Citation
Static and dynamic neutronic analysis of the uranium tetra-fluoride, ultrahigh temperature, vapor core reactor system

Material Information

Title:
Static and dynamic neutronic analysis of the uranium tetra-fluoride, ultrahigh temperature, vapor core reactor system
Creator:
Kahook, Samer Dakhlallah, 1961- ( Dissertant )
Dugan, Edward T. ( Thesis advisor )
Lear, William E. ( Reviewer )
Person, Willis B. ( Reviewer )
Phillips, Winfred M. ( Degree grantor )
Lockhart, Madelyn M. ( Degree grantor )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1991
Language:
English
Physical Description:
xx, 372 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
ATMs ( jstor )
Boilers ( jstor )
Cooling ( jstor )
Coupling coefficients ( jstor )
Geometry ( jstor )
Inlets ( jstor )
Liquids ( jstor )
Neutrons ( jstor )
Reactivity ( jstor )
Vapors ( jstor )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF
Nuclear Engineering Sciences thesis Ph. D
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
An Ultrahigh Temperature Vapor core Reactor (UTVR) system is investigated in this research. The UTVR can be characterized as a thermal, high power density (hundreds of MW^^/m ), externally-moderated, 235 coupled core, highly-enriched U , circulating-fuel, steady-state, burst power reactor. The investigated reactor system includes two types of fissioning regions: (1) the central Ultrahigh Temperature Vapor Core region (UTVC) which contains a vapor mixture of highly-enriched uranium tetrafluoride (UF^) fuel and a metal fluoride working fluid at an average temperature of «3000 K and an average pressure of «50 atm; and (2) the Boiler COLumn region (BCOL) which contains highly enriched liquid UF^ fuel. The combination of three features differentiates the UTVR from other nuclear reactor concepts. These three features are as follows: 1. the multi-core configuration resulting in a coupled-core system by means of direct neutron transport through the media; 2. the circulating fuel and the associated neutronic and mass flow coupling between the UTVC and boiler cores; and 3. the employment of a two-phase fissioning fuel, i.e., a liquid-vapor combination. Static and dynamic neutronic analysis of this novel system indicates distinct advantages over other existing or conceptual nuclear power systems. These include a unique combination of some very effective inherent negative reactivity feedbacks such as the vapor-fuel density power coefficient of reactivity, the direct neutronic coupling among the multiple fissioning core regions, and the mass flow coupling feedback between the two types of fissioning cores. Static neutronic analysis is performed using multidimensional discrete ordi nates and Monte Carlo neutron transport codes. Parameters such as the UTVC and boiler column reactivities and reaction rates, core-to-core neutronic coupling coefficients, and neutron lifetimes as a function of vapor core density and boiler core liquid volume are obtained from the static neutronic analysis. The dynamic behavior of the UTVR is examined using a non-linear model, which incorporates circulating-fuel, coupled-core, point reactor kinetics and energetics equations. These equations are solved using a system analysis code. The dynamic analysis indicates that the unique and strong negative reactivity feedbacks of the UTVR are capable of stabilizing the UTVR safely and quickly even when large reactivity insertions are imposed {6p - $ 1.00). The analysis also shows that the system exhibits good dynamic performance even when an inherent negative reactivity feddback is suppressed (e.g., the vapor fuel density power coefficient of reactivity). However, due to the strength of the UTVR's inherent negative reactivity feedbacks, it is found that external reactivity insertions alone are inadequate for bringing about power level changes during normal operations. Additional methods of reactivity control, such as variations in the mass flow rate of the fuel and/or working fluid or variations in the inlet pressure of the fuel/working fluid entering the boiler columns, are needed to achieve the desired power level control.
Thesis:
Thesis (Ph. D.)--University of Florida, 1991.
Bibliography:
Includes bibliographical references (leaves 366-370)
Additional Physical Form:
Also available on World Wide Web
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Samer Dakhlallah Kahook.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
026382369 ( AlephBibNum )
25138701 ( OCLC )
AJA1243 ( NOTIS )

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Full Text









STATIC AND DYNAMIC NEUTRONIC ANALYSIS
OF THE URANIUM TETRA-FLUORIDE, ULTRAHIGH
TEMPERATURE, VAPOR CORE REACTOR SYSTEM















By

SAMER DAKHLALLAH KAHOOK


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1991
























Dedicated to

my parents,

Mr. and Mrs.

Dakhlallah Kahook,

asking Allah

to reward them,

have mercy on them,

and grant them paradise

as they raised and cherished me in my childhood.













ACKNOWLEDGEMENTS


The author would like to express his appreciation and sincere

thanks to the members of his supervisory committee, Dr. Edward T. Dugan,

Dr. Nils J. Diaz, Dr. Alan M. Jacobs, Dr. Samim Anghaie, Dr. William E.

Lear, Jr., and Dr. Willis B. Person for their guidance and assistance

during the course of this research.

Special thanks are extended to Dr. Dugan, chairman of the author's

supervisory committee for his patience and enduring support. The author

recognizes that much of his knowledge in reactor physics and computer

programming was realized while researching under the guidance and

direction of Dr. Dugan.

Support for this research has been provided, in part, by the Air

Force Wright Aeronautical Laboratories (AFWAL), the Frederick Hauck

Fund, and the University of Florida. The AFWAL work was performed for

the Innovative Science and Technology Directorate of the Strategic

Defense Initiative within the Innovative Nuclear Space Power Institute

(INSPI). This support is greatly appreciated.

Funding for the computer analysis was provided for by the National

Science Foundation at the San Diego Supercomputer Center and the

University of Florida and the International Business Machines (IBM)

Corporation through their Research Computing Initiative at the North

East Regional Data Center. The author is grateful for these funds.


iii







Thanks are also due to the fellow students whose friendships,

comments, and encouragements have also facilitated in this research.

The author would like to express his love and respect to his

parents Mr. and Mrs. Dakhlallah Kahook, to his brothers Nofal and

Mohammed, and to his sisters for their love, understanding, and patience

throughout the author's stay at the University of Florida. The

financial support provided to the author by his family is gratefully

acknowledged.

Finally, the author would like to express his love and deepest

appreciation to his wife, Layali, whose understanding, patience, and

support provided the motivation needed to finish this research.













TABLE OF CONTENTS
Page

ACKNOWLEDGEMENTS .... ............................................ iii

LIST OF TABLES.................................................. x

LIST OF FIGURES ................................................... xiv

ABSTRACT........................................................ xviii

CHAPTER

I INTRODUCTION................................................ 1

Introduction ........................................... 1
Description of the Ultrahigh Temperature Vapor Core
Reactor.... ........................................ 2
Dissertation Objectives ................................ 6
Dissertation Organization............................... 7

II PREVIOUS RESEARCH ON RELATED CONCEPTS........................ 11

Introduction .......................................... 11
Previous Research on Gas Core Reactors................... 12
Previous Research on Coupled Core Reactors............... 13
Previous Research on Circulating Fuel Reactors............ 14
Remarks............................................... 18

III DESIGN OF THE URANIUM TETRA-FLUORIDE, ULTRAHIGH
TEMPERATURE VAPOR CORE REACTOR .......................... 19

Introduction.. ........................................ 19
Preliminary Design Considerations........................ 20
Choice of Materials................................... 21
The Moderator-Reflector Material...................... 21
The Fissioning Fuel Material ......................... 24
The Working Fluid Material...... .................... 29
Description of a Uranium Tetra-Fluoride, UTVR/Disk
MHD-Rankine Power Cycle.............. ............... 29
Neutronic Analysis of the Ultrahigh Temperature Vapor
Core Reactor......................................... 37
Static Neutronic Calculations......................... 37
Dynamic Neutronic Calculations........................ 39









IV STATIC, ONE-DIMENSIONAL, UTVR NUCLEAR CHARACTERIZATION
AND CONFIGURATION OPTIMIZATION .......................... 40

Introduction ........................................... 40
Scoping Calculations.............. .. .................. 43
Geometric Variations.................................. 43
UTVC radius....................................... 43
Inner BeO moderator-reflector region thickness..... 50
Outer BeO moderator-reflector region thickness..... 55
UF4 boiler region thickness....................... 57
UF boiler core volume............................. 59
Fuel Density Variations.............................. 61
UF4 partial pressure and mole fraction (UF4:NaF)
2. in the UTVC ............................. ...... 61
U23 enrichment in UF,............................. 62
U23 as the fissile i otope....................... 62
Average density of the UF4 in the boiler region.... 66
Material Variations.......................... ...... 68
Choice of metal fluoride in UTVC................... 68
Wall cooling region..................... ......... 70
Other metal fluoride working fluids................ 70
NaF mass flow rate to the boiler region............. 73
UF /NaF inlet velocity to the boiler............... 76
Addition of Li F poison to the boiler.............. 76
BeO in the annular boiler region................... 79
Reactivity effects of liner materials.............. 81
One-Dimensional Results................................. 84
The Neutron Multiplication Factor..................... 86
Power Sharing Factor.................................. 87
Spherical "Mock-up" Comments ............................ 90

V STATIC, TWO-DIMENSIONAL, UTVR NUCLEAR CHARACTERIZATION
AND CONFIGURATION OPTIMIZATION .......................... 94

Introduction .......................................... 94
Scoping Calculations in R-0 Geometry..................... 96
Geometric Variations.. .............................. 98
UTVC radius variations............................. 98
Inner BeO moderator-reflector region thickness
variations .................................. 104
Variation in the area of the boiler columns........ 107
Variation in the number of boiler columns.......... 109
Fuel/Working-Fluid Density Variations................. 110
UF4 partial pressure in the UTVC................... 112
Average UF density in the boiler columns.......... 115
Varying the UF4 average density in the UTVC as a
function of the radial distance from the
center line ................................... 116
Scoping Calculations in R-Z Geometry .................... 121
Geometric Variations.. .............................. 124


PaRe


CHAPTER









V MBEO region height................................. 124
(cont.) TBEO region height................................. 129
First OBEO region height............................ 131
Boiler: subcooled and saturated liquid region
height ...................................... 134
Material Variation.................................... 136
Poisoning the boiler feedline walls................ 136
Comments on Power Sharing ............................ 140
Two-Dimensional Results................................. 144
The Neutron Multiplication Factor.................... 144
The Power Sharing Factor............................. 146
Remarks.............................................. 148

VI STATIC, THREE-DIMENSIONAL NEUTRONIC ANALYSIS OF THE UTVR.... 151

Introduction .. ......................................... 151
Description of the UTVR Geometry in MCNP................. 152
Description of the Boiler Column...................... 156
Reactivity Worths of the Boiler Feedlines, UTVC Inlet
Plenums, and the MHD Duct Regions..................... 158
Reducing the Uncertainty in Parameters Associated with
the Boiler Columns in MCNP Calculations............... 161
Performance of Variance-Reduction Techniques.......... 165
Nuclear and Physical Characteristics of the UTVR...... 166
Energy Cutoff........... ................... ........ 168
Implicit Capture and Weight Cutoff.................... 168
Weight Windows.................................... 173
Boiler-to-UTVC Symmetry.............................. 178
Neut ~o Transport Coupling Coefficients.................. 185
Obtained Directly from MCNP...................... 187
f i Obtained Indirectly from MCNP.................... 189
Isolation of secondary coupling effets...... ..... 190
Neutron Multiplicatign Factor of the j Core, keff... 197
Reactivity of the j Core, p ... ................... 198
Prompt Neutron Generation Time, A (t)................ 198
Results of Density Variations in the UTVC and Boiler
Columns............................................ 199

VII KINETIC EQUATIONS OF A FOUR-BOILER COLUMN UTVR SYSTEM....... 209

Introduction .. ......................................... 209
The Four-Boiler Column UTVR System Coupled Core Point
Reactor Kinetics Equations ........................... 209
Core-to-Core Fuel-Flow Coupling....................... 211
Core-to-Core Neutron Transport Coupling............... 214
Steady-State Solution ................................ 218
The Linearized UTVR CC-PRK Equations.................. 221
Inherent Reactivity Feedbacks of the UTVR............... 229
Reactivity Feedback of the Boiler Columns, 6p (t)..... 233
Reactivity Feedback of the UTVC, Sp (t)............... 251


vii


CHAPTER


Page









VIII DYNAMIC ANALYSIS OF THE UTVR............................... 264

Introduction .. ......................................... 264
The Unperturbed UTVR Configuration....................... 265
Results of the Dynamic Analysis ......................... 269
Boiler Column Reactivity Perturbation................. 269
UTVC Reactivity Perturbation ......................... 276
Variations in Core-to-Core Direct Neutron Transport
Delay Times...................................... 283
Variations in the Coupling Coefficients............... 287
Variations in the UTVC Fuel Mass Reactivity Feedback
Coefficient....................................... 291
Concluding Remark................................ ....... 296

IX SUMMARY OF RESULTS, CONCLUSIONS, AND RECOMMENDATIONS
FOR FURTHER RESEARCH... ................................ 300

Introduction .. ......................................... 300
Summary of Results...................................... 300
Results from the Static Neutronic Analysis............ 300
Results from the Dynamic Neutronic Analysis........... 302
Comments and Conclusions............................... 303
Recommendations for Further Research..................... 305
Static Neutronic Analysis ............................ 305
Dynamic Neutronic Analysis ........................... 307

APPENDICES

A DESCRIPTION OF THE COMPUTER CODES .......................... 309

Introduction .. ......................................... 309
Description of Nuclear Codes ............................ 309
AMPX: A Modular Code System for Generating Coupled
Multigroup Neutron-Gamma Libraries from ENDF/B..... 309
The AMPX-DRIVER module ............................... 311
The XLACS module................................... 311
The NITAWL module................................. 312
The XSDRNPM module ................................ 312
DOT-4: A One- and Two-Dimensional Neutron/Photon
Transport Code....... .............................. 315
GIP................................................ 316
MCNP-A General Monte Carlo Code for Neutron and
Photon Transport................................ 317
Description of the EASY5 Engineering Analysis Program.... 318

B BENCHMARK CALCULATIONS OF XSDRNPM AND DOT-4 WITH MCNP....... 320

Comparison of XSDRNPM with MCNP ......................... 320
Comparison of DOT-4 with MCNP............................ 324
Conclusion .. ........................................... 329


viii


CHAPTER


Paqe







APPENDICES Page

C DESCRIPTION OF THE ISOLATOR OF SECONDARY COUPLING
EFFECTS CODE......................................... 331

Introduction .. ......................................... 331
Description of the ISCE Code............................ 331
The MAIN Module.................................. .. 331
The REED Module...................................... 332
The ERIN Module..................................... 332
The NOUT Module............................... ...... 333
The ESTM Module.................................. .. 333
The RITE Module ................................... .. 337
Input Data Format ..................................... 337
Input Data File ..................................... 337
List of Input Data Files.............................. 339
Comparison of Results Obtained from ISCE with Results
Obtained Directly from MCNP .......................... 340

D CIRCULATING-FUEL, COUPLED CORE POINT REACTOR
KINETICS EQUATIONS....................................... 345

Description and Definition of Symbols, Parameters, and
Terms used in the Circulating-Fuel, Coupled Core
Point Reactor Kinetics Equations...................... 350
Definition of Superscripts and Subscripts............. 350
Definition of Integral P rameters..................... 351
Neutron population, NJ(t).......................... 351
Reactivity, pJ(t)................................. 354
Effective delayed neutron fractio A(t)........... 354
Prompt neutron generation time, A^(t)............. 358
Effective dejlyed neutron precursor concentration
for the i delayed neutron grQu, Ci(t)........ 358
Effective coupling coefficient, i (t)............ 359
Interpretation of Equations (D-1) and (D-4)........... 361
Equation (D-1) ................................... 361
Equation (D-4).................................... 365

LIST OF REFERENCES.............................................. 366

BIOGRAPHICAL SKETCH....... ... ....... ...... .............. ..... 371













LIST OF TABLES


Table Page

3-1 Properties of Selected Metal Fluoride Working Materials..... 30

3-2 200 MW UF /UTVR Power Cycle Therm9dynamic Operating
ChaFacteristics for NaF, KF, Li F, and RbF Working
Fluids.............................................. .. 34

3-3 Energy Balance7Data for 200 MW UF,/UTVR Power Cycle with
NaF, KF, Li F, and RbF Workng Fluids................... 35

3-4 PUTX/PgC as a function of the Metal Fluoride Mass Flow
te- Cb the Boiler Region as Required on the Basis of
Thermodynamic/Flow Considerations ....................... 36

4-1 kfA as a function of Voided UTVC Radius for the UF4/NaF
efankine Cycle System.................................. 48

4-2 kefr as a function of the Liquid UF4 Core Volume for a
Two Region Reactor.................................... 60

4-3 keff as a function of UTVC Radius and Metal Fluoride Type... 69

4-4 kef, as a function of NaF Entrance Velocity and Average
Density in the Wall Cooling Region....................... 72

4-5 kefr as a function of Metal Fluoride Type and Wall
Cooling Region Thickness................................ 74

4-6 k e and P /P as a function of NaF Diverted Flow
Rate toUtlf B9CT1r Region............................... 75

4-7 k f and PUTV/Pol as a function of UF Average Density
end the "ck -bp" Number of Boiler Columns in the
Annular Boiler Region................................. 80

4-8 Reactivity Penalty (6k/k) as a function of UTVC Liner
Material Thickness............................... ... .. 82

4-9 Reactivity Penalty (6k/k) as a function of Boiler Region
Liner Material Thickness .............................. 83

4-10 Reactivity Penalty (6k/k) as a function of Both the UTVC
and Boiler Region Liner Material Thickness............... 85

x







Table Page
5-1 kef and P Tvr/P RO as a function of UTVC Radius for the
'6F /NaFURAiki f0Cycle System in R-O Geometry for a
Four-Boiler and an Eight-Boiler Column UTVR
Configuration ............................................ 101
5-2 kef and P /Pn as a function of the Number of UF4
oiler Clumn Ch R-O Geometry.......................... 111
5-3 UF4 Temperature and Density Profiles in the UTVC as a
function of Radial Distance for a Four-Boiler Column
UTVR Configuration in the R-0 Coordinate System.......... 120
5-4 UTVR Dimensions of the Reference R-Z Cylindrical
Configuration ............................................ 125
5-5 k e, P ITC/Pg p, and P P hl as a function of
hGe -Mo tor-RefpYgei r lifane Separator Slab
Region Height ...................................... 127
5-6 keff, P T /P C and P /nr/Ph le r versus the Top BeO
Mode-ator-Reector Rh H ...................... 130
5-7 kef, P TVP/P r, and PJ as a function of
ethe Frt OW r BeO MggetatoPo sector Region
(OBEO#1) Height ...................................... 132
5-8 k ef, P TVC/PBnL, and P /Pho as a function of
ef he Wj ht b the SubM8 ed adlturated Liquid
Region of the Boiler Column............................. 135
5-9 kf P /P and P P "e as a function of
kef oly Num BRckness sPP un gehe Boiler Feedlines
Region............................................... 138
6-1 Description of UTVR Regions Employed in the Three-
Dimensional MCNP Monte Carlo Calculations................ 155
6-2 Reactivity Worths of the Boiler Feedlines, UTVC Inlet
Plenums, and MHD Duct Regions........................... 159
6-3 Selected UTVR Results from a 30-Minute MCNP Monte Carlo
Analog Calculation Performed on a CRAY X-MP/48
Supercomputer ............................................ 162
6-4 UTVR Fission Rate as a function of Neutron Energy........... 169
6-5 Effect of Employing Energy Cutoff on the UTVC and Boiler
Column FOM Tallies............................. ...... 169







Table Page

6-6 Effect of Employing Implicit Capture and Weight Cutoff on
the UTVC and Boiler Column FOM Tallies................... 171

6-7 Effect of Employing Weight Windows on the UTVC and Boiler
Column FOM Tallies.................................... 175

6-8 Effects of Employing Variance-Reduction Techniques in
MCNP Monte-Carlo Calculations on Uncertainties of
Selected UTVR Parameters............................... 177

6-9 Effects of Employing Variance-Reduction Techniques and
utilizing Boiler-to-UTVC Symmetry in MCNP Monte-Carlo
Calculations on Uncertainties of Selected UTVR
Parameters ............................................... 184

6-10 Integral Kinetics Parameters as a function of the UF4
Partial Pressure in the UTVC .......................... 201

6-11 Integral Kinetics Parameters as a function of Saturated
Liquid Cone Region Height for Two Different H Values.. 203

6-12 Integral Kinetics Parameters as a function of Saturated
Liquid Cone Region Height at UF4 Partial Pressures of
2.5 and 7.5 atm in the UTVC ........................... 204

6-13 Integral Kinetics Parameters as a function of HSUB and HSAT
in Boiler Column at a UF4 Partial Pressure of 5 atm in
the UTVC.................. .......................... 206

6-14 Integral Kinetics Parameters as a function of Vapor Cone
Region Density at a UF4 Partial Pressure of 5 atm in
the UTVC ............................................... 208

8-1 Values of Selected UTVR Parameters at the Initial,
Unperturbed Steady State Condition ...................... 267

8-2 Relevant Properties for the UF Fuel, NaF Working Fluid,
and the UF4/NaF Fuel/Working Fluid Mixture............... 268

8-3 Final Equilibrium Conditions as a Result of $ 1.00 Positive
and Negative Reactivity Step Insertions Imposed on the
Boiler Columns....................................... 278

8-4 Final Equilibrium Conditions as a Result of $ 0.20 Positive
and Negative Reactivity Step Insertions Imposed on the
UTVC................................................. 282

8-5 Final Equilibrium Conditions Following a Positive Step
Reactivity Insertion of $ 0.20 Imposed on the UTVC
with Normal and Reduced Coupling Coefficients............ 289

xii







Table Page

8-6 Final Equilibrium Conditions Following a Positive Step
Reactivity Insertion of $ 0.20 Imposed on the UTVC
with Normal and Reduced UTVC Fuel Loading
Coefficients of Reactivity.............................. 293

B-1 XSDRNPM and MCNP Benchmark Calculations on a Five-Region
Spherical "Mock-up" of the UTVR......................... 322

B-2 DOT-4 and MCNP Benchmark Calculations on the Cylindrical
"Mock-up" of the UTVR in both the R-9 and R-Z
Coordinate Systems..................................... 326

C-1 Comparison of Results obtained from ISCE with Results
obtained Directly using MCNP for Two Different UTVR
Fuel Loadings........................................ 343

D-1 The Six Delayed Neutron Groups Energy Spectra, Decay
Constants, Yied, and Fractions Data for Thermal
Fission in U ......................................... 356


xiii













LIST OF FIGURES


Figure Page

1-1 Side View Schematic of the Ultrahigh Temperature
Vapor Core Reactor...................................... 3

1-2 Top View Schematic of the Ultrahigh Temperature
Vapor Core Reactor...................................... 5

3-1 UF6 and UF4 Saturation Vapor Curves......................... 25

3-2 Uranium Metal and UF4 Saturation Vapor Curves............... 27

3-3 Partial Pressures of Constituent Species of the Uranium-
Fluorine System at One Atmosphere ....................... 28

3-4 Schematic of a 200 MWe UF4/KF UTVR MHD-Rankine Cycle Power
System ................................................. 31

4-1 Four Region, One-Dimensional Spherical "Mock-up" of
the UTVR............................................. 41

4-2 keff and PUTVC/PBCOL as a function of the UTVC Radius....... 45

4-3 kef and Fission Rates of the UTVC and the Boiler as a
function of the UTVC Radius............................. 46

4-4 k and P T/P as a function of the Inner BeO
Moderate r-Ref Ttor Region Thickness..................... 51

4-5 kef as a function of the Inner BeO Moderator-Reflector
Region Thickness........................................ 53
4-6 kefc and PRTVr/Pr as a function of the Outer BeO
Moderate r-RefT~E or Region Thickness..................... 56

4-7 ke and PUT/PRrn as a function of the UF4 Inlet
Velocity t te.Boiler Region....................... 58

4-8 keff and PuTVC/Prni as a function of the UF4 Partial
e Pressure 1in tf UTVC.................................... 63

4-9 k as a function of the U235 Enrichment at Different
eF4 Partial Pressures in the UTVC...................... 64

xiv







Figure Page

4-10 kef as a function of the Fissile Fuel (U235 and U233)
Enrichment ............................................. 65

4-11 kef and PUTv /PBC as a function of the UF4 Average
Density in the Biler Region.............. .............. 67

4-12 Five Region, One-Dimensional Spherical "Mock-up" of the
UTVR................................................. 71

4-13 kef and P v/PR i as a function of the UF4/NaF Inlet
Velocity t 0 tfl Boiler Region............................ 77

4-14 k and P /PC as a function of the Li6F Mass Flow
Rate to tUf B/p r Region............................... 78

5-1 Six Region, Two-Dimensional R-B Representation of a
UTVR with Six-Boiler Columns ............................ 97

5-2 kf and P TVC/PCnL as a function of the UTVC Radius for a
eour- ad n Eht-Boiler Column UTVR Configuration...... 100

5-3 kf and P /P as a function of the Inner BeO
eIoderatTr-Reft or Region Thickness for a Four-
Boiler Column UTVR ........... .......................... 106

5-4 k f and P /PBC as a function of the UF Inlet Velocity
efto the HBYer ion for a Four-Boiler CoTumn............ 108

5-5 kff and PTvr/P o as a function of the UF Partial
Pressure in t ObTVC for a Four-Boiler Cotumn
UTVR System............................................ 113

5-6 kff and P C/PC as a function of the UF4 Average
Density iu the Biler Region for a Six-Boiler Column
UTVR System............................................ 117

5-7 Thermal Neutron Flux and Vapor Fuel Temperature Profile as a
function of Radial Position from the Centerline of the
UTVC for a Four-Boiler Column UTVR System................ 118

5-8 Representation of the UTVR in the R-Z Coordinate System..... 122

5-9 The Horizontal Boiler Configuration of the UTVR in the R-Z
Coordinate System................................. .. .. 143

6-1 Side View Schematic of the Four-Boiler Column UTVR on the
y-z Plane at x=0.0................................... 153

6-2 Side View Schematic of a Boiler Column ..................... 157







Figure Page

6-3 Top View Schematic of a Four-Boiler Column UTVR System...... 180

7-1 Schematic of the Core-to-Core Circulating Fuel Coupling..... 212

7-2 Schematic of Boiler-to-UTVC Neutron Transport Coupling for
a Four-Boiler Column UTVR System........................ 215

7-3 Schematic of Boiler-to-Boiler and UTVC-to-Boiler Neutron
Transport Coupling for a Four-Boiler Column UTVR System.. 217

7-4 Block Diagram of the UTVC Transfer Function................. 228

7-5 Block Diagram of the Boiler Column Transfer Function........ 230

7-6 Block Diagram of the UTVR Transfer Function................. 231

7-7 Fuel/Working Fluid Density Profile in the Boiler Column
due to Boiling in Space (=zero gravity).................. 234

7-8 Side View Schematic of the UTVC............................ 251

8-1 UTVC and Boiler Column Regions Power Levels as a function
of Time Following a $ 1.00 Positive Reactivity Step
Insertion Imposed on the Boiler Columns at t=0 sec....... 270

8-2 UTVC Pressure, U235 Loading, and UF /NaF Inlet and Outlet
Mass Flow Rates, as a function of Time Following a
$ 1.00 Positive Reactivity Step Insertion Imposed on
the Boiler Columns at t=O sec........................... 273

8-3 Boiler Column Outlet Mass Flow Rate and U235 Loading as a
function of Time Following a $ 1.00 Positive Reactivity
Step Insertion Imposed on the Boiler Columns at t=0 sec.. 274

8-4 UTVC and Boiler Column Regions Power Levels as a function
of Time Following a $ 1.00 Negative Reactivity Step
Insertion Imposed on the Boiler Columns at t=0 sec....... 277

8-5 UTVC and Boiler Column Regions Power Levels as a function
of Time Following a $ 0.20 Positive Reactivity Step
Insertion Imposed on the UTVC at t=0 sec................. 279

8-6 UTVC and Boiler Column Regions Power Levels as a function
of Time Following a $ 0.20 Negative Reactivity Step
Insertion Imposed on the UTVC at t=0 sec................. 281

8-7 UTVC and Boiler Column Regions Power Levels as a function
of Time Following a $ 0.20 Positive Reactivity Step
Inqrtion_6 posed on the UTVC at t=0 sec with
t = 10 sec .. .................................... 284

xvi







Figure Page

8-8 UTVC and Boiler Column Regions Power Levels as a function
of Time Following a $ 0.20 Positive Reactivity Step
Insertion _posed on the UTVC at t=O sec with
7T = 10 sec.................. .................. .. 285
t
8-9 UTVC and Boiler Column Regions Power Levels as a function
of Time Following a $ 0.20 Positive Reactivity Step
Insertion Imposed on the UTVC at t=O sec with the
Coupling Coefficients Reduced by One Order in Magnitude.. 288

8-10 UTVC and Boiler Column Regions Power Levels as a function
of Time Following a $ 0.20 Positive Reactivity Step
Insertion Imposed on the UTVC at t-0 sec with the
UTVC Fuel Mass Reactivity Feedback Coefficient Reduced
by a Factor of Five.................................. 292

8-11 UTVC and Boiler Column Regions Power Levels as a function
of Time Following a $ 0.20 Positive Reactivity Step
Insertion Imposed on the UTVC at t=0 sec with the
UTVC Fuel Mass Reactivity Feedback Coefficient Increased
by a Factor of Two..................................... 295

A-I Schematic of the Flow between the AMPX System Code Modules.. 314

C-1 Example of the ISCE Code Input Data File.................... 339

C-2 Input Data Files List Format............................... 340

C-3 Output File as obtained from ISCE.......................... 341

D-1 Schematic of Neutrons and Neutron Interactions in the UTVR.. 348

0-2 Top View Schematic of the Plasma Core Assembly (PCA)........ 357

D-3 Probability Distribution function of Delay Times for the
Transport of Neutrons from Core k to Core j.............. 364


xvii













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

STATIC AND DYNAMIC NEUTRONIC ANALYSIS
OF THE URANIUM TETRA-FLUORIDE, ULTRAHIGH
TEMPERATURE, VAPOR CORE REACTOR SYSTEM

By

Samer Dakhlallah Kahook

May, 1991

Chairman: Dr. Edward T. Dugan
Major Department: Nuclear Engineering Sciences

An Ultrahigh Temperature Vapor core Reactor (UTVR) system is

investigated in this research. The UTVR can be characterized as a

thermal, high power density (hundreds of MWth/m ), externally-moderated,

coupled core, highly-enriched U235, circulating-fuel, steady-state,

burst power reactor. The investigated reactor system includes two types

of fissioning regions: (1) the central Ultrahigh Temperature Vapor Core

region (UTVC) which contains a vapor mixture of highly-enriched uranium

tetrafluoride (UF4) fuel and a metal fluoride working fluid at an

average temperature of =3000 K and an average pressure of =50 atm; and

(2) the Boiler COLumn region (BCOL) which contains highly enriched

liquid UF4 fuel. The combination of three features differentiates the

UTVR from other nuclear reactor concepts. These three features are as

follows:

1. the multi-core configuration resulting in a coupled-core system by

means of direct neutron transport through the media;

xviii







2. the circulating fuel and the associated neutronic and mass flow

coupling between the UTVC and boiler cores; and

3. the employment of a two-phase fissioning fuel, i.e., a liquid-vapor

combination.

Static and dynamic neutronic analysis of this novel system

indicates distinct advantages over other existing or conceptual nuclear

power systems. These include a unique combination of some very

effective inherent negative reactivity feedbacks such as the vapor-fuel

density power coefficient of reactivity, the direct neutronic coupling

among the multiple fissioning core regions, and the mass flow coupling

feedback between the two types of fissioning cores.

Static neutronic analysis is performed using multidimensional

discrete ordinates and Monte Carlo neutron transport codes. Parameters

such as the UTVC and boiler column reactivities and reaction rates,

core-to-core neutronic coupling coefficients, and neutron lifetimes as a

function of vapor core density and boiler core liquid volume are

obtained from the static neutronic analysis.

The dynamic behavior of the UTVR is examined using a non-linear

model, which incorporates circulating-fuel, coupled-core, point reactor

kinetics and energetic equations. These equations are solved using a

system analysis code. The dynamic analysis indicates that the unique

and strong negative reactivity feedbacks of the UTVR are capable of

stabilizing the UTVR safely and quickly even when large reactivity

insertions are imposed (6p = $ 1.00). The analysis also shows that the

system exhibits good dynamic performance even when an inherent negative

reactivity feddback is suppressed (e.g., the vapor fuel density power


xix







coefficient of reactivity). However, due to the strength of the UTVR's

inherent negative reactivity feedbacks, it is found that external

reactivity insertions alone are inadequate for bringing about power

level changes during normal operations. Additional methods of

reactivity control, such as variations in the mass flow rate of the fuel

and/or working fluid or variations in the inlet pressure of the

fuel/working fluid entering the boiler columns, are needed to achieve

the desired power level control.












CHAPTER I
INTRODUCTION

Introduction


The concept of Vapor Core Reactors (VCRs) has emerged at the

University of Florida (UF) as a consequence of extensive theoretical and

experimental studies performed on their predecessors, the Gaseous Core

Reactors (GCRs). Unlike GCRs (where the fuel is supplied to the reactor

in gaseous form), the working fluid and/or fuel undergo a liquid-to-

vapor phase change in VCRs. Studies performed on VCRs and GCRs indicate

that gaseous-fueled (or vapor-fueled) reactor concepts have distinct

advantages over other existing or proposed nuclear power systems. These

advantages include high operating temperatures and efficiency, rapid

startup capabilities, simple geometry, and an assortment of efficient

power control methods [1-9].

The Ultrahigh Temperature Vapor Core Reactor (UTVR)/Disk

Magnetohydrodynamic (MHD) Generator Power System is being studied for

the Strategic Defense Initiative Organization (SDIO) as a possible

source for space power. The SDI space power systems are required to

operate in at least one of the three following power modes:

Station-keeping mode (base load). This mode may be required to

produce up to a few Megawatts electric (MWe) for a period of about 7

years.









Alert mode (enhanced surveillance mode). The power requirements

for this mode range from 10's of MWe up to =100 MWe. The power system
must be capable of functioning for periods of a few hours to a few days.

Burst power mode (defense mode). The power level for this mode

ranges from =100 MWe up to =1 Gigawatt electric (GWe) for operating

times of about 30 minutes; a burst power system must be capable of

achieving this power level in less than 100 seconds.

The UTVR/MHD Generator Power System is a burst power mode concept.

At burst power levels, the UTVR can operate at very high temperatures

which provides an efficient heat rejection capability and a high

thermodynamic efficiency. This and other features appear to make the

UTVR/MHD Generator Power System an exceptional concept for burst power

operations. The UTVR/MHD Generator Power System is the concept examined

in this research.

Description of the Ultrahiqh
Temperature Vapor Core Reactor

The UTVR/Disk MHD Generator Power System is a highly enriched

(>85%), BeO externally-moderated, circulating fuel reactor with uranium

tetra-fluoride (UF4) as the fissioning fuel. The working fluid is in

the form of a metal fluoride such as NaF, KF, RbF, and Li7F. Shown in

Figure 1-1 is a side view schematic of the UTVR.

The UTVR includes two types of fissioning regions: (1) the central

Ultrahigh Temperature Vapor Core regions (UTVC) which contain a vapor
mixture of highly-enriched UF4 and a metal fluoride working fluid at an

average temperature of =3000 K and an average pressure of =50 atm, and

(2) the boiler column regions (BCOL) which contain highly enriched UF4









TBEO


.. ....... :y.. WallI
.. ... ..-1- Cool ant
'i ^iiiiiiii~ii!!~ i\'''i.'-







...- Rejection
... .System
::: :.:.: . ...- ......... DOB E O







MHD Duct
V, po r *^I BEO ^ ... To Heat

Region
I BED



p. .i.-i. i;^ Boiler
S. ... ...... -.... Column
. . ... .: '.. r. .." ,






Figure 1-1. Side View Schematic of the Ultrahigh Temperature
Vapor Core Reactor









fuel. This reactor has symmetry about the midplane with identical top

and bottom vapor core and boiler column regions separated by a BeO slab

(mid-plane BeO Region MBEO) and the MHD ducts where power is extracted.

The UTVC is surrounded in the radial direction by the wall cooling

region. The wall cooling region contains a subcooled liquid metal

fluoride. By tangentially injecting the metal fluoride into the UTVC,

the UTVC walls are maintained at the desired low temperatures (=2000 K).

As the metal fluoride is injected into the UTVC, an annular buffer zone

is obtained which aids in maintaining the UF4 away from the UTVC walls.

This reduces the possibility of condensation of uranium or uranium

compounds on the UTVC walls. Beyond this buffer zone, the metal

fluoride vaporizes and mixes with the UF4 in the UTVC.

The UF4 is vaporized in the boiler columns prior to its entrance to

the UTVC. The boiler region, which includes a number of boiler columns,

is connected to the UTVC via the UTVC inlet plenums, as shown in Figure

1-1. The UF4 liquid is supplied to the boiler columns by means of

feedlines. Each boiler column consists of three distinct regions: the

subcooled liquid region, the saturated liquid-vapor region, and the

superheated vapor region.

Shown in Figure 1-2 is a top view schematic of the UTVR. Figure 1-

2 shows three distinct BeO regions: the inner BeO region (IBEO) which

separates the UTVC walls from the boiler columns in the radial

direction, the annular boiler BeO region (BBEO) with a radial thickness

equal to the diameter of the boiler columns, and the outer BeO region

(OBEO) surrounding the boiler columns and the BBEO region. Three other

BeO regions are shown in Figure 1-1. These are the mid-plane BeO region









OBEO


Figure 1-2. Top View Schematic of the Ultrahigh Temperature
Vapor Core Reactor









(MBEO) mentioned previously, the lower BeO region (LBEO) separating the
boiler feedlines from the MHD duct, and the top BeO region (TBEO) above

the UTVC.

Use of the UF4 as the vapor fuel and metal fluorides as the working

fluid in the UTVR/MHD Generator Power System allows for operation on a

direct, closed Rankine type cycle and leads to space power systems with

high efficiency (=20%), small radiator size (=5 m2/MWe), and high

specific power (=5 kwe/kg). A description of an example UF4-Metal

Fluoride UTVR/MHD Generator Rankine Cycle Power System is furnished in

Chapter III.

Dissertation Objectives

A goal of this research is the nuclear design and analysis of the

UF4-Metal Fluoride UTVR/MHD Generator Rankine Cycle Power System for
space power applications. Complete characterization of this innovative

system requires an integrated and thorough investigation of its

neutronic, heat transfer, and mass flow behavior. Although this

research focuses on the nuclear aspects of the proposed system, it

incorporates results from auxiliary and supporting thermodynamic, heat

transfer, and fluid flow calculations, thus, assuring a reliable and

integrated nuclear analysis.

The nuclear design of the UTVR incorporates results from the static
and dynamic neutronic analysis performed on the UTVR. The static

neutronic analysis establishes basic neutronic characteristics and

obtains reference reactor configurations that are optimized for the

static neutronic characteristics while also considering other important









parameters like specific power (kw/kg) for the UTVR. Applicable UTVR

parameters that are needed for the dynamic neutronic studies such as

reactivity, neutron generation time, and core-to-core coupling

coefficients are also obtained from the static analysis.

The dynamic neutronic analysis focuses on characterizing the UTVR

with respect to stability and dynamic response. Effects of core-to-core

neutronic coupling (by means of direct neutron transport through the

media and by delayed neutron emission from the decay of the delayed

neutron precursors which are carried in the fuel that circulates between

the UTVC and boiler columns) and effects of other important reactivity

feedback phenomena such as fuel density and mass flow related feedback

for the vapor and boiler cores are included in the dynamic analysis.

Thus, the primary objective of this research is the development and

application of the methods and the models needed for the nuclear design

and analysis of this unique reactor concept.

It is recognized that acoustic phenomena are inherent to the UTVR

and their effects are potentially very significant. However, acoustic

effects are not included in this research and are recommended for future

work when the necessary tools for treating these effects are available.

Recommended future work will require coupled space-time neutron field-

gas density field calculations.

Dissertation Organization


A brief summary of previous work performed on related reactor

concepts such as gas core reactors, coupled core reactors, and

circulating fuel reactors is presented in Chapter II.









A section addressing preliminary design considerations for the UF4

UTVR reactor system is presented in Chapter III. It includes point

design conditions for the UTVR from preliminary thermodynamic, heat

transfer, and fluid flow calculations. A description of an example UF4-

Metal Fluoride UTVR/MHD Generator Rankine Cycle Power System is also

presented in Chapter III. A section in Chapter III discusses the plan

used in the nuclear design and analysis of the UTVR system.

The results of the static one- and two-dimensional neutronic

calculations are presented in Chapters IV and V, respectively. These

calculations are performed with XSDRNPM [10], a one-dimensional discrete

ordinates (Sn) neutron transport code, and with DOT-4 [11], a one- and
two-dimensional Sn neutron transport code. The static analysis examines

effects of variations in geometry and fuel/working fluid loadings on the

neutron multiplication factor (keff) and power sharing factor (i.e.,

power distribution between the UTVC and the UF4 boiler columns,

PUTVC/PBCOL) Basic neutronic characteristics of the UTVR such as fuel
density reactivity coefficients, optimum BeO region thicknesses, optimum
number of UF4 boiler columns, and a reference UTVR configuration for

three-dimensional analysis are obtained from the static neutronic

analysis results presented in Chapters IV and V.

The results obtained from static three-dimensional neutronic

calculations are presented in Chapter VI. These calculations are

performed using MCNP [12], a three-dimensional Monte Carlo neutron

transport code. Parameters such as UTVC and boiler core reactivities

and reaction rates, core-to-core coupling coefficients, and neutron

lifetimes as a function of vapor core density and boiler core liquid









volume are obtained from the results of calculations performed with

MCNP. The methods and models used in obtaining the core-to-core neutron

transport coupling coefficients and the reactivities of the vapor and

boiler cores are derived and described in Chapter VI.

The circulating-fuel, coupled core, point reactor kinetics

equations for a four-boiler column UTVR are derived in Chapter VII. A

section in Chapter VII contains a detailed discussion of significant

UTVR inherent reactivity feedbacks such as the vapor fuel density

feedback of the UTVC and the liquid fuel/working fluid volume feedback

of the boiler region. Energetics equations relating the power levels

and the neutron population levels of the vapor and boiler cores to

fuel/working fluid temperature, density, and liquid volume and flow

rates are also included in Chapter VII.

The dynamic neutronic analysis and performance studies are included

in Chapter VIII. The dynamic analysis examines the behavior of core

power levels, reactivities, fuel densities, and total system power

during full power transients. Effects of the core-to-core circulating

fuel and neutron transport coupling and fuel density variations in the

vapor core and boiler cores are included in the dynamic analysis.

The conclusions obtained from this research are included in Chapter

IX. Suggestions and recommendations are made for further research which

are needed before the technical feasibility of the UTVR/MHD Generator

Power System can be realized.

A brief description of the nuclear and system analysis computer

codes used in this research is presented in Appendix A. Appendix B

contains the results of benchmark calculations performed with XSDRNPM









and MCNP on a reference UTVR in spherical coordinates. Results from

benchmark calculations performed in R-8 and R-Z cylindrical coordinates

with DOT-4 and MCNP are also included in Appendix B. A description of

the Isolator of Secondary Coupling Effects (ISCE) code is presented in

Appendix C. The ISCE code, a special code developed as a part of this

research, incorporates the models derived in Chapter VI with results

obtained from the MCNP code to obtain parameters needed for the dynamic

analysis and performance studies such as core-to-core neutron transport

coupling coefficients and the reactivities of the UTVC and boiler cores.

Appendix D contains a description of the circulating fuel, coupled core,

point reactor kinetics equations.












CHAPTER II
PREVIOUS RESEARCH ON RELATED CONCEPTS

Introduction

The UTVR can be characterized as a thermal, high power density

(hundreds of MWth/m3), externally-moderated, coupled-core (vapor and

boiling cores), highly-enriched, U235 circulating fuel, steady state

reactor. The combination of three features differentiates the UTVR from

other nuclear reactor concepts. These features are the following:

1. The multi-core configuration resulting in a coupled-core system by

means of direct neutron transport through the media.

2. The circulating fuel and the associated neutronic and mass flow

coupling between the UTVC and boiler cores. This feature provides

additional neutronic coupling between the cores by means of delayed

neutron emission from the decay of the delayed neutron precursors

which are carried in the fuel/working fluid mixture. The mass flow

coupling between the vapor and boiler cores is an inherently

stabilizing phenomenon. For example, an increase in the power level

of the boiler core increases the voiding and decreases the density of

the fuel/working fluid mixture in the boiler core. This leads to a

decrease in the boiler core power level. Additionally, the density

of the fuel/working fluid exiting the boiler core and entering the

vapor core decreases. This causes a decrease in the reactivity of

the vapor core resulting in a decrease in the vapor core power level.

11









The decrease in the vapor core power level causes a decrease in the

number of neutrons directly transported to the boiler cores through

the media and a decrease in the delayed neutron precursor

concentration decaying in the boiler cores. This causes a further

decrease in the boiler core power level.

3. The employment of a two-phase fissioning fuel, i.e., a liquid-vapor

combination.

Studies on reactors combining all three of these key features have

never been reported. However, studies and research pertaining to

coupled-core reactors, circulating fuel reactors, or gaseous (vapor)

core reactors have been reported. Therefore, the following sections of

this chapter briefly summarize previous research on reactors that

possess one of these key features or aspects of the UTVR.

Previous Research on Gas Core Reactors


Research on gas core reactors has been reported as early as 1955 by

George Bell [13]. The reactors examined by Bell employed gaseous UF6

fuel and beryllium (Be), D20, and graphite reflectors in spherically

symmetric geometries. The analysis was done using age theory to

describe neutron slowing down in the moderator-reflector region and

diffusion theory to describe neutron diffusion into the core and the

fissions in the core. The reactor was considered to be strictly a

thermal reactor.

Since then, different analytical methods and models have been used

in studying this and other gas core reactor concepts. This includes the

Nuclear Piston Engine and Pulsed Gaseous Core Reactor Power Systems









examined by E.T. Dugan [2] and the Heterogeneous Gas Core Reactor

examined by K.I. Han [5]. Summaries of previous work on gas core

reactors can be found in the studies reported by Dugan and Han.

Previous Research on Coupled Core Reactors

The initial work on the kinetics of coupled core reactors was

reported in 1958 by Robert Avery [14]. Avery investigated the dynamic

characteristics of coupled fast-thermal breeder reactors. The analysis

incorporates the point reactor kinetics equations for each core. The

equations include terms accounting for the neutronic interaction

(coupling) between the cores. The coupling terms along with integral

parameters used in the point reactor kinetics equations are obtained

from steady state analysis of the interacting cores.

Neutron kinetics studies using the coupled core treatment have been

applied to nuclear reactor systems other than fast-thermal reactors.

These include modular cores of large thermal power reactors, clustered

reactors, and Argonaut-type reactors. Research has also been performed

on coupled gas core reactors. This includes the work performed by M.M.

Panicker [6] on the Coupled Multiple Chamber Gaseous Core Reactor Power

System.

The differences in the various approaches used in the analysis of

coupled core reactor systems lie in the choice of the weighting function

(neutron flux, importance function, or average fission density); the

choice of suitable phase-space regions for the averaging process; and

the selection of how to incorporate the coupling effects (e.g., as a

source term or reactivity effect) into the pertinent dynamic neutronics









equations. Detailed discussions of these differences and their

applications to various reactor systems are reported by Adler et al.

[15] and Panicker [6].

Previous Research on Circulating Fuel Reactors


The kinetics of circulating fuel reactors is affected by the loss

of a fraction of delayed neutrons due to the decay of the delayed

neutron precursors outside the core. The fraction of delayed neutrons

that is lost depends mainly on the time the fuel spends in the core

relative to the time the fuel remains outside the core. Various methods

have been used in approximating the effects of circulating fuel.

The impact of various methods used for approximating the effect of

circulating fuel on the kinetics of nuclear reactors has been

investigated by John MacPhee [16]. In comparing the approximate

methods, MacPhee employed an "exact" model ("exact" with respect to the

method of treating the effect of the circulating fuel on the delayed

neutrons). The "exact" model employed the following assumptions:

1. Point reactor kinetics equations are valid in the sense that the

reactor kinetics effects are considered to be spatially independent.

2. Reactor power level is low enough such that the effect of neglecting

reactivity feedback due to temperature and radiolytic gas formation

is valid.

3. One delayed neutron group is used.

4. Perfect mixing in the core vessel occurs.

5. Fission occurs only in the core.

6. Fuel mass flow rate is constant.









The reactor kinetic equations employed in the "exact" model are

dN(t) p(t) N(t) + C(t) (2-1)
dt A

d(t) N(t) (t) + e- (2-2)
dt A Tc Tc
where

N(t) = neutron population level in the core at time t;

p(t) = core reactivity at time t;
P = fraction of delayed neutrons;

A = prompt neutron generation time;
A average decay constant of delayed neutrons;
t = delayed neutron precursor concentration;

7c time fuel remains in the core;

7T = time fuel spends in the loop outside the core.

Equation (2-1) describes the time dependent behavior of the neutron

population level and Equation (2-2) describes the time dependent

variation of the concentration of the delayed neutron precursors. The

effect of the circulating fuel is accounted for in the last two terms in

Equation (2-2).

MacPhee compared two approximate methods with the "exact" model.

In the approximate methods, modified versions of Equation (2-2) are

employed. The first method employs reduced values for # as shown by

dC(t)_ f N(t) C(t) (2-3)
dt A

where f is the fraction of delayed neutrons lost as a result of the fuel

circulating and is given by









7C
f = (2-4)
Tc + Tj


The second method neglects the delay time associated with the

delayed neutron precursors re-entering the core, i.e., C(t-T,) = C(t).

With this assumption, the equation describing the delayed neutron

precursors concentration is

dC(t) = N(t) - C(t) (2-5)
dt A aD

where aD is the delayed neutron attenuation factor, obtained from steady

state conditions imposed on Equation (2-2), and is given by

rTc
aD = -T (2-6)
XTc + 1 e

MacPhee analyzed the "exact" model by linearizing Equations (2-1)

and (2-2), taking the Laplace transform of the linearized equations, and

computing the frequency response of the linear system. The results of

MacPhee's investigation and comparisons include the following

conclusions:

1. The frequency response of the "exact" model predicts a peak when fast

reactivity changes are introduced. The approximate methods do not

predict the peaking found by the "exact" model. Thus, for fast

reactivity changes the approximate methods are not valid.

2. Although the frequency response indicates peaking, circulating fuel

reactors do not exhibit self-sustained oscillations as a result of

the feedback produced by the delayed neutron precursors re-entering

the core, i.e., the peaking is finite.









The peaking is due to the coupling of the delayed neutron decay

constants with the loop circulation period and occurs for small values

of aD. The reason the peaking is finite is because aD is greater than

zero for all practical reactor configurations. Equation (2-6) indicates

that aD approaches unity as rT approaches zero for all values of T ,

i.e., all delayed neutrons are emitted in the core. However, as 7T

approaches infinity (fuel does not re-enter core), aD approaches zero if

and only if Tc approaches zero such that no delayed neutrons are emitted

in the core. For such cases, the velocity of the fuel in the core is

required to be infinite and an infinite amount of fuel is required to

maintain the reactor critical. Since aD is always larger than zero, the

peaking is therefore finite.

It should be noted that an inherent assumption in MacPhee's

analysis is that the employed fuel is incompressible. Thus, some of

these conclusions do not pertain to the UTVR.

M.A. Schultz [17] indicates that a number of smaller peaks would

occur in the frequency response of circulating fuel reactors if more

than one delayed neutron group is included in MacPhee's "exact" model.

The fact that the mixing of the circulating fuel in the external loop of

an actual reactor will smooth over the peaks and reduce any tendency

toward sustained oscillations is pointed out by Schultz.

The effect of fuel temperature reactivity feedback in circulating

fuel reactors has been investigated by W.K. Ergen [18]. The analysis

indicates damped power oscillations for circulating fuel reactors occurs

with negative fuel temperature feedback. Ergen also concludes that the

decrease in damping of oscillations due to the loss of delayed neutrons








18

is compensated to some extent by the damping effect caused by the

circulation itself.

Remarks


In deriving the models needed to analyze the UTVR/MHD Generator

Power System (see Chapters VI and VII and Appendix D), references to

previous work are also made. Where applicable, modifications to and

comparisons with previous models are indicated.












CHAPTER III
DESIGN OF THE URANIUM TETRA-FLUORIDE,
ULTRAHIGH TEMPERATURE VAPOR CORE REACTOR

Introduction

In the design of nuclear power reactors, the choice of materials

for fuel, moderator, coolant/working fluid, and structure and the

selection of the power extraction system are based on the application

and the required performance of these reactors. Once the appropriate

materials and a suitable power extraction system are selected, a

reference reactor configuration can be chosen. Then, a complete

characterization of the reference reactor power system is required to

determine its overall performance and feasibility.

Although this research focuses on the nuclear aspects of the UTVR,

a section in this chapter addresses preliminary design considerations

that led to the reference UTVR configuration. Another section in this

chapter discusses considerations involved in the materials selection for

the UTVR. A detailed description of an example UF4-Metal fluoride

UTVR/MHD Generator Rankine Cycle Power System is also given. This is

followed by a section discussing the plan followed in this research for

the neutronic analysis of the UTVR system.









Preliminary Design Considerations

Since the UTVR is being developed for SDI's Burst Power Mode for

space power applications, the following issues need be realized:

1. The size and mass of the power system are important constraints.

This is due to the following: (a) the expense and logistics involved

in the deployment of the power system into space, (b) the need to

constantly maneuver and relocate the defense system, and (c) the need

for defense systems to be inconspicuous.

2. The required power level for this system ranges from =100 MWe up to

=1 GWe for operating times of =30 minutes. Such power levels when

considered with the size requirement demand a high power density

system.

3. The system is required to achieve the Burst Power Mode in less than

100 seconds. Thus, the system needs to be designed to withstand

thermal stresses and shocks caused by a rapid transition from the

alert mode.

4. The power system is required to be able to operate during a seven

year period. This requires the power system to be tested

periodically; thus, the system needs to be designed to operate at

full power for a total time of about three hours (assuming two-annual

tests during the seven-year period lasting about ten minutes each

plus the 30 consecutive minutes of operation).

5. The system needs to be operated at high temperatures to provide

compact radiators for heat rejection in space and high power cycle

efficiency.







21

The above issues are the primary considerations applied during the

preliminary design of the UTVR.

Choice of Materials

The UTVR is a BeO externally-moderated, circulating fuel reactor

with UF4 as the fissioning fuel and a metal-fluoride working fluid.

Research is being conducted to select and develop suitable structural

materials that are compatible with the fluoride fuel/working fluid

mixture and the high temperature environment of the UTVR. The choice of

BeO as the moderator-reflector material, UF4 as the fuel, and metal

fluoride as the working fluid is based on the following considerations.

The Moderator-Reflector Material

For thermal reactors, moderator-reflector materials used in nuclear

reactors have low mass numbers and relatively large scattering and

relatively small absorption cross sections. Moderators used in nuclear

reactors include ordinary water (H20), heavy water (D20), beryllium (Be)

or beryllium-oxide (BeO), and graphite. The choice depends largely on

the intended application of the reactor system; and on the nuclear,

mechanical, physical, and chemical properties; and the cost of the

moderator material. Since the size and mass of the power system are

significant constraints, and since high temperatures are needed for

efficient heat rejection in space, the moderator-reflector material is

required to have a high melting temperature (or high boiling temperature

if a liquid moderator is used) and relatively good neutronic properties

(high slowing-down power and small capture cross section for neutrons).







22

For space power reactors, beryllium or BeO is superior to graphite

as a moderator and reflector material from a neutronics standpoint. In

the study of Highly Enriched Heterogeneous Gas Core Reactors (HGCRs),

S.D. Kahook [7] has shown that the use of Be as the moderator and

reflector material provides a higher reactivity (=30% Sk/k) than

graphite (total size of the HGCR was fixed). This is mainly due to the

higher slowing-down power (exls = average logarithmic energy loss per

collision x macroscopic scattering cross section) of 16 m-I for Be

versus 6.5 m-1 for graphite [19] and the (n,2n) reaction of Be. Another

drawback of graphite is its larger thermal diffusion length, LT, (=54 cm
versus =21 cm) and its larger slowing-down length, TT, (=192 cm versus

=100 cm) compared to Be. The larger values of LT and TT require

graphite-moderated reactors to have a larger size compared to beryllium-

moderated reactors, an important design criterion for the space power

system under investigation. Although the melting temperature of

graphite is higher then that of BeO (=4000 K versus =2800 K), BeO is a

better choice than graphite for the power system under investigation.

The drawback of the lower melting temperature of BeO can be compensated

for by the use of auxiliary coolant channels in the moderator-reflector

regions to maintain BeO at safe operating temperatures (=1600 K to =2000

K). Also, due to the low heat conductivity of the vapor fuel and the

fact that the fuel is the working fluid (most of the energy generated is

directly deposited in and removed by the fuel/working fluid mixture),

the temperature of moderator-reflector regions can be considerably

cooler then the temperature of the vapor fuel.







23

E.T. Dugan [2] examined the effect of using H20, D20, Be, BeO, and

graphite as moderator-reflector materials for the Nuclear Piston Engine

which employed an externally-moderated UF6-fueled gas core reactor for

terrestrial power generation. The study indicates that the use of H20

and graphite results in relatively low keff values. This is due to the

relatively high thermal absorption cross section of H20 and high LT and

TT of graphite. However, the large slowing-down power as well as the
(n,2n) reaction of Be and the small thermal absorption cross section of

D20, cause Be, BeO, and D20 to be excellent choices for moderator-

reflector materials as proven by Dugan.

The relatively large LT of D20 of =97 cm requires that the size of

reactors employing D20 as the moderator to be quite large.

Additionally, the high temperature environment of the UTVR, the chemical

incompatibility between H20 or D20 and UF6 or UF4, the normal

deterioration of D20 into H20 in time (small amounts can have large

effects on neutronics), and the added complications involved with a

liquid moderator versus a solid moderator in space all aid in rejecting

D20 as the moderator-reflector material for the UTVR. The ceramic

nature of BeO with a high melting temperature of =2800 K and its

exceptional resistance to thermal shock [20] make this an especially

well-suited moderator-reflector material for the high temperature

environment of the power system under investigation. Although per unit

mass Be is neutronically superior to BeO as a moderator, the anticipated

moderator temperature range of 1600 K to 2000 K for this burst power

system precludes the use of Be (melting point of Be is 1728 K).









The Fissioning Fuel Material

The advantages and key features of vapor-fueled reactors are more

than adequate to justify the study of a fuel in the vapor state.

However, uranium exists in a gaseous state in various forms such as UF4,

UF6, or uranium metal vapor. Reactors employing uranium in these forms

have all been investigated at the University of Florida. The choice of

the fuel along with the working fluid are dictated by the type of power

cycle, e.g., Brayton or Rankine cycle. It is appropriate to compare

features of these cycles in order to select a suitable fuel.

The Brayton cycle is simpler in design than the Rankine cycle.

However, it generally has a lower thermodynamic efficiency. Due to this

lower efficiency, more heat has to be rejected into space which implies

that a larger radiator is needed. In addition, the heat rejection to

space is done at a varying (decreasing) temperature rather than a

constant temperature thereby decreasing the effective temperature of

heat rejection and further increasing the required radiator size. The

greater pumping power required for gas compression in the Brayton cycle

demand larger and more massive compressors as compared to the pumps in a

Rankine type of cycle. Since size and mass are significant constraints,

and since a Brayton type of cycle requires larger radiators and more

massive compressors than a Rankine type of cycle, a Rankine type of

cycle appears to be the better choice, especially for high power

systems.

For space power Rankine cycle systems, the most desirable fuel

choice is UF4. This can be seen from Figure 3-1 where the UF6 and UF4

saturation vapor curves are shown and from the uranium metal and UF4




































































L Il I IO
o o o0 i
M i0 0


(prI) afnssaJd JodPA









saturation vapor curves given in Figure 3-2. For the UF6 to be in the

gaseous state, at pressures required for criticality in the core, its

temperature need only be in the 400 to 500 K range. This implies that

the UF6 must be 400 K or less to achieve a liquid state at the exhaust

pressures when a gas turbine or MHD generator is used for power

conversion. This low heat rejection temperature can easily be achieved

on earth, but is unrealistic in a space environment. Thus, one is

restricted to a Brayton type of cycle when UF6 is the fissioning fuel

fluid in a space power system.

When uranium metal vapor is used as the fissioning fuel and working

fluid, the difficulty is not in achieving a liquid state at the heat

rejection end of the cycle as with UF6. For example, at an exhaust

pressure of 1 atm one need "cool down" to only about 4000 K to achieve

liquid uranium. The obstacle with uranium metal vapor is the extremely

high temperatures of the vapor in the core. The fluid temperature needs

to be at least 6000 K at all locations in the core to ensure the vapor

state at pressures needed for criticality. This indicates that the peak

gas temperature in the core will be at least 8000 K or 9000 K.

