THE PROPAGATION OF ZIRCONIUM RUNAWAY OXIDATION
OF SPENT FUEL CLADDING FOLLOWING LOSS
OF WATER DURING STORAGE
by
Nicola Anthony Pisano
B.S.
University of Notre Dame, Notre Dame
(1980)
Submitted to the Department of
Nuclear Engineering
in Partial Fulfillment of the
Requirements of the
Degree of
MASTER OF SCIENCE IN NUCLEAR ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 1982
©) Massachusetts
Institute of Technology 1982
Signature of Author
Department of Nuclear Engineering
August 1, 1982
Certified by_
Thesis Supervisor
Certified by
I
Thesis-Reader
K-)
'Acc
e
p
y
Chairman, De
#
JAN I
W'o'nmittee on Graduate Students
16i
"
Archives
THE PROPAGATION OF ZIRCONIUM RUNAWAY OXIDATION
OF SPENT FUEL CLADDING FOLLOWING LOSS
OF WATER DURING STORAGE
by
Nicola Anthony Pisano
Submitted to the Department of Nuclear Engineering
on August 1, 1982 in partial fulfillment of the
requirements for the Degree of Master of Science in
Nuclear Engineering
ABSTRACT
A phenomenological model of kinetic and diffusion controlled reaction is used in the modelling of the energetic
oxidation of zirconium cladding at elevated temperatures in
air at one atmosphere. Employing models developed in NUREG/
CR0649, a numerical analysis of the propagation of vigorous
oxidation in a spent fuel storage pool following hypothetical drainage is performed. Energy transfer via conventional
heat transfer mechanisms, levitation and convection of particulate, and combustion of zirconium vapor is addressed.
Numerical simulation of data obtained at Sandia National
Laboratories is performed.
The results of the pool-wide analysis indicate that
propagation of runaway zirconium oxidation from recently
discharged to older fuel assemblies is highly dependent on
the fuel storage configuration and minimum spent fuel decay
time. It is shown that the minimum decay time to prevent
carryover from spent fuel stored adjacent to recently discharged fuel (\90 day decay time) is approximately 2 years.
Propagation via zirconium particulate is found to be a
minor effect. It is concluded that the generation of zirconium vapor is highly unlikely, and that at worst it could
only intensify the oxidation reaction within a given fuel
holder, without leading to widespread burning. Numerical
simulation of experimental data yields unsatisfactory results, for a variety of reasons related to both computer
modelling and experiment conditions.
Thesis Supervisor: Dr. Frederick Best
Title: Post Doctoral Research Associate
Department of Nuclear Engineering
DEDICATION
To the memory of my father
NICOLA PISANO,
4 June 1921 -
SR.
6 July 1982
ACKNOWLEDGEMENTS
The author wishes to thank Dr. Allen Benjamin of Sandia
National Laboratories who acted as sponsor and coordinator
for the research performed in this report, Dr. K. T. Stalker
of Sandia
National Laboratories for his assistance in pro-
viding the experimental data of Chapter 5, Dr. I-Wei Chen
and Dr. John Meyer for their assistance in various areas of
computer modelling, Dr. Frederick Best for his indefatigable
patience and guidance in this research, my wife, Raisa
for her assistance in preparing the illustrations in this
report, and Wynter Snow for a timely and expert typing service.
5
TABLE OF CONTENTS
page
ABSTRACT.........
2
DEDICATION.......
3
ACKNOWLEDGEMENTS.
4
TABLE OF CONTENTS
5
LIST OF FIGURES..
8
10
LIST OF TABLES...
CHAPTER 1. INTRODUCTION.....
1.1 Preface.............
1.2 Background..........
Problem Statement...
Summary of Contents.
CHAPTER 2. POOL-WID E PROPAGA TION OF ZI RCONIUM
BURNING. .........
21
21
2.1 Description of SFUEL Program.
26
Zircaloy Clad Oxidation.......
Extension of SFUEL Program to Assess
0
Pool-Wide Burning............. .......
..
0.....0
0...
2.3.1 Inter-Holder Conduction.
2.3.2
Revised Fluid Dynamics Calculations......
2.3.3 Maximum Driving Force Approximations .....
2.3.4
Revision of Clad Thickness Model.........
2.3.5
Limit to Maximum Rate of Change of
Temperature..............................
TABLE OF CONTENTS (Continued)
page
2.3.6 Fuel Relocation Models...................
2.3.7 Computer Operating System Changes........
2.4 Modified SFUEL Analysis of Burning
Propagation....................................
2.4.1 Characteristics of Revised SFUEL Code....
2.4.2 Pool-Wide Propagation Results............
2.5 Propagation of Zirconium Burning Via
Conventional Heat Transfer Mechanisms:
Summary........................................
CHAPTER 3. PROPAGATION OF RUNAWAY OXIDATION VIA
PARTICULATE TRANSPORT.......................
83
3.1 Introduction...................................
83
3.2 Particulate Propagation Potential Algorithm....
85
3.2.1 PARTICLE Model Description...............
85
3.2.2 Analytical Evaluation of PARTICLE
Results..................................
93
3.3 Energy Transfer Via Burning Particulate:
Results........................................
97
3.4 Propagation Via Particulate: Summary...........
101
CHAPTE
4. ZIRCONIUM VAPOR ANALYSIS....................
105
4.
Mechanisms for Formulation of Zirconium
Vapor..........................................
105
4.2
Quantitative Analysis of Vapor Generation......
110
4.3
Zirconium Vapor Analysis: Summary..............
112
TABLE OF CONTENTS (Continued-3)
page
CHAPTER 5. COMPUTER SIMULATION OF SANDIA
EXPERIMENTS.................................
113
5.1 Introduction...................................
113
5.2 Sandia Experiments on Zirconium Burning........
113
5.2.1 Experimental Configuration...............
113
5.2.2 Experimental Data: Air Test #4...........
115
5.3 Computer Simulation of Experimental Tests......
121
5.3.1 Introduction............................. 121
5.3.2 Experimental Simulation: CLAD............ 121
5.4 Experiment Simulation: Results.................
126
5.5 Experiment Simulation: Summary.................
126
CHAPTER 6. PROPAGATION OF ZIRCONIUM BURNING:
CONCLUSIONS.................................
130
APPENDIX A. HEAT TRANSFER COEFFICIENTS AND SKIN
FRICTION CORRELATIONS USED IN SFUEL........
133
APPENDIX B. SFUELlW INPUT, OUTPUT AND PROGRAM
LISTING.................................... 136
APPENDIX C. PARTICLE INPUT, OUTPUT AND PROGRAM
LISTING................................... . 173
APPENDIX D. THE VAPOR PROGRAM, INPUT, OUTPUT
AND PROGRAM LISTING........................
182
APPENDIX D. CLAD INPUT, OUTPUT AND PROGRAM LISTING.....
196
REFERENCES.............................................
217
LIST OF FIGURES
page
CHAPTER 1
1.1
Typical Design Characteristics of PWR Fuel
Analyzed in this Report............................
17
CHAPTER 2
2.1
Elevation View of Spent Fuel Pool Showing
Modes of Heat Transfer Addressed in SFUEL........... 22
2.2
Spent Fuel Storage Racks Analyzed in this
Report............................................
24
2.3
Correlations for Zirconium Oxidation in Air.......
28
2.4
Cubic Plot of Oxidation of Zirconium -1.5%
Tin Alloy at 600* -800C...........................
30
2.5
Cubic Plot of Oxidation of Zirconium -1.5%
...........................31
Tin Alloy at 8250 -900C
2.6
Finite Difference Mesh for One-Dimensional
Fluid Flow..........................................
41
Maximum Clad Temperature for Original and
Revised SFUEL.................................. .....
57
Comparison of Original and Revised Codes
during Runaway Oxidation............................
61
Peak Clad Temperature During Runaway..............
63
2.10 Comparison of Peak Clad Temperature, Melt
Options 1 and 4.....................................
64
2.11 Peak Clad Temperatures in Pool Sections 6
....................
and 5, Melt Option 4...........
67
2.12 Peak Clad Temperatures in Pool Sections 6
....... .......
and 5, Melt Option 1.........
68
2.7
2.8
2.9
...
2.13 Temperature in Pool Section 6, Axial Location 16...........................................
LIST OF FIGURES (Continued)
page
2.14 Temperature in Pool Section 5, Axial Location 16...........................................
2.15 Temperature in Pool Section 4, Axial Location 16...........................................
2.16 Temperature in Pool Section 3, Axial Location 16...........................................
2.17 Composite View of Pool-Wide Propagation
Results...........................................
2.18 Pool-Wide Propagation Sensitivity, Case 3.........
2.19 Results of Propagation Sensitivity Analysis,
High Density Storage..............................
CHAPTER 3
102
3.1
Spatial Variation of Effective Heat Flux..........
3.2
Temperature Rise due to (q/A)f.................. 103
CHAPTER 4
4.1
Postulated Mechanism of Zirconium Vapor
Generation........................................
109
CHAPTER 5
5.1
Configuration of Zirconium Burning Apparatus......
114
5.2
Experimental Data for Zircaloy-2 Oxidation
in Air............................................
117
Experimental Data for Zircaloy-2 Oxidation
in Air............................................
118
5.4
Experiment Simulation Model.......................
123
5.5
Comparison of Pre-Oxidation Heat-Up of
Experimental Assembly.............................
127
5.3
LIST OF TABLES
page
CHAPTER 1
1-1
Decay Heat Generation Rates versus Time
from Discharge....................................
18
CHAPTER 2
2-1
2-2
Oscillatory Instability of Interholder
Space Interface Temperatures......................
39
PWR Spent Fuel 17 x 17 Array, Cylindrical
Storage Rack Configuration, Full Discharge
Loading, 3" Baseplate Hole........................
58
2-3
Steady-State Nitrogen Mass Flow Rates............... 59
2-4
Comparison of SFUEL Results Under Runaway
Conditions........................................
62
2-5
Maximum Clad Node Temperature for Different
Fuel Relocation Options (Modified SFUEL)............ 66
2-6
Decay Heats Used in Assessing Pool-Wide
Propagation.......................................
77
Assessment of Sensitivity of Pool-Wide
Propagation.......................................
79
2-7
CHAPTER 3
3-1
Terminal Velocity of Settling Particulates.......... 91
3-2
Particle Size Frequency Distribution................ 95
3-3
Input Values to the PARTICLE Program................. 98
3-4
Particle Distributions for Ramp in Figure 2.3.....
99
3-5
Effective Energy Fluxes to Horizontal
Structures........................................
100
3-6
Thermophysical Properties of 5% Cr-Steel........... 100
11
LIST OF TABLES
(Continued)
page
CHAPTER 4
4-1
Thermophysical Properties of Zirconium and
its Oxide......................................... 107
CHAPTER 5
5-1
5-2
Experimental Data for Zircaloy Oxidation
in Air............................................
119
Comparison of Pre-Oxidation Assembly Heat-Up......
128
12
CHAPTER 1
INTRODUCTION
1.1
PREFACE
The majority of commerical power reactors currently in
operation were designed with the expectation of a viable
fuel reprocessing system being available.
In consonance
with this fuel life-cycle, spent fuel storage pools were
designed to accommodate the amount of spent fuel expected
to be awaiting shipment to a reprocessing facility.
The
prohibition on spent fuel reprocessing in this country has
resulted in the accumulation of large inventories of spent
Light Water Reactor (LWR) fuel.
While the fate of fuel re-
processing is as yet undecided, the need for increased storage capacity both on-site and away-from-reactor has continued to grow.
The interim solution of this dilemma has
been a re-design of the spent fuel pools into longer-term
storage facilities.
This has been achieved primarily by
re-designing the fuel storage racks, thereby allowing for
denser packing.
Public acceptance of high density storage arrangements
and licensing of interim on-site fuel storage has been difficult to achieve.
Demands have been made to assure that
the modified storage arrangements do not increase the health
hazard to on-site workers or the community.
At hearings conducted to determine whether the reactor
licensees should be granted permission to modify their spent
fuel storage pools to accommodate higher density storage
arrangements, a common concern is in regard to the potential
risk to the public if the water were inadvertently drained
from the pool.
It is known that, at elevated temperatures, zirconiumthe major constituent of LWR cladding-undergoes vigorous
exothermic oxidation upon exposure to air.
One scenario
advanced by citizen-advocate groups contends that a loss of
pool water could lead to a widespread burning of the zirconium cladding, eventually engulfing the entire pool area.
The radioactive fission product release from the degraded
fuel rods would then be quite large.
The purpose of the research described in this report
is to examine, refine, and extend existing analyses of this
topic and to investigate aspects of the problem not previously addressed.
1.2
BACKGROUND
The investigation by A. S. Benjamin, et al.,
(Bl),
NUREG/CR0649, indicates that a self-sustaining clad oxidation reaction could initiate if a high density storage configuration were to lose its coolant, provided that some
fuel had been recently discharged from the reactor.
The
analysis addresses various pool racking arrangements, fuel
types, and fuel pool building characteristics.
The report
indicates which storage configurations are likely to result
in clad temperatures susceptible to vigorous exothermic oxidation.
Several recommendations are made relating to stor-
age rack design and minimum cooling times required prior to
fuel storage in the various racking configurations.
However,
the analysis does not predict whether a self-sustaining zirconium cladding oxidation reaction, once initiated, will
either extinguish within a localized region, or propagate to
other sections of the fuel pool.
The computer code developed to analyze the spent fuel
behavior up to the initiation of runaway oxidation, SFUEL,
has as its parameters of primary importance the fuel-cladding
temperature and extent of Zircaloy oxidation.
The code is
based on phenomenological models for zirconium clad oxidation.
In particular, the transient non-isothermal burning
of zirconium in gas of variable oxygen mass fraction is
modelled by a parabolic oxidation kinetics formula derived
from isothermal constant oxygen mass fraction experimental
data.
A search of the current literature reveals that there
is a dearth of information on zirconium oxidation in a dry
air environment, and all of that which exists is obtained
under isothermal conditions.
Additionally, it is to be
noted that the Sandia study is the only one available in
the open literature which addresses numerical simulation of
large-scale zirconium oxidation in a dry air environment.
15
1.3
PROBLEM STATEMENT
The scenario for burning propagation of zirconium clad-
ding, and the associated processes investigated in this report, may be outlined as follows:
1) At the initiation of the accident, the spent fuel
pool is instantaneously drained of coolant.
2) The resulting heat-up of the fuel rods,due to the
reduced heat transfer in the absence of liquid
coolant,initiates a natural convection circulation
of air throughout the fuel pool.
3a) For low density storage arrangements, the natural
convection cooling is sufficient to allow the fuel
rods to achieve a steady-state temperature below
the runaway oxidation value.
3b) For certain high density storage arrangements, the
natural circulation is insufficient to achieve a
steady-state temperature below the value for vigorous oxidation ( %800*C).
4) The oxidation reaction energy leads to even more
vigorous oxidation, potentially resulting in local
clad melting, vaporization and spallation with subsequent levitation of particulate.
The natural convection heat-up of the fuel is
in the study mentioned earlier.
However,
analyzed
the SFUEL code was
limited by numeric stability,and other problems, to calculations prior to the initiation of runaway oxidation.
The
research described in this report details steps taken to extend the code to analyze the propagation of the oxidation
reaction throughout the fuel pool.
Propagation in this
analysis occurs by conventional heat transfer mechanisms:
conduction, convection, and radiation.
In addition, two
other propagation mechanisms, involving simultaneous mass
transfer as cited in (4) above are examined.
In performing
these analyses, a host of phenomena occurring concurrently
in the fuel pool and impacting on the propagation potential
needed to be examined.
Typical design characteristics for
fuel analyzed in this report are shown in Figure 1.1, while
the decay heats for discharged fuel are given in Table 1-1.
Experiments have been performed at Sandia National Laboratories to observe the behavior of heated Zircaloy-2 clad
exposed to air in a forced convection environment.
Concur-
rently, a computer code employing the oxidation models in
SFUEL has been developed to allow analytical prediction of
the experimental results.
1.4
SUMMARY OF CONTENTS
As outlined in the Problem Statement, the SFUEL code
has been revised and extended to analyze pool wide propagation of zirconium runaway oxidation.
This objective neces-
sitated the refinement of many of the SFUEL models, as well
176.8"
FUEL
ROD
PWR ASSEMBLY
GRID
Figure 1.1
Typical Design Characteristics of PWR Fuel
Analyzed in this Report (after NUREG/CRO649).
Design Properties of Fuel Assemblies Used in the Analysis
Older PWR
Newer PWR
15 x 15
17 x 17
Number of fuel rods per assembly
Numer of non-fuel rods per assembly
208
264
17
25
Active fuel height (in.)
Rod center-to-center pitch (in.)
144
144
0.558
0.496
0.420
0.026
0.374
0.023
0.456
0.461
177
193
Rod array
Fuel rod outside diameter (in.)
Clad thickness (in.)
Channel thickness (in.)
Metric tons uranium per assembly
(MTU)
Number of assemblies per core,
typical reactors
18
TABLE l-l*
Thermal Decay Power of PWR Spent Fuel as a
Function of Decay Time and Discharge Cycle
Standard Case:
3 cycles @ 3.3% wt. enrichment.
Operating power = 37.3 MW/MTU.
Total Burnup = 33,000 MWD/MTU.
30-day down time between cycles.
35-day down time within cycles.
295 operating days per cycle.
Decay Power, KW/MTU
Cycle
1
Cycle
2
Cycle
3
10 days
75.4
81.1
86.6
30 days
43.4
48.6
53.2
90 days
21.3
26.0
30.0
180 days
11.4
15.6
19.2
Decay Time
11.0
1 year
5.05
8.20
2 years
2.22
4.07
5.90
3 years
1.25
2.46
3.76
5 years
0.607
1.30
2.12
10 years
0.368
0.779
1.28
*After NUREG/CRO649.
as incorporating several new analytic capabilities.
Fea-
tures of the current SFUEL code which were examined and/or
revised were:
1) mass flow calculations
2) oxidation kinetics models
3) all modes of heat transfer
4) methods of fuel rod and fluid temperature calculation
The revised code was validated for low temperature
performance by comparison with the original SFUEL results.
The SFUEL program, revisions and results are explicated in
Chapter 2.
The spallation and levitation of burning zirconium particulate entails a two-part analysis.
The first portion in-
volves the numerical solution of the equations for particle
temperature and transport.
The second portion employs a
semi-infinite body conduction model to assess the ability of
the burning particle to ignite a receiving surface.
The
model equations and evaluation algorithm are described in
Chapter 3.
The particulate model requires a number of assumptions,
yet for these cases, quantitative as well as qualitative information can be obtained.
However, for the zirconium vapor-
ization analysis, there are too many unknowns to allow a
quantitative assessment at this time.
Rather, the zirconium
vaporization analysis, described in Chapter 4, gives the
conservative estimates for the intensity of propagation of
vigorous zirconium burning via this mechanism relative to
those discussed above.
The fifth chapter of this report concerns the simulation of the experiments conducted at Sandia.
The computer
code is described and predictions are compared to the experimental findings.
Discrepancies are explored and ramifica-
tions are discussed.
The final chapter of this report summarizes the significant results of the previous chapters.
The relationships
of the various propagating mechanisms are discussed and the
results superimposed to give a global view of the pool-wide
behavior.
The relevant observations obtained from the ex-
periment numerical simulation are applied, and an overall
summary is given for the propagation of zirconium runaway
oxidation in a drained spent fuel pool.
CHAPTER 2
POOL-WIDE PROPAGATION OF ZIRCONIUM BURNING
2.1
DESCRIPTION OF SFUEL PROGRAM
The SFUEL computer code developed at Sandia is a two-
dimensional model of a spent fuel pool.
The model
corresponds to the elevation view of the fuel pool, as shown
in Fig. 2.1.
The code models six adjacent pool sections,
where the first section is bounded by the pool liner and the
sixth section corresponds to one-half of the center pool section.
The primary assumptions embodied in the pool analysis
are as follows:
1) The water drains instantaneously, leaving the pool
completely devoid of water.
2) The geometry of the fuel assemblies and racks remains
undistorted.
3) Temperature variations across the fuel rods are neglected, and all rods in a particular pool section have the
same axial temperature distribution.
4) The air flow patterns are one-dimensional and involve
a Boussinesq approximation.
5) Radiation view factors are based on projected areas.
All radiating surfaces are gray bodies.
6) All decay heat emanates from the fuel rods.
Oxida-
tion of fuel pool structural materials is not addressed.
HOLDER
CHANNEL
(IF PRESENT)
qdec
chem
- DECAY HEAT INPUT TO FUEL RODS
- CHEMICAL OXIDATION HEAT INPUT
TO FUEL RODS
- AXIAL CONDUCTION IN FUEL RODS
- RADIATION FROM OUTERMOST HOLDER
TO SIDEWALL LINER
CONVECTION FROM AIR STREAM 3 TO
qa I
3 1 SIDEWALL LINER
qlwW - CONDUCTION INTO CONCRETE SIDEWALL
gh
RADIATION FROM RODS TO CHANNEL WALL
-
CONVECTION FROM RODS TO AIR STREAM 1
ach
-
CONVECT ION FROM AIR STREAM 1 TO
CHANNEL WALL
RADIATION FROM CHANNEL WALL TO HOLDER
qca
-
CONVECTION FROM CHANNEL WALL TO AIR
STREAM 2
CONVECTION FROM AIR STREAM 2 TO HOLDER
ghh
.
RADIATION FROM HOLDER TO ADJACENT HOLDER
qha
- CONVECTION FROM HOLDER TO AIR STREAM 3
qra I
qa 1c
a2
qa h'3
ty
1
-RADIATION
FROM UPPER TIE PLATE
I - RADIATION FROM LOWER TIE PLATE
2 2 TO FLOOR LINER
CONVECTION FROM LOWER TIE
qta2a4
PLATE TO AIR STREAM 4
CONVECTION FROM AIR STREAM 4
q12
TO FLOOR LINER
CONDUCTION INTO CONCRETE FLOOR
ci2 -
CONVECTION FROM AIR STREAM 3 TO ADJACENT
HOLDER
Figure 2.1
Elevation View of Spent Fuel Pool Identifying
Heat Transfer Modes Considered in the SFUEL
Program (after NUREG/CRO649).
7) The spent fuel is arranged such as to have the hottest elements in the center pool section, and the cooler
elements progressively toward the ends of the pool.
8) Axial heat conduction is negligible for channel
walls, holder walls and liner.
Axial conduction is consid-
ered for the fuel rods only.
9) The spaces between adjacent racks are closed to air
flow; heat transmission through these regions is by conduction through stagnant air.
As observed in NUREG/CR0649, the assumption that temperature variations within a pool section are negligible (3) is
found to be adequate, since the heatup time is on the order
of hours.
When vigorous oxidation is in progress, the as-
sumption may still be assumed to hold on an assembly-byassembly basis within a given pool section.
This topic is
discussed later in this chapter.
The assumption that the inter-holder spaces are closed
to air flow is a conservative representation of the fact that
the air flow in these spaces is retarded by support structures.
Typical spent fuel holder designs analyzed in this
report are shown in Fig. 2.2.
The philosophy adopted in this research is to investigate the propagation of burning in configurations where it is
thought most likely to occur and not duplicate the results of
the earlier (NUREG/CR0649) analysis.
For similar reasons,
the analyses performed here are carried out under the assumption that the fuel pool building ventilation rate remains
12.75"
HIGH DENSITY
CYLINDRICAL
SQUARE
0.25" SS
12.5"1
8.4"
8.4"1
0.13"
SS
0.25"
Ss 5
3. 5 OPEN
O31
HIGH DENSITY
Figure 2.2
9.5
8.4"
CYLINDRICAL
SQUARE
Spent Fuel Storage Racks Analyzed in this Report
25
constant throughout the duration of the accident, such that
the room air conditions above the pool remain at the ambient
those external to the fuel pool building.
conditions, i.e.,
As suggested in Assumption (4) above, the air flowing
through the heated channels is treated using a Boussinesq approximation.
In essence, the air is a thermally-expandable
but incompressible flow.
The fluid dynamics calculations are
thus performed in a steady-state fashion, where the compressibility effects in the energy equation and the mass
accumulation term in the continuity term are neglected.
Thus, the rigorous set of fluid conservation equations
(R1),
Eqn. 2.1.
1 Dp +
2.la
p Dt + VV = 0
PC
DT
(Pvx) =-7(pvxV)
[
= V(kVT)
D
+ [y (
-
VP +
V 6V
6
.
DP
+ q + TSP
5v
P[+
[2
2.lb
+ p
2 6v
+
6v
5v
)~
6v x
( 6v+
+
+ 6
+ X
l
2.lc
and similar equations of the form 2.lc in the other two coordinate directions are reduced to the one-dimensional flow
equations (2.2):
6u
6x
0
2.2a
ST
6ST
-
w
Tt
f)
2.2b
= pg + orifice and friction pressure loss
pg+6r±.c
2.2c
P~
cpupcpgg+
qwhere q= hA(Tw
-T)22
In equations 2.2 the molecular effects of heat and momentum
transfer are described by phenomenological engineering models.
The SFUEL Program is formulated as a semi-explicit code.
In particular the fluid dynamics calculations are performed
in an implicit fashion, employing a Newton-Raphson iterative
method to obtain consistent values of temperatures and mass
flows.
All other parameters in the code are calculated in a
fully explicit manner.
The requirement for an explicit re-
presentation of the fuel clad temperature arises from the
highly non-linear character of the zirconium oxidation reaction model.
2.2
ZIRCALOY CLAD OXIDATION
Oxidation of zirconium in air at elevated temperatures
occurs by the following highly energetic exothermic reaction:
02 + Zr
-*
ZrO
2
liberating approximately 262 kcal per mole of Zr.
The oxida-
tion reaction may be either kinetically controlled or
diffusively controlled.
The kinetic-controlled reaction is
modelled using a parabolic reaction rate law:
27
2w
where
dw
= KO exp (-E/RT)
2.3
2
w = weight gain per unit surface area (mgO2 /cm
t = time
(seconds)
Ea = activation energy (cal/mole)
R = gas constant
(*K)
KO = reaction rate constant ([mg/cm2 2/s)
T = temperature (*K)
The parabolic kinetic model is derived from experimental
data obtained under isothermal, constant free stream oxygen
mass fraction conditions.
Thus, the parabolic formulation
inherently incorporates these assumptions.
As suggested
earlier, the actual processes are neither isothermal nor occur with constant oxygen mass fraction.
The empirically
derived coefficients for Eqn.2.3 are:
KO=1.15x103
(mgO2 /cm2 ) 2/s
Ea= 2 7 4 3 0 (cal/mole) (T<9200 C)
K 0 =5.76x103
(mgO2 /cm2 ) 2/s
Ea =52990 (cal/mole) (920 0 C<T11550 c)
K 0 =6.20x103
(mgO2 /cm2 ) 2/s
Ea =29077 (cal/mole) (T>1155 0 C)
This data is presented graphically in Fig. 2.3.
The data of
Hayes and Roberson (Hl) and White (Wl) is obtained for pure
zirconium oxidation in moist and dry air, respectively.
The
data of Lestikow (Ll) is obtained from oxidizing heated
Zircaloy-4 tubes in dry air.
There is little data available
+6
+4
6. 20 x 10 4 exp (-29077/RT)
+2
U
0
C)
0
5. 76 x 107 exp (-52990/RT)
-2
E
MONO-TETRAGONAL
PHASE CHANGE OF
ZrO
-4
2
1. 15 x 10
3
exp (-27340/RT)
-6
a-
-8
B
PH ASE CHANGE
OF Zr -02 SOLID
SOL UTIONS
0 LEISTIKOW
-10
-12
(1975)
(1967)
0
WHITE
>
HAYES AND ROBERSON (1945)
5I
I
8
5
7
8
6
9
10 /T
Figure 2.3
10
11
12
(*K)
Correlations for Zirconium Oxidation in Air
(After NUREG/CR0649)
13
in the open literature which has been derived for Zircaloy-2,
the alloy from which reactor-grade LWR fuel cladding is made.
It is suggested in Biederman, et al.,
(B2) that the alloying
of zirconium has a profound effect on the oxygen diffusivity
in the oxide phase.
The typical composition range for
Zircaloy-2 according to American specification PDS.11538-4
(Ml) is given in weight per cent as:
0.03-0.08
Tin
1.3 -1.6
Nickel
Iron
0.07-0.20
Average (Fe+Cr+Ni) 0.23-0.32
Zirconium
Chromium 0.05-0.16
0.68-0.77
Investigation of high temperature oxidation of a 1.5 wt % tin
alloy by Mallett and Albrecht (1955) (M2) showed that the
oxidation rate followed a cubic law, Eqn.2.4, in the temperature range 600-900*C at 1 atmosphere oxygen (Figures 2.4 and
2.5):
2dw
3Wdt = K0 exp( Ea/RT)
2.4
where K 0 = reaction rate constant [(ml/cm2 3S]
Ea = activation energy (cal/mole)
and
K0 = 5.34x10
K
= 87.2
4
Ea = 38400±1100
600 < T < 800 0 C
Ea = 22600±1400
825 < T < 900 0 C
A study of oxidation of zirconium and zircaloy in dry
0.50
0.45
0.40
U
0.35
0.30
0.25
Co
0
0.20
0.15
>1
X
0
0.10
0.05
0
Figure 2.4
5
10
15
20
Time,
min
25
30
35
Cubic Plot of Oxidation of Zirconium-1.5% Tin
Alloy at 6000 - 800*C (after Reference M2).
38
36
34
32
CN
30
>428
'-
26
900 0 C
24
22
20
-2
91
18
Q)
T
o
m16
14
U
S12
13
10
o
8
6
4
2
Ally82*
a
0
Figure 2.5
20
40
nd
00*
60
80
Time,
min
(ate
100
Reerece
120
2)
140
Cubic Plot of Oxidation of Zirconium- 1.5% Tin
Alloy at 825' and 900'C (after Reference M2).
air performed by Kendall (1955) (Kl), covered the temperature range of 500-700 0 C.
His study indicated that the
reaction proceeds in two stages:
initially an approximately
cubic dependence with time, and at higher exposures, a linear
The rate constants calculated from his
function of time.
data and applicable with equation 2.4 for the initial reaction are:
Zircaloy-2
KO = 1.1x109
Zirconium
KO
=
1.8x9
(mg/cm2) 3/hr
Ea = 39400 cal/mole
(mg/cm2 3/hr
Ea = 41400 cal/mole
In the linear regime which conforms to Eqn.2.5:
dw
K exp(-Ea/RT)
2.5
the appropriate constants obtained from his data are:
Zircaloy-2
K
= 8.5x106
(mg/cm2)/hr
Ea = 31000 cal/mole
Zirconium
KO = 7.4x6
(mg/cm2)/hr
Ea = 29800 cal/mole
In the research performed in this report, only the first
three correlations have been implemented.
Additional tests
performed by Mallett and Albrecht for the oxidation of Zr -2.5
wt % Sn alloy in 1 atmosphere of 02 showed a parabolic rate
0
dependence in the temperature range of 500-900 C.
While
there is evidence that alloying components have a profound
effect on the oxidation rate, the parabolic formulation was
employed because it is the only formulation for which high
temperature (>900*C) data are available.
Biederman, et al.,
(B2) showed that the empirical formu-
lation Eqn.2.3 given above may be used for the calculation of
oxide thickness under transient heating conditions (assuming
02 concentration at the wall constant) by dividing the time
interval into a series of small intervals and calculating the
oxide formation between specified initial and final temperatures.
The calculational method of Biederman, et al., does
not attempt to reconcile the energy liberated during the interval by chemical oxidation with the specified temperature
at the end of the interval.
All of the codes proposed in the
present report do account for the synergistic effect of chemical reaction energy and temperature.
The spent fuel holders are closed channels in which air
is admitted at the base and exits at the top of the assembly.
The oxygen mass fraction of the gases is depleted as the gas
moves up the channel.
At some point in the channel, the
chemical oxidation reaction may become limited by the quantity of 02 available for reaction; this is a diffusion-limited
condition.
components:
In this regime, the reaction is limited by two
1) the ability of the oxygen to diffuse through
the gas to the oxide surface, and 2) the ability of the oxygen to diffuse through the oxide layer to the unreacted
zirconium metal.
by Eqn.2.6:
This is represented for a planar geometry
d W0
C
dW 2 =
where
W
=
02
W
- C
-
2
( O
2.6a
2
weight gain of oxygen per unit surface area
2
(mgO2 /cm )
C 0 2 (x),
C0
2(o)
=
Oxygen concentrations at the oxidation boundary and free stream,
respectively (mgO2 /cm )
HT = overall mass transfer coefficient
and HT is defined as:
HT
where
H
D
+h
2.6b
D
HD = mass transfer coefficient for 02 through the
oxide layer (cm/s)
hD = mass transfer coefficient for 02 through the gas
to clad outer surface (cm/s)
The value of HD is determined primarily by the diffusivity of
oxygen in zirconium oxidation.
Depending on temperature, the
zirconium dioxide can be in either a monoclinic (800
0 C)
or
tetragonal (1200*C) crystalline form, with either preferential
or non-preferential orientation.
The diffusivity of oxygen is
profoundly influenced by these factors; however, because of
this complexity (and as a conservative assumption), no attempt has been made to model the oxygen concentration
The value of hD is deter-
gradient through the oxide layer.
mined for laminar flow using the Chilton-Colburn analogy (C2):
f
h
.
where
H
2.7a
T
D
IH
Pr2/
PCP
2.7b
p
hD
D
2/
2.7c
(Sc)
h = heat transfer coefficient (W/cm2
p = gas density (gm/cm3)
c
= gas specific heat
V = gas velocity
(J/gm *K)
(cm/s)
Pr = Prandtl number = v/a
Sc = Schmidt number = v/D
f = friction factor
and D = diffusivity of oxygen in air.
The values of Schmidt number are well documented for air, and
the heat transfer coefficient is
lations of Appendix A.
of oxygen in
For lack of data on the diffusivity
the various forms of zirconium dioxide,
transfer coefficient is
the mass
composed entirely of the oxygen
through air diffusion portion.
sponds
calculated using the corre-
This approximation corre-
to the physical situation of the oxide layer spalling
as quickly as it appears.
The reaction rate used in the code
is determined by the relative magnitudes of the values calculated for the reaction rate:
if the rate calculated by using
Eqn.2.3 is greater than the result of Eqn.2.6, the reaction
is kinetically controlled, otherwise it is diffusion limited.
The oxidation thickness in either reaction regime is
calculated explicitly using the temperature of the clad at
the beginning of the time interval for which the change in
thickness is to be calculated, thus:
W=
where
[ dt(At)
= KV
0 exp (-E/RT) A+
p2
p = density of Zr
K0 =
Aw
22.8
(gm/cm3)
(mgZr/cm2 ) 2/s = Kxf
f = stoichiometric ratio of Zr to 02 in ZrO
2
Aw = reduction in clad thickness at the beginning of
the time step (cm)
However, since for the non-isothermal case the temperature is
also a function of time, Eqn.2.3 should be:
w =
J-t
K
exp(-E/RT(t))
dt
dt
/
+ Aw2},
2.9
Equation 2.9 must be solved using non-linear techniques,
since neither the new oxidation thickness nor temperature are
known.
Additionally,
the clad temperature is
a function of
decay heat generation rate, and external flow parameters.
While the resulting system of equations can be solved implicitly, the computational effort is large and must employ
an iterative approach by making successive approximations to
the new-time clad temperature.
The computational complexity
as well as potential for non-convergence are the basis for
the use of old-time temperatures in calculation of new-time
oxidation thicknesses in this report.
References (Gl) and
(W3) describe implicit evaluations for stiff systems of equations such as that presented above; however, this refinement
was beyond the scope of this project.
2.3
EXTENSION OF SFUEL PROGRAM TO ASSESS POOL-WIDE BURNING
Calculation of fuel pool conditions subsequent to initia-
tion of runaway oxidation at one or more locations in the
pool necessitated the revision of the SFUEL computer code.
A
careful review of models in the code was undertaken to delineate both the physics-related and computational limits of the
original code.
The revised code was developed in an evolu-
tionary manner, such that only models or programming
structures which led to non-physical results or computational
instability were adjusted.
Models were then implemented for
mechanisms such as burnout of entirely oxidized clad, fuel/
clad relocation, etc., which had not appeared in the original
program.
The following is a discussion of the individual
models which were revised or implemented:
38
2.3.1
Inter-Holder Conduction
Initial program runs with the original SFUEL code indicated that the code is numerically unstable when any of the
fluid passages are blocked, such as occurs in the interholder air spaces.
