DEVELOPMENT OF MODELS FOR THE
TWO-DIMENSIONAL, TWO-FLUID CODE FOR
SODIUM BOILING NATOF-2D
by
R. G. Zielinski and M. S. Kazimi
Energy Laboratory Report No. MIT-EL 81-030
September 1981
mmmuwIhn
Energy Laboratory
and
Department of Nuclear Engineering
Massachusetts Institute of Technology
Cambridge, Mass. 02139
DEVELOPMENT OF MODELS FOR THE
TWO-DIMENSIONAL, TWO-FLUID CODE FOR
SODIUM BOILING NATOF-2D
by
R. G. Zielinski and M. S. Kazimi
September 1981
Topical Report of the
MIT Sodium Boiling Project
sponsored by
U. S. Department of Energy,
General Electric Co. and
Hanford Engineering Development Laboratory
Energy Laboratory Report No. MIT-EL 80-030
go
N
-i-
REPORTS IN REACTOR THERMAL HYDRAULICS RELATED TO THE
MIT ENERGY LABORATORY ELECTRIC POWER PROGRAM
A.
Topical Reports (For availability check Energy Laboratory Headquarters,
Headquarters, Room E19-439, MIT, Cambridge,
Massachusetts 02139)
A.1
A.2
A.3
A.4
General Applications
PWR Applications
BWR Applications
LMFBR Applications
A.1
J.E. Kelly, J.
Assessment and
COBRA-IIIC/MIT
MIT-EL-78-026,
Loomis, L. Wolf, "LWR Core Thermal-Hydraulic Analysis-Comparison of the Range of Applicability of the Codes
and COBRA-IV-1," MIT Energy Laboratory Report No.
September 1978.
M. S. Kazimi and M. Massoud, "A Condensed Review of Nuclear Reactor
Thermal-Hydraulic Computer Codes for Two-Phase Flow Analysis," MIT
Energy Laboratory Report No. MIT-EL-79-018, February 1979.
J.E. Kelly and M.S. Kazimi, "Development and Testing of the Three
Dimensional, Two-Fluid Code THERMIT for LWR Core and Subchannel
Applications," MIT Energy Laboratory Report No. MIT-EL-79-046.
J.N. Loomis and W.D. Hinkle, "Reactor Core Thermal-Hydraulic Analysis-Improvement and Application of the Code COBRA-IIIC/MIT," MIT Energy
Laboratory Report No. MIT-EL-80-027, September 1980.
D.P. Griggs, A.F. Henry and M.S. Kazimi, "Development of a ThreeDimensional Two-Fluid Code with Transient Neutronic Feedback for LWR
Applications," MIT Energy Laboratory No. MIT-EL-81-013, April 1981.
J.E. Kelly, S.P. Kao and M.S. Kazimi, "THERMIT-2: A Two-Fluid Model
for Light Water Reactor Subchannel Transient Analysis," MIT Energy
Laboratory Report No. MIT-EL-81-014, April 1981.
H.C. No and M.S. Kazimi, "The Effect of Virtual Mass on the
Characteristics and the Numerical Stability in Two-Phase Flows,"
MIT Energy Laboratory Report No. MIT-EL-81-023, April 1981.
J.W. Jackson and N.E. Todreas, "COBRA IIIC/MIT-2: A Digital Computer
Program for Steady State and Transient Thermal-Hydraulic Analysis of
Rod Bundle Nuclear Fuel Elements," :IT-EL-81-018, June 1981.
J.E. Kelly, S.P. Kao, and M.S. Kazimi, "User's Guide for THERMIT-2:
A Version of THERMIT for both Core-Wide and Subchannel Analysis of
Light Water Reactors," 1MIT Energy Laboratory Report No. !'IT-EL 81-029,
August 1981.
-ii-
A.2
P. Moreno, C. Chiu, R. Bowring, E. Khan, J. Liu, and N. Todreas,
"Methods for Steady-State Thermal/Hydraulic Analysis of PWR Cores,"
MIT Energy Laboratory Report No. MIT-EL 76-006, Rev. 1, July 1977,
(Orig. 3/77).
J. Liu and N. Todreas, "Transient.Thermal Analysis of PWR's by a
Single Pass Procedure Using a Simplified Model'Layout," MIT Energy
Laboratory Report No. MIT-EL 77-008, Final, February 1979 (Draft,
June 1977).
J. Liu and N. Todreas, "The Comparison of Available Data on PWR
Assembly Thermal Behavior with Analytic Predictions," MIT Energy
Laboratory Report No. MIT-EL 77-009, Final February 1979, (Draft,
June 1977).
A.3
L. Guillebaud, A. Levin, W. Boyd, A. Faya, and L. Wolf, "WOSUB-A
Subchannel Code for Steady-State and Transient Thermal-Hydraulic
Analysis of Boiling Water Reactor Fuel Bundles," Vol. II, Users
Manual, MIT-EL 78-024, July 1977.
L. Wolf; A. Faya, A. Levin, W. Boyd, L. Guillebaud, "WOSUB-A Subchannel
Code for Steady State and Transient Thermal-Hydraulic Analysis of
Boiling Water Reactor Fuel Pin Bundles," Vol. III, Assessment and
Comparison, MIT-EL 78-025, October 1977.
L. Wolf, A. Faya, A. Levin, L. Guillebaud, "WOSUB-A Subchannel Code
for Steady-State Reactor Fuel Pin Bundles," Vol. I, Model Description,
MIT-EL 78-023, September 1978.
A. Faya L. Wolf and N. Todreas, "Development of a Method for BWR
Subchannel Analysis," MIT-EL 79-027, November 1979.
A. Faya, L. Wolf and N. Todreas, "CANAL User's Manual," MIT Energy
Laboratory No. MIT-EL 79-028, November 1979.
A.4
W.D. Hinkle, "Water Tests for Determining Post-Voiding Behavior in
the LMFBR," MIT Energy Laboratory Report MIT-EL-76-005, June 1976.
W.D. Hinkle, Ed., "LMFBR Safety and Sodium Boiling - A State of the
Art Report," Draft DOE Report, June 1978.
M.R. Granziera, P. Griffith, .W.D. Hinkle, M.S. Kazimi, A. Levin,
M. Manahan, A. Schor, N. Todreas, G. Wilson, "Development of
Computer Code for Multi-dimensional Analysis of Sodium Voiding in the
LMFBR," Preliminary Draft Report, July 1979.
M. Granziera, P. Griffith, W.D. Hinkle, M.S. Kazimi, A. Levin,
M. Manahan, A. Schor, N. Todreas, R. Vilim, G. Wilson,."Development
of Computer Code Models for Analysis of Subassembly Voiding in the
LIFBR," Interim Report of the MTI Sodium Boiling Project Covering
Work Through September. 30, 1979, MIT-EL-80-005.
~
lllllllW10
YIIII
IY
IIIi ^
~
61411111,i
A. Levin and P. Griffith, "Development of a Model to Predict Flow
Oscillations in Low-Flow Sodium Boiling, "MIT-EL-80-006, April 1980.
M. R. Granziera and M. S. Kazimi, "A Two-Dimensional, Two-Fluid
Model for Sodium Boiling in LMFBR Assemblies," MIT-EL-80-011, May 1980.
G. Wilson and M. Kazimi, "Development of Models 'for the Sodium Version
of the Two-Phase Three Dimensional Thermal Hydraulics Code THERMIT,"
MIT-EL-80-010, May 1980.
R.G. Zielinski and M.S. Kazimi, "Development of Models for the
Two-Dimensional, Two-Fluid Code for Sodium Boiling NATOF-2D,"
MIT Energy Laboratory Report No MIT-EL-81-030, September 1981.
-------- -----
I
_______________________________InIIIIIIIIIIEEIIIIEEIIYhIhIIIIIYI,,,IIL
-iv-
B.
Papers
B.1
B.2
B.3
B.4
General Applications
PWR Applications
BWR Applications
LMFBR Applications
B.1
J.E. Kelly and M.S. Kazimi, "Development of the Two-Fluid MultiDimensional Code THERMIT for LWR Analysis," Heat Transfer-Orlando
1980, AIChE Symposium Series 199, Vol. 76, August 1980.
J.E. Kelly and M.S. Kazimi, "THERMIT, A Three-Dimensional, Two-Fluid
Code for LWR Transient Analysis," Transactions of American Nuclear
Society, 34, p. 893, June 1980.
B.2
P. Moreno, J. Kiu, E. Khan, N. Todreas, "Steady State Thermal
Analysis of PWR's by a Single Pass Procedure Using a Simplified
Method," American Nuclear Society Transactions, Vol. 26.
P. Moreno, J. Liu, E. Khan, N. Todreas, "Steady-State Thermal Analysis
of PWR's by a Single Pass Procedure Using a Simplified Nodal Layout,"
Nuclear Engineering and Design, Vol. 47, 1978, pp. 35-48.
C. Chiu, P. Moreno, R. Bowring, N. Todreas, "Enthalpy Transfer between
PWR Fuel Assemblies in Analysis by the Lumped Subchannel Model,"
Nuclear Engineering and Design, Vol. 53, 1979, 165-186.
B.3
L. Wolf and A. Faya, "A BWR Subchannel Code with Drift Flux and Vapor
Diffusion Transport," American Nuclear Society Transactions, Vol. 28,
1978, p. 553.
S.P. Kao and M.S. Kazimi, "CHF Predictions In Rod Bundles," Trans. ANS,
35, 766 June 1981.
B.4
W.D. Hinkle, (MIT), P.M. Tschamper (GE), M.H. Fontana (ORNL), R.E.
Henry (ANL), and A. Padilla (HEDL), for U.S. Department of Energy,
"LMFBR Safety & Sodium Boiling," paper presented at the ENS/ANS
International Topical Meeting on Nuclear Reactor Safety, October 16-19,
1978, Brussels, Belgium.
M.I. Autruffe, G.J. Wilson, B. Stewart and M. Kazimi, "A Proposed
Momentum Exchange Coefficient for Two-Phase Modeling of Sodium
Boiling," Proc. Int. Meeting Fast Reactor Safety Technology, Vol. 4,
2512-2521, Seattle, Washington, August 1979.
M.R. Granziera and M.S. Kazimi, "NATOF-2D: A Two Dimensional Two-Fluid
Model for Sodium Flow Transient Analysis," Trans. ANIS, 33, 515,
November 1979.
I ii IliYI
"---
InmIhufI
I
411ll0
419u
lIM0II ill
DEVELOPMENT OF MODELS FOR THE
TWO-DIMENSIONAL, TWO-FLUID CODE FOR
SODIUM BOILING NATOF-2D
ABSTRACT
Several features were incorporated into NATOF-2D, a twodimensional, two fluid code developed at M.I.T. for the purpose
of analysis of sodium boiling transients under LMFBR conditions.
They include improved interfacial mass, momentum and energy
exchange rate models, and a cell-to-cell radial heat conduction
mechanism which was calibrated by simulation of Westinghouse
Blanket Heat Transfer Test Program Runs 544 and 545.
Finally,
a direct method of pressure field solution was implemented into
NATOF-2D, replacing the iterative technique previously available,
and resulted in substantially reduced computational costs.
The models incorporated into NATOF-2D were tested by
running the code to simulate the results of the THORS Bundle 6A
Experiments performed at Oak Ridge National Laboratory, and
four tests from the W-1 SLSF Experiment performed by the Hanford
Engineering Development Laboratory.
The results demonstrate
the increased accuracy provided by the inclusion of these
effects.
EIIIIImmII,,IIIIYIYLIIIIIYIIIIhA
111Wi
NOTICE
This report was prepared as an account of work
sponsored by the United States Government and
two of its subcontractors. Neither the United
States nor the United States Department of Energy,
nor any of their employees, nor any of their contractors, subcontractors, or their employees,
makes any warranty, express or implied, or assumes
any legal liability or responsibility for the
accuracy, completeness or usefulness of any
information, apparatus, product or process disclosed, or represents that its use would not
infringe privately owned rights.
,6
MIYM
ACKNOWLEDGEMENTS
Funding for this project was provided by the United
States Department of Energy.
This support was deeply
appreciated.
A very special thanks is due to Andrei L. Schor, whose
enthusiasm for Sodium Boiling provided a constant source of
information and inspiration.
The work described in this report was performed primarily
by the principal author, Robert G. Zielinski, who submitted
this work in partial fulfillment for the M.S. degree in
Nuclear Engineering at M.I.T.
-4-
Table of
Contents
Page
TITLE PAGE
ABSTRACT
.
.
..
.
.
.
..
. .
ACKNOWLEDGEMENTS .
.
TABLE OF CONTENTS
.
List of Figures
. .
List of Tables .
. .
Nomenclature .
. .
.
0
.
.
.
.
•
0
0
.
2
*
.
.
.
S
0
.
0
0
.
0.
3
.
6
"
6
4
.
.
.
.•
8
12
.
Chapter 1: INTRODUCTION
.
.
Sthe
.
.
.
S
0
.
.
.
6
0
0
17
SCod .
.
1.1
Description of
the Code
1.2
Scope of Work
-~CI1II
-
1.2.1
Chapter 2:
17
-l~~~-
21
~
Interfacial Mass, Energy,
and Momentum Exchange Models
.
.
.
1.2.3
Direct Solution of the Pressure
1.2.4
Comparison to Experiments on Boiling
.
.
.
.
.
.
.
.
.
.
22
iel
23
.
INTERFACIAL MASS, ENERGY AND MOMENTUM
EXCHANGE MODELS
.
.
.
. .
.
. .
.
.
. .
21
22
Fluid Conduction Model
.
.
. . .
1.2.2
Behavior
13
24
.
2.1
Introduction .
.. .
24
2.2
Conservation Equations Used in NATOF-2D
26
. .
. . . .
. . . .. .. ..
. . .
2.4 Energy Exchange Rate ........
. .
. . .
2.5 Interfacial Momentume Transfer . . . . .
. . .
.0 .
. .
2.6
Programming Information .0
. . .
Chapter 3: FLUID CONDUCTION MODEL . . . . .
. .
2.3
3.1
Interfacial Mass Exchange
Introduction .
. .
.
. . .
....
. .
.
.
. .
.
.
29
40
47
54
i55
-5-
Page
.
.
.
.
.
.
.
.
.
.
.
.
.
.
57
.
. .
.
.
.
.
.
.
.
.
.
.
.
61
.
.
.
.
.
.
.
.
.
65
3.5
Experimental Calibration .
.
. .
.
.
.
68
3.6
A Comparison with Effective Conduction
.
3.2
Formulation
3.3
Intercell Areas
3.4
Implementation Form
.
Mixing Lengths
3.7
Chapter 4:
.
.
.
.
.
Programming Information
.
.
.
.
.
.
.
.
.
.
.
.
.
.
DIRECT SOLUTION OF THE PRESSURE FIELD
.
.
.
.
..
.
.
.
.
4.1
Introduction
4.2
Direct Method Solution Techniques
4.3
A Comparison of Direct and Iterative
Methods
4.4
Chapter 5:
in
NATOF-2D
.
.
Programming Information
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
5.3
On the Modelling of Sodium Reactors
5.4
The Mass Exchange Rate
........
5.5
Varying Mesh Spacing .
.
Chapter 6: VERIFICATION OF MODELS
.
.
.
.
6.2
THORS Bundle 6A Experiment, Test 71H,
Run 101
6.2.1
*
*
*
*
*
* a * a a a a * * * * * ,
.
.
.
88
. . .
97
101
103
* .
.
106
.
.
.
109
*
* .
113
*
* * .
*
.
115
* • 115
* *
*
116
*
*
116
Description of the THORS Bundle
6A Experiment .
6.2.2
*
87
* *
.......
Introduction .
.
...
.
.
6.1
.
..
102
Double versus Single Precision .
.
..
. *
5.2
.
..
102
Introduction .
.
..
.
5.1
.
..
92
.
.
79
.
.
.
.
88
.
.
.
.
.
EXPERIENCES WITH NATOF-2D
.
.
.
.
Simulation Results
.
.....
.
.
.
a
* 4 & 117
IIY
i
-6-
Page
6.3
6.4
The W-1 SLSF Experiinent
.
.
. . .
.
.
.
.
.
.
6.3.1
Test Objecti ve
6.3.2
Test Apparat Ius and Procedure
6.3.3
Tests Chosen for Simulation
.
W-1 SLSF Simulation Results
6.4.1
.
.
.
. .
.
.
6.4.3
BWT 2'
.
.
.
.
6.4.4
BWT 7B' .
.
. .
7.1
Conclusions
7.2
Recommendations
Apperidix A:
126
127
.
.
a
*
0
.
133
.
.
.
.S
0
0
6
0
.
. .
. . e*
133
*
*
a
.
0
.
.
.
138
*
.
145
*
*S
151
151
.
.
S
,
S
.
*
*
*
.0
a
S
0
*
0
6
0
*
0
0
153
0
**
.
.S
.
.
155
0
ow
SPECIFIED INLET VELOCITY AND MASS FL
157
157
Introduction .
A.2
Treatment of the Momentum Equations
A.3
Inlet Velocity Boun dary Condition
A.4
Specified Inlet Mas s Flow Rate Boundary
*
0
0
Condition . . . •
. o
S
Programming Informa tion
.
.
.
. .....
a
159
165
..
0
167
0
173
.
174
. .
.
Appendix B:
NATOF-2D INPUT DESCRIPTION
Appendix C:
INPUT FILES FOR NATOF-2D TEST CASES
C.1
.
127
A.1
A.5
.
..
CONCLUSIONS AND RECOMMENDATIONS
.
126
133
LOPI 4
Refer ences
.
LOPI 2A . .
6.4.2
Chapt:er 7:
.
.
179
Westinghouse Blanket Heat Transfer Test
Program Run 544
.
.
.
.
...
*.
.
0
C.2
Westinghouse Blanket Heat Transfer Test
..
. . . . . . . .
Program Run 545
C.3
THORS Bundle 6A Test 71H Run 101
. .
.
0
.
179
180
.
181
-7-
Page
W-1
SLSF LOPI 2A .
.
C.5
W-1
SLSF LOPI 4
..
186
C.6
W-1
SLSF BWT 2'
.......
188
C.7
W-1
SLSF BWT 7B'
.
.
.
C.4
.
.
.
.
.
.
.
.
.
.
Appendix D:
NATOF-2D HEXCAN MODEL .
.
.
.
Appendix E:
NATOF-2D SPACER PRESSURE DROP MODEL
Appendix F:
SAMPLE OUTPUT
Appendix G:
LISTING OF NEW NATOF-2D SUBROUTINES
.
.
.
191
.
.
.
.
.
*..
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.S
0
.
.
0
.
.
.
194
196
198
202
---
--
m'millinumiI
IIIflf
-8-
List of Figures
Number
Page
1.1
Typical Arrangement of Cells Used in NATOF-2D
19
2.1
Condensation Coefficient as a Function of
Pressure . . . . . . . . . . . . . . .
33
2.2
2.3
2.4
2.5
2.6
3.1
3.2
.
A Comparison of New and Old MasS Exchange
Rates for a Superheat of 2 C . . . . .
A Comparison of New and Old Mass Exchange
Rates for a Superheat of 20 C . . .
.
38
.
39
Variations between Vapor and Saturation
Temperatures for an Interfacial Heat
Transfer Nusselt Number of 0.006 ....
45
Vapor and Saturation Temperatures for an
Interfacial Heat Transfer Nusselt Number
of 6.0 . . . . . . . . . . . . . . . . .
46
Values of r and K as a Function of Void
Fraction for Typical Operating
Conditions . . . . . . . . . . . . .
.
51
Top View of Fluid Channels Showing the Radial
Heat Transfer Between Them . . . . . . .
58
Top View of Fluid Channels Showing Radial
Cell Boundary Numbering Scheme . . . . .
62
.
3.3
Unit Cell Used in NATOF-2D
. .
63
3.4
Radial Temperature Profile at the End of the
Heated Zone for Various Effective
Nusselt Numbers . . . . . . . . . . . .
71
A Comparison Between Westinghouse Run 544 and
NATOF-2D Radial Temperature Profiles at
the Heated Zone Midplane for Nul = 22 and
Nu 2 = 28 . . . . . . . . . . . . . . . .
72
A Comparison Between Westinghouse Run 544
NATOF-2D Radial Temperature Profiles
the End of the Heated Zone for Nu =
and Nu 2 = 28 . . . . . . . . . . . .
73
3.5
3.6
.
.
. . .
.
.
and
at
22
. .
-9-
List of Figures (continued)
Page
Number
3.7
A Comparison Between Westinghouse Run 544 and
NATOF-2D Radial Temperature Profiles 25
Inches Downstream of Heated Zone for
Nul = 22 and Nu 2 = 28
3.8
..........
3.11
*
.
.
.
.
3.13
.
.
.
.
.
74
e .
.
*
*
*
*
75
*
.
.
.
.
.
.
76
and
at
22
.
.
77
A Comparison Between Westinghouse Run 545 and
NATOF-2D Radial Temperature Profiles 25
Inches Downstream of the Heated Zone for
. . . . . . . . .
78
Possible Temperature Distributions Which Yield
the Same Cell Averaged Temperature
. .
.
81
19-Pin Cell Geometry Used for the Calculation
.
of Effective Mixing Lengths
4.1
.
A Comparison Between Westinghouse Run 545
NATOF-2D Radial Temperature Profiles
the End of the Heated Zone for Nul =
. .
..
and Nu 2 = 28 . . . .
Nul = 22 and Nu 2 = 28
3.12
.
A Comparison Between Westinghouse Run 545 and
NATOF-2D Radial Temperature Profiles at
the Heated Zone Midplane for Nul = 22
and Nu 2 = 28 .
3.10
.
Normalized Heat Input per Rod for Westinghouse
Blanket Heat Transfer Test Program
Run 545
3.9
.
. .
.
.
.
83
Arrangement of Cells for Pressure Field
Matrix Shown in Figure 4.2 .
.
.
.
.
.
.
.
89
.
. .
.
. .
.
.
. .
90
.
.
.
.
.
.
.
.
.
93
.
.
.
.
.
.
...
4.2
Pressure Field Matrix .
4.3
Upper Triangular
4.4
Lower Triangular Matrix .
4.5
A Comparison of Steady State CPU Usage
Between the Direct and Iterative
Techniques
(10 Axial Levels, 5 Radial Nodes)
.
.
98
A Comparison of Transient CPU Usage
Between the Direct and Iterative
Techniques
(10 Axial Levels, 5 Radial Nodes)
. . .
99
4.6
Matrix
.
93
.
-10-
List of Figures (continued)
Number
6.1
Page
Location of Cells Used in the NATOF-2D
Simulation of the THORS Bundle 6A
.
.
119
NATOF-2D Predicted Inlet Mass Flow Rate of
. .
THORS Bundle 6A Test 71H Run 101
.
121
Experiment
6.2
6.3
6.4
6.5
6.6
6.7
.
.
6.9
6.11
.
.
.
.
.
.
.
.
.
122
123
A Comparison of NATOF-2D Predicted Liquid and
Vapor and Velocities
(THORS Bundle 6A, Test 71H, Run 101)
.
125
Location of Cells Used in the NATOF-2D
Simulation of the W-1 SLSF Experiments .
130
W-1
SLSF Test LOPI 2A Experiment Inlet Mass
5
/
.
.
134
A Comparison Between Experiment and NATOF-2D
Predicted Temperature Histories at the
End of the Heated Zone for the Central
Channel (W-I SLSF Test LOPI 2A)
. . . .
135
.
.
.
.
.
.
.
.
A Comparison of NATOF-2D Central and Edge
Channels Temperature Histories at the
End of the Heated Zone
W-1
.
SLSF Test LOPI 2A)
.
.
.
.
.
.
.
SLSF Test LOPI 4 Inlet Mass Flow Rate
/
5
/
.
•a .
a. • •
a• . . . •
136
137
A Comparison Between W-1 SLSF Test LOPI 4 and
NATOF-2D Temperature Histories at the
End of the Heated Zone for the Central
Channel
6.12
.
NATOF-2D Predicted Temperature Histories at
End of the Heated Zone for the Central
and Edge Channels
. .
(THORS Bundle 6A, Test 71H,Run 101)
(W-1
6.10
.
NATOF-2D Temperature Profiles of Central
Channel At Various Points in Time
(THORS Bundle 6A, Test 71H, Run 101)
Flow Rate /
6.8
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
139
A Comparison Between W-1 SLSF Test LOPI 4 and
NATOF-2D Temperature Histories at the
End of the Heated Zone for the Edge
Channel
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
140
-11-
List of Figures (continued)
Page
Number
6.13
NATOF-2D Void Maps for W-1 SLSF Test LOPI 4
for the Central and the Middle Channels
.
(Void Fraction = 0.1)
.
.
. .
.
.
. .
6.14
W-1 SLSF Test BWT 2' Inlet Mass Flow Rate . .
6.15
A Comparison Between W-1 SLSF Test BWT 2'
and NATOF-2D Temperature Histories at
the End of the Heated Zone for the
Central
6.16
.
.
.
. .
.
.
.
.
.
SLSF TEst 7B')
.
.
.
.
.
.
.
.
.
.
.
143
.
.
.
.
.
.
.
.
.
144
147
.
.
.
.
.
.
.
..
148
.
.
.
.
.
.
.
.
.
149
.
.
.
.
.
.
.
.
.
150
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
161
.
.
.
.
168
.
.
170
.
.
171
Example Cell Numbering Scheme for Flow
.
.
.
.
.
Cell Configuration for Matrix Shown in
Figure A.4
A.4
.
Positions Used for the Evaluation of
Boundary Calculation .
A.3
.
A Comparison Between NATOF-2D and W-1 SLSF
BWT 7B' Temperature Histories at the End
of the heated Zone for the Central
Variables
A.2
.
A Comparison of NATOF-2D Edge Channel Void
Maps for CF = 1.0 and CF = 0.01
Channel
A.1
.
A Comparison of NATOF-2D Central Channel Void
Maps for CF = 1.0 and CF = 0.01
(W-1
6.20
.
. . . . . . . . . . . . . . . . .
(W-1 SLSF Test 7B')
6.19
.
A Comparison Between W-1 SLSF Test BWT 7B'
and NATOF-2D Predicted Inlet Mass Flow
Rate
6.18
.
142
A Comparison Between W-1 SLSF Test BWT 2' and
NATOF-2D Temperature Histories at the
End of the Heated Zone for the Edge
Channel
6.17
.
Channel
141
.
.
.
.
.
.
.
.
.
.
.
.
Pressure Field Matrix With Flow Boundary
Calculation
.
.
.
.
.
.
.
.
.
.
.
.
-------
Ill
IIIYII IIYIIYYI
IIIIYI
iY
ii
-12-
List of Tables
Page
Number
2.1
Parameters Used in K versus r Comparison
.
.
52
3.1
Test Parameters for Westinghouse Blanket
Heat Transfer Test Program Bundle . .
.
.
69
3.2
Conduction Mixing Length Theory Results .
.
.
84
5.1
Convergence Criteria Versus Minimum Timestep
.
104
5.2
A Comparison of PWR and LMFBR Properietes . .
107
5.3
A Comparison of Water and Sodium Properties .
107
6.1
Description of THORS Bundle 6A
6.2
Geometric Parameters of W-1 SLSF Test
(Single
Bundle
6.3
.
Precision)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.a
....
.
118
.
.
.
.
.
128
.
.
.
.
.
129
Power and Flow Rates of the W-I SLSF
Experiment
.
.
.
.
.
.
.
.
.
.
-13-
Nomenclature
Definition
Letter
Units(SI)
a
Fuel pellet radius
m
A
Area
m
A
Matrix of pressure coefficients
Ar
Volumetrically averaged
Radial area between cells
Ar*
Radial area constant
b
Fuel rod radius
CF
Condensation factor
d
Wire wrap diameter
D
Diameter
Dc
Conductive diameter
De
Equivalent diameter
f
friction factor
g
Gravity constant
m/s2
hfg
Enthalpy of vaporization
J/kg
h
Enthalpy
J/kg
H
Heat transfer coefficient
w/m 2-
J
Mass flux
kg/m2-s
K
Momentum exchange
coefficient
kg/m 3-s
L
2
oK
Lower triangular matrix
9ij
..
ij
Effective conduction mixing
length
Centroid-to-centroid distance
of adjacent cells
m
Momentum exchange rate
kg/m 2-s
-
IIIIIYIIIYIIIYIIIYIIiril
Nomenclature (continued)
Units (SI)
Letter
kg/mole
M
Molecular weight of particles
n
Row number
Nu
Nusselt Number
P
Pressure
N/m2
P
Fuel pitch
m
N/m 2
P
Perimeter
m
q
Heat flux
W/m 2
q'"
Power density
W/m3
q"
Heat flux
W/m2
Q
Heat generation rate
W/m
r
Radial spacial coordinate
m
R
Universal gas constant
J/mole- OK
S ij
Length of common cell boundary
m
SBT
Stable boiling timestep
s
t
Time
s
T
Temperature
OK
U
Velocity
m/s
U
Upper triangular matrix
V
Volume
w
Bandwidth of matrix
W
Total inlet mass flow rate
kg/s
z
Axial spacial coordinate
m
m
-15-
Nomenclature (continued)
Definition
Greek
a
Void fraction
a
Thermal diffusivity
6
Increment
A
Increment,
r
Mass exchange rate
Units(SI)
2
m2/s
spacing
kg/m 3-s
Interfacial velocity
weighting factor
3
p
Density
a
Mass exchange coefficient
Subscripts
r
Radial position
z
Axial position
v
Vapor
Liquid
i
Interface
e
Evaporation
c
Condensation
s
Saturation
T
Total
int
Interface
c
Conduction
w
Wall
kg/rn
kg/m 3
-16-
Nomenclature (continued)
Superscripts
n+1
New time step
n
Old time step
Ili,.
-17-
Chapter 1
INTRODUCTION
1.1
Description of the Code
computer
The
NATOF-2D
code
developed
was
Institute of Technology for the simulation of
Massachusetts
Liquid
both steady state and transient conditions in
Reactors
Breeder
Fast
the
at
I
/
/.
Metal
code uses the two
The
fluid model of conservation equations, and a two-dimensional
r-z geometry which takes advantage of the symmetry found
bundles.
LMFBR
two
The
effects
calculation allows the multidimensional
be
to
boiling
observed,
nature
dimensional
the
sodium
corresponding high
the
without
of
of
in
computational costs of a three dimensional code.
The
model
coupled
separately,
is
assumption
treats
made
the
and
liquid
about
two
the
the
relationship
phases,
generality.
The method thus requires the
and
between
allows
which
of
momentum
phases
by only the exchange coefficients.
properties
mass,
vapor
solution
No
the
greater
of
the
energy conservation equations for each
phase.