The choice of UF4 as the fuel rather than UF6 or uranium metal

vapor is justified by examining the saturation vapor curves, Figures 3-1

and 3-2, and the mole fraction of constituent species versus temperature

curve of the uranium-fluorine system, Figure 3-3 [21]. At pressures

required for criticality in the core, the temperature of UF4 need be

only about 2000 K to guarantee the vapor state. On the other hand, one

need cool down to only 1700 K in order to obtain liquid UF4. The

extremely high uranium metal vapor temperature in the core and the




































































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extremely low UF6 heat rejection temperatures are avoided. Thus, UF4

has a saturation vapor pressure-temperature behavior that is highly

desirable for a direct Rankine cycle burst power system for a space

environment. Also, Figure 3-3 indicates that in the expected gas

temperature operating range of 2500 K to 4000 K, UF4 is the predominant

uranium-fluorine specie.

The Working Fluid Material

It has been shown in the previous section that a Rankine type of

cycle is more appropriate for a burst mode space power system, and on

this basis the fuel is selected to be UF4. Therefore, a working fluid

that is compatible with the UF4 fuel and suitable for a Rankine type of

cycle is needed.

Preliminary chemical and material studies [22,23] indicate that a

working fluid in the form of a metal fluoride should be compatible with

the UF4 fuel. These working fluids include Li7F, KF, NaF, and RbF.

Table 3-1 list relevant properties of these materials.

Description of a Uranium Tetra-Fluoride,
UTVR/Disk MHD-Rankine Power Cycle


An example UF4/KF UTVR MHD-Rankine cycle power system schematic is

shown in Figure 3-4. This system is capable of producing 200 MWe with a

thermodynamic efficiency of z26%. The mass flow rates of UF4 and KF are

59 and 209 kg/sec, respectively.

For the system illustrated in Figure 3-4, about 40 MW is required

to vaporize the liquid UF4 in the UF4-boiler. The UF4 vapor is then

directed to the UTVC where it is mixed with the KF. In the UTVC, 30 MW























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of the fissioning power is deposited in the UF4 to raise its temperature

from 2350 K to 4000 K. The KF flows around the UTVC cooling the vapor

core wall region where about 35 MW is added to it in the form of

sensible heat to raise its temperature from 1920 K to 2300 K. Then the

KF is injected into the vapor core where it is mixed with the UF4. In

the UTVC, 115 MW is added to the KF to raise its temperature from 2300 K

to 2665 K and 367 MW is added in the form of latent heat to vaporize it

at 2665 K. Another 180 MW of heat is then required to raise the KF

temperature from 2700 K to 4000 K. A total of 697 MW of thermal power

is therefore added to the KF fluid.

The UF4/KF mixture is then passed through the nozzle, the MHD

generator, and the diffuser. In the MHD generator, about 10 MW of
energy is produced by fission and 200 MWe is extracted. Waste heat in

the amount of 526 MW is rejected to space via a 720 m2 primary

condensing radiator which allows the separation of the vapor mixture

into UF4 vapor and KF liquid. The UF4 vapor is then passed through a 56

m2 secondary condensing radiator in which 31 MW is rejected to space.

Both the UF4 and the KF are then compressed via separate pumps.

For the purpose of generating this cycle, it is assumed that UF4

and KF are completely separable; this may not be the case. In a real

system many species, including KxU Fz species, may be present as shown

by Hildenbrand and Lau [23].

The system described above has the potential to be extremely

reliable since the only components with moving parts are the UF4 and KF

liquid pumps.









Table 3-2 lists 200 MWe UF4/UTVR power cycle operating

characteristics for KF, NaF, Li7F, and RbF working fluids. Table 3-3

presents the energy balance data for a 200 MWe UF4/UTVR power cycle with

KF, NaF, Li7F, or RbF as the working fluids. Table 3-2 indicates that
if none of the metal fluoride is vaporized in the boiler columns, then

the required power sharing (or power ratio) of the UTVC to the boiler

core (PUTVC/PBCOL) based on thermodynamic/flow considerations is =20.

It also indicates that =90% of the total fission power produced is

deposited in the metal fluoride. The power sharing ratio, PUTVC/PBCOL'

places another restriction on the power system. That is, the UTVR needs

to be configured so that the power sharing between the UTVC and boiler

columns based on nuclear analysis matches what is obtained from

thermodynamic and flow considerations. One method of controlling

PUTVC/PBCOL is to divert part of the metal fluoride from the UTVC wall
cooling region to the boiler region. The power sharing as a function of

metal fluoride mass flow rate to the boiler region as determined on the

basis of thermodynamic and mass flow requirements is listed in Table 3-

4.

Table 3-4 indicates that the required power sharing ratio decreases

by a factor of =3 for NaF, KF, and Li7F and by a factor =2 for RbF when

only 10% of the metal fluoride is diverted to the boiler region. This

is due to the large amount of power needed to vaporize the metal

fluoride as compared to the UF4.


















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Neutronic Analysis of the
Ultrahiqh Temperature Vapor Core Reactor


Static Neutronic Calculations

A number of computer codes existing at the University of Florida,

San Diego SuperComputer Center, and the Air Force SuperComputer Center-

Kirtland are used in analyzing the static neutronic behavior of the

UTVR.

In the static neutronic analysis, the fuel is assumed to be

stationary. Thus, the effect of the loss of delayed neutrons outside

the cores is ignored. This assumption over-estimates the contribution

of the delayed neutrons inside the cores, i.e., keff is somewhat larger

for the stationary system than for the corresponding circulating system.

This effect is corrected for in the dynamic model and is discussed in

detail in Appendix D.

One of the computer packages used at University of Florida is the

AMPX [24] modular code system which is described in Appendix A.

Weighted multigroup neutron cross-sections files are generated from

ENDF/B [25] data using the XLACS [26] code, an AMPX module. Self-

shielding calculations are performed on the weighted multigroup neutron

library created by XLACS using NITAWL [27]. NITAWL produces a 123-

neutron group AMPX library. This 123-neutron group library is then

collapsed first to 27- and then to four-neutron groups using XSDRNPM

[10].

Four-neutron group, 1-D, spherical geometry, discrete ordinates

(Sn) calculations are performed using XSDRNPM. Group dependent neutron
flux distributions in space, region reactions rates, and eigenvalues are









obtained from the 1-D calculations. Basic static neutronic

characteristics of the UTVR are obtained from the 1-D calculations.

These include keff and PUTVC/PBCOL behavior as a function of moderator-

reflector regions thickness, fuel density and enrichment, and types of

working fluids. Reactivity penalties as a function of different liner

materials and their thicknesses are also obtained from the 1-D neutronic

static calculations.

Two-dimensional Sn cylindrical geometry calculations in the R-O and

the R-Z coordinate systems are performed using DOT-4 [11]. The four-

neutron group cross-section library produced by XSDRNPM is converted to

DOT-4 format using GIP [28]. In R-9 geometry, the boiler region can be

accurately modeled as a number of boiler columns separated by BeO

moderator. The nozzle, disk-MHD generator, and diffuser regions can all

be modeled in the R-Z geometry. Results obtained from the 1-D spherical

"mock-up" of UTVR are compared with results obtained from calculations

performed in the R-9 and the R-Z coordinate systems. These comparisons

are necessary to determine the reliability of the obtained results.

The reference UTVR configuration for the 3-D analysis is obtained

from 1- and 2-D static neutronic calculation results. The 3-D

calculations are performed using MCNP [12], a 3-D Monte Carlo neutron

transport code. Integral parameters for the dynamic neutronic analysis

are calculated from MCNP results by using ISCE, a special code developed

as a part of this research. Parameters obtained from ISCE include core-

to-core coupling coefficients, and the reactivity and neutron

multiplication factors of individual cores.









Dynamic Neutronic Calculations

The over-estimate of keff obtained from the static neutronic

calculations due to the assumption of a stationary fuel is corrected for

in the UTVR kinetic model.

The dynamic analysis in the time domain is performed using

circulating-fuel, coupled-core, point reactor kinetics models. Inherent

reactivity feedback effects such as vapor fuel density and boiler column

liquid volume changes are included in the dynamic model. Dynamic and

stability analysis studies are performed with the Engineering Analysis

System code, EASY5 [29].

The computer codes mentioned above are described in Appendix A.













CHAPTER IV
STATIC, ONE-DIMENSIONAL, UTVR NUCLEAR
CHARACTERIZATION AND CONFIGURATION OPTIMIZATION

Introduction

The initial one-dimensional (1-D) spherical "mock-up" configuration

used to perform the preliminary nuclear characterization of the

Ultrahigh Temperature Vapor Core Reactor (UTVR) is shown in Figure 4-1.

It consists of four regions (the wall cooling region is neglected in the

initial calculations). The first is the Ultra High Temperature Vapor

Core (UTVC) region which contains the fuel mixture that consists of

highly enriched UF4 vapor and a metal fluoride vapor at 3000 K and 50

atm. The second is the inner beryllium oxide (BeO) moderator-reflector

region (IBEO) which contains only BeO. The third is the boiler column

(BCOL) region where the UF4 is vaporized. The fourth is the outer BeO

moderator-reflector region (OBEO).

The actual cylindrical reactor system is converted to the 1-D

spherical geometry by conserving the volumes of the UTVC and the boiler

cores and by conserving the thicknesses of the inner and outer BeO

moderator-reflector regions. Neutronic calculations are performed using

XSDRNPM [10]. XSDRNPM is capable of computing the system's neutron

multiplication factor (keff), region average and local neutron fluxes

and currents, and the fission rate in each region (power produced).

XSDRNPM is described in detail in Appendix A.


















Outer BeO Moderator-
Reflector Region

UF4 Boiler Region

Inner BeO Moderator-
Reflector Region -




Ultrahigh Temperature
Vapor Core

Region 1




Region 2 -- -


Region 3 -- -



Region 4


Figure 4-1. Four Region,
"Mock-up" of


One-Dimensional Spherical
the UTVR









Modeling the UTVR in the 1-D spherical geometry is expected to

result in excessively high values for keff. The keff values are

expected to be quite large due to the following:

1. Neutron leakage from the UTVR is underestimated since spherical

configurations provides the smallest surface-to-volume ratio.

2. Neutron streaming from the MHD duct is not accounted for since the

MHD duct regions is not included in the 1-D spherical "mock-up."

3. Reactivity worth of the boiler column is overestimated since the

boiler column is treated as a spherical shell surrounding the UTVC.

The boiler column in the actual reactor system consists of a number

of boiler columns separated by BeO moderator. By configuring the

boiler region as a spherical shell surrounding the UTVC, the

probability for neutrons interacting with the boiler region is

relatively large. Additionally, thermal neutron flux depression in

the boiler region is underestimated since the thickness of the boiler

region in 1-D is small compared to the thickness of the actual boiler

columns.

4. Core protective materials (liners and cladding), structural support

members, and piping are not included in the analysis in order to

simplify the scoping analyses. That is, the 1-D spherical "mock-up"

represents a "clean" UTVR system.

Detailed three-dimensional neutronic analysis using MCNP [12]

(Chapter VI) indicate that, when neutron leakage and streaming are

accounted for, actual boiler configuration is modeled, and structural

and liner materials are employed, keff values of =1.05 is obtained.









Therefore, the high keff values obtained in this preliminary stage of

analysis are "reasonable" and needed.

Scoping Calculations

To commence the nuclear characterization of the reactor system,

numerous 1-D scoping calculations are performed. These calculations

examine the effect of variations in geometry, fuel density, fuel

enrichment, mole fraction, and materials. The results of these studies

are described in this section.

Geometric Variations

As mentioned previously, size and mass are significant constraints

on space power systems. In addition to the total power requirement, an

important design consideration for this system is the power sharing or

the amount of power generated in each fissioning region (UTVC regions

and the boiler regions). Of these constraints (power sharing, total

power production, and size and mass of the reactor system), the power

sharing is expected to be the most restrictive. To determine which

configurations are capable of meeting these constraints, the effects of

variations of the following parameters are examined:

UTVC radius

While maintaining the inner and outer BeO region thicknesses at 20

cm and 35 cm, respectively, and the boiler region volume at 8.5 x 10-03

m3 (the boiler region contains an equal mixture by volume of liquid and

vapor UF4 with an inlet velocity of 2 m/sec at a mass flow rate of 68

kg/sec), the radius of the vapor core (UTVC) is varied from 40 to 150









cm. The fuel mixture in the UTVC region is maintained at 3000 K with

the partial pressures of the UF4 and the NaF fixed at 5 and 45 atm,

respectively. The results, keff and PUTVC/PBCOL' are given in Figure 4-
2 as a function of UTVC radius.

Figure 4-2 indicates that keff increases from 1.462 to 1.479 and

PUTVC/PBCOL increases from 0.23 to 0.53 as the UTVC radius increases
from 40 cm to about 70 cm. However, for UTVC radii above 70 cm keff

decreases while PUTVC/PBCOL continues to increase.

The interpretation of the behavior of keff and PUTVC/PBCOL as a

function of the UTVC radius requires two further sets of calculations.

The first consists of varying the UTVC radius from 40 to 150 cm while

the boiler region is voided. The second consists of varying the UTVC

radius over the same range but with a voided UTVC and a loaded boiler.

The results are given in Figure 4-3, where values of keff and fission

rate versus the UTVC radius are given for both cases.

The fission power produced in a region is a function of the thermal

neutron flux, the macroscopic fission cross section, and the volume of

the fissioning region. When the boiler is voided and the UTVC radius is

varied from 40 to 150 cm, the most significant change is an increase in

the UTVC volume. This increase in the UTVC volume results in an

increase in PUTVC. On the other hand, when the UTVC is voided and the

UTVC radius is varied from 40 to 150 cm, two phenomena occur: (1)

neutronic de-coupling of the annular boiler region, and (2) reduced

number of mean free paths that a neutron encounters when passing through

the boiler. That is, as the UTVC radius increases and the boiler volume

is fixed, the boiler region in the 1-D spherical "mock-up" becomes






















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thinner and more separated in space from itself. Thus, fewer fissions

occur and a decrease in PBCOL is observed.

Thus, the observed increase in PUTVC/PBCOL as the UTVC radius

increases, when both cores are loaded (Figure 4-2) is due to the

increase in PUTVC and the decrease of PBCOL as shown in Figure 4-3.

Although not shown in Figure 4-3, when the vapor core is voided an

optimum value for keff is obtained at a UTVC radius between 0 and 40 cm

for an inner BeO thickness (IBEO) of 20 cm. This optimum UTVC radius is

due to optimum neutronic coupling from one segment of the boiler region

to the other. Calculations for the voided UTVC configuration have been

performed as its radius is varied from 0 to 80 cm at IBEO thickness of

0, 5, 10, 15, and 20 cm. The results are given in Table 4-1 for this

type of variation.

For the case where the IBEO is 20 cm, keff peaks at a voided UTVC

radius of about 30 cm. At this radius, the strongest neutronic coupling

for the boiler region exists. However, for smaller IBEO thicknesses of

15, 10, 5, and 0 cm, a larger voided UTVC radius (beyond 40 cm) is

needed to show the peak in keff. This indicates that as the IBEO

thickness decreases, the voided UTVC region radius required for optimum

boiler coupling increases. However, when the UTVC is loaded, the radius

required for optimum boiler coupling is different as is shown in the

following section.

As mentioned previously, for an IBEO of 20 cm and UTVC radii

greater than 40 cm, keff increases as the loaded UTVC radius increases

when the boiler is voided and keff decreases as the unloaded UTVC radius

increases when the boiler is loaded (as shown in Figure 4-3). However,













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the overall or net effect for the reactor system when both cores are

loaded is an increase in keff until the UTVC radius is about 70 cm; keff
then decreases for radii beyond 70 cm. This can be explained as
follows: as the UTVC radius increases from 40 to 70 cm, the gained

reactivity from the "larger" UTVC core overrides the lost reactivity due

to the de-coupling of the boiler region. For vapor core radii beyond 70

cm, there is a diminishing gain in reactivity due to the larger UTVC and

the boiler region de-coupling effects dominate. A justification for

this assertion is as follows: in the range of 40 to 70 cm for the UTVC

radius, from the data used to generate Figure 4-3, the gain in 6k/k for

the loaded UTVC is about 0.28 (or 9.3 x 10-03 6k/k per cm of IBEO) and

the loss in 6k/k for the loaded boiler is only 0.08 (2.7 x 10-04 6k/k
per cm of vapor core radius); thus, a net increase in keff is obtained.

However, when the UTVC radius increases from 70 to 150 cm, the gain in

6k/k for the loaded UTVC is only about 0.18 (or 2.3 x 10-03 6k/k per cm

of vapor core radius) and the loss in 6k/k for the loaded boiler is

about 0.37 (or 4.6 x 10-03 6k/k per cm of IBEO); thus, a net decrease in

keff is obtained. The observed decrease in the rate of increase of keff

as the UTVC radius increases beyond 70 cm for the case of the loaded

UTVC and voided boiler is due to approaching infinite reactor

configuration; this is seen in Figure 4-3 where the vapor core keff

clearly begins to saturate as the UTVC radius increases beyond =100 cm.

It is concluded from the above discussions, that the UTVC radius

along with the IBEO thickness are the determining parameters that

influence neutronic coupling for the UTVC/boiler regions and the

neutronic coupling from one segment of the boiler to the other. That









is, varying the UTVC radius for a fixed IBEO thickness, or varying the

IBEO thickness for a fixed UTVC radius, will lead to an optimum

configuration with respect to neutronic coupling. Thus, for a given

UTVC radius, an IBEO thickness can be selected to yield optimum overall

neutronic coupling between the boiler and the UTVC. A value of 60 cm is

selected for the UTVC region radius for further analysis. This value is

based on the fact that the UTVC is in reality a cylinder and is expected

to be 100 cm in height with a radius of about 55 cm. These dimensions

appear to be in a range that is acceptable with respect to thermo-

hydraulics and acoustic calculations performed on the system [30].

Inner BeO moderator-reflector region thickness

With the UTVC radius fixed at 60 cm, the vapor fuel temperature set

at 3000 K, the UF4 partial pressure set at 5 atm, the NaF partial

pressure set at 45 atm, the boiler region volume fixed at 8.5 x 10-03

m3, and the OBEO region thickness held at 35 cm, the IBEO thickness is

varied from 5 to 50 cm. The results are given in Figure 4-4 where keff

and PUTVC/PBCOL are plotted as a function of IBEO.

The results indicate that the optimum neutronic coupling between

the vapor core and the boiler region occurs at an IBEO thickness of

about 16 cm where keff is greatest with a value of about 1.507. Beyond

a BeO thickness of 16 cm, keff decreases. This is due to the de-

coupling phenomenon for the boiler and the decreased thickness of the

boiler at the higher IBEO thicknesses.

Figure 4-4 also indicates that PUTVC/PBCOL initially increases as

the IBEO thickness increases from 5 cm to about 16 cm and remains at a

constant level of 0.39 as the IBEO thickness increases from about 16 to




















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45 cm. This ratio then undergoes a further increase as IBEO increases

beyond 45 cm. Three relevant phenomena occur as the IBEO thickness is

increased, they are: (1) a decrease in the number of mean free paths

that a neutron encounters when passing through the boiler due to the

decrease in the boiler thickness as IBEO increases; (2) neutronic de-

coupling of the annular boiler region; and (3) an increase and then a

decrease in the neutronic coupling between the UTVC and boiler cores.
In order to explain the observed behavior of k ff and PUTVC/PBCOL two

additional sets of calculations are needed. The first involves varying

the IBEO thickness from 5 to 60 cm for a fully loaded UTVC with a radius

of 60 cm and with the boiler voided. The second set involves varying

the IBEO thickness over the same range for a voided vapor core with a

radius of 60 cm and with a loaded boiler (volume fixed at 8.5 x 10-03

m3). The results are shown in Figure 4-5.
The results indicate that for a 60 cm radius voided UTVC, optimum

neutronic coupling of the boiler column occurs at an IBEO thickness of

=12 cm. As the IBEO increases beyond 12 cm a decrease in keff occurs.

This decrease in keff of the boiler region translates to a decrease in

PBCOL* However, for the case where the boiler is voided and the UTVC is
loaded, keff continues to increase as IBEO increases. From the data

used to generate Figure 4-5, as IBEO increases from 10 to 45 cm, the

values of 6k/k per cm of IBEO are +1.9 x 10-03 and -4.8 x 10-03 for the
loaded UTVC and for the loaded boiler cases, respectively. These values

reflect the change in reactivity expected if the only phenomena that are

affected by the variation in IBEO are those listed in items 1 and 2






















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above, and the net result should be a continuous increase in PUTVC/PBCOL

and a continuous decrease in keff

However, neutronic coupling between the UTVC and the boiler cores

is also affected by variations in the IBEO thickness. As the IBEO

thickness increases from 5 to =12 cm, PUTVC increases (due to increase

in the UTVC neutron reflection rate, which can be inferred from the keff

curve shown in Figure 4-5, and due to an increase in boiler-to-UTVC

neutronic coupling) and PBCOL increases (due to enhanced neutronic

coupling of the boiler region, as shown in Figure 4-5). However, the

increase in PUTVC is larger than the increase in PBCOL" This causes

PUTVC/PBCOL to increase. As the IBEO region increases from 12 to =16
cm, PUTVC continues to increase while PBCOL begins to decrease, thus a

further increase in PUTVC/PBCOL is obtained. Although Figure 4-5

indicates that an increase in PUTVC should occur in the IBEO thickness

range of 16 to 45 cm for the case the vapor core is loaded and the

boiler region is voided, the actual result when both cores are loaded is

a decrease in PUTVC* For IBEO thicknesses above 16 cm, a decrease in

the boiler-to-UTVC neutronic coupling occurs which causes PUTVC to

decrease. In this IBEO thickness range, PUTVC is decreasing at about

the same rate PBCOL is decreasing. The net result is a constant

PUTVC/PBCOL behavior over an IBEO range from about 16 to 45 cm.
From the data used to generate Figure 4-5, for the IBEO thickness

range from 45 to 60 cm, the values of 6k/k per cm of IBEO are about -5.1
x 10-03 and +3.0 x 10-04 for the loaded boiler cases and the loaded

UTVC, respectively. Above an IBEO thickness of 45 cm, the rate of









decrease in PBCOL is greater than the rate of decrease of PUTVC* This

leads to an increase in PUTVC/PBCOL and a further decrease in keff'

The combined neutronic coupling between the UTVC and the boiler

region (i.e., the combined boiler-to-UTVC, UTVC-to-boiler, and annular

boiler region neutronic coupling) is an optimum at an IBEO thickness of

=16 cm for this configuration. For future calculations, a thickness of

15 cm is selected for IBEO.

Outer BeO moderator-reflector region thickness

Maintaining the UTVC radius at 60 cm, the IBEO thickness at 15 cm,

and the boiler volume at 8.5 x 10-03 m3 (0.12 cm thick), the OBEO region

thickness is varied from 10 to 100 cm in order to obtain the optimum

outer BeO thickness. The results are shown in Figure 4-6 where keff and

PUTVC/PBCOL are plotted as a function of OBEO thickness.
Figure 4-6 clearly indicates that keff saturates at an OBEO

thickness of about 40 cm. At this thickness and beyond, keff is around

1.52 and PUTVC/PBCOL is 0.36. Increasing the OBEO thickness beyond 40

cm does not enhance the system neutronically, i.e., the value of keff*

It only increases the size and the mass of the system. This is very

undesirable since the system is intended for space power production.

The results also indicate that as the OBEO thickness decreases below 40

cm, PUTVC/PBCOL increases since PBCOL decreases. This is anticipated

since the OBEO has a direct effect on the boiler region and less of an

effect on the UTVC. Thus, a method to increase PUTVC/PBCL is to reduce

the thickness of the OBEO. However, this will cause a greater number

of- and more energetic neutrons to leak out of the reactor which will

require the use of heavier and thicker shielding.























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For all further calculations, a thickness of 40 cm is selected for

the OBEO.

UF4 boiler region thickness

Increasing the UF4 boiler region thickness in the "mocked-up"

spherical geometry is analogous to increasing the cross sectional flow

area of the actual boiler column configuration. This in turn results in
a decrease in the inlet velocity of the UF4 liquid to the boiler

(assuming a fixed mass flow rate is required). The inlet velocity of

the UF4 liquid to the boiler dictates the amount of liquid UF4 present

in that region at a given power level. Thus, the reactivity worth of

the boiler is strongly influenced by the inlet UF4 velocity. This

velocity also impacts on the boiler region friction and acceleration

pressure losses. The lower the inlet velocity the lower the pressure

losses, but if the velocity is too low, then the size (area) of the

boiler region will be great. Knowledge of the neutronic behavior of the

system as a function of the inlet velocity of the UF4 liquid to the

boiler is obviously essential. The velocity of the UF4 liquid is varied

from 0.5 m/sec to 6.0 m/sec which corresponds to annular boiler region

thicknesses from about 0.48 cm to about 0.04 cm. The keff and

PUTVC/PBCOL results as a function of UF4 inlet velocity are plotted in
Figure 4-7.

Figure 4-7 indicates that as the inlet velocity of the UF4 liquid

increases, keff decreases and PUTVC/PBCOL increases. Both behaviors are

due to the decrease of the boiler area (volume) as the velocity

increases. A decrease in the amount of fissile material in the boiler

causes PBCOL to decrease thus increasing PUTVC/PBCOL and decreasing























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keff, respectively. It should be noted that selecting an inlet UF4

velocity based only on results from these 1-D calculations is not
reasonable since the boiling and non-boiling regions (amount of vapor

versus liquid UF4) cannot be accurately modeled. Two- and three-

dimensional modeling are needed in order to select a suitable operating

velocity. For these preliminary studies, a flow rate of 2 m/sec is

chosen. This flow rate, 2 m/sec, corresponds to a boiler volume of 8.5

x 10-03 m3 (100 cm in height) at a mass flow rate of 68 kg/sec.

UF4 boiler core volume

A safety consideration in the design of the UF4/metal-fluoride

nuclear power system is the unwanted possibility of self-criticality in

a UF4 boiler region. That is, the size of the boiler columns and the

amount of the liquid UF4 present in the boiler region should be chosen

so that the region cannot become self-critical even under extreme

conditions. For this study, a two region core, in 1-D spherical

geometry, is used to mockup a 50 cm in height UF4 boiler core surrounded

by 40 cm of BeO reflector or 40 cm of Be reflector. The spherical

radius of the core is varied from 4 cm to 11 cm which corresponds to an

equivalent cylindrical radius of 1.3 cm to about 6.0 cm. The boiler

contains 100% enriched U235 in completely liquid UF4. Values for keff

range from 0.491 for the spherical radius of 4 cm to 1.101 for an 11 cm

spherical radius, as shown in Table 4-2. The boiler is found to be

critical (keff = 1.0) at a spherical radius of about 9.8 cm, which

corresponds to a cylinder with a radius of about 5 cm.

The results indicate that the UF4 boiler columns will not become

self-critical since in reality they will not contain 50 cm of liquid.



























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If the reactor system has only two boiler columns at a total UF4 mass

flow rate of 62 kg/sec, then the required cross sectional flow area per

column for an entrance velocity of about 2 m/sec for the UF4 liquid will

be about 4.2 x 1003 m2. This corresponds to a cylindrical radius of
about 3.7 cm which is lower than the critical radius obtained in the 1-D

calculations; a larger number of columns would result in even smaller

boiler region radii (for an inlet velocity of 2 m/sec).

Fuel Density Variations

During reactor startup and power level changes, there will be

changes in the UF4/MF vapor pressure and temperature and, thus, in the

density. Also, the amount of liquid and the void volume fraction of the

UF4 in the boiler region will change depending on the power level. An

insight into the effect of density changes will help in predicting the

response of the reactor to power level changes and/or perturbations

introduced into the system. The effects of variations in the following

parameters are therefore studied.

UF4 partial pressure and mole fraction (UF4:NaF) in the UTVC
Preliminary calculations and analysis of the MHD generator indicate

that a mole fraction of about 10% for UF4 and 90% for NaF results in

efficient energy extraction [31]. Maintaining the UTVC radius at 60 cm,

the IBEO thickness at 20 cm, the boiler volume at 8.5 x 10-03 m3, and

the OBEO thickness at 35 cm (these calculations were performed prior to

obtaining the optimum IBEO and OBEO thicknesses of 15 and 40 cm,

respectively), the UF4 partial pressure is varied from 1 to 20 atm at









NaF partial pressures of 20, 40, and 60 atm. The result of these

variations are shown in Figure 4-8.

The results indicate that the system is essentially unaffected

neutronically by the NaF partial pressure. Thus, the UTVC can be

operated with a fuel mixture that is optimum with respect to the demands
of the MHD generator. Figure 4-8 indicates that keff saturates at UF4

partial pressures above =10 atm. This corresponds to a U235 density of

2.5 x 10-05 atoms/barn-cm. Beyond a UF4 partial pressure of 10 atm or a

U235 density of 2.5 x 10-05 atoms/barn-cm, the UTVC is becoming black to

neutrons. The results, as shown in Figure 4-8, indicate that

PUTVC/PBCOL increases as the UF4 partial pressure increases up to about
10 atm and remains at about a constant level as the UF4 partial pressure
further increases. For all further analysis, partial pressures of 5 atm
for UF4 and 45 atm for NaF are used.

U235 enrichment in UF4

The U235 enrichment is varied from 80% to 100% at UF4 partial

pressures in the UTVC of 1, 2, 3, 4, 5, 6, and 7 atm. The results, as

shown in Figure 4-9, indicate that keff increases as the enrichment

increases and as the UF4 partial pressure increases. The U235

enrichment is fixed at 100% for all further analysis.

U233 as the fissile isotope

The U235 fissile isotope in UF4 is replaced with U233. The U233

enrichment is varied from 80% to 100% at UF4 partial pressures of 3 and

5 atm. The results, as shown in Figure 4-10, indicate the same behavior

as obtained in Figure 4-9 for U235 with the exception that keff is

higher when U233 is the fissile fuel. This is due to the lower thermal



























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capture cross section, ac, (48 barns versus 99 barns) and the higher

number of neutrons liberated per thermal fission, v, (2.49 versus 2.42)

in U233 versus U235; this corresponds to a higher number of fission
neutrons liberated per thermal neutron absorbed in the fuel, R, (about
2.29 for U233 and 2.07 for U235).

Average density of the UF4 in the boiler region

By varying the effective density of the UF4 in the boiler region,

the boiler column is simulated as a mixture of liquid and vapor with

some average quality. An examination of the effects of such density

changes aids in determining the reactor response due to power level

changes and UF4 inlet velocity changes. While maintaining the UTVC
radius at 60 cm, and the UF4 and NaF vapor partial pressures at 5 and 45
atm, respectively, the IBEO thickness at 20 cm, the boiler volume at 8.5

x 1003 m3, and the OBEO thickness at 35 cm, the "overall" density of

the UF4 in the boiler region is varied from 0.20 g/cm3 to 4.0 g/cm3 to

simulate the presence of both liquid and vapor UF4. A value of 0.20

g/cm3 for the density of UF4 reflects a mixture composed of about 5
volume percent liquid at 5 atm and 95 volume percent vapor, and a

density of =4 g/cm3 reflects a mixture of pure UF4 liquid.

The results, as shown in Figure 4-11, indicate an increasing

behavior for keff and a decreasing behavior for PUTVC/PBCOL as the
3
density of the UF4 increases to about 1.6 g/cm3. For densities above

1.6 g/cm3 the rate of increase of keff decreases and PUTVC/PBCOL levels
off. This indicates that above this density, the boiler starts to

become black to neutrons and begins to saturate.
























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Material Variations

The types and location of structural materials needed in

constructing the UTVR are not known at this point. These structural

materials include piping, spacers, and protective coatings. Also, the

over-all cycle analysis may prove that a metal fluoride other than NaF

(e.g., KF, RbF, or Li7F) provides better over-all system performance
than the UF4/NaF does. By including some sample structural and

protective coating materials at selected surfaces or regions (e.g., the

UTVC wall) and by examining other metal fluorides, reactivity penalties

due to these materials can be estimated. The effects of variations of

the following materials are therefore studied.

Choice of metal fluoride in UTVC

At inner UTVC core radii of 40, 80, and 120 cm, while maintaining
the IBEO thickness at 20 cm, the boiler volume at 8.5 x 10-03 m3, and

the OBEO thickness at 35 cm, calculations are performed to examine the

reactivity effect of using NaF, Li7F, or KF as the working fluid in the

vapor fuel mixture. The results, as shown in Table 4-3, indicate that,

with regard to the UTVC only (the wall cooling region is not included in

these calculations), the use of Li7F as the working fluid results in the

highest value for keff, followed by NaF and then by KF. As the UTVC

radius increases, the difference in keff as a function of selected metal

fluoride working fluid becomes greater. At these larger radii, the

reactivity contribution of the boiler region to keff decreases at the

same time the reactivity contribution of the UTVC to keff increases.

This explains the behavior of the differences in keff for the different

fuel mixtures at the higher UTVC radii. Since the type of metal













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fluoride has a small effect on keff, especially at a UTVC radius of 60

cm, NaF is used as the working fluid for further analysis.

Wall cooling region

Figure 4-12 shows the five region 1-D spherical "mock-up" of the

UF4/NaF UTVR system which includes the wall cooling region. To simulate

the variation of the NaF velocity in the wall cooling region, the

thickness of this region is varied from about 0.35 cm to about 3.3 cm.

This correspond to a NaF inlet velocity range of 0.5 to 5.0 m/sec.

Also, at each NaF velocity, the effective density of the NaF is varied
3 3 3
from 0.1 g/cm3 to 2.2 g/cm3. A value of 0.1 g/cm for the density of

NaF reflects a mixture composed of =4 volume percent liquid at 45 atm

and 96 volume percent vapor, and a density of 2.2 g/cm3 reflects a

mixture of pure NaF liquid. The results, shown in Table 4-4, indicate a

maximum penalty of about 10% 6k/k for a wall cooling region thickness of

3.33 cm and a NaF density of 2.2 g/cm3. However, the expected velocity

of the NaF is about 3 m/sec which corresponds to a wall cooling region

thickness of 0.58 cm. The reactivity penalty of the liquid NaF region

is then about 2.25% 6k/k for a NaF density of 2.2 g/cm3. As the inlet

velocity of the NaF increases in the wall cooling region and/or as the

effective density decreases, PUTVC/PBCOL increases.

Other metal fluoride working fluids

The NaF in the UTVC and wall cooling region is replaced by Li7F and

KF to examine the reactivity penalty or gain if other liquid metal

fluorides are used instead of NaF. The neutron multiplication factor,

keff, and the average PUTVC/PBCOL are obtained for the different metal
fluorides at wall cooling region thicknesses of 0.44, 0.87, and 3.33 cm

















Outer BeO Moderator-
Reflector Region ---


UF4 Boiler Region -- '. ',

Inner BeO Moderator-
Reflector Region
Wall Cooling Region-- -




Ultrahigh Temperatuce .. ...

Region 1


Region 2 --


Region 3

Region 4


Region 5


Figure 4-12. Five Region,
"Mock-up" of


One-Dimensional Spherical
the UTVR












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which reflect metal fluoride inlet velocities of 4.0, 2.0, and 0.5

m/sec, respectively. The average density of the metal fluoride is also

varied and results are presented for two different cases, as shown in

Table 4-5.

The results indicate that the use of Li7F as the metal fluoride

results in the highest values for both keff and PUTVC/PBCOL' followed by
NaF and then by KF. As the thickness of the wall cooling region

increases from 0.00 to 3.33 cm, maximum reactivity penalties are about

2%, 10%, and 22% Sk/k for LiF, NaF, and KF, respectively. Comparing

these results with those in Table 4-3 where the wall cooling region is

not treated, a greater difference in keff is noticed from one type of

metal fluoride to the other. This is due to the much greater density of
the (liquid) metal fluoride in the wall cooling region which in effect

results in a higher overall parasitic absorption.

NaF mass flow rate to the boiler region

To account for the possibility that complete separation of the

UF4/NaF mixture into pure UF4 and pure NaF cannot be achieved, and to

attempt to decrease the required PUTVC/PBCOL (on the basis of

thermodynamic and flow considerations) from its present value of 21, the

NaF mass flow rate to the boiler is varied from 0 kg/sec to 158 kg/sec;

this range corresponds to diverting 0% to 100% of the NaF from the wall

cooling region to the boiler region. The results, as shown in Table 4-

6, indicate a slight increase in keff from 1.548 to 1.554 as the NaF is

diverted to the boiler.

Table 4-6 also indicates that as the NaF is diverted to the boiler

region, the thermodynamic requirement for PUTVC/PBCOL decreases while













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the value of PUTVC/PBCOL obtained from the neutronic calculations shows
a slight increase.
UF4/NaF inlet velocity to the boiler
Building on the results obtained in Table 4-6, an attempt is made

to further increase UTVC/PBCOL by varying the inlet velocity of the

liquid UF4/NaF into the boiler region. The velocity is varied from 0.5
to 6.0 m/sec. The mass flow rate of the NaF is set at 94.8 kg/sec to

the boiler and 63.2 kg/sec to the wall cooling region.
Thermodynamically, this requires that PUTVC/PBCOL be 1.25. As the inlet
velocity of the mixture increases from 0.5 to 6.0 m/sec the amount of
the mixture in the boiler decreases; thus, the amount of fissile

material in the boiler decreases. This, as seen from Figure 4-13,
causes a decrease in keff from 1.577 to 1.468 and an increase in

PUTVC/PBCOL from 0.18 to 0.41.
Addition of Li6F poison to the boiler

An attempt is made to decrease the reactivity of the boiler region
by adding Li6 poison to the boiler region in order to obtain the
required PUTVC/PBCOL. Li6 is added to the boiler region in the form of
Li6F. The UF4/NaF-Li6F inlet velocity is fixed at 2 m/sec. The Li6F

mass flow rate is varied from 5.2 x 10-03 to 3.9 kg/sec. This

corresponds to an atom ratio variation from 0.1% to 75.0% (the atom

ratio being the ratio of the Li6 atoms to that of the U235). As the
mass flow rate of the Li6F increases, a decrease in keff from 1.550 to
0.957 and an increase in PUTVC/PBCOL from 0.266 to 0.574 is observed, as

shown in Figure 4-14. However, the required PUTVC/PBCOL value of 1.25

is not achieved in the 1-D configuration.










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BeO in the annular boiler region

The boiler region in the actual reactor system is made of a number

of cylindrical boiler columns separated by BeO moderator. To account

for this, i.e., the presence of BeO in the boiler region, the boiler

region in 1-D spherical geometry is modeled as a region that contains a

homogenized mixture of liquid and vapor UF4 and BeO. The one-

dimensional boiler region annular thickness is varied from 2.0 to 4.3 cm
to simulate the actual boiler region configuration that contains from 2

to 8 boiler columns. The volume of the UF4 and the total volume of the

boiler region are conserved when converting from the true cylindrical

configuration to the 1-D spherical geometry. As the annular boiler

region thickness is varied (i.e., as the number of boiler columns is

varied), the total mass flow rate and the inlet velocity of the UF4 in
the boiler region are kept constant (i.e., the total UF4 cross sectional
flow area is fixed) but the volume of the BeO and the average UF4

density are varied. The results, keff and PUTVC/PBCOL, are given in

Table 4-7.

The results indicate that keff decreases and PUTVC/PBCOL increases

as the simulated number of boiler columns increases. This is due to the

decrease in the amount of moderator present in the boiler region; thus,

less neutron thermalization is occurring in the boiler region. This

results in a decrease in the average thermal neutron flux in the boiler

region causing keff to decrease and PUTVC/PBCOL to increase. The

results also indicate that keff increases and PUTVC/PBCOL decreases as

the UF4 density in the boiler increases. This is due to the increase in

the amount of fissile material present in the boiler region.




Full Text

PAGE 1

STATIC AND DYNAMIC NEUTRONIC ANALYSIS OF THE URANIUM TETRA-FLUORIDE, ULTRAHIGH TEMPERATURE, VAPOR CORE REACTOR SYSTEM By SAMER DAKHLALLAH KAHOOK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1991

PAGE 2

'.A it. ^^ » !• , <) Aj ^ '.01. ..

PAGE 3

Dedicated to my parents, Mr. and Mrs. Dakh1a1lah Kahook, asking A11ah to reward them, have mercy on them, and grant them paradise as they raised and cherished me in my childhood. ,: i

PAGE 4

ACKNOWLEDGEMENTS The author would like to express his appreciation and sincere thanks to the members of his supervisory committee, Dr. Edward T. Dugan, Dr. Nils J. Diaz, Dr. Alan M. Jacobs, Dr. Samim Anghaie, Dr. William E. Lear, Jr., and Dr. Willis B. Person for their guidance and assistance during the course of this research. Special thanks are extended to Dr. Dugan, chairman of the author's supervisory committee for his patience and enduring support. The author recognizes that much of his knowledge in reactor physics and computer programming was realized while researching under the guidance and direction of Dr. Dugan. Support for this research has been provided, in part, by the Air Force Wright Aeronautical Laboratories (AFWAL), the Frederick Hauck Fund, and the University of Florida. The AFWAL work was performed for the Innovative Science and Technology Directorate of the Strategic Defense Initiative within the Innovative Nuclear Space Power Institute (INSPI). This support is greatly appreciated. Funding for the computer analysis was provided for by the National Science Foundation at the San Diego Supercomputer Center and the University of Florida and the International Business Machines (IBM) Corporation through their Research Computing Initiative at the North East Regional Data Center. The author is grateful for these funds. iii

PAGE 5

Thanks are also due to the fellow students whose friendships, comments, and encouragements have also facilitated in this research. The author would like to express his love and respect to his parents Mr. and Mrs. Dakhlallah Kahook, to his brothers Nofal and Mohammed, and to his sisters for their love, understanding, and patience throughout the author's stay at the University of Florida. The financial support provided to the author by his family is gratefully acknowledged. Finally, the author would like to express his love and deepest appreciation to his wife, Layali, whose understanding, patience, and support provided the motivation needed to finish this research. y -h .• " ^ ' ; r . " ' ' i ' I '• 'i. y ' X'.,. ^' ^ V ': ' I?--l t' -^-.: ' " ' ' ' -<*' 5 .1 .' iv

PAGE 6

TABLE OF CONTENTS i Page ACKNOWLEDGEMENTS i i i LIST OF TABLES x LIST OF FIGURES xiv ABSTRACT xvi i i CHAPTER I INTRODUCTION 1 Introduction 1 Description of the Ultrahigh Temperature Vapor Core Reactor 2 Di ssertati on Objecti ves 6 Dissertation Organization 7 II PREVIOUS RESEARCH ON RELATED CONCEPTS 11 Introduction 11 Previous Research on Gas Core Reactors 12 Previous Research on Coupled Core Reactors 13 Previous Research on Circulating Fuel Reactors 14 Remarks 18 III DESIGN OF THE URANIUM TETRA-FLUORIDE, ULTRAHIGH TEMPERATURE VAPOR CORE REACTOR 19 Introduction 19 Prel iminary Design Considerations 20 Choice of Materials 21 The Moderator-Reflector Material 21 The Fissioning Fuel Material 24 The Working Fluid Material 29 Description of a Uranium Tetra-Fluoride, UTVR/Disk MHD-Rankine Power Cycle 29 Neutronic Analysis of the Ultrahigh Temperature Vapor Core Reactor 37 Static Neutronic Calculations 37 Dynamic Neutronic Calculations 39

PAGE 7

CHAPTER Piae IV STATIC, ONE-DIMENSIONAL, UTVR NUCLEAR CHARACTERIZATION AND CONFIGURATION OPTIMIZATION 40 Introducti on 40 Scoping Calculations 43 Geometric Variations 43 UTVC radius 43 Inner BeO moderator-reflector region thickness 50 Outer BeO moderatorreflector region thickness 55 UF^ boiler region thickness 57 UF? boiler core volume 59 Fuel Density Variations 61 UF^ partial pressure and mole fraction (UF^rNaF) Jlin the UTVC 61 U"5 enrichment in UF« 62 u""* as the fissile isotope 62 Average density of the UF^ in the boiler region 66 Material Variations 68 Choice of metal fluoride in UTVC 68 Wall cooling region 70 Other metal fluoride working fluids 70 NaF mass flow rate to the boiler region 73 UF^/NaF inlet velocity to the boiler 76 Addition of Li F poison to the boiler 76 BeO in the annular boiler region 79 Reactivity effects of liner materials 81 One-Dimensional Results 84 The Neutron Multiplication Factor 86 Power Sharing Factor 87 Spherical "Mock-up" Comments 90 V STATIC, TWO-DIMENSIONAL, UTVR NUCLEAR CHARACTERIZATION AND CONFIGURATION OPTIMIZATION 94 Introduction 94 Scoping Calculations in R-^ Geometry 96 Geometri c Vari ati ons 98 UTVC radius variations 98 Inner BeO moderator-reflector region thickness variations 104 Variation in the area of the boiler columns 107 Variation in the number of boiler columns 109 Fuel/Working-Fluid Density Variations 110 UF4 partial pressure in the UTVC 112 Average UF^ density in the boiler columns 115 Varying the UF^ average density in the UTVC as a function of the radial distance from the center 1 ine 116 Scoping Calculations in R-Z Geometry 121 Geometric Variations 124 vi

PAGE 8

CHAPTER Page V MBEO region height 124 (cont.) TBEO region height 129 First OBEO region height 131 Boiler: subcooled and saturated liquid region height 134 Material Variation 135 Poisoning the boiler feedline walls 136 Comments on Power Shari ng 140 Two-Dimensional Results 144 The Neutron Multiplication Factor 144 The Power Sharing Factor 146 Remarks 148 VI STATIC, THREE-DIMENSIONAL NEUTRONIC ANALYSIS OF THE UTVR.... 151 Introduction 151 Description of the UTVR Geometry in MCNP 152 Description of the Boiler Column 156 Reactivity Worths of the Boiler Feedlines, UTVC Inlet PI enums , and the MHD Duct Regi ons 158 Reducing the Uncertainty in Parameters Associated with the Boiler Columns in MCNP Calculations 161 Performance of Variance-Reduction Techniques 165 , ' Nuclear and Physical Characteristics of the UTVR 166 r V Energy Cutoff 168 ' * Implicit Capture and Weight Cutoff 168 Weight Windows 173 Boiler-to-UTVC Symmetry 178 Neutron Transport Coup! i ng Coef f i ci ents 185 187 MCNP 189 Isolation of secondary coupling effects , 190 Neutron Multiplication Factor. of the j Core, k^ff... 197 Reactivity of the j^" Core, p^..., :I1... 198 Prompt Neutron Generation Time, A'^(t) 198 Results of Density Variations in the UTVC and Boiler Col umns 199 VII KINETIC EQUATIONS OF A FOUR-BOILER COLUMN UTVR SYSTEM 209 Introduction 209 The Four-Boiler Column UTVR System Coupled Core Point Reactor Kinetics Equations 209 Core-to-Core Fuel -Flow Coupling 211 Coreto-Core Neutron Transport Coupling 214 . Steady-State Solution 218 The Linearized UTVR CC-PRK Equations 221 Inherent Reactivity Feedbacks of the UTVR « 229 Reactivity Feedback of the Boiler Columns, Sp {t) 233 Reactivity Feedback of the UTVC, 8p^{t) 251 vii B01 ler-to-uivt symmetry itron Transport Coupling Coefficie ir. Obtained Directly from MCNP. €1*'^ Obtained Indirectly from MC^

PAGE 9

CHAPTER r^ :-. ^aae VIII DYNAMIC ANALYSIS OF THE UTVR 264 Introduction 264 The Unperturbed UTVR Configuration 265 Results of the Dynamic Analysis 269 Boiler Column Reactivity Perturbation 269 UTVC Reactivity Perturbation 276 Variations in Core-to-Core Direct Neutron Transport Delay Times 283 Variations in the Coupling Coefficients 287 Variations in the UTVC Fuel Mass Reactivity Feedback Coefficient 291 Concluding Remark 296 IX SUMMARY OF RESULTS, CONCLUSIONS, AND RECOMMENDATIONS FOR FURTHER RESEARCH 300 Introduction 300 Summary of Resul ts 300 Results from the Static Neutronic Analysis 300 Results from the Dynamic Neutronic Analysis 302 Comments and Conclusions 303 Recommendati ons for Further Research 305 ' Static Neutronic Analysis 305 Dynamic Neutronic Analysis 307 APPENDICES .. J ' ; . 4 '< ' ^* :^w, A DESCRIPTION OF THE COMPUTER CODES 309 Introduction 309 Description of Nuclear Codes 309 AMPX: A Modular Code System for Generating Coupled Multigroup Neutron-Gamma Libraries from ENDF/B 309 The AMPX-DRIVER module 311 The XLACS module 311 The NITAWL module 312 The XSDRNPM module 312 DOT-4: A Oneand Two-Dimensional Neutron/Photon Transport Code 315 GIP 316 MCNP-A General Monte Carlo Code for Neutron and Photon Transport 317 Description of the EASY5 Engineering Analysis Program 318 B BENCHMARK CALCULATIONS OF XSDRNPM AND DOT-4 WITH MCNP 320 Comparison of XSDRNPM with MCNP 320 Comparison of DOT-4 with MCNP 324 Conclusion 329 • • • Vlll

PAGE 10

APPENDICES Page C DESCRIPTION OF THE ISOLATOR OF SECONDARY COUPLING EFFECTS CODE 331 Introduction 331 Description of the ISCE Code 331 The MAIN Module 331 The REED Module 332 The ERIN Module 332 The NOUT Module 333 The ESTM Module 333 The RITE Module 337 Input Data Format 337 Input Data File 337 List of Input Data Files 339 Comparison of Results Obtained from ISCE with Results Obtained Directly from MCNP 340 D CIRCULATINGFUEL, COUPLED CORE POINT REACTOR KINETICS EQUATIONS 345 Description and Definition of Symbols, Parameters, and Terms used in the Circulating-Fuel, Coupled Core Point Reactor Kinetics Equations 350 Definition of Superscripts and Subscripts 350 Definition of Integral Parameters 351 Neutron population, N"^{t) 351 Reactivity, /)J(t) 354 Effective delayed neutron fraction, ^(t) 354 Prompt neutron generation time, A'^{t) 358 Effective delayed neutron precursor concentration for the i delayed neutron grouo, tj{t) 358 Effective coupling coefficient, c:3 (t) 359 Interpretation of Equations (D-l) and (D-4) 361 Equation (D-l) 361 Equation (D-4) 365 LIST OF REFERENCES 366 BIOGRAPHICAL SKETCH 371 ix

PAGE 11

,.. . y. . J-**.' ^. • ,* / • 1 " , t h !• ; ".^ ; ? V :v . ,'* .,, i.> ... ' V> LIST OF TABLES Table Page 3-1 Properties of Selected Metal Fluoride Working Materials 30 3-2 200 MW UF./UTVR Power Cycle Thermodynamic Operating Characteristics for NaF, KF, Li F, and RbF Working Fluids 34 Energy Balance,Data for 200 MW UF./UTVR Power Cycl( NaF, KF, Li'F, and RbF Working Fluids 3-3 Energy Balance,Data for 200 MW„ UF>,/UTVR Power Cycle with 35 ^'^ ^\il)lC^^BCOl *^ ^ function of the Metal Fluoride Mass Flow Rates 10 the Boiler Region as Required on the Basis of Thermodynamic/Flow Considerations 36 unction of Voided UTVC Radius for the UF./NaF Cycle System 48 4-2 kg^r as a function of the Liquid UF^ Core Volume for a Two Region Reactor 60 4-3 kg^^ as a function of UTVC Radius and Metal Fluoride Type... 69 ^f as a function of NaF Entrance Velocity and Average Density in the Wall Cooling Region 72 4-4 k. 4-5 kg^£ as a function of Metal Fluoride Type and Wall Cooling Region Thickness 74 4-6 k ff and PiiTur/PRrni ^^ ^ function of NaF Diverted Flow ^^Rate to^tftg BSTTfer Region 75 4-7 kg^x and PuTUc/'^BCOL ^^ ** function of UF. Average Density and the Mockea-up" Number of Boiler Columns in the Annular Boiler Region 80 4-8 Reactivity Penalty (5k/k) as a function of UTVC Liner Material Thickness 82 4-9 Reactivity Penalty (6k/k) as a function of Boiler Region Liner Material Thickness 83 4-10 Reactivity Penalty (5k/k) as a function of Both the UTVC and Boiler Region Liner Material Thickness 85 X

PAGE 12

Table Paqe 5-1 k ff and Putvc/''rC0L ^^ ^ function of UTVC Radius for the oF./NaF Rahkine cycle System in R-^ Geometry for a Four-Boiler and an Eight-Boiler Column UTVR Configuration 101 5-2 k^£ and PnTyr/PRroL ""^ ^ function of the Number of UF, feoiler cAlumns Tn R-e Geometry ? m 5-3 UF^ Temperature and Density Profiles in the UTVC as a function of Radial Distance for a Four-Boiler Column UTVR Configuration in the R-^ Coordinate System 120 5-4 UTVR Dimensions of the Reference R-Z Cylindrical Configuration 125 Region Height 127 '-' '«ffed:ras^'iS9t4c?S? RiavsR'H^^jiifv.":!"!.!'':"^'^". 130 (0BE0#1) Height 132 '-' '^fCheWt«S94hrSub?8e?E^'Siiil!!:tSra?ed"uSi?3 °' Region of the Boiler Column 135 MolyDdehum Thickness sdrrounaTng the Boiler Feedlines Region 138 6-1 Description of UTVR Regions Employed in the ThreeDimensional MCNP Monte Carlo Calculations 155 6-2 Reactivity Worths of the Boiler Feedlines, UTVC Inlet Plenums, and MHD Duct Regions 159 6-3 Selected UTVR Results from a 30-Minute MCNP Monte Carlo Analog Calculation Performed on a CRAY X-MP/48 Supercomputer 162 6-4 UTVR Fission Rate as a function of Neutron Energy 169 6-5 Effect of Employing Energy Cutoff on the UTVC and Boiler Column FOM Tallies 169 -<'., xi

PAGE 13

Table Page 6-6 Effect of Employing Implicit Capture and Weight Cutoff on the UTVC and Boiler Column FOM Tallies 171 6-7 Effect of Employing Weight Windows on the UTVC and Boiler Column FOM Tallies 175 6-8 Effects of Employing Variance-Reduction Techniques in MCNP Monte-Carlo Calculations on Uncertainties of Selected UTVR Parameters 177 6-9 Effects of Employing Variance-Reduction Techniques and utilizing Boiler-to-UTVC Symmetry in MCNP Monte-Carlo Calculations on Uncertainties of Selected UTVR Parameters 184 6-10 Integral Kinetics Parameters as a function of the UF^ Partial Pressure in the UTVC ? 201 6-11 Integral Kinetics Parameters as a function of Saturated Liquid Cone Region Height for Two Different H^""* Values.. 203 6-12 Integral Kinetics Parameters as a function of Saturated Liquid Cone Region Height at UF^ Partial Pressures of 2.5 and 7.5 atm in the UTVC ? 204 6-13 Integral Kinetics Parameters as a function of H^^^ and H^'^^ in Boiler Column at a UF^ Partial Pressure of 5 atm in the UTVC : 206 6-14 Integral Kinetics Parameters as a function of Vapor Cone Region Density at a UF^ Partial Pressure of 5 atm in the UTVC ? 208 8-1 Values of Selected UTVR Parameters at the Initial, Unperturbed Steady State Condition 267 8-2 Relevant Properties for the UF^ Fuel, NaF Working Fluid, and the UF^/NaF Fuel/Working Fluid Mixture 268 8-3 Final Equilibrium Conditions as a Result of $ 1.00 Positive and Negative Reactivity Step Insertions Imposed on the Boi 1 er Col umns 278 8-4 Final Equilibrium Conditions as a Result of $ 0.20 Positive and Negative Reactivity Step Insertions Imposed on the UTVC 282 8-5 Final Equilibrium Conditions Following a Positive Step Reactivity Insertion of $ 0.20 Imposed on the UTVC with Normal and Reduced Coupling Coefficients 289 xii

PAGE 14

Table Page 8-6 Final Equilibrium Conditions Following a Positive Step Reactivity Insertion of $ 0.20 Imposed on the UTVC with Normal and Reduced UTVC Fuel Loading Coefficients of Reactivity 293 B-1 XSDRNPM and MCNP Benchmark Calculations on a Five-Region Spherical "Mock-up" of the UTVR 322 B-2 DOT-4 and MCNP Benchmark Calculations on the Cylindrical "Mock-up" of the UTVR in both the R-S and R-Z Coordi nate Systems 326 C-1 Comparison of Results obtained from ISCE with Results obtained Directly using MCNP for Two Different UTVR Fuel Loadi ngs 343 D-1 The Six Delayed Neutron Groups Energy Spectra, Decay Constants, Yield, and Fractions Data for Thermal Fission in U"^ 356 Xlll

PAGE 15

LIST OF FIGURES Figure Page 1-1 Side View Schematic of the Ultrahigh Temperature Vapor Core Reactor 3 1-2 Top View Schematic of the Ultrahigh Temperature Vapor Core Reactor 5 3-1 UFg and UF^ Saturation Vapor Curves 25 3-2 Uranium Metal and UF^ Saturation Vapor Curves 27 3-3 Partial Pressures of Constituent Species of the UraniumFl uori ne System at One Atmosphere 28 3-4 Schematic of a 200 MW UF./KF UTVR MHD-Rankine Cycle Power System ?. . . ? 31 4-1 Four Region, One-Dimensional Spherical "Mock-up" of the UTVR 41 4-2 kg^^ and PuTVc/'^BCOL ^^ ** function of the UTVC Radius 45 ... 46 4-3 k rr and Fission Rates of the UTVC and the Boiler as a function of the UTVC Radius 4-4 kg^£ and Pmjvc/''bCOL ^^ *' function of the Inner BeO Moderator-Reflector Region Thickness 51 4-5 kg^r as a function of the Inner BeO Moderator-Reflector Region Thickness 53 4-6 kg^jF and Pmjvc/Prcol ^^ ^ function of the Outer BeO Moderator-Ref lector Region Thickness 56 4-7 k ff and PiiTur/PRrni ^s a function of the UF^ Inlet ^^Velocity t6 tRrfeoiler Region ? 58 *"^ '^eff *"^ ''uTVc/'^RfOl ^^ ^ function of the UF* Partial Pressure iH tne DTVC ? 63 4-9 k rj: as a function of the U Enrichment at Different OF^ Partial Pressures in the UTVC 64 xiv

PAGE 16

Figure vv Page 4-10 k^^ as a function of the Fissile Fuel (U^^^ and U^^^) Enri chment 65 4-11 k ff and PiiTwr/PRrni ^^ a function of the UF* Average ^^Density^lrthrBbiler Region 7 67 4-12 Five Region, One-Dimensional Spherical "Mock-up" of the UTVR 71 4-13 k /Tf and PiiTur/PRroL ^^ * function of the UF./NaF Inlet Velocity *0 tne Boiler Region ? 77 *"^* '^eff *"^ ^UTVc/^BCOL ^^ ^ function of the Li°F Mass Flow Kate to trie BoTTer Region 78 5-1 Six Region, Two-Dimensional R-8 Representation of a UTVR with Six-Boiler Columns 97 5-2 k ff and Pmtvc/'^BCOL ^^ ^ function of the UTVC Radius for a Fourand an Eight-Boiler Column UTVR Configuration 100 5-3 k rr and PijTVc/'^BCOL *^ * function of the Inner BeO Moderator-RefTector Region Thickness for a FourBoiler Column UTVR 106 ^"* '^eff ^"^ ''uTVc/'^BCOL ^^ * function of the UF^ Inlet Velocity to the Boiler Region for a Four-Boiler CoTumn 108 •pff *"*^ ''uTVc/'^BCOL *^ * function of the UF^ Partial Pressure in tne DTVC for a Four-Boiler CoTumn 5-5 k,, UTVR System 113 5-6 k rr and Piitvc/''bcQL ^^ ^ function of the UF^ Average Density in the Boiler Region for a Six-Boiler Column UTVR System 117 5-7 Thermal Neutron Flux and Vapor Fuel Temperature Profile as a function of Radial Position from the Center! ine of the UTVC for a Four-Boiler Column UTVR System 118 5-8 Representation of the UTVR in the R-Z Coordinate System 122 5-9 The Horizontal Boiler Configuration of the UTVR in the R-Z Coordinate System 143 6-1 Side View Schematic of the Four-Boiler Column UTVR on the y-z Plane at x=0.0 153 6-2 Side View Schematic of a Boiler Column 157 > XV

PAGE 17

Figure Page 6-3 Top View Schematic of a Four-Boiler Column UTVR System 180 7-1 Schematic of the Core-to-Core Circulating Fuel Coupling 212 7-2 Schematic of Boiler-to-UTVC Neutron Transport Coupling for a Four-Boiler Column UTVR System 215 7-3 Schematic of Boiler-to-Boiler and UTVC-to-Boiler Neutron Transport Coupling for a Four-Boiler Column UTVR System.. 217 7-4 Bl ock Di agram of the UTVC Transfer Functi on 228 7-5 Block Diagram of the Boiler Column Transfer Function 230 7-6 Block Diagram of the UTVR Transfer Function 231 7-7 Fuel/Working Fluid Density Profile in the Boiler Column due to Boiling in Space (»zero gravity) 234 7-8 Side View Schematic of the UTVC 251 8-1 UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 1.00 Positive Reactivity Step Insertion Imposed on the Boiler Columns at t=0 sec 270 8-2 UTVC Pressure, U^"'^ Loading, and UF./NaF Inlet and Outlet Mass Flow Rates, as a function of Time Following a $ 1.00 Positive Reactivity Step Insertion Imposed on the Boi 1 er Col umns at t=0 sec 273 8-3 Boiler Column Outlet Mass Flow Rate and U Loading as a function of Time Following a $ 1.00 Positive Reactivity Step Insertion Imposed on the Boiler Columns at t=0 sec. 274 8-4 UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 1.00 Negative Reactivity Step Insertion Imposed on the Boiler Columns at t=0 sec 277 8-5 UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 0.20 Positive Reactivity Step Insertion Imposed on the UTVC at t=0 sec 279 8-6 UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 0.20 Negative Reactivity Step Insertion Imposed on the UTVC at t=0 sec 281 8-7 UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 0.20 Positive Reactivity Step Insertion Imposed on the UTVC at t»0 sec with t{ J = lO""-* sec 284 xvi