The oscillatory instability, reflected in
the interfacial node temperatures of the inter-holder spaces,
is initiated at the lowest boundary node for the non-flow
channels, and leads to the appearance of negative temperatures
which progress up the channel, increasing in amplitude with
increasing holder temperature.
tions are shown in Table 2-1.
Examples of these oscillaThese oscillations exist at all
times, although they do not halt the computation until the
equation:
TBARNC = TW -
0.38*(TW- TA)
takes on negative (Kelvin) values.
2.10
Examination of the finite
differencing scheme for the fluid dynamics calculations indicates that an oscillatory instability is to be expected for
non-flow channels.
This will be shown by performing a Von
Neumann stability analysis on the linearized discretization of
the energy equation employed in SFUEL.
The essence of this
method of stability analysis is to express the error of each
term of the discretized equation with a finite Fourier Series
representation.
The decay or amplification of each mode is
then considered separately to determine stability or
TABLE 2-1
Oscillatory Instability of Interholder
Space Interface Air Temperature
Case Analyzed:
Time = 550 sec.
PWR, CYLINDRICAL STORAGE RACK
3.0" BASEPLATE HOLE
Interface Air Temperature, C
POOL
SECTION
2
POOL
SECTION
4
POOL
SECTION
6
0.388
0.623
1.183
19
0.355
0.564
1.051
18
0.732
1.168
2.199
17
0.674
1.069
2.000
16
1.022
1.627
3.056
15
0.923
1.466
2.740
14
1.231
1.958
3.672
13
1.080
1.717
3.210
12
1.340
2.130
3.989
11
1.130,
1.796
3.360
1.134
2.128
3.979
1.407
1.697
3.173
1.235
1.956
3.645
0.897
1.426
1.031
1.631
2.665
2.243
0.634
0.752
1.005
1.877
1.186
0.467
2.186
0.671
-0.1384
1.232
Axial
Location
(top)
0.298
0.424
1 (bottom)-0.074
0.869
-0.280
The values presented above are the difference between the
temperature at the location and the initial room temperature. Note that negative values correspond to temperatures
below room temperature. The air interfaces in pool sections
1, 3 and 5 behave in a similar fashion and have been omitted
for clarity.
instability.
The one-dimensional fluid conservation equations are
solved in discretized form in SFUEL on a grid as shown in
Fig. 2.6.
The dotted lines represent the outlines of the
control volumes of fluid with indices I-1, I, and I+1.
The
points (O's) located on the boundaries of the control volumes
represent interfacial fluid values.
The energy equation is
solved for temperature on a grid such as Fig. 2.6, employing
the value of temperature at the previous time step and the
value of temperature at the fluid interface behind it.
The equation for the average fluid node temperature in
SFUEL is:
2*GCPA*TA3 n+
TAVE3 n+1
I
I--
_
2*GCPA +
+ PCA + HAT
PCA
TAVE3n
T
+ WKl
2.11
1
The variables in Eqn.2.11 are defined as:
(The final equality states the equation in more traditional
variables.)
GCPA =
GOXO*CPOX + GNI*CPNI| = 'n CPOX
+
1
2 CpNI
mC
PCA = PROOM*CP*AXA3(J)*DELX/(RA*DELT) = P0 Cp AAX/(RAt)
2.12
2.13
41
///////////////////////////
I-
6
I-1i
o0
Figure 2.6
I-
2
I+7
I+1
0
Finite Difference Mesh for One-Dimensional
Fluid Flow
6
42
Employing the ideal gas law
p0TO = PO/Ro
Eqn.2.13 becomes:
PCA = p 0 T0 CpAAX/At
2.14
WKl = HWA3(I,J)*AWA3 + HWJMl(I,J)*WZDX = hA 1
2.15
HAT = HWA3(I,J)*AWA3*TW(I,J) + HWJMl(I,J)*WZDX*TWJMl(I)=hA 1 T
2.16
In Eqns.2.14 and 2.15 the wall temperatures and heat transfer
coefficients have been assumed constant.
2.12 -
2.16 into 2.11, we obtain:
2C TA3
TAVE 3 n+1=
I
+ p0 C A
+ hA,T
0C
A Ax
p At + hA
TAVE3 n
1
PO T
2mC +
p
where A
Substituting Eqns.
0
2.17
= heat transfer area and A is the flow cross-
sectional area.
The temperature of the interface fluid node is calculated by
using:
TA3I+
2 - TAVE31 - TA3q
=
2.18
Since the temperatures showing the oscillation are the interface nodal temperatures, expand Eqn. 2.18 using 2.11.
(The nomenclature TA3 and TAVE3 will be dropped in favor of
T, where the location of the point is denoted by the subscript as in Fig. 2.3.)
n+1
TIn+
=
2mCT
2
PT0 CA
1+
p I--_
Ax
+ hA Tw
-
n
Tn~
2.19
PO TOC A A
I+4
2mC
p
+
n
+At
hA
1
Eqn. 2.19 is nonlinear, incorporating variables m, Cp, p0 '
T 0,
h, Tw,
nator.
and the old-time level value of T in the denomi-
In order to perform a stability analysis, Eqn. 2.19
To do this, let the ratio Tn/T0
must first be linearized.
Also, let p0 = p; m= puA; and
1, or equivalently, T = T
p, u, CpI Tw,
The analysis may be
A, A 1 and h be constant.
further simplified by neglecting the assumed constant heat
source hA Tw
Tn+l
I+42
Equation 2.19 reduces to:
2
2puACT n
p
J
+ pT nCC A An+1
+ pC A
2puAC
p
+ hA
At
1
T
1
Dividing pC pA through the brackets in the above equation
20
yields:
'
2
n+l
+4
2uTn+l + Ax Tn
At I
12u +
+ hAl/pC A
At
ip
1
Tn+
Tn+
1
-
2.21
The Fourier component of the solution may be written as:
n ei(mIAx)
Tn _
2.22
where $ n is the amplification function at time-level n of
the particular component whose wave number is m, and i = V1.
Substitution into Eqn. 2.21 gives:
Ax
AtL
2.23
hx
A
A+
hCA
cos
+
2iu sin-
p
The amplification factor $ is given by the complex Eqn. 2.23.
For stability
: 1, then:
l
Ax 2
$ 2
hA1
At+
pCCA
p
2
1
2m
cos
2
2.24
+ 4u sin
2m
s
1, the choice of m will rehA
For stability
, and u.
strict the values of Ax, At, pC
p 1
It is apparent that for I$l
S<
1:
S<
2
hAn
+ pC A
22
os
2 + 4u
sin m
2.25
For
This equation must be satisfied for all wave numbers m.
= 0, Eqn. 2.25 becomes:
the case sin }= 1, cos
22
2
At
1 <
I 2A
)[
4u
2
Ax
u > 2At
2.26
Equation 2.26 represents the stability criterion for the
SFUEL finite difference scheme.
Note that this condition
is not met for any of the SFUEL cases which involve blocking
one or more of the flow paths, since for these cases
u = 10-10 cm/s.
Because the air in the inter-holder spaces is actually
stagnant, the old SFUEL scheme was replaced by a fully implicit two-dimensional conduction scheme, assuming that
there are no buoyancy-driven recirculation flows in this
region.
The conduction equation is:
Tn -+1
T
fTI
At
1
+T
- 2T
I +
TW+
(Ax) 2
TW 2 -2T
2
2.27
(Ay) 2
where Tn is the previous time interval value and all other
values are determined as the new time step.
The new-time
values of the holder wall temperatures, TW1 and TW 2 , are
evaluated explicitly as in the original SFUEL, while the
thermal diffusivity is evaluated at the previous node-average air temperature.
The interface air temperatures are
determined as the arithmetic averages of the average air
temperatures determined using Eqn. 2.27.
The set of simul-
taneous equations (2.27) for all axial nodes are solved with
a standard tridiagonal matrix inversion routine.
For the
steady-state cases reported in NUREG/CR0649, where the amplitude of the inter-holder oscillations is small, this conduction model produces essentially identical results.
How-
ever, the model's utility is not in confirming previous
work, but in allowing extension of the code to the calculation of conditions which could not be assessed previously.
2.3.2
Revised Fluid Dynamics Calculations
Examination of the code results subsequent to the revision described above revealed another difficulty with the
assembly out-flow boundary evaluation, independent of the
interholder non-flow approximation.
This problem was mani-
fest in the appearance of large negative values of air temperature at the top-most node.
The air temperature differ-
encing scheme presented in Sec. 2.3.1 incorporates an upwind
scheme for the first half of the node control volume, followed by an extrapolation to obtain the node out-flow interface temperature (see Eqns. 2.11 and 2.18).
This differenc-
ing scheme is thought to have undesirable characteristics,
especially with regard to boundary conditions.
The stabi-
lity criteria of the upwind-extrapolation at the boundary
conditions cannot be obtained with Von-Neumann analysis and
requires other analytical techniques.
Consequently, a for-
ward differencing scheme was substituted, which provides
physically real values of air temperatures at all points in
the calculational domain.
The forward differencing in the
revised SFUEL program consists of the use of Eqn. 2.11 over
n+1
is replaced by the node
the entire node length, where TA3 n_
average temperature of the previous node divided by 2:
TAVE3n 1
I
Sn+l
=GCPA * TAV+ PCA+ HAT
=CA2.28
GCPA +
PCA
+ WK1
TAVE3n
where:
TAV = TA4AVE
if I= 1
average temperature of base
node
= TAVE3
if I > 1
average temperature of pre-
vious node
This scheme requires a fictitious top-most convective node
in order to determine the top-most interface air temperature.
The interfacial air temperatures are calculated as
the arithmetic average of the node average temperatures of
the two adjacent nodes, as for the non-flow channels.
The
change of differencing scheme was motivated partly by the
appearance of the large negative outflow temperatures, and
partly by a non-convergence of mass flows.
While this re-
vision did not resolve the mass flow divergence, it does
result in physical temperature fields and has also been retained for its better known computational characteristics.
48
2.3.3
Maximum Driving Force Approximations
The mass flow divergence alluded to above comes about
due to the orifice pressure drop at the base of the fuel
assembly holders.
Since the pressure drop is calculated
in terms of reduced pressure, under certain conditions the
pressure drop calculated at the holder inlet is sufficiently large to be outside the range of convergence of the Newton Raphson scheme.
This obstacle has been overcome by
using an approximation based on purely physical grounds.
All cases investigated during this research achieve steadystate mass flows while the temperatures of the fuel assemblies are still relatively low.
Further increases in gas
temperatures produce decreases in the channel gas density
which is already very low compared to the room air density.
Therefore, the buoyancy force, which is proportional to the
difference bewteen the room air density and the channel air
density, approaches a constant value:
Buoyancy
Force
-
lim
pchannel
0
P
room
g(l _
channel
proom
_
=
room g
2.29
This is the physical basis for making the approximation that
the mass flows do not change from their steady values once
runaway oxidation initiates.
In the SFUEL code, this appro-
ximation is implemented by by-passing mass flow calculations
after the initiation of runaway oxidation, typically several
hours after the steady-state mass flow distribution has been
attained.
For fuel pool analyses in which the holder base-
plate holes are greater than 5 inches, it has been observed
that this approximation is unnecessary, since the mass flow
calculations do not diverge subsequent to the initiation of
runaway oxidation.
Cases run in which the mass flow calcu-
lations are continued during pool-wide burning indicate
that the constant mass flow approximation is slightly nonconservative, since it provides larger mass flows (lO10%)
after fuel/clad relocation than would be sustained by the
buoyant driving force.
The use of steady-state mass flow
values for pool-wide burning calculations is thought to be
better than the alternate approximation of assuming that
all channels are connected to an infinite reservoir, since
in the physical situation, the total pool-wide mass flow is
limited by that which can flow between the liner and the
first holder assembly.
It is thought that this approxima-
tion results in a slightly reduced mass flow relative to
the infinite reservoir approximation and is therefore the
more conservative of the two approximations.
2.3.4
Revision of Clad Thickness Model
It has been observed that the original SFUEL model did
not account for total consumption by oxidation of the clad
in a node.
That is, it was possible for clad to "regener-
ate," especially when the node experiences repeated transi-
tions from the kinetic-controlled to diffusion-controlled
regimes.
Several minor programming changes were required
to assure that a fully oxidized clad node did not continue
to generate chemical oxidation energy.
2.3.5
Limit to Maximum Rate of Change of Temperature
The fuel temperature calculation scheme in the original
SFUEL is fully explicit.
By definition then, the new time
step fuel temperature is given by the sum of the previous
temperature plus the difference bewteen the new-time energy
input minus the old-time energy losses, multiplied by the
ratio of the time-step size to the thermal capacitance of
the fuel/clad node.
When the energy input to the node dur-
ing the new time step is far greater than the previous time
step energy input for which the losses were calculated, i.e.,
when runaway oxidation initiates, the new time temperature
can become very large in a very short time.
Use of Hamming's stability analysis (H4),(S3) for the
relevant equations illustrates this point.
The fuel/clad
temperature is determined as:
TRI = TRI + QRIJ* DELT/CR
where:
TR
= rod temperature at axial location I
CR
= thermal capacitance of fuel/clad node
= (RHOF*AF*CR+ RHOC*AC*CC)*DELX = (pcPA)TAx
2.30
RHOF
density of fuel
RHOC
density of clad
AC
=
cross sectional area of clad
AF
=
cross sectional area of fuel
CC
=
specific heat of clad
CF
=
specific heat of fuel
DELX
=
length of fuel/clad node = Ax
DELT
=
time interval size = At
and
QRIJ = QDECAY I +QCHEMI + QCOND
- QCONDI+1+
QR
2.31
where:
QDECAY1
= heat generation due to radioactive decay
QCHEM
= chemical oxidation energy input
QCOND
= energy conducted into node from node below
1
=
_1 - TRI)/DELX
energy conducted out of node from node above
QCOND I+l
=
(AF*SMKF
(AF*SMKF + AC*SMKC)*(TR
(AF*SMKF + AC*SMKC)*(TRI - TRI+ 1 ) /DELX
+ AC*SMKC) =
(kA)T
SMKF
= thermal conductivity of fuel
SMKC
= thermal conductivity of clad
QR
= radiative and convective loss from the node
) + radiative losses
= HRAl*AR (TAVEln
Iz-TR IR
52
HRAl
= convective heat transfer coefficient = h
AR
= surface area of the clad node = A
TAVEl
= temperature of the air at node I
All of the above heat fluxes are determined using temperatures at the previous time step.
To simplify the analysis,
neglect QCHEM and the radiative losses.
Then QRIJ is given
by:
QRIJ = hA(TAVEl
-
TR )
+ kAT(TR
+ + TRn_
T
n~
I
It
2TR IT~/D~
)/DELX
2.32
follows that TRn+l is equal to:
I
TRn+1 = TR
+
+
_
phAt
p 1lT
Ax 2 (TR 1+
TR
-2TR)
(TAVEl2- TRn)
2.33
For TAVEln constant, the difference equation will be convergent only if the spectral radius of its matrix of coefficients exceeds unity, but is less than 1+ O(t):
A
= 1+
(
At
(Ax)2
hpAt
pA PT
k
pA
sin 2 (N;-l)lr
T
2N
2.34
The term (hAAt/pcpA2 )T allows the convergence
criterion
to be satisfied for the restriction:
At
(Ax) 2
-
2
2.35
pcpA
k
2.35
Equation 2.35 is the stability criteria for conductive and
convective stability of the rod temperature scheme outlined
above.
Similar, although more restrictive stability cri-
teria result from the non-linearities introduced by radiative heat transfer and chemical oxidation energy input.
The strategy adopted to maintain stability has been to
limit the temperature rise during any interval to a value
in concert with the fuel numerical stability limits.
This
value is not attainable in closed form for the chemical oxidation energy, since this varies non-linearly.
Instead,
a value of temperature rise which conservatively satisfies
the calculable stability criteria is used (100*C/s) and
the results carefully monitored for the appearance of irregularies.
If the temperature rise during an interval fails
to meet the above criterion, the time interval is reduced,
all values initialized to the previous values, and the calculations repeated.
This strategy is quite successful in
containing the exponential growth observed in the original
SFUEL program.
When the clad experiences vigorous oxida-
tion, the time steps are typically reduced by 2 orders of
magnitude to satisfy the stability limit.
2.3.6
Fuel Relocation Models
Several possibilities exist for modelling fuel relocation when a node reaches the melting point of zirconium
dioxide, approximately 2740*K, assuming that molten zirconium, if any, will be contained by the oxidized clad:
1) Ignore clad change to liquid phase; allow fuel/clad
temperature to continue to climb.
2) Hold node temperature at melting point for all successive calculations.
3) Hold node temperature at melting point; calculate
energy deposited in melting zirconium dioxide; zirconium solid remains intact.
4) Let node disappear (rod relocate) when node clad
temperature reaches the melting point.
Since only
a few rods are restrained at the top, it is reason-
able that all nodes above the melted node should
relocate as well.
5) Hold node temperature at melting point until mass
is completely melted; followed by relocation as in
(4).
Options (1) and (4) have been investigated in this report,
as they represent the extreme situations.
Options (2),
(3)
and (5) require a kinetic model for melting, which is a
function of the grain structure at a given radial location
in the clad.
Such a refinement is beyond the scope of this
project.
2.3.7
Computer Operating System Changes
In addition to the modelling revisions noted above,
programming changes related to system compatibility with
the researcher's computer operating system have been made.
A restart option has been added to the program to allow the
program to be interrupted and restarted at a later time to
complete the calculations.
The feature was required to keep
the computational costs to a minimum, as well as to allow
parametric evaluation of certain code variables.
2.4
MODIFIED SFUEL ANALYSIS OF BURNING PROPAGATION
2.4.1
Characteristics of Revised SFUEL Code
Following the series of revisions described above, a
sample case was analyzed with both the original SFUEL code
(except for operating system re-programming) and the revised
version of SFUELlW.
The case chosen is one which achieves
steady-state:
PWR SPENT FUEL 17 x17 ARRAY
CYLINDRICAL STORAGE RACK CONFIGURATION
FULL CORE DISCHARGE LOADING, 3" BASEPLATE HOLE
The plot of temperature vs. time for this case is displayed
in Fig. 12 of NUREG/CR0649.
(1 8 th node,
Figure 2.7.
Table 2-2.
6 th
The maximum clad temperature
pool section) for the two codes is shown in
Values depicted in Figure 2.7 are tabulated in
A comparison of the steady-state nitrogen mass
flow distribution in the pool sections is presented in Table
The agreement is good and it indicates that the re-
2-3.
vised code is functioning properly and also that the prerunaway temperature calculations obtained with the initial
SFUEL code are satisfactory despite the objections noted
earlier.
The variation between the two codes is expected to
become for pronounced at higher rates of change of temperatures.
This is shown for the case of:
PWR SPENT FUEL 17 x 17 ARRAY
CYLINDRICAL STORAGE RACK CONFIGURATION
FULL CORE DISCHARGE LOADING, 1.5" BASEPLATE HOLE
350
2
300
:94
4
6
6
66
250
original SFUEL
200
A
revised SFUEL
150
100
50
0
0
5
10
15
20
25
30
35
40
45
Time from initiation of accident x 103 seconds
Figure 2.7
Maximum Clad Temperature for Original and Revised
SFUEL.
TABLE 2-2
PWR Spent Fuel 17 x 17 Array
Cylindrical Storage Rack Configuration
Full Discharge Loading, 3" Baseplate Hole
MAXIMUM CLAD TEMPERATURE
(18 NODE, 6 SECTION)
x103 secs
Original
SFUEL
Modified
SFUEL
0.0
10
10
3.6
93.9
92.0
7.2
197.0
188.9
10.8
276.6
264.0
14.4
311.2
302.7
18.0
317.9
316.5
21.6
317.4
320.2
25.2
316.8
320.9
28.8
316.7
321.1
32.4
316.7
321.1
36.0
316.7
321.2
39.6
316.7
321.2
43.2
316.8
321.2
Time
TABLE 2-3
Steady State Nitrogen Mass Flow Rates
Pool Section
Fuel
Assemblies
Original
SFUEL
(gm/s)
Modified
SFUEL
1
6.79
6.54
2
33.40
33.74
3
36.31
36.68
4
37.92
38.25
5
41.03
41.35
6
43.68
43.93
Liner Holder Space
199.1
200.5
Figure 2.8 illustrates the effect of the revised fluid dynamics and holder heat transfer calculations during runaway,
and suggests that the results obtained with the original
SFUEL are accurate up to runaway.
The effect of the fuel
temperature stability limit is particularly well depicted
by the behavior of the original SFUEL subsequent to sustaining oxidation in the 2nd kinetically controlled regime
(correlation in the center of Figure 2.3).
The revised
version is not as subject to the accumulation of fuel temperature error and growth as the original version, by virtue of the time step reduction capability.
Consequently,
the temperature rise of the peak clad node predicted by
SFUELlW is smoother and occurs over one and one-half hours
later than predicted in the original code.
The values
graphed in Figure 2.8 are tabulated in Table 2-4.
The re-
sults presented thus far indicate that results calculated
by the original SFUEL are conservative with respect to both
temperature rise during oxidation and time to initiation
of runaway oxidation.
The results given above validate the new code's ability
to predict fuel pool conditions up to the point of initiation of runaway oxidation.
Figures 2.9 and 2.10 illustrate
the code's predictions for the maximum node clad temperature (node 18, section 6) up to the time at which the node
either burns out (upper curve) or melts and falls unimpeded
2200
+ original SFUEL
2000
A revised SFUEL
1800
1600
1400
1200
(1)
1000
800
++
A
+
600
(1) Revised SFUEL
reaches runaway conditions at t = 42500 sec.
400
200
5
10
15
20
25
30
35
40
Time from initiation of accident x 10 3 seconds
Figure 2.8
Comparison of Original and Revised Codes during
Runaway Oxidation.
45
62
TABLE 2-4
Comparison of SFUEL Results
Under Runaway Conditions
Time
Original Version
Modified Version
secs
TR (18,6) *C
TR (18,6) *C
10.0
10.0
1800
43.89
43.61
3600
77.94
76.59
0
5400
115.4
112.4
9000
200.2
192.7
10800
246.2
235.9
12600
293.6
280.9
14400
341.7
325.4
16200
389.9
370.4
18000
437.8
414.7
19800
485.4
458.3
21600
533.8
501.4
23400
583.9
544.7
25200
636.4
588.5
27000
691.4
633.1
28800
748.4
678.1
30600
806.8
723.5
32400
865.2
768.7
34200
921.3
813.5
36000
995.2
857.4
37150
2012.3
37800
x
899.5
39600
code breaks down
942.0
41400
1014.0
3500
+
3000
clad melt, Option 4
0 clad melt, Option 1
2500
u
0
5
2000
ci)
ro1500
P
1000
500
Expanded View
Fig. 2
-
0
50
100
150
200
250
300
350
Time from initiation of accident x 102 seconds
Figure 2.9
Peak Clad Temperature during Runaway.
400
450
3500
+clad
melt, Option 4
Oclad
melt, Option 1
3000
U 2500
w
02000
4I
1500
1000
-
+-.
----
+-+
500
0
412
414
416
418
420
422
424
426
428
430
432
Time from Initiation of accident x 102 seconds
Figure 2.10
Comparison of Peak Clad Temperatures, Melt Options 1 and 4
434
65
to the pool floor.
Note that Figure 2.10 is an enlarged
view of the oxidizing time interval depicted in Figure 2.9.
The numerical values in Table 2-5 corresponding to Figures
2.9 and 2.10 reveal that the total time from initiation of
runaway to clad melting is approximately 8n,10 minutes.
The decrease in clad temperature for option 1 occurs because the clad has been completely oxidized.
The purpose of the foregoing discussion was to demonstrate the capability of the revised SFUEL code to predict
the behavior of the peak clad node temperature from initiation of runaway oxidation to melting/burnout.
With this
tool available, the question of propagation of the vigorous
zirconium oxidation from the fresh fuel pool section to the
older fuel sections of the pool is addressed.
Figures 2.11
and 2.12 present the peak clad node temperatures' histories
for the fuel assemblies in pool sections 5 and 6 for the
1.5 inch baseplate hole case described in Figures 2.9 and
2.10.
Figure 2.11 illustrates the impact of the "total"
fuel relocation model (option 4)-where all nodes above the
melted node disappear-while Figure 2.12 gives the result
for the case where the high temperature node neither melts
nor relocates (option 1).
The lower curve in each figure
depicts the behavior of the peak clad node temperature in
the adjacent fuel pool section.
In Figure 2.12, it is evi-
dent that sufficient energy is transferred to the adjacent
TABLE 2-5
Maximum Clad Node Temperature for Different
Fuel Relocation Options (Modified SFUEL)
Time
x10
3
secs
Peak Clad Temperature (*C)
0.0
10.0
3.6
76.6
9.0
192.7
12.6
280.4
16.2
370.4
19.8
458.3
23.4
544.7
27.0
633.1
30.6
723.5
34.2
813.5
37.8
899.5
39.2
942.0
41.4
1014
42.3
1117
42.54
1656
42.61
2117
42.67
2465
42.73
2747
42.79
994
2981
42.86
42.98
43.97
1002
1002
3194
43.11
1009
Option 4
3428
3361
43.13
3293
43.19
4237
43.20
1014
1
Option 1
2800
-
2600
.
Clad Melt Option 4
+
peak node, Section 6
peak node, Section 5
2400
2200
-p
2000
a)
1800
-
1600
-
.1400
a>
1200
-
1000
--
4-
---
----
800
600
424
423
425
426
Time from initiation
Figure 2.11
427
428
429
430
431
432
of accident x 102 seconds
Peak Clad Temperatures for Pool Sections 5 and 6,
Melt Option 4
-
3500
Clad Melt Option 1
+
-
3000
O
peak node, Section 6
5
peak node, Section
3
W 2500-
0-
2500
1 000
100
423
424
425
426
427
428
429
430
431
432
Time from initiation of accident x 102 seconds
Figure 2.12
Peak Clad Temperatures for Pool Sections 5 and 6,
Melt Option 1
pool section to engender runaway zirconium oxidation.
The
fuel relocation model, Figure 2.11, not only results in
little energy transfer to the adjacent pool section, but
the subsequently cooler air temperatures at the burned out
axial position cools the wall and, indirectly, the adjacent
pool section.
Unless a physical situation can be envi-
sioned in which the fuel behaves as in Figure 2.12, the propagation of runaway zirconium oxidation to the adjacent
fuel pool section for this configuration might be ruled
out.
Examination of the approximations involved in calcu-
lating Figure 2.11 (instantaneous phase change and relocation upon achieving the melting point of ZrO 2 ) indicates
that the actual situation may yield a curve for the adjacent pool section peak clad temperature which lies slightly
above that presented.
2.4.2
Pool-Wide Propagation Results
A pool-wide analysis was performed for the case:
PWR SPENT FUEL 17 x 17 ARRAY
HIGH DENSITY STORAGE CONFIGURATION
90 DAY MINIMUM DECAY TIME, 5.0" BASEPLATE HOLE
to determine whether the runaway oxidation would propagate
across fuel pool sections.
This case was run with contin-
ually-calculated mass flows and clad melt option 4 (complete
relocation upon reaching melting point of ZrO 2 ).
Propaga-
tion was observed to occur across four pool sections.
The
temperature profile for the initially hottest node in pool
section 6, the first to undergo runaway oxidation, is shown
in Figure 2.13.
The arrow in the figure denotes the time
at which the fuel/clad relocates; the subsequent temperature profile is for air occupied by the fuel/clad node.
The temperature at the location is observed to increase and
decrease as the remaining fuel/clad nodes beneath it undergo
vigorous oxidation.
These oscillations would be expected
to be damped in the actual pool, as relocated clad and uranium dioxide block the lower channel and reduce the available air flow and therefore reduce the oxygen available for
chemical interaction.
Figures 2.14, 2.15 and 2.16 display
the temperature-time history for the peak clad temperatures
in the three adjacent pool sections which experience vigorous burning.
Figure 2.17 is a composite of the four preced-
ing figures, and is intended to show the waiting times before the various sections of the pool ignite.
It may be
inferred from the figure that chemical oxidation energy
liberated in pool section 6 has an immediate effect upon
the oxidation rate in section 5; however, the decay heat
generated within these two sections is approximately the
same.
Sections 4 and 3, which are modelled as much older
fuel, and have lower decay heat generation, require significantly longer times to experience runaway oxidation.
2800
POOL SECTION 6
CLAD
RELOCATES
2600
2400
2200
I
I
I
I
I
2000
1800
1600
1400
1200
I'
11
21
1000
_
800
220
230
240
250
260
270
280
290
Time from initiation of accident x 102 seconds
Figure 2.13
Temperature in Pool Section 6, Axial Location 16
300
310
2800
POOL SECTION 5
CLAD
RELOCATES
2600
U
0
2400
0
2200
1
I
-H
4-)
0
1-4
2000
I
1800
I
1600
I
-)
(1)
4
I
1400
+
E-4
I
1200
V
1000
-
sa
%.. + ..+ .+ f+++
800
2
220
Figure 2.14
230
240
250
260
270
280
290
Time from initiation of accident x 102 seconds
Temperature in Pool Section 5, Axial Location 16
300
310
2800
POOL SECTION 4
CLAD RELOCATES
2600
2400
2200
2000
1800
1600
1400
1200
1000
800
I
220
min
230
240
250
260
270
280
Time from initiation of accident x 10
Figure 2.15
2
2
Temperature in Pool Section 4, Axial Location 16
290
seconds
300
310
2800
POOL SECTION 3
2600
2400
2200
2000
1800
1600
1400
1200
1000
800
220
230
240
250
260
270
280
290
Time from initiation of accident x 102 seconds
Figure 2.16
Temperature in Pool Section 3, Axial Location 16.
300
310
PWR HIGH DENSITY, 90-DAY MINIMUM DECAY TIME, LARGE BASEPLATE HOLES
2800
Pool Section
Symbol
2600
2400
2200
A
6
+
5
o
4
0
3
2000
1800
1600
1400
1200
1000
--
800
220
230
240
250
260
270
280
290
Time from initiation of accident x 102 seconds
Figure 2. 17
Composite View of Pool Wide Propagation Results
300
310
In an effort to assess the sensitivity of the burning
propagation relative to the decay heat, a series of analyses were performed in which the center pool section was
given a high decay heat generation rate, and the remaining
sections were given a substantially lower decay heat rate;
these values are reported in Table 2-6.
The results for
the third case are shown in Figure 2.18; the values used
in this plot are tabulated in Table 2-7.
Examination of
Figure 2.18 shows that the highest clad temperature in
pool section 5 peaks approximately 2 hours after runaway
oxidation occurs in the adjacent (6 th) pool section.
The
peak temperatures in the other pool sections are quite low
and achieve steady-state as the remaining clad in the sixth
pool section burns out.
This trend is clearly evident in
Figure 2.18, where the air temperature in the sixth pool
section is observed to be generally decreasing (between the
intervals in which the remaining clad in section 6 ignites).
Of the cases shown in Table 2.6, cases 1 through 4 attained peak temperatures in pool section 5 and subsequently
cooled (without experiencing runaway) as the 6th pool section
burned out.
5 th
Cases 5 and 6 sustained runaway oxidation in the
pool section.
By plotting the peak temperatures reached
in pool section 5 versus the decay heat generation rate, a
curve of burning propagation is established, shown in Figure
2.19.
This curve indicates that the minimum decay time for
TABLE 2-6
Decay Heats Used in Assessing Poolwide Propagation
PWR SPENT FUEL 17 x17 ARRAY
HIGH DENSITY STORAGE CONFIGURATION
> 5.0" BASEPLATE HOLE
POOL SECTION
(1 fuel assembly per section)*
1
2
3
Case 1
1.0
1.0
1.0
Case 2
2.0
2.0
2.0
Case 3
3.0
3.0
Case 4
4.0
Case 5
Case 6
4
5
6
1.0
30.0
2.0
2.0
30.0
3.0
3.0
3.0
30.0
4.0
4.0
4.0
4.0
30.0
5.0
5.0
5.0
5.0
5.0
30.0
6.0
6.0
6.0
6.0
6.0
30.0
*Units of decay heat are KW/MTU. The decay time for
these heat generation rates may be estimated from Table
1-1.
1200
A Pool Section 6
CLAD RELOCATES
1150
-
o 1100
-
Q)
-
0 Pool Section 5
A
1050
1000-
(0i
a)
950
-g
900-
850
800
18
20
22
24
26
28
30
Time from initiation of accident x 103 seconds
Figure 2.18
Poolwide Propagation Sensitivity, Case 3
32
34
TABLE 2-7
Assessment of Sensitivity of Pool Wide Propagation
PWR, High Density, Section 6 @ 30 KW/MTU,
All Others @ 3.0 KW/MTU
Time From
Initiation of
Accident
x103 secs
Temperature at Axial Location, *C
POOL SECTION 6
NODE 15
18.
853.7
20.
913.6
22.
987.9
POOL SECTION 5
NODE 15
23.29
1081.3
23.42
1140.
23.54
2104.
23.55
2741.
23.67
1090.
23.79
1041.
23.92
1056.
24.04
1074.
24.17
1098.
24.29
1136.
24.42
2263.
24.54
1032.
28.17
1002.
28.29
1007.
851.
28.79
1033.
857.6
29.29
1076.
864.3
29.79
928.
876.8
30.29
929.
931.4
30.79
932.
935.3
31.29
936.
934.5
31.79
940.3
925.9
32.29
944.7
915.8
32.79
949.3
906.0
33.17
953.1
899.0
I
I
I
If
I
I
1000
Ii
975
4
' '--
950
/'I
/'I
I
GLr)
900 -
E4J
(o
4j
925 -
4
>1)
875 0
0
/
850 _
/
/
/
/
/
/
/
A
INITIATION OF
RUNAWAY OXIDATION
It,
825 -
/
800 -
/
I
I
/
DECAY HEAT
Figure 2.19
(KW/MTU)
Results of Propagation Sensitivity Analysis,
High Density Storage.
81
spent fuel to be stored adjacent to recently discharged
fuel (,\90 days) is approximately 2 years (see Table 1-1).
For the cases presented in Table 2-6, the vigorous oxidation reaction propagated only between pool sections 6 and
5.
The peak clad temperatures in section 4, even while
section 5 was experiencing rapid oxidation (case 6, Table
2-6), did not exceed 750*C.
Thus, any storage configura-
tion which uses fuel of the same or lower decay heats than
shown in Table 2-6, cases 1- 4, is not susceptible to propagation of vigorous zirconium burning.
2.5
PROPAGATION OF ZIRCONIUM BURNING VIA CONVENTIONAL HEAT
TRANSFER MECHANISMS: SUMMARY
The SFUEL code described in NUREG/CR0649 has been re-
vised and extended to analyze the poolwide propagation of
vigorous oxidation of zirconium alloy clad.
Correlations
for zirconium oxidation have been presented which suggest
a variety of reaction rate laws may apply; linear, parabolic or cubic.
An explicit formulation of the parabolic
reaction rate model is used exclusively in this report, as
it is the most commonly cited phenomenological model for
zirconium oxidation.
It is shown that it is possible for widespread burning
of zirconium to occur in a spent fuel pool of high density
or cylindrical storage configuration, following loss of
water.
A curve is given (Figure 2.19) which indicates the
82
minimum decay heat generation rate for spent fuel which is
to be stored adjacent to freshly discharged fuel, if runaway
oxidation is to be confined to the freshly discharged assembly.
Future work on this topic should focus on removing some
of the conservative approximations made in this analysis.
In particular, the conduction scheme through the stagnant
air in the interholder spaces should be re-modelled to allow
buoyancy induced recirculating flows.
Also, axial conduc-
tion of heat in the holder walls would tend to flatten the
axial temperatures along the fuel rod length during the temperature excursions engendered by zirconium burning by increasing the radiative heat transfer.
Finally, it is recom-
mended that a larger database be developed for oxidation of
Zircaloy-2 in air, especially with respect to the effect of
zirconium alloying on the amount of chemical energy released
during the oxidation reaction.
CHAPTER 3
PROPAGATION OF RUNAWAY OXIDATION
VIA PARTICULATE TRANSPORT
3.1
INTRODUCTION
The burning particulate model developed during this
project simulates the behavior of an oxidizing sphere of
zirconium metal levitated by convective air currents.
The
burning particle model requires the simultaneous solution
of mass, energy, and momentum conservation for the particle.