For
calculational
purposes,
the
fuel
assembly
is
divided into a finite number of axial and radial mess cells.
There
is
no
constraint as to the positioning or number of
axial levels other than
at
each
level
the
mesh
spacing
-18-
remains
the
constant.
However, the boundaries between cells in
direction
radial
the
between
fall
must
pin
fuel
centerlines, and so the number of radial cells is limited to
the
of
number
Figure 1.1 shows a typical
pin rows.
fuel
arrangement of cells used by NATOF-2D.
The fluid properties of
of
average
volumetric
a
cell
treated
are
as
the
properties in that cell, which
the
necessitates the use of sufficiently small cells in order to
detailed
obtain
evaluated
at
velocities
fluid
each
cell
face.
calculation are P, a, Tv, TZ, Uvz
,
unknowns
The
Uvr
,
Utz
of
the
and Utr.
,
NATOF-2D uses a partially implicit method to solve
dynamics
fluid
The
equations.
velocity and interfacial exchange
However,
for
are
faces of the cell, and are assumed to be
the
across
uniform
The
information.
terms
are
involving
treated
the
sonic
implicitly.
the convective terms, only the velocities are
treated implicitly, while all other factors are evaluated at
the previous
timestep.
This
method
imposes
a
timestep
limitation such that
At -< Az
UZ
z
In most cases, this is not a detrimental
this
constraint,
since
timestep is usually the same order of magnitude as the
time at which information is
required.
The equations are
by
Iteration
solved
reduction
to
a
Newton
problem, in which the unknowns become linearized.
SIIIW IIIII
l
ll IYwl
JIJlil
-19-
Figure 1.1
Typical Arrangement of Cells
NATOF-2D
Used in
-20-
the
only
involving
equations
a
to
reduced
These equations are further
pressures
set
of
linear
a cell.
of
The
pressure field is then solved for by either an iterative
The
a direct technique, and all variables are then updated.
of
advantage
a Newton Iteration technique is that a
using
solution can always be attained
small
timestep.
or
The
by
taking
a
sufficiently
heat conduction equations are solved
implicitly and coupled to the fluid dydnamics equations.
The code has the capability to operate
velocity
or
flow
boundary
with
pressure,
conditions at the inlet, and a
pressure boundary condition at the outlet.
The velocity and
flow inlet boundary conditions are new features incorporated
into NATOF-2D, and are described in Appendix A.
NATOF-2D is able
boiling
conditions,
covered
in
this
to
handle
including
thesis
the
flow
addresses
most
severe
reversal.
some
of
The
the
sodium
work
major
difficulties encountered in past sodium boiling simulations.
------ -------
YYII
-21-
Scope of Work
1.2
Interfacial Mass, Energy and Momentum Exchange Models
1.2.1
The constituative equations used for the calculation of
the interfacial mass, energy
have
improved
been
to
of
ability
more
momentum
physically
the
two-fluid
model
boiling transients, since one of the
this
exchange
rates
for the
account
The terms have a pronounced
observed phenomena.
the
and
effect
on
simulate sodium
to
assumptions
major
of
is that for void fractions below 0.957 the vapor
work
phase does not come in contact with the
wall.
Thus
these
often represent the only source of mass, momentum and
terms
energy for the vapor phase.
The mass exchange rate, which has the strongest
of
any
constituative
effect
relation on the running of the code,
has been implemented in a more basic form than before, using
It
the kinetic theory of condensation.
fully
implicit
independent
so
manner
variables
are
that
all
accounted
is
treated
dependencies
for.
The
in
a
on the
momentum
been
modified to take into account the
effects of mass exchange.
Finally, the energy exchange rate
exchange
has been
rate
has
modelled
subcooled
transients.
vapor
to
or
prevent
superheated
the
appearance
liquid
of
highly
in two phase flow
-22-
Fluid Conduction Model
1.2.2
The high conductivity of liquid sodium coupled with the
the only mechanism available in NATOF-2D for
of
radial
small
the
temperature differences to exist
large
allowed
velocities
However,
transfer.
mass
to
solely
modelling
the
energy exchange between cells due
was
phenomenon
this
Previously,
gradients across the core.
temperature
radial
small
in
turbulence found in LMFBR bundles usually results
between internal channels and the edge channel.
incorporated
conduction
heat
Therefore, a radial
into NATOF-2D.
model
of
vapor
the
been
The model is applied only when
single phase liquid is present in adjacent cells
conductivity
has
phase is very low.
since
the
Presently,
only radial conduction has been employed in the code,
since
any
axial
effects
convection
axial
tend
to
dominate
conduction effects.
Calibration of the model is accomplished by
of
Westinghouse
two
experiments /
2
Blandket
/.
Heat
simulation
Transfer Test Program
The model developed is also compared
to analytic results based on conduction mixing length theory
/
3
1.2.3
1.
Direct Solution of the Pressure Field
The
computer
time
usage
of
NATOF-2D
is
strongly
dependent on the solution technique used for the calculation
of
the
pressure
field.
A more efficient method has been
--- -
YIIIIIIIIYIIIYIYIIYIIIYIIYYYIIIIIIIIIIII
uI
lI MY
M
-23-
solve
the
a
uses
implemented into NATOF-2D which
to
method
direct
pressure field matrix, rather than the iterative
The advantages of
technique previously employed.
substantially
reduced
are
this
time, and the capability of
running
using smaller axial mesh cell spacings.
1.2.4
Comparison to Experiments on Boiling Behavior
The
major
are
NATOF-2D
experiences
documented
in
Chapter
5
to
serve
foundation for future work, and also provide an
for
as
a
explanation
changes deemed necessary to the previously derived
any
models, especially the mass exchange rate.
the
running
while
encountered
difficulties
Also,
some
of
sodium boiling codes in general are
with
discussed.
Chapter 6
discusses
results
the
transients performed by NATOF-2D.
of
the
Thors
4
/,
One test was a simulation
while the other four are from
the SLSF W-1 experiments done
at
Development Laboratory /
/.
Finally Chapter
NATOF-2D.
five
Bundle 6A experiments conducted at Oak Ridge
National Laboratory /
thesis,
for
obtained
7
5
summarizes
the
the
Hanford
Engineering
findings
of
this
and makes recommendations for future development of
-24-
Chapter 2
INTERFACIAL MASS, ENERGY, AND MOMENTUM
EXCHANGE MODELS
2.1
Introduction
In the two fluid model NATOF-2D, each phase in the flow
energy,
field is described by a set of mass,
momentum
of these equations takes into account the
Each
equations.
and
occur
between
phases.
the
This
is
interactions
which
accomplished
by the use of empirical correlations or simple
physical models, that describe the mass, energy and momentum
exchange rates at the liquid/vapor interface.
One of the requirements of two phase flow modelling
mass, energy, or momentum be gained or lost at the
no
that
is
interface.
This is the so called "jump
This
interface.
requirement
equations of each phase can
be
is
met
summed
condition"
at
the
if the conservation
and
the
NATOF-2D
was
together,
interface exchange terms cancel each other.
For the sodium boiling transients
which
designed to simulate, these exchange rates take on a special
One of the basic assumptions of this work is
significance.
that only the liquid phase is in contact with the
values
of
void
fraction
up
to
0.957.
Thus,
applications, the vapor phase is entirely dependent
liquid
phase
as
a
mass,
energy
wall
for
for many
on
the
or momentum source, and
-25-
of
thereby dependent on the accuracy
the
exchange
models
incorporated into this code.
This
chapter
will
cover
the
models
developed
for
interfacial transport exchange, and compare the results with
those previously used in NATOF-2D /
1
/.
I
_
MINIIIIIIYIIIIiIUIIN
1 41
II I
-262.2
Conservation Equations Used in NATOF-2D
Since this chapter deals
interfacial
enegy
mass,
in this section.
the
modelling
used
form
by
are
NATOF-2D
Since NATOF-2D is a two-phase,
two-dimensional R-Z code, for each phase there will
mass
the
of
momentum exchange rates, the
and
conservation equations in the
summarized
with
one
be
and one energy conservation equation, and two momentum
equations (one for each
below
are
the
eight
at
direction)
conservation
each
node.
equations
Given
written
in
control volume form.
Mass Conservation
liquid phase:
j
- f(l-a)p dV +
- (l-a)p
Az+ Az_
v
=
Ar+
-
(1-a)p
Ar-
dV
U
rdA
(2.1)
ap vUv
-vz dA
vU
apv dV
v
+
V
vapor phase:
at
UzdA
AZ+
Az_
=
Ar+
F dV
f ap Uvr dA
Ar-
(2.2)
-27-
Energy Conservation
liquid phase:
+
(eZ + U2 /2)dV
(1-a)p
Az+
Ar+
- Ar( l-a)pzUkr(ek
v
P TF dV
Ar+
V
j
-
qZidA
(2.3)
Ai
+
+ U2/2)dV
V
+ U /2)dA
aPv Uvr(e
AZ-
v
- fa PVUz(eV
Az+
S-
k
+ U /2)dA = fQdV- f(l-a) p gU zdV
k
P•n-U dA +
i U" f*zdA
A
P,
w
vapor phase
ap v(e
+ U /2)dA
(1-a) pU(e
=
v
+ U 2/2)dA
V
Az-
f
Q dV -
p gU vdV
V
Ar-
fP
-
-f dA + jP'n*U dA
at
dV
+
I qVi dA
(2.4)
Ai
Momentum Conservation--Axial
Direction
liquid phase
t
(l-)pUtzdV+
I
-
(-)
Az+ - Az-
IAr+-
I(1-a)Pk UzU
rdA -
(1-a)p g
P-k*-
AR
Ar-
dV - fM
zdV
2)dA
p U£zdA
dA =
fkzdA
-
Aw
(2.5)
_~_~~I _
I __EIEIIIHIIwI
,IuIIIl
WNNNO
-28-
vapor phase
-4apvUvzdV
2c
PdA
ap U
A
Az+
V
f
J
+
Pek*n
-
Aw
-
vUvz U vr dA -
A
Ar+
AzAz
_
dA = -fvzdA
A
+
f
A r-
+
pvgdV
f MvzdV
v
V
(2.6)
Momentum Conservation--Radial Direction
liquid phase
-
_lt-(l-a)p ZUrdV +
v
Az+
-
Ar+
z-
-
f(l-a)p Ur dA
(1l-a)ptUrUtzdA +
A
SPr*n
dA
=
Ar_
-f f rdA + f MrdV
(2.7)
v
Aw
vapor phase
a
PvUvrdV +
V
-
P.r-n dA
Av
v
-
-
fvUvz vrdA +
Az+
A z_
=
fvr dA
Aw
w
Ar+
-
i
ap vU2 dA
Ar-
MvrdV
v
(2.8)
-29-
Interfacial Mass Exchange
2.3
2.1 and 2.2) r represents the mass
(equations
phase
vapor
exchange
rate
positive
for
and
liquid
In the mass conservation equations for the
phases,
between
At the
1 has units of kg/m3-s.
evaporation.
rate
present time, the accepted model for the mass exchange
is
This model
based on the kinetic theory of condensation.
particles departing from the interface.
are
particles
The
be arriving from the vapor phase, and departing
to
assumed
When the arrival
from the liquid phase.
condensation is
occurring.
the
exceeds
rate
In
the reverse
departure
rate,
situation,
evaporation takes place and when the net flux
an
zero,
the
/
mass
6
condition exists.
equilibrium
rate
exchange
is
essentially
is
The derivation of
due
to
Schrage
/.
Using a Maxwell-Boltzmann distribution, it is
to
a
particles arriving at the interface, and a flux of
of
flux
between
difference
the
views the interaction simply as
as
defined
be
will
and
show
that
in
a
possible
stationary container the mass flux of
particles passing in either direction through the
interface
is given by:
M= 2 p
i
2rR
T
(2.9)
where
ji
= mass flux of phase i (kg/m
2 -s)
M = molecular weight of particles
____
^_
_
~ I_
I I11
-30-
R = universal gas constant
P = pressure exerted by the particles
T = temperature of the particles
If there exists a progress velocity on the
=
towards the interface such that Jv
M
2 P
side
vapor
PvVp then
T
(2.10)
where
S= e-2
+
(2.11)
+ erf#)
or(1
V
P
2R/
(2.12)
(.
p v(2RT/M)
(2RT/M) T
Vp = progress velocity
not
At the liquid-vapor interface
striking
the
striking
defined as the fraction of molecules
which
actually
do
represents the ratio
condense.
of
the
In
flux
a
of
c
Therefore,
condense.
will
surface
molecules
the
all
similar
surface
the
manner,
molecules
is
oe
actually
leaving the interface to the flux given by equation 2.9.
At the condensing surface, molecules are arriving at
progress
flow
rate
pvVp,
surface at a rate equivalent
a
and molecules are departing the
to
that
of
molecules
in
a
-31-
Thus the net flux towards the surface
stationary container.
is given by:
P
i
.
M
2JR
2
P
(2.13)
v
TTeT
T
C
If it is assume that P << 1, or in other words that the
be
can
Y
condensation rate is low,
approximated
by
the
following expression:
T
+ 1
/
Pv( 2RT /M) T
(2.14)
Substituting this into equation 2.13 yields
eT
a 2fRT 2
v
2
c
e
(2.15)
When the two phases are in equilibrium, the net flux, j,
to
equal
zero,
and
ac
=
Since
e*.
the values of the
individual coefficients in non-equilibrium systems have
been determined, it is justified to set a
is
= ae
=
not
a,. Using
this approximation the net flux becomes:
J=
2a
M 21
1J
v
2
(2.16)
and the mass exchange rate is thus
F
.2a
= -jA = A T_
M
2Mv
FP
-
v -1
w(2.17)
where
A = the interfacial
area per unit volume
-32-
The literature shows a wide variation in the value of a
for sodium, ranging
a = 0.001
/,
7
of
presence
the
pressures
figure
however, attributes
this
variation
non-condensible
gases
which
tend
to
to
additional
These gases add an
on
conducted
Tests
to
2.1).
(See
condensation.
to
low
at
pressures
congregate at the interface.
resistance
1.0
a =
atmospheric
at
Rohsenow /
from
nearly
gas-free systems where the flow was high show that any gases
present are swept away from the interface, and
a = 1.0
for
all pressures.
In the models developed for
that
only
NATOF-2D,
it
assumed
is
the liquid phase is in contact with the wall for
values of void fraction up to adryout .
Below
this
value,
all heat gains to the vapor phase are solely from the liquid
the
liquid
phase
is
evaporating,
phase.
When
phase is
entering the system at the saturation
Similarly,
condenses
a < adryout,
temperature.
condensation occurs when the liquid phases loses
The
heat to the wall, and becomes subcooled.
again
the vapor
at
vapor
the saturation temperature.
phase
Thus, for
it is justified to set T v to T s and Pv to Ps in
equation 2.17.
For values of a > adryout' the liquid becomes entrained
in
the
vapor
experiences
T
phase,
and
then
it
the heat losses and gains.
= Ts, and PP = P s in equation 2.17.
is
the
vapor
which
Thus for this case,
In order
to
obtain
-33-
1.0
0.1
0.01
0.001
I
I
L
0.01
0.1
1.0
Pressure (bar)
Figure 2.1
Condensation Coefficient as a Function of
Pressure
- ---
I
YIIIIYIYIYIYIYYIIYYU
rl~
-34-
the
correct
behavior
of this relation, it is necessary to
a
reverse the sign of equation 2.17 so that in
vapor
superheated
environment, the entrained liquid evaporates, instead
of condensing.
section
Equation 2.30 of the next
confirms
this behavior.
sodium
For the range of temperatures in which
and
condensation
occurs,
f
Ts
Ts .
and T2
approximation, the final form of the mass exchange
boiling
With this
rate
is
arrived at.
S< adryout
n+l
S>
2a
2 - a
L
3HL
n+1
s
_
M
yTT
i
(2.18)
s
dryout
n+l
n+1 2- a
M
(2.19)
P
s
where
PV = pressure corresponding to a saturation
temperature of T
Pk = pressure corresponding to a saturation
temperature of TZ
Ps = system pressure
T s = saturation temperature
A
= interfacial area calculated implicitly
The formulations previously used for the mass
rate in NATOF-2D were:
exchange
-35For evaporation
{P
A
I'=
2
h
ngfT
Tn+i
S
(2.20)
For condensation
(p v h fg n, T v - T s n+1
n+
R
n+ (l_)n
S=
relations
These
AT/T s << 1,
were
where
AT = T, - T s .
discussed
be
will
in
that
assumption
Simulations by NATOF-2D,
detail
furthler
results
These
large.
quite
however, show that AT can be
S
the
on
based
(2.21)
T2
s-
in Chapter 5.
The
interfacial areas in equations 2.20 and 2.21 were calculated
explicitly, and the term a(1 to zero for single
treats
all
terms
eliminated the a(1 /
8
phase
a) was added to force F to go
The
flc)w.
present
implicitlyr, including the areas, and has
a) term.
The
areas
and depend on the flow regime.
/,
formulation
are
from
Wilson
The following is
a summary of the equations us(
a < cm
A
3a
1 rm
m
23
mM
/D3 (P/D)2-
r m = 6. x lO1
7/2
(2.22)
- -
----
--
IYIIIIIYIIUIII
-36-
(a
m
<
a < 0.55
A2
2
D
-
2/3 (P/D)
(2.23)
0.55 < a < 0.65
a + b.a
A3
3
2
+ c
+ d-t
3
where
-
- 0.55
a
S-0.65 -
a = A2
0.55
b =
3A2
3a
aA2
c = 3(A 4 - A3 ) - a3A 4
aA 2
-+
-a-- + 2(A
d
So
ol
22
-
2*A
-
A4
(2.24)
)
0.65 < a < 0.957
4
D
4
2/3r(P/D)2
(2/3(P/D)
ITra
-
2
2/3(P/D)
-
T
(2.25)
0.957 < a < 1.0
A5
SA
4
"
1 -
0.957
(2.26)
-37-
A transition regime area, A3 ,
between A2 and A4,
and
their
is
which
has been added in order to keep the areas
derivatives
with
to a
respect
comparison was made between the previous
exchange
rate
formulations.
2.2
and
respectively.
predicts
a
term,
the
expected.
present
results
are
shown
that
the
new
mass
in
formulation
rapid vapor production especially in high
Even discounting the effects of the
a(1 - a)
present mass exchange rate still is 2 to 4 times
greater than the
vigorous
and
A
for liquid superheats of 20C and 200C
The results show
more
void regions.
2.3
continuous.
The system pressure used for
this comparison was 2 bars, and the
figure
fit
polynomial
a
and
one
previously
sustained
implemented.
Thus
more
boiling for the same superheats is
Ell
I
--
-38-
U
W
103
J
4W3
I
0)
C
'U
X
U,
L&
U)
U)
'U
102
0.1
0.2
0.3
0.4
Void
Figure 2.2
0.5
0.6
0.7
0.8
0.9
1.0
Fraction
A Comparison of New and Old Mass Exchange
Rates for a Superheat of 20 C
-39-
u
C-,
C,
!
E
o,
w
4.)
,
C.)
a
u-J
C,
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Void Fraction
Figure 2.3
A Comparison of New and Old Mass Exchange
Rates for a Superheat of 200 C
_
I~~
II
I
11111U111
-40-
Energy Exchange Rate
2.4
Reliable constituative relations
for
transfer are not available at the present time.
in
heat
interphase
This is due
part to the insufficient attention which this phenomenon
has recieved until only recently, and also
to
the
extreme
difficulty in gathering useful data on the subject.
Starting. with
equations,
two
the
phase
2.3 and 2.4, one can define an energy
equations
exchange due to the difference in
and
phase
conservation
energy
temperature
between
the interface, and an energy exchange associated
With
with the heat transferred by virtue of mass exchange.
the
premise,
this
the
energy
exchange
from the liquid/vapor
interface to the vapor become
qiv = r.hvs + AiHiv(Ti -
energy
Similarly, the
exchange
TV)
from
(2.27)
the
liquid
to
the
liquid/vapor interface is:
.i
=
r*hts + AiH£I(Tz - Ti)
where
r
= mass exhange rate
hvs = enthalpy of the vapor at the saturation
temperature
hts = enthalpy of the liquid at the
saturation temperature
A.
= interfacial area
(2.28)
-41-
H.
= interface to vapor phase heat transfer
coefficient
HPi = liquid phase to interface heat transfer
coefficient
the
Since the "jump condition" at
interface
requires
that
qiv
£i
we have
-*h
+ A.H i(T
-
+ AiH i(T
Tv) = r'hs
for
Equation 2.29 can be used to solve
the
-
Ts)
mass
(2.29)
exchange
rate to yield
H.
ivAi(T
-
Ti
)
+ H i.A(T
- Ti )
hfg
(2.30)
The above relationship shows that if Hiv and H£i were known,
and if Ti was defined,
determined.
exchange
mass
the
Unfortunately,
2.28 and the
interfacial
would
be
there is a lack of data on the
interface heat transfer coefficients at
Therefore,
rate
the
time.
present
an alternative is to use either equation 2.27 or
given
formulation
energy
in
exchange rate.
section
2.3
for
the
One cannot use equation
2.27 for the vapor energy equation and equation 2.28 for the
liquid energy equation simultaneously since there
would
be
no guarantee that the jump condition was being satisfied.
Attempts to define
an
interface
temperature
with
a
-42-
value
somewhere
in
temperatures
have
temperature
based
the range between the liquid and vapor
proven
For
fruitless.
an
interface
on two infinite bodies in contact, Ti is
given by the relation
T
- Ti
T
-
(kpc )
_
T
(kpc
)
(2.31)
Since the conductivity and density of the liquid phase is so
much greater than that
vapor
TR.
Ti
yields
2.31
equation
the
of
phase,
This
solution
of
would
be
result
acceptable is T, stayed near the saturation temperature when
but
both phases are present,
attaining
a
sodium
high
difficulties
experienced
condensation
vapor
rate
resulted in vapor coexisting with liquid which is
in
have
subcooled
by as much as 100 0 C.
Therefore, the decision was made to set the interfacial
temperature to the saturation temperature.
temperature
chosen
was
it
since
temperature for a two-phase mixture,
for
is
The
the
saturation
equilibrium
As previously
stated,
values of a < adryout' the vapor gains heat solely from
the liquid.
state
In an evaporating
the
assumption
that
Ti = Ts implies that all the liquid superheat is utilized as
latent
where
heat
Tk < Ts,
temperature,
virtue
for
of
and
mass
evaporation.
And
is
kept
the
all
vapor
losses
heat
transfer
to
the
in a condensing state
at
from
liquid
the
saturation
the vapor are by
phase.
For
-43-
With
the roles of each phase will be reversed.
a >adryout
this understanding, the final form of the interfacial energy
exchange rate becomes:
C <
dryout
S=Svs
n+h
(n+1
+ An+1
s
iv
i
(2.32)
+1)
-
C > adryout
n+1 hs
qi=
+ An+lHH
(2.33)
(Tn+1 - Tn+1)
where
H.
Nu
iv
De
k
= Nu
H
V
e
previous
The
formulation
of
the
heat
interfacial
exchange rate was
i
(T
AiHI(T
+ AiH
+ r h
= F h
ehvs + chs
+
- TV)
-
(2.34)
Tv)
This formulation effectively kept T V equal to T,, and led to
situations of the vapor phase being subcooled by as much
100 0 C.
as is
as
The present formulation has eliminated this problem
shown in
figure 2.5.
The nusselt number
transfer
temperature
coefficients
of
the
chosen
has
phases.
a
for
the
pronounced
To
interfacial
effect
illustrate
this,
on
heat
the
three
-44of a sodium boiling transient were run in which
simulations
only the interfacial nusselt number was
varied.
these
In
cases there was no switch in correlations at a=adryout.
temperatures
correspond to those found at the top of
given
As can
the heated section of the fuel bundle.
figure
been
(Nu
2.4,
plotted
= 0.006)
temperature
time,
versus
to
leads
to
a
small
saturation
seconds
in
number
nusselt
At
temperature.
boiling inception, at a
after
to
void fraction corresponding
began
seen
quite a variation between the vapor
the
0.55
be
vapor and liquid temperatures have
the
where
and
approximately
The
the
adryout
phase
vapor
The liquid temperature
superheat to high levels.
stayed very close to the saturation temperature.
When
Nu
indicates.
=
6,
This
Tv
Ts
and
T
as
figure
2.5
test case also showed that the saturation
temperature was more stable with time,
wild
= Ts
fluctuations.
For
and
less
prone
to
Nu = 6000, the results were about
the same.
Based on these simulations, a
recommended.
A
value
of
2.33.
term
=
10
is
value in this range will keep the vapor at
the saturation temperature, but not make the
contribution
Nu
sensible
heat
the dominating one in equations 2.32 and
-45-
1120
-
Saturation Temperature
1100
-
A
r
1060
1040
/
I
I
1
!
I
r
1020
I
cIII
'
1,
000 -
I
I I
I II
980 a
I
II
II
S960
940
920
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (sec)
Figure 2.4
Variations Between Vapor and Saturation
Temperatures for an Interfacial Heat
Transfer Nusselt Number of 0.006
-46-
1120
1100
1080
1060
1040
0
03
5-
1020
4L
543
1000
E
03
980
960
940
920
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (sec)
Figure 2.5
Vapor and Saturation Temperatures for
an Interfacial Heat Transfer Nusselt
Number of 6.0
-47-
Interfacial Momentum Transfer
2.5
the
The interfacial momentum exchange rate, similar to
energy exchange rate, is composed of two terms.
interfacial
The first term takes into account the momentum gain
exchange
mass
interface,
the
across
due
to
and the second term
accounts for the effect of shear stresses at the
interface.
section will show how these terms can be combined into
This
a single term which contains both of these effects.
phase
The momentum conservation equation for the vapor
in the z-direction written in differential form is
t
+
U
Vv
(p
a
=
3Z
z
(ap U 2
vz
f
- ap v.35)
g - Mi
-fwz
(ap UvrUvz)
vz v r
r
+
+
- Ui r
(2.35)
where
Miz = shear stress contribution
U.i
= contribution due to mass exchange which is
traveling at an interfacial velocity
i
= rlnu
+
factor (0 I
(1 -
r)Uv, where n is a weighting
n < 1)
In order to facilitate the implimentation of the finite
difference scheme utilized by NATOF-2D, it is
cast
equation
2.35
into
non-conservative
necessary
form.
to
This is
accomplished by applying the product rule of differentiation
to the following terms:
-48-
Uv
-a-(apv Uvz )_ = a+ vat
at
a (ap U2 ) = ap
z
v vz
v vz
3r-(ap v Uvz Uvr
(2.36)
U-(ap
a
vapv
vzat
U aVzp
z
(2.38)
+
a (apU vr
+r Uvz 3r
ap v Uvr au
2.35,
equation
into
Substituting these values
(2.37)
v vz
z
the
vapor
momentum equation becomes:
vz
apvz
vvr
-f wz
(ap v ) + ap vUz
vz -t
at
+U vz
a pv T-
-
ap
a (aPUv)
U
vz Dr
vr
+
g
-
Miz
+U vz az (apU vz ) +
aP
a-p
3z
+
-
(2.39)
Uizr
+
The vapor mass conservation equation is given by:
v(() + -(ap
v UV
r
(ap Uv
+
(2.40)
and this can be substituted into equation 2.39 to yield
non-conservative
auvz
+
pV
v at
-f
Next Mz
vz
-
the
form
auvz
acp vUvz 7t
ap g
-
M
+
+
ap U
-a vz
v vr ar
-
Uiz
Uv
+
ap
a aaz
(2.41)
is defined such that
M'
vt
=
-Miz
Vziz
+
U iz
i.
-
Uvz
v2)
(2.42)
-49-
where
(2.43)
- j, z )
vz
V
izZ :
K = interfacial momentum exchange coefficient
be rearranged by the following procedure:
M'
VZ can
Mz
vz
(1-n)U~vz - Uvz]F
= -K(Uz - U z) + [ nUz
= -K(U
Uvz
Ezz)
-
+
[ n(U
-
Ezvz
+ U v-
U v)
vz
Uv
vz
= -(K + Jr)*(Uvz - Uz)
(2.44)
One can follow the
same
the
for
procedure
liquid
phase
momentum equation to obtain the non-conservative form, which
is:
(l-)pt
(l-c)fr
t+
3tz
3t
S
+ (l-a)P Uz
Z az
R
-f wz -
(l-)p
(l-)j
+ (1U-a)p
r
-z z
+ Mizz)- Uizr
+ URz
iz
(2.45)
Defining
where
Mz = Miz Miz
Uizr + Uzr
= K*(Uvz - UZ)
one can simplify MIz to obtain
Mz = (K PIZ
(l-n)F)*(U vz -
U Ezz)
(2.46)
In order to better interpret these results, consider
a
-50-
situation
where
n = 0.5
terms
MIz
and
evaporating
an
where UV> Uk. For
both
MIz
U
that
so
= (Uz
condition
+
Uv)/2, and
(F > 0),
The vapor phase bulk
decrease.
(Ui < UV )
momentum decreases by picking up slower particles
and
the
liquid
momentum
bulk
phase
the
decreases by losing
particles traveling at Ui > UY.
In a condensing condition, both MI
vapor
The
and
M
increase.
phase bulk momentum increases by losing its slow
particles and the liquid phase gains momentum
by
receiving
fast particles.
A comparison of K and nr verses void fraction was
in
order
to
access the importance of this phenomenon.
can be seen in figure 2.6, for values of
term is the dominating one.
a > 0.88,
made
As
the
This is a desired result, since
as the liquid becomes entrained in the vapor phase, the slip
ratio
should
decrease as the liquid particles become borne
in the vapor phase.
Parameters used for this comparison are
given in Table 2.1.
To determine what effect this modification actually has
on NATOF-2D simulations, a sodium boiling transient was
with the new correlation (with n = 0.5),
same
transient
insignificant
without
difference
The
it.
for
the
run
and compared to the
results
full
range
showed
of
an
void
fractions.
Simulations were also run in which n was varied in the
range from 0.0 to 1.0.
The only noticeable
difference
was
-51-
1
v,
t
C
v
4
SL
C
rCI
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Void Fraction
Figure 2.6
Values of r and K as a Function of Void
Fraction for Typical LMFBR Transient
C,onditions
1.0
-52-
Table 2.1
Parameters Used in K versus nr Comparison
Pressure (N/m2
)
Saturation Temperature (OK)
2.
x 105
1235.59
Fuel Pin Diameter (m)
5.842 x 10
Hydraulic Diameter (m)
4.223 x 10-3
Pitch/Diameter
1.25
Vapor Density (kg/m 2 )
0.53
Vapor Velocity (m/s)
Liquid Velocity (m/s)
T
- Tsat (OK)
25.0
5.9
2.0
-53-
that
for
n = 0.0
the
vapor
was lower than for
velocity
n = 1.0, and for n = 0.0 the liquid velocity was higher than
for n = 1.0.
condensation
Since these results are
is
occurring
Refering to equation 2.44, the
when
n = 1.