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Figure Page 8-8 UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 0.20 Positive Reactivity Step Insertion Imposed on the UTVC at t=0 sec with Tx ~ lU sec. 285 8-9 UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 0.20 Positive Reactivity Step Insertion Imposed on the UTVC at t=0 sec with the Coupling Coefficients Reduced by One Order in Magnitude.. 288 8-10 UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 0.20 Positive Reactivity Step Insertion Imposed on the UTVC at t-0 sec with the UTVC Fuel Mass Reactivity Feedback Coefficient Reduced by a Factor of Fi ve 292 8-11 UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 0.20 Positive Reactivity Step Insertion Imposed on the UTVC at t=0 sec with the UTVC Fuel Mass Reactivity Feedback Coefficient Increased by a Factor of Two 295 A-1 Schematic of the Flow between the AMPX System Code Modules.. 314 C-1 Example of the ISCE Code Input Data File 339 C-2 Input Data Files List Format 340 C-3 Output File as obtained from ISCE 341 D-1 Schematic of Neutrons and Neutron Interactions in the UTVR.. 348 D-2 Top View Schematic of the Plasma Core Assembly (PCA) 357 D-3 Probability Distribution function of Delay Times for the Transport of Neutrons from Core k to Core j 364 xvii

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STATIC AND DYNAMIC NEUTRONIC ANALYSIS OF THE URANIUM TETRAFLUORIDE, ULTRAHIGH TEMPERATURE, VAPOR CORE REACTOR SYSTEM "By Samer Dakhlallah Kahook May, 1991 Chairman: Dr. Edward T. Dugan Major Department: Nuclear Engineering Sciences ' An Ultrahigh Temperature Vapor core Reactor (UTVR) system is investigated in this research. The UTVR can be characterized as a thermal, high power density (hundreds of MW^^/m ), externally-moderated, 235 coupled core, highly-enriched U , circulating-fuel, steady-state, burst power reactor. The investigated reactor system includes two types of fissioning regions: (1) the central Ultrahigh Temperature Vapor Core region (UTVC) which contains a vapor mixture of highly-enriched uranium tetrafluoride (UF^) fuel and a metal fluoride working fluid at an average temperature of «3000 K and an average pressure of «50 atm; and (2) the Boiler COLumn region (BCOL) which contains highly enriched liquid UF^ fuel. The combination of three features differentiates the UTVR from other nuclear reactor concepts. These three features are as follows: 1. the multi-core configuration resulting in a coupled-core system by means of direct neutron transport through the media; xviii

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2. the circulating fuel and the associated neutronic and mass flow coupling between the UTVC and boiler cores; and 3. the employment of a two-phase fissioning fuel, i.e., a liquid-vapor combination. Static and dynamic neutronic analysis of this novel system indicates distinct advantages over other existing or conceptual nuclear power systems. These include a unique combination of some very effective inherent negative reactivity feedbacks such as the vapor-fuel density power coefficient of reactivity, the direct neutronic coupling among the multiple fissioning core regions, and the mass flow coupling feedback between the two types of fissioning cores. Static neutronic analysis is performed using multidimensional discrete ordi nates and Monte Carlo neutron transport codes. Parameters such as the UTVC and boiler column reactivities and reaction rates, core-to-core neutronic coupling coefficients, and neutron lifetimes as a function of vapor core density and boiler core liquid volume are obtained from the static neutronic analysis. The dynamic behavior of the UTVR is examined using a non-linear model, which incorporates circulating-fuel, coupled-core, point reactor kinetics and energetics equations. These equations are solved using a system analysis code. The dynamic analysis indicates that the unique and strong negative reactivity feedbacks of the UTVR are capable of stabilizing the UTVR safely and quickly even when large reactivity insertions are imposed {6p $ 1.00). The analysis also shows that the system exhibits good dynamic performance even when an inherent negative reactivity feddback is suppressed (e.g., the vapor fuel density power xix

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coefficient of reactivity). However, due to the strength of the UTVR's inherent negative reactivity feedbacks, it is found that external reactivity insertions alone are inadequate for bringing about power level changes during normal operations. Additional methods of reactivity control, such as variations in the mass flow rate of the fuel and/or working fluid or variations in the inlet pressure of the fuel/working fluid entering the boiler columns, are needed to achieve the desired power level control. XX

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CHAPTER I INTRODUCTION Introduction The concept of Vapor Core Reactors (VCRs) has emerged at the University of Florida (UF) as a consequence of extensive theoretical and experimental studies performed on their predecessors, the Gaseous Core Reactors (GCRs). Unlike GCRs (where the fuel is supplied to the reactor in gaseous form), the working fluid and/or fuel undergo a liquid-tovapor phase change in VCRs. Studies performed on VCRs and GCRs indicate that gaseous-fueled (or vapor-fueled) reactor concepts have distinct advantages over other existing or proposed nuclear power systems. These advantages include high operating temperatures and efficiency, rapid startup capabilities, simple geometry, and an assortment of efficient power control methods [1-9]. The Ultrahigh Temperature Vapor Core Reactor (UTVR)/Disk Magnetohydrodynamic (MHD) Generator Power System is being studied for the Strategic Defense Initiative Organization (SDIO) as a possible source for space power. The SDI space power systems are required to operate in at least one of the three following power modes: Station-keeping mode (base load) . This mode may be required to produce up to a few Megawatts electric (MW ) for a period of about 7 years. ,

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2 Alert mode (enhanced surveillance mode) . The power requirements for this mode range from 10' s of MWg up to ^lOO MW^. The power system must be capable of functioning for periods of a few hours to a few days. Burst power mode (defense mode) . The power level for this mode ranges from -100 MW up to ~1 Gigawatt electric (GW ) for operating times of about 30 minutes; a burst power system must be capable of achieving this power level in less than 100 seconds. ' • . The UTVR/MHD Generator Power System is a burst power mode concept. At burst power levels, the UTVR can operate at very high temperatures which provides an efficient heat rejection capability and a high thermodynamic efficiency. This and other features appear to make the UTVR/MHD Generator Power System an exceptional concept for burst power operations. The UTVR/MHD Generator Power System is the concept examined in this research. Description of the Ultrahigh Temperature Vapor Core Reactor The UTVR/Disk MHD Generator Power System is a highly enriched {>85%), BeO externally-moderated, circulating fuel reactor with uranium tetra-fluoride (UF^) as the fissioning fuel. The working fluid is in the form of a metal fluoride such as NaF, KF, RbF, and Li F. Shown in Figure 1-1 is a side view schematic of the UTVR. The UTVR includes two types of fissioning regions: (1) the central Ultrahigh Temperature Vapor Core regions (UTVC) which contain a vapor mixture of highly-enriched UF^ and a metal fluoride working fluid at an average temperature of -3000 K and an average pressure of «50 atm, and (2) the boiler column regions (BCOL) which contain highly enriched UF^

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TBEO UTVC Inlet P1 enums SSS • • • • • Wall Coolant OBEO LBEO To Heat Reject! on System MHD Duct Region IBEO UF4 Boi 1 er Column Figure 1-1. Side View Schematic of the Ultrahigh Temperature Vapor Core Reactor

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fuel. This reactor has symmetry about the midplane with identical top and bottom vapor core and boiler column regions separated by a BeO slab (mid-plane BeO Region MBEO) and the MHD ducts where power is extracted. The UTVC is surrounded in the radial direction by the wall cooling region. The wall cooling region contains a subcooled liquid metal fluoride. By tangentially injecting the metal fluoride into the UTVC, the UTVC walls are maintained at the desired low temperatures (=2000 K). As the metal fluoride is injected into the UTVC, an annular buffer zone is obtained which aids in maintaining the UF^ away from the UTVC walls. This reduces the possibility of condensation of uranium or uranium compounds on the UTVC walls. Beyond this buffer zone, the metal fluoride vaporizes and mixes with the UF^ in the UTVC. The UF. is vaporized in the boiler columns prior to its entrance to the UTVC. The boiler region, which includes a number of boiler columns, is connected to the UTVC via the UTVC inlet plenums, as shown in Figure 1-1. The UF^ liquid is supplied to the boiler columns by means of feedlines. Each boiler column consists of three distinct regions: the subcooled liquid region, the saturated liquid-vapor region, and the superheated vapor region. Shown in Figure 1-2 is a top view schematic of the UTVR. Figure 12 shows three distinct BeO regions: the inner BeO region (IBEO) which separates the UTVC walls from the boiler columns in the radial direction, the annular boiler BeO region (BBEO) with a radial thickness equal to the diameter of the boiler columns, and the outer BeO region (OBEO) surrounding the boiler columns and the BBEO region. Three other BeO regions are shown in Figure 1-1. These are the mid-plane BeO region

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OBEO Wa11 Cooling Region Figure 1-2. Top View Schematic of the Ultrahigh Temperature Vapor Core Reactor

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(MBEO) mentioned previously, the lower BeO region (LBEO) separating the boiler feedlines from the MHD duct, and the top BeO region (TBEO) above the UTVC. . „ .^ . .,,. > . -^ , i , . i , Use of the UF^ as the vapor fuel and metal fluorides as the working fluid in the UTVR/MHD Generator Power System allows for operation on a direct, closed Rankine type cycle and leads to space power systems with 2 high efficiency (»20%), small radiator size (=5 m /MW^), and high specific power (=s5 kw /kg). A description of an example UF^-Metal Fluoride UTVR/MHD Generator Rankine Cycle Power System is furnished in Chapter III. Dissertation Ob.iectives A goal of this research is the nuclear design and analysis of the UF^-Metal Fluoride UTVR/MHD Generator Rankine Cycle Power System for space power applications. Complete characterization of this innovative system requires an integrated and thorough investigation of its neutronic, heat transfer, and mass flow behavior. Although this research focuses on the nuclear aspects of the proposed system, it incorporates results from auxiliary and supporting thermodynamic, heat transfer, and fluid flow calculations, thus, assuring a reliable and integrated nuclear analysis. The nuclear design of the UTVR incorporates results from the static and dynamic neutronic analysis performed on the UTVR. The static neutronic analysis establishes basic neutronic characteristics and obtains reference reactor configurations that are optimized for the static neutronic characteristics while also considering other important

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^ parameters like specific power (kw/kg) for the UTVR. Applicable UTVR parameters that are needed for the dynamic neutronic studies such as reactivity, neutron generation time, and core-to-core coupling coefficients are also obtained from the static analysis. The dynamic neutronic analysis focuses on characterizing the UTVR with respect to stability and dynamic response. Effects of core-to-core neutronic coupling (by means of direct neutron transport through the media and by delayed neutron emission from the decay of the delayed neutron precursors which are carried in the fuel that circulates between the UTVC and boiler columns) and effects of other important reactivity feedback phenomena such as fuel density and mass flow related feedback for the vapor and boiler cores are included in the dynamic analysis. Thus, the primary objective of this research is the development and application of the methods and the models needed for the nuclear design and analysis of this unique reactor concept. It is recognized that acoustic phenomena are inherent to the UTVR and their effects are potentially very significant. However, acoustic effects are not included in this research and are recommended for future work when the necessary tools for treating these effects are available. Recommended future work will require coupled space-time neutron fieldgas density field calculations. Dissertation Organization A brief summary of previous work performed on related reactor concepts such as gas core reactors, coupled core reactors, and circulating fuel reactors is presented in Chapter II.

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at A section addressing preliminary design considerations for the UF^ UTVR reactor system is presented in Chapter III. It includes point design conditions for the UTVR from preliminary thermodynamic, heat transfer, and fluid flow calculations. A description of an example UF^Metal Fluoride UTVR/MHD Generator Rankine Cycle Power System is also presented in Chapter III. A section in Chapter III discusses the plan used in the nuclear design and analysis of the UTVR system. The results of the static oneand two-dimensional neutronic calculations are presented in Chapters IV and V, respectively. These calculations are performed with XSDRNPM [10], a one-dimensional discrete ordinates (S ) neutron transport code, and with DOT-4 [11], a oneand two-dimensional S neutron transport code. The static analysis examines effects of variations in geometry and fuel/working fluid loadings on the neutron multiplication factor (kg^^) and power sharing factor (i.e., power distribution between the UTVC and the UF^ boiler columns, Pinyr/^DrQi) ' Basic neutronic characteristics of the UTVR such as fuel density reactivity coefficients, optimum BeO region thicknesses, optimum number of UF^ boiler columns, and a reference UTVR configuration for three-dimensional analysis are obtained from the static neutronic analysis results presented in Chapters IV and V. The results obtained from static three-dimensional neutronic calculations are presented in Chapter VI. These calculations are performed using MCNP [12], a three-dimensional Monte Carlo neutron transport code. Parameters such as UTVC and boiler core reactivities and reaction rates, core-to-core coupling coefficients, and neutron lifetimes as a function of vapor core density and boiler core liquid

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5 volume are obtained from the results of calculations performed with MCNP. The methods and models used in obtaining the core-to-core neutron transport coupling coefficients and the reactivities of the vapor and boiler cores are derived and described in Chapter VI. The circulating-fuel, coupled core, point reactor kinetics equations for a four-boiler column UTVR are derived in Chapter VII. A section in Chapter VII contains a detailed discussion of significant UTVR inherent reactivity feedbacks such as the vapor fuel density feedback of the UTVC and the liquid fuel/working fluid volume feedback of the boiler region. Energetics equations relating the power levels and the neutron population levels of the vapor and boiler cores to fuel/working fluid temperature, density, and liquid volume and flow rates are also included in Chapter VII. The dynamic neutronic analysis and performance studies are included in Chapter VIII. The dynamic analysis examines the behavior of core power levels, reactivities, fuel densities, and total system power during full power transients. Effects of the core-to-core circulating fuel and neutron transport coupling and fuel density variations in the vapor core and boiler cores are included in the dynamic analysis. The conclusions obtained from this research are included in Chapter IX. Suggestions and recommendations are made for further research which are needed before the technical feasibility of the UTVR/MHD Generator Power System can be realized. A brief description of the nuclear and system analysis computer codes used in this research is presented in Appendix A. Appendix B contains the results of benchmark calculations performed with XSDRNPM

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s.i « . 1 ^ ' ' r.. } . . 10 and MCNP on a reference UTVR in spherical coordinates. Results from benchmark calculations performed in R-^ and R-Z cylindrical coordinates with DOT-4 and MCNP are also included in Appendix B. A description of the Isolator of Secondary Coupling Effects (ISCE) code is presented in Appendix C. The ISCE code, a special code developed as a part of this research, incorporates the models derived in Chapter VI with results obtained from the MCNP code to obtain parameters needed for the dynamic analysis and performance studies such as core-to-core neutron transport coupling coefficients and the reactivities of the UTVC and boiler cores. Appendix D contains a description of the circulating fuel, coupled core, point reactor kinetics equations.

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CHAPTER II PREVIOUS RESEARCH ON RELATED CONCEPTS Introduction ' The UTVR can be characterized as a thermal, high power density 3 (hundreds of MW^i^/m ), externally-moderated, coupled-core (vapor and 235 boiling cores), highly-enriched, U circulating fuel, steady state reactor. The combination of three features differentiates the UTVR from other nuclear reactor concepts. These features are the following: 1. The multi-core configuration resulting in a coupled-core system by means of direct neutron transport through the media. 2. The circulating fuel and the associated neutronic and mass flow coupling between the UTVC and boiler cores. This feature provides additional neutronic coupling between the cores by means of delayed neutron emission from the decay of the delayed neutron precursors which are carried in the fuel/working fluid mixture. The mass flow coupling between the vapor and boiler cores is an inherently stabilizing phenomenon. For example, an increase in the power level of the boiler core increases the voiding and decreases the density of the fuel/working fluid mixture in the boiler core. This leads to a decrease in the boiler core power level. Additionally, the density of the fuel/working fluid exiting the boiler core and entering the vapor core decreases. This causes a decrease in the reactivity of the vapor core resulting in a decrease in the vapor core power level.

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12 The decrease in the vapor core power level causes a decrease in the number of neutrons directly transported to the boiler cores through the media and a decrease in the delayed neutron precursor concentration decaying in the boiler cores. This causes a further decrease in the boiler core power level. > , s^-*^ 3. The employment of a two-phase fissioning fuel, i.e., a liquid-vapor combination. Studies on reactors combining all three of these key features have never been reported. However, studies and research pertaining to coupled-core reactors, circulating fuel reactors, or gaseous (vapor) core reactors have been reported. Therefore, the following sections of this chapter briefly summarize previous research on reactors that possess one of these key features or aspects of the UTVR. Previous Research on Gas Core Reactors Research on gas core reactors has been reported as early as 1955 by George Bell [13]. The reactors examined by Bell employed gaseous UFg fuel and beryllium (Be), DJQy ^^^ graphite reflectors in spherically symmetric geometries. The analysis was done using age theory to describe neutron slowing down in the moderator-reflector region and diffusion theory to describe neutron diffusion into the core and the fissions in the core. The reactor was considered to be strictly a thermal reactor. Since then, different analytical methods and models have been used in studying this and other gas core reactor concepts. This includes the Nuclear Piston Engine and Pulsed Gaseous Core Reactor Power Systems

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13 examined by E.T. Dugan [2] and the Heterogeneous Gas Core Reactor examined by K.I. Han [5]. Summaries of previous work on gas core reactors can be found in the studies reported by Dugan and Han. Previous Research on Coupled Core Reactors The initial work on the kinetics of coupled core reactors was reported in 1958 by Robert Avery [14]. Avery investigated the dynamic characteristics of coupled fast-thermal breeder reactors. The analysis incorporates the point reactor kinetics equations for each core. The equations include terms accounting for the neutronic interaction (coupling) between the cores. The coupling terms along with integral parameters used in the point reactor kinetics equations are obtained from steady state analysis of the interacting cores. Neutron kinetics studies using the coupled core treatment have been applied to nuclear reactor systems other than fast-thermal reactors. These include modular cores of large thermal power reactors, clustered reactors, and Argonaut-type reactors. Research has also been performed on coupled gas core reactors. This includes the work performed by M.M. Panicker [6] on the Coupled Multiple Chamber Gaseous Core Reactor Power System. The differences in the various approaches used in the analysis of coupled core reactor systems lie in the choice of the weighting function (neutron flux, importance function, or average fission density); the choice of suitable phase-space regions for the averaging process; and the selection of how to incorporate the coupling effects (e.g., as a source term or reactivity effect) into the pertinent dynamic neutronics

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\ ' y V V? > ^ ^'^^r -^^-v^^ v^ . -f •'-. ? V 14 equations. Detailed discussions of these differences and their applications to various reactor systems are reported by Adler et al . [15] and Panicker [6]. Previous Research on Circulating Fuel Reactors The kinetics of circulating fuel reactors is affected by the loss of a fraction of delayed neutrons due to the decay of the delayed neutron precursors outside the core. The fraction of delayed neutrons that is lost depends mainly on the time the fuel spends in the core relative to the time the fuel remains outside the core. Various methods have been used in approximating the effects of circulating fuel. The impact of various methods used for approximating the effect of circulating fuel on the kinetics of nuclear reactors has been Investigated by John MacPhee [16]. In comparing the approximate methods, MacPhee employed an "exact" model ("exact" with respect to the method of treating the effect of the circulating fuel on the delayed neutrons). The "exact" model employed the following assumptions: 1. Point reactor kinetics equations are valid in the sense that the reactor kinetics effects are considered to be spatially independent. 2. Reactor power level is low enough such that the effect of neglecting reactivity feedback due to temperature and radiolytic gas formation is valid. 3. One delayed neutron group is used. 4. Perfect mixing in the core vessel occurs. 5. Fission occurs only in the core. 6. Fuel mass flow rate is constant.

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15 The reactor kinetic equations employed in the "exact" model are dN(t) . />(t) -^N(t) .Ae(t) (2-1) dt A ^(t) = t N(t) Ae(t) ^ . ^^^^H^ e-''« (2-2) dt A Tc Tc where N(t) = neutron population level in the core at time t; p(t) = core reactivity at time t; P ' fraction of delayed neutrons; A * prompt neutron generation time; X * average decay constant of delayed neutrons; C » delayed neutron precursor concentration; r » time fuel remains in the core; Tf time fuel spends in the loop outside the core. Equation (2-1) describes the time dependent behavior of the neutron population level and Equation (2-2) describes the time dependent variation of the concentration of the delayed neutron precursors. The effect of the circulating fuel is accounted for in the last two terms in Equation (2-2). MacPhee compared two approximate methods with the "exact" model. In the approximate methods, modified versions of Equation (2-2) are employed. The first method employs reduced values for ^ as shown by ^(t) = Z^N(t) -\t{t) ' (2-3) dt A where / is the fraction of delayed neutrons lost as a result of the fuel circulating and is given by

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/ = t : '

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The peaking is due to the coupling of the delayed neutron decay constants with the loop circulation period and occurs for small values of arj. The reason the peaking is finite is because arj is greater than zero for all practical reactor configurations. Equation (2-6) indicates that arj approaches unity as Tg approaches zero for all values of t , i.e., all delayed neutrons are emitted in the core. However, as r« approaches infinity (fuel does not re-enter core), a^ approaches zero if and only if r approaches zero such that no delayed neutrons are emitted in the core. For such cases, the velocity of the fuel in the core is required to be infinite and an infinite amount of fuel is required to maintain the reactor critical. Since ap is always larger than zero, the peaking is therefore finite. It should be noted that an inherent assumption in MacPhee's analysis is that the employed fuel is incompressible. Thus, some of these conclusions do not pertain to the UTVR. M.A. Schultz [17] indicates that a number of smaller peaks would occur in the frequency response of circulating fuel reactors if more than one delayed neutron group is included in MacPhee's "exact" model. The fact that the mixing of the circulating fuel in the external loop of an actual reactor will smooth over the peaks and reduce any tendency toward sustained oscillations is pointed out by Schultz. The effect of fuel temperature reactivity feedback in circulating fuel reactors has been investigated by W.K. Ergen [18]. The analysis indicates damped power oscillations for circulating fuel reactors occurs with negative fuel temperature feedback. Ergen also concludes that the decrease in damping of oscillations due to the loss of delayed neutrons

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18 is compensated to some extent by the damping effect caused by the circulation itself. Remarks In deriving the models needed to analyze the UTVR/MHD Generator Power System (see Chapters VI and VII and Appendix D), references to previous work are also made. Where applicable, modifications to and comparisons with previous models are indicated. 0'

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CHAPTER III DESIGN OF THE URANIUM TETRA-FLUORIDE, ULTRAHIGH TEMPERATURE VAPOR CORE REACTOR Introduction [, In the design of nuclear power reactors, the choice of materials for fuel, moderator, coolant/working fluid, and structure and the selection of the power extraction system are based on the application and the required performance of these reactors. Once the appropriate materials and a suitable power extraction system are selected, a reference reactor configuration can be chosen. Then, a complete characterization of the reference reactor power system is required to determine its overall performance and feasibility. Although this research focuses on the nuclear aspects of the UTVR, a section in this chapter addresses preliminary design considerations that led to the reference UTVR configuration. Another section in this chapter discusses considerations involved in the materials selection for the UTVR. A detailed description of an example UF^-Metal fluoride UTVR/MHD Generator Rankine Cycle Power System is also given. This is followed by a section discussing the plan followed in this research for the neutronic analysis of the UTVR system. ., -, /^'«-. 19

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20 Preliminary Design Considerations Since the UTVR is being developed for SDI's Burst Power Mode for space power applications, the following issues need be realized: 1. The size and mass of the power system are important constraints. This is due to the following: (a) the expense and logistics involved in the deployment of the power system into space, (b) the need to constantly maneuver and relocate the defense system, and (c) the need for defense systems to be inconspicuous. 2. The required power level for this system ranges from »100 MW up to »1 GW for operating times of ss30 minutes. Such power levels when considered with the size requirement demand a high power density system. 3. The system is required to achieve the Burst Power Mode in less than 100 seconds. Thus, the system needs to be designed to withstand thermal stresses and shocks caused by a rapid transition from the alert mode. 4. The power system is required to be able to operate during a seven year period. This requires the power system to be tested periodically; thus, the system needs to be designed to operate at full power for a total time of about three hours (assuming two-annual tests during the seven-year period lasting about ten minutes each plus the 30 consecutive minutes of operation). 5. The system needs to be operated at high temperatures to provide compact radiators for heat rejection in space and high power cycle efficiency. ' • ' " ' * ' '

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.. 21 The above issues are the primary considerations applied during the preliminary design of the UTVR. Choice of Materials The UTVR is a BeO externally-moderated, circulating fuel reactor with UF4 as the fissioning fuel and a metal -fluoride working fluid. Research is being conducted to select and develop suitable structural materials that are compatible with the fluoride fuel/working fluid mixture and the high temperature environment of the UTVR. The choice of BeO as the moderator-reflector material, UF* as the fuel, and metal fluoride as the working fluid is based on the following considerations. The Moderator-Reflector Material For thermal reactors, moderator-reflector materials used in nuclear reactors have low mass numbers and relatively large scattering and relatively small absorption cross sections. Moderators used in nuclear reactors include ordinary water (H^O), heavy water (DpO), beryllium (Be) or beryllium-oxide (BeO), and graphite. The choice depends largely on the intended application of the reactor system; and on the nuclear, mechanical, physical, and chemical properties; and the cost of the moderator material. Since the size and mass of the power system are significant constraints, and since high temperatures are needed for efficient heat rejection in space, the moderator-reflector material is required to have a high melting temperature (or high boiling temperature if a liquid moderator is used) and relatively good neutronic properties (high slowing-down power and small capture cross section for neutrons).

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'-.::, 22 For space power reactors, beryllium or BeO is superior to graphite as a moderator and reflector material from a neutronics standpoint. In the study of Highly Enriched Heterogeneous Gas Core Reactors (HGCRs), S.D. Kahook [7] has shown that the use of Be as the moderator and reflector material provides a higher reactivity (»30% 5k/k) than graphite (total size of the HGCR was fixed). This is mainly due to the higher slowing-down power {^xS^ = average logarithmic energy loss per collision x macroscopic scattering cross section) of 16 m" for Be versus 6.5 m" for graphite [19] and the (n,2n) reaction of Be. Another drawback of graphite is its larger thermal diffusion length, Lj, (=54 cm versus »21 cm) and its larger slowing-down length, Tj, (=192 cm versus =100 cm) compared to Be. The larger values of Lj and Tj require graphite-moderated reactors to have a larger size compared to berylliummoderated reactors, an important design criterion for the space power system under investigation. Although the melting temperature of graphite is higher then that of BeO (=4000 K versus =2800 K), BeO is a better choice than graphite for the power system under investigation. The drawback of the lower melting temperature of BeO can be compensated for by the use of auxiliary coolant channels in the moderator-reflector regions to maintain BeO at safe operating temperatures (=1600 K to =2000 K). Also, due to the low heat conductivity of the vapor fuel and the fact that the fuel is the working fluid (most of the energy generated is directly deposited in and removed by the fuel/working fluid mixture), the temperature of moderator-reflector regions can be considerably cooler then the temperature of the vapor fuel.

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E.T. Dugan [2] examined the effect of using H2O, D2O, Be, BeO, and graphite as moderator-reflector materials for the Nuclear Piston Engine which employed an externally-moderated UFg-fueled gas core reactor for terrestrial power generation. The study indicates that the use of H2O and graphite results in relatively low kg^^ values. This is due to the relatively high thermal absorption cross section of H2O and high Lj and Tj of graphite. However, the large si owing-down power as well as the (n,2n) reaction of Be and the small thermal absorption cross section of D2O, cause Be, BeO, and D2O to be excellent choices for moderatorreflector materials as proven by Dugan. The relatively large Lj of D2O of «97 cm requires that the size of reactors employing D2O as the moderator to be quite large. Additionally, the high temperature environment of the UTVR, the chemical incompatibility between H2O or D2O and UFg or UF^, the normal deterioration of D2O into H2O in time (small amounts can have large effects on neutronics), and the added complications involved with a liquid moderator versus a solid moderator in space all aid in rejecting D2O as the moderator-reflector material for the UTVR. The ceramic nature of BeO with a high melting temperature of «2800 K and its exceptional resistance to thermal shock [20] make this an especially well -suited moderator-reflector material for the high temperature environment of the power system under investigation. Although per unit mass Be is neutronically superior to BeO as a moderator, the anticipated moderator temperature range of 1600 K to 2000 K for this burst power system precludes the use of Be (melting point of Be is 1728 K).

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> "... ; 24 The Fissioning Fuel Material The advantages and key features of vapor-fueled reactors are more than adequate to justify the study of a fuel in the vapor state. However, uranium exists in a gaseous state in various forms such as UF^, UFg, or uranium metal vapor. Reactors employing uranium in these forms have all been investigated at the University of Florida. The choice of the fuel along with the working fluid are dictated by the type of power cycle, e.g., Brayton or Rankine cycle. It is appropriate to compare features of these cycles in order to select a suitable fuel. The Brayton cycle is simpler in design than the Rankine cycle. However, it generally has a lower thermodynamic efficiency. Due to this lower efficiency, more heat has to be rejected into space which implies that a larger radiator is needed. In addition, the heat rejection to space is done at a varying (decreasing) temperature rather than a constant temperature thereby decreasing the effective temperature of heat rejection and further increasing the required radiator size. The greater pumping power required for gas compression in the Brayton cycle demand larger and more massive compressors as compared to the pumps in a Rankine type of cycle. Since size and mass are significant constraints, and since a Brayton type of cycle requires larger radiators and more massive compressors than a Rankine type of cycle, a Rankine type of cycle appears to be the better choice, especially for high power systems. For space power Rankine cycle systems, the most desirable fuel choice is UF^. This can be seen from Figure 3-1 where the UFg and UF< saturation vapor curves are shown and from the uranium metal and UF^

PAGE 46

>,-.' -Viv*25

PAGE 47

^ '•• .: '^ -^ __,. ' ^/ 1../ 26 saturation vapor curves given in Figure 3-2. For the UFg to be in the gaseous state, at pressures required for critical ity in the core, Its temperature need only be in the 400 to 500 K range. This implies that the UFg must be 400 K or less to achieve a liquid state at the exhaust pressures when a gas turbine or MHD generator is used for power conversion. This low heat rejection temperature can easily be achieved on earth, but is unrealistic in a space environment. Thus, one is restricted to a Brayton type of cycle when UFg is the fissioning fuel fluid in a space power system. When uranium metal vapor is used as the fissioning fuel and working fluid, the difficulty is not in achieving a liquid state at the heat rejection end of the cycle as with UFg. For example, at an exhaust pressure of 1 atm one need "cool down" to only about 4000 K to achieve liquid uranium. The obstacle with uranium metal vapor is the extremely high temperatures of the vapor in the core. The fluid temperature needs to be at least 6000 K at all locations in the core to ensure the vapor state at pressures needed for critical ity. This indicates that the peak gas temperature in the core will be at least 8000 K or 9000 K. The choice of UF^ as the fuel rather than UFg or uranium metal vapor is justified by examining the saturation vapor curves. Figures 3-1 and 3-2, and the mole fraction of constituent species versus temperature curve of the uranium-fluorine system. Figure 3-3 [21]. At pressures required for critical ity in the core, the temperature of UF. need be only about 2000 K to guarantee the vapor state. On the other hand, one need cool down to only 1700 K in order to obtain liquid UF^. The extremely high uranium metal vapor temperature in the core and the

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27 0) I. 3 to > iio a. s. o c (O sCO (U iCVJ o o o evj

PAGE 49

28 3 a. o I E 3 •* l« '••

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29 extremely low UFg heat rejection temperatures are avoided. Thus, UF^ has a saturation vapor pressure-temperature behavior that is highly desirable for a direct Rankine cycle burst power system for a space environment. Also, Figure 3-3 indicates that in the expected gas temperature operating range of 2500 K to 4000 K, UF^ is the predominant uranium-fluorine specie. The Working Fluid Material It has been shown in the previous section that a Rankine type of cycle is more appropriate for a burst mode space power system, and on this basis the fuel is selected to be UF^. Therefore, a working fluid that is compatible with the UF^ fuel and suitable for a Rankine type of cycle is needed. Preliminary chemical and material studies [22,23] indicate that a working fluid in the form of a metal fluoride should be compatible with the UF^ fuel. These working fluids include Li^F, KF, NaF, and RbF. Table 3-1 list relevant properties of these materials. Description of a Uranium Tetra-Fluoride. UTVR/Disk MHD-Rankine Power Cycle An example UF^/KF UTVR MHD-Rankine cycle power system schematic is shown in Figure 3-4. This system is capable of producing 200 MW^ with a thermodynamic efficiency of «26%. The mass flow rates of UF^ and KF are 59 and 209 kg/sec, respectively. For the system illustrated in Figure 3-4, about 40 MW is required to vaporize the liquid UF^ in the UF^-boiler. The UF^ vapor is then directed to the UTVC where it is mixed with the KF. In the UTVC, 30 MW

PAGE 51

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PAGE 52

31 B u i) l/t — 00 • ^ * ^ CM ii (A at >.?s U "s-;^ (dm fl X ooin o o> ^ fH u) m U. 00 oo CM LO 9\ jaiioa -^jn C DAin tBk -. a* ^ ^^ t^ ^ (0 O) ^^ o o ^ o Ln 00 u. o vo 3 * CM (U +-> >> 00 3 o c c a> I ^ (O > m o ii. vo ^ CM o o CM « O n (U u CO

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of the fissioning power is deposited in the UF^ to raise its temperature from 2350 K to 4000 K. The KF flows around the UTVC cooling the vapor core wall region where about 35 MW is added to it in the form of sensible heat to raise its temperature from 1920 K to 2300 K. Then the KF is injected into the vapor core where it is mixed with the UF^. In the UTVC, 115 MW is added to the KF to raise its temperature from 2300 K to 2665 K and 367 MW is added in the form of latent heat to vaporize it at 2665 K. Another 180 MW of heat is then required to raise the KF temperature from 2700 K to 4000 K. A total of 697 MW of thermal power is therefore added to the KF fluid. The UF^/KF mixture is then passed through the nozzle, the MHD generator, and the diffuser. In the MHD generator, about 10 MW of energy is produced by fission and 200 MW^ is extracted. Waste heat in the amount of 526 MW is rejected to space via a 720 m primary condensing radiator which allows the separation of the vapor mixture into UF^ vapor and KF liquid. The UF^ vapor is then passed through a 56 2 m secondary condensing radiator in which 31 MW is rejected to space. Both the UF^ and the KF are then compressed via separate pumps. For the purpose of generating this cycle, it is assumed that UF^ and KF are completely separable; this may not be the case. In a real system many species, including K^^U F^ species, may be present as shown by Hildenbrand and Lau [23]. The system described above has the potential to be extremely reliable since the only components with moving parts are the UF, and KF liquid pumps. ,.'

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33 Table 3-2 lists 200 MW^ UF^/UTVR power cycle operating characteristics for KF, NaF, Li F, and RbF working fluids. Table 3-3 presents the energy balance data for a 200 MW UF^/UTVR power cycle with KF, NaF, Li F, or RbF as the working fluids. Table 3-2 indicates that if none of the metal fluoride is vaporized in the boiler columns, then the required power sharing (or power ratio) of the UTVC to the boiler core (PyTvc/'^BCOL^ based on thermodynamic/flow considerations is «20. It also indicates that =90% of the total fission power produced is deposited in the metal fluoride. The power sharing ratio, PijTVc/^BCOL' places another restriction on the power system. That is, the UTVR needs to be configured so that the power sharing between the UTVC and boiler columns based on nuclear analysis matches what is obtained from thermodynamic and flow considerations. One method of controlling ''uTVc/'^BCOL ^^ *° divert part of the metal fluoride from the UTVC wall cooling region to the boiler region. The power sharing as a function of metal fluoride mass flow rate to the boiler region as determined on the basis of thermodynamic and mass flow requirements is listed in Table 34. • .'/: ' : , Table 3-4 indicates that the required power sharing ratio decreases by a factor of =3 for NaF, KF, and Li F and by a factor «2 for RbF when only 10% of the metal fluoride is diverted to the boiler region. This is due to the large amount of power needed to vaporize the metal fluoride as compared to the UF^.

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34 o c o u i0) 4-> u i(-> a. O c o o E a> u >» o i. 2 o JC OX» o O UO J2 CVJ OC CO a>

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35 I B 1m I 8 Mr 49 <« O 0) » C i-^ 0) sc o I •

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36 s-

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37 Neutronic Analysis of the Ultrahigh Temperature Vapor Core Reactor Static Neutronic Calculations A number of computer codes existing at the University of Florida, San Diego Supercomputer Center, and the Air Force Supercomputer CenterKirtland are used in analyzing the static neutronic behavior of the UTVR. In the static neutronic analysis, the fuel is assumed to be stationary. Thus, the effect of the loss of delayed neutrons outside the cores is ignored. This assumption over-estimates the contribution of the delayed neutrons inside the cores, i.e., kg^^ is somewhat larger for the stationary system than for the corresponding circulating system. This effect is corrected for in the dynamic model and is discussed in detail in Appendix D. -' ' ^'^' One of the computer packages used at University of Florida is the AMPX [24] modular code system which is described in Appendix A. Weighted multigroup neutron cross-sections files are generated from ENDF/B [25] data using the XLACS [26] code, an AMPX module. Selfshielding calculations are performed on the weighted multigroup neutron library created by XLACS using NITAWL [27]. NITAWL produces a 123neutron group AMPX library. This 123-neutron group library is then collapsed first to 27and then to four-neutron groups using XSDRNPM [10]. Four-neutron group, 1-D, spherical geometry, discrete ordinates (S ) calculations are performed using XSDRNPM. Group dependent neutron flux distributions in space, region reactions rates, and eigenvalues are

PAGE 59

' 38 obtained from the 1-D calculations. Basic static neutronic characteristics of the UTVR are obtained from the 1-D calculations. These include k^^ and PuTVc/^BCOL ^6*^^^"'°'^ ^^ * function of moderatorreflector regions thickness, fuel density and enrichment, and types of working fluids. Reactivity penalties as a function of different liner materials and their thicknesses are also obtained from the 1-0 neutronic static calculations. ' * r '"'' J Two-dimensional S cylindrical geometry calculations in the R-9 and the R-Z coordinate systems are performed using DOT-4 [11]. The fourneutron group cross-section library produced by XSDRNPM is converted to DOT-4 format using GIP [28]. In R-9 geometry, the boiler region can be accurately modeled as a number of boiler columns separated by BeO moderator. The nozzle, disk-MHO generator, and diffuser regions can all be modeled in the R-Z geometry. Results obtained from the 1-0 spherical "mock-up" of UTVR are compared with results obtained from calculations performed in the R-6 and the R-Z coordinate systems. These comparisons are necessary to determine the reliability of the obtained results. The reference UTVR configuration for the 3-0 analysis is obtained from 1and 2-0 static neutronic calculation results. The 3-D calculations are performed using MCNP [12], a 3-0 Monte Carlo neutron transport code. Integral parameters for the dynamic neutronic analysis are calculated from MCNP results by using ISCE, a special code developed as a part of this research. Parameters obtained from ISCE include coreto-core coupling coefficients, and the reactivity and neutron multiplication factors of individual cores.

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39 Dynamic Neutronic Calculations The over-estimate of k^^: obtained from the static neutronic calculations due to the assumption of a stationary fuel is corrected for in the UTVR kinetic model. The dynamic analysis in the time domain is performed using circulating-fuel, coupled-core, point reactor kinetics models. Inherent reactivity feedback effects such as vapor fuel density and boiler column liquid volume changes are included in the dynamic model. Dynamic and stability analysis studies are performed with the Engineering Analysis System code, EASY5 [29]. The computer codes mentioned above are described in Appendix A.

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CHAPTER IV STATIC, ONE-DIMENSIONAL, UTVR NUCLEAR CHARACTERIZATION AND CONFIGURATION OPTIMIZATION Introduction The initial one-dimensional (1-D) spherical "mock-up" configuration used to perform the preliminary nuclear characterization of the Ultrahigh Temperature Vapor Core Reactor (UTVR) is shown in Figure 4-1. It consists of four regions (the wall cooling region is neglected in the initial calculations). The first is the Ultra High Temperature Vapor Core (UTVC) region which contains the fuel mixture that consists of highly enriched UF^ vapor and a metal fluoride vapor at 3000 K and 50 atm. The second is the inner beryllium oxide (BeO) moderator-reflector region (IBEO) which contains only BeO. The third is the boiler column (BCOL) region where the UF^ is vaporized. The fourth is the outer BeO moderator-reflector region (OBEO). The actual cylindrical reactor system is converted to the 1-D spherical geometry by conserving the volumes of the UTVC and the boiler cores and by conserving the thicknesses of the inner and outer BeO moderator-reflector regions. Neutronic calculations are performed using XSDRNPM [10]. XSDRNPM is capable of computing the system's neutron multiplication factor (kgffr), region average and local neutron fluxes and currents, and the fission rate in each region (power produced). XSDRNPM is described in detail in Appendix A. 40

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41 Outer BeO ModeratorReflector Region UFBoiler Region Inner BeO ModeratorReflector Region Ultrahigh Temperature Vapor Core Region 1 Region 2 Region 3 Region 4 Figure 4-1. Four Region, One-Dimensional Spherical "Mock-up" of the UTVR

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': ^1 i ' [ > '. ' % ''.J \.., . ^ : ' ' . '' * '-•^ ' '' 42 Modeling the UTVR in the 1-D spherical geometry is expected to result in excessively high values for kg^f. The k^^^ values are expected to be quite large due to the following: 1. Neutron leakage from the UTVR is underestimated since spherical configurations provides the smallest surface-to-volume ratio. 2. Neutron streaming from the MHD duct is not accounted for since the MHD duct regions is not included in the 1-D spherical "mock-up." 3. Reactivity worth of the boiler column is overestimated since the boiler column is treated as a spherical shell surrounding the UTVC. The boiler column in the actual reactor system consists of a number of boiler columns separated by BeO moderator. By configuring the boiler region as a spherical shell surrounding the UTVC, the probability for neutrons interacting with the boiler region is relatively large. Additionally, thermal neutron flux depression in the boiler region is underestimated since the thickness of the boiler region in 1-D is small compared to the thickness of the actual boiler columns. 4. Core protective materials (liners and cladding), structural support members, and piping are not included in the analysis in order to simplify the scoping analyses. That is, the 1-D spherical "mock-up" represents a "clean" UTVR system. Detailed three-dimensional neutronic analysis using MCNP [12] (Chapter VI) indicate that, when neutron leakage and streaming are accounted for, actual boiler configuration is modeled, and structural and liner materials are employed, k^^^ values of «1.05 is obtained.

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43 Therefore, the high k rr values obtained in this preliminary stage of analysis are "reasonable" and needed. Scoping Calculations To commence the nuclear characterization of the reactor system, numerous 1-D scoping calculations are performed. These calculations examine the effect of variations in geometry, fuel density, fuel enrichment, mole fraction, and materials. The results of these studies are described in this section. Geometric Variations As mentioned previously, size and mass are significant constraints on space power systems. In addition to the total power requirement, an important design consideration for this system is the power sharing or the amount of power generated in each fissioning region (UTVC regions and the boiler regions). Of these constraints (power sharing, total power production, and size and mass of the reactor system), the power sharing is expected to be the most restrictive. To determine which configurations are capable of meeting these constraints, the effects of variations of the following parameters are examined: ^' UTVC radius While maintaining the inner and outer BeO region thicknesses at 20 -03 cm and 35 cm, respectively, and the boiler region volume at 8.5 x 10 m (the boiler region contains an equal mixture by volume of liquid and vapor UF^ with an inlet velocity of 2 m/sec at a mass flow rate of 68 kg/sec), the radius of the vapor core (UTVC) is varied from 40 to 150

PAGE 65

44 cm. The fuel mixture in the UTVC region is maintained at 3000 K with the partial pressures of the UF^ and the NaF fixed at 5 and 45 atm, respectively. The results, kg^^ and PutVC^^'bCOL' ^^^ given in Figure 42 as a function of UTVC radius. Figure 4-2 indicates that kg^^ increases from 1.462 to 1.479 and ''utvc/''bC0L increases from 0.23 to 0.53 as the UTVC radius increases from 40 cm to about 70 cm. However, for UTVC radii above 70 cm k^^^ decreases while PuTVc/^BCOL continues to increase. The interpretation of the behavior of kg^^ and PyTVC^^'BCOL ^'^ * function of the UTVC radius requires two further sets of calculations. The first consists of varying the UTVC radius from 40 to 150 cm while the boiler region is voided. The second consists of varying the UTVC radius over the same range but with a voided UTVC and a loaded boiler. The results are given in Figure 4-3, where values of k rr and fission rate versus the UTVC radius are given for both cases. The fission power produced in a region is a function of the thermal neutron flux, the macroscopic fission cross section, and the volume of the fissioning region. When the boiler is voided and the UTVC radius is varied from 40 to 150 cm, the most significant change is an increase in the UTVC volume. This increase in the UTVC volume results in an increase in PyTurOn the other hand, when the UTVC is voided and the UTVC radius is varied from 40 to 150 cm, two phenomena occur: (1) neutronic de-coupling of the annular boiler region, and (2) reduced number of mean free paths that a neutron encounters when passing through the boiler. That is, as the UTVC radius increases and the boiler volume is fixed, the boiler region in the 1-D spherical "mock-up" becomes

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46 (oas/suoLSSLj.) a^e^ uoisslj o CO 8 S 8 -• ^ ^ O "^^^51 'joaaej uoiq-BDitditiLnw uoj5.naN o U 0> o 01 c le <_> > => O) O ro o -o •1« (/> a: •r(_> T3 13 C ID O) 0)4^ O I 0) iK / • .

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47 thinner and more separated in space from itself. Thus, fewer fissions occur and a decrease in Pgroi ^^ observed. Thus, the observed increase in PuTVC^'^BCOL' "^^ ^^^ ^^^^ radius increases, when both cores are loaded (Figure 4-2) is due to the increase in PnTwr and the decrease of Pgrni ''^ shown in Figure 4-3. Although not shown in Figure 4-3, when the vapor core is voided an optimum value for k r^ is obtained at a UTVC radius between and 40 cm for an inner BeO thickness (IBEO) of 20 cm. This optimum UTVC radius is due to optimum neutronic coupling from one segment of the boiler region to the other. Calculations for the voided UTVC configuration have been performed as its radius is varied from to 80 cm at IBEO thickness of 0, 5, 10, 15, and 20 cm. The results are given in Table 4-1 for this type of variation. For the case where the IBEO is 20 cm, k^^^ peaks at a voided UTVC radius of about 30 cm. At this radius, the strongest neutronic coupling for the boiler region exists. However, for smaller IBEO thicknesses of 15, 10, 5, and cm, a larger voided UTVC radius (beyond 40 cm) is needed to show the peak in Kff This indicates that as the IBEO thickness decreases, the voided UTVC region radius required for optimum boiler coupling increases. However, when the UTVC is loaded, the radius required for optimum boiler coupling is different as is shown in the following section. As mentioned previously, for an IBEO of 20 cm and UTVC radii greater than 40 cm, k r^ increases as the loaded UTVC radius increases when the boiler is voided and k rr decreases as the unloaded UTVC radius increases when the boiler is loaded (as shown in Figure 4-3). However,

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48

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49 the overall or net effect for the reactor system when both cores are loaded is an increase in k r^-r until the UTVC radius is about 70 cm; k r.^ then decreases for radii beyond 70 cm. This can be explained as follows: as the UTVC radius increases from 40 to 70 cm, the gained reactivity from the "larger" UTVC core overrides the lost reactivity due to the de-coupling of the boiler region. For vapor core radii beyond 70 cm, there is a diminishing gain in reactivity due to the larger UTVC and the boiler region de-coupling effects dominate. A justification for this assertion is as follows: in the range of 40 to 70 cm for the UTVC radius, from the data used to generate Figure 4-3, the gain in 5k/k for the loaded UTVC is about 0.28 (or 9.3 x 10"°^ 5k/k per cm of IBEO) and the loss in 6k/k for the loaded boiler is only 0.08 (2.7 x 10'°* 5k/k per cm of vapor core radius); thus, a net increase in k^r is obtained. However, when the UTVC radius increases from 70 to 150 cm, the gain in 6k/k for the loaded UTVC is only about 0.18 (or 2.3 x 10 ""^ 5k/k per cm of vapor core radius) and the loss in 5k/k for the loaded boiler is 03 about 0.37 (or 4.6 x 10 5k/k per cm of IBEO); thus, a net decrease in '^eff ^^ obtained. The observed decrease in the rate of increase of kg^^ as the UTVC radius increases beyond 70 cm for the case of the loaded UTVC and voided boiler is due to approaching infinite reactor configuration; this is seen in Figure 4-3 where the vapor core k^^^ clearly begins to saturate as the UTVC radius increases beyond *100 cm. It is concluded from the above discussions, that the UTVC radius along with the IBEO thickness are the determining parameters that influence neutronic coupling for the UTVC/boiler regions and the neutronic coupling from one segment of the boiler to the other. That

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'"" 50 is, varying the UTVC radius for a fixed IBEO thickness, or varying the IBEO thickness for a fixed UTVC radius, will lead to an optimum configuration with respect to neutronic coupling. Thus, for a given UTVC radius, an IBEO thickness can be selected to yield optimum overall neutronic coupling between the boiler and the UTVC. A value of 60 cm is selected for the UTVC region radius for further analysis. This value is based on the fact that the UTVC is in reality a cylinder and is expected to be 100 cm in height with a radius of about 55 cm. These dimensions appear to be in a range that is acceptable with respect to thermohydraulics and acoustic calculations performed on the system [30]. Inner BeO moderator-reflector region thickness With the UTVC radius fixed at 60 cm, the vapor fuel temperature set at 3000 K, the UF^ partial pressure set at 5 atm, the NaF partial -0"? pressure set at 45 atm, the boiler region volume fixed at 8.5 x 10 3 m , and the OBEO region thickness held at 35 cm, the IBEO thickness is varied from 5 to 50 cm. The results are given in Figure 4-4 where k rr and PuTVC^^'bCOL ^^^ plotted as a function of IBEO. The results indicate that the optimum neutronic coupling between the vapor core and the boiler region occurs at an IBEO thickness of about 16 cm where k^^^ is greatest with a value of about 1.507. Beyond a BeO thickness of 16 cm, k^^^ decreases. This is due to the decoupling phenomenon for the boiler and the decreased thickness of the boiler at the higher IBEO thicknesses. Figure 4-4 also indicates that PuTVC^'^BCOL """iti^lly increases as the IBEO thickness increases from 5 cm to about 16 cm and remains at a constant level of 0.39 as the IBEO thickness increases from about 16 to

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51 lOOajyOAlflj «jo^3Bj SuueilS JBMOd o o in m I o

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.. 52 45 cm. This ratio then undergoes a further increase as IBEO increases beyond 45 cm. Three relevant phenomena occur as the IBEO thickness is increased, they are: (1) a decrease in the number of mean free paths that a neutron encounters when passing through the boiler due to the decrease in the boiler thickness as IBEO increases; (2) neutronic decoupling of the annular boiler region; and (3) an increase and then a decrease in the neutronic coupling between the UTVC and boiler cores. In order to explain the observed behavior of k^^^r and PuTVc/'^BCOL ^^° additional sets of calculations are needed. The first involves varying the IBEO thickness from 5 to 60 cm for a fully loaded UTVC with a radius of 60 cm and with the boiler voided. The second set involves varying the IBEO thickness over the same range for a voided vapor core with a -03 radius of 60 cm and with a loaded boiler (volume fixed at 8.5 x 10 m ). The results are shown in Figure 4-5. The results indicate that for a 60 cm radius voided UTVC, optimum neutronic coupling of the boiler column occurs at an IBEO thickness of «12 cm. As the IBEO increases beyond 12 cm a decrease in kg^^ occurs. This decrease in k r^ of the boiler region translates to a decrease in PgPQ. . However, for the case where the boiler is voided and the UTVC is loaded, k /r^ continues to increase as IBEO increases. From the data used to generate Figure 4-5, as IBEO increases from 10 to 45 cm, the values of 5k/k per cm of IBEO are +1.9 x 10"°^ and -4.8 x 10'°^ for the loaded UTVC and for the loaded boiler cases, respectively. These values reflect the change in reactivity expected if the only phenomena that are affected by the variation in IBEO are those listed in items 1 and 2

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53

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54 above, and the net result should be a continuous increase in Pyyvc/^'BCOL and a continuous decrease in kg^f However, neutronic coupling between the UTVC and the boiler cores is also affected by variations in the IBEO thickness. As the IBEO thickness increases from 5 to «12 cm, P^jvc increases (due to increase in the UTVC neutron reflection rate, which can be inferred from the k^^^ curve shown in Figure 4-5, and due to an increase in boiler-to-UTVC neutronic coupling) and Pgroi increases (due to enhanced neutronic coupling of the boiler region, as shown in Figure 4-5). However, the increase in PiiTur is larger than the increase in PgcoL" ^^^^ causes ''uTVc/^'bCOL ^° increase. As the IBEO region increases from 12 to »16 cm, PuTur continues to increase while PRrni begins to decrease, thus a further increase in Putvc/''bCOL ^^ obtained. Although Figure 4-5 indicates that an increase in Piijur should occur in the IBEO thickness range of 16 to 45 cm for the case the vapor core is loaded and the boiler region is voided, the actual result when both cores are loaded is a decrease in Piitvc ^°^ ^^^^ thicknesses above 16 cm, a decrease in the boiler-to-UTVC neutronic coupling occurs which causes Pnt^q to decrease. In this IBEO thickness range, PnTwr is decreasing at about the same rate Pgrni ^^ decreasing. The net result is a constant P,j-ryQ/PnpQl behavior over an IBEO range from about 16 to 45 cm. From the data used to generate Figure 4-5, for the IBEO thickness range from 45 to 60 cm, the values of 5k/k per cm of IBEO are about -5.1 X 10" and +3.0 x 10' for the loaded boiler cases and the loaded UTVC, respectively. Above an IBEO thickness of 45 cm, the rate of

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55 decrease in PgQQL ^^ greater than the rate of decrease of PnTwrThis leads to an increase in PuTVc/''bC0L ^^^ ^ further decrease in kg^f. The combined neutronic coupling between the UTVC and the boiler region (i.e., the combined boiler-to-UTVC, UTVC-to-boiler, and annular boiler region neutronic coupling) is an optimum at an IBEO thickness of «16 cm for this configuration. For future calculations, a thickness of 15 cm is selected for IBEO. Outer BeO moderator-reflector region thickness * Maintaining the UTVC radius at 60 cm, the IBEO thickness at 15 cm, -03 3 and the boiler volume at 8.5 x 10 m (0.12 cm thick), the OBEO region thickness is varied from 10 to 100 cm in order to obtain the optimum outer BeO thickness. The results are shown in Figure 4-6 where k cf and ''uTVc/'^BCOL ^^^ plotted as a function of OBEO thickness. Figure 4-6 clearly indicates that k rxr saturates at an OBEO thickness of about 40 cm. At this thickness and beyond, k rr is around 1.52 and PijTVc/^BCOL ^^ 0-36. Increasing the OBEO thickness beyond 40 cm does not enhance the system neutronically, i.e., the value of k /^r. It only increases the size and the mass of the system. This is yery undesirable since the system is intended for space power production. The results also indicate that as the OBEO thickness decreases below 40 cm, PijTVc/^BCOL increases since PorQi decreases. This is anticipated since the OBEO has a direct effect on the boiler region and less of an effect on the UTVC. Thus, a method to increase Pmtvc/''bCOL ^^ ^° reduce the thickness of the OBEO. However, this will cause a greater number ofand more energetic neutrons to leak out of the reactor which will require the use of heavier and thicker shielding.

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56 lOOajyOAiflj «jo^.3Bj SuueMS jaMOj S o U) s o 19 o CO

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57 For all further calculations, a thickness of 40 cm is selected for the OBEO. UF^ boiler region thickness Increasing the UF^ boiler region thickness in the "mocked-up" spherical geometry is analogous to increasing the cross sectional flow area of the actual boiler column configuration. This in turn results in a decrease in the inlet velocity of the UF^ liquid to the boiler (assuming a fixed mass flow rate is required). The inlet velocity of the UF^ liquid to the boiler dictates the amount of liquid UF^ present in that region at a given power level. Thus, the reactivity worth of the boiler is strongly influenced by the inlet UF^ velocity. This velocity also impacts on the boiler region friction and acceleration pressure losses. The lower the inlet velocity the lower the pressure losses, but if the velocity is too low, then the size (area) of the boiler region will be great. Knowledge of the neutronic behavior of the system as a function of the inlet velocity of the UF^ liquid to the boiler is obviously essential. The velocity of the UF. liquid is varied from 0.5 m/sec to 6.0 m/sec which corresponds to annular boiler region thicknesses from about 0.48 cm to about 0.04 cm. The k rr and ^UTVC^^'bCOL "^^^^^^^ ^^ * function of UF^ inlet velocity are plotted in Figure 4-7. Figure 4-7 indicates that as the inlet velocity of the UF^ liquid increases, k ^r^ decreases and PutVC^^'bCOL increases. Both behaviors are due to the decrease of the boiler area (volume) as the velocity increases. A decrease in the amount of fissile material in the boiler causes PRroi ^° decrease thus increasing PuTVC^'^BCOL ^^^ decreasing

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58 1009j^0Ainj «jo^3pj Suueqs jaMoj u e (U (4o u m

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59 '^eff '^fispectively. It should be noted that selecting an inlet UF^ velocity based only on results from these 1-D calculations is not reasonable since the boiling and non-boiling regions (amount of vapor versus liquid UF^) cannot be accurately modeled. Twoand threedimensional modeling are needed in order to select a suitable operating velocity. For these preliminary studies, a flow rate of 2 m/sec is chosen. This flow rate, 2 m/sec, corresponds to a boiler volume of 8.5 -03 3 X 10 m (100 cm in height) at a mass flow rate of 68 kg/sec. UF ^ boiler core volume A safety consideration in the design of the UF^/metal -fluoride t nuclear power system is the unwanted possibility of self-critical ity in a UF^ boiler region. That is, the size of the boiler columns and the amount of the liquid UF^ present in the boiler region should be chosen so that the region cannot become self -critical even under extreme conditions. For this study, a two region core, in 1-D spherical geometry, is used to mockup a 50 cm in height UF^ boiler core surrounded by 40 cm of BeO reflector or 40 cm of Be reflector. The spherical radius of the core is varied from 4 cm to 11 cm which corresponds to an equivalent cylindrical radius of 1.3 cm to about 6.0 cm. The boiler ?35 contains 100% enriched U in completely liquid UF^. Values for kg^^ range from 0.491 for the spherical radius of 4 cm to 1.101 for an 11 cm spherical radius, as shown in Table 4-2. The boiler is found to be critical (kg^^ = 1.0) at a spherical radius of about 9.8 cm, which corresponds to a cylinder with a radius of about 5 cm. The results indicate that the UF^ boiler columns will not become self-critical since in reality they will not contain 50 cm of liquid.

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8« O O 60

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61 If the reactor system has only two boiler columns at a total UF^ mass flow rate of 62 kg/sec, then the required cross sectional flow area per column for an entrance velocity of about 2 m/sec for the UF^ liquid will be about 4.2 x 10 m . This corresponds to a cylindrical radius of about 3.7 cm which is lower than the critical radius obtained in the 1-D calculations; a larger number of columns would result in even smaller boiler region radii (for an inlet velocity of 2 m/sec). Fuel Density Variations „ ^ , ^ -, w' During reactor startup and power level changes, there will be changes in the UF^/MF vapor pressure and temperature and, thus, in the density. Also, the amount of liquid and the void volume fraction of the UF^ in the boiler region will change depending on the power level. An insight into the effect of density changes will help in predicting the response of the reactor to power level changes and/or perturbations introduced into the system. The effects of variations in the following parameters are therefore studied. UF. partial pressure and mole fraction (UF ^ rNaF) in the UTVC Preliminary calculations and analysis of the MHD generator indicate that a mole fraction of about 10% for UF^ and 90% for NaF results in efficient energy extraction [31]. Maintaining the UTVC radius at 60 cm, 03 3 the IBEO thickness at 20 cm, the boiler volume at 8.5 x 10 "'^ m , and the OBEO thickness at 35 cm (these calculations were performed prior to obtaining the optimum IBEO and OBEO thicknesses of 15 and 40 cm, respectively), the UF. partial pressure is varied from 1 to 20 atm at

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62 NaF partial pressures of 20, 40, and 60 atm. The result of these variations are shown in Figure 4-8. , The results indicate that the system is essentially unaffected neutronically by the NaF partial pressure. Thus, the UTVC can be operated with a fuel mixture that is optimum with respect to the demands of the MHD generator. Figure 4-8 indicates that k^^^ saturates at UF^ 235 partial pressures above =10 atm. This corresponds to a U density of 2.5 X 10"^ atoms/barn-cm. Beyond a UF^ partial pressure of 10 atm or a U^^^ density of 2.5 x 10"^ atoms/barn-cm, the UTVC is becoming black to neutrons. The results, as shown in Figure 4-8, indicate that '*UTVc/''bC0L increases as the UF^ partial pressure increases up to about 10 atm and remains at about a constant level as the UF^ partial pressure further increases. For all further analysis, partial pressures of 5 atm for UF^ and 45 atm for NaF are used. 235 U enrichment in UF ^ The U^^^ enrichment is varied from 80% to 100% at UF^ partial pressures in the UTVC of 1, 2, 3, 4, 5, 6, and 7 atm. The results, as shown in Figure 4-9, indicate that k^^^ increases as the enrichment 235 increases and as the UF^ partial pressure increases. The U enrichment is fixed at 100% for all further analysis. 233 U as the fissile isotope The U^^^ fissile isotope in UF^ is replaced with U^^^. The U^^^ enrichment is varied from 80% to 100% at UF^ partial pressures of 3 and 5 atm. The results, as shown in Figure 4-10, indicate the same behavior 235 as obtained in Figure 4-9 for U with the exception that k^rf is 233 higher when U is the fissile fuel. This is due to the lower thermal

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63 :('' lOOajyOAinj «jo^3igj Buueqs J3M0d 00 00 •r 8 «^j I LO o

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64 .^''^ 4* P S. a* 4-» «9 I m CO CM (U (U o c *^ )( 'Jo^DBj uo!.ae3LLd!.q.LnH uojinBn o

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:^f^ 65 4 »^,. ."