Employing initial conditions determined from SFUEL, the code
calculates the temperature, velocity, distance, mass gain
and heat generation rates as functions of time and oxidation
rate for a variety of allowable particle diameters.
Parti-
cle mass gain via oxidation, heat transferred by convection
and radiation from the particle during flight are accounted
for in the computer program.
The oxidation model in the
program, developed in spherical coordinates, assumes a kinetically controlled reaction, based on the parabolic models
presented in Sec. 2.2.
Diffusion limited kinetics are not
included because the particle is in motion and will always
be moving into a region of fresh air.
The distributions of
particles determined from the PARTICLE code are employed in
determining an effective heat flux to the receiving surface
(upon which the particulate comes to rest).
These evalua-
tions are performed using closed form analytic transient
conduction equations.
The primary assumptions embodied in
the burning particulate analysis are:
1) Gas velocity in the subassemblies is uniform
throughout a pool section;
2) Air properties for the subchannel are assumed to be
those of the hottest fuel node;
3) Interaction of particles in the flow is neglected;
4) The amount of zirconium metal spalling into the air
flow is equal to one half of the amount oxidized in
place on the fuel rod;
5) The initial temperature of the particle is equal to
that of the hottest clad node in the assembly;
6) To account for turbulent convection at the subassembly exit and subsequent lateral dispersion of the
particles, a horizontal component of velocity equal
in magnitude to the vertical particle velocity is
assigned;
7) Subsequent to exiting the subassembly, the air temperature is assumed to be room air temperature;
8) Room air pressure is used in all calculations;
9) Air velocity exiting from inter-holder spaces is
assumed negligible, in agreement with Sec. 2.1,
thus the particle descent is in stagnant air unless
it enters another holder/pool section;
10) All particles are assumed to land on a horizontal
surface at the same height as the origin;
11) The heat flux to the receiving surface is approximated as that conducted in from the projected area
of the particle, while the rest of the energy is
lost by radiation and convection.
There are many interacting effects which would need to
be quantified in an exact analysis; however, the threat due
to particle levitation is insufficient to warrant a more
detailed analysis.
Assumptions 4 and 5 are approximations,
since there is as yet no formulation for the spallation rate
of unreacted zirconium nor is there a quantitative database
to develop a phenomenological model.
3.2
PARTICULATE PROPAGATION POTENTIAL ALGORITHM
3.2.1
Particle Model Description
The PARTICLE code outlined above calculates time-temperature histories for spheres of unreacted Zr metal assumed to spall from the oxidizing fuel rods.
The initial
conditions taken from SFUEL are:
TR(I)
= temperature of hottest fuel/clad node
TAVEl(I) = temperature of air flow at the assembly
exit
GNIl(I)
= mass flow rate of nitrogen in pool section
AXAl
= cross-sectional flow area of pool section
PROOM
= room pressure
TROOM
= room temperature
The range of particle sizes investigated in the code
are:
D largest possible particle that can be entrained
by the air flow (cm)
DMIN
smallest particle that will not be extinct subsequent to exiting a specified upper plenum
distance (cm)
The parabolic kinetic oxidation formula is (see Sec.
2.2):
W = /K 0 t exp(-E/RT)
where
W
= mass gain in oxygen (mgO2 /cm2 )
T
= temperature *K
K
= Arrhenius pre-exponentiation constant
2 ?
(mgO 2 /cm ) '/s
E
= reaction activation energy (cal/mole)
R
= gas constant
(cal/mole
3.1
4K)
Differentiating with respect to time (constant temperature) yields:
K 0 exp (-E/RT)
dw _
dt
3.2
-r
= mass gain per time (mg/cm2)/sec.
where
7'
For a spherical particle of mass m =
3
3pD
, the deriva-
tive of mass with respect to time is:
pD2 dD
dm
=
2
3.3
dt
Multiplying Eqn. 3.2 by the surface area of a sphere
and equating to Eqn. 3.3 yields:
dD
-f
/ K 0exp(-E/RT)
3.4
PZr dt-
pZr = density of Zirconium (mg/cm3)
where
f
=
stoichiometric mass of Zr to mass of O2' and
the minus sign indicates particle oxygen mass
gain.
Integrating (with constant temperature) from D
D <D
0
to D,
and time from 0 to t gives upon rearranging:
D = D
0
2f /K0exp(-E/RT) /
03.5
PZr
.
(PZrDO) 2
0 < t
for
<
2
4f K0 exp (-E/RT)
D
where
= outer diameter at time t= 0.
The mass of oxygen added to the particle is then:
M
02
(t)
=
f
Zr
D03_ D0
2f v/K 0 exp (-E/RT)
3.6
PZr
during
(pD0 ) 2
0 < t <
4f2 K 0 exp (-E/RT)
and constant
(C)
M~~
D3
PZr Tr
0
f02(
thereafter.
Heat generated in the particle during the interval of
oxidation is:
QOXID = AHm
where
0
f
QOXID
=
heat generated (J)
AH
=
heat of combustion 12.03 J/mg.
3.7
89
As was discussed in Sec. 2.2, the oxidation is evaluated
isothermally during a given time step.
The temperature of
the particle is obtained from an energy balance, assuming
the particle is thermally thin:
m
tCptt dT
mtot
where
tot dt
q
O0XID
~ UCONV
3.8
~ ERAD
mttCptt dT = heat storage in particle,
= total mass of particle at time t, (mg)
mtot
= total specific heat of particle (J/*K)
Cptot
= heat generated by oxidation, Eqn. 3.7,
qOXID
(W)
qCONV
= hTrD 0 2 Tw -Tair
h
= heat transfer coefficient, (W/cm2K)
Tw
= temperature of particle, (*K)
T.
air
= ambient temperature,
q RAD
= Eo i7DO (T
0
2
(J)
4
4
w - T air ),
(*K)
(W)
= emissivity of particle
= Stefan-Bolztmann constant
The equations of motion for the particle are dealt with
in rectilinear coordinates:
d y
dt2tot
CD
air
2
2
y(O) = 0
y'(0) = V 0
3.9
90
d 2X
27D 2
D Pair
2
dt
8m
x
totrD
(0) = 0
x' (0)
= 0
3.10
where
d 2 = net acceleration in x, y directions
dt2' dt
g
=
acceleration due to gravity (980 cm/s 2
CD
=
drag coefficient as function of particle
Reynolds number, based on the magnitude
of velocity, V
(mg/cm )
pair
density of room air,
mtot
total mass of particle at time t, (mg)
y
dt' dt
=
velocities in respective coordinate positions
V
=
=
(
dt
)2+ ( dx)2
'/2
dtI
vector velocity,
(cm/s)
Terminal velocity of the particle is calculated from
one of several formulas shown in Table 3-1 (Pl), and is
determined by the particle Reynolds number.
The convective heat transfer coefficient from the burning sphere is calculated using the following equation developed by Whitaker (Wl):
Nu = 2+ (0.4 Red
+ 0.06 Re d
w)
4
which is valid for the range 3.5 < Re d < 8 x10~.
3.11
For Reynolds'
TABLE 3-1
Terminal Velocity of Settling Particulate
Particle
Diameter
Reynolds
Number
mL
2 x10
3
- 10
6
5 x10
2 - 10 6
Laws of Settling
2
Newton' s
C =
Law
p
gD
p
tp
-
2 x 10 3
2.0 - 5 x 10 2
cr
(p - p)
s
K cr =
2,360
Intermediate Law
C = 18.5NRe-0.8
0.153g
u
3 - 10 2
10~4 - 2.0
t
0.71
1.14
D
(p -p)
s
p
0.29 0.43
0.71
=
p
p-
K cr = 43.5
Stoke's Law
C = 24NRe 1
gD 2(p
t
18P
See next page for explanation of terms.
- p)
K cr = 3.3
-13
gp (ps - P) -
0.44
u= 174
102
Critical Particle
Diameter
TABLE 3-1 (Cont'd.)
C
= overall drag coefficient
D
= diameter of spherical particle, (cm)
Dp, crit = critical particle diameter above which law will not
apply, (cm)
g
= local acceleration due to gravity (cm/s )
NRe
= Reynolds number, DPput/.J
p
= fluid density, (gm/cm )
p5
= particle density, (gm/cm )
93
numbers less than 3.5, the value of Nusselt number for conduction from a sphere to an infinite, stagnant medium is
employed:
Nu = 2.0
3.12
The equations for energy and momentum conservation,
3.8- 3.10, are integrated numerically as simultaneous ordinary differential equations, using a third-order Runge Kutta
scheme.
A listing of the PARTICLE program is given in Ap-
pendix C.
3.2.2
Analytical Evaluation of PARTICLE Results
The output generated by particle consists of the time temperature - displacement histories of a series of allowed
particle sizes.
A spallation rate equal to one-half the
oxidation rate at the fuel/clad node is used to estimate the
total mass of particulate.
A normal distribution is as-
signed to the distribution of particle sizes, where the mean
is given by:
DMAX+ DMIN
2
3.13
and the standard minimum and maximum sizes are taken as representative of a 2-a deviation from the mean, thus:
a =
(D2- p)/2
3.14
The continuous distribution of particle sizes is approxiThe frequency of occurrence for
mated by 5 particle sizes.
the individual particle sizes (representing discrete segments of the continuous distribution) are given in Table 3-2.
It is estimated that subsequent to arriving on the receiving surface, the particles will lose half of their enerby by convection and radiation to the room.
The effective
heat flux to the horizontal structures within range of the
particulate is then calculated as:
(q/A)eff
4
1
x.
al
l
m
Cp (Tp - T)f
.
p
3.15
= mass of particle upon arrival, (gm)
m
p.
C
= specific heat of particle, (J/gm *C)
p
= frequency of occurence, (1/sec)
f
where
T
p
= temperature of particle of size i upon arrival,
(*C)
Ax.
= range within which particle will fall, (cm)
L
= width of holder,
(cm)
Particles of a given size are assumed to have a uniform
spatial distribution between zero and their maximum possible
range.
The total energy input to the receiving surface is
equal to the arrival rate times the time during which the
clad continues to oxidize vigorously, on the order of 8 r 10
minutes.
If the fuel/clad axial location reaches the melting
95
TABLE 3-2
Particle Size Frequency Distribution
y=
(D MIN + DMAX)/2
a
(DMA
-
p)/2
Frequency
P
+
Ia
yi
y
p -a
y - 2a
p + a
+(7
y + 2a
Diameter
Frequency
Diameter
P
-
y~ -
P
2 a < D <p
3
cy
-
<
D
<
p
-
< y +
3
0.044
1
0.242
1
3
a < D < p + 2 a
y +
y +
C < D
-
3
< D <
p +
0.383
0.242
0.044
temperature of ZrO 2 , spallation ceases.
The temperature rise in the receiving surface (assumed
to be stainless steel) is calculated by using the solution
For a
to the semi-infinite transient conduction equations.
constant surface heat flux, the appropriate equation is (H2):
T = T. + 2(/A)
k
Tr
-x 2O4aT
(q/A).
k
X
1-erf
2 /a
3.16
where
a
=
thermal diffusivity of the receiving surface,
(cm2/s)
x
=
depth into surface, (cm)
T
=
time,
k
=
conductivity of the receiving surface,
(sec)
(W/cm *C)
(q/A)
=
surface heat flux, (W/cm 2
The temperature rise is calculated using equation 3.16,
where (q/A) is taken as the maximum value of (q/A)eff given
by Eqn. 3.15.
This maximum value of heat flux is assumed to
be constant during the entire interval in which the clad
node oxidizes.
The receiving surface peak temperature is
the variable of interest and it is obtained by setting x= 0
in Eqn. 3.16, giving:
T
=
T=+2
(q/A3
T.
+ 2 (/)
1
k
aT3.17
'IT
.1
The values of receiving surface peak temperature thus
obtained may be used in evaluating the potential for exothermic oxidation of the receiving structural surface; however, such analysis is beyond the scope of this project.
For the purposes of this report, the temperature rises are
employed qualitatively, so as to assess the potential for
energy transfer via the levitation and convection mechanism.
3.3
ENERGY TRANSFER VIA BURNING PARTICULATE: RESULTS
The energy transfer via burning particulate has been
analyzed for the pool configuration case illustrated in
Figure 2.10.
For this case, spallation is assumed to occur
for the duration of the "spike" (lower curve) in Figure 2.10.
This is the period of the most rapid oxidation of the clad,
and is thought to be the most likely time at which spallation could occur.
The PARTICLE code is run for a series of
discrete values representative of the oxidation period,
shown in Table 3-3.
The PARTICLE results for the values in Table 3-3 are
presented in Table 3-4.
The frequencies given in Table 3-2
are used, together with total zirconium oxidized in the interval and Eqn. 3.15 to obtain effective energy fluxes to
horizontal structures.
These values are tabulated in Table
3-5 for the first and last oxidation intervals presented in
Table 3-3.
It is apparent that the effective heat fluxes,
which were all calculated relative to a receiving surface
98
TABLE 3-3
Input Values to the PARTICLE Program
Time From
Initiation
of Accident
(seconds)
Peak Clad
Temperature
(*C)
Peak Air
Temperature
(*C)
41400
1014
1005
4. 57
x 10-
42300
1117
1097
6. 35
x 10 3
42540
1656
1574
1.259 x 10-2
42610
2117
2006
1.194 x 10-2
42670
2465
2344
2. 584 x 10-2
42730
------- clad relocates--------------
Reduction
in Clad
Thickness
(cm)
3
TABLE 3-4
Particle Distributions for Ramp in Figure 2.3
Time From
Initiation
of Accident
(seconds)
Diameter
(cm)
42300
.001665
42540
42610
42670
Time
(seconds)
Distance
(cm)
Temperature
(0 C)
1.11
44.32
10.4
.002773
.475
34.44
379.8
.004617
.250
21.64
916.
.007687
.160
12.13
1070.
.01280
.001
.612
1170.
.001744
1.0
61.11
10.1
564.3
.003041
.500
49.81
.005304
.301
36.72
1395.
.00925
.202
21.54
1671.
.01613
.007
.641
1767.
1.15
69.63
10.
.00324
.650
69.04
419.
.00584
.350
50.87
1672.
.01052
.251
33.17
2079.
.01896
.257
.032
2232.
.001801
.001840
1.15
74.7
10.
426.6
.003384
.650
76.37
.006226
.350
57.88
1912.
.01145
.251
38.49
2400.
.02107
.0016
.022
2582.
100
TABLE 3-5
Effective Energy Fluxes to Horizontal Structures
Time from initiation of accident = 42300 seconds
Distance (cm)
(/A) eff
(W/cm ) per assembly
44.32
0.0
34.44
1.2 x 10~
21.64
3.48 x 10-3
12. 13'
2.39 x 10-2
3.89 x 101
0.625
Time from initiation of accident = 42670 seconds
Distance (cm)
(/A)ff
(W/cm ) per assembly
76.4
3.4 x 10 2
57.9
1.51
38.5
3.09
4.127 x 102
0.022
TABLE 3-6
Thermophysical properties of 5% Cr Steel
a = 0.011 cm 2/s
k = 0. 4 W/cm 0 C
(Hl)
101
temperature of 10*C are relatively low.
The spatial energy
variation of the effective energy input rate is plotted in
Figure 3.1.
The amount of decay heat radiated and con-
ducted to the upper tie plate by the
tops of the fuel pins
is approximately 0.103 w/cm 2, where the decay heat generation is taken as 5 kw/MTU.
For simplicity, the conservative
approximation is made that the maximum heat flux (for TIME=
42670) exists for the entire episode of vigorous oxidation.
The maximum possible temperature rise is then calculated
using Eqn. 3.17 and values of (q/A)eff estimated from Figure
3.1,
and properties of steel given in Table 3-6.
The re-
sulting temperature rise of the adjacent tie plates for the
dispersion of burning particulate is presented in Figure
3.2.
The large temperature rise apparent within the first
10 centimeters should be viewed in light of the conservative
nature of the evaluation, and would be lower if the analysis
were extended to a larger number of particles.
3.4
PROPAGATION VIA PARTICULATE: SUMMARY
The analysis presented above indicates that it is pos-
sible to transfer large amounts of heat within relatively
short distances via levitation and convection of burning
particulate.
Figure
3.2
shows that the majority of the
energy will be deposited within a short distance of the
burning assembly.
At distances greater than the vicinity of
the burning assembly, the energy transfer and resultant temperature rise are quite low.
First-hand observation (B3) of
20
18
16
14
12
(q/A) eff
(W/cm2
10
8
6
4
2
I
0
a
I
30
40
a
a-II
50
Distance (cm)
Figure 3.1
Spatial Variation of Effective Heat Flux
70
240
220
200
180
DISTANCE TO ADJACENT HOLDER
U
o
160
w
uo
o
140
Q
High Density Storage
0
Square Storage Array
120
-P
(a
100
.-
80
-
60
-
40
-
20
-
Q)
0
0
10
20
30
Distance
Figure 3. 2
50
40
60
70
(cm)
Temperature Rise of Steel Tie Plates due to Particulate
80
104
the spallation of vigorously oxidizing clad suggests that
the use of a spallation rate of 50% of the oxidation rate
is an order of magnitude too high.
Additionally, the as-
sumption that all clad which spalls from the rod will be
buoyed by the air flow is not supported by experimental
data, which show that much of the spalled metal leaves the
clad in large fragments.
Further investigation in this area should examine mechanisms for the formation of unreacted Zircaloy particles.
Additionally, the particulate obtained in recent experiments
should be sized to provide a basis for the present analysis
or future revisions.
105
CHAPTER 4
ZIRCONIUM VAPOR ANALYSIS
4.1
MECHANISMS FOR FORMATION OF ZIRCONIUM VAPOR
The objective of this portion of the analysis is to
ascertain whether unreacted zirconium in a pool experiencing
runaway oxidation could vaporize, and if so, what would be
its subsequent behavior.
It is believed that a cloud of
zirconium vapor, generated in an oxygen depleted region of
the burning fuel pool, may be diffuse or be convected to a
region of higher oxygen concentration, where it could burn
or explode.
Experience gained with metal fires indicates
that it is indeed possible to sustain dust explosions (H3).
Fires of this nature have occurred in the past at manufacturers' scrap piles, where millings from pyrophoric metals
have been improperly discarded.
For situations such as
these, the metal dust is present at (and responsible for)
the explosion and ensuing conflagration.
The mechanism pro-
posed for fuel pool analysis is radically different in two
respects:
1) The fuel for the reaction is a gaseous phase, rather
than a finely divided solid, and
2) The reacting substance must be generated as a result
of the oxidation process, that is, it does not preexist, as metal dust or millings.
106
Pertinent thermodynamic properties of zirconium and its
oxide are listed in Table 4-1. compiled from references (B4),
(Pl),
(Kl),
(Ml),
(L2),
(S7), and (Ql).
It is observed from
Table 4-1 that the density of zirconium vapor (pZr = 0.311
kg/m ) is greater than three times the density of air at the
same temperature (pair,3578K = 0.0948 kg/m3 ).
Thus, it is
expected that a cloud of Zirconium vapor would fall through
the rising hot air.
As the zirconium descends down the
channel, it will either enter a region of higher oxygen content and oxidize or cool and condense on the surrounding
rods and structures.
The analysis of the vaporization of zirconium consists
of two portions: a part related to the vaporization mechanism and a part related to the subsequent behavior of the
vapor.
Quantitative analysis of this mechanism is beyond
the scope of this report; however, observations related to
the following aspects of the problem are pertinent.
1)
Any method to assess the ability
of zirconium to
vaporize must entail a reaction kinetics formulation.
Two
possible modes of vapor generation can occur: the vapor is
generated in the unreacted portion of a piece of clad undergoing vigorous oxidation on its external surface, or the
vapor is generated by an unreacted piece of Zircaloy in an
oxygen depleted environment.
A description of the first
mode of vapor generation must include the migration of the
zirconium vapor through the oxidizing layer without itself
107
TABLE 4-1
Thermophysical Properties of Zirconium and its Oxide
Zirconium
Zirconium
Dioxide
ZrO 2 -
Melting Point, *C
1855.
2700.
Boiling Point, *C
3578.
5000.
Heat of Fusion,
kcal/mole
Heat of Vaporization,
kcal/mole
20.8
5.5
142.150
Heat of Transition
((ct + 6) , kcal/mole
0.920
Vapor Density (kg/m3
@ 3578 0 K, 1 atm
0.31073*
N/A
Vapor Pressure for zirconium, 1949 < T< 2045 0 K is given by:
log P = - (31066/T)
+ 7. 3351 -
2. 415 x 104 T
where P is in atmospheres, T is in degrees Kelvin.
At
higher temperatures, the following measurements have been
made for pure zirconium:
Temperature, *K
Vapor Pressure,
atmospheres
2450
2700
3000
10-2
3850
1.0
*Calculated by ideal gas approximation.
108
undergoing the oxidation reaction.
Neglecting considera-
tions of meling and boiling points, it might be thought
possible that the zirconium vapor could diffuse through the
zirconia matrix to the oxide-film gas interface where the
reaction occurs; however, studies (M2) show that oxygen is
the diffusive ion and that the reaction occurs at the metaloxide film interface.
It is thus unlikely that zirconium
vapor could be released at the surface by diffusing through
the oxide layer.
2) The second possible mechanism, that of vapor generation by zirconium in an oxygen depleted environment, is only
thought to be possible in a configuration such as shown in
Figure 4.1.
The reaction flame front and the depleted oxy-
gen region are depicted in the figure.
It is conceivable
that the clad adjacent to the flame front may experience
sufficiently elevated temperatures for the Zircaloy to vaporize.
However, a mechanism must then be proposed which
explains why the clad remains in the oxygen depleted region
sufficiently long to undergo vaporization, while it is possible for the clad to melt and relocate at a temperature
thousands of degrees lower than the boiling point.
It is
considered unlikely that unreacted clad would be subjected
to heat fluxes due to chemical oxidation high enough to
cause the thin outer layer of clad to sublimate.
None of
the computer studies performed during the course of this
109
FUEL
ASSEMBLY
Figure 4.1
Postulated Mechanism of Zirconium Vapor Generation
110
research have indicated that clad in the oxygen depleted
region reach temperatures at which vaporization is possible.
No temperatures recorded in the experiments performed at
Sandia (discussed in Chapter 5) could have resulted in zirconium vaporization.
3) Despite the above observations, it is conceivable
that the scenario depicted in Figure
4.1
may exist.
It
may be hypothesized that the vapor generated in the unreacted region could be carried up to the assembly exit via
the convective air flows present in the holder, rather than
sinking through the hot air as postulated previously.
If
oxygen is present at the assembly exit, then a diffusion
flame could result, analogous to a Bunsen burner.
If for
some reason there were no oxygen available at the assembly
exit, then the subsequent behavior of the vapor would be
governed by the same plume dispersion mechanics as the particulate, as discussed in Sec. 3.3.
If it is assumed that
a cloud of zirconium vapor has the same ignition characteristics as a cloud of very fine zirconium dust particles,
then the minimum explosive concentration of zirconium at
20*C is 0.045 mg/cm 3, as reported in reference B4.
4.2
QUANTITATIVE ANALYSIS OF VAPOR GENERATION
A one-dimensional implicit radial conduction model of
the cross-section of a fuel rod was developed to assess the
feasibility of vapor generation.
The code is based upon
111
premise (1) discussed in Sec. 4.1, namely, that the vapor
is generated in the inner portion of the clad undergoing
vigorous oxidation.
In the code, the fuel, gap, and clad
temperatures are monitored as the oxidation front sweeps
through the clad thickness.
The oxidation model employed
is that of Chapter 2, a parabolic kinetics formulation which
is evaluated explicitly for each time interval, while all
other parameters are calculated implicitly.
The VAPOR code, listed in Appendix D, is programmed to
vary the thermophysical properties of clad elements as functions of temperature and constitution (Zr or ZrO2).
A con-
stant heat input is assigned, and the external surface of
the clad is subjected to radiative and convective cooling.
However, the code was not programmed to model the behavior
of the oxidation reaction within the clad for the diffusion
controlled regime, for the reasons outlined in Sec. 2.2.
Restricted by limited thermophysical property data at elevated temperatures, the lack of properly characterized data
for fuel and gap properties and the absence of diffusion
controlled reaction kinetics, the VAPOR code can offer
little quantitative information about the potential for
vapor generation.
The one-dimensional nature of the code
(no axial conduction) ignores an important heat sink for
the chemical oxidation energy during a rapid oxidation
reaction.
112
4.3
ZIRCONIUM VAPOR ANALYSIS: SUMMARY
The discussion of zirconium vapor generating mechanisms
in Sec. 4.1 indicates that there is not sufficient data
available from which to formulate quantitative models.
There
is no evidence to suggest that vaporization of zirconium can
occur by the mechanisms discussed above.
Still, it is con-
ceivable that a holder basket may be blocked at a certain
axial level, forming a crucible, so that heat input from the
oxidation reaction in adjacent regions could vaporize a portion of the molten clad.
However, blockage of assemblies
would be expected to reduce the amount of oxygen available
for oxidation, thereby reducing the chemical energy input
and limiting the runaway.
113
CHAPTER 5
COMPUTER SIMULATION OF SANDIA EXPERIMENTS
5.1
INTRODUCTION
In conjunction with this research, experiments were
performed at Sandia National Laboratories in which Zircaloy-2 tubes, heated in an inert environment, were suddenly
exposed to air.
High-speed movies and luminosity
ments of the oxidation reaction were taken.
measure-
Additionally,
temperatures of various components of the assembly were recorded by thermocouples.
This chapter describes modelling
efforts to simulate the behavior of the clad tubes observed
during the experiment.
5.2
SANDIA EXPERIMENTS ON ZIRCONIUM BURNING
5.2.1
Experimental Configuration
The zirconium burning experiments performed at Sandia
National Laboratories employed nine silicon-carbide heaters
of which three or more were sheathed with Zircaloy-2 tubing.
Figure 5.1 illustrates the configuration of the experiment
apparatus.
The assembly consists of a circular chamber 18 cm in
diameter filled witn alumina fluff insulation.
A trapezoi-
dal duct of 26 cm 2 cross sectional area is located along
the axis of the chamber.
Inside the duct, nine silicon-
114
\GAS
INLET
Figure 5.1
Configuration of Zirconium Burning Apparatus.
115
carbide electric resistance heaters of radius 0.635 cm are
arranged, several of which are enclosed in Zircaloy-2 cladding of inner and outer diameters of 0.546 and 0.484 respectively.
33 cm.
The length of the heater rods is approximately
In several tests, a stainless steel holder was pre-
sent to restrain lateral movement of the rods.
The gas flowing through the assembly, either helium or
air at 1 atmosphere, was introduced at the base of the
assembly, and was vented to the room atmosphere through a
short length of tube.
5.2.2
Experimental Data: Air Test #4
The primary objective of the experiments was to serve
as a subject for the refinement of in-pile visual diagnostic
methods under development at Sandia.
The need to make ex-
tensive use of thermocouples in obtaining temperature measurements was subordinate to the need for maintaining good
visibility.
Consequently, there are few tests which yielded
appropriate data.
Air Test #4.
The most useful data were obtained during
In this test, three heaters were clad, while
the remaining heaters were left bare.
the following procedure
The experiment used
(Sl):
1) Heaters were raised to a steady-state temperature
of approximately 10000C in a helium flow at 12
liters per minute (lpm).
2) Individual heater power was set at 175 watts; the
116
airflow was then changed from 12 lpm He to 12 lpm
air; temperatures were recorded at 1 minute intervals.
The time temperature history for the central pin in the
assembly was recorded at the following axial locations:
1) approximately 3 - 5 cm below the top of the pin
2) at the mid-point of the pin
The temperature histories for these locations are presented
in Figure 5.2, along with the temperatures recorded at the
mid-height on the alumina fluff side of the zirconia liner,
which separates the airflow from the alumina fluff insulation.
Figure 5.3 shows the temperature histories for the
following mid-height locations:
1) the centrally located, clad tube
2) an off-center clad tube
3) an off-center bare-heating element
These data are tabulated in Table 5-1.
The temperature rise shown in Figure 5.2 for the time
2 to 11 minutes represents the simultaneous occurrence of
three transient phenomena:
1) the temperature rise due to increased heater input
power, since steady-state was attained for a lower
heater input power,
1600
1500
[
1400
*
1300
$-4
4
1200
0
1100
1000
A
top of center pin
0
mid-height of center pin
900
mid-height of inside of liner
80
0
-
2
4
6
8
10
12
14
16
18
Time from Introduction of Oxygen
Figure 5.2
22
20
24
(minutes)
Experimental Data for Zircaloy-2 Oxidation in Air
a
a
26
28
-
-A
p
L
p_
-A
30
1500
1400
0
o
1300
0)
a) 1200
A
mid-height, center clad pin
0
mid-height, off center clad pin
<
mid-height, off center base heater
1100
1
1000
0
2
4
6
8
10
12
14
14
a
16
1
12
20
20
22
22
Time from introduction of oxygen (minutes)
Figure 5.3
Experimental Data for Zircaloy-2 Oxidation in Air
A
24
26
2
28
119
TABLE 5-1
Experimental Data for Zircaloy Oxidation in Air
Time From
Introduction
of 02,
minutes
0
1
2
3
4
5
6
7
8
9
10
11
12
12.25
13
14
15
16
17
18
19
20
21
22
23.5
24
25
26
27
28
29
30
Temperature, *C
TOP
PIN
MID
PIN
BOT
PIN
LINER
MIDCLAD
MIDBARE
1035
1039
1057
1136
1287
1396
1438
1440
1490
1435
1471
1518
1551
1571
1558
1531
1505
1486
1467
1453
1440
1422
1460
1471
1525
1460
1385
1329
1299
1258
1225
1196
1119
1119
1130
1204
1293
1348
1390
1397
1384
1375
1390
1428
1454
1466
1473
1480
1473
1453
1440
1422
1409
1416
1448
1467
1518
1447
1366
1305
1275
1234
1199
1164
932
937
948
997
1057
1102
1136
1147
1142
1139
1139
1153
1164
1171
1182
1193
1199
1187
1176
1168
1159
1153
1171
1171
1199
1159
1096
1052
1024
986
948
911
868
868
874
905
970
1041
1096
1119
1130
1142
1183
1182
1222
1227
1246
1252
1252
1246
1240
1222
1216
1204
1216
1222
1246
1246
1217
1176
1136
1102
1069
1041
1080
1085
1110
1193
1264
1317
1354
1354
1342
1342
1360
1390
1422
1427
1435
1441
1441
1428
1406
1390
1378
1378
1396
1406
1454
1427
1348
1293
1240
1199
1164
1130
1041
1041
1057
1130
1199
1252
1284
1281
1275
1269
1293
1326
1360
1360
1366
1372
1375
1372
1354
1342
1330
1330
1336
1342
1375
1366
1305
1252
1205
1170
1136
1102
120
2) the temperature rise due to energy liberated by
the oxidation reaction,
3) the melting of the stainless steel spacer.
It has become apparent that the coupled nature of these
phenomena limit the usefulness of the data.
The temperature rise during the time 20 to 24 minutes
is attributed to the fact that the heater input in the operable heaters which remained, was increased during this period.
The following observations have been made (B3) with
respect to the temperature measurements.
The temperatures
measured by the thermocouples may be considerably lower than
the true temperatures at a given axial location.
cies may arise due to two mechanisms..
Discrepan-
Since the thermocou-
ple is in contact with the outer surface of the clad and not
welded to it, the thermocouple is insulated by the ZrO
2
oxide film, which is a much poorer thermal conductor than
zirconium.
As the oxidation front moves into the clad, the
insulating thickness of the oxide layer leads to steeper
temperature gradients in the clad.
If the oxidized clad
beneath the thermocouple fragments, the insulating characteristic is further exacerbated by the presence of a gas
filled gap between the fragmented oxide and reacting metal
oxide film interface.
The second mechanism which may lead
to erroneous readings is the fact that the thermocouples
are not attached to the clad surface, but merely rest
121
against the clad surface when the apparatus is assembled.
Bowing of the clad surface, which is evident in the movies
taken of the experiment, could add large resistances to the
thermocouple-clad system.
It is conceivable that the bow-
ing mechanism could result in apparently large, rapid clad
temperature variations.
COMPUTER SIMULATION OF EXPERIMENTAL TESTS
5.3
5.3.1
Introduction
The experiment modelling effort focused on two sets of
control volume boundaries.
In the first approach, the be-
havior of the rods, canister and gas were modelled, using
the CLAD code.
The second approach employed the VAPOR code
mentioned in Chapter 4, in an attempt to simulate the detailed processes taking place in the oxidizing clad.
How-
ever, this effort was later abandoned because the VAPOR code
does not have the capability to calculate the diffusion controlled behavior of the oxidation reaction front within the
clad.
5.3.2
Experiment Simulation Code: CLAD
The stand alone SFUEL code used to analyze open frame
spent fuel holders was revised in an effort to simulate the
zirconium burning experiments performed at Sandia.
However,
the essence of the code is still that of its predecessor:
the array of fuel rods is modelled as a single component.
122
In the new code, the model has been extended to calculate
the temperature and oxidation potential for a central rod.
The principal components modelled by the CLAD code are depicted in Figure 5.4.
The major characteristics of the
code are as follows:
1) the original air temperature calculation scheme
has been replaced by a donor-cell method;
2) axial heat conduction is calculated for the fuel
bundle only;
3) in the new code, all temperature calculations,
except evaluation of fuel temperature, are performed
implicitly; thus, the stability criteria of Sec. 2.2.5
applies to the fuel/clad nodes;
4) the rods are assumed to remain intact while undergoing vigorous oxidation;
5) the gas mass flow and heater power are input and
constant;
6) the principal mode of convective heat transfer is
by forced convection; assuming that natural convection is negligible, the energy equation is uncoupled
from the momentum equation and solved independently;
7) radiation, conduction and convection modes are included;
8) convection heat transfer models are: forced convection for rod bundle, constant Nusselt number and
123
ALUMINA
FLUFF
ALUMINA
OUTER SHELL
INPUT GAS FLOWS
Figure 5.4
Experiment Simulation Model
124
natural convection from the exterior of the canister;
9) the oxidation thickness is evaluated explicitly,
using the parabolic kinetics formulation of Sec.
2.2;
10) The diffusion-limited oxidation reaction is governed
by the Sherwood number calculated using the ChiltonColburn analogy for heat and mass transfer.
The natural convection heat transfer correlation used
for the exterior of the canister is the same as that for
the global SFUEL code (see Appendix A).
Examination of the
ratio of assembly length to hydraulic diameter (approximately 25) shows that the velocity profile is not expected
to become fully developed at any point in the assembly.
For
the Reynolds numbers encountered in this experiment (on the
order of 10 % 100), no experimental data on heat transfer
coefficients for turbulent developing flow parallel to an
Even the use of the Reynolds ana-
array of rods was found.
logy to obtain a value of Nusselt number from the available
fully developed friction factor data for laminar flows parallel to rod arrays is not strictly correct.
Kayes (K3) gives
the following solution for Nusselt number in the entrance
region of a circular tube:
Nu = Nu
+
K.[(D/x)RebPr1
1 + K2[(D/x)Re bPr]n
125
where for uniform wall heat flux and developing velocity
distribution:
Nu, = 4.36
K1
= 0.023
K2
= 0.0012
n
= 1.0
The fully developed friction factor for laminar flow parallel to a rod array with the same pitch to diameter ratio as
the experiment rod configuration is obtained from the work
of Sparrow and Loeffler (S2).
Use of Eqn. 2.7 yields an
approximate value of Nu= 10.02, for 10< Re<100.
Evaluation
of Eqn. 5.1 shows that the Nusselt number does not vary appreciably over the entire rod length.
On this basis, a
constant value of Nu= 10.02 is employed in the CLAD program.
It is quite possible that the variation of Nu with entrance
length for a rod array may be substantially different from
that of Eqn. 5.1: the fully developed value is thought to
be a good approximation given the lack of experimental data.
The value of 10.02 is also used for the Sherwood number when
calculating the mass transfer coefficient for oxygen diffusion in nitrogen for the diffusion-limited oxidation reaction regime, as described in Sec. 2.2.
Simulation of the experiment required that the code be
provided with accurate thermophysical property data for the
fuel, clad, air and canister materials.
Thermophysical pro-
126
perties for which experimental values at elevated temperatures were not available were input to the code as nominal
values, or the upper limits of available property-temperature correlations.
A listing of the CLAD program is pre-
sented in Appendix E of this report.