(r < 0),
this
term
(K + qr)
region
was
is
Refering to equation 2.46,
where
expected.
smallest
the
term
is smallest for r = 1, and so the liquid phase
is not dragged as much by the vapor phase.
velocity.
a
Thus the vapor phase isn't slowed down by the
liquid phase as much.
(K - (1-)r)
for
Hence, the lower
-542.6
Programming Information
Both the new mass exchange
and
rate
energy
exchange
rate were incorporated into subroutine NONEQ.
The
momentum
subroutine WS.
since
it
must
exchange
was
rate
incorporated
into
Since r is required in this formulation, and
be evaluated at the previous time step, the
value of the mass exchange
rate
is
stored
ONESTP for use in the following time step.
in
subroutine
I
I
YIIillllll
-55-
Chapter 3
FLUID CONDUCTION MODEL
3.1 Introduction
Some
of
the
previous
boiling
sodium
transients
simulated with NATOF-2D have shown a large difference in the
fluid termperature between the central channels and the edge
channel.
A
small
variation
is
expected
since the edge
channel experiences heat losses to the hexcan container, and
since there is usually a lower power to flow
ratio
ouside
the
channel.
However,
whereas
experiments a
radial
temperature
for
steady
state
reported
variation
In LMFBR bundles, the fuel
spacer
between
wires.
fuel
transversely
These
rods,
and
1
wires
tend
and good mixing of the
coolant.
developed,
to
unable
are
pressure
gradient,
act
to
small in magnitude.
the
9
SLSF
was
/, NATOF-2D
wound
as a spacing agent
sweep
the
coolant
This results in turbulence
NATOF-2D,
simulate
which
10 0C
of
helically
this
radial velocities found in NATOF-2D are due
radial
W-1
the
/.
rods
around the bundle.
is
/
operation
predicted a difference of 600C /
with
in
in
in
as
originally
phenomenom.
solely
to
The
the
most cases is rather
Since mass transfer between
cells
was
only mechanism available for energy exchange, the large
temperature gradients persisted.
When boiling
occurs,
the
-56-
effects
sweeping
mentioned
previously
negligible
become
compared with the expansion of the vapor phase.
to
Therefore
radial
profile,
the
temperature
observed
heat conduction has been incorporated into
The heat transfer between cells has been modelled
the code.
in terms of "effective"
conduction
Besides
cells.
adjacent
effects,
for
account
the
between
the
modelling
the formulation will also be used
pure
has
been
neglected
since
the
phase
makes
any
for
vapor-liquid
calculating
or
dominate
any
vapor-vapor radial heat
This chapter will present
radial
offer typical values for the effective
conduction.
Axial heat
Also, the low conductivity of the vapor
transfer effects negligible.
methodology
for
the high axial
velocities allow the effects of convection to
conductive effects.
in
conduction
account
to
mixing and diffusive effects in the fuel bundle.
conduction
fluid
heat
the
conduction, and
nusselt
number
for
... .....
..
-
illll
ii
iii
Ii4 l
-57-
3.2
Formulation
figure
For the arrangment of cells shown in
heat
total
In
the
transfer rate to cell i can be expressed as the
sum of the heat transfer
faces.
3.1,
this
rates
through
each
of
its
two
formulation, the heat flux is given by an
effective heat transfer coefficient times the difference
the
temperature of the adjacent cells.
in
Written explicitly,
this becomes:
qiT
qi-,i
+
(3.1)
i+1,i
where
q-,i = Ai-1,ihi-1, 1i-1
i+li
Ai+liJi+li
Ti)
i+l - T i
(3.2)
(3.3)
and
qiT = total heat transfer rate to cell i
qi-1,i = heat from cell i-I to cell i
heat from cell i+1 to cell i
qi+1,i =
hi1,i =1effective heat tranxfer coefficient between
cell i-1 and cell i
Ti = temperature of cell i
Ai+1, i = intercell area
On either side of the interface seperating two adjacent
cells, a heat transfer coefficient has been defined with the
form:
-58-
Figure 3.1
Top View of Fluid Channels Showing the
Radial Heat Transfer Between Them
^
I
I
-59-
KR
Nu/
i
2
(3.4)
D
where
Nu = effective nusselt number
KX = conductivity of the liquid in cell i
D = conductive diameter of cell i
4*A
-
P
pc = perimeter of fluid-fluid conduction
Conservation of energy.requires that the heat flux from
and
cell i to the interface of cells i
opposite
to
be
i+1
equal
and
heat flux from cell i+1 to the interface,
the
and so an interface temperature, Tin t , can be
defined
such
that
hi(Tint - T i
)
=
(3.5)
- Ti+l)
-hi+(Tint
Solving for the interface temperature yields
TTint
hiT i + hi+lTi+ 1
h
+
h i + hi+l
(3.6)
Since the heat flux to the interface from cell i is the same
as the heat flux between cells i and
side
of
equation
3.5
can
be
i+1,
the
right
hand
equated to equation 3.3 to
yield:
hi(Tint - T i )
=
hi+li(Ti+l, i - Ti
(37)
I_
-60-
solved for.
can
h
Substituting in equation 3.6 for Tint,
now
be
The result is:
i+1 hi
1+1,h
h
+ h.
hi+ 1 + h i
Considering the case where h i
(3.8)
= hi+1 ,
equation
3.8
reduces
to
1/2h
h
1/2hi+
hi+,i
KR
= Nu---D
as one would expect.
In summary, the methodology
of
this
approach
is
to
calculate h as given by equation 3.4 for each cell, and then
use
these
values
to
solve
for
hi+l, i .
Once
this
accomplished, equation 3.1 can be evaluated for each cell.
is
-61-
3.3
Intercell Areas
NATOF-2D is structured in such a way that the
between
cells
at
lies
centerlines
as
one
illustration
of
this,
boundary
plane connecting the fuel pin
the
along
with
the
An
outward.
radially
travels
numbering
of the
boundaries is given in figure 3.2.
Treating the bundle as a porous body, the
transfer
rn
heat
radial
area becomes dependent only on the radial distance
Thus
Ar
=
Ar
*
r
(39)
where
Ar
= volumetrically averaged radial area between
cell boundaries
Ar
= radial area constant
r
= //2* n-p
n
= row number (1,2,3,.....)
p
= fuel rod pitch (m)
Considering for the moment the unit
3.3, Ar
cell
shown
in
figure
can be solved for by requiring that
Ar.dr
Vcell =
r.
=
Ar *rndr
*
2
= Ar *r
/2
1
(3.10)
-62-
Figure 3.2
Showing Radial
Top View of Fluid Channels
Cell Boundary Numbering Sc heme
----
1111
I
-63-
4I
D
Figure
3.3
Unit Cell Used in NATOF-2D
Figure 3.3
Unit Cell Used in NATOF-2D
-64-
The volume of the unit cell is given by
V_
cell
i
2
2
Az
Apn
pin
2
(3.11)
where
Api
= area of the pin and wirewrap
n
2
=-(D2 +d
D = fuel pin diameter
d = wire wrap diameter
Az = axial height of cell
Using the relation in equation 3.10 yields:
Ar
Ar
4 (V3
-
P2
Az
(3.12)
The final form of the radial heat transfer area is
Ar
1
=
2
Ai2
P
/
When this formulation is
nPAz
then
(n = 1,2,...)
(3.13)
implimented
in
NATOF-2D,
it
is
necessary to divide the total heat transferred by the volume
of
the
cell
so that the term will appear as a heat source
term in the energy conservation equation.
boundaries
lie at nj_
1
For a cell
whose
and nj, excluding the edge cell, the
total volume is given by
2
V =
P2*
2
Apin
pin2
2
2
(3.14)
(3.14)
~-I-
"
IYIY
IYIIIIYIIIYIYYIIIIIIIYIYIYIVVIII
-65Implementation Form
3.4
Up until this point, no mention has been
time
step
discretinization
In this
formulation.
NATOF-2D
and
the
used
section,
in
the
code
options
of
the
for
this
available
in
limitations of each are
and
advantages
the
made
covered.
The first option is to treat the calculation in a fully
explicit manner such that
n
n+1
iT = Ai-,ihi-
n
n
(Ti-1-Ti)
n
n
n
A i+l,ihi+l,i (Ti+,i-Ti)
+
(3.15)
where the superscript (n+1) refers to the present time step,
on
the
step.
are low.
terms
of equation 3.15 are known values, this option
RHS
requires that the calculation be
time
all
Since
and (n) refers to the previous time step.
performed
only
once
per
Thus the cpu costs for the explicit calculation
It also ensures strict energy conservation since
q1-1,i
=
q
,i -
1"
A fourier -stability analysis performed on equation 3.15
shows that this formulation limits the time step size to
Ar 2
At <
- 2-a*Nu
(3.16)
where
pc p
In most cases,
more
the convective
restrictive
than
the
time
limit
conductive
(At
limit.
< Az/U)
However,
is
a
-66-
calculates
feature has been implimented into the code which
limitation when the explicit calculation is
step
time
the
and
utilized,
maintains
a
the
below
value
step
time
conductive limit.
The second option available is to treat the radial heat
transfer
calculation
semi-implicitly.
The
of
form
the
calculation is:
n+l
n
n
n+1
=
) + Ai+1
Ai-l,ihi-,1(Ti-l-Ti
qiT
n
n
h i+,i(Ti+T
n+1
-T
(3.17)
A stability analysis applied to equation 3.17 shows that the
therefore
poses
no
scheme is unconditionally stable,
and
constraint to the time step size.
However, it does have two
The first is that it fails to conserve energy
limitations.
since the relation
n
h
n
1-1,i
(T
1-1 1
n
n
n+l
-T
) = -h
n+1
(3.18)
)
(T -T
11.i-1 1 1-1
(3.18)
The second limitation
will not be satisfied in general.
is
this calculation needs to be performed once per newton
that
iteration, instead of once per time
step.
Thus,
the
cpu
usage will be greater than the explicit method.
A fully implicit calculation of the form
n
n+1
iT
=
n+1
n
n+l
Ai-1,ih-1,1(Ti-1-Ti
)
+
n+1
n+1
Ai+1,ihi+l,i (Ti+1-Ti
(3.19)
cannot be utilized by NATOF-2D since the solution scheme
of
----
--- --"
----
""' ~---
111
11liii~i1
-67-
the
code requires that the only linkage between cells be by
the pressures of the cells.
A
such
formulation
of
this
would also link the cell temperatures.
As can be seen in figure 3.2, all
similar,
and
therefore
has
a
quite
nusselt number, Nu 2,
is
are
we are somewhat justified in using
the same effective nusselt number Nul.
however,
cells
interior
different
used
effects of any differences.
to
shape,
take
edge
The
and
into
channel,
so a second
account
the
-68-
3.5
Experimental Calibration
effective
In order to obtain a practical value for the
experiment
sodium
phase
single
for radial heat conduction, a steady-state,
number
nusselt
Westinghouse
Heat Transfer Test Program /
Blanket
The heat transfer test section was
assembly.
blanket
mockup
of
/.
2
LMFBR
an
section consisted of 61 rods
test
The
a
the
from
chosen
was
axial
Each rod delivered an
contained in a hexagonal duct.
heat output approximating a chopped cosine distribution with
a 1.4 maximum-to-average ratio over a 114.3 cm.
test
the total bundle power was 440 kw, and the
544,
No.
radial power distribution was uniform.
Table
in
given
In
length.
3.1,
and
the
are
Test parameters
for
input
NATOF-2D
the
simulation is given in appendix C.
The
test
procedure
to
was
adjust
loop
test
the
operating parameters until the desired sodium flow and inlet
temperature
was
achieved.
At
this
power to the
point,
power
bundle was gradually increased until the test section
gradient and temperature rise attained operating conditions.
The
test section was then allowed to achieve a steady state
configuration, at which point data was collected.
No.
544,
different
For
test
the temperature profile across the bundle at three
axial
levels
was
recorded.
These
levels
corresponded to the heated zone midplane, the outlet of
the
heated zone, and 25 inches downstream of the heated zone.
In the NATOF-2D simulation, the proper flow
and
total
-69-
Table
3.1
Westinghouse Blanket Heat Transfer Test Program
Rod Bundle Test Section Design /
2
/
PARAMETER
Number of Rods
Rod Diameter (cm)
61
1.32
Length of Heated Zone (cm)
114.3
Total Bundle Length (cm)
265.
.094
Wire Wrap Spacer Diameter (cm)
Triangular Rod Pitch
1.43
Wire Wrap Pitch (cm)
10.16
Pitch to Diameter Ratio
Duct Inside Diameter (cm)
1.082
11.4
Axial Power Distribution, Cosine
Max/Avg
Sodium Inlet Temperature (oC)
1.40
316.
Sodium Flow Rate (m 2 /hr)
Run 544
13.5
Run 545
12.0
Test Bundle Power (kw)
440
-70-
the
nusselt numbers were varied until
as
Then the
the core was established.
through
rise
enthalpy
profile
temperature
as possible the experimental
closely
obtained
matched
results.
As the effective nusselt number was increased, the
section
radial temperature profile at the end of the heated
became
as
flatter,
shown
figure
in
A comparison
3.4.
between the experimental results and the NATOF-2D simulation
3.6' and
3.5,
for different elevations is given in figures
3.7.
From this experiment, the recommended
for
the
tests
was
values
effective nusselt number are
Nu
= 22
= 28
2
A second experiment from the same series of
Nu
simulated
NATOF-2D in order to verify the generality of
by
the previous results.
This was Run No.
the same total power was used
power
was
distribution
as
varied
In this test,
545.
before,
but
the
to give a power skew which
peaked at the edge pins and was at a minimum at the
The
normalized
heat
The results for three
figures 3.9,
3.10,
radial
center.
input per rod is shown in figure 3.8.
different
and 3.11.
elevations
are
shown
in
-71-
450
S-II~L
440
d-------
9
-4--430
420
CL
410
cr
E
I-
400
A
N'u
~aS1
390 1-
380 -
Nu2
0
0
7
14
A
15
20
*
22
28
*
p
p
Channel Number
Figure 3.4
Radial Temperature Profiles at the End of the
Heated Section for Various Effective Nusselt
Numbers
-72-
390
380
0
S-
370
r-
E
-j
..
50
350
2
1
3
4
5
Channel Number
Figure 3.5
A Comparison Between Westinghouse Run 544 and
NATOF-2D Radial Temperature Profiles at the
Heated Zone Midplane for Nul = 22 and Nu2 = 28
/
2
/
-73-
Experiment
- NATOF-2D
450 1.
440
0
4 430
SQ.
i-
* 420
0r
-J
410 1-
2
i
i
I
a
1
2
3
4
5
Channel Number
Figure 3.6
A comparison Between Westinghouse Run 544 and
NATOF-2D Radial Temperature Profiles at the
End of the Heated Zone for Nul = 22 and Nu 2 = 28
/
2
/
-74-
440
430
= 420
E
-~
410
*r0
-J
400
1
2
3
4
5
Channel Number
Figure 3.7
A Comparison Between Westinghouse Run 544 and
NATOF-2D Radial Temperature Profiles 25 Inches
Downstream of Heated Zone for Nul = 22 and
/
2
/
Nu2 = 28
-75-
Figure 3.8
Normalized Heat Input per Rod for
Westinghouse Blanket Heat Transfer Test
Program Run 545
/
/
2
-76-
400 1-
390 Ii
0
S.
380 1
S*,..0.I
I-
370
-J
Experiment
360
NATOF-2D
1
I
i
I
I
1
2
3
4
5
Channel Number
Figure 3.9
A Comparison Between Westinghouse Run 545 nad
NATO.F-2D Radial Temperature Profiles at the
Heated Zone Midplane for Nul = 22 and Nu2 = 28
/
2
/
-77-
450
440
430
E
I-,
.-
*
= 420
Experiment
NATOF-2D
410
I
I
1
2
I
3
4
5
Channel Number
Figure 3.10
A Comparison Between Westinghouse Run 545 and
NATOF-2D Radial Temperature Profiles at the
End of the Heated Zone for NuI = 22 and
/
/
2
Nu 2 = 28
-78-
Experiment
NATOF-2D
460 -
450 0.. )
0
v
I-
440
0-
**
r-
CT
430
.. 1
420 F
I
1
2
3I
4
I
3
4
5
Channel Number
Figure 3.11
A Comparison Between Westinghouse Run 545 and
NATOF-2D Radial Temperature Profiles 25 Inches
Downstream of the Heated Zone for Nul = 22
and Nu 2 = 28
/
2
/
-79-
A Comparison with Effective Conduction Mixing Lengths
3.6
In this section,
a
is
comparison
made
between
the
fluid-to-fluid conduction model implemented in NATOF-2D, and
an
analytic
mixing
model
for
lengths
subcahnnel
developed
codes.
energy
The
to
by
transport
model
the effective
determine
used
mixing lengths is from M. R. Yeung
conduction
for the evaluation of
/
3
1.
Before making the comparison, a brief outline is
of
the
method
of
M.
in
given
R. Yeung to calculate the effective
conduction mixing lengths.
In this model, the heat transfer
rate due to conduction between subcahnnels i and j is
given
by the relation:
S
ij
(T
- T
i iiJ
)
S
KZ
*
(T
1
- T )L1
(3.20)
ij
where
= the length of the common boundary
S
mixing length
ij - the effective conduction
i
*=
ij
the centroid-to-centroid distance of
adjacent coolant channels
Ki = conductivity of the liquid phase
LiJ = ratio of the effective conductive mixing
length to the centroid-to-centroid distance
The effective conduction mixing length
takes
into
account
-80-
the fact that the actual heat flux due to conduction,
.T
this,
illustration
As an
attempting to model a localized effect.
of
3.20
equation
with bulk temperatures while
deal
codes
subchannel
by
given
that
may be quite different than
since
(3.21)
)interface
q
possible temperature distributions which yield the
same bulk temperature are shown in figure 3.12.
As
can
be
seen, 3T/3x)int can vary widely.
Equation 3.20 can be rearranged to give:
(T
S
L
ij
=K
*
R
1_
- T
)
Qij
(3.22)
ij
Since the total heat transfer can be expressed as
q
ijQ
ij
ds
s ds
(3.23)
equation 3.23 can be substituted
also
into
equation
3.20,
and
the dimensionless group q'''a /2k and the rod radius b
can multiply and divide 3.20 to yield a form
analytically
which
can
by evaluating each quantity.
determined
be
This
form is:
S
(T i
- T)
q"'a
2 /2K
L j
jij
qss
s
q"a
2 /2b
b
(3.24)
where
a = fuel pellet radius
-81-
1
2
Channel Number
Figure 3.12
Possible Temperature Distributions which
Yield the Same Cell Averaged Temperature
-82-
b = fuel rod radius
qs = heat flux at common boundary
S
q"' = power density of the fuel
The method of evaluating each of these terms
temperature
local
from
the
and power distributions and the geometry
For the purposes
/.
3
is given in reference /
this
of
comparison, it will suffice to give the results for a 19-pin
hexagonal
geometry
The
bundle.
used for calculating the
effective mixing lengths is shown in figure 3.13, where
dashed
denote
lines
cell
the
The results of the
boundaries.
calculation are give in Table 3.2.
To compare the NATOF-2D formulation with the conductive
mixing
the
results,
length
heat
of
transfer
both
formulations are equated such that
2S. .Az
Ar*hij(T
i
-
T)
= K
- T
L
ij
where the factor 2 has been added to
(3.25)
the
RHS
of
equation
3.25 to take into account the fact that NATOF-2D divides the
core
into 6 symmetrical volumes, while Yeung's work divides
the core into 12.
Thus the area used in NATOF-2D
is
twice
as large.
Assuming
that k. is
the same for both
formulations,
hij is then given by
2-Nu*K,
Dcl + D c2
(3.26)
go,,141
-83-
Effective
Wall
Distance
(w)
Figure 3.13
19-pin Cell Geometry Used for
of Effective Mixing Lengths
Calculation
-84Table 3.2
S
12
/1
S23/
*
12
1
-
1P
- 1
2
1P
/3 D
1)
23
3 D
(P/D -
S34 /34
1
P
+ 1 (2W
2. p +
2 3 D
1 P 2W
2 D D
1
16
1 ( 2W)
12 - D
1
4J
2W
~
D
12
Effective Mixing Lengths
2W/D
L2
3
Fuel Bundle
1.10
.85
.86
1.20
.75
.27
.57
1.30
.79
.77
.79
1.40
.791
.79
.81
1.04
.69
.69
.645
1.08
.69
.69
.68
1.10
.69
.69
.69
1 .20
.69
.69
.70
1.12
Blanket Bundle
-
i
Ii
,
II i
1.1III
-85-
Rearranging
equation
in
3.25
dimensionless
form,
and
substituting in 3.26 yields:
1
DC + Dc2S
DNu
J
N=
ij
Ar
(3.27)
ij
As can be seen, both formulations are of the same form,
and
differ only by a constant multiple.
of
First, considering cells 1-2, each term of equation
3.27 can be evaluated to get:
12
12 =2
3DD
2
r
-
Ar =
P-Az
4[
c1
J) 2
Substituting these values into equation 3.27 yields
=
Nu
From Table 3.2, L 1 2 has
blanket
bundle
of
3
L12
a
(3.28)
value
equal
to
0.69.
For
a
the type simulated, this relation shows
that the Nusselt number, due to conduction only, should be:
Nu = 4.348
For the edge cell, the complex geometry
each
be
term
numerically evaluated.
requires
For this comparison,
typical dimensions of a blanket assembly were used:
D
D
= 1.320 x 10-2 meters
= 1.320 x 10
meters
that
-86P
= 1.426 x 10
-2
meters
P/D = 1.08
W
= P - D/2 = 7.66 x 10
-3
meters
The result is that
Nu = 4.6322 L
34
= 6.617
The results show
calculated
from
nusselt
effective
the
that
length theory is much smaller than
mixing
This is to be
that required for the experiment calibration.
expected for two reasons.
theory
length
only
takes
is
first
The
into
accounting
for
turbulence
reason is that the mixing length
that
account
and
mixing.
results
are
the
mixing
fluid-to-fluid
nusselt
conduction effects, while the effective
also
number
number
is
The second
specifically
for 19 pin bundles, while the simulations were conducted for
61
pin
bundles.
It
is
expected that for smaller bundle
sizes the effective nusselt numbers will also decrease.
-- ---
I
__
II
IIIII l~inn,
,m10lM
-87-
3.7
Programming Information
Two additional subroutines have been added to NATOF-2D.
QCOND
The first is subroutine
per
transferred
unit
which
which
the
heat
volume, and its derivative (when the
implicit formulation is required).
HTRAN,
calculates
effective
the
calculates
The second is subroutine
transfer
heat
coefficient.
The user
performed
by
specifies
the
specifying
which are a user input.
the
type
of
calculation
to
be
sign of the nusselt numbers,
A negative nusselt number refers to
a semi-implicit calculation, while a positive nusselt number
refers to a fully explicit calculation.
The nusselt numbers given in
used
as
this
chapter
should
be
a gauge for the ones actually used, which can best
be determined by calibration to steady state results of
experiment being simulated.
the
IIIIIIIIIIIYIYII
iilliY
Yr
lii~
ii ~~~~ ,
-88Chapter 4
DIRECT SOLUTION OF THE PRESSURE FIELD
4.1
Introduction
In the solution scheme employed by NATOF-2D,
conservation
equations,
the
equation
equations governing the exchange
single
equation
for
each
terms
cell
which
of
are
the
state,
eight
and the
to
reduced
a
involves
only the
pressure of a cell and its (up to four) neighbors.
The form
of the equation is:
ai Pij_- + biJPi-lJ + ci
PiJ + diJPi+lJ + eijPiJ+1 = fij
(4.1)
As can be seen, the pressure of a cell is influenced only by
the pressure of the cells directly in contact with it.
written out in matrix form,
is
this large system
bands.
the
diagonal
and
contained
in
five
example, the resulting matrix for the solution
For
of a problem with four axial levels and three
(figure 4.1)
equations
a matrix whose non-zero
a five-stripe band matrix, i.e.
components are near
of
When
radial
nodes
is shown in figure 4.2.
NATOF-2D
Previously,
used
an
iterative
solution
technique known as block-tri-diagonal, which is an extension
of the Gauss-Siedel iterative technique.
methods,
and
this
proceeded
Like all iterative
scheme started from an initial approximation
to
calculate
a
sequence
of
further
-89-
fictitious cell
fictitious cell
I
Figure 4.1
I
Arrangement of Cells for Pressure Field
Matrix Shown in Figure 4 .2
-
'~"
--
^
.m
II
"inIImYYY
mmrIIrl.
-90-
0
0
0
0
0
0
0
0
a3,7
0
0
0
0
0
0
a4,8
0
0
0
0
a5, 5
0
0
a5,9
0
0
a5 ,6
0
a6,5
a6,7
U
0
a6,10 0
0
0
a7,
7
a7,8
0
0
a7,11
a8,7
a8,8
0
0
0
0
a ,9
0
a10,9
al,l
al,2
0
0
al,5
0
0
a2,1
a2,2
a2,3
0
0
a2,6
0
a3,2
a3,3
a3,4
0
0
0
a4,3
a4,4
0
a5,1
0
0
a6,2
0
0
0
0
0
0
a8,4
0
0
0
0
0
a9,5
0
0
0
0
810,6 0
0
0
0
0
a,6
0
all,7 0
0
0
0
0
a7,3
0
0
0
0
0
Figure 4.2
Pressure Field Matrix
0
a12,8 0
ag910 0
0
a8,12
0
a10,10 10,110
alO,10alO,11ll
a11,10
all,11 all,12
0
a12,11al2,12
-91-
eventually gave the required solution
which
approximations
within a user defined convergence.
method
The iteration
on
depended
dominant, i.e. the terms bij
diagonally
4.1 being much smaller than the terms
situation
the
allowed
and dij
aij
field
pressure
axial
be
until
direction
This
solved for
were
iterations
directly in the radial direction, and then
the
of equation
eij.
and
to
being
matrix
the
performed
in
converged.
Diagonal dominance could only be
the
solution
maintained
by
having an axial mesh spacing which was much greater than the
could be
used
limited
the
number of cells which
simulations,
and
thus
prevented
high
the
pressure
field
This
spacing.
radial
in
resolution.
Since
solution
the
has
method
employed
for
a strong influence on the running time of the
code, a more efficient technique would result in drastically
reduced costs.
i.e.
a
The method now employed is a direct
method
without
any
section
will
which
calculates
intermediate
give
a
the
required
approximations.
The
method,
solution
following
background on direct methods and the
solution technique employed in NATOF-2D.
___
-92Direct Method Solution Techniques
4.2
in
If a matrix is of the form shown
accomplished
easily
is
solution
by
the
4.3,
figure
what is called "back
The n-th equation gives xn directly (bn/un),
substitution".
and then the (n-1)th equation can be solved for xn- 1since
is
terms,
In matrix
determined.
can
procedure
This
known.
continued until xI is
be
the
system
situation
similar
when
occurs
of equations Ax
A is a lower triangular
matrix (figure 4.4) and "forward substitution" is
The
form
triangular
Gaussian
Elimination,
interchanges
rows to
and
eliminate
technique
for
addition
all
matrix
can
example,
be
which
employed.
obtained by
uses
row
and subtraction of multiples of
terms
below
the
diagonal.
This
can be used for small matrices, but for the large
occurring
systems
the
of
matrix.
triangular
b is easy to solve when A is an upper
A
xn
in
most
calculations,
NATOF-2D
the
computational costs become prohibitive.
In order
by
required
performed
on
factoring
the
to
the
avoid
number
of
row
interchanges
Gaussian elimination, triangularization can be
the
matrix.
matrix
into
Triangulazization
refers
to
a lower and a upper triangular
such that
A = LU
where
L = lower triangular matrix
U = upper triangular matrix
(4.2)
-93-
u
11
u
12
u
0
U
0
0
u
0
0
u
0
0
0
0
Figure 4.3
22
25
35
45
u
55
0
u
U
U
U
u
16
26
b
b
36
b3
46
b
56
bs
U66
b
Upper Triangular Matrix
0
0
bi
122
0
bb24
32
0
42
0
'52
162
Figure 4.4
U
lb
b5
5
b6
6
b,
Lower Triangular Matrix
--
iII
IihhY
i IIYllU
IUIMiIIl
,ill
lhi,
l,,ii,
'
Willilili
-94-
Thus the system of equations
Ax = b
becomes
LUx = b
The
factorization,
Defining
y
is
/
unique.
/
10
= Mx, the system Ly = b can be solved for y by
Then, U. = y can be solved for
solution
x
by
This is the basic technique used for
backward substitution.
direct
possible,
when
forward substitution.
the
(4.3)
of
the
pressure
field
employed by
NATOF-2D.
Since the pressure field matrix is in
of
number
reduced
to
true
especially
the
if
the
w
if
This
is
number
of
zeros.
bandwidth
is
much less than the
large
dimension of the matrix.
value
the
form,
required for the LU factorization is
operations
due
band
a
The bandwidth of a matrix A has
aj' = 0 whenever li-jl 2 w.
For example, the
matrix in figure 4.2 has a bandwidth of 5.
Taking into account the
presence
of
the
the
zeros,
terms of the upper triangular matrix are given by /
11
/:
i-1
uij = aij -
1
uikUkj
k = max(1,j-w+1)
where
j = i,...
min(i+w-l,n)
n = dimension of the matrix A
(4.4)
-95-
The terms of the lower triangular matrix are
by
given
the
relation:
.. =
-j
j-1
-1
..
33
Skj ik )
(4.5)
k = max(l,i-w+l)
where
i = j+1,.. . min(j+1-w,n)
A comparison between
count
of
operations
using
elimination and using a LU factorization of a band
Gaussian
matrix
the
given
is
by
Franklin
/
11
/.
