered as isentropic.
P VY = const.
(C.1)
By applying the logaritmic derivative, we obtain,
dV,
V
dP
dV
P
V
(C.2)
I
is the change in volume due to the pressure change.
For a constant
cross-sectional area and a small displacement x, which is the relative
change
in volume,
dV
(C.2) can be expressed
A x
V
V
'
as:
(C.3)
Hence,
dP
P
A x
Y
V
(C.4)
104
the pressure force acting on the volume V is given as
dF = dp
and the equation
A,
of motion
m
for the volume
of gas in the tube
is,
= dF
-dx
dt2
pAL
dXZ
P
= -
A
(C.5)
rearranging the last equation (C.5), we obtain,
x
+
+
p
PA
=
(C.6)
= 0
whose characteristical natural frequency or Helmholtz frequency is:
(1/2 '
A
1/2
= a ( A )1/2
(C.7)
where
a is the speed of sound,
A is the tube
area,
V is the discharge volume,
A
A
V
L is the tube length,
A
P is the cell pressure,
If
I
p is the gas density.
L
Figure
1.C
105
10.
BIOGRAPHY
Ibrahim Sinan Akmandor is the son of Dr. Neset-Ayten Akmandor.
was born in Ankara-Turkey
and he is from Bolu,
a forest
covered
He
country
beside the Black Sea.
I. Sinan Akmandor has graduated Summa Cum-laude and/or with honors
from the following academical institutions:
1.
Lyc~e Edmont-Rostant, France (June 1976)
2.
Bosphorus University, Istanbul, Turkey, (Mechanical Eng. and
Math)
(June
1980)
Also known as "Koc" (pronounced "coach"), he published "Yaprak-The
Voice of Turkish Students" between 1983-1984.
I. S. Akmandor will be teaching Aerodynamics and Fluid Mechanics in
Universities in Turkey.
106
Numerical
model
Atua
Process
Experimental
Theoretical
data
model
FI
G
1
: APPRCH TO THEPROBLEM
107
FIG
2
The flame shape
photograph at 0.2Atm
38 cc/min SiH4
108
x
!_r
ume
flame
REGION 3
J Laser
REGION2
REGION
I
nozzle 0.621 dia.
<Silane
.I
Argon nozzle
19.7 dia.
FIG
3:
GEOMETRY
OF THE
PROBLEM AND DEFINITION
SUB-REGIONS.
-
OF
109
symmetric velocity
profile
I
;quire solution platched
inflecti
to the Poiseuille profile
at the inflection point
po
iseuille profile
REGIO1
In Region , the Jet Preserves Its Initial Profile (the parabolic profile,
also called Poiseu.lle profile)
FIG
4
OF 2 VELOCITYPROFILES
PATCHING
IN A NON-SIMILARREGION
110
-
A
umum
U
c,; J
0
('tJ
r
'
° ,~.IT
'ID
. . Cd
m
'@.
_l-CD
w
-Z
0
-Q
u.0
-j
-a;
0:
,x
- 0
-J
_D
LD
N
CD
r-
re)
S/IV
cJ
AIDOIJA
3NI1931N30
v
LO
,kl:13A3
-
0
n
v
e
C- b
,fie Cf
g e°
F- W
0
e
U-
L11
8
E
In
l
l
U
0
J
-3o
UJ
3
O
RADIUS (METER
(METER) YOx
R<ADIUS
c
O---
PROFILES
THEORETIC AL VELOCITY OCATION S
T AXIAL L
AT DIFFEREN
FIG 6
: PADIAL VELOCITY
PROFILES
RU CONDITINS:
unfocused, Gaussian profile
Laser power: 180 H
cc/in
Argon flow rate :1000
c/min
Silane flow rate: 38
Atm.
0.2
Cell pressure:
112
c'J
N.~
N
°
IZ
ID-
II
X
Ix
E <
tE> lL.
111u c
Z )
F <
-LJ
LAJ
J E
0 < Lj
X~ F
z.
C)
N
0
(
(D
'
,
0
C
(0
O
,
NJ
No 38nliV83dI31 3N1831N30
Li
0r
CMZ
I-
113
1600
A
A
__
I
I
....
1400
..
X;O.O
X=0.0
X = 0.0065
-X-0.0()6
v
I
o
^
II 9n
-W W
o
1000
ILL
°. I
800
Q_
CLl
il
600
400
200
0
I
I
_
O
0.1
RADIUS (METER)
0.2
0.3
x IO-2
THEORETICAL TEMPERATURE PROFILES
AT DIFFERENT AXIAL LOCATIONS
SILANE MASS FLUX 38cc/min
RADI AL TEMPERATURE
IG PROFILES
RUNC0NDITIOaNS:
Laser power: 180 W unfocused, Gaussian profile
Argon flow rate :1000 cc/min
Silane flow rate: 38 cc/min
Cell pressure: 0.2 Atm.
114
!
0,
L
-J
E
0I--
LL
AD
cni
0n
_I-
I
__
(WW )X
o
UwL
z00I
u
115
0
'::"1.::.:.
U
: LA:
a.)
en
or)
E
I
4I
/
:1
.1
' .:['
I ' ...S'.
'- ':1
-I'
.C: 2
-
a)
-
=
a1)
I
U
63
0
0
:
I
Computed points
a:.B'..''.'..
E...
I.
1-\
a)--3
0
I
I
I.:.::::
I
,
'"
:.,:.'. :
I,', '.-:
_
_i
IB..'''."
1:' '-..: ::1
r'-i.'::'--1--'
0.005
0.010 0.015
I 'T----!e
0.020 0.025
Axial distance (meters)
FIG 10
THEORETICAL
AXIALDISTRIBUTION
OFVELOCITY
RLNCI!ITIONS:
Laserpower:18014unfocused,
Tophat profile
Argon flow rate :1000c/min
Silaneflow rate: 38cc/mAin
Cell pressure:0.2 Atm.
Nozzlediameter:148 m
116
,."#%
A
--
0
a)
-W
0,
00
0.005
0
0.010 0.015 0.020 0.025
Axial distance (meters)
FiG
11
THEORETICAL
D EXPERIMENTAL
OFTEMPERATURE
DISTRIBUTION
RUNCONDITIONS:
Tophatprofile
180Wunfocused,
Laser
power:
cc/min
Argonflowrate:1000
38cc/min
Silane
flowrate:
Cellpressure:
0.2Atm.
Nozzle
diameter:
1.48
mm
117
I
U
a)
U)
0
U
0
(0
1. _
0
0
0.1
0.2
Radial distance (meterss)
FIG 12
RADIALELOCITY
PROFILES
FROM MLYTICAL
RESULTS
RU CNDITIONS:
Laserper: 180Wunfocused,
Tophat profile
ArSon flow rate :1000ec/min
Silane flow rate: 38 c/min
Cell pressure:0.2 Atm.
Nozzle diameter:1.48 g
0.3
x o10- 2
118
40-
V
(
C(
-4
20
=v0
suvo
v.vv
X106
U')
Rodiol distance (meters
°.U~
FRKf*LYTImL
PROFILES
TEMPERATUE
RADIAL
FIG
13
Tophat pofiles
hat ofile
Top
unfocusd,
Laser power:180W
cc/mir,
:1000
Arfon flow rateSilane flowrate 38 cc/min
Cell pressure: 0.2 Ate.
dianeter: 1.48 m
Nozzle
R4 cMDITSd
119
OPTICAL
DETECTOR
OSCILLOSCOPE
I
CO2
WATER COOLED
150W
LASER
BEAM
CHOPPER
SOURCE
DISTURBS
THE LASER
LUUlt.
BEAM STOPPER
BEAM
/
SPLITTER
at
/
|
*RMIRROR
He -Ne
0.005
W
LASER SOURCE
EXPERIMENTAL
FIG 14
SET- UP
TOP VIEW
D(PERIMENTAL
SETUPFORVELOCITY
MEASURBETS
BYAPERTUTION
METHOD
120
n *
*
cr
3
i
;
w
L
_
>~
0
en
L~
Z
ONI
n-
Xr
H~z L
Lii
T
I
I
A_
Lmv
tz Kt
~~~~~~~
v oe
Q.,
< z
-a
_9
-t)
L
121
UNSTEADY FLAME
STEADY
FLAME WITH
A PERTURBATION
X= 4.64
MM
X= 6.59 MM
X= 7.59 MM
X= 8.59
MM
X= 9,59 MM
X= 11.09 MM
X= 12.09 MM
X= 14.09
MM
X= 16.09 MM
X= 19.09 MM
FIG 16
SINALSDISPLAYED
BYTHEPHOTODETECTORS:
PROP1ATION
OFTHEDISTUR/CE
FRONT
AT
DIFFERENT
AXIAL
LOCATIONS
122
I
i
I
I
I
I
I
.