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66 capture cross section, a , (48 barns versus 99 barns) and the higher number of neutrons liberated per thermal fission, v, (2.49 versus 2.42) in U versus U ; this corresponds to a higher number of fission neutrons liberated per thermal neutron absorbed in the fuel, r?, (about 2.29 for U^^^ and 2.07 for U^^^). Average density of the UF ^ in the boiler region By varying the effective density of the UF^ in the boiler region, the boiler column is simulated as a mixture of liquid and vapor with some average quality. An examination of the effects of such density changes aids in determining the reactor response due to power level changes and UF^ inlet velocity changes. While maintaining the UTVC radius at 60 cm, and the UF^ and NaF vapor partial pressures at 5 and 45 atm, respectively, the IBEO thickness at 20 cm, the boiler volume at 8.5 -03 3 X 10 m , and the OBEO thickness at 35 cm, the "overall" density of 3 3 the UF^ in the boiler region is varied from 0.20 g/cm to 4.0 g/cm to simulate the presence of both liquid and vapor UF^. A value of 0.20 3 g/cm for the density of UF^ reflects a mixture composed of about 5 volume percent liquid at 5 atm and 95 volume percent vapor, and a 3 density of «4 g/cm reflects a mixture of pure UF^ liquid. The results, as shown in Figure 4-11, indicate an increasing behavior for k^^^ and a decreasing behavior for Putvc/''bCOL ^^ *^® 3 density of the UF^ increases to about 1.6 g/cm . For densities above 1.6 g/cm"^ the rate of increase of k^^^ decreases and PuTVC^'^BCOL ^^^^^^ off. This indicates that above this density, the boiler starts to become black to neutrons and begins to saturate.

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67 lOOaj^OAinj «jo:^oBj BuucMS Ja«od i r

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68 Material Variations The types and location of structural materials needed in constructing the UTVR are not known at this point. These structural materials include piping, spacers, and protective coatings. Also, the over-all cycle analysis may prove that a metal fluoride other than NaF (e.g., KF, RbF, or Li F) provides better over-all system performance than the UF^/NaF does. By including some sample structural and protective coating materials at selected surfaces or regions (e.g., the UTVC wall) and by examining other metal fluorides, reactivity penalties due to these materials can be estimated. The effects of variations of the following materials are therefore studied. Choice of metal fluoride in UTVC At inner UTVC core radii of 40, 80, and 120 cm, while maintaining -03 3 the IBEO thickness at 20 cm, the boiler volume at 8.5 x 10 m , and the OBEO thickness at 35 cm, calculations are performed to examine the reactivity effect of using NaF, Li F, or KF as the working fluid in the vapor fuel mixture. The results, as shown in Table 4-3, indicate that, with regard to the UTVC only (the wall cooling region is not included in these calculations), the use of Li F as the working fluid results in the highest value for k^^^, followed by NaF and then by KF. As the UTVC radius increases, the difference in k^^^ as a function of selected metal fluoride working fluid becomes greater. At these larger radii, the reactivity contribution of the boiler region to k rr decreases at the same time the reactivity contribution of the UTVC to k^^ increases. This explains the behavior of the differences in k -^ for the different fuel mixtures at the higher UTVC radii. Since the type of metal

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69 V V

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70 fluoride has a small effect on k «, especially at a UTVC radius of 60 cm, NaF is used as the working fluid for further analysis. Wall cooling region > -* Figure 4-12 shows the five region 1-D spherical "mock-up" of the UF^NaF UTVR system which includes the wall cooling region. To simulate the variation of the NaF velocity in the wall cooling region, the thickness of this region is varied from about 0.35 cm to about 3.3 cm. This correspond to a NaF inlet velocity range of 0.5 to 5.0 m/sec. Also, at each NaF velocity, the effective density of the NaF is varied 3 3 3 from 0.1 g/cm to 2.2 g/cm . A value of 0.1 g/cm for the density of NaF reflects a mixture composed of «4 volume percent liquid at 45 atm and 96 volume percent vapor, and a density of 2.2 g/cm reflects a mixture of pure NaF liquid. The results, shown in Table 4-4, indicate a maximum penalty of about 10% 5k/k for a wall cooling region thickness of 3 3.33 cm and a NaF density of 2.2 g/cm . However, the expected velocity of the NaF is about 3 m/sec which corresponds to a wall cooling region thickness of 0.58 cm. The reactivity penalty of the liquid NaF region 3 is then about 2.25% 5k/k for a NaF density of 2.2 g/cm . As the inlet velocity of the NaF increases in the wall cooling region and/or as the effective density decreases, PuTVC^^BCOL increases. Other metal fluoride working fluids The NaF in the UTVC and wall cooling region is replaced by Li F and KF to examine the reactivity penalty or gain if other liquid metal fluorides are used instead of NaF. The neutron multiplication factor, kg^^, and the average Pyyvc/^'BCOL ^^^ obtained for the different metal fluorides at wall cooling region thicknesses of 0.44, 0.87, and 3.33 cm

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71 Outer BeO ModeratorReflector Region UF. Boiler Region Inner BeO ModeratorReflector Region Wall Cooling Region Ultrahigh Temperatucfi_^»_ Vapor Core Region 1 Region 2 Region 3 Region 4 Region 5 Figure 4-12. Five Region, One-Dimensional Spherical "Mock-up" of the UTVR

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73 which reflect metal fluoride inlet velocities of 4.0, 2.0, and 0.5 m/sec, respectively. The average density of the metal fluoride is also varied and results are presented for two different cases, as shown in Table 4-5. The results indicate that the use of Li F as the metal fluoride results in the highest values for both k^^^ and PuTVC^^'bCOL' ^°^l°wed by NaF and then by KF. As the thickness of the wall cooling region increases from 0.00 to 3.33 cm, maximum reactivity penalties are about 2%, 10%, and 22% 5k/k for LiF, NaF, and KF, respectively. Comparing these results with those in Table 4-3 where the wall cooling region is not treated, a greater difference in k^^^ is noticed from one type of metal fluoride to the other. This is due to the much greater density of the (liquid) metal fluoride in the wall cooling region which in effect results in a higher overall parasitic absorption. NaF mass flow rate to the boiler region To account for the possibility that complete separation of the UF^/NaF mixture into pure UF^ and pure NaF cannot be achieved, and to attempt to decrease the required PuTVC^^'bCOL ^°" ^^^ basis of thermodynamic and flow considerations) from its present value of 21, the NaF mass flow rate to the boiler is varied from kg/sec to 158 kg/sec; this range corresponds to diverting 0% to 100% of the NaF from the wall cooling region to the boiler region. The results, as shown in Table 46, indicate a slight increase in k^^^ from 1.548 to 1.554 as the NaF is diverted to the boiler. Table 4-6 also indicates that as the NaF is diverted to the boiler region, the thermodynamic requirement for PutVC^^'bCOL d^^^'^^^^s while

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74

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76 the value of Pmtvc/^BCOL o^^^'i^^^ ^^^^ ^^^ neutronic calculations shows a slight increase. UF^ /NaF inlet velocity to the boiler Building on the results obtained in Table 4-6, an attempt is made to further increase PuTVC^'^BCOL ^^ varying the inlet velocity of the liquid UF^/NaF into the boiler region. The velocity is varied from 0.5 to 6.0 m/sec. The mass flow rate of the NaF is set at 94.8 kg/sec to the boiler and 63.2 kg/sec to the wall cooling region. Thermodynamically, this requires that Pmtvc/''bCOL ^^ 1«25. As the inlet velocity of the mixture increases from 0.5 to 6.0 m/sec the amount of the mixture in the boiler decreases; thus, the amount of fissile material in the boiler decreases. This, as seen from Figure 4-13, causes a decrease in keff from 1.577 to 1.468 and an increase in ''uTVc/^'bCGL ^'^°'" 0-18 to 0.41. Addition of Li F poison to the boiler An attempt is made to decrease the reactivity of the boiler region by adding Li poison to the boiler region in order to obtain the required PuTVC^'^BCOL' ^^ added to the boiler region in the form of Li^F. The UF^/NaF-Li^F inlet velocity is fixed at 2 m/sec. The Li^F -03 mass flow rate is varied from 5.2 x 10 to 3.9 kg/sec. This corresponds to an atom ratio variation from 0.1% to 75.0% (the atom ratio being the ratio of the Li atoms to that of the U ). As the mass flow rate of the Li F increases, a decrease in k^x^ from 1.550 to 0.957 and an increase in PuTVc/'^BCOL ^^°^ 0.266 to 0.574 is observed, as shown in Figure 4-14. However, the required PuTVC^^'bcOL ^^^"^ ^^ 1*25 is not achieved in the 1-D configuration.

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77 lOOflj^OAinj .J013BJ BuiJBMS JSMOd m o • O

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* « V ! * '"» * * J ! '-' < 78 lOOaj^OAinj «joiDBj BuLJCMS J9«o
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, : 79 BeO in the annular boiler region The boiler region in the actual reactor system is made of a number of cylindrical boiler columns separated by BeO moderator. To account for this, i.e., the presence of BeO in the boiler region, the boiler region in 1-D spherical geometry is modeled as a region that contains a homogenized mixture of liquid and vapor UF. and BeO. The onedimensional boiler region annular thickness is varied from 2.0 to 4.3 cm to simulate the actual boiler region configuration that contains from 2 to 8 boiler columns. The volume of the UF. and the total volume of the boiler region are conserved when converting from the true cylindrical configuration to the 1-D spherical geometry. As the annular boiler region thickness is varied (i.e., as the number of boiler columns is varied), the total mass flow rate and the inlet velocity of the UF^ in the boiler region are kept constant (i.e., the total UF^ cross sectional flow area is fixed) but the volume of the BeO and the average UF^ density are varied. The results, k^^^ and PuTVC^^'bCOL' ^^^ given in Table 4-7. The results indicate that k^^ decreases and Phtuc/^bcol """creases as the simulated number of boiler columns increases. This is due to the decrease in the amount of moderator present in the boiler region; thus, less neutron thermal ization is occurring in the boiler region. This results in a decrease in the average thermal neutron flux in the boiler region causing k rr to decrease and PijTVc/^BCOL ^° increase. The results also indicate that k^^ increases and Putvc/'^BCOL ^^^^reases as the UF^ density in the boiler increases. This is due to the increase in the amount of fissile material present in the boiler region.

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81 Reactivity effects of liner materials At this stage of analysis, the type and amount (thickness) of liner materials needed to protect the UTVC and boiler walls from the corrosive characteristics of the fluoride-fuel (UF*) and metal fluoride working fluid and from the erosion effects due to high temperature and high velocity of the fuel and working fluids are not known. However, by including protective coatings of some selected materials on the surfaces of the UTVC and boiler walls, estimates of likely reactivity penalties are obtained. The selected materials are aluminum (Al), niobium (Nb), titanium (Ti), nickel (Ni), and molybdenum (Mo). The thickness of the liner material is varied from to 50 mils (0.0254 to 0.127 cm) and is also examined at 393.7 mils (1 cm). Table 4-8 shows the results of including the liner materials at the different thicknesses on the UTVC walls only. This analysis indicates that the use of Al (or AlpOj) as the protective material results in the highest value for k r:^ (lowest reactivity penalty 5k/k), followed by Nb, Ti, Ni, and then by Mo. The reactivity penalty for Al is negligible when compared to the other materials. This is due to the low thermal absorption cross section for Al which is 0.23 barns (at 2200 m/sec). The highest reactivity penalty is for Mo which is about 8% 5k/k at a thickness of 50 mils. Table 4-9 contains the results of placing the liner materials on the boiler walls only. The results are qualitatively in agreement with those in Table 4-8. That is, Al had the highest value for k^^^ followed by Nb, Ti, Ni, and then by Mo. However, except for Al at 10 mils, the reactivity penalty is higher roughly by a factor of about 2 for all

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84 thicknesses and all materials (due to placing the coating on both sides of the annular boiler region in the 1-D mock-up calculations). At 50 mils, the reactivity penalty for Al is less than 1% 5k/k while the value for Mo is about 19% 5k/k. "a; Table 4-10 contains the results for the case where the liner materials are placed around both the UTVC and the boiler regions. The results indicate that the reactivity penalty for the case where both regions are lined is roughly equal to the sum of the reactivity penalty of the lined UTVC and the reactivity penalty of the lined boiler. The results shown in Tables 4-9 and 4-10 give an over-estimate of the reactivity penalty for using liner materials around the boiler region in the 1-D spherical "mock-up." By simulating the boiler region as an annular shell in 1-0 (thickness of about 0.1 cm) and then placing a protective coating on both sides of the annular region, the amount of liner materials added to the boiler region is over-estimated. That is, the ratio of the volume of the boiler region (or UF^) to the volume of the liner material is much too low. For example, for a thickness of 30 mils this ratio is about 0.7 in the 1-D spherical approximation and about 16 for a 4-column boiler region configuration in the actual cylindrical geometry. More reliable estimates for liner reactivity penalties require 2-D and 3-D calculations. One-Dimensional Results The general behavior of k^^^, P^J^y(., P^^^^, and Putvc/^BCOL obtained from 1-D scoping calculations are summarized in this section. Reference reactor configurations for 2and 3-D analysis are given based

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86 on results obtained from the 1-D scoping calculations. Conclusions concerning the 1-D spherical "mock-up" are also presented. The Neutron Multiplication Factor The following summarizes the general behavior of k
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87 9. Increases as the OBEO thickness increases from to about 40 cm. For OBEO thicknesses beyond 40 cm, k^^^ begins to level off and the system saturates. 10. Increases as the average UF^ density in the boiler regions increases. \ , ^ >., ' ' ' f ti 11. Increases as the thickness of the boiler region increases or as the UF^ inlet velocity to the boiler decreases. 12. Increases slightly as the metal fluoride is diverted from the wall cooling region to the boiler region. 13. Decreases as the mass flow rate of Li F increases to the boiler region. 14. Increases as the simulated number of boiler columns decreases, i.e., as the volume fraction of BeO in the boiler region increases while maintaining the UF^ volume constant. 15. Decreases as the thickness of UTVC and boiler liner materials increases. Power Sharing Factor The following is a summary of the behavior of Piijwr* PrcoL' ^"^ PyjyQ/PgQQL as a function of variations performed on the UTVC: 1. As the UTVC radius increases to 70 cm, both PiiTwr and Pgroi increases with P^jy^ increasing at a higher rate. For UTVC radii above 70 cm PnTur increases and Pgroi decreases. The net result is an increase in PijTVc/^BCOL *^ *^® ^^^^ radius increases. 2. As the UF^ partial pressure increases to about 10 atm P^jvc increases and PgroL i^^'nains at about a constant level. For partial

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88 pressures above 10 atm, Pyyvc ^®9^"^ ^° level off and PgQQL '^^main at about a constant level. Thus, PuTVC^'^BCOL increases as the UF. partial pressure increase to about 10 atm and begins to level off at pressures higher than 10 atm. 3. As the U^^^ enrichment increases from 80% to 100% at UF^ partial pressures below 4 atm, P^jvc increases slightly and P3Q0L '^e'^ains at about a constant value. This translates to a slight increase in ''uTVc/'^BCOL* However, at UF. partial pressures above 4 atm 235 '*UTVc/''bCOL ^^9^"^ ^0 level off as the U enrichment increases above 80%. ?35 233 4. When U is replaced with U at a UF^ partial pressure of 3 atm, PyjyQ remains at about a constant level and Pg^QL increases. At a UFj partial pressure of 5 atm the gain in PorQi is reduced. The 235 233 result is a decrease in Piitvc/''bCOL *'^®" ^^ replaced with U ; this decrease is smaller at higher UF^ partial pressures. 5. As the thickness of the wall cooling region increases and as the average density of the NaF in the wall cooling region increases, a decrease in Piijur occurs. The result is a decrease in Pijtvc/''bC0L* 6. As Li F is replaced with NaF and when NaF is replaced with KF, a decrease in PuTwr occurs which result in a decrease in Pijtvc/''rcol* 7. As the IBEO thickness increases from to about 16 cm both PnTwr snd ^BCOL increase with PnTur increasing at a higher rate. For IBEO thicknesses ranging from about 16 to about 40 am, Piijyr ^nd PDrQi remain at about a constant level. However, as the IBEO thickness increases beyond 40 cm, only PyjyQ increases. This translates in ''uTVC^^'bCOL increasing as the IBEO thickness increases from to

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%: , 89 about 16 cm, remaining constant as the IBEO thickness increases from 16 to 40 cm, and then increasing for IBEO thicknesses beyond 40 cm. 8. As the OBEO thickness increases from 10 to about 50 cm, PyTur remains at about a constant level and Pgroi increases. As OBEO J increases beyond 50 cm Pjjjvc *"^ ''bCOL '^®'"*^" **• about constant levels. Thus, a decrease in Pmtvc/''bCOL ^^^"'"^ *^ ^^^^ increases to 50 cm and beyond 50 cm Pijtvc/''bCOL ^^^®^^ °^^ ^^^ remains at about a :± constant level . • 9. As the average UF* density increases in the boiler region from about 0.4 to about 1.6 g/cm , Pyjur remains constant and PgQQL increases. 3 As the UF^ average density increases from 1.6 to 4.0 g/cm , PgcoL begins to level off. Thus, PutvC^'^BCOL decreases as the UF^ average 3 density increases from 0.4 to 1.6 g/cm and then begins to level off 3 as the UF^ average density increases above 1.6 g/cm . 10. As the size or thickness of the UF^ boiler column increases (or the inlet liquid velocity of the UF^ to the boiler decreases) an increase in PgQQL ^^^^^^ which results in a decrease in Putvc^^'bcOL* 11. As the metal fluoride is diverted from the wall cooling region to the boiler region a slight increase in Pyjy(^ and a slight decrease In PgQQL occurs. Thus, PyTVc/'^BCOL ^"creases slightly by about 6% when 158 kg/sec of NaF is diverted to the boiler region. 12. As the mass flow rate of the Li F increases, Pg^Q^ decreases. Thus, an increase in Putvc^'^BCOL °^^"'^^* 13. As the simulated number of boiler columns decreases, or the BeO volume fraction in the boiler region increases, Pg^Q^ increases. Thus, a decrease in Putvc^^'bCOL occurs. V

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90 14. As the thickness of the liner material surrounding the UTVC increases, a decrease in PjjyQ occurs. However, as the thickness surrounding the boiler region increases, a greater relative decrease occurs in Pg^QL* ^^® "®* effect, is a gain in Pyyvc/^'BCOL *'^®" ^^® thickness of the liner material surrounding both cores increases by the same amount. Spherical "Mock-up" Comments One-dimensional calculations are relatively inexpensive, fast, and can give reliable qualitative behavior (and sometimes even reliable quantitative behavior) depending on the actual configuration, type of modeling, assumptions made, and sought results. The cases presented in this chapter are performed using XSDRNPM on an IBM 3090/4000 mainframe computer. The convergence levels of the -05 -04 point wise neutron flux and k ^^ ranged from 5 x 10 to 1 x 10 . The required computer time for each problem ranged from fractions of a second to a maximum of three seconds. In contrast, as is shown in the next chapter, 2-D calculations, at a convergence level of 5 x 10' , require about 300 to 600 seconds on the IBM 3090/400, and sometimes even more. Three-dimensional calculations (e.g., Monte Carlo calculations with MCNP) require an average of about 1-hour for the UTVR system on a -03 Cray X-MP/48 supercomputer for an uncertainty level of 5 x 10 . Thus, 1-D calculations are inexpensive and extremely fast as compared to 2and 3-D calculations. Analyzing the system in the 1-D spherical "mock-up" provides useful results pertaining to the relative global properties of the UTVR system.

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, 91 The general behavior of k^^r and PuTVC'''''bCOL ^"® ^° *^® variations performed on certain system's parameters are obtained. However, modeling the UTVR in the 1-D spherical geometry resulted in relatively high values for k^^^ and relatively low values for '*UTVc/'*BCOL* Tfisse are due for the following reasons: 1. Spherical configurations provide the optimum (greatest) volume-tosurface area ratio. Thus, the neutron leakage is underestimated and '^eff ^^ over-estimated. 2. The boiler region is treated as a spherical shell that surrounds the UTVC and contains a homogenized mixture of liquid and vapor UF^ or a homogenized mixture of UF^ and BeO. In reality, the boiler region consists of a number of cylindrical columns separated by BeO. These boiler columns are made up of four distinct regions. They are: (a) a subcooled and saturated liquid region, (b) a region containing a saturated liquid and vapor mixture, (c) a saturated vapor region, and (d) a superheated vapor region. In the 1-D spherical "mock up," the probability that the neutrons leaving the UTVC will interact with the boiler region is larger than that of the actual configuration. Also, the thickness of the boiler region in 1-D is small compared to the actual boiler columns. This also causes the average thermal neutron flux to be higher (i.e., not to be properly depressed) in the 1-D configuration as compared to the actual configuration. Thus, the fission rate and reactivity worth of the boiler region in 1-D are over estimated. Thus, the 1-D spherical "mock-up" yields a boiler region with a fission rate or a power level which is higher than that in the actual

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' ,-' r-.-''^ ' -'' 92 three-dimensional system. This results in relatively high kg^r values and low Putvc/PbCOL^ r . ^x . To accurately account for neutron leakage and to accurately represent the boiler region, two-dimensional {2-D) and three-dimensional {3-D) calculations are needed. In 2-D, R-^ cylindrical geometry, the boiler region can be simulated as cylindrical columns separated by BeO moderator containing the UF^ fuel at an average density. However, since the UTVC contains vapor (or gaseous) fuel, perpendicular leakage from the UTVC in the axial direction cannot be accounted for by means of buckling. Consequently, the R-^ calculations will result in high k -^ values. On the other hand, relative k ^r^ and Piitvc/''bCOL ^^^^^^^ ""^^ optimum configuration(s) for boiler-to-boiler and UTVC-to-boiler coupling can be obtained in the R-^ coordinate system as a function of the following: 1. The number of boiler columns in the boiler region. In R-$ geometry, the boiler region can be accurately modeled as a number of boiler columns separated by BeO. 2. The average UF^ density in the boiler columns. The boiling of the UF^ in the boiler columns can be simulated by varying the effective density of the UF^ in the boiler region. That is, the UF^ in the boiler columns can be simulated as a mixture of liquid and vapor with some average quality. 3. The UF^/MF density in the radial direction in the UTVC to account for the varying temperature in the radial direction in the UTVC. In 2-D, R-Z cylindrical geometry, the UF^ density in the boiler region can be varied axially and in the UTVC it can be varied radially

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93 and axially, the MHD disk with its nozzle and diffuser can be modelled and neutron leakage and neutron streaming can be accounted for. However, the boiler region will be simulated as a cylindrical shell surrounding the UTVC, but it need not be the same height as the UTVC region and can be simulated by shorter columns. In the 3-D geometry, the UTVR can be modelled accurately in (R,^,Z) cylindrical coordinates. •* V

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CHAPTER V STATIC, TWO-DIMENSIONAL, UTVR NUCLEAR CHARACTERIZATION AND CONFIGURATION OPTIMIZATION Introduction The results obtained from the 1-D spherical scoping calculations provide reference configurations that are used in the 2-D R-9 and R-Z cylindrical calculations. In the 2-D calculations, only one of the two symmetric reactors that constitute the UTVR is analyzed (refer to Chapter I, pages 2-4 for details). The mass flow rates of the UF^ and the NaF in each of the reference symmetric reactors are 31 kg/sec and 79 kg/sec, respectively. The following describes a reference cylindrical geometry configuration used for each symmetric reactor of the UTVR system: 1. The UTVC region contains UF^/NaF at 3000 K with the partial pressures of the UF^ and the NaF fixed at 5 and 45 atm, respectively. The UTVC's height is 100 cm and the radius is 53.67 cm. 2. The wall cooling region has liquid NaF with a mass flow rate of 79 kg/sec at an inlet velocity of 2 m/sec. 3. The thicknesses of the IBEO and OBEO regions are fixed at 15 and 40 cm, respectively. 4. The boiler region contains UF^ with a mass flow rate of 31 kg/sec at an inlet velocity of 2 m/sec. 94

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95 Neutronic calculations are performed using DOT-4 [11], a 1-D and 2D S neutron/photon transport code. DOT-4 is capable of computing the system's k «, region average and local neutron fluxes and currents, and the fission rate in each region. DOT-4 is described in detail in Appendix A. The analysis in cylindrical geometry consists of calculations performed for both the R-6 and the R-Z coordinate systems. There are advantages and disadvantages involved with modeling the UTVR in each of these two coordinate systems. The advantages of modeling the UTVR in the R-Z coordinate system include the following: 1. The volume of the boiler region is conserved. 2. The neutron leakage in all directions and neutron streaming from the MHD duct are accounted for. 3. Variations in the boiler's height, axial inlet locations of the boiler feedlines and of the UTVC inlet plenums can all be modeled. 4. Temperature and density variations in the radial and axial directions can be accounted for. A major limitation of the R-Z coordinate system, however, is that the boiler region is treated as an annular shell surrounding the UTVC region. On the other hand, the advantages of modeling the UTVR in the R-^ coordinate system include the following: 1. The boiler region is modeled as a number of boiler columns separated by BeO moderator. 2. Temperature and density variations in the radial and 6 directions can be accounted for.

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96 A major limitation of the R-^ coordinate system is the inability to model or simulate important neutron phenomena (e.g., leakage) in the axial direction from the vapor regions. From these differences, it is expected that kg^^ values obtained from calculations performed in the R-Z coordinate system more accurately represent the actual system, and Puyyc/'^BCOL ^*^"®^ obtained from calculations performed in the R-6 coordinate system are more typical of the actual system. Thus, for a complete study in 2-D, modeling the UTVR in both the R-0 and the R-Z coordinate systems is required. For both of these coordinate systems, variations in geometry, fuel density, and material are performed. The results of these studies are described in this chapter. Scoping Calculations in R-0 Geometry The R-0 configuration is shown in Figure 5-1. It consists of six regions. The first is the UTVC region containing the fuel/working fluid vapor mixture. The second is the wall cooling region which contains liquid NaF with a mass flow rate of 79 kg/sec at an inlet velocity of 2 m/sec. The third is the IBEO region. The fourth is the BBEO region with a radial thickness equal to the diameter of the boiler columns. The fifth is the boiler region where the UF^ fuel or the UF^/MF fuel mixture is vaporized. The sixth is the OBEO region. The boiler columns in the actual system are conical (see Chapter VI). However, in 2-D, R-^ geometry the boiler columns are simulated as trapezoidal columns of infinite height (there is no variation in the axial direction in R-$ geometry). For the results to be meaningful and

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97 UTVC UF^ Boiler Column Hall Cooling Region Figure 5-1. Six Region, Two-Dimensional R-9 Representation of a UTVR with Six-Boiler Columns

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98 valuable, the design of these trapezoidal columns should represent the actual boiler columns as accurately as possible. Geometric Variations All of the calculations performed on the 1-D spherical "mock-ups" cannot be repeated in two dimensions due to the expense involved. However, it is necessary to compare and benchmark selected results from the 1-D spherical calculations against those from the 2-D cylindrical calculations. These comparisons will help to assure that selected configurations, including those chosen for 3-D studies, are capable of meeting the design constraints which include the size and mass of the reactor system, the total power requirement, and the power sharing (''UTVC^'^BCOL^ '^®^"^'^®'''^"^* ^" addition, some results that are not possible in the 1-D calculations are obtained from some of the 2-D calculations. These include some of the results from the following calculations: UTVC radius variations Although results of varying the UTVC radius are available from the 1-D spherical calculations, varying the UTVC radius in the R-$ geometry is needed in order to determine the extent of neutronic coupling among the separated boiler columns and between the boiler columns and the UTVC. While maintaining the wall cooling region cross sectional flow area (A^^) at 1.52 x 10 m^; the IBEO and OBEO thicknesses at 15 cm and -03 2 40 cm, respectively; and the total boiler region A^ at 4.19 x 10 m (the boiler region contains a mixture of 25 volume percent vapor and 75 volume percent liquid UF^ with an inlet velocity of 2 m/sec at a mass

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99 flow rate of 31 kg/sec), the UTVC radius is varied from 40 to 140 cm first for a 4-column boiler configuration and then for an 8-column boiler configuration. The fuel mixture in the UTVC is UF^/NaF at 3000 K with partial pressures of 5 and 45 atm for the UF^ and NaF, respectively. The k^^^ and Putvc/''bCOL "^^s"^*^ *"^® given in Figure 5-2 and Table 5-1 as a function of the UTVC radius. Figure 5-2 and Table 5-1 indicate that for a four-column boiler configuration, kg^^ increases from 1.19 to 1.40 and PuTVC^'^BCOL increases from 1.51 to 5.45 as the UTVC radius increases from 40 to 100 cm. However, for UTVC radii above 80 or 90 cm, k^^^ remains essentially constant at 1.40 while PuTVC^'^BCOL continues to increase at a constant rate to a value of 8.8 at a UTVC radius of 140 cm. The results for an eight-column configuration are as follows: k^^^ increases from 1.275 to 1.40 and Piitvc/'^BCOL ^ "greases from 1.13 to 2.96 as the UTVC radius increases from 40 to 80 cm. However, for UTVC radii above 80 cm, k^^^ remains essentially constant while Putvc^'^BCOL continues to increase up to a value of 6.25 at a UTVC radius of 140 cm. The interpretation of the behavior of k^^^ and Putvc^^'bCOL ^^ *^ follows: as the UTVC radius increases, the distance separating the boiler columns increases (shown in Table 5-1), resulting in a decrease in boiler-to-boiler neutronic coupling. However, at 40 cm tangential separation distance, this coupling is already weak. Thus, the decreased boiler-to-boiler coupling can cause no more than a small decrease in the boiler's fission rate resulting in a small decrease in P3QQL. This tends to cause Pijtvc/''bC0L *° increase. Also, as the boiler columns become further apart, the probability that the neutrons exiting the UTVC

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•)-> O) c I— o (J .f0) 3 C C (« o a: o 10 > CM CO E E U\J ox o e 00 •-< u ^ • X o CM in in H H II H 101 0) in CO CM 3 ••O . — O -1ca C7) 4-> (U ceo:*/) O) -rfcO f >) c c o •->•<-a^ •f-^ r—
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102 return to the UTVC or that neutrons leaving the boiler columns reach the UTVC increases. This, along with the increased UTVC volume, causes the fission rate in the UTVC to increase, i.e., PyjyQ increases. This causes a further increase in PuTVC^^'bCOL" ^^o'*'^^®^' ^^^ increase in Pti-ryr ^s greater than the decrease in PgQpL i^esulting in an increase in •^eff Figure 5-2 indicates that as the UTVC radius increases to about 80 cm, both k ff and Pmtuc/Pbcol ^^^ increasing at faster rates for the 4column boiler than the 8-column boiler configuration. This is explained as follows: the greater tangential separation distance between the boiler columns along with the smaller surface area of the boiler columns in the 4-column configuration causes the neutronic coupling of the 4column boilers to be weaker relative to the 8-column boiler configuration. This weaker neutronic coupling along with the greater flux depression in the boiler columns of the 4-column boiler configuration causes the thermal neutron flux to be lower in the boiler columns of the 4-column configuration relative to the 8-column configuration. This results in a lower fission rate in the boiler region of the 4-column boilers relative to the 8-column boilers. Thus, the boiler region in the 8-column configuration is worth more than the 4-column configuration. This explains the larger k x^ and the lower ''uTVC^^'bCGL ^*^"®^ °^ *^® 8-column configuration relative to the 4column configuration. Now, as the UTVC radius increases, the average thermal flux to in the boiler region is decreasing due to the larger UTVC and the greater separation distance between the boiler columns. However, since the surface area of the boiler region in the 8-column

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103 boiler configuration is larger than the surface area of the 4-column configuration, the thermal neutron flux in the boiler region decreases more rapidly for the 4-column configuration than for the 8-column configuration. Thus, Pg^QL <^6creases more rapidly resulting in ''uTVc/'^BCOL increasing at a higher rate for the 4-column configuration than for the 8-column configuration. This can be seen by observing that the rate of change (slope) of PuTVC^^'bCOL ^^^ *^® 4-boiler system is higher than that of the 8-column boiler, as seen in Figure 5-2. Also, as the UTVC radius increases, the fission rate of the UTVC increases at a higher rate for the 4-column configuration than for the 8-column configuration due to a greater decrease in the average thermal neutron flux in the boiler region. Thus, k^^^ and P^jvc increase more rapidly in the 4-column configuration relative to the 8-column configuration. i» The results obtained in the 1-D spherical "mock-up," shown in Figure 4-2, seem different than the results shown here. That is, k^^ reaches a maximum value at a UTVC radius of about 70 cm and then decreases while the Pyyvc/'^BCOL ^*^"^^ obtained from the 2-D R-^ calculations, are larger than those obtained from the 1-D spherical calculations. This is due to the fact that in the 1-D approximation, the boiler region is treated as a spherical shell surrounding the UTVC. Conservation of boiler region volume causes this region to be quite 235 thin. However, since the density of the U is =400 times greater in the boiler than in the UTVC, the large surface area of this shell leads to the probability of a neutron having an interaction in the boiler region being much greater in the 1-D mockup than in the 2-D R-^ configuration. In R-6 geometry, the boiler region is modeled as

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104 trapezoidal columns separated by BeO with the column radial widths almost equal to their average arc lengths. Also, in R-^ geometry, the boiler region surface area is much less then the surface area of the UTVC. This implies that the probability that a neutron will have an interaction with the UTVC is much greater than in the 1-D mockup. This explains why the obtained PuTVC^^'bCOL ^*^"®^ ^^^ greater in 2-D, R-^ than the 1-D spherical configuration. The peaking behavior of kg^^ in the 1-D mockup is explained in Chapter IV on pages 45 through 50. The reason k^^ in the R-0 geometry does not reach a maximum value is seen clearly from the Pmtvc/''bC0L ^^^^^^^*^ ^^' ''uTVC^'^BCOL ^^ la'^Qei' than 1.0 which indicates that the UTVC is actually worth more than the boiler regions. Thus, an increase in the UTVC radius causes a larger increase in the UTVC's reactivity than the decrease in the boiler reactivity. A UTVC radius of 60 cm is chosen for all further calculations. At this radius, the neutron multiplication factor is quite adequate (k rr «1.28 for a 4-column boiler configuration) with a power density of «700 W^j,/cm^ in the UTVC. Inner BeO moderator-reflector region thickness variations By varying the IBEO thickness at a fixed UTVC radius in R-$ geometry, the effect of the neutronic coupling as a function of the separation distance among the boiler columns and between the UTVC and boiler columns is obtained. Thus, with the UTVC radius fixed at 60 cm, the vapor fuel at 3000 K, the UF^ at 5 atm, the NaF partial pressure at 0? 7 45 atm, the wall cooling region A at 1.52 x 10 m , the boiler A at -03 2 4.19 X 10 m , the number of boiler columns fixed at 4, and the OBEO region thickness fixed at 40 cm, the IBEO region thickness is varied

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105 from 5 to 40 cm. The k^^^ and PuTVc/'^BCOL ''^^"1^^ ^^^ given in Figure 5-3. The results indicate that the optimum IBEO thickness is about 16 cm, for a UTVC radius of 60 cm and a four-column boiler configuration. That is, at an IBEO thickness of 16 cm, k^^^ has a maximum value of about 1.32. Beyond an IBEO thickness of about 16 cm, k ^^ decreases. This implies that the combined boiler-to-boiler, boiler-to-UTVC, and UTVC-to-boiler neutronic coupling is optimum at an IBEO thickness of about 16 cm. This can be seen from Figure 5-3 where PuTVC^'^BCOL increases as IBEO increases from 5 to 10 cm, then decreases as IBEO increases from 10 to about 16 cm, and then increases as IBEO further increases. The behavior of k^^^ and PuTVC^'^BCOL ^^ explained as follows: as IBEO increases from 5 to 10 cm, neutrons leaving the boiler regions are now thermal ized at a higher rate and enter the UTVC as thermal neutrons. Thus, the average thermal flux increases in the UTVC and decreases in the boiler columns. This causes an increase in PuTwr 3"d a decrease in '^BCOL* ^^^^ results in an increase in Putvc/^BCOL* ^^"^® l^eff ^^ increasing, this implies that the increase in P^jvc ^^ greater than the decrease in PgcoL' As IBEO increases from about 10 to 16 cm, the UTVC is no longer shielding the boilers from one another. Thus, the neutronic coupling among the boiler columns increases and fewer neutrons from the boiler columns are entering the UTVC. The average thermal neutron flux increases in the boiler columns and decreases in the UTVC region as IBEO increases from 10 to 16 cm. Thus, the optimum neutronic coupling for a

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y 107 4-column boiler configuration at a UTVC radius of 60 cm is at a boiler separation distance of »116 cm. This corresponds to an IBEO thickness of about 16 cm. As the IBEO increases from 10 to 16 cm, there is a slight decrease in the UTVC's fission rate and an increase in the boiler's fission rate. This results in a decrease in Pijtvc/''bC0L* Since k x^ is increasing, then the decrease in the UTVC's fission rate is less then the increase in the boiler fission rate. As IBEO further increases beyond 16 cm, the neutronic coupling among the boiler columns begins to decrease due to the greater separating distance. As this the separation distance between the boilers increases, the probability for the neutrons to interact with other boiler columns decreases. This causes the thermal neutron flux in the boiler columns to decrease resulting in a decrease in Pgroi Also, the increased BeO moderator region surrounding the UTVC leads to an increase in the thermal flux in the UTVC causing P^y^ur to increase. This results in a higher PuTVC^^'bCOL ^''uTVC increases and PgQQL decreases). Since k rr decreases as the IBEO thickness increases above 16 cm, the fission rate in the boiler region decreases at a higher rate than it increases in the UTVC. For all further calculations, an IBEO thickness of 15 cm is used. Variation in the area of the boiler columns The UF^ inlet velocity to the boiler region is varied from 0.5 to 4 m/sec for a four-column boiler reactor (A^^ of each boiler decreases from 4.2 X 10'°^ to 5.2 X 10'°* m^). The kg^^ and PyTVc/'^BCOL '^^^"l^s ^^ * function of UF^ inlet velocity are shown in Figure 5-4. . .d-v'

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108 lOOaj^OAlflj «jo:^3Bj 6uLj«MS JaMOj m u o I/) 6 in -^ o J >» u o 0)
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109 Figure 5-4 indicates that as the UF. inlet velocity increases from 0.5 to about 2 m/sec, kg^^ decreases from 1.396 to 1.313 and Piitvc/''bC0L increases from 1.39 to 2.83. As the inlet velocity further increases beyond 2 m/sec, k^^^ remains essentially constant while PuTVc/'^BCOL continues to increase. The behavior of kg^^ and PuTVc/'^BCOL ^^ explained as follows: as the UF^ inlet velocity increases from 0.5 to 2 m/sec, the boiler region area decreases. This in turn increases the thermal neutron flux in the UTVC. The result is an increase in P^jvc ^"® ^° *^® higher thermal flux in the UTVC. Although the average thermal neutron flux in the boiler region also increases, there is a loss of fuel in the boiler which leads to a decrease in PbcoL' ^^® decrease in PgQQL ^^ greater than the Increase in Putvc ^^® "®^ result is a decrease in k rr and an increase ^" '*UTVc/''bC0L' ^^ *^® ^^4 ^"^®^ velocity further increases above 2 m/sec, the thermal neutron flux continues to increase in the system; thus, there is still an increase in P^jy^ and a decrease in P3Q0L ^^^^ to the loss of fuel in the boiler). However, the reactivity worth of the boiler columns, at these high UF^ velocities (relatively small boiler areas), is small. That is, at these high UF^ inlet velocities, the decrease in the boiler's fission rate is compensated by the increase in the UTVC's fission rate. This results in an essentially constant '^eff ^^^ **" increase in Pmtvc/''bC0L* Variation in the number of boiler columns With the UTVC radius fixed at 53.67 cm (this run was performed prior to varying the UTVC radius); the wall cooling region thickness set at 0.6 cm; the IBEO and OBEO thicknesses fixed at 15 and 40 cm,

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110 respectively; and the total A^ of the boiler columns fixed at 4.19 x 10' m , the number of boiler columns is varied from two to eight. The results are shown in Table 5-2. Table 5-2 indicates that as the number of boiler columns increases, '^eff increases and PuTVC^'^BCOL ^^^creases. This behavior is explained as follows: as the number of columns increases, the size of each column decreases which results in a decrease in the thermal flux depression in the boiler columns, i.e., an increase in the average thermal neutron flux in the boilers. Also, as the number of columns increases, the surface area of the boilers relative to the UTVC increases. In addition, since the separating distance between the boilers decreases, there is stronger neutronic coupling among the boilers. These effects result in an increase in the average thermal neutron flux in the boilers. The effect is an increase in the boiler's fission rate, i.e., an increase in PRroi • ^^^o, as the number of boiler columns increases, the probability that a neutron leaving the UTVC will have an interaction with the boiler columns increases and the probability of that neutron returning to the UTVC decreases. This causes the thermal neutron flux in the UTVC to decrease. Thus, the fission rate in the UTVC decreases resulting in a decrease in Pujyc" ^^® increase in Pgroi *"^ ^^^ decrease in P^jy^ results in a decrease in Putvc/^BCOL' ^^^°' ^^^ increase in the boiler's fission rate is greater than the decrease in the UTVC's fission rate. This results in an increase in k rr.

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Ill CO u (J (/) g CO U) ^ ^ CNi >»

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112 Fuel/Workino-Fluid Density Variations Knowledge of the effects of fuel density changes due to power level changes and/or perturbations introduced to the system, in both the vapor and boiler cores, is essential for estimating and predicting reactor behavior in terms of stability, safety, and dynamic response. Therefore, the density of the fuel in both the UTVC and the boiler columns is varied and compared to the 1-D spherical "mock-up" results. These comparisons are crucial in determining the contribution of the boiler region (annular boiler region versus separated boiler columns) to the reactor's dynamic response during such perturbations. That is, they are needed to establish which boiler configuration yields k rr and Pittvc/^'bcol '^^^'^^''O'^ ^^^^ result in stable systems with "good" dynamic response. {' i : ';.,... ^ UF^ partial pressure in the UTVC With the UTVC radius fixed at 60 cm; the wall cooling region thickness fixed at 0.31 cm; the IBEO, BBEO, and OBEO thicknesses fixed at 15, 3.25, and 40 cm, respectively; and the boiler region A fixed at -03 2 4.19 X 10 m , the number of boiler columns fixed at four, and the UF. density in the boiler columns fixed at 3.7 g/cm , the UF^ partial pressure in the UTVC is varied from to 15 atm at a NaF partial pressure of 45 atm. The results of this variation are shown in Figure 5-5. As shown in Figure 5-5, V. cr increases with the UF^ partial pressure until it begins to saturate at a UF. partial pressure =8 atm or a U^^^ density of »1.5 x 10"°^ atoms/barn-cm. Thus, the UTVC is becoming black to neutrons for UF^ partial pressures beyond »8 atm. The

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113 lOOajyOAiflj 'jo^DBJ BULJ^MS -•9«0d in o • • xet (/> ro (VJ (NJ o 6^. u I c 10 4-> *o in m o I o ^ CVJ c o S^ < CO CSJ o o 8 !-> I/) •— >» M QC i~ => «> I— O. =3 (U O f O t-> i~ **0) Oi— •^ c o O OQ 4-> (. U 3 C O 3 U. *n> &. (/> o 0) »— a. •!c u _: .H ^ -^ " *^^3i 'j(ri3Bj uoi^BOiidi^inW uojanaN I m o>

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114 results also indicate that the rate at which PuTVC^^'bCOL ^^ increasing decreases as the UF. partial pressure increases and does not appear to level off. The behavior of k^^^ and PutVc/^'bCOL *^ * function of UF^ partial pressure in the 2-D R-6 calculation, shown in Figure 5-5, does not agree with the 1-D results, shown in Figure 4-9. The dissimilarities are: (1) k r* saturates at UF^ partial pressures above »10 and »8 atm in the 1-D and the 2-D configurations, respectively; (2) for UF^ partial pressures above »10 atm, Pijtvc/''bCOL ^®9^"^ *° level off in the 1-D configuration but continues to increase in the 2-D configuration. The reasons for these disagreements are as follows: in the 1-D configuration the fission rate of the boiler region is greater than that of the UTVC. This is due to the way the boiler region is modeled in the 1-D spherical "mock-up," i.e., as a spherical shell surrounding the UTVC. As the UF^ partial pressure increases, the fission rate increases in the UTVC and decreases in the boiler. The UTVC's fission rate is increasing due to the increase in it's fissile density. As the pressure of the UFj increases in the UTVC, the number of neutrons interacting with the boiler region decrease as a consequence of the increased absorption rate in the UTVC. This causes the thermal neutron flux to decrease in the boiler region, and results in neutronic de-coupling of opposed segments in the annular boiler region and between the UTVC and the boiler region. Thus, the boiler's fission rate decreases. As the UF^ partial pressure increases to about «10 atm, the increase in the UTVC's fission rate is higher than the decrease in the boiler's fission rate. The result is a noticeable increase in both k ^^ and Pntvc/'^BCOL*

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115 As the UF^ partial pressure increases above «10 atm, the rate of increase in the UTVC's fission rate decreases and the rate of decrease of the boiler's fission rate also decreases. This translates to small gains in k^^^ and Putvc/^BCOLHowever, in the 2-D R-$ geometry, the boiler region is modeled as four separate columns rather than as a spherical shell surrounding the UTVC. In this configuration, the boiler's contribution to the system is small and is very sensitive to changes in the UTVC. This is seen at low UF^ partial pressures where most of the fissioning is already occurring in the UTVC (Pg^QL ^ ^UTVC^* ^^ *^® ^'^4 partial pressure increases to about s8 atm, the fission rate in the UTVC is increasing at a higher rate than it is decreasing in the boiler columns. This causes both k rr and PuTVc/'^BCOL *° increase. As the UF^ partial pressure increases above »8 atm, the fission rate is increasing in the UTVC at about the same rate it is decreasing in the boiler region which causes PijTur/PRcnL to further increase. Thus, k r^ begins to saturate and Pijtvc/''bC0L continues to increase, but at a decreasing rate. Average UF ^ density in the boiler columns The boiling of the UF^ in the boiler columns in the R-e configuration is treated by employing an average density for the UF.. This average density simulates a mixture of liquid and vapor UF^ at some average quality. Maintaining the UTVC radius at 53.67 cm (this run is performed prior to fixing the UTVC radius at 60 cm); the UF. and NaF partial pressures at 5 and 45 atm, respectively; the wall cooling region thickness at 0.6 cm; the IBEO and OBEO thicknesses at 15 and 40 cm, -03 ? respectively; and the total A^^ of all six boilers at 4.19 x 10 m ,

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116 the average density of the UF* in the boiler columns is varied from 1.4 to 3.7 g/cm . The k^^^ and PuTVc/^BCOL '^^s"''*^ *^ ^ function of the UF^ average density in the boiler columns are shown in Figure 5-6. The results indicate that as the density of the UF^ increases from 1.4 to 3.7 g/cm^, k^^^ increases slightly {»0.4% 5k/k) and Pujvc/'*BCOL decreases from 2.4 to 1.9. This is explained as follows: as the density of the UF* in the boiler columns increases, the average thermal neutron flux in the boiler decreases due to the increase in the absorption. However, the decrease in the flux is dominated by the increase in the fuel density and the fission rate in the boiler columns increases resulting in an increase in Pgroi • ^^® ^^^^ ^^ ^^^ ^^^^ ^^^° decreases resulting in a decrease in its fission rate. Thus, PiiTwr decreases. The result is a decrease in PuTVC^'^BCOL' ^^® reason that k^^ increases slightly is due to the fact that the fission rate in the boiler columns is increasing at a greater rate then it is decreasing in the UTVC. Varying the UF ^ average density in the UTVC as a function of the radial distance from the center line One of the characteristics of the UTVR is the large temperature gradient in the vapor core near the surface walls. This allows the temperature of the fuel/working fluid mixture to be considerably hotter than the surrounding structure. This is shown in Figure 5-7, where the temperature profile of the fuel mixture in a vapor core relative to the surroundings is plotted. Due to this temperature behavior, the density of the fuel near the surface wall is considerably higher than its average density in the vapor core. The effect of the fuel's density profile in the vapor core is studied.

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117 lOOfljyOAinj «jo5,3Bj Suueqs J9mo<| in t CVJ o t CNJ <>J 00 CNi M e 00 u • 91 in CM if) CNi CM s CSJ **^5| 'jcnoBj uoi^BOtidiiinw uoj^naN

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()l) 3jrr).euadmai 118 o U) m (O (_> 0) I— a> o o. o O) O) sc a. c

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119 With the UTVC radius fixed at 60 cm; the UF^ and NaF average partial pressures fixed at 5 and 45 attn, respectively; the wall cooling region thickness fixed at 0.31 cm; the IBEO and OBEO thicknesses fixed at 15 and 40 cm, respectively; and the total A^ of the four-boiler -03 2 columns fixed at 4.19 x 10 m , two R-6 calculations are made. In the first calculation, a flat temperature profile of 3000 K is assumed in the UTVC. This corresponds to a U^^^ atomic density of 1.22 x 10"°^ atoms/barn-cm. In the second calculation, the UTVC is divided into five sub-regions. The fuel temperature is assumed to be 5000 K in the first sub-region and 2500 K in the fifth sub-region. The temperature and density values of the UF^ in the vapor core sub-regions are presented in Table 5-3. In both calculations, the fuel loading in the UTVC is conserved (7.14 kg of UF. in a 100 cm high core). Results from these two calculations are also given in Table 5-3. The thermal neutron flux and temperature profile as a function of radial position from the centerline of the UTVC are plotted in Figure 5-7. Table 5-3 indicates that both k^^^ and Pujvc/''bCOL ^^^ essentially independent of the examined temperature profiles. Figure 5-7 indicates that the thermal neutron flux is lower near the surface walls for the case where a varying fuel temperature profile is assumed. This is due to the higher neutron absorption as a result of the higher fuel density near the surface walls. This causes the fission rate to be higher in the regions near the surface walls (regions 4 and 5). Note that the higher fission rates in regions 4 and 5 will yield an increase in temperature in these regions.

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120 U U E o m o csj o <— I ^ II H ^

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121 Scoping Calculations in R-Z Geometry The initial R-Z reference configuration used in the preliminary R-Z scoping studies is shown in Figure 5-8. It consists of fifteen regions. The description of each is as follows: Region one (Rl) . This is the mid-plane BeO slab (MBEO) separating the two symmetric UTVR reactors. The MBEO lower boundary is reflected. It is bounded by 0.0 < z < Zj and 0.0 < r < rg. Region two (R2) . This is the nozzle region located in front of the MHD duct which contains the UF^/NaF fuel mixture at an average temperature of a3000 K and an average pressure of «25 atm. It is bounded by z, < z < z^ and 0.0 < r < r^. Region three (R3) . This is the MHD generator region with the fuel mixture at an average temperature of ^2500 K and at an average pressure of »2 atm to =3 atm. It is bounded by Zj < z < Z2 and r^ < r <, r^. Region four (R4) . This is the diffuser region which follows the MHD duct. It contains the fuel mixture at an average temperature of s2000 K and an average pressure of «Z atm to ^3 atm. It is bounded by Zj ^ z < Z2 and r^ < r < rg. Region five (R5) . This is the UTVC region containing the UF^/metal fluoride fuel mixture at an average temperature of «3000 K and an average pressure of =50 atm. It is bounded by Zg < z < Zg and 0.0 < r < "3Region six (R6) . This is the lower BeO region (LBEO) containing only BeO. It separates the MHD duct from the wall cooling region feedlines. It is bounded by Z2 < z < z^ and rj < r < rg.

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122 TBEO (R15) '11 ^10-" Z9 UTVC Inlet Plenums (R13) A^ 7ZZ2ZZZZZZZ A. UTVC (R5) ^.n Nozzle (R2) u. '"a '•4 '•5 ''e Hall Cooling •Region (R7) Li quid/Vapor UF^ Column (R12) IBEO (R8) 0BE0#2 (R14) Liquid UF. Column (RID Boiler Column Feedline (RIO) -0BE0#1 (R9} LBEO (R6) Oiffuser (R4) MBEO (Rl) '*7 ''8 MHD Duct (R3) Figure 5-8. Representation of the UTVR in the R-Z Coordinate System

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123 Region seven a fR7a) . This the wall cooling region containing liquid metal fluoride. It surrounds the UTVC only in the radial direction. It is bounded by the region contained in Zj < z < Zjq and r^ < r < r^. Region seven b (R7b) . This the wall cooling region feedlines containing liquid metal fluoride. It is bounded by the region contained in Zg ^ z < ZjQ and r^ < r < rg. Region eight (R8) . This is the inner BeO region (IBEO) separating the wall cooling region from the boiler column regions and the inner portion of the wall cooling region feedlines from the UTVC inlet plenum region. It is bounded by z^ < z < Zy and r^ < r < r^. Region nine (R9) . This is the lower outer BeO region (0BE0#1) separating the boiler column feed line from the wall cooling region feed lines. It is bounded by Z3 < z < z^ and r^ < r < rg. Region ten (RIO) . This is the boiler feedline containing liquid UF^. It is bounded by z^ < z < Zg and rg < r < rg. Region eleven (RID . This is the subcooled and saturated liquid region of the boiler column containing liquid UF^. It is bounded by z^ < z < Zg and r^ < r < rg. Region twelve (R12) . This is the saturated liquid/vapor and vapor region of the boiler column with an axially varying UF^ density. It is bounded by Zg < z < Zg and r^ < r < rg. Region thirteen (R13) . This is the UTVC inlet plenum region. It connects the boiler column with the UTVC core and is bounded by z^ < z < Zg and r^ < r < r^.

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124 Region fourteen (R14) . This is the upper outer BeO region (0BE0#2). It is bounded by Zg < z < Zg and rg < r < rg. Region fifteen (R15) . This is the top BeO moderatorreflector region (TBEO). It is bounded by Zg < z < Zjj and 0.0 < r < rg. Table 5-4 contains the dimensions of this initial reference R-Z configuration. Geometric Variations / ^ , ! "') 4 ? * ' * . '' /s J t. «'.*'-* Configuring the UTVR in the 2-D R-Z coordinate system permits the modeling of regions that could not be treated in either the 1-D spherical or the 2-D R-^ cylindrical geometries. These include the TBEO, MBEO, nozzle, diffuser, and the MHO duct. By studying the effects of variations in the dimensions of these regions, UTVR configurations are obtained that are capable of meeting the design constraints which include size and mass of the reactor system, the total power requirement, and the power sharing (PutVC^^'bCOL^ requirement. Effects of variations of the following parameters are examined: MBEO region height In order to optimize the neutronic coupling between the UTVR's two symmetric reactors, the MBEO height is varied from 2.5 to 15 cm while maintaining the thicknesses and heights of regions 2 through 15 at the values shown in Table 5-4. The UF. average quality in region R12 is fixed at 0.5 to simulate boiling (i.e., the UF^ fuel enters region R12 as a saturated liquid and exists R12 with a quality of 0.9. The UF^ average quality in region R13 is fixed at 0.95 which assumes the UF^

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125 5' V

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126 fuel enters R13 with a 0.9 quality and exists as saturated vapor. The results are shown in Table 5-5. The results, as shown in Table 5-5, indicate that the optimum neutronic coupling between the two symmetric reactors occurs at an MBEO thickness of about 7.5 cm. Since the MBEO has a reflected lower surface, then the optimum total separation distance between the two symmetric reactors is about 15 cm. As the MBEO thickness increases to about 7.5 cm, k^^^ increases to about 1.187 and PijTVc/'*BCOL ^^^ ^vapor/^'boiler ^"crease to 0.81 and 0.78, respectively. As the MBEO thickness increases beyond 7.5 cm, k r^^ decreases and PijtVc/''bC0L """^ ''vapor/'^boiler ^®^^" *° ^®^®^ °^^' Results obtained from the R-Z "mockup" indicate that values for Pyyvc/'^BCOL (*^-^) ^^^ quite low compared to those obtained from the R-0 "mock-up" (average of «3). This is due to the way the boiler region is modeled in the R-Z "mock-up," i.e., as an annular shell surrounding the UTVC. Thus, in the R-Z "mock-up," the importance of the boiler region is exaggerated which results in an overestimate of ^qqqiAdditionally, the R-^ "mock-up" does not account for neutron streaming from the MHD-duct regions. This causes Pnyur to be underestimated in the R-d "mock-up." The interpretation of the behavior of k^^^ and Pyaoor/^'boiler *^ ® function of MBEO region thickness is as follows: as the MBEO region ; thickness increases to about 7.5 cm, the fission rates in the UTVC, nozzle, MHD-duct, and diffuser regions increase indicating increased neutron thermal ization in the MBEO region. Therefore, P„,„„^ increases. vapor The neutron leakage rate from the vapor regions decreases due to increased neutron absorption in these regions. Since the vapor regions

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127 a>

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128 leakage rate decreases, the rate of which neutrons are transported to the boiler regions decreases, causing a decrease in the boiler regions thermal flux. This causes a decrease in Pboiier" ^o^'^ver, the gain in P is greater than the loss in P^oiier' ^^^^ translates to an increase in both k^^^ and Pyapor/^boiler" ^^^^ ^^ shown in Table 5-5. As the MBEO thickness increases above 7.5 to about 12.5 cm, Py-pQincreases less than Pkoiier decreases. This yields a decreased rate of increase in Pvapor/^boiler ^"^ ^ decrease in k^^^. The fission rate in the MHD duct does not increases as the MBEO thickness increases above =10 cm. This implies that the MBEO region is acting as a reflector for the MHD duct and it no longer transfers neutrons between the top and bottom regions efficiently. The UTVC fission rate decreases above MBEO thicknesses of 12.5 cm due to the decreased coupling between the UTVC's on either side of the MBEO region. The net result is a decrease in P„-_--. Since the thermal neutron flux reaching the boilers is still decreasing, P^oiier ^'^'^^^^'^ decreases. The decrease in Phoiier ^^ about the same as the decrease in Pwannr *'"^ ''vaoor/'^boiler ^^ "°*' essentially constant while k^^^ continues to decrease. > > For all further calculations, the MBEO height is fixed at 7.5 cm. This is translates to 15 cm of BeO separating the two symmetric reactors. There appears to be about an 7% 5k/k reactivity penalty for R-Z geometry relative to R-^. For example, an average value for k o:^ is »1.18 for the R-Z geometry (from Table 5-5) versus 1.28 for R-^ (Table 5-3). This is due mainly to two reasons. The first is that neutron

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129 leakage in the axial direction is not accounted for in the R-^ geometry. The second is the way the MHD duct is modeled in the R-Z geometry. In the present R-Z configuration, the nozzle, MHD duct, and diffuser regions (R2, R3, and R4) act as a sink for neutrons, not by absorbing them, but by allowing them to stream out of the system. In the actual configuration, the nozzle section in front of the MHD duct will have a converging section which will cut down on the streaming. Also, for optimum MHD performance, analysis has shown that the height of the disk MHD generator should be decreasing as R increases [31]. Thus, neutron streaming can be greatly reduced, resulting in a higher k rr^. However, for this preliminary analysis, the nozzle, MHD duct, and diffuser are modeled as in Figure 5-8. TBEO region height With the thicknesses and heights of regions 2 through 14 fixed at the values given in Table 5-4, the thickness of the MBEO fixed at 7.5 cm, and the UF^ average quality in regions 12 and 13 fixed at 0.5 and 0.9, respectively, the height of the TBEO region (R15) is varied from 20 to 70 cm. The results are given in Table 5-6. Table 5-6 indicates that as the TBEO thickness increases from 20 to about 40 cm both kg^^ and Pyanor/^boiler ^"C'"®^^^^^ ^^ c""' ^Qff i^ about 1.187 and P„,„„VPk«
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130 «L ^
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131 Table 5-6 indicates that as the TBEO thickness increases, the fission rate in all regions increases except in RIO and Rll (the liquid UF. regions); in these two regions, the fission rate decreases. This translates to a greater increase in P„,^«^ than in Pu„^i„„. The result vapor DO! ler is an increase in both k^^^ and Pvapor/^boiler"^^^ ''"^o" ^hat P^^^^^ is increasing at a higher rate than P^oiler' ^^ ^"® *° ^^® ^*^^ ^^^^ *^® liquid UF* regions are already surrounded (in the axial direction) with about 70 to 80 cm of BeO, namely 0BE0#2. Thus, the TBEO region has less of an effect on the liquid UF^ regions and more of an effect on the vapor core regions. For all further calculations, the TBEO thickness is fixed at 50 cm. First OBEO region height With the thicknesses and heights of regions 2 through 14 fixed at the values given in Table 5-4, the thickness of the MBEO fixed at 7.5 cm, and the TBEO height fixed at 50 cm, the height of 0BE0#1 (R9) is varied from 5 to 80 cm. This is done in order to obtain the optimum separation distance between the boiler region and the MHD duct. However, varying the height of 0BE0#1 requires varying the height of R12, i.e., the amount of fuel in R12 will vary. In order to eliminate the interference of R12 fuel variation, both regions R12 and R13 are voided so that the amount of fissionable material in the boilers is conserved and so that the results obtained are only due to 0BE0#1 height variations. The k^^^ and Pvanor/''boiler '^^^"^ts as a function of 0BE0#1 height are shown in Table 5-7. Table 5-7 indicates that as the 0BE0#1 height increases from 5 to about 40 cm, k x^ increases from a value of about 0.91 to about 0.96 and

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132 £0)0) a X3 T3 O O O O o — I ^ :» II II II 11 V.

