AN INVESTIGATION OF THE PHYSICAL AND
NUMERICAL FOUNDATIONS OF TWO-FLUID
REPRESENTATION OF SODIUM BOILING
WITH APPLICATIONS TO LMFBR EXPERIMENTS
by
Hee Cheon No and Mujid S. Kazimi
Energy Laboratory Report No. MIT-EL 83-003
March 1983
i1
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i
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In
Will
Energy Laboratory
and
Department of Nuclear Engineering
Massachusetts Institute of Technology
Cambridge, Mass. 02139
AN INVESTIGATION OF THE PHYSICAL AND
NUMERICAL FOUNDATIONS OF TWO-FLUID
REPRESENTATION OF SODIUM BOILING
WITH APPLICATIONS TO LMFBR EXPERIMENTS
by
Hee Cheon No and Mujid S. Kazimi
March 1983
Topical Report of the
MIT Sodium Boiling Project
sponsored by
U.S. Department of Energy
Energy Laboratory Report No. MIT-EL 83-003
AN INVESTIGATION OF THE PHYSICAL AND
NUMERICAL FOUNDATIONS OF TWO-FLUID
REPRESENTATION OF SODIUM BOILING
WITH APPLICATIONS TO LMFBR EXPERIMENTS
by
Hee Cheon No and Mujid S. Kazimi
ABSTRACT
This work involves the development of physical models
for the constitutive relations of a two-fuid, three-dimensional
sodium boiling code, THERMIT-6S.
The code is equipped with a
fluid conduction model, a fuel pin model, and a subassembly
wall model suitable for simulating LMFBR transient events.
Mathematically rigorous derivations of time-volume
averaged conservation equations are used to establish the
differential equations of THERMIT-6S.
These equations are
then discretized in a manner identical to the original
THERMIT code.
A virtual mass term is incorporated in
THERMIT-6S to solve the ill-posed problem.
Based on a simplified flow regime, namely cocurrent
annular flow, constitutive relations for two-phase flow of
sodium are derived.
The wall heat transfer coefficient is
based on momentum-heat transfer analogy and a logarithmic
law for liquid film velocity distribution.
A broad literature
review is given for two-phase friction factors.
It is
concluded that entrainment can account for some of the
discrepancies in the literature.
Mass and energy exchanges
are modelled by generalization of the turbulent flux concept.
iii
Interfacial drag coefficients are derived for annular flows
with entrainment.
Code assessment is performed by simulating three
experiments for low flow-high power accidents and one
experiment for low flow/low power accidents in the LMFBR.
While the numerical results for pre-dryout are in good
agreement with the data, those for post-dryout reveal the
need for improvement of the physical models.
The benefits
of two-dimensional non-equilibrium representation of sodium
boiling are studied.
iv
ACKNOWLEDGEMENTS
We are greatly indepted to our co-workers on the Sodium
Boiling Project.
Professor Neil E. Todreas, Robert Zielinski
and Kang Y. Huh, for useful discussions on several aspects of
this work.
Very special thanks are given to Professor Andrei
Schor, who shared tough times and provided invaluable
information on the original THERMIT.
Recognition is extended
to Gail Jacobson for her skillful typing of this manuscript.
Funding for this project was partially provided by
the United States Department of Energy.
Specially acknowledged
is the financial support of the Korean Government to the first
author throughout the period he stayed at M.I.T.
This report is based on the thesis submitted by the first
author in partial fulfillment of the requirements of the
Ph.D. degree in Nuclear Engineering at M.I.T.
TABLE OF CONTENTS
page
ABSTRACT
ACKNOWLEDGEMENTS
iv
TABLE OF CONTENTS
v
LIST OF TABLES
x
LIST OF FIGURES
xi
NOMENCLATURE
Chapter 1
INTRODUCTION
xvii
1
1.1 Background
1
1.2 Previous Work
3
1.3 Reseach Objective and Summary of Study
9
Chapter 2
TWO-PHASE FLOW MODEL
18
2.1 Introduction
19
2.2 Local Instant Balance Equation
20
2.3 General Time Averaged Balance Equation
24
2.4 General Time-Volume Averaged Balance Equation
27
2.5 Time-Volume Averaged Conservation Equations
31
2.6 Interfacial Jump Conditions
40
2.7 Comparison
41
Chapter 3
DESCRIPTION OF THERMIT-6S EQUATIONS
49
3.1 Differential Equations of THERMIT-6S
49
3.2 Difference Equations of THERMIT-6S
54
3.3 Solution Procedures of THERMIT-6S
61
Chapter 4
STUDIES OF VIRTUAL MASS
66
4.1 Introduction
67
4.2 Physical Significance of Virtual Mass
69
vi
4.3 Ill-posed Problems and Mathematical Stability
72
4.4 Characteristics and Stability Analysis
76
of the Two-Fluid Model
4.5 Numerical Stability
86
4.6 Application of Thermodynamics Second Law
93
to Drew's Virtual Mass
4.7 Numerical Results and Discussion
97
4.8 Conclusions
103
Chapter 5 CONSTITUTIVE RELATIONS
5.1 Constitutive Relations in Single Phase Region
5.1.1 Wall Friction in Single Phase Region
105
105
105
5.1.1.1 Axial Wall Friction in Liquid Region
105
5.1.1.2 Axial Wall Friction in Vapor Region
108
5.1.1.3 Transverse Wall Friction in Liquid Region
108
5.1.1.4 Transverse Wall Friction in Vapor Region
109
5.1.2 Wall Heat Transfer in Single Phase Region
110
5.1.2.1 Wall Heat Transfer in Liquid Region
110
5.1.2.2 Wall Heat Transfer in Vapor Region
111
5.2 Constitutive Relations in Two-Phase
112
Pre-Dryout Region
5.2.1 Axial Pressure Drop Coefficient in
113
Two-Phase Flow
5.2.1.1 Literature Review
113
5.2.1.2 Effect of Entrainment on Pressure Drop
121
5.2.2 Transverse Pressure Drop Coefficient in
127
Two-Phase Flow
5.2.3 Wall Heat Transfer Coefficient in
129
- IIYIYU,
ld'
__II
I
'll
vii
Two-Phase Flow
129
5.2.3.1 Literature Review
5.2.3.2 Model Development
5.2.3.3 Comparison to Experiments
5.2.4 Interfacial Mass and Momentum Exchange
0
132
143
148
5.2.4.1 Literature Review
149
5.2.4.2 Model Development
150
5.2.4.3 Model Validation
161
5.2.5 interfacial Momentum Exchange
166
5.2.5.1 Analytical Models
166
5.2.5.2 Model Validation
170
5.3 Constitutive Relations in Post-Dryout
Chapter 6
CODE ASSESSMENT
6.1 W-1 Experiments
173
186
187
6.1.1 Test Apparatus and Procedures
189
6.1.2 Numerical Simulation of W-7b'
190
6.1.2.1 2-D Analysis of W-7b'
190
6.1.2.2 Results
196
6.2 P3A Experiment
210
6.2.1 Test Apparatus and Procedures
210
6.2.2 Numerical Simulation of P3A
211
6.2.2.1 2-D Analysis of P3A
211
6.2.2.2 Results
214
6.3 OPERA 15-Pin Experiment
219
6.3.1 Test Apparatus and Procedures
219
6.3.2 Numerical Simulation of OPERA 15-Pin
220
6.3.2.1 2-D Analysis of OPERA 15-Pin
220
hI.ImlWM
viii
6.3.2.2 Results
222
228
6.4 THORS
6.4.1 Test Apparatus and Procedures
228
6.4.2 Numerical Simulation of the test 72B, run 101
230
6.4.2.1 2-D Analysis of the test 72B, run 101
230
6.4.2.2 Results
231
6.5 Sensitivity Analysis
239
6.5.1 Interfacial Mass and Energy Exchange Models
239
6.5.2 Interfacial Momentum Exchange Model
240
6.5.3 Boundary Positions
242
6.5.4 Normalization of Heat Source
245
Chapter 7
SUMMARY AND RECOMMENDATIONS
251
7.1 Summary
251
7.2 Recommendations
260
References
Appendix A
263
CONDUCTION MODELS
278
A.1 Fuel Conduction Model
278
A.2 Structural Conduction Model
281
A.3 Fluid Conduction Model
286
A.3.1 Fully Explicit Formulation
287
A.3.2 Partially Implicit Formulation
291
Appendix B
PROPERTIES
293
B.1 Sodium Properties
293
B.2 Properties of Fuel and Structure
299
Appendix C
DERIVATION OF Eq (5.2.1-13)
307
Appendix D VALVE MODEL
309
Appendix E
311
GENERALIZED NORMALIZATION OF HEAT SOURCE
1111IYI
IY IIY
I
IIIuil1. 11111"'111111110Y
J, IIEE*EIIIIhlIIIkII1, i i1ii wlllil'.
ix
Appendix F
INPUT INFORMATION
312
F.1 Added or Modified Input Description
312
F.2 Input
313
LIST OF TABLES
No.
page
1.1
Features of several sodium boiling codes
16
1.2
Main Drawbacks of Various
17
Modelling Approaches
5.1
Constitutive Relations of THERMIT-6S
106
5.2
Main Parameters of Sodium Boiling Experiments
115
5.3
THERMIT-6S Wall Friction and
181
Heat Transfer Models
5.4
THERMIT-6S Interfacial Exchange Models
184
6.1
Experiments and Codes Selected for THERMIT-6S
188
Code Assessmsnt and Comparison
~"I--~I---------~~
~~
--- h
xi
LIST OF FIGURES
page
No.
1.1 Possible Accident Paths and Lines of Assurance
10
For a Potential CDA
1.2 Key Events and Potential Accident Paths for
11
Unprotected Loss of Flow Accident
1.3 Key Events and Potential Accident Paths for
12
Loss of Flow Accident
1.4 Key Events and Potential Accident Paths for
13
Unprotected Transient Overpower Accident
1.5 Key Events and Potential Accident Paths for
14
Inadequate Natural Circulation Decay Heat
1.6 Key Events and Potential Accident Paths for
15
Local Subassembly Accident
2.1 Configuration of Interface
23
3.1 A Typical Fluid Mesh Cell Showing Location
56
of Variable and Subscripting Conventions
4.1 Configuration of the Solid Sphere
70
Accelerating in Fluid
4.2 Characteristics Map with Real Charateristics
85
for Given Values of
4.3 Stability Diagram of Eq (4.50)
92
4.4 Vapor Velocity Transients with a =0.8
98
4.5 Vapor Velocity Transients with a =0.9
101
4.6 Restriction on X to Insure that AS > 0
102
for Interfacial Momentum Exchange
xii
5.1 Experimental Two-Phase Friction Pressure Drop
116
with Low Mass Flux or Low Quality Flow
5.2 Experimental Two-Phase Friction Pressure Drop
119
under the Diabatic Condition
5.3 Experimental Two-Phase Friction Pressure Drop
120
with High Mass Flux and High Quality Flow
5.4 Experimental Two-Phase Friction Pressure Drop
122
in Seven-Pin Bundle
5.5 Ideal Liquid Film in the Tube
133
5.6 Comparison between Data and Correltions
144
5.7 Comparison of Correlations with GE data
146
5.8 Simple Representation of Mass and Heat Transfer
151
around the Vapor-Liquid Interface
5.9 Movement of an Eddy in a Fluid
154
5.10 Comparison between Analytic Solution and Results
162
Calculated according to the Suggested Model
5.11 Comparison of Void Volume in the Fuel Bundle
164
between Estimated Experimental Results and
Results from the Suggested Model
5.12 Interface Drag Coefficient to
171
Dimensionless Film Thickness
6.1 SLSF Loop Steady-State Hydraulics at 145 MW ETR
Power and 4.29 lb/sec Test Section Sodium Flow
191
6.2 Radial Meshes of the Fuel bundle
193
for the 2-D Numerical Simulation of W-7b'
6.3 Forcing Function of W-7b' Numerical Simulation
6.4 Comparison between Temperature Data, and
195
197
-
---
oil
xiii
I-D and 2-D Predictions by THERMIT-6S
at 33.7
(from the bottom of the fuel) in the
Innermost Channel for the W-7b' Test
6.5 Comparison between Temperature Data of
198
Hex Can-Inner Face, and 1-D and 2-D Predictions
by THERMIT-6S at 33.9 (from the bottom of
the fuel) for the W-7b'. Test
6.6 Comparison between Temperature Data and
199
2-D Predictions by THERMIT-6S at the Midplane
of Heated Section in the Intermediate Channel
6.7 Comparison between Inlet Flow Data and
202
Predictions by THERMIT-6S, SAS-3D, and
NATOF-2D for the W-7b' Test
6.8 Profile of Voided Region in the Innermost
204
Channel for the W-7b' Test (2-D Simulation)
6.9 Profile of Voided Region in the Outermost
205
Channel for the W-7b' Test (2-D Simulation)
6.10 Comparison between Inlet Flow Data and 1-D
Predictions by THERMIT-6S for the W-7b' Test
209
6.11 Radial Temperature Gradients at the Last Cell
of Heated Section Predicted by THERMIT-6S
212
for the W-7b', P3A, and OPERA-15 Pin
6.12 Radial Meshes of the Fuel Bundle for
213
the 2-D Numerical Simulation of P3A
6.13 Comparison among Temperature Data,
COBRA-3M Predictions and 2-D Precdictions
by THERMIT-6S at 35.7 (from the bottom of the
215
xiv
fuel)
in the Innermost Channel for P3A Test
Channel for the P3A test
6.14 Comparison among Temperature Data, COBRA-3M
216
Predictions, and 2-D Predictions by THERMIT-6S
at 21.7 (from the bottom of the fuel) in the
Outermost Channel for the P3A Test
6.15 Comparison between Inlet Flow Data and
218
Predictions by NATOF-2D and THERMIT-6S
for the P3A Test
6.16 Geometry of OPERA-15 Pin Bundle
221
6.17 Comparison of Temperature Transients of
224
OPERA-15 Pin with Pretest Predictions by
THERMIT-6S at 35.5
-(above fuel bottom)
in Innermost and Outermost Channels
6.18 Axial Void Profile in Innermost Channel Predicted
225
by THERMIT-6S for the OPERA-15 Pin Test
6.19 Axial Void Profile in Outermost Channel Predicted
226
by THERMIT-6S for the OPERA-15 Pin Test
6.20 Comparison of Flow Transients of OPERA-15 Pin
227
with Pretest Predictions by THERMIT-6S
6.21 Temperature vs Radius with Time as a Parameter
232
at Z=33 in for the test 72B, run 101 of THORS
6.22 Comparison between Predictions by THERMIT-6S
and Temperature Data at 31
233
(from the bottom of
the fuel) in the Intermediate Channel for the
test 723, run 101 of THORS
6.23 Comparison between Inlet Flow Data and Predictions
235
-~~---
II
----
YMI,
xv
by THERMIT-6S for the test 728, run 101 of THORS
6.24 Comparison between Inlet Flow Data and Predictions
236
by THERMIT-6S for the test 723, run 101 of THORS
6.25 Axial Void Profile in the Innermost Channel
237
Predicted by THERMIT-6S for the test 728,
run 101 of THORS
6.26 Axial Void Profile in the Outermost Channel
238
Predicted by THERMIT-6S for the test 728,
run 101 of THORS
6.27 Comparison of Void Fractin in the Last Cell of
241
Heated Section Predicted with Various
Interfacial Momentum Exchange Models
6.28 Comparison of Vapor and Liquid Velocities
243
at the Last Cell of the Heated Section
Predicted with Various Interfacial
Momentum Exchange Relations
6.29 Comparison of CPU Time for the W-7b' Test
244
Using the Multics Computer
6.30 Comparison between Inlet Flow Data and
Predictions for A Boundary and for B Boundary
246
6.31 Comparison of Temperature and the Rate of Wall
Heat Transfer in the Last Cell of Innermost
248
Channel for Case A and Case B
of Power Normalization
6.32 Comparison between Centerline Fuel Temperature
Data in the Last Cell of Innermost Channel
and Predictions for Case A and Case B
250
xvi
of Power Normalization
A.1
Thermit Model of Hex Can with
282
Associated Structure
A.2
Hex Can with Associated Structure
283
A.3
Top View of Fluid Channels
289
010111111111111111,
1Ijil~ III
Iii
xvii
NOMENCLATURE
A
: Area
C
: Constant
Cd
: Drag Coefficient
D
: Diameter
dp
: Pressure drop
e
: Internal energy
E
: Entrainment fraction, Effectiveness of heat transfer
by eddies or Total energy
f
: Friction factor
F
: Force or Rate of momentum transfer per unit volume
g
: Graviatational acceleration
G
: Mass flux
H
: Heat transfer coefficient
h
: Heat transfer coefficient or Enthalpy
j
: Volumetric flux
k
: Thermal conductivity
L
: Length
P
: Pressure or Perimeter
p
: Pressure
Q
: Rate of heat transfer per unit volume
q"
: Heat flux
Q
: Volumetric heat generation rater
R
: Radius
s
: Slip ratio
i----
--
r
xviii
S
: Entropy
t
: Time
T
: Temperature
At
: Time interval
U
: Velocity
: Velocity vector
V
: Volume
x
: Quality
X
:Lockhort-Martinelli parameter
W
: Mass Flow rate
Az
: Axial mesh size
c
: Void fraction
a
: Surface tension
S
: Two-phase multiplier
r
: Vapor generation rate
p
: Density
: Viscosity
: Shear force
: Thickness of liquid film
Subscripts
b
: Bulk
d
: Droplets
e
: Equivalent
f
: Saturated liquid or Fluid
g
: Saturated vapor
i
: Interfacial
i,j,k : Nodal locations
__~__
__
-
~
xix
1
:Liquid
1d
: Liquid droplet
if
: Liquid film
r
:radial
s
: Saturation
t
: Transverse
tt
: Turbulent-turbulent flow
v
: Vapor
vv
: Viscous-viscous flow
w
: Wall
z
: Axial
14
: Single-phase
24
: Two-phase
Superscripts
n
: Time step
s
: Spatial fluctuation
t
: Turbulent fluctuation
x,y,z : Spatial directions
Dimensionless Group
Nu
: Nusselt number
Pe
: Peclet number
pr
: Prandtl number
Re
: Reynolds number
We
: Weber number
YLI
Chapter 1
INTRODUCTION
1.1 Background
The growing public concern about the
places
industry
nuclear
an increasingly large emphasis on the safety aspects
In order to
of nuclear reactors.
margins
low, the U.S. Fast Breeder Reactor Safety
provide
is to
safety
sufficient
that the public risk will be acceptably
assure
to
offer
Program
approach
four levels of protection, which are mainly
aimed at reducing both the probability and consequences of a
Core Disrutive Accident (CDA).
These levels
of
protection
are referred to as Lines of Assurance (LOA).
Figure 1.1
Lines
illustrates
accident
possible
of Assurance for a potential CDA.
of LOA.1 is to reduce the probability
serious accident.
control,
and
paths
The main objective
of
occurrence
of
a
The special emphasis is placed on quality
inspection,
and
monitoring
at
the
of
construction and operation, and on
providing
redundant plant protection system.
The LOA.2 is based on an
assumption
that
low
on
an
assurance
through
design,
LOA.2
places
that accidents can be contained
within a limited number of fuel assemblies.
providing,
multilevel
probable but mechanistically possible
events involving failure of LOA.1 will occur.
emphasis
a
level
ways
to
This is done by
control
neutronic
reactivity and reactor coolability backed up by experimental
and analytical works.
The failure of LOA's 1 and 2 leads to
core disruption and potential release of significant amounts
N__--I
of
radioactivity
environment,
without
from
which
the
containment
should
be
harm to the public.
building
mitigated
and
to
the
terminated
The objective of LOA'-s 3 and 4
is to ensure that the consequences of a postulated
CDA
can
be sufficiently limited by the plant containment capability.
The present work is related to LOA 2, which in
requires
an
understanding
of
the
mechanisms
general
and design
conditions which enable potential accidents to be terminated
with limited core damage.
this
work
is to
multidimensional
Specifically
develop
and
an
the
objective
understanding
non-equilibrium
of
of the role
effects
of
sodium
boiling in determining the sequence of events during various
accidents.
These accidents include
1) Unprotected loss of flow (LOF):
power
to
the
primary
rundown and loss
operating
of
at power;
accident
of
electrical
coolant pumps, resulting in pump
core
flow
while
the
reactor
is
coupled with a simultaneous failure
of the plant protection
This
loss
system
corresponds
to
to
scram
the
the
reactor.
low flow/high power
condition.
2) Loss of pipe integrity (LOPI):
leak
in
a
reactor
coolant
pipe
undetected defect
resulting
decrease in core flow and partial loss
scram.
This
accident
power condition.
corresponds
of
in
coolant
or
rapid
with
to the low flow/low
3
3)
Unprotected transient overpower
of
plant
or operator error
system
control
reactivity
in
resulting in a sudden increase
malfunction
(TOP):
and
reactivity
core
power--coupled with a failure to scram.
Inadequate natural circulation
4)
removal:
heat
decay
postulated inadequate heat removal by natural circulation
of
loss
and
shutdown
following
forced cooling in the
primary loops.
5)
Local subassembly
internal
or
flow
inlet
fault:
subassembly blockage resulting in cooling disturbance and
potential for fuel failure propagation with scram.
can
accidents
The key events relating to each of these
be
shown in Figs 1.2-1.6.
1.2 Previous Work
the
The first code to analyze voiding in the LMFBR was
(2) which used a homogeneous equilibrium
code
Transfugue-1
model and a momentum integral method (3).
The Transfugue-II
forerunner of SAS 1A (5).
of
improvement
Transfugue-I,
compressibility, variable vapor
determined
the
by
This code was the
sonic
included
densities
velocity
code
of
(4),
an
sectional
but
time
the
homogeneous
steps
equilibrium two-phase flow.
SAS codes of ANL represent a long development from
single-channel
(7), SAS
3D
SAS
(8),
1A
and
version
through SAS 2A (6),
SAS
(9) which
4A
is
now
the
SAS 3A
under
_
YIIP-P--- l
aP--sY__Yi_~==~_I;_~_LI;~
_
_
___i I__il_ ~~_
Accident Progression
not Controlled
Containment
Limited Release
Limits Release
to Environment
I
LOA4
-
-i--
"to Environment
li
i
c -
Containment Fails
Uncontrolled Release to Environmen
Figure 1.1
Possible Accident Paths and Lines of Assurance
For a Potential CDA
(From Reference 1)
~-~
...
i.-.O.i-r^l--~
--
I
-- -D~-- --I-- -- ----
ii
---------- ; I.
Fault Occurs
Leading to Pump/
Flow Coastdown &
Core Heatup
Reactor Scram;
Failure to Scram
Boiling
Flow
Transfer;
& Power
Reactivity &
Power Decrease
Two-phase
& Beat
Reactivity
Increase
C
No Damage
D
Flow/Heat Transfer
Instability
Fuel Pin Failure &
Fission Gas/Molten
Fuel Release
Reactivity/Power
Reactivity/Power
Decrease Due to
Fuel Motion
Increase Due to Fuel
Motion
mm~m
immmmmumem~t
MFCI & Subassembly
Voiding
Mechanical
Disassembly
Some Clad/Fuel
Melting but Adequate
Cooling Restored
J
C
m1
Bulk Subassembly
Voiding; Clad
Melting & Relocation
Limited
Core Damage
i
~
•
Core Disruption
Accident
Core Geometry
Not Coolable;
Gradual Meltdown
Core Disruption
Accident
Figure 1.2
Key Events & Potential Accident Paths
For Unprotected Loss of Flow Accident
----
;
--
-~L --
I
Fault Develops
with Potential for
Loss of Pipe
Integrity
1
Fault Undetected
Loss of Pipe Integrity
Fault Detected
Operator Action
No Damage
I
Flow Decay
and Beatup
Reactor Scram
Reactivity and
Power Decrease
Boiling, Two Phase
Flow and Heat
Transfer
uel Pin Failure
and Fission Gas
Release
W
Flow/Boat Transfer
Adequate Cooling
Instability and CHF
No CHF
Bulk Subassembly
Voiding; Clad
Melting and
Relocation
Some Clad/Fuel Melting
and Relocation but
Adequate Cooling
Restored
-
Limited
Core Damage
Core Geometry not
Coolable; Gradual
Meltdown
CDA 3
Figure 1.3
Key Events and Potential Accident Path
For Loss of Pipe Integrity Accident
(From Reference 1)
--
~-"~~-----~ II
.-
--
--l---L-~i~-
--... --
Figure 1.4
Key Events and Potential Accident Paths
For Unprotected Transient Overpower Accident
I~ --
-~-
--
~~~
--
I 1
L-
0111
IN
P1111-
Reactor Shutdown;
Power & Flow Decrease;
Cooldown
Loss of Forced
Cooling in Primary
Loop
Single-Phase Natural
Circulation Flow &
Reat Transfer
F
I
Adequate Heat
Removal with no
Coolant Boiling
Boiling Natural
Circulation Flow &
Heat Transfer
No Damage
Fuel Pin Failure
& Fission Gas
Release
I I
i
II
!
I
I
,
|
Inadequate Heat Removal;
Flow/Heat Transfer
Instability & CUF
dequate Heat Removal
Until Forced Cooling
Restored
Bulk Subassembly
Voiding; Clad/Fuel
Melting & Relocation
No or Minor
Core Damage
Core Geometry
Not Coolable;
Gradual Meltdown
Core Disruption
Accident
Figure 1.5
Key Events and Potential Accident Paths for
Inadequate Natural Circulation Decay Heat Removal Accident
I
__
__
:_~ _I__ _
~I
__
_
~1_; __I
Local Fault Due to Clad
Defect, Fission Gas/
Molten Fuel Release &/or
SBlocka e Formation
Local Flow Restriction
& Increased Coolant
Temperatures
in
WakeI
Fuel Pin Failure & FTssion
Gas/Molten Fuel Release;
Gradual Blockage Propagation
A
Local Boiling
in Wake
nwMMM=f I
ault Tolerated or
Detected; Operator
or Control Action
Flow/Heat Transfer
Instability & CHF
E
Pin-to-Pin
Failure Propagation
i
S
No or Minor
Core Damage
I
Bulk Subassembly
Voiding; Clad/Fuel
Melting & Relocation
Fault Tolerated or
Detected; Belated
perator or Control Action
Core Geometry
Coolable; Adequate
Cooling Restored
Molten Steel-Sodium
Interaction; Subassembly
Wrapper Failure
Subassembly to
Subassembly Propagation
FCCi
Limited
DamageCore
Core Disrup t ion
Accident
Figure 1.6
Key Events and Potential Accident Paths
For Local Subassembly Accident
•11
development.
_Inn
__
miil
I
SAS 1A was a single-channel code that included
kinetics and a combination of slip two-phase flow and
point
single-bubble slug-boiling models.
The sodium boiling model
was later completely rewritten as a multibubble slug-boiling
model for the multichannel
expanded
version
of
SAS
SAS
2A
2A,
contained
3A,
SAS
version.
an
to allow
models
SAS
complete calculations of cladding and fuel motions.
3D
used the same models as SAS 3A but restructured the code for
better numerical efficiency.
and
The SAS coolant dynamics model calculates temporal
spatial
different
void distribution and uses two different models for
times
boiling
of
A
expulsion.
multibubble
slug-boiling model is used for the initial portion of bubble
formulation.
Voiding
is assumed
filling the cross section of
the
to
result from bubbles
coolant
except
channel,
that a liquid film is left on the clad and structure.
Vapor
pressure and temperature are assumed to be spatially uniform
during initial bubble growth. When the bubble length exceeds
a user-specified maximum value, the vapor bubble calculation
is switched
to
a different model where axial variation in
vapor pressure within the bubble is accounted for.
In this
model, the vapor pressure, temperatures, and mass-flow rates
are
calculated at fixed nodes within the bubble and also at
the lower and upper interfaces.
solution
of
the
vapor
This
cells
uses
momentum
equations with the momentum equations for the
setting
the
boundary
conditions
at
the
a
simultaneous
and
continuity
liquid
slugs
interfaces.
To
11
obtain numerical stability for reasonably large time
steps,
SAS's
main
limitations are the restriction from a slug model, its
lack
differencing
implicit
an
model,
mixing
a
of
its
and
used.
is
scheme
incorporate the radial heat losses and the
adequately
to
inability
multidimensional
effect of voiding.
The
(10)
code
BLOW
in Germanny
developed
the SAS 3A, in order to describe sodium
between
difference
main
BLOW
momentum
vapor
the
and SAS is as follows:
assumption
obtained
a
by
film
liquid
the
equation,
velocity in the BLOW is directly calculated on
the
phenomena.
voiding
while the vapor velocity along a bubble can be
solving
a
slug-boiling model, which is similar to that in
multibubble
The
uses
basis
of
of an universal velocity profile across the
liquid film thickness.
The MOC code (11) was developed by Siegmann in 1971
to
predict the behavior of sodium during boiling and expulsion.
In
the
two-phase
the liquid
and
vapor
temperature.
The
method
of
the
characteristic
slip
flow,
additional
region a slip flow model is assumed with
in
equilibrium
conservation
slopes
numerical
at
saturation
equations are solved using
When
characteristics.
appeared
the
imaginery
in the two-phase flow with
difficulties
were
encounted.
The
limitations of this code are its incapability to
consider flow reversal and the use of small time step
sizes
limited by the sonic velocity.
The COBRA-3M code (12) is a homogenous equilibrium code
111
111111111111111
,111
01EIMI'1
12
with models such as turbulent mixing and wire-wrap mixing on
as well as axial sodium voiding and temperature
radial
for
allowed
network
model
The subchannel
a subchannel basis.
distribution.
A major limitation of COBRA-3M is its lack of
capability to
consider flow reversal, recirculation, and
To
condition.
capabilities, an ACE method
scheme was
developed
a
temporally
explicit
code
COBRA-IV
(13);
pressure-velocity was selected which imposed no
on
flow
boundary
these
provide
boundary
pressure-pressure
a
for
the
transient
restriction
and could accept either flow or pressure
velocity
However,
conditions.
this
code
retains
a
limitation of a homogeneous equilibrium model.
The main features of the BACCHUS code (14) developed at
model,
equilibrium
marching
homogeneous
the
Grenoble are similar to those of COBRA-3M;
and mixing model.
technique,
Their main differences are that the BACCHUS uses the
porous
body analysis and the assumption that the flow is steady.
A
transient version is now under development.
developed
(15)
The THERMIT-4S code
nonhomogeneous-equilibrium
at
is a
M.I.T.
3-D code for which the numerical
scheme and solution procedures are based on the THERMIT code
(16).
With the assumption of
line
saturation
code
this
equations.
constitutive
exchange
are
the
and
vapor
and
liquid
This assumption eliminates the need of
equations
which
on
solves four equations; mixture
mass and internal energy equations,
momentum
equilibrium
thermal
for
interfacial
required
for
the
and
energy
two-fluid
model.
mass
However, as the assumption of thermal equilibrium requires
complete condensation of vapor in the subcooled region,
numerical difficulties may be encountered due to large
pressure fluctuation caused by complete condensation of
vapor.
The NATOF code
(17) developed at M.I.T. adopts a
two-dimensional two-fluid model for the simulation of sodium
boiling transients.
The numerical scheme and solution
procedure are based on THERMIT (16).
The code has the
structural model of heat sink through the can, the radial
liquid conduction model, and the fuel conduction model.
As
the code solves the conservation equations for both vapor
and liquid, the interfacial mass, momentum, and energy
exchange must be-given throuqh the constitutive equations.
The two-fluid approach has the advantage of accounting for
non-equilibrium phenomena and slip flow, but heavily depends
on the physical models.
Although an advanced two-phase flow
model and numerical scheme are used in this code, the range
of validity of its physical models is limited and the illposedness problem of its equations remains unresolved.
These are the motivation for development of the present code,
THERMIT-6S.
The main features of several codes, that were designed
for analysis of sodium boiling in the LMFBR, are given in
Table 1.1.
Limitations of the codes
(see Table 1.2) are
attributed to the characteristics of sodium and of the LMFBR
such as the high density ratio of liquid to vapor, the
--
""-"' " ""' 11111
11111111
111111111
111
1111
14
Table 1.1 : Feature of Several Sodium Boiling Codes
1-D
H-EM
Transfugue-I,II
2-D
BACCHUS
3-D
COBRA-3M
COBRA-IV
Multi-bubble
slug model
Slip-EM
SAS-3A,D
BLOW
MOC
(3 Eq)
Non H-EM
THERMIT-4S
(4 Eq)
Two-Fluid
Model
NATOF-2D
THERMIT-6S
Table 1.2 Main Drawbacks of Various Modelling Approaches
I. Two-Phase Model
I.1 HOMOGENEOUS MODEL : No slip High void fraction -
High pressure build-up Early flow reversal
1.2 EQUILIBRIUM MODEL : Temperature equilibrium Large mass exchange -
Rapid pressure transients Numerical difficulties
1.3 SLIP MODEL : Need correlation for slip in annular
or reversed flow
1.4 TWO-FLUID MODEL : Nonhomogeneous-Nonequilibrium Need more correlations
II. Geometrical Representation
II.1 1-D REPRESENTATION : No radial temperature gradient
and vapor propagation
II.2 2-D REPRESENTATION : No mixing in thee direction
11.3 3-D REPRESENTATION : Large CPU time
'11
-
MI III
existence of the highly subc6oled region, reversed flow, and
sharp radial temperature gradients.
The high density ratio induces high slip between liquid
possibility
the
and vapor and, as a result, eliminates
of
model which assumes no slip.
The
existence of a highly subcooled region in the bundle of
the
use
the
of
homogeneous
The
difficulties and large pressure fluctuations.
the
numerical
causes
condensed,
is
vapor
where
LMFBR,
use
of
equilibrium approach, in which vapor must be completely
in the
condensed
subcooled
turn
in
and
fluctuations
pressure
enlarges
region,
numerical
difficulties.
The
non-equilibrium approach instead of the equilibrium approach
Correlations
is better suited to represent real situations.
in annular
for slip
Hence,
available.
Sharp
work.
multi-dimensional
or
reversed
flow
readily
not
are
the slip model is avoided in the present
temperature
radial
behavior
voiding
multi-dimensional model.
lead
gradients
to
us
A 3-D model is chosen
favor
because
and
a
it
can be transformed into 1-D or 2-D models.
According to the above brief review,
a
two-fluid
3-D
is selected for the analysis of sodium boiling in the
model
LMFBR.
The THERMIT code is taken as the basic tool for
development
of
a two-fluid
the
3-D code due to its following
main features:
1. Two-fluid 3-D code
2. Advanced numerical scheme
3. General Boundary conditions
Although (r, 8, z) coordinates are fitted in the geometry of
the hexagonal bundle of the LMFBR, (x, y, z) coordinates are
used in
THERMIT-6S
to
due
adoptation
easy
the
the
of
original THERMIT code which uses (x, y, z) coordinates.
1.3 Research Objective and Summary of Study
multidimensional
work
present
The objective of the
provide
to
is
a
for understanding and prediction of
model
the
in
melting
the behavior of two-phase flow before fuel
LMFBR subassembly under unprotected loss of flow and loss of
is, under
that
intergrity,
pipe
information
provide
non-equilibrium behavior
code,
which
In
accidents.
accidents and low flow/low power
power
flow/high
low
to
order
and
multidimensional
on
the
of
sodium
boiling,
the
THERMIT
is a two-fluid and three dimensional code, is
taken as a basic tool for the development of the code.
The present work
mathematical,
In Chapter
The
is developed.
The
the
of
of
mathematical
formulation is
the
virtual
mass
on the numerical stability are discussed in Chapter 4.
physical
developed
are
2,
the
of
foundations
numerical
discussed in Chapter 3. The effects
term
investigation
physical
and
numerical,
two-phase flow of sodium.
formulation
the
includes
models
of
in Chapter 5.
compared
to
the
relations
constitutive
are
In Chapter 6, the code predictions
experimental
results.
recommendations are presented in Chapter 7.
A
summary
and
18
Chapter 2
TWO-PHASE FLOW MODEL
General time-volume averaged conservation equations and
here.
jump conditions for two-phase flows are derived
time-averaged
equations
two-phase flow
are
equations
by
turbulent
flow
integrating
a
the
for
obtained
a
from
technique
often
equations.
The
phase
single
local
region
instant
used
for
results
The
balance
phase
single
by
obtained
time-averaged equations over a flow volume
over
are spatially averaged twice; first, they are averaged
a
single
phase
region of the k-th phase and then averaged
over the total volume of the k-th phase, in a
The
mass,
momentum,
advantages
of
the
averaged
time-volume
present
model
comparing it with Ishii's model (18)
(19).
flow
volume.
and energy conservation equations are
obtained from the general
The
in
are
and
equations.
explained
Banerjee's
by
model
Finally, the assumptions and approximate terms of the
equations of the THERMIT-6S are clarified.
2.1
Introduction
The most important characteristic of two phase flows is
the presence of moving internal interfaces separating phases;
a two-phase system consists of a number of single phase
regions bounded by moving interfaces.
Therefore, a formula-
tion based on these local instant variables and moving
interfaces results in a multi-boundary problem which are not
known a priori.
In order to overcome these difficulties, in
the formulation of two-phase flow equations,
we do not
explicitly integrate the microscopic phenomena but preserve
their effect on macroscopic phenomena.
Three approaches have been used widely to develop a
two-phase flow model:
1) Mixture Model Approach
2) Control Volume Approach
3) Averaging Approach
Ishii (18,20) provides an excellent description of these
three approaches.
The first and second approaches are mainly
based on hypothesis, physical intuition, and assumed
similarity with a single phase flow system.
In these two
approaches, even though easy formulation and practicality in
simple cases are main advantages, we encounter the difficulty
of identifying the effect of the microscopic scales of
phenomena on the macroscopic behavior.
On the other hand,
the averaging approach enables us to set up mathematically
rigorous equations.
The averaging approach can be classified
into three main groups:
Eulerian,
-
I
~I=T~-L --; r----~-~3~i~ii~_
20
Lagrangian, and Boltzmann statistical averages.
In an Bulerian description, time and space
are
taken
variables
as
coordinates.
variables and various dependent
independent
express
As
their
the
changes
particle
with
respect
the
these
Eulerian
the Lagrangian average is taken by following a
certain particle and observing it in a time
Boltzmann
to
coordinate in a Lagrangian
description displaces the spatial variable of
description,
coordinates
statistical
averaging
with
a
interval.
concept
particle number density is widely used when
the
of
The
the
collective
mechanics of large number of particles are in question.
The
Eulerian average has been considered as the most widely used
method,
due
observations
time-volume
to
and
its
close
relation
instrumentations,
averaged
flows are derived on
conservation
a
basis
of
to
experimental
Therefore,
the
of two phase
equations
the
here
Eulerian
averaging
approach.
2.2 Local Instant Balance Equation
A two-phase flow is considered to
be
subdivided
into
several single phase regions separated by moving interfaces.
Each
separate
conditions.
feature
of
Jump
phase
can
conditions
be
connected
constitute
a
through
jump
characteristic
moving boundaries and provide relations between
the phase interaction terms.
of continuum mechanics will be
conditions will be derived.
Therefore, the standard method
first
used
and
then
jump
material
of
the
a
k-th phase with the volume and surface, Vmk
we can set up the general balance equation.
and Amk
i
dt
Pyk dV
-
+Ik
dA +
k
(2.1)
kkdv
Vmk
Amk
Vmk
where ok is the density of k-th phase,
Wk
is
quantity
any
conserved in the k-th phase, Jk and Sk are the efflux
being
and source of Tk,
for
For
equation .
Let us start from a general balance
the
k-th
and
phase.
nk is an outward unit
normal
vector
Using the Reynolds transport theorem
(20), we have
d
kJ
dt
kk dV =
--Vmk
Vmk
k kOk+
dV +
at
Note
(2.2)
Amk
where vk denotes the velocity of
phase.
v k n+ dA
0k kVk
kk
a
material
of
the
k-th
that, while the Reynolds transport theorem is
derived for a material volume, the Leibnitz rule (21) (given
later by Eq (2.24)) is a purely geometric theorem.
The Gauss's theorem (22) gives a transformation between
a volume and surface integral.
(PkTkVk
Amk
dV
+Jk )k
+ Jk) n(k
Vmk
Substituting Eqs (2.2) and (2.3) into Eq (2.1), we have
(2.3)
II
__
--
lllii
22
T
By
the
+ V-(Pk'kk+
of
axiom
3
k)
continuum,
PkSk
we
get
dV=0
the
(2.4)
local
instant
differential balance equation
+t
k
The local instant
(2.5)
0
(2.5)
kkVk
differential
balance
equation,
interfacial
volume
balance
analysis,
equation
neglecting
based
by
on
the
surface tension.
shown in Fig 2.1, control volumes, V 1 and V 2 are
Eq
Let
can be applied in each phase up to an interface.
us derive an
control
PkSk
"J
Pk
at
As
surrounded
the areas of each phase, Al and A2 , with an interfacial
area Ai. As the control volumes, V 1 and V 2, are very
thin,
-
we put nl
1 -n2 '
where n11 and n 2 are outward unit normal vectors from
surfaces, Al and A 2, respectively. Now we have the general
balance equation for the control volumes, V and V .
