AN ABSTRACT OF THE THESIS OF
Abd Y. Lafi for the degree of Doctor of Philosophy in Nuclear Engineering
presented on June 6. 1990.
Title: A General Theory of Flooding Implementing The Cuspoid Catastrophe.
Abstract Approved:
Redacted for Privacy
Tr-
J Jose N. Reyes, Jr.
The flooding phenomenon can be defined as the maximum attainable flow
condition beyond which the well defined countercurrent flow pattern can no longer
exist.
Thus the countercurrent flow limit (CCFL) or the flooding limit may be thought
of as the flow condition at which the strong interaction between the two phases
Occurs.
Considerable effort has been devoted to understanding and analyzing the
flooding transition in many fields. For example; the flooding phenomenon is one of the
important phenomena encountered in the safety analysis of light water reactors
(pressurized water reactors and boiling water reactors). Accurate predictions of
flooding behavior are particularly important in the assessment of emergency core
cooling system (ECCS) performance.
Currently, the
postulated loss-of-coolant
accident (LOCA) is considered the design basis accident. A physical understanding of
the flooding phenomenon will help assess core refill during the course of a LOCA.
Understanding the physical mechanisms of the flooding phenomenon might help
establish more reliable equations and correlations which accurately describe the
thermal hydraulic behavior of the system. The models can provide best-estimate
capability to the design codes used in the evaluation of ECCS performance.
The primary concern of this study was to:
1.
Understand the physical mechanisms involved in the flooding phenomenon in
order to derive a suitable analytical model.
2. Show that the combination of:
a. Linear Instability Theory
b. Kinematic Wave Theory
c. Catastrophe Theory
can provide a general model for flooding phenomenon.
The theoretical model derived using the aforementioned combination of theories
indicates good agreement between the experimental and the predicted values.
Comparisons have been made using a large volume of air-water flooding data.
A GENERAL THEORY OF FLOODING IMPLEMENTING
THE CUSPOID CATASTROPHE
by
Abd Y. Lafi
A Thesis
submitted to
Oregon State University
in partial
fulfillment of
the requirements of the
degree of
Doctor of Philosophy
Completed June 6, 1990
Commencement June 1991
Approved:
Redacted for Privacy
rof4sor of Nuclear
ngineering
in Charge of Major
Redacted for Privacy
Head of Department of Nuclear Engineering
Redacted for Privacy
Dean of Gradu
School
1
1
Date thesis is presented
June 6, 1990
Typed by Abd Y. Lafi for
Abd Y. Lafi
© Copyright by Abd Y. Lafi
June 6, 1990
All Rights Reserved
DEDICATION
This work is dedicated to the memory of my
father
and to my mother whom I owe all what I am.
" My Lord! Bestow on them
Thy mercy, even as they cherished
me in childhood"
ACKNOWLEDGEMENT
Sincere gratitude is expressed to Dr. Jose Reyes for serving as my major
professor throughout this work. His modesty, encouragement, support, and guidance
were greatly appreciated.
Deep respect and appreciations are extended to Drs: J. C. Ring le, L. R. Davis,
A. C. Klein, and T. Plant for serving as my committee members.
Amongst the people who will always be remembered are my mother, sisters,
brothers, relatives, and friends for their prayers, encouragement, and support. For
always, I manifest my special heartfelt thanks to my brother Emad for his sacrifices and
for being a never-ending source of cooperation.
Special thanks go to my devoted wife Sidoof who deserves the deepest appreciation
for her patience, understanding, and indispensable aid. In addition to her endeavors
during my studies she provided me with a very lovely daughter and son Ola and Amru.
To Ola and Amru as an integral and meaningful part of my life, I give my love. I look
to the world through your eyes.
TABLE OF CONTENTS
Page
1. INTRODUCTION
1
1.1
Two Phase Flow Pattern
1
1.2
Annular Two-Phase Flow
3
1.3 Onset of the Flooding Phenomenon
5
1.4 Phenomenon Description
5
1.5
Significance
7
1.6
Objectives
9
11
2. REVIEW
2.1
11
Introduction
2.2 Modeling the Flooding Phenomenon
2.2.1
2.2.2
Interfacial Instability
Potential Flow Model
14
B.
Viscous Laminar Flow Model
18
C.
Finite-Amplitude Wave Model
18
Limiting Condition Criterion
B.
20
Separated Cylinders Model
20
Drift-Flux Model
22
C. Separated-Flow Model
23
Static Equilibrium Theory
25
A.
2.3
14
A.
A.
2.2.3
12
Stationary Wave Model
25
B. Hanging Film Model
26
C. Roll Wave Model
26
Flooding Correlations
28
1.
Wallis Parameter
28
2.
Kutateladze Parameter
28
TABLE OF CONTENTS (Continued)
Page
3. FLOODING ANALYTICAL MODEL
3.1
Introduction
30
30
3.2 Equations of Motion
32
3.3 Boundary Conditions
40
3.4 Connection to Kinematic Wave Theory
50
3.5 Connection to Catastrophe Theory
55
1. Case of the Cusp Catastrophe
2.
3.6
Case of the Swallow-Tail Catastrophe
Entrainment Effect
56
59
64
4. COMPARISON BETWEEN THE EXPERIMENTAL
VALUES AND THE MODEL PREDICTION
4.1
Flooding in Tubes
67
67
Data of EPRI NP-1283
67
4.1.2 Data of EPRI NP-1284
73
4.1.3 Data of EPRI NP-1336
73
4.1.4 Data of EPRI NP-2262
75
4.1.5 Data of NUREG/CR-0312
81
4.1.1
4.2 Flooding in Annuli
81
4.2.1
Data of NUREG/CR-0312
81
4.2.2
Data of NUREG/CR-0526
87
4.3 Summary
90
5. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK
95
REFERENCES
98
APPENDIX(A): AIR-WATER FLOODING DATA IN TUBES AND ANNULI
102
APPENDIX (B): CATASTROPHE THEORY
140
LIST OF FIGURES
Page
Figure
1.1
Major Two-Phase Flow Patterns.
2
1.2
Schematic Diagram of Countercurrent Flow.
4
1.3
Physical Flooding Stages.
6
1.4
ECC Bypass Phenomenon During a PWR Cold Leg Break LOCA.
8
2.1
Basic Design Geometry for the Flooding Experiment.
2.2
The Coordinate System Used in the Flooding Analysis Based
13
on Interfacial Instability.
15
3.1
The Schematic Diagram Used in The Analytical Model.
31
3.2
The Cusp Catastrophe and Its Bifurcation Set.
58
3.3
The Bifurcation Set of the Flooding Catastrophe Based on the
Physical Parameters Indicating the Flooding Boundaries.
4.1
Flooding Data of EPRI NP-1283 for Different Tube Sizes with
Nozzle Air Supply, Tapered Liquid Inlet, and Sharp Edge Exit.
4.2
Flooding Data of EPRI NP-1284
72
Dukier and Smith Data) for a
Tube Size of .0508m.
4.6
71
Flooding Data of EPRI NP-1283 for Different Tube Sizes with
Indirect Air Supply, Sharp Edge Liquid Inlet, and Tapered Exit.
4.5
70
Flooding Data of EPRI NP-1283 for Different Tube Sizes with
Indirect Air Supply, Tapered Liquid Inlet, and Sharp Edge Exit.
4.4
69
Flooding Data of EPRI NP-1283 for Different Tube Sizes with
Nozzle Air Supply, Sharp Edge Liquid Inlet, and Tapered Exit.
4.3
62
74
Flooding Data of EPRI NP-1336 Compared with the Exact
Theoretical Model.
76
LIST OF FIGURES (Continued)
Page
Figure
4.7
Flooding Data of EPRI NP-1336 Compared with the Theoretical
Model Using an Empirical Coefficient of 1.5.
4.8
Flooding Data of EPRI NP-2262 for a Tube Size of .0127m with
a Stub Entry.
4.9
77
78
Flooding Data of EPRI NP-2262 for a Tube Size of .0127m with
a Plate Entry.
79
4.10 Flooding Data of EPRI NP-2262 for a Tube Size of .0127m with
Stub and Plate Entry.
4.11
Flooding Data of EPRI NP-2262 for a Tube Size of .0305m and
Different Lengths.
4.12
80
82
Flooding Data of EPRI NP-2262 for Tube Sizes of .0127m and
.0305m, and Different Lengths.
83
4.13
Flooding Data of NUREG/CR-0312 for a Tube Size of .0508m.
84
4.14
Flooding Data of NUREG/CR-0312 for a Tube Size of .1524m.
85
4.15
Flooding Data of NUREG/CR-0312 for a Tube Size of .254m.
86
4.16 Flooding Data of NUREG/CR-0312 for an Annulus with Gap
Width =.0254m.
88
4.17 Flooding Data of NUREG/CR-0312 for an Annulus with Gap
Width =.0508m.
89
4.18 Flooding Data of NUREG/CR-0526 for Steady State Case in 1/15
Scale Model.
91
LIST OF FIGURES (Continued)
Figure
Page
4.19 Flooding Data of NUREG/CR-0526 for Steady State Case in 2/15
Scale Model.
92
4.20 Flooding Data of NUREG/CR-0526 Plenum Filling Case in 1/15
Scale Model.
4.21
93
Flooding Data of NUREG/CR-0526 for Steady State and Plenum
Filling in 1/15 and 2/15 Scale Models.
94
LIST OF APPENDIX TABLES
Page
Table
A.1
Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0159m (Tapered
Inlet-Sharp Edge Exit Geometry and Nozzle Air Supply)
A.2
1 04
Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0318m (Tapered
Inlet-Sharp Edge Exit Geometry and Nozzle Air Supply)
A.3
1 05
Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .046m (Tapered
Inlet-Sharp Edge Exit Geometry and Nozzle Air Supply)
A.4
1 06
Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0699m (Tapered
Inlet-Sharp Edge Exit Geometry and Nozzle Air Supply)
A.5
1 07
Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0159m (Sharp
Edge Inlet-Tapered Exit Geometry and Nozzle Air Supply)
A.6
1 08
Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0318m (Sharp
Edge Inlet-Tapered Exit Geometry and Nozzle Air Supply)
A.7
1 09
Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .046m (Sharp
Edge Inlet-Tapered Exit Geometry and Nozzle Air Supply)
A.8
110
Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0699m (Sharp
Edge Inlet-Tapered Exit Geometry and Nozzle Air Supply)
111
LIST OF APPENDIX TABLES (Continued)
Page
Table
A.9
Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0159m (Tapered
Inlet-Sharp Edge Exit Geometry and Indirect Air Supply)
112
A.10 Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0318m (Tapered
Inlet-Sharp Edge Exit Geometry and Indirect Air Supply)
A.11
113
Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .046m (Tapered
Inlet-Sharp Edge Exit Geometry and Indirect Air Supply)
114
A.12 Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0699m (Tapered
Inlet-Sharp Edge Exit Geometry and Indirect Air Supply)
115
A.13 Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0159m (Sharp
Edge Inlet-Tapered Exit Geometry and Indirect Air Supply)
116
A.14 Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0318m (Sharp
Edge Inlet-Tapered Exit Geometry and Indirect Air Supply)
117
A.15 Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .046m (Sharp
Edge Inlet-Tapered Exit Geometry and Indirect Air Supply)
118
A.16 Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1283 for a Tube Size of .0699m (Sharp
Edge Inlet-Tapered Exit Geometry and Indirect Air Supply)
119
LIST OF APPENDIX TABLES (Continued)
Page
Table
A.17 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1284 ( Dukier and Smith Data ) for a Tube Size of .0508m
120
A.18 Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1336 for a Tube Size of .0508m
121
A.19 Calculated Flooding Parameters Using the Experimental Values
of EPRI NP-2262 for a Tube Size of .0127m with a Stub Entry
122
A.20 Calculated Flooding Parameters Using the Experimental Values
of EPRI NP-2262 for a Tube Size of .0127m with a plate Entry
123
A.21 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-2262 for a Tube Size of .0305m and Different Lengths
124
A.22 Calculated Flooding Parameters Using the Experimental
Values of NUREG/CR-0312 for a Tube size of .0508m
125
A.23 Calculated Flooding Parameters Using the Experimental
Values of NUREG/CR-0312 for a Tube size of .1524m
126
A.24 Calculated Flooding Parameters Using the Experimental
Values of NUREG/CR-0312 for a Tube size of .254m
127
A.25 Calculated Flooding Parameters Using the Experimental Values
of NUREG/CR-0312 for an Annulus with Gap Width..0254m
129
A.26 Calculated Flooding Parameters Using the Experimental Values
of NUREG/CR-0312 for an Annulus with Gap Width =.0508m
132
A.27 Calculated Flooding Parameters Using the Experimental Values
of NUREG/CR-0526 for Steady State Case in 1/15 Scale Model
134
A.28 Calculated Flooding Parameters Using the Experimental Values
of NUREG/CR-0312 for Steady State Case in 2/15 Scale Model
137
LIST OF APPENDIX TABLES (Continued)
Ewe
Table
A.29 Calculated Flooding Parameters Using the Experimental Values
of NUREG/CR-0526 for Plenum Filling Case in 1/15 Scale Model
139
NOMENCLATURE
a
amplitude of the disturbance
A
amplitude of the disturbanc defined by equation 2.12
A
constant defined in equation 2.43
B
constant defined in equation 2.43
B0
Bond number
c
wave speed
c
constant defined in equation 2.47
c'
constant defined in equation 2.48
D
tube diameter
E
entrainment
f
friction factor
g
gravitational acceleration
h
average liquid film thickness
i
complex part of the wave velocity
J
superficial velocity
k
wave number
K
Kutateladze number
I
mixing length
m
empirical constant defined in equation 2.47
m' empirical constant defined in equation 2.48
n
constant
P
pressure
p
ampitude of the perturbed pressure
NOMENCLATURE (Continued)
op pressure drop
Q
volumetric flow rate
R
tube radius
r
radial coordinate
Re Reynolds number
s
sheltering coefficient
t
time
T
temperature
u
axial velocity
u
mean axial velocity
u'
perturbed axial velocity
radial velocity
mean radial velocity
v' perturbed radial velocity
W mass flow rate
We Weber number
x
axial coordinate
Greek symbols
a
13
void fraction
volumetric equlity
i
disturbance of the interface
X.
wave length
NOMENCLATURE (Continued)
p
density
a
surface tension
dimensionless wave number
ti
shear stress
viscosity
v
stream function
(I)
amplitude of perturbed stream function
v
kinematic viscosity
Superscripts
*
dimensionless
'
perturbed
mean
Subscripts
g
gas
i
liquid
j
index indicates either I or
I
liquid entrained
gc
gas critical
gI
drift flux
cr
critical relative
NOMENCLATURE (Continued)
w
wall
i
interfacial
A GENERAL THEORY OF FLOODING IMPLEMENTING
THE CUSPOID CATASTROPHE
1.
INTRODUCTION
In general, the flooding is one of the most important effects caused by the
strong interaction between the two phases in a two phase flow system.
Prior to
clarifying the nature of this interaction and its relation to the flooding phenomenon,
different flow patterns within the general framework of two phase flow will be
characterized.
1.1 Two Phase Flow Patterns;
Figure (1.1) illustrates the major two phase flow patterns that can be
observed in upward two phase flow. Bubbly flow is established when the gas phase is
dispersed as bubbles in the liquid continuum. In this flow configuration, the general
shape of the bubbles are spherical when they are small but deviate from this shape
when they enlarge. As the gas flow rate increases, bubble coalescence results in big
bullet-shaped slugs. The flow pattern of slugs separated by small bubbles flowing in
the core and the liquid flowing as a film along the wall is called slug or plug flow. Upon
increasing the gas flow rate further, the slugs may break down into smaller
oscillating slugs. This flow configuration is called churn flow and it can be considered
as an introduction to an annular two phase flow pattern.
2
6
Bubbly
Slug
Figure 1.1
Churn
Major Two-Phase Flow Patterns.
Annular
3
1.2 Annular Two Phase Flow;
An annular two phase flow pattern can be established if the relative velocity
between the phases is significant. Both phases in this pattern are continuous, the
liquid flows along the wall and the gas flows in the central core. Liquid may also be
dispersed in the gas core in the form of droplets. These droplets are generated by
shearing-off of the crests of the wavy liquid-gas interface.
Annular two phase flow
may be concurrent upward flow or concurrent downward flow if the two phases are
flowing upward or downward simultaneously. If they are flowing against each other,
countercurrent flow is established. Within the general framework of annular two
phase flow, countercurrent flow has both simple and complex features. The physical
situation is represented in Figure (1.2)
.
The liquid film is flowing down under the
action of gravity along the wall of a long tube while a gas is blown upward at a greater
speed. The simplicity of this system is due to the existence of the two components of
the flow in separate configurations. The complexity is due to the waviness of the
separating interface at high gas flow rates. The waviness is non-steady and
multidimensional in nature.
Therefore it is a very complicated task, if not impossible,
to analyze the interface without some idealizations and approximations.
One of these approximations is to neglect the complex structure of the liquid
film thickness due to non-steady and nonuniform fluctuations and to assume an
average value for the film thickness at a certain level of the tube over a sufficiently
long period of time (1).
4
O
4o
c
0
0
0
0
<
o
t1
0
w
III
a
ti
t 3
0
Ugo
Q
o
E3
Liquid film
gi
Gas core
0
0
e.
o
O
G
Q7
-..-
Figure 1.2
Radius -°-
Schematic Diagram of Countercurrent Row.
5
1.3
Onset of the Flooding Phenomenon;
During countercurrent flow, if the liquid film is maintained at a constant flow
rate and the gas flow rate is varied, there occurs a certain value of gas flow rate at
which the flooding phenomenon can take place. The onset of the flooding phenomenon
can be defined as the limiting condition of the countercurrent flow pattern. The
limiting condition is defined as the maximum attainable flow condition beyond which
mixing between the two phases will occur and a well defined countercurrent flow
pattern can no longer exist. Thus the countercurrent flow limit (CCFL) or the flooding
limit may be thought of as the flow condition at which the strong interaction between
the two phases occurs. It is possible to observe this phenomenon experimentally in a
flow channel by increasing the upward gas flow rate gradually with respect to a fixed
downward liquid flow rate or vice versa until the countercurrent flow identity is lost.
Due to the waviness of the liquid-gas interface caused by the significant
relative velocity between the two phases, the flooding phenomenon is usually
associated with a large interfacial shear stress compared to that at the wall. It is
also characterized by a significant increase in the gas pressure gradient (2).
In
addition, the flooding phenomenon is accompanied by the continuous shearing off of the
crests of the wavy interface in the form of droplets which flow in the gas core
stream as a dispersed liquid phase.
1.4 Phenomenon Description;
To physically describe the flooding phenomenon, let's consider Figure (1.3). The
gas enters the tube at the lower part and flows upward while the liquid is introduced
6
aq
00t)0 cro
MDDI11111...
J
00111E.
[
)..
0°
GI
q
4
a
,
]
D
4
S'
]
t
,...DT......_.".IS
Figure 1.3
Physical Flooding Stages.
7
to the tube at some level and flows downward countercurrent to the gas core. The
liquid film will be under the action of gravity and gas flow.
Upon increasing the gas
flow rate, large waves are generated at the interface of the two phases. As the gas
flow rate increases further, a flooding transition occurs where part of the liquid film
is carried upward above the injection point and the other part continues to flow down.
At a certain gas flow rate, the liquid film may be completely prevented from flowing
down. Another transition can be reached if the gas flow rate is reduced. This
transition results in
the liquid film flowing down and is called "flow reversal" (3).
1.5 Significance;
Considerable effort has been devoted to understanding and analyzing the
flooding transition in many fields. For example; the flooding phenomenon is one of the
important phenomena encountered in the safety analysis of light water reactors
(pressurized water reactors and boiling water reactors).
Particularly in the
assessment of emergency core cooling system (ECCS) performance. Currently, the
postulated loss-of-coolant accident (LOCA) is considered the design basis accident. A
physical understanding of the flooding phenomenon will help assess core refill during
the course of a LOCA. When a break in the cold leg occurs, the emergency core
coolant system (ECCS) attempts to inject subcooled water into the core. The steam
upflow generated by the core may oppose the flow of the injected water.
For certain
steam flow rates, the liquid might be prevented from penetrating down into the lower
plenum and be swept out of the break. This phenomenon is called emergency core
coolant bypass. Figure (1.4 ) depicts the path of the steam and the injected water
during the course of a LOCA (4).
Vessel
ccumulalor
ECC bypass
Injection
cold water
ECC 'Nee don
Steam 1110
\____,,,___)
WAIL
as
afte..
Condensation
Steam plugging
does not occur
Break
ECC penetration
Hot wall
Delay effects
Steam
CON
Figure 1.4
Lower plenum
entrainment
ECC Bypass Phenomenon During a PWR Cold Leg Break LOCA.
00
9
Understanding the physical mechanisms of the flooding phenomenon might help
establish more reliable equations and correlations which accurately describe the
thermal hydraulic behavior of the system. The models can provide best-estimate
capability to the design codes used in the evaluation of ECCS performance. No
analytical model can take into consideration all aspects of the flooding phenomenon
without some idealizations and approximations.
Therefore before implementing any
analytical model, the applicability of these assumptions for various flow conditions
should be examined. A comparison should also be made between the predicted values
and those derived from the experimental data for the purpose of evaluation.