FIG 17
THEBASIC
STRUCTURE
OFTHECMtPUATIAL CODES
123
FLOW CHART OF THE ELLIPTICAL
ALGORITHM:
READ DATA
GRID
[_CALCULATE
INITIAL GUESS AND BOUNDARY
CONDITIONS
. ADVANCE:CALCULATE UV)H,
CONCENTRATIONS
TEMPERATUREj
COMPLETE:CALCULATE
DENSITY, VISCOSITY
oRRECT
ADJUST:
FOR CELL-WISE
CONTINUITY AND
OVERALL CONTINUITY.
P
PRESSURE
v DETERMINE
__j7,,
DECIDE:IS
PRN
ALL DOMAIN
COVERED
?
CONVERGENCE
PRINT OUTPUT
FIG 18
124
FIG 19
THE GRID AND THE FORWARD STEP
FOR THE ELLIPTICAL ALGORITHM.
x
AT START u'S
AND P2'S ARE "KNOWN".
SOLVE MOMENTUM EQUATIONS FOR V2'S
GUESS P4'S AND SOLVE FOR u 3 'S
CHECK OVERALL CONTINUITY AT
.
3,
MAKE UNIFORM ADJUSTMENT TO u3 'S, FOR
OVERALL CONTINUITY.
0
MAKE UNIFORM ADJUSTMENT TO
.
TO
P4'S,
*
SATISFY OVERALL MOMENTUM,
CHECK CELL-WISE CONTINUITY AT
ADJLST
p 2 'S
1
3
s
,
VIA PFESSUFE CORFECTION EQUATIONS TO
SATISFY CELL-WISECNTINUITY.
u
2
APPRPRIATELY. ME
AND TO u3'S,
FOR OvERL
.
.
ADJUST-u1 's, v's,
UNIFORMADJUS
S TO
u
1
's
1
2
CONTINUITY. ITERATE AS NECESARY.
3
4 5
125
,.t
CQC
.4
o*
',.
,
,
CD
N
C
O
ED
z 3q--sS', @-_r. .
--
o
1_
|_
CS''
_
_
O@C
D
0[
~
~~
C
~
~
7
,XQ CD
C
sb..~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
W
_
._
7-
W
i
gdi -- .
CD
CD -
q
II1
u_
C.D
LL
4
CDI
J
LUI
I
LL
4
__J
LU
I
.H
41
_J
LU
4
ri
>
&I
E
I
LjJ
LO
I
!
iI
'OI
iI
I
I
.
-I
I,
:
::
iI --
CD
CD
m.
~3
L,)
<
CD
L)
U
126
el
._
a
o
C
ao .u
I ._
-'o
0
(
,-4
U5
tS
.
; .5 .1 ;;
a - tu
3' -I.CD0:3
n
C Z)
-, *1 ,
0
a o
.
"
a,
CD 2
*
a
U
CD
(DO
-..
-
FI
.A
(lo
I.-
-
3-
Wa
w
o
t
V. Ia
.o
.
C
'-1
Ft m Z
CD
.NgU.
-_J
-J
2
Lii
11-
_
LL
mr-lj
>-
LLi
lil
0
-LJ
LU
F-
2
c~
Qo
0
<}
CDJ
Ln
09
127
i
t_~~~~~~C
L
%
Lh
Ih
I
14
II
-- Lb
CD x0 --
X
L-n
rAi _
L~~~~~~~~O
C
CD
N
C,
C)
._
a:
.3_ tS =
I
.LL
CE
_J
LUJ
LiL
PNJ
U-'
llJ
,.,
Z
CE
I
0
t
I
i
. - -
CD
I
i
I
i
0
- '
....
-
-
i
C
O
CD
0
I
"~~~
I
0C
i
I
<--
C
~~~H
0
~
CD
C
C
<
C~:i-H
C
128
§
-J
Lb
W
0
0
O0
Atoo~~~~~~~~~~~~~~~~~~~~;
(X
C
bIS
,
O~~~~~~~
ndU
.
_! m I
0
I- _
Ui 0
CN'
..CN
.
0
LU
N
U-
Li
rr -
Z
0
ZI
AD
LU
EU
._
m
Lj0
_JI
n
0aw
CD
0'
C3
i
LO
CD
0
CD
<5
O~~
C
O
LD
O0
CD <-CD*
Z
129
Z
._ _
z
M.
Gt ba . OSJ
.
_0.
_
C~
_
=
_
C)3
=
M
_
F ,
o
z,
"g !!_- ;-
4
CD~~~~~~m
a
b
~ o .-
IC.
r'H
M
y
.
t.
~
,-
.
.
w
C)
-
-xJ
L.W
I
C-
I
Ill
LU
I1
LuLLJ
,_
-J
LU
0
I
CZD
I
III
2 HF
n-
iI
1I
f1
t8
L
Et
L
11
44
I
0
C
CD
U
I
*
CD
N
C
CO
130
-5
-5
CD
CD
C)
co
CD
0
O)
cv~
o
CD
-§
i.
CD
UC)
_-J
LLJ
LC
0
c-i
F-D
A
J
._
riCc
U')
fI
iI
iI
CD
05
c5
i
i
Co
C)
.iI
u
CD
-
131
§
en
C~
.
!2
CD
(j
I
..
_,,_
!
LO
;d E
I-)
CC
D5; I
LL
Clb
CM
C
-J
ULL
N
L
LLJ
LU
C1
C
[_)
a:
-J
?_
(D9
CD
m
Ln
T
E5--SI
Co
L)
I .
C
'b~~~~~~~~~X
Ln
CD
132
-J
Lfl
C)
C,
-a -
; *
;;
. I
¢e9 .-
co
-
rS)
l
CiDL'
O g,Z
C)
a, . Zi
Cfl .--cb,,,
o
,
90 .01
'
.- a
_
-s
-i 1 a'
&
Z
*E
Lt.X
cD9
i
I
L
o
I-Q
;I
_
I
I
_:
11
C)
__J
Lu
(N
__J
Lu
-L
IL
LU
I
tI
ZD
11
>I--
cr
,DoC
_J
Lu
>
F--
Vn
E
-
\
\
ll
\
EI
I
E
U
I
rxI
LO
° a
Lr)
CD
I
I
/
11
Co
133
lbm~
I¢~~~ ~~
~
CD
C)~~~~~L
-
e
~~~~C)
r~~~~~~j
C)
. mi .
C)
CO
d
0._
I
ZU-E_X
C)
w
I
I
LU
rb.
LL
:
I
O0
Z 2:
m-
LiyJ
._ I
LI_
CZ
U-
t
i
I
0
CD
o d~~
O
C
CD
CD
CD
CD
C)
I
L.
o
o)
C~~
L
_W
i34
-5
-5~~~~
-.
CiD
dULI)
Loq
Xn_
~~~~
w
.)m
LL.
Fad
CO
t _.
&
- C~%
.p
.8
fi
__J
LU
b_
LL
Z
CiD
uO
I
>-i
,_
U_)
t
CD
fiD
iO)
i <f°
I
U)
O
-
8
. (1,
135
.-
.,
z
L
"
'v
il
u
I
.,
e
-
w
E
CD)
_
C, _CNrJ
Iu
Z
LO
w9
ur)
9
C)
L
I
My
io
'o
%Re
b
.-
c m
ga
CD
I
I
11
2 - C:)
UU
LJ
-J
LU
,h_
I
I
Ili
b_
Vl
FOu
C0
Lu
U_J
U,
I
:
i
l
LU
III
111
DU
\
\
I
I
f\01
CD
· ·
o~~~C
ULD
o
°
136
cD'~~~~~~~~~~~~~l
uL)
c'~
LO
c3)
w_
LU)
wZ
I--
;
I-
5-
._
._
-1
La
c
Uz5
a,
C
O
w
LU
-_J
r
uJ
En
@
L:
1
L1_
Z
r-
cr
m
C
U
@_.
en
'
137
I
__5 CD
--LI I|
LbCD
1
N'-4
II
L
(\
IIl
*
~
X~Z
;
,
,
S
'0C
- "
'
C.4
CD
i.,,
-11
_J
LU
-J
-
LL
L_
N
cZ
C9
re'
z
CD
...
uC')
m
"D
Cn
t
t
I~
CD
Ln
CD
o 1~1
CD
CD
U
cD
CD
138
I~
Lfl
co
1
C
-4~
1~~~~~_
o.-rIuI
,XE
a.