1
2
2
k=l
d
SpkVk
2
dtdV
T
dv
-
k=11
Vk
Vk
IAk
f nk
J
Pk(V
k k kk
)+k]
k]
i+
A
23
n22
v1
Al
1-phase
r .
2-phase
A
V2
1
Fig. 2.1:
A2
Configuration of Interface
-1
.1 W
6i,
_
0li.
24
We let the volumes, V
and V2 shrink down to
the
interface
so that the volumes, V1 and V2 vanish, while the interfacial
area
remains finite in the limit.
A.
The volume integrals
vanish and Ak is equal to Ai, in the limit.
Then
we
have
the general interfacial balance equation.
2
Z (;k'k + nk*.k
k=1
(2.7)
i) n k
Pk(vk
Vk
where
0
)
(2.8)
mk is the local instant mass flux through the interface.
2.3 General Time Averaged Balance Equation
In a single phase turbulent
averaging
is
essential
and
flow,
the
Eulerian
time
a well-known technique.
As a
two-phase flow consists of several single phase regions, the
technique used in a single phase turbulent flow is taken
to
obtain the general time-averaged balance equation for one of
single phase regions.
Let us integrate the local instant balance equation, Eq
(2.5), over
should
be
a
time
interval
than
smaller
phase to pass through a
macroscopic
Also
fluid.
the
At
which
as
position
At should be large enough to
is
single phase regions
However,
time
interval
\t
and
than
the
of the unsteadiness of the bulk
local variations of properties.
used
The
transport time of one single
reference
constant
time
Lt.
smooth
out
the
Ishii (18) and Delhaye (23)
time interval sufficient for several
a
to
pass
through
a
local
position.
turbulence originated in a single phase region
_ _
-
------i--iT-i--
25
will be confined in that phase
effect
by
the
interface
and
the
turbulence in a phase can be transmitted through
of
the other phase,
At should be smaller than a time
interval
for a single phase region to pass through a reference point.
As a result, we get a time-averaged equation
which
is
identical to that made in analyzing a single phase turbulent
flow.
aPkk
where
F
In order
(PkVk)+ VJk-pkSk m 0,
(2.9)
F dt
(2.10)
+
at
to
obtain
Eq
(2.9),
we
used
the
following
relationships which come from the Leibnitz rule.
aF
-~
at
aF and
-at
VF = V.*F
(2.11)
By including a non-zero scalar weighting function w, we
define the general weighted mean value of a function F as:
F
(2.12)
and
In general, the volume, momentum, energy,
considered
to
be
extensive variables.
quantity per unit volume
of the
entropy are
If F is taken as a
extensive
characteristic,
IIIIIIIIII
I
10
IIIIllIllllIll
l
--
I I
Ill
26
it
be expressed in terms of the variable per unit mass T
can
as:
F - PY
(2.13)
The appropriate mean value for T should be weighted
by
the
density as
-.
(2.14)
=
P
let
Now
us
caused
variables
fluctuating
introduce
by
components
of
In general, as they are
turbulence.
defined as a difference between a local instant variable and
its weighted mean value, we have
F
't
(2.15)
=F-F
Since the mean values of T and v are weighted by
mass,
the
fluctuating components are given as follows:
k'k1k
Using
Eqs
k
ic
k
(2.14)
and
2.16)
(2.16)
vk
k
Vk = vk +v k
and
and
(2.15),
we
have
the
additional
relationships:
, 'j
Tk
k k 't =
,
= 0,
*'t
PkVk
-0
vk
= 0
= 0,
(2.17)
27
From Eqs (2.16) and (2.17), the convective flux term becomes
*
-1.
't -'t
(2.18)
k
k kk
kk k
Pkk k =
Now we obtain the time-averaged balance equation
-V
at
-t
t
where
+ V.j+k
+t___T
-
-10
k k
k
-
-t
+V(Jk
-0
k
kS=
,
0+Jkkk
,
't 't
k k Vk
Jk
(2.20)
In the same manner, we
the time interval,
l m
2
can
equation
balance
interfacial
(2.19)
obtain
the
time-averaged
by integrating Eq (2.7) over
At.
+ nk
= 0,
(2.21)
k=1
= mk k
where
T
(2.22)
through
is weighted by the time averaged mass flux
the
interface.
2.4 General Time-Volume Averaged Balance Equation
Many practical problems of two-phase
with
using
time-volume
flows
averaged equations.
averaged
dealt
A time-volume
averaged equation can be obtained by integrating
time
are
the
local
equation, Eq (2.19), over a flow volume Vf ,
11111
---
1
1__I~i
28
which consists of several single phase regions.
I
I
-
i1 Vf f
at
t(J
+V
ki
PkS k
]
dV = 0,
(2.23)
where I is the maximum number of the k-th phase regions in the
volume
Vf
Vki
and
is the volume of the i-th single phase
region of the k-th phase.
The
theorems
we
will
use
are
given below:
Leibnitz rule
at
Vki
(2.24)
F vi.nk dA,
-at v+
FdV=
Vki
Aki
Gauss' theorem
SV.FdV =
V
F dV +
inkF
(2.25)
dA,
Aki+AWi
Vki
Vki
where Aki is the area of the i-th single phase region of
ki
the k-th phase in the volume Vf and vi-nk
is the surface
-.
displacement velocity of Aki .
For a solid interface vw =0.
Using these theorems, we obtain the equation
4
t
SV.
kadV + 7P pk~kdV +
k
Vki
k
fa
Vki
k
-
-t
(Jk
kf
Vki
k)dVJ- kk
kSk dV
Vki
-t
+
k k ,(,k
k .e(Vp
k
Aki
k
i ) +Jk]dA
i
nk 3kdA
+
= 0,
(2.26)
Aki+Akwj
where Akwj
denotes the area of the j-th single phase region
of the k-th phase contacting the solid surface in the volume
29
Vf .
The fifth term in Eq (2.26) can be simplified by using
the definition of the turbulent flux, Eq (2.20).
B= [ k
k] *nk
+
i
k
"t
'k
-k
k k
S(PkkVk
0
- 0
i)n k
(2.27)
From Eqs (2.14) and (2.18), Eq (2.27) becomes
B = (kVk
kPkVi).n
-
Then, we have the following
(2.28)
k
equation
which
is
consistent
with the jump condition, Eq (2.7).
B
=
(2.29)
mK k
Let us introduce spatial averaged relationships
for
a
single phase region.
F dV
<F> =
Vki
<F>ai =
dV
(2.30)
dA
(2.31)
Vki
F dA
Aki
<F>w j
f
Aki
F dA
Akwj
/
dA
(2.32)
ikwj
Substituting Eqs (2.30), (2.31) and (2.32) into Eq. (2.26),
WIIN
I
we obtain the equation averaged for a single phase region.
1i [±
> +V'aki
k
ki k
1 <n+ "$'wj
k>]
L
k
akiki
- V-'
f
(2.33)
= 0,
wj
Vfi
where
1
Aki
L i
Vf
Vf
Lw
(2.34)
Vf
wJ
f
Then Eq (2.33) is averaged over the total
the
kJk
k ai
k
k a
Lai
k
+Lw
-t
*-
S1
-
> + Vki
<kk
k
k
volume,
total surface, of k-th phase in the volume Vf
or
over
by using
the following relationships:
<<F>> =
<<F>>
a
i
ki
<F>
-
Z a
i ki
=
1
L ai
ai
i
<F>
a
(2.35)
Lk
1
a
1
<<F>>
=
j
Lak
1
w3
L
ek =i
aki'
<F>
w3
wj
(2.37)
Lwk
Lwj
IAki
where
(2.36)
1
Lai
i L.
1
--- <F>
Lw j
ai
a
wherei
V
Lak
1
Lwk
Y Akj
(2.38)
Vf
Substituting Eqs (2.35), (2.36), and (2.37)
into
Eq(2.33),
31
we have the general time-volume averaged equation.
-t
a
Lak
1
+ 1
+
Jk>>
+--- <<nk
>
<<n k
+L
k
k
Lwk
k Jk a
Lak
(2.39)
= 0
Let us obtain the time-volume averaged interfacial jump
Integrating Eq
condition from Eq (2.21).
(2.21)
over
the
interfacial surface area Aki , we get the jump condition.
SJ
(I2k
k + nk.jk )dA
k=1 i
2
-
1
l L=
k=1
<<mk "'k
+ n k J k > > a=
0
(2.40)
ak
2.5 Time-Volume Averaged Conservation Equations
The forms
equations
for
of
the
time-volume
averaged
conservation
mass, momentum, and energy of the k-th phase
can be written by specifying Eq (2.39) as follows:
Mass
In this case,
Tk = 1, Jk = 0,
Sk = 0
(2.41)
--------
YI
We have
Pk> + V
t 'k
k
k
-@tkVk
where
-t
.k
and
rk
rk
i2.42)
rk
<Pk Vk>>
"
PkVk t - 0
1
->>
(2.43)
a
ak
is the time-volume averaged interfacial mass
flow
rate
per unit volume of k-th phase.
Momentum
In this case,
k.
k
'?k ~VkF
(2.44)
Sk
Sk = Fkk
Tk
k
We have
4#-
at ak <<
>
k
+ V'k <<
>
k
- V.ak <<k
-
k
where
Lwk
-t
Tk
==
*
.'t
PkVk
k
ki
.>>
a
V.k<<-
L ak
<<-
>>
>k
.4
n>>
a
= 0
<Tknk>>
k
kVkVk > > + Vak<<k
ak<<PkFk >>-
7*.
>> +
+ 1 <<mV
a Lak
Lakk
ak
1
0.
4-
(2.45)
w
*'t
Vk
and Pki is the local interfacial pressure.
The time-averaged local interfacial pressure pki can be
(2.46)
33
.divided into three terms as follows:
where
+APki +
<<Pk
Pki
Apki
<
APki
Pki -
<Pki >>a -
<
(2.47)
Pki
(2.48)
<Pk > '
(2.49)
<Pki>a'
and Apki is the difference between the average interfacial
and Apki is the difference
and average phase pressures
between the local and average interfacial pressures.
Using
Eq (2.25) ,
1
-<<
Lak
k
we obtain the following relationship:
>>= -
a
(<<p >>+
k
Pki
)
1 <<'p'.>>(2.50)
ki a
Lak
k-
The second term on the right-hand side in Eq (2.50)
term
which
Substituting
leads
to
Eq
(2.50)
is
the
the virtual mass for inviscid flows.
into
Eq
(2.45),
we
obtain
the
momentum conservation equation.
at
<< kVkVk
k <<PkVk>> + Vk
- a k <<PkFk>>
k <<Tkk>>
>> +
k < <pk >
k <<Tknk>>
- PkiV' k
4-0
<<
=
ak
kn
i
Lwk
k
w(2.51)
11
34
Energy
In this case,
2
Tk
Ek
ek +
(2.52)
- )vk
q
k
'
Sk w Fk'vk +Qk
where ek and Qk are the specific internal energy and
volumetric internal energy generation rate.
In solving two-phase problems, it is often
separate
the
mechanical
energy equation.
equation
derived
and
to
thermal effects in the total
From dotting the
by
useful
local
instant
momentum
combining Eqs (2.5) and (2.44) by the
velocity of k-th phase vk , we have
the
mechanical
energy
balance equation.
2-l +
2
1
a 1
-t
PkVk) +V(i Pkvkvk) +VoPkVk-PkV' k
+ V-(TkOk) - Tk:Vk - PkFk Vk = 0
(2.53)
Using Eqs (2.5) and (2.52), we have the local instant energy
balance equation.
a--
at
V2+0+Vqk
(ek 1v)2 + V-Pkk (ek+ 1 V
kk
k
+ V (PkVk) + 7 ("k- k
k
-k(k k
)
- k
k
k
v + Jk ) = 0
(2.54)
Subtracting Eq (2.53) from Eq
(2.54),
we
can
obtain
the
internal energy equation.
a
e +V4
kek
at Pkek +
k
+ V qk + pkV.Vk + Tk:VV k -PkQk = 0
(2.55)
the
compare
If we
Eqs
energy,
and
that
and
they
(Tk :Vk)
appear
interconversion
we
(2.55),
opposite
these
thermal
out
find
are common to both
with
Therefore,
equations.
and
mechanical
and
(2.53)
that (pkV-vk)
of
equations
signs
equations
in
the
two
the
describe
terms
of mechanical and thermal energy.
The term
can be either positive or negative, depending on whether the
fluid
is expanding
represents
a
or
reversible
contracting.
mode
of
As
a
result,
interchange.
it
Using the
continuity equation, we have
k
PkV-
It
+
k
k
=
-Pk
Pk
Dt
For a fluid of constant density, the term (pkV. k)
zero.
On the othere
positive
and
hand,
therefore
the
term (-Tk:V k)
represents
an
2.56)
becomes
is always
irreversible
degradation of mechanical to thermal energy.
Let us get the time averaged internal
(2.55).
energy
from
Eq
1111111111111111191
WIIIIIM
-
ihlili
36
.e
Pkek
"
V. Pkekvk
.+ ++ V•qk
- PkQk
+
Pkvk + k Vvkk
(2.57)
0
Using Eqs (2.16), (2.17), and (2.18), we can obtain the time
averaged internal energy equation.
-
at Pkek
(Pkekk) +
+ pk V
where
-t0a
k:Vk
(k
= 0,
(2.58)
t
't *I't
qk= Pkek Vk
The time-volume averaged internal
obtained
energy
equation
can
be
by integrating Eq (2.58) over the flow volume Vf .
In order to
a
k
special
perform
that ,
manipulation
the
as
term
follows.
pressure can be divided into two
(pkVk) requires
The
components:
local
instant
time-averaged
local pressure and local time fluctuation of pressure.
't
Pk
Then
(pkV
=
(2.59)
Pk +k
k)
k
becomes
P
pkV v_ = Pk'v
k
k + Pk
t
V Vk
Let us integrate Eq (2.60) over the flow volume Vf .
(2.60)--
(2.60)
Then,
we have
S i
pk
i=1 Skf
Vki
-
-
V=
kdV
Z
f
(PkV
-
k + Pk Vvk)dv (2.61)
Vki
Also the time averaged local pressure can
two
components:
time-space
averaged
be
divided
pressure
into
and
time
averaged space fluctuation of pressure.
Pk= <<Pk
>
's
(2.62)
+ Pk
Substituting Eq (2.62) into Eq (2.61), Eq,(2.61) becomes
S
i
<<pk>>
f
Vk
dV
(
ki
+
(pkSV'Vk +P
k
(2.63)
)dV
Vki
Using Eq (2.25), the first term in Eq (2.63) is expressed as
follows:
dV
1
f
Vki
=
<<k
1
-
1
Vk dV+ <<Pk > > Vf
Vki
Then, taking the procedure
nk vk dA
(2.64)
Aki
described
in Section
2.4
and
38
vk is equal to vk , we have
assuming that
1
i
I
-
*
j
-
PkV VkdV =<<Pk>>Vak<<Vk> >
Vki
1 <
ak <<
+ <<Pk
'tV
's
k
ak<Pk Vk +k
k
Applying the same procedures
(2.58),
we
have
the
for
other
terms
265)
(2.6
in
Eq
time-volume averaged internal energy
equation.
Lak
+
ak<<Tk:Vvk> >
+
k
4-0
<<k's
vk
1
S<<n
ek+nk*qk >>a
a
kk
mk k
Lak
.
k't
k>>
1
>>w
k >>
Lw <<nkqk
The term (mkek ) can be expressed in terms
of
(2.66)
enthalpy,
h.
The enthalpy is defined as follows:
(2.67)
h = e + R
Pk _
;k k =
k
+
k
-0 +V
i)*nk
(2.68)
By Eq (2.68), we have
+
-<mkh
<PkVikVk
k<k
-
<<<<k
-"
>
k
+
k
-i)*n
k"
k
2.6
(2.69)
Using the definition Eq (2.62), we have
<<Pk(vk -vi)
nk>>
+
--
4s-
+
i)*n k
= <<Pk>><<vk
<<Pk (k -vi)*n
k >>
(2.70)
Combining the second term in Eq (2.65) and the first term in
Eq (2.70), then we obtain
1
L
-I+
-
+
.
V
<<pk>><<nk
k
[<<Pk>><<nk Vk>
ak
1
-
+
>
Lak
(2.71)
nk>>
<<k >><<
L
i) >>]
k
Applying Leibnitz rule to Eq (2.71), Eq (2.71) becomes
-
1
ak
L
Lak
+
k >
ik
i
-
> = <k
k
at
>
(2.72)
Then we obtain the final internal energy equation:
-
at
ak<Pkek > > +V ak<< Pkekk
+ <<pk>>V*ak<<Vk>>+ <<pk
k
't
Pk Vvk>
-
1
= ak <<mkhk +n
k
a
>>
<<nk'qk
Lwk
(2.73)
w
Using the definition of enthalpy,
enthalpy
V-ak <k +qk
at
's k
<<Pk
k
+ CLk <<k:VVk> > +ak
1
>> +
-t
energy
equation
from
Eq
we
can
(2.73).
obtain
the
The first and
40
second terms on the left-hand side in Eq (2.73) become
a
at
and
a
<<P
k Pkek
at 'k «Tkr
at
ak<<
k
>>
(2.74)
k < <Pkekk >
V
- Vak<<
k>
k
44.
<<k k>>VPkk>>
4-
-
<<Vk>>*.Vak
k>>-
Vak<
k>>
(2.75)
Then, we have the final enthalpy. energy equation
?--- ak<<~k~i~i,>
+
>> + V-ek<<qk+
- <<vk>>Vak<Pk
>> -
a
at
k
k:
k>> -V-ak
1
Lak <<k
ak
ka >
+nkqk
k
>
-
<<Pk
'S
+ ak
k>
'S
Vk>> +k
k
't
k
+k
1
wk <<nko kw
wk
Vk>
(2.76)
2.6 Interfacial Jump Condition
From Eq (2.40)
we
conditions.
Mass
In this case,
k = 1, Jk = 0
can
obtain
the
interfacial
jump
41
we have
k
(2.77)
rkr = 0
Momentum
In this case,
=k Vk'
k = PkI
k
we have
2
SL
k=1
<<k
-+
k+nk (PkI
(2 78)
Tk)>>a
ak
Energy
In this case,
2
E
+q-
2'
k
2 Lak <<mkEk+nk
k=1
k
K
k
ek
)J-V
.e
k -k
We have
k
(P
>>a= 0
(2.79)
O
(2.80)
The equation can be written in enthalpy form
z
-1
<<mk
+
+ n*k
pk
i
+T
'k
)> >
= 0
k=1 Lak
2.7 Comparison
Ishii (20) derived time averaged equations by averaging
the local
several
equations
single
over
time
interval
At
in which
phase regions with singular interfaces pass
through a reference point.
limiting forms
a
of
Therefore, he was forced to
use
the Leibniz rule derived by Delhaye and
42
Achard
(23),
which lead.
about
their
derivation
interpretation
phenomena.
to a question about the validity
describing
terms
of
an
and
ambiguous
physical
interfacial- transfer
Ishii's time-averaged conservation equations are
as follows:
Mass
a
at
kk
(2.81)
O k k vkI rki
Momentum
atkk- v k +-tVa k Pk-v k v k +V akPk
- V'ak(k +k) -ak k k- Mki
(2.82)
Energy
+-
a-t kEk
+7
kak k
- ckk(Fkvk +Qk)
where
Qk
is
rki, Mki, and Eki
the
-t
+ V kk+
(
Jk)
of
source
body
are
(2.83)
= Eki
the
rate
of
energy,
and
interfacial
mass,
43
momentum, and energy transfer, respectively:
1
I
rki
*1 nkPkvk -vi)
-
1
1
-+
)vk
JAtv ni nk[Pk(v k
Mki
i )
(e k + 2
(2.86)
Tk ) vk + qk] }
- (pk
.i
(2.85)
2
Vk
kn[Pk(Vk
(I
j Vni
Eki
niWe
may
(k-k)
1
1
where v . =
(2.84)
ni
t j
:displacement velocity of the interface
are
We may notice that Ishii's time averaged equations
basically
in
the
same
equations derived here.
equations
as the time-volume averaged
forms
However, it is noticed that Ishii's
time-volume
obtaining
in
difficulties
leave
averaged equations which give clear physical meaning for two
First, he introduced the concept of time-averaged
reasons.
equations
void fraction when he derived local time-averaged
and
natural
the
is how
question
this
concept
can
be
transformed to the concept of volume-averaged void fraction.
(2.84)-(2.86)
Secondly,
Eqs
transfer
phenomena
as I Fk-nk dA
A
the
commutativity
are
which
interfacial
describe
well-defined
not
forms
As a result, Delhaye and Archard
of
operators
averaging
such
assumed
to express Eqs
(2.84)-(2.86) into the well-defined forms as follows:
SAt
V
(F *n ).
dV
=f
Ai
ndA
Fkk
(2.87)
II
-
-
III
nm
llll
44
These difficulties mainly
come
from
their
averaging
the
local equations over a time interval in which several single
phase
with singular interface are allowed to pass through a
reference point.
Banerjee and Chan (1q) derived one
fluid
volume Vf which
regions.
As
volume
by averaging the local equations over a
equations
averaged
dimensional
their
of
consists
several
single
equations are not averaged over a time
interval, turbulent contribution terms do not appear in
equations.
conservation
phase
the
for these turbulent terms
Except
and the use of instantaneous properties, their equations are
the same as the equations derived here.
Let
compare
us
equations,
the
(2.42),
Eqs
conservation equations of
time-averaged
(2.51),
the
and
THERMIT-6S.
conservation
(2.73)
The
with
the
following
equations are used in the THERMIT-6S.
Mass
)
S(kp
k ) +V(kPkk
S(ak k+V.(ak k k
rk
(2.88)
k(.
Momentum
a
at
444
k
rki-k
vk k
+ V (akPkvkvk) + k VPk
-Fwk
-F
kkg
wk + akk
sk vk
(2.89)
45
Energy
at akPkek + V* (akPkekvk) + pV* (akk)
+
k_
at = Qik
+ Qwk
(2.90)
(2.90
+ Qk k
where
force
Fsk , Fvk , and Qik : interfacial standard drag
and virtual mass force, energy exchange rate, per unit
volume, respectively,
Fwk and Qwk :
wall
friction
force
and
wall
heat
transfer rate, per unit volume, respectively,
:
Qu
liquid energy conduction
rate
per
unit
volume,
: gravitational force
g
Note that QVV
equation
of
the
is put to be zero for the
THERMIT-6S.
vapor
energy
The relationships between Eq
(2.89) and Eq (2.51), and between Eq (2.90)
and
Eq
(2.73)
ImimamnilmIIrllMrmiNNIIMInI
IurmmilImYII
ulgplll
Imm
,nII
Ml h
illiu,,, i
, i ih I I llllll
I11
i
iillYII
46
are as follovs:
vi
<<V>>ar
1
*
Fsk sk
F
Fvk
Fk
rk a
1
<<
- ---«Pki >
L
ak
-
ik
1 <
Lak
wk
1
Lwk
Implicitly
terms
a
<<n *kTk>>w
--Lwk
w
can
>>
k* ke
k
Q = - V-c (q
volume.
-
4
L
-<nk
Lak
*1
F k
Fwk .
THERMIT-6S
k k
k
w
+q)
each property
be
a
k
considered
in the
to
equations
be averaged in time and
Therefore, the THERMIT-6S neglects the
from
properties.
the
nonuniform
the
of
spatial
Properties can be expressed as
fluctuating
distribution
the
sum
of
of
a
47
spatial average and spatial fluctuating component:
Pk
pk
<<Pk > >
k
+
k
k
>>>
<<k k>> and <<PkVk
Then,
-
>> +
<<
andy
P
become
k's
Pk
h,
<<P
k
k
'S
<<
<< k
_
3
3-
+
k
s
's
a
's
>>+ <<vk>><<pk
+ <<k >><<pk'sS vks
+ <<p 's
k
s
Yk'S >>
's v 's >>
k
k
An adequate assessment should be given for all terms in the
by
equations
assessing
their
thereby justify the neglect
of
comparative
these
magnitudes and
spatial
fluctuating terms.
In addition to neglecting the
terms,
distribution
in
the momentum equation of the THERMIT-6S, several assumptions
are made:
1)
Negligible turbulent contribution term, V7ak < < knk>>
2) Negligible phase shear force, V-ak<<Tk*nk>>
3)
The average interfacial pressure <<ki
the phase pressure <<Pk>>.
a
equals
As a result, we have
Apki = 0
Similarily, beside the
terms
the
following
omission
assumptions
energy equation of the THERMIT-6S:
of
the
distribution
are made in the internal
48
1)
Negligible
energy,
2)
conversion
to
<<k: vk >
Negligible vapor energy conduction rate,
V-' <<q
4
irreversible
+
>>
internal
49
Chapter 3
DESCRIPTION OF THERMIT-6S Equations
3.1 Differential Equations of THERMIT-6S
The mathematically rigorous differential equations
of two-phase flow derived in-Chapter 2
several
with
order
in
assumptions
simplified
are
derive
to
Eqs
differential equations of the THERMIT-6S.
the
(2.42),
and (2.73) for conservation equations, and Eqs
(2.51),
(2.77), (2.78), and (2.80) for jump conditions are used
to obtain the equations of the THERMIT-6S.
Conservation Equations
1.
Vapor Mass
( v vv v ) = Tv
Ct ( v v +
2.
Liquid Mass
at
3.
(
=
p )+
(3.2)
Vapor Energy
-t
(a vPev) +V. (avpev
) +pV. (av v)
(3.3)
= Qiv+Q
+ p -
4.
(3.1)
Liquid Energy
+-,
S(-+
S(oap
e)
+V*(at
+ pV.(c- v)
)
+p
te v
- =
) +OY
+
(q
+q
t
)
(3.4)
50
Vapor Momentum
5.
t (avPvV v
rV
i
+V(avPvVVVV) + aVP
)
- Fiv - FWV +aPg
(3.5)
Liquid Momentum
6.
-P. -10
-0
at (a ZP1v
=
v
p y
+*(pva
.
X) + c~Vp
+ jPg
- Fw
(3.6)
Jump Condition
1.
Mass
2.
(3,7)
=
v =-
v+ £ = 0 :
Energy
.iv+Qi = 0 : QiQ
-Q
(3.8)
-F i
(3.9)
3. Momentum
= 0 : F.i
F.
iv+F
and
Here Qi
exchange
and Q,
WV
rates
denote
Fi
rate,
'
energy
interfacial
per
, FWV , and FwA. are
and
unit
the
wall
momentum
volume,
heat
transfer
and wall friction force of vapor and liquid, per unit
volume, respectively.
is
i v
different from
Note that vi
in the above
equations
Vi.nk in Eq (2.24); v i and vi'nk are the
interfacial material velocity and displacement
velocity
of
51
the interfacial surface, respectively.
For reasons involved with the selection of a difference
strategy for solving these equations, momentum equations are
by
form
non-conservative
a
in
rewritten
mass
using
Then Eqs (3.5) and (3.6) become
equations.
av
p y *V Vv +ca vVp
atva+a vvv
v
p
w-+
) -.
r(P.~-
=
v
+ ap
y -Vv£
+a
g
X
w +c0
+i-F
= - F(v -v)
(3.10)
vg
(3.11)
, then
If we assume that V. is equal to v
) = r (V - Vv )
r (vi
= 0
F(vi -v)
In THERMIT-6S
Fi
of
consists
two
drag force (Fs ) and virtual mass force (Fv
standard
unit volume.
s
v
),
per
Both terms are defined as follows
- K(v -v
F
the
components:
v
v
(3.12)
)
avS
v
+ (v v-V
3t
)
a 9
at
4
v
v
[(X-2)Vv + (1 -
-v£
)V1]}
(3.13)
52
For details of the virtual mass force, refer to
Eqs
Introducing
(3.12)
(3.11), we get the
and
final
(3.13)
momentum
into
Chapter
Eqs
equation
4.
(3.10) and
used
in the
THERMIT-6S.
av
(avv
p +v avPCv)
Vt
atv
-(r+K)(v
a
iC
) - F
-
LCP C )
SK(;v -
a
+E Vv+Evm
v +aVp
+ apg
(3.14)
av
av
(aP p+
av
-
pPaC
) -FW
-2
+E
-E
+aVp
+ap g
(3.15)
where
Ev = avPV
E
=
*Vv
(3.16)
Qv-v
oa
Evm = a v.v [X
-t
4
Xv *-V
(3.17)
-1)v v *Vv
.C v - (X - 2)v 2*Vvv
- (1-
,)vt VV]
As shown in Eq (2.59), Qiv and Qi£ can be divided
(3.18)
into
two
components contributed by mass exchange and interfacial heat
flux. These are defined as follows:
= L
iv Q
+q
(gmh iv+
- 1= (g h.
iv
+ q !") '
(3.19)
(3.20)
where hit and hiv are the enthalpy on the liquid and vapor
side of the interface, respectively.
_ aa~-~--
_
_
I --
--u
--;r
_
.~ -~-... ~
..ri_:--:---rr.
53
q"i
and qiv are the heat flux on the liquid and vapor
side of the interface,
gm and La
L are the interfacial mass flux and the ratio of
interfacial area to fluid volume in the control volume.
'k'
=
< <n k
k
k
-ik
gm
*4k
>>a
a
= -<<>a
of g
m
expressed
7 can be
The mass exchange rate
in terms
1
and
La
S=
gm
_La
(3.21)
jump
the
condition,
According
to
neglected
V
term <<pk.ei+Tkk
>>
,
Eq
(2.67)
the
with
the following relation
can be obtained
ghi.
=
+i
(3.22)
g hiv. +q
Assuming the saturation state at the
the
interface,
we
obtain
relationship between the interfacial mass flux and heat
flux.
gm
(3.23)
h
h
As the interfacial
heat
flux
in
the
gas
phase,
c'.:
IV
is
usually much smaller than qi', a simplified relationship can
be obtained by neglecting q v:
gm
qj /hfg
(3.24)
54
, then
If we set up a physical model of q!i
rate
can
Substituting
by
obtained
be
(3.24)
Eq
Eq
into
(3.21)
Eqs
using
mass
(3.19),
exchange
and (3.24).
we
have
the
interfacial energy transfer rate per unit volume,
i
Q L1
a
hh
fg
T
(3.25)
In addition to these conservative equations
there
are
four equations of state:
Pv M Pv(P,TV)
P£
ev = e (P,T ),
e, = e (p,T2 )
These equations of
be
can
(3.26)
to
used
eliminate
pressure. For the
which
closure
represent
of
the
the
conservation
liquid
and
equations
of mass, energy, or
transfer
momentum at the interface between the solid and
between
the
and internal energies in terms of temperature and
densities
terms,
state
P X(pTZ)
fluid,
and
vapor, must be explicitly expressed in
terms of primary variables and properties.
These terms
are
as follows:
F, Fi , Qi' Fwv~, F,
The
physical
models
of
Qwv
the
and Qwz
above
terms,
so
constitutive equations, are described in Chapter 5.
3.2 Difference Equations of THERMIT-6S
called
55
THERMIT-6S
The difference equations of
those
(17) and can be obtained by discretizing
THERMIT
of
a first-order spatial
by
is characterized
treats implicitly-sonic terms and terms
fastest
but
A
conditions.
method, on the other hand, allows large time steps
leads
compromise
a
is
method
size
extremes; reasonable time step
per
and
step
time
code
increased
greatly
to
semi-implicit
time
the
basis, but requires
Courant
to
extremely small time steps due
implicit
step
time
per
a
on
method
local
represents
method
A fully explicit
convective terms.
to
relating
exchange processes, and explicitly
the
as
such
phenomena
scheme
semi-implicit
first-order
a
discretization
For
layout.
scheme, donor cell method, and staggered mesh
time
spatial
The
the proceeding set of differential equations.
discretization
basically
are
simpler
but
structure.
these two
between
less
The
computation
code structure than the
implicit method.
are
We begin with the mass and energy equations, which
3.1) The subscripts
positions
along
centers of the mesh cells. (See Fig
the
about
differenced
the
i,
j, and
x,
y,
k
are
to
used
indicate
and z axes, respectively. The
the
superscripts n or n+1 refers to the time level at which
variables
are
evaluated.
A superscript n+ / is used only
for the exchange terms which are functions of both
old
time
values of variables.
A and V represents the flow
a:eas of the cell faces and the flow volumes
respectively.
and
new
of
the
cell,
With these definition, the mass and energy
111^-1 ~- -
"~~'IIIIYYIIYIYIIIIIIY-~
-
^'~IIIIIIYIYII
IIYIYIIYI~~
'
pll
NMIN.
56
S(v)
,
a,'
I
/
(VX 3,; 1
't
(VZ)
Iy
Fig. 3.1:
k -h
cell conter
at"p qD st'P L'
L*
"'
A Typical Fluid Mesh Cell Showing
Location of Variables and
Subscripting Conventions
(i,4,k):
difference equations are as follows:
Vapor Mass:
n
n+1
(ap) - (ap) n
At
1-{[A(cvp ) n (vx ) n+1] i+
v
v
nt+
-[A(a vp)
n-A( n + l i(vv)
I
+ [A(vPv n (v)y n+l
p )
v
]j+
+ [A(a p )n(v)Z n+l
-[A(a vP)n(v )n+lj_
(vzn+ 1
v)n
[A(a
v
v v
(3.27)
n+
}k-=
Liquid Mass:
Same as Eq.
(3.27) with a v replaced by aZ,
subscript
(3.28)
v by Z and r by -r
Vapor Energy:
(actoe )
n+1
-(
p ev )
n
1
A(a
+ -[A(c
V
t
e
e
v vv
+)n
+p)
x n+l
(v
v
i+
-[A(av pe + P) n (vX) n+i[A
+[A(~ v pv ev +p)
n+1
+n
n+
w£
i
e +p)n(vy n+l
(vzv ) n+l] k+
-[A(a vPvev + P)n(v ) n+l]k
Sn+
-A(L
+ pn
n+l
n
vAt
v
(3.29)
Liquid Energy:
Same as Eq. (3.29) with a v replaced by a., subscript v by k
and Qi by -Qi,
and added Q££
(3.30)
lilil
IIIU
YIIYIIYIIIYIIIIII
~~~~'^^'IIIIII11I1IYYIIYYIY
1. dIi
i
i I
H1I1
iiiiMI'i
iH
58
term
The liquid conduction
0in
asQ
models for it will be presented in
and
(3.30)
Eq
expressed
is
V.*a(q +q )
The properties transferred through the cell
Appendix A.
faces must be explicitly expressed in terms of cell centered
uses
a
as
such
quantities
ev and e. . The THERMIT-6S
a, pv, P1
relationship
particular
often
as
to
referred
differencing; the properties will be transferred
donor-cell
through the cell faces with all centered quantities upstream
of that cell.
cell-centered
If C stands for any
quantity,
is determined as
the value of Ci
Ci
n+l
if
v+
if
n+l
vi+
< 0
(3.31)
Ci+
Ci
0
For the donor cell decisions updated
velocities
used.
are
The primary reason to use updated velocities is to solve the
packing
problem
occurred
which
if
old
velocities
for
of
the
donor-cell criteria were used.
Now let us consider only
vapor
momentum.
permutation of the
The
a
single
component
additional equations are obtained by
and
axes
indices.
According
to
the
staggered mesh scheme, each momentum equation is differenced
about
the
center
of
a
face of a mesh cell.
Based on Eq
(3.14), the one-dimensional vapor momentum equation in the x
direction is differenced at the point (i+,j,k) as follows:
xn+1
(avp v + av p aCv) n
v
where
(v
n
x n+1
V xi+
n
vi+h
(Pi+1
xi+=Pi
)n+
n
n
xn
vt
At
)i+I
(aPi+)n
v v i+
vv
En
(apC )
[(X - 1)Av
X~v nvi
vv
(
-x
Av n
n
zi+
- (X- 2)Av
n
£1
i+
v
\
i+
Ay
v
(
i+
Az
Vv i+
v
n
x
IxV
\
Ax /i+
+vy
+ Vy n
ki+
x
n
yv
\y
i+
n
)i+ subscripts ^ replaced by
(Avz
: same zi+
as Av
with
v and v by 2.
Av
(3.32)
xxn
vi+
xn
n+
+1
i+
1
- (1 -)v]
vzz
n
n
Evm i+
n
Ev i+
K (v -
E i+
v i+;
-
i+
At
(avP C )
-
xn
vv
: same as Av
by I.
with subscript v replaced
60
In the above equation we have used an expression of the
A vx
x v
form -to
designate
the
difference approximation
afvX
of
a
associated with the point (i+ ,j,k).
Similar
ax
expressions have been written for the y and z directions.
In Eq
(3.32)
variables
defined.
are
we
appearing
again
at
faced
with
the
of
locations other than those already
We first consider the quantity a:
i+l Axi+
ai+
+
i+l
+
Ax
1
+a.iAx.i
ax
(3.33)
+Ax.
The quantity Pvi+ can be expressed in the
(3.33).
problem
Every
velocity
except vi+
same
way
is at
the
as
Eq
wrong
position, so an averaging must be performed. We define
vY
vi+
1
4
vi,j-
+vy.
+vy
v1,j+
vi+1,j-
+vy
vi+1,j+
(3.34)
z
Vvi+
1 z
vvi,k-
z
z
+Vvi,k+ +vi+l,k-
z
+vi+l,k+
(3.35)
Finally, according to the donor cell scheme, the
difference
61
(
approximations of the convective derivatives are given by;
=
xV)
Ax)i+
x
Axi
vi+h - vVi-1
X
vi+,
yi+xi+hj-1
SV
vXAZ'k+l-Vvi+ ' k
ZVvi+
(iv
vi+
if
XAz
,k
vi+,k-
and Azk+
The mesh spacings Lyj+
vi+x
> 0
if v
j/1
vi+h 0j/,
j+1
X
vi+,j -vi+,j-
Yvi+
<0
(3.36)
x
V
if VX
AX
X
vi+
<0
(3.37)
Ayj_
j-
iff
vY
vi+ > 00
Az
A k+
if
Vv+
<0
vi+
(3.38)
Azk_
if V(
v
if
>0
appearing in the
above
expressions are evaluated as;
AYj+
= (Ayj + AYj+1)/2
(3.39)
AZk+
= (Lzk +Az
(3.40)
k+
)/2
We have now completed our specification
equations,
of
the
difference
with the exception of the exchange terms.
These
terms are discussed in Chapter 5.
3.3 Solution Procedure of THERMIT-6S
The most important
equations
lies
in
the
characteristic
treatment
examination of the mass and energy
of
of
our
difference
coupling terms.
equations
reveals
An
that
62
cell-centered
pn+l, and en+lare
as a +l,
in these equations such
appearing
variables
only
coupled
to
velocities vn + l on the faces of the mesh cell.
The momentum
velocities only to the pressure in
these
relates
equation
new-time
the
cells.
the mesh cell in question and in the six surrounding
see that coupling between mesh cells at time level
we
Thus
Hence,
n+1 is through the pressure variable.
we
need
to
solve only the reduced pressure equation.
Let us describe
Denote
by
principal
solution
is
accomplished.
system of two-fluid finite difference equations
the
F(x) =0
this
how
.
Here,
x
is a
vector
variables.
unknown
containing
shall
We
use
ten
the
a Newton
iteration to solve the non-linear system,
F(x) = 0
(3.41)
We first linearize the old iterate xm as
F(x)
F(x
+l
) =F(xm ) +J(x m )(x
+l
- xm ) = 0
where the Jacobian matrix J(xm ) is defined as
(3.42)
( x)xm
We then solve the following linear system for
l
J(xm)A x m+
where Axm+l is the
= -F(x
m)
iteration
(3.43)
error
(xm+l-xm).
and F(xm)
is
called a residual error.
This linear system is solved either by the direct method
iteration
or
method. (For detail refer to Ref 15) This process
is called as the inner iteration.
This
Newton
iteration,
63
called as the outer iteration, continues until the iteration
error Axm + 1
F(xm
error
residual
the
or
)
is sufficiently
small.
in
equations
momentum
exchange
momentum
interfacial
of
terms
pressure.
the
Because
the
treated
been
has
term
in
velocities
express
The first procedure is to
implicitly, the liquid and vapor velocities are coupled with
each
At
other.
face of each mesh cell, the momentum
the
equations can be put in the form
m+l
am+1
(3.44)
Sa
x
m+1
b~p m+1 +g
velocity
By solving the above 2x2 matrix, we determine each
in terms of the pressure difference as
vm+1 = c Apm+1 + f
(3.45)
m+1 = d apm+1 + g'
The above
velocities
equations
appearing
can
now
be
used
in
the
mass
and
finally obtain a set of four equations for
involving
the
four unknowns, p,
a,
TV
center of the mesh cells and pressures, at
to
eliminate
Then we
energy.
each
mesh
, and T.
the
all
cell
, at the
centers
of
'Im.1.
64
the six adjacent cells.
Xp
XXX
+
x
x
x
x
x
x
T
Lx
x
x
x.
.T .
xx
V
X
X
XX1
x XX
X
X
x
x
x
x
x
x
x x
x
.x
x
x
x
x
x.
x
x
I
I
x
x
=
I
x Lp6 1
(3.46)
By inverting the 4x4 matrix that appears above, we
can
put
this system in the reduced form:
0
SP
I
a
+
T
S
L TZ
-0
P
rx
-x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
- x
x
x
x
x
x.
x-
p6
(3.47)
The first equation in the above set is an equation involving
its
only the pressure in a cell
and
remaining
relate
three
equations
temperature to the pressure.
solved
iteratively
an inner iteration.
other
pressure
convergence
is
iterations,
the
void
The
fraction and
can
be
with each complete sweep referred to as
Once the pressure field is
T.