1.6
Objectives;
The purpose of this study can be summarized as follows:
1. To understand the physical mechanisms involved in the flooding phenomenon
in order to derive a suitable analytical model.
2.
To show that Kelvin- Helmholtz instability theory, kinematic wave theory,
and catastrophe theory can be combined to formulate a general model for flooding
behavior.
3. To develop an analytical model based on the stated combination of theories
and to evaluate this model against the experimental data.
This thesis is divided into five chapters.
Chapter 2 reviews the different
models proposed for the flooding phenomenon. The review is classified into three
flooding model categories:
10
1. the instability of the gas-liquid interface.
2. countercurrent flow limit condition.
3. static equilibrium theory.
Chapter 3 represents the theoretical approach to the phenomenon and presents the
proposed analytical model. The purpose of this model is to predict the flooding
condition. This model is based on:
1. Linear Instability Theory
2. Kinematic Wave Theory
3. Catastrophe Theory.
The combination above will be shown to provide a general model for flooding
phenomena. Chapter 4 will compare the available experimental data for air-water
flooding with the analytical model prediction. Chapter 5 is devoted to conclusions and
suggestions for future work.
11
2. REVIEW
2.1
Introduction:
There have been a considerable number of theoretical models proposed over the
years for the onset of flooding. Bankoff and Lee in their fairly comprehensive review
on flooding mentioned the efforts done in studying this phenomenon, among others, by
Imura, et al.
H. J.
,
Zvirin, et al.
,
,
Wallis, G. B.
,
Bharathan, et al.
,
Dobran, F.
,
Shearer,
Wallis, et al., and Richter, H. J. (5). Additional experimental studies on
flooding have been carried out by Kamei, et al. , Feind , Wallis , English, et al.
Hewitt, et al.
,
Hewitt and Wallis, Shires and Pickering , Clift, et al.
,
,
Tobilevich, et al.
,
Grolmes, et al. and Alekseev, et al. (6). The basic experimental design for such
studies consists of an upper plenum, a lower plenum, and a connecting test section.
The liquid is introduced into the system via the upper plenum, while the gas or the
steam is introduced through the lower plenum as shown in Figure (2.1). It is worth
mentioning that some studies have limited applicability to reactor analyses because of
their use of atypical air/water systems for the sake of understanding the main
characteristics of flooding. However, in the case of steam/water systems like those
encountered in the scaled experiments aimed at improving reactor safety analyses,
the condensation effects need to be taken into account.
In spite of the different methods employed in these treatments, none of them can
be generalized to coincide with the extensive experimental data. This is because of three
reasons:
1. Some of these models are empirical and therefore can be used only in their range of
validity. Even those which have a theoretical basis still rely on the experiments to
correlate some of their coefficients ( e.g., the dimensionless wave number and the liquid
12
film thickness in (7)).
2. The definitions and criteria employed to determine the onset of flooding are not
unified. Some studies are based on flow visualization and consider the generation of large
waves at the liquid-gas interface and the appearance of a chaotic flow pattern as the
starting point of flooding. Other studies consider the entrainment of the liquid in the gas
field as an indication of the onset of flooding. Other criteria are based on the sudden
increase in the pressure drop at a certain flow rate (8,9).
3. The geometrical conditions (inlet and outlet conditions), the method of liquid film
introduction, the stabilizing effect caused by the surface tension, and the destabilizing
effect caused by the viscosity which have a very significant impact on the flooding are
frequently ignored.
2.2
Modeling the Flooding Phenomenon;
Different approaches for modeling flooding phenomenon in vertical annular two
phase flow have been developed (10). Modeling the instability of the liquid gas interface
is one of these approaches. In another approach the limiting condition criterion for
countercurrent flow has been employed. The third approach is to consider the static
equilibrium theory.
The first approach includes potential flow, viscous laminar flow, and finiteamplitude wave models. The separate-cylinders, the drift flux, and the separatedflow models can be categorized under the second approach. Last, but not least, the
third approach includes the stationary wave, hanging film, and roll wave models.
13
Air out
Water in
Upper plenum
Water out
Test section
Air in
Lower plenum
Water out
Figure 2.1
Basic Design Geometry for the Flooding Experiment.
14
2.2.1
Interfacial Instability;
A. Potential Flow Model',
Based on the assumption that the two fluids are inviscid and the flow for both is
potential flow, Imura, et al. (11) employed the instability concept to predict the onset
of flooding in a tube of constant cross section as shown in Figure (2.2). In this
analysis the continuity and Bernoulli equations along with appropriate boundary
conditions were used. The disturbance at the interface was expressed as:
= A(t) sin k(x-ct)
(2.1)
where A(t), k, and c are the the amplitude of the disturbance, the wave number, and
the wave velocity respectively.
The final equation for the flooding was expressed in terms of the relative velocity
between the two phases that leads to unstable waves on the liquid film surface as
ug + ui-=V
pg
k-
1
R-h
1
(2.2)
where R, h, cr , and Pg are the radius of the tube, the mean film thickness, the liquid
surface tension, and the gas density respectively.
By expressing the gas and liquid velocities in terms of the corresponding area
averaged local volumetric fluxes (superficial velocities)
Ja
U9=
===
g
a
(2.3)
and
Ji
u1= 1-a
(2.4)
15
Tube well
tUg
Gas
Figure 2.2
The Coordinate System Used in the Flooding Analysis Based
on Interfacial Instability.
16
and using the following dimensionless parameters:
r
13g
g ApD
JI =J
17:
g b.pD
Pg
p
PI
= kh
Equation (2.2) can be written as
Jg
D
JI
a
1-a
a= (
2h )
1 -D
1.
P
R-h
h
(2.5)
But
2
(2.6)
and thus,
D
2
R-h
(2.7)
.
Therefore equation (2.5) can be written as
Jg
a
where Jg
1-a
r
.12
c
D
17-7
1
(2.8)
J1*, D*, p*, C are the dimensionless parameters for gas superficial
velocity, liquid superficial velocity, tube diameter, density, and the wave number.
17
Equation (2.8) is the nondimensionalized form of the flooding equation based on the
potential flow model. It can be used to evaluate the gas flooding velocity for a certain
liquid flux if the nondimensionalized wave number
and the void fraction a are known.
Imura and his coworkers used the following experimentally developed correlation for
the nondimensionalized wave number:
.12
.046 D
a
pg
)
(2.9)
Since D is known, the only parameter required to find a is the mean liquid film
thickness. The mean liquid film thickness was found using the following correlations
400
h* = 1.442 Ref' /3
for Re,
h* = .532 Reif /2
for 400 < Rei 5 2000
(2.10)
(2.11)
where
"P r
since pi
g PI2
11
pg
As it is noted this model has a theoretical basis. The liquid and gas viscosities, the
tube diameter, and the surface tension effects were included through the
dimensionless wave number and the liquid film thickness correlations.
18
B.
Viscous Laminar Flow Model; (12)
Another model based on interfacial instability assumes laminar flow in the liquid
film.
The model is developed by imposing a small perturbation to the interface
between the phases. This perturbation is represented by:
(2.12)
x, t ). A e ik(x-ct)
Expressing the perturbed velocities in terms of the stream function Air and introducing
them into the Navier-Stokes equations leads to the Orr-Sommerfeld equation after
some linearization and elimination of pressure drops. Under suitable boundary
conditions and some approximations the Orr-Sommerfeld equation can be solved to
obtain the amplification rate kci. The amplification rate determines the wave growth
rate as a function of gas frictional velocity for various liquid Reynolds numbers. The
flooding velocity for each liquid Reynolds number can be chosen to be the minimum
frictional velocity for each curve in the unstable region kci > 0. Therefore the flooding
velocity can be expressed as a function of Re, kci, and the fluid properties.
C.
Finite-Amplitude Wave Model: (13)
Zvirin, et al. obtained the flooding curve in the Jog* /2
1
J1.
1
/2 plane using the
global momentum balance equation expressed in nondimensionalized form as
.2
2 fiJg
a5/2
2
2 fw..11
=1
-a
(1-a)2
along with the stability criteria given by
(2.13)
19
Jg
a
J1
+ = u a,
1 -a
(2.14)
or
J p
Js
=u
--,=+
a 1-a
a
ps
I
gApD
(2.15)
Equations (2.13 and 2.15) were used to eliminate a.
The critical relative velocity used in this analysis was given by Jeffery as
4 1.11 c k
ucr=c+V s pgtanh (kh)
(2.16)
where
C=
aktanh (1<h)
(2.17)
and
h=4
7tA.vi
(2.18)
s= .3 (dimensionless sheltering coefficient).
For the interfacial and wall friction factors the following correlations had been used
fw = .005
(2.19)
fi=.005+14.6(1-a) 1.87
(2.20)
The unreliability of this model is due to using Jeffery's analysis for the
critical velocity.
In this analysis the wave was considered to move in the direction of
the air. Therefore a modification to include the countercurrent flow pattern should be
made.
20
2.2.2
Limiting Condition Criterion;
A.
Separated Cylinders Model: (14)
This model is considered as the simplest model for separated two phase flow.
In
this model each phase is represented by a cylinder. These cylinders are arranged in
such a way that their cross sections add up to the cross sectional area of the real
tube. Each phase flows separately without interacting with the other in its imaginary
cylinder. If we assume the radius of the liquid phase cylinder is R1 and that of the gas
phase cylinder is R9 then
R:
(2.21)
R2
and,
R12
1- a =
,
R'
(2.22)
where R is the physical tube radius.
By taking into account the frictional effect only, the pressure drop in each cylinder
can be assumed to be the same as in the actual flow. Wallis expressed the
nondimensionalized flux for the gas and the liquid in terms of a nondimensionalized
pressure drop and a constant mixing length for both phases ( Ig and li ) as follows:
J
g
=7R6- In,,,
ar a
J*
.
7Rl
7/4
47-7* (1co'
m
(2.23)
(2.24)
21
It was argued that at the time of flooding, the mixing length extends over the
whole tube, thus a value of the mixing length was taken to be = (R/7) according to
Nikuradse for each phase.
4--AP a 7/4
J
(2.25)
J1=41----7
1-AP (1-a)
7/4
(2.26)
Elimination of AP from the equation above results in
Jr: 2
ji* 2
(1
a)
7/2
+
a 72
1
(2.27)
This equation represents a family of curves of the form
f (Ji , Jg a)=0
By using this equation and its derivative with respect to
(2.28)
a set equal to zero, a can be
eliminated to get the envelope equation as
J,*419 + J9 4'9
(2.29)
This equation represents the flooding line that describes the upper limit of maximum
allowable liquid and gas flow rates.
22
B.
Drift -flux Model
Generally the drift- flux model considers the relative motion between the two
phases instead of the individual motion for each phase.
The drift flux can be
expressed in terms of the relative velocity Lig' and the void fraction a as
Jo=u4 cx ( 1-a)
(2.30)
.
But
J0=Jg(1-a)-aJi
.
(2.31)
For the purpose of generality, equation (2.31) can be written as
Jeugia(1-a)n
(2.32)
where n is flow condition dependent.
Substituting equation (2.32) into equation (2.31) yields
Jg(1-a)-aJt= ugl a (1-a) n
(2.33)
Equation (2.33) represents a family of curves for various values of a. The locus of
the tangents to this equation determines the maximum limit of operating conditions of
the countercurrent flow regime.
23
C.
Separated - Flow Model
Unlike the homogeneous model which disregards the detailed information about
the behavior of each component, and the separated cylinders model which neglects the
interaction between the two components, the separated-flow model treats each phase
separately and takes into consideration the interaction between them. This model has
been used by Barathan et al. (15) for the flooding analysis of an air/water pair. It
was assumed that there was a steady, one - dimensional flow inside a tube with a
constant cross section. The cross-sectional area occupied by the liquid phase was
labeled Af, that of the gas phase was labeled Ag and the total area of the pipe was
labeled At. By using an average value of the liquid film thickness, the global
momentum equation for the gas phase (assuming a constant velocity and negligible
compressibility) can be written as
4.;
dP
dx +Pgg+7727=0
(2.34)
But
Ag .(D-2h )2
At
D
D-2 h=D-rd ,
thus equation (2.34) can be written as
dP
717
+
Pg g
BZwe
=0
(2.35 )
24
where tj is the interfacial shear stress which is a function of the interfacial friction
factor (0, the gas superficial velocity J9, and the void fraction a.
The global momentum balance equation for the whole flow can be written as
dP
dx
where ..cw
4 tw
+.. a+(1-a)pi1g---D-.0
41w
(2.36)
is the wall shear stress which is a function of the wall frictional factor
(fw), the liquid superficial velocity J1, and the liquid fraction (1-a).
From equation (2.34)
dP
-42i
dx =37 -Pgg
(2.37)
To relate the interfacial and wall shear stresses substitute equation (2.37) in equation
(2.36) to get
(1-a)gAP=5 CtiN+.7aw
J
(2.38)
where
AP=Pi-Pg
By expressing the shear stresses in terms of the corresponding friction
factors and superficial velocities, equation (2.38) can be written in the same form of
equation (2.28) namely
f (Jg 0.11,a)=0
25
This represents a family of curves in Jg*, J1 plane for various values of a . An
envelope for this family can be found by differentiating equation (2.5) with respect to
a and setting the result equal to zero, i.e.,
df
da
=0
(2.39)
Elimination of a from equation (2.38) and equation (2.39) leads to an envelope equation
that represents the locus of tangents to the operating line in the Jg*, J1 plane.
Therefore the envelope determines the limit of the maximum possible operating
conditions under the countercurrent flow regime or the upper limit counter-current
flow.
In other words, for any value of J1* on the envelope curve there exist a
maximum possible Jg* and vice versa.
2.2.3
A.
Static Equilibrium Theory:
Stationary Wave Model;
Shearer and Davidson (16) assumed a stationary wave on the surface of the
laminar liquid film maintained by the pressure gradient due to the gas drag on the
wave front. Implementing a numerical solution and using suitable boundary conditions
at the crest and the trough of the stationary wave indicated that the wave is unstable
except at some values of gas flow rates. The instability of these waves leads to
bridging of the liquid film. This model was in a good agreement with the experiment
for Re, < 250.
26
B.
Hanging Film Model.,
The hanging film condition may take place when the film is supported against the
gravity by the gas flowing up in such a way that no liquid flows down. This model is
useful in finding the critical gas velocity at which the hanging liquid film can exist.
As
indicated in (17) this model uses Bernoulli's equation for the liquid and gas phases to
find the continuity of the pressure at the interface of the film and the dimensionless
velocity potential. The critical gas velocity emerges from this analysis was found in
terms of Kutateladze number Kg = 1.87.
However; another value for Kutateladze
number under this condition = 3.2 was reported in (18).
C.
Roll Wave Model.,
Similar to the separated - flow model, Richter in his analysis used equation
(2.38) to express the global force balance for steady, one dimensional, annular twoHe coupled this equation with a stability criteria developed by balancing
phase flow.
the drag force induced by the gas and the surface tension force. This criteria can be
expressed mathematically as follows:
2
Vg la g
a
2
4h
The correlation proposed by Wallis (19) for the interfacial friction factor was used:
(
fi=fw 1 +
300 h
)
The following equation for flooding was derived:
(2.40)
27
3
.25
.6 .2
fw Bo Jg
2
.4
+ fwBoJg + 150 fwJg =1
.
(2.41)
From this equation the dimensionless critical gas superficial velocity at the condition
of hanging film ,i.e; at zero liquid down flow rate at= 0) is given by
J*2.-75
ge
+V( 75)2
L Bo
Bo
)
1
Bo fw
(2.42)
where Bo is the Bond number (D2).
It
is noted that the interfacial friction factor correlation of equation (2.40) was
deduced for concurrent flow.
However, measurements indicate that the friction
factor correlation for countercurrent flow is. much higher than that of concurrent flow
(20).
The major consideration that must be taken into account for countercurrent
flow flooding analysis is the gravitational effect.
The correlation for the interfacial
friction factor given by equation (2.40) did not account for this effect. The following
correlation was proposed by Bharathan for the countercurrent flow of an air/water
pair at atmospheric pressure for a wide range of tube diameters(.64-15.2 cm):
.8
f1= .005 +A h
(2.43)
where
A= .2754 e2"844/D
B=1.63 +
4.74
74
D
h.= h
a
The importance of using the constitutive correlation given in equation (2.43) rather
28
than that of equation (2.40) was demonstrated in (21).
2.3
Flooding Correlations;
Many correlations were presented over the years to characterize the flooding
phenomenon. Depending upon the characteristic length scale, two dimensionless
parameters have been chosen to correlate flooding behavior.
1. Wallis parameter:
By using the tube diameter as a characteristic length which is a good choice when
D* ranges from 3 to 20 (22) then
P
gi pD
(2.44)
where the subscript j refers either to g or I for the gas or liquid phase respectively.
This parameter physically signifies the ratio of the inertial forces to the buoyancy
forces.
2. Kutateladze parameter;
If the characteristic length is chosen to be the Taylor instability wave length or
the natural characteristic length
[a/( g Ap)]1/2 (which is the criteria when D'
(23), and justified through successful application in two-phase flow (24)) then
30
29
or
2
)
1/4
CI
g
(2.45)
Apo)
This parameter signifies the ratio of the inertial forces acting on capillary waves of
natural characteristic wavelengths.
It is clear from the above definitions that Jk. and
Kk can be connected via the square root of the nondimensionalized diameter D* as
follows
K.
.
J.
(2.46)
1
0/ D
The most popular form that can adequately correlate the flooding experimental data
in single channels was given either in terms of the Wallis parameter
,1n
J9
112
+mJI
=c
,
(2.47)
or in terms of Kutateladze parameter
K
1/2+
m K11/2= c
'
(2.48)
where m, m', c, and c' are determined from the experiment and depend on the fluid
properties, the geometry, and the inlet conditions. Equations (2.47 and 2.48) indicate
that the sum of the square roots of the nondimensionalized superficial velocities is
generally constant.
Thus the gas-liquid flow rates are connected by the above relations. They can be
used to predict the critical value of the gas flow rate that results in the hanging film
phenomenon by setting J1 1 /2 and K11 /2 equal to zero (25).
30
3.
3.1
FLOODING ANALYTICAL MODEL
Introduction:
Due to the complexity of the flooding phenomenon, formulating the governing
equations and determining the appropriate boundary and interfacial conditions for the
analytical model is not an easy task. It requires significant mathematical efforts and
the extensive application of idealizations and approximations that must be justified in
light of flow conditions.
In order to direct these efforts, one must first understand
the physical situation prior to building the analytical model.
To illustrate the physical situation we consider Figure (2.1) in which the liquid
is introduced into the upper plenum while the gas is blown upward from the lower
plenum. A well defined countercurrent flow in the test section can be established at a
certain level of gas and liquid flow rates. This regime can experience a catastrophic
change that leads to destroying the interface between the two phases. This drastic
change can occur when the relative velocity between the phases exceeds some limited
value.
As shown in Figure (3.1), a thin liquid film flows countercurrently along the
wall of a long vertical tube enclosing a cylindrical gas core.
In order to derive the
analytical model, it is assumed that the flooding phenomenon is caused initially by the
action of the unstable growth of infinitesimal waves generated on the liquid-gas
interface. This growth is due to the relative velocity between the two phases. The
drag induced by the gas flow can lead to shearing-off of the crests of these waves in
the form of droplets causing liquid entrainment.
31
Liquid
Gas core
film
Jgc
;
x
rl
I
I
I
I
I
I
I
I
I
I
I
I
I
I
a
R-h
Figure 3.1
h
The Schematic Diagram Used in The Analytical Model.
32
The ingredients of this model will involve three theories: the linear instability
theory of Kelvin-Helmholtz described in (26), the kinematic wave theory (27), and
Catastrophe Theory (28).
3.2 Equations of Motion;
In cylindrical coordinates (r,9,x), the Navier-Stokes equations of motion and the
continuity equation using constant fluid properties and ignoring the gravity terms may
be written as
1. Axial momentum eauation;
a u.
at
+
w. au.
+J
1
I ar r a e
au.
a P.
lax
pi ax
au.
au.
a2u.1+_.1+
.1
ar2
r ar
a 2u.
a2u.
1
+
1+
1
ax2
r2 ae2
2. Tangential momentum eayation;
aw
at
a w w- a w
ar r ae
+
a2wi
i
a IN;
a Pi
1
1
r
lax
a2w;
a 2Wi
ae
2 a vi
WI
+ a X2
ar2 +7 ar + r2 -,e2
0
1
3.
1
aw1.
1
1
1
1
2+
r
2a
r
JR adial momentum eauation;
ay.
a v.
w. ay.
1
1
1
1
w.2
i a p.1+
pi ar
ay.
1
1
at ±viTr".7 -SW +-7-+ui a x
a2v.
1 ar
2' +r
a2vi
a
r
+ r2
ae
2+
a
a Zvi
ax
2
r
2
r
2
....._
A
33
4.
Continuity equation.,
ay.
vi
1 aw
au.
ar
r
r
ax
a0
where
v, w, and u are the radial, tangential, and axial components of the velocity
p is the density and .1) is the kinematic viscosity
j is either I for the liquid or g for the gas phases.
Being that the fluid motion is in the axial direction, and due to axial symmetry, the
tangential component of the velocity and all the derivatives with respect to 0 can be
set equal to zero. Therefore the tangential component of the momentum may be
omitted completely.
Moreover, the viscosity term can be disregarded. This is
justified since our primary concern is the wavy interface region.