Lb
~~~~~~~~ ~ ~ ~~ ~~~
°
~~~~~~~~~~~~~~~~~~~~I
-O
_
E
Lf) i
__
ot , _
C~
.
--
_~.= _
..
_
I0_
9Ia
l 2
8
ao
- .0i
c
,
.
Z
X
(D
Ir9
1C
LU
o
LL
CD
O
LLJ
LL
LU
_
LU
(_
-_J
LU
L
g>
CE
C_
Z
C)
-
L)
cD<
Lnf
<
°-
139
L
X
D
X
Ic
C
I
_
.
a
CnM9
C)
C)
Iii
;= Ed a
-.
§W 4
C)
_J
.
(D
.LL
U-
U
CDI
-J
2-
WN
LLU
rl-_i
ULU
LL
z
1
LID
CD
C)
(o k
co
o
CD
C)
O)
om
o
0C
CD
O
C)
CD
C)
140
§
to
2
£2
CD,
CD
v\
0
9
M,
£g !
CD
CD
£2
·., ,-i IC,
CiD
it t
B
UN~£2
m
Z3_
I
LU
ullJ
NCD
2
c~ Pli£2
LL-
z
0
LL
p14
L
I
1-
Z
w
_J
cn
tO
O-
f?
CD
UL
CD
CD
CD <*
rn
H
Ln
~~~CDC
~~~~~~~~~~bO
CD£<-£
141
-J
_b
C,
-
,--
CD
_
-
Ci
.-
C0
L'J6
O
St__J
C) *g
aLU
*-
11
z.--.. _ N_ W
-O
LO
,-
U4.
I
-Loon
U
I
r
I
Z
LU
0 *
C3
E
__-J
LU
t3 0E f
LUW
I
S.
I
I
111
r'.i
LL
LU
I
I
lIt
E
f
r
E
\p i
Co
I\
CD
0
F_J
LL
Ln
>-
ULr
CD
142
lb~~~~~~~~CD
CD
__
~
L
m
(-
C
C
(%J
_
o~~
CD~~~~~~D-
r-.
CL
LU
--
fL
is!E
.
tN L
C
-J
-
IHI
tU
L
NJ
C-j
E
z*,.C
_j
U-
JU
0
z
a:
cm
C
LLr
O <I
o 4
O <0
_ 4
o
143
CD
Lf)
un
C)
C)
C)
Cd
c. ii>u
oni
CD
La
= I
M
_
I
_g
E
_
Edc _ -._
. r,
00
CD
l_
LU
U-
Z
L
L
tz
Lo
CD
u
CD
n-
_.J
I
/
>-)
j
I
CD
Lf
C)
Ln
CD
C)
C)
CD
C)
144
e-'
ORDINATE:
emi'
CROSSTRaN
PL-S, XU 4.800E-03 STEP 100
emax ° egin
1.0 ....
4..........
0.9
. t t~~~~~~4
.
.
.
+
0.8
X\'
0.7
i
0.6
4.
\
+
\
:
0.5
0.3
4.
+
EtOO
EtoO
0.2
!"°
11
ji
0.1
ARGON
.
0.0
.2
.3
.4
,4
,3
'.2"
Y(I) MIN=5.97E-04MAX=
8.65E-03
vABSCISSA
i
n
I
1
.
.5
.5
,6
.6
.7
.7
.8
.8
ABSCISSA:
FIG 39
RADIAL
VELOCITY
NDTEMPERATURE
PROFILES
OFAR6ONEAR
PIPE IT X= 0.004
CODE
FRMTHEPAAOLICAL
cell pressure: 0.7 Ato
Argon
massflux: 1000c-c/in at S.T.P.
1
.9
X-
min
Xmax - Xmi n
145
CROSs-STRI'PL
X 8.000E-03ISTEP-200
ORDINATE: 6;Bmin
~max. .u~n~..
1.0
....... ..................
...............................
0 +*/
0.8+
,
0.6t
/
os +
/|X;
/+
AXIAL VELOCITY
TEMPERATURE;
03;
[IN
4;
I1 /U
02
*~
,,.
T
SH4
Si
H2
AR
O.OOE+O0
3.OQE+2O.OOE+SO
.OOE+O0O0OE+O OOE+00O
f/ /
ARSON
0.*0
ABSCISSA
IS
.'.+
-f .
.1
.2
t
'...
.3
.........
.4
.
.5
.............
T................... f...........
.6
.7
Y(I) MIN-5.97E-04MAX=
9.00E-03
.8
.9
x - Xmin
ABSC I SSA:
FIG
40
+
0
3.0E-01
3.00E1,20.00+00 O.00E+00
.OOE+a1.00EfOO
.OOEi00
OE+00 OOE+OO
MAX 3.61£-0 3£+02 O.OOE+mO
l
RADMIL
ELOCITY
M TEPERATUIRE
PROFILESABSCISSA:
OFAR [N
PIPE IT X 0.008FROM
THEFASOLICAL.
CODE
cell pressure: 0.7 At
Argon
mass
flux :1000 cc/minat S.T.P.
Xmin
1
146
ORDI NATE:
0 'o3
"
CROSS-STRE,1
PLOT,
X- 1.000E-02ISTEP=
250
emax.....
e3in
..
1.0 +..
.......
.......
0.9 +
0.8 +
0.7 +
0.6 +
0.5 +
0.3
E+00+
0.2 +
+
10E+00
0.1 +
~
D~~fOG
;·
~
0. weT
ABSCISSIS
ARGON
T-.
.-.
-...............
.T.."
...6r.......'...............
..........
.5
.
.7
.8
.9
1
.1
'.2
.3
.4
Y() MIN=5.97E-04MAX=
8.71E-03
x - Xmin
ABSCISSA:
Xmax - Xmin
FIG 41
RMDIAL
VELOCITY
AD TEHPERATURE
PROFILES
OFARGON
NEAR
PIPE EXITX=0.01 m
FROM
THEPARABOLItL
CODE
147
ORDINATE:
CROSS-STREEM
PLOT,XU%
1.200E-02ISTEP=300
'min
0max
din
1.0
,P
0.5
E
0.8
U
T
SiH 4
Si
AR
H2
*
0
0.00E+00 3.00E+02 O.OOE+OO
0.00E400 O.OOE+tOO
1.00E+00
3.56E-01 3.00E+02 O.OOE+tOO
0.OOE+00O
0.OOE+001.00E+00
0.7
·
IPERATURE
0.6
-
0.5
kAXIAL VELOCITY
0.3
0.2
*
0.1
ARGON
.....
...
0
ABSCISSIS
~~~~~~~~~~~~~~~~.
....... ....T7
.5
.6
.
·
,
.I
I
....
T. t-....-.
....
.
'.'J · ......-.
.1
.2
.3
.4
Y(1) MIN=5.97E-04MAX=
2.62E-02
·--...
7..
.~ ·
.
-
.8
..........................
.8
.9
x - Xmin
ABSCISSA:
Xmax - Xmin
FIG 42 : RADIALELOCITY
NDT'mERTURE
PROFILES
OFARGON
AFTER
THEPIPEEXITX=0.012 m
FROM
THEPARBOLICAL
CODE'
L
148
ORDI NATE:
0
400
CROSS-STRPLOT,XU 1.600E-02ISTEP-
mi
max eiin
.
.
.
.
.
.
.
.
.
1.0
........
..................
+.........+.........+.........+........
0.9
0.8
AR
SiH 4
H2
Si
T
U
A
'
1
*
1.00E+00
Et00 3.00E+02 0.00E+00 0.00E+00 0.OOE+O0
8IE-013.00E+02 0.OOE+000.OOE+000.0OE+001.00E+00
U
0.7
0.6
0.5
i
+
+
TURE
0.3
(XIALVELOCITY
t
0.2
0.1
ARGON
_0
Mr-
0.0
-
P
......., ;t..
........
T....r....t...%..t
........
;
......
.........
........
....
;.
0
.1
.2
.3
.4
A8SCISSAIS
Y(1)MIN=
5.97E-04 t- 5.30E-02
.5
.6
.7
.8
.9
1
x - xmXn
ABSC I SSA:
FIG
43
: RADIAL
VELOCITY TPERATURE
PROFILES
OFTHEMtR AR6 JETX=0.016 m
FROM
THEPARABOLICAL
CODE
xma x - Xmi n
149
-min
ORDINATE:
'DNT
8Xm
C-,OSS-STREAM
PLOT,XU=2.000E-02ISTEP=500
t^+
I..........+..............................................
tt. ..........