The above procedure
iteration
the
neighbors.
The pressure equation
unknowns a, Tv , and
substitution.
six
error,
not
time
procedure is repeated.
is
be
found
repeated
by
until
the
back
the
Ap m+is sufficiently small.
If
number
of
attained
step
can
found,
in
size
a
is
specified
reduced and the same
The method will always converge,
if
65
the time step size is small enough.
A crucial property of the reduced pressure
its
diagonal
dominance.
problem
is
In the one-dimensional flow the
pressure equations can be expressed as follows:
m+l1
Pi+
-
2+
a
t
Ax)
m+1
m+1
Pi
+ pi-i
=
where a is the sound speed
a -2 =
p
p
The amount by which the
sum
of
any
row
of
coefficients
exceeds zero can always be expressed in the form _
)2
ot1 2
:.l--.lm---~n _
__
-- -~------I_
-~-
~-1
66
Chapter 4
STUDIES OF VIRTUAL MASS
It is known that the
model
of
the
two-phase
characteristics, exhibits
short-wavelength
value problem.
which
stability.
profound
flow
unbounded
two-fluid
possesses
complex
instabilities
in
the
suggestions
to
overcome
virtual
mass
provide
represents
real
effect
upon
dissipation
the
of
the
virtual
for
numerical
of
the
concluded
that
the
suggested
virtual
mass
quantitative
numerical
Second
finite
has
a
bounds
on
mass terms are suggested for
hyperbolicity,
mathematical
-satisfaction
these
mathematical characteristic and
Here,
stability.
coefficients
the
physical
Also, it was found that the virtual mass
numerical
stability,
Law of Thermodynamics.
difference
model
convective stability conditions
values.
equation
one model for the virtual mass force terms is
studied here. The
effects
six
limit and constitutes an ill-posed initial
Among
difficulties,
typical
is
with
scheme
restricted
the
above
with
and
It is
the
only by the
suggested
67
4.1 Introduction
two-fluid
model
the
offers
description of two-phase flow.
But it was reported in early
(24)
Jarvis
solve the two-phase equations
to
tried
process
using the two-fluid model for modeling the cooldown
in cryogenics and faced instability problems.
the
that,
He attributed
to the fact that the system was found to
instabilities
be nonhyperbolic.
In
problems.
work that the model has inherent instability
1965
general
and
detailed
most
the
-flow
two-phase
of
models
several
the
Among
In 1967 Richtmyer and Morton (25)
showed
if the initial value problem (IVP) is ill-posed, then
no difference scheme that is consistent with the problem can
instabilities
MOC.
transient sodium boiling code called
in his
(27),
Conference
In 1976
pressure
oscillations
demonstrated
that
any
Bryce
with
Heat
the instability problems in the
(28)
the
its
by
caused
two-fluid model were shown to be
equations.
International
Fifth
the
during
severe
In a round
GEVATRAN code.
in the
difficulties
discussion
Transfer
ill-posed
large-scale
experienced
code.
RELAP-UK
He
simple two-fluid model with complex
characteristics does not give solutions
the
numerical
In 1973 Boure (26) appears to have encountered
stability
table
encountered
(11)
Siegmann
In 1971
be stable.
which
mesh size and time step size are refined.
that, though solutions may be obtained
by
converge
as
He concluded
using
numerical
technique, the significance of these solutions is not clear.
In
1976
Lyczkowski
(29)
did
a sample
calculation
~-~ ~~IIIIII
IUI
I
YIYIYIYIYIIIIYYIIY
I1IYII I IIIIY11
IIYI
68
illustrating error growth caused by complex characteristics.
reported his attempt to achieve stability by
(30)
Anderson
found
was
It
using virtual mass terms in the RISQUE code.
that
safety,
reactor
In the 1977 ANS topical meeting on thermal
the virtual mass has a profound effect on the dynamics
His code was used to
of two-phase flow.
calculations
numerical
perform
It was
the behavior of interfacial waves.
on
grows
observed that for no virtual mass, the wave amplitude
but
rapidly,
the computed result with virtual mass shows a
stable oscillatory behavior for the wave amplitude.
In 1979
Rivard and Travis (31) successfully dealt with critical flow
the
with
two-fluid
code,
But
K-FIX.
for
the
results
presented, the value of the interfacial drag coefficient was
chosen
sufficiently large that there was no relative motion
In 1980 investigators
between the phases.
showed
that
their
(32)
R.P.I.
without virtual mass it was more costly to run
the problem, and even could not
using
at
GEAR.
code,
run
Computer
the
complete
time without
running
virtual mass was forty times longer than that
problem
with
virtual
with
complex
mass.
It may well be argued that equation sets
characteristics
may
phenomena if the
dissipation
to
still
numerical
damp
the
be
adequate
method
high
a
for
introduces
frequency
range
sufficient
instabilities.
Obviously there are real physical effects to accomplish
dissipation
needed
for numerical stability.
candidates suggested for numerical stability,
of
the
Among several
Drew's
model
69
(33,38) with the virtual mass terms will be studied here.
4.2 Physical Significance of Virtual Mass
an
Let us consider a solid sphere motion in
and
inviscid,
a
In this perfect fluid
incompressible fluid.
there is no dissipation of energy.
infinite,
about
For inviscid flow
sphere with the radius r, the pressure distribution along
the surface is given by Lamb (34) as
P
p
S= 1 r dU
-- cos e + 9 U22 cos 26
16
dt
2
-
U2
(4.1)
16
where U is the instantaneous relative velocity
the
between
sphere and fluid, 6 is the polar angle measured with respect
to
direction
the
of
U,
and
the time derivative is made
following the particle, that is,
dU = 9
dt
at
+ U.VU
(4.2)
the same for surface elements in the positions
so
that,
when
are
(4.1)
Eq
The last two terms on the right-hand side of
e and
t-6
U is constant, the pressures on the various
elements of the anterior half of the sphere are balanced
equal
pressures
posterior half.
;
on
the
elements
corresponding
But when the sphere is
being
of
by
the
accelerated,
there is an excess of pressure on the anterior, and a defect
on the posterior half.
The reverse holds when the motion is
being retarded.
Integrating the
local
pressure,
Eq
(4.1)
over
the
70
fluid
solid sphere
the direction of motion
of sphere
\
Fig. 4.1:
Configuration of the Solid
Sphere Accelerating in Fluid
71
surface area of the sphere, we obtain the resultant force in
the
direction
motion
the
of
of
sphere
the
due
to
acceleration of the body.
-! PdA
A
fw2wr2 sin6 cos 8Pded
- 2
o
o
2
-ir
3
P
S -dU
where
M' =
where2
r 33p ,
dU
- 1 M dU
M=
(4.2)
r 33P a
M: mass of the fluid disp3aced by the sphere
Then, it appears that the effect of the fluid pressure is
equivalent simply to an addition to the inertia of the solid,
the increment being half the mass of the fluid displaced.
Also, note that Eq (4.2) specifies the equivalent momentum
transmitted to the fluid in unit time, F.
The external
force f must be equal to the sum of the time derivative
of the total momentum of the sphere and F:
M dU + F = f
s dt
(4.3)
where Ms is the mass of the sphere.
Using Eq (4.2), we then obtain
(Ms( + M') dU W=f
(4.4)
(4.4)
72
This is the equation for a body immersed in a perfect fluid.
The presence of the liquid effectively increases the mass of
a moving sphere from M to M+M'. M' is the virtual mass to be
considered for accelerating objects in the fluid.
Eq
2.5,
in Section
If we use the notation described
(4.2) can be expressed as
L
where
S<<P>>
a
M' -/
dt
Vf
f
-
2
(4.5)
(1 P
b Zt
3
4
ab
=
a
r /Vf
4
Referring to Eq (3.13), the coefficient of the virtual
mass
for a sphere, Cv is 0.5.
4.3 Ill-posed Problems and Mathematical Stability
physical
Quantitative
reality.
be
idealizations
are
laws
of
As knowledge grows, a given physical situation can
idealized
mathematically in a number of different ways.
It is therefore important to characterize
idealized
formulations.
Hadamard
those
reasonable
examined
(35)
this
problem, asserting that a physical formulation is well-posed
continuously
if its solution exists, is unique, and depends
on
the
uniqueness
external
are
determination
repeated with
continuous
an
(boundary)
Existence
and
the
principle
of
of
affirmation
without
the
conditions.
which
experiments
expectation
dependence
of
criterion
stability of the solution; a small
could
consistent
is an
change
not
data.
be
The
expression of the
in any
of
the
73
problem's
data
should produce only a correspondingly small
change in the solution.
The general one-dimensional problem can be
represented
in matrix form, as follows:
A(x)
+
at
(x)
az +
C(x) = 0,
where x is a column vector of
and
(4.6)
independent
variables,
A(x)
B(x) are the coefficient matrices, and C(x) is a source
or sink vector.
The Initial Value Problem consideration is to find a
solution of Eq (4.6) in some region
a < z < b
subject to the initial condition
x(O,z)
(4.7)
-. u o (z)
and the value of x or
its
derivatives
prescribed
on
the
boundaries,
z = b
z = a and
(4.8)
When the definition of a well-posed problem by Hadamard
(35) is applied to the
well-posed
if
the
above
system,
(4.7)
(4.6)
is
and
the
defined
characteristics
of
is
to
said
be
solution of Eq (4.6) exists, is unique,
and depends continuously on both of the
Eq
it
initial
condition,
boundary conditions Eq (4.8).
as
hyperbolic
Eq
(4.6)
if
are
all
Also
Eq
of
the
values
distinct
and
real.
ml
0 1116,U
I
74
According to Lax (36) the requirement of a well-posedness in
linear partial differential equations
of
Eq
like
form
a
(4.6) is the same as that of hyperbolicity.
indeterminate; characteristics may separate
Therefore,
solution.
the
in
discontinuities
of
typical
imaginary
two
and
real
The
system.
physical
travel
of
information
characteristics
vapor.
indicate
interfacial
of
velocities
The
complex
convective
The two
may
relative
two
represent
wave propagation to both liquid
values
of
characteristics
the
an instability of interfacial waves.
Characteristics
related
a
real
The
characteristics.
of liquid and vapor through the system.
imaginary
in
two-fluid models have four
vapor, and two relative sound velocities to both
and
Physically
are two convective velocities of liquid and
characteristics
velocities
are
characteristics
the
trajectories of discontinuities in the solution.
they represent the speed
are
equation
differential
partial
a
in
derivatives
highest-order
the
curves,
characteristic
Along
only
and
stability
can
be
shown
to
be
in the limit of large frequency, as discussed
in the following.
The characteristics, u, of Eq
(4.6)
are
defined by the equation
Det (vA-B) = 0
To examine the local linear stability behavior of Eq
x
is
(4.9)
(4.6),
replaced by (xo+x) and the result is linearized with
ex , so that
respect to
A
0
(6x)
at
+x
x)aA(
9(6x)
B(x
_
ax
0
Xo/ az+
x
solution,
a
uniform
(4.10)
6x
, about
an
steady-state unperturbed
solution is assumed, which means that
z and t.
at
x)= 0
As Eq (4.10) describes small perturbations,
unperturbed
axo
x Xo
+ 6x
xo
)
x
of
is independent
x,
We take a wave form for the perturbed amount, 6x
6x = 6xo exp[i(kz-wt)]
(4.11)
Then Eq (4.10) becomes
-iw A(x )
x +ik
B(x )6x
+x
(C
= 0
(4.12)
Eq (4.12) is a homogeneous linear equation in the components
of 6x . For a nontrivial solution the
determinant
of
the
coefficient matrix must vanish:
Det(-iwA + ikB + D) = 0,
(4.13)
where
D =
and the superscipt T denoting the matrix transpose.
For nonzero k Eq (4.13) can be rewritten as
A - B +
Det
(
(k
D
= 0
(4.14)
i imll
nmnImmAI
AnIh
OllllllII
i,,l,
1hiIIU
IAnai
i
76
Inspection
stability
of
is
Eq
(4.11)
shows
that
the
condition
for
the imaginary part,(Im(w)),of w must be
that
v.
less than or equal to zero for all roots of
Comparison
of Eq (4.9) and Eq (4.14) shows that, if D * 0 or k o *, the
dispersion
relations
can
be
long
instability' at
Helmholtz instability (37).
complex
characteristics
to
u.
Physically
wavelenths
is
the
well-known
But the system which
for
behavior
this
possesses
unphysical and unbounded
exhibits
instabilities in the short-wavelength
reasons
lies
in
limit.
the
One
of
pressure.
This
the
choice leads to an equation set with
all real characteristics when the two fluid
equal.
the
choice that the
interfacial pressure is spatially constant and equal to
bulk
the
w/k
characteristics simply by equating
the
from
obtained
velocities
are
The spatial interfacial pressure distribution may be
accounted for by the implementation of the virtual mass term
and that motivated the present study.
4.4 Characteristic and Stability Analysis of the Two-Fluid Model
The effect of virtual mass on the
be
illustrated
by
adding
the
characteristics
virtual
can
mass terms to the
one-dimensional momentum equations of both liquid and vapor.
Here Drew's model (33,38) for
the
virtual
mass
terms
is
considered.
The conservation equations for one dimensional flow are
as follows:
77
Conservation of vapor mass:
S(yPv) +
r
(p Vv)
(4.15)
Conservation of liquid mass:
a
az (L p
)
-
(4.16)
Conservation of vapor momentum:
aV
V v +av
v
.V
V V at
+
VvVv az
Tr(Vv)cv
v
az
-
where Vi
,
Fs , Fv and Fwv are
standard
drag
g - F
F
rF+
tvo
Fs+ FV]
the
"
,
(4.17)
v
interfacial
velocity,
the
force, the virtual mass force, and the vapor
wall friction force, per unit volume, respectively,
and F
and, Fv are defined as follows:
(4.18)
F s = K(V -V)
v
v zv
at
+ (V -V
v at
av
at
)[(X - 2)
(V
V
av
+ (1- X)
(4.19)
F - aCpg - Fwt ,
(4.20)
Conservation of liquid momentum
aV
Va
+
[
-----
--
111
11111I
113
I
lil
1111
_
-
O
78
where FL
is
vall
liquid
the
friction
force
per
unit
volume.
For convenience by summing up Eq (4.17) and
Zq
(4.20),
we
get the momentum equation:
8V
%P at +
aV
a at
PV
Sy multiplying Eq (4.17) by
then
subtracting
_V + aV
-a (%vP+ 1I)
* r(L- V,)-
the
av
av
at
first
3z
g - (FW+ Fw)
(4.21)
and Eq (4.20) by
and
av
term from the second term,-we
obtain:
aV1
3V
a (a
+P
tCV)
t
+ {[f(
vPv + VPCv(A- 1)]
(-LvPL+ avPCv(1"
ar(V1-aVvL-
PIC)
L
-
v
at
- aVp C (- Z)VL ) a
(4.22)
)]vL + 1vP CVXV) a
aV) -
Fs+ a (oj
cFwv+ avFw
- p )g-
Conservation of vapor energy:
at (p
where Q
e )+ a(a
and Qi
e
Sv + Qi
)+ P L(V )+ P
are the heat transfer
per
unit
(4.23)
volume
from wall to vapor and from the interface, respectively.
Conservation of liquid energy:
a
a
a (aLe )+ i (a ~e V )+ P
(
Vi)wL
it
-
- Qj
,(
4)
(4.24)
79
where QO,
is the heat transfer per unit volume from wall to
liquid.
The above conservation equations can be put together in
phe following matrix form:
ax
A(x) I
ax
+ B(x)
+ C(x)
= 0,
(4.25)
where x is a column vector of independent variables:
X = (a,P,Vv,V£,evet) T
It is assumed that the density of both liquid and vapor is a
function of
spatial
Vapor
pressure.
and
time
and
liquid
with
densities
derivatives can be expressed in terms of
pressure by using the following
definitions
of
the
sound
velocities of vapor and liquid:
a 2.-.
v
-p
'
a-2
aaR
("ap
Then, the matrices of A(x) and B(x)
constructed as follows:
in Eq
(4.25)
can
be
i
s0
7
IPv
%a;2
~I
L
_
_
_
_
_
_
_
_
_ _
_
_
_
_
__2_
(4.26)
pvev +
y
p
aICPLvap
_______
-LV+pLCv
____2
P0
Py
czvva
ccV v
aBa
(Pl1)Jv-2
2
VV V ae Va w
+ PVV VVv
Pi
(a,)e,,V.aL,
a pe +Pc
VVv
V
(4.27)
(1-cvv
~pV
V
c~l0leC a v
%V
Vy
CILPXVI
LPLV
81
In the same way as Eq
u of
characteristics
the
(4.9)
Eq
(4.25) is defined by the following equation;
Det (WA- B)
-
(4.28)
0
from
Reducing the terms related to the energy equations
(4.28),
Eq
we get
Cvp ( - V ) x aep (V- V )x
Det
p (ii- V )
-pV(u-
V)
P
VI)
V)
Cav - 2
caa 2 ( - V )
-LvPv
-2
0
-1
0
0
0
-ap
(a p +pC)(U-V
) -(ap +pjCvX-V,)
-pMC v(-2)(V-V
)
+p~Cv A(V -V
)
(4.29)
From
terms
the
related
obtained - = Vv ,VQ.
to
energy
the
means
this
Physically
equations
we
that energy is
transferred only by convection.
For
a >>V1
simplicity
.
we
assume
that av
>> VV
v
Then, we can neglect
a L'2(p-V)
and
vav
(-Vv )
These terms are related to the sonic velocities
through
and
liquid
and
vapor.
Now
we
get
the
transferred
simplified
- m- will
11 111
IN
16--
82
determinant form
"avP v
P (U-Vv)
*0
Det
(LPv+PCV) (-VV)
.PLc(x-2)(v4.vl)
- (aLPL+pLCv)(U-V )
+pLC I (V -v L)
(4.30)
From Eq (4.30) we obtain an algebraic equation of the form
aq2 + bq + c * 0,
where
a
a 2 PV + va1pP
b
C(y(2-)
In
order
+y
+ 2avPL Vr
CE
v tLL+ a iv(1 )J]Vr
q =
-
Vr
for
(4.31)
2
VV
VV - V1
Eq
(4.31)
to
have
real
and
distinct
83
characteristics, the following must be satisfied;
b 2 -4ac
= [y2(X2+4(1-)aQ) + 4y(l-X) a v a 2P
-4va a3 P
(4.32)
V >0
]
)
Now, let us consider special cases;
i) There is no virtual mass; C, = 0 and hence Y = 0
Then,
b
2
4ac = -4av
3
O PvP
the system with no virtual mass always has two
Therefore,
complex characteristics except in the single phase region
(a
=
0 or a
ii)
= 0)
If a *0,
then b 2 -4ac
= y2 (X-2) 2
Therefore, if Cv # 0 and X # 2, then
lir
a v0
iii)
(b 2 - 4ac) > 0
If at +0,
then b 2 -4ac
V
Therefore, if C,
= 72 2
0 and X # 0, then
lim (b 2 -4ac) > 0
a k0
iv)
If X=1, then b2
4ac =
2 -4
a a vP P.
Therefore,
if C
ifCv
>
, then b2 -4ac > 0
-
I
mll",
84
v)
0 < X <1,
If
2
then
22
C 2 [ 2 +4(1-X)Q] +4a
P
--
a 2 y(1
v
) (1
V)
.
Therefore, if (Cv ) > 4
4
, then b 2
This relationship is drawn in Fig. 4.2.
vi)
If
1 < X <2,
then-
3 v
2
2
2
Cv (X-2) 2 - 4a v 2 Cv > 4a v a 3 Pv
Therefore, if [C v (X-2)] 2 > 4a2vR
[C+a
v
Pv
P
.
4ac >0.
85
0.8
-
: C v X=
Real Characteristic
Region
4act v7
3-
p
-v
t = 2000
Pv
2000
0.6
C
= 0.02
0.4
S= 0.05
0.2
X = 0.1
0
0.5
Fig. 4.2:
Characteristics Map with Real
Characteristics for Given Values
of X
-
I
I I
~L-=-CIL-
86
The effect of virtual mass on stability can
illustrated
by
be
simply
using the characteristics analysis.
If the
mass exchange rate, standard drag force, wall friction,
the
and
energy equations are neglected in order to clearly know
the effect of virtual mass on
stability,
D
in
Eq
(4.14)
reduced to zero.
As
shown u in
characteristics
Section
V
in
4.3,
if
D
0,
-
the
Eq (4.9) become equal to w/k in Eq
(4.14)
w = k P
(4.34)
Then, when the conditions for all real
satisfied,
characteristics
we have all real values, w.
As all real values,
w satisfy the stability condition, Im(w)
w,
the
stability
are
<
0 for all
values
condition is equivalent to the condition
for real and distinctive characteristics, Eq (4.32).
4.5 Numerical Stability
Here we examine the numerical stability of the equation system
including virtual mass.
similar
We form the difference scheme in a
way to the THERMIT (17).
the stability analysis, the
drag
force,
and
wall
mass
friction
For the same reason as in
rate,
standard
neglected.
Also the
exchange
are
energy equations are dropped because they only represent
convective
properties
characteristic analysis
as
we
found
in
the
preceeding
87
vapor mass
n+1
t
z
+(-,n
p
V
v v v )J+1/2
-( (a n
V
v vv
0
-1/2
(4.35)
liquid mass
Sn+1
_.n
(=
(aP.
"',LP
-
At
'+
1 .n n n++l
n n+l
L(aP V
j+1/2" (a'
V )jl/2
Az
£ 1 R
e+/1-/
] = 0
(4.36)
momentum equations
v v
+ (a pV
L L-
n
v )j+1f2
+
At
AV
n)
(n+l
Vn+ n
nv
+1/2
n L
n+
+(vPvv
at
AV
(P~+i P )n+
=0
/ +-) + z
Az
Az j+1/2
v [a0p + PC ]
n (n+1
V
+,/2
a
n
_n(
j+1/2
(4.37)
Vn+1n)+/2
[ap +
AVv
+ {<c(,aLvp
p Cv()-2)V>
+ v ,LCv()-1)Vv- aP
AV
-
<[aevj
vpCv (1-))]V 1+ av j Cv)V > -}
n
0
(4.38
"
-IM-1
11N1
II
I,
I
lI l l llll
m
, ,
88
The above numerical scheme can be modified as follows:
vapor mass
n+I
v
n+I
vj
(an*
(
n
v
n
n
- avd-I
n
+ vj-1
(n+.
j
)+
n
vJ-l
-
n
v
+t
Z
vn+l
vJ-1
n
Pn Vn+1
Vj
vj+l/2
n
vj" v3-1
vJ+1/2
/
n
n+1
Vvj .1/2 )]
(vJ+1/2
0
(4.39)
0
lituid mass
Same as Eq. (4.31) with %v replaced by ac, subscript v by I.
(4.40)
momentum equations
+A.
(
Pv)
+ (Pj+1/2
[tlp
+
+{<[.aPv+
- <[a
a0
(VL.
- [ aj +P Cv ]
v-
(Vn+ -n
PCv,(X-I)]V v - pLCv(.-2)V >
+ PLCV(lX)]V
o-
(4.41)
0
Pj-1/2)n]
P Cn (Vn+1 -
n+ (.LPtVL)n
V
Vn
+ PCvXVv > A (V-
A0
(Vvj - VvJ-1)
Vj>1 >n
(4.42)
(4.42)
89
Treating the coefficients as constants and using Von Neumann
local stability
jU=
analysis,
3
nexp(ikjAz)
,
we
obtain
the determinant form for a nontrivial solution.
pV (C-1+ V ) a
v(U-1+ VV)
vpvcq
0
Det
- (CL PLp Cv) ( -I)
(aQv+P Cv)(;-1)
+(p (Pv+PC2)Vt-)Ivv,-Vv
-pICv(X-2)
VL
+PLCV XV
}
4
(4.43)
where
= At V(1l-exp(-ikdz))
q
t
sin (-)
AZ
2
* 21 -
k = nAz : nzl'
:J: the number of axial mesh
For simplicity we assume in the same way as in the previous analysis
that a >>Vv , a z>>V
Then the determinant reduces to the 3 x3 determinant
pv (-l + Vv )
avPv q
Det -p(;-1 + v)
0
=0
aQ pZq
)(
(aP +p-C )(v-1+v)
-(zpz+pC
pC V(X-2)(VV-Vz)
-pCv X(V
v- v )
-1+V)
(4.44)
90
As we notice
that
rq
appears
as
a
common
coefficient,
resolving the determinant we obtain the modified determinant
-avp v
PV(U-Vv )
0
pL( )
0
1Pf
Det
(X-2)(VV)
(4.45)
= 0
+-pC (V.- V )
+PCv
where I -(€-1)
The above determinant, Eq (4.45) is in the exact
as Eq (4.30).
same
form
Therefore we can use Eq (4.31) as follows:
)
2
where a = a, pv
+
acvIIPpt + P C
b a pLCv(2-X) + 2Cat a
c
a vP~+
aL vptCv(1
X)-
q a 1 Vv
Vr a VV M Vz
__
_L_ _____
_
91
For the numerical stability the absolute value of the growth
factor,
IlI
,must be less than 1. From Eq (4.46)
-b± b/ ac ~
2a
Vr
q
(4.47)
qb 2 - 4 a c ) (
(Vv
2a
(-b
{-
+
= -
-1 = -
where
Assuming that
= -(V
j
)
,
(4.48)
+q)
1 and
X
, we have
u=-(5-1)
From Eq(4.47) and the relationship
(4.49)
C2 >>4 a 3 p/p
V
v z v 2
,
each root in
Eq
(4.48) can be simplified separatively as follows:
i)
For the root with + b2-4ac in Eq (4.48),
0 < (a
V+ (avaQ
+p C )
we have
V
)
< (a2pv+ PCv+ avaPt)
(4.50)
=-P + p exp(-ikAz) ,
2At
where
p
(a 2p+ PQCV)
2A
V
aP
v
At
t- Vv+ ((va P
V
A- V
Z z
+ aspe+ p C
V
Fig (4.3) shows the stability region
(4.50).
For
the
from
Eq
stability p in Eq (4.50)
numerical
must satisfy the condition,
derived
<p < 1
.
From the above
stability condition we have
(PV
+ PC)V
+
2 + v t°
ai1
.tVo
t
(v ap P)v
+
PCv
PC
Lv
(4.51)
IIiiI 111iiil ll~
ll lllYYY
ilMMMIYIYI
i I viin
92
Z Plane
y
stability region
c= c(xy)
Fig. 4.3:
Stability Diagram of Eq. (4.50)
93
Ifiv
at
<
and
1
At
ii)
always
met.
(4.48),
we have
For
+
i2v ACv
b -4ac
in Eq
t
A )
aPC
avaP
is
condition
above
the root with -
+ (PCv +
(aP) A
(4.52)
A
-r + r exp (-ikAz) ,
a
(a2
where r =
For the
the
,
zV<1
) tV + (PC + aaOPL) A
aP +
numerical
C+ avaLPL
stability,
satisfy the condition, O<r< 1.
r
Eq
in
(4.52)
must
The above condition is
equivalent to the followinb inequality;
)
0 < (a
V+(C +
< ( 2p I
+ VP+
p)
V LL
I CaQ
(4.53)
(.3
V<1 ,the
t v < 1 and
Az
Az v
is always met.
Also, if
Therefore,
and Cv>4a
be
can
it
P
,
.the
that,
concluded
finite
above
difference
inequality
if X =
1
scheme
is
restricted only by the convective stability cond itions.
4.6 Application of the Second Law of Thermodynamics
to Drew's Virtual Mass
--
I. W11111
1111111111411
IN MIYN
IINIM
'1
WfiI
94
Drew et al. (38) derived a
virtual
mass
by
applying
constitutive
the
equation
general principles such as
coordinate invariance and objectivity to the two-phase
situation.
It
first
order
combination
derivative
of
material
acceleration
However, it was found that
terms.
may
their neglect of the Second Law of Thermodynamics
important and universal law.
to violation of this
Drew
Lahey
and
meaningful
impossible.
(39)
tried
quantitative
implications,
flow
found that their virtual mass model is
was
consistent with any
for
Although
general approach to give a
a
result
complexities
lead
the
for
second
law
involved in the problem made it
Here, a simple procedure is suggested to obtain
a quantitative range of the viable form of the virtual
mass
term.
Ishii
has
(18)
shown
that
the
rate
average
of
production of entropy at the interface is given by,
AS = rTvhvi ('
z
i
2
qki
L
a
k=1
T
-V
+ Fi. [V
where
at
AS
the
rate, Tk
)
v
vi
T
T
T
Tk
T
i
(4.54)
+ (V i-V i)]
is the volumetric rate of production
interface,
is
the
r
of
entropy
is the volumetric vapor generation
average
temperature,
hki
is
the
95
interfacial
enthalpy
of
energy of phase-k, qki
phase-k,
is the
gk
is the
interfacial
Gibbs free
heat
flux
to
phase-k,
-1 is the interfacial area density, Vk is the
a
average velocity of phase-k, and Fi is the volumetric interfacial
drag force on the liquid phase.
this thesis isothermal flow without phase
slip
condition
at
the
interface
are
For the purpose of
change
and
assumed.
a
no
Then, we
obtain the result that the volumetric interfacial mechanical
dissipation rate is non-negative:
AS = F.*(V
-v
1
V
) >- 0
(4.55)
The physical meaning of Eq(4.54)
interfacial
drag
force
liquid in order to satisfy
entropy
production.
is
if Vv >V. ,the
that,
should be transfered from vapor to
the
Taking
Drew's virtual mass force for
condition
the
of
standard
volumetric
non-negative
drag force and
interfacial
drag
force, we obtain
F.
= F +F
1
s
(4.56)
v
Refer to Eqs (4.18) and (4.19) for F
S
the
coefficient
of
virtual
and F
V
.
Here
Cv
is
mass, and X is the arbitrary
coefficient but must be given in order to satisfy Eq (4.55).
Fi
is always positive, that is, a dissipation term.
For simplicity we assume steady state,
and incompressible two-phase flow.
one-dimensional
MI
ll
mlllllnMAi i
M IilliliMMM
I
Iinli lHIl
ithdi
Continuity equation
aV
V
aao
(4.57)
o v az
az
az
£
(4.58)
z
Momentum equation
aV
(4.59)
=
v-1PV
+
-a
(4.60)
a Pg - Fw
F
Q and Eq (4.60) by av
Multiplying Eq (4.59) by
, and then
subtracting each other, we get the equation
£1 z£V
S
v) +
-P
p
F. = a (pV
vvV -z3 X) +
aV
z
)g+a F
(p -p
vt £
.va
F
v
v w
(4.61)
Using the continuity equations we obtain
PV
S
2
2
pgV
a(
V
First we take the case,
.V-
V
ii)
a
+ ---- +
(p
-
If-
F.12.
always satisfies the Second Law because (
If
z<
az
negative.
(4.62)
>V >0
i)
z-
F
- P )g +
> 0,
2 P V2
, it is not clear whether F.
2
is
Oa.z
+
a.
non-negative
Therefore, we have to assure that F.
aa
)> 0
or
> 0 in the
v > 0. Being used Eq (4.59) and
> 0 and
case of V > V
az
v
i
(4.60), Eq (4.56) becomes the equation
pC
Fi = K(Vv - V)+ (avv
-
[V VQ (a +a
+a V ) ]3z
vz
Sv
> 0
)+(X-)
(V - V)
(4.63)
97
In order to satisfy the inequality, Eq
0, C,
>
V v > V,
>
0 and
in case
(4.63)
of
v
-* < 0, we simply obtain the
inequality,
+a V)
V V (a v+a z)+(X-l) (Vv - V) (a Vv
>0
(4.64)
in case
Notice that we obtain the same result as Eq (4.63)
of
V >V
Vv
If
> 0 and C > 0.
=
2
V.,
Rv then Vv
(av +a)
>
0
V
If
a v +0,
if
O-*1, X >
X>- 1
Vv
Vv
In a general case
> 1 -
where
V
s is a slip ratio, V
Fig 4.4 shows that the range of
satisfy
AS > 0.
THERMIT-6S.
Law, ?
(4.65)
s
(s-1) (C s + Cv)
must
on
X based
Conservatively,
X is
(4.65)
Eq
as
set
In conclusion, in order to satisfy
be
specified
within
the
the
range
inequality, Eq (4.65).
4.7 Numerical Results and Discussion
to
1 in the
Second
of
the
SI
0 YII
lN
-l
l0
98
1.41
1.2
1
0.8
0.6
0
0.2
Fig. 4.4:
0.4
0.6
0.8
1.0
Restriction on X to Insure that
AS > 0 for Interfacial Momentum
Exchange
99
An one-dimensional numerical simulation is performed by
the THERMIT-6S
model.
For
code
the
containing
numerical
the
stated
virtual
mass
simulation the geometry of the
THORS test (40) and the same axial mesh size, 4" are used in
order to see the effects of virtual mass
stability
in
the
real
geometry.
heat transfer, but wall friction
on
the
numerical
No mass exchange and no
and
interfacial
friction
are included in the computation.
Initially, a constant pressure difference
inlet
and
outlet
is applied
mixture of sodium with
channel.
constant
Physically,
velocities are
nearly
to
void
it may
linealy
the
be
between
the
stagnant two-phase
fraction
expected
accelerated
along
that
with
the
vapor
time
as
Hancox et al.'s results (41) showed.
Their studies involved the acceleration from
rest
of a stratified steam-water mixture with a constant pressure
difference
was
done
interfacial
between
with
the inlet and outlet.
wall
mass,
heat
and
Their simulation
momentum
transfer,
heat, and momentum transfer set to zero.
The steam velocities were obtained with 8, 16, 32,
128
no
equal mesh sizes.
indication
accelerated
of
vapor
space increments
and
and
The steam velocity transients showed
solution
instability
velocities with time.
were
64,
used,
the
but
linearly
However, when 128
oscillation
of
rapidly
increasing amplitude developed.
The present analysis illustrates
that
in
some
cases
vapor velocities are initially in growing modes but they are
100
damped soon as shown in Figs 4.5 and 4.6.
The reason for this
behavior of vapor velocities can be explained as follows: while
the initial small interfacial friction force can not stabilize
a growing instability, later the interfacial friction force,
which increases with time, becomes enough to damp growing
oscillations.
These results are consistent with Stewart's
conclusion (42) that for a given mesh size, a minimum interfacial force is required to stabilize growing modes.
Bryce (28) noticed the same type of damping oscillations
when the LOCA code RELAP-UK was used for the numerical
simulation of Edward's simple blowdown experiment.
Note that,
as interfacial momentum transfer in Hancox et al.'s studies
was set to zero, steam velocities did not damp but grew
with time unlike the present results, in which interfacial
momentum is allowed to be transferred between vapor and liquid.
Even though the present stability criterion was derived
for negligible compressibility, it is noted in Figs 4.5 and
4.6 that the derived criterion predicts well the cut-off
stability limits.
Ardon ('43) revealed the same result,
that the boundary of the unstable region is virtually
unaltered by compressibility.
With the much lower coefficient of interfacial force,
the same simulation is performed.
In this simulation
oscillations do not occur but higher acceleration of both
vapor and liquid with lower slip ratios is noticed.
Considering that vapor reaches sonic velocities, this
101
60
a a0.8
Cv =0
C = 10 - 3
v
C = 5 x 10 - 3
40
-C
v
C
-
= 7 x 10
3
= (4a3
Pv
-3
= 5x10
20
I
I
%
I
0
tI
-
I
2
4
6
8
Time (ms)
Fig.
d
4.5:
Vapor Velocity Transients with a= 0.8
10
--------
-
102
60
= 0.9
Cv
0
Cv = 10
- 3
2x 10- 3
Cv
Cv = 5 x 10 -33
40
Cv= (4ava 3
)
-) 2 x 10 -3
u
20
I
'
0
2
I4
I
6
8
time (ms)
Fig. 4.6:
Vapor Velocity Transients with a =0.9
10
103
phenomena
may
subsonic
flows
is
instability
the
wavelength
long
a
is driven by-pressure
which
instability,
Kelvin-Helmholtz
At
explained by the following argument.
be
reduction at a wave crest caused by the constriction of
supersonic
at
However,
flow.
vapor
velocities,
zero
the
with
Therefore,
growth.
area
opposes
reduction causes a recovery in vapor pressure which
wave
the
virtual
mass
coefficient, the flow can be stabilized.
on virtual mass coefficient ; higher C
velocities,
as
expected.
depend
velocities
Fig 4.5 and Fig 4.6 show that vapor
produces lower vapor
That indicates that virtual mass
has a considerable physical effect in the sodium flow due to
the high acceleration of the flow resulting
wavy
prediction
critical flow for the annular flow with
of
with
or
interfaces
high
the
Including virtual mass in
density ratio of liquid to vapor.
the
from
droplets,
Henry
(44)
found
a
considerable physical effect for virtual mass on the critical
His models generally described the data well for the
flow.
high void fraction flow regime.
4.8 Conclusions
The results from this
can
Chapter
be
summarized
as
follows;
1.
If Cv >4a va
difference
scheme
Pv/p
,
the
is
only
restricted
semi-implicit
by
finite
the convective
stability conditions.
2.
Numerical
simulation
showed
that
the
above
stated
___~
/1
1
104
numerical stability predicts well the onset
for
inequality
of instability.
A growing instability may be
friction
interfacial
is not
enough
possible,
if
to stabilize growing
This growing instability was always damped with time
modes.
due to an increase in interfacial friction.
At
supersonic
of vapor, all of wave modes can be stabilized by
velocities
a recovery in vapor pressure from the
constriction
of
the-
vapor flow.
that
3. It is noticed
virtual
mass
has
a
considerable
effect in the sodium flow due to high acceleration
physical
resulting from high density ratio.
4. It is recommended that virtual mass be included
model
two-fluid
in the
it functions as a stabilizer with
because
real physical effects, and we can not assure that sufficient
always
interfacial force for stability
virtual
including
mass
exists.
Moreover,
in THERMIT-6S neither changes the
numerical scheme nor requires large coding and computational
effort.
and
5.
Future effort must be focussed on the evaluation of Cv
to recommend specific values in terms of flow parameters.
For
satisfied,
inequality,
-V
second
the
x must
*>-
-
be
law
of
specified
s+
(s-l) (as+
)
thermodynamics
within
to
be
the range of the
105
Chapter 5
CONSTITUTIVE RELATIONS
While the two fluid model provides more applicability
and flexibility in the analysis of two-phase flow than other
models do, its accuracy strongly depends on its constitutive
As shown in Table 5.1, the constitutive equations
equations.
mass
of THERMIT-6S can be subdivided into seven categories:
exchange, energy exchange, momentum exchange, and wall
friction and wall heat transfer, for each phase.
5.1
Constitutive Relations in Single Phase Region
The THERMIT-6S code requires two kinds of constitutive
equations in the single phase region:
and wall heat transfer coefficient.
wall friction factor
(Refer to Table 5.1)
The wall friction factor can be subdivided into the axial wall
friction factor and transverse wall friction factor.
The
models adopted here are those of Wilson and Kazimi (45).
The
transverse friction factor holds for bare rod geometry only
while that of the axial wall friction is for wire-wrapped
rod bundles.
5.1.1
5.1.1.1
Wall Friction in Single Phase Region
Axial Wall Friction in Liquid Region
In order to solve the momentum equation for the axial
direction of fluid flow it is necessary to calculate the
pressure drop due to friction.
This change in pressure is
defined in terms of the friction factor, f as:
APz
z
f
A
D
2Vz
2
(5.1.1-1)
*
TABLE 5.1
Constitutive Relations of THERMIT-6S
Wall Friction
Single Phase
Liquid Region
Pre-Dryout
Region
(0<a<0.957)
X
)
Post-Dryout
Region
(0.957<a<1)
Single Phase
Vapor Region
X
Force F
Wall Heat
X
X
Transfer Qw
Interfacial
Mass Exchange r
0
Interfacial Energy
Exchange Qi
0
Interfacial
Momentum
Exchange F i
X:
0:
G:
previous work
simple extension of present work
present work
0
107
where Apz =pressure drop in the axial direction
Az = node length in the axial direction
De = equivalent diameter for the flow
v z = velocity in the axial direction
of
Then, the rate
volume, Fw
wall
momentum
is expressed as
(5.1.1-2)
Fw = Apz/Az
For the axial wall friction factor, f in
liquid
unit
per
transfer
the
single
phase
region, THERMIT-6S uses a combination of the laminar
of
correlation
Novendstern's
Markely
and
Engel
(46)
turbulent correlation (47).
and
simplified
In laminar flow,
Markley et al. suggested the following correlation.
f=L
(PD
32
r--
1
Re
for Re < 400
(5.1.1-3)
where H=wire wrap lead length
P/D= pitch to diameter ratio
In turbulent flow, Novendstern's correlation simplified
by Wilson (45) is used in THE,IT-6S:
fT
T
M
0.316
Re 0.25
, for 2600 < Re < 200,000
(5.1.1-4)
where
1.034
0.124
(I/D
In the transition
region,
29.7(P/D) 6.94 Re0
2.239
(H/D)
85
(5.1.1-5)
the
following
relationship
is
108
employed without any verification:
fTt
7
where
=
T1-W
+ fL
fT
(5.1.1-6)
, for 400 < Re < 2600
Re - 400
2200
5.1.1.2 Axial Wall Friction in Vapor Region
is
The Markley-Novendstern correlation described above
implemented
with vapor properties being substituted for the
liquid properties.