For
Gaussian
elimination there are:
2
C1 = n
+ (n-1)n(n+1)/3
C2 = n(n-1) + (n-1)n(2n+1)/ '6
multiplications or
divisions
additions or
subtractions
For the LU factorization and solution there are
C 1 = w(w-1)(3n-2w+1)/3 + (2w-1)n - w(w-1)
multiplications or
divisions
C 2 = w(w-1)(3n-2w+1)/3
additions or
subtractions
IYY
IIIYIIYIYIYIIIYlllylII
i I ii
dIIll u
-96-
For the matrix given in figure 4.2 the count would be
Cl
C2
Gaussian Elimination
1616
1480
LU Factorization and
Solution
384
260
As can be seen, the saving is substantial.
dependence on the bandwidth of the
be
operations
for the LU factorization and solution has a strong
required
can
of
number
The above result shows that the
drastically
reduced by reordering the numbering of
is
In NATOF-2D, the numbering scheme
the cells.
bandwidth
The
matrix.
count
to
from the bottom to the top for each cell in a channel.
a
with
problem
twelve axial levels and three radial nodes
has a bandwidth of thirteen.
so
numbering
Thus
that
by
However,
rearranging
the
cells are numbered across for each
the
axial level, the 'bandwidth is reduced to four.
be
Since there will
diagonal
elements,
employed to
achieved
reduce
a
number
of
divisions
partial
pivoting
strategy
cumulative
rounding
error.
a
by
the
is also
This
is
by reordering the rows of the matrix such that the
largest elements appear on the diagonal. /
12
/
.
-97-
4.3
A Comparison of Direct and Iterative Methods in
NATOF-2D
Unit (CPU) time usage of the direct and
Processing
Central
The results are plotted as
calculations.
NATOF-2D
most
for
size
typical
of
matrix
a
gave
which
nodes,
radial
5
comparison had 12 axial levels and
time
CPU
the
for
used
case
test
The
calculations.
transient
and
state
steady
both
for
methods
solution
iterative
the
between
made
is
comparison
a
section
In this
versus
time into the simulation.
4.5,
For the steady state calculation, shown in figure
the
CPU
usage for the direct solution is about 100 seconds
less than for the iterative solution.
in
the
CPU
for
expected,
since
converging to
to
state calculations
solution
may
in
the
of
fact
in
the
into
longer
is
to
be
pressure per time step is
convergence
a
This
solution.
therefore
and
zero,
meet
change
the
time
the
CPU usage per Newton iteration
iterative
the
decreases
required
the
increases,
As
down.
settling
is
calculation
difference
occurs at the start of the calculation, when
usage
system
The major
iterations
fewer
criterion.
duration,
are
For steady
iterative
the
be quicker, since the direct method
takes a fixed amount of time to
solve
the
pressure
field
matrix regardless of the pressure change increment.
For the single phase transient case,
4.6,
shown
in
figure
the direct solution used only half the CPU time of the
III uiI
-98-
400
U 300
E
I-
200
100
1.0
2.0
3.0
4.0
Time into Simulation (sec)
Figure 4.5
A Comparison of Steady State CPU Usage
Between the Direct and Iterative Techniques
(10 Axial Levels, 5 Radial Nodes)
-99-
200
S150
100
50
0.5
1.0
1.5
Time into Simulation (sec)
Figure 4.6
A Comparison of Transient CPU Usage Between
the Direct and Iterative Techniques
(10 Axial Levels, 5 Radial Nodes)
-100-
iterative solution.
CPU requirements
during
This is a considerable savings when the
are
large.
Experience
has
shown
that
boiling transients, the large pressure changes cause
the iterative solution to have
an
greater
even
CPU
time
usage relative to the direct method.
A transient was also run with a decreased mesh spacing,
so that there were now 42
cells
the
in
axial
direction,
before.
The
iterative technique took four times as much CPU time as
the
covering
direct
approximately
solution,
thus
the
same
length
demonstrating
the
as
important
diagonal dominance plays in the iterative technique.
role
-101-
4.4
Programming Information
Two subroutines have been incorporated into NATOF-2D to
DIRECT
which
The second is subroutine
commercial subroutine
at
M.I.T.
MULTICS
computer
which performs Lower-Upper Factorization
The basic algorithum
and the solution of Band Matrices.
this subroutine can be found in /
laws,
the
on
available
LEQT1B
Subroutine LEQT1B is a
performs the direct solution.
system
subroutine
is
rearranges the pressure field matrix in order
to minimize the bandwidth.
which
one
The first
perform the direct solution.
10
of
Due to copyright
/.
this subroutine cannot be disseminated outside of the
Massachusetts
subroutine
of
Institute
similar
of
form
Technology.
should
be
However,
a
available on most
computer systems.
The user has the option to specify either a
iterative
direct
or
solution technique by setting the input parameter
indgs to either 0 or 1 respectively.
'
'
T
,
i
11lilnli1
,H
ii
I II Jllll
-102-
Chapter 5
EXPERIENCES WITH NATOF-2D
5.1
Introduction
The original development of NATOF-2D
until
June
of
1980,
running the code was of
limited
extremely
low
computational
the
six
months
past
completed
therefore prior experience with
and
a
wasn't
have
nature.
However,
the
cost available at M.I.T. over
allowed
numerous
testing
simulations to be made which yielded information in the area
of code capability and constraints.
the
experiences
In this chapter some of
encountered will be documented in order to
provide a foundation for future work, and
explanation
relations.
for
any
changes
made
to
also
the
provide
an
constituative
-1035.2
Double versus Single Precision
to
effort
an
In
in
digits
significant
in
digits
significant
it
are
there
9
precision, and 18 in double
single
At the time,
precision.
MULTICS
On
computed.
is
which a variable
variable is stored and to
a
which
of
number
the
Single precision refers to
precision code.
to a single
converted
was
NATOF-2D
costs,
computational
and
storage
memory
both
reduce
that
felt
was
out
carrying
calculations to the 18th place was being excessive.
value
The results showed exact agreement in the
precision.
of
computed in both single and double
transient
phase
single
same
the
A comparison was made between the results of
This was
to the eigth significant digit.
up
variables
costs
computational
encouraging since the single precision
were 25% less.
However, in two
was
size
timestep
encountered
boiling
phase
convergence
obtaining
in
were
problems
smaller,
considerably
the
where
transients
of
the
Newton
Iterations for the single precision version of NATOF-2D.
To
explore this problem, the code was modified so that the user
could
a
impose
calculation.
calculation
of decreasing time step sizes on a
series
The procedure was
to
reach
a
steady
gradually reduce the timestep
already
at
a
steady
always be attained.
state
This
to
is
a
allow
state
size.
phase
solution, and then
Since
solution,
single
the
code
was
convergence should
particularly
true
for
the
'
IIII
iI.' , ,
-104iteration
scheme
since as At+O,
used,
results showed that
for
small
actually
and
had
diverged,
AP-0.
timestep
difficulty
However, the
sizes,
in
the
code
reaching
the
convergence criteria (in NATOF-2D, convergence is assumed if
6Pmax <
user
relaxed,
the
input).
code
could
run to slightly smaller timestep
sizes, but it still wasn't converging on zero
state
solution
implies.
was
criteria
convergence
the
If
Table
5.1
as
lists
a
the
steady
smallest
timestep size for various convergence criterias.
Table 5.1
Newton Iteration Convergence
Minimum Timestep Size
(N/m2 )
(seconds)
-2
10 2
0.01
The
problem
conservation
-3
0.1
10
1.0
10
10.0
10
-6
experienced
equation.
-4
The
requires that the quantity
n+1
(1 - c)pe
At
was
traced
solution
to
the
scheme
energy
employed
n
(1 - ac)pk e
(5.1)
-105be evaluated at each timestep.
and/or
timesteps
sizes
in a steady state configuration, the term pke, has a
However, the magnitude of
very small variation.
is
small
For
typically
term
this
of the order of 109, which is the same as the
In any single precision computation,
machine precision.
it
is reasonable to assume that the 1st 8 digits are valid, but
that
the
9th
digit is subject to "noise" fluctuations and
computational roundoff errors.
Yet for small timesteps, the
calculation was relying on this term to solve the
Any
error
would
to the
proportional
inversely
be
then
equation.
timestep size.
When
the
code
10-10
timesteps
of
stability
problems.
was
seconds
From
to
returned
could
this
be
precision,
double
attained
experience
it
with
has
no
been
concluded that double precision is a necessity for NATOF-2D.
. .1010 11,L,
5.3
On the Modelling of Sodium Reactors
Numerous
problems
phenomenological
have
modelling
been
of
encountered
sodium
which were not encountered in PWR and
for
reason
this
difficulty
boiling
BWR
be
can
between
typical
the
transients
modelling.
traced
characteristics of LMFBRs and the properties of
comparison
in
The
to
the
sodium.
A
PWR and LMFBR characteristics,
and water and sodium properties is given in
Table
5.2
and
Table 5.3 respectively.
As shown in Table 5.2, the most striking difference
core
properties
unit length.
is the temperature rise of the coolant per
For a PWR this is 9.7 0 C/m, while for an
125.4 C/m.
is
it
In
the
numerical
scheme
LMFBR
employed in
NATOF-2D, the core is divided into a finite number of
axial
Unless the number is large, there will always exist
levels.
a
in
substantial temperature difference from cell to cell.
some of the loss-of-flow transients simulated, this
as much as 150 0 C.
model
for
can
In
be
Combined with the higher power density, a
LMFBR
transient
analysis
experiences
rapid
temperature changes throughout its length not experienced by
PWR codes.
The
pj/pv,
ratio,
psi,
and
1000
which
for
is
is
difference
major
second
approximately
sodium
at
44
the
6 for water at 2200
psi.
At
pressure, the ratio for sodium increases to 3000.
density
ratio
for
sodium
density
atmospheric
The large
leads to a rapid voiding of the
-107-
Table 5.2
A Comparison of PWR and LMFBR properties /
13
Proposed
PWR
LMFBR
Core Thermal Power (MWth)
3,411
3,800
Core Diameter (m)
3.4
3.11
Core Height (m)
3.7
1.22
Core Power Density (kw/liter)
98
395.7
Reactor Inlet Temperature (OC)
289
385
Reactor Outlet Temperature (OC)
325
538
System Flow Rate (total 10 6 1lb/hr)
136
136.8
Table 5.3
A comparison of Water and Sodium Properties /
Water
Pressure (psi)
2250
Saturation Temperature
Liquid Density (kg/m
Vapor Density (kg/m
3
( C)
)
3)
Liquid Specific Heat (J/kg-OK)
Liquid Conductivity (W/m-oK)
Vapor Specific Heat (J/kg-oK)
Vapor
Conductivity
(W/m-oK)
346.
593.4
102.
9211.
0.4074
7709.4
0.1061
Sodium
44
1016.
707.5
0.748
1324.
46.03
281.8
.073
-
*mmmommirnim
iii,
illklllll1,,I, ,I ,
ll
-108core for
extremely
small
superheats.
this
Accompanying
is the expulsion of the liquid phase, and a mass
phenomenon
depletion of the core often resulting in flow reversal.
numerical difficulty experienced during the
of
boiling
initial
The
stages
is attributable to the fact that the density of
the cell varies
by
a
factor
of
1000
in
very
a
short
timespan.
The boiling transients which the code
simulate
are
is
required
thus of an extremely harsh nature.
to
For these
reasons, the mass exchange rate plays a critical role in the
calculations.
-109-
The Mass Exchange Rate
5.4
in
used
Of all the constituative relations
NATOF-2D,
exchange
basic physical requirements of any mass
that
the
model
prevent
essence,
In
vapor.
superheated liquid or
highly
with
situation
two-phase
subcooled
is
vapor production rate should not exceed the limit
established by equilibrium, and also that it should
a
The
mass exchange rate is probably the most important.
the
it
toward
tend
should
equilibrium.
and
the high density ratio leads to void fractions of
As the void travels
0.9 in as little as 1/10th of a second.
into subcooled liquid regions, it is
quickly,
transient,
required
cells
condense
to
remains
condensation
where
occurring
and
low
throughout
the
is
occurring
are
is
evaporation
density
mixture
the
required
contains a negligible amount of energy.
it
since
However, unlike cells where
thus
vapor
of
The high power density of the
evaporation and condensation.
core
rate
the
The mass exchange rate determines
to experience rapid density changes throughout the
transient.
Since at even void fractions of 0.95, the
vapor
phase
the
cell,
rapid
only
represents
2%
of
the
mass
in
condensation requires that either the pressure of
decrease
(in
cell
order to lower the saturation temperature) or
else that the mass flux into the cell
The
the
requirements
of
a
be
extremely
large.
large condensation rate have often
--
---
~'
MUMMI
-110-
lead to code failure on a negative cell pressure error.
either
case,
the
change
In
pressure for the timestep is
in
large.
is
The pressure of a cell
the
thus effect the stability of
change
pressure,
in
if
determines
6P,
convergence
variable
key
the
in
Large variations in pressure
scheme of NATOF-2D.
numerical
the
of
code,
a
since
Newton
has
is
it
the
which
iteration,
the
Defining
occurred.
convergence criteria in relative terms such that
6P
convergence - p
a convergence of 10
can easily be attained in single phase
However,
calculations.
convergence
-10
must
be
in
relaxed
boiling
to
10
or 10
proven a necessity if timesteps are to be
computational
within
time
the
transients,
4
This has
which
taken
10 -
limitations (i.e.
3
are
or 10 -
4
sec)
Small timestep sizes, well below the convective
are
necessary,
since
the
effect of reducing the timestep
size is to reduce the magnitude of I,
kg/s-m.
One
noticeable
limit,
phenomenon
which
has
units
of
during sodium boiling
siumlations, is the appearance of a stable boiling timestep,
or SBT.
code
The SBT refers to the timestep size
can attain a reasonable convergence (10
which
at
-3
-
10
single Newton Iteration, but above which convergence
be
obtained
the
-4
) in a
cannot
regardless of the number of Newton Iterations.
The
SBT
appears
to
be
a
function
the
of
convergence
the mesh cell spacing distance (whose effects are
criteria,
covered in the next section) and the condensation rate.
The condensation rate, as previously mentioned,
profound
corresponding to a
reducing
the
small
change.
energy
rate (i.e.
By
numerically
multiplying it by a small number,
which will be designated CF), the calculation proceeds
essentially
What
quickly.
a
it requires a large density change
since
effect,
has
occurs
is
code is.
the
that
more
allowed to operate in a highly nonequilibrium, two-phase low
the
density mode which prevents
density
The
change.
need
to
the
make
effect is so pronounced that setting
the condensation rate to zero allows the code to
timestep
large
run
at
a
limited by any boiling effects (except during
not
the short period of boiling inception) but by the convective
-2
sec due to the high vapor velocities), while
limit (10
setting
the
timestep
sizes
manipulation
multiplicative
of
10
-7
factor
seconds.
to
In
one
necessitates
the
past,
this
has been justified by reference to experiments
which showed the condensation rate to be slightly lower than
the evaporation
rate.
However,
the
fact
that
analytic
results done with NATOF-2D show vapor coexisting with liquid
subcooled
by
hundreds
of
However, in the experiments
covered
in
degrees
simulated
negates
in
this premise.
this
work,
and
chapter 6, this manipulation has been necessary
in order to obtain results of a time spanning any reasonable
lImMilM
-112duration.
In the previous formulation used for the mass
rate, the term a(l
- a) was added.
exchange
This term had the effect
of inhibiting condensation at small void fractions, and thus
allowed
the
code
to
run
smoother
than
the formulation
presently implimented, which requires higher condensation.
-1135.5
Varying Mesh Spacings
In
order
gradients in the axial direction, the number of
temperature
decreasing AT from cell to cell, the mass exchange
by
detailed
rate decreases, and also the results are of a more
the inaccuracies caused
Smaller mesh cells reduce
nature.
mentioned
As
by volume averaging of the fluid properties.
in
Chapter 4, NATOF-2D could not use small mesh spacing due
order
to the necessity of maintaining diagonal dominance in
to
are
this
of
The two advantags
axial cells was increased.
that
large
the
of
effects
the
minimize
to
However, the direct
obtain the pressure field solution.
solution technique does not have
and
constraint,
this
so
small mesh spacing becomes possible.
limit
There is a practical
axial
will
Iteration
be
much.
as
times
ten
roughly
per
steps
computational
of
by a factor of 10, the number
Newton
of
number
example, if the number of levels is increased
For
levels.
the
to
However, the convective time limit, Az/v, would decrease the
time step size
also
by
need
a
factor
about
computation
would
compute the
same
time
length
a
time
step
doubtful
that
greater than
before
could
in
ten,
so
that
the
times more cpu time to
100
size
be
of
single
phase.
It
is
two orders of magnitude
taken
during
the
boiling
transient.
For testing purposes, the number of
increased
axial
levels
was
by a factor of 4, from 10 levels to 40 levels, to
I------~-~I-~
IY
U IIIIii
,
-114-
see whether any increased timesteps could
boiling.
taken
The decreased mesh spacing allowed timesteps to be
which
varied
from 2 x 10
seconds to 10
with a condensation factor of = 0.01.
roughly
during
taken
be
three
times
greater
case, which had a CF = 0.002.
than
The
cpu
for the
Increasing CF to 0.1,
times greater, and a SBT of 2 x 10
-4
cpu
usage
decreasing
6
seconds.
drawn from this is that higher accuracy
results can be attained without the corresponding cpu
by
was
usage
for the 10 axial level
same 40 axial level calculation resulted in a
The conclusion
seconds,
the mesh spacing.
costs
For the experimental test
simulations covered in Chapter 6, 40 axial levels were used.
__
__
Ii
iiIWI ',
Chapter 6
VERIFICATION OF MODELS
6.1
Introduction
In order to test the capability
the
and
NATOF-2D
of
validity of the models described in the previous chapters, a
The
total of five sodium boiling transients were simulated.
first one was THORS Bundle 6A Run 101 conducted at Oak Ridge
4
National Laboratory /
W-1
SLSF
Experiments,
Development Laboratory,
/.
done
The other four were from the
by
the
include
and
Hanford
two
Loss
Engineering
of
Integrity transients and two Boiling Window Tests /
The tests cover a wide range of conditions under
the
code
will
be
previous simulations /
to
drastically
to
required
increase
1
operate.
number
improved the quality of the results,
/.
5
which
In contrast to
been
made
of mesh cells.
This
/, the decision has
the
Piping
but
also
constrained
the length of the calculation due to the large CPU usage.
In the next sections, a description of the runs will be
given, and the results will be presented.
-116-
THORS Bundle 6A Experiment, Test 71H Run 101
6.2
Description of the THORS Bundle 6A Experiments
6.2.1
/
/
4
was
THORS
the
of
purpose
The
Experiments
6A
Bundle
to investigate the extent of dynamic boiling
stability at low flow conditions.
conducted
The tests were
in the THORS Facility, an engineering-scale high-temperature
facility
sodium
the thermo-hydraulic testing of LMFBR
for
subassemblies.
The test section used was a full-length simulated LMFBR
units
simulator
pin
fuel
spaced
heated
electrically
19
It consisted of
fuel subassembly.
by
wire-wrap
helical
The heated length of Bundle 6A was 0.9 meters, and
spacers.
axial
1.3
a
had variable pitch heater windings to produce
peak-to-mean chopped cosine power distribution.
Appreciable
fabricating
effort
was
expended
in
designing
a low thermal inertia bundle housing.
However,
a posttest analysis revealed that sodium had penetrated
entire
and
the
housing region, and thus the housing had a high heat
capacity.
The selected run from this series
101.
The
bundle
power
level
was
was
Test
127 kw.
71H,
Run
This was the
lowest power run that exhibited dryout.
At 3.2 seconds into
the transient, the initial flow of 0.39
1/s
was
over a period of 3.6 seconds to a flow of 0.12 1/s.
inception
occurred
at
13.7
seconds.
Flow
decreased
Boiling
oscillations
__i
M
il020Ii i
Mii
il11
llii
9,li
W
lliliMilllNWI
HIM, ",
-117-
of
occurred at a frequency
1.5
to
1.1
A
Hz.
In Figure
of the bundle is given in Table 6.1.
description
the
6.1 the locations of the axial levels used in
geometry
NATOF-2D
simulation are shown.
pressure boundary conditions during the single phase
outlet
This allowed the exact flow rate
part of the transient.
be
boundary
the
inception,
cells.
four
or
At
the
were
conditions
of
point
to
cells
axial
established
permitted the proper flow splits to be
three
of
number
large
the
while
maintained,
first
velocity,
inlet
The simulation was carried out under
in
the
boiling
switched
to
pressure/pressure, and the transient was continued.
fluid-to-fluid
Nu
simulation,
this
For
was
0.01,
and
the
heat conduction nusselt numbers were
radial
Nu 2 = 13.
and
13
CF
The
input
parameters
for
the
simulation are given in Appendix C.3.
6.2.2
Simulation Results
of
The time
simulation
boiling
inception
approximately
was
THORS experiment.
2.7
hexcan
is
a
heat
capacitance.
NATOF-2D
seconds sooner than the
In NATOF-2D, the hexcan is
During
transients
the
heated up and cooled off by the coolant, with no
losses to the environment.
heat
the
This can be attributed to underestimating
the heat capacity of the hexcan.
modelled as
during
In principal, by
adjusting
the
capacity of the hexcan, the boiling inception time can
~iYll
rn
-118-
Table 6.1
Description of THORS Bundle 6A
Number of Pins
Pin Diameter (m)
19
5.84x10 -
3
Pitch/Diameter Ratio
Wire Wrap Diameter (m)
1.42 x 10-3
Distance from fuel pin to the wall (m)
0.71 x 10-
Heated Length (m)
0.9
Axial Power Distribution, Cosine
peak/mean
Radial Power Distribution, peak/mean
Total Bundle Power (kw)
1.3
1.0
127
Pin Power (kw/pin)
6.7
Inlet Flow, Steady State (kg/s)
0.3344
Inlet Flow, Boiling Inception (kg/s)
0.1029
Inlet Liquid Temperature (OK)
660.91
3
~_
_
~___~
11
,I",.
-119-
Axial
Cell Level
Height (m)
2.245
40
Gas
Plenum
1.594
31
1.391
Blanket
-1.238
Active
Fuel
0.324
- 0.0
Figure 6.1
Location of Cells used in the NATOF-2D
Simulation of THORS Bundle 6A Experiment
-120-
calculation,
of the can.
a
a
the value is estimated based on the properties
The results of this test indicate the need
for
discusses
the
model.
sophisticated
more
of
start
the
to
Prior
value.
desired
any
assume
Appendix
D
sensitivity of the boiling inception time to the hexcan heat
capacity.
the
At
almost
followed
boiling
of
time
immediately.
The first is that
this.
allows
channel
the
to
There
void
two reasons for
are
rate
small
the
reversal
flow
inception,
condensation
of
rapidly, which increases the
fluid pressure and forms a flow blockage.
The second reason
is that information about pressure drops at the inlet due to
accurately
be
not
with
As
simulated.
could
effect
valve adjustments was not documented, so this
the
hexcan heat
drop
capacity, it is possible to adjust the spacer pressure
feature of NATOF-2D to obtain the experiment's flow reversal
time.
Appendix
E
typical
gives
values
of
spacer
the
pressure drop, and its effect on flow reversal.
Figure 6.2 shows the inlet mass flow
a frequency of
of
about
3
Hz.
oscillations
was
This
found
in
the
Figure 6.3, the temperature profile of the
at
various
during
the
The flow oscillations were quite severe and had
transient.
frequency
rate
about
twice
experiment.
central
the
In
channel
points of time during the transient is plotted.
Figure 6.4 shows the temperature history at the end
of
heated section for both the central and edge channels.
the
This
-121-
0.3
0.2
a, 0.1
0.0
I-"
-0.1
-0.2
- -0.3
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
time into simulation (sec)
0.3
0.2
0.1
0.0
U-
L -0.1
-0.2
-0.3
10.
11.
12.
13.
14.
time into simulation (sec)
Figure 6.2
NATOF-2D Predicted Inlet Mass Flow Rate of
THORS Bundle 6A Test 71H, Run 101
-122-
900 -
10.8 sec
800
o
0
0)
L
700
3.0 sec
0)
E
600
-
*r-
..I- J
*1
0r
500 1-
400
0.0
0.5
1.0
1.5
Distance above start of fuel (m)
Figure 6.3
NATOF-2D Temperature Profi
Channel at Various Poi nts
(THORS Bundle 6A Test 71H,
es of Central
n Time
Run 101)
_ _ I_
~ _~___
I--C
I
IIRIIIIIII
m.
-123-
1000
900
0
w
oJ
5-
800
Cr
4-,
EI
I"0
-.J
700
3.0
6.0
9.0
12.0
time into simulation (sec)
Figure 6.4
NATOF-2D Predicted Temperature Histories at
End of the Heated Zone for the Central and
Edge Channels (THORS Bundle 6A Test 71H, Run 101)
-124-
figure
demonstrates
the
of
effects
conduction model in reducing radial
the
temperature
radial
heat
variations
to levels comparable with the test results.
Figure 6.5 compares the vapor
The
and
liquid
velocities.
large difference in velocities indicates the need for a
separated flow model.
II I llill
I iioal
il i
1j
,lli,
111
l lI
l ll l ,,,,,
-125-
26
20
vapor
-
10
liquid
0
12.
11.
13.
time into transient (sec) -- End of the Heated Zone
5
4 --
A-
vapor
liquid
3
*1
liquid
only
r 0-1*
0
£
*
0
*
*
L
-I
11.
0
I
12.
13.
time into transient (sec) -- Midplane of Heated Zone
Figure 6.5
A Comparison of NATOF-2D Predicted Liquid and
Vapor Velocities
(THORS Bundle 6A Test 71H, Run 101)
,, IJ,
1 ilI
-126-
The W-1 SLSF Experiments
6.3
Test Objective
6.3.1
The W-1
provide
pin
"stability" in a fuel
under
on
data
experimental
5
/
Experiment
SLSF
sodium
(LOPI)
The
accident
first
was
flow
transients
The test was divided into
LMFBR accident conditions.
two groups.
to
boiling and boiling
during
bundle
designed
was
/
the
Loss-of-Piping
Integrity
The objective of this series
simulation.
was to determine the heat transfer characteristics from fuel
pins to sodium
double
coolant
a
during
transient
simulating
a
ended pipe break at the primary vessel inlet nozzle.
The difference in heat transfer characteristics as a
result
of fuel conditions were studied.
The second group of tests was the Boiling Window Tests.
The objective of this series was to establish the family
flow/heat-flux
combinations
that
will
boiling in the bundle for a given
inlet
produce
of
incipient
temperature.
The
test runs were designed to determine whether or not there is
a
regime of boiling beyond the onset that persists and does
not immediately lead to dryout for low flow and intermediate
to high heat fluxes.
The SLSF W-1 boiling window
three
operating
phases:
approach
boiling and dryout with fuel
boiling
tests
were
tests
designed
pin
to
to
were
conducted
boiling,
failure.
determine
The
in
incipient
incipient
low heat flux
__
iIbYYImgkII
,YIYIIIIYIII
i liij I MIlI
i
-127-
combinations for
achieved.
the
which
onset
of
sodium
boiling
is
One dryout test was performed to identify the far
end of the "boiling window" at the highest heat flux tested.
Test Apparatus and Procedure
6.3.2
The tests were carried out on the
(SLSF)
Facility
Sodium
Loop
Safety
the Engineering Test Reactor under the
at
direction of the Hanford Engineering Development Laboratory.
Each LOPI was initiated from steady
full-flow
conditions.
Over
the
state
full
first 0.5 seconds of the
transient the inlet flow was dramatically reduced.
seconds
into
power
At
0.65
the transient, the reactor was scrammed.
The
test section was returned to full flow after approximately 3
seconds from time zero.
The
Boiling
steady-state
Window
flow
of
Tests
1.95
kg/sec.
a
at
initiated
were
The flow was linearly
reduced to its "low flow" value in 0.5 seconds, where it was
held for a specific time, and then linearly returned to
state in 0.5 seconds.
initial
The geometric parameters of the
characteristics
respectively.
used in
6.3.3
its
fuel
of the tests are given in
Figure 6.6 shows
the
axial
bundle
and
the
table 6.2 and 6.3
cell
locations
the NATOF-2D simulations.
Tests Chosen For Simulation
From the available results,
a total of four
test
were
-128-
Table 6.2
Geometric Parameters of the W-1 SLSF Bundle
19
Number of Pins
Pin Diameter (m)
5.842 x 10 -
Fuel Pellet Diameter (m)
4.94 x 10
Wire Wrap Diameter (m)
inner pins
1.422 x 10 -
outer pins
7.11 x 10
3
4
Fuel Pitch (m)
7.264 x 10
Pitch/Diameter Ratio
1 .25
Flat-to-Flat (m)
3.26 x 10-2
Length of Active Fuel
0.9144
3
Axial Power Distribution, Cosine
1.4
peak/mean
Radial Power Distribution, Cosine
peak/mean
Fuel
Fill Gas
Inlet Liquid Temperature (OC)
1.0
Uranium-Plutonium mixed
oxide,-Pu 25% of total mass
10% Helium-Neon
388.
-
lIlm imli
i dlIii
i ur
Nlnnd
Uij
I I l i ll~
iiI lll,dJii IIILHH
-129-
Table 6.3
Power and Flow Rates of the W-1 SLSF Tests
LOPI 2A
Power (kw)
Steady State Flow Rate (kg/s)
Low Flow Rate (kg/s)
661.8
1 .95
.65
LOPI 4
Power (kw)
Steady State Flow Rate (kg/s)
Low Flow Rate (kg/s)
705.3
1 .95
.65
BWT 2'
Power (kw)
Steady State Flow Rate (kg/s)
Low Flow Rate (kg/s)
348.3
1.95
.47
BWT TB'
Power (kw)
Steady State Flow Rate (kg/s)
Low Flow Rate (kg/s)
661.8
1.98
.74
,111 0
-130-
Axial
Cell Level
Height (m)
39
2.4511
Gas
Pl enum
29
1.4478
Blanket
24
-.1938
Active
Fuel
0.2794
6
0.0
Figure 6.6
Location of Cells used in the NATOF-2D
Simulation of W-l SLSF Experiment
-131chosen
for
simulation
NATOF-2D.
Two
tests
and two from
the
BWT
with
selected from the LOPI tests,
were
tests.
tests were selectively chosen to provide a full range
These
a
to
calculation
phase
single
of transients: from
full
dryout simulation.
LOPI simulation is
The first
the
LOPI 2A.
this
In
test,
maximum coolant temperature reached 9410C and showed no
indication
of
This
boiling.
provide
will
test
some
calibration information for the future test of the series.
The second LOPI simulation
about 0.5
seconds
thermocouples
used
4.
LOPI
The
maximum
reached was 9560C, and the data showed
temperature
coolant
is
of
boiling.
Failure
one
of
of
the
for the test section power calculations
resulted in LOPI 4 being run at approximately
5%
overpower
This transient will test the sensitivity of
(15.12 kw/pin).
the code to such an occurrence.