..~~~~~~~~~~
1~~~
~ ~ ~ ~ ~~.
1.0i i
0.9+
+
U
l
0.8
SIH4
'
t
SI
H2
AR
*'
I
IORDINATE
~' @
08
T
.OOE
00 100E+00
O.OOE+
MIN 0.00E+003.00E+020.OOE+000.OOE+00t
f1s
MA 3.40E-01 3.00E+02O.OOE+OO.00E+00O.00E+001.OOE+00
+
+
0.7+
.*
p
TEMPERATURE
+
0.5
+
03
1
A
AXIAL VELOCITY
0.2
\~~~~~~~~~~~~~~~~~~~
''
0.!
0.0
0
0.1~ +
\*
......... ................
.....................
.3
.4
.2
.1
Y(I)MN= 5.97E-04tiX= 7.64E-02
IS
ABSCISSA
+
ARGON
.5
.6
.7
.8
......... ....
1
.9
x - Xmin
ABSC I SSA:
FIG
44
PR0FILESXmax
A TEPERATURE
ELOCITY
RADIAL
JETX= 0.02m
OFTHE UM ARGON
COE
FRCM
THEPARABmOLICAL
-
Xmin
150
0
ORDINATE:-
mim
max
_
1.0
CROSS-STRCAI
PLOT,XlI 2.005E-02 ISTEP-502
3min
__
__
. . .... ..r.... .....r.......... r..
0.9:
......t.........7 .........t....
.t .........t .........
ARGON
f
0.8
0.7 1
0.6 4
+
AXIAL VELOCITY
U
ORDINATE
*
0.5 ;
T
SIH 4
H2
I
I{
AR
SI
t
MIN 0.OOE+003.00E+020.00E+00 0.00E+00 000E+00
0.OOE+00
MAX 7.50E-01 3.00E+021.00E+0OO.OOE+O
00.00E+00 1.00E+00
0.3 +
k
I
II
t
0.2 ;1
0.2+'
I
SILANE
;
0.
:AA
A
............
+.....+..........
....
0.1. +IN '%\
t'
1
O
.Q
0
.1
ABSCISSA
IS
Y()
el^.^.
.2
.3
.4
IN- O.OOE+00
MX= 7.65E-02
..
.5
....
....
.6
^
.7
0t0.t.
^
.8
..
.,.ttt,
.9
X -
X4
1
ABSCISSA:
FIG
45
Xmax :RADIAL
ELOCITY
ANDTIPERATURE
PROFILES
OF SILE
NDTHEAN R AR6 JErTS
ATTHESIL4NE
NOZZLE
EIT X= 0.02005
THERADIAL
MSSCONCETRATION
OFSLAE AND
AR6ON
AREALSOCALCULATED
BYTHEPARABOLICAL
CODE
Xmin
151
·I
ORDINATE :- 0 min
i.'
-- max - dini
---1.0
..
PLOT,XU-2.397E-02ISTEP 600
ROSS-STREI
L
..
. ' . . . ...
r . . . .....
. . .r . . . ..t........
t
t .t.
.
.
1
t
. ... . ...
.
0.9
MIXIN6
OFTHESILANE
ANDAR64W
JET
t
0.8
A TEMPEATURE
PROFILES
R4DIAL
VELOCITY
AND
THEWUA ARGNJETS
OFSILANE
ATX= 0.02397a
ARGON
MASS
CNCETRATI OFSILE
THERADIAL
CODE
CALCULATED
BYTHEPARABOLICAL
AREALSO
t
+
0.7
+
0.6
0.5
0.3
0.2
H
Si
U
SIHQ
T
~~S' AR
~
ORDINTE *
6.25E-01
MIN O.OOE+O0
3.00E+02
-4.77E-07 O.OOE+OO.0.OOE+0
MX 4.29E-019.12E+02 3.75E-01 O.OOE+00
1.00E+00
O.OOE+00
.
GON
+
t
SILANE
t
AXIAL VEKOCITY
0.1
0.0
+
.......................
0
.1
.3
.4
.2
MtX=7.54E-02
ABSCISSA
IS
Y() MIN O.OOE+00
.5
+.
...................................
....
.6
.7
.8
x
ABSCISSA:
F I G 4.6 MIXN6OFT
SIE
L
mm
) AR4JET BEFORE
1
.9
Xmin
Xmax - Xmin
-
1
152
CROSS-STREWI
PLOT,X
ORDINATE: 3G
2.797E-02ISTEP=700
miA
emax .- ein
E. .
1.6
".kk%~'..
T-...
j.'T...
*T......~.....¥..,..;,....
1.0
,.......lg"~*..".."....
.
;...'. J.,T .......* *........
*
0.9
^
+
1.
.f
0.8t
...
/ AXIAL.
-1
0.7;
VELOCITY
;
I
I
ARGON
-
F1'
0.6t+
*
r
+
·~~~~~~~~~~~~~~~~~~~~~~~~~~
ORDI~
NT
'....
p
U
0.3 +
·
|
0.3
IQII
I2
SI
AR
+
A
KIN 0.00E+00 3.00E+02-9.54E-07 .00E+00 .OOE+008.53E-01
0AX 3.GE-01
3.-5E-02
1.00E+.0
S
AR.
T
HH2185E-02 1.29E-01
U 81E"02
*I~~~
I ~ %.
+
. +
. +
. +.*+
+
.8
A
~~\
.,
0.1
S H4
T
T;
V
..
0
.1
ABSCISSA
IS
FIG
47
.2
.
.3
...... ......... .
.4
.5
.6
.......................
......................
.7
.8
=
1
x - Xmin
Y(I) HIN O.OOE+00
MAX=
7.86E-02
THEREACTIO
ZONE:
RADIAL
VELOCITY
AN TEiPERATURE
PROFILES
OFTHEINNER
RECTiN6
JETm THE aR
MG6N
JETATX=0.02797a
TIERADIALASS'CONCWRATION
OF SILNE, AR6GN
WD SILICON
AREALSOCALCULATED
BYTHE
AMRALICAL
CODE
.9
ABSCISSA:
Xmax - Xmin
153
NATE:
0RDI
cROSS-SrREM
PLOT,XU=3.197E-02ISTEP=800
-
fin
max
l^ -1t
.... ............
z--".T--..........
*Z8t-@§--sT
* .,**e-.r
@e-.eT~e
t .........t ...............
.........
.
1.0
.1
T
~~ORDINATE 0
0.9
.}
SIH4
H2
SI
AR
+
MIN O.OOE+O0
3.00E+02
-1.19E-06O.OOE+OO
O.OOEtOO
9.13E-01
0.8 t
|
lMX 3.29E-018.22E+022.32E-021.06E-027.37E-02 l.OOE+OQ
|
0.8 ! .*vt I+
ARGON
0.7+ x I
0.6+
+
0.5
+
0.3 ++
0.2; 0
SILICON
.1
I
0.1
0.0 ........ ..
0
AWSISEA
IS
+i
+
%S
'....:.. "....+............,......... .....
.1
.2
.3
.4
MAX=
9.58-02
Y(I) IN=-O.00E+OO
.5
.+.......... .......... ......... +.........
.6
-
.7
.8
.9 I
x - Xmin
ABSCI SSA:
FIG 8
ZE:
: THEEND
OF HEREACTIN
PROFILES
VELOCITY TIPERATURE
4RFIG
RADIAL
THEgtuLR ARGNJET
REACTING
JETAND
OF THEINNER
ATX=0.03197m
OFSILANE,
ARGON
MASS
CONCENTRATION
THERADIAL
Xmax -
mi n
154
ORDINATE:-
8o 'emin
emaxc
XU=3.597E-02ISTEP=900
PLOT,
CROSS-STREIM
amin
1.
;
U
0.
*
ORDINATE
MIN 0.00E+00
3.26E-01
0
I
0
I
I
I
0
f
0]
I
0
I
I
I
.
A
.Z
.1
0
9.47E-02
W;X5
Y(I) MIN-0.00E+00
IS
'ABSCISSA
^
^
FIG 49
s
,,/
n
,
,^
,,
.,,
.