5.1.1.3 Transverse Wall Friction in Liquid Region
The momentum equations in the x and y
require
a
friction
pressure drop term.
directions
also
In this case, the
geometry is different from the axial direction:
flow along rods, we have flow across rods.
instead
of
The single phase
friction factor in the transverse direction is defined by:
t
Pt PV1vm
DV
2
(5.1.1-7)
max
where
Apt
=
frictional
pressure
drop
in
the
transverse
direction
At= length in the transverse direction
Vmax= transverse velocity at the point of maximum flow
constriction between rods
Dv = volumetrically
defined,
transverse
diameter
4 x volume of fluid in tube bank
Exposed surface area of tubes
hydraulic
109
The correlation employed in THERMIT-6S was developed by
Gunter and Shaw
the
transverse
the single phase liquid region.
in
factor
for
(48),
define both a laminar and a turbulent
factor,
friction
f
for
occurring
the
at a
Thus,
Reynolds number of 202.5.
, for Re<202.5
180/Re
(5.1.1-8)
f =
1.92/Re0.
Re =
where
friction
Gunter and Shaw
correlation
transition
the
with
wall
145
for Re > 202.5
P]VmaxDv
Note that this correlation was obtained, based on bare-rod data.
5.1.1.4 Transverse Wall Friction in Vapor Region
The Gunter and Shaw correlation described above is used
with vapor
properties
being
substituted
for
the
liquid
properties.
5.1.2 Wall Heat Transfer in Single Phase Region
In order to solve
specify
the
energy
equation,
we
the heat flux from the wall to the fluid.
need
to
The wall
heat flux is defined as:
qww
=
h(T w - Ts)
s
(5.1.2-1)
"lIInIM
nnll
I
110
Then, the rate of wall heat transfer per unit volume, QOw
is
expressed as
Q
= q./Lw
(5.1.2-2)
where
- is the wall area density defined as the ratio
w
the wall surface area to the fluid volume.
of
The wall heat transfer coefficient, h can be obtained by
an
empirical formulation.
5.1.2.1 Wall Heat Transfer in Liquid Region
The wall heat transfer coefficient in the single
liquid
region
is
recommendation (49).
based
These
on
and
Kazimi
correlations
are
phase
Carelli's
selected
reasonably conservative in the range specified.
as
Recommended
correlations for bare-rod bundle heat transfer are:
i)
P
For 1.15 < -<
1.3;
DNu=4.0+0.33
()3.8
10 < Pe < 5000,
(
O0.86+
0.16 (E)5.0
(5.1.2-3)
ii)
P
For 1.05 < D
1.15;
150 < Pe < 1000,
Nu=Pe 0 .3[-16.15+24.96 (
iii)
8.55)
2];
(5.1.2-4)
For Pe < 150,
Nu=4.50[-16.15+ 24.96
L
.
(E)
- 8.55
D
()
D)D
(5.1.2-5)
111
5.1.2.2 Wall Heat Transfer in Vapor Region
As the liquid metal in the single
its
high
thermal
single phase
liquid
phase
region
loses
conductivity and low Nusselt number, the
correlation
can
not
be
used.
The
Dittus-Boelter correlation is recommmended:
Nu = 0.023 Re.8
Pr04
V
V
(5.1.2-6)
Note that this correlation was obtained, based on circulartube data.
112
5.2 Constitutive Relations in Two-Phase Pre-Dryout Region
The fundamental nature of the
of
, i.e. ,flow regimes, makes the analysis of
flow
two-phase
structures
internal
physical
two-phase flow more complicated from numerical and
points of view.
Here we shall use a simplified flow regime based on the
following assumptions:
1. The exchange mechanism is dominantly controlled
by
local phenomena.
2. While vapor appears in the form of a
bubble
large
remain
on
the
cladding and hence an annular flow is dominant
on
the
a subassembly,
within
liquid
films
scale of a subchannel.
important.
Numerically, use of a simple flow regime is very
If several flow regimes are implemented in the code, abrupt
equations
makes
often
to
due
properties
change of flow
different
constitutive
it difficult to obtain convergence
with reasonable time step sizes.
The
equations
for
greatest
lies
two-phase
discrepancies
obstacle
to
obtaining
in the limited experimental data available
flow
between
of
sodium.
Moreover,
be
exist
there
data of different authors.
main reasons of these discrepancies will
constitutive
constitutive
Here the
clarified
relations for two-phase flow of sodium will be
derived within our present capabilities, based on the
assumptions.
and
above
113
5.2.1 Axial Pressure Drop Coefficient in Two-Phase Flow
By
several
reviewing
discrepancies
two-phase
in the
drop
pressure
data
between
of
frictional
on
publications
of liquid metals,
flow
authors
different
become
evident. It is also found that acceleration loss by droplets
has
a
effect on the hydraulic models, due to
considerable
the high density ratio of the two fluids and
ratio.
the
high-slip
The objective of the following investigation of the
friction pressure drop multiplier is to identify the primary
reasons of these discrepancies.
5.2.1.1 Literature Review
important
The
drop
pressure
questions
in
the
area
of
two-phase
of liquid metals have often
characteristics
been stated as
1.
Can the two-phase pressure drop of liquid metals be
satisfactorily predicted by the correlations now in use
for ordinary fluids?
2.
If not,
what
or
modified
new
correlations
are
required?
3.
How well can friction
for
adiabatic
two-phase
pressure
flow
drop
correlations
be used to predict the
two-phase pressure drop for heated channels?
To
answer
the
above
question,
the
Lockhart-Martinelli
-
--
---
YI
114
correlation. (50) is chosen to represent non-metalic fluids.
Several experiments on liquid-metal, two-phase pressure drop
are cited in Table 5.2.
The
low-quality
investigated
by
Lurie
sodium
and
the
data
drop
pressure
(51).
of
sodium
comparison between these
A
potassium
data
obtained
at
University of Michigan showed excellent agreement.
range
the
the
pressure
Over the
dependencies
parameter,
between
discerned
Xtt .
It
was
also
that the high scatter in the low-quality and high
region, as seen in Fig
variations
be
could
drop ratio was plotted as a function of
Lockhart-Martinelli
concluded
Xtt
the
of flow rates and pressure covered in the experiment,
no additional parametric
when
was
and
the
errors.
5.1,
is
However,
due
most
Lockhart-Martinelli
to
of
experimental
the data fall
predictions
for
viscous-viscous and turbulent-turbulent flow.
The pressure drop data of Fauske and Grolmes (52)
obtained
for
forced
tube under both the
convection
subsonic
and
were
flashing sodium flow in a
sonic
flow
conditions.
Friction factor data reduced from experiments under subsonic
conditions
slightly
are
below
displayed
the
in Fig 5.1.
Lockhart-Martinelli
The data fall
correlation
for
turbulent-turbulent flow.
Russian data (53) with low-mass flux of potassium under
adiabatic conditions are shown in Fig 5.1.
good
agreement
Results
are
in
with the Lockhart-Martinelli correlation in
the high-quality region.
TABLE 5.2
Main Parameters of Sodium Boiling Experiments
Range of Parameters
Classification
Fuid
Geometry*
Condition
Lurie, 1965 [51]
Na
S
Fauske, 1970 [52]
Na
Quality
Pressure
Mass 2 Flux
,
A
0.110.7
160q850
14
S
F
0.35%0.55
640%1815
0.92"2.5
K
S
A
1ll.5
1052177
1.5q88
K
S
A
0.037%1.075 110 763
K
S
A
0.95^l.16
Zeigarnick, 1980 [57]
Na
S
D
Chen, 1970 [581
K
S
Kaiser, 1974
[59]
Na
Kaiser, 1977
[60]
Na
Author
[53]
Alad'yev, 1969
Smith, 1966
[54]
[55]
Baroczy, 1968
*S = Single tube
78 = 7 pin bundle
**A - Adiabatic condition
D - Diabatic condition
F - Flashing condition
Bar
kg/m
sec
%
%37.84
165"608
1%9.4
1-2
150x400
t45
A
0.62.9
2001330
131
S
A
0.5^1.7
743345
\86
1 2.04
7881523
"35
7B
A
0.961.73
231233
0.2"87
116
10
-
00
101
100
10"
10-
Xtt
a
*
x
-
: Potassium data from Ref.
: Sodium data from Ref. 52
: Data of Boroczy
5ss5
---Data range of Lurie (sii
53
: Lockhart-Martinelli correlation
---- : Smith correlation ts41
Fig. 5.1:
b v
Experimental Two-Phase Friction Pressure
Drop with Low Mass Flux or Low Quality Flow
117
Smith et al. (54) measured the
in the
drop
pressure
horizontal two-phase flow of potassium. They estimated that
exceed
never
losses
acceleration
This
3%.
evaluation
drops
demonstrated that the experimental potassium pressure
in
a horizontal
correlation
Lockhart-Martinelli
frictional.
essentially
were
duct
The
substantially
predicts
greater potassium'pressure gradients than shown by the data.
in terms of the
evaluated
are
results
If their
Lockhart-Martinelli multiplier,
an
obtain
we
multiplier,
In Fig
between
approximate
02
IS
2
2
i,S
Z£,L
5.1
02 L , and parameter Xtt
for Smith's
relationship
this
the
Smith
(11.58
0.444
1
1+43.5 X1.1
tt
relationship shows
the
that
deviation
correlation and the Lockhart-Martinelli
But,
correlation in the high-quality region is small.
quality
As
decreases, the deviation becomes larger and larger.
expected,
calibration
closed
as
the
decreasing
accuracy
with
correspondence
influence
increasing
between
the
of
quality
data
heat
loss
results in
and
the
Lockhart-Martinelli correlation at high quality.
Pressure drop
data
for
two-phase
potassium
flowing
through a straight horizontal tube were presented by Baroczy
(55).
Ref. 55 demonstrated that his data were better predicted
by the Lockhart-Martinelli correlation than that by the Baroczy
correlation (56).
As Fig 5.1 shows, the data are typically
lower than the Lockhart-Martinelli correlation.
r
II I
III
- E~----~ii~-
118
As
the
two-phase multiplier,
for total flow, the
flow
was
two-phase
2to in Ref 55, was evaluated
multiplier,
2
*to
to by (l-x)
obtained by dividing
18
for
liquid
implying a
0.2
(f-c/Re
) was
Blausius-type single-phase friction factor
.,
assumed. We next discuss diabatic data in the literature.
Most recently Zeigarnick and
data
on
the
pressure
drop
Litvinov
(57)
presented
in sodium with heat addition.
They showed that
the
acceleration
entrained droplets can be neglected in the
of
range investigated.
experimental
for
data
adiabatic
experimental
In
Fig
drop
5.2
associated
a
comparison
non-metallic
flow
of
the
their
showed
that
the
lie significantly below the correlation.
The average discrepancy is about -35%.
this
with
with the Lockhart-Martinelli correlation
and
data
pressure
They explained
that
is associated with a "push aside" effect of the vapor
flow from the
interface
and
the
change
of
liquid
film
characteristics.
Chen and Kalish (58) reported quite
from
an
results
experimental -investigation of two-phase pressure
drop for potassium without
fairly
different
vaporization.
Their
data
lie
above the Lockhart-Martinelli correlation (Fig 5.3).
Although they
corresponded
reported
to
the
that
all
data
from
their
annular-dispersed
flow
regime, they
neglected the momentum-acceleration effect of
the
study
droplets
in the vapor core.
Kaiser et al. (59) investigated the
pressure
drop
in
tube flow under adiabatic conditions, as well as the type of
O
I
10
\
100
10'
10-2
101
Xtt
Fig. 5.2:
Experimental Two-Phase Friction Pressure
Drop under the Diabatic Condition
1
0
I
1
t4,
10111V
-----
Chen correlation
a581
correlotion from
,
101
Ref. 59
Lockhart - Martinelli
correlat ion
Xtt
Fig. 5.3:
Experimental Two-Phase Friction Pressure Drop
with High Mass Flux and High Quality Flow
121
flow
and
critical
heat
flux
of
reported that the flow patterns of
boiling
pure
annular-dispersed flow were dominating.
sodium.
annular
They
flow
and
Most of the data in
Fig 5.3 lie fairly above even Chen's results.
Kaiser
and
Peppler
(60)
evaluated
the
two-phase
pressure drop of sodium in seven-pin bundle geometries under
adiabatic conditions.
of
conditions,
as
Experiments were done over two ranges
given in Table 5.2.
In Fig 5.4 data of
the pin bundle experiment is seen to fall slightly below the
Lockhart-Martinelli correlation.
the
results
from
displacement
by
the
a
single
factor
However,
tube
of
0.6
comparison
experiments
toward
with
shows
lower
a
values.
Although an attempt was made to explain these differences by
investigating
resistance
the
influence
coefficients
explanation
for
of
void
the
fractions
single-phase
and
of
flow,
no
was found for deviations in the experiments.
systematic failure seems unlikely as
the
same
A
measurement
principles were employed in both sets.
5.2.1.2 Effect of Entrainment on Pressure Drop
The above review indicates that data are
two
groups.
Each
group
may
pattern: "pure annular" flow
While
the
data
which
"annular-dispersed"
the
number
flow.
below the Lockhart-Martinelli
correlation, were taken in annular flow without
small
into
be characterized by the flow
or
fall
divided
or
with
a
of droplets, the data which fall fairly above
Lockhart-Martinelli
correlation,
were
taken
in
io2
10
100
I0 Fig.5.4:
XI
Xtt
00
Experimental Two-Phase Friction Pressure Drop in
Seven-Pin Bundle
101
123
drop
pressure
groups
two
the difference between the
acceleration
it can be argued that
Therefore,
flow.
annular-dispersed
of
is
droplets
by
caused
the
in the gas core,
pressure
which was neglected in the evaluation of the
drop
in annular-dispersed flow.
Entrainment effects on pressure drop can be included in
the following equation:
2
dP
dP2
f + d (
ca
f
PU f+
2
td PRU2
2
+
a P
)
(5.2.1-1)
+ (apZ +agPg)g dz
where
dP2$
=total pressure drop
dP2$f
=frictional pressure dr.op
a Zfada'g =volumetric fraction of liquid film, liquid
droplets, and
gas
in
the
flow
channel,
respectively
Uzf
=liquid film velocity
Utd
=liquid droplet velocity
UZ
=total liquid average velocity
Ug
=gas average velocity
Obviously,
and
at + ag
aZd +Of
=1
= a
Given values for G, x, and
akf , the velocities UZ
, Ukf
124
, and Ug
can be evaluated by the following equations:
U
= G(1-x)/(p
=
Uif
(5.2.1-3)
G(l-x)/(p zaf)
= G x/(ag p ),
U
where
(1 -E)
(5.2.1-2)
a)
(5.2.1-4)
E = (aQd pI U d)/(CP
(5.2.1-5)
U )
From the definition of entrainment fraction, E, we obtain
core liquid fraction,
cad , in terms of aQf
a
and E as
EUz
=
ad
We assume flow conditions
Ud= Ug .
and
x and
As both U
the
such
gas
and U
functions
are
of
"adcan be expressed in terms of
x, then
iteration
by
solved
fraction,
within
the
that
the gas and of the droplets are equal, i.e.,
of
velocities
(5.2.1-6)
kf UZd-EU
Also
E.
expressed in terms of
the
if
we
know
the
two-phase multiplier,
a
,
G,
d ' G, and
entrainment
£ , can be
aif and E.
By definition
W
w2
(5.2.1-7)
X,
where
T
=
2
z2
f
(5.2.1-8)
.2
T£ = f £
j
and where
(5.2.1-9)
2
(5.2.1-10)
= Uat,
r
is the
wall
shear
liquid flows along the channel.
stress
when
the
total
125
Eqs
Substituting
(5.2.1-8)
and
(5.2.1-9)
into
Eq
(5.2.1-7),we obtain
Ulf
( f)
2
2
(5.2.1-11)
Using the definition of the Reynolds number and assuming the
same Blausius-type relation for fQf
f
f
f rU
a U
and
f,, we obtain
0.2
(5.2.1-12)
Substituting Eq (5.2.1-12)into Eq (5.2.1-11) and
definition
of
the
entrainment
fraction, E, we obtain the
02
i
final form of
(1
2
E) 1.8
(5.2.1-13)
12
A detailed derivation
Appendix C .
of
Equation
Eq
(5.2.3-33)
is described
droplets
than in annular flow with no droplets
fraction.
liquid
the
Q
liquid
(5.2.1-12).
film
is lower
for the same void
If the single-phase pressure drop for
flow,
in
(5.2.1-13) shows that the two-phase
2
2£ , in annular flow with
multiplier,
the
using
the
total
in Eq (5.2.1-7), is replaced by that for
flow,
f f is equal
to
ft
from
Eq
Therefore, we obtain the relationship suggested
by Turner and Wallis (61):
2
1
1(2
(5.2.1-14)
if
Note that for no entrainment,
to
42
in Eq (5.2.1-13) is equal
f2 in Eq (5.2.1-14).
So far the only
unknown
variable
is the
entraiment
126
fraction, E.
For Reynolds numbers between 200 and 3000, the
amount of entrainment is a function of both vapor and liquid
flow
rate.
Steen
and
Wallis
(62)
turbulent flow above a film Reynolds
amount
of
entrainment
showed that in fully
number
of
3000,
the
and the conditions for the onset of
entrainment are largely independent of the liquid film
rate
and
depend
primarily
on
the
vapor
Unfortunately, this approach fails to take
general
velocity.
account
a
increase in the fraction of liquid entrained as the
channel diameter increases.
for
into
flow
liquid
metals,
the
As there are no data
Steen-Wallis
available
relationship can be
taken to represent the entrainment fraction, E, here and
to
provide a rough estimation of acceleration loss by droplets.
If
we
apply
this
entrainment fraction
approach
from
to
the
data
Steen
and
Wallis
of Ref 63, the
leads
to
a
negative friction pressure drop for most of the experiments.
This
implies
that
the
relationship from Steen and Wallis
overpredicts the entrainment fraction, and the
pressure* drop by the droplets.
Conversely, the entrainment
fraction can be calculated when the friction
is known.
predict
acceleration
pressure
drop
If the Lockhart-Martinelli correlation is used to
the
friction pressure drop, the results imply, for
most of the experiments, a smaller entrainment fraction than
predicted by Steen and Wallis.
The large acceleration effect
of
droplets
in
liquid
metals is primarily due to the high ratio of liquid-to-vapor
densities
and
the
high-slip
ratio.
Assuming
complete
127
a
homogenization, a gas void fraction of 0.9,
of
fraction
and
0.001,
a
density
ratio
core
of
3000
acceleration loss by droplets is about three times
vapor.
liquid
that
the
by
In conclusion, acceleration loss by droplets.should
of
not be neglected in the evaluation
frictional
pressure
drop in the annular-dispersed flow.
5.2.2 Transverse Pressure Drop Coefficient in Two-Phase Flow
data
As there are few
coefficients
transverse
for
pressure
drop
in two-phase flows, the following correlations
recommended by Wilson and Kazimi
(45)
are
in
the
liquid
and
adopted
present version.
For transverse two-phase
vapor
flow
with
both
Reynolds number in the laminar regime, the Gunter and
Shaw correlation (48) for single phase flow
is
transformed
as follows:
PL = f
where
2
At P Vmax
for Re
< 202.5
f = 180/Re£
Re
R
li
IV
vmax
If either liquid or vapor is in the
the
(5.2.2-1)
Ishihara
correlation
(64)
turbulent
regime,
is applied to the friction
MEM
-
m lI,l
128
pressure drop to yield the following equation:
2
SAt G
APT
where
2p
D
(5.2.2-2)
Ito
G = G +G v
=
f
LPZVkmax
v Pvvmax
1.92
Re 0.
14 5
Re = G Dv/I
2
8
1
to
tt
2
tt
1
2
tt
e
x
1-x
r
)
1.855
p(
v P(V
0.145
129
5.2.3 Wall Heat Transfer Coefficient in Two-Phase Flow
5.2.3.1 Literature Review
Analogies between heat and momentum transport processes
flow
have been used successfully for single-phase
(65) remarked on the similarity between the
Reynolds
since
While Reynolds assumed the entire flow field
two processes.
to
be
extended
(69)
Martinelli
velocity
successfully
on
concentrated
momentum
the
core.
turbulent
a
this
approach
liquid
in
for
metals.
concept
have
expressions
for
analogy
improved
incorporating
diffusivities in the well-known
eddy
energy
and
and
applied
of
adaptations
recent
(68)
Karman
profile to three zones-a
transfer
heat
predicting single-phase
More
zone,
buffer
laminar sublayer, a
Von
core.
turbulent
highly
universal
the
(67)
Taylor
based on two zones-a laminar zone near
analogies
the wall and a
and
(66)
Prandtl
turbulent,
highly
developed
analysis
equations for momentum flux and heat flux:
=
+
=
-
+ E p
- =(a
(
hq"
h)
-
In his falling film
velocity
(5.2.3-1)
v dV
(5.2.3-2)
dT
p dy
analysis,
Dukler
the
viscosity
used
the
form of Deissler for both laminar and buffer zones
and the form of von Karman for a turbulent
that
(70)
core.
Assuming
ratio of the eddy thermal diffusivity to the eddy
is
unity
and
the
molecular
conductivity
is
130
negligible,
a
was
profile
temperature
above assumption results in an overestimated
heat
coefficient at low values of the Prandtl number.
al.
(71)
modified
the
heat
Freon-12 and -22.'
can
be
transfer
Traviss et
Dukler film analsis to include the
Their prediction method for
effect of total pressure drop.
condensation
The
constructed.
transfer has been verified for data with
As their derivation is quite general,
extended to liquid metals.
it
Applying the Traviss et
al.'s analytical method to annular flow evaporation
without
nucleation, Bjorge (72) obtained a correlation that appeared
to
be
in
better
with water data than the Chen
agreement
correlation (73).
are
There
transfer
for
correlations
few
liquid
metals.
Chen (74) following the same
non-metalic
contribute
to
total
micro-convection
S, and
two-phase
momentum
with
following
water
the
heat
measure
transfer,
and
as
approach
heat
organic
method
his
for
two
mechanisms
transfer-the
macro-and
postulated
wall
boiling
A first attempt was made by
mechanisms.
function,
data
He
flows.
flow
for
that
He
used
of
the
suppression
the
effectiveness
of
F, which were obtained from
fluids.
He
proposed
the
modified form of the Lyon-Martinelli equation for
the evaluation of the macro-convective contribution to total
wall heat transfer:
hmac -
[6 + 0.024 (F Re0.8 )(
Pr)
D1
(5.2.3-3)
131
For the two-phase flow, he recommended the values 7, 1,
and
and
for. 6, 8, Y,
0.8
a,
respectively.
1,
The NATOF
code (16) used a modified Chen correlation:
h
hmac = F0
transfer
heat
liquid
single-phase
is the
h
where
(5.2.3-4)
,
coefficient at the same inlet mass flow.
the
During
two
last
experimental
several
decades,
investigations of heat transfer and pressure drop for sodium
boiling
data
were
results
accumulated,
As experimental
have been made.
condensation
and
emerged
discrepancies
give
that
the
Recently Zeigarnick
obtained by different authors.
and Litvinov (57) provided data
among
better
insight
the
heat
transfer
and
friction pressure loss characteristics.
The
accuracy
and
into
mechanisms
the
determining
reliability of the two-phase alkali-metal heat transfer data
depend
on
of
accuracy
the
the
determination
measured
thermocouples.
saturaticn
the
and
Zeigarnick
temperature
saturation
temperature
with
observed
that
Litvinov
measurements
in
alkali-metal,
two-phase flow with thermocouples are inherently
with
uncertainties
caused
by
significant
drops around the thermocouple location.
the
Previously,
saturation temperature and the flow stability.
investigators
of
To
associated
local pressure
overcome
this
problem they measured the saturation pressure directly, thus
determining more accurately the saturation temperature.
alkali-metal
flow,
the
occurrence
of
For
a significant wall
132
superheat is known to be probable.
boiling
flow
In
their
experiments,
stabilization over a sufficiently long period
was achieved either by drilling artificial double-reentrant,
angle-type cavities in the heating surface or
of
a
small
injection
by
amount of inert gas at the test tube entrance.
It was also found that the phase change in sodium occurs
the vapor-liquid interface without bubble
from
evaporation
This suggested that the macroscopic
generation at the wall.
contribution to the wall heat transfer in
two-phase
forced
dominant
over
nucleate
In the present work, a macroscopic
heat
transfer
is highly
in sodium
convection
boiling.
by
developed
be
will
correlation
on
the
basis
of
the
above-stated observation.
5.2.3.2 Model Development
With the aid of Fig 5.5, an approximate distribution of
shear stress in the liquid film can
steady-state
differential
momentum
be
obtained
from
the
equation in the liquid
film, assuming negligible radial velocity and incompressible
liquid flow:
1 3
r Dr (r T)rz
aP
- az + p0g
Z
(5.2.3-5)
As the right-hand side is a function of z alone, it is taken
to be constant in the r direction. Integrating Eq
from
the
boundary
wall,
R,
conditions,
to
(5.2.3-5)
r in the liquid film and using the
= T
and
R
T.
we
=
R-6
133
vapor-liquid interface
film
wall
Fig. 5.5:
Ideal Liquid Film
in the Tube
134
obtain the equation
2-1
R
rz
r
w
2-r
R2 (R 6) 2
2
2(R-6)
Assuming a thin film, R>> 6 , we
j
get
(5.2.3-6)
i
the
5-T Wv,
distribution
of
shear stress in the thin liquid film:
+
S= Tw (1--- 6+ M),
rz
where
M
= 1-Ti/Tw
Y
6
Y U/
=
6U /V
1
w/
/Pd)
UT -(go
Traviss
(5.2.3-7)
et
distribution
al.
(71)
assumed
a
linear
shear
stress
in the thin liquid film and then obtained the
same expression as Eq (5.2.3-7).
Assuming that the
analogy
von
Karman
momentum-heat
transfer
is applicable to the liquid layer, the shear stress
and heat flux are written as
p
q*.
-pL Cp
dV
(5.2.3-8)
(5.2.3-9)
(
+ ch) dy
135
Rearranging Eq (5.2.3-8) and solving for
L
r1
L
Cm
U2
(dV)
1
1
=
Vz
,
we obtain
V
(5.2.3-10)
i
dy ')
where
Em
V Zr/U
velocity
universal
The von Karman
the
for
distributions
laminar and the buffer zones are assumed to be
0<y
+
the
=
V
<30;
5<y
For
<5;
Vz
=
which
(5.2.3-12)
-3.05+5 In y
the
zone,
turbulent
distribution,
(5.2.3-11)
y
was
from
obtained
universal
Karman
von
single-phase flow
in the
data, is not directly applicable to the liquid film
the
Therefore,
flow.
two-phase
applicability
of
a
logarithmic law is assumed as
y
The
> 30;
value
of
V
a
(5.2.3-13)
= a + b In y
is determined
by
the
requirement
of
continuity between the buffer and the turbulent zones, which
yields
(5.2.3-14)
a = -3.05 + (5 -b) In 30
The only empirically determined coefficient, b, is obtained
by
applying
the
suggested
Zeigarnick and Litvinov data,
correlation to a subset of the
namely
the
data
at
q"=l.0
136
MW/m.
The results are b=6 and a=-6.45.
Now let us obtain an expression of
for each
em
zone.
For the laminar zone, we obtain
0
0<y+ <5; Cm
(5.2.3-15a)
For the buffer zone, the eddy viscosity is of the same order
of magnitude as the kinematic
viscosity
and
Tr/w
1.
Therefore,
--
5<y <30; Em=mv
(5.2.3-15b)
>>1
For the turbulent zone, assuming Cm /
and
using
Eq
(5.2.3-7).
++
+
>30; C My
Since (q/A) %(q/A)
&
[ 1M
(5.2.3-15c)
, Eq (5.2.3-9) can
be
integrated
from
the wall to the interface in the liquid layer to yield
1
h
-
Tw
T6
6
o
where
and
T
Ch
/
cm
ICp
T
at + Em) U
are
liquid
interfacial
Em=
vL
the
(5.2.3-16)
dy ,
wall
temperature,
and
the
respectively,
and
temperature
"
Substituting Eq (5.2.3-15) into Eq (5.2.3-16) and completing
I-VhNW--- ~Liil--ru-y_~l.__U
u_~---r-
137
the integration we obtain
hDh p, Cp- UT Dh
kF 2
Nu = - -=
kL
k F
where
6+ > 30;
F2 = 5Pr
S
Ey
0< 6
5 <6
where
<
+
y=
Using
5;
< 30;
1+
the
5
+ E In
(1+ 5 EPrQ)+
2M +y- 1
1+y - 2M
F2 = 6
F 2.=
(5.2.3-17)
x
+1- 60M/6
y- 1+ 60M/6
+
Prz
5 PrL +
ln 1+EPr-
-1)
in
the
lOM
E6+ Pr.
continuity
of
mass
approximating, with an error of <4%, we can obtain
film
and
6+ as an
explicit function of Re
< Re< 50; 6
=
0.7071 Rez 0 . 5
50< Re < 1125; 6
Re. >1125; 6
=
= 0.4818 Re0.585
61 4
0.133 Re 0. 7
(5.2.3-18a)
(5.2.3-18b)
(5.2.3-18c)
For the value of E in Eq (5.2.3-17) the following simplified
138
Rohsenow and Cohen correlation (75) is used
E = 27.73 Pr
(5.2.3-19)
(More recent correlations, while available, do not alter
the results significantly.)
approach
in Eq (5.2.3-7), the
*Ti /Tw
To approximate
of Wallis (62) can be applied in the following manner.
The momentum of both phases is governed by
dP
dz
4
D
(5.2.3-20)
W
the momentum of the vapor phase is governed by
dP
4/
(5.2.3-21)
z +(-D
from Eqs (5.2.3-20) and (5.2.3-21) we obtain
(5.2.3-22)
w
For computational purposes, it is necessary
the
relationship
and the
derive
to
between the bulk liquid temperature, Tm,
interfacial
liquid
temperature,
T6 .
From
the
definition of the bulk liquid temperature, we have
JAL (T-T
T .
w
Az
where A
w)
V z dAL
T
m
and dA
(5.2.3-23)
dA
are the area and the differential area to
be occupied by liquid, respectively.
For simplicity a linear temperature distribution is assumed
along
the radial direction in the liquid layer.
excellent approximation for liquid metals because
This is an
of
their
139
Therefore,
high thermal conductivity.
w 6
o
(5.2.3-24)
z
Thus, in the laminar zone we obtain
=
0< 6 < 5;
T,
2
(T w -T 66)
€T
3
w m.
(5.2.3-25a)
In the buffer zone we obtain
5< 6+ <30; V+ = -3.05 + 5 In y
Tw - Tm
In the case of
case
of
6 = 30, T
- T
- Tm = 0.613(T
in
)
the
From the above
).
T - Tm = 0.66(T - T
wis approximated in this zone as
) is approximated in this zone as
6+= 5,
result, (Tw - T m
T w - Ti
m
10.45- 2.775.6+2+ 2.56+2 In6 +
12.31- 8.05 6 + 5 6 + n 6
Tw- T 6
6+
(5.2.3-25b)
0.64(T w - T )
In the tirbulent zone, we obtain
V+
6+ >30; V+z = -6.45 + 6 In y
TW -T
6+
Tw - Tm =
30,
234.9- 4.725 6
42.27 -12.456
+
T - T = 0.5(T - T
w
m
w
).
T - T = 0.613(T - T
6
w
w
m
In the case of 6+
of 6+
6
o,
+ 36+2 n 6
+ 66+ In 6+
); in the
From
the
case
above
140
result, (Tw --Tm
) is approximated in this zone as
Tw-Tm % 0.55(Tw -TS)
(5.2.3-25c)
T, to
From Eq (5.2.3-17), and relating
the
two-phase
friction pressure drop, we obtain
\
1p
h -
1/2 D
v24
$2
)J
IEz
1/2
(5.2.3-26)
Using the Blausius-type correlation for [-(dP/dz)]f, it can
be easily recognized that for turbulent flow
0.2 G1 .8 x1.8
Substituting
rearranging
V
0.092
0.092
dPz
g
~~-
Eq
the
(5.2.3-27)
1.2
o Pv Dh
(5.2.3-27)
result,
we
Eq
into
derive
(5.2.3-26)
and
the final form of the
Nusselt number correlation as
Re0.9
Nu = 0.152 Pr
2
= 0.152 Pr
Pr
F
xtt
Re0.9
Re L
1
(5.2.3-28)
where
Pr- = C 'Et/kL
ReL = G(l-x)Dh A
x -(
.X t
) 0.9(P)
L
L x-X
0.1
IMV
141
by the Lockhart and Martinelli definition.
the
Considering the definition of
Nusselt
number
in
terms of the bulk liquid temperature, we obtain
N"b"
S
Dh
T
k
Tw- T 6
W Nu*
= Nu F 1
(5.2.3-29)
Therefore, we get the final form of the new correlation:
0.9
1
Nub = 0.152 p- Pr, Re£
2
*
(5.2.3-30)
where
i)
Re
< 50; Fl = 1.5
F 2 = 0.7071 Re 0
ii)
5
Pr £
50 < Re. < 1125;
F
1.563
F2 = 5Pr
where
+
5
-
In [1+ EPr±(
6
- - 1)],
E = 0.00375 Pet [1-exp(-0.00375 Pe,)]
585
6+ = 0.4818 Re 0 "
142
iii) Re
> 1125;
F 1 - 1.818.
F2 =
2
5PrL
+ 5(1+ 5EPr ) +
a
E
1
6
2M+YM
EY
1+y2K
0 " 60.
where
B+Y-I
M
6+
- 0.133
I
6
M1
Pr
0.133 Ret0.7614
1-/-
'
143
The
*,
parameter
in
is
(5.2.3-28)
Eqs
taken
relation between the Zeigarnick and Litvinov work
empirical
and the Lockhart and Martinelli correlation.
frictional
that
found
Litvinov
two-phase
alkali-metal,
losses
an
in
flow with heat input are less than
effect of the vapor flow from
on
interface
the
the
main
the change of liquid film characteristics.
and
relationship
empirical
pressure
Ziegarnick and
They explained this by a "push-aside"
in an adiabatic flow.
stream
the
from
between
their
results
and
The
the
Lockhart and Martinelli correlation is
0.88
L' Ltt
(5.2.3-31)
'
where
x
ltt
tt
2
xtt
tt
5.2.3.3 Comparison to Experiments
with
the
and Litvinov data and the data from Ref 76.
The
Several correlations are now to be
Zeigarnick
former experiments were
heat
parameters:
flux
made
in
the
compared
following
higher
pressure
forced
coefficient
0.1
convection
than
of
at the wall up to 1.1 MW/m.s, mass
flow rate 150 to 400 kg/m.s, flow quality up
operating
range
to
0.2 MPa.
boiling
to
0.45,
and
They reported a much
sodium
heat
transfer
the previous results as shown in Fig 5.6.
According to several experiments (e.g., Ref
77),
the
heat
transfer correlations for alkali metals in forced convection
144
%w
o
1: Sodium curve by Zeigai
o
o
W
C
0
2: Potassium pool boilin
-
data curve
_/
-12
(Ref. 77
Suggested correla'
51--A--A Chen correlation
e
e-- NATOF correlation
-
E
8-
A
0.4
0.2
0.6
Heat flux rate, q (MW/mrn
Fig. 5.6:
a-
1.0
0.8
2
)
Comparison between Data and
Correlations
145
were
results
previous
whole
the results from several correlations over the
in
and
boiling
unstable
to
subject
Also
saturation-temperature measurement.
in
uncertainties
It seems that the
boiling coincide.
pool
in
and
boiling
range
the Zeigarnick and Litvinov experiments are shown in Fig
While
5.6.
with
agreement
the data over the whole range, the Chen and
transfer
heat
lower
a
predict
correlations
NATOF
excellent
in
is
correlation
suggested
the
coefficient.
Comparison of several correlations with the data of Ref
76 is given in Fig 5.7 where
are
the
correlation
same
inlet
wall
and
the
NATOF
macroscopic contribution.
experiment
is
in
hl,
and
transfer,
Chen
microscopic
suggested
the
included only the
correlation
in
lower than the Zeigarnick and Litvinov
transfer
coefficient.
also
showed
The
the
Hoffman and
that
the
heat
coefficient in unstable boiling is much lower than
stable
single-phase
heat
data in Fig 5.7.
of
the
The heat transfer coefficient
Krakoviak (78) potassium data
that
and
Furthermore, some data are seen to drop below
single-phase .heat
transfer
h2,
While
flow.
macroscopic
heat
the
correlation
mass
both
included
contributions to
results.
parameters
heat transfer coefficient with boiling and without
the
boiling at
this
the
boiling
and
could
be
equal
to
the
transfer coefficient, as are most of the
Therefore, it can be concluded
that
most
the data of Ref 76 were obtained in the unstable boiling
regime.
While the NATOF correlation is
generally
in
good
146
10
suggested
correorrelatiation
hl
I
NATOF
Chenlation
correlation
ao
I I0.3
2
10
I00
I/Xtt
Fig. 5.7:
Comparison of Correlations
with GE Data
147
with Ref 76, the suggested correlation is in good
agreement
agreement with the high heat transfer coefficient of some of
the latter.
correlation
To evaluate the suggested correlation, the
for
E
Sensitivity
parameters
change
and
analysis
method
showed
that
for
Tri / Tw
the
were
used.
of
these
effect
the heat transfer coefficient is small.
on
in the
Ti/ Tw
Wallis
the
and
coefficient
were
separately
was
within
changed
over
10%
the
range of 0.0001 to 1.0 and 0.001 to 1i,respectively.
when
The
E
possible
I
I
r
I
L -----~-1----I;-
---------il--------~----~L
-
-
148
5.2.4 Interfacial Mass and Energy Exchange
The exchange
liquid-vapor
two-fluid
of
mass,
interfaces
model.
non-equilibrium
As
energy,
be
must
the
phenomena,
and
explicitly
interfacial
slip
momentum
across
given in the
exchange
controls
ratio, and mass transfer,
accurate and reasonable physical models are
very
important
in analysis of LMFBR transients.
A survey of the literature on models of mass and energy
exchange is given.
have
It is found that most of
models,
which
been developed for the analysis of non-metallic flows,
must be modified for application to metallic
flows.
Here,
it is assumed that mass and energy exchange at the interface
is controlled
either
by the rate at which the latent heat
for mass exchange is transported
interface,
away
from
described
by
a simple kinetic
The
latter
theory.
the
mechanism
of
because of its easy generalization.
mechanism,
of modified mixing theory, which
eddies,
is
The turbulent flux
model is taken as the basis for description
of
into
or by the rate at which vapor is supplied to the
interface, whichever is smaller.
loss
or
accounts
for
the
former
The concept
the
energy
is introduced to generalize the turbulent
flux model.
The proposed model is verified by two methods:
the
comparison
volume
in
is
of the predicted bulk liquid temperature to
analytical results, and the other is the comparison
void
one
the
bundle
estimated
from
of
the
experimental
results with that according to the suggested model.
149
5.2.4.1 Literature Review
basis
Among the models proposed to provide a
exchange
mass
rate,
for
the
best known are the'stagnant film
the
model, the penetration model, the Reynolds analogy, and
the
turbulent flux model.
(79)
theory
film
The main limitation of the stagnant
lies in the assumed simple spatial dependence of parameters;
pressure, velocity, temperature, and composition depend
the
only on the distance from the wall, and become constant when
the distance exceeds the corresponding film thickness.
The penetration model, which was
in 1935,
(80)
presents
contacting the phase
transient
which
the
fluid
elements
reservoir.
are
suggested
by
Higbie
a picture of small fluid elements
interface
for
transport processes occur.
replaced
by
fluid
fresh
during
periods
brief
Then, the
the
from
The penetration model leads us to the conclusion
that the transport flux of properties should be proportional
to
square
the
root
of
the
diffusivity
of the material
true
This has been found to be approximately
involved.
in
non-metallic flow systems.
The Reynolds
(65),
was
based
analogy,
on
Reynolds'
postulate
suggested by von Karmann (68) and established by
Chilton and Colburn (81).
It is well known
that
Reynolds'
postulate breaks down for metallic flows.
The turbulent flux model
adopted
developed
by
Spalding
(82)
a simple visualization of the Prandtl mixing-length
theory for
turbulent
motion
in the
neighborhood
of
an
150
interface.
An eddy is visualized, which originates around
the interface, travels transversely at a constant temperature
over a distance of one mixing-length, and then is mixed with
the fluid there.
However, in sodium, it is expected that
there are considerable energy losses from the eddies due to
the high conductivity of sodium.
Nevertheless, we shall use
this model as the basis for a generalized model, because of
its easy application and flexibility.
5.2.4.2
Model Development
The turbulent flux model assumes that mass exchange is
determined by the effectiveness of energy transfer by eddies.
This model uses a simple approach to describe the movement of
an eddy around the vapor-liquid interface, as shown in
Fig. 5.8.