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133 ''vapof/'^boiler decreases from about 2.5 to about 1.8. As the height of 0BE0#1 further increase to about 55 cm, an increase in k r.^ occurs and '*vapor/'*boiler '^®'"*^"s essentially constant. For 0BE0#1 heights above about 55 cm, k^^ decreases and there is a slight increase in ''vapor/^boiler' ^^"^' ^^® optimum value of k^^^ occurs at an 0BE0#1 height of about 55 cm. The values of k^^^ appear to be low compared to k^^ values obtained from other R-Z configuration calculations, 0.96 versus 1.19 (Table 5-5) for TBEO and MBEO thicknesses of 40 and 7.5 cm, respectively. This is due to the fact that regions R12 and R13 are voided and contain no fissionable materials. Table 5-6 indicates that, on the average, R12 and R13 produce about 46% of the total fissioning in the system. Thermodynamic and flow considerations require that regions R12 and R13 only produce roughly 3% of the total fission of the system (»29 MW is required to fully vaporize 31 kg/sec of saturated liquid UF^). The reason that the 2-D neutronic calculations indicate that «40% of the total fission power is produced in R12 (Table 5-6 for TBEO thickness of 40 cm) is due to the error involved in approximating the boiler region as an annular region in the R-Z geometry. The fission rates in the vapor and boiler cores are shown in Table 5-7. Table 5-7 explains the behavior of k^^^ and Pyaoor/^'boiler" ^^ the height of OBEOIl increases from 5 to about 55 cm, the boiler regions are being moved to a region where the average thermal flux is higher, i.e., a region of higher importance. This causes more fissions to occur in the boiler regions resulting in an increase in P[,Qj]e^' Since more thermal neutrons are now being absorbed in the boiler regions, the

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134 thermal neutron flux in the UTVC decreases causing Pygpor ^° decrease. The fission rate increases from about 2.6 to about 3.5 fissions/sec in the boiler regions and decreases from about 6.5 to about 6.2 fission/sec in the vapor regions. This is a gain of about 0.9 fissions/sec in the boiler regions and a loss of only about 0.3 fissions/sec. The result is an increase in \K.^ff and a decrease in Pyapor^'^boiler' ^^^^ ^^ shown in Table 5-7. '^^ As the GBEGIl height increase above about 55 cm, the boiler regions are moving to a relatively less important region where the average thermal flux is less. This results in Pygpo^ increasing and Pfjoiier decreasing with P^giier decreasing at a higher rate than P^^pQ^ is increasing. The overall outcome is a decrease in k r:^ and an increase ^" '*vapor/'*boiler* For all further calculations, the height of 0BE0#1 is fixed at 55 cm. \^';;,. -;: ''^' ' ' ,:;. Boiler; subcooled and saturated liouid region height With the height of 0BE0#1 fixed at 55 cm (z^ fixed at 87.86 cm), and the UF« average quality in regions R12 and R13 fixed at 0.4 and 0.9, respectively, the height of the liquid UF^ in region Rll is varied from 2.5 cm to 10 cm. The k^r^ and Pyaoor/'^boiler "^^^^^^^ ^^ * function of region Rll height are given in Table 5-8. "**' The results, shown in Table 5-8, indicate that k r^r increases and ''vaDor/^'boiler decreases as the liquid boiler region height increases. This behavior is explained as follows: as the amount of fissile material in the region is increasing, the average thermal flux in that region is decreasing due to increased absorption. However, the increased fuel ^rt

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135

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136 more than compensates for the decreased flux and the fission rate in the liquid boiler region increases. The average thermal flux in all other regions decreases due to the increased absorption rate in region Rll. Thus, the fission rate decreases in the vapor core regions and in all other boiler regions. However, the fission rate in region Rll is increasing at a faster rate than it is decreasing in all of the other boiler regions and vapor core regions combined, as seen in Table 5-8. This causes an increase in P^oiier ^"^ * decrease in P^gporThus, k^^^ increases and Pvapor/^boiler decreases. ^ Table 5-8 indicates that the reactivity worth of each additional cm of UF* liquid is decreasing as the height of the liquid boiler region is increasing. For example, the reactivity worth of increasing the height of Rll from 2.5 to 5 cm is about 0.15% Sk/k per cm height of UF^ liquid but only about 0.08% Sk/k per cm of UF^ liquid when the height is increased from 7.5 cm to 10 cm. Material Variation ^^ Poisoning the boiler feedline walls Tables 5-5 through 5-8 indicate that about 10% of all fissions are occurring in the inlet feedlines (region RIO) to the boiler regions. This value is undesirably high. The reason that these feedlines are producing 10% of all the fissions in the system is due to the way they are modeled in the R-Z geometry. In R-Z, they are modeled as an annular disk with a height of 0.025 cm and a radial thickness of 40 cm. In reality, the boiler's feedlines will likely be a number (e.g., from two to eight) of horizontal pipes. Each pipe will have a radius ranging

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137 from about 0.75 cm to about 1.5 cm depending on the number of boiler columns in the system. This implies that these feedlines will not be producing a high percentage of the total power. In order to reduce the amount of fissioning which is predicted in the feedlines (region RIO), this region is enclosed with molybdenum (Mo). In order for this to be possible, two more regions, R16 and R17, are added to the R-Z configuration. Region R16 is defined as the molybdenum region below RIO; it is Mj cm high and has a radial thickness equal to that of the feedline region. Region R17 is defined as the molybdenum region above RIO; it is M2 cm high and also has a radial thickness equal to that of the feedline region. With the dimensions and densities of all other regions fixed, the thicknesses of R16 and R17 are varied from cm to 2 cm. The results are given in Table 5-9. Table 5-9 indicates that k r^ decreases from about 1.11 when there is no Mo liner to 0.966 when the boiler feedline is surrounded by 0.5 cm of Mo from below and 2.0 cm of Mo from above. This is a penalty of about 14% 5k/k. This amount of Mo reduces the boiler feedline relative fission rate from about 1.38 to 0.24 fissions/sec, i.e., it reduces its contribution to the total fissions in the system from about 12.4% to 2.5%. The results, as shown in Table 5-9, indicate that as the thicknesses of R16 and R17 increase from to 0.5 cm, the relative fission rates in all vapor core regions and in the UTVC inlet plenum region increase; the fission rate decreases not only in the boiler feedline region, but also in the boiler liquid and vapor regions. The

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« e

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139 result is an increase in P.,-_-„ and a greater decrease in Pu«;i„^. This Vapor Doi I er translates to an increase in PvaDor/^'boiler *"^ ^ decrease in k^^ as shown in Table 5-9. The reason that the fission rate in the boiler feedline decreases as the Mo thickness increases is due to the shielding effect of the Mo regions. As the Mo thickness increases, less neutrons are entering the feedline region causing a decrease in its fission rate. The reasons that the fission rate in both the liquid and vapor boiler regions is decreasing and increasing in all vapor regions as the Mo thickness increases is explained as follows: as the Mo thickness increases, the absorption rates of the thermal and epithermal neutrons increase at a higher rate than that of the more energetic neutrons. This causes the neutron spectrum to shift to a harder spectrum especially in regions nearby the boiler feedline region. This requires '/ ^ the neutrons to have more collisions in order to become thermal, i.e., ' travel a longer distance. Consequently, the regions that are further away from the boiler's feedline experience a higher thermal neutron flux level than the regions nearby. Since the liquid and vapor boiler regions are relatively closer to the feedlines than the vapor regions, the thermal neutron flux in these regions (Rll and R12) decreases while it increases in the vapor regions. The results are an increase in the vapor regions fission rate and a decrease in the liquid and vapor boiler region fission rate.

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140 Comments on Power Sharing The results obtained from neutronics calculations performed in the R-0 and the R-Z coordinate systems indicate that current UTVR configurations do not accomplish the power sharing, PuTVc/'^BCOL* required by the thermodynamic and flow calculations. For example, for the initial R-Z reference UTVR configuration given in Figure 5-9 and Table 5-4, results presented in Tables 5-5, 5-6, and 5-8 show a ''uTVc/'^BCOL ^^^"^ which is typically in the range of only 0.7 to 1.3. The R-^ results shown in Figure 5-5 indicate that a value of «2.8 for '^UTVc/^'bCOL ^^ obtained for a UF. partial pressure of 5 atm. However, the required PutVC^^'bCOL ^^^"^» based on thermodynamic and flow considerations, are listed in Table 3-2. For the 200 MW UF^/NaF system. Table 3-2 list a value of a22 for PuTVc/^'bcol ^^^^^ assumes UF. and NaF mass flow rates of 62 and 158 kg/sec, respectively, with all the UF^ flowing through the boiler cores and all the NaF flowing through the UTVC wall cooling region. In order for the neutronic calculation results to be in agreement with the thermodynamic requirements, the UTVR flow needs to be redistributed such that the required thermodynamic power sharing value (PuTVc/'^BCOL^ ^^ reduced and/or the fission rate in the UTVC is increased and/or decreased in the boiler columns. Thermodynamic analysis of this UF^/NaF system reveals that about 90% of the total power generated in the UTVR is added to the metal fluoride. Thus, the required PuTVC^'^BCGL ^*" ^^ reduced by having a fraction of the metal fluoride vaporized with the UF^ in the boiler region. Thermodynamic analysis indicates that about 30% of the NaF '

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141 needs to be diverted from the wall cooling region to the boiler region in order to obtain a PuTVC^^'bCOL ^*'^"^ °^ "^ ^^^ ^^^ UF^/NaF system. The fission rate of the UTVC can be increased either by increasing the volume of the UTVC and/or by increasing the pressure of the UF«. However, the UF^ pressure for the reference configuration is 5 atm, and Figure 5-5 indicates that the system saturates at UF^ pressures above 8 atm. Thus, if the UF* pressure is increased to 8 atm, the result is a relatively small increase in PuTVC^'^BCOL* ^^^^ ^'^ *° °"^^ ^'^ (•'esults from R-^ calculations), at the cost of loosing the most important inherent power control mechanism, namely, the vapor core density. The size of the UTVC can be increased by increasing the its radius. However, according to Table 5-1 and Figure 5-2, k rr increases to a maximum value at a UTVC radius of 80 cm and then saturates while ''utvc/^'bcol continues to increase as the UTVC radius increases. Increasing the UTVC radius from 60 to 80 cm results in an increase in '*UTVc/'*BCOL ^^^"'^ ~^'^ ^° =4.3, and increasing the UTVC to 140 cm causes ^UTVc/'^BCOL ^° increase to =8.8; these results are based on R-6 calculations. The penalties involved in increasing the size of the UTVC are the additional weight added to the system and the reduction in the boiler's contribution to reactor reactivity control. If the required reflector thickness is 55 cm (the combined thicknesses of the IBEO and OBEO regions), then increasing the UTVC radius from 60 to 80 cm causes the mass of the UTVR to increase by =23%, and increasing the UTVC radius from 60 to 140 cm causes the mass of the UTVR to increase by «90%. Whether these penalties are acceptable depends on the system mass

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142 fraction of the UTVR/shield combination and on the importance of the boiler's contribution to reactor reactivity control. PgPQi can be decreased by reducing the amount of fissile material in the boiler cores. This can be done by reducing the boiler's cross sectional flow area and, hence, the boiler core volume. The cross sectional flow area of the boilers can be reduced by increasing the inlet velocity of the UF* (the reference UF. velocity is 2 m/sec). The increased velocity will result in increased frictional losses. A reduction in the boiler cross sectional flow area reduces the amount of free surface available for boiling which makes the boiling of the UF. more difficult. Reducing the cross sectional areas of the boiler regions also results in reduced boiler column surface area. This in turn reduces the neutronic coupling and, hence, the degree of the inherent negative reactivity feedback resulting from the boiling or voiding in the boiler regions. For the reference configuration from Table 5-4, the distance that the UF^ fluid travels in the boiler region from the boiler feedline inlet to the UTVC's inlet plenum is »80 cm. This distance is equal to the sum of the radial length of RIO added to the sum of the axial heights of Rll and R12. PgCOL ^^" ^^^° ^^ reduced by reducing the boiler flow path. This can be achieved in a number of ways. 1. Reduce the axial heights of Rll and R12 by increasing the inlet axial location of the inlet boiler's feed lines, i.e., Zm. This method can be taken to the extreme of eliminating Rll and R12 by requiring the UF^ to flow in the radial direction only, as shown in Figure 5-9. If this extreme is selected, i.e., Rll and R12 are eliminated and since

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143 TBEO 7'5 ^4-1 UTYC ••::;;-:\--:.V.>^'.:Vr:r-vv;i? • • •-•..••'••..•;••••.. • •.*.•••••-•> •.•:•.-•/;•: :^-=::••.:.'.••'. •.••••:•:•> ....•••_ • ^ r • ' * -! .' • . '^ . ^ZZZZZZ22ZZZZ Zi-^mimfm^^^^zz^ ' •' '• .'! '• ' ' 1 ' '(•"••'•'••. /.•>•. • :••••• ^ ^' •.•.••.:•. Nozzle • • ••.•••:.• J "•-« •'»i. .,•«.,.• ••vi ^ JU "Ur I I I i i MHD Duct UTVC Inlet Plenums 'Liquid/ Vapor UF. Column Wall Cooling Feedlines Boiler Feedlines — Wall Cool i ng Region •OBEO LBEO Diffuser MBEO Figure 5-9. The Horizontal Boiler Configuration of the UTVR in the R-Z Coordinate System

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144 RIO will be at a relatively less important region, the result is a significant reduction in the boiler region's relative fission rate from about 3.4 fissions/sec (from Table 5-9 at Mj and M2 thicknesses of 0.5 cm) to less than 0.5 fissions/sec. This design corresponds to * '^vaoor^'^boiler ^*^"® °^ roughly 3.6. However, this configuration will also reduce the extent of the boiler-to-UTVC neutronic coupling and, thus, reduce the extent of the inherent negative reactivity control . 2. Reduce the boiler feedlines flow path by increasing the IBEO region thickness. This, however, causes the UTVC-to-boiler neutronic coupling to decrease. 3. A combination of items 1 and 2 above. Two-Dimensional Results The general behavior of k^^^, PyTVc/'^BCOL* ''uTVC *"^ ^BCOL obtained from the two-dimensional scoping calculations performed in both the R-$ and R-Z coordinate systems are summarized in this section and compared to the one-dimensional spherical results where possible. , ? Reference reactor configurations for three-dimensional analysis are given based on results obtained from the 2-D scoping calculations. Conclusions concerning the 2-D R-^ and R-Z calculations are also presented in this section. The Neutron Multiplication Factor The following summarizes the general behavior of kg^^ due to variations performed on the UTVR in the 2-D R-6 coordinate system:

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145 1. k^^ increases as the UTVC radius increases and saturates at a value of «1.40 at UTVC radii of »80 and «100 cm for 8and a 4-boiler column configurations, respectively. 2. k rr increases as the IBEO thickness increases until a maximum value is obtained at an IBEO thickness of about 16 cm. 3. k ff decreases as the cross sectional flow area of the boiler columns -03 decreases and levels off as the A^^ decreases below about 4.2 x 10 o m (or as the UF^ inlet velocity to the boiler columns increases above 2 m/sec). 4. k re increases as the number of boiler columns increases. 5. k^^ increases as the UF^ partial pressure in the UTVC increases to about 8 atm. For UF^ partial pressures beyond »8 atm, k^^^ begins to level off and the system saturates. 6. k fr increases slightly (=0.4% 5k/k increase) as the UF^ density in the boiler region increases from 1.4 to 3.7 g/cm . .^ The general behavior of k r^ due to variations performed on the UTVR in the 2-D R-Z coordinate system is as follows: 1. k ff increases as the MBEO thickness increases until a maximum value is obtained at an MBEO thickness of »7.5 cm. For MBEO thicknesses above 7.5 cm, kg^^ decreases. 2. k ff increases as the TBEO thickness increases from 20 to about 50 cm. For TBEO thicknesses above 50 cm, k^^^ begins to level off and the system saturates. 3. k re increases to a maximum value as the BeO moderator thickness between the MHD disk and the boiler columns increases to about 60 cm;

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146 ' '^eff ^^®" decreases with further increases in the BeO separation distance. ^ 4. k rr increases as the height of the liquid UF^ in the boiler region increases. 5. k rr decreases as the molybdenum thickness surrounding the boiler region feedlines increases. As expected, modelling the UTVR in R-8 geometry results in k^r; values which are high relative to values from the calculations performed using the R-Z coordinate system. The relatively higher k ^^ values obtained from calculations performed in the R-0 coordinate system are due mainly to the inability to simulate neutron phenomena in the axial direction; i.e., neutron leakage in the axial direction is not accounted for in the R-6 coordinate system. Thus, the kg^^ values obtained from calculations performed in the R-Z coordinate system are more typical of the actual system as a result of conserving the fuel loadings in both the vapor and boiler regions and due to accounting for neutron leakage in all directions and neutron streaming from the MHD duct. The Power Sharing Factor The general behavior of the fission rate in the UTVC (Pmtvc) ^ind the boiler columns (PgQQL^ ^"^ ^^® general behavior of the power ratio ^''uTVC^'^BCOL^ *'^ * function of variations performed on the UTVR in the 2-D R-^ coordinate system are as follows: 1. As the UTVC radius increases, Pyjyr increases while Pgroi decreases causing PuTVC^'^BCOL ^° increase.

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147 2. As the IBEO thickness increases from 5 to «10 cm, Pn-rwr increases and PgQQL decreases causing PuTVc/''bCOL *° increase; as IBEO increases from aslO to «16 cm, Piijyr decreases and Pgroi increases causing ''uTVc/'^BCOL *° decrease; as the IBEO further increases above »16 cm, Pyjy£ increases and PgQQL decreases causing Putvc/^'bcOL ^° increase. 3. As the cross sectional flow area of the boiler columns decreases (or as the UF^ inlet velocity to the boiler columns increases), PiiTur increases and Pg^QL ^^^creases causing Putvc/'^BCOL *° increase. 4. As the number of boiler columns increases, PyjyQ decreases and Pgrni increases causing Pyjvc/PBCOL *° decrease. 5. As the UF^ partial pressure in the UTVC increases, Pujvc increases and Pgrni decreases causing Putvc/^'bCOL *° increase. 6. As the UF^ density in the boiler columns increases from 1.4 to 3.7 g/cm , PiiTur decreases and PgQQL increases, causing PutvC^'^BCOL ^° decrease. The following summarizes the general behavior of PyjyQ, PgcOL' ^"^ ''utvc/^'bcol *^ * function of variations performed on the UTVR in the 2-D R-Z coordinate system: 1. As the MBEG thickness increases to =7.5 cm, PyjyQ increases and Pg^QL decreases causing Putvc/'^BCOL ^° increase. As the MBEG thickness further increases above «7.5 cm, PjjyQ increases and Pg^QL '^^c'^^*^®^ at decreasing rates causing Putvc/^BCOL ^° increase at a decreasing rate. 2. As the TBEO thickness increases, both PyjyQ and PgQpL increase with PyjYQ increasing at a higher rate. Thus, Putvc/''bCOL ^"creases as the TBEO thickness increases.

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148 3. As the BeO moderator thickness between the boiler feedlines and the MHD duct increases to about 60 cm, Piixyr decreases and Porni increases causing PuTVc/'^BCOL *° decrease; Pujvc ^^®" increases and PgQQi decreases causing Putvc^^'bCOL ^° increase as the BeO thickness further increases. 4. As the height of the UF^ liquid in the boiler column increases, Piirwr decreases and Pornt increases causing Pijtvc/''bCOL ^° decrease. 5. As the molybdenum thickness surrounding the boiler region feedlines increases, P^jyc increases and P^qq^ decreases causing PyTVc/^BCOL ^° increase. The results also indicate that the Pijtvc/''bCOL ^*^"®^ obtained from the R-0 calculations are higher than those obtained from the R-Z calculations. The higher Putvc/^BCOL ^^^''^^ ^^^^ calculations performed in the R-^ coordinate system are due to the ability to model the boiler , region in R-6 geometry as separated boiler columns whereas in the R-Z coordinate system, the boiler region is modeled as an annular cylindrical region surrounding the UTVC. Thus, the P||tvc/''bCOL ^^^^^^ obtained from calculations performed in the R-6 coordinate system are more typical of the actual system due to the ability to more accurately represent the boiler region as separated boiler columns in R-0 geometry. Remarks The two-dimensional calculations presented in this chapter are performed using the DOT-4 S^ transport theory code on an IBM 3090/4000 mainframe computer. The required computer time for an average problem ranged from »300 to «600 seconds for convergence levels of »5 x 10' .

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149 Thus, the 2-D calculations require at least two order of magnitudes greater computer time at a factor of five lower convergence level compared to the 1-D calculations. These 2-D calculations provide useful information pertaining to general behavior of global properties of the UTVR system such as the k^^^ and PuTVC^'^BCOL ^*^''^^' ^" insight is also gained to the behavior of local parameters such as fission rates and neutron fluxes in the UTVR regions as a function of variations performed on the UTVR system. Modeling the UTVR in the 2-D R-^ cylindrical geometry results in relatively high k r^^ values due to the inability to account for neutron leakage in the axial direction; and the PuTVc/'^BCOL ^^^'^^^ obtained from calculations performed in the 2-D R-Z coordinate system are underestimated due to the modeling of the boiling region as an annular shell surrounding the UTVC. However, by careful examination and crosscomparison of the results of calculations performed in both the 2-D R-0 and the R-Z geometries, reference reactor configurations for 3-D analysis can be obtained. v. The following are conclusions pertaining to the UTVR system as obtained from calculations performed in the 2-D geometry from both the R-$ and R-Z coordinate systems: 1. The system's k^^^ saturates at UTVC radii above »80 cm. 2. The optimum IBEO thickness is at »16 cm. 3. A desirable OBEO thickness is 40 cm. 4. The optimum MBEO thickness is at «7.5 cm. ,:^ 5. The optimum TBEO height is at »50 cm. . *

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150 6. About 30% of the NaF needs to be diverted from the wall cooling region to the boiler region so that the nuclear design provides power sharing between the vapor cores and boiler cores (PuTVc/'^BCOL^ which matches the power sharing requirement on the basis of thermodynamic considerations. •

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CHAPTER VI STATIC, THREE-DIMENSIONAL NEUTRONIC ANALYSIS OF THE UTVR U Introduction Results of the three-dimensional static neutronic analysis performed on a four-boiler column UTVR are presented in this chapter. These results include the UTVC and boiler column reactivities {p and B U B p ), prompt neutron generation time (A and A ), and the direct core-to_-iV|c core neutronic transport coupling coefficients {ei ) as a function of U R fuel loadings in the UTVC and boiler columns. These parameters (p , p , U B i'^-k A , A , and if ) are used in the dynamic neutronic and performance studies presented in Chapter VIII. Three-dimensional neutronic calculations are performed using MCNP [12], a general purpose, continuous energy, generalized geometry, time dependent, coupled neutron-photon Monte Carlo transport code. The MCNP code is described in detail in Appendix A. A description of the UTVR configuration used in the 3-D analysis is presented in this chapter. The boiler column configuration used for modeling boiling under =zero gravity conditions is also described in this chapter. This is followed by a section describing the variancereduction techniques, employed in the 3-D analysis of the UTVR, that reduce the required computation time by a factor of =13. A section describing the derivation of the methods and models used for obtaining 151

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152 the reactivities of the UTVC and boiler columns and the direct neutronic coupling coefficients is also included in this chapter. Description of the UTVR Geometry in MCNP Only one of the two symmetric reactors that constitute the UTVR is analyzed in the 3-D calculations (refer to Chapter I, pages 2-4 for ' details). A cross-sectional side view of the UTVR on the y-z plane is shown in Figure 6-1. The underlined numbers and numbers within parentheses denote the surface and cell numbers, respectively. The MCNP code uses surfaces to define zones or regions in the ordinary Cartesian coordinate system; the basic unit of MCNP geometry is the cell. Planes, cylinders, and cones are the only surfaces used in defining the geometry of the UTVR in MCNP. The surfaces used in defining the cells of the UTVR are the following (dimensional units in cm): 1. Surfaces 1 through 11 are planes normal to the z-axis at z=0.0, 7.5, 15.0, 25.0, 27.5, 40.0, 82.0, 87.0, 95.0, 140.0, and 190.0, respectively. The asterisk following the number "1" is used to define surface 1 as a reflected surface. 2. Surfaces 20 through 25 are cylinders on the z-axis with radii equal to 35.0, 50.0, 65.0, 65.312, 105.0, and 126.0, respectively. 3. Surfaces 30 through 35 are cylinders parallel to the z-axis at x=0.0. Surfaces 30 and 31 have radii equal to 2.3 cm each and are centered at y=68.0 and -68.0, respectively; surfaces 32 and 33 have radii equal to 5.2 cm each and are centered at y=85.0 and -85.0, respectively; and surfaces 34 and 35 have radii equal to 3.0 cm each and are centered at y»85.0 and -85.0, respectively.

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153 190.0 140.0 132.5 125.0 105.095.087.085.082.011 10 (200) 30 ;;(2i)\ '"> 22 52 23 25

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154 4. Surfaces 40 through 45 are analogous to surfaces 30 through 35 with the exception that their x and y locations are reversed. 5. Surfaces 50 through 53 are cylinders parallel to the y-axis at x=0.0 with radii equal to 1.6, 2.6, 2.3, and 10.0 cm, respectively, and are centered at z=85.0, 85.0, 95.0, and 130.0, respectively. 6. Surfaces 60 through 63 are analogous to surfaces 50 through 53 except they are parallel to the x-axis at y-0.0. 7. Surfaces 70 through 77 are circular cones parallel to the z-axis with vertices intersecting the z-axis at z=-150.0, -130.0, -30.83, 17.33, 19.66, 50.38, 330.0, and 500.0; and slopes equal to 0.44, 0.44, 0.73, 8.22, 2.6, 0.75, 0.16, and 0.07, respectively. 8. Surfaces 80 through 83 are cones parallel to the z-axis with {x,y,z) vertices at (0.0,-85.0,92.11), (0.0,85.0,92.11), (0.0,-85.0,130.0), and (0.0,85.0,130.0); and slopes equal to 0.07, 0.07, 0.014, and 0.014, respectively. 9. Surfaces 90 through 93 are analogous to surfaces 80 through 83 with the exception that their x and y locations are reversed. Combinations of sense-signed surfaces allow the construction or modeling of the UTVR for MCNP input. For example. Figure 6-1 indicates that the UTVC, which is represented by cell 200, is the formed by the space above surface 4, inside surface 73, inside surface 22, and below surface 10. A description of the UTVR cells (regions) is given in Table 6-1. This includes the volumes of the fissile regions and their contents.

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,'' '

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^ 156 Description of the Boiler Column A side view schematic of one of the four boiler columns is shown in Figure 6-2. Figure 6-2 indicates that the boiler column is made of three regions: 1. The first is the subcooled liquid region (cell 3) which is the region formed by the union of two subregions. The first subregion is formed by all space above surface 7 (plane normal to the z-axis at z=82.0), below surface 8 (plane normal to the z-axis at z»87.0) and inside surface 34 (cylinder parallel to the z-axis at x=0.0 and y^SS.O with a radius of 3.0 cm). The second subregion is formed by a11 space above surface 8, below surface 9 (plane normal to z at { z»95.0), and inside surface 32 (same as surface 34 except it has a 5.2 cm radius). 2. The second is the saturated liquid region (cell 4), or liquid cone region, formed by the space below the inverted cone (surface 83) and above the plane normal to the z-axis at z=95.0 (surface 9). 3. The third is the vapor cone region (cell 11) which is the region formed by the space above the inverted cone (surface 83), inside the upward cone (surface 81), above the plane at z=95.0 (surface 9), inside the cylinder with the 5.2 cm radius (surface 32), and below the plane at z=140.0 (surface 10). The boiler column model shown in Figure 6-2 is configured to allow for boiling in space (=zero gravity). This configuration is based on the following consideration: As the fuel flows in the +z-direction, fission heat remaining in D the fuel raises the fuel's temperature to saturation (H-x).

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157 140.0 U ^• .s* '< y-135.0 y*. jvi 4'' Figure 6-2. Side View Schematic of a Boiler Column

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158 For the reference configuration, this is assumed to occur at z=95.0 cm. Due to the depression of the thermal neutron flux in the liquid region, the majority of fission occurs at the outer boundaries. Since the fuel is at 1°^^^, the fuel in the outer regions vaporizes and accelerates away from the liquid region causing the radius of the saturated liquid region to decrease as the fuel flows axially. The shape of the saturated liquid region is assumed to be a cone converging in the +z-direction. Since the volume of the vapor fuel is «150 times greater than the liquid fuel volume (at about 3000 K and 50 atm), then to reduce pressure losses due to boiling and friction, the volume of the boiler column is increased in the +z-di recti on. The cone diverging in the +z-direction (surface 81) provides the extra volume needed for boiling. ^ > Reactivity Worths of the Boiler Feedlines. UTVC Inlet Plenums, and the MHD Duct Regions For the purpose of obtaining the direct core-to-core neutronic transport coupling coefficients among the multiple fissioning core regions (i.e., the UTVC and the surrounding boiler columns), the boiler feedlines, UTVC inlet plenum, nozzle, MHD duct, and diffuser regions are not included in the analysis; they are replaced by BeO. However, the reactivity worths of replacing the contents of these regions with BeO, for the reference configuration given Figure 6-1 and Table 6-1, are obtained and presented in Table 6-2. Table 6-2 indicates that when the contents of the boiler feedline and UTVC inlet plenums regions are replaced with BeO, k /r^ decreases

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c o a> O) u 3 o c 0) a. •M 0) 0) c T3 0) 0) O) o CO 0) C •fj (4o (/I .c I. o >> u to 0) CNJ I O) jQ (/>

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f* 160 from 1.050 to 1.047 (=0.3% 5k/k decrease) and the fission rates of the UTVC, nozzle, MHD duct, diffuser, boiler liquid column, boiler liquid cone, and boiler vapor region increase by «1.2%, «2.8%, «4.4%, «5.1%, =6.3%, «2.3%, and =15.6%, respectively. Since the total increase in the -03 fission rates of these regions (=9.3 x 10 fissions/sec) is less than the contribution of the boiler feedlines and UTVC inlet plenums to -02 fission (=1.4 x 10 fissions/sec), k^^^ decreases. When the nozzle, MHD duct, and diffuser regions contents are replaced with BeO, the results indicate that kg^:^ increases from 1.050 to 1.145 which translates to a reactivity increase of =8.6% 8k/k. Table 6-2 also indicates that the fission rates of the UTVC, UTVC inlet plenums, boiler feedlines, and boiler core regions increase by =15%, =12%, =3.4%, and =5.7%, respectively. The large increase in k^^^ (mainly due to the large increase in the fission rate of the UTVC) is caused by the decrease in the neutron leakage rate from the MHD duct regions. Although the MHD duct regions are designed in a manner to reduce neutron streaming, the results indicate that neutron streaming from the MHD duct regions remains significant. When the contents of the boiler feedlines, UTVC inlet plenums, nozzle, MHD duct, and diffuser regions are replaced with BeO, Table 6-2 indicates that k^^ increases from 1.050 to 1.152, or a reactivity increase of =9.3% 5k/k. This result is in disagreement with the previous results. That is, since the effects of replacing the contents of the boiler feedlines and UTVC inlet plenums with BeO is a reactivity decrease of =0.3% 5k/k and the effect of replacing the MHD duct regions with BeO is a reactivity increase of =8.6% 5k/k, then the effect of

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161 replacing the contents of the boiler feedlines, UTVC inlet plenums, and MHD duct regions with BeO should be a reactivity increase of »8.3% Sk/k and not 9.3% Sk/k. The reason for the discrepancy is due to the relatively large uncertainty levels in the k^^^ values, as shown in Table 6-2. These uncertainty levels are larger than some of the observed changes in k^^^. For example, when only the boiler feedlines and UTVC plenums are removed, the uncertainty in k rr is «0.8% which is greater than the »0.3% 6k/k reactivity decrease (k^^ decreases from 1.050 to 1.047). The uncertainty in fission rates of the boiler column regions are quite large. The uncertainty of other important parameters associated with the boiler columns is even greater. For example, the uncertainty in the fission rate of a boiler column due neutrons that escaped the boundaries of an adjacent boiler column (Fg ) is found to be «85%. This parameter, Fg , is used in obtaining neutronic coupling coefficients. Since a primary objective of performing 3-D Monte Carlo calculations with MCNP is to obtain neutronic coupling coefficients among the interacting cores, the uncertainty in parameters associated with the boiler columns needs to be reduced. Reducing the Uncertainty in Parameters Associated with the Boiler Columns in MCNP Calculations Listed in Table 6-3 are results obtained from a 30 minute, MCNP calculation performed on the reference UTVR configuration with the boilers feedlines, UTVC inlet plenums, and MHD duct regions replaced by BeO. Table 6-3 indicates that the uncertainty in the boiler column neutron flux (^Rrni^ ^^ ~^ times greater than the uncertainty in the

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•a a> e o 0) a. o o 0) 00 ^ II H CT

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163 UTVC neutron flux (*UTVC^* ^^^^ ^^ ^^^ *° ^^^ relatively small size of the boiler region compared to the size of the UTVC. Thus, neutrons have a greater probability of entering the UTVC than entering the UF^ boiler columns. Table 6-3 also indicates that the uncertainty in Pg or Fg (Bo is the opposite boiler) is quite large (>80%). Since these parameters are used in obtaining the neutronic coupling coefficients which are needed for the dynamic neutronic analysis of the UTVR, the large uncertainty in parameters associated with the boiler columns needs to be reduced. The uncertainty (relative error), R. is calculated in MCNP from R = «^/ ^^-^^ where is the Monte Carlo estimated mean for the process x and is given by NS = JT X4 (6-2) where x^ is the value of x for the i history (particle) and NS is the number of histories examined in the problem. The quantity a^^^ in Equation (6-1) is the relative standard deviation of (square root of the variance of ) given by ?"' •', a = ffx / vNS t ' ^^'^J where a is the estimated standard deviation of the population of x (the square root of the variance of x) given by NS — I (xi-)^ « -2 . (6-4) •-1 i=l ^ '

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164 Using Equations (6-3) and (6-4) in (6-1) results in the following: 2 ^ °x ^ ^ (6-5) 7 2 NS '^ NS ^ According to Equation (6-5), the uncertainty, R^^^^, is inversely proportional to the square root of the number of histories examined, NS, and directly proportional to the square root of the history variance, a. Thus, the uncertainty of a given parameter can be reduced by increasing the number of histories examined (i.e., by increasing computation time and/or the rate at which particles are examined) and/or by reducing the history variance. Increasing the rate at which particles are examined or reducing the history variance can be achieved by the use of variance reduction techniques. The uncertainty, hence, can be reduced by increasing the computation time and/or employing variance-reduction techniques. Unless variance-reduction techniques are employed, a reduction in the uncertainty of Fg"^" or Fg^^° from »80% to »10% (reliable confidence intervals are generated when the uncertainty is »10% or less [32]) would require an increase in computer time by a factor of «64, (i.e., each problem would require «1,900 minutes or «30 hours of Cray X-MP/48 Supercomputer time) which is prohibitive. In order to decrease R, o^ needs to be decreased or NS needs to be increased (note that if the computation time is fixed, increasing NS implies decreasing the time per particle which increases a ). Unfortunately, these two goals usually conflict since decreasing a normally requires better information which requires more computation

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165 time and increasing NS normally increases a^ because there is less time per history to obtain information. However, it is sometimes possible to 'l decrease a substantially without decreasing NS substantially or increase NS substantially without increasing a^ substantially. Variance-reduction techniques attempt to decrease R^^^ by either destroying and/or producing particles. By destroying the "less" important particles (i.e., particles that do not contribute to a given tally), an increase in NS per unit time is obtained. By producing and examining a greater number of the "more" important particles, a decrease in a is obtained. Variance-reduction techniques in Monte Carlo calculations can often reduce the computation time required to obtain results with an acceptable uncertainty. The MCNP code employs a variety of variance-reduction techniques [12,32-33] which include geometry splitting and Russian Roulette, implicit capture and weight cutoff, time and energy cutoffs, forced collision, and weight windows. The choice of the variance-reduction technique or the combination of different techniques employed depends on the geometry and configuration of the problem and on the parameter whose variance needs to be reduced. Therefore, prior to implementing any variance-reduction technique, the following considerations need be realized: Performance of Variance-Reduction Technicues >"> The Figure Of Merit (FOM) is a measure of efficiency for MCNP -. * calculations defined as -1 FOM = ""CPU R (6-6)

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166 where Trpu i^ the computer time for the calculation in minutes. For a well -sampled problem, the FOM should be roughly constant because R^^ is (on the average) inversely proportional to yR5, the number of histories examined, and Tppy is (on the average) directly proportional to NS; therefore, the product remains ^constant. The utility of the FOM is that it can be used to examine the effects of the employed variance-reduction technique without having to run MCNP for a great amount of time. An increase in the FOM of a given tally indicates a decrease in the relative error of that tally and an increase in the efficiency of the MCNP calculation. The FOM can also be used to estimate the amount of computer time (Tppn) required to achieve a desired relative error. According to Equation (6-6), if the FOM increases by a factor of two, then the required TQpy decreases by a factor of two for a fixed R, or R^^ decreases by a factor of J2 for a fixed T^py. Nuclear and Physical Characteristics of the UTVR Knowledge of how neutrons interact in the UTVR regions aids in determining what the variance-reduction techniques need to accomplish and, consequently, which variance-reduction technique to use. Therefore, prior to employing any variance reduction technique, the following needs to be realized: *> BeO regions . Fast neutrons above the (n,2n) threshold (E^«1.84 MeV) are very important in producing secondary neutrons from the Be(n,2n)2He interaction. However, a great amount of computer time is wasted following low energy neutrons in the extremely high scattering

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167 BeO regions, especially in BeO regions that are far away from the UTVC or boiler columns. UTVC region . Due to its relatively large size (»20% of the UTVR volume) neutrons have a large probability of entering the UTVC. 19 235 3 However, due to its low vapor density {«lxlO U -atoms/cm ), a large amount of time in the UTVC is wasted following neutrons that are above thermal energies. Boiler columns . Due to its relatively small size {«0.4% of the UTVR volume), the probability for neutrons to reach the boiler columns is relatively small. However, due to its relatively high fuel density (»140 times greater than the density of the UTVC), the probability that a neutron will have an interaction once it enters the boiler region is extremely large. Therefore, in order to decrease the uncertainties of parameters associated with the boiler columns, the employed variance-reduction techniques need to accomplish the following: 1. Reduce the number of neutrons as their energies decrease when entering BeO regions that are far away from the UTVC or the boiler columns. Also, terminate all neutrons in the far away BeO regions if their energy is «thermal . 2. Increase the importance of neutrons if their energy is greater than 1.84 MeV in BeO regions. 3. Increase the number of neutrons as they approach the boiler columns. 4. Reduce the number of fast neutrons entering the UTVC.

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168 The above is accomplished by employing the following variancereduction techniques: energy cutoff, implicit capture and weight cutoff, and weight windows, as follows: Energy Cutoff Energy cutoff is a variance reduction technique that terminates particles when their energies fall below the specified energy cutoff (E^y^). Energy cutoff is generally used when it is "known" that lowenergy particles are either of zero importance or »zero importance. Since the neutronic coupling coefficients are a measure of the contribution to the fission rate of a given core due to neutrons originating in other cores, and since thermal neutrons have relatively high importance in causing fission, then energy cutoff must be used cautiously. Table 6-4, which lists the UTVR fission rate as a function of neutron energy, indicates that neutrons with energies below 10" MeV produce less than «0.001% of the fission at an uncertainty level of 100%. Thus, it can be assumed that neutrons with energies less than 10" 09 MeV have »zero importance. Results of employing an energy cutoff of -OQ 10 MeV are given in Table 6-5. The results indicate that an average increase of -7% in the FOM of the boiler column and UTVC flux and fission rate tallies is obtained. The increase in the FOM is due to increasing NS per unit time (=8%) without increasing a^ as a consequence of terminating the un-important particles. ',, r T ^ , v

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169 0) i0) c c o s!-> 3 c o o 0) re su 0) (« QC u re i«/) > O) I. « CO CO ^— • ^ CM •— I CM — < ^H O CM »« r^ >5 >5 *« »5 s5 CD csj O) LO ot in o o ^H o vo C3 o o 9« o CD »fi CO ^H CM CO CO CD CD CD CD ^ ^ I I I I I I CD O O CD O CD X X X X X X m in CO ^ o» i-H o> o 00 1^ r^ •— I CM m CO CM CO OJ CT» 00 lO CO •— I o o o o o I I I I I o o o o o I— I f— I t— I I— I t-H ^ X X X X X in *o CO — • ^ ^ 00 m CM i-H r— t f-.! m 00 0» 00 VO CO •— I o CD o o o CD O O CD O CD CD X X X X X lO CO •-< o o 00 m CM ^H •-• ^H m 00 I X t—t CSJ CD in o o )-> re c O) o> L. Q. O c o a> o O) N O o OQ o re 0) re >> a* C7> o o 0) in a> re Percent Change

PAGE 191

V' Implicit Capture and Weight Cutoff Implicit capture, survival biasing, and absorption by weight reduction are synonymous. With implicit capture, the particle always survives the collision and is followed with a reduced weight. Implicit capture is applied in MCNP after selecting the collision nuclide. After the collision, the weight of the particle, WGT , is reduced by (1 ol/o^); where a^ and a^ are the microscopic absorption and total cross sections for the i nuclide, respectively. In weight cutoff, Russian roulette is played if WGT falls below R^xWC2, the weight cutoff; WC2 is a user-specified weight cutoff and R^ J J is equal to the source cell importance divided by the j cell importance (the cell where the particle is colliding). If WGT falls below RjXWC2, with probability WGTy(R.xWCl) the particle survives with new weight R^xWCl, where WCl is a user-specified weight; otherwise the particle is terminated. Weight cutoff is normally employed with implicit capture and geometry splitting. Shown in Table 6-6 are results of an MCNP calculation performed with WCl and WC2 set to the default values of 0.50 and 0.25, respectively. The results indicate employing implicit capture and OQ weight cutoff over the entire neutron energy range (10' MeV < E < 15 MeV) causes «7% and al2% reduction in the flux and fission rate FOM tallies of the UTVC and boiler column, respectively. Since the neutron interaction rate increases as neutrons become ^thermal , and since employing implicit capture causes these neutrons to always survive these collisions as long as their weight is larger than R^xWC2, then a great amount of time is spent colliding, surviving, and following thermal

PAGE 192

o 0) o -o c
o o u a> I Ol .a (U J•!-> a. o M O 'q. E ia.

PAGE 193

e, ^ .3 172 neutrons. This causes a reduction in a since more information is obtained. However, the increase in computation time (or reduction in NS per minute), as a result of acquiring more information, is greater than the obtained decrease in o. Thus the FOM tallies decrease. The reason for a larger decrease in the FOM tallies of the boiler column, relative to the UTVC, is due to its relatively high fuel density (»140 times greater than the density of the UTVC). The higher fuel density of the boiler column increases the rate at which neutrons interact. Therefore, more time per neutron is spent in the boiler column colliding and surviving neutrons. The aim of employing variance-reduction techniques is to increase the precision (i.e., reduce the uncertainty) of the results by reducing a and/or increasing NS per unit time. This can be accomplished by allowing neutrons to survive every collision until they cause fission. The probability for a neutron to cause fission increases as its energy decreases. A substantial reduction in a without a substantial increase in computation time is possible if fast neutrons are allowed to survive every collision until they are wthermal . This can be accomplished by employing implicit capture only when neutrons are above the thermal energies. Since Table 6-4 indicates that =90% of the fission is caused by neutrons with energies less than 1.86 x 10" MeV, the neutron energy range of 1.86 x 10' Mev < E^ < 15 MeV is selected to be the range where implicit capture is employed. The results of this calculation are also shown in Table 6-6. The results indicate that the UTVC and boiler column FOM tallies increase by »6.5% and »14%, respectively. This

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173 indicates that utilizing partial implicit capture yields a reduction in a which is substantially larger than the decrease in NS per unit time. Weight Windows The weight window is a space-energy-dependent splitting and Russian roulette variance-reduction technique. Each space-energy phase-space cell is assigned a lower weight bound, W. , and an upper weight bound, Wy. The upper weight bound is equal to CyxWj^, where Cjj is a userspecified constant. These weight bounds define a window of acceptable weights. If WGT is above the upper weight bound, the particle is split into a number of particles such that all the split particles have weights within the window. If WGT^ is below the lower weight bound, Russian roulette is played and the particle is either terminated or its weight raised to W^, where W^ is the survival weight equal to C^xW^ (C^ is a user-specified constant less than Cm). No action is taken if Wj^ < WGT„
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174 Since the importance of high energy neutrons is greater in BeO regions than in the fissile regions (especially for the UTVC), and since the importance of low energy neutrons is greater in the fissile regions and in BeO regions that are near the fissile regions than in BeO regions that are far away from the fissile regions, then the use of weight windows is ideal for the UTVR. That is, the UTVR is characterized by having an importance function that is space-energy-dependent which can be described with weight windows. For the purpose of utilizing weight windows, 170 cells and fourneutron energy groups are used to represent the space and energy phases of the UTVR, and the fission rate of the boiler columns is the selected tally for obtaining optimized weight windows. The results of employing weight windows are shown in Table 6-7. The results indicate that using the weight window generator causes about a 20% reduction in the UTVC and boiler column FOM tallies and »19% reduction in NS per unit time. Usually, the weight window generator slows calculations by 20-50% [32]. However, the generator can be turned off once an acceptable weight window is obtained. Table 6-7 shows that, when the third generated weight windows are employed and the particle's weight is checked at surfaces and collisions, the UTVC flux and fission rate FOM tallies decrease by «63% and »48%, respectively, while the boiler column flux and fission rate FOM tallies increase by =81% and »178%, respectively. However, when the particle's weight is checked only at surfaces, the UTVC flux and fission rate FOM tallies decrease by 60% and 46%, respectively, while the boiler column flux and fission rate FOM tallies increase by »75% and «200%,

PAGE 196

«-> 175 0) 0) (« o O) o T3
PAGE 197

176 respectively. The reason that the UTVC FOM tallies decrease and the boiler column FOM tallies increase is due to the fact that the boiler column fission rate tally is the tally selected for obtaining the optimized weight windows. The cost of reducing the variance of the boiler column tallies is an increase in the variance of the UTVC tallies. This is what usually occurs when variance reduction techniques are employed, i.e., reducing the variance of certain parameters is achieved at a cost of increasing the variance of other parameters. However, due to the size of the UTVC and the initial or previously available uncertainty levels in UTVC parameters, the obtained increases in the uncertainty of parameters associated with UTVC are acceptable. This can be seen in Table 6-8, where the effects of employing energy cutoff, implicit capture and weight cutoff, and weight windows on the uncertainty levels for 30-minute MCNP calculations are shown. Table 6-8 indicates that employing the weight window variance-09 reduction technique with an energy cutoff of 10 MeV and partial implicit capture reduce the uncertainty of the boiler column parameters by an average of about 40%. Table 6-8 also indicates that employing these variance-reduction techniques causes the uncertainty levels in IJ4JJ M^-D '^eff ^''"^^ ^''"^' ^"^^^ fission rate, and in the F^ and Fg parameters to increase from 1.4% to 1.8%, 0.95% to 1.40%, 1.44% to 1.81%, 1.48% to 2.20, and from 12.0% to 17.4%, respectively. Although 8:40% average reduction in the uncertainty levels of the Fg^ " and p^ parameters is obtained, the obtained uncertainty levels remain quite large {k50%). To reduce the uncertainty of these parameters from »50% to »10% would require «750 Cray X-MP/48

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c o «J c o •r•-> u I Z (/> 0) =3 CT C .c u (U h«/) C O) O -M •^ O) •M E U (Q 3 I"O «> 0) O. OC I Q^ a> > o t— c ^ •<-o (O •-> > u a> oil— c a> •f«/> ^^ I— o Q. B (/> UJ O) *4J o c •^» «/) (O •-> 4J u s0) a> <4o 00 I

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.:' •, 'Y. 178 Supercomputer minutes. Although a reduction in the required computation time from »1,900 to «750 minutes is achieved through the variancereduction methods, the 750 minutes required to achieve the desired uncertainty levels remain unacceptably large. However, by taking advantage of the symmetric alignment of the UF^ boiler columns around the UTVC, a further reduction in the uncertainty levels of boiler column parameters is possible. This is discussed in detail in the following section. Boiler-to-UTVC Symmetry , ivt' Due to the symmetric alignment of the boiler columns around the UTVC, the uncertainty levels in parameters associated with the boiler columns can be reduced. This statement is based on the following argument: ^ Since the UTVR employs boiler columns that are symmetric with respect to the UTVC and since the power level is the same for all boiler columns, then as the number of histories examined (NS) approaches infinity, the reaction rates or tallies (e.g., flux and fission rate) of any boiler column should be the same for all boiler columns. However, since NS is relatively small (»20,000 source particles), the tallies obtained for each boiler column are different. Since the estimated value for any process X (e.g., fission) in each boiler column is statistically averaged and since the boiler columns are equivalent, then each boiler column tally reflects an independent measurement of NS samples each. Then combining the tallies of all boiler columns

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179 reflects a tally of n x NS histories, where n is the number of independent measurements. Since the uncertainty, R^j,^, is inversely proportional to the square root of the number of histories examined and since averaging the score of all boilers reflects a measurement with n x NS histories (i.e., the number of histories examined increases by a factor equal to n), then, according to Equation (6-5), the uncertainty level of that tally decreases. The equations that describe the modified tally as a consequence of combining the tallies of all boiler columns are as follows: n n NS . = 4 ^ '"'' =Tinjr ^ ^ ^i ^'-'^ where is the modified average value of the tally describing the > process x (averaged over all boiler columns), x^ is the value of x for the i history for the j measurement set, is the average score for the process x for the j measurement set, and n is the number of different sets of measurement. With reference to Figure 6-3, where a schematic of a four-boiler column UTVR system is shown, and utilizing Equation (6-7), the following equations are used to obtain the modified scores (tallies): 4 (6-8) (6-9) ?=

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•^^ Boiler Bl o 180 o Boiler B3 o Boiler B2 \ '-'i •L o Boiler B I -. '^' ' > Figure 6-3. Top View Schematic of a Four-Boiler Column UTVR System ?*^ ...',."

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181 e A ^ e J=l where f? and F? are the average flux and fission rate of all four boiler columns, respectively; fj-^ and Fj"^ are the j boiler column flux and fission rate, respectively; F^ is the average fission rate in the boiler column due to neutrons escaping the boundaries of the UTVC; F^-^ is the fission rate in the j boiler column due to neutrons escaping U'^-B the boundaries of the UTVC; Fg is the average fission rate in the UTVC due to neutrons escaping the boundaries of any boiler column; and F^ "^ is the fission rate in the UTVC due to neutrons escaping the boundaries of the j boiler column. The average fission rate in the boiler column due to neutrons escaping the boundaries of an opposite boiler column, Fg , and due to neutrons escaping the boundaries of an adjacent boiler column, Fg , are given by the following: ^B^Bo

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182 m n X NS (6-14) where a^ is the standard deviation of all x measurements for all boiler columns given

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183 R 1 n 1 1 7 ^ n^ NS j=l K>/) 4/ o^ / + 1 1 (6-20) The relative error of the j measurement set is VxJ ^ = aj /nS ^ . (6-21) Substituting Equation (6-21) into (6-20) yields 1 <%> n I n*^ NS j=l NS R ^ 1 1 (6-22) Note that if NS is large enough such that » o.O (for all j measurements), then / « 1. With this assumption, Equation (622) can be simplified to the following: j=l (6-23) Results of utilizing boiler-to-UTVC symmetry are listed in Table 69. Table 6-9 indicates that the uncertainty in the boiler column flux B<-Bo B*-U U'«-B and fission rate and in the F" , Fg , and F^ parameters is reduced by »50% and the uncertainty in the F^ parameter is reduced by «64%, as a consequence of utilizing boiler-to-UTVC symmetry without any increase in the uncertainty of other parameters. Table 6-9 also lists the expected uncertainty levels of various parameters as a function of computation time. According to Table 6-9, to obtain results with reliable confidence intervals (no more than 10% uncertainty) will require «120 to »150 minutes on the Cray X-MP/48 Supercomputer. Thus, ,.+ v.* ,<'.

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^ I o I U a> 1 — o>e c: •fCO c ^(»u o a> t«/» u ra a

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185 by employing energy cutoff, implicit capture and weight cutoff, and weight windows and utilizing boiler-to-UTVC symmetry, the required computation time is reduced by a factor of =13 (required computation time is reduced from =1,900 to =150 minutes on the Cray X-MP/48 Supercomputer). ....' ^^: ' *' ^ • . Neutron Transport Coupling Coefficients One of the unique inherent reactivity control mechanisms of the UTVR is the direct neutronic coupling among the multiple fissioning core regions (i.e., the UTVC and the surrounding boiler columns). The direct neutronic coupling can be defined as the contribution to the fission rate of a given core from neutrons originating in other cores. The j core total fission rate, FJ, is defined as follows: . NC NC . . . 1 =1 i =1 where F"^^^ is fission rate in core j due to neutrons generated in core i given by oo FJ"^ = f r^(E) $J*"^E) dE , (6-25) 1 where *'^^^(E) is the neutron flux in core j due to neutrons generated in core i, and 2"i(E) is the macroscopic fission cross-section of core j. The quantity NC that appears in Equation (6-24) is the number of cores in the reactor system and si is the total number of fission neutrons generated in the i core given by

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186 sL = f vUe) TA£) #t(E) dE (6-26) i} = \v\£) 2:^(E) #j(E) dE where i/^(E) is the number of neutrons liberated per fission in core i and *x(E) is the total neutron flux in core i due to neutrons generated in core i, #^*"^(E), and neutrons transported from all j cores other than core i, #^*''^{E), given by NC NC ' V I -. ^'^ k^i k=l v-i The quantity /^^^ in Equation (6-24) is the probability for transporting the i core neutrons to core j where they cause fission, i.e., the neutronic coupling probability, and is given by i J-i F^^ ft = -J' ' (6-28) s. The neutronic coupling probability, /^^\ is a measure of the effectiveness of the i core neutrons in causing fission in core j. It can also be expressed by where Pg » probability for the i core neutrons to escape absorption in core i, i.e., i core neutron leakage probability from core i. This parameter depends on the contents of core i (the type and density of the materials) and the size of core i. The i core

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187 leakage probability decreases as the macroscopic absorption cross section and/or the size of core i increase, p:^^^ conditional probability for the i core neutrons to be transported from the outer surface of core i to core j where they have an interaction. This parameter depends on the distance and the type of media separating core i from core j and depends also on the size and contents of core j. p4^^ conditional probability for the i core neutrons incident on the surface of core j to cause fission in core j. This parameter depends on the contents and size of core j. Once the neutronic coupling probability is obtained, the neutronic coupling coefficient, ft^\ can be obtained by using ^ i . . „ , -. The effective coupling coefficient is defined as the fraction of neutrons in core j that originated in core i (by (n,/), {n,2n), (7,n), or delayed neutron emission) and are transported to the j core where they cause fission. The neutronic coupling coefficient can be obtained either directly or indirectly from MCNP. These methods are described below. ej^*^ Obtained Directly from MCNP To obtain the coupling coefficients directly from MCNP for a system composed of NC distinct cores (i.e., fj""^ t fij*"^ for all i and j cores), two sets of NC calculations are needed for each given configuration.