In this region, the
temporal and convective term in the equations of motion are larger than the viscous
term as shown below by the following simple scaling analysis.
the temporal term :
7 -7 ,
au
av
the convective term:
au
au
ax
ar
the viscous term
V2u
av
ax
:
- v2v v2
where w = the period,
= the wavelength, and
av
ar
-Vw
34
V = the characteristic velocity
For large A and small .t)
V 03o
Vv
A2
or
Thus the viscous term can be neglected. For the case of high viscosity liquids, the
viscosity effect might be taken into consideration in the interfacial boundary condition
or in the correlation for the liquid film thickness. Therefore the set of the equations
above may be reduced to
Axial momentum equation:
au,.
at
au.
au,.
1
+ V.
+U.-=-ar
Jax
aP.
ax
(3.1)
ap.
'
p- ar
(3.2)
Radial momentum equation:
ay.
at
+
ay.
ay.
Jar
Jax
1
Continuity equation:
,
_
+ au;
_=
ar r ax
a v;
v,.
(3.3)
By perturbing the interface between the phases when imposing some disturbance,
the velocity can be considered as a sum of a mean velocity (unperturbed) which is
assumed to be a function of r only and a perturbed velocity which is function of r, x,
35
and t as follows
u=u-1 (r)+ u. (r,x,t)
(3.4)
"
v=v(r x t)
(3.5)
where the radial mean velocity
Vi= 0
because the flow is in the axial direction.
Similarly, the pressure can be written as
P1 -= Nx) +
1
'x '
(3.6)
Now we introduce the Stokes stream function w that can be written in a similar
fashion as
(3.7)
where
Nx)
,
x,
(r)
ivj(r,x,t)
is the mean pressure
is the perturbed pressures.
is the mean stream function
is the perturbed stream function
Assuming that the perturbed stream function and the perturbed pressure exhibit
periodic behavior in the direction of flow and in time, we can write
w;(r,x ,t) = (11(r) e
ik(x-ct)
(3.8a)
36
(r,x,t) = pio) e ik(x-ct)
(3.8b)
where
4j
is the amplitude of the perturbed stream function
WO is the amplitude of the perturbed pressure
k is the wave number = 27c/X.
c is the wave velocity which is complex and defined by:
cR +
where (cR) is the real part of the wave velocity. The imaginary part of the wave
velocity (c1) determines the instability of the system.
will remain constant.
If ci = 0 the wave amplitude
If CI > 0 the wave will grow resulting in an unstable interface.
If CI < 0, the wave will decay.
To express the perturbed velocity in terms of the stream function we use the
continuity equation (equation 3.3 ) along with the definitions given in equations (3.4
and 3.5):
a[rvi(r,x,Ol
r
ar
+
a(ui(r)+u'fr,x,t)]
ax
=0
However,
because
ui(r)
is a function of r only.
Therefore, the equation above can be written as
37
a[ rvi(r,x,t)]
aui(r,x,t)
ar
ax
r
=0
(3.9)
To satisfy equation (3.9), the perturbed velocities v' and u' must be defined in
terms of the stream function ly as
U.=1 am
ar
v.=--r
(3.10)
alg;
(3.11)
ax
Using equations (3.7, 3.10, and 3.11) we may write the terms of equation (3.1) as
follows
The first term:
au.
at
=
a u.
at
a u.
+
at
a (1
=
at
alitr,x,t)
-ikc a4);
imx-ct)
are
r
ar
r
The second term :
aut
Jar
a u;
a
,
=V
+
ar
ar
a u;
=
ar
J
=
_k
r
a
(r)
ar
eik(x-ct)
The third term :
,
au, _
= (u + u
J ax
u
J
J
,a
_ a u; - au;
,au;
au,
+
u.
+
u
+
u
+
=
u
lax
J
ax
Jax
ax
ax
ax
a u;
_au
=u
k
= "-r
ax
where the mean velocity
ik(x-t)
actl
ar
e
u-
i(r)
is a function of r only and all the terms with
powers higher than the first and all the products of small quantities have been set
38
equal to zero.
The fourth term:
pfr) i k
d Pi
R dx
pi
where the mean pressure is assumed to be constant along the x-axis.
Now equation (3.1) can be written, after rearranging terms and dividing by
ik elk( x-ct) as follows
(u; -c) ao
ar
r
au;
1
4)-(r)
r
- p(r)
ar
(3.12)
131
The same procedure can be applied term by term to equation (3.2) to write the radial
momentum equation in the following form:
(uic)44r) k2
1
a pi
(3.13)
R ar
r
From equation (3.12)
pfr)
(uj-c)t.
a pi(r)
1
ar
= Pj
(u; -c
41(0
_
ar
a24r, 1
aui,_ 1 et:,
r2
7a
ar
1
(r)-7
(3.14)
Substituting equation (3.14) in equation (3.13) we get after some arrangement
a2r)
1
2
a
----1-4a4(r)
41( 0
-k
r
1
a r2
ctr(r) a uj
'
ar
c
ai
4)i(r)=0
(3.15)
39
If we consider the mean velocity to be constant then equation (3.15) can be written
as
a2(6
art -7 -77 -1(2 41(0=0
(3.16)
The solution of equation (3.16) can be found by assuming
(r)=fi(r) r
a4 (r) r)
a
=fi(r)+r
f(r)
air
and
D244r)
a fi(r)
a2f; (r)
art
Dr
art
Upon substitution in equation (3.16) we get.
2
a2
a r2
(r)
a f(r)
Dr
-
+k2
r2)
1
=0
This equation is recognized as the modified Bessel equation of first order whose two
independent solutions are li(kr) and Kl(kr).
fj (r)= A1 I1 (k r)+A2 Kl(k r)
Thus,
4(0= A1rl1(kr)+A2rK1(k r)
(3.17)
Two equations representing the solution in each domain can be derived from equation
(3.17):
40
For the gas core ( 0 < r < R-h )
09(0= A1 rI1(k r)+A2rKi (k r)
(3.18)
For the liquid film( R-h < r < R )
oi(r). A3rl1 (kr)+A4rKi (kr)
where Al
A2
,
,
A3
,
(3.19)
A4 are constants to be determined from the following boundary
conditions:
3.3 Boundary Conditions*,
To determine the constants above, the following boundary conditions may be
applied:
1. v9
2.
v1
=0
centerline boundary condition
1=0
wall boundary condition
141
3. v9
1
=
rA3-1,
DT1
Dt
=
at
+CI al
g ax
(3.20)
(3.21)
(3.22a)
kinematic interfacial boundary condition
4.
v1
I
=
al II .311
Dt
at
ax
kinematic interfacial boundary condition
(3.22b)
41
where
R is the tube radius,
h is the average thickness of the liquid film,
is the substantial derivative.
D/Dt
is the assumed interficial disturbance which can be expressed
mathematically as:
(3.23)
(x , t )=a eik(x-ct)
Here (a) is the amplitude of the disturbance at the interface, (x), is the distance in the
direction of flow and (t), is the time.
In this expression, the disturbance is assumed to propagate in the direction of the
flow (axial propagation). It is noted that the conditions stated in equations (3.20,
3.21, 3.22a, and 3.22b) are based on the following arguments:
a.
The disturbance vanishes far away from the wavy surface at the center line.
b. No slip condition at the wall.
c.
The kinematics of the wavy interface surface.
Now applying the boundary condition (1) stated in equation (3.20) to equations (3.8a,
3.11 and 3.18) results in:
k
,
v
r4
1
(r) elk(x-ct)
,
i k e ik6(-c°
i
(k r)+A 2 Kl(k r) I =0
r4
Due to the finiteness of the fluctuation and since the function K1(0) is singular
42
(K1 (0)= 00), A2 should be set to zero.
(Og (r)= Al r11 (k r)
To find A1, apply the boundary condition (3) stated in equation (3.22a) to get
v
=-
g
k og(r)
ik a eikix-c0(u-c)=-ikAi lifk(r-h)leik(x-ct)
Dividing by ikeik(x-ct) results in
a ( c-u )
Al
k (R-h)]
a(c--6)
r li(kr)
(3.24)
(1)g(r)- 11[k(R -h)]
The same result can be obtained if we apply another boundary condition that is:
Va
Is finite or if we apply the symmetrical boundary condition, namely:
r=0
a 4g (r)
0
ar
nstead of the boundary condition stated in equation (3.20) along
instead
with the kinematic boundary condition stated in equation (3.22a).
Now to find A3, we apply the boundary condition (2) stated in equation (3.21) to
equations (3.8a, 3.11, and 3.19):
k
r=R
Therefore
K1(kR)
A3 =
AA
li(kR)
eik6')
I = ik [A311 (kR)+ A4 K (kR) 1e ik(xt),- 0
43
41(r)=A4[ r K low
K 10(R) li(kr) r
11(kR)
To find A4, we apply the boundary condition (4) stated in equation (3.22b) to get
A4
a(c+ti1)11(kR)
(R-h)]
11((R) k l[k (1:1-h)] -ki(kR)
a( c + ui ) Ili(kR).Ki(kr)-ki(kR). li(kr)] r
01(0
li(kR)ki[k(R-h)] ki(kR)Ii[k(R-h)]
(3.25)
Note: The coordinate in the previous analysis was chosen in such a way that the
x-axis lies on the center line as shown below:
Lr
CL
if we choose the x-axis to lie on the tube surface as shown below:
ci
then g(r) and 01(r) can be written as
a( c - u9) [11(kR) .Ki(kr)-ki(kR). li(kr)] r
li(kR)ki[k(R-h)] ki(kR) li[k(R-h)]
44
a(c+u?
4)1(011[k(R-h)] r li(kr)
In the rest of the derivation we will use equations (3.24) and (3.25) for 4(r) and
o(r). The next step is to find the gas and liquid fluctuating pressures. The fluctuating
gas pressure can be found from equation (3.13) and equation (3.24) as follows
R-h
,
Pg
Pg
.1 0
a(c-ug)2k2 r R-h
li [k(R-h)]
p
Pg 4-Pgb-
Pg 1 = P
g
R-h
R-h
a(c-Ug)2k
gli
[ k(R-h)]
-
.*.
li(kr)a r
J0
2
a(c-ug) k
1°(kr)
lo
10[k(R-h)1-1
li[k(R_h)]
where the gas pressure fluctuation was assumed to vanish at the center line.
For a large argument: (29)
ex
10 (x) -11(x) - .4 -1.x
4
thus for a large k(R-h), which is a usual case for annular two phase flow, we have
k (R-h)
10[k(R-h)] -l1[k(R-h)]..4
therefore;
10[k(R-h)]-1
11 [k(R-h)]
vk(R-h)
45
pg4= pg a( c-Lg )2k
(3.26)
Similarly, the liquid fluctuating pressure can be found from equation (3.13) with
equation (3.25) taking into consideration that the liquid is moving opposite to the wave
propagation as follows
pi a (c +42k2
R
[ li(kR)ki(kr)-Ki(kR) li(kr)
r
R-h
a Pi=
R-h
1
(kR)Kl[k(R-h)] -Ki(kR)li[k(R-h)]
pia( c+ ji)2k Il1(kR)EK0(kR)-Ko[k(R-h)B+K1(kR)00(kR)- lo[k(R-h)]]}
PI
1.1(kR)Ki[k(R-h)1-Ki(kR)Ii[k(R-h)]
RI h
where
Pi IR=0
Dividing by K1 (R) 11 (R) results in.
R-h
,2. N1-N2+N3-N4
t.I
akutI)
K
N5 - N6
where
N1=-
Ko(kR)
K1(kR)
N2-
Ko[k(R-h)]
K1(kR)
lo(kR)
N3=11(kR) =1
T
-ekh,\FI-h
46
lo[k(Rh)]
_kliNrg-h
e
R
11(kR)
K1[k(R h)]
N5
Ki(kR)
-e
kh\p"-R-h
li[k(R-h)]
N6-
li(kR)
e_
kh
R-I1
where for a large argument: (30)
Ko(x)=Ki (x) - 1'25
ex Nrx
1
pi
2
I
R-h
=pia(c+ui) k
eh,F-h
+1
e41/.[Fr
R-h
ekh,FR _e-khrFr
R-h
R-h
2 -( ekh + e-kh
)fri7h
=pia(c+61)2k
(e1' -e4h)ri--3
R-h
For a shallow liquid film
\F1
R-h
Now we have two cases
Case (1);
For small kh
1
sinh (kh)
.% pi Imo=
-coth (kh) - -
kh
2
pla(c+Co2k22 h
1
(3.27)
47
Case (2);
For large kh
1
sinh (kh)
-coth (kh). -1
2
Pi
I
R-h
= Pia(C+UI) k
.
(3.28)
Now these fluctuating pressures can be related through the following dynamic
boundary condition at the interface:
+R2+g Afq
,13-=a(
Ri
1
1
are the radii of curvatures,
where R1 and R2
or
=al +R-h+n
+ )g
aZn
1
axe
By using equation (3.23) we can write
k2 a elk(m-
Upon expanding
(
1+
a eik(x-ct)
R-h
+
ae
R-h
ik(x-co
R-h
)
l+gepae ik(x-co
48
we get
a eildx-ct)
R-h
a ei
since
wx-co
is small.
R-h
Neglecting the constant 1/(R-h) which is immaterial in the fluctuating region (31) we
can write
,
,
(PiP )
=
cs( k2
g
)±g 4]. e
(Rh)
1
ik(x-co
2
or
(p1-139)
4=a[ a( k2-
1
(Fi-h)2
)+g AP]
(3.29)
by using equation (3.8b) and dividing by eik(x -ct). Here, the left hand side represents
the destabilizing force while the right hand side represents the stabilizing force. The
last term of the right hand side represents the hydrostatic pressure due to the
perturbation. The gravitational effect was added to account for the stabilizing
enhancement due to gravity since in the countercurrent flow this effect becomes
comparable to the shear force (32). This is in contrast with concurrent flow in which
the gravitational forces are exceeded by shear forces.
From equations (3.26, 3.27, and 3.29 and by taking into account that the gas at the
interface is acting outward and perpendicular to the interface we may write:
49
2 k2h
--2+pg(c-ug) 2 k=a(k2-
pi(C
1
(R-h)
2
)±g AP
Dividing by k:
pt(c+ui)
2 kh
2
+pg (c-u g )2 =Cri k -
1
(3.30)
(R-h)k
or from equations (3.26, 3.28, and 3.29)
pl(c+ul) k +p (c-u9)2 k=a(k22
1
(R-h)
2
+gAp
Dividing by k:
Pi(C+02+ pg(c-ug) 2 =a(k-
1
(R-h)
)+2-'61:
(3.31)
The right hand side, which represents the stabilizing effect on the system, will be
minimum when
r
a(k- (R-hrk
)+2A-P,,
^
That is;
a+
a
(R-11)2k2
--2-LP =0
k2
Solve for k to get
k=\/ g
a
(R-h)2
1.0
50
Thus equation (3.30) becomes
7,2 h
Piku-hut)
7
g Ap_
a
1
(R-h)2
1
2
2g
+pg (c-ug )2 =
[(R-h)g ep- R4ia
(R-h)
a
(3.32)
Equation (3.31) becomes
pi(c+u1)2 + pg(c-ug)2=2
0:14021
[(R-h)g ep-
1
a
(3.33)
Equations (3.32 and 3.33) have been derived from the Kelvin-Helmholtz Instability
Theory. In the next section, kinematic wave theory will be applied to these equations.
3.4
Connection to Kinematic Wave Theory;
Kinematic or continuity wave theory applies to situations where there is a functional
relationship between the flow rate and concentration. There are many examples from
every day experience such as the dependence of water flow rate in a river on the
depth of the river and the dependence of car flow rate on a highway on the traffic
density (33).
In the problem being addressed here, a functional relationship exists between the
flow velocity (which represents the flow rate) and the depth of the flow (which
represents the concentration). Since the longitudinal travelling dynamic wave is a
special case of kinematic waves (34), the dynamic wave velocity (c) can be
51
considered as a kinematic wave velocity. When the kinematic wave velocity is equal to
zero, the wave cannot propagate leading to the flooding transition. By allowing c = 0,
equations (3.32 and 3.33) can be written as
-U2
h
2
/gam
a
2
-2
1
2
P+ g ug
(R-h)
[(R- h)g
2
Ft_hcr
(R-h) g Ap
(3.34)
a
_2
_2
PI ui +139 ug=2
1
(R-h)
2g
iv
[(R-h)g
R_her
(3.35)
a
In terms of the superficial velocities defined by
J =au
J1=(1-a)ui
equations (3.34 and 3.35) can be written as
2
2
h J1
PI
2
2(1-a)
Jg
1
a
(R-h)
a =
[ (R-h) g
1
(R-h)2 g Ap
2 + Pg
1
(3.36a)
52
12
JJ2
i
PI
"g
(1-0)2
rs a2
-2
2
[ (R-h)gAp---R-h
1
(R-h)2g Ap
a
(3. 36b)
From the void fraction expression:
2
R-h ) .(1_h )2.1_2h
for h
R
Therefore the liquid film thickness h can be expressed in terms of a as
h=R(1-a)
2
thus equations (3.36a and 3.36b) can be written as
J2
,2
PI
1
1
4(1-a)V
2
74-pg
(1-a)
2-1LP R-1
R2 -1
a
+A
-1 =2
a2
\/ gAP-(4)
J2
42.gApa-(T)
a
2
CY
(3.37a)
2
(3.37b)
53
Defining
gAp R2
q=7
1
a
x=pis.112
J12
2
z
g
boa-4
Equations (3.37a and 3.37b) can be written as.
xq +-Y=2z
(1a)
a2
(1 a)2
(3.38a)
+Y =2z
(3.38b)
a2
And by defining :
A=x/z=
Ki2(mod.)where Ki (mod.) is the modified liquid Kutateladze number
131,42
2
Kkmod)=
B=y/z= Kg2(mod.)where Kg (mod.) is the modified gas Kutateladze number
2
Pg Jg
Kg2(mod.)--
54
Thus equation (3.38a and 3.38b) can be written as
Aq
B
(1 a) a2
(3.39a)
+=2
A
(1 a)2 a2
(3.39b)
It is noted here that the modified Kutateledze number whether for the liquid or the gas
takes into account both the effects of the geometry and the surface tension while the
existing definition for the Kutateladze number takes into account the surface tension
only.
That is,
K
2
Rs Jts
Moreover, the Wallis parameter takes into account the geometry only
12
'2 PI vis
J =
kg
'VD
Therefore the new definition for the Kutateladze number may have a wider range of
applications than the existing definitions.
Equations (3.39a and 3.39b) have been derived by applying kinematic wave theory to
equations (3.32 and 3.33). In the next section, the connection to catastrophe theory
is made apparent.
55
3.5
Connection to Catastrophe Theory;
Catastrophe Theory is applied to situations where smooth (continuous) changes
result in discontinuous or drastic changes in the system behavior. Furthermore,
Catastrophe Theory attempts to characterize those changes in the causes that lead to
discontinuous changes in their effects. This characterization requires some
mathematics (35). Appendix B provides some useful background information on
Catastrophe Theory.
A system can be characterized by two types of parameters:
1. Dependent parameters which are also called state parameters since they determine
the state or the behavior of the system.
2. Independent parameters or control parameters because they control the qualitative
properties of the system. The control parameters may be classified as:
a. Mathematical control parameters which are those that appear in the canonical
forms of catastrophe classes.
b. Physical control parameters which are those responsible for actual control of the
system.
Physical control parameters can be rearranged to form an appropriate
set of mathematical control parameters.
Using the basic concepts of Catastrophe Theory, it can be shown that equations (3.39a
and 3.39b) represents equilibrium response surfaces. These response surfaces fall
within the category of the cuspoids (either cusp or swallowtail) catastrophe as shown
below.
56
1. Case of the cusp catastrophe.,
The response surface in the case of small kh is represented by equation
(3.39a) which can be rewritten as follows
Aqa2+B(1-a)=2a2(1-a)
(3.40)
Or
a3 +a2(212-1)-aL +2
2
/
2
(3.41)
This form is equivalent to the standard model or the canonical form of the cusp
catastrophe represented by
X
3
+ax+b=0
(3.42)
where x, a and b are the mathematical state and control parameters (36). The
physical state parameter for the system under consideration is a, and the physical
control parameters are A and B defined earlier.
To phrase equation (3.41) in the language of the canonical cusp catastrophe,
the following expression for the mathematical state parameter can be used:
1-A2c1
(3.43)
Upon substituting equation (3.43) in equation (3.42) and comparing coefficients with
equation (3.41) we get the following relationship between the mathematical and
physical control parameters.
57
a
46B+(Aq-2)21
12
(3.44)
9B(Aq +4)+ (Aq -2)3
108
(3.45)
Bifurcation set.,
It is noted that equation (3.42) determines the critical points of the system
( given by setting the derivative of the standard function with respect to the state
parameter equal to zero), and defines the 2-dimensional manifold in 3-dimensional
space whose coordinates are x, a, and b. Although equation (3.41) is equivalent to
equation (3.42), the latter one will be used for convenience. Thus the mathematical
response surface in x,a, and b space is represented by equation (3.42). The two-fold
degenerate critical points or the singularity points which form the fold curve can be
determined by setting the gradient of equation (3.42) equal to zero:
3 x2+a=0
(3.46)
Eliminating x from equations (3.42 and 3.46) results in
27 b2+4 a3=0
(3.47)
which represents the bifurcation set of the system or the projection of the fold curve
onto the control space ( a-b plane ). This bifurcation set has a cusp and defines the
catastrophe boundaries which determine the relation between a and b in the control
surface defined by these parameters as shown in Figure (3.2).