In Fig. 5.8, the solid lines, I.G. and I.L. rep-
resent the gas side interface and the liquid side interface.
It is assumed that each eddy with bulk properties enters
each region across the B.G. or B.L. lines, which are located
at a distance of one mixing length in each phase, Rg or ZR,
from I.G. or I.L.
Denote: hbg and hbt are the bulk specific
enthalpy in the gas and liquid regions, respectively.
T
and T
are the bulk temperatures of the vapor and liquid
regions, respectively.
hmg and hm£ are the specific enthalpy transported by eddies
originated near the interface as they cross B.G. or B.L.,
respectively.
G Region (Vapor Region)
9
I
9mhig
it
Vapor-Liquid
Interface
L
g, h b t
Fig. 5.8:
Region (Liquid Region)
T
q
g-
gm, hm
Simple Representation of Mass and Heat Transfer
around the Vapor-Liquid Interface
1
1i0,
152
,
are the heat flux
and qi
and qi
and q"
,
q"
bg
bi
1
through B.G and B.L, and through I.G and I.L, respectively.
and
are the turbulent mass flux of each region,
g
and g,
gm
is the mass flux transferred from one phase to the other
across the interface.
First, let us establish the
gm
flux,
qQ
considering
the
heat
interface
' by
relationship
the
, and the mass flux,
and qig
steady
between
balance
energy
state
equation in the region surrounded by I.G and I.L.
saturation
the
Assuming
(5.2.4-1)
= gmh.g +q'! ig
g mh iL + q 1
state
the
at
and
interface
rearranging Eq (5.2.4-1), we obtain the relationship
g
where hfg
=
(5.2.4-2)
(qZ-q g)/hfg
is the latent heat of vaporization.
As the interfacial heat flux in the
usually
much smaller than q"
i 9
gas
phase,
q'
19
,
is
, a simplified expression of
Eq (5.2.4-2) may be obtained by neglecting q"
gm = q
This
simplified
evaluation of q11
(5.2.4-3)
/hfg
equation
enables
us
to
focus
on
the
.
The steady state liquid energy balance equation in
the
L region leads to the equation:
-h sk )
q!
bt +g Z (hbt -hmmZ ) + g m (hMI
"it = q"
(5.2.4-4)
153
.
the saturation temperature of the above system, T s
a
Assuming
and T
temperature
linear
the
in
distribution
can be expressed in terms of T
iQ , q11
length,
mixing
to
is the specific liquid enthalpy corresponding
where hst
s
qb" = k (T - T
where k
is the molecular conductivity of liquid.
non-metallic
For
losses
(5.2.4-5)
)/ Z
is equal to
Therefore, hm
heat
negligible
are
z
travel the distance
Then
hsz .
we
get
a
.
simple
for non-metallic flows by substituting
of qi,
relationship
they
while
eddies
from
there
flows,
Eq (5.2.4-5) in Eq (5.2.4-4):
q'
= Hi (T - Ts)
Hi
= kz /Z
(5.2.4-6)
where
and C
pZ
(5.2.4-7)
+ gzCpZ
is the specific heat of liquid
However, for metallic flows such as sodium, heat losses
from eddies require a modification of the
Deissler's
modified
mixing-length
above
treatment.
theory (83) is taken to
account for heat losses from eddies.
Fig 5.9 illustrates an eddy movement.
An eddy of fluid
originates at the surface 1, where its temperature is
It travels at the velocity v,
the
surface
2,
a distance
T
, with the temperature T',
z£
away, where T2
and T 2
to
are
~
-~-I~iIIu
""""~~'-'
--
154
,
1
T1
eddy of temperature T'
VI
,
2
Fig. 5.9:
Movement of an Eddy in a Fluid
T2
155
the temperature of the eddy and of the fluid.
As
breaks up and mixes with the fluid.
out
of
the
eddy
different from T1
.
between
surfaces
the
Based on
heat
the
above
Then the eday
is
conducted
and 2, T' is
2
picture, Deissler
1
expressed the effectiveness of energy transfer by an eddy as
E
T' -T
E =
2
(5.2.4-8)
2
For the value of E the
following
simplified
Rohsenow
and
Cohen correlation (75) is used.
E = 27.73 Pr
(5.2.4-9)
where Pr is the Prandtl number.
Now E can be expressed in terms of
parameters
defined
in Fig 5.8.
(5.2.4-10)
hbt - hmt = E(hbbt-h il)
Combining Eq (5.2.4-10)
with
Eq
(5.2.4-4),
qit
can
be
expressed as follows:
k
q
= k
-T
[gZE + gm(-E) I(T
s+C
S
Replacing Eq (5.2.4-3) with Eq
(5.2.4-11)
-Ts)
and
(5.2.4-11)
rearranging
the results, we get the equation.
m=
Z E)(T -T s
(kI/a +
hfg -C (1-E) (T -T)
PR
)
s
(5.2.4-12)
I
~----
----
iViili
156
Since hfg
C
is
(1-E)(Ti -Ts ),
than
greater
much
for
the
practical
all
value,
purposes,
Eq
(5.2.4-12) can be simplified:
(5.2.4-13)
gm = Hi (TQ- Ts)/hfg,
Hi
where
-
+C
k /I
(5.2.4-14)
gE
Let us compare Eqs (5.2.4-6)
(5.2.4-13)
and
(5.2.4-14).
and
with
Eqs
notice that the mass flux
We
obtained by Eqs (5.2.4-13) and
(5.2.4-7)
(5.2.4-14)
is
always
less
than that obtained by Eqs (5.2.4-6) and (5.2.4-7), because E
is
always
less
1.
than
Physically this implies
that
energy loss from an eddy lessens the effectiveness of energy
transfer by the eddy.
The unknown variables in Eq
are
defined
by
(5.2.4-14),
g
and
Q
the Prandtl momentum transfer theory.
rate of momentum transfer by eddies
and turbulent mass flux, g
dV
per
unit
area,
t
, are expressed as follows:
,
The
'
2
(5.2.4-15)
St
dV
-
(5.2.4-16)
where
axial
p
and V z
flow
aTe the liquid density and
velocity,
and
v-
turbulence when it is divided by V z .
and (5.2.4-16), g.
g
= /
t
is
the
From
averaged
time
intensity
Eq
can be expressed in terms of
of
(5.2.4-15)
t
(5.2.4-17)
524
157
Tt
can be approximated by
the
interfacial
stress,
.
For an annular flow of sodium, the correlation of Eq.
(5.2.5-19) can be used :
Ti
:i
Cfi
fi
Pv2
2
(5.2.4-18)
Cfi= 0.005(1+234.3(1- ~/) 1 .15)
where
V
and
=
a and
= V -V
pv are the volumetric fraction of vapor and
the
vapor density.
The simplest relationship for
I2
suggested by Prandtl (66)
would be one of direct proportionality to the distance
from
the solid wall, y.
z = E y
where
5
(5.2.4-19)
is a constant value.
In a turbulent pipe flow, he suggested 0.4 for
In summary the interfacial heat flux, q
.
and the mass
158
, can be calculated as follows:
flux, gm
q!
it
where
H (T - T)
.
1
i
(5.2.4-20)
gm
i
q/hfg
Hi
kL/I,L+Cp9gE
E
" 27.73 Pr
I
-
0.46
-2
pV
f
Cfi
i
2
2
Cfi - 0.005(1+234.3(1 - Y)115)
Kowalchuk and Sonin (84) reasoned that mass exchange in
the interface is controlled either by the rate at which
the
latent heat for mass exchange is tranpsorted away from the
interface, or by the rate at which the vapor is supplied to the
whichever
interface,
is smaller.
These two mechanisms are
in competition with each other; when the amount of
supplied
latent
heat
required for phase transformation into the bulk fluid,
mass
vapor
exceeds
exchange
is
the
to
capability
controlled
mechanisms are referred to
the
by
as
the
transport
former
latent
mechanism.
heat
These
transport
159
control
mechanism
and
vapor
supply
control
mechanism,
respectively.
The vapor supply control mechanism can be viewed as the
a
net effusive molecular flow between two quantitiesof
arrival
of
molecules
interface and a rate of
from
departure
the vapor space toward the
of
surface of liquid into the vapor space.
flux
model
rate
molecules
from
the
While the turbulent
represents the latent heat transport mechanism,
the vapor supply control mechanism can
be
described
by
a
kinetic theory.
From a simple kinetic theory it can be shown that, in a
stationary container of molecules, the mass rate of flow
molecules passing through an
j
where
interface is given by
P
(5.2.4-21)
j:
efflux of molecules moving in one direction
M:
molecular weight
R:
universal gas constant
P and T:
of
pressure and temperature
The net molecular flux through the interface of liquid-vapor is
difference between the fluxes in the either direction
qm
(2)
-?
(5.2.4-22)
where v and a denote vapor and liquid.
Following Schrage (85) who introduced Y
to account for
-
i--~-Irl---l
I
=;
~ff~-ur
i--
; -R
._-I
160
the net motion of the vapor towards
superimposed
on
the
assumld
the
surface
Maxwellian
which
is
distribution, Eq
(5.2.4-22) is rewritten in the form
(5.2.4-23)
Pv
aer P
(T-
TI
Ti
where oe
a,'are
and
coefficients
evaporation
molecules
condense.
condensation
which are usually taken to be the same and are
written generally as
the
and
a. Y
striking
is defined as that fraction
the
surface
For a very small ratio
characteristic
molecular
of
velocity
which
overall
actually
to
speed
of
do
a
Silver and Simpson (86)
suggested a simplified form of Eq (5.2.4-23).
a
-m
=---
S(M l--
2 -a
(P
-
21rF/\T
Pv
(5.2.4-24)
T
v
TI
Assuming small differences in pressure and
temperature
and
using the Clausius-Clapeyon relationship, we can arrive at a
final
form
for
a
heat transfer coefficient, Hit
for the
vapor supply control mechanism
2 2
Hi
a
2-a
v fg
M
(5.2.4-25)
p /-\
Experiments with potassium and sodium vapor indicate
considerable
reduction
in
the
coefficient from near unity at low
near
atmospheric
pressure.
value
a
of the condensation
pressure
to
below
0.1
Wilcox and Rohsenow (87) have
161
which
considered the precision* with
condensation
the
coefficient
be
should
value
be made.
in
reduction
the
the result of
is
pressure
with
the
They conclude that
the
and
unity
coefficient
condensation
can
of
measurements
increasing random errors with pressure. We adopt this here.
.5.2.4.3 Model Validation
Two methods are designed
model.
suggested
Numerical
for
the
results
validation
for
of
the
the methods are
obtained by a 2-D numerical simulation of the P3A test
(88)
for which THERMIT-6S with the suggested model is used.
methods
Indirect
because
model
suggested
must
of
be
to
developed
verify
the
the lack of experimental data
concerning the mass exchange of sodium.
One of them
is
to
compare the bulk liquid temperature from the suggested model
with
the
bulk liquid
temperature
of
definition
derived
in
the
can
analytically.
The analytical
be
by
predicted
5.2.3.
(T - Tt
From
Eq
(5.2.3-25)
we
may
) for the turbulent zone in terms of
(5.2.4-26)
0.55(T - T)
(5.2.4-26),
Fig 5.10 shows results, calculated from Eq
THERMIT-6S
with the suggested model, and according to
the assumption of thermal equilibrium.
state,
the
) as follows:
Tw - Tt
from
using
bulk liquid temperature in the same way
Section
approximate
(T - T
determined
values
liquid
temperature
is
In
forced
equilibrium
the
to
be
equal
to
-20
-10
x
it
10
20
40
30
(T, - Ts) (K)
-10
Fig. 5.10:
Temperature difference between
wall and saturation temperature
Comparison between Analytic Solution and Results
Calculated According to.the Suggested Model
163
to
saturation temperature, if vapor temperature is assumed
be
As, in the equilibrium state,
in the saturation state.
the
represent
with
region
evaporation
state
equilibrium
the dashed line to
),
(Tw - Ts
negative
the
with
region
is, the
that
region,
no vapor can exist in the subcooled
) in Fig 5.10.
positive (T
w - Ts
results
In both the evaporation and subcooled regions, most
the suggested model fall between results from
by
predicted
the
in
only
appears
the equilibrium state and analytical results.
is to
The second method to verify the suggested model
the void volume in the bundle as a function of time
compare
after boiling.
flowmeter
readings
to
led
and
inlet
of
A comparison
outlet
bundle
void
volume
of
prediction
production rates before the voids reached the exit flowmeter
and upset further
flowmeter
outlet
flowmeter
inlet
To
readings.
FE
FE
2-1
1-1
bundle
the
for
account
reading 4% lower than the bundle
at
of
start
the
P3A
the
LOF
transient, two normalization methods were used to obtain the
void
volume.
(Ref
88)
In method
readings were made to coincide at
inception
the
the
8.8
two flowmeter
state were made to agree.
boiling
second
time by adjusting FE 2-1 gain to zero.
B, the flowmeter zeros and flowmeter
uncertainty
A,
amplitudes
In method
at
steady
These two methods should span the
in determination
of the void volume, which is
proportional to the enclosed area between the two
flowmeter
curves.
Fig 5.11 shows the void volumes
in the
bundle
as
a
,ll
N
_
iildli
164
150
3
(in )
Estimated experimental
results
Results calculated from
suggested model
100
Method A
/
4
Method B
0
o
r4/
/
500
0/
0
0.5
1
1.2
St
initial boiling time
Fig. 5.11:
1.5
Time (sec)
flow reversal
Comparison of Void Volume in the Fuel
Bundle between Estimated Experimental
Results and Results from the Suggested
Model
2
165
function
of
time
boiling, which are estimated from
after
experimental
void
volume
indeterminate
rupturing
of
calculational
estimation
was
undoubtedly
was
also
pins.
The
is
less
amount
fuel
results
Some of the
given in Ref 88.
was
but
an
sodium
vapor,
fission
gas released from
void
the
from
volume
than that by method A, but
up
greater than that by method B,
The
respectively.
experimental and.calculational results,
to
1.75
when
seconds,
failure of fuel pin clad and release of fission gas into the
sodium vapor occurred.
At 1.1 seconds just before flow reversal takes place in
the computation, as the test section is completely voided in
the radial direction, a high rate
of
vapor
generation
is
As a result, pressure built up in the bundle causes
noted.
the inlet flow to be
reversed.
These
physical
processes
result in the rapid propagation of the void in the bundle in
both
the
reaches the
downward
upper
and
upward
plenum.
directions
Afterwards,
the
until the void
void
calculated by the present code is almost flattened.
volume
~
__ ~_~1_ __ _s_
i
I_
_ _
_ _
__^:i~~ _ _
166
5.2.5 Interfacial Momentum Exchange
ratio,
Interfacial momentum exchange controls the slip
one of the most important factors in the analysis
is
which
the
and droplets for the evaluation of interfacial
film
liquid
both
of
effects
the
included
work
As his
from Ref 63.
coefficient of sodium by using data
drag
vapor-liquid
predicts
drag coefficient, his correlation
a
much
higher
Here,
interfacial drag coefficient than other correlations.
methods
calculate momentum transfer by droplets will be
to
presented so
liquid
for
Autruffe (89) derived a correlation
of two-phase flow.
film
be
can
drag
interfacial
the
that
derived.
coefficient
for
The results are compared to
experimental data and correlations, for non-metallic flows.
5.2.5.1 Analytical Models
A one-dimensional, steady-state flow model is now to be
developed
for
vertical
annular-dispersed
upward
adiabatic
For
flow.
flow, the vapor momentum balance equation
exchange
should account for a momentum
between
vapor
and
droplets, and can be stated as:
U2) + a - = -Ffi- Fdi- ap g,
d (a
g g
di
fi
g dz
9
ggg
dz
where Ffi
unit
and Fdi
volume,
from
are the
vapor
momentum
to
exchange
(5.2.5-1)
rates,
per
liquid film and to droplets,
respectively.
Assuming uniform dispersion of
same
diameter, Dd
, Fdi
spherical
droplets
of
the
is expressed in an empirical form
I~__
~__~_IL
_____ ~__j______l__l__ _~_____
_^_~P
167
as
g -U
2 (U
Fdi = AiCd
can be expressed in terms
The quantity Ai
Cad as
(5.2.5-3)
coefficient
drag
the
correlated
for
R d0.2 1 7
C = 0.271 Re
d
400 < Re
droplets
falling
undergoing distortion of shape, is used for C d
who
(90),
McManus
and
An expression obtained by Carofano
:
(5.2.5-4)
< 10,000
is the Reynolds number for a droplet.
Red =
As an
droplet
the
of
A. = 6 ad/Dd
where Red
unit
is the drag coefficient.
volume, and Cd
fraction,
per
droplets
of
area
interfacial
is the
where Ai
(5.2.5-2)
)2
£rUdD
U
d
U
appropriate
description
the
of
droplet
entrained
velocity and diameter as a function of the flow condition is
not
Dd
available,
U d
is assumed to be one-half of Ug
, and
is determined by the work of Smith (91) as
(5.2.5-5)
Dd = 0.0122 cm
The criterion used to establish a change in the droplet area
is that used by Rabin et
considered
to
occur
al.
when
(92).
the
Droplet
following
break-up
is
expression
is
__ __
_ _ I
~1111~_~
~_ -_l~i--~~~__IB__--LPs
I -
168
satisfied:
We
.5
> 1.0
(5.2.5-6)
Red
where We is the Weber number.
It is also assumed that once droplet shatter occurs, the
original droplet diameter will be reduced to one-third of the
initial value, as Martindale and Smith suggested (93).
The
liquid
film
momentum
balance
equation
can
be
written as
d
d(
2
+
U f) +
( 21f 01 kf
where Fwj
is
the
dP F dP
F
- alf P
X
Ffi 00 FwX
liquid
wall
friction
g,
force
(5.2.5-7)
per
unit
volume.
Equation (5.2.1-3) is used to calculate Uzf
and
f , and Fdi
for given G, x,
is obtained by adding Eq (5.2.5-1) to Eq
(5.2.5-7):
( p
( aP
Fdi
di' -dz-
(a
+9g-
)g - Fw
- (aZP +agPg)g
where Fwz
Fwl-
= 02 F
i
w10
(acUf )2
F f
Q
2D
--
ff a 0.184/Re0 "2
Re£ =
DP .aefUtfe
D: tube diameter
-Fk
)
(a +
)dP
f T*
(5.2.5-8)
(5.2.5-9)
(5.2.5-10)
(5.2.5-11)
(5.2.5-12)
169
Using the definition of the liquid entrainment fraction,
E,
and the following relation for G', we obtain
G'=(1-E) (1-x)G
(5.2.5-13)
Eqs (5.2.5-10) and (5.2.5-12) can be rewritten in
terms
of
G' as
F
wl0
Re
The
G 2
f
G'2
e2DP
(5.2.5-14)
DG'
(5.2.5-15)
two-phase
multiplier,
,
0
was
evaluated
by
the
Lockhart-Martinelli correlation.
As, the liquid entrainment fraction, E, which is needed
to calculate the acceleration loss of both vapor and liquid,
and the wall friction
(5.2.1-6),
d is
loss,
iterated
are
functions
until
Fdi
in
of
Eq
Cad in
(5.2.5-2)
matches F
in Eq (5.2.5-8). For given G, x, and
di
Ffi
can be calculated by the following procedure:
1.
Guess
2.
Calculate acceleration
vapor
and
Eq
a
kf
,
a.d
pressure
loss
by
both
liquid
and
loss by gravity and wall friction
loss.
3.
Calculate Fdi
from Eq (5.2.5-8).
4.
Calculate
5.
Reiterate the procedure until the change in
aid from Eqs (5.2.5-2) and (5.2.5-3).
sufficiently small.
6.
Calculate Ffi
from Eq (5.2.5-7).
%ad is
170
5.2.5.2 Model Validation
The
Ffi
(63).
above
procedure
the
from
now
applied
to
determine
boiling experiments of Kaiser et al.
sodium
The term Ffi
is
in Eq
interfacial friction, Ti
(5.2.5-1)
is
related
to
the
, per unit area as
4 Ga-
where
Ff
Ffi
=i--
Ti,
'i'
(5.2.5-16)
T.
(Ug-UUf )2
= Ci1
fi 2 g (Ug
(5.2.5-17)
1
D
In Fig 5.12, data for Cfi
from
our
analysis
of
Kaiser
in Eq
(5.2.5-17)
obtained
et al. (63) are compared to
those for Cfi
of Wallis (62) and of Moeck (94), which
derived
air-water
from
experiments.
Moeck
were
obtained the
air-water data in Fig 5.12 by using his film flow model
droplet
(95-97).
interchange
model
to
analyze
He calculated Cfi
by
subtracting
acceleration
of
net
entrainment,
and
three sets of data
droplet
drag,
vapor acceleration, and
vapor static head from the overall vapor pressure drop.
The
important result that we note is that the data cluster
into
a single band, regardless of
1.
vapor Reynolds number
2.
tube diameter
3.
total liquid flow rate.
Earlier work by Autruffe included the effects
of
both
liquid film and droplets for the evaluation of Cfi :
+ Fdi
Fi2. = Ff
fii
di
(5.2.5-18)
~u~ura---i~-^. ilYIL WC--~I~. i
-YIIIL-~--_~-_L-I1_I1_1IL
171
0.08
0.05
Cfi
0.01
-
x
0.001
0.005
0.01
0.05
B/D
Wallis correlation
Cw=0.005 (1+ 300 -
Moeck correlation
C =0.005(1+1458 (1)1.42
M
0
)
Suggested correlation Cs 0.005 ( I+ 520 (Autruffe correlation
C= 4.36
All0
Air-Water Flow Data
o 3 1.75 mm tube (Ref. (95) )
* 9.53 mm tube ( Ref. (96))
A 12.7 mm tube (Ref.(97))
Fig. 5.12:
(1+300 - 0.95/
Sodium Data
x 9mm tube (Ref.(63))
Interface Drag Coefficient to
Dimensionless Film Thickness
-
-
NilY
172
where F i
momentum
is total
liquid
between
exchange
and
vapor.
As
a
his
result,
interfacial
correlation
predicts
coefficient
than
drag
Several problems can be raised when
is
applied.
First,
as
a
correlations.
other
Autruffe's
correlation
is
derived for
correlation
his
annular-dispersed, two-phase flow of sodium, it may
appropriate
for
the
early
flow
annular
is dominant.
where
accidents
LMFBR
not
be
boiling transients,
of
stage
especially low-flow/low-power
higher
much
pure
Second, since his correlation
predicts an average liquid velocity that is much higher than
the real
liquid
coefficient,
film
which
velocity,
the
is a function
wall
of
heat
transfer
liquid
the
film
velocity, will be mispredicted.
As the sodium data lead to values of Cfi
that have
an
upper
bound
represented
by
the
Wallis correlation and a
lower
bound
represented
by
the
Moeck
correlation,
intermediate correlation is recommended for Cfi
Cfi = (0.005(1 + 520 (6).15)
D
an
of sodium:
(5.2.5-19)
... . .. .
173
5.3 Constitutive Relations in Post-Dryout Region
After flow reversal the subcooled
from
entering channels.
the hot wall is rapidly
occur.
will
long
evaporated
Especially,
behavior.
enough
to
is
prevented
Then, the liquid film remaining on
in
and
have
eventually
low
the
transients, the region and period of
and
inlet
dryout
flow/high
dryout
considerable
are
power
extensive
effects on system
Unfortunately, few studies have been done on
post-dryout
region.
Therefore,
the
a simple dryout criterion
and post-dryout models are implemented in the present code.
The experiments analyzed
by
Autruffe
(89)
indicated
that vapor did not come into contact with the wall until the
void
fraction
exceeded 0.957.
This value of void fraction
0.957 is used for a dryout criterion.
for
a
void
0.957
to
1.0,
As the void
assumed
from 0.0 to 1.0.
fraction
increases
contact
fraction
Therefore, the wall contact areas of
liquid and vapor after dryout are estimated in terms of
void fraction
the
and the criterion of dryout 0.957.
Cf= 1.0,
1.0- i
Cf= 0.043
0043
where Cfz
that
the liquid contact fraction decreases
linearly from 1.0 to 0.0, while the vapor
goes
is
fraction below 0.957 the entire surface of the
rod is coated with liquid.
from
It
and Cfv
Cf
= 0.0,
Cf v
for 0 < a< 0.957
- 0.957
0.043
, for 1.0 >
are the fractions of
area of liquid and vapor, respectively.
the
(5.3.1)
> 0.957
(5.3.2)
(5.3.2)
wall
contact
_ ..
,Mlnle
n
. .
11
l1
174
These contact area fractions multiplied by
area
Aw
in
cell become
a
.
and vapor contact area Awv
the
total
wall
the liquid contact area Awl
If these
contact
areas
are
divided by the fluid volume, we obtain the wall contact area
1
density
1- for the LMFBR geometry.
w
1
=
WV
f
(5.2.3)
w
4
4
D[2v(P/D)2 -]
4
De
De
Aw
1
=
Lw
Vf
where
1
Awl = Cf
(5.2.5)
The total heat flow Qwt" through the fuel surface
A
per
unit
volume is contributed by the heat transferred
to or from liquid and vapor, Qw
Qwt = QwL
Using
Eqs
area
and Qwv
+ Qw v
(5.2.6)
and
(5.2.3)
(5.2.4)
each
component
can
be
expressed as follows:
Q
=
(5.2.7)
a
iv(5.2.8)
1
WV
The
heat
two-phase
transfer
coefficient
for
the
liquid
film
in
flow, which is described in Sections 5.2.3.2,
is used for q"
.
For
v
the
Dittus-Boelter
correlation
175
is
used.
=
q"
Nu£
(5.3-9)
- (Tw - T)
e
k
t = Nu v -Y
q wq
Dee (T w -T )v
F
where
= 0.152 - 2 Pr
Nu
Nu
1 .
v
=
0.023 Re
(5.3-10)
Re0.9
2
(5.3-11)
2
0.8 Pr 0.4
v
(5.3-12)
v
= G(1-x)De/ 1
Re
Re v = GxD
De/
v
Refer to Eqs (5.2.3-30) and (5.2.3-31) for
Fl', F 2 , and.z
For the axial wall liquid friction in
region,
the
correlation
of
the
the
post-dryout
wall friction factor for
annular flow is assumed to hold true.
Accounting
for
the
reduced contact area, we have
(5.3-13)
Fwzkd = Cf 2 FwzZa
where
force
zd
per
and Fza
unit
volume
:
for
liquid
axial
wall
friction
post-dryout and annular flow,
IYIIIII_ ~_
.rI.... ~l--L1---
-r~----ill.----*~
-(-PU-~x--._rr_-r
-I------1---I~YlhluT
176
respectively.
.2
2
1)
a 2D
22
Fwzia
(5.3-14)
e
1
2
Xtt
Xtt
tt
0.88
Zvi1V1
it
In Eq
20
2+
= (1+
where
(5.3.14)
fZ
is
calculated
described in Section 5.1.1, but Re.
using
correlations
is defined as
De
Rep
Reg
=
Ua
In the post-dryout region, we can
which
by
contacts
vapor
the
wall
consider
the
region
in
as a single-phase region.
Therefore, for this contacting area, we have
(5.3-15)
Fwzvd = Cfv Fwz
v
where Fwzvd
is vapor axial wall
volume for post-dryout, and Fwv z
flow friction force given by
friction
force
per
unit
is the single phase vapor
2
F wzv
Here, fv
f
v
e
is calculated by the same correlation
5.2.2, but Rev
(5.3-16)
2D
is defined as
Re = P DeU /]v
in
Section
177
For the transverse friction factor in post-dryout flows with
regime,
numbers
is multiplied
(5.2.2-1)
Eq
Reynolds
vapor
and
both liquid
by
in
the
laminar
the wall contact
areas, respectively.
2
U
z£ max
2
At
D
Apt = Cf .f
v
(5.3-17)
2
Ap
where
f
= Cf
f
180
R =180
Re
180
Re
v
V
v Uvmax
2
At
D
Re£ = ap
(5.3-18)
IUtmax Dv/
Re v = avPvUvmaxDv/ v
If either liquid or vapor is in the turbulent
flow
regime,
the Ishihara correlation (64) is used,which was described in
Section 5.2.2.
Fwt ka
= f
1
v
G
2p
R
where G and f are the total
2
o
mass
(5.3-19)
flow
rate
and
friction
factor, respectively.
For the
transverse
friction
factor
in post-dryout
flows Eq (5.3.19) is split into two components in such a way
that
the
proper limits are maintained as the void fraction
.. .
..
.. .
-nl
fyb.i.L
UII~-_llMI I-N
,
....
I
-
178
Thus we have
approaches zero or one.
Cf
Fwtd
f1
G2
G
v
8
(5.3-20)
(1+tt
9
2
1
G
F wtvd = Cf v f Dv
1 2PI
1
X2
(5.3-21)
tt
In the
same
and
liquid
smooth
can
be
, if we assume that a very
corrected by the fraction of Cft
thin
area
interfacial
the
manner,
after dryout remains on the
film
wall.
AA-= Ait
it Cf
where
Ai
and
(5.3-22)
Ait :.
corrected
interfacial
area,
and
area for a given void fraction when the wall is
interfacial
totally covered by liquid.
The
complexity
difficult
to
of
the
interfacial
determine
the
in THERMIT-6S.
assumed to be in the shape of
channel.
By
this
In
the
cylinder
slug
model
which
= A
void is
fills
up
a
assumption the void fraction can be the
axial length filled up with vapor,
Vf
it
area, Ait interfacial area is
functions of axial length of the control volume
a =
makes
interfacial
Therefore, a simple slug model of the
implemented
structure
Af Az
= -
bz
Az
Az and
the
.
(5.3-23)
179
:
and Vf
where V,
vapor and tbtal flow volume
: total flow area
1
1 can be
Then, the area density
a
geometry and void fraction.
Af
0.5 D
A--c=
f
Af
D2
P 2
)2 -]
Af = -[2/(
where
energy
and
volume,
After
respectively.
most of liquid is in the form of droplets which are
momentum
condition,
equations
terms
L- will be multiplied by interfacial
ap
flux to get the rates of interfacial mass
almost homogenized with vapor.
by
in
(5.3-26)
and energy exchange per unit
dryout,
1
is expressed
ap
:
-
= - Cf
La
This area density
of
(5.3-25)
of the contact area fraction, CfU
ap
terms
(5.3-24)
(5.3-24)
The area density of post-dryout
for
in
AVit
1
La
La
mass
expressed
the
to
implement
this
exchange after dryout is not accounted
interfacial
refer
In order to
Table
area
5.3
fraction.
and
5.4.
For
complete
Note that most
physical models for post-dryout are based on the liquid film
models which are developed for
the
wall
pre-dryout.
However,
when
is hot enough, there is little liquid left. on the
wall and the droplets play an important role in transferring
mass and energy.
As, presently, effects of droplets are not
180
considered in most of models for the
work for them is recommended.
present
code,
future
TABLE 5.3
THERMIT-6S Wall Friction and Heat Transfer Models
TABLE 5.3 (continued)
2
Liquid
Fwtl'1 -t7Dt
I
ma x
Fwtt
V
Vapor
1)
f l
Transverse
Wall
Friction
Force
per unit
volume
Apt
Fwt
At
For Ret< 2 0
2
.5
i)
For Ret<202.5
i)
For Ret<202.5
and Re <202.5
fL-180/Ret
22
Fw 3Cf Fa
ii) For Re.?202.5
Re
Re v
t
Pt
- 0
2D
v
v4
v3
0
Fwtv2
Fwtvl - 0
4
4 Pv
2
vtmax-
v max
1.92
11
eT
T Re
0.145
Re
Re- "
Fwv 3 =Cf *Fwtv 4
D
RetutPtIvtmax I t
it
fv4f L
Pt Ivtmaxlv
Fb
wtt2
t2
G=a Pt v tmax
2
x )1.
85 5 1
x(1)
ff2
=
Re =
D
x PV
v+a
vvmax
1
2ttX
tt
1
Re v=vPv IvvmnaxI
2
P t Imax
2D
8
tt
ii) For Re >202.5
or Re >202.5
V
Fwt3=Cft(1+ Xtt
tt
)
xft2
t2 2DVPt
0.145
Fwv=Cv (- )
Xtt
2
G
xf
12 2Dv t
fT
CGD
v
I 9=
I
f
fv4
L
ii)For Rev>202.5
ii)For Re I 202.5
2
to
i) For Re (<202.5
I
f v4"f
--
Liquid
Owl I'1/l'w
Vapor
Owvl = 0
r
Owv2
Owv 3";t
w
wv2 =w2
e
2 /.w
OwID
Owv3 =qZv4 Iwv
0
I
(~
1I =h
('rw-Tt )
q"
2=
h12 (Tw-T
'Owv4wv4wLywv
I
qw
! )
3 =h
2(T-T)
qwv4=hv4 |w
k
h
Rate of
all
=Nul
tj
I
1) For 1.15
Heat Transfe
ht2=Nu2
?< 1.3
- D-
0.9
F
Nu =0.152 -21 Pr Re .9I2
t
F2
2
i)
Nu =4.0+0.33(E)3.8 ()0 .8c
volume
h
Tv4w-Tvl
Qw=q"/L W
1.05<
ii)For
1.5
F 2 = 0.7071 Re
+ 0. 16Q ) 5.0
D
p<1.15
0
Pr 1
ii)For 50<Re,<1125
and
= 1.563
F
150Pe< 1000
F 2 = 5Pr + Z InllE
Nu =Pe
0 .
-8.55(
+
)
3 -16.15+24.96(
Pr,
-1)I
6
(I
)21
E= 27.73 Prt
+=0.48i8 Re0.
For Pe< 150
iii)
Nul=4.501-16.15+24.96()
-8.55(-
D
21
5 85
For Ret> 1125
iii)
F
= 1.818
5
F 2 = 5Pr + -(1+5EPr
I+l-
6 Int-1+
P=60
A
,
=l-
=0.133 Ro0.
20
Xtt
7 6 14
0. 88
X
Lt
IL
___________
_
_
_
__
_
_
_
_
j
D
R0.8
x Re V
For Re <50
F
=Nu(
v
kv
Nu 4 =0.023
10<Pe 5000
and
per unit
v3=h
f
Dh
= 0
I
-
--
0.4
v
v
TABLE 5.4
TIIERHIT-6S Interfacial Exchange Models
Post-Dryout Regime
Pre-Dryout Regime
(10
(1 > a > 0.967)
a < 0.957)
4It
D123 I
2 -
41v
1
1
Dn
e
La2
"&1
E
rqt
1.73 Pr
2i(v v
.005(1+234.3
in
h -2
2
(1-
11 " 15)
1.O-a)
1T. -0.957
rAIII,.8
5.4
(continued)
Interfacial Heat
Flux
q
= II
(T-T
q9 = Ii1 (T-Tv)
v )
h9
Rate of Interfacial
Energy Exchange per
unit volume
ii
Lal h
h
q?
itg
0
i2
=
I
h
qit
a2 fg
Oi =S_1, h g_ q
qt
a
fq
Rate of Interfacial
Momentum Exchange
per unit volume
F
=Fs +F
v
Standard Drag Force
Fs =lv-VI
a
i =Cfi SPv(vv-t
2
C
= 0.005(1+ 234.3(1 -
K=
L - Cfi
al
)
1
.
15
Fs2 = Fsl
v Vt
Fsl= K(v -V-)
Virtual Mass Force
Fv
F =
F
v1
Iv
p
+
C
Pqv
(Vv-V)
,t
v
t
-+
v *vV(v
-v
v
v
I ((A-2)Vv + (l-A)Vv
Fv
t
1)
v
= Fv2
-- ----
YmuI
186
Chapter 6.
The
"code assessment
ascertaining
well
how
processes measured in
within
CODE ASSESSMENT
which
is
the
defined
as
the
task
of
code can simulate the physical
experiments,
quantifying
the
error
the code is capable of predicting certain key
parameters, and determining the code's limitations.
In LMFBR analysis, the important physical processes are
temperature transients, incipient boiling, vapor propagation
in the axial and radial
directions,
flow
coastdown,
flow
reversal,
flow oscillation, and dryout.
to affect
reactor
incipient
boiling, of flow reversal, and of initial dryout,
wall and
behavior
temperature
fluid
are
The key parameters
therefore
the
and
distribution,
times
of
inlet
and
by
the
outlet flow rate.
The
capability
range
validation
derived
for
the
code
of
the
it
restricted
equations
constitutive
were
forced convection flow, based on an assumption
of the predominance of the cocurrent
Although
is
of the constitutive equations implemented
Most
in THERMIT-6S.
of
is
questionable
annular
whether
these
flow
regime.
constitutive
equations can be applied to natural convection flow,
it
is
expected that the differences in the local processes between
two-phase
flow
under
natural
convection
convection may be much smaller than those
flow.
and
in single
forced
phase
Also, the code can only be applied to predictions of
phenomena
before
fuel
melting
occurs.
The
code
was
187
to be applicable to a component system
designed
originally
but it can be extended to
system
of
analysis
the
loop
whole
a
the future in order to account for the effect of
in
loop system and to obtain accurate boundary conditions (98).
sodium' boiling
of
predictions
in
relation to experimental results will
LMFBR subassemblies in
be
are
This
given.
voiding behavior and temperature distribution in
bundles
model
2-D
vs
was
the need to determine whether two-dimensional
by
motivated
1-D
of
assessment
an
Additionally,
large
pin
This work includes comparison with
important.
predictions by SAS (7 and 8) and NATOF (16 and 99).
SAS
is
considered to be one of the best 1-D codes developed so far.
between
Comparison
insight into
the present code and SAS will give some
effects
2-D
future
and
development
needs.
Also, comparison with NATOF-2D, which uses the same solution
scheme
but
different
physical models, will illustrate the
effect of improvement in the physical models of the
A
code.
description
brief
of
experiments
present
selected for
assessment is shown in Table 6.1.
6.1 W-1 Experiments
The Sodium Loop Safety Facility (SLSF) W-1
(100)
Experiments
were 19-pin bundle in-pile tests performed by Hanford
Engineering Development Laboratory and coordinated with
Argonne
utilized
National
SLSF
as
Laboratory P series.
a
packaged
Engineering Test Reactor (ETR).
testing
the
The W-1 Experiments
facility
in
the
-U--~
LI~
LI-~ --
PIIS-Wr-~
~---i_-I-~LI~-r_ ~-c---~-_~
-I~YI-~L-L~\I-_LI~-U-~---~-~--- U--
188
TABLE 6.1
Experiments and Codes Selected for THERMIT-6S
Code Assessment and Comparison
# of Pins
in Assembly
19
37
61
P3A
OPERA
-15 Pin
Type s of
Accident
W-1
(W-7b')
Low Flow/
*
High Power
*
*
*
*
*
(LOF)
NATOF-2D
SAS-3D
THORS
Low Flow/
Low Power
(LOPI)
(test 72B,
run 101)
.
*
SAS-3A
*:
1-D Analysis by THERMIT-6S
**: 2-D Analysis by THERMIT-6S
NATOF-2D: 2-D Code
SAS-3D and SAS-3A: 1-D Codes
NATOF-2D
SAS-3A
-
(P-tet)
*
189
tests
The W-1 Experiments consisted of many individual
to
provide
experimental data on sodium boiling and boiling
stability in a fuel pin bundle during flow transients in the
The
LMFBR.
fundamental
were
tests
W-1
also
the rate at which sodium boiling
information:
develops, the time required to
channels,
coolant
time
the
provide
to
designed
dryout
establish
local
in
fuel pins can sustain without
power
pin
cladding failure, the operating conditions (fuel
and flow) which produce stable boiling in a pin array.
The tests were divided into two parts.
first
The
was
the protected LMFBR Loss-Of-Piping Integrity (LOPI) accident
simulation
the heat transfer characteristics
determine
to
The
from fuel pins to coolant and boiling behavior.
was
phase
the
Boiling
Window
flow-power combinations which
boiling
stable
temperature.
phases:
The
approach
zones
BWT
in
will
were
to establish
(BWT)
produce
bundle
the
tests
Tests
for
second
incipient
a
subdivided
and
given inlet
into
three
to boiling, incipient boiling, and dryout
with fuel pin failure.
6.1.1 Test Apparatus and Procedures
The test train includes the
channel,
flow
flow
divider,
test
and
subassembly,
test
instrumentation.
Downward coolant flow in the loop is outside the test
flow
bypass
train
divider, which is insulated over much of its length to
minimize radial heat transfer.
portion
of
the
test
Upward
flow
in
the
lower
train occurs in parallel through the
----
I_~
II
190
test assembly and
subassembly
Test
channel,
bypass
is metered at both the test subassembly inlet
flow
coolant
the
The two parallel coolant streams mix above
and outlet.
the
subassembly outlet flowmeter and upward flow continues
test
heat
the
reaching
to
prior
flowmeters
through two loop
exchanger inlet.
The BWT tests were initiated at a steady state flow
1.95
Kg/sec.
of
The flow was linearly reduced to its low flow
value in 0.5 seconds, where it was held for a specific time,
to
and then linearly returned
interesting
most
because
code,
oscillation,
flow
and
and
2.0
occurred
section
test
boiling
accident.
flow
Boiling
clad
dryout.
before
had 38% of full flow for a period of 3.0
This test simulates a high power/low
seconds.
of
seconds
in this test.
occurred at a pin power of 14.4 Kw/ft
The
is
demanding test to be run by the
reversal,
dryout
W-7b'
The
W-7b') is selected for numerical simulation.
the
0.5
tests, BWT-7b' (simply called as
BWT
the
Among
seconds.
in
state
initial
its
The
thermal
and
flow
hydraulic
sodium
operating
conditions in W-7b' are shown in Fig 6.1.