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188 The first set of NC calculations is used to generate neutron fission sources in each of the NC cores. The second set is needed to transport neutrons (generated from the first set of calculations) from each distinct core to all other interacting cores. For example, if a reactor system is composed of three distinct cores, namely cores A, B, and C, then in order to obtain the coupling coefficients from core C to cores B and A (i.e., £^ and «* ) two calculations need to be performed, as follows: \w , * 1. The first calculation generates a neutron fission source in core C. This is accomplished by voiding out cores A and B or by given zero importance to cores A and B, thereby core C is the only fissioning core. This calculation will be performed for a number of cycles (equal to KCT) and NSRCK neutron source points will be generated for each cycle, where KCT and NSRCK are input parameters for the MCNP code. The magnitude of NSRCK depends on the heterogeneity and size of core C. Both KCT and NSRCK need to be large enough so that a fairly good fission neutron source distribution is obtained. Typical values range from three to five cycles for KCT and 300 to a3000 for NSRCK. " 2. The neutron fission source generated in core C from the first calculation is used in this calculation. Cores A and B are no longer voided and are each assigned appropriate importance values. Since fission occurs in cores A and B, only one cycle (i.e., KCT=1) will be performed. However, the number of neutron source points needs to be increased so that a large number of neutrons are followed from core C to cores A and B (the uncertainty in €^ and

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189 in £^ decreases as the number of histories examined increases) The number of neutron source points can be increased by increasing the NSRCK parameter in the MCNP input file. The MCNP code will start NSRCKVnSRCK° particles at each source location (NSRCK"^ is the requested number of neutrons to be followed and NSRCK^ is the original number of source points generated in the first calculation). The obtained neutron flux and fission rate from the results of this calculation for cores A and B are due to neutrons that originated in core C. If the results are normalized to one-fission source neutron, then f^ and /^ are equal to the fission rates of core A and core B, respectively. Then using Equation (6-30), values for 6^ and «? are obtained. To obtain the coupling coefficients from core A to cores B and C and the coupling coefficients from core B to cores A and C, four more MCNP calculations need be performed. The previous example illustrates the expense and difficulty involved in obtaining the coupling coefficients from MCNP directly. To obtain all coupling coefficients, the required number of different MCNP calculations is equal to twice the number of the distinct cores. This process is tedious and can be prohibitively expensive, especially for a reactor system that is composed of a large number of distinct cores. Therefore, an alternate method is highly desirable. gj^^ Obtained Indirectly from MCNP One of the features of the MCNP code is the vast number of user defined particle tallies. An additional feature is the flagging 4 «<

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,W -. 190 capability in MCNP. That is, particles can be "flagged" when they leave designated surfaces or regions. The contribution of these flagged particles to a tally is listed separately in addition to the normal tally. This is used to determine the tally contribution from particles that pass through a specific region. For example, if the user defines a tally to be the neutron flux and fission rate in one of the boiler columns, and if the UTVC is flagged for that tally, then in addition to the total neutron flux and fission rate, the neutron flux and fission rate due to neutrons that passed through the UTVC are also calculated for that boiler column. However, the flagged tallies are not only due to neutrons that originated in the UTVC; they may also be due to neutrons that originated in another region but passed through the UTVC prior to entering the specified boiler column. To obtain the neutron flux and reaction rates in the boiler region due to neutrons that originated only in the UTVC, the contribution from neutrons that originated in regions other than the UTVC and passed through the UTVC need to be eliminated. The following section describes a method, developed as part of this research, for eliminating these second order effects. This method, called Isolator of Secondary Coupling Effects (ISCE), calculates the neutronic transport coupling coefficients (c*^^) indirectly from MCNP using the flagged tallies. Isolation of secondary coupling effects If the interacting cores are separated by "many" mean free paths, such that the probability for a neutron born in core i to pass through or interact with another core other than core j prior to causing fission ..r"

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191 in core j is negligible, then the coupling probability from core i to j, /t^\ is given by JH pJ"^ '^e /J = -L_ . -L(6-31) where F*^^^ is the fission rate in core j due to neutrons generated in core i and F^^^ is the fission rate in core j due to neutrons that escaped the boundaries of core i. The approximation portion of Equation (6-31) is only valid when the probability for the i core neutrons to pass through or interact with any core other than core j is =zero. However, if the probability that neutrons will have an interaction in more than one core is significant, then the fission rate in core j due to neutrons that passed through core i can be expressed by NC pJ^i =pJH ^ Y PJ"^""^ (6-32) where F"^^^ is the fission rate in core j due to neutrons that originated in core k but passed through core i prior to causing fission in core j, and is given by where fi*^^ is the probability for k core neutrons to escape absorption in core k, pass through core i, escape absorption in core i, and then be transported to core j where they cause fission. It is given by the following: i^k d k i^k"

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192 where pL is the probability that the k core neutrons incident on core i will escape absorption in core i, p^j^^ is the probability for the k core neutrons to have an interaction in core j having escaped absorption in core i, and p^t^ is the probability that the k core neutrons incident on core j from core i will cause fission in core j. The product of the first two terms on the RHS of Equation (6-34) (i.e., Pg pi ) is the probability for the k neutrons to be transported to core i given by i^k .k „i^k h_ k Pg Pt = -V (6-35) V where s2 is the rate at which k core neutrons are transported to core i . The product of the last three terms on the RHS of Equation (6-34) (i.e., Pgi^ pi^^ P/k^) ^^ ^^® probability for the k core neutrons in core i to escape absorption in core i, be transported to core j, and then cause fission in core j. This product can be estimated by the following: The last term in Equation (6-36), namely /t^\ is the probability or the effectiveness of the i core neutrons in causing fission in core j. This term is multiplied by aj^ which is needed to correct for the fact that the LHS of Equation (6-36) is the probability for the k core neutrons passing through core i to cause fission in core j, and not for the i core neutrons to cause fission in j. What distinguishes the k

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193 core neutrons from the i core neutrons in core 1 is that the average energy of the k core neutrons is lower. The average energy of the k^ core neutrons is lower than that of the i core neutrons in core i as a result of si owing-down in the moderatorreflector region separating core k from core i. This causes the following: (1) the probability for escaping absorption in core i is lower for the k core neutrons compared to the i core neutrons, (2) the probability to be transported to core j from core i is lower for the k core neutrons compared to the i core neutrons, and (3) the probability for causing fission once in core j is higher for the k core neutrons. The above dissimilarities between the k core neutrons and the i core neutrons are corrected for by a|^. The aj^ coefficient is used to account for the energy importance of the k core neutrons in core i relative to the energy importance of the i core neutrons in core i. One way of estimating a^ is by (6-37) are the k core neutron and the i core neutron scattering rates in core i, respectively. Equation (6-37) gives the ratio of the k core neutron scattering rate per k core neutron in core i to the i core neutron scattering rate per i core neutron in core i . From Equation (6-37), the ai^ coefficient is a measure of the effectiveness of the k core neutrons passing through core i to cause fission in core j relative to the i core neutrons. Due to slowing1 a. =

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194 down in the moderator-reflector region separating core k from core i, the scattering rate per k core neutron in core i is lower than that for the i core neutrons. That is, since the average energy of the k core neutrons in core i is less than the average energy of the i core neutrons in core i, the total number of interactions (e.g., scattering and absorption) that the k core neutrons undergo in core i, before being absorped in core i or before leaking out of core i, is less than that for the i core neutrons in core i. This causes ail to be less than unity, i.e., the probability for the k core neutrons in core i to cause fission in core j is less than the probability for the i core neutron in core i to cause fission in core j. : v Substituting Equation (6-37) into (6-36) and then substituting the resultant equation and Equation (6-35) into Equation (6-33) yields Since f^*'^ is equal to F-^^Vsl, Equation (6-38) becomes ,-jH^k FJ^"" ^i^k H-k ^ ji S' s'*-' (6-39) Substituting Equation (6-39) into Equation (6-32) and solving for F"^"*-^ results in the following: PJH ^ pj-i _ F^ y 3i.k ^ (6.40J According to Equation (6-40), the terms S^"'^, S^'"^ and F^"^ need to be known in order to solve for F'^^\ However, since the ratio ''e'"VSe'"^ is a relatively "good" estimate of fJ^Vs^""^ and since S^"^ is

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195 equal to T^'"'^ A^""^ = T^^*^ A^""^ (T^"'^ and A^'"'^ are the total interaction and absorption rates for the k core neutrons in core i and subscript "e" denotes reaction rates due to neutrons that are flagged escaping the boundaries of the k core), then F]^*'\ Sg*"\ and Sg can be used as initial guesses for F'^*'\ S^*"\ and S^ , respectively. For example, for a four-boiler column UTVR system in which the UTVC contains 235 ?35 «10.2 kg of U and each boiler column contains »3.6 kg of U , values for FJ^VSg""^ and fJ^Vs^"^ are 1.54 and 1.47, respectively. Then, Equation (6-40) can be re-written as follows: Ff , Ff !ill I Sj: , „, .,., (6-41) .«^:: 'II ''" ._ , ..: w . I. where subscript i in Equation (6-41) denotes the iteration number. For the first iteration (i.e., when £=1), the quantities F^^^J and S^^j on the RHS of Equation (6-41) are replaced with F^""^ and S^*""^, respectively, and sJ^J is replaced with S^t^ For £ > 1, the value calculated from the previous iteration for F^"^ is used in Equation (6-41) for F;^^j. Equations (6-42) and (6-43), which are derived in the same manner as Equation (6-41), are used to obtain values for 7;^"^ and A;j*'\ respectively, as follows: T^"^' NC Tf ' = Tf ' -Id I Sj'j V (6-42) aJ"^! NC A= A ^-^ 2. 5p-l • {6-A3) Values for Sj""^ and S;^*"^ are then obtained by using

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sf = Tf' Af : . , ;r^ '" (6-45) where iV"^ and A^J"^ are the total interaction and absorption rates in core j due to the i core neutrons, respectively, and T^^^ and aJ^^ are the total interaction and absorption rates in core i due to the i core neutrons, respectively, obtained from the £ iteration and are given by NC . . kfi AJ^^ = a| X AJ^"^ . (6-47) kfti Equations (6-41) through (6-47) are solved iteratively until converged values for ff'\ sf'\ ij^'V ^f'\ F^, s\^\ Vf\ and A^ are obtained. The coupling coefficients (e**^^) are then calculated using These equations are incorporated in ISCE (Isolator of Secondary Coupling Effects), a code developed as part of this research and described in detail in Appendix C. Other parameters needed for the dynamic neutronic and performance studies are also obtained indirectly from MCNP. These include the neutron multiplication factor of the j core (k^ff)» reactivity of the j core (p'^), and the prompt neutron generation time in the j* core

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197 (A"^). The equations and models used for obtaining these parameters are also incorporated in ISCE and are described in the following sections: Neutron Multiplication Factor of the i^" Core, k;^^^ The neutron multiplication factor, kg^^r, obtained from MCNP is the neutron multiplication factor for the entire reactor system. Since the dynamic neutronic analysis is performed using coupled-core point reactor kinetics models, the neutron multiplication factor of the individual, cores needs to be obtained. To obtain the neutron multiplication factor of the j*^ core, k^^^, neutrons supplied to the j core from all other cores in the reactor system need to be treated as extraneous neutron sources. That is, since k^^^ is the total number of neutrons produced in a generation in the j*^ core divided by the j^" core neutrons produced in a preceding generation, then the effect of neutrons supplied to the j^^ core from all other cores needs to be eliminated prior to obtaining k;^^^. Then, k^^^^r is given by oo
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198 < sJ k=l (6-51) Reactivity of the i^^ Core, d^ The reactivity of the j core, p'^(t), can be considered as the ratio of net production of neutrons to the total production of neutrons in the j core. It accounts for all the interactions that neutrons These interactions include production by fission and losses due to absorption and leakage. The reactivity of the j core is expressed by undergo in the j "core j,.. Production"^ (Absorption"^ + Leakage"^) '' p"'(t) = : . Production"^ In terms of kj^^r^, /»"^(t) is given by f ^/ pHt) = keff(t)-l «k^ff(t) 'eff (t) ^eff^ krrr(t) (6-52) (6-53) Prompt Neutron Generation Time. A"^(t) The prompt neutron generation time of the j core, A"^(t), describes the average time for two birth events in successive . generations, i.e., it is the mean time it takes a neutron to be absorbed causing fission from birth. The j core prompt neutron generation time, A"^, is related to the j core neutron removal time, i^, by the following expression:

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199 AJ(t) = «J(t)/k^^f(t) (6-54) where, £"^(t) is the average time it will take a prompt neutron generated in the j core to be removed from the reactor system by absorption or leakage. The neutron removal time obtained from the MCNP code is the average time it will take a prompt neutron to be removed from the reactor system given by NC . . «sys. ^ J «Js^ (6-55) where i^^^ is the neutron removal time from the entire reactor system and s^^^ is the total number of neutrons generated in the reactor system given by ^ ,^ s^^ = I s^ . ' ' ^ f'^iy(6-56) j=l 7 Results of Density Variations in the UTVC and Boiler Columns The dynamic analysis studies are performed to examine system behavior during full power transients, e.g., core power levels. Since the power levels of the UTVC and boiler columns dictate the amount of fuel present in the UTVC and boiler columns, and since the amount of fuel in the UTVC and boiler columns affects core power level behavior, then, the effect of variations in fuel loadings in the UTVC and boiler U R U B i'«-k columns on /) , p , A , A , and c^ are needed for the dynamic analysis.

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200 Values for these parameters as a function of fuel loadings in the UTVC and boiler columns, as obtained using ISCE, are listed in this section. For this analysis, the dimensions of the UTVR regions are fixed at the values shown in Figure 6-1. The fuel loading in the UTVC is given in terms of an average pressure at an average temperature of 3000 K. The fuel loading in the boiler column is given in terms of the heights SUB of the subcooled liquid region (H ) and saturated liquid region (H ). Unless otherwise stated, the vapor cone region (third region of 235 the boiler column) contains U at a homogenized atom density of 2.1 x -05 10 atoms/barn-cm. Listed in Table 6-10 are the coupling coefficients and reactivities of the UTVC and boiler columns as a function of UF^ partial pressure in the UTVC for two boiler column configurations. The results indicate that as the UF^ partial pressure increases for both boiler column fuel loadings, p and 6^ increase while /) , «^ , €^ , and l^ U B decrease. Table 6-10 also indicates that both A and A decrease. As the UTVC vapor fuel density increases, the neutron mean free path in the UTVC decreases (i.e., the probability that neutrons will have an interaction in the UTVC increases). Thus, neutrons originating in the UTVC are more likely to be absorbed in the UTVC causing p to increase. With respect to the boiler regions, the increased UTVC density acts as a poison to the boiler columns since neutrons born in a given boiler column will more likely be absorbed in the UTVC rather than be reflected back to that boiler column or transported to other boiler columns. Thus, p , ej , and cj decrease. The higher fuel loading in the UTVC enhances the reactivity worth of the UTVC and lessens the

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a> 0) sa. a> .c *O c o u M u a> 0) a> E

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202 ll'«-R reactivity worth of the boiler columns. This in effect reduces eY due to reduced neutronic influence of the boiler columns on the UTVC and increases e^ due to increased neutronic influence of the UTVC on the boiler columns. As the UTVC loading increases, both the UTVC and boiler column neutron removal time decrease. Since p increases, A decreases. Since the reactivity of the boiler column decreases at a lower rate than i is decreasing, A also decreases. However, the decrease in A is greater p than the decrease in A . Table 6-10 indicates that the rates at which fi and e? increase and the rates at which A , A , /> , c* , £? ", and ei*' decrease as the UF^ pressure in the UTVC increases. This is due to reduced influence of the UTVC pressure on these parameters at higher UTVC fuel loading. The effect of varying the height of the saturated liquid cone SAT region (H ) on these integral kinetics parameters is shown in Table 6SUB 11 for two H values of 8.0 cm and 18.0 cm, respectively. The UF^ vapor fuel pressure in the UTVC is fixed at 5.0 atm for both cases. As SAT H increases, the general behavior that is expected, based on physical U R4-U U R considerations, is a decrease in /> , fj , A , and A and an increase in p .||«-R B<-Rn R^-Ro /) , e^ , e^ , and e^ . These expectations are due to the decrease in the neutron mean free path in the boiler columns as a result in increased fuel loadings in the boiler columns. However, Table 6-11 indicates fluctuation in the behavior of these parameters. Similar behavior is observed as H is varied for UF^ pressures of 2.5 atm and 7.5 atm; these results are shown in Table 6-12. ~ n '. "'"" ., t-/" ' " ''•''-

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s 203 .o 0) a> as: a> c o o I o I. a. 3 u > •CQ an rO O) cn (u C -rI to 0) ^ <«

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iO CM V) $m 3 M « C-0 o 0) o c o ** u e vt I I. a. o u : Q) O) . ^ c c • I

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205 The fluctuations exhibited in these parameters can be attributed to the relatively large uncertainties in the quantities obtained from MCNP which are used to calculate these parameters. The reason that fluctuations are observed when the boiler column fuel loading is varied and are hardly observed as the UF^ pressure in the UTVC is varied is due to 1. The relatively large effect that the UTVC generally has compared to the effect of the boiler columns (i.e., on the average, the UTVC contributes »80% of the fission power while each boiler column contributes only «5%). 2. The variations in the UF^ vapor pressure in the UTVC are larger than SAT the variations in H for the boiler column. Varying the UF^ pressure from 2.5 atm to 5.0 atm and then to 7.5 atm reflects fuel loading variations in the UTVC of 100% and then 50%, respectively, CAT while varying H from 10 cm to 30 cm and then to 50 cm reflects a fuel loading variation in the boiler column of «70% and then «40%. Table 6-13 lists the effects of varying H^^^ and H^'^^ at a UF^ vapor pressure of 5.0 atm in the UTVC. The results indicate that as the U B'«-U B fuel loading in the boiler column increases, p , f* , and A decrease while /) , and c^ increase. The results also indicate that fluctuations occur in the behavior of A , £^ , and i^*'°° as the fuel loading in the boiler column increases. ^ > While maintaining the UF^ vapor pressure at 5.0 atm in the UTVC and H^^^ and H^'^^ at 8.0 cm and 40.0 cm, respectively, the U^^^ density in -05 the vapor cone region is varied from 1.5 x 10 atoms/barn-cm to 3.1 x -05 10 atoms/barn-cm. The expected results, based on physical

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i-' 206 < CO •a c CO oo O 0) l*-

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207 considerations, are decreases in p and £^ and increases in p , cL , £? , and £°^°° as the vapor cone fuel content increases. However, the results, as shown in Table 6-14, indicate fluctuations in the behavior of these parameters. The fluctuations exhibited in these parameters are also attributed to the relatively large uncertainties in the quantities used to calculate these parameters.

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208 O)
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CHAPTER VII KINETIC EQUATIONS OF A FOUR-BOILER COLUMN UTVR SYSTEM Introduction The Circulating-fuel, Coupled-core Point Reactor Kinetics (CC-PRK) equations for a four-boiler column UTVR are derived in this chapter. This is followed by a section discussing inherent reactivity feedbacks and the associated energetic equations of the UTVC and boiler columns. These equations are used in constructing the model used in the dynamic analysis studies presented in Chapter VIII. The UTVR inherent reactivity feedback, such as vapor fuel density and boiler column liquid volume changes, are included in the dynamic model. The Four-Boiler Column UTVR System Coupled Core Point Reactor Kinetics Equations The CC-PRK equations for a four-boiler UTVR configuration are derived in this section. Since the derivation of the boiler column equations is almost identical to that of the UTVC, only the derivation of the UTVC CC-PRK equations is shown. The boiler CC-PRK equations are, however, presented at the end of this section, and where applicable, the differences between the UTVC and boiler column equations are noted. The equations governing the time dependent behavior of the neutron population level in the j core and the time dependent variation of the concentration of the i delayed neutron precursor group in the j core 209

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210 for a circulating-fuel reactor system composed of NC cores are derived in Appendix D. These equations, namely Equations (0-26) and (D-4), are j ND . d_NJ(t) = ^eJlLli NJ(t) . X XiC^(t) <^* AJ(t) i=l 't ^^^ : — hr k^'j AJ{t) A'^(t-Tj ) , J 7J-^(t) "^' ^ t ' (7-1) 1 cJ(t) = -f^ NJ(t) Xi C^{t) (1 / rj) e^(t) where N^(t) * neutron population level in the j core at time t; pj(t) = j*" core reactivity at time t; effective delayed neutron fraction of the i delayed neutron ? » effective delayed neutron fraction; X. ' decay constant for the i^ delayed neutron group; AJ(t) = prompt neutron generation time of the j core at time t; t^t) ' effective delayed neutron precursor concentration for the i delayed neutron group in the j core at time t; £J'"k(t) = effective coupling coefficient from the k^^ core to the j*" core at time t; ri*"^ = delay time for the transport of neutrons through the media from the k^^ core to the j^" core;

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211 t;^*™ = effective delay time for the fuel flow transport of precursors from the m core to the j core; Tj* , effective precursors residence time in the j* core; /"^*™ = m core and the j core loop connection coefficient; equal to the fraction of fuel flowing from the m* core to the j*^ core; , NO ' number of delayed neutron groups; NC number of cores in the reactor system. To obtain the CC-PRK equations for a UTVR with four boiler columns, the third term on the RHS of Equation (7-1), which describes the coreto-core neutron coupling by means of neutron transport through the media, and the third term on the RHS of Equation (7-2), which describes the core-to-core neutron coupling by means of fuel flow, need to be modified, as follows: J'" IT ' i ' . : ' }' ' ' ' , » -,.'-.(' ' Core-to-Core Fuel -Flow Coupling v *. 4.' ; :. For the reactor system under investigation, there are two types of fissioning regions: the vapor core region and the boiler region that consists of four boiler columns where the fuel/working fluid mixture is vaporized. Shown in Figure 7-1 is a schematic of core-to-core circulating fuel coupling. Figure 7-1 indicates that each boiler column receives only 1/4 of the fuel/working fluid mixture (which contains the delayed neutron precursors) and the UTVC receives the fuel/working fluid mixture from all four boiler columns. It is also assumed that all boiler columns are at the same power level and are behaving and have behaved in an

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1

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.V 213 identical manner. Because of this assumption, this treatment cannot handle any imbalance among the boiler columns. Additionally, it is assumed that the i delayed neutron precursor concentration exiting any boiler column at time t are delayed by the same amount of time, namely, T« , where Tp is the delay time for the transport of fuel from any boiler column to the UTVC. With the above assumptions, the third term on the RHS of Equation (7-2) describing the inflow of the i delayed neutron precursor group to the UTVC from the boiler columns becomes (4/r^)e?,t-rr^e-'''""' . ,. : (7-3) For any boiler column, the third term on the RHS of Equation (7-2) becomes '. ; " B'H) (I/4r°)e';(t-rP)e''' . ' • (7-4) B , *U,^ BMJ, "^i "^i Using Equation (7-3) in (7-2) and using Equation (7-4) in (7-2) yields U^B ^c^(t) = ^N^(t) :(A,.iA^)e^(t) .i^e?(t-rr) (^-^) ^* A"(t) t" , , -, ., ^-^ '-„/: ::';^^" "• • ' ^ C?(t) AnB,!) (Vi.l/rB) cB(t) . lli^ cV-r) • ('-^' "^ AB(t) . , , . 4 7^ Equations (7-5) and (7-6) describe the time dependent behavior of the effective delayed neutron precursor concentration for the i

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214 delayed neutron group in the UTVC and any boiler column, respectively. Equations (7-5) and (7-6) have the following added assumptions: 1. Precursor transport through the connecting loops is a pure time delay. No fissioning occurs in the fuel outside the UTVC and boiler columns. 2. Slug flow outside the core (in the loop) is assumed. Core-to-Core Neutron Transport Coupling Shown in Figure 7-2 is a schematic of the neutron transport coupling from the four-boiler columns to the UTVC. Under the assumption that the four boiler columns are always producing the same amount of power, then the power and neutron levels are the same for all boilers at any time. As a consequence of the above assumption, and due to the symmetric alignment of the boiler columns around the UTVC, as shown in Figure 7-2, the following are true: 1. The effective boiler-to-UTVC neutron coupling coefficient, fj (t), is the same for all boilers. That is -U*€,^, -U^Bl,^, -U^B2,., -U^B3,., ,, ,, £^ (t) = €^ (t) = e^ (t) = €^ (t) . (7-7) 2. If an average delay time for the boiler-to-UTVC neutron transport process is used, then the transport delay times, t^ , from each boiler to the UTVC are all equal and all transport or directlyLI+-B coupled neutrons generated in any boiler column at time t-Tx which ultimately cause fission in the UTVC, enter the UTVC after being delayed a time t^ .

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G .U*B3 .U*B3 Boiler 83 Boiler Bl 215 o .Ih-Bl U*B1 Boiler B2 Boiler B Figure 7-2. Schematic of Boiler-to-UTVC Neutron Transport Coupling for a Four-Boiler Column UTVR System

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216 The above imply that only one neutron transport coupling term is needed to account for the boiler columns' neutron contribution to the UTVC. This term is "' N«(t-rP) . (7-8) * ^t ^^^ ,.B.. U-B A^t) It is assumed in Equation (7-8) that A^{t) » A^(t-T^^^). Then, the equation describing the time dependent behavior of the neutron population level in the UTVC is -U^B, ^t < A"(t) A^{t) ' 1=1 ^ N",t, = l<^.^t) . !!l-^ NB,t -.;;'«, . I A,e^(t, . (7-9, Equation (7-9) assumes that six delayed neutron precursor groups are required for describing the effect of the delayed neutron precursor decay. To obtain the equation describing the time dependent behavior of the neutron population level in the boiler column, the following needs to be done to Equation (7-9): 1. For all terms but the second term on the RHS, replace all superscripts "U" with "B" and "B" with "U." 2. For the second term on the RHS, refer to Figure 7-3 where it is noted that boiler B is receiving neutrons by means of transport from the UTVC and boiler columns Bl, B2, and B3. Since one of the assumptions is that all boiler columns are always producing the same amount of power, and since columns B2 and B3 are symmetric with respect to core B, then at time t, cores B2 and B3 supply boiler B

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217 .^k-1^: Boiler Bl Boiler B2 Boiler B Figure 7-3. Schematic of Boiler-to-Boiler and UTVC-to-Boiler Neutron Transport Coupling for a Four-Boiler Column UTVR System Via 2 t

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218 the same number of neutrons that were produced at time t-r? or at p^po R'«-R1 time t-T? . For boiler Bl, neutrons produced at time t-T^ cause fission in boiler B at time t while for the UTVC, neutrons Bm produced at time t-r^ cause fission in boiler B at time t. With the assumptions that A^^(t) « A^(t-Tj^^^), /fi^{t) « A^Ct-rJ^^^), and U U B+4J A (t) « A (t-r^ ), then the terms that account for neutron transport through the media from the UTVC and from the Bl, B2, and B3 boiler columns to boiler B are the following: \-B^B2,., „B,. B^B2, -B^Bl,., ^B,. B^Bl, 2£^ {t)ti°{t-T^ )+ €^ (t)N''(t-r^ ) + f^. (t) N"(t-T^ ) /^ (t) . (7-10) Employing the above modifications to Equations (7-9) and replacing subscript "Bl" with "Bo" and subscript "B2" with "Bn" to denote adjacent and opposing boilers, respectively, yields _ -B^Bn -B^Bo ^* h^{t) A^Ct) A^Ct) A (t) i=l Steady-State Solution To obtain the steady state conditions for the UTVC and boiler columns, the UTVC and boiler columns are assumed to be in subcritical equilibrium (p^ ' P^ < and p^ » pg < 0). Subcritical equilibrium is achieved in the UTVC and boiler columns through the steady state neutron

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;".{*; 219 sources, S^ and S^. S^ is the steady state neutron source supplied to ' the UTVC from all boiler columns by means of neutron transport through the media, S^ , and the transport of the delayed neutron precursors from the boiler columns through the connecting loop, Sp ; and S° is the steady state neutron source supplied to any boiler column by means of neutron transport through the media from all other boiler columns and from the UTVC, 2 sj^^^ (where i» 1, 2, and 3) and S^^, and from the transport of the delayed neutron precursors from the UTVC through the connecting loop, S^^V. Setting Equations (7-5), (7-6), (7-9), and (7-11) to zero, as a first step in obtaining the steady state values for the neutron population and precursor concentration in the UTVC and boiler columns, yields the following: --U^B U f. U ^^0 B A ° ^y ^0=^^^^ I/i^io (7-12) „ ' . U-B fl. T "^i ^p e" .-JiI^n" . JLi_!_e» (7-13) 10 U u ° U 10 Ao(l+AiT^) (l+A^r^) B^Bn -B^Bo -BHJ A® ^ N„ = N„ + N^ + > \i C.^ (7-14) _R 0_rO _R'^'10 ^ ' B 1 r^^ e? = — -E N^ + -I e*:* (7-15) 10 D R Rio A^d+AiO 4(l+AiT)

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''^ , . ",''. , 220 where the subscript "o" denotes values at a steady state condition. Solving for C^^ and 0?^ in terms of N^ and N^ by use of Equations (7-13) and (7-15) yields U B .:;; B U^B 10 u Ob ' ' .^ ^0 ^oi ^0 ^oi and B,, , U, U , BMJ C. = — _S -SN„ + — e *^ n" (7-17) 10 B ° U ° Aq ^oi 4^0 ^oi , , B^ U-B, U B, -^i(''£ ^ ^0 ) ... (7-18) where S?oi = d ^^i^c^(l + ^i t e * ,^ , 5, . f ' ' ? ' \ » Substituting Equations (7-16) and (7-17) into Equations (7-12) and (7-14) and solving for N^ and N^ yields • ' ' n"=4-? ° °^ N^ (7-19) ^ ^0 ^c "^ and -BHJ , ,B,. ,U, U ,BMJ N„ = n" (7-20) B ^-B^Bn -B^Bo B „U ° ^-^-2^0 -^0 -^c^^ where K^ = J ^ (l+^iT^) (7-21) 6 , in^j Lin-j ^ y ^i^i g"^i ^« (7-22) ^
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221 U B required. A value for N^ or N^ is selected that corresponds to the UTVC or boiler column steady state power level by use of ?l = ( dJ„ NJ (7-23) where P^ and N^ are the j core steady state power and neutron population levels, respectively, ^ is the appropriate conversion factor, and d^ is the density of the fuel in core j at the steady state B U condition. Then a value for N^ or N^ is calculated from Equation (7-19) or (7-20). Then by use of Equations (7-16) and (7-17), values for cV and C^„ are obtained. \ 10 i "^ ' , ' ' The Linearized UTVR CC-PRK Equations It is customary to obtain the system response to small external and/or internal perturbations as a procedure for examining the system's inherent stability [34-36]. If such perturbations are sufficiently small, then the response of the system (reactivity, neutron and power levels, temperature ...etc), due to these perturbations, can be adequately determined by using linear approximations. These linear approximations greatly reduce the expense associated with repeating complicated calculations. For perturbation about some initial equilibrium condition, the following relations can be written: N^t) = N^ + 5N"(t) (7-24) N^(t) = Nq + 5N^t) (7-25) pU(t) = ,;; . 5pU(t) (7-26)

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222 e';(t)= C^„. «>) (7-28) C?(t)= C^„. «f(t) . . (7-29) where 8 denotes fluctuation of a parameter about the initial steady state condition. Substituting the above relations into Equations (7-5), (7-6), (7-9), and (7-11) and eliminating the steady state terms by use of Equations (7-12) through (7-15) yields the following non-linear CCPRK equations for the UTVC and boiler columns: L 5N^t) = A. 5pU(t) . 'll SH^it) . ifVli!^ ^^ A^t) A^t) A^t) A^(t) i=l 4:iif1 ^ U £_5cV(t) = — — sr{t) scJt) V M4/r^)e"'^'"^'6C?(t-r^^^ . [ " (7-31) B B — L SHht) -!!^ «/(t) . '-^ SH\t) . '""^'^ ^""'t) "** A8(t) A^(t) A^(t) A"(t) A^Ct) .I^5NB,t-xr°) * I MSt'.W (7-32, A^(t) i=l

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223 ^ .C?(t) . J^ ^NB(t) '^''''-« dt , BMJ + (1/4 T) e B 6C.(t) 6c:{t-Tg^) (7-33) It is assumed in Equations (7-30) through (7-33) that l^^{t) « l^^ and AJ(t) « AJ. Great emphasis has been focused on linearfeedback theory for stability and control analysis. As a consequence, powerful methods of linear-feedback theory for stability and control analysis have been developed. If the various terms in Equations (7-30) through (7-33) are linearized, then the use of the well -developed methods of linear feedback theory such as Root Locus, and Bode and Nyquist plots is possible. Therefore, if all perturbations about the initial equilibrium condition are small, and if the UTVR is at full power or at a level such that N (t) and N (t) are large relative to the perturbations 6N (t) and 5N (t), then the following relations can be written: 5p^(t) 5N"(t) I (( p^ 5N^(t) 5p^(t) 6N^(t) «/(t) 6N^(t) ({ « No Sp^{t) pI 5N^t) 6p^(t) 5N^(t) I « \w\ 5p^(t) (7-34) (7-35) (7-36) ., ., (7-37) If the above relations hold, then the stability of the UTVR system can be examined by using the methods of linear-feedback theory in the

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224 Laplace and frequency domain. This is advantageous because the methods of stability analysis for a linearized systems of equations are easily applied in the frequency domain; Equations (7-30) through (7-33) can be converted from a set of 14 differential equations to a set of 14 linear algebraic equations (six equations for the six delayed neutron precursor groups and one equation for the neutron population for each core) by neglecting second order perturbation terms by the use of Equations (734) through (7-37). Neglecting second order perturbations in Equations (7-30) through (7-33) and transforming these Equations from the time domain to the Laplace domain yields 5N"(s)=_^5pU(s) ^ LJ. _6N^s) + -±Y h^t%) (7-38) (s-Uq) (s-Uq) (S-Uq) ^-^j 1 II U^B 5e"(s) = -—16n"(s) . (4/r^ ^St]{s) (7-39) 1 (s+u^-) c (s+u^) ^ BHJ b B "^''t 6 6N^(s)= ^—Sf>^{s) + L-1 5N^(s) + ! y Ai5C?(s) (7-40) s-b(s) s-b(s) s-b(s) -ti ^ *S^^) = 7— !-r ^N^^s) . l/(4r^) \ St\{s) . (7-41) ^ (s+bi) c (s+bi) 1 The coefficients that appear in the above Equations used to reduce the complexity of analytical representations are defined as follows: bi =^i +1/V 5-^^. ,.,.., .. bi =N^/a^

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225 U u Si = Ai + s, Uq =(Pq-^)/A", € =€^ /h , ^^ =Pi /a"*, ,U ,7U*-B / U oU /,U B*-Bn B^Bo B <,-B^Bn -ST't -B-Bo "^''t , , ,B ,-, s.y^ and b(s) ={^0 -^ ^2£t e + £^ e j/A'^. (7-42) The initial conditions in Equations (7-38) through (7-41) are set to zero. Transfer functions demonstrate characteristics information about the system and, therefore, are independent of initial conditions. Solving for 5cV(s) and 5C?(s) from Equations (7-39) and (7-41) yields B U.U . B.B [}^ ~U ''c^'c^i U *'^c^i -^i''£ B Jci(s) *'i(sj B U ^B U .U BHJ «B '^c'^cP\ R ^c^'i "^i'"; II B B U U -(s+Ai)(Tfl +T« ) ?i(s)=(lH-T^-Hr^i)(l+r"s+T"Ai)-e / « .(7-45) Substituting Equation (7-43) into (7-38) and solving for 5N (s) in terms of 6p^(s) and 6N^(s) yields 5N^(s) =|ui Uo(s)|6p^s) + |[Bt"^s)+ B^'^s) Uo(s)|5N^s) (7-46) where

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226 Uo(s) = Brcs) 6 \ o f s -uo(T^/A") X _— 1 + r^ s-Ai -1 -U-B ^-s''t ^^U «" e ^ = 4 6^ e B^^s)=(4r^VA'^) I B,.B. ^ ^i^i ^-(s+Ai)T^ U^B i=l «i(^) (7-47) (7-48) (7-49) Equation (7-46) yields the fluctuation in the neutron population level of the UTVC as a function of fluctuation in UTVC reactivity, neutron transport coupling, and fuel flow coupling. The -juj Uq(s)[term is the UTVC zero power reactivity transfer function; -JB^ (s) Uq{s)[^ 5N (s) is the UTVC response due to fluctuation in boiler-to-UTVC neutron transport coupling (neutron transport source term); -JB^ (s) Uq(s)[' 5N^(s) is the UTVC response due to fluctuation in boiler-to-UTVC fuel flow coupling (precursor fuel flow source term); and -{uj Ug(s)[5p (s) is the UTVC response due to fluctuation in UTVC reactivity (external and/or internal). ' ; t'': ' ''''• * : There are two terms in Equation (7-46) that affect the response of U B the UTVC, namely 5p (s) and 6N (s). One of these terms can be viewed as a forcing function and the other can be viewed as a disturbance function. Since Equation (7-46) is for a linear system, then the principle of superposition applies. Thus, the response 5N (s) due to U B the influence of both 8p {s) and 5N (s) can be obtained by summing the responses that occur when these terms are separated. That is 6n"(s) = 5n"(s) + 6Sg(s) *v,>, . (7-50)

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where 5N^(s) = { ui Uo(s)}5/)^s) Uo(s)} 6N^(s) . 227 (7-51) (7-52) A block diagram for the UTVC transfer functions is presented in Figure 7-4. Substituting Equation (7-44) into (7-40) and solving for 5N (s) in terms of 8p^{s) and 6N^(s) yields 5N^(s) = Ul Bo(s) j 6/(s) rr.BHJ, , ,,BHJ, . |[Ut (s)^U^ (s) Jo(s)j 6N^(s) (7-53) where Bo(s) = • s -b(s) (T^/A^) X ^^ 11+^ BHJ i^ «i(s) BMJ c(-^i)) -1 U?^(s) = 6'^ e -ST. B -'"t = ??^ e""t /4aB i=i ^ivs; r , . *«/. .. V. , *'r. (7-54) (7-55) (7-56) Equation (7-53) can also be partitioned into boiler column reactivity and source (from neutron transport coupling and fuel flow coupling) transfer functions as follows: 5N^s) = 5N^s) + 5sJ(s) where 5N^s) = { bj Bo(s)} 5p^(s) (7-57) (7-58)

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228 ^k _^

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229 «sg(s) . {[u^-^(s) . il^U) j B„(s)} 6n"(s) . . (7-59) A block diagram for the boiler column transfer functions is presented in Figure 7-5. Figure 7-6 presents a block diagram showing the interaction between the UTVC and boiler columns along with the inherent reactivity feedback of the UTVC (U ) and boiler columns (B ). Inherent feedbacks of the UTVR are discussed in the following section. Although a linear form of the UTVR's CC-PRK equations are derived in this section, the dynamic analysis in this research is restricted to examining the transient response of the UTVR's non-linear model. Since results obtained from linearized models can often be useful, stability analysis using the linearized CC-PRK equations are recommended for future work. Inherent Reactivity Feedbacks of the UTVR Reactivity feedback is the phenomenon that occurs when the state (reactivity) of the reactor system is altered as a result of a power level change. This alteration will affect the nuclear characteristics of the system which in turn will modify the original reactivity. Thus, inherent reactivity feedback is the response of the system to the effects of power level changes. Inherent feedback is either prompt or delayed. Feedbacks are considered prompt if they occur essentially instantaneously with no time delay after a perturbation, such as the fuel temperature reactivity feedback. Feedbacks are considered delayed if the physical changes that

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230 ;«• , I M I— i o eao^ rO: M a

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231 ., f:r:e o u c 3 0) (4o u o i3

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232 affect the reactivity when the system is perturbed occur after a time delay, such as the moderator temperature reactivity feedback. The reactivity of the j core, />"^{t), can be expressed by the following: . ^ pj(t) = pI + Spirit) + 8p\^^it) (7-60) where p^ « reactivity of the j core at some steady state; 5p;^ (t) = externally imposed reactivity on the j core at time t. This may be viewed as a perturbation imposed on the j core by adding positive reactivity such as control rod withdrawal or adding negative reactivity such as control rod insertion; 5/)^l^(t) inherent reactivity feedback of the j core at time t in response to a power fluctuation. It includes e.g., temperature and density feedbacks. The inherent reactivity of the j core can be expressed by Sp\^{t) = 6/)j(t) + Sp^j^it) + 5/)^„{t) (7-61) where 5pi(t) fluctuation in the j core reactivity due to fluctuation in the mass of the fissile fuel present in the j core; 5PT^(t) = fluctuation in the j core reactivity as a result of Doppler broadening due to fluctuation in the average fuel temperature of the j core; 5p^M(t) = fluctuation in the j core reactivity due to fluctuation in 'TM' the temperature of the moderator surrounding the j core.

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233 In the sections that follow, inherent reactivity feedbacks of the UTVC and boiler column regions are discussed. Energetics equations associated with the UTVC and boiler columns are also derived in the following sections. In deriving the energetics equations of the UTVR, the following assumptions are made: 1. Pressure losses due to friction, boiling, and shock (flow area contractions and expansions and restrictions) are neglected. 2. Fuel/working fluid inlet pressure (P?-) and temperature (T^p) to the boiler columns are fixed. D Reactivity Feedback of the Boiler Column, Sp {t) ;,>, , Shown in Figure 7-7 is a schematic of the boiler column model that is configured to allow for boiling in space {»zero gravity). According to Figure 7-7, the boiler column is composed of three distinct regions: the subcooled liquid region, the saturated liquid region, and the vapor cone region. R The total power produced in the boiler column at time t, Pj(t), is P^(t) = p5"^t) ^ pS^^Ct) ^ pVAP(t) . (7-62) where Clip P (t) power produced in the subcooled liquid region at time t; P (t) » power produced in the saturated liquid region at time t; VAP P (t) = power produced in the vapor cone region at time t. The power produced in the boiler column at time t can also be expressed by

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234 "TK H^^^(t) 7\ HS"B{t) 2iL hut^^^ r, ^ Vapor Cone ^Region Saturated Liquid Region Subcooled Liquid Region Figure 7-7. Fuel/Working Fluid Density Profile in the Boiler Column due to Boiling in Space {»zero gravity)

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235 P?(t) = P^o H5P?{t) (7-63) D where Pjq is the initial steady-state power level of the boiler column D and 6Pj{t) is the fluctuation in power level of the boiler column at time t. The fluctuation in the boiler column power level can be due to externally imposed reactivity changes, such as control rod withdrawal or insertion, and/or due to inherent system response. D Power fluctuations in the boiler column, 5Pj(t), induce the following: 1. The mass of fuel/working fluid in the boiler column fluctuates due to changes in the volumes (heights) of the liquid regions and due to changes in the density of the fuel in the vapor cone region. 2. The outlet and average fuel temperature fluctuate. 3. The temperature of the moderatorreflector region surrounding the boiler core fluctuates. Then, the boiler column inherent reactivity feedback due to power fluctuations can be expressed by 6p\^{t) = 6p^^{t) + 8py{t) + Spj^it) (7-64) or Sp]^{t) = 4f6f\^^{t) + aT^5T^(t) + aTM«TM(t) (7-65) where e^f = boiler column fuel loading coefficient of reactivity; D SHAt) = fluctuation in boiler column fuel loading at time t;

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236 Oj, = boiler column fuel temperature (Doppler) coefficient of reactivity; , <* * • 5T^{t) = fluctuation in the boiler column fuel temperature at time t; ttj « moderator temperature coefficient of reactivity; 6T (t) = fluctuation in the moderator temperature du6 to boiler column power fluctuations. ^/ A power fluctuation in the boiler column produces a fluctuation in the temperature of the fuel only in the vapor cone region. That is, since the inlet and saturation temperature of the fuel/working fluid are fixed, then the average temperature of the fuel in the liquid regions is fixed. Consequently, a boiler column power fluctuation will alter the temperature of the fuel only in the vapor cone region. However, due to 19 235 3 the relatively low density of the vapor fuel («10' U atoms/cm in the vapor cone region) and the high enrichment of the employed fuel 235 («85% U ), the temperature (Doppler) coefficient of reactivity of the vapor fuel is quite small {a^r » -10 ^'^eff^'^eff ^^^ ^ U])Therefore, reactivity feedbacks due to fluctuations in the boiler column fuel temperature are neglected. Reactivity feedback due to fluctuation in the temperature of the moderator-reflector region surrounding the boiler column is also neglected in this research. This assumption is based on the following considerations: "^ >^v^ Because the temperature of fuel/working fluid fluctuates only in the vapor cone region, and due to the low density and the high velocity of the vapor fuel , the rate of heat transfer to the moderatorreflector region from the vapor cone region is relatively small.

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237 Then, fluctuations in the temperature of the moderator region are also small. Additionally, the time required for fluctuations in the moderator temperature to produce an appreciable change in the boiler column reactivity is large {»100 sec), especially when compared to the time duration for which the transient is to be followed («10's of sec or less). The dominant reactivity feedback in the boiler column arises from changes in the amount of fissioning-fuel present in the boiler column. Changes in the fuel loading are due to variations in the volume (height) of the non-boiling regions and variations in the density of the fuel in the vapor cone region. The ultimate effect of these fuel loading changes is a variation in the fission rate due to an alteration in the B B macroscopic cross section of the boiler column, 2^. Alterations in 2^ are caused by changes in the boiler column microscopic cross section. Op as a result of variations in the self-shielding and changes in the B B density of the fuel/working fluid, N^ in the boiler column. Then, 5pj« can be expressed by 5p?N(t) = 5p^{t) = Of]5^ 5M^(t) . : \ (7-66) A power fluctuation will cause a variation in the amount of fuel in the boiler column due to SUB 1. Fluctuation in the subcooled liquid region volume, V (fluctuation SUB in the subcooled liquid region height, H ). SAT 2. Fluctuation in the saturated liquid region volume, V (fluctuation SAT in the saturated liquid region height, H ). VAP 3. Fluctuation in the volume of the vapor cone region, V .

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238 4. Fluctuation in the average density of the fuel in the vapor cone region, dj. Then, the fluctuation in boiler column reactivity feedback due to fluctuations in the amount of fissile fuel present in the boiler column, D ^fi^A^)* can be expressed by 5pg^{t) = SpI^ H^%t),H^^t),d5(t)] . (7-67) That is, pL(t) is a function of H^^^(t), H^'^^Ct), V^^'^(t), and 6 dy(t). The equations needed to relate the heights of the subcooled and saturated liquid regions and the average fuel density in the vapor cone region as function of the boiler column power level (or neutron population level) are derived below. D In terms of the boiler column neutron population level, N (t), or B B the boiler column neutron density, n (t), Pj(t) can be expressed by P?(t) = ^ d/(t) N^t) = e V^ d^(t) n^t) (7-68) Dp where ^ is the appropriate conversion factor and d^(t) and V are the density of the fuel/working fluid mixture and volume of the boiler column, respectively. Similarly, the power levels in the subcooled liquid, saturated liquid, and the vapor cone regions are P^"^t) = e dj N^^^t) = e d^ V^^^t) n^^^(t) (7-69) P^'^^(t) = ^ d^ H^'^'^it) = e d^ y^^\t) n^A^(t) .! V; (7-70) P'^^t) = e d^t) NVAP(t) = e dS(t) vVAP(t) nVAP(t) '" (7-71) i^ »f i-:'-*.i : •: ^ .'-. rf^c

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239 where superscripts "SUB," "SAT," and "VAP" denote quantities for the subcooled liquid, saturated liquid, and vapor cone regions, D respectively; d£ is the average density of the liquid fuel/working fluid in the liquid regions of the boiler column (assumed to be constant and B the same for both liquid regions) and dy(t) is the average density of the vapor fuel/working fluid in the vapor cone region of the boiler column at time t. A power fluctuation in the boiler column region can be caused by a fluctuation in the neutron population level in the boiler column region. In order to obtain H^^^(t), H^'^^Ct), and djct) as a function of the boiler column power level, fluctuations in neutron flux depression in the liquid regions are neglected. That is, although a power fluctuation in the boiler column causes the volume of the fuel in the subcooled and saturated liquid regions to fluctuate, which in turn causes the axial neutron flux profile (shape function of the angular neutron flux ^(j;>E,Q,t)) in the liquid regions to fluctuate, the time dependent behavior of ift(jj,E,Q,t) is neglected in this research. Fluctuations in the radial neutron flux profile at a given z, which are caused by fluctuations in the density of the fuel in the vapor cone region of the Ol ID OAT boiler column and/or by changes in H and H , are also neglected in this research. The assumption of constant neutron flux profiles (i.e., neutron density profiles) in the boiler column is consistent with the assumption upon which the point reactor kinetics equations are based, i.e., that l^(»;,E,Q,t) is not changing drastically with respect to time. These assumptions imply that all regions in the boiler column and the i w •' '

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240 boiler column as a whole must all have the same relative time dependent behavior of the neutron density, i.e., nV) B SUB,., n (t) SUB SAT... n (t) SAT VAP,., n (t) VAP (7-72) where the subscript "o" denotes values at a steady state condition. Solving for nS^^{t), n^^^Ct), and n^^^'ct) from the above expression in B terms of n (t) yields SUB n"-{t) = SAT n (t) n^^^^t) = SUB, B ' SAT, B ' VAP, B n^(t) n^t) n^(t) Substituting Equation (7-73) into (7-69), (7-74) into (7-70), and into (7-71), yields pSUB(t, . „SUB vSUB(,j ^Bj,, pS'^^(t) = w^^^ vS^^(t) n^(t) pVAP(t) = (jVAP' jjB^tj V^^^(t) n^(t) . SUB . .B SUB, B where w "'^ = ^ d^ n^ /n^ ^„ , SAT . .B SAT, B w = e Dq /n^ 7-73) 7-74) 7-75) 7-75) 7-76) 7-77) 7-78) 7-79) 7-80) 7-81) The time rate of change of the energy content in the subcooled liquid region can be expressed by

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241 1 E^UB,,, ^ pSUB(,, , d_ ^SUB,^, _ d_ ^SUB,^, _ -SUB,,, ^ ^SUB,,, ,,.33, where SUB E (t) = energy content of the fuel/working fluid mixture in the subcooled liquid region at time t; SUB EQjjJ(t) = energy content of the fuel/working fluid mixture exiting the subcooled liquid region at time t; SUB E^Jj (t) = energy content of the fuel/working fluid mixture entering the subcooled liquid region at time t; SUB W (t) = work done on the fuel/working fluid mixture in the subcooled liquid region at time t; SUB Q (t) = rate of heat transfer out of the subcooled liquid region to the moderatorreflector region. ^ In this research, heat transfer across the internal boundaries separating the subcooled liquid region from the saturated liquid region is neglected. However, the rate of heat transfer to the moderatorCI ID reflector region surrounding the subcooled liquid region, Q (t), is assumed to be given by ' ''')' •SUB SUB SUB SUB ,, „-. Q (t) = /q A^^^t) P^^'^Ct) (7-83) SUB where /q » the fraction of power transferred to the moderator-reflector region per unit surface area, is assumed to be constant and A^ (t) is the surface area of the subcooled liquid region in contact with the Clip moderator-reflector region. With reference to Figure 7-7, A^ (t) is AfV) = 2irri (zj zj . 2^r2 (zSUB(t) -zj) . (7-84) t'M-t -"k. »>*»' il ^ " (." V j' 'i

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242 and V^^^(t) is vSUB(t)= .rl(z,-z^)..rl(z''\t)-z,) . (7-85) Solving for z^^^(t) from Equation (7-85) yields zSUB(t) = zi (zi -Zo)(ri/r2f ^ V^^^t) / ,r r^ . (7-86) Substituting Equation (7-86) into (7-84) yields A^V) = 2^ri (zi -Zo) (l riAz) 2vSUB(t)/r2 . , (7-87) In applying Equation (7-82), the types of energy associated with the fuel/working fluid within, entering, or exiting any UTVR region (i.e., boiler column or UTVC) due to mechanical, electrical, magnetic, and gravitational field effects are neglected (i.e., W(t)= 0). Thus, the types of energy associated with the fuel/working fluid within, entering, or exiting any UTVR region are restricted to internal, kinetic, and heat transfer forms. However, in the liquid regions of the boiler column, the magnitude of the kinetic energy changes are negligible when compared to internal energy changes. For example, the ratio of the change in kinetic energy to the change in internal energy of the fuel/working fluid in the subcooled liquid region is less than =3 x 10" for the reference boiler column configuration at steady state conditions. Employing these restricitions to Equation (7-82) yields: 1 U^^^t) = r5"B,t, pSUB,t, , ,B_^ ,B_^(„ _ ,B .SAT,,, . ,,.33, ; where

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243 SUB U (t) = internal energy content of the fuel/working fluid mixture in the subcooled liquid region at time t given by uS"B(t) = )"B H]"^t) = ^'"' of yS'-^t) (7-89) or 1 uSU^Ct) = f^ d? 1 vS"B(t, (7-90) dt •' * dt SUB SUB where ^ and M^ (t) are the average specific internal energy and mass of the fuel/working fluid mixture in the subcooled liquid region, respectivley; Clip r (t) » fraction of fission power remaining in the fuel/working fluid mixture in the subcooled liquid region at time t given by m^ (t) = mass flow rate of the fuel/working fluid entering the ^ subcooled liquid region (equal to the mass flow rate of the fuel/working fluid entering the boiler column) at time t; m (t) = mass flow rate of the fuel/working fluid exiting the subcooled liquid region (equal to the mass flow rate of the fuel/working fluid entering the saturated liquid region) at time t; h^ specific enthalphy of the subcooled liquid fuel/working fluid mixture entering the subcooled liquid region; p h« specific enthalphy of the saturated liquid fuel/working fluid mixture exiting the subcooled liquid region. Substituting Equations (7-90) into (7-88) yields r

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SUB .B d ,,SUB/4., SUB,4.> nSUB/4.x u^ B ... .B .B ... 244 (7-92) The mass flow rate of the fuel/working fluid exiting the subcooled liquid region, m (t), is given by •SAT,^, .B ,^, d uSUB,., .8 ... .B d .,SUB,xx m (t) = m.^{t) M^ (t) = m.^Ct) d^ ^ V (t) (7-93) Substituting Equations (7-93) for m^'^^(t) and (7-76) for P^"^(t) into (7-92) and rearranging terms yields 1 vSUB(t, = dt SUB SUB,., ,,SUB,4.x B,4., w r (t) V (t) n (t) + ^n ^ ^lit) .B SUB ^B ^ h^ (7-94) Once Equation (7-94) is integrated, a value for the subcooled SUB SUR liquid region height, H (t), is obtained by substituting H (t) + Zq SUB for Zci|R(t) in Equation (7-85) and solving for H (t) in terms of V^^^t), i.e.. H^^^t) = (zjZo)(l (ri/r2)2) ^ y^^^{t) / Kr, (7-95) The time rate of change of the energy content in the saturated liquid region can be expressed by 1 E«T(t) = pSAT(t, , 4 E^AT,^, _ d^ ^SAT^^^ dt dt dt "out' (7-96) Since heat transfer across the internal boundaries of the boiler H •^ column regions (i.e., subcooled liquid, saturated liquid, and vapor cone regions) is neglected in this research, and since the saturated liquid region is bounded only by the subcooled liquid and vapor cone regions.

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245 the term describing the rate of heat transfer out of the saturated liquid region is omitted from Equation (7-96). Although the fuel/working fluid is vaporized in the saturated liquid region, the change in kinetic energy due to the acceleration of the vapor fuel is small relative to the change in internal energy («0.2 %). This is due to the high heat of vaporization of the fuel/working fluid mixture («1.6 MJ/kg) as a result of diverting 30% of the metal fluoride working fluid from the wall cooling region to the boiler columns where it is vaporized and due to the small density variation between the liquid and vapor phases of the fuel/working fluid mixture (a factor of «140). Neglecting kinetic energy changes and heat transfer across the boundaries of the saturated liquid region results in the following: / * dt * y where in'''^P(t) = mass flow rate of the fuel/working fluid exiting the saturated liquid region at time t (equal to the mass flow rate of the fuel/working fluid mixture entering the vapor cone region); h specific enthalpy of the saturated vapor fuel/working fluid • mixture in the saturated liquid region; u^'^^ = specific internal energy of the saturated liquid fuel/working ,. * •-* « fluid mixture in the saturated liquid region; V^'^^(t) = volume of the saturated liquid region at time t. The mass flow rate of the fuel/working fluid exiting the saturated liquid region, lii (t), is given by

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246 .VAP,^, SAT,., d „SAT,., -SAT B d ySAT,., ,, Qp, m (t) = m (t) — M, (t) = m (t) d« — V (t) (7-98) dt -' dt where M^'^^(t) is the mass of the fuel/working fluid in the saturated liquid region at time t. Substituting Equation (7-98) into (7-97) and using Equation (7-77) for P^'^^(t) yields 1 vSAT(t) = "'"' V'"'^^) n\t) 'g" '^ mSAT(t) . (7-99) dt ^B , SAT ^B , .B , SAT . B , d^(u^ -hg) d^(u^ -hg) CAT The height of the saturated liquid region, H (t), is then obtained from SAT H (t) = 3 V^^\t)/irr^ . (7-100) The time rate of change of the energy content of the fuel/working fluid mixture in the vapor cone region can be expressed by 1 E«^t) = pVA^t) . 1 E'^^t) 1 iZit) q«^t) . (7-101) The rate of heat transfer out of the vapor cone region to the surrounding moderator-reflector region is assumed to behave according to Q^'^^t) = f^^^^ A3^^t) p^'^^t) (7-102) VAP where /I , the fraction of power transferred to the moderator-reflector VAP region per unit surface area, is assumed to be constant and A' (t) is the surface area of the vapor cone region in contact with the moderatorVAP reflector region. A' (t) is given by Af^t) = A^ A^^t) ; f::v ; % ' '\ (7-103) ;/"vr. % l).v/, M*

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247 where A^ is the total surface area of the boiler column in contact with the moderatorreflector region. Due to the relatively low power and the expanding cross-sectional flow area (in the +z-direction) of the vapor cone region, the change in kinetic energy of the fuel/working fluid relative to the change in its internal energy is »-3% to «-5%. Although such changes are relatively small, the effects of kinetic energy changes in the vapor cone region are included in the analysis. When the effects of kinetic energy changes are included, Equation (7-101) becomes dt B 2 hg v^(t)/2 f^out^t) ^ ^e^t )/2J w .VAP,^, m (t) (7-104) where r^^''(t) fraction of fissioning power remaining in the vapor cone region at time t given by r«^t, = 1 f^ A«-(t, (7-105) •^out(t) C(t) VeCt) mass flow rate of the fuel/working fluid exiting the vapor cone region at time t (equal to the mass flow rate of fuel/working fluid exiting any boiler column at time t); specific enthalpy of the fuel/working fluid mixture exiting the vapor cone region at time t; linear velocity of the vapor fuel/working fluid mixture exiting the vapor cone region at time t; > a • ,i ' f ^'

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248 v.(t) = linear velocity of the vapor fuel/working fluid mixture , entering the vapor cone region at time t; ^ U^'^''{t) = internal energy content of the vapor fuel/working fluid mixture in the vapor cone region at time t given by • ' U^^t) = ;(t) Hfw ;(t) djct) V^^^t) (7-106)' where ^{t) is the average specific internal energy, dy(t) VAP is the average density, and M^ (t) is the mass of the vapor fuel/working fluid in the vapor cone region at time t; and V (t) is the volume of the vapor cone region at time t. (7-107) Since ];(t)= (1/2) cj^^ ^ll^^{t) ^ T^^^ and assuming the vapor fuel/working fluid in the vapor cone region p behaves as an ideal gas, then, dy(t) is given by '' <(t)-2p;^>,.t,)/RB_. .;tf^ (7-.08) and thus, -jmiw-. ^ > ri ,.j. '. * •-( U'^^tj^ (cS^^pJ/R^V^^t) < (7-109) where ' Cw u ' specific heat of the vapor fuel/working fluid mixture at V, V constant volume in the vapor cone region; p R universal gas constant divided by the molecular weight of the fuel/working fluid mixture (assumed to be constant); p p^ » pressure of fuel/working fluid in the vapor cone region. Substituting Equation (7-109) into (7-104) yields

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249 B B C D U \l r XllI^ ^ vVAP(t) = rVAP(t) pVAP(t) . h^ . Vi2(t)/2 ^B dt t 9 ^ ' .VAP,^. m (t) hout^t) * ^e D j^ ^A.*^4> •s««4> -i-u^.* \/VAr (7-110) Since the volume of the boiler column, V , is constant, then V (t) is yVAP^tj ^ yB _ ySUB^tj _ ySAT^^j (7-111) and 1 vVAP(t) = 1 vSUB(t) 1. vSAT(t) . dt dt dt (7-112) The mass flow rate of the fuel/working fluid exiting the vapor cone :B region, inout(t), is given by •B ,*. VAP,., d „VAP,., %ut^*) = "" ^*^ " dt / ^^ (7-113) or iL •B ... VAP,., .B,., d uVAP,^» uVAP,., d .B %ut^^) ="» (t) ^v^*^ dt ^ ^ ^ ^ dt v^^ • (7-114) Using Equation (7-108) for d5(t) in (7-114) yields • B ... •»"' /j.\ 2 P^R^ Tout't) * 'sat B dt d_ ^VAP,t, 2 P^R" houtC) * iat V'^^t) lT^„,(t, (7-115) Realizing that

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. B _ B B "g S»v 'sat 250 (7-116) (7-117) where T^^ is the saturation temperature of the fuel/working fluid in D the boiler column and T°y^(t) is the temperature of the fuel/working fluid exiting the boiler column at time t. Substituting Equations (7115), (7-116), and (7-117) into (7-110) yields ^^out(t) = 2r^^^(t) (l^Ct)) nVAP g^ e^^t) vVAP(t) eo(t) P'"^(t) 1 (T^t))^ ^VAP gB vVAP(t) m'"^(t) 2r(t) d_ yVAP(t) vVAP(t) dt (7-118) where T^t) = .B ,B 'sat ^ 'out' t:.. -h T„..,(t) / «s(t)= Cp',v4t^(t)/2 and g^ = pV R^ . (7-119) (7-120) (7-121) (7-122) Substituting Equation (7-108) into (7-78) and solving for P^^P^^j yields P^AP(t) = VAP^ „B B "' vVAP(t) nB(t) = g" "'"' vVAP(t) n\t) (7-123) 2« P„/R" w.,^ n -B..VAP^ ^out(t)-4t T^(t) Substituting Equation (7-123) into (7-118) yields r

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.VAP,^^ „VAP^ „ 1 e:!(t) /e^Ct) 251 0' . 2 y,y^ ^"'/''o''" 4vVAP(t) . (7-124) 1 -c: ..T'^{t)/e^(t) D Once a value for TQjj^(t) is obtained by integrating Equation (7124), the density of the fuel/working fluid in the vapor cone region is calculated by use of Equation (7-108). Reactivity Feedback of the UTVC. 5/)^(t) Shown in Figure 7-8 is a side view schematic of the UTVC. As shown in Figure 7-8, power generated in the UTVC, P (t), is removed by two fluids. The first is the vapor fuel/working fluid mixture from the boiler columns, which enters the UTVC with a mass flow rate equal to mj^(t) and a temperature equal to T|j^(t) = ^^^^(t-T^ ). The second is the wall cooling region working fluid, referred to as the coolant, which enters the UTVC with a mass flow rate equal to m"^{t) and a temperature u equal to T"^. Power fluctuations in the UTVC, 5P (t), induce the following: 1. 5M4{t); the mass of the vapor fuel/working fluid mixture from the boiler columns in the UTVC fluctuates due to changes in the average density of the vapor fuel in the UTVC. 2. 5T4(t): the average temperature of the vapor fuel fluid fluctuates. 3. 5Tm(t): the temperature of the moderator-reflector region surrounding the UTVC fluctuates. -;-

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252 ilit„(t) at Tf^(t) *«„(t) at T«„ I *y„(t) at T«„ I "!. ' *Lt't> 'C't' »,,-i.f; <* •^ ^ ,,«,./ ,^^ Figure 7-8. Side View Schematic of the UTVC

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253 4. 5Mrj(t): the mass of the wall coolant in the UTVC fluctuates due to changes in the density and volume occupied by the liquid wall coolant. Since the time required for fluctuations in the moderator temperature to produce an appreciable change in the UTVC reactivity is large («100 sec) relative to the duration for which the transient is to be followed («10's sec), and since the fuel temperature (Doppler) coefficient of reactivity is quite small relative to the fuel density coefficient of reactivity, effects of fluctuations in moderator and fuel temperature are neglected in this research and are recommended for future work. The efftecs of the mass of the wall coolant on the reactivity of the UTVC is also neglected in this research and is recommended for future research. Then, the inherent reactivity feedback of the UTVC, 5/)j«(t), due to power fluctuations can be expressed by Spirit) ^ «/'M/(t) = «M/«M^(t) <7-125) where 5pu^(t) is the UTVC reactivity feedback due to fluctuations in the fuel/working fluid mixture loading in the UTVC at time t, a^r is the UTVC fuel loading coefficient of reactivity, and SViAt) is the fluctuation in the fuel/working fluid loading in the UTVC at time t. The UTVC fuel loading at time t, My{t), is given by M^(t) = V" d^(t) ' ^ (7-126) where d4(t) is the average density of the vapor fuel/working fluid mixture from the boiler column in the UTVC at time t and V is the volume of the UTVC. Since the vapor fuel fluid in the UTVC is at

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254 superheat temperatures, then, under the assumption that the vapor fuel behaves as an ideal gas, d4(t) is approximated by U U U 2p,(t)/R, dy{t) « L L (7-127) Tout(t) Tf„(t) where Vf{^) " pressure of the vapor fuel/working fluid mixture from the boiler columns in the UTVC at time t; r4 = universal gas constant divided by the molecular weight of the p vapor fuel/working fluid mixture, equal to R ; Tq x{t) = temperature of the fluid exiting the UTVC at time t; T| (t) temperature of the vapor fuel/working fluid mixture entering the UTVC at time t given by Tf„{t)= Ot-r^) Vr-;,;: (7-128) where Tgy^(t-T^ ) is the temperature of the fuel/working U<-B U'^-R fluid exiting the boiler column at time t-Tp and 7g is the ,. ' delay time for the transport of fuel between the outlet of the boiler column and the inlet of the UTVC. The equations needed to relate ViAt) as a function of the UTVC power level are derived below. In terms of the neutron population level in the UTVC, N (t), the power level of the UTVC, P (t), can be expressed by P"(t) = e d^(t) N"{t) = e V" d^{t) n^t) (7-129)

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255 where ^ is the appropriate conversion factor and n (t) is the neutron density level in the UTVC at time t. The time rate of change of the energy content in the UTVC can be expressed by: ^ E^t) = P"(t) Q"(t) ^ 1 E^„{t) 1 E^yt(t) ^ W"{t) (7-130) where E (t) » energy content of the UTVC at time t given by E^t) = E^(t) . E[j(t) (7-131) where E4(t) and Ejj(t) are the energy content of the vapor fuel/working fluid (from the boiler columns) and of the wall coolant in the UTVC at time t, respectively; eV (t) « energy content of the fluid entering the UTVC at time t given ..-' ' v» 4 * * by where E-[ (t) is the energy content of the vapor fuel/working u fluid entering the UTVC from the boiler columns and E^ (t) is the energy content of the wall coolant entering the UTVC at time t; Eq x(t) energy content of the fluid exiting the UTVC at time t given by .U 'out^"'' " "-out^"' " "out E!!..(t) = E{,(t) . E;;..,(t) (7-133)

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256 where ^Q^,A^) is the energy content of the vapor fuel/working fluid mixture (boiler column fluid) exiting the UTVC and E" ^(t) is the energy content of the wall coolant exiting the UTVC at time t; Q^(t) rate of heat transfer out of the UTVC at time t; W^{t) « work done on the UTVC fluid at time t. In applying Equation (7-130), the following assumptions and restrictions are made: 1. Heat transfer out of the UTVC, Q (t), is assumed to be deposited into the wall cooling region, which is removed by the wall coolant prior to its mixing with the vapor fuel/working fluid mixture, i.e., Q (t) is included in the third and fourth terms on the RHS of Equation (7-130). That is, there is no heat transfer acoss thermodynamic boundaries since wall coolant region and vapor core are both part of the thermodynamic system. 2. Types of energies due to mechanical, electrical, magnetic, and gravitational field effects are neglected, i.e., W (t) 0. 3. Effects of kinetic energy changes are neglected since the ratio of kinetic energy changes to internal energy changes is less than «0.1 %. 4. Perfect mixing of the vapor fuel/fuel working fluid mixture with the vaporized wall coolant is assumed to occur in the UTVC. This assumes that both fuel/working fluid mixture and coolant exit the UTVC at the same temperature. 5. The liquid coolant inlet temperature to the wall coolant region is fixed.