58
Figure 3.2
The Cusp Catastrophe and Its Bifurcation Set
59
2. Case of the swallowtail catastrophe;
The response surface in the case of large kh is represented by equation (3.39b)
which can be rewritten as follows
Aa2+B(1
_02.2a2(1 _a)2
(3.48)
Or
a4 -2a3+a2(
B
A
2
=0
(3.49)
This form is equivalent to the canonical form of the swallowtail or dovetail
catastrophe represented by: (37,38)
X
4
+ax2 +bx+c=0
(3.50)
This equation defines a 3-dimensional manifold in 4-dimensional space whose
coordinates are x, a, b, and c. Unlike the 2-dimensional manifold of the cusp
catastrophe, this manifold cannot be visualized because it needs four dimensions.
To prove that equation (3.49) is an equivalent form of the swallowtail canonical form
we express the mathematical state parameter as
x = a - .5
(3.51)
60
Upon substituting equation (3.51) in equation (3.49) and comparing coefficients with
equation (3.50), the mathematical control parameters can be written as
a = - .5 (1+A+B)
b = .5 (B-A)
c = .0625 (1-2B-2A)
Bifurcation set;
As in the previous case, the critical set or the response surface is represented
by equation (3.49) or equation (3.50). Catastrophe Theory concentrates on the
behavior of the state parameters relative to changes in the physical parameters
(a, A, B) or the mathematical parameters (x, a, b, c). For the present case, and for
the purpose of evaluating the bifurcation set, it is easier to use the physical
parameters. This set can be determined by setting the gradient of equation (3.48)
equal to zero i.e.,
2 Aa -2B (1- a)=4 a2 (1- a)+4 a(1- a) 2
(3.52)
Now we solve equation (3.48) and its derivative equation (3.52) simultaneously:
A a2 + B (1- a)2=2 a2(1- a) 2
(3.48)
2Aa -2B(1- a) = -4a2 (1- a)+4 a(1- a)2
(3.52)
61
Multiplying equation (3.52) by (1- a) and adding it to equation (3.48) yields
A a2 + A a (1-a)- 2 a(1- a)3=0
Solving for A results in
A = 2 (1- a)3
Thus a can be written as
a = 1- (.5 A)3
(3.53)
By substituting into equation (3.52) we get
A1/3 +B1/3.21/3
(3.54)
Equation (3.54) represents the bifurcation set or the catastrophe boundaries which
determine the relationship between K1 and Kg in the control surface defined by these
parameters as shown in Figure (3.3). Thus the bifurcation set can be visualized as a
cusp in 2-dimensional space determined by the physical parameters. This set can also
be visualized as a cusped surface determined by the mathematical parameters (a, b,
and c) when using the canonical form of the swallowtail catastrophe.
62
1.600
1.400 1.200 -
Kg
1.000 -
Flooding
Line
0.800 -
Strong interaction
I Jump 3
0.600 -
0.400 -
Flooding
Weak interaction
I area
0.200 -
Line
1
0.000
-1.600
1
-1.200
-0.800
-0.400
Countercurrent
Figure 3.3
i
0.000
KI
A
4
0.400
0.800
1.200
1.600
Concurrent
The Bifurcation Set of the Flooding Catastrophe Based on the
Physical Parameters Indicating the Flooding Boundaries
63
The bifurcation set depicted in Figure (3.3) provides significant insight into the
flooding process. The central region within the cusp boundaries represents the "weak
interaction" region. Within this region, the liquid and gas phase remain separate and
the annular two-phase flow pattern is maintained. The region outside of the cusp
boundaries represents the "strong interaction" region where the separated two-phase
pattern can no longer be maintained. A flooding process can be described as follows
for a vertical countercurrent flow system. That is, the liquid flows downward
countercurrent to the gas phase.
negative.
For countercurrent flow, the values of K1 are
Assuming a constant negative value for K1, an increasing value of Kg will
take the system from point 1 to 2. Along this path, the two phases interact very
weakly. At point 2, the system reaches a catastrophe boundary. A strong interaction
between the two phases develops and the separated flow pattern is destroyed.
Instead
of the uniform annular flow pattern that was originally present, a chaotic dispersed
flow pattern may develop. This new flow pattern may be conducive to a very rapid
liquid flow reversal. This is represented by the "jump" transition from point 2 to
point 3. During the jump transition, the liquid flow rate passes through zero and
reverses direction until it reaches the catastrophe boundary at point 3. This is in
agreement with the physical observations of the oscillatory behavior that occurs
during the flooding process. A variety of flow system configurations can be described
with this diagram.
64
3.6
Entrainment Effect;
At the liquid-gas interface, an interaction between the two phases represented
by continuous entrainment and deposition processes usually takes place. A fraction of
the total liquid flow rate can enter the gas core. This fraction can be defined as the
ratio of the entrained liquid flow rate (W1e) to the total liquid flow rate (W1) and can
be expressed mathematically as
(3.55)
Wie/ WI
Different mechanisms for entrainment have been proposed. The shearing-off of the
wave crests can be considered as the main mechanism when a low viscosity liquid
such as water is involved (39). This mechanism represents the competition between
three forces: the viscous, surface tension, and shear forces. In the present analysis,
the entrainment effect can be taken into consideration through using pgc and Jgc
where
pgc and Jsc are the homogeneous core density and the core superficial
velocity respectively.
For the homogeneous density one can use the weighted (by the
superficial velocity) density, namely:
Jie
Pg c
Jg
7+J
+ Vg
9
(3.56)
where Jie is the superficial velocity of the entrained liquid in the gas. This superficial
velocity can be defined as
(3.57)
Also the homogeneous density can be defined in terms of the volumetric quality (3)
which is defined as the ratio of the volumetric flow rate of the gas to the total
65
volumetric flow rate in the gas core
Qg
13=
(3.58)
%...g+%.(141,
But
01.=E al
Therefore
0,,
13 =
1
ag E
=
E al
1+
(3.59)
Qg
A correlation reported in (40) can be used for the quasi-equilibrium condition (far
away from the entrance):
(3.60)
E =tanh (7.25 x 10-7We 125 Rei25)
= 7.25 x 10-7 wel .25 Re1.25
for small argument
where
We is the entrainment Weber number defined by
2
We =
pgJg D
6
).3
1/
C.
Pg )
Re is the total liquid Rynolds number defined by
Rei=
By using equations (3.56 and 3.58) the homogeneous gas core density can be written
as
Pgc = (1-13) PI 4-13 Pg
(3.61)
66
Of « Qg and since E is a fraction that lies between (0 and 1) then,
In the case of
E
Qs
«1
(3.62)
Thus by expanding equation (3.59) and neglecting the higher order terms we can
write
E Qi
13= 1
-
(3.63)
`dg
Pgc can be written as
Pgc = Pg + 7.25 x 10-7
(VVe R
)1'25
Reg
Pg
(
qpg)
A
(3.64)
With respect to the homogeneous gas core velocity it can be expressed in terms of the
homogeneous gas core superficial velocity as follows
Jgc =Jle + Jg= E J1 + ulg
(3.65)
Thus the modified gas Kutateladze number expression
2
K
Pg
4
g %know
can be written for the gas core as
2
j2gc
Kgc(mod.)=
(3.66)
This definition will be used in the flooding equation (3.54) to perform the flooding
analyses.
67
4.
COMPARISON BETWEEN THE EXPERIMENTAL
VALUES AND THE MODEL PREDICTION
In spite of the many experiments conducted to clarify the nature of the flooding
phenomenon, no consistent results have been reached. The reason for this may be
related to the lack of systematic experimental procedure and the different criteria
adopted to determine the flooding condition. Many functional relationships were
proposed either in terms of Wallis or Kutateladze parameters to fit the extensive
experimental data for flooding. Since they are empirical relationships, it can be
expected that they change from one experiment data set to another. In this chapter,
a large volume of experimental data is examined against the theoretical model
prediction (equation 3.54).
Some of the data is described well using the exact theoretical model. The
remainder of the data can be reasonably described by varying only one empirical
coefficient.
These data may be classified as
1. Data for flooding in tubes
2.
Data for flooding in annuli.
4.1 Flooding in Tubes;
4.1.1. Data of EPRI NP-1283;
This data set consists of air-water flooding data obtained for different flow
conditions and tube geometries (41).
The tube diameters were .0159, .0318, .0460,
and .0699 m, and the tube length was .94 m. The inlet was either straight or tapered
at an angle of 450. The experiment was conducted using a system made of upper and
68
lower plena and a connecting test section.
Liquid was introduced into the system
through the upper plenum to flow downward under the action of gravity. The air was
introduced either directly through a nozzle aligned with the tube axis, or indirectly
through the lower plenum.
Two criteria were used to determine the onset of the flooding.
The first
criterion depends on the observation of a chaotic flow pattern in the tube section. The
second criterion is the sudden change of pressure in the tube. The flooding condition
was reached by fixing the liquid flow rate and increasing the air flow until one of the
criteria above was met.
The experimental data was classified into the following categories based on the
inlet-exit geometry and the method of air introduction for all of the different sizes
used.
1.
Data for tapered inlet, sharp edge exit, and nozzle air supply or direct air supply
(TI, SE, NAS).
2.
Data for sharp edge inlet, tapered exit, and nozzle air supply (SI, TE, NAS).
3.
Data for tapered inlet, sharp edge exit, and indirect air supply (TI, SE, IAS).
4.
Data for sharp edge inlet, tapered exit, and indirect air supply (SI, TE, IAS).
These data were used to evaluate the numerical values shown in tables (A.1 to A.16)
of the Appendix for the purpose of comparison with the model prediction. Each data
category above was plotted along with the model prediction for all tube sizes. A closer
look at Figures (4.1 to 4.4) indicates a good agreement between the experimental and
predicted data. It is worth mentioning here that equation (3. 54 ) with the right hand
constant equal to (2 or 2.26) was used to evaluate the predicted values.
69
0.5
1.5
1
2
25
A-1/3
Figure 4.1
Flooding Data of EPRI NP-1283 for Different Tube Sizes with
Nozzle Air Supply, Tapered Liquid Inlet, and Sharp Edge Exit.
70
Figure 4.2
Flooding Data of EPRI NP-1283 for Different Tube Sizes with
Nozzle Air Supply, Sharp Edge Liquid Inlet, and Tapered Exit.
71
Figure 4.3
Flooding Data of EPRI NP-1283 for Different Tube Sizes with
Indirect Air Supply, Tapered Liquid Inlet, and Sharp Edge Exit.
72
.
2.5
2
IAS.
Nis
SL TE
model
ix
I
a
11
I
lill
0.5
0
0
Figure 4.4
0.5
25
Flooding Data of EPRI NP-1283 for Different Tube Sizes with
Indirect Air Supply, Sharp Edge Liquid Inlet, and Tapered Exit.
73
4.1.2.
Data of EPRI NP-1284:
The second set of the onset of air-water flooding data are those of Dukler and
Smith reported in (42) for a tube size of .0508m. The criterion implemented here is
the characterization of the point at which the pressure drop against flow rate curve
showed a sudden change in slope.
The numerical values of A1/3 = Ki(mod.)2/3 and
B1/3. Kgc(rnod.)2/3 derived from
the reported values of the nondimensionalized liquid and gas core superficial velocity
are presented in table (A.17). The comparison between the analytical model
prediction and the experimental values is graphically represented in Figure (4.5).
Very good agreement was obtained using equation (3.54) with the right hand side
constant set at 2.
4.1.3.
Data of EPRI NP-1336.,
The third set of air-water flooding data are those reported in (43). The test
facility consisted of a vertical tube of 2" diameter and 60" length between two
plenums allowing for the introduction or extraction of air and water to or from the
facility. This data set characterizes two conditions.
The first condition describes how
increasing the air flow rate causes the transition from the smooth countercurrent
flow to a flow pattern characterized by the appearance of rough surges on the air-
water interface near the bottom of the test section.
As the air flow rate increases
further, these surges propagate up until they reach the upper plenum. The second
condition describes the condition when part of the liquid film penetrates downward
into the lower plenum and the rest accumulates in the upper plenum. The values of the
74
A"1/3
Figure 4.5
Flooding Data of EPRI NP-1284 ( Dukier and Smith Data) for a
Tube Size of .0508m.
75
modified liquid and gas core Kutateladze numbers corresponding to the liquid and gas
core nondimensionalized superficial velocities for the latter condition are tabulated in
(A.18) and graphically compared to the model prediction in Figure (4.6). As shown in
this figure, the general trend of the flooding behavior can be predicted by the model
equation. However, using a value of 1.5 as an empirical coefficient, correlates the
data better as shown in Figure (4.7)
4.1.4. Data of EPRI NP-2262.,
This section examines another set of air-water flooding data that used the basic
experiment design consisting of two plenums and different connecting test section
sizes and lengths. The test section sizes range from .0127m to .0305m while the
lengths range from .5" to 10" (i.e., from .0127m to .254m). The onset of flooding
was determined when the transition from separated flow to an active mixing condition
takes place. The experimental procedure was performed by increasing air flow until
the mixing between the two-phases occurs. Two cases of entry condition were
treated, stub and direct entries. The stub entry is related to those test sections
whose path inlets have been raised above the entrance, while the direct or plate entry
is related to those which have path inlets flush with the entrance. The numerical
values for the air and water Kutateladze number with the entrainment included were
derived from the values reported in (44) and tabulated in tables (A.19 - A.21). It is
noted (in spite of large experimental data scattering) that the model equation with the
same theoretical value of the constant (i.e., 1.26) predicts the data for tube size .5"
regardless of the entry condition as shown in Figures (4.8-4.10). A trial to evaluate
the model against the experimental data for a tube size of .0305m and different
76
Figure 4.6
Flooding Data of EPRI NP-1336 Compared with the Exact
Theoretical Model.
77
Figure 4.7
Flooding Data of EPRI NP-1336 Compared with the Theoretical
Model Using an Empirical Coefficient of 1.5.
78
1.4
1.3
1.2
1.1
1
model
c,Z 0.9
t&I 0.8 -
0.7
i
0.6
0.5
0.4
0.3
0.3
-MI
0.4
0.5
0.6
0.7
08
A^1/3
Figure 4.8
Flooding Data of EPRI NP-2262 for a Tube Size of .0127m.with
a Stub Entry.
79
= 08
7
0.7
0.6
0.50.4
0.3
D.3
0.5
D.4
0.6
A"1 /3
Figure 4.9
Flooding Data of EPRI NP-2262 for a Tube Size of .0127m.with
a Plate Entry.
80
1.4
1.3
1.2
1.1
17' 0
N
ria
9
08
0.7
0.6
0.5
0.4
0.3
03
0.4
0.5
0.6
0.7
0.8
A-1 /3
Figure 4.10
Flooding Data of EPRI NP-2262 for a Tube Size of .0127m.with
Stub and Plate Entry.
81
lengths (.061m, .122m, and .254m) indicates good agreement as shown in Figure
(4.11). Good agreement can also be noted through the graphical representation of
Figure (4.12) for the predicted and experimental values for all data of tube sizes
(.0127m and .0305m), and tube lengths ranging from .0127m to .254m.
4.1.5.
Data of NUREG/CR-0312;
As reported in (45), the experimental data presented graphically on the J g*-1/2
J x'1 /2
plane for air-water flooding in different tube sizes of (2", 6", and 10") were
used to derive the numerical values shown in tables (A.22 to A.24) for the sake of
comparison.
These experimental values were extracted from the measurements
obtained using a test facility consisting of vertical, transparent, and square ended
tubes of 40 - 48 inches in length connected to upper and lower plena.
A closer look at Figures (4.13
4.15) shows that the model equation (3.54 )
correlates the experimental data well for 2" tubes. it also correlates data for 6" and
10" sizes when using 1.6 as an empirical coefficient in the model equation.
4.2 Flooding in Annuli;
4.2.1 Data of NUREG/CR-0312;
The test facility used to conduct the flooding experiments consists of upper and
lower plena with 40" diameter. The plexiglass tube for the annulus has an inner
diameter of 17.5". One and two inch annular gaps were constructed using two
interchangeable inner tubes with 15.5" and 13.5" outside diameters respectively.
82
1.4
1.2
1c,) 0.8
N
0.6
0.4
0.2
0
0.1
0.2
0.3
0.4
05
A-1/3
Figure 4.11
Flooding Data of EPRI NP-2262 for a Tube Size of .0305m and
Different Lengths.
83
1.4
1.2
II
WI
MI
go
II dm
NI
73 0.3
model
IN
VI
1
111
I.
El
w
= 0.E.
Mr
U-
MI
0.4
0.7.
0
01
0.2
0.3
0.4
0
0.6
07
A-1/3
Figure 4.12
Flooding Data of EPRI NP-2262 for Tube Sizes of .0127m and
.0305m, and Different Lengths.
84
0
0
0.2
0.4
0.8
0.6
1
1.2
14
A-1 /3
Figure 4.13
Flooding Data of NUREG/CR-0312 for a Tube Size of .0508m.
85
2
1.8
1.6
1.4
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
A-1 /3
Figure 4.14
Flooding Data of NUREG/CR-0312 for a Tube Size of .1524m.
86
2
1.8
1.6
..
1.4
.......
......_............._
du
............_____
--e.m.--
1.2
_
..........
.............
_
model
...._.............
..........___.
..
1
Is
0.8
or N.
0.6
0.4
...........
0.2
0
0.2
0.4
0.8
0.6
1.2
14
A-1/3
Figure 4.15
Flooding Data of NUREG/CR0312 for a Tube Size of .254m.
87
The calculated values shown in tables (A.25 and A.26) are based on the
experimental values reported in (46). Using the gap width in the definitions of A
and B instead of the radius, and a constant of 1.8 in equation (3.54) resulted in good
agreement between the predicted and the experimental values as shown in Figures
(4.16 and 4.17).
4.2.2.
Data of NUREG/CR-0526:
As reported in (47,48), extensive test programs have been conducted at the Battelle
Columbus Laboratories (BCL) using 1/15 and 2/15 scale models of pressurized water
reactor vessels to study emergency core coolant penetration into the lower plenum.
The 1/15 vessel scale inner diameter is 12.1" while that of the 2/15 scale is 24.35".
The downcomer gap widths are .59" and 1.23" for the 1/15 and 2/15 scale models
respectively.
The annulus circumference (Ca) ) was used as a characteristic length in
stead of D in the definition of the nondimensionalized liquid and gas superficial
velocities.
Jj =J1
That is,
#7
1
g
D
where the subscript j refers either to g or I for the gas or liquid phase respectively
This was done because the previously used hydraulic diameter (2 x gap) did not
adequately correlate the test results of Creare, Inc.. The numerical values shown in
tables (A.27 and A.28) were calculated using the liquid and gas superficial velocities
88
2.4
2.2 Sr
-1046.
2
model
E mw
1.8
sr-
1.6
7.1 1.4
pis
-w
1 1.2
Me
W
m...
c4
1-
0.8
IN
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
14
A^1/3
Figure 4.16
Flooding Data of NUREG/CR-0312 for an Annulus with Gap
Width =.0254m.
89
2.4
2.2
2'
1.8
1.6
1.4
1.2
0
0:2
0:4
0.8
0.6
1.2
14
A-1/3
Figure 4.17
Flooding Data of NUREG/CR-0312 for an Annulus with Gap
Width =.0508m.
90
derived from the definitions above. Again using the gap width instead of the radius in
the definitions of A and B in equation (3.54 ) with the constant equal to 1.8 - 2
correlates well the steady state (s. s.) data for 1/15 and 2/15 scales as shown in
Figures (4.18 and 4.19).
Another set of data was reported in (49) for air-water plenum filling studies
using 1/15 scale at different flow rates.
As before, the numerical values shown in
table (A.29) showed some success in correlating the data when the constant was set
equal to 1.8, however good agreement between the theoretical model and the
experiment can be achieved if 1.6 is chosen for the constant as shown in Figure
(4.20).
Generally using 1.8 as an empirical coefficient in the model equation may
correlate all of the data of the 1/15 and 2/15 scale models as shown in Figure
(4.21).
4.3. Summary:
In summary, a total of seven data sets have been examined against the theoretical
model that has been developed. In most cases good agreement can be obtained by
empirically varying only one coefficient.
91
2.2
2
1.8
1.6
1.4 -
0.8
0.6
0.4
0.2
0
0.2
Figure 4.18
0.4
0.6
0.8
1
1.2
1.4
1.6
k1/3
Flooding Data of NUREG/CR-0526 for Steady State Case in 1/15
Scale Model.
92
2
1.8
1.61.6
1.4-
cp
0.8
0.6 0.4
0.2 0
0
0.2
0.4
0.6
0.8
1
1.2
1:4
1.6
1.8
2
A-1/3
Figure 4.19
Flooding Data of NUREG/CR-0526 for Steady State Case in 2/15
Scale Model.
93
2.2
2
1.8
1.6
1.4
N 1.2
-1
<
=
1-
0.8
0.6
0.4-4
0.2
0
0
0.2
0:4
0.6
0.8
1.2
1:4
16
A-1/3
Figure 4.20
Flooding Data of NUREG/CR-0526 Plinum Filling Case in 1/15
Scale Model.