,
_
.
x - Xmin
ABSC I SSA:
ZONE:
THEPOST-REACTION
xma
PROFILES
NDTEMPERATURE
VELOCITY
RADIAL
JETS
ARGON
JETANDTHE9M
REACTING
OFTHEINNER
ATX=0.03597
ARGON
OFSIIlANE,
CONCEMTRATION
MASS
THERADIAL
COOD
BYTHEPARABOLICAL
CALCULATED
AREALSO
SILICON
AND
R
- Xmin
155
CROSS-STREW
PLOT,XU=3.997E-02ISTEP1000
e- 'emin
ORDINATE:
3m0 -. 0,
max
ffin
0.9
W
·
0.8+
ARGON
:^1e
z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A
'.J'IORDINATE
*@A
+
0.7+
U
,
·
I
SIH4
H2
SI
AR
+
7H|
9.52E-01
O.OOEtOOO.OOE+00
3.00E+02-5.96E-07
ltIN .OOOE+00
IAMX
*
M
',
0.6
..... t ....-..
;..+'..........
.... T..... ....t
ARGON;4-.
X........T......
-.
1.0
3.23E-01 7.51E+022.33E-024.39E-033.06E-02 1.OOE+00
0.5'
+
.
J
t
e
0.3
*
*
t
I
0.2
+
0.1i,
... ..........
.0 .
0
ABSCISSA
IS
;
SILICON
AXIAL
s
.
VELOCITY
+
X
.
+.:,,^,.,,.......
.
.
.
..
...
.1
.2
.3
.4
Y(I) MIN 0.OOEO0
I'MX=
9.27E-02
...........
.5
...
.6
.7
..
.8
.
......
.........
.9
1
I-
,
ABSCISSA: FIG
50
THEPOST-REACTON
ZONE:
Xmax
RADIAL
VELOCITY
AND
TEMPERATURE
PROFILES
OFTHEINNER
REACTING
JETANDTHEAULARARG60N
JETS
ATX=0.03997m
THERADIAL
MASSCONCENTRATION
OFSILANE,
ARGON
AND
SILICON
AREALSOGCALCULATED
BYTHEPARABOLICAL
CODE
Xmin
156
ORDINATE: e-emiu
DW-STREM
PLOT,
XU=3.997E-02ISTEP=1000
emax
o .- ei
'e
'**
.........
.........
^
.........
,........
.........
1.0
I*
A
I
AA
A
0.9 +
8
~~~~~+
S~~~~S
^~~~~~
·
*
\
JrmM
A
*
0.8+
~
MHMWM
e^ ~
*
t
4!A.
r~
--
1 *
0.7+
*
'
AA
""
0.6 +
A
t
^"
'- **
A"
e
^^
"^**
eee
ffeee
~~***i*^^ ff
0.5 +
Qi!
*m****
Q
^^
-'
JJ
0.3 +
-
U
ORDINATE *
4
MIN 0.OOE+003.OOE+O2
0.OOE+00
MX 7.50E-01 1.33E+031.00E+OO
0.2+
n2
0.1 +
0.0
SI
AR
4
4
-"
4s
.5
.6
+
AAAA
AAAAAAA
4
4
1
.2
.3
.4
XU MIN=4.00E-05
MAX=
4.00E-02
AAA
e~~
'~e . . .. ^
-
O.OOE+OO
O.OOE+OO
O.OOE+OO
4.51E-02 3.15E-011.OOE+00OO
3
4
+
*MAAAA
-
AAAAAAAAAAAAAAAAAAAAAAAAAA.7
2
O ,
,4SCISSqIS
-
ifif
ff
&IAA
-
SIH 4
T
+~~A.
AA
@1
6
.AAAAAAAA8 9
.7
.8
1
.9
1
x -
Xmin
ABSCISSA:
Xmax - Xmin
FI G 51
: IAL DISTRIBUTION
OFUELOCITY,
TEMPERATURE
AD SS CONCENTRATION
AS CALCULATED
BYTHE
PARABOLICAL
CODEAT r=O.0
RUNCON4DITIOS:
Laserpower:180W unfocused,Gaussian
profile
Argon flow rate :1000cc/min
157
ORDINATE'
0
mian
e max
*"et
-
CRSS-STREm
PLTor,XU=1.000E-01ISTEP-254
ein
-
1.0.^^^..-.
......
. ^~
, ^^e
f
f
-....... 4 ....
t #....... -.. ...t..-.-#............
f
0.9*^e
.
-
ORDINATE:
t
^Q
#
0.8 -
m
emax-
I
nmin
o_
U
P
Si
T
AR
O0RDI'TE *
+
8
HIN 0.00E+003.00E+02 1.06E+059.41E-010.00E+00
MX 1.53E-01 7.12E+02 1.06E+05 1.00E+00 4.84E-02
t
0.70
t
0.6 -
*
*0e
0.5 -
0.3 -
..#~~
^
*^e
t
0.2-
*
*
0
A
^ e
.
0.1-
^
8
. t
0.0#ti
I
...
#
*
^* e
^* e
~~~00
* * * * * * * *
^ .. ^- ^
... ^
:I.+.H.+.+...-....^.-.A.^.-.^^-^..
0
.1
.2
.3
.4
ABSCISSA
IS
Y(I) MIN=0.00E+00
MAX=
1.09E-01
.5
.6
.7
*
*
.8
.A..
.9
X -
1
Xmin
ABscISSA:
FIG 52 : RADIAL
VELOCITY,
TEMPERATURE,
ANDCCENTRATIN
PROFILES
ATA POST-REACTIGN
POSITION
X=0.20
LaRUN
COITIONS:
Laser power:180 unfocused, Gaussianprofile
Xm
x - xmjn
xmax - Xmin
158
ORD
I O~~
NAT
DIX4-STREM
PLOT,XUS1.OOOE-01
ITEP= 254
0.9:
**if
eee
1.0 ***** .**....
A
ORDIN0ORDITE
1.0 ; ;
-
.
-
-
U
*
-
-
T
.
-
,
P+
SI:
A
-
OAAA
ff n
l1
cman
r..ni
xIJ.... ..................................................................................-
A^ M,
0.9 -
·
U
'Jr,"(~(~
ns1n
Ie4f?
Mfif
~~.8:~~~
eX*l*^
0.8-
*^
*
AA
T
P
z~~ORDINATE
,
4CUL
-
Si-
+
A
.UUCU-x I.UuOLl'U
U.UULtUU
7.48E-01 1.16E+031.06E+056.25E-01
A
OAA~~~M
0.70.7 -.
ff*"
A
*
*^
AA
^^*
AA
^^
A*
^*
e *
,
0,6 .
0.5 -
e
eeeee
eeeeeee
@eeeee
^^ **
e
0.2 -
eeeeee
"f*
^^ *
e
0.3-
e tM
.
AAAA*_
^^ **
e
^^^ ** AAA
0
^^^^
A
^^^^^^^^^*
·*~
AAAAA*
e
Q
^^^^^^
*
0
0
ABSCISSA
IS
-
.....
.1
.2
.3
.4
XU HIN- 3.56E-05
MAX:
1.00E-01
.5
A.,111tLt1..,.1A....
.6
AA.A
.7
AXIAL
VELOCITY,
TiPERATURE,
NDCONCTRATIN
FIG
53
PROFILES
AT r= 0.0
RUNCONITIONS:xmx-Xi
Laserpower:180Wunfocused, ussian
profile
Argon flow rate :1000cc/in
Silane flow rate: 30 cc/min
Cell pressure:1.0 Atm.
.8
1
.9
xx
-
-
min
min
ABSC I SSA:
Xmax
-
Xmin
159
FLOW CHART FOR THE HYPERBOLICAL
ALGORITHM
READ DATA
CALCULATE
STEP SIZE
DETERMINE BOUNDARY CONDITIONS
FOR THE AXIAL
LOCATION
DETERMINE RADIAL GRID WIDTH
CALCULATE
ADVANCE:
U,H,CONCENTRATIONS
V
EVALUATE
USE SIMPLE TO CORRECT
UV AND H
0~~~~~~~~~
.
CALCULATE
COMPLETE:
TEMPERATURE,
DENSITY, VISCOSITY
DECI
PRI
IS THE WHOLE DOMAIN SWEPT ?
FIG
OUTPUT
.I
U
,,,.I
MUN UIRKI U
~,,~
li
I
54
UIP.........
b14I
b~RtPbUNIL UMWLK UUM
-
- OA-
160
THE GRID
t
(-PI
Y/YE
or
)
(E -I)
FIG
55
THE SIMPLE ALGORITHM
THE DOWNSTREAM PRESSURE IS GUESSED
U,H AND THE EQUATIONS OF SPECIES ARE SOLVED
V IS FOUND BY EXTRAPOLATION
CALCULATE CHANGE IN V FROM THE TRUNCATED Y-MOMENTUM
EVALUATE THE STREAMLINE SLOPES (NEW)
FIND U,V AND P BY USING THE PRESSURE CORRECTION RQUATION
CALCULATE THE TEMPERATURE AND
DENSITY FIELD AND THE
CONCENTRATION OF SILICON AND ARGON.