The first BWT simulation
approximately
0.8
seconds
is
of
BWT
boiling
coolant temperature reached 953oC at
kw/ft.
2'.
a
In
this
test
The
was observed.
pin
power
7.5
of
The test section was held at 24% of full flow for a
duration of 4 seconds.
of
This test offers the opportunity
simulating a low-power/low-flow sodium boiling transient.
The second BWT to be simulated is BWT7B'.
most
interesting
at
the
and demanding test to be run by the code.
Approximately 2.0 seconds of boiling
dryout
This is
a pin power of 14.4 kw/ft.
occurred
before
clad
The test section had
-132-
38% of full flow for a period of 3.0
oscillations,
flow
reversal
hypothesized for sodium boiling
and
seconds.
Inlet
flow
dryout is the worst case
codes.
Ability
to
model
this sequence of events will severly test the limitations of
NATOF-2D.
The input files used in the NATOF-2D simulation of
W-1 SLSF Tests are given in Appendix C.
the
~_
O MM
l im immlllimilllY
I
IIdIY
l
-1336.4
W-1 SLSF Simulation Results
6.4.1
LOPI 2A
The
LOPI
2A
was
simulation
carried
out
under
velocity/pressure boundary conditions in order to accurately
the experiment's inlet flow rate (shown in Figure
duplicate
the experiment's result is shown in Figure
and
temperature
The shape of the two curves
6.8.
close,
fairly
are
slightly
predicting
NATOF-2D
liquid
predicted
NATOF-2D
A comparison between the
6.7).
with
temperatures.
higher
Differences are to be expected however, since
average
cell
temperatures are being compared with temperatures taken at a
point.
simulation shows that the fuel pin properties
This
model
the
Figure 6.9 compares the temperatures at the end of
the
accurately
used in the code are quite good, and
response of real fuel pins during a reactor scram.
heated .zone for the central and edge channels.
shows how the edge channel takes longer to
This figure
respond
to
the
transient due to the effects of heat losses to the can.
6.4.2
LOPI 4
The
4
LOPI
velocity/pressure
shown
in
approximately
boundary
0.5
seconds
The
experiment,
of
boiling.
flow rate is
there
The
predicted temperature at the end of the heated zone
matched
the
experiment
for
the
central
under
run
also
conditions.
the
in
As
6.10.
was
simulation
was
NATOF-2D
closely
channel (Figure
-134-
2.0
1.5
1.0
0.5
0.0
0.
1.
2.
3.
4.
5.
6.
7.
time into transient (sec)
Figure 6.7
W-1 SLSF Test
Flow Rate
LOPI 2A Experiment
Inlet Mass
I
-
I
AiilII
-135-
1100
1000
F*
Z 900
0o
4J
S-
NATOF-2D
s800
I-
C" 700
Experiment
600
500
0.
1.
2.
3.
4.
5.
6.
7.
time into simulation (sec)
Figure 6.8
A Comparison Between Experiment and NATOF-2D
Predicted Temperature Histories at the End of
the Heated Zone for the Central Channel
(W-l SLSF Test LOPI 2A)
-136-
1100
1000
Central
SChannel
900
c)
4-
800 I-
' Edge
Channel
S-
700
**-
600
500
I
1.0
-
2.0
3.0
--
4.0
time into simulation (sec)
Figure 6.9
A Comparison of NATOF-2D Central and Edge
Channel Temperature Histories at the End of
the Heated Zone (W-l SLSF Test LOPI 2A)
I~
I
--
--
----
IIIEIIEIIIIEI~IEEIIUIEIYIIIiiI
I.YYIUY
__
-137-
2.0
1.5
1.0
0.5
0.0
0.0
1.0
2.0
3.0
4.0
time into transient (sec)
Figure 6.10
W-1 SLSF Test LOPI 4 Experiment Inlet Mass
Flow Rate
-138-
6.11), but not as well for the edge channel (Figure 6.12).
Void maps for the central and middle channels are shown
Although these maps are for voids
in Figure 6.13.
they
nearly
the
overlap
the
regions
In and near the
traveled.from the center of the bundle to
demonstrates
the
value
of
voided
were large, as the coolant
velocities
radial
A negligible
void maps for 0.9.
void was found in the edge channel.
0.1,
of
a
the
edge.
two-dimensional
This
model
in
simulating boiling transients.
6.4.3
BWT 2'
The
BWT
transient.
2'
The
in Figure 6.14.
simulation
was
a
low
power-low
flow
test section inlet mass flow rate is shown
The temperature histories at
the
midplane
and end of the heated zone for the central channel are shown
in
Figures 6.15 and 6.16.
Unlike the experiment, which had
0.8 seconds of boiling, no sodium boiling
The
simulation.
liquid
reached
in
occurred
the
a maximum temperature of
931 0 C, approximately 20 0 C below saturation.
The exact reason for this
possibly
may
be
associated
capacity of the fuel rods.
rate
also
could
have
result
to
The
been
is
not
known,
overestimating
reported
too
high.
inlet
In
a
but
the
heat
mass
flow
low power
transient such as this, the necessity of using cell averaged
temperatures, results
in
lower
peak
temperatures.
Even
-
- - -an-
Ilimilili
I41I0
I
i I,
II
-139-
1100
1000
900
NATOF-2D
0
o
800
S.
w
E
I-03
Experiment
of,=
4.J
700
0*
-J
600
500
Ik
I
0.0
I
I
a
1
1.0
I
1
2.0
1
1
3.0
4.0
5.0
time into simulation (sec)
Figure 6.11
A Comparison Between W-1 SLSF Test LOPI 4
and NATOF-2D Temperature Histories at the
End of the Heated Zone for the Central
Channel
1100
1000
Experiment
900
0
I
800
NATOF-2D
I
c0
*1l
0L
700
-J
600
500
I
a
0.0
1
I
t
1.0
2.0
a
3.0
I
4.0
I
I
5.0
time into simulation (sec)
Figure 6.12
A Comparison Between W-1 SLSF Test LOPI 4
and NATOF-2D Temperature H 4 stories at the End
of the Heated Zone for the lMidplane Channel
......
YJiYiliiim~lll
Y
Yiiill
iu
-141-
-0.3
Li.
G 0.2
(.
< 0.1
40
0.0
LU
.2
-. 3
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.7
1.8
time into simulation (sec) -- Central Channel
0.3
LL 0.2
U
0.1
4-
o 0.0
0
-. 1
-.2
u
-. 3
1.1
1.2
1.3
1.4
1.5
1.6'
time into simulation (sec) -- Middle Channel
Figure 6.13
NATOF-2D Void Maps for W-1 SLSF Test LOPI 4
for the Central and Middle Channels
Void = 0.1
,MIN
-142-
2.0
1.5
1.0
0.5
0.0
-. 5
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
time into transient (sec)
Figure 6.14
W-1 SLSF Test BWT 2'
Inlet Mass
Flow Rate
--
--
---
~
~
~
---
U1111hl
Ii
-143-
1100
1000
900
o
.
800
E
700
I
e-
-J
600
500
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
time into simulation (sec)
Figure 6.15
A Comparison Between W-l SLSF Test BWT 2' and
NATOF-2D Temperature Histories at the End of
the Heated Zone for the Central Channel
-144-
900
800
700
I
-o
*1a
-J
600
500
400
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
time into simulation (sec)
Figure 6.16
A Comparison Between W-l SLSF Test BWT 2'
and NATOF-2D Temperature Histories at the
end of the Heated Zone for the Central
Channel
- ---
using
cells
5
------
*IYliill
cm in length resulted in axial cell to cell
temperature differences of 120 C.
Further simulations of this
that
have
experiment
revealed
this result is not caused by overestimating the radial
fluid-to-fluid heat transfer or the hexcan heat capacity.
6.4.4
BWT 7B'
BWT 7B' was a high power boiling window transient.
simulation was carried out under velocity/pressure
until
conditions
the
point
of
boiling
code
code
When
when
the
the dryout point, at which time the switch is
reached
made in interfacial mass and energy
the
boundary
inception,
pressure/pressure boundary conditions were used.
The
reduced
to
exchange
correlations,
extremely small timesteps (10-7 sec)
which were below an acceptable level.
The reason for this can again be traced to the problems
of boiling and condensation.
Even at high
void
fractions,
the liquid phase is often subcooled to a large extent.
the
switch
temperature.
in
correlations
requires
drastic
Thus
changes in
The vapor phase must act as the source of this
heat, and since it obeys the perfect gas law, large pressure
changes are necessary.
In order to obtain results,
was
relaxed (6P
= 10-2).
the
convergence
criteria
It was found that with this large
convergence criteria it was possible to set CF = 1.0 and run
-146for 0.7 seconds during boiling, until small
The
required.
simulation
offers
timesteps
interesting
some
observations when compared to the same case
were
which
was
run
with CF = 0.01.
For CF = 0.01,
after
boiling
flow reversal occurred at
CF = 1.0).
6.18
seconds
inception, while for CF = 1.0, flow reversal
didn't occur until 0.45 seconds.
experiment's
0.25
flow
rate
Figure 7.17
compares
the
to that predicted by NATOF-2D (for
A comparison of void maps is
given
and 6.19 for the central and edge channels.
in
figures
As can be
seen the large condensation rate keeps the void centralized.
Figure 6.20 compares the temperatures at the end of the
heated zone for the central channel.
-147-
2.0
1.5
U
4J
1.0 I-
Experiment
r,,
ou,
0)
4
'a
3J
0.51-
LIu)
u)
'U
4.
43
NATOF- 2D
-4
0.0 -
-. 51I
0.0
I
I
1.0
2.0
I
3.0
I
4.0
1
5.0
time into simulation (sec)
Figure 6.17
A Comparison Between W-1 SLSF BWT 78B' and
NATOF-2D Predicted Inlet Mass Flow Rates
-148-
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-.1
-. 2
-. 3
-. 4
-.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time after boiling inception (sec)
Figure 6.18
A Comparison of NATOF-2D Central Channel Void
Maps for CF = 1.0 and CF = 0.01 -- Void = 0.9
(W-l SLSF Test BWT 7B')
-149-
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-.1
CF = 1.0
-. 2
-. 3
-. 4
CF = 0.01
-. 5
I
0.0
'
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time after boiling inception (sec)
Figure 6.19
A Comparison of NATOF-2D Edge Channel Void
Maps for CF = 1.0 and CF = 0.01 -- Void = 0.9
(W-1 SLSF Test BWT 7B')
-150-
1100 1-
1000 I
0
w
54=)
900
S..
NATOF-2D
0a
I-
-o
Experiment
800
-J
700
600
0.0
I
I
II
1.0
2.0
3.0
4.0
I
5.0
6.0
7.0
time into simulation (sec)
Figure 6.20
A Comparison Between NATOF-2D and W-1 SLSF
BWT 7B' Temperature Histories at the End of
Heated Zone for the Central Channel
- C ~-...I~.-~--~I---
YIIYYIIYIUIYllllr
-151-
Chapter 7
CONCLUSIONS AND RECOMMENDATIONS
7.1
Conclusions
The
models
implemented
in
NATOF-2D
simulations
described
particular, the fluid heat
conduction
during
the
temperature
improved
gradients found in the
technique
detailed
performed
Chapter
in
model
6.
gave
a
to
comparable
distribution
experiments.
The
direct
well
In
much
the
solution
allowed the simulations to be performed in a more
fashion,
and
the
within
imposed
limits
by
computational costs.
Little more can be said about the interfacial
and
exchange
energy
The effects of
Chapter 2.
correlate
rates
with
than
these
'experimental
sodium
described
in
are
difficult
to
results, since two-phase flow
interactions are almost never directly
the
is
what
terms
momentum
However,
measured.
boiling simulations showed that the models gave
physically reasonable results.
Problems were encountered during boiling
simulations,
and
most
in
the
of these were covered in Chapter 5.
For this reason, it is difficult to judge the effect of
mass
exchange
model of the code.
the
However, it was possible
to achieve full condensation with the model
convergence criteria.
test
for
a
relaxed
-152-
The
simulation,
core,
and
more
the
better
smaller
volume
changes
upon
attained.
the
smaller
of
axial
mesh
confined
condensation
is
is
cell
spacing
in
the
the voided areas of the
achieved.
Since
the
the cell leads to reduced pressure drop
boiling,
increased
convergence
could
be
__
-
YIIYLIIIIII
-153-
Recommendations
7.2
NATOF-2D, in its present form. is an adequete model for
However, before extensive use of
single phase calculations.
necessary, to
the code can occur, it will be
of
difficulties
a
rates
behaved, since they effect the void fraction, flow
well
reversal and heat transfer from the fuel
has
is
condensation
requirement to have the evaporation and
be
It
condensation.
and
boiling
the
overcome
NATOF-2D
occurred,
can
a
be
Once
pins.
this
valuable tool in LMFBR
accident analysis.
Throughout the present effort, there has persisted
of
problem
the transition between the vapor and
modelling
to
liquid phases without resorting
small
very
of constituative relations in the belief that
be
could
overcome.
cause
which
the
by
of
the
being
the
NATOF-2D,
The
particularly
small
condensation,
Newton Iteration
behaved
well
scheme
numerical
the
functions.
properties of sodium and LMFBRs have severely
tested the limits of the method, and have
either
relations
constituative
simulated.
method, is extremely powerful for
However,
problem
the difficulties, but rather the properties of
transients
utilized
this
However, continued work has shown that
properties
it is not the
timesteps.
focus of this work had been the development
the
Initially,
the
timesteps
or
else
must
the
be
taken
relaxed
posed
to
a
choice:
achieve
convergence
associated with larger timesteps must be accepted.
full
criteria
-154-
The results of
improving
the
this
work
capability
to
effective
of
value in this line
nature
semi-implicit
for
Rather
would
be
develop a method which is able to solve
than
iterations
it
sizes,
for the unknowns of the problem at each
more
need
of the iteration scheme.
than the simple reduction of timestep
cost
the
demonstrate
development
of
the
would
use
the
to
cause
the
6A
Test
be
calculation
if
Of particular
are required.
before
even
timestep,
to
derivatives of the problem to be well behaved.
The NATOF-2D simulation of THORS Bundle
Run
101
demonstrated
71H,
the need for improving the model for
heat losses to the hexcan, and for including heat losses
to
the environment.
Another area for future work would be
of
a
mechanism
the
development
for assuming a temperature gradient in the
used
cell, replacing the assumed flat profile presently
the code.
in
the
This would be necessary to achieve greater detail
simulation
results
since further decreases in the
axial mesh spacing beyond those used in the simulations
not
in
possible
due to computational constraints.
gradient would allow boiling inception to occur
are
An assumed
in
a
more
localized manner, reducing the effects of full cell boiling.
-155-
References
1.
Granziera, M.R., and Kazimi, M.S., "A Two Dimensional,
Two Fluid Model for Sodium Boiling in LMFBR
Assemblies," MIT Energy Laboratory Report
MIT-EL-80-011, May 1980
2.
Engel, F.C., Markley, R.A., and Minushkin, B. "The
Effect of Heat Input Patterns on Temperature
Distributions in LMFBR Blanket Assemblies,"
Westinghouse Electric Corp. Advanced Reactor
Division, Madison, Pennsylvania (1978)
3.
Yeung.M.R., "A Multicell SJug Flow Heat Transfer
Analysis for Finite LMFBR Bundles," PhD. Thesis,
Department of Nuclear Engineering, MIT, (1978)
4.
Ribando, R.J., et al.,"Sodium boiling in a Full-Length
19-Pin Simulated Fuel Assembly (THORS Bundle 6A)",
ORNL/TM-6553 (1979)
5.
Rothrock, R.B. and Henderson, J.M.,"Description and
Results of the W-1 SLSF Experiment Fuel Pin
Heat Release Characteristics and Sodium Boiling
Stability", Hanford Engineering Development
Laboratory, Energy Contract No. EY-76-C-14-2170
6.
Schrage, R.W., A Theoretical Study of Interphase Mass
Transfer, Columbia University Press, N.Y. (1953)
7.
Collier, J.G., Convective Boiling and Condensation,
McGraw-Hill, United Kingdom, 1972
8.
Wilson, G., and Kazimi, M.S., "Development of Models
for the Sodium Version of the Two Phase, Three
Dimensional Thermal Hydraulics Code THERMIT,"
MIT-EL-80-010
9.
Knight,D.D., "SLSF W-1 LOPI Experiments Preliminary
Evaluation of Data," ST-TN-80015, October 1979
10.
Martin,R.S., and Wilkinson, J.H., "Solution of
Symmetric and Unsymmetric Band Equations and the
Calculation of Eigenvectors of Band Matrices,"
Numerische Mathematic 9(4) 1967
11.
Franklin, Joel N., Matrix Theory, Prentice Hall. Inc.,
Englewood Cliffs, N.J. (1968)
-156-
12.
Bell, W.W., Matrices for Scientist and Engineers, Van
Nostrand Reinhold Co., New York, N.Y. (1975)
13.
Nero, Anthony V., A Guidebook to Nuclear Reactors,
University of California Press, Berkley and Los
Angeles, California (1979)
14.
Weast, Robert C. et al., CRC Handbook of Chemistry and
Physics, The Chemical Rubber Company, Cleveland,
Ohio (1969)
-157-
Appendix A
SPECIFIED INLET VELOCITY AND MASS FLOW RATE
A.1
Introduction
it
In the simulation of sodium boiling transients,
necessary
to
establish
the
proper
flow rate through the
bundle in order to obtain a temperature
corresponds
to
the
distribution
experimental results.
could only be accomplished by specifying
the
is
which
Until now, this
the
pressures
at
inlet and at the outlet, and allowing the AP across the
core to
determine
associated
with
the
this
mass
method
flow
rate.
One
difficulty
is that the pressures at the
inlet and outlet are not always provided, and when provided,
they often do not specify the effects of pressure drops
to
valve
throttling
and
fuel
pin
spacers.
due
Another
difficulty is that the flow rate is very sensitive to the AP
across the core, and thus
specified
pressures
any
can
small
lead
inaccuracies
to
large
in
flow
discrepencies.
This has resulted in the use of a trial
error
to
process
the
rate
and
determine the necessary inlet and outlet
pressures.
An alternative to this method would be to
velocity
of
the
fluid
specify
the
or the total mass flow rate at the
inlet, and infer the required inlet pressure from this.
advantage of these methods is
that
the
proper
flow
The
rate
-158could always be maintained.
This
appendix
will
describe
how
the
/
relationship
between pressures and fluid velocity is treated in NATOF-2D,
and
show
how these equations can be modified in order that
an inlet velocity or an inlet mass flow rate can be imposed.
A.2
Treatment of the Momentum Equation
As described in the introduction to Chapter Four,
to
function
of
This
cells.
between
pressure field solution is accomplished by
a
velocities
treating the
relies on only the
NATOF-2D
interactions
pressures relating the
reduction
by
employed
scheme
solution
the
at
a
cell
as
only
boundary
a
pressures in the two neighboring cells in
the
the momentum equation, and then substituting these relations
into the
mass
section
will
describe
equations to obtain this result.
appendix
inlet,
is
only
the
on
Since the
calculations
of
liquid
z-direction
the
of
treatment
the
This
equations.
conservation
energy
and
the
focus
momentum
this
of
the velocity at the
and
vapor
momentum
equations need be considered.
The time discretinized, finite difference form
momentum
equation
for
the
of
the
vapor and liquid phases in the
z-direction are:
vapor phase
n
(Un+l- U )
+J
2vz
rvz-
(
+
(A Un
v
Uvz i+j
n+l
n
(ap v)i+j
g
=
-(wz
+ Mvz i+j
+
pn
(A.1)
-160-
liquid phase
+n
n
n+l
((l
(
-
-
(ArU z)n
+U
tr i+&
Ar
+ ((1 - a)p )n
un(A)
)
+
+
g
Un)(AU
=
(-_)n
-(Mwz
z
(pn
i
+
_ pnj)
Azi+1
- M£v
(A.2)
where the interface momentum exchange terms are given by:
Mvt
z
=
(K + pr)n(Uvz-
M vz
=
(K-
Uz
(1-n)r)n(Uvz-
) n+ l
Utz
'
(A.3)
)n+
1
(A.4)
and the wall friction terms are given by:
M
=
n n .un+l
fV *Uvz U vz
(A-5)
M
=
n n
fn*U
*Un+l
(A.6)
wz
A detailed description of the donor cell technique used to
n
evaluate the terms AUv+ 4 , etc. is given in Reference 1.
Since
these
terms
focus of this section
variables,
the
are
is
technique
treated
the
used
explicitly while the main
treatment
of
the
implicit
need not be repeated.
locations used to evaluate the terms
are
shown
in
The
figure
A.1.
For greater clarity, equation A.1 can be rearranged
so
ELIlliI
-161-
1+1
2
ij-I 2
Figure A.1
1+1 i+
+,j+
+:
Positions Used for the Evaluation of
Variables
-162-
that
When
all implicit terms appear on the left hand side.
the relations given by equations A.3 and A.5 are substituted
the result is
into equation A.1
At
(v
___
+ f vUvz + (K+nr)
V
At
-(K
n
Uvz i+
+
j
n
ai++j
n+l
+ nr)Unz i+ j
has
_ p
i+1LAzi+
iL
(A.7)
= EXPLICIT TERMS
As can be seen, the specific choice of
discretinization
n+l
n+l
-&jAz1+
V
VZ
(p
the
time
step
made the velocities only dependent on
equation
n+1
corresponding to A.7 for the liquid phase, the term Utz i++j
the
pressures.
Since
there
is
also
an
n+l
n+l
the
eight
Pi+lj,
be replaced with a term only dependent on Uv,
n+1
. Thus knowledge of any two variables in equation
and P
can
A.7 allows the calculation of the third.
The iteration scheme for
conservation
the
solution
equatons (2 mass, 2 energy, and 4 momentum) is
an extension of the Newton iteration solution
A.1
of
algebraic
The equations are cast in a form similar to that
equations.
of
of
and
A.2,
and
then
a vector X of the unknowns is
defined such that
n+1
X
=
, P, T , T v , Uvz, Uvr, Uz, UZr
(A.8)
The finite difference equations can be written in the form
F (X) = 0
where
i
I
---
III
i
-163p = 1,2,...8
At an iteration k,
be
F (X ) will not equal zero in general, so this is
obtained.
not
will
solution
approximate
an
A Taylor expansion around the point
an exact solution.
Xk is made to obtain
8 aF,
3 PX
Fp(Xk) +
F (Xk+1)
is
8 aF
k
X)k
k+l
q
(A.10)
q
q=1
If Xk +
(X
required to be the solution of A.9 then
k+l
-X) k
*(Xq
xqX
q
q
k+l
Defining 6X = Xq
q
k
Xq
-
-F (Xk )
=
p
(A.11)
be
can
A.11
equation
written
explicitly for the vapor and liquid momentum equations.
The
result is:
vapor phase
(apv) i+J.6U
v
At
aUv
+
+
aMu
vz
z
*Z U
(i+lj
z
1
£3,
-
(
J)
-
=
aMw v6U
+ auvz
-F 1
(A.12)
6a
liquid phase
((l-a))i+
+
aMv
a AvUz
, *6Z
U
aM
P
vz
-
i+ lj
Az)
iz
At
=
) + M
Pij
-F 2
aZ
(A.13)
Solving equations A.12 and A.13 for the velocities yields
6Uvzi+fJ
= Wvi+
(6Pi
vziaji+lj
-
6Pij)
+ Rvz
(A.14)
-164-
U z
=W
( 6 Pi+lj - 6Pij
i
) +
Rz
(A.15)
Rzi+}j
where
Wv(l-)p
vzi+ltj
=
a
+
-A-z At
(
=v +4+
At
aM
M
aM
(-wz
Wz
U
Uv
At
Mvz
3z
U9kz
zz
+ 3Mwzi +
(1-a)p
Mv
MMwz
VMv +(z
Uvz
z
aUz
Vz
z
Mvz
Uzz
(1-a)
z
J
AzMv
+
z
Mv z ]
3Mv
-M
aUz
3Uvz
Rvz =
M
[
aU z
At
+ z
+auV
vz
M vz
wzR +
r(l-+)
F
Mvz
+
vz
au
_
aMv
z
mvz mvz
(l-a)P, + Mwz
aU
At
+U
£z
+Mtvz)
z
Uv----)
and similarly for the liquid phase.
Equation A.14 and A.15 are the
equations
which
form
of
the
will be used in the following two sections
to descriibe the inlet velocity and mass flow rate
conditions.
momentum
boundary
Inlet Velocity Boundary Condition
A.3
An inlet velocity boundary condition refers to
is
across
constant
user
velocity, given as a function of time, which
fluid
defined
a
inlet.
bundle
the
the
Only
inlet
velocity and the outlet pressure need to be specified, since
the
pressure
inlet
two
these
With
the
calculate
no longer enters into the calculation.
parameters,
iteration is performed,
the
generate
field
pressure
the
After
distribution.
necessary
pressure
inlet
will
there
the
to
However,
specified velocity can be inferred.
since a constant velocity across the bundle is
general
can
scheme
iteration
the
assumed,
in
be a different inlet pressure for each
cell.
The inlet velocity
Wvz i+4j and Wkz
i+jj in
condition
is
imposed
by
setting
equation A.14 and A.15 to zero, and
then setting
Rvz
=
Uinlet
-
Uvz 1++j
(A.16)
Rz
=
Uinlet
-
UIz 1++j
(A.17)
Since Rv, and Rtz represent the error term from the previous
iteration,
Newton
one can see that as Rvz and Rz go to
iteration converges on the exact solution.
zero,
the
Equation
A.7 is then used to update the boundary pressure, Plj, since
P2.1 , Uvz l+aj, and Ukz 1+.I are known.
The inlet velocity boundary
condition
calculation
is
-166simple to perform, but it has the disadvantage of preventing
localized
flow
reversal
during sodium boiling transients.
The second method, inlet mass flow rate, does not have
restriction.
this
________________________Mi
I...1 i
-167A.4
Specified Inlet Mass Flow Rate Boundary Condition
The method for specifying the inlet mass flow rate
by Andrei L. Schor at the Massachusetts Institute
developed
For
of Technology.
figure
was
A.2,
cell
the
in
and A.2 can be written for the
A.1
equations
shown
scheme
numbering
bottom row of real cells in the form:
P0
)
+
rcj
(A.18)
- P0
)
+
rvj
(A.19)
Uj
=
a j.(Pj -
Uvj
=
avj'(P
j
J = 1,..nJ
The mass flow rate of each phase into the cell is given by:
W
=
Aj.(1 - c)pU
(A.20)
W
=
Aj*a*pU
(A.21)
where
W
=
liquid mass flow rate into cell j
Wvj = vapor mass flow rate into cell j
Aj
= flow area of cell j
The total mass flow rate into the cell, Wj, is:
Wj
W
WYj
=
+
A (1 -
A
ap
(a
Wvj
a)p,(a2 j(Pj -
(A.22)
PO)
(Pj - P0 ) + rVj)
+ r
j)
+
(A.23)
-168-
I
I
U
"i
---a
1*I>
IL
I
r
I
I
1v
>1
03
CI0
I--
---
I
Figure A.2
A
Example Cell Numbering Scheme for Flow
Boundary Cal culation
-169-
Wj
=
A ((1 -
a)paj
+ aPvavj)(Pj -
Aj((1 - a)ptrij
P0)
+
+ apvrvj)
(A.24)
And the total mass flow rate into the bundle then becomes:
nJ
WT
-
W
(A.26)
j=1
Defining
aj = A ((1 - a)pa
rj = A ((1 - a)pr
j
+
ap aj)
(A.27)
+
apvrvj)
(A.28)
WT is then given by:
nJ
WT =
nJ
I a
(P
-
)
P
+
(A.29)
J=1
j=1
nJ
=
I rj
nj
,
ajPj
-
P
nj
,
aj
+
j=1
j=1
I rj
(A.30)
j=1
Rearranging equation A.30 yields
-jaj1
P0 + alP +
"
+ a P
=
rj
(A31)
j=1
j=1
When written in incremental form, equation A.31 becomes
nj
-a
J
6P
j=1
Thus,
condition,
+
aSP
for
the
there
equation to solve.
field
+
1
1 1
a 26P
2
2
2
specified
will
be
an
6P
+
a.
mass
flow
nj
nj
additional
This equation is added to
matrix described in Chapter 4.
=0
rate
(A.32)
boundary
pressure
the
field
pressure
For the configuration
of cells shown in figure A.3, the resulting matrix is
shown
in figure A.4.
The coefficients c 0 1,c
0 2
and c 0 3 are given by
equation
-170-
7
8
9
4
5
6
1
2
3
0
Figure A.3
Cell Configuration for Matrix Shown in
Figure A.4
-171-
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.
0
0
0
a4 , 7
0
0
a5, 5
a5 ,6
0
a5 , 8
0
a6,
a6,
0
0
0
a7, 4
0
a7, 7
a7, 8
0
0
0
a8 , 5
a8 , 7
a8,
0
0
0
0
0
a9 , 8
0
1
C01
C0 2
C10
al,1
a 1 ,2
C20
a2,1
a2 , 2
a2, 3
0
a2, 5
C30
0
a3,
a3 ,3
0
0
0
a4 , 1
0
0
a4 , 4
a4,
0
0
a5 , 2
a5 , 4
0
0
0
a6 ,3
0
0
0
0
0
0
0
Figure A.4
C0 3
2
al,
4
a3,
5
5
6
6
0
a9 , 6
a6,
8
9
a8 , 9
ag.
9
Pressure Field Matrix with Flow Boundary
Calculation
-172-
and c30 are the coefficients
The coefficients c1 0 , c
A.8.
Previously, these
of the momentum equation at the boundary.
were
terms
used,
not
a
during
constant
condition,
With
timestep.
these
however,
terms
pressure in the boundary cell to
cells,
and
in
this
boundary
the
since
are
the
the
flow
used
bottom
was
pressure
boundary
to relate the
row
real
of
way the boundary cell pressure can be
updated.
Note that the bandwidth of the matrix remains the same,
and
therefore
the
additional
cpu
are
requirements
negligible.
This method offers an advantage over a specified
velocity,
it
since
allows the boundary pressure to adjust
itself to the conditons prevalent in the bundle.
principle,
it
is
inlet
possible
to
Thus,
in
have flow reversal in some
channels, while still maintaining a net positive flow.
.
.
.
.
..
. .
-I i illlllll ll iIIIhIl
ii lllii
-173-
A.5
Programming Information
inlet
The boundary condition at the
specified
is
by
setting the input parameter, itbd, to -1, 0 or 1 to indicate
a velocity, pressure, or flow boundary condition.
For the velocity boundary condition,
the
inlet
subroutine
is
and
input,
BC.