6.1.2 Numerical Simulation of W-7b'
6.1.2.1 2-D Analysis of W-7b'
To save computer time
nature
of
sodium
and
probe
the
two-dimensional
boiling, a 2-D analysis instead of a 3-D
analysis is done by considering one-sixth of the
test
fuel
191
*
e~
r .;
17.0 psi@
0***
C
S
Cover
Gas
TOTAL LOOP FLOW
45972 b/h
RESERVOIR
Heat
Exchanger
Outlet
Boundary
asi
PUMP
Test
Section
Primary
Vessel
Inlet
Boundary
TEST SECTION FLOW a 15444 b/h
Fig. 6.1:
BYPASS FLOW - 30528 b/h
SLSF Loop Steady-State Hydraulics at 145 MW
ETR Power and 4.29 lb/sec Test Section
Sodium Flow (Fuel Pin Peak Linear Power
of 14.4 kW/ft)
I
I
~*~LL"---=i- =SE --rz-~I~Elll
192
Since the present code uses a x-y-z coordinate, the
bundle.
assumption adopted for the 2-D analysis is that there are no
in the y direction,
exchange
mass, energy, or momentum
net
which can be achieved by assigning a zero flow area in the y
Then the x-z coordinate will be transformed into
direction.
the
direction.
radial
Seventeen
direction are considered: one in
8"),
nine
the
three
shown in Fig 6.2, the fuel bundle consists of
in
in
assumed and swirl flow is not considered.
is
direction
power
As a result, uniform
the r-z coordinate.
the
As
meshes
in the axial
meshes
lower
a
blanket
(each
the fuel (each 4"), two in the upper blanket
in
(each 6"), and five in the gas plenum (each 8").
measured
inlet
condition.
After
Up to 1.5 seconds into transients, the
flow
is
used
the
for
inlet
boundary
boiling starts, the pressure-pressure boundary condition
order
in
needed
to
flow
predict
coastdown
oscillation.
Unfortunately, as there are no
inlet
outlet
and
pressures
of
the
test
the
for
section,
an
The positions of
estimation of the pressures is inevitable.
the inlet and outlet boundary are shown
flow
and
data
is
in
Fig
6.1.
The
reason for selecting these positions as the inlet and outlet
boundary
test
positions is that, while the mass flow rate in the
section
boiling,
is
time
at
substantially
with
time
after
in the total mass flow rate in the loop is
change
relatively small.
with
varied
Therefore,
the
positions
conditions is relatively small.
the
variation
selected
for
of
the
pressures
boundary
The cover gas region is not
193
Fig. 6.2:
Radial Meshes of the Fuel Bundle
for the 2-D Numerical Simulation
of W-7b'
.
.,lll
IIi
II
Ii i
,
194
the
selected as the outlet boundary because
between
the
of
outlet
drop
pressure
the test section and the cover gas
rate
in
the loop which can not be predicted in the present code.
To
region requires information on the total mass flow
for
account
drops between the inlet boundary and
pressure
boundary
outlet of the test section, and between the outlet
and
outlet
test section, a valve model is designed and
of
described in Appendix D. The
importance
of
boundary
the
will be discussed in detail in Section 6.5.3.
positions
the velocity-pressure
it is not
boiling,
outlet pressures.
boundary
condition
is used
As
before
necessary to know the exact inlet and
However,
use
of
the
pressure-pressure
boundary condition after boiling requires information on the
values
of inlet and outlet pressures.
is estimated
pressure
temperature
in the
so
last
as
cell
boiling inception, which can be
Therefore, the inlet
give
to
of
the
estimated
saturation
the
heated section at
by
experimental
Note that the fluid temperature ultimately reaches a
data.
saturation temperature which is fairly constant over a
time
Then the outlet pressure is determined so as to give
span.
the experimental inlet flow
at
1.5
Figure
seconds.
6.3
shows the forcing function used for the boundary conditions;
the
solid
represents
line
the pressure-pressure boundary
boundary
condition and the velocity-pressure
condition
is
used in the time period of the dashed line, 1.5 seconds.
In
order
pressures,
it
to
obtain
is
the
inevitable
correct
to
use
inlet
some
and
outlet
iteration.
195
I
12
I
I
8-
I
I
I
Inlet Pressure
S4
Outlet Pressure
0
1
0
1
t 2
Incipient
Boiling
I
I
I
+ 4
Scram
1
6
1
Time (sec)
Fig. 6.3:
Forcing Function for W-7b' Numerical
Simulation (Dashed line represents
the period when the inlet velocity
boundary condition was used)
8
1LI~
c-P-~Llrr~L.-O~i~LPi~i
-.L-i --.-.-lir----p~ ~..~IP~-.ru.r-r~--r~lu~
196
Therefore, loop simulation is recommended for future work to
in
eliminate the uncertainties
inlet
the
estimating
and
outlet pressures.
direction
Uniform power distribution in the radial
used
for
numerical simulation during transients.
the
is
The
constant Nusselt number for liquid thermal conduction during
Zielinski
transients is taken to equal 22 as recommended by
Various
(99).
of
types
fuel rods for the W-1 tests were
To
used to obtain uniform radial power.
for
the
radial geometries, a generalized normalization of
different
heat source is needed. (See Appendix E)
this
account
of
normalization
heat
source
The
importance
of
be discussed in
will
Section 6.5.4.
6.1.2.2 Results
The numerical results for the W-7b' are
in Fig
the computation, incipient boiling starts at
In
6.4-6.10.
shown
1.55 seconds and inlet flow reversal occurs at 2.65 seconds.
Dryout occurs within 0.15 seconds after flow reversal.
All
predictions
temperature
(Figs
6.4-6.6)
present code are higher than experimental data.
by
the
Especially,
it is observed that temperature differences between data and
predictions
very
are
large
at the midplane of the heated
section. (See Fig 6.6) One of the plausible explanations
these
large
temperature differences is geometry distortion
due to high power.
actual
events
of
and
Also there is a time delay
between
the
measurement of events by thermocouples,
197
1600
I
I
I
I
Temperature data
or wire-wrap
SThermocouple at
33.7" from the
bottom of the fuel in the
x
X
15001-d%
1400
I
-,//
m/
/
\innermost
S
1300
lx
S
1e
2001
channel
i
-- 2-D predictions
by THERMIT-6S
--- I-D predictions
\K
by THERMIT-6S
/
'C
1100
-/
1000
-I
900,
x
8 0 0..
70C
i
0
1
3.
Scram
7
Time (sec)
Fig. 6.4:
Comparison between Temperature Data,
and 1-D and 2-D Predictions by
THERMIT-6S at 33.7" (from the
bottom of the fuel) in the Innermost
Channel for the W-7b' Test
8
ouLl~,,~u-*r~uco;~.,Y;IL
o~,_~_____
^ ~_
198
1500
I
I
I
F
X
14001-
x
Temperature
data of Hex-
x
can innerface
thermocouples
at 33.9" from.
the bottom of
the fuel
1300
3
//
1200
---
by THERMIT-6S
/
---
1100
-
1000
"
900
2-D predictiom
1
1-D prediction.
by THERMIT-6S-
I
800
_ _
7001
I
0
1
Fig. 6.5:
2
3
I
+ 4
Scram
Time (sec)
I
6
7
Comparison between Temperature Data
of Hex-Can Innerface, and 1-D and
2-D Predictions by THERMIT-6S at
33.9" (from the bottom of the fuel)
for the W-7b' Test
8
199
1550
- -I : Experimental
data at the
midplane of
heated section
in the intermediate channel
I
1400
V
--- : 2-D predictions
by THERMIT-6S
#
1200
-
//
/.
1000
//
/
800
0
1
2
3
4
5
6
7
Scram
Time (sec)
Fig. 6.6:
Comparison between Temperature Data
and 2-D Predictions by THERMIT-6S
at the Midplane of Heated Section
in the Intermediate Channel for the
W-7b' Test
8
200"
which is not mentioned in Ref 100.
After boiling inception, rapid vapor production in
from
axially
vapor
and
liquid to vapor,
to the high density ratio of
due
occurs
region
saturated
is
transferred
radially
and
saturated region to the subcooled region
the
surrounding the saturated region.
condensed in
the
the
subcooled
This transferred vapor is
Some
region.
of
the
vapor
produced is trapped in that cell for a short time due to the
drag
interfacial
between
force
liquid
and
vapor.
The
in
that
trapped vapor increases void fraction and pressure
This
cell.
causes
inlet
rapid
flow coastdown and strong
radial fluid transfer from the center channel to
channel.
Rapid
region.
outer
flow .coastdown increases power-flow
inlet
mismatch and in turn
the
fluid
temperature
subcooled
in the
The large radial velocity escalates energy transfer
due to vapor transferred from the inner channel to the outer
channel.
If the homogenous model, which assumes that an infinite
interfacial drag force exists, is used, more vapor
in the
saturation
region is trapped in that region.
more trapped vapor results in more rapid
fraction
and
This
pressure.
increase
the
Newton
iteration.
size to achieve the
requires
much
CPU time.
that the computation may
be
in void
convergence
The reduction of the time step
convergence
larger
This
is the main reason that the
homogeneous model often brings about failure of
of
produced
of
the
Newton
iteration
Even though it is assumed
successful,
much
more
rapid
201
flow coastdown and earlier flow reversal is expected.
inlet
For details refer to Section 6.5.2.
the
last
fluid
the
At 1.0 seconds after boiling inception,
in
mesh of the heated section in the outmost channel
reaches saturation temperature and vapor production in that
cell occurs instead of vapor condensation.
rate
end
of
of
generation and pressure build-up around the
vapor
the
As a result, the
heated
this
accelerated" by
coastdown
Flow
increase.
section
increase in pressure and vapor
sharp
generation can be seen in Fig 6.7.
As shown in Fig 6.7, the
rate of inlet flow coastdown is more rapid than that in the
experiment. This can be explained by the following reasons:
1. While the constant pressure boundary
condition
is
used in the computation, rapid inlet flow coastdown, in
reality, may cause an increase in the inlet pressure
due to the hydraulic resistance of the loop.
2.
The
condensation
experiment
rate
may
be
in
higher
the
due to shattered droplets around interfaces
produced by
instability.
Kelvin-Helmholtz
instability
and
Taylor
Slower flow coastdown can be predicted if
a model to account for this effect
is implemented
in
the present code.
As a result, flow reversal occurs 0.2 seconds ahead of
that
in the experiment.
Voiding of the rest of the bundle proceeds rapidly from
this time on, with the
channel response.
whole
bundle
indicating
a single
As the gas plenum region of the bundle is
202
I
I1
I"
I
I
i
2
-----
S"--*
Inlet Flow D
2-D predicti
THERMIT-6
Xby
A - A SAS-3D
1
m-m NATOF-2D
I
It
II
-1
0
1
I
Ii
II I
2
3
II
+
4
5
&
7
Scram
Time (sec)
Fig. 6.7:
Comparison between Inlet Flow Data and
Predictions by THERMIT-6S, SAS-3D, and
NATOF-2D for the W-7b' Test
203
subcooled due to a lower heat capacity than those in
mildly
region,
blanket
the heated section and
vapor
rapid
more
propagation in the gas plenum occurs until vapor reaches the
end of pin at around 2.8 seconds.
dryout occurs.
reversal,
flow
Within 0.15 seconds after
It seems that the incidence
in
of flow reversal may be the most important single factor
the occurrence of dryout in low flow/high power accidents.
After flow reversal the liquid film
hot
wall is rapidly evaporated due to high power and little
liquid supply from subcooled liquid
Dryout
occurs.
sharp
were
experiment
The
obtained
from
thermocouples may be closer
to
the
heat
transfer
the
cladding
in
rise
in
data
temperature
the
thermocouples.
wire-wrap
wire-wrap
the
by
to
than
temperature
wall
dryout
eventually
measured
temperature
dryout,
After
place.
takes
temperature
and
reduces
dramatically
coefficient and as a result,
vapor
the
on
remaining
temperature due to the low heat transfer coefficient.
Thus, predictions before and after boiling in
fluid
represent
Figs
6.4-6.6
After
and wall temperature, respectively.
rapidly
accelerated
to
due
high
void
dryout, vapor
is
fraction
this high vapor velocity dramatically reduces
and
the time step size, and in turn
this
reason
increases
CPU
time.
For
development of an advanced numerical scheme is
recommended as future work.
Liquid
reversal,
reentry
followed
occurs
by
inlet
at
seconds
0.2
flow
after
oscillation.
6.8-6.9 show voiding oscillation at the
lower
flow
Figures
boundary
of
--
E
h hiEY
O 1i1111
I MI
II0I inII1
IIhIl
EI i
nIUl,
204
Length (in)
70
62
G
hm
Gas
Plenum
+
-/
Upper
Blanket
+
36
32
Top of
Fuel
-m
24
16
8
-
: 10% void
fraction
"
: dryout region
(a > 0.957)
0
Bottom of
Fuel
L
Scram
6
5
Time (sec)
Fig. 6.8: Profile of Voided Region in the
Innermost Channel for the W-7b'
Test (2-D Simulation)
205
Length (in)
70
I
II
62
54
Gas
Plenum
- 46
--
----
,-
Upper
Blanket
-
36
Top of
Fuel 32
24
16
--
8
"
: 10% void
fraction
: dryout region
(a >0 .957)
I ~II
I
S0
Bottom
of Fuel
Scram
Scram
Fig. 6.9: Profile of Voided Region in the
Outermost Channel for the W-7b'
Test (2-D Simulation)
6
Time (sec)
206
by
void
liquid
The
reentry.
5 cycles/sec, compared to
oscillation is
of
frequency
flow
the
cycles/sec
3
in
the experiment and the amplitude is much larger than that in
the
is
It
experiment.
to identify the reasons for
hard
these results, but several reasons can be considered.
1.
again,
Here
are
pressures
the
in
changes
in
condition
contrast with the constant boundary
outlet
oscillation, in
flow
during
expected
and
inlet
the
computation.
2.
highly
The constitutive relations after dryout are
In
questionable.
However,
neglected.
are
post-dryout
model, effects of droplets are
our
the
large
region
and
to
enough
of
duration
affect
the
flow
behavior as shown in Figs 6.8 and 6.9.
3. Large radial velocities
effect
on
numerical
the
control
stability.
have
a
multi-dimensional
of
the
time
This
may
cause
step
size for
some
error
accumulation.
seconds.
3.5
Scram and flow recovery are simulated at
Due
to lack of information on the scram, it is assumed that
the
scram
a
causes
corresponding
to
about
negative
twenty
reactivity
dollars
of
-0.13
worth of negative
reactivity which gives a prompt drop of power:
q(t) - 0.054 x q(t )
After this prompt drop in power, the test
power
decay with a period with 80 seconds.
section
goes
on
With rapid flow
recovery, vapor condensation takes place starting
from
the
207
is
liquid
subcooled
directly
and
Then the hot walls are rewetted by liquid
transported.
the
which
into
cells
lower
reduced
is rapidly
region
dryout
shown in Figs
as
6.4-6.6.
At about 4.6 seconds vapor
channel
innermost
in the
disappears and 0.13 seconds later is quenched in the outmost
This delay results from heat transfer from the hex
channel.
can
to
fluid in the outmost channel.
immediately
occur
drop does
not
cladding
temperature
The wall temperature
after
the
The
to
0.25
up
increase
continuously
scram.
This
seconds after the scram and then is sharply lessened.
delayed
time
be
can
a
measure
of how fast the cladding
temperature will respond to the scram.
Fig 6.7 shows
predictions
by
SAS-3D,
NATOF-2D,
and
THERMIT-6S as well as the experimental inlet flow transient.
Results by SAS are characterized by early flow reversal (1.7
seconds)
and large flow oscillation.
The NATOF code, which
has the same numerical logic but different physical
models,
predicts that cladding dryout at 1.76 seconds and early flow
reversal (1.95 seconds) occurs.
the
occurrence
of
experimental results
primarily
present
predictions.
This
was
caused by use of the Autruffe correlation for the
interfacial drag
intefacial
before flow reversal unlike the
dryout
and
Note that NATOF-2D predicts
coefficient
which
predicts
force than the present model does.
much
higher
As a result,
more vapor is trapped in the cell and then rapid increase in
void fraction takes place.
NATOF-2D calculation stopped
at
208
seconds
2.2
to
due
of convergence of the Newton
failure
iteration.
subject.
interesting
Both
and
predictions
2-D
predictions
1-D analysis predicts 0.28
present code.
boiling.
incipient
1-D
between
Comparison
(Fig
were
is
an
done by the
of
delay
second
This can be understandable
6.10)
because fluid temperature in the 1-D calculation is averaged
Therefore, at the same axial
over the x-y plane.
innermost
in
predicted
temperature
cell
1-D
higher
but
in
cell,
experiments
lower
in the
cell
outmost
than
time delay of incipient boiling in the 1-D
calculation is quite natural.
of
the
is
As boiling is initiated in the
predictions in 2-D analysis.
innermost
analysis
position,
with
steep
In the
radial
numerical
simulation
temperature gradients,
incipients boiling in the 1-D analysis can be
substantially
delayed.
As radial temperature gradients in transients will
much
steeper than at steady states, caution is required
be
when 1-D simulation is performed.
Initial dryout in the 1-D
analysis occurs at 1.78 seconds after boiling,
compared
to
1.55 seconds in the 2-D analysis.
In the 1-D analysis there
is
transferred
no
room
to
condense
vapor
in the the r
direction.
However, in the 2-D analysis strong
caused
an
by
increase
pressure build-up.
flow
cross
flow
in pressure in inner cells reduces
The frequency
and
amplitude
of
inlet
oscillation in the 1-D analysis are higher than in the
2-D case; the frequency in the I-D is about
compared to 5 cycles/sec in the 2-D.
6
cycles/sec,
209
2
I
l
I
I
I1
I
1
2
--- Inlet Flow Data
*
o
*--.1-D predictions
by THERMIT-6S
1
.4-I
-1
0
1
Il
I11 'ell
2
3
I
t
4
Scram
I
5
I
6
7
Time (sec)
Fig. 6.10:
Comparison between Inlet Flow Data
and 1-D Predictions by THERMIT-6S
for the W-7b' Test
I"L L iC
~*
~-~-sLLIIS;ICrPTr~Li--Ui
rri
~iW~T
T~-hl-~- i--~ L-----
210
6.2 P3A Experiment
The Sodium Loop Safety Facility (SLSF)
(88)
an
was
in-pile
performed by Argonne National
test
Laboratory in 1977.
Coolant boiling was
section
and
flowmeters
10
by
test
at 0.8 seconds.
followed
seconds,
by
Gas passing through the bundle exit
inlet flow oscillation.
10.8 seconds, was indicative of clad failure.
at
flowmeter
at
detected
sensors
acoustic
Inlet flow reversal occurred
experiment
P3A
at
Extensive clad melting and motion occurred
12
seconds.
ETR scram occurred at 12.9 seconds, as planned.
6.2.1 Test Apparatus and Procedures
SLSF test
apparatus
has
already
described
been
in
Section 6.1.1.
The P3A experiment was designed to operate
rates
flow
of
bundle
at
test
assembly
and exit temperatures of 720 F and
inlet
conditions
These operating
1216 F, respectively.
the
coolant
9.2 lb/sec in the test subassembly and 19.3
lb/sec in the loop while removing 1240 Kw of
power
at
simulate
design, center subassembly, beginning-of-life operating
conditions in the FFTF core.
radial
direction
predicted values.
non-uniform
A temperature gradient in the
was observed which did not agree with the
This
radial
discrepancy
power
was
distribution
attributed
in the bundle.
estimated radial power profile is as follows:
Power Factor
Fuel Pins
center pin
:
0.9
to
a
An
211
1st row
:
0.95
2nd row
:
1.07
1.156
3rd row
The
pump
unprotected
coastdown
t
at
0
-
while the reactor power was
second,
reached
rate
flow
seconds when boiling
decay
curve
the
in
42%
the initial value at 8.8
of
started.
condition
Without
was
flow
the
boiling
was designed to follow the
the
Afterwards,
projected ETR LOF curve up to 13 seconds.
flow
inlet
The
Kw.
maintained at the initial full power, 1240
mass
flow
start
to
caused
trip
held constant at 2.76 lb/sec corresponding to 30%
of the initial value.
6.2.2 Numerical Simulation of P3A
6.2..2.1 2-D Analysis of P3A
subchannel
be justified by Fig 6.11 which
shows
very
uniform
temperature gradients in the central two rings.
Fig
6.12,
the
are
considered:
one
mesh
zones
in
the
plenum in
blanket
the
P3A
the
upper
(each 6"), and five in the gas plenum (each 8").
upper
As shown in
axial
in the lower blanket (each
8"), nine in the fuel (each 4"), two in
the
radial
fuel bundle consists of three mesh zones in
the radial direction. Seventeen
direction
This can
are agglomerated into one mesh.
rings
two
central
the
In the 2-D analysis of the P3A test,
blanket
In effect,
in the W-1 test is replaced by the gas
test.
Up
to
8.8
seconds
into
the
Y__YI_~~Y~liU~L-.__
~Y^(---- --~LWU~CLIY
U--F~
-^-"~L~YLIIL-~
212
1250
1150
1050
45
(d
950
850
1
2
3
4
5
Central
Subchannel
Fig. 6.11:
Subchannel
Radial Temperature Gradients at the
Last Cell of Heated Section Predicted
by THERMIT-6S for the W-7b', P3A, and
OPERA-15 Pin
213
Fig. 6.12:
Radial Meshes of the Fuel Bundle
for the 2-D Numerical Simulation
of P3A
-L
~-ZLi
LI~-*
L-II~ r~l--~~*--L-----
--rrr --,l-rr.ul
~I11_-
-------
u~.-DI--1 --
.~__1~1~i^^l
I
214
when boiling started, the measured inlet flow is
transients
W-7b',
the
shown in Fig
6.1.
so
estimated
The
inlet
and
pressures
outlet
section
at
8.8
seconds.
constant pressure drop between the
observe
8.8
After
thermal
liquid
conduction,
used
is
the
to
The estimated
Nusselt
constant
Nu=22,
of
seconds the
boundaries
flow coastdown and flow oscillation.
nonuniform power distribution and
for
are
to give the same inlet mass flow rate and
as
saturation temperature as those measured at the end
heated
the
outlet boundaries are positioned as
and
inlet
to
Similar
condition.
boundary
inlet
used for the
number
used for the
are
numerical simulation during transients.
6.2.2.2 Results
Numerical
results
Incipient
6.13-6.15.
for
the
boiling
P3A
are
starts
at
inlet flow reversal occurs at 10.6 seconds.
within 0.2 seconds after flow reversal.
Dryout
occurs
Figs 6.13-6.14 show
(12) as well as data at various positions
predictions
distribution.
agreement
9.4 seconds and
transients predicted by THERMIT-6S and COBRA-3M
temperature
COBRA-3M
in Figs
shown
with
were
with
The
time.
obtained by using uniform power
The predictions by THERMIT-6S are
in closer
experimental results, partly because of use
of uniform power distribution for COBRA-3M analysis.
Incipient boiling starts in the last cell of the heated
section of the
central
channel,
delayed
0.6
seconds
comparison with experimental incipient boiling time.
in
In the
-.
~~YI
215
I
1700
I
I
2-D predictions by THERMIT-6S
-
1500
I
Temperature data at 35.7"
(from the bottom of the fuel)
at 35.7" in the innermost
channel
X
1600 -
I
-
--
COBRA-3M predictions
1400
1300
4
1200
E-4
1100
1000
900
800
700I
0
2
I
I
4
6
I
8
10
12
Time (sec)
Fig. 6.13:
Comparison among Temperature Data, COBRA-3M
Predictions, and 2-D Predictions by
THERMIT-6S at 35.7" (from the bottom of the
fuel) in the Innermost Channel for the P3A Test
i
lli
Ell
II
I
I
i
216
1700
X
1600
Temperature data at 21.7"
in the outermost channel
2-D predictions by THERMIT-6S
-
-
1500
COBRA-3M predictions
1400
1300
e'
1200
I
°
/
) S1100ll,
1000 --
900
800
700
0
Fig. 6.14:
2
4
6
8
10
12
Time (sec)
Comparison among Temperature Data, COBRA-3M
Predictions by THERMIT-6S at 21.7" (from
the bottom of the fuel) in the Outermost
Channel for the P3A Test
217
initial stage of boiling, THERMIT-6S predicts a higher inlet
rate than the experiment because, while pump power was
flow
preset to decrease slowly
drop
pressure
to
up
constant
condition is used in the numerical
boundary
As shown in Fig 6.15, the rate
simulation.
a
seconds,
13
flow
inlet
of
coastdown in the computation after 0.8 seconds is more rapid
than
This can be explained by the
in the experiment.
that
reasons described in Section 6.1.2.2.
seconds
that
confirms the conclusion from the W-7b' simulation
flow
of
incidence
factor in the occurrence of dryout in low
to
high
void
flow/high
power
dryout, vapor is rapidly accelerated due
After
accidents.
the
the most important single
is
reversal
This
occurs.
dryout
reversal,
flow
after
Within
experiment.
almost at the same time as that in the
0.2
more
the data, flow reversal occurs
than
coastdown
flow
rapid
by
followed
As initial slower flow coastdown is
this
and
fraction
vapor
high
velocity
dramatically reduces the time step size.
Liquid
reentry
occurs
0.2
at
seconds
reversal, followed by inlet flow oscillation.
of
flow
oscillation
cycles/sec in the
than
larger
that
is
6
experiment
in
the
cycles/sec,
and
the
after
flow
The frequency
to
compared
amplitude
is
3
much
experiment due to the plausible
reasons described in Section 6.1.2.2.
Fig 6.15 shows predictions by the
16)
flow
and
NATOF-2D
code
(Ref
the present code as well as the experimental inlet
rates.
The
predictions
by
NATOF-2D
are
highly
0.6
0.4
''I 1
11
0.M2m1*
0
H
"
*-*
2-D predictions by THERMIT-6S
0
0.5
oBoiling Time
6.15:
JV-
A-A NATOF-2D
S-0.6
Fig.
a
1.'
41
1
1.5
2
2.5
Time(sec)
Comparison between Inlet Flow and Predictions by NATOF-2D and
THERMIT-6S for the P3A Test
THERMIT-6S for the P3A Test
3
219
and
irregular
In Ref 88
terminated at around 1.6 seconds.
the predictions by
SAS-3A
were
to
reported
in good
be
agreement with data.
6.3 OPERA 15-Pin Experiment
OPERA is an Out-of-Pile expulsion and Reentry Apparatus
OPERA 15-Pin (101) is a
at ANL.
simulates
the
sodium
which
experiment
initiating phase of a pump coastdown without
The objective
the scram of a LMFBR, specifically, the FFTF.
is to
of the OPERA 15-Pin pump-coastdown test
investigate
sodium boiling initiation and the subsequent two-dimensional
nature
of
sodium
voiding in large pin bundles approaching
the size of current LMFBR subassemblies.
bundle
the
steep
radial
In
large
a
pin
coolant temperature profile will
affect the behavior of vapor generation, vapor condensation,
onset of flow reversal, radial voiding, clad dryout, and the
axial void oscillations.
This
experiment
simulators
utilized
15
fuel
electric
pin
in a triangular thin walled duct to represent a
high powered sub-assembly in which each pin produces 33
Kw.
The 15-Pin triangular test bundle was fabricated to simulate
the one-sixth sector of a 61-pin hexagonal bundle.
6.3.1 Test Apparatus and Procedures
The OPERA facility is a
Steady-state
coolant
once-through
sodium
system.
flow with the inlet temperature 588 K
through the test section is generated by first
pressurizing
220
the
argon
with
vessel
blowdown
opening the sodium
gas,
control valves at the inlet to the test section, and at
the
the receiver vessel to atmosphere.
The
same
venting
time
pump-coastdown flow transient is generated by depressurizing
nozzles,
the blowdown vessel through a valve and two
which
can be preset by calibration procedures, to give the desired
flow decay.
The pin power is maintained at 33
3
until
Kw
heater
10% from the full 33
pins fail or their power output varies
Kw at which time all pin power is terminated.
Full
size
pins
interior
wraps
wire
where the outside
wall
O.D.)
are
on
the
pins on the one side of the bundle
the
and
(0.0142cm
conditions
are
to
be
simulated.
Small wire wraps (0.079 cm O.D.) and fillers (0.158 cm O.D.)
are used on the pins and walls at the other two sides of the
bundle where the interior part of the 61-pin bundle is to be
simulated.
This arrangement ensures that the power to flow
ratio of each subchannel is the same
subchannel
in a
hexagonal
61-Pin
as
the
corresponding
The
bundle.
thermal
capacitance of the wall is minimized by using a thin wall of
0.5 mm thickness.
6.3.2 Numerical Simulation of OPERA-15 Pin
6.3.2.1 2-D Analysis of OPERA-15 Pin
A 2-D
numerical
analysis
simulation.
is performed
As
shown
for
the
in Fig
OPERA-15
6.16,
Pin
the test
section consists of five mesh zones in the radial direction.
~-- -;-r_ ._iii~ ~ __
-.--II------~-221
Fig. 6.16:
Geometry of OPERA-15
Pin Bundle
222
Twenty
nine
mesh
zones
in
the
axial
direction
are
considered: 18 in the heated section (each 2') and 11 in the
(each 2" for low 3 mesh zones and each 6"
section
unheated
for the next 8 mesh zones).
outlet
Inlet and
outlet
of
the
boundaries
blowdown
vessel, respectively.
vessel
are
at
positioned
the
and inlet of the receiver
is
As the receiver vessel
connected
to the atmosphere, the outlet pressure is set to be equal to
one
over
atmosphere
the transients.
The inlet velocities
inlet
boundary
OPERA-15
Pin flow
measured in the experiment are used for the
condition
up
to
9
seconds
into
the
transients when boiling starts in the numerical
simulation.
After boiling the outlet pressure estimated by the figure of
depressurization
given in Ref 101 is used
for the outlet boundary condition.
Uniform radial and axial
blowdown
vessel
power distributions, and constant Nusselt number for
liquid
heat conduction, Nu-22, are used.
6.3.2.2 Results
The results by the present code were obtained by
the
using
pretest information package for the OPERA-15 Pin, which
contains
only
description,
and
experiment
apparatus,
operating
procedures.
interesting to compare predictions by the
test
section
Therefore, it is
code
to
transients
in
present
experimental data (102) which were published later.
Boiling first starts at 9 seconds
the
hottest
channel,
into
compared to 9.4 seconds in the test.
223
Fig 6.17 illustrates one of
the
reasons
primary
the
for
prediction of this earlier boiling: temperature predicted by
the present code is higher than data over transients.
Up to 2 seconds after boiling starts, flow coastdown is
very slow.
Figs 6.18 and 6.19 reveal
the
reason
of
this
slow flow coastdown: it takes about 1.8 seconds for vapor to
reach
the
large
inlet
channel due to the large bundle size,
outermost
hydraulic
and
resistance,
temperature gradients shown in Fig 6.11.
radial
large
In the test it was
estimated that the sodium void progresses from the innermost
channel to the outermost channel in about 1.5 seconds.
Rapid flow coastdown follows and flow reversal
at
the
inlet occurs at 3.1 seconds from incipient boiling, compared
to 2.6 seconds in the experiment. (See Fig 20) This delay of
flow
reversal
is attributed
to
larger
temperature drop
laterally across the bundle in the computation which is 144
K at boiling inception, compared to 109 K in the experiment.
Note
that
for
all
three
experiments
simulating the low
flow/high power accidents, the W-7b', P3A, and OPERA-15 Pin,
rapid flow
reaches
coastdown
around
occurred
when
the
outmost
channel
saturation temperature and vapor production
in some cells of the outmost channel takes place instead
of
vapor condensation.
The outlet flow is almost constant up
to
when vapor reaches the end of the test section.
the
outlet
3.2
seconds
Afterwards,
flow rapidly drops due to high void fraction at
the end of the test section, but never becomes the
negative
224
1400
1200
1000
800
Flow
Boiling
Time
Reversal
Time into transients (sec)
Fig. 6.17:
Comparison of Temperature
Transients of OPERA-15 Pin with
Pretest Predictions by THERMIT-6S
at 35.5" (above fuel bottom) in
Innermost and Outermost Channels
/
48
36
-
Void Fraction
--24 --
:
o:
0.1
0.5
x--x:
0.7
A--A:
dryout
16
O
0
Gas
Plenum
/
/
8
End Lt
of
Heated
Section0
Heated -8
Zone
0
'
C)
A
*
-
-12
0
1
+ Incipient Boiling
Fig. 6.18:
Time (sec)
2
3 Flow
Reversal
Axial Void Profile in Innermost Channel Predicted by THERMIT-6S for
the OPERA-15 Pin Test
48
'
0
36
Void Fraction
c
-
: 0.1
o-o
: 0.5
I
"
24
x--x : 0.7
tA--A
: dryout
16
O
I
Gas
Plenum 8
End
0
of
Heated
Section
*
0
Heated-8
Zone0
-12
Incipient Boiling
Fig. 6.19:
1
2
3 Flow Reversal
Axial Void Profile inOutermost Channel Predicted by THERMIT-6S
for the OPERA-15 Pin Test
4
0.75
- W0 . 5
r.
41
00
4J
o:3
0
cId
'U 0
0
H
-4
U)
U)
0)
UI)
0r
-0.5
9
9.4
10
+
11
12
SIncipient boiling time in the test
Time into transients (sec)
Predicted incipient boiling time
Fig. 6.20:
Comparison of Flow Transients of OPERA-15 Pin with Pretest Predictions
by THERMIT-6S
228
flow
because the interfacial force by high vapor velocities
drags the liquid along.
Within 0.05 seconds
predicted.
However,
experiment was
inlet
flow
after
the
substantially
flow
reversal,
occurrence
delayed;
of
in the
dryout
elasped
is
dryout
from
time
reversal to dryout is around 2 seconds.
In the
computation liquid reentry occurs at 0.14 seconds after flow
reversal and is followed by inlet flow oscillation of
which
the frequency is 3.5 cycles/sec, compared to 2 cycles/sec in
the experiment.
6.4 THORS
During
certain
loss-of-pipe-integrity
postulated
in a loop-type liquid-metal fast breeder reactor
transients
(LMFBR), the flow-power mismatch, even with
may
be
of
Experimental
boiling
and
analytical
sodium boiling in an electrically heated 19-pin
simulated fuel assembly have been
National
scram,
to boiling conditions of several seconds
conducive
duration in the reactor core.
studies
reactor
(ORNL) to
Laboratory
stability
at
the
low
made
at
the
investigate
Ridge
Oak
the extent of
flow/low power
transient
condition.
6.4.1 Test Apparatus and Procedures
The THORS Bundle 6A experiments (40) were performed
1978
at
simulated
ORNL.
LMFBR
The
fuel
test
section
assembly.
in
used was a full-length
It
consists
of
19
229
hexcan is a 0.02"
bundle
The
pins.
heated
electrically
thick 316 stainless steel backed by 1" of Marimet insulation
contained in a stainless steel jacket.
Early in the test program, a comparison of experimental
duct-wall temperatures with analytical predictions indicated
higher values for hexcan thermal inertia
and
that
suggested
was
it
insulation may have occurred.
some
anticipated,
than
sodium leakage into the
Posttest examination revealed
that sodium had indeed penetrated the entire region.
The heated length of Bundle 6A
distribution.
chopped-cosine power
variable
to produce a 1.3 axial peak-to-mean
windings
heater
pitch
with
is 36"
the
inside
core
The
and the insulation between the winding and the type
winding
plenum
fission-gas
simulated
nitride.
heated length, a nickel reflector and a
the
of
Downstream
boron
compacted
316 stainless steel sheath is of
have
length
and
simulate
the
same
the
thermal inertia as a FFTF fuel assembly.
to
The test-section inlet value was set
of
drop
nominal-flow
pressure
configuration.
The test-section
was
set
at
F.
730
bundle power, a
sodium
inlet
reactor
the
temperature
inlet
For each run, at a specified constant
sodium
flow
was
to
established
give
a
test-section bulk outlet temperature of 1300 F.
The test 72B, run 101 is
simulation
because
available in Ref 40.
power
10.0
kw/pin.
selected
for
the
numerical
predictions by SAS-3A for this test are
This test was a dryout
Before
flow
reduction
run
the
with
began
at
3
230
seconds, the flow was about 9.3 gpm (0.58 l/s) and the
was
up
reduced
Incipient boiling
3.7 gpm (0.23 1/s).
to
flow
occurred at 12 seconds and afterwards, stable boiling lasted
until 22 seconds, when a period of dryout and
At
rewet
began.
seconds, permanant dryout began and power was cut at
30
31.5 secondss
6.4.2 Numerical Simulation of the test 72B, run 101
6.4.2.1 2-D Analysis of the test 72B, run 101
A 2-D analysis of THORS was performed,
with
the
test
section divided in the same way as the W-7b' analysis.
Inlet and outlet boundaries are positioned at the inlet
and outlet of the test section because the inlet and
pressures
the
of
test
boiling
condition is
started,
used.
condition
boundary
the
After
is
into
the
the
and
kept
transients,
velocity-pressure
boiling
used
and
measured
Up to 12 seconds
constant after boiling.
when
were
section
outlet
boundary
pressure-pressure
inlet
and
outlet
pressures are kept at a constant value which gives the
same
mass flow rate as the experimental value at 12 seconds.
The
1.3
axial
peak-to-mean
uniform radial
number
for
distribution,
power
liquid
and
constant
Nusselt
conduction, Nu=22, are used.
heat
properties of Marimet
sodium,
chopped-cosine power distribution,
insulation
degraded
by
leakage
The
of
which were estimated in Ref 40 by assuming that all
the void space was full
Appendix B)
of
sodium,
are
used.
(Refer
to
231
6.4.2.2 Results
Strong radial temperature gradients
in
are
test
the
in Fig 6.21. The temperature difference between the
evident
just
centerline and periphery of the bundle is almost 160 K
the
On
boiling.
before
other
hand,
code
temperature gradients are predicted by the present
is the
use
liquid conduction model.
of
a high Nusselt number for the
The Nusselt number 22 was obtained
liquid
by comparison to a steady state experiment with high
As
(99).
velocity
as
One of the plausible explanation on this
shown in Fig 6.21.
discrepancy
radial
milder
the
72 B, run 101 is a low flow
test
transient, it is expected that turbulent contribution on the
the
6.22,
Fig
In
effective liquid conduction will be reduced.
predictions are in good agreement with the data inspite
radial
of the strong
temperature
gradients,
because
the
difference between the predictions and the data
temperature
in the intermediate channel is relatively small, compared to
that in the outermost and innermost channels.
This mild predicted radial temperature gradient can
one
of
reasons
the
boiling.
Strong
influence
the
simulation
as
be
for the 1.5 second delay of incipient
radial
temperature
highly
gradients
prediction of incipient boiling time for 1-D
explained
in
Section
Actually
6.1.2.2.
SAS-3A, with 100% of structpre heat capacities which implies
that all the void space in the Marimet insulation was filled
with
liquid
inceptions
sodium,
were
predicted
delayed
11
that
seconds
boiling
and
and
24-26
dryout
seconds,
232
Z- 33.00IN.
0,
: Data
-
: Predictions by
THERMIT-6S
o
10.0
of THORS)
o0.0 101
SECS.
a-10.0
o
0.0
0.2
0.1
0.6
0.8
1.0
RADIUS (IN.)
Fig.
6.21:
Temperature vs. Radius with Time as a
Parameter at Z = 33" (test 72B, run
101 of THORS)
14100
Time
Fiq.
6.22:
(sec)
Conmarison between Predictions by TtIERMIT-6S and
Te',mpe(,rature( I)Data at 11" (from the I)ottoI
of the fuel)
in th
It fmreldiate Chalnel for the test
720, rul 101
of 'Ti)i1S
234
0% of structure heat capacities SAS-3A
With
respectively.
was satisfactory in predicting boiling inception.
In the initial
oscillation
and
low
enough
resistance.
hydraulic
which
have
non-condensible
out-of-pile
boiling,
inlet
large
a
for
such
experiments
as
nucleation
of
boiling,
THORS are likely to have
is possible in out-of-pile experiments.
boiling
in
in-pile
such as fuel to provide
source
gas
Unlike
superheat
little non-condensible gas and as a result, large
incipient
flow
in the test takes place due to large superheat
inlet
experiments
of
stage
As no superheat for
the present analysis is assumed, Fig
shows the small amplitude of inlet flow oscillation in
6.23
6.23
the initial stage of boiling. Figs
approximate
boundaries
of
predicted
much
6.24
present
the inlet mass flow rate in the
reading
in
experiment because of a difficulty
SAS-3A
and
data.
The
larger oscillation than the present
code does.