TREAT THE NEWLY FOUND PRESSURE AS THE GUESS PRESSURE
FOR THE NEXT STEP AND MARCH UNTIL THE WHOLE DOMAIN IS
COVERED.
161
OWg-STREW
PLOT,XU-3.760E-02ISTEP1000
P
U
ORDINATE
%
ORDINATE'
MIN 1.27E+03 1.47E+05
Mt 1.56E+03 6.16E+05
10 .....
.. - ................. .
*
·
-;"mis
emax
efin
.............. -.............................................+.
0.9
0.8
f*
**
4
**
4.44
*
*
1k
4.4.444,4&,
44
44
* ******
**** ***
*
0.7
**
**
*
0.6
*-
*+ +.
++
++
0.5
+
+4***
+*
0.3
+
*
++
+
0.2
+14
+1.
4+
4+
0.1 -
.
0.. 0..0
4+l+H+l+H4**.
4+14
4+
*-
+ 1+I-H+f
4+
4+
4+1+14+
*.
**
*.-- ...
......1
.2
.3
.4
--
.5
.6
.7
.8
ABSCISSA
IS
XU HIN 3.76E-05MX=3.76E-02
.9
1
x - Xmin
ABSC I SSA:
FIG
56
-
t
CD
Xmax - Xmin
NUa
I
S
oNERSTINC
JET
DISTRIBUTIONS.
A PRESSURE
'AXIALVELOCITY
KIKGU'S RESULTS
ItIPARISNWITH
(WCFINED
JET)
162
CROSS-STREM
PLOT,XU-3.760E-02
IEP-1000
U
P
ORDINATE
*
+
MIN O.00E+002.08E+05
MAX 2.64E+032.39E+05
1,0
-.........
0.9
-
-..........I..I
ORDINATE:' "omin
e -e
Omax
,A,
,,,
A-, ,
,
timt
-i,
A-
r
,,A
7+
0.8-
0·7
0.6-
0.3-
:
0*PRE
· +
|
A
+I
VELOCITY PROFILE
PROFILE
'+,, +
\+
0.2-
I
~~~~~~~~~~~~+
0.1-
0.0.......-..............................................
0
ABSCISSA
IS
.1
.2
.3
.4
.5
.6
................
.7
.8
Y(I) MIN=O.OOE+00
AX=3.76E-03
................
.9
1
x - Xmin
ABSCISSA:
Xmax
FIG 57
UEREPNDED
NEACTIN6 SPERSONIC
JET
RADIAL
ELOCITY
AND
PRESSURE
DISTRIBUTIONS.
(CONFINED
JET)
Xmin
163
DIN-STREM
PLOT,XU=3.760E-02ITEP=1000
U
P
ORDIITE *
+
MIN 1.06E+03 1.26E+05
AX 1.33E+033.08E+05
1.0***
........-..
.........
.. ................
....................
. ***,.
*+t
0.9,
,
.
+
*
*+
*t
*
+
+
+
**.H.
0.?++4f**
+$
0.8-
***
+
*
. *+ t
0. -
*
t
* .*
+t
+t
** *+ t
* *tt*H:::"
*
* ft
0.6- *4-I.
*
.
t
*
*1111+H1111111
ft
t
*
**
+t*f**f
0.2-+
fhAA4~L.A
. t**
*
.
..
I....
t~~~~~~~~~~~~~~~llrillll
+
+*******
t
+******
t
ft
ft
t
+*
* ft* *
+4+1-1
.
*
+
*
**t+
f
-
*+
t
**
.t.+
****t
********t
t
+
**
f
*
**
f*
-+
*
t
*
0.5- +
, **
. * ***
0.3 _ * *
f
** * +H.*
* ++
t-
+****
t
******
-
****
ft* +*** ****
*****ft- +***
f
*
ft + * ***
**
.1-f**
+ -H.**
0.1- t**
*
. tI/dr
*
.H*
0.0 ftI.*
....0
ABSCISSA
IS
.-.-..-.-.-
.1
.2
.3
.4
.5
.6
.7
.8
XIU MIN=3.76E-05 X=3.76E-02
FIG 58
"OOMMR NNRE4CTN6
SLPERSONIC
JEr
AXIAL
VELOCITY
AND
PEURE DISTRIIONS.
(CONFINED
JET)
.9
1
164
V2 --,,
treams
,,,
Expansionwave
,
4,Z/Slip
" Shock wave
line
v
v, (Pi> P2)
p1
Coaxial supersonic jets in a cylinder
.~~
I
i-~~~~
Pi
(Pi >P2)
Coaxial supersonic free jet
FIG 59 SCTiC O TE u
CONFINED
PM FREE
JETS:RELATIVE
DISPLACMSO
OF HE IF"
N I
ESAND
SHOCK
OMES
(NONREWIN6
JETS)
165
D14N-SRWPLOT,
XiJ 3.760E-02ISTEP=1000
U
T
P
Si
t
A
ODINATE *
9
MIN 1.54E+037.13E+025.45E+04 .0E400
!X 1.75E+038.33E+023.23E+05 .00E+00
ORDINATE:
0OOm18
maX
gin
i
*999 e'i
H
*
H
iie
.
+*e
*.**
e
e
e+
.- ** -
1.0
....
**,,,,,,,,,,..
..
,.
..
,...........................................................
***
*
*
0.9
*
*
+e
f
frk~~~~~~~~~~~1
+
4t
+4
i4
4+
*
*
*
*
*
*
**
*
0.8*
*
9999
* *
0.79*
*,*
9*
#
·
0.1
nn
UtU
+9
e+
***
+
te*
t·
*
*i*
**
***
*
***
**
*
***
9+4
.**
e
444
IFIIIIII.:IL::
l::: Lt:i:I:I:
II:I: *
e
49 ee**4
+9*14
*
*i ***
***-
*e*
A +**
**
.
+
.5
.6
-A
@i-i}ee+
99999+*
+,
4
14
.
.
AAAAAAAAAAAAAAA~AAAAAAAA&AAAAAAAAAAAAAAAAAAAA~
.1
.2
.3
.4
XU MIN=3.76E-05AX=3.76E-02
*+-.
*H+
+
e9+
++1
**
*
.
999*~~~~~~~~~~"
~ + e*ee
.
e9 9 94*
ee
+4+
+-
**
44*+
*4.
AAAAAAAAAA^AAAA
0
ABSCISSIS
,*
**
+99
,
-
*
* +49
9+ 44
99
+ **Q
99 e9
ee
99999999999
*+, 99999999
9 9 e+*
+
*t499e
9H
*+9
I eeeeee99ee
+*** * * .* eee*
99 .
+
i}
t**
+
9
+* *
*
9
+ee
AAAAAA~AAAAAAAAAAAAAAAAA
.7
.8
.9
1
- Xmin
ABSCISSA:
FIG 60
I
SUPERSONIC
SILE-AR60NJET
LDEREX ,WDED
AXIAL
VELOCITY,TEMPETlRE,PRESSUE
AND
ATION
DISTRI8UTINS
(CONFINED
JET)
CWJWR
i
i
*
**
*e+ + 99 ,*
0.6::H e*
0.2-
-4
sii************** +4t99 ***
** +9 9
**
,e
*9 t+
**
**i
0.50.3-
II
U4
Xmax
Xmin
166
--..-*.*-.
0,9
+ **
**
+e
** +, *
DO-STREI PLOT,XU 3.804E-02ISTEP1000
U
T
P
SI
ORDINATE:
ORDINTE * O
0
+
MIN 1.60E+03 ?.04E+02 4.68E+04 0.00E+00
1.77E+03 8.01E+022.06E+050.OOE+00
'min
a
-
max
fi n
1.0
..... ........ ......... ......................................
..
* +
,**
**
*
+0
4*
*
+
**
**
***
+t
*
*
Mt ,*
*
0.8+ ** *
**
+*
*
*
+H*
*+* **
.
0.7-
·
0.6-
+*
t0*.*
*
*
+0t
*
t+*
+
*
**
+*
·
0.3-
0+
+·
+*
+40
0
0
.........
.
GAl
0.1
*te
*t
..
..
.
.I-
0+
+Ot**
*
**
**
*0+*
.
.H~
*.A.
..
..
..
.'.-
*
4*
**
*- e
"
**
*
*
0
.1
.2
.3
.4
ABSCISSA
IS
XU MIN=3.16E-05
MX= 3.8DE-02
.
