LOOP SIMULATION CAPABILITY
FOR SODIUM-COOLED SYSTEMS
by
Oluwole A. Adekugbe, Andrei L. Schor and Mujid S. Kazimi
Energy Laboratory Report No. MIT-EL 84-013
July 1984
Energy Laboratory
and
Department of Nuclear Engneering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
LOOP SIMULATION CAPABILITYU
FOR SODIUM-COOLED SYSTEMS
by
Oluwole A. Adekugbe
Andrei L. Schor
Mujid S. Kazimi
July 1984
Topical Report of the
MIT Sodium Boiling Project
sponsored by
Oak Ridge National Laboratory
Energy Laboratory Report No. MIT-EL 84-013
LOOP SIMULATION CAPABILITY
FOR SODIUM-COOLED SYSTEMS
by
Oluwole A. Adekugbe
Andrei L. Schor
Mujid S. Kazimi
ABSTRACT
A one-dimensional loop simulation capability has been implemented
in the thermal-hydraulic analysis code, THERMIT-4E.
This code had been
used to simulate and investigate flow in test sections of experimental
sodium loops and of LMFBR fuel assemblies.
Such analyses had required
the use of boundary conditions specified at the inlet and outlet.
The
new code, THERMIT-4E/L simulates the entire primary coolant loop and
therefore eliminates the need to specify such boundary conditions.
additions and modifications to the THERMIT-4E code include:
The
constant
temperature heat sinks, implicit heat transfer to environment and
generalized body force field specification.
To date, applications have
been focused on natural circulation.
A series of experiments performed in the Sodium Boiling Test
Facility (SBTF) at the Oak Ridge National Laboratory have been simulated
with the loop code.
The results of single-phase calculations are
generally in good agreement with the experimental data.
However, we
have not as yet been able to obtain a stabilized flow configuration when
a significant amount of boiling takes place in the heated section.
It
appears that the extremely violent condensation n the plena loads to the
.noted calculational difficulty.
iii
An analytical treatment approximating the single-phase loop
behavior has also been developed.
The results are quite general and can
be applied to other loop systems.
Approximate expressions have been
obtained for the frequency and damping coefficient of a flow oscillation
in a loop.
The analysis has also yielded a criterion for stability,
dependent on the input power, difference between the upper and lower
plena temperatures, and a modified Stanton number.
ACKNOWLEDGEMENTS
The authors wish to express their appreciation for the support
provided by the Oak Ridge National Laboratory and the United States
Department of Energy.
Thanks are due to Mrs. Rachel Morton for her help in computer
related matters.
Dr. Sorel Kaiserman's suggestions and comments on the
flow oscillation modeling are greatly appreciated.
The work described in this report is based on the thesis submitted
by the first author for the M.S. degree in Nuclear Engineering at M.I.T.
The fellowship provided to him by the. Center for Energy Research and
Development at the University of Ife, Nigeria is gratefully
acknowledged.
TABLE OF CONTENTS
ABSTRACT
i.
ACKNOWLEDGEMENTS
iv
NOMENCLATURE
ix
LIST OF TABLES
xii
LIST OF FIGURES
xv
1. INTRODUCTION
1
1.1.
Fast Breeder Reactor Safety Issues
1
1.1.1.
Introduction
1
1.1.2.
Safety Design and the U.S. Ground Rules
2
1.1.3.
The Scope and Limitation of Numerical
Models in Safety Designs
12
The Scope and Limitations of THERMIT in
Safety Design
13
1.2.
Outline of the Present Investigation
14
1.3.
Organization of Report
1.1.4.
2. THE THERMIT CODE
15
17
2.1.
Introduction
17
2.2.
Mathematical and Physical Models in THERMIT
18
2.2.1.
The Two-Phase Flow Model
2.2.1.1.Introduction
18
2.2.1.2.The Six-Equation Model
19
2.2.2.
Mixture Models
22
2.2.2.1.The Four-Equation Model
22
2.2.2.2.The Homogeneous Equilibrium Model (HEM)_26
2.2.3.
The Exchange Terms and the Interfacial Jump
Conditions
27
vi
2.3.
The Physical Models in THERMIT _9
2.3.1.
Wall Friction
79
2.3.2.
Interfacial Momentum Exchange
34
2.3.3
Wall Heat Transfer __
2.4.
Problems with THERMIT Physical Models and Loop
Simulation
40
2.4.1.
Forced and Natural Convection
40
2.4.2.
Condensation Modeling
45
2.5.
The Numerical Methods
49
2.5.1.
Introduction
49
2.5.2
The Numerical Methods for Fluid Dynamics _o
2.5.3.
The Solution Scheme _o
2.5.4.
The Jacobian Matrix and the Pressure Problem
A2
2.5.5.
The Numerical Method for Fuel Rod and Hexagonal
Can Conduction _
65
Overall Solution Scheme and the Hierachy of
Subroutines in THERMIT
65
2.5.6.
3. THERMAL-HYDRAULIC SINGLE PHASE LOOP ANALYSIS
67
3.1.
Introduction
67
3.2.
Typical Flow Loops
69
3.2.1.
A Natural Convection Loop
69
3.2.2.
A Forced Convection Loop
73
3.3.
One-Dimensional Loop Analysis
75
3.3.1.
Mathematical Model
75
3.3.1.1.The Governing Equations
75
3.3.1.2.Functional Dependence of Mass Flow Rate on
Heat Input for the ORNL Sodium Boiling Test
Facility (SBTF) Loop
78
vii
3.3.1.3.Comparison of Analytical Results with the
Codes Calculations
R8
3.3.1.4.Correction for Form Losses in the Actual Loop
91
3.3.2.
Loop Flow Oscillation
3.3.2.1.First Order Perturbation Theory Applied to
Flow Oscillation
93
94
3.3..2.2.Stability Boundary
109
3.3.2.3.Numerical Experiments
111
4. IMPLEMENTATION OF ONE-DIMENSIONAL LOOP CAPABILITY IN THERMIT
1
i21
4.1.
Introduction
11
4.2.
Loop Component Models
171
4.2.1.
The Heater
121
4.2.2.
Constant Temperature Sinks (The Plena)
172
4.2.2.1.Numerical Scheme for Plenum Heat Transfer
122
4.2.3.
Treatment of the Body Force
12
4.2.4.
The Expansion Tank
1 15
4.3.
Implementation in THERMIT-4E/L
1-6
5. THERMIT SIMULATION OF NATURAL CONVECTION LOOP EXPERIMENTS
142
5.1.
Introduction
5.2.1.
Geometry Transformation
143
5.2.2.
Non-Uniform Flow Cross-Sectional Areas
145
5.2.3.
Single-Phase Calculations
148
4.42
5.2.3.1.Simulation of the Single-Phase Test:
ORNL/TM-7018; 10742
148
5.2.3.2.Uniform Cross-Sectional Loop Calculations
154
5.2.4.
Two-Phase Calculations
6. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK
154
16
1
viii
6.1.
Summary and Conclusions
161
6.2.
Recommendations for Future Work
164
APPENDIX A.
167
APPENDIX B.
169
APPENDIX C.
173
APPENDIX D.
186
REFERENCES
189
ix
NOMENCLATURE
flow area
specific heat
J/Kg.K
contact fraction
diameter
m
internal energy per unit mass
J/Kg
force
N
friction factor
mass flux, pu
Kq(m2*sec)
gravitational acceleration
m/sec 2
enthalpy per unit mass
J/Kg
heat transfer coefficient
W/(m2 K)
heat transfer coefficient
W/(m2 K)
friction coefficient
N.s/m 4
thermal conductivity
W/(m.K)
Nusselt number, hD/K
perimeter
pressure
Prandtl number, iCp /K
peclet number, Re Pr
heat source
heat source
Reynolds number, pUD/i
nucleate boiling supression factor
Stanton number, HD/WC
temperature
K
time
sec
U
Velocity
m/sec
u
velocity
m/sec
V
fluid volume
m3
W
mass flow rate
Kg/sec
x
quality
loop patial coordinate
void (vapor) fraction
thermal diffusivity, k/pC
m2/sec
phase change rate
Ka/(m3.sec)
decrement (or change) in....
time step size
sec
mesh spacings
m
weighting factor for interfacial velocity
viscocity
N.sec/m 2
P
At
density
Kg/m 3
T
shear stress
Pa
angular frequency
rad/sec
Subscripts
a
phase
e
equivalent
i
interfacial
liquid
p
at constant pressure
sat
saturation
v
vapor
xi
w
wall
w
wetted
t
turbulent
z
laminar
xii
LIST OF FIGURES
FIG.
1.1.
FIG. 1.2.
Possible Accident Paths and Lines of Assurance
for a Potential CDA
Key Events and Potential Accident Paths for
Unprotected Loss of Flow Accident
7
Key Events and Potential Accident Path for
Loss of Pipe Integrity Accident
8
Key Events and Potential Accident Paths for
Unprotected Transient Overpower Accident
9
Key Events and Potential Accident Paths for
Inadequate Natural Circulation Decay Heat
Removal Accident
10
Key Events and Potential Accident Paths for Local
Sub-Assembly Accident
11
FIG. 2.1.
The Fluid-Wall Interaction
29
FIG. 2.2.
Heat Transfer Selection Logic_6
FIG. 2.3.
Hex Can with Associated Structure Actual
Representative
41
Hex Can with Associated Structure Equivalent
Representation
42
Velocity Profiles for Forced and Natural Convection
in High and Low Pr Fluids
44
Vapor and Saturation Temperatures for an Interfacial
Heat Transfer Nusselt Number of 6.0
47
Variations Between Vapor and Saturation Temperatures
for an Interfacial Heat Transfer Nusselt Number
of 0.006
48
FIG. 2.8.
Minimum Computational Effort
so
FIG. 2.9.
A Typical Fluid Staggered Grid Showing
Locations of Variables and Subscripting Convections
53
Sodium Internal Energy Per Unit Volume Versus
Internal Energy
61
Summary of the Solution Technique in THERMIT 4
66
FIG. 1.3.
FIG. 1.4.
FIG. 1.5.
FIG. 1.6.
FIG. 2.4.
FIG. 2.5.
FIG. 2.6.
FIG. 2.7.
FIG. 2.10.
FIG. 2.11.
xii
FIG. 3.1.
Sodium Boiling Test Facility - Loop
70
FIG. 3.2.
Sodium Boiling Test Facility - Test
Section
71
FIG. 3.3.
Core Sub-Assembly (EBR-II)
72
FIG. 3.4.
Loop C.F. Na (French Experiment)
74
FIG. 3.5.
Test Section of Loop C.F. Na
74
FIG. 3.6.
Loop Modeling Geometry
87
FIG. 3.7.
Marginal Stability Curves Using AT
Parameter
FIG. 3.8.
as a
122
Flow Oscillation Leading to Flow Reversal at
Input Power of 100W and Reduced Loop Resistance
117
FIG. 3.9.
Flow Oscillations at Higher Input Powers
118
FIG. 4.1.
Heater Rod in Loop Geometry
123
FIG. 4.2.
The Plenum Heat Conduction Discretization Grid
126
FIG. 4.3.
Setting Up the Acceleration Due to Gravity
Array for a Typical Loop
134
FIG. 4.4.
The Expansion Tank and the Boundary Cells
137
FIG. 4.5.
Modified Subroutines
141
FIG. 5.1.
Schematic Diagram for Geometry Transformation
144
FIG. 5.2.
Staggered Mesh Arrangement with Suddent Flow
Area Change
147
FIG. 5.3.
The Actual Loop Calculation Cells
149
FIG. 5.4.
Relative Error in Test Section Power Determination
as a Function of Test Section Power
150
Loop Temperature Profile at Input Power of
270 watts
152
Pressure Profile Round the Loop at Input Powers
of Zero and 270W
153
Functional Dependence of the Mass Flow Rate on
Input Power
155
FIG.
5.5.
FIG. 5.6.
FIG.
5.7.
xiv
FIG. 5.8.
Condensation Near the Upper Plenum; Pressure
and Void Profiles
157
Condensation Near the Lower Plenum; Pressure
and Void Profiles
158
Low Quality Boiling at Input Power of 450 watts;
Temperature and Void Distributions
159
FIG. Al.
The Loop/Tank Interaction Modeling Geometry
167
FIG. D1.
Hierachy of Subroutines in THERMIT-4E
187
FIG. 5.9.
FIG. 5.10
xv
LIST OF TABLES
TABLE 2.1
Two-Phase Flow Models
23
TABLE 3.1
Comparison of Analytical Predictions with
Codes Results for the Loop Mass Flow
Rate
90
1. INTRODUCTION
1.1. Fast Breeder Reactor Safety Issues
1.1.1. Introduction
The usual procedure in the safety analysis of a power plant
(nuclear of fossil) is to postulate certain abnormal events of
increasing damaging potential to the plant and to the environment.
Calculations are then performed such that after a particular event,
the plant can resume operation with no significant damage or the
accident may result in limited or severe damage to the plant but no
off-site consequences.
Probabilistic Risk Assessment (PRA) and engineer-
ing judgement are often used in the postulation of these events.
In the liquid metal fast breeder reactor, typically there are
several critical masses of fuel present
in the core.
This fact, in
conjunction with the core high power density led people to conjecture
accidents in which the fuel melted and slumped into a super critical
configuration.
This slumping of the fuel will eliminate the sodium
coolant and cause fast neutron spectrum hardening.
The core will event-
ually meltdown, and possibly cause vessel failure.
This concern about meltdown led to an emphasis on hypothetical core-disruptive accidents (CDA) for the liquid metal fast breeders.
These accidents are considered hypothetical because they would occur only
when the built-in reactor safety systems fail to operate and there is a
sustained inability to remove heat from the fuel at a rate commensurate
with its generation.
For example, if the reactor control system fails to
control the power level, the safety system will scram the reactor.
But
if both systems fail, then a major heat up excursion will occur and
lead to a CDA.
Likewise CDA will occur when the heat transport from
the core deteriorates due to pump failures or extreme pipe leakage but
again, only when coupled with the failure of the safety system to scram.
Other possibilities are associated with the malfunctions of
the redundant heat removal systems.
Design and construction faults
leading to the disintegration of welded parts and bowing of the fuel
rods in the operational temperature field could also lead to severe
accidents.
In fact, the fuel rod bowing initiated the meltdown of
EBR-I and the clogging of a coolant channel by a disintegrated member
caused the ENRICO FERMI reactor accident.
The other events that could
lead to severe accidents of the CDA type are related to an inherent
safety parameter - reactivity coefficient of the reactor.
Unfortunately,
the liquid-metal fast breeder core designs that are most desirable in
terms of performance and economics also have undesirable positive reactivity
during severe heatup transients.
For instance, core structural materials
and coolant both absorb neutrons and moderate neutron's energies.
The
loss of these materials contributes to the positive reactivity by increasing the available higher energy neutrons.
This happens if sodium boils
around the fuel pins, or if the stainless-steel cladding melts and flows
from the core.
1.1.2. Safety Design and the U.S. Ground Rules
The detailed mechanistic analyses of the severe accidents of an
LMFBR plant had been deemed an impossible task by many knowledgeable
scientists and engineers.
An appropriate model for the description of the
---
---
Yi
,,
IIu NMMI
3
thermodynamic, the fluid-dynamic, and the thermal and neutronic behavior
especially at the advanced stages of these accidents
beyond the state of the art in the various fields.
has been judged
As a result, the
hypothetical CDA has been dealt with in two major ways.
First, engineers have attempted to design reliable systems with
very low probabilities of even entering the severe accident regime.
Second, the complexity of the problem has been side-stepped by
basing design on a highly conservative estimate of the "damage potential".
This is the potential for a neutronically heated core materials to produce
high pressures damaging to the containment structures and it is typically
based on the assumption of isentropic (reversible and adiabatic) expansion
of the fuel.
While this approach has been found to work for small scale
experimental breeder reactors, e.g. the EBR-II developed at Argonne
National Laboratory, its consideration for large commercial reactors still
stands to answer some questions.
density level and the reactor size
Damage potential depends on the energyin a way that may not make it work
well for large cores [22] .
The U.S. Fast Breeder Reactor Safety Development Program recognizes
the difficulties involved in the safety analysis of the severe accidents
of the LMFBR plants.
However, the Program's fundamental objective is to
develop a program to make sure that LMFBR power plants are designed,
constructed and operated to assure that the public risk from any accident
that may occur from these plants will be acceptably low.
Thus a compromise
has to be sought between an almost impossible task and a very important
constraint that
borders on the integrity of our lives, our equipments
and the eco-system in general.
To provide this assurance, the U.S. Breeder Reactor Safety
Development Program is based on four levels of protection aimed at
reducing both the probability and consequences of a postulated core
disruptive accident.
These levels of protection, referred to as lines
of assurance (LOA's), have been defined as follows [1]:
* LOA-1 : Prevent Accident
* LOA-2 :
Limit Core Damage
* LOA-3 :
Control Accident Progression
* LOA-4 :
Attenuate Radiological Products
Fig. 1.1 illustrates possible accident paths and lines of
assurance for a potential CDA.
In Fig. 1.1, it has been estimated [22] that LOA-1 and LOA-2 fall
within the first phase of the CDA Scenario (10-30 sec.).
LOA-3 falls
within the second phase with a duration of about 1 sec. during an accident,
and LOA-4 falls into the third phase which may last for milliseconds for
highly fueled core or a few days for the highly depleted cores.
For the purpose of our analysis, our interest shall be primarily
on LOA-2.
LOA-2 becomes operative upon the failure of LOA-1 to terminate
an accident in progress.
The design and control equipment should make
the probability of entering LOA-2 very low.
anistically
possible.
However, this is still mech-
As a goal, a failure probability of 10- 2 or less
has been set for LOA-2 for all accidents identified under it (LOA-2).
These accidents are:
UIIri
I
Reactor Operator
~
r
...
..
.I...
--
Normal Condition
Fault Occurs
Inherent Response
or Fault Detection
and Control Action
-
ae
No Da=age
t
ill
PPS Action Required
•
l
II
II
li
1
II
I
I1
II
PPS
1
IIII
No Damage
Action
NNEEMONOW
LOAI
PPS Failure, Some
Clad/FTdel Melting
and Relocation
Inherent Response
and
Intact
10A2
Numa
Aiiw
2
1
--
I
i
m
-
-I
S
II
I
I
0
No
I•iI
Acciden I Progression
Controlled,
Containment Intact
or Di spersal
LOA3
Core Damage
Coolable
0 __ --
1
Core Mel tdo-w-
Liited
Core'and I-S
ILeaves
Con rol.ed Rel eas
to Environent
-
ll
l
Progression
not Co ntroll.d
Accident
naireease
Lite
Limits Release
to Environment
l
LOAI
_
JLII
-
-
to Enviro=-."ant
N
I
I-
I
1
I
Uncontrolled ReContainment Fails
Figure 1.1
lease to Environmen
Possible Accident Paths and Lines of Assurance
For a Potential CDA (From Reference 2)
1. Loss of Flow Without Scram - loss of electrical power
to motors driving the primary coolant pumps, resulting in pump rundown and loss of core flow while the
reactor is operating at power - coupled with the
failure of the safety system to scram the reactor.
2. Loss of Piping Integrity (LOPI) - leak in a reactor
coolant pipe, resulting in double-ended guillotine
rupture at the inlet nozzle of the reactor vessel
followed by rapid decrease in core flow and partial
loss of liquid - with scram.
3. Transient Over Power Without Scram - malfunction of
plant reactivity control system or operator error,
resulting in a sudden increase in core reactivity
and power coupled with the failure to scram.
4. Loss of Shutdown Heat Removal System - loss of forced
cooling to the core and failure of shutdown heat removal
system following shutdown.
5. Local Sub Assembly Fault - Inlet flow blockage or
internal sub assembly fault resulting in cooling and
disturbance and potential for fuel failure propagation
- with scram (once condiction is detected).
Figures 1.2 through 1.6 illustrate the paths for these accidents.
In these accidents, the occurrence of sodium boilina assumes
a major role in dictating the paths, the rate of progression and the
final states of the events taking place.
Consequently, research and
development work relating to LOA-2 has focused on the understanding
BI
ll IIM IIri
iWill
l
IluMlllll
Fault Occurs
Leading to Pump/
Flow Coastdorn &
Core Heatup
Reactor Scra=;
Reactivity &
Pcwer Decrease
Failure to Scram
Boiling
Flow
Transfer;
& Power
Two-phase
& Eeat
Reactivity
Increase
IIIIII
C
No Damage
II
LI
I
I
Flow/Heat Transfer
Instability
& CHF
I
Fuel Pin Failure &
Fission Gas/Moltae
Fuel Release
Fissio
Pe M
I
Reactivity/Power
Increase Due to Fuel
Motion
-'CI & Subasse=bly
Vciding
Mechanical
Disasse.-bly
Disruption
SCoreAccident
Figure 1.2
.1
eactivity/?cver
Decrease Due to
Fuel Motion
Some Clad/Fuel
Melting but Adequate
Cooling Restored
C
Limited
Core Damage
Bulk Subasse=nbly
Voiding; Clad
Melting & Relocation
Core Geometry
Not Coolable;
Gradual Mel!tdown
Core Disrution
Accide
Key Events and Potential Accident Paths for Unprotected Loss
of Flow Accident (From Reference 2)
Fault Develops
with Potential for
Loss of Pipe
Integrity
I
rili
I
I|
I
Fault Detected
Operator Action
No Damage
Flow/Hcat Transfer
Instability and CMF
Adequate Cooling
No CHF
Core Dam'age
Bulk Subassembly
Voiding; Clad
Melting and
Relocation
Some Clad/Fuel Melting
and Relocation but
Adequate Cooling
Restored
Limited
Core Damage
Fault Undetected
Loss of Pipe Integrity
qilIll
flow Decay
and Heatup
Reactor Scram
Reactivity and
Power Decrease
iEii___ii
- --
Boiling, Two Phase
Flow and Heat
Transfer
I
Fuel Pin iailure
and Fission Gas
Release
Core Geometry not
Coolable; Gradual
Meltdo-wn
Limite
CDA 3
I
Figure 1.3
Key Events and Potential Accident Path For Loss of Pipe
Integrity Accident (From Reference 2).
Faul: Cc=urs Leacing
to Tranrser.t Over;ower
esatup
and Core
Reactcr Ecr.
Failure to Scr-=
PFoer De--y
uel Pin Failure
and Fissicn Cas
Pelease
No Dage
C
V
Predcninan:ly Eish
Failure
Aial
Location
Subs:tantial Nunber
ridplace
of AxIl
Failures
i
Fuel Relocation
to TFrm ?ar:ial
Blockaege
M'CI and Scoasse.bly
Voidin
Mechanical
Disasse=bly
(
CD
to Tor=
No .locka.e
artial
.1
eaac:lviy azc
Reactivcy ana
Power Decrease
Adequate Cooling
Without Eoiling
"it d
FPoer InCrease
Eoiling in Wake
of Blockage
g
nr. cCuUa Ie Cooin
i
(
CF.0 and
1
Adequate Cooling
No Blockagee
Propagation
I
.or f-CI:
Elockage P:.-pa~atiCn
Dr.-ge
)
Li ited
SCore Damage
Figure 1.4
Fuel Relocatio.
Fuel S-ee: t
-
-
i
-
Reactivity and
Power Increase
)
1
M-MM"
Sul'k Sutasse=:iy
Vcdin~g and
XeltdorVn
d0 W"'
Y.-!1t
CCA D
Key Events and Potential Accident Paths for Unprotected Transient
Overpower Accident (From Reference 2).
10
Reactor Shutdown;
Power & Flow Decrease;
Cooldown
Loss of Forced
Cooling in Primary
Loop
Single-Phase Natural
Circulation Flow &
Eeat Transfer
I1
1
Adequate Heat
Removal with no
Coolant Boiling
Boiling Natural
Circulation Flow &
BEeat Transfer
C
No Da=age
=m
Fuel Pin Failure
6 Fission Gas
Release
I
Inadequate Heat Removal;
Flow/Eeat Transfer
Instability & CEF
Adequate Heat Reoval
Bulk Subasse=bly
Voiding; Clad/Fuel
Melting & Relocation
No or Minor
Until Forced Cooling
Restored
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 (From
Reference 2).
~
__
___
Local.Fault Due to Clad
Defect, Fission Gas/
Molten Fuel Release &/or
lock&ate Formation
Local Flo Restriction
& Increased Coolant
Temperatures in Wake
ue ?in Failure 4 Fission
Gas/Molten Fuel Release ;
Gradual Blockage PropagatioI
Local Boiling
in Wake
Flov/Heat Transfer
Instability & CRT
.ault Tolerated or
Detected; Operator
or Control Action
wiI
Minor
No or
Core Damage
Bulk Subassembly
Voiding; Clad/Fuel
Melting & Relocation
~~I.
Fault Tolerated or
Detected; Belated
perator or Control Action
Molten Steel-Sodium
Interaction; Subassembly
Wrapper Failure
Core Geometry
Coolable; Adequate
Cooling Restored
Subassembly Propagation
Limited Core
Damage
Core Disruption
Accident
Figure 1.6
Subassembly to
Key Events and Potential Accident Paths for Local SubAssembly Accident.
I
of the two-phase sodium boiling and heat transfer processes during the
accidents.
Such understanding should help in the designs of systems
that will terminate all postulated accidents with limited core damage
as required in LOA-2.
1.1.3. The Scope and Limitation of Numerical Models in Safety Designs
Engineering systems are designed to perform certain functions usually of transferring mass, energy, information or any combinations
Normally, a system is designed to perform within a range in
of these.
which its behavior can be adequately modeled and safe performance
guaranteed.
At the design stage, experimental data can be obtained
from a prototype of the system.
Such data provide the standard against
which numerical model (code) results for the system are tested.
agreement of these results confirms a good numerical model.
A good
Henceforth,
the numerical model becomes a very useful tool for the design because
of its flexibility, it
is capable of simulating further experiments,
performing controlled examinations of isolated phenomena, and predicting
results in a variety of the system's configurations.
These results
become useful design parameters.
In safety design, however, the probability of some abnormal
events occurring during the life of the system, such that its performance
goes out of the safe range becomes an input in the design.
problem then becomes much more complex than described above.
The design
The behavior
of most systems outside their performance range are often not well understood.
It is therefore difficult or i.n some cases impossible to formulate
any worthwhile mathematical model for the system in these regimes.
Even
r-u
-,
Mu
il
I
'u
if good engineering guesses are made to come out with some empirical
laws and constitutive equations and thus a numerical model, experimental
data are often not available against which the
be compared.
numerical results could
The state of affairs is not completely satisfactory
at this point.
However, in many systems that had failed, the failure
paths had agreed well enough with the numerical predictions.
Again, that
depends on how good the engineering guesses and the empirical laws are
in the first place.
1.1.4. The Scope and Limitations of THERMIT In Safety Design
THERMIT is a component code for the thermal - hydraulic
calculations of a nuclear reactor core.
It was originally written for
water coolant and later revised for sodium coolant.
Essentially, THERMIT attempts to satisfy the LOA-2 demands.
The results of the simulations of sodium boiling experiments in the test
sections of various experimental loops and the core conditions of reactor
plants have shown that the physical models and the constitutive relations
in the code are satisfactory within the steady boiling regimes.
Post dry-out conditions, core variable geometry,multiphase,
multicomponent fluid dynamics and variety of heat transfer processes that
characterize the LOA-3 and LOA-4 are beyond the scope of THERMIT.
THERMIT being a
component
Also,
code cannot simulate the entire coolant
loop of the reactor and thus cannot adequately predict the results of
transients that may originate outside the reactor core, unless the core
boundary conditions can be accurately specified.
The present work initiates
the process of building the loop capability in THERMIT.
14
1.2.
Outline of the Present Investigation
Recent workers on THERMIT, especially Hee Cheon No - MIT
-EL 83-003 (1983) and Kan Yuh Hul - MIT-EL 82-023 has suggested
that implementing the loop capability in THERMIT could serve very usefully
in explaining some salient discrepancies between experimental results and
the code's predictions.., which may be inherent in the boundary conditons
that are used.
Some of the advantages of the loop capable code are:
(1) guessing the boundary conditions at the inlet and outlet plena would
not be necessary and (2) the code would be able to simulate a larger
portion of the actual power system and therefore treat transients that
may originate from outside the core e.g. feed-ump failure, heat exchanger
malfunction and pipeline rupture on line.