94
Figure 4.21
Flooding Data of NUREG/CR-0526 for Steady State and Plenum
Filling in 1/15 and 2/15 Scale Models.
95
5.
CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK
In this study, a general picture for the flooding phenomenon was drawn via an
examination of different theoretical and experimental efforts devoted to clarifying the
nature of this phenomenon. Based on this picture, it can be said that at low flow
rates, the two phases can coexist in a smooth countercurrent annular flow.
By
increasing the flow rate of one phase and retaining the flow rate of the second
constant, deviation from this regime occurs.
For example, by increasing the gas flow
rate in a tube and maintaining the liquid flow rate constant, a strong interaction
between the two phases becomes more noticeable. The interaction is due to the drag
induced by the gas on the interface between the two phases. The drag helps increase
the waviness of the interface.
Droplets form as a result of shearing-off of the crests
of the wavelets generated on that surface. Further increasing the gas flow rate may
change the liquid's direction from down to up flow while entrainment continues.
Therefore, the onset of entrainment does not mean the onset of flooding but rather
flooding is always associated with entrainment. Thus flooding, in short, may be
thought of as a result of the very strong interaction between the two phases.
Furthermore, entrainment should be taken into account in any relevant analysis.
The primary concern of this study was to find some functional
form for
flooding expressed in terms of nondimensionalized superficial velocities for the gas
and liquid phases that leads to or initiates the flooding transition. With the
entrainment taken into consideration, this functional
relationship was theoretically
found to be linear if plotted on the K9c2 /3(mod.) and K 12/ 3(mod.) plane.
96
The major findings of this study are as follows:
1. A theoretical model based on Kelvin-Helmholtz Instability Theory , Kinematic Wave
Theory, and Catastrophe Theory has been developed and shown to be in general
agreement with a wide range of data.
2. The class of the elementary catastrophe that describes the flooding phenomenon is
the cuspoids. Within the cuspoids, the classification depends on the liquid film
thickness. When the liquid film thickness is thin, the analytical model reveals a
cusp catastrophe while in the case of a thick liquid film thickness, the flooding
represents a swallowtail catastrophe. By using the physical parameters, the
bifurcation set can be represented as a cusp defining the boundaries of the flooding
which is treated as a swallowtail catastrophe.
3.
Taking into consideration the entrainment effect is important in order to represent
the actual situation at the time of flooding. It was sufficient to use the
homogeneous core concept with a homogeneous density and a superficial velocity.
It is felt that a detailed gas core representation regarding the droplet distribution,
the droplets velocities, and the entrainment and deposition fractions is required
for more accurate results.
4. The dimensionless number ( modified Kutateladze Number ) used in the theoretical
model serves to unify the Wallis and Kutateladze scaling theories.
5. Good agreement with flooding data from tube geometries can be obtained by
modifying the model to include one empirical coefficient with values between
1.26-2.26. This deviation from the theoretical value of 1.26 may be due to
differences in entrance geometry, experimental procedure and measurement
techniques.
97
6.
Good agreement with flooding data from annular geometries ( reactor vessel scale
models ) is obtained by setting the empirical coefficient to a value of 1.8 and using
the annulus gap width instead of the radius in the definition of modified Kutateladze
number.
7.
Many experimental studies use different criteria to characterize the onset of
flooding. This generates significant uncertainty in the data which hampers the
modeling effort.
The following suggestions for future research in this area should be considered:
1.
Study the effect of variable velocity in both the gas and liquid regions.
2.
Although neglecting the effect of viscosity in the early stage of deriving the
air-water flooding model is justified, and some effects of viscosity have been
taken into consideration in the entrainment fraction which depends on the Reynolds
number, further justification of this procedure or a more formal treatment of this
effect in the derivation may be important when treating highly viscous fluids.
3.
Air-water systems have been made the focus of this study in order to:
a.
Derive a simple functional relationship for the flooding.
b.
Explore the hydrodynamic nature of the flooding and
c.
Eliminate the condensation effect.
Since the flooding phenomenon is encountered in reactor systems especially with
regard to LOCA analysis, the separation between the hydrodynamic and condensation
effects is not acceptable. Therefore the condensation effect should be included in the
steam-water system. One way to include this effect is to modify the proposed model
by introducing the steam core Kutateladze number instead of an air modified
Kutateladze number and subtracting the amount of steam condensed (50).
98
REFERENCES
1.
Bharathan, D. , G. B. Wallis, and H. J. Richer. Air-Water Countercurrent
Annular Flow in Vertical Tubes, Electric Power Research Institute, Interim
Report, EPRI NP-786, May 1978.
2.
Bankoff, S. G. and S. C. Lee. A review of the Flooding Phenomenon with
Particular Reference to Steam-Water Flow, Department of Chemical Engineering
and Mechanical and Nuclear Engineering, Northwestern University, 1982.
3.
Delhaye, J. M., M. Giot, and M. L. Riethmuller. Thermohydraulics of Two-Phase
Systems for Industrial Design and Nuclear Enaineering, McGraw-Hill Book
Company, New York, NY, 1981.
4. Compendium of ECCS Research for Realistic LOCA Analysis, NUREG-1230, 1988.
5.
Reference 2, pg 1.
6.
Imura, H.
,
H. Kusuda, and S. Funatsu," Flooding Velocity in a Countercurrent
Annular Two-Phase Flow", Chemical Engineering Science, Pergamon Press, Vol.
32, 1977, pp. 79-87.
7.
Reference 6, pg 84
.
8. Tien, C. L. , K. S. Chung, and C. P. Liu. Flooding in Two-Phase Countercurrent
Flows, EPRI NP-1283, Topical Report, December 1979.
9.
Reference 3, pg 62.
10.
Reference 2, pg 9.
11.
Reference 6, pg 86.
12.
Reference 2, pg 13.
13.
Reference 2, pg 42.
99
14. Wallis, G. B. One Dimensional Two-Phase Flow, McGraw-Hill Book Company,
New York, NY,1969.
15.
Bharathan, D., G. B. Wallis, and H. J. Richter. Air-Water Countercurrent
Annular Flow, EPRI NP-1165, 1979.
16.
Reference 2, pg 21.
17.
Wallis, G. B. and J. T. Kuo," The Behavior of Gas-Liquid Interface in Vertical
Tubes", Int. J. Multiphase Flow, 1976.
18. Dilber,I.
,
S. G. Bankoff, R. S. Tankin, and M. C. Yuen. Countercurrent
Steam/Water Flow Above a Perforated Plate-Vertical Injection of Water,
NUREG/CR-2323, 1981.
19.
Reference 14, pg 320.
20. McCarthy, G. E. and H. M. Lee. Review of Entrainment Phenomenon and
Application to Vertical Two-Phase Countercurrent Flooding, EPRI NP-1284,1979.
21.
Ostrogorsky,A. G., R. R. Gay, and R. T. Lahey, Jr. The Analysis of
Countercurrent Two-Phase Flow Pressure Drop and CCFL Breakdown in Diabatic
and Adiabatic Conduits, NUREG/CR-2386, 1981.
22. Reference 18, pg 5.
23. Wallis, G. B. , D. C. deSieyes, R. J. Rosselli, J. Lacombe. Countercurrent
Annular Flow Regimes for Steam and Subcooled Water in a Vertical Tube, EPRI NP1 336,1 980.
24.
Reference 18, pg 4.
25. Reference 21, pg 2.
26. Hewitt, G. F and N. S. Hall-Taylor.
New York 1970.
Annular Two-Phase Flow", Pergamon Press,
100
27. Reference 8, pg 2-7.
28. Gilmore, Robert. Catastrophe Theory for Scientists and Engineers, John Wiley
& Sons, New York 1981
29. Abramowitz, M. and I. A. Stegun, eds. Handbook of Mathematical Functions,
Ams55, U.S. Department of Commerce, 1972.
30.
Reference 29, pg 379 .
31. Levich, Veniamin G. Physicochemical Hydrodynamics, Prentice Hall,Inc.,
Englewood Cliffs, N.J, 1962.
32. Reference 8, pg 2-4.
33. Reference 14, pg 123.
34.
Reference 8, pg 2-7.
35. Sinha, D. K. Catastrophe Theory and Applications, John Wiley & Sons, New
York, 1981.
36. Zuber, N., " A method of Scaling Limiting Processes and Phenomenon in Single
and Two-Phase Systems", Proceeding of the Joint NRC/ANS Meeting on Basic
Thermal Hydraulic Mechanaisms in LWR Analysis, Topical Meeting, Bethesda, MD.
September 14-15, 1982, Included in the Errata Sheet for NUREG/CP-0043.
37. Bakker Th. and L. Lander. Differentiable Germs and Catastrophes", Cambridge
University Press, 1975.
38.
Reference 36.
39.
Ishii, M. and K. Mishima," Droplet Entrainment Correlation in Annular Two-Phase
Flow", Jnt. J. Heat Mass Transfer 32, 1835 1989.
40.
Reference 35, pg 1842.
41.
Reference 8, pg 7-1.
101
42.
Reference 20, pg 3-10.
43.
Reference 23, pg 5-3.
44. Liu, C. P, C. L Tien, and G. E. McCarthy. flooding in Vertical Gas-LiaiJid
Countercurrent Flow Through Parallel Paths, EPRI NP-2262 1982.
45.
Richter, H. J., G. B. Wallis, and M. S. Speers. Effect of Scale on Two-Phase
Countercurrent Flow Flooding, NUREG/CR-0312, 1978.
46. Reference 45, pp 32-37.
47. Collier, Robert P, et. al.
Stearrt.Water Mixing and System Hydrodynamics
Program, NUREG/CR-0526, 1978.
48. Beckner, W. D., J. N. Reyes, Jr., and R. Anderson. Analysis of ECC Bypass
Data, NUREG-0573, 1979.
49. Cudnik, R. A, et al. steam-Water Mixing and System Hydrodynamics Proaratrk,
NUREG/CR-0147, BBI -2003, 1978.
50.
Reference 48, pg 7.
51. Arnold, V. I. Catastrophe Theory, Springer-Verlag, 1984.
APPENDICES
102
APPENDIX (A)
AIR-WATER FLOODING DATA IN TUBES AND ANNULI
AA1/3
is the cubic root of A defined by:
p
KIncd)
AP
BA1/3eE is the cubic root of B defined by:
2
g Apcs
where:
+ Jg
= Jie + Jg = E
E =tanh (7.25 x 10-7We125Rei25)
= 7.25 x 10-7 Wei.25
JIB
j
Pgc-Pi
We =
pgJgD
11/3
Pg
.pg +7.25x107
g
+
Re1.25
.-"4
(We Rei ) 125
Reg
pg
q µg
(
)
g
103
pt J1 D
Rei=
A
Reg =
pg Jg D
Pg
B^1 /3p is the cubic root of B predicted using the model equation (3.54).
The above equations were used directly in the analysis of flooding in tubes. The same
equations were used in the case of annuli except that the gap width is used instead of R
in the definitions of A and B and twice the gap width is used instead of D in the
definitions of Reynolds number.
104
Table A.1
Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0159m
(Tapered Inlet-Sharp Edge Exit Geometry and Nozzle Air Supply)
Gas 4en.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.0011
998
1.21
AA1/3
13^1/3 p
Rel
Reg
We
0.7246
0.7246
1.0743
1.0743
1.0743
1.3564
1.3564
1.5354
1.5354
1.1857
1.1857
1.1857
0.9036
0.9036
1551.4
1551.4
2800.7
2800.7
2800.7
3973.4
3973.4
9071.2
8602
7209.3
6771.1
6187.5
4741.3
4519
177.91
E
0.003
159.98 0.0026
112.37 0.0019
99.128 0.0016
82.777 0.0013
48.605 0.0007
44.154 0.0007
rohgc
1.2439
1.2414
1.2603
1.2558
1.25
1.2516
1.2487
Jgc
8.4873
8.0483
6.7453
6.3353
5.7892
4.4362
4.2282
D(m)
0.0159
BA1 /3e E
1.5288
1.4746
1.3175
1.262
1.1866
0.994
0.962
105
Table A.2 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0318m
(Tapered Inlet-Sharp Edge Exit Geometry and Nozzle Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
1.19
0.072
997
0.0009
A^113
BA1/3 p
Rel
Reg
We
E
rohgc
0.3574
0.3574
0.479
0.479
0.479
0.7317
0.7317
0.9361
1.9026
1.9026
1096.3
1096.3
1701
1701
1701
3211.7
3211.7
4647.9
4647.9
6144.4
6144.4
6144.4
7181.1
7181.1
11718
23840
22915
21927
20918
19850
16247
17544
13285
14239
7694.5
8262.2
8024.6
7473.8
7320.3
5083.4
636.77
588.31
538.67
490.23
441.45
295.73
344.83
197.72
227.15
0.0133
0.0121
0.0121
0.0107
0.0094
0.0067
0.0081
0.0044
0.0053
0.0012
0.0015
0.0013
0.0012
0.0011
0.0005
1.2305
1.2282
1.2519
1.2477
1.2433
1.2774
1.288
1.2925
1.3038
1.2541
1.2613
1.2582
1.2645
1.2622
1.2671
0.9361
1.1276
1.1276
1.1276
1.2511
1.2511
1.7341
1.781
1.781
1.781
1.5283
1.5283
1.3239
1.3239
1.1324
1.1324
1.1324
1.0089
1.0089
0.5259
66.331
76.478
72.144
62.58
60.036
28.951
Jgc
11.34
10.9
10.431
9.9506
9.4425
7.7287
8.3458
6.3197
6.7738
3.6602
3.9303
3.8173
3.5553
3.4823
2.4182
D(m)
0.0318
BA1/3eE
1.848
1.7987
1.7578
1.7016
1.6412
1.4491
1.5295
1.2721
1.3362
0.875
0.9193
0.9009
0.8606
0.8483
0.6661
106
Table A.3
Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .046m
(Tapered Inlet-Sharp Edge Exit Geometry and Nozzle Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.21
AA1/3
BA1/3 p
Rel
0.2905
0.2905
0.4439
0.4439
0.5678
0.6837
0.6837
0.7589
1.0517
1.9695
1.9695
1171.6
1171.6
2212.8
2212.8
3201.9
4230.4
4230.4
4946.6
8070.6
11195
14319
1.3081
1.5413
1.8161
1.8161
1.6922
1.5763
1.5763
1.5011
1.2083
0.9519
0.7187
Reg
35684
37787
34138
33551
31479
28649
25016
26491
22422
18368
15950
We
E
0.0224
0.0259
870.97 0.0235
841.24 0.0225
740.56 0.0211
613.41 0.0178
467.69 0.0127
524.47 0.0153
375.71 0.0114
252.14 0.0075
190.12 0.0056
951.64
1067.1
rohgc
1.2594
1.2639
1.3124
1.3098
1.3539
1.387
1.3544
1.4013
1.4847
1.5166
1.5475
Jgc
11.541
12.221
11.041
10.851
10.181
D(m)
0.046
13^1/3eE
1.8842
1.9598
1.8547
1.8321
1.7755
1.6809
9.2666
8.0912 1.5235
1.498
7.8243
7.253 1.4604
5.9418 1.2877
1.18
5.1597
107
Table A.4 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0699m
(Tapered Inlet-Sharp Edge Exit Geometry and Nozzle Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.2
AA1/3
B ^1/3 p
Rel
0.2543
0.2543
0.3916
0.3916
0.4347
0.4347
0.6024
0.7493
0.7493
1.0807
1.194
1.194
2.0057
2.0057
1.8684
1.8684
1.8253
1.8253
1.6576
1.5107
1.5107
1.1793
1.066
1.066
1460.8
1460.8
2792.5
2792.5
3265.5
3265.5
5327.6
7389.7
7389.7
12801
14866
16840
Reg
We
E
rohgc
Jgc
53171
63469
49256
53171
47159
1405.9
2003.2
1206.5
1405.9
1106
1306.7
1006.2
844.65
644.27
563.55
377.57
330.96
0.0386
1.2706
11.411
13.621
10.572
51260
44983
41213
35994
33664
27555
25798
1.2921
0.0375 1.3415
0.0601
0.0454
0.035
0.0431
0.0351
0.0306
0.0218
0.0212
0.0133
0.0117
11.412
10.122
11.002
9.6557
8.8472
7.7263
1.7361 7.2279
1.6786 5.9158
1.7067 5.5388
1.3587
1.3612
1.3826
1.4768
1.5654
1.4983
D(m)
0.0699
13^1/3eE
1.8755
2.1224
1.815
1.918
1.7717
1.8828
1.7642
1.6969
1.5279
1.535
1.3281
1.2782
108
Table A.5 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0159m
(Sharp Edge Inlet-Tapered Exit Geometry and Nozzle Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
1.21
998
AA1/3
BA1/3 p
Rel
Reg
We
0.7246
0.7246
1.0743
1.0743
1.3564
1.3564
1.5354
1.5354
1.1857
1.1857
0.9036
0.9036
1551.4
1551.4
2800.7
2800.7
3973.4
3973.4
6599
7220
5035.3
4525.4
3918.3
3482.3
94.153
E
rohgc
0.0013 1.2311
112.71 0.0017 1.2341
54.819 0.0008 1.2394
44.28 0.0006
1.235
33.196 0.0005 1.2412
26.218 0.0003 1.2362
Jgc
D(m)
0.0159
13^1 /3eE
6.1741 1.2323
6.7552 1.3096
4.7111 1.0313
4.2341 0.9594
0.873
3.6661
3.2581 0.8058
109
Table A.6 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0318m
(Sharp Edge Inlet-Tapered Exit Geometry and Nozzle Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.072
997
0.0009
1.19
AA1/3
13^1/3 p
Rel
0.3574
0.3574
0.479
0.479
0.7317
0.7317
0.7317
0.9361
0.9361
1.1276
1.1276
1.1276
1.9026
1.9026
1096.3
1096.3
1701
1701
3211.7
3211.7
3211.7
4647.9
4647.9
6144.4
6144.4
6144.4
7181.1
7181.1
11718
1.2511
1.2511
1.7341
1.781
1.781
1.5283
1.5283
1.5283
1.3239
1.3239
1.1324
1.1324
1.1324
1.0089
1.0089
0.5259
Reg
We
E
0.0068
0.0084
0.0076
0.0062
0.0012
8087.7 73.282 0.0012
8333.6 77.807 0.0013
7612.5 64.925 0.0011
18242
372.81
19831 440.61
18242 372.81
314.3
16749
8234.8 75.973
7192.1
57.951
0.001
6550.9
6286
6025.3
5316.8
5430.3
3040
48.078
44.269
40.673
31.67
33.037
10.354
0.0008
0.0007
0.0007
0.0005
0.0005
0.0001
rohgc
1.2171
1.2207
1.237
1.2313
1.2215
1.2207
1.2221
Jgc
8.6772
9.4333
8.6774
7.9673
3.9171
3.8471
3.9641
3.6212
3.4211
3.1162
2.9901
2.8661
2.5291
2.5831
1.2345
1.2308
1.2403
1.2373
1.2344
1.2347
1.2362
1.2257 1.4461
D(m)
0.0318
13^1/3eE
1.5403
1.6302
1.5487
1.4608
0.9075
0.8965
0.9149
0.8643
0.8313
0.7831
0.7613
0.7395
0.6804
0.6903
0.4675
110
Table A.7 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .046m
(Sharp Edge Inlet-Tapered Exit Geometry and Nozzle Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.21
AA1/3
13^113 p
Rel
0.2905
0.4439
0.5678
0.5678
0.6837
0.6837
0.7589
0.7589
1.0517
1.9695
1.8161
1.6922
1.6922
1.5763
1.5763
1171.6
2212.8
3201.9
3201.9
4230.4
4230.4
4946.6
4946.6
8070.6
11195
11195
14319
1.5011
1.5011
1.2083
1.3081 0.9519
1.3081 0.9519
1.5413 0.7187
Reg
35684
31479
27923
25393
24985
25016
24190
20709
20709
16976
15297
12369
We
951.64
740.56
582.69
481.9
466.54
467.69
437.33
320.5
320.5
215.38
174.88
114.34
E
0.0224
0.0192
0.0156
0.0123
0.0127
0.0127
0.0122
0.0082
0.0093
0.0062
0.0047
0.003
rohgc
Jgc
1.2594 11.541
1.3007 10.181
1.3302 9.0311
1.3142 8.2129
1.3541 8.0812
1.3544 8.0912
1.3769 7.8243
1.3422 6.6979
1.4538 6.6986
1.4824 5.4915
1.443 4.9482
1.4405 4.0009
D(m)
0.046
BA1/3eE
1.8842
1.7518
1.6295
1.5234
1.5221
1.5235
1.498
1.3391
1.3753
1.2126
1.1211
0.9724
111
Table A.8
Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0699m
(Sharp Edge Inlet-Tapered Exit Geometry and Nozzle Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.2
AA1/3
13^1/3 p
Rel
0.2543
0.2543
0.3916
0.3916
0.4347
0.4347
0.6024
0.6024
0.7493
0.7493
0.8829
0.8829
1.0807
1.194
1.194
1.2975
1.2975
2.0057
2.0057
1.8684
1.8684
1.8253
1.8253
1.6576
1.6576
1.5107
1.5107
1460.8
1460.8
2792.5
2792.5
3265.5
3265.5
5327.6
5327.6
7389.7
7389.7
9452.5
9452.5
12801
14866
14866
16840
16840
1.3771
1.3771
1.1793
1.066
1.066
0.9625
0.9625
Reg
We
E
53264 1410.8 0.0388
58530 1703.6 0.0491
53171 1405.9 0.0454
49256 1206.5 0.0375
0.039
49256 1206.5
44983 1006.2 0.0311
44904 1002.7 0.0349
42611 902.93 0.0307
40183 802.97 0.0287
37597 702.93 0.0243
34820 602.91 0.0214
37597 702.93 0.0259
33757 566.68 0.0213
26240 342.41 0.0118
27075 364.53 0.0128
24870 307.59 0.0106
23859 283.09 0.0096
rohgc
Jgc
1.2708 11.431
1.2815 12.561
1.3587 11.412
1.3415 10.572
1.372 10.572
1.3501 9.6545
1.4761 9.6387
1.4552 9.1463
8.626
1.5518
1.5184 8.0706
1.5861 7.4749
1.6332 8.0715
1.7384 7.2479
1.6448 5.6335
1.6662 5.8127
1.6797 5.3396
1.6507 5.1223
D(m)
0.0699
13^1/3eE
1.8778
2.0053
1.918
1.815
1.8287
1.7121
1.7618
1.6933
1.6637
1.58
1.5232
1.6189
1.5385
1.2769
1.3094
1.2407
1.1998
112
Table A.9 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0159m
(Tapered Inlet-Sharp Edge Exit Geometry and Indirect Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
1.21
0.073
998
0.001
AA1/3
BA1/3 p
Rel
Reg
We
0.7246
0.7246
1.0743
1.0743
1.3564
1.3564
1.2754
1.2754
0.9257
0.9257
0.6436
0.6436
1551.4
1551.4
2800.7
2800.7
3973.4
3973.4
4797.6
3794.8
3394.1
3220.3
2940.4
2400.9
50.319
31.482
25.185
E
0.0006
0.0003
0.0003
22.671 0.0003
18.902 0.0002
12.602 0.0001
rohgc
1.2131
1.2092
1.2163
1.2151
1.2204
1.2151
Jgc
4.5261
3.58
3.2021
3.038
2.7741
2.265
D(m)
0.0159
13^1/3eE
0.997
0.8518
0.7923
0.7647
0.7208
0.6288
113
Table A.10 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0318m
(Tapered Inlet-Sharp Edge Exit Geometry and Indirect Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.21
AA1/3
13^1/3 p
Rel
Reg
We
E
0.3566
0.3566
0.3566
0.478
0.478
0.478
0.7302
0.7302
0.7302
0.9342
0.9342
1.1253
1.1253
1.2485
1.2485
1.7305
1.7305
1.6434
1.6434
1.6434
1.522
1.522
1.522
1.2698
1.2698
1.2698
1.0658
1.0658
0.8747
0.8747
0.7515
0.7515
0.2695
0.2695
1097.4
1097.4
1097.4
1702.7
1702.7
1702.7
3214.9
3214.9
3214.9
4652.6
4652.6
6150.5
6150.5
7188.3
7188.3
11730
11730
8316.8
8596.6
8386.7
8081.4
7572.6
8117.5
6879.4
7494.2
7784.6
7288.6
6552.9
6432.1
4386.3
5327.6
5919
3674
4110.7
75.608
80.782
76.885
71.39
62.684
72.028
51.732
61.392
66.242
58.069
46.938
45.223
0.0009
21.031
31.025
38.297
14.755
18.471
0.001
0.001
0.001
0.0008
0.001
0.0008
0.0009
0.001
0.001
0.0007
0.0008
0.0003
0.0005
0.0006
0.0002
0.0003
rohgc
1.2082
1.2086
1.2083
1.2136
1.2123
1.2136
1.2236
1.2268
1.2284
1.2408
1.2348
1.2479
1.227
1.2439
1.2514
1.2464
1.2549
Jgc
3.923
4.055.