These
the
values
velocities
are
velocity
the
passed
at
are updated in
subroutine
to
ONESTP, where the differnece between Ub and Uj (i = 1,nj) is
used
to
calcualte
AU, rather than AP in the boundary cell
momentum equations.
For the flow boundary condition, the desired flow input
for the timestep is used along with
previous
timestep
to
get
a
the
first
pressures
estimate
of
for
subroutine BC (that is, solving equation A.31 for PO).
new
PO
PO
in
The
is passed to subroutine ONESTP, where it is treated
the same as a pressure boundary condition.
is
the
The
difference
that 6Pois calculated in subroutine DIRECT, and added to
PO to get
a
better
estimate.
convergence is attained.
This
is
continued
until
This option is only available when
the direct method pressure field solution is used.
-174-
Appendix B
**********
N A T 0 F - 2 D -
INPUT DESCRIPTION
**********
SECTION I
The following cards are read via namelist input. A
total of four namelists are used: "restrt", "unos", "duos",
and "tres".
The input should look like:
$"namelist" fl,f2,f3,...,fn,$end
for each namelist, where each fI is a field consisting of:
all blanks, or
name = constant, or
of constants.
name = list
The order of input is immaterial; as many cards as
needed may be used; the $end signifying the end of the
namelist input should appear only on the last card, for
each namelist.
For additional details on the use of namelist input,
the user is referred to a standard fortran manual.
Group
No.
1
Contents
Format
namelist
$restrt, nres
nres = steady state or transient indicator
(1/0)
2
namelist
,ncf
,nj
,ni
igauss, dtmax ,dtmin ,epsl
tset ,indgs ,sprint, itbd
$unos
,ncld
,itml
,eps2
,$end
,nset
ni
= number of mesh cells in the axia 1
nj
= number of mesh cells in the radi al
direction
ncf
ncld
itml
=
=
=
igauss =
direction
number of mesh cells in the fuel
number of mesh cells in the clad
maximum number of iterations in the
Newton iterative solution
maximum number of iterations in the
pressure problem solution
(only necessary when indgs = 1)
NIN
I
-175-
indgs
= indicator for direct or gausian
dtmax
solution (0/1)
= maximum value of the time step
dtmin
= minimum time step increment
epsl
= convergence criterion for the Newton
increment
iteration (N/m-2)
= convergence criterion for the
pressure problem (N/m-2)
sprint = time interval between monitoring
prints
itbd
= inlet boundary condition number
-1 = velocity boundary condition
0 = pressure boundary condition
1 = flow boundary condition
eps2
The following two corresponding cards can be
incremented from i = 1 to 40, and control
The code will print nset
the printed output.
times the flow map at an increment of tset.
nset(i) = number of printouts
tset(i) = time between printouts
3
namelist
$duos
apu
tinit
$end
,nrow
,dil
,ntcd
,pitch
,radr
,ip
,d
,e
,ad
,
,thc
,thg
,iss
,
,rnusll,rnusl2,alpdry,
= number of rows of fuel pins in the
fuel assembly
pitch = distance between fuel pin
centerlines (m)
= diameter of. the fuel pin (m)
d
e
= minimum distance between fuel pin
surface and hex can wall (m)
ad
= fraction of theoretical density
of fuel
apu
= fraction of plutonium in the fuel
dil
= fraction of helium gas in gas
compositon
= fuel pin outside radius (m)
radr
the
= clad thickness (m)
thg
= gap thickness (m)
= transient or steady state
iss
indicator (0/1)
tinit = initial starting time (sec)
ntcd
= number of boundary condition cards
= partial or full boundary
lp
calculation (0/1)
rnusll = effective nusselt number for radial
nrow
-176heat conduction -- inner cells
0 > implicit calculation
0 = bypass calculation
0 < explicit calculation
rnusl2 = effective nusselt number for radial
heat conduction -- edge cells
alpdry = dryout void fraction
The following cards are required only for an initial
start, and appear at the very end of the input file.
3
namelist
,pout
,tin
,tav
,$end
$tres
,pin
pin
= initial inlet pressure to the fuel
assembly (N/m2)
= initial outlet pressure to the fuel
assembly (N/m2)
= initial inlet temperature of the
pout
tin
tav
coolant (K)
= average temperature of the fuel
assembly (K)
SECTION II
The following cards are read via NIPS free-format
input processor. Fields are separated by blanks. Entry
(or group of entries) repetition is allowed; for example
n(a b m(c d e ) f ) where: a,b,c,d,e,f are entries (integer
or real) and n,m are integers representing the number of
repetitions; note that no blanks must appear between a left
parenthesis and the integer preceding it. Up to 10 levels
of nesting are permitted.
The end of a group is marked by a $-sign.
The following cards govern the boundary conditions of
the problem as a function of time. These cards are always
required. The order of input must be maintained.
Those cards marked with a * are necessary only for a full
1)
boundary calculation (lp
calculation (lp = 0), the
boundary
For a partial
follows:
as
boundary is calculated
X = X1(L)*dtime + X2(L)
and for a full boundary calculation:
X = (X1(L)*dtime + X2(L))*exp(OMX(L)*dtime)
+ X3(L)
~I~
--
I
IIIYIY
iiIII1~
-177where
dtime =
L =
tb(L) =
X1, X2,
time - tb(L-1)
index of currect time segment ,
time at the end of segment L
X3, OMX = input parameters
-
= time at the end of a time segment
1
tb(ntcd)
2
bnbl(ntcd) = velocity at inlet (m/s) (itbd = -1)
= pressure at bottom of fuel
assembly (N/m2) (itbd = 0)
= flow at inlet (kg/s) (itbd = 1)
bnb2(ntcd)
bnb3(ntcd)
omb(ntcd)
3
pntl(ntcd)
pressure at top of fuel assembly
(N/m2)
pnt2(ntcd)
pnt3(ntcd)
omt(ntcd)
4
albl(ntcd)
void fraction at the inlet of the
fuel assembly
alb2(ntcd)
alb3(ntcd)
oma(ntcd)
*
*
tvbl(ntcd) = vapor temperature at the inlet (K)
tvb2(ntcd)
tvb3(ntcd)
omv(ntcd)
*
*
tlbl(ntcd) = liquid temperature at the inlet(K)
tlb2(ntcd)
tlb3(ntcd)
oml(ntcd)
*
*
hnbl(ntcd) = power density in
hnb2(ntcd)
hnb3(ntcd)
omh(ntcd)
5
6
7
the fuel pins (w/m2)
The following cards are always required for
a steady state calculation, but not for a transient
calculation.
The dimensions are given by:
ni
= number of axial cells
nj
= number of radial cells
npin = ncf + ncld + 2
nn
= ni*nj
8
n(19)
= row numbers where the boundary
-178-
between cell J and cell J + 1 lies
(n
9
10
11
12
13
dz(ni)
tcan(ni)
shape(nn)
sppd(nn)
ppp(npin)
14
iplnm(ni)
= 2,...)
=
=
=
=
=
axial mesh spacing of cells (m)
heat capacity of hex can per unit area
power density shape in fuel assembly
spacer pressure drop
radial power profile inside fuel pin
(fuel, gap and clad)
= axial composition of the fuel pin
0 = gas compositon
1 = mixed oxide U,Pu02
RESTART
OPTION
For a restart of a previous calculation, the namelists
restrt,unos and duos are required, for a selected number of
cards.
For namelist restrt, the following previously defined
card is required:
nres = 0
For namelist unos, the following previously defined
cards are required:
nset(i)=
tset(i)=
The following cards are optional inputs:
epsl
,eps2 ,dtmax ,igauss,itml
,indgs
dtmin ,sprint,itbd
For namelist duos, the following previously defined
cards are required
lss
tinit
= 0
=
ntcd
Ip
=
rnusll
rnusl2 =
alpdry =
Boundary condition cards are always required.
As of September 1, 1981
-
lII
ll
-179Appendix C.1:
Westinghouse Blanket Heat Transfer
Test Program Run 544
Srestrt
ntype
rPes
=
0
= 1
Seand
Sunos
sprint = 0.1
ni
= 14
nj
= 5
ncf
ncld
= 4
= 2
itmi
= 8
indgs
= 0
igauss
dtmax
dtmin
epsl
eps2
nset
tset
Send
$duos
rnusl1
rnusl2
tsr
nrow
pitch
d
a 1
S
ad
apu
dil
=
=
=
=
a 1.0
a 1.e-8
a 10.
= 0.001
3 1
= 3.0
= 22.0
z 28.0
= 0.0
= 5
= 1.4216e-2
= 1.32e-2
98.60e-4
0.95
0.0
0.9
radr
= 6.60e-3
the
a 8.61e-4
thg
• 1.36e-4
= 1
Iss
tinit = 0.0
a 1
ntcd
Ip
a 0
Send
$tb
3.5
Spnbl
.0
Spnb2
1.888ae5
Spntl
0.0
$pnt2
1.5e5
Salb1
0.0
$alb2
0.0
Stvbl
0.0
Stvb2
569.14
Stlbi
0.0
Stlb2
589.14
Shnwl
0.0
Shnw2
a.930e7
Sn
2 3 4 5 15(0)
Sdz
14(.1633)
Stcan
14(6500.)
5(2(0.0) .3a .71 .92 1.0 .92 .71 .38 5(0.0)) Sshape
Ssppd
4(0.0 61.0 12(0.0)) 0.0 89. 12(0.0)
4(1.0) 4(0.0)
Sppp
Slplmn
9(1) 5(0)
Stres
pin
= 1.888e5
pout
a
tin
tav
Send
= 589.14
= 700.0
1.5e5
-180Appendix C.2:
Westinghouse Blanket Heat Transfer
Test Program Run 545
Srestrt
ntype
z 0
nres
Send
Sunos
= 1
sprint = 0.1
ni
nj
ncf
ncld
= 14
= 5
= 4
= 2
itmi
indgs
igauss
dtmax
epsl
=
=
=
=
=
=
eps2
= .001
nset
tset
=
a
dtmin
8
0
100
1.0
1.e-8
10.
1
3.0
Send
$duos
rnusll
= -22.0
rnusl2 a -23.0
tsr
= 0.0
nrow
= 5
pitch
d
e
ad
apu
dil
radr
the
thg
Iss
tinit
ntcd
=
=
=
=
=
=
=
=
=
=
=
Ip
Send
3.5
0.0
1.833e5
0.0
1.421E5e-2
1.32e- 2
8.60e--4
0.95
0.0
0.9
6.60e--3
8.61e--4
1.36e--4
1
0.0
a 1
= 0
1.5e5
0.0
0.0
0.0
589.14
0.0
589.14
0.0
8.3765e7
2 3 4 5 15(0)
14(.1633)
14(6500.
2(0.0) .31 .58
.76 .82
.76
2(0.0) .34 .64
.83 .91
.83
2(0.0) .39 .72
.93 1.02
.93
2(0.0) .42 .79 1.02 1.11 1.02
2(0.0) .46 .87 1.12 1.22 1.12
5(0.0 61.0 12(0.0))
4(1.0) 4(0.0)
9(1) 5(o)
Stres
pin
;out
tin
tav
Send
=
=
=
=
1.838e5
1.5e5
5a9.14
700.0
Stb
$pnbl
Spnb2
$pntl
$pnt2
Salbl
Salb2
Stvbl
$tvb2
Stlbl
$tlb2
Shnwl
Shnw2
Sn
Sdz
Stcan
.31 5(0.0)
.34 5(0.0)
.39 5(0.0)
.42 5(0.0)
.58
.64
.72
.79
.87 .46 5(0.0)
$shape
Ssppd
Sppp
Slplmn
Appendix C.3:
THORS Bundle 6A Test 71H Run 101
- Steady State
Srestrt
nres
Send
$unos
sprint
ni
nj
ncf
ncld
itmi
indgs
igauss
dtmax
epsi
eps2
nset
tset
itbd
1
a
a
=
*
*
a
*
*
*
-
=
i
*
=
8.
42
3
4
2
8
0
0
1.0
11.
0.002
5
1.2
-1
Send
Sduos
rnusll a -13.
rnusl2 = -13.
nrow
U 3
pitch
-
7.265e-3
d
e
ad
apu
dil
a
*
*
a
5.842e-3
7.113e-4
0.95
0.0
0.9
radr
w
2.921e-3
the
thg
*
0.381e-3
0.6e-4
1
0.0
I
0
a
a
Iss
tinit
ntcd
Ip
*
a
*
Send
11.0
0.0
1.02504
0.0
1.4445e5
0.0
0.0
0.0
660.91
0.0
660.91
0.0
4.90971831e8
2 3 17(0)
7(.05398) 25(.0508) 10(.07232)
7(0.0) 21(650.) 14(0.0)
3( 7(0.0) .43 .515 .64 .73 .815 .885 .94 .98
1.0 1.0 .98 .94 .885 .815 .73 .64 .515 .43
17(0.0))
2(0.0 50. 40(0.0)) 0.0 50. 40(0.0)
Stb
Sfbl
$fb2
Spntl
Spnt2
Salbi
$alb2
Stvbl
$tvb2
StlbI
$tlb2
Shnwl
Shnw2
Sn
$dz
Stcan
$shape
Ssppd
4(1.0) 4(0.0)
Sppp
28(1) 14(0)
Stres
pin
a
1.7277e+5
pout
a
1.4445e+5
$Sp1na
tin
tav
Send
=
*
660.91
900.0
-182-
THORS Bundle 6A Test 71H Run 101
- Transient (0.0 -
11.0 sec)
$restrt
* 0
nres
$end
$unos
= 5
itml
igauss a 100
dtmax
=
1.0
dtmin
= 1.Oe-04
nset(1) a 12
nset(2) - 0
tset(1) = 1.0
tset(2) = 0.0
indgs
sprint
epsi
eps2
itbd
=
a
=
a
=
0
10.0
250.
1.e-2
-1
Send
$duos
rnusi1
= -13.0
rnus12
=
Iss
= 0
tinit
- 0.0
-13.0
ntcd
a 3
a 0
Ip
Send
3.2 6.8 30.0
0.0 -0.197185 0.0
1.02504 1.02504 .315174
0.0 -3347.222 0.0
1.4445e5 1.4445e5 1.324e5
3(0.0)
3(0.0)
3(0.0)
3(660.91)
3(0.0)
3(660.91)
3(0.0)
3(4.90971831e8)
$tb
Sfbl
$fb2
Spntl
$pnt2
Salbi
Salb2
Stvbl
$tvb2
Stlbi
Stlb2
$hnbi
Shnb2
___*
----
Y
-183-
THORS Bundle 6A Test 71H Run 101
-
Transient (11.0 -
20.0 sec)
Srestrt
O
nres
Send
$unos
itmi
igauss
dtmax
dtmin
*
5
*
100
1.0
=
*
nset(1) *
nset(2)
tset(1) tset(2) =
indgs
sprint
*
epsi
eps2
*
itbd
*
1.0e-08
50
O
0.1
0.0
0
15.
250.
1.e-2
0
Send
$duos
rnusl1
rnusl2
Isa
tinit
ntcd
Ip
a -13.0
-13.0
0
0.0
a 3
a 0
a
*
*
Send
5.0 10. 25.
3(0.0)
3(1.6113e5)
3(0.0)
3(1.324e5)
3(0.0)
3(0.0)
3(0.0)
3(660.91)
3(0.0)
3(660.91)
3(0.0)
3(4.90971831e8)
Stb
Spnbl
Spnb2
$pntl
Spnt2
Salbi
Salb2
Stvbi
Stvb2
Stlbi
Stlb2
Shnbl
Shnb2
The flow map output for this problem at Time = 11.3415 sec
appears in Appendix F.
Appendix C.4:
W-1 SLSF LOPI 2A
- Steady State
$restrt
nres
= 1
Send
$unos
sprint
= 0.1
ni
= 40
nj
a 3
ncf
ncld
itmi
= 4
= 2
= 6
indgs
igauss
dtmax
= 0
= 500
= 1.0
epsl
= 10.0
eps2
= 0.005
nset(1) = 1
tset(1) u 6.0
itbd
= -1
Send
$duos
rnus11
rnusl2
= -13.
= -13.
nrow
pitch
3
= 0.7264d-2
d
e
= 0.584d-2
0.711d-3
ad
= 0.954
apu
dil
=
radr
the
thg
Iss
tinit
ntcd
=
=
=
=
=
=
=
0.25
0.9
0.2921d-2
0.381d-3
0.6d-4
I
0.0
1
= 0
lp
Send
8.0
$tb
0.0
Sfbl
5.97742777
$fb2
0.0
Spntl
2.776e+5
$Spnt2
-0.0
Salbl
0.0
$alb2
0.0
$tvbl
661.14
$tvb2
0.0
Stlbi
661.14
$tlb2
0.0
$hnwl
1.907849e+9
$hnw2
2 3 17(0)
$n
6(.05948) 23(.0508) 11(.10033)
$dz
Stcan
29(0.85e+4) 11(0.0)
3( 6(0.0) .66 .835 .98 1.11 1.22 1.3 1.37 .141 1.42
1.41 1.38 1.32 1.25 1.14 1.02 .85 .7 .5 16(0.0)) $shape
2(0.0 0.0 38(0.0)) 0.0 0.0 38(0.0)
$sppd
4(1.0) 4(0.0)
Sppp
$1plnm
29(1) 11(0)
Stres
pin
= 6.0d+5
pout
= 2.776d+5
tin
tav
= 661.14
= 900.0
Send
OMNiYlbh
-185-
W-1 SLSF LOPI 2A
- Transient (0.0 -
4.5 sec)
$restrt
nres
=0
Send
Sunos
Itml
a 5
igauss
= 100
dtmax
a 1.0
dtmin
a 1.0e-06
nset(l) - 45
nset(2) - 0
tset(1) - 0.1
tset(2) = 0.0
indgs
sprint
epsl
eps2
itbd
= 0
a 0.1
= 70.0
= 1.e-2
= -1
Send
Sduos
rnusl1
a -13.0
rnusl2
Iss
x -13.0
a 0
tinit
* 0.0
ntcd
= 7
a0
Ip
Send
0.33 0.5 0.65 0.85 1.33 3.0 6.0
-13.8911754 -.81961177 1.85778667 1.393341
0.29027917 0.0 0.4180020
5.9774277 1.3933398 1.2540058 1.5326738
1.811342 1.950676 1.950676
2(-2.1356e5) 5(0.0)
2.776e5 2.071252e5 5(1.7082e5)
7(0.0)
7(0.0)
7(0.0)
7(661.14)
7(0.0)
7(661.14)
3(0.0) -8.346839e9 3(0.0)
4(1.907849e9) 3(2.384811e8)
$tb
$vbI
$vbl
Spntl
$pnt2
Salbl
Salb2
Stvbl
Stvb2
Stlbi
Stlb2
Shnbl
Shnb2
-186Appendix C.5:
W-1 SLSF LOPI 4
- Steady State
$restrt
nres
= 1
Send
$unos
sprint
= 0.1
ni
nj
ncf
ncld
itml
=
=
=
=
=
40
3
4
2
6
indgs
igauss
dtmax
epsi
a
=
z
=
0
500
1.0
10.0
eps2
a 0.005
nset(l) = i
tset(1) = 5.0
a -1
itbd
Send
$duos
rnusll
rnusl2
= -13.
= -13.
nrow
pitch
= 3
= 0.726ad-2
d
e
= 0.584d-2
= 0.711d-3
ad
apu
radr
a
=
=
-
the
thg
a 0.381d-3
- 0.6d-4
Iss
a i
tinit
- 0.0
dil
0.954
0.25
0.9
0.2921d-2
ntcd
= 1
lp
=0
Send
8.0
Stb
0.0
$fbi
5.97742777
$fb2
0.0
Spntl
2.776e+5
$pnt2
0.0
Salbl
0.0
Salb2
0.0
Stvbl
661.14
Stvb2
0.0
Stlbi
661.14
Stlb2
0.0
$hnwl
2.0332515e+9
$hnw2
2 3 17(0)
Sn
6(.05948) 23(.0508) 11(.10033)
$dz
29(0.85e+4) 11(0.0)
$tcan
3( 6(0.0) .66 .835 .98 1.11 1.22 1.3 1.37 .141 1.42
1.41 1.38 1.32 1.25 1.14 1.02 .85 .7 .5 16(0.0)) Sshape
2(0.0 0.0 38(0.0)) 0.0 0.0 38(0.0)
$sppd
4(1.0) 4(0.0)
$ppp
29(1) 11(0)
$1plnm
$tres
pin
= 600000.0
pout
a 277600.0
tin
tav
- 661.14
= 900.0
Send
I
1IIiiwill
illI lliiJl,,
dI g
-187-
W-1 SLSF LOPI 4
- Transient (0.0 -
5.0 sec)
$restrt
nres
a 0
Send
Sunos
itmi
Igauss
a 5
* 100
dtmax
dtmin
a 1.0
* 1.0e-08
nset(1) = 30
nset(2) * 6
tset(1)
*
0.1
tset(2) * 0.333333333
indgs
sprint
epsI
eps2
itbd
= 0
* 10.
= 700.
= 1.e-2
*
-1
Send
$duos
rnusl1
rnusl2
Iss
tinit
-13.0
= -13.0
*
0
- 0.0
ntcd
7
=
Ip
Send
0.33 0.5 0.65 0.85 1.33 3.0 6.0
-13.8911754 -.81961177 1.85778667 1.393341
0.29027917 0.0 0.4180020
5.9774277 1.3933398 1.2540058 1.5326738
1.811342 1.950676 1.950676
2(-2.1356e5) 5(0.0)
2.776e5 2.071252e5 5(1.7082e5)
$tb
Svbl
$vbl
Spnti
$pnt2
7(0.0)
Salbi
7(0.0)
7(0.0)
7(661.14)
7(0.0)
7(661.14)
3(0.0) -8.8954752e9 3(0.0)
4(2.0332515e9) 3(2.5415644e8)
Salb2
Stvbl
Stvb2
Stlbi
Stlb2
Shnbl
$hnb2
-188Appendix C.6:
W-1 SLSF BWT 2'
- Steady State
$restrt
nres
= i
$end
$unos
sprint = 6.0
ni
= 40
nj
3
ncf
=4
ncld
= 2
itml
6
indgs
igauss
dtmax
epsi
eps2
nset(1)
tset(1)
itbd
Send
$duos
rnusll
rnusl2
nrow
pitch
= 0
= 500
1.0
= 1000.0
= 0.005
1i
= 3.0
= -1
=
=
=
=
-13.
-13.
3
0.7264d-2
d
e
ad
apu
dil
=
=
=
=
=
0.584d-2
0.711d-3
0.954
0.25
0.9
radr
the
thg
Iss
tinit
ntcd
Ip
Send
8.0
= 0.2921d-2
O0.381d-3
= 0.6d-4
= I
= 0.0
= 1
= 0
$tb
0.0
5.97742777
$fbl
$fb2
0.0
2.776e+5
0.0
0.0
0.0
661.14
$pnti
$pnt2
Salbl
$alb2
$tvbl
$tvb2
0.0
$tlbi
661.14
$tlb2
0.0
$hnwl
9.3999172e+8
$hnw2
2 3 17(0)
Sn
6(.05948) 23(.0508) 11(.10033)
$dz
29(100.) 11(1.0)
$tcan
3(6(0.0) .66 .835 .98 1.11 1.22 1.3 1.37 1.41 1.42
1.41 1.38 1.32 1.25 1.14 1.02 .85 .7 .5 16(0.0)) $shape
2(0.0 30. 38(0.0)) 0.0 30.0 38(0.0)
$sppd
4(1.0) 4(0.0)
Sppp
29(1) 11(0)
$1plnm
$tres
pin
= 600000.0
pout
tin
tav
Send
= 277600.0
= 661.14
= 900.0
^ YYIIIIII
-189-
W-1 SLSF BWT 2'
-
Transient (0.0 - 3.0 sec)
$restrt
a 0
nres
Send
Sunos
ital
z 5
Igauss
dtmax
dtmin
= 100
= 1.0
* 1.0e-06
nset(1)
nset(2)
tset(1)
tset(2)
indgs
sprint
a 27
= 0
- 0.11111111111
= 0.0
a 0
=. 4.0
epsl
a 700.
eps2
= 1.e-2
itbd
Send
Sduos
a -1
rnusll
rnusl2
*
Iss
a 0
tinlt
* 0.0
=
-9.0
-9.
ntcd
a 2
a 0
lp
Send
0.5 5.0
-9.07342871 0.0
5.9774277 1.44071334
-2.1356e5 0.0
2.776e5 1.7082e5
2(0.0)
2(0.0)
2(0.0)
2(661.14)
2(0.0)
2(661.14)
2(0.0)
2(9.3999172e+8)
Stb
$Vb 1
Svb2
Spntl
Spnt2
Salbl
Salb2
Stvbl
Stvb2
Stlbl
Stlb2
$hnbl
Shnw2
-190-
W-1 SLSF BWT 2'
- Transient (3.0 -
Srestrt
= 0
nres
Send
$unos
itml
=5
igauss a 100
dtmax
m 1.0
dtmin
a 1.Oe-09
nset(l) z 23
nset(2) - 0
tset(1)
.11111111
tset(2) x 0.0
0
indgs
sprint x 10.
epsl
- 700.
eps2
= 1.e-2
itbd
$end
$duos
=0
rnusll
rnusl2
= -11.0
= -11.0
hss
a 0
tinit
= 0.0
ntcd
= 3
Ip
='0
Send
2.5 3.0 7.0
0.0 6.9528e+5 0.0
2.1087e5 2.1087e5 5.5878e5
0.0 2.1356e5 0.0
1.7082e5 1.7082e5 2.776e5
3(0.0)
3(0.0)
3(0.0)
3(661.14)
3(0.0)
3(661.14)
3(0.0)
3(9.399917e+8)
$tb
Spnbl
Spnb2
Spntl
$pnt2
SalbI
$alb2
Stvbl
Stvb2
StlbI
$tlb2
$hnbi
Shnb2
5.5 sec)
-191Appendix C.7:
W-1 SLSF BWT 7B'
-
Steady State
Srestrt
nres
= 1
Send
Sunos
sprint a 10.
ni
= 40
nj
ncf
ncld
Itml
indgs
a 3
= 4
a 2
6
a 0
igauss
a 500
dtmax
= 1.0
epsl
= 1000.0
eps2
= 0.005
nset(1)
1I
tset(1) - 3.0
itbd
=
-1
Send
$duos
rnusli
rnusl2
a
nrow
pitch
d
a 3
a 0.7264d-2
a 0.584d-2
e
ad
apu
- 0.711d-3
= 0.954
- 0.25
a 0.9
dil
radr
the
thg
Iss
-13.
= -13.
a 0.2921d-2
m 0.381d-3
w 0.6d-4
0 1
tinit
U 0.0
U 1
ntcd
a 0
Ip
Send
8.0
0.0
6.06976171
0.0
2.776e+5
Stb
$fbi
Sfb2
Spntl
Spnt2
0.0
Salb
*
0.0
$alb2
0.0
$tvbl
661.14
$tvb2
0.0
Stlbi
661.14
Stlb2
0.0
Shnwl
1.7860674e+9
$hnw2
Sn
2 3 17(0)
$dz
6(.05948) 23(.0508) 11(.10033)
Stcan
29(100.) 11(1.0)
3(6(0.0) .66 .835 .98 1.11 1.22 1.3 1.37 1.41 1.42
1.41 1.38 1.32 1.25 1.14 1.02 .85 .7 .5 16(0.0)) $shape
Ssppd
2(0.0 30. 38(0.0)) 0.0 30.0 38(0.0)
4(1.0) 4(0.0)
Sppp
$1pinm
29(1) 11(0)
Stres
pin
a 600000.0
pout
* 277600.0
a 661.14
* 900.0
tin
tav
Send
-192-
W-1 SLSF BWT 7B'
- Transient (0.0 -
$restrt
nres
$end
Sunos
itml
igauss
dtmax
=0
= 5
= 100
a 1.0
dtmin
a 1.0e-06
nset(1)
nset(2)
tset(1)
tset(2)
indgs
sprint
=
15
=
=
=
0
epsl
a 700.
eps2
= 1.e-2
itbd
$end
$duos
rnus11
z -1
=
-13.0
rnusl2
=
-13.0
Iss
= 0
tinit
= 0.0
0.1
0.0
0
= 2.
ntcd
= 2
= 0
Ip
$end
0.5 5.0
-7.6025298 0.0
6.06976171 2.2684968
-2.1356e5 0.0
2.776e5 1.7082e5
2(0.0)
2(0.0)
2(0.0)
2(661.14)
2(0.0)
2(661.14)
2(0.0)
2(1.7860674e+9)
$tb
Svbl
$vb2
$pntl
Spnt2
$albl
Salb2
$tvbl
$tvb2
$tlbi
$tlb2
$hnbl
$hnw2
1.5 sec)
0 4-111il,
U
-193-
W-1 SLSF BWT 7B'
-
Transient (1.5 -
5.0 sec)
Srestrt
nres
= 0
Send
Sunos
itmi
a 5
igauss
dtmax
dtmin
nset(1)
nset(2)
tset(1)
tset(2)
indgs
sprint
epsi
eps2
a 100
a 1.0
a 1.06-09
* 45
- 0
a 0.05
= 0.0
= 0
= 10.
- 2000.
* 1.e-2
itbd
Send
$duos
a 0
rnusl1
rnusl2
a -11.0
u -11.0
alpdry • .0.957
= 0
Iss
tinit
a 0.0
ntcd
= 3
Ip
a 0
Send
2.0 2.5 7.0
0.0 6.6306e+5 0.0
2(2.3684e5) 5.6837e5
0.0 2.1356e5 0.0
1.7082e5 1.7082e5 2.776e5
3(0.0)
3(0.0)
3(0.0)
3(661.14)
3(0.0)
3(661.14)
0.0 -3.214921e9 0.0
2(1.7860674e+9) 1.7860674e+8
Stb
Spnbl
Spnb2
$pntl
Spnt2
$albl
Salb2
Stvbl
Stvb2
$tlbI
Stlb2
Shnbl
Shnb2
-194-
Appendix D
NATOF-2D HEXCAN MODEL
In NATOF-2D, the user specifies the
the
hexcan
per
heat
capacity
of
area (J/m2 -OK) for each axial level.
unit
Presently, this value is estimated by the user based on
properties
of the can.
limitations
properties
since
or
the
However, this simple model has many
it
cannot
dimensions
take
into
account
varying
of the hexcan, or heat losses to
the environment.
The value chosen for the hexcan
heat
capacity
has
a
pronounced effect of the temperature profile of the coolant.