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257 With these assumptions, Equation (7-130) becomes 1 E^it) . 1 ElJct) = P"(t) . hf„(t) m{„(t) . h^„ <„(t) Kut^^^ ""out^*) (7-134) where h/ (t) specific enthalpy of the fuel/working fluid mixture entering the UTVC from the boiler columns at time t; li/ (t) = mass flow rate of the fuel/working fluid mixture entering the UTVC from the boiler columns at time t given by *{„(t) = 4

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258 hQy^(t) = specific enthalpy of the UTVC fluid exiting the UTVC at time t; iiiQy^{t) = mass flow rate of the UTVC fluid exiting the UTVC at time t given by %ut^t) = m{,^{t) . r^l^^^t) (7-139) V -, where %ut{t) and mgy^Ct) are the mass flow rates of the vapor fuel/working mixture from the boiler columns and wall coolant exiting the UTVC at time t, respectively. Substituting Equation (7-139) into (7-134) yields f -J w .w ^ut^^^ %ut^^^ " ^ut^*^ ""out^*^ (7-140) where ^^^(t) and ^^^^(t) are the specific enthalpies for the vapor fuel/working fluid mixture from the boiler columns and the vapor wall coolant exiting the UTVC at time t, respectively. The energy content of the vapor fuel/working fluid mixture from the boiler column in the UTVC, E4(t), is given by / *• E"(t) = "(t) M"(t) (7-141) where 4(t) is the average specific internal energy of the vapor fuel/working fluid mixture from the boiler columns in the UTVC at time t and is assumed to be "(t) ' cl. i(t) ^ ^IM h "-"^'

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259 where c{ „ is the specific heat of the fuel/working fluid mixture from the boiler columns at constant volume. Substituting Equation (7-142) into (7-141) yields E>) J 11 v,v' Tin(t) T^ut(^) My(t) or ^E/ft)(^{,v/2' K"(t)* ^LC) f (c{v/2) M"(t) ^ M"(t) dt / d_ dt Tin(t) T^ut(t) (7-143) (7-144) where 'mV) dt / -"{„(*) "if„t(t) "out' Substituting Equation (7-145) into (7-144) yields ^ E^(t)= {c{v/2) dt ^in^t) ^ "^out^t) "^^n^t) ^r..^^"^ out' ,U ^ ^; M;(t) (7-148)

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260 or k '> J ^ Mj(t) = J I m^nCt) mj{t) (7-149) where MJ? is total mass of the subcooled and saturated liquid coolant in the wall cooling region, m^(t) is the mass flow rate u of the saturated vapor wall coolant at time t, and fl is the average specific internal energy of the liquid subcooled and saturated liquid coolant in the wall cooling region given by ..W <">£WWW "in ^ "sat sub + u sat sat (7-150) E!!(t) U W where u" and u" ^ are the inlet and saturation specific internal energy of the wall cooling region coolant, WW W respectively, and V^^j^, V^^^, and V are the volumes of the subcooled liquid, saturated liquid, and total volume occupied by the wall coolant region. energy content of the vapor coolant at time t given by Ev(t) = J|(t) MjJ(t) (7-151) W where M"(t) is the mass of the vapor coolant in the UTVC at time W t and ''(t) is average specific internal energy of the vapor coolant in the UTVC at time t given by /z (7-152) W W <%(t) =. c^^ w u 'b-

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261 u where c" is the specific heat of the vapor coolant fuel at V, V U constant volume and T^^^ saturation temperature of the coolant in the UTVC. Substituting Equation (7-152) into (7-151) yields EjJ(t)= (cJJ^y/2) W U Tsat ^ W(^) MU(t) (7-153) or ^Ej(t)= (cJ_,/2) w w u ^at " ^out' T. <<*' W,.. d ^U ^ (c„ ./2) MJt) ^T„..,(t) v,v dt (7-154) where 5t <<*' "•v^*) ""out^^J (7-155) Substituting Equation (7-155) into (7-154) yields ^Ej(t)= (c;;^,/2) ^ t" (t) .W sat out d .U "v^^^ %ut(^^ + (c. ./2)MJt) ^T .(t) 'V,V dt (7-156) Substituting Equations (7-149) and (7-156) into (7-147) yields ^E[J(t) == (c!;„/2) -v,v' W U Tsat * T„„t(t) w w (Cv,v/2)M^t) ^T^^,(t).uJ "out' W .W m^„(t) my(t) (7-157) Substituting Equations (7-146) and (7-157) into (7-140) and rearranging terms yields

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262 Yc(t)iOt)= 2P"(t)2 5uj;ri,^„(t). dt W ni;{t) W u U "sat »^ ^outlt) *out(t) £ ^n 7-158) 7-159) 7-160) 7-161) 7-162) 7-163) In order to simplify Equation (7-158), the mass flow rate ratio of the wall coolant-to-the fuel/working fluid mixture from the boiler columns is assumed to be constant throughout the loop, i.e.,

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263 Y!!(t)4TL(t)= 2pV). dt out ^ pT{„(t)-c;^^T^^,(t)] m{„ .VAP,^. m (t) (t) (7-165) ^^v,v"/(t)^4(t) In order to obtain the mass flow rate of the vapor fuel/working fluid mixture from the boiler columns exiting the UTVC, in{u^(t), the flow is assumed to be choked at the core exit. This assumption requires that the back pressure of the exit nozzle remains low enough (relative to the UTVC pressure) such that choked flow conditions are obtained at all times. Then, m^,,t(t) can be expressed by m^...(t) = "out' but' y + 1 y h) Tin(t) ^ "^out^^) U, At Py(t) (7-166) where a/=(Y/-l)(7/-l)/2 (7-167) Ax = throat area of the outlet nozzle (assumed to be critical throat area). Once %ut(t) is obtained, the loading of the vapor fuel/working fluid mixture from the boiler columns in the UTVC is obtained by integrating Equation (7-145).

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CHAPTER VIII DYNAMIC ANALYSIS OF A FOUR-BOILER COLUMN UTVR Introduction Results of the dynamic neutronic analysis performed on a fourboiler column UTVR are presented in this chapter. The examined model incorporates the circulating-fuel, coupled-core, point reactor kinetics and energetic equations derived in Chapter VII. This lumped parameter model is analyzed using the Engineering Analysis System code, EASY5 [29]. EASY5 is an interactive program that has the capability to model, analyze, and design large complex dynamic systems defined by algebraic, differential, and/or difference equations. By integrating the differential -difference equations for a period of time and resolving the algebraic equations, EASY5 effectively simulates the behavior of the non-linear system. EASY5 is described in Appendix A. The dynamic analysis is conducted in order to determine the transient behavior of the UTVR and to examine the stability and controllability of the UTVR. For example, following a finite perturbation, can the UTVR achieve an equilibrium condition without exceeding design limits when going from one equilibrium condition to another? Additionally, the dynamic analysis provides an insight into how the different reactivity feedback phenomena of the UTVR (e.g., coreto-core neutronic and mass flow coupling) interact with one another. 264

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265 In this chapter, the behavior of core power levels, reactivities, fuel loadings, and total system power is examined during full power transients. Full power transients are achieved by perturbing the reactivities of the UTVC and boiler columns from an initial steady state condition. The unperturbed, initial steady-state configuration of the UTVR is described in the following section. The Unperturbed UTVR Configuration In examining the dynamic behavior of the UTVR, only one of the two symmetric reactors that constitute the UTVR is examined (refer to Chapter I, pages 2-4 for details). Each reactor, which is composed of four boiler columns that symmetrically surround the central vapor core (UTVC), is configured to produce 455 MW^j^ of fission power. Of this, 116 MW^u is produced by the boiler regions (29 MW^^ produced by each boiler column). Thus, the initial power sharing factor, PuTVc/^BCOL* ^^ «2.9. The mass flow rate of the UF^ fuel is 31 kg/sec (7.75 kg/sec per boiler column) and the mass flow rate of the NaF is 79 kg/sec. About 70% of the NaF working fluid flows to the wall cooling region and the remaining 30% is diverted to the boiler columns where it is vaporized, B^-U i.e., «6 kg/sec per boiler column. During the analysis, ^ (fuel circulation time from the UTVC to the boiler columns) and tJ (fuel circulation time from the boiler columns to the UTVC) are fixed at 1.0 sec and 0.008 sec, respectively, and the associated delay time for the direct transport of neutrons among the interacting cores is assumed to be fixed and the same for all cores (i.e., tJ"""^ 10"" sec). The

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266 effect of t2*""^ on the dynamic behavior of the UTVR is also examined. Listed in Table 8-1 are the conditions for the initial steady-state configuration of the UTVR (fuel loading, power levels, mass flow rates, temperatures, etc). In this research, complete separation of the fuel/working fluid mixture into UF^ fuel and NaF is assumed to be possible. Table 8-2 lists thermodynamic properties used in this research for the UF^ and NaF fluids and for the UF^/NaF mixture. Thermodynamic properties for the UF./NaF mixture are obtained using [38] (8-1) M y =

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o c: o o 0) •-> to »-» . a to 0) o 0) Xi u 0) o. 0) 0) -> 0) E a. •a a> 4-> u O) 0) I/) 0) 00 0)

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268 c i. o 0) (O 0) c o> o O) 0) o Q. o a. C= 0) > 3 rX li. to 3

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269 fluid mixture, and for the purpose of examining the dynamic behavior of the UTVR, the analysis in this research is performed using the UF*/NaF fuel/working fluid mixture. ^ Results of the Dynamic Analysis Boiler Column Reactivity Perturbation Shown in Figure 8-1 are the power levels of the UTVC and boiler column regions as a function of time in response to a positive D reactivity step insertion of $ 1.00 (5/)gj(= 0.006502) imposed on the boiler columns at time t-0 sec. The results indicate that the power levels of the UTVC and boiler columns initially start to increase as soon as the reactivity insertion is applied. These results also show that the UTVC power level response lags the boiler column's power behavior by «0.023 sec. Additionally, Figure 8-1 indicates that the UTVC and boiler column power levels oscillate with a «0.4 sec period, which reflects a time constant of =0.065 sec. The results also indicate that these oscillations are damped and die out after «2.5 sec. Once the oscillations die out, the boiler column region power level has increased by »10 kW (»0.009%), or =2.5 kW per boiler column, and the UTVC power level has increased by =500 kW (=0.15%). Although the perturbation is imposed on the boiler columns, the results indicate that the effect on the UTVC is much greater. As soon as the reactivity of the boiler column is increased, the neutron population level in the boiler column increases, and thus, the boiler column power level increases causing a decrease in the amount of fuel/working fluid in the boiler column and an increase in the

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270 119 118 ill7 CO a. 115 114 » Time (sec) A

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271 temperature of the fuel/working fluid exiting the boiler columns. These have an immediate effect on the UTVC and a delayed effect on the UTVC as described below. There are two immediate (10" to »10" sec) effects. The first is an increase in the number of neutrons transported to the UTVC from the boiler columns as a consequence of the increase in the boiler column region's power level. The second, although a relatively small effect, is an increase in the reactivity of the UTVC as a result of a decrease in the fuel loading in the boiler columns. These two effects cause an increase in the UTVC's neutron population level, i.e., an increase in the UTVC's power level. The increase in the UTVC's power level causes the average temperature and pressure of the UTVC to increase. The increase in the UTVC pressure causes an increase in the mass flow rate of the fuel/working fluid exiting the UTVC and a decrease in the mass flow rate of the fuel/working fluid entering the UTVC. Thus, the UTVC's fuel loading decreases causing a decrease in the UTVC's reactivity. The delayed effects {»0.008 sec) include increases in the inlet temperature ofand in the delayed neutron precursors carried bythe fuel/working fluid entering the UTVC from the boiler columns. The increased temperature of the fuel/working fluid entering the UTVC causes a further increase in the UTVC pressure and consequently a further decrease in the UTVC loading. The increase in the neutron precursors entering the UTVC causes additional increases in the neutron population and power levels of the UTVC. The increase in the UTVC neutron and power levels consequently increases the boiler column power levels by transporting more neutrons

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272 to the boiler columns. The fuel loading in the boiler columns further decreases, causing a further decrease in the boiler column reactivity. The power levels of the UTVC and boiler column regions continue to increase until their fuel loadings, i.e., reactivities, have dropped to levels that are low enough to make the entire system subcritical. This first occurs at t»0.075 sec for the boiler columns and at t«0.098 sec for the UTVC. After this, the UTVC and boiler columns power levels decrease causing their fuel loadings to increase. The rates at which the power levels are decreasing decrease as the fuel loadings in the UTVC and boiler columns increase. The fuel loadings in the UTVC and boiler columns continue to increase until their reactivities are high enough so that the power levels begin to rise again. The behavior of 235 the UTVC pressure, U loading, and inlet and outlet mass flow rates of the UF^/NaF working/fluid mixture from the boiler columns as a function of time are given in Figure 8-2. The behavior of the boiler column outlet mass flow rate (the inlet mass flow rate of the UF^/NaF to the boiler is held constant at 13.675 kg/sec per boiler column) and U loading as a function of time are shown in Figure 8-3. The time constant (or period) at which the power levels of the UTVC and boiler columns are oscillating reflects the outcome of the interaction of several short time constant phenomena (included in the dynamic model). The time constants associated with the UTVR range from -04 «10 sec (direct core-to-core neutronic coupling delay time) to »80 sec (time constant for the longest lived delayed neutron precursor emitter). Other important time constants are the fuel residence time in U B the UTVC (t^) and in the boiler columns {t°) and the delay times

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273 Q. 49.88 6.804 *H. 6.788 Time (sec) s Tine (sec) ^ 54.75 n-_o 54.60 54.80 54.56 Time (sec) J\r^-" -f Time (sec) mass flow rates in units of kg/sec Figure 8-2. UTVC Pressure, U^^^ Loading, and UF^/NaF Inlet and Outlet Mass Flow Rates, as a function of Time Following a $ 1.00 Positive Reactivity Step Insertion Imposed on the Boiler Columns at t=0 sec

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274 14.0

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4,; 275 associated with transporting fuel in the loop from the UTVC to the boiler columns (t^ ) and from the boiler columns to the UTVC (t* ). Effects of the direct core-to-core neutron transport delay times on the dynamic behavior of the UTVR are examined and described in a later section of this chapter. '' The reason that the fuel loadings in the boiler column and UTVC decrease is a result of the system's inherent response to compensate for the positive reactivity step insertion imposed on the boiler column. That is, by discharging fuel from both cores, the system compensates for the added reactivity in an attempt to achieve a new equilibrium 235 configuration. In the final equilibrium configuration, the U loadings decrease by =5 gm in the UTVC and by =10 gm in each boiler column. The fact that the UTVC discharges less fuel than the boiler columns is due to two main reasons. The first is the fact that the perturbation is imposed on the boiler column. The second is the higher fuel density in the boiler columns, which causes the reactivity of the 235 boiler columns to be less affected by U fuel loading than the 235 reactivity of the UTVC (i.e., the U reactivity worth per gm is much smaller in the boiler then in the UTVC). Thus, the boiler columns need to discharge more fuel than the UTVC in order to compensate for the added reactivity. The results, as shown in Figures 8-1, 8-2, and 8-3, indicate that, in addition to the oscillations being damped, the amplitude of the oscillations are very modest, with a maximum of less than «2% during the initial phase of the transient. •**<"^,

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276 Shown in Figure 8-4 is the behavior of the UTVC and boiler column power levels following a negative $ 1.00 reactivity step insertion imposed on the boiler columns at t=0 {Sp^^= -0.006502). The period and amplitude of the oscillations shown in Figure 8-3 are similar to those of Figure 8-1. The differences are (1) the power levels initially decrease rather than increase and (2) the final equilibrium conditions have lower power levels and higher fuel loadings in the UTVC and boiler columns. The final equilibrium conditions for these two perturbations are listed in Table 8-3. UTVC Reactivity Perturbation The behavior of the UTVC and boiler column power levels as a function of time due to a positive, external, reactivity step insertion of $ 0.20 imposed on the UTVC [Sp^^ » 0.0013) are presented in Figure 85. The results indicate that the power levels of the UTVC and boiler columns increase following the positive reactivity step insertion and oscillate with about a 0.4 sec period. The results also indicate that a new equilibrium configuration is reached after »3 sec in which the power level of the boiler region increased by «30 kW {«0.03%) and the power level of the UTVC increased by 3.19 MW («0.9%). Initially, the power level response of the boiler column lags that of the UTVC. This is due to the fact that the perturbation is introduced to the UTVC at time t=0 sec, and the effect of this perturbation is detected by the boiler columns at t«10 sec, i.e., B*4J -04 after a time delay equal to t^ (10 sec). However, once the first oscillation occurs, the UTVC power level response lags that of the ^. >

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277 118 117 116 a. 115 114 Time (sec) 12 3 4 Time (sec) Figure 8-4. UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 1.00 Negative Reactivity Step Insertion Imposed on the Boiler Columns at t=0 sec

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279 356 352 348 5 344 it 340 336 332 12 3 4 122 Time (sec) Time (sec) Figure 8-5. UTVC and Boiler Column Regions Power Levels as a function of Time Following a $ 0.20 Positive Reactivity Step Insertion Imposed on the UTVC at t=0 sec

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280 boiler column by =0.020 sec; and by the time the third oscillation occurs, the UTVC power level response lags that of the boiler column by «0.023 sec, a lag time which is equal to the lag time obtained when the perturbation is introduced in the boiler columns. Shown in Figure 8-6 are the UTVC and boiler column power level behavior following a negative $ 0.20 reactivity step insertion imposed on the UTVC at time t=0, i.e., Sp^^ -0.0013. The period («0.4 sec) and amplitude (maximum of =5%) of the oscillations of Figure 8-6 are similar to those of Figure 8-5. The final equilibrium conditions for the positive and negative reactivity step insertions imposed on the UTVC at time t«0, i.e., 8p^^ = 0.0013 and 8p^^ « -0.0013, are listed in Table 8-4. In comparing the results when the reactivity of the boiler column is perturbed (Figures 8-1, 8-2, and 8-3 and Table 8-3) with the results when the reactivity of the UTVC is perturbed (Figures 8-4 and 8-5 and Table 8-4), the following similarities are observed: 1. The UTVC and boiler columns oscillate with about a 0.4 sec period, which reflects a time constant of =0.065 sec. 2. The oscillations are damped and new equilibrium configurations are achieved in less than =3 sec. 3. During the transients, the power level response of the UTVC lags that of the boiler columns. 4. At the new equilibrium configurations, the change in the power level of the UTVC is greater than the change in the power level of the boiler columns, and the change in the fuel loading in the UTVC is less than the change in the fuel loading in the boiler columns.

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281 348

PAGE 303

u (U u a> (/> CO CO u I I f > M P «V -** o H H T 00 O O'

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283 The main difference is that both cores (UTVC and boiler columns) are more influenced by perturbations imposed on the UTVC than by perturbations imposed on the boiler column region. For example, when a positive reactivity step insertion of $ 1.00 is imposed on each boiler column {5/>gj( " 0.006502), the power level of the UTVC increases by «500 kW and the power level of the boiler region increases by »10 kW; but when a positive reactivity step insertion of only $ 0.20 is imposed on the UTVC (5pgj^ « 0.0013), the power level of the UTVC increases by »3200 kW and the power level of the boiler region increases by »30 kW. Variations in Core-to-Core Direct Neutron Transport Delay Times Although the delay time associated with the core-to-core direct neutron transport coupling (t^^"^) is a fraction of the neutron life time, the effect of artificial variations in this parameter on the dynamic behavior of the UTVR is examined. While maintaining the boilerto-UTVC fuel circulation loop time fixed at 0.008 sec and UTVC-to-boiler fuel circulation loop time fixed at 1.0 sec, a positive reactivity step insertion of $ 0.20 is imposed on the UTVC for t^^"^ values of 10 sec and 10' sec. The UTVC and boiler column power levels as a function of time for tI*""^ of 10' sec and 10' sec are shown in Figures 8-7 and 88, respectively. It should be noted that having a rj""^ of 10'°^ sec is highly unlikely and having a tI*""^ of 10' sec is impossible since neutron 03 removal life times are «10 sec for this reactor system. The analysis performed in this section (i.e., varying rj'"'^) is a mathematical exercise conducted to help obtain a better or more detailed

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v4«i284 J3U 352

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285 352.5 § 350.0 2 347.5 345.0 342.5 340.0 337.5 119 118 S 116 (^ 115 114 12 3 Time (sec) A

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286 understanding of the effects of the direct coreto-core neutron transport coupling. The results indicate that increasing r|^"^ from 10" sec (Figure 85) to 10 sec (Figure 8-7) has no significant effect on the behavior of the UTVC and boiler region power levels, especially when compared to increasing t^^"^ to 10" sec (Figure 8-8). This indicates that as long as tV'^ is less than the prompt neutron generation time, variations in t2^"^ have no significant effect on system behavior during transients. Additionally, the results indicate that decreasing t^**"^ below 10" sec should have no effect on the system behavior. The consequences of artificially increasing t2*^^ from 10" sec -(\7 (Figure 8-5) to 10 (Figure 8-8) include: 1. The amplitudes of the oscillations decrease from a maximum of »5% to a maximum of «3% and the periods of these oscillations increase from «0.4 sec to «0.5 sec. 2. The oscillations are more damped which causes a reduction in the time required to reach the final equilibrium configuration, i.e., from «3 sec, to «2.5 sec, and then to «2 sec. 3. The new equilibrium power level of the UTVC is lower by »30 kW, i.e., the increase is =3.16 MW when t]*""^ is artificially increased to 10" sec versus «3.19 MW when a value of 10" sec is used for T^*""^; and the new equilibrium power level of the boiler columns is lower by «10 kW, i.e., the increase is only =20 kW when tI*'^ is 02 / artificially increased to 10 sec versus »30 kW when a value of .-04 10' sec is used for t^^'^ .

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287 Variations in the Coupling Coefficients In order to determine the effect of the direct coreto-core neutronic coupling coefficients on the dynamic behavior of the UTVR, the magnitudes of the coupling coefficients are artificially reduced by one order of magnitude. Since results for the current configuration indicates that the system is tightly coupled (e.g., e^ is »50%), then, by artificially reducing the magnitudes of the coupling coefficients by an order of magnitude, an insight is gained into whether a tightlycoupled or a loosely-coupled core system is desirable. Results of static neutronic analysis, presented in Chapters IV and V, indicate that the degree of direct coreto-core neutronic coupling is influenced by design factors such as the distance separating the interacting cores. Then, to some extent, the reactor system can be designed so as to obtain coupling coefficients that enhance the system dynamic behavior. Results of imposing a positive reactivity step insertion of $ 0.20 on the UTVC with the reduced coupling coefficients are shown in Figure 8-9. The final equilibrium conditions following a Sp^^ reactivity step insertion of 0.0013 with normal and reduced coupling coefficients are listed in Table 8-5. Comparison of the results of Figure 8-9 (reduced coupling coefficients) with the results of Figure 8-5 (normal coupling coefficients) indicate that the effects of reducing the coupling coefficients for a 6p^^ = 0.0013 include the following: 1. The length of the transient is prolonged from «3 sec to »6 sec. 2. The periods at which the power in both cores are oscillating are noticeably different. When normal coupling coefficients are

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r 'i 288 360 355 ^350 = 345 :r 340 a. 335 330 119 ^118 § ^117 k

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e

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290 employed, the power in both cores oscillates with a «0.4 sec period. However, when reduced coupling coefficients are employed, the UTVC power oscillates with a »0.45 sec period while the period of power oscillation for the boiler columns remains unchanged at about 0.4. This causes the time that the UTVC power level response lags that of the boiler column to increase with time. For example, at t=0.1 sec, the UTVC power level response lags that of the boiler columns by «0.04 sec, while at t=0.5 sec, the UTVC power level response lags that of the boiler columns by =0.08 sec. 3. The amplitudes of the oscillations increase from a maximum of «4.6% to a maximum of =5.7% for the UTVC and decrease from a maximum of =3.2% to a maximum of =1.5% for the boiler columns. 4. The new equilibrium power level of the UTVC is higher by «1.3 MW (i.e., the increase is »4.5 MW when reduced coupling coefficients are employed versus =3.2 MW when normal coupling coefficients are , employed); in contrast, the new equilibrium power level of the boiler columns is lower by 28 kW (i.e., the increase is only =2 kW when reduced coupling coefficients are employed versus =30 kW when normal coupling coefficients are employed). Additionally, reducing the coupling coefficients causes the fuel loading in the UTVC to be more affected and the fuel loading in the boiler columns to be less affected by a reactivity perturbation imposed on the UTVC. For example. Table 8-5 indicates that imposing a positive reactivity 235 step insertion of $ 0.20 on the UTVC causes the U loading in the UTVC to be reduced by =36 gm when reduced coupling coefficients are

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291 employed, versus =26 gm when nonnal coupling coefficients are employed. The results of this analysis indicate that having a tightly coupled-core reactor system aids in stabilizing the UTVR. Variations in the UTVC Fuel Mass Reactivity Feedback Coefficient There are a number of factors that influence the inherent reactivity feedback due to a given variation in the fuel loading, i.e., 5pj|^ due to a given fissile fuel mass variation. These include types, enrichments, and loadings of fissile fuel; types of working fluid; and size and configuration of the reactor system. It is, therefore, desirable to examine the effects of changes in the magnitude of the fuel mass coefficient of reactivity on the dynamic behavior of the UTVR. For this purpose, the effects of reducing the magnitude of the UTVC fuel mass coefficient of reactivity by a factor of five is investigated. The UTVC and boiler columns power level behavior as a function of time due to a positive reactivity step insertion of $ 0.20 imposed on the UTVC are plotted in Figure 8-10. The final equilibrium conditions for this analysis are shown in Table 8-6. The results indicate that reducing the magnitude of the fuel mass reactivity feedback of the UTVC causes the power levels of the UTVC and boiler columns to oscillate with increased periods (the periods increase from 0.41 sec to 0.47 sec) and increased amplitudes (maximum amplitudes of oscillations increase from «4.7% to «5.3%). Although the amplitudes and periods of the oscillations have increased, the results indicate that the damping of these oscillations has also increased causing the

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292 0.8 1.2 Time (sec) 2.0 1.CC. 120 1 2 118 CL

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0) o o. CSJ o u a> u » o > >»o •-» •^ > •-> •M (U u • (-> w c > '•fT3 fo _l o a.1— a> Li. o> c u I— -o O 0) liu c w oac •fc c Or3 O •r3 3 ero ai I 00 0) u (/) 00 o u 0)

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294 UTVC and boiler columns to achieve the new equilibrium configuration in a shorter time. That is, the length of the transient is shortened from ~3 sec to «1.2 sec, which is a desirable behavior. The increases in the amplitudes and periods of the oscillations when the magnitude of the UTVC fuel mass coefficient of reactivity is reduced are due to the relative increase in the importance of the boiler column fuel mass reactivity feedback, i.e., reducing the magnitude of the UTVC fuel mass coefficient of reactivity causes the boiler columns fuel mass coefficient of reactivity to be the dominant control 235 mechanism. Since the U reactivity worth in the UTVC is reduced, then more fuel is required to be discharged from the system which causes the period of the oscillations and thus, the amplitudes to increase. Since the boiler column fuel mass coefficient of reactivity becomes the dominant feedback control mechanism, and since the boiler columns have a larger damping effect due to their liquid fuel, the damping of the oscillations increases. Table 8-6 indicates that the UTVC and boiler column steady-state power levels increase and fuel loadings decrease when the magnitude of the UTVC fuel mass coefficient of reactivity is reduced. This is expected, since a reduced fuel mass reactivity coefficient implies that more fuel needs to be discharged in order to compensate for the excess reactivity that is imposed on the system. Shown in Figure 8-11 are the UTVC and boiler columns power levels, in response to imposing a $ 0.20 positive step reactivity insertion on the UTVC, when the magnitude of the UTVC fuel mass coefficient of reactivity is increased by a factor of two. The results indicate that

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295 .'*'^ 425 400 375 i ^350 ^325 a. 300 275 140 130 £120 2 110 CO a. 100 90 %:i ' . .'.

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296 undamped oscillations are obtained, i.e., the system becomes unstable. < For the studied UTVR reference configuration, sustained oscillations are found to occur when the magnitude of the UTVC fuel mass coefficient of reactivity is increased by a factor of »1.7. It can be seen from Figures 8-10 and 8-11 that the dynamic behavior of the UTVR can be highly dependent on the operating conditions. The range of conditions for which the UTVR exhibits dynamically desirable behavior can be determined by performing a series of calculations such as those presented in Figures 8-10 and 8-11. Research performed on other gas core reactor concepts, such as that conducted by Kiratadas Kutikkad [37], indicate that reducing the magnitude of the vapor fuel mass coefficient of reactivity beyond a certain limit also causes the system to become unstable, i.e., undamped oscillations are obtained. This behavior is not observed with the UTVR. It is found that eliminating the UTVC fuel mass coefficient of reactivity does not cause the UTVR to become unstable. This is due to other strong and effective negative reactivity feedbacks that are inherent to the UTVR, such as the boiler column fuel mass coefficient of reactivity. Determining the ranges of conditions for which the UTVR exhibits desirable dynamic performance is recommended for future work. Concluding Remarks The analysis performed in this chapter indicates that the UTVR's inherent reactivity feedbacks are capable of transitioning the UTVR safely and quickly from one equilibrium configuration to another. The analysis also indicates that there should be a wide range of operating

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297 conditions for which the system exhibits good dynamic behavior. However, it should be noted that the effects of acoustic phenomena have been neglected. It is recognized that acoustic phenomena are inherent to the UTVR and their effects are potentially very significant. Analysis of acoustic effects is strongly recommended for future work when the necessary tools for treating these effects are available. Such analysis will require coupled space-time neutron field-gas density field calculations. In addition to neglecting effects of acoustic phenomena, the energetics equations employed in the model used for analyzing the dynamic behavior of the UTVR contain the following assumptions and restrictions: 1. Inlet temperature, pressure, and mass flow rate of the fuel/working fluid mixture entering the boiler column regions are fixed. 2. Pressure in the boiler column regions is assumed to be constant. 3. A surge tank in the loop connecting the boiler column with the UTVC is used. This is needed in order to maintain the boiler columns at a constant pressure. 4. Heat transfer across the internal boundaries of the boiler column regions is neglected. 5. The vapor fuel/working fluid mixture is assumed to behave as an ideal gas. 6. Perfect mixing of the vapor fuel/working fluid mixture with the vaporized wall coolant is assumed to occur in the UTVC. This implies that both fuel/working fluid mixture and coolant exit the UTVC at the same temperature. * . ;.

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298 7. Types of energies due to mechanical, electrical, magnetic, and gravitational field effects are neglected, i.e., W(t) 0. The assumptions, restrictions, and limitations in the circulating fuel, coupled-core point reactor kinetics equations include the following: 1. The effects of photoneutrons are neglected. Photoneutrons are the result of the Be{'r,n)2He reaction. 2. The time-dependent behavior of the function describing the shape of the angular neutron flux is neglected, i.e., l&'^(];,E,g,t) » 3. The microscopic cross sections of the fuel, working fluid, and moderator are assumed to be constant with respect to time. Thus, reactivity feedbacks due to changes in the fuel and moderator temperature are neglected. 4. The time-dependent behavior of the density distribution of the fuel/working fluid in the UTVC and the vapor regions of the boiler columns is neglected in this research, i.e., "!(];, t) =*"1q{j;). Additionally, in obtaining the integral parameters used in the CCPRK equations (e.g., A and p), a uniform fuel/working fluid density is assumed in the UTVC and in the vapor regions of the boiler columns. 5. The delay time associated with transporting the coupling-neutrons i«-k through the media, namely r^ , is the same for all neutrons. 6. Precursor transport through the connecting loops is a pure time delay. No fissioning occurs in the fuel outside the UTVC and boiler columns.

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299 7. Slug flow outside the core (in the loop) is assumed. 8. The effects of fluctuation in the thickness of the wall cooling region on the reactivity of the UTVC and boiler columns and on the coupling coefficients are neglected. .>.T

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CHAPTER IX SUMMARY OF RESULTS, CONCLUSIONS, AND RECOMMENDATIONS FOR FURTHER RESEARCH Introduction A summary of results obtained from the static and dynamic neutronic analysis performed on the Ultrahigh Temperature Vapor core Reactor (UTVR) is presented in this chapter. Another section in this chapter highlights some of the important conclusions drawn from this research. Suggestions and recommendations for further research are also identified in this chapter. Summary of Results Results from the Static Neutronic Analysis Results obtained from 1-, 2-, and 3-D static neutronic analysis are summarized in this section. The focus of the 1and 2-D static neutronic analysis was to obtain optimum UTVR geometric configurations for the 3-D analysis. During the analysis, behavior of the system's neutron multiplication factor, k *^, and power sharing factor, ''UTVC^'^BCOL' ^^^ observed. Results obtained from the 1and 2-D static neutronic analysis can be summarized as follows: 1. k r^ saturates at vapor core radii above =s80 cm and UF^ partial pressures above »8 atm (at 3000 K). 2. The optimum outer BeO region radial thickness is a40 cm. 300

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301 3. An inner BeO region radial thickness of =15 cm results in highest k fr values and in highest "overall" neutronic coupling among the interacting cores. 4. A height of «7.5 cm for the mid-plane BeO slab results in highest k f f values and maximum neutronic coupling between the upper and lower reactors. 5. The optimum height for the top BeO region is »50 cm. 6. As the number of boiler columns increases, k^r increases while PyjYQ/PgQQL decreases . > .^ ; « *. ^ 7. A fraction of the metal fluoride needs to be diverted from the wall cooling region to the boiler region so that the nuclear design provides a PuTVc/^'bCOL ^^^^^ matches the PuTVc/''bCOL ''^^"i'^s*^ o" ^^^ basis of thermodynamic considerations. For the NaF working fluid system, this fraction is found to be «30%. Results of the 3-D static neutronic analysis include the following: 1. Neutron streaming from the UTVC's exit nozzle, MHD duct, and diffuser regions is significant. The UTVC exit nozzle, MHD duct, and diffuser regions are found to impose a large penalty on the system's reactivity {«15% 6k/k). 2. A factor of «13 reduction in required computation time for Monte Carlo calculations is achieved by employing selected variancereduction techniques (i.e., energy cutoff, implicit capture and weight cutoff, and weight windows) and utilizing boiler-to-UTVC symmetry.

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302 3. The boiler-to-boiler neutronic coupling is relatively small (c° and c^"*"°° <1%), especially when compared with the UTVC-to-boiler D4JJ neutronic coupling (e^ «50%). 4. The boiler-to-UTVC neutronic coupling effect is significant. The neutronic coupling of the four boiler columns to the UTVC ranges from »8% to «18%, (i.e., e^ «2% to »4.5% for each boiler column). Results from the Dynamic Neutronic Analysis i t ' ,.''. 'i ' _ » V wi The following is a summary of results obtained from the dynamic neutronic analysis performed on the four-boiler column UTVR system: 1. Perturbations imposed on the UTVC have a greater impact on the UTVC and on the boiler columns than do perturbations imposed on the boiler columns. 2. Perturbations imposed on the boiler columns and on the UTVC have a greater impact on the UTVC than they do on the boiler columns. 3. Effects of variations in the magnitudes of the fuel mass reactivity feedback coefficients indicate that the UTVR can be operated over a wide range of conditions that yield good dynamic performance. 4. Effects of variations in the coupling coefficients indicate that the dynamic capabilities of the UTVR are exceptional. 5. The combination of the strong inherent negative reactivity feedbacks of the UTVR provide for rapid, self-stabilizing transition of the UTVR from one equilibrium configuration to another, even when relatively large reactivity insertions are imposed on the reactor system (£p $ 1.00). Maximum amplitudes of oscillations in core power levels incurred for the examined transients are «5%.

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303 Comments and Conclusions Static and dynamic neutronic analysis of a highly enriched, 235 externally moderated, U circulating fuel, coupled-core (vapor and boiler cores), burst power, ultrahigh temperature vapor core reactor system reveal some attractive features for this novel reactor concept. These features include compact reactor configurations, flexibility in reactor design, inherent reactivity control mechanisms, high operating temperatures and efficiency, and simple core geometry. Configuring the reactor system such that maximum direct neutronic coupling among the interacting cores is obtained leads to dimensions for the critical reactor of «2.5 m in diameter and «1.9 m in height. Static neutronic analysis indicates that the size of the reactor system can be further reduced by the following: " 233 1. Using other possible fissile fuels such as U . The analysis 235 233 indicates that the reactivity worth of replacing U with U is «12% 5k/k. 2. Increasing the fuel loading in the UTVC (the current configuration employs 5 atm of UF^ at 3000 K). Static neutronic analysis indicates that k r^ saturates above a UF^ partial pressure of jsS atm and consequently, above a UF^ partial pressure of =58 atm in the UTVC, the vapor core fuel mass reactivity feedback is significantly reduced. However, dynamic analysis indicates that the other reactivity control mechanisms of the UTVR (e.g., boiler column fuel mass reactivity feedback) are sufficient and adequate for rapid, self-stabilizing transitions of the UTVR from one equilibrium configuration to another.

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304 3. Increasing the number of boiler columns. 4. Careful design of the vapor core exit nozzle, MHD duct, and diffuser regions in order to minimize neutron streaming losses. One of the restricitions imposed on the reactor system is that that the power distribution or power sharing between the central vapor core and the surrounding boiler columns obtained from the nuclear design matches that required on the basis of thermodynamic and flow considerations. By employing four boiler columns and diverting a fraction of the metal fluoride to the boiler column regions, this objective was achieved. This assumes that separation of the UF^/NaF fuel/working fluid mixture into UF^ fuel and NaF working fluid is possible. Thus, separation of the fuel/working fluid mixture into fuel and working fluid is desirable. The dynamic neutronic analysis indicates that the unique and effective negative reactivity feedbacks of the UTVR are capable of stabilizing the UTVR safely and quickly. The analysis also indicates that the system exhibits relatively good dynamic performance even when an inherent reactivity control mechanism is suppressed (e.g., the vapor fuel mass coefficient of reactivity). That is, the dynamic neutronic analysis indicates that there should be a wide range of operating conditions for which the system exhibits relatively "good" dynamic behavior. Due to the restriction of constant pressure in the boiler columns, a surge tank for the boiler columns was required in order to make the system dynamically stable. By employing a surge tank, the fuel/working fluid inlet mass flow rate to the boiler columns was kept constant and

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305 the inlet mass flow rate to the UTVC was determined on the basis of the pressure in the UTVC and temperature of the fuel/working fluid mixture exiting the boiler columns. The significant amount of research performed on the UTVR, as presented in this dissertation, has furnished a considerable amount of insight into the neutronic characteristics of this novel reactor concept. However, due to the preliminary nature of this research, the results obtained from this work are not to be considered as representative of an optimized UTVR configuration. The dynamic model employed in this research is a relatively simple model. For example, the time-dependent behavior describing the shape of the angular neutron flux, l&(jj,E,Q,t), has been neglected in this research. It is recognized that more sophisticated models, which can better represent the actual system, are needed for further studies. Therefore, recommendations in areas that require further research are presented and outlined in the section that follows. Recommendations for Further Research Static Neutronic Analysis One of the major problems encountered in analyzing externallymoderated vapor core reactors is the relatively large computation times required to achieve acceptable convergence levels. This is true for 2-D S„ calculations, and even more so for 3-D Monte Carlo calculations, n Additionally, since the 1and 2-D S^ calculations are unable to properly model the UTVR due to its complex geometry, 3-D Monte Carlo modelling is crucial for accurately treating and reliably estimating

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306 essential parameters such as reactivity worths of liquid volume variations in the boiler columns and neutronic coupling coefficients among the interacting cores. In this research, an attempt was made to reduce the computation times for 3-D Monte Carlo neutronic calculations. The use of energy cutoff, implicit capture and weight cutoff, and weight windows variance-reduction techniques and boiler-to-UTVC symmetry, significantly reduced Monte Carlo computation times (by a factor of »13). However, the uncertainty levels in parameters obtained from MCNP need to be further reduced. The uncertainty levels in these parameters can be decreased by increasing the computation time and/or by decreasing the history variance further. Methods for reducing the history variance include the following: 1. Increase the number of the space-cells and energy-phases of the UTVR with the weight window technique. Currently, 170 cells and fourneutron energy groups are employed to represent the space and energy phases of the UTVR. OQ 2. Employ a higher cutoff energy. Currently, a cutoff energy of 10 MeV is employed. 3. Employ source biasing so that a greater number of neutrons are initiated in the boiler columns. This may require modifications to the MCNP code. 4. Examine other variance-reduction techniques such as DXTRAN [13,32] spheres and exponential transforms.

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307 Utilizing the above, so that the uncertainty levels in parameters obtained from MCNP can be further reduced, is recommended for future work. Future research should focus on reducing neutron streaming from the vapor core exit nozzle and MHO duct/diffuser regions. Dynamic Neutronic Analysis Determining the range of conditions for which the UTVR exhibits desirable dynamic performance is recommended for future work. This can be determined by performing a series of calculations such as those performed in Chapter VIII. Dynamic and stability analysis studies in the Laplace and frequency domains should be investigated. The dynamic analysis studies presented in this dissertation indicate that the amplitudes of the oscillations are quite small (maximum of »5%) and the final equilibrium conditions are not far from the initial conditions (e.g., variation in U loadings in the UTVC and boiler column are less than 1%), even when large perturbations are introduced to the system {Sp $ 1.00). This indicates that results obtained from linearized models should agree with results obtained from the non-linear model. By employing the linearized circulating fuel, coupled-core point reactor kinetics equations derived in Chapter VII and linearizing the energetics equations, the use of the well -developed methods of linear feedback theory such as Root Locus, and Bode and Nyquist plots is possible. Enhancements and modifications to the dynamic model developed in this research should be employed. These include the following:

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308 1. The ba1ance-of -plant loop should be included in the dynamic model. By employing BOP modules (e.g., the MHD duct regions, space radiators, and pumps), the time-dependent behavior of the inlet temperature, pressure, and mass flow rate of the fuel/working fluid mixture entering the boiler column regions can be accounted for. 2. The thermodynamic/fluid flow model employed in this research is rather simple. Future models should allow the pressure in the B B boiler columns to be time-dependent (i.e., T^^ » ^sat^*^^* ^^^^ may eliminate the need for surge tanks. Also, the effects of different models for simulating boiling in space should be investigated. Future work should focus on power level control methods. Possible methods include: 1. controlling the mass flow rate of the fuel and/or working fluid; 2. time-varying reactivity insertions; and/or 3. designing the reactor system such that the number of loaded boiler columns depend on the required power level. . Different start-up scenarios should be devised and investigated.

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APPENDIX A DESCRIPTION OF THE COMPUTER CODES Introduction The static and dynamic nuclear design and analysis of the UTVR requires the use of computer codes. These codes are used in characterizing and investigating the novel features and the stability of the UTVR. The codes that are used in this research, which include nuclear and system analysis codes, are described in this Appendix. Description of Nuclear Codes Static nuclear codes have been used in this research for the following purposes: (1) to read and manipulate the Evaluated Nuclear Data Files (ENDF/B) [25]; (2) to collapse multigroup nuclear crosssection data; and (3) to compute fluxes and currents, reaction rates, and eigenvalues. Codes used in this research for the static neutronic analysis of the UTVR are described in this section. AMPX: A Modular Code System for Generating Coupled Multigroup NeutronGamma Libraries from ENDF/B AMPX is a modular code system [24] that is designed to produce coupled multigroup neutron-gamma cross-section sets. Basic neutron and gamma cross-section data for AMPX are obtained from ENDF/B data files. AMPX is flexibly dimensioned; neutron group structures, gamma group 309

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^ n^ 310 structures, and expansion orders to represent anisotropic processes are all arbitrary and limited only by available computer core and budget. The AMPX modular code system is designed to retain as much generality as possible in creating the standard data interfaces (cross-section libraries). The AMPX system can be used to 1. generate multigroup neutron cross-sections; 2. generate multigroup gamma cross-sections; 3. generate gamma yields for gamma -producing neutron interactions; 4. combine neutron cross-sections, gamma cross-sections, and gamma yields into final "coupled cross-sections sets;" 5. perform one-dimensional discrete ordinates (S ) transport or diffusion theory calculations for neutrons ana gammas and, on option, collapse the cross-sections to a broad-group structure, using the 1-0 results as weighting functions; 6. plot cross-sections, on option, to facilitate the "evaluation" of a particular multigroup set of data; 7. update and maintain multigroup cross-section libraries in such a manner as to make it not only easy to combine new data with previously processed data, but also to do it in a single pass on the computer; and 8. output multigroup cross-sections in convenient formats that are suitable for other codes. AMPX consists of a number of modules that form the AMPX modular code system. Each module is capable of performing at least one of the above listed processes. Since the nuclear design and analysis of the UTVR in this research focuses on neutron interactions, not all AMPX modules are used. The AMPX modules that are used in this research include the AMPX-DRIVER [39], XLACS [26], NITAWL [27], and XSDRNPM [10] modules, these are described as follows:

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^ . 311 The AMPX-DRIVER module ^ The AMPX-ORIVER module [39] manipulates the execution sequence of •, ^\ .-. ' iz.. ; AMPX modules. Different AMPX modules can be executed in one computer batch job and/or the same AMPX module can be executed more than once by calling that module from the DRIVER, as requested by the user. The XLACS module The XLACS module [26] calculates weighted multigroup neutron crosssections from ENDF/B files. XLACS is designed to produce full -energyrange neutron cross-section libraries. Provisions are included for treating fast, resonance, and thermal ENDF/B data in a single calculation. Energy group structure and expansion orders used for representing differential cross-sections can be arbitrarily specified by the user. Smooth cross-sections can be averaged over arbitrary usersupplied weighting functions or over any of several built-in weighting functions. The ENDF/B format is very general, allowing data to be specified in several ways for practically any nuclear process [25]. A corresponding generality is required on the part of the processing codes which use the data. The XLACS program attempts not only to accommodate this generality, but also to allow new processing methods to be easily added as modifications and improvements in data representation occur. XLACS features include the following: 1. processes point, resonance, and S(a,^) data; 2. treats discrete inelastic levels to an arbitrary order of anisotropy; 3. produces anisotropic matrices for "continuum" inelastic processes; 4. allows the use of several built-in weighting options; and

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< 312 5. handles all ENDF/B data in files 1, 2, 3, 4, 5, and 7 [40]. The NITAWL module .-; ':»,"' '^ , ^\ The NITAWL module [27], which is an acronym for Nordhiem's [41] Integral Treatment And Working Library production, is designed to read the AMPX created cross-section libraries, to perform resonance selfshielding calculations, and to collect data into "workable" arrangements such as the format required by DOT-4 [11]. In particular, NITAWL is designed to output the working libraries in two forms: first, it can produce output on cards, tape, or disk for the ANISN [42], DOT-4 [11], or MORSE [43] codes; second, it can produce output on tape or disk for the XSDRNPM code [10]. Of most importance, NITAWL is specifically designed to process resonance nuclides prior to the transport calculation in XSDRNPM. The actual neutron resonance self-shielding calculation employs the Nordhiem Integral Treatment, although the narrow resonance, narrow resonance infinite mass, and infinite dilution treatments are available as alternate methods. The XSDRNPM module The XSDRNPM module is provided in the AMPX system for two purposes: first, to provide a 1-D S^ transport or diffusion theory calculation for computing reaction rates, eigenvalues, or critical dimensions; and second, to permit spatial and energy weighting. XSDRNPM is an updated version of the XSDRN code [44] with the following added features: 1. performs coupled neutron-gamma calculations; 2. represents mixtures to an arbitrary order of anisotropy; 3. performs adjoint calculations;

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313 4. stores data more efficiently so that larger problems run with less computer core storage; 5. employs improved thermal flux scaling techniques for better problem convergence; 6. uses more defaults which make the XSDRNPM module easier to use; 7. calculates S constants for any even order for any of the three available standard 1-0 geometries (slab, sphere, or cylinder); and 8. calculates mixture-dependent fission spectra and uses them in XSDRNPM, which takes into account all fissionable nuclides in a problem. The collapsed cross-sections from XSDRNPM are written as an AMPX weighted library which can be directly used by XSDRNPM for transport or diffusion calculations. The collapsed cross-sections can also be output on cards or binary format suitable for the ANISN [42], DOT-4 [11], or MORSE [43] codes. There are four weighting options available in XSDRNPM. They are: " v > >^ -* ^ 1. Cell weighting, which generates cross-sections consistent with mocking up a cellular configuration as a homogenized region. 2. "Inner" cell weighting, which performs a cell weighting as in the above item but only over the N regions selected by the input. This allows one to mock-up a calculation with non-zero leakage at the outer boundary of the cell. 3. Zone weighting which produces a set of cross-sections weighted over each material region in which a nuclide occurs. 4. "Region" weighting, which produces one set of cross-sections for a nuclide but weighted over a composite spectrum made of all spectra from zones where the nuclide is present. Shown in Figure A-1 is a schematic of the computation flow path for the AMPX modules that are used in this research. Figure A-1 indicates that the resonance data are processed in the NITAWL module rather than the XLACS module. The figure also indicates that a 123-neutron group (30 thermal groups) cross-section AMPX working library is obtained from

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314 •*(/) M C « U 1^ U U 0) 0) u •I-P K (. (/) (O O IQ 3 L. U t. r— a f U) I. ^. o •«-> e -o D) O C C 'rC C -pO O X -r«/l -M »J,O >^(S 3 3 C U 1 09) 01 U O 3 •-) JM L. e C N ^ ^^

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315 NITAWL. This 123-neutron group file is then collapsed first to 27(six thermal groups) and then to four-neutron groups (one thermal group) using XSDRNPM. The resultant four-neutron group library is produced in two formats-AMPX and ANISN formats. DOT-4; A Oneand Two-Dimensional Neutron/Photon Transport Code DOT-4 [11] is a oneand two-dimensional discrete ordinates (S ) transport code. The principal application of DOT-4 is to the deeppenetration transport of neutrons and photons. DOT-4 computes the neutron and gamma flux or fluence throughout the system due to sources that are either produced in the system (fission and scattering) or incident on the system (external). Critical ity (kg^^ and search) problems are also a feature of the DOT-4 code. The DOT-4 code is an updated version of the DOT code [45]. The general features of DOT-4 include the following: 1. Solves the Boltzmann transport equation in oneand two-dimensional geometries using the discrete ordinates or, as an option, the diffusion theory approximation. 2. Employs a vast number of boundary conditions; these include void, reflected, periodic, albedo, and fixed boundary sources. 3. Allows sources to be specified at internal or external boundaries, distributed by space and energy or determined from an input flux file. 4. Expresses the output edits of the results in a number of ways and transfers results to output files for subsequent analysis. 5. Prints the distributed flux moments and boundary directional fluxes to a single file; this output file along with the original input data provide an "exact" restart. 6. Expresses anisotropic cross-sections in a Legendre expansion of arbitrary order. .'5 '-ir ^.i'.i 'i \J'

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316 7. Employs several flux solution models; these include the linear zero model, step model, ordinary weighted difference model, and 9weighted model . 8. Employs several methods for removing the effects of negative fluxes caused by the finite difference approximation and of negative scattering sources due to truncation of the cross-section expansion. 9. Allows a varying space mesh; the space mesh can be described such that the number of first I-dimensional intervals varies with the second J-dimension. 10. Allows the user to vary the number of directions across the space mesh and with energy. 11. Allows the user to select direction sets that are biased, with discrete directions concentrated so as to give fine detail to streaming phenomena. Asymmetric direction sets can be used when streaming is primarily either upwards or downward. Figure A-1 indicates that a four-neutron group cross-section file in ANISN format is produced from the XSDRNPM module. This file contains cross-section data for individual nuclides. Although DOT-4 is capable of calculating mixture cross-section data from the nuclide data, it is preferred that the mixture dependent cross-sections are obtained prior to running DOT-4. By collapsing the data from nuclide data to mixture data, computer input/output manipulation and storage in DOT-4 is greatly reduced allowing DOT-4 to run faster and more efficiently. This mixture cross section data can be generated with the GIP code [28] which is described in the following section. GIP GIP (Group-organized cross-section Input Program) [28] accepts nuclide organized microscopic cross-section data either from the input stream in ANISN card-image format or from a data library prepared by the ACLl program [46], GIP prepares a group-organized file of microscopic and/or macroscopic cross-sections for use by D0T-*4, ANISN, or related \.»t; I 'i' ' : " . . „ ^vi t •

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317 codes. Adjoint cross-sections can be prepared by GIP. The total upscatter cross-section can be inserted into the data format as required by DOT-4 and ANISN. GIP treats each component of a Legendre expansion as a separate nuclide. A simple sequential file sorting technique is used. The result is a "GIP" cross-section file. . * • T > ; "T. Advantages of GIP are its extreme simplicity and high compatibility with DOT and ANISN. Disadvantages are its inability to translate complex input files such as ISOTX and AMPX, and its inefficiency for very large problems, since it treats each component of a Legendre expansion as a separate nuclide. MCNP-A General Monte Carlo Code for Neutron and Photon Transoort MCNP [12] is a general purpose, continuous energy, generalized geometry, time dependent, coupled neutron-photon Monte Carlo transport code. MCNP solves neutral and charged transport problems and can be operated in any of the following modes: neutron transport only; photon transport only; coupled neutron-gamma transport, where the photons are produced by neutron interactions; or coupled photon-electron transport. General features of the MCNP code include the following: 1. Employs a variety of nuclear data tables which include continuous energy and thinned continuous energy, discrete reaction, and multigroup reaction neutron interaction data; neutron dosimetry cross-sections; neutron S(a,^) data; neutron-photon interaction data; photon interaction data; and coupled electron-photon interaction data. 2. Allows input in either horizontal or vertical format. 3. Uses surfaces to create zones or regions; the basic unit of MCNP geometry is the cell. Combinations of sense-signed surfaces are used to define regions of space encompassed by a cell in the

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318 ordinary Cartesian coordinate system. There are 26 standard surfaces (planes, spheres, cylinders, cones etc..) defined in MCNP by first-, second-, and fourth-order equations that describe three dimensional geometries. 4. Utilizes a variety of variance reduction techniques [32,33] which include geometry splitting and Russian Roulette, implicit capture and weight cutoff, time and energy cutoffs, forced collision, and weight window. 5. Allows re-start runs. ^ ,: , « 6. Prints output in user defined particle tallies such as flux, current, heating etc., as a function of cell or surface, energy, time, angle, etc. Tally modifiers allow easy tally conversion to the quantity of interest which include reaction rates such as absorption and fission and fission heating; surface and/or cell flagging allow core-to-core coupling coefficients to be obtained. 7. Plots results using two modules; the first is PLOT which is an interactive plotter used to plot two-dimensional slices of the problem geometry specified in the input file; the second is MCPLOT which draws ordinary two-dimensional x-y plots, contour plots, and three-dimensional surface plots. 8. Employs extensive error analysis procedures that give statistical error or uncertainty associated with the results. 9. Calculates k^r^ and neutron removal lifetime. The calculation is performed as 4 series of cycles or generations of neutrons in which both k rr and removal lifetimes are estimated in each cycle and are also averaged over all previous "active" cycles. 10. Employs a repeated structure capability which allows the user to describe only once the cells and surfaces of any structure that appears more than once in a geometry. This reduces the required amount of input data and computer memory. The amount of time required to set a problem is greatly reduced. Description of the EASYS Engineering Analysis Program EASY5 Engineering Analysis [29] is an interactive program that has the capability to model, analyze, and design large complex dynamic systems defined by algebraic, differential, and/or difference equations. EASYS contains generic building blocks, called standard components, that comprise the General Purpose Component Library. These standard

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319 components are used to construct a block diagram of the mathematical model. EASY5 allows the user to define building blocks to model special effects or subsystems that are not available in the General Purpose Component Library. EASY5 reduces the mathematical model of the system to a set of first-order differential/difference equations and algebraic equations. By integrating the differential/difference equations for a period of time and resolving the algebraic equations, EASY5 effectively simulates the behavior of the non-linear system. The general features of the EASY5 program are described as follows: 1. The ability to model and analyze any type of continuous, discontinuous, and/or multi-rate digital system. 2. The ability to construct and interactively display the block diagram of the modeled system. 3. The ability to use output from any standard component as input to more than one component. 4. The ability to simulate the dynamic behavior of the non-linear system by resolving the algebraic equations and integrating the difference/ differential equations for a period of time. Available integration algorithms include variableor fixed-step Runge-Kutta methods. 5. The ability to locate a steady-state operating point for the nonlinear system. 6. The ability to perform non-linear and linear analysis on the same model. Linear analysis include frequency response, Root Locus, eigenvalue sensitivity, stability margin, and power spectral density analysis. 7. The ability to optimize the system using parameter optimization capabilities. 8. The ability to perform 1-, 2-, and 3-D table look-ups. 9. The ability to display plots of results on the screen and/or print and plot hardcopy results. Plots include time-history, Bode, Nyquist, Root Locus, and power spectral density plots. A hardcopy output of the system's block diagram can also be generated.