3.956
3.8121
3.572
3.8291
3.2451
3.5351
3.6721
3.4381
3.0911
3.0341
2.0691
2.5131
2.7921
1.7331
1.9391
D(m)
0.0318
BA1/3eE
0.9051
0.9254
0.9102
0.8893
0.8513
0.8919
0.801
0.8487
0.8709
0.8363
0.7778
0.7709
0.5939
0.6792
0.73
0.5305
0.573
114
Table A.11 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .046m
(Tapered Inlet-Sharp Edge Exit Geometry and Indirect Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
998
1.21
0.001
0.073
AA1/3
E3^1/3 p
0.2905
0.2905
0.2905
0.4439
0.4439
0.5678
0.5678
0.6837
0.6837
0.7589
0.7589
1.0517
1.7095
1.7095
1.7095
1.5561
1.5561
1.4322
1.4322
1.3163
1.3163
1.2411
1.2411
0.9483
0.6919
0.6919
0.4587
1.3081
1.3081
1.5413
Rel
Reg
1171.6 21500
1171.6 22804
1171.6 23653
2212.8 18084
2212.8 17480
3201.6 15704
3201.6 16315
4230.4 15063
4230.4 14367
4946.6 13187
4946.6 14505
8070.6 13656
11195 12070
11195 12880
14319 9160.1
We
E
rohgc
Jgc
0.0064 1.2232 7.0112
0.0074 1.2254 7.4362
0.0081 1.2268 7.7132
0.0049 1.2397 5.8972
0.0045 1.2377 5.7002
0.0038
1.251 5.1213
0.0041
1.254 5.3203
0.0036 1.2678 4.9123
0.0032 1.2632 4.6853
0.0027 1.2676 4.3003
0.0034 1.2779 4.7304
0.0033 1.3313 4.4536
110.1 0.0027 1.3642 3.9366
125.36 0.0031
1.381 4.2008
63.406 0.0014 1.3477 2.9874
349.32
392.95
422.77
247.13
230.89
186.37
201.13
171.47
155.98
131.4
158.99
140.92
D(m)
0.046
13^1/3eE
1.3385
1.3929
1.4278
1.198
1.1705
1.0937
1.1228
1.0686
1.0341
0.9778
1.0448
1.0174
0.9447
0.9905
0.7828
115
Table A.12 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0699m
(Tapered Inlet-Sharp Edge Exit Geometry and Indirect Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.2
M1/3
E3^1/3 p
Rel
0.2543
0.2543
0.2543
0.3916
0.3916
0.3795
0.3795
0.3795
0.6024
0.7493
0.7493
0.8829
0.8829
1.0807
1.0807
1.0807
1.194
1.194
1.2975
1.2975
1.2975
1.7457
1.7457
1.7457
1.6084
1.6084
1.6205
1.6205
1.6205
1.3976
1.2507
1.2507
1460.8
1460.8
1460.8
2792.5
2792.5
2663.4
2663.4
2663.4
5327.6
7389.7
7389.7
9452.5
9452.5
12801
12801
12801
14866
14866
16840
16840
16840
1.1171
1.1171
0.9193
0.9193
0.9193
0.806
0.806
0.7025
0.7025
0.7025
Reg
49769
47392
53543
42900
42970
42970
40523
42825
35053
34997
34875
34759
31739
26912
20187
21902
19278
20383
18486
We
1231.8
1116.9
1425.7
915.21
918.2
918.2
816.62
912.04
611.01
609.06
604.85
600.82
500.96
360.15
202.66
238.55
184.82
206.6
169.94
162.4
18071
17200 147.12
E
rohgc
0.0327
0.0289
0.0393
0.0265
0.0266
0.0263
0.0227
1.2639
1.2594
1.2713
1.315
1.3153
1.3087
1.2995
0.0261
1.3081
0.0188 1.3904
1.486
0.0203
0.0202 1.4845
0.0213 1.5851
1.536
0.0169
0.0121 1.5832
1.449
0.0059
0.0072 1.4814
0.0055 1.4801
0.0063 1.5045
0.0051 1.5074
0.0048 1.4971
0.0042 1.4759
Jgc
10.681
10.171
11.491
9.2071
9.2221
9.222
8.6969
9.191
7.5234
7.5122
7.4861
7.4619
6.8133
5.7772
4.3331
4.7013
4.1382
4.3753
3.9682
3.8792
3.692
D(m)
0.0699
13^1/3eE
1.7915
1.7319
1.8847
1.6443
1.6462
1.6434
1.5767
1.6395
1.4641
1.4954
1.4915
1.5211
1.4167
1.2821
1.0275
1.093
1.0036
1.0473
0.9819
0.9649
0.9292
116
Table A.13 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0159m
(Sharp Edge Inlet-Tapered Exit Geometry and Indirect Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.21
M1/3
BA1/3 p
0.7103 1.2897
0.7103 1.2897
1.0531 0.9469
1.0531 0.9469
1.0531 0.9469
1.3296 0.6704
Rel
Reg
1551.4
1551.4
2800.7
2800.7
2800.7
3973.4
5837.4
4026.9
We
E
0.001
74.496
35.452 0.0004
2757.1 16.618 0.0002
3128.1 21.391 0.0002
2393.5 12.524 0.0001
2393.5 12.524 0.0001
rohgc
Jgc
1.2176 5.5071
1.2101
3.799
2.601
1.212
2.951
1.2145
2.258
1.2097
1.215
2.258
D(m)
0.0159
13^1/3eE
1.1377
0.8864
0.6889
0.7499
0.6266
0.6275
117
Table A.14 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0318m
(Sharp Edge Inlet-Tapered Exit Geometry and Indirect Air Supply)
Gas Den.(Kg/mA3) Liq.Den.(Kg/mA3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
1.2
998
AA1/3
BA1/3 p
0.3566
0.3566
0.3566
0.478
0.478
0.478
0.1573
0.1573
0.1573
0.2013
0.2013
0.2424
0.2424
0.2424
0.269
0.269
0.269
0.3728
0.3728
1.6434
1.6434
1.6434
1.522
1.522
1.522
1.8427
1.8427
1.8427
1.7987
1.7987
1.7576
1.7576
1.7576
Rel
1097.4
1097.4
1097.4
1702.7
1702.7
1702.7
321.49
321.49
321.49
465.26
465.26
615.05
615.05
615.05
1.731 718.83
1.731 718.83
1.731 718.83
1.6272
1173
1.6272
1173
Reg
We
8957 87.697
8446.1 77.977
8371.9 76.613
9033.3 89.198
8647.5 81.741
8838.3 85.387
8053.9 70.904
7848.2 67.329
8000.9 69.974
7063.8 54.543
6932.4 52.532
5431.4 32.247
5741 36.027
5590.4 34.163
5079.5 28.204
4954.4 26.832
5431.4 32.247
3052.8 10.187
3167.3 10.966
E
rohgc
0.0011
0.001
0.0009
0.0013
0.0011
0.0012
0.0006
0.0006
0.0006
0.0005
0.0005
0.0003
0.0003
0.0003
0.0002
0.0002
0.0003
8E-05
1.2091
1.2084
1.2083
1.216
1.215
8E-05
1.2021
1.2155
1.2017
1.2016
1.2017
1.2022
1.2021
1.2021
1.2023
1.2022
1.2023
1.2022
1.2025
1.202
Jgc
4.225
3.984
3.949
4.2611
4.0791
4.1691
3.799
3.702
3.774
3.332
3.27
2.562
2.708
2.637
2.396
2.337
2.562
D(m)
0.0318
BA1/3eE
0.9512
0.9145
0.9091
0.9585
0.9307
0.9445
0.8844
0.8692
0.8805
0.8104
0.8003
0.6802
0.7058
0.6934
0.6505
0.6398
0.6802
1.44 0.4632
1.494 0.4747
118
Table A.15 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .046m
(Sharp Edge Inlet-Tapered Exit Geometry and Indirect Air Supply)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
1.21
998
0.001
0.073
AA1/3
13^1/3 p
0.2905 1.7095
0.2905 1.7095
0.2905 1.7095
0.4439 1.5561
0.4439 1.5561
0.4439 1.5561
0.5678 1.4322
0.5678 1.4322
0.6837 1.3163
0.6837 1.3163
0.6837 1.3163
0.7589 1.2411
1.0517 0.9483
1.0517 0.9483
1.3081 0.6919
1.3081 0.6919
1.5413 0.4587
1.5413 0.4587
Rel
1171.6
1171.6
1171.6
2212.8
2212.8
2212.8
3201.9
3201.9
4230.4
4230.4
4230.4
4946.6
8070.6
8070.6
11195
11195
14319
14319
Reg
We
24297 446.11
28361 607.79
25846 504.79
21917
363
20820 327.55
23307 410.48
20062 304.15
19075 274.94
18648 262.79
18605 261.58
18572 260.64
18133 248.47
16312 201.06
16925 216.46
14392 156.52
15063 171.47
11555
100.9
11196 94.729
E
rohgc
0.0087
0.0128
1.2279
1.2352
1.2306
1.253
1.249
1.2581
1.2736
1.2682
1.2934
1.2931
1.2929
1.309
1.3714
1.3812
1.4138
1.429
1.4093
1.3996
0.0101
0.0079
0.0069
0.0092
0.0069
0.0061
0.0062
0.0062
0.0061
0.006
0.0052
0.0057
0.0041
0.0046
0.0025
0.0023
Jgc
7.9232
9.2483
8.4283
7.1474
6.7893
7.6004
6.5425
6.2204
6.0816
6.0676
6.0566
5.9136
5.3199
5.52
4.694
4.9131
3.7688
3.6517
D(m)
0.046
BA1/3eE
1.454
1.6151
1.5163
1.3667
1.3192
1.4257
1.2955
1.2508
1.2402
1.2382
1.2367
1.2222
1.1567
1.1884
1.075
1.1121
0.9276
0.9062
119
Table A.16 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1283 for a Tube Size of .0699m
(Sharp Edge Inlet-Tapered Exit Geometry and Indirect Air Supply)
Gas Den.(Kg/mA3) Liq.Den.(Kg/mA3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.072
0.001
998
1.2
AA1/3
BA1/3 p
1.7452
1.7452
1.6075
1.6075
1.5643
1.5643
1.3962
1.3962
1.249
0.751
1.249
0.751
0.8849 1.1151
0.8849 1.1151
0.8849 1.1151
1.0832 0.9168
1.0832 0.9168
1.0832 0.9168
1.1968 0.8032
1.1968 0.8032
0.2548
0.2548
0.3925
0.3925
0.4357
0.4357
0.6038
0.6038
Rel
1460.8
1460.8
2792.5
2792.5
3265.5
3265.5
5327.6
5327.6
7389.7
7389.7
9452.5
9452.5
9452.5
12801
12801
12801
14866
14866
Reg
57225
49582
42914
42844
35006
40402
28831
28398
27564
27932
27033
26352
26240
24782
24484
24381
23645
22848
We
E
1651.1
1239.5
0.0472
0.033
0.027
0.0269
0.0169
0.0242
0.0117
0.0113
0.0114
0.0118
0.0115
0.0108
0.0107
928.53
925.51
617.85
823.02
419.11
406.61
383.07
393.37
368.45
350.14
347.17
309.65
302.24
299.71
281.88
263.21
0.01
0.0097
0.0096
0.0092
0.0085
rohgc
1.2802
1.2647
1.317
1.3168
1.3049
1.33
1.3445
1.3413
1.4034
1.4074
1.4687
1.4586
1.457
1.5445
1.5383
1.5362
1.5871
1.5677
Jgc
12.281
10.641
9.2101
9.1951
D(m)
0.0699
BA1/3eE
1.9747
1.7874
1.6455
1.6436
7.5128 1.4321
8.6711 1.5858
1.271
6.1879
6.0949 1.2572
5.9162 1.2512
5.9952 1.2636
1.254
5.8026
5.6565 1.2301
5.6325 1.2261
5.3198 1.2035
5.2558 1.1922
5.2338 1.1883
5.076 1.1771
4.9048 1.1457
120
Table A.17 Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-1284 (Dukler and Smith Data) for a Tube Size of .0508m
Gas Den.(Kg/m^3) Liq.Den.(Kg /m "3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.2
AA1/3
13^1/3p
0.0937
0.1823
0.3083
0.5220
1.9063
1.8177
1.6917
1.4780
Rel
237.11
643.46
1415.2
3118.8
Reg
We
E
43335
1285 0.0219
37575 966.08 0.0197
33787 781.12 0.0184
24531 411.77 0.0100
rohgc
1.2080
1.2224
1.2512
1.2851
D(m)
0.0508
Jgc 13^1/3eE
12.7959
11.0952
9.9769
7.2441
1.9906
1.8172
1.7062
1.3906
121
Table A.18 Calculated Flooding Parameters Using the Experimental
Values of EPRI NP-1336 for a Tube Size of .0508m
Gas Den.(Kg/m^3) Liq.Den.(Kg/mA3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
1.2
998
0.001
0.073
0.0508
Jgc
BA1/3eE
AA1/3
0.084
0.030
0.055
0.085
0.121
0.148
0.205
0.268
0.305
0.333
0.381
0.443
0.534
0.591
0.646
0.687
0.769
0.831
13^1/3p
1.176
1.230
1.205
1.175
1.139
1.112
1.055
0.992
0.955
0.927
0.879
0.817
0.726
0.669
0.614
0.573
0.491
0.429
Rel
Reg
13.538 35971
43.27 35346
106.85 32213
206.22 28991
348.75 27664
468.86 25538
765.92 24351
1144.4 21541
1393.1 20318
1592.4 19018
1946 17550
2441.1 15311
3228.5 12655
3760.1 11529
4290.5 9281.6
4704 8545.9
5578.2 7848.9
6266.4 6517.2
We
E
885.35 0.0067
854.88 0.0086
710.03 0.0085
575.11 0.0077
523.65 0.0078
446.25 0.0069
405.76 0.0069
317.5 0.0057
282.46 0.0051
247.5 0.0045
210.76 0.0039
160.42 0.0029
109.59 0.0019
90.943 0.0016
58.948 0.0010
49.973 0.0008
42.154 0.0007
29.063 0.0004
rohgc
1.200 10.621
1.201 10.437
1.202
9.512
1.204
8.560
1.207
8.168
1.208
7.541
1.215
7.190
1.220
6.361
1.223
5.999
1.225
5.616
1.229
5.182
1.231
4.521
1.233
3.737
1.235
3.404
1.230
2.741
1.229
2.523
1.232
2.318
1.228
1.924
D(m)
1.754
1.734
1.631
1.521
1.475
1.399
1.358
1.253
1.206
1.155
1.096
1.001
0.882
0.829
0.717
0.678
0.641
0.566
1 22
Table A.19 Calculated Flooding Parameters Using the Experimental Values
of EPRI NP-2262 for a Tube Size of .0127m with a Stub Entry
Gas Den.(Kg /m "3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
1.2
998
0.001
0.0127
rohgc
Jgc
13"1/3eE
1.202
3.452
2.516
1.774
1.379
1.129
3.278
3.169
AA1/3
0.390
0.390
0.471
0.538
0.619
0.477
0.540
BA1/3p
0.870
0.870
0.789
0.722
0.641
0.783
0.720
Rel
Reg
480.31 2922.5
480.31 2130.6
637.24 1501.8
776.74 1167.6
958.97 955.68
648.56 2775.1
781.36 2682.7
We
23.377
12.425
6.1732
3.7313
2.4998
21.079
19.698
E
.00017
.00008
.00004
.00002
.00001
1.201
1.201
1.201
1.201
.00017
.00016
1.203
1.203
D(m)
0.830
0.672
0.532
0.450
0.394
0.802
0.784
123
Table A.20 Calculated Flooding Parameters Using the Experimental Values
of EPRI NP-2262 for a Tube Size of .0127m with a plate Entry
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
1.2
998
0.001
0.0127
rohgc
Jgc
13^1/3eE
1.202
1.202
3.676
3.528
1.956
4.030
3.765
AA1/3
0.354
0.354
0.478
0.478
0.542
13^1/3p
0.906
0.906
0.782
0.782
0.718
Rel
415.6
415.56
650.18
650.18
785.29
Reg
3112.7
2986.8
1655.7
3412.1
3187.5
We
26.519
24.418
7.5036
31.865
27.808
E
.00020
.00018
.00005
.00028
.00025
1.201
1.204
1.204
D(m)
0.865
0.842
0.568
0.920
0.880
Table A.21
124
Calculated Flooding Parameters Using the Experimental Values of
EPRI NP-2262 for a Tube Size of .0305m and Different Lengths
Gas Den.(Kg/mA3) Liq.Den.(Kg/mA3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.2
length=2.4"
Run 17.2
AA1/3
0.196
0.196
0.246
0.256
0.298
0.290
0.320
0.353
0.377
D(m)
0.0305
BA1/3eE
rohgc
Jgc
.00067
.00077
.00093
.00074
.00076
.00061
.00041
.00037
.00047
1.202
1.203
1.204
1.204
1.205
1.204
1.204
1.205
1.206
3.859
4.085
4.253
1.070 409.46 11331 146.43 .00166
1.077 385.83 10001 114.07 .00120
1.014 491.54 11592 153.25 .00184
1.004 563.63 10324 121.56 .00143
1.000 653.72 10048 115.14 .00138
0.970 646.16 9987.4 113.76 .00136
9764 108.72 .00131
0.940 694.74
0.907 839.55 10866 134.66 .00179
0.883 906.06 10729 131.29 .00177
5.576
4.922
5.705
1.143
5.081
1.075
1.056
.00183
.00124
.00176
1.204
1.203
1.205
1.205
1.206
1.206
1.206
1.209
1.210
1.210
1.209
1.212
1.073 397.95 10532 126.49 .00137
1.034 529.92 9377.8 100.29 .00110
0.982 721.54 11136 141.44 .00183
0.943 882.43 10023 114.57 .00148
0.874 1185.8 8231.2 77.267 .00097
0.913 1008.7 9935.4 112.58 .00150
0.948 862.02 10084 115.96 .00149
1.203
1.204
1.208
1.209
1.209
1.210
1.209
5.183
4.615
1.088
1.008
5.481
1.131
4.933
1.055
0.925
1.049
1.059
Rel
BA1/3p
Reg
We
1.064 427.23 7840.5 70.106
1.064 427.23 8300.8 78.58
1.014 601.77 8641.6 85.165
1.004 638.65 7845.3 70.193
0.962 803.32 7734.1 68.216
0.970 771.18 7122.2 57.85
0.940 894.21 5997.6 41.022
0.907 1036.5 5691.6 36.944
0.883 1142.7 6164.3 43.335
E
3.861
3.806
3.505
2.952
2.801
3.034
0.894
0.928
0.954
0.895
0.886
0.839
0.748
0.722
0.762
Run 17-3, Lenrth=4.8"
0.190
0.183
0.215
0.235
0.260
0.258
0.320
0.353
0.377
0.329
0.357
0.369
0.931
931.2
0.903
1053.1
0.891
1106.3
10860
134.51
9164 95.773
10501
125.75
4.945
4.915
4.805
5.348
5.280
5.345
4.510
5.168
1.051
1.161
1.051
1.036
1.113
1.104
1.113
0.994
1.089
Run 18-1,Lenrth=10"
0.187
0.226
0.278
0.317
0.386
0.347
0.312
4.051
4.890
4.962
125
Table A.22 Calculated Flooding Parameters Using the Experimental
Values of NUREG/CR-0312 for a Tube size of .0508m
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
1.2
998
0.0508
Jgc
BA1/3eE
5.396
1.121
4.841
4.296
1.044
0.964
0.779
0.592
0.547
0.317
0.126
A^1/3
13^1/3P
0.251
1.009 1041.7 18273 228.48 .003659
0.956 1384.6 16394 183.91 .002995
0.907 1731.9 14548 144.81 .002349
0.763 2895.7 10550 76.167 .001197
6982 33.356 .000496
0.516 5307.3
0.444
6092 6198.3 26.289 .000381
0.178 9309.6 2754.8 5.1928 .000056
-0.072 12716 688.7 0.3246 .000002
0.304
0.353
0.497
0.744
0.816
1.082
1.332
Rel
Reg
We
E
rohgc
1.214
1.217
1.219
1.222
1.225
1.225
1.213
1.202
3.115
2.062
1.830
0.813
0.203
D(m)
126
Table A.23 Calculated Flooding Parameters Using the Experimental
Values of NUREG/CR-0312 for a Tube size of .1524m
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.2
AA1/3
B^1 /3P
0.002
0.002
0.018
0.010
0.018
0.053
1.598
1.598
1.582
1.590
1.582
1.547
1.499
1.336
1.018
0.984
0.867
0.674
0.101
0.264
0.582
0.616
0.733
0.926
Rel
Reg
145756
129862
117332
111824
99577
80742
73306
54570
38102
32811
27063
22162 22259
1.784
1.784
60.187
26.75
60.187
307.71
801.85
3380.5
11051
12013
15623
We
4845.7
3846.5