As
example,
an
different
two
values
simulation
chosen
cases
of
the
for
D.1
gives
the
run
hexcan
the
Chapter 6) and the input
Table
were
can
test
be
heat
The
Nu 1 = 13.
radial
capacity.
for
The
found
in
Appendix
C.6.
liquid temperature at the end of the
heat
and Nu 2 = 13.
NATOF-2D
was BWT 2' (described in
heated zone for the central channel
time.
with
conduction
at
various
nusselt
points
of
numbers were
__
_.__
illililillhl
1"11
-195Table D.1
Liquid Temperature (oC)
Time (sec)
at Various Points in Time
HCAN = 100
HCAN = 8500
0.0
538.
538.
0.5
580.
578.
1.0
661.
649.
1.5
735.
708.
2.0
790.
756.
2.5
832.
794.
As can be seen, an inaccurate choice of the hexcan
capacity
can
cause
large
heat
liquid temperature differences.
Thus, in two-phase transients, boiling inception time can be
drastically altered.
-196-
Appendix E
SPACER PRESSURE DROP MODEL
The spacer pressure drop feature of NATOF-2D allows the
user to simulate pressure drops which occur
due
to
valve
throttling,
spacer
in
the
wires, etc.
bundle
The spacer
pressure drop is calculated as follows:
AP = SPPD * p
U 2
2
where SPPD is specified for each cell.
Flow reversal occurs in NATOF-2D when the
the
first
real
pressure
cell exceeds the pressure at the boundary.
Thus, by specifying a large boundary pressure, and
value
of
SPPD,
in
it
is
possible
a
large
to prevent flow reversal
during boiling transients.
As a sample test of this feature,
in
which
a
constant
flow
simulations were
of 2 kg/sec was imposed on the
bundle, with an outlet pressure of 2.5 bars.
parameters
used
were
from
run
the
The
geometric
W-1 SLSF Experiment.
value of SPPD for the bottom row of cells was varied over
The
a
wide range, and the inlet pressure necessary to maintain the
flow rate was inferred.
Table E.1 gives values of SPPDs and
the corresponding inlet pressure.
_I_~ _____I__
__
__
IM111W
116
id
-197-
Table E.1
Inlet Pressure vs. SPPD for Constant Flow Boundary Condition
Inlet Pressure (bars)
SPPD
0.0
5.16
10.0
5.26
100.0
6.12
500.0
9.96
As can be
reversal
drop.
time
seen,
it
drops
possible
are
experimental
accurately
determine
flow
outlet
and
inlet
known, the spacer pressure
drop feature can be used to obtain the
rate.
to
by the correct choice of the spacer pressure
Furthermore, if the
pressure
is
correct
inlet
flow
flow map at time =
11.3415 sec.
number of time steps u
161
number of iterations •
206
time step size = 0.26120-02 sec.
cpu time =
712.16
inlet mass flow rate = 0.577033D-01 kg/sec
outlet mass flow rate = 0.7576690+00 kg/sec
total heat transfered x 0.3617300+05 watt
inlet enthalpy flow * 0.503501D+05 watt
outlet enthalpy flow v 0.982316D+06 watt
channel
iz
p
void
(bar)
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
1.3240
1.3502
1.3756
1.4131
1.4484
1.4785
1.4979
1.5115
1.5224
1.5321
1.5408
1.5479
1.5552
1.5629
1.5710
1.5795
1.5882
1.5972
1.6058
1.6139
1.6210
1.6266
1.6303
1.6325
1.6312
1.6293
1.6280
1.6267
0.0025
0.0025
0.0219
0.1406
0.4837
0.7753
0.8795
0.9099
0.9376
0.9447
0.9480
0.9494
0.9494
0.9491
0.9488
0.9485
0.9484
0,9482
0.9472
0,9458
0.9435
0.9397
0.9271
0.7176
0.0169
0.0000
0. 0001
0.0000
number
tv
tl
tsat
twall
---------(degree celsius) -------914.304
914.304
916.493
919.664
922.590
925.046
926.604
927.685
928.546
929.312
929.992
930.547
931.117
931.712
932.334
932.985
933.656
934.336
934.990
935.601
936.137
936.559
936.833
936.993
936.900
939.245
939.215
939.198
723.717
723.717
731.763
739.740
747.438
755.750
764.217
774.945
795.077
825.230
858.958
880.073
892.759
899.557
904.565
913.784
924.738
934.664
935.600
936.620
937.665
938.932
941.047
938.730
915.544
887.411
853.287
812.318
912.005
914.303
916.490
919.664
922.590
925.046
926.604
927.685
928.546
929.312
929.992
930.547
931.117
931.712
932.334
932.985
933.656
934.336
934.990
935,601
936.137
936.559
936.833
936.993
936.900
936.761
936.658
936.565
722.192
722.192
730.203
738.455
746.526
755.160
763.753
773.502
784.533
796.585
809.545
819.322
829.350
839.675
837.319
861.088
887.445
949.574
960.766
971.042
974.239
972.108
957.825
939.721
917.795
890.151
856.202
815.276
uvz
(m/sec)
2.80873
2.93854
3.63601
4.88741
6.84971
9.26300
11.15557
12.64466
14.59305
16.13365
16.77430
17.27383
17.66196
18.03033
18.38645
18.62471
18.68256
18.07009
16.99696
15.25347
12.62162
8.62277
2.39687
-0.19455
0.21688
0.25595
0.23974
0.22233
ulz
(m/sec)
2.78910
2.86145
3.14366
3.54369
3.65853
3.24684
2.81298
2.55882
2.45305
2.53376
2.54972
2.60706
2.68181
2.76169
2.83970
2.89438
2.90893
2.83700
2.70355
2.46241
2.06388
1.41161
0.37249
0.06971
0.22406
0.25595
0.23974
0.22233
uvr
(m/sec)
0.00000
0.00185
0.00474
-0.00620
-0.00175
0.01743
-0.00296
-0.01069
0.00918
0.00266
-0.00716
-0.00413
-0.00271
0.00079
0.00354
0.00364
0.00143
-0.00168
-0.00413
-0.00784
-0.01515
-0.02123
0.00124
0.05584
0.01677
0.00227
-0.00069
-0.00076
J
ulr
(m/sec)
0.00000
0.00185
0.00474
-0.00620
-0.00174
0.01726
-0.00291
-0.01046
0.00892
0.00258
-0.00693
-0.00400
-0.00262
0.00076
0.00343
0.00352
0.00138
-0.00163
-0.00400
-0.00760
-0.01472
-0.02069
0.00122
0.05571
0.01077
0.00227
-0.00069
-0.00076
i-
M
H
X
w
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1.6280
1.6267
1.6255
1.6243
1.6231
1.6220
1.6209
1.6198
1.6187
1.6176
1.6165
1.6154
1.6142
1.6131
1.6120
1.6113
0.000(
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
939.215
939.198
939.136
939.004
938.794
938.507
938.191
937.718
937.517
935.479
935. 645
935.666
935.616
935.541
935.457
387.770
853.287
812.318
765.934
716.213
664.447
611.886
560.093
509.717
462.993
416.119
398.087
390.952
388.616
387.967
387.808
387.770
936.658
936.565
936.474
936.384
936.296
936.210
936.125
936.043
935.962
935.860
935.796
935.712
935.628
935.543
935.457
935.406
channel number
iz
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
p
(bar)
1.3240
1.3502
1.3754
1.4133
1.4484
1.4778
1.4980
1,5119
1.5220
1,5320
1,5410
1.5480
1.5553
1.5629
1.5709
1.5793
1.5882
1.5972
1.6059
1.6141
1.6215
void
0.0032
0.0032
0.0246
0.1954
0.5861
0,8179
0,8626
0,9151
0,9400
0,9403
0.9494
0,9498
0,9505
0,9502
0.9492
0.9484
0.9479
0.9479
0.946!
0.9452
0.9421
tv
-----
856.202
015.276
768.824
718.979
667.045
614.262
562.240
511.548
464.089
416.466
398.212
390.990
388.626
387.969
387.008
387.808
912.005
914.297
916.469
919.684
922.596
924.992
926.613
927.716
928.520
929.305
930.011
930,557
931.124
931.710
932.326
932.977
933.653
934.340
935.000
935.620
936.172
0.23974
0.22233
0.21106
0.20355
0.19773
0.19274
0.18825
0.18441
0.18099
0.18042
0.18100
0.18249
0.18417
0.18467
0.18292
0.18292
-0.00069
-0.00076
-0.00039
-0.00016
-0.00007
-0.00005
-0.00004
-0.00003
-0.00002
0.00001
0.00005
0.00009
0.00010
0.00003
-0.00010
0.00000
-0.00069
-0.00076
-0.00039
-0.00016
-0.00007
-0.00005
-0.00004
-0.00003
-0.00002
0.00001
0.00005
0.00009
0.00010
0.00003
-0.00010
0.00000
uvr
(m/sec)
ulr
(m/sec)
0.00000
0.00251
-0.00313
-0.00152
0.03048
0.02046
-0.06563
0.00142
0.02238
-0.00466
0,00442
0.00128
0.00143
0.00406
0.00571
0.00460
0.00092
-0.00240
-0.00554
-0.01208
-0.02705
0.00000
0.00251
-0.00313
-0.00152
0.03039
0.02028
-0.06462
0.00138
0.02176
-0.00452
0.00427
0.00124
0.00139
0.00392
0,00553
0.00445
0.00089
-0.00232
-0.00537
-0.01173
-0.02630
2
.tI
tsat
twa I
.(degr'ee celsius) --------
914.297
724.332
914.297
724.332
916.472
733.267
919.684
742.515
922.596
751.942
924.992
762.102
926.613
771.577
927.716
783.069
928.520
806.522
929.305
826.719
930.011
864.631
930.557
882.380
931.124
894.599
931.710 ' 902.509
932.325
907.412
932.976
915.767
933.653
925.415
934.340
934.675
935.000
935.603
935.620
936.669
936.172
937.913
0.23974
0.22233
0.21106
0.20355
0.19773
0.192740.18825
0.18441
0.18099
0.18042
0.18100
0.18249
0.18417
0.18467
0.18292
0.18292
722.080
722. 680
731.532
741.160
751.028
.761.498
770. 883
781.281
792.532
805.750
819.832
829.535
839.339
848.931
844.384
866.811
891 .275
948.597
958.983
968.288
90.,950
uv
(m/sec)
2.76187
2.89537
3.70797
5.23751
7.86026
10.16014
10.84887
12.66733
14.66586
15.96186
16.77386
. 17.34419
17.78258
18.14940
18.45993
18.64648
18.67160
18.05545
17.01460
15.26988
12.35471
ulz
(m/sec)
2.73827
2.79878
3.08188
3.54216
3.73747
3.32169
2.80917
2.49359
2.48910
2.57973
2.53586
2.59818
2.66169
2.74732
2.84136
2.90881
2.92745
2.85070
2.71860
2.47962
2.03568
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1.5882
1.5972
1.6059
1.6141
1.6215
1.6273
1.6303
1.6306
1,.6306
1.6293
1.6280
1.6267
1.6255
1.6243
1.6231
1.6220
1.6209
1.6198
1.6187
1.6176
1.6165
1.6153
1.6142
1.6131
1.6120
1.6113
0,9479
0.9479
0.94e
0.9452
0.9421
0.9364
0.9242
0.5223
0.0039
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
933.653
934.340
935.000
935.620.
936.172
936.609
936.830
936.854
936.856
938.798
938.750
938.733
938.668
938.537
938..343
938.088
937.810
937.402
937.206
935.520
935.653
935.665
935.615
935.540
935.457
387.770
925.415
934.675
935.609
936.669
937.913,
939.654
941.021
935.666
913.266
884.788
850.373
809.189
762.942
713.496
661.989
609.711
558.244
508.289
462.128
416.149
396.270
391.063
388.660
387.980
387.811k
387.770
933.653
934.340
935.000
935.620
936.172
936.609
936.830
936.854
936.856
936.755
936.660
936.567
936.475
936.385
936.296
936.210
936.125
936.043
935.962
935.880
935.795
935.711
935.627
935.543
935.458
935.406
channel
iz
42
41
40
39
38
37
36
p
(bar)
1.3240
1.3501
1.3755
1.4134
1.4475
1.4772
1.4999
void
0.0001
0.0001
0.0015
0.0152
0.1216
0.4855
0.8437
tv
tl
-------------914.291
914.291
916.486
919.689
922.519
924.942
926.765
724.102
724.102
734.048
746.893
761.580
776.337
789.181
number
891.275
948.597
958.983
968.288
970.950
968.521
954.083
937. 146
915.593
887.539
853,351
812.186
765.048
716.278
664. G01
612.099
560,399
510.122
463.820
416.502
398.402
391.105
388.672
387.983
387'. 812
387.812
18.67160
18.05545
17.01460
15.26988
12.35471
7.23057
0.56214
0.23920
0.25300
0.23780
0.22265
0.21310
0.20640
0.20090
0.19599
0.19151
0.18741
0.18383
0.18056
0.17989
0.18018
0.18147
0.18371
0.18544
0.18290
0.18290
2.92745
2.85070
2.71860
2.47962
2.03568
1.22169
0.17885
0.23592
0.25495
0.23780
0.22265
0.21310
0.20640
0.20090
0.19599
0.19151
0.18741
0.18383
0.18056
0.17989
0.18018
0.18147
0.18371
0.18544
0.18290
0.18290
0.00092
-0.00240
-0.00554
-0.01208
-0.02705
-0.04532
-0.00820
0.01277
0.00997
-0.00004
-0.00131
-0.00081
-0.00036
-0.00015
-0.00007
-0.00005
-0.00005
-0.00005
-0.00003
0.00001
0.00008
0.00017
0.00025
0.00017
-0.00027
0.00000
3
tcan
tsat
twall
(degree celsius)--------------
uvz
(m/sec)
ulz
(m/sec)
724.102
724.102
734.048
746.893
761.580
776.337
789.181
2.34813
2.32161
2.43372
2.82541
3.63084
5.11298
7.82457
2.34786
2.31818
2.38073
2.53235
2.69108
2.56616
2.14141
912.005
914.290
916.483
919.688
922.519
924.942
926.765
722.663
722.663
732.156
744.564
759.823
775.609
788.647
.
0.00089
-0.00232
-0.00537
-0.01173
0.02630
-0.04420
-0.00814
0.01276
0.00997
-0.00004
-0.00131
-0.00081
-0.00036
-0.00015
-0.00007
-0.00005
-0.00005
-0.00005
-0.00003
0.00001
0.00008
0.00017
0.00025
0.00017
-0.00027
0.00000
I
ru
O
1
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1.5118
1.5214
1.5321
1.5409
1.5480
1 5553
1.5628
1.5707
1.5792
1.5882
1.5973
1.6061
1.6144
1.6221
1.6284
1.6305
1.6303
1.6303
1.6293
1.6280
1.6268
1.6255
1.6243
1.6231
1.6220
1.6209
1.6198
1.6187
1.6176
1.6165
1.6153
1.6142
1.6131
1.6120
1.6113
0.9269
0.9180
0.9375
0.9439
0.9453
0.9472
0.9468
0.9458
0.9447
0.9443
0.9447
0.9442
0.9412
0.9333
0,9067
0.7020
0.2680
0.0019
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
927.713
928.472
929.315
930.002
930.555
931.121
931.702
932.314
932.968
933.651
934.344
935.010
935.642
936,222
936.693
936.847
936.831
936.036
937.786
937.716
937.595
937.452
937.308
937.153
936.976
936.790
936.545
936.377
935.680
935.700
935.675
935.615
935.539
935.458
387.770
806.221
817.640
841.307
868.943
881,914
894.130
899.128
899.633
909.964
921.601
934.564
935.556
936.799
938.643
941.081
937.777
924.139
904.095
874.505
838.230
797.743
753.030
704.672
654.042
602.703
552.252
503.598
458.816
415.774
398.497
391.241
388.736
388.004
387.816
387.770
927.712
928.472
929.315
930.002
930.555
931.121
931.702
932.314
932.968
933.651
934.345
935.010
935.642
936.222
936.693
936.847
936.831
936.636
936.755
936.662
936.568
936.476
936.385
936.297
936.210
936.126
936.043
935.962
935.880
935.795
935.711
935.626
935.542
935.459
935.406
801.344
814.924
829.134
843.314
853.109
861.994
870.457
862.147
880.900
899.784
943.994
953.065
960.568
959.247
946.747
939.050
925.918
906.129
876.635
840.426
800.003
755.304
706.882
656.122
604.603
553.963
505.048
460.152
416.088
398.636
391.291
388.751
388.008
387.817
387.817
806.221
817.640
841.307
868.943
881.914
894.130
899.128
878.998
896.462
913.215
930.023
932.410
933.121
934.261
940.342
937.468
923.782
903.736
874.105
837.852
797.409
752.745
704.431
653.842
602.545
552.133
503.527
458.768
415.774
398.497
391.241
388.736
388.004
387.816
387.816
9.66715
10.81656
12.20306
12.85329
13.38528
13.80544
14.14821
14.47696
14.68455
14.76841
14.38129
13.55669
11.89976
8.51273
2.10520
0.22167
0.28323
0.18297
0.18005
0.18869
0.19229
0.19148
0.18871
0.18541
0.18221
0.17933
0.17694
0.17464
0.17395
0.17292
0.17110
0.16851
0.16678
0.16960
0.16960
1.99827
2.13968
2.12147
2.11556
2.15590
2.19861
2.27296
2.36172
2.43020
2.45266
2.39341
2.29605
2.11027
1.67252
0.65164
0.29299
0.27942
0.18343
0.18005
0.18869
0.19229
0.19148
0.18871
0.18541
0.18221
0.17933
0.17694
0.17464
0.17395
0.17292
0.17110
0.16851
0.16678
0.16960
0.16960
C
-
1
2
3
4 c
5 c
6 c
7 c
8 c
9 c
10
11
12
13
14
15
16
17
18
19
20
21,
22
23 *
24,
25
26
27
28
29
30
31
32
33
34
35 '
36
37
38
39
40
41
42
43
44
45 c
46
47
48
*
*
subroutine read2(p.tv.tl.alfa,uvz,ulz,uvr,ulr.dh,dv,
qst,tr,dtr.tw.sppd.tcan.tinit.dtmax,
np,ntr,npin,npml,nn,.ican,nres.itbd)
this subroutine reads all other information, controls the
writing of input data for a restart, and calculates
parameters which will remain constant throughout the
problem.
implicit real*8 (a-h,o-z)
common /number/ zero,one,big.small
common /bcond/ tb(51),pnbl(51),pnb2(51).pnb3(51),omp(51).
pntl(51),pnt2(51),pnt3(51),omt(51),albi(51),
*
*
alb2(51),alb3(51),oma(51).tvbl(51),tvb2(51),
*
tvb3(51),omv(51),tlbi(51),tlb2(51),tlb3(51),
*
oml(51),hnwl(51).hnw2(51).hnw3(51).omh(51),
*
vbl(51).vb2(5i),vb3(51).omvb(51).flbi(51),
*
flb2(51),flb3(51).omfb(51),1max,lp
common /pshape/ shape(500)
common /dim/ dz(150),dzl(150),dro(150),dr(150),dr2(150)dr3(150),
*
dr4(150),dr5(150),dr6(150),ni,nj.nimlnim2,nJmlnni,
*
nnj,nnjj
common /pinO/ rodr(20),vp(20),vm(20),radr,ppp(20)
common /gconst/ dil,radfu,radcl
common /cconst/ caO,cal,ca2.ca3,cbO,cbl,cb2,cb3
common /fconst/ faO,fal,fa2,fa3,fbO,fbl,fb2,ad,apu,lplnm(150)
common /fconst/ncf,ncc,ng
common /pd/ d4.pod2
common /dryout/ alpdry
common /poverd/ r
common /hxcn/ acov
common /stst/ tafp,lss
common /eccof/ cefl,ccfv
common /extra/ con
common /nussy/ rnusll,rnusl2
common /tbound/ tbc(50).tbmax,tsr
dimension p(nn),tv(nn),tl(nn),alfa(nn),uvz(nn),ulz(nn),
*
uvr(nn),ulr(nn).dh(nn),dv(nn).qsi(nn),tr(ntr),
*
dtr(ntr),tw(np),sppd(nn),tcan(ncan)
dimension rad(20),xin(5),n(20)
namelist/duos/nrow,pitch,d,e,ad,apu,dil,radr,thc,thg.lss,
*
tinit.ntcd,lp,cefl,ccfv,con,rnusll,rnus2,
*
alpdry,tsr,itbc
namelist/tres/pin,pout,tav,tin
faO = 1.81d+06
fal = 3.72d+03
fa2 = -2.510d0
1
2
3-4
5
6
7
8
9
10
ii
12
13
14
15.
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
H
-
H
Z
Z
(D
R
C
I
H
0
0Q)
0
Ca
C
t
O
H
H
49
50
51
52
53
54
55
56
57
58
59
60
.61
62
63
64
65
66
67
fa3 = 6.59d-04
fbO = 10.80d0
fbi = -8.84d-03
fb2 = 2.25d-06
c
caO
4.,28d+06
cal a 3.75d+02
ca2 = -7.45d-03
ca3 = zero
cbO = 16.27d0
cbl = zero
cb2 = zero
cb3 = zero
alpdry = 0.957dO
c
tb(t)
zero
read(5,duos)
c
68
69
79
71
72
73
74
75
76
77
78
79.
80
81.
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
=
c
41
42
43
2
3
call nips(4h tb,4h
if(itbd) 41,42,43
continue
call nips(4h
v,4hbi
call nips(4h
v,4hb2
if(lp.eq.0) go to 2
call nips(4h
v,4hb3
call nips(4h om,4hvb
go to 2
continue
call nips(4h pn,4hbi
call nips(4h pn,4hb2
if (lp.eq.0) go to 2
call nips(4h pn.4hb3
call nips(4h omp,4h
go to 2
continue
call nips(4h fl,4hbi
call .nips(4h fl,4hb2
if,(Ip.eq.0) go to 2
call nips(4h fl,4hb3
call nips(4h om,4hfb
continue
call nips(4h pn,4hti
call nips(4h pn,4ht2
if (Ip.eq.0) go to 3
call nips(4h pn,4ht3
call nips(4h omt,4h
continue
,tdum,tb(2),ntcd,ierr,0)
,idum,vbi(2) ,ntcd,ierr,0)
,idum,vb2(2) ,ntcd,ierr,O)
4950
51
52
53
54
55
56
57
58
59
60
*61,
62
63
64
65
66
67
68
69
70
71
72
73
,idum,vb3(2) ,ntcd,ierr,0)
,idum,omvb(2),ntcd.lerr,0)
,tdum,pnbi(2),ntcd,ierr.0)
,idum,pnb2(2),ntcd,ierr,0)
,idum,pnb3(2),ntcd,ierr,O)
,tdum,omp(2),ntcd,ierr,0)
,idum,flbl(2),ntcd, err,0)
,idum,flb2(2),ntcd,ierr,0)
,idum,flb3(2),ntcd,ierr,0)
0idum,omfb(2),ntcd,ierr,0)
,idum,pntl(2),ntcdierr.0)
,idum,pnt2(2),ntcd,lerr,0)
,idum,pnt3(2),ntcd,.err.0)
,idum,omt(2).ntcd,ierr,0)
77
78
79
80
81
82
83
8485
86
87
88
89
90
91
92
93
94
95
96
I
Ij
CD
97
call nips(4h al,4hbI
,idum,alb1(2) ntcd.ierr.0)
98
call nips(4h al,4hb2
,idum,alb2(2),ntcd,ierr,O)
99
if (lp.eq.0) go to 4
100
call nips(4h al,4hb3
,idum,alb3(2),ntcd,ierr,O)
101
call nips(4h oma,4h
,idum,oma(2),ntcd,ierr,0)
102
4 continue
103
tv,4hbl
.idum,tvbl(2),ntcd,ierr.0)
call nips(4h
tv,4hb2
104
call nips(4h
,idum,tvb2(2),ntcd.ierr,O)
105
if (Ip.eq.0) go to 5
106
call nips(4h
tv,4hb3
.idum,tvb3(2).ntcd. err.O0)
call nips(4h omv,4h
107
,idum,omv(2),ntcd.ierr,0)
5 continue
108
109
call nips(4h
tl,4hbi
,idum,tlbi(2),ntcd.ierr,O0)
110
call nips(4h
tl ,4hb2
.idum,tlb2(2).ntcd,ierr.0)
111
if (lp.eq.0) go to 6
112
tl,4hb3
,idum,tlb3(2).ntcd,ierr,O)
call nips(4h
113
call nips(4h oml,4h
,idum,oml(2),ntcd, err,O)
114
6 continue
115,
call nips(4h
hn,4hw 1
.idum,hnwi(2),ntcd,lerr.0)
116
call nips(4h
hn.4hw2
,idum,hnw2(2),ntcd,ierr.0)
117
if (lp.eq.0) go to 7
hn,4hw3
118
call nips(4h
,idum,hnw3(2),ntcd,ierr,O)
119
call nips(4h omh.4h
,idum,omh(2),ntcd, err,O)
120
7 continue
121
if(itbc.eq.0) go to 225
122
,idum,tbc(1),ltbc,ierr,0)
call nips(4h
tb,4hc
123
itbcmi = itbc - 1
124
tbmax = tsr
125
do 22 1 = 1,itbcml,2
126
tbmax = tbmax + tbc(i)*tbc(i+l)
127
22 continue
128
225 continue
129 C
130
if (nres.eq.1) go to 23
131
read(7.1003)nrow.pitch,d.e
132
write(8,1003)nrow,pitch,d,e
133
call reduml(n,dz,tcan,shape,sppd.ppp,lplnm,ncan,nn.npin,
134
ad,apu,dil,radr.thc,thg.ni)
135
go to 24
136
137
23 continue
138 c
139
call nips(4h
n,4h
,n(1),rdum, 19.ierr,1)
140
call nips(4h dz.4h
,Idum,dz(1).ni,ierr.0)
141
call nips(4h tc,4han
.Idum,tcan(1),ni,lerr,0)
142
call nips(4h sh,4hape .Idum,shape(1),nn. ierr.0)
call nips(4h sp,4hpd ,idum,sppd(1),nn,ierr.0)
143
144
call nips(4h ppp.4h
,Idum,ppp(1),npin.ierr,O)
9798
99
100
101
102
103
104
105
106
107
108
109.
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
,lplnm(1),rduT,ni,ierr,1)
call nips(4h Ipl,4hnm
145
24 continue
146
147 c
Imax=ntcd + 2
148
do 25 ko=l,nn
149
qsi(ko) = (4.*d/(pitch - d))**2
150
25 continue
151
152 c
povd a pitch/d
153
pod2 = povd*povd
154
d4 = 4./d
155
r = -16.15 + 24.96*povd - 8.55*povd*povd
156
1.57 c
= dz(i)
dzi(i)
158
do 111 i - 2.ni
159
dzl(i) = (dz(i) + dz(i-1))/2.0
160
111 continue
161
162 c
al = dsqrt(3.OdO)/2.0
163,
a2 = 3.1415927/4.0
164
w = pitch - d
165 .
16r c
x = (pitch*pitch*al - (d*d + w*w)*a2)/a2/d .
167
168
xi = 4.0/x
xix = x/ai/pitch*6.
169
170 c
do 8 j = 1,njml
171
do 8 1 = 1,ni
172
ko = (J-I)*ni + 1
173
dh(ko) = x
174
dv(ko) m xi
175.
8 continue
176
dr5(1) = xix*(n(1) - 1)*d*a2
177
178 c
do 9 j = 2,njmi
179
180 c
n41 = n(j) - 1
181
n42 = n(j-1) - 1
182
,dn4 = n41*n41 - n42*n42
183
dr4(j) = dn4*x*a2*d*3.0
184
dr5(j) = xix*d*a2*n41
185
dr6(j) = dr4(j)/1.5/(n41 +n42)/w
186
187 c
nx = n(j) - n(J-1)
188
nxl = 2*n41
189
nx2 = (2*n42 + nx)*nx
190
dnxi = nxl
191
drl(j) = dnxi/nx2/pitch/al
192
145146
147
148
149
150
151
152
153
154
155
156
157.
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180'
181
182
183
184
185
186
187
188
189
190
191
192
Ao
C
193
194
195
196 c
197
198
199
200 c
201
202
203
204 c
205
206
207
208 c
209
210
211,
212
213
214,
215
216
217
218 c
219
220
221
222
223224 c
225
226
227
228
229
230 C
231
232
233
234
235
236 c
237
238
239
240
dr2(j)
dro(J)
9 continue
= 2.0*n42/nx2/pitch/al
= pitch*al*nx
dn4 = (n(1) - 1)*(n(1) - 1)
dr4(1) = dn4*x*a2*d*3.0
dr6(1) = dr4(1)/1.5/(n(i) - i)/w
drl(1) = 2.0/pitch/ai/(n(1)-1)
dr2(1) = 0.0
dro(1) = pitch*al*(n(1)-1)
bi = (n(njml) + nrow - 2)
b2 = (nrow - n(njmil))
b3 = (nrow - 1)
xx = bl*b2/2.0 + b3/2.0 + 1.0/6.0
pt = b3*pitch + (d/2.0 + e)/al + a2*d*xx*4.0
ac = (bl*pitch + (d/2.0 + e)/al)*(b2*ptch*al + d/2.0 + e)*
0.50 - a2*(d*d + e*e)*xx
*
y = 4.0*ac/pt
pp = a2*d*xx*4.0
yy = pp/ac
arm = (one - a2/al*(d*d + w*w)/(pitch*pitch))*
(n(njml) - 1)*pitch
*
drl(nj)
dr2(nj)
dro(nj)
dr4(nj)
acov =
= zero
= arm/ac
= b2*pitch + d/2.0 + e
= ac*6.0
(b3*pitch + (d/2.0 + e)/al)/ac
do 10 1 = 1,ni
ko = njml*nt + I
dh(ko) = y
dv(ko) = yy
10 continue
dr3(nj) =
dr6(nj) =
do 11 j =
dr3(j)
11 continue
radfu
radcl
ncld
drf
dro(nj)
dr4(nj)/i.5/(n(njml) -1)/w
1.njml
= (dro(j) + dro(j+l))/2.0
radr - thg - the
radfu + thg
npin - ncc
radfu/ncf
193 194
195
196
197
198
199
200
201
202
203
204
205.