In the present work, a loop capability is implemented in the
four-equation model of THERMIT.
was developed by A. L. Schor. [1].
This four equation version,THERMIT-4E,
One-dimensional flow is deemed sufficient,
within the scope of this investigation, to describe the flow within the
reactor's primary loop sections.
Usually, in nuclear power reactors and in the experimental loop
that is simulated in this work, the plena are maintained at constant
temperatures by cooling or heating during operations.
A numerical scheme
for structural heat conduction to keep the plena at constant temperatures
during transients is implemented in the code.
In order to be able to simulate loop flows while preserving the
loop geometry, it becomes necessary to be able to input the acceleration
due to gravity with the appropriate signs and magnitudes at the different
sections of the loop.
built in the code.
To achieve this, an input array for gravity is
------
LIY,
A typical sodium test loop (as well as actual LMFBR plant) is
provided with an expansion tank of sodium pressurized with an argon cover.
During the loop transients, mass is rejected into or withdrawn from this
expansion tank.
In the case of natural convection loop, the argon cover
pressure sets the pressure level in the loop.
We have assumed an infinite
mass expansion tank connected to the loop at the fictitious boundary cells
(discussed in Chapter 4).
In this context, the thermodynamic state of
the expansion tank would remain constant curing the loop transients.
A series of natural convection experiments performed in the sodium
Boiling Test Facility at the Oak Ridge National Laboratory are simulated.
1.3.
Organization of Report
In Chapter 2 of this report, the mechanics of the code THERMIT-4E
is discussed.
The mathematical and physical models in the code are reviewai
Explanation of the code from the numerical point of view is given and the
hierachy of the code's subroutine is illustrated.
Some of the problems
especially in the physical models are discussed.
In Chapter 3, the analysis of a single-phase thermal hydraulic loop
is given. Typical flow loops for both natural and forced convection are
illustrated.
A one-dimensional loop analysis is made. An expression is de-
ived for the mass flow rate as a-function of power input for the SBTF(and
can be applied for any vertical rectangular loop with upper and lower
plena).
Formulas are also obtained for the single-phase flow oscillation
frequency and damping coefficient.
A numerical experiment is performed
using the code to verify typical flow oscillation frequency and
agreement was obtained.
good
In Chapter 4, the methods leading to the implementation of
the one-dimensional loop capability in THERMIT are given.
The modeling
geometry, the constant temperature plena problem including the numerics,
the treatment of the body force, the expansion tank and the boundary
conditions are treated.
The modifications made in the affected sub-
routines are discussed in.detail.
In Chapter 5, the simulation of a series of natural convection
loop experiments are made.
A brief discussion about the geometry,
especially non-uniform cross-sectional flow area and sloping sections
are given.
The method of setting up the gravitational acceleration
arrays for various loop geometries is illustrated.
Results are obtained
for both single-phase and two-phase calculations.
In Chapter 6, conclusion and recommendations for future work
are given.
~ ----
1
.. _~-l..li
17
2. THE THERMIT CODE
2.1.
Introduction
Much work has been done and is still being done in the area
of sodium boiling simulation in LMFBR plants in the United States and
abroad.
Sodium boiling plays an extremely important role in the initia-
tion of some transients of interest in safety research.
At various
research centers and institutions, efforts have been directed towards
building two - or three - dimensional computational tools for sodium
boiling simulation.
Some of the products of such efforts include the
following:
- HEV-2D code, an equilibrium, equal-velocity two-phase flow model
developed at Purdue University.
- NATOF-2D code: recently Zienlinski and Kazimi [3] improved this code
to its present status with significant increase in the reliability
of its predictions over its original form.
- COMMIX-2 code: uses three-dimensional, two-fluid two-phase flow
model.
It is the product of a major on going effort at Argonne
National Laboratory.
- CAPRICORN code: the preliminary version of this code was recently
released by Hanford Engineering Development Laboratory. It employs
a more implicit scheme than NATOF-2D or THERMIT [1].
- BACCUS code: uses homogeneous equilibrium two-dimensional (r-z)
geometry.
in France.
It has been under development at Grenoble Research Center
18
- TOPFRES: A two-fluid, three-dimensional two-phase flow code being
developed in Japan.
At MIT, efforts have been directed towards the development
of the code THERMIT.
Originally THERMIT was written for water and
later modified for sodium coolant.
Currently there are two versions of
the sodium coolant THERMIT viz; THERMIT-4E developed by A. L. Schor[1] and
THERMIT-6S developed by HeeCheon No [2]. THERMIT-4E uses mixture mass,
mixture energy and separate phasic momentum equations resulting in a
4-equation mixture model with thermal equilibrium assumed between the
coexistirg phases at saturation.
THERMIT-6S use, the general two-'luid
(six-equation) model.
2.2.
Mathematical and Physical Models In THERMIT
2.2.1. The Two-Phase Flow Model
2.2.1.1. Introduction
Mathematical models for vapor-liquid flows are usually derived
starting from the local instantaneous differential conservation laws of
mass, momentum and energy and the interfacial jump conditions.
of varying sophistication
Models
result from the specific choices for the
averaging procedures and the assumptions made about the nature of the
mechanical and thermal coupling between the vapor and the liquid phases.
The most general model is the two-fluid, six-equation model
(also referred to as the separated-phase model).
by an average temperature and velocity.
It describes each phase
It could in theory provide the
maximum in capability and physical consistency among the two-phase flow
19
models.
Various two-phase mixture models exist.
These mixture models
use less than six equations and consequently require additional assumptions to be made about the thermal and mechanical coupling between the
phases.
2.2.1.2.
The Six-Equation Model
The detailed derivation of the volume-averaged two-phase equa-
tions is given in [1].
The working forms of these conservation equations,
written for one-dimension that is relevant to our loop flows are given
below.
S is the only spatial co-ordinate that runs round the loop with
unit vector e.
Vapor mass equation
a- (O
) +
(apvUv)
=
r
(2.1 a)
Liquid mass equation
-a [(1-a)p] +
s [(1-a)pu
]
(2.1 b)
= -r
momentum equation
Vapor
3U v
v at
pv
v+
aUv
s
+
aas
-F
wv
-F. + c eg
v
iv
.c)
Liquid momentum equation
(1-+
(-a)o
U
+ (l-c)
+ (1-a)
-F
e*g
- F.
(2.1 d)
20
Vapor internal energy equation
-t
(ap e ) +
v v
-
(aUv ) + P
(dc e U ) + P v
s
v v v
s
= Q'
t
+ Qiv +Qkv
(2.1 e)
Liquid internal energy equation
t (-a)p
[(1-a)U
[(1-a). e Uc ] + P
e] +
= wk
+
QiZ
+
Qkk
(2.1 f)
where
F.
1V
Qiv
=
F.
+
rU
(2.2 a)
=
F.
+
?U
(2.2 b)
SQw
+
F
U
F
zU
1
1
(2.2 c)
wv v
(.2.2 d)
=
2
U /2
+
F.U
+ Qi1
(2.2 e)
=
TU 2/2
+
F.U 1 1+ Qi
1
(2.2 f)
Sv
21
Qkv
(2.2 g)
as (q v)
k
(1-(X)
-
(2.2 h)
q]
It should be noted that the internal energy equations are not
They are the reduced forms of the total energy
conservation equations.
conservation equations, obtained by subtracting the mechanical energy
from the total energy (reference [1]). This is done for numerical
convenience. Also the momentum equations are written in the non-conservative forms for the same convenience reason.
P
There are 8 unknowns in equations(2.1), These are: a, Pv' F
P, ev , e., Uv and UZ.
'
The effective fluid conduction sources, equation
(2.2 (g) & (h)) are assumed to depend, via constitutive relations,on
these variables and the phase temperatures Tv , T., and hence provide
two additional unknowns.
Thus we have a total of 10 unknowns.
Equations
(2.1) and (2.2) are equivalent to 6 equations, hence we must provide
4 additional equations for closure.
These are the equations of state
given in the forms:
Tv)
v(P,
(2.3 a)
pZ
P (P, T)
(2.3 b)
ev
ev(P, TV )
(2.3 c)
e (P, T )
(2.3 d)
P
e
=
=
v
2.2.2.
Mixture Models
As mentioned earlier on, a mixture model is a degenerate form
of the six-equation model and we should expect consistent results from
all models by
activating the appropriate constraints or assumptions
that led to each model.
Table 2.1
gives the summary of the two-phase
flow models.
The four-equation model shall be discussed in greater detail
because of its relevance to THERMIT-4E that is used in this work.
The
homogeneous equilibrium model (HEM) shall also be discussed because it
provides an easy analytical tool for the loop analysis that is the subject of chapter 3.
2.2.2.1.
The Four-Equation Model
The detail of the considerations leading to the adoption of
the four-equation model in THERMIT-4E has been given in reference [1].
Importantly, the code is developed for the particular applications of
the analysis of two-phase sodium coolant flows.
The very high conductivity
of the liquid sodium precludes significant temperature gradients in the
vicinity of the liquid-vapor interface and thus makes the assumption of
thermal equilibrium at saturation of the coexisting phases a reasonable
one.
The assumption of mechanical equilibrium cannot be a good one
however, because the enormous liquid-vapor density ratio of sodium at
near atmospheric pressure coupled with the prevalent low flow conditions
lead to substantial slip ratios.
It will therefore be necessary to
write separate momentum equations for the two phases in any worthwhile
mixture model.
TABLE 2.1
Two-Phase Flow Models
(General assumption: pt= pv
)
----------------
Two-PhaseFlow Model
(suggested
nomenclature)
Implosed
Restrictions
Conservation
Equations
Required Constitutive Relations
External
Interfacial
Total
1
i
E
.
.
'rota ]
T
a
U
Total
i-
-
--
Qw
Fw
r
Oi
F.
-.
3C
1
1
1
3
2
1
3
1
1
0
0
0
2
4C2M
2
1'
1
4
1
1
2
1
1
1
0
0
3
4C2E
1
2
1
4
1
1
2
2
1
1*
1
0
5
4C2K
5C 2 MK
1
1
2
4
2
0
2
1
2
1*
0
1
5
2
2
1
5
0
1
1
2
1
1
1
0
5
5 C 1E
2
1
2
5
1
0
1
1
2
1.
0
1
5
5C1M
1
2
2
5
-1 0 I
1
2
2
1*
1
1
7
2
2
2
6
0
2
2
1
11
6C
I,egend:
-
0
7
M = Conservation of Mass
E = Conservation ofi Energy
K = Conservation of Momentum
Ta = Phase "a" temperature; a = v or
U = Relative velocity = i -Ui
cv
Q
*note that the interface mass exchangel,
r,
is
.
needed whenever Q0
and/or F. are needed.
24
the parameters pv'
In the 6-equation model,
P'
ev, ez are
functions of Tv or T and P (eqn. 2.3), but with the assumption of
thermal equilibrium at saturation, Tv = T = Ts , these parameters
all become functions only of Ts .
Thus, the equations of state become:
pv ( p)
(2.4 a)
=
p( p)
(2.4 b)
S
ev( P)
(2.4 c)
=
e( p)
(2.4 d)
=
TS (p)
(2.4 e)
S
Hence the 3 unknowns Tv , T, P in (2.3) reduce to only 1
unknown Ts in (2.4).
The number of conservation equations is also
reduced by two, from six to four, yielding the four-equation model as
follows:
Mixture mass equation
a- o
+
[o
v
(1-)
U
= 0
(2.5 a)
Momentum equations
(identical to 2.1 (c) & (d))
(2.5 b,c)
-
Mixture internal energy equation
-
(Pmem)
[apvevUv + (1-a)p e U]
+ P s [Uv + (1-)
U]
= Q + Qim
(2.5 d)
+ Q
where
(1-a) p9
(2.6 a)
=
[p e v + (1-a) ezp]/pm
(2.6 b)
-
mixture wall heat source
-ap
+
Q'w
wQ sourcwve due to interfacial effects
mixture+ heat
Qim
--
mixture heat source due to interfacial effects
Qiv
-
Qik
mixture conduction heat transfer rate
Qkv
p
+
+
Qk
and em are 2 additional unknowns to the 10 unknowns counted
in the six-equation model.
Thus we have a total of 12 unknowns.
(2.5) and definitions (2.6) represent a total of 6 equations.
Equations
The 4
equations of state (2.4), and the 2 equations implied in the assumption
of thermal equilibrium at saturation:
Tv
TZ
Tsat (P)
(2.7).
provide the additional 6 equations required for closure.
By using the four equations (2.5), the two definitions (2.6)
and the two constraints (2.7), we shall be able to calculate the following eight quantities
em.
c, Tv , TZ
p ,P
, p , ev, eZ for any given P and
This is a very important step in the solution technique in THERMIT.
As shall be shown in a later section, reduction of conservation equations
to pressure problem is a dominant feature of the numerical method in the
code.
The following terms are neglected in THERMIT-4E calculations
because of their relatively very low magnitudes: (i) contribution of
interfacial effects to mixture heat source, (ii)work terms due to the
interfacial momentum exchange,i.e.
FiUv and FiUZ, (iii) the kinetic
energy transport via interfacial mass exchange,i.e.
Uv 2/2 and
U 2/2,
thus Qim = 0 and (iv) the pseudo work terms due to wall forces i.e.
F U and F U in the wall heat source.
2.2.2.2.
The Homogeneous Equilibrium Model (HEM)
The HEM or the three-equation mixture model is obtained by
assuming thermal equilibrium of the co-existing phase at saturation
and equal phase velocities.
Equilibrium drift flux model would result
if a correlation for relative velocity were used.
The resulting HEM conservation equations are given below:
27
Mixture mass equation
--
m
+
+
~s
mUm)
m--
=
0
(2.8 a)
Mixture momentum equation
Um
m at
+
(PU ) aUm
mm
- Fw
+ aP
as
+ Pm e*g
(2.8 b)
Mixture internal energy equation
a p
~meS
+
+
7_s mU
+p
+P
mUm)
=,
(2.8 c)
+ Qk
where
Um = the mixture velocity
Um = Uv = Uz
2.2.3.
The Exchange Terms and the Interfacial Jump Conditions
The wall and the interfacial exchange terms are the mass,
momentum and energy exchanges that take place at the fluid-wall and
the fluid-fluid interface respectively. The interfacial jump conditions
are essentially the equations of conservation of mass, momentum and
energy at the fluid-fluid interface.
The definitions of the exchange terms and the interfacial jump
conditions have been given in Reference [1].
- i---;" -L--2-"-
-~ --_-_----
ERRATA SHEET
for
LOOP SIMULATION CAPABILITY
FOR SODIUM-COOLED SYSTEMS
by
Oluwole A. Adekugbe
Andrei L. Schor
Mujid S. Kazimi
1. There is no page 28
- .-l E-L_---.l~;rst----
_-T--
2.3.
2.3.1.
The Physical Models in THERMIT
Wall Friction
The fluid-solid interaction at the wall lead to momentum
dissipation Fwa [force per mixture unit volume] of the phase "a"
forming interface with the solid (Fig.
2.1).
The Fluid-Wall Interaction
Figure 2.1.
In Fig. 2.1
?wa represents the average wall shear for the
phase "a" and Awa represents the average area 'wetted' by the phase "a".
Aw
wa
By analogy
to
V
wa
(2.9)
single-phase flow, 7wa can be related to the kinetic
energy of phase "a" through a'Darcy-type relation.
-1 f
wa
8 wa
P
a IU
U
a
(2.10)
where
fwa
friction factor for phase a.
The wetted area per unit volume for phase "a" is given
as:
Awa,
wa
AL
V
w C
A
fa
(2.11)
-
fa
De Cfa
where
wetted perimeter for phase "a"
Pwa
=
L
= 'length' of the control volume
A
=
Pw
= total wetted perimeter
De
=
equivalent hydraulic diameter
=
4A/P
=
contact fraction of phase a = Pwa/Pw
Cfa
total flow area
31
.Combining (2.11), (2.10) and (2.9) we obtain the final forms
of the wall frictional force per unit volume for phase a as:
Cfa
Fwa
fwa
e
=
a IUa
(2.12 a)
Ua
(2.12 b)
Kwa Ua
We shall refer to Kwa as the wall friction coefficient for phase a.
The factor Cfa and fwa must be defined with proper considerations
to the two-phase situations.
An assumption which has been deemed adequate is that whenever twophase flow exists,an annular flow regime prevails, with the liquid coating the solid surfaces.
contact is allowed.
At very high void fractions, some vapor wall
Accordingly, Cfa is prescribed as:
1.0
Cfa
; ~a
10(0.99-a);
i
0.0
;
4 0.89
0.89-,< a < 0.99
a > 0.99
(2.13)
and
1 - Cf
Cfv
For fwa'
the following postulate is made by analogy to the
single-phase flows.
YI
MINII.
C Re-b
=
fwa
(2.14)
a
The Reynold's number Rea of the phase "a" is defined to.take into
account the actual flow area of phase "a".
PaUaDe,a
Pa
Rea
(2.15)
where
4Aa
De,a
P
4aA
P
(2.16)
aaD e
We shall now provide the working form correlation (equation (2.14))
for the axial flow condition that is relevant to our I-D loop flow problem.
Actually the correlations that follow were formulated for wire-wrapped rod
bundle flow-channels but by proper adjustment of parameters, essentially
by letting H/D
+..,
they have been found to
work for circular pipe loop
flows.
Axial Flow
(fwa) laminar
(fwa) turbulent
32
P 1.5
FH D
0.316M
Re0.25
Rea
Re
Rea
for
Rea
a
400
(2.17 a)
for
Rea > 2600
(2.17 b)
.
33
( fwa ) turbulent
( fwa ) transition
"
x
+
(fwa) laminar
, for 400 < Rea < 2600
(2.17 c)
S0.885
where
94
66.94
1.034 1 2 4
M
(P/D)
=
H
0
+
Rea (0.086)
29.7(P/D)
(H/D) 2 .239
(H/0)
(Rea - 400)/2200
= wire-wrap lead length (meters)
P/D
= Pitch-to-diameter ratio ,
H/D
=
helical pitch-to-diameter ratio.
The laminar flow correlation was proposed by Engel et al, and
the correlation used in turbulent flow is a slightly modified version
of the correlation due to Novendstern.
tions for bare rods (i.e. H
-
c),
correlation by requiring flaminar
To avoid unrealistic situa-
a cut-off is imposed on the laminar
Re >, 60.
The hydraulic diameter
has been recommended to be calculated thus [1]:
D
= 4 x A (bundle)/Pw (rods + ducts)
(2.18)
_
2.3.2.
_
_1
1
1111
Interfacial Momentum Exchange
The interfacial momentum exchange Fia in (2.2) is made up of
two components, one due to interfacial mass exchange, the other due to
form and shear drag at the interface.
The form of the correlation used in THERMIT-4E for Fia are
given below
F.iv
=
Kiv (U - U )
Fit
=
Ki
Kiv
=
nr
Ki
= (1-n)r
"
(Uv - U)
(2.19)
where
n
+
Ki
+
Ki
(2.20.)
is a weighting factor defined (empirically) for the present by a
donor-like formulation [1].
n =
1 , if r > 0 (evaporation)
n
0 , if r < 0
=
(condensation)
r and Ki must be specified in (2.19) in order to obtain the momentum
exchange coefficients Kiv and Ki
in (2.20).
I is obtained from the equation of conservation of mass on any
one of the phases.
Thus for the vapor phase;
35
?
(ac )
=
+
(aUv)
'
(2.21)
'
The following correlations for K. are obtained using the Wallis
Ill
for friction factor.
correlation
0.01
D
e
(Kturbulent
(Ki)turbulent
(Ki)laminar
1 + 150 (1-)]
IU
vUr
I
321v
- 3
De
(2.22)
(2.23)
where
Ur
2.3.3
= relative velocity
= Uv - U
Wall Heat Transfer
The heat transfer correlations between the fluid and the solid
surfaces (heater or fuel rods and the hex can) that are used in the code
are given in this section.
Fuel or Heater Rods
The heat transfer regime selection logic is presented in Fig. 2.2
adapted from Schor and Todreas [1].
The correlation for single-phase
liquid in triangular - arrayed bundle due to Schad is adopted.
Nu
=
Nu
(Pe/150)0 .3
Pe > 150
(2.24)
=Nu
Pe < 15
Figure 2.2
Heat Transfer Selection Logic (adapted from Reference 1)
37
where
=
Nu
and Pe =
4.5 [-16.15 + 24.96 (P/D) - 8.55 (P/D)2
Re.*Pr
The single-phase vapor heat transfer correlation used is the well-known
Dittus-Boelter's correlation.:
Nu
= 0.023 Re0.8 Pr0.4
(2.25)
For two-phase fluid heat transfer, the total heat transfer
coefficient for two-phase flow boiling with no liquid deficiency is given by
hTP TPc=
h
(2.26)
+hNBNB
as suggested by Manahan [1].
The convective component hc could be represented by the Schad's
correlation in which the Peclet number for two-phase (Pe
Pe
TP
) is given by
TP
(2.27)
= ReTp Pr
and the two-phase Reynold's number (ReTP) is obtained through the factor
F defined as
=
(2.28)
(ReTP/Re )0.8
F depends on the Martinelli's parameter, Xtt
.
101,
x
0Xtt.9 0.5.
=
__i 0 .1
I
(2.29)
The heat transfer correlation for nucleate boiling
due to
Forster-Zuber's analysis istl :
0.79
hNB
= 0.00122
0.
K
45
CP p
0. 4 9
.
AT0.
24
sat
S0.5Z 0.29 hfg0.24 p0.24
AP
0.75 S
sat
(2.30)
where
ATsa t
t
S
= wall superheat ,
= pressure difference corresponding to
=
nucleate boiling suppression factor ,
=
0.99
(ATsate/ATsa )099 ,
sat
~sat,e
sat '
ATsa it,e = effective wall superheat
The following fits for F and S are given in reference [1].
-1
; Xtt
1.0
,
0.10
(2.31)
F =
-1
2.3 5
(Xtt
0.
736
+ 0.213)0;3
-1
Xtt
> 0.10
39
S=
]
[1.0 + 0.12 (ReTp)l.14
-
;
ReTP
<
[1.0 + 0.42 (ReTP)078
-1
;
32.5
& ReTP
-
;
ReTp
> 70.0
0.1
32.5
< 70.0
(2.32)
where
= ReTP (10 - 4 )
ReTp
At high void regimes, (0.89 < a < 0.99), film begins to blanket
the surface.
Heat transfer decreases and is approximated by
2hTP,c
hfilm
%TPIc
+ (1-
2)
hvapor
(2.33)
where
=
10(0.99-a)
The Hex Can
The hexagonal can of the LMFBR assembly provides a heat transfer
medium between the assembly and the adjacent materials, broadly referred
to as the environment.
The heat loss through the hex-can can be signifi-
cant for consideration in the general energy balance and also during
transients, this structural material plays the role of heat source or
sink affecting the fluid heat up and cool down behavior.
The actual representation and the equivalent annular representation of the hex can with the associated structure are illustrated in
1.......__.
MlWi
d IIIYII
I i
ll hl
iihl
40
figures
2.3
and 2.4 respectively.
In Chapter 4, it will become clear how the hex can heat removal
capability that is already built in the code is adapted for the constant temperature plena problem in the implementation of loop capability in THERMIT-4E.
In this regard, the hex can heat transfer process becomes an important
aspect of the present work.
The heat transfer correlation used is due to Dwyer[l]for liquid
sodium flowing in an annulus, transferring heat only through its outer
boundary.
Nu
=
A +
A
=
5.54 + 0.023 (r2 /r1 )
C
=
0.0189 + 0.316 x 10-
CPe
+ 0.867 x 10S
=
4
(r2/r
0.758 (r 2 /rl)-0.
2
(r 2 /r
1
)
(2.34)
1)2
0 204
rl, r2 = outer and inner radius respectively.
2.4.
2.4.1.
Problems with THERMIT Physical Models and Loop Simulation
Forced and Natural Convection
The physical models in THERMIT especially the wall friction and
the wall heat transfer are derived for forced convections.
The shapes
and the thicknesses of the momentum and the thermal boundary layers
peripheral mesh cell
Figure 2.3 Hex Can with Associated Structure Actual
Representation
______________________
___________
U I,
I,, ldllllikllll
l,, ,
W
Imi
,,
,
42
imaginary sodium annulus.
e
Figure 2.4 Hex can with Associated Structure Equivalent
Representation.
43
determine the extent of the frictional resistance and the heat transfer
rate between the fluid and the wall of the channel.
In a natural convection flow, the heat supplied create a
bouyancy force field which drives the fluid.
For higher Prandtl
number (>0.7) fluids, the resultant thermal and momentum boundary layer
profiles are different from the forced flow case.
In a high Prandtl
number fluid, typically the thermal conductivity is low, and the
temperature profile across the boundary layer has a relatively steep
grandient.
The flow field responds to the temperature profile, resulting
in higher flows near the wall than near the channel center.
In the forced
flow case, on the other hand, the flow field responds to the driving
pressure drop regardless of the temperature profile.
We would therefore
expect significant level of inconsistency whenever forced flow correlations
are used for natural convections for the high Prandtl number fluids.
Little work has been done in the area of natural convection
in the liquid sodium.
However, sodium (low Prandtl number (X0.001))
has high thermal conductivity which precludes steep temperature gradients
in the boundary layer.
profile is close
Thus for natural convection in sodium, the velocity
to a unform
to the forced convection flows.
distribution which is similar
Accordingly,
the forced flow correlations for wall friction factor used in THERMIT should
be applicable to natural convection loop flows.
This agreement has been
obtained in our calculations as will be seen in chapter 5. The heat
transfer correlation, however, should depend on the Rayleigh number
Ra [24] for the natural convection flows.
Figures (2.5 a, b, c, & d) show
the velocity profiles for forced and natural convections for both high and
low Pr fluids.
_I
~II_~
I..
I
..
. ... ..
l-
I
I II[FI
Il
44
(a) Forced ConvectionHigh Pr Fluid
(c) Forced Convection,Low Pr Fluid
Figure 2.5
(b) Natural Convection,High
Pr Fluid
(d) Natural Convection,Low
Pr Fluid
Velocity Profiles for Forced and Natural Convection in
High and Low Pr Fluids
Condensation Modeling
2.4.2.
In a typical two-phase flow natural convection.loop, a singlephase liquid is heated to boiling in a section.
The two-phase fluid
flows from the heated section through an adiabatic hot leg to an upper
plenum
where it condenses and returns through an adiabatic cold
leg to the heated section as a single-phase liquid again.
tion modeling in th_ upper
Thus condensa-
plays an in-portant role in two-
lenum
phase loop simulation.
Zielinski and Kazimi [3] have obtained that for the range of
temperatures in which sodium boiling and condensation occurs, the form
of the mass exchange rate is given by the following approximate relations.
S<
dryout
n+1
S=
n+1 2 a
An+l
2-a
M) 1/2
-2R
P -P
s
Ts 1/2
S> adryout
(2.37 a)
- n+1
=
An+l 2o.,
n+1 20
1/2
Pv-P
TMs
(2.37 b)
-1-1
1-1
h illI
miEEogaggggminmM*.lk
*
I,
46
where
Pv = pressure corresponding to a saturation temperature of Tv
Pz = pressure corresponding to a saturation temperature of T
Ps = pressure corresponding to the saturation temperature
Ts
Ts = saturation temperature
A = interfacial area calculated implicitly
R = universal gas constant
M = molecular weight of a particle
a
= mass exchange coefficient
In THERMIT-4E, the assumption of thermal equilibrium at saturation
of the coexisting phases leads to the situation whereby heat is extracted from the vapor phase as latent heat during condensation.
Suc-
cessful low quality condensation has been obtained in some of our
calculations.
At high quality and void regimes, the interfacial mass transfer must be adequately modeled.
Thus, the phasic temperatures must
be able to have different values according to the Nigmatulin
model [4] for instance.
Figures (2.6) and (2.7) illustrates the time variations of the
vapor and saturation temperatures for low and high ranges of interfacial heat transfer Nusselt number.