5.4
EXPERIMENT SIMULATION: RESULTS
The CLAD program was run for the heat input and mass
flow rates of Air Test #4, performed at Sandia (see Sec.
5.2.2).
Results for the steady-state calculations (prior
to introduction of air) compare favorably with the time= 0
experimental data.
A plot of the predicted vs. observed
temperatures for the heatup of the mid-point of an offcenter clad rod is shown in Fig. 5.5, and tabulated in Table
5-2.
However, the results for the oxidation portion of the
code compare poorly, with the code predicting temperature
rises several hundred degrees higher than those observed.
The divergence of the observed and predicted values is attributed to the causes discussed in Sec. 5.2.2, and well as
the limitations of the phenomenological oxidation model employed in the clad code.
Temperatures recorded during other
experiments have been observed to exceed 2400*C.
5.5
EXPERIMENT SIMULATION: SUMMARY
Experimental tests were performed for the vigorous oxi-
dation of Zircaloy-2 in an air environment.
For reasons
related to experimental assembly configuration and the test
1000
+
++
800
600
measured values for
mid-height, off center clad
400
value predicted by clad
200 1-
2
4
6
8
a
a
10
12
M
14
16
a
18
a
20
a
22
a
24
Time
Figure 5.5
Comparison of Pre-Oxidation Experimental Assembly Heat-Up.
1
26
~
A
28
is
30
128
TABLE 5-2
Comparison of Pre-Oxidation Assembly Heat-Up
Time
(minutes)
Measured,
Off-Center
Clad Rod,
Mid Height
Predicted
Average Rod
Mid-Height
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
22
24
26
28
30
93
260
343
443
510
610
650
704
760
810
848
871
882
898
916
927
932
936
940
945
950
955
960
982
982
128
253
369
476
571
654
724
781
827
863
892
914
931
945
955
962
969
973
977
980
982
984
988
988
988
129
objectives, limited data of a quantitative nature have been
obtained.
Computer simulation of the test results is suc-
cessful for the non-oxidizing heat-up period of the assembly; however, satisfactory results for the oxidation results
could not be obtained.
The cause of this discrepancy is
attributable to both insufficient modelling of the assembly
(assumptions 1 to 10, Sec. 5.3.2) and the phenomena described in Sec. 5.2.2,which are not amenable to quantitative
analysis.
There is insufficient evidence to indicate that
the oxidation model is itself at fault for the deviation
between experimentally measured and predicted values.
Future work in computer simulation of the oxidation
effect must be based on an appropriate database.
It is re-
commended that oxidation models, other than the parabolic
model, be developed to incorporate non-uniformity of oxygen
concentration and variable temperature capabilities.
130
CHAPTER 6
PROPAGATION OF ZIRCONIUM BURNING: CONCLUSIONS
The SFUEL code has been revised and extended to calculate the propagation of runaway zirconium oxidation in
a spent fuel pool following loss of water.
Analysis of
cylindrical and high-density storage arrangements have
engendered the following conclusions:
1) The high density storage configuration is most susceptible to propagation of zirconium runaway oxidation.
2) The minimum decay time for spent fuel to be stored
adjacent to recently discharged fuel (%o90 days decay time)
is approximately two years (decay power < 4.0 KW/MTU), if
propagation of zirconium runaway oxidation is to be prevented for a high density storage configuration.
3) Levitation and dispersion of particulate does not
play a major role in the propagation of energetic zirconium
oxidation.
It can at most increase the likelihood of burn-
ing in fuel holders immediately adjacent to that containing
fuel experiencing runaway oxidation.
Even for this situa-
tion, the effective heat flux due to particulate is small
relative to the radiated and convected heat fluxes from the
assembly experiencing the zirconium burning.
131
4) There are no feasible mechanisms for the generation
of zirconium vapor in a spent fuel pool in which assemblies
are vigorously oxidizing.
In the event that zirconium vapor
is generated, it will either suppress the natural convection
in the holder, reducing chemical energy release via oxygen
starvation, or sink through the rising air and subsequently
burn at the oxygen diffusion front.
Alternatively, if the
zirconium vapor descends to lower, cooler portions of the
fuel pool prior to oxidizing, it may cool sufficiently to
condense.
5) Fuel relocation as modelled in this report acts to
contain runaway oxidation in adjacent fuel holders by removing the heat source to lower, cooler portions of the spent
fuel pool.
6) Fuel relocated to the base of the holder may obstruct the baseplate hole causing extinguishment of the oxidation reaction via oxygen starvation.
It is suggested in
NUREG/CR0649 that coolability of spent fuel during a postulated accident, prior to runaway,may be maintained for a
nearly complete drainage of the pool (water blocking baseplate holes) by drilling holes at various elevations in the
lower part of the holders.
It is apparent that this design
change could exacerbate the vigorous oxidation situation
by: a) allowing hot reaction gases to enter the inter-holder
spaces, resulting in larger convective heat fluxes to adja-
132
cent fuel holders; b) providing more oxygen for the oxidation reaction and thereby reducing the regions of diffusioncontrolled reaction; and c) negating the ability of relocated fuel to cause extinguishment of runaway oxidation via
obstruction of the baseplate holes.
133
APPENDIX A
HEAT TRANSFER COEFFICIENTS AND SKIN
FRICTION CORRELATIONS USED IN SFUEL
(AFTER NUREG/CR0649)
TABLE A-i
Equations Used for Nusselt Number and Skin Friction Coefficient*
Flow Geometry
1.
Forced Convection Parallel
to a Flat Plate
Turbulent Flow
Laminar Flow
(NuD )l= 0.332 Re 0.5Pr 0
(cf) 1= 0.664 Re
-33-
-0.5
Re
(NuD 2 = 7.54 + 0.0234 ReDPr
(cf)2=
ReD
-
24
Pr
0.6
DH
5 x10 5
--
]
(NuD) 2 = 0. 023 ReD 0
8 Pr 0
.4
(cf 2= 0.0014+ 0.125 Red 0.32
/ReD
ReD
3000
Poiseuille Solution
0.8
"Power Law" Solution [El]
Blasius Solution [El]
2a. Forced Convection Between
Parallel Plates
(Applied Outside Fuel
Element)
e
(c f) 1 = 0.0592 Re -0.2
5 x 105
Re
(N)00296R
uDl
[El]
3000
Correlation
[El,S4]
TABLE A-i
(Cont'd.)
Flow Geometry
Laminar Flow
Turbulent Flow
2b. Longitudinal
Forced Convection Between
Parallel Tubes
in an Infinite
Array (Applied
Inside Fuel
Element)
(NuD)2 = 8
Assumed to be same as
(c) 2
= 25
(2a).
/ReD
3000
ReD
Sparrow-Loeffler [S5]
3.
Free Convection
Past a Vertical
Plate
(NuD
Gr
3
= 0. 36 Gr 025
1x10
Correlation
,
DH
Pr=0.71
[El]
(NuD
Gr
x-
u.33([rD H
0. 116 Gr
l x10 9 ,
Correlation
Pr= 0.71
[El]
*To obtain Nusselt number for a particular condition, take the maximum of
(NuD)2 ' and (NuD)3.
and
(cf)2 '
(NuD)l'
To obtain skin friction coefficient, take the maximum of (cf)
136
APPENDIX B
SFUELlW INPUT, OUTPUT AND PROGRAM LISTING
CONTENTS
page
Section
B.l
SFUELlW Input
137
B.2
SFUELlW Output
140
B.3
SFUELlW Program Listing
142
B-1
Sample Input Listing
139
B-2
Sample Output Listing
141
Table
137
SFUELlW INPUT, OUTPUT AND PROGRAM LISTING
SFUELlW INPUT
B.1
The SFUEL program was developed on a Control Data computer operating system.
To implement this program on the
Information Processing System's Multics computer, it was
necessary to incorporate a restart capability in the program.
A namelist output file, under the heading $RESTRT,
is used rather than a BLOC data transfer to facilitate program debugging and allow variables to be altered prior to
restart.
The first card of the SFUEL1W input contains two
entries:
IRESRT = 1
0
Program begins calculations from time = 0.
Program is being restarted with a $RESTRT
data file.
TTEST
Time at which restart file is written; file
is updated during each printout.
The format for the first card is:
1
2 00 0 0
or I10,I10.
The remaining input is identical to that for the original
SFUEL program, and is entered under the namelist heading
$INPUT.
The interested reader is referred to NUREG/CRO649,
138
Appendix D,for a list of the names, dimensions and units of
the input variables.
For the code presented in this appen-
dix, the following variables described in NUREG/CR0649 have
been rendered inoperable in the course of revising the SFUEL
code.
ASINK
DAMP
IPLOT
TIMWON
CSINK
DMWTR
PRMAX
VENT
CPS
EPS
TIMWOF
Additionally, it is recommended that the following variables only be input with these specified values:
FSTR = 1.0
Use of
IBLOCK = 3
other than these values will lead to computational
instability.
A sample input listing is shown in Table B-l.
This
particular case corresponds to the curve in Fig. 2.7, run
for the modified SFUEL code.
139
TABLE B-1
Sample Input Listing
32000
1
PWR CYLINDRICAL RACK
$INPUT
DELT
=
50.
366.
FL
NPRNEW
=
VROOM
=
36.
4.25e9
FSTR
=
1.0
WS
=
21.4
IBLOCK
=
3.
=
28.0
ICHEM
=
1.
WW
XW
NCEND
=
1.
XWL
=
40.6
NSECT
=
6.
XS
=
0.0
NROD
=
XWW
=
3.69
RHOW
=
7.82
CPW
=
0.30
FMULT
=
-1.0
0.4614
NDECAY
=
6.
RCI
0.418
NASS (1)
RCO
0.475
SKBOT
=
RF
0.401
TRDELT
=
0.0
UL
=
TRMAX
=
EPW
POWO
=
289.
ROWS
=
TIMEX
= 43200.
NPRINT
=
FDECAY(l)
=
0.351
0.460
1,
5 * 4
0.0
2232.
0.
6000.
-36.
2.72,
3.76, 5.05, 5.90,
8.20,
11.04
For variables not appearing in the above list, the default
values are used.
140
B.2
SFUELlW OUTPUT
The user has the choice of long or short output format,
depending upon the sign of the input quantity NPRINT.
The
short output format provides output for the hottest pool
section only.
A short-format output is shown in Table B-2,
these results corresponding to the input of Table B-1, with
an elapsed real time of 10 hours.
The variables shown in
Table B-2 are defined in Appendix B of NUREG/CR0649, in order of their appearance in the output.
The following variables have been replaced in the
printout:
has been replaced by TAI+l, which is the tem-
TS
perature at the interface of the airflow control volumes
(*C).
has been replaced by TA3I, which is the tem-
TAVE2
perature of the interholder air control volume
interfaces (*C).
is not used.
12
In addition, the output variable IS has been redefined
as:
IS = 1
oxidation reaction is kinetics rate-limited
2
oxidation reaction is diffusion rate-limited
3
clad is completely oxidized
4
clad has melted and relocated
***
KIT-
i
I
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
i8
19
20
DPFRAC= 9.342E-03
DGFRAC= 5.839E-03
TIME =
3.600E+04
TRMAX= 3.230E+02
***
TROOM=
1.000E+01
PROOM= l.013E+06
DELT= 500.00000
TL(I)
J
TB(J)
TA4AVE(J)
PA4AVE(J)
FOXAVB(J)
GNIAVB(J)
GNI(J,1)
GNI(J,2)
GNI(J.3)
1.122E+01
1.133E+01
1.147E+01
1
2
3
4
5
6
1. 122E+01
1.263E+01
1.264E+01
1.265E+01
1.267E+01
1.271E+01
1.276E+01
4.376E+02
4.375E+02
4.372E+02
4. 370E+02
4.369E+02
4. 369E+02
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
9.697E+01
1.771E+02
1.4 19E+02
1.044E+02
6.460E+01
2. 196E+01
6.544E+00
3. 374E+01
3.668E+01
3.825E+01
4. 135E+01
4.393E+01
1 .000E-10
1.000E- W
i .000E-10
1.000E-10
1.000E- 10
1.000E-10
-2.005E+02
-2.OOOE- 10
-2.OOOE-10
-2.000E-10
-2.000E- 10
-2.000E-10
1.165E+01
1.186E+01
1.209E+01
1.234E+01
1.260E+01
1.285E+01
1.310E+01
1.332E+01
1.351E+01
1.366E+01
1.375E+01
1.378E+01
1.374E+01
1.363E+01
1.344E+01
1.317E+01
1.220E+01
1. 151E+01
1. 153E+01
1. 151E+01
1. 159E+0i
1.171E+01
.- 6
TR
2. 101E+01
3. 129E+01
4.473E+01
6.075E+01
7.897E+01
9.903E+01
1.205E+02
1.430E+02
1.660E+02
1.891E+02
2. 118E+02
2. 336E+02
2.541E+02
2.727E+02
2.891 E+02
3.029E+02
3.137E+02
3.212E+02
3.230E+02
2.958E+02
TAVEI
i.679E+01
2.396E+01
3.439E+01
4.785E+01
6.404E+01
8.261E+01
1.031E+02
1.251E+02
1.481E+02
1.716E+02
1.949E+02
2. 177E+02
2.394E+02
2.595E+02
2.775E+02
2.930E+02
3.057E+02
3. 152E+02
3.200E+02
3.050E+02
11
TAiI+l
2.038E+01
2.9 1BE+01
4. 112E+01
5.595E+01
7.332E+01
9. 286E+01
1.141E+02
1.366E+02
1.598E+02
1.833E+02
2.063E+02
2.286E+02
2. 494E+02
2.685E+02
2.853E+02
2.994E+02
3.105E+02
3. 176E+02
3. 125E+02
3.050E+02
TA31
2.018E+01
2.846E+01
3.958E+01
5.331E+01
6.932E+01
8.726E+01
1.067E+02
1.272E+02
1.483E+02
1.695E+02
i.904E+02
2.103E+02
2.290E+02
2.459E+02
2.607E+02
2.730E+02
2.825E+02
2.883E+02
2.813E+02
1.413E+02
12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
TW
1.730E+01
2.489E+01
3.579E+01
4.975E+01
6.645E+01
8.551E+01
1.065E+02
1.289E+02
1.522E+02
1.759E+02
1.994E+02
2.221E+02
2.436E+02
2.633E+02
2.808E+02
2.957E+02
3.077E+02
3. 164E+02
3. 199E +02
3.003E+02
TAVE3
1.678E+01
2.358E+01
3.334E+01
4.583E+01
6.078E+01
7.785E+01
9.666E+01
1. 168E+02
1.377E+02
1.590E+02
1.801E+02
2.006E+02
2.200E+02
2.379E+02
2.538E+02
2.675E+02
2.785E+02
2.865E+02
2.900E+02
2. 727E+02
13
RCT
1.538E-04
1.538E-04
1.538E-04
1.538E-04
1.538E-04
1.538E-04
1.538E-04
1.538E-04
1.538E-04
i.538E-04
1.539E-04
1.539E-04
1.539E-04
1.540E-04
1.542E-04
1.545E-04
1.549E-04
1.552E-04
1.552E-04
i.543E-04
FOX
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.3001-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
2.300E-01
EGEN=
2.301E+09
ELINRS- 7.461E+04
ECONV3' 4.007E+05
2.939E+04
ECHEM=
ELINRB- 8.207E+04
ESTAIR= 7.546E+05
EFUEL=
4.358E+08
ECONCS= 9.305E+05
EREMDR= 9.090E+05
ESTR=
0.000E+00
ECONCB= 1.041E+06
EROOM= 0.000E+00
EHOLDR= 1.293E+08
ECONVi= 1.645E+09
ESINK= 0.000E+00
ERAD=
8.631E+07
ECONV2- 0.000E+00
ELOSS= 0.000E+00
PGEN=
PLINRS=
PCONV3=
1.366E400
PCHEM=
PLINRB= 9.773E-01
PSTAIR=-5.045E-03
PFUEL= -5.904E+00
PCONCS= 2.731E+01
PREMDR= 3.425E401
PSTR=
0.000E+00
PCONCB= 2.948E+01
QRooM=
0.000E+0()
PHOLOR=-2.256E+00
PCONVI= 6.040E404
PRAD=
3.415E+03
PCONV2= 0.0(X)E+00
OSINK=
01-OSS=
6.391E+04
8.453E-01
1.548E+01
0.000E+00
0
000E+00
IS
OM
000
000
OU0
'rrrlrrlr'IN3AI'IbViSI'SI'YONI'GGNI'CGNI'ZONI'IGNI'riliI
liwrMH'CVMH'ZVMH'dW3iH' ViH4ZVSH'IVSH'WbH'ZVbH'lVbH'inOH
$WHOH'SbGH'NOH' VSH'iVH'inowNoloxoglioaxoolexoolxoolqolND
'MNO /ibiS3b/ IS113WVN
ZOIND'IOIND'XVWIND'GIND'SAVIND'INO'NID'19'Nlid3D'9d3D6VdOD
'ZdOD'IdOD'dOD'HISJ'WOObJ'IdJ'OXOJ'SAVYOJ'AVXOJ'XOJ'OJ'M3NJ
,iwrvisovj'J'bIS3'biVIS3'NNIS3'WOON3'bGW3b3'GVb3'Md3'id3'Sd3
'ld3'3d3'SS013'SbNI13'8bNI13'13'HOlOH3'N393'13nJ3'CAN003'ZAN003
'TAN003'SON033'80NO03'W3HO3,xazaxvwioovbjiombasbaiba
lebanVidG'qOdG'COdG'ZOdG'lOdG'YVWdG'OVbAdO'da'biMWG'iOVJIG'HO
'INDG'XVWOO'OVbJDO'ZI30'AI30'YI30'01130'il3G'30'dWVG'MMD'tAD
SSO'NNISO'SO'bO'MdO'SdO'WOObd3'YOdO'OdO'lNdO'M3NdO'IdO'NO3d34d3
'ldSV'NNISV'SV'bV'ldV'Hd-IV'INDV
'JV'SOV'OV'GV'3WI.L')(VWII'il3G /IHIS3b/ iSI13WVN
iO8XOO 'GXOD 'SIND 'IND 'SdOO
'GAVXOJ /MOIJ/
NOWWOO
THdlV'C3AVI'iwrmilmi /0018/ NOWWOO
NOISN3WIG
NOlSN3WICI
NOISN3WICI
NOISN3WI0
NOISN3WI0
(TZ)SGI'(CZ)bIVO'(CZ)THdlV
(T)ldA '(T)GVlA '(I)IdX '(09VIX
(iz)x (iz)iwrmi '(9'IZ)Mi
'(9'IZ)SI
'(9'TZ)Hi
(IZ)10111 '(IZ)li '(5)31ili '(9)03WIi '(IZ)10001
090
(9)10ig.L '(9)81 '(9)OAVqVI '(9)3AMI '(L) Vi NOISN3WICI
OLCO
(9'TZ)CVI '(9'IZ)ZVi '(9'IZ)IVI '(9'TZ)Oi3AVI NOiSN3WIG
090
(9 1Z)OAVI '(9'lZ)OZ3AVI '(9'IZ)Z3AVI '(9'IZ)013AVI NOISN3WICI
050
(9'IZ)13AVI l(L)qVOHb 6(TZ)CVOHld '(IZ)ZVOHb NOISN3Wia
O io
(IZ)IVOHH '(9'lZ)1Ob '(9'IZ)Mb '(9)dOllb NOISN3WIG
Offo
(9) 108ib ' (9' 1Z) Sb ' (9' 1 Z)Hb "(I Z)lb NOISN3WIG
OZO
(9'IZ)AV03ab '(TZ)GNOob '(iz)ioiiob '(iz)iob NO ISN3Wia
(9'; 1Z)W31-lob 6 (9) ioisob 1 (9) sob ';(9) ab ' (I Z)b NO ISN3W 10
Olzo
000
(9)AvnVId I(L)nVid
MMMcl I(L) Vcl NO ISN3W 10
OLZO
(I Z) CVd '(TZ)ZVd '(IZ)lVd '(9'IZ)WXO '(QiON '(9)SSVN NO ISN3W 10
09ZO 9 11 1 Z)SI '(9'IZ)iGNI '(9'lZ)ZGNI ';(9'TZ)IGNI '(9)AV3301 NO ISN3W 10
OLZO
(9'iz)iwrMH '(9'TZ)CVMH '(9'IZ)ZVMH 'M VIH NO ISN3W 10
09ZO
(9'IZ)ZVSH (9'TZ)lVSH '(9'lZ)ZVHH '(9'IZ)IVNH NO ISN3W 10
05ZO
(9'IZ)NGH 6(9) VSH '(P9)IOGXOD '(L)exoo '(iz)xoo NOISN3WI0
WO
(1 49) OINO'(P9)COINO '(P9) ZO INONO ISN3W 10
(P9) TOINO ' (L)SINO ' (9) SAVIND " (P9) IND (P9) SdOO NO ISN3W 10
OQO
OZZO (9)SAVXOA '(9'TZ)XOA '(TZ)HMd0J '(IZ)AV030A (i z)Hme aj NOI SN3W 10
Ol ZO
(9)NNSIVA '(9)SIVA '(9)MSVA '(9)SOVA 1(91 1z)mba NOI SN3Wia
OOZO
(9'1 Z) SbG "(I Z) IN ' (9) Gba ' (C9)OdCl 4 W9) OdO NO ISN3W 10
OLTO
(P9)ZOdG '(P9)TOda '(P9)dG "(Q NiMWG ' W9) INOG NO ISN3Wia
0910
(9) A13 0 ' (9) MO ' (9) SO ' (C)ZAA303 6 K) I AA300 NOI SN3W 10
OLTO
(9) SO "(9) Cvxv 1 (9) zVxV 1,(9) 1vxV 1(9) IdMV '(9)MV NOISN3Wia
0910
(9) IV ' (9) 1 dSV ' (9) S V ' (9) b V ' (9) AV I (9)OV '(9)SV NOISN3WIa
0510
OfT 10
*H31VM ON HIM
OCTO
100d 13ni IN3dS NMd NO HMS V NI 3SIN 3HnivH3dW3I
SNIISIa WVU90Ud MTqanaS
US
OTLO
ooLo
0890
OLo
HOI*q '(03S)
3WIiHOI/
GVIX
VIVO
OZLO
/9*01 6501/ six lix vivo
/509 I*o 163*1 163*1/ GX 'NOMWI1 'JOMWIi 'INdNi VIVO
/63*1 163'T
'*0009C '5ZO*/ XVWNi 'i13ONi 'XVWII '8WS VIVO
/0
o69o
/93151*1 '1 '9C 'TZ/ XVWNd
6*04C 6*1 "05 '*0/
iOldl
'HiMWO
'M3NNdN
'iN1NdN
'N VIVO
410VA10 '1-130 4dWVO VIVO
'C-319,L 'Z-35 *T 'Z-365*z 'Z-3z6oC 'Z-38L*
0590
/*0*5 '5-R *C '5-3oL*5
' -310*1 ' -319*1
' -396,z ' -351*5 ' -350*9 'C-31 *l 'C-3ZC'Z 'C-39L*C
OL90
0990
'Z-3CC*5
'*0*5 65-366,1 '5-308*L ' -38Z'l ' -3C6*i
090
' -359*9 'C-3Wi 'C-36Coz 'C-39L*C
'Z-365*z
'Z-R6*
'Z-38L*
'Z-3CC*5
IL399 o6
'L311C*9
6*045 '83551'C '83SL5oi
IL3551'C IL3555#1 '939LL*L '93z65*z '53 9*8
'53*1
0190
0090
'C-319*L 'Z-35 *T
OC90
OZ90
6
8
L
9
5
C
z
o65o
'3*1 '3*1 'Z3*1 '13*1
'03'T
41-3*1
1
NMdOJ 'Nmeoi 4ioooi Iiinwi 'AV330N VIVO
'00*1 69T/
'*0
'*CSZ/ dOINX 'iOSNX '01 VIVO
0850
Po
oL5o
0950
0550
050
OC50
OZ50
0150
OM
0 O
OM
/ZIO* "086
'ZI-3L9*5
'6L*Zl NOONWS
'DWS 'DIS 'MOHH VIVO
/599 'zS*L
1 *oi
' C*Z/
SOHN '1OHN
'JOHN 'NOOOHH
VIVO
/5*9 '931LS*Z '93CTO*1 1 59z65i i*C/ OOHb 'VN 'Od 41d VIVO
/51 'z* IL, ILV
xvwN
'Mc13 Id3
'Sc13 VIVO
/Co 'Le 4C99*
'%*/
IcI3 'Od3 'MdO
'SdO VIVO
/oWi IoWi 'oW
'LWU XOdO 'INdO 'IdO 'NOOdO VIVO
L
IND lo.LndNI 'dOiNX 'IOSNX '01 '10001 'NOONWS 'MOHN 'SOHN 'IOHU
o6 o
'JOHN 'NOOOHN
'OOHb
'AV030N 'XVWN
Iiinwi
'AV3303
'Mc13
OPO
'Id3 'Sc13 'IcI3 'Od3
'MdO 'SdO
'XOdO
'INdJ
'IdO
'NOOdO
5
oL o
lmmx 41mx lmx laix lsx lix lax mm
4WOObA 'AM
'in'NOMWIi 'JOMWI.L,'INdN1 'XVWNI '1130U '03WI1
05 0
'XVWII 'SWS 'SMOH 'AN 'OON 'ION 'XVWNd
'OMOd '103SN
4OONN
z
6M3NNdN 'ININdN 'ON30N 'SSVN 'N '.LO-IdI 'W3HOI 'NOO-181 'NiSJ
I
'IA
4N.LMWa 'IOVAla 41130 4dWVO 'NNISO 4NNISV /.LndNI/ IS113WVN
iwrmmx6MMX'IMX'MX'81X'MSX'SX'XVWdX'IdX'HiVdX'ONX'IX'dOINX 2
'IOSNX'SX'X'XGZM'IdMM'XaMM'MM'IdSM'DWSM'SM'TNM'SOM'WOOHA 2
6IN3A'Idn6inlriMi'MigONNISI'NNisigriSi$Si'iNd I.L'OldNi'IdHi 2
lowoouilwooidilxvw 8ilrINI'1130111' li'Oi4XVWi4iOili4M3NIllriii4ii 2
'NOWII'AOWII'03WII'33WIi'3WII'XVWII'1003i'10igi'M3NSI'risi 2
'81'OC3AVI' 3AVI'OZ3AVI'Z3AVi'013AVI'13AV.Llrl Vi'OAV Vi 2
63AV Vi' vilriCv.L6CvilrizvilzvilrilVi'lVi'TANWS'ANWS'NOONWS 2
$13NWS'ONWS40WS4XJWS'CVM3WS'ZVMJWS'W31AWS'
VIAWS
2
4ZVSAWS /ibIS3H/ IS113WVN
ivsiws'zvNjwslivHjws$iwrjws' vejwslviwslawsg vxvwslzvxvws I
'IVXVWS'IdMVWS'MVWS'.LVWS'IdSVWS'SVWS'NVWS'AVWS43VWS48VWS'SIS I
'SMO 8'XOHNOMOH i$SOHN'WbOHb'OOHH'M3NOHH'IOHH'AOH 8'NOOOHH 2
4OOHH /IHIS3*d/ IS113WVN
VOHN4CVOHtl'ZVOHH'TVOHH'A'd' 3H'C3H6Z3N'13H'3 1'10 l'OO 8'N3H4l3H 2
,vH'iwrmblrimb4mb'IN3Ab4doiib'1091b'NNisbsrisblsbliwrmbb 2
I.LNblmsNb'NNisHblsHNblwoobbIriNb'IW3bb'CW3Hb'ZW3Nb 2
'IW3HbeNbNblinob'NNlsiblssoibiiblib'OiNlb'WON30b'AV33ab 2
4aNOObI.Lo.Li3b'iob'W3Hob'10183b'SOb'3b68b4NNisvb'SSOIVb'b4Md
'AvnVidgnVid'HiSd4HlViSd'WOObd$XVWHd'NOW3Nd'aVHd'OMOd'Od
'H3wnNd'NlWd'SIJNlld'BHNlld'id'HalOHd'N30d'13nAd'WON30d'CANOOd
6ZANOOd'IAN03d4S3NO3d'SONOOd'W3HOd'VOd'3AV Vd$ Vd'iVd
'ZVd$lVd$IOIWXO'NWXO'OWX06WXO'i33SN'aONN'VINNdN'M3NNdN'INIHdN
gIdN'dNgCiON'iON'TWNgAV03aN'ON30N'SSVN'N4rixvwlilxvwN'iiN$N
2
2
2
2
2
2
0OU1
0011
G0IN*03803'.' I d-SM*~SM=1 VXVWS
z13a;czI3a=8VWS (o*1*3*1±SA) JI
Z130.c (MMX+IdMM) =GVWS
0611
0911
IdSM*'1dSM=1VWS
OLIr
0911
0511
011
0OM
0111
0111
0011
X13G,1cdMM*Ct1=IdMVWS C0*1*b3'1Si)
Al
X130*Zl30aO l=1dMVWS
X13J'.MM** 1 =MVWS
X13OG'1dSMW '=IdSVWS
Xl 30*SM** 1=SVWS
00UN* (I 31d* I Ob-030h'03 ) *' I d=OVWS
00bN*cd*c Id=3VWS
aObN*cXI30*,03bc ld**l=bVWS
X130*cIdMM=XGMM (01*O3*IdSJ) Al
XOZO=XOMM
0601
0801
oLoi
X13a0czI3a=Xozo 56
0901
MMX+IdMMA=ZI30 (*0oov0ilMMx)
0501
iI
IdMM=Z130
0101
56 01 09 (5'o
3*ISJ) 31
tAX+MM=Z1 30
1/ (IdSM-MM) =MSX
0o1
0N01
SX'* 1+SM=IdSM
(1-N) /1A=X13J 06
0660
MX*C l+MM=IdMM
0101
0001
0860
oL60
0960
og6o
S3fllVA IVIIINI
3
3
MM=SMA
06 010o (o03N*SX) 31
0T6o
0Z60
0160
0060
0680
0=1UVIS I
(o0)AdooaH IIV3
3
£8 0± 09 (Otn3*1ldl) 11
£8 01 09 (0~b3*IbV1SI) Al
(±ndNI'S)Ov3H 58
18 01 09
3
1=3031
3
666 01 09) (I1b3*031) Al Z8S
58 '18 ((S)A03) Al
3
0880
oLgo
0980
0580
08y0
(
Mot
(1)31111 (666=Pua'o68V5')aV3H 18
0=3031 08 3
0180
061S 1 09 V3
18 01 09 (1~3*IHS3'81))
0080
o6Lo
OSLO
oLLo
o9Lo
ogLo
0=1OldN
1=Ibv1I
3
/ooiT
'81
(0)
'
"'001 '81Z/ XVWdA 'XVWdX( 'XVWA 'XVWX Viva
/HOlslZ
)V3dHOT/ GVIA viva
NO0T 'Ifl1Vb3dW31H~t " OVIO
1 3
3
Vt1T
oZL
OUT
i
Od-=NIWd
oi=xvw±
OILI
3flNIINOO 51t
ooLi
C't-Loo/'1+±d3/' 1)/CC)8v= (r)
NNSIVJ
o691
( t-1d3/ 1+id3/ 1)/Cr)8v= Cr) 8±vd
0291
(*1-Md3/* +((r) IdSV*Sd3) /(r)mv) /(r)mvNisj=
(r)msvI zii
OL91
CI -Md3/*1+(SOV~cd3) /r)mv) /r)mvcusj=
(r)so vi
0991
(r) SSVN*cX130*SJtA*'T=S3V
0S9 1
ZIT 0± 09 C'O3N'SX) JI
0~91
C 1-Sd3/ 1+3d3/ 1)/Cr) sv.tisj= Cr) sovi
o001
33sN'i=r 511 00
OZ91 i
C -Md3/'Z) /XGMM=MMVJ
0191
C't-1d3/ 1+CXOMM).'Md3) /XOZO) /XaZo=IMVd
0091
3flNIINOO 011
0651
ex*' Cr)
SV)AldO).I0Hb= Cr) so
oS51 Cr) SSVNcX13G*~ CMM-*.MM- CMY,% C USJ-e 1)3'c* -IdMM) ).cdMM) *MdO).CM0Hb= Cr) MO
oLM
(r)
SSVN3.X130* (SM),SM-IdSM).cldSM)).cSdO),cS0H1=(Cr) so
0951
.L3sN'1=r 011 00
0551
IX*aXI30.'Z130*.cdO*Ic1Hb=10
0'7t
mmx~'s*+CMMX-1MX) +18=18
OMS
MMX'sS'+CIJ3SN) A130= C±J3SN) A13a
0Z51
MMX-1MX+ I)A13a= CI) A13Q
0151
3flNI±NOO 501
0051
Cr) A130+18=18
06W I
r) SSVN).c MMX+1dMM)= Cr) A130
OPT 1
I3SN'i=r 501 00
OLMi
'0-18
o9
1
I-N=IWN
05 i1
VIN~dN=dN
0'71
(INIbdN) SSVI=VINIdN
oPi
1
3flNIINOD 001
0rM
MMX*Z13a*5,+Cr)0vxv=Cr)0vxv (1D3SN'03*r) ji
01'rI
CMMX-IMX)*z13G+Cr)0vxv=Cr)0vxv Cvb3*r) ji
Cr) SSVN)AVXVWS= (r)0£xv
0 0 y1
offi 1(r) SSVN).cZVXVWS= Cr) zvxv
Offl I
r) SSVN.c1VXVWS= Cr) tvxv
oLM
MMXz13*5+(r)ev=(r)9v C33SNw03or) ji
09l(MtAX-IMx)*zi30+
Cr)sv= Cr) sv CI?3or) j i
Oscl I
r) SSVN~c8VWS= Cr) 8v
Orcl 1
r) SSVN*IVWS= Cr) iv
OUT
Cr) SSVN'*1dMVWS= Cr) 1dMV
OM 1
r) SSVN*MVWS= Cr) mv
01C Li
r) SSVN*ICdSVWS= Cr) 1dSV
ooL t
(r) SSVN*SVWS= Cr) SY
o6zi
( r) SSVN*OVWS= Cr) Dv
o8Zt
Cr) SSVN*C.VWS= Cr) jv
Uzi
(r) SSVN).tVWS= Cr) uv
o9Z I
.L33SN'i=r 001 00
05Z 1
Z13i 38x=qivxv
0Tzl
1dMM*cIdMM-Z13G*.ZI30=LVXVWS C0'i~b3*iSJ) Al
OU1
Zi 30*cMtX=LVXVWS
IdSM*IcdSM-MM*cMM=ZVXVWS
0ZZ1
146
WSMG=SMG*DELX/RA
RHOO=PO/(RA*TO)
F0=.23
CPO=FO*CPOX+(1.-FO)*CPNI
FPL=SMG*RHOO*FL
UPL=SMG*RHOO*UL
APL=FPL+UPL
PW=4.*WS+2.*PI*RCO*NROD
DE=4.*SMAXA1/PW
EL=FL*(1.+2.*SMB)/PI
QDENOM=COS (SMB*FL/EL)-COS ((1 .+SMB)*FL/EL)
EGEN=0.
ECHEM=0.
EFUEL=0.
ESTR=0.
EHOLDR=0.
ERAD=0.
ELINRS=0.
ELINRB=0.
ECONCS=0.
ECONCB=0.
ECONV1=0.
ECONV2=0.
ECONV3=0.
ESTAIR=0.
EREMDR=O.
EROOM=0.
ESINK=0.
ELOSS=0.
PCONCS=0.
PCONCB=0.
QSINK=0.
QASINK=0.
QLSINK=0.
QLOSS=0.
QOUT=0.
DELTO=DELT
TIME=0.
X (1)=0.
DO 120 I=1,NM1
X(+1)=I*DELX
Q(I)=(COS((X(I)+SMB*FL)/EL)-COS((X(I+1)+SMB*FL)/EL))/QDENOM
120 CONTINUE
DO 130 I=1,NM1
TL(I)=TO+1.E-10
TLTOT(I)=0.
QCL (I)=0.
QCLTOT(I)=0.
130 CONTINUE
DO 140 J=1,NSECT
TB(J)=TO+1.E-10
TBTOT(J)=0.