0e4*
+40e
0*
*#*
,
~,
"..-
.5
*
*
-.............-.
.6
..
.7
NoRE
susoNIC
SILANER
T
AXItAL
VELOCITY,TEPERATUREPRESSURE
AND
CONCEITRTION
DISTRIBUTINS
(FEE JET)
.
..
.8
x -
ABSC I SSA
FIG 61
*4*
*+
++1
**++ +.
* +1
4t*
+t
0+
..
.
ff
** k
+** ***
+
+*
+*
+40
*eet+
+4
..
ee
+
e***i**
+H+
+H-4+
.++H1+H++H+1-.
...
I-i-tlt
· * *40i
. ..
o0
+
+
.*+etH.
0.0
+
+
0
0
* te0+
.Te e*
-*
+t
0+
00
*40+
*
0.2-
*t
***
.
-
+
+ 0.
*t
0
eee
0+4
,40
*
*+
00000+Nt
m
.
*,+
400
*l
t
**
0000 *
0
+0
4 +t"
+0
*4+
+
*+
44-f
*
+*o+
t.
*
**
000
.
004+40
*
**
**#0
+0
+e
.5-
*
**
*
**
*
+*
..
,
.oe0-
..
..
,9
1
Xmi n
-
Xmax -
Xmin
167
DN#4STRE4M
PLOT,XU-3.84E-02 ISTEP1000
**
t **
*++
^^
***
+
+*
*
*0
*
^
*
4*
* *A**+
*0.eee.
^^
.*-
T
U
SI
P
ORDIN.TE *
+ ^
e
HIN 1.60E+037.04Et02 4.68E+04 O.00E+00
MQX 1.77E.03 8.98E+02 2.06E+05 3.15E-01
1.0
-
ORDINATE:
H*......... ...-.-............................
·
0.9 ·
0.8-
0.7-
0.6-
+*
+*
f
efeeFf
0.5-
^^
* ++
.
+
**
++ **
+
*
+ **
+ ***
+ * *
+ * *
+
*
+* *
+*
**
+*
*
+*
6min
****
** *
** **
*
,
*
*
*
**
***
ft****
***
*
**
**
*
**
,
ff +
**t
iA
r
++ ^e * *
* ++ + ^e* *
**
*
.H+ ^f* * **
*
*^^
A*
I*.
*+
^
ii .
*,
+
^ *#
f
* *
f
*
^^+** t.
*+
^A+*
^^
^
*+
*t
+
***
t***
**
ef ^
0
e+
A
++
^
+
+4.-
+++ +
-
-
**
^ t
+
**A**
e
e+
0
-
max-
* ++
^ *
^^ *
+-
.
U0............
::":':
,:
.,.,.:,, ., ** .
ll4.||S2' *-4.
4+-1-
+4*
ft
ft
·
*t
-+efeeeeOeetee
eeee*
0.2*t
A
* + +
* f +
** e*+
.1
.2
.3
.4
XU MIN=
3.16E-05MAX=
3.85E-02
t*
+*
^ *t
^ *
* *A^
**
* ^^ *
* ^ *
^t*
t
+++
*+
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA^AA
0
A8SCISSA
IS
+* *
eetee
· iN-*
-
0.10*0
***Fl *
*
t +e
A *
*I .^
* *
+0
·
Ai^
^*AAAAAAAAAAAAAAAAAAAAAAAA
AAAAAAAA**
.5
.6
.7
.8
.9
X - Xmin
ABSCISSA:
Xmax - Xmin
FIG %162
-
UtNDEREXPADED
REACTING
SUPERSONIC
SILAE-AR60N
JET
AXIALELOCITY,TIPERATURE,PRESSURE
AM
CNCNTRATION
DISTRIBtIONS
(FREEJET)
1
168
o5
w
i
65
'I
x
u-wn
Z
0
c5I
Z
w
U,
w
-j-j
0
C)~~~C
CD
- W Z
a.
z0 _
im
< wDJ
Z
L UJ
LL aLU<
C . . -nULC. xwc Q
a:
D
C
0.
Xw
L
169
0
C'4
CN
CD
i_
.9
C=
.
4a
.
0=
L
-dd-- ,
I
U%
C:)
.
C=)
i
E
- Z
170
I
ii
AL~t
4
jLt
%
- o
o::
-r
:i
. c
I
.0
o00
U- I.
kk.
CD
ICD
171
DW-STREAM
PLOT,
XU=3.760E-02ISTEP=1000
ORDINATE:
mi
max
1.0
LASER FOCUS
.i
n
it Et*
+t* A
*
.................................................................
Ii,,
. AAAAAAAA
·- :U
T
P
S
09 1ORDINATE 1.2E0 T
SI
+
MIN 1.52E+03 7.13E+025.45E+040.0DE+00
MX 1.85E+039.43E+023.09E+053.35E-01
e
AA
AAAAAAAAAAAA .AA
AAAAAAA
^
-1
A*
**
+
+
** ***
*
A
+ ^ ,A
*~~~~~~~~~~~+
e^*
~~
~
~
eAe **
*
Att
e
0.7
-0.7
( ++ ^,^^AQee**,
***
......
· **
0.6l l l
0.6+ *t
tt
t
A
^
e+ t ^
+. *
e
+t
ee
^
et.4
*
ft *
0.5-
A
e
**
*
*
eee
t
eee
eff
e
04+
*8
**e+
0.2******* +
.
:+4++
HH
*
eef
++
+
.
+-
H+
+ ++*+ .
+**
.
A
eeeee
0.1-
e
ee
eeee
^* +
*
e
ee
i
^***
e.+* *^
"|:::',':::: ;;;;t;;;;;; ++880+++if
*Fl
^^*
^^
AH
^A
if
ee
A+**
+
e +t
+
+
8e
eeeeeeeet
e
e
^**
t ,ee
A*LA&AA*iAi***A***&AL* t
*
ere eeeee"~~
t**8rq
Ut
0.3-
*8*it**
^
+
.*
t+
**+1
t
*t 4+*HWH 44
44
44+4+14
44
+
44
+
+4
*
4H-,1
+
0.0
AAAAAAAAAAAA.
0
IBSCISSA
IS
FIG 66
.
AAAAAAAAAAAAAAAAAA
AAAA
AAAAAAA
.1
.2
.3
.4
XU MIN=3.79E-05IMX=3.76E-02
.5
.6
.7
REACTI FOCUSED
ONTHESHOC
FORAN
UIERWND SI E JETCINED
IN ACYLINDRICAL
TUE.
ABSCISSA:
.8
.9
x - Xmin
Xmax - Xmi n
1
172
1DN-STREAI
PLOT,XU=3.760E-02ISTEP=1000
ORDINATE:
e-
**
**
**
e
~e
!
+
*0+
A
++
+
+ieee
,,e
max
1.0
..........
Imin
-
U
0.9 -
T
.
.-.
-
.
*AAZAAAAUAAAU^AAAA
0.6-
**
,
**
********
i 1* 0 i
+
e--fi,---
f
10 HH
4+1+
+
*+
*
+
+
*++
.1
ABSCISSA
IS
XU
67
eet
AA
i if*
A
ifA
'***H
A
0Ii + +
14+*
**
^
*
A
.A
IAA
.2
.3
.4
IN=3.76E-05MtX=3.76E-02
REACTINOCUSED
BEFORE
THESCK
FORANUNDERE(P
EDSILANE
JET
CONFINED
INA CYLINDRICAL
TUBE.
+HXt+l.
A
,,,,.6
.6
eee*
~~*0eeee
e
0
^^
AA
.5
**4
+ ++1-H491+H4
+*
*++
^^
,1
+H
+
++
0+ +H. +.I-I.++
AA
+1
+
+*F-
*e*e+
#q++ + +
^
+
e
e4**
* *e*e+t
^^*4A+4
*^
^^
, +
A
+
*+
f,
,
H
e
e+
^A
t+
**4 *****Af
+ *
*+4** ^WA*+ +
4.1~
***+-
*+
(} AAA^^^^^AA^AAAAAAAAAAAAAAAAAAAAAAAAAAAAA/~i
FIG
^
AA
+e
0
e+**
A.++A
4rt**,!.iN^^
-.* II
00
+
0.1-
0
t!:
',0
lo'1
.
.
^ if+**
++
". .
+
*
*
**
t* **
^^
.
+
I
.;;?
0.3+
#eee +
· 00.
if000
A
++4-14
.','~/I
tO
+
+
,
r..