1120 -
Vapor and Saturation
Temperature
1100
1080 1060
1040
o
1020
S-
E
1000
980
960 940 920 -
0.0
0.1
Figure 2.6
0-.4
0.3
0.2
Time (sec)
0.5
0.6
0.7
0.8
Vapor and Saturation Temperatures for an Interfacial
Heat Transfer Nusselt Number of 6.0 (from Reference 3)
-IIIYIIYIII
----
IIYIYIYIIIIIII
48
1120
1100
10 E0
1060
1040
1020
1000
980
960
940
020
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (sec)
Figure 2.7 Variations between Vapor and Saturation Temperatures
for an Interfacial Heat Transfer Nusselt Number of 0.006
(from reference 3)
,,1101110
00
2.5. The Numerical Methods
2.5.1. Introduction
THERMIT is a lumped parameter component code that can handle
up to three-dimensional two-phase flows.
is used for the fluid dynamics.
An Eulerian numerical approach
This approach follows the evolution of
the volume- (and time-) averaged values of material parameters and
other quantities of interest at fixed points in space.
The reactor
is divided by a mesh into a collection of cells and the parameters and
quantities are calculated at each cell as a function of time.
The
smearing of transported entities within the cells due to this technique
is minimized by reducing the sizes of the averaging volumes wherever
there is a strong spatial variation of the quantity being averaged.
The numerical method in THERMIT is a modified form of the
successful I.C.E. (Implicit Continuum Eulerian) technique.
Like the I.C.E.
method, it uses a staggered grid, treats sonic propagation implicitly and
convective transport explicitly and obtains a pressure-field solution
from which the other variables are inferred.
In THERMIT, all the equations
(mass, momentum and energy) are blended simultaneously to obtain the
pressure-field solution while in the I.C.E., the energy equation is treated
explicitly.
This choice of treatment is necessary in THERMIT because the
change in density with energy can no longer be assumed a small correction
to the flow field in two-phase flows as can be done in single-phase flows F8].
The next subsection gives a review of the numerical method used
in the four-equation model - THERMIT-4 , adapted for one-dimensional
closed-loop flow situations.
The detail of the analysis for multi-
50
dimensional flows has been given in Schor and Todreas[1].
2.5.2.
The Numerical Methods For Fluid Dynamics
The Finite Difference Equations
The choice of the method of treatment of the time discretization
of a system of partial differential equations can be obtained from a
spectrum of schemes, ranging from fully explicit to fully implicit ones.
Whatever the choice, stability and consistency must be ascertained in
order to guarantee convergence (Lax Equivalence Theorem [15]).
judicious choice can be
A
qualitatively inferred from the curve of minimum
computational efforts (fiqure 2.8 [19]) and from the Knowledge of the
time scales of the phenomenainvolved.
3
1.0
Fully Implicit
Figure 2.8
Fully Explicit
Minimum Computational Effort
51
LEGEND
0
Computational time per time step (normalized to a fully
implicit scheme).
-This is low for the fully explicit scheme because
the linear equations of the fully explicit shceme are
solved directly while the non-linear equations of the
fully implicit scheme are solved iteratively.
Total number of time steps required for one time step of
the fully implicit scheme.
-This is larger for a fully explicit schene because of the
full courant criterion required for numerical stability
leading to very small time steps
At < minimum over all
AS
v+c
For the semi- implicit scheme, the more relaxed stability
criterion is
At <
minimum over all
-
v
where c is the speed of sonic propagation and v is the transport
speed.
Total computation effort.
The phenomena respresented by our equations are associated with
different time scales.
i)
We have:
Local pheomena (couplings); here we have the interphase
momentum exchanqe and the fluid-wall interaction.
Generally
the implied time constants could vary widely, from very short
__I_
52
to moderate;
ii)
sonic propagation; the very high sound speed in liquid makes
the transient time for a pressure pulse quite small (10-5 to
106 sec), for the grid. size of interest in our applications;
iii)
transport by convection; as long as the phase convective
velocities are well below their sonic counterparts, the time
constants involved will be considerably larger than above;
iv)
transport by diffusion; in our applications, the time constant
of this pheomenon is of the same order of magnitude as that
associated with convection.
An optimum scheme would allow acceptable time steps (on the
scale of the transients under consideration) and would not lead to a
prohibitively complex and expensive algorithm.
In the light of the above,
we seek a numerical method that treats the first two types of phenomena
in a fully or highly implicit manner, while describing explicitly the
two transport mechanisms.
In space, a full donor-cell differencing is used accompanied by
additional averaging whenever quantities are required at the locations
other than those at which they are originally defined.
The widely used
staggered-mesh approach is adopted, whereby the scalar quantities are
defined at the cell center while the fluxes are defined at the cell faces
to which they are normal (Fig. 2.9).
The discrete analogs of the partial differential equations
describing our two phase model will now be presented.
_1_1~_ __
53-
i-1/2
i+1/2
i+1
i+3/2
Ui+3/2
cell center (i):
a, p, Pv' P'
Pm'
em, ev, e., Tv , Tl
Figure 2.9 A typical fluid staggered grid showing
locations of variables and subscripting convections
The Mixture Mass Equation
v(n+1
m
n)/At
m
-{A[[aov)n(Vv)n+1
+ {A[ap)
n(U)
+ ((1-a) ))n(U
n+1
)n(U1)n+} i+1/2
+ ((1-a)p zi1/
)n+l] i-1/
2
-
0
(2.38)
In the above, the convected quantities are needed at cell faces, where
fluxes are defined.
quantities.
Full donor-cell differencing is used to define these
Let C stand for any cell-centered quantity (see Fig. 2.9)
and consider the face (i + 1/2), normal to the direction of flow in the
loop.
The quantity Ci+ 1/2 is then
determined as:
YIIIIYI
54
C.
, if
Ci+
i+1
'
1
(U)
i+1/2
> 0
>
i+1/2
if (U)n
i+1/2
<
0
It is important to note that donor-cell decisions are made
only with regard to quantities at time level n, using velocities at
the same time level.
As a result no difficulty arises even if a velocity
sign change occurs during a time step.
The Mixture Energy Equation
A number of variants for the finite difference energy equation
exists.
The conservative/semi-implicit convection (CSIC) scheme is
given below:
V[(P e )n+1
(Pe)n]/t
[pn +
ven
][An(Un+1/2
n+1/
Sn+2(pe]) n
][A(n(U ) i-1/2
+[Pn+
(Pe p z)+1/2
n+ i+1/2(Ui+i/2
+1/2[[A(1-a)An ( n+l]
-
[pn + (pev)n-1 /2[An(Uv) n+l]i
2
[pn + (P e )ni/2][A(1_.)(U)nl
i-i/2
: n+1/2
+ Qkn+1/2
(2.39)
IUI MWM
The difference forms of energy and the mass equations,
equations(2.38)and(2.39)are a strict adaptation of the scheme used for
a six-equation model [1], [2] to a four-equation "mixture" model.
The
schemes for both models are equivalent for single-phase, either liquid
or vapor.
For two-phase however, the four-equation adaptation suffers
a subtle flaw, namely the lack of monotonicity of the mixture internal
energy density (me m) with respect to em . This feature is undesirable
for the Newton method used to solve our system of equations.
To avoid the problem raised by the product pmem
,
a non-
conservative/semi-implicit convection (NCSIC) form of the energy equation
is used.
To this end, the mass equation is multiplied by em and then
subtracted from the conservative form of the energy equation.
The
resulting difference equation is
)
n+l - (em)n]/At
V(Qm)n[(em
+ [conv e - conVm
=
(Q+
n + 1/ 2
Qk)n+1/2
(2.40)
where convmnn+1/2 and conve n+1/2 stand for the semi-implicit convective
terms in the mass and energy equations, respectively.
The heat sources
appear with superscript n+1/2, indicating a combination of implicit /
explicit components in the constitutive relations used for them
The Phasic Momentum Equations
The momentum equations are used in the non-conservative form,
particularly convenient to our method.
The control volume for which the
oilillilI
61
' 1 hUIJN1
56
momentum equation is written is offset by half mesh with respect to
that used for the scalar quantities (Fig. 2.9).
The momentum equations
are presented below:
Vapor Momentum Equation
(apv)n
i+1/2
(U )n]
nl
[ (Uv)
(U)i+/
2
-
At
+
+ (c
n
+ ai+1/
S-
vn
]AUn
) i+1/2] n
5
+1/2(Uv)i+i/2(.
/v)
v
2
i)n
(Pi+1
ASi+ 1/2
n
n+1/2
(F n+1/2
(Fwvi+1/2 - (Fiv )+1/2-
(Mp)
^ +
eg
(2.41 a)
Liquid Momentum Equation
(similar to(2.41a))
(2.41 b)
AU
In the above equations (- )i+1/2 represents a difference
approximation for the spatial derivatives 3U a/S
i+1/2, where
evaluated at the point
a = k or v.
Again the cell-centered quantities a, Qv',
at the cell faces.
are now needed
Donor-cell differencing can be used in case of single-
phase liquid where the properties in the adjacent cells are not greatly
different.
Things are different, however, once the face in question
separates a liquid cell and a two-phase cell.
In this case the mixture
57
density (mainly through a) may vary by as much as two orders of
In such a situation a change in the sign of the velocity(ies)
magnitude.
at the face, for donor-cell scheme, would lead to very large changes in
terms of the momentum equations, which in turn could generate large
pressure spikes and even ruin the solution, by imposing an impractically
short time steps.
As a result, a weighted average scheme is adopted.
Let C be a cell-centered quantity, then its value at the cell surface
is specified as:
Ci+/2
.
(C ASi + Ci+ AS i+)/(ASi + AS
)i+
)
(2.42)
For the product caa for instance, we define
(aaCa i+1/2
(2.43)
S(a)i+1/2 (Pa)i+1/2
The difference approximation of the convective derivatives are
defined through a donor-cell logic-:
,
\
(Uv)i+3/2 - (Uv)i+1/2
if (Uv ) i+1/2
< 0
if (Uv
v ) i+1/2
i+I / 2
>-
i+1/2
0
AS.
L i+1
(Uv i+1/2 - (Uv)i-1/2
(2.43)
and the mesh spacing (LS)i+1/2 needed in the pressure gradient is given by:
53
(AS) i+1/2
(2.44)
= (ASi + ASi+1)/2
In the momentum equations, the wall and the interfacial
exchange terms havealinear dependence on the new time phase velocities
or they can be linearized in these new time velocities about the old
time velocities [1].
The following forms of constitutive relations
are adopted in our calculations.
n
n+1/2
(Fwa)
(Kwa)
i+1/2
n+1/2
(Fia)i+1/2
n+1
(Ua) i/2
i+1/2
i+1/2
n
(2.45)
n+1
= (Kia)
(U
ia 1+1/2
- U) i+/2
(2.46)
The coefficients Kwa and Kia can be complex functions of any variables,
the only requirement being its evaluation using old time quantities.
With equations(2.45)and(2.461 the momentum equations(2.41 a)
and(2.41 b) can be written in the form:
Un+1
v
=
a APn + 1
+ b
Un+1
=
a APn +
+ b
v
v
where the coefficients av , aZ, by and b
APn+1
(2.47)
contain old time quantities only.
(Pi+1 - Pi )n+1 is the pressure drop over the interface
at i+1/2.
The spatial subscripts have been dropped in(2.47)with the under-
standing that the velocities are evaluated at the faces of a node while
the AP's are the total pressure drops across the faces.
by, bk are defined below [1]
The quantities av3 a,
(2.48 a)
-)]/d
ae 2 + At k (/iv
--
-[(1-a)el
t
(2.48 b)
+ At kia]/d
(2.48 c)
(fle 2 + At kivf 2 )/d
=
(f2 e1
Sapv
=
(l-a)p
S
c
=
S
+ At ki fl)/d
(2.48 d)
+ At(kwv + k.iv )
(2.49 a)
iZ
+
1~l
t(kw
(2.49 b)
+ ki )
(2.49 c)
[Uv - at (conv\ + e.g )]
(1-a)p[U
ee 2
- At(conv z + e-
- (At)
2
In equations (2.48) and (2.49),
kiv ki
)]
(2.49.d)
(2.49 e)
everything is evaluated at the old time.
Consequently the coefficients a's and b's can be calculated only once at
the beginning of the current time step and stored.
2.5.3. The Solution Scheme
The finite difference equations described in the preceding
section combined with the equations of state (equations 2.3) form a
large system of non-linear equations.
The following seven new time
variables appear as unknowns for all cells in the domain of the problem:
n+1
pn+l
n+l
n+l
n+l
n+l
n+1
Pm
M
9
, em
e
, T,' Tv ,' Uv
and U
The new time temperatures appear from the fully-implicit treatment of the
heat sources and sinks that is adopted for our loop flow model.
The high
heat transfer coefficient and the low heat capacity of the plenum material
that are required to keep the plenum temperature constant during transients may give rise to instabilities for the fully explicit or the semiimplicit treatment, hence our decision to use a fully-implicit treatment.
n+l
n+1
Note also that Pm
and em
now appear as separate unknowns due
to the non-conservative form of the energy equation adopted (equation 2.40).
This splitting of the product (mem )n+1 ., which otherwise would appear as
an unknown from the conservative form (equation 2.39), is highly desirable.
The product pmem is a non-monotonic function of em for sodium (and also
for water at low pressure, (Fig.2 10).
This behavior have the tendency
of ruining the Newton-Raphson method adopted to solve our non-linear
system.
extremum
Generally, the Newton-Raphson method is destroyed when an
point
exists between the guess and the solution.
61
,
4--
P = 1.5 bars
10-
io2
E
S
101
10 0
0
1
2
3
4
5
67
Internal Energy (MJ/kg)
Figure 2.10
Sodium Internal Energy Per Unit Volume versus Internal
Energy
YY
62
The Jacobian Matrix and the Pressure Problem
2.5.4.
The new time velocities that appear in the mass and the energy
equations are eliminated in favor of the new time pressures using the
momentum equations in the form of(2.47). Thus for each cell we now have
two scalar conservation equations namely the mass and the energy equations.
The appropriate equation of state is combined with these scalar conservation equations for closure.
In our one-dimensional loop flow treatment, the. elimination of
the new time velocities leads to the appearance of the new time local and
two neighboring pressures in the mass and the energy equations for each
node.
In three dimensions up to six neighboring new time pressures will
appear.
The resulting mass and energy together with the state equations
can be written in functional form for node 'i' as follows [19].:
Rmi ( mi'
Pi-'
Pi'
Rei (0mi' emi' Pi-l
Cmi - Imi ( P i . emi)
Pi+l)
Pi'.,
Pi+)
= 0
(2.50 a)
= 0
(2.50 b)
= 0
(2.50 c)
where
Rmi refers to the mass equation for node 'i'
Rei refers to the mass equation for node 'i'
All the quantities inside the parentheses are new time quantities
Equations(2.50)are generally highly non-linear,
non-linearity being mainly the state equation.
the source of
63
The pressure P and the mixture internal energy em are taken
as the main variables and the mixture density pm is eliminated
through the equation of state.
Consequently we obtain two non-linear
scalar equations in P's and em for each node.
These equations can be
written symbolically as:
i(U)
(2.51)
=
0
=
]T
[Rm 1 , Rel........, RmN, ReN
=
[P
where
em........' PN' emN T
Applying Newton's method to solve 2.51 we have
J(U)U
(2.52)
= - R(U)
where the jacobian J(U) is given by
Let k be the counter for the Newton iteration.
Then the scheme becomes:
J(Uk) (Uk+1 -uk)
(2.53)
The entries of the jacobian matrix for a particular node 'i'
are obtained from the following partial derivatives
- MVM
-II
MMIIIIY,
64
i
aRmi
Pi-1
i
aRei
aemi
aRei
aPi1
aP
i-1
aRmi
aRei
e m
i+1
aRei
aP
i+1
We denote these generally non-zero entries by "x" and thus
obtain a matrix form for equation(2.52), for cell i:
k +1
6P
i-1
6
x
o
x
x
x
0
x
0
x
x
x
0
emi-1
-
6P.
6 emi
Rmi
LRei
Pi+1
6emi+1
Lemi+l
(2.54)
Equation(2.54)forms a total of 2N equations, where N is the total
number of nodes.
The full 2 x 2 block in(2.54)provide local (within cell)
coupling while the sparce 2 x 2 blocks provides spatial coupling,
indicating a field couplinq throuqh pressure only.
The next step in the solution is to solve the main diagonal
block to eliminate 6emi
in favor of the neighboring pressures.
This
procedure effectively reduces the problem to a pure pressure problem in
N equations.
The pressure
problem in matrix form becomes:
IIMM
65
k+1
6P.1i1
P.
xx
=
R k.
(2.55)
6P
i+1
Equation(2.55) when written for the N-cell domain gives rise
to an N x N tridiagonal jacobian matrix in the
right hand side becomes an N x
left hand side while
1 vector.
The pressure increments are solved in(2.55)
by a direct
technique (i.e., LU decomposition).
The increment 6e k+1 is then obtained from the second equation ofC2.54)
mi
in each cell.
This completes a Newton iteration.
The process is then
repeated until successive changes in the main variables become very small.
2.5.5
The Numerical Method for Fuel Rod and Hexagonal Can Conduction
This is similar to the numerical method for plenum heat conduction
given in section 4.2.2.1.
2.5.6.
Overall Solution Scheme and the Hierachy of Subroutines in THERMIT
The solution scheme in THERMIT is summarized in the block diagram
shown in Fig. 2.11. The hierachy of the subroutine for THERMIT-4E is
depicted in the chart THERMIT4, (Fig. Dl, Appendix D).
.~~..milli
1 SUMMARY OF THE SOLUTION TECHNIQUE IN THERMIT 4
Vol. averaged differential equations of conservation
Area averaged interfacial and wall exchange terms
Area averaged interfacial jump conditions
Equations of state
Difference forms of the conservation equations
Difference forms of the state equations*
Difference forms of the constitutive equations*
Use the momentum equations to eliminate UV ? in the mixture
Mass and energy equations
Pick P and em as main var. and eliminate pm through
equation of state
1
Obtain the Jacobian of the resulting 2 scalar equations
Reduce to pressure problem
Do innter iteration on pressure problem
Back substitute the converged P to obtain the other
variables
* Algebraic equations
differenced only in
time
re the
chances in
main variabl
small
Yes
Stop
Figure 2.11
67
3. THERMAL-HYDORAULIC. SINGLE PHASE LOOP ANALYSIS
3.1. Introduction
The laws of conservation can be summarized in the vector
equation thus:
S
t
+
=
(3.1)
B
where
P
is the vector of the conserved flow properties of
mass, momentum and enerqy ,
e
isa vector representing the fluxes of the conserved
quantities,
B
is a vector representing the interface and wall transfers
of mass, momentum and energy, and
s
is the spatial coordinate in our one-dimensional representation of the loop.
If we let 7o represent the rest (non-flow) condition of the loop
and 6T
a given element of T applied to the rest condition, then the loop
will change from the state of rest characterized by To to a state of
flow characterized by T , where
=-T
In practice, 5T
+
6T
(3.2)
is supplied by providing 'holes' in the loop for access
to the environment from where 67 can be applied.
Since a real loop
cannot be an infite sink (or source) of T, at some other hole(s), 6T
68
(or some fraction of it) must be rejected to the environment. The
continuous transport of 6W added to the loop at some points to the points
of rejection maintains the loop flow [20].
The steady state configuration of the loop flow is specified
mathematically by
3.3
0,
S
looo
while the transient condition is given by
S~0
,
loop
<
t
<
T
+
T.
-
Where
t is the time,
T is the period through which the transient
lasts and T is the time at the onset of the transient.
During steady state, there is no net exchanqe of T between
the loop and the environment.
During transient, the loop has storage
capability for some fraction of 6-T.
This storage capability depends on
the inertias associated with transport of energy, momentum and mass
within the loop.
A large environment would maintain its thermodynamic state
constant despite the disturbance from a loop during transient.
In this
context, the environment would serve as an infinite source or sink for
the loop.
An important aspect of the loop flow that has drawn the attention
of many investigators is the issue of flow instability.
A thermal-hydraulic
69
loop (thermosyphon) may become unstable; three types of instabilities
have been found both experimentally and theoretically.
One is associated
with the onset of motion in the whole loop (not in just one local cell).
The second is the existence of multiple steady state (meta-stable equilibrium) solutions and the third is that of oscillation growth [12].
The
third instability in which small amplitude oscillations grow, may lead
to flow reversals.
Both analytical and numerical methods have been used
to investigate these phenomena and to find stability boundaries (or
marginal stability curves) [14].
In this work, a first order perturbation
theory is used to derive an approximate relation for the natural frequency
of oscillation and to obtain the marqinal stability curves for the particular
loop being simulated.
The expressions obtained can be applied to loops
of similar geometry.
3.2.
3.2.1.
Typical Flow Loops
A Natural Convection Loop [6]
A typical natural convection loop is the Sodium Boiling Test
Facility (SBTF) at Oak-Ridge National Laboratory.
A series of single
channel sodium boiling experiments [ONRL/TM-7018], designed to simulate
Fast Test Reactors (FTR) natural convection boiling behavior was conducted
in this facility.
The SBTF is shown in Fig. 3.1 with the axial dimensions of the
tubular test section shown in Fig. 3.2.
These dimensions are approximately
equal to those of a full-scale FTR fuel sub-assembly, Fig. 3.3.
The heated
length is 0.97m and the simulated fission gas plenum region downstream
of the heated zone is 1.50m.
The inside diameter of the Hastalloy X
70
ORNL-DWG 79-16755 ETD
EOUALIZER
LINE
1.83 m
Figure 3.1.
i
Soditm Boiling Test facility - loop.
71
ORNL-DWG 79-6142
-
r - -
-
ETD
T
AYLOR PRESSUPE
TRANSDUCER (PT-7)
" UPPER PLENUM TANK
'CT..FLOWMETER (FESB)
VOLTAGE CONNECTION FOR
VOID DETECTOR SYSTEM
z
O
w
Ye
wE
'
THERMOCOUPLES AND
VOID DETECTOR
VOLTAGE TAPS
4
wo.
\i
VOLTAGE CONNECTION FOR
VOID DETECTOR SYSTEM
Z
O
U-o
w,,.
w
SENSOTEC
PRESSURE
TRANSDUCER
(PE-2B)
I(FE-1B)
NNECTION FOR
TOR SYSTEM
LOWER PLENUM
TAYLOR PRESSURE
TRANSDUCER (PT-14A)
Figure 3.2.Sodium Boiling Test facility -
test section.
9153)/4 OVERALL LENGTH
-
I-
041.000
4
PPE ..- LA-KET
Su
UPP(A BLANKET
.
43/ 6 -
2It
ECTION
HEXAGONAL TUBING
SECTION
14 22
CORE SECTION
LOWER BLANKET SECTION
LOWER ADAPTER
CENTRAL TIE ROo
I
test section is 3.25mm which corresponds to the average hydraulic diameter
of the fuel sub-assembly.
A coil upstream of the heated section simulates
the sub-assembly inlet module hydraulic resistance [c/f EBR-Idesign] and
accommodate the thermal growth of the test section.
Tanks at the top and bottom of the test section simulate the
reactor inlet and outlet 0lena.
these tanks and completes the loop.
A 51.0 mm I D return line connects
An a-celectromagnetic pump in the
return leg of the loop is available for forced flow testing.
Oxide control
is accomplished by a zirconium foil trap in the facility dump tank.
The heat source is a 15.2kw radiant furnace.
The reflector
of the furnace is water-cooled, and the power input to the test section
is computed from an energy balance on the furnace.
A guard heater assembly
is used to approximate adiabatic conditions in the simulated fission qas
plenum region of the test section.
The upper and the lower plenum
temperatures are maintained at 590 0 C and 420 0 C respectively, by sodium-to-air
heat exchangers located inside the plenums.
These temperatures correspond
to the rated FTR operating conditions.
3.2.2
A Forced Convection Loop [7]
A typical forced convection loop is Sodium Boiling Test loop
[Loop C.F. Na) at the Centre D'Etudes Nucleaires in Grenoble, France.
The schematic of this loop is illustrated in Fig. 3.4 with the detail of
the test section illustrated in Fig. 3.5.
This loop is designed so that one can perform tests with pressure
difference, inlet subcooling, outlet mixing temperature, fluid velocity
and specific power the same as in an LMFBR plant.
This is important
03 ,MBP2 ,
02,
Figure 3.4
LOOP C.F. Na
No
t
COOL
FLOW
U
w~
=
Figure 3z5
MSE
The Test Section
because the stability of boiling and the voiding of channel, for'instance
depend on the inlet subcooling and the pressure difference between the
inlet and the outlet of the test section.
The loop uses a mechanical pump with vertical shaft and gas cover.
The pressure head is 100 meters of sodium for a flow rate of up to
10 m3/hr.
The pump is maintained at maximum temperature of 5500C.
The
main bypass valve (VR2) is used to keep the pressures at the inlet of
the heated bypass (VR04) and test section constant even if there are flow
oscillations within the test section.
Throttling valve VR4 is used to
simulate flow blockage at the bottom of the fuel assembly by its pressure
drop.
The test section consists of a heated rod 6.6mm diamter, 600mm
heated length, whose power of 30kw is dissipated in electrically heated
wires insulated by boron nitride.
The outside diameter of the flow
annulus is 8.6 mm.
3.3.
One-Dimensional Loop Analysis
3.3.1.
Mathematical Model
In this work, a one-dimensional flow approximation has been deemed
sufficient to represent the state of flow within the sections of the exoerimental loops.
For ease of manipulations, we shall adopt the three equation
homogeneous equilibrium model (HEM) for the loop analysis that is done in
the next sub-section.
3.3.1.1.
The Governing Equations
Summing the two phasic momentum equations (1.2) we have:
--'IIYY
at [ayUv + (1-a)pU]1 z
s [ap U
+
(I-a)p zU ]
+
(-a)tj!
v
as
Pas as e.v[
k
= 0
- [apy + (1-a)p ] g*e
(3.4 a)
where s denotes the only spatial coordinate that runs around the loop and e
is the unit vector in the direction of s.
We define:
Mixture mass flux
Gm
+ (1-a)pzUe
= aPyUv
,
(3.4 b)
Mixture velocity
Um
= Uv
=
U
, and
(3.4 c)
Mixture stress tensor
t
=
CT
+
(1-c)
(3.4 d)
From equations (2.8) and (3.4) we obtain the one-dimensional
HEM aoverning equations for a loop as:
Mixture Mass Equatioh
;Pm
Gm
-t
=
-s
(3.E a)
0
Mixture Momentum Equation
(
+s
2
m
PM
BP
as
fIGm Gm
2DePm
(3.5 b)
Mixture Energy Equation
L (p h - p)
-t mm
q"PH
A
)
+ -as (hG
mm
G
P
PM
TP
+
f
lG m
JG
2e m
(3.5 c)
2e mI
Where,
Se
m
P
+ -
is the mixture specific enthalpy,
PM
PH
= heated perimeter ,
A
=
q"
= heat flux
flow cross-sectional area, and
The stress tensor T has been written in the usual wall
frictional dissipation form.
78
The Steady State Loop Flow Model
The steady state loop flow model is easily obtained by neglecting
the terms containing time derivatives in(3.5). We have:
aGm
a
as
(3.6 a)
0
-
as
2
Gm
Pm
a
as
flGmlGm
2DePm
+
g
+ Pm
q"PH
Gm
aP
Tas (hmG
mmm)
A
+
Pm
^
e
(3.6 b)
fG m
2e Pm
(3.6 c)
-1.
3.3.1.2.
Functional Dependence of Mass Flow Rate on Heat Input For
the ORNL Sodium Boiling Test Facility (SBTF) Loop
We neglect friction and form loss terms in the energy equation
(3.6 c) compared to the source and the sink terms.
Then the steady state
governing equation for the single phase liquid reduces as:
dG
ds
d
-ads
G2)
dP
ds
p0
d (hG)
ds
(3.7.a)
0
-
PH
A
f)GIG
2 DeP
q
source
+
p(s) -e
PH qin
A
sink
(3.7 b)
(3.7 c)
We have neglected the effect of spatial variation in density
in both the acceleration and frictional dissipation terms in(3.7 b).
This is in compliance with the Bousinesque approximation in which it
effect of spatial density variation is only important
is assumed that
in the head loss term (bouyancy).
This approximation as well as the
neglect of the friction and form loss terms in the energy equation will
be valid for single-phase sodium flows in which large spatial density
variation does not exist in the loop.
We shall base our derivation on uniform cross-sectional flow
area and make correction for the non-uniform cross-section of the actual
loop by a k-factor for the local losses.