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
I
#I
241
242
243 c
244
245
246
247
248
249
250
251
252
253
254
255
256 c
257
258 c
259,
260
261
262
263
264
265
266
267
268
269
270
274272
273
274 c
275
276 c
277
278
279
280
281
282
283
284
285 c
286 c
287 c
288 c
drc
tafp
= thc/ncld
= radfu*radfu/d
rad(1) = zero
do 14 k = 1.ncf
rad(k+1) = rad(k) + drf
14 continue
rad(ng+I) = rad(ng) + thg
do 15 k = nccnpml
rad(k+1) = rad(k) + drc
15 continue
do 16 k = 1.npmi
if(k.eq.ng) rodr(k) - (rad(k+i ) + rad(k))/2.0
if(k.ne.ng) rodr(k) - (rad(k+1 )+rad(k))/(rad(k+I)-rad(k))/2.0
16 continue
call
inecho(nrow.pitch,d,e,ad,apu,dil.thc,thg,lsstinit,lp)
vm(1) - zero
vp(1) = drf*drf/8.0
rm
= (radr + rad(npml))/2.0
vm(npin) = (radr*radr + w*w/4.0- rm*rm)/2.0
vp(npin) = zero
do 17 k = 2,npml
rp = (rad(k+1) + rad(k))/2.0
rm = (rad(k) + rad(k-1))/2.0
vp(k) = (rp*rp -rad(k)*rad(k))/2.0
vm(k) - (rad(k)*rad(k) - rm*rm)/2.0
17 continue
if(nres.eq.1) go to 18
call redum2(tv.tl,p,alfauvz,ulz,uvr,ulr,tr,tcantw,nn,
*
ntr.ncan.np.ni.nim2,nj,npin,tinit,lss)
go to 20
18 continue
read(5.tres)
qpp = hnw2(2)*radfu*radfu/radr/2.0
call stead(pin,pout.tin,tav,qpp,p,tv,tl,uvz,ulz.uvr.ulr,
*
alfa,twtr.dtr,dh,dv,nn,np,ntr.npin.npml)
write(8,1003)nrow.pitch.d.e
call redum3(n,dz,tcan,shape.sppd.ppplplnm,ncan.nn,npin,
*
ad.apu,dil,radr,thc,thg,ni)
20 continue
computes the time step limitation imposed when the explicit
radial heat conduction option is utilized
241242
243
244
245
246
247
248
249
250
251
252
253.
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276277
278
279
280
281
282
283
284
285
286
287
288
W
/
289
290
291
292
293
294
295
296
297
N
if(rnusll.le.zero) return
drmin = dr6(1)
do 37 1 = 2,nj
37
drmln = dminl(drmin,dr6(i))
drmin = drmin*drmin/dmaxl(rnusli,rnusl2)*1.8623d+03
dtmax = dminl(dtmax.drmin)
return
1003 format(15,3d15.9)
end
289290
291
292
293
294
295
296
297
o
cx)
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316,
317
318 .
319 c
320 c
321 c
322 c
323 c
324 c
325 c
326 c
327
328.
329
330 -c
331
332
333
334
335 c
336 c
337 c
338
339
340
341 .
342
343 c
344
345
subroutine ws(po,tvo,tlo.alfao,alfazaalfar.rhov,
rhol.rhovz.rholz,rhovr,rholr.hv,hl,
uvzo.ulzo.uvroulro,
wev.wel.,wz3.wz4.wz5.wz6,wz7.wz8,wz9,
wzlO.wzli,wr3,wr4,wr5.wr6,wr7,wr8,wr9.
wrlO,wril.dh,dv,qsi,sppd,gamo,nn)
implicit real*8 (a-h.o-z)
common /dim/ dz(150).dzl(150)dro(1
50),drdri(150),dr2(150),dr3(150)
*
dr4(150),dr5(150).dr6(150),ni.nj.nimi.nim2.nJml.nni,
*
nnj,nnJJ
common /tempo/ time.dt.dto,dtls,sprint,ndt,nres
common /number/ zero.one.big,small
common /flbdry/ fbftr(20).explm(20),expvm(20)
dimension po(nn).tvo(nn).tlo(nn).alfao(nn).alfaz(nn).
*
alfar(nn),rhov(nn),rhol(nn),rhovz(nn),rholz(nn).
*
rhovr(nn),rholr(nn).hv(nn),hl(nn).uvzo(nn).
*
ulzo(nn),uvro(nn),ulro(nn)wev(nn).wel(nn),
*
wz3(nn),wz4(nn),wz5(nn),wz6(nn,wz7(,
nn).wz(nn).
*
wz9(nn).wz10(nn),wzll(nn).wr3(nn).wr4(nn),
*
wr5(nn),wr6(nn),
(nn)nn)wr8(nn),wr9(nn).wrlO(nn).
*
wrtl(nn),dh(nn),dv(nn),qsi(nn),sppd(nn),gamo(nn)
*
*
*
*
*
subroutine ws complete the evaluation of the explicit terms
involved in the solution of the problem stated with subroutine
donor. here are set the terms containing the time Increment
dt.it is written separately from subroutine donor in order to
allow a change in the value of dt when the problrm does not
converge with the previous dt.(see next coment in this subroutine.)
do 3 jo a 1,nj
do 3 io - 2,ni
ko = (jo-1)*ni+io
wwzl
wwz2
wwrl
wwr2
a alfaz(ko)*rhovz(ko)
a (one - alfaz(ko))*rholz(ko)
- alfar(ko)*rhovr(ko)
= (one - alfar(ko))*rholr(ko)
calculate the interfacial and wall
*
*
*
*
friction terms
call coeff(tvo(ko).tlo(ko).uvzo(ko).uvro(ko).ulzo(ko).ulro(ko).
alfaz(ko),alfar(ko),rhovz(ko).rhovr(ko).
rholz(ko),rholr(ko).dh(ko),dv(ko).qsi(ko).
sppd(ko).wwzl,wwz2,wwri.wwr2,
fvz,flz,fvr,flrciz,clr)
*
wev(ko) * -(rhov(ko)*hv(ko)+po(ko))*alfao(ko)/dt
wel(ko) = -(rhol(ko)*hl(ko)+po(ko))*(one-alfao(ko))/dt
12
3
4
5
6
7
8
9
10
11
12
13.
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
3637
38
39
40
41
42
43
44
45
46
47
48
0
346 c
347
348 c
349 c
350 c
351 c
352 c
353 c
354 c
355 c
356 c
357 c
358 c
359 c
360
361
362
363
364
365 c
366 ,
367
368
369
370 c
371
372
373
374
375
376.c
377
378 c
379
380
381
382
383
384
385
386
387 c
388
389
390
391
392
393
If(ndt.ne.0) go to 1
since the program allows a change in the value of the time increment
dt,even if the time step is not completed,we put a check here to know
if such a change did occur (in this case ndt would be different than
zero) In case the test be true,we subtract the terms which have the
old dt and add them back with the new value of dt.
here we have added the effects of mass exchange on the
interfacial momentum exchange coefficient. We assume
the interfacial velocity is given by ui = eta*uv +
(1 - eta)*ul
where 0 < eta < 1
eta = 0.5
wz4(ko) =
wz6(ko) =
wr4(ko) =
wr6(ko) =
clz
clz
cir
cdr
+
+
-
eta*gamo(ko)
(i. - eta)*gamo(ko)
eta*gamo(ko)
(1. - eta)*gamo(ko)
wz3(ko)
wz5(ko)
wr3(ko)
wr5(ko)
=
=
=
=
wz4(ko)
wz6(ko)
wr4(ko)
wr6(ko)
+
+
+
+
alfaz(ko)*rhovz(ko)/dt + fvz
(one-alfaz(ko))*rholz(ko)/dt + flz
alfar(ko)*rhovr(ko)/dt +.fvr
(one-alfar(ko))*rholr(ko)/dt + flr
wz7(ko)
wz8(ko)
wr7(ko)
wr8(ko)
go to 2
=
=
=
=
wz7(ko)
wz8(ko)
wr7(ko)
wr8(ko)
-
uvzo(ko)/dt*alfaz(ko)*rhovz(ko)
ulzo(ko)/dt*(one-alfaz(ko))*rholz(ko)
uvro(ko)/dt*alfar(ko)*rhovr(ko)
ulro(ko)/dt*(one-alfar(ko))*rholr(ko)
1 dtc = one/dto - one/dt
wz7(ko)
wz8(ko)
wr7(ko)
wr8(ko)
wz3(ko)
wz5,(ko)
wr3(ko)
wr5(ko)
=
=
=
=
=
=
=
=
uvzo(ko)*alfaz(ko)*rhovz(ko)*dtc + wz7(ko)
ulzo(ko)*(one-alfaz(ko))*rholz(ko)*dtc + wz8(ko)
uvro(ko)*alfar(ko)*rhovr(ko)*dtc + wr7(ko)
ulro(ko)*(one-alfar(ko))*rholr(ko)*dtc + wr8(ko)
wz3(ko) - alfaz(ko)*rhovz(ko)*dtc
wz5(ko) - (one-alfaz(ko))*rholz(ko)*dtc
wr3(ko) - alfar(ko)*rhovr(ko)*dtc
wr5(ko) - (one-alfar(ko))*rholr(ko)*dtc
2 continue
wzll(ko) = wz3(ko)*wz5(ko)-wz4(ko)*wz6(ko)
wzlO(ko) = -(alfaz(ko)*wz6(ko)+(one-alfaz(ko))*wz3(ko))/
/
dzl(io)/wz11(ko)
wz9(ko)
= -(alfaz(ko)*wz5(ko)+(one-alfaz(ko))*wz4(ko))/
/
dzi(to)/wzi1(ko)
49 50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
N)
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
3 continue
c
c
c
radial direction equations
do 4 jo a 1,njmi
do 4 io a 2,niml
ko = (jo-1)*ni + io
c
wrll(ko) = wr3(ko)*wr5(ko) - wr4(ko)*wr6(ko)
wrlO(ko) = -(alfar(ko)*wr6(ko)+(one-alfar(ko))*wr3(ko))/
dr3(jo)/wrll(ko)
/
wr9(ko)
- -(alfar(ko)*wr5(ko)+(oneralfar(ko))*wr4(ko))/
/
dr3(Jo)/wrii(ko)
4 continue
c
c
c
c
c
these terms are only used for the flow boundary condition
they are the explicit terms of the non-discretized liquid
and vapor momentum equations
do 5 i - 1.nj
ko - (i-1)*ni +.2
explm(i) - -(wz7(ko)*wz6(ko) + wz3(ko)*wz8(ko))/wzll(ko)
expvm(i) - -(wz7(ko)*wz5(ko) + wz4(ko)*wz8(ko))/wz11(ko)
5 continue
return
end
9798
99
100
101
102
103
104
105
106
107
108
109.
110
111
112
113
114
115
116
117
118
119
120
121
122
H
W
420
421
422 c
423 c
424 c
425 c
426 c
427 c
428 c
429 c
430 c
431
432
433
434
435
436
437 c
438,
439
440 .
441
442
443
444
445
446
447
448
449
450.
451
452453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
*
subroutine qcond(t1,tc,tcn,tr,all.alp,alr,qici,qic2,
indl,numr)
this subroutine calculates the radial heat transfer between
cells.
If vapor is present in a cell, no heat transfer is
assumed. Both a partially Implicit and a full explicit
calculation is possible.
qici =
qic2 =
heat transfered per unit volume
the derivative of qici with respect to tl
implicit real*8 (a-h,o-z)
common /dim/ dz(150).dzl(150).dro(150),dri(150),dr2(150).dr3(150).
dr4(150).dr5(150).dr6(150).ni,njntml,nim2,nJmlnni,
*
nnj,nnjj
*
common /nussy/ rnusli,rnusl2
common /number/ zero,one,big,small
10
30
40
50
qici = zero
qic2 = zero
rnui = dabs(rnusli)
ti
= tc
ti = ten
If(indl.eq.1)
if(numr.eq.nj) go to 50
go to 30
if(numr.ne.1)
continue
if(alc.ne.zero.or.alr.ne.zero) return
call htran(tc,tr.h,rnul.1,2)
qic2 = -onesdr5(1)*h/dr4(i)
qicl = qic2*(ti - tr)
return
continue
if(alc.ne.zero) return
if(all.ne.zero) go to 40
call htran(tc,tl.h,rnul,numr,(numr-1))
qic2 = -one*dr5(numr - 1)*h/dr4(numr)
qici = qic2*(ti - tl)
continue
if(alr.ne.zero) return
if(numr.ne.nJml) rnul = dabs(rnusl2)
call htran(tc,tr.h.rnul,numr,(numr + 1))
qic2 = qic2 - h*dr5(numr)/dr4(numr)
qici = qici + h*dr5(numr)*(tr - ti)/dr4(numr)
return
continue
if(alc.ne.zero.or.all.ne.zero) return
rnu2 = dabs(rnusl2)
call htran(tc,tl,h,rnu2,nj.njml)
12
3
4
5
6
7
8
9
10
11
12
13.
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
toc
468
469
470
471
qlc2 = -one*dr5(njml)*h/dr4(nj)
qict * qic2*(ti - tl)
return
end
4950
51
52
52
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
subroutine htran(ti.t2.h,rnu,nui,nu2)
c
c
c
c
this subroutine calculates the intercell heat transfer coefficient
for subroutine qcond
implicit real*8 (a-ho-z)
common /dim/ dz(150),dzl(150),dro(150),drl(150),dr2(150),dr3(150),
*
dr4(150),dr5(15
(15,dr6(150),ni,nnim,nim2nmnni
*
nnj,nnjJ
c
convi = condl(tl)/dr6(nul)
conv2 = condl(t2)/dr6(nu2)
c
h = 2.*rnu*conv*conv2/(convi + conv2)
return
end
.
I2
3
4
5
6
7
8
9
10
11
12
13.
14
15
16
0
488
489
490 c
491 c
492 c
493 c
494 c
495 c
496
497
498
499
500
501'
502
503
504
505
506,
507
508
509,
510
511 c
512
513
514
515
516
517
518.
519
520
521
522
523
524
525
*526
527
528
529
530
531
532
533
534
535
*
subroutine bc(p,tv,tl,alfa,alfaz.rhov.z,rholz,uv,ul.wz9,wzlO,
time,itbd,nn)
this subroutine calculates the boundary conditions as a function
of time. The inlet boundary condtion indicator, itbd, can be
either -1,0.1 to indicate a velocity, pressure, or flow boundary
condition at the inlet.
implicit real*8 (a-h,o-z)
common /bcx/ ulo
,pnbl(51),pnb2(51),pnb3(51),omp(51) ,
common /bcond/ tb(51)
*
pntl(51).pnt2(51).pnt3(51),omt(51) ,albl(51),
alb2(51),alb3(51),oma(51) ,tvbl(51).tvb2(51).
*
tvb3(51),omv(51) ,tlbl(51).tlb2(5i),tlb3(5i),
*
oml(51) ,hnwi(51),hnw2(51).hnw3(51),omh(51) ,
*
vbl(51) ,vb2(51) ,vb3(51) ,omvb(51),flbi(51),
*
flb2(51) ,flb3(51) ,omfb(51),lmax,lp
*
common /dim/ dz(150),dzi(150).dro(150),dri(150),dr2(150).dr3(150),
dr4(150),dr5(150),dr6(150),ni,nj,niml,nim2,njml.nni,
*
nnj,nnjj
*
common /flbdry/ fbf.tr(20),explm(20),expvm(20)
dimension p(nn),tv(nn).tl(nn),alfa(nn),alfaz(nn),rholz(nn),
*
rhovz(nn).uv(nn),ul(nn),wz9(nn),wz40(nn)
•
1 a 2
1 continue
if(time.le.tb(l)) go to 2
1 = 1 + 1
if(l.gt.lmax) return
go to 1
2 continue
dtime = time - tb(l-1)
if(itbd) 3.5,7
3 continue
vb
= vbt(l)*dtime + vb2(l)
If(lp.eq.1) vb = dexp(omvb(l)*dtime)*vb + vb3(1)
do 4 j = 1,(nn-nimi),ni
ul(j) = vb
uv(j) = vb
4 continue
go to 100
5 continue
pnb a pnbi(l)*dtime + pnb2(1)
if(lp.eq.1) pnb = dexp(omp(l)*dtime)*pnb + pnb3(1)
do 6 j = 1,(nn-niml),ni
p(j) = pnb
6
go to 100
7 continue
.
12
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
3G
37
38
39
40
41
42
43
44
45
46
47
48
a
N)
H
{
flb = flbl(1)*dtime + flb2(1)
536
if(lp.eq.1) flb = dexp(omfb(1)*dtime)*flb + flb3(1)
537
fbftrl = O.dO
538
fbftr2 = O.dO
539
do 8 j = t,nJ
540
ko = (J-i)*ni + 2
541
fbftr(J).= dr4(j)*((1.dO-alfa(ko))*rholz(ko)*wzlO(ko) +
542.
alfa(ko)*rhovz(ko)*wz9(ko))
+
543
fbftrl = fbftrl + fbftr(j)
544
fbftr2 = fbftr2 + dr4(J)*((1.dO-alfa(ko))*rholz(ko)*
545
explm(j) + alfa(ko)*rhovz(ko)*expvm(j))
*
546
8 continue
547
fib = (fbftr2 - flb)/fbftrl
548
do 9 j = 1,nj
549
ko = (j-1)*ni + 2
550
fbftr(j) = -fbftr(j)/fbftri
551
fib = flb - fbftr(j)*p(ko)
552
553
9 continue
,nj
do 10 J =
554,
ko = (J-1)*ni + 1
555
p(ko) = flb
556 .
10 continue
557,
100 continue
558
pnt = pntl(l)*dtime + pnt2(1)
559
alb = albl(i)*dtime + alb2(1)
560
tvb = tvbl(1)*dtime + tvb2(1)
561
tib = tlbl(l)*dtime + tlb2(l)
562
if(lp.eq.0) go to 11
563
564 c
pnt = dexp(omt(l)*dtime)*pnt + pnt3(l)
565
alb = dexp(oma(])*dtime)*alb + alb3(1)
566.
tvb = dexp(omv(1)*dtime)*tvb + tvb3(1)
567
tlb = dexp(oml(1)*dtlme)*tlb + tlb3(l)
568
569 c
11 continue
570
do 12 j = ni,nn,ni
571
572
573
574
575
576
577
578
579
ko = j - nimi
p(J)
,alfa(ko)
tv(ko)
tl(ko)
12 continue
return
end
=
=
=
=
pnt
alb
tvb
tlb
4950
51
52
53
54
55
56
57
58
59
60
61.
62
63
64
65
66
67
.68
69
70
71
72
73
74
75
76
77
78
. 79
80
81
82
83
84
85
86
87
88
89
90
91
92
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594 c
595 c
596 c
597 c
598,c
599 c
600 a
601, d
602 c
603 c
604 c
605 c
606
607
608
609 c
610.
611
612 613 c
614 c
615 c
616
617
618
619
620
621 c
622 c
623 c
624
625
626
627
subroutine noneq(alfao,alfa,tv,tl,p,rhov,rhol,ts,s,iflag)
implicit real*8 (a-h,o-z)
common /error/ ierr
common /number/ zero,one,big,small
common /pd/ d4,pod2
common /dryout/ alpdry
dimension s(5,2)
data an,rgas /6.d-04,2.09882d-02/.half /0.50dO/
data ai,bl /12020.0,21.9358/
data pi,sr3,cadry.adry /3.141592654,3.464101616,0.043,0.957/
data hO,hi /5.089d+06.-.O143d+04/
data rnu /10.0/
data hlO.hli,hl2.hl3 /-6.75075d+04,1.63014d+03,
*
-.416720.1.54272d-04/
14
subroutine noneq calculates the mass and energy exchange rates
and its derivatives.
rgas = square root of gas constant for sodium over 2*pi
pod2 a pitch to diameter ratio squared
s(t,
s(2,
s(3,
s(4,
s(5,
)
)
)
)
)
exchange rate
- d/dtv
a d/dtl
= d/dp
* d/dalfa
i2
3
4
5
6
7
8
9
10
11
12
13.
s(
s(
ax = alfa
if(alfa.lt.1.d-4) ax * 1.e-4
if(alfa.gt.0.9999) ax = 0.9999
xx - one/(sr3*pod2 - pi)
ann = an*an*d4*d4*pi*xx/12.
if(alfa.gt.ann) go to 10
incipient boiling
area = 3.*ax/an
darda = 3./an
go ,to 60
10 continue
if(ax.gt.0.55) go to 20
bubbly flow correlation
area - d4*dsqrt(3.*pi*xx*ax)
O0.5*d4*dsqrt(3.*pi*xx)/dsqrt(ax)
darda
go to 60
20 continue
,1) - mass
.2) - energy
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36 37
38
39
40
41
42
43
44
45
46
47
48
628
629 c
630 c
631 c
632
633
634 c
635
636
637
638 c
639
640
641
642
643 c
644 c
645 c
646
647
648 ,
649 '
650
651 c
652 c
653 c
654
655
656
657
658.
659
660 c
661
662
663
664
665
666 c
667
668 c
669
670
671
672
673
674
675
if(ax.gt.0.65) go to 30
slug/churn flow transition
arl = d4*dsqrt(3.*pi*xx*ax)
dardal = 0.5*d4*dsqrt(3.*pi*xx)/dsqrt(ax)
ar3 = sr3*pi*pod2*xx*xx - pt*ax*xx
ar2 = d4*dsqrt(ar3)
darda2 = -0.5*d4/dsqrt(ar3)*pi*xx
call poly(arl,dardal,ar2,darda2,area,darda,alfa)
go to 60
30 continue
if(ax.gt.0.957) go to 40
annular flow correlation
arl = sr3*pi*pod2*xx*xx - pi*ax*xx
area = d4*dsqrt(arl)
darda = -0.5*d4/dsqrt(arl)*pI*xx
go to 60
40 continue
dryout correlation
arl
=
area
=
dardal =
darda2 =
darda =
60 continue
sr3*pi*pod2*xx*xx - pi*ax*xx
d4*dsqrt(arl)*dsqrt(l. - ax)*4.822
-2.411*d4/dsqrt(arl)*pi*xx*dsqrt(1. -2.411*d4*dsqrt(ari)/dsqrt(1. - ax)
dardal + darda2
ts
hlg
ftr
pstl
srts
=
=
=
=
=
coef
= rgas*ftr/(i.0 - O.5*ftr)
sat(p)
hl*ts + hO
0.1
dexp(bl -al/tl)
dsqrt(ts)
ce = 0.0
cc = 0.0
if(alfa.eq.0.0) go to 70
if(alfa.gt.alpdry) go to 85
If(tl.gt.ts) ce = 1.0
if(tl.le.ts) cc = 1.0
go to 80
ax) •
4950
51
52
53
54
55
56
57
58
59
60
61.
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
ro
co
v
676
677
678
679 c
680 c
681 c
682'
683
684
685 c
686
687
688
689
690
691
692
693
694,
695
696 c
697
698 c
699
700
701
702 c
703
704
705
706.
707
708
709 c
710 c
711 c
712
713
714
715
716
717
718
719
720
721
722
723
70 continue
if(tl.gt.ts) ce *
80 continue
1.0
mass exchange rate
cel = (pstl - p)/srts
ce2 = area*coef*(ce + cc)
ddp a dtsdp(p)
s(1.1) = cei*ce2
s(2.1) = 0.0
s(3.1) = ce2*al*pstl/ti/tl/srts
= -ce2*(0.5*cel*ddp/ts + 1./srts)
s(4.,)
s(5.1) = darda*coef*(ce + cc)*cel
go to 87
85 continue
if(tv.gt,ts) ce - 1.0
If(tv.le.ts) cc a 0.01
pstv = dexp(bl - al/tv)
mass exchange rate
cel = (pstv - p)/srts
ce2 = area*coef*(ce + cc)
ddp = dtsdp(p)
s(1,1) =
s(2,1) =
s(3,1) a
s(4.1) =
s(5,1) =
87 continue
cel*ce2
ce2*ai*pstv/tv/tv/srts
0.0
-ce2*(0.5*cel*ddp/ts + 1./srts)
darda*coef*(ce + cc)*cei
energy exchange rate
u
= condl(tv)*rnu*d4
((hl3*ts + h12)*ts + hli)*ts + hO
hi
a hi + hlig
hv,
dhldp - ((3.*h13*ts + 2.*h12)*ts + hli)*ddp
dhvdp = dhldp + hI*ddp
if(alfa.gt.alpdry) go to 90
s(1,2) = s(1.,)*hv + area*u*(ts - tv)
s(2,2) = s(2,1)*hv - u*area
s(3,2) = s(3,i)*hv
s(4,2) - s(4,1)*hv + s(1.1)*dhvdp + area*u*ddp
s(5.2) = s(5,1)*hv + darda*u*(ts - tv)
return
9798
99
100
101
102
103
104
105
106
107
108
109.
110
1I1
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132133
134
135
136
137
138
139
140
141
142
143
144
%q
724
725
726
727
728
729
730
731
732
90 continue
u
=
s(1,2) =
s(2,2) =
s(3.2) =
s(4,2) =
s(5.2) =
return
end
condv(tl)*rnu*d4
s(1,1)*hl + area*u*(t) - ts)
s(2,1)*hl
s(3.1)*hl + area*u
s(4,1)*hl + s(1,1)*dhldp - area*u*ddp
s(5,1)*hl + darda*u*(tl - ts)
145146
147
148
149
150
151
152
153
a
.4
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
c
c
c
c
subroutine poly(one,two,three,four,area,darda.alfa)
this subroutine performs a polynomial fit for the area and the
the derivative of the area in the bubbly/annular flow transition
Implicit real*8 (a-h,o-z)
a =a one
two
b
c = 3*(three - one) - four - 2.*two
d = four +two + 2*(one - three)
c
x = 10.*(alfa - .55)
c
area - a +b*x + c*x*x + d*x*x*x
darda * b + 2.*c*x + 3.*d*x*x
c
return
end
2
3
4
5
6
8
9
10
11
12
13.
14
15
16
17
I
751
subroutine direct(al,a2,a3,a4,f.x.nc,zi.z2.z3.z4,bbb.
aaa,xl,dpbd,itbd,nbandi,nband2,ndds)
752
*
753
implicit real*8 (a-h,o-z)
754
common /number/ zero,one,big,small
755
common /gauss/ nznr,nzmi
756
common /error/ ierr
757
common /cntrl/ epsi.eps2,res,itl,it2,it3.itmi.imttm2,igauss,indgs
758
common /flbdry/ fbftr(20),explm(20),expvm(20)
759
dimension al(nc).a2(nc).a3(nc),a4(nc),f(nc),x(nc),
xl(nddsnband2).zl(ndds),z2(ndds),z3(ndds),
*
760
z4(ndds),bbb(ndds).aaa(ndds,nbandi)
761
*
762 c
763
dpbd = zero
764
10 continue
765 c
766 c
this subroutine solves the pressure problem by use of a
direct solution, using library subroutine leqtlb.
767 c
768 c
769, c
rearrange numbering of cells to minimize the
770 c
bandwidth.
771 c
772
do 30 J=l,nr
773
do 20 I=i,nz
774 c
775
incl = nr*(i-1) + J
776
inc2 = nz*(J-1) + 1
777 c
778
z1i(nci) = al(Inc2)
779
z2(incl) = a2(inc2)
780
z3(incl) = a3(inc2)
781'
z4(incl) = a4(inc2)
782
bbb(incl) = f(inc2)
783 'c
784
20
continue
785
30 continue
786 c
787
if(itbd.ne.1) go to 36
788
do 35 1 = 1,(ndds-1)
789
,ko = ndds + 1 - i
790
zl(ko) - zl(ko - 1)
791
z2(ko) = z2(ko - 1)
792
z3(ko) = z3(ko - 1)
793
z4(ko) = z4(ko - 1)
794
bbb(ko) = bbb(ko - 1)
795
35 continue
796
zi(1) = zero
797
z2(1) = zero
798
z3(1) = zero
4.
12
3
4
5
6
7
8
9
10
11
12
13*
14
15
16
17
18
19
20
21
22
23
24"
25
26
27
28
29"
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
it41
799
800
801
802 c
803 c
804 C
805
806
807
808
809
810
811
812
813 c
814 c
815 c
816
817,
818
819 :
820
821 c
822 c
823 c
824
825
826
827
828
829
830 c
831 c
832 c
833
834
835
836 '
837
838
839
840 c
841 c
842 c
843 c
844 c
845
846
z4(1) = zero
bbb(1) = zero
36 continue
set values for
ier
ijob
n
nlc
nuc
ia
ib
m
.
input data
= O
0
= ndds
Inr
- nic
- ndds
= ndds
= 1
a
set up matrix aaa
JO
ji
j2
j3
j4
= nr + 1
1
- nr
= nr + 2
= 2*nr + 1
=
initialize matrix aaa to zero
nband a 2*nr + 1
do 50 j a 1,ndds
do 40 k a 1,nband
aaa(j.k) a zero
40
continue
50 continue
input band components
do 60 1 a 1,ndds
aaa(i.ji) a z2(i)
aaa(i.j2) a zi(i)
aaa(i.j3) a z4(i)
,aaa(I.j4) - z3(t)
aaa(i.JO) a one
60 continue
for a flow boundary condition, it is necessary to add an
additional equation to the pressure field matrix, in
order to update the boundary pressure
if(itbd.ne.1) go to 70
do 63 j a 1,nr
*
4950
51
52
53
54
55
56
57
58
59
60
61.
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84'
85
86
87
88
89
90
91
92
93
94
95
96
ru
847
aaa(i.(nband2+j)) = fbftr(J)
848
63 continue
849
do 68 J = 2,nr
850
nbi = (2-j) + nr
851
aaa(j,nbl) = aaa(j,1)
852
aaa(j,1) = O.OdO
853
68 continue
854
70 continue
855 c
856 c at this point a call to the library subroutine
857 c
leqtlb is made
858 c
859
call leqtlb(aaa,n,nlc.nuc,ia,bbb,m,ib.ijob,xlier)
860 c
861 c
check the results
862 c
863
if (ier.eq.129) ierr - 1
864 c
865
if(itbd.ne.1) go to 90
866
dpbd = bbb(1)
867.
do 80 i = 2,ndds
868'
bbb(i-i) = bbb(i)
869
80 continue
870
90 continue
871 c
872 c
now convert back the results to the old numbering scheme
873 c
874
do 110 j - 1,nr
875
do 100 i = 1,nz
876 c
877
inci = nr*(i - 1) + j
878
inc2 = nz*(j - i) + 1
879. c
880
x(inc2) - bbb(incl)
881
100
continue
882
110 continue
883
res = O.OdO
884 1
do 120 1 = 1,nc
885
,xx = dabs(x(l))
886
if(xx.gt.res) res - xx
887
120 continue
888
return
889
end
04
4.
*
9798
99
100
101
102
103
104
105
106
107
108
10%
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
rN
4=