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^ v-i APPENDIX B BENCHMARK CALCULATIONS OF XSDRNPM AND DOT-4 WITH MCNP One-, two-, and three-dimensional static neutronic analysis has been performed on the UTVR with the XSDRNPM [10], DOT-4 [11], and MCNP [12] computer codes, respectively. Although results obtained from the 1-D spherical and the 2-D cylindrical (in the R-Z and R-9 coordinate systems) calculations are used to obtain basic neutronic characteristics and reference configurations for the 3-D analysis of the UTVR, it is desirable to compare the results obtained from these different computer codes for at least a few reference configurations. Such comparisons aid in determining the accuracy and reliability of the obtained results. Therefore, results obtained from steady state neutronic calculations using XSDRNPM and DOT-4 are compared with MCNP results in this Appendix. Comoarison of XSDRNPM with MCNP Both XSDRNPM and MCNP calculations are performed on a reference UTVR system in spherical geometry. The selected system for this benchmark calculation is a five region configuration as shown in Figure 4-14. The first region is a 60 cm UTVC at 3000 K with UF^ fuel at 5 atm and NaF at 45 atm. The second region is the wall cooling region containing liquid NaF with a radial thickness of =0.38 cm. The third region is the IBEO region which contains only BeO and is 15 cm thick. 320

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321 The fourth region Is the boiler core region containing a homogenized liquid UF^/NaF mixture with a radial thickness of «0.48 cm. The fifth region is the OBEO region and is 40 cm thick. Results obtained from MCNP and from XSDRNPM using 123, 26-, and four-neutron groups are given in Table B-1. Table B-1 indicates that results obtained from calculations performed with XSDRNPM show the following differences relative to MCNP: 1. The 123-neutron group calculation underestimates k rx by only «0.07% and the 26and four-neutron group calculations overestimate k r^ by «0.26% and «1.51%, respectively. 2. The 123-, 26-, and four-neutron group calculations overestimate the average thermal neutron flux (#tu) in the vapor core by »2.5%, «1.9%, and «1.7%, respectively, and overestimate #xu in the boiler region by «1.0%, »1.4%, and »3.0%, respectively. 3. The 123-, 26-, and four-neutron group calculations all overestimate the fission rate in the vapor core by »2.3%, »2.3%, and «1.5%, respectively. However, the 123-neutron group underestimates the boiler core fission rate by «0.2% and the 26and four-neutron groups overestimate the boiler core fission rate by =0.2% and «1.2% respectively. 4. The 123-, 26-, and four-neutron group calculations all overestimate the absorption rate in the vapor core by =1.9%, «1.9%, and «1.2%, respectively. However, the 123and 26-neutron group calculations underestimate the boiler core absorption rate by »0.6% and «0.3%, respectively, and the four-neutron group calculations overestimates the absorption rate in the boiler region by =1.3%. < .:y. : ^r^j-,: : ." ^-'H : ^ -..i. •

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' W^.i'v

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323 Table B-1 indicates that using 123and 26-neutron groups in XSDRNPM provides results that are more in agreement with MCNP than fourneutron groups. The larger number of groups provide a better resolution in the high and low energy ranges. Using a larger number of neutron groups in the high neutron energy range allows a better treatment of important phenomena such as the (n,2n) reaction in the beryllium moderator, e.g., 93 and 20 fast groups for the 123and 26-neutron group structures, respectively, compared with only three-fast groups for the four-neutron group structure; and using a larger number of thermal groups provides a better representation of thermal neutron scattering, e.g., 30 and 6 thermal groups in the 123and 26-group structures, respectively, compared to only one thermal group in the four-group structure. Table B-1 also indicates that a higher value for k rr is obtained with the four-neutron group structure relative to the 123and 26-group structures. This tendency is almost always observed when thermal reactors are analyzed using a collapsed group structure rather than an uncollapsed group structure. Using fewer groups, especially fewer thermal groups, overemphasizes the importance of the "more" important energy groups (thermal neutrons). This can be seen from Table B-1 where the total fission rate of the UTVR (UTVC and boiler cores combined fission rates) is =0.632, «0.634, and «0.638 fissions/sec for the 123-, 26-, and four-neutron groups calculations, respectively. Note that the as the number of groups decreases, the fission rate of the UTVR increases.

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324 The reasons for the disagreements between results obtained from XSDRNPM and MCNP calculations include the following: Solution method . Monte Carlo transport theory is used in MCNP while XSDRNPM solves the 1-D Boltzmann transport theory equation using the discrete ordinates (S^) method. Theoretically, both methods should provide the same answers. However, results obtained with MCNP are statistically averaged results with uncertainties that are inversely proportional to the number of particles (or histories) examined. Thus, results obtained from MCNP depend on the number of particles examined. On the otherhand, the order of quadrature used to represent the directional -neutron flux (n in S ), the number of mesh points used in representing the geometry, and the selected convergence level all affect the accuracy of results obtained from XSDRNPM. Neutron cross-section library . The MCNP code uses a continuous energy neutron cross-section library (although discrete or multi -group cross-section libraries are available in MCNP) while XSDRNPM uses a neutron cross-section library that has a maximum of "only" 123-groups. Another factor that contributes to the discrepancy is the lower energy cutoff in the neutron cross-section library employed by both methods, i.e., 0.005 eV for XSDRNPM and 0.001 eV for MCNP. Comoarison of DOT-4 with MCNP For the benchmark study of DOT-4 with MCNP, reference UTVR configurations in both the R-0 and R-Z cylindrical coordinate systems are used. A symmetric four-boiler column UTVR system is used in the R-e calculation as shown in Figure 5-1. The vapor core has a 60 cm radius

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325 and contains 5 atm of UF^ and 45 atm of NaF at 3000 K. The wall cooling region contains liquid NaF and is »0.38 cm thick. The IBEO and 06E0 regions are 15 and 40 cm thick, respectively. The boiler region contains only UF^ at a density of «2.8 gm/cm . The cross sectional flow -03 2 area of each boiler column is 1.05 x 10 m . The reference UTVR configuration in the R-Z coordinate system is shown in Figure 5-9. It employs a vapor core with a radius of 60 cm and a height of 100 cm. The vapor core contains 5 atm of UF^ and 45 atm of NaF at 3000 K. The thicknesses of the MBEO, LBEG, IBEO, OBEO, and TBEO regions are 7.5 cm, 55 cm, 15 cm, 40 cm, and 50 cm, respectively. The annular boiler region has a radial thickness of «0.1 cm. The height of the liquid UF. column is 10 cm and the height of the saturated liquid/vapor and vapor region of the boiler column is 30 cm. Results of the DOT-4 and MCNP calculations in the R-9 and R-Z coordinate systems are listed in Table B-2. Table B-2 indicates that DOT-4 overestimates k rr by »1.5% and «0.7%, underestimates the #^^ in the UTVC by =12.8% and »10.4%, overestimates the #^^ in the boiler by «0.0% and »8.5%, underestimates the fission rate in the UTVC by »1.9% and »0.0%, overestimates the fission rate in the boiler by «2.9% and «4.4%, underestimates the absorption rate in the UTVC »2.8% and »0.4%, and overestimates the absorption rate in the boiler by »1.2% and «6.0% relative to MCNP for the R-Z and R-0 configurations, respectively. The discrepancies in the DOT-4 and MCNP results are partially due to the reasons listed in the previous section where XSDRNPM and MCNP calculations for a spherical "mock-up" of the UTVR are compared.

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o (4O U o u ic o 0) o (/> c o o2 r; o 0) o c I I t— 0) CSi I CO a>

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327 However, these discrepancies are greater for the 2-D cylindrical "mockup" than the 1-D spherical "mock-up," especially for the reaction rates of the boiler region in the R-9 configuration. The larger discrepancies are due to the greater uncertainty in the MCNP calculation when estimating the i^^^ in the boiler region {«2.15% for the R-0 cylindrical configuration compared to »0.9% for the R-Z cylindrical and «0.7% for the 1-D spherical configurations). The uncertainty in the MCNP calculations is inversely proportional to the total number of neutrons that contribute to a given interaction. For the R-8 cylindrical configuration, the surface area of the boiler region is small compared to the surface area of the vapor core. Consequently this, a neutron has a greater probability of having an interaction in the vapor core than it does in the boiler region. This is apparent from the results shown in Table B-2 where the uncertainty of the i^u is «0.83% in the vapor core compared to «2.15% in boiler region for the R-9 configuration. For the R-Z cylindrical configuration, the boiler core is treated as an annular region surrounding the vapor core. For this configuration, the surface area of the boiler region is comparable to that of the vapor core and the high density of the fluid in the boiler core relative to the vapor core result in comparable probabilities for neutron interaction in the boiler core and the vapor core regions. This is seen from the relative fission rates of the vapor core and boiler core of «0.26 and «0.20, respectively. The final result is that the uncertainty of f^i^ in both the boiler and vapor cores is about the same (0.91% versus 0.76%).

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For the 1-D spherical "mock-up," since the boiler region surrounds the vapor core (modeled as a spherical shell surrounding the vapor core) and since the density of the boiler region fluid is greater than that of the vapor core, a neutron has a greater probability of having an interaction in the boiler core than it does in the vapor core. This is seen in Table B-1 where the fission rate of the boiler core is about four times greater than that of the vapor core. Therefore, the uncertainty of the $^u in the boiler region is smaller than that of the vapor core. The previous discussion explains why the uncertainty of the #^u and of the reaction rates is greater in the boiler core than in the vapor core for the 2-D calculations, especially for the R-9 calculation. The reason that the discrepancy between the DOT-4 and the MCNP calculations in estimating k^^ is higher for the R-Z configuration than the R-9 configuration is due to the greater discrepancies between the DOT-4 and MCNP calculations in calculating the fission rate of the vapor core for the R-Z configuration than for the R-0 calculation. Additionally, the fission rate of the vapor core is about three times greater than that of the boiler core for the R-0 configuration. This implies that the vapor core has more of an effect on k^^ than does the boiler core. However, the fission rate of the vapor core is only about 25% greater than that of the boiler core for the R-Z configuration. This implies that both the vapor core and boiler core have about an equal effect on k rr. Since the vapor core fission rate obtained from both the DOT-4 and MCNP calculations for the R-0 configuration is the same and in better agreement than for the R-Z configuration, the discrepancy in k rr for

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329 the R-8 configuration calculated from both codes should be smaller than for the R-Z configuration. Another reason why the discrepancies are greater for the 2-D cylindrical configurations than for the 1-D spherical configuration is the poorer convergence levels achieved in the DOT-4 calculations compared to the convergence levels obtained in XSDRNPM, »10" versus -05 «10 , respectively. The much faster convergence of the 1-D calculation compared to the 2-D calculations allows for superior convergence levels in XSDRNPM as compared to DOT-4. Conclusion The comparisons presented in this Appendix indicate that results obtained from XSDRNPM, DOT-4, and MCNP will be in more agreement if: 1. A large number of neutron groups (such as 123or 26-neutron groups) is used in XSDRNPM and DOT-4 calculations. 2. Good convergence levels in XSDRNPM and DOT-4 (»10'°^) and low uncertainty levels («0.5% or less) in MCNP are obtained. 3. A similar lower energy cutoff is used in the neutron cross-section library. Although the 1-D spherical calculations using XSDRNPM and the 2-D cylindrical calculations in the R-9 and R-Z coordinate systems using DOT-4 are performed with only four-neutron groups in Chapters IV and V, the results obtained from these calculations are adequate for this research since these are only preliminary or scoping calculations. That is, these results are only used to obtain basic neutronic characteristics and reference configurations for the 3-D analysis of the

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< f 330 UTVR. Since the MCNP code is a 3-D code which uses a continuous energy neutron cross-section library with a lower energy cutoff which is lower than the value used in the XSDRNPM and DOT-4 codes, results obtained from MCNP should more accurately describe the UTVR than results from the XSDRNPM and DOT-4 codes. Therefore, the 3-D analysis is performed using MCNP to obtain parameters that are needed for the dynamic neutronic analysis and performance studies of the UTVR.

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APPENDIX C DESCRIPTION OF THE ISOLATOR OF SECONDARY COUPLING EFFECTS CODE Introduction Isolator of Secondary Coupling Effects (ISCE) calculates the reactivities (p), neutron life and reproduction times {i and A), and neutron multiplication factors (kgrr) of the individual cores and the coupling coefficients among the interacting cores (c^ ) of a four boiler column UTVR system. The solution method is based on the models derived in Chapter VI, namely Equations (6-40) through (6-56). A description of the ISCE code is included in the following section. This is followed by a section describing the input data to ISCE and the output obtained from ISCE. A section is also included in this appendix that compares results obtained from the ISCE code with results obtained directly from MCNP [12]. Description of the ISCE Code ISCE is written in standard Fortran-77 and contains six modules: MAIN, ERIN, ESTM, NOUT, REED, and RITE. These modules are described below. , ; . ' ' -.'*'' The MAIN Module "' ' The MAIN module controls the execution sequence of the other modules and prompts the user to enter the name of the input data file or 331

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332 the name of the file containing the list of input data files. MAIN then checks if the input file exists on the current drive. If the input file does not exist, the user is then prompted to enter "Q" to quit or any other key to re-enter the name of the input file; otherwise, the file is assigned to unit 10 and the REEO module is then called. The REED Module The REED module reads in all necessary input data from unit 10. If an end-of-file is reached prior to reading in all needed data, execution is terminated. The ERIN Module ERIN performs the following tasks: 1. Checks if the input data are capture rates or absorption rates and reads in the data to their associated variables. 2. Calculates the scattering rates by using Equations (6-43) and (644). 3. Checks input data for the following input errors: h< At T^ < C' e e i ri NC . . NC . . e NC . . F{< I F^ ji'i

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333 where. si is the fission neutron source rate in the i core, TJ, AJ, and FJ ire the total^interaction, absorption, and fission rates in core 1, and ^q , '^e ' ^"^ '^e ' *"^® ^^® total interaction, absorption, and fission rates in core i due to neutrons flagged escaping the boundaries of core j. If any error is detected, execution is terminated. However, prior to terminating execution, a list of all detected errors is printed in the error list file and the user is informed that such a file has been created. 4. Checks input data if the minimum and maximum number of iterations is less than zero. If so, correction is made. 5. Checks if the maximum number of iterations is less than the minimum number of iterations. If so, the maximum number of iterations is changed to twice the number of minimum iterations. The NOUT Module NOUT selects a name for either the output file or the error list data file from the first eight characters of the title line card in the input data file. The name of the output file or the error list file need not be eight characters long. However, if all eight characters are blank characters, ISCE assigns the default name "OUTF" to be the name of the output file. ISCE then checks if a file with the same name as the output file exists on the current drive. If so, the extension is indexed to the next number (i.e., ".ROO" to ".ROl" up to ".RSg" for the output file or ".EOO" to ".EOl" .... up to ".E99" for the error list file). If there are 100 files on the current disk drive, ISCE aborts and prints a message on the screen informing the user to either move such files from the current drive or change the name of the output file on the title card. If an output file name is selected, ISCE prints a message on the screen indicating the name of the output file.

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334 The ESTM Module U B ESTM calculates the reactivities {p and p ), neutron U B multiplication factors (kg^f. '^eff^' "'"^ neutron life and reproduction times («^, «^, A^, and A^) of the UTVC and boiler column by solving Equations (6-40) through (6-56) for a four-boiler column UTVR system. 1. Calculates r„ and r„ if the input data are normalized to the total The following are the steps executed by ESTM: Calculates r^ and rz if the input data ar_ — number of neutrons generated in the system using r" = (4nB.n^/n" (C-l) r^ = ( 4n^ + n" ) /rfi (C-2) 2. Calculates neutron total interaction, fission, absorption, and scattering rates in the UTVC due to neutrons originating in the UTVC, prior to performing the first iteration, using UHJ U U*-B ,r ->» RXq = RXy 4 RXg (C-3) where RX is replaced with T for total interaction rate, F for fission rate, A for absorption rate, or S for scattering rate. Superscripts "U," "U^J," and "U^B" are used to denote total reaction rates in the UTVC, reaction rates in the UTVC due to neutrons originating in the UTVC, and reaction rates in the UTVC due to neutrons originating in any boiler column, respectively, and subscripts "0" and "e" are used to denote reaction rates used in the zero iteration number (i.e., £ = 0) and reactions rates due to neutrons escaping the specified cores, respectively. 3. Calculates neutron total interaction, fission, absorption, and scattering rates in the boiler column due to neutrons originating in that boiler column, prior to performing the first iteration, using 8<-B B B*-Bn B«-Bo BHJ //» *» RXq = RXy 2 RX° RXg RX" (C-4) where superscripts "Bn" and "Bo" are used to denote adjacent boiler and opposite boiler, respectively. 4. Initiates iteration counter.

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335 5. Estimates the fission, absorption, total interaction, and scattering rates in the UTVC first due to neutrons originating in any boiler column and then due to neutrons originating in the UTVC using RX RX £ U^ = RX U*-B l
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336 10. Calculates the coupling coefficients among all interacting cores using ?^"^ F i-j /pi (C-12) 11. Calculates the neutron multiplication factor and reactivities of the UTVC and boiler column using {C-13) {C-14) (C-15) (C-16) ^eff

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337 a". «" where /Cf (^-"' sys U . B ff. 5-. s / = s , + 4 s , . (C-Z3) The RITE Module The RITE module outputs the calculated kinetics parameters and the title card to the selected output file assigned to unit 11. Input Data Format The user has the option of entering the name of the input data file or the name of a file containing a list of input data files for multiple data processing. These two types of input files are described below. Input Data File The input data, contained in the input data file, consists of a title card and three data blocks (A, B, and C), as follows: Title card . The format for the title card is 80A1 which is stored in the TC array. The title card is the first card in the input file and is used to label the problem output. The first eight characters of the title card are used to assign a name for the output file or the error list file. Data block A . This block contains general problem information such as number of iterations and convergence criteria. It contains the ' ,»^, V-'" following variables: ' K < ' -'.'^'•' " r ' .*

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338 ITN Minimum number of iterations. The problem will perform ITN iterations even if the problem reached full convergence. ITM Maximum number of iterations. After ITM iterations, the problem will be forced into the termination phase and the program will continue as if full convergence was attained. A message is printed if this happens. B inputgdata normalization factor. If NRM is^equal to one, then r„ and r„ are both equated to one; otherwise r|: and r„ are NRM = Block then — _. _ — , — — , .. — . — calculated using Equations (C-1) and (C-2), respectively. FCC Required convergence level of calculated parameters. For £ > ITN, ISCE checks if the convergence level of the reaction rates, given by Equation (C-11) is less than or equal to FCC; if so, the problem will be forced into the termination phase; otherwise, the program continues execution until convergence or ITM iterations are performed. £^y^ = Neutron lifetime for the entire reactor system (msec). Data block B . This block contains specific problem information such as the fission and capture rate of the UTVC and any boiler column. There are eight variables, as follows: Sr = Rate at which neutrons are generated in the UTVC (neutrons/sec). Fj = Fission rate of the UTVC (fissions/sec). Xj « If Xj < 0, then Xj Cj: neutron capture rate in UTVC I ,-_i..-..,--., -i.-....i.. «o .0 _.... ^—ion rate in UTVC If Xj < 0, then Xj Cj: neutron capture rate in \iV (captures/sec); otherwise, Xj = Aj neutron absorpti( (absorptions/sec) Tj Total neutron interaction rate in the UTVC (interactions/sec). p s^ Rate at which neutrons are generated in any boiler column •' (neutrons/sec). g Fj = Fission rate of any boiler column (fissions/sec). '^ If X? < 0, then x5 = C?: (captures/sec); otherwise, aj boiler column (absorptions/sec) g Tj * Total neutron interaction rate in any boiler column (interactions/sec). Xj = If Xj < 0, then Xj = Cji neutron capture rate in any boiler column (captures/sec); otherwise, Xj = Aj neutron absorption rate in any

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339 Data block C . This block contains three variables repeated four times. The first three variables are for the BHJ process, as follows: Fg = Fission rate in any boiler column due to neutrons escaping the boundaries of the UTVC (fissions/sec). x|^ If Xg < 0, capture rate in any boiler colugnudue to neutrons escaoing the boundaries of the UTVC, i.e., C^ (captures/sec); If Xg > 0, absorption rate in any boiler column due to neutrons escaping the boundaries of the UTVC, i.e., Ag (absorptions/sec). Fg Neutron total interaction rate in any boiler column due to neutrons escaping the boundaries of the UTVC (interactions/sec). The above input is repeated for the B-^Bn, B^Bo, and U-^B processes. An example input file is shown in Figure C-1. 25A10C8L UF4»2.5 ATM, [BCOL COL= 8.0 cm, CONE10 cm, VAP-DEN-3.4-04] 100 200 8 1.0-06 2.136 7.929-01 3.278-01 -7.035-02 7.103-01 4.087-02 1.685-02 -3.930-03 8.335-02 1.000-02 -2.118-03 1.847-02 1.903-04 -3.800-05 2.980-04 2.166-04 -4.362-05 3.390-04 1.193-02 -2.456-03 2.046-02 Figure C-1. Example of the ISCE Code Input Data File List of Input Data Files If there is more than one case to be processed, the user has an option of listing the input data file names of these cases in an auxiliary file. There is no limit to the number of cases that can be processed by ISCE, provided that there is sufficient disk space

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340 available for storing the output files. The name of each input data file is entered in columns 1 through 12, and the list is terminated by entering "*END" in columns 1 through 4 on the last line of the file containing the list of input data files. An example file listing the names of input data files is shown in Figure C-2. CASE.l CASE. 2 CASE. 3 CASE. 4 CASE. 5 -END Figure C-2. Input Data Files List Format Shown in Figure C-3 is the output file created by ISCE for the example input data file given in Figure C-1. Comparison of Results Obtained from ISCE with Results Obtained Directly from MCNP The coupling coefficients can be obtained either directly or indirectly from MCNP [12]. Obtaining the coupling coefficients directly from MCNP is tedious and can be prohibitively expensive. For these reasons, ISCE has been developed. It is desirable to compare results obtained from ISCE with results obtained directly from MCNP. Such comparison aid in determining the accuracy of results obtained using ISCE. Two configurations are selected for the comparison. The first configuration employs 2.5 atm of UF^ at 3000 K in the UTVC and each 235 boiler column contains 0.9 kg of U (the heights of the subcooled liquid and the saturated liquid regions, H^ ^^ and H^ x., are 8.0 cm and

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341

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342 235 10.0 cm, respectively, and the vapor cone region contains U at a homogenized atom density of 2.1 x 10 atoms/barn -cm). The second configuration employs 7.5 atm of UF^ at 3000 K in the UTVC and each boiler column contains 2.96 kg of U^^^ (^^?ub" ^^'^ ^"'' "sat' ^^'^ ^"'' 235 and the vapor cone region contains U at a homogenized atom density of 2.1 X 10"^ atoms/barn -cm). Results of these calculations are listed in Table C-1. Table C-1 indicates that results obtained from calculations performed with ISCE show the following differences relative to results obtained directly from MCNP: 1. For the lower UTVR fuel loading, ISCE underestimates k^^^ by 2.3% and overestimates k"^^ by 2.6%, while for the higher fuel loading it U B underestimates k^^^ by 1.6% and k^^^ by 1.9%. 2. For the lower UTVR fuel loading, ISCE overestimates A^ by 1.6% and R U A by 2.8%, while for the higher fuel loading it overestimates A by p 1.6% and underestimates A by 2.1%. 3. For the lower UTVR fuel loading, ISCE underestimates e^ by 1.5% and ij"^" by 7.1% and overestimates ej^^^ by 8.3% and e^"^ by 6.1%, B+4J while for the higher fuel loading ISCE overestimates e^ by 2.5% and e^"^ by 5.2% and underestimates cj^^" by 4.3% and £^*"^° by 2.2%. The discrepancies in the ISCE and MCNP results can be attributed to the relatively large uncertainties associated with the obtained parameters. For example, the highest discrepancy between the ISCE and MCNP calculations is 8.3% for e^*'°^ for the lower fuel loading. However, the uncertainty in c°^°° is 9.5% for the ISCE calculation and 8.9% for the MCNP calculation. Since the discrepancies between the ISCE

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343 f 3 >» I IS e e (A i 9 C (/) •f— o> (0 c •-> T^ T3 O (0 O 3 0) 0) Li. oe: oi— o fI 5

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344 and MCNP results are less than the uncertainties in these parameters, it can be assumed that results obtained using ISCE are comparable with results obtained directly using MCNP. Additionally, the impact of discrepancies between the two different methods is reduced when the effects of the coupling coefficients and the reactivities are combined. That is, the neutron population level in a given core is a function of the reactivity of that core and the neutronic contribution to that core from all other cores. Then, if the reactivity level of that core is underestimated, the overall impact involved is reduced if the coupling coefficients to that core are overestimated. For example, ISCE underestimates \f}^^^ by 2.3% and overestimates e^ by 6.1% for the lower fuel loading configuration. Since the effect of the reactivity of the UTVC (or k^^^) on the UTVC neutron population is much greater than the effect of the four boilers combined on the UTVC (the four boiler columns provide «12% of the UTVC neutron source), then, the effect of underestimating k^^^ is compensated -U*-B to some extent by the overestimation of ci: .

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»^>...> i ^ ,-'--' APPENDIX D CIRCULATING-FUEL, COUPLED CORE POINT REACTOR KINETICS EQUATIONS Coupled Core Point Reactor Kinetics (CC-PRK) equations have been used in modelling the time dependent neutronic behavior of several types of coupled core reactors by many authors [47-54]. These equations are derived from either the time dependent Boltzmann's transport equation or the time dependent diffusion equation. A complete derivation of the coupled core point reactor kinetics equations is not shown in this appendix. Instead, modifications to the original equations derived by Plaza and Kohler [47], are presented. These modifications are needed to account for the circulating fuel and the circulating fuel coupling between the "boiler" cores and the "vapor" core. The coupled core point reactor equations derived by Plaza and Kohler are the following: — ND iNJ(t) = ^^^^^ -^^^^ NJ(t) + X ^iCi(t) ^^ AJ(t) i=l ; NC t * u .X 7f Nt) ihll r ^him P^-^ir) dr (D-1) m AJ(t) i A'^{t-T) le-jct) =^il^NJ{t) x^tht) (D-2) /It ' AJ{t) where the terms shown in Equations (D-1) and (D-2) are described later in this Appendix. 345

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' ^^V , 346 Equation (D-1) describes the time dependent behavior of the neutron population level in the j* core and Equation (D-2) describes the time dependent variation of the concentration of the i delayed neutron precursor group in the j core. In this research, the word "core" is used to denote both the core and its associated moderator-reflector regions for either the vapor or boiler cores. ; y' Equation (D-2) is derived for a core with stationary fuel. To account for the fact that the UTVR employs circulating fuel, two additional terms need to be added to Equation (D-2). They are NC _\ ^^^ m^j The first term in the above expression describes the loss of the i delayed neutron group precursor concentration from the j core due to the out-flow of fuel. The second term describes the gain of the i delayed neutron group precursor concentration entering the j core due to the in-flow of the fuel that exited core(s) m t^*^ seconds ago. The first term can be viewed as a time-dependent neutron sink and the second term can be viewed as a time-dependent neutron source. These two terms account for loss of a fraction of the delayed neutron precursors via decay outside the j core after exiting core(s) j or m. This loss fraction of the delayed neutron precursors depends mainly on three parameters: the time the fuel remains in core j, namely t^, the time the fuel spends travelling from core j to core(s) m, namely t'?^'^, and the time the fuel spends travelling from core(s) m to core j, namely rj"^. By controlling these parameters, control can be achieved over the

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«.> 347 fraction of the precursors which decay in the cores which can in effect control the reactivity of the j core. There are several classes of neutrons that are present in the j core. They can be divided into the following categories: 1. Neutrons^that are produced immediately or instantaneously by fission in the j core and are referred to as the prompt fission neutrons. 2. Neutrons that are produced by the decay of the delayed neutron precursors in the j core which exited core(s) m H seconds ago, and are referred to as the fuel flow coupled-neutrons. 3. Neutrons that are produced by the decay of the delayed. neutron precursors in the j core which are produced in the j core, referred to as the delayed neutrons. 4. Neutrons that are produced by the Be(7,n)2He interaction and are transported to the j core, referred to as the photoneutrons. The incident gamma (7) particles are produced either in core j or core(s) m. 5. Neutrons that are produced in the j core which escape the .. boundaries of the j core and are then reflected back to the j^" core, referred to as the reflected neutrons. 6. Neutrons that are produced by the Be(n,2n)2He interaction and are transported to the j core, referred to as the {n,2n) neutrons. The threshold energy of the incident neutron causing the Be(n,2n)2He interaction is «1.84 MeV [55]. Due to this relatively high threshold energy, this reaction is exhibited by the prompt fission neutrons that were produced either in core j or core{s) m. 7. Neutrons that are produced in all "cores" other than the j core by fission (n,/), (n,2n), {7,n), or by the decay of the delayed neutron precursors and are transported to core j after being delayed by a time T2 , referred to as the transport coupled-neutrons. A schematic showing the neutron classes and their interactions in a four-boiler column UTVR system is given in Figure D-1. The prompt fission neutrons produced in core j and the reflected neutrons are already described in the first term on the LHS of Equation (D-1). The transport coupled-neutrons are accounted for in the last term on the RHS of Equation (D-1). The fuel flow coupled-neutrons and

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348 Boiler B3 Boiler Bl Boiler B O Be(T.n)2He O Be{n,2n)2He Q scattering interaction ^ B"decay '''t/} fission o Boiler B2 -« ,I'.t. ^neutron flight , , , , path '"''"^fission fragment migration path ^>.y^y^y^ T-pay flight path numbers refer to type of neutron interaction Figure D-1. Schematic of Neutrons and Neutron Interactions in a Four-Boiler Column UTVR System

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349 the delayed neutrons are accounted for in second term on the RHS of Equation (D-1), and the rates of change of the concentrations of the precursors are described in Equation (D-2). The two terms in (D-3) also account for the fuel flow coupled-neutrons. The neutrons produced by the Be(n,2n)2He interaction are directly accounted for in the transport coupling term or in ^^ff The effect of the presence of beryllium in the moderator-reflector contributes to the category of the photoneutrons. These photoneutrons are the result of the Be(7,n)2He reaction. The threshold energy of the gamma -particle for this type of reaction is »1.6 MeV; and the obtained neutrons have energies equal to the difference between the incident energy of the gamma particle and the 1.6 MeV threshold energy [19]. Dugan [2] has determined that the effect of the photoneutrons for a gaseous core reactor with D2O, Be, or BeO moderator is small compared to the delayed neutrons produced from the decay of the delayed neutron precursors. Dugan has also determined that the fraction of the gamma "ays (7) that escape the boundaries of the gaseous core to the beryllium moderator and eventually produce photoneutrons is at most «0.3 for the Pulsed Gaseous Core Reactor (PGCR). The effect of photoneutrons, as found by Dugan, is relatively small. Neglecting the photoneutrons in the PGCR was found to cause only a «0.5% 6k/k decrease. On the other hand, neglecting both the delayed neutrons and photoneutrons in the PGCR was found to cause a »3.5% decrease in 6k/k. It should be emphasized that these results are for a pulsed system. The reactivity worth of these neutrons is significantly smaller in a steady state system. Since the contribution of the photoneutrons for a beryllium moderated system

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i i 350 is relatively small compared to the delayed neutrons, and since the system at hand employs beryllium-oxide rather than beryllium as the moderator (lower beryllium atomic density of 0.0728 atoms/barn-cm versus 0.124 atoms/barn-cm for BeO and Be, respectively), then the contribution or the effect of the photoneutrons should be even smaller for a BeO moderated system. Also, the system being studied is "steady state" rather than pulsed. The effects of photoneutrons are, therefore, neglected in this research and research in this area is recommended for future work. Adding the two terms of {D-3) to Equation (D-2) yields — J Mr i*1II ^q(t) =4iNJ(t)_Ec^(t) + X L_c'!'(t-rJ^)e '^ . (D-4) "^^ AJ(t) ri m*i T^ c c Description and Definition of Symbols. Parameters, and Terms used in the Circulating-Fuel, Coupled Core Point Reactor Kinetics Equations The following section describes all symbols, parameters, and terms used in Equations (D-1) and (D-4), and, where applicable, the method(s) used in obtaining the values for these terms are described. Definition of Superscripts and Subscripts The superscripts and subscripts in Equations (D-1) through (D-4) are used exclusively to identify quantities corresponding to the following: j = the j core; m ' the core(s) where the precursors were prior to entering core j;

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351 j-^ fuel transport through the loop from core m to core j; j^k neutron transport through the media from core k to core j; i » the i^ delayed neutron precursor group; k -all cores other than core j (not to be confused with k^^^, the neutron multiplication factor); i = the loop connecting core(s) m with core j or the loop connecting core j with core{s) m; c either core j or core m. Definition of Integral Parameters Integral parameters used in Equations (D-l) through (D-4), namely N^{t), P"^(t), ^(t), AJ(t), C^(t), and £:j'"'^(t), are defined below; ND is the number of the delayed neutron precursor groups and NC is the total number of cores in the reactor system. Neutron pooulation. N-^Ct) The neutron population in the j*" core, denoted by N"J(t), is also called the amplitude time function or the amplitude factor. The amplitude time function accounts for most of the growth and decay of the forward neutron angular flux, *'^(i;,E,g,t). The relation between N'^(t) and •^(i;,E,g,t) is given by Equation (D-5), as follows: *'^(r,E,n,t) = NJ(t) ^(r,E,Q,t) / (D-5) where ^^(!:,E,9,t) is the function describing the shape of the angular neutron flux in core j, referred to as the shape function. Equation (D5) implies that "most" of the time dependency of f"^{»;,E,g,t) is contained in N-^Ct) and its shape is expressed in i&"^(r^,E,Q,t) with less time dependency. Thus, a crucial assumption in the derivation of the

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352 coupled core point kinetics equations is that <>^(i;,E,g,t) is not changing drastically with respect to time. Another equation is needed in order to relate the neutron population in core j, N'^(t), to the power level of core j, namely P^(t), as follows: pj{t) = ey JJJ 2^{r,E,t) fJ(r,E,fl,t) dV^ dO dE (D-6) or pj(t) = ey NJ(t) JJJ r^(r,E,t) lfrJ(r,E,fi,t) dV^ dfl dE (D-7) where e^ is the average energy release per fission. Since the reactor system at hand employs a fuel that exists in both the liquid and vapor states, and since the power level dictates the volume occupied by the liquid and vapor phases of the fuel in the boiler or the vapor density in the UTVC, then the macroscopic fission cross section of core j, 2J{>:,E,t), is rj(r,E,t) = D^{r,t) a^{E) (D-8) where a4(E) is the microscopic fission cross section (cm ) and DJ{j;,t) is the fissile fuel density in core j at time t, (atoms/barn-cm). It is implicitly assumed that a4{E) is constant with respect to time. The fissile fuel density of core j is also space-time dependent and can be expressed by D^(r,t) = d"^(t) ^j(r,t) (D-9) where jd^jt) is the shape function describing the density distribution of the fissile fuel in space in core j, and d^(t) is the amplitude ti r I me

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353 function that describes the fuel density of core j at time t. By use of Equations (D-9) and (D-8), Equation (D-7) becomes pj(t) = e^NJ(t)d^(t) JJJ a^{E) i>j(r,t) ^J(r,E,Q,t) dV^ dfl dE . (D-10) Although 4(1^, t) is changing in time during transients, its time dependence is neglected in this research due to the following: 1. The effect and behavior of 4(1;, t) are not currently known. 2. The treatment of 'i(i^,t) requires the coupling of space-time thermal hydraulics studies-'with space-time neutron kinetics studies. Currently, tools for such studies pertaining to vapor core reactors do not exist. For example, a 10 sec transient for a four-group, two-region vapor core reactor in one-dimensional spherical coordinates using S transport theory with the diffusion acceleration method is estimated to require about one hour on a CRAY X-MP super computer; two-dimensional calculations are estimated to require about 30 hours. Thus, powerful acceleration methods that are unique and specific to vapor core reactors need be developed. 3. A primary goal and objective of this research is the investigation of the unique feedback features of this novel reactor concept (e.g., fuel flow coupling and direct core-to-core neutron transport coupling, liquid fuel volume, vapor fuel density, ..., etc.) and their impact on system behavior including key parameters such as total power, vapor and boiler core power sharing, vapor density, temperature, pressure ..., etc. With this assumption, i.e., "i{i;,t) = 4q{i;), Equation (D-IO) is normalized by requiring the following to be true: e/ JJJ a^(E) ^j^Cr) <^(r,E,fl,t) dV^ dfl dE = 1 . (D-11) Then, the power level of core j is pj{t) = e"^ NJ(t) diet) (D-12) where ^^ is the appropriate conversion factor. An initial value for d4(0) is assumed (or obtained for an initial reference state) and an «*"-\. -. r i

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354 appropriate value for N'^{0) is computed which corresponds to the initial power level of core j at t-0, P"^(0). The effect of including the time dependence of 'i(»;,t) is recommended for future work. Reactivity. p^[t) The reactivity of the j core, />"^(t), that appears in Equation (D1) can be considered as the ratio of net production of neutrons to the total production of neutrons in the j core. It accounts for all the interactions that neutrons undergo in the j "core." These interactions include production by fission and losses due to absorption and leakage. The reactivity of the j core is expressed by i Production -(Absorption'' + Leakage"') P^(t) = . (D-13) Production"^ Expression (D-13) for p'^{t) is analogous to that for the relative change in the effective neutron multiplication factor, ^iff That is . ki.At) 1 Ski.At) pJ(t) = -!II = —HI — . (D-14) "^eff^*^ '^eff^^^ From the expression given in (D-14), a value for k^^r can be computed at time t for a given p"'(t). A detailed discussion of the UTVR system inherent feedbacks is presented in Chapter VII, and methods for obtaining k^^^ and p"'(t) are derived and discussed in Chapter VI. Effective delayed neutron fraction. B{t) The effective delayed neutron fraction, ^, appearing in Equation (D-1) is equal to the sum of the effective fractions of all delayed neutrons groups. It is given by ,..

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V }i^'' 355 ND _ , .,1=1 . '' Table D-1 on the following page lists delayed neutron data for thermal fission in U [56]. Diaz, Dugan, Oliver, and Carroll [1] have found how the delayed neutron effectiveness factor, 7^ varies with the gaseous fuel loading for the Plasma Core Assembly. The delayed neutron effectiveness factor is defined as follows: yJ = ^ /^ = ^/ 0.006502 (D-16) 235 where the fi value of 0.006502 is for pure U fuel. A top view schematic of the PCA is shown in Figure D-2. The reactor includes two fuel regions, gaseous UFg fuel at the center and solid driver fuel elements in an outer annular region. The gaseous UFg core is 93% enriched with U and is surrounded with «17 cm Be that separates the gaseous core from the solid driver fuel annular region. The solid driver fuel employs UOg pellets that are 92.7% enriched with U^^^ and are «1.2 cm in diameter. About 51 cm of Be moderatorreflector surrounds the annular solid fuel region. As the gaseous fuel loading increases, 7^ decreases due to the decrease in the leakage probability of the more-energetic prompt neutrons and due to the enhanced (n,2n) interaction in Be that occurs for fast neutrons. At high gas fuel loadings (1.7 atm), 1^ becomes less than unity. Values for 7^ ranged from «0.99 to «1.02 which corresponds to UFg fuel loadings of 2.445 kg and 0.203 kg, respectively. Since the UTVR system employs two fissioning regions, a vapor core and 2 to 8 boiler columns that surround the vapor core, and since »15 cm

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Ol 356 s e o u 10 o e o o >. u <« U « o. to >. o> i. 0) c o. 3 O iC9 an an U\J M = 3 0> C o

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357 UFg Gas Core UOg Driver Core Be Moderator /Reflector Be Moderator/ Ref 1 ector Figure D-2. Top View Schematic of the Plasma Core Assembly (PCA)

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^v i 358 of BeO separates the two fissioning regions and »40 cm of BeO is used in the outer regions, the importance of the prompt neutrons relative to the delayed neutrons in the UTVR should be similar to that of the PCA. Therefore, for the reactor system at hand, a value of 1.0 is assumed for 7^, i.e., the effective delayed neutron fraction is set equal to the total delayed neutron fraction (^=0.006502) and the effective fractions of the delayed neutron groups are set equal to the delayed neutron fraction groups listed in Table D-1 (for UF^ fuel enriched with 100% U235,. Prompt neutron generation time. A'^(t) The prompt neutron generation time of the j core, A"^(t), that appears in Equations (D-1) through (D-4) describes the average time for two birth events in successive generations, i.e., it is the mean time it takes a neutron to be absorbed causing fission from birth. The prompt neutron generation time, A'^, is related to the neutron removal time, £"^, by the following expression: AJ(t) = «J(t)/k^ff(t) (D-17) where, £'^(t) is the average time it will take a prompt neutron to be removed from the j core by absorption or leakage. The neutron removal time and the neutron multiplication factor of core j are both obtained from the MCNP code. Values for A"^(t) are then obtained by the use of Equation (D-17). Effective delayed neutron orecursor concentration for the i^^ delayed neutron group. C'|(t) . The delayed neutron precursor concentration for the i* delayed neutron group, c"J(t), is defined as the concentration of those fission

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'"tit \ r * y r i^' s 1 359 products in the j core at time t that will eventually undergo a fi' decay, with a decay constant A^, into a delayed neutron emitter. The emitter then undergoes an essentially instantaneous transmutation or decay by (delayed) neutron emission. The effective delayed neutron precursor concentration, C^(t), is calculated from Equation (0-4). It includes an accounting for the relative energy and spatial importance of the delayed neutrons. The effective delayed neutron precursor concentration can be related to the actual delayed neutron precursor concentration by: t\it) . '. JJJ ^ #r ^i(r't) dV^ dQ dE (D-18) where F^Ct) = JJJJJ X(E) V 2^(r,E\t) ^ *j* dV^ dfl dE dO^ dE^ (D-19) V neutron speed; V » total number of neutrons liberated per fission; X.(E) energy spectrum for the emission of the i delayed neutron group; X(E) = energy spectrum of all neutrons (prompt and delayed); *i* " *o (1J>^'Q)» adjoint (angular) flux for a reference critical or • steady state configuration which is assumed to satisfy the boundary condition of zero outgoing importance; i^ ' ^(!;.E',Q',t); shape function describing the shape of the angular neutron flux. Effective coupling coefficient, e^ (t) -i«-k The effective coupling coefficient, £^ (t), that appears in Equation (D-1) is defined as the fraction of neutrons in core j that are

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360 generated in the k "core" (by (n,/), (n,2n), (7.n), or delayed neutron emission) and are transported through the media to the j core where they cause fission at time t. The effective coupling coefficient is given by the following: A*'^^^) = ^r" ^n(t) (D-20) where 6^ is the fraction of neutrons that are generated in core k, transported through the media toand cause fission in core j at an unperturbed steady state condition with an effective fissile fuel density of d"i . Since this parameter depends only on the geometry, the media, and the fuel density of the particular system containing the interacting cores, it is termed the static coupling coefficient, the quantity T (t) in Equation (D-20) is an amplitude function describing -i«-k the time behavior of ei (t) as a function of the effective fissile fuel densities of the interacting cores. Since ei (t) is dependent on the fissile fuel density of the interacting cores, which depends on the power level of core j, and on H , the static coupling coefficient, then e:^ (t) is viewed as the kinetic coupling coefficient. It is the fraction of neutrons in core j that originated in the k core which cause fission in the j core at time t. ^ \ * • . , i^-k Values for e^i are obtained by the use of j^k j-^k / j 7o / Vo /4o ^^-21)

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361 where s^ is the fission neutron source rate in "core" j due to neutrons that originated in core k for an unperturbed steady state condition and s^ is the total fission neutron source rate in "core" j. Values for s^ are obtained by use of ISCE, a special code developed for computing core-to-core coupling coefficients, and the reaction rates, reactivity, and k^^ of individual cores. The equations used in ISCE are derived in Chapter VI and ISCE is discussed in detail Appendix C. Interpretation of Equations (D-1) and (D-4) Equation (D-1) Equation (D-1) describes the time dependent behavior of the neutron population level of the j^ core at time t. The terms of Equation (D-l) are described as follows: • -' ^* *^ i dt (rate of change ^ of the neutron r population level . J pHt) Ht) AJ{t) ND . 2 A.C^(t) ^ i=l ' 1 ^^ NJ(t) ^1 prompt neutron production rate less neutron losses from leakage and absorption. ' rate of neutron production through the effective delayed neutron precursor decay. NC Z k^j cf^t) AJ(t). r .k N-^Ct-T) . . -r PJ ^T) dr A^t-T) rate of neutrons entering core j by means of transport from all other k cores (k^j), The second term on the RHS of Equation (D-1) describes the decay of the effective delayed neutron precursors in core j. It includes the core-to-core neutron coupling by means of the fuel flow. The third term

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: 362 on the RHS of Equation (D-1) describes the core-to-core neutron coupling by means of neutron transport through the media. The function P'^ (t) dr is defined as the probability distribution function for the delay times of neutrons transported from core k to core j. The choice of P^ {t) should reflect the physical and nuclear features of the system under investigation. Choices for P^ (t) include the following: 1. The model used by Avery [57] assumes that all neutrons produced in core k at time t which are transported to core j are delayed by the same time, namely r^ . 2. The model used by C.G. Chezem and H.H. Helmick [58] defines P'^'"'^(t) as follows: P^^Ht) = (l/rfS e^ ^^ (D-22) where r^f is the effective interaction time for neutrons entering core j from core k. This function does not provide for a minimum time delay, i.e., it assumes that the probability that a neutron born in core k at time t will reach core j with no time delay is not zero. 3. The model suggested by Adler [15] requires a minimum delay time, A^ , associated with the most energetic coupling neutrons; an i'^k average delay time, ri , associated with the average energy coupling neutrons; and a half-width, r*^ , which is characteristic of the coupling neutron pulse shape. These conditions require P*^ (^)=0 for T
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^j^k _ ^j*-k e e 363 H{t-AfS (D-23) where H and ^"^ are parameters that are characteristics of the i«-k delayed transport coupling process and, thus, dependent on ri and A:/ . The function H{t-A:^ ) is the Heaviside step function. Figure D-3 is a plot of the P^ (t) distribution function given by Equation (D-23). , . Panicker [7] has determined that, for the Multiple Chamber Gaseous Core Reactor Power System, reducing the core-to-core delay time by three orders of magnitude, ri , has no significant effect on the core power i'^k level. However, increasing t^ by two to three orders of magnitude (to the extent that they are comparable to or larger than the neutron generation time), causes a significant reduction in the core power level . In this research, the model suggested by Avery is used for P*' (t). That is, neutrons produced in the k core which are transported to the j core where they cause fission are all delayed by a time equal to i<-k ^ ri , as shown by the following: ,. P^^^{T) = 8{t-rl^^) . (D-24) Then, the term of Equation (D-1) describing the core-to-core transport coupling-neutrons is modified as follows: NC A'^(t) Nl^Ct -rfS J ?J "(t) 1— . (D-25) ki'j AJ(t) A'^{t-Tj ) -H';-

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364

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365 Using {D-25) in Equation (D-l) results in the following: ND rJ, lNJ(t)= ^eJiLliill NJ(t) . X AiC^(t) A^(t) NC j^,^ A*^(t)'N'^(t-Tp) (t) k^j j-k, {D-26) AJ{t) A^t-r^ ") Equation (D-4) Equation (0-4) describes the time dependent behavior of the effective delayed neutron precursor concentration for the i delayed neutron group in the j^^ core at time t. The terms of Equation (D-4) are described as follows: q(t) dt ^ 1? rate of change of the effective precursor concentration for the ' delayed group in core j. ^i(t) AJ(t) NJ(t) { XiCJ(t) = e^{t) ^c rate of formation of the delayed neutron precursors for the i delayed group in core j. rate of loss of the delayed neutron precursors for the i delayed neutron group through decay in core j. rate of loss of the delayed neutron precursors for the i delayed neutron group through outflow from core j. NC 2 m^j / j-nn c^(t,j*in. rate of gain of delayed neutron ' precursors for the i delayed group through the inflow of the precursors that exited core m n seconds ago. I th where /^"^ is equal to the fraction of fuel flowing from the m''" core to the j core.

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K ':. LIST OF REFERENCES 1. N.J. Diaz, E.T. Dugan, C.C. Oliver, E.E. Carroll Jr., and E.D. Whitney, "Basic Feasibility Studies of Pulsed Gaseous Core Nuclear Systems," Final Report, NSF Grant Number 77-28034, University of Floridla (February 1983). 2. E.T. Dugan, "The Nuclear Piston Engine and Pulsed Gaseous Core Reactor Power System," Ph.D. Dissertation, University of Florida (1976). 3. N.J. Diaz, E.T. Dugan, C.C. Oliver, and R.A. Gater, "Heterogeneous Gas Core Reactor and Dual Fluid Closed Cycle Power Conversion System", Final Report to US-DOE, EN-77-S-05-5546, University of Florida (December 1977). 4. N.J. Diaz, E.T. Dugan, and E.E. Carroll Jr., "Gas Core Reactor Neutronics-Theoretical Modeling and Experimental Verification," Nuclear Technology . Vol. 69, pp. 134-153 (1985). 5. E.T. Dugan, N.J. Diaz, and C.C. Oliver, "Neutronics and Energetics of Pulsed Gaseous Core Nuclear Systems," NSF Final Report , Grant Number ENG-75-01437, University of Florida (April 1978). 6. K.I. Han, "Heterogeneous Gas Core Reactor," Ph.D. Dissertation, University of Florida (1977). 7. M.M. Panicker, "Neutronics of a Multiple Chamber Gaseous Core Reactor Power System," Ph.D. Dissertation, University of Florida (1989). 8. S.D. Kahook, "Neutronic Analysis of Highly Enriched, Graphite And Beryllium Moderated Heterogeneous Gas Core Reactors," Master's Project, University of Florida (1986). 9. W.E. Kerrick, "Graphite-Moderated, Uranium Hexaflouride-Fueled Unit Cells: Neutronic Analysis With Applications To Fission-Fusion Blankets," Master's Project, University of Florida (1978). 10. L.M. Petrie and N.M. Greene, "XSDRNPM: AMPX Module with OneDimensional S^ Cabability for Spatial Weighting," AMPX: A Modular Code System for Generating Coupled Multiqroup Neutron-Gamma Libraries from ENDF/B . ORNL-TM-3706, Oak Ridge National Laboratory, Oak Ridge (March 1976). 366

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367 11. W.A. Rhoades and R.L. Childs, "An Updated Version of the DOT-4 Oneand Two-Dimensional Neutron/Photon Transport Code," US-DOE Report ORNL-5851, Oak Ridge National Laboratory (April 1982). 12. J.F. Briemeister, Editor, "MCNP-A General Monte Carlo Code for Neutron and Photon Transport, Version SA," LA-7396, Rev. 2, Los Alamos National Laboratory (1986). 13. G.I. Bell, "Calculation of the Critical Mass of UFf. as a Gaseous Core, with Reflectors of D^O, Be and C," USAEC Report LA-1874, Los Alamos Scientific Laboratory (February 1955). 14. R. Avery, "Theory of Coupled Reactors," Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy . Geneva, 1958, Vol. 12, pp. 182-191, United Nations, New York, New York (1958). 15. F.T. Adler, G.C. Hopkins, and S.J. Gage, "Spatial and Spectral Coupling Effects in Multi-Core Reactor System," Coupled Reactor Kinetics-Proceedings of the National Topical Meeting on Coupled Reactor Kinetics at Texas A and M University . College Station, Texas, January 1967, CONF-670107, pp. 41-63 (1967). 16. John Macphee, "The Kinetics of Circulating Fuel Reactors," Nuclear Science and Engineering , Vol. 4, no. 6, pp. 588-597 (1958). 17. M.A. Schultz, Control of Nuclear Reactors and Power Plants . 2 Ed., McGraw-Hill Book Company, Inc., New York, New York (1961). 18. W.K. Ergen, "Kinetics of Circulating-Fuel Reactor," Journal of Apllied Physics . Vol. 25, no. 6, pp. 702-711 (1954). 19. S. Glasstone and A. Sesonske, Nuclear Reactor Engineering . 3 Ed., Van Nostrand Reinhold Company, New York, New York (1981). 20. W.D. Wilkinson and W.F. Murphy, "Nuclear Reactor Metallurgy," D. Van Nostrand Company, Inc., Princeton, New Jersey (1958). 21. H.A. Hassan and J.E. Deese, "Thermodynamic Properties of UFg at High Temperatures," Report NASA-CR-2373, North Carolina State University, Raleigh, N.C. (1973). 22. M.M. El-Wakil, Nuclear Energy Conversion . American Nuclear Society, 2^^ Ed., La Grange Park, Illinois (1982). 23. D.L. Hildenbrand and K.H. Lau, "Chemistry of Working Fluids for Space Reactor; The K-U-F System," Final Report to INSPI, INSPI 1.5SRI-001, Contract #F 33615-88-C-2881, Standford Research Institute (SRI) International, Melno Park, California (1988).

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,. ,.^. 368 24. N.M. Greene, J.L. Lucius, L.M. Petrie, W.E. Ford, III, J.E. White, and R.Q. Wright, AMPX: A Modular Code System for Generating Coupled MultiqrouD Neutron-Gamma Libraries From ENDF/B . ORNL-TM-3706, Oak Ridge National Laboratory, Oak Ridge, Tennessee (March 1976). 25. H.C. Honek, •'ENDF/B--Specifications for an Evaluated Nuclear Data File for Reactor Applications," BNL-50066, Brookhaven National Laboratory, Brookhaven, New York (May 1966). 26. N.M. Greene, J.L. Lucius, and J.E. White, "XLACS: A Program to Produce Weighted Multigroup Neutron Cross Sections From ENDF/B," AMPX; A Modular Code System for Generating Coupled Multiqroup Neutron-Gamma Libraries From ENDF/B . ORNL-TM-3706, Oak Ridge National Laboratory, Oak Ridge, Tennessee (March 1976). 27. L.M. Petrie, N.M. Greene, J.L. Lucius, and J.E. White, "NITAWL: AMPX Module for Resonance Self -Shielding and Working Library Production," AMPX: A Modular Code System for Generating Coupled Multigroup Neutron-Gamma Libraries from ENDF/B . Oak Ridge National Laboratory, Oak Ridge (March 1976). 28. W.A. Rhoades "GIP: Group-Organized Cross Section Input Program," ORNL-TM-8362, Oak Ridge National Laboratory. Oak Ridge, Tennessee (April 1982). 29. J. Harrison, P. Kamber, and R. Hammond, "EASY5 Engineering Analysis System," User's Guide, The Boeing Company, 20491-0516-R2a, Seattle, Washington (October 1988). 30. Innovative Nuclear Space Power Institute, "Acoustical Gas Core Reactor," Proceedings of the second Investigator's Working Group Meeting, INSPI-IWG-86-001, University of Florida (1986). 31. G.E. Welch, "The Analysis of the Magnetohydrodynamic Flow of a Fissioning Gas In A Disk MHD Generator," Ph.D. Dissertation, University of Florida, in progress. 32. T.E. Booth, "A Sample Problem for Variance Reduction in MCNP," LA10363-MS, VC-32, Los Alamos National Laboratory (October 1985). 33. J.S. Hendricks and T.E. Booth, "MCNP Varience Reduction Overview," Monte Carlo Methods and Applications in Neutronics. Photonics, and Statistical Phvsica-Lecture Notes in Physics 240 . pp. 83-92, Springer Verlag, New York, New York (1985). 34. D.L. Hetrick, Dynamics of Nuclear Reactors . The University of Chicago Press, Chicago, Illinois (1971). 35. J.J. Duderstadt and L.J. Hamilton, Nuclear Reactor Analysis . John Wiley and Sons Inc., New York, New York (1976).

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369 36. G.I. Bell and S. Glasstone, Nuclear Reactor Theory . Robert E. Krieger Publishing Company, Malabar, Florida (1970). 37. Kiratadas Kutikkad, "Startup and Stability of a Gaseous Core Nuclear Reactor System," Ph.D. Dissertation, University of Florida (1990). 38. K. Wark, Thermodynamics . McGraw-Hill Book Company, New York, New York (1983). 39. L.M. Petri e, "DRIVER: The AMPX Module Manager," AMPX; A Modular Code System for Generating Coupled Multiorouo Neutron-Gamma Libraries from ENDF/B . ORNL-TM-3706, Oak Ridge National Laboratory, Oak Ridge (March 1976). 40. D. Garber, C. Dunford, and S. Pearl stein "Data Formats and Procedures for the Evaluated Nuclear Data File, ENDF," BNL-NCS50496 (ENDF 102), Brookhaven National Laboratory, Brookhaven, New York (October 1975). 41. L.W. Nordheim, "The Theory of Resonance Absorption," Proceedings of Symposia in Aoolied Mathematics . Vol. 11, p. 48 (1951). 42. W.W. Engle, Jr., A User Manual for ANISN: A One-Dimensional Discrete Ordinates Transport Code With Anisotropic Scattering . Union Carbide Report K-1693, Oak Ridge, Tennessee (March 1967). 43. E.A. Straken, P.N. Stevens, D.C. Irving, and V.R. Cain, The Morse Code A Multjgroup Neutron and Gamma-Ray Monte Carlo Transport Code . ORNL-4585, Oak Ridge National Laboratory, Oak Ridge, Tennessee (September 1970). 44. N.M. Greene and C.W. Craven, "XSDRN: A Discrete Ordinates Spectral Averging Code," 0RNL-TM-2500, Oak Ridge National Laboratory, Oak Ridge (July 1969). 45. F.R. Mynatt, DOT; A Two-Dimensional Discrete Ordinates Transport Code . ORNL K-1694, Oak Ridge National Laboratory, Oak Ridge, Tennessee (June 1970). 46. W.A. Rhoades, "The ACLl Program for Cross Section Management," ORNL/TM-4015, Oak Ridge National Laboratory, Oak Ridge (December 1972). .1 > . 47. H. Plaza and W.H. Kohler, "Coupled-Reactors Kinetics Equations," Nuclear Science and Engineering . 26, 3, pp. 419-423 (1966). 48. S. Kaplan, A.F. Henry, S.G. Margolis, and J.J. Taylor, "Space-Time Reactor Dynamics," Proceedings of Third International Confeference on Peaceful Uses of Atomic Energy . Geneva 1964, pp. 41-50, United Nations, New York, New York (1964).

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370 49. M. Becker, "A Generalized Formulation of a Point Nuclear Reactor Kinetics Equations," Nuclear Science and Engineering . M. PP. 458464 (1968). 50. R.G. Cockrell and R.B. Perez, "Kinetic Theory of Spatial and Spectral Coupling of the Reactor Neutron Field," Proceeding Symposium Neutron Dynamics and Control University of Arizona, Tucson, Arizona (April 1965). 51. S.J. Gage, F.T. Adler, and P.N. Powers, "Investigations on Nonlinear Stability of Coupled Nuclear Systems," Proceeding Symposium Neutron Dynamics and Control . University of Arizona, Tucson, Arizona (April 1965). 52. F.C. Difilippo and R.M. Waldman, "The Kinetics of a Coupled TwoCore Nuclear Reactor," Nuclear Science and Engineering . 61, pp. 6071 (1976). 53. C.E. Cohn, "Reflected-Reactor Theory," Nuclear Science and Engineering . 13, pp. 12-17 (1962). 54. A.F. Henry and N.J. Curlee, " Verification of a Method for Treating Neutron Space-Time Problems," Nuclear Science and Engineering . 4, pp. 727-744 (1958). 55. A.B. Chilton, J.K. Shultis, ans R.R. Faw, Principles of Radiation Shielding . Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1984. 56. J.R. Lamarsh, Introduction to Nuclear Reactor Theory 2 ^ Ed., Addi son-Wesley Publishing Company, Inc., Reading, Massachusetts, 1983. 57. R. Avery, "Coupled Fast-Thermal Power Breeder," Nuclear Science Engineering . 3, pp. 120-144 (1958). 58. C.G. Chezem and H.H. Helmick, "Pulsed Neutron Analysis in the Los Alamos Coupled Reactor Experiments," LA-3263-MS, Los Alamos National Laboratory (1965).

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BIOGRAPHICAL SKETCH Samer Dakhlallah Kahook was born on August 22, 1961, in El-Blreh, a city located about six miles north of Jerusalem in the Holy Land of AlAqsa. After the June 1967 war, he emigrated with his family to the USA where he had his first four years of schooling. In the Summer of 1971, he returned to El-Bireh. He attended the Mughtaribeen Elementary School, El-Bireh Al-Jadeedah Middle School -Agriculture Division, and then Al-Hashimeyah Scientific High School. In 1978, he came to West Palm Beach, Florida, to join his family. He graduated from Twin Lakes High School in June, 1979. Samer D. Kahook enrolled at the University of Florida in June, 1980, and graduated with a degree in nuclear engineering in 1984. In January, 1985, he was admitted to the graduate program in nuclear engineering at the University of Florida and was awarded a departmental research assistantship in neutron activation analysis. In December, 1986, he completed the requirements for the Master of Science degree with research on heterogeneous gas core reactors. In March, 1987, the author entered the Ph.D. program at the University of Florida with his studies being supported by funds from the Innovative Nuclear Space Power Institute, the University of Florida, the National Science Foundation, and the Frederick A. Hauck Fund. 371

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372 The author is a member of the American Nuclear Honor Society, the Phi Kappa Phi Honor Society, Phi Eta Sigma Honor Society, and Alpha Lambda Delta Honor Society. Samer D. Kahook has been married to Layalee F. Shihadeh since December, 1983. Allah (SWT ) has blessed them with Khalid, their sixyear-old son, and Noor, their two-year-old daughter. w .% .^^. --A, 1. (SWT) is an abbreviation for Siibh'anaho Wa Ta'ala which means "Praise the One above all."

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. • William E. Lear, Jr. Assistant Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 'Willis B. Person Professor of Chemistry This dissertation was submitted to the Graduate Faculty of the College Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. , . . '. , May, 1991 /li)«^Winfred M. Phillips ' Dean, College of Engineering 'V.i-\^\yMadelyn M. Lockhart Dean, Graduate School

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Edward T. Dugan, Chaira Associate Professor of Nuclear Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Nils J. uia Professor of Sciences Nuclear Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Alan m! Jacob/ Professor of Nuclear Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Samim Anghaie Associate Professor of Nuclear Engineering Sciences


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