Form(3.7 a) G is constant, hence equation(3.7'b) and(3.7 c)
further reduces as
dp
ds
+
p
e
()e
-
f
2D Po
=
0
(3.8 a)
and
Gdh
ds
P
AH q source
P
AH
,
sink
(3.8 b)
Where in 3.8,
po
is the initial (reference) density ,
A
is the heated area (both for source and sink),
PH
is the heated perimeter in the source and the cooled
perimeter in the sink
q"
is heat flux
010iI
li,
,ill
,
80
Integrating (3.8 a) round the loop we have:
loop
dP
ds
p(s) ge ds
+
f GIG ds = 0
loop 2Depo
(3.9)
-
loop
Now,
dP
loop
p(s)
Sds =
, and
(3.10)
= pO (1 - a(T(s) - T0 ))
where
is a reference temperature corresponding to P and
is the coefficient of thermal expansion
B
Equation(3.10)is valid in the range of temperature of interest for liquid
sodium flows.
Using(3.10)in (3.9)we have :
flIGI
T(s) ds
-
= loop
loop
2
ds
(3.11 a)
Dep o
The friction factor f in(3.11 a) is given in the usual form as:
:
where 'a' and 'b'
aRe
1-I
)
(3.11 b)
are flow regime dependent positive numbers, and
Re is the Reynold's number.
becomes:
= a(
GDe -b
The right hand side intearal
in(3.11 a)
ml
a G(2 -b)ibL
2D + b )
fIGIG
loop 2Dep o
O
2e
a W(2 -b) bL
2
2D (lb)A( -b)
where L is the total length of the loop.
Again, the variation of
viscosity ipwith temperature has been neglected.
-
Po
(3.11 .c)
Equations (3.11) yield:
(3.12)
a W(2-b) bL
g. T(s)ds
D
2
20,
loop
To evaluate the integral
(1+b)A
A(2-b)PP,
in(3.12)we re-write the energy equation
(3.8.b) using temperature as the dependent variable in the form appropriate
to
the SBTF loop.
Thus we have:
LH
H
2
LH
<S<
L
LE1
4H
W
dT
Tw1 -T(s))
e
A p ds
4H(Tw -T(s))
e
2
L
LE2
2
(heated section)
E1
<S
<
<S <
otherwise
1 (upper plenum)
2
_
2
(lower plenum)
(cold & hot legs)
(3.13)
*$*Now"l
--
1IL
llil
I
I
I Ih
82
where
W
is the mass flow rate
is the heat flux
is the heat transfer coefficient between the
liquid sodium and the plenum wall.
H
Twl, Tw2 are the constant temperatures of the upper plenum
and the lower plenum respectively.
Other geometrical parameters
that appear in(3.13)are as depicted in Fig. 3.6.
Solving 3.13, we obtain
44 1
T(s)
+
k1
+
k
k2 e- as
LE1
S2-<
k3e -as
2LE
2
-pe
C WL
CpWLH
LH
T(s)
= Twl
+
T(s)
= Tw2
+
'
LH
-2
-s
LE1
s
<
s < LE
22
2
(3.14 a)
(3.14 b)
(3.14 c)
LE
2
T(s)
=
LE
-1
2
<
s
<
LE
<
s
<
22
LH
LE
2
2
2
<
S
<
(3.14 d)
where
4HA
DeCp
and
H
W
H =
The constants k1 , k2 . . .
k
(3.14 e)
are to be determined by matching the inlet and
the outlet boundary conditions of the various sections.
At steady state T(-LH/ 2 ) - Tw2 is the inlet temperature to the
heated section
From(3.14 a):
k
=
1
Tw
L
(H)
L11cW 2
LHEquation
3.14(a)
becomespW
+
2
Equation 3.14(a) becomes
T(s)
S
+
= Tw2
CpL
L
H
((cL
2
+
LH
s)-
LH
2
<
-
(3.15 a)
2
2
From 3.14(b)
k2
=
T(-
-
Tw
exp(-aLE /2)
At steady state, T(4.E /2) = T(LH/2), hence
1
[Tw2 - Twl + Q
k2
T(s)=
Twl
+
Tw2 - Twl
+
exp(-L
E
/2)
,
and
exp(-aLE /2 + s)
LE
<
21 -< s -2
LE
E1
(3.15 b)
.1
i11m1iii,
84
From equation (3.14 c)
LEexp(-
k3=
(T(- --
T(s) = Tw2 +
T
w2
Tw2) exp(-aLE 2/2)
= T(LE 1 /2), hence
T(-LE 2 /2)
At steady state;
K3
)-
LE2
Tw 2
+
- Twl +
Twl-Tw2 + (Tw2 - Twl +
exp(-aLE /2)
L
T(s) =
K4
LH
(3.15 c)
2
-
< < LE
< s<-
T
=Tw2
exp(-aLE2/2 + s)
L
2
S.
)exp( -aLE 1), and
(3.15 d)
L
T(s) =
K5
T
+
Tw2 - Twl +
Q
L
exp(-aL E;
2
s <
(3.15 e)
T(s) =
K6
- Tw2
+
(Tw2 -
= Tw2 + ITwl
LE 2
2
< s
<
) exp(-aLE
LH
2
exp(-aLE )
(3.15 f)
We now express the loop integral in eqn. 3.12 as integrals over the sections
to have:
B8o
g*e T(s)ds = poq
loop
- JTds
LH
-
Tds -
Lh
fTds + Tds
LE
1
Lc
-
1
1
-Tds
E2
2
-
Tds
85
The top and bottom plena have the same dimensions and heat transfer
is the effective vertical
L'
2 = L
=
coefficient, hence
A 900 turn has been assumed in the
height of the return (cold) leg.
plena, thus making the effective elevation change in them 1/2 LE.
Evaluating the indicated integrals above using equations 3.15 we have:
o loope Tds =
o 1oopg
l
e
-
+
[
LH +L
pog *C W
+'Lh
p
BPo~
0
[- L ,
+
(exp(-2aLE)- 1)]
2p
E
I
Lc
exp(-LE)]exp(-aLE)
BP
(Tw2 - Tw ) exp(-aLE ) (exp(-aLE)-l)
c0
+
6p g [Tw 2 (LH + Lh, + L
) + 1 (Twl+
Tw2 )LE
+ 1 (T - Tw2)
- Twlc
w2
w1 c 1 a w1
+ Bog
[(Tgw2 - Twl)exp)(aLE
+ Lc2 (Twl
' c I1~ + LcR 2 exp(-aLE)
Tw 2 ) exp(-2aLE)
For practical values of H, A, De , Cp and LE ' exp(-aLE)<< 1. (For example
exp(-aLE)
0.006 for the SBTF loop).
The above equation simplifies, after
neglecting terms containing exp(-aLE) or exp(- 2aLE), and noting that
L'
= LE + Lc
+ LH + Lh
as:
.
-Sc o
eog Tds
loop
=agpo CpW
+
...--
-
-
lIN1
+ Lh
hZ +2
gp0 (Tw2 - Twl)(LH + Lh
+--
+ L
-
1)
(3.16)
Substituting (3.16) into (3.12)we have:
3 b)
aW(
De(l+b)
(2b b)
2
- W(Tw 2 - LTwl)
2D (1+b)AH(2-b)
(LH + Lh
w2
-
C
(
+
L
H 1 h
2
+
L
h
+)
20
2
+ Lz
2
-
c2
= 0
(3.17)
Equation (3.17)gives the analytical relationship between the
mass flow rate and the power input for the SBTF loop assuming constant area.
It is interesting to note in(3.17)that for Twl = Tw2 and
remains stagnant as expected.
If Q = 0 and Twl
= 0,
0 the loop
Tw2 , a positive flow
field develops as have been obtafned in our calculations for Twl >
Tw2 *
Equation(3.17)is cast in dimensionless form by dividing through
by Q/C De
to obtain:
aW(3-b )
2 De ( 2 +b)
A
b
C
2
L
aW
)po2 g
WC
L
- -- (T 2 - T)(LH + L + E + L
c 2
2
h
H
w1
w2
Q De
LH
SH ++ 22Lh
L
2D e
+1 I
2t
0
D
De )
St
(3.181
Upper
P1 enum
Lower
P1 enum
Fiqure 3.6.
Loop Modeling Geometry
where the dimensionless number
St = C
e
4HA
WCp
is the Staton number defined with respect to the
plena heat convection.
It should be noted,that equation(3.18)can be used for all constant cross
sectional area loops of geometry similar to Figure 3.6
3.3.1.3.
Comparison of Analytical Results with the Codes Calculations
The values of the following parameters that appear in(3.17)are
used.
Geometrical Parameters:
LH
=
0.97 m
LhZ
=
1.5 m
=
3.0885 m
= 0.6175 m
= 3.25 x 10-3m
= 8.295 x 10-6m2
= 11.07 m
Fluid Properties:
C
p
11
= 1.2573 x 10
p
= 850.14 kg/m 3
H
= 1.0 x 106
B
= 0.274 x 10- 3
J/kg - K
(see Table 3.1)
Watt/m2 - K
K-1
Plena Temperatures:
Twl
= 863.15 K (upper plenum)
Tw2
= 693.15 K (lower plenum)
=
-H = 8.121 m
DC
-aLE
e
0.006
For the range of power calculated (150 W to 370 W) for the
constant area geometry case, the flows fall within the transition regime
400 <Re< 2600 as prescribed in the code.
Equation(3.17)written for transition regime, using the friction
correlation in Equation(2.17 c) becomes:
eff
eff (3-bt) bt
L
iaturbW
b
L
aam W(3-bt)
lam
+
(I+b ) (2-b )
2
A
2poBgD
e
o
- W [(Tw2
Twl)(LH + L
LH
pL + Lh
+
2P
L+
+ Lc
(1-bt) ( 2-bt)
BgD
A
o
A
2
L1
1
+
(3.18)
=
p
Where
1
=
eff
ala m
eff
and aturb.
for laminar flow,
0.25 for turbulent flow, and
are as given in Table 3.1.
The value of M that
appears in(2.17 c) has been taken as unity since the value of H/D (helical
-
--I
------ IYYIYIYIIIYIII
90
pitch-to-diameter ratio) used for the calculation is very large 1010
TABLE 3.1
Comparison of Analytical Predictions with Codes
Results for the Loop
Mass Flow Rate
Code's
W(g/s)
.316AV
eff
a
1am
60/1-
Analytic.
W(g/s)
eff
turb
Q
(W)
114
(x10 - 4 )
.373t
150
2.1137
738.1
0.1537
.1239
55.197
.375
.481*
230
2.103775
895.8
.2254
.1511
52.8068
.503
.495*
240
2.1001
923.5
.23797
.1552
52.3766
.519
.509*
250
2.096
951.5
.25068
.1593
51.93796
.534
.522*
260
2.090
978.5
.26295
.1632
51.5215
.548
.534*
270
2.0892
99816
.2743
.1674
51.1320
.561
.549*
280
2.086
1031.2
.28691
.1705
50.66679
.575
.30982
.1771
49.84624
.605
-
e
290
.574*
300
2.0793
1081.6
.633+
330
2.0723
1196.79 0.36218
.19017
47.918
.636
.656t
350
2.0653
1244.48 0.3839
.19578
47.097
.659
.678+
370
2.0583
1290.59 0.4048
.2011
46.289
.681
---
*: Results obtained for cut-point on top.of the lower plenum.
-:
Results obtained for cut-point beneath the upper plenum.
91
In the actual experiment the upper plenum is connected to the'
pressurizer line, thus the results obtained for cut-point beneath the
upper plenum are more dependable.
The discrepancies of about 0.5%
between the analytical results and the code's calculation can be
explained in terms of the approximations made in obtaining equation
(3.17)
3.3.1.4.
Correction for Form Losses in the Actual Loop
Equation (3.17) or (3.18) cannot predict accurately the
mass flow rate for a given Q in the actual loop.
There is pressure
loss due to contraction and expansion and elbows in the non-uniform
cross-sectional area loop.
This extra drop is corrected for by the
k-factor. The ratio of the flow cross-sectional area in the cold
return leg to that of the test section is about 250:1.
For all practical
purposes, the fluid in this return leg can be regarded as stagnant
compared to the flow in the test section.
Thus we neglect the
frictional losses in the return leg in the actual geometry calculation.
Accordingly the value of L in(3.17)and(3.18)becomes the length of the test
section including the plena only.
The form pressure drop is given as:
AP
form
KIGIG
2Po De
2
KW
2A2 oDe
Adding(3.19)to the friction pressure drop in(3.8 a),
of(3.17)that is valid for the actual loop as:
(3.19)
we obtain the form
_
2A
2 2g
gD
oe
2P0
2 gD (1+b )A(2 -b)
gDe
A
H + Lh
- W[(T w2 - Twl)(L
hZ
H
W1
+ E
7E
L2T
H
+
C
,
-IM
aW( 3-) bbL
33
KW
2A p
____^_
_^1_-
1-]
L +
ha-Za
LE
1
cz2
(3.20)
= 0
where
Lv = LH + LhZ
:
is the height of the test section
including the plena
+ 2 LE + Lc
1
The value of K is obtained by calibrating with equation(3.20)and
the code's calculations for the actual geometry.
The consistent value of
K obtained is
K = 0.216
Hence for the actual SBTF loop, equation(3.18)is modified as:
0.108 C W3
A p2
gD e
aW( 3- b) b c L
WC
(Tw2- Twl)
(2+b) (2- b) 2
A
O
6
e
o Bg
2D e
x (LH + Lhk
LE
2
+ L
ct2
L2 -
e
e
St
LH + 2 Lh
+ 1
= 0
(3.21)
93
It should be noted that equation (3.21)is valid only for the
SBTF loop (Fig. 3.1).
Equation(3.18
however, is general for all uniform
cross sectional vertical rectangular natural convection loops in which
the heated section is between two plena from where heat is rejected.
The only condition that should be met is that the dimensionless number
L
(St. D- ) associated with the plenal heat convection, the thermal
e
capacity of the fluid and the loop geometrical factors be greater than
LE
two (St. E > 2) so that the approximation made to obtain(3.16)can still
e
be valid. This condition is easily met by
3.3.2.
most practical loops.
Loop Flow Oscillation
In this section, an analytical investigation is made of the
loop flow oscillation for the single-phase liquid sodium in a vertical
rectangular loop in which heat is added between an upper and a lower plenum
both of which serve as constant temperature heat sinks.
Creveling et al [10], using horizontal toroidal loop had
obtained single-phase water oscillations with frequencies ranging between
0.007 Hz and 0.018 Hz, and Kaizerman et al
[13], using a vertical toroid,
had obtained frequencies ranging between 0.005 Hz and 0.02 Hz, also for
single-phase water.
The report of the Oak Ridge experiments [6] indicates that the
single-phase frequencies are not available.
The analysis done in this
section has shown that the effective resistance of the loop for the
range of input powers corresponding to single-phase flows, are such that
the initial surges are quickly damped off.
By reducing the loop hydraulic
-
i
ttIIt,
_lltii
resistance,flow oscillation has been numerically obtained for the loop
which has yielded a value of frequency that is in good agreement with the
analytical prediction.
3.3.2.1.
First Order Perturbation Theory Applied to Flow Oscillation
We represent the dynamic state of the loop flow by the steady
state solution plus in general, a space-time dependent perturbation of the
flow properties.
Accordingly, equations (3.7) for the single-phase
liquid becomes:
at
at
+ a (G + G')
as
+
= 0
)
-- (
S(p+p)
a (G+G')2
S
(3.22'a)
as
p
1
')
+
Sq
((h'
+ h)(G + G'))
as
+
(p+p')g e
t
Ctank
h
at (P0
(G+G)
2 Depo
=
(t') dt'
G'
tank
source
A
H
(3.22 b)
qsink PH
A
(3.22 c)
Where in equations (3.22) we have assumed the following:
- The effect of the perturbation in the density is negligible
in the acceleration and the frictional loss terms and in the
95
enthalpy density (p h).
- The steady state correlations for the friction factor f and
the heat transfer coefficient (inherent in (3.22 c ) are
employed even under dynamic conditions.
Welander [14] has
pointed out that this is true whenever the advection time is
large in comparison with the time for momentum or energy to
diffuse across the tube cross-section.
The primes are used to denote the perturbations in the various flow
quantities.
We shall assume that the perturbation in the mass flux G' is
This assumption can only be valid for the uniform
only time dependent.
cross-section loops and in single-phase flows in which the density does
not vary appreciably around the loop.
The term
j
G'an(t')dt'
where Ctank is the gravity-tank capacitance of the expansion tank, and
G'tank(t) is the mass flux into the tank, is a new addition to the
momentum equation.
expansion tank.
could take place.
It represents the pressure drop (or rise) due to the
In transient conditions, net flow into the expansion tank
Under steady-state conditions, there is no net flow
into the tank and hence, the term does not appear in equation(3.7 b).
Its inclusion in transient situation is important because it affects
the frequency of the flow oscillation that may take place. In our model
mm
mouse a
l
umm
i l
ulillYiij,
,kll
,
96
we do not solve for Gtank'
A separate mementum equation must
be written for the tank's fluid which will be coupled to the loop fluid
flow through the pressure at the tank junction.
The modeling of the
expansion tank-loop fluid interaction has not been done in this work.
An infinite mass expansion tank has been assumed.
tank does not alter the junction pressure.
Flow into such a
Thus in our calculations,
the gravity tank capacitance does not come into play.
equation(3.22 b) is just for completeness.
Its inclusion in
The gravity tank capacitance
is given by [23]:
Ctank
Atank
Pog
We re-write equations(3.22)using the temperature T as the dependent
variable with the corresponding perturbation T'.
dependent.
T' is both space and time
The right hand side of equation(3.22)is written as in equation
(3.13).
Steady state heat addition and extraction has been assumed, hence
there is no perturbations in the heat source nor in the wall temperatures
of the heat sinks (plena).
The dynamic state of flow is due to some
perturbation in the mass flux at the initial time.
With the above, and neglecting terms containing second and higher
orders of perturbations, we have the following equations of perturbations
for the momentum and the energy equations obtained by subtracting equations
(3.7)from equations (3.22). The momentum equation is integrated round the
loop prior to this subtraction.
We have:
97
d G'L
dt
-
aub (2b)G(1b)L G'
2De
e*g T'ds
- po
loop
o+b)P
o
Ct
tank
Gtank(t') dt'
4HT'
De
dT
BT'
aT'
PC T + G C T + G'Cp ds
p
Ts
o p 3t
0
(3.23 a)
(upper and lower plenum)
;
otherwise
(3.23b)
With the initial conditions:
G'(0)
T'(O)
=
0
(3.23a)
Equation(3.23 b) must be solved for T' which must then be used
in equation (3.23 a) to evaluate G'.
We let
R'
= _ au b(2-b)G (1-b) L
-2D (+b)
e
o
(3.24)
IC
-~I"L
I1I
98
and
4H
DePoC p
aW
-o
0
where a is as defined in equation(3.14 e).
Next we define Laplace
transforms as:
(z)
, {G'(t)}
T (s,z) .
f{T'(s,t)} =
Then taking the Laplace transforms of equations(3.23)we have:
zG L - G0L
- p0
SR
T ds
(3.25 a)
tank
loop
-
zT +
The
W
dT
G dT
pO ds
I A0WT
0OcJ
; (upper and lower plena)
;
otherwise
(3.25b)
dT
term ds in (3.25b) can be obtained by differentiating the steady-
state temperature profile (Equation (3.15)).
The perturbation in the buoyancy pressure drop ( 6 PB) driving
the flow perturbation (G ) round the loop is given by:
SPB
=-
Po
loop
e. T ds
(3.26)
The close loop integral in equation (3.26) simplifies for the loop in
which the upper and the lower plena are maintaned at constant tempera-
99
tures even during transients.
Then the magnitude of T' (and thus of
T) is small in the return leg compared to the test section.
Thus
the integral in equation (3.26) can be replaced by the integral over the
Hence, equation 3.26 reduces as:
test section only.
PB
-
o
(3.27)
T-s T ds,
where T-s is used to indicate that the integration is performed over
the heated section and the adiabatic hot section only.
integral in Equation (3.27) into two components and
PB = Pog
LH/ 2
-L
H/2
We split the
have:
T1ds +fLH/2+LhZ T2ds}
H/2
(3.28)
We must obtain the temperature distributions in the heated section and the adiabatic hot section, respectively.
We must obtain the temperature distributions T1 and T2, perform the integrations indicated in Equation (3.28) and use the result
obtained in-Equation (3.25a).
This will give the perturbation in the
mass flux as a function of the frequency domain variable, z.
The steady state temperature gradients at the different loop
sections are obtained from Equation
(3.15) thus:
1.
-
llili 1
100
-LH
CpWLH
- a(T
2
< s < LH/ 2
H/2
/22
- TWI + WC
exp(-ca(
LE
LE
<s<
+ s)); -~-9- 2 -
dT
-a(Twl
ds
- TW
2 + (TW-T
-cL
E
+ W
2
LE
)exp(-a(-
p
;
LE
- LE2
2
+ s))
L
s
E
2
otherwise
(3.29)
From Equations (3.29) and (3.25b) we have:
zT
1
+ W
Ap 0
dT
1d
ds
G
p C WLH
dT2
ds -0;
2
zT 3 +
W dT3
AP
o
ds
-
= 0 ;
- L /2 < s < LH/2
H
H
LH /2 < s <
H
(3.30a)
L
(3.30b)
El
Gac (Tw - TW +
) exp(-a LE/2 + s))
2
WCp
E
Po
p
WT 3
Ap
LE
s <
2
LE
2
(3.30c)
zT +
W
Ap
zT
4
0
dT
4
ds
Ga
(TW1 - TW
2 + (Tw2-TW1 + WCp '-)exp(-aLE))
P
0
x exp(-a(LE/ 2 +
LE2
2
aWT 4
po 4
LE2
2
(3.30d)
lmIwb
101
zT A s
L
zT +
L
-3
(3.30e)
s < - LH /2
(3.30f)
<S
0 ; +
0
W dT
zT6 + A o ds 6 - 0
o
Solving Equations (3.30) we have:
=-
W
+ K exp(- ApoZ
1
GQ
oCpWLHz
(3.31a)
APoz
W s)
= K2 exp (-
(3.1lb)
AApo
=K 3 exp(-
(z + p
W
)s)
0
+ po
Gz (T 2 - Tw
w
APo
T (z
4 = K4 exp( -
+
p0
z
(Tw1
T5 = K5 exp(-
SAp
T6 = K6 exp( -
+
)
exp
LE
+
2-1( s))
aW
Apo
2
Tw2 ) exp (-a(LE/ + s))
Ap z
W s)
oZ
W
s)
(3.31c)
(3.31d)
(3.31e)
(3.31f)
. , ..k
. . .. ..
.
. ...
.. ..
I
I
Ii,
102
In obtaining (3.31d), the second term in the square bracket in Equation
(3.30d) has been neglected compared to the first term in that bracket.
The neglected term is small because of the exp(-aLE) multiplying it
which is usually a small number.
Using the boundary conditions of the common temperatures at the
six interfaces between adjacent sections round the loop, we eliminate K
2
through K6 in Equations (3.31) and solve for K1 to obtain:
K
1
Q6G~
Ap z LH
Po PWLH z exp (- W 2H
-G (Tw
cG
+ -P
SApo
T ) exp(-aLE ) exp(
Ao z LE
(T 1 - Tw2 ) exp(-L E) exp(2 )
3aL
Te
--
z LE
W e
pWLZ
exp(-2L
)E
exp (
W
Ao z LE
)
G
.
ApoZ L
G (T - T + - )exp(-2aL E ) exp(- W
2)
p Z w2
w
WC
E
W
2
(3.32)
103
Considering the order of magnitudes, the fourth and the sixth
.
terms are negligible compared to the second and third terms, and the fifth
term is negligible compared to
the first term.
This is satisfactory for
aLE >> 2. This condition can always be satisfied for practical values
of heat transfer coefficient and loop dimensions (for the SBTF loop,
aLE = 8.121).
Then,
QG
exp (- Apoz LH)
pC WL Hz
W
2
K1 K
=
2~G
+
APz LE
(Twl - Tw2) exp(-aL E) sinh(- -)
(3.33)
Now at s = LH/ 2 ; T1 = T2 , hence
K2 exp(-
K2
W
W
+
-Q
22) = PCpWLHz
WoCpLHZ exp (
W
2)
+
K1
1
exp(-W
2
(3.34)
Substituting Equations (3.33) and (3.34) into Equations (3.31a)
and (3.31b) respectively, gives the required temperature distributions,
T and T .
2
Using Equations (3.31a) and (3.31b) in Equation (3.28) and performing the integration we obtain:
-
-- -
In.~.-~. -
I ~lnluuuulv lu
104
QG
PB
Po9
oC Wz
pC
g
SA
+
W
Ao z L
2K1+2K
( W
z sinh
S
w
2H
2
ApoZ
ApoZ LH
W
z K2 exp (- W
2) (exp ( W
Lh )-
1)}
(3.35)
Substituting for K1 and K2 in Equation (3.35) we obtain:
-Q
6PB = Gp0oB
oCpWz
4Wa
+42
+
2Q
2
Ap
Apnh LH
Ap o LH
2)
exp (- W
W
2
(T
Apoz LE
(Twl
Tw2) exp(-aLE) sinh (W
2)
APZ
Apoz
Sxsinh
2
2 (exp (AP C LHz
Lh ) - 1)
ApoZ
2 exp(2
ApO CpLHz
ApoZ
LH) (exp (-
2cW
- 2
2
)
2
ApW
+
LH
(Twl - Tw2 ) exp(-aLE) exp(ApoZ
x sinh
(
W
X
Lh ) - 1
AQoz LE
W
2 ) (exp(
LE
22 ) }
Apo z
Lh)-
(3.36)
105
Equation (3.36) can be written in the compact form:
6PB = GF(z)
(3.37)
Substituting Equation (3.37) into Equation (3.25a) gives:
G{ zL - R' + z
ZCtank
- F(z)} = GoL
or
GH(z) = GoL
(3.38)
We wish to obtain the contributions to the resistance, the inertia
the capacitance terms due to function F(z).
and
To this end, we expand the
exponential and the hyperbolic functions in F(z) (Equation (3.36)), in
Taylor series about z=O (corresponding to the expansion about steady state
in time)
such that the resulting form of the function H(z) in Equa-
tion (3.38) will be a quadratic polynomial in z. This is equivalent to
the truncation of the system to a second order, which is plausible
in view of the low frequencies that we anticipate for the single-phase
flow oscillations.
The Taylor series expansion has been done under
Appendix B. The resulting form of Equation (3.38) becomes:
[L -
QgP02QBgp2
A 2LH
6W3Cp
+QgpALH
2W C
p
-
2
A
A 24g9
-2
24W
T
24W 2
gpoA
2W
2 2
2
2
2
)(LH+LE) aLEexp(-aLE)]z
Twl - Tw2 )LH LEexp(-LE)]z
~^---~--~
'~ 1111~
106
+ gA/ATank +
-_ g (Twl
pWCp
Tw2) aLE exp(-aLE
= zGoL
(3.39)
In Equation (3.39), the coefficients of z 2 are related to the flow inertia where the first term, L is related to the hydraulic inertia of
the fluid inthe entire loop, usually given by:
I = pL/A.
The second and the third terms are related to the inertia
due to heat
addition and heat extraction in the heated section and the plena respectively.
The coefficients of z are related to the resistances, where R'
is related to the hydraulic resistance in the loop while the first and last
terms give some kind of fluid resistance due to heat addition and heat
extraction at the heated section and the plena respectively.
The constant coefficients are related to the inverses of flow
capacitances.
For our uniform cross-section, non-deformable liquid sodium
loop, the'capacitances due to acoustic volume and pipe flexibility do
not exist.
There are two forms of capacitance present.
One is the
gravity tank capacitance of the expansion tank and, which for our
own loop does not contribute to loop pressure drop (or rise) due to
the assumption
of very large tank.
The second type of capacitance
is the liquid-volume capacitance due to the fluid volume changes that
take place in the heated section and in the plena.
These are the two
terms represented by the second and third constants in Equation 3.39.