^
*+
.E
,
^^ *
^ +*t**
^ e***
.
e
+
101
'LASEF
0.7 -
**
VE~~~~~~~~~~~-1
^Q **********
A
*** * *****
*
R
0.2*
_
+
+
*
H
0.8 -
AAA
11
P
00RDINTE
+
HIN 9.64E+02587E+02 5.45E+040. )OE+00+
2.18E+03 1.29E+03 3.76E+05 7.08?-01
0.5+
AAAAAAAA-AAAAA
.~~~~~~~~~~~
AAA
-
00
,
00149
#1-
e
ee~
eel-
.7,, ,,
.~7
.,.8
.9
.9
*
1
x - xmin
Xmax -
Xmin
173
..
istinq pressure
..
rolled valve
manual
valve (kept closed)
sure probe
pressure controlled
adjusting
schematic
of the reaction cell and gas supply
lines and analog circuit
FIG 68
valve
174
Top view
FIG 69
Experimental set-up of
high speed movie camera
camera settings
oscilloscope
I) 25 frames/sec
2) 100 frames/sec
3) 150 frames /sec
graph recorder
ed movie camera
flow settings
Silane:SiH4 20cc/min
Argon: 1000cc/min
cell pressure: .7 atm
Iight
mi
time interval
generator
chronometer
15 0 watts C021 aser
source
175
-
1.6
E
1.2
E-0.8
i,, 0.4
U
en
o
0
30
60
90
Frequency ( Hz)
FI G 70
ANALYSISOFTHE OSCILLATIONS
FOURIER
OF TYPE1 LARGEAMPLITUDE,FIXED
OFTHEREACTION
FLUCTUATIONS
FREQUENCY
FLAME
120
176
20mV
T+
*-5Omsec
E 0.1
I
I
-I
E
ii
-B
C
G5N
I
IIII
0
10
50
90
130
170
210
250
Frequency ( Hz)
FIG 71
FOURIERALYSIS OF THEOSCILLATIONS
OFTYPE2: SALL AMPLITUDE
FLUCTUATIONS
OF THEREACTION
FLAPME
OVER
A BROAD
I
l*
177
CL©
-ssY-%i:
i,;<_
- r-e
,f
a)
-t9:
r_ ~~E
O__
9
a)
(/
_
6 a)_
a) E
>>
- 37T--
U
C~~~~~~--
0C
-
o
_ Ctj
11
_
.
Q)s
C:-
0
0
C\J
E
_ -.M
-5-_
=_ _~~
;-~~
Q)
_
~
1(--.
f
~_
_
o~~~~~~~~~~~~~~~~~~~~~~~~
_
~
C~
_
_
~~~~~4
0
a)
OE
E
ODFE
- 0
O
9
o:
9
9
q
tt
0I
iN
)
_
---
Ul,
f-,
U-
-
178
I A
E
IS
n
0(
a)
C
C
20
0
40
60
80
Mass flow rate (cc/min)
FIG
73
STABLE AND UNSTABLE
REGIONS
ClhiYWPrirY*LaYuI-I1YIWYC(I--
-^ I
FOR A REACTION
100
120
179
-'
U-b
0.5
cV 0.4
en
0.3
--
g 0.2
0.1
n
0
20 40 60 80. 100 120 140 160
Time (msec)
FIG 74
Flame velocity
vs time
.
180
·-
0
E E
4
*c
a) 3
-C
LL I
0
20 40 60
80
100 120 140 160 180
Time (msec)
FIG 75
:FLAMEHEIGHTAS A FNCTION OF TIME
181
.-
r- 0
I
a)
No
N
0
a)
C
-
.4-
E
0
or)
C
4-
0 0
40
80
120
Time (msec)
FIG 76
:FLMF Mu
-OF
- T-OF TIME
n
TT?emmO
tl.E
160
-r
n rLULIIL"
200
182
* *
UPS1
I
o -ov' o' o . °
"-'".LASER"':
I
HEATED:
|: ". -REGION,
Threshold
value to st(
y
a)
Q
* ..
from prop
upstream
I(
::."'"-: :
4--
".'".'
.....
...
(
0
2
1
3
4
Axial distance (mm)
g(O)
T(x,O)
(
K
FI G 77
_UY_I·
E^_·IIL_
II
pCpU
K
KX
-1)
J
DISTRIBUTION
:AXIAL TEMPERATURE
ANDFLAMEPROPAGATION
LIMIT
+Tw
:.
183
a) cell pressure:
SiH4
Argon
horizontal axis
vertical
:
c) cell pressure:
horizontal
vertical
axl.
axis
b) cell pressure: 533 torr
: 50 msec/unit
axis
SiH4
Argon
760 torr
: 20 cc/min
:1000 cc/min
20
60
horizontal
volt/unit
torr
: 10 cc/min
:1000cc/min
: 0.2 sec/unit
: 20 mvolt/unit
SiH4
Argon
axis
d) cell pressure: 533 torr
SiH 4
Argon
horizontal axis
vertical axis
oscillations of the first type
78
: 50 msec/unit
vertical axis : 10 mvolt/unit
oscilloscopesignalof the light meteroutput.
FIG
: 20 cc/min
:1000 cc/min
OFTYPE1:
SCILLATIONS
OSCILLOSCOPE
SIQILSFROM
LIGHTMETERS
FOCUSED
THEFME.
: 10
:1000
: 50
: 20
cc/min
ccimin
msec/unit
mvolt/unit
184
cell pressure:760 torr
SiH4
: 50 cc/=in
Lin
:1000 cc/=:
Argon
horizontalaxis : 50 msec/unit
verticalaxis
20 mvolt/unit
ceullpressure:760 torr
: 0 cc/min
SiH4
Argon
horizontal axis
:1000 cc/min
: 20 msec/unit
verticalaxis :
5 mvolt/unit
oscilloscopeoutputof the lightmeter signal
and of the microphonesignal
oscillationsof the secondtype
FIG 79
OSCILLATIONS
OFTYPE
2:
OSCILLOSCOPE
SIBWLSFRO A
LIGHTMETER
(UPPER
CURVE)
FOCUSED
ON
THEFLAEAND
FROM
A MICROPHONE
(LIOWER
CURVE)
PLACED
INTHECELL
WHEREONLY
ARSONJETSAREFLOWING.
185
line
A
5 times more sensitive
data recording than
line
C
experimental set-up
of the hot wire
anemometer
FIG
80
SETUPFOR
: PERIMENTAL
A HOT
WIRE
MEASUREENS
WITH
VELOCITY
186
D
0'
-0
4-
OC
.
0~
Q
a)
FIG 81
DISTURB S DUE TO A FULL FILTERMD
THE RESULTING
VELOCITYFLUCTUATiONS
AS MEASUREDWITH A HOT WIRE.
187
, r%_
I-
0
-tC
0
°
a)
4.
81
6
"
I
II
11
"--. Pressure signal
A
n6
4
i
Velocity signal
2
_
I
I
I
10
0
I
20
I
I
I
30
Time (seconds)
F IG
82
SLO RESPINSE
OF THEPRESSURE
CONTROLLED
VALVETO
INDUCED
DISTURBANCE.
THESHARP ELOCITY
FLUCTUATIONS
AREMEASURED
WITH
I
40
I
188
co
co
'e-4
0
0-
04
E-
U)
co
0-X
z
to
z
~
ND
PQ
0
I
I..
E
-
i !ii?
n
2
I
-0
'
*-4
0
189
_
__
·· ·I__ __I
0 ¢
z-
OD
r-
Z
0
C2 N
4<
*o6
0.o
6 Li
0
C)
0o= a
U
0
a°
.j
0
0
P.
EU
EU
oi
T
_.
-
N
0
r:
C)
o
a-
0 0
,_
0
;=
0
a.
_k u
;
I-
- N~
°, i
0
0
._
0d
0
.
C)
U)
m
z
U)
-4Q
0
0-
(.
;
oo
E..D
I.._
0
CD
to
S/IN A£JIDOrA
m
~Z
0C2
NIrhIULN.D
'4
0
J
f
Z
3
190
14J
"J
Vc. C6
5
C;
0.
c
0
_-4
0C66
.0
N
E--
co
0
UD
0
o-
0co 000
O
A
r-
L4
0)
LJ.,
0.
mo
0 _
00
0
O
t.0m
L- M
0
r
Ct)
mL.A
N
;E N
[-,
z
E
O
L0 o
V0
0
L)L
f ,
co
E--
to
0-
ui)
00
L
U-
oo
Co
1-4
00
0
~~~~~~~-4
YI ZfnLVHSdNSL
d
00
In
NIRLNSISO
0
-
L)
_
E