While dryout in low flow/high power accidents
the
such
as
W-7b', P3A, and OPERA-15 Pin occurred immediately after
flow reversal, dryout in the test 72
after
3
B, run
101
occurred
seconds from flow reversal. (see Fig 6.25) Another
distinctive characteristic of the test 72 B, run 101 is
periodical
occurrence
of
dryout
bundle. (See Figs 6.21, 6.25, and
power
the
and rewet over the whole
6.26)
In
low
flow/high
accidents this phenomenon did not occur or lasted for
a very short time.
:
Approximate Boundaries
of Inlet Flow Data
:
Predictions by TIIERMIT-6S
-3
0
1
2
3
4
5
6
tincipient Boiling
Fig.
6.23:
Flow Data and Predictions
Comparison between Inlet
by TIIERMIT-6S for the Test 72U, Run 101 of TIIORS
-:
Approximate
Boundaries
of Inlet
Flow Data
-- : Predictions
by SAS-3A
I Incipient.
Fiq.
6.24:
r) il inq
Time (sec)
low lata and Predict ions
('Comparismo) bheween Inletl
for t li Tv';I 7211, Riun 101 of TIIIS
by ;A; - IA
Gas
Plenum
Upper Blanket
t
End
of
Hleat
Sect ion
Time
Fiq.
6.25:
(set)
Axial Void Profile in the Innermost Channel Predicted by
TIIERMI'T-6S for the Test 72D, Run 101 of T'IIORS
I
Gas
Plenum
t-
Upper
.8 Blanket
End
t
4
of
r
Heated
4 Section
Heated
Sect ion
16
8
Bottom of
Ileated
0
Section
0
Fig.
1
6.26:
2
3
Time
(sec)
Axial Void Profile
in the Outermost Channal Predicted by
TIHEMIT-6S for the Test 7213, Run 101 of TIIORS
.
239
6.5 Sensitivity Analysis
W-7b'
The sensitivity analysis reported here is donefor the
simulation.
6.5.1 Interfacial Mass and Energy Exchange Models
the
5.2.4,
Section
in
described
model
equilibrium
model
and
is
the
model
The kinetic
kinectic model are considered.
model,
exchange
mass
For comparison with the present
theory
adopted in NATOF-2D based on a simple kinetic
with the condensation coefficient set to unity.
The
equilibrium
temperature
assumes
model
With an
(17),
infinite heat transfer coefficient assumed in THERMIT
both
vapor
While the liquid film at a
temperature.
in
is
zone
equilibrium
forced to have the saturation
are
liquid
and
reality
model
it
at
is
no
is
there
vapor and liquid.
between
difference
that
a
saturated
superheated
state,
at
saturated
always
a
boiling
in
the
state.
Therefore, the superheat energy contained in the liquid film
has
to
be transferred to vapor by mass exchange to achieve
the saturation state.
the
difference
Note that mass exchange is driven
between liquid and saturation temperatures.
high
Then in the heated section with
temperature
by
gradient
in
the
from
region will
be
the
saturated
condensed.
or
high
liquid film, the equilibrium
model predicts higher mass exchange rates.
transferred
flux,
heat
The
region
kinetic
Also
to
all
vapor
the subcooled
model
almost the same results as the equilibrium model.
produces
'I~--------~
-
I------ ------ ---
--
-I
---------~-;;-~
--
-
240
As a result, the equilibrium model and
predict
higher
kinetic model
condensation and evaporation rates than the
present model, which allows a nonequilibrium
vapor
and
may
cause
iteration.
failure
greater
of
pressure
convergence
As expected, severe numerical
encounted;
calculation
with
both
due
to
failure
of
fluctuation
of
the
Newton
difficulties
are
homogeneous
and
the
kinectic models stopped within 0.1 seconds
boiling
between
liquid. (See Fig 5.10) These larger condensation
and evaporation rates induce
which
state
after
convergence
of
incipient
the
Newton
iteration.
6.5.2 Interfacial Momentum Exchange Models
The
present
compared
with
model
described
in
Section
5.2.4
is
the homogeneous model and the Autruffe model
which was implemented in the old version of THERMIT.
In the
homogeneous model vapor and liquid velocities are forced
be
equal
by
an
infinite momentum exchange.
The Autruffe
model was developed with data obtained in annular flow
large
amount
result,
with
of droplets and this included the interfacial
momentum exchange by both the liquid film and droplets.
a
to
As
his model predicts much larger drag coefficients
than the present model does.
The larger interfacial drag force, the more vapor
be
trapped
in the cells.
Therefore, the homogeneous model
predicts the largest void fraction
Notice
will
as
seen
in
rig
6.27.
that the homogeneous and Autruffe models predict the
----------
241
Dryout
0.957
S0.5
*.4
0
0.5
1 1.1
Boiling Time
Time (sec)
Fig. 6.27:
Comparison of Void Fraction in
the Last Cell of Heated Section
Predicted with Various Interfacial
Momentum Exchange Models
-I
mov
niiiniiiwi
i
Ohl"
1111I
ii IIlllI
li
ill1i1ili
I.
m
EIi
n
mMM
mlkI
242
causes larger pressure build-up in channels,
vapor
trapped
rapid
and in turn more
flow
velocities.
large
have
velocities
(See
6.28)
flow
earlier
Also
large
the
dramatically reduce time step sizes and increase
The
CPU time as illustrated in Fig 6.29.
the
and
coastdown
This large void fraction drives vapor and liquid
reversal.
to
more
this
Also
occurrence of dryout before flow reversal.
homogeneous
computation
with
model stopped at around 0.7 seconds due to
failure of convergence of the Newton iteration.
6.5.3 Boundary Positions
Let us investigate the importance of selecting boundary
W-7b'
positions by comparing inlet flow rates of
predicted
with two choices of boundary positions: the inlet and outlet
of
the
section
test
(called A boundary) and the boundary
positions shown in Fig 6.1 (called
cases
pressure
the
drops
boundary).
B
the
between
inlet
In
and outlet
boundaries are set to give the same inlet mass flow rate
the
experimental
to 3.5 seconds.
both
as
value at 1.5 seconds and kept constant up
In the
B boundary
case
the
hydraulic
resistances between the inlet boundary and inlet of the test
section,
between
and
the
outlet
of the test section and
outlet boundary are estimated by the valve
model
described
in Appendix D.
In the
pressure
B boundary
drops
case
flow
coastdown
decreases
between the inlet boundary and inlet of the
test section and between the outlet of the test section
and
243
1
150
100
-
Homogeneous Model
A-A : Vapor Velocities by
Autruffe Correlation
A-A : ILiquid Velocities by
Autruffe Correlation
Vapor Velocities by
the Present Model
o-o
Liquid Velocities by
the Present Model
1
/as
,6
i
!
rI
____ ____ ___L
0
Boiling Time
Fig. 6.28:
0.5
Time (sec)
1
Comparison of Vapor and Liquid
Velocities Predicted at the Last
Cell of the Heated Section with
Various Interfacial Momentum
Exchange Relations
1.1
244
3500
*00*.. Homogeneous Model
-
3000
Autruffe Correlation
- - Present Model
2000
1000
0
0
Boiling Time
Fig. 6.29:
0.5
Time (sec)
1
Comparison of CPU Time for the
W-7b' Test Using the MULTICS
Computer
1.1
245
boundary.
outlet
As
result,
a
flow
pressure at the inlet of the heated
the
at
pressure
of
outlet
reduction increases
decreases
but
section
the heated section, while the
are
inlet and outlet pressures in the A boundary case
in the pressure drop between the
increase
This
constant.
kept
inlet and outlet of the test section
substantially
retards
flow reversal as shown in Fig 6.30.
6.5.4 Normalization of Heat Source
Q(t)
In order to partition the total power
into
each
mesh in the fuel rods, the volumetric heat generation rate q"'
in THERMIT is obtained by using the following equation
= Q(t)
q'" (i,j,k)
where
qz
qt ,
'
distribution
in
and
qz ( j) ) qt(i)N) qr(k)
(k)
N Nt Nr
qr:
axial
and
(6.1)
transverse
power
the reactor, and radial power distribution
within a fuel rod
N,
z
N
Nt and N :
t
r
=
Nt =
N
=
normalization factors
q (j) dz(j +1)
(6.2)
qt(i) rn(i)
qr (k)a(k)
dz, rn, and a in Eq (6.2) are mesh spacing of j+1'th cell in
the z direction, number of fuel rods in
the
i'th
channel,
246
-
: Data
*-* : Predictions
B Boundary
I.
A Boundary
0
r
1
I
0
1
2
-
3
4
6
Time (sec)
Fig. 6.30:
Comparison between Inlet Flow Data
and Predictions for A Boundary
and B Boundary
247
and
horizontal
plane cross-sectional area of the k'th mesh
cell in the fuel rod, respectively.
The assumption involved in Eq (6.1) is that
at
the
all
same positions have the same geometry, that is, the
same thickness of cladding, gap distance and radius
center
fuels
fuel
different.
void
region.
of
the
The actual geometry of fuels is
Therefore, the
general
normalization
of
heat
source described in Appendix E is needed.
Let us
investigate
normalization
by
the
comparing
(6.1) (called as Case B).
gap
importance
it
of
this
(called as Case A) with Eq
The W-7b' test, in which
several
distances were used, is selected for comparison between
Case A and Case B. In the real geometry of the
gap
general
W-7b'
test,
distances and radii of center fuel void were 0.05588 mm
and 0.8763 mm for the center fuel, and 0.07974 mm and 0.0 mm
for seven fuels in the second ring, respectively.
B,
0.07974
mm
On
distances and
channels
innermost
cases at
different
the
channel,
the
other
radii
in Case
intermediate
Case
for the uniform gap distance and 0.0 mm for
the uniform radius of center fuel
channels.
In
of
A;
and
channel.
same
volumes
hand,
we
center
mm
0.07974
Then
fuel.
are
the
used
for
all
can take different gap
fuel
0.05588
positions
of
void
void
and
mm
and
wall
for
0.8763
0.0
different
mm
mm
in the
in the
heat flux for both
are
different
due
to
Fig
6.31 verifies the above
the
explanation; Case B has the larger amount of fuel and higher
heat flux than Case A does.
~---- -
I IIII
Y
I1MII
IIII
248
10,000
b
9500
9
1-3 0
0
X
IDo
5000
I
0
Fig. 6.31:
0.5
I
1.0
Time (sec)
Comparison of Temperature and the Rate of
Wall Heat Transfer in the Last Cell of the
Innermost Channel for Case A and Case B
of Power Normalization
2500
1.5
249
Also as expected, the centerline fuel temperature
the
larger
with
gap distance is higher by about 400 K than that
with the smaller gap distance. (Fig 6.32) The wall heat flux
of the last cell of the
channel
for
Case
B
heated
section
in the
innermost
is higher than that for Case A. As a
result, Case B predicts higher fluid temperature and earlier
boiling inception.
mm-
-1111
__
II
250
3200
S3000
2800
-4
ra
0
-
2600
-
.
-
*"----
*-I
w
2400
: Data
A-A
-
2200
0
Fig. 6.32:
: Predictions for Case B
: Predictions for Case A
0.5
Time (sec)
Comparison between Centerline Fuel
Temperature Data in the Last Cell of
Innermost Channel and Predictions for
Case A and Case B of Power Normalization
251
Chapter 7 SUMMARY AND RECOMMENDATIONS
7.1 Summary
limitations
the
reveals
work
Review of previous
of
models involved in several codes which were designed for the
of
can be attributed to the characteristics
such
These limitations
of sodium boiling in the LMFBR.
analysis
the LMFBR such as the
existence
and
reversal,
flow
region,
itself
high density ratio of liquid to vapor, and of
the
as
sodium
of
temperature
radial
sharp
subcooled
highly
the
gradients.
The high density ratio induces high slip between liquid
use
of
homogeneous
the
possibility
the
and vapor and, as a result, eliminates
model which assumes no slip.
The
of
the
bundle
existence of highly subcooled region in the
LMFBR,
where
is
vapor
of
numerical
causes
condensed,
difficulties due to large pressure fluctuations.
Use of the
be
completely
equilibrium approach, in which
condensed
the
in
subcooled
must
vapor
region,
Therefore,
fluctuations and in turn numerical difficulties.
the
non-equilibrium
approach
instead
pressure
enlarges
of
the equilibrium
approach is needed to represent the physical conditions.
are
flow
correlations for slip in annular or reversed
As
not
readily available, the slip model is difficult to apply.
Sharp
radial
multi-dimensional
voiding
multi-dimensional model.
gradients
temperature
behavior
lead
us
The 3-D representation
and
to favor the
is chosen
-
I
I
-pl---.i~-ii~i--i-
Y
252
because
a
3-D
model
can
be
used to simulate 1-D or 2-D
geometries.
According to the above brief review, the two-fluid
3-D
model is more suitable for the analysis of sodium boiling in
The THERMIT code.(17) is taken as the basic tool
the LIFBR.
for
the
of
development
the two-fluid 3-D code due to its
following features:
1.
It is based on a two-fluid 3-D geometry
2.
It has an advanced numerical scheme
3.
It allows flexible boundary conditions
Although cylindrical (r, 0, z) coordinates are better fitted
for the geometry of
rectangular
(x,
the
hexagonal
of
bundle
the
LMFBR,
y, z) coordinates are used for the present
code due to the easy adoption of the original THERMIT code.
To investigate the main
equations
differential
rigorous derivations of
assumptions
of
involved
mathematically
THERMIT-6S,
time-volume
the
in
averaged
conservation
equations for the two-fluid model were done by starting from
the
integral
general balance equation.
THERMIT-6S includes all terms to
It was found that
dominantly
to
contribute
the analysis of sodium boiling.
The
THERMIT-6S
discretized
equations.
differential
were
equations
then
in the same way as THERMIT to obtain difference
Spatial discretization
is
characterized
by
a
first-order spatial scheme, donor cell method, and staggered
mesh
layout.
For
time
discretization,
a
first-order
semi-implicit scheme treats implicitly sonic terms and terms
253
related to local phenomena, and treats explicitly convective
Therefore, the
terms.
step
time
in THERMIT-6S
size
is
limited by the convective limit:
At
<
AzI
IAz
z max
The most important characteristic of the present
lies in the treatment of coupling terms. Coupling
equations
time
between mesh cells at the new
the
is through
level
Hence, we need only to solve the reduced
pressure variable.
pressure
difference
which form a NxN matrix (N is the total
equations
number of fluid cells) in 3-D, compared to a 10N x0ON matrix
implicit .method.
of equations obtained by the ordinary
the
of
property
crucial
reduced
A
pressure problem is its
diagonal dominance.
It is known that the typical two-fluid
constitutes
To
problem.
ill-posed
an
virtual
the
represents
mass
accomplish dissipation
found
that
the
numerical
for
virtual
real
these
overcome
difficulties, a virtual mass term was studied
physical
here
because
effects
stability.
It
a
real
physical
effect.
Quantitative
Drew
et
al.
(33)
was
were
suggested
for
well
bounds on the
coefficients of the virtual mass terms in a model
by
to
mass has a profound effect on the
mathermatical characteristic and numerical stabiliy as
as
and
limit
short-wavelength
unbounded instabilities in the
as
exhibits
characteristics,
complex
possesses
THERMIT
such
model
proposed
mathermatical
--;--;
-----------
-~~P. i-i i_.
_ 1 ilX-pR
=_-_l--1PI-
254
hyperbolicity, numerical stability, and satisfaction of
the
thermodynamics second law.
For the closure
functions,
of
the
conservation
equations,
the
which represent the transfer of mass, energy, or
momentum at solid-fluid and vapor-liquid interfaces, must be
explicitly expressed
in
properties.
the
While
terms
of
two-fluid
primary
model
provides
applicability and flexibility in the analysis
flow
variables
of
and
wider
two-phase
than other models do, its accuracy strongly depends on
these constitutive equations.
The constitutive relations for pre-dryout were derived,
based on two assumptions:
1.
The exchange mechanism is
dominantly
controlled
by
local phenomena.
2.
The cocurrent annular flow regime in
two-phase
flow
of sodium is dominant on the scale of the subchannel.
These two assumptions lead to a simplified flow regime which
facilitates not only developing
physical
models
obtaining
convergence of the Newton iteration.
obstacles
to
discrepancies
obtaining
between
constitutive
clarified
also
Significant
equations
lie
in
data of different authors as well as
limited experimental data available for
sodium.
but
two-phase
flow
of
Here, the main reasons of these discrepancies were
and
physical
models
were
derived
within
the
present capabilities, based on the above two assumptions.
By reviewing several publications on frictional pressure
drop in the two-phase flow of liquid metals,
it
was
found
255
discrepancies between data of different authors become
that
It
flow.
annular-dispersed
or
pattern: pure annular
flow
the
by
characterized
are
which
groups,
two
into
divided
The review indicates that the data can be
evident.
is
argured that the difference between the two groups is caused
by
of
evaluation
the
in
commonly neglected
drop of droplets, which was
pressure
acceleration
the
Lockhart-Martinelli
correlations
turbulent-turbulent
flow,
fall much
the
below
for
and
viscous-viscous
diabatic annular flow
for
data
the
between
fall
flow
annular
adiabatic
frictional
while most of data
pressure drop in annular-dispersed flow.
for
the
Lockhart-Martinelli
for
correlation
turbulent-turbulent flow.
Discrepancies between data of
been
to
due
reported
boiling
different
have
authors
Recently
instability.
Zeigarnick and Litvinov provided data for wall heat transfer
Their
which were obtained with complete boiling stability.
data
were
used to validate wall heat transfer coefficients
for both boiling and condensation in the
flow
sodium.
of
wall
The
convective
transfer coefficient in
heat
two-phase flow was derived by using
forced
momentum-heat
transfer
analogy and logarithmic law for velocity distribution in the
liquid
film.
Only
one
in the logarithmic form
constant
The
needs to be empirically determined.
suggested
with their data over
proposed
were
correlation
correlation
a
broad
from
the
to be in good agreement
found
range
predicts
results
of
higher
parameters.
heat
The
transfer
256
coefficients than previous correlations do.
Two control mechanisms, "vapor supply" and 'latent heat
energy
transport", were proposed for the models of mass and
The rate of exchange was assumed to be controlled
exchange.
The
values.
by whichever mechanism yields smaller
former
mechanism is described by a simple kinectic theory approach.
The
flux model is taken as the basis for a model
turbulent
of the
A
mechanism.
latter
modified
When
introduced to account for the energy losses of eddies.
the
liquid
bulk
values,
they
found
were
of
time,
predicted
In a
be in close agreement.
to
comparison of the void volume
function
analytically
with
compared
was
proposed model
according to the
calculated
temperature
in
a
calculational
19
is
theory
mixing
as
bundle
pin
fall
results
a
around
into
the
A one-dimensional steady-state model was developed
for
experimental results until release of fission gas
sodium vapor from the pins occurred in the test.
vertical
upward
annular-dispersed
It
momentum tranfer by droplets.
obtained
from
the
model
suggested
correlation and the Moeck
was
correlation
flow
to
account
that
shown
fall
for
results
near the Wallis
which
were
derived
from air-water data.
Pre-dryout
generally
predictions
in good
by
agreement
the
with
present
data.
code
were
Discrepancies
between data and predictions emerge after dryout. The higher
frequency and larger amplitude of flow oscillation than data
were predicted.
Actually, the present physical
models
for
257
post-dryout
are
highly questionable because they have been
derived by a simple extension
Therefore,
pre-dryout.
of
liquid
film
models
for
development of physical models for
post-dryout is recommended.
the
can
code
simulate
the
experiments
and
to
assess
capability
of
the
present
process measured in
physical
the
is
The
capability.
code's
code
well
how
A code assessment was performed to determine
the
by
restricted
followings:
1.
validation
The
implemented
in
the
of
range
--
code
equations
constitutive
Most
of the constitutive
on
based
equations for two-phase flow are derived,
the
it
is
questionable whether these constitutive equations can
be
annular
convective
forced
applied
to
Though
flow.
natural convection flow, it is expected that
two-phase
the differences in the local processes between
flow
under
natural convection and forced convection may
be much smaller than those in single phase flow.
2.
No fuel melting model -- The code can be
applied
to
simulation
is
predictions of phenomena before fuel melting.
3.
A component code -- However,
a
loop
possible, if the code is slightly modified.
The W-7b' (in-pile 19 pin bundle), P3A (in-pile 37
OPERA-15
Pin
(out-of-pile
bundle) were chosen for
the
numerical
bundle),
and
flow/high
power
simulations
present
accidents.
a
series
These
of
simulator
simulation
three
following
pin
of 61
of
low
numerical
physical
258
phenomena:
1. After incipient boiling rapid vapor production in the
and
saturated region occurs due to high power
density
ratio,
axially
from
and
the
the
high
is transferred radially and
vapor
region
saturated
to
subcooled
the
Strong cross flow caused by rise in pressure in
region.
As a result, mild
inner cells reduces pressure build-up.
flow coastdown in the initial stage of boiling occurs.
2.
When
build-up
and
rapid
coastdown
Flow
place.
take
reaches
channel
expulsion
vapor
temperature,
saturation
pressure
outermost
in the
fluid
is
by these sharp increases in vapor generation
accelerated
and pressure.
3. Rapid flow coastdown leads to
this
time
rapidly,
flow
reversal.
From
on voiding of the rest of the bundle proceeds
with
the
whole
bundle
indicating
a
single
channel response.
4.
dryout
In a short time after flow reversal,
It turns
occurs.
out that the incidence of flow reversal is the
dryout
most important single factor in the occurrence of
in
the
low
flow/high
power
Dryout
accidents.
dramatically reduces wall heat transfer coefficients
sharp
rise
in the
cladding
Then, vapor is rapidly
fraction,
takes place.
temperature
accelerated
due
and
to
high
void
and this high vapor velocity reduces time step
sizes and in turn increases CPU time.
5. Liquid reentry occurs in a
short
time
after
flow
259
reversal, followed by inlet flow oscillation.
amplitude of the flow oscillation
larger
and
frequency
The higher
than data are predicted.
For the numerical
accidents,
the
power
flow/low
low
of
simulation
test 72B, run 101 of THORS was chosen.
are
accidents
distinctive characteristics of this type
The
as
follows:
1. While dryout in the low flow/high power accidents was
predicted
to
occur
immediately
flow
after
reversal,
dryout in THORS was significantly delayed.
2. The periodical dryout and rewet over the whole bundle
occurred for a long time until a sustained dryout began.
analysis
Comparison between 1-D and 2-D
performed
by
the present code reveals the following facts:
1.
Incipient
substantially
in
boiling
analysis
1-D
can
be
delayed when experiments with steep radial
temperature gradients are simulated.
2. 1-D analysis predicts more rapid flow coastdown
2-D
analysis
does.
While
in 1-D analysis 'there is no
room to condense vapor transferred in the
strong
cross
flow
caused
than
r direction,
by rise in pressure in inner
cells in 2-D analysis reduces pressure build-up.
Sensitivity
analysis
confirms
the
review
of
the
limitations of the models described at the beginning of this
chapter.
Severe
numerical difficulties in the equilibrium
model were encounted; calculation stopped within 0.1 seconds
after incipient boiling due to failure of convergence of the
260
Newton iteration.
void
fraction
homogeneous
The
and
larger
model
pressure
build-up,
result, earlier dryout and flow reversal
higher
predicts
and
the
than
as a
present
model does.
7.2 Recommendations
Based
on
the
experience,
present
following
the
recommendations for future work are made:
1. The development of a loop
recommended
to
account
simulation
capability
for the effect of a loop system
and to obtain accurate boundary conditions.
Section
is
As shown
in
6.5.3, boundary conditions have great effects on
flow transients.
As the code is originally
designed
to
be applicable to a component system, boundary values have
to
be
data
by
provided
experiments measured the inlet and
the
section,
test
usually
boundary
needed,
which
values.
To
simple loop model is
hydraulic
outlet
estimation
rough
leaves
by
pressures
to
few
of
iteration was
uncertainties
eliminate
needed
Since
estimation.
or
in
the
these uncertainties, a
account
for
the
loop
including a simple pump model for
resistance,
pump trip.
2.
Development
recommended
to
of
an
reduce
advanced
CPU time.
numerical
sizes
are
limited
by
the
is
After dryout vapor is
rapidly accelerated due to high void fraction.
step
scheme
convective
As
time
stability
condition, this high vapor velocity (more than 100 m/sec)
261
sizes
dramatically reduces time step
less
(often
seconds) and in turn increases CPU time.
10~
than
Moreover,
consider a multidimensional effect on'the control
if we
unacceptable.
totally
scheme
numerical
10-*
to
up
reduced
be
will
sizes
time
stability,
numerical
of time step sizes for
seconds, which is
unless
Therefore,
an
advanced
capability of the
the
is developed,
step
present code is limited to a few second predictions after
Studies of the two-step method or fully implicit
dryout.
method are suggested.
questionable.
highly
models which were
post-dryout
are
They are based on the liquid film
developed
for
However,
pre-dryout.
the wall is hot enough, there is little liquid left
when
Then, droplets would play an important role
on the wall.
in transferring mass, momentum,
region
the
to
for
models
physical
present
3. The
physical
energy.
Moreover,
and duration of post-dryout are large enough
flow
affect
and
Therefore,
behavior.
development
models for post-dryout is recommended.
of
Because
there are little data for post-dryout, it is inevitable
that
physical
model for the LWR would be used, based on
analogy of dryout mechanisms between water and sodium.
4. The liquid conduction model is important in prediting
accurate radial temperature gradients, incipient
time, and flow reversal time.
which
was
obtained
boiling
A constant Nusselt number,
from a high flow test, was used for
liquid radial conduction.
In the numerical simulation of
IIIIIWIIIMIAIIIIIYYY
IIIIMI
HYII,
262
THORS, which are
low
flow
experiments,
milder
radial
temperature gradients than the data were predicted due to
use of the high Nusselt number.
improved
liquid
conduction
Therefore, a study of an
model
is recommended
account for effects of various flow variables
conduction.
on
to
liquid
263
References
1.
W.D. Hinkle, "LMFBR Safety and Sodium Boiling,
A
State
of the Art Report," draft, M.I.T. (Dec. 1977)
2. R.C. Noyes, J.G. Morgan, and H.H. Cappel, "Transfugue-1,
A
Digital
Code
for
Two-Phase
Transient
and
Flow
Heat
Transfer," NAA-SR-11008 (July 1965)
3. J.E. Meyer, "Hydrodynamic Models for
the
Treatment
of
Reator Thermal Transients," Nuclear Sci. and Eng., 10 (1961)
4.
J.A.
Transient
Landomi,
Two-Phase
"Transfugue-IIa,
Flow
A
Single
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Code
Digital
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Heated
NAA-SR-12503 (March 1968)
5. D.R. MacFarlane, Ed. "SASIA, A
Analysis
Computer
Code
for
the
of Fast Reator Power and Flow Transients," Argonne
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6. F.E. Dunn, et.al., "The SAS2A
LMFBR
Accident
Analysis
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7. M.G. Stevenson, et.al., "Current Status and Experimental
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Beverly
Hills,
California,
USAEC-CONF 740401, p.1303 (1974)
8.
J.E. Cahalan, et.al.,
Computer Code," ANL
"SAS3D
LMFBR
Accident
Analysis
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h i
I
I
bI
ll i
264
9.
J.E. Cahalan, et.al., "The Status and Experimental Basis
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10.
P. Wirtz, "Ein Beitrag zur
des
Siedens
unter
theoretischen
Beschreibung
Storfallbedingungen in natriumgekuhlten
Schnellen Reaktoren," KFK 1858 (Oct. 1973)
11.
E.R. Siegmann, "Theoretical Study of
in Reator
Boiling
Sodium
a Compressible Flow
Utilizing
Coolant
Transient
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12.
W.W. Marr,
"COBRA-3M:
Thermal
Analyzing
A
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Digital
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Behavior
Code
for
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Argonne Nat. Lab. ANL-8131 (Mar. 1975)
13.
C.W. Stewart, et.al., "COBRA-IV : The
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and
the
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14.
G.
Basque,
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Analysis
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IAEA
D.
Grand,
Experimental
Incoherency
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and
B.
Menant,
Evidence
in
of
"Theoretical
Three
Undisturbed
Meeting
Fast Breeder Reator Fuel Sub-Assemblies
Types
of
Cluster
on Thermodynamics of
under
Nominal
and
non-Nominal Operating Conditions, Karlsruhe (Feb. 1979)
15.
A. L. Schor, "A Four-Equation Two-Phase Flow Model
Sodium
for
Boiling Simulation of LMFBR Fuel Assemblies," Ph. D.
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265
16.
M.R. Granziera and
Kazimi,
"A
Two-Dimensional,
Fuel
LMFBR
in
Boiling
Sodium
for
Model
Two-Fluid
M.S.
Assemblies," MIT-EL-80-011 (May 1980)
17.
W.H.
Program
Reed
and
H.B.
Stewart,
"THERMIT--A
Computer
Three-Dimensinal Thermal Hydraulic Analysis of
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Light Water Reator Codes," M.I.T.
18.
M. Ishii,
Dynamic
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Two-Phase
of
Theory
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19.
S. Banerjee and A.M.C. Chan, "Seperated Flow Models-i,"
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20.
Models
M. Ishii, "Foundation of Various Two-phase
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R. Aris, Vectors, Tensors and the
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F.B. Hildebrand, Advanced
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J.T. Delhaye and J.L. Achard, Invited
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24.
S.
Javis,
Evaluation
of
Jr.,
a
"On
the
Formulation
and
Numerical
Set of Two-Phase Flow Equations Modelling
the Cooldown Process," Technical Note 301,
National
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266
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for
R.D. Richtmyer and K.W. Morton, Difference Methods
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A
J.A. Boure, "GEVATRAN:
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Two-Phase Flow Dynamics," CONF-720686-3 (June 1972)
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D. Gidaspow, Proc. 5th Int'l. Heat Trensfer Conference,
Tokyo, Japan, CONF-740925, V.II (1974)
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the
Hydrodynamic
RELAP-UK,"
OECD/NEA
Hyperbolic
Modeled
Equations
in the
Meeting
Specialists
on
Regime
LOCA
of
Code
Transient
Two-Phase Flow, Toronto (Aug. 1976)
29.
R.W. Lyczkowski, D. Gidaspow, C.W.
and
Solbrig,
E.D.
Hughes, "Characteristics and Stability Analyses of Transient
One-Dimensional
Two-Phase
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278
Appendix A
CONDUCTION MODELS
A.1 Fuel Conduction Model
The fuel conduction model in THERMIT-6S is based on the
water version THERMIT (17).
Only radial heat conduction
is
considered:
aT
1 a (rk aT)= q"'
;r
PCDT
p at
(A.1)
r Dr
The fuel and clad are divided into mesh cells and
is
assumed
for
the gap.
one
cell
Fuel temperatures are located at
the boundaries of mesh cells, designated by the subscript k.
Fuel pin properties are evaluated
cells
denoted
by
the
differencing of Eq-(A.1)
in the
subscript
is
center
k+
obtained
.
by
of
The
mesh
spatial
integrating
the
equation between the centers of two adjacent mesh cells:
rk+
rk+
T ]
r k+
k+ a
(A.2)
q"' rdr
(rk - ] dr=
[rpC
p at ar
ar
4
rkrkThe terms in Eq (A.2) can be approximated as follows:
r
k+
rk+
rC
T
St
2
r
-r 2k2
<pCp>k-
+
2
rk
t
p >k+
rk-
J
1aT k
2
rk+
-rk2
J(A.
ar (rk -)dr
r
ar
T
-T
Tk+l
= (rk)
k+
(rk)
rk+l -rkk-
3
)
Tk-Tk
kkrk-rkl
(A.4)
rk+
f
rk-
2 2r
2 _rk+
"' dr = k
q "'>k-
2
+
2
2
<q
A-rk
(A.5)
279
to
There are four positions
For
boundary
require
conditions.
the center of the fuel pin, Eq (A.2) is integrated from
and the resulting equation is:
r=rk =0 to r=rl
r
<pC>
p 1
2
T
n+1
1
T
n
1
At
8
2
1
kr
Tn+)
(Tn+
2
2
=q
(A.6)
8
1
1
For the aroundner and outer surface around the gap, the term
k is related to the gap heat transfer coefficient hgap
as
follows:
kk-
= hgap d
k
= h
k+
*d
gap
(A.7)
(A.8)
gap
where d
is the gap width.
gap
For the clad outer surface node, Eq (A.2) is integrated from
r-rN_
to r=rN
=outside fuel pin
the clad surface heat flux q"
w
2 2
N N-
(r 2-r 2
pNN-
p
2
and
introduciLng
we obtain the equation
n
n+1
)
radius,
rrN
(T +1-T )
N +
N
rN-rNAt
2 2
(r -r
+ q"rN
2 N-
)
n+1
n+1
(T N -T N-
)
(A.9)
where
n
n+l
q"
w = h R (TNN
T and T :
£
v
h and h :
2
v
n+l
-Tnz
n
n+l
) +h(T
v N
n+l
-T v
)
(A.10)
liquid and vapor temperature
liquid and vapor heat transfer coefficient
280
The fully implicit difference scheme used for the
conduction
model
was
proven to be unconditionally stable.
In this way we ensure that a time
convective
fuel
step
determined
by
the
stability condition will not cause any stability
As in the fuel
problem for the fuel conduction solutions.
conduction equations the temperature at a cell k is coupled
with its neighbors k+l and k-1, the fuel conduction equations
must be solved simultaneously or iteratively.
Both
Tw
and
in Eq (A.10) are treated implicitly in order to overcome
Tf
the
plausible
large
oscillation of fluid temperature. Since
solving for Tl+lin the fluidequations requires information about
wall heat flux, the fluid equations
fuel
conduction
equations.
and Tf
is as follows:
for T,
1.
are
coupled
with
the
The overall solution procedure
Calculate hn using wall and fluid properties
at
the
previous time.
2. With the assumption
T w+l 'o
guess
of
Tw
n+1
Tnf + 1
for
T nf+
=
T nf
find
the
and
Tn+l
w
by the forward elimination
a
first
variation
of
the
rod conduction problem.
3.
Solve the fluid equations using
q
= h (Tn +lO-Tn+l) +h
w
w
f
and find Tn+1
f
+1
n+l
aT f
(T
f
-T
f
(A.11)
281
4.
Set
n+1
DTn +
n+1,0
w
the
compute
backward
T)
(A. 12)
substitution
rod
the
of
get
the
remaining
putting
Eqs
(A.11)
to
problem
conduction
(Tn+
Tf
S
and
1
new
rod
temperatures.
As we obtain Eq (A.10) by
(A.12)
and
together, this procedure does not contain any approximation.
For
a
detailed description of this procedure refer to Ref.
17.
A.2 Structural Conduction Model
structure
surrounding
significant.
the
systems
PWR
Unlike
the
radial
system
heat
in the
loss
the
to
may
LMFBR
be
The primary reasons for this difference lie in
the larger thermal conductivity of
sodium,
radial
steeper
temperature profile, and presence of a can in the LMFBR.
The model employed for structural conduction is based
on the fuel conduction model, but geometrical
transformation is needed to
Figure A.1
A.2.
shows
keep
the
same
the transformed structure
volume.
from
Fig.
The hex can is transformed into an annulus, and sodium
in the fluid channels contacting the hex can is formed to an
imaginary annulus.
is
determined
by
The inner radius of the
setting
annulus equal to the sum
of
sodium
annulus
the cross-sectional area of the
the
cross-sectional
area
of
282
"imaginary" sodium annulus
stainless steel wall
Fig. A.1:
THERMIT Model of Hex Can with
Associated Structure
283
hex can
Fig.
A.2:
stainless steel wall
Hex Can with Associated Structure
IMININNYII111
M
111MOM
1111415MI1011MINN*
Mil
284
sodium in each of the fluid cells adjacent to the structure.
The heat flux on the outer boundary, q"ut
q"ut
=
hout
(T -T
will be
)
ut
(A.13)
: constant temperature outside the structure
where T
:
T
w out
temperature
at
the
outer
boundary
of
the
structure
If an adiabatic condition at
hout
is
set
as
the
the
outer
zero value.
wall
is desired,
For the inner boundary
condition, the Dwyer correlation (103), which was
developed
for liquid sodium flowing in an annulus, is used.
Nu = A + C(
where
Pe)
,
A
= 5.54 + 0.023 r 12
C
= 0.0189 + 0.00316 r1 2 + 0.0000867 r12
8
= 0.758 r 12
(A.14)
-0.0204
r12 = r2/r 1
r1 and r2: inner and outer radius of sodium annulus
T is assumed to be 1.0.
The heat flux on the inner boundary of the structure can
be
calculated as follows:
q'.'
in
=
)
- T
Nu - (T
sodium
win
D
e
(A.15)
285
equivalent diameter of sodium annulus
where De :
T
sodium
sodium
temperature in the
and T in:
win
annulus
and temperature at the inner boundary of the structure
This completes the list of boundary conditions necessary
to
solve the radial heat conduction equation which is described
in the fuel conduction model.
no
The original model (45) assumes that there is
heat
transfer between the structure and fluid, if vapor exists in
However,
structure.
the
contacting
cells
it takes
as
considerable time for the cells contacting the structure
significant amount of
structure.
appears
vapor
after
dryout
reach
in the single phase region.
single
the
cells,
the
to
the
transferred
After dryout little energy exchange between the
Therefore, Eq (A.14) is used
structure and fluid may occur.
the
be
can
energy
in
to
As Eq (A.14) is valid
only
in
phase region, the correlation for the wall heat
transfer coefficient in two-phase flows, which is described
in
Section
two-phase flow.
is
assumed
is used
5.2.3,
In the post-dryout region
there
that
structure and fluid.
comes
from
the
the
is no
heat
it
transfer between the
The primary reason of this
This instability of liquid
assumption
temperature
instabilities of structure heat flux and vapor
temperature and at last
significant
which result in numerical problems.
caused
(a >0.957),
experience that there is an instability of
liquid temperature.
induces
account for the effect of
to
pressure
oscillation,
These instabilities are
by the small heat capacity of fluid after dryout and
286
use of fluid temperature at the old time for the
of
the
heat
evaluation
on the inner boundary of the structure.
flux
Moreover, for liquid metal, as the heat transfer coefficient
understandable.
is so large, these instabilities are
as
an
Also
attempt to include the Tsodium term implicitly would
couple the fluid cells to each other
through
temperatures,
the above assumption is adopted.
A.3 Fluid Conduction Model
In a bundle of LMFBR,
profile
a
radial
non-flat
temperature
in steady state conditions and the gradient
exists
of radial temperature profile sharply increases with time in
This steep radial temperature profile will
flow transients.
affect
the
voiding
initiating
profile
temperature
exchange
flow
behavior,
between
is
time
of
boiling,
controlled
primarily
channels.
ignored.
that
than
of
This radial
reversal, and dryout.
energy
by
Expecially for liquid sodium,
where its thermal conductivity is two
greater
two-dimensional
orders
of
magnitude
water, conduction effects cannot be
by
The conduction effects can be represented
in Eq (2.73).
term V-k<< -+qt>>
the
This term includes
energy exchange by both molecular and turbulent conductions.
Two options
inclusion
have
of
been
heat
developed
conduction
in
THERMIT-6S
between
adjacent
channels: fully explicit formulation and partially
formulation.
are as follows:
Three
for
the
fluid
implicit
assumptions made for these two options
287
and
liquid
are
vapor
or
vapor,
and
The conduction effects between vapor
1.
This is justified,
negligible.
because the thermal conductivity of vapor sodium is three
orders of magnitude lower than that of liquid sodium.
in
velocity
Clinch River Breeder Reactor the
low
extremely
axial conduction become significant.
will
For the
the
core
reduced by at least three orders of magnitude
be
should
of
Only in case
conduction if desired.
flow
the inclusion of axial
permits
model
the
though
now,
code
the
in
2. Only radial conduction is incorporated
before axial conduction effects become as large as 2%
of
the amount of energy transferred by convection (104).
3.
effective
The
practical value for
heat
radial
the
Transfer Test Program
with
the
NATOF-2D by Zielinske (99).
section
is a
containing
experiment,
61
the
mockup
rods
of
in a
recommended
order to obtain a
number
numerically
for
simulated
The heat transfer test
LMFBR
blanket
hexagonal
duct.
a
is
Westinghouse Blanket Heat
was
(105)
conduction
Nusselt
effective
conduction,
for
In
transients.
throughout
constant
number
Nusselt
value
the
for
assembly
From this
effective
Nusselt number is 22.
A.3.1 Fully Explicit Formulation
The net rate of heat flow into a given
fluid
cell
is
expressed as the sum of the heat flows from each of the four
288
sides.
For the configuration of Fig A.3.
n+1
S
=
n+1
n+1
1-0 +20
n+1
n+1
3-0 +4-0
(A.16)
n+l
n
n
n
n
q 0 = A1-0 h-0 (T 1 -T 0 )
where
qT:
(A.17)
total heat flow into channel 0
ql-0: heat flow from channel I to channel 0
A1-0 : heat flow area between channel 1 and 0
h-0 :
effective conduction heat transfer coefficient
between channels 1 and 0
T land T0 :
liquid temperature in channels 1
and
0,
respectively.
As heat flow entirely depends on quantities evaluated at the
previous time n, this method is called
the
fully
explicit
After boiling the liquid contact area between
channels
formulation.
will
be
reduced.
To account for it in a simple way, A 1-0
is multiplied by the average void fraction of channels 0 and
1.