107
We re-write Equation(3.39)in the compact form:
4 (ez2 + fz + c) = zG'L
(3.40)
0
where
0g 2A2 2
e
=L -
6W33C LH
2
- R' -
2W Cp
24W
(T -T
(Twl-Tw2)(
3 LH+
)Cexp(-aL
LE)~LEexp(-LE
(3.41a)
gp oA
QBgPo ALH
f
0
2
(Twl
- Tw2) LHaLE exp(-aLE)
(3.41b)
c = gA/ATank
gQ -g(Twl
WG"p
- Tw)
w2 aLEexp(-aLE
(3.41c)
In bond graph notation, our second order system is simply given
by [23]:
R:f
-C:c
SF
IG'
I:e
Next we define the transfer function from Equation (3.40) as:
I-*131~
L101116111
,1411,I
IIIIh1,
1,
ii
,IiII161 ii 1
llililll liliNII ,, ,
108
zL
zL
Y(zy =
ez2+fz+c
O
zL/e
(z+f/2e)2+c/e-(f/2e) 2
or
G
zL/e
W22
2
O
(z+r) +W
(3.42a)
Where
a = f/2e
(3.42b)
2 = c/e - W2
(3.42c)
Multiplying Equation (3.42a) through by Go and taking the inverse
Laplace transform (the inverse Laplace transform of equation (3.42a)
can be found in table A-i of reference [26J]we have:
(-IG(z)] = G'(t) = Ae-tsin(wt + )
(3.43)
where
p = tan-1 (-~
A = GL7e(cosec¢)
It should be noted that the mass flowrate G'(t) as given by equation
(3.43) does not reproduce the initial value G. at t=0.This is due to the
109
order truncation of our system to second order in obtaining the Taylor
series expansion in Appendix B.
Equation (3.43) gives the time response of the perturbation in
the mass flux G' due to an initial perturbationin the mass flux Go .
The stability and the frequency are determined by the sign of a and
the value of w respectively.
If we use the initial flowrate (a fictitious steady-state) in
Equation (3.41) with the corresponding value of the resistance R'
obtained using Equation (3.24), we will obtain the initial flow response
which will in general be oscillatory.
Depending on the value of the re-
sistance, the initial surge could be either
slowly damped.
quickly damped or
If R' is low enough, the initial surge may grow and
it may be impossible to attain steady state.
3.3.2.2
Stability Boundary
For stability, the damping factor must be greater than or
equal to zero, marginal stability being provided by the equality.
Hence,
a>0
or
f/2e > 0
110
For marginal stability;
f = 0 (e
0)
(3.44)
Thus, by using the expression for f in Equation (3.40),
we will be able to obtain the marginal stability curves for
the mass flowrate against the input power using ATw (Twl - Tw2)
as a parameter.
Figure 3.7 is an example of such curves.
For the
steady state flows, the inertia term e is negative, hence the
condition for stability will only be satisfied for f less
than zero.
Thus, in Figure 3.7, the region above the marginal
stability curve corresponding to a particular ATw is the region
of unstable operating points while the region below is the region
of stability.
Information obtained from the stability maps such as Figure
3.8 could be useful in the stability control of the natural convection loop flows.
During
a reactor cooldown for instance, the de-
cay heat addition to the core cannot be controlled.
This type of
analysis can then be used to assess the probability of obtaining
self-natural convection cooling during shutdown.
Another set of marginal stability contours can be obtained by
using Equation (3.40) in conjunction with mass flowrate Equation
(3.18).
By using the stability criterion Equation (3.44), and
111
the mass flowrate equation, we shall be able to eliminate the mass
flowrate W between the two equations and evaluate the input power Q
for varying values of the dimensionless number St.LE/D e
curve obtained by plotting Q against St. LE/D
e
.
The
is a marginal stab-
ility curve.
3.3.23
Numerical Experiments
In order to obtain oscillation in the loop and thus be able to
verify the predictions of Equations (3.40) and (3.41), the frictional
resistance in the loop was reduced.
A new friction factor was im-
plemented in THERMIT-4E/L (only for the purpose of obtaining oscillation in the loop).
The new correlation which is obtained by retain-
ing the turbulent part of the correlation in Equation (2.17c) is
given as:
f = 0.1099 Re-0 .25 ;
Re > 1000
(3.45)
The correlation for laminar regime as given in Equations (2.17) was
retained for flow regimes or Re < 1000.
At the initial flow conditions,
the Reynold's number was greater than 1000, hence the wall friction
was indeed evaluated using Equation (3.45).
The correlation of Equation
(3.45) effectively reduces the frictional resistance in the loop by a
factor of about ten.
Flow reversal was obtained for input power of 150 Watts
0111
I_I
112
ATww = Twwll - Tw2
w2
AT =170K
1.4
ATw=100K
ATw= 50K
AT =0
1.2
1.0
ATW,= 170K
0.8
Locus of Operating
Points
0.4
I
150
p
200
Input Power
Figure 3.7
300
250
Q
350
(Watts)
;4arginal Stability Curves Using AT, as a Parameter.
and flow oscillations were obtained for 250, 300 and 350 Watts.respectively, usinr. the unfirom cross-section representation of the SBTF
loop.
An initial mass flowrate of 0.845 g/sec wasused for all calcula-
tions.
and 3.9.
The results of these calculations are shown in Figures 3.8
At an input power of 150 Watts, no oscillation in the sense
of flow amplitude variation in a single direction of motion was obtained.
Rather flow reversal of approximately constant period of 45
seconds predominated the flow in the loop.
Result of calculation to
be shown later in this section indicates that a0O and w2 > 0 for this
case.
For calculations at input powers of 250 Watts, 300 Watts and
350 Watts, the general trend of increasing damping with increasing
At 250 Watts (Fig.
input power can be observed from Figure 3.9.
3.9 (a)), the flow reversal tendency is still obvious with the inFigure 3.9(c) shows
let flow reversing after 65 seconds of flow.
that steady state flow is attained within less than seventy seconds
of flow at the input power of 350 Watts.
Calculations using Equations (3.40) and (3.41) are given below.
The resistance in the loop corresponding to the initial flowrate
is obtained by using the definition of Equation (3.24) in Equation
(3.17) to obtain the form:
-R'W
(2-b)Ap
g
+ W(T
(Tw
- T
w2
(L +
(LH
+
L L1/)
E
Lh
L
(H) +Lh+
C
2
h
1
za
) = 0
(3.46)
~31111C----
-
-
YI
YIIUYIIIYIIIIYII
..
lultin
114
Using the knowivalues in Equation (3.44) we have:
Q = 150 (Watts)
R' = 2.6413 (m/sec)
Q = 250 (Watts)
R' = 0.1004 (m/sec)
Q = 300 (Watts)
R' =.1.1699 (m/sec)
Q = 350 (Watts)
R'
=
-2.44 (m/s'ec)
Then from Equations (3.40) and (3.41) we obtain:
For
Q = 150 (Watts)
Wo = 0.845 x 10-3 (kg/sec)
e = 6.1697 (m)
f = -1.1071 (m/sec)
c = 0.3641 (m/sec2 )
a =-0.0897
(sec'l )
S= 0.2257 (rad/sec)
F = 0.035 Hz
For Q = 250 (Watts)
W = 0.845 x 10-3 (kg/sec)
0
e = 3.41 (m)
f = 2.7534 (m/sec)
ll
115
c = 0.6169 (m/sec2 )
a = 0.'4037 (s-1)
w = 0.134 (rad/sec)
F = 0.022 (Hz)
For Q = 300 (Watts)
Wo = 0.845 x 10-3 (kg/sec)
e = 2.0304 (m)
f = 4.1929 (m/sec)
c = 0.7433 (m/sec2 )
a = 1.032 (sec-)
= -0.700
For Q = 350 (Watts)
W
=
0.845 x 10- 3 (kg/sec)
e = 0.6507 (m)
f = 5.959 (m/sec)
c = 0.86966 (m/sec2 )
a = 4.578 (sec-l)
2 = -20.295
The results of these calculations show that the flow is unstable
at Q = 150 Watts with a = - 0.0897.
At the other (higher) powers,
Equations (3.40), predict that the flow is stable, with damping
factors of 0.4037 sec -I at 250 Watts and 4.578 sec-1 at 350 Watts.
The predictions about damping of Equations (3.40) are in good agreement
with the results of the code's calculations. The general trend of
increasing damping with increasing input power predicted from the above
calculations can be observed from the results of the code's calculations
III~
--------
I.IIYllllillI ,
II
il
niMM m
iiiia,
116
as depicted in figures 3.8 and 3.9. It should be noted that no further
approximations are made in obtaining the resistance term in the Taylor
series expansion of Appendix B.
The frquencies predicted by equations (3.40) show some inconsistencies with the results of the code's calculations. The analytical
predictions about frequencies are 0.035Hz for input power of 150 Watts,
and 0.022Hz for 250 Watts. The code's results for these input powers are
0.022Hz and 0.026Hz respectively. At 300 Watts and 350 Watts,the analytical expressions predict that the flow is damped and non-oscillatory with
w2< 0. The results of the code's calculations (Figures 3.9(b)&(c)) show
still increasing frequencies with increasing input powers of 0.033Hz at
300 Watts and 0.083Hz at 350 Watts.It should be noted that there are some
errors involved in the determinations of the frequencies from the figures.
The approximations made in obtaining the capacitance term from the Taylor
series expansion in Appendix B is probably not valid for the range of
values calculated. In summary,further investigations should be made about
the contributions to the capacitance term in the loop to be able to predict
the frequencies to a desirable accuracy.
sq
0D
U,
0d
C
C
Un
H
(D
(D
LQ
C
o*2
P
-4.0
-3.0
-2.0 -1.0
0
Mass Flowrate
1.0
2.0
(g/sec)
117
3.0
4.0
Ou let
.,
SInlet
(a)
r4
0I,
10
20
30
TIME
Figure 3.9 ().
40
50
60
(SEC)
Flow Oscillation at
Input Power of 250 Watts
S*Outlet
o
(b)
Inl et
M
Ln
10
30
20
TIME
40
50
60
(SEC)
Figure 3.9(b). Flow Oscillation at Input Power of 300 Watts
S
;
I
ZOutlet
(c)
Inlet
Ln
10
20
'30
TIME
40
!150
60
(SEC)
Figure 3.9(c). Flow Oscillation at Input Power of 350 Watts
121
4. IMPLEMENTATION OF ONE-DIMENSIONAL LOOP CAPABILITY IN THERMIT
4.1. Introduction
In this chapter, the steps leading to the implementation of
a one-dimensional whole loop simulation capability in THERMIT are
presented.
The four equation thermal equilibrium version - THERMIT-4E
In this work, applications have been limited to a natural
is used.
convection loop, hence a pump model which will be required for a forced
convection loop simulation has not been provided.
However, with a
minor modification in the solution scheme, the natural convection loop
capable THERMIT-4E/L can easily be adapted to forced convection loops.
A typical natural convection loop consists of the following
basic features for which appropriate models should be provided: Heat
addition at the heated section; heat extraction at the constant
temperature sink(s) (heat exchanger or plena); mass and energy exchanges
and mechanical balance between the loop and the supression tank - this
is particularly important for transients; and the adiabatic hot and
cold legs.
The next section presents these models.
4.2. Loop Component Models
4.2.1.
The Heater
The existing capability in the code for fuel rod (heater rod)
heat transfer to the fluid in a flow channel is used.
The heat conduction
in the heater rod and the heat transfer model between the heater rod and
the moving fluid have been given in references [1] and [ 5].
For the
specific application to the experimental loop being simulated in this
work, the loop has been modeled as a one-dimensional annular channel with
122
a central heater rod running around the loop.
Heat is applied only to
that section of the loop that corresponds to the heated section in the
actual experiment (Fig. 4.1).
4.2.2
Constant Temperature Sinks (The Plena)
In order to keep the temperature of the plenum constant in time
and spatially flat while transferring heat through its wall, it is
required that the thermal inertia be very low to avoid any significant
heat storage during transients and that the conductivity be high.
The
values of pc . 10- 5 and k - 104, in consistent S.I. units have been used
for the fictitious plena material in our simulations.
4.2.2.1.
Numerical Scheme for Plenum Heat Transfer
By analogy with the rod heat transfer capability already
implemented in the code, the structure nodes are counted starting from
the boundary adjacent to the environment.
However, since the perimeter
of the channel in contact with the structure is an input to the code,
the structure is discretized with increasing radius from the channel
center line (Fig. 4.2).
The transient heat conduction equation, with no sources, is
given below:
pc
5tT
1 Tr
r
(rK
-)
;r
= 0
(4.1)
We integrate (4.1)over the control volume bounded by the interfaces
at i + 1/2 and i - 1/2 to obtain:
123
Upper
Plenum
QE
0
-o
Heated Section
Lower
Plenum
Figure 4.1
Heater Rod in Loop Geometry
1___1__~~ __
_
124
r2
C2
i
Ti
at i - 1/2
+
2
r
pc --
i + 1/2
i
T
rK
1/2
-
I - 1/2
Temperatures are evaluated at the main grid points while the
properties pc and
K
are evaluated at half mesh points.
The values of quantities at i-1/2 and i+1/2 are replaced by their
weighted averages over [i-l,i] and [i, i+1] respectively.
A linear implicit difference scheme is used.
thermal inertia pc and the thermal conductivity
K
Thus the
are treated explicitly,
whereas the temperatures are determined implicitly.
We have:
2
1
r2 )(pc)
[(i+1/2
i
n
-
[( r
(Tn+l
[(
i+1/2 i+l
+ (r2
-
r2
i+1/2
Tn+l)
n
-
)i n
T
+1/ 2 )(pc) i _ 1/2
(Kr)
(Tn+l
1/2
n+l - Tn
l
1t
Tn+l
i-i
= 0
(4.2)
125
where
=ri+
(r)i+1/2
(-)
-
(i)
ri
(i-1
(i-I)
At the half-cells adjacent to the external surfaces, we
represent the net heat flux with its convective and conductive
components.
Thus, at i=1 and i=I, equation (4.2) takes on special forms
as follows:
Si=I:
1 [(r 2 - r I_
2
n /2
-1
2I
I-1/22 )(pc) I-1/2
Tn+1 - T n
I
I
q,
At -q 'sfRI
At
(< r
Ar
I-1/2
(Tn+1
I-1
(4.3)
where
RI is the radius of the fluid channel (see Fig.4.2)
*
1
i=1:
22
- [(r
2
3/2
2
n
- r )(pc)
1
3/2 ]
Tn+l -ITn
1
1
At
n
(n+1
n+l1 _ q sIR
(Ar 3/2 (T2 - T1 )
(4.4)
where R1 is the outer plenum radius.
We now relate the convective heat fluxes to the appropriate
temperature differences.
126
RI
1
2
i-i
Figure 4.2
i-1
1
i+1
I
I
R
The Plenum Heat Conduction Discretization Grid
127
q
= hT
hn(Tn+l
q"
e 1
q
- Tn(+I
)
(4.5)
-Te)
(4.6)
e
where Tf and Te refer to the fluid and the enviror ment temperatures,
respectively, and he and hf are the corresponding heat transfer coefficients.
The reader should note that we have used a fully implicit
coupling with the fluid temperature.
This has been done in order to
avoid the numerical instability which could otherwise appear for high
values of heat transfer coefficients and very low thermal inertia.
Note that both extremes are needed to keep the plenum temperature
constant during transients.
The heat transfer coefficients are
determined explicitly because of their generally weak dependence on
temperature.
In our case, these coefficients may be actually given
arbitrarily large, fixed values.
Using (4.5) in (4.3) and (4.6) in (4.4) and with (4.2), we have
the following finite difference equations for the structure heat
transfer:
[2
1
2,t
2 2 (c)n
(r/2- r (pc)3/2
3/2
1
3/2 +
r
n+1
+ h e R] T
- (
Ar
n+1
3/2 T2
2
Tn
hnR T + 1 (r2 - r2)(c)
e 1 e 2 t 3/2
1
3/2 1
1
[226t
2
(rIi+1/22
-
n
2
2
n
2
(r - r./2)(c)i-/2
+
r.)(pc)
1
i+12ic1/2
i-1/2
+
< rn
(ar )i+
i+1/2
.
I
.l
l-
l
128
n
+ ("-)
Ar
i-1/2
1
(r2
< r
n+1
Ar i+1/2 i+l
I
)(c)n
i+1/2
,r 2 - rl_1/2)(pc)I-1/2
2
(P<nr
1
A-
-
n
.n
Tn+l
I
<nrr)
Tn+l
T 1
+
+
hf I
=
R n+l +
= hhnRT
r-1/2 I-
r2
i-1/2
1
+
1
i-
1/
Tn+l
2 i-I
)(pc)n
i-1/2
(( -- ) I-/2
- r
(r
t
f If
r
(<
Ar
Tn
(l<i<I)
n+1
I
)(pc)
I
Tn
I-1/2 I
n
In matrix form, this system of equations can be written as:
(dl + Rlhn)
-c 2
7F
-al
d2
Sn+l
f l Tn+R1 hnTe
fT
2 2
-a2
-aI-
-cI-1
-CI
d +Rh fn
where the entries are defined as:
I-1
f
e
Tn
f Tn+R hnTn+l
I I ff
129
i
i+1/2
Ar
i
r2
+ (r2
n
i+1/2
r2
1 [(r2
i+1/2- i
2A
)(pc)
n
+ a + c.
i-1/2
-1/2
n
C
i
Ar
.I
2At
1
)
i-1/2
2
-
i+1/2
r)(pc)n
+ (r
i
r
i+1/2
The symmetry of the problem provides ci+ 1 = a i
-
r2
)(c)n
1-1/2
1<i<I
1i-1/2
.
Using the Gauss elimination procedure, we perfom the forward
elimination of the above matrix and obtain the form:
1
n+1
x
1
b1
b2
1
x
bl-
f
d +R hn
b
bI
1
where b.'s are the modified values of the righ-hand-side vector components.
From the last step of the forward elimination, i.e. at i=I, we obtain a
relationship between the wall temperature and the fluid temperature,
both at the new time level:
IIIY ll"
--
IIYI r
130
n n+
n+l
n n
,T)
(d+R hf)T 1 = RIhf-f + g(TeT1,T2,...T
(4.7)
(4.7)
The surface heat source (Qws) is a function of the fluid temperature directly and indirectly through the wall temperature. The implicit
treatment here provides additional terms in the derivatives of the heat
flux with respect to the main fluid dynamic variable p and e (p - pressure,
em - mixture energy).
Hitherto, heat flux derivatives had been obtained
for fuel tod conduction and fluid conduction only.
With Aws being the structure (plenum) heat transfer area, we have:
(4.8)
Qws = Awq
Thus,
(-,'ws
=
;e
-ws
f
aQws "Tws
3Tws
;Tf
;T
e
(4.9)
em
and
(6Q\e p
Qws
TTp
where
Tws
= Tn+1
I
from (4.5) and (4.8):
;T ws
Tf /
Tf
e
(4.10)
p
IMMON
3Qws
Tf
r
Qws
ws f
Tws
(4.11)
and from (4.7):
aTws
DTf
= Rhn /(d+Rl
If
hn)
f
(4.12)
It can be shown that in the limit of very large hf and he , the
wall heat source derivatives reduce to zero.
4.2.3.
Treatment of the Body Force
The treatment of the body force in the momentum equations (2.1 c&d)
is generalized by providing the acceleration due to gravity, g, as an
imput array.
In this manner, proper sign and magnitude of g can be
specified for each node depending on the direction of flow relative to
gravity.
g is specified at the faces of each node with its magnitude and
sign determined by the magnitude and the sign of the product g-*e respectively.
e is the unit vector in the direction of flow at the particular
surface in consideration.
Fig. 4.3 illustrates how the g-array can be
set up for a typicalloop.
Note that the interfaces between the boundary
fictitions cells and the first and last real cells are considered in
-)
setting up the g-array.
These boundary cells are a single cell and the
x-x section in Fig. 4.3 is meant to show an imaginary interface.
132
Sloping Sections and Corner-Cells
The SBTF loop as well as many other loops contain slopinq
sections and corner-cells (-the plena in the case of the SBTF) in which
the flow exhibits up to 900 turn.
Dividing such loops into calculational
cells often leads to a situation whereby some interfaces are between
cells with different values of the gravitational force.
the magnitude of g
Specifying
for these interfaces and the corner-cells require
a spatial averaging of the gravitational force over the two adjacent cells.
Generally the distance-averaged value gi of the acceleration
due to gravity at the interface at the point Si should be specified,
The averaging distance is that of the momentum grid in the direction
of flow about the interface at Si(i.e. Si+/2-
g
SASi+ 1/2 ASi
1/2 ASi_
g i-
Si-1/2)
Hence;
1e/ 1-1/2
k + i+1 i-1
i+1 2
i2a
ei+l k
i+1
or
A
g.i ASi = qi-1 ASi-
A
A
ei-1 k + gi+
ASi
A
ei+1 lk
(4,13)
Sj1/2
where
gjej.k =
J
-
j+1/2- Sj-1/2
g(s) e(s) * kds
(4,14)
Sj-1/2
is the magnitude of the body force at the interface
at the point S.
133
e. is the unit vector in the direction of flow
at the interface at point Sj,
and
is the unit vector in the positive z - axis.
For the case where either the cell i-i or i having the common
interface at Si is a corner-cell, the integral in (4.14) vanishes for
half of the cell when flow is horizontal.
Thus for this case, (4.33)
reduces as:
T-ilaS.
e. ok
i-1 1-1 1-1
-
i
i
+ (1/2 g) ASi
i
(4.15)
where cell i is the corner-cell.
Equation (4.15) shows that only one-half of the magnitude of
g for a corner-cell should be specified in setting up the g-array,
For a closed loop:
N+1
oi
i
-i
=
0
(4,16)
i=1
where
N
=
total number of cells
N+1
=
total number of interfaces
Pi
= the density of
node 1.
At rest (no heat input) and other situations when the density
of the fluid in the loop is uniform, (4.16) reduces to:
134
-A1-"--T- -A
Mesh Size Array
AZ (m): 0.5 4(0.8) 0.5 4(0.8) 0.5 4(0.8)
8(0.5)
Gravitational Acceleration Array
2 ): -9,8
GRAV (M/S
1.9325 4(2.5364)
5(9.8) 4(2.5364) 1.9325
8(-9.8)
'b
J
Figure 4.3
Setting Up the Acceleration
Due to Gravity Array for a
Typical Loop.
135
N+1
giS
=
(4.17)
0
Equation(4.17)provides a formal way of checking the proper
setting up of the g-array.
Fig. 5.6 shows the hydrostatic pressure
profile round the SBTF loop obtained from the code's calculation.
4.2.4.
The Expansion Tank
A typical sodium loop (and also an LMFBR primary loop) is provided
with an expansion tank.
by an argon gas cover.
The expansion tank :consists of sodium pressurized
In a natural convection loop, the pressure of the
argon gas sets the pressure level in the loop.
During a loop transient,
sodium is ejected into or withdrawn from the tank.
The volume as well
as the thermodynamic state of the expansion tank fluid, and thus it;
pressure, changes during the loop transients.
A thorough loop model
should therefore incorpora::' the changing thermodynamic state of the
suppression tank leading to the changing system's pressure,
Mass and
energy exchanges that takes place between the loop fluid and the expansio.
tank during transients should also be well modeled.
In this work we have restricted ourselves to a one-dimensional
treatment and steady state calculations.
Hence the loop-tank interaction
does not pose any problem in our calculations.
We have assumed an infinite
mass expansion tank whose thermodynamic state and thus its pressure stay
136
constant.
The loop is "cut" at the expansion tank junction and the
constant pressure of the tank is imposed as a boundary condition.
For
all our single-phase calculations, the boundary updating capability of
the code was turned on .
The boundary updating subroutine in the code
uses a donor-cell logic to reset the boundary conditions depending on
whether flow is into or out of the boundary cell,
Whenever flow is into
a boundary cell, the conditions of that boundary cell is automatically
set to the conditions of the cell from which flow is exiting.
By turning
on this boundary updating we are able to infer how much energy is lost into
the tank from the difference between the inlet and the outlet conditions.
Figure 4.4 illustrates these concepts.
A better way of modeling the loop-expansion tank interaction is
to write a separate momentum equation for the tank fluid.
The details
of this method are given in appendix A.
4.3.
Implementation ir THERMIT-4E/L
The changes made in THERMIT-4E leading to the one-dimensional
loop capable THERMIT-4E/L consist of making modifications in some
subroutines and providing additional array locations, inputs and flags.
The block diagram shown in Fig. 4.5 illustrates the hierarchy of the
affected subroutines.
Three levels of changes are made according to the nature of
the modifications required in the subrouti es.
The first level consists
of those subroutines that require some fundamental changes in them.
The
second level are those prompted by calls made to the modified subroutines
137
Expansion
Tank
Pl enum
Boundary Cells
Figure 4.4
The Expansion Tank and the Boundary Cells
138
in the first level.
The third level consists of the changes required
in the global array locations and the common blocks due to the new variables
added.
The details of the changes made in the subroutines are given below.
First Level
INITSC
- modified to calculate the structural mesh radii increasing from
the face adjacent to the fluid channel.
The perimeter of the
structure (plena) in contact with the fluid is the input for
this calculation.
STEMPF
- modified to be able to provide temporary storage for the rhs
and the upper diaqonal elements after the forward elimination
of the structural heat conduction matrix.
These stored
quantities are to be used later for the final stage of
the structure temperature calculation.
QLOSS
- split into QLOSSO and QLOSS1.
QLOSSO, which calls
STEMPF, performs the forward elemination state of the
structure temperature calculation for all axial levels.
QLOSSI,
which is called in TRANS does the final stage of the temperature calculation after the fluid Jynamics calculation is
completed.
QWK4EQ
- Expanded to include structure wall heat source derivative
calculation.
UPD4EQ
- Expanded to update the structure wall temperature iteration
after each Newton iteration.
MINI
III.
llH
I nll
illnII J,Illh
139
- overridden by providing a flag (IHTS = 3) and providLing
HLOSS
the fluid-structure heat transfer coefficient as an input
through the array HLSS.
- modified to use the input array GRAVN for the gravitational
MOMENT
acceleration in the momentum equations.
2nd Level
TRANS
- calls QLOSSO and QLOSS1 if the structure temperature calculation
is desired.
INPUT
- expanded to read and provide storage locations for the added
arrays in the global array.
JAC4EQ, JACOBG, NEWTON
- the changes in these consist only of expanding the argument
lists in the subroutine, dimension and the relevant call
statements.
The new real variable AREAS is added to the
common LOCAL in JACOBG.
3rd Level
COMMON /POINT/
- the following arrays are added and pointers are provided for
them in the array pointer common block POINT: HOUT, TOUT,
DTSTR, GRAVN, DTWS.
140
COMMON /RC/
NAMELIST /REALIN/
NAMELIST /RSTART/
- the real variables HOUT and TOUT (now being provided as arrays)
are removed from the above common block and namelists.
141
Figure 4.5. Modified Subroutines
142
5. THERMIT SIMULATION OF NATURAL CONVECTION LOOP EXPERIMENTS
5.1.
Introduction
In this chapter, the details of the whole loop simulation of
a series of experiments performed in the sodium boiling test facility
(SBTF) loop at the Oak Ridge National Laboratory (ORNL/TM-7018)
presented.
are
The one-dimensional loop capable version (THERMIT-4E/L) of
the code that is developed in this work is used for the simulations.
Results of the sinqle-phase calculations in the actual loop qeometry
have shown agreement of the mass flowrate to within 10% of the experimental
data.
Calculations performed in the uniform cross-section equivalent loop
have also shown agreement to within less than 1% of the analytical predictions
for the flow rate developed in chapter 3 or this thesis.
The agreement of
the results both with the experimental data and with analytical predictions,
indicate that the numerical models in THERMIT are dependable at least within
the range of conditions covered.
It has been pointed out in chapter 2 that condensation modeling is
It has turned out that this
essential to two-phase boiling loop simulation.
is even more necessary for a sodium loop in which boiling takes place explosively
due to the very large density ratio between the liquid and the vapor phases.
This often leads to flow reversal in some cells.
Whenever flow reverses in
a cell that is adjacent to a subcooled liquid sodium cell (such as the
plena), subcooled liquid flows into the boiling cell and causes rapid
condensation in the case of the upper plenum.
At the lower plenum, the
two-phase sodium from the boiling adjacent cell flows into the plenum
(inlet module) and condenses rapidly.
In this work the problem
of condensation modeling as pointed out in chapter 2 has not been solved.
143
It has therefore not been possible to perform steady state two-phase
calculations in the loop.
At some initial flow situations, however, the
quality is low and calculations have been obtained. The results of the
various attempts and the modes of failure of some of our two-phase calculations are also presented in this chapter.