1740
1750
1760
1770
1780
1790
1800
1810
1820
1830
1840
1850
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
2020
2030
2040
2050
2060
2070
2080
2090
2100
2110
2120
2130
2140
2150
2160
2170
2180
2190
2200
2210
2220
2230
2240
2250
05zz
OLz
OUZ
OZLZ
OILZ
OOLZ
101dl=ldl
O=IdN
01=NNISI
013*1=XVWIND
'O=XVWDO
'O=XVWdO
Od=WOOldd
NNUNOO 591
NNUNO) Z91
*o=(rl)ZT8H
*o=(rll)ZVSH
*o=(rll)ZVMH
099Z
059Z
09z
o0z
OZ9Z
019Z
OOHN=WKH*d
OA=WOOUA
oi=wMli
069Z
089Z
OL9Z
oi=(rli)ivi
oi=(ri)zvi
oi=(rli)0i
oW
OW
Mz
oo=(rll)W3HOb
eo=(rll)AV030b
*o=(rli)wxo
0T-3*i+oi=(ri) u
01-3'1+01=(ri)si
0T-3'T+01=(rli)mi
oi=(rli)13AVi
oi=(rll)Z3AVi
oi=(rll)0AVI
085Z
oL5z
095Z
055Z
05Z
ocsz
ozsz
015Z
005Z
o=(rgi)si
*ooS9/*i=(rli)ioN
009Z
o6gz
09 z
os z
Oz
oc z
OM
OOZ
oo z
offz
o9cz
oM
09cz
05cz
Oiz
OUZ
ozU
OlU
OOR
o6zz
o8zz
oLzz
o9zz
IWN'I=l Z91 00
103SN'i=r 591 oa
3nNIINOO 091
*o=(Pr)INOG
*o=(zlr)INDa
oo=(ilr)INDO
oo=(zlr)ioexos
3nNIINOO 191
0T-3*Z-=(Pr)IN!)
01-3*t=(zlr)IN!D
01-3*1=(Tlr)IND
191 01 00 (0*10*51ndNI)AI
103SN'i=r o91 oa
3nNIINOO 0 1
(r) AV3301
oj=(r)SAVXOA
ldV=(r)3A0Vd
oi=(r)MOVI
oi=(r)qvi
,o=(r)eb
*o=(r)iosib
so-(Nioisob
oo=(r)eob
LVT
OLZC
o9z£
Os?£
Ozc
4,it =flwj i / 1*013',,
=Md3 it/ £0013',i
=.Ld3 it/ C*013
c
f 1 =Sd3 it / £*013',
=1d3 it/ P013'i
=3d3 it/ £*013
z
',1
=MdJ i / P013'i,
=SdO it / £0l3'ii
=XOdO i / P013
I
1', =INdO i / C*013',,
=1dO i / C*013dT', =NO~dO ,/)ivwboj o16 r
O~cdO±IX '±0S)4X '01 'N03>IWS 'MOHbJ 'SOHbI '1OHb
z
Ozzc'J0Hb 'NOOOHN 'OOHb 'AV330N 'XVWN '±1flwA 'Md3 'Id3 'Sd3
I
o1U£ '1d3 '3d3 'MdO 'SdO 'XOdO 'INdO '1dO 'NO~d) (oC6Vr9)3±;liM
061C
(103SN'w1='(r)o3wI±
'(r)SSVN
'r)
(oZ6V19)3±Iw'
021£
C/,,3w1
SSVN
r
,/)ivwb0d
oI6i
OLI £
(o16Vi9)
3IlbM
091£
(C*013',,
=MMX it/ £013'1,
=IMX 1
051C
/ £01 3'it
=MX it/ £013'i
=S.LX i/
£*013',
=SX
£
OTI£
/ £13'il
=IX it/ £o13'i,
=SX it/ £013'i
=MM
11
z
Mc1
/ £013'
=SM it / P013', =W00OIA it / C*013 ' , =±N3A it
I
011£
P 0131 1
=infi / £013'i =iNdbit
/ £013', =XVWb± it +
011£
/ £0T3'
±13GU± it/ P013' =N0MW1±
/ £o13' =AoMWli
6
11
11
11
001C
P 013',, =XVWI± it / P013',
=SWS it / Po13'i,
=SM0b
it
8
060£
/ 013'i
=A i / P£0i
=03N i / PO13'ii
=lob
11
L
080£
/ 013', =XVW~d it/ PO13'i, =OMOd it/
51i',t
=103SN
it
9
OLOC
/ 1 i =CObN it/ 51' =M3N~dN
it
/
S
I',,=IN
I
dN
11/
Si
1
S
11
090£
11 =0N30N
it/
5 iu
=N
it/
5 i0 =101d I i/ S i,
=W3H1I 1
050£
Si',,1=N3071131i / £013'i,
=bISA
it
/
£013't,
=I11
c
O17oc
/ Pov3'XS/£'o13'x8/P0T3',,
=b-1MW0
it
z
Ooc
it'1',=~n /
£'o13',,
=.I130
it / P013'i, =dWVO it
1
010£ / £013'ii =NNIS3 it / Po13d1',,
=NiNISV iI/ Lv' 11111)ivwboj 0067
ooo£'lx 'SX
'MM
'SOM
'W00NA
'.LN3A
'in
'INdN±
'XVWbLL
'±130U.L
$NOMWIJ.
£
0661 6A0MWli 'XVWIJ. 'SWS 'SM0Oi 'Jb '03Ni 610b 'XVW~id 'OMOd '±J3SN
Z
o86z'GObN 'M3NbdN 4±NIbdN 'QN3JN 'N 'L0ldi 'W3HO1 ')iJ019I 'i±SA '1:1 1
oLR 'bI±MW0 '.LOVAIG '113a 'dWVO ')ANISO ')INISV '(1)31111 (0o6V79)3Iim
691
0S6z
oM6
iflNI 3±INIM3
016z
0161
oo6z
068z
088Z
±1flwj(A3al)bMdaJ=(A3Ol)AV3A
891
AV330N'I=AO01 891 00 L91
691 0± 09
±1flwjc(Aoa i) bMeA= (Aoai) AV03ClJ 991
AV030N'I=ADaI 991 00
L91 0± 09 (001*19*ObN) Al
0L81
691 0±i 09(e0±1±1fliwj)
ji
0=(I)±ON
t=±N3AI (XVW~d3S*W00d) JI
=±N3A I
08£z I
1=£±ON (Cob3sNO0181) l
o-£±ON
(Z~3D01sI0lbo*o3*SX) 31
0=(Z)±ON
0981
058Z
0OR8
0t81
0181
1=(Z)±ON
009Z
o6Lz
oLLz
09LZ
01=1 dNI
81'T
149
4
E1O.3 / " KMAX= ",15 / " NDECAY=",
5
15 / " RHOC= ",E1O.3 / " RHOCON=",E1O.3 / " RHOF= ",
6
E10.3 / " RHOL= ",E 10.3
" RHOS= ",E1O.3 / " RHOW= ",
7
/ TO=
E1O.3 / " SMKCON=",E10.3
",E1O.3 / " XKBOT= ",
8
E1O.3 / " XKTOP= ",E 10.3)
WRITE (6,4940)
TCOOL
I
FDECAY"/
4940 FORMAT(/"
WRITE(6,4950) (1, TCOOL(I), FDECAY(I), 1=1,NDECAY)
4950 FORMAT(14, 1P2E12.3)
START TIME LOOP
170 DO 174 J=1,NSECT
DO 172 I=1,NM1
TAVElO(IJ)=TAVE1(I,J)
TAVE20(I ,J)=TAVE2(IJ)
TAVE30(I,J)=TAVE3(I,J)
172 CONTINUE
TA4AVO(J)=TA4AVE(J)
174 CONTINUE
DO 178 J=1,NSECT
DO 176 L=1,3
AGNI=AXA1 (J)
IF (L.EQ.2) AGNI=AXA2(
IF (L.EQ.3) AGNI=AXA3(
GNIO1 (JL)=-200.*AGNI
DP01 (JL)=-PO
GNI02 (J,L)=GN 101 (J,L)
DPO2 (J,L)=DPo1 (J,L)
GNI03 (J,L)=200.*AGNI
DP03(JL)=PO
GNI04 (J,L)=GNI03 (JL)
DPO4(J,L)=DP03(J,L)
176 CONTINUE
178 CONTINUE
180 DO 550 K=1,KMAX
KIT=K
PCONV1=0.
PCONV2=0.
PCONV3=0.
PSTAIR=O.
IDIREC=-1
START LOOP THROUGH SECTIONS OF POOL
190 CONTINUE
INPUTG=0
IFLG=1
761 CONTINUE
DO 325 J=1,NSECT
TRMAX=8500.
IF(IRESRT.NE.1) NPRNTA=40
IF(IRESRT.NE.1) NPRNEW=5
IF(IRESRT.NE.1) NPRINT=+1
START DETERMINATION
OF AIR PROPERTIES
3280
3290
3300
3310
3320
3330
3340
3350
3360
3370
3380
3390
3400
3410
3420
3430
3440
3450
3460
3470
3480
3490
3500
3510
3520
3530
3540
3550
3560
3570
3580
3590
3600
3610
3620
3630
3640
3650
3660
3670
3680
3690
3700
3710
3720
3730
3740
3750
o6z
OW
otz
o9z
o5z
O Z
Oczq
ozz
Olzq
ooz
o61q
xozm;c(rli)iwrMH+ VMV:,c(rll) VMH=INM ZCZ
EVMV/ 0(0MM),.W31JWS+ (YOMM-CVMV) ;cCVMAWS) =CVMAWS
VMV/(XGMM) dW31H+(XQMM-CVMV)),t(rll)iVMH)=(rll)CVMH
ZCZ 01 OD (1*3N*r) ji
XGMM+ VMV=CVMV (i33SN*b3or) ji
xazm-(r)IdMV=CVMV
5T-3*i=(rll)CVMH (*O*b3*(rll)iVMH) JI
WII)CON1 l(rl0HGH
Z
loxoj 1(r) vxv 1 vmjws l(rll)CVMH ' 3H 'WOOU '(N3VGI)CVOK8
I
'(r'Novei) vi l(rli)mi 'HO '19 'IJ 'ONX 'HlVdX 'I)dONdV 11VO
MMX*OZ=HO
09lq
((rlI)CGNl l(rll)bOH 'OXOJ l(r)Cvxv
z
oLiq
'W31JWS 'dWUH 'CH 'WOUd '(N3V91)CVOHb l(rNov9i)Cvi
I
091
4(rli)mi
'HO '19 'IJ 'ONX 'HiVdX 'T)dObdV 11VO (I*b3*r) ji
05lq
((rll)CONI l(rll)bGH
Z
oliq
loxoj l(r)Cvxv liwrjws l(ri)iwrMH 'C3b 'WOObd '(N:)VGI)CVMI
I
ocl
(r'Novei)Cvi l(i)twrmi
'Ha '10 '13 'ONX 'HiVdX 'I)dONdV
11VO
iwrmtix**Z=HG OQ
XGMM=XGZM
mmx=iwrmmy
Ozlq
oll
00147
imx=iwrmmx
(i)u=(i)iwrmi
09oq
05oq
(i-rli)mi=(i)iwrmi ozz
OQ 01 09
XGZG=XGZM
o60
080q
OLoq
O Oq
OiOq
OzOq
olo
000
o6GE
oLff
o96C
OZZ 01 09 (103N*r) ji
03bial*HiVdX+(Z/(33NIGI-1))*IA=ONX
X13a*(5*-il)=HiVdX
Z/(33Niai+i)+i=amji
Z/(03NIaI-I)+I=N3V8I
OT8101*1
1+ (Z/ (33b 10 1-1) ) ;cN= I
1+11=11
0=11
ITT 01 09 (I'3N-r)ji
I
o56C
oq6C
Tqz
((r) vxv*(r)0XV*WOONd**Z)/
I
(l+*b3*03Niai) ji
((r6N)CVI*Vb*ID*19),cdOINX)-(N)CVd=(N) Vd (1-*b3*03HICII) 31
oC15C
(11300N) /X1303.c (r) iVXV*d3*WOObd=V3d
lNdO*(OXOA-*I)+XOdO*OXOA=dO
oz6C
(r)
oiff
vxv* (r) CVXV3..WOObd3.c *Z) /
oxo9+(Pr)IND=ID
oo6C
ME
ME
USE
(d39)SGV=Vd39
INd3*(Vr)IND+XOd3*OXOD=d3D
(Pr)IN5*(OXOJ--I)/OXOJ=OXOD OIZ
(r)SAVX03=OXOJ
ME
(r)3AV Vl=(r'i)Cvi
ME
ME
ME
OZSE
OISE
Ooss
(r)3AMd=(I)EVd OOZ
Olz 01 00
wOObj=OxOj
woobi=(r'N)EVl
Idn+,o=(N)EVd
ooz oi oo (*oi9*(Pr)IND) Al
MU
oSLC
5 Z 01 00
oLU
OLE
(*O*.Ll*331JI0I*(Pr) IND) Al
3NnSS3bd WOOH 3AOQV 3bnSS3Nd SI d
SN3alOH N33MI39 'i 13NNVHO
OST
o9S
QSS'i
05q
OC5
OZ5
~o6z
oo5r
o6T17
wooi= Cr'N) ZV±
Idl+*0= (N)ZYd
oSz o± 09 (,o*±
(z'r) 1ND) JI
0± 09 (o±*J1*3l31IaI(z'r) IND) 31
0~6Z 01 09 (003*SX) Al5q
13010H- ONV 3fl13flb±S N33MI39 4Z 13NNVHO
(ro)3V)tr X~(ZNV+AOdCN~
(P*3Nr) i
CWon~- Cr 'N)0lv)*'d+L4AN0Jd=CAN0~d
Osq1
CAN0~d=CN03d (t~b391DI) JI
OL r
o9'Pr
(r) Cvxv*. Cr) CVXV*WOO~d*
oq
(r)Lvxv.,c(r)CVXV-*WOObd*
* )
(,1
0x09= CL'r) ±08x09
v*N.c1*1O.lG
-
*Z)/( r'N) LVl*Vc
I D*IDd~)X)+
z
1fl inc ( (Cr' N)Li.v)/W00O1d) '.cWS+
(N)CVd-(r)3AVqVd=(L'r)da (*o±i9*Lr) 1ND) JI
Ozqq
I
3flNIiNO3 L'iZ
TqZ 01 09 (TWN1II)JI
3flNI1NOO 0tT
((r)Lvxv/(r' I)AVi*r/(xzm).twrws+vmv. Lcvmws)+
z
((r'move1)LCvi- (rIai1 V)*dV
r vv r VVWOd
I
(VU*1I9.cI9) -03bilaI*r' OL3AV±/W00bd cOWSM(NVl)CVd=(0mII)LVd
01 qT
oqr
o6L'i
o8Lci
oLL'r
091q
OSL'T
OqcqCr' I)OL3AV/((r' I)oCAVI-
0=HdIV
I*I=HdlV (0OOLC*3D*0U)
Al
0*=HdIV 5Q~
Cr' ;)AVI)~V~d+tllviSd= diViSd
3flNI1NO3
ZZ
3flNIIN0o ML
C(r) Cvxv~c Cr) LYXV*W00bd) / I
CVub*I~19) -33bl i U i C~ 3AVI/W00bd*c9WSM-(COVa I)LVd= (amj )LVd
Cr'1) 0L3AV1/(C r'1) 0L3AV1- Cr'1i)L3AV±) sVd+HIV1Sd= IY1Sd
xOzm/ CxazM- Cr) 1dMV); Cr'i)
LVMH=(r'1)
iwMH
C(XOZM- Cr) ldMV) * /mmx) / Cxamm'* Cl) tiI
vo)
=
(r'Il)
LVMH
(r
It)i3V*b
d*Ob*()1HdIV=C() lI VO
oe z/C( r11)C3AVI+ (rI +i) ]AV) =C(r Iam I)LCvi
30Cii*1+ CZ/C031dIUI-t))*N=l
IWN't=ll LLR oa
Cwoob±' Cr) 3AV1 VI'MMX'X3'±13a'r) 1NVHO liVO
3flNI1NOO LZZ
(i-rli)mi= (iii) twfmi
3flNIINOO IIt
ZZ 01 09 tb3*r)JI
r'ivei)Lv±-(r' I)L3AVI±'*Z=Cr'ami'u)Lvj
OzcqCINM+ I
oL~r
oLi C r'1)0L3AVi/Vd+Vd0D*Z) / IVH+V~d+C(r INova I)Lv±vdD** Z)=(r11)C3AVI
ooLq
Ci)
tiwixazIW Cr' i) wrMH+ Cr' i)m±*Cvmv*
r' I) LMH=1VH
TST
0805
oLoS
0905
0505
005
OC05
OZ05
0105
0005
o66
086
oL6
o96
in* (((r'N) ZVi*V I) /W0011d) *DWS+
(N)ZVd-(r)3AV Vd=(zlr)dG (,o*iD*(zlr)IND) Al
3nNIiN03 oLz
((r)zvxv/(rll)Z3AViicVAWS+
z
(r'Nove i) zvi- (r I ami i) ZVi) *HdIV) * ( (r) zvyv).c (r) ZVXV*W00lJd) /
1
(T8* I D*ID) -018101*
(r, 1)
Z3AVI/W00bd*5WSM- (NOV 8 1) ZVd= (OM A1) ZVd
*O=HdIV
lol=HdIV (*000 *3D*ZRI) JI
O*Z=HdIV 59Z
(rll)OZ3AVI/((r'I)OZ3AVI-(r"I)Z3AVi)*VOd+lllViSd=blViSd
(rNovei)zvi-(r'I)Z3AVI**Z=(rlamji)zvi
(INM+
I
(rll)OZ3AVi/VOd+VdODkc*Z)/(IVH+VOd+(rl)4ovei)ZVI).cVdOD).c'Z)=(rll)Z3AVi Z9Z
o56
o /((r)IdSV+(r)mv)).czv ijws+*z/vjws=vjws
o6
z/ (r I
0 61
i) bi).c (r) IdSV+
(r) mv)
*z/ (r) I dSV+(r)mv)
(r' 1) MIH+* Z/IVH=iVH
(r- 1) ZVbH+'Z/
I MM= I NM
loxoj l(r)zvxv Izvbjws l(r,'I)ZVbH 'Z3b 'WOObd '()40VSI)ZVOHH
I
oo6 '(r'Novei)zvi
l(rli)bi
'HO '19 'IJ "ONX 'HIVdX 'T)dUdV IIVO
w6
Z
oz6
(zi l(rll)bOH
o6P
Z9Z 01 09 (0*T*b3*biSJ) JI
08P
*z/((r)IdSV*ZVSJWS+(r)mv).tzvmjws)=vjws
oLB
(r I i) si* (r) IdSV*
(r 1 1) ZVSH+(r' i) mi* (r) mv).,. (r 1 1) ZVMH=IVH
(r)IdSV;,.(rll)ZVSH+(r)mv*(rll)ZVMH=INM
51-3*i=(rll)ZVMH (*O*b3'(rll)ZVMH) JI
((rll)ZONI 6(rll)bGH
Z
loxoj g(r)zvxv Izvsjws 11(rll)ZVSH 'Z31d 'WOObd '(NOVSI)ZVOHb
I
ozsq
l(rl)iovs0zvi I(r'i)si 'HO '10 'IJ 'ONX 'HiVdX 'I)dObdV
IIVO
((rll)ZONI l(rll)bOH
Z
loxoj l(r)zvxv Izvmjws g(r,'I)ZVMH 'Z3b "WOOlid '(-AOVSI)ZVOHH
I
o6Lq
l(rl'Aovsi)zvi l(rli)mi 'Ha '19 '11 'ONX
'HlVdX
'I)dObdV
IIVO
(MM+IdSM)/ZVXVWS=HG
33bl0l*HiVdX+(Z/(03 liai-i))*ij=ONX
X13G).c(5*-11)=HlVdX
Z/(03bl0l+l)+I=aMAI
Z/(33bl0l-l)+I=)13VSI
oCLIT
03bl0l*ll+(Z/(O3blGI-1))*N=l
IWN'1=11 oLz oa
((r)zvxv*(r)ZVXV*WOOHd**Z)/
I
((ri)zvi).cvd*io*io*ioeNx)-(T)ZVd=(I)ZVd (T+*b3*O3bl0l) Al
098
OSP
OP
M
olsq
oosq
oSLq
oLLq
o9L
oSLq
oqLq
ozL
oiL
ooO
o69q
((r)zvxv*(r)ZVXV*WOObd**Z)/
OM
oL91
099
1
((r'N)ZVI*Vb).419*10*dOINX)-(N)ZVd=(N)ZVd (1-*b3*33bl0l) A[
(1130)AVb)/Xl3a*(r)ZVXV;dO*WOOHd=VOd
INdO*(0X0A-*0+X0dO*0X0J=d3
059
oxoo+(zlr)IND=ID
01M
Mq
(dOD)SSV=VdOD
INdO*(zlr)IN9+X0dO*0X09=d39
(r)3AVqVd=(I)ZVd 05Z
o6Sq
(z'lr)IN9*(0X0A-*T)/0X0J=0X09 09Z
(r)GAVX0A=0X0J
(r)3AVqVl=(rli)zvi
OZ917
0191
009q
oss
oL5q
09Z 01 00
woobj=oxoj
153
C
C
C
C
2
+(XKTOP*GI*GI*RA*TA2(NJ))/(2.*PROOM*AXA2(J)*AXA2(J))
IF (GNI (J,2) .LT.0.) DP(J,2)=PA2(1)-0.
1 -(XKBOT*GI*GI*RA*TA2(1,J))/(2.*PROOM*AXA2(J)*AXA2(J))
GOXBOT(J,2)=GOXO
PCONV2-PCONV2+GCP*TA2(N,,J)
CHANNEL 1, WITHIN ASSEMBLY
290 IF (GNI (J,1)*IDIREC.LT.0.) GO TO 325
IF (GNI(J,1).GT.o.) GO TO 300
PA1(N)=O.+UPL
TA1(NJ)=TR00M
FOX(N,J)=FROOM
GOX(N)=FOX(NJ)/(1.-FOX(NJ))*GNI(J,1)
GI=GOX(N)+GNI (J,1)
GO TO 310
300 PA1(1)=PA4AVE (J)
TA1(1,J)=TA4AVE(J)
FOX(1,J)=FOXAVB(J)
GOX(1)=FOX(1,J)/(1.-FOX(1,J))*GNI (J,1)
GI=GOX(1)+GNI (J,1)
310 IF (IDIREC.EQ.-1) PA1(N)-PA1(N)-(XKTOP*GI*GI*R A*TA1(N,J))
I /(2.*PROOM*AXA1(J)*AXA1(J))
DO 320 li=1,NM1
I=N*((1-IDIREC)/2)+II*IDIREC
IBACK=I+(1-IDIREC)/2
IFWD=I+(1+IDIREC)/2
XPATH=(Il-.5)*DELX
XNC=FL*((1-IDIREC)/2)+XPATH*IDIREC
GOX(IFWD)=GOX(IBACK)-OXM(I,J)*IDIREC
IF (GOX(IFWD)*GOX(IBACK).LE.o.) GOX(IFWD)=0.
FOX(IFWDJ)=GOX(IFWD)/(GOX(IFWD)+GNI (J,1))
FOXAV=(FOX(IBACKJ)+FOX(IFWD,J))/2.
Gl=.5*(GOX(IBACK)+GOX(IFWD))+GNI (J,1)
CALL APROP(2, XPATH, XNC, FL, GI, DE, TS(I,J), TA1(IBACK,J),
1
RHOA1(IBACK), PROOM, RE1, HSA1(1,J), SMFSA1 , AXA1(J),
2
FOX(IBACK,J), HDR(I,J), IND1(I,J))
CALL APROP(2, XPATH, XNC, FL, GI, DE, TR(I,J), TAl(IBACK,J),
1
RHOA1(IBACK), PROOM, REl, HRA1(I,J), SMFRA1 , AXA1(J),
2
FOX(IBACK,J), HDR(I,J), IND1(I,J))
GCP1=GOX(IBACK)*CPOX+GNI (J,1)*CPNI
GCP2=GOX(IFWD)*CPOX+GNI (J,1)*CPNI
GCP=(GCP1+GCP2)/2.
GCPA=ABS(GCP)
CP=FOXAV*CPOX+(1.-FOXAV)*CPNI
PCA=PROOM*CP*AXA1(J)*DELX/(RA*DELT)
IF (HRA1(1,J).EQ.0.) HRA1(I,J)=1.E-15
WK1=HRA1(1,J)*AR(J)+HSA1(1,J)*AS(J)
5090
5100
5110
5120
5130
5140
5150
5160
5170
5180
5190
5200
5210
5220
5230
5240
5250
5260
5270
5280
5290
5300
5310
5340
5350
5360
5370
5380
5390
5400
5410
5420
5430
5440
5450
5460
5470
5480
5490
5500
5510
5520
5530
5540
5550
5560
5570
5580
0809
oLo9
Cr' 1)ZVMH+ ( (r'i)m.-
0909
00o9
0 09
OZ09
0109
0009
o865
oL65
Cr, 1)C3AVi) ( XQZM- Cr) ldMV) * (r' 1)CVMH= Cr' 1)Mb
('i (r, mi)- **Cr, i)si)' *91s'.(r) msvd=ms~ib 5c
iwm+
-lm~orim~
5U 01 09
£c* i) iwrmi1,c!9i s.Acwrvj)A,-x azw.,'t(r,i)1WrMH-= (1))10
C')iob-iwrMb=(I) T
OU 0100 (P'3Nr) ji
1wrmuO+((i) iwrmi- Cr' )i3AVi)*xazm*(r i)1wrMH=1wrmb
0665o (~,~i) iwrm1- * cr,'i) mi)-*coi svwrvj=iwrmbb Sz
XaMM=xOZM
mmvj=iwrvi
oS65
0t65
oC65
szc 01 09
xazo=x ozm
itMvj=iwrvd
01 6g
9ZC 01 09 (I'3N-r) ji
TWN '1=I 0I 00
i33SN'1=r o5C 00
oo65
o685
0885
oL85
S1N3W313 Hfl1Jflis
1VOUt1I3A 01 S3xfl1A 1V3H
0985
0585
0
3
3
ZZ 01 09 (1-*O3*33blaI)
31
3flNI1NOO 5Z
oq85
OC8s
3
ozSS
01 SS
o6LS
OBLS
OLL5
Cwoou- Cr'N) lvi) *cINd3) CI 'r)IN9+XdO*(N) X09)+IANO~d=1ANO~d
IANO3d=INO~d C1~b*!1dI)31
(i)xo9= (1'r) 109x0
0085
o- ((r) I vxv* (r) I VXVktW00d*
- )/((r'1) I v±i
I
V~I9cI *I 9*cNX) -C(1)1Vd= (I'Ir) da (oo 11(1'r) INO) Al
inw(((r'N) I V±i
z
Vb) /WOObd) *cows+ (Cr) I vxv3., (r)I VXV*CW00d)r* /)C(r'N) tv± V t
VIc9I.cl9d01)4X)+(N)1Vd-(r)3AVl Vd=C1'r)da C*oi9,(i'r) 1N9) Al
09L5
05L5
3flNI1N03 0Z
oqL5
(Cr) tvxv/(r' i)13AVI*VAWS+
Z
oCL5
((r INva i)I vi- (r'o I i)I VI) *~HdV) * ((r) I vxv*c (r) I VXV*~WO0~d) /
I
oZL5
(vb).i9).io) -33blak r'I)I3AVI/W00OId'A9WSM- CN3VSI) IVd-(aMdl) l~d
(C(r) Ivxv), Cr) IVXV*W00~d*~ 0 ) / I
(r1 )1v )cNc1*19.i e )-()IV =()1V
(10b3* 3N 1aI) JI
0O=HdIV
I=Hd1V (*OOOC30*I3b)
Al
01ILS
OOLS
o695
0*Z=HdIV SIC
0895
(r'I)oI3AV±/(Cr'I)ot3AV±-(r',)Imv±I)*V~d+NVISd=ulViSd
(r'rwN)I3AV±=(r'N)IVj. (IWNb3*)Al
z/((r' 1I)
I3AVI+AVi) =(r 'Nove ) lvi
0995
C11M+
I
Cr'I) 013AV1/V~d+Vd39) /(jVH+V~d+AVI*~Vd9) - Cr' I) 3AVi
(r'i-i)I3AVl=AVi. (1*3N*)dI
Cr) 3AV~Vl=AVJ. ZI
o 95
z/()s*((
o005
)s*(
or/ Cr) sv* (ivsws-iv~uws) +vdws=viws
)IVH-(
)NLs( 11
N)+V=V
ri Cr) sv3. (Cr'') VSH- Cr')I) VNH)+TMM=INM
OZ95
01 9S
ZI
0095
0±09 (O'3*USA) Al
oz/(C r)Nv~iv~dws+ Cr) sv'ivsdws) =v.ws
(r' I i) bi*(r) Nv~c(r I i) I VH+(r I ) s.L*r) sv* (r ) I VSH=iV H
o655
0959
0o99
0SS9
OtrS9
0£ S9
0159
005S9
ooS9
09'19
0O=H1VdX O t,
9ztr 01 09
rrr=r
rrr=rr
sn i 0±0 CP±i*rr-r)) ji
Cr)m01je 11vo
9zqi 01 09
r= rr
enr 010 Cv19CiDrr-r)) ji
9zq i
010 (0%L1C1+r)G1ND) JI
O£'r 01 09 (1:)SN19or) ji
r=rr r
i+r=r 9zq
o=r r
01 9
00~9
060
oS80
OM£
0909
oS50
otr£
oM£
OM£
o=r
1
3flNI INOO '7Z
o l+r) 81No+ (r) 8 1NO) *5S' (r) GA 1ND
(C'r) IN9-(z'r) 1N9-C1r) 1N9-Cr)51N9=(t+r)51N9
103SN'i=r 'TZn 00
0o=(i)nfVid
0= CI+1J3SN) 5X09
.0= (T+1D3SN) 81N9
0o=(I)Gx09
0=(T) SIND
i01 0 (01i9*ofldNt)JI
3flNI1N03
oL'
01M£
0o£0
o6z9
SM01.A SSVW 'S31A3dOU M01A 3SV8
0899
tTt
01 09
oLz9
09Z9
OSZ9
otrr9
oZZr9
ol Z9
00Z9
0619
0819
oLt19
0919
05t 9
05C
((r,' I)N-Cr' i) I3AV) * (r) sv* Cr' 1)1VHH+ ((r, o uL- (r'6)Z3AV1) *
sbb-((r' I)u-(r'Ii)I3AVI) *(r)tv* (r'Ii)ItYH= (r'1)N
Cr' I)sba+(r' Imba=(r' i)mta
Cr' I)sb+ (r I) MO= Cr' 0)mb
SU£ 01 09 (0O3N*SX) Al
bis '*(C(r) sv* Cr' Ii) IVsH+ (r) IdSV*. Cr'II ) ZV SH) -= Cr' Ii) sba
msb-sb~3-isI'*(((r i)si-Cr' I) 3AVI)*
I
I) I VSH+ ((r' I)SI-Cr' 4)Z3AV1) * Cr) IdSV'* Cr' I)ZVSH) = Cr' I)sb
Cr) svx. (r'
1y'* Cr' i)u±)* 91s*Cr) s ov j= s~du
(iy*~ Cr' Ii) siCxazM- Cr) ldMV) *' r'I) VMH- Cr' i)ma=
Cr' I)mOO
(Cr i ) mi- r II)
C3AVI) *(X0ZM- (r)
IdMV) *(r 11)£VMH+ (r i)mb=V
OZt9
at 19
0019
0609
Ii)m
LM 01 09 C±3SN*3r) ji
Ek*rim~i*irj(~sj*
t
usj*. Cr) mv'S Cr' 4)ZVMH- CXOZM- Cr) ldMV) * (r'
1)£VMH-= Cr' I)0mb
twrm~b-msu?+((r' l)m1-(r' I) 3AVI)*N±SA(r)mv*
I
090t
050L
0O=WON3Od
0=H3wflNd
oLtr 0± 09 C0±oifDoicNl)31
HfNliN03
TC7
00oL
o0oL
TofsSThd 'S3l±H3d0~d M013 3SVS
OZOL
o1oL
000£
0669
0869
oL69
o969
o569
ot,69
oC69
oz69
ot 69
0069
0689
0999
oL99
0999
0599
0~99
M~9
OZ99
0199
0099
o6L9
o9L9
OLL9
09 £9
05L9
o1TL9
oUL9
ozL9
ot L9
ooL9
0699
0899
oL99
061 0± 09
1+0I101
3flNIINOJ Zi
((i+r) flVid+ Cr) flVid) *5*= (r) AVlid
lVidG+ (r)nfVid= Ct+r)nfVid
a '' (vxv/ Cr) 3AV TV''z/ (Cr)8v).Otv83ws+ Cr)±v'.'qv±3ws)+ +
(r) 'rvi- (I+r) 17Vl) *'HdIV) *c(qTVXV).c 1 VXV*.W00Ud) / CVb*~ I 9k' 19) -=flVidO
*0=HdIV
I=HdIV (*OQO 3Doq3H) AlI
0 Z=HdIV
(r)vi- Cr)3AVqVl*cZ=
C14r)qvi~
Cr) 0AV Vi/(C
r)0AV1ViCr)
3AV
1V1)
cV~d+
I
VISd=
I
ViSd
Ci~m+Cr)OAV Vl/
1
V~d+dO'.NI 9+dJ9*cZ) / iVH+V~d+NlidJ9+Cr) tVi,.dJ9D) Z=
Cr) 3AVqIV
oz/C(r't) £v±*cCCr)8dJ9-(Clr)Sd39) sev)+Cr't)zvi±tCCz'r)8dJ9(zCI'r) SdIO) sev) +C(r'I) i vi* ( (1 r)Sd39-C I'r)SdO) sGV) )=N I dOD
*/CClr)iosx09-CPr) 1ND(Clr)±oexo9+Clr) lN)sav+Cz'r)±oex0)- Czr) IN9- CCz'r)±oax09+
C'r) IN9)S8V+CI 'r)±oaxos-CI 'r)1N9-CCi'r)18x09+CI'r) 1N9)Sgv)=NI9
Cr) ai* Cr) ev.-. Cr)VGH+ Cr' I)UJ.' Cr) iv*' Cr)
'ViH=iVH
1TViH=I NIM
Cr) 9avc Cr) qVSH+ Cr) iv.,. Cr)
51-Pi=Cr)rV81H C0o3*Cr) VSH) 31
C±13a*VH) /±X)c Cr) 8V'.dJ*W00bd=V~d
INd3O' C r)SAVXOJ-' I)+X~dOc Cr) 8AVXO3=d3
*Zi(Zd39+T d39) =d39
lNdO. C+r) 8I N9+Xdc
CI+r) 8X09=Zd39
INdJ' Cr) SINS+X0d3*' r)8X05=td3
CBONI lamJ0
Z
'Cr)SAVX03 ' vxv '6vs3ws ' Cr) VSH 6 - 'W00Od ' Cr) 'V0HH
T
'Cr)qvi 'Cr)si 'HO '10 '18 '3NX 'HlVdX 'T-)d0~idV 11VO
CBONI '8UH
Z
'Cr) 8AVXO3 6 vxv 'Tvi.3ws ' Cr) V.H ' 3U 'WOQ~d ' Cr)Wl-l0H I
'Cr)qvi 'Cr'T)Hi 'HO 'ID '18 '3NX "H.LVdX 'T-)d0OIdV 11VO
SIX's' =HO
t-sII(o0*1vI) A1
0999
0599
0~99
o9
oZ99
ot199
0099
o659
0859
i= a
5T-301=10 COo3I90) Al
((I+r)sxo9+CI+r)aIN9+Cr)sxos+Cr)SIN9)*5'=1D
(T-r)A13a*cS'+HlVdX=HlvdX CI'3NC) 31
(r)Al30*5*+HlVdXwHlVdX
133sN'I=r zL~i 00
Ct I )Lvl= (T)
'ivJ
*0=3NX
013' I=H.LVdX
9ST
0o5L
05SL
OMS
oz5L
01 SL
005L
0Etr
o -o= (r) 80
o00 = (r) ±08±0b
OP~ 0± 09 (0b3*9flndNI0dI
VGH- (r) sba
ai±o s.%. Cr) e±v j* - Cr) ev. Cr)
(c r)
(r) eob-ab0+ (C(r) si- Cr) 3AV~i) * (r) av*. (r) tV9H= Cr) aO
abb- (C(r 0 bi- Cr) 3AV V) *(r)±iv's(r) V±H=Cr)±08±ib
(r) 91) ).c9 i s*C Cr) sivi-abbi
o*(r' n) b±- **.