3140
2852.2
2261.6
1487
1225.7
679.22
331.13
245.56
167.05
113.01
E
.03388
.02538
.04747
.03437
.03149,
.02804
.02798
.01917
.01050
.00738
.00487
.00326
rohgc
Jgc
1.200 14.346
1.200 12.782
1.202 11.548
1.201 11.006
1.201
9.801
1.207
7.947
1.220
7.215
1.279
5.371
1.403
3.751
1.380
3.230
1.387
2.664
1.416
2.191
D(m)
0.1524
13^1/3eE
2.144
1.985
1.856
1.797
1.663
1.449
1.363
1.137
0.923
0.831
0.732
0.647
127
Table A.24 Calculated Flooding Parameters Using the Experimental
Values of NUREG/CR-0312 for a Tube size of .254m
D(m)
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.2
0.254
Jgc
13^1/3eE
AA1/3
13^1/3P
0.010
0.015
0.015
0.035
0.056
0.065
0.090
0.133
1.590
1.585
1.585
1.565
1.544
1.535
1.510
1.467
1.459
1.444
1.388
1.366
1.380
1.338
1.338
1.308
1.300
1.275
1.244
1.208
1.185
1.185
1.197
1.132
1.160
1.123
1.067
1.083
1.127
0.999
0.956
0.141
0.156
0.212
0.234
0.220
0.262
0.262
0.292
0.300
0.325
0.356
0.392
0.415
0.415
0.403
0.468
0.440
0.477
0.533
0.517
0.473
0.601
0.644
Rel
41.585
76.119
76.119
270.2
544.2
688.35
1122.3
2003.5
2200.5
2545.3
4045.1
4688
4265.1
5565.4
5565.4
6548.7
6817.4
7667.1
8802.4
10168
11094
11094
10594
13283
12115
13650
16148
15402
13473
19310
21421
Reg
We
191114
181495
168574
158009
146843
133572
133572
135956
119591
122522
110252
110252
99127
99127
108916
111538
98630
104219
98630
98630
98630
98630
85230
85230
74522
74522
74527
86825
86825
86825
82486
4998.5
4508
3889
3416.8
2950.9
2441.7
2441.7
2529.6
1957.3
2054.4
1663.5
1663.5
1344.7
1344.7
1623.4
1702.5
1331.3
1486.4
1331.3
1331.3
1331.3
1331.3
994.11
994.11
760.02
760.02
760.12
1031.7
1031.7
1031.7
931.12
E
.07738
.07910
.06577
.07679
.07616
.06374
.07202
.08702
.06464
.07122
.06143
.06373
.04771
.05099
.06453
.07133
.05298
.06261
.05647
.05854
.05983
.05983
.04106
.04345
.03035
.03127
.03262
.04722
.04567
.04997
.04511
rohgc
1.201
1.202
1.202
1.209
1.219
1.222
1.240
1.285
1.279
1.299
1.350
1.380
1.337
1.391
1.420
1.479
1.444
1.507
1.536
1.602
1.648
1.648
1.540
1.651
1.529
1.581
1.671
1.758
1.672
1.940
1.980
11.286
10.718
9.955
9.331
8.672
7.888
7.888
8.030
7.063
7.236
6.512
6.512
5.855
5.855
6.433
6.589
5.826
6.157
5.827
5.827
5.827
5.827
5.035
5.036
4.402
4.403
4.403
5.130
5.130
5.131
4.875
1.827
1.766
1.681
1.613
1.540
1.447
1.455
1.490
1.365
1.394
1.317
1.327
1.223
1.239
1.328
1.368
1.250
1.316
1.276
1.294
1.307
1.307
1.159
1.186
1.057
1.069
1.089
1.227
1.206
1.268
1.233
Table A.24 (Continued)
0.641
0.481
0.561
0.612
0.660
0.904
0.962
0.981
1.035
1.067
1.154
0.959
1.119
1.039
0.988
0.940
0.696
0.638
0.619
0.565
0.533
0.446
21273
13813
17409
19841
22243
35638
39133
40292
43659
45717
51428
1 28
74379
62894
59637
62411
62411
40450
37437
38981
38981
38981
38981
757.09
541.34
486.72
533.06
533.06
223.91
191.8
207.95
207.95
207.95
207.95
.03477
.02052
.01904
.02204
.02268
.00863
.00728
.00811
.00828
.00837
.00862
1.862
1.500
1.570
1.667
1.738
1.706
1.707
1.758
1.817
1.854
1.957
4.395
3.715
3.523
3.687
3.688
2.390
2.212
2.303
2.303
2.304
2.304
1.128
0.938
0.919
0.967
0.980
0.730
0.693
0.719
0.727
0.732
0.745
129
Table A.25 Calculated Flooding Parameters Using the Experimental Values
of NUREG/CR-0312 for an Annulus with Gap Width=.0254m
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.2
Gap (m)
Jgc
BA1/3eE
AA1/3
0.553
0.275
0.000
0.514
0.959
0.231
0.377
0.453
0.594
0.689
0.753
0.792
0.824
0.859
1.072
0.554
0.380
0.208
0.126
0.129
0.553
0.000
0.010
0.166
0.006
0.093
0.068
0.038
B^1 /3P
1.247
1.525
1.800
1.286
0.841
1.569
1.423
1.347
1.206
1.111
1.047
1.008
0.976
0.941
0.728
1.246
1.420
1.592
1.674
1.671
1.247
1.800
1.790
1.634
1.794
1.707
1.732
1.762
Reg
Rel
3402.4
1192.4
0
12976
22440
54373
12924
496.18
27641
22231
19646
16659
14572
11970
10458
3045.9
7765.7
920.59
1916.8
2523.8
3785.6
4732.1
5408.1
5824.3
6180.8 8211.1
6583.8 6798.5
9177.6 586.34
3410.6 13214
1935.5 20066
785.4 26610
370.78 30413
382.37 37862
3397.4 12716
0
52736
7.8837 51070
557.79 33063
3.7765 36588
232.76 38130
145.43 40920
60.096 44282
We
E
115.21 .002090
344.57 .006325
2023 .000000
114.3 .002013
0.1685.000001
522.79 .009983
338.17 .006957
264.1 .005471
189.91 .004009
145.3 .003034
98.042 .001918
74.837 .001394
46.134 .000773
31.627 .000490
0.2352 .000001
119.47 .002188
275.52 .005398
484.54.008725
632.91 .010099
980.92 .017600
110.65 .001986
1903 .000000
1784.7 .014092
748.02 .013783
916.03.005093
994.86 .015822
1145.8 .016783
1341.8 .016393
rohgc
1.236
1.222
1.200
1.232
1.201
1.222
1.240
1.247
1.261
1.266
1.258
1.252
1.239
1.232
1.201
1.238
1.235
1.217
1.208
1.212
1.235
1.200
1.200
1.215
1.200
1.206
1.204
1.201
3.832
6.626
16.055
3.816
0.147
8.162
6.565
5.801
4.919
4.303
3.535
3.088
2.425
2.008
0.173
3.902
5.925
7.858
8.980
11.180
3.755
15.572
15.080
9.763
10.804
11.259
12.083
13.075
0.0254
0.898
1.289
2.311
0.894
0.101
1.481
1.287
1.187
1.068
0.978
0.856
0.781
0.662
0.583
0.113
0.909
1.200
1.442
1.572
1.821
0.886
2.264
2.216
1.666
1.774
1.827
1.914
2.016
130
Table A.25 (Continued)
0.022
0.715
0.515
0.478
0.106
0.038
0.439
0.649
0.675
0.913
1.012
1.168
1.326
1.405
0.092
0.197
0.255
0.456
0.580
1.659
0.566
0.472
0.847
0.541
0.480
1.168
1.225
1.364
1.405
0.031
1.659
0.242
0.059
0.030
0.444
0.091
0.000
0.000
0.681
0.720
0.802
1.899
0.649
1.778
1.085
1.285
1.322
1.694
1.762
1.361
1.151
1.125
0.887
0.788
0.632
0.474
0.395
1.708
1.603
1.545
1.344
1.220
0.141
1.234
1.328
0.953
1.259
1.320
0.632
0.575
0.436
0.395
1.769
0.141
1.558
1.741
1.770
1.356
1.709
1.800
1.800
1.119
1.080
0.998
27.371
5000.1
3059.3
2728.4
286.82
62.255
2403.7
4326.5
4588.8
7210.8
8412.8
10443
12619
13766
232.48
724.53
1066.4
2545
3648.9
17667
3521.7
2680.3
6443.7
3291.9
2753.2
10443
11217
13168
13766
44.145
17667
986.56
117.95
43.022
2442.4
227.83
0
0
4649.2
5047.8
5938.3
-0.099 21633
1.151 4326.5
49566 1681.1 .017850
12715 110.63 .002187
14102 136.08 .002506
16171 178.93 .003429
40593 1127.5 .019493
940.67 0.6055 .000001
21136 305.68 .006488
19743 266.72 .006338
16716 191.21 .004243
14730 148.48 .003463
13458 123.94 .002871
10846 80.491 .001767
48.82 .000992
8446.7
879.56 0.5294 .000004
41006 1150.6 .018971
36167 895.04 .018414
30250 626.16 .012977
23695 384.18 .008758
12900 113.88 .002096
1128.7 0.8718 .000007
16678 190.34 .003949
21758 323.93 .007169
20730 294.06 .007910
20854 297.59 .006788
21958 329.91 .007384
18732 240.09 .006926
15953 174.14 .004720
14516 144.18 .003880
11919 97.207 .002397
8826.7 53.311 .000269
1128.7 0.8718 .000007
32920 741.57 .015724
43960 1322.3 .019052
49731 1692.3 .020153
26688 487.35 .011671
40280 1110.2 .018051
54088 2001.8 .000000
53310 1944.6 .000000
13277 120.62 .002393
16588 188.29 .004262
13127 117.91 .002473
1382.2 1.3074 .000012
22279 339.63 .008573
1.201
1.257
1.236
1.239
1.209
1.200
1.249
1.292
1.278
1.313
1.320
1.313
1.299
1.204
1.207
1.225
1.230
1.263
1.239
1.207
1.256
1.259
1.364
1.271
1.262
1.457
1.421
1.434
1.384
1.200
1.207
1.231
1.203
1.201
1.271
1.207
1.200
1.200
1.256
1.286
1.274
1.213
1.311
14.636
3.755
4.164
4.775
11.986
0.278
6.241
5.830
4.936
4.350
3.974
3.203
2.494
0.260
12.108
10.679
8.932
6.997
3.809
0.333
4.925
6.425
6.122
6.158
6.484
5.532
4.712
4.287
3.520
2.606
0.333
9.721
12.980
14.684
7.881
11.894
15.971
15.741
3.921
4.899
3.876
0.408
6.579
2.173
0.891
0.949
1.040
1.906
0.155
1.247
1.206
1.075
0.997
0.940
0.813
0.686
0.148
1.918
1.773
1.576
1.351
0.895
0.175
1.067
1.275
1.268
1.243
1.284
1.212
1.080
1.017
0.881
0.688
0.175
1.668
2.007
2.178
1.466
1.895
2.302
2.280
0.916
1.072
0.914
0.200
1.313
1 31
Table A.25 (Continued)
0.543
1.257
1.290
0.510
0.953
0.847
1.384
0.416
1.405
0.395
1.497
0.303
1.549
0.251
1.659
0.141
1.238
0.562
1.490
0.310
0.183
1.617
1.714
0.086
1.762
0.038
0.703
1.097
1.008
0.792
0.650
1.150
1.766
0.034
1.800
0.000
1.257
0.543
1.008
0.792
0.901
0.899
1.290
0.510
1.899 -0.099
1.899 -0.099
2.165 -0.365
1.800
0.000
1.326
0.474
1.747
0.053
1.663
0.137
1.220
0.580
1.526
0.274
0.872
0.928
1.002
0.798
1.825 -0.025
1.995 -0.195
11648
12114
6443.7
13461
18994
20498
22007
17455
14829
11713
13766
15143
15940 9416.1
17667 1128.7
3481.2 25035
1428.6 32981
647.11 38917
208.56 42660
60.89 47553
4877 22571
5824.3 18798
10192 15692
52.579 50362
54009
0
11648 16105
5824.3 22636
7043.2 20314
12114 18401
21633 17300
21633 14923
26336 12903
49538
0
12619 15370
101.98 46506
417.76 40842
3648.9 25746
15588 15045
6730.3 21541
8297.3 19053
20385 16080
23297 14388
246.86 .007370
287.51 .009005
331.39 .009184
208.47 .006186
150.46 .004138
93.877 .002350
60.669 .001379
0.8718 .000007
428.87 .010869
744.32 .017329
1036.4 .021501
1245.3 .020381
1547.3 .019654
348.6 .009126
241.8 .006039
168.5 .004422
1735.5 .021869
1996 .000000
177.47 .004879
350.6 .009609
282.36 .007687
231.69 .006875
204.8 .006812
152.39 .004708
113.92 .003437
1679.2 .000000
161.64 .004429
1479.9 .021148
1141.4 .021745
453.57 .011795
154.88 .004426
317.52 .008802
248.39 .006823
176.93 .005590
141.65 .004377
1.501
1.554
1.379
1.518
1.456
1.402
1.355
1.207
1.301
1.250
1.224
1.207
1.202
1.331
1.325
1.391
1.202
5.610
6.055
6.499
5.156
4.380
3.459
2.781
0.333
7.393
9.739
11.492
12.597
14.041
6.666
5.551
4.634
14.871
1.200 15.948
4.756
1.435
1.365
6.685
1.377
5.999
5.435
1.501
5.111
1.767
4.409
1.654
3.812
1.667
1.200 14.627
4.539
1.442
1.203 13.732
1.215 12.060
7.603
1.311
4.444
1.505
1.383
6.362
1.398
5.627
4.750
1.672
4.250
1.672
1.235
1.315
1.324
1.172
1.037
0.875
0.748
0.175
1.415
1.678
1.861
1.969
2.114
1.331
1.176
1.060
2.196
2.300
1.090
1.345
1.255
1.209
1.226
1.086
0.989
2.171
1.058
2.084
1.917
1.446
1.058
1.307
1.209
1.146
1.064
132
Table A.26 Calculated Flooding Parameters Using the Experimental Values
of NUREG/CR-0312 for an Annulus with Gap Width=.0508m
Gas Den.(Kg/mA3) Liq.Den.(Kg/mA3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
1.2
998
Gap (m)
Jgc
BA1/3eE
AA1/3
BA1/3P
0.644
0.243
0.196
0.140
0.036
0.000
0.150
0.306
0.242
0.280
0.347
0.430
0.510
1.156
1.557
1.604
1.660
1.764
1.800
1.650
1.494
1.558
1.520
1.453
1.370
1.290
1.249
0.551
0.599
1.073
0.253
0.224
0.133
0.024
0.000
0.283
0.330
0.388
0.452
0.551
0.624
0.810
1.207
0.307
0.246
0.026
Rel
8574.4
1983.9
1432.9
868.93
111.93
0
966.97
2803.5
1977.9
2461.1
3393.7
4672.4
6037.4
6787.2
1.201 7675.9
0.727 18423
1.547 2107.3
1.576 1761.9
1.667 804.01
1.776 63.147
1.800
0
1.517 2499.1
1.470 3145.4
1.412 4005.1
1.348 5037.5
1.249 6787.2
1.176
8162
0.990 12075
0.593 21982
1.493 2815.7
1.554 2021.4
1.774 67.935
Reg
We
547.8
47710
53576
60192
72224
89285
55678
40046
46595
43397
36046
32234
27623
23208
15719
1176.9
48073
53412
61011
71845
89253
43278
38873
35487
31422
27859
23730
16703
1404.4
47625
53733
70348
0.1027
778.78
982.03
1239.6
1784.7
2727.4
1060.6
548.67
742.79
644.33
444.53
355.49
261.05
184.28
84.537
0.4739
790.68
976.06
1273.5
1766
2725.5
640.79
516.99
430.85
337.79
265.54
192.66
95.457
0.6748
776.01
987.81
1693.2
E
rohgc
.00000
.01991
.02452
.02895
.02735
.00000
.02447
.01401
.01875
.01658
.01130
.00925
.00671
.00447
.00174
.00000
.02060
.02563
.02937
.02340
.00000
.01653
.01338
.01132
.00885
.00705
.00495
.00227
1.200
0.081
1.255
7.044
7.910
8.887
10.663
13.182
8.220
5.913
6.879
6.407
5.322
4.759
4.079
3.427
.00001
.02163
1.206
.02692
.02261
1.267
1.244
1.228
1.203
1.200
1.228
1.265
1.253
1.263
1.271
1.289
1.298
1.287
1.257
1.203
1.260
1.256
1.226
1.201
1.200
1.264
1.272
1.285
1.294
1.314
1.313
1.309
1.285
1.201
2.321
0.174
7.098
7.886
9.008
10.607
13.177
6.390
5.740
5.240
4.639
4.114
3.504
2.466
0.207
7.032
7.934
10.386
0.0508
0.068
1.354
1.459
1.570
1.760
2.026
1.490
1.208
1.332
1.274
1.128
1.052
0.951
0.845
0.646
0.113
1.363
1.461
1.583
1.753
2.025
1.272
1.187
1.121
1.036
0.961
0.863
0.682
0.127
1.363
1.471
1.729
Table
A.26 (Continued)
0.000
1.800
0.869
0.931
1 33
0
0.634
13433
1.166 8374.2
0.516
1.284
0.416
1.384
0.371
0.049
3749
1.481 2985.2
1.482 2971.3
1666
1.584
1.556 2002.6
1.751 177.52
0.000
1.800
0
0.324
1.476
0.340
1.460
0.410
1.390
0.503
1.297
0.551
1.249
3050.2
3289.7
4356.6
5915.4
6787.2
8266.9
13159
7411.3
2014.9
4214.3
497.6
72.085
0.319
0.318
0.216
0.244
6141.1
4446.9
1.429
0.629
0.857
1.171
0.585
0.245
1.215
1.555
0.401
1.399
0.097
1.703
0.027
0.000
1.773
0.998
0.802
0.943
1.800
0
0.233
16533
0.865 14995
0.906 14017
11514
1.016
1.149 8713.4
1.211 7497.5
1.277 6260.3
1.567 1858.2
0.388
1.412
0.935
0.894
0.784
0.651
0.589
0.523
0.151
1.649
0.039
1.761
4005.1
976.88
126.5
0.000
1.800
0
0.599
1.201
0.733
1.067
7675.9
10400
14329
14654
16705
0.908
0.892
0.921
0.879
1.005
0.795
89093 2715.7 .00000
17001 98.892 .00243
22216 168.86 .00422
284.6 .00750
28842
33466 383.18 .01004
38304 501.98 .01348
41461 588.12 .01552
47107 759.2 .02133
55474 1052.9 .02778
53739 988.04 .02687
70514 1701.2 .02891
93893 3016.2 .00000
46085 726.64 .02033
40873 571.57 .01534
36449 454.54 .01236
32855 369.32 .01029
28695 281.71 .00759
194.7 .00503
23855
17967 110.44 .00278
47610 775.51 .02753
61968 1313.8 .03842
1123 .03797
57291
68675 1613.6 .03502
80959 2242.4 .03260
94280 3041.1 .00000
18262 114.11 .00307
24614 207.28 .00631
31192 332.87 .01122
35157 422.87 .01440
41077 577.28 .01982
44041
663.6 .02272
718.9 .02401
45839
57871 1145.8 .03173
52302 935.91 .02986
65353 1461.2 .03662
81288 2260.7 .03790
93923 3018.1 .00000
42643 622.13 .02108
39237 526.74 .01848
35180 423.43 .01524
31332 335.88 .01147
22224 168.98 .00502
2.023
1.306
13.153
2.510
3.280
4.259
1.289
4.941
1.079
1.288
1.180
1.290
5.656
6.122
6.955
1.256
8.191
1.498
1.267
7.934
1.471
1.205
10.411
1.733
1.200
13.862
2.095
1.290
6.805
1.335
1.282
1.230
1.298
6.035
5.382
1.323
4.851
1.075
1.320
4.237
0.981
1.316
3.522
2.653
0.867
7.031
1.431
1.624
1.582
1.217
9.150
8.460
10.139
1.200
1.328
1.306
1.274
1.336
1.485
1.283
1.386
0.694
0.824
0.981
1.240
1.355
1.145
0.721
1.709
1.202
11.953
1.899
1.200
13.919
2.101
1.385
2.697
3.635
4.607
5.192
6.066
6.504
6.769
8.545
7.723
0.738
9.649
12.001
1.662
1.453
13.867
6.297
2.095
1.526
5.795
1.269
1.613
1.202
1.557
5.196
4.628
1.451
3.282
0.854
1.456
1.536
1.514
1.480
1.458
1.418
1.268
1.352
1.236
1.204
1.200
0.915
1.091
1.176
1.295
1.350
1.374
1.545
1.476
1.905
1.320
1.100
134
Table A.27 Calculated Flooding Parameters Using the Experimental Values