5.2.1.
Geometry Transformation
The geometry in the code is a rod-bundle geometry which is
different from the simple tube of the experiment.
It is therefore necessary
to transform the simple tube geometry to a rod-bundle geometry.
The constitutive relations for the rod-bundle geometry are used
directly for the circular tube.
The justifications for this in the case of the
wall friction correlation has been discussed in section 2.4.
For the heat
transfer correlation, the range of the Rayleigh number involved makes
the code's correlation applicable.
It should be noted that the heat
transfer coefficient is prescribed at hiqh fixed values for the plenum heat
transfer calculations.
The tube is taken as a single fuel rod flow channel (Figure 5.1(c)).
The dashed lines in the figure is used to show the boundary of the flow
channel associated with a fuel rod.
In the rod-bundle geometry, a pitch-to-diameter ratio and a wirewrap-to-diameter ratio are necessary to calculate the wall heat transfer and
wall friction.
Since there is no wire wrap in the tube geometry, a very
large value of the wire wrap-to-diameter ratio is prescribed as an inout.
An equivalent pitch-to-diameter ratio for the circular tube geometry is
calculated using the invariants of the hydraulic channel transformation.
144
(a)
SS S
S
S S
I
I
II
i
I
i
4
Figure 5.1.
Equivalent
Equivalent Single
Rodded Channel
(b)
(c)
Equivalent Channel
In a Fuel Assembly
D --
Simple Circular Tube
Schematic Diagram for Geometry Transformation
h 1I
l
Idl
WIII
145
These in variants are:
(i)
the flow area, and
(ii) the equivalent hydraulic diameter.
The preservation of flow area provides:
P2
2
lD
2
2D
4
4
e
(5.1)
The preservation of the equivalent hydraulic diameter provides:
2
T)D
2-
)
4(P
D
where
=
De
(5.?)
e
P is the pitch-to-diameter ratio
D is the fuel rod diameter
De is the circular tube diameter
Eliminating De between equations(5.1)and(5.2)we obtain:
P/D
5.2.2
=
Vr/2
= 1.2533
Non-Uniform Flow Cross-Sectional Areas
The SBTF loop flow cross-section is section-wise uniform.
Consequently, the staggered mesh arrangements at some points in the loop
are such that the two halves of the scalar grid (control volume) belong
to two sections of differing flow cross-sectional area (Figure 5.2).
146
Khalil and Schor [18] have implemented a non-uniform flow area correction
in THERMIT-4E.
The correction procedure is based on the volume-averaged
areas given in reference [181.
The volume averaged areas are calculated as follows:
volume of cell i
The flow area at the face i; A.
1
mesh size of cell i
A1
(5.3)
AS
The flow area at the face i
.+
1/2;
Ai+1/2
_ (volume of cell i + volume of cell i+1)
(mesh size of cell i + mesh size of cell i+1)
V.
1
+
V
i+1
AS.1 + ASi+l
(5.4)
The flow area at the face i + 1:
Ai+
volume of cell i+1
1
mesh size of cell i+1
Vi+ 1
ASi+1
(5.5)
Equations(5.3), (5.4)and (5.5)are used in preparing the inputs
for the simulation of the loop experiments.
---
-
h
.--- Y
147
i+1/2
Fiqure 5.2.
i+1
Staggered Mesh Arrangement with Sudden Flow Area Change
148
5.2.3. .Single-Phase Calculations
5.2.3.1.
Simulation of the Single-Phase Test: ORNL/TM-7018; 107R2
The SBTF loop is divided by a mesh into calculational cells as
shown in Figure 5.3.
The heated section and the adiabatic simulated
fission gas region downstream of the heated section (adiabatic hot leg)
are each divided into five uniform meshes.
Thus a mesh in the heated
section is 0.194 m long and a mesh in the adiabatic hot leg is 0.3m long.
The mesh sizes in the heated section are smaller than in the other sections
of the loop because of the strong spatial variation of the flow properties
in this section.
The upper and the lower plena are each taken as a single
cell with an effective length of 0.6175m.
The heated section inlet module
is taken as a single cell of length 0.6175m.
The vertical portion of the
return leg is divided into six uniform meshes of length 0.6175m each.
Boundary Conditions
Due to the purely one-dimensional treatment of the loop flow, it
is necessary to specify the inlet, and the outlet boundary condition at the
cut point.
For all our calculations, the constant pressure of the infinite
mass expansion tank that we assume was used as the inlet and the outlet
boundary conditions.
condition.
This corresponds to a pressure-pressure boundary
A pressure of 1 atmosphere has been specified for the boundaries.
The other flow property required at the boundary is the mixture
internal energy (or the temperature for this case of single-phase).
In this
calculation and all other single-phase calculations, the boundary condition
updating discussed in section 4.2.4 has been applied.
In single phase,
local flow reversals do not take place, hence the inlet temperature is
maintained at the constant temperature of the expansion tank while the
149
QE
Upper Plenum
'cut-point'
Heated Section
Lower Plenum
Fiqure 5.3.
The Actual Loop Calculation Cells
150
ORNL-DWG 79-6162 ETD
0.7
I1 I
I
I
I
0.6 t
0.5-
0.4-
0.3
0,~
0.2
)
I
1
0.4
0.6
j
10
0.1
0
0 0
0.2
0.8
1.0
1.2
1.4
4.6
4.8
QTS (kW)
Relative error in test section power determination as
Figure 5.4.
function of test section power. (from Reference 6)
2.0
151
outlet temperature varies as the flow develops.
Input Power
The input power to the test section was not measured directly
but was calculated based on the heat balance over the coolant, the furnace,
the clamshell heater of the simulated adiabatic fission gas region and the
ambient air [6].
The uncertainties involved in the measurements of these
quantities of heat plus the error involved due to the shifting of the center
of focus of the radiant furnace due to the slight bending that occurred
during heating made it impossible to know precisely the actual input power
during the test.
Figure 5.4 gives the relative error in the input power as
a function of the reported test section power.
The test section power of 300 watts (Figure 5.4) involves an
over estimation of about 38%.
However, at the low test section powers, the
relative errors have been over estimated [25] and an input power of 270W
has been deemed appropriate for the 107R2 test.
Results
The result
of this calculation shows that the inlet volumetric
flow rate is 0.8 ml/sec (data. 0.7 ml/sec).
The discrepancy lay be
attributed to the uncertainty in the input power.
Figure 5.5 illustrates the temperature profile round the loop for
this test.
A maximum temperature rise of 341Ko was obtained with the
maximum temperature of 1034K occurring at the simulated adiabatic fission gas
section.
Figure 5.6 gives the pressure profiles for the rest condition
(hydrostatic head) and for the steady state flow at the power of 270 watts.
It can be observed from this figure that the pressure drop across the large
152
1200
1034.34K
1000
863.15K
800
Upper Plenu
693.15K.
v
Lower Plenum
600
600
L-
-
400
200
Heated Length
,
2
4
6
8
10
DISTANCE ROUND THE LOOP (m)
Figure 5.5.
Loop Temperature Profile at Input Power of 270 watts.
-
~--
- -Ylu~--i-r-
_..
_ ~_n
:n-------
n~--- ,~_~___~_~
153
.3 1.4
Figure 5.6 Pressure Profile Around the Loop at Input Powers of
Zero and 270W.
_~__
154
cross-sectional area return leg is due mainly to the hydrostatic head
drop even for the steady state flow at 270 watts.
5.2.3.2.
Uniform Cross-Sectional Loop Calculations
A series of calculations with input power ranging from 150 watts
to 370 watts were calculated for the uniform flow cross section loop.
The
results of these calculations are compared with the predictions of the analytical
expression (equation 3.18) derived in chapter 3 (Table 3.1).
Figure 5.7
illustrates the functional dependence of mass flow rate on the input power
both for the code's calculations and the analytical predictions.
5.2.4.
Two-Phase Calculations
Initial attempts were made to obtain boiling in the actual loop
geometry and letting condensation to occur in the upper plenum.
The input
power was gradually increased from an initial single-phase flow value.
The 'cut-point'
for this calculation was at the upper plenum corresponding
to the position of the pressurizer junction in the actual experiment.
With
this set-up, the boiling fluid exiting the last cell of the adiabatic hot
leg will condense in the first half of the plenum.
Successful steady boil-
ing in this set-up was ruined by the flow reversal that occurred locally at
the cell adjacent to the upper plenum leading to rapid condensation in that
cell.
The long time delay in the large cross-sectional area return leg
coupled with the short time steps for the two-phase calculations make the
actual geometry two-phase calculations very time consuming and expensive.
Hence, the uniform cross-sectional area geometry equivalent loop has been
1111
155
0.6
0.5
W
0.4
(g/sec)
0.3
A Analytical
0.2
0
Numerical
U-
0.1
200
250
300
350
INPUT POWER (watts)
Figure 5.7
Functional Dependence of the Mass Flow Rate on
Input Power
used for all the two-phase calculations.
Since the 'cut-point' is provided
just below the upper plenum in this geometry, condensation will occur in
the expansion tank outside the calculation domain.
Condensing in the
expansion tank of a fixed pressure is feasible in the light of the infinite
mass expansion tank that we assumed.
Condensation Near the Upper Plenum
Even with the 'cut-point' provided at the inlet to the upper plenum
as described above, flow reversal in the last boiling cell, at the end of
the adiabatic hot leg had caused sub-cooled liquid from the expansion tank
to flow into the cell ard cause rapid condensation there.
The distribution
of void and pressure in the loop is shown in Figure 5.8.
In order to avoid condensation taking place in the last boiling
cell as described above, the boundary update algorithm was turned on again.
Thus, even when flow reversal takes place, the conditions of the fluid in
the cell does not change.
Condensation in the last cell was successfully
avoided by this way.
Condensation Near the Lower Plenum
By turning on the boundary update algorithm as described above,
we were able to obtain boiling up through the simulated adiabatic fission
gas section with condensation taking place in the expansion tank out of the
domain of the problem.
section
As voiding developed in the rest of the heated
however, local flow reversal that has occurred in the first heated
cell had caused two-phase sodium to flow into the inlet module cell and
cause rapid condensation there
and thus terminate the calculation.
The
void distribution in the loop with the accompanying pressure profile is
illustrated in Figure 5.9.
ilI
C
I
157
1.6
I'
ressure
1.4
1.2
0.8
s-
0.6
-o
0.4
0.2
1.0
0.0
0.8
0.6
Void
0.4
Upper 0.2
Plenum
2.0
4.0
6.0
8.0
10.0
DISTANCE ROUND THE LOOP (m)
Figure 5.8
Condensation Near the Upper Plenum at 640 Watts; Pressure
and Void Profiles
°
158
1.4
1.2
1.0
0.8
0.6
0.4
0.2
1.0
0.0
0.8
0.6
0.4
0.2
2.0
Figure 5.9
4.0
6.0
8.0
10.0
DISTANCE ROUND THE LOOP (m)
Condensation Near the Lower Plenum at 910 Watts; Pressure
and Void Profiles
o
mn111IuIYImmrnYY
iiii IIll,I -
1ill
H11i
l iI
'llU
I."
159
1200
sat
sat
.-
.
A
1000
863..15K
Lr-
Upper
P1
Plenum
. 800
Lower
P1 enum
"'
600
Void
Heated Length
2.0
4.0
6.0
8.0
10.0
DISTANCE ROUND THE LOOP (m)
Figure 5.10.
Low Quality Boiling at Input power of 450 watts; Temperature
and Void Distributions
.2
160
Low Quality Non-Steady State Boiling
Figure 5.10 illustrates the void distribution and the temperature
profile in the loop for some initial low quality boiling in the loop at an
input power of 450 watts.
Steady state two-phase was not reached at this
power level because as the flow developed, the voids collapsed and singlephase was the final state of the loop flow.
I
161
6. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK
6.1.
Summary and Conclusions
One-dimensional loop capability has been implemented in the
four equation, thermal equilibrium model of the sodium version of THERMIT.
The resulting loop code has given good results for the single-phase loop
simulations.
Good agreement has been obtained among the code's results,
the experimental data and the analytical predictions.
The implementation
consists of providing models for certain loop components.
These components
include: constant temperature heat extraction inthe plena during transients;
heat transfer to the fluid from the heater rod; the body force, and the expansion tank.
The existing models in the code for the exchange of mass,
momentum and energy between the wall and the fluid as well as between the
phases in the cases of two-phase flows are used directly.
Since our applica-
tions have been limited to the simulation of a natural convection loop
experiments in this work, we have not implemented a model for the pump
which is an essential component of a forced convection loop.
With some
minor modifications in the solution scheme however, a pump model can easily
be included in this loop capable version (THERMIT-4E/L)of the code.
In the Oak Ridge experiments (ORNL/TM-1078) that are simulated
in this work, the plena are kept at constant temperatures during the tests.
This is along the line of the practice in the operation of Fast Test
Reactors (FTR) for which these experiments are designed to simulate.
Constant temperature plenum heat conduction has been achieved numerically
by treating the coupling between the fluid and the plenum wall temperatures
fully implicitly in the numerical scheme for the plenum heat conduction.
162
This way, the numerical scheme has remained stable despite the use of high
heat transfer coefficient and low thermal inertia for the fictitious plenum
material.
It should be noted that these extremes of values are required
to keep the plenum temperature constant during transients.
A spatial averaging procedure is used to provide the body force
(gravity) at the various sections (sloping and vertical) of the loop.
The body force is set up to be provided as an input array in the code.
This way, it has been possible to simulate the experiments while preserving
the geometry and other conditions.
The model for the expansion tank - loop fluid interaction requires
a two-dimensional treatment, at least locally at the junction of the tank.
We have restricted ourselves to a pure one-dimensional treatment and have
treated the expansion tank as of infinite mass.
The mass and energy exchanges
that may take place between the loop and the tank during transient will
therefore lead to no changes in the tank's junction pressure.
The loop has
been cut at the expansion tank's junction and treated as a straight channel
with the equal and constant pressure of the junction used as the inlet and
the outlet boundary conditions.
The model for the heating of the sodium coolant by the fuel rod that
already exists in the code is used directly.
Two modes of heating are avail-
able; one is the steady heat ejection to the fluid by the heater rod.
This
mode of heating does not recognize the feedback effect of the fluid temperature.
The second mode of heating goes through the transient heat conduction
in the heater rod and accounts for the thermal intertia in the rod as well
as the feedback from the fluid temperature.
The numerical scheme of THERMIT-4E, have been found to work well
163
* for the loop simulation of two-phase boiling under the condition of
reduced density ratio of the liquid to the vapor phases [4].
For the
naturally high density ratio of sodium at atmospheric pressure, the code
fails to converge for the condition of condensation.
The problem of rapid
condensation has been encountered in our two-phase calculations.
At the
inception of boiling, the local flow reversals that have taken place in the
cells have led to rapid condensation at the inlet
first and the last boiling
module to the heated section and at the last boiling cell.
Condensation
have not been observed to occur in the plena in any of our attempts.
The single-phase thermal-hydraulic loop analysis that is done
in this work has yielded an expression for the functional dependence of
the mass flow rate on the input power for the loop.
The dimensionless
form of the equation can be applied to other loops of similar geometry
(that is vertical loop with heated section between constant temperature
plena).
The predictions of the equation have shown agreement to within
1% of the code's results.
The single-phase loop flow oscillations that
has also been investigated have given the result for the damping of
oscillations that are in good agreement with the result of the one-dimensional
loop code - THERMIT-4E/L.
Flow oscillations in the loop have been obtained
numerically by reducing the frictional resistance through the wall friction
correlation in the code.
The criterion for stability of flow due to an
initial surge from the prescribed initial flow condition has been found
to depend on the steady power input, the difference between the upper
and the lower plenum temperature, and the modified Stanton number
- St LE/De .
This dimensionless group arises from the heat convection at the walls of
164
the plena.
Experimentally, the initial surge will arise from the heat
addition at the initial time and will either grow or decay depending on
the effective resistance in the loop.
In view of its high level of flexibility, THERMIT-4E/L can easily
accept developments that will be made in the future.
With more studies done
on the numerical models - especially towards obtaining a faster calculational
tool, the physical and the loop component models, the code might be improved
to the state that it can handle a wide range of real LMFBR accident
situations.
6.2.
Recommendations for Future Work
The following items are recommended for future work:
1. Condensation Modeling
The problem of rapid condensation failure mode of the code should
be solved.
The Nigmatulin model that has been incorporated into the six-
equation version of the THERMIT [2] may be further developed to effectively
solve the problem of rapid condensation.
However, the ultimate model
should incorporate the transient phenomena taking place at the interface
during condensation.
2. The Expansion Tank
The better treatment of the loop-expansion tank interaction
should be provided.
In particular the energy exchange taking place between
the loop fluid and the tank fluid should be accounted for.
process can be developed for this.
A proper mixing
The changing thermodynamic state of
the tank content leading to the changing pressure of the argon gas cover,
165
and thus the changing pressure at the loop-tank junction should be well
modeled.
-
A separate momentum equation can be written for the tank fluid.
The common pressure at the tank junction provides the coupling between
the loop momentum equation and that of the tank.
This treatment requires
some extra storage locations in the pressure field solution matrix and more
details on this can be found under Appendix A.
3. Two-Phase Oscillation
Mathematical model for the two-phase oscillations in the loop should
be provided.
In general a model for flow inertia, resistance and capacitance
can be developed.
Insight regarding the contributions to these quantities
due to heat addition at the heated section and heat extraction in the plena
can be obtained from the expressions obtained for them in the case of the
single-phase analysis of chapter 3. THERMIT-4E/L can then be used directly
to obtain information on oscillation with this lumped parameter treatment.
4. More Loop Components
Other components that typify the primary loop of the LMFBR should
be modeled.
These components include:
a. Primary coolant pump
b. Bypass channels
c. Heat exchanger
d. Valves
With the accomplishment of the models for this components, a wide range of
transients under LOA-2 can be simulated.
166
5. Improvement of the Overall Numerical Scheme
Efforts should be directed towards implementing a fully implicit
scheme in THERMIT.
The present level of implicatness in the numerical
scheme of the code leads to serious limitations on time-steps giving rise
to very short time steps under certain conditions and consequently, long
and expensive computational time.
The situation with loop simulation
is even worsened by the time delay caused by the adiabatic return leg.
167
APPENDIX A. Treatment of the Expansion Tank and the Loop-Fluid Interaction
A better way of modeling the loop-expansion tank interaction is
to write a separate momentum equation for the tank fluid.
The pressure at
the loop-tank junction provides a coupling between the loop fluid and the
expansion tank fluid flows.
pressure-field solution.
This junction pressure becomes part of the
Once the pressure is obtained, the flow into
or from the tank can be used to determine the energy exchange between the
loop and the tank.
This method requires two dimensional treatment at
least at the junction. Also extra storage is required for the additional
entries in the pressure matrix.
The example below illustrates how
the pressure problem can be set up and shows the additional entries in the
forward eliminated matrix.
expansion tank
Figure Al. Loop/Tank Interaction Modeling Geometry
168
The pressure matrix for the loop and the expansion tank in figure Al
can be set up thus:
x
x
0
0
0
x
x
x
0
X
X
X
X
X
X
X
x
After performing the forward Gauss elimination of the matrix we obtain
the form:
0
0
00
0
0
1
x
0
0
1
0
0
0
1
x
0
0
0
1
P1
0
The entries encircled are additional entries that come into the pressure
matrix due to the expansion tank junction.
-
--
IIYIYU
169
APPENDIX B. Obtaining the Inertia, Resistance and Capacitance
Components from SPD
The expression for the Laplace transform of the bouyancy pressure
drop in the loop is obtained as:
O3gG
+
oCp Wz
F~~
2 L
APo2CpLHZ
exp (-
ApoZ
LH)sinh
W T
sinh
LE
APo0 z
+
(exp
CpLH z
A opH
2
-
(
ApWZLh
exp.
(Twl - Tw2 ) exp (-aLE)
APo2Z
x exp
(- AZW
L
LH
W
z
LH
sinh (AP
inh
---- T)
)-1)
LH exp
H
Ap0 CpLHz
2aW
oW
(Twl - Tw2 ) exp (-aLE) sinh
+
Po
-1
A
Lhi)
exp A z LH
exp W
2)
sinhAp z LE
We expand the exponential and the hyperbolic components of 6PB
in Taylor series about z = 0 (corresponding to expansion about steady state
in time).
Term by term, we obtain:
170
Ap z L2
2QG
exp
Aoo2 CpLH z
'-
QG AL
=-
2
QGA L
H
oWCpZ
W (Twl
+
@(z2)
6W3C
2 2 (Twl - Tw2 ) exp (-aLE)
Apo z
=G (T
2
H z
2W2C
Aoz L
sinh
sinh
A
- Tw2) LHa LE exp (-caLE)
2
) s i nh
z
L
+ @ (z2)
Aq2C0 L
02
(exp (- Ap
pH
APO CPLH z
- LhZ
-
Lh)
+
WC LHz
-1
QG A2(L+ 3
Q A(Lhk2)
6W3C LH
p H
2W2C pLH
p(
QG
p~2T.,exp
Ac z L
(z 2
exp
A WZ L
z +
(
-
Ao CpLHz
QG(Lh-LH)
-WCL
h
H
0 p LH
GA
2
Q A Lh 2
2W2C LH
PH
+
QG A2
o
3C L
6W
L 3 z
h
+
+
(z2
Z2)
-IY
S- --- IIYII
lilYlli1mm
lill1ii.
II1HIIIkl,1,
171
where LH2 and LH3 have been neglected- compared to the Lht2
and LhZ 3 respectively.
2WaG
(Twl - Tw2) exp (-aLE) exp (-
2
-A
0
- Ap
exp
H)
Lh)
- 1
z
sinh(Apoz LE
(
-2WG
W\72
SAPOZ
(Twl - Tw2) exp (-aLE) exp
Apo2 z
- Tw2)
w
0OS(Twl1
A2
+
LE exp (-a
W
H+LE)
=
o
Apz ()LE)
- Tw2) LHaLE exp(-aLE)
(Twl
W
2
(Twl
- Tw2 ) (3LH 2 + LE2 )aLE exp(-aLE) z +
Summing terms we have finally:
OPB
- exp-
o
24W
g
g
2
(Twl -Tw2)
[
oWC
O
aLE exp (-aLE
Z
(z 2 )
172
+
+ -W (Twl - Tw2) LHcLE exp (-aLE)]
2W2
P
22
+
oALH
3
cp
2
o+ (T_
2
w1
- Tw2 )(3L 2 + L 2 )aL
w2
H
E
E
exp(-aL
E
173
APPENDIX C. Typical Computer Inputs and Outputs
Some of the important computer inputs and outputs are published
here.
The inputs are those for the actual loop geometry and for the uniform
cross-section loop calculations.
The outputs for the simulation of the
test ORNL/TM-7018; 107R2 at 270 Watts in the actual loop geometry and the
single-phase calculations at the input powers of 150 Watts, 250 Watts and
370 Watts in the uniform cross-section representation are included.
The
results for the low quality two-phase boiling, and the two-phase calculations
at the input powers of 640 Watts and 910 Watts that failed due to rapid
condensation at the upper and lower plena are also reported,
Typical
results for the plenum heat conduction and hydrostatic pressure profile
calculations for the loop are included.
.
INPUT FOR ACTUAL LOOP GEOMETRY CALCULATIONS
2
SINGLE PHASE MEASUREMENT FOR SODIUM NATURAL CONVECTION IN A
VERTICAL CHANNEL:ORNL/ M-70 1
SINTGIN NC-1.NZ*30.NR-l.NARF.I.NXI,.NRZS-t.IHTF-1,
IHTS3,ISSI.IXFL*O.IDUMP*I.1B*-O.
ISIRPR-I.ISIPR-tIO1II.NITMAX-2.IPFSOL-34.
NEO-4.NUMOER-O.1HIRPR-I
$
SREALIN
D-3.25E-3.25E-3POR-.2533.HDRI.OE0.OELPR-O.5.
RNUSS-7.O.RAOF*l.625E-4.WINLET*S.50E-4.GRAV*O.ODO
s
SROOINP 00-210.0
S
ISPJCR
OSINDENT
ISIFCAR
ISHRZF
l ?nIMAF
3$MNRZF
7$MNRZS
4$NRMZS
4.07327E-3
SOX
4.01327E-3
SOY
I.OE-6 0.604 0.02 4(0.4764331) 0.02 5(0.604) 0.02 4(0.4164331)
0.02 0.604 0.65 510.194) 5(0.3) I.OE-6
s$0
SARX
30(0.0)
30(0.0)
SARY
8.29577E-6 18(2.04262E-3) 12(8.29577E-6)
SARZ
11.1046E-3 40.8564E-6 4(973.267E-6) 40.8564E-6 5(I.23386E-3)
40.8564E-6 4(973.267E-6) 40.8564E-6 11.1048E-3 5.39225E-6
5(l.60938E-6) 5(2.4885E-6)
$VOL
3.25E-3
SHEOZ
3.25E-3
SWEOZ
SP
32(i.01325E+05)
3210.0)
$ALP
32(693.15)
STEMP
31(12.OE-2)1
VEL
-9.8 0.0 3.22756 3(3.3630454) 3.22756 0.0 4(9.8) 0.0
3.22756 343.3630454) 3.22756 0.0 12(-9.0)
SGRAV
30(693.5)
STWF
20(0.0) 5(1.0) 5(0.0)
$OZ
1.0
SOT
1.0
SOR
1.0
IRN
1.625E-3
SORZF
I.62929E-2
$PCX
10RZ5
2.03E-2
I.OE*6 1710.0) 2(1.OE6) 10(0.0)
SHOUT
863.15 171500.0) 2(693.15) 10(500.0)
STOUT
30(693.15)
STWS
30(2.5E406)
SHLSS
SIMO l IE1*I)-IOO.O.DIMIN-I.OE-6.DTMOE-6.
O.UTSP20.O.DTLP20.0.IREDMX20
STIMDAI IENO*O.0
s
0
SINGLE-PHASE TEST ORNL/TM-7018; 107R2 SIMULATION IN ACTUAL GEOMETRY AT POWER OF 270W.
TIME
StEP
O80* 232680
laImTlER Of NFION ITERATIONS
NUM.BER OF INNER IIERATIONS
*
2
I
O
10O1
REACIOR POWER
TOTAL IIEAl IRANSfER
FLOW ENIlIALPY RISE
ILOW ENERGY RISE *
0.270
0.270
0. 137
0. 137
Kw
KW
KW
KW
w
v
REAL
TIME
lIME
"*********** SEC
SIEP SIZE
I
I
I
I
I
I
ZIMMI
PIBARI
S
0 0
2 302 0
3 614 0
4 862 2
5 1338 6
6 IsIS I
7 2291.5
8 2539 7
9 2851 7
to 3455 7
II 4059 7
I2 4663.7
13 5267.7
14 5579 7
IS 5827.9
I6 6304 4
17 6700 6
18 7257.2
19 1505 5
20 7817 5
21 6444.5
22 6866 5
23 9060 5
24 9254 5
25 9448 5
26 9642 5
27 9689.5
28 1019 5
2910469 5
3010789.5
3111089 5
3211239 5
1 01325
0.99901
0.98901
0.99551
I 00950
I 02149
1.034A46
1.04091
1.04097
1.086997
1.13697
1.18499
IZ
1.23299
1.23299
1.23949
1.25250
1.26551
1.21853
I 28505
1.28505
1.23217
1. 19679
1.18091
1.16534
I.I5007
1.13511
1. 11626
1.09337
I 07048
1.04159
1.02470
1.01325
VOID
O.
OO0000*0
0
0
INLET FLOW RAlE
OUTLET FLOW RATE
1OTAL SYSTEM MASS
GLOBAL MASS ERROR
0.626
0.625
29923.699
-0.2190-09
MAXIMUM
IN
IN
IN
MAXIMUM RELATIVE CHANGES OVER Il1E TIME StEP
IN PRESSURE:
0.2390-07
IN MIXTURE DENSITY: 0.8560-06
IN MIXTURE ENERGY: 0.3370-05
IC
*
QUAL(%)
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0 0000 0.000
0.0000 0 000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000oo
0.0000 0.000
0.0000 0.000
0.000
O.wOO
0.0000 0.000
0.0000 O
0.000
ooo
0.0000 0.000
0.00O00 0.000
0.0000 0.000
0.0000 0 000
00000 0.000
0.0000 0.000
0.0000 0.000
O 0000 0.000
0.0000 0 000
0.0000 0.000
0 0000 0.000
0.0000 0 000
0.0000 0.000
0.0000 0.000
0.0000
EM
ROM
1127823.