This
satisfies
the
n+l
n+l
Wilson's
)
(i.e. ql0 = -q 0 1
there is no
results
conduction
in a
sudden
after
energy
model
conservation
(45)
boiling.
assumes that
This
assumption
reduction of energy transfer between
channels after boiling, even
though
large
liquid
contact
areas due to large mesh sizes and low void fraction exist in
the initial stage of boiling.
In Eq (A.17) h1-0
transfer
can be expressed in terms
coefficients within each channel, h I
of
and ho .
heat
By
289
Fig. A.3:
Top View of Fluid Channel.s
290
the boundary condition
)
q"-0 = h (Ti-T
(A.18)
= h 1(T1- T i)
where Ti:
To
interface temperature between two channels
define
ho
and
h1 ,
the
constant
Number
Nusselt
approximation is used
h0 = Nu k0/De0
(A.19)
h i = Nu kl/Del
thermal conductiviy of sodium
where k:
De:
Combining
equivalent diameter
Eqs
(A.16)
and
(A.18)
and
rearranging
the
resulting, we have
h1-0
hlh 0
hh= 0
hl+h0
(A.20)
These steps are repeated for each face of the channel.
The major advantages of the explicit method are shorter
computer time and strict conservation of
this
method
introduces
energy.
However,
a stability limit on the time step
291
size.
At < (Ax)
4a
where
=
a
2(x)
(A.21)
k/PC
Ax = radial mesh size
The minimal radial mesh size can be the radial channel size.
-3
With typical values for k, P, and Cp ( Ax=6.3x10
m,
k=67w/m.K,
Cp
than
0.6
less
corresponds
to
=1260J/kg -. K,
In
seconds.
),
p=800kg/m
the
W-1
At
must
this
Experiment
of the steady-state velocity.
2.8%
Surely
boiling will occur at 2.8% of the steady-state velocity
in
turn
be
and
the time step size will be reduced due to the high
velocity of vapor.
Therefore, in most of cases we
can
use
the fully explicit formulation without more reduction of the
time step size.
A.3.2 Partially Implicit Formulation
The partially implicit formulation was developed at MIT
by Andrei Schor (15) in order
to
overcome
problem introduced by the explicit method.
the
stability
Eq (A.17) can be
modified as follows:
n+1
n+1
n
= A-0 h 1-00 (T
q-01-01-0
n
n+
- T R1
10
)2
(A.22)
29.2
The primary reason to introduce this method instead
of
the
fully implicit method is to avoid coupling of temperature of
adjacent
cells,
which
requres
enormously increased computer
adjacent
a
new solution scheme and
When
time.
temperature
in
cells at the previous time is used like Eq (A.22),
the decoupling of
temperature
can
be
achieved.
It
was
proved that this method will not introduce the limitation of
the time step size.
lack
of
energy
conditions.
The main drawback of this method is the
conservation, which can be serious in some
Therefore, if a user assures that the time step
size limited by convection is
conduction
small
enough
than
that
by
throughout transients, use of the fully explicit
formulation is recommended.
293
Appendix B
PROPERTIES
B.1 Sodium Properties
The present version uses least squares polynomial
derived
from
sodium properties correlations in ANL-RDP-72,
and
ANL-RDP-74,
conductivity
(106)
for
tension.
surface
and
Units.
in S.I.
converted
and
HEDL-TME-77-27,
in ANL-7323
presented
fits
These
the
correlations
enthalpy, vapor
liquid
were
properties
All
properties
include the
following approximation.
1.
phase
was
A constant sonic speed, 2100 m/sec was taken for
the
gas
A perfect
behavior
for
the
vapor
assumed.
2.
estimation of liquid compressibility.
3. The subcooled effect of liquid was neglected.
These approximations can be examined as
equation
of
state
The
follows.
a real gas can be expressed in the
for
following form:
Pv
1+
RT
+
v2
v
Stone et. al. fitted
v
Eq
. ...
3
(B.1)
(B.1)
to
their
P-v-T
data
and
obtained the following expressions for the coefficients with
their
average
derivation
of
the
equations
from
their
294
data, ± 0.26%.
B = -4.44734 x 10-5T exp(15554.65/T) ft3/mole
C = 0.243388 exp(2495.8/T)
D = -0.811895 exp(31175/T)
With typical values in the accident analysis
of
the
LMFBR
the deviation of an ideal gas behavior from Eq (B.1) is less
than
10- 3
The
first
same
condition the percentage error
-4
introduced by the second approximation is less than 10 .
In
.
and
the
second assumptions produce tolerable errors
which are less than average deviations of the equations from
data.
In order to analyze the effect of the third
assumption
on liquid enthalpy, we use Maxwellian reltionship.
dh(TP) =
dT+
dP
(B.2)
= C (T,P)dT + [v-T)
]daP
p
\(T]P
Integrating Eq (B.2) from
the
system
pressure
P
corresponding
sat
(p]
temperature T and assuming that [v-T 7v
saturation
we
obtain
pressure
the
expression
of
the
enthalpy
P
the
is
to
the
system
constant,
of subcooled
295
liquid.
h hk(.T,P) = h (T
, P sat)
sat
+ [v-T(*-] (P-P
"Tp]
sat
)
At T=661 K and P=1.5 bar
h (T,P) - h (T,Psat) = 180 J/m3
assumption
The percentage error introduced by the third
less
than
which
10 -3 %.
It
error than one
lower
can
that
conclude
be
in the present version represent well real
used
properties
data.
of
deviation
standard
is much
is
properties.
Saturation Temperature
T
sat
=
-11484.6
683538
+/142665998.6 - 1.367076P
for 0.73 < P < 2.2x10 6
Liquid Density
density,
none
shows
a
pressure
dependence
negligible compressibility for the liquid
as
described
the
for
Among all correlations we have viewed
due
phase.
liquid
to
the
However,
in Section 3.3, the diagonal dominance of the
present numerical scheme in the single phase
is provided by the compressibility of liquid.
liquid
region
Therefore, a
296
the
term to represent
included
compressibiliy
liquid
of
must
the correlation of the liquid density.
in
presnt version a
simple
expression
is
used
due
be
In the
to
the
inaccuracy of the estimation of liquid compressibility.
(9p/3P) = 1/a
2
where a is the speed of sound whiah is taken, equal to 2,100
m/sea at 900 C.
Then we have the liquid density
p
= 1004.23 + 2.26x10
0.2139 T
-7
-
1.1046x10
5
T2
P
< 1644.4
for 550 < T
Vapor Density
For the vapor density the expression
density
in
which
gives
saturation conditions is used and a perfect gas
behavior in the superheated zone is assumed.
Pv =
the
P
(4144.4/Tv -
-5
1.0834x10
-13
-2
2
-9 T 3 T v + 3.8903x10
v
4.922x10 13 Tv
for T
7.4461 + 1.3738x10
4
> 550
Internal Energies
T
-
297
The
of
source
expressions
properties
sodium
the enthalpies.
for
gives
only
The internal energies can
be calculated by the following equation
e = h - P /p
For the liquid enthalpy
+
= -67507.5 + 1630.4 T, - 0.41672 T
3
-3
T
0.15427x10
h
for 550 < T
< 1644.4
For vapor enthalpy
h
v
= h6
h£ + 5.3139x10
R
- 2.0296x10 3 T
+
-4 T 3
2
1.0625 T v + 3.3163xl0
v
for T
> 550
Liquid Thermal Conductivity
k
= 110.45 - 6.5112x10
3
-9
T
2.4617x10
for 550 < T
-2
T
the
+ 1.543x10
< 1250
Vapor Thermal Conductivity
-5
T£
2
298
k v = 2.83662x10
+ Ta (6.88299x10
- 1.67826xl0
for 360 < T v < 1644 ,
where Ta = 1.8 T
- 459.67
Liquid Viscosity
5
-5=
F1 E = 3.6522x10
3.6522x
2.8733x10 4 -45.6877T +0.16626T 2
4
T
for 550 < T, < 1250
Vapor Viscosity
v
= 1.261x10
+ 6.085x10
for 360 < T
Tv
< 1644
Surface Tension
a = 0.2067 - 10
-4 TZ + 0.027314 T 2
for 550 < TR < 1644
Liquid Specific Heat
Ta)
299
C
P&
+ 4.6281x10
= 1.6301x10 3 - 0.83344 T
-4
2
Tz 2
for 373 < T£ < 1773
Vapor Specific Heat
C
pv
= 0.5871x10
3
- 0.83344 T
v
for 873 < T
v
+ 4.6281x10
-4
T
2
v
< 1473
B.2 Properties of Fuel and Structure
1)
UO,
Fuel
The properties of UO, fuel based on Ref
approximated
by
polynomial
fits
(107).
have
been
whose maximum errors are
within one standard deviation in experimental data.
PC
P
= F (1.78x10
k = Pf (10.8 -
where P
P :
F:
= 1 -
-6
+ 3.62x10
8.84x10
-3
3
T - 2.61 T
T + 2.25x10
-6
2
+ 6.59x0
--4
T2 )
(2.74 - 5.8x10 - 4 T) (1-F)
porosity
fraction of theoretical density
P, Cp, and k:
2)
fuel density, capacity and conductivity
Mixed Oxide Fuel (UO PuO
3
T )
300
The following expressions
Tang
from
come
et.
al.'s
recommendation (108).
p = F (10.96x10
C
P
3
F
+ 11.45x10
uo2
= 196.6 + 0.2666 T -
1.834x10
3
F
puOa
-4 T 2 + 4.814x10 -8 T 3
k = 113.3 (1/(0.78+0.02935T) + 6.6x10
where F
and F
uo2
)
(1-P )
+10P
•+Pf+loP
f
1 3 T 3 )1+P
: mass fraction of UO
puo2
and PuO 2 in
2
2
the fuel
3) Gap (Ref. 107)
The gap heat transfer coefficient is divided
in
three
parts:
a) Open gap component, based on
the
conductivity
of
a
mixture of four noble gases
b) Radiation component
c) Closed gap
d) Contribution from partial fuel-clad contact
The thermal conductivity equations of the
are calculated as follows:
He : k
= 3.366x10
-3 T 0.668
T
Ar : k 2 = 3.421x10
Zr : k 3 = 4.029x10
-5 T 0.872
Kr : k4 = 4.726x10 -5
T0.923
T
individual
gases
301
where T is the gas temperature.
The relationship for calculation the thermal conductivity of
a monatomic gas mixture is based
(109).
Brokaw
k
1
n
k
of
work
the
on
mix
=
in
Y.. x. / X
1+E
j=1
j4i
where
(xi-xj) (xi-0.142xj)
[1 + 2.41
= Q..
T..
13
13
1/
[1 + (k./k.)
(x.+x.)2
2
(x./x.)1 /4 ]
2
/2
23/2 (1 + xi/x)
The
n:
number of components in mixture
M:
Molecular weight
x:
mole fraction
thermal
important
role
in
the
of
conductivity
determining
gas
mixture
so-called
plays
an
contact
gap
fuel
rod
conductance as well as open gap conductance in a
when the ratio of the mean fuel path of the gas molecules to
the characteristic dimension (gap) of the body exceeds about
0.01.
Heat
transfer from a solid to a gas in this knudsen
domain is then dependent
which
is
defined
as
on
the
an
accommodation
ratio
of
the
coefficient
actual
energy
-302
to
interchange
interchange
energy
possible
maximum
surface and a gas.
a
between
the
Generally, the accommodation
is near
coefficient of a heavey gas such as argon or xenon
unity
But the effect can
and this effect can be neglected.
not be neglected for the light
following
factor
should
gas
such
as
The
helium.
be divided when temperature drops
are calculated in the Knudsen domain.
kT1/2
f = 1 + C
where P:
gas pressure
r:
characteristic dimension
C:
empirical constant
The characteristic dimension,
, is essentially equal to the
gap dimension and during pellet-to-cladding
be
equal
should
to the root mean square of the surface roughness,
approximately 4.389xi0 -6
rods.
contact
meters
in most
commercial
fuel
0.2103 NJWT /W for C suggested in Ref. (107).
If a nominal open hot gap is calculated, a fraction
the
fuel
cladding
pellet
at
zero
is assumed
contact
to
be
pressure
in contact with the
and
an
effective
fuel-to-cladding gap conductance is calculated as follows:
h gap= (1-F) h
where h gap:
net
of
+ Fh
gap conductance
ho:
nomical open gap conductance
hc:
contact gap conductance (zero contact pressure)
303
fraction of pellet circumference in contact
F:
with
the cladding.
The nominal open gap conductance is determined from
hO = kmix /
(dr + 6)
nominal hot radial gap width
where dr:
6 :
fuel
root mean square of the
and
the
cladding
surfaces roughness.
The fraction of pellet circumference in high
the
F,
cladding,
is assumed
to
contact
with
be an inverse power law
function of the ratio of the nominal hot radial gap and
the
hot pellet radius as follows:
1
100 dr
F =
+ 1.429
al
where R :
al and a 2
+ 0.3
a2
hot radius of the fuel pellet
: empirical constants
The relationships for a 1 and a 2 as a function of
greater
than
600
Mwd/Mtu
are
equations, respectively:
a
100 - 98 F
a2 = 4 - 0.5 F
where
F
1-
(x-600
1000 /
x : burnup in Mwd/Mtu
burnup
is
described by the following
304
When the burnup is less than or equal to 600
then
Mwd/Mtu,
al -100 and a 2 -4
When the fuel and cladding are in contact, the
contact
gap conductance can be calculated
hc=C 1
+Jmix /
where P is the pellet-cladding contact pressure in N/m2
when
is zero
estimated
F
is less
one,
than
the
and
C1 is
constant
be 4x10 - 4 for stainless steel-uranium dioxide
to
pairs.
The value 'of the pressure exponent, n, is governed
behavior
of
the
contact points.
by
the
fuel and cladding at the interface of the
The following
values
are
recommended
in
Ref. 107.
n=l for 0 > P > 1000 psi
n=2 for P > 1000 psi
The radiation heat transfer
contribution
on
the
gap
heat transfer coefficient is accounted for as follows:
h
=h
+h
gap
gap,o
gap,r
where
hgap,d
the gap heat transfer coefficient without
the
radiation heat transfer
h
gap,r
: the gap heat transfer coefficient
contributed
305
by the radiation heat transfer
gap ,r
(T2 + T )
fs
cs
= 5.279x10 8 (T fs + Tc)
cs
F
r
F r = max (1.1485, min (2.451, -0.154 + 0.0013025 Tfs)
+ 0.33 Rfs / Rcs
Tfsand Ts
fuel and clad surface temperature
:
Rfs and Rs
:
fuel and clad surface radii.
Niobium (110)
pC
P
= 1.808x10 6 + 826.7 T
k = 41 + 0.01455 T
Boron Nitride (108)
pC P = 2.84x10
6 + 6.68x10 2 T
k = 13.85
Copper (110)
pC
P
= 3.151x10
k = 429.7 -
6
3
+ 1.0286x10
T
0.0962
316 Stainless Steel (111)
p = 8084.2 - 0.42086 T C
= 4.187x10
3
3.8942x10 - 5 T 2
(0.1097 + 3.17x0
k = 9.248 + 1.571x0-2 T
-5
T)
306
Zircaloy (108)
1096 > T > 843 : pC
6 + 570 T
= 1.892x10
1230 > T > 1096: PCp = 2.3322x0
: pC
T > 1230
= 4.4941x106
k = 9.724 + 0.00917 T
304 Stainless Steel (111)
- 4
p = 7984.1 - 0.26506 T - 1.158x0
C
P
= 4.187x10
3
( 0.1122 + 3.222x10
k = 8.116 + 1.618x10
-2
-5
T2
T )
T
Marimet Insulation (40)
Degraded marimet
experiments.
Sodium
insulation
was
used
in the
leakage into the insulation occurred.
Assuming that all void space was full
of
sodium,
Ref.
estimated the effective conductivity and heat capacity.
pC
P
THORS
= 1.32x10
k = 26.1
6
40
307
Appendix C
DERIVATION OF Eq (5.2.1-13)
The derivation of Eq (5.2.1-13) is primarily based on
analogy between shear stress in single-phase flow and that
in
two-phase flow.
In single-phase flow shear stress can be expressed
as follows:
T
where
w
pz U
2
= f
2
,
(C.1)
f = c / Ren
Re = pU zDe /
D
= D for a tube
U
= average liquid velocity
According to analogy, shear stress in two-phase flow can be
obtained as follows:
U 2f
p
Tw2~
where
f
2
Re2
D
= c /
pU
2
f2
(C.2)
Ren
De
= 4 Af / P
1
= 46
A
= flow area of liquid film
D
= perimeter of liquid film
6
= liquid filr
thickness
U f = liquid film velocity
The two-phase flow multiplier, P. 2 , is defined by the following
equation:
lil|IIYll liIkIIllai~Wi-
.
...
308
2
where Twl
T,w2
/
wl
(C.3)
is the single-phase shear stress at the same liquid
mass flow rate as that in two-phase flow.
S
wl~
where
f
= f
0
2
2
(C.4)
= c / Re n
Rel#
= PjlDe / "I
j
=a IUI
atf
2z
+ azd
Substituting Eqs (C.2) and (C.4) and into Eq (C.3), then we
have
z2
*2
sf
Uf
J(
MMM
2~
ffl22
I
SD n
(C.5)
2-n
(C.6)
(46(
Equation (C.6) can be expressed interms of liquid film volumetric
fraction, a2 f, and entrainment fraction, E.
#
where
= (/a
)
(.-E)2-n
n=0.2 for turbulent flow
n=l
for laminar flow
309
VALVE MODEL
Appendix D
A simplified valve model is needed to account for
pressure drops between the inlet boundary and inlet of test
section, and between the outlet of the test section and
outlet boundary. (See Fig 6.1)
From the definition of
pressure drop, we have
W
Ap
= f
Az
(D.1)
2D epA 2
Considering the constant inlet temperature except in the
case of flow reversal and assuming that the Blausius
relationship, (f=c/Re0
. 2 ),
is applicable, we obtain a
simplified equation:
Ap/Az = C1 W1.
where
C
= 0.5 c
0.2
8
-1
p
(D.2)
De
-1.2
A
-1.8
As C1 is a function of geometry and properties, C 1 is set
to be constant throughout transients.
C1 can be calculated
for a given pressure drop by using Eq (D.2).
In the code Eq (D.2) is written as follows:
Ap/Az = fwl * vn+l
(D.3)
where fwl = spd * ( tarea * rol ) **1.8 * v ** 0.8,
and spd, tarea, and rol correspond to C1 , A, and p,
respectively.
111
___ _
___
Yi
310
Note that v in Eq (D.3) is a numerically predicted velocity
in a channel. When one-sixth of the fuel bundle is taken
for 2-D analysis, then A and W are set to be one-sixth of
the bundle area and of the bundle mass flow rate, respectively.
For example, let us calculate C1 for the restricted area
of the P3A test positioned between the inlet boundary and
inlet of the test section by using Eq (D.2).
In Ref. 88 the test mass flow rate and pressure drop at that
position, at steady state and boiling inception, are
9.46 lb/sec and 11.3 psi, and 3.87 lb/sec and 2 psi,
respectively.
Using the axial mesh length of the fictitious
inlet mesh, Az=8 , one-sixth of the test mass flow rate for W
in 2-D analysis, and S.I. units, we obtain the values of C
1
5
5
at steady state and boiling inception: 7.1x10 and 6.4x10 .
The ratio of both cases is 1.11, which confirms the assumption
of the consatant value, C1.
Appendix E
311
GENERALIZED NORMALIZATION OF
HEAT SOURCE
The total power in the subassembly, Qt,
can be expressed
as follows:
where
( E.1
Z q"(i,j,k) AV (i,j,k)
Qt =
i j k
q
= volumetric heat generation rate in a cell
AV
= volume of a cell
i,j,k = indicators of positions of a cell
Multipling Eq (E.1) by a normalization factor, Nf, we have
qz (
SQt
i j k
[TZ Z qz(j
i j k
where
and qz'
Nf =
qt'
j)
)
r
t(i)
(k) AV (i,j,k)
=
i
(i,j,k)
q
i j k
qt(i) qr(k) AV (i,j,k)] AV (i,j,k)I ,(E.2)
qt(i) qr(k) AV (i,j,k),
Z qz(j)
i j k
and qr represent the axial power distribution,
transverse power shape in the subassembly, and radial power
distribution within a fuel rod.
From Eq (E.2) we can obtain the following relationship:
Qt qz ( j ) qt(i) qr(k) = q
(i,j,k)
Nf
(E.3)
Therefore, the normarized volumetric heat generation rate can
be expressed as follows:
q . (i,j,k)
= Qt qz
(j )
qt(i)
r(k) / Nf
(E.4)
-
IlIhImIIYII
in
-----
312
Appendix F
INPUT INFORMATION
F.1 Added or Modified Input Description
The description of input, which is added to or modified
from the previous version (45), is as follows:
nrzf (narf,nc)
= number of radial zones in each axial region of fuel
nrmzf (nrzfmx,narf,nc)
= number of radial meshes per zones in the fuel
mnrzf (nrzfmx,narf,nc)
-
material
in
each
radial
(1/2/3/4/5/6/7/8/9/10/11/12
steel/304
stainless
insulation/air
or
steel/liquid
zone
of
the
fuel/gap/316
stainless
sodium/degraded
Marimet
void/zircaloy/niobium/tungsten/boron
nitride/copper)
drzf.(nrzfmx,narf,nc)
= thickness of each radial zone in the fuel (m)
cv
fuel
- virtual mass coefficient
alpdry = void fraction when dryout occurs
pin
= inlet pressure at steady state (Pa)
pout
= outlet pressure at steady state (Pa)
vin
= inlet velocity at steady state (m/sec)
vout
= outlet velocity at steady state (m/sec)
spdl
= spatial pressure drop coefficient at inlet
spd2
= spatial pressure drop coefficient at outlet
tarea
= total axial flow area
313
F.2 Input
Input for 2-D Analysis of the W-7b' Test at Steady State
2d17w
Stidat
ntc=3
Send
SLSF Bundle (19 pin), W1 Test, Run 7b',
Steady state, 17 Channels
Heat Loss
$intdat
nc=3
nz= 17
nr=1
ni tmax=2
iitmax=30
narf=4
nx 1
itam= l
iht=33
ihtpr=l
ishpr=1100
i tb=O
ibb=1
iwft=l
nrzs=l
$end
$readat
epsn=1.e-5
epsi=l.e-5
rnuss=22.
pin=8.6e5
pout= .4952e5
vin=6.
vout=6.
hdt=3.258e-3
pdr=1.2435
velx=1.98
tdelay=80.
radf=2.921e-3
hdr=52.23
spdl= 130.75e5
spd2=O.
tarea=64.162e-6
Send
$roddat
qO=1.103e5
ftd=0.9357
fpuo2=O. 25
pgas=l.e5
gmix (1) 1.
Send
Sncr
3
puss
Ot1ndsip
SMIS (11BL)91 OZ99
C-'rOWPO 8-~9-01)z
~zaS(K-seco £-'StLU0o
U.aS (50)lr (500t U9991001
(090)9Z (ot1)9 (00O)Zt
((000)9z (o90L (oo0)11)z
.jbS
zb$ (-0)L
9o0 WTo zol 5cl 5001 6so
JMI$
AIS
9zot SoT SL* 6
(("009)9t (Z99)t)
ZAA$ (*9) IrS
((*oS)Lt (*Z99)Z)C
dIES (obo)LS
dS (5aZ56qVt (Sa~q)Lt Sa9~q)j
zPS (rOro)9 (LZv*o)r (910100)6 (Z~oro)Z
(A-S9)
£j-Zqrq9
xp$
[OAS (9-6ESLOS (9-61*t)Z (9-0SLB*o)6 (9-8SSLtI)i
(9-0SCL0)6 (9-*LLqes)
(9-'tL0VS) s (9-.srVo)
(9-sizzosS)s (9-060900r (9-0tt6*r)6 (9-scurse) I
Lies (9-s'59,9) 91
(9-955609r) 9
(9-OCS909r) S
A'je$ (0*0)t1S
xies (S-9SL~oo)S (C-o919o)z (C-95L9zro)6 (S-*SLSo)r
szjuws C
XU!s I
4z.juwS SL L L(KZ IL)Z SL LL
6L L L K~ r I L)r c r I 0Z
J4-U$
(( s t 9 1)'r)c
4q4, 11) c
J4z.1us
Vjspu!s
0
315
Iput for 1-D Analysis for the W-7b' at steady state
ldl7w
Stidat
ntc-3
Send
SLSF Bundle (19 pin) ,W1 Test, Run 7b',
Steady state, 17 Channels
Heat Loss
$intdat
nc-1
nz=17
nr=1
ni tmax=2
iitmax 1l
narf-4
nx= 1
it am-0
iht=33
ihtpr l
ishpr= 1100
itb=O
ibb=1
iwft=O
nrzs-1
Send
$readat
epsnl .e-5
eps i-l .e-5
rnuss=O
hdt=3.258e-3
pdr-1.2435
velx 1.98
vely"5.78
tdelay=80.
radf=2.92 le-3
hdr-52.174
pin=8.6e5
pout 1.4952e5
vin=6.
vout=6.
spd = 130.75e5
spd2=1.734e4
tarea=64.162e-6
Send
$roddat
qO=1.103e5
ftd=0.937
fpuo2=0.25
pgas1l.7e5
hgap=0.285
gm ix (1) 1.
Send
316
1
0
1 2 11 13
4 4 4 4
$ncr
$Sindent
$ifcar
$nrzf
4(1 6 1 3) $nrmzf
7 7 7 3 2(1 1 2 3) 7 7 7 3 $mnrzf
1 $inx
3 $mnrzs
4 $nrmzs
17(0.0) $arx
17(0.0) $ary
18 (64. 162e-6) $arz
13.058e-6 9(6.529e-6) 2(8.161e-6) 5(13.058e-6) Svol
4.39e-3 $hedz
3.37e-3 $wedz
16.e-3 $dx
18.82e-3 Sdy
2(0.2032) 9(0.1016) 2(0.127) 6(0.2032) $dz
(4.6e5 17(4.e5) 1.4952e5) $p
19(0.0) $alp
(2(662.) 17(800.)) $tv
18(6.) $vvz
(1(662.) 16(800.)) Stwf
0. .7475 1.05 1.26 1.39 1.415 1.35 1.2 0.94 0.6
7(0.) $qz
(4(1.0)) $qt
12(0.) 6(1.0) 26(0.) $qr
4(3.16667) Srn
4(0.889e-3 1.659e-3 0.0823e-3 0.38e-3) $drzf
18.82e-3 $pcx
1.016e-3 $drzs
662. 16(780.) $tws
$timdat
tend=2.
dtminml.e-4
dtmaxml .e+1
dtlp=2.
dtspl10.
Send
317
Input for Analysis of the W-7b' Test in Transients
2dw
$tidat
ntc--2
Send
Srestart
ni tmax-6
cv=1.e-2
iss=0
iitmax=40
tO00.4
itb=0
ibb=1
ihtpr=l
ishpr=1100
epsn l.e-4
epsi=l.e-5
tdelay=2.
rnuss--22.
ntabls=2
itbt=2
ibbt= 1
natc=1
Send
continue W-7b' transient, v-p bc
2
0. 1. 0.5 0.38 $vin
2
0. 1. 4.
1. Spout
Stimdat
tend=3.5
dtmi n-.e-4
dtmax5 .e-1
dtlp=0.5
dtsp= 30.
$end
(0-1.5 sec)
318
Input for Analysis of the W-7b' Test in Transients (1.5-3.5 sec)
2dwl
Stidat
ntc--2
Send
$restart
ni tmax=-10
issm0
iitmax-40
to-.4
itb=0
ibb=0
ihtpr-0
ishpr-1100
epsn*- .e-4
epsill.e-5
alpdry=0.957
tdelay-2.
rnuss--22.
spd2-1.743e4
tarea-6.46e-5
ntabls-2
itbtn2
ibbta1
natc=l
Send
continue W-7b' transient, p-p bc
2
0. 1. 1.5 0.64 Spin
2
0. 1. 1.5 1. Spout
Stimdat
tend=4.7
dtmin-l.e-4
dtmax5.e-I
dtlp-0.1
dtsp- 0.02
Send
319
Input for Analysis of the W-7b' Test in Transients
2dw2
Stidat
ntc=-2
qO=0.05956e5
$end
Srestart
ni tmaxm-10
iss=0
iitmax=40
tO=0. 4
itb=O0
ibb=O0
ihtpr=0
ishpr=l100
epsnl .e-4
epsi=l.e-5
alpdry=0.957
tdelay=2.
rnuss=-22.
spd2= 1.743e4
tarea=6.46e-5
ntabls=2
itbt=2
ibbt=l
natc= 1
Send
continue W-7b' transient, p-p bc
2
0. 1. 3.5 0.64 4.0 1. Spin
2
0. 1. 1.5 1. Spout
$timdat
tend= 10.
dtmin=l.e-4
dtmax=5.e-I
dtlp=0.
dtsp= 0.02
Send
(3.5-8 sec)
320
Input for 2-0 Analysis of the P3A Test at Steady State
p3a17
Stidat
ntc=3
Send
SLSF In-Reactor Experiment P3A
(37 pin)
Steady state, 2-0, 17 axial cells
Heat Loss & Gain
$intdat
ncw3
nz=17
nr=1
ni tmax=1l
iitmax-10
narf=3
nx=1
itam 1
iht=33
ihtpral
ishpr=1100
itb-0
ibbiwft=1
nrzs-1
Send
Sreadat
epsn-1.e-5
eps i1.e-5
rnuss=-22.
alpdry=0.957
pin5.78e+5
pout=2. 776e+5
vin=6.755
hdt=3.609e- 3
pdr1 .2435
velx=2.014
vely-5.78
tdelay-80.
radf=2.92085e-3
hdr=52.23
spdl7. le5
spd2=O.
tar ea=120.914e-6
Send
Sroddat
qO=206.67e3
ftd=0 .9357
fpuo2=0.25
pgasl1.e5
gmix(1)l1.
Send
3
$ncr
321
Sindent
0
Sifcar
1 2 11
$nrzf
3(3 3 3)
$nrmzf
3))
3(3(10 1
3(7 7 3 1(1 2 3) 7 7 3) Smnrzf
1 Sinx
9 $mnrzs
4 $nrmzs
17(0.0) 1.7672e-3 9(0.8836e-3) 2(1.1045e-3) 5(1.7672e-3)
1.1438e-3 9(0.572e-3) 2(0.715e-3) 5(1.1438e-3) Sarx
51(0.0) Sary
18(41.438e-6) 18(44.86e-6) 18(34.616e-6) $arz
8.42e-6 9(4.21e-6) 2(5.2625e-6) 5(8.42e-6)
9.12e-6 9(4.558e-6) 2(5.6975e-6) 5(9.12e-6)
7.034e-6 9(3.517e-6) 2(4.396e-6) 5(7.034e-6) $vol
5.419e-3 3.911e-3 3.772e-3 $hedz
2.749e-3 3.32e-3 3.034e-3 $wedz
3.6322e-3 6.2912e-3 12.5824e-3 $dx
3(25.99e-3) $dy
6(0.2032) $dz
2(0.2032) 9(0.1016) 2(0.127)
3(5.78e5 17(4.e5)
57(0.0) Salp
3(2(695.) 17(800.))
54(6.755)
2.776e5)
$p
Sty
Svvz
3(1(695.) 16(800.)) Stwf
0. 0.7475 1.05 1.26 1.39 1.415 1.35 1.2 0.94 0.6 7(0.0)
3(1.156) 3(1.07) 3(0.95) $qt
3(14(0.0) 10(1.0) 18(0.0)) $qr
3(1.6667) 3(2.5) 3(2.) Srn
3(3(2.47e-3 0.06985e-3 0.381e-3))
25.99e-3 Spcx
3.048e-3 $drzs
695. 16(780.) Stws
$timdat
tend=2.
dtmi n-.e-4
dtmaxl .e+1
dtlp=2.
dtsp=2.
clm=0.95
Send
$drzf
Sqz
111
111WHINMI
wa.....W.M
Mlhlli11111I11
1411IM16111
01
1141111MA
11
11
J1
1'1I001110-101
322
Input for Analysis of the P3A Test in Transients (0-8.8 sec)
2dp
$tidat
ntc--2
Send
$restart
ni tmax-2
iss-0
iitmax=20
itb=O
ibb1l
ihtpr=O0
epsn1 .e-4
cpumax=10000.
epsi=l.e-4
tdelay=2.
rnuss--22.
ntabl s=2
itbt=2
ibbtl
natc=l
Send
continue P3A transient, v-p bc
10
0. 1. 1. 0..883 2. 0.777 3. 0.69 4. 0.622 5. 0.553
6. 0.51 7. 0.468 8. 0.425 8.8 0.42 $vin
2
0. 1. 8.8 0.5126 Spout
$timdat
tend= 10.8
dtmin=l.e-4
dtmax5 .e-I
clm-0.95
dtlp=2.
dtsp- 10.
Send
323
Input for Analysis of the P3A Test in Transients (8.8-30 sec)
2dpl
Stidat
ntc--2
Send
$restart
ni tmax=-5
cv=1 .e-2
iss-O
alpdry=0.957
iitmax=40
ishpr=0100
itb=0
ibb=O0
ihtpr=O0
epsn l.e-4
epsi=t.e-4
spd2 1.734e4
tdelay=2.
rnuss=-22.
ntabls=2
itbt=2
ibbt=
natc=1
Send
continue P3A transient, p-p bc
2
0. 1. 8.8 0.455 Spin
4
0. 1. 8.8 0.5131 10.2 0.5131 11.6 0.5131 Spout
Stimdat
tend- 30.
dtmi n- 1.e-4
dtmax=5.ec m=0.95
dt p=0.2
dtsp= 0.02
Send
324
Input for 2-0 Analysis of the OPERA 15-Pin at Steady State
opera
Stidat
ntc-3
Send
OPERA out of Pile 15-Pin
2-0, 30 axial cells
Steady state,
Heat Loss & Gain
Sintdat
nc-5
nz-29
nr-l
nitmax=2
iss1l
iitmaxl10
narf=4
nx-5
itam-1
ihtm33
ihtpr-1
ishpr=1100
itb=O
ibb1l
iwf t 1
nrzsml
Send
Sreadat
epsnl.e-5
eps iml.e-5
rnuss--22.
alpdryO.957
pi n6.2885e+5
poutl.013e+5
vi n4 .3
hdt=3.98e-3
pdr m 1.2435
velx=2.161
vely5.78
tdelay=80.
radf=2.92085e-3
hdrm52.23
spdl 1 .206e5
spd217.09e5
tarea=320.24e-6
Send
Sroddat
q0=495.e3
Send
5
0
1 2 20 23
5(4 4 4 4)
$ncr
Sindent
$Sifcar
$nrzf
puss
56*omwto
"Z=dsjp
"Zwd t lp
T+8*j=XeWjP
-0-1=ulwzp
. Z=PUD-4
lepwils
sMIS
((0008)LZ
51*895
51*99S)5
szjps C-090560
4zipS
xodS Z-aqZ*Z (Z-aS5 -j)E Z-aglr-q
(S-oTWo S-s5LiEoo S-seLLiso C-001700Z
-aigCoo C-950ooo E-a590*o C-soz
K-81K90 C-85000 E-99EOO c-95z5*1)Z)5
u.'S KUS8*T)q (5*Z)q (5*0q (5* )l U9999*Z)
jbS ((090)5Z 0*1 o*j (O*O)SI)5
IbS ((ooj) )5
zbS (0*0)01 (0*1)81 ooo
4m*4S ((,,ooe)Lz 51*985 51*895)5
ZAAS
AZS
(j*q)051
((0009)6Z 51*885 91*985)5
djeS (ooo)551
dS (59STool (50* )R 5;a598Z*9)5
ZPS (ITZ510O)s (9050*0)lz (Lzl*o)z
ApS K-09L5096Z)5
xpS (S-sz16z*9)q E-aqil eq
zpams C-aiizoz E-;aL5*z E-sOez E-aqL*z S-95i*E
ZP04S E-a0o C-slioq E-sgio E-a6o*q -aLlr*9
[OAS (9-QM*5)L (9-OT16-01Z 9-;aSLL-q
(9-0985*L)L (9-;06ZSOZ)IZ 9-;aIZE09
(9-;aqzzllol)L (9-2801901Z 9-OZ508 (9-a98*ZT)L
(9-;aL8Z* )IZ 9-09IL-01 (9-01-ZOL (9-aUl"T)TZ 9-09ff-ol
zies (9-az9otc)oc
(9-85LL06I)oE (9-aggo*MOE (9-85WMOE (9-059COIB)OE
xjeS
(C-0650*1)L
AjeS (o,*o)5jj
(i-aiffoo)IZ E-OZ884'o
K-090"OL (i-QZ0*o)IZ C-091"T
(C-05MOL (E-8911990)IZ E-86Z5*1
( -aM*Z)t (E-;aZ6L*o)TZ E-386*1 (0*0)6Z
szwjus q
SZJUW$ j
xu.'s 6 1 E z I
E 11 01 It C Z ZI ZT)5
4zjuwS K Z L S Z I
jzwJUS ((q I z Qq)5
326
Input for Analysis of the OPERA 15-Pin in Transients (0-9 sec)
2dop
Stidat
ntc--2
Send
$restart
ni tmax=4
cv1 .e-2
iss-0
iitmax=30
pout= 1.013e5
itb=0
ibb-I
i htpr-O
istrpr-0
epsn- 1.e-4
eps i-1.e-4
tdel ay2.
rnuss--22.
ntabls-2
itb t-2
ibb t- 1
natc-1
Send
continue OPERA transient, v-p bc
7
0. 1. 0.86 0.93 2.86 0.78 3.86 0.719
5.86 0.59 10.86 0.48 15.86 0.349 $vin
2
0. 1. 20. 1. Spout
Stimdat
tend= 11.1
dtminl1.e-4
dtmax-5.e-i
clm-0.95
dtlp=2.
dtsp- 10.
Send
327
Input for Analysis of the OPERA 15-Pin in transients
(9-20 sec)
2dopl
Stidat
ntc--2
Send
$restart
ni tmax=-5
iss=0
alpdry=0.957
iitmax=30
cpumax= 10000.
ishpr=0110
itb=0
ibb=O
ihtpr=0
epsn=1.e-4
epsi-l.e-4
tdelay=2.
rnuss=-22.
ntabls=2
itbt=2
ibbt=l
natc=1
Send
continue OPERA transient, p-p bc
5
0. 1. 5.86 0.5083
2
0. 1. 20. 1. Spout
Stimdat
tend= 20.
dtmi n-l.e-5
dtmax=5.e-1
clm=0.95
dtlp=O. 2
dtsp= 0.05
Send
9.51 0.44 10.86 0.424 15.86 0.35 Spin
Jill
1111i
,
ll -
i
i
328
Input for 2-D Analysis of the test 718, run 101 at Steady State
2dinl
$tidat
ntc=3
Send
Thors Bundle 6A (19 pin), Test 72B, Run 101,
Steady state, Heat losses to Sodium Soaked
Insulation, Gas Plenum Represented, 17 Channels
$intdat
nc-3
nz= 17
nr=1
ni tmax=2
iitmax=30
iflashul
narf-3
nx= 1
itam- 1
iht-33
i tb0O
ibb= 1
iwft=1
nrzs-5
Send
$readat
epsn=1.e-5
epsi-1.e-5
rnuss-22.
pin-2. 137e5
pout= 1.617e5
vin-1.572
vout, 1.572
spd l=0.
spd2=0.
hdt-3.2e-3
pdr-1.2435
velx=2.76
vely=5.78
tdelay=80.
radf=2.92le-3
hdr-52.174
Send
Sroddat
qO=0.316667e5
$end
$ncr
3
$indent
0
$ifcar
1 11 12
Snrzf
4)
3(4 4
$nrmzf
3))
3(3(1 2 1
3(3(7 3 2 3 )) Smnrzf
1 Sinx
Ai
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ilild llow l
puas
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330
Input for Analysis of the test 72B, run 101 in Transients (0-9 sec)
2dppl
Stidat
ntc--2
Send
$restart
ni tmax-5
iss-0
i itmax=40
iflash=0
itb-0
ibb 1
ihtpr=0
istrpr-1
epsnl .e-4
epsi-i.e-4
tdelay=8.
ntabls-2
itbt-2
ibbt=1
natc=1
Send
continue 728, 101 transient, v-p bc
2
0. 1. 5.3 0.4 $vin
2
0. 1. 5.4 0.9 Spout
$timdat
tend- 17.
dtmi n-.e-6
dtmax5 . e+2
cm-0.9
dtlp-2.
dtsp- 0.2
Send
331
Input for Analysis of the test 728, run 101 in Transients
2dpp2
$tidat
ntc=-2
Send
$restart
nitmaxn-5
iss=0
iitmax=40
iflash=0
itb=0
ibb=0
ihtpr=O0
epsn1l.e-4
epsi=l.e-4
alpdry=0.957
tdelay=8.
ntabis=2
itbt=2
ibbt 1
natc=1
Send
continue 72B, 101 transient, p-p bc
2
0. 1. 4.3 0.8243 Spin
2
0. 1. 4.3 0.9 Spout
Stimdat
tend=40.
dtmin=l.e-6
dtmax=5.e+2
clm=0 .9
dtlp=0.5
dtsp= 0.025
Send
(9-40 sec)