(r)d0±0-NN ISO=N I Sb
A- Cr) dollO
S-*. (r' IWN) u±) *9IS C*r)ANSiV
033sN'T=r o9P 00
(~
I
*0=)NIS8-O
OPiL
3
os'iL
3
oLtiL
o9 iL
3
S±N3W313 1Vnfl±JflUs
IV±N0ZIbiH 0± S3xflii ±V3H
3flNI±03 oL'i
3flNIN03 091
((Cr) 3AV 1TVd- (1 r)do) SeV'XVWdO) 1XVWV=XVWd0
o9tr01 ±09 (vb3*()±ON) Al
C'I=i 09qi 00
01Qt~
oC L
oz L
oiTL
oo iL
(Ct+r) Vd+ Cr) Vd) 3cS*= Cr) 3AV 1Vd
C1+r)nvld+T) qVd=(1+r) Vd
£ON=(C)±ON (1*3Nr) ji
o6 L
o8CL
oLM £
±33SN'i=r oL 00
0= (i)ION
'd
WON30d/b3wflNd= (1)
3nlNO
o9CL
05CL
O'TU
oUL
5
(C)13303+(Z) 1A303+(1) tA303+lI3wflNd=b3wflNd
oMt
3lI±NO3 0'7
ooU
o6zL
*0=(l)TAA303
*0=(l)Z.A303 GVT
o r 0± 09
((r)AvnVid-(1'r)da)).()ThA03=(1)t.A303
o8zL
oLzL
o9zL
(1'r) OdO=(i'r) ZOdG
O'zL
0±l 09 o ((r) 3Av~d * 1(1'1r) 0dG) AI
(1'r) da= (1r)LOda
(1'r) IN9=(i'r)i0IND 9C'T
MV 0± 09
'Tdo=(1'r)C0d0
ozzL~~8P
oizL
oozL
061£
ML(1'r)
MVi 0±0
091 L
((r)3Av'Td±0(i'r)0dO) Ai
OSi £ (i'r) da=(1'r) Z~d0
(1'r) IN9-(1'r) 0N9
0'T I
oti
ozL £f 6
ottL
001£
o~oL
9C7 0±09o ((r)3A0V'd±I9*(1r)da)
01 0±0 (I1'M0*(1)±ION)
i't=1 O0r
CI0N=(C)±0N C1*3Nr)
±33SN'i=r OS'T
Al
Al
00
Ai
00
0O=XVWda
oLoL
LST
008
0109
0009
'i@013'=XVWJI.
H6'C*013'=3V8U9
HO!'
*013'=3V Ad0
HOT
I
I
1130 'W00~d 'OW0~li
Oos
'31dNI 'OVbADO '3VHdG '11N1 '3W11 '(1)31111 (oZoS'9)31lbiM
oCLZ-ldbi=3ldbI.
£LM-woao~.=woobl
3flNIJ.N03
086L
*LZ-NNlIS=3NIS
066£
K
(L)ON IMb~
SC 01 09 (±S3ii±±13W11)dI
Al
(*0i9*3Wl)
(01-XVW1)/3Wu3'sxvw1=ovbjia
oL6L
o96L
oSGL
0*1=ovbU1O
XVW IND/XVWD9OVU.A9O
0~6£
oM6
oz6L
o16GL
oo6L
099
01 09 (*MLoil1XVW1
o99L
595
ldV/XVWdGOVHdO
*M-ldb1= (ldN) idA
o09/3W 11= (ldN) 1dX
T+ldN=IdN
S9S 01 09 (o~b3*dI) Al
0OV1NbJdN6IldN) Al
1+dN=dN 095
o£9L
at
9£
3
1fl01Nl~d
098L
08rL
oU9L
oz9L
009L
I
'HOT
(CO013 d=3VbA9O 'H01'C*0!3'=3VH~dO
'i*0T3dT'3WlI IV 30N3ANOO 01 0311VA SM01A SSVW HSC/).1VWb0J 0105
ovb9oa 'OVU.~dG '3W11 (0105'9)311W'
XVW IN9/XVW9O=3VbA9G
ldV/XVWd0=3V 1Ada
3lI1NOJ 055
o6LL
OCS
(dWVG-*I)*c(1r) IN9G+(1'r) IN9=(1'r) 1N9 oZ5
o9L £3flNI.NOO
OLLL
13sN'i=r oC5 00
0%S 01 09 U0o3*3W1) Al
095 01 09 (XVWIN9ktc10011'XVWGOONVIdV'*1001V'XVWdO) Ai
3flNINO3 015
3flNIINO) 005
C((1' r) IN90) Say'xVw90) 1XVWV=XVW90
(((1'r) 1N9)SSV'XVWIN9)IXVWV=XVWIN9
I
((i1'r)Zod0-(1r)C0d0)/
05LL
orLL
oCLL
oz £
01 LL
ooLL
o69L
09£
(I)10N) Al
oo5 01o! 09 (
P1=1 005 00
C10N=(C)10N (1*3Nr) ji
133SN't=' oi5 00
0=K()ION
0=Xvwoa
0-=XVWINO
095 01 09 (CLI*9XVW03IUH~*3*S3I)iI
o95 01 09 (oI.9*oifdNl)JI
099£
o59L
ol9£
oM9
oz9L
at 9£
009£
3
o65L
3
SM01A SSVW 30 33N3Sb3AN03
095L
3
oLSL
3flINO3 OP 1
o9SL
oo=r' b
891
OZ58
0158
QoS
'rimi '(r'I)ZoNI
3flN I NOJ 019
3flNIiNOO 009
'rizvi 'risi
I
£CLz-(r' I)C3Av1=ri~v±
09119
05119
£Lz- (ri I)M±v.=r Is
o Lz-(r' 1) v±v=rIisvi
£CLz-(r' 0i~=ribi
IWN'1=I
00 79
06Q
oB£9
OL£9
0908
0509
Or£8
I
009 00
I
'X i ICVii 'x(L
ith 'X'T ,C3AV1,, 'XL 11Mi
1i 'XOTlii
1 xr/)1vwb0j 0805
,i~tv,1'x(L tilh, 'x'i 113AViii 'XL ,,U.lii 'XOTI
(0905'9) 311HM
r (oL05'9)3.I~M
019 0± 09 (1 SNw11-r0No011NI~1dN) Al
019 01 09 (£ Z0T*1o10HlJII
3flNI1N03 z6S
0M£
01 £
00£M
06z~8
09 z9
(£'i) IND
'
OLZS
o9zs9
05s9
O'79
0£QR
ozze
0tZ9
00zs
0619
0919
TWN'Z=l z65 00
(r'1 1) .ld=101
103SN'i=r 019 00
3flNI1N03 £65
3flNIN03 065
(CoZ138dT'WIX9Z)1VWb0J 0905
I
(ri)SAVIND ' (ri)SAVX0J
(z'riI 1N9 (TIN1N
'ItlviI'risi 'Ni (o9oS'9)311tfM
6(rO3AV11Vd
*£Lz- Cr) 3AV'7v1=rli'
£Lr-(rsi)8=risi oS
065 01 09
(£1Z3d'£t'X)1VW8i0J 0505
riu 'Ni (oSoS'9)3iIHM
*£Lz-(ri)i±=rii oLS
o65 01 09
(C*Z138'£ilx6'Zl3d'i'X)VWNOA005o1o
(C'nIr )I NO z
£Lz- Cr,1) 3AVvv=r I 7v1
*Lr-(rnii=rii
oL1 9
0919
05 19
0181
095 01 09 (lWoio9ri) fi
oLS 01 09 (133Soso9i) ji
rixvws1=nI o65 00
(133SN' IWN) oxvw=r ixvw
o£e1
1
Me
oi19
V/ 11 (CAr) I NO,,
0019
0609
'x£ ,g(r)SAVX0A,,
0909
'x£
i1(r)MV11d,, 'x£
'x£ ,,(r)si 1i IxL ,1r,, 'xiiiuo
oLo8
(50013 '=113a
H6'i*013 I00OOd
,,(r)3AVq1Vi,,
I
"xL ,11,, 'xV1/)ivw~oj
o0o5
(005'9) 31IPIM
£65 01 09
0909
0509
H64C*ot3'=W00LL
z
H6
6ST
o~o6
ozo6
0106
ooo6
0668
0868
oL68
0968
o568
0r68
oC68
OZ68
0168
0068
0688
088
0L88
098
0588
0$88
OC8s
oZ88
0188
0088
06UR
OUR8
OUR8
09L8
oSL8
O'rL8
OURL
OZLS
01 L8
ooL8
0698
oL98
0998
0598
oMT9
0C99
0Z98
0198
0098
o658
0858
oL59
0959
0559
0578
',,f
',1,
(0lW*9§~Wb
(JoMWlr*31
I I -N=1
IWN'1=1t o69 00
3WIlaNVwN0MWI.V*103WI1) 31
*0=IW3bIO
*O=IdbJ.
*0=xvwj
o=xvwia
*0=101wx0
0=8SiN lid
0O=SbNllid
0O=aVbd
*0=bOI0Hd
*0=ULSd
0=i3flld
* 0=W3H~d
* 0=N3Dd
I b-0Z3U=lob
S3 inionbi.s aNV SllVM HolHl S3ufl1YU3dW31 M3N 31vinlUV
1130.cb IV1Sd+b IV1S3=b IVlS3
ii 3G*AN0~d+CANO03AN0J3
Ii 30*ZAN03d+ZAN03 3=ZAN0 33
113a*T1ANO~d+TANO03=TANO03
113 0+3 W 1=3W 11
3W I.=S3W I. 589
3W11 30NVAGV
(o$6)aN1Old llV3
1+J.OidN=L1dN
(xvwA '00 "1 'XVWX '10 "1 'Z 'SV1A
'SVIX '3111i. 4 HT '0 '1 '1 'ldN 'idA 'ldX '0 '0)Ndi01d 11VO
0=ldl
Al
S89 0± 00 CXVWdX±11 oo%9/3wIVaNV*XVWdA*11 * LZ-dL)
S89 01L 00 (o0b31dl) Al
(M3NldN) SGVI=VlN)JdN (.lNd~l39*XVW) AlI
M3NbdN~.LN I dN=IN I~dN (XVW~.U 3D*XVwl) AlI
006 0± 09 (XVWb.w39*XVWI) Al
006 0± 09 (XVWI±L3903W11) Al 089
0=d N
(1 0t03
9
'=SSO13
llHWH
61*'01 3'11 =)INIS3
'i0T3',,=HIVlS3
,~'C03'i,
,,
'03'
=W00H3
11-4AN03
,,'C0t3
5
,,/C*013
-0VU3
,, lP0t3,,-=I010H3
,
'013,,
=H1S3
11 '1*013
I
=13flJ3
,,'C*013'ii =W3HO3
11 'C*013dlii, =N303 is//)1VWMI0A 0009
Z
'NII1S3
SS013 ')INIS3 'W00N3 'OIfWU
I
'iAN033 'ZAN033 '1AN033 'GON033 'SON033 ISHN113 'SbINI13
'0VHI3 balO1H3 6HIS3 '13n3I3 'W3H33 'N393 (000949)311~M
09T
0£ S6
oz56
0156
0056
Cr)
so'.±sj=sso
(Cr' i)mb'rimO) NlWV=rlMb
C(I- (MMO/113.c(r,0)MbOa)dX3) /C(r, i)mba.*(r, i)m1-*L) =rimb
oL 6
o9'6
ZIL 01 00 (03*SX) ~i
Cr'I) Mb+b010Hd=bQ10Hd
C1- (MM3/1130),,.(r'0)Mba) dX3) it(r Imba/rimb+(r'1m= (rI)OM1
Cr'i)M±=Cr'Usmi (IvO3*DIt)di
rimb+ Cr'I) MO-W~=£WT3
o6 6
01~t6
ozr6
o6C6
oM£
oLC6
o9C6
oM£
o~zS
T10/
I l-N=l
oC 61WN'1=ll OTL 00
(11nwj)sev'~'= OIL
oiL 01. 09
1
C(I-A3a1) AV030-A30) AV30) .c( (I-A30 1)10031CAO I)10031) /CC -AO I)100J1-33W II)+ (1-Ak0I) AV03a=j
0109o C01~91-A01)100J1) ji 9oL
SOL
OIL 01 09
o5C6
0*16
ozC6
at £6
ooC6
o6z6
oLz6
o9z6
oSz6
Oz16
CI) AV330J=J
90L 0± 09 C16190A001) Al *1oL
ZOL 01 09
I+C(r) A0330 1=C(r) AV3301
4IOL 01 09 CAV03NOb3A,3QI) 31
'roL 01 09 CCAk30lI1031*31*3Wi) J1
(r)AV33ai,Aoai zoL
1130-3Wu1+(r)O3Wli=33Wll ioL
OIL 01 09
Cr),AV030J=J
ioL 01L 09) (C01911fiwj) ji
CN)
0£ Q6
ozz6
0N03b+0V~dd=aT~d
r)doiib-= CN) QN0~
Cr) ±oab.L= Cl)
oiz6
ooz6
0616
oM 6
OOL 01 09
oL16
0916
0N0
Cr) SSVN~s(I)b Mwa985Z=tW3bb
Cr) SSVN~rc (Z)b1MWG* 0 985Z=ZW3j
Cr) SSVN'.' C£)MWo**9S5Z=£3b
CdMWI1193WI1N0NMWI1*31*3WI1) 31
OOL
*=Wmb
*o=iW3b
103SN'1-r QSL o
3flNI1N0O 069
Ci) 1b+stN11d=SbNtld
M3N11=ClI 1
016
0£C16
oz 16
0116
0016
10111
o6o6o Z/113a* C3W I Ds'. (01-M3N11) + (1130-3W 11) * (01- (1) 1±) + (1) 1011= C(1)
(1.-(13/113a*CI)O)dx3)'.(I)1ba/I1b+CI)11=M3N11
0806
I b+(lI 1-lW3Ni=lW3bb
oLo6
C' 1J ± 3a
0*106
i)iba d3)(I 1o)CI
- l£)= lI
T9T
oz66
lll+13ld=13l~d
3flNIiN03 tOL
3flNIINO3 z2rL
'T=(r'ww) SI
(ri-i)t3AVI= (r'Iww) u1
IWN'I=WW iziL oa
(u'aO
o886
r%tU+ibb-W3Hb=IW3Hb
oo66
0986
o586
oi86
oU86
oz86
Nw~I
(r'I)b+(I+i) ONO~b- C') aN03+(r' I)W3H0b+Cr' 0AV33a0=iUOb
Al
U(r)mo+*/(r)so+Ho=Ho C5*o b3*HSJ)
X130'* (C(r) 3V*003O3HH+ Cr) AV 3,..cA*0HH) =H3
CI 'Iws 'IIAws "n 'jo '(r'I)HI)d0~dJ iivo oUL
'o Cr)
£=(r'i)si
(5ooo soi * O-
ot 86
0086
o6L6
o8L6
oLL6
bv= Cr'I)W3H3b
sev) j i
r 1in)
N3b=Cr' i).L3
1130/((Cr' I)IO-NOH) *c13RL=3?b
59o /o6+rI)±3H=N3H
(rH/1a.GX*
awxo=Cr' i)wxo
z=(r'i)si
031lW1l N0isnliii
o9L6
oSL6
oIL6
otL6
oZL6
ooL6
o696
--
z=si
3
ozL 0± 09 (awxo±1i*wxo) Ai
Nwx0o= Cr' wxO
t=(r' Osi
GU.IWII SOI13NI)N OIIOGVVd
--
I=SI
3
Nx
L3/("1 O-~)*()H*(z1/(0*Z
(ob 'NOH 'Cr't)13H '031d '13H '1130 '(r'I)Hu±)w3H3 I1VO
otLl 0109
o896
oL96
o996
o596
oIT96
M~6
oz96
ot96
0096
o656
oLS6
o956
o556
(*39G(ri)si) Hi
oUL 01 09 (0ti3*W3H3I) Al
j*ib()SNO~*00=rlA0o
X13a/ ((rI )u±- (rI -i)uI)*
v)=(I) 0N03b
z(i~sAw)(~
(o/i~sows*ro
(0 'i3oWs I)i~ws '33 'o '(r't-i)l~l)d0OIdl 11V3
(0 '3)IWS 'ANWS '33 'jo '(r'I)YLL)d~lidJ 11VO
SUL 01 09 (vb3oI) Ai CUL
(r'1 1)i.= (r'iS1)s ziL
CUL 01 09
(r'I)sb+HiSd=UiSd
StL
risb+ Cr'1) sb-ZW3=ZW3Nb
ZW3H-(r'i)sb=risb (ZW3bb*L!D(ris-(r'Iosb)) ii
(Cr' i)sO'risb) tNiwv=risb
Z9T
5**(
(C*,s3W I
06101
02101
0910ot
05101
0101
I*I d)/AO 3WS.'N0Jd3.'cNOJ0H 1),*XOZOG (1)10ll1i**l) = (1)1 ob
(I)13b+S3N0Jd=S3N0~d
IWN'1=I OWL 00
*0=93NO~d
*0=S3NO~d
002 01 09 (o3V030HOOI) Al
31hJON03 OINI xfl'l 1V3H 3NlW83130
0 101
i130.'8bN I1d+GbN113=SbN 113
I130*S JN I1d+SbN I13=SbN 113
11 30'0Vbd+0Vb3=GVb3
1130 0G10Hd+b010H3=H010H3
11 3G' ISd+N1S3= 1S3
113 0.13fl~d+13lJ3=13l.3
113G).W3H~d+W3H33=W3HJ3
I130*N3Dd+N3D3=N393
3flNI.LNOJ
01101
00101
06001
08001
OLool
09001
"St
19L 01 09
* Z/1130=1130
S3W11=3W11
0=91J I
CN0Jd=CANOd
I N0~d=TAN0d
OV.LSd=)d1I Sd
SSL
95L
017001
Cr' i i)sbi= (ri')b
Cr' i i)sin~= Cr, ii o
r
ssi=Cr
s
Cr 1 )S11= Cr 1
1WN't=II 9SL 00
.L03SN't=r S5L 00
17SL 01 09 (0o10±1*010)AI
( (9'1 ) SbLL- (9'6)) 1) =Iola
1TSL 01 09
CV03C91)SOI)1
TWN'1=I 'TL 00
3flNI1N03
OTo
Cr) 80+8bN1I1d=dbN I d
0C001
M3NS1= Cr) 81
Cr)8±=(r)s8±L C10b39511I)A
01 001 0Z/' i13G*~(3W1I" C0-M3N8I)+. 1130-3W 11) (oCa- Cr) 81) ) + Cr) 1018±- Cr) 10181
00001
13a*( r)800) dX3)*( r) 00/ Cr) 0+Cr) 81=M3N81
3flNIINOO
0666
0866
oL66
0966
05L
O1L
C(r61'I"1L'dbI)TXVWV=1du.l (103SN03or) Ai
C(r' I i) u11xvwi) i xvwv-xvw±
CCb/i~b) s8v'xvw10)ixvwv=xvw1O
Cr' I)WXO+io1wxo=IO1WXO
Cr' DW3Ho0+W3H3d=W3H~d
Cr'I) AV300+N30d=N39d
01766
OC66
C9T
1130'3WIl 16665 ININd
3
OTLOT
0
o0oi
08901
o0oi
09901
05901
0 901
OC901
OZ901
01901
00901
06501
08501
0
00LOT
3,dnsoio
in*Wd0H l*9WS=Idn
I=IN3AI (XVAld*3D'WO&8d) l
WOONI*Vb*WNOHb=WOONd
508 01 09 (T'b3*IN3AI) Al
M3NdO=WOObdO
M3NOHN=W fOHb
M3NJ=WOONJ
(M3NdO*WOOIdA).cM3NOHld)/(1130),tWOOllb+WOOHI).tWOONdO*WOObA cWNOHH)=WoobI
il3O*WOObb+WOOb3=WOON3
SSOIVb-IN3Ab-ANISVb-OlNlb=WOONb
lNdO .t(M3NA-'T)+XOdO),,M3NA=M3NdO
1
(WOOIdA).cM3NOHb) / (1130%.c ( (OJ).cOOK8-WOONJ*
OLSOT
09501
05501
0501
0 501
OZ501
OTSOT
00501
o6qoi
OPOT
Mol
05 01
0 01
WNOHN)*IN3A+inowNo*wooNj+smoN*ioiwxo)-WOObA*WldOHII*WOONJ)=M3NJ
1
WOONA/(il3(1*
((OOHld-WNOHN)*IN3A+inowNg+smoN),tioiwxo)-WOOIJA).cWIIOHN)=M3NOHb
I113G*SSOlb+SS013=SS013
IN3Ab+NNISlb+SSOIVb=SSOlb
WOONI*WOONdo*inowN9=ssoivb
(WOObi*WOONdO)/(iN3Ab-NNIsvb-OiNib)=inow'do (T*b3*IN3AI) Al
*o=inowbo
(01*OdO*OOHH-WOONi*WOObdO*WNOHN)*iN3A=IN3Ab
SMON*( ANOOd+ZANOOd+IAN00d)=OiNib Z08
OL 01
NNIS3/113G*NNISb+NNISI=NNISI
1130*NNISb+NNIS3=NNIS3
NNISIb-SMON*NNISNb+NNISVb=NNISb
(NNISI-WOONI)).cNNISV*WNH=NNISVb
OMI
OMI
(( **01- **NNISI)*DIS*L*+(OI-NNISI)*inOH)*NNISV=NNisib
OMI
0001
(XONI $WNaH 'SZOO 'NNISV 4XJWS sinOH '3b 'Od 'XOHN
'01 'NNISI 4013*1 601-3*1 '013*1 6013*1 '013*1 $i)dONdV IIVO
offoi
OKOT
(XGNI 'WNOH 'WOONA 6NNISV 'XJWS 'WbH 43b 'WOObd
6WOONi 'NNISi '013*1 601-3*1
oLM
09M
05col
01COT
OCCOT
OZSOT
01M
00COT
NMI
OLZOT
09ZOT
05ZOT
'XOHb
IIVO
'013*1 4013*1 '013*1 4i)dONdV
ZOS Ol 09 (*Oeb3*NNISV*NNIS3) JI
INdO*(WOONA-*I)+XOdOkcWOOIdA=WOObdO 009
S311b3dObd NIV WOOU 3NIWH3i3O
3
113a*G3NO3d+83N003=S3NO03
113a*S3NO3d+S3N003=SON033
3nNIINOO OLL
il 3a* (r) sob+ (r) ioisob= (r) ioisob
o6zoi
i
113a/ ( (r) iolsob5***((S**3WI1.*Id)/NOONWS*NOOdO*NOOOHb)*(r)av*(r)ioiei*" )=(r)e:)b
(r)G3b+93NO3d=S3NO3d
103SN't=r oLL oa
3nNIINOO 09L
il3a* 0) iob+ Mioilob= 0) io.Liob
1130/ ( (010110b-
O Zol
OCZ01
OZZOT
t9T
165
5999 FORMAT(3X,"TIME =",2X,F12.2,3X,"DELT =",E1O.4)
805 EREMDR=EGEN+ECHEM-EFUEL-ESTR-EHOLDR-ERAD-ELINRS-ELINRB
1 -ECONCS-ECONCB-ECONV1-ECONV2-ECONV3-ESTAIR
PREMDR=PGEN+PCHEM-PFUEL-PSTR-PHOLDR-PRAD-PLINRS-PLINRB
1 -PCONCS-PCONCB-PCONV1-PCONV2-PCONV3-PSTAIR
IF (NP.EQ.O) WRITE(6,6010) PGEN, PCHEM, PFUEL, PSTR, PHOLDR,
1 PRAD, PLINRS, PLINRB, PCONCS, PCONCB, PCONV1, PCONV2,
PCONV3, PSTAIR, PREMDR, QROOM, QSINK, QLOSS
2
PFUEL=
PCHEM= ",E10.3, "
6010 FORMAT(/" PGEN= ",1PE10.3, "
PRAD=
PHOLDR=",E1O.3, "
PSTR= ",E10.3, "
1
E1O.3, "
2
E1O.3/ " PLINRS=",E1O.3, "
PLINRB=",E1O.3, "
PCONCS=",
PCONV2="
PCONV1=",E1O.3,
PCONCB=",E1O.3,
E1O.3, "
3
PREMDR=",
PSTAIR=",EIO.3, "
E1O.3/ " PCONV3=",E1O.3, "
4
QLOSS= ",
QSINK= ",E10.3, "
QROM= ",E1O.3, "
E10.3, "
5
6
EIO.3)
GO TO 170
900 IF (IPL.EQ.0) GO TO 910
TITLE, XLAB,
CALL PLOTPR(O, 0, XPL, YPL, NPL, 1, 1, 0, 1H
C
YLAB, 2, 1, 0., XMAX, 1, 0., YMAX)
C
NPLOT=NPLOT+1
C
CALL PLOTND(940)
910 IF (NCEND.EQ.0) GO TO 81
10720
10730
10740
10750
10760
10770
10780
10790
10800
10810
10820
10830
10840
10850
10860
10870
10880
10890
10900
10910
10920
999 CONTINUE
C 999 IF (NPLOT.GT.0) CALL EXTFLM(O)
STOP
END
10930
10940
10950
i 01 09 (iNOJ'I1) JI
ol900UdV
7101 09 ((I) iJ'1*1) JI Z
009ObdV
065OObdV
) /30) =d
(*9/* 1) ** (0139' 1/bd,'c 19) *((1X
095008dV
OZ 01 00 (ZOb3'IGNI) 11
OL50OoUV
S313130 1N3W31V1S DNIM0110J
N013VA NOUOHN03
095OObdV
05500~dV
NOli13ANOJ iv~fl1VN 1N3-fl8wfl 'S31V-ld 1311VNVd NO N0I133OO
0O1TOOdV
0C500dV
Z=IGNI
OZ00bdV
.c091 I 0=1 fNO 01
(3NX/30)* C C0.c*Xb9)
01 S00bdV
OOSOO~dV
V NO N01133ANOO iv~fl1VN IN31flefl
31V~d IVOI1NM
061T00bdV
021I00dV
0z 01 09
OV 0obcdV
1=10N1
091T00bcV
(ONX/3O)*~(5z0o't'X~9)*oc9C~0=1fNOI
0S1100bdV
0VT00bcdV
IVOI1NM V NO N01103ANOO iv~fl1VN NVNIWVI
31V~d
OMJ00cIV
0Z1T00dV
01 01 09 (6+i319DXN!9) Al
O01T0dv
O04T00dV
''(x/N)19=9
o6CoobdV
(Z.c*3Nlwx) / (vi-mi)SV.'(C**1X)*(Z'.c',c3N0HU)*~Vfl8,.cWS=1I39
09C?00~dV
3NNV91/*1-VI39
OLCOObdV
(9S*6oi+NtlV81)/IS.1'*u4c3NV81).d-39'I 0=3Nflwx
09CO0bdV
(3N)JV81cVt) /W00bd=3N0Hb
05C00bdV
(Vl-mI)*WcO -mi=3NNV9I
04 100MV
Oz 01 09 (0*11*01) if
0ff0ObdV
o=tnlNO
0ZC00~dV
01 LOQ~dV
(OScV0HN) /lwx=Aia
00C00~dV
Nd/lwX,.d 3=WS
06Z00~dV
Moq*rLZ*=d3
08Z00~dV
OLZoo~dV
05Z00UdV
0qZ00dV
0Q00bdV
0OZ00OUV
01 ZOO~dV
00Z00~dV
o61 0OHdV
oLTOOudV
09T~OOdV
'58Z
'Sty* 'S 1* 'SoP
(8S*6oI+NVsL)
l
3
3
3
3
3
3
3
3
3
3
3
3
PL4=3s
3a/H1VdX*c3N=X3N
(v'~fwx) /3Q.c (9) SIEV=3b
/5*1**IVg±&y-39 t o=nwx
(V1cVN)/W00OId=V0HH
Z/ (VI+MI) -11val
oo96=ows
HiI VWV/L+3q7T1-S=vH
/W6'98*
'oLo*
'6L
'SWo
6UL
6W
4z5- 6-o
I
'* 'LI WH 'HJ 'AN viva
(L)WH '(L)HJ
O5TOdV
NOISN3WI0
3
0ZT0OOdV
OTT00~dV
QOOO~d
3
3
07100OdV
OCt0OOdV
S3I1lbd0~d )iIV
(aNI 'UH $X0J 'V 'JWS 'H 43NI 'WOO~d
'V0HN 'Vi 'MI 63a '9 '1X '3NX 'HiVdX 6aI~d0IV MuounI~lfs
I
99T
oL600bidV
096O~dV
0S600bdV
01760O0UV
0C600bdV
oz600NdV
01 6OObdV
00600bdV
MOM8od
02800Od
oLSoo~dV
0980o0M
05800Md
0r800Od
MO8ooM
orSo~bdV3
01 80o0M
0080o0M
oELoo~dV
oRLooMd
08 01 09
1=CGN I
W4W
IX/3GObd3bOrCr0+''L=nNa
S31V~d 131IVbVd N33M13S M01J O33b0J NVNIWVI
3
3
3
05 01 09 (*OOOC03N) Al
09 01 09 (vOV~(i)sev1) Ai o r
3
Z=ZaNI
(0*.tX3b)/zSo*=zAws
(HiVdX/3a) *c(9'0**bcd) c(840*ccX~b) .96Z0'=rfNO 0
31Y~d V .LSVd M01J
G33bO0I1N31lSbIl
3
3
0 01 09
t=raNI
(50*.'X3b)/r99o=JWS
(HiVdx/3a)*~(tC*o*3'ud) k (S00*c'*X3) '.Zr=ZnNa
OLL00OM
o9Loo~dV
OSLOOMid
o rL~odv
oULoo~dV
31V~d V 1SVd M01J 033N0A NVNIWVI
3
0C 01 09 (5+3*S*1D9x3b) Al SZ
001 01 09
OZLoo~dV
1flNa=fNG
01 LooMd
SZ 01 09 (C"Nai) ji or
OOLooMd
o690oodV
O NOud
oL90O~dV
o990oodV
oS900udV
01 T90V
Al aH/iinWH~1 fNO=1 fNO
1T=AIGH (*I±9DAiaH) Al
(lX/JNX)*ffo-jLL+W=AIaH
*i11flWH (*ij.9'1lnWH) Al
0o=11lWH (01V11~inWH) Al
(I -1)WH- (1)WH)c(I - ) -(1)1 )/(I--1)1 -) + (I-) WH=1flWH 'TI
ri
oC9oo0M
01 09
I+1=1l
or9008dV
L9T
168
C
C
C
TURBULENT FORCED FLOW BETWEEN PARALLEL PLATES
50 DNU3=.023*(RE**.8)*(PR**.4)
SMF3=.00140+.125/(RE**.32)
IND3=2
GO TO 80
C
60 IF (RE.GE.3000.) GO TO 70
C
C
C
LAMINAR FORCED FLOW THROUGH AN ARRAY OF TUBES
DNU3=8.
SMF3=25./RE
IND3=1
GO TO 80
C
C
C
TURBULENT FORCED FLOW THROUGH AN ARRAY OF TUBES
70 DNU3=.023*(RE**.8)*(PR**.4)
SMF3=.00140+.125/(RE**.32)
IND3=2
C
80 DNU=DNU1
IND=3*INDI-2
IF (DNU1.GT.DNU2) GO TO 90
DNU=DNU2
IND=3*IND2-1
IF (DNU2.GT.DNU3) GO TO 100
90 IF (DNU1.GT.DNU3) GO TO 100
DNU=DNU3
IND=3*IND3
100 SMF=AMAX1(SMF2,SMF3)
H=SMK*DNU/DE
HD=DIF*DNU/DE
RETURN
END
APR00980
APR00990
APRO1000
APRO1010
APRO1020
APRO1030
APRO1040
APRO1050
APRO1060
APRO1070
APRO1080
APRO1090
APR01100
APRO1110
APR01120
APR01130
APR01140
APRO1150
APR01160
APRO1170
APR01180
APRO1190
APRO1200
APRO1210
APRO1220
APRO1230
APR01240
APRO1250
APRO1260
APRO1270
APRO1280
APRO1290
APRO1300
APRO1310
APRO1320
APRO1330
APR01340
OMOIAS
011001:18
0000-1:18
offoolls
offoolAs
oLcoolis
09001AS
oscool is
O Cooljs
oicoolis
ozcoolje
olcoolis
o0coolis
06ZOOIJS
o9zoolis
OLZOOIAS
o9zool is
OSZOOIAG
Ozoolis
oQ00lis
ozzoolis
ol z6ol is
oozoolis
06100138
081001AS
oLloolis
091001 Aa
05100118
O Tooljs
ocloolls
0zloolle
011001 As
001001:18
aN3
NiinlH
3nNIiNO3
XOdO*(ir)IOGXOD+lNdO),c(ir)IND=(ir)8d3D
(ir)IND*((r)SAVXOA-oi)/(r)SAVXOA=(ir)ioaxoo
0 01 09 (00011*01r)IND) Al
C11=1 os 00
(i+r)SIND*
('0'39' (i+r)SIND) Al
((r)SAVXOJ-*I)/(r)SAVXOJ=(i+r)exo!)
(r)SIND*((r)GAVXOJ-*i)/(r)SAVXOA=(r)exoo (*0031*(r)SIND) Al
(NIXOD+NIIND)/NIXOD=(r)SAVXOJ
(i+r)8XOD-NIXOD=NIXO9
o+r)SIND-NIIND=NIIND
O 01 00 (*0*3D0(t+r)SINS) Al
(r)SXOD+NIXOD=NIXOD
(r)SIND+NIIND.=NIIND
OZ 01 00 (*0*31*(r)SIND) Al
3nNIINOO
(-ilr)lO8XO9-NIXO9=NIxoD
(i'r)IN9-NIIN9=NIIND
01 01 00 (00*30*(16r)IND) 31
05
0
OC
OZ
Pi=i oi oa
OWT=XOdO
OWT=INd3
*O=NIXOD
*O=NIIND
IND 49d39 4GAVXOJ /MOIA/ NOWWOO
10SX09 'SXOD 49IND
(Pioaxoo "(L)SXOD NOISN3WICI
(P9)IND "(P9)8d3D '(9)SAVXOA NOISN3Wia
(L)GIND
SMOIJ SSVW 3SV9
(r)MOIAS
Minoan
69T
OT031-3
01 '003HO
001T03H-3
o6 0O3HO
09CO03HO
OLCO03HO
o90oo3HO
05CO03HO
070031-3
OU003H-3
OZC003HO
at £003HO
00C003H43
06Z003HO
OSZ003HO
OLZO03HO
ON3
N~fn13N
3flNIN03 0~
1130/ (lON-N3OI)*~Hl3G03
JI
ION=NON (ION*39*NO)
3lNI1NO3 OZ
(M)l~bS=N3Oi
OZ 01 09 (*031*M) 31
13ld)13l+ (NZ0HtlNZ0HN) /(1130G*)31VN) =M
ION=NOH
(31! (O1/C3+O) -) dX3*ct3=)I31Vb
'T+38*L=Hl13o
C+359=NZ0HN 01
o09 1T1=zo
53W5O=10
t1Y31) Ai
01 01 09 (o6
8389*?=13
05Z003H3
04T003H-3
O0Z003HO
'oL99z=.o
09ZO03HO
0tZ003HO
0OZ0031$3
061 0031-3
091003H43
oL1003H43
091003H43
O0510031-13
0110031-13
0T03HO
0Z1003HO
011003143
001003H43
01 01 09
A~1~3) i
04
o36i
9+31 r9=43
C+3809=ZO
'+3L*T-10
NOISS3NdX3 1VNIWON
3
3
3
3
0T 01 00 (1313*39*13H) Al
I3N-OHI3
ob
3
3
3
NOII33UN 3aix0 wflIN03IZ A0 S01I3NIN
(30 'NON '13OI 'ONI $ION '1130 '31) W3HO MuNotifsfl
171
SUBROUTINE FPROP(T, CF, CC, SMKF, SMKC, ICALL)
C
C
C
FUEL AND CLAD PROPERTIES
IF (ICALL.EQ.0) GO TO 20
IF (T.GE.3200.) GO TO 4
THETA=535.285
EO=37.6946
CAPK1=19.1450
CAPK2=7.84733E-4
CAPK3=5.64373E+6
R=1.9865E-3
THETT=THETA/T
EXPT=EXP(THETT)
CF=(CAPK1*THETT*THETT*EXPT/((EXPT-1.)**2)+2.*CAPK2*T+((CAPK3
1 *EO)/(R*T*T))*EXP(-EO/(R*T)))*4.184/270.13
GO TO 6
4 CF=51.*4.184/270.13
6 CONTINUE
IF (T.GT.1223.) GO TO 10
CC=(7.1E-2+1.7E-5*T-0.89E+3/(T*T))*4.184
RETURN
10 CC=0.087*4.184
RETURN
20 SMKF=0.030
SMKC=0.30
RETURN
END
FPROO100
FPROO110
FPROO120
FPROO130
FPROO140
FPROO150
FPROO160
FPROO170
FPROO180
FPROO190
FPROO200
FPROO210
FPROO220
FPROO230
FPROO240
FPROO250
FPROO260
FPROO270
FPROO280
FPR00290
FPROO300
FPROO310
FPROO320
FPR00330
FPR00340
FPR00350
FPROO36O
FPR00370
0N3
woon=(riN)fli
(N)V39/rNC3Ai*()O-N~vwOO= (r'i-N) 3AVI
O
1-'Z'OZ=N OZ 00
(i-Pr) vwwv9= (r'oz)3AV.L
NOiiflOS 3.LfdWOJ
0
bOI03A
3
(rr)VjA/(ot-rr) vwwvo).c(rr) v-(rr) a) =(rr) vwwv
(i -Pr) VAS8/ (i -Pr) o* (PP) v-(Pr) 9= (Pr) V.13E
i'tz'c=rr 51 oa
51
0
VWWVD I2VAS.3 3.LdWOO
Z)a= (I ) a
W00oL+ (I
MTVi+ (Z) 0= (z) 0
0'0=(IZ) 3
0*0=(Z) V
3flNIINQ3
(I-W)IHdIVlcZWV9)/(r, t-w) C3AV.L+
01
((i-w) iwrmL+(rPI'I-W) MI).c~WV= (W)0
IZ'Z=W 01 00
Z**(MM/Z130) =mWV
((Z13a~cZ130)/*t) ci13a=ZWVD
Zl~azl30/*T*11/MM=MMV
((MM MM)T+
3flNII.N03
3AVL-3968*7=(N) tHdlV
L6tz-(r'N)AVI~c-3t+(Pi+()
ozlm5 00
TMd1V' 3AVI' IWPmiL'M /0018/ NOWWO3
(W00OLL"TAVi'MMX'Z13a'J.130'P)INVHO Muo3n~ilfs
3
ZL
173
APPENDIX C
PARTICLE INPUT, OUTPUT AND PROGRAM LISTING
CONTENTS
page
Section
C. 1
PARTICLE Input and Output
174
C.2
PARTICLE Program Listing
177
C-1
Sample Input Listing
175
C-2
Sample Output Listing
176
Table
174
PARTICLE INPUT, OUTPUT AND PROGRAM LISTING
C.1
PARTICLE INPUT AND OUTPUT
The particle program is an interactive code programmed
in FORTRAN IV.