1128251.
1128251.
1121251.
610.94
010.63
110.113
1128251.
112251.
1128251.
1126211.
1128248.
1128226.
1128134
1127840.
112074.
1127040.
1125924.
1124000.
1120909.
1116237.
111039.
913565.
913565
999519.
1085473.
1171421.
1257360.
1343333.
1343331.
1343329.
1343327.
1343325.
1343323.
1343323.
810.85
StO 86
810 88
310.88
910.94
611 00
811. 10
611.29
1i1.30
811.88
812 47
613.35
13.39
850.4 1
850.36
034.75
81.94
802.96
76 91
770.79
770.70
770.75
170.73
170.71
170.69
770.67
I
VAP
662.61
863. 15
663. 15
063. 15
863.15
063. 15
663.15
03 IS
863 IS
863. 13
863.06
862.62
862.21
062.19
861.30
859.76
857.30
853.58
853.42
693.16
693.16
760.89
829. 10
897.5S
966.01
1034.24
1034.24
1034.24
1034.24
1034.24
1034.24
1014.24
10
962..01I
063. IS
863. 15
863. 15
863.15
063. 15
663.15
663. IS
863.15
863. 13
663 06
062.02
802.21
662.19
661.30
659.76
857.30
653.58
653.42
693.16
693.16
760.89
829.10
897.55
966.01
1034.24
1034.24
I034.24
1034.24
1034.24
I034.24
1034.24
1
914.25 SEC
CPU TIME *
SEC
O
TIME STEP REDUCTIONS DUE 10 ERROR
LAST PRINT
REDUCED
1IME
STEPS
SINCE
0
MAXIMUM TEMPERATURES
ROD*
1035.33 AT
1034 45 AT
WALL:
1034.24 At
LIOUID
G/S
0/S
0
0
IC
I
1
I
IZ
25
25
25
RELATIIVE LINEARIZATION ERRORS
PRESSURE:
0 3010-01
0.1630-12
MASS/VOLUME:
0.170t-17
ENERGY/VOLUME:
SAT
1158.70
1156. 10
1156. 10
1156.82
11560.26
I159 68
1161.09
161.79
I161. 79
1166.095
171.74
176.46
111.04
181 .04
11 1.65
1182.86
1184.06
1165.25
1185.04
1185.84
1180.96
I177 60
1176.07
1174.5s
1173.04
T171.55
1169.65
1 67. 3 1
1164.92
1162 50
1160.03
1158.17
VVZ
VLZ
0.093
0.000
0.000
0.000
0.000
O 000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0000
0.000
0.000
0.000
0.000
0.089
0.009
0.091
0.092
0.094
0.096
0.098
0.098
0.098
0 090
0.098
0.098
0.093
0.000
0.000
0.000
0.000
0000
0 000
O 000
0.000
0.000
0.000
0 006
0.000
0.000
0 000
0.000
0.000
0 000
0 000
0.089
0.089
0.09 1
0 092
0.094
0.096
0.098
0.090
0.098
0.090
0 090
ROV
O 2173
0 267
0 2678
0.2694
0 2727
0 2759
0.2792
0 2808
0.2808
0 29217
0.3046
0.3165
0.3283
0 3283
0.3299
0.3331
0 3363
0.3395
0 3411
0.3411
0 3781
0.3194
0 3155
0 3117
0 3019
0.3042
0 2995
0.29386
0 2881
0.2024
0 2767
0.2730
ROL
FLOWIG/S
0.626
O 626
O 626
0.626
O 626
O 626
0.626
0 676
0 626
0 626
0 676
0 621
0 627
0 627
0 627
0 627
0 620
0.628
0.628
0 6211
0.628
0.628
O 628
0.628
0 628
O 628
0.620
0 628
0 620
0 628
0 628
CONDENSATION NEAR LOWER PLENUM AT INPUT POWER OF 910W .
lIME SIEP NO * 24322
REAL TIME -000-***.**t
fIIIMER OF NFWIII
ITERAIIONS tAIMBER OF INJNER ITERATIONS I
O
O0
loiAL REACTOR POWER a
IO1AL IlEAl TRANSFER *
FrOW E[NIIALPY RISE
FlOW ENERGY RISE •
0.910 K(W
Q)UALI%
zIMMI
P(BARI
VOID
I
2
3
4
0 0
3)2 0
614 0
862 2
1.01325
O 05663
0.85641
O 86274
0 0000
O 0000
0.0000
0.0000
0 000
0.000
0.000
O 000
1127823
1128251.
I12P75I.
1128251.
810 94
810.70
810.70
10O 71
EM
ROIM
5 1338.6
6 I8s5 1
0.67541
08810
0.0000
O 0000
0.000
0 000
1128251.
1128251.
810.72
810 13
7
8
O
O
90082
90719
0.0000
O 0000
O 000
0 000
1128251.
1128251.
010.74
9 2851 7 0.90'04
IO 3455.7 0.95477
II 4059 7
1.00257
12 4663 7
1.05041
1.09930
13 5267.7
14 5579 7
1.09825
15 5827 9
1.10472
16 6304 4
I 11769
17 6780.8
1.13073
10 7257 2
. 14400
15100
19 7505 ~
20 7817 5
1.15 72
21 8444 5 10.00000
22 8866 5 O 41657
23 9060 5 O 41549
24 9254 5 O 41936
25 9448 5 O 41910
26 9642 5 O 40994
27 9009 5 0 35763
2810189 5 0.33666
29101489 5 O 32562
3060789 5 O 30683
311101d 5 6.06986
3211239 5
I 01325
O 0000
0.0000
0.0000
0.0000
0 0000
0.0000
O 0000
O 0000
0.0000
0 0000
0.0000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1128248.
1128226.
t129134.
1127842.
11270718.
1127053.
1125931.
1124013.
1120931.
1116272.
1116066.
9365
926474.
1423053.
1397138.
1391629.
1407678.
1394196.
1370073.
1375919.
1360044
1351348.
1342795
13,13324.
810 75
810 s0
810.87
810.97
811.16
611.16
811.38
011.75
012.33
813.21
813.25
-85026856 0B
13 99
41.13
91 31
24.26
47 8
169 21
27 92
123.43
284 08
775 83
770.67
0-00--0"0
0 0000
0.A018
O 9462
O 8803
O 9683
O 9373
0.1790
0.9637
O 9393
O 6305
0.0000
0.0000
0.000
O 842
0.2175
0. 16
O 482
O 231
0.048
0.339
O 065
O 020
0.000
0.000
STEP
a
&
*
SIZE
810 75
*
0.623540-09 SEC
CPU lIME 5u3 28 SEC
I TIME SIEP REDUCIIONS DUE TO ERROR
4
1030 REDUCFD
lIME STEPS SINCE LAST PRINT
19 757 G/S
-53.958 G/S
29906 436 G
-0
MAXIMUM
IN
IN
IN
I VAP
T
LIO
862.801
063. i5
863 15
063. 15
863.15
063. 15
863.15
863. 15
863.15
063. IS
863 IS
863 IS
063.15 863. 15
863.13
863. 13
863.06 663.06
962 82 962.82
862.22
862.22
862 20 862.20
861.30
861.30
059 77 859.77
857 32 857.32
8953 61 853.61
853.45
853.45
'693.23'
693.23
703.29
703.29
I067.88 1067 88
O061.63 1067.63
1068.51 1068 51
1068.45 1068.45
1066.37 1066 31
1053 73 1053.73
10.18.22 1048 22
6045 21 1045.21
1039 89 1039 89
1033 82 1033.82
1034.24 1034.24
062.81
863.15
863 15
063.15
MAXIMUM TEMPERAIURES
ROD1113 89 AT
WALl.:
1110 90 At
1068 51 AT
LIOUID'
4600-07 G
SIEP
z2
2291 5
2539 7
TIME
INLEI FlOW RATE
OUlET FLOW RAIE
TOTAL SvSIEM MASS
GLOBAL MASS ERROR
0.910 KW
-94.775 KW
-94 765 kW
MAXIMUM RELATIVE CHANGES OVER TilE TIME
IN PRESSIJRE:
0.7040-06
IN MIXTIRE DENSITY: O 1000-09
IN MIXTURE ENERGY:
0.
D000-09
IC
SEC
I
RELATIIVE LINEARIZAIION ERRORS
PRESSURE
O 7040O00
MASS/VOIUMF:
0.1080-06
ENERGY/VOLUME:
0.1470-07
SAT
1158.78
1140.41
1140.38
6141.18
1142.75
1144.31
1145.85
1146 62
1146 60
1152 21
1157.60
1162 80
1167 82
1167.01
168.48
1169.80
1171.11
172.44
1173.14
1173 21--1487.34
VVZ
2 937
O Oil
0.011
0.011
0.01
0.011
0.012
0 012
0.012
0 013
0.014
0.015
O 016
0 016
0.016
0.017
0.0I0
0.018
0 018
4 538
94 878
1067.88
(41j393'
1067 63
IO68 56
1060 45
1066 37
1053.73
040 22
1045 21
1039 89
1400 38
1158.711
-24 637
-2 620
31.372
64.912
43 699
35 031
27 060
-151 472
-8.440
VLZ
2 937
0.01
0.011
0.011
0.011
0.011
0.012
0 012
0.012
0 013
O 014
O 015
0 016
0 016
0.016
0 OIl
0 019
0.00
0.018
4.539
0.209
-1 025
-O 570
-0 207
O 356
1.792
O 805
0 410
-0 019
-9 529
-8.440
ROV
0 2738
0 2344
0 2344
0.2360
0.2392
0 2424
0 2456
0 2472
0 2472
O 2592
O 2762
0 2831
0.2950
O 2950
O 2966
0.2999
0.3031
0 3064
O 3081
0 3083
2 2421
0
1200
0. 1197
O 1207
0 1207
0. II82
0. 0O41
0 0984
O 0954
0 0902
I 4194
0.2738
ROL
IO 94
810 70
810 70
610.71
810.72
810.73
SIO 74
860.75
810 75
80I 80
010 87
810.97
811. 16
ill. 16
611.38
III
75
812.33
813.21
813 25
850 26
856 80
762. 10
762 16
761 95
761 97
762 45
765 40
766 69
767 39
768 63
775.83
770 67
FLOWIG/S)
19 757
18 389
18.388
18 431
IB 623
IS 968
19.467
19 491
20 347
21 428
22.708
24 151
25 120
25 773
27 068
28 380
29 649
30 695
30.721
32 009
I 480
-0 389
-0 453
-0 044
0 102
0.770
1 159
0 122
-0 026
-61 332
-53.958
.INPUT FOR UNIFORM CROSS-SECTION LOOP CALCULATIONS
2
SINGLE PlIASE MEASUREMENT FOR SODIUM NATURAL CONVECTION IN A
VERTICAL CHANNEL:ORNL/IM-701
SINIGIN NC I.NZ*26.NR*t.NARFut.NX I.NRZS., .ITF5l.
IHIS-3.ISS-l.IXFL-O.IDUMPt.198"O.
ISIRPR-I.ISHPROtIt.NITMAX-2.1PFSOL134.
NEQ4.NUMOER*0. IHRPR.s
SREALIN tOiD3.25E-3.POR*I.2533.HODRaI.OE#0.OELPR-O.5.
1
RNUSS*7.O.RADF*I.625E-4.WINLETl.60OE-4.GRAV0O.00
$
SRODINP 00-150.0
tSNCR
OS INDENT
ISIFCAR
ISNRZF
ISNRMAF
3$MNRZF
ItSINX
7SMNRZS
4$NRMZS
SOX
4.07327E-3
toy
4.07327E-3
1.OE-6 0.617'5 4(0.4575) 6(0.6175) 4(0.4575) .6175 5(0.194)
5(0.3) 5.OE- 6
Soz
SARX
2610.0)
SARY
26(0.0)
SARZ
27(8.285E-6)
5.122162E-6 4(3.794962E-6) 6(S.22,'2E-6) 4(3.794962E-6)
%VOL
5.122162E-6 5(1.60923E-6) 5(2.48851E-6)
SIEDZ
3.25E-3
3.25E-3
SWEOZ
sP
28(1.01325E,05)
SALP
26(0.0)
281693. 5)
SVEL
275 12.OE-2)
-9.6 5(0.0) 5(9.6) 5(0.0)
26(693.5)
1610.0) 541.0) 5(0.0)
5.0
SOf
STEMP
1f(-9.1)
SGRAV
STWF
SOZ
$OR
$RN
SDRZF
$PCX
SORZS
2.03E -2
SHOUT
I.OE*6 14(0.0) 5.0846 10(0.0)
STOUT
863.15 14(500.0) 693.15 10(500.0)
STWS
26(693.15)
SHLSS
26(2.5E*06)
SflMDAf TENOD300.0.OTMIN-I.OE-6.OTMAXAI.O.DTSP-S.O.DTLP*O.0.IREODMXS S
STIMOAT TENDOO.0
$
0
1.0
i.625E-3
I.62929E-2
HYDROSTATIC PRESSURE CALCULATION IN UNIFORM CROSS-SECTION REPRESENTATION
TIME STEP NO 45
nrIMOER OF NEWION IlERATIONS
NtIMRER OF INNER IIERATIONS *
IOIAI
REACTOR POWER TOTAL IlEAI TRANSFER *
FlOW ENIIIIALPY RISE FLOW ENERGY RISE -
-
REAL TIME 2
I
O
0
0.000
-0.000
-0.000
-0.000
45.000000 SEC
TIME SIEP
IZ
ZIMMI
P(BAR)
KW
KMW
KMW
MW
1.01325
I
0O
2
3
91.0
291 0
O
4
485 0
0.97284
5 679.0
6 073 O
7 1120 0
8 1420 0
9 17120 0
10 2020 0
It 2320 0
12 277 7
13 3316 2
14 3771137
15 4231 2
16 4688.6
17 5226 3
Is 5843 a
19 6461 3
20 70170
21 7696 3
22 8313.8
23 8851 2
24 9300 6
25 9766 0
2610223 4
2710760 9
28 11069.6
1.00517
0
98901
95668
O 94052
0 91994
0 89495
0 86996
0.84491
0.81998
0.70171
0.73701
0.73701
0.73701
0.13701
O 78177
O 03321
0.88464
0.93608
0.99753
1.03097
I 08375
I 09375
1.08375
I 09375
I 03897
I 01325
VOID
OUAL(%)
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0 0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0 0000
O 0000
0.0000
O 0000
0.0000
0 0000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
O 000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0 000
0 000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.100000401
O
INLET FlOW RATE a
OUTLEI FLOW RATE e
TOTAL SYSTEM MASS a
GLOBAL MASS ERROR -
EM
913656.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
913556.
ROM
850. 14
850.13
850.12
850.10
650 08
850.07
050 05
050.02
950.00
849.97
049.95
049.91
849.87
649.87
849.67
649.67
849.91
849.96
050.01
850 06
950. 12
050. 17
950.21
850.21
050.21
850.21
850. 17
850. 14
SEC
CPU TIME 21.50 SEC
TIME STEP REDUCTIONS DUE 10 ERROR
O
0 REDUCED TIME STEPS SINCE LAST PRINT
-0.0oo00G/S
-0 000 G0/
73.224 0
MAXIMUM TEMPERATURES
ROD:
693.15 At
WALL:
693.15 At
LIOUID:
693.15 AT
-0.1540-14 0
MAXIMUM RELATIVE CHANGES OVER TIlE TIME STEP
IN PRESSURE:
0.1000-09
IN MIXTURE DENSITY: O.ICD-09
IN MIXTURE ENERGY:
0. 1000-09
IC
SIZE -
MAXIMUM
IN
IN
IN
T VAP.
T LIQ
693. I5
693.15
693.15
693.15
693. IS
693. IS
693. IS
693. IS
693.15
693. 15
693. 15
693. 15
693. IS
693.1S
693. I5
693.15
693. IS
693. 15
693. IS
693. 15
693. 15
693. IS
693. IS
693 15
693 15
693.15
693.15
693. IS
693. IS
693.15
693.15
693. 15
693. 15
693. 15
693.15
693. 15
693. IS
693.15
693. 15
693. 15
693. 15
693.15
693. 15
693.15
693. 1
693.15
693.15
693. IS
693. 15
693. IS
693.15
693. IS
693. 15
693. 15
693. 15
693. 1I
1158.78
1157.89
1156.09
1154.27
1152 43
1150.50
1148.14
145. 14
1142.00
1138.94
1135.73
1130.66
1124.46
1124.40
1124.46
1124.46
1130.66
1137.44
143.109
1150.04
1155.93
1166.31
1166.31
1166.31
1166.31
1161 .5
1158.78
IZ
22
22
22
RELATIVE LINEARIZATION ERRORS
PRESSURE:
0.9020-14
0.7450-15
MASS/VOLUME:
ENERGY/VOLUME:
0.5630-t9
T SAT
I161.50
IC
I
I
1
VVI
-0.000
-0.000
-0 000
-0.000
-0.000
-0.000
-0 000
-0.000
-0.000
-0.000
-0 000
-0.000
-0.000
-0 000
-0.000
-0.000
-0.000
-0 000
-0.000
-0 000
-0 000
-0.000
-0 000
-0 000
-0.000
-0 000
-0.000
VLZ
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0 000
-0.000
-0 000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0 000
-0.000
-0.000
-0 000
-0.000
-0.000
-0.000
-0 000
-0.000
-0.000
-0 000
-0.000
ROV
0.2738
0.2710
0.2678
0.2637
0 2591
0.2556
0 2504
0.2441
0.2370
0.2315
0.2251
0.2154
0.2039
0.2039
0.2039
0 2039
0.2154
0.2265
0.2415
0.2545
0.2674
0.2803
0.2914
0.2914
0.2914
O 2914
0.2803
0.2738
ROL
650. 14
650. 13
850. 12
850. 10
850 08
850.07
850 05
850 02
650 00
849.97
849.95
849 91
049.87
849.67
849 07
849 67
849.91
849 96
850.01
850 06
850. 12
850. 17
850 21
650.21
850.21
850 21
850 I1
850. 14
FLOW(G/S)
-0.000
-0.000
-0 000
-0 000
-0 000
-0 000
-0 000
-0 000
-0 000
-0 000
-0.000
-O 000
-0.000
-0.000
-0 000
-0 000
-0.000
-0 000
-0 000
-0.000
-0 000
-0 000
-0 000
-0.000
-0 000
-0 000
-O 000
SINGLE-PHASE CALCULATION AT 150W
lIME STEP NO *
502
REAL TIME *
0.5000403 SEC
NUMBER OF NEWTON IIERATIONS *
2
NUMBER OF INNER ITERATIONS *
I
O
O0
REACIOR POWER
TOTAL
TOIAL
IEA
0
TRANSFER *
FLOW ENIIIALPY RISE
FLOW ENERGY RISE -
0. ISO
0. 150
0. ISO
0. ISO
0. 150
KW
K
NW
KW
CPU TIME *
219.88 SEC
TINE STEP SIZE - 0.IOOOOo*OI SEC
O TIME STEP REOUCIIONS DUE TO ERROR 0
0
INLET FlOW RATE
OUTIE
FLOW RATE
TOTfAL SYSTEM MASS
GLOBAL MASS ERROR
KW
0.373
0.373
74.193
-0.3760-14
MAXIMUM RELAlTIVE CIHANGES OVER IllE TINE STEP
IN PRESSURE:
0.1000-09
IN MIXTIIRE DENSITY: 0.1000-09
IN MIXTURE ENERGY:
0. 1000-09
IC
IZ
Z4(MM)
I
O 0
2 308 7
3 846.2
4 1303 6
5 1761.3
6 2218.8
7 2756.2
8 3373.
9 3991 2
1O 4608 7
II 5226.2
12 5843.0
13 6361 3
14 6838 a
IS 7296 3
s16 7753 8
17 5291.3
18 8691.0
19 8091 0
20 90S.0
O
21 9279.0
22 9473.0
23 9720 0
2410020 0
2510320.0
2616620 0
21170920 0
2811070.0
PIBAR)
I.01325
VOID
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
I.08527 0.0000
1.13410 0.0000
1.16293 0.0000
0.0000
1.23176
1.23155 0.0000
1.23137 0.0000
1.231168 0.0000
1.23100 0.0000
1.23076 0.0000
I.19691 0.0000
1.8100 0.0000
1.16553 0.0000
I.IS27
1.15021 0.0000
1.13530 0.0000
I.11642 0.0000
1.09349 0.0000
I 01057 0.0000
I 04764 0.0000
1.02471 0.0000
1.01325 0.0000
0.98060
0 9838
0.98820
0.98802
0.987084
0.90762
1.03644
DUAL(%)
EM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
913558.
1128246.
1128248.
118246.
1128246.
1128248.
1123248.
1120246.
1128246.
1128246.
1128248.
1020246.
1128240.
1128246.
1128248.
1128246.
1128246.
913562.
993889.
101421S.
1 b4544.
12348071.
1315198.
13151998.
1315198.
1315191.
1315198.
1315198.
1315198.
RONM
050.14
810.83
910.83
010.03
610.83
8010.83
810.93
810.88
110.93
110.,98
111.03
111.08
811.00
811.08
811.07
611.07
850.36
035.77
821.02
806.12
191.12
776.07
776.05
1776.03
1778.00
715.980
775.96
775.95
REDUCED TIME
STEPS SINCE LAST
PRINT
MAXIMUM TEMPERAIURES
1012.55 AT
ROD:
WALL:
1012 08 At
LIOUIDO: 1011.95 At
O/S
0/5S
a
0
MAXIMUM RELATIVE LINEARIZATION ERRORS
V VAP
693.15
863.15
63.1I
183.
IS
863.IS
683.15
093.15
063.15
663.
I5
063.15
T LIO
IN PRESSURE:
IN MASS/VOLUME:
0.1210-12
0.3400-15
IN ENERGY/VOLUME:
0.1510-18
T SAT
693. S1 1158.78
863. 15 1156.05
863.15 1156.03
063.15 1156.00
11955.98
083.IS
883. I5 1155.96
863.15 1155.84
1181 30
063.IS
1
863.15 663.15 1166.47
1171.45
863.15
063 15
tS 1116.27
863. 15 863.
063. 19 063.15 1180 92
e63.16
863.15
1500.909
863.15
1000.07
1110.10
063.1!
063.15 1180.69
063.1!
662.1I5
893.19 1110.69
100 83
893.19
1160.61
111.11
756.44
756.44
820.IS 620.15 1176.09
804. 10O 84.10 1174.51
940.09 948.09 1173.06
1011.95 1011.95 171.57
1011.95 ot1011.95
1169.67
1011.95 1011.95 1167.32
1011.95 1011.95 1164.93
10o1. 95 1011.95 1162.51
ot1011.95 101.95 1160.03
1011.95 1011.95 1I58.7
VVZ
VLZ
0.053
0.056
0.056
0.056
0.050
0.05
0.0586
0.056
0.056
0.056
0.0568
0.056
0.056
0.056
0 056
0.056
0.053
0.054
0.055
0.056
0.057
0.0568
0.058
0.058
0.058
0.050
0.058
0.053
0.056
0.056
0.056
0.056
0.056
0.0586
0.056
0.056
0.056
0 0586
0.056
0.056
0.056
0.0586
0.056
0.053
0.054
0.055
0.056
0.057
0.058
0 058
0.056
0.056
0.056
0.058
ROV
0.2738
0.2677
0.2676
0.2676
0.2675
0.2675
0.2674
0.2796
0.2910
0.3039
0.3160
0.3280
0.3280
0.3279
0.3279
0.3270
0.3278
0.3194
0.3155
0 3t17
0 3019
0.3042
0.2995
0.2938
0.2881
0.2824
0.2767
0.2738
ROL
050.14
110.183
110.83
810.83
To 03
810.63
110.13
610 as
10.93
10.90
811.03
811.00
111.08
111.08
611.01
1111.07
650.36
035.77
021.02
806.12
791.12
776.07
17786.05
776.03
776 00
775.90
715.96
775.95
FLOWIG/S)
0.373
0.373
0 373
0.373
0.373
0.373
0.373
0 373
0.373
0.373
0.313
0 313
0 373
0 313
0.313
0.373
0.313
0.373
0 373
0 373
0.313
0.373
0.373
0.373
0 373
0 373
0.373
?
A0
SINGLE-PHASE CALCULATION AT 250W
IINE STEP NO *
t13
NIIMDER OF NEWION ITERATIONS NUIMBER OF INNER ITERATIONS IDIAL REACIOR POWER TOIAL HEAT TRANSFER FlO
ENIIIALPY RISE FLOW ENERGY RISE -
0
0
-0
-0
REAL TIME v 150.710295 SEC
2
I O
0
MAXIMUM RELATIVE CHANGES OVER
IN PRESSURE:
IN MIXIIIRE DENSITY:
IN MIXTURE ENERGY:
IC
IZ
ZIMM)
PIBAR)
I
0.0
2
97.0
3 291 O
4 465.0
5 679 O
6 073.0
7 1120 O
8 1420.0
9 1720.0
10 2020.0
11 2320.0
12 27178.7
13 3316.2
14 3773.7
15 4231.2
16 4608.0
17 5226 3
10 5843.
19 6461 3
20 70718
21 7696.3
22 8313.8
23 8851 2
24 9308.6
25 9166.0
2610223 4
27"10760 9
2811069.6
I 01325
1.00515
O 90914
0.97334
0.95778
O 94245
0.92309
0 69957
0.87605
0.85253
0.82902
0.79113
0.745711
0.74519
0.74466
0.74413
0.78830
0 83906
0.00983
0 94060
0.99138
1.04216
I.0,8637
I.08505
1.08532
I 08480
1.03936
1.01325
VOID
THiE
TIME
EM
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
913556.
979439.
1045321.
1111203.
1177085.
1242968.
1242968.
12429?7.
1242967.
1242966.
1242966.
909755.
909755.
909755.
909755.
909?55.
909755.
909755.
909755.
909755.
909755.
909755.
909755.
909755.
909755.
9097%5.
900443.
908443.
O 0000
0.0000
O 0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0 0000
O 0000
0 0000
O 0000
0.0000
0.0000
0.0000
0.0OOO.
0.0000oo
0.0000
0 0000
0.0000
0.000
0 000
0.000
0.000
O.OOO
TIME STEP REOUCTIONS OUtE tO ERROR
0
0 REDUCED TIME STEPS SINCE LAST PRINT
w
*
o
*
0.759
0 759
72.224
-0.3660-14
STEP
MAXIMUM
IN
IN
IN
0.3000-00
0.3990-07
0.1350-06
OUAL(X)
ROM
650.14
638.21
026. 15
813.99
001.72
789.40
789.36
709.36
76119.36
789.34
769.31
789.29
850.60
650.56
050.56
850.56
650.56
050.60
650 65
950.70
950 75
950.60
850.85
850.90
850 90
950.90
650.90
651.09
851.06
85.67 SEC
O
INLET FLOW RATE
OUTLET FLOW RATE
TOTAL SYSTEM MASS
GLOOAL MASS ERROR
.250 KW
250 KW
.004 KW
.004 KM
CPU TIME -
TIME STEP SIZE * 0.011210*00 SEC
I VAP
T LIO
693.15
745 02
797.19
849.57
902.06
954.54
954.54
954.54
954.54
954.54
954.54
690.1 7
690.1 7
690.17
690. 17
690.17
690.11
690.I17
690. I7
690. 17
690. 17
690.17
690. 17
690. 11
690.17
690.17
689. 14
689. 14
693.15
745.02
797.19
649.57
902.06
954.54
954.54
954.54
954.54
954.54
954.54
690. I1
690. IT
690. 11
690. 17
690.1 1
690. 17
690.17
690.17
690. I7
690. 17
690. 17
690. 17
690. 17
690. I1
690. 17
689. 14
609. 14
T
O/S
G/S
0
0
MAXIMUM TEMPERATURES
ROD:
WALL:
LIQUID:
955.54 AT
954.72 AT
954.54 AT
IC
I
I
I
IZ
5
S
5
RELATIVE LINEARIZATION ERRORS
PRESSURE:
0.2390-08
MASS/VOLUME:
0.2870-15
ENERGY/VOLUME:
0.9380-16
SAT
1156.786
1157.69
115-.11
I154.33
I152.56
1150. 76
1148.51
1145.70
1142 83
1139.90
1136.90
1131.92
1125.69
1125.61
1125.54
1125.46
1131.54
1138.19
1144.52
1150 57
1156.36
1161.92
1166.58
1166.53
1166.47
1966.42
1161.62
1158.78
VVZ
VLZ
0. 108
0. 109
0. III
0.112
0.114
0.016
0. 116
0 li6
0. I16
0. 116
0. 116
0. IOI
0. 108
0.106
0. lOB
0. 10
0. 108
0. 108
0. 106
0. 108
0. 108
0. 109
0.111
0. 112
0.114
0. 116
0. II6
0. 116
0. 116
0.116
0.116I
0. 10
0. 198IO
0.100
0. 108
0 109
0.106
0.