of NUREG/CR-0526 for Steady State Case in 1/15 Scale Model
Gas Den.(Kg/mA3) Liq.Den.(Kg/mA3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.2
Gap(m)
0.015
Set 352
AA1/3
0.543
0.559
0.533
0.354
0.319
0.291
0.179
0.149
0.098
Set 353
0.708
0.712
0.682
0.548
0.586
0.366
0.308
0.498
set 354
0.445
0.442
0.414
0.403
0.374
0.372
0.409
0.366
0.344
B^1 /3p
Reid
Reg
We
E
rohgc
Jgc
BA1/3eE
0.360
0.514
0.659
0.863
1.029
1.334
1.558
1.728
1.896
1939.4 2019.6 4.7303
1.241 2026.8 3433.3 13.671
1887 4981.6 28.781
1.267
1.446 1022.1 7472.4 64.758
1.481 873.63 9727.6 109.74
237.9
1.509 760.05 14322
381.03
1.621 366.92 18126
1.651 279.56 21172 519.86
1.702 148.52 24370 688.75
.00003
.00013
.00032
.00075
.00140
.00356
.00534
.00736
.00893
1.202
1.205
1.208
1.207
1.208
1.213
1.207
1.206
1.204
1.092
1.088
1.118
1.252
1.214
1.434
1.492
1.302
2891.7 5099.4 30.159
2917.9 7152.7 59.334
2734.4 8330.7 80.489
1965.7 10182 120.24
2175.3 10552 129.14
1074.6 13060 197.81
829.94 15214 268.45
1703.6 16594 319.36
.00038
.00088
.00126
.00192
.00216
.00308
.00423
.00629
1.214
1.224
1.228
1.225
1.230
1.217
1.215
1.243
2.552
3.580
4.169
5.096
6.536
7.614
8.305
0.670
0.842
0.933
1.066
1.093
1.256
1.390
1.484
1.355
1.358
1.386
1.397
1.426
1.428
1.391
1.434
1.456
1441.5
1424
1293
1240.5
1109.5
1100.8
1266.8
1074.6
978.46
17789
19708
21761
21172
23915
25076
22872
26743
29906
.00718
.00924
.01156
.01068
.01409
.01583
.01302
.01848
.02387
1.239
1.244
1.246
1.242
1.244
1.246
1.248
1.249
1.252
8.903
9.863
10.891
10.596
11.969
12.550
11.447
13.384
14.968
1.552
1.664
1.779
1.745
1.893
1.955
1.840
2.043
2.202
1.257
367.01
450.44
549.19
519.86
663.31
729.29
606.69
829.42
1037.3
1.011
1.718
2.493
3.740
4.868
7.168
9.071
10.596
12.196
5.281
Table A.27 (Continued)
Set 355
0.956
0.546
0.340
0.308
1
.00069
.00165
.00304
.00532
.00920
1.233
1.222
1.215
1.218
1.236
3.108
4.801
6.578
8.347
10.023
0.768
1.024
1.260
1.478
1.679
601.34 .01314
724.4 .01617
24992
802.5 .01818
26305
28779 960.55 .02242
1075 .02529
30445
1.253
1.253
1.255
1.258
1.257
11.397
12.508
13.165
14.404
15.238
1.837
1.955
2.023
2.150
2.232
.00034
.00109
.00019
.00053
.00085
137.51 .00199
217.66 .00348
289.64 .00566
1.234
1.229
1.224
1.223
1.217
1.214
1.218
1.244
2.266
3.875
1.786
2.855
3.630
5.450
6.856
7.909
0.623
0.889
0.530
0.724
0.849
370.49
493.75
593.37
677.38
732.23
928.24
928.24
.00743
.01032
.01292
.01495
.01606
.02177
.02144
1.244
1.247
1.252
1.252
1.249
1.260
1.256
8.945
10.327
1.559
1.717
1.828
1.911
1.959
2.127
2.125
8.2005
17.377
28.781
58.499
132.46
.00009
.00023
.00042
.00089
.00220
1.218
1.227
1.231
1.228
1.228
1.331
44.728
106.73
200.37
0.844
1.254
1.460
1.492
1.409
4534.1
1956.9
960.99
829.94
1188.1
6210.2
9593
13144
16678
20027
1371.6
1240.5
1188.1
1118.2
1030.9
22771
0.403
0.391
0.376
0.356
1.369
1.397
1.409
1.424
1.444
Set 357
1.236
0.748
1.235
0.842
0.594
0.384
0.368
0.366
0.564
1.052
0.565
0.958
1.206
1.416
1.432
1.434
6665.8
3136.3
6657
3747.9
2219
1153.2
1083.3
1825.9
4527.2
7741.7
3567.9
5705.3
7253.6
10889
13699
15803
23.77
69.509
14.764
37.751
61.021
0.409
0.382
0.390
0.374
1.326
1.364
1.369
1.391
1.418
1.410
1.426
1581.3
1397.8
1371.6
1266.8
1144.4
1179.4
1109.5
17873
20633
22619
24168
25127
28291
28291
Set 359
1.363
1.246
1.092
0.775
0.568
0.437
0.554
0.708
1.025
1.232
7722.8 2659.1
6744.4 3870.8
5538.8 4981.6
3311 7102.2
2079.2 10687
0.391
35
322.61
465.18
Set 356
0.431
Set 358
0.474
0.436
0.431
11.321
12.096
12.576
14.159
14.159
1.937
2.493
3.554
5.349
1.111
1.297
1.436
0.435
0.560
0.663
0.839
1.102
Table A.27 (Continued)
136
Set 360
249.78
409.86
552.59
582.82
762.95
.00477
.00786
.01178
.01230
.01697
1930.7
1196.9
0.401
0.386
1.259
1.407
1.373
1.399
1.414
1231.8
1161.9
14676
18799
21828
22417
25649
Set 361
1.394
1.244
0.887
0.691
0.422
0.308
0.406
0.556
0.913
1.109
1.378
1.492
7984.9
6735.7
4053.6
2786.9
1327.9
829.94
2490.8 7.1953 .00008
3837.2 17.076 .00023
5116.2 30.358 .00041
8347.6 80.815 .00128
12521 181.83 .00292
15820 290.26 .00466
1.401
1223.1
1.358
1.605
1.627
1.553
1.618
1424
419.34
349.45
594.07
375.66
0.541
0.393
0.427
Set 362
0.399
0.442
0.195
0.173
0.247
0.182
1354.1
19068
19775
20398
21054
22367
20431
421.68
453.53
482.54
514.09
580.2
484.13
.00819
.00932
.00742
.00767
.01019
.00725
7.345
1.242
9.409
1.233
1.249 10.925
1.245 11.220
1.251 12.837
1.366
1.608
1.784
1.814
1.988
1.216
1.247
1.920
2.561
4.178
6.267
7.918
0.416
0.556
0.673
0.935
1.222
1.427
1.235
1.245
1.210
1.208
1.218
1.209
9.543
9.897
10.208
10.537
11.194
10.225
1.624
1.668
1.687
1.722
1.798
1.688
1.217
1.227
1.222
1.228
1.221
137
Table A.28 Calculated Flooding Parameters Using the Experimental Values
of NUREG/CR-0312 for Steady State Case in 2/15 Scale Model
Set:48.7
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
0.073
0.001
998
1.2
AA1/3
1.509
1.449
1.256
1.032
0.950
0.876
0.712
0.643
0.527
0.481
0.354
0.386
0.429
0.491
0.551
0.704
0.863
1.174
1.415
1.932
1.302
1.166
0.968
0.755
0.592
0.503
0.434
0.408
0.380
B^1 /3p
Reid
Reg
18867 7861.4
17754 10210
14336 13272
0.968 10679 17356
1.050 9433.6 17918
1.124 8347.1 19653
1.288 6121.2 21951
1.357 5246.8 23686
1.473 3895.3 25371
1.519 3391.8 27872
1.646 2146.4 35070
1.614 2437.9 33232
1.571 2861.9 29812
1.509 3497.8 28842
1.449 4160.3 27413
1.296 6015.2 24095
1.137 8161.6 20777
0.826 12958 11690
0.585 17145 13068
0.068 27347 6687.3
0.698 15131 13170
0.834 12825 16897
1.032 9698.6 20011
1.245 6677.7 21389
1.408 4637.3 27158
1.497 3630.3 27209
1.566 2914.9 32569
1.592 2649.9 34151
1.620 2384.9 33028
0.491
0.551
0.744
We
34.381
57.988
97.999
167.58
178.6
214.88
268.05
312.11
358.09
432.18
684.21
614.38
494.43
462.78
418.05
322.97
240.14
76.023
95.007
24.878
96.497
158.83
222.76
254.51
410.3
411.84
590.09
648.82
606.85
E
rohgc
Jgc BA1/3eE
1.887
.0007
.0013
.0024
.0044
.0047
.0057
.0070
1.313
1.355
1.376
1.382
1.364
.0081
1.319
3.187
4.167
4.302
4.719
5.270
5.687
.0089
.0109
.0173
.0156
.0124
.0120
.0110
.0087
.0065
.0017
.0025
.0005
.0024
.0044
.0062
.0067
.0110
.0104
.0155
.0170
.0153
1.291
6.091
1.288
1.270
1.276
1.279
1.297
6.692
8.420
7.978
7.157
6.925
1.311
1.345
6.581
1.361
1.329
1.370
1.328
1.415
1.341
1.386
1.420
1.400
1.338
1.326
1.293
1.292
1.288
1.273
Gap(m)
0.0312
2.451
5.785
4.988
2.807
3.138
1.606
3.162
4.057
4.805
5.135
6.520
6.532
7.819
8.199
7.929
0.560
0.673
0.806
0.965
0.981
1.043
1.114
1.169
1.215
1.293
1.500
1.449
1.349
1.325
1.286
1.190
1.085
0.732
0.805
0.506
0.804
0.957
1.066
1.098
1.283
1.274
1.436
1.480
1.442
138
Table A.28 (Continued)
0.394
0.424
0.476
0.493
0.544
0.590
0.723
0.909
1.176
1.534
1.682
2.015
1.576
1.524
1.507
1.456
1.410
1.277
2517.4
2808.9
3338.8
3524.3
4080.8
4610.8
6253.7
1.091
8824.1
0.824
0.466
0.318
12984
19344
22206
29122
1..606
0.015
35274
27668
33794
29914
28689
27464
24809
21338
16999
11843
6942.5
3726.5
692.2
425.87
635.32
497.82
457.88
419.6
342.41
253.3
160.75
78.028
26.813
7.7254
.0182
.0102
.0176
.0131
.0123
.0113
.0095
.0071
.0044
.0020
.0005
.0001
1.287
1.269
1.316
1.303
1.316
1.327
1.359
1.396
1.425
1.416
1.315
1.263
8.469
6.642
8.114
7.182
6.888
6.594
5.957
5.123
4.082
2.844
1.667
0.895
1.512
1.280
1.480
1.360
1.327
1.293
1.218
1.111
0.962
0.754
0.515
0.336
139
Table A.29 Calculated Flooding Parameters Using the Experimental Values
of NUREG/CR-0526 for Plenum Filling Case in 1/15 Scale Model
Gas Den.(Kg/m^3) Liq.Den.(Kg/m^3) Liq.Vis.(Kg/m/s) Surf.Tens(N/m)
1.2
998
0.001
0.073
Gap(m)
0.015
Set:49
AA1/3
0.789
0.682
0.600
0.503
0.418
0.358
B ^1/3p
Reid
Reg
We
E
rohgc
3398.4
3854 17.227
2734.4 5301.4 32.595
2254
7018 57.121
1729.8 8381.2 81.468
1310.4 10620 130.79
1039.6 12942 194.26
.00019
1.211
.00041
.00078
6150.3 3803.5 16.778
3826.5 6378.5 47.185
3625.5
6227 44.971
2044.3 9222.7 98.648
1.371 1362.9 11680 158.21
1.456 978.46 13329 206.05
.00022
0.525
6989 3584.7 14.903
0.856 4446.8 6546.8 49.708
1.022 3328.5
8701 87.803
1.124 2699.5 10300 123.04
1.241 2026.8 12370 177.46
1.328 1572.5 13380 207.62
1.011
1.118
1.200
1.297
1.382
1.442
Jgc
BA1/3e E
1.214
1.217
1.216
1.216
1.216
1.929
2.653
3.512
4.195
5.315
6.477
0.556
0.688
0.830
0.934
1.094
1.248
1.904
3.192
3.116
4.616
5.845
6.671
0.553
.00247
.00317
1.223
1.228
1.225
1.222
1.219
1.215
.00019
.00078
.00148
.00214
.00315
.00360
1.225
1.235
1.238
1.237
1.234
1.228
1.794
3.277
4.355
5.155
6.191
6.696
0.532
0.796
0.963
1.078
1.217
1.280
.00114
.00193
.00299
Set:50
1.171
0.854
0.823
0.562
0.429
0.344
Set:51
1.275
0.944
0.778
0.676
0.559
0.472
0.629
0.946
0.977
1.238
.00071
.00066
.00152
0.781
0.768
0.997
1.166
1.272
140
APPENDIX (B)
CATASTROPHE THEORY
Catastrophe Theory is a special way of looking at nature, and in the language of
causality or cause-effect relationship it provides an explanation as to how continuous
causes can lead to discontinuous effects. To a catastrophe theorist, it means a
powerful mathematical tool for a new way of thinking. According to the French
mathematician Rend Thom (the founder of catastrophe theory), some classes of
discontinuities called elementary catastrophes can occur for a gradient dynamical
system.
For non-gradient dynamical systems, other types of catastrophes which are
called non-elementary catastrophes can occur (see reference 35).
The difference between the classical mathematics and catastrophe theory
is
that the classical mathematics deals with the smooth and continuous processes, while
catastrophe theory deals with jump transition, discontinuities, and sudden qualitative
changes (51).
Mathematical Background;
Definition: Catastrophe Theory (see reference 47) is a mathematical program to
study how the qualitative nature of the solutions of equations depends on the
parameters involved in the equations.
Define a space RN ( N-dimensional space) with general coordinates
x1, x2, x3,
form:
, XN . Consider a general set of integro-differential equations in the
141
t
n(ai; ri 't,
da;
dt
d2 s2;
;
xi
.
,
d t2
a aI
a2u.
a)q
a)y axm
1
1 dxi
,
=0 (B1)
1 5 y5 x
where
1 <i <n
1 51, m 5N
Now we seek solutions that describe the state of some system in the form of:
(t ,x
;
ry )
t ,X
where t and x are defined as state parameters and ry are defined as control
parameters. The major simplifications of the set above can be made by:
1.
Eliminating the integral parts to get a set of nonlinear differential equations.
2. Eliminating the space and spatial gradients (of any order) dependency. Using the
simplifications of 1 and 2 leads one to write equation (B1) as:
n
3.
(ai ; ry ;
t
-arai
d
1 =0
(B2)
Eliminating the time derivatives higher than the first order and assuming that the
time derivatives occur in a canonical way that leads to a dynamical system defined
142
by:
=0=
4.
dt
(
'
B3)
;t)
;
leads to:
Eliminating the time dependency of f in the dynamical system
dQi
°
dt
-
B4)
)
(11i ;
This system is called an autonomous dynamical system.
dynamic system, if f is assumed to be derived as the negative
5. In an autonomous
gradient with respect to Q of some potential V( Q ; I-1) then
- a V (0
;
)
fi
a
Ci
and
av
d Qj
;
)
=0
an;
dt
dQ
For the equilibrium condition:
a0
dt
get:
which means that the state does not change with time, we
(B.5)
143
a V (0
;
=o
(B6)
an;
Catastrophe Theory concentrates on studying the behavior of this function as the
control parameters change.
Applications of Catastrophe Theory;
Catastrophe Theory was applied in a variety of fields such as:
Economics, Linguistics, Biomechanical studies, Experimental psychology,
Heart beat and nerve impulse studies, Embryology, Optics (geometrical and physical),
Thermodynamics, Hydrodynamics, Aerodynamics, Geology, Elementary particles, and
Quantum mechanics, etc .
Elementary Catasjrophes;
A system can be described by two types of parameters:
1.
State parameters.
2.
Control parameters.
The control parameters influence the state parameters and both govern the standard
function of the system. From the standard function, the standard model or the
canonical form of the catastrophe can be derived. In general, catastrophe theory
characterizes qualitatively how the small changes in the control parameters can lead
to drastic or catastrophic changes in the state parameters or the system behavior.
According to Thom's classification theorem, a system that has no more than
(2) state parameters, and no more than (4) control parameters can experience only
144
seven types of catastrophes. These catastrophes are called elementary catastrophes.
The elementary catastrophes along with their standard functions are listed below.
Catastrophe class
Standard function
a. Cuspoids
1. Fold
b.
x3/3 + a x
2.
Cusp
x4/4 + a x2 /2 + b x
3.
Swallowtail
x5/5 + a x3 /3 + b x2 /2 + c x
4.
Butterfly
x6 /6 + a x4 /4 + b x3 /3 +c x2 /2 + d x
Umbilics:
1.
Hyperbolic
x3 + y3+ a x+ b y+ c x y
2.
Elliptic
x3 -xy2+ax+by+c(x2+y2)
3.
Parabolic
x2y + y4 +ax+by+cx2+
d y2
where x and y are state parameters and a, b, c, d are control parameters.
As it is noted, the standard function of a system is governed by both the control and
state parameters.
These catastrophes are arranged according to the number of the state and
control parameters involved. The cuspoids include 1 state parameter and different
control parameters ranging from 1 to 4. The fold catastrophe which is the simplest
one includes 1 control parameter while the rest of the cuspoids ( cusp, swallowtail,
and butterfly) includes 2, 3, and 4 control parameters respectively.
The umbilics
include 2 state parameters and 3 control parameters for hyperbolic and elliptic
catastrophes, and 4 control parameters for the parabolic case.
145
The standard models or the canonical forms of the elementary catastrophes
which represent the equilibrium response surface of the system can be obtained by
differentiating the standard functions with respect to the state parameters and setting
the results equal to zero. Therefore, it is apparent that the choice of the fractions
that appear in the standard functions is just for convenience. For example, the
canonical form of the cuspoids are as follows:
Fold
x2 + a = 0
Cusp
x3 +ax+b=0
Swallowtail
x4 + a x2 + b x + c = 0
Butterfly
x5 +ax3+bx2+cx+d=0