Input is entered via a free format READ
statement, or through data statements.
Input variables are
given below in order of input.
Input
Variable
Name
Definition
TR
Temperature of zirconium particle
Nominal
Value
at its point of origin (*C)
TAVEl
Temperature of airflow into which
particle is spalled at its origin
(*C)
TROOM
Temperature of air in the spent
10.
fuel building above the fuel pool
(*C)
GNI
Nitrogen mass flow rate in the assembly where the particle originates (mg/s)
AXAl
Cross-sectional area of holder
from which the particle originates
(cm2)
579.
175
PROOM
Room pressure in fuel pool building
1.0315e5
(Pa)
UL
Unheated upper length of fuel assem-
0.
bly (cm)
5.
The number of particle sizes to be
N
analyzed
The input listed in Table C-1 was employed in the analysis
of Chapter 3 for the first four variables described above;
the other variables appear in data statements.
A portion of the output appears in Table C-2.
The out-
put is self explanatory, with the following exception:
W = total airflow exiting the channel (nitrogen and
oxygen), and is given in units of mg/s.
TABLE C-l
Sample Input Listing
TR
TAVEl
TROOM
GNI
1390
1370
283
14800
176
TABLE C-2
Sample Output Listing
TR=
1390.0 K
W=.1820E+05 MG/S
MAXIMUM PARTICLE SIZEw
PARTICLE DIAMETER=
TIME,SEC
0. 1000E-O
0. 2000E -01
0. 3000E -01
0.4000E-01
0. 5000E -01
0.6000E-01
0.7000E-01
0.8000E-01
0.9000E-01
0. 1000E+00
0. 1100E+00
0. 1200E+00
0. 1300E+00
0. 1400E+00
0. 1500E+00
0. 1600E+00
0.2150E+00
0.3000E+00
0.3900E+00
0.4800E+00
0.5700E+00
0.6600E+00
0.7500E+00
0.8400E+00
0.9300E+00
0. 1020E+01
0. 11 10E+01
TAVEI=
1370.0 K
0.1313E-01 CM
0.1665E-02
TEMPERATURE,
0.1333E+04
0.1273E+04
0.1215E+04
0.1160E+04
0.1108E+04
0.1059E+O4
0.1012E+04
0.9684E+03
0.9271E+03
0.8881E+03
0.8513E+03
0.8166E+03
0.7838E+03
0.7528E+03
0.7236E+03
0.6961E+03
0.5706E+03
0.4433E+03
0.3666E+03
0.3255E+03
0.3043E+03
0.2936E+03
0.2883E+03
0.2857E+03
0.2844E+03
0.2837E+03
0.2834E+03
CM
TROOM=
283.0
MINIMUM PARTICLE SIZE=
PARTICLE ABS. VELOCITY=
HEAT GEN.,
0.2727E-05
0.1297E-05
0.6713E-06
0.3653E-06
0.2508E-06
0.1736E-06
0.1206E-06
0.8371E-07
0.5799E-07
0.4003E-07
0.2750E-07
0.1879E-07
0.1277E-07
0.8623E-08
0.5790E-08
0.3866E-08
0.386iE-09
0.1022E-10
0.3350E-12
0.2678E-13
0.5400E-14
0.2148E-14
0.1288E-14
0.9678E-15
0.8187E-15
0.7370E-15
0.6856E-15
J
DISTANCE, CM
0.1171E+01
0.2287E+01
0.3352E+01
0.4371E+01
0.5348E+01
0.6286E+01
0.7187E+01
0.8054E+01
0.8891E+01
0.9698E+0i
0.1048E+02
0.1123E+02
0.1196E+02
0.1267E+02
0.1336E+02
0.1403E+02
0.1739E+02
0.2179E+02
0.2568E+02
0.2901E+02
0.3192E+02
0.3450E+02
0.3683E+02
0.3894E+02
0.4087E+02
0.4266E+02
0.4432E+02
0.OOOOE+00 CM
0.1200E+03
HEIGHT, CM
0.1123E+01
0.2102E+01
0.2948E+01
0.3669E+01
0.4274E+01
0.4767E+01
0.5154E+01
0.5439E+01
0.5622E+01
0.5707E+01
0.5694E+01
0.5639E+01
0.5583E+01
0.5528E+01
0.5472E+01
0.5417E+01
0.5112E+01
0.4640E+01
0.4141E+01
0.3643E+01
0.3144E+01
0.2645E+01
0.2146E+01
0.1647E+01
0.1148E+01
0.6488E+00
0.1499E+00
CM/S
ILLS
0
C(086*0HH0*'O0
DA/ll=ew i
o49 tTI-
=Z3
83WoS=1 0
*oL99z- =Z3
9399*1=10
o9L9i-
=zo
3flNIiNO3
C) / (SOH )c.c*~p9A) =XVWQ
OZ 01 00 ('005*13)iI
01
snwv/xvwa*DA*M H= 3
((L o** (0HbG*' 086) *C51 0) / (DAIi"
0**Snwv*c6Z
0.*cS0HN) ) =XVWO
SnfwW/xvw0a.cA*~S0HH=3 N
C(o86*0HIIO) /
(9A).sfwv* 81) )iids=xvwa
(1VXV.'S0HN) /01 ND=DA
(swi i i)nfwv=jnwv
0 ZI (13AVi+i) =swI1i
(13Av1)nfwv=Sfwv
(w00o±) nwv=NJnwv
S0HII- *0059=OHUO
(13AV13c Z+30L8*z) /W00Oid=S0H8
(W00NI* Z+3oL8 z)/W00~d=N0HN
/0qT*0'O 4T*O/ZOZd3'Zd3 VIVO
(I L j'xi ,=wootu,
xi'xi I't Lj'xi ',,=I 3AV111
IND*CZ*1=0IND
e £L +woo.=woou
*CLZ+T 3AV1=t 3AV1
lND'WO01't3AVI'Nl 'aV31I
(8S *o60+vi) / (5* I *Vi*i -39'l' '0) = (vi) nwv
(i **c (mnwv/iinwv) 3.0 *0Ud*c (99 *0**03N).90 00+
5*~o3Nkcq0)+ *Z))c a/JIVO =(dmwliw"3~liv)I
'2
(Z0Zd3*WvV 1*+ZdD~ (wv- **a.c'7 £0q7)) / '2
(((WOO~II-Mi) *±H~c9Tqt + ( **WOONI-q*Mi) *ci -399 Z 1)*a~a- I2
±~aWa*C~OO Z = (WOONI.'O'IH'mi
Iiawa ' WV)
iLi
(wv+4c*aVqoiQ/
(i ** ( (oi' (OMI/ Z ) d Xhct13) ib s
(01) I± S / ( ( (oM±/ Z3) dX3 *13)iabOs)
* ( ( (OMI/ Z ) dX3
q2
-2
'2
ONIISII WV'd9Odd arID.lUVd
zoo
LLT
,r
O=DVI
3
1
0
SSVW NI 30NVHO 31ndWOO
I
(IWO 'iH913H,,'XV ,iWO '30NVISIO,,'XZ
Cz
'*N30 iV3H,,'XZ',,N '311niVb3dW3i,,'X5',,03S'3WII,,'X9'/)IVW 80A
(CZ'90)31lbM
R lNlbd
(iS/WO,,'XZ' *013'XZ',,=,kilOO13A 'S8V 3101A Vdi, '2
IX5111wollxz' *013'XZ611=N3i3WVIO 3131AVd,,'X5'/)iVA10J
zz
OZ'a (ZZ'90) 311 M
OZ60'ZZ lNlbd
z=rr
500=Aao
5*o=xao
bi=oMi
( iL*olmnwvlsnwv'03b'O'IdIVO)OIH=IH
snwv/o*oz* o*l*SOHb=03b
OZ=OA
IA-DA=OZ OL
(SOHb/OHbO*O* *696z)lNbS=iA
OL 01 09 (*005*il*3b)Jl 09
snwv/O*IA*SOHH=3b
(C o*).csnwv*6z*O ,c*SOH'd)/(IL*O).,.),t(#o863,cOHbO)).c 51*0* 1*1**G)=iA
09 01 09 (*Z*11*31d)Jl
snwv/G*iA*SOHN=RJ
(snwv**9i) / (oo86*OHbG*a*O) =.LA
1130+3WI.L=3WIl
W11=113a (113a*il*Wli)Al
((Idi/ZO)dX3)P*OZkclO)/Z).c).cG*9300 *I=WlI
500*0=il3G
C-30*1=0 (C-30*1*il*0)J1
(o
(r) ivoi j*aa) ***oi =a
3nNIINOO
GMqT-=z3
83qO*5=13
LC oi oo (o6zqi*ii*u)ji
LC
5C
OL99z-=zo
8389*q=13
5C oi oo
009LCI-=zo
*oqC6=i3
O'O=OA
oo86=to
O'O=OX
000*0=3WIl
N4i=r oS oa
SNOuvinolV3 31311HVd N1038
(,,W3,,'X' *0T3'XZ',,=3ZIS 313111IVd wnwlNIWi
SX519 ,W3,,"X" *0T3'XZ",,=3ZIS 31311HVd
wnwixvw,,,IX541///)IVWHOA
0
3
'2
TZ
Niwa4lxvwa (TZ'90)311bM
NIWG'XVWa'IZ INIbd
(C-30*1/XVWG)01001*((N)ivoij/66*)=ao
L*08ZZ/(SWII*(bi/ZO)dX3*13)iubS=Niwa
8LT
0 #0=1 0
0 *O=AOO
(OA *il *,k) J I
iOZ-=AA
091 01 09 (iOZ-*ID*AA)JI
3nNIiNOO 051
(80HVIMP.0 0*696z) idbs=ioz
i LLS (C **bnwv*6z **bOHb) /I L Okc* (o
051 01 09 (*005*11*RI)JI
0O=iOZ
o86).ciWd) 2.0 1 *I;c*.G*C5T
051 01 00 (*Z*11*311)31
bnwv/AA*O*UOHb=3b
NnwWW 5.*IOHb*G*Q=IOZ
'dMl- (C3.c*a*9CZ5 *0) / (TWV+C**O;M'CO 0 =iOHH
AiI3013A IVNIWb3i 03A31HOV SVH 3101INVd JI MlWbU30
AAO+OZ=AA
AO+OA=A
0 *9/ KAI+ZAI* * +Ikl) =,kAa
AN* * +l AN) =Aa
KAN+z
o *9/
(80HId'ID'O',kGO'ZAI**Z+IAI-iO.LA'CWV)XAJ*il3Q= AI
(ZAI**Z+TAI-OZ),*il3O4AN
(bOH8'19'0'AGO'O*Z/TAI+iO-LA'ZWV)XAA*il3G=ZAI
(0* Z/Ikl+OZ) Ac1130=ZkN
(b0HId'T9'O',kGO'iO-LA' TWV) XAhc1130=TAI
OZ*1130=IAN
(AA*AA+XA3.cXA)I'dbS=lOiA
30NViSIO A UndWOO
qwv=cwv
Nv=zwv
NV=TWV
59 oi 09 (o4olq*((Twv)sev-Nv))jl
0
0
o6 oi ou (T*b3*DVlJl)AI
3nNIINOO
0*041awa
O*O=Ziowa
o*o=iiawa
qwv=cwv
qwv=zwv
NV=TWV
q8 01 09 (Z*3N*TDVIAI)AI
(a'Z3'T3'OMi'il3G+3WII)ziawa=Ciawa
(G'ZO'13'OMI'OOZ/.LI30+3WII)ziawa=ziawo
(alzo,113"OMI"3WII)ZIOWO=TIOWa
C**a*zo 6u=Nv
(a'ZO"10'OMI'1130+3WII)ZOWO=CWV
(a"zo4iolomi"*Z/il3O+3WIi)ZOWG=ZWV
(G'Z34T0'OMi"3WII)ZOWO=TWV
r*z-oz=rr (oooii*omi)ji
r*z-61=rr (*oo5*ivomi)ji
r*z-Ci=rr (oooVivomi)ji
05 01 09 (T9b3*9VIJI*ONV*Z*b3*TSVIJI)JI
W11=113a (i*b3*OVIJI)J1
0001/((I"0-((OMI)DOIV/*I))**001)dX3=WlI
OqT
lmiNlbdN
0=19VIAl
6LT
180
160
C
80
C
90
CONTINUE
IF(Y.LE.O.O) IFLAG=1
IF(IFLAG.EQ.1) PRINT 111,TIME,TWO,QOXD,XO,YO
IF(IFLAG.EQ.0) GO TO 80
VY=0.0
Y=0.0
VX=0.0
X=XO
HT=O.O
DELT=TIM
GO TO 140
COMPUTE X DISTANCE
KX1=DELT*VO
LX1=DELT*FYX(AM1,VTOTCDX,DO.,RHOR)
KX2=DELT*(VO+LX1/2.0)
LX2=DELT*FYX(AM2,VTOT+LX1/2.0,CDXD,O.,RHOR)
KX3=DELT*(VO-LX1+2.*LX2)
LX3=DELT*FYX(AM3,VTOT-LX1+2.*LX2,CDX,D,O.,RHOR)
DX=(KX1+4.*KX2+KX3)/6.0
DVX=(LX1+4.*LX2+LX3)/6.0
X=XO+DX
VX=VO+DVX
COMPUTE TEMPERATURE CHANGE
KT1=DELT*FT(AM1,DMDT1,TWO,HTD,TROOM)
KT2=DELT*FT(AM2,DMDT2,TWO+KT1/2.0,HT,D,TROOM)
KT3=DELT*FT(AM3,DMDT3,TWO-KT1+2.*KT2,HT,D,TROOM)
DT=(KT1+4.*KT2+KT3)/6.0
TW=TWO+DT
C
C
COMPUT DRAG COEFF FOR NEXT TIME STEP
VTOT=SQRT(VX*VX+VY*VY)
RETOT=RHOR*VTOT*D/AMU(TW)
CD=0.44
IF(RETOT.GT.500.) GO TO 93
CD=18.5/(RETOT**0.8)
IF(RETOT.GT.2.0) GO TO 93
CD=24./RETOT
93
C
C
C
116
100
CONTINUE
COMPUTE HEAT TRANSFER COEFF
IF(IFLAG.EQ.1) GO TO 116
AMUW=AMU(TW)
HT=HTC(CAIR,D,RETOT,AMUS,AMUW,0.714)
DETERMINE NEW OXIDATION PARAMETERS
CONTINUE
C1=9340.
C2= -13760.
IF(TW.LT.1193.) GO TO 100
C1=4.68E8
C2= -26670.
IF(TW.LT.1493.) GO TO 110
C1=5.04E8
OS 01 09
ON3
dOIS
3nNIINOO
01 ol 09
05
SS
r*z-ii=rr (T*b3*9VlAl)AI
MI=OMI
X=OX
A=OA
XA=OA
AA=OZ
113G+3WIi=3WIl
OS 01 00 (0-0-31-A)AI
(0*0*b3oA*GNV*O*Z*31*(WOO81-Mi))AI
3
T+iNI 8dN=INI IdN OZI
O=iNiHdN
3nNIINOO 11
55 oi oo (rrWIN18dWAI
((*0T3'XS)9)IVW80A
III
A6X'aXOb'MI43WIi (111'90)UIM
A'X'OXOb'MI'3WIl'ITl AINd ZT1
11 oi 09
aXOb'Ml'3WIl (111'90)31lbM
axob'MI'M14111 lNlbd
ZIT 01 09 (O*b3*9VIAI)AI
on oi oo (rr*3N*INIHdN)AI
d3lS 3WIl IVNIA ONV SON033S O*Z A SIMUNI iv inOiNlbd
50*0=1130 (0*0031*axob)Ai
0
Z=IDVIAI (0*0*31*GXOb)Al
(Twv4wv)*98z* i=axob
3nNIiNO3
Oll
OOC9 1- =zo
T8T
182
APPENDIX D
THE VAPOR PROGRAM
INPUT, OUTPUT AND PROGRAM LISTING
CONTENTS
page
Section
D. 1
VAPOR Model Equations
183
D. 2
VAPOR Input
186
D. 3
VAPOR Output
189
D. 4
VAPOR Program Listing
191
D-1
Sample Input Listing
188
D-2
Sample Output Listing
190
Radial Conduction Model
Control Volumes Used in
VAPOR
185
Table
Figure
D. 1
183
THE VAPOR PROGRAM
INPUT, OUTPUT AND PROGRAM LISTING
D.l
VAPOR MODEL EQUATIONS
The VAPOR code was developed to assess the potential
for generation of zirconium gas within the inner portions
of a piece of Zircaloy-2 clad undergoing rapid oxidation.
The postulated mechanism is that the chemical oxidation
energy deposited in the unreacted clad may exceed the rate
at which the oxidized outer thickness (which has a much
lower thermal conductivity) can either conduct away the
heat or ablate, with resultant vaporization (Cl).
The one-dimensional radial conduction model employed
in the code accounts for a fuel, gap and clad cross-section.
A finite difference grid for the model equations is shown
in Fig. D.l.
The finite difference equations for transient
conduction are derived from energy balances performed over
the control volumes depicted in Fig. D.l.
For example, the
energy balance for the center fuel/heater node may be written as:
-kAdT
q"'V + pcV dT = 0
D.1
A finite difference representation of this equation gives:
184
2T ~T'
2
k (2rAr)
1 J+
2
q"' Ar
=
1
_l_~__l
D. 2
T
aAt
Ar/2
which is finally reduced to:
4
T2 - 2T,
+
Ar
qm
T, - T,
=
R
-D.3
a
At
where:
q"' = decay heat/input power volumetric generation
rate, (W/cm3
a
= thermal diffusivity of fuel/heater node, (cm2/S)
k
= thermal conductivity of fuel/heater node,
(W/cm K)
(K)
T
= temperature,
Ar
= radial distance, (cm)
and the superscript "lo" refers to old time temperature
values; new temperatures are not superscripted.
The remain-
ing equations corresponding to Fig. D.1, though more complex, are derived in a similar manner and may be inferred
from the program listing.
The chemical oxidation energy deposited in the clad
during any time period is calculated explicitly, using the
isothermal parabolic reaction law.
The VAPOR program was not used for quantitative analysis in this report due to lack of modelling of axial conduction effects, mass transfer through the oxide layer, and
185
FUEL
GAP
CLAD
I
N
N
N
N
N
N
N
N
N
N
0
0
0
0
0
0
0
>2
N
-HAr3 K-
KAr -A]
1>
FUEL (HEATER)
CENTERLINE
Figure D.l
/
ADVANCING
OXIDATION
FRONT
Radial Conduction Model Control Volumes used
in VAPOR.
186
thermophysical property data.
It was beyond the scope of this project to formulate
mechanistic models for clad oxidation.
Should more data
on non-isothermal zirconium oxidation become available,
it is recommended that the heat transfer equations described in this program be joined to the one-dimensional
radial mass-transfer equations described in reference Bl.
The mechanistic equations described in that report, which
neglect energy release via oxidation reaction, yielded
predictions in good agreement with experimental data.
D.2
VAPOR INPUT
The version of the VAPOR code in this appendix was
used in attempts to simulate the experimental data presented in Chapter 5.
Input to the code is performed using
free format READ statements and DATA statements.
Input
variables are defined below in order of input.
Input
Variable
Name
Definition
timax
Termination time for experimental
simulation, (sec)
nprntl
Number of time intervals between
printouts during assembly heat-up
nprnt2
Number of time intervals between
printouts during oxidation reaction
Nominal
Value
187
tburn
Time at which oxygen is introduced
-
into the test assembly, (sec)
xcl
NOT USED
0
xc2
NOT USED
0
delt
Computational Time Step, (sec)
tempr
Temperature of assembly at time= 0,
283
(K)
troom
Temperature to which assembly ini-
283
tially loses energy, (K)
h
Convective heat transfer coefficient, (W/cm2 K)
n
Number of fuel volumes
5
m
Number of clad volumes
10
qf
Resistance heating power of heaters,
-
(W/cm3)
Additional input values related to thermophysical properties
of the assembly have been given mnemonic variable names and
may be inferred from the program listing.
A sample input
for the experimental assembly configuration is given in
Table D.l.
Results for this case, at time = 360 seconds
(since the introduction of oxygen) are shown in Table D.2.
188
TABLE D.l
VAPOR Input
timax
= 3800.
Values input via DATA
Statements are:
nprntl = 60.
= 0.002
nprnt2 = 1.
h
tburn = 3600.
qf = 20.
delt
= 60.
189
D.3
VAPOR OUTPUT
The following is a list of output variables, as they are in
order in the VAPOR output of Table D.2.
Output
Variable
Name
Definition
time
Elapsed time since introduction of oxygen,
(sec)
rin
Outer radius of the unreacted Zircaloy clad,
(cm)
qf
heater input linear power, (W/cm)
qtot
Total energy released by the oxidation reaction, (W)
first
column
Radial node number:
heater nodes
1- n
(n+ 1) - (n+
3)
(n + 3) - (n + m + 3)
gap nodes
clad nodes
second
column
Radial distance of node from center, (cm)
third
column
Temperature at nodal locations,
fourth
column
Energy released by oxidation reaction during
time step,
(W)
(K)
190
TABLE D-2
Sample Output Listing
time=
qf-w/cm=20.00
120.0
rin=0.544589
qtot=
0.000000
1
2243.71
2
2243.57
0.070444
2243.02
0.140889
3
2242.14
4
0.211333
2240.94
0.281778
5
2240.19
6
0.317000
7
1594.73
0.400500
0.484000
8
1171.99
0.OOOOE+00
9
0.487263
1171.98
0.0000E+00
0
0.493789
1171.96
0.OOOOE+00
1
0.500316
1171.94
0.OOOOE+00
2
0.506842
1171.93
0.0000E+00
0.513368
1171.91
3
0.0000E+00
4
0.519895
1171.89
0.OOOOE+00
0.526421
1171.88
5
0.0000E+00
6
0.532947
1171.86
0.OOOOE+00
0.539474
1171.84
7
8
0.546000
1171.83
0.6287E+01
0.3772E+03
0
0
0
0
0
0
0
0
0
0
191
D.4
VAPOR PROGRAM LISTING
dimension aa(20) ,bb(20) ,cc(20),dd(20),to(20) ,t(20),r (20)
dimension condg(3),alphg(3),condc(20),alphc(20),qchem(20),iflg(20)
print," input: timax, nprnti, nprnt2, tburn, xci, xc2, delt"
read, timax,nprnti,nprnt2,tburn,xci,xc2,delt
nprnt=nprnt1
iflg2-0
tempr=283.
troom=283.
h=0.002
iflg3=0
time=0.
qtot=0.
n=5
m-10
nn=0
condf=0. 16297
alphf=0.042513
do 4 i=1,3
alphg (i)=0. 1554
4 condg(i)=2.13e-4
do 6 1=1,20
condc (1)=0.30
alphc (1)=0.7118
if1g (1)=0
6 qchem(1)=0.0
npi=n+1
np2=n+2
np3=n+3
np4=n+4
np5=n+5
nmp1=n+m+3
do 5 j=1,nmpi
t(j)=tempr
5 to(j)=tempr
tai r=troom
sig=5.67e-12
eps=0.25
rci=0.484
rco=0.546
rcl=rco-rci
rf=0.317
qf=20.
r in=rco
rct=0.
drf=rf/ (n-0.5)
drc=(rco-rci)/(m-0.5)
drg=rc i-rf
r (1)=0.0
do 2 i=2,n
2 r(i)=r(i-1)+drf
r (npl)=r (n)+drf/2.
192
r(np2)=r(npl)+drg/2.
r(nP3)=r(np2)+drg/2.
r(nP4)=r(nP3)+drc/2.
do 3 i=nP5,nmpl
3 r(i)=r(i-l)+drc
al=4./drf**2
bl=qf/condf
cl=l./(alphf*delt)
a2=1./drf**2
a3p=(drf/drg)*(4.*rf+drg)/(4.*rf+drf)
c3=2.*al
e3=8./drg**2
ag=l./drg**2
bg=l./(2.*(rf+drg/2.)*drg)
a5P=(drg/drc)*(4.*rci+drc)/(4.*rci+drg)
c5=e3
e5=8./drc**2
a6=1./drc**2
b6=1./(2.*(rci+drc/2.)*drc)
alp=(rco-drc/2.)/drc
cl=h*rco
d]P=0.5*(rco*drc-drc**2/4.)
elp=(rco*drc-drc**2/4.)/(delt*2.)
if(time.eq.0.) go to 37
60 nn=nn+l
61 a3=a3p*condf/condg(l)
d3=1./(alphg(l)*delt)
f3=a3*cl
g3=a3*c3
cg=l./(2.*alphg(2)*delt)
a5=a5p*condg(3)/condc(l)
b5=d3
d5=1./(alphc(l)*delt)
f5=a5*b5
g5=a5*c5
c6=qchem(2)/(condc(2)*r(np4)*drc*6.28)
d6=1./(alphc(2)*delt)
al=alp*condc(m)
bl=sig*eps*rco*(troom**2+t(nmpl)**2)*(troom+t(nmpl))
cl=h*rco
dl=qchem(m)/6.28
el=elp*condc(m)/alphc(m)
bb(l)=l.
zz=al+cl
cc(l)=-al/zz
dd (1) = (cl *to (1)+bl)
/zz
do 10 i=2,n-1
zz=2.*a2+cl
aa(i)=-(a2*(I.-O-5/float(i)))/zz
193
bb (i)=1.0
cc (i)=-(a2*(1.+0.5/float(i)))/zz
10 dd(i)=(cl*to(i)+bl)/zz
zz=a2*(3.+0.5/float(n))+c1
aa(n)=-(a2*(1.-0.5/float(n)))/zz
bb(n)=1.0
cc(n)=-(2.*a2*(1.+0.5/float(n)))/zz
dd (n)=(cl*to(n)+b1)/zz
zz=f3+g3+d3+e3
aa(npl)=-g3/zz
bb(npl)=1.0
cc(npl)=-e3/zz
dd (npl)=((f3+d3)*to(np1))/zz
zz=2.*ag+cg
aa(np2)=-(ag-bg)/zz
bb(np2)=1.0
cc(np2)=-(ag+bg)/zz
dd (np2)=(cg*to(np2))/zz
zz=f5+g5+d5+e5
aa(np3)=-g5/zz
bb(np3)=1.0
cc(np3)=-e5/zz
dd (np3) = ((f 5+d5) *to (np3) )/zz
zz=3.*a6-b6+d6
aa(np4)-(2.*(a6-b6))/zz
bb(np4)=1.0
cc(np4)=-(a6+b6)/zz
dd (np4)=(d6*to(np4)+c6)/zz
do 20 j=np5,nmpl-1
i-j-np3
c6=qchem(i)/(condc(i)*r (j)*drc*6.28)
d6=1./(alphc(i)*delt)
zz=2.*a6+d6
aa(j)=-(a6-1./(2.*(rci+(float(i)+0.5)*drc)*drc))/zz
bb (j) =1.0
cc(j)=-(a6+1./(2.*(rci+(float(i)+0.5)*drc)*drc))/zz
20 dd(j)=(d6*to(j)+c6)/zz
zz=al+bl+cl+el
aa(nmpl)=-al/zz
bb(nmpl)=1.0
dd(nmpl)=(el*to(m)+bl*troom+cl*tair+dl)/zz
call tridag(l,nmpl,aa,bb,cc,dd,t)
go to 62
if(iflg3.eq.1) go to 62
if(iflg2.ne.1) go to 62
iflg3=1
go to 61
62 continue
time=time+delt
194
if(nn.ne.nprnt) go to 37
print," "
print 28, time,rin,qf,qtot
28 format(lx,"time=",f7.1,3x,"rin=",f8.6,3x,"qf-w/cm=",f5.2,3x,"qtot=",3xel0.4)
print," "
do 30 j=1,np3
print 32, j,r(j),(t(j)-273.)
32 format(3x,i5,5x,f8.6,3x,flO.2)
30 continue
nn=0
do 34 j=np4,nmpl
print 36, j,r(j),(t(j)-273.),qchem(j-np3),iflg(j-np3)
36 format(3x,i5,5x,f8.6,3x,flO.2,3x,elO.4,3x,i4)
34 continue
37 continue
if(time.gt.timax) go to 70
do 50 j=1,nmpl
50 to(j)=t(j)
rhozr=6500.
do 80 i=1,3
alphg(i)=4.896e-7*t(n+i)+1.325e-3*t(n+i)-.2197
80 condg(i)=alphg(i)*0.38783/t(n+i)
j=nmp1+1
90 continue
if(t(nmpl).gt.292.) trom=0.90*t(nmpl)
if(t(nmpl).gt.292.) tar=0.96*t(nmpl)
if(t(nmpl).gt.1300.) trom=0.89*t(nmpl)
if(t(nmpl).gt.1560.) trom=0.88*t(nmpl)
if(troom.lt.trom) troom=trom
if(tair.lt.tar) tair=tar
if(time.lt.tburn.and.iflg2.ne.1) go to 120
if(iflg2.eq.0) nn=0
if(iflg2.eq.0) time=delt
iflg2=1
if(t(nmpl).gt.1323.) eps=0.25+4.5e-4*(t(nmpl)-1323.)
nprnt=nprnt2
j=j-1
qchem(j-np3)=0.
xc1=9340.
xc2=13760.
if(t(j).lt.1193.) go to 100
xcl=4.68e8
xc2=26670.
if(t(j).lt.1428.) go to 100
xc1=5.04e5
xc2=14630.
100 continue
if(iflg(j-np3).eq.1) go to 90
if(j.1t.np4) go to 120
rct=rco-rin
ratek=xcl*exp(-xc2/t(i))
w=(ratek*delt)/(rhozr*rhozr)+rct*rct
rcn=sqrt(w)
195
rinn=rco-rcn
if(rinn.gt.(r(j)-drc/2.)) go to 110
rinn=r (i)-drc/2.
iflg(j-np3)=1
110 qchem(j-np3)=(rin**2-rinn**2)*7.8e4*3.1416/delt
rin=rinn
if(iflg(j-np3).eq.1) go to 90
120 continue
xnu-4.2
tfilm-(tair+t(nmpl))/2.
h=xnu*(1.Oe-4+5.913e-7*tfilm)/(2.*rco)
do 130 1=1,m
ts=0.001*(1.8*t(np3+1)-32.)
condc(1)=((((.20635*ts-.5567)*ts+.6748)*ts-.13153)*ts+.68244)*.20768
if(iflg(1).eq.1) condc(1)=0.80*condc(1)
ccl=4.184*(7.le-2+1.75e-5*t(np3+1)-0.89e3/t(np3+1)**2)
if(t(np3+1).gt.1223.) ccl=0.364
alphc(1)=condc(1)/(6.5*ccl)
if(iflg(l).ne.1) go to 130
tcon=2.667e-3*t(np3+1)-3.12507
condc(1)=1.531e-3+6.027e-4*tcon+1.0434e-4*tcon**2
go to 133
ccl=((3.764e-9*t(np3+1)-1.667e-5)*t(np3+1)+0.02457)*t(np3+1)-0.81896
133 continue
alphc(1)=condc(1)/(6.5*ccl)
130 continue
do 140 k=1,m
140 qtot=qtot+qchem(k)*delt
go to 60
70 continue
end
subroutine tridag(if,1,a,b,c,d,v)
dimension a(1),b(1),c(1),d(1),beta(35),gam(35),v(35)
beta(if)=b(if)
gam(if)=d(if)/beta(if)
ifpl=if+1
do 1 i=ifpl,1
*c (i-1) /beta (i-1)
beta (i) =b (i) -a (i)
*gam(i-1) )/beta (i)
1 gam(i)=(d (i) -a (i)
v (1)=gam(1)
last=1-if
do 2 k=1,Iast
i=1-k
2 v(i)=gam(i)-c(i)*v(i+1)/beta(i)
return
end
196
APPENDIX E
CLAD INPUT, OUTPUT AND PROGRAM LISTING
CONTENTS
page
S ection
E.1
CLAD Input
197
E.2
CLAD Output
202
E.3
CLAD Program Listing
205
E-1
Sample Input Listing
200
E-2
Sample Output Listing
201
Table
197
CLAD INPUT, OUTPUT AND PROGRAM LISTING
E.1
CLAD INPUT
The CLAD code, used to analyze the experimental test
assembly, is based on the stand alone SFUEL code which was
developed to analyze open frame spent fuel holders in
NUREG/CR0649.
Input is entered under the heading $INPUT
and is defined below:
Input
Variable
Name
Description
DELH
Energy released by chemical
Nominal
Value
7.8e4
oxidation reaction, (J/cm3Zr)
DELT
Computational Time Step
FL
Active fuel
(heater) length,
33.
(cm)
FRAD*
Radiative view factor (a dif-
0.0
ferent formulation of radiative
heat transfer is employed in
this code)
G
Nitrogen mass flow rate (gm/s)
HWOUT*
Initial value of heat transfer
coefficient from exterior of
assembly, (W/cm2 K)
0.0
198
IH20*
Flag indicating presence of water
0.0
in the storage assembly
N
Number of axial node interfaces
NPRINT
Number of time intervals between
11
-
printout
NROD
Number of rods in assembly
NTIME
Number of time steps for program
9
-
run
POWO
Total power input to heater
1.35
array, (KW)
RC
Clad outer radius, (cm)
0.546
RCI
Clad inner radius,
0.484
RF
Radius of heater element, (cm)
0.317
SMB*
Axial power profile parameter
0.0
TIMEO*
Decay heat parameter
4.0e6
TO
Initial temperature of assembly,
(cm)
283.
(*K)
TSAO
Ambient temperature outside the
283.
canister, (*K)
W
Width of the zirconia duct in
5.08
which the rod array is placed,
(cm)
XCS*
Distance between clad and struc-
0.0
ture, (cm)
XOX*
Initial oxide thickness, (cm)
0.0
199
XS
Thickness of zirconia liner,
0.254
(cm)
XW
Radial average thickness of
5.84
alumina fluff, (cm)
Note that the asterisked (*) quantities are input in the
present version only to satisfy the namelist read statement; otherwise, these variables are not used in the program.
A sample input listing for the experimental simula-
tion is given in Table E-l.
200
TABLE E-1
Sample Input Listing
$INPUT
RF
=
0.317
RC
=
0.546
w
=
5.08
xS
=
0.254
FL
=
HWOUT
=
0.0
Powo
=
TIMEO
=
4.0e6
TSAO
=
SMB
=
0.025
FRAD
=
0.0
TO
=-283.
RCI
=
0.484
xox
=
0.0
xCS
=
0.0
NROD
=
9.
N
=
11.
NPRINT
=
6.
NTIME
=
360.
IH20
=
0.
DELT
=
10.
xw
=
5.84
G
=
12.0
38.1
1.35
283.
TIME
=
0.i0OE+02 PEXIT -
JT
36 DELT
TCENT
0.33706E+03
0.43768E+03
0.52304E+03
0.58551E+03
0.62817E+03
0.65504E+03
0.66863E+03
0.66422E+03
0.60642E+03
0.32986E+03
RIN(I)
1
2
3
4
5
6
7
8
9
10
EGEN ORAD PGEN POGEN
0.54600E+00
0.54600E+00
0.54600E+00
0.54600E+00
0.54600E+00
0.54600E+00
0.54600E+00
0.54600E+00
0.54600E+00
0.54600E+00
*
0.000E+00 G -
0.252E+00 PHI -
0.000E+00 KIT a
0.100E+02
TCAVG
0.33679E+03
0.43735E+03
0.52262E+03
0.58504E+03
0.62767E+03
0.65453E+03
0.66813E+03
0.66377E+03
0.60622E+03
0.32986E+03
OF(I)
0.13500E+03
0.13500E+03
0.13500E+03
0.13500E+03
0.13500E+03
0.13500E+03
0.13500E+03
0.13500E+03
0.13500E+03
0.13500E+03
TAVE( I)
0.12851E+03
0.31371E+03
0.42830E+03
0.51054E+03
0.56841E+03
0.60695E+03
0.63010E+03
0.63812E+03
0.61626E+03
0.49024E+03
QCHEM(I)
0.OOOOOE+00
0.00000E+00
0.00000E+00
0.00000E+00
0.OOOOOE+00
0.00000E+00
0.00000E+00
0.00000E+00
0.OOOOOE+00
0.00000E+00
TSIN(I)
0.13170E+03
0.30691E+03
0.41443E+03
0.48974E+03
0.54122E+03
0.57440E+03
0.59339E+03
0.59798E+03
0.57179E+03
0.44325E+03
OCOND(I)
-0.38587E+01
-0.26516E+02
-0.22487E+02
-0.16458E+02
-0.11241E+02
-0.70829E+01
-0.35876E+01
0.11488E+01
0.15178E+02
0.72876E+02
TSOUT( I)
0.10130E+03
0.22635E+03
0.29522E+03
0.33863E+03
0.36540E+03
0.38098E+03
0.38887E+03
0.38912E+03
0.37051E+03
0.28353E+03
OT(I)
0.14596E+03
0.10975E+03
0.95152E+02
0.82205E+02
0.70012E+02
0.59349E+02
0.49604E+02
0.35272E+02
-0.74332E+01
-0.15678E+03
TWIN(
I)
RHO(I)
TWOUT(I)
0.10130E+03
0.22635E+03
0.29522E+03
0.33863E+03
0.36540E+03
0.38098E+03
0.38887E+03
0.38912E+03
0.37051E+03
0.28353E+03
O.10000E+O2
0.12867E+02
0.13709E+02
0.14192E+02
0.14472E+02
0.14631E+02
0.14719E+02
0.14729E+02
0.14479E+02
0.13076E+02
0.12471E-02
0.67867E-03
0.54012E-03
0.47107E-03
0.43149E-03
0.40803E-03
0.39434E-03
0.38730E-03
0.38739E-03
0.40682E-03
EXI(I)
EX2(I)
FOX(I)
0.29943E+01
0.88247E+01
0.14818E+02
0.20831E+02
0.26287E+02
0.30716E+02
0.33876E+02
0.35480E+02
0.34250E+02
0.25763E+02
0.00000E+00
0.20750E+00
0.24016E+00
0.25245E+00
0.25433E+00
0.25112E+00
0.24549E+00
0.23686E+00
0.21417E+00
0.13022E+00
0.23000E+00
0.23000E+00
0.23000E+00
0.23000E+00
0.23000E+00
0.23000E+00
0.23000E+00
0.23000E+00
0.23000E+00
0.23000E+00
0.486E+06 ECHEM =
0.000E+00 EFUEL 0.303E+06 ECONV *
0.172E+05 DELE
0.201E+05
0.766E+05 OSTR 0.687E+05 ECHOX 0.000E+00
0.135E+04 PCHEM 0.0O0E+00 PCONV 0.107E+03 PRAD *
0.307E+03 POUT *
0.234E+03 DELP =
0.15E+02 POCHEM 0.000E+00 POCONV 0.302E+01
0.594E+01 PDOUT a
0.549E+00 PDRAD -
0. 702E+03
I5(I)
0
0
0
0
0
0
0
0
0
0
202
E.2
CLAD OUTPUT
A sample output for the experiment simulation is shown
in Table E-2, corresponding to a step in the transient heat
shown in Fig. 5.4.
The output corresponds to the pre-oxi-
dation conditions of the experiment, for an elapsed time of
360 seconds.
As described below, the output variable "TIME"
gives time from the introduction of oxygen, while the total
elapsed time for the experiment is given by:
TOTAL ELAPSED TIME = JT * DELT
(seconds)
The output is defined below in order of appearance in Table
E-2.
Output
Variable
Name
Description
TIME
Time from introduction of airflow (sec);
during pre-oxidation heat-up, this equals
DELT
PEXIT
Not used in this program version
G
Total mass flow rate
PHI
Not used in this program version
(02+ N2),
(gm/s)
203
KIT
Number of successive substitution iterations
performed to properly account for radiative
heat transfer
JT
Number of time intervals elapsed
DELT
Computational Time Step
I
Axial nodal location, where I= 1 is the base
of the assembly and I= 10 is
TCENT
the top
Temperature of center rod in the array at
axial location I, (*C)
TCAVG
Average temperature of all rods at location
I, (*C)
TAVE (I)
Average airflow temperature at location I,
(*C)
TSIN (I)
Temperature of air-side of zirconia liner,
(*C)
TSOUT (I)
Temperature of alumina fluff side of zirconia liner, (*C),
(also equal to TWOUT(I))
RHO (I)
Density of gas flow at location I, (gm/cm )
RIN (I)
Outer radius of unreacted clad, (cm)
QF(I)
Heater input power in the rod array at axial
location I, (W)
QCHEM(I)
Energy released by chemical oxidation, (W)
QCOND(I)
Energy conducted into heater/clad at location I, (W)
QT (I)
Total energy exchange between rod array,
airflow and zirconia liner,
(W)
204
EXl(I)
Energy absorbed by zirconia liner, (W)
EX2(I)
Energy removed by natural convection from
the exterior of the canister, (W)
FOX
Oxygen mass fraction at axial location I
IS(I)
Flag indicating regime of oxidation kinetics
1 - kinetics rate-limited
2 - diffusion rate-limited
EGEN
PGEN
PDGEN
Energy, power and heat flux input to system
2
via heaters, (J), (W), and (W/cm ) respectively
ECHEM
PCHEM
PDCHEM
Energy, power and heat flux via oxidation
2
reaction, (J), (W), and (W/cm ), respectively
EFUEL
Energy stored in the heater and clad, (J)
ECONV
PCONV
PDCONV
Energy, power and heat flux input to airflow
and convected out of the assembly, (J),
and (W/cm2),
DELE
DELP
(W),
respectively
Energy and power imbalances-present version
does not account for convective loss from
outside of canister, (J),
QRAD
PRAD
PDRAD
respectively
Energy, power and heat flux out of the ends
of the rod assembly by radiation, (J),
(W/cm2),
QSTR
POUT
PDOUT
(W),
(W),
respectively
Energy, power and heat flux to zirconia liner
2
and insulating canister, (J), (W), (W/cm2)
respectively
ECHOX
Energy stored in oxidized clad layers,
(J)
(113a',c Z) / (*i/ (ZO*
-Zda *LW)=dSJ
Z-da/ CZ/Zba-LWI =5V
(Zba-%.9b Z)/P1 =98
(~0~~Z) P' I=58
Z~~) /I'
i=
~~ait ~
(Za'cbc /t0~C1 / (0+W
1=V
~
±130!' t=dZO
(±130* Z)/(I
(TG0' ItM) +t U* I b) =d 10
*l/IbO+lb=dtJ
lbG+Tb=9b
'1 /M= IN~
S*m~b
S.i/sx=tba
*'OZi l3W1
O=W3HON
ClI I , 11H1 1I X8 / Z II lVlIZ 39 / W13wIN,, XL 11 INI dN,i X9 AiN1 C
XII 11
AMN 1 X8 ,1SOX,, X6 ,,xox, x6
,iJiobi x6 ,1130,1 X8 Ali
XQI Z
,,avbj 11 xS /
711301 / 11ews,, x6 110VSI 1i X9 ,03W1.I 11 xL A1 MUd 1 , Xg I
,a1flMH,, X8 O,,
X01 g,SX i XOI
Mll XIi ,OH
i0
Ji XOi)IVWNO
91
1
11
imWoi infdNi 3Hi SI SIHJ.
QIHI '3WIiN'1NIlJdN'N'0ObiN'SOX'XOXI
'I'13'J.a'0V'8WJ'OYS'OS3WII'OMOdnOMH'13'SXM'3OI'AJ
(9i'9) 31Ibm
(indNI '5) 0V3N
H13a ' OV3bI
UA'MX'9'1130'OZHI '3WIiLN'.LNI~dN'N'aObIN'SJX'XOX'1 Ii
'O.L'GVbU'8WS'OVSI'O3WIL'OMOdlflOMH'1I
'SX'M'JU'
AU
/.LldN
I/i.SI13WVN
13SVOS 11VO
H13a'Il3al'b'I~b4Xl3'~V NOWWOO
1 T1i/Id VILVO
/
Ci ) ~~iNQISN3WIQ
(Ti)1X3'Cii)iX3 NOISN3WI0
'(Ti)N31' (1i) 3AVI'" (ti.fl
(ii)NIMI' Ciu) sob" (it)Az
(Ii)OHNU(1)1 (ii)
iO.nosi' Cii)NIS1' Cu) sbs (tii'
CII) 0N03bi
'(1)01' ii)W3H?~ C~)~ Ci~d'Ci)±'(113± (l~xNOISN3WIO
CL)
ao' L)033' L) se' L)
CH'L) 010± NOISN3Wia
C.ndino'±ndN0)13fljs WVM0~d
vv'(hi'L)M3N±'
DNI1lsirI WVHDOdd
3
vqVTD
Id-=IdV
in*OOHb*DWS-Id=Id
IJ*OMJ*9WS-=Id
(Oi,%cVb)/Od=OOHb
*06=ows
OMOd*'0001=OMOd 91
99Z'O=MOdVO
SU*O=MOdV
91 01 00 (9+3* *11*03WIi)JI
OWO=MOdVO
iSZ*O=MOdV
((*I-Sd3/*1)3.tSV/(GObNY/*I-*I)).'.bV+3d3/*I)/5*0=S3d3
M*M=Gvbv
GObN*3b'%e3b*Id-M*M=V
OV+JV=IV
HIM* ( IOb* IOb-O J*Ob) * Id=OV
GObN*Jb*J I*Id=JV
GObN*XI30*0b*Id**Z=HV
SO=OSO
SOS*SOHb*XI30*SX*M**q=SO
X13a*M** *biSJ=SV
(I -N)
(1) X** Z-1 A) =X130
OZ-3*1=(I)X
5*9=OOHb
Lz*C=AOHb
(91*8z/LL*0+00*ZC/Czo)*L+3q IC*8=V I
9+35MO'T=Od
ZI-3L9*5=ois
01-3*1+01=031
W'ZT39 / I
llmxll XOT
lox,, XOT oxoNws,, xL ,mNws,, xg ,sNws,, )(8 ,Hisj,, x8)ivwboA
Li
MX'DX'XONWS'MNWS'SNWS'HiSA (LI49)31lbM
CZ500=mos
C0Z'O=SOHH
q8001=s3s
96o*O=MOHb
(01-(Sd3/*I)+(XOd3/*I))/*I=S3d3 (*0*10*XOX) Al
(*I-(Sd3/*T)+(3d3/0T))/500=S3d3
((OT-3d3/"1)3.c(*I-GObNX)/*T+3d3/*I)/5*0=iN30d3
(aOHN)ivoij=aOHNX
59*=XOd3
Z*O=Sd3
0O=3d3
qiLeo=ox
Z9000=XONWS
U100=MNWS
6zioo=sNws
O*I=HISA
9*i5=in
5=M3NHdN
OZ+3*T=INd3I
*9=M3NMS
*000c=MS31
OZ+3*1=XVWOI
90Z
3flNIINOO 5Z
0=(I) WHOb
0= (i)wxo
£Zeo=(I)xoj
o00=(I)W3H3b
o0o=(i)mb
ooo=(i) sb
oo=(i)ibL
0*0=(I) 0N0Jb
Ml~
0OHbJ= (I1)
0*0=(I)d
oi=(i)inomi
o±=(I) NIMi
O±=(i) .LfOSJ
Oi=(I) NNIS.L
O±=(I) NISI
01= (I) I
oi=Ci'r)aioi ot
L'i=r
01 00
001.=(I)NXOIi
0J.L=(I) NINJJ.
03=(I) JNOI
Oli= (I)N1U
0=(1)113W
01= () 011
001=(I) NOI
00.1= (1)31
X13a+(I) X= (1+1) X
(xox'qoos9/oGT1) tVwV=0)iOH
TWN'I=I 5Z 00
I -N=TWN
* I+MOdV-=IMOdV
VHi/X13~0*0d*DWS=DWSM
T rz3'UM0d 31AV)VUH~TI'
HI)IYWH0A ZZ
(*
IM (ZZ69)llI1IM
OMOd/1 A*S=IM
( ZT3i i,1Dws, x6 gAM,~g x6 o,~3 x6 / *TI301 / ,3, XOi C
,,Mdii X01 il1dit XOI i,00H~i X8 Ai X11 013t XOI OM~dii XS 1 1SV,i X01 Z
1
1
1OT3I01
, Xt1Xl3a,, xS i
Xu T IVX01
/ 11OV,i XOT
,Vii XIT ,i.V,X0I/
DWS'3)IX63)IX'3aMd'ld00HHI I
IVW80:1 3718VI
V dfl13S 31-1 SI SIHI
0*0=3NX
00O=ONX
Mdl/V*c 'T] 0
G0H~N*OH*~I d*s +M* 'T=Md
LOZ
208
C
C
TC (N)=0.
TCN (N)=0.
TCX (N)=O.
TCXN (N)=0.
TSIN (N)=O.
TSINN(N)=o.
TSOUT(N)=0.
TWIN (N)=O.
TWOUT(N)=0.
RHO (N) =0.
QF (N)=O.
QT(N)=O.
QS(N)=O.
QW(N)=O.
QCHEM(N)=0.
FOX(N)=0.23
EGEN=0.0
ECHEM=0.0
CHOX=O.
EFUEL=0.0
ESTR=0.0
ECONV=0.0
QRAD=0.0
QSTR=O.0
TIME=O.0
DELTO=DELT
NP=O
F=1.0
G=G*0.021
START TIME LOOP
DO 70 J=1,NTIME
JT=J
TIME=TIME+DELT
DO 30 I=1,NM1
T (I)=T I0(1)
TC (I)=TCN (I)
TCX (I)=TCXN(I)
30 TSIN(I)=TSINN(I)
START DETERMINATION OF AIR PROPERTIES
GNI=0.77*G
GOX(1)=0.23*G
GI=G
DO 38 I=1,NM1
KIT=O
11=1
FOX(1+1)=GOX(1+1)/(GoX(1+1)+GNI)
Gl=.5*(GOX(I)+GOX(1+1))+GNI
45 KIT=KIT+1
CALL APROP(GI,DE,TSIN(I),T(I),P(I),RHO(I),CP,RE(I),HSIN,SMF,A
1 ,FOX(I),CPOX,CPNI,RA,HDR(I))
CALL APROP(GI,DETCX(I),T(I),P(I),RHO(I),CP,RE(I),HR,SMF,A
1 ,FOX(l),CPOX,CPNI,RA,HDR(I))
IF (HSIN.EQ.0.) HSIN=1.E-20
WK1=RHO(I)*CPNI*A*DELX/DELT
+(1) N3±i( 1) NOI)
CD- M~ vv
(ZG-ZV) -
(Z)vv
1) I)1O.Lc IO+(1) NODL'I S+(I)3AVic IV= (1)00
o3-=(I) 33
GHdIV/dS3=53
8>Iws/L~i1flOMH=L9
(il3O*.8HdIV) / i=£
(1130.c'VHd1V) / t=£
dCV*c(SNWS/VNWS) =Cv
VHdl V/dZO=ZO
dlG*.SOS*cS0Hbi=l0
dlO*.VNWS=t 3
1 ) NIS.LI (1I)N3.L) I bhS3d3*c9 I S*I d/ * frt1
tINISH=IV
MNWS=8)IWS
SNWS=V)4WS
(M3ScM0Hb)/M)IWS=SHdIV
(S3S*.SOHb) /SNWS=VHdIV
-3686+C-3Z9 t=M)IWS
ii1*L1* t -3£9I 'Z+i1
LOO=MOs (00£P±9*(I)NIM)dI
Vro+(I) NIMI*c'-356*i=moS
1** 3 0*1iTo
+i7 -3LZ9+-3lSI=SNWS
96812o- (I)NIMI~c (LSZO*O+(i) NIMIb.*(S-3L991- (I)NlMIAC-39L£))=S3S
OZ-3*i=ifOMH (*O~3.lOtH) Al
(±nOMH'OVSI'(i)imM' (I)X)ANO3M 11VO
3bufli.f1s aNV IIVM HoflObHl S3~flJV~dW3.L M3N 31LVin3lV3
3
3flNI.LNOO M£
(N)I*~ (I Nd3*~ I ND+X~d~.c (N)X09) =ZN03
£ 01 09 (IWN'l.VII) Al
3fliNO3 ££
(I).*~(INdO.cIN9+X~d3O (I)X09) =NOJ3
CC 01 09 (1
Ho, ) AI
(I)1= (1)0 I11
HiivN3dW3.L HIV M3N U1vinlflV
)M+4m+NM+l NtM=SNM
INd3S9=T)M
Xl 30AcV'tH=0)M
Xl30acSV*cN ISH=ZNM
3
6OZ
31* (3NWS*3V+JNWS),'V)
/(C1)
-
(1-1)01*
Xl 301
(NWS*3V+JNWSJV) ) =(1) ONOJo
0c, =i)WS
L6z9i 'o=jiws
P1 01 09 (POb3xt)l1
I=XI
IWN't=t 8P 00
0=11v31
31 010 HilM Xfl1
IV3H NOul20NO3
3
31 11V031i
3NlNINO3 Si
Mb= (I) X3
(I)
(1) sb= C1) 1 X3
(1) S30+(ONISi- (I)3AVi))vNISHv.SV=(I) sb
(1) SiO+((I) 3AV- (1)X3i) bHvbV= (I)A
(ONI Sl-(1)
X31)vS3Hv-(O0bNX/'* I-'* )v.bV=(1)
sob
ISl+Zv.( I)X31) * (ON ISi+ (I)X31) * IS*S3d3=S3H
(I)M
Idv'Z=
(OVSi- () ifoMi)10MHX130Ll
(iI'rr) M3N1= (I'rr)01O± 171
L't=rr 'i 00
(I'L)M3N1=(I)1noMi
(I)NIMI= (I)iloSi
(I'C)M3N1= (I)NIMi
3flNIINOi r'
Si 01 00 (ZOO'9'(ONISi-(I)NISi)SV)I1
Ol1 019O (501q9i)JI
(I ')M3NI=(I)NISi
(rr)Wbdl=CI 'rr)M3NI £1
L'i=rr it 00
(I)NISI=ONISi
(EvvON
3
00
(I 'L)10i1v5+0VSi''Le= (L)
0'0= L)33
5v-= WL vv
(I'9) 010n13= (9)00
(9G+%V)-=(9)33
17i+'TV~ = (9) 68
(91-1)V) -= (9) VV
0S8+%7v
M=
CS0
3
00
= )(
(I'E) 010i+ ZS-7)-=
CM
vv
(I
'17)* 010a+C17= (1)00
C3-= C0)30
t3+toGt9+t3= Ct)
1130/
*0=(I)W3H3O
*0=(I)WXO ill
oo59/oti=(i±,
6li
DNI1V3H IVOIW3HJ ON
3
05t 01 0D
ONC8V= (1) W3H3b
1()
1J1I-NO I)'.' (H130) =Ob
u8V/1130.c0WX04 ((*9*
)/Z 16) +(1I±J=NOJU
GWxO=(I)wxO
Z= (I)S I
0311W11 N0IsfliOi - z=si
6l i 01 0 (awxo±1ii*wxo)i
3
11 30/ (NN I Id- (I1) NIN)*H130=0b
£iTI 01 09 (o031'NWXO) JI
NWX0=(a)wxO
T==() S I
Q311WII S313NIN 3I109VHVd - T=SI
(1)X0J.c (I1)
0HHI~ (1)HOH~cuJV=GWX0
~bN3=(I)1U
V,
1130/ ( (1)I-t NOH) 3.4H130=1 Ob
INOH-OH=(I) 3NIHJ
u8V/O0uNX*c1130,.cWXO*9S8V1.0+
(I).L=t NOu
OwxO=(I) IWXO
L'i 01 09 (awx0.L1-Nwx0)ni
INNIH=(I)3NIH (aWXO*11*NWXO)31
0=(i)lWxO C*OJ(I*Nxod)1
)1wx0=(I) TWX0
(1)X0J* (1)MIu.* (1)HH*IV=aWX0
0=tDb (o031xwx)4WX1
113/ IN a -11)3a/N
IH (I)oNx/II)*HIv=iob
(1a)XOd'INN I H(1)
O N'lb'T NON '(1)Id' (1)1NOI) W3H3 11V3
6qi 01 09 (1b3*oZHIvuO'O~3*W3H3N) JI
Pi 0109D (50ooo0o1 *(I t- (1)N11) s~V)A
(I)NTN3±=(a) tNO1
X13aOV*30
HHA~cO
O =3
Miot=jo
lf9*o=t33 (*i.LTID*a)NO)JI
IWN'tIl o5 oa
(I-N) 3±*50*ONWS
cV+
W
- V(N)
0N03b
(N.c0o- * (1) 31) *OTV*
Sd3*.9I S-= (I)GN03b
saoH Ao SON3 WObJ xnii 1V3H N011VIaVN 31Vifl1vo
3flNIINO3 O
3
TTZ
sv/lfl~d=iflOGd
11 30'.W3H~d+W3H3 3=W3HJ3
IV/W3H~d=W3HOad
(i)SO+±id=±flOd
((I) W3H~b- (I)WHb) + (I)W3H30+W3H~d=W3H~d
IWN'I=I £S 00
0o=LfOd
O=W3H~d
5'(9)'rZ0 / (+38L',tZZ 16) )+XOH3=XO0H3
113'.' (N)X09- (1)X09)
11 3a,,c vMd+0Vll=0Vib
(MV~. Z) /GT~d=0V~0d
()0GN030- (N)G0O3=0V~d
11 30*.ANO3d+ANO3 3=ANO033
(SV+ V) /ANO3d=ANO3Od
I NO33-ZN003=AN03d
1130'.N3Dd+N3O3=N3O3
WJVN30d=N3 D0d
J*0M0d=N30d
swflS ADOi33 31Viflo1vo
3flNIINOJ 05
(INO+ (1+1) X09) / (1+1) X09= (1+1) X0J
0=UT+I)X09 (o0'31*(I+I)XOD)Hl
1I) twxo-( 1)wxo- (I) xOo=(1+t) X09
q*Czo= (I)xoo
0 0+d.LS3=U.LS3
(I) NIS.L=(I) NNISJ.
3bfl.Vb3dW31 3bfl13flis M3N 31vifl1vo
0
031=(I)NxOI
(1)N31= (1)NI NO
(00±11iI)Nx3I)
Al
05Z 01 09 (o031*xOX) 31
(I) N3i= (I) NX3J.
1 *(1) NI N3) A
3flNIINOJ Z5
( (1)ND
(I)NOI=NNOI
Z5 0± 00 ((I)W3HJO±1i*I)WH3O)AI
(1'()
01L- **. (1)
1 N31L) *'00MN/MV'
I
((0)lb- (1+1) 0N03b- (1)aN03b+( I) W3H3b+(I) J0) * (0/1130)+(1) 01=(1) N3J.
3flNIINO3 P1T
05Z 0± 09
(I)INOI=(I)NINOI
0
(1)01= (1)NO1
P'T 0± 09 (-0±1(I()3QNbZ)JI
PIr 01 09 (±13W±11-(1)0±)iI
T==() 113W (113W±LD* (1)I) 31
a0N.cX13'.c
d'.'(Z**10
I3
''TI'
CIN3
3nN IiNO3 oL
3flNI INOO
oLt
3flNI iNOO
I=W3H-3N (oz±9±~r)Ai
000=3W11 (I3JHN) Al
9=iNI~dN (Ovioor)Ai
zL 0±0o (XVW3Ir39*(I)N3I) Al
IWN'I=I oLT oa
3fliNO3 69
0=dN
(C.Z13,,= iflOCd 1 1 'ZI 0 = 0d Jt
'~Ih=N390d 1'xi)i.vwnA 691
' *Mot= ANO00d it'tMii= W3HJ13d
ifl0d'CIVbCd'ANQJGd'W3H30d'N30d (69T'9) 3.IbM
d-1304inl~d'GVbd'AN0Jd'W3H~d'N39d (S9'9) 31IJM
GVbb I'XI)I.VWb0J
( Z3, XOH33 i'C'Z3,,= IdLSO II~T3,
£9
X0H-3'Lusb'0V CL9'9) 3itbM
313a'ANO03'-I3nU3'W3H33'N393
(IWN'T=I
(99'9)3i;bM
'(I) SI' (I) X0AI
((505138Z
HI)!!
11(I)0HH,,
x6
, 1(l)inOMI,, XL ii(I)NIM.Lo Xg
11(i)inosi,,
xLt
(IWN't=I'(I)0HbJ'(I)30MI I
iVWbI0A 319VIbVA ifidiflO 31-11 SI SIHJ.
(io,,=() LvwoA 9
(ioZT3,= 113
AV.() 3AVlb
11Lag- (I)
t~(I, XJ±=(I) JX1L
S50
Wn'tINO
69 0± 0 (NI=NL)3N) I
*Uz-(~inoi=(dNodN
~±SO0V~OAN0J-U±S-13lZJ-W3HJN 9=l3130
idW3inos =d1 30s
(lNSI()3S
ifl~d-0V
11
214
SUBROUTINE CHEM(TCRCT,RCTTQC,RIN,RINNFOX)
COMMON AR,DELX,RCI,RC,DELT,DELH
QC=0.
RCL=RC-RCI
IF (RCT.GE.RCL) GO TO 40
C1=9340
C2=13760.
C3=0.0
IF(TC.LT.1193.) GO TO 10
C1=4.68E8
C2=29000.
IF(TC.LT.20000.) GO TO 10
CI=5.04E5
C2=15630.
10 CONTINUE
RHOZR=6.5E+3
RCT=RC-R IN
RATEK=C1*EXP(-C2/TC)
W=(RATEK*DELT)/(RHOZR*RHOZR)*1. + RCT*RCT
RCN=SQRT (W)
IF(RCN.GT.RCL) QC=O.
IF(RCN.GT.RCL) GO TO 41
RINN=RC-RCN
RCTT=RIN-RINN+1./6500.
IF(RINN.GT.RCI) GO TO 40
41 CONTINUE
RINN=RCI
RIN=RCI
RCTT=RCT
QC=O.0
40 CONTINUE
RETURN
END
215
SUBROUTINE APROP(GDETC,T,P,RHOCPREH,SMF,A
1 ,FOXCPOXCPNIRA,HD)
AMOX=32.00
AMNI=28.16
AMAIR=1./(FOX/AMOX+(1.-FOX)/AMNI)
RU=8.3144E+7
RA=RU/AMAIR
P0=1.01325E+6
TBAR=(TC+T)/2.
RHO-PO/(RA*T)
CPOX=0.27*4.184
CPNI=0.27*4.184
CP=FOX*CPOX+(1.-FOX)*CPNI
XMU=0.146E-4*(TBAR**1.5)/(TBAR+109.58)
RE= (G*DE) / (XMU*A)
PR=0.714
SMK=CP*XMU/PR
SC=0.748
DIF=XMU/(RHO*SC)
IF(RE.GE.3000.) GO TO 10
XNU=10.02
SMF=25./(RE+1.E-5)
GO TO 15
10 XNU=.023*(RE**.8)*(PR**.4)
SMF=.00140+.125*(RE**(-.32))
15 H=SMK*XNU/DE
HD=DIF*XNU/(DE*2.)
RETURN
END
216
SUBROUTINE WCONV(XTWTOHW)
SIG=5.67E-12
FWR=O.
EPW=0.4
EPO=0.7
EPWO=1./((1./EPW)+(1./EPO)-1.)
SMG=980.
P0=1.01325E+6
RA=2.768E+6
C
C
10
20
1
2
TBAR=TW-0.38*(TW-TO)
BETA=1./TBAR
RHO=PO/(RA*TBAR)
XMU=0.146E-4*(TBAR**1.5)/(TBAR+109.58)
IF(X.LT.1.E-10) GO TO 20
GRX=SMG*BETA*(RHO**2.)*(X**3.)*(TW-TO)/(XMU**2.)
IF(GRX.GT.1.E+9) GO TO 10
LAMINAR BOUNDARY LAYER ON WALL
IF(GRX.LT.O.O) GO TO 20
XNU=0.360*(GRX**0.25)
GO TO 20
TURBULENT BOUNDARY LAYER ON WALL
XNU=0.1160*(GRX**0.333333)
CONTINUE
CP=0.28*4.184
PR=0.714
SMK=CP*XMU/PR
HW=SMK*XNU/X
HWR=FWR*SIG*EPWO*(TW+TO)*(TW*TW+TO*TO)
HW=HW+HWR
RETURN
END
SUBROUTINE TRIDAG(A,B,C,D,V)
DIMENSION A(1),B(1),C(1) ,D(1),V(7),BETA(10),GAM(10)
BETA (1) =B (1)
GAM(1)=D(1)/B(1)
DO 1 1=2,7
BETA (I)=B (I)-A (I)*C (I-1)/BETA (I-1)
GAM(I)=(D (I)-A (I)*GAM(I-1) ) /BETA (I)
V (7) =GAM (7)
DO 2 K=1,6
1=7-K
V(I)=GAM(I) -C(I) *V(1+1)/BETA (I)
RETURN
END
217
REFERENCES
B-1
Benjamin, A. S., et al., "Spent Fuel Heatup Following
Loss of Water During Storage," NUREG/CR-0649, (March
1979).
B-2
Biederman, R. R., Ballinger, R. G., and Dobson, W. G.,
"A Study of Zircaloy-4-Steam Oxidation Reaction Kinetics," EPRI NP-225, (September 1976).
B-3
Best, F. R., personal communication, Massachusetts
Institute of Technology, August 15, 1982.
B-4
Baumeister, T., and Marks, L. S., Mechanical Engineer's
Handbook, McGraw Hill Book Company, (1958).
C-i
Cook, B. A., and Hobbins, R. R., "Fuel and Cladding
Structures Formed During Severe High Temperature Transients," Proceedings, Am. Nucl. Soc. Topical Meeting,
Sun Valley, Idaho, (August 2 - 6, 1981).
C-2
Chilton, T. H., and Colburn, A. P.,
26, 1183 (1934).
E-1
Edwards, D. K., Denny, V. E., and Mills, A. F., Transfer Processes, Holt, Rhinehart and Winston, (1973).
G-1
Gear, W. C., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, Inc., (1971).
H-1
Hayes, E. T., and Roberson, A. H., "Some Effects of
Heating Zirconium in Air, Oxygen and Nitrogen," J. Electrochem. Soc., 96, 142, (1949).
H-2
Holman, J. P.,
(1976).
H-3
Hartman, T., Nagy, J., and Jacobson, M., "Explosive
Characteristics of Titanium, Zirconium, Thorium and
Uranium," Bureau of Mines, Investigation 4835, (December, 1951).
H-4
Hamming, R. W., Numerical Methods for Scientists and
Engineers, McGraw-Hill Book Company, (1962).
K-i
Kendall, L. F., "Reaction Kinetics of Zirconium and
Zircaloy-2 in Dry Air at Elevated Temperatures," Hanford Atomic Products Operation, Wash., Contract No.
W-31-109-Eng.-52, (September 1955).
Ind. Eng. Chem.,
Heat Transfer, McGraw-Hill Book Company,
218
K-2
Kanury, A. M., Introduction to Combustion Phenomena,
Gordon and Breach, (1977).
K-3
Kayes, W. M., Trans. ASME, 77, 1265, (1955).
L-1
Lestikow, S., et al., "Study on High Temperature Steam
Oxidation of Zircaloy-4 Cladding Tubes," Nuclear Safety Project Second Semiannual Report, 1975, KfK-2262,
Karlsruhe, 233, (1976).
L-2
Lustman, B., and Kerze, F., The Metallurgy of Zirconium, National Nuclear Energy Series, McGraw-Hill Book
Company, (1956).
M-1
Miller, G. L.,
M-2
Mallett, M. W., and Albrecht, W. M., "High Temperature
Oxidation of Two Zirconium-Tin Alloys," J. Electrochem.
Soc., 102, 407, (1955).
N-l
Norris, R. H., et al., Heat Transfer Data Book, General
Electric Co., Schenectady, N.Y., (1978 update).
P-1
Perry, J. H., Chemical Engineer's Handbook, McGraw-Hill
Book Company, (1952).
Q-1
Quill, L. L., The Chemistry and Metallury of Miscellaneous Metals, New York, (1950).
R-1
Rohsenow, W. M., and Choi, H., Heat, Mass and Momentum
Transfer, Prentice Hall, Inc., (1961).
S-1
Stalker, K. T., personal communication, Sandia National
Laboratories, letter dated July 6, 1982.
S-2
Sparrow, E. M., and Loeffler, A. L., AIChe Journal,
5:325, (1959).
S-3
Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon
Press, (1978).
S-4
Schlichting, H., Boundary Layer Theory, 4th ed., McGrawHill Bood Company, (1960).
S-5
Sparrow, E. M., and Loeffler, A. L., "Heat Transfer to
Longitudinal Laminar Flow Between Cylinders," J. Heat
Transfer, Trans. ASME, 83:415, (1961).
Zirconium, Academic Press, Inc.,
(1957).
219
S-6
Siegel, R., and Norris, R. H., "Tests of Free Convection in a Partially Enclosed Space Between Two Heated
Vertical Plates," Trans. ASME, 79, 663, (1957).
S-7
Skinner, G. B., Edwards, J. W., and Johnstone, H. L.,
"The Vapor Pressure of Inorganic Substances, V. Zirconium between 1949 and 2045K," J. Amer. Chem. Soc., 73,
174, (1951).
W-l
Whitaker, S., "Forced Convection Heat-Transfer Correlations for Flow in Pipes, Past Flat Plates, Single Cylinders, Single Spheres and Flow in Packed Beds and Tube
Bundles," AIChe J., 18, (1962).
W-2
White, J. H., reported in AEC Fuels and Materials
Development Program, Progess Report No. 67, GEMP-67,
General Electric Co., 151, (1967).
W-3
Willoughby, R. A., ed., Stiff Differential Systems,
Plenum Press, Inc., (1973).