IOU
0. 0l8
0.108
0. 10
0. 108
0. 10S
. 107
0.
0.
0.
0.
0.
0.
108
o00
IO8
o00
10
108
100
0. 108
0.108
0. 100
0. 17
ROV
0.2130
0.2718
0 2678
0.2639
0.2599
0 2561
0.2512
0.2453
0.2393
0.2334
0.2274
0.216
0.2061
0.2060
0.2059
0.2057
0.2170
0.2300
0 2428
0.2556
0 2664
0.2811
0.2921
0.2919
0.2918
0.2907
0.2004
0.2738
ROL
850.14
838.31
026.15
013.90
801.72
789.40
789 38
719 36
789.34
7189.31
789.29
050.60
850 56
850.56
050 56
050.56
050.60
850.65
650 70
850 75
850.80
050.05
850.90
050.90
050 90
850.90
851.09
851.06
FLOW(G/S)
0.759
0.759
0.759
0.759
0.759
0.759
0.759
0.759
O 759
0.759
0.759
0.159
0.759
O 759
0.759
0.759
0.759
0.159
0 759
0 759
0.759
0 759
0.759
0.759
0.759
0.759
0.759
SINGLE-PHASE CALCULATION AT 370W
TIME STEP NO *
852
tAIMBER OF NEWTON ITERATIONS *
NUMOER OF INNER ITERATIONS *
TOIAL REACIOR POWER a
ItITAL IEAT TRANSFER a
FlOW ENTIIALPY RISE
FLOW ENERGY RISE -
REAL TIME *
2
1
0
0
0.370
0.370
0.370
0.370
0.0500403 SEC
TIME STEP SIZE * 0.100000*01 SEC
O
0
MW
KV
KW
KV
INLET FLOW RATE *
OUILEI FLOW RAlE
TOTAL SYSTEM MASS
GLOBA MASS ERROR
MAXFMUM RELATIVE CHANGES OVER TH4E
TIME STEP
IN PRESSURE:
0.1210-5O
IN MIXTURE DENSITY: 0.2960-04
IN MIXIURE ENERGY:
0.6180-04
IC
Ii
ZIMM)
P(BAR)
I
0.0
2 308.7
3 846.2
4 1303.8
5 1761.3
6 2218.6
7 2756.2
0 3373.7
9 3991.2
10 4608.7
It 5226.2
12 56843.8
13 6381.3
14 6030 6
15 7296.3
16 7753.8
17 8291.3
I 8691.0
19 8891.0
20 9085 0
21 9279.0
22 9473.0
23 9720.0
2410020.0
2510320.0
2610620 0
2710920.0
7811070 0
1.01325
0.90646
0.96791
0.90756
0.971 S
0.98674
0.98625
1.03471
1.06320
1.13180
1.16033
1.22885
1.22837
1.22796
1.227155
1.22713
1.22661
1. 19259
1.17662
1.16144
. 14644
1.131984
I 11350
1.09122
1.06894
I 04667
1.02439
1.01325
VOID
QUAL(%)
EM
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000,
0.000
0.000
0.000
913556.
1120241.
1126241.
1123241.
1120241.
1128241.
1128241.
1121241.
1123241.
1128242.
1120242.
1121242.
1120242.
1128242.
1128242.
1128242.
913566.
1022699.
1131845.
1241019.
1350228.
1459470.
1459720.
1459941.
1400047.
1459972.
1459710.
1459110.
ROM
50. 14
110.93
110.13
110.63
010.63
610.13
610.13
1110.66
810.93
1010.99
S11.02
8IT.07
311.07
611.07
611.07
611.07
350.35
630.50
910.35
769.99
769.51
149.05
746.99
740.92
748.86
740.07
746.90
748.89
T VAP
693. IS
863.14
03. 14
063.14
663.14
663.14
663. 14
863.14
663. 14
963.14
863.14
863.14
863.14
663.14
863.14
663.14
693.16
719.25
666.01
952.99
1039.70
1125.69
1 125.8
1126.05
1126. 13
1126.03
1125.17
1125.01
T LI
MAXIMUM TEMPERATURES
ROD:
1127.20 AT
1126.13 AT
WALL:
LIOUID: 1126.13 AT
0.67
0.870
73.726
0.4490-09
MAXIMUM
IN
IN
IN
CPU TIME *
24.86 SEC
TIME SIEP REOUCIONS OUE TO ERROR
0
REDUCED TIME SIEPS SINCE LAST PRIUT
IC
I
I
I
IT
21
24
24
RELATIVE LINEARIZATION ERRORS
0.1290-05
PRESSURE:
MASS/VOLUME:
0.9020-1O
0.2120-13
ENERGY/VOLUME:
T SAT
693.15 1153.70
6083.14 1156.03
663. 14 1155.98
863. 14 1155.93
063. 14 1155.19
963.14 1155.14
863.14 1155.79
863.14 1161.12
063.14 11606.26
063.14 1171.22
176.01
663. 14
063.14 1130.65
663.14 1100.60
363.14 1100.57
663.14 1180.53
063.14 1180.49
693.16 1180.44
779.25 1171.20
866.01 1115.61
952.99 1174.17
1039.70 1172.6
1125.69 1171.23
1169.37
S125.8
1126.05 1167.09
1126.13 1164.76
1126.06 1162.40
1125.67 1160.00
1125.87 1158.70
VVZ
VLZ
0.096
0.101
0. 10
0.101
0. 101
0.101
0.101
0.1 01
0. 01
0. 01
0. 101
0.101
0. 101
0. 101
0.O101
0.101
0.096
0.099
0.101
0. 104
0. 106
0.109
0. 109
0. 109
0. 109
0.109
0. 109
0.096
0. 101
0.1 01
0. 101O
0.10
0.101
0.101
0. 101
0.101
0. 101
0. lOt
0. 101
0. 101
0. 101
0.101
0.l01
0.096
0.099
0. 01
0.104
0. 106
0. 109
0. 109
0. 109
0. 109
0.109
0. 109
ROV
ROL
0.2736
0.2676
0.2675
0.2674
0.2673
0.2672
0.2671
0.2792
0.2913
0.3034
0.3 154
0.3273
0.3272
0.3271
0.3270
0.3269
0.3263
0.3164
0.3145
0.3107
0.3070
0.3034
0 2988
0.2933
0. 21771
0.2622
0.2766
0.2738
850.14
810.63
610.03
11to.13
910.83
010.03
810.83
110.63
610.66
010.93
10.90
811.02
311.01
11.07
11.017
811.07
1111.01
050.35
830.50
810.35
789.99
169.51
749.05
740.99
148.92
748 66
740.87
748.90
740.89
FLOvIG/S)
0.678
0 67
0.678
0 670
0.676
0.678
0 676
0.678
0 676
0.67
0.670
0 6s
0 678
0.676
0 670
0.678
0.676
0.678
0.678
0.678
0.673
0.678
0.678
0.678
0.67
0.670
0.678
CONDENSATION NEAR UPPER PLENUM AT 640W IN UNIFORM CROSS-SECTION
IME SIEP
ImBIfIGIR OF
tI iltR4 OF
NDO
5496
NIWIOI
IIElAIINS
INNFR IIRATIONS •
fIOlia
REACIIIR POIWER
IOIAI IIEAE IRANSFER
1floW tJIIIAIPY RISE
flOW FNERGY RISf
REAL
1IME
1)
I
0
*
1150*04 SEC
O 640 KW
O 640 KW
*
STEP SIZE
INtEl FlOW RAIE •
OUIltt flOW RATE •
TOlAL SYStEM MASS *
GlORAI MASS tRRUR
-835 765 KW
-835 607
lIME
KW
MAXIMUM RfIAIIVF CIIANGES OVER IIIE TIME
IN PRESSIllR:
0. 3720 04
IC
IZ
ZIMMI
P(BARI
I
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
O
308.7
8,46 2
1303 8
1.01325
O 97706
O 95677
0 93949
0 92219
O 904189
O 80455
0 91024
O 93592
O 96160
O 90129
1 01297
0 99261
O 97529
O 95797
0.94066
O 91951
0 81051
0.84766
0 82548
0 80391
0 78504
O 79969
0 78339
0.76502
O 75714
10 00023
1 01325
1761.3
2218 8
2756 2
3371 7
3991 2
4606 7
5226 2
5843 8
638I 3
6838 8
7296 3
7753 8
8291 3
IB
11691 O
19 8891 O
20 9011)R50
21 9279 O
22 9413 O
23 9120.0
24 1(020 0
2510320 0
2610620 0
2710920 0
2011070 0
MIXIURE ENERGY:
VOID
OUALt)
0.0000 0.000
0.0000 O 000
0.0000 0.000
O 0000 0 000
0.0000 0.000
0.0000 0 000
O 0000 0 (X)O
0.0000 0 000
0.0000 .0.000
0.0000 0. O(X)
O 000
0.0000
0 0000 O 000
0.0000 0.000
0o. (K)00
0000 0.000
O 000
0.0000
0 000
0.000
O 0000
0.010
O 000(
0.0000
O 0ro0
0.000
0.0000 O 000
0.0000 0.0(00
0 1920
O 007
0.6494 0 055
O 7291 0 078
0.8258 0. 134
0 880O 0 219
0.0000 0 000
0.oo0000 0.000
878520-09 SEC
CPU TIME 159.37 SEC
I lIME STEP REOUCIIONS ODUE TO ERROR
4
343 REDUCED TIME STEPS SINCE LAST PRINT
0
4.698
-512.656
61.766
-0.5070-06
SIEP
MAXIMUM
It
IN
IN
IN MIxIIIRE DFNSIIv: 0. 1000 -09
IN
*
0
0.
OOT)D09
EM
1128251.
1128251.
1128251.
11282-1.
1120251.
(128252
1 ?82752
1120252.
1120252.
1128292.
1120252.
1128250.
1121781.
913626.
993749.
1110146.
124:1096.
1379532.
14667-5.
1471357.
1469601.
1469904.
1471341.
1412523.
145062.
RUM
810.86
110 82
8 to 80
810 78
810.76
810.75
810.73
810.75
810.78
810.60
10 86
810.84
81082
810.80
BIO 87
850.03
835.47
813.94
789 26
763.68
603.98
262 00
202 66
130 52
84 73
766.70
750 69
I VAP
1
LIQ
863.15 863.15
863.15 863.18
863.15 863.15
8063.15 063.15
863.15 863.15
803.15 863.15
863.15 863 15
863.15 663.15
863.15 863.15
803.15 063.15
803.15 863.15
863.15 803.15
863.1S
863 15
063.15 863 15
003.15 863 15
862.78 862.78
693.21 693.20
750.33- 756.33
849.21 849.21
954.64
954 64
1062.86.1062 8G
1131.10 1131 10
1133.06 1133 06
1130.88 1130 68
1128:37 1128.37
1127 28 1127.28
1060.85 1068.85
1118 32 1118.32
T
G/S
G/S
0
0
RELATIVE LINEARIZAIION ERRORS
PRESSURE
0.3720102
MASS/VOLUME:
0.4810-06
ENERGV/VOLUME:
0.1620-07
SAI
1158.78
1154.75
1152 44
1190.44
1148.41
1146.34
1143 87
1146 98
1150.02
1152.99
1155.90
1158.75
1156.50
1154.55
1152 58
1150.57
1148.09
1142.14
1139.28
1136 44
1133.62
1131.10
1133.06
1130 88
1128 37
1127.28
1,181.34
1158.78
MAXIMUM TEMPERAIURES
ROD1133 71 AT
MALL:
1133.06 AT
LIOUID: 1133 06 AT
VVZ
VLZ
0.699
0.699
0.699
0.699
0.699
0 699
0.700
0.700
0.700
0 700
0.700
0.700
0 700
0.700
0.701
0.701
0.701
0.701
0.702
0.702
0.702
0.702
0.703
0.703
0.703
0 703
0.704
0.704
0.704
0 704
0 705
0 705
0 674
0 674
O 678
0.678
0.682
0.682
0 686
0.686
O 736
0.691
-3.577
0.088
I7.130
0.988
23.967
1.214
19 528
0.610
158 374
-8.011
-92.075 -92.0175
ROV
O 2730
0.2648
0.2597
0 2553
0.2510
0.2466
0 2415
0 2480
0.2545
O 2609
0.2614
0 2138
0 2687
0 2643
0 2600
0 2556
0 2503
0.2379
0.2321
0.2265
0.2210
0.2162
O 2199
0.2158
0.2111
0.2091
2 2427
0.2738
ROL
FLOWIG/SI
810 86
61O 02
81to 80
O1078
810 76
810 75
010.73
810 75
610 78
4.698
4.698
4 698
4 699
4.700
4 701
4 703
4.706
4 709
4.713
4.717
4.721
4 725
4.728
4 732
4 737
4 744
4 691
4 598
4 488
4 373
O 435
2. 164
2.069
0 687
-50 884
-572 656
8 to 83
8I0
8to 86
84
810 82
1810 O
810 80
810 87
850.03
875 47
813 94
789 26
763 68
741.41
746.96
747 46
748 04
740 29
766.70
750 69
LOW QUALITY BOILING AT 450WOIN ACTUAL LOOP GEOIIETRY
TIME STEP NO *
888
REAL TIME *
MIMItER OF NEWTON ITERAIONS *
2
NUMBER OF INNER IIERATIONS
1
O
O
TOTAL REACIOR POWER a
lOYAL tlEAl IRANSFER •
FLOW ENIIIALPY RISE
FLOW ENERGY RISE &
0.8700*03 SEC
0.450 NWv
0.450 KMW
0.450 KM
0.450 KW
TIME STEP SIZE * 0.100000401 SEC
CPU IME *
0.12 SEC
O TIME STEP REDUCTIONS OUE tO ERROR 0
0 REDUCED TIME SIEPS SINCE LAST PRINT
INLET FLOW PATE
OUILEI FLOW RATE
IOIAL SYSIEM MASS
OLOBAL MASS ERROR
0.763 0/5S
0.763 0/5s
72.333 0
-0.1230,01 0
MAXIMUM RELATIVE CHIANGES OVER IE TIME STEP
IN PRESSURE:
0.3900-04
IN MIXTURE DENSITY: 0.31211*00
IN MIXTURE ENERGY: 0.1650-02
IC
I
I
2
3
4
5
6
7
Z(MM)
0.0
300.7
846.2
1303 8
1761.3
2211.9
2756.2
8 3373 7
9 3991.2
o104606.7
I1 5226.2
12 5041.6
13 63801.3
14 6830.8
IS 7296.3
16 7753.8
17 8291.3
Is 6691 0
19 8891.0
20 9085.0
21 9219 O0
22 9473 0
23 9720.0
2410020.0
2510320.0
2610620 0
2710920 0
26811070.0
P(BARI
VOID
1.01325
0.98841
0.97833
O 960734
0.98685
0.90636
O 98579
I 03420
1.08261
1.13102
1.17944
1.22786
1.22728
1.22679
1.22631
1.22502
1.22519
1.19113
1.17537
1. 16005
1.14516
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0. 1772
0.312I
0. 1009
0.0000
0.0000
1.13070
1.11255
1.09052
1.06647
I 04640
t.02431
I.01325
Q UALI()
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000O
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.009
0.018
0.006
0.000
0.000
EM
9135568.
1129240.
1128240.
1128240.
1128241.
1128241.
1128241.
1128240.
1128241.
1128241.
1128241.
1123241.
1120242.
1128242.
1128242.
1128242.
913568.
1032003.
1151152.
1271161.
1391866.
1512695.
1514135.
1512883.
1510235.
1506904.
1501793.
1501793.
ROM
850. 14
310.3
310.83
010.83
oIO. 83
310.03
310.3
8 I0,83
810.13
910.338
810.93
610.98
111.02
311.07
611.07
811.07
811.071
811.07
850.35
928.80
806.76
784.34
761.71
739.13
738.04
608.20
SOS.88
606.33
741.09
741.04
MAXIMUM
IN
IN
IN
7
VAP
693.15
863 14
063.14
863. 14
863.14
863.14
863. 14
863.14
063.14
863.14
863.14
863.14
863.14
063.14
863.14
863.14
693.16
736.63
801.39
978.97
1072.59
1167.17
1163.29
1167.01
1164.71
1162.37
1158.70
1158.70
T LIO
I SAT
893.15
863.14
063.14
863.14
863.14
863.14
963. 14
963.14
363.14
863.14
663.14
663.14
863.14
863.14
863.14
863.14
693.16
706.63
081.39
976.97
1072.59
1167.17
1163.29
1167.01
1164.71
1162.37
1153.70
1158.70
1153.76
1156.03
1155.96
1155.91
1155.85
1155.80
1155.172
1161.00
1166.19
1171.14
1175.93
1180.56
1180.50
1180.46
1180.41
1100.36
11980.31
11117.06
0175.53
1174.03
1172.56
117 1. II
1169.20
1167.01
1164.71
1162.37
1159.99
1159.70
MAXIMUM TEMPERAIURES
ROD:
1169 02 AT
WALL:
1168.29 AT
LIQUID: 1161.29 At
IC
I
I
I
Iz
21
22
22
RELATIVE LINEARIZATION ERRORS
PRESSURE:
0 3890-04
MASS/VOLUME:
0 4530D00
0.1350-11
ENERGY/VOLUME:
VVZ
VLZ
0. 108
0.114
0.114
0.114
0. 114
0.114
0. 114
0. 114
0. 114
0. 114
0.114
0.I114
0. 114
0. 114
0.114
0.114
0. 108
0. III
0.114
0. I17
ll4
0.
0.121
0.124
0.124
0.124
0.124
0. 124
0. 124
0. 108
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0. 114
0.114
0. 114
0.114
0.100
0. 111
0.114
0. 117
0.121
0.124
0.124
0.124
0.124
0. 124
0.124
ROV
ROL
0.2731
0.2676
0.2675
0.2674
0.2672
0.2671
0.2670
0.2791
0.2911
0.3032
0.3151
0.3271
0.3269
0.3260
0.3267
0.3266
0.3264
0.3100
0.3141
0.3103
0.30671
0.3031
0.2986
0.2931
0.2076
0.2821
0.2766
0.2738
850. 14
610.03
6 0. 83
010.03
010.03
810.93
310.03
810.80
610.93
810.96
311.02
811.07
11 .07
11.07
811.07
3t.01
850.35
623:00
306.76
784.34
761.71
739. 13
738.34
739.12
739.65
740. 19
741 05
741.04
FLOWI0/SI
0.763
0.763
0.763
0.763
O 763
0.763
0.763
0.763
O 763
0.763
0.763
0.763
0.763
0.763
0 763
0.763
0.763
0.763
0.763
0.762
0 762
0.761
0.761
0.627
0 524
0 625
0.763
,
it
PLENUM HEAT CONDUCTION CALCULATION RESULT AT 250W
02
IZ
HEAT
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
178.123
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-178.116
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
SOURCE
(W)
H-STRUCTURE
(W/M.*2)
0.25000+07
0.25000+07
0.25000407
0.25000407
0.25000+07
0.2500007
0.25000+01
0.25000407
0.25000407
0.25000+07
0.25000+07
0.25000+07
0.25000+07
0.25000407
0.25000+07
0.25000+07
0.25000+07
0.25000407
0.25000+07
0.25000+07
0.25000+07
0.25000D+07
0.25000407
0.2500007
0.25000*07
0.25000407
STRUCTURE TEMPERATURES (DEG K)
863.148
863.141
863.141
863.141
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
693. 152
786.625
881.393
976.972
1072.588
1167.169
1168.287
1167.013
1164.713
1162.373
1158.696
863. 148
863.141
863.141
863.141
863.142
863.142
C63.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
693.152
786.625
881.393
976.972
1072.588
1167.169
1168.287
1167.013
1164.713
1162.373
1158.696
863.148
863.141
863.141
863.141
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
693.152
786.625
881.393
976.972
1072.588
1167.169
1168.207
1167.013
1164.713
1162.373
1158.696
863. 148
863.141
863.141
863.141
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
693.152
706.625
881.393
976.972
1072.580
1167.169
1168.287
1167.013
1164.713
1162.373
1158.696
863.140
863.141
863.141
863.141
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
693. 152
786.625
881.393
976.972
1072.588
1167.169
1168.287
1167.013
1164.713
1162.373
1158.696
PLENUM HEAT CONDUCTION CALCULATION RESULT AT 4501-
IZ
HEAT SOURCE
(W)
1
158.588
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-158.582
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
-0.000
2
3
4
6
6
7
8
9
10
IIt
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
H1-STRUCTURE
(W/M**2)
0.25OOD007
0.25000107
0.25000107
0.25000407
0.2500D07
0.2500007
0.25000407
0.25000+07
0.25000407
0.25000+07
0.25000407
0.25000407
0.2500D07
0.25000+07
0.250OD007
0.25000D07
0.25000407
0.25000407
0.2500D+07
0.25000407
0.2500D07
0.2500007
0.25000407
0.2500007
0.25000407
0.25000407
STRUCTURE TEMPERATURES
863.148
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
693.152
779.248
866.013
952.989
1039.703
1125.689
1)25.879
1126.051
1126.134
1126.075
1125.870
863.148
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
693.152
779.248
866.013
952.989
1039.703
1125.689
1125.879
1126.051
1126.134
1126.075
1125.870
m
R
m
(DEG K)
863.148
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
*863.142
863.142
863.142
863.142
863.142
863.142
693.152
779.248
866.013
952.989
1039.703
1125.689
1125.879
1126.051
1126.134
1126.075
1125.870
863.148
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863. 142
863.142
863.142
863.142
863.142
863.142
693.152
779.248
866.013
952.989
1039.703
1125.689
1125.879
1126.051
1126.134
1126.075
1125.870
863.148
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
863.142
693.152
779.248
866.013
952.989
1039.703
1125.689
1125.879
1126.051
1126.134
1126.075
1125.870
186
APPENDIX D. Hierachy of Subroutines. in T~PMIT-4,
The hierachy of the subroutines in the four equation version
:of the Sodium
colant version of THERMIT is illustrated in the chart
TEMIT4 in figure D1 below.
L87
TH E?-i1TI
mPI:-.
i
I Np*T
T L'.:;C
i
I
T
TFL.A S
:NIT
1F
?J.U.
EX.,j
Dt. P
-
IEL
I
&0
%0VIC
DFA
IS
I
I 41PCayl
I
EDiTn
STAk,,
IT
--
MAPPER
--
S
I
SETICC
ACTARV
I
INITSC
IN;ITRC
I
1OV
STATEG
HOVE
I
EDITOR
HP2
TlING
DIP
STEMPF
Iow.ir
PFSOLS
JACOBC
PRPOPUP
QCMNGE
NEWERR
STATEG
JA' EQ
MC!~NTI
TND
I
FWALL
ITR
FINTER
I
1.1: LINN,
EDITOR
E TIMING
T.LIZ
KP2
MP 3
CVECX
JAC4EO
PFSOLS
STATEG
PGUESS
PSCLID
PSOL2D
PSOL2R
-PSOL3L
" PSOL3P
L- PSL3DS
SBSTAT :
- QtK4EQ
tP 4
MP 6
FIGURE D1
CVECY
L
Fpseg31
• w v w..,, j..
1
CECZ
188
PSOLZD
PSOL2R
PS L3L
LEOTIS
0GOPT
OMGOIPT
PSOL3P
PSOLDS
PSOL3D
LECTIB
LEQTIB
-LUDAP R
- REBEMAX
LUE LP
OMGOPT
FIGURE Dl (CONTINUED)
II
- -' I
~" "'
111~
189
REFERENCES
1.
A. L. Schor and N. E. Todreas., "A Four-Equation Two-Phase Flow
Model for Sodium Boiling Simulation of UMFBR Assemblies",
MIT Energy Laboratory Report: MIT-EL 82-039.
2.
H. C. No and M. S. Kazimi,"An Investigation of the Physical and
Numerical Foundations of Two-Fluid Representation of
Sodium Boiling",MIT Energy Laboratory Report:MIT-EL
83-003.
3.
4.
R. G. Zielinski and M. S. Kazimi, "Development of Models for the
Two-Dimensional ,Two-Fluid Code for Sodium Boiling
NATOF-2D",MIT Energy Laboratory Report:MIT-EL 81-030.
Kang Y. Hul, "Simulation of Sodium Boiling Experiments with Thermit
Sodium Version",MIT Energy Laboratory Report:MIT-EL
82-023.
5.
A. L. Schor, "Numerical Method for Fuel (Heater) Rod Conduction",
Course Notes, Nuclear Engineering Department,MIT.
6.
P. W. Garrison,R. H. Morris and B. H. Montgcmery,"Dryout Measurements for Sodium Natural Convection In a Vertical
7.
J. Costa
Channel", Oak Ridge National Laboratory;ORNL/IM-7018.
and P. Charlety,"Forced Convection Boiling of Sodium
In a Narrow Channel",Centre D'Etude Nuclearies,Grenoble,
France. ASME,Liquid-Metal Heat Transfer & Fluid D namics,
7
Winter Annual Meeting,New York,NY, Nov. 19 0,p 172-8.
8.
A. L. Schor, "Numerical Method in THERMIT/TRAC",
9.
Nuclear Engineering Department,MIT.
M. S. Kazimi,"Flow Loops",Course Notes, Nuclear Engineering
Course Notes,
Department,MIT.
10.
H. F. Creveling et al,"Stability Characteristics of a singlePhase Free Convetion Loop",J.Fluid Mech. 67(1975)
65-84.
11.
E. Wacholder, S. Kaizerman and E. Elias,"Numerical Analysis
of the Stability and transient Behavior of Natural
Convection Loops", Letters in Applied and Engineering
Sciences, Pergamon Press.
1
190
12.
Y. Zvirin and Y. Rabinoviz, "On The Behavior of Natural Circulation
Loops in Parallel Channels", Mechanical Engineering
Department,Technion-Israel Institute of Technology,
Haifa, Israel.
13.
S. Kaizerman,E. Wacholder and E. Elias,"Stability and Transient
Behavior of a Vertical Toroidal Thermosyphon" ,Nuclear
Engineering Department, Technion-Israel Institute of
Technology,Haifa, Israel.
14.
Y. Zvirin,"A Review of Natural Circulation Loops in PWR and Other
Systems", Nuclear Engineering and Design 67(1981) 203225.
15.
Roache, C=.putational Fluid Dyrzmics, Harmosa
Publishers,
Albuquerque.
16.
S. V. Patankar, Numerical Heat Transfer and Fluid Flow , McGrawHill (1980).
17.
El-Wakil, Nuclear Power Engineering, McGraw-Hill (1962)
18.
Yehia F. Khalil, "A Sodium Experiment Simulation Using 4-Equation
Two-Phase,One-Dimensional Flow Model",Course 22.904
(Spring 1983),
MIT.
Notes MIT Course 22.43 (Spring 1983).
19
Lecture
20.
Oluwole A. Adekugbe,"Assembly Flow Distribution Control", MIT
Course 22.32 Project, (Fall 1983)
21.
0. Adekugbe and A. L. Schor,MIT Sodium Boiling Project,Progress
Report: July 1
- September 30,1983. Nuclear Enginee-
ring Department,MIT;Oak Ridge National Laboratory
Oak Ridge,TN
22.
Charles R. Bell,"Breeder Reactor Safety -
Modeling the Impossible",
Los Alamos Science.
23.
R. C. Rosenberg and D. C. Karnopp, Introduction to Physical System
Dynamics,
McGraw-Hill, 1983.
24.
Rohsenow and Choi, Heat Mass and Momentum Transfer, Prentice-Hall.
25.
Personal Discussion with Dr. Allen Levine of Oak Ridge National
Laboratory